LMFDB - Embedded newform 105.2.b.a.41.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [105,2,Mod(41,105)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("105.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	105.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.838429221223$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	105.41
Dual form			105.2.b.a.41.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205i q^{2} -1.73205i q^{3} -1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +(2.00000 + 1.73205i) q^{7} -1.73205i q^{8} -3.00000 q^{9} +1.73205i q^{10} +3.46410i q^{11} +1.73205i q^{12} +(3.00000 - 3.46410i) q^{14} +1.73205i q^{15} -5.00000 q^{16} +6.00000 q^{17} +5.19615i q^{18} -3.46410i q^{19} +1.00000 q^{20} +(3.00000 - 3.46410i) q^{21} +6.00000 q^{22} -3.46410i q^{23} -3.00000 q^{24} +1.00000 q^{25} +5.19615i q^{27} +(-2.00000 - 1.73205i) q^{28} +6.92820i q^{29} +3.00000 q^{30} +3.46410i q^{31} +5.19615i q^{32} +6.00000 q^{33} -10.3923i q^{34} +(-2.00000 - 1.73205i) q^{35} +3.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +1.73205i q^{40} +6.00000 q^{41} +(-6.00000 - 5.19615i) q^{42} -8.00000 q^{43} -3.46410i q^{44} +3.00000 q^{45} -6.00000 q^{46} -12.0000 q^{47} +8.66025i q^{48} +(1.00000 + 6.92820i) q^{49} -1.73205i q^{50} -10.3923i q^{51} +9.00000 q^{54} -3.46410i q^{55} +(3.00000 - 3.46410i) q^{56} -6.00000 q^{57} +12.0000 q^{58} -12.0000 q^{59} -1.73205i q^{60} +6.92820i q^{61} +6.00000 q^{62} +(-6.00000 - 5.19615i) q^{63} -1.00000 q^{64} -10.3923i q^{66} +8.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +(-3.00000 + 3.46410i) q^{70} -3.46410i q^{71} +5.19615i q^{72} -6.92820i q^{73} +3.46410i q^{74} -1.73205i q^{75} +3.46410i q^{76} +(-6.00000 + 6.92820i) q^{77} +8.00000 q^{79} +5.00000 q^{80} +9.00000 q^{81} -10.3923i q^{82} +(-3.00000 + 3.46410i) q^{84} -6.00000 q^{85} +13.8564i q^{86} +12.0000 q^{87} +6.00000 q^{88} +6.00000 q^{89} -5.19615i q^{90} +3.46410i q^{92} +6.00000 q^{93} +20.7846i q^{94} +3.46410i q^{95} +9.00000 q^{96} -6.92820i q^{97} +(12.0000 - 1.73205i) q^{98} -10.3923i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{5} - 6 q^{6} + 4 q^{7} - 6 q^{9} + 6 q^{14} - 10 q^{16} + 12 q^{17} + 2 q^{20} + 6 q^{21} + 12 q^{22} - 6 q^{24} + 2 q^{25} - 4 q^{28} + 6 q^{30} + 12 q^{33} - 4 q^{35} + 6 q^{36} - 4 q^{37}+ \cdots + 24 q^{98}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^5 - 6 * q^6 + 4 * q^7 - 6 * q^9 + 6 * q^14 - 10 * q^16 + 12 * q^17 + 2 * q^20 + 6 * q^21 + 12 * q^22 - 6 * q^24 + 2 * q^25 - 4 * q^28 + 6 * q^30 + 12 * q^33 - 4 * q^35 + 6 * q^36 - 4 * q^37 - 12 * q^38 + 12 * q^41 - 12 * q^42 - 16 * q^43 + 6 * q^45 - 12 * q^46 - 24 * q^47 + 2 * q^49 + 18 * q^54 + 6 * q^56 - 12 * q^57 + 24 * q^58 - 24 * q^59 + 12 * q^62 - 12 * q^63 - 2 * q^64 + 16 * q^67 - 12 * q^68 - 12 * q^69 - 6 * q^70 - 12 * q^77 + 16 * q^79 + 10 * q^80 + 18 * q^81 - 6 * q^84 - 12 * q^85 + 24 * q^87 + 12 * q^88 + 12 * q^89 + 12 * q^93 + 18 * q^96 + 24 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/105\mathbb{Z}\right)^\times$.

$n$	$22$	$31$	$71$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.73205i		−	1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$		−	1.73205i		−	1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$		−	1.73205i		−	1.00000i
$3$		−	1.73205i		−	1.00000i
$4$	−1.00000			−0.500000
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$	−3.00000			−1.22474
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		−	1.73205i		−	0.612372i
$9$	−3.00000			−1.00000
$10$			1.73205i			0.547723i
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$			1.73205i			0.500000i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	3.00000	−	3.46410i	0.801784	−	0.925820i
$15$			1.73205i			0.447214i
$16$	−5.00000			−1.25000
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$			5.19615i			1.22474i
$19$		−	3.46410i		−	0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$		−	3.46410i		−	0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	1.00000			0.223607
$21$	3.00000	−	3.46410i	0.654654	−	0.755929i
$22$	6.00000			1.27920
$23$		−	3.46410i		−	0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$		−	3.46410i		−	0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$	−3.00000			−0.612372
$25$	1.00000			0.200000
$26$		0			0
$27$			5.19615i			1.00000i
$28$	−2.00000	−	1.73205i	−0.377964	−	0.327327i
$29$			6.92820i			1.28654i	0.765641	+	0.643268i	$0.222422\pi$
$29$			6.92820i			1.28654i	−0.765641	+	0.643268i	$0.777578\pi$
$30$	3.00000			0.547723
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$			5.19615i			0.918559i
$33$	6.00000			1.04447
$34$		−	10.3923i		−	1.78227i
$35$	−2.00000	−	1.73205i	−0.338062	−	0.292770i
$36$	3.00000			0.500000
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$			1.73205i			0.273861i
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$	−6.00000	−	5.19615i	−0.925820	−	0.801784i
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		−	3.46410i		−	0.522233i
$45$	3.00000			0.447214
$46$	−6.00000			−0.884652
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$			8.66025i			1.25000i
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		−	1.73205i		−	0.244949i
$51$		−	10.3923i		−	1.45521i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	9.00000			1.22474
$55$		−	3.46410i		−	0.467099i
$56$	3.00000	−	3.46410i	0.400892	−	0.462910i
$57$	−6.00000			−0.794719
$58$	12.0000			1.57568
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		−	1.73205i		−	0.223607i
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$	6.00000			0.762001
$63$	−6.00000	−	5.19615i	−0.755929	−	0.654654i
$64$	−1.00000			−0.125000
$65$		0			0
$66$		−	10.3923i		−	1.27920i
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$	−6.00000			−0.727607
$69$	−6.00000			−0.722315
$70$	−3.00000	+	3.46410i	−0.358569	+	0.414039i
$71$		−	3.46410i		−	0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$		−	3.46410i		−	0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$			5.19615i			0.612372i
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$			3.46410i			0.402694i
$75$		−	1.73205i		−	0.200000i
$76$			3.46410i			0.397360i
$77$	−6.00000	+	6.92820i	−0.683763	+	0.789542i
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$	5.00000			0.559017
$81$	9.00000			1.00000
$82$		−	10.3923i		−	1.14764i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−3.00000	+	3.46410i	−0.327327	+	0.377964i
$85$	−6.00000			−0.650791
$86$			13.8564i			1.49417i
$87$	12.0000			1.28654
$88$	6.00000			0.639602
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		−	5.19615i		−	0.547723i
$91$		0			0
$92$			3.46410i			0.361158i
$93$	6.00000			0.622171
$94$			20.7846i			2.14377i
$95$			3.46410i			0.355409i
$96$	9.00000			0.918559
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$	12.0000	−	1.73205i	1.21218	−	0.174964i
$99$		−	10.3923i		−	1.04447i
$100$	−1.00000			−0.100000
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$	−18.0000			−1.78227
$103$			3.46410i			0.341328i	0.985329	+	0.170664i	$0.0545913\pi$
$103$			3.46410i			0.341328i	−0.985329	+	0.170664i	$0.945409\pi$
$104$		0			0
$105$	−3.00000	+	3.46410i	−0.292770	+	0.338062i
$106$		0			0
$107$		−	10.3923i		−	1.00466i	−0.864675	−	0.502331i	$-0.832476\pi$
$107$		−	10.3923i		−	1.00466i	0.864675	−	0.502331i	$-0.167524\pi$
$108$		−	5.19615i		−	0.500000i
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−6.00000			−0.572078
$111$			3.46410i			0.328798i
$112$	−10.0000	−	8.66025i	−0.944911	−	0.818317i
$113$			6.92820i			0.651751i	0.945413	+	0.325875i	$0.105659\pi$
$113$			6.92820i			0.651751i	−0.945413	+	0.325875i	$0.894341\pi$
$114$			10.3923i			0.973329i
$115$			3.46410i			0.323029i
$116$		−	6.92820i		−	0.643268i
$117$		0			0
$118$			20.7846i			1.91338i
$119$	12.0000	+	10.3923i	1.10004	+	0.952661i
$120$	3.00000			0.273861
$121$	−1.00000			−0.0909091
$122$	12.0000			1.08643
$123$		−	10.3923i		−	0.937043i
$124$		−	3.46410i		−	0.311086i
$125$	−1.00000			−0.0894427
$126$	−9.00000	+	10.3923i	−0.801784	+	0.925820i
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$			12.1244i			1.07165i
$129$			13.8564i			1.21999i
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$	−6.00000			−0.522233
$133$	6.00000	−	6.92820i	0.520266	−	0.600751i
$134$		−	13.8564i		−	1.19701i
$135$		−	5.19615i		−	0.447214i
$136$		−	10.3923i		−	0.891133i
$137$		−	20.7846i		−	1.77575i	−0.460086	−	0.887875i	$-0.652181\pi$
$137$		−	20.7846i		−	1.77575i	0.460086	−	0.887875i	$-0.347819\pi$
$138$			10.3923i			0.884652i
$139$		−	17.3205i		−	1.46911i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$		−	17.3205i		−	1.46911i	0.678551	−	0.734553i	$-0.262608\pi$
$140$	2.00000	+	1.73205i	0.169031	+	0.146385i
$141$			20.7846i			1.75038i
$142$	−6.00000			−0.503509
$143$		0			0
$144$	15.0000			1.25000
$145$		−	6.92820i		−	0.575356i
$146$	−12.0000			−0.993127
$147$	12.0000	−	1.73205i	0.989743	−	0.142857i
$148$	2.00000			0.164399
$149$		−	6.92820i		−	0.567581i	−0.958886	−	0.283790i	$-0.908408\pi$
$149$		−	6.92820i		−	0.567581i	0.958886	−	0.283790i	$-0.0915919\pi$
$150$	−3.00000			−0.244949
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$	−6.00000			−0.486664
$153$	−18.0000			−1.45521
$154$	12.0000	+	10.3923i	0.966988	+	0.837436i
$155$		−	3.46410i		−	0.278243i
$156$		0			0
$157$		−	13.8564i		−	1.10586i	−0.833227	−	0.552931i	$-0.813509\pi$
$157$		−	13.8564i		−	1.10586i	0.833227	−	0.552931i	$-0.186491\pi$
$158$		−	13.8564i		−	1.10236i
$159$		0			0
$160$		−	5.19615i		−	0.410792i
$161$	6.00000	−	6.92820i	0.472866	−	0.546019i
$162$		−	15.5885i		−	1.22474i
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$	−6.00000			−0.468521
$165$	−6.00000			−0.467099
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$	−6.00000	−	5.19615i	−0.462910	−	0.400892i
$169$	13.0000			1.00000
$170$			10.3923i			0.797053i
$171$			10.3923i			0.794719i
$172$	8.00000			0.609994
$173$	18.0000			1.36851			0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000			1.36851			0.684257	+	0.729241i	$0.260127\pi$
$174$		−	20.7846i		−	1.57568i
$175$	2.00000	+	1.73205i	0.151186	+	0.130931i
$176$		−	17.3205i		−	1.30558i
$177$			20.7846i			1.56227i
$178$		−	10.3923i		−	0.778936i
$179$		−	10.3923i		−	0.776757i	−0.921500	−	0.388379i	$-0.873035\pi$
$179$		−	10.3923i		−	0.776757i	0.921500	−	0.388379i	$-0.126965\pi$
$180$	−3.00000			−0.223607
$181$			20.7846i			1.54491i	0.635071	+	0.772454i	$0.280971\pi$
$181$			20.7846i			1.54491i	−0.635071	+	0.772454i	$0.719029\pi$
$182$		0			0
$183$	12.0000			0.887066
$184$	−6.00000			−0.442326
$185$	2.00000			0.147043
$186$		−	10.3923i		−	0.762001i
$187$			20.7846i			1.51992i
$188$	12.0000			0.875190
$189$	−9.00000	+	10.3923i	−0.654654	+	0.755929i
$190$	6.00000			0.435286
$191$			10.3923i			0.751961i	0.926628	+	0.375980i	$0.122694\pi$
$191$			10.3923i			0.751961i	−0.926628	+	0.375980i	$0.877306\pi$
$192$			1.73205i			0.125000i
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$	−12.0000			−0.861550
$195$		0			0
$196$	−1.00000	−	6.92820i	−0.0714286	−	0.494872i
$197$			13.8564i			0.987228i	0.869681	+	0.493614i	$0.164324\pi$
$197$			13.8564i			0.987228i	−0.869681	+	0.493614i	$0.835676\pi$
$198$	−18.0000			−1.27920
$199$		−	10.3923i		−	0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$		−	10.3923i		−	0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$		−	1.73205i		−	0.122474i
$201$		−	13.8564i		−	0.977356i
$202$			10.3923i			0.731200i
$203$	−12.0000	+	13.8564i	−0.842235	+	0.972529i
$204$			10.3923i			0.727607i
$205$	−6.00000			−0.419058
$206$	6.00000			0.418040
$207$			10.3923i			0.722315i
$208$		0			0
$209$	12.0000			0.830057
$210$	6.00000	+	5.19615i	0.414039	+	0.358569i
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	−6.00000			−0.411113
$214$	−18.0000			−1.23045
$215$	8.00000			0.545595
$216$	9.00000			0.612372
$217$	−6.00000	+	6.92820i	−0.407307	+	0.470317i
$218$			3.46410i			0.234619i
$219$	−12.0000			−0.810885
$220$			3.46410i			0.233550i
$221$		0			0
$222$	6.00000			0.402694
$223$			17.3205i			1.15987i	0.814664	+	0.579934i	$0.196921\pi$
$223$			17.3205i			1.15987i	−0.814664	+	0.579934i	$0.803079\pi$
$224$	−9.00000	+	10.3923i	−0.601338	+	0.694365i
$225$	−3.00000			−0.200000
$226$	12.0000			0.798228
$227$	−24.0000			−1.59294			−0.796468	−	0.604681i	$-0.793301\pi$
$227$	−24.0000			−1.59294			−0.796468	+	0.604681i	$0.793301\pi$
$228$	6.00000			0.397360
$229$			6.92820i			0.457829i	0.973447	+	0.228914i	$0.0735176\pi$
$229$			6.92820i			0.457829i	−0.973447	+	0.228914i	$0.926482\pi$
$230$	6.00000			0.395628
$231$	12.0000	+	10.3923i	0.789542	+	0.683763i
$232$	12.0000			0.787839
$233$		−	6.92820i		−	0.453882i	−0.973909	−	0.226941i	$-0.927128\pi$
$233$		−	6.92820i		−	0.453882i	0.973909	−	0.226941i	$-0.0728724\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$	12.0000			0.781133
$237$		−	13.8564i		−	0.900070i
$238$	18.0000	−	20.7846i	1.16677	−	1.34727i
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		−	8.66025i		−	0.559017i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$			1.73205i			0.111340i
$243$		−	15.5885i		−	1.00000i
$244$		−	6.92820i		−	0.443533i
$245$	−1.00000	−	6.92820i	−0.0638877	−	0.442627i
$246$	−18.0000			−1.14764
$247$		0			0
$248$	6.00000			0.381000
$249$		0			0
$250$			1.73205i			0.109545i
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$	6.00000	+	5.19615i	0.377964	+	0.327327i
$253$	12.0000			0.754434
$254$			6.92820i			0.434714i
$255$			10.3923i			0.650791i
$256$	19.0000			1.18750
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$	24.0000			1.49417
$259$	−4.00000	−	3.46410i	−0.248548	−	0.215249i
$260$		0			0
$261$		−	20.7846i		−	1.28654i
$262$			20.7846i			1.28408i
$263$			24.2487i			1.49524i	0.664127	+	0.747620i	$0.268803\pi$
$263$			24.2487i			1.49524i	−0.664127	+	0.747620i	$0.731197\pi$
$264$		−	10.3923i		−	0.639602i
$265$		0			0
$266$	−12.0000	−	10.3923i	−0.735767	−	0.637193i
$267$		−	10.3923i		−	0.635999i
$268$	−8.00000			−0.488678
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$	−9.00000			−0.547723
$271$		−	24.2487i		−	1.47300i	−0.676435	−	0.736502i	$-0.736476\pi$
$271$		−	24.2487i		−	1.47300i	0.676435	−	0.736502i	$-0.263524\pi$
$272$	−30.0000			−1.81902
$273$		0			0
$274$	−36.0000			−2.17484
$275$			3.46410i			0.208893i
$276$	6.00000			0.361158
$277$	14.0000			0.841178			0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000			0.841178			0.420589	+	0.907251i	$0.361823\pi$
$278$	−30.0000			−1.79928
$279$		−	10.3923i		−	0.622171i
$280$	−3.00000	+	3.46410i	−0.179284	+	0.207020i
$281$		−	13.8564i		−	0.826604i	−0.910594	−	0.413302i	$-0.864375\pi$
$281$		−	13.8564i		−	0.826604i	0.910594	−	0.413302i	$-0.135625\pi$
$282$	36.0000			2.14377
$283$		−	17.3205i		−	1.02960i	−0.857311	−	0.514799i	$-0.827867\pi$
$283$		−	17.3205i		−	1.02960i	0.857311	−	0.514799i	$-0.172133\pi$
$284$			3.46410i			0.205557i
$285$	6.00000			0.355409
$286$		0			0
$287$	12.0000	+	10.3923i	0.708338	+	0.613438i
$288$		−	15.5885i		−	0.918559i
$289$	19.0000			1.11765
$290$	−12.0000			−0.704664
$291$	−12.0000			−0.703452
$292$			6.92820i			0.405442i
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$	−3.00000	−	20.7846i	−0.174964	−	1.21218i
$295$	12.0000			0.698667
$296$			3.46410i			0.201347i
$297$	−18.0000			−1.04447
$298$	−12.0000			−0.695141
$299$		0			0
$300$			1.73205i			0.100000i
$301$	−16.0000	−	13.8564i	−0.922225	−	0.798670i
$302$			13.8564i			0.797347i
$303$			10.3923i			0.597022i
$304$			17.3205i			0.993399i
$305$		−	6.92820i		−	0.396708i
$306$			31.1769i			1.78227i
$307$			24.2487i			1.38395i	0.721923	+	0.691974i	$0.243259\pi$
$307$			24.2487i			1.38395i	−0.721923	+	0.691974i	$0.756741\pi$
$308$	6.00000	−	6.92820i	0.341882	−	0.394771i
$309$	6.00000			0.341328
$310$	−6.00000			−0.340777
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			20.7846i			1.17482i	0.809291	+	0.587408i	$0.199852\pi$
$313$			20.7846i			1.17482i	−0.809291	+	0.587408i	$0.800148\pi$
$314$	−24.0000			−1.35440
$315$	6.00000	+	5.19615i	0.338062	+	0.292770i
$316$	−8.00000			−0.450035
$317$		−	27.7128i		−	1.55651i	−0.627950	−	0.778253i	$-0.716106\pi$
$317$		−	27.7128i		−	1.55651i	0.627950	−	0.778253i	$-0.283894\pi$
$318$		0			0
$319$	−24.0000			−1.34374
$320$	1.00000			0.0559017
$321$	−18.0000			−1.00466
$322$	−12.0000	−	10.3923i	−0.668734	−	0.579141i
$323$		−	20.7846i		−	1.15649i
$324$	−9.00000			−0.500000
$325$		0			0
$326$			27.7128i			1.53487i
$327$			3.46410i			0.191565i
$328$		−	10.3923i		−	0.573819i
$329$	−24.0000	−	20.7846i	−1.32316	−	1.14589i
$330$			10.3923i			0.572078i
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$		0			0
$333$	6.00000			0.328798
$334$		−	20.7846i		−	1.13728i
$335$	−8.00000			−0.437087
$336$	−15.0000	+	17.3205i	−0.818317	+	0.944911i
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		−	22.5167i		−	1.22474i
$339$	12.0000			0.651751
$340$	6.00000			0.325396
$341$	−12.0000			−0.649836
$342$	18.0000			0.973329
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$			13.8564i			0.747087i
$345$	6.00000			0.323029
$346$		−	31.1769i		−	1.67608i
$347$			17.3205i			0.929814i	0.885360	+	0.464907i	$0.153912\pi$
$347$			17.3205i			0.929814i	−0.885360	+	0.464907i	$0.846088\pi$
$348$	−12.0000			−0.643268
$349$		−	6.92820i		−	0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$		−	6.92820i		−	0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$	3.00000	−	3.46410i	0.160357	−	0.185164i
$351$		0			0
$352$	−18.0000			−0.959403
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$	36.0000			1.91338
$355$			3.46410i			0.183855i
$356$	−6.00000			−0.317999
$357$	18.0000	−	20.7846i	0.952661	−	1.10004i
$358$	−18.0000			−0.951330
$359$		−	3.46410i		−	0.182828i	−0.995813	−	0.0914141i	$-0.970861\pi$
$359$		−	3.46410i		−	0.182828i	0.995813	−	0.0914141i	$-0.0291387\pi$
$360$		−	5.19615i		−	0.273861i
$361$	7.00000			0.368421
$362$	36.0000			1.89212
$363$			1.73205i			0.0909091i
$364$		0			0
$365$			6.92820i			0.362639i
$366$		−	20.7846i		−	1.08643i
$367$		−	10.3923i		−	0.542474i	−0.962513	−	0.271237i	$-0.912567\pi$
$367$		−	10.3923i		−	0.542474i	0.962513	−	0.271237i	$-0.0874327\pi$
$368$			17.3205i			0.902894i
$369$	−18.0000			−0.937043
$370$		−	3.46410i		−	0.180090i
$371$		0			0
$372$	−6.00000			−0.311086
$373$	14.0000			0.724893			0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000			0.724893			0.362446	+	0.932005i	$0.381942\pi$
$374$	36.0000			1.86152
$375$			1.73205i			0.0894427i
$376$			20.7846i			1.07188i
$377$		0			0
$378$	18.0000	+	15.5885i	0.925820	+	0.801784i
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		−	3.46410i		−	0.177705i
$381$			6.92820i			0.354943i
$382$	18.0000			0.920960
$383$	12.0000			0.613171			0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000			0.613171			0.306586	+	0.951843i	$0.400813\pi$
$384$	21.0000			1.07165
$385$	6.00000	−	6.92820i	0.305788	−	0.353094i
$386$			24.2487i			1.23423i
$387$	24.0000			1.21999
$388$			6.92820i			0.351726i
$389$		−	6.92820i		−	0.351274i	−0.984455	−	0.175637i	$-0.943802\pi$
$389$		−	6.92820i		−	0.351274i	0.984455	−	0.175637i	$-0.0561985\pi$
$390$		0			0
$391$		−	20.7846i		−	1.05112i
$392$	12.0000	−	1.73205i	0.606092	−	0.0874818i
$393$			20.7846i			1.04844i
$394$	24.0000			1.20910
$395$	−8.00000			−0.402524
$396$			10.3923i			0.522233i
$397$		−	13.8564i		−	0.695433i	−0.937600	−	0.347717i	$-0.886957\pi$
$397$		−	13.8564i		−	0.695433i	0.937600	−	0.347717i	$-0.113043\pi$
$398$	−18.0000			−0.902258
$399$	−12.0000	−	10.3923i	−0.600751	−	0.520266i
$400$	−5.00000			−0.250000
$401$			27.7128i			1.38391i	0.721940	+	0.691956i	$0.243251\pi$
$401$			27.7128i			1.38391i	−0.721940	+	0.691956i	$0.756749\pi$
$402$	−24.0000			−1.19701
$403$		0			0
$404$	6.00000			0.298511
$405$	−9.00000			−0.447214
$406$	24.0000	+	20.7846i	1.19110	+	1.03152i
$407$		−	6.92820i		−	0.343418i
$408$	−18.0000			−0.891133
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$			10.3923i			0.513239i
$411$	−36.0000			−1.77575
$412$		−	3.46410i		−	0.170664i
$413$	−24.0000	−	20.7846i	−1.18096	−	1.02274i
$414$	18.0000			0.884652
$415$		0			0
$416$		0			0
$417$	−30.0000			−1.46911
$418$		−	20.7846i		−	1.01661i
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$	3.00000	−	3.46410i	0.146385	−	0.169031i
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		−	6.92820i		−	0.337260i
$423$	36.0000			1.75038
$424$		0			0
$425$	6.00000			0.291043
$426$			10.3923i			0.503509i
$427$	−12.0000	+	13.8564i	−0.580721	+	0.670559i
$428$			10.3923i			0.502331i
$429$		0			0
$430$		−	13.8564i		−	0.668215i
$431$			10.3923i			0.500580i	0.968171	+	0.250290i	$0.0805259\pi$
$431$			10.3923i			0.500580i	−0.968171	+	0.250290i	$0.919474\pi$
$432$		−	25.9808i		−	1.25000i
$433$			6.92820i			0.332948i	0.986046	+	0.166474i	$0.0532382\pi$
$433$			6.92820i			0.332948i	−0.986046	+	0.166474i	$0.946762\pi$
$434$	12.0000	+	10.3923i	0.576018	+	0.498847i
$435$	−12.0000			−0.575356
$436$	2.00000			0.0957826
$437$	−12.0000			−0.574038
$438$			20.7846i			0.993127i
$439$			17.3205i			0.826663i	0.910581	+	0.413331i	$0.135635\pi$
$439$			17.3205i			0.826663i	−0.910581	+	0.413331i	$0.864365\pi$
$440$	−6.00000			−0.286039
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$		0			0
$443$		−	10.3923i		−	0.493753i	−0.969047	−	0.246877i	$-0.920596\pi$
$443$		−	10.3923i		−	0.493753i	0.969047	−	0.246877i	$-0.0794043\pi$
$444$		−	3.46410i		−	0.164399i
$445$	−6.00000			−0.284427
$446$	30.0000			1.42054
$447$	−12.0000			−0.567581
$448$	−2.00000	−	1.73205i	−0.0944911	−	0.0818317i
$449$			13.8564i			0.653924i	0.945037	+	0.326962i	$0.106025\pi$
$449$			13.8564i			0.653924i	−0.945037	+	0.326962i	$0.893975\pi$
$450$			5.19615i			0.244949i
$451$			20.7846i			0.978709i
$452$		−	6.92820i		−	0.325875i
$453$			13.8564i			0.651031i
$454$			41.5692i			1.95094i
$455$		0			0
$456$			10.3923i			0.486664i
$457$	−38.0000			−1.77757			−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000			−1.77757			−0.888783	+	0.458329i	$0.848448\pi$
$458$	12.0000			0.560723
$459$			31.1769i			1.45521i
$460$		−	3.46410i		−	0.161515i
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$	18.0000	−	20.7846i	0.837436	−	0.966988i
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		−	34.6410i		−	1.60817i
$465$	−6.00000			−0.278243
$466$	−12.0000			−0.555889
$467$	24.0000			1.11059			0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000			1.11059			0.555294	+	0.831654i	$0.312606\pi$
$468$		0			0
$469$	16.0000	+	13.8564i	0.738811	+	0.639829i
$470$		−	20.7846i		−	0.958723i
$471$	−24.0000			−1.10586
$472$			20.7846i			0.956689i
$473$		−	27.7128i		−	1.27424i
$474$	−24.0000			−1.10236
$475$		−	3.46410i		−	0.158944i
$476$	−12.0000	−	10.3923i	−0.550019	−	0.476331i
$477$		0			0
$478$	18.0000			0.823301
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	−9.00000			−0.410792
$481$		0			0
$482$		0			0
$483$	−12.0000	−	10.3923i	−0.546019	−	0.472866i
$484$	1.00000			0.0454545
$485$			6.92820i			0.314594i
$486$	−27.0000			−1.22474
$487$	4.00000			0.181257			0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000			0.181257			0.0906287	+	0.995885i	$0.471112\pi$
$488$	12.0000			0.543214
$489$			27.7128i			1.25322i
$490$	−12.0000	+	1.73205i	−0.542105	+	0.0782461i
$491$		−	24.2487i		−	1.09433i	−0.837025	−	0.547165i	$-0.815707\pi$
$491$		−	24.2487i		−	1.09433i	0.837025	−	0.547165i	$-0.184293\pi$
$492$			10.3923i			0.468521i
$493$			41.5692i			1.87218i
$494$		0			0
$495$			10.3923i			0.467099i
$496$		−	17.3205i		−	0.777714i
$497$	6.00000	−	6.92820i	0.269137	−	0.310772i
$498$		0			0
$499$	−20.0000			−0.895323			−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000			−0.895323			−0.447661	+	0.894203i	$0.647743\pi$
$500$	1.00000			0.0447214
$501$		−	20.7846i		−	0.928588i
$502$		−	20.7846i		−	0.927663i
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$	−9.00000	+	10.3923i	−0.400892	+	0.462910i
$505$	6.00000			0.266996
$506$		−	20.7846i		−	0.923989i
$507$		−	22.5167i		−	1.00000i
$508$	4.00000			0.177471
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$	18.0000			0.797053
$511$	12.0000	−	13.8564i	0.530849	−	0.612971i
$512$		−	8.66025i		−	0.382733i
$513$	18.0000			0.794719
$514$		−	10.3923i		−	0.458385i
$515$		−	3.46410i		−	0.152647i
$516$		−	13.8564i		−	0.609994i
$517$		−	41.5692i		−	1.82821i
$518$	−6.00000	+	6.92820i	−0.263625	+	0.304408i
$519$		−	31.1769i		−	1.36851i
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$	−36.0000			−1.57568
$523$		−	17.3205i		−	0.757373i	−0.925525	−	0.378686i	$-0.876376\pi$
$523$		−	17.3205i		−	0.757373i	0.925525	−	0.378686i	$-0.123624\pi$
$524$	12.0000			0.524222
$525$	3.00000	−	3.46410i	0.130931	−	0.151186i
$526$	42.0000			1.83129
$527$			20.7846i			0.905392i
$528$	−30.0000			−1.30558
$529$	11.0000			0.478261
$530$		0			0
$531$	36.0000			1.56227
$532$	−6.00000	+	6.92820i	−0.260133	+	0.300376i
$533$		0			0
$534$	−18.0000			−0.778936
$535$			10.3923i			0.449299i
$536$		−	13.8564i		−	0.598506i
$537$	−18.0000			−0.776757
$538$		−	31.1769i		−	1.34413i
$539$	−24.0000	+	3.46410i	−1.03375	+	0.149209i
$540$			5.19615i			0.223607i
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$	−42.0000			−1.80405
$543$	36.0000			1.54491
$544$			31.1769i			1.33670i
$545$	2.00000			0.0856706
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$			20.7846i			0.887875i
$549$		−	20.7846i		−	0.887066i
$550$	6.00000			0.255841
$551$	24.0000			1.02243
$552$			10.3923i			0.442326i
$553$	16.0000	+	13.8564i	0.680389	+	0.589234i
$554$		−	24.2487i		−	1.03023i
$555$		−	3.46410i		−	0.147043i
$556$			17.3205i			0.734553i
$557$			27.7128i			1.17423i	0.809504	+	0.587115i	$0.199736\pi$
$557$			27.7128i			1.17423i	−0.809504	+	0.587115i	$0.800264\pi$
$558$	−18.0000			−0.762001
$559$		0			0
$560$	10.0000	+	8.66025i	0.422577	+	0.365963i
$561$	36.0000			1.51992
$562$	−24.0000			−1.01238
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		−	20.7846i		−	0.875190i
$565$		−	6.92820i		−	0.291472i
$566$	−30.0000			−1.26099
$567$	18.0000	+	15.5885i	0.755929	+	0.654654i
$568$	−6.00000			−0.251754
$569$			27.7128i			1.16178i	0.813982	+	0.580891i	$0.197296\pi$
$569$			27.7128i			1.16178i	−0.813982	+	0.580891i	$0.802704\pi$
$570$		−	10.3923i		−	0.435286i
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$	18.0000			0.751961
$574$	18.0000	−	20.7846i	0.751305	−	0.867533i
$575$		−	3.46410i		−	0.144463i
$576$	3.00000			0.125000
$577$		−	34.6410i		−	1.44212i	−0.692870	−	0.721062i	$-0.743654\pi$
$577$		−	34.6410i		−	1.44212i	0.692870	−	0.721062i	$-0.256346\pi$
$578$		−	32.9090i		−	1.36883i
$579$			24.2487i			1.00774i
$580$			6.92820i			0.287678i
$581$		0			0
$582$			20.7846i			0.861550i
$583$		0			0
$584$	−12.0000			−0.496564
$585$		0			0
$586$			10.3923i			0.429302i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−12.0000	+	1.73205i	−0.494872	+	0.0714286i
$589$	12.0000			0.494451
$590$		−	20.7846i		−	0.855689i
$591$	24.0000			0.987228
$592$	10.0000			0.410997
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$			31.1769i			1.27920i
$595$	−12.0000	−	10.3923i	−0.491952	−	0.426043i
$596$			6.92820i			0.283790i
$597$	−18.0000			−0.736691
$598$		0			0
$599$			24.2487i			0.990775i	0.868672	+	0.495388i	$0.164974\pi$
$599$			24.2487i			0.990775i	−0.868672	+	0.495388i	$0.835026\pi$
$600$	−3.00000			−0.122474
$601$		−	13.8564i		−	0.565215i	−0.959236	−	0.282607i	$-0.908801\pi$
$601$		−	13.8564i		−	0.565215i	0.959236	−	0.282607i	$-0.0911993\pi$
$602$	−24.0000	+	27.7128i	−0.978167	+	1.12949i
$603$	−24.0000			−0.977356
$604$	8.00000			0.325515
$605$	1.00000			0.0406558
$606$	18.0000			0.731200
$607$			17.3205i			0.703018i	0.936185	+	0.351509i	$0.114331\pi$
$607$			17.3205i			0.703018i	−0.936185	+	0.351509i	$0.885669\pi$
$608$	18.0000			0.729996
$609$	24.0000	+	20.7846i	0.972529	+	0.842235i
$610$	−12.0000			−0.485866
$611$		0			0
$612$	18.0000			0.727607
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$	42.0000			1.69498
$615$			10.3923i			0.419058i
$616$	12.0000	+	10.3923i	0.483494	+	0.418718i
$617$			6.92820i			0.278919i	0.990228	+	0.139459i	$0.0445365\pi$
$617$			6.92820i			0.278919i	−0.990228	+	0.139459i	$0.955464\pi$
$618$		−	10.3923i		−	0.418040i
$619$		−	17.3205i		−	0.696170i	−0.937463	−	0.348085i	$-0.886832\pi$
$619$		−	17.3205i		−	0.696170i	0.937463	−	0.348085i	$-0.113168\pi$
$620$			3.46410i			0.139122i
$621$	18.0000			0.722315
$622$		0			0
$623$	12.0000	+	10.3923i	0.480770	+	0.416359i
$624$		0			0
$625$	1.00000			0.0400000
$626$	36.0000			1.43885
$627$		−	20.7846i		−	0.830057i
$628$			13.8564i			0.552931i
$629$	−12.0000			−0.478471
$630$	9.00000	−	10.3923i	0.358569	−	0.414039i
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		−	13.8564i		−	0.551178i
$633$		−	6.92820i		−	0.275371i
$634$	−48.0000			−1.90632
$635$	4.00000			0.158735
$636$		0			0
$637$		0			0
$638$			41.5692i			1.64574i
$639$			10.3923i			0.411113i
$640$		−	12.1244i		−	0.479257i
$641$		−	27.7128i		−	1.09459i	−0.836940	−	0.547295i	$-0.815658\pi$
$641$		−	27.7128i		−	1.09459i	0.836940	−	0.547295i	$-0.184342\pi$
$642$			31.1769i			1.23045i
$643$		−	31.1769i		−	1.22950i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$		−	31.1769i		−	1.22950i	0.788723	−	0.614749i	$-0.210743\pi$
$644$	−6.00000	+	6.92820i	−0.236433	+	0.273009i
$645$		−	13.8564i		−	0.545595i
$646$	−36.0000			−1.41640
$647$	12.0000			0.471769			0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000			0.471769			0.235884	+	0.971781i	$0.424201\pi$
$648$		−	15.5885i		−	0.612372i
$649$		−	41.5692i		−	1.63173i
$650$		0			0
$651$	12.0000	+	10.3923i	0.470317	+	0.407307i
$652$	16.0000			0.626608
$653$			41.5692i			1.62673i	0.581754	+	0.813365i	$0.302367\pi$
$653$			41.5692i			1.62673i	−0.581754	+	0.813365i	$0.697633\pi$
$654$	6.00000			0.234619
$655$	12.0000			0.468879
$656$	−30.0000			−1.17130
$657$			20.7846i			0.810885i
$658$	−36.0000	+	41.5692i	−1.40343	+	1.62054i
$659$		−	10.3923i		−	0.404827i	−0.979300	−	0.202413i	$-0.935122\pi$
$659$		−	10.3923i		−	0.404827i	0.979300	−	0.202413i	$-0.0648785\pi$
$660$	6.00000			0.233550
$661$			48.4974i			1.88633i	0.332323	+	0.943166i	$0.392168\pi$
$661$			48.4974i			1.88633i	−0.332323	+	0.943166i	$0.607832\pi$
$662$		−	48.4974i		−	1.88491i
$663$		0			0
$664$		0			0
$665$	−6.00000	+	6.92820i	−0.232670	+	0.268664i
$666$		−	10.3923i		−	0.402694i
$667$	24.0000			0.929284
$668$	−12.0000			−0.464294
$669$	30.0000			1.15987
$670$			13.8564i			0.535320i
$671$	−24.0000			−0.926510
$672$	18.0000	+	15.5885i	0.694365	+	0.601338i
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$			24.2487i			0.934025i
$675$			5.19615i			0.200000i
$676$	−13.0000			−0.500000
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		−	20.7846i		−	0.798228i
$679$	12.0000	−	13.8564i	0.460518	−	0.531760i
$680$			10.3923i			0.398527i
$681$			41.5692i			1.59294i
$682$			20.7846i			0.795884i
$683$			17.3205i			0.662751i	0.943499	+	0.331375i	$0.107513\pi$
$683$			17.3205i			0.662751i	−0.943499	+	0.331375i	$0.892487\pi$
$684$		−	10.3923i		−	0.397360i
$685$			20.7846i			0.794139i
$686$	27.0000	+	17.3205i	1.03086	+	0.661300i
$687$	12.0000			0.457829
$688$	40.0000			1.52499
$689$		0			0
$690$		−	10.3923i		−	0.395628i
$691$		−	31.1769i		−	1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$		−	31.1769i		−	1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$	−18.0000			−0.684257
$693$	18.0000	−	20.7846i	0.683763	−	0.789542i
$694$	30.0000			1.13878
$695$			17.3205i			0.657004i
$696$		−	20.7846i		−	0.787839i
$697$	36.0000			1.36360
$698$	−12.0000			−0.454207
$699$	−12.0000			−0.453882
$700$	−2.00000	−	1.73205i	−0.0755929	−	0.0654654i
$701$			20.7846i			0.785024i	0.919747	+	0.392512i	$0.128394\pi$
$701$			20.7846i			0.785024i	−0.919747	+	0.392512i	$0.871606\pi$
$702$		0			0
$703$			6.92820i			0.261302i
$704$		−	3.46410i		−	0.130558i
$705$		−	20.7846i		−	0.782794i
$706$		−	10.3923i		−	0.391120i
$707$	−12.0000	−	10.3923i	−0.451306	−	0.390843i
$708$		−	20.7846i		−	0.781133i
$709$	22.0000			0.826227			0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000			0.826227			0.413114	+	0.910679i	$0.364441\pi$
$710$	6.00000			0.225176
$711$	−24.0000			−0.900070
$712$		−	10.3923i		−	0.389468i
$713$	12.0000			0.449404
$714$	−36.0000	−	31.1769i	−1.34727	−	1.16677i
$715$		0			0
$716$			10.3923i			0.388379i
$717$	18.0000			0.672222
$718$	−6.00000			−0.223918
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−15.0000			−0.559017
$721$	−6.00000	+	6.92820i	−0.223452	+	0.258020i
$722$		−	12.1244i		−	0.451222i
$723$		0			0
$724$		−	20.7846i		−	0.772454i
$725$			6.92820i			0.257307i
$726$	3.00000			0.111340
$727$			3.46410i			0.128476i	0.997935	+	0.0642382i	$0.0204617\pi$
$727$			3.46410i			0.128476i	−0.997935	+	0.0642382i	$0.979538\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$	12.0000			0.444140
$731$	−48.0000			−1.77534
$732$	−12.0000			−0.443533
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$	−18.0000			−0.664392
$735$	−12.0000	+	1.73205i	−0.442627	+	0.0638877i
$736$	18.0000			0.663489
$737$			27.7128i			1.02081i
$738$			31.1769i			1.14764i
$739$	−4.00000			−0.147142			−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000			−0.147142			−0.0735712	+	0.997290i	$0.523440\pi$
$740$	−2.00000			−0.0735215
$741$		0			0
$742$		0			0
$743$		−	31.1769i		−	1.14377i	−0.820334	−	0.571885i	$-0.806212\pi$
$743$		−	31.1769i		−	1.14377i	0.820334	−	0.571885i	$-0.193788\pi$
$744$		−	10.3923i		−	0.381000i
$745$			6.92820i			0.253830i
$746$		−	24.2487i		−	0.887808i
$747$		0			0
$748$		−	20.7846i		−	0.759961i
$749$	18.0000	−	20.7846i	0.657706	−	0.759453i
$750$	3.00000			0.109545
$751$	16.0000			0.583848			0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000			0.583848			0.291924	+	0.956441i	$0.405705\pi$
$752$	60.0000			2.18797
$753$		−	20.7846i		−	0.757433i
$754$		0			0
$755$	8.00000			0.291150
$756$	9.00000	−	10.3923i	0.327327	−	0.377964i
$757$	−2.00000			−0.0726912			−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000			−0.0726912			−0.0363456	+	0.999339i	$0.511572\pi$
$758$		−	6.92820i		−	0.251644i
$759$		−	20.7846i		−	0.754434i
$760$	6.00000			0.217643
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$	12.0000			0.434714
$763$	−4.00000	−	3.46410i	−0.144810	−	0.125409i
$764$		−	10.3923i		−	0.375980i
$765$	18.0000			0.650791
$766$		−	20.7846i		−	0.750978i
$767$		0			0
$768$		−	32.9090i		−	1.18750i
$769$			41.5692i			1.49902i	0.661991	+	0.749512i	$0.269712\pi$
$769$			41.5692i			1.49902i	−0.661991	+	0.749512i	$0.730288\pi$
$770$	−12.0000	−	10.3923i	−0.432450	−	0.374513i
$771$		−	10.3923i		−	0.374270i
$772$	14.0000			0.503871
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$		−	41.5692i		−	1.49417i
$775$			3.46410i			0.124434i
$776$	−12.0000			−0.430775
$777$	−6.00000	+	6.92820i	−0.215249	+	0.248548i
$778$	−12.0000			−0.430221
$779$		−	20.7846i		−	0.744686i
$780$		0			0
$781$	12.0000			0.429394
$782$	−36.0000			−1.28736
$783$	−36.0000			−1.28654
$784$	−5.00000	−	34.6410i	−0.178571	−	1.23718i
$785$			13.8564i			0.494556i
$786$	36.0000			1.28408
$787$			24.2487i			0.864373i	0.901784	+	0.432187i	$0.142258\pi$
$787$			24.2487i			0.864373i	−0.901784	+	0.432187i	$0.857742\pi$
$788$		−	13.8564i		−	0.493614i
$789$	42.0000			1.49524
$790$			13.8564i			0.492989i
$791$	−12.0000	+	13.8564i	−0.426671	+	0.492677i
$792$	−18.0000			−0.639602
$793$		0			0
$794$	−24.0000			−0.851728
$795$		0			0
$796$			10.3923i			0.368345i
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$	−18.0000	+	20.7846i	−0.637193	+	0.735767i
$799$	−72.0000			−2.54718
$800$			5.19615i			0.183712i
$801$	−18.0000			−0.635999
$802$	48.0000			1.69494
$803$	24.0000			0.846942
$804$			13.8564i			0.488678i
$805$	−6.00000	+	6.92820i	−0.211472	+	0.244187i
$806$		0			0
$807$		−	31.1769i		−	1.09748i
$808$			10.3923i			0.365600i
$809$		−	55.4256i		−	1.94866i	−0.225122	−	0.974331i	$-0.572278\pi$
$809$		−	55.4256i		−	1.94866i	0.225122	−	0.974331i	$-0.427722\pi$
$810$			15.5885i			0.547723i
$811$			38.1051i			1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$811$			38.1051i			1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$812$	12.0000	−	13.8564i	0.421117	−	0.486265i
$813$	−42.0000			−1.47300
$814$	−12.0000			−0.420600
$815$	16.0000			0.560456
$816$			51.9615i			1.81902i
$817$			27.7128i			0.969549i
$818$		0			0
$819$		0			0
$820$	6.00000			0.209529
$821$		−	20.7846i		−	0.725388i	−0.931908	−	0.362694i	$-0.881857\pi$
$821$		−	20.7846i		−	0.725388i	0.931908	−	0.362694i	$-0.118143\pi$
$822$			62.3538i			2.17484i
$823$	44.0000			1.53374			0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000			1.53374			0.766872	+	0.641800i	$0.221812\pi$
$824$	6.00000			0.209020
$825$	6.00000			0.208893
$826$	−36.0000	+	41.5692i	−1.25260	+	1.44638i
$827$		−	38.1051i		−	1.32504i	−0.749042	−	0.662522i	$-0.769486\pi$
$827$		−	38.1051i		−	1.32504i	0.749042	−	0.662522i	$-0.230514\pi$
$828$		−	10.3923i		−	0.361158i
$829$		−	34.6410i		−	1.20313i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$		−	34.6410i		−	1.20313i	0.798823	−	0.601566i	$-0.205456\pi$
$830$		0			0
$831$		−	24.2487i		−	0.841178i
$832$		0			0
$833$	6.00000	+	41.5692i	0.207888	+	1.44029i
$834$			51.9615i			1.79928i
$835$	−12.0000			−0.415277
$836$	−12.0000			−0.415029
$837$	−18.0000			−0.622171
$838$		−	20.7846i		−	0.717992i
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$	6.00000	+	5.19615i	0.207020	+	0.179284i
$841$	−19.0000			−0.655172
$842$			17.3205i			0.596904i
$843$	−24.0000			−0.826604
$844$	−4.00000			−0.137686
$845$	−13.0000			−0.447214
$846$		−	62.3538i		−	2.14377i
$847$	−2.00000	−	1.73205i	−0.0687208	−	0.0595140i
$848$		0			0
$849$	−30.0000			−1.02960
$850$		−	10.3923i		−	0.356453i
$851$			6.92820i			0.237496i
$852$	6.00000			0.205557
$853$		−	41.5692i		−	1.42330i	−0.702533	−	0.711651i	$-0.747948\pi$
$853$		−	41.5692i		−	1.42330i	0.702533	−	0.711651i	$-0.252052\pi$
$854$	24.0000	+	20.7846i	0.821263	+	0.711235i
$855$		−	10.3923i		−	0.355409i
$856$	−18.0000			−0.615227
$857$	30.0000			1.02478			0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000			1.02478			0.512390	+	0.858753i	$0.328760\pi$
$858$		0			0
$859$			38.1051i			1.30013i	0.759879	+	0.650065i	$0.225258\pi$
$859$			38.1051i			1.30013i	−0.759879	+	0.650065i	$0.774742\pi$
$860$	−8.00000			−0.272798
$861$	18.0000	−	20.7846i	0.613438	−	0.708338i
$862$	18.0000			0.613082
$863$		−	17.3205i		−	0.589597i	−0.955559	−	0.294798i	$-0.904747\pi$
$863$		−	17.3205i		−	0.589597i	0.955559	−	0.294798i	$-0.0952525\pi$
$864$	−27.0000			−0.918559
$865$	−18.0000			−0.612018
$866$	12.0000			0.407777
$867$		−	32.9090i		−	1.11765i
$868$	6.00000	−	6.92820i	0.203653	−	0.235159i
$869$			27.7128i			0.940093i
$870$			20.7846i			0.704664i
$871$		0			0
$872$			3.46410i			0.117309i
$873$			20.7846i			0.703452i
$874$			20.7846i			0.703050i
$875$	−2.00000	−	1.73205i	−0.0676123	−	0.0585540i
$876$	12.0000			0.405442
$877$	38.0000			1.28317			0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000			1.28317			0.641584	+	0.767052i	$0.278277\pi$
$878$	30.0000			1.01245
$879$			10.3923i			0.350524i
$880$			17.3205i			0.583874i
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$	−36.0000	+	5.19615i	−1.21218	+	0.174964i
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$		−	20.7846i		−	0.698667i
$886$	−18.0000			−0.604722
$887$	−36.0000			−1.20876			−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000			−1.20876			−0.604381	+	0.796696i	$0.706579\pi$
$888$	6.00000			0.201347
$889$	−8.00000	−	6.92820i	−0.268311	−	0.232364i
$890$			10.3923i			0.348351i
$891$			31.1769i			1.04447i
$892$		−	17.3205i		−	0.579934i
$893$			41.5692i			1.39106i
$894$			20.7846i			0.695141i
$895$			10.3923i			0.347376i
$896$	−21.0000	+	24.2487i	−0.701561	+	0.810093i
$897$		0			0
$898$	24.0000			0.800890
$899$	−24.0000			−0.800445
$900$	3.00000			0.100000
$901$		0			0
$902$	36.0000			1.19867
$903$	−24.0000	+	27.7128i	−0.798670	+	0.922225i
$904$	12.0000			0.399114
$905$		−	20.7846i		−	0.690904i
$906$	24.0000			0.797347
$907$	−32.0000			−1.06254			−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000			−1.06254			−0.531271	+	0.847202i	$0.678286\pi$
$908$	24.0000			0.796468
$909$	18.0000			0.597022
$910$		0			0
$911$		−	17.3205i		−	0.573854i	−0.957952	−	0.286927i	$-0.907366\pi$
$911$		−	17.3205i		−	0.573854i	0.957952	−	0.286927i	$-0.0926337\pi$
$912$	30.0000			0.993399
$913$		0			0
$914$			65.8179i			2.17706i
$915$	−12.0000			−0.396708
$916$		−	6.92820i		−	0.228914i
$917$	−24.0000	−	20.7846i	−0.792550	−	0.686368i
$918$	54.0000			1.78227
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$	6.00000			0.197814
$921$	42.0000			1.38395
$922$		−	31.1769i		−	1.02676i
$923$		0			0
$924$	−12.0000	−	10.3923i	−0.394771	−	0.341882i
$925$	−2.00000			−0.0657596
$926$		−	34.6410i		−	1.13837i
$927$		−	10.3923i		−	0.341328i
$928$	−36.0000			−1.18176
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$			10.3923i			0.340777i
$931$	24.0000	−	3.46410i	0.786568	−	0.113531i
$932$			6.92820i			0.226941i
$933$		0			0
$934$		−	41.5692i		−	1.36019i
$935$		−	20.7846i		−	0.679729i
$936$		0			0
$937$			6.92820i			0.226335i	0.993576	+	0.113167i	$0.0360996\pi$
$937$			6.92820i			0.226335i	−0.993576	+	0.113167i	$0.963900\pi$
$938$	24.0000	−	27.7128i	0.783628	−	0.904855i
$939$	36.0000			1.17482
$940$	−12.0000			−0.391397
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$			41.5692i			1.35440i
$943$		−	20.7846i		−	0.676840i
$944$	60.0000			1.95283
$945$	9.00000	−	10.3923i	0.292770	−	0.338062i
$946$	−48.0000			−1.56061
$947$			31.1769i			1.01311i	0.862207	+	0.506557i	$0.169082\pi$
$947$			31.1769i			1.01311i	−0.862207	+	0.506557i	$0.830918\pi$
$948$			13.8564i			0.450035i
$949$		0			0
$950$	−6.00000			−0.194666
$951$	−48.0000			−1.55651
$952$	18.0000	−	20.7846i	0.583383	−	0.673633i
$953$		−	34.6410i		−	1.12213i	−0.827771	−	0.561066i	$-0.810391\pi$
$953$		−	34.6410i		−	1.12213i	0.827771	−	0.561066i	$-0.189609\pi$
$954$		0			0
$955$		−	10.3923i		−	0.336287i
$956$		−	10.3923i		−	0.336111i
$957$			41.5692i			1.34374i
$958$		0			0
$959$	36.0000	−	41.5692i	1.16250	−	1.34234i
$960$		−	1.73205i		−	0.0559017i
$961$	19.0000			0.612903
$962$		0			0
$963$			31.1769i			1.00466i
$964$		0			0
$965$	14.0000			0.450676
$966$	−18.0000	+	20.7846i	−0.579141	+	0.668734i
$967$	20.0000			0.643157			0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000			0.643157			0.321578	+	0.946883i	$0.395787\pi$
$968$			1.73205i			0.0556702i
$969$	−36.0000			−1.15649
$970$	12.0000			0.385297
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$			15.5885i			0.500000i
$973$	30.0000	−	34.6410i	0.961756	−	1.11054i
$974$		−	6.92820i		−	0.221994i
$975$		0			0
$976$		−	34.6410i		−	1.10883i
$977$		−	6.92820i		−	0.221653i	−0.993840	−	0.110826i	$-0.964650\pi$
$977$		−	6.92820i		−	0.221653i	0.993840	−	0.110826i	$-0.0353498\pi$
$978$	48.0000			1.53487
$979$			20.7846i			0.664279i
$980$	1.00000	+	6.92820i	0.0319438	+	0.221313i
$981$	6.00000			0.191565
$982$	−42.0000			−1.34027
$983$	−60.0000			−1.91370			−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000			−1.91370			−0.956851	+	0.290578i	$0.906153\pi$
$984$	−18.0000			−0.573819
$985$		−	13.8564i		−	0.441502i
$986$	72.0000			2.29295
$987$	−36.0000	+	41.5692i	−1.14589	+	1.32316i
$988$		0			0
$989$			27.7128i			0.881216i
$990$	18.0000			0.572078
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$	−18.0000			−0.571501
$993$		−	48.4974i		−	1.53902i
$994$	−12.0000	−	10.3923i	−0.380617	−	0.329624i
$995$			10.3923i			0.329458i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$			34.6410i			1.09654i
$999$		−	10.3923i		−	0.328798i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.2.b.a.41.1	✓	2
3.2	odd	2		105.2.b.b.41.2	yes	2
4.3	odd	2		1680.2.f.b.881.2		2
5.2	odd	4		525.2.g.b.524.4		4
5.3	odd	4		525.2.g.b.524.1		4
5.4	even	2		525.2.b.a.251.2		2
7.2	even	3		735.2.s.b.521.1		2
7.3	odd	6		735.2.s.f.656.1		2
7.4	even	3		735.2.s.d.656.1		2
7.5	odd	6		735.2.s.a.521.1		2
7.6	odd	2		105.2.b.b.41.1	yes	2
12.11	even	2		1680.2.f.c.881.2		2
15.2	even	4		525.2.g.c.524.2		4
15.8	even	4		525.2.g.c.524.3		4
15.14	odd	2		525.2.b.b.251.1		2
21.2	odd	6		735.2.s.f.521.1		2
21.5	even	6		735.2.s.d.521.1		2
21.11	odd	6		735.2.s.a.656.1		2
21.17	even	6		735.2.s.b.656.1		2
21.20	even	2	inner	105.2.b.a.41.2	yes	2
28.27	even	2		1680.2.f.c.881.1		2
35.13	even	4		525.2.g.c.524.1		4
35.27	even	4		525.2.g.c.524.4		4
35.34	odd	2		525.2.b.b.251.2		2
84.83	odd	2		1680.2.f.b.881.1		2
105.62	odd	4		525.2.g.b.524.2		4
105.83	odd	4		525.2.g.b.524.3		4
105.104	even	2		525.2.b.a.251.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.b.a.41.1	✓	2	1.1	even	1	trivial
105.2.b.a.41.2	yes	2	21.20	even	2	inner
105.2.b.b.41.1	yes	2	7.6	odd	2
105.2.b.b.41.2	yes	2	3.2	odd	2
525.2.b.a.251.1		2	105.104	even	2
525.2.b.a.251.2		2	5.4	even	2
525.2.b.b.251.1		2	15.14	odd	2
525.2.b.b.251.2		2	35.34	odd	2
525.2.g.b.524.1		4	5.3	odd	4
525.2.g.b.524.2		4	105.62	odd	4
525.2.g.b.524.3		4	105.83	odd	4
525.2.g.b.524.4		4	5.2	odd	4
525.2.g.c.524.1		4	35.13	even	4
525.2.g.c.524.2		4	15.2	even	4
525.2.g.c.524.3		4	15.8	even	4
525.2.g.c.524.4		4	35.27	even	4
735.2.s.a.521.1		2	7.5	odd	6
735.2.s.a.656.1		2	21.11	odd	6
735.2.s.b.521.1		2	7.2	even	3
735.2.s.b.656.1		2	21.17	even	6
735.2.s.d.521.1		2	21.5	even	6
735.2.s.d.656.1		2	7.4	even	3
735.2.s.f.521.1		2	21.2	odd	6
735.2.s.f.656.1		2	7.3	odd	6
1680.2.f.b.881.1		2	84.83	odd	2
1680.2.f.b.881.2		2	4.3	odd	2
1680.2.f.c.881.1		2	28.27	even	2
1680.2.f.c.881.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	105.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -1.73205i q^{3} -1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +(2.00000 + 1.73205i) q^{7} -1.73205i q^{8} -3.00000 q^{9} +1.73205i q^{10} +3.46410i q^{11} +1.73205i q^{12} +(3.00000 - 3.46410i) q^{14} +1.73205i q^{15} -5.00000 q^{16} +6.00000 q^{17} +5.19615i q^{18} -3.46410i q^{19} +1.00000 q^{20} +(3.00000 - 3.46410i) q^{21} +6.00000 q^{22} -3.46410i q^{23} -3.00000 q^{24} +1.00000 q^{25} +5.19615i q^{27} +(-2.00000 - 1.73205i) q^{28} +6.92820i q^{29} +3.00000 q^{30} +3.46410i q^{31} +5.19615i q^{32} +6.00000 q^{33} -10.3923i q^{34} +(-2.00000 - 1.73205i) q^{35} +3.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} +1.73205i q^{40} +6.00000 q^{41} +(-6.00000 - 5.19615i) q^{42} -8.00000 q^{43} -3.46410i q^{44} +3.00000 q^{45} -6.00000 q^{46} -12.0000 q^{47} +8.66025i q^{48} +(1.00000 + 6.92820i) q^{49} -1.73205i q^{50} -10.3923i q^{51} +9.00000 q^{54} -3.46410i q^{55} +(3.00000 - 3.46410i) q^{56} -6.00000 q^{57} +12.0000 q^{58} -12.0000 q^{59} -1.73205i q^{60} +6.92820i q^{61} +6.00000 q^{62} +(-6.00000 - 5.19615i) q^{63} -1.00000 q^{64} -10.3923i q^{66} +8.00000 q^{67} -6.00000 q^{68} -6.00000 q^{69} +(-3.00000 + 3.46410i) q^{70} -3.46410i q^{71} +5.19615i q^{72} -6.92820i q^{73} +3.46410i q^{74} -1.73205i q^{75} +3.46410i q^{76} +(-6.00000 + 6.92820i) q^{77} +8.00000 q^{79} +5.00000 q^{80} +9.00000 q^{81} -10.3923i q^{82} +(-3.00000 + 3.46410i) q^{84} -6.00000 q^{85} +13.8564i q^{86} +12.0000 q^{87} +6.00000 q^{88} +6.00000 q^{89} -5.19615i q^{90} +3.46410i q^{92} +6.00000 q^{93} +20.7846i q^{94} +3.46410i q^{95} +9.00000 q^{96} -6.92820i q^{97} +(12.0000 - 1.73205i) q^{98} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{5} - 6 q^{6} + 4 q^{7} - 6 q^{9} + 6 q^{14} - 10 q^{16} + 12 q^{17} + 2 q^{20} + 6 q^{21} + 12 q^{22} - 6 q^{24} + 2 q^{25} - 4 q^{28} + 6 q^{30} + 12 q^{33} - 4 q^{35} + 6 q^{36} - 4 q^{37}+ \cdots + 24 q^{98}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^5 - 6 * q^6 + 4 * q^7 - 6 * q^9 + 6 * q^14 - 10 * q^16 + 12 * q^17 + 2 * q^20 + 6 * q^21 + 12 * q^22 - 6 * q^24 + 2 * q^25 - 4 * q^28 + 6 * q^30 + 12 * q^33 - 4 * q^35 + 6 * q^36 - 4 * q^37 - 12 * q^38 + 12 * q^41 - 12 * q^42 - 16 * q^43 + 6 * q^45 - 12 * q^46 - 24 * q^47 + 2 * q^49 + 18 * q^54 + 6 * q^56 - 12 * q^57 + 24 * q^58 - 24 * q^59 + 12 * q^62 - 12 * q^63 - 2 * q^64 + 16 * q^67 - 12 * q^68 - 12 * q^69 - 6 * q^70 - 12 * q^77 + 16 * q^79 + 10 * q^80 + 18 * q^81 - 6 * q^84 - 12 * q^85 + 24 * q^87 + 12 * q^88 + 12 * q^89 + 12 * q^93 + 18 * q^96 + 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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