LMFDB - Embedded newform 624.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

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:	yes
Analytic conductor:	$4.98266508613$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +2.82843 q^{15} +7.65685 q^{17} +2.82843 q^{19} +2.82843 q^{21} +4.00000 q^{23} +3.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +1.17157 q^{31} -2.00000 q^{33} +8.00000 q^{35} -7.65685 q^{37} +1.00000 q^{39} +5.17157 q^{41} +1.65685 q^{43} -2.82843 q^{45} +11.6569 q^{47} +1.00000 q^{49} -7.65685 q^{51} -2.00000 q^{53} -5.65685 q^{55} -2.82843 q^{57} -7.65685 q^{59} +13.3137 q^{61} -2.82843 q^{63} +2.82843 q^{65} -6.82843 q^{67} -4.00000 q^{69} -2.00000 q^{71} +0.343146 q^{73} -3.00000 q^{75} -5.65685 q^{77} +11.3137 q^{79} +1.00000 q^{81} -3.65685 q^{83} -21.6569 q^{85} -2.00000 q^{87} +14.8284 q^{89} +2.82843 q^{91} -1.17157 q^{93} -8.00000 q^{95} +3.65685 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 4 q^{17} + 8 q^{23} + 6 q^{25} - 2 q^{27} + 4 q^{29} + 8 q^{31} - 4 q^{33} + 16 q^{35} - 4 q^{37} + 2 q^{39} + 16 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} - 4 q^{59} + 4 q^{61} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 12 q^{73} - 6 q^{75} + 2 q^{81} + 4 q^{83} - 32 q^{85} - 4 q^{87} + 24 q^{89} - 8 q^{93} - 16 q^{95} - 4 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 4 * q^17 + 8 * q^23 + 6 * q^25 - 2 * q^27 + 4 * q^29 + 8 * q^31 - 4 * q^33 + 16 * q^35 - 4 * q^37 + 2 * q^39 + 16 * q^41 - 8 * q^43 + 12 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 - 4 * q^59 + 4 * q^61 - 8 * q^67 - 8 * q^69 - 4 * q^71 + 12 * q^73 - 6 * q^75 + 2 * q^81 + 4 * q^83 - 32 * q^85 - 4 * q^87 + 24 * q^89 - 8 * q^93 - 16 * q^95 - 4 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.82843	0.730297
$16$	0	0
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	1.17157	0.210421	0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157	0.210421	0.105210	+	0.994450i	$0.466448\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	8.00000	1.35225
$36$	0	0
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	5.17157	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$41$	5.17157	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.65685	−1.07217
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	−2.82843	−0.374634
$58$	0	0
$59$	−7.65685	−0.996838	−0.498419	−	0.866936i	$-0.666086\pi$
$59$	−7.65685	−0.996838	−0.498419	+	0.866936i	$0.666086\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	2.82843	0.350823
$66$	0	0
$67$	−6.82843	−0.834225	−0.417113	−	0.908855i	$-0.636958\pi$
$67$	−6.82843	−0.834225	−0.417113	+	0.908855i	$0.636958\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	0.343146	0.0401622	0.0200811	−	0.999798i	$-0.493608\pi$
$73$	0.343146	0.0401622	0.0200811	+	0.999798i	$0.493608\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	−5.65685	−0.644658
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.65685	−0.401392	−0.200696	−	0.979654i	$-0.564320\pi$
$83$	−3.65685	−0.401392	−0.200696	+	0.979654i	$0.564320\pi$
$84$	0	0
$85$	−21.6569	−2.34902
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	14.8284	1.57181	0.785905	−	0.618347i	$-0.212197\pi$
$89$	14.8284	1.57181	0.785905	+	0.618347i	$0.212197\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	−1.17157	−0.121486
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	7.65685	0.761885	0.380943	−	0.924599i	$-0.375599\pi$
$101$	7.65685	0.761885	0.380943	+	0.924599i	$0.375599\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	0	0
$107$	11.3137	1.09374	0.546869	−	0.837218i	$-0.315820\pi$
$107$	11.3137	1.09374	0.546869	+	0.837218i	$0.315820\pi$
$108$	0	0
$109$	5.31371	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$109$	5.31371	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$110$	0	0
$111$	7.65685	0.726756
$112$	0	0
$113$	−5.31371	−0.499872	−0.249936	−	0.968262i	$-0.580410\pi$
$113$	−5.31371	−0.499872	−0.249936	+	0.968262i	$0.580410\pi$
$114$	0	0
$115$	−11.3137	−1.05501
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−21.6569	−1.98528
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−5.17157	−0.466305
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	0	0
$129$	−1.65685	−0.145878
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	2.82843	0.243432
$136$	0	0
$137$	−10.8284	−0.925135	−0.462567	−	0.886584i	$-0.653072\pi$
$137$	−10.8284	−0.925135	−0.462567	+	0.886584i	$0.653072\pi$
$138$	0	0
$139$	7.31371	0.620341	0.310170	−	0.950681i	$-0.399614\pi$
$139$	7.31371	0.620341	0.310170	+	0.950681i	$0.399614\pi$
$140$	0	0
$141$	−11.6569	−0.981684
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−5.65685	−0.469776
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−9.17157	−0.751365	−0.375682	−	0.926749i	$-0.622592\pi$
$149$	−9.17157	−0.751365	−0.375682	+	0.926749i	$0.622592\pi$
$150$	0	0
$151$	3.51472	0.286024	0.143012	−	0.989721i	$-0.454321\pi$
$151$	3.51472	0.286024	0.143012	+	0.989721i	$0.454321\pi$
$152$	0	0
$153$	7.65685	0.619020
$154$	0	0
$155$	−3.31371	−0.266163
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−11.3137	−0.891645
$162$	0	0
$163$	−18.8284	−1.47476	−0.737378	−	0.675480i	$-0.763936\pi$
$163$	−18.8284	−1.47476	−0.737378	+	0.675480i	$0.763936\pi$
$164$	0	0
$165$	5.65685	0.440386
$166$	0	0
$167$	3.65685	0.282976	0.141488	−	0.989940i	$-0.454811\pi$
$167$	3.65685	0.282976	0.141488	+	0.989940i	$0.454811\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	−11.6569	−0.886254	−0.443127	−	0.896459i	$-0.646131\pi$
$173$	−11.6569	−0.886254	−0.443127	+	0.896459i	$0.646131\pi$
$174$	0	0
$175$	−8.48528	−0.641427
$176$	0	0
$177$	7.65685	0.575524
$178$	0	0
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	−13.3137	−0.984178
$184$	0	0
$185$	21.6569	1.59224
$186$	0	0
$187$	15.3137	1.11985
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	5.31371	0.382489	0.191245	−	0.981542i	$-0.438748\pi$
$193$	5.31371	0.382489	0.191245	+	0.981542i	$0.438748\pi$
$194$	0	0
$195$	−2.82843	−0.202548
$196$	0	0
$197$	0.485281	0.0345749	0.0172874	−	0.999851i	$-0.494497\pi$
$197$	0.485281	0.0345749	0.0172874	+	0.999851i	$0.494497\pi$
$198$	0	0
$199$	−21.6569	−1.53521	−0.767607	−	0.640921i	$-0.778553\pi$
$199$	−21.6569	−1.53521	−0.767607	+	0.640921i	$0.778553\pi$
$200$	0	0
$201$	6.82843	0.481640
$202$	0	0
$203$	−5.65685	−0.397033
$204$	0	0
$205$	−14.6274	−1.02162
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	5.65685	0.391293
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	−4.68629	−0.319602
$216$	0	0
$217$	−3.31371	−0.224949
$218$	0	0
$219$	−0.343146	−0.0231876
$220$	0	0
$221$	−7.65685	−0.515056
$222$	0	0
$223$	12.4853	0.836076	0.418038	−	0.908429i	$-0.362718\pi$
$223$	12.4853	0.836076	0.418038	+	0.908429i	$0.362718\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	17.3137	1.14915	0.574576	−	0.818452i	$-0.305167\pi$
$227$	17.3137	1.14915	0.574576	+	0.818452i	$0.305167\pi$
$228$	0	0
$229$	−1.31371	−0.0868123	−0.0434062	−	0.999058i	$-0.513821\pi$
$229$	−1.31371	−0.0868123	−0.0434062	+	0.999058i	$0.513821\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	0	0
$233$	6.97056	0.456657	0.228328	−	0.973584i	$-0.426674\pi$
$233$	6.97056	0.456657	0.228328	+	0.973584i	$0.426674\pi$
$234$	0	0
$235$	−32.9706	−2.15076
$236$	0	0
$237$	−11.3137	−0.734904
$238$	0	0
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	0.343146	0.0221040	0.0110520	−	0.999939i	$-0.496482\pi$
$241$	0.343146	0.0221040	0.0110520	+	0.999939i	$0.496482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.82843	−0.180702
$246$	0	0
$247$	−2.82843	−0.179969
$248$	0	0
$249$	3.65685	0.231744
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	0	0
$255$	21.6569	1.35620
$256$	0	0
$257$	−4.34315	−0.270918	−0.135459	−	0.990783i	$-0.543251\pi$
$257$	−4.34315	−0.270918	−0.135459	+	0.990783i	$0.543251\pi$
$258$	0	0
$259$	21.6569	1.34569
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	5.65685	0.347498
$266$	0	0
$267$	−14.8284	−0.907485
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	27.7990	1.68867	0.844334	−	0.535817i	$-0.179996\pi$
$271$	27.7990	1.68867	0.844334	+	0.535817i	$0.179996\pi$
$272$	0	0
$273$	−2.82843	−0.171184
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	1.17157	0.0701402
$280$	0	0
$281$	21.1716	1.26299	0.631495	−	0.775380i	$-0.282442\pi$
$281$	21.1716	1.26299	0.631495	+	0.775380i	$0.282442\pi$
$282$	0	0
$283$	−28.9706	−1.72212	−0.861061	−	0.508502i	$-0.830199\pi$
$283$	−28.9706	−1.72212	−0.861061	+	0.508502i	$0.830199\pi$
$284$	0	0
$285$	8.00000	0.473879
$286$	0	0
$287$	−14.6274	−0.863429
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	−3.65685	−0.214369
$292$	0	0
$293$	2.14214	0.125145	0.0625724	−	0.998040i	$-0.480070\pi$
$293$	2.14214	0.125145	0.0625724	+	0.998040i	$0.480070\pi$
$294$	0	0
$295$	21.6569	1.26091
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	−4.00000	−0.231326
$300$	0	0
$301$	−4.68629	−0.270113
$302$	0	0
$303$	−7.65685	−0.439875
$304$	0	0
$305$	−37.6569	−2.15623
$306$	0	0
$307$	22.8284	1.30289	0.651444	−	0.758697i	$-0.274164\pi$
$307$	22.8284	1.30289	0.651444	+	0.758697i	$0.274164\pi$
$308$	0	0
$309$	2.34315	0.133297
$310$	0	0
$311$	10.6274	0.602626	0.301313	−	0.953525i	$-0.402575\pi$
$311$	10.6274	0.602626	0.301313	+	0.953525i	$0.402575\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	8.00000	0.450749
$316$	0	0
$317$	8.48528	0.476581	0.238290	−	0.971194i	$-0.423413\pi$
$317$	8.48528	0.476581	0.238290	+	0.971194i	$0.423413\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	−11.3137	−0.631470
$322$	0	0
$323$	21.6569	1.20502
$324$	0	0
$325$	−3.00000	−0.166410
$326$	0	0
$327$	−5.31371	−0.293849
$328$	0	0
$329$	−32.9706	−1.81773
$330$	0	0
$331$	−26.1421	−1.43690	−0.718451	−	0.695578i	$-0.755148\pi$
$331$	−26.1421	−1.43690	−0.718451	+	0.695578i	$0.755148\pi$
$332$	0	0
$333$	−7.65685	−0.419593
$334$	0	0
$335$	19.3137	1.05522
$336$	0	0
$337$	9.31371	0.507350	0.253675	−	0.967290i	$-0.418361\pi$
$337$	9.31371	0.507350	0.253675	+	0.967290i	$0.418361\pi$
$338$	0	0
$339$	5.31371	0.288601
$340$	0	0
$341$	2.34315	0.126888
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	11.3137	0.609110
$346$	0	0
$347$	8.68629	0.466305	0.233152	−	0.972440i	$-0.425096\pi$
$347$	8.68629	0.466305	0.233152	+	0.972440i	$0.425096\pi$
$348$	0	0
$349$	3.65685	0.195747	0.0978735	−	0.995199i	$-0.468796\pi$
$349$	3.65685	0.195747	0.0978735	+	0.995199i	$0.468796\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−33.4558	−1.78067	−0.890337	−	0.455301i	$-0.849532\pi$
$353$	−33.4558	−1.78067	−0.890337	+	0.455301i	$0.849532\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	0	0
$357$	21.6569	1.14620
$358$	0	0
$359$	−34.9706	−1.84568	−0.922838	−	0.385189i	$-0.874136\pi$
$359$	−34.9706	−1.84568	−0.922838	+	0.385189i	$0.874136\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−0.970563	−0.0508016
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	0	0
$369$	5.17157	0.269221
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−0.485281	−0.0249272	−0.0124636	−	0.999922i	$-0.503967\pi$
$379$	−0.485281	−0.0249272	−0.0124636	+	0.999922i	$0.503967\pi$
$380$	0	0
$381$	5.65685	0.289809
$382$	0	0
$383$	−30.9706	−1.58252	−0.791261	−	0.611479i	$-0.790575\pi$
$383$	−30.9706	−1.58252	−0.791261	+	0.611479i	$0.790575\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	1.65685	0.0842226
$388$	0	0
$389$	−26.9706	−1.36746	−0.683731	−	0.729734i	$-0.739644\pi$
$389$	−26.9706	−1.36746	−0.683731	+	0.729734i	$0.739644\pi$
$390$	0	0
$391$	30.6274	1.54890
$392$	0	0
$393$	−8.00000	−0.403547
$394$	0	0
$395$	−32.0000	−1.61009
$396$	0	0
$397$	30.9706	1.55437	0.777184	−	0.629273i	$-0.216647\pi$
$397$	30.9706	1.55437	0.777184	+	0.629273i	$0.216647\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	26.1421	1.30548	0.652738	−	0.757584i	$-0.273620\pi$
$401$	26.1421	1.30548	0.652738	+	0.757584i	$0.273620\pi$
$402$	0	0
$403$	−1.17157	−0.0583602
$404$	0	0
$405$	−2.82843	−0.140546
$406$	0	0
$407$	−15.3137	−0.759072
$408$	0	0
$409$	−34.9706	−1.72918	−0.864592	−	0.502475i	$-0.832423\pi$
$409$	−34.9706	−1.72918	−0.864592	+	0.502475i	$0.832423\pi$
$410$	0	0
$411$	10.8284	0.534127
$412$	0	0
$413$	21.6569	1.06566
$414$	0	0
$415$	10.3431	0.507725
$416$	0	0
$417$	−7.31371	−0.358154
$418$	0	0
$419$	−14.6274	−0.714596	−0.357298	−	0.933990i	$-0.616302\pi$
$419$	−14.6274	−0.714596	−0.357298	+	0.933990i	$0.616302\pi$
$420$	0	0
$421$	37.3137	1.81856	0.909279	−	0.416186i	$-0.136634\pi$
$421$	37.3137	1.81856	0.909279	+	0.416186i	$0.136634\pi$
$422$	0	0
$423$	11.6569	0.566776
$424$	0	0
$425$	22.9706	1.11424
$426$	0	0
$427$	−37.6569	−1.82234
$428$	0	0
$429$	2.00000	0.0965609
$430$	0	0
$431$	−8.34315	−0.401875	−0.200938	−	0.979604i	$-0.564399\pi$
$431$	−8.34315	−0.401875	−0.200938	+	0.979604i	$0.564399\pi$
$432$	0	0
$433$	−21.3137	−1.02427	−0.512136	−	0.858905i	$-0.671146\pi$
$433$	−21.3137	−1.02427	−0.512136	+	0.858905i	$0.671146\pi$
$434$	0	0
$435$	5.65685	0.271225
$436$	0	0
$437$	11.3137	0.541208
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	25.9411	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$443$	25.9411	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$444$	0	0
$445$	−41.9411	−1.98820
$446$	0	0
$447$	9.17157	0.433801
$448$	0	0
$449$	−31.7990	−1.50069	−0.750344	−	0.661048i	$-0.770112\pi$
$449$	−31.7990	−1.50069	−0.750344	+	0.661048i	$0.770112\pi$
$450$	0	0
$451$	10.3431	0.487040
$452$	0	0
$453$	−3.51472	−0.165136
$454$	0	0
$455$	−8.00000	−0.375046
$456$	0	0
$457$	−7.65685	−0.358173	−0.179086	−	0.983833i	$-0.557314\pi$
$457$	−7.65685	−0.358173	−0.179086	+	0.983833i	$0.557314\pi$
$458$	0	0
$459$	−7.65685	−0.357391
$460$	0	0
$461$	5.17157	0.240864	0.120432	−	0.992722i	$-0.461572\pi$
$461$	5.17157	0.240864	0.120432	+	0.992722i	$0.461572\pi$
$462$	0	0
$463$	24.4853	1.13793	0.568964	−	0.822363i	$-0.307344\pi$
$463$	24.4853	1.13793	0.568964	+	0.822363i	$0.307344\pi$
$464$	0	0
$465$	3.31371	0.153670
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	19.3137	0.891824
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	3.31371	0.152364
$474$	0	0
$475$	8.48528	0.389331
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−25.3137	−1.15661	−0.578306	−	0.815820i	$-0.696286\pi$
$479$	−25.3137	−1.15661	−0.578306	+	0.815820i	$0.696286\pi$
$480$	0	0
$481$	7.65685	0.349123
$482$	0	0
$483$	11.3137	0.514792
$484$	0	0
$485$	−10.3431	−0.469658
$486$	0	0
$487$	7.79899	0.353406	0.176703	−	0.984264i	$-0.443457\pi$
$487$	7.79899	0.353406	0.176703	+	0.984264i	$0.443457\pi$
$488$	0	0
$489$	18.8284	0.851451
$490$	0	0
$491$	−30.6274	−1.38220	−0.691098	−	0.722761i	$-0.742873\pi$
$491$	−30.6274	−1.38220	−0.691098	+	0.722761i	$0.742873\pi$
$492$	0	0
$493$	15.3137	0.689695
$494$	0	0
$495$	−5.65685	−0.254257
$496$	0	0
$497$	5.65685	0.253745
$498$	0	0
$499$	−26.1421	−1.17028	−0.585141	−	0.810931i	$-0.698961\pi$
$499$	−26.1421	−1.17028	−0.585141	+	0.810931i	$0.698961\pi$
$500$	0	0
$501$	−3.65685	−0.163376
$502$	0	0
$503$	7.31371	0.326102	0.163051	−	0.986618i	$-0.447866\pi$
$503$	7.31371	0.326102	0.163051	+	0.986618i	$0.447866\pi$
$504$	0	0
$505$	−21.6569	−0.963717
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	11.7990	0.522981	0.261491	−	0.965206i	$-0.415786\pi$
$509$	11.7990	0.522981	0.261491	+	0.965206i	$0.415786\pi$
$510$	0	0
$511$	−0.970563	−0.0429352
$512$	0	0
$513$	−2.82843	−0.124878
$514$	0	0
$515$	6.62742	0.292039
$516$	0	0
$517$	23.3137	1.02534
$518$	0	0
$519$	11.6569	0.511679
$520$	0	0
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	0	0
$523$	15.3137	0.669622	0.334811	−	0.942285i	$-0.391328\pi$
$523$	15.3137	0.669622	0.334811	+	0.942285i	$0.391328\pi$
$524$	0	0
$525$	8.48528	0.370328
$526$	0	0
$527$	8.97056	0.390764
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−7.65685	−0.332279
$532$	0	0
$533$	−5.17157	−0.224006
$534$	0	0
$535$	−32.0000	−1.38348
$536$	0	0
$537$	−23.3137	−1.00606
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	−14.0000	−0.600798
$544$	0	0
$545$	−15.0294	−0.643790
$546$	0	0
$547$	−23.3137	−0.996822	−0.498411	−	0.866941i	$-0.666083\pi$
$547$	−23.3137	−0.996822	−0.498411	+	0.866941i	$0.666083\pi$
$548$	0	0
$549$	13.3137	0.568215
$550$	0	0
$551$	5.65685	0.240990
$552$	0	0
$553$	−32.0000	−1.36078
$554$	0	0
$555$	−21.6569	−0.919282
$556$	0	0
$557$	−7.79899	−0.330454	−0.165227	−	0.986256i	$-0.552836\pi$
$557$	−7.79899	−0.330454	−0.165227	+	0.986256i	$0.552836\pi$
$558$	0	0
$559$	−1.65685	−0.0700775
$560$	0	0
$561$	−15.3137	−0.646545
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	15.0294	0.632293
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	−42.9706	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$569$	−42.9706	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$570$	0	0
$571$	12.9706	0.542801	0.271401	−	0.962466i	$-0.412513\pi$
$571$	12.9706	0.542801	0.271401	+	0.962466i	$0.412513\pi$
$572$	0	0
$573$	3.31371	0.138432
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	−31.9411	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$577$	−31.9411	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$578$	0	0
$579$	−5.31371	−0.220830
$580$	0	0
$581$	10.3431	0.429106
$582$	0	0
$583$	−4.00000	−0.165663
$584$	0	0
$585$	2.82843	0.116941
$586$	0	0
$587$	10.9706	0.452804	0.226402	−	0.974034i	$-0.427304\pi$
$587$	10.9706	0.452804	0.226402	+	0.974034i	$0.427304\pi$
$588$	0	0
$589$	3.31371	0.136539
$590$	0	0
$591$	−0.485281	−0.0199618
$592$	0	0
$593$	−20.4853	−0.841230	−0.420615	−	0.907239i	$-0.638186\pi$
$593$	−20.4853	−0.841230	−0.420615	+	0.907239i	$0.638186\pi$
$594$	0	0
$595$	61.2548	2.51120
$596$	0	0
$597$	21.6569	0.886356
$598$	0	0
$599$	23.3137	0.952572	0.476286	−	0.879290i	$-0.341983\pi$
$599$	23.3137	0.952572	0.476286	+	0.879290i	$0.341983\pi$
$600$	0	0
$601$	−0.627417	−0.0255929	−0.0127964	−	0.999918i	$-0.504073\pi$
$601$	−0.627417	−0.0255929	−0.0127964	+	0.999918i	$0.504073\pi$
$602$	0	0
$603$	−6.82843	−0.278075
$604$	0	0
$605$	19.7990	0.804943
$606$	0	0
$607$	−41.9411	−1.70234	−0.851169	−	0.524892i	$-0.824106\pi$
$607$	−41.9411	−1.70234	−0.851169	+	0.524892i	$0.824106\pi$
$608$	0	0
$609$	5.65685	0.229227
$610$	0	0
$611$	−11.6569	−0.471586
$612$	0	0
$613$	−47.6569	−1.92484	−0.962421	−	0.271561i	$-0.912460\pi$
$613$	−47.6569	−1.92484	−0.962421	+	0.271561i	$0.912460\pi$
$614$	0	0
$615$	14.6274	0.589834
$616$	0	0
$617$	−34.8284	−1.40214	−0.701070	−	0.713093i	$-0.747294\pi$
$617$	−34.8284	−1.40214	−0.701070	+	0.713093i	$0.747294\pi$
$618$	0	0
$619$	−23.7990	−0.956562	−0.478281	−	0.878207i	$-0.658740\pi$
$619$	−23.7990	−0.956562	−0.478281	+	0.878207i	$0.658740\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−41.9411	−1.68034
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	−5.65685	−0.225913
$628$	0	0
$629$	−58.6274	−2.33763
$630$	0	0
$631$	−43.1127	−1.71629	−0.858145	−	0.513408i	$-0.828383\pi$
$631$	−43.1127	−1.71629	−0.858145	+	0.513408i	$0.828383\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	16.0000	0.634941
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	30.2843	1.19616	0.598078	−	0.801438i	$-0.295931\pi$
$641$	30.2843	1.19616	0.598078	+	0.801438i	$0.295931\pi$
$642$	0	0
$643$	−22.8284	−0.900265	−0.450133	−	0.892962i	$-0.648623\pi$
$643$	−22.8284	−0.900265	−0.450133	+	0.892962i	$0.648623\pi$
$644$	0	0
$645$	4.68629	0.184523
$646$	0	0
$647$	−11.3137	−0.444788	−0.222394	−	0.974957i	$-0.571387\pi$
$647$	−11.3137	−0.444788	−0.222394	+	0.974957i	$0.571387\pi$
$648$	0	0
$649$	−15.3137	−0.601116
$650$	0	0
$651$	3.31371	0.129874
$652$	0	0
$653$	−25.3137	−0.990602	−0.495301	−	0.868721i	$-0.664942\pi$
$653$	−25.3137	−0.990602	−0.495301	+	0.868721i	$0.664942\pi$
$654$	0	0
$655$	−22.6274	−0.884126
$656$	0	0
$657$	0.343146	0.0133874
$658$	0	0
$659$	47.3137	1.84308	0.921540	−	0.388283i	$-0.126932\pi$
$659$	47.3137	1.84308	0.921540	+	0.388283i	$0.126932\pi$
$660$	0	0
$661$	−34.9706	−1.36020	−0.680099	−	0.733121i	$-0.738063\pi$
$661$	−34.9706	−1.36020	−0.680099	+	0.733121i	$0.738063\pi$
$662$	0	0
$663$	7.65685	0.297368
$664$	0	0
$665$	22.6274	0.877454
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	−12.4853	−0.482709
$670$	0	0
$671$	26.6274	1.02794
$672$	0	0
$673$	16.6274	0.640940	0.320470	−	0.947259i	$-0.396159\pi$
$673$	16.6274	0.640940	0.320470	+	0.947259i	$0.396159\pi$
$674$	0	0
$675$	−3.00000	−0.115470
$676$	0	0
$677$	26.6863	1.02564	0.512819	−	0.858497i	$-0.328601\pi$
$677$	26.6863	1.02564	0.512819	+	0.858497i	$0.328601\pi$
$678$	0	0
$679$	−10.3431	−0.396934
$680$	0	0
$681$	−17.3137	−0.663463
$682$	0	0
$683$	−47.9411	−1.83442	−0.917208	−	0.398408i	$-0.869563\pi$
$683$	−47.9411	−1.83442	−0.917208	+	0.398408i	$0.869563\pi$
$684$	0	0
$685$	30.6274	1.17021
$686$	0	0
$687$	1.31371	0.0501211
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	5.85786	0.222844	0.111422	−	0.993773i	$-0.464460\pi$
$691$	5.85786	0.222844	0.111422	+	0.993773i	$0.464460\pi$
$692$	0	0
$693$	−5.65685	−0.214886
$694$	0	0
$695$	−20.6863	−0.784676
$696$	0	0
$697$	39.5980	1.49988
$698$	0	0
$699$	−6.97056	−0.263651
$700$	0	0
$701$	5.02944	0.189959	0.0949796	−	0.995479i	$-0.469721\pi$
$701$	5.02944	0.189959	0.0949796	+	0.995479i	$0.469721\pi$
$702$	0	0
$703$	−21.6569	−0.816804
$704$	0	0
$705$	32.9706	1.24174
$706$	0	0
$707$	−21.6569	−0.814490
$708$	0	0
$709$	−4.62742	−0.173786	−0.0868931	−	0.996218i	$-0.527694\pi$
$709$	−4.62742	−0.173786	−0.0868931	+	0.996218i	$0.527694\pi$
$710$	0	0
$711$	11.3137	0.424297
$712$	0	0
$713$	4.68629	0.175503
$714$	0	0
$715$	5.65685	0.211554
$716$	0	0
$717$	2.00000	0.0746914
$718$	0	0
$719$	29.9411	1.11662	0.558308	−	0.829634i	$-0.311451\pi$
$719$	29.9411	1.11662	0.558308	+	0.829634i	$0.311451\pi$
$720$	0	0
$721$	6.62742	0.246818
$722$	0	0
$723$	−0.343146	−0.0127617
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	10.3431	0.383606	0.191803	−	0.981433i	$-0.438567\pi$
$727$	10.3431	0.383606	0.191803	+	0.981433i	$0.438567\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.6863	0.469219
$732$	0	0
$733$	−36.6274	−1.35286	−0.676432	−	0.736505i	$-0.736475\pi$
$733$	−36.6274	−1.35286	−0.676432	+	0.736505i	$0.736475\pi$
$734$	0	0
$735$	2.82843	0.104328
$736$	0	0
$737$	−13.6569	−0.503057
$738$	0	0
$739$	18.1421	0.667369	0.333685	−	0.942685i	$-0.391708\pi$
$739$	18.1421	0.667369	0.333685	+	0.942685i	$0.391708\pi$
$740$	0	0
$741$	2.82843	0.103905
$742$	0	0
$743$	−2.00000	−0.0733729	−0.0366864	−	0.999327i	$-0.511680\pi$
$743$	−2.00000	−0.0733729	−0.0366864	+	0.999327i	$0.511680\pi$
$744$	0	0
$745$	25.9411	0.950409
$746$	0	0
$747$	−3.65685	−0.133797
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	0.970563	0.0354163	0.0177082	−	0.999843i	$-0.494363\pi$
$751$	0.970563	0.0354163	0.0177082	+	0.999843i	$0.494363\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.94113	−0.361795
$756$	0	0
$757$	51.9411	1.88783	0.943916	−	0.330185i	$-0.107111\pi$
$757$	51.9411	1.88783	0.943916	+	0.330185i	$0.107111\pi$
$758$	0	0
$759$	−8.00000	−0.290382
$760$	0	0
$761$	32.4853	1.17759	0.588795	−	0.808282i	$-0.299602\pi$
$761$	32.4853	1.17759	0.588795	+	0.808282i	$0.299602\pi$
$762$	0	0
$763$	−15.0294	−0.544102
$764$	0	0
$765$	−21.6569	−0.783005
$766$	0	0
$767$	7.65685	0.276473
$768$	0	0
$769$	42.0000	1.51456	0.757279	−	0.653091i	$-0.226528\pi$
$769$	42.0000	1.51456	0.757279	+	0.653091i	$0.226528\pi$
$770$	0	0
$771$	4.34315	0.156415
$772$	0	0
$773$	34.1421	1.22801	0.614004	−	0.789303i	$-0.289558\pi$
$773$	34.1421	1.22801	0.614004	+	0.789303i	$0.289558\pi$
$774$	0	0
$775$	3.51472	0.126252
$776$	0	0
$777$	−21.6569	−0.776935
$778$	0	0
$779$	14.6274	0.524082
$780$	0	0
$781$	−4.00000	−0.143131
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	28.2843	1.00951
$786$	0	0
$787$	40.7696	1.45328	0.726639	−	0.687020i	$-0.241081\pi$
$787$	40.7696	1.45328	0.726639	+	0.687020i	$0.241081\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	15.0294	0.534385
$792$	0	0
$793$	−13.3137	−0.472784
$794$	0	0
$795$	−5.65685	−0.200628
$796$	0	0
$797$	24.3431	0.862278	0.431139	−	0.902285i	$-0.358112\pi$
$797$	24.3431	0.862278	0.431139	+	0.902285i	$0.358112\pi$
$798$	0	0
$799$	89.2548	3.15761
$800$	0	0
$801$	14.8284	0.523937
$802$	0	0
$803$	0.686292	0.0242187
$804$	0	0
$805$	32.0000	1.12785
$806$	0	0
$807$	−18.0000	−0.633630
$808$	0	0
$809$	18.6863	0.656975	0.328488	−	0.944508i	$-0.393461\pi$
$809$	18.6863	0.656975	0.328488	+	0.944508i	$0.393461\pi$
$810$	0	0
$811$	30.1421	1.05843	0.529217	−	0.848487i	$-0.322486\pi$
$811$	30.1421	1.05843	0.529217	+	0.848487i	$0.322486\pi$
$812$	0	0
$813$	−27.7990	−0.974953
$814$	0	0
$815$	53.2548	1.86544
$816$	0	0
$817$	4.68629	0.163953
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	−23.7990	−0.830590	−0.415295	−	0.909687i	$-0.636322\pi$
$821$	−23.7990	−0.830590	−0.415295	+	0.909687i	$0.636322\pi$
$822$	0	0
$823$	−15.0294	−0.523893	−0.261947	−	0.965082i	$-0.584364\pi$
$823$	−15.0294	−0.523893	−0.261947	+	0.965082i	$0.584364\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	26.0000	0.904109	0.452054	−	0.891990i	$-0.350691\pi$
$827$	26.0000	0.904109	0.452054	+	0.891990i	$0.350691\pi$
$828$	0	0
$829$	−17.3137	−0.601330	−0.300665	−	0.953730i	$-0.597209\pi$
$829$	−17.3137	−0.601330	−0.300665	+	0.953730i	$0.597209\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	7.65685	0.265294
$834$	0	0
$835$	−10.3431	−0.357939
$836$	0	0
$837$	−1.17157	−0.0404955
$838$	0	0
$839$	−43.2548	−1.49332	−0.746661	−	0.665204i	$-0.768344\pi$
$839$	−43.2548	−1.49332	−0.746661	+	0.665204i	$0.768344\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−21.1716	−0.729188
$844$	0	0
$845$	−2.82843	−0.0973009
$846$	0	0
$847$	19.7990	0.680301
$848$	0	0
$849$	28.9706	0.994267
$850$	0	0
$851$	−30.6274	−1.04989
$852$	0	0
$853$	3.65685	0.125208	0.0626042	−	0.998038i	$-0.480059\pi$
$853$	3.65685	0.125208	0.0626042	+	0.998038i	$0.480059\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	−49.5980	−1.69423	−0.847117	−	0.531406i	$-0.821664\pi$
$857$	−49.5980	−1.69423	−0.847117	+	0.531406i	$0.821664\pi$
$858$	0	0
$859$	0.686292	0.0234160	0.0117080	−	0.999931i	$-0.496273\pi$
$859$	0.686292	0.0234160	0.0117080	+	0.999931i	$0.496273\pi$
$860$	0	0
$861$	14.6274	0.498501
$862$	0	0
$863$	28.3431	0.964812	0.482406	−	0.875948i	$-0.339763\pi$
$863$	28.3431	0.964812	0.482406	+	0.875948i	$0.339763\pi$
$864$	0	0
$865$	32.9706	1.12103
$866$	0	0
$867$	−41.6274	−1.41374
$868$	0	0
$869$	22.6274	0.767583
$870$	0	0
$871$	6.82843	0.231372
$872$	0	0
$873$	3.65685	0.123766
$874$	0	0
$875$	−16.0000	−0.540899
$876$	0	0
$877$	42.2843	1.42784	0.713919	−	0.700228i	$-0.246918\pi$
$877$	42.2843	1.42784	0.713919	+	0.700228i	$0.246918\pi$
$878$	0	0
$879$	−2.14214	−0.0722524
$880$	0	0
$881$	−25.5980	−0.862418	−0.431209	−	0.902252i	$-0.641913\pi$
$881$	−25.5980	−0.862418	−0.431209	+	0.902252i	$0.641913\pi$
$882$	0	0
$883$	−27.5980	−0.928746	−0.464373	−	0.885640i	$-0.653720\pi$
$883$	−27.5980	−0.928746	−0.464373	+	0.885640i	$0.653720\pi$
$884$	0	0
$885$	−21.6569	−0.727987
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	32.9706	1.10332
$894$	0	0
$895$	−65.9411	−2.20417
$896$	0	0
$897$	4.00000	0.133556
$898$	0	0
$899$	2.34315	0.0781483
$900$	0	0
$901$	−15.3137	−0.510174
$902$	0	0
$903$	4.68629	0.155950
$904$	0	0
$905$	−39.5980	−1.31628
$906$	0	0
$907$	12.9706	0.430680	0.215340	−	0.976539i	$-0.430914\pi$
$907$	12.9706	0.430680	0.215340	+	0.976539i	$0.430914\pi$
$908$	0	0
$909$	7.65685	0.253962
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	−7.31371	−0.242048
$914$	0	0
$915$	37.6569	1.24490
$916$	0	0
$917$	−22.6274	−0.747223
$918$	0	0
$919$	−3.31371	−0.109309	−0.0546546	−	0.998505i	$-0.517406\pi$
$919$	−3.31371	−0.109309	−0.0546546	+	0.998505i	$0.517406\pi$
$920$	0	0
$921$	−22.8284	−0.752222
$922$	0	0
$923$	2.00000	0.0658308
$924$	0	0
$925$	−22.9706	−0.755267
$926$	0	0
$927$	−2.34315	−0.0769590
$928$	0	0
$929$	11.7990	0.387112	0.193556	−	0.981089i	$-0.437998\pi$
$929$	11.7990	0.387112	0.193556	+	0.981089i	$0.437998\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	0	0
$933$	−10.6274	−0.347926
$934$	0	0
$935$	−43.3137	−1.41651
$936$	0	0
$937$	−21.3137	−0.696289	−0.348144	−	0.937441i	$-0.613188\pi$
$937$	−21.3137	−0.696289	−0.348144	+	0.937441i	$0.613188\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	34.1421	1.11300	0.556501	−	0.830847i	$-0.312144\pi$
$941$	34.1421	1.11300	0.556501	+	0.830847i	$0.312144\pi$
$942$	0	0
$943$	20.6863	0.673638
$944$	0	0
$945$	−8.00000	−0.260240
$946$	0	0
$947$	21.0294	0.683365	0.341682	−	0.939815i	$-0.389003\pi$
$947$	21.0294	0.683365	0.341682	+	0.939815i	$0.389003\pi$
$948$	0	0
$949$	−0.343146	−0.0111390
$950$	0	0
$951$	−8.48528	−0.275154
$952$	0	0
$953$	40.3431	1.30684	0.653421	−	0.756994i	$-0.273333\pi$
$953$	40.3431	1.30684	0.653421	+	0.756994i	$0.273333\pi$
$954$	0	0
$955$	9.37258	0.303290
$956$	0	0
$957$	−4.00000	−0.129302
$958$	0	0
$959$	30.6274	0.989011
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	11.3137	0.364579
$964$	0	0
$965$	−15.0294	−0.483815
$966$	0	0
$967$	−18.1421	−0.583412	−0.291706	−	0.956508i	$-0.594223\pi$
$967$	−18.1421	−0.583412	−0.291706	+	0.956508i	$0.594223\pi$
$968$	0	0
$969$	−21.6569	−0.695718
$970$	0	0
$971$	15.3137	0.491440	0.245720	−	0.969341i	$-0.420976\pi$
$971$	15.3137	0.491440	0.245720	+	0.969341i	$0.420976\pi$
$972$	0	0
$973$	−20.6863	−0.663172
$974$	0	0
$975$	3.00000	0.0960769
$976$	0	0
$977$	42.1421	1.34825	0.674123	−	0.738619i	$-0.264522\pi$
$977$	42.1421	1.34825	0.674123	+	0.738619i	$0.264522\pi$
$978$	0	0
$979$	29.6569	0.947837
$980$	0	0
$981$	5.31371	0.169654
$982$	0	0
$983$	−25.3137	−0.807382	−0.403691	−	0.914895i	$-0.632273\pi$
$983$	−25.3137	−0.807382	−0.403691	+	0.914895i	$0.632273\pi$
$984$	0	0
$985$	−1.37258	−0.0437341
$986$	0	0
$987$	32.9706	1.04946
$988$	0	0
$989$	6.62742	0.210740
$990$	0	0
$991$	−4.68629	−0.148865	−0.0744325	−	0.997226i	$-0.523715\pi$
$991$	−4.68629	−0.148865	−0.0744325	+	0.997226i	$0.523715\pi$
$992$	0	0
$993$	26.1421	0.829596
$994$	0	0
$995$	61.2548	1.94191
$996$	0	0
$997$	−39.2548	−1.24321	−0.621607	−	0.783330i	$-0.713520\pi$
$997$	−39.2548	−1.24321	−0.621607	+	0.783330i	$0.713520\pi$
$998$	0	0
$999$	7.65685	0.242252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.2.a.k.1.1		2
3.2	odd	2		1872.2.a.w.1.2		2
4.3	odd	2		39.2.a.b.1.2	✓	2
8.3	odd	2		2496.2.a.bf.1.2		2
8.5	even	2		2496.2.a.bi.1.2		2
12.11	even	2		117.2.a.c.1.1		2
13.12	even	2		8112.2.a.bm.1.2		2
20.3	even	4		975.2.c.h.274.2		4
20.7	even	4		975.2.c.h.274.3		4
20.19	odd	2		975.2.a.l.1.1		2
24.5	odd	2		7488.2.a.co.1.1		2
24.11	even	2		7488.2.a.cl.1.1		2
28.27	even	2		1911.2.a.h.1.2		2
36.7	odd	6		1053.2.e.m.352.1		4
36.11	even	6		1053.2.e.e.352.2		4
36.23	even	6		1053.2.e.e.703.2		4
36.31	odd	6		1053.2.e.m.703.1		4
44.43	even	2		4719.2.a.p.1.1		2
52.3	odd	6		507.2.e.h.22.1		4
52.7	even	12		507.2.j.f.361.3		8
52.11	even	12		507.2.j.f.316.2		8
52.15	even	12		507.2.j.f.316.3		8
52.19	even	12		507.2.j.f.361.2		8
52.23	odd	6		507.2.e.d.22.2		4
52.31	even	4		507.2.b.e.337.2		4
52.35	odd	6		507.2.e.h.484.1		4
52.43	odd	6		507.2.e.d.484.2		4
52.47	even	4		507.2.b.e.337.3		4
52.51	odd	2		507.2.a.h.1.1		2
60.23	odd	4		2925.2.c.u.2224.3		4
60.47	odd	4		2925.2.c.u.2224.2		4
60.59	even	2		2925.2.a.v.1.2		2
84.83	odd	2		5733.2.a.u.1.1		2
156.47	odd	4		1521.2.b.j.1351.2		4
156.83	odd	4		1521.2.b.j.1351.3		4
156.155	even	2		1521.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.a.b.1.2	✓	2	4.3	odd	2
117.2.a.c.1.1		2	12.11	even	2
507.2.a.h.1.1		2	52.51	odd	2
507.2.b.e.337.2		4	52.31	even	4
507.2.b.e.337.3		4	52.47	even	4
507.2.e.d.22.2		4	52.23	odd	6
507.2.e.d.484.2		4	52.43	odd	6
507.2.e.h.22.1		4	52.3	odd	6
507.2.e.h.484.1		4	52.35	odd	6
507.2.j.f.316.2		8	52.11	even	12
507.2.j.f.316.3		8	52.15	even	12
507.2.j.f.361.2		8	52.19	even	12
507.2.j.f.361.3		8	52.7	even	12
624.2.a.k.1.1		2	1.1	even	1	trivial
975.2.a.l.1.1		2	20.19	odd	2
975.2.c.h.274.2		4	20.3	even	4
975.2.c.h.274.3		4	20.7	even	4
1053.2.e.e.352.2		4	36.11	even	6
1053.2.e.e.703.2		4	36.23	even	6
1053.2.e.m.352.1		4	36.7	odd	6
1053.2.e.m.703.1		4	36.31	odd	6
1521.2.a.f.1.2		2	156.155	even	2
1521.2.b.j.1351.2		4	156.47	odd	4
1521.2.b.j.1351.3		4	156.83	odd	4
1872.2.a.w.1.2		2	3.2	odd	2
1911.2.a.h.1.2		2	28.27	even	2
2496.2.a.bf.1.2		2	8.3	odd	2
2496.2.a.bi.1.2		2	8.5	even	2
2925.2.a.v.1.2		2	60.59	even	2
2925.2.c.u.2224.2		4	60.47	odd	4
2925.2.c.u.2224.3		4	60.23	odd	4
4719.2.a.p.1.1		2	44.43	even	2
5733.2.a.u.1.1		2	84.83	odd	2
7488.2.a.cl.1.1		2	24.11	even	2
7488.2.a.co.1.1		2	24.5	odd	2
8112.2.a.bm.1.2		2	13.12	even	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.82843 q^{5} -2.82843 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} +2.82843 q^{15} +7.65685 q^{17} +2.82843 q^{19} +2.82843 q^{21} +4.00000 q^{23} +3.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +1.17157 q^{31} -2.00000 q^{33} +8.00000 q^{35} -7.65685 q^{37} +1.00000 q^{39} +5.17157 q^{41} +1.65685 q^{43} -2.82843 q^{45} +11.6569 q^{47} +1.00000 q^{49} -7.65685 q^{51} -2.00000 q^{53} -5.65685 q^{55} -2.82843 q^{57} -7.65685 q^{59} +13.3137 q^{61} -2.82843 q^{63} +2.82843 q^{65} -6.82843 q^{67} -4.00000 q^{69} -2.00000 q^{71} +0.343146 q^{73} -3.00000 q^{75} -5.65685 q^{77} +11.3137 q^{79} +1.00000 q^{81} -3.65685 q^{83} -21.6569 q^{85} -2.00000 q^{87} +14.8284 q^{89} +2.82843 q^{91} -1.17157 q^{93} -8.00000 q^{95} +3.65685 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} + 4 q^{17} + 8 q^{23} + 6 q^{25} - 2 q^{27} + 4 q^{29} + 8 q^{31} - 4 q^{33} + 16 q^{35} - 4 q^{37} + 2 q^{39} + 16 q^{41} - 8 q^{43} + 12 q^{47} + 2 q^{49} - 4 q^{51} - 4 q^{53} - 4 q^{59} + 4 q^{61} - 8 q^{67} - 8 q^{69} - 4 q^{71} + 12 q^{73} - 6 q^{75} + 2 q^{81} + 4 q^{83} - 32 q^{85} - 4 q^{87} + 24 q^{89} - 8 q^{93} - 16 q^{95} - 4 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^11 - 2 * q^13 + 4 * q^17 + 8 * q^23 + 6 * q^25 - 2 * q^27 + 4 * q^29 + 8 * q^31 - 4 * q^33 + 16 * q^35 - 4 * q^37 + 2 * q^39 + 16 * q^41 - 8 * q^43 + 12 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 - 4 * q^59 + 4 * q^61 - 8 * q^67 - 8 * q^69 - 4 * q^71 + 12 * q^73 - 6 * q^75 + 2 * q^81 + 4 * q^83 - 32 * q^85 - 4 * q^87 + 24 * q^89 - 8 * q^93 - 16 * q^95 - 4 * q^97 + 4 * q^99

Introduction

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