LMFDB - Embedded newform 3700.2.d.k.149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3700,2,Mod(149,3700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3700.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3700.d (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:	$29.5446487479$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 15x^{10} + 83x^{8} + 216x^{6} + 277x^{4} + 165x^{2} + 36 $ x^12 + 15x^10 + 83x^8 + 216x^6 + 277x^4 + 165*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.2
Root			$2.45978i$ of defining polynomial
Character	$\chi$	$=$	3700.149
Dual form			3700.2.d.k.149.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.85250i q^{3} +2.47048i q^{7} -5.13675 q^{9} +1.45305 q^{11} +0.691767i q^{13} -1.07378i q^{17} -2.15136 q^{19} +7.04705 q^{21} -5.00668i q^{23} +6.09508i q^{27} -4.85250 q^{29} -3.84313 q^{31} -4.14481i q^{33} +1.00000i q^{37} +1.97326 q^{39} -8.90340 q^{41} -1.25195i q^{43} -0.737297i q^{47} +0.896719 q^{49} -3.06297 q^{51} -0.399453i q^{53} +6.13675i q^{57} -5.08846 q^{59} +1.84313 q^{61} -12.6902i q^{63} +10.5297i q^{67} -14.2816 q^{69} -8.84326 q^{71} +7.90223i q^{73} +3.58972i q^{77} -1.16218 q^{79} +1.97595 q^{81} -4.08426i q^{83} +13.8417i q^{87} -15.4744 q^{89} -1.70900 q^{91} +10.9625i q^{93} -10.4542i q^{97} -7.46394 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 18 q^{9} - 16 q^{11} - 4 q^{19} + 20 q^{21} - 26 q^{29} - 6 q^{31} - 24 q^{39} + 44 q^{41} - 64 q^{49} - 10 q^{51} - 26 q^{59} - 18 q^{61} - 76 q^{69} + 40 q^{71} - 28 q^{79} + 60 q^{81} - 32 q^{89}+ \cdots - 26 q^{99}+O(q^{100}) $ 12 * q - 18 * q^9 - 16 * q^11 - 4 * q^19 + 20 * q^21 - 26 * q^29 - 6 * q^31 - 24 * q^39 + 44 * q^41 - 64 * q^49 - 10 * q^51 - 26 * q^59 - 18 * q^61 - 76 * q^69 + 40 * q^71 - 28 * q^79 + 60 * q^81 - 32 * q^89 - 56 * q^91 - 26 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$	$1851$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 2.85250i	− 1.64689i	−0.567395	−	0.823446i	$-0.692049\pi$
$3$	− 2.85250i	− 1.64689i	0.567395	−	0.823446i	$-0.307951\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.47048i	0.933754i	0.884322	+	0.466877i	$0.154621\pi$
$7$	2.47048i	0.933754i	−0.884322	+	0.466877i	$0.845379\pi$
$8$	0	0
$9$	−5.13675	−1.71225
$10$	0	0
$11$	1.45305	0.438110	0.219055	−	0.975713i	$-0.429703\pi$
$11$	1.45305	0.438110	0.219055	+	0.975713i	$0.429703\pi$
$12$	0	0
$13$	0.691767i	0.191862i	0.995388	+	0.0959308i	$0.0305827\pi$
$13$	0.691767i	0.191862i	−0.995388	+	0.0959308i	$0.969417\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.07378i	− 0.260431i	−0.991486	−	0.130215i	$-0.958433\pi$
$17$	− 1.07378i	− 0.260431i	0.991486	−	0.130215i	$-0.0415669\pi$
$18$	0	0
$19$	−2.15136	−0.493556	−0.246778	−	0.969072i	$-0.579372\pi$
$19$	−2.15136	−0.493556	−0.246778	+	0.969072i	$0.579372\pi$
$20$	0	0
$21$	7.04705	1.53779
$22$	0	0
$23$	− 5.00668i	− 1.04397i	−0.852956	−	0.521983i	$-0.825192\pi$
$23$	− 5.00668i	− 1.04397i	0.852956	−	0.521983i	$-0.174808\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	6.09508i	1.17300i
$28$	0	0
$29$	−4.85250	−0.901086	−0.450543	−	0.892755i	$-0.648770\pi$
$29$	−4.85250	−0.901086	−0.450543	+	0.892755i	$0.648770\pi$
$30$	0	0
$31$	−3.84313	−0.690246	−0.345123	−	0.938558i	$-0.612163\pi$
$31$	−3.84313	−0.690246	−0.345123	+	0.938558i	$0.612163\pi$
$32$	0	0
$33$	− 4.14481i	− 0.721519i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	0	0
$39$	1.97326	0.315975
$40$	0	0
$41$	−8.90340	−1.39048	−0.695239	−	0.718779i	$-0.744701\pi$
$41$	−8.90340	−1.39048	−0.695239	+	0.718779i	$0.744701\pi$
$42$	0	0
$43$	− 1.25195i	− 0.190921i	−0.995433	−	0.0954604i	$-0.969568\pi$
$43$	− 1.25195i	− 0.190921i	0.995433	−	0.0954604i	$-0.0304323\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.737297i	− 0.107546i	−0.998553	−	0.0537729i	$-0.982875\pi$
$47$	− 0.737297i	− 0.107546i	0.998553	−	0.0537729i	$-0.0171247\pi$
$48$	0	0
$49$	0.896719	0.128103
$50$	0	0
$51$	−3.06297	−0.428901
$52$	0	0
$53$	− 0.399453i	− 0.0548691i	−0.999624	−	0.0274345i	$-0.991266\pi$
$53$	− 0.399453i	− 0.0548691i	0.999624	−	0.0274345i	$-0.00873378\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.13675i	0.812832i
$58$	0	0
$59$	−5.08846	−0.662462	−0.331231	−	0.943550i	$-0.607464\pi$
$59$	−5.08846	−0.662462	−0.331231	+	0.943550i	$0.607464\pi$
$60$	0	0
$61$	1.84313	0.235988	0.117994	−	0.993014i	$-0.462354\pi$
$61$	1.84313	0.235988	0.117994	+	0.993014i	$0.462354\pi$
$62$	0	0
$63$	− 12.6902i	− 1.59882i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.5297i	1.28640i	0.765697	+	0.643201i	$0.222394\pi$
$67$	10.5297i	1.28640i	−0.765697	+	0.643201i	$0.777606\pi$
$68$	0	0
$69$	−14.2816	−1.71930
$70$	0	0
$71$	−8.84326	−1.04950	−0.524751	−	0.851256i	$-0.675842\pi$
$71$	−8.84326	−1.04950	−0.524751	+	0.851256i	$0.675842\pi$
$72$	0	0
$73$	7.90223i	0.924887i	0.886649	+	0.462443i	$0.153027\pi$
$73$	7.90223i	0.924887i	−0.886649	+	0.462443i	$0.846973\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.58972i	0.409087i
$78$	0	0
$79$	−1.16218	−0.130756	−0.0653779	−	0.997861i	$-0.520825\pi$
$79$	−1.16218	−0.130756	−0.0653779	+	0.997861i	$0.520825\pi$
$80$	0	0
$81$	1.97595	0.219550
$82$	0	0
$83$	− 4.08426i	− 0.448306i	−0.974554	−	0.224153i	$-0.928038\pi$
$83$	− 4.08426i	− 0.448306i	0.974554	−	0.224153i	$-0.0719615\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	13.8417i	1.48399i
$88$	0	0
$89$	−15.4744	−1.64028	−0.820142	−	0.572160i	$-0.806106\pi$
$89$	−15.4744	−1.64028	−0.820142	+	0.572160i	$0.806106\pi$
$90$	0	0
$91$	−1.70900	−0.179152
$92$	0	0
$93$	10.9625i	1.13676i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.4542i	− 1.06146i	−0.847539	−	0.530732i	$-0.821917\pi$
$97$	− 10.4542i	− 1.06146i	0.847539	−	0.530732i	$-0.178083\pi$
$98$	0	0
$99$	−7.46394	−0.750154
$100$	0	0
$101$	−8.90623	−0.886203	−0.443102	−	0.896471i	$-0.646122\pi$
$101$	−8.90623	−0.886203	−0.443102	+	0.896471i	$0.646122\pi$
$102$	0	0
$103$	− 1.26689i	− 0.124830i	−0.998050	−	0.0624151i	$-0.980120\pi$
$103$	− 1.26689i	− 0.124830i	0.998050	−	0.0624151i	$-0.0198803\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.8794i	1.24509i	0.782582	+	0.622547i	$0.213902\pi$
$107$	12.8794i	1.24509i	−0.782582	+	0.622547i	$0.786098\pi$
$108$	0	0
$109$	−13.2464	−1.26878	−0.634389	−	0.773014i	$-0.718748\pi$
$109$	−13.2464	−1.26878	−0.634389	+	0.773014i	$0.718748\pi$
$110$	0	0
$111$	2.85250	0.270747
$112$	0	0
$113$	11.9640i	1.12547i	0.826636	+	0.562737i	$0.190252\pi$
$113$	11.9640i	1.12547i	−0.826636	+	0.562737i	$0.809748\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.55343i	− 0.328515i
$118$	0	0
$119$	2.65276	0.243178
$120$	0	0
$121$	−8.88866	−0.808060
$122$	0	0
$123$	25.3970i	2.28997i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.73744i	− 0.686586i	−0.939228	−	0.343293i	$-0.888458\pi$
$127$	− 7.73744i	− 0.686586i	0.939228	−	0.343293i	$-0.111542\pi$
$128$	0	0
$129$	−3.57119	−0.314426
$130$	0	0
$131$	−2.17162	−0.189735	−0.0948677	−	0.995490i	$-0.530243\pi$
$131$	−2.17162	−0.189735	−0.0948677	+	0.995490i	$0.530243\pi$
$132$	0	0
$133$	− 5.31489i	− 0.460860i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.82578i	− 0.497730i	−0.968538	−	0.248865i	$-0.919942\pi$
$137$	− 5.82578i	− 0.497730i	0.968538	−	0.248865i	$-0.0800575\pi$
$138$	0	0
$139$	5.00813	0.424784	0.212392	−	0.977185i	$-0.431875\pi$
$139$	5.00813	0.424784	0.212392	+	0.977185i	$0.431875\pi$
$140$	0	0
$141$	−2.10314	−0.177116
$142$	0	0
$143$	1.00517i	0.0840564i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.55789i	− 0.210971i
$148$	0	0
$149$	−5.82399	−0.477120	−0.238560	−	0.971128i	$-0.576675\pi$
$149$	−5.82399	−0.477120	−0.238560	+	0.971128i	$0.576675\pi$
$150$	0	0
$151$	12.4489	1.01308	0.506539	−	0.862217i	$-0.330925\pi$
$151$	12.4489	1.01308	0.506539	+	0.862217i	$0.330925\pi$
$152$	0	0
$153$	5.51576i	0.445923i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.86082i	0.228318i	0.993462	+	0.114159i	$0.0364173\pi$
$157$	2.86082i	0.228318i	−0.993462	+	0.114159i	$0.963583\pi$
$158$	0	0
$159$	−1.13944	−0.0903633
$160$	0	0
$161$	12.3689	0.974808
$162$	0	0
$163$	8.32188i	0.651820i	0.945401	+	0.325910i	$0.105671\pi$
$163$	8.32188i	0.651820i	−0.945401	+	0.325910i	$0.894329\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.87274i	− 0.686593i	−0.939227	−	0.343297i	$-0.888456\pi$
$167$	− 8.87274i	− 0.686593i	0.939227	−	0.343297i	$-0.111544\pi$
$168$	0	0
$169$	12.5215	0.963189
$170$	0	0
$171$	11.0510	0.845091
$172$	0	0
$173$	− 0.0633788i	− 0.00481860i	−0.999997	−	0.00240930i	$-0.999233\pi$
$173$	− 0.0633788i	− 0.00481860i	0.999997	−	0.00240930i	$-0.000766905\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.5148i	1.09100i
$178$	0	0
$179$	−0.304169	−0.0227346	−0.0113673	−	0.999935i	$-0.503618\pi$
$179$	−0.304169	−0.0227346	−0.0113673	+	0.999935i	$0.503618\pi$
$180$	0	0
$181$	0.546542	0.0406241	0.0203121	−	0.999794i	$-0.493534\pi$
$181$	0.546542	0.0406241	0.0203121	+	0.999794i	$0.493534\pi$
$182$	0	0
$183$	− 5.25751i	− 0.388647i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.56026i	− 0.114097i
$188$	0	0
$189$	−15.0578	−1.09529
$190$	0	0
$191$	19.3098	1.39721	0.698604	−	0.715509i	$-0.253805\pi$
$191$	19.3098	1.39721	0.698604	+	0.715509i	$0.253805\pi$
$192$	0	0
$193$	− 2.41197i	− 0.173618i	−0.996225	−	0.0868089i	$-0.972333\pi$
$193$	− 2.41197i	− 0.173618i	0.996225	−	0.0868089i	$-0.0276669\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.8862i	− 0.918107i	−0.888409	−	0.459054i	$-0.848189\pi$
$197$	− 12.8862i	− 0.918107i	0.888409	−	0.459054i	$-0.151811\pi$
$198$	0	0
$199$	−21.8555	−1.54930	−0.774648	−	0.632393i	$-0.782073\pi$
$199$	−21.8555	−1.54930	−0.774648	+	0.632393i	$0.782073\pi$
$200$	0	0
$201$	30.0358	2.11856
$202$	0	0
$203$	− 11.9880i	− 0.841393i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	25.7181i	1.78753i
$208$	0	0
$209$	−3.12602	−0.216232
$210$	0	0
$211$	−0.678051	−0.0466790	−0.0233395	−	0.999728i	$-0.507430\pi$
$211$	−0.678051	−0.0466790	−0.0233395	+	0.999728i	$0.507430\pi$
$212$	0	0
$213$	25.2254i	1.72842i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 9.49437i	− 0.644520i
$218$	0	0
$219$	22.5411	1.52319
$220$	0	0
$221$	0.742808	0.0499666
$222$	0	0
$223$	19.4839i	1.30474i	0.757901	+	0.652370i	$0.226225\pi$
$223$	19.4839i	1.30474i	−0.757901	+	0.652370i	$0.773775\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.68509i	0.443705i	0.975080	+	0.221852i	$0.0712103\pi$
$227$	6.68509i	0.443705i	−0.975080	+	0.221852i	$0.928790\pi$
$228$	0	0
$229$	12.8099	0.846503	0.423251	−	0.906012i	$-0.360889\pi$
$229$	12.8099	0.846503	0.423251	+	0.906012i	$0.360889\pi$
$230$	0	0
$231$	10.2397	0.673722
$232$	0	0
$233$	10.7733i	0.705780i	0.935665	+	0.352890i	$0.114801\pi$
$233$	10.7733i	0.705780i	−0.935665	+	0.352890i	$0.885199\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3.31512i	0.215340i
$238$	0	0
$239$	−4.29404	−0.277758	−0.138879	−	0.990309i	$-0.544350\pi$
$239$	−4.29404	−0.277758	−0.138879	+	0.990309i	$0.544350\pi$
$240$	0	0
$241$	12.9639	0.835078	0.417539	−	0.908659i	$-0.362893\pi$
$241$	12.9639	0.835078	0.417539	+	0.908659i	$0.362893\pi$
$242$	0	0
$243$	12.6488i	0.811423i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.48824i	− 0.0946943i
$248$	0	0
$249$	−11.6503	−0.738311
$250$	0	0
$251$	7.22620	0.456114	0.228057	−	0.973648i	$-0.426763\pi$
$251$	7.22620	0.456114	0.228057	+	0.973648i	$0.426763\pi$
$252$	0	0
$253$	− 7.27494i	− 0.457372i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 13.5222i	− 0.843492i	−0.906714	−	0.421746i	$-0.861417\pi$
$257$	− 13.5222i	− 0.843492i	0.906714	−	0.421746i	$-0.138583\pi$
$258$	0	0
$259$	−2.47048	−0.153508
$260$	0	0
$261$	24.9261	1.54289
$262$	0	0
$263$	− 6.00269i	− 0.370141i	−0.982725	−	0.185071i	$-0.940749\pi$
$263$	− 6.00269i	− 0.370141i	0.982725	−	0.185071i	$-0.0592514\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	44.1407i	2.70137i
$268$	0	0
$269$	−25.0161	−1.52526	−0.762629	−	0.646836i	$-0.776092\pi$
$269$	−25.0161	−1.52526	−0.762629	+	0.646836i	$0.776092\pi$
$270$	0	0
$271$	7.31221	0.444185	0.222093	−	0.975026i	$-0.428711\pi$
$271$	7.31221	0.444185	0.222093	+	0.975026i	$0.428711\pi$
$272$	0	0
$273$	4.87491i	0.295043i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 28.2585i	− 1.69789i	−0.528483	−	0.848944i	$-0.677239\pi$
$277$	− 28.2585i	− 1.69789i	0.528483	−	0.848944i	$-0.322761\pi$
$278$	0	0
$279$	19.7412	1.18187
$280$	0	0
$281$	−12.9144	−0.770406	−0.385203	−	0.922832i	$-0.625869\pi$
$281$	−12.9144	−0.770406	−0.385203	+	0.922832i	$0.625869\pi$
$282$	0	0
$283$	14.4542i	0.859214i	0.903016	+	0.429607i	$0.141348\pi$
$283$	14.4542i	0.859214i	−0.903016	+	0.429607i	$0.858652\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 21.9957i	− 1.29837i
$288$	0	0
$289$	15.8470	0.932176
$290$	0	0
$291$	−29.8206	−1.74812
$292$	0	0
$293$	− 18.7935i	− 1.09793i	−0.835846	−	0.548964i	$-0.815022\pi$
$293$	− 18.7935i	− 1.09793i	0.835846	−	0.548964i	$-0.184978\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	8.85643i	0.513902i
$298$	0	0
$299$	3.46346	0.200297
$300$	0	0
$301$	3.09292	0.178273
$302$	0	0
$303$	25.4050i	1.45948i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.8018i	1.13015i	0.825039	+	0.565076i	$0.191153\pi$
$307$	19.8018i	1.13015i	−0.825039	+	0.565076i	$0.808847\pi$
$308$	0	0
$309$	−3.61380	−0.205582
$310$	0	0
$311$	14.5105	0.822816	0.411408	−	0.911451i	$-0.365037\pi$
$311$	14.5105	0.822816	0.411408	+	0.911451i	$0.365037\pi$
$312$	0	0
$313$	27.6930i	1.56530i	0.622461	+	0.782651i	$0.286133\pi$
$313$	27.6930i	1.56530i	−0.622461	+	0.782651i	$0.713867\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.80676i	− 0.494637i	−0.968934	−	0.247318i	$-0.920451\pi$
$317$	− 8.80676i	− 0.494637i	0.968934	−	0.247318i	$-0.0795494\pi$
$318$	0	0
$319$	−7.05091	−0.394775
$320$	0	0
$321$	36.7383	2.05053
$322$	0	0
$323$	2.31009i	0.128537i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	37.7854i	2.08954i
$328$	0	0
$329$	1.82148	0.100421
$330$	0	0
$331$	−19.5580	−1.07500	−0.537501	−	0.843263i	$-0.680632\pi$
$331$	−19.5580	−1.07500	−0.537501	+	0.843263i	$0.680632\pi$
$332$	0	0
$333$	− 5.13675i	− 0.281492i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 33.9345i	− 1.84853i	−0.381750	−	0.924266i	$-0.624678\pi$
$337$	− 33.9345i	− 1.84853i	0.381750	−	0.924266i	$-0.375322\pi$
$338$	0	0
$339$	34.1272	1.85353
$340$	0	0
$341$	−5.58424	−0.302404
$342$	0	0
$343$	19.5087i	1.05337i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.21881i	− 0.226478i	−0.993568	−	0.113239i	$-0.963877\pi$
$347$	− 4.21881i	− 0.226478i	0.993568	−	0.113239i	$-0.0361225\pi$
$348$	0	0
$349$	−17.6555	−0.945076	−0.472538	−	0.881310i	$-0.656662\pi$
$349$	−17.6555	−0.945076	−0.472538	+	0.881310i	$0.656662\pi$
$350$	0	0
$351$	−4.21637	−0.225053
$352$	0	0
$353$	− 21.7023i	− 1.15510i	−0.816357	−	0.577548i	$-0.804010\pi$
$353$	− 21.7023i	− 1.15510i	0.816357	−	0.577548i	$-0.195990\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 7.56700i	− 0.400488i
$358$	0	0
$359$	−2.87247	−0.151603	−0.0758015	−	0.997123i	$-0.524152\pi$
$359$	−2.87247	−0.151603	−0.0758015	+	0.997123i	$0.524152\pi$
$360$	0	0
$361$	−14.3717	−0.756403
$362$	0	0
$363$	25.3549i	1.33079i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.6572i	0.660701i	0.943858	+	0.330351i	$0.107167\pi$
$367$	12.6572i	0.660701i	−0.943858	+	0.330351i	$0.892833\pi$
$368$	0	0
$369$	45.7346	2.38085
$370$	0	0
$371$	0.986841	0.0512342
$372$	0	0
$373$	− 5.09508i	− 0.263813i	−0.991262	−	0.131907i	$-0.957890\pi$
$373$	− 5.09508i	− 0.263813i	0.991262	−	0.131907i	$-0.0421099\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.35680i	− 0.172884i
$378$	0	0
$379$	24.5053	1.25875	0.629377	−	0.777100i	$-0.283310\pi$
$379$	24.5053	1.25875	0.629377	+	0.777100i	$0.283310\pi$
$380$	0	0
$381$	−22.0710	−1.13073
$382$	0	0
$383$	2.87538i	0.146925i	0.997298	+	0.0734626i	$0.0234050\pi$
$383$	2.87538i	0.146925i	−0.997298	+	0.0734626i	$0.976595\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.43096i	0.326904i
$388$	0	0
$389$	10.8073	0.547954	0.273977	−	0.961736i	$-0.411661\pi$
$389$	10.8073	0.547954	0.273977	+	0.961736i	$0.411661\pi$
$390$	0	0
$391$	−5.37610	−0.271881
$392$	0	0
$393$	6.19455i	0.312474i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 28.5482i	− 1.43279i	−0.697694	−	0.716396i	$-0.745791\pi$
$397$	− 28.5482i	− 1.43279i	0.697694	−	0.716396i	$-0.254209\pi$
$398$	0	0
$399$	−15.1607	−0.758986
$400$	0	0
$401$	28.3691	1.41669	0.708343	−	0.705868i	$-0.249443\pi$
$401$	28.3691	1.41669	0.708343	+	0.705868i	$0.249443\pi$
$402$	0	0
$403$	− 2.65855i	− 0.132432i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.45305i	0.0720248i
$408$	0	0
$409$	−7.46524	−0.369132	−0.184566	−	0.982820i	$-0.559088\pi$
$409$	−7.46524	−0.369132	−0.184566	+	0.982820i	$0.559088\pi$
$410$	0	0
$411$	−16.6180	−0.819707
$412$	0	0
$413$	− 12.5710i	− 0.618577i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 14.2857i	− 0.699573i
$418$	0	0
$419$	11.1288	0.543679	0.271839	−	0.962343i	$-0.412368\pi$
$419$	11.1288	0.543679	0.271839	+	0.962343i	$0.412368\pi$
$420$	0	0
$421$	−38.1030	−1.85703	−0.928513	−	0.371300i	$-0.878912\pi$
$421$	−38.1030	−1.85703	−0.928513	+	0.371300i	$0.878912\pi$
$422$	0	0
$423$	3.78731i	0.184145i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.55341i	0.220355i
$428$	0	0
$429$	2.86724	0.138432
$430$	0	0
$431$	27.2146	1.31088	0.655441	−	0.755247i	$-0.272483\pi$
$431$	27.2146	1.31088	0.655441	+	0.755247i	$0.272483\pi$
$432$	0	0
$433$	− 11.9302i	− 0.573329i	−0.958031	−	0.286665i	$-0.907453\pi$
$433$	− 11.9302i	− 0.573329i	0.958031	−	0.286665i	$-0.0925466\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.7712i	0.515255i
$438$	0	0
$439$	−13.8417	−0.660626	−0.330313	−	0.943871i	$-0.607154\pi$
$439$	−13.8417	−0.660626	−0.330313	+	0.943871i	$0.607154\pi$
$440$	0	0
$441$	−4.60622	−0.219344
$442$	0	0
$443$	29.7010i	1.41114i	0.708642	+	0.705568i	$0.249308\pi$
$443$	29.7010i	1.41114i	−0.708642	+	0.705568i	$0.750692\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.6129i	0.785765i
$448$	0	0
$449$	6.61844	0.312343	0.156172	−	0.987730i	$-0.450085\pi$
$449$	6.61844	0.312343	0.156172	+	0.987730i	$0.450085\pi$
$450$	0	0
$451$	−12.9371	−0.609182
$452$	0	0
$453$	− 35.5105i	− 1.66843i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.3188i	1.27792i	0.769240	+	0.638960i	$0.220635\pi$
$457$	27.3188i	1.27792i	−0.769240	+	0.638960i	$0.779365\pi$
$458$	0	0
$459$	6.54479	0.305485
$460$	0	0
$461$	−21.5771	−1.00494	−0.502472	−	0.864593i	$-0.667576\pi$
$461$	−21.5771	−1.00494	−0.502472	+	0.864593i	$0.667576\pi$
$462$	0	0
$463$	1.20958i	0.0562137i	0.999605	+	0.0281069i	$0.00894787\pi$
$463$	1.20958i	0.0562137i	−0.999605	+	0.0281069i	$0.991052\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.8812i	0.642345i	0.947021	+	0.321172i	$0.104077\pi$
$467$	13.8812i	0.642345i	−0.947021	+	0.321172i	$0.895923\pi$
$468$	0	0
$469$	−26.0133	−1.20118
$470$	0	0
$471$	8.16048	0.376015
$472$	0	0
$473$	− 1.81914i	− 0.0836443i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.05189i	0.0939495i
$478$	0	0
$479$	−27.1852	−1.24212	−0.621061	−	0.783762i	$-0.713298\pi$
$479$	−27.1852	−1.24212	−0.621061	+	0.783762i	$0.713298\pi$
$480$	0	0
$481$	−0.691767	−0.0315418
$482$	0	0
$483$	− 35.2823i	− 1.60540i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10.2171i	0.462982i	0.972837	+	0.231491i	$0.0743604\pi$
$487$	10.2171i	0.462982i	−0.972837	+	0.231491i	$0.925640\pi$
$488$	0	0
$489$	23.7381	1.07348
$490$	0	0
$491$	−41.5453	−1.87491	−0.937455	−	0.348105i	$-0.886825\pi$
$491$	−41.5453	−1.87491	−0.937455	+	0.348105i	$0.886825\pi$
$492$	0	0
$493$	5.21053i	0.234671i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 21.8471i	− 0.979977i
$498$	0	0
$499$	−14.2455	−0.637714	−0.318857	−	0.947803i	$-0.603299\pi$
$499$	−14.2455	−0.637714	−0.318857	+	0.947803i	$0.603299\pi$
$500$	0	0
$501$	−25.3095	−1.13074
$502$	0	0
$503$	22.2356i	0.991437i	0.868483	+	0.495719i	$0.165095\pi$
$503$	22.2356i	0.991437i	−0.868483	+	0.495719i	$0.834905\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 35.7174i	− 1.58627i
$508$	0	0
$509$	−43.5439	−1.93005	−0.965024	−	0.262163i	$-0.915564\pi$
$509$	−43.5439	−1.93005	−0.965024	+	0.262163i	$0.915564\pi$
$510$	0	0
$511$	−19.5223	−0.863617
$512$	0	0
$513$	− 13.1127i	− 0.578940i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.07133i	− 0.0471169i
$518$	0	0
$519$	−0.180788	−0.00793571
$520$	0	0
$521$	28.0979	1.23099	0.615495	−	0.788141i	$-0.288956\pi$
$521$	28.0979	1.23099	0.615495	+	0.788141i	$0.288956\pi$
$522$	0	0
$523$	6.53967i	0.285960i	0.989726	+	0.142980i	$0.0456685\pi$
$523$	6.53967i	0.285960i	−0.989726	+	0.142980i	$0.954332\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.12669i	0.179761i
$528$	0	0
$529$	−2.06689	−0.0898649
$530$	0	0
$531$	26.1382	1.13430
$532$	0	0
$533$	− 6.15908i	− 0.266779i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.867640i	0.0374414i
$538$	0	0
$539$	1.30297	0.0561231
$540$	0	0
$541$	−6.13904	−0.263938	−0.131969	−	0.991254i	$-0.542130\pi$
$541$	−6.13904	−0.263938	−0.131969	+	0.991254i	$0.542130\pi$
$542$	0	0
$543$	− 1.55901i	− 0.0669035i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 35.6966i	− 1.52628i	−0.646236	−	0.763138i	$-0.723658\pi$
$547$	− 35.6966i	− 1.52628i	0.646236	−	0.763138i	$-0.276342\pi$
$548$	0	0
$549$	−9.46768	−0.404071
$550$	0	0
$551$	10.4395	0.444736
$552$	0	0
$553$	− 2.87115i	− 0.122094i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 44.1070i	− 1.86887i	−0.356129	−	0.934437i	$-0.615904\pi$
$557$	− 44.1070i	− 1.86887i	0.356129	−	0.934437i	$-0.384096\pi$
$558$	0	0
$559$	0.866058	0.0366304
$560$	0	0
$561$	−4.45063	−0.187906
$562$	0	0
$563$	− 25.6867i	− 1.08257i	−0.840840	−	0.541283i	$-0.817939\pi$
$563$	− 25.6867i	− 1.08257i	0.840840	−	0.541283i	$-0.182061\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.88155i	0.205006i
$568$	0	0
$569$	0.874988	0.0366814	0.0183407	−	0.999832i	$-0.494162\pi$
$569$	0.874988	0.0366814	0.0183407	+	0.999832i	$0.494162\pi$
$570$	0	0
$571$	−7.50834	−0.314214	−0.157107	−	0.987582i	$-0.550217\pi$
$571$	−7.50834	−0.314214	−0.157107	+	0.987582i	$0.550217\pi$
$572$	0	0
$573$	− 55.0812i	− 2.30105i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	33.9087i	1.41164i	0.708392	+	0.705819i	$0.249421\pi$
$577$	33.9087i	1.41164i	−0.708392	+	0.705819i	$0.750579\pi$
$578$	0	0
$579$	−6.88015	−0.285929
$580$	0	0
$581$	10.0901	0.418608
$582$	0	0
$583$	− 0.580423i	− 0.0240387i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 46.4075i	− 1.91544i	−0.287695	−	0.957722i	$-0.592889\pi$
$587$	− 46.4075i	− 1.91544i	0.287695	−	0.957722i	$-0.407111\pi$
$588$	0	0
$589$	8.26794	0.340675
$590$	0	0
$591$	−36.7580	−1.51202
$592$	0	0
$593$	− 4.79573i	− 0.196937i	−0.995140	−	0.0984685i	$-0.968606\pi$
$593$	− 4.79573i	− 0.196937i	0.995140	−	0.0984685i	$-0.0313944\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	62.3428i	2.55152i
$598$	0	0
$599$	1.54282	0.0630377	0.0315189	−	0.999503i	$-0.489966\pi$
$599$	1.54282	0.0630377	0.0315189	+	0.999503i	$0.489966\pi$
$600$	0	0
$601$	28.7100	1.17110	0.585552	−	0.810635i	$-0.300878\pi$
$601$	28.7100	1.17110	0.585552	+	0.810635i	$0.300878\pi$
$602$	0	0
$603$	− 54.0882i	− 2.20264i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 27.3875i	− 1.11162i	−0.831308	−	0.555811i	$-0.812408\pi$
$607$	− 27.3875i	− 1.11162i	0.831308	−	0.555811i	$-0.187592\pi$
$608$	0	0
$609$	−34.1958	−1.38568
$610$	0	0
$611$	0.510038	0.0206339
$612$	0	0
$613$	− 30.1125i	− 1.21623i	−0.793848	−	0.608117i	$-0.791925\pi$
$613$	− 30.1125i	− 1.21623i	0.793848	−	0.608117i	$-0.208075\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.51196i	− 0.382937i	−0.981499	−	0.191469i	$-0.938675\pi$
$617$	− 9.51196i	− 0.382937i	0.981499	−	0.191469i	$-0.0613250\pi$
$618$	0	0
$619$	−30.7584	−1.23628	−0.618142	−	0.786067i	$-0.712114\pi$
$619$	−30.7584	−1.23628	−0.618142	+	0.786067i	$0.712114\pi$
$620$	0	0
$621$	30.5161	1.22457
$622$	0	0
$623$	− 38.2292i	− 1.53162i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	8.91698i	0.356110i
$628$	0	0
$629$	1.07378	0.0428146
$630$	0	0
$631$	22.9168	0.912302	0.456151	−	0.889902i	$-0.349228\pi$
$631$	22.9168	0.912302	0.456151	+	0.889902i	$0.349228\pi$
$632$	0	0
$633$	1.93414i	0.0768751i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.620320i	0.0245780i
$638$	0	0
$639$	45.4256	1.79701
$640$	0	0
$641$	−42.5989	−1.68256	−0.841278	−	0.540603i	$-0.818196\pi$
$641$	−42.5989	−1.68256	−0.841278	+	0.540603i	$0.818196\pi$
$642$	0	0
$643$	8.44016i	0.332847i	0.986054	+	0.166424i	$0.0532219\pi$
$643$	8.44016i	0.332847i	−0.986054	+	0.166424i	$0.946778\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 19.2212i	− 0.755664i	−0.925874	−	0.377832i	$-0.876670\pi$
$647$	− 19.2212i	− 0.755664i	0.925874	−	0.377832i	$-0.123330\pi$
$648$	0	0
$649$	−7.39377	−0.290231
$650$	0	0
$651$	−27.0827	−1.06145
$652$	0	0
$653$	− 48.5978i	− 1.90178i	−0.309530	−	0.950890i	$-0.600172\pi$
$653$	− 48.5978i	− 1.90178i	0.309530	−	0.950890i	$-0.399828\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 40.5918i	− 1.58364i
$658$	0	0
$659$	26.8285	1.04509	0.522545	−	0.852612i	$-0.324983\pi$
$659$	26.8285	1.04509	0.522545	+	0.852612i	$0.324983\pi$
$660$	0	0
$661$	−25.9141	−1.00794	−0.503972	−	0.863720i	$-0.668128\pi$
$661$	−25.9141	−1.00794	−0.503972	+	0.863720i	$0.668128\pi$
$662$	0	0
$663$	− 2.11886i	− 0.0822896i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.2949i	0.940704i
$668$	0	0
$669$	55.5779	2.14876
$670$	0	0
$671$	2.67815	0.103389
$672$	0	0
$673$	− 3.87956i	− 0.149546i	−0.997201	−	0.0747730i	$-0.976177\pi$
$673$	− 3.87956i	− 0.149546i	0.997201	−	0.0747730i	$-0.0238232\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.68947i	− 0.333964i	−0.985960	−	0.166982i	$-0.946598\pi$
$677$	− 8.68947i	− 0.333964i	0.985960	−	0.166982i	$-0.0534021\pi$
$678$	0	0
$679$	25.8270	0.991147
$680$	0	0
$681$	19.0692	0.730733
$682$	0	0
$683$	4.76231i	0.182225i	0.995841	+	0.0911125i	$0.0290423\pi$
$683$	4.76231i	0.182225i	−0.995841	+	0.0911125i	$0.970958\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 36.5403i	− 1.39410i
$688$	0	0
$689$	0.276328	0.0105273
$690$	0	0
$691$	−34.4542	−1.31070	−0.655350	−	0.755325i	$-0.727479\pi$
$691$	−34.4542	−1.31070	−0.655350	+	0.755325i	$0.727479\pi$
$692$	0	0
$693$	− 18.4395i	− 0.700459i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.56033i	0.362123i
$698$	0	0
$699$	30.7307	1.16234
$700$	0	0
$701$	37.8935	1.43122	0.715610	−	0.698500i	$-0.246149\pi$
$701$	37.8935	1.43122	0.715610	+	0.698500i	$0.246149\pi$
$702$	0	0
$703$	− 2.15136i	− 0.0811401i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 22.0027i	− 0.827496i
$708$	0	0
$709$	2.07659	0.0779880	0.0389940	−	0.999239i	$-0.487585\pi$
$709$	2.07659	0.0779880	0.0389940	+	0.999239i	$0.487585\pi$
$710$	0	0
$711$	5.96984	0.223886
$712$	0	0
$713$	19.2413i	0.720593i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.2487i	0.457437i
$718$	0	0
$719$	−18.1139	−0.675533	−0.337767	−	0.941230i	$-0.609671\pi$
$719$	−18.1139	−0.675533	−0.337767	+	0.941230i	$0.609671\pi$
$720$	0	0
$721$	3.12982	0.116561
$722$	0	0
$723$	− 36.9795i	− 1.37528i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.5608i	0.688382i	0.938900	+	0.344191i	$0.111847\pi$
$727$	18.5608i	0.688382i	−0.938900	+	0.344191i	$0.888153\pi$
$728$	0	0
$729$	42.0086	1.55588
$730$	0	0
$731$	−1.34433	−0.0497217
$732$	0	0
$733$	11.3513i	0.419270i	0.977780	+	0.209635i	$0.0672276\pi$
$733$	11.3513i	0.419270i	−0.977780	+	0.209635i	$0.932772\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.3001i	0.563586i
$738$	0	0
$739$	33.9071	1.24729	0.623646	−	0.781707i	$-0.285651\pi$
$739$	33.9071	1.24729	0.623646	+	0.781707i	$0.285651\pi$
$740$	0	0
$741$	−4.24520	−0.155951
$742$	0	0
$743$	− 30.6460i	− 1.12429i	−0.827038	−	0.562146i	$-0.809976\pi$
$743$	− 30.6460i	− 1.12429i	0.827038	−	0.562146i	$-0.190024\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	20.9798i	0.767612i
$748$	0	0
$749$	−31.8182	−1.16261
$750$	0	0
$751$	−35.0742	−1.27988	−0.639938	−	0.768427i	$-0.721040\pi$
$751$	−35.0742	−1.27988	−0.639938	+	0.768427i	$0.721040\pi$
$752$	0	0
$753$	− 20.6127i	− 0.751169i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	13.8724i	0.504200i	0.967701	+	0.252100i	$0.0811212\pi$
$757$	13.8724i	0.504200i	−0.967701	+	0.252100i	$0.918879\pi$
$758$	0	0
$759$	−20.7518	−0.753242
$760$	0	0
$761$	−1.38470	−0.0501954	−0.0250977	−	0.999685i	$-0.507990\pi$
$761$	−1.38470	−0.0501954	−0.0250977	+	0.999685i	$0.507990\pi$
$762$	0	0
$763$	− 32.7251i	− 1.18473i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.52003i	− 0.127101i
$768$	0	0
$769$	36.3472	1.31071	0.655357	−	0.755319i	$-0.272518\pi$
$769$	36.3472	1.31071	0.655357	+	0.755319i	$0.272518\pi$
$770$	0	0
$771$	−38.5721	−1.38914
$772$	0	0
$773$	17.9396i	0.645244i	0.946528	+	0.322622i	$0.104564\pi$
$773$	17.9396i	0.645244i	−0.946528	+	0.322622i	$0.895436\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.04705i	0.252811i
$778$	0	0
$779$	19.1544	0.686278
$780$	0	0
$781$	−12.8497	−0.459797
$782$	0	0
$783$	− 29.5764i	− 1.05697i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 22.3205i	− 0.795641i	−0.917463	−	0.397821i	$-0.869767\pi$
$787$	− 22.3205i	− 0.795641i	0.917463	−	0.397821i	$-0.130233\pi$
$788$	0	0
$789$	−17.1227	−0.609583
$790$	0	0
$791$	−29.5567	−1.05092
$792$	0	0
$793$	1.27501i	0.0452770i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.2845i	0.860201i	0.902781	+	0.430101i	$0.141522\pi$
$797$	24.2845i	0.860201i	−0.902781	+	0.430101i	$0.858478\pi$
$798$	0	0
$799$	−0.791698	−0.0280083
$800$	0	0
$801$	79.4882	2.80858
$802$	0	0
$803$	11.4823i	0.405202i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	71.3584i	2.51193i
$808$	0	0
$809$	−22.4785	−0.790303	−0.395152	−	0.918616i	$-0.629308\pi$
$809$	−22.4785	−0.790303	−0.395152	+	0.918616i	$0.629308\pi$
$810$	0	0
$811$	−11.1407	−0.391202	−0.195601	−	0.980684i	$-0.562666\pi$
$811$	−11.1407	−0.391202	−0.195601	+	0.980684i	$0.562666\pi$
$812$	0	0
$813$	− 20.8581i	− 0.731525i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.69340i	0.0942301i
$818$	0	0
$819$	8.77869	0.306752
$820$	0	0
$821$	−36.5507	−1.27563	−0.637814	−	0.770190i	$-0.720161\pi$
$821$	−36.5507	−1.27563	−0.637814	+	0.770190i	$0.720161\pi$
$822$	0	0
$823$	16.4494i	0.573391i	0.958022	+	0.286696i	$0.0925568\pi$
$823$	16.4494i	0.573391i	−0.958022	+	0.286696i	$0.907443\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0790i	0.420028i	0.977698	+	0.210014i	$0.0673510\pi$
$827$	12.0790i	0.420028i	−0.977698	+	0.210014i	$0.932649\pi$
$828$	0	0
$829$	−30.3250	−1.05323	−0.526616	−	0.850104i	$-0.676539\pi$
$829$	−30.3250	−1.05323	−0.526616	+	0.850104i	$0.676539\pi$
$830$	0	0
$831$	−80.6073	−2.79624
$832$	0	0
$833$	− 0.962882i	− 0.0333619i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 23.4241i	− 0.809657i
$838$	0	0
$839$	5.85098	0.201998	0.100999	−	0.994887i	$-0.467796\pi$
$839$	5.85098	0.201998	0.100999	+	0.994887i	$0.467796\pi$
$840$	0	0
$841$	−5.45325	−0.188043
$842$	0	0
$843$	36.8382i	1.26877i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 21.9593i	− 0.754529i
$848$	0	0
$849$	41.2307	1.41503
$850$	0	0
$851$	5.00668	0.171627
$852$	0	0
$853$	− 44.8679i	− 1.53625i	−0.640302	−	0.768124i	$-0.721191\pi$
$853$	− 44.8679i	− 1.53625i	0.640302	−	0.768124i	$-0.278809\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.1121i	0.447901i	0.974600	+	0.223951i	$0.0718955\pi$
$857$	13.1121i	0.447901i	−0.974600	+	0.223951i	$0.928105\pi$
$858$	0	0
$859$	12.6423	0.431349	0.215675	−	0.976465i	$-0.430805\pi$
$859$	12.6423	0.431349	0.215675	+	0.976465i	$0.430805\pi$
$860$	0	0
$861$	−62.7427	−2.13827
$862$	0	0
$863$	37.0117i	1.25989i	0.776638	+	0.629947i	$0.216923\pi$
$863$	37.0117i	1.25989i	−0.776638	+	0.629947i	$0.783077\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 45.2035i	− 1.53519i
$868$	0	0
$869$	−1.68870	−0.0572854
$870$	0	0
$871$	−7.28406	−0.246811
$872$	0	0
$873$	53.7007i	1.81749i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.6811i	1.33994i	0.742389	+	0.669969i	$0.233692\pi$
$877$	39.6811i	1.33994i	−0.742389	+	0.669969i	$0.766308\pi$
$878$	0	0
$879$	−53.6085	−1.80817
$880$	0	0
$881$	41.8698	1.41063	0.705316	−	0.708894i	$-0.250805\pi$
$881$	41.8698	1.41063	0.705316	+	0.708894i	$0.250805\pi$
$882$	0	0
$883$	26.0807i	0.877685i	0.898564	+	0.438843i	$0.144611\pi$
$883$	26.0807i	0.877685i	−0.898564	+	0.438843i	$0.855389\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 31.2959i	− 1.05081i	−0.850851	−	0.525407i	$-0.823913\pi$
$887$	− 31.2959i	− 1.05081i	0.850851	−	0.525407i	$-0.176087\pi$
$888$	0	0
$889$	19.1152	0.641103
$890$	0	0
$891$	2.87115	0.0961871
$892$	0	0
$893$	1.58619i	0.0530799i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 9.87951i	− 0.329867i
$898$	0	0
$899$	18.6488	0.621971
$900$	0	0
$901$	−0.428926	−0.0142896
$902$	0	0
$903$	− 8.82256i	− 0.293596i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 44.5322i	− 1.47867i	−0.673340	−	0.739333i	$-0.735141\pi$
$907$	− 44.5322i	− 1.47867i	0.673340	−	0.739333i	$-0.264859\pi$
$908$	0	0
$909$	45.7491	1.51740
$910$	0	0
$911$	40.0369	1.32648	0.663240	−	0.748407i	$-0.269181\pi$
$911$	40.0369	1.32648	0.663240	+	0.748407i	$0.269181\pi$
$912$	0	0
$913$	− 5.93462i	− 0.196407i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.36495i	− 0.177166i
$918$	0	0
$919$	34.1438	1.12630	0.563150	−	0.826355i	$-0.309589\pi$
$919$	34.1438	1.12630	0.563150	+	0.826355i	$0.309589\pi$
$920$	0	0
$921$	56.4847	1.86124
$922$	0	0
$923$	− 6.11747i	− 0.201359i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.50769i	0.213740i
$928$	0	0
$929$	5.63909	0.185013	0.0925063	−	0.995712i	$-0.470512\pi$
$929$	5.63909	0.185013	0.0925063	+	0.995712i	$0.470512\pi$
$930$	0	0
$931$	−1.92917	−0.0632258
$932$	0	0
$933$	− 41.3912i	− 1.35509i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.0293i	1.40571i	0.711335	+	0.702853i	$0.248091\pi$
$937$	43.0293i	1.40571i	−0.711335	+	0.702853i	$0.751909\pi$
$938$	0	0
$939$	78.9943	2.57788
$940$	0	0
$941$	45.7393	1.49106	0.745529	−	0.666473i	$-0.232197\pi$
$941$	45.7393	1.49106	0.745529	+	0.666473i	$0.232197\pi$
$942$	0	0
$943$	44.5765i	1.45161i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.16489i	0.265323i	0.991161	+	0.132662i	$0.0423524\pi$
$947$	8.16489i	0.265323i	−0.991161	+	0.132662i	$0.957648\pi$
$948$	0	0
$949$	−5.46650	−0.177450
$950$	0	0
$951$	−25.1213	−0.814613
$952$	0	0
$953$	9.73686i	0.315408i	0.987486	+	0.157704i	$0.0504092\pi$
$953$	9.73686i	0.315408i	−0.987486	+	0.157704i	$0.949591\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.1127i	0.650151i
$958$	0	0
$959$	14.3925	0.464758
$960$	0	0
$961$	−16.2304	−0.523561
$962$	0	0
$963$	− 66.1580i	− 2.13191i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	20.9448i	0.673538i	0.941587	+	0.336769i	$0.109334\pi$
$967$	20.9448i	0.673538i	−0.941587	+	0.336769i	$0.890666\pi$
$968$	0	0
$969$	6.58954	0.211687
$970$	0	0
$971$	27.8694	0.894371	0.447186	−	0.894441i	$-0.352426\pi$
$971$	27.8694	0.894371	0.447186	+	0.894441i	$0.352426\pi$
$972$	0	0
$973$	12.3725i	0.396644i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 56.1379i	− 1.79601i	−0.439985	−	0.898005i	$-0.645016\pi$
$977$	− 56.1379i	− 1.79601i	0.439985	−	0.898005i	$-0.354984\pi$
$978$	0	0
$979$	−22.4850	−0.718625
$980$	0	0
$981$	68.0436	2.17247
$982$	0	0
$983$	9.70948i	0.309684i	0.987939	+	0.154842i	$0.0494869\pi$
$983$	9.70948i	0.309684i	−0.987939	+	0.154842i	$0.950513\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 5.19577i	− 0.165383i
$988$	0	0
$989$	−6.26813	−0.199315
$990$	0	0
$991$	−44.7952	−1.42297	−0.711483	−	0.702703i	$-0.751976\pi$
$991$	−44.7952	−1.42297	−0.711483	+	0.702703i	$0.751976\pi$
$992$	0	0
$993$	55.7891i	1.77041i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 50.2907i	− 1.59272i	−0.604822	−	0.796361i	$-0.706756\pi$
$997$	− 50.2907i	− 1.59272i	0.604822	−	0.796361i	$-0.293244\pi$
$998$	0	0
$999$	−6.09508	−0.192840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3700.2.d.k.149.2		12
5.2	odd	4		3700.2.a.m.1.1	✓	6
5.3	odd	4		3700.2.a.n.1.6	yes	6
5.4	even	2	inner	3700.2.d.k.149.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3700.2.a.m.1.1	✓	6	5.2	odd	4
3700.2.a.n.1.6	yes	6	5.3	odd	4
3700.2.d.k.149.2		12	1.1	even	1	trivial
3700.2.d.k.149.11		12	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.85250i q^{3} +2.47048i q^{7} -5.13675 q^{9} +1.45305 q^{11} +0.691767i q^{13} -1.07378i q^{17} -2.15136 q^{19} +7.04705 q^{21} -5.00668i q^{23} +6.09508i q^{27} -4.85250 q^{29} -3.84313 q^{31} -4.14481i q^{33} +1.00000i q^{37} +1.97326 q^{39} -8.90340 q^{41} -1.25195i q^{43} -0.737297i q^{47} +0.896719 q^{49} -3.06297 q^{51} -0.399453i q^{53} +6.13675i q^{57} -5.08846 q^{59} +1.84313 q^{61} -12.6902i q^{63} +10.5297i q^{67} -14.2816 q^{69} -8.84326 q^{71} +7.90223i q^{73} +3.58972i q^{77} -1.16218 q^{79} +1.97595 q^{81} -4.08426i q^{83} +13.8417i q^{87} -15.4744 q^{89} -1.70900 q^{91} +10.9625i q^{93} -10.4542i q^{97} -7.46394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 18 q^{9} - 16 q^{11} - 4 q^{19} + 20 q^{21} - 26 q^{29} - 6 q^{31} - 24 q^{39} + 44 q^{41} - 64 q^{49} - 10 q^{51} - 26 q^{59} - 18 q^{61} - 76 q^{69} + 40 q^{71} - 28 q^{79} + 60 q^{81} - 32 q^{89}+ \cdots - 26 q^{99}+O(q^{100}) \) 12 * q - 18 * q^9 - 16 * q^11 - 4 * q^19 + 20 * q^21 - 26 * q^29 - 6 * q^31 - 24 * q^39 + 44 * q^41 - 64 * q^49 - 10 * q^51 - 26 * q^59 - 18 * q^61 - 76 * q^69 + 40 * q^71 - 28 * q^79 + 60 * q^81 - 32 * q^89 - 56 * q^91 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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