LMFDB - Embedded newform 1530.2.d.b.919.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1530,2,Mod(919,1530)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1530, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1530.919");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1530.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:	$12.2171115093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 170)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			919.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1530.919
Dual form			1530.2.d.b.919.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{8} +(-2.00000 - 1.00000i) q^{10} +6.00000 q^{11} -3.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +7.00000 q^{19} +(1.00000 - 2.00000i) q^{20} +6.00000i q^{22} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +3.00000 q^{26} -5.00000 q^{29} +5.00000 q^{31} +1.00000i q^{32} +1.00000 q^{34} +8.00000i q^{37} +7.00000i q^{38} +(2.00000 + 1.00000i) q^{40} +4.00000i q^{43} -6.00000 q^{44} +8.00000 q^{46} -3.00000i q^{47} +7.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +3.00000i q^{52} +9.00000i q^{53} +(-6.00000 + 12.0000i) q^{55} -5.00000i q^{58} +5.00000 q^{59} -3.00000 q^{61} +5.00000i q^{62} -1.00000 q^{64} +(6.00000 + 3.00000i) q^{65} -2.00000i q^{67} +1.00000i q^{68} +15.0000 q^{71} +11.0000i q^{73} -8.00000 q^{74} -7.00000 q^{76} -8.00000 q^{79} +(-1.00000 + 2.00000i) q^{80} +4.00000i q^{83} +(2.00000 + 1.00000i) q^{85} -4.00000 q^{86} -6.00000i q^{88} -1.00000 q^{89} +8.00000i q^{92} +3.00000 q^{94} +(-7.00000 + 14.0000i) q^{95} -9.00000i q^{97} +7.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^5	$ 2 q - 2 q^{4} - 2 q^{5} - 4 q^{10} + 12 q^{11} + 2 q^{16} + 14 q^{19} + 2 q^{20} - 6 q^{25} + 6 q^{26} - 10 q^{29} + 10 q^{31} + 2 q^{34} + 4 q^{40} - 12 q^{44} + 16 q^{46} + 14 q^{49} + 8 q^{50} - 12 q^{55} + 10 q^{59} - 6 q^{61} - 2 q^{64} + 12 q^{65} + 30 q^{71} - 16 q^{74} - 14 q^{76} - 16 q^{79} - 2 q^{80} + 4 q^{85} - 8 q^{86} - 2 q^{89} + 6 q^{94} - 14 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^5 - 4 * q^10 + 12 * q^11 + 2 * q^16 + 14 * q^19 + 2 * q^20 - 6 * q^25 + 6 * q^26 - 10 * q^29 + 10 * q^31 + 2 * q^34 + 4 * q^40 - 12 * q^44 + 16 * q^46 + 14 * q^49 + 8 * q^50 - 12 * q^55 + 10 * q^59 - 6 * q^61 - 2 * q^64 + 12 * q^65 + 30 * q^71 - 16 * q^74 - 14 * q^76 - 16 * q^79 - 2 * q^80 + 4 * q^85 - 8 * q^86 - 2 * q^89 + 6 * q^94 - 14 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1530\mathbb{Z}\right)^\times$.

$n$	$307$	$1261$	$1361$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−2.00000	−	1.00000i	−0.632456	−	0.316228i
$11$	6.00000			1.80907			0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000			1.80907			0.904534	+	0.426401i	$0.140219\pi$
$12$		0			0
$13$		−	3.00000i		−	0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$		−	3.00000i		−	0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$	7.00000			1.60591			0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000			1.60591			0.802955	+	0.596040i	$0.203260\pi$
$20$	1.00000	−	2.00000i	0.223607	−	0.447214i
$21$		0			0
$22$			6.00000i			1.27920i
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	3.00000			0.588348
$27$		0			0
$28$		0			0
$29$	−5.00000			−0.928477			−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000			−0.928477			−0.464238	+	0.885710i	$0.653672\pi$
$30$		0			0
$31$	5.00000			0.898027			0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000			0.898027			0.449013	+	0.893525i	$0.351776\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	1.00000			0.171499
$35$		0			0
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$			7.00000i			1.13555i
$39$		0			0
$40$	2.00000	+	1.00000i	0.316228	+	0.158114i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$	8.00000			1.17954
$47$		−	3.00000i		−	0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$		−	3.00000i		−	0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$		0			0
$52$			3.00000i			0.416025i
$53$			9.00000i			1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$			9.00000i			1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$		0			0
$55$	−6.00000	+	12.0000i	−0.809040	+	1.61808i
$56$		0			0
$57$		0			0
$58$		−	5.00000i		−	0.656532i
$59$	5.00000			0.650945			0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000			0.650945			0.325472	+	0.945552i	$0.394477\pi$
$60$		0			0
$61$	−3.00000			−0.384111			−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000			−0.384111			−0.192055	+	0.981384i	$0.561515\pi$
$62$			5.00000i			0.635001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	6.00000	+	3.00000i	0.744208	+	0.372104i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$			1.00000i			0.121268i
$69$		0			0
$70$		0			0
$71$	15.0000			1.78017			0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000			1.78017			0.890086	+	0.455792i	$0.150644\pi$
$72$		0			0
$73$			11.0000i			1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$			11.0000i			1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	−8.00000			−0.929981
$75$		0			0
$76$	−7.00000			−0.802955
$77$		0			0
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	+	2.00000i	−0.111803	+	0.223607i
$81$		0			0
$82$		0			0
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$	2.00000	+	1.00000i	0.216930	+	0.108465i
$86$	−4.00000			−0.431331
$87$		0			0
$88$		−	6.00000i		−	0.639602i
$89$	−1.00000			−0.106000			−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000			−0.106000			−0.0529999	+	0.998595i	$0.516878\pi$
$90$		0			0
$91$		0			0
$92$			8.00000i			0.834058i
$93$		0			0
$94$	3.00000			0.309426
$95$	−7.00000	+	14.0000i	−0.718185	+	1.43637i
$96$		0			0
$97$		−	9.00000i		−	0.913812i	−0.889515	−	0.456906i	$-0.848958\pi$
$97$		−	9.00000i		−	0.913812i	0.889515	−	0.456906i	$-0.151042\pi$
$98$			7.00000i			0.707107i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−3.00000			−0.294174
$105$		0			0
$106$	−9.00000			−0.874157
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	−9.00000			−0.862044			−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000			−0.862044			−0.431022	+	0.902342i	$0.641847\pi$
$110$	−12.0000	−	6.00000i	−1.14416	−	0.572078i
$111$		0			0
$112$		0			0
$113$		−	13.0000i		−	1.22294i	−0.791269	−	0.611469i	$-0.790579\pi$
$113$		−	13.0000i		−	1.22294i	0.791269	−	0.611469i	$-0.209421\pi$
$114$		0			0
$115$	16.0000	+	8.00000i	1.49201	+	0.746004i
$116$	5.00000			0.464238
$117$		0			0
$118$			5.00000i			0.460287i
$119$		0			0
$120$		0			0
$121$	25.0000			2.27273
$122$		−	3.00000i		−	0.271607i
$123$		0			0
$124$	−5.00000			−0.449013
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$			1.00000i			0.0887357i	0.999015	+	0.0443678i	$0.0141274\pi$
$127$			1.00000i			0.0887357i	−0.999015	+	0.0443678i	$0.985873\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	−3.00000	+	6.00000i	−0.263117	+	0.526235i
$131$	−6.00000			−0.524222			−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000			−0.524222			−0.262111	+	0.965038i	$0.584419\pi$
$132$		0			0
$133$		0			0
$134$	2.00000			0.172774
$135$		0			0
$136$	−1.00000			−0.0857493
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$			15.0000i			1.25877i
$143$		−	18.0000i		−	1.50524i
$144$		0			0
$145$	5.00000	−	10.0000i	0.415227	−	0.830455i
$146$	−11.0000			−0.910366
$147$		0			0
$148$		−	8.00000i		−	0.657596i
$149$	−2.00000			−0.163846			−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000			−0.163846			−0.0819232	+	0.996639i	$0.526106\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		−	7.00000i		−	0.567775i
$153$		0			0
$154$		0			0
$155$	−5.00000	+	10.0000i	−0.401610	+	0.803219i
$156$		0			0
$157$			14.0000i			1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$			14.0000i			1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$		−	8.00000i		−	0.636446i
$159$		0			0
$160$	−2.00000	−	1.00000i	−0.158114	−	0.0790569i
$161$		0			0
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$		0			0
$166$	−4.00000			−0.310460
$167$			6.00000i			0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$			6.00000i			0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$		0			0
$169$	4.00000			0.307692
$170$	−1.00000	+	2.00000i	−0.0766965	+	0.153393i
$171$		0			0
$172$		−	4.00000i		−	0.304997i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$	6.00000			0.452267
$177$		0			0
$178$		−	1.00000i		−	0.0749532i
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$	−8.00000			−0.589768
$185$	−16.0000	−	8.00000i	−1.17634	−	0.588172i
$186$		0			0
$187$		−	6.00000i		−	0.438763i
$188$			3.00000i			0.218797i
$189$		0			0
$190$	−14.0000	−	7.00000i	−1.01567	−	0.507833i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	26.0000i		−	1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$		−	26.0000i		−	1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$	9.00000			0.646162
$195$		0			0
$196$	−7.00000			−0.500000
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	−7.00000			−0.496217			−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000			−0.496217			−0.248108	+	0.968732i	$0.579809\pi$
$200$	−4.00000	+	3.00000i	−0.282843	+	0.212132i
$201$		0			0
$202$			12.0000i			0.844317i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		−	3.00000i		−	0.208013i
$209$	42.0000			2.90520
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		−	9.00000i		−	0.618123i
$213$		0			0
$214$	−4.00000			−0.273434
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$		0			0
$218$		−	9.00000i		−	0.609557i
$219$		0			0
$220$	6.00000	−	12.0000i	0.404520	−	0.809040i
$221$	−3.00000			−0.201802
$222$		0			0
$223$			15.0000i			1.00447i	0.864730	+	0.502237i	$0.167490\pi$
$223$			15.0000i			1.00447i	−0.864730	+	0.502237i	$0.832510\pi$
$224$		0			0
$225$		0			0
$226$	13.0000			0.864747
$227$			9.00000i			0.597351i	0.954355	+	0.298675i	$0.0965448\pi$
$227$			9.00000i			0.597351i	−0.954355	+	0.298675i	$0.903455\pi$
$228$		0			0
$229$	−18.0000			−1.18947			−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000			−1.18947			−0.594737	+	0.803921i	$0.702744\pi$
$230$	−8.00000	+	16.0000i	−0.527504	+	1.05501i
$231$		0			0
$232$			5.00000i			0.328266i
$233$			27.0000i			1.76883i	0.466702	+	0.884414i	$0.345442\pi$
$233$			27.0000i			1.76883i	−0.466702	+	0.884414i	$0.654558\pi$
$234$		0			0
$235$	6.00000	+	3.00000i	0.391397	+	0.195698i
$236$	−5.00000			−0.325472
$237$		0			0
$238$		0			0
$239$	10.0000			0.646846			0.323423	−	0.946254i	$-0.395166\pi$
$239$	10.0000			0.646846			0.323423	+	0.946254i	$0.395166\pi$
$240$		0			0
$241$	−20.0000			−1.28831			−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000			−1.28831			−0.644157	+	0.764894i	$0.722792\pi$
$242$			25.0000i			1.60706i
$243$		0			0
$244$	3.00000			0.192055
$245$	−7.00000	+	14.0000i	−0.447214	+	0.894427i
$246$		0			0
$247$		−	21.0000i		−	1.33620i
$248$		−	5.00000i		−	0.317500i
$249$		0			0
$250$	2.00000	+	11.0000i	0.126491	+	0.695701i
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$		−	48.0000i		−	3.01773i
$254$	−1.00000			−0.0627456
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	28.0000i		−	1.74659i	−0.487190	−	0.873296i	$-0.661978\pi$
$257$		−	28.0000i		−	1.74659i	0.487190	−	0.873296i	$-0.338022\pi$
$258$		0			0
$259$		0			0
$260$	−6.00000	−	3.00000i	−0.372104	−	0.186052i
$261$		0			0
$262$		−	6.00000i		−	0.370681i
$263$			27.0000i			1.66489i	0.554107	+	0.832446i	$0.313060\pi$
$263$			27.0000i			1.66489i	−0.554107	+	0.832446i	$0.686940\pi$
$264$		0			0
$265$	−18.0000	−	9.00000i	−1.10573	−	0.552866i
$266$		0			0
$267$		0			0
$268$			2.00000i			0.122169i
$269$	−19.0000			−1.15845			−0.579225	−	0.815168i	$-0.696645\pi$
$269$	−19.0000			−1.15845			−0.579225	+	0.815168i	$0.696645\pi$
$270$		0			0
$271$	−6.00000			−0.364474			−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000			−0.364474			−0.182237	+	0.983255i	$0.558334\pi$
$272$		−	1.00000i		−	0.0606339i
$273$		0			0
$274$	12.0000			0.724947
$275$	−18.0000	−	24.0000i	−1.08544	−	1.44725i
$276$		0			0
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	33.0000			1.96861			0.984307	−	0.176462i	$-0.0564652\pi$
$281$	33.0000			1.96861			0.984307	+	0.176462i	$0.0564652\pi$
$282$		0			0
$283$		−	11.0000i		−	0.653882i	−0.945045	−	0.326941i	$-0.893982\pi$
$283$		−	11.0000i		−	0.653882i	0.945045	−	0.326941i	$-0.106018\pi$
$284$	−15.0000			−0.890086
$285$		0			0
$286$	18.0000			1.06436
$287$		0			0
$288$		0			0
$289$	−1.00000			−0.0588235
$290$	10.0000	+	5.00000i	0.587220	+	0.293610i
$291$		0			0
$292$		−	11.0000i		−	0.643726i
$293$		−	19.0000i		−	1.10999i	−0.831853	−	0.554996i	$-0.812720\pi$
$293$		−	19.0000i		−	1.10999i	0.831853	−	0.554996i	$-0.187280\pi$
$294$		0			0
$295$	−5.00000	+	10.0000i	−0.291111	+	0.582223i
$296$	8.00000			0.464991
$297$		0			0
$298$		−	2.00000i		−	0.115857i
$299$	−24.0000			−1.38796
$300$		0			0
$301$		0			0
$302$		−	8.00000i		−	0.460348i
$303$		0			0
$304$	7.00000			0.401478
$305$	3.00000	−	6.00000i	0.171780	−	0.343559i
$306$		0			0
$307$		−	24.0000i		−	1.36975i	−0.728659	−	0.684876i	$-0.759856\pi$
$307$		−	24.0000i		−	1.36975i	0.728659	−	0.684876i	$-0.240144\pi$
$308$		0			0
$309$		0			0
$310$	−10.0000	−	5.00000i	−0.567962	−	0.283981i
$311$	4.00000			0.226819			0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000			0.226819			0.113410	+	0.993548i	$0.463823\pi$
$312$		0			0
$313$			22.0000i			1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$			22.0000i			1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	8.00000			0.450035
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$	−30.0000			−1.67968
$320$	1.00000	−	2.00000i	0.0559017	−	0.111803i
$321$		0			0
$322$		0			0
$323$		−	7.00000i		−	0.389490i
$324$		0			0
$325$	−12.0000	+	9.00000i	−0.665640	+	0.499230i
$326$	−4.00000			−0.221540
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	7.00000			0.384755			0.192377	−	0.981321i	$-0.438380\pi$
$331$	7.00000			0.384755			0.192377	+	0.981321i	$0.438380\pi$
$332$		−	4.00000i		−	0.219529i
$333$		0			0
$334$	−6.00000			−0.328305
$335$	4.00000	+	2.00000i	0.218543	+	0.109272i
$336$		0			0
$337$			13.0000i			0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$			13.0000i			0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$			4.00000i			0.217571i
$339$		0			0
$340$	−2.00000	−	1.00000i	−0.108465	−	0.0542326i
$341$	30.0000			1.62459
$342$		0			0
$343$		0			0
$344$	4.00000			0.215666
$345$		0			0
$346$		0			0
$347$		−	5.00000i		−	0.268414i	−0.990953	−	0.134207i	$-0.957151\pi$
$347$		−	5.00000i		−	0.268414i	0.990953	−	0.134207i	$-0.0428487\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$		0			0
$352$			6.00000i			0.319801i
$353$		−	8.00000i		−	0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$		−	8.00000i		−	0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$		0			0
$355$	−15.0000	+	30.0000i	−0.796117	+	1.59223i
$356$	1.00000			0.0529999
$357$		0			0
$358$		−	12.0000i		−	0.634220i
$359$	−28.0000			−1.47778			−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000			−1.47778			−0.738892	+	0.673824i	$0.764651\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$			2.00000i			0.105118i
$363$		0			0
$364$		0			0
$365$	−22.0000	−	11.0000i	−1.15153	−	0.575766i
$366$		0			0
$367$		−	10.0000i		−	0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$		−	10.0000i		−	0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$		−	8.00000i		−	0.417029i
$369$		0			0
$370$	8.00000	−	16.0000i	0.415900	−	0.831800i
$371$		0			0
$372$		0			0
$373$			14.0000i			0.724893i	0.932005	+	0.362446i	$0.118058\pi$
$373$			14.0000i			0.724893i	−0.932005	+	0.362446i	$0.881942\pi$
$374$	6.00000			0.310253
$375$		0			0
$376$	−3.00000			−0.154713
$377$			15.0000i			0.772539i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$	7.00000	−	14.0000i	0.359092	−	0.718185i
$381$		0			0
$382$		0			0
$383$		−	27.0000i		−	1.37964i	−0.723983	−	0.689818i	$-0.757691\pi$
$383$		−	27.0000i		−	1.37964i	0.723983	−	0.689818i	$-0.242309\pi$
$384$		0			0
$385$		0			0
$386$	26.0000			1.32337
$387$		0			0
$388$			9.00000i			0.456906i
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$		−	7.00000i		−	0.353553i
$393$		0			0
$394$	2.00000			0.100759
$395$	8.00000	−	16.0000i	0.402524	−	0.805047i
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		−	7.00000i		−	0.350878i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	10.0000			0.499376			0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000			0.499376			0.249688	+	0.968326i	$0.419672\pi$
$402$		0			0
$403$		−	15.0000i		−	0.747203i
$404$	−12.0000			−0.597022
$405$		0			0
$406$		0			0
$407$			48.0000i			2.37927i
$408$		0			0
$409$	31.0000			1.53285			0.766426	−	0.642333i	$-0.222033\pi$
$409$	31.0000			1.53285			0.766426	+	0.642333i	$0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−8.00000	−	4.00000i	−0.392705	−	0.196352i
$416$	3.00000			0.147087
$417$		0			0
$418$			42.0000i			2.05429i
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	20.0000			0.974740			0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000			0.974740			0.487370	+	0.873195i	$0.337956\pi$
$422$			16.0000i			0.778868i
$423$		0			0
$424$	9.00000			0.437079
$425$	−4.00000	+	3.00000i	−0.194029	+	0.145521i
$426$		0			0
$427$		0			0
$428$		−	4.00000i		−	0.193347i
$429$		0			0
$430$	4.00000	−	8.00000i	0.192897	−	0.385794i
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$		−	20.0000i		−	0.961139i	−0.876957	−	0.480569i	$-0.840430\pi$
$433$		−	20.0000i		−	0.961139i	0.876957	−	0.480569i	$-0.159570\pi$
$434$		0			0
$435$		0			0
$436$	9.00000			0.431022
$437$		−	56.0000i		−	2.67884i
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$	12.0000	+	6.00000i	0.572078	+	0.286039i
$441$		0			0
$442$		−	3.00000i		−	0.142695i
$443$			10.0000i			0.475114i	0.971374	+	0.237557i	$0.0763467\pi$
$443$			10.0000i			0.475114i	−0.971374	+	0.237557i	$0.923653\pi$
$444$		0			0
$445$	1.00000	−	2.00000i	0.0474045	−	0.0948091i
$446$	−15.0000			−0.710271
$447$		0			0
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$		0			0
$452$			13.0000i			0.611469i
$453$		0			0
$454$	−9.00000			−0.422391
$455$		0			0
$456$		0			0
$457$		−	12.0000i		−	0.561336i	−0.959805	−	0.280668i	$-0.909444\pi$
$457$		−	12.0000i		−	0.561336i	0.959805	−	0.280668i	$-0.0905560\pi$
$458$		−	18.0000i		−	0.841085i
$459$		0			0
$460$	−16.0000	−	8.00000i	−0.746004	−	0.373002i
$461$	20.0000			0.931493			0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000			0.931493			0.465746	+	0.884918i	$0.345786\pi$
$462$		0			0
$463$		−	31.0000i		−	1.44069i	−0.693615	−	0.720346i	$-0.743983\pi$
$463$		−	31.0000i		−	1.44069i	0.693615	−	0.720346i	$-0.256017\pi$
$464$	−5.00000			−0.232119
$465$		0			0
$466$	−27.0000			−1.25075
$467$			14.0000i			0.647843i	0.946084	+	0.323921i	$0.105001\pi$
$467$			14.0000i			0.647843i	−0.946084	+	0.323921i	$0.894999\pi$
$468$		0			0
$469$		0			0
$470$	−3.00000	+	6.00000i	−0.138380	+	0.276759i
$471$		0			0
$472$		−	5.00000i		−	0.230144i
$473$			24.0000i			1.10352i
$474$		0			0
$475$	−21.0000	−	28.0000i	−0.963546	−	1.28473i
$476$		0			0
$477$		0			0
$478$			10.0000i			0.457389i
$479$	−37.0000			−1.69057			−0.845287	−	0.534313i	$-0.820570\pi$
$479$	−37.0000			−1.69057			−0.845287	+	0.534313i	$0.820570\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$		−	20.0000i		−	0.910975i
$483$		0			0
$484$	−25.0000			−1.13636
$485$	18.0000	+	9.00000i	0.817338	+	0.408669i
$486$		0			0
$487$		−	2.00000i		−	0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$		−	2.00000i		−	0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$			3.00000i			0.135804i
$489$		0			0
$490$	−14.0000	−	7.00000i	−0.632456	−	0.316228i
$491$	13.0000			0.586682			0.293341	−	0.956008i	$-0.405233\pi$
$491$	13.0000			0.586682			0.293341	+	0.956008i	$0.405233\pi$
$492$		0			0
$493$			5.00000i			0.225189i
$494$	21.0000			0.944835
$495$		0			0
$496$	5.00000			0.224507
$497$		0			0
$498$		0			0
$499$	28.0000			1.25345			0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000			1.25345			0.626726	+	0.779240i	$0.284395\pi$
$500$	−11.0000	+	2.00000i	−0.491935	+	0.0894427i
$501$		0			0
$502$		−	12.0000i		−	0.535586i
$503$		−	42.0000i		−	1.87269i	−0.351085	−	0.936344i	$-0.614187\pi$
$503$		−	42.0000i		−	1.87269i	0.351085	−	0.936344i	$-0.385813\pi$
$504$		0			0
$505$	−12.0000	+	24.0000i	−0.533993	+	1.06799i
$506$	48.0000			2.13386
$507$		0			0
$508$		−	1.00000i		−	0.0443678i
$509$	−12.0000			−0.531891			−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000			−0.531891			−0.265945	+	0.963988i	$0.585684\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	28.0000			1.23503
$515$		0			0
$516$		0			0
$517$		−	18.0000i		−	0.791639i
$518$		0			0
$519$		0			0
$520$	3.00000	−	6.00000i	0.131559	−	0.263117i
$521$	−28.0000			−1.22670			−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000			−1.22670			−0.613351	+	0.789810i	$0.710179\pi$
$522$		0			0
$523$		−	14.0000i		−	0.612177i	−0.952003	−	0.306089i	$-0.900980\pi$
$523$		−	14.0000i		−	0.612177i	0.952003	−	0.306089i	$-0.0990204\pi$
$524$	6.00000			0.262111
$525$		0			0
$526$	−27.0000			−1.17726
$527$		−	5.00000i		−	0.217803i
$528$		0			0
$529$	−41.0000			−1.78261
$530$	9.00000	−	18.0000i	0.390935	−	0.781870i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−8.00000	−	4.00000i	−0.345870	−	0.172935i
$536$	−2.00000			−0.0863868
$537$		0			0
$538$		−	19.0000i		−	0.819148i
$539$	42.0000			1.80907
$540$		0			0
$541$	−34.0000			−1.46177			−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000			−1.46177			−0.730887	+	0.682498i	$0.760893\pi$
$542$		−	6.00000i		−	0.257722i
$543$		0			0
$544$	1.00000			0.0428746
$545$	9.00000	−	18.0000i	0.385518	−	0.771035i
$546$		0			0
$547$			13.0000i			0.555840i	0.960604	+	0.277920i	$0.0896450\pi$
$547$			13.0000i			0.555840i	−0.960604	+	0.277920i	$0.910355\pi$
$548$			12.0000i			0.512615i
$549$		0			0
$550$	24.0000	−	18.0000i	1.02336	−	0.767523i
$551$	−35.0000			−1.49105
$552$		0			0
$553$		0			0
$554$	2.00000			0.0849719
$555$		0			0
$556$		0			0
$557$		−	11.0000i		−	0.466085i	−0.972467	−	0.233042i	$-0.925132\pi$
$557$		−	11.0000i		−	0.466085i	0.972467	−	0.233042i	$-0.0748681\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$		0			0
$562$			33.0000i			1.39202i
$563$			32.0000i			1.34864i	0.738440	+	0.674320i	$0.235563\pi$
$563$			32.0000i			1.34864i	−0.738440	+	0.674320i	$0.764437\pi$
$564$		0			0
$565$	26.0000	+	13.0000i	1.09383	+	0.546914i
$566$	11.0000			0.462364
$567$		0			0
$568$		−	15.0000i		−	0.629386i
$569$	−21.0000			−0.880366			−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000			−0.880366			−0.440183	+	0.897908i	$0.645086\pi$
$570$		0			0
$571$	−10.0000			−0.418487			−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000			−0.418487			−0.209243	+	0.977864i	$0.567100\pi$
$572$			18.0000i			0.752618i
$573$		0			0
$574$		0			0
$575$	−32.0000	+	24.0000i	−1.33449	+	1.00087i
$576$		0			0
$577$			16.0000i			0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$			16.0000i			0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$		−	1.00000i		−	0.0415945i
$579$		0			0
$580$	−5.00000	+	10.0000i	−0.207614	+	0.415227i
$581$		0			0
$582$		0			0
$583$			54.0000i			2.23645i
$584$	11.0000			0.455183
$585$		0			0
$586$	19.0000			0.784883
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$	35.0000			1.44215
$590$	−10.0000	−	5.00000i	−0.411693	−	0.205847i
$591$		0			0
$592$			8.00000i			0.328798i
$593$			22.0000i			0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$			22.0000i			0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$		0			0
$595$		0			0
$596$	2.00000			0.0819232
$597$		0			0
$598$		−	24.0000i		−	0.981433i
$599$	−32.0000			−1.30748			−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000			−1.30748			−0.653742	+	0.756717i	$0.726802\pi$
$600$		0			0
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$		0			0
$604$	8.00000			0.325515
$605$	−25.0000	+	50.0000i	−1.01639	+	2.03279i
$606$		0			0
$607$		−	12.0000i		−	0.487065i	−0.969893	−	0.243532i	$-0.921694\pi$
$607$		−	12.0000i		−	0.487065i	0.969893	−	0.243532i	$-0.0783062\pi$
$608$			7.00000i			0.283887i
$609$		0			0
$610$	6.00000	+	3.00000i	0.242933	+	0.121466i
$611$	−9.00000			−0.364101
$612$		0			0
$613$			17.0000i			0.686624i	0.939222	+	0.343312i	$0.111549\pi$
$613$			17.0000i			0.686624i	−0.939222	+	0.343312i	$0.888451\pi$
$614$	24.0000			0.968561
$615$		0			0
$616$		0			0
$617$			29.0000i			1.16750i	0.811935	+	0.583748i	$0.198414\pi$
$617$			29.0000i			1.16750i	−0.811935	+	0.583748i	$0.801586\pi$
$618$		0			0
$619$	−14.0000			−0.562708			−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000			−0.562708			−0.281354	+	0.959604i	$0.590783\pi$
$620$	5.00000	−	10.0000i	0.200805	−	0.401610i
$621$		0			0
$622$			4.00000i			0.160385i
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−22.0000			−0.879297
$627$		0			0
$628$		−	14.0000i		−	0.558661i
$629$	8.00000			0.318981
$630$		0			0
$631$	−42.0000			−1.67199			−0.835997	−	0.548734i	$-0.815110\pi$
$631$	−42.0000			−1.67199			−0.835997	+	0.548734i	$0.815110\pi$
$632$			8.00000i			0.318223i
$633$		0			0
$634$		0			0
$635$	−2.00000	−	1.00000i	−0.0793676	−	0.0396838i
$636$		0			0
$637$		−	21.0000i		−	0.832050i
$638$		−	30.0000i		−	1.18771i
$639$		0			0
$640$	2.00000	+	1.00000i	0.0790569	+	0.0395285i
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$			44.0000i			1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$			44.0000i			1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$		0			0
$645$		0			0
$646$	7.00000			0.275411
$647$		−	17.0000i		−	0.668339i	−0.942513	−	0.334169i	$-0.891544\pi$
$647$		−	17.0000i		−	0.668339i	0.942513	−	0.334169i	$-0.108456\pi$
$648$		0			0
$649$	30.0000			1.17760
$650$	−9.00000	−	12.0000i	−0.353009	−	0.470679i
$651$		0			0
$652$		−	4.00000i		−	0.156652i
$653$			30.0000i			1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$			30.0000i			1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$		0			0
$655$	6.00000	−	12.0000i	0.234439	−	0.468879i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.0000			0.740135			0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000			0.740135			0.370067	+	0.929005i	$0.379335\pi$
$660$		0			0
$661$	−16.0000			−0.622328			−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000			−0.622328			−0.311164	+	0.950356i	$0.600719\pi$
$662$			7.00000i			0.272063i
$663$		0			0
$664$	4.00000			0.155230
$665$		0			0
$666$		0			0
$667$			40.0000i			1.54881i
$668$		−	6.00000i		−	0.232147i
$669$		0			0
$670$	−2.00000	+	4.00000i	−0.0772667	+	0.154533i
$671$	−18.0000			−0.694882
$672$		0			0
$673$			35.0000i			1.34915i	0.738206	+	0.674575i	$0.235673\pi$
$673$			35.0000i			1.34915i	−0.738206	+	0.674575i	$0.764327\pi$
$674$	−13.0000			−0.500741
$675$		0			0
$676$	−4.00000			−0.153846
$677$			24.0000i			0.922395i	0.887298	+	0.461197i	$0.152580\pi$
$677$			24.0000i			0.922395i	−0.887298	+	0.461197i	$0.847420\pi$
$678$		0			0
$679$		0			0
$680$	1.00000	−	2.00000i	0.0383482	−	0.0766965i
$681$		0			0
$682$			30.0000i			1.14876i
$683$			45.0000i			1.72188i	0.508709	+	0.860939i	$0.330123\pi$
$683$			45.0000i			1.72188i	−0.508709	+	0.860939i	$0.669877\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$		0			0
$687$		0			0
$688$			4.00000i			0.152499i
$689$	27.0000			1.02862
$690$		0			0
$691$	−16.0000			−0.608669			−0.304334	−	0.952565i	$-0.598434\pi$
$691$	−16.0000			−0.608669			−0.304334	+	0.952565i	$0.598434\pi$
$692$		0			0
$693$		0			0
$694$	5.00000			0.189797
$695$		0			0
$696$		0			0
$697$		0			0
$698$			2.00000i			0.0757011i
$699$		0			0
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$			56.0000i			2.11208i
$704$	−6.00000			−0.226134
$705$		0			0
$706$	8.00000			0.301084
$707$		0			0
$708$		0			0
$709$	−23.0000			−0.863783			−0.431892	−	0.901926i	$-0.642154\pi$
$709$	−23.0000			−0.863783			−0.431892	+	0.901926i	$0.642154\pi$
$710$	−30.0000	−	15.0000i	−1.12588	−	0.562940i
$711$		0			0
$712$			1.00000i			0.0374766i
$713$		−	40.0000i		−	1.49801i
$714$		0			0
$715$	36.0000	+	18.0000i	1.34632	+	0.673162i
$716$	12.0000			0.448461
$717$		0			0
$718$		−	28.0000i		−	1.04495i
$719$	−27.0000			−1.00693			−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000			−1.00693			−0.503465	+	0.864016i	$0.667942\pi$
$720$		0			0
$721$		0			0
$722$			30.0000i			1.11648i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$	15.0000	+	20.0000i	0.557086	+	0.742781i
$726$		0			0
$727$		−	29.0000i		−	1.07555i	−0.843088	−	0.537775i	$-0.819265\pi$
$727$		−	29.0000i		−	1.07555i	0.843088	−	0.537775i	$-0.180735\pi$
$728$		0			0
$729$		0			0
$730$	11.0000	−	22.0000i	0.407128	−	0.814257i
$731$	4.00000			0.147945
$732$		0			0
$733$			14.0000i			0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$			14.0000i			0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	10.0000			0.369107
$735$		0			0
$736$	8.00000			0.294884
$737$		−	12.0000i		−	0.442026i
$738$		0			0
$739$	1.00000			0.0367856			0.0183928	−	0.999831i	$-0.494145\pi$
$739$	1.00000			0.0367856			0.0183928	+	0.999831i	$0.494145\pi$
$740$	16.0000	+	8.00000i	0.588172	+	0.294086i
$741$		0			0
$742$		0			0
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$		0			0
$745$	2.00000	−	4.00000i	0.0732743	−	0.146549i
$746$	−14.0000			−0.512576
$747$		0			0
$748$			6.00000i			0.219382i
$749$		0			0
$750$		0			0
$751$	11.0000			0.401396			0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000			0.401396			0.200698	+	0.979653i	$0.435679\pi$
$752$		−	3.00000i		−	0.109399i
$753$		0			0
$754$	−15.0000			−0.546268
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	45.0000i		−	1.63555i	−0.575536	−	0.817776i	$-0.695207\pi$
$757$		−	45.0000i		−	1.63555i	0.575536	−	0.817776i	$-0.304793\pi$
$758$			20.0000i			0.726433i
$759$		0			0
$760$	14.0000	+	7.00000i	0.507833	+	0.253917i
$761$	−14.0000			−0.507500			−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000			−0.507500			−0.253750	+	0.967270i	$0.581664\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	27.0000			0.975550
$767$		−	15.0000i		−	0.541619i
$768$		0			0
$769$	−37.0000			−1.33425			−0.667127	−	0.744944i	$-0.732476\pi$
$769$	−37.0000			−1.33425			−0.667127	+	0.744944i	$0.732476\pi$
$770$		0			0
$771$		0			0
$772$			26.0000i			0.935760i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		0			0
$775$	−15.0000	−	20.0000i	−0.538816	−	0.718421i
$776$	−9.00000			−0.323081
$777$		0			0
$778$			30.0000i			1.07555i
$779$		0			0
$780$		0			0
$781$	90.0000			3.22045
$782$		−	8.00000i		−	0.286079i
$783$		0			0
$784$	7.00000			0.250000
$785$	−28.0000	−	14.0000i	−0.999363	−	0.499681i
$786$		0			0
$787$		−	47.0000i		−	1.67537i	−0.546154	−	0.837685i	$-0.683909\pi$
$787$		−	47.0000i		−	1.67537i	0.546154	−	0.837685i	$-0.316091\pi$
$788$			2.00000i			0.0712470i
$789$		0			0
$790$	16.0000	+	8.00000i	0.569254	+	0.284627i
$791$		0			0
$792$		0			0
$793$			9.00000i			0.319599i
$794$	2.00000			0.0709773
$795$		0			0
$796$	7.00000			0.248108
$797$		−	54.0000i		−	1.91278i	−0.292096	−	0.956389i	$-0.594353\pi$
$797$		−	54.0000i		−	1.91278i	0.292096	−	0.956389i	$-0.405647\pi$
$798$		0			0
$799$	−3.00000			−0.106132
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$		0			0
$802$			10.0000i			0.353112i
$803$			66.0000i			2.32909i
$804$		0			0
$805$		0			0
$806$	15.0000			0.528352
$807$		0			0
$808$		−	12.0000i		−	0.422159i
$809$	10.0000			0.351581			0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000			0.351581			0.175791	+	0.984428i	$0.443752\pi$
$810$		0			0
$811$	−32.0000			−1.12367			−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000			−1.12367			−0.561836	+	0.827249i	$0.689905\pi$
$812$		0			0
$813$		0			0
$814$	−48.0000			−1.68240
$815$	−8.00000	−	4.00000i	−0.280228	−	0.140114i
$816$		0			0
$817$			28.0000i			0.979596i
$818$			31.0000i			1.08389i
$819$		0			0
$820$		0			0
$821$	27.0000			0.942306			0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000			0.942306			0.471153	+	0.882051i	$0.343838\pi$
$822$		0			0
$823$			54.0000i			1.88232i	0.337959	+	0.941161i	$0.390263\pi$
$823$			54.0000i			1.88232i	−0.337959	+	0.941161i	$0.609737\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			32.0000i			1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$			32.0000i			1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$		0			0
$829$	−18.0000			−0.625166			−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000			−0.625166			−0.312583	+	0.949890i	$0.601194\pi$
$830$	4.00000	−	8.00000i	0.138842	−	0.277684i
$831$		0			0
$832$			3.00000i			0.104006i
$833$		−	7.00000i		−	0.242536i
$834$		0			0
$835$	−12.0000	−	6.00000i	−0.415277	−	0.207639i
$836$	−42.0000			−1.45260
$837$		0			0
$838$		0			0
$839$	31.0000			1.07024			0.535119	−	0.844776i	$-0.320267\pi$
$839$	31.0000			1.07024			0.535119	+	0.844776i	$0.320267\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$			20.0000i			0.689246i
$843$		0			0
$844$	−16.0000			−0.550743
$845$	−4.00000	+	8.00000i	−0.137604	+	0.275208i
$846$		0			0
$847$		0			0
$848$			9.00000i			0.309061i
$849$		0			0
$850$	−3.00000	−	4.00000i	−0.102899	−	0.137199i
$851$	64.0000			2.19389
$852$		0			0
$853$		−	30.0000i		−	1.02718i	−0.858036	−	0.513590i	$-0.828315\pi$
$853$		−	30.0000i		−	1.02718i	0.858036	−	0.513590i	$-0.171685\pi$
$854$		0			0
$855$		0			0
$856$	4.00000			0.136717
$857$			57.0000i			1.94708i	0.228510	+	0.973541i	$0.426614\pi$
$857$			57.0000i			1.94708i	−0.228510	+	0.973541i	$0.573386\pi$
$858$		0			0
$859$	−33.0000			−1.12595			−0.562973	−	0.826475i	$-0.690342\pi$
$859$	−33.0000			−1.12595			−0.562973	+	0.826475i	$0.690342\pi$
$860$	8.00000	+	4.00000i	0.272798	+	0.136399i
$861$		0			0
$862$			16.0000i			0.544962i
$863$		−	8.00000i		−	0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$		−	8.00000i		−	0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$		0			0
$865$		0			0
$866$	20.0000			0.679628
$867$		0			0
$868$		0			0
$869$	−48.0000			−1.62829
$870$		0			0
$871$	−6.00000			−0.203302
$872$			9.00000i			0.304778i
$873$		0			0
$874$	56.0000			1.89423
$875$		0			0
$876$		0			0
$877$		−	10.0000i		−	0.337676i	−0.985644	−	0.168838i	$-0.945999\pi$
$877$		−	10.0000i		−	0.337676i	0.985644	−	0.168838i	$-0.0540015\pi$
$878$		−	16.0000i		−	0.539974i
$879$		0			0
$880$	−6.00000	+	12.0000i	−0.202260	+	0.404520i
$881$	−38.0000			−1.28025			−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000			−1.28025			−0.640126	+	0.768270i	$0.721118\pi$
$882$		0			0
$883$			26.0000i			0.874970i	0.899226	+	0.437485i	$0.144131\pi$
$883$			26.0000i			0.874970i	−0.899226	+	0.437485i	$0.855869\pi$
$884$	3.00000			0.100901
$885$		0			0
$886$	−10.0000			−0.335957
$887$			14.0000i			0.470074i	0.971986	+	0.235037i	$0.0755211\pi$
$887$			14.0000i			0.470074i	−0.971986	+	0.235037i	$0.924479\pi$
$888$		0			0
$889$		0			0
$890$	2.00000	+	1.00000i	0.0670402	+	0.0335201i
$891$		0			0
$892$		−	15.0000i		−	0.502237i
$893$		−	21.0000i		−	0.702738i
$894$		0			0
$895$	12.0000	−	24.0000i	0.401116	−	0.802232i
$896$		0			0
$897$		0			0
$898$		−	6.00000i		−	0.200223i
$899$	−25.0000			−0.833797
$900$		0			0
$901$	9.00000			0.299833
$902$		0			0
$903$		0			0
$904$	−13.0000			−0.432374
$905$	−2.00000	+	4.00000i	−0.0664822	+	0.132964i
$906$		0			0
$907$			27.0000i			0.896520i	0.893903	+	0.448260i	$0.147956\pi$
$907$			27.0000i			0.896520i	−0.893903	+	0.448260i	$0.852044\pi$
$908$		−	9.00000i		−	0.298675i
$909$		0			0
$910$		0			0
$911$	−8.00000			−0.265052			−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000			−0.265052			−0.132526	+	0.991180i	$0.542309\pi$
$912$		0			0
$913$			24.0000i			0.794284i
$914$	12.0000			0.396925
$915$		0			0
$916$	18.0000			0.594737
$917$		0			0
$918$		0			0
$919$	6.00000			0.197922			0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000			0.197922			0.0989609	+	0.995091i	$0.468448\pi$
$920$	8.00000	−	16.0000i	0.263752	−	0.527504i
$921$		0			0
$922$			20.0000i			0.658665i
$923$		−	45.0000i		−	1.48119i
$924$		0			0
$925$	32.0000	−	24.0000i	1.05215	−	0.789115i
$926$	31.0000			1.01872
$927$		0			0
$928$		−	5.00000i		−	0.164133i
$929$	−34.0000			−1.11550			−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000			−1.11550			−0.557752	+	0.830008i	$0.688336\pi$
$930$		0			0
$931$	49.0000			1.60591
$932$		−	27.0000i		−	0.884414i
$933$		0			0
$934$	−14.0000			−0.458094
$935$	12.0000	+	6.00000i	0.392442	+	0.196221i
$936$		0			0
$937$		−	44.0000i		−	1.43742i	−0.695311	−	0.718709i	$-0.744734\pi$
$937$		−	44.0000i		−	1.43742i	0.695311	−	0.718709i	$-0.255266\pi$
$938$		0			0
$939$		0			0
$940$	−6.00000	−	3.00000i	−0.195698	−	0.0978492i
$941$	5.00000			0.162995			0.0814977	−	0.996674i	$-0.474030\pi$
$941$	5.00000			0.162995			0.0814977	+	0.996674i	$0.474030\pi$
$942$		0			0
$943$		0			0
$944$	5.00000			0.162736
$945$		0			0
$946$	−24.0000			−0.780307
$947$			9.00000i			0.292461i	0.989251	+	0.146230i	$0.0467141\pi$
$947$			9.00000i			0.292461i	−0.989251	+	0.146230i	$0.953286\pi$
$948$		0			0
$949$	33.0000			1.07123
$950$	28.0000	−	21.0000i	0.908440	−	0.681330i
$951$		0			0
$952$		0			0
$953$		−	44.0000i		−	1.42530i	−0.701520	−	0.712650i	$-0.747495\pi$
$953$		−	44.0000i		−	1.42530i	0.701520	−	0.712650i	$-0.252505\pi$
$954$		0			0
$955$		0			0
$956$	−10.0000			−0.323423
$957$		0			0
$958$		−	37.0000i		−	1.19542i
$959$		0			0
$960$		0			0
$961$	−6.00000			−0.193548
$962$			24.0000i			0.773791i
$963$		0			0
$964$	20.0000			0.644157
$965$	52.0000	+	26.0000i	1.67394	+	0.836970i
$966$		0			0
$967$		−	16.0000i		−	0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$		−	16.0000i		−	0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$		−	25.0000i		−	0.803530i
$969$		0			0
$970$	−9.00000	+	18.0000i	−0.288973	+	0.577945i
$971$	−37.0000			−1.18739			−0.593693	−	0.804691i	$-0.702331\pi$
$971$	−37.0000			−1.18739			−0.593693	+	0.804691i	$0.702331\pi$
$972$		0			0
$973$		0			0
$974$	2.00000			0.0640841
$975$		0			0
$976$	−3.00000			−0.0960277
$977$		−	38.0000i		−	1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$		−	38.0000i		−	1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$		0			0
$979$	−6.00000			−0.191761
$980$	7.00000	−	14.0000i	0.223607	−	0.447214i
$981$		0			0
$982$			13.0000i			0.414847i
$983$			30.0000i			0.956851i	0.878128	+	0.478426i	$0.158792\pi$
$983$			30.0000i			0.956851i	−0.878128	+	0.478426i	$0.841208\pi$
$984$		0			0
$985$	4.00000	+	2.00000i	0.127451	+	0.0637253i
$986$	−5.00000			−0.159232
$987$		0			0
$988$			21.0000i			0.668099i
$989$	32.0000			1.01754
$990$		0			0
$991$	−19.0000			−0.603555			−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000			−0.603555			−0.301777	+	0.953378i	$0.597580\pi$
$992$			5.00000i			0.158750i
$993$		0			0
$994$		0			0
$995$	7.00000	−	14.0000i	0.221915	−	0.443830i
$996$		0			0
$997$			28.0000i			0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$			28.0000i			0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$			28.0000i			0.886325i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1530.2.d.b.919.2		2
3.2	odd	2		170.2.c.a.69.1	✓	2
5.2	odd	4		7650.2.a.s.1.1		1
5.3	odd	4		7650.2.a.cb.1.1		1
5.4	even	2	inner	1530.2.d.b.919.1		2
12.11	even	2		1360.2.e.b.1089.2		2
15.2	even	4		850.2.a.h.1.1		1
15.8	even	4		850.2.a.d.1.1		1
15.14	odd	2		170.2.c.a.69.2	yes	2
60.23	odd	4		6800.2.a.g.1.1		1
60.47	odd	4		6800.2.a.r.1.1		1
60.59	even	2		1360.2.e.b.1089.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
170.2.c.a.69.1	✓	2	3.2	odd	2
170.2.c.a.69.2	yes	2	15.14	odd	2
850.2.a.d.1.1		1	15.8	even	4
850.2.a.h.1.1		1	15.2	even	4
1360.2.e.b.1089.1		2	60.59	even	2
1360.2.e.b.1089.2		2	12.11	even	2
1530.2.d.b.919.1		2	5.4	even	2	inner
1530.2.d.b.919.2		2	1.1	even	1	trivial
6800.2.a.g.1.1		1	60.23	odd	4
6800.2.a.r.1.1		1	60.47	odd	4
7650.2.a.s.1.1		1	5.2	odd	4
7650.2.a.cb.1.1		1	5.3	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:	\( N \)	\(=\)	\( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1530.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} -1.00000i q^{8} +(-2.00000 - 1.00000i) q^{10} +6.00000 q^{11} -3.00000i q^{13} +1.00000 q^{16} -1.00000i q^{17} +7.00000 q^{19} +(1.00000 - 2.00000i) q^{20} +6.00000i q^{22} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +3.00000 q^{26} -5.00000 q^{29} +5.00000 q^{31} +1.00000i q^{32} +1.00000 q^{34} +8.00000i q^{37} +7.00000i q^{38} +(2.00000 + 1.00000i) q^{40} +4.00000i q^{43} -6.00000 q^{44} +8.00000 q^{46} -3.00000i q^{47} +7.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +3.00000i q^{52} +9.00000i q^{53} +(-6.00000 + 12.0000i) q^{55} -5.00000i q^{58} +5.00000 q^{59} -3.00000 q^{61} +5.00000i q^{62} -1.00000 q^{64} +(6.00000 + 3.00000i) q^{65} -2.00000i q^{67} +1.00000i q^{68} +15.0000 q^{71} +11.0000i q^{73} -8.00000 q^{74} -7.00000 q^{76} -8.00000 q^{79} +(-1.00000 + 2.00000i) q^{80} +4.00000i q^{83} +(2.00000 + 1.00000i) q^{85} -4.00000 q^{86} -6.00000i q^{88} -1.00000 q^{89} +8.00000i q^{92} +3.00000 q^{94} +(-7.00000 + 14.0000i) q^{95} -9.00000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^5	\( 2 q - 2 q^{4} - 2 q^{5} - 4 q^{10} + 12 q^{11} + 2 q^{16} + 14 q^{19} + 2 q^{20} - 6 q^{25} + 6 q^{26} - 10 q^{29} + 10 q^{31} + 2 q^{34} + 4 q^{40} - 12 q^{44} + 16 q^{46} + 14 q^{49} + 8 q^{50} - 12 q^{55} + 10 q^{59} - 6 q^{61} - 2 q^{64} + 12 q^{65} + 30 q^{71} - 16 q^{74} - 14 q^{76} - 16 q^{79} - 2 q^{80} + 4 q^{85} - 8 q^{86} - 2 q^{89} + 6 q^{94} - 14 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^5 - 4 * q^10 + 12 * q^11 + 2 * q^16 + 14 * q^19 + 2 * q^20 - 6 * q^25 + 6 * q^26 - 10 * q^29 + 10 * q^31 + 2 * q^34 + 4 * q^40 - 12 * q^44 + 16 * q^46 + 14 * q^49 + 8 * q^50 - 12 * q^55 + 10 * q^59 - 6 * q^61 - 2 * q^64 + 12 * q^65 + 30 * q^71 - 16 * q^74 - 14 * q^76 - 16 * q^79 - 2 * q^80 + 4 * q^85 - 8 * q^86 - 2 * q^89 + 6 * q^94 - 14 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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