LMFDB - Embedded newform 8044.2.a.b.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$0$
Dimension:	$87$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16825 q^{3} -3.99117 q^{5} +4.93889 q^{7} +1.70132 q^{9} +O(q^{10})$	$q-2.16825 q^{3} -3.99117 q^{5} +4.93889 q^{7} +1.70132 q^{9} +0.942122 q^{11} -3.45573 q^{13} +8.65387 q^{15} +0.867434 q^{17} +0.466305 q^{19} -10.7088 q^{21} +6.36046 q^{23} +10.9295 q^{25} +2.81587 q^{27} +0.836382 q^{29} -0.211744 q^{31} -2.04276 q^{33} -19.7120 q^{35} -3.65013 q^{37} +7.49289 q^{39} +9.50589 q^{41} -8.58522 q^{43} -6.79025 q^{45} -10.4805 q^{47} +17.3926 q^{49} -1.88082 q^{51} -2.36811 q^{53} -3.76017 q^{55} -1.01107 q^{57} +12.0375 q^{59} +7.92424 q^{61} +8.40262 q^{63} +13.7924 q^{65} -6.12750 q^{67} -13.7911 q^{69} +3.58023 q^{71} +15.5288 q^{73} -23.6978 q^{75} +4.65304 q^{77} -15.6239 q^{79} -11.2095 q^{81} +5.21222 q^{83} -3.46208 q^{85} -1.81349 q^{87} +17.4235 q^{89} -17.0675 q^{91} +0.459113 q^{93} -1.86110 q^{95} -12.6506 q^{97} +1.60285 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	$ 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) $ 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * 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
$3$	−2.16825	−1.25184	−0.625920	−	0.779887i	$-0.715277\pi$
$3$	−2.16825	−1.25184	−0.625920	+	0.779887i	$0.715277\pi$
$4$	0	0
$5$	−3.99117	−1.78491	−0.892454	−	0.451139i	$-0.851018\pi$
$5$	−3.99117	−1.78491	−0.892454	+	0.451139i	$0.851018\pi$
$6$	0	0
$7$	4.93889	1.86673	0.933363	−	0.358935i	$-0.116860\pi$
$7$	4.93889	1.86673	0.933363	+	0.358935i	$0.116860\pi$
$8$	0	0
$9$	1.70132	0.567106
$10$	0	0
$11$	0.942122	0.284061	0.142030	−	0.989862i	$-0.454637\pi$
$11$	0.942122	0.284061	0.142030	+	0.989862i	$0.454637\pi$
$12$	0	0
$13$	−3.45573	−0.958446	−0.479223	−	0.877693i	$-0.659082\pi$
$13$	−3.45573	−0.958446	−0.479223	+	0.877693i	$0.659082\pi$
$14$	0	0
$15$	8.65387	2.23442
$16$	0	0
$17$	0.867434	0.210384	0.105192	−	0.994452i	$-0.466454\pi$
$17$	0.867434	0.210384	0.105192	+	0.994452i	$0.466454\pi$
$18$	0	0
$19$	0.466305	0.106978	0.0534889	−	0.998568i	$-0.482966\pi$
$19$	0.466305	0.106978	0.0534889	+	0.998568i	$0.482966\pi$
$20$	0	0
$21$	−10.7088	−2.33684
$22$	0	0
$23$	6.36046	1.32625	0.663124	−	0.748509i	$-0.269230\pi$
$23$	6.36046	1.32625	0.663124	+	0.748509i	$0.269230\pi$
$24$	0	0
$25$	10.9295	2.18589
$26$	0	0
$27$	2.81587	0.541915
$28$	0	0
$29$	0.836382	0.155312	0.0776561	−	0.996980i	$-0.475256\pi$
$29$	0.836382	0.155312	0.0776561	+	0.996980i	$0.475256\pi$
$30$	0	0
$31$	−0.211744	−0.0380303	−0.0190151	−	0.999819i	$-0.506053\pi$
$31$	−0.211744	−0.0380303	−0.0190151	+	0.999819i	$0.506053\pi$
$32$	0	0
$33$	−2.04276	−0.355599
$34$	0	0
$35$	−19.7120	−3.33193
$36$	0	0
$37$	−3.65013	−0.600078	−0.300039	−	0.953927i	$-0.597000\pi$
$37$	−3.65013	−0.600078	−0.300039	+	0.953927i	$0.597000\pi$
$38$	0	0
$39$	7.49289	1.19982
$40$	0	0
$41$	9.50589	1.48457	0.742286	−	0.670084i	$-0.233742\pi$
$41$	9.50589	1.48457	0.742286	+	0.670084i	$0.233742\pi$
$42$	0	0
$43$	−8.58522	−1.30923	−0.654617	−	0.755961i	$-0.727170\pi$
$43$	−8.58522	−1.30923	−0.654617	+	0.755961i	$0.727170\pi$
$44$	0	0
$45$	−6.79025	−1.01223
$46$	0	0
$47$	−10.4805	−1.52873	−0.764367	−	0.644782i	$-0.776948\pi$
$47$	−10.4805	−1.52873	−0.764367	+	0.644782i	$0.776948\pi$
$48$	0	0
$49$	17.3926	2.48466
$50$	0	0
$51$	−1.88082	−0.263367
$52$	0	0
$53$	−2.36811	−0.325284	−0.162642	−	0.986685i	$-0.552002\pi$
$53$	−2.36811	−0.325284	−0.162642	+	0.986685i	$0.552002\pi$
$54$	0	0
$55$	−3.76017	−0.507022
$56$	0	0
$57$	−1.01107	−0.133919
$58$	0	0
$59$	12.0375	1.56715	0.783573	−	0.621299i	$-0.213395\pi$
$59$	12.0375	1.56715	0.783573	+	0.621299i	$0.213395\pi$
$60$	0	0
$61$	7.92424	1.01459	0.507297	−	0.861771i	$-0.330645\pi$
$61$	7.92424	1.01459	0.507297	+	0.861771i	$0.330645\pi$
$62$	0	0
$63$	8.40262	1.05863
$64$	0	0
$65$	13.7924	1.71074
$66$	0	0
$67$	−6.12750	−0.748594	−0.374297	−	0.927309i	$-0.622116\pi$
$67$	−6.12750	−0.748594	−0.374297	+	0.927309i	$0.622116\pi$
$68$	0	0
$69$	−13.7911	−1.66025
$70$	0	0
$71$	3.58023	0.424895	0.212447	−	0.977172i	$-0.431857\pi$
$71$	3.58023	0.424895	0.212447	+	0.977172i	$0.431857\pi$
$72$	0	0
$73$	15.5288	1.81750	0.908751	−	0.417338i	$-0.137037\pi$
$73$	15.5288	1.81750	0.908751	+	0.417338i	$0.137037\pi$
$74$	0	0
$75$	−23.6978	−2.73639
$76$	0	0
$77$	4.65304	0.530263
$78$	0	0
$79$	−15.6239	−1.75783	−0.878913	−	0.476983i	$-0.841730\pi$
$79$	−15.6239	−1.75783	−0.878913	+	0.476983i	$0.841730\pi$
$80$	0	0
$81$	−11.2095	−1.24550
$82$	0	0
$83$	5.21222	0.572116	0.286058	−	0.958212i	$-0.407655\pi$
$83$	5.21222	0.572116	0.286058	+	0.958212i	$0.407655\pi$
$84$	0	0
$85$	−3.46208	−0.375515
$86$	0	0
$87$	−1.81349	−0.194426
$88$	0	0
$89$	17.4235	1.84688	0.923442	−	0.383737i	$-0.125363\pi$
$89$	17.4235	1.84688	0.923442	+	0.383737i	$0.125363\pi$
$90$	0	0
$91$	−17.0675	−1.78916
$92$	0	0
$93$	0.459113	0.0476078
$94$	0	0
$95$	−1.86110	−0.190945
$96$	0	0
$97$	−12.6506	−1.28447	−0.642237	−	0.766506i	$-0.721993\pi$
$97$	−12.6506	−1.28447	−0.642237	+	0.766506i	$0.721993\pi$
$98$	0	0
$99$	1.60285	0.161092
$100$	0	0
$101$	−14.4599	−1.43881	−0.719405	−	0.694591i	$-0.755585\pi$
$101$	−14.4599	−1.43881	−0.719405	+	0.694591i	$0.755585\pi$
$102$	0	0
$103$	−9.48475	−0.934560	−0.467280	−	0.884109i	$-0.654766\pi$
$103$	−9.48475	−0.934560	−0.467280	+	0.884109i	$0.654766\pi$
$104$	0	0
$105$	42.7405	4.17105
$106$	0	0
$107$	−5.53499	−0.535087	−0.267544	−	0.963546i	$-0.586212\pi$
$107$	−5.53499	−0.535087	−0.267544	+	0.963546i	$0.586212\pi$
$108$	0	0
$109$	−11.3139	−1.08367	−0.541836	−	0.840484i	$-0.682271\pi$
$109$	−11.3139	−1.08367	−0.541836	+	0.840484i	$0.682271\pi$
$110$	0	0
$111$	7.91441	0.751203
$112$	0	0
$113$	−7.10342	−0.668233	−0.334117	−	0.942532i	$-0.608438\pi$
$113$	−7.10342	−0.668233	−0.334117	+	0.942532i	$0.608438\pi$
$114$	0	0
$115$	−25.3857	−2.36723
$116$	0	0
$117$	−5.87929	−0.543540
$118$	0	0
$119$	4.28416	0.392728
$120$	0	0
$121$	−10.1124	−0.919310
$122$	0	0
$123$	−20.6112	−1.85845
$124$	0	0
$125$	−23.6655	−2.11671
$126$	0	0
$127$	6.54633	0.580892	0.290446	−	0.956891i	$-0.406196\pi$
$127$	6.54633	0.580892	0.290446	+	0.956891i	$0.406196\pi$
$128$	0	0
$129$	18.6149	1.63895
$130$	0	0
$131$	16.0782	1.40476	0.702379	−	0.711803i	$-0.252121\pi$
$131$	16.0782	1.40476	0.702379	+	0.711803i	$0.252121\pi$
$132$	0	0
$133$	2.30303	0.199698
$134$	0	0
$135$	−11.2386	−0.967268
$136$	0	0
$137$	4.27213	0.364992	0.182496	−	0.983207i	$-0.441582\pi$
$137$	4.27213	0.364992	0.182496	+	0.983207i	$0.441582\pi$
$138$	0	0
$139$	2.15800	0.183039	0.0915197	−	0.995803i	$-0.470828\pi$
$139$	2.15800	0.183039	0.0915197	+	0.995803i	$0.470828\pi$
$140$	0	0
$141$	22.7243	1.91373
$142$	0	0
$143$	−3.25572	−0.272257
$144$	0	0
$145$	−3.33815	−0.277218
$146$	0	0
$147$	−37.7116	−3.11040
$148$	0	0
$149$	1.07841	0.0883469	0.0441734	−	0.999024i	$-0.485935\pi$
$149$	1.07841	0.0883469	0.0441734	+	0.999024i	$0.485935\pi$
$150$	0	0
$151$	−22.8824	−1.86214	−0.931072	−	0.364836i	$-0.881125\pi$
$151$	−22.8824	−1.86214	−0.931072	+	0.364836i	$0.881125\pi$
$152$	0	0
$153$	1.47578	0.119310
$154$	0	0
$155$	0.845105	0.0678805
$156$	0	0
$157$	−13.5770	−1.08356	−0.541780	−	0.840520i	$-0.682249\pi$
$157$	−13.5770	−1.08356	−0.541780	+	0.840520i	$0.682249\pi$
$158$	0	0
$159$	5.13465	0.407204
$160$	0	0
$161$	31.4136	2.47574
$162$	0	0
$163$	8.72074	0.683061	0.341531	−	0.939871i	$-0.389055\pi$
$163$	8.72074	0.683061	0.341531	+	0.939871i	$0.389055\pi$
$164$	0	0
$165$	8.15300	0.634710
$166$	0	0
$167$	−17.9526	−1.38922	−0.694609	−	0.719388i	$-0.744423\pi$
$167$	−17.9526	−1.38922	−0.694609	+	0.719388i	$0.744423\pi$
$168$	0	0
$169$	−1.05795	−0.0813809
$170$	0	0
$171$	0.793333	0.0606677
$172$	0	0
$173$	10.4469	0.794263	0.397131	−	0.917762i	$-0.370006\pi$
$173$	10.4469	0.794263	0.397131	+	0.917762i	$0.370006\pi$
$174$	0	0
$175$	53.9794	4.08046
$176$	0	0
$177$	−26.1003	−1.96182
$178$	0	0
$179$	19.1486	1.43124	0.715618	−	0.698492i	$-0.246145\pi$
$179$	19.1486	1.43124	0.715618	+	0.698492i	$0.246145\pi$
$180$	0	0
$181$	−8.86330	−0.658804	−0.329402	−	0.944190i	$-0.606847\pi$
$181$	−8.86330	−0.658804	−0.329402	+	0.944190i	$0.606847\pi$
$182$	0	0
$183$	−17.1817	−1.27011
$184$	0	0
$185$	14.5683	1.07108
$186$	0	0
$187$	0.817229	0.0597617
$188$	0	0
$189$	13.9073	1.01161
$190$	0	0
$191$	20.2819	1.46754	0.733772	−	0.679396i	$-0.237758\pi$
$191$	20.2819	1.46754	0.733772	+	0.679396i	$0.237758\pi$
$192$	0	0
$193$	−2.07637	−0.149460	−0.0747302	−	0.997204i	$-0.523810\pi$
$193$	−2.07637	−0.149460	−0.0747302	+	0.997204i	$0.523810\pi$
$194$	0	0
$195$	−29.9054	−2.14157
$196$	0	0
$197$	16.3156	1.16244	0.581219	−	0.813747i	$-0.302576\pi$
$197$	16.3156	1.16244	0.581219	+	0.813747i	$0.302576\pi$
$198$	0	0
$199$	−12.8579	−0.911473	−0.455736	−	0.890115i	$-0.650624\pi$
$199$	−12.8579	−0.911473	−0.455736	+	0.890115i	$0.650624\pi$
$200$	0	0
$201$	13.2860	0.937120
$202$	0	0
$203$	4.13080	0.289925
$204$	0	0
$205$	−37.9397	−2.64982
$206$	0	0
$207$	10.8212	0.752123
$208$	0	0
$209$	0.439316	0.0303882
$210$	0	0
$211$	−10.0716	−0.693358	−0.346679	−	0.937984i	$-0.612691\pi$
$211$	−10.0716	−0.693358	−0.346679	+	0.937984i	$0.612691\pi$
$212$	0	0
$213$	−7.76284	−0.531901
$214$	0	0
$215$	34.2651	2.33686
$216$	0	0
$217$	−1.04578	−0.0709920
$218$	0	0
$219$	−33.6702	−2.27522
$220$	0	0
$221$	−2.99761	−0.201641
$222$	0	0
$223$	16.9196	1.13302	0.566509	−	0.824056i	$-0.308294\pi$
$223$	16.9196	1.13302	0.566509	+	0.824056i	$0.308294\pi$
$224$	0	0
$225$	18.5945	1.23963
$226$	0	0
$227$	3.24039	0.215072	0.107536	−	0.994201i	$-0.465704\pi$
$227$	3.24039	0.215072	0.107536	+	0.994201i	$0.465704\pi$
$228$	0	0
$229$	29.3055	1.93656	0.968280	−	0.249867i	$-0.0803868\pi$
$229$	29.3055	1.93656	0.968280	+	0.249867i	$0.0803868\pi$
$230$	0	0
$231$	−10.0890	−0.663805
$232$	0	0
$233$	−3.23475	−0.211915	−0.105958	−	0.994371i	$-0.533791\pi$
$233$	−3.23475	−0.211915	−0.105958	+	0.994371i	$0.533791\pi$
$234$	0	0
$235$	41.8294	2.72865
$236$	0	0
$237$	33.8765	2.20052
$238$	0	0
$239$	10.6407	0.688291	0.344146	−	0.938916i	$-0.388169\pi$
$239$	10.6407	0.688291	0.344146	+	0.938916i	$0.388169\pi$
$240$	0	0
$241$	21.7040	1.39807	0.699037	−	0.715085i	$-0.253612\pi$
$241$	21.7040	1.39807	0.699037	+	0.715085i	$0.253612\pi$
$242$	0	0
$243$	15.8573	1.01725
$244$	0	0
$245$	−69.4171	−4.43489
$246$	0	0
$247$	−1.61142	−0.102532
$248$	0	0
$249$	−11.3014	−0.716198
$250$	0	0
$251$	−1.92927	−0.121775	−0.0608873	−	0.998145i	$-0.519393\pi$
$251$	−1.92927	−0.121775	−0.0608873	+	0.998145i	$0.519393\pi$
$252$	0	0
$253$	5.99233	0.376735
$254$	0	0
$255$	7.50666	0.470085
$256$	0	0
$257$	20.2568	1.26359	0.631794	−	0.775137i	$-0.282319\pi$
$257$	20.2568	1.26359	0.631794	+	0.775137i	$0.282319\pi$
$258$	0	0
$259$	−18.0276	−1.12018
$260$	0	0
$261$	1.42295	0.0880784
$262$	0	0
$263$	−16.0907	−0.992197	−0.496098	−	0.868266i	$-0.665234\pi$
$263$	−16.0907	−0.992197	−0.496098	+	0.868266i	$0.665234\pi$
$264$	0	0
$265$	9.45152	0.580602
$266$	0	0
$267$	−37.7785	−2.31201
$268$	0	0
$269$	−17.7528	−1.08241	−0.541205	−	0.840891i	$-0.682032\pi$
$269$	−17.7528	−1.08241	−0.541205	+	0.840891i	$0.682032\pi$
$270$	0	0
$271$	−14.8698	−0.903276	−0.451638	−	0.892201i	$-0.649160\pi$
$271$	−14.8698	−0.903276	−0.451638	+	0.892201i	$0.649160\pi$
$272$	0	0
$273$	37.0065	2.23974
$274$	0	0
$275$	10.2969	0.620926
$276$	0	0
$277$	−23.1736	−1.39237	−0.696183	−	0.717865i	$-0.745120\pi$
$277$	−23.1736	−1.39237	−0.696183	+	0.717865i	$0.745120\pi$
$278$	0	0
$279$	−0.360243	−0.0215672
$280$	0	0
$281$	−13.8035	−0.823449	−0.411725	−	0.911308i	$-0.635073\pi$
$281$	−13.8035	−0.823449	−0.411725	+	0.911308i	$0.635073\pi$
$282$	0	0
$283$	−6.07366	−0.361042	−0.180521	−	0.983571i	$-0.557778\pi$
$283$	−6.07366	−0.361042	−0.180521	+	0.983571i	$0.557778\pi$
$284$	0	0
$285$	4.03534	0.239033
$286$	0	0
$287$	46.9486	2.77129
$288$	0	0
$289$	−16.2476	−0.955739
$290$	0	0
$291$	27.4297	1.60796
$292$	0	0
$293$	21.5351	1.25810	0.629048	−	0.777366i	$-0.283445\pi$
$293$	21.5351	1.25810	0.629048	+	0.777366i	$0.283445\pi$
$294$	0	0
$295$	−48.0437	−2.79721
$296$	0	0
$297$	2.65290	0.153937
$298$	0	0
$299$	−21.9800	−1.27114
$300$	0	0
$301$	−42.4014	−2.44398
$302$	0	0
$303$	31.3526	1.80116
$304$	0	0
$305$	−31.6270	−1.81096
$306$	0	0
$307$	−1.92740	−0.110002	−0.0550012	−	0.998486i	$-0.517516\pi$
$307$	−1.92740	−0.110002	−0.0550012	+	0.998486i	$0.517516\pi$
$308$	0	0
$309$	20.5653	1.16992
$310$	0	0
$311$	−8.34226	−0.473046	−0.236523	−	0.971626i	$-0.576008\pi$
$311$	−8.34226	−0.473046	−0.236523	+	0.971626i	$0.576008\pi$
$312$	0	0
$313$	9.25750	0.523265	0.261632	−	0.965168i	$-0.415739\pi$
$313$	9.25750	0.523265	0.261632	+	0.965168i	$0.415739\pi$
$314$	0	0
$315$	−33.5363	−1.88956
$316$	0	0
$317$	19.4152	1.09047	0.545234	−	0.838284i	$-0.316441\pi$
$317$	19.4152	1.09047	0.545234	+	0.838284i	$0.316441\pi$
$318$	0	0
$319$	0.787974	0.0441181
$320$	0	0
$321$	12.0012	0.669844
$322$	0	0
$323$	0.404489	0.0225064
$324$	0	0
$325$	−37.7692	−2.09506
$326$	0	0
$327$	24.5313	1.35658
$328$	0	0
$329$	−51.7619	−2.85372
$330$	0	0
$331$	35.3988	1.94570	0.972848	−	0.231445i	$-0.0743452\pi$
$331$	35.3988	1.94570	0.972848	+	0.231445i	$0.0743452\pi$
$332$	0	0
$333$	−6.21004	−0.340308
$334$	0	0
$335$	24.4559	1.33617
$336$	0	0
$337$	−6.71393	−0.365731	−0.182866	−	0.983138i	$-0.558537\pi$
$337$	−6.71393	−0.365731	−0.182866	+	0.983138i	$0.558537\pi$
$338$	0	0
$339$	15.4020	0.836522
$340$	0	0
$341$	−0.199488	−0.0108029
$342$	0	0
$343$	51.3281	2.77146
$344$	0	0
$345$	55.0426	2.96340
$346$	0	0
$347$	−1.13877	−0.0611324	−0.0305662	−	0.999533i	$-0.509731\pi$
$347$	−1.13877	−0.0611324	−0.0305662	+	0.999533i	$0.509731\pi$
$348$	0	0
$349$	32.8461	1.75821	0.879107	−	0.476624i	$-0.158140\pi$
$349$	32.8461	1.75821	0.879107	+	0.476624i	$0.158140\pi$
$350$	0	0
$351$	−9.73089	−0.519396
$352$	0	0
$353$	24.2186	1.28903	0.644513	−	0.764594i	$-0.277060\pi$
$353$	24.2186	1.28903	0.644513	+	0.764594i	$0.277060\pi$
$354$	0	0
$355$	−14.2893	−0.758398
$356$	0	0
$357$	−9.28914	−0.491633
$358$	0	0
$359$	12.2451	0.646270	0.323135	−	0.946353i	$-0.395263\pi$
$359$	12.2451	0.646270	0.323135	+	0.946353i	$0.395263\pi$
$360$	0	0
$361$	−18.7826	−0.988556
$362$	0	0
$363$	21.9262	1.15083
$364$	0	0
$365$	−61.9779	−3.24407
$366$	0	0
$367$	19.2314	1.00387	0.501935	−	0.864905i	$-0.332622\pi$
$367$	19.2314	1.00387	0.501935	+	0.864905i	$0.332622\pi$
$368$	0	0
$369$	16.1725	0.841909
$370$	0	0
$371$	−11.6958	−0.607217
$372$	0	0
$373$	15.5667	0.806014	0.403007	−	0.915197i	$-0.367965\pi$
$373$	15.5667	0.806014	0.403007	+	0.915197i	$0.367965\pi$
$374$	0	0
$375$	51.3128	2.64978
$376$	0	0
$377$	−2.89031	−0.148858
$378$	0	0
$379$	21.0732	1.08246	0.541229	−	0.840875i	$-0.317959\pi$
$379$	21.0732	1.08246	0.541229	+	0.840875i	$0.317959\pi$
$380$	0	0
$381$	−14.1941	−0.727185
$382$	0	0
$383$	33.6987	1.72193	0.860963	−	0.508668i	$-0.169862\pi$
$383$	33.6987	1.72193	0.860963	+	0.508668i	$0.169862\pi$
$384$	0	0
$385$	−18.5711	−0.946470
$386$	0	0
$387$	−14.6062	−0.742474
$388$	0	0
$389$	−19.7958	−1.00369	−0.501843	−	0.864959i	$-0.667344\pi$
$389$	−19.7958	−1.00369	−0.501843	+	0.864959i	$0.667344\pi$
$390$	0	0
$391$	5.51728	0.279021
$392$	0	0
$393$	−34.8616	−1.75853
$394$	0	0
$395$	62.3577	3.13755
$396$	0	0
$397$	−4.80115	−0.240963	−0.120481	−	0.992716i	$-0.538444\pi$
$397$	−4.80115	−0.240963	−0.120481	+	0.992716i	$0.538444\pi$
$398$	0	0
$399$	−4.99355	−0.249990
$400$	0	0
$401$	−10.3223	−0.515471	−0.257735	−	0.966216i	$-0.582976\pi$
$401$	−10.3223	−0.515471	−0.257735	+	0.966216i	$0.582976\pi$
$402$	0	0
$403$	0.731728	0.0364500
$404$	0	0
$405$	44.7389	2.22310
$406$	0	0
$407$	−3.43887	−0.170459
$408$	0	0
$409$	33.2230	1.64277	0.821387	−	0.570372i	$-0.193201\pi$
$409$	33.2230	1.64277	0.821387	+	0.570372i	$0.193201\pi$
$410$	0	0
$411$	−9.26305	−0.456912
$412$	0	0
$413$	59.4518	2.92543
$414$	0	0
$415$	−20.8029	−1.02117
$416$	0	0
$417$	−4.67909	−0.229136
$418$	0	0
$419$	−7.11962	−0.347816	−0.173908	−	0.984762i	$-0.555640\pi$
$419$	−7.11962	−0.347816	−0.173908	+	0.984762i	$0.555640\pi$
$420$	0	0
$421$	33.4141	1.62851	0.814253	−	0.580511i	$-0.197147\pi$
$421$	33.4141	1.62851	0.814253	+	0.580511i	$0.197147\pi$
$422$	0	0
$423$	−17.8306	−0.866953
$424$	0	0
$425$	9.48059	0.459876
$426$	0	0
$427$	39.1369	1.89397
$428$	0	0
$429$	7.05921	0.340822
$430$	0	0
$431$	14.0947	0.678918	0.339459	−	0.940621i	$-0.389756\pi$
$431$	14.0947	0.678918	0.339459	+	0.940621i	$0.389756\pi$
$432$	0	0
$433$	8.21265	0.394675	0.197337	−	0.980336i	$-0.436771\pi$
$433$	8.21265	0.394675	0.197337	+	0.980336i	$0.436771\pi$
$434$	0	0
$435$	7.23794	0.347033
$436$	0	0
$437$	2.96592	0.141879
$438$	0	0
$439$	−5.41842	−0.258607	−0.129304	−	0.991605i	$-0.541274\pi$
$439$	−5.41842	−0.258607	−0.129304	+	0.991605i	$0.541274\pi$
$440$	0	0
$441$	29.5904	1.40907
$442$	0	0
$443$	0.304121	0.0144492	0.00722462	−	0.999974i	$-0.497700\pi$
$443$	0.304121	0.0144492	0.00722462	+	0.999974i	$0.497700\pi$
$444$	0	0
$445$	−69.5401	−3.29652
$446$	0	0
$447$	−2.33827	−0.110596
$448$	0	0
$449$	−8.20892	−0.387403	−0.193701	−	0.981061i	$-0.562049\pi$
$449$	−8.20892	−0.387403	−0.193701	+	0.981061i	$0.562049\pi$
$450$	0	0
$451$	8.95571	0.421708
$452$	0	0
$453$	49.6148	2.33111
$454$	0	0
$455$	68.1192	3.19348
$456$	0	0
$457$	−0.748373	−0.0350074	−0.0175037	−	0.999847i	$-0.505572\pi$
$457$	−0.748373	−0.0350074	−0.0175037	+	0.999847i	$0.505572\pi$
$458$	0	0
$459$	2.44258	0.114010
$460$	0	0
$461$	29.0915	1.35493	0.677463	−	0.735557i	$-0.263080\pi$
$461$	29.0915	1.35493	0.677463	+	0.735557i	$0.263080\pi$
$462$	0	0
$463$	−14.0903	−0.654830	−0.327415	−	0.944881i	$-0.606178\pi$
$463$	−14.0903	−0.654830	−0.327415	+	0.944881i	$0.606178\pi$
$464$	0	0
$465$	−1.83240	−0.0849756
$466$	0	0
$467$	17.4936	0.809509	0.404754	−	0.914425i	$-0.367357\pi$
$467$	17.4936	0.809509	0.404754	+	0.914425i	$0.367357\pi$
$468$	0	0
$469$	−30.2631	−1.39742
$470$	0	0
$471$	29.4383	1.35644
$472$	0	0
$473$	−8.08832	−0.371901
$474$	0	0
$475$	5.09647	0.233842
$476$	0	0
$477$	−4.02890	−0.184471
$478$	0	0
$479$	−8.71162	−0.398044	−0.199022	−	0.979995i	$-0.563777\pi$
$479$	−8.71162	−0.398044	−0.199022	+	0.979995i	$0.563777\pi$
$480$	0	0
$481$	12.6139	0.575143
$482$	0	0
$483$	−68.1127	−3.09923
$484$	0	0
$485$	50.4907	2.29267
$486$	0	0
$487$	34.7869	1.57634	0.788172	−	0.615455i	$-0.211028\pi$
$487$	34.7869	1.57634	0.788172	+	0.615455i	$0.211028\pi$
$488$	0	0
$489$	−18.9088	−0.855084
$490$	0	0
$491$	−26.1412	−1.17974	−0.589868	−	0.807500i	$-0.700820\pi$
$491$	−26.1412	−1.17974	−0.589868	+	0.807500i	$0.700820\pi$
$492$	0	0
$493$	0.725506	0.0326751
$494$	0	0
$495$	−6.39725	−0.287535
$496$	0	0
$497$	17.6824	0.793162
$498$	0	0
$499$	34.5972	1.54879	0.774393	−	0.632705i	$-0.218056\pi$
$499$	34.5972	1.54879	0.774393	+	0.632705i	$0.218056\pi$
$500$	0	0
$501$	38.9259	1.73908
$502$	0	0
$503$	−33.2931	−1.48447	−0.742233	−	0.670142i	$-0.766233\pi$
$503$	−33.2931	−1.48447	−0.742233	+	0.670142i	$0.766233\pi$
$504$	0	0
$505$	57.7118	2.56814
$506$	0	0
$507$	2.29391	0.101876
$508$	0	0
$509$	−33.5334	−1.48634	−0.743170	−	0.669103i	$-0.766679\pi$
$509$	−33.5334	−1.48634	−0.743170	+	0.669103i	$0.766679\pi$
$510$	0	0
$511$	76.6948	3.39278
$512$	0	0
$513$	1.31306	0.0579728
$514$	0	0
$515$	37.8553	1.66810
$516$	0	0
$517$	−9.87388	−0.434253
$518$	0	0
$519$	−22.6515	−0.994291
$520$	0	0
$521$	−6.53836	−0.286451	−0.143225	−	0.989690i	$-0.545747\pi$
$521$	−6.53836	−0.286451	−0.143225	+	0.989690i	$0.545747\pi$
$522$	0	0
$523$	31.2169	1.36502	0.682510	−	0.730876i	$-0.260888\pi$
$523$	31.2169	1.36502	0.682510	+	0.730876i	$0.260888\pi$
$524$	0	0
$525$	−117.041	−5.10809
$526$	0	0
$527$	−0.183673	−0.00800094
$528$	0	0
$529$	17.4555	0.758935
$530$	0	0
$531$	20.4796	0.888738
$532$	0	0
$533$	−32.8498	−1.42288
$534$	0	0
$535$	22.0911	0.955081
$536$	0	0
$537$	−41.5190	−1.79168
$538$	0	0
$539$	16.3860	0.705795
$540$	0	0
$541$	−31.2684	−1.34433	−0.672166	−	0.740400i	$-0.734636\pi$
$541$	−31.2684	−1.34433	−0.672166	+	0.740400i	$0.734636\pi$
$542$	0	0
$543$	19.2179	0.824718
$544$	0	0
$545$	45.1556	1.93425
$546$	0	0
$547$	−18.5156	−0.791668	−0.395834	−	0.918322i	$-0.629544\pi$
$547$	−18.5156	−0.791668	−0.395834	+	0.918322i	$0.629544\pi$
$548$	0	0
$549$	13.4816	0.575382
$550$	0	0
$551$	0.390009	0.0166150
$552$	0	0
$553$	−77.1647	−3.28138
$554$	0	0
$555$	−31.5878	−1.34083
$556$	0	0
$557$	16.4231	0.695871	0.347935	−	0.937518i	$-0.386883\pi$
$557$	16.4231	0.695871	0.347935	+	0.937518i	$0.386883\pi$
$558$	0	0
$559$	29.6682	1.25483
$560$	0	0
$561$	−1.77196	−0.0748121
$562$	0	0
$563$	4.85745	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$563$	4.85745	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$564$	0	0
$565$	28.3510	1.19273
$566$	0	0
$567$	−55.3624	−2.32500
$568$	0	0
$569$	−33.0227	−1.38438	−0.692192	−	0.721713i	$-0.743355\pi$
$569$	−33.0227	−1.38438	−0.692192	+	0.721713i	$0.743355\pi$
$570$	0	0
$571$	17.4214	0.729062	0.364531	−	0.931191i	$-0.381229\pi$
$571$	17.4214	0.729062	0.364531	+	0.931191i	$0.381229\pi$
$572$	0	0
$573$	−43.9762	−1.83713
$574$	0	0
$575$	69.5165	2.89904
$576$	0	0
$577$	−22.3592	−0.930825	−0.465412	−	0.885094i	$-0.654094\pi$
$577$	−22.3592	−0.930825	−0.465412	+	0.885094i	$0.654094\pi$
$578$	0	0
$579$	4.50210	0.187101
$580$	0	0
$581$	25.7426	1.06798
$582$	0	0
$583$	−2.23104	−0.0924004
$584$	0	0
$585$	23.4653	0.970169
$586$	0	0
$587$	3.32617	0.137286	0.0686429	−	0.997641i	$-0.478133\pi$
$587$	3.32617	0.137286	0.0686429	+	0.997641i	$0.478133\pi$
$588$	0	0
$589$	−0.0987371	−0.00406839
$590$	0	0
$591$	−35.3763	−1.45519
$592$	0	0
$593$	−20.2556	−0.831797	−0.415899	−	0.909411i	$-0.636533\pi$
$593$	−20.2556	−0.831797	−0.415899	+	0.909411i	$0.636533\pi$
$594$	0	0
$595$	−17.0988	−0.700984
$596$	0	0
$597$	27.8792	1.14102
$598$	0	0
$599$	−6.54619	−0.267470	−0.133735	−	0.991017i	$-0.542697\pi$
$599$	−6.54619	−0.267470	−0.133735	+	0.991017i	$0.542697\pi$
$600$	0	0
$601$	39.1688	1.59773	0.798865	−	0.601511i	$-0.205434\pi$
$601$	39.1688	1.59773	0.798865	+	0.601511i	$0.205434\pi$
$602$	0	0
$603$	−10.4248	−0.424532
$604$	0	0
$605$	40.3604	1.64088
$606$	0	0
$607$	20.7097	0.840582	0.420291	−	0.907389i	$-0.361928\pi$
$607$	20.7097	0.840582	0.420291	+	0.907389i	$0.361928\pi$
$608$	0	0
$609$	−8.95661	−0.362940
$610$	0	0
$611$	36.2176	1.46521
$612$	0	0
$613$	22.9700	0.927748	0.463874	−	0.885901i	$-0.346459\pi$
$613$	22.9700	0.927748	0.463874	+	0.885901i	$0.346459\pi$
$614$	0	0
$615$	82.2628	3.31715
$616$	0	0
$617$	−4.04617	−0.162893	−0.0814463	−	0.996678i	$-0.525954\pi$
$617$	−4.04617	−0.162893	−0.0814463	+	0.996678i	$0.525954\pi$
$618$	0	0
$619$	−25.2525	−1.01498	−0.507491	−	0.861657i	$-0.669427\pi$
$619$	−25.2525	−1.01498	−0.507491	+	0.861657i	$0.669427\pi$
$620$	0	0
$621$	17.9103	0.718714
$622$	0	0
$623$	86.0526	3.44763
$624$	0	0
$625$	39.8059	1.59224
$626$	0	0
$627$	−0.952549	−0.0380411
$628$	0	0
$629$	−3.16625	−0.126247
$630$	0	0
$631$	10.9763	0.436958	0.218479	−	0.975842i	$-0.429890\pi$
$631$	10.9763	0.436958	0.218479	+	0.975842i	$0.429890\pi$
$632$	0	0
$633$	21.8378	0.867974
$634$	0	0
$635$	−26.1275	−1.03684
$636$	0	0
$637$	−60.1042	−2.38142
$638$	0	0
$639$	6.09110	0.240960
$640$	0	0
$641$	47.3222	1.86912	0.934558	−	0.355810i	$-0.115795\pi$
$641$	47.3222	1.86912	0.934558	+	0.355810i	$0.115795\pi$
$642$	0	0
$643$	−21.0877	−0.831617	−0.415808	−	0.909452i	$-0.636501\pi$
$643$	−21.0877	−0.831617	−0.415808	+	0.909452i	$0.636501\pi$
$644$	0	0
$645$	−74.2953	−2.92538
$646$	0	0
$647$	3.65910	0.143854	0.0719271	−	0.997410i	$-0.477085\pi$
$647$	3.65910	0.143854	0.0719271	+	0.997410i	$0.477085\pi$
$648$	0	0
$649$	11.3408	0.445165
$650$	0	0
$651$	2.26751	0.0888707
$652$	0	0
$653$	1.22355	0.0478813	0.0239406	−	0.999713i	$-0.492379\pi$
$653$	1.22355	0.0478813	0.0239406	+	0.999713i	$0.492379\pi$
$654$	0	0
$655$	−64.1708	−2.50736
$656$	0	0
$657$	26.4193	1.03072
$658$	0	0
$659$	43.4028	1.69073	0.845367	−	0.534185i	$-0.179382\pi$
$659$	43.4028	1.69073	0.845367	+	0.534185i	$0.179382\pi$
$660$	0	0
$661$	32.7565	1.27408	0.637040	−	0.770831i	$-0.280159\pi$
$661$	32.7565	1.27408	0.637040	+	0.770831i	$0.280159\pi$
$662$	0	0
$663$	6.49958	0.252423
$664$	0	0
$665$	−9.19179	−0.356442
$666$	0	0
$667$	5.31978	0.205983
$668$	0	0
$669$	−36.6859	−1.41836
$670$	0	0
$671$	7.46560	0.288206
$672$	0	0
$673$	43.8615	1.69074	0.845368	−	0.534184i	$-0.179381\pi$
$673$	43.8615	1.69074	0.845368	+	0.534184i	$0.179381\pi$
$674$	0	0
$675$	30.7760	1.18457
$676$	0	0
$677$	−35.7309	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$677$	−35.7309	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$678$	0	0
$679$	−62.4799	−2.39776
$680$	0	0
$681$	−7.02598	−0.269236
$682$	0	0
$683$	26.0305	0.996029	0.498015	−	0.867169i	$-0.334063\pi$
$683$	26.0305	0.996029	0.498015	+	0.867169i	$0.334063\pi$
$684$	0	0
$685$	−17.0508	−0.651477
$686$	0	0
$687$	−63.5417	−2.42427
$688$	0	0
$689$	8.18353	0.311768
$690$	0	0
$691$	8.05233	0.306325	0.153163	−	0.988201i	$-0.451054\pi$
$691$	8.05233	0.306325	0.153163	+	0.988201i	$0.451054\pi$
$692$	0	0
$693$	7.91629	0.300715
$694$	0	0
$695$	−8.61296	−0.326708
$696$	0	0
$697$	8.24573	0.312329
$698$	0	0
$699$	7.01375	0.265284
$700$	0	0
$701$	−33.9468	−1.28215	−0.641077	−	0.767477i	$-0.721512\pi$
$701$	−33.9468	−1.28215	−0.641077	+	0.767477i	$0.721512\pi$
$702$	0	0
$703$	−1.70208	−0.0641950
$704$	0	0
$705$	−90.6966	−3.41583
$706$	0	0
$707$	−71.4156	−2.68586
$708$	0	0
$709$	4.57230	0.171716	0.0858582	−	0.996307i	$-0.472637\pi$
$709$	4.57230	0.171716	0.0858582	+	0.996307i	$0.472637\pi$
$710$	0	0
$711$	−26.5812	−0.996873
$712$	0	0
$713$	−1.34679	−0.0504376
$714$	0	0
$715$	12.9941	0.485953
$716$	0	0
$717$	−23.0718	−0.861631
$718$	0	0
$719$	−24.2384	−0.903940	−0.451970	−	0.892033i	$-0.649279\pi$
$719$	−24.2384	−0.903940	−0.451970	+	0.892033i	$0.649279\pi$
$720$	0	0
$721$	−46.8441	−1.74457
$722$	0	0
$723$	−47.0596	−1.75017
$724$	0	0
$725$	9.14121	0.339496
$726$	0	0
$727$	21.9879	0.815487	0.407743	−	0.913097i	$-0.366316\pi$
$727$	21.9879	0.815487	0.407743	+	0.913097i	$0.366316\pi$
$728$	0	0
$729$	−0.754301	−0.0279371
$730$	0	0
$731$	−7.44711	−0.275441
$732$	0	0
$733$	−14.5099	−0.535935	−0.267967	−	0.963428i	$-0.586352\pi$
$733$	−14.5099	−0.535935	−0.267967	+	0.963428i	$0.586352\pi$
$734$	0	0
$735$	150.514	5.55178
$736$	0	0
$737$	−5.77286	−0.212646
$738$	0	0
$739$	−28.9109	−1.06350	−0.531752	−	0.846900i	$-0.678466\pi$
$739$	−28.9109	−1.06350	−0.531752	+	0.846900i	$0.678466\pi$
$740$	0	0
$741$	3.49397	0.128354
$742$	0	0
$743$	1.22598	0.0449767	0.0224883	−	0.999747i	$-0.492841\pi$
$743$	1.22598	0.0449767	0.0224883	+	0.999747i	$0.492841\pi$
$744$	0	0
$745$	−4.30412	−0.157691
$746$	0	0
$747$	8.86764	0.324450
$748$	0	0
$749$	−27.3367	−0.998861
$750$	0	0
$751$	10.8826	0.397111	0.198556	−	0.980090i	$-0.436375\pi$
$751$	10.8826	0.397111	0.198556	+	0.980090i	$0.436375\pi$
$752$	0	0
$753$	4.18315	0.152443
$754$	0	0
$755$	91.3276	3.32375
$756$	0	0
$757$	−5.82103	−0.211569	−0.105784	−	0.994389i	$-0.533735\pi$
$757$	−5.82103	−0.211569	−0.105784	+	0.994389i	$0.533735\pi$
$758$	0	0
$759$	−12.9929	−0.471612
$760$	0	0
$761$	12.2437	0.443833	0.221917	−	0.975066i	$-0.428769\pi$
$761$	12.2437	0.443833	0.221917	+	0.975066i	$0.428769\pi$
$762$	0	0
$763$	−55.8779	−2.02292
$764$	0	0
$765$	−5.89009	−0.212957
$766$	0	0
$767$	−41.5983	−1.50203
$768$	0	0
$769$	−16.2747	−0.586881	−0.293441	−	0.955977i	$-0.594800\pi$
$769$	−16.2747	−0.586881	−0.293441	+	0.955977i	$0.594800\pi$
$770$	0	0
$771$	−43.9219	−1.58181
$772$	0	0
$773$	39.3305	1.41462	0.707309	−	0.706904i	$-0.249909\pi$
$773$	39.3305	1.41462	0.707309	+	0.706904i	$0.249909\pi$
$774$	0	0
$775$	−2.31424	−0.0831301
$776$	0	0
$777$	39.0884	1.40229
$778$	0	0
$779$	4.43265	0.158816
$780$	0	0
$781$	3.37301	0.120696
$782$	0	0
$783$	2.35514	0.0841660
$784$	0	0
$785$	54.1880	1.93405
$786$	0	0
$787$	−15.2773	−0.544575	−0.272288	−	0.962216i	$-0.587780\pi$
$787$	−15.2773	−0.544575	−0.272288	+	0.962216i	$0.587780\pi$
$788$	0	0
$789$	34.8887	1.24207
$790$	0	0
$791$	−35.0830	−1.24741
$792$	0	0
$793$	−27.3840	−0.972434
$794$	0	0
$795$	−20.4933	−0.726822
$796$	0	0
$797$	36.4837	1.29232	0.646160	−	0.763202i	$-0.276374\pi$
$797$	36.4837	1.29232	0.646160	+	0.763202i	$0.276374\pi$
$798$	0	0
$799$	−9.09111	−0.321620
$800$	0	0
$801$	29.6429	1.04738
$802$	0	0
$803$	14.6300	0.516281
$804$	0	0
$805$	−125.377	−4.41897
$806$	0	0
$807$	38.4926	1.35500
$808$	0	0
$809$	51.3142	1.80411	0.902056	−	0.431619i	$-0.142057\pi$
$809$	51.3142	1.80411	0.902056	+	0.431619i	$0.142057\pi$
$810$	0	0
$811$	27.4995	0.965637	0.482818	−	0.875721i	$-0.339613\pi$
$811$	27.4995	0.965637	0.482818	+	0.875721i	$0.339613\pi$
$812$	0	0
$813$	32.2415	1.13076
$814$	0	0
$815$	−34.8060	−1.21920
$816$	0	0
$817$	−4.00333	−0.140059
$818$	0	0
$819$	−29.0372	−1.01464
$820$	0	0
$821$	40.7774	1.42314	0.711570	−	0.702615i	$-0.247984\pi$
$821$	40.7774	1.42314	0.711570	+	0.702615i	$0.247984\pi$
$822$	0	0
$823$	−44.1072	−1.53748	−0.768741	−	0.639560i	$-0.779116\pi$
$823$	−44.1072	−1.53748	−0.768741	+	0.639560i	$0.779116\pi$
$824$	0	0
$825$	−22.3263	−0.777300
$826$	0	0
$827$	−25.1450	−0.874377	−0.437189	−	0.899370i	$-0.644026\pi$
$827$	−25.1450	−0.874377	−0.437189	+	0.899370i	$0.644026\pi$
$828$	0	0
$829$	31.8341	1.10564	0.552822	−	0.833299i	$-0.313551\pi$
$829$	31.8341	1.10564	0.552822	+	0.833299i	$0.313551\pi$
$830$	0	0
$831$	50.2462	1.74302
$832$	0	0
$833$	15.0870	0.522732
$834$	0	0
$835$	71.6521	2.47962
$836$	0	0
$837$	−0.596243	−0.0206092
$838$	0	0
$839$	33.6002	1.16001	0.580004	−	0.814614i	$-0.303051\pi$
$839$	33.6002	1.16001	0.580004	+	0.814614i	$0.303051\pi$
$840$	0	0
$841$	−28.3005	−0.975878
$842$	0	0
$843$	29.9295	1.03083
$844$	0	0
$845$	4.22247	0.145257
$846$	0	0
$847$	−49.9441	−1.71610
$848$	0	0
$849$	13.1692	0.451967
$850$	0	0
$851$	−23.2166	−0.795853
$852$	0	0
$853$	−14.3998	−0.493040	−0.246520	−	0.969138i	$-0.579287\pi$
$853$	−14.3998	−0.493040	−0.246520	+	0.969138i	$0.579287\pi$
$854$	0	0
$855$	−3.16633	−0.108286
$856$	0	0
$857$	−14.0732	−0.480730	−0.240365	−	0.970683i	$-0.577267\pi$
$857$	−14.0732	−0.480730	−0.240365	+	0.970683i	$0.577267\pi$
$858$	0	0
$859$	14.6755	0.500721	0.250361	−	0.968153i	$-0.419451\pi$
$859$	14.6755	0.500721	0.250361	+	0.968153i	$0.419451\pi$
$860$	0	0
$861$	−101.796	−3.46921
$862$	0	0
$863$	12.0055	0.408674	0.204337	−	0.978901i	$-0.434496\pi$
$863$	12.0055	0.408674	0.204337	+	0.978901i	$0.434496\pi$
$864$	0	0
$865$	−41.6954	−1.41769
$866$	0	0
$867$	35.2288	1.19643
$868$	0	0
$869$	−14.7196	−0.499329
$870$	0	0
$871$	21.1750	0.717487
$872$	0	0
$873$	−21.5227	−0.728432
$874$	0	0
$875$	−116.881	−3.95131
$876$	0	0
$877$	18.9017	0.638266	0.319133	−	0.947710i	$-0.396608\pi$
$877$	18.9017	0.638266	0.319133	+	0.947710i	$0.396608\pi$
$878$	0	0
$879$	−46.6936	−1.57494
$880$	0	0
$881$	−9.50899	−0.320366	−0.160183	−	0.987087i	$-0.551208\pi$
$881$	−9.50899	−0.320366	−0.160183	+	0.987087i	$0.551208\pi$
$882$	0	0
$883$	15.7695	0.530687	0.265343	−	0.964154i	$-0.414515\pi$
$883$	15.7695	0.530687	0.265343	+	0.964154i	$0.414515\pi$
$884$	0	0
$885$	104.171	3.50166
$886$	0	0
$887$	27.2950	0.916475	0.458238	−	0.888830i	$-0.348481\pi$
$887$	27.2950	0.916475	0.458238	+	0.888830i	$0.348481\pi$
$888$	0	0
$889$	32.3316	1.08437
$890$	0	0
$891$	−10.5607	−0.353796
$892$	0	0
$893$	−4.88710	−0.163540
$894$	0	0
$895$	−76.4255	−2.55462
$896$	0	0
$897$	47.6582	1.59126
$898$	0	0
$899$	−0.177098	−0.00590656
$900$	0	0
$901$	−2.05418	−0.0684345
$902$	0	0
$903$	91.9370	3.05947
$904$	0	0
$905$	35.3750	1.17590
$906$	0	0
$907$	−15.0013	−0.498108	−0.249054	−	0.968490i	$-0.580120\pi$
$907$	−15.0013	−0.498108	−0.249054	+	0.968490i	$0.580120\pi$
$908$	0	0
$909$	−24.6008	−0.815957
$910$	0	0
$911$	−5.09007	−0.168642	−0.0843208	−	0.996439i	$-0.526872\pi$
$911$	−5.09007	−0.168642	−0.0843208	+	0.996439i	$0.526872\pi$
$912$	0	0
$913$	4.91055	0.162515
$914$	0	0
$915$	68.5753	2.26703
$916$	0	0
$917$	79.4084	2.62230
$918$	0	0
$919$	−4.15260	−0.136982	−0.0684909	−	0.997652i	$-0.521818\pi$
$919$	−4.15260	−0.136982	−0.0684909	+	0.997652i	$0.521818\pi$
$920$	0	0
$921$	4.17908	0.137705
$922$	0	0
$923$	−12.3723	−0.407239
$924$	0	0
$925$	−39.8940	−1.31171
$926$	0	0
$927$	−16.1366	−0.529994
$928$	0	0
$929$	11.7684	0.386110	0.193055	−	0.981188i	$-0.438160\pi$
$929$	11.7684	0.386110	0.193055	+	0.981188i	$0.438160\pi$
$930$	0	0
$931$	8.11028	0.265804
$932$	0	0
$933$	18.0881	0.592179
$934$	0	0
$935$	−3.26170	−0.106669
$936$	0	0
$937$	−40.7603	−1.33158	−0.665790	−	0.746139i	$-0.731905\pi$
$937$	−40.7603	−1.33158	−0.665790	+	0.746139i	$0.731905\pi$
$938$	0	0
$939$	−20.0726	−0.655044
$940$	0	0
$941$	38.7919	1.26458	0.632289	−	0.774732i	$-0.282115\pi$
$941$	38.7919	1.26458	0.632289	+	0.774732i	$0.282115\pi$
$942$	0	0
$943$	60.4619	1.96891
$944$	0	0
$945$	−55.5064	−1.80562
$946$	0	0
$947$	5.44595	0.176969	0.0884847	−	0.996078i	$-0.471798\pi$
$947$	5.44595	0.176969	0.0884847	+	0.996078i	$0.471798\pi$
$948$	0	0
$949$	−53.6631	−1.74198
$950$	0	0
$951$	−42.0971	−1.36509
$952$	0	0
$953$	18.4581	0.597916	0.298958	−	0.954266i	$-0.403361\pi$
$953$	18.4581	0.597916	0.298958	+	0.954266i	$0.403361\pi$
$954$	0	0
$955$	−80.9484	−2.61943
$956$	0	0
$957$	−1.70853	−0.0552288
$958$	0	0
$959$	21.0996	0.681340
$960$	0	0
$961$	−30.9552	−0.998554
$962$	0	0
$963$	−9.41677	−0.303451
$964$	0	0
$965$	8.28716	0.266773
$966$	0	0
$967$	−47.5163	−1.52802	−0.764011	−	0.645203i	$-0.776773\pi$
$967$	−47.5163	−1.52802	−0.764011	+	0.645203i	$0.776773\pi$
$968$	0	0
$969$	−0.877034	−0.0281744
$970$	0	0
$971$	−30.3182	−0.972959	−0.486479	−	0.873692i	$-0.661719\pi$
$971$	−30.3182	−0.972959	−0.486479	+	0.873692i	$0.661719\pi$
$972$	0	0
$973$	10.6581	0.341684
$974$	0	0
$975$	81.8932	2.62268
$976$	0	0
$977$	47.5007	1.51968	0.759840	−	0.650110i	$-0.225277\pi$
$977$	47.5007	1.51968	0.759840	+	0.650110i	$0.225277\pi$
$978$	0	0
$979$	16.4150	0.524627
$980$	0	0
$981$	−19.2485	−0.614556
$982$	0	0
$983$	−50.0362	−1.59591	−0.797954	−	0.602718i	$-0.794084\pi$
$983$	−50.0362	−1.59591	−0.797954	+	0.602718i	$0.794084\pi$
$984$	0	0
$985$	−65.1183	−2.07484
$986$	0	0
$987$	112.233	3.57241
$988$	0	0
$989$	−54.6060	−1.73637
$990$	0	0
$991$	11.8872	0.377610	0.188805	−	0.982015i	$-0.439539\pi$
$991$	11.8872	0.377610	0.188805	+	0.982015i	$0.439539\pi$
$992$	0	0
$993$	−76.7536	−2.43570
$994$	0	0
$995$	51.3181	1.62689
$996$	0	0
$997$	−18.8274	−0.596271	−0.298136	−	0.954524i	$-0.596365\pi$
$997$	−18.8274	−0.596271	−0.298136	+	0.954524i	$0.596365\pi$
$998$	0	0
$999$	−10.2783	−0.325191

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.b.1.15	✓	87

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.b.1.15	✓	87	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.16825 q^{3} -3.99117 q^{5} +4.93889 q^{7} +1.70132 q^{9} +O(q^{10})\)	\(q-2.16825 q^{3} -3.99117 q^{5} +4.93889 q^{7} +1.70132 q^{9} +0.942122 q^{11} -3.45573 q^{13} +8.65387 q^{15} +0.867434 q^{17} +0.466305 q^{19} -10.7088 q^{21} +6.36046 q^{23} +10.9295 q^{25} +2.81587 q^{27} +0.836382 q^{29} -0.211744 q^{31} -2.04276 q^{33} -19.7120 q^{35} -3.65013 q^{37} +7.49289 q^{39} +9.50589 q^{41} -8.58522 q^{43} -6.79025 q^{45} -10.4805 q^{47} +17.3926 q^{49} -1.88082 q^{51} -2.36811 q^{53} -3.76017 q^{55} -1.01107 q^{57} +12.0375 q^{59} +7.92424 q^{61} +8.40262 q^{63} +13.7924 q^{65} -6.12750 q^{67} -13.7911 q^{69} +3.58023 q^{71} +15.5288 q^{73} -23.6978 q^{75} +4.65304 q^{77} -15.6239 q^{79} -11.2095 q^{81} +5.21222 q^{83} -3.46208 q^{85} -1.81349 q^{87} +17.4235 q^{89} -17.0675 q^{91} +0.459113 q^{93} -1.86110 q^{95} -12.6506 q^{97} +1.60285 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9}+O(q^{10}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9	\( 87 q + 13 q^{3} - 2 q^{5} + 8 q^{7} + 98 q^{9} + 36 q^{11} - q^{13} + 16 q^{15} + 31 q^{17} + 35 q^{19} - 3 q^{21} + 39 q^{23} + 93 q^{25} + 55 q^{27} - 5 q^{29} + 46 q^{31} + 25 q^{33} + 68 q^{35} - 11 q^{37} + 54 q^{39} + 83 q^{41} + 28 q^{43} - 14 q^{45} + 48 q^{47} + 103 q^{49} + 77 q^{51} + 3 q^{53} + 35 q^{55} + 14 q^{57} + 122 q^{59} - 13 q^{61} + 39 q^{63} + 41 q^{65} + 32 q^{67} - 10 q^{69} + 100 q^{71} + 34 q^{73} + 97 q^{75} + 4 q^{77} + 52 q^{79} + 131 q^{81} + 67 q^{83} - 2 q^{85} + 89 q^{87} + 68 q^{89} + 75 q^{91} + 138 q^{95} + 36 q^{97} + 107 q^{99}+O(q^{100}) \) 87 * q + 13 * q^3 - 2 * q^5 + 8 * q^7 + 98 * q^9 + 36 * q^11 - q^13 + 16 * q^15 + 31 * q^17 + 35 * q^19 - 3 * q^21 + 39 * q^23 + 93 * q^25 + 55 * q^27 - 5 * q^29 + 46 * q^31 + 25 * q^33 + 68 * q^35 - 11 * q^37 + 54 * q^39 + 83 * q^41 + 28 * q^43 - 14 * q^45 + 48 * q^47 + 103 * q^49 + 77 * q^51 + 3 * q^53 + 35 * q^55 + 14 * q^57 + 122 * q^59 - 13 * q^61 + 39 * q^63 + 41 * q^65 + 32 * q^67 - 10 * q^69 + 100 * q^71 + 34 * q^73 + 97 * q^75 + 4 * q^77 + 52 * q^79 + 131 * q^81 + 67 * q^83 - 2 * q^85 + 89 * q^87 + 68 * q^89 + 75 * q^91 + 138 * q^95 + 36 * q^97 + 107 * q^99

Introduction

L-functions

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