LMFDB - Embedded newform 3528.2.bl.b.1097.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3528,2,Mod(521,3528)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3528, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3528.521"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.bl (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1097.6
Root			$0.991445 + 0.130526i$ of defining polynomial
Character	$\chi$	$=$	3528.1097
Dual form			3528.2.bl.b.521.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.765367 - 1.32565i) q^{5} +(0.507306 - 0.292893i) q^{11} -2.16478i q^{13} +(2.93015 + 5.07517i) q^{17} +(4.52607 + 2.61313i) q^{19} +(1.94218 + 1.12132i) q^{23} +(1.32843 + 2.30090i) q^{25} -5.41421i q^{29} +(-3.74952 + 2.16478i) q^{31} +(-2.00000 + 3.46410i) q^{37} -8.92177 q^{41} +10.4853 q^{43} +(-3.69552 + 6.40083i) q^{47} +(4.68885 - 2.70711i) q^{53} -0.896683i q^{55} +(-2.86976 - 1.65685i) q^{65} +(4.82843 + 8.36308i) q^{67} -4.58579i q^{71} +(10.9269 - 6.30864i) q^{73} +(1.17157 - 2.02922i) q^{79} +13.5140 q^{83} +8.97056 q^{85} +(2.93015 - 5.07517i) q^{89} +(6.92820 - 4.00000i) q^{95} +8.28772i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{67} + 64 q^{79} - 128 q^{85}+O(q^{100}) $ 16 * q - 24 * q^25 - 32 * q^37 + 32 * q^43 + 32 * q^67 + 64 * q^79 - 128 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.765367	−	1.32565i	0.342282	−	0.592851i	−0.642574	−	0.766224i	$-0.722133\pi$
$5$	0.765367	−	1.32565i	0.342282	−	0.592851i	0.984856	+	0.173373i	$0.0554667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.507306	−	0.292893i	0.152958	−	0.0883106i	−0.421567	−	0.906797i	$-0.638520\pi$
$11$	0.507306	−	0.292893i	0.152958	−	0.0883106i	0.574526	+	0.818487i	$0.305187\pi$
$12$		0			0
$13$		−	2.16478i		−	0.600403i	−0.953876	−	0.300202i	$-0.902946\pi$
$13$		−	2.16478i		−	0.600403i	0.953876	−	0.300202i	$-0.0970540\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.93015	+	5.07517i	0.710666	+	1.23091i	0.964607	+	0.263690i	$0.0849396\pi$
$17$	2.93015	+	5.07517i	0.710666	+	1.23091i	−0.253941	+	0.967220i	$0.581727\pi$
$18$		0			0
$19$	4.52607	+	2.61313i	1.03835	+	0.599492i	0.919366	−	0.393404i	$-0.128703\pi$
$19$	4.52607	+	2.61313i	1.03835	+	0.599492i	0.118985	+	0.992896i	$0.462036\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.94218	+	1.12132i	0.404973	+	0.233811i	0.688628	−	0.725115i	$-0.258213\pi$
$23$	1.94218	+	1.12132i	0.404973	+	0.233811i	−0.283654	+	0.958927i	$0.591547\pi$
$24$		0			0
$25$	1.32843	+	2.30090i	0.265685	+	0.460181i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	5.41421i		−	1.00539i	−0.864463	−	0.502697i	$-0.832341\pi$
$29$		−	5.41421i		−	1.00539i	0.864463	−	0.502697i	$-0.167659\pi$
$30$		0			0
$31$	−3.74952	+	2.16478i	−0.673433	+	0.388807i	−0.797376	−	0.603483i	$-0.793779\pi$
$31$	−3.74952	+	2.16478i	−0.673433	+	0.388807i	0.123943	+	0.992289i	$0.460446\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−2.00000	+	3.46410i	−0.328798	+	0.569495i	−0.982274	−	0.187453i	$-0.939977\pi$
$37$	−2.00000	+	3.46410i	−0.328798	+	0.569495i	0.653476	+	0.756948i	$0.273310\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.92177			−1.39335			−0.696673	−	0.717389i	$-0.745337\pi$
$41$	−8.92177			−1.39335			−0.696673	+	0.717389i	$0.745337\pi$
$42$		0			0
$43$	10.4853			1.59899			0.799495	−	0.600672i	$-0.205100\pi$
$43$	10.4853			1.59899			0.799495	+	0.600672i	$0.205100\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.69552	+	6.40083i	−0.539047	+	0.933656i	0.459909	+	0.887966i	$0.347882\pi$
$47$	−3.69552	+	6.40083i	−0.539047	+	0.933656i	−0.998956	+	0.0456902i	$0.985451\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.68885	−	2.70711i	0.644063	−	0.371850i	−0.142115	−	0.989850i	$-0.545390\pi$
$53$	4.68885	−	2.70711i	0.644063	−	0.371850i	0.786178	+	0.618000i	$0.212057\pi$
$54$		0			0
$55$		−	0.896683i		−	0.120909i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.86976	−	1.65685i	−0.355949	−	0.205507i
$66$		0			0
$67$	4.82843	+	8.36308i	0.589886	+	1.02171i	0.994247	+	0.107113i	$0.0341608\pi$
$67$	4.82843	+	8.36308i	0.589886	+	1.02171i	−0.404361	+	0.914600i	$0.632506\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.58579i		−	0.544233i	−0.962264	−	0.272116i	$-0.912276\pi$
$71$		−	4.58579i		−	0.544233i	0.962264	−	0.272116i	$-0.0877236\pi$
$72$		0			0
$73$	10.9269	−	6.30864i	1.27890	−	0.738371i	0.302251	−	0.953229i	$-0.402262\pi$
$73$	10.9269	−	6.30864i	1.27890	−	0.738371i	0.976645	+	0.214858i	$0.0689287\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.17157	−	2.02922i	0.131812	−	0.228306i	−0.792563	−	0.609790i	$-0.791254\pi$
$79$	1.17157	−	2.02922i	0.131812	−	0.228306i	0.924375	+	0.381485i	$0.124587\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.5140			1.48335			0.741676	−	0.670759i	$-0.234031\pi$
$83$	13.5140			1.48335			0.741676	+	0.670759i	$0.234031\pi$
$84$		0			0
$85$	8.97056			0.972994
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.93015	−	5.07517i	0.310595	−	0.537967i	−0.667896	−	0.744255i	$-0.732805\pi$
$89$	2.93015	−	5.07517i	0.310595	−	0.537967i	0.978491	+	0.206288i	$0.0661382\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.92820	−	4.00000i	0.710819	−	0.410391i
$96$		0			0
$97$			8.28772i			0.841490i	0.907179	+	0.420745i	$0.138231\pi$
$97$			8.28772i			0.841490i	−0.907179	+	0.420745i	$0.861769\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.15640	−	14.1273i	−0.811592	−	1.40572i	−0.911749	−	0.410748i	$-0.865268\pi$
$101$	−8.15640	−	14.1273i	−0.811592	−	1.40572i	0.100157	−	0.994972i	$-0.468066\pi$
$102$		0			0
$103$	−12.8017	−	7.39104i	−1.26138	−	0.728260i	−0.288042	−	0.957618i	$-0.593004\pi$
$103$	−12.8017	−	7.39104i	−1.26138	−	0.728260i	−0.973342	+	0.229357i	$0.926338\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.3199	+	6.53553i	1.09433	+	0.631814i	0.934727	−	0.355367i	$-0.115644\pi$
$107$	11.3199	+	6.53553i	1.09433	+	0.631814i	0.159607	+	0.987181i	$0.448977\pi$
$108$		0			0
$109$	4.82843	+	8.36308i	0.462479	+	0.801038i	0.999084	−	0.0427961i	$-0.0136266\pi$
$109$	4.82843	+	8.36308i	0.462479	+	0.801038i	−0.536604	+	0.843834i	$0.680293\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	20.2426i		−	1.90427i	−0.305682	−	0.952134i	$-0.598884\pi$
$113$		−	20.2426i		−	1.90427i	0.305682	−	0.952134i	$-0.401116\pi$
$114$		0			0
$115$	2.97297	−	1.71644i	0.277231	−	0.160059i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.32843	+	9.22911i	−0.484402	+	0.839010i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.7206			1.04832
$126$		0			0
$127$	16.9706			1.50589			0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706			1.50589			0.752947	+	0.658081i	$0.228632\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.69552	−	6.40083i	0.322879	−	0.559243i	−0.658202	−	0.752841i	$-0.728683\pi$
$131$	3.69552	−	6.40083i	0.322879	−	0.559243i	0.981081	+	0.193599i	$0.0620160\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.57321	−	4.94975i	0.732459	−	0.422885i	−0.0868620	−	0.996220i	$-0.527684\pi$
$137$	8.57321	−	4.94975i	0.732459	−	0.422885i	0.819321	+	0.573335i	$0.194351\pi$
$138$		0			0
$139$			10.4525i			0.886570i	0.896381	+	0.443285i	$0.146187\pi$
$139$			10.4525i			0.886570i	−0.896381	+	0.443285i	$0.853813\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.634051	−	1.09821i	−0.0530220	−	0.0918368i
$144$		0			0
$145$	−7.17738	−	4.14386i	−0.596049	−	0.344129i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−14.0665	−	8.12132i	−1.15238	−	0.665324i	−0.202911	−	0.979197i	$-0.565040\pi$
$149$	−14.0665	−	8.12132i	−1.15238	−	0.665324i	−0.949465	+	0.313873i	$0.898373\pi$
$150$		0			0
$151$	6.07107	+	10.5154i	0.494056	+	0.855731i	0.999977	−	0.00684943i	$-0.00218026\pi$
$151$	6.07107	+	10.5154i	0.494056	+	0.855731i	−0.505920	+	0.862580i	$0.668847\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			6.62742i			0.532327i
$156$		0			0
$157$	−14.3548	+	8.28772i	−1.14563	+	0.661432i	−0.947819	−	0.318807i	$-0.896718\pi$
$157$	−14.3548	+	8.28772i	−1.14563	+	0.661432i	−0.197814	+	0.980240i	$0.563384\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3.17157	−	5.49333i	0.248417	−	0.430270i	−0.714670	−	0.699462i	$-0.753423\pi$
$163$	3.17157	−	5.49333i	0.248417	−	0.430270i	0.963087	+	0.269191i	$0.0867564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−13.5140			−1.04574			−0.522871	−	0.852412i	$-0.675139\pi$
$167$	−13.5140			−1.04574			−0.522871	+	0.852412i	$0.675139\pi$
$168$		0			0
$169$	8.31371			0.639516
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.2179	+	19.4299i	−0.852879	+	1.47723i	0.0257200	+	0.999669i	$0.491812\pi$
$173$	−11.2179	+	19.4299i	−0.852879	+	1.47723i	−0.878599	+	0.477560i	$0.841521\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.37706	+	1.94975i	−0.252413	+	0.145731i	−0.620869	−	0.783914i	$-0.713220\pi$
$179$	−3.37706	+	1.94975i	−0.252413	+	0.145731i	0.368455	+	0.929645i	$0.379887\pi$
$180$		0			0
$181$		−	14.4107i		−	1.07114i	−0.844492	−	0.535568i	$-0.820098\pi$
$181$		−	14.4107i		−	1.07114i	0.844492	−	0.535568i	$-0.179902\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.06147	+	5.30262i	0.225084	+	0.389856i
$186$		0			0
$187$	2.97297	+	1.71644i	0.217405	+	0.125519i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	19.6830	+	11.3640i	1.42421	+	0.822267i	0.996655	−	0.0817226i	$-0.0260421\pi$
$191$	19.6830	+	11.3640i	1.42421	+	0.822267i	0.427554	+	0.903990i	$0.359375\pi$
$192$		0			0
$193$	2.82843	+	4.89898i	0.203595	+	0.352636i	0.949684	−	0.313210i	$-0.101404\pi$
$193$	2.82843	+	4.89898i	0.203595	+	0.352636i	−0.746089	+	0.665846i	$0.768071\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.3848i		−	1.02487i	−0.858725	−	0.512436i	$-0.828743\pi$
$197$		−	14.3848i		−	1.02487i	0.858725	−	0.512436i	$-0.171257\pi$
$198$		0			0
$199$	−2.97297	+	1.71644i	−0.210748	+	0.121675i	−0.601659	−	0.798753i	$-0.705493\pi$
$199$	−2.97297	+	1.71644i	−0.210748	+	0.121675i	0.390911	+	0.920428i	$0.372160\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−6.82843	+	11.8272i	−0.476918	+	0.826046i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.06147			0.211766
$210$		0			0
$211$	24.8284			1.70926			0.854630	−	0.519238i	$-0.173784\pi$
$211$	24.8284			1.70926			0.854630	+	0.519238i	$0.173784\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.02509	−	13.8999i	0.547306	−	0.947962i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.9867	−	6.34315i	0.739042	−	0.426686i
$222$		0			0
$223$		−	3.43289i		−	0.229883i	−0.993372	−	0.114942i	$-0.963332\pi$
$223$		−	3.43289i		−	0.229883i	0.993372	−	0.114942i	$-0.0366681\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.06147	−	5.30262i	−0.203197	−	0.351947i	0.746360	−	0.665542i	$-0.231800\pi$
$227$	−3.06147	−	5.30262i	−0.203197	−	0.351947i	−0.949557	+	0.313595i	$0.898466\pi$
$228$		0			0
$229$	19.9790	+	11.5349i	1.32025	+	0.762247i	0.983769	−	0.179442i	$-0.0574293\pi$
$229$	19.9790	+	11.5349i	1.32025	+	0.762247i	0.336483	+	0.941690i	$0.390763\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.09450	−	2.36396i	−0.268240	−	0.154868i	0.359848	−	0.933011i	$-0.382829\pi$
$233$	−4.09450	−	2.36396i	−0.268240	−	0.154868i	−0.628087	+	0.778143i	$0.716162\pi$
$234$		0			0
$235$	5.65685	+	9.79796i	0.369012	+	0.639148i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	17.0711i		−	1.10424i	−0.833766	−	0.552118i	$-0.813820\pi$
$239$		−	17.0711i		−	1.10424i	0.833766	−	0.552118i	$-0.186180\pi$
$240$		0			0
$241$	−14.6764	+	8.47343i	−0.945391	+	0.545822i	−0.891646	−	0.452733i	$-0.850449\pi$
$241$	−14.6764	+	8.47343i	−0.945391	+	0.545822i	−0.0537446	+	0.998555i	$0.517116\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5.65685	−	9.79796i	0.359937	−	0.623429i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−28.2960			−1.78603			−0.893015	−	0.450026i	$-0.851415\pi$
$251$	−28.2960			−1.78603			−0.893015	+	0.450026i	$0.851415\pi$
$252$		0			0
$253$	1.31371			0.0825921
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.05309	−	15.6804i	0.564716	−	0.978117i	−0.432360	−	0.901701i	$-0.642319\pi$
$257$	9.05309	−	15.6804i	0.564716	−	0.978117i	0.997076	−	0.0764157i	$-0.0243476\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.82655	−	3.36396i	0.359281	−	0.207431i	−0.309485	−	0.950904i	$-0.600157\pi$
$263$	5.82655	−	3.36396i	0.359281	−	0.207431i	0.668765	+	0.743474i	$0.266823\pi$
$264$		0			0
$265$		−	8.28772i		−	0.509111i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.29610	+	3.97696i	0.139996	+	0.242480i	0.927495	−	0.373836i	$-0.121958\pi$
$269$	2.29610	+	3.97696i	0.139996	+	0.242480i	−0.787499	+	0.616316i	$0.788624\pi$
$270$		0			0
$271$	14.3548	+	8.28772i	0.871989	+	0.503443i	0.868009	−	0.496549i	$-0.165400\pi$
$271$	14.3548	+	8.28772i	0.871989	+	0.503443i	0.00398032	+	0.999992i	$0.498733\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.34784	+	0.778175i	0.0812777	+	0.0469257i
$276$		0			0
$277$	−4.17157	−	7.22538i	−0.250646	−	0.434131i	0.713058	−	0.701105i	$-0.247310\pi$
$277$	−4.17157	−	7.22538i	−0.250646	−	0.434131i	−0.963704	+	0.266974i	$0.913976\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	9.89949i		−	0.590554i	−0.955412	−	0.295277i	$-0.904588\pi$
$281$		−	9.89949i		−	0.590554i	0.955412	−	0.295277i	$-0.0954120\pi$
$282$		0			0
$283$	21.0772	−	12.1689i	1.25291	−	0.723369i	0.281225	−	0.959642i	$-0.409259\pi$
$283$	21.0772	−	12.1689i	1.25291	−	0.723369i	0.971687	+	0.236273i	$0.0759260\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.67157	+	15.0196i	−0.510093	+	0.883506i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.7151			0.625985			0.312992	−	0.949756i	$-0.398669\pi$
$293$	10.7151			0.625985			0.312992	+	0.949756i	$0.398669\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.42742	−	4.20441i	0.140381	−	0.243147i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	26.1313i		−	1.49139i	−0.666288	−	0.745695i	$-0.732118\pi$
$307$		−	26.1313i		−	1.49139i	0.666288	−	0.745695i	$-0.267882\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	14.7821	+	25.6033i	0.838214	+	1.45183i	0.891386	+	0.453245i	$0.149734\pi$
$311$	14.7821	+	25.6033i	0.838214	+	1.45183i	−0.0531718	+	0.998585i	$0.516933\pi$
$312$		0			0
$313$	−7.49903	−	4.32957i	−0.423870	−	0.244722i	0.272861	−	0.962053i	$-0.412030\pi$
$313$	−7.49903	−	4.32957i	−0.423870	−	0.244722i	−0.696732	+	0.717332i	$0.745363\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.4576	−	7.19239i	−0.699687	−	0.403965i	0.107544	−	0.994200i	$-0.465702\pi$
$317$	−12.4576	−	7.19239i	−0.699687	−	0.403965i	−0.807231	+	0.590236i	$0.799035\pi$
$318$		0			0
$319$	−1.58579	−	2.74666i	−0.0887870	−	0.153784i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			30.6274i			1.70416i
$324$		0			0
$325$	4.98096	−	2.87576i	0.276294	−	0.159518i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.414214	−	0.717439i	0.0227672	−	0.0394340i	−0.854417	−	0.519587i	$-0.826086\pi$
$331$	0.414214	−	0.717439i	0.0227672	−	0.0394340i	0.877185	+	0.480153i	$0.159419\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	14.7821			0.807631
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.26810	+	2.19642i	−0.0686715	+	0.118943i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.2042	+	8.77817i	−0.816207	+	0.471237i	−0.849107	−	0.528222i	$-0.822859\pi$
$347$	−15.2042	+	8.77817i	−0.816207	+	0.471237i	0.0329000	+	0.999459i	$0.489526\pi$
$348$		0			0
$349$			16.5754i			0.887263i	0.896209	+	0.443631i	$0.146310\pi$
$349$			16.5754i			0.887263i	−0.896209	+	0.443631i	$0.853690\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.131316	−	0.227446i	−0.00698926	−	0.0121057i	0.862510	−	0.506041i	$-0.168891\pi$
$353$	−0.131316	−	0.227446i	−0.00698926	−	0.0121057i	−0.869499	+	0.493935i	$0.835558\pi$
$354$		0			0
$355$	−6.07917	−	3.50981i	−0.322649	−	0.186281i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	24.5819	+	14.1924i	1.29739	+	0.749046i	0.979952	−	0.199235i	$-0.0638457\pi$
$359$	24.5819	+	14.1924i	1.29739	+	0.749046i	0.317433	+	0.948281i	$0.397179\pi$
$360$		0			0
$361$	4.15685	+	7.19988i	0.218782	+	0.378941i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	19.3137i		−	1.01093i
$366$		0			0
$367$	−13.5782	+	7.83938i	−0.708776	+	0.409212i	−0.810608	−	0.585589i	$-0.800863\pi$
$367$	−13.5782	+	7.83938i	−0.708776	+	0.409212i	0.101831	+	0.994802i	$0.467530\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.65685	+	4.60181i	−0.137567	+	0.238273i	−0.926575	−	0.376110i	$-0.877261\pi$
$373$	−2.65685	+	4.60181i	−0.137567	+	0.238273i	0.789008	+	0.614383i	$0.210595\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−11.7206			−0.603642
$378$		0			0
$379$	16.8284			0.864418			0.432209	−	0.901773i	$-0.357734\pi$
$379$	16.8284			0.864418			0.432209	+	0.901773i	$0.357734\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−14.1480	+	24.5051i	−0.722930	+	1.25215i	0.236890	+	0.971536i	$0.423872\pi$
$383$	−14.1480	+	24.5051i	−0.722930	+	1.25215i	−0.959820	+	0.280615i	$0.909461\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.630399	−	0.363961i	0.0319625	−	0.0184536i	−0.483934	−	0.875105i	$-0.660792\pi$
$389$	0.630399	−	0.363961i	0.0319625	−	0.0184536i	0.515896	+	0.856651i	$0.327459\pi$
$390$		0			0
$391$			13.1426i			0.664647i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.79337	−	3.10620i	−0.0902340	−	0.156290i
$396$		0			0
$397$	−7.49903	−	4.32957i	−0.376366	−	0.217295i	0.299870	−	0.953980i	$-0.403057\pi$
$397$	−7.49903	−	4.32957i	−0.376366	−	0.217295i	−0.676236	+	0.736685i	$0.736390\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−27.3286	−	15.7782i	−1.36473	−	0.787924i	−0.374477	−	0.927236i	$-0.622178\pi$
$401$	−27.3286	−	15.7782i	−1.36473	−	0.787924i	−0.990249	+	0.139312i	$0.955511\pi$
$402$		0			0
$403$	4.68629	+	8.11689i	0.233441	+	0.404331i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			2.34315i			0.116145i
$408$		0			0
$409$	10.9269	−	6.30864i	0.540300	−	0.311942i	−0.204900	−	0.978783i	$-0.565687\pi$
$409$	10.9269	−	6.30864i	0.540300	−	0.311942i	0.745201	+	0.666840i	$0.232354\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	10.3431	−	17.9149i	0.507725	−	0.879406i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.65914			0.423027			0.211513	−	0.977375i	$-0.432161\pi$
$419$	8.65914			0.423027			0.211513	+	0.977375i	$0.432161\pi$
$420$		0			0
$421$	−14.9706			−0.729621			−0.364810	−	0.931082i	$-0.618866\pi$
$421$	−14.9706			−0.729621			−0.364810	+	0.931082i	$0.618866\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.78498	+	13.4840i	−0.377627	+	0.654070i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.87039	−	5.12132i	0.427272	−	0.246685i	−0.270912	−	0.962604i	$-0.587325\pi$
$431$	8.87039	−	5.12132i	0.427272	−	0.246685i	0.698184	+	0.715919i	$0.253992\pi$
$432$		0			0
$433$			33.8937i			1.62883i	0.580284	+	0.814414i	$0.302941\pi$
$433$			33.8937i			1.62883i	−0.580284	+	0.814414i	$0.697059\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.86030	+	10.1503i	0.280336	+	0.485557i
$438$		0			0
$439$	−2.97297	−	1.71644i	−0.141892	−	0.0819213i	0.427373	−	0.904075i	$-0.359439\pi$
$439$	−2.97297	−	1.71644i	−0.141892	−	0.0819213i	−0.569265	+	0.822154i	$0.692772\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.43551	+	4.29289i	0.353272	+	0.203962i	0.666125	−	0.745840i	$-0.267952\pi$
$443$	7.43551	+	4.29289i	0.353272	+	0.203962i	−0.312854	+	0.949801i	$0.601285\pi$
$444$		0			0
$445$	−4.48528	−	7.76874i	−0.212623	−	0.368273i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			9.41421i			0.444284i	0.975014	+	0.222142i	$0.0713049\pi$
$449$			9.41421i			0.444284i	−0.975014	+	0.222142i	$0.928695\pi$
$450$		0			0
$451$	−4.52607	+	2.61313i	−0.213124	+	0.123047i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.65685	−	9.79796i	0.264616	−	0.458329i	−0.702847	−	0.711341i	$-0.748088\pi$
$457$	5.65685	−	9.79796i	0.264616	−	0.458329i	0.967463	+	0.253012i	$0.0814213\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.7151			−0.499054			−0.249527	−	0.968368i	$-0.580275\pi$
$461$	−10.7151			−0.499054			−0.249527	+	0.968368i	$0.580275\pi$
$462$		0			0
$463$	−29.6569			−1.37827			−0.689135	−	0.724633i	$-0.742010\pi$
$463$	−29.6569			−1.37827			−0.689135	+	0.724633i	$0.742010\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.4161	+	26.7015i	−0.713373	+	1.23560i	0.250211	+	0.968191i	$0.419500\pi$
$467$	−15.4161	+	26.7015i	−0.713373	+	1.23560i	−0.963584	+	0.267407i	$0.913833\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.31925	−	3.07107i	0.244579	−	0.141208i
$474$		0			0
$475$			13.8854i			0.637105i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−11.7206	−	20.3007i	−0.535528	−	0.927562i	−0.999138	−	0.0415223i	$-0.986779\pi$
$479$	−11.7206	−	20.3007i	−0.535528	−	0.927562i	0.463609	−	0.886040i	$-0.346554\pi$
$480$		0			0
$481$	7.49903	+	4.32957i	0.341926	+	0.197411i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.9867	+	6.34315i	0.498878	+	0.288027i
$486$		0			0
$487$	−19.7279	−	34.1698i	−0.893957	−	1.54838i	−0.835090	−	0.550114i	$-0.814584\pi$
$487$	−19.7279	−	34.1698i	−0.893957	−	1.54838i	−0.0588679	−	0.998266i	$-0.518749\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			26.5269i			1.19714i	0.801069	+	0.598571i	$0.204265\pi$
$491$			26.5269i			1.19714i	−0.801069	+	0.598571i	$0.795735\pi$
$492$		0			0
$493$	27.4781	−	15.8645i	1.23755	−	0.714500i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	3.58579	−	6.21076i	0.160522	−	0.278032i	−0.774534	−	0.632532i	$-0.782016\pi$
$499$	3.58579	−	6.21076i	0.160522	−	0.278032i	0.935056	+	0.354500i	$0.115349\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19.6369			0.875566			0.437783	−	0.899081i	$-0.355764\pi$
$503$	19.6369			0.875566			0.437783	+	0.899081i	$0.355764\pi$
$504$		0			0
$505$	−24.9706			−1.11118
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.09494	−	8.82469i	0.225829	−	0.391147i	−0.730739	−	0.682657i	$-0.760824\pi$
$509$	5.09494	−	8.82469i	0.225829	−	0.391147i	0.956568	+	0.291510i	$0.0941576\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−19.5959	+	11.3137i	−0.863499	+	0.498542i
$516$		0			0
$517$			4.32957i			0.190414i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.93015	−	5.07517i	−0.128372	−	0.222347i	0.794674	−	0.607037i	$-0.207642\pi$
$521$	−2.93015	−	5.07517i	−0.128372	−	0.222347i	−0.923046	+	0.384689i	$0.874309\pi$
$522$		0			0
$523$	−7.49903	−	4.32957i	−0.327910	−	0.189319i	0.327003	−	0.945023i	$-0.393961\pi$
$523$	−7.49903	−	4.32957i	−0.327910	−	0.189319i	−0.654913	+	0.755705i	$0.727295\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.9733	−	12.6863i	−0.957172	−	0.552624i
$528$		0			0
$529$	−8.98528	−	15.5630i	−0.390664	−	0.676651i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			19.3137i			0.836570i
$534$		0			0
$535$	17.3277	−	10.0042i	0.749143	−	0.432518i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.17157	+	14.1536i	−0.351323	+	0.608510i	−0.986482	−	0.163872i	$-0.947602\pi$
$541$	−8.17157	+	14.1536i	−0.351323	+	0.608510i	0.635158	+	0.772382i	$0.280935\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.7821			0.633194
$546$		0			0
$547$	6.34315			0.271213			0.135607	−	0.990763i	$-0.456702\pi$
$547$	6.34315			0.271213			0.135607	+	0.990763i	$0.456702\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.1480	−	24.5051i	0.602726	−	1.04395i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−22.6037	+	13.0503i	−0.957750	+	0.552957i	−0.895480	−	0.445102i	$-0.853167\pi$
$557$	−22.6037	+	13.0503i	−0.957750	+	0.552957i	−0.0622698	+	0.998059i	$0.519834\pi$
$558$		0			0
$559$		−	22.6984i		−	0.960039i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−22.8072	−	39.5032i	−0.961207	−	1.66486i	−0.719477	−	0.694516i	$-0.755619\pi$
$563$	−22.8072	−	39.5032i	−0.961207	−	1.66486i	−0.241730	−	0.970344i	$-0.577715\pi$
$564$		0			0
$565$	−26.8347	−	15.4930i	−1.12895	−	0.651797i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.67423	+	2.12132i	0.154032	+	0.0889304i	0.575035	−	0.818129i	$-0.304988\pi$
$569$	3.67423	+	2.12132i	0.154032	+	0.0889304i	−0.421003	+	0.907059i	$0.638322\pi$
$570$		0			0
$571$	10.4853	+	18.1610i	0.438795	+	0.760016i	0.997597	−	0.0692856i	$-0.0220720\pi$
$571$	10.4853	+	18.1610i	0.438795	+	0.760016i	−0.558801	+	0.829301i	$0.688739\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			5.95837i			0.248481i
$576$		0			0
$577$	3.74952	−	2.16478i	0.156094	−	0.0901212i	−0.419918	−	0.907562i	$-0.637941\pi$
$577$	3.74952	−	2.16478i	0.156094	−	0.0901212i	0.576013	+	0.817441i	$0.304608\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.58579	−	2.74666i	0.0656766	−	0.113755i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.85483			0.200380			0.100190	−	0.994968i	$-0.468055\pi$
$587$	4.85483			0.200380			0.100190	+	0.994968i	$0.468055\pi$
$588$		0			0
$589$	−22.6274			−0.932346
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.66205	−	2.87875i	0.0682522	−	0.118216i	−0.829880	−	0.557942i	$-0.811591\pi$
$593$	1.66205	−	2.87875i	0.0682522	−	0.118216i	0.898132	+	0.439726i	$0.144924\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9.71092	−	5.60660i	0.396777	−	0.229080i	−0.288315	−	0.957536i	$-0.593095\pi$
$599$	9.71092	−	5.60660i	0.396777	−	0.229080i	0.685093	+	0.728456i	$0.259762\pi$
$600$		0			0
$601$		−	33.8937i		−	1.38255i	−0.722590	−	0.691277i	$-0.757049\pi$
$601$		−	33.8937i		−	1.38255i	0.722590	−	0.691277i	$-0.242951\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.15640	+	14.1273i	0.331605	+	0.574357i
$606$		0			0
$607$	4.52607	+	2.61313i	0.183707	+	0.106064i	0.589033	−	0.808109i	$-0.299509\pi$
$607$	4.52607	+	2.61313i	0.183707	+	0.106064i	−0.405326	+	0.914172i	$0.632842\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.8564	+	8.00000i	0.560570	+	0.323645i
$612$		0			0
$613$	−0.656854	−	1.13770i	−0.0265301	−	0.0459515i	0.852456	−	0.522800i	$-0.175112\pi$
$613$	−0.656854	−	1.13770i	−0.0265301	−	0.0459515i	−0.878986	+	0.476848i	$0.841779\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12.2426i		−	0.492870i	−0.969159	−	0.246435i	$-0.920741\pi$
$617$		−	12.2426i		−	0.492870i	0.969159	−	0.246435i	$-0.0792592\pi$
$618$		0			0
$619$	−16.5512	+	9.55582i	−0.665248	+	0.384081i	−0.794274	−	0.607560i	$-0.792148\pi$
$619$	−16.5512	+	9.55582i	−0.665248	+	0.384081i	0.129026	+	0.991641i	$0.458815\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2.32843	−	4.03295i	0.0931371	−	0.161318i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−23.4412			−0.934662
$630$		0			0
$631$	−39.5980			−1.57637			−0.788185	−	0.615438i	$-0.788979\pi$
$631$	−39.5980			−1.57637			−0.788185	+	0.615438i	$0.788979\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12.9887	−	22.4971i	0.515441	−	0.892770i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−16.3419	+	9.43503i	−0.645468	+	0.372661i	−0.786718	−	0.617313i	$-0.788221\pi$
$641$	−16.3419	+	9.43503i	−0.645468	+	0.372661i	0.141250	+	0.989974i	$0.454888\pi$
$642$		0			0
$643$			13.8854i			0.547586i	0.961789	+	0.273793i	$0.0882784\pi$
$643$			13.8854i			0.547586i	−0.961789	+	0.273793i	$0.911722\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.79337	+	3.10620i	0.0705045	+	0.122117i	0.899123	−	0.437697i	$-0.144206\pi$
$647$	1.79337	+	3.10620i	0.0705045	+	0.122117i	−0.828618	+	0.559814i	$0.810872\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−35.6917	−	20.6066i	−1.39672	−	0.806399i	−0.402676	−	0.915343i	$-0.631920\pi$
$653$	−35.6917	−	20.6066i	−1.39672	−	0.806399i	−0.994048	+	0.108944i	$0.965253\pi$
$654$		0			0
$655$	−5.65685	−	9.79796i	−0.221032	−	0.382838i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			47.4142i			1.84700i	0.383604	+	0.923498i	$0.374683\pi$
$659$			47.4142i			1.84700i	−0.383604	+	0.923498i	$0.625317\pi$
$660$		0			0
$661$	−32.4590	+	18.7402i	−1.26251	+	0.728910i	−0.973559	−	0.228434i	$-0.926639\pi$
$661$	−32.4590	+	18.7402i	−1.26251	+	0.728910i	−0.288950	+	0.957344i	$0.593306\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	6.07107	−	10.5154i	0.235073	−	0.407158i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−13.6569			−0.526433			−0.263217	−	0.964737i	$-0.584783\pi$
$673$	−13.6569			−0.526433			−0.263217	+	0.964737i	$0.584783\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	22.9385	−	39.7306i	0.881597	−	1.52697i	0.0320325	−	0.999487i	$-0.489802\pi$
$677$	22.9385	−	39.7306i	0.881597	−	1.52697i	0.849565	−	0.527484i	$-0.176865\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−30.0753	+	17.3640i	−1.15080	+	0.664414i	−0.949082	−	0.315030i	$-0.897985\pi$
$683$	−30.0753	+	17.3640i	−1.15080	+	0.664414i	−0.201716	+	0.979444i	$0.564652\pi$
$684$		0			0
$685$		−	15.1535i		−	0.578985i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.86030	−	10.1503i	−0.223260	−	0.386697i
$690$		0			0
$691$	27.1564	+	15.6788i	1.03308	+	0.596448i	0.917865	−	0.396893i	$-0.129912\pi$
$691$	27.1564	+	15.6788i	1.03308	+	0.596448i	0.115213	+	0.993341i	$0.463245\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.8564	+	8.00000i	0.525603	+	0.303457i
$696$		0			0
$697$	−26.1421	−	45.2795i	−0.990204	−	1.71508i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			28.9289i			1.09263i	0.837580	+	0.546315i	$0.183970\pi$
$701$			28.9289i			1.09263i	−0.837580	+	0.546315i	$0.816030\pi$
$702$		0			0
$703$	−18.1043	+	10.4525i	−0.682815	+	0.394224i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	12.1421	−	21.0308i	0.456007	−	0.789828i	−0.542738	−	0.839902i	$-0.682612\pi$
$709$	12.1421	−	21.0308i	0.456007	−	0.789828i	0.998746	+	0.0500739i	$0.0159457\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.70967			−0.363630
$714$		0			0
$715$	−1.94113			−0.0725940
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.4525	+	18.1043i	−0.389813	+	0.675175i	−0.992424	−	0.122859i	$-0.960794\pi$
$719$	−10.4525	+	18.1043i	−0.389813	+	0.675175i	0.602611	+	0.798035i	$0.294127\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.4576	−	7.19239i	0.462663	−	0.267119i
$726$		0			0
$727$		−	6.12293i		−	0.227087i	−0.993533	−	0.113544i	$-0.963780\pi$
$727$		−	6.12293i		−	0.227087i	0.993533	−	0.113544i	$-0.0362201\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	30.7235	+	53.2146i	1.13635	+	1.96821i
$732$		0			0
$733$	−5.62427	−	3.24718i	−0.207737	−	0.119937i	0.392522	−	0.919743i	$-0.371603\pi$
$733$	−5.62427	−	3.24718i	−0.207737	−	0.119937i	−0.600259	+	0.799805i	$0.704936\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.89898	+	2.82843i	0.180456	+	0.104186i
$738$		0			0
$739$	−20.1421	−	34.8872i	−0.740940	−	1.28335i	−0.952068	−	0.305888i	$-0.901047\pi$
$739$	−20.1421	−	34.8872i	−0.740940	−	1.28335i	0.211127	−	0.977459i	$-0.432287\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	47.6985i		−	1.74989i	−0.484224	−	0.874944i	$-0.660898\pi$
$743$		−	47.6985i		−	1.74989i	0.484224	−	0.874944i	$-0.339102\pi$
$744$		0			0
$745$	−21.5321	+	12.4316i	−0.788876	+	0.455458i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6.75736	+	11.7041i	−0.246580	+	0.427088i	−0.962575	−	0.271017i	$-0.912640\pi$
$751$	−6.75736	+	11.7041i	−0.246580	+	0.427088i	0.715995	+	0.698105i	$0.245973\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18.5864			0.676428
$756$		0			0
$757$	−27.5980			−1.00307			−0.501533	−	0.865139i	$-0.667230\pi$
$757$	−27.5980			−1.00307			−0.501533	+	0.865139i	$0.667230\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−14.6508	+	25.3759i	−0.531090	+	0.919874i	0.468252	+	0.883595i	$0.344884\pi$
$761$	−14.6508	+	25.3759i	−0.531090	+	0.919874i	−0.999342	+	0.0362791i	$0.988449\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	33.8937i		−	1.22224i	−0.791539	−	0.611119i	$-0.790720\pi$
$769$		−	33.8937i		−	1.22224i	0.791539	−	0.611119i	$-0.209280\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.15640	−	14.1273i	−0.293365	−	0.508124i	0.681238	−	0.732062i	$-0.261442\pi$
$773$	−8.15640	−	14.1273i	−0.293365	−	0.508124i	−0.974603	+	0.223938i	$0.928109\pi$
$774$		0			0
$775$	−9.96192	−	5.75152i	−0.357843	−	0.206601i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−40.3805	−	23.3137i	−1.44678	−	0.835300i
$780$		0			0
$781$	−1.34315	−	2.32640i	−0.0480615	−	0.0832450i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			25.3726i			0.905586i
$786$		0			0
$787$	−33.1023	+	19.1116i	−1.17997	+	0.681256i	−0.956008	−	0.293341i	$-0.905233\pi$
$787$	−33.1023	+	19.1116i	−1.17997	+	0.681256i	−0.223963	+	0.974598i	$0.571899\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22.4357			−0.794715			−0.397357	−	0.917664i	$-0.630073\pi$
$797$	−22.4357			−0.794715			−0.397357	+	0.917664i	$0.630073\pi$
$798$		0			0
$799$	−43.3137			−1.53233
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.69552	−	6.40083i	0.130412	−	0.225880i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.12372	+	3.53553i	−0.215299	+	0.124303i	−0.603772	−	0.797157i	$-0.706336\pi$
$809$	−6.12372	+	3.53553i	−0.215299	+	0.124303i	0.388473	+	0.921460i	$0.373003\pi$
$810$		0			0
$811$			12.2459i			0.430011i	0.976613	+	0.215005i	$0.0689769\pi$
$811$			12.2459i			0.430011i	−0.976613	+	0.215005i	$0.931023\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.85483	−	8.40882i	−0.170057	−	0.294548i
$816$		0			0
$817$	47.4571	+	27.3994i	1.66031	+	0.958582i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	34.0827	+	19.6777i	1.18950	+	0.686755i	0.958192	−	0.286126i	$-0.0923677\pi$
$821$	34.0827	+	19.6777i	1.18950	+	0.686755i	0.231303	+	0.972882i	$0.425701\pi$
$822$		0			0
$823$	−10.8284	−	18.7554i	−0.377455	−	0.653772i	0.613236	−	0.789900i	$-0.289867\pi$
$823$	−10.8284	−	18.7554i	−0.377455	−	0.653772i	−0.990691	+	0.136128i	$0.956534\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			55.0122i			1.91296i	0.291796	+	0.956481i	$0.405747\pi$
$827$			55.0122i			1.91296i	−0.291796	+	0.956481i	$0.594253\pi$
$828$		0			0
$829$	9.37379	−	5.41196i	0.325565	−	0.187965i	−0.328305	−	0.944572i	$-0.606477\pi$
$829$	9.37379	−	5.41196i	0.325565	−	0.187965i	0.653870	+	0.756607i	$0.273144\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−10.3431	+	17.9149i	−0.357939	+	0.619969i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−51.7373			−1.78617			−0.893084	−	0.449890i	$-0.851463\pi$
$839$	−51.7373			−1.78617			−0.893084	+	0.449890i	$0.851463\pi$
$840$		0			0
$841$	−0.313708			−0.0108175
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.36304	−	11.0211i	0.218895	−	0.379137i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.76874	+	4.48528i	−0.266309	+	0.153753i
$852$		0			0
$853$			29.5641i			1.01226i	0.862458	+	0.506129i	$0.168924\pi$
$853$			29.5641i			1.01226i	−0.862458	+	0.506129i	$0.831076\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.93015	−	5.07517i	−0.100092	−	0.173365i	0.811630	−	0.584171i	$-0.198580\pi$
$857$	−2.93015	−	5.07517i	−0.100092	−	0.173365i	−0.911722	+	0.410807i	$0.865247\pi$
$858$		0			0
$859$	−12.0251	−	6.94269i	−0.410291	−	0.236882i	0.280624	−	0.959818i	$-0.409459\pi$
$859$	−12.0251	−	6.94269i	−0.410291	−	0.236882i	−0.690915	+	0.722936i	$0.742792\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	17.8278	+	10.2929i	0.606866	+	0.350374i	0.771738	−	0.635941i	$-0.219388\pi$
$863$	17.8278	+	10.2929i	0.606866	+	0.350374i	−0.164872	+	0.986315i	$0.552721\pi$
$864$		0			0
$865$	17.1716	+	29.7420i	0.583851	+	1.01126i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1.37258i		−	0.0465617i
$870$		0			0
$871$	18.1043	−	10.4525i	0.613440	−	0.354170i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	−0.723298	−	0.690536i	$-0.757375\pi$
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	0.959671	+	0.281126i	$0.0907079\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	39.0112			1.31432			0.657160	−	0.753751i	$-0.271758\pi$
$881$	39.0112			1.31432			0.657160	+	0.753751i	$0.271758\pi$
$882$		0			0
$883$	−4.14214			−0.139394			−0.0696970	−	0.997568i	$-0.522203\pi$
$883$	−4.14214			−0.139394			−0.0696970	+	0.997568i	$0.522203\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.02509	+	13.8999i	−0.269456	+	0.466712i	−0.968721	−	0.248151i	$-0.920177\pi$
$887$	−8.02509	+	13.8999i	−0.269456	+	0.466712i	0.699265	+	0.714862i	$0.253511\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−33.4523	+	19.3137i	−1.11944	+	0.646309i
$894$		0			0
$895$			5.96909i			0.199525i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.7206	+	20.3007i	0.390904	+	0.677066i
$900$		0			0
$901$	27.4781	+	15.8645i	0.915427	+	0.528522i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19.1035	−	11.0294i	−0.635023	−	0.366631i
$906$		0			0
$907$	5.10051	+	8.83433i	0.169359	+	0.293339i	0.938195	−	0.346108i	$-0.112497\pi$
$907$	5.10051	+	8.83433i	0.169359	+	0.293339i	−0.768835	+	0.639447i	$0.779163\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	20.1838i		−	0.668718i	−0.942446	−	0.334359i	$-0.891480\pi$
$911$		−	20.1838i		−	0.668718i	0.942446	−	0.334359i	$-0.108520\pi$
$912$		0			0
$913$	6.85572	−	3.95815i	0.226891	−	0.130996i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	18.8995	−	32.7349i	0.623437	−	1.07982i	−0.365404	−	0.930849i	$-0.619069\pi$
$919$	18.8995	−	32.7349i	0.623437	−	1.07982i	0.988841	−	0.148975i	$-0.0475975\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−9.92724			−0.326759
$924$		0			0
$925$	−10.6274			−0.349427
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.0586	+	17.4219i	−0.330010	+	0.571595i	−0.982514	−	0.186191i	$-0.940386\pi$
$929$	−10.0586	+	17.4219i	−0.330010	+	0.571595i	0.652503	+	0.757786i	$0.273719\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.55082	−	2.62742i	0.148828	−	0.0859257i
$936$		0			0
$937$			3.58673i			0.117173i	0.998282	+	0.0585867i	$0.0186594\pi$
$937$			3.58673i			0.117173i	−0.998282	+	0.0585867i	$0.981341\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−11.2179	−	19.4299i	−0.365692	−	0.633397i	0.623195	−	0.782067i	$-0.285834\pi$
$941$	−11.2179	−	19.4299i	−0.365692	−	0.633397i	−0.988887	+	0.148669i	$0.952501\pi$
$942$		0			0
$943$	−17.3277	−	10.0042i	−0.564268	−	0.325780i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.2919	−	12.2929i	−0.691894	−	0.399465i	0.112427	−	0.993660i	$-0.464138\pi$
$947$	−21.2919	−	12.2929i	−0.691894	−	0.399465i	−0.804321	+	0.594195i	$0.797471\pi$
$948$		0			0
$949$	−13.6569	−	23.6544i	−0.443320	−	0.767853i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			53.2132i			1.72374i	0.507125	+	0.861872i	$0.330708\pi$
$953$			53.2132i			1.72374i	−0.507125	+	0.861872i	$0.669292\pi$
$954$		0			0
$955$	30.1294	−	17.3952i	0.974963	−	0.562895i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−6.12742	+	10.6130i	−0.197659	+	0.342355i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	8.65914			0.278748
$966$		0			0
$967$	−29.6569			−0.953700			−0.476850	−	0.878985i	$-0.658222\pi$
$967$	−29.6569			−0.953700			−0.476850	+	0.878985i	$0.658222\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	11.7206	−	20.3007i	0.376132	−	0.651480i	−0.614364	−	0.789023i	$-0.710587\pi$
$971$	11.7206	−	20.3007i	0.376132	−	0.651480i	0.990496	+	0.137543i	$0.0439205\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−15.5014	+	8.94975i	−0.495934	+	0.286328i	−0.727033	−	0.686603i	$-0.759101\pi$
$977$	−15.5014	+	8.94975i	−0.495934	+	0.286328i	0.231099	+	0.972930i	$0.425768\pi$
$978$		0			0
$979$		−	3.43289i		−	0.109716i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	9.29319	+	16.0963i	0.296407	+	0.513391i	0.975311	−	0.220835i	$-0.0708783\pi$
$983$	9.29319	+	16.0963i	0.296407	+	0.513391i	−0.678904	+	0.734227i	$0.737545\pi$
$984$		0			0
$985$	−19.0692	−	11.0096i	−0.607597	−	0.350796i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	20.3643	+	11.7574i	0.647548	+	0.373862i
$990$		0			0
$991$	−13.9289	−	24.1256i	−0.442467	−	0.766376i	0.555405	−	0.831580i	$-0.312563\pi$
$991$	−13.9289	−	24.1256i	−0.442467	−	0.766376i	−0.997872	+	0.0652046i	$0.979230\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			5.25483i			0.166589i
$996$		0			0
$997$	46.8138	−	27.0279i	1.48261	−	0.855984i	0.482802	−	0.875730i	$-0.339619\pi$
$997$	46.8138	−	27.0279i	1.48261	−	0.855984i	0.999805	+	0.0197461i	$0.00628577\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.bl.b.1097.6		16
3.2	odd	2	inner	3528.2.bl.b.1097.3		16
7.2	even	3		504.2.k.a.377.3	✓	8
7.3	odd	6	inner	3528.2.bl.b.521.3		16
7.4	even	3	inner	3528.2.bl.b.521.5		16
7.5	odd	6		504.2.k.a.377.6	yes	8
7.6	odd	2	inner	3528.2.bl.b.1097.4		16
21.2	odd	6		504.2.k.a.377.5	yes	8
21.5	even	6		504.2.k.a.377.4	yes	8
21.11	odd	6	inner	3528.2.bl.b.521.4		16
21.17	even	6	inner	3528.2.bl.b.521.6		16
21.20	even	2	inner	3528.2.bl.b.1097.5		16
28.19	even	6		1008.2.k.c.881.5		8
28.23	odd	6		1008.2.k.c.881.4		8
56.5	odd	6		4032.2.k.f.3905.4		8
56.19	even	6		4032.2.k.e.3905.3		8
56.37	even	6		4032.2.k.f.3905.5		8
56.51	odd	6		4032.2.k.e.3905.6		8
84.23	even	6		1008.2.k.c.881.6		8
84.47	odd	6		1008.2.k.c.881.3		8
168.5	even	6		4032.2.k.f.3905.6		8
168.107	even	6		4032.2.k.e.3905.4		8
168.131	odd	6		4032.2.k.e.3905.5		8
168.149	odd	6		4032.2.k.f.3905.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.k.a.377.3	✓	8	7.2	even	3
504.2.k.a.377.4	yes	8	21.5	even	6
504.2.k.a.377.5	yes	8	21.2	odd	6
504.2.k.a.377.6	yes	8	7.5	odd	6
1008.2.k.c.881.3		8	84.47	odd	6
1008.2.k.c.881.4		8	28.23	odd	6
1008.2.k.c.881.5		8	28.19	even	6
1008.2.k.c.881.6		8	84.23	even	6
3528.2.bl.b.521.3		16	7.3	odd	6	inner
3528.2.bl.b.521.4		16	21.11	odd	6	inner
3528.2.bl.b.521.5		16	7.4	even	3	inner
3528.2.bl.b.521.6		16	21.17	even	6	inner
3528.2.bl.b.1097.3		16	3.2	odd	2	inner
3528.2.bl.b.1097.4		16	7.6	odd	2	inner
3528.2.bl.b.1097.5		16	21.20	even	2	inner
3528.2.bl.b.1097.6		16	1.1	even	1	trivial
4032.2.k.e.3905.3		8	56.19	even	6
4032.2.k.e.3905.4		8	168.107	even	6
4032.2.k.e.3905.5		8	168.131	odd	6
4032.2.k.e.3905.6		8	56.51	odd	6
4032.2.k.f.3905.3		8	168.149	odd	6
4032.2.k.f.3905.4		8	56.5	odd	6
4032.2.k.f.3905.5		8	56.37	even	6
4032.2.k.f.3905.6		8	168.5	even	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.765367 - 1.32565i) q^{5} +(0.507306 - 0.292893i) q^{11} -2.16478i q^{13} +(2.93015 + 5.07517i) q^{17} +(4.52607 + 2.61313i) q^{19} +(1.94218 + 1.12132i) q^{23} +(1.32843 + 2.30090i) q^{25} -5.41421i q^{29} +(-3.74952 + 2.16478i) q^{31} +(-2.00000 + 3.46410i) q^{37} -8.92177 q^{41} +10.4853 q^{43} +(-3.69552 + 6.40083i) q^{47} +(4.68885 - 2.70711i) q^{53} -0.896683i q^{55} +(-2.86976 - 1.65685i) q^{65} +(4.82843 + 8.36308i) q^{67} -4.58579i q^{71} +(10.9269 - 6.30864i) q^{73} +(1.17157 - 2.02922i) q^{79} +13.5140 q^{83} +8.97056 q^{85} +(2.93015 - 5.07517i) q^{89} +(6.92820 - 4.00000i) q^{95} +8.28772i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{67} + 64 q^{79} - 128 q^{85}+O(q^{100}) \) 16 * q - 24 * q^25 - 32 * q^37 + 32 * q^43 + 32 * q^67 + 64 * q^79 - 128 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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