LMFDB - Embedded newform 1296.5.e.e.161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,5,Mod(161,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1296.e (of order $2$, degree $1$, not 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:	$133.967472157$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.221456830464.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 $ x^8 - 4x^7 + 38x^6 - 100x^5 + 449x^4 - 736x^3 + 1900x^2 - 1548*x + 2307
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.5
Root			$0.500000 + 2.20403i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.5.e.e.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.20976i q^{5} -43.9826 q^{7} +O(q^{10})$	$q+2.20976i q^{5} -43.9826 q^{7} +212.195i q^{11} -177.413 q^{13} +43.3642i q^{17} -528.015 q^{19} +597.013i q^{23} +620.117 q^{25} +439.437i q^{29} -1512.94 q^{31} -97.1911i q^{35} +702.206 q^{37} -597.597i q^{41} +92.2250 q^{43} +1720.90i q^{47} -466.533 q^{49} -1311.69i q^{53} -468.900 q^{55} -3947.12i q^{59} +4224.68 q^{61} -392.040i q^{65} -645.184 q^{67} +6397.63i q^{71} -3196.15 q^{73} -9332.88i q^{77} +9565.38 q^{79} +3550.36i q^{83} -95.8247 q^{85} +502.000i q^{89} +7803.06 q^{91} -1166.79i q^{95} -7428.84 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 52 q^{7}+O(q^{10}) $ 8 * q - 52 * q^7	$ 8 q - 52 q^{7} - 20 q^{13} - 100 q^{19} - 1588 q^{25} - 2956 q^{31} - 32 q^{37} - 136 q^{43} - 4884 q^{49} + 3996 q^{55} + 8956 q^{61} + 15008 q^{67} + 20716 q^{73} - 12100 q^{79} + 32184 q^{85} + 45868 q^{91} - 62672 q^{97}+O(q^{100}) $ 8 * q - 52 * q^7 - 20 * q^13 - 100 * q^19 - 1588 * q^25 - 2956 * q^31 - 32 * q^37 - 136 * q^43 - 4884 * q^49 + 3996 * q^55 + 8956 * q^61 + 15008 * q^67 + 20716 * q^73 - 12100 * q^79 + 32184 * q^85 + 45868 * q^91 - 62672 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.20976i	0.0883905i	0.999023	+	0.0441953i	$0.0140724\pi$
$5$	2.20976i	0.0883905i	−0.999023	+	0.0441953i	$0.985928\pi$
$6$	0	0
$7$	−43.9826	−0.897604	−0.448802	−	0.893631i	$-0.648149\pi$
$7$	−43.9826	−0.897604	−0.448802	+	0.893631i	$0.648149\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	212.195i	1.75368i	0.480786	+	0.876838i	$0.340352\pi$
$11$	212.195i	1.75368i	−0.480786	+	0.876838i	$0.659648\pi$
$12$	0	0
$13$	−177.413	−1.04978	−0.524889	−	0.851171i	$-0.675893\pi$
$13$	−177.413	−1.04978	−0.524889	+	0.851171i	$0.675893\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	43.3642i	0.150049i	0.997182	+	0.0750246i	$0.0239035\pi$
$17$	43.3642i	0.150049i	−0.997182	+	0.0750246i	$0.976096\pi$
$18$	0	0
$19$	−528.015	−1.46265	−0.731323	−	0.682031i	$-0.761097\pi$
$19$	−528.015	−1.46265	−0.731323	+	0.682031i	$0.761097\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	597.013i	1.12857i	0.825581	+	0.564284i	$0.190848\pi$
$23$	597.013i	1.12857i	−0.825581	+	0.564284i	$0.809152\pi$
$24$	0	0
$25$	620.117	0.992187
$26$	0	0
$27$	0	0
$28$	0	0
$29$	439.437i	0.522517i	0.965269	+	0.261258i	$0.0841375\pi$
$29$	439.437i	0.522517i	−0.965269	+	0.261258i	$0.915863\pi$
$30$	0	0
$31$	−1512.94	−1.57434	−0.787170	−	0.616736i	$-0.788455\pi$
$31$	−1512.94	−1.57434	−0.787170	+	0.616736i	$0.788455\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 97.1911i	− 0.0793397i
$36$	0	0
$37$	702.206	0.512933	0.256467	−	0.966553i	$-0.417442\pi$
$37$	702.206	0.512933	0.256467	+	0.966553i	$0.417442\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 597.597i	− 0.355501i	−0.984076	−	0.177750i	$-0.943118\pi$
$41$	− 597.597i	− 0.355501i	0.984076	−	0.177750i	$-0.0568820\pi$
$42$	0	0
$43$	92.2250	0.0498783	0.0249392	−	0.999689i	$-0.492061\pi$
$43$	92.2250	0.0498783	0.0249392	+	0.999689i	$0.492061\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1720.90i	0.779039i	0.921018	+	0.389520i	$0.127359\pi$
$47$	1720.90i	0.779039i	−0.921018	+	0.389520i	$0.872641\pi$
$48$	0	0
$49$	−466.533	−0.194308
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1311.69i	− 0.466958i	−0.972362	−	0.233479i	$-0.924989\pi$
$53$	− 1311.69i	− 0.466958i	0.972362	−	0.233479i	$-0.0750111\pi$
$54$	0	0
$55$	−468.900	−0.155008
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 3947.12i	− 1.13390i	−0.823751	−	0.566952i	$-0.808123\pi$
$59$	− 3947.12i	− 1.13390i	0.823751	−	0.566952i	$-0.191877\pi$
$60$	0	0
$61$	4224.68	1.13536	0.567681	−	0.823249i	$-0.307841\pi$
$61$	4224.68	1.13536	0.567681	+	0.823249i	$0.307841\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 392.040i	− 0.0927905i
$66$	0	0
$67$	−645.184	−0.143726	−0.0718628	−	0.997415i	$-0.522894\pi$
$67$	−645.184	−0.143726	−0.0718628	+	0.997415i	$0.522894\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6397.63i	1.26912i	0.772874	+	0.634560i	$0.218819\pi$
$71$	6397.63i	1.26912i	−0.772874	+	0.634560i	$0.781181\pi$
$72$	0	0
$73$	−3196.15	−0.599766	−0.299883	−	0.953976i	$-0.596948\pi$
$73$	−3196.15	−0.599766	−0.299883	+	0.953976i	$0.596948\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 9332.88i	− 1.57411i
$78$	0	0
$79$	9565.38	1.53267	0.766334	−	0.642442i	$-0.222079\pi$
$79$	9565.38	1.53267	0.766334	+	0.642442i	$0.222079\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3550.36i	0.515367i	0.966229	+	0.257683i	$0.0829591\pi$
$83$	3550.36i	0.515367i	−0.966229	+	0.257683i	$0.917041\pi$
$84$	0	0
$85$	−95.8247	−0.0132629
$86$	0	0
$87$	0	0
$88$	0	0
$89$	502.000i	0.0633759i	0.999498	+	0.0316879i	$0.0100883\pi$
$89$	502.000i	0.0633759i	−0.999498	+	0.0316879i	$0.989912\pi$
$90$	0	0
$91$	7803.06	0.942285
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1166.79i	− 0.129284i
$96$	0	0
$97$	−7428.84	−0.789546	−0.394773	−	0.918779i	$-0.629177\pi$
$97$	−7428.84	−0.789546	−0.394773	+	0.918779i	$0.629177\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 916.675i	− 0.0898613i	−0.998990	−	0.0449307i	$-0.985693\pi$
$101$	− 916.675i	− 0.0898613i	0.998990	−	0.0449307i	$-0.0143067\pi$
$102$	0	0
$103$	3554.25	0.335022	0.167511	−	0.985870i	$-0.446427\pi$
$103$	3554.25	0.335022	0.167511	+	0.985870i	$0.446427\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	550.477i	0.0480808i	0.999711	+	0.0240404i	$0.00765304\pi$
$107$	550.477i	0.0480808i	−0.999711	+	0.0240404i	$0.992347\pi$
$108$	0	0
$109$	−2561.93	−0.215633	−0.107816	−	0.994171i	$-0.534386\pi$
$109$	−2561.93	−0.215633	−0.107816	+	0.994171i	$0.534386\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 20321.4i	− 1.59146i	−0.605650	−	0.795731i	$-0.707087\pi$
$113$	− 20321.4i	− 1.59146i	0.605650	−	0.795731i	$-0.292913\pi$
$114$	0	0
$115$	−1319.26	−0.0997548
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1907.27i	− 0.134685i
$120$	0	0
$121$	−30385.7	−2.07538
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2751.41i	0.176090i
$126$	0	0
$127$	−21354.6	−1.32399	−0.661995	−	0.749508i	$-0.730290\pi$
$127$	−21354.6	−1.32399	−0.661995	+	0.749508i	$0.730290\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 30837.9i	− 1.79698i	−0.438996	−	0.898489i	$-0.644666\pi$
$131$	− 30837.9i	− 1.79698i	0.438996	−	0.898489i	$-0.355334\pi$
$132$	0	0
$133$	23223.5	1.31288
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10997.3i	− 0.585930i	−0.956123	−	0.292965i	$-0.905358\pi$
$137$	− 10997.3i	− 0.585930i	0.956123	−	0.292965i	$-0.0946419\pi$
$138$	0	0
$139$	−16945.3	−0.877038	−0.438519	−	0.898722i	$-0.644497\pi$
$139$	−16945.3	−0.877038	−0.438519	+	0.898722i	$0.644497\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 37646.0i	− 1.84097i
$144$	0	0
$145$	−971.051	−0.0461856
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16608.8i	0.748113i	0.927406	+	0.374056i	$0.122033\pi$
$149$	16608.8i	0.748113i	−0.927406	+	0.374056i	$0.877967\pi$
$150$	0	0
$151$	11030.6	0.483778	0.241889	−	0.970304i	$-0.422233\pi$
$151$	11030.6	0.483778	0.241889	+	0.970304i	$0.422233\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3343.24i	− 0.139157i
$156$	0	0
$157$	19934.6	0.808738	0.404369	−	0.914596i	$-0.367491\pi$
$157$	19934.6	0.808738	0.404369	+	0.914596i	$0.367491\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 26258.2i	− 1.01301i
$162$	0	0
$163$	7936.22	0.298702	0.149351	−	0.988784i	$-0.452282\pi$
$163$	7936.22	0.298702	0.149351	+	0.988784i	$0.452282\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19187.0i	0.687979i	0.938974	+	0.343989i	$0.111778\pi$
$167$	19187.0i	0.687979i	−0.938974	+	0.343989i	$0.888222\pi$
$168$	0	0
$169$	2914.22	0.102035
$170$	0	0
$171$	0	0
$172$	0	0
$173$	52188.1i	1.74373i	0.489746	+	0.871865i	$0.337090\pi$
$173$	52188.1i	1.74373i	−0.489746	+	0.871865i	$0.662910\pi$
$174$	0	0
$175$	−27274.3	−0.890591
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12197.1i	0.380672i	0.981719	+	0.190336i	$0.0609578\pi$
$179$	12197.1i	0.380672i	−0.981719	+	0.190336i	$0.939042\pi$
$180$	0	0
$181$	6870.48	0.209715	0.104858	−	0.994487i	$-0.466561\pi$
$181$	6870.48	0.209715	0.104858	+	0.994487i	$0.466561\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1551.71i	0.0453385i
$186$	0	0
$187$	−9201.66	−0.263138
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 41513.0i	− 1.13794i	−0.822360	−	0.568968i	$-0.807343\pi$
$191$	− 41513.0i	− 1.13794i	0.822360	−	0.568968i	$-0.192657\pi$
$192$	0	0
$193$	59736.0	1.60369	0.801847	−	0.597530i	$-0.203851\pi$
$193$	59736.0	1.60369	0.801847	+	0.597530i	$0.203851\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12371.6i	− 0.318783i	−0.987215	−	0.159391i	$-0.949047\pi$
$197$	− 12371.6i	− 0.318783i	0.987215	−	0.159391i	$-0.0509532\pi$
$198$	0	0
$199$	16836.3	0.425149	0.212575	−	0.977145i	$-0.431815\pi$
$199$	16836.3	0.425149	0.212575	+	0.977145i	$0.431815\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 19327.6i	− 0.469013i
$204$	0	0
$205$	1320.55	0.0314229
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 112042.i	− 2.56501i
$210$	0	0
$211$	45213.4	1.01555	0.507776	−	0.861489i	$-0.330468\pi$
$211$	45213.4	1.01555	0.507776	+	0.861489i	$0.330468\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	203.795i	0.00440877i
$216$	0	0
$217$	66543.1	1.41313
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 7693.36i	− 0.157518i
$222$	0	0
$223$	−50665.9	−1.01884	−0.509420	−	0.860518i	$-0.670140\pi$
$223$	−50665.9	−1.01884	−0.509420	+	0.860518i	$0.670140\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 40035.6i	− 0.776952i	−0.921459	−	0.388476i	$-0.873002\pi$
$227$	− 40035.6i	− 0.776952i	0.921459	−	0.388476i	$-0.126998\pi$
$228$	0	0
$229$	−26767.5	−0.510430	−0.255215	−	0.966884i	$-0.582146\pi$
$229$	−26767.5	−0.510430	−0.255215	+	0.966884i	$0.582146\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	39898.5i	0.734927i	0.930038	+	0.367464i	$0.119774\pi$
$233$	39898.5i	0.734927i	−0.930038	+	0.367464i	$0.880226\pi$
$234$	0	0
$235$	−3802.78	−0.0688597
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 50240.2i	− 0.879539i	−0.898111	−	0.439770i	$-0.855060\pi$
$239$	− 50240.2i	− 0.879539i	0.898111	−	0.439770i	$-0.144940\pi$
$240$	0	0
$241$	−69260.8	−1.19249	−0.596243	−	0.802804i	$-0.703341\pi$
$241$	−69260.8	−1.19249	−0.596243	+	0.802804i	$0.703341\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1030.93i	− 0.0171750i
$246$	0	0
$247$	93676.5	1.53545
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 117510.i	− 1.86521i	−0.360904	−	0.932603i	$-0.617532\pi$
$251$	− 117510.i	− 1.86521i	0.360904	−	0.932603i	$-0.382468\pi$
$252$	0	0
$253$	−126683.	−1.97914
$254$	0	0
$255$	0	0
$256$	0	0
$257$	61587.5i	0.932451i	0.884666	+	0.466225i	$0.154386\pi$
$257$	61587.5i	0.932451i	−0.884666	+	0.466225i	$0.845614\pi$
$258$	0	0
$259$	−30884.8	−0.460411
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 25636.0i	− 0.370629i	−0.982679	−	0.185314i	$-0.940670\pi$
$263$	− 25636.0i	− 0.370629i	0.982679	−	0.185314i	$-0.0593303\pi$
$264$	0	0
$265$	2898.52	0.0412747
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 70454.4i	− 0.973652i	−0.873499	−	0.486826i	$-0.838155\pi$
$269$	− 70454.4i	− 0.973652i	0.873499	−	0.486826i	$-0.161845\pi$
$270$	0	0
$271$	677.821	0.00922946	0.00461473	−	0.999989i	$-0.498531\pi$
$271$	677.821	0.00922946	0.00461473	+	0.999989i	$0.498531\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	131586.i	1.73998i
$276$	0	0
$277$	−105928.	−1.38055	−0.690276	−	0.723546i	$-0.742511\pi$
$277$	−105928.	−1.38055	−0.690276	+	0.723546i	$0.742511\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 723.756i	− 0.00916599i	−0.999989	−	0.00458300i	$-0.998541\pi$
$281$	− 723.756i	− 0.00916599i	0.999989	−	0.00458300i	$-0.00145882\pi$
$282$	0	0
$283$	21575.2	0.269390	0.134695	−	0.990887i	$-0.456995\pi$
$283$	21575.2	0.269390	0.134695	+	0.990887i	$0.456995\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26283.8i	0.319099i
$288$	0	0
$289$	81640.5	0.977485
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 104257.i	− 1.21443i	−0.794539	−	0.607213i	$-0.792287\pi$
$293$	− 104257.i	− 1.21443i	0.794539	−	0.607213i	$-0.207713\pi$
$294$	0	0
$295$	8722.20	0.100226
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 105918.i	− 1.18475i
$300$	0	0
$301$	−4056.29	−0.0447710
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9335.55i	0.100355i
$306$	0	0
$307$	−3762.72	−0.0399232	−0.0199616	−	0.999801i	$-0.506354\pi$
$307$	−3762.72	−0.0399232	−0.0199616	+	0.999801i	$0.506354\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 83899.1i	− 0.867434i	−0.901049	−	0.433717i	$-0.857202\pi$
$311$	− 83899.1i	− 0.867434i	0.901049	−	0.433717i	$-0.142798\pi$
$312$	0	0
$313$	−35434.9	−0.361695	−0.180848	−	0.983511i	$-0.557884\pi$
$313$	−35434.9	−0.361695	−0.180848	+	0.983511i	$0.557884\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 152500.i	− 1.51758i	−0.651335	−	0.758791i	$-0.725791\pi$
$317$	− 152500.i	− 1.51758i	0.651335	−	0.758791i	$-0.274209\pi$
$318$	0	0
$319$	−93246.2	−0.916326
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 22897.0i	− 0.219469i
$324$	0	0
$325$	−110017.	−1.04158
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 75689.5i	− 0.699269i
$330$	0	0
$331$	37146.9	0.339052	0.169526	−	0.985526i	$-0.445776\pi$
$331$	37146.9	0.339052	0.169526	+	0.985526i	$0.445776\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1425.71i	− 0.0127040i
$336$	0	0
$337$	−42065.2	−0.370393	−0.185197	−	0.982702i	$-0.559292\pi$
$337$	−42065.2	−0.370393	−0.185197	+	0.982702i	$0.559292\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 321038.i	− 2.76088i
$342$	0	0
$343$	126121.	1.07201
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 106897.i	− 0.887783i	−0.896081	−	0.443892i	$-0.853598\pi$
$347$	− 106897.i	− 0.887783i	0.896081	−	0.443892i	$-0.146402\pi$
$348$	0	0
$349$	189690.	1.55737	0.778686	−	0.627413i	$-0.215886\pi$
$349$	189690.	1.55737	0.778686	+	0.627413i	$0.215886\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 215389.i	− 1.72851i	−0.503050	−	0.864257i	$-0.667789\pi$
$353$	− 215389.i	− 1.72851i	0.503050	−	0.864257i	$-0.332211\pi$
$354$	0	0
$355$	−14137.3	−0.112178
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 116777.i	− 0.906085i	−0.891489	−	0.453043i	$-0.850339\pi$
$359$	− 116777.i	− 0.906085i	0.891489	−	0.453043i	$-0.149661\pi$
$360$	0	0
$361$	148479.	1.13933
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 7062.74i	− 0.0530137i
$366$	0	0
$367$	−126204.	−0.936999	−0.468500	−	0.883464i	$-0.655205\pi$
$367$	−126204.	−0.936999	−0.468500	+	0.883464i	$0.655205\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	57691.3i	0.419144i
$372$	0	0
$373$	111465.	0.801161	0.400581	−	0.916261i	$-0.368808\pi$
$373$	111465.	0.801161	0.400581	+	0.916261i	$0.368808\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 77961.6i	− 0.548527i
$378$	0	0
$379$	−200558.	−1.39624	−0.698121	−	0.715980i	$-0.745980\pi$
$379$	−200558.	−1.39624	−0.698121	+	0.715980i	$0.745980\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	75627.6i	0.515565i	0.966203	+	0.257782i	$0.0829917\pi$
$383$	75627.6i	0.515565i	−0.966203	+	0.257782i	$0.917008\pi$
$384$	0	0
$385$	20623.5	0.139136
$386$	0	0
$387$	0	0
$388$	0	0
$389$	274489.i	1.81395i	0.421181	+	0.906977i	$0.361616\pi$
$389$	274489.i	1.81395i	−0.421181	+	0.906977i	$0.638384\pi$
$390$	0	0
$391$	−25889.0	−0.169341
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21137.2i	0.135473i
$396$	0	0
$397$	3509.10	0.0222646	0.0111323	−	0.999938i	$-0.496456\pi$
$397$	3509.10	0.0222646	0.0111323	+	0.999938i	$0.496456\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16311.6i	0.101440i	0.998713	+	0.0507200i	$0.0161516\pi$
$401$	16311.6i	0.101440i	−0.998713	+	0.0507200i	$0.983848\pi$
$402$	0	0
$403$	268415.	1.65271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	149004.i	0.899519i
$408$	0	0
$409$	−7485.89	−0.0447504	−0.0223752	−	0.999750i	$-0.507123\pi$
$409$	−7485.89	−0.0447504	−0.0223752	+	0.999750i	$0.507123\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	173604.i	1.01780i
$414$	0	0
$415$	−7845.46	−0.0455535
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 109003.i	− 0.620884i	−0.950592	−	0.310442i	$-0.899523\pi$
$419$	− 109003.i	− 0.620884i	0.950592	−	0.310442i	$-0.100477\pi$
$420$	0	0
$421$	331272.	1.86905	0.934525	−	0.355897i	$-0.115825\pi$
$421$	331272.	1.86905	0.934525	+	0.355897i	$0.115825\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	26890.9i	0.148877i
$426$	0	0
$427$	−185812.	−1.01911
$428$	0	0
$429$	0	0
$430$	0	0
$431$	270347.i	1.45535i	0.685922	+	0.727675i	$0.259399\pi$
$431$	270347.i	1.45535i	−0.685922	+	0.727675i	$0.740601\pi$
$432$	0	0
$433$	−7609.14	−0.0405845	−0.0202922	−	0.999794i	$-0.506460\pi$
$433$	−7609.14	−0.0405845	−0.0202922	+	0.999794i	$0.506460\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 315232.i	− 1.65070i
$438$	0	0
$439$	99551.1	0.516555	0.258278	−	0.966071i	$-0.416845\pi$
$439$	99551.1	0.516555	0.258278	+	0.966071i	$0.416845\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 222038.i	− 1.13141i	−0.824608	−	0.565705i	$-0.808604\pi$
$443$	− 222038.i	− 1.13141i	0.824608	−	0.565705i	$-0.191396\pi$
$444$	0	0
$445$	−1109.30	−0.00560183
$446$	0	0
$447$	0	0
$448$	0	0
$449$	353910.i	1.75550i	0.479122	+	0.877749i	$0.340955\pi$
$449$	353910.i	1.75550i	−0.479122	+	0.877749i	$0.659045\pi$
$450$	0	0
$451$	126807.	0.623433
$452$	0	0
$453$	0	0
$454$	0	0
$455$	17242.9i	0.0832891i
$456$	0	0
$457$	−257982.	−1.23526	−0.617629	−	0.786470i	$-0.711907\pi$
$457$	−257982.	−1.23526	−0.617629	+	0.786470i	$0.711907\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	212205.i	0.998513i	0.866454	+	0.499257i	$0.166394\pi$
$461$	212205.i	0.998513i	−0.866454	+	0.499257i	$0.833606\pi$
$462$	0	0
$463$	69090.4	0.322297	0.161148	−	0.986930i	$-0.448480\pi$
$463$	69090.4	0.322297	0.161148	+	0.986930i	$0.448480\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 126113.i	− 0.578264i	−0.957289	−	0.289132i	$-0.906633\pi$
$467$	− 126113.i	− 0.578264i	0.957289	−	0.289132i	$-0.0933667\pi$
$468$	0	0
$469$	28376.9	0.129009
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19569.7i	0.0874704i
$474$	0	0
$475$	−327431.	−1.45122
$476$	0	0
$477$	0	0
$478$	0	0
$479$	287163.i	1.25158i	0.779993	+	0.625788i	$0.215223\pi$
$479$	287163.i	1.25158i	−0.779993	+	0.625788i	$0.784777\pi$
$480$	0	0
$481$	−124580.	−0.538467
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 16416.0i	− 0.0697884i
$486$	0	0
$487$	−274661.	−1.15808	−0.579040	−	0.815299i	$-0.696573\pi$
$487$	−274661.	−1.15808	−0.579040	+	0.815299i	$0.696573\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 317517.i	− 1.31706i	−0.752556	−	0.658528i	$-0.771179\pi$
$491$	− 317517.i	− 1.31706i	0.752556	−	0.658528i	$-0.228821\pi$
$492$	0	0
$493$	−19055.8	−0.0784032
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 281384.i	− 1.13917i
$498$	0	0
$499$	195123.	0.783622	0.391811	−	0.920046i	$-0.371849\pi$
$499$	195123.	0.783622	0.391811	+	0.920046i	$0.371849\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 21373.6i	− 0.0844777i	−0.999108	−	0.0422388i	$-0.986551\pi$
$503$	− 21373.6i	− 0.0844777i	0.999108	−	0.0422388i	$-0.0134490\pi$
$504$	0	0
$505$	2025.64	0.00794289
$506$	0	0
$507$	0	0
$508$	0	0
$509$	79865.5i	0.308265i	0.988050	+	0.154132i	$0.0492582\pi$
$509$	79865.5i	0.308265i	−0.988050	+	0.154132i	$0.950742\pi$
$510$	0	0
$511$	140575.	0.538352
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7854.04i	0.0296128i
$516$	0	0
$517$	−365166.	−1.36618
$518$	0	0
$519$	0	0
$520$	0	0
$521$	142732.i	0.525830i	0.964819	+	0.262915i	$0.0846839\pi$
$521$	142732.i	0.525830i	−0.964819	+	0.262915i	$0.915316\pi$
$522$	0	0
$523$	232027.	0.848271	0.424136	−	0.905599i	$-0.360578\pi$
$523$	232027.	0.848271	0.424136	+	0.905599i	$0.360578\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 65607.5i	− 0.236229i
$528$	0	0
$529$	−76583.1	−0.273666
$530$	0	0
$531$	0	0
$532$	0	0
$533$	106021.i	0.373197i
$534$	0	0
$535$	−1216.43	−0.00424989
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 98995.8i	− 0.340753i
$540$	0	0
$541$	−376322.	−1.28577	−0.642887	−	0.765961i	$-0.722264\pi$
$541$	−376322.	−1.28577	−0.642887	+	0.765961i	$0.722264\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 5661.27i	− 0.0190599i
$546$	0	0
$547$	275034.	0.919204	0.459602	−	0.888125i	$-0.347992\pi$
$547$	275034.	0.919204	0.459602	+	0.888125i	$0.347992\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 232029.i	− 0.764257i
$552$	0	0
$553$	−420710.	−1.37573
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 318602.i	− 1.02692i	−0.858112	−	0.513462i	$-0.828363\pi$
$557$	− 318602.i	− 1.02692i	0.858112	−	0.513462i	$-0.171637\pi$
$558$	0	0
$559$	−16361.9	−0.0523612
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 47565.1i	− 0.150062i	−0.997181	−	0.0750311i	$-0.976094\pi$
$563$	− 47565.1i	− 0.150062i	0.997181	−	0.0750311i	$-0.0239056\pi$
$564$	0	0
$565$	44905.5	0.140670
$566$	0	0
$567$	0	0
$568$	0	0
$569$	313777.i	0.969161i	0.874747	+	0.484580i	$0.161028\pi$
$569$	313777.i	0.969161i	−0.874747	+	0.484580i	$0.838972\pi$
$570$	0	0
$571$	−324095.	−0.994031	−0.497015	−	0.867742i	$-0.665571\pi$
$571$	−324095.	−0.994031	−0.497015	+	0.867742i	$0.665571\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	370218.i	1.11975i
$576$	0	0
$577$	−292524.	−0.878636	−0.439318	−	0.898332i	$-0.644780\pi$
$577$	−292524.	−0.878636	−0.439318	+	0.898332i	$0.644780\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 156154.i	− 0.462595i
$582$	0	0
$583$	278333.	0.818894
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36965.4i	0.107280i	0.998560	+	0.0536400i	$0.0170824\pi$
$587$	36965.4i	0.107280i	−0.998560	+	0.0536400i	$0.982918\pi$
$588$	0	0
$589$	798856.	2.30270
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 5939.58i	− 0.0168906i	−0.999964	−	0.00844532i	$-0.997312\pi$
$593$	− 5939.58i	− 0.0168906i	0.999964	−	0.00844532i	$-0.00268826\pi$
$594$	0	0
$595$	4214.62	0.0119049
$596$	0	0
$597$	0	0
$598$	0	0
$599$	156326.i	0.435689i	0.975983	+	0.217845i	$0.0699026\pi$
$599$	156326.i	0.435689i	−0.975983	+	0.217845i	$0.930097\pi$
$600$	0	0
$601$	141638.	0.392131	0.196066	−	0.980591i	$-0.437183\pi$
$601$	141638.	0.392131	0.196066	+	0.980591i	$0.437183\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 67145.1i	− 0.183444i
$606$	0	0
$607$	−267638.	−0.726392	−0.363196	−	0.931713i	$-0.618314\pi$
$607$	−267638.	−0.726392	−0.363196	+	0.931713i	$0.618314\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 305309.i	− 0.817819i
$612$	0	0
$613$	191952.	0.510825	0.255413	−	0.966832i	$-0.417789\pi$
$613$	191952.	0.510825	0.255413	+	0.966832i	$0.417789\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	383833.i	1.00826i	0.863628	+	0.504130i	$0.168187\pi$
$617$	383833.i	1.00826i	−0.863628	+	0.504130i	$0.831813\pi$
$618$	0	0
$619$	−121238.	−0.316416	−0.158208	−	0.987406i	$-0.550572\pi$
$619$	−121238.	−0.316416	−0.158208	+	0.987406i	$0.550572\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 22079.3i	− 0.0568864i
$624$	0	0
$625$	381493.	0.976622
$626$	0	0
$627$	0	0
$628$	0	0
$629$	30450.6i	0.0769653i
$630$	0	0
$631$	455676.	1.14445	0.572226	−	0.820096i	$-0.306080\pi$
$631$	455676.	1.14445	0.572226	+	0.820096i	$0.306080\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 47188.7i	− 0.117028i
$636$	0	0
$637$	82768.8	0.203980
$638$	0	0
$639$	0	0
$640$	0	0
$641$	572907.i	1.39434i	0.716907	+	0.697168i	$0.245557\pi$
$641$	572907.i	1.39434i	−0.716907	+	0.697168i	$0.754443\pi$
$642$	0	0
$643$	449264.	1.08663	0.543313	−	0.839530i	$-0.317170\pi$
$643$	449264.	1.08663	0.543313	+	0.839530i	$0.317170\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 502638.i	− 1.20073i	−0.799725	−	0.600366i	$-0.795021\pi$
$647$	− 502638.i	− 1.20073i	0.799725	−	0.600366i	$-0.204979\pi$
$648$	0	0
$649$	837558.	1.98850
$650$	0	0
$651$	0	0
$652$	0	0
$653$	687812.i	1.61303i	0.591211	+	0.806517i	$0.298650\pi$
$653$	687812.i	1.61303i	−0.591211	+	0.806517i	$0.701350\pi$
$654$	0	0
$655$	68144.5	0.158836
$656$	0	0
$657$	0	0
$658$	0	0
$659$	269083.i	0.619607i	0.950801	+	0.309803i	$0.100263\pi$
$659$	269083.i	0.619607i	−0.950801	+	0.309803i	$0.899737\pi$
$660$	0	0
$661$	−401251.	−0.918361	−0.459181	−	0.888343i	$-0.651857\pi$
$661$	−401251.	−0.918361	−0.459181	+	0.888343i	$0.651857\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	51318.4i	0.116046i
$666$	0	0
$667$	−262349.	−0.589696
$668$	0	0
$669$	0	0
$670$	0	0
$671$	896456.i	1.99106i
$672$	0	0
$673$	−108737.	−0.240075	−0.120038	−	0.992769i	$-0.538301\pi$
$673$	−108737.	−0.240075	−0.120038	+	0.992769i	$0.538301\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 95053.7i	− 0.207392i	−0.994609	−	0.103696i	$-0.966933\pi$
$677$	− 95053.7i	− 0.207392i	0.994609	−	0.103696i	$-0.0330669\pi$
$678$	0	0
$679$	326740.	0.708700
$680$	0	0
$681$	0	0
$682$	0	0
$683$	704385.i	1.50997i	0.655741	+	0.754986i	$0.272356\pi$
$683$	704385.i	1.50997i	−0.655741	+	0.754986i	$0.727644\pi$
$684$	0	0
$685$	24301.5	0.0517906
$686$	0	0
$687$	0	0
$688$	0	0
$689$	232710.i	0.490203i
$690$	0	0
$691$	−883081.	−1.84946	−0.924729	−	0.380627i	$-0.875708\pi$
$691$	−883081.	−1.84946	−0.924729	+	0.380627i	$0.875708\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 37445.0i	− 0.0775219i
$696$	0	0
$697$	25914.3	0.0533426
$698$	0	0
$699$	0	0
$700$	0	0
$701$	713342.i	1.45165i	0.687879	+	0.725825i	$0.258542\pi$
$701$	713342.i	1.45165i	−0.687879	+	0.725825i	$0.741458\pi$
$702$	0	0
$703$	−370775.	−0.750240
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40317.7i	0.0806599i
$708$	0	0
$709$	78489.9	0.156143	0.0780713	−	0.996948i	$-0.475124\pi$
$709$	78489.9	0.156143	0.0780713	+	0.996948i	$0.475124\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 903245.i	− 1.77675i
$714$	0	0
$715$	83188.8	0.162725
$716$	0	0
$717$	0	0
$718$	0	0
$719$	146207.i	0.282820i	0.989951	+	0.141410i	$0.0451636\pi$
$719$	146207.i	0.282820i	−0.989951	+	0.141410i	$0.954836\pi$
$720$	0	0
$721$	−156325.	−0.300717
$722$	0	0
$723$	0	0
$724$	0	0
$725$	272502.i	0.518435i
$726$	0	0
$727$	−705233.	−1.33433	−0.667166	−	0.744909i	$-0.732493\pi$
$727$	−705233.	−1.33433	−0.667166	+	0.744909i	$0.732493\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3999.26i	0.00748420i
$732$	0	0
$733$	131096.	0.243996	0.121998	−	0.992530i	$-0.461070\pi$
$733$	131096.	0.243996	0.121998	+	0.992530i	$0.461070\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 136905.i	− 0.252048i
$738$	0	0
$739$	−82938.7	−0.151869	−0.0759344	−	0.997113i	$-0.524194\pi$
$739$	−82938.7	−0.151869	−0.0759344	+	0.997113i	$0.524194\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	838373.i	1.51866i	0.650707	+	0.759329i	$0.274473\pi$
$743$	838373.i	1.51866i	−0.650707	+	0.759329i	$0.725527\pi$
$744$	0	0
$745$	−36701.6	−0.0661261
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 24211.4i	− 0.0431575i
$750$	0	0
$751$	−308245.	−0.546533	−0.273266	−	0.961938i	$-0.588104\pi$
$751$	−308245.	−0.546533	−0.273266	+	0.961938i	$0.588104\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24375.1i	0.0427614i
$756$	0	0
$757$	298072.	0.520151	0.260075	−	0.965588i	$-0.416253\pi$
$757$	298072.	0.520151	0.260075	+	0.965588i	$0.416253\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 237538.i	− 0.410170i	−0.978744	−	0.205085i	$-0.934253\pi$
$761$	− 237538.i	− 0.410170i	0.978744	−	0.205085i	$-0.0657471\pi$
$762$	0	0
$763$	112680.	0.193553
$764$	0	0
$765$	0	0
$766$	0	0
$767$	700269.i	1.19035i
$768$	0	0
$769$	693972.	1.17352	0.586758	−	0.809762i	$-0.300404\pi$
$769$	693972.	1.17352	0.586758	+	0.809762i	$0.300404\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	106414.i	0.178091i	0.996028	+	0.0890454i	$0.0283816\pi$
$773$	106414.i	0.178091i	−0.996028	+	0.0890454i	$0.971618\pi$
$774$	0	0
$775$	−938201.	−1.56204
$776$	0	0
$777$	0	0
$778$	0	0
$779$	315540.i	0.519972i
$780$	0	0
$781$	−1.35754e6	−2.22562
$782$	0	0
$783$	0	0
$784$	0	0
$785$	44050.7i	0.0714848i
$786$	0	0
$787$	−96723.1	−0.156164	−0.0780820	−	0.996947i	$-0.524880\pi$
$787$	−96723.1	−0.156164	−0.0780820	+	0.996947i	$0.524880\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	893787.i	1.42850i
$792$	0	0
$793$	−749512.	−1.19188
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 546425.i	− 0.860229i	−0.902774	−	0.430115i	$-0.858473\pi$
$797$	− 546425.i	− 0.860229i	0.902774	−	0.430115i	$-0.141527\pi$
$798$	0	0
$799$	−74625.4	−0.116894
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 678207.i	− 1.05180i
$804$	0	0
$805$	58024.3	0.0895402
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 467941.i	− 0.714981i	−0.933917	−	0.357490i	$-0.883633\pi$
$809$	− 467941.i	− 0.714981i	0.933917	−	0.357490i	$-0.116367\pi$
$810$	0	0
$811$	1.15071e6	1.74954	0.874771	−	0.484537i	$-0.161012\pi$
$811$	1.15071e6	1.74954	0.874771	+	0.484537i	$0.161012\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17537.2i	0.0264025i
$816$	0	0
$817$	−48696.2	−0.0729543
$818$	0	0
$819$	0	0
$820$	0	0
$821$	474735.i	0.704312i	0.935941	+	0.352156i	$0.114551\pi$
$821$	474735.i	0.704312i	−0.935941	+	0.352156i	$0.885449\pi$
$822$	0	0
$823$	1.19867e6	1.76970	0.884849	−	0.465878i	$-0.154261\pi$
$823$	1.19867e6	1.76970	0.884849	+	0.465878i	$0.154261\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 227743.i	− 0.332993i	−0.986042	−	0.166496i	$-0.946755\pi$
$827$	− 227743.i	− 0.332993i	0.986042	−	0.166496i	$-0.0532454\pi$
$828$	0	0
$829$	17171.5	0.0249862	0.0124931	−	0.999922i	$-0.496023\pi$
$829$	17171.5	0.0249862	0.0124931	+	0.999922i	$0.496023\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 20230.8i	− 0.0291557i
$834$	0	0
$835$	−42398.8	−0.0608108
$836$	0	0
$837$	0	0
$838$	0	0
$839$	756669.i	1.07493i	0.843285	+	0.537467i	$0.180619\pi$
$839$	756669.i	1.07493i	−0.843285	+	0.537467i	$0.819381\pi$
$840$	0	0
$841$	514176.	0.726976
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6439.75i	0.00901894i
$846$	0	0
$847$	1.33644e6	1.86287
$848$	0	0
$849$	0	0
$850$	0	0
$851$	419226.i	0.578880i
$852$	0	0
$853$	−562876.	−0.773596	−0.386798	−	0.922164i	$-0.626419\pi$
$853$	−562876.	−0.773596	−0.386798	+	0.922164i	$0.626419\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	869774.i	1.18425i	0.805845	+	0.592127i	$0.201711\pi$
$857$	869774.i	1.18425i	−0.805845	+	0.592127i	$0.798289\pi$
$858$	0	0
$859$	−422771.	−0.572952	−0.286476	−	0.958087i	$-0.592484\pi$
$859$	−422771.	−0.572952	−0.286476	+	0.958087i	$0.592484\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.33087e6i	1.78696i	0.449100	+	0.893482i	$0.351745\pi$
$863$	1.33087e6i	1.78696i	−0.449100	+	0.893482i	$0.648255\pi$
$864$	0	0
$865$	−115323.	−0.154129
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.02972e6i	2.68780i
$870$	0	0
$871$	114464.	0.150880
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 121014.i	− 0.158059i
$876$	0	0
$877$	−1.27682e6	−1.66009	−0.830045	−	0.557696i	$-0.811686\pi$
$877$	−1.27682e6	−1.66009	−0.830045	+	0.557696i	$0.811686\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	404060.i	0.520588i	0.965529	+	0.260294i	$0.0838194\pi$
$881$	404060.i	0.520588i	−0.965529	+	0.260294i	$0.916181\pi$
$882$	0	0
$883$	−728109.	−0.933845	−0.466923	−	0.884298i	$-0.654637\pi$
$883$	−728109.	−0.933845	−0.466923	+	0.884298i	$0.654637\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	109520.i	0.139202i	0.997575	+	0.0696010i	$0.0221726\pi$
$887$	109520.i	0.139202i	−0.997575	+	0.0696010i	$0.977827\pi$
$888$	0	0
$889$	939232.	1.18842
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 908660.i	− 1.13946i
$894$	0	0
$895$	−26952.8	−0.0336478
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 664842.i	− 0.822620i
$900$	0	0
$901$	56880.2	0.0700667
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15182.1i	0.0185368i
$906$	0	0
$907$	1.42932e6	1.73746	0.868731	−	0.495284i	$-0.164936\pi$
$907$	1.42932e6	1.73746	0.868731	+	0.495284i	$0.164936\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	562860.i	0.678210i	0.940749	+	0.339105i	$0.110124\pi$
$911$	562860.i	0.678210i	−0.940749	+	0.339105i	$0.889876\pi$
$912$	0	0
$913$	−753368.	−0.903786
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.35633e6i	1.61297i
$918$	0	0
$919$	−290385.	−0.343829	−0.171915	−	0.985112i	$-0.554995\pi$
$919$	−290385.	−0.343829	−0.171915	+	0.985112i	$0.554995\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.13502e6i	− 1.33229i
$924$	0	0
$925$	435450.	0.508926
$926$	0	0
$927$	0	0
$928$	0	0
$929$	324055.i	0.375480i	0.982219	+	0.187740i	$0.0601163\pi$
$929$	324055.i	0.375480i	−0.982219	+	0.187740i	$0.939884\pi$
$930$	0	0
$931$	246336.	0.284203
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 20333.5i	− 0.0232589i
$936$	0	0
$937$	−672276.	−0.765717	−0.382859	−	0.923807i	$-0.625060\pi$
$937$	−672276.	−0.765717	−0.382859	+	0.923807i	$0.625060\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 54389.2i	− 0.0614234i	−0.999528	−	0.0307117i	$-0.990223\pi$
$941$	− 54389.2i	− 0.0614234i	0.999528	−	0.0307117i	$-0.00977737\pi$
$942$	0	0
$943$	356773.	0.401207
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 335302.i	− 0.373884i	−0.982371	−	0.186942i	$-0.940142\pi$
$947$	− 335302.i	− 0.373884i	0.982371	−	0.186942i	$-0.0598576\pi$
$948$	0	0
$949$	567038.	0.629622
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1.29405e6i	− 1.42484i	−0.701753	−	0.712421i	$-0.747599\pi$
$953$	− 1.29405e6i	− 1.42484i	0.701753	−	0.712421i	$-0.252401\pi$
$954$	0	0
$955$	91734.0	0.100583
$956$	0	0
$957$	0	0
$958$	0	0
$959$	483690.i	0.525933i
$960$	0	0
$961$	1.36547e6	1.47855
$962$	0	0
$963$	0	0
$964$	0	0
$965$	132002.i	0.141751i
$966$	0	0
$967$	−990491.	−1.05925	−0.529624	−	0.848233i	$-0.677667\pi$
$967$	−990491.	−1.05925	−0.529624	+	0.848233i	$0.677667\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.17055e6i	− 1.24152i	−0.784002	−	0.620758i	$-0.786825\pi$
$971$	− 1.17055e6i	− 1.24152i	0.784002	−	0.620758i	$-0.213175\pi$
$972$	0	0
$973$	745296.	0.787233
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 525152.i	− 0.550169i	−0.961420	−	0.275084i	$-0.911294\pi$
$977$	− 525152.i	− 0.550169i	0.961420	−	0.275084i	$-0.0887058\pi$
$978$	0	0
$979$	−106522.	−0.111141
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.52002e6i	1.57304i	0.617562	+	0.786522i	$0.288120\pi$
$983$	1.52002e6i	1.57304i	−0.617562	+	0.786522i	$0.711880\pi$
$984$	0	0
$985$	27338.4	0.0281774
$986$	0	0
$987$	0	0
$988$	0	0
$989$	55059.5i	0.0562911i
$990$	0	0
$991$	1.13839e6	1.15916	0.579580	−	0.814915i	$-0.303217\pi$
$991$	1.13839e6	1.15916	0.579580	+	0.814915i	$0.303217\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	37204.3i	0.0375792i
$996$	0	0
$997$	−793430.	−0.798212	−0.399106	−	0.916905i	$-0.630679\pi$
$997$	−793430.	−0.798212	−0.399106	+	0.916905i	$0.630679\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.5.e.e.161.5		8
3.2	odd	2	inner	1296.5.e.e.161.4		8
4.3	odd	2		162.5.b.c.161.7		8
9.2	odd	6		432.5.q.b.17.3		8
9.4	even	3		432.5.q.b.305.3		8
9.5	odd	6		144.5.q.b.65.2		8
9.7	even	3		144.5.q.b.113.2		8
12.11	even	2		162.5.b.c.161.2		8
36.7	odd	6		18.5.d.a.5.4	✓	8
36.11	even	6		54.5.d.a.17.2		8
36.23	even	6		18.5.d.a.11.4	yes	8
36.31	odd	6		54.5.d.a.35.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.5.d.a.5.4	✓	8	36.7	odd	6
18.5.d.a.11.4	yes	8	36.23	even	6
54.5.d.a.17.2		8	36.11	even	6
54.5.d.a.35.2		8	36.31	odd	6
144.5.q.b.65.2		8	9.5	odd	6
144.5.q.b.113.2		8	9.7	even	3
162.5.b.c.161.2		8	12.11	even	2
162.5.b.c.161.7		8	4.3	odd	2
432.5.q.b.17.3		8	9.2	odd	6
432.5.q.b.305.3		8	9.4	even	3
1296.5.e.e.161.4		8	3.2	odd	2	inner
1296.5.e.e.161.5		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.20976i q^{5} -43.9826 q^{7} +O(q^{10})\)	\(q+2.20976i q^{5} -43.9826 q^{7} +212.195i q^{11} -177.413 q^{13} +43.3642i q^{17} -528.015 q^{19} +597.013i q^{23} +620.117 q^{25} +439.437i q^{29} -1512.94 q^{31} -97.1911i q^{35} +702.206 q^{37} -597.597i q^{41} +92.2250 q^{43} +1720.90i q^{47} -466.533 q^{49} -1311.69i q^{53} -468.900 q^{55} -3947.12i q^{59} +4224.68 q^{61} -392.040i q^{65} -645.184 q^{67} +6397.63i q^{71} -3196.15 q^{73} -9332.88i q^{77} +9565.38 q^{79} +3550.36i q^{83} -95.8247 q^{85} +502.000i q^{89} +7803.06 q^{91} -1166.79i q^{95} -7428.84 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 52 q^{7}+O(q^{10}) \) 8 * q - 52 * q^7	\( 8 q - 52 q^{7} - 20 q^{13} - 100 q^{19} - 1588 q^{25} - 2956 q^{31} - 32 q^{37} - 136 q^{43} - 4884 q^{49} + 3996 q^{55} + 8956 q^{61} + 15008 q^{67} + 20716 q^{73} - 12100 q^{79} + 32184 q^{85} + 45868 q^{91} - 62672 q^{97}+O(q^{100}) \) 8 * q - 52 * q^7 - 20 * q^13 - 100 * q^19 - 1588 * q^25 - 2956 * q^31 - 32 * q^37 - 136 * q^43 - 4884 * q^49 + 3996 * q^55 + 8956 * q^61 + 15008 * q^67 + 20716 * q^73 - 12100 * q^79 + 32184 * q^85 + 45868 * q^91 - 62672 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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