LMFDB - Embedded newform 1600.3.b.s.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1151,1600)]

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, 0]))

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

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

chi := DirichletCharacter("1600.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.b (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:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$-0.309017 - 0.951057i$ of defining polynomial
Character	$\chi$	$=$	1600.1151
Dual form			1600.3.b.s.1151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.80423i q^{3} -8.50651i q^{7} -5.47214 q^{9} +O(q^{10})$	$q-3.80423i q^{3} -8.50651i q^{7} -5.47214 q^{9} +1.79611i q^{11} +0.472136 q^{13} +23.8885 q^{17} -9.40456i q^{19} -32.3607 q^{21} -16.1150i q^{23} -13.4208i q^{27} -6.94427 q^{29} -47.4468i q^{31} +6.83282 q^{33} +26.3607 q^{37} -1.79611i q^{39} -41.4164 q^{41} +2.00811i q^{43} -35.3481i q^{47} -23.3607 q^{49} -90.8774i q^{51} -21.6393 q^{53} -35.7771 q^{57} +73.8644i q^{59} +26.1378 q^{61} +46.5488i q^{63} -88.8693i q^{67} -61.3050 q^{69} +39.4144i q^{71} -137.554 q^{73} +15.2786 q^{77} +113.703i q^{79} -100.305 q^{81} -21.2412i q^{83} +26.4176i q^{87} +67.4427 q^{89} -4.01623i q^{91} -180.498 q^{93} +39.1672 q^{97} -9.82857i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 16 q^{13} + 24 q^{17} - 40 q^{21} + 8 q^{29} - 80 q^{33} + 16 q^{37} - 112 q^{41} - 4 q^{49} - 176 q^{53} - 128 q^{61} - 120 q^{69} - 264 q^{73} + 240 q^{77} - 276 q^{81} - 88 q^{89} - 400 q^{93} + 264 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 - 16 * q^13 + 24 * q^17 - 40 * q^21 + 8 * q^29 - 80 * q^33 + 16 * q^37 - 112 * q^41 - 4 * q^49 - 176 * q^53 - 128 * q^61 - 120 * q^69 - 264 * q^73 + 240 * q^77 - 276 * q^81 - 88 * q^89 - 400 * q^93 + 264 * q^97

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
$3$	− 3.80423i	− 1.26808i	−0.773302	−	0.634038i	$-0.781396\pi$
$3$	− 3.80423i	− 1.26808i	0.773302	−	0.634038i	$-0.218604\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 8.50651i	− 1.21522i	−0.794237	−	0.607608i	$-0.792129\pi$
$7$	− 8.50651i	− 1.21522i	0.794237	−	0.607608i	$-0.207871\pi$
$8$	0	0
$9$	−5.47214	−0.608015
$10$	0	0
$11$	1.79611i	0.163283i	0.996662	+	0.0816415i	$0.0260162\pi$
$11$	1.79611i	0.163283i	−0.996662	+	0.0816415i	$0.973984\pi$
$12$	0	0
$13$	0.472136	0.0363182	0.0181591	−	0.999835i	$-0.494219\pi$
$13$	0.472136	0.0363182	0.0181591	+	0.999835i	$0.494219\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	23.8885	1.40521	0.702604	−	0.711581i	$-0.252020\pi$
$17$	23.8885	1.40521	0.702604	+	0.711581i	$0.252020\pi$
$18$	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
$23$	− 16.1150i	− 0.700650i	−0.936628	−	0.350325i	$-0.886071\pi$
$23$	− 16.1150i	− 0.700650i	0.936628	−	0.350325i	$-0.113929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 13.4208i	− 0.497066i
$28$	0	0
$29$	−6.94427	−0.239458	−0.119729	−	0.992807i	$-0.538203\pi$
$29$	−6.94427	−0.239458	−0.119729	+	0.992807i	$0.538203\pi$
$30$	0	0
$31$	− 47.4468i	− 1.53054i	−0.643708	−	0.765271i	$-0.722605\pi$
$31$	− 47.4468i	− 1.53054i	0.643708	−	0.765271i	$-0.277395\pi$
$32$	0	0
$33$	6.83282	0.207055
$34$	0	0
$35$	0	0
$36$	0	0
$37$	26.3607	0.712451	0.356225	−	0.934400i	$-0.384064\pi$
$37$	26.3607	0.712451	0.356225	+	0.934400i	$0.384064\pi$
$38$	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
$43$	2.00811i	0.0467003i	0.999727	+	0.0233502i	$0.00743326\pi$
$43$	2.00811i	0.0467003i	−0.999727	+	0.0233502i	$0.992567\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 35.3481i	− 0.752087i	−0.926602	−	0.376044i	$-0.877284\pi$
$47$	− 35.3481i	− 0.752087i	0.926602	−	0.376044i	$-0.122716\pi$
$48$	0	0
$49$	−23.3607	−0.476749
$50$	0	0
$51$	− 90.8774i	− 1.78191i
$52$	0	0
$53$	−21.6393	−0.408289	−0.204145	−	0.978941i	$-0.565441\pi$
$53$	−21.6393	−0.408289	−0.204145	+	0.978941i	$0.565441\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−35.7771	−0.627668
$58$	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
$63$	46.5488i	0.738869i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 88.8693i	− 1.32641i	−0.748439	−	0.663204i	$-0.769196\pi$
$67$	− 88.8693i	− 1.32641i	0.748439	−	0.663204i	$-0.230804\pi$
$68$	0	0
$69$	−61.3050	−0.888478
$70$	0	0
$71$	39.4144i	0.555132i	0.960707	+	0.277566i	$0.0895277\pi$
$71$	39.4144i	0.555132i	−0.960707	+	0.277566i	$0.910472\pi$
$72$	0	0
$73$	−137.554	−1.88430	−0.942152	−	0.335186i	$-0.891201\pi$
$73$	−137.554	−1.88430	−0.942152	+	0.335186i	$0.891201\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.2786	0.198424
$78$	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
$83$	− 21.2412i	− 0.255919i	−0.991779	−	0.127959i	$-0.959157\pi$
$83$	− 21.2412i	− 0.255919i	0.991779	−	0.127959i	$-0.0408427\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	26.4176i	0.303650i
$88$	0	0
$89$	67.4427	0.757783	0.378892	−	0.925441i	$-0.376305\pi$
$89$	67.4427	0.757783	0.378892	+	0.925441i	$0.376305\pi$
$90$	0	0
$91$	− 4.01623i	− 0.0441344i
$92$	0	0
$93$	−180.498	−1.94084
$94$	0	0
$95$	0	0
$96$	0	0
$97$	39.1672	0.403785	0.201893	−	0.979408i	$-0.435291\pi$
$97$	39.1672	0.403785	0.201893	+	0.979408i	$0.435291\pi$
$98$	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
$103$	− 35.7721i	− 0.347302i	−0.984807	−	0.173651i	$-0.944444\pi$
$103$	− 35.7721i	− 0.347302i	0.984807	−	0.173651i	$-0.0555565\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	121.099i	1.13177i	0.824485	+	0.565884i	$0.191465\pi$
$107$	121.099i	1.13177i	−0.824485	+	0.565884i	$0.808535\pi$
$108$	0	0
$109$	197.469	1.81164	0.905821	−	0.423660i	$-0.139255\pi$
$109$	197.469	1.81164	0.905821	+	0.423660i	$0.139255\pi$
$110$	0	0
$111$	− 100.282i	− 0.903441i
$112$	0	0
$113$	−81.2786	−0.719280	−0.359640	−	0.933091i	$-0.617100\pi$
$113$	−81.2786	−0.719280	−0.359640	+	0.933091i	$0.617100\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.58359	−0.0220820
$118$	0	0
$119$	− 203.208i	− 1.70763i
$120$	0	0
$121$	117.774	0.973339
$122$	0	0
$123$	157.557i	1.28095i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.84616i	− 0.0145367i	−0.999974	−	0.00726834i	$-0.997686\pi$
$127$	− 1.84616i	− 0.0145367i	0.999974	−	0.00726834i	$-0.00231361\pi$
$128$	0	0
$129$	7.63932	0.0592195
$130$	0	0
$131$	225.609i	1.72221i	0.508428	+	0.861105i	$0.330227\pi$
$131$	225.609i	1.72221i	−0.508428	+	0.861105i	$0.669773\pi$
$132$	0	0
$133$	−80.0000	−0.601504
$134$	0	0
$135$	0	0
$136$	0	0
$137$	52.8328	0.385641	0.192820	−	0.981234i	$-0.438236\pi$
$137$	52.8328	0.385641	0.192820	+	0.981234i	$0.438236\pi$
$138$	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
$143$	0.848009i	0.00593013i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	88.8693i	0.604553i
$148$	0	0
$149$	−132.971	−0.892420	−0.446210	−	0.894928i	$-0.647227\pi$
$149$	−132.971	−0.892420	−0.446210	+	0.894928i	$0.647227\pi$
$150$	0	0
$151$	151.221i	1.00146i	0.865603	+	0.500732i	$0.166936\pi$
$151$	151.221i	1.00146i	−0.865603	+	0.500732i	$0.833064\pi$
$152$	0	0
$153$	−130.721	−0.854388
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−36.7477	−0.234062	−0.117031	−	0.993128i	$-0.537338\pi$
$157$	−36.7477	−0.234062	−0.117031	+	0.993128i	$0.537338\pi$
$158$	0	0
$159$	82.3209i	0.517741i
$160$	0	0
$161$	−137.082	−0.851441
$162$	0	0
$163$	− 302.854i	− 1.85800i	−0.370079	−	0.929000i	$-0.620669\pi$
$163$	− 302.854i	− 1.85800i	0.370079	−	0.929000i	$-0.379331\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	99.3839i	0.595113i	0.954704	+	0.297557i	$0.0961717\pi$
$167$	99.3839i	0.595113i	−0.954704	+	0.297557i	$0.903828\pi$
$168$	0	0
$169$	−168.777	−0.998681
$170$	0	0
$171$	51.4631i	0.300954i
$172$	0	0
$173$	−181.639	−1.04994	−0.524969	−	0.851121i	$-0.675923\pi$
$173$	−181.639	−1.04994	−0.524969	+	0.851121i	$0.675923\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	280.997	1.58755
$178$	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
$183$	− 99.4340i	− 0.543355i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	42.9065i	0.229447i
$188$	0	0
$189$	−114.164	−0.604043
$190$	0	0
$191$	324.095i	1.69683i	0.529328	+	0.848417i	$0.322444\pi$
$191$	324.095i	1.69683i	−0.529328	+	0.848417i	$0.677556\pi$
$192$	0	0
$193$	−181.777	−0.941850	−0.470925	−	0.882173i	$-0.656080\pi$
$193$	−181.777	−0.941850	−0.470925	+	0.882173i	$0.656080\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	140.525	0.713324	0.356662	−	0.934234i	$-0.383915\pi$
$197$	140.525	0.713324	0.356662	+	0.934234i	$0.383915\pi$
$198$	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
$203$	59.0715i	0.290993i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	88.1833i	0.426006i
$208$	0	0
$209$	16.8916	0.0808213
$210$	0	0
$211$	93.9455i	0.445240i	0.974905	+	0.222620i	$0.0714608\pi$
$211$	93.9455i	0.445240i	−0.974905	+	0.222620i	$0.928539\pi$
$212$	0	0
$213$	149.941	0.703949
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−403.607	−1.85994
$218$	0	0
$219$	523.287i	2.38944i
$220$	0	0
$221$	11.2786	0.0510346
$222$	0	0
$223$	214.035i	0.959797i	0.877324	+	0.479899i	$0.159327\pi$
$223$	214.035i	0.959797i	−0.877324	+	0.479899i	$0.840673\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	41.4225i	0.182478i	0.995829	+	0.0912389i	$0.0290827\pi$
$227$	41.4225i	0.182478i	−0.995829	+	0.0912389i	$0.970917\pi$
$228$	0	0
$229$	−73.2786	−0.319994	−0.159997	−	0.987117i	$-0.551148\pi$
$229$	−73.2786	−0.319994	−0.159997	+	0.987117i	$0.551148\pi$
$230$	0	0
$231$	− 58.1234i	− 0.251616i
$232$	0	0
$233$	307.050	1.31781	0.658905	−	0.752227i	$-0.271020\pi$
$233$	307.050	1.31781	0.658905	+	0.752227i	$0.271020\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	432.551	1.82511
$238$	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
$243$	260.796i	1.07323i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.44023i	− 0.0179767i
$248$	0	0
$249$	−80.8065	−0.324524
$250$	0	0
$251$	− 221.169i	− 0.881152i	−0.897715	−	0.440576i	$-0.854774\pi$
$251$	− 221.169i	− 0.881152i	0.897715	−	0.440576i	$-0.145226\pi$
$252$	0	0
$253$	28.9443	0.114404
$254$	0	0
$255$	0	0
$256$	0	0
$257$	257.056	1.00022	0.500108	−	0.865963i	$-0.333293\pi$
$257$	257.056	1.00022	0.500108	+	0.865963i	$0.333293\pi$
$258$	0	0
$259$	− 224.237i	− 0.865781i
$260$	0	0
$261$	38.0000	0.145594
$262$	0	0
$263$	− 164.168i	− 0.624212i	−0.950047	−	0.312106i	$-0.898966\pi$
$263$	− 164.168i	− 0.624212i	0.950047	−	0.312106i	$-0.101034\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 256.567i	− 0.960926i
$268$	0	0
$269$	35.4752	0.131878	0.0659391	−	0.997824i	$-0.478996\pi$
$269$	35.4752	0.131878	0.0659391	+	0.997824i	$0.478996\pi$
$270$	0	0
$271$	− 298.950i	− 1.10314i	−0.834130	−	0.551568i	$-0.814030\pi$
$271$	− 298.950i	− 1.10314i	0.834130	−	0.551568i	$-0.185970\pi$
$272$	0	0
$273$	−15.2786	−0.0559657
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−457.246	−1.65071	−0.825354	−	0.564616i	$-0.809024\pi$
$277$	−457.246	−1.65071	−0.825354	+	0.564616i	$0.809024\pi$
$278$	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
$283$	169.918i	0.600418i	0.953874	+	0.300209i	$0.0970563\pi$
$283$	169.918i	0.600418i	−0.953874	+	0.300209i	$0.902944\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	352.309i	1.22756i
$288$	0	0
$289$	281.663	0.974611
$290$	0	0
$291$	− 149.001i	− 0.512030i
$292$	0	0
$293$	−26.8591	−0.0916694	−0.0458347	−	0.998949i	$-0.514595\pi$
$293$	−26.8591	−0.0916694	−0.0458347	+	0.998949i	$0.514595\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	24.1052	0.0811624
$298$	0	0
$299$	− 7.60845i	− 0.0254463i
$300$	0	0
$301$	17.0820	0.0567510
$302$	0	0
$303$	379.999i	1.25412i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	118.031i	0.384466i	0.981349	+	0.192233i	$0.0615730\pi$
$307$	118.031i	0.384466i	−0.981349	+	0.192233i	$0.938427\pi$
$308$	0	0
$309$	−136.085	−0.440405
$310$	0	0
$311$	− 121.835i	− 0.391753i	−0.980629	−	0.195877i	$-0.937245\pi$
$311$	− 121.835i	− 0.391753i	0.980629	−	0.195877i	$-0.0627552\pi$
$312$	0	0
$313$	−219.548	−0.701431	−0.350716	−	0.936482i	$-0.614062\pi$
$313$	−219.548	−0.701431	−0.350716	+	0.936482i	$0.614062\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	366.859	1.15728	0.578642	−	0.815582i	$-0.303583\pi$
$317$	366.859	1.15728	0.578642	+	0.815582i	$0.303583\pi$
$318$	0	0
$319$	− 12.4727i	− 0.0390993i
$320$	0	0
$321$	460.689	1.43517
$322$	0	0
$323$	− 224.661i	− 0.695546i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 751.217i	− 2.29730i
$328$	0	0
$329$	−300.689	−0.913948
$330$	0	0
$331$	− 162.846i	− 0.491981i	−0.969272	−	0.245990i	$-0.920887\pi$
$331$	− 162.846i	− 0.491981i	0.969272	−	0.245990i	$-0.0791132\pi$
$332$	0	0
$333$	−144.249	−0.433181
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−17.1084	−0.0507666	−0.0253833	−	0.999678i	$-0.508081\pi$
$337$	−17.1084	−0.0507666	−0.0253833	+	0.999678i	$0.508081\pi$
$338$	0	0
$339$	309.202i	0.912101i
$340$	0	0
$341$	85.2198	0.249911
$342$	0	0
$343$	− 218.101i	− 0.635863i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 167.498i	− 0.482703i	−0.970438	−	0.241351i	$-0.922409\pi$
$347$	− 167.498i	− 0.482703i	0.970438	−	0.241351i	$-0.0775906\pi$
$348$	0	0
$349$	483.495	1.38537	0.692687	−	0.721239i	$-0.256427\pi$
$349$	483.495	1.38537	0.692687	+	0.721239i	$0.256427\pi$
$350$	0	0
$351$	− 6.33644i	− 0.0180525i
$352$	0	0
$353$	−307.994	−0.872504	−0.436252	−	0.899825i	$-0.643694\pi$
$353$	−307.994	−0.872504	−0.436252	+	0.899825i	$0.643694\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−773.050	−2.16540
$358$	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
$363$	− 448.039i	− 1.23427i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 517.325i	− 1.40960i	−0.709404	−	0.704802i	$-0.751036\pi$
$367$	− 517.325i	− 1.40960i	0.709404	−	0.704802i	$-0.248964\pi$
$368$	0	0
$369$	226.636	0.614190
$370$	0	0
$371$	184.075i	0.496159i
$372$	0	0
$373$	−88.3545	−0.236875	−0.118438	−	0.992961i	$-0.537789\pi$
$373$	−88.3545	−0.236875	−0.118438	+	0.992961i	$0.537789\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.27864	−0.00869666
$378$	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
$383$	− 431.612i	− 1.12692i	−0.826142	−	0.563462i	$-0.809469\pi$
$383$	− 431.612i	− 1.12692i	0.826142	−	0.563462i	$-0.190531\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10.9887i	− 0.0283945i
$388$	0	0
$389$	296.354	0.761837	0.380918	−	0.924609i	$-0.375608\pi$
$389$	296.354	0.761837	0.380918	+	0.924609i	$0.375608\pi$
$390$	0	0
$391$	− 384.963i	− 0.984560i
$392$	0	0
$393$	858.269	2.18389
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−86.1904	−0.217104	−0.108552	−	0.994091i	$-0.534621\pi$
$397$	−86.1904	−0.217104	−0.108552	+	0.994091i	$0.534621\pi$
$398$	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
$403$	− 22.4014i	− 0.0555865i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	47.3467i	0.116331i
$408$	0	0
$409$	63.4102	0.155037	0.0775186	−	0.996991i	$-0.475300\pi$
$409$	63.4102	0.155037	0.0775186	+	0.996991i	$0.475300\pi$
$410$	0	0
$411$	− 200.988i	− 0.489022i
$412$	0	0
$413$	628.328	1.52138
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−478.768	−1.14812
$418$	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
$423$	193.430i	0.457280i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 222.341i	− 0.520705i
$428$	0	0
$429$	3.22602	0.00751986
$430$	0	0
$431$	− 375.882i	− 0.872117i	−0.899918	−	0.436058i	$-0.856374\pi$
$431$	− 375.882i	− 0.872117i	0.899918	−	0.436058i	$-0.143626\pi$
$432$	0	0
$433$	368.164	0.850263	0.425132	−	0.905131i	$-0.360228\pi$
$433$	368.164	0.850263	0.425132	+	0.905131i	$0.360228\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−151.554	−0.346806
$438$	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
$443$	279.181i	0.630205i	0.949058	+	0.315102i	$0.102039\pi$
$443$	279.181i	0.630205i	−0.949058	+	0.315102i	$0.897961\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	505.850i	1.13166i
$448$	0	0
$449$	756.079	1.68392	0.841959	−	0.539542i	$-0.181403\pi$
$449$	756.079	1.68392	0.841959	+	0.539542i	$0.181403\pi$
$450$	0	0
$451$	− 74.3885i	− 0.164941i
$452$	0	0
$453$	575.279	1.26993
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−285.672	−0.625103	−0.312551	−	0.949901i	$-0.601184\pi$
$457$	−285.672	−0.625103	−0.312551	+	0.949901i	$0.601184\pi$
$458$	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
$463$	− 630.603i	− 1.36199i	−0.732286	−	0.680997i	$-0.761547\pi$
$463$	− 630.603i	− 1.36199i	0.732286	−	0.680997i	$-0.238453\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 496.010i	− 1.06212i	−0.847334	−	0.531060i	$-0.821794\pi$
$467$	− 496.010i	− 1.06212i	0.847334	−	0.531060i	$-0.178206\pi$
$468$	0	0
$469$	−755.967	−1.61187
$470$	0	0
$471$	139.796i	0.296808i
$472$	0	0
$473$	−3.60680	−0.00762537
$474$	0	0
$475$	0	0
$476$	0	0
$477$	118.413	0.248246
$478$	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
$483$	521.491i	1.07969i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	626.363i	1.28617i	0.765796	+	0.643084i	$0.222345\pi$
$487$	626.363i	1.28617i	−0.765796	+	0.643084i	$0.777655\pi$
$488$	0	0
$489$	−1152.13	−2.35608
$490$	0	0
$491$	22.3013i	0.0454201i	0.999742	+	0.0227100i	$0.00722945\pi$
$491$	22.3013i	0.0454201i	−0.999742	+	0.0227100i	$0.992771\pi$
$492$	0	0
$493$	−165.889	−0.336488
$494$	0	0
$495$	0	0
$496$	0	0
$497$	335.279	0.674605
$498$	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
$503$	780.853i	1.55239i	0.630492	+	0.776196i	$0.282853\pi$
$503$	780.853i	1.55239i	−0.630492	+	0.776196i	$0.717147\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	642.066i	1.26640i
$508$	0	0
$509$	288.950	0.567683	0.283841	−	0.958871i	$-0.408391\pi$
$509$	288.950	0.567683	0.283841	+	0.958871i	$0.408391\pi$
$510$	0	0
$511$	1170.11i	2.28984i
$512$	0	0
$513$	−126.217	−0.246036
$514$	0	0
$515$	0	0
$516$	0	0
$517$	63.4891	0.122803
$518$	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
$523$	− 367.962i	− 0.703560i	−0.936083	−	0.351780i	$-0.885577\pi$
$523$	− 367.962i	− 0.703560i	0.936083	−	0.351780i	$-0.114423\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1133.44i	− 2.15073i
$528$	0	0
$529$	269.308	0.509089
$530$	0	0
$531$	− 404.196i	− 0.761198i
$532$	0	0
$533$	−19.5542	−0.0366870
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−992.551	−1.84833
$538$	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
$543$	600.220i	1.10538i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	97.8499i	0.178885i	0.995992	+	0.0894423i	$0.0285085\pi$
$547$	97.8499i	0.178885i	−0.995992	+	0.0894423i	$0.971492\pi$
$548$	0	0
$549$	−143.029	−0.260527
$550$	0	0
$551$	65.3078i	0.118526i
$552$	0	0
$553$	967.214	1.74903
$554$	0	0
$555$	0	0
$556$	0	0
$557$	896.302	1.60916	0.804580	−	0.593845i	$-0.202391\pi$
$557$	896.302	1.60916	0.804580	+	0.593845i	$0.202391\pi$
$558$	0	0
$559$	0.948103i	0.00169607i
$560$	0	0
$561$	163.226	0.290955
$562$	0	0
$563$	− 771.186i	− 1.36978i	−0.728647	−	0.684890i	$-0.759850\pi$
$563$	− 771.186i	− 1.36978i	0.728647	−	0.684890i	$-0.240150\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	853.245i	1.50484i
$568$	0	0
$569$	8.74767	0.0153738	0.00768688	−	0.999970i	$-0.497553\pi$
$569$	8.74767	0.0153738	0.00768688	+	0.999970i	$0.497553\pi$
$570$	0	0
$571$	511.138i	0.895164i	0.894243	+	0.447582i	$0.147715\pi$
$571$	511.138i	0.895164i	−0.894243	+	0.447582i	$0.852285\pi$
$572$	0	0
$573$	1232.93	2.15171
$574$	0	0
$575$	0	0
$576$	0	0
$577$	713.712	1.23694	0.618468	−	0.785810i	$-0.287754\pi$
$577$	713.712	1.23694	0.618468	+	0.785810i	$0.287754\pi$
$578$	0	0
$579$	691.521i	1.19434i
$580$	0	0
$581$	−180.689	−0.310996
$582$	0	0
$583$	− 38.8666i	− 0.0666666i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	422.169i	0.719198i	0.933107	+	0.359599i	$0.117086\pi$
$587$	422.169i	0.719198i	−0.933107	+	0.359599i	$0.882914\pi$
$588$	0	0
$589$	−446.217	−0.757584
$590$	0	0
$591$	− 534.588i	− 0.904548i
$592$	0	0
$593$	308.663	0.520510	0.260255	−	0.965540i	$-0.416193\pi$
$593$	308.663	0.520510	0.260255	+	0.965540i	$0.416193\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−640.000	−1.07203
$598$	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
$603$	486.305i	0.806476i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	630.403i	1.03856i	0.854605	+	0.519278i	$0.173799\pi$
$607$	630.403i	1.03856i	−0.854605	+	0.519278i	$0.826201\pi$
$608$	0	0
$609$	224.721	0.369001
$610$	0	0
$611$	− 16.6891i	− 0.0273144i
$612$	0	0
$613$	812.525	1.32549	0.662745	−	0.748846i	$-0.269392\pi$
$613$	812.525	1.32549	0.662745	+	0.748846i	$0.269392\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	437.935	0.709781	0.354891	−	0.934908i	$-0.384518\pi$
$617$	437.935	0.709781	0.354891	+	0.934908i	$0.384518\pi$
$618$	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
$623$	− 573.702i	− 0.920870i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 64.2597i	− 0.102487i
$628$	0	0
$629$	629.718	1.00114
$630$	0	0
$631$	875.496i	1.38747i	0.720228	+	0.693737i	$0.244037\pi$
$631$	875.496i	1.38747i	−0.720228	+	0.693737i	$0.755963\pi$
$632$	0	0
$633$	357.390	0.564597
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−11.0294	−0.0173146
$638$	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
$643$	1153.20i	1.79348i	0.442563	+	0.896738i	$0.354069\pi$
$643$	1153.20i	1.79348i	−0.442563	+	0.896738i	$0.645931\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 355.751i	− 0.549847i	−0.961466	−	0.274924i	$-0.911347\pi$
$647$	− 355.751i	− 0.549847i	0.961466	−	0.274924i	$-0.0886526\pi$
$648$	0	0
$649$	−132.669	−0.204420
$650$	0	0
$651$	1535.41i	2.35854i
$652$	0	0
$653$	−557.915	−0.854387	−0.427194	−	0.904160i	$-0.640498\pi$
$653$	−557.915	−0.854387	−0.427194	+	0.904160i	$0.640498\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	752.715	1.14569
$658$	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
$663$	− 42.9065i	− 0.0647157i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	111.907i	0.167776i
$668$	0	0
$669$	814.237	1.21710
$670$	0	0
$671$	46.9464i	0.0699648i
$672$	0	0
$673$	695.378	1.03325	0.516625	−	0.856212i	$-0.327188\pi$
$673$	695.378	1.03325	0.516625	+	0.856212i	$0.327188\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−820.237	−1.21158	−0.605788	−	0.795626i	$-0.707142\pi$
$677$	−820.237	−1.21158	−0.605788	+	0.795626i	$0.707142\pi$
$678$	0	0
$679$	− 333.176i	− 0.490686i
$680$	0	0
$681$	157.580	0.231396
$682$	0	0
$683$	− 335.508i	− 0.491227i	−0.969368	−	0.245613i	$-0.921011\pi$
$683$	− 335.508i	− 0.491227i	0.969368	−	0.245613i	$-0.0789894\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	278.769i	0.405777i
$688$	0	0
$689$	−10.2167	−0.0148283
$690$	0	0
$691$	336.568i	0.487074i	0.969892	+	0.243537i	$0.0783077\pi$
$691$	336.568i	0.487074i	−0.969892	+	0.243537i	$0.921692\pi$
$692$	0	0
$693$	−83.6068	−0.120645
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−989.378	−1.41948
$698$	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
$703$	− 247.911i	− 0.352647i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	849.703i	1.20184i
$708$	0	0
$709$	−1224.60	−1.72722	−0.863609	−	0.504162i	$-0.831801\pi$
$709$	−1224.60	−1.72722	−0.863609	+	0.504162i	$0.831801\pi$
$710$	0	0
$711$	− 622.197i	− 0.875101i
$712$	0	0
$713$	−764.604	−1.07238
$714$	0	0
$715$	0	0
$716$	0	0
$717$	163.226	0.227651
$718$	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
$723$	513.883i	0.710764i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 152.843i	− 0.210238i	−0.994460	−	0.105119i	$-0.966478\pi$
$727$	− 152.843i	− 0.210238i	0.994460	−	0.105119i	$-0.0335224\pi$
$728$	0	0
$729$	89.3808	0.122607
$730$	0	0
$731$	47.9709i	0.0656237i
$732$	0	0
$733$	−761.286	−1.03859	−0.519295	−	0.854595i	$-0.673805\pi$
$733$	−761.286	−1.03859	−0.519295	+	0.854595i	$0.673805\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	159.619	0.216580
$738$	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
$743$	495.247i	0.666551i	0.942830	+	0.333275i	$0.108154\pi$
$743$	495.247i	0.666551i	−0.942830	+	0.333275i	$0.891846\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	116.235i	0.155602i
$748$	0	0
$749$	1030.13	1.37534
$750$	0	0
$751$	− 800.059i	− 1.06533i	−0.846328	−	0.532663i	$-0.821191\pi$
$751$	− 800.059i	− 1.06533i	0.846328	−	0.532663i	$-0.178809\pi$
$752$	0	0
$753$	−841.378	−1.11737
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−276.367	−0.365082	−0.182541	−	0.983198i	$-0.558432\pi$
$757$	−276.367	−0.365082	−0.182541	+	0.983198i	$0.558432\pi$
$758$	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
$763$	− 1679.77i	− 2.20154i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	34.8740i	0.0454681i
$768$	0	0
$769$	−835.430	−1.08639	−0.543193	−	0.839608i	$-0.682785\pi$
$769$	−835.430	−1.08639	−0.543193	+	0.839608i	$0.682785\pi$
$770$	0	0
$771$	− 977.898i	− 1.26835i
$772$	0	0
$773$	213.522	0.276225	0.138112	−	0.990417i	$-0.455897\pi$
$773$	213.522	0.276225	0.138112	+	0.990417i	$0.455897\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−853.050	−1.09788
$778$	0	0
$779$	389.503i	0.500004i
$780$	0	0
$781$	−70.7926	−0.0906436
$782$	0	0
$783$	93.1976i	0.119026i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	370.182i	0.470371i	0.971951	+	0.235185i	$0.0755697\pi$
$787$	370.182i	0.470371i	−0.971951	+	0.235185i	$0.924430\pi$
$788$	0	0
$789$	−624.531	−0.791547
$790$	0	0
$791$	691.397i	0.874080i
$792$	0	0
$793$	12.3406	0.0155619
$794$	0	0
$795$	0	0
$796$	0	0
$797$	274.426	0.344323	0.172162	−	0.985069i	$-0.444925\pi$
$797$	274.426	0.344323	0.172162	+	0.985069i	$0.444925\pi$
$798$	0	0
$799$	− 844.414i	− 1.05684i
$800$	0	0
$801$	−369.056	−0.460744
$802$	0	0
$803$	− 247.063i	− 0.307675i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 134.956i	− 0.167232i
$808$	0	0
$809$	665.214	0.822266	0.411133	−	0.911575i	$-0.365133\pi$
$809$	665.214	0.822266	0.411133	+	0.911575i	$0.365133\pi$
$810$	0	0
$811$	− 360.665i	− 0.444717i	−0.974965	−	0.222358i	$-0.928624\pi$
$811$	− 360.665i	− 0.444717i	0.974965	−	0.222358i	$-0.0713755\pi$
$812$	0	0
$813$	−1137.27	−1.39886
$814$	0	0
$815$	0	0
$816$	0	0
$817$	18.8854	0.0231156
$818$	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
$823$	− 122.433i	− 0.148764i	−0.997230	−	0.0743822i	$-0.976302\pi$
$823$	− 122.433i	− 0.148764i	0.997230	−	0.0743822i	$-0.0236985\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1532.98i	− 1.85366i	−0.375477	−	0.926832i	$-0.622521\pi$
$827$	− 1532.98i	− 1.85366i	0.375477	−	0.926832i	$-0.377479\pi$
$828$	0	0
$829$	195.475	0.235796	0.117898	−	0.993026i	$-0.462384\pi$
$829$	195.475	0.235796	0.117898	+	0.993026i	$0.462384\pi$
$830$	0	0
$831$	1739.47i	2.09322i
$832$	0	0
$833$	−558.053	−0.669931
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−636.774	−0.760781
$838$	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
$843$	21.4532i	0.0254487i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1001.85i	− 1.18282i
$848$	0	0
$849$	646.407	0.761375
$850$	0	0
$851$	− 424.801i	− 0.499179i
$852$	0	0
$853$	1055.28	1.23714	0.618570	−	0.785730i	$-0.287712\pi$
$853$	1055.28	1.23714	0.618570	+	0.785730i	$0.287712\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−155.378	−0.181304	−0.0906521	−	0.995883i	$-0.528895\pi$
$857$	−155.378	−0.181304	−0.0906521	+	0.995883i	$0.528895\pi$
$858$	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
$863$	− 930.702i	− 1.07845i	−0.842162	−	0.539225i	$-0.818717\pi$
$863$	− 930.702i	− 1.07845i	0.842162	−	0.539225i	$-0.181283\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1071.51i	− 1.23588i
$868$	0	0
$869$	−204.223	−0.235009
$870$	0	0
$871$	− 41.9584i	− 0.0481727i
$872$	0	0
$873$	−214.328	−0.245508
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−33.5217	−0.0382231	−0.0191115	−	0.999817i	$-0.506084\pi$
$877$	−33.5217	−0.0382231	−0.0191115	+	0.999817i	$0.506084\pi$
$878$	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
$883$	542.308i	0.614166i	0.951683	+	0.307083i	$0.0993529\pi$
$883$	542.308i	0.614166i	−0.951683	+	0.307083i	$0.900647\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	714.720i	0.805773i	0.915250	+	0.402886i	$0.131993\pi$
$887$	714.720i	0.805773i	−0.915250	+	0.402886i	$0.868007\pi$
$888$	0	0
$889$	−15.7044	−0.0176652
$890$	0	0
$891$	− 180.159i	− 0.202199i
$892$	0	0
$893$	−332.433	−0.372266
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−28.9443	−0.0322679
$898$	0	0
$899$	329.484i	0.366500i
$900$	0	0
$901$	−516.932	−0.573731
$902$	0	0
$903$	− 64.9839i	− 0.0719645i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 347.233i	− 0.382837i	−0.981509	−	0.191418i	$-0.938691\pi$
$907$	− 347.233i	− 0.382837i	0.981509	−	0.191418i	$-0.0613087\pi$
$908$	0	0
$909$	546.604	0.601324
$910$	0	0
$911$	− 1427.54i	− 1.56701i	−0.621386	−	0.783504i	$-0.713430\pi$
$911$	− 1427.54i	− 1.56701i	0.621386	−	0.783504i	$-0.286570\pi$
$912$	0	0
$913$	38.1517	0.0417871
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1919.15	2.09286
$918$	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
$923$	18.6089i	0.0201614i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	195.750i	0.211165i
$928$	0	0
$929$	1535.96	1.65335	0.826675	−	0.562680i	$-0.190230\pi$
$929$	1535.96	1.65335	0.826675	+	0.562680i	$0.190230\pi$
$930$	0	0
$931$	219.697i	0.235980i
$932$	0	0
$933$	−463.489	−0.496773
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−338.721	−0.361496	−0.180748	−	0.983529i	$-0.557852\pi$
$937$	−338.721	−0.361496	−0.180748	+	0.983529i	$0.557852\pi$
$938$	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
$943$	667.424i	0.707766i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 656.135i	− 0.692856i	−0.938077	−	0.346428i	$-0.887394\pi$
$947$	− 656.135i	− 0.692856i	0.938077	−	0.346428i	$-0.112606\pi$
$948$	0	0
$949$	−64.9443	−0.0684344
$950$	0	0
$951$	− 1395.62i	− 1.46752i
$952$	0	0
$953$	−436.675	−0.458211	−0.229105	−	0.973402i	$-0.573580\pi$
$953$	−436.675	−0.458211	−0.229105	+	0.973402i	$0.573580\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−47.4489	−0.0495809
$958$	0	0
$959$	− 449.423i	− 0.468637i
$960$	0	0
$961$	−1290.20	−1.34256
$962$	0	0
$963$	− 662.671i	− 0.688132i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 903.436i	− 0.934267i	−0.884187	−	0.467133i	$-0.845287\pi$
$967$	− 903.436i	− 0.934267i	0.884187	−	0.467133i	$-0.154713\pi$
$968$	0	0
$969$	−854.663	−0.882005
$970$	0	0
$971$	− 1866.89i	− 1.92265i	−0.275420	−	0.961324i	$-0.588817\pi$
$971$	− 1866.89i	− 1.92265i	0.275420	−	0.961324i	$-0.411183\pi$
$972$	0	0
$973$	−1070.56	−1.10026
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1073.95	1.09923	0.549615	−	0.835418i	$-0.314774\pi$
$977$	1073.95	1.09923	0.549615	+	0.835418i	$0.314774\pi$
$978$	0	0
$979$	121.135i	0.123733i
$980$	0	0
$981$	−1080.58	−1.10151
$982$	0	0
$983$	534.114i	0.543351i	0.962389	+	0.271675i	$0.0875777\pi$
$983$	534.114i	0.543351i	−0.962389	+	0.271675i	$0.912422\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1143.89i	1.15895i
$988$	0	0
$989$	32.3607	0.0327206
$990$	0	0
$991$	− 520.419i	− 0.525146i	−0.964912	−	0.262573i	$-0.915429\pi$
$991$	− 520.419i	− 0.525146i	0.964912	−	0.262573i	$-0.0845710\pi$
$992$	0	0
$993$	−619.502	−0.623869
$994$	0	0
$995$	0	0
$996$	0	0
$997$	457.680	0.459057	0.229528	−	0.973302i	$-0.426282\pi$
$997$	457.680	0.459057	0.229528	+	0.973302i	$0.426282\pi$
$998$	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.b.s.1151.1		4
4.3	odd	2	inner	1600.3.b.s.1151.4		4
5.2	odd	4		1600.3.h.n.1599.2		8
5.3	odd	4		1600.3.h.n.1599.8		8
5.4	even	2		320.3.b.c.191.4		4
8.3	odd	2		100.3.b.f.51.3		4
8.5	even	2		100.3.b.f.51.4		4
15.14	odd	2		2880.3.e.e.2431.4		4
20.3	even	4		1600.3.h.n.1599.1		8
20.7	even	4		1600.3.h.n.1599.7		8
20.19	odd	2		320.3.b.c.191.1		4
24.5	odd	2		900.3.c.k.451.1		4
24.11	even	2		900.3.c.k.451.2		4
40.3	even	4		100.3.d.b.99.3		8
40.13	odd	4		100.3.d.b.99.5		8
40.19	odd	2		20.3.b.a.11.2	yes	4
40.27	even	4		100.3.d.b.99.6		8
40.29	even	2		20.3.b.a.11.1	✓	4
40.37	odd	4		100.3.d.b.99.4		8
60.59	even	2		2880.3.e.e.2431.3		4
80.19	odd	4		1280.3.g.e.1151.8		8
80.29	even	4		1280.3.g.e.1151.2		8
80.59	odd	4		1280.3.g.e.1151.1		8
80.69	even	4		1280.3.g.e.1151.7		8
120.29	odd	2		180.3.c.a.91.4		4
120.53	even	4		900.3.f.e.199.4		8
120.59	even	2		180.3.c.a.91.3		4
120.77	even	4		900.3.f.e.199.5		8
120.83	odd	4		900.3.f.e.199.6		8
120.107	odd	4		900.3.f.e.199.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.3.b.a.11.1	✓	4	40.29	even	2
20.3.b.a.11.2	yes	4	40.19	odd	2
100.3.b.f.51.3		4	8.3	odd	2
100.3.b.f.51.4		4	8.5	even	2
100.3.d.b.99.3		8	40.3	even	4
100.3.d.b.99.4		8	40.37	odd	4
100.3.d.b.99.5		8	40.13	odd	4
100.3.d.b.99.6		8	40.27	even	4
180.3.c.a.91.3		4	120.59	even	2
180.3.c.a.91.4		4	120.29	odd	2
320.3.b.c.191.1		4	20.19	odd	2
320.3.b.c.191.4		4	5.4	even	2
900.3.c.k.451.1		4	24.5	odd	2
900.3.c.k.451.2		4	24.11	even	2
900.3.f.e.199.3		8	120.107	odd	4
900.3.f.e.199.4		8	120.53	even	4
900.3.f.e.199.5		8	120.77	even	4
900.3.f.e.199.6		8	120.83	odd	4
1280.3.g.e.1151.1		8	80.59	odd	4
1280.3.g.e.1151.2		8	80.29	even	4
1280.3.g.e.1151.7		8	80.69	even	4
1280.3.g.e.1151.8		8	80.19	odd	4
1600.3.b.s.1151.1		4	1.1	even	1	trivial
1600.3.b.s.1151.4		4	4.3	odd	2	inner
1600.3.h.n.1599.1		8	20.3	even	4
1600.3.h.n.1599.2		8	5.2	odd	4
1600.3.h.n.1599.7		8	20.7	even	4
1600.3.h.n.1599.8		8	5.3	odd	4
2880.3.e.e.2431.3		4	60.59	even	2
2880.3.e.e.2431.4		4	15.14	odd	2

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.80423i q^{3} -8.50651i q^{7} -5.47214 q^{9} +O(q^{10})\)	\(q-3.80423i q^{3} -8.50651i q^{7} -5.47214 q^{9} +1.79611i q^{11} +0.472136 q^{13} +23.8885 q^{17} -9.40456i q^{19} -32.3607 q^{21} -16.1150i q^{23} -13.4208i q^{27} -6.94427 q^{29} -47.4468i q^{31} +6.83282 q^{33} +26.3607 q^{37} -1.79611i q^{39} -41.4164 q^{41} +2.00811i q^{43} -35.3481i q^{47} -23.3607 q^{49} -90.8774i q^{51} -21.6393 q^{53} -35.7771 q^{57} +73.8644i q^{59} +26.1378 q^{61} +46.5488i q^{63} -88.8693i q^{67} -61.3050 q^{69} +39.4144i q^{71} -137.554 q^{73} +15.2786 q^{77} +113.703i q^{79} -100.305 q^{81} -21.2412i q^{83} +26.4176i q^{87} +67.4427 q^{89} -4.01623i q^{91} -180.498 q^{93} +39.1672 q^{97} -9.82857i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 16 q^{13} + 24 q^{17} - 40 q^{21} + 8 q^{29} - 80 q^{33} + 16 q^{37} - 112 q^{41} - 4 q^{49} - 176 q^{53} - 128 q^{61} - 120 q^{69} - 264 q^{73} + 240 q^{77} - 276 q^{81} - 88 q^{89} - 400 q^{93} + 264 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 - 16 * q^13 + 24 * q^17 - 40 * q^21 + 8 * q^29 - 80 * q^33 + 16 * q^37 - 112 * q^41 - 4 * q^49 - 176 * q^53 - 128 * q^61 - 120 * q^69 - 264 * q^73 + 240 * q^77 - 276 * q^81 - 88 * q^89 - 400 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

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