LMFDB - Embedded newform 2496.2.a.bl.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,2,Mod(1,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,-2,0,0,0,3,0,4,0,-3,0,-2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1248)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	2496.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.96239 q^{5} +3.35026 q^{7} +1.00000 q^{9} +1.61213 q^{11} -1.00000 q^{13} +2.96239 q^{15} +2.00000 q^{17} +3.35026 q^{19} +3.35026 q^{21} -6.70052 q^{23} +3.77575 q^{25} +1.00000 q^{27} -2.00000 q^{29} +6.57452 q^{31} +1.61213 q^{33} +9.92478 q^{35} -7.92478 q^{37} -1.00000 q^{39} +6.96239 q^{41} +0.775746 q^{43} +2.96239 q^{45} +2.38787 q^{47} +4.22425 q^{49} +2.00000 q^{51} -11.9248 q^{53} +4.77575 q^{55} +3.35026 q^{57} -0.312650 q^{59} -14.6253 q^{61} +3.35026 q^{63} -2.96239 q^{65} -8.12601 q^{67} -6.70052 q^{69} -4.31265 q^{71} +0.0752228 q^{73} +3.77575 q^{75} +5.40105 q^{77} +12.0000 q^{79} +1.00000 q^{81} +8.31265 q^{83} +5.92478 q^{85} -2.00000 q^{87} +8.88717 q^{89} -3.35026 q^{91} +6.57452 q^{93} +9.92478 q^{95} -7.92478 q^{97} +1.61213 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 2 q^{5} + 3 q^{9} + 4 q^{11} - 3 q^{13} - 2 q^{15} + 6 q^{17} + 13 q^{25} + 3 q^{27} - 6 q^{29} + 8 q^{31} + 4 q^{33} + 8 q^{35} - 2 q^{37} - 3 q^{39} + 10 q^{41} + 4 q^{43} - 2 q^{45} + 8 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 2 * q^5 + 3 * q^9 + 4 * q^11 - 3 * q^13 - 2 * q^15 + 6 * q^17 + 13 * q^25 + 3 * q^27 - 6 * q^29 + 8 * q^31 + 4 * q^33 + 8 * q^35 - 2 * q^37 - 3 * q^39 + 10 * q^41 + 4 * q^43 - 2 * q^45 + 8 * q^47 + 11 * q^49 + 6 * q^51 - 14 * q^53 + 16 * q^55 + 20 * q^59 - 2 * q^61 + 2 * q^65 - 16 * q^67 + 8 * q^71 + 22 * q^73 + 13 * q^75 - 24 * q^77 + 36 * q^79 + 3 * q^81 + 4 * q^83 - 4 * q^85 - 6 * q^87 - 6 * q^89 + 8 * q^93 + 8 * q^95 - 2 * q^97 + 4 * q^99

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
$5$	2.96239	1.32482	0.662410	−	0.749141i	$-0.269534\pi$
$5$	2.96239	1.32482	0.662410	+	0.749141i	$0.269534\pi$
$6$	0	0
$7$	3.35026	1.26628	0.633140	−	0.774037i	$-0.281766\pi$
$7$	3.35026	1.26628	0.633140	+	0.774037i	$0.281766\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.61213	0.486075	0.243037	−	0.970017i	$-0.421856\pi$
$11$	1.61213	0.486075	0.243037	+	0.970017i	$0.421856\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.96239	0.764885
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	3.35026	0.768603	0.384301	−	0.923208i	$-0.374442\pi$
$19$	3.35026	0.768603	0.384301	+	0.923208i	$0.374442\pi$
$20$	0	0
$21$	3.35026	0.731087
$22$	0	0
$23$	−6.70052	−1.39716	−0.698578	−	0.715534i	$-0.746183\pi$
$23$	−6.70052	−1.39716	−0.698578	+	0.715534i	$0.746183\pi$
$24$	0	0
$25$	3.77575	0.755149
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	6.57452	1.18082	0.590409	−	0.807104i	$-0.298966\pi$
$31$	6.57452	1.18082	0.590409	+	0.807104i	$0.298966\pi$
$32$	0	0
$33$	1.61213	0.280635
$34$	0	0
$35$	9.92478	1.67759
$36$	0	0
$37$	−7.92478	−1.30283	−0.651413	−	0.758724i	$-0.725823\pi$
$37$	−7.92478	−1.30283	−0.651413	+	0.758724i	$0.725823\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.96239	1.08734	0.543671	−	0.839298i	$-0.317034\pi$
$41$	6.96239	1.08734	0.543671	+	0.839298i	$0.317034\pi$
$42$	0	0
$43$	0.775746	0.118300	0.0591501	−	0.998249i	$-0.481161\pi$
$43$	0.775746	0.118300	0.0591501	+	0.998249i	$0.481161\pi$
$44$	0	0
$45$	2.96239	0.441607
$46$	0	0
$47$	2.38787	0.348307	0.174154	−	0.984719i	$-0.444281\pi$
$47$	2.38787	0.348307	0.174154	+	0.984719i	$0.444281\pi$
$48$	0	0
$49$	4.22425	0.603465
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−11.9248	−1.63799	−0.818997	−	0.573798i	$-0.805470\pi$
$53$	−11.9248	−1.63799	−0.818997	+	0.573798i	$0.805470\pi$
$54$	0	0
$55$	4.77575	0.643961
$56$	0	0
$57$	3.35026	0.443753
$58$	0	0
$59$	−0.312650	−0.0407036	−0.0203518	−	0.999793i	$-0.506479\pi$
$59$	−0.312650	−0.0407036	−0.0203518	+	0.999793i	$0.506479\pi$
$60$	0	0
$61$	−14.6253	−1.87258	−0.936289	−	0.351231i	$-0.885763\pi$
$61$	−14.6253	−1.87258	−0.936289	+	0.351231i	$0.885763\pi$
$62$	0	0
$63$	3.35026	0.422093
$64$	0	0
$65$	−2.96239	−0.367439
$66$	0	0
$67$	−8.12601	−0.992750	−0.496375	−	0.868108i	$-0.665336\pi$
$67$	−8.12601	−0.992750	−0.496375	+	0.868108i	$0.665336\pi$
$68$	0	0
$69$	−6.70052	−0.806648
$70$	0	0
$71$	−4.31265	−0.511817	−0.255909	−	0.966701i	$-0.582375\pi$
$71$	−4.31265	−0.511817	−0.255909	+	0.966701i	$0.582375\pi$
$72$	0	0
$73$	0.0752228	0.00880416	0.00440208	−	0.999990i	$-0.498599\pi$
$73$	0.0752228	0.00880416	0.00440208	+	0.999990i	$0.498599\pi$
$74$	0	0
$75$	3.77575	0.435986
$76$	0	0
$77$	5.40105	0.615506
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.31265	0.912432	0.456216	−	0.889869i	$-0.349204\pi$
$83$	8.31265	0.912432	0.456216	+	0.889869i	$0.349204\pi$
$84$	0	0
$85$	5.92478	0.642632
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	8.88717	0.942038	0.471019	−	0.882123i	$-0.343886\pi$
$89$	8.88717	0.942038	0.471019	+	0.882123i	$0.343886\pi$
$90$	0	0
$91$	−3.35026	−0.351203
$92$	0	0
$93$	6.57452	0.681745
$94$	0	0
$95$	9.92478	1.01826
$96$	0	0
$97$	−7.92478	−0.804639	−0.402320	−	0.915499i	$-0.631796\pi$
$97$	−7.92478	−0.804639	−0.402320	+	0.915499i	$0.631796\pi$
$98$	0	0
$99$	1.61213	0.162025
$100$	0	0
$101$	−15.4010	−1.53246	−0.766231	−	0.642566i	$-0.777870\pi$
$101$	−15.4010	−1.53246	−0.766231	+	0.642566i	$0.777870\pi$
$102$	0	0
$103$	−4.62530	−0.455744	−0.227872	−	0.973691i	$-0.573177\pi$
$103$	−4.62530	−0.455744	−0.227872	+	0.973691i	$0.573177\pi$
$104$	0	0
$105$	9.92478	0.968559
$106$	0	0
$107$	−14.5501	−1.40661	−0.703305	−	0.710889i	$-0.748293\pi$
$107$	−14.5501	−1.40661	−0.703305	+	0.710889i	$0.748293\pi$
$108$	0	0
$109$	15.4010	1.47515	0.737576	−	0.675264i	$-0.235970\pi$
$109$	15.4010	1.47515	0.737576	+	0.675264i	$0.235970\pi$
$110$	0	0
$111$	−7.92478	−0.752187
$112$	0	0
$113$	−9.47627	−0.891452	−0.445726	−	0.895169i	$-0.647055\pi$
$113$	−9.47627	−0.891452	−0.445726	+	0.895169i	$0.647055\pi$
$114$	0	0
$115$	−19.8496	−1.85098
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	6.70052	0.614236
$120$	0	0
$121$	−8.40105	−0.763732
$122$	0	0
$123$	6.96239	0.627777
$124$	0	0
$125$	−3.62672	−0.324383
$126$	0	0
$127$	5.29948	0.470252	0.235126	−	0.971965i	$-0.424450\pi$
$127$	5.29948	0.470252	0.235126	+	0.971965i	$0.424450\pi$
$128$	0	0
$129$	0.775746	0.0683007
$130$	0	0
$131$	−9.14903	−0.799355	−0.399677	−	0.916656i	$-0.630878\pi$
$131$	−9.14903	−0.799355	−0.399677	+	0.916656i	$0.630878\pi$
$132$	0	0
$133$	11.2243	0.973266
$134$	0	0
$135$	2.96239	0.254962
$136$	0	0
$137$	0.513881	0.0439038	0.0219519	−	0.999759i	$-0.493012\pi$
$137$	0.513881	0.0439038	0.0219519	+	0.999759i	$0.493012\pi$
$138$	0	0
$139$	13.9248	1.18108	0.590542	−	0.807007i	$-0.298914\pi$
$139$	13.9248	1.18108	0.590542	+	0.807007i	$0.298914\pi$
$140$	0	0
$141$	2.38787	0.201095
$142$	0	0
$143$	−1.61213	−0.134813
$144$	0	0
$145$	−5.92478	−0.492026
$146$	0	0
$147$	4.22425	0.348411
$148$	0	0
$149$	12.8872	1.05576	0.527879	−	0.849320i	$-0.322987\pi$
$149$	12.8872	1.05576	0.527879	+	0.849320i	$0.322987\pi$
$150$	0	0
$151$	13.2750	1.08031	0.540154	−	0.841566i	$-0.318366\pi$
$151$	13.2750	1.08031	0.540154	+	0.841566i	$0.318366\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	19.4763	1.56437
$156$	0	0
$157$	−21.0738	−1.68187	−0.840936	−	0.541134i	$-0.817995\pi$
$157$	−21.0738	−1.68187	−0.840936	+	0.541134i	$0.817995\pi$
$158$	0	0
$159$	−11.9248	−0.945696
$160$	0	0
$161$	−22.4485	−1.76919
$162$	0	0
$163$	21.9003	1.71537	0.857683	−	0.514178i	$-0.171903\pi$
$163$	21.9003	1.71537	0.857683	+	0.514178i	$0.171903\pi$
$164$	0	0
$165$	4.77575	0.371791
$166$	0	0
$167$	−17.4617	−1.35123	−0.675613	−	0.737257i	$-0.736121\pi$
$167$	−17.4617	−1.35123	−0.675613	+	0.737257i	$0.736121\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.35026	0.256201
$172$	0	0
$173$	−21.8496	−1.66119	−0.830595	−	0.556876i	$-0.812000\pi$
$173$	−21.8496	−1.66119	−0.830595	+	0.556876i	$0.812000\pi$
$174$	0	0
$175$	12.6497	0.956230
$176$	0	0
$177$	−0.312650	−0.0235002
$178$	0	0
$179$	21.9248	1.63873	0.819367	−	0.573269i	$-0.194325\pi$
$179$	21.9248	1.63873	0.819367	+	0.573269i	$0.194325\pi$
$180$	0	0
$181$	18.6253	1.38441	0.692204	−	0.721702i	$-0.256640\pi$
$181$	18.6253	1.38441	0.692204	+	0.721702i	$0.256640\pi$
$182$	0	0
$183$	−14.6253	−1.08113
$184$	0	0
$185$	−23.4763	−1.72601
$186$	0	0
$187$	3.22425	0.235781
$188$	0	0
$189$	3.35026	0.243696
$190$	0	0
$191$	21.7743	1.57554	0.787768	−	0.615972i	$-0.211237\pi$
$191$	21.7743	1.57554	0.787768	+	0.615972i	$0.211237\pi$
$192$	0	0
$193$	19.2506	1.38569	0.692844	−	0.721087i	$-0.256357\pi$
$193$	19.2506	1.38569	0.692844	+	0.721087i	$0.256357\pi$
$194$	0	0
$195$	−2.96239	−0.212141
$196$	0	0
$197$	4.51388	0.321601	0.160800	−	0.986987i	$-0.448592\pi$
$197$	4.51388	0.321601	0.160800	+	0.986987i	$0.448592\pi$
$198$	0	0
$199$	−9.14903	−0.648558	−0.324279	−	0.945962i	$-0.605122\pi$
$199$	−9.14903	−0.648558	−0.324279	+	0.945962i	$0.605122\pi$
$200$	0	0
$201$	−8.12601	−0.573164
$202$	0	0
$203$	−6.70052	−0.470285
$204$	0	0
$205$	20.6253	1.44053
$206$	0	0
$207$	−6.70052	−0.465719
$208$	0	0
$209$	5.40105	0.373598
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	−4.31265	−0.295498
$214$	0	0
$215$	2.29806	0.156727
$216$	0	0
$217$	22.0263	1.49525
$218$	0	0
$219$	0.0752228	0.00508308
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	18.4241	1.23377	0.616883	−	0.787054i	$-0.288395\pi$
$223$	18.4241	1.23377	0.616883	+	0.787054i	$0.288395\pi$
$224$	0	0
$225$	3.77575	0.251716
$226$	0	0
$227$	−1.61213	−0.107001	−0.0535003	−	0.998568i	$-0.517038\pi$
$227$	−1.61213	−0.107001	−0.0535003	+	0.998568i	$0.517038\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	5.40105	0.355363
$232$	0	0
$233$	0.0752228	0.00492801	0.00246400	−	0.999997i	$-0.499216\pi$
$233$	0.0752228	0.00492801	0.00246400	+	0.999997i	$0.499216\pi$
$234$	0	0
$235$	7.07381	0.461444
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	9.08840	0.587880	0.293940	−	0.955824i	$-0.405033\pi$
$239$	9.08840	0.587880	0.293940	+	0.955824i	$0.405033\pi$
$240$	0	0
$241$	15.7743	1.01611	0.508057	−	0.861323i	$-0.330364\pi$
$241$	15.7743	1.01611	0.508057	+	0.861323i	$0.330364\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	12.5139	0.799483
$246$	0	0
$247$	−3.35026	−0.213172
$248$	0	0
$249$	8.31265	0.526793
$250$	0	0
$251$	15.2243	0.960946	0.480473	−	0.877009i	$-0.340465\pi$
$251$	15.2243	0.960946	0.480473	+	0.877009i	$0.340465\pi$
$252$	0	0
$253$	−10.8021	−0.679122
$254$	0	0
$255$	5.92478	0.371024
$256$	0	0
$257$	−4.07522	−0.254205	−0.127103	−	0.991890i	$-0.540568\pi$
$257$	−4.07522	−0.254205	−0.127103	+	0.991890i	$0.540568\pi$
$258$	0	0
$259$	−26.5501	−1.64974
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−12.7757	−0.787786	−0.393893	−	0.919156i	$-0.628872\pi$
$263$	−12.7757	−0.787786	−0.393893	+	0.919156i	$0.628872\pi$
$264$	0	0
$265$	−35.3258	−2.17005
$266$	0	0
$267$	8.88717	0.543886
$268$	0	0
$269$	−21.8496	−1.33219	−0.666095	−	0.745867i	$-0.732036\pi$
$269$	−21.8496	−1.33219	−0.666095	+	0.745867i	$0.732036\pi$
$270$	0	0
$271$	−0.499293	−0.0303299	−0.0151649	−	0.999885i	$-0.504827\pi$
$271$	−0.499293	−0.0303299	−0.0151649	+	0.999885i	$0.504827\pi$
$272$	0	0
$273$	−3.35026	−0.202767
$274$	0	0
$275$	6.08698	0.367059
$276$	0	0
$277$	13.8496	0.832139	0.416070	−	0.909333i	$-0.363407\pi$
$277$	13.8496	0.832139	0.416070	+	0.909333i	$0.363407\pi$
$278$	0	0
$279$	6.57452	0.393606
$280$	0	0
$281$	−14.4387	−0.861338	−0.430669	−	0.902510i	$-0.641722\pi$
$281$	−14.4387	−0.861338	−0.430669	+	0.902510i	$0.641722\pi$
$282$	0	0
$283$	−6.55008	−0.389362	−0.194681	−	0.980867i	$-0.562367\pi$
$283$	−6.55008	−0.389362	−0.194681	+	0.980867i	$0.562367\pi$
$284$	0	0
$285$	9.92478	0.587893
$286$	0	0
$287$	23.3258	1.37688
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−7.92478	−0.464559
$292$	0	0
$293$	14.4387	0.843515	0.421758	−	0.906709i	$-0.361413\pi$
$293$	14.4387	0.843515	0.421758	+	0.906709i	$0.361413\pi$
$294$	0	0
$295$	−0.926192	−0.0539250
$296$	0	0
$297$	1.61213	0.0935451
$298$	0	0
$299$	6.70052	0.387501
$300$	0	0
$301$	2.59895	0.149801
$302$	0	0
$303$	−15.4010	−0.884767
$304$	0	0
$305$	−43.3258	−2.48083
$306$	0	0
$307$	−15.8740	−0.905977	−0.452988	−	0.891516i	$-0.649642\pi$
$307$	−15.8740	−0.905977	−0.452988	+	0.891516i	$0.649642\pi$
$308$	0	0
$309$	−4.62530	−0.263124
$310$	0	0
$311$	8.25202	0.467929	0.233964	−	0.972245i	$-0.424830\pi$
$311$	8.25202	0.467929	0.233964	+	0.972245i	$0.424830\pi$
$312$	0	0
$313$	−3.40105	−0.192239	−0.0961193	−	0.995370i	$-0.530643\pi$
$313$	−3.40105	−0.192239	−0.0961193	+	0.995370i	$0.530643\pi$
$314$	0	0
$315$	9.92478	0.559198
$316$	0	0
$317$	−10.4387	−0.586293	−0.293147	−	0.956067i	$-0.594702\pi$
$317$	−10.4387	−0.586293	−0.293147	+	0.956067i	$0.594702\pi$
$318$	0	0
$319$	−3.22425	−0.180524
$320$	0	0
$321$	−14.5501	−0.812106
$322$	0	0
$323$	6.70052	0.372827
$324$	0	0
$325$	−3.77575	−0.209441
$326$	0	0
$327$	15.4010	0.851680
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−6.57452	−0.361368	−0.180684	−	0.983541i	$-0.557831\pi$
$331$	−6.57452	−0.361368	−0.180684	+	0.983541i	$0.557831\pi$
$332$	0	0
$333$	−7.92478	−0.434275
$334$	0	0
$335$	−24.0724	−1.31522
$336$	0	0
$337$	31.4010	1.71052	0.855262	−	0.518196i	$-0.173396\pi$
$337$	31.4010	1.71052	0.855262	+	0.518196i	$0.173396\pi$
$338$	0	0
$339$	−9.47627	−0.514680
$340$	0	0
$341$	10.5990	0.573965
$342$	0	0
$343$	−9.29948	−0.502125
$344$	0	0
$345$	−19.8496	−1.06866
$346$	0	0
$347$	21.2506	1.14079	0.570396	−	0.821370i	$-0.306790\pi$
$347$	21.2506	1.14079	0.570396	+	0.821370i	$0.306790\pi$
$348$	0	0
$349$	19.9248	1.06655	0.533274	−	0.845942i	$-0.320961\pi$
$349$	19.9248	1.06655	0.533274	+	0.845942i	$0.320961\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−6.43866	−0.342695	−0.171348	−	0.985211i	$-0.554812\pi$
$353$	−6.43866	−0.342695	−0.171348	+	0.985211i	$0.554812\pi$
$354$	0	0
$355$	−12.7757	−0.678066
$356$	0	0
$357$	6.70052	0.354629
$358$	0	0
$359$	−28.5647	−1.50759	−0.753793	−	0.657112i	$-0.771778\pi$
$359$	−28.5647	−1.50759	−0.753793	+	0.657112i	$0.771778\pi$
$360$	0	0
$361$	−7.77575	−0.409250
$362$	0	0
$363$	−8.40105	−0.440941
$364$	0	0
$365$	0.222839	0.0116639
$366$	0	0
$367$	−19.3258	−1.00880	−0.504400	−	0.863470i	$-0.668286\pi$
$367$	−19.3258	−1.00880	−0.504400	+	0.863470i	$0.668286\pi$
$368$	0	0
$369$	6.96239	0.362447
$370$	0	0
$371$	−39.9511	−2.07416
$372$	0	0
$373$	25.6991	1.33065	0.665325	−	0.746554i	$-0.268293\pi$
$373$	25.6991	1.33065	0.665325	+	0.746554i	$0.268293\pi$
$374$	0	0
$375$	−3.62672	−0.187283
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	24.7513	1.27139	0.635695	−	0.771941i	$-0.280714\pi$
$379$	24.7513	1.27139	0.635695	+	0.771941i	$0.280714\pi$
$380$	0	0
$381$	5.29948	0.271500
$382$	0	0
$383$	21.6121	1.10433	0.552164	−	0.833735i	$-0.313802\pi$
$383$	21.6121	1.10433	0.552164	+	0.833735i	$0.313802\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	0	0
$387$	0.775746	0.0394334
$388$	0	0
$389$	−25.3258	−1.28407	−0.642035	−	0.766675i	$-0.721910\pi$
$389$	−25.3258	−1.28407	−0.642035	+	0.766675i	$0.721910\pi$
$390$	0	0
$391$	−13.4010	−0.677720
$392$	0	0
$393$	−9.14903	−0.461508
$394$	0	0
$395$	35.5487	1.78865
$396$	0	0
$397$	3.92478	0.196979	0.0984895	−	0.995138i	$-0.468599\pi$
$397$	3.92478	0.196979	0.0984895	+	0.995138i	$0.468599\pi$
$398$	0	0
$399$	11.2243	0.561916
$400$	0	0
$401$	−17.4109	−0.869459	−0.434729	−	0.900561i	$-0.643156\pi$
$401$	−17.4109	−0.869459	−0.434729	+	0.900561i	$0.643156\pi$
$402$	0	0
$403$	−6.57452	−0.327500
$404$	0	0
$405$	2.96239	0.147202
$406$	0	0
$407$	−12.7757	−0.633270
$408$	0	0
$409$	−15.9248	−0.787430	−0.393715	−	0.919233i	$-0.628810\pi$
$409$	−15.9248	−0.787430	−0.393715	+	0.919233i	$0.628810\pi$
$410$	0	0
$411$	0.513881	0.0253479
$412$	0	0
$413$	−1.04746	−0.0515422
$414$	0	0
$415$	24.6253	1.20881
$416$	0	0
$417$	13.9248	0.681899
$418$	0	0
$419$	−36.9986	−1.80750	−0.903750	−	0.428062i	$-0.859197\pi$
$419$	−36.9986	−1.80750	−0.903750	+	0.428062i	$0.859197\pi$
$420$	0	0
$421$	39.4010	1.92029	0.960145	−	0.279503i	$-0.0901698\pi$
$421$	39.4010	1.92029	0.960145	+	0.279503i	$0.0901698\pi$
$422$	0	0
$423$	2.38787	0.116102
$424$	0	0
$425$	7.55149	0.366301
$426$	0	0
$427$	−48.9986	−2.37121
$428$	0	0
$429$	−1.61213	−0.0778342
$430$	0	0
$431$	−35.6385	−1.71664	−0.858322	−	0.513111i	$-0.828493\pi$
$431$	−35.6385	−1.71664	−0.858322	+	0.513111i	$0.828493\pi$
$432$	0	0
$433$	33.0738	1.58943	0.794713	−	0.606986i	$-0.207621\pi$
$433$	33.0738	1.58943	0.794713	+	0.606986i	$0.207621\pi$
$434$	0	0
$435$	−5.92478	−0.284071
$436$	0	0
$437$	−22.4485	−1.07386
$438$	0	0
$439$	32.4749	1.54994	0.774970	−	0.631998i	$-0.217765\pi$
$439$	32.4749	1.54994	0.774970	+	0.631998i	$0.217765\pi$
$440$	0	0
$441$	4.22425	0.201155
$442$	0	0
$443$	−9.14903	−0.434684	−0.217342	−	0.976096i	$-0.569739\pi$
$443$	−9.14903	−0.434684	−0.217342	+	0.976096i	$0.569739\pi$
$444$	0	0
$445$	26.3272	1.24803
$446$	0	0
$447$	12.8872	0.609542
$448$	0	0
$449$	−24.3634	−1.14978	−0.574891	−	0.818230i	$-0.694955\pi$
$449$	−24.3634	−1.14978	−0.574891	+	0.818230i	$0.694955\pi$
$450$	0	0
$451$	11.2243	0.528529
$452$	0	0
$453$	13.2750	0.623716
$454$	0	0
$455$	−9.92478	−0.465281
$456$	0	0
$457$	29.4763	1.37884	0.689421	−	0.724361i	$-0.257865\pi$
$457$	29.4763	1.37884	0.689421	+	0.724361i	$0.257865\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	7.11283	0.331278	0.165639	−	0.986186i	$-0.447031\pi$
$461$	7.11283	0.331278	0.165639	+	0.986186i	$0.447031\pi$
$462$	0	0
$463$	−42.7974	−1.98896	−0.994481	−	0.104918i	$-0.966542\pi$
$463$	−42.7974	−1.98896	−0.994481	+	0.104918i	$0.966542\pi$
$464$	0	0
$465$	19.4763	0.903190
$466$	0	0
$467$	−12.6253	−0.584229	−0.292115	−	0.956383i	$-0.594359\pi$
$467$	−12.6253	−0.584229	−0.292115	+	0.956383i	$0.594359\pi$
$468$	0	0
$469$	−27.2243	−1.25710
$470$	0	0
$471$	−21.0738	−0.971030
$472$	0	0
$473$	1.25060	0.0575027
$474$	0	0
$475$	12.6497	0.580410
$476$	0	0
$477$	−11.9248	−0.545998
$478$	0	0
$479$	−34.3390	−1.56899	−0.784494	−	0.620136i	$-0.787077\pi$
$479$	−34.3390	−1.56899	−0.784494	+	0.620136i	$0.787077\pi$
$480$	0	0
$481$	7.92478	0.361339
$482$	0	0
$483$	−22.4485	−1.02144
$484$	0	0
$485$	−23.4763	−1.06600
$486$	0	0
$487$	21.5271	0.975484	0.487742	−	0.872988i	$-0.337821\pi$
$487$	21.5271	0.975484	0.487742	+	0.872988i	$0.337821\pi$
$488$	0	0
$489$	21.9003	0.990368
$490$	0	0
$491$	−26.0263	−1.17455	−0.587276	−	0.809387i	$-0.699800\pi$
$491$	−26.0263	−1.17455	−0.587276	+	0.809387i	$0.699800\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	4.77575	0.214654
$496$	0	0
$497$	−14.4485	−0.648104
$498$	0	0
$499$	1.42548	0.0638135	0.0319067	−	0.999491i	$-0.489842\pi$
$499$	1.42548	0.0638135	0.0319067	+	0.999491i	$0.489842\pi$
$500$	0	0
$501$	−17.4617	−0.780130
$502$	0	0
$503$	−1.67276	−0.0745847	−0.0372924	−	0.999304i	$-0.511873\pi$
$503$	−1.67276	−0.0745847	−0.0372924	+	0.999304i	$0.511873\pi$
$504$	0	0
$505$	−45.6239	−2.03024
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−6.58910	−0.292057	−0.146028	−	0.989280i	$-0.546649\pi$
$509$	−6.58910	−0.292057	−0.146028	+	0.989280i	$0.546649\pi$
$510$	0	0
$511$	0.252016	0.0111485
$512$	0	0
$513$	3.35026	0.147918
$514$	0	0
$515$	−13.7019	−0.603780
$516$	0	0
$517$	3.84955	0.169303
$518$	0	0
$519$	−21.8496	−0.959089
$520$	0	0
$521$	26.3733	1.15543	0.577717	−	0.816237i	$-0.303944\pi$
$521$	26.3733	1.15543	0.577717	+	0.816237i	$0.303944\pi$
$522$	0	0
$523$	3.32582	0.145428	0.0727141	−	0.997353i	$-0.476834\pi$
$523$	3.32582	0.145428	0.0727141	+	0.997353i	$0.476834\pi$
$524$	0	0
$525$	12.6497	0.552080
$526$	0	0
$527$	13.1490	0.572781
$528$	0	0
$529$	21.8970	0.952044
$530$	0	0
$531$	−0.312650	−0.0135679
$532$	0	0
$533$	−6.96239	−0.301575
$534$	0	0
$535$	−43.1030	−1.86350
$536$	0	0
$537$	21.9248	0.946124
$538$	0	0
$539$	6.81003	0.293329
$540$	0	0
$541$	−33.8496	−1.45531	−0.727653	−	0.685945i	$-0.759389\pi$
$541$	−33.8496	−1.45531	−0.727653	+	0.685945i	$0.759389\pi$
$542$	0	0
$543$	18.6253	0.799288
$544$	0	0
$545$	45.6239	1.95431
$546$	0	0
$547$	−25.7743	−1.10203	−0.551015	−	0.834495i	$-0.685759\pi$
$547$	−25.7743	−1.10203	−0.551015	+	0.834495i	$0.685759\pi$
$548$	0	0
$549$	−14.6253	−0.624193
$550$	0	0
$551$	−6.70052	−0.285452
$552$	0	0
$553$	40.2031	1.70961
$554$	0	0
$555$	−23.4763	−0.996512
$556$	0	0
$557$	4.88717	0.207076	0.103538	−	0.994626i	$-0.466984\pi$
$557$	4.88717	0.207076	0.103538	+	0.994626i	$0.466984\pi$
$558$	0	0
$559$	−0.775746	−0.0328106
$560$	0	0
$561$	3.22425	0.136128
$562$	0	0
$563$	−32.2228	−1.35803	−0.679015	−	0.734124i	$-0.737593\pi$
$563$	−32.2228	−1.35803	−0.679015	+	0.734124i	$0.737593\pi$
$564$	0	0
$565$	−28.0724	−1.18101
$566$	0	0
$567$	3.35026	0.140698
$568$	0	0
$569$	−18.2228	−0.763941	−0.381971	−	0.924174i	$-0.624754\pi$
$569$	−18.2228	−0.763941	−0.381971	+	0.924174i	$0.624754\pi$
$570$	0	0
$571$	36.9986	1.54834	0.774171	−	0.632976i	$-0.218167\pi$
$571$	36.9986	1.54834	0.774171	+	0.632976i	$0.218167\pi$
$572$	0	0
$573$	21.7743	0.909636
$574$	0	0
$575$	−25.2995	−1.05506
$576$	0	0
$577$	−1.10299	−0.0459179	−0.0229589	−	0.999736i	$-0.507309\pi$
$577$	−1.10299	−0.0459179	−0.0229589	+	0.999736i	$0.507309\pi$
$578$	0	0
$579$	19.2506	0.800028
$580$	0	0
$581$	27.8496	1.15539
$582$	0	0
$583$	−19.2243	−0.796187
$584$	0	0
$585$	−2.96239	−0.122480
$586$	0	0
$587$	−37.0884	−1.53080	−0.765401	−	0.643554i	$-0.777459\pi$
$587$	−37.0884	−1.53080	−0.765401	+	0.643554i	$0.777459\pi$
$588$	0	0
$589$	22.0263	0.907580
$590$	0	0
$591$	4.51388	0.185676
$592$	0	0
$593$	39.8397	1.63602	0.818010	−	0.575204i	$-0.195077\pi$
$593$	39.8397	1.63602	0.818010	+	0.575204i	$0.195077\pi$
$594$	0	0
$595$	19.8496	0.813752
$596$	0	0
$597$	−9.14903	−0.374445
$598$	0	0
$599$	22.0263	0.899972	0.449986	−	0.893036i	$-0.351429\pi$
$599$	22.0263	0.899972	0.449986	+	0.893036i	$0.351429\pi$
$600$	0	0
$601$	−35.4010	−1.44404	−0.722019	−	0.691873i	$-0.756786\pi$
$601$	−35.4010	−1.44404	−0.722019	+	0.691873i	$0.756786\pi$
$602$	0	0
$603$	−8.12601	−0.330917
$604$	0	0
$605$	−24.8872	−1.01181
$606$	0	0
$607$	−10.0752	−0.408941	−0.204470	−	0.978873i	$-0.565547\pi$
$607$	−10.0752	−0.408941	−0.204470	+	0.978873i	$0.565547\pi$
$608$	0	0
$609$	−6.70052	−0.271519
$610$	0	0
$611$	−2.38787	−0.0966030
$612$	0	0
$613$	−14.3733	−0.580532	−0.290266	−	0.956946i	$-0.593744\pi$
$613$	−14.3733	−0.580532	−0.290266	+	0.956946i	$0.593744\pi$
$614$	0	0
$615$	20.6253	0.831692
$616$	0	0
$617$	38.6615	1.55645	0.778227	−	0.627984i	$-0.216119\pi$
$617$	38.6615	1.55645	0.778227	+	0.627984i	$0.216119\pi$
$618$	0	0
$619$	8.12601	0.326612	0.163306	−	0.986575i	$-0.447784\pi$
$619$	8.12601	0.326612	0.163306	+	0.986575i	$0.447784\pi$
$620$	0	0
$621$	−6.70052	−0.268883
$622$	0	0
$623$	29.7743	1.19288
$624$	0	0
$625$	−29.6225	−1.18490
$626$	0	0
$627$	5.40105	0.215697
$628$	0	0
$629$	−15.8496	−0.631963
$630$	0	0
$631$	−22.5745	−0.898677	−0.449339	−	0.893362i	$-0.648340\pi$
$631$	−22.5745	−0.898677	−0.449339	+	0.893362i	$0.648340\pi$
$632$	0	0
$633$	−12.0000	−0.476957
$634$	0	0
$635$	15.6991	0.623000
$636$	0	0
$637$	−4.22425	−0.167371
$638$	0	0
$639$	−4.31265	−0.170606
$640$	0	0
$641$	−17.8496	−0.705015	−0.352508	−	0.935809i	$-0.614671\pi$
$641$	−17.8496	−0.705015	−0.352508	+	0.935809i	$0.614671\pi$
$642$	0	0
$643$	2.47295	0.0975234	0.0487617	−	0.998810i	$-0.484473\pi$
$643$	2.47295	0.0975234	0.0487617	+	0.998810i	$0.484473\pi$
$644$	0	0
$645$	2.29806	0.0904861
$646$	0	0
$647$	−22.9525	−0.902357	−0.451179	−	0.892434i	$-0.648996\pi$
$647$	−22.9525	−0.902357	−0.451179	+	0.892434i	$0.648996\pi$
$648$	0	0
$649$	−0.504032	−0.0197850
$650$	0	0
$651$	22.0263	0.863281
$652$	0	0
$653$	19.4010	0.759222	0.379611	−	0.925146i	$-0.376058\pi$
$653$	19.4010	0.759222	0.379611	+	0.925146i	$0.376058\pi$
$654$	0	0
$655$	−27.1030	−1.05900
$656$	0	0
$657$	0.0752228	0.00293472
$658$	0	0
$659$	−15.8496	−0.617411	−0.308705	−	0.951158i	$-0.599896\pi$
$659$	−15.8496	−0.617411	−0.308705	+	0.951158i	$0.599896\pi$
$660$	0	0
$661$	−30.8773	−1.20099	−0.600494	−	0.799629i	$-0.705029\pi$
$661$	−30.8773	−1.20099	−0.600494	+	0.799629i	$0.705029\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	33.2506	1.28940
$666$	0	0
$667$	13.4010	0.518891
$668$	0	0
$669$	18.4241	0.712316
$670$	0	0
$671$	−23.5778	−0.910212
$672$	0	0
$673$	16.3272	0.629369	0.314684	−	0.949196i	$-0.398101\pi$
$673$	16.3272	0.629369	0.314684	+	0.949196i	$0.398101\pi$
$674$	0	0
$675$	3.77575	0.145329
$676$	0	0
$677$	−32.0752	−1.23275	−0.616375	−	0.787452i	$-0.711400\pi$
$677$	−32.0752	−1.23275	−0.616375	+	0.787452i	$0.711400\pi$
$678$	0	0
$679$	−26.5501	−1.01890
$680$	0	0
$681$	−1.61213	−0.0617768
$682$	0	0
$683$	−15.1344	−0.579103	−0.289552	−	0.957162i	$-0.593506\pi$
$683$	−15.1344	−0.579103	−0.289552	+	0.957162i	$0.593506\pi$
$684$	0	0
$685$	1.52232	0.0581647
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	11.9248	0.454298
$690$	0	0
$691$	19.7235	0.750319	0.375160	−	0.926960i	$-0.377588\pi$
$691$	19.7235	0.750319	0.375160	+	0.926960i	$0.377588\pi$
$692$	0	0
$693$	5.40105	0.205169
$694$	0	0
$695$	41.2506	1.56472
$696$	0	0
$697$	13.9248	0.527439
$698$	0	0
$699$	0.0752228	0.00284519
$700$	0	0
$701$	7.92478	0.299315	0.149657	−	0.988738i	$-0.452183\pi$
$701$	7.92478	0.299315	0.149657	+	0.988738i	$0.452183\pi$
$702$	0	0
$703$	−26.5501	−1.00136
$704$	0	0
$705$	7.07381	0.266415
$706$	0	0
$707$	−51.5975	−1.94053
$708$	0	0
$709$	−15.5515	−0.584049	−0.292024	−	0.956411i	$-0.594329\pi$
$709$	−15.5515	−0.584049	−0.292024	+	0.956411i	$0.594329\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	−44.0527	−1.64979
$714$	0	0
$715$	−4.77575	−0.178603
$716$	0	0
$717$	9.08840	0.339412
$718$	0	0
$719$	18.5501	0.691801	0.345901	−	0.938271i	$-0.387573\pi$
$719$	18.5501	0.691801	0.345901	+	0.938271i	$0.387573\pi$
$720$	0	0
$721$	−15.4960	−0.577100
$722$	0	0
$723$	15.7743	0.586654
$724$	0	0
$725$	−7.55149	−0.280455
$726$	0	0
$727$	−11.9511	−0.443243	−0.221621	−	0.975133i	$-0.571135\pi$
$727$	−11.9511	−0.443243	−0.221621	+	0.975133i	$0.571135\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.55149	0.0573840
$732$	0	0
$733$	−8.80209	−0.325113	−0.162556	−	0.986699i	$-0.551974\pi$
$733$	−8.80209	−0.325113	−0.162556	+	0.986699i	$0.551974\pi$
$734$	0	0
$735$	12.5139	0.461581
$736$	0	0
$737$	−13.1002	−0.482550
$738$	0	0
$739$	23.8251	0.876421	0.438211	−	0.898872i	$-0.355612\pi$
$739$	23.8251	0.876421	0.438211	+	0.898872i	$0.355612\pi$
$740$	0	0
$741$	−3.35026	−0.123075
$742$	0	0
$743$	7.53690	0.276502	0.138251	−	0.990397i	$-0.455852\pi$
$743$	7.53690	0.276502	0.138251	+	0.990397i	$0.455852\pi$
$744$	0	0
$745$	38.1768	1.39869
$746$	0	0
$747$	8.31265	0.304144
$748$	0	0
$749$	−48.7466	−1.78116
$750$	0	0
$751$	44.6253	1.62840	0.814200	−	0.580584i	$-0.197176\pi$
$751$	44.6253	1.62840	0.814200	+	0.580584i	$0.197176\pi$
$752$	0	0
$753$	15.2243	0.554803
$754$	0	0
$755$	39.3258	1.43121
$756$	0	0
$757$	11.6728	0.424254	0.212127	−	0.977242i	$-0.431961\pi$
$757$	11.6728	0.424254	0.212127	+	0.977242i	$0.431961\pi$
$758$	0	0
$759$	−10.8021	−0.392091
$760$	0	0
$761$	−19.8397	−0.719189	−0.359594	−	0.933109i	$-0.617085\pi$
$761$	−19.8397	−0.719189	−0.359594	+	0.933109i	$0.617085\pi$
$762$	0	0
$763$	51.5975	1.86796
$764$	0	0
$765$	5.92478	0.214211
$766$	0	0
$767$	0.312650	0.0112891
$768$	0	0
$769$	−35.1002	−1.26574	−0.632872	−	0.774256i	$-0.718124\pi$
$769$	−35.1002	−1.26574	−0.632872	+	0.774256i	$0.718124\pi$
$770$	0	0
$771$	−4.07522	−0.146766
$772$	0	0
$773$	34.2882	1.23326	0.616631	−	0.787253i	$-0.288497\pi$
$773$	34.2882	1.23326	0.616631	+	0.787253i	$0.288497\pi$
$774$	0	0
$775$	24.8237	0.891694
$776$	0	0
$777$	−26.5501	−0.952479
$778$	0	0
$779$	23.3258	0.835734
$780$	0	0
$781$	−6.95254	−0.248781
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−62.4288	−2.22818
$786$	0	0
$787$	31.8251	1.13444	0.567221	−	0.823565i	$-0.308018\pi$
$787$	31.8251	1.13444	0.567221	+	0.823565i	$0.308018\pi$
$788$	0	0
$789$	−12.7757	−0.454829
$790$	0	0
$791$	−31.7480	−1.12883
$792$	0	0
$793$	14.6253	0.519360
$794$	0	0
$795$	−35.3258	−1.25288
$796$	0	0
$797$	−16.0752	−0.569414	−0.284707	−	0.958615i	$-0.591896\pi$
$797$	−16.0752	−0.569414	−0.284707	+	0.958615i	$0.591896\pi$
$798$	0	0
$799$	4.77575	0.168954
$800$	0	0
$801$	8.88717	0.314013
$802$	0	0
$803$	0.121269	0.00427948
$804$	0	0
$805$	−66.5012	−2.34386
$806$	0	0
$807$	−21.8496	−0.769141
$808$	0	0
$809$	1.32582	0.0466135	0.0233067	−	0.999728i	$-0.492581\pi$
$809$	1.32582	0.0466135	0.0233067	+	0.999728i	$0.492581\pi$
$810$	0	0
$811$	−12.3488	−0.433627	−0.216813	−	0.976213i	$-0.569566\pi$
$811$	−12.3488	−0.433627	−0.216813	+	0.976213i	$0.569566\pi$
$812$	0	0
$813$	−0.499293	−0.0175110
$814$	0	0
$815$	64.8773	2.27255
$816$	0	0
$817$	2.59895	0.0909259
$818$	0	0
$819$	−3.35026	−0.117068
$820$	0	0
$821$	54.1378	1.88942	0.944711	−	0.327905i	$-0.106343\pi$
$821$	54.1378	1.88942	0.944711	+	0.327905i	$0.106343\pi$
$822$	0	0
$823$	−15.7283	−0.548254	−0.274127	−	0.961694i	$-0.588389\pi$
$823$	−15.7283	−0.548254	−0.274127	+	0.961694i	$0.588389\pi$
$824$	0	0
$825$	6.08698	0.211922
$826$	0	0
$827$	−27.4880	−0.955852	−0.477926	−	0.878400i	$-0.658611\pi$
$827$	−27.4880	−0.955852	−0.477926	+	0.878400i	$0.658611\pi$
$828$	0	0
$829$	−15.6728	−0.544337	−0.272169	−	0.962250i	$-0.587741\pi$
$829$	−15.6728	−0.544337	−0.272169	+	0.962250i	$0.587741\pi$
$830$	0	0
$831$	13.8496	0.480436
$832$	0	0
$833$	8.44851	0.292723
$834$	0	0
$835$	−51.7283	−1.79013
$836$	0	0
$837$	6.57452	0.227248
$838$	0	0
$839$	29.8641	1.03102	0.515512	−	0.856882i	$-0.327602\pi$
$839$	29.8641	1.03102	0.515512	+	0.856882i	$0.327602\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−14.4387	−0.497294
$844$	0	0
$845$	2.96239	0.101909
$846$	0	0
$847$	−28.1457	−0.967098
$848$	0	0
$849$	−6.55008	−0.224798
$850$	0	0
$851$	53.1002	1.82025
$852$	0	0
$853$	−34.2228	−1.17177	−0.585884	−	0.810395i	$-0.699253\pi$
$853$	−34.2228	−1.17177	−0.585884	+	0.810395i	$0.699253\pi$
$854$	0	0
$855$	9.92478	0.339420
$856$	0	0
$857$	17.6267	0.602117	0.301059	−	0.953606i	$-0.402660\pi$
$857$	17.6267	0.602117	0.301059	+	0.953606i	$0.402660\pi$
$858$	0	0
$859$	−6.80209	−0.232084	−0.116042	−	0.993244i	$-0.537021\pi$
$859$	−6.80209	−0.232084	−0.116042	+	0.993244i	$0.537021\pi$
$860$	0	0
$861$	23.3258	0.794942
$862$	0	0
$863$	32.4142	1.10339	0.551696	−	0.834045i	$-0.313981\pi$
$863$	32.4142	1.10339	0.551696	+	0.834045i	$0.313981\pi$
$864$	0	0
$865$	−64.7269	−2.20078
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	19.3455	0.656252
$870$	0	0
$871$	8.12601	0.275339
$872$	0	0
$873$	−7.92478	−0.268213
$874$	0	0
$875$	−12.1504	−0.410760
$876$	0	0
$877$	26.8773	0.907582	0.453791	−	0.891108i	$-0.350071\pi$
$877$	26.8773	0.907582	0.453791	+	0.891108i	$0.350071\pi$
$878$	0	0
$879$	14.4387	0.487004
$880$	0	0
$881$	−33.4763	−1.12784	−0.563922	−	0.825828i	$-0.690708\pi$
$881$	−33.4763	−1.12784	−0.563922	+	0.825828i	$0.690708\pi$
$882$	0	0
$883$	−26.3996	−0.888418	−0.444209	−	0.895923i	$-0.646515\pi$
$883$	−26.3996	−0.888418	−0.444209	+	0.895923i	$0.646515\pi$
$884$	0	0
$885$	−0.926192	−0.0311336
$886$	0	0
$887$	13.7743	0.462497	0.231248	−	0.972895i	$-0.425719\pi$
$887$	13.7743	0.462497	0.231248	+	0.972895i	$0.425719\pi$
$888$	0	0
$889$	17.7546	0.595471
$890$	0	0
$891$	1.61213	0.0540083
$892$	0	0
$893$	8.00000	0.267710
$894$	0	0
$895$	64.9497	2.17103
$896$	0	0
$897$	6.70052	0.223724
$898$	0	0
$899$	−13.1490	−0.438545
$900$	0	0
$901$	−23.8496	−0.794544
$902$	0	0
$903$	2.59895	0.0864877
$904$	0	0
$905$	55.1754	1.83409
$906$	0	0
$907$	−3.74798	−0.124450	−0.0622249	−	0.998062i	$-0.519820\pi$
$907$	−3.74798	−0.124450	−0.0622249	+	0.998062i	$0.519820\pi$
$908$	0	0
$909$	−15.4010	−0.510820
$910$	0	0
$911$	1.42075	0.0470714	0.0235357	−	0.999723i	$-0.492508\pi$
$911$	1.42075	0.0470714	0.0235357	+	0.999723i	$0.492508\pi$
$912$	0	0
$913$	13.4010	0.443510
$914$	0	0
$915$	−43.3258	−1.43231
$916$	0	0
$917$	−30.6516	−1.01221
$918$	0	0
$919$	−55.5487	−1.83238	−0.916191	−	0.400743i	$-0.868752\pi$
$919$	−55.5487	−1.83238	−0.916191	+	0.400743i	$0.868752\pi$
$920$	0	0
$921$	−15.8740	−0.523066
$922$	0	0
$923$	4.31265	0.141953
$924$	0	0
$925$	−29.9219	−0.983828
$926$	0	0
$927$	−4.62530	−0.151915
$928$	0	0
$929$	49.4636	1.62285	0.811424	−	0.584458i	$-0.198693\pi$
$929$	49.4636	1.62285	0.811424	+	0.584458i	$0.198693\pi$
$930$	0	0
$931$	14.1524	0.463825
$932$	0	0
$933$	8.25202	0.270159
$934$	0	0
$935$	9.55149	0.312367
$936$	0	0
$937$	−5.07381	−0.165754	−0.0828770	−	0.996560i	$-0.526411\pi$
$937$	−5.07381	−0.165754	−0.0828770	+	0.996560i	$0.526411\pi$
$938$	0	0
$939$	−3.40105	−0.110989
$940$	0	0
$941$	−18.8119	−0.613252	−0.306626	−	0.951830i	$-0.599200\pi$
$941$	−18.8119	−0.613252	−0.306626	+	0.951830i	$0.599200\pi$
$942$	0	0
$943$	−46.6516	−1.51919
$944$	0	0
$945$	9.92478	0.322853
$946$	0	0
$947$	−15.2652	−0.496052	−0.248026	−	0.968753i	$-0.579782\pi$
$947$	−15.2652	−0.496052	−0.248026	+	0.968753i	$0.579782\pi$
$948$	0	0
$949$	−0.0752228	−0.00244183
$950$	0	0
$951$	−10.4387	−0.338497
$952$	0	0
$953$	5.47627	0.177394	0.0886969	−	0.996059i	$-0.471730\pi$
$953$	5.47627	0.177394	0.0886969	+	0.996059i	$0.471730\pi$
$954$	0	0
$955$	64.5040	2.08730
$956$	0	0
$957$	−3.22425	−0.104225
$958$	0	0
$959$	1.72164	0.0555945
$960$	0	0
$961$	12.2243	0.394331
$962$	0	0
$963$	−14.5501	−0.468870
$964$	0	0
$965$	57.0278	1.83579
$966$	0	0
$967$	−46.0216	−1.47996	−0.739978	−	0.672632i	$-0.765164\pi$
$967$	−46.0216	−1.47996	−0.739978	+	0.672632i	$0.765164\pi$
$968$	0	0
$969$	6.70052	0.215252
$970$	0	0
$971$	−18.4485	−0.592041	−0.296020	−	0.955182i	$-0.595660\pi$
$971$	−18.4485	−0.592041	−0.296020	+	0.955182i	$0.595660\pi$
$972$	0	0
$973$	46.6516	1.49558
$974$	0	0
$975$	−3.77575	−0.120921
$976$	0	0
$977$	−0.160295	−0.00512828	−0.00256414	−	0.999997i	$-0.500816\pi$
$977$	−0.160295	−0.00512828	−0.00256414	+	0.999997i	$0.500816\pi$
$978$	0	0
$979$	14.3272	0.457901
$980$	0	0
$981$	15.4010	0.491718
$982$	0	0
$983$	28.8167	0.919109	0.459555	−	0.888149i	$-0.348009\pi$
$983$	28.8167	0.919109	0.459555	+	0.888149i	$0.348009\pi$
$984$	0	0
$985$	13.3719	0.426063
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−5.19791	−0.165284
$990$	0	0
$991$	40.2228	1.27772	0.638860	−	0.769323i	$-0.279406\pi$
$991$	40.2228	1.27772	0.638860	+	0.769323i	$0.279406\pi$
$992$	0	0
$993$	−6.57452	−0.208636
$994$	0	0
$995$	−27.1030	−0.859222
$996$	0	0
$997$	12.8021	0.405446	0.202723	−	0.979236i	$-0.435021\pi$
$997$	12.8021	0.405446	0.202723	+	0.979236i	$0.435021\pi$
$998$	0	0
$999$	−7.92478	−0.250729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.a.bl.1.3		3
3.2	odd	2		7488.2.a.cx.1.1		3
4.3	odd	2		2496.2.a.bk.1.3		3
8.3	odd	2		1248.2.a.p.1.1	yes	3
8.5	even	2		1248.2.a.o.1.1	✓	3
12.11	even	2		7488.2.a.cy.1.1		3
24.5	odd	2		3744.2.a.ba.1.3		3
24.11	even	2		3744.2.a.z.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1248.2.a.o.1.1	✓	3	8.5	even	2
1248.2.a.p.1.1	yes	3	8.3	odd	2
2496.2.a.bk.1.3		3	4.3	odd	2
2496.2.a.bl.1.3		3	1.1	even	1	trivial
3744.2.a.z.1.3		3	24.11	even	2
3744.2.a.ba.1.3		3	24.5	odd	2
7488.2.a.cx.1.1		3	3.2	odd	2
7488.2.a.cy.1.1		3	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.96239 q^{5} +3.35026 q^{7} +1.00000 q^{9} +1.61213 q^{11} -1.00000 q^{13} +2.96239 q^{15} +2.00000 q^{17} +3.35026 q^{19} +3.35026 q^{21} -6.70052 q^{23} +3.77575 q^{25} +1.00000 q^{27} -2.00000 q^{29} +6.57452 q^{31} +1.61213 q^{33} +9.92478 q^{35} -7.92478 q^{37} -1.00000 q^{39} +6.96239 q^{41} +0.775746 q^{43} +2.96239 q^{45} +2.38787 q^{47} +4.22425 q^{49} +2.00000 q^{51} -11.9248 q^{53} +4.77575 q^{55} +3.35026 q^{57} -0.312650 q^{59} -14.6253 q^{61} +3.35026 q^{63} -2.96239 q^{65} -8.12601 q^{67} -6.70052 q^{69} -4.31265 q^{71} +0.0752228 q^{73} +3.77575 q^{75} +5.40105 q^{77} +12.0000 q^{79} +1.00000 q^{81} +8.31265 q^{83} +5.92478 q^{85} -2.00000 q^{87} +8.88717 q^{89} -3.35026 q^{91} +6.57452 q^{93} +9.92478 q^{95} -7.92478 q^{97} +1.61213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 2 q^{5} + 3 q^{9} + 4 q^{11} - 3 q^{13} - 2 q^{15} + 6 q^{17} + 13 q^{25} + 3 q^{27} - 6 q^{29} + 8 q^{31} + 4 q^{33} + 8 q^{35} - 2 q^{37} - 3 q^{39} + 10 q^{41} + 4 q^{43} - 2 q^{45} + 8 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 2 * q^5 + 3 * q^9 + 4 * q^11 - 3 * q^13 - 2 * q^15 + 6 * q^17 + 13 * q^25 + 3 * q^27 - 6 * q^29 + 8 * q^31 + 4 * q^33 + 8 * q^35 - 2 * q^37 - 3 * q^39 + 10 * q^41 + 4 * q^43 - 2 * q^45 + 8 * q^47 + 11 * q^49 + 6 * q^51 - 14 * q^53 + 16 * q^55 + 20 * q^59 - 2 * q^61 + 2 * q^65 - 16 * q^67 + 8 * q^71 + 22 * q^73 + 13 * q^75 - 24 * q^77 + 36 * q^79 + 3 * q^81 + 4 * q^83 - 4 * q^85 - 6 * q^87 - 6 * q^89 + 8 * q^93 + 8 * q^95 - 2 * q^97 + 4 * q^99

Introduction

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