LMFDB - Embedded newform 420.2.d.b.41.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(41,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("420.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.d (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:	$3.35371688489$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	420.41
Dual form			420.2.d.b.41.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.50000 + 0.866025i) q^{3} +1.00000 q^{5} +(2.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} -5.19615i q^{11} +1.73205i q^{13} +(1.50000 + 0.866025i) q^{15} -3.00000 q^{17} +3.46410i q^{19} +(4.50000 - 0.866025i) q^{21} +1.00000 q^{25} +5.19615i q^{27} -5.19615i q^{29} +10.3923i q^{31} +(4.50000 - 7.79423i) q^{33} +(2.00000 - 1.73205i) q^{35} -8.00000 q^{37} +(-1.50000 + 2.59808i) q^{39} -6.00000 q^{41} +10.0000 q^{43} +(1.50000 + 2.59808i) q^{45} -3.00000 q^{47} +(1.00000 - 6.92820i) q^{49} +(-4.50000 - 2.59808i) q^{51} +10.3923i q^{53} -5.19615i q^{55} +(-3.00000 + 5.19615i) q^{57} -6.00000 q^{59} -6.92820i q^{61} +(7.50000 + 2.59808i) q^{63} +1.73205i q^{65} +2.00000 q^{67} -10.3923i q^{71} -6.92820i q^{73} +(1.50000 + 0.866025i) q^{75} +(-9.00000 - 10.3923i) q^{77} -13.0000 q^{79} +(-4.50000 + 7.79423i) q^{81} +12.0000 q^{83} -3.00000 q^{85} +(4.50000 - 7.79423i) q^{87} +(3.00000 + 3.46410i) q^{91} +(-9.00000 + 15.5885i) q^{93} +3.46410i q^{95} +1.73205i q^{97} +(13.5000 - 7.79423i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 2 q^{5} + 4 q^{7} + 3 q^{9} + 3 q^{15} - 6 q^{17} + 9 q^{21} + 2 q^{25} + 9 q^{33} + 4 q^{35} - 16 q^{37} - 3 q^{39} - 12 q^{41} + 20 q^{43} + 3 q^{45} - 6 q^{47} + 2 q^{49} - 9 q^{51} - 6 q^{57}+ \cdots + 27 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 2 * q^5 + 4 * q^7 + 3 * q^9 + 3 * q^15 - 6 * q^17 + 9 * q^21 + 2 * q^25 + 9 * q^33 + 4 * q^35 - 16 * q^37 - 3 * q^39 - 12 * q^41 + 20 * q^43 + 3 * q^45 - 6 * q^47 + 2 * q^49 - 9 * q^51 - 6 * q^57 - 12 * q^59 + 15 * q^63 + 4 * q^67 + 3 * q^75 - 18 * q^77 - 26 * q^79 - 9 * q^81 + 24 * q^83 - 6 * q^85 + 9 * q^87 + 6 * q^91 - 18 * q^93 + 27 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.50000	+	0.866025i	0.866025	+	0.500000i
$3$	1.50000	+	0.866025i	0.866025	+	0.500000i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$7$	2.00000	−	1.73205i	0.755929	−	0.654654i
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$		−	5.19615i		−	1.56670i	−0.621582	−	0.783349i	$-0.713510\pi$
$11$		−	5.19615i		−	1.56670i	0.621582	−	0.783349i	$-0.286490\pi$
$12$		0			0
$13$			1.73205i			0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$13$			1.73205i			0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$	1.50000	+	0.866025i	0.387298	+	0.223607i
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	4.50000	−	0.866025i	0.981981	−	0.188982i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			5.19615i			1.00000i
$28$		0			0
$29$		−	5.19615i		−	0.964901i	−0.875923	−	0.482451i	$-0.839747\pi$
$29$		−	5.19615i		−	0.964901i	0.875923	−	0.482451i	$-0.160253\pi$
$30$		0			0
$31$			10.3923i			1.86651i	0.359211	+	0.933257i	$0.383046\pi$
$31$			10.3923i			1.86651i	−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$	4.50000	−	7.79423i	0.783349	−	1.35680i
$34$		0			0
$35$	2.00000	−	1.73205i	0.338062	−	0.292770i
$36$		0			0
$37$	−8.00000			−1.31519			−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000			−1.31519			−0.657596	+	0.753371i	$0.728427\pi$
$38$		0			0
$39$	−1.50000	+	2.59808i	−0.240192	+	0.416025i
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$		0			0
$45$	1.50000	+	2.59808i	0.223607	+	0.387298i
$46$		0			0
$47$	−3.00000			−0.437595			−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000			−0.437595			−0.218797	+	0.975770i	$0.570213\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$	−4.50000	−	2.59808i	−0.630126	−	0.363803i
$52$		0			0
$53$			10.3923i			1.42749i	0.700404	+	0.713746i	$0.253003\pi$
$53$			10.3923i			1.42749i	−0.700404	+	0.713746i	$0.746997\pi$
$54$		0			0
$55$		−	5.19615i		−	0.700649i
$56$		0			0
$57$	−3.00000	+	5.19615i	−0.397360	+	0.688247i
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$		−	6.92820i		−	0.887066i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$		−	6.92820i		−	0.887066i	0.896258	−	0.443533i	$-0.146275\pi$
$62$		0			0
$63$	7.50000	+	2.59808i	0.944911	+	0.327327i
$64$		0			0
$65$			1.73205i			0.214834i
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.3923i		−	1.23334i	−0.787222	−	0.616670i	$-0.788481\pi$
$71$		−	10.3923i		−	1.23334i	0.787222	−	0.616670i	$-0.211519\pi$
$72$		0			0
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$		0			0
$75$	1.50000	+	0.866025i	0.173205	+	0.100000i
$76$		0			0
$77$	−9.00000	−	10.3923i	−1.02565	−	1.18431i
$78$		0			0
$79$	−13.0000			−1.46261			−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−13.0000			−1.46261			−0.731307	+	0.682048i	$0.761089\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	−3.00000			−0.325396
$86$		0			0
$87$	4.50000	−	7.79423i	0.482451	−	0.835629i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	3.00000	+	3.46410i	0.314485	+	0.363137i
$92$		0			0
$93$	−9.00000	+	15.5885i	−0.933257	+	1.61645i
$94$		0			0
$95$			3.46410i			0.355409i
$96$		0			0
$97$			1.73205i			0.175863i	0.996127	+	0.0879316i	$0.0280257\pi$
$97$			1.73205i			0.175863i	−0.996127	+	0.0879316i	$0.971974\pi$
$98$		0			0
$99$	13.5000	−	7.79423i	1.35680	−	0.783349i
$100$		0			0
$101$	−18.0000			−1.79107			−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000			−1.79107			−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$		−	1.73205i		−	0.170664i	−0.996353	−	0.0853320i	$-0.972805\pi$
$103$		−	1.73205i		−	0.170664i	0.996353	−	0.0853320i	$-0.0271951\pi$
$104$		0			0
$105$	4.50000	−	0.866025i	0.439155	−	0.0845154i
$106$		0			0
$107$			10.3923i			1.00466i	0.864675	+	0.502331i	$0.167524\pi$
$107$			10.3923i			1.00466i	−0.864675	+	0.502331i	$0.832476\pi$
$108$		0			0
$109$	−11.0000			−1.05361			−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000			−1.05361			−0.526804	+	0.849987i	$0.676610\pi$
$110$		0			0
$111$	−12.0000	−	6.92820i	−1.13899	−	0.657596i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.50000	+	2.59808i	−0.416025	+	0.240192i
$118$		0			0
$119$	−6.00000	+	5.19615i	−0.550019	+	0.476331i
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$	−9.00000	−	5.19615i	−0.811503	−	0.468521i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$	15.0000	+	8.66025i	1.32068	+	0.762493i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	6.00000	+	6.92820i	0.520266	+	0.600751i
$134$		0			0
$135$			5.19615i			0.447214i
$136$		0			0
$137$		−	20.7846i		−	1.77575i	−0.460086	−	0.887875i	$-0.652181\pi$
$137$		−	20.7846i		−	1.77575i	0.460086	−	0.887875i	$-0.347819\pi$
$138$		0			0
$139$		−	6.92820i		−	0.587643i	−0.955860	−	0.293821i	$-0.905073\pi$
$139$		−	6.92820i		−	0.587643i	0.955860	−	0.293821i	$-0.0949270\pi$
$140$		0			0
$141$	−4.50000	−	2.59808i	−0.378968	−	0.218797i
$142$		0			0
$143$	9.00000			0.752618
$144$		0			0
$145$		−	5.19615i		−	0.431517i
$146$		0			0
$147$	7.50000	−	9.52628i	0.618590	−	0.785714i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−5.00000			−0.406894			−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000			−0.406894			−0.203447	+	0.979086i	$0.565214\pi$
$152$		0			0
$153$	−4.50000	−	7.79423i	−0.363803	−	0.630126i
$154$		0			0
$155$			10.3923i			0.834730i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	−9.00000	+	15.5885i	−0.713746	+	1.23625i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.00000			0.626608			0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000			0.626608			0.313304	+	0.949653i	$0.398564\pi$
$164$		0			0
$165$	4.50000	−	7.79423i	0.350325	−	0.606780i
$166$		0			0
$167$	15.0000			1.16073			0.580367	−	0.814355i	$-0.302909\pi$
$167$	15.0000			1.16073			0.580367	+	0.814355i	$0.302909\pi$
$168$		0			0
$169$	10.0000			0.769231
$170$		0			0
$171$	−9.00000	+	5.19615i	−0.688247	+	0.397360i
$172$		0			0
$173$	9.00000			0.684257			0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000			0.684257			0.342129	+	0.939653i	$0.388852\pi$
$174$		0			0
$175$	2.00000	−	1.73205i	0.151186	−	0.130931i
$176$		0			0
$177$	−9.00000	−	5.19615i	−0.676481	−	0.390567i
$178$		0			0
$179$			10.3923i			0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$			10.3923i			0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$		0			0
$181$			17.3205i			1.28742i	0.765268	+	0.643712i	$0.222606\pi$
$181$			17.3205i			1.28742i	−0.765268	+	0.643712i	$0.777394\pi$
$182$		0			0
$183$	6.00000	−	10.3923i	0.443533	−	0.768221i
$184$		0			0
$185$	−8.00000			−0.588172
$186$		0			0
$187$			15.5885i			1.13994i
$188$		0			0
$189$	9.00000	+	10.3923i	0.654654	+	0.755929i
$190$		0			0
$191$			15.5885i			1.12794i	0.825795	+	0.563971i	$0.190727\pi$
$191$			15.5885i			1.12794i	−0.825795	+	0.563971i	$0.809273\pi$
$192$		0			0
$193$	16.0000			1.15171			0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000			1.15171			0.575853	+	0.817554i	$0.304670\pi$
$194$		0			0
$195$	−1.50000	+	2.59808i	−0.107417	+	0.186052i
$196$		0			0
$197$			10.3923i			0.740421i	0.928948	+	0.370211i	$0.120714\pi$
$197$			10.3923i			0.740421i	−0.928948	+	0.370211i	$0.879286\pi$
$198$		0			0
$199$		−	24.2487i		−	1.71895i	−0.511182	−	0.859473i	$-0.670792\pi$
$199$		−	24.2487i		−	1.71895i	0.511182	−	0.859473i	$-0.329208\pi$
$200$		0			0
$201$	3.00000	+	1.73205i	0.211604	+	0.122169i
$202$		0			0
$203$	−9.00000	−	10.3923i	−0.631676	−	0.729397i
$204$		0			0
$205$	−6.00000			−0.419058
$206$		0			0
$207$		0			0
$208$		0			0
$209$	18.0000			1.24509
$210$		0			0
$211$	25.0000			1.72107			0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000			1.72107			0.860535	+	0.509390i	$0.170129\pi$
$212$		0			0
$213$	9.00000	−	15.5885i	0.616670	−	1.06810i
$214$		0			0
$215$	10.0000			0.681994
$216$		0			0
$217$	18.0000	+	20.7846i	1.22192	+	1.41095i
$218$		0			0
$219$	6.00000	−	10.3923i	0.405442	−	0.702247i
$220$		0			0
$221$		−	5.19615i		−	0.349531i
$222$		0			0
$223$			5.19615i			0.347960i	0.984749	+	0.173980i	$0.0556628\pi$
$223$			5.19615i			0.347960i	−0.984749	+	0.173980i	$0.944337\pi$
$224$		0			0
$225$	1.50000	+	2.59808i	0.100000	+	0.173205i
$226$		0			0
$227$	27.0000			1.79205			0.896026	−	0.444001i	$-0.146441\pi$
$227$	27.0000			1.79205			0.896026	+	0.444001i	$0.146441\pi$
$228$		0			0
$229$		−	10.3923i		−	0.686743i	−0.939200	−	0.343371i	$-0.888431\pi$
$229$		−	10.3923i		−	0.686743i	0.939200	−	0.343371i	$-0.111569\pi$
$230$		0			0
$231$	−4.50000	−	23.3827i	−0.296078	−	1.53847i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	−3.00000			−0.195698
$236$		0			0
$237$	−19.5000	−	11.2583i	−1.26666	−	0.731307i
$238$		0			0
$239$		−	25.9808i		−	1.68056i	−0.542156	−	0.840278i	$-0.682392\pi$
$239$		−	25.9808i		−	1.68056i	0.542156	−	0.840278i	$-0.317608\pi$
$240$		0			0
$241$		−	6.92820i		−	0.446285i	−0.974786	−	0.223142i	$-0.928369\pi$
$241$		−	6.92820i		−	0.446285i	0.974786	−	0.223142i	$-0.0716315\pi$
$242$		0			0
$243$	−13.5000	+	7.79423i	−0.866025	+	0.500000i
$244$		0			0
$245$	1.00000	−	6.92820i	0.0638877	−	0.442627i
$246$		0			0
$247$	−6.00000			−0.381771
$248$		0			0
$249$	18.0000	+	10.3923i	1.14070	+	0.658586i
$250$		0			0
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−4.50000	−	2.59808i	−0.281801	−	0.162698i
$256$		0			0
$257$	−18.0000			−1.12281			−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000			−1.12281			−0.561405	+	0.827541i	$0.689739\pi$
$258$		0			0
$259$	−16.0000	+	13.8564i	−0.994192	+	0.860995i
$260$		0			0
$261$	13.5000	−	7.79423i	0.835629	−	0.482451i
$262$		0			0
$263$			31.1769i			1.92245i	0.275764	+	0.961225i	$0.411069\pi$
$263$			31.1769i			1.92245i	−0.275764	+	0.961225i	$0.588931\pi$
$264$		0			0
$265$			10.3923i			0.638394i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	12.0000			0.731653			0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000			0.731653			0.365826	+	0.930683i	$0.380786\pi$
$270$		0			0
$271$			6.92820i			0.420858i	0.977609	+	0.210429i	$0.0674861\pi$
$271$			6.92820i			0.420858i	−0.977609	+	0.210429i	$0.932514\pi$
$272$		0			0
$273$	1.50000	+	7.79423i	0.0907841	+	0.471728i
$274$		0			0
$275$		−	5.19615i		−	0.313340i
$276$		0			0
$277$	8.00000			0.480673			0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000			0.480673			0.240337	+	0.970690i	$0.422742\pi$
$278$		0			0
$279$	−27.0000	+	15.5885i	−1.61645	+	0.933257i
$280$		0			0
$281$			5.19615i			0.309976i	0.987916	+	0.154988i	$0.0495340\pi$
$281$			5.19615i			0.309976i	−0.987916	+	0.154988i	$0.950466\pi$
$282$		0			0
$283$			25.9808i			1.54440i	0.635382	+	0.772198i	$0.280843\pi$
$283$			25.9808i			1.54440i	−0.635382	+	0.772198i	$0.719157\pi$
$284$		0			0
$285$	−3.00000	+	5.19615i	−0.177705	+	0.307794i
$286$		0			0
$287$	−12.0000	+	10.3923i	−0.708338	+	0.613438i
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	−1.50000	+	2.59808i	−0.0879316	+	0.152302i
$292$		0			0
$293$	3.00000			0.175262			0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000			0.175262			0.0876309	+	0.996153i	$0.472070\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$		0			0
$297$	27.0000			1.56670
$298$		0			0
$299$		0			0
$300$		0			0
$301$	20.0000	−	17.3205i	1.15278	−	0.998337i
$302$		0			0
$303$	−27.0000	−	15.5885i	−1.55111	−	0.895533i
$304$		0			0
$305$		−	6.92820i		−	0.396708i
$306$		0			0
$307$		−	25.9808i		−	1.48280i	−0.671063	−	0.741400i	$-0.734162\pi$
$307$		−	25.9808i		−	1.48280i	0.671063	−	0.741400i	$-0.265838\pi$
$308$		0			0
$309$	1.50000	−	2.59808i	0.0853320	−	0.147799i
$310$		0			0
$311$	−30.0000			−1.70114			−0.850572	−	0.525859i	$-0.823744\pi$
$311$	−30.0000			−1.70114			−0.850572	+	0.525859i	$0.823744\pi$
$312$		0			0
$313$			22.5167i			1.27272i	0.771393	+	0.636358i	$0.219560\pi$
$313$			22.5167i			1.27272i	−0.771393	+	0.636358i	$0.780440\pi$
$314$		0			0
$315$	7.50000	+	2.59808i	0.422577	+	0.146385i
$316$		0			0
$317$			10.3923i			0.583690i	0.956466	+	0.291845i	$0.0942691\pi$
$317$			10.3923i			0.583690i	−0.956466	+	0.291845i	$0.905731\pi$
$318$		0			0
$319$	−27.0000			−1.51171
$320$		0			0
$321$	−9.00000	+	15.5885i	−0.502331	+	0.870063i
$322$		0			0
$323$		−	10.3923i		−	0.578243i
$324$		0			0
$325$			1.73205i			0.0960769i
$326$		0			0
$327$	−16.5000	−	9.52628i	−0.912452	−	0.526804i
$328$		0			0
$329$	−6.00000	+	5.19615i	−0.330791	+	0.286473i
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$	−12.0000	−	20.7846i	−0.657596	−	1.13899i
$334$		0			0
$335$	2.00000			0.109272
$336$		0			0
$337$	−20.0000			−1.08947			−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000			−1.08947			−0.544735	+	0.838608i	$0.683370\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	54.0000			2.92426
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		−	17.3205i		−	0.927146i	−0.886059	−	0.463573i	$-0.846567\pi$
$349$		−	17.3205i		−	0.927146i	0.886059	−	0.463573i	$-0.153433\pi$
$350$		0			0
$351$	−9.00000			−0.480384
$352$		0			0
$353$	−21.0000			−1.11772			−0.558859	−	0.829263i	$-0.688761\pi$
$353$	−21.0000			−1.11772			−0.558859	+	0.829263i	$0.688761\pi$
$354$		0			0
$355$		−	10.3923i		−	0.551566i
$356$		0			0
$357$	−13.5000	+	2.59808i	−0.714496	+	0.137505i
$358$		0			0
$359$			10.3923i			0.548485i	0.961661	+	0.274242i	$0.0884271\pi$
$359$			10.3923i			0.548485i	−0.961661	+	0.274242i	$0.911573\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	−24.0000	−	13.8564i	−1.25967	−	0.727273i
$364$		0			0
$365$		−	6.92820i		−	0.362639i
$366$		0			0
$367$		−	29.4449i		−	1.53701i	−0.639844	−	0.768505i	$-0.721001\pi$
$367$		−	29.4449i		−	1.53701i	0.639844	−	0.768505i	$-0.278999\pi$
$368$		0			0
$369$	−9.00000	−	15.5885i	−0.468521	−	0.811503i
$370$		0			0
$371$	18.0000	+	20.7846i	0.934513	+	1.07908i
$372$		0			0
$373$	20.0000			1.03556			0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000			1.03556			0.517780	+	0.855514i	$0.326758\pi$
$374$		0			0
$375$	1.50000	+	0.866025i	0.0774597	+	0.0447214i
$376$		0			0
$377$	9.00000			0.463524
$378$		0			0
$379$	−8.00000			−0.410932			−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000			−0.410932			−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$	3.00000	+	1.73205i	0.153695	+	0.0887357i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$	−9.00000	−	10.3923i	−0.458682	−	0.529641i
$386$		0			0
$387$	15.0000	+	25.9808i	0.762493	+	1.32068i
$388$		0			0
$389$			15.5885i			0.790366i	0.918602	+	0.395183i	$0.129319\pi$
$389$			15.5885i			0.790366i	−0.918602	+	0.395183i	$0.870681\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−13.0000			−0.654101
$396$		0			0
$397$			5.19615i			0.260787i	0.991462	+	0.130394i	$0.0416241\pi$
$397$			5.19615i			0.260787i	−0.991462	+	0.130394i	$0.958376\pi$
$398$		0			0
$399$	3.00000	+	15.5885i	0.150188	+	0.780399i
$400$		0			0
$401$		−	5.19615i		−	0.259483i	−0.991548	−	0.129742i	$-0.958585\pi$
$401$		−	5.19615i		−	0.259483i	0.991548	−	0.129742i	$-0.0414148\pi$
$402$		0			0
$403$	−18.0000			−0.896644
$404$		0			0
$405$	−4.50000	+	7.79423i	−0.223607	+	0.387298i
$406$		0			0
$407$			41.5692i			2.06051i
$408$		0			0
$409$			3.46410i			0.171289i	0.996326	+	0.0856444i	$0.0272949\pi$
$409$			3.46410i			0.171289i	−0.996326	+	0.0856444i	$0.972705\pi$
$410$		0			0
$411$	18.0000	−	31.1769i	0.887875	−	1.53784i
$412$		0			0
$413$	−12.0000	+	10.3923i	−0.590481	+	0.511372i
$414$		0			0
$415$	12.0000			0.589057
$416$		0			0
$417$	6.00000	−	10.3923i	0.293821	−	0.508913i
$418$		0			0
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$	−13.0000			−0.633581			−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000			−0.633581			−0.316791	+	0.948495i	$0.602605\pi$
$422$		0			0
$423$	−4.50000	−	7.79423i	−0.218797	−	0.378968i
$424$		0			0
$425$	−3.00000			−0.145521
$426$		0			0
$427$	−12.0000	−	13.8564i	−0.580721	−	0.670559i
$428$		0			0
$429$	13.5000	+	7.79423i	0.651786	+	0.376309i
$430$		0			0
$431$			5.19615i			0.250290i	0.992138	+	0.125145i	$0.0399396\pi$
$431$			5.19615i			0.250290i	−0.992138	+	0.125145i	$0.960060\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	4.50000	−	7.79423i	0.215758	−	0.373705i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	20.7846i		−	0.991995i	−0.868324	−	0.495998i	$-0.834802\pi$
$439$		−	20.7846i		−	0.991995i	0.868324	−	0.495998i	$-0.165198\pi$
$440$		0			0
$441$	19.5000	−	7.79423i	0.928571	−	0.371154i
$442$		0			0
$443$		−	31.1769i		−	1.48126i	−0.671913	−	0.740630i	$-0.734527\pi$
$443$		−	31.1769i		−	1.48126i	0.671913	−	0.740630i	$-0.265473\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			25.9808i			1.22611i	0.790041	+	0.613054i	$0.210059\pi$
$449$			25.9808i			1.22611i	−0.790041	+	0.613054i	$0.789941\pi$
$450$		0			0
$451$			31.1769i			1.46806i
$452$		0			0
$453$	−7.50000	−	4.33013i	−0.352381	−	0.203447i
$454$		0			0
$455$	3.00000	+	3.46410i	0.140642	+	0.162400i
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$		−	15.5885i		−	0.727607i
$460$		0			0
$461$	12.0000			0.558896			0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000			0.558896			0.279448	+	0.960161i	$0.409849\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$	−9.00000	+	15.5885i	−0.417365	+	0.722897i
$466$		0			0
$467$	9.00000			0.416470			0.208235	−	0.978079i	$-0.433228\pi$
$467$	9.00000			0.416470			0.208235	+	0.978079i	$0.433228\pi$
$468$		0			0
$469$	4.00000	−	3.46410i	0.184703	−	0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	51.9615i		−	2.38919i
$474$		0			0
$475$			3.46410i			0.158944i
$476$		0			0
$477$	−27.0000	+	15.5885i	−1.23625	+	0.713746i
$478$		0			0
$479$	12.0000			0.548294			0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000			0.548294			0.274147	+	0.961688i	$0.411605\pi$
$480$		0			0
$481$		−	13.8564i		−	0.631798i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			1.73205i			0.0786484i
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$	12.0000	+	6.92820i	0.542659	+	0.313304i
$490$		0			0
$491$			5.19615i			0.234499i	0.993102	+	0.117250i	$0.0374077\pi$
$491$			5.19615i			0.234499i	−0.993102	+	0.117250i	$0.962592\pi$
$492$		0			0
$493$			15.5885i			0.702069i
$494$		0			0
$495$	13.5000	−	7.79423i	0.606780	−	0.350325i
$496$		0			0
$497$	−18.0000	−	20.7846i	−0.807410	−	0.932317i
$498$		0			0
$499$	13.0000			0.581960			0.290980	−	0.956729i	$-0.406019\pi$
$499$	13.0000			0.581960			0.290980	+	0.956729i	$0.406019\pi$
$500$		0			0
$501$	22.5000	+	12.9904i	1.00523	+	0.580367i
$502$		0			0
$503$	3.00000			0.133763			0.0668817	−	0.997761i	$-0.478695\pi$
$503$	3.00000			0.133763			0.0668817	+	0.997761i	$0.478695\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$	15.0000	+	8.66025i	0.666173	+	0.384615i
$508$		0			0
$509$	−24.0000			−1.06378			−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000			−1.06378			−0.531891	+	0.846813i	$0.678518\pi$
$510$		0			0
$511$	−12.0000	−	13.8564i	−0.530849	−	0.612971i
$512$		0			0
$513$	−18.0000			−0.794719
$514$		0			0
$515$		−	1.73205i		−	0.0763233i
$516$		0			0
$517$			15.5885i			0.685580i
$518$		0			0
$519$	13.5000	+	7.79423i	0.592584	+	0.342129i
$520$		0			0
$521$	−42.0000			−1.84005			−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000			−1.84005			−0.920027	+	0.391856i	$0.871833\pi$
$522$		0			0
$523$			17.3205i			0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$			17.3205i			0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$		0			0
$525$	4.50000	−	0.866025i	0.196396	−	0.0377964i
$526$		0			0
$527$		−	31.1769i		−	1.35809i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−9.00000	−	15.5885i	−0.390567	−	0.676481i
$532$		0			0
$533$		−	10.3923i		−	0.450141i
$534$		0			0
$535$			10.3923i			0.449299i
$536$		0			0
$537$	−9.00000	+	15.5885i	−0.388379	+	0.672692i
$538$		0			0
$539$	−36.0000	−	5.19615i	−1.55063	−	0.223814i
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	−15.0000	+	25.9808i	−0.643712	+	1.11494i
$544$		0			0
$545$	−11.0000			−0.471188
$546$		0			0
$547$	32.0000			1.36822			0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000			1.36822			0.684111	+	0.729378i	$0.260191\pi$
$548$		0			0
$549$	18.0000	−	10.3923i	0.768221	−	0.443533i
$550$		0			0
$551$	18.0000			0.766826
$552$		0			0
$553$	−26.0000	+	22.5167i	−1.10563	+	0.957506i
$554$		0			0
$555$	−12.0000	−	6.92820i	−0.509372	−	0.294086i
$556$		0			0
$557$			20.7846i			0.880672i	0.897833	+	0.440336i	$0.145141\pi$
$557$			20.7846i			0.880672i	−0.897833	+	0.440336i	$0.854859\pi$
$558$		0			0
$559$			17.3205i			0.732579i
$560$		0			0
$561$	−13.5000	+	23.3827i	−0.569970	+	0.987218i
$562$		0			0
$563$	−36.0000			−1.51722			−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000			−1.51722			−0.758610	+	0.651546i	$0.774121\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.50000	+	23.3827i	0.188982	+	0.981981i
$568$		0			0
$569$		−	20.7846i		−	0.871336i	−0.900107	−	0.435668i	$-0.856512\pi$
$569$		−	20.7846i		−	0.871336i	0.900107	−	0.435668i	$-0.143488\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$	−13.5000	+	23.3827i	−0.563971	+	0.976826i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	1.73205i		−	0.0721062i	−0.999350	−	0.0360531i	$-0.988521\pi$
$577$		−	1.73205i		−	0.0721062i	0.999350	−	0.0360531i	$-0.0114785\pi$
$578$		0			0
$579$	24.0000	+	13.8564i	0.997406	+	0.575853i
$580$		0			0
$581$	24.0000	−	20.7846i	0.995688	−	0.862291i
$582$		0			0
$583$	54.0000			2.23645
$584$		0			0
$585$	−4.50000	+	2.59808i	−0.186052	+	0.107417i
$586$		0			0
$587$	36.0000			1.48588			0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000			1.48588			0.742940	+	0.669359i	$0.233431\pi$
$588$		0			0
$589$	−36.0000			−1.48335
$590$		0			0
$591$	−9.00000	+	15.5885i	−0.370211	+	0.641223i
$592$		0			0
$593$	21.0000			0.862367			0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000			0.862367			0.431183	+	0.902264i	$0.358096\pi$
$594$		0			0
$595$	−6.00000	+	5.19615i	−0.245976	+	0.213021i
$596$		0			0
$597$	21.0000	−	36.3731i	0.859473	−	1.48865i
$598$		0			0
$599$		−	25.9808i		−	1.06155i	−0.847514	−	0.530773i	$-0.821902\pi$
$599$		−	25.9808i		−	1.06155i	0.847514	−	0.530773i	$-0.178098\pi$
$600$		0			0
$601$		−	31.1769i		−	1.27173i	−0.771799	−	0.635866i	$-0.780643\pi$
$601$		−	31.1769i		−	1.27173i	0.771799	−	0.635866i	$-0.219357\pi$
$602$		0			0
$603$	3.00000	+	5.19615i	0.122169	+	0.211604i
$604$		0			0
$605$	−16.0000			−0.650493
$606$		0			0
$607$		−	8.66025i		−	0.351509i	−0.984434	−	0.175754i	$-0.943764\pi$
$607$		−	8.66025i		−	0.351509i	0.984434	−	0.175754i	$-0.0562365\pi$
$608$		0			0
$609$	−4.50000	−	23.3827i	−0.182349	−	0.947514i
$610$		0			0
$611$		−	5.19615i		−	0.210214i
$612$		0			0
$613$	26.0000			1.05013			0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000			1.05013			0.525065	+	0.851062i	$0.324041\pi$
$614$		0			0
$615$	−9.00000	−	5.19615i	−0.362915	−	0.209529i
$616$		0			0
$617$		−	41.5692i		−	1.67351i	−0.547575	−	0.836757i	$-0.684449\pi$
$617$		−	41.5692i		−	1.67351i	0.547575	−	0.836757i	$-0.315551\pi$
$618$		0			0
$619$			6.92820i			0.278468i	0.990260	+	0.139234i	$0.0444640\pi$
$619$			6.92820i			0.278468i	−0.990260	+	0.139234i	$0.955536\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	27.0000	+	15.5885i	1.07828	+	0.622543i
$628$		0			0
$629$	24.0000			0.956943
$630$		0			0
$631$	17.0000			0.676759			0.338380	−	0.941010i	$-0.390121\pi$
$631$	17.0000			0.676759			0.338380	+	0.941010i	$0.390121\pi$
$632$		0			0
$633$	37.5000	+	21.6506i	1.49049	+	0.860535i
$634$		0			0
$635$	2.00000			0.0793676
$636$		0			0
$637$	12.0000	+	1.73205i	0.475457	+	0.0686264i
$638$		0			0
$639$	27.0000	−	15.5885i	1.06810	−	0.616670i
$640$		0			0
$641$			20.7846i			0.820943i	0.911873	+	0.410471i	$0.134636\pi$
$641$			20.7846i			0.820943i	−0.911873	+	0.410471i	$0.865364\pi$
$642$		0			0
$643$			15.5885i			0.614749i	0.951589	+	0.307374i	$0.0994504\pi$
$643$			15.5885i			0.614749i	−0.951589	+	0.307374i	$0.900550\pi$
$644$		0			0
$645$	15.0000	+	8.66025i	0.590624	+	0.340997i
$646$		0			0
$647$	36.0000			1.41531			0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000			1.41531			0.707653	+	0.706560i	$0.249754\pi$
$648$		0			0
$649$			31.1769i			1.22380i
$650$		0			0
$651$	9.00000	+	46.7654i	0.352738	+	1.83288i
$652$		0			0
$653$			10.3923i			0.406682i	0.979108	+	0.203341i	$0.0651801\pi$
$653$			10.3923i			0.406682i	−0.979108	+	0.203341i	$0.934820\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	18.0000	−	10.3923i	0.702247	−	0.405442i
$658$		0			0
$659$		−	15.5885i		−	0.607240i	−0.952793	−	0.303620i	$-0.901805\pi$
$659$		−	15.5885i		−	0.607240i	0.952793	−	0.303620i	$-0.0981953\pi$
$660$		0			0
$661$			48.4974i			1.88633i	0.332323	+	0.943166i	$0.392168\pi$
$661$			48.4974i			1.88633i	−0.332323	+	0.943166i	$0.607832\pi$
$662$		0			0
$663$	4.50000	−	7.79423i	0.174766	−	0.302703i
$664$		0			0
$665$	6.00000	+	6.92820i	0.232670	+	0.268664i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−4.50000	+	7.79423i	−0.173980	+	0.301342i
$670$		0			0
$671$	−36.0000			−1.38976
$672$		0			0
$673$	−32.0000			−1.23351			−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000			−1.23351			−0.616755	+	0.787155i	$0.711553\pi$
$674$		0			0
$675$			5.19615i			0.200000i
$676$		0			0
$677$	−27.0000			−1.03769			−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000			−1.03769			−0.518847	+	0.854867i	$0.673639\pi$
$678$		0			0
$679$	3.00000	+	3.46410i	0.115129	+	0.132940i
$680$		0			0
$681$	40.5000	+	23.3827i	1.55196	+	0.896026i
$682$		0			0
$683$		−	20.7846i		−	0.795301i	−0.917537	−	0.397650i	$-0.869826\pi$
$683$		−	20.7846i		−	0.795301i	0.917537	−	0.397650i	$-0.130174\pi$
$684$		0			0
$685$		−	20.7846i		−	0.794139i
$686$		0			0
$687$	9.00000	−	15.5885i	0.343371	−	0.594737i
$688$		0			0
$689$	−18.0000			−0.685745
$690$		0			0
$691$		−	41.5692i		−	1.58137i	−0.612225	−	0.790684i	$-0.709725\pi$
$691$		−	41.5692i		−	1.58137i	0.612225	−	0.790684i	$-0.290275\pi$
$692$		0			0
$693$	13.5000	−	38.9711i	0.512823	−	1.48039i
$694$		0			0
$695$		−	6.92820i		−	0.262802i
$696$		0			0
$697$	18.0000			0.681799
$698$		0			0
$699$		0			0
$700$		0			0
$701$			36.3731i			1.37379i	0.726756	+	0.686896i	$0.241027\pi$
$701$			36.3731i			1.37379i	−0.726756	+	0.686896i	$0.758973\pi$
$702$		0			0
$703$		−	27.7128i		−	1.04521i
$704$		0			0
$705$	−4.50000	−	2.59808i	−0.169480	−	0.0978492i
$706$		0			0
$707$	−36.0000	+	31.1769i	−1.35392	+	1.17253i
$708$		0			0
$709$	13.0000			0.488225			0.244113	−	0.969747i	$-0.421503\pi$
$709$	13.0000			0.488225			0.244113	+	0.969747i	$0.421503\pi$
$710$		0			0
$711$	−19.5000	−	33.7750i	−0.731307	−	1.26666i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	9.00000			0.336581
$716$		0			0
$717$	22.5000	−	38.9711i	0.840278	−	1.45540i
$718$		0			0
$719$	12.0000			0.447524			0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000			0.447524			0.223762	+	0.974644i	$0.428166\pi$
$720$		0			0
$721$	−3.00000	−	3.46410i	−0.111726	−	0.129010i
$722$		0			0
$723$	6.00000	−	10.3923i	0.223142	−	0.386494i
$724$		0			0
$725$		−	5.19615i		−	0.192980i
$726$		0			0
$727$			51.9615i			1.92715i	0.267445	+	0.963573i	$0.413821\pi$
$727$			51.9615i			1.92715i	−0.267445	+	0.963573i	$0.586179\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	−30.0000			−1.10959
$732$		0			0
$733$		−	36.3731i		−	1.34347i	−0.740792	−	0.671735i	$-0.765549\pi$
$733$		−	36.3731i		−	1.34347i	0.740792	−	0.671735i	$-0.234451\pi$
$734$		0			0
$735$	7.50000	−	9.52628i	0.276642	−	0.351382i
$736$		0			0
$737$		−	10.3923i		−	0.382805i
$738$		0			0
$739$	11.0000			0.404642			0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000			0.404642			0.202321	+	0.979319i	$0.435152\pi$
$740$		0			0
$741$	−9.00000	−	5.19615i	−0.330623	−	0.190885i
$742$		0			0
$743$			10.3923i			0.381257i	0.981662	+	0.190628i	$0.0610525\pi$
$743$			10.3923i			0.381257i	−0.981662	+	0.190628i	$0.938947\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	18.0000	+	31.1769i	0.658586	+	1.14070i
$748$		0			0
$749$	18.0000	+	20.7846i	0.657706	+	0.759453i
$750$		0			0
$751$	−11.0000			−0.401396			−0.200698	−	0.979653i	$-0.564321\pi$
$751$	−11.0000			−0.401396			−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$	36.0000	+	20.7846i	1.31191	+	0.757433i
$754$		0			0
$755$	−5.00000			−0.181969
$756$		0			0
$757$	−8.00000			−0.290765			−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000			−0.290765			−0.145382	+	0.989376i	$0.546441\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$		0			0
$763$	−22.0000	+	19.0526i	−0.796453	+	0.689749i
$764$		0			0
$765$	−4.50000	−	7.79423i	−0.162698	−	0.281801i
$766$		0			0
$767$		−	10.3923i		−	0.375244i
$768$		0			0
$769$		−	17.3205i		−	0.624593i	−0.949985	−	0.312297i	$-0.898902\pi$
$769$		−	17.3205i		−	0.624593i	0.949985	−	0.312297i	$-0.101098\pi$
$770$		0			0
$771$	−27.0000	−	15.5885i	−0.972381	−	0.561405i
$772$		0			0
$773$	15.0000			0.539513			0.269756	−	0.962929i	$-0.413057\pi$
$773$	15.0000			0.539513			0.269756	+	0.962929i	$0.413057\pi$
$774$		0			0
$775$			10.3923i			0.373303i
$776$		0			0
$777$	−36.0000	+	6.92820i	−1.29149	+	0.248548i
$778$		0			0
$779$		−	20.7846i		−	0.744686i
$780$		0			0
$781$	−54.0000			−1.93227
$782$		0			0
$783$	27.0000			0.964901
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	50.2295i		−	1.79049i	−0.445577	−	0.895244i	$-0.647001\pi$
$787$		−	50.2295i		−	1.79049i	0.445577	−	0.895244i	$-0.352999\pi$
$788$		0			0
$789$	−27.0000	+	46.7654i	−0.961225	+	1.66489i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	12.0000			0.426132
$794$		0			0
$795$	−9.00000	+	15.5885i	−0.319197	+	0.552866i
$796$		0			0
$797$	9.00000			0.318796			0.159398	−	0.987214i	$-0.449045\pi$
$797$	9.00000			0.318796			0.159398	+	0.987214i	$0.449045\pi$
$798$		0			0
$799$	9.00000			0.318397
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−36.0000			−1.27041
$804$		0			0
$805$		0			0
$806$		0			0
$807$	18.0000	+	10.3923i	0.633630	+	0.365826i
$808$		0			0
$809$		−	36.3731i		−	1.27881i	−0.768871	−	0.639404i	$-0.779181\pi$
$809$		−	36.3731i		−	1.27881i	0.768871	−	0.639404i	$-0.220819\pi$
$810$		0			0
$811$			20.7846i			0.729846i	0.931038	+	0.364923i	$0.118905\pi$
$811$			20.7846i			0.729846i	−0.931038	+	0.364923i	$0.881095\pi$
$812$		0			0
$813$	−6.00000	+	10.3923i	−0.210429	+	0.364474i
$814$		0			0
$815$	8.00000			0.280228
$816$		0			0
$817$			34.6410i			1.21194i
$818$		0			0
$819$	−4.50000	+	12.9904i	−0.157243	+	0.453921i
$820$		0			0
$821$		−	36.3731i		−	1.26943i	−0.772747	−	0.634714i	$-0.781118\pi$
$821$		−	36.3731i		−	1.26943i	0.772747	−	0.634714i	$-0.218882\pi$
$822$		0			0
$823$	2.00000			0.0697156			0.0348578	−	0.999392i	$-0.488902\pi$
$823$	2.00000			0.0697156			0.0348578	+	0.999392i	$0.488902\pi$
$824$		0			0
$825$	4.50000	−	7.79423i	0.156670	−	0.271360i
$826$		0			0
$827$		−	41.5692i		−	1.44550i	−0.691108	−	0.722752i	$-0.742877\pi$
$827$		−	41.5692i		−	1.44550i	0.691108	−	0.722752i	$-0.257123\pi$
$828$		0			0
$829$		−	45.0333i		−	1.56407i	−0.623233	−	0.782036i	$-0.714181\pi$
$829$		−	45.0333i		−	1.56407i	0.623233	−	0.782036i	$-0.285819\pi$
$830$		0			0
$831$	12.0000	+	6.92820i	0.416275	+	0.240337i
$832$		0			0
$833$	−3.00000	+	20.7846i	−0.103944	+	0.720144i
$834$		0			0
$835$	15.0000			0.519096
$836$		0			0
$837$	−54.0000			−1.86651
$838$		0			0
$839$	−12.0000			−0.414286			−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000			−0.414286			−0.207143	+	0.978311i	$0.566417\pi$
$840$		0			0
$841$	2.00000			0.0689655
$842$		0			0
$843$	−4.50000	+	7.79423i	−0.154988	+	0.268447i
$844$		0			0
$845$	10.0000			0.344010
$846$		0			0
$847$	−32.0000	+	27.7128i	−1.09953	+	0.952224i
$848$		0			0
$849$	−22.5000	+	38.9711i	−0.772198	+	1.33749i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	13.8564i		−	0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$		−	13.8564i		−	0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$		0			0
$855$	−9.00000	+	5.19615i	−0.307794	+	0.177705i
$856$		0			0
$857$	6.00000			0.204956			0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000			0.204956			0.102478	+	0.994735i	$0.467323\pi$
$858$		0			0
$859$			6.92820i			0.236387i	0.992991	+	0.118194i	$0.0377103\pi$
$859$			6.92820i			0.236387i	−0.992991	+	0.118194i	$0.962290\pi$
$860$		0			0
$861$	−27.0000	+	5.19615i	−0.920158	+	0.177084i
$862$		0			0
$863$			31.1769i			1.06127i	0.847599	+	0.530637i	$0.178047\pi$
$863$			31.1769i			1.06127i	−0.847599	+	0.530637i	$0.821953\pi$
$864$		0			0
$865$	9.00000			0.306009
$866$		0			0
$867$	−12.0000	−	6.92820i	−0.407541	−	0.235294i
$868$		0			0
$869$			67.5500i			2.29148i
$870$		0			0
$871$			3.46410i			0.117377i
$872$		0			0
$873$	−4.50000	+	2.59808i	−0.152302	+	0.0879316i
$874$		0			0
$875$	2.00000	−	1.73205i	0.0676123	−	0.0585540i
$876$		0			0
$877$	2.00000			0.0675352			0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000			0.0675352			0.0337676	+	0.999430i	$0.489249\pi$
$878$		0			0
$879$	4.50000	+	2.59808i	0.151781	+	0.0876309i
$880$		0			0
$881$	−36.0000			−1.21287			−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000			−1.21287			−0.606435	+	0.795133i	$0.707401\pi$
$882$		0			0
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$		0			0
$885$	−9.00000	−	5.19615i	−0.302532	−	0.174667i
$886$		0			0
$887$	−48.0000			−1.61168			−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000			−1.61168			−0.805841	+	0.592132i	$0.798286\pi$
$888$		0			0
$889$	4.00000	−	3.46410i	0.134156	−	0.116182i
$890$		0			0
$891$	40.5000	+	23.3827i	1.35680	+	0.783349i
$892$		0			0
$893$		−	10.3923i		−	0.347765i
$894$		0			0
$895$			10.3923i			0.347376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	54.0000			1.80100
$900$		0			0
$901$		−	31.1769i		−	1.03865i
$902$		0			0
$903$	45.0000	−	8.66025i	1.49751	−	0.288195i
$904$		0			0
$905$			17.3205i			0.575753i
$906$		0			0
$907$	−32.0000			−1.06254			−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000			−1.06254			−0.531271	+	0.847202i	$0.678286\pi$
$908$		0			0
$909$	−27.0000	−	46.7654i	−0.895533	−	1.55111i
$910$		0			0
$911$		−	31.1769i		−	1.03294i	−0.856306	−	0.516469i	$-0.827246\pi$
$911$		−	31.1769i		−	1.03294i	0.856306	−	0.516469i	$-0.172754\pi$
$912$		0			0
$913$		−	62.3538i		−	2.06361i
$914$		0			0
$915$	6.00000	−	10.3923i	0.198354	−	0.343559i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−11.0000			−0.362857			−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000			−0.362857			−0.181428	+	0.983404i	$0.558072\pi$
$920$		0			0
$921$	22.5000	−	38.9711i	0.741400	−	1.28414i
$922$		0			0
$923$	18.0000			0.592477
$924$		0			0
$925$	−8.00000			−0.263038
$926$		0			0
$927$	4.50000	−	2.59808i	0.147799	−	0.0853320i
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	24.0000	+	3.46410i	0.786568	+	0.113531i
$932$		0			0
$933$	−45.0000	−	25.9808i	−1.47323	−	0.850572i
$934$		0			0
$935$			15.5885i			0.509797i
$936$		0			0
$937$		−	22.5167i		−	0.735587i	−0.929907	−	0.367794i	$-0.880113\pi$
$937$		−	22.5167i		−	0.735587i	0.929907	−	0.367794i	$-0.119887\pi$
$938$		0			0
$939$	−19.5000	+	33.7750i	−0.636358	+	1.10221i
$940$		0			0
$941$	−24.0000			−0.782378			−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000			−0.782378			−0.391189	+	0.920310i	$0.627936\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	9.00000	+	10.3923i	0.292770	+	0.338062i
$946$		0			0
$947$			10.3923i			0.337705i	0.985641	+	0.168852i	$0.0540061\pi$
$947$			10.3923i			0.337705i	−0.985641	+	0.168852i	$0.945994\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$		0			0
$951$	−9.00000	+	15.5885i	−0.291845	+	0.505490i
$952$		0			0
$953$			51.9615i			1.68320i	0.540102	+	0.841599i	$0.318386\pi$
$953$			51.9615i			1.68320i	−0.540102	+	0.841599i	$0.681614\pi$
$954$		0			0
$955$			15.5885i			0.504431i
$956$		0			0
$957$	−40.5000	−	23.3827i	−1.30918	−	0.755855i
$958$		0			0
$959$	−36.0000	−	41.5692i	−1.16250	−	1.34234i
$960$		0			0
$961$	−77.0000			−2.48387
$962$		0			0
$963$	−27.0000	+	15.5885i	−0.870063	+	0.502331i
$964$		0			0
$965$	16.0000			0.515058
$966$		0			0
$967$	2.00000			0.0643157			0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000			0.0643157			0.0321578	+	0.999483i	$0.489762\pi$
$968$		0			0
$969$	9.00000	−	15.5885i	0.289122	−	0.500773i
$970$		0			0
$971$	−6.00000			−0.192549			−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000			−0.192549			−0.0962746	+	0.995355i	$0.530693\pi$
$972$		0			0
$973$	−12.0000	−	13.8564i	−0.384702	−	0.444216i
$974$		0			0
$975$	−1.50000	+	2.59808i	−0.0480384	+	0.0832050i
$976$		0			0
$977$			51.9615i			1.66240i	0.555976	+	0.831198i	$0.312345\pi$
$977$			51.9615i			1.66240i	−0.555976	+	0.831198i	$0.687655\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−16.5000	−	28.5788i	−0.526804	−	0.912452i
$982$		0			0
$983$	57.0000			1.81802			0.909009	−	0.416777i	$-0.136840\pi$
$983$	57.0000			1.81802			0.909009	+	0.416777i	$0.136840\pi$
$984$		0			0
$985$			10.3923i			0.331126i
$986$		0			0
$987$	−13.5000	+	2.59808i	−0.429710	+	0.0826977i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	28.0000			0.889449			0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000			0.889449			0.444725	+	0.895667i	$0.353302\pi$
$992$		0			0
$993$	6.00000	+	3.46410i	0.190404	+	0.109930i
$994$		0			0
$995$		−	24.2487i		−	0.768736i
$996$		0			0
$997$			15.5885i			0.493691i	0.969055	+	0.246846i	$0.0793941\pi$
$997$			15.5885i			0.493691i	−0.969055	+	0.246846i	$0.920606\pi$
$998$		0			0
$999$		−	41.5692i		−	1.31519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.d.b.41.2	yes	2
3.2	odd	2		420.2.d.a.41.2	yes	2
4.3	odd	2		1680.2.f.a.881.1		2
5.2	odd	4		2100.2.f.d.1049.3		4
5.3	odd	4		2100.2.f.d.1049.2		4
5.4	even	2		2100.2.d.a.1301.1		2
7.6	odd	2		420.2.d.a.41.1	✓	2
12.11	even	2		1680.2.f.d.881.1		2
15.2	even	4		2100.2.f.c.1049.4		4
15.8	even	4		2100.2.f.c.1049.1		4
15.14	odd	2		2100.2.d.e.1301.1		2
21.20	even	2	inner	420.2.d.b.41.1	yes	2
28.27	even	2		1680.2.f.d.881.2		2
35.13	even	4		2100.2.f.c.1049.3		4
35.27	even	4		2100.2.f.c.1049.2		4
35.34	odd	2		2100.2.d.e.1301.2		2
84.83	odd	2		1680.2.f.a.881.2		2
105.62	odd	4		2100.2.f.d.1049.1		4
105.83	odd	4		2100.2.f.d.1049.4		4
105.104	even	2		2100.2.d.a.1301.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.d.a.41.1	✓	2	7.6	odd	2
420.2.d.a.41.2	yes	2	3.2	odd	2
420.2.d.b.41.1	yes	2	21.20	even	2	inner
420.2.d.b.41.2	yes	2	1.1	even	1	trivial
1680.2.f.a.881.1		2	4.3	odd	2
1680.2.f.a.881.2		2	84.83	odd	2
1680.2.f.d.881.1		2	12.11	even	2
1680.2.f.d.881.2		2	28.27	even	2
2100.2.d.a.1301.1		2	5.4	even	2
2100.2.d.a.1301.2		2	105.104	even	2
2100.2.d.e.1301.1		2	15.14	odd	2
2100.2.d.e.1301.2		2	35.34	odd	2
2100.2.f.c.1049.1		4	15.8	even	4
2100.2.f.c.1049.2		4	35.27	even	4
2100.2.f.c.1049.3		4	35.13	even	4
2100.2.f.c.1049.4		4	15.2	even	4
2100.2.f.d.1049.1		4	105.62	odd	4
2100.2.f.d.1049.2		4	5.3	odd	4
2100.2.f.d.1049.3		4	5.2	odd	4
2100.2.f.d.1049.4		4	105.83	odd	4

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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 0.866025i) q^{3} +1.00000 q^{5} +(2.00000 - 1.73205i) q^{7} +(1.50000 + 2.59808i) q^{9} -5.19615i q^{11} +1.73205i q^{13} +(1.50000 + 0.866025i) q^{15} -3.00000 q^{17} +3.46410i q^{19} +(4.50000 - 0.866025i) q^{21} +1.00000 q^{25} +5.19615i q^{27} -5.19615i q^{29} +10.3923i q^{31} +(4.50000 - 7.79423i) q^{33} +(2.00000 - 1.73205i) q^{35} -8.00000 q^{37} +(-1.50000 + 2.59808i) q^{39} -6.00000 q^{41} +10.0000 q^{43} +(1.50000 + 2.59808i) q^{45} -3.00000 q^{47} +(1.00000 - 6.92820i) q^{49} +(-4.50000 - 2.59808i) q^{51} +10.3923i q^{53} -5.19615i q^{55} +(-3.00000 + 5.19615i) q^{57} -6.00000 q^{59} -6.92820i q^{61} +(7.50000 + 2.59808i) q^{63} +1.73205i q^{65} +2.00000 q^{67} -10.3923i q^{71} -6.92820i q^{73} +(1.50000 + 0.866025i) q^{75} +(-9.00000 - 10.3923i) q^{77} -13.0000 q^{79} +(-4.50000 + 7.79423i) q^{81} +12.0000 q^{83} -3.00000 q^{85} +(4.50000 - 7.79423i) q^{87} +(3.00000 + 3.46410i) q^{91} +(-9.00000 + 15.5885i) q^{93} +3.46410i q^{95} +1.73205i q^{97} +(13.5000 - 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 2 q^{5} + 4 q^{7} + 3 q^{9} + 3 q^{15} - 6 q^{17} + 9 q^{21} + 2 q^{25} + 9 q^{33} + 4 q^{35} - 16 q^{37} - 3 q^{39} - 12 q^{41} + 20 q^{43} + 3 q^{45} - 6 q^{47} + 2 q^{49} - 9 q^{51} - 6 q^{57}+ \cdots + 27 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 2 * q^5 + 4 * q^7 + 3 * q^9 + 3 * q^15 - 6 * q^17 + 9 * q^21 + 2 * q^25 + 9 * q^33 + 4 * q^35 - 16 * q^37 - 3 * q^39 - 12 * q^41 + 20 * q^43 + 3 * q^45 - 6 * q^47 + 2 * q^49 - 9 * q^51 - 6 * q^57 - 12 * q^59 + 15 * q^63 + 4 * q^67 + 3 * q^75 - 18 * q^77 - 26 * q^79 - 9 * q^81 + 24 * q^83 - 6 * q^85 + 9 * q^87 + 6 * q^91 - 18 * q^93 + 27 * q^99

Introduction

L-functions

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Database

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