LMFDB - Embedded newform 3872.2.c.i.1937.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3872,2,Mod(1937,3872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3872, 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("3872.1937");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3872 = 2^{5} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3872.c (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:	$30.9180756626$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} - 2 x^{16} - 2 x^{15} - 4 x^{14} - 4 x^{13} + 12 x^{12} + 16 x^{11} + 32 x^{9} + \cdots + 1024 $ x^20 - x^18 - 2x^16 - 2x^15 - 4x^14 - 4x^13 + 12x^12 + 16x^11 + 32x^9 + 48x^8 - 32x^7 - 64x^6 - 64x^5 - 128x^4 - 256*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1937.8
Root			$-0.549610 - 1.30305i$ of defining polynomial
Character	$\chi$	$=$	3872.1937
Dual form			3872.2.c.i.1937.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.612207i q^{3} -1.46030i q^{5} +4.42813 q^{7} +2.62520 q^{9} -5.57312i q^{13} -0.894002 q^{15} +2.55806 q^{17} +3.01989i q^{19} -2.71093i q^{21} +3.70783 q^{23} +2.86754 q^{25} -3.44379i q^{27} -3.65947i q^{29} -1.49166 q^{31} -6.46637i q^{35} +1.48880i q^{37} -3.41190 q^{39} -3.14210 q^{41} +6.25559i q^{43} -3.83357i q^{45} +2.65892 q^{47} +12.6083 q^{49} -1.56606i q^{51} -9.79852i q^{53} +1.84880 q^{57} +2.57763i q^{59} +5.58617i q^{61} +11.6247 q^{63} -8.13840 q^{65} +11.5059i q^{67} -2.26996i q^{69} -1.12369 q^{71} -14.1221 q^{73} -1.75553i q^{75} +0.766251 q^{79} +5.76730 q^{81} +8.56399i q^{83} -3.73552i q^{85} -2.24035 q^{87} -8.84290 q^{89} -24.6785i q^{91} +0.913204i q^{93} +4.40994 q^{95} -0.298285 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 10 q^{7} - 10 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{23} - 2 q^{25} - 2 q^{31} + 28 q^{39} + 2 q^{41} - 2 q^{47} - 2 q^{49} - 22 q^{57} - 30 q^{63} + 18 q^{65} + 34 q^{71} - 2 q^{73} - 58 q^{79} - 12 q^{81}+ \cdots + 10 q^{97}+O(q^{100}) $ 20 * q + 10 * q^7 - 10 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^23 - 2 * q^25 - 2 * q^31 + 28 * q^39 + 2 * q^41 - 2 * q^47 - 2 * q^49 - 22 * q^57 - 30 * q^63 + 18 * q^65 + 34 * q^71 - 2 * q^73 - 58 * q^79 - 12 * q^81 + 34 * q^87 - 8 * q^89 - 92 * q^95 + 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times$.

$n$	$485$	$1695$	$2785$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 0.612207i	− 0.353458i	−0.984260	−	0.176729i	$-0.943448\pi$
$3$	− 0.612207i	− 0.353458i	0.984260	−	0.176729i	$-0.0565516\pi$
$4$	0	0
$5$	− 1.46030i	− 0.653064i	−0.945186	−	0.326532i	$-0.894120\pi$
$5$	− 1.46030i	− 0.653064i	0.945186	−	0.326532i	$-0.105880\pi$
$6$	0	0
$7$	4.42813	1.67368	0.836838	−	0.547451i	$-0.184402\pi$
$7$	4.42813	1.67368	0.836838	+	0.547451i	$0.184402\pi$
$8$	0	0
$9$	2.62520	0.875068
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 5.57312i	− 1.54571i	−0.634586	−	0.772853i	$-0.718829\pi$
$13$	− 5.57312i	− 1.54571i	0.634586	−	0.772853i	$-0.281171\pi$
$14$	0	0
$15$	−0.894002	−0.230830
$16$	0	0
$17$	2.55806	0.620420	0.310210	−	0.950668i	$-0.399601\pi$
$17$	2.55806	0.620420	0.310210	+	0.950668i	$0.399601\pi$
$18$	0	0
$19$	3.01989i	0.692811i	0.938085	+	0.346406i	$0.112598\pi$
$19$	3.01989i	0.692811i	−0.938085	+	0.346406i	$0.887402\pi$
$20$	0	0
$21$	− 2.71093i	− 0.591573i
$22$	0	0
$23$	3.70783	0.773136	0.386568	−	0.922261i	$-0.373661\pi$
$23$	3.70783	0.773136	0.386568	+	0.922261i	$0.373661\pi$
$24$	0	0
$25$	2.86754	0.573508
$26$	0	0
$27$	− 3.44379i	− 0.662757i
$28$	0	0
$29$	− 3.65947i	− 0.679546i	−0.940507	−	0.339773i	$-0.889650\pi$
$29$	− 3.65947i	− 0.679546i	0.940507	−	0.339773i	$-0.110350\pi$
$30$	0	0
$31$	−1.49166	−0.267910	−0.133955	−	0.990987i	$-0.542768\pi$
$31$	−1.49166	−0.267910	−0.133955	+	0.990987i	$0.542768\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 6.46637i	− 1.09302i
$36$	0	0
$37$	1.48880i	0.244757i	0.992483	+	0.122379i	$0.0390522\pi$
$37$	1.48880i	0.244757i	−0.992483	+	0.122379i	$0.960948\pi$
$38$	0	0
$39$	−3.41190	−0.546341
$40$	0	0
$41$	−3.14210	−0.490713	−0.245357	−	0.969433i	$-0.578905\pi$
$41$	−3.14210	−0.490713	−0.245357	+	0.969433i	$0.578905\pi$
$42$	0	0
$43$	6.25559i	0.953968i	0.878912	+	0.476984i	$0.158270\pi$
$43$	6.25559i	0.953968i	−0.878912	+	0.476984i	$0.841730\pi$
$44$	0	0
$45$	− 3.83357i	− 0.571475i
$46$	0	0
$47$	2.65892	0.387844	0.193922	−	0.981017i	$-0.437879\pi$
$47$	2.65892	0.387844	0.193922	+	0.981017i	$0.437879\pi$
$48$	0	0
$49$	12.6083	1.80119
$50$	0	0
$51$	− 1.56606i	− 0.219292i
$52$	0	0
$53$	− 9.79852i	− 1.34593i	−0.739674	−	0.672965i	$-0.765020\pi$
$53$	− 9.79852i	− 1.34593i	0.739674	−	0.672965i	$-0.234980\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.84880	0.244879
$58$	0	0
$59$	2.57763i	0.335579i	0.985823	+	0.167789i	$0.0536629\pi$
$59$	2.57763i	0.335579i	−0.985823	+	0.167789i	$0.946337\pi$
$60$	0	0
$61$	5.58617i	0.715235i	0.933868	+	0.357618i	$0.116411\pi$
$61$	5.58617i	0.715235i	−0.933868	+	0.357618i	$0.883589\pi$
$62$	0	0
$63$	11.6247	1.46458
$64$	0	0
$65$	−8.13840	−1.00944
$66$	0	0
$67$	11.5059i	1.40566i	0.711356	+	0.702832i	$0.248081\pi$
$67$	11.5059i	1.40566i	−0.711356	+	0.702832i	$0.751919\pi$
$68$	0	0
$69$	− 2.26996i	− 0.273271i
$70$	0	0
$71$	−1.12369	−0.133358	−0.0666788	−	0.997774i	$-0.521240\pi$
$71$	−1.12369	−0.133358	−0.0666788	+	0.997774i	$0.521240\pi$
$72$	0	0
$73$	−14.1221	−1.65287	−0.826436	−	0.563031i	$-0.809635\pi$
$73$	−14.1221	−1.65287	−0.826436	+	0.563031i	$0.809635\pi$
$74$	0	0
$75$	− 1.75553i	− 0.202711i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.766251	0.0862100	0.0431050	−	0.999071i	$-0.486275\pi$
$79$	0.766251	0.0862100	0.0431050	+	0.999071i	$0.486275\pi$
$80$	0	0
$81$	5.76730	0.640811
$82$	0	0
$83$	8.56399i	0.940020i	0.882661	+	0.470010i	$0.155750\pi$
$83$	8.56399i	0.940020i	−0.882661	+	0.470010i	$0.844250\pi$
$84$	0	0
$85$	− 3.73552i	− 0.405174i
$86$	0	0
$87$	−2.24035	−0.240191
$88$	0	0
$89$	−8.84290	−0.937346	−0.468673	−	0.883372i	$-0.655268\pi$
$89$	−8.84290	−0.937346	−0.468673	+	0.883372i	$0.655268\pi$
$90$	0	0
$91$	− 24.6785i	− 2.58701i
$92$	0	0
$93$	0.913204i	0.0946949i
$94$	0	0
$95$	4.40994	0.452450
$96$	0	0
$97$	−0.298285	−0.0302862	−0.0151431	−	0.999885i	$-0.504820\pi$
$97$	−0.298285	−0.0302862	−0.0151431	+	0.999885i	$0.504820\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.63142i	0.261836i	0.991393	+	0.130918i	$0.0417924\pi$
$101$	2.63142i	0.261836i	−0.991393	+	0.130918i	$0.958208\pi$
$102$	0	0
$103$	2.26365	0.223044	0.111522	−	0.993762i	$-0.464427\pi$
$103$	2.26365	0.223044	0.111522	+	0.993762i	$0.464427\pi$
$104$	0	0
$105$	−3.95876	−0.386335
$106$	0	0
$107$	− 6.04559i	− 0.584450i	−0.956350	−	0.292225i	$-0.905604\pi$
$107$	− 6.04559i	− 0.584450i	0.956350	−	0.292225i	$-0.0943956\pi$
$108$	0	0
$109$	11.9381i	1.14346i	0.820440	+	0.571732i	$0.193728\pi$
$109$	11.9381i	1.14346i	−0.820440	+	0.571732i	$0.806272\pi$
$110$	0	0
$111$	0.911454	0.0865113
$112$	0	0
$113$	12.4098	1.16741	0.583706	−	0.811965i	$-0.301602\pi$
$113$	12.4098	1.16741	0.583706	+	0.811965i	$0.301602\pi$
$114$	0	0
$115$	− 5.41452i	− 0.504907i
$116$	0	0
$117$	− 14.6306i	− 1.35260i
$118$	0	0
$119$	11.3274	1.03838
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1.92361i	0.173446i
$124$	0	0
$125$	− 11.4889i	− 1.02760i
$126$	0	0
$127$	−8.91108	−0.790730	−0.395365	−	0.918524i	$-0.629382\pi$
$127$	−8.91108	−0.790730	−0.395365	+	0.918524i	$0.629382\pi$
$128$	0	0
$129$	3.82971	0.337187
$130$	0	0
$131$	8.30263i	0.725404i	0.931905	+	0.362702i	$0.118146\pi$
$131$	8.30263i	0.725404i	−0.931905	+	0.362702i	$0.881854\pi$
$132$	0	0
$133$	13.3725i	1.15954i
$134$	0	0
$135$	−5.02894	−0.432823
$136$	0	0
$137$	−16.5315	−1.41238	−0.706192	−	0.708020i	$-0.749589\pi$
$137$	−16.5315	−1.41238	−0.706192	+	0.708020i	$0.749589\pi$
$138$	0	0
$139$	16.8902i	1.43261i	0.697788	+	0.716305i	$0.254168\pi$
$139$	16.8902i	1.43261i	−0.697788	+	0.716305i	$0.745832\pi$
$140$	0	0
$141$	− 1.62781i	− 0.137086i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.34391	−0.443787
$146$	0	0
$147$	− 7.71890i	− 0.636644i
$148$	0	0
$149$	19.0320i	1.55916i	0.626303	+	0.779580i	$0.284567\pi$
$149$	19.0320i	1.55916i	−0.626303	+	0.779580i	$0.715433\pi$
$150$	0	0
$151$	−10.1613	−0.826914	−0.413457	−	0.910524i	$-0.635679\pi$
$151$	−10.1613	−0.826914	−0.413457	+	0.910524i	$0.635679\pi$
$152$	0	0
$153$	6.71542	0.542909
$154$	0	0
$155$	2.17826i	0.174962i
$156$	0	0
$157$	− 16.7488i	− 1.33670i	−0.743848	−	0.668349i	$-0.767001\pi$
$157$	− 16.7488i	− 1.33670i	0.743848	−	0.668349i	$-0.232999\pi$
$158$	0	0
$159$	−5.99872	−0.475729
$160$	0	0
$161$	16.4187	1.29398
$162$	0	0
$163$	− 6.97143i	− 0.546045i	−0.962008	−	0.273022i	$-0.911977\pi$
$163$	− 6.97143i	− 0.546045i	0.962008	−	0.273022i	$-0.0880233\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.99355	0.773324	0.386662	−	0.922222i	$-0.373628\pi$
$167$	9.99355	0.773324	0.386662	+	0.922222i	$0.373628\pi$
$168$	0	0
$169$	−18.0597	−1.38920
$170$	0	0
$171$	7.92783i	0.606257i
$172$	0	0
$173$	− 3.55549i	− 0.270319i	−0.990824	−	0.135159i	$-0.956845\pi$
$173$	− 3.55549i	− 0.270319i	0.990824	−	0.135159i	$-0.0431546\pi$
$174$	0	0
$175$	12.6978	0.959866
$176$	0	0
$177$	1.57804	0.118613
$178$	0	0
$179$	− 5.39181i	− 0.403003i	−0.979488	−	0.201501i	$-0.935418\pi$
$179$	− 5.39181i	− 0.403003i	0.979488	−	0.201501i	$-0.0645820\pi$
$180$	0	0
$181$	− 4.01052i	− 0.298100i	−0.988830	−	0.149050i	$-0.952379\pi$
$181$	− 4.01052i	− 0.298100i	0.988830	−	0.149050i	$-0.0476215\pi$
$182$	0	0
$183$	3.41989	0.252805
$184$	0	0
$185$	2.17409	0.159842
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 15.2495i	− 1.10924i
$190$	0	0
$191$	−0.569284	−0.0411920	−0.0205960	−	0.999788i	$-0.506556\pi$
$191$	−0.569284	−0.0411920	−0.0205960	+	0.999788i	$0.506556\pi$
$192$	0	0
$193$	−7.11655	−0.512261	−0.256130	−	0.966642i	$-0.582448\pi$
$193$	−7.11655	−0.512261	−0.256130	+	0.966642i	$0.582448\pi$
$194$	0	0
$195$	4.98238i	0.356796i
$196$	0	0
$197$	− 1.44252i	− 0.102775i	−0.998679	−	0.0513876i	$-0.983636\pi$
$197$	− 1.44252i	− 0.102775i	0.998679	−	0.0513876i	$-0.0163644\pi$
$198$	0	0
$199$	−0.847005	−0.0600426	−0.0300213	−	0.999549i	$-0.509558\pi$
$199$	−0.847005	−0.0600426	−0.0300213	+	0.999549i	$0.509558\pi$
$200$	0	0
$201$	7.04396	0.496843
$202$	0	0
$203$	− 16.2046i	− 1.13734i
$204$	0	0
$205$	4.58839i	0.320467i
$206$	0	0
$207$	9.73380	0.676546
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 15.1087i	− 1.04012i	−0.854128	−	0.520062i	$-0.825909\pi$
$211$	− 15.1087i	− 1.04012i	0.854128	−	0.520062i	$-0.174091\pi$
$212$	0	0
$213$	0.687931i	0.0471363i
$214$	0	0
$215$	9.13500	0.623002
$216$	0	0
$217$	−6.60526	−0.448395
$218$	0	0
$219$	8.64567i	0.584220i
$220$	0	0
$221$	− 14.2564i	− 0.958986i
$222$	0	0
$223$	18.7075	1.25275	0.626373	−	0.779523i	$-0.284539\pi$
$223$	18.7075	1.25275	0.626373	+	0.779523i	$0.284539\pi$
$224$	0	0
$225$	7.52787	0.501858
$226$	0	0
$227$	− 27.1008i	− 1.79874i	−0.437186	−	0.899371i	$-0.644025\pi$
$227$	− 27.1008i	− 1.79874i	0.437186	−	0.899371i	$-0.355975\pi$
$228$	0	0
$229$	− 11.6235i	− 0.768101i	−0.923312	−	0.384051i	$-0.874529\pi$
$229$	− 11.6235i	− 0.768101i	0.923312	−	0.384051i	$-0.125471\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.1177	1.71103	0.855514	−	0.517780i	$-0.173242\pi$
$233$	26.1177	1.71103	0.855514	+	0.517780i	$0.173242\pi$
$234$	0	0
$235$	− 3.88281i	− 0.253287i
$236$	0	0
$237$	− 0.469104i	− 0.0304716i
$238$	0	0
$239$	−14.1891	−0.917817	−0.458909	−	0.888483i	$-0.651759\pi$
$239$	−14.1891	−0.917817	−0.458909	+	0.888483i	$0.651759\pi$
$240$	0	0
$241$	−17.2505	−1.11120	−0.555602	−	0.831448i	$-0.687512\pi$
$241$	−17.2505	−1.11120	−0.555602	+	0.831448i	$0.687512\pi$
$242$	0	0
$243$	− 13.8621i	− 0.889257i
$244$	0	0
$245$	− 18.4119i	− 1.17629i
$246$	0	0
$247$	16.8302	1.07088
$248$	0	0
$249$	5.24293	0.332257
$250$	0	0
$251$	− 20.3207i	− 1.28263i	−0.767277	−	0.641316i	$-0.778389\pi$
$251$	− 20.3207i	− 1.28263i	0.767277	−	0.641316i	$-0.221611\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.28691	−0.143212
$256$	0	0
$257$	15.0884	0.941190	0.470595	−	0.882349i	$-0.344039\pi$
$257$	15.0884	0.941190	0.470595	+	0.882349i	$0.344039\pi$
$258$	0	0
$259$	6.59260i	0.409644i
$260$	0	0
$261$	− 9.60685i	− 0.594649i
$262$	0	0
$263$	−10.8242	−0.667448	−0.333724	−	0.942671i	$-0.608305\pi$
$263$	−10.8242	−0.667448	−0.333724	+	0.942671i	$0.608305\pi$
$264$	0	0
$265$	−14.3087	−0.878979
$266$	0	0
$267$	5.41368i	0.331312i
$268$	0	0
$269$	− 19.5769i	− 1.19363i	−0.802381	−	0.596813i	$-0.796434\pi$
$269$	− 19.5769i	− 1.19363i	0.802381	−	0.596813i	$-0.203566\pi$
$270$	0	0
$271$	−25.8557	−1.57062	−0.785311	−	0.619102i	$-0.787497\pi$
$271$	−25.8557	−1.57062	−0.785311	+	0.619102i	$0.787497\pi$
$272$	0	0
$273$	−15.1083	−0.914398
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 24.5001i	− 1.47207i	−0.676943	−	0.736035i	$-0.736696\pi$
$277$	− 24.5001i	− 1.47207i	0.676943	−	0.736035i	$-0.263304\pi$
$278$	0	0
$279$	−3.91591	−0.234440
$280$	0	0
$281$	−9.32504	−0.556285	−0.278143	−	0.960540i	$-0.589719\pi$
$281$	−9.32504	−0.556285	−0.278143	+	0.960540i	$0.589719\pi$
$282$	0	0
$283$	18.9916i	1.12893i	0.825456	+	0.564466i	$0.190918\pi$
$283$	18.9916i	1.12893i	−0.825456	+	0.564466i	$0.809082\pi$
$284$	0	0
$285$	− 2.69979i	− 0.159922i
$286$	0	0
$287$	−13.9136	−0.821295
$288$	0	0
$289$	−10.4563	−0.615079
$290$	0	0
$291$	0.182612i	0.0107049i
$292$	0	0
$293$	− 5.91337i	− 0.345463i	−0.984969	−	0.172731i	$-0.944741\pi$
$293$	− 5.91337i	− 0.345463i	0.984969	−	0.172731i	$-0.0552593\pi$
$294$	0	0
$295$	3.76410	0.219154
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 20.6642i	− 1.19504i
$300$	0	0
$301$	27.7005i	1.59663i
$302$	0	0
$303$	1.61097	0.0925478
$304$	0	0
$305$	8.15745	0.467094
$306$	0	0
$307$	− 13.6216i	− 0.777423i	−0.921360	−	0.388711i	$-0.872920\pi$
$307$	− 13.6216i	− 0.777423i	0.921360	−	0.388711i	$-0.127080\pi$
$308$	0	0
$309$	− 1.38582i	− 0.0788365i
$310$	0	0
$311$	−14.1842	−0.804312	−0.402156	−	0.915571i	$-0.631739\pi$
$311$	−14.1842	−0.804312	−0.402156	+	0.915571i	$0.631739\pi$
$312$	0	0
$313$	−2.16820	−0.122554	−0.0612770	−	0.998121i	$-0.519517\pi$
$313$	−2.16820	−0.122554	−0.0612770	+	0.998121i	$0.519517\pi$
$314$	0	0
$315$	− 16.9755i	− 0.956464i
$316$	0	0
$317$	14.1656i	0.795619i	0.917468	+	0.397809i	$0.130229\pi$
$317$	14.1656i	0.795619i	−0.917468	+	0.397809i	$0.869771\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.70115	−0.206578
$322$	0	0
$323$	7.72506i	0.429834i
$324$	0	0
$325$	− 15.9811i	− 0.886474i
$326$	0	0
$327$	7.30860	0.404166
$328$	0	0
$329$	11.7740	0.649124
$330$	0	0
$331$	10.7862i	0.592865i	0.955054	+	0.296433i	$0.0957970\pi$
$331$	10.7862i	0.592865i	−0.955054	+	0.296433i	$0.904203\pi$
$332$	0	0
$333$	3.90840i	0.214179i
$334$	0	0
$335$	16.8019	0.917988
$336$	0	0
$337$	6.28480	0.342355	0.171177	−	0.985240i	$-0.445243\pi$
$337$	6.28480	0.342355	0.171177	+	0.985240i	$0.445243\pi$
$338$	0	0
$339$	− 7.59734i	− 0.412631i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	24.8344	1.34093
$344$	0	0
$345$	−3.31481	−0.178463
$346$	0	0
$347$	− 17.3557i	− 0.931701i	−0.884863	−	0.465851i	$-0.845748\pi$
$347$	− 17.3557i	− 0.931701i	0.884863	−	0.465851i	$-0.154252\pi$
$348$	0	0
$349$	24.8420i	1.32976i	0.746950	+	0.664880i	$0.231517\pi$
$349$	24.8420i	1.32976i	−0.746950	+	0.664880i	$0.768483\pi$
$350$	0	0
$351$	−19.1926	−1.02443
$352$	0	0
$353$	17.9063	0.953056	0.476528	−	0.879159i	$-0.341895\pi$
$353$	17.9063	0.953056	0.476528	+	0.879159i	$0.341895\pi$
$354$	0	0
$355$	1.64092i	0.0870911i
$356$	0	0
$357$	− 6.93471i	− 0.367024i
$358$	0	0
$359$	14.7140	0.776577	0.388288	−	0.921538i	$-0.373066\pi$
$359$	14.7140	0.776577	0.388288	+	0.921538i	$0.373066\pi$
$360$	0	0
$361$	9.88024	0.520013
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.6225i	1.07943i
$366$	0	0
$367$	31.4422	1.64127	0.820635	−	0.571453i	$-0.193620\pi$
$367$	31.4422	1.64127	0.820635	+	0.571453i	$0.193620\pi$
$368$	0	0
$369$	−8.24865	−0.429407
$370$	0	0
$371$	− 43.3891i	− 2.25265i
$372$	0	0
$373$	20.5892i	1.06607i	0.846094	+	0.533034i	$0.178948\pi$
$373$	20.5892i	1.06607i	−0.846094	+	0.533034i	$0.821052\pi$
$374$	0	0
$375$	−7.03360	−0.363213
$376$	0	0
$377$	−20.3947	−1.05038
$378$	0	0
$379$	13.0063i	0.668087i	0.942558	+	0.334044i	$0.108413\pi$
$379$	13.0063i	0.668087i	−0.942558	+	0.334044i	$0.891587\pi$
$380$	0	0
$381$	5.45542i	0.279490i
$382$	0	0
$383$	−37.1361	−1.89757	−0.948783	−	0.315929i	$-0.897684\pi$
$383$	−37.1361	−1.89757	−0.948783	+	0.315929i	$0.897684\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.4222i	0.834787i
$388$	0	0
$389$	− 3.54956i	− 0.179970i	−0.995943	−	0.0899849i	$-0.971318\pi$
$389$	− 3.54956i	− 0.179970i	0.995943	−	0.0899849i	$-0.0286819\pi$
$390$	0	0
$391$	9.48483	0.479669
$392$	0	0
$393$	5.08293	0.256400
$394$	0	0
$395$	− 1.11895i	− 0.0563006i
$396$	0	0
$397$	35.1583i	1.76454i	0.470742	+	0.882271i	$0.343986\pi$
$397$	35.1583i	1.76454i	−0.470742	+	0.882271i	$0.656014\pi$
$398$	0	0
$399$	8.18672	0.409849
$400$	0	0
$401$	14.2497	0.711597	0.355799	−	0.934563i	$-0.384209\pi$
$401$	14.2497	0.711597	0.355799	+	0.934563i	$0.384209\pi$
$402$	0	0
$403$	8.31320i	0.414110i
$404$	0	0
$405$	− 8.42196i	− 0.418491i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.3906	−0.513783	−0.256892	−	0.966440i	$-0.582698\pi$
$409$	−10.3906	−0.513783	−0.256892	+	0.966440i	$0.582698\pi$
$410$	0	0
$411$	10.1207i	0.499218i
$412$	0	0
$413$	11.4141i	0.561650i
$414$	0	0
$415$	12.5060	0.613893
$416$	0	0
$417$	10.3403	0.506367
$418$	0	0
$419$	− 20.7586i	− 1.01412i	−0.861910	−	0.507062i	$-0.830732\pi$
$419$	− 20.7586i	− 1.01412i	0.861910	−	0.507062i	$-0.169268\pi$
$420$	0	0
$421$	0.165219i	0.00805227i	0.999992	+	0.00402614i	$0.00128156\pi$
$421$	0.165219i	0.00805227i	−0.999992	+	0.00402614i	$0.998718\pi$
$422$	0	0
$423$	6.98021	0.339389
$424$	0	0
$425$	7.33533	0.355816
$426$	0	0
$427$	24.7363i	1.19707i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.8135	−1.62874	−0.814370	−	0.580345i	$-0.802918\pi$
$431$	−33.8135	−1.62874	−0.814370	+	0.580345i	$0.802918\pi$
$432$	0	0
$433$	−5.39449	−0.259243	−0.129621	−	0.991564i	$-0.541376\pi$
$433$	−5.39449	−0.259243	−0.129621	+	0.991564i	$0.541376\pi$
$434$	0	0
$435$	3.27157i	0.156860i
$436$	0	0
$437$	11.1972i	0.535637i
$438$	0	0
$439$	0.0455090	0.00217203	0.00108601	−	0.999999i	$-0.499654\pi$
$439$	0.0455090	0.00217203	0.00108601	+	0.999999i	$0.499654\pi$
$440$	0	0
$441$	33.0994	1.57616
$442$	0	0
$443$	0.0874300i	0.00415393i	0.999998	+	0.00207696i	$0.000661118\pi$
$443$	0.0874300i	0.00415393i	−0.999998	+	0.00207696i	$0.999339\pi$
$444$	0	0
$445$	12.9132i	0.612147i
$446$	0	0
$447$	11.6515	0.551097
$448$	0	0
$449$	29.9107	1.41157	0.705787	−	0.708424i	$-0.250594\pi$
$449$	29.9107	1.41157	0.705787	+	0.708424i	$0.250594\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	6.22081i	0.292279i
$454$	0	0
$455$	−36.0379	−1.68948
$456$	0	0
$457$	21.5081	1.00611	0.503053	−	0.864255i	$-0.332210\pi$
$457$	21.5081	1.00611	0.503053	+	0.864255i	$0.332210\pi$
$458$	0	0
$459$	− 8.80940i	− 0.411188i
$460$	0	0
$461$	7.88518i	0.367250i	0.982996	+	0.183625i	$0.0587832\pi$
$461$	7.88518i	0.367250i	−0.982996	+	0.183625i	$0.941217\pi$
$462$	0	0
$463$	−5.12007	−0.237950	−0.118975	−	0.992897i	$-0.537961\pi$
$463$	−5.12007	−0.237950	−0.118975	+	0.992897i	$0.537961\pi$
$464$	0	0
$465$	1.33355	0.0618418
$466$	0	0
$467$	18.6252i	0.861873i	0.902382	+	0.430936i	$0.141817\pi$
$467$	18.6252i	0.861873i	−0.902382	+	0.430936i	$0.858183\pi$
$468$	0	0
$469$	50.9494i	2.35262i
$470$	0	0
$471$	−10.2537	−0.472466
$472$	0	0
$473$	0	0
$474$	0	0
$475$	8.65966i	0.397332i
$476$	0	0
$477$	− 25.7231i	− 1.17778i
$478$	0	0
$479$	−39.2942	−1.79540	−0.897698	−	0.440611i	$-0.854762\pi$
$479$	−39.2942	−1.79540	−0.897698	+	0.440611i	$0.854762\pi$
$480$	0	0
$481$	8.29726	0.378323
$482$	0	0
$483$	− 10.0517i	− 0.457366i
$484$	0	0
$485$	0.435584i	0.0197788i
$486$	0	0
$487$	18.2826	0.828466	0.414233	−	0.910171i	$-0.364050\pi$
$487$	18.2826	0.828466	0.414233	+	0.910171i	$0.364050\pi$
$488$	0	0
$489$	−4.26796	−0.193004
$490$	0	0
$491$	4.10219i	0.185129i	0.995707	+	0.0925646i	$0.0295065\pi$
$491$	4.10219i	0.185129i	−0.995707	+	0.0925646i	$0.970494\pi$
$492$	0	0
$493$	− 9.36113i	− 0.421604i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.97585	−0.223197
$498$	0	0
$499$	9.37476i	0.419672i	0.977737	+	0.209836i	$0.0672930\pi$
$499$	9.37476i	0.419672i	−0.977737	+	0.209836i	$0.932707\pi$
$500$	0	0
$501$	− 6.11811i	− 0.273337i
$502$	0	0
$503$	−14.4704	−0.645202	−0.322601	−	0.946535i	$-0.604557\pi$
$503$	−14.4704	−0.645202	−0.322601	+	0.946535i	$0.604557\pi$
$504$	0	0
$505$	3.84264	0.170995
$506$	0	0
$507$	11.0562i	0.491025i
$508$	0	0
$509$	5.70909i	0.253051i	0.991963	+	0.126526i	$0.0403826\pi$
$509$	5.70909i	0.253051i	−0.991963	+	0.126526i	$0.959617\pi$
$510$	0	0
$511$	−62.5347	−2.76637
$512$	0	0
$513$	10.3999	0.459165
$514$	0	0
$515$	− 3.30559i	− 0.145662i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.17669	−0.0955462
$520$	0	0
$521$	33.9942	1.48931	0.744655	−	0.667449i	$-0.232614\pi$
$521$	33.9942	1.48931	0.744655	+	0.667449i	$0.232614\pi$
$522$	0	0
$523$	34.5786i	1.51201i	0.654563	+	0.756007i	$0.272853\pi$
$523$	34.5786i	1.51201i	−0.654563	+	0.756007i	$0.727147\pi$
$524$	0	0
$525$	− 7.77369i	− 0.339272i
$526$	0	0
$527$	−3.81575	−0.166217
$528$	0	0
$529$	−9.25201	−0.402261
$530$	0	0
$531$	6.76680i	0.293654i
$532$	0	0
$533$	17.5113i	0.758498i
$534$	0	0
$535$	−8.82835	−0.381683
$536$	0	0
$537$	−3.30090	−0.142444
$538$	0	0
$539$	0	0
$540$	0	0
$541$	29.0659i	1.24964i	0.780769	+	0.624820i	$0.214828\pi$
$541$	29.0659i	1.24964i	−0.780769	+	0.624820i	$0.785172\pi$
$542$	0	0
$543$	−2.45527	−0.105366
$544$	0	0
$545$	17.4332	0.746755
$546$	0	0
$547$	13.4006i	0.572968i	0.958085	+	0.286484i	$0.0924866\pi$
$547$	13.4006i	0.572968i	−0.958085	+	0.286484i	$0.907513\pi$
$548$	0	0
$549$	14.6648i	0.625879i
$550$	0	0
$551$	11.0512	0.470797
$552$	0	0
$553$	3.39306	0.144288
$554$	0	0
$555$	− 1.33099i	− 0.0564974i
$556$	0	0
$557$	0.917195i	0.0388628i	0.999811	+	0.0194314i	$0.00618560\pi$
$557$	0.917195i	0.0388628i	−0.999811	+	0.0194314i	$0.993814\pi$
$558$	0	0
$559$	34.8631	1.47455
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.4995i	1.03253i	0.856429	+	0.516265i	$0.172678\pi$
$563$	24.4995i	1.03253i	−0.856429	+	0.516265i	$0.827322\pi$
$564$	0	0
$565$	− 18.1219i	− 0.762395i
$566$	0	0
$567$	25.5383	1.07251
$568$	0	0
$569$	−3.69958	−0.155094	−0.0775472	−	0.996989i	$-0.524709\pi$
$569$	−3.69958	−0.155094	−0.0775472	+	0.996989i	$0.524709\pi$
$570$	0	0
$571$	− 22.3627i	− 0.935849i	−0.883768	−	0.467925i	$-0.845002\pi$
$571$	− 22.3627i	− 0.935849i	0.883768	−	0.467925i	$-0.154998\pi$
$572$	0	0
$573$	0.348519i	0.0145596i
$574$	0	0
$575$	10.6323	0.443399
$576$	0	0
$577$	−33.8245	−1.40813	−0.704065	−	0.710135i	$-0.748634\pi$
$577$	−33.8245	−1.40813	−0.704065	+	0.710135i	$0.748634\pi$
$578$	0	0
$579$	4.35680i	0.181062i
$580$	0	0
$581$	37.9224i	1.57329i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−21.3649	−0.883332
$586$	0	0
$587$	− 44.1660i	− 1.82293i	−0.411380	−	0.911464i	$-0.634953\pi$
$587$	− 44.1660i	− 1.82293i	0.411380	−	0.911464i	$-0.365047\pi$
$588$	0	0
$589$	− 4.50466i	− 0.185611i
$590$	0	0
$591$	−0.883119	−0.0363267
$592$	0	0
$593$	13.4081	0.550605	0.275303	−	0.961358i	$-0.411222\pi$
$593$	13.4081	0.550605	0.275303	+	0.961358i	$0.411222\pi$
$594$	0	0
$595$	− 16.5414i	− 0.678129i
$596$	0	0
$597$	0.518542i	0.0212225i
$598$	0	0
$599$	38.1205	1.55756	0.778780	−	0.627297i	$-0.215839\pi$
$599$	38.1205	1.55756	0.778780	+	0.627297i	$0.215839\pi$
$600$	0	0
$601$	−5.00586	−0.204193	−0.102097	−	0.994774i	$-0.532555\pi$
$601$	−5.00586	−0.204193	−0.102097	+	0.994774i	$0.532555\pi$
$602$	0	0
$603$	30.2052i	1.23005i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−29.4921	−1.19705	−0.598523	−	0.801106i	$-0.704245\pi$
$607$	−29.4921	−1.19705	−0.598523	+	0.801106i	$0.704245\pi$
$608$	0	0
$609$	−9.92056	−0.402002
$610$	0	0
$611$	− 14.8185i	− 0.599492i
$612$	0	0
$613$	− 18.9375i	− 0.764879i	−0.923981	−	0.382439i	$-0.875084\pi$
$613$	− 18.9375i	− 0.764879i	0.923981	−	0.382439i	$-0.124916\pi$
$614$	0	0
$615$	2.80904	0.113272
$616$	0	0
$617$	−22.1947	−0.893524	−0.446762	−	0.894653i	$-0.647423\pi$
$617$	−22.1947	−0.893524	−0.446762	+	0.894653i	$0.647423\pi$
$618$	0	0
$619$	41.5178i	1.66874i	0.551205	+	0.834370i	$0.314168\pi$
$619$	41.5178i	1.66874i	−0.551205	+	0.834370i	$0.685832\pi$
$620$	0	0
$621$	− 12.7690i	− 0.512401i
$622$	0	0
$623$	−39.1575	−1.56881
$624$	0	0
$625$	−2.43953	−0.0975814
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.80844i	0.151852i
$630$	0	0
$631$	−21.5614	−0.858346	−0.429173	−	0.903222i	$-0.641195\pi$
$631$	−21.5614	−0.858346	−0.429173	+	0.903222i	$0.641195\pi$
$632$	0	0
$633$	−9.24963	−0.367640
$634$	0	0
$635$	13.0128i	0.516397i
$636$	0	0
$637$	− 70.2677i	− 2.78411i
$638$	0	0
$639$	−2.94992	−0.116697
$640$	0	0
$641$	−10.0707	−0.397770	−0.198885	−	0.980023i	$-0.563732\pi$
$641$	−10.0707	−0.397770	−0.198885	+	0.980023i	$0.563732\pi$
$642$	0	0
$643$	30.6956i	1.21052i	0.796029	+	0.605259i	$0.206930\pi$
$643$	30.6956i	1.21052i	−0.796029	+	0.605259i	$0.793070\pi$
$644$	0	0
$645$	− 5.59251i	− 0.220205i
$646$	0	0
$647$	−41.2455	−1.62153	−0.810764	−	0.585373i	$-0.800948\pi$
$647$	−41.2455	−1.62153	−0.810764	+	0.585373i	$0.800948\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.04379i	0.158488i
$652$	0	0
$653$	24.2798i	0.950141i	0.879948	+	0.475071i	$0.157577\pi$
$653$	24.2798i	0.950141i	−0.879948	+	0.475071i	$0.842423\pi$
$654$	0	0
$655$	12.1243	0.473735
$656$	0	0
$657$	−37.0735	−1.44637
$658$	0	0
$659$	2.24471i	0.0874413i	0.999044	+	0.0437207i	$0.0139212\pi$
$659$	2.24471i	0.0874413i	−0.999044	+	0.0437207i	$0.986079\pi$
$660$	0	0
$661$	− 21.5996i	− 0.840129i	−0.907494	−	0.420064i	$-0.862007\pi$
$661$	− 21.5996i	− 0.840129i	0.907494	−	0.420064i	$-0.137993\pi$
$662$	0	0
$663$	−8.72783	−0.338961
$664$	0	0
$665$	19.5278	0.757254
$666$	0	0
$667$	− 13.5687i	− 0.525382i
$668$	0	0
$669$	− 11.4528i	− 0.442793i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	21.8507	0.842283	0.421142	−	0.906995i	$-0.361630\pi$
$673$	21.8507	0.842283	0.421142	+	0.906995i	$0.361630\pi$
$674$	0	0
$675$	− 9.87519i	− 0.380096i
$676$	0	0
$677$	− 39.6499i	− 1.52387i	−0.647653	−	0.761935i	$-0.724249\pi$
$677$	− 39.6499i	− 1.52387i	0.647653	−	0.761935i	$-0.275751\pi$
$678$	0	0
$679$	−1.32084	−0.0506893
$680$	0	0
$681$	−16.5913	−0.635779
$682$	0	0
$683$	5.22481i	0.199922i	0.994991	+	0.0999610i	$0.0318718\pi$
$683$	5.22481i	0.199922i	−0.994991	+	0.0999610i	$0.968128\pi$
$684$	0	0
$685$	24.1409i	0.922377i
$686$	0	0
$687$	−7.11597	−0.271491
$688$	0	0
$689$	−54.6083	−2.08041
$690$	0	0
$691$	0.0821400i	0.00312475i	0.999999	+	0.00156238i	$0.000497320\pi$
$691$	0.0821400i	0.00312475i	−0.999999	+	0.00156238i	$0.999503\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.6647	0.935585
$696$	0	0
$697$	−8.03766	−0.304448
$698$	0	0
$699$	− 15.9894i	− 0.604776i
$700$	0	0
$701$	21.9415i	0.828719i	0.910113	+	0.414360i	$0.135994\pi$
$701$	21.9415i	0.828719i	−0.910113	+	0.414360i	$0.864006\pi$
$702$	0	0
$703$	−4.49602	−0.169571
$704$	0	0
$705$	−2.37708	−0.0895261
$706$	0	0
$707$	11.6523i	0.438228i
$708$	0	0
$709$	40.8478i	1.53407i	0.641604	+	0.767036i	$0.278269\pi$
$709$	40.8478i	1.53407i	−0.641604	+	0.767036i	$0.721731\pi$
$710$	0	0
$711$	2.01157	0.0754396
$712$	0	0
$713$	−5.53082	−0.207131
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.68667i	0.324410i
$718$	0	0
$719$	42.5909	1.58837	0.794187	−	0.607674i	$-0.207897\pi$
$719$	42.5909	1.58837	0.794187	+	0.607674i	$0.207897\pi$
$720$	0	0
$721$	10.0237	0.373303
$722$	0	0
$723$	10.5609i	0.392764i
$724$	0	0
$725$	− 10.4937i	− 0.389725i
$726$	0	0
$727$	26.7043	0.990408	0.495204	−	0.868777i	$-0.335093\pi$
$727$	26.7043	0.990408	0.495204	+	0.868777i	$0.335093\pi$
$728$	0	0
$729$	8.81541	0.326497
$730$	0	0
$731$	16.0021i	0.591861i
$732$	0	0
$733$	− 23.8158i	− 0.879655i	−0.898082	−	0.439827i	$-0.855040\pi$
$733$	− 23.8158i	− 0.879655i	0.898082	−	0.439827i	$-0.144960\pi$
$734$	0	0
$735$	−11.2719	−0.415769
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0.704285i	0.0259075i	0.999916	+	0.0129538i	$0.00412343\pi$
$739$	0.704285i	0.0259075i	−0.999916	+	0.0129538i	$0.995877\pi$
$740$	0	0
$741$	− 10.3036i	− 0.378511i
$742$	0	0
$743$	1.25332	0.0459800	0.0229900	−	0.999736i	$-0.492681\pi$
$743$	1.25332	0.0459800	0.0229900	+	0.999736i	$0.492681\pi$
$744$	0	0
$745$	27.7923	1.01823
$746$	0	0
$747$	22.4822i	0.822581i
$748$	0	0
$749$	− 26.7707i	− 0.978179i
$750$	0	0
$751$	−32.0736	−1.17038	−0.585192	−	0.810895i	$-0.698981\pi$
$751$	−32.0736	−1.17038	−0.585192	+	0.810895i	$0.698981\pi$
$752$	0	0
$753$	−12.4405	−0.453356
$754$	0	0
$755$	14.8385i	0.540027i
$756$	0	0
$757$	31.0673i	1.12916i	0.825379	+	0.564580i	$0.190962\pi$
$757$	31.0673i	1.12916i	−0.825379	+	0.564580i	$0.809038\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.3256	−0.555553	−0.277776	−	0.960646i	$-0.589597\pi$
$761$	−15.3256	−0.555553	−0.277776	+	0.960646i	$0.589597\pi$
$762$	0	0
$763$	52.8635i	1.91379i
$764$	0	0
$765$	− 9.80649i	− 0.354555i
$766$	0	0
$767$	14.3654	0.518706
$768$	0	0
$769$	42.6201	1.53692	0.768460	−	0.639898i	$-0.221023\pi$
$769$	42.6201	1.53692	0.768460	+	0.639898i	$0.221023\pi$
$770$	0	0
$771$	− 9.23723i	− 0.332671i
$772$	0	0
$773$	34.6702i	1.24700i	0.781823	+	0.623501i	$0.214290\pi$
$773$	34.6702i	1.24700i	−0.781823	+	0.623501i	$0.785710\pi$
$774$	0	0
$775$	−4.27739	−0.153649
$776$	0	0
$777$	4.03603	0.144792
$778$	0	0
$779$	− 9.48880i	− 0.339972i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−12.6024	−0.450374
$784$	0	0
$785$	−24.4582	−0.872949
$786$	0	0
$787$	− 40.0266i	− 1.42679i	−0.700761	−	0.713397i	$-0.747156\pi$
$787$	− 40.0266i	− 1.42679i	0.700761	−	0.713397i	$-0.252844\pi$
$788$	0	0
$789$	6.62664i	0.235915i
$790$	0	0
$791$	54.9521	1.95387
$792$	0	0
$793$	31.1324	1.10554
$794$	0	0
$795$	8.75990i	0.310682i
$796$	0	0
$797$	− 36.1405i	− 1.28016i	−0.768308	−	0.640081i	$-0.778901\pi$
$797$	− 36.1405i	− 1.28016i	0.768308	−	0.640081i	$-0.221099\pi$
$798$	0	0
$799$	6.80167	0.240626
$800$	0	0
$801$	−23.2144	−0.820241
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 23.9762i	− 0.845050i
$806$	0	0
$807$	−11.9851	−0.421896
$808$	0	0
$809$	6.43465	0.226230	0.113115	−	0.993582i	$-0.463917\pi$
$809$	6.43465	0.226230	0.113115	+	0.993582i	$0.463917\pi$
$810$	0	0
$811$	9.56683i	0.335937i	0.985792	+	0.167968i	$0.0537207\pi$
$811$	9.56683i	0.335937i	−0.985792	+	0.167968i	$0.946279\pi$
$812$	0	0
$813$	15.8290i	0.555148i
$814$	0	0
$815$	−10.1803	−0.356602
$816$	0	0
$817$	−18.8912	−0.660920
$818$	0	0
$819$	− 64.7860i	− 2.26381i
$820$	0	0
$821$	− 17.1960i	− 0.600146i	−0.953916	−	0.300073i	$-0.902989\pi$
$821$	− 17.1960i	− 0.600146i	0.953916	−	0.300073i	$-0.0970110\pi$
$822$	0	0
$823$	38.3934	1.33831	0.669155	−	0.743123i	$-0.266656\pi$
$823$	38.3934	1.33831	0.669155	+	0.743123i	$0.266656\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	17.6011i	0.612050i	0.952024	+	0.306025i	$0.0989991\pi$
$827$	17.6011i	0.612050i	−0.952024	+	0.306025i	$0.901001\pi$
$828$	0	0
$829$	28.5872i	0.992874i	0.868073	+	0.496437i	$0.165359\pi$
$829$	28.5872i	0.992874i	−0.868073	+	0.496437i	$0.834641\pi$
$830$	0	0
$831$	−14.9991	−0.520314
$832$	0	0
$833$	32.2528	1.11749
$834$	0	0
$835$	− 14.5935i	− 0.505030i
$836$	0	0
$837$	5.13696i	0.177559i
$838$	0	0
$839$	−53.0362	−1.83101	−0.915506	−	0.402303i	$-0.868210\pi$
$839$	−53.0362	−1.83101	−0.915506	+	0.402303i	$0.868210\pi$
$840$	0	0
$841$	15.6083	0.538217
$842$	0	0
$843$	5.70885i	0.196623i
$844$	0	0
$845$	26.3724i	0.907239i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11.6268	0.399030
$850$	0	0
$851$	5.52022i	0.189231i
$852$	0	0
$853$	4.06897i	0.139319i	0.997571	+	0.0696595i	$0.0221913\pi$
$853$	4.06897i	0.139319i	−0.997571	+	0.0696595i	$0.977809\pi$
$854$	0	0
$855$	11.5770	0.395924
$856$	0	0
$857$	−23.5570	−0.804692	−0.402346	−	0.915488i	$-0.631805\pi$
$857$	−23.5570	−0.804692	−0.402346	+	0.915488i	$0.631805\pi$
$858$	0	0
$859$	− 21.8683i	− 0.746138i	−0.927804	−	0.373069i	$-0.878306\pi$
$859$	− 21.8683i	− 0.746138i	0.927804	−	0.373069i	$-0.121694\pi$
$860$	0	0
$861$	8.51800i	0.290293i
$862$	0	0
$863$	−22.4232	−0.763294	−0.381647	−	0.924308i	$-0.624643\pi$
$863$	−22.4232	−0.763294	−0.381647	+	0.924308i	$0.624643\pi$
$864$	0	0
$865$	−5.19206	−0.176535
$866$	0	0
$867$	6.40144i	0.217404i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	64.1235	2.17274
$872$	0	0
$873$	−0.783058	−0.0265025
$874$	0	0
$875$	− 50.8744i	− 1.71987i
$876$	0	0
$877$	− 18.2771i	− 0.617173i	−0.951196	−	0.308587i	$-0.900144\pi$
$877$	− 18.2771i	− 0.617173i	0.951196	−	0.308587i	$-0.0998560\pi$
$878$	0	0
$879$	−3.62021	−0.122107
$880$	0	0
$881$	−6.65131	−0.224088	−0.112044	−	0.993703i	$-0.535740\pi$
$881$	−6.65131	−0.224088	−0.112044	+	0.993703i	$0.535740\pi$
$882$	0	0
$883$	25.5001i	0.858147i	0.903270	+	0.429073i	$0.141160\pi$
$883$	25.5001i	0.858147i	−0.903270	+	0.429073i	$0.858840\pi$
$884$	0	0
$885$	− 2.30441i	− 0.0774618i
$886$	0	0
$887$	22.5084	0.755760	0.377880	−	0.925855i	$-0.376653\pi$
$887$	22.5084	0.755760	0.377880	+	0.925855i	$0.376653\pi$
$888$	0	0
$889$	−39.4594	−1.32343
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.02966i	0.268702i
$894$	0	0
$895$	−7.87363	−0.263186
$896$	0	0
$897$	−12.6507	−0.422396
$898$	0	0
$899$	5.45869i	0.182057i
$900$	0	0
$901$	− 25.0652i	− 0.835042i
$902$	0	0
$903$	16.9585	0.564342
$904$	0	0
$905$	−5.85654	−0.194678
$906$	0	0
$907$	13.7034i	0.455014i	0.973776	+	0.227507i	$0.0730574\pi$
$907$	13.7034i	0.455014i	−0.973776	+	0.227507i	$0.926943\pi$
$908$	0	0
$909$	6.90800i	0.229124i
$910$	0	0
$911$	44.3015	1.46778	0.733888	−	0.679271i	$-0.237704\pi$
$911$	44.3015	1.46778	0.733888	+	0.679271i	$0.237704\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	− 4.99405i	− 0.165098i
$916$	0	0
$917$	36.7651i	1.21409i
$918$	0	0
$919$	44.6532	1.47297	0.736486	−	0.676453i	$-0.236484\pi$
$919$	44.6532	1.47297	0.736486	+	0.676453i	$0.236484\pi$
$920$	0	0
$921$	−8.33920	−0.274786
$922$	0	0
$923$	6.26247i	0.206132i
$924$	0	0
$925$	4.26919i	0.140370i
$926$	0	0
$927$	5.94254	0.195178
$928$	0	0
$929$	−37.8438	−1.24161	−0.620807	−	0.783964i	$-0.713195\pi$
$929$	−37.8438	−1.24161	−0.620807	+	0.783964i	$0.713195\pi$
$930$	0	0
$931$	38.0758i	1.24788i
$932$	0	0
$933$	8.68366i	0.284290i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17.0665	−0.557539	−0.278770	−	0.960358i	$-0.589927\pi$
$937$	−17.0665	−0.557539	−0.278770	+	0.960358i	$0.589927\pi$
$938$	0	0
$939$	1.32739i	0.0433176i
$940$	0	0
$941$	4.66455i	0.152060i	0.997106	+	0.0760299i	$0.0242245\pi$
$941$	4.66455i	0.152060i	−0.997106	+	0.0760299i	$0.975776\pi$
$942$	0	0
$943$	−11.6504	−0.379388
$944$	0	0
$945$	−22.2688	−0.724404
$946$	0	0
$947$	52.8254i	1.71659i	0.513153	+	0.858297i	$0.328477\pi$
$947$	52.8254i	1.71659i	−0.513153	+	0.858297i	$0.671523\pi$
$948$	0	0
$949$	78.7044i	2.55485i
$950$	0	0
$951$	8.67227	0.281218
$952$	0	0
$953$	38.7779	1.25614	0.628069	−	0.778157i	$-0.283845\pi$
$953$	38.7779	1.25614	0.628069	+	0.778157i	$0.283845\pi$
$954$	0	0
$955$	0.831323i	0.0269010i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−73.2038	−2.36387
$960$	0	0
$961$	−28.7749	−0.928224
$962$	0	0
$963$	− 15.8709i	− 0.511433i
$964$	0	0
$965$	10.3923i	0.334539i
$966$	0	0
$967$	38.0583	1.22387	0.611937	−	0.790907i	$-0.290391\pi$
$967$	38.0583	1.22387	0.611937	+	0.790907i	$0.290391\pi$
$968$	0	0
$969$	4.72933	0.151928
$970$	0	0
$971$	5.71228i	0.183316i	0.995791	+	0.0916578i	$0.0292166\pi$
$971$	5.71228i	0.183316i	−0.995791	+	0.0916578i	$0.970783\pi$
$972$	0	0
$973$	74.7920i	2.39772i
$974$	0	0
$975$	−9.78375	−0.313331
$976$	0	0
$977$	0.568676	0.0181936	0.00909678	−	0.999959i	$-0.497104\pi$
$977$	0.568676	0.0181936	0.00909678	+	0.999959i	$0.497104\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	31.3400i	1.00061i
$982$	0	0
$983$	−2.19510	−0.0700129	−0.0350065	−	0.999387i	$-0.511145\pi$
$983$	−2.19510	−0.0700129	−0.0350065	+	0.999387i	$0.511145\pi$
$984$	0	0
$985$	−2.10650	−0.0671187
$986$	0	0
$987$	− 7.20815i	− 0.229438i
$988$	0	0
$989$	23.1946i	0.737547i
$990$	0	0
$991$	−10.7399	−0.341165	−0.170582	−	0.985343i	$-0.554565\pi$
$991$	−10.7399	−0.341165	−0.170582	+	0.985343i	$0.554565\pi$
$992$	0	0
$993$	6.60340	0.209553
$994$	0	0
$995$	1.23688i	0.0392117i
$996$	0	0
$997$	− 5.24854i	− 0.166223i	−0.996540	−	0.0831115i	$-0.973514\pi$
$997$	− 5.24854i	− 0.166223i	0.996540	−	0.0831115i	$-0.0264858\pi$
$998$	0	0
$999$	5.12711	0.162215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3872.2.c.i.1937.8		20
4.3	odd	2		968.2.c.i.485.14		20
8.3	odd	2		968.2.c.i.485.13		20
8.5	even	2	inner	3872.2.c.i.1937.13		20
11.2	odd	10		352.2.w.a.81.5		40
11.6	odd	10		352.2.w.a.113.6		40
11.10	odd	2		3872.2.c.h.1937.8		20
44.3	odd	10		968.2.o.d.493.10		40
44.7	even	10		968.2.o.j.269.8		40
44.15	odd	10		968.2.o.d.269.3		40
44.19	even	10		968.2.o.j.493.1		40
44.27	odd	10		968.2.o.i.245.2		40
44.31	odd	10		968.2.o.i.565.6		40
44.35	even	10		88.2.o.a.37.5	✓	40
44.39	even	10		88.2.o.a.69.9	yes	40
44.43	even	2		968.2.c.h.485.7		20
88.3	odd	10		968.2.o.d.493.3		40
88.13	odd	10		352.2.w.a.81.6		40
88.19	even	10		968.2.o.j.493.8		40
88.21	odd	2		3872.2.c.h.1937.13		20
88.27	odd	10		968.2.o.i.245.6		40
88.35	even	10		88.2.o.a.37.9	yes	40
88.43	even	2		968.2.c.h.485.8		20
88.51	even	10		968.2.o.j.269.1		40
88.59	odd	10		968.2.o.d.269.10		40
88.61	odd	10		352.2.w.a.113.5		40
88.75	odd	10		968.2.o.i.565.2		40
88.83	even	10		88.2.o.a.69.5	yes	40
132.35	odd	10		792.2.br.b.37.6		40
132.83	odd	10		792.2.br.b.685.2		40
264.35	odd	10		792.2.br.b.37.2		40
264.83	odd	10		792.2.br.b.685.6		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.o.a.37.5	✓	40	44.35	even	10
88.2.o.a.37.9	yes	40	88.35	even	10
88.2.o.a.69.5	yes	40	88.83	even	10
88.2.o.a.69.9	yes	40	44.39	even	10
352.2.w.a.81.5		40	11.2	odd	10
352.2.w.a.81.6		40	88.13	odd	10
352.2.w.a.113.5		40	88.61	odd	10
352.2.w.a.113.6		40	11.6	odd	10
792.2.br.b.37.2		40	264.35	odd	10
792.2.br.b.37.6		40	132.35	odd	10
792.2.br.b.685.2		40	132.83	odd	10
792.2.br.b.685.6		40	264.83	odd	10
968.2.c.h.485.7		20	44.43	even	2
968.2.c.h.485.8		20	88.43	even	2
968.2.c.i.485.13		20	8.3	odd	2
968.2.c.i.485.14		20	4.3	odd	2
968.2.o.d.269.3		40	44.15	odd	10
968.2.o.d.269.10		40	88.59	odd	10
968.2.o.d.493.3		40	88.3	odd	10
968.2.o.d.493.10		40	44.3	odd	10
968.2.o.i.245.2		40	44.27	odd	10
968.2.o.i.245.6		40	88.27	odd	10
968.2.o.i.565.2		40	88.75	odd	10
968.2.o.i.565.6		40	44.31	odd	10
968.2.o.j.269.1		40	88.51	even	10
968.2.o.j.269.8		40	44.7	even	10
968.2.o.j.493.1		40	44.19	even	10
968.2.o.j.493.8		40	88.19	even	10
3872.2.c.h.1937.8		20	11.10	odd	2
3872.2.c.h.1937.13		20	88.21	odd	2
3872.2.c.i.1937.8		20	1.1	even	1	trivial
3872.2.c.i.1937.13		20	8.5	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:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3872.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.612207i q^{3} -1.46030i q^{5} +4.42813 q^{7} +2.62520 q^{9} -5.57312i q^{13} -0.894002 q^{15} +2.55806 q^{17} +3.01989i q^{19} -2.71093i q^{21} +3.70783 q^{23} +2.86754 q^{25} -3.44379i q^{27} -3.65947i q^{29} -1.49166 q^{31} -6.46637i q^{35} +1.48880i q^{37} -3.41190 q^{39} -3.14210 q^{41} +6.25559i q^{43} -3.83357i q^{45} +2.65892 q^{47} +12.6083 q^{49} -1.56606i q^{51} -9.79852i q^{53} +1.84880 q^{57} +2.57763i q^{59} +5.58617i q^{61} +11.6247 q^{63} -8.13840 q^{65} +11.5059i q^{67} -2.26996i q^{69} -1.12369 q^{71} -14.1221 q^{73} -1.75553i q^{75} +0.766251 q^{79} +5.76730 q^{81} +8.56399i q^{83} -3.73552i q^{85} -2.24035 q^{87} -8.84290 q^{89} -24.6785i q^{91} +0.913204i q^{93} +4.40994 q^{95} -0.298285 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 10 q^{7} - 10 q^{9} + 4 q^{15} - 2 q^{17} + 4 q^{23} - 2 q^{25} - 2 q^{31} + 28 q^{39} + 2 q^{41} - 2 q^{47} - 2 q^{49} - 22 q^{57} - 30 q^{63} + 18 q^{65} + 34 q^{71} - 2 q^{73} - 58 q^{79} - 12 q^{81}+ \cdots + 10 q^{97}+O(q^{100}) \) 20 * q + 10 * q^7 - 10 * q^9 + 4 * q^15 - 2 * q^17 + 4 * q^23 - 2 * q^25 - 2 * q^31 + 28 * q^39 + 2 * q^41 - 2 * q^47 - 2 * q^49 - 22 * q^57 - 30 * q^63 + 18 * q^65 + 34 * q^71 - 2 * q^73 - 58 * q^79 - 12 * q^81 + 34 * q^87 - 8 * q^89 - 92 * q^95 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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