LMFDB - Embedded newform 1600.3.h.n.1599.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1599,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1600.1599");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.h (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:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1599.7
Root			$-0.951057 + 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1600.1599
Dual form			1600.3.h.n.1599.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.80423 q^{3} -8.50651 q^{7} +5.47214 q^{9} +O(q^{10})$	$q+3.80423 q^{3} -8.50651 q^{7} +5.47214 q^{9} -1.79611i q^{11} -0.472136i q^{13} +23.8885i q^{17} -9.40456i q^{19} -32.3607 q^{21} +16.1150 q^{23} -13.4208 q^{27} +6.94427 q^{29} +47.4468i q^{31} -6.83282i q^{33} +26.3607i q^{37} -1.79611i q^{39} -41.4164 q^{41} -2.00811 q^{43} -35.3481 q^{47} +23.3607 q^{49} +90.8774i q^{51} +21.6393i q^{53} -35.7771i q^{57} +73.8644i q^{59} +26.1378 q^{61} -46.5488 q^{63} -88.8693 q^{67} +61.3050 q^{69} -39.4144i q^{71} +137.554i q^{73} +15.2786i q^{77} +113.703i q^{79} -100.305 q^{81} +21.2412 q^{83} +26.4176 q^{87} -67.4427 q^{89} +4.01623i q^{91} +180.498i q^{93} +39.1672i q^{97} -9.82857i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 80 q^{21} - 16 q^{29} - 224 q^{41} + 8 q^{49} - 256 q^{61} + 240 q^{69} - 552 q^{81} + 176 q^{89}+O(q^{100}) $ 8 * q + 8 * q^9 - 80 * q^21 - 16 * q^29 - 224 * q^41 + 8 * q^49 - 256 * q^61 + 240 * q^69 - 552 * q^81 + 176 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	3.80423	1.26808	0.634038	−	0.773302i	$-0.281396\pi$
$3$	3.80423	1.26808	0.634038	+	0.773302i	$0.281396\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−8.50651	−1.21522	−0.607608	−	0.794237i	$-0.707871\pi$
$7$	−8.50651	−1.21522	−0.607608	+	0.794237i	$0.707871\pi$
$8$	0	0
$9$	5.47214	0.608015
$10$	0	0
$11$	− 1.79611i	− 0.163283i	−0.996662	−	0.0816415i	$-0.973984\pi$
$11$	− 1.79611i	− 0.163283i	0.996662	−	0.0816415i	$-0.0260162\pi$
$12$	0	0
$13$	− 0.472136i	− 0.0363182i	−0.999835	−	0.0181591i	$-0.994219\pi$
$13$	− 0.472136i	− 0.0363182i	0.999835	−	0.0181591i	$-0.00578053\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	23.8885i	1.40521i	0.711581	+	0.702604i	$0.247980\pi$
$17$	23.8885i	1.40521i	−0.711581	+	0.702604i	$0.752020\pi$
$18$	0	0
$19$	− 9.40456i	− 0.494977i	−0.968891	−	0.247489i	$-0.920395\pi$
$19$	− 9.40456i	− 0.494977i	0.968891	−	0.247489i	$-0.0796053\pi$
$20$	0	0
$21$	−32.3607	−1.54098
$22$	0	0
$23$	16.1150	0.700650	0.350325	−	0.936628i	$-0.386071\pi$
$23$	16.1150	0.700650	0.350325	+	0.936628i	$0.386071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−13.4208	−0.497066
$28$	0	0
$29$	6.94427	0.239458	0.119729	−	0.992807i	$-0.461797\pi$
$29$	6.94427	0.239458	0.119729	+	0.992807i	$0.461797\pi$
$30$	0	0
$31$	47.4468i	1.53054i	0.643708	+	0.765271i	$0.277395\pi$
$31$	47.4468i	1.53054i	−0.643708	+	0.765271i	$0.722605\pi$
$32$	0	0
$33$	− 6.83282i	− 0.207055i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	26.3607i	0.712451i	0.934400	+	0.356225i	$0.115936\pi$
$37$	26.3607i	0.712451i	−0.934400	+	0.356225i	$0.884064\pi$
$38$	0	0
$39$	− 1.79611i	− 0.0460542i
$40$	0	0
$41$	−41.4164	−1.01016	−0.505078	−	0.863074i	$-0.668536\pi$
$41$	−41.4164	−1.01016	−0.505078	+	0.863074i	$0.668536\pi$
$42$	0	0
$43$	−2.00811	−0.0467003	−0.0233502	−	0.999727i	$-0.507433\pi$
$43$	−2.00811	−0.0467003	−0.0233502	+	0.999727i	$0.507433\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−35.3481	−0.752087	−0.376044	−	0.926602i	$-0.622716\pi$
$47$	−35.3481	−0.752087	−0.376044	+	0.926602i	$0.622716\pi$
$48$	0	0
$49$	23.3607	0.476749
$50$	0	0
$51$	90.8774i	1.78191i
$52$	0	0
$53$	21.6393i	0.408289i	0.978941	+	0.204145i	$0.0654413\pi$
$53$	21.6393i	0.408289i	−0.978941	+	0.204145i	$0.934559\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 35.7771i	− 0.627668i
$58$	0	0
$59$	73.8644i	1.25194i	0.779848	+	0.625970i	$0.215297\pi$
$59$	73.8644i	1.25194i	−0.779848	+	0.625970i	$0.784703\pi$
$60$	0	0
$61$	26.1378	0.428488	0.214244	−	0.976780i	$-0.431271\pi$
$61$	26.1378	0.428488	0.214244	+	0.976780i	$0.431271\pi$
$62$	0	0
$63$	−46.5488	−0.738869
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−88.8693	−1.32641	−0.663204	−	0.748439i	$-0.730804\pi$
$67$	−88.8693	−1.32641	−0.663204	+	0.748439i	$0.730804\pi$
$68$	0	0
$69$	61.3050	0.888478
$70$	0	0
$71$	− 39.4144i	− 0.555132i	−0.960707	−	0.277566i	$-0.910472\pi$
$71$	− 39.4144i	− 0.555132i	0.960707	−	0.277566i	$-0.0895277\pi$
$72$	0	0
$73$	137.554i	1.88430i	0.335186	+	0.942152i	$0.391201\pi$
$73$	137.554i	1.88430i	−0.335186	+	0.942152i	$0.608799\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.2786i	0.198424i
$78$	0	0
$79$	113.703i	1.43928i	0.694350	+	0.719638i	$0.255692\pi$
$79$	113.703i	1.43928i	−0.694350	+	0.719638i	$0.744308\pi$
$80$	0	0
$81$	−100.305	−1.23833
$82$	0	0
$83$	21.2412	0.255919	0.127959	−	0.991779i	$-0.459157\pi$
$83$	21.2412	0.255919	0.127959	+	0.991779i	$0.459157\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	26.4176	0.303650
$88$	0	0
$89$	−67.4427	−0.757783	−0.378892	−	0.925441i	$-0.623695\pi$
$89$	−67.4427	−0.757783	−0.378892	+	0.925441i	$0.623695\pi$
$90$	0	0
$91$	4.01623i	0.0441344i
$92$	0	0
$93$	180.498i	1.94084i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	39.1672i	0.403785i	0.979408	+	0.201893i	$0.0647092\pi$
$97$	39.1672i	0.403785i	−0.979408	+	0.201893i	$0.935291\pi$
$98$	0	0
$99$	− 9.82857i	− 0.0992785i
$100$	0	0
$101$	−99.8885	−0.988995	−0.494498	−	0.869179i	$-0.664648\pi$
$101$	−99.8885	−0.988995	−0.494498	+	0.869179i	$0.664648\pi$
$102$	0	0
$103$	35.7721	0.347302	0.173651	−	0.984807i	$-0.444444\pi$
$103$	35.7721	0.347302	0.173651	+	0.984807i	$0.444444\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	121.099	1.13177	0.565884	−	0.824485i	$-0.308535\pi$
$107$	121.099	1.13177	0.565884	+	0.824485i	$0.308535\pi$
$108$	0	0
$109$	−197.469	−1.81164	−0.905821	−	0.423660i	$-0.860745\pi$
$109$	−197.469	−1.81164	−0.905821	+	0.423660i	$0.860745\pi$
$110$	0	0
$111$	100.282i	0.903441i
$112$	0	0
$113$	81.2786i	0.719280i	0.933091	+	0.359640i	$0.117100\pi$
$113$	81.2786i	0.719280i	−0.933091	+	0.359640i	$0.882900\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 2.58359i	− 0.0220820i
$118$	0	0
$119$	− 203.208i	− 1.70763i
$120$	0	0
$121$	117.774	0.973339
$122$	0	0
$123$	−157.557	−1.28095
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.84616	−0.0145367	−0.00726834	−	0.999974i	$-0.502314\pi$
$127$	−1.84616	−0.0145367	−0.00726834	+	0.999974i	$0.502314\pi$
$128$	0	0
$129$	−7.63932	−0.0592195
$130$	0	0
$131$	− 225.609i	− 1.72221i	−0.508428	−	0.861105i	$-0.669773\pi$
$131$	− 225.609i	− 1.72221i	0.508428	−	0.861105i	$-0.330227\pi$
$132$	0	0
$133$	80.0000i	0.601504i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	52.8328i	0.385641i	0.981234	+	0.192820i	$0.0617635\pi$
$137$	52.8328i	0.385641i	−0.981234	+	0.192820i	$0.938236\pi$
$138$	0	0
$139$	− 125.852i	− 0.905407i	−0.891661	−	0.452703i	$-0.850460\pi$
$139$	− 125.852i	− 0.905407i	0.891661	−	0.452703i	$-0.149540\pi$
$140$	0	0
$141$	−134.472	−0.953703
$142$	0	0
$143$	−0.848009	−0.00593013
$144$	0	0
$145$	0	0
$146$	0	0
$147$	88.8693	0.604553
$148$	0	0
$149$	132.971	0.892420	0.446210	−	0.894928i	$-0.352773\pi$
$149$	132.971	0.892420	0.446210	+	0.894928i	$0.352773\pi$
$150$	0	0
$151$	− 151.221i	− 1.00146i	−0.865603	−	0.500732i	$-0.833064\pi$
$151$	− 151.221i	− 1.00146i	0.865603	−	0.500732i	$-0.166936\pi$
$152$	0	0
$153$	130.721i	0.854388i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 36.7477i	− 0.234062i	−0.993128	−	0.117031i	$-0.962662\pi$
$157$	− 36.7477i	− 0.234062i	0.993128	−	0.117031i	$-0.0373376\pi$
$158$	0	0
$159$	82.3209i	0.517741i
$160$	0	0
$161$	−137.082	−0.851441
$162$	0	0
$163$	302.854	1.85800	0.929000	−	0.370079i	$-0.120669\pi$
$163$	302.854	1.85800	0.929000	+	0.370079i	$0.120669\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	99.3839	0.595113	0.297557	−	0.954704i	$-0.403828\pi$
$167$	99.3839	0.595113	0.297557	+	0.954704i	$0.403828\pi$
$168$	0	0
$169$	168.777	0.998681
$170$	0	0
$171$	− 51.4631i	− 0.300954i
$172$	0	0
$173$	181.639i	1.04994i	0.851121	+	0.524969i	$0.175923\pi$
$173$	181.639i	1.04994i	−0.851121	+	0.524969i	$0.824077\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	280.997i	1.58755i
$178$	0	0
$179$	− 260.907i	− 1.45758i	−0.684735	−	0.728792i	$-0.740082\pi$
$179$	− 260.907i	− 1.45758i	0.684735	−	0.728792i	$-0.259918\pi$
$180$	0	0
$181$	−157.777	−0.871697	−0.435848	−	0.900020i	$-0.643552\pi$
$181$	−157.777	−0.871697	−0.435848	+	0.900020i	$0.643552\pi$
$182$	0	0
$183$	99.4340	0.543355
$184$	0	0
$185$	0	0
$186$	0	0
$187$	42.9065	0.229447
$188$	0	0
$189$	114.164	0.604043
$190$	0	0
$191$	− 324.095i	− 1.69683i	−0.529328	−	0.848417i	$-0.677556\pi$
$191$	− 324.095i	− 1.69683i	0.529328	−	0.848417i	$-0.322444\pi$
$192$	0	0
$193$	181.777i	0.941850i	0.882173	+	0.470925i	$0.156080\pi$
$193$	181.777i	0.941850i	−0.882173	+	0.470925i	$0.843920\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	140.525i	0.713324i	0.934234	+	0.356662i	$0.116085\pi$
$197$	140.525i	0.713324i	−0.934234	+	0.356662i	$0.883915\pi$
$198$	0	0
$199$	− 168.234i	− 0.845397i	−0.906270	−	0.422698i	$-0.861083\pi$
$199$	− 168.234i	− 0.845397i	0.906270	−	0.422698i	$-0.138917\pi$
$200$	0	0
$201$	−338.079	−1.68198
$202$	0	0
$203$	−59.0715	−0.290993
$204$	0	0
$205$	0	0
$206$	0	0
$207$	88.1833	0.426006
$208$	0	0
$209$	−16.8916	−0.0808213
$210$	0	0
$211$	− 93.9455i	− 0.445240i	−0.974905	−	0.222620i	$-0.928539\pi$
$211$	− 93.9455i	− 0.445240i	0.974905	−	0.222620i	$-0.0714608\pi$
$212$	0	0
$213$	− 149.941i	− 0.703949i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 403.607i	− 1.85994i
$218$	0	0
$219$	523.287i	2.38944i
$220$	0	0
$221$	11.2786	0.0510346
$222$	0	0
$223$	−214.035	−0.959797	−0.479899	−	0.877324i	$-0.659327\pi$
$223$	−214.035	−0.959797	−0.479899	+	0.877324i	$0.659327\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	41.4225	0.182478	0.0912389	−	0.995829i	$-0.470917\pi$
$227$	41.4225	0.182478	0.0912389	+	0.995829i	$0.470917\pi$
$228$	0	0
$229$	73.2786	0.319994	0.159997	−	0.987117i	$-0.448852\pi$
$229$	73.2786	0.319994	0.159997	+	0.987117i	$0.448852\pi$
$230$	0	0
$231$	58.1234i	0.251616i
$232$	0	0
$233$	− 307.050i	− 1.31781i	−0.752227	−	0.658905i	$-0.771020\pi$
$233$	− 307.050i	− 1.31781i	0.752227	−	0.658905i	$-0.228980\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	432.551i	1.82511i
$238$	0	0
$239$	42.9065i	0.179525i	0.995963	+	0.0897625i	$0.0286108\pi$
$239$	42.9065i	0.179525i	−0.995963	+	0.0897625i	$0.971389\pi$
$240$	0	0
$241$	−135.082	−0.560506	−0.280253	−	0.959926i	$-0.590418\pi$
$241$	−135.082	−0.560506	−0.280253	+	0.959926i	$0.590418\pi$
$242$	0	0
$243$	−260.796	−1.07323
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.44023	−0.0179767
$248$	0	0
$249$	80.8065	0.324524
$250$	0	0
$251$	221.169i	0.881152i	0.897715	+	0.440576i	$0.145226\pi$
$251$	221.169i	0.881152i	−0.897715	+	0.440576i	$0.854774\pi$
$252$	0	0
$253$	− 28.9443i	− 0.114404i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	257.056i	1.00022i	0.865963	+	0.500108i	$0.166707\pi$
$257$	257.056i	1.00022i	−0.865963	+	0.500108i	$0.833293\pi$
$258$	0	0
$259$	− 224.237i	− 0.865781i
$260$	0	0
$261$	38.0000	0.145594
$262$	0	0
$263$	164.168	0.624212	0.312106	−	0.950047i	$-0.398966\pi$
$263$	164.168	0.624212	0.312106	+	0.950047i	$0.398966\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−256.567	−0.960926
$268$	0	0
$269$	−35.4752	−0.131878	−0.0659391	−	0.997824i	$-0.521004\pi$
$269$	−35.4752	−0.131878	−0.0659391	+	0.997824i	$0.521004\pi$
$270$	0	0
$271$	298.950i	1.10314i	0.834130	+	0.551568i	$0.185970\pi$
$271$	298.950i	1.10314i	−0.834130	+	0.551568i	$0.814030\pi$
$272$	0	0
$273$	15.2786i	0.0559657i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 457.246i	− 1.65071i	−0.564616	−	0.825354i	$-0.690976\pi$
$277$	− 457.246i	− 1.65071i	0.564616	−	0.825354i	$-0.309024\pi$
$278$	0	0
$279$	259.635i	0.930593i
$280$	0	0
$281$	−5.63932	−0.0200688	−0.0100344	−	0.999950i	$-0.503194\pi$
$281$	−5.63932	−0.0200688	−0.0100344	+	0.999950i	$0.503194\pi$
$282$	0	0
$283$	−169.918	−0.600418	−0.300209	−	0.953874i	$-0.597056\pi$
$283$	−169.918	−0.600418	−0.300209	+	0.953874i	$0.597056\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	352.309	1.22756
$288$	0	0
$289$	−281.663	−0.974611
$290$	0	0
$291$	149.001i	0.512030i
$292$	0	0
$293$	26.8591i	0.0916694i	0.998949	+	0.0458347i	$0.0145947\pi$
$293$	26.8591i	0.0916694i	−0.998949	+	0.0458347i	$0.985405\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	24.1052i	0.0811624i
$298$	0	0
$299$	− 7.60845i	− 0.0254463i
$300$	0	0
$301$	17.0820	0.0567510
$302$	0	0
$303$	−379.999	−1.25412
$304$	0	0
$305$	0	0
$306$	0	0
$307$	118.031	0.384466	0.192233	−	0.981349i	$-0.438427\pi$
$307$	118.031	0.384466	0.192233	+	0.981349i	$0.438427\pi$
$308$	0	0
$309$	136.085	0.440405
$310$	0	0
$311$	121.835i	0.391753i	0.980629	+	0.195877i	$0.0627552\pi$
$311$	121.835i	0.391753i	−0.980629	+	0.195877i	$0.937245\pi$
$312$	0	0
$313$	219.548i	0.701431i	0.936482	+	0.350716i	$0.114062\pi$
$313$	219.548i	0.701431i	−0.936482	+	0.350716i	$0.885938\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	366.859i	1.15728i	0.815582	+	0.578642i	$0.196417\pi$
$317$	366.859i	1.15728i	−0.815582	+	0.578642i	$0.803583\pi$
$318$	0	0
$319$	− 12.4727i	− 0.0390993i
$320$	0	0
$321$	460.689	1.43517
$322$	0	0
$323$	224.661	0.695546
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−751.217	−2.29730
$328$	0	0
$329$	300.689	0.913948
$330$	0	0
$331$	162.846i	0.491981i	0.969272	+	0.245990i	$0.0791132\pi$
$331$	162.846i	0.491981i	−0.969272	+	0.245990i	$0.920887\pi$
$332$	0	0
$333$	144.249i	0.433181i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 17.1084i	− 0.0507666i	−0.999678	−	0.0253833i	$-0.991919\pi$
$337$	− 17.1084i	− 0.0507666i	0.999678	−	0.0253833i	$-0.00808063\pi$
$338$	0	0
$339$	309.202i	0.912101i
$340$	0	0
$341$	85.2198	0.249911
$342$	0	0
$343$	218.101	0.635863
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−167.498	−0.482703	−0.241351	−	0.970438i	$-0.577591\pi$
$347$	−167.498	−0.482703	−0.241351	+	0.970438i	$0.577591\pi$
$348$	0	0
$349$	−483.495	−1.38537	−0.692687	−	0.721239i	$-0.743573\pi$
$349$	−483.495	−1.38537	−0.692687	+	0.721239i	$0.743573\pi$
$350$	0	0
$351$	6.33644i	0.0180525i
$352$	0	0
$353$	307.994i	0.872504i	0.899825	+	0.436252i	$0.143694\pi$
$353$	307.994i	0.872504i	−0.899825	+	0.436252i	$0.856306\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 773.050i	− 2.16540i
$358$	0	0
$359$	23.2494i	0.0647615i	0.999476	+	0.0323807i	$0.0103089\pi$
$359$	23.2494i	0.0647615i	−0.999476	+	0.0323807i	$0.989691\pi$
$360$	0	0
$361$	272.554	0.754998
$362$	0	0
$363$	448.039	1.23427
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−517.325	−1.40960	−0.704802	−	0.709404i	$-0.748964\pi$
$367$	−517.325	−1.40960	−0.704802	+	0.709404i	$0.748964\pi$
$368$	0	0
$369$	−226.636	−0.614190
$370$	0	0
$371$	− 184.075i	− 0.496159i
$372$	0	0
$373$	88.3545i	0.236875i	0.992961	+	0.118438i	$0.0377886\pi$
$373$	88.3545i	0.236875i	−0.992961	+	0.118438i	$0.962211\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.27864i	− 0.00869666i
$378$	0	0
$379$	19.3332i	0.0510112i	0.999675	+	0.0255056i	$0.00811956\pi$
$379$	19.3332i	0.0510112i	−0.999675	+	0.0255056i	$0.991880\pi$
$380$	0	0
$381$	−7.02321	−0.0184336
$382$	0	0
$383$	431.612	1.12692	0.563462	−	0.826142i	$-0.309469\pi$
$383$	431.612	1.12692	0.563462	+	0.826142i	$0.309469\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.9887	−0.0283945
$388$	0	0
$389$	−296.354	−0.761837	−0.380918	−	0.924609i	$-0.624392\pi$
$389$	−296.354	−0.761837	−0.380918	+	0.924609i	$0.624392\pi$
$390$	0	0
$391$	384.963i	0.984560i
$392$	0	0
$393$	− 858.269i	− 2.18389i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 86.1904i	− 0.217104i	−0.994091	−	0.108552i	$-0.965379\pi$
$397$	− 86.1904i	− 0.217104i	0.994091	−	0.108552i	$-0.0346214\pi$
$398$	0	0
$399$	304.338i	0.762752i
$400$	0	0
$401$	442.997	1.10473	0.552365	−	0.833602i	$-0.313725\pi$
$401$	442.997	1.10473	0.552365	+	0.833602i	$0.313725\pi$
$402$	0	0
$403$	22.4014	0.0555865
$404$	0	0
$405$	0	0
$406$	0	0
$407$	47.3467	0.116331
$408$	0	0
$409$	−63.4102	−0.155037	−0.0775186	−	0.996991i	$-0.524700\pi$
$409$	−63.4102	−0.155037	−0.0775186	+	0.996991i	$0.524700\pi$
$410$	0	0
$411$	200.988i	0.489022i
$412$	0	0
$413$	− 628.328i	− 1.52138i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 478.768i	− 1.14812i
$418$	0	0
$419$	− 435.678i	− 1.03980i	−0.854226	−	0.519902i	$-0.825968\pi$
$419$	− 435.678i	− 1.03980i	0.854226	−	0.519902i	$-0.174032\pi$
$420$	0	0
$421$	582.912	1.38459	0.692294	−	0.721615i	$-0.256600\pi$
$421$	582.912	1.38459	0.692294	+	0.721615i	$0.256600\pi$
$422$	0	0
$423$	−193.430	−0.457280
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−222.341	−0.520705
$428$	0	0
$429$	−3.22602	−0.00751986
$430$	0	0
$431$	375.882i	0.872117i	0.899918	+	0.436058i	$0.143626\pi$
$431$	375.882i	0.872117i	−0.899918	+	0.436058i	$0.856374\pi$
$432$	0	0
$433$	− 368.164i	− 0.850263i	−0.905131	−	0.425132i	$-0.860228\pi$
$433$	− 368.164i	− 0.850263i	0.905131	−	0.425132i	$-0.139772\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 151.554i	− 0.346806i
$438$	0	0
$439$	− 483.549i	− 1.10148i	−0.834677	−	0.550739i	$-0.814346\pi$
$439$	− 483.549i	− 1.10148i	0.834677	−	0.550739i	$-0.185654\pi$
$440$	0	0
$441$	127.833	0.289870
$442$	0	0
$443$	−279.181	−0.630205	−0.315102	−	0.949058i	$-0.602039\pi$
$443$	−279.181	−0.630205	−0.315102	+	0.949058i	$0.602039\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	505.850	1.13166
$448$	0	0
$449$	−756.079	−1.68392	−0.841959	−	0.539542i	$-0.818597\pi$
$449$	−756.079	−1.68392	−0.841959	+	0.539542i	$0.818597\pi$
$450$	0	0
$451$	74.3885i	0.164941i
$452$	0	0
$453$	− 575.279i	− 1.26993i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 285.672i	− 0.625103i	−0.949901	−	0.312551i	$-0.898816\pi$
$457$	− 285.672i	− 0.625103i	0.949901	−	0.312551i	$-0.101184\pi$
$458$	0	0
$459$	− 320.603i	− 0.698482i
$460$	0	0
$461$	99.1146	0.214999	0.107500	−	0.994205i	$-0.465716\pi$
$461$	99.1146	0.214999	0.107500	+	0.994205i	$0.465716\pi$
$462$	0	0
$463$	630.603	1.36199	0.680997	−	0.732286i	$-0.261547\pi$
$463$	630.603	1.36199	0.680997	+	0.732286i	$0.261547\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−496.010	−1.06212	−0.531060	−	0.847334i	$-0.678206\pi$
$467$	−496.010	−1.06212	−0.531060	+	0.847334i	$0.678206\pi$
$468$	0	0
$469$	755.967	1.61187
$470$	0	0
$471$	− 139.796i	− 0.296808i
$472$	0	0
$473$	3.60680i	0.00762537i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	118.413i	0.248246i
$478$	0	0
$479$	− 579.090i	− 1.20896i	−0.796621	−	0.604478i	$-0.793382\pi$
$479$	− 579.090i	− 1.20896i	0.796621	−	0.604478i	$-0.206618\pi$
$480$	0	0
$481$	12.4458	0.0258749
$482$	0	0
$483$	−521.491	−1.07969
$484$	0	0
$485$	0	0
$486$	0	0
$487$	626.363	1.28617	0.643084	−	0.765796i	$-0.277655\pi$
$487$	626.363	1.28617	0.643084	+	0.765796i	$0.277655\pi$
$488$	0	0
$489$	1152.13	2.35608
$490$	0	0
$491$	− 22.3013i	− 0.0454201i	−0.999742	−	0.0227100i	$-0.992771\pi$
$491$	− 22.3013i	− 0.0454201i	0.999742	−	0.0227100i	$-0.00722945\pi$
$492$	0	0
$493$	165.889i	0.336488i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	335.279i	0.674605i
$498$	0	0
$499$	627.362i	1.25724i	0.777714	+	0.628619i	$0.216379\pi$
$499$	627.362i	1.25724i	−0.777714	+	0.628619i	$0.783621\pi$
$500$	0	0
$501$	378.079	0.754649
$502$	0	0
$503$	−780.853	−1.55239	−0.776196	−	0.630492i	$-0.782853\pi$
$503$	−780.853	−1.55239	−0.776196	+	0.630492i	$0.782853\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	642.066	1.26640
$508$	0	0
$509$	−288.950	−0.567683	−0.283841	−	0.958871i	$-0.591609\pi$
$509$	−288.950	−0.567683	−0.283841	+	0.958871i	$0.591609\pi$
$510$	0	0
$511$	− 1170.11i	− 2.28984i
$512$	0	0
$513$	126.217i	0.246036i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	63.4891i	0.122803i
$518$	0	0
$519$	690.997i	1.33140i
$520$	0	0
$521$	−602.984	−1.15736	−0.578680	−	0.815555i	$-0.696432\pi$
$521$	−602.984	−1.15736	−0.578680	+	0.815555i	$0.696432\pi$
$522$	0	0
$523$	367.962	0.703560	0.351780	−	0.936083i	$-0.385577\pi$
$523$	367.962	0.703560	0.351780	+	0.936083i	$0.385577\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1133.44	−2.15073
$528$	0	0
$529$	−269.308	−0.509089
$530$	0	0
$531$	404.196i	0.761198i
$532$	0	0
$533$	19.5542i	0.0366870i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 992.551i	− 1.84833i
$538$	0	0
$539$	− 41.9584i	− 0.0778449i
$540$	0	0
$541$	616.885	1.14027	0.570134	−	0.821551i	$-0.306891\pi$
$541$	616.885	1.14027	0.570134	+	0.821551i	$0.306891\pi$
$542$	0	0
$543$	−600.220	−1.10538
$544$	0	0
$545$	0	0
$546$	0	0
$547$	97.8499	0.178885	0.0894423	−	0.995992i	$-0.471492\pi$
$547$	97.8499	0.178885	0.0894423	+	0.995992i	$0.471492\pi$
$548$	0	0
$549$	143.029	0.260527
$550$	0	0
$551$	− 65.3078i	− 0.118526i
$552$	0	0
$553$	− 967.214i	− 1.74903i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	896.302i	1.60916i	0.593845	+	0.804580i	$0.297609\pi$
$557$	896.302i	1.60916i	−0.593845	+	0.804580i	$0.702391\pi$
$558$	0	0
$559$	0.948103i	0.00169607i
$560$	0	0
$561$	163.226	0.290955
$562$	0	0
$563$	771.186	1.36978	0.684890	−	0.728647i	$-0.259850\pi$
$563$	771.186	1.36978	0.684890	+	0.728647i	$0.259850\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	853.245	1.50484
$568$	0	0
$569$	−8.74767	−0.0153738	−0.00768688	−	0.999970i	$-0.502447\pi$
$569$	−8.74767	−0.0153738	−0.00768688	+	0.999970i	$0.502447\pi$
$570$	0	0
$571$	− 511.138i	− 0.895164i	−0.894243	−	0.447582i	$-0.852285\pi$
$571$	− 511.138i	− 0.895164i	0.894243	−	0.447582i	$-0.147715\pi$
$572$	0	0
$573$	− 1232.93i	− 2.15171i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	713.712i	1.23694i	0.785810	+	0.618468i	$0.212246\pi$
$577$	713.712i	1.23694i	−0.785810	+	0.618468i	$0.787754\pi$
$578$	0	0
$579$	691.521i	1.19434i
$580$	0	0
$581$	−180.689	−0.310996
$582$	0	0
$583$	38.8666	0.0666666
$584$	0	0
$585$	0	0
$586$	0	0
$587$	422.169	0.719198	0.359599	−	0.933107i	$-0.382914\pi$
$587$	422.169	0.719198	0.359599	+	0.933107i	$0.382914\pi$
$588$	0	0
$589$	446.217	0.757584
$590$	0	0
$591$	534.588i	0.904548i
$592$	0	0
$593$	− 308.663i	− 0.520510i	−0.965540	−	0.260255i	$-0.916193\pi$
$593$	− 308.663i	− 0.520510i	0.965540	−	0.260255i	$-0.0838067\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 640.000i	− 1.07203i
$598$	0	0
$599$	462.196i	0.771612i	0.922580	+	0.385806i	$0.126077\pi$
$599$	462.196i	0.771612i	−0.922580	+	0.385806i	$0.873923\pi$
$600$	0	0
$601$	355.358	0.591277	0.295639	−	0.955300i	$-0.404468\pi$
$601$	355.358	0.591277	0.295639	+	0.955300i	$0.404468\pi$
$602$	0	0
$603$	−486.305	−0.806476
$604$	0	0
$605$	0	0
$606$	0	0
$607$	630.403	1.03856	0.519278	−	0.854605i	$-0.326201\pi$
$607$	630.403	1.03856	0.519278	+	0.854605i	$0.326201\pi$
$608$	0	0
$609$	−224.721	−0.369001
$610$	0	0
$611$	16.6891i	0.0273144i
$612$	0	0
$613$	− 812.525i	− 1.32549i	−0.748846	−	0.662745i	$-0.769392\pi$
$613$	− 812.525i	− 1.32549i	0.748846	−	0.662745i	$-0.230608\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	437.935i	0.709781i	0.934908	+	0.354891i	$0.115482\pi$
$617$	437.935i	0.709781i	−0.934908	+	0.354891i	$0.884518\pi$
$618$	0	0
$619$	− 770.250i	− 1.24435i	−0.782880	−	0.622173i	$-0.786250\pi$
$619$	− 770.250i	− 1.24435i	0.782880	−	0.622173i	$-0.213750\pi$
$620$	0	0
$621$	−216.276	−0.348270
$622$	0	0
$623$	573.702	0.920870
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−64.2597	−0.102487
$628$	0	0
$629$	−629.718	−1.00114
$630$	0	0
$631$	− 875.496i	− 1.38747i	−0.720228	−	0.693737i	$-0.755963\pi$
$631$	− 875.496i	− 1.38747i	0.720228	−	0.693737i	$-0.244037\pi$
$632$	0	0
$633$	− 357.390i	− 0.564597i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 11.0294i	− 0.0173146i
$638$	0	0
$639$	− 215.681i	− 0.337529i
$640$	0	0
$641$	842.571	1.31446	0.657232	−	0.753689i	$-0.271727\pi$
$641$	842.571	1.31446	0.657232	+	0.753689i	$0.271727\pi$
$642$	0	0
$643$	−1153.20	−1.79348	−0.896738	−	0.442563i	$-0.854069\pi$
$643$	−1153.20	−1.79348	−0.896738	+	0.442563i	$0.854069\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−355.751	−0.549847	−0.274924	−	0.961466i	$-0.588653\pi$
$647$	−355.751	−0.549847	−0.274924	+	0.961466i	$0.588653\pi$
$648$	0	0
$649$	132.669	0.204420
$650$	0	0
$651$	− 1535.41i	− 2.35854i
$652$	0	0
$653$	557.915i	0.854387i	0.904160	+	0.427194i	$0.140498\pi$
$653$	557.915i	0.854387i	−0.904160	+	0.427194i	$0.859502\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	752.715i	1.14569i
$658$	0	0
$659$	284.157i	0.431194i	0.976482	+	0.215597i	$0.0691697\pi$
$659$	284.157i	0.431194i	−0.976482	+	0.215597i	$0.930830\pi$
$660$	0	0
$661$	−716.735	−1.08432	−0.542160	−	0.840275i	$-0.682393\pi$
$661$	−716.735	−1.08432	−0.542160	+	0.840275i	$0.682393\pi$
$662$	0	0
$663$	42.9065	0.0647157
$664$	0	0
$665$	0	0
$666$	0	0
$667$	111.907	0.167776
$668$	0	0
$669$	−814.237	−1.21710
$670$	0	0
$671$	− 46.9464i	− 0.0699648i
$672$	0	0
$673$	− 695.378i	− 1.03325i	−0.856212	−	0.516625i	$-0.827188\pi$
$673$	− 695.378i	− 1.03325i	0.856212	−	0.516625i	$-0.172812\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 820.237i	− 1.21158i	−0.795626	−	0.605788i	$-0.792858\pi$
$677$	− 820.237i	− 1.21158i	0.795626	−	0.605788i	$-0.207142\pi$
$678$	0	0
$679$	− 333.176i	− 0.490686i
$680$	0	0
$681$	157.580	0.231396
$682$	0	0
$683$	335.508	0.491227	0.245613	−	0.969368i	$-0.421011\pi$
$683$	335.508	0.491227	0.245613	+	0.969368i	$0.421011\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	278.769	0.405777
$688$	0	0
$689$	10.2167	0.0148283
$690$	0	0
$691$	− 336.568i	− 0.487074i	−0.969892	−	0.243537i	$-0.921692\pi$
$691$	− 336.568i	− 0.487074i	0.969892	−	0.243537i	$-0.0783077\pi$
$692$	0	0
$693$	83.6068i	0.120645i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 989.378i	− 1.41948i
$698$	0	0
$699$	− 1168.09i	− 1.67108i
$700$	0	0
$701$	429.364	0.612502	0.306251	−	0.951951i	$-0.400925\pi$
$701$	429.364	0.612502	0.306251	+	0.951951i	$0.400925\pi$
$702$	0	0
$703$	247.911	0.352647
$704$	0	0
$705$	0	0
$706$	0	0
$707$	849.703	1.20184
$708$	0	0
$709$	1224.60	1.72722	0.863609	−	0.504162i	$-0.168199\pi$
$709$	1224.60	1.72722	0.863609	+	0.504162i	$0.168199\pi$
$710$	0	0
$711$	622.197i	0.875101i
$712$	0	0
$713$	764.604i	1.07238i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	163.226i	0.227651i
$718$	0	0
$719$	− 496.022i	− 0.689877i	−0.938625	−	0.344939i	$-0.887900\pi$
$719$	− 496.022i	− 0.689877i	0.938625	−	0.344939i	$-0.112100\pi$
$720$	0	0
$721$	−304.296	−0.422047
$722$	0	0
$723$	−513.883	−0.710764
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−152.843	−0.210238	−0.105119	−	0.994460i	$-0.533522\pi$
$727$	−152.843	−0.210238	−0.105119	+	0.994460i	$0.533522\pi$
$728$	0	0
$729$	−89.3808	−0.122607
$730$	0	0
$731$	− 47.9709i	− 0.0656237i
$732$	0	0
$733$	761.286i	1.03859i	0.854595	+	0.519295i	$0.173805\pi$
$733$	761.286i	1.03859i	−0.854595	+	0.519295i	$0.826195\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	159.619i	0.216580i
$738$	0	0
$739$	− 183.975i	− 0.248951i	−0.992223	−	0.124476i	$-0.960275\pi$
$739$	− 183.975i	− 0.248951i	0.992223	−	0.124476i	$-0.0397249\pi$
$740$	0	0
$741$	−16.8916	−0.0227957
$742$	0	0
$743$	−495.247	−0.666551	−0.333275	−	0.942830i	$-0.608154\pi$
$743$	−495.247	−0.666551	−0.333275	+	0.942830i	$0.608154\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	116.235	0.155602
$748$	0	0
$749$	−1030.13	−1.37534
$750$	0	0
$751$	800.059i	1.06533i	0.846328	+	0.532663i	$0.178809\pi$
$751$	800.059i	1.06533i	−0.846328	+	0.532663i	$0.821191\pi$
$752$	0	0
$753$	841.378i	1.11737i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 276.367i	− 0.365082i	−0.983198	−	0.182541i	$-0.941568\pi$
$757$	− 276.367i	− 0.365082i	0.983198	−	0.182541i	$-0.0584322\pi$
$758$	0	0
$759$	− 110.111i	− 0.145073i
$760$	0	0
$761$	891.207	1.17110	0.585550	−	0.810636i	$-0.300879\pi$
$761$	891.207	1.17110	0.585550	+	0.810636i	$0.300879\pi$
$762$	0	0
$763$	1679.77	2.20154
$764$	0	0
$765$	0	0
$766$	0	0
$767$	34.8740	0.0454681
$768$	0	0
$769$	835.430	1.08639	0.543193	−	0.839608i	$-0.317215\pi$
$769$	835.430	1.08639	0.543193	+	0.839608i	$0.317215\pi$
$770$	0	0
$771$	977.898i	1.26835i
$772$	0	0
$773$	− 213.522i	− 0.276225i	−0.990417	−	0.138112i	$-0.955897\pi$
$773$	− 213.522i	− 0.276225i	0.990417	−	0.138112i	$-0.0441035\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 853.050i	− 1.09788i
$778$	0	0
$779$	389.503i	0.500004i
$780$	0	0
$781$	−70.7926	−0.0906436
$782$	0	0
$783$	−93.1976	−0.119026
$784$	0	0
$785$	0	0
$786$	0	0
$787$	370.182	0.470371	0.235185	−	0.971951i	$-0.424430\pi$
$787$	370.182	0.470371	0.235185	+	0.971951i	$0.424430\pi$
$788$	0	0
$789$	624.531	0.791547
$790$	0	0
$791$	− 691.397i	− 0.874080i
$792$	0	0
$793$	− 12.3406i	− 0.0155619i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	274.426i	0.344323i	0.985069	+	0.172162i	$0.0550752\pi$
$797$	274.426i	0.344323i	−0.985069	+	0.172162i	$0.944925\pi$
$798$	0	0
$799$	− 844.414i	− 1.05684i
$800$	0	0
$801$	−369.056	−0.460744
$802$	0	0
$803$	247.063	0.307675
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−134.956	−0.167232
$808$	0	0
$809$	−665.214	−0.822266	−0.411133	−	0.911575i	$-0.634867\pi$
$809$	−665.214	−0.822266	−0.411133	+	0.911575i	$0.634867\pi$
$810$	0	0
$811$	360.665i	0.444717i	0.974965	+	0.222358i	$0.0713755\pi$
$811$	360.665i	0.444717i	−0.974965	+	0.222358i	$0.928624\pi$
$812$	0	0
$813$	1137.27i	1.39886i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.8854i	0.0231156i
$818$	0	0
$819$	21.9773i	0.0268344i
$820$	0	0
$821$	666.899	0.812301	0.406151	−	0.913806i	$-0.366871\pi$
$821$	666.899	0.812301	0.406151	+	0.913806i	$0.366871\pi$
$822$	0	0
$823$	122.433	0.148764	0.0743822	−	0.997230i	$-0.476302\pi$
$823$	122.433	0.148764	0.0743822	+	0.997230i	$0.476302\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1532.98	−1.85366	−0.926832	−	0.375477i	$-0.877479\pi$
$827$	−1532.98	−1.85366	−0.926832	+	0.375477i	$0.877479\pi$
$828$	0	0
$829$	−195.475	−0.235796	−0.117898	−	0.993026i	$-0.537616\pi$
$829$	−195.475	−0.235796	−0.117898	+	0.993026i	$0.537616\pi$
$830$	0	0
$831$	− 1739.47i	− 2.09322i
$832$	0	0
$833$	558.053i	0.669931i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 636.774i	− 0.760781i
$838$	0	0
$839$	1325.97i	1.58041i	0.612840	+	0.790207i	$0.290027\pi$
$839$	1325.97i	1.58041i	−0.612840	+	0.790207i	$0.709973\pi$
$840$	0	0
$841$	−792.777	−0.942660
$842$	0	0
$843$	−21.4532	−0.0254487
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1001.85	−1.18282
$848$	0	0
$849$	−646.407	−0.761375
$850$	0	0
$851$	424.801i	0.499179i
$852$	0	0
$853$	− 1055.28i	− 1.23714i	−0.785730	−	0.618570i	$-0.787712\pi$
$853$	− 1055.28i	− 1.23714i	0.785730	−	0.618570i	$-0.212288\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 155.378i	− 0.181304i	−0.995883	−	0.0906521i	$-0.971105\pi$
$857$	− 155.378i	− 0.181304i	0.995883	−	0.0906521i	$-0.0288951\pi$
$858$	0	0
$859$	226.033i	0.263136i	0.991307	+	0.131568i	$0.0420011\pi$
$859$	226.033i	0.263136i	−0.991307	+	0.131568i	$0.957999\pi$
$860$	0	0
$861$	1340.26	1.55664
$862$	0	0
$863$	930.702	1.07845	0.539225	−	0.842162i	$-0.318717\pi$
$863$	930.702	1.07845	0.539225	+	0.842162i	$0.318717\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1071.51	−1.23588
$868$	0	0
$869$	204.223	0.235009
$870$	0	0
$871$	41.9584i	0.0481727i
$872$	0	0
$873$	214.328i	0.245508i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 33.5217i	− 0.0382231i	−0.999817	−	0.0191115i	$-0.993916\pi$
$877$	− 33.5217i	− 0.0382231i	0.999817	−	0.0191115i	$-0.00608376\pi$
$878$	0	0
$879$	102.178i	0.116244i
$880$	0	0
$881$	−933.850	−1.05999	−0.529994	−	0.848001i	$-0.677806\pi$
$881$	−933.850	−1.05999	−0.529994	+	0.848001i	$0.677806\pi$
$882$	0	0
$883$	−542.308	−0.614166	−0.307083	−	0.951683i	$-0.599353\pi$
$883$	−542.308	−0.614166	−0.307083	+	0.951683i	$0.599353\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	714.720	0.805773	0.402886	−	0.915250i	$-0.368007\pi$
$887$	714.720	0.805773	0.402886	+	0.915250i	$0.368007\pi$
$888$	0	0
$889$	15.7044	0.0176652
$890$	0	0
$891$	180.159i	0.202199i
$892$	0	0
$893$	332.433i	0.372266i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 28.9443i	− 0.0322679i
$898$	0	0
$899$	329.484i	0.366500i
$900$	0	0
$901$	−516.932	−0.573731
$902$	0	0
$903$	64.9839	0.0719645
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−347.233	−0.382837	−0.191418	−	0.981509i	$-0.561309\pi$
$907$	−347.233	−0.382837	−0.191418	+	0.981509i	$0.561309\pi$
$908$	0	0
$909$	−546.604	−0.601324
$910$	0	0
$911$	1427.54i	1.56701i	0.621386	+	0.783504i	$0.286570\pi$
$911$	1427.54i	1.56701i	−0.621386	+	0.783504i	$0.713430\pi$
$912$	0	0
$913$	− 38.1517i	− 0.0417871i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1919.15i	2.09286i
$918$	0	0
$919$	569.162i	0.619327i	0.950846	+	0.309664i	$0.100216\pi$
$919$	569.162i	0.619327i	−0.950846	+	0.309664i	$0.899784\pi$
$920$	0	0
$921$	449.017	0.487532
$922$	0	0
$923$	−18.6089	−0.0201614
$924$	0	0
$925$	0	0
$926$	0	0
$927$	195.750	0.211165
$928$	0	0
$929$	−1535.96	−1.65335	−0.826675	−	0.562680i	$-0.809770\pi$
$929$	−1535.96	−1.65335	−0.826675	+	0.562680i	$0.809770\pi$
$930$	0	0
$931$	− 219.697i	− 0.235980i
$932$	0	0
$933$	463.489i	0.496773i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 338.721i	− 0.361496i	−0.983529	−	0.180748i	$-0.942148\pi$
$937$	− 338.721i	− 0.361496i	0.983529	−	0.180748i	$-0.0578518\pi$
$938$	0	0
$939$	835.210i	0.889468i
$940$	0	0
$941$	−1439.77	−1.53004	−0.765022	−	0.644004i	$-0.777272\pi$
$941$	−1439.77	−1.53004	−0.765022	+	0.644004i	$0.777272\pi$
$942$	0	0
$943$	−667.424	−0.707766
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−656.135	−0.692856	−0.346428	−	0.938077i	$-0.612606\pi$
$947$	−656.135	−0.692856	−0.346428	+	0.938077i	$0.612606\pi$
$948$	0	0
$949$	64.9443	0.0684344
$950$	0	0
$951$	1395.62i	1.46752i
$952$	0	0
$953$	436.675i	0.458211i	0.973402	+	0.229105i	$0.0735801\pi$
$953$	436.675i	0.458211i	−0.973402	+	0.229105i	$0.926420\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 47.4489i	− 0.0495809i
$958$	0	0
$959$	− 449.423i	− 0.468637i
$960$	0	0
$961$	−1290.20	−1.34256
$962$	0	0
$963$	662.671	0.688132
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−903.436	−0.934267	−0.467133	−	0.884187i	$-0.654713\pi$
$967$	−903.436	−0.934267	−0.467133	+	0.884187i	$0.654713\pi$
$968$	0	0
$969$	854.663	0.882005
$970$	0	0
$971$	1866.89i	1.92265i	0.275420	+	0.961324i	$0.411183\pi$
$971$	1866.89i	1.92265i	−0.275420	+	0.961324i	$0.588817\pi$
$972$	0	0
$973$	1070.56i	1.10026i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1073.95i	1.09923i	0.835418	+	0.549615i	$0.185226\pi$
$977$	1073.95i	1.09923i	−0.835418	+	0.549615i	$0.814774\pi$
$978$	0	0
$979$	121.135i	0.123733i
$980$	0	0
$981$	−1080.58	−1.10151
$982$	0	0
$983$	−534.114	−0.543351	−0.271675	−	0.962389i	$-0.587578\pi$
$983$	−534.114	−0.543351	−0.271675	+	0.962389i	$0.587578\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1143.89	1.15895
$988$	0	0
$989$	−32.3607	−0.0327206
$990$	0	0
$991$	520.419i	0.525146i	0.964912	+	0.262573i	$0.0845710\pi$
$991$	520.419i	0.525146i	−0.964912	+	0.262573i	$0.915429\pi$
$992$	0	0
$993$	619.502i	0.623869i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	457.680i	0.459057i	0.973302	+	0.229528i	$0.0737184\pi$
$997$	457.680i	0.459057i	−0.973302	+	0.229528i	$0.926282\pi$
$998$	0	0
$999$	− 353.781i	− 0.354135i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.h.n.1599.7		8
4.3	odd	2	inner	1600.3.h.n.1599.2		8
5.2	odd	4		320.3.b.c.191.1		4
5.3	odd	4		1600.3.b.s.1151.4		4
5.4	even	2	inner	1600.3.h.n.1599.1		8
8.3	odd	2		100.3.d.b.99.4		8
8.5	even	2		100.3.d.b.99.6		8
15.2	even	4		2880.3.e.e.2431.3		4
20.3	even	4		1600.3.b.s.1151.1		4
20.7	even	4		320.3.b.c.191.4		4
20.19	odd	2	inner	1600.3.h.n.1599.8		8
24.5	odd	2		900.3.f.e.199.3		8
24.11	even	2		900.3.f.e.199.5		8
40.3	even	4		100.3.b.f.51.4		4
40.13	odd	4		100.3.b.f.51.3		4
40.19	odd	2		100.3.d.b.99.5		8
40.27	even	4		20.3.b.a.11.1	✓	4
40.29	even	2		100.3.d.b.99.3		8
40.37	odd	4		20.3.b.a.11.2	yes	4
60.47	odd	4		2880.3.e.e.2431.4		4
80.27	even	4		1280.3.g.e.1151.7		8
80.37	odd	4		1280.3.g.e.1151.1		8
80.67	even	4		1280.3.g.e.1151.2		8
80.77	odd	4		1280.3.g.e.1151.8		8
120.29	odd	2		900.3.f.e.199.6		8
120.53	even	4		900.3.c.k.451.2		4
120.59	even	2		900.3.f.e.199.4		8
120.77	even	4		180.3.c.a.91.3		4
120.83	odd	4		900.3.c.k.451.1		4
120.107	odd	4		180.3.c.a.91.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.3.b.a.11.1	✓	4	40.27	even	4
20.3.b.a.11.2	yes	4	40.37	odd	4
100.3.b.f.51.3		4	40.13	odd	4
100.3.b.f.51.4		4	40.3	even	4
100.3.d.b.99.3		8	40.29	even	2
100.3.d.b.99.4		8	8.3	odd	2
100.3.d.b.99.5		8	40.19	odd	2
100.3.d.b.99.6		8	8.5	even	2
180.3.c.a.91.3		4	120.77	even	4
180.3.c.a.91.4		4	120.107	odd	4
320.3.b.c.191.1		4	5.2	odd	4
320.3.b.c.191.4		4	20.7	even	4
900.3.c.k.451.1		4	120.83	odd	4
900.3.c.k.451.2		4	120.53	even	4
900.3.f.e.199.3		8	24.5	odd	2
900.3.f.e.199.4		8	120.59	even	2
900.3.f.e.199.5		8	24.11	even	2
900.3.f.e.199.6		8	120.29	odd	2
1280.3.g.e.1151.1		8	80.37	odd	4
1280.3.g.e.1151.2		8	80.67	even	4
1280.3.g.e.1151.7		8	80.27	even	4
1280.3.g.e.1151.8		8	80.77	odd	4
1600.3.b.s.1151.1		4	20.3	even	4
1600.3.b.s.1151.4		4	5.3	odd	4
1600.3.h.n.1599.1		8	5.4	even	2	inner
1600.3.h.n.1599.2		8	4.3	odd	2	inner
1600.3.h.n.1599.7		8	1.1	even	1	trivial
1600.3.h.n.1599.8		8	20.19	odd	2	inner
2880.3.e.e.2431.3		4	15.2	even	4
2880.3.e.e.2431.4		4	60.47	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.80423 q^{3} -8.50651 q^{7} +5.47214 q^{9} +O(q^{10})\)	\(q+3.80423 q^{3} -8.50651 q^{7} +5.47214 q^{9} -1.79611i q^{11} -0.472136i q^{13} +23.8885i q^{17} -9.40456i q^{19} -32.3607 q^{21} +16.1150 q^{23} -13.4208 q^{27} +6.94427 q^{29} +47.4468i q^{31} -6.83282i q^{33} +26.3607i q^{37} -1.79611i q^{39} -41.4164 q^{41} -2.00811 q^{43} -35.3481 q^{47} +23.3607 q^{49} +90.8774i q^{51} +21.6393i q^{53} -35.7771i q^{57} +73.8644i q^{59} +26.1378 q^{61} -46.5488 q^{63} -88.8693 q^{67} +61.3050 q^{69} -39.4144i q^{71} +137.554i q^{73} +15.2786i q^{77} +113.703i q^{79} -100.305 q^{81} +21.2412 q^{83} +26.4176 q^{87} -67.4427 q^{89} +4.01623i q^{91} +180.498i q^{93} +39.1672i q^{97} -9.82857i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 80 q^{21} - 16 q^{29} - 224 q^{41} + 8 q^{49} - 256 q^{61} + 240 q^{69} - 552 q^{81} + 176 q^{89}+O(q^{100}) \) 8 * q + 8 * q^9 - 80 * q^21 - 16 * q^29 - 224 * q^41 + 8 * q^49 - 256 * q^61 + 240 * q^69 - 552 * q^81 + 176 * q^89

Introduction

L-functions

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