LMFDB - Embedded newform 8624.2.a.ct.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8624,2,Mod(1,8624)] 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("8624.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,2,0,-4,0,0,0,14,0,0,0,0,0,0,0,10,0,-6,0,0,0, 0,0,0,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.8629867032$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.9248.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 2 $ x^4 - 5*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4312)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.662153$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35829 q^{3} +2.35829 q^{5} +2.56155 q^{9} -1.00000 q^{11} +6.04090 q^{13} +5.56155 q^{15} -7.36520 q^{17} -1.32431 q^{19} +8.68466 q^{23} +0.561553 q^{25} -1.03399 q^{27} +8.24621 q^{29} +5.75058 q^{31} -2.35829 q^{33} -9.80776 q^{37} +14.2462 q^{39} +2.06798 q^{41} +4.00000 q^{43} +6.04090 q^{45} +7.36520 q^{47} -17.3693 q^{51} -10.0000 q^{53} -2.35829 q^{55} -3.12311 q^{57} -0.290319 q^{59} +10.7575 q^{61} +14.2462 q^{65} -4.68466 q^{67} +20.4810 q^{69} +13.5616 q^{71} +10.0138 q^{73} +1.32431 q^{75} -3.12311 q^{79} -10.1231 q^{81} -3.39228 q^{83} -17.3693 q^{85} +19.4470 q^{87} -1.61463 q^{89} +13.5616 q^{93} -3.12311 q^{95} -6.33122 q^{97} -2.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9} - 4 q^{11} + 14 q^{15} + 10 q^{23} - 6 q^{25} + 2 q^{37} + 24 q^{39} + 16 q^{43} - 20 q^{51} - 40 q^{53} + 4 q^{57} + 24 q^{65} + 6 q^{67} + 46 q^{71} + 4 q^{79} - 24 q^{81} - 20 q^{85} + 46 q^{93}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 - 4 * q^11 + 14 * q^15 + 10 * q^23 - 6 * q^25 + 2 * q^37 + 24 * q^39 + 16 * q^43 - 20 * q^51 - 40 * q^53 + 4 * q^57 + 24 * q^65 + 6 * q^67 + 46 * q^71 + 4 * q^79 - 24 * q^81 - 20 * q^85 + 46 * q^93 + 4 * q^95 - 2 * 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
$3$	2.35829	1.36156	0.680781	−	0.732487i	$-0.261641\pi$
$3$	2.35829	1.36156	0.680781	+	0.732487i	$0.261641\pi$
$4$	0	0
$5$	2.35829	1.05466	0.527331	−	0.849660i	$-0.323193\pi$
$5$	2.35829	1.05466	0.527331	+	0.849660i	$0.323193\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.56155	0.853851
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.04090	1.67544	0.837722	−	0.546098i	$-0.183887\pi$
$13$	6.04090	1.67544	0.837722	+	0.546098i	$0.183887\pi$
$14$	0	0
$15$	5.56155	1.43599
$16$	0	0
$17$	−7.36520	−1.78632	−0.893162	−	0.449735i	$-0.851518\pi$
$17$	−7.36520	−1.78632	−0.893162	+	0.449735i	$0.851518\pi$
$18$	0	0
$19$	−1.32431	−0.303817	−0.151908	−	0.988395i	$-0.548542\pi$
$19$	−1.32431	−0.303817	−0.151908	+	0.988395i	$0.548542\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.68466	1.81088	0.905438	−	0.424478i	$-0.139542\pi$
$23$	8.68466	1.81088	0.905438	+	0.424478i	$0.139542\pi$
$24$	0	0
$25$	0.561553	0.112311
$26$	0	0
$27$	−1.03399	−0.198991
$28$	0	0
$29$	8.24621	1.53128	0.765641	−	0.643268i	$-0.222422\pi$
$29$	8.24621	1.53128	0.765641	+	0.643268i	$0.222422\pi$
$30$	0	0
$31$	5.75058	1.03283	0.516417	−	0.856337i	$-0.327265\pi$
$31$	5.75058	1.03283	0.516417	+	0.856337i	$0.327265\pi$
$32$	0	0
$33$	−2.35829	−0.410526
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.80776	−1.61239	−0.806193	−	0.591652i	$-0.798476\pi$
$37$	−9.80776	−1.61239	−0.806193	+	0.591652i	$0.798476\pi$
$38$	0	0
$39$	14.2462	2.28122
$40$	0	0
$41$	2.06798	0.322963	0.161482	−	0.986876i	$-0.448373\pi$
$41$	2.06798	0.322963	0.161482	+	0.986876i	$0.448373\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	6.04090	0.900524
$46$	0	0
$47$	7.36520	1.07433	0.537163	−	0.843479i	$-0.319496\pi$
$47$	7.36520	1.07433	0.537163	+	0.843479i	$0.319496\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−17.3693	−2.43219
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	−2.35829	−0.317992
$56$	0	0
$57$	−3.12311	−0.413665
$58$	0	0
$59$	−0.290319	−0.0377964	−0.0188982	−	0.999821i	$-0.506016\pi$
$59$	−0.290319	−0.0377964	−0.0188982	+	0.999821i	$0.506016\pi$
$60$	0	0
$61$	10.7575	1.37735	0.688677	−	0.725068i	$-0.258192\pi$
$61$	10.7575	1.37735	0.688677	+	0.725068i	$0.258192\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.2462	1.76703
$66$	0	0
$67$	−4.68466	−0.572322	−0.286161	−	0.958182i	$-0.592379\pi$
$67$	−4.68466	−0.572322	−0.286161	+	0.958182i	$0.592379\pi$
$68$	0	0
$69$	20.4810	2.46562
$70$	0	0
$71$	13.5616	1.60946	0.804730	−	0.593641i	$-0.202310\pi$
$71$	13.5616	1.60946	0.804730	+	0.593641i	$0.202310\pi$
$72$	0	0
$73$	10.0138	1.17203	0.586014	−	0.810301i	$-0.300696\pi$
$73$	10.0138	1.17203	0.586014	+	0.810301i	$0.300696\pi$
$74$	0	0
$75$	1.32431	0.152918
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.12311	−0.351377	−0.175688	−	0.984446i	$-0.556215\pi$
$79$	−3.12311	−0.351377	−0.175688	+	0.984446i	$0.556215\pi$
$80$	0	0
$81$	−10.1231	−1.12479
$82$	0	0
$83$	−3.39228	−0.372351	−0.186176	−	0.982516i	$-0.559609\pi$
$83$	−3.39228	−0.372351	−0.186176	+	0.982516i	$0.559609\pi$
$84$	0	0
$85$	−17.3693	−1.88397
$86$	0	0
$87$	19.4470	2.08494
$88$	0	0
$89$	−1.61463	−0.171150	−0.0855750	−	0.996332i	$-0.527273\pi$
$89$	−1.61463	−0.171150	−0.0855750	+	0.996332i	$0.527273\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	13.5616	1.40627
$94$	0	0
$95$	−3.12311	−0.320424
$96$	0	0
$97$	−6.33122	−0.642838	−0.321419	−	0.946937i	$-0.604160\pi$
$97$	−6.33122	−0.642838	−0.321419	+	0.946937i	$0.604160\pi$
$98$	0	0
$99$	−2.56155	−0.257446
$100$	0	0
$101$	12.8255	1.27618	0.638090	−	0.769961i	$-0.279725\pi$
$101$	12.8255	1.27618	0.638090	+	0.769961i	$0.279725\pi$
$102$	0	0
$103$	−7.36520	−0.725715	−0.362857	−	0.931845i	$-0.618199\pi$
$103$	−7.36520	−0.725715	−0.362857	+	0.931845i	$0.618199\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.3693	1.29246	0.646230	−	0.763142i	$-0.276345\pi$
$107$	13.3693	1.29246	0.646230	+	0.763142i	$0.276345\pi$
$108$	0	0
$109$	−2.87689	−0.275557	−0.137778	−	0.990463i	$-0.543996\pi$
$109$	−2.87689	−0.275557	−0.137778	+	0.990463i	$0.543996\pi$
$110$	0	0
$111$	−23.1296	−2.19536
$112$	0	0
$113$	−14.6847	−1.38142	−0.690708	−	0.723134i	$-0.742701\pi$
$113$	−14.6847	−1.38142	−0.690708	+	0.723134i	$0.742701\pi$
$114$	0	0
$115$	20.4810	1.90986
$116$	0	0
$117$	15.4741	1.43058
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.87689	0.439735
$124$	0	0
$125$	−10.4672	−0.936212
$126$	0	0
$127$	−9.36932	−0.831392	−0.415696	−	0.909504i	$-0.636462\pi$
$127$	−9.36932	−0.831392	−0.415696	+	0.909504i	$0.636462\pi$
$128$	0	0
$129$	9.43318	0.830545
$130$	0	0
$131$	17.5420	1.53266	0.766328	−	0.642450i	$-0.222082\pi$
$131$	17.5420	1.53266	0.766328	+	0.642450i	$0.222082\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.43845	−0.209868
$136$	0	0
$137$	3.56155	0.304284	0.152142	−	0.988359i	$-0.451383\pi$
$137$	3.56155	0.304284	0.152142	+	0.988359i	$0.451383\pi$
$138$	0	0
$139$	−3.97292	−0.336979	−0.168489	−	0.985703i	$-0.553889\pi$
$139$	−3.97292	−0.336979	−0.168489	+	0.985703i	$0.553889\pi$
$140$	0	0
$141$	17.3693	1.46276
$142$	0	0
$143$	−6.04090	−0.505165
$144$	0	0
$145$	19.4470	1.61498
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.2462	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$149$	12.2462	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$150$	0	0
$151$	4.87689	0.396876	0.198438	−	0.980113i	$-0.436413\pi$
$151$	4.87689	0.396876	0.198438	+	0.980113i	$0.436413\pi$
$152$	0	0
$153$	−18.8664	−1.52525
$154$	0	0
$155$	13.5616	1.08929
$156$	0	0
$157$	7.07488	0.564637	0.282319	−	0.959321i	$-0.408896\pi$
$157$	7.07488	0.564637	0.282319	+	0.959321i	$0.408896\pi$
$158$	0	0
$159$	−23.5829	−1.87025
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	−5.56155	−0.432966
$166$	0	0
$167$	0.580639	0.0449312	0.0224656	−	0.999748i	$-0.492848\pi$
$167$	0.580639	0.0449312	0.0224656	+	0.999748i	$0.492848\pi$
$168$	0	0
$169$	23.4924	1.80711
$170$	0	0
$171$	−3.39228	−0.259414
$172$	0	0
$173$	6.04090	0.459281	0.229640	−	0.973276i	$-0.426245\pi$
$173$	6.04090	0.459281	0.229640	+	0.973276i	$0.426245\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.684658	−0.0514621
$178$	0	0
$179$	−3.31534	−0.247800	−0.123900	−	0.992295i	$-0.539540\pi$
$179$	−3.31534	−0.247800	−0.123900	+	0.992295i	$0.539540\pi$
$180$	0	0
$181$	−9.72350	−0.722742	−0.361371	−	0.932422i	$-0.617691\pi$
$181$	−9.72350	−0.722742	−0.361371	+	0.932422i	$0.617691\pi$
$182$	0	0
$183$	25.3693	1.87535
$184$	0	0
$185$	−23.1296	−1.70052
$186$	0	0
$187$	7.36520	0.538597
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0540	−1.30634	−0.653170	−	0.757211i	$-0.726561\pi$
$191$	−18.0540	−1.30634	−0.653170	+	0.757211i	$0.726561\pi$
$192$	0	0
$193$	0.246211	0.0177227	0.00886134	−	0.999961i	$-0.497179\pi$
$193$	0.246211	0.0177227	0.00886134	+	0.999961i	$0.497179\pi$
$194$	0	0
$195$	33.5968	2.40591
$196$	0	0
$197$	−16.2462	−1.15749	−0.578747	−	0.815507i	$-0.696458\pi$
$197$	−16.2462	−1.15749	−0.578747	+	0.815507i	$0.696458\pi$
$198$	0	0
$199$	20.9343	1.48400	0.741998	−	0.670402i	$-0.233879\pi$
$199$	20.9343	1.48400	0.741998	+	0.670402i	$0.233879\pi$
$200$	0	0
$201$	−11.0478	−0.779252
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.87689	0.340617
$206$	0	0
$207$	22.2462	1.54622
$208$	0	0
$209$	1.32431	0.0916042
$210$	0	0
$211$	15.1231	1.04112	0.520559	−	0.853826i	$-0.325724\pi$
$211$	15.1231	1.04112	0.520559	+	0.853826i	$0.325724\pi$
$212$	0	0
$213$	31.9821	2.19138
$214$	0	0
$215$	9.43318	0.643337
$216$	0	0
$217$	0	0
$218$	0	0
$219$	23.6155	1.59579
$220$	0	0
$221$	−44.4924	−2.99288
$222$	0	0
$223$	−3.10196	−0.207723	−0.103861	−	0.994592i	$-0.533120\pi$
$223$	−3.10196	−0.207723	−0.103861	+	0.994592i	$0.533120\pi$
$224$	0	0
$225$	1.43845	0.0958965
$226$	0	0
$227$	−27.5559	−1.82895	−0.914474	−	0.404646i	$-0.867395\pi$
$227$	−27.5559	−1.82895	−0.914474	+	0.404646i	$0.867395\pi$
$228$	0	0
$229$	−7.65552	−0.505891	−0.252946	−	0.967481i	$-0.581399\pi$
$229$	−7.65552	−0.505891	−0.252946	+	0.967481i	$0.581399\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.4924	−1.47353	−0.736764	−	0.676150i	$-0.763647\pi$
$233$	−22.4924	−1.47353	−0.736764	+	0.676150i	$0.763647\pi$
$234$	0	0
$235$	17.3693	1.13305
$236$	0	0
$237$	−7.36520	−0.478421
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−9.43318	−0.607644	−0.303822	−	0.952729i	$-0.598263\pi$
$241$	−9.43318	−0.607644	−0.303822	+	0.952729i	$0.598263\pi$
$242$	0	0
$243$	−20.7713	−1.33248
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−0.870958	−0.0549744	−0.0274872	−	0.999622i	$-0.508751\pi$
$251$	−0.870958	−0.0549744	−0.0274872	+	0.999622i	$0.508751\pi$
$252$	0	0
$253$	−8.68466	−0.546000
$254$	0	0
$255$	−40.9620	−2.56514
$256$	0	0
$257$	−18.8664	−1.17685	−0.588425	−	0.808551i	$-0.700252\pi$
$257$	−18.8664	−1.17685	−0.588425	+	0.808551i	$0.700252\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	21.1231	1.30749
$262$	0	0
$263$	−11.1231	−0.685880	−0.342940	−	0.939357i	$-0.611423\pi$
$263$	−11.1231	−0.685880	−0.342940	+	0.939357i	$0.611423\pi$
$264$	0	0
$265$	−23.5829	−1.44869
$266$	0	0
$267$	−3.80776	−0.233031
$268$	0	0
$269$	−0.743668	−0.0453423	−0.0226711	−	0.999743i	$-0.507217\pi$
$269$	−0.743668	−0.0453423	−0.0226711	+	0.999743i	$0.507217\pi$
$270$	0	0
$271$	9.43318	0.573025	0.286512	−	0.958077i	$-0.407504\pi$
$271$	9.43318	0.573025	0.286512	+	0.958077i	$0.407504\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.561553	−0.0338629
$276$	0	0
$277$	−27.3693	−1.64446	−0.822231	−	0.569153i	$-0.807271\pi$
$277$	−27.3693	−1.64446	−0.822231	+	0.569153i	$0.807271\pi$
$278$	0	0
$279$	14.7304	0.881886
$280$	0	0
$281$	10.8769	0.648861	0.324431	−	0.945910i	$-0.394827\pi$
$281$	10.8769	0.648861	0.324431	+	0.945910i	$0.394827\pi$
$282$	0	0
$283$	−3.39228	−0.201650	−0.100825	−	0.994904i	$-0.532148\pi$
$283$	−3.39228	−0.201650	−0.100825	+	0.994904i	$0.532148\pi$
$284$	0	0
$285$	−7.36520	−0.436277
$286$	0	0
$287$	0	0
$288$	0	0
$289$	37.2462	2.19095
$290$	0	0
$291$	−14.9309	−0.875263
$292$	0	0
$293$	−13.4061	−0.783193	−0.391596	−	0.920137i	$-0.628077\pi$
$293$	−13.4061	−0.783193	−0.391596	+	0.920137i	$0.628077\pi$
$294$	0	0
$295$	−0.684658	−0.0398624
$296$	0	0
$297$	1.03399	0.0599980
$298$	0	0
$299$	52.4631	3.03402
$300$	0	0
$301$	0	0
$302$	0	0
$303$	30.2462	1.73760
$304$	0	0
$305$	25.3693	1.45264
$306$	0	0
$307$	29.0432	1.65758	0.828792	−	0.559557i	$-0.189029\pi$
$307$	29.0432	1.65758	0.828792	+	0.559557i	$0.189029\pi$
$308$	0	0
$309$	−17.3693	−0.988106
$310$	0	0
$311$	0.580639	0.0329250	0.0164625	−	0.999864i	$-0.494760\pi$
$311$	0.580639	0.0329250	0.0164625	+	0.999864i	$0.494760\pi$
$312$	0	0
$313$	−1.03399	−0.0584444	−0.0292222	−	0.999573i	$-0.509303\pi$
$313$	−1.03399	−0.0584444	−0.0292222	+	0.999573i	$0.509303\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.43845	−0.249288	−0.124644	−	0.992202i	$-0.539779\pi$
$317$	−4.43845	−0.249288	−0.124644	+	0.992202i	$0.539779\pi$
$318$	0	0
$319$	−8.24621	−0.461699
$320$	0	0
$321$	31.5288	1.75977
$322$	0	0
$323$	9.75379	0.542715
$324$	0	0
$325$	3.39228	0.188170
$326$	0	0
$327$	−6.78456	−0.375187
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−17.1771	−0.944138	−0.472069	−	0.881562i	$-0.656493\pi$
$331$	−17.1771	−0.944138	−0.472069	+	0.881562i	$0.656493\pi$
$332$	0	0
$333$	−25.1231	−1.37674
$334$	0	0
$335$	−11.0478	−0.603606
$336$	0	0
$337$	−0.246211	−0.0134120	−0.00670599	−	0.999978i	$-0.502135\pi$
$337$	−0.246211	−0.0134120	−0.00670599	+	0.999978i	$0.502135\pi$
$338$	0	0
$339$	−34.6307	−1.88088
$340$	0	0
$341$	−5.75058	−0.311411
$342$	0	0
$343$	0	0
$344$	0	0
$345$	48.3002	2.60039
$346$	0	0
$347$	2.63068	0.141222	0.0706112	−	0.997504i	$-0.477505\pi$
$347$	2.63068	0.141222	0.0706112	+	0.997504i	$0.477505\pi$
$348$	0	0
$349$	−0.163030	−0.00872678	−0.00436339	−	0.999990i	$-0.501389\pi$
$349$	−0.163030	−0.00872678	−0.00436339	+	0.999990i	$0.501389\pi$
$350$	0	0
$351$	−6.24621	−0.333398
$352$	0	0
$353$	−26.6849	−1.42029	−0.710147	−	0.704053i	$-0.751372\pi$
$353$	−26.6849	−1.42029	−0.710147	+	0.704053i	$0.751372\pi$
$354$	0	0
$355$	31.9821	1.69744
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.4924	1.50377	0.751886	−	0.659293i	$-0.229144\pi$
$359$	28.4924	1.50377	0.751886	+	0.659293i	$0.229144\pi$
$360$	0	0
$361$	−17.2462	−0.907695
$362$	0	0
$363$	2.35829	0.123778
$364$	0	0
$365$	23.6155	1.23609
$366$	0	0
$367$	11.6284	0.607000	0.303500	−	0.952831i	$-0.401845\pi$
$367$	11.6284	0.607000	0.303500	+	0.952831i	$0.401845\pi$
$368$	0	0
$369$	5.29723	0.275763
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.8769	0.563184	0.281592	−	0.959534i	$-0.409137\pi$
$373$	10.8769	0.563184	0.281592	+	0.959534i	$0.409137\pi$
$374$	0	0
$375$	−24.6847	−1.27471
$376$	0	0
$377$	49.8145	2.56558
$378$	0	0
$379$	12.6847	0.651567	0.325784	−	0.945444i	$-0.394372\pi$
$379$	12.6847	0.651567	0.325784	+	0.945444i	$0.394372\pi$
$380$	0	0
$381$	−22.0956	−1.13199
$382$	0	0
$383$	−18.4130	−0.940861	−0.470430	−	0.882437i	$-0.655901\pi$
$383$	−18.4130	−0.940861	−0.470430	+	0.882437i	$0.655901\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.2462	0.520844
$388$	0	0
$389$	31.1771	1.58074	0.790370	−	0.612629i	$-0.209888\pi$
$389$	31.1771	1.58074	0.790370	+	0.612629i	$0.209888\pi$
$390$	0	0
$391$	−63.9643	−3.23481
$392$	0	0
$393$	41.3693	2.08681
$394$	0	0
$395$	−7.36520	−0.370584
$396$	0	0
$397$	12.8255	0.643691	0.321846	−	0.946792i	$-0.395697\pi$
$397$	12.8255	0.643691	0.321846	+	0.946792i	$0.395697\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.4924	−1.12322	−0.561609	−	0.827403i	$-0.689817\pi$
$401$	−22.4924	−1.12322	−0.561609	+	0.827403i	$0.689817\pi$
$402$	0	0
$403$	34.7386	1.73045
$404$	0	0
$405$	−23.8733	−1.18627
$406$	0	0
$407$	9.80776	0.486153
$408$	0	0
$409$	−7.36520	−0.364186	−0.182093	−	0.983281i	$-0.558287\pi$
$409$	−7.36520	−0.364186	−0.182093	+	0.983281i	$0.558287\pi$
$410$	0	0
$411$	8.39919	0.414302
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−9.36932	−0.458817
$418$	0	0
$419$	37.5697	1.83540	0.917700	−	0.397275i	$-0.130044\pi$
$419$	37.5697	1.83540	0.917700	+	0.397275i	$0.130044\pi$
$420$	0	0
$421$	24.7386	1.20569	0.602844	−	0.797859i	$-0.294034\pi$
$421$	24.7386	1.20569	0.602844	+	0.797859i	$0.294034\pi$
$422$	0	0
$423$	18.8664	0.917314
$424$	0	0
$425$	−4.13595	−0.200623
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−14.2462	−0.687814
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	−29.3335	−1.40968	−0.704840	−	0.709367i	$-0.748981\pi$
$433$	−29.3335	−1.40968	−0.704840	+	0.709367i	$0.748981\pi$
$434$	0	0
$435$	45.8617	2.19890
$436$	0	0
$437$	−11.5012	−0.550175
$438$	0	0
$439$	16.7984	0.801743	0.400871	−	0.916134i	$-0.368707\pi$
$439$	16.7984	0.801743	0.400871	+	0.916134i	$0.368707\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.80776	−0.370958	−0.185479	−	0.982648i	$-0.559384\pi$
$443$	−7.80776	−0.370958	−0.185479	+	0.982648i	$0.559384\pi$
$444$	0	0
$445$	−3.80776	−0.180505
$446$	0	0
$447$	28.8802	1.36599
$448$	0	0
$449$	−19.1771	−0.905022	−0.452511	−	0.891759i	$-0.649472\pi$
$449$	−19.1771	−0.905022	−0.452511	+	0.891759i	$0.649472\pi$
$450$	0	0
$451$	−2.06798	−0.0973771
$452$	0	0
$453$	11.5012	0.540371
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.8769	−0.883024	−0.441512	−	0.897255i	$-0.645558\pi$
$457$	−18.8769	−0.883024	−0.441512	+	0.897255i	$0.645558\pi$
$458$	0	0
$459$	7.61553	0.355462
$460$	0	0
$461$	−34.9211	−1.62644	−0.813218	−	0.581959i	$-0.802286\pi$
$461$	−34.9211	−1.62644	−0.813218	+	0.581959i	$0.802286\pi$
$462$	0	0
$463$	−4.19224	−0.194830	−0.0974149	−	0.995244i	$-0.531057\pi$
$463$	−4.19224	−0.194830	−0.0974149	+	0.995244i	$0.531057\pi$
$464$	0	0
$465$	31.9821	1.48314
$466$	0	0
$467$	−26.5219	−1.22729	−0.613643	−	0.789584i	$-0.710297\pi$
$467$	−26.5219	−1.22729	−0.613643	+	0.789584i	$0.710297\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.6847	0.768788
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−0.743668	−0.0341218
$476$	0	0
$477$	−25.6155	−1.17285
$478$	0	0
$479$	−22.6762	−1.03610	−0.518052	−	0.855349i	$-0.673343\pi$
$479$	−22.6762	−1.03610	−0.518052	+	0.855349i	$0.673343\pi$
$480$	0	0
$481$	−59.2477	−2.70146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.9309	−0.677976
$486$	0	0
$487$	−35.8078	−1.62260	−0.811302	−	0.584627i	$-0.801241\pi$
$487$	−35.8078	−1.62260	−0.811302	+	0.584627i	$0.801241\pi$
$488$	0	0
$489$	28.2995	1.27975
$490$	0	0
$491$	−33.8617	−1.52816	−0.764079	−	0.645122i	$-0.776806\pi$
$491$	−33.8617	−1.52816	−0.764079	+	0.645122i	$0.776806\pi$
$492$	0	0
$493$	−60.7350	−2.73537
$494$	0	0
$495$	−6.04090	−0.271518
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0.492423	0.0220439	0.0110219	−	0.999939i	$-0.496492\pi$
$499$	0.492423	0.0220439	0.0110219	+	0.999939i	$0.496492\pi$
$500$	0	0
$501$	1.36932	0.0611766
$502$	0	0
$503$	−29.4608	−1.31359	−0.656796	−	0.754068i	$-0.728089\pi$
$503$	−29.4608	−1.31359	−0.656796	+	0.754068i	$0.728089\pi$
$504$	0	0
$505$	30.2462	1.34594
$506$	0	0
$507$	55.4021	2.46049
$508$	0	0
$509$	15.6014	0.691518	0.345759	−	0.938323i	$-0.387621\pi$
$509$	15.6014	0.691518	0.345759	+	0.938323i	$0.387621\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.36932	0.0604568
$514$	0	0
$515$	−17.3693	−0.765384
$516$	0	0
$517$	−7.36520	−0.323921
$518$	0	0
$519$	14.2462	0.625339
$520$	0	0
$521$	6.33122	0.277376	0.138688	−	0.990336i	$-0.455712\pi$
$521$	6.33122	0.277376	0.138688	+	0.990336i	$0.455712\pi$
$522$	0	0
$523$	7.20217	0.314929	0.157465	−	0.987525i	$-0.449668\pi$
$523$	7.20217	0.314929	0.157465	+	0.987525i	$0.449668\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−42.3542	−1.84498
$528$	0	0
$529$	52.4233	2.27927
$530$	0	0
$531$	−0.743668	−0.0322725
$532$	0	0
$533$	12.4924	0.541107
$534$	0	0
$535$	31.5288	1.36311
$536$	0	0
$537$	−7.81855	−0.337395
$538$	0	0
$539$	0	0
$540$	0	0
$541$	31.3693	1.34867	0.674336	−	0.738425i	$-0.264430\pi$
$541$	31.3693	1.34867	0.674336	+	0.738425i	$0.264430\pi$
$542$	0	0
$543$	−22.9309	−0.984058
$544$	0	0
$545$	−6.78456	−0.290619
$546$	0	0
$547$	18.2462	0.780152	0.390076	−	0.920783i	$-0.372449\pi$
$547$	18.2462	0.780152	0.390076	+	0.920783i	$0.372449\pi$
$548$	0	0
$549$	27.5559	1.17606
$550$	0	0
$551$	−10.9205	−0.465230
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−54.5464	−2.31537
$556$	0	0
$557$	−5.50758	−0.233364	−0.116682	−	0.993169i	$-0.537226\pi$
$557$	−5.50758	−0.233364	−0.116682	+	0.993169i	$0.537226\pi$
$558$	0	0
$559$	24.1636	1.02201
$560$	0	0
$561$	17.3693	0.733333
$562$	0	0
$563$	−14.3128	−0.603212	−0.301606	−	0.953433i	$-0.597523\pi$
$563$	−14.3128	−0.603212	−0.301606	+	0.953433i	$0.597523\pi$
$564$	0	0
$565$	−34.6307	−1.45693
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	26.2462	1.09837	0.549185	−	0.835701i	$-0.314938\pi$
$571$	26.2462	1.09837	0.549185	+	0.835701i	$0.314938\pi$
$572$	0	0
$573$	−42.5766	−1.77866
$574$	0	0
$575$	4.87689	0.203381
$576$	0	0
$577$	−36.1181	−1.50362	−0.751808	−	0.659382i	$-0.770818\pi$
$577$	−36.1181	−1.50362	−0.751808	+	0.659382i	$0.770818\pi$
$578$	0	0
$579$	0.580639	0.0241305
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.0000	0.414158
$584$	0	0
$585$	36.4924	1.50878
$586$	0	0
$587$	−34.9211	−1.44135	−0.720673	−	0.693275i	$-0.756167\pi$
$587$	−34.9211	−1.44135	−0.720673	+	0.693275i	$0.756167\pi$
$588$	0	0
$589$	−7.61553	−0.313792
$590$	0	0
$591$	−38.3134	−1.57600
$592$	0	0
$593$	12.9885	0.533373	0.266687	−	0.963783i	$-0.414071\pi$
$593$	12.9885	0.533373	0.266687	+	0.963783i	$0.414071\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	49.3693	2.02055
$598$	0	0
$599$	24.9848	1.02085	0.510427	−	0.859921i	$-0.329488\pi$
$599$	24.9848	1.02085	0.510427	+	0.859921i	$0.329488\pi$
$600$	0	0
$601$	16.2177	0.661535	0.330768	−	0.943712i	$-0.392692\pi$
$601$	16.2177	0.661535	0.330768	+	0.943712i	$0.392692\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	2.35829	0.0958783
$606$	0	0
$607$	−3.22925	−0.131071	−0.0655357	−	0.997850i	$-0.520876\pi$
$607$	−3.22925	−0.131071	−0.0655357	+	0.997850i	$0.520876\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	44.4924	1.79997
$612$	0	0
$613$	5.12311	0.206920	0.103460	−	0.994634i	$-0.467009\pi$
$613$	5.12311	0.206920	0.103460	+	0.994634i	$0.467009\pi$
$614$	0	0
$615$	11.5012	0.463771
$616$	0	0
$617$	4.24621	0.170946	0.0854730	−	0.996340i	$-0.472760\pi$
$617$	4.24621	0.170946	0.0854730	+	0.996340i	$0.472760\pi$
$618$	0	0
$619$	−45.3882	−1.82431	−0.912154	−	0.409849i	$-0.865582\pi$
$619$	−45.3882	−1.82431	−0.912154	+	0.409849i	$0.865582\pi$
$620$	0	0
$621$	−8.97983	−0.360348
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−27.4924	−1.09970
$626$	0	0
$627$	3.12311	0.124725
$628$	0	0
$629$	72.2362	2.88024
$630$	0	0
$631$	−2.43845	−0.0970730	−0.0485365	−	0.998821i	$-0.515456\pi$
$631$	−2.43845	−0.0970730	−0.0485365	+	0.998821i	$0.515456\pi$
$632$	0	0
$633$	35.6647	1.41755
$634$	0	0
$635$	−22.0956	−0.876837
$636$	0	0
$637$	0	0
$638$	0	0
$639$	34.7386	1.37424
$640$	0	0
$641$	11.9460	0.471840	0.235920	−	0.971773i	$-0.424190\pi$
$641$	11.9460	0.471840	0.235920	+	0.971773i	$0.424190\pi$
$642$	0	0
$643$	23.2926	0.918571	0.459286	−	0.888289i	$-0.348105\pi$
$643$	23.2926	0.918571	0.459286	+	0.888289i	$0.348105\pi$
$644$	0	0
$645$	22.2462	0.875944
$646$	0	0
$647$	16.3450	0.642590	0.321295	−	0.946979i	$-0.395882\pi$
$647$	16.3450	0.642590	0.321295	+	0.946979i	$0.395882\pi$
$648$	0	0
$649$	0.290319	0.0113960
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.9309	−0.506024	−0.253012	−	0.967463i	$-0.581421\pi$
$653$	−12.9309	−0.506024	−0.253012	+	0.967463i	$0.581421\pi$
$654$	0	0
$655$	41.3693	1.61643
$656$	0	0
$657$	25.6509	1.00074
$658$	0	0
$659$	29.3693	1.14407	0.572033	−	0.820231i	$-0.306155\pi$
$659$	29.3693	1.14407	0.572033	+	0.820231i	$0.306155\pi$
$660$	0	0
$661$	−43.9009	−1.70755	−0.853773	−	0.520645i	$-0.825692\pi$
$661$	−43.9009	−1.70755	−0.853773	+	0.520645i	$0.825692\pi$
$662$	0	0
$663$	−104.926	−4.07500
$664$	0	0
$665$	0	0
$666$	0	0
$667$	71.6155	2.77296
$668$	0	0
$669$	−7.31534	−0.282827
$670$	0	0
$671$	−10.7575	−0.415288
$672$	0	0
$673$	12.6307	0.486877	0.243439	−	0.969916i	$-0.421725\pi$
$673$	12.6307	0.486877	0.243439	+	0.969916i	$0.421725\pi$
$674$	0	0
$675$	−0.580639	−0.0223488
$676$	0	0
$677$	−25.4879	−0.979579	−0.489790	−	0.871841i	$-0.662926\pi$
$677$	−25.4879	−0.979579	−0.489790	+	0.871841i	$0.662926\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−64.9848	−2.49022
$682$	0	0
$683$	−7.50758	−0.287269	−0.143635	−	0.989631i	$-0.545879\pi$
$683$	−7.50758	−0.287269	−0.143635	+	0.989631i	$0.545879\pi$
$684$	0	0
$685$	8.39919	0.320917
$686$	0	0
$687$	−18.0540	−0.688802
$688$	0	0
$689$	−60.4090	−2.30140
$690$	0	0
$691$	−28.5899	−1.08761	−0.543805	−	0.839212i	$-0.683017\pi$
$691$	−28.5899	−1.08761	−0.543805	+	0.839212i	$0.683017\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.36932	−0.355398
$696$	0	0
$697$	−15.2311	−0.576917
$698$	0	0
$699$	−53.0438	−2.00630
$700$	0	0
$701$	−25.1231	−0.948887	−0.474443	−	0.880286i	$-0.657351\pi$
$701$	−25.1231	−0.948887	−0.474443	+	0.880286i	$0.657351\pi$
$702$	0	0
$703$	12.9885	0.489870
$704$	0	0
$705$	40.9620	1.54272
$706$	0	0
$707$	0	0
$708$	0	0
$709$	23.5616	0.884873	0.442436	−	0.896800i	$-0.354114\pi$
$709$	23.5616	0.884873	0.442436	+	0.896800i	$0.354114\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	49.9418	1.87033
$714$	0	0
$715$	−14.2462	−0.532778
$716$	0	0
$717$	56.5991	2.11373
$718$	0	0
$719$	−17.2517	−0.643381	−0.321690	−	0.946845i	$-0.604251\pi$
$719$	−17.2517	−0.643381	−0.321690	+	0.946845i	$0.604251\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−22.2462	−0.827345
$724$	0	0
$725$	4.63068	0.171979
$726$	0	0
$727$	−41.4153	−1.53601	−0.768004	−	0.640444i	$-0.778750\pi$
$727$	−41.4153	−1.53601	−0.768004	+	0.640444i	$0.778750\pi$
$728$	0	0
$729$	−18.6155	−0.689464
$730$	0	0
$731$	−29.4608	−1.08965
$732$	0	0
$733$	−1.90495	−0.0703608	−0.0351804	−	0.999381i	$-0.511201\pi$
$733$	−1.90495	−0.0703608	−0.0351804	+	0.999381i	$0.511201\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.68466	0.172562
$738$	0	0
$739$	−21.3693	−0.786083	−0.393042	−	0.919521i	$-0.628577\pi$
$739$	−21.3693	−0.786083	−0.393042	+	0.919521i	$0.628577\pi$
$740$	0	0
$741$	−18.8664	−0.693073
$742$	0	0
$743$	−34.7386	−1.27444	−0.637218	−	0.770683i	$-0.719915\pi$
$743$	−34.7386	−1.27444	−0.637218	+	0.770683i	$0.719915\pi$
$744$	0	0
$745$	28.8802	1.05809
$746$	0	0
$747$	−8.68951	−0.317933
$748$	0	0
$749$	0	0
$750$	0	0
$751$	30.5464	1.11465	0.557327	−	0.830293i	$-0.311827\pi$
$751$	30.5464	1.11465	0.557327	+	0.830293i	$0.311827\pi$
$752$	0	0
$753$	−2.05398	−0.0748510
$754$	0	0
$755$	11.5012	0.418570
$756$	0	0
$757$	−2.49242	−0.0905886	−0.0452943	−	0.998974i	$-0.514423\pi$
$757$	−2.49242	−0.0905886	−0.0452943	+	0.998974i	$0.514423\pi$
$758$	0	0
$759$	−20.4810	−0.743413
$760$	0	0
$761$	−20.0276	−0.726001	−0.363001	−	0.931789i	$-0.618248\pi$
$761$	−20.0276	−0.726001	−0.363001	+	0.931789i	$0.618248\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−44.4924	−1.60863
$766$	0	0
$767$	−1.75379	−0.0633256
$768$	0	0
$769$	5.29723	0.191023	0.0955115	−	0.995428i	$-0.469551\pi$
$769$	5.29723	0.191023	0.0955115	+	0.995428i	$0.469551\pi$
$770$	0	0
$771$	−44.4924	−1.60236
$772$	0	0
$773$	−3.39228	−0.122012	−0.0610060	−	0.998137i	$-0.519431\pi$
$773$	−3.39228	−0.122012	−0.0610060	+	0.998137i	$0.519431\pi$
$774$	0	0
$775$	3.22925	0.115998
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.73863	−0.0981217
$780$	0	0
$781$	−13.5616	−0.485271
$782$	0	0
$783$	−8.52648	−0.304712
$784$	0	0
$785$	16.6847	0.595501
$786$	0	0
$787$	−10.7575	−0.383463	−0.191731	−	0.981447i	$-0.561410\pi$
$787$	−10.7575	−0.383463	−0.191731	+	0.981447i	$0.561410\pi$
$788$	0	0
$789$	−26.2316	−0.933868
$790$	0	0
$791$	0	0
$792$	0	0
$793$	64.9848	2.30768
$794$	0	0
$795$	−55.6155	−1.97248
$796$	0	0
$797$	43.9009	1.55505	0.777525	−	0.628852i	$-0.216475\pi$
$797$	43.9009	1.55505	0.777525	+	0.628852i	$0.216475\pi$
$798$	0	0
$799$	−54.2462	−1.91909
$800$	0	0
$801$	−4.13595	−0.146137
$802$	0	0
$803$	−10.0138	−0.353380
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.75379	−0.0617363
$808$	0	0
$809$	−51.4773	−1.80984	−0.904922	−	0.425577i	$-0.860071\pi$
$809$	−51.4773	−1.80984	−0.904922	+	0.425577i	$0.860071\pi$
$810$	0	0
$811$	−9.59621	−0.336968	−0.168484	−	0.985704i	$-0.553887\pi$
$811$	−9.59621	−0.336968	−0.168484	+	0.985704i	$0.553887\pi$
$812$	0	0
$813$	22.2462	0.780209
$814$	0	0
$815$	28.2995	0.991289
$816$	0	0
$817$	−5.29723	−0.185327
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.26137	−0.253423	−0.126712	−	0.991940i	$-0.540442\pi$
$821$	−7.26137	−0.253423	−0.126712	+	0.991940i	$0.540442\pi$
$822$	0	0
$823$	−49.6695	−1.73137	−0.865685	−	0.500589i	$-0.833117\pi$
$823$	−49.6695	−1.73137	−0.865685	+	0.500589i	$0.833117\pi$
$824$	0	0
$825$	−1.32431	−0.0461064
$826$	0	0
$827$	24.4924	0.851685	0.425842	−	0.904797i	$-0.359978\pi$
$827$	24.4924	0.851685	0.425842	+	0.904797i	$0.359978\pi$
$828$	0	0
$829$	−36.5357	−1.26894	−0.634469	−	0.772949i	$-0.718781\pi$
$829$	−36.5357	−1.26894	−0.634469	+	0.772949i	$0.718781\pi$
$830$	0	0
$831$	−64.5449	−2.23904
$832$	0	0
$833$	0	0
$834$	0	0
$835$	1.36932	0.0473872
$836$	0	0
$837$	−5.94602	−0.205525
$838$	0	0
$839$	−11.9545	−0.412715	−0.206358	−	0.978477i	$-0.566161\pi$
$839$	−11.9545	−0.412715	−0.206358	+	0.978477i	$0.566161\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	25.6509	0.883465
$844$	0	0
$845$	55.4021	1.90589
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	−85.1771	−2.91983
$852$	0	0
$853$	−1.32431	−0.0453434	−0.0226717	−	0.999743i	$-0.507217\pi$
$853$	−1.32431	−0.0453434	−0.0226717	+	0.999743i	$0.507217\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	14.7304	0.503181	0.251591	−	0.967834i	$-0.419046\pi$
$857$	14.7304	0.503181	0.251591	+	0.967834i	$0.419046\pi$
$858$	0	0
$859$	−48.0368	−1.63900	−0.819498	−	0.573082i	$-0.805748\pi$
$859$	−48.0368	−1.63900	−0.819498	+	0.573082i	$0.805748\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.7386	1.45484	0.727420	−	0.686192i	$-0.240719\pi$
$863$	42.7386	1.45484	0.727420	+	0.686192i	$0.240719\pi$
$864$	0	0
$865$	14.2462	0.484386
$866$	0	0
$867$	87.8375	2.98312
$868$	0	0
$869$	3.12311	0.105944
$870$	0	0
$871$	−28.2995	−0.958893
$872$	0	0
$873$	−16.2177	−0.548887
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.4924	1.56994	0.784969	−	0.619535i	$-0.212679\pi$
$877$	46.4924	1.56994	0.784969	+	0.619535i	$0.212679\pi$
$878$	0	0
$879$	−31.6155	−1.06637
$880$	0	0
$881$	−15.1838	−0.511554	−0.255777	−	0.966736i	$-0.582331\pi$
$881$	−15.1838	−0.511554	−0.255777	+	0.966736i	$0.582331\pi$
$882$	0	0
$883$	0.492423	0.0165713	0.00828567	−	0.999966i	$-0.497363\pi$
$883$	0.492423	0.0165713	0.00828567	+	0.999966i	$0.497363\pi$
$884$	0	0
$885$	−1.61463	−0.0542751
$886$	0	0
$887$	8.85254	0.297239	0.148620	−	0.988894i	$-0.452517\pi$
$887$	8.85254	0.297239	0.148620	+	0.988894i	$0.452517\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.1231	0.339137
$892$	0	0
$893$	−9.75379	−0.326398
$894$	0	0
$895$	−7.81855	−0.261345
$896$	0	0
$897$	123.723	4.13101
$898$	0	0
$899$	47.4205	1.58156
$900$	0	0
$901$	73.6520	2.45370
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22.9309	−0.762248
$906$	0	0
$907$	−51.2311	−1.70110	−0.850550	−	0.525895i	$-0.823731\pi$
$907$	−51.2311	−1.70110	−0.850550	+	0.525895i	$0.823731\pi$
$908$	0	0
$909$	32.8531	1.08967
$910$	0	0
$911$	32.9848	1.09284	0.546418	−	0.837512i	$-0.315991\pi$
$911$	32.9848	1.09284	0.546418	+	0.837512i	$0.315991\pi$
$912$	0	0
$913$	3.39228	0.112268
$914$	0	0
$915$	59.8283	1.97786
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−24.9848	−0.824174	−0.412087	−	0.911145i	$-0.635200\pi$
$919$	−24.9848	−0.824174	−0.412087	+	0.911145i	$0.635200\pi$
$920$	0	0
$921$	68.4924	2.25690
$922$	0	0
$923$	81.9239	2.69656
$924$	0	0
$925$	−5.50758	−0.181088
$926$	0	0
$927$	−18.8664	−0.619652
$928$	0	0
$929$	35.0841	1.15107	0.575536	−	0.817776i	$-0.304793\pi$
$929$	35.0841	1.15107	0.575536	+	0.817776i	$0.304793\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.36932	0.0448294
$934$	0	0
$935$	17.3693	0.568037
$936$	0	0
$937$	15.0565	0.491873	0.245937	−	0.969286i	$-0.420904\pi$
$937$	15.0565	0.491873	0.245937	+	0.969286i	$0.420904\pi$
$938$	0	0
$939$	−2.43845	−0.0795757
$940$	0	0
$941$	29.6238	0.965710	0.482855	−	0.875700i	$-0.339600\pi$
$941$	29.6238	0.965710	0.482855	+	0.875700i	$0.339600\pi$
$942$	0	0
$943$	17.9597	0.584847
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.9309	0.875136	0.437568	−	0.899185i	$-0.355840\pi$
$947$	26.9309	0.875136	0.437568	+	0.899185i	$0.355840\pi$
$948$	0	0
$949$	60.4924	1.96367
$950$	0	0
$951$	−10.4672	−0.339421
$952$	0	0
$953$	−58.1080	−1.88230	−0.941151	−	0.337988i	$-0.890254\pi$
$953$	−58.1080	−1.88230	−0.941151	+	0.337988i	$0.890254\pi$
$954$	0	0
$955$	−42.5766	−1.37775
$956$	0	0
$957$	−19.4470	−0.628632
$958$	0	0
$959$	0	0
$960$	0	0
$961$	2.06913	0.0667461
$962$	0	0
$963$	34.2462	1.10357
$964$	0	0
$965$	0.580639	0.0186914
$966$	0	0
$967$	4.87689	0.156830	0.0784152	−	0.996921i	$-0.475014\pi$
$967$	4.87689	0.156830	0.0784152	+	0.996921i	$0.475014\pi$
$968$	0	0
$969$	23.0023	0.738941
$970$	0	0
$971$	50.1048	1.60794	0.803970	−	0.594670i	$-0.202717\pi$
$971$	50.1048	1.60794	0.803970	+	0.594670i	$0.202717\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	8.00000	0.256205
$976$	0	0
$977$	22.6847	0.725747	0.362873	−	0.931839i	$-0.381796\pi$
$977$	22.6847	0.725747	0.362873	+	0.931839i	$0.381796\pi$
$978$	0	0
$979$	1.61463	0.0516037
$980$	0	0
$981$	−7.36932	−0.235284
$982$	0	0
$983$	50.8485	1.62181	0.810907	−	0.585174i	$-0.198974\pi$
$983$	50.8485	1.62181	0.810907	+	0.585174i	$0.198974\pi$
$984$	0	0
$985$	−38.3134	−1.22076
$986$	0	0
$987$	0	0
$988$	0	0
$989$	34.7386	1.10462
$990$	0	0
$991$	61.4773	1.95289	0.976445	−	0.215767i	$-0.0692252\pi$
$991$	61.4773	1.95289	0.976445	+	0.215767i	$0.0692252\pi$
$992$	0	0
$993$	−40.5086	−1.28550
$994$	0	0
$995$	49.3693	1.56511
$996$	0	0
$997$	19.2840	0.610729	0.305365	−	0.952235i	$-0.401222\pi$
$997$	19.2840	0.610729	0.305365	+	0.952235i	$0.401222\pi$
$998$	0	0
$999$	10.1411	0.320850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.ct.1.4		4
4.3	odd	2		4312.2.a.ba.1.1	✓	4
7.6	odd	2	inner	8624.2.a.ct.1.1		4
28.27	even	2		4312.2.a.ba.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4312.2.a.ba.1.1	✓	4	4.3	odd	2
4312.2.a.ba.1.4	yes	4	28.27	even	2
8624.2.a.ct.1.1		4	7.6	odd	2	inner
8624.2.a.ct.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35829 q^{3} +2.35829 q^{5} +2.56155 q^{9} -1.00000 q^{11} +6.04090 q^{13} +5.56155 q^{15} -7.36520 q^{17} -1.32431 q^{19} +8.68466 q^{23} +0.561553 q^{25} -1.03399 q^{27} +8.24621 q^{29} +5.75058 q^{31} -2.35829 q^{33} -9.80776 q^{37} +14.2462 q^{39} +2.06798 q^{41} +4.00000 q^{43} +6.04090 q^{45} +7.36520 q^{47} -17.3693 q^{51} -10.0000 q^{53} -2.35829 q^{55} -3.12311 q^{57} -0.290319 q^{59} +10.7575 q^{61} +14.2462 q^{65} -4.68466 q^{67} +20.4810 q^{69} +13.5616 q^{71} +10.0138 q^{73} +1.32431 q^{75} -3.12311 q^{79} -10.1231 q^{81} -3.39228 q^{83} -17.3693 q^{85} +19.4470 q^{87} -1.61463 q^{89} +13.5616 q^{93} -3.12311 q^{95} -6.33122 q^{97} -2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9} - 4 q^{11} + 14 q^{15} + 10 q^{23} - 6 q^{25} + 2 q^{37} + 24 q^{39} + 16 q^{43} - 20 q^{51} - 40 q^{53} + 4 q^{57} + 24 q^{65} + 6 q^{67} + 46 q^{71} + 4 q^{79} - 24 q^{81} - 20 q^{85} + 46 q^{93}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 - 4 * q^11 + 14 * q^15 + 10 * q^23 - 6 * q^25 + 2 * q^37 + 24 * q^39 + 16 * q^43 - 20 * q^51 - 40 * q^53 + 4 * q^57 + 24 * q^65 + 6 * q^67 + 46 * q^71 + 4 * q^79 - 24 * q^81 - 20 * q^85 + 46 * q^93 + 4 * q^95 - 2 * q^99

Introduction

L-functions

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