LMFDB - Embedded newform 7803.2.a.by.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7803,2,Mod(1,7803)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7803, 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("7803.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.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:	$62.3072686972$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 6 x^{14} - 3 x^{13} + 76 x^{12} - 69 x^{11} - 354 x^{10} + 523 x^{9} + 720 x^{8} - 1437 x^{7} + \cdots - 51 $ x^15 - 6x^14 - 3x^13 + 76x^12 - 69x^11 - 354x^10 + 523x^9 + 720x^8 - 1437x^7 - 529x^6 + 1752x^5 - 111x^4 - 902x^3 + 258x^2 + 135x - 51
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.54787$ of defining polynomial
Character	$\chi$	$=$	7803.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54787 q^{2} +4.49165 q^{4} +1.21129 q^{5} -0.847998 q^{7} -6.34841 q^{8} +O(q^{10})$	$q-2.54787 q^{2} +4.49165 q^{4} +1.21129 q^{5} -0.847998 q^{7} -6.34841 q^{8} -3.08621 q^{10} +2.31282 q^{11} +4.97825 q^{13} +2.16059 q^{14} +7.19164 q^{16} +5.83217 q^{19} +5.44069 q^{20} -5.89276 q^{22} -7.34076 q^{23} -3.53278 q^{25} -12.6839 q^{26} -3.80891 q^{28} +5.67331 q^{29} +8.65552 q^{31} -5.62655 q^{32} -1.02717 q^{35} -8.23904 q^{37} -14.8596 q^{38} -7.68976 q^{40} -5.81186 q^{41} -3.96985 q^{43} +10.3884 q^{44} +18.7033 q^{46} -4.15345 q^{47} -6.28090 q^{49} +9.00107 q^{50} +22.3606 q^{52} -12.9203 q^{53} +2.80149 q^{55} +5.38344 q^{56} -14.4549 q^{58} -9.54544 q^{59} +1.81320 q^{61} -22.0531 q^{62} -0.0475468 q^{64} +6.03010 q^{65} -10.6015 q^{67} +2.61710 q^{70} -7.23931 q^{71} -7.90050 q^{73} +20.9920 q^{74} +26.1961 q^{76} -1.96126 q^{77} +0.364759 q^{79} +8.71115 q^{80} +14.8079 q^{82} -6.09259 q^{83} +10.1147 q^{86} -14.6827 q^{88} -9.42028 q^{89} -4.22155 q^{91} -32.9722 q^{92} +10.5825 q^{94} +7.06444 q^{95} +13.3186 q^{97} +16.0029 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 6 q^{2} + 12 q^{4} + 3 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) $ 15 * q - 6 * q^2 + 12 * q^4 + 3 * q^5 + 3 * q^7 - 18 * q^8	$ 15 q - 6 q^{2} + 12 q^{4} + 3 q^{5} + 3 q^{7} - 18 q^{8} - 6 q^{11} - 3 q^{13} - 3 q^{14} + 6 q^{16} - 3 q^{19} - 6 q^{20} - 12 q^{22} + 3 q^{23} + 6 q^{25} + 24 q^{26} + 9 q^{28} + 6 q^{29} - 42 q^{32} - 33 q^{35} - 36 q^{38} + 15 q^{40} - 3 q^{43} + 18 q^{44} + 12 q^{46} - 24 q^{47} + 18 q^{49} - 42 q^{50} - 12 q^{52} - 48 q^{53} - 3 q^{55} + 15 q^{56} - 12 q^{58} - 18 q^{59} - 63 q^{62} + 30 q^{64} - 24 q^{65} + 12 q^{67} + 51 q^{70} + 21 q^{71} - 18 q^{73} + 48 q^{74} + 60 q^{76} - 54 q^{77} - 39 q^{79} - 6 q^{80} + 39 q^{82} - 51 q^{83} - 48 q^{86} - 48 q^{88} - 36 q^{89} + 9 q^{91} + 3 q^{92} + 36 q^{94} - 39 q^{95} + 3 q^{97} - 15 q^{98}+O(q^{100}) $ 15 * q - 6 * q^2 + 12 * q^4 + 3 * q^5 + 3 * q^7 - 18 * q^8 - 6 * q^11 - 3 * q^13 - 3 * q^14 + 6 * q^16 - 3 * q^19 - 6 * q^20 - 12 * q^22 + 3 * q^23 + 6 * q^25 + 24 * q^26 + 9 * q^28 + 6 * q^29 - 42 * q^32 - 33 * q^35 - 36 * q^38 + 15 * q^40 - 3 * q^43 + 18 * q^44 + 12 * q^46 - 24 * q^47 + 18 * q^49 - 42 * q^50 - 12 * q^52 - 48 * q^53 - 3 * q^55 + 15 * q^56 - 12 * q^58 - 18 * q^59 - 63 * q^62 + 30 * q^64 - 24 * q^65 + 12 * q^67 + 51 * q^70 + 21 * q^71 - 18 * q^73 + 48 * q^74 + 60 * q^76 - 54 * q^77 - 39 * q^79 - 6 * q^80 + 39 * q^82 - 51 * q^83 - 48 * q^86 - 48 * q^88 - 36 * q^89 + 9 * q^91 + 3 * q^92 + 36 * q^94 - 39 * q^95 + 3 * q^97 - 15 * q^98

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 $

$2$	−2.54787	−1.80162	−0.900809	−	0.434216i	$-0.857026\pi$
$2$	−2.54787	−1.80162	−0.900809	+	0.434216i	$0.857026\pi$
$3$	0	0
$3$	0	0
$4$	4.49165	2.24583
$5$	1.21129	0.541705	0.270852	−	0.962621i	$-0.412695\pi$
$5$	1.21129	0.541705	0.270852	+	0.962621i	$0.412695\pi$
$6$	0	0
$7$	−0.847998	−0.320513	−0.160257	−	0.987075i	$-0.551232\pi$
$7$	−0.847998	−0.320513	−0.160257	+	0.987075i	$0.551232\pi$
$8$	−6.34841	−2.24450
$9$	0	0
$10$	−3.08621	−0.975945
$11$	2.31282	0.697340	0.348670	−	0.937246i	$-0.386633\pi$
$11$	2.31282	0.697340	0.348670	+	0.937246i	$0.386633\pi$
$12$	0	0
$13$	4.97825	1.38072	0.690359	−	0.723467i	$-0.257452\pi$
$13$	4.97825	1.38072	0.690359	+	0.723467i	$0.257452\pi$
$14$	2.16059	0.577442
$15$	0	0
$16$	7.19164	1.79791
$17$	0	0
$17$	0	0
$18$	0	0
$19$	5.83217	1.33799	0.668996	−	0.743266i	$-0.266724\pi$
$19$	5.83217	1.33799	0.668996	+	0.743266i	$0.266724\pi$
$20$	5.44069	1.21657
$21$	0	0
$22$	−5.89276	−1.25634
$23$	−7.34076	−1.53065	−0.765327	−	0.643641i	$-0.777423\pi$
$23$	−7.34076	−1.53065	−0.765327	+	0.643641i	$0.777423\pi$
$24$	0	0
$25$	−3.53278	−0.706556
$26$	−12.6839	−2.48753
$27$	0	0
$28$	−3.80891	−0.719817
$29$	5.67331	1.05351	0.526753	−	0.850018i	$-0.323409\pi$
$29$	5.67331	1.05351	0.526753	+	0.850018i	$0.323409\pi$
$30$	0	0
$31$	8.65552	1.55458	0.777288	−	0.629145i	$-0.216595\pi$
$31$	8.65552	1.55458	0.777288	+	0.629145i	$0.216595\pi$
$32$	−5.62655	−0.994643
$33$	0	0
$34$	0	0
$35$	−1.02717	−0.173623
$36$	0	0
$37$	−8.23904	−1.35449	−0.677245	−	0.735758i	$-0.736826\pi$
$37$	−8.23904	−1.35449	−0.677245	+	0.735758i	$0.736826\pi$
$38$	−14.8596	−2.41055
$39$	0	0
$40$	−7.68976	−1.21586
$41$	−5.81186	−0.907659	−0.453830	−	0.891088i	$-0.649943\pi$
$41$	−5.81186	−0.907659	−0.453830	+	0.891088i	$0.649943\pi$
$42$	0	0
$43$	−3.96985	−0.605396	−0.302698	−	0.953087i	$-0.597887\pi$
$43$	−3.96985	−0.605396	−0.302698	+	0.953087i	$0.597887\pi$
$44$	10.3884	1.56611
$45$	0	0
$46$	18.7033	2.75765
$47$	−4.15345	−0.605843	−0.302922	−	0.953015i	$-0.597962\pi$
$47$	−4.15345	−0.605843	−0.302922	+	0.953015i	$0.597962\pi$
$48$	0	0
$49$	−6.28090	−0.897271
$50$	9.00107	1.27294
$51$	0	0
$52$	22.3606	3.10085
$53$	−12.9203	−1.77473	−0.887366	−	0.461065i	$-0.847468\pi$
$53$	−12.9203	−1.77473	−0.887366	+	0.461065i	$0.847468\pi$
$54$	0	0
$55$	2.80149	0.377752
$56$	5.38344	0.719392
$57$	0	0
$58$	−14.4549	−1.89802
$59$	−9.54544	−1.24271	−0.621355	−	0.783529i	$-0.713418\pi$
$59$	−9.54544	−1.24271	−0.621355	+	0.783529i	$0.713418\pi$
$60$	0	0
$61$	1.81320	0.232157	0.116079	−	0.993240i	$-0.462968\pi$
$61$	1.81320	0.232157	0.116079	+	0.993240i	$0.462968\pi$
$62$	−22.0531	−2.80075
$63$	0	0
$64$	−0.0475468	−0.00594335
$65$	6.03010	0.747942
$66$	0	0
$67$	−10.6015	−1.29518	−0.647590	−	0.761989i	$-0.724223\pi$
$67$	−10.6015	−1.29518	−0.647590	+	0.761989i	$0.724223\pi$
$68$	0	0
$69$	0	0
$70$	2.61710	0.312803
$71$	−7.23931	−0.859148	−0.429574	−	0.903032i	$-0.641336\pi$
$71$	−7.23931	−0.859148	−0.429574	+	0.903032i	$0.641336\pi$
$72$	0	0
$73$	−7.90050	−0.924684	−0.462342	−	0.886702i	$-0.652991\pi$
$73$	−7.90050	−0.924684	−0.462342	+	0.886702i	$0.652991\pi$
$74$	20.9920	2.44027
$75$	0	0
$76$	26.1961	3.00490
$77$	−1.96126	−0.223507
$78$	0	0
$79$	0.364759	0.0410385	0.0205193	−	0.999789i	$-0.493468\pi$
$79$	0.364759	0.0410385	0.0205193	+	0.999789i	$0.493468\pi$
$80$	8.71115	0.973936
$81$	0	0
$82$	14.8079	1.63526
$83$	−6.09259	−0.668748	−0.334374	−	0.942440i	$-0.608525\pi$
$83$	−6.09259	−0.668748	−0.334374	+	0.942440i	$0.608525\pi$
$84$	0	0
$85$	0	0
$86$	10.1147	1.09069
$87$	0	0
$88$	−14.6827	−1.56518
$89$	−9.42028	−0.998548	−0.499274	−	0.866444i	$-0.666400\pi$
$89$	−9.42028	−0.998548	−0.499274	+	0.866444i	$0.666400\pi$
$90$	0	0
$91$	−4.22155	−0.442538
$92$	−32.9722	−3.43758
$93$	0	0
$94$	10.5825	1.09150
$95$	7.06444	0.724796
$96$	0	0
$97$	13.3186	1.35230	0.676152	−	0.736762i	$-0.263646\pi$
$97$	13.3186	1.35230	0.676152	+	0.736762i	$0.263646\pi$
$98$	16.0029	1.61654
$99$	0	0
$100$	−15.8680	−1.58680
$101$	14.9344	1.48603	0.743016	−	0.669273i	$-0.233394\pi$
$101$	14.9344	1.48603	0.743016	+	0.669273i	$0.233394\pi$
$102$	0	0
$103$	4.63203	0.456408	0.228204	−	0.973613i	$-0.426715\pi$
$103$	4.63203	0.456408	0.228204	+	0.973613i	$0.426715\pi$
$104$	−31.6040	−3.09903
$105$	0	0
$106$	32.9191	3.19739
$107$	−16.3063	−1.57639	−0.788194	−	0.615427i	$-0.788984\pi$
$107$	−16.3063	−1.57639	−0.788194	+	0.615427i	$0.788984\pi$
$108$	0	0
$109$	4.90988	0.470281	0.235140	−	0.971961i	$-0.424445\pi$
$109$	4.90988	0.470281	0.235140	+	0.971961i	$0.424445\pi$
$110$	−7.13783	−0.680565
$111$	0	0
$112$	−6.09849	−0.576253
$113$	−8.00319	−0.752877	−0.376439	−	0.926442i	$-0.622851\pi$
$113$	−8.00319	−0.752877	−0.376439	+	0.926442i	$0.622851\pi$
$114$	0	0
$115$	−8.89178	−0.829163
$116$	25.4825	2.36599
$117$	0	0
$118$	24.3206	2.23889
$119$	0	0
$120$	0	0
$121$	−5.65088	−0.513717
$122$	−4.61981	−0.418258
$123$	0	0
$124$	38.8776	3.49131
$125$	−10.3357	−0.924449
$126$	0	0
$127$	12.8696	1.14199	0.570996	−	0.820953i	$-0.306557\pi$
$127$	12.8696	1.14199	0.570996	+	0.820953i	$0.306557\pi$
$128$	11.3742	1.00535
$129$	0	0
$130$	−15.3639	−1.34750
$131$	−4.18314	−0.365482	−0.182741	−	0.983161i	$-0.558497\pi$
$131$	−4.18314	−0.365482	−0.182741	+	0.983161i	$0.558497\pi$
$132$	0	0
$133$	−4.94567	−0.428844
$134$	27.0113	2.33342
$135$	0	0
$136$	0	0
$137$	−8.63334	−0.737596	−0.368798	−	0.929510i	$-0.620231\pi$
$137$	−8.63334	−0.737596	−0.368798	+	0.929510i	$0.620231\pi$
$138$	0	0
$139$	−4.39737	−0.372980	−0.186490	−	0.982457i	$-0.559711\pi$
$139$	−4.39737	−0.372980	−0.186490	+	0.982457i	$0.559711\pi$
$140$	−4.61369	−0.389928
$141$	0	0
$142$	18.4448	1.54786
$143$	11.5138	0.962831
$144$	0	0
$145$	6.87201	0.570689
$146$	20.1295	1.66593
$147$	0	0
$148$	−37.0069	−3.04195
$149$	−6.05888	−0.496363	−0.248181	−	0.968714i	$-0.579833\pi$
$149$	−6.05888	−0.496363	−0.248181	+	0.968714i	$0.579833\pi$
$150$	0	0
$151$	−5.54763	−0.451460	−0.225730	−	0.974190i	$-0.572477\pi$
$151$	−5.54763	−0.451460	−0.225730	+	0.974190i	$0.572477\pi$
$152$	−37.0250	−3.00313
$153$	0	0
$154$	4.99705	0.402674
$155$	10.4843	0.842121
$156$	0	0
$157$	−14.8291	−1.18349	−0.591745	−	0.806125i	$-0.701561\pi$
$157$	−14.8291	−1.18349	−0.591745	+	0.806125i	$0.701561\pi$
$158$	−0.929358	−0.0739358
$159$	0	0
$160$	−6.81537	−0.538803
$161$	6.22495	0.490595
$162$	0	0
$163$	20.5167	1.60699	0.803495	−	0.595311i	$-0.202971\pi$
$163$	20.5167	1.60699	0.803495	+	0.595311i	$0.202971\pi$
$164$	−26.1048	−2.03845
$165$	0	0
$166$	15.5231	1.20483
$167$	−7.96215	−0.616130	−0.308065	−	0.951365i	$-0.599681\pi$
$167$	−7.96215	−0.616130	−0.308065	+	0.951365i	$0.599681\pi$
$168$	0	0
$169$	11.7830	0.906384
$170$	0	0
$171$	0	0
$172$	−17.8312	−1.35961
$173$	17.8866	1.35989	0.679947	−	0.733261i	$-0.262003\pi$
$173$	17.8866	1.35989	0.679947	+	0.733261i	$0.262003\pi$
$174$	0	0
$175$	2.99579	0.226460
$176$	16.6329	1.25375
$177$	0	0
$178$	24.0017	1.79900
$179$	−19.7329	−1.47491	−0.737453	−	0.675398i	$-0.763972\pi$
$179$	−19.7329	−1.47491	−0.737453	+	0.675398i	$0.763972\pi$
$180$	0	0
$181$	−3.11617	−0.231623	−0.115812	−	0.993271i	$-0.536947\pi$
$181$	−3.11617	−0.231623	−0.115812	+	0.993271i	$0.536947\pi$
$182$	10.7560	0.797285
$183$	0	0
$184$	46.6022	3.43556
$185$	−9.97985	−0.733733
$186$	0	0
$187$	0	0
$188$	−18.6559	−1.36062
$189$	0	0
$190$	−17.9993	−1.30581
$191$	5.25069	0.379926	0.189963	−	0.981791i	$-0.439163\pi$
$191$	5.25069	0.379926	0.189963	+	0.981791i	$0.439163\pi$
$192$	0	0
$193$	−26.7341	−1.92436	−0.962182	−	0.272407i	$-0.912180\pi$
$193$	−26.7341	−1.92436	−0.962182	+	0.272407i	$0.912180\pi$
$194$	−33.9342	−2.43633
$195$	0	0
$196$	−28.2116	−2.01512
$197$	−11.0820	−0.789560	−0.394780	−	0.918776i	$-0.629179\pi$
$197$	−11.0820	−0.789560	−0.394780	+	0.918776i	$0.629179\pi$
$198$	0	0
$199$	−7.70739	−0.546362	−0.273181	−	0.961963i	$-0.588076\pi$
$199$	−7.70739	−0.546362	−0.273181	+	0.961963i	$0.588076\pi$
$200$	22.4275	1.58587
$201$	0	0
$202$	−38.0511	−2.67726
$203$	−4.81095	−0.337663
$204$	0	0
$205$	−7.03983	−0.491683
$206$	−11.8018	−0.822273
$207$	0	0
$208$	35.8018	2.48241
$209$	13.4887	0.933036
$210$	0	0
$211$	8.72756	0.600830	0.300415	−	0.953809i	$-0.402875\pi$
$211$	8.72756	0.600830	0.300415	+	0.953809i	$0.402875\pi$
$212$	−58.0333	−3.98574
$213$	0	0
$214$	41.5463	2.84005
$215$	−4.80863	−0.327946
$216$	0	0
$217$	−7.33986	−0.498262
$218$	−12.5097	−0.847266
$219$	0	0
$220$	12.5833	0.848366
$221$	0	0
$222$	0	0
$223$	−20.9828	−1.40511	−0.702555	−	0.711630i	$-0.747957\pi$
$223$	−20.9828	−1.40511	−0.702555	+	0.711630i	$0.747957\pi$
$224$	4.77130	0.318796
$225$	0	0
$226$	20.3911	1.35640
$227$	5.34936	0.355049	0.177525	−	0.984116i	$-0.443191\pi$
$227$	5.34936	0.355049	0.177525	+	0.984116i	$0.443191\pi$
$228$	0	0
$229$	26.5316	1.75326	0.876630	−	0.481165i	$-0.159786\pi$
$229$	26.5316	1.75326	0.876630	+	0.481165i	$0.159786\pi$
$230$	22.6551	1.49383
$231$	0	0
$232$	−36.0165	−2.36460
$233$	17.8329	1.16827	0.584137	−	0.811655i	$-0.301433\pi$
$233$	17.8329	1.16827	0.584137	+	0.811655i	$0.301433\pi$
$234$	0	0
$235$	−5.03103	−0.328188
$236$	−42.8748	−2.79091
$237$	0	0
$238$	0	0
$239$	−9.46372	−0.612157	−0.306079	−	0.952006i	$-0.599017\pi$
$239$	−9.46372	−0.612157	−0.306079	+	0.952006i	$0.599017\pi$
$240$	0	0
$241$	−19.3394	−1.24576	−0.622880	−	0.782317i	$-0.714038\pi$
$241$	−19.3394	−1.24576	−0.622880	+	0.782317i	$0.714038\pi$
$242$	14.3977	0.925521
$243$	0	0
$244$	8.14429	0.521384
$245$	−7.60798	−0.486056
$246$	0	0
$247$	29.0340	1.84739
$248$	−54.9488	−3.48925
$249$	0	0
$250$	26.3339	1.66550
$251$	−2.22520	−0.140454	−0.0702268	−	0.997531i	$-0.522372\pi$
$251$	−2.22520	−0.140454	−0.0702268	+	0.997531i	$0.522372\pi$
$252$	0	0
$253$	−16.9778	−1.06739
$254$	−32.7901	−2.05743
$255$	0	0
$256$	−28.8850	−1.80531
$257$	−1.58427	−0.0988242	−0.0494121	−	0.998778i	$-0.515735\pi$
$257$	−1.58427	−0.0988242	−0.0494121	+	0.998778i	$0.515735\pi$
$258$	0	0
$259$	6.98669	0.434132
$260$	27.0851	1.67975
$261$	0	0
$262$	10.6581	0.658459
$263$	3.68858	0.227448	0.113724	−	0.993512i	$-0.463722\pi$
$263$	3.68858	0.227448	0.113724	+	0.993512i	$0.463722\pi$
$264$	0	0
$265$	−15.6501	−0.961381
$266$	12.6009	0.772613
$267$	0	0
$268$	−47.6183	−2.90875
$269$	3.87199	0.236079	0.118040	−	0.993009i	$-0.462339\pi$
$269$	3.87199	0.236079	0.118040	+	0.993009i	$0.462339\pi$
$270$	0	0
$271$	−30.0854	−1.82756	−0.913780	−	0.406209i	$-0.866851\pi$
$271$	−30.0854	−1.82756	−0.913780	+	0.406209i	$0.866851\pi$
$272$	0	0
$273$	0	0
$274$	21.9966	1.32887
$275$	−8.17067	−0.492710
$276$	0	0
$277$	−6.14426	−0.369173	−0.184586	−	0.982816i	$-0.559095\pi$
$277$	−6.14426	−0.369173	−0.184586	+	0.982816i	$0.559095\pi$
$278$	11.2039	0.671967
$279$	0	0
$280$	6.52090	0.389698
$281$	0.386348	0.0230476	0.0115238	−	0.999934i	$-0.496332\pi$
$281$	0.386348	0.0230476	0.0115238	+	0.999934i	$0.496332\pi$
$282$	0	0
$283$	−5.25013	−0.312088	−0.156044	−	0.987750i	$-0.549874\pi$
$283$	−5.25013	−0.312088	−0.156044	+	0.987750i	$0.549874\pi$
$284$	−32.5165	−1.92950
$285$	0	0
$286$	−29.3356	−1.73465
$287$	4.92844	0.290917
$288$	0	0
$289$	0	0
$290$	−17.5090	−1.02816
$291$	0	0
$292$	−35.4863	−2.07668
$293$	2.21366	0.129323	0.0646617	−	0.997907i	$-0.479403\pi$
$293$	2.21366	0.129323	0.0646617	+	0.997907i	$0.479403\pi$
$294$	0	0
$295$	−11.5623	−0.673182
$296$	52.3048	3.04016
$297$	0	0
$298$	15.4372	0.894256
$299$	−36.5442	−2.11340
$300$	0	0
$301$	3.36642	0.194037
$302$	14.1347	0.813358
$303$	0	0
$304$	41.9429	2.40559
$305$	2.19631	0.125761
$306$	0	0
$307$	12.3443	0.704526	0.352263	−	0.935901i	$-0.385412\pi$
$307$	12.3443	0.704526	0.352263	+	0.935901i	$0.385412\pi$
$308$	−8.80931	−0.501957
$309$	0	0
$310$	−26.7127	−1.51718
$311$	13.9458	0.790794	0.395397	−	0.918510i	$-0.370607\pi$
$311$	13.9458	0.790794	0.395397	+	0.918510i	$0.370607\pi$
$312$	0	0
$313$	11.0115	0.622408	0.311204	−	0.950343i	$-0.399268\pi$
$313$	11.0115	0.622408	0.311204	+	0.950343i	$0.399268\pi$
$314$	37.7826	2.13220
$315$	0	0
$316$	1.63837	0.0921655
$317$	23.3409	1.31096	0.655478	−	0.755214i	$-0.272467\pi$
$317$	23.3409	1.31096	0.655478	+	0.755214i	$0.272467\pi$
$318$	0	0
$319$	13.1213	0.734652
$320$	−0.0575928	−0.00321954
$321$	0	0
$322$	−15.8604	−0.883864
$323$	0	0
$324$	0	0
$325$	−17.5871	−0.975555
$326$	−52.2739	−2.89518
$327$	0	0
$328$	36.8961	2.03724
$329$	3.52212	0.194181
$330$	0	0
$331$	−3.66104	−0.201229	−0.100615	−	0.994925i	$-0.532081\pi$
$331$	−3.66104	−0.201229	−0.100615	+	0.994925i	$0.532081\pi$
$332$	−27.3658	−1.50189
$333$	0	0
$334$	20.2865	1.11003
$335$	−12.8415	−0.701605
$336$	0	0
$337$	4.81459	0.262268	0.131134	−	0.991365i	$-0.458138\pi$
$337$	4.81459	0.262268	0.131134	+	0.991365i	$0.458138\pi$
$338$	−30.0216	−1.63296
$339$	0	0
$340$	0	0
$341$	20.0186	1.08407
$342$	0	0
$343$	11.2622	0.608100
$344$	25.2022	1.35881
$345$	0	0
$346$	−45.5728	−2.45001
$347$	24.1518	1.29654	0.648269	−	0.761412i	$-0.275493\pi$
$347$	24.1518	1.29654	0.648269	+	0.761412i	$0.275493\pi$
$348$	0	0
$349$	23.4984	1.25784	0.628920	−	0.777470i	$-0.283497\pi$
$349$	23.4984	1.25784	0.628920	+	0.777470i	$0.283497\pi$
$350$	−7.63289	−0.407995
$351$	0	0
$352$	−13.0132	−0.693604
$353$	−7.14130	−0.380093	−0.190046	−	0.981775i	$-0.560864\pi$
$353$	−7.14130	−0.380093	−0.190046	+	0.981775i	$0.560864\pi$
$354$	0	0
$355$	−8.76889	−0.465404
$356$	−42.3126	−2.24256
$357$	0	0
$358$	50.2769	2.65722
$359$	36.8992	1.94746	0.973732	−	0.227695i	$-0.0731191\pi$
$359$	36.8992	1.94746	0.973732	+	0.227695i	$0.0731191\pi$
$360$	0	0
$361$	15.0142	0.790223
$362$	7.93961	0.417297
$363$	0	0
$364$	−18.9617	−0.993864
$365$	−9.56979	−0.500906
$366$	0	0
$367$	1.77678	0.0927473	0.0463737	−	0.998924i	$-0.485234\pi$
$367$	1.77678	0.0927473	0.0463737	+	0.998924i	$0.485234\pi$
$368$	−52.7921	−2.75198
$369$	0	0
$370$	25.4274	1.32191
$371$	10.9563	0.568825
$372$	0	0
$373$	−1.53457	−0.0794572	−0.0397286	−	0.999211i	$-0.512649\pi$
$373$	−1.53457	−0.0794572	−0.0397286	+	0.999211i	$0.512649\pi$
$374$	0	0
$375$	0	0
$376$	26.3678	1.35982
$377$	28.2431	1.45460
$378$	0	0
$379$	−25.3334	−1.30129	−0.650645	−	0.759382i	$-0.725501\pi$
$379$	−25.3334	−1.30129	−0.650645	+	0.759382i	$0.725501\pi$
$380$	31.7310	1.62777
$381$	0	0
$382$	−13.3781	−0.684482
$383$	31.6291	1.61617	0.808087	−	0.589064i	$-0.200503\pi$
$383$	31.6291	1.61617	0.808087	+	0.589064i	$0.200503\pi$
$384$	0	0
$385$	−2.37565	−0.121075
$386$	68.1151	3.46697
$387$	0	0
$388$	59.8227	3.03704
$389$	22.9734	1.16480	0.582398	−	0.812904i	$-0.302115\pi$
$389$	22.9734	1.16480	0.582398	+	0.812904i	$0.302115\pi$
$390$	0	0
$391$	0	0
$392$	39.8737	2.01393
$393$	0	0
$394$	28.2355	1.42249
$395$	0.441828	0.0222308
$396$	0	0
$397$	31.6336	1.58764	0.793822	−	0.608151i	$-0.208088\pi$
$397$	31.6336	1.58764	0.793822	+	0.608151i	$0.208088\pi$
$398$	19.6374	0.984336
$399$	0	0
$400$	−25.4065	−1.27032
$401$	−2.66922	−0.133294	−0.0666472	−	0.997777i	$-0.521230\pi$
$401$	−2.66922	−0.133294	−0.0666472	+	0.997777i	$0.521230\pi$
$402$	0	0
$403$	43.0893	2.14643
$404$	67.0803	3.33737
$405$	0	0
$406$	12.2577	0.608339
$407$	−19.0554	−0.944540
$408$	0	0
$409$	0.690964	0.0341660	0.0170830	−	0.999854i	$-0.494562\pi$
$409$	0.690964	0.0341660	0.0170830	+	0.999854i	$0.494562\pi$
$410$	17.9366	0.885825
$411$	0	0
$412$	20.8055	1.02501
$413$	8.09451	0.398305
$414$	0	0
$415$	−7.37988	−0.362264
$416$	−28.0104	−1.37332
$417$	0	0
$418$	−34.3676	−1.68097
$419$	1.99950	0.0976818	0.0488409	−	0.998807i	$-0.484447\pi$
$419$	1.99950	0.0976818	0.0488409	+	0.998807i	$0.484447\pi$
$420$	0	0
$421$	−18.6055	−0.906775	−0.453387	−	0.891314i	$-0.649785\pi$
$421$	−18.6055	−0.906775	−0.453387	+	0.891314i	$0.649785\pi$
$422$	−22.2367	−1.08247
$423$	0	0
$424$	82.0231	3.98339
$425$	0	0
$426$	0	0
$427$	−1.53759	−0.0744094
$428$	−73.2421	−3.54029
$429$	0	0
$430$	12.2518	0.590833
$431$	29.9586	1.44305	0.721527	−	0.692386i	$-0.243441\pi$
$431$	29.9586	1.44305	0.721527	+	0.692386i	$0.243441\pi$
$432$	0	0
$433$	−41.1474	−1.97742	−0.988708	−	0.149852i	$-0.952120\pi$
$433$	−41.1474	−1.97742	−0.988708	+	0.149852i	$0.952120\pi$
$434$	18.7010	0.897678
$435$	0	0
$436$	22.0535	1.05617
$437$	−42.8126	−2.04800
$438$	0	0
$439$	−21.5654	−1.02926	−0.514629	−	0.857413i	$-0.672070\pi$
$439$	−21.5654	−1.02926	−0.514629	+	0.857413i	$0.672070\pi$
$440$	−17.7850	−0.847866
$441$	0	0
$442$	0	0
$443$	39.3828	1.87113	0.935567	−	0.353149i	$-0.114889\pi$
$443$	39.3828	1.87113	0.935567	+	0.353149i	$0.114889\pi$
$444$	0	0
$445$	−11.4107	−0.540918
$446$	53.4614	2.53147
$447$	0	0
$448$	0.0403196	0.00190492
$449$	24.3794	1.15054	0.575269	−	0.817965i	$-0.304898\pi$
$449$	24.3794	1.15054	0.575269	+	0.817965i	$0.304898\pi$
$450$	0	0
$451$	−13.4418	−0.632947
$452$	−35.9476	−1.69083
$453$	0	0
$454$	−13.6295	−0.639663
$455$	−5.11351	−0.239725
$456$	0	0
$457$	−6.52188	−0.305081	−0.152540	−	0.988297i	$-0.548745\pi$
$457$	−6.52188	−0.305081	−0.152540	+	0.988297i	$0.548745\pi$
$458$	−67.5992	−3.15871
$459$	0	0
$460$	−39.9388	−1.86216
$461$	−17.1999	−0.801080	−0.400540	−	0.916279i	$-0.631177\pi$
$461$	−17.1999	−0.801080	−0.400540	+	0.916279i	$0.631177\pi$
$462$	0	0
$463$	−18.2588	−0.848559	−0.424279	−	0.905531i	$-0.639473\pi$
$463$	−18.2588	−0.848559	−0.424279	+	0.905531i	$0.639473\pi$
$464$	40.8004	1.89411
$465$	0	0
$466$	−45.4360	−2.10478
$467$	7.12524	0.329717	0.164858	−	0.986317i	$-0.447283\pi$
$467$	7.12524	0.329717	0.164858	+	0.986317i	$0.447283\pi$
$468$	0	0
$469$	8.99005	0.415122
$470$	12.8184	0.591269
$471$	0	0
$472$	60.5984	2.78927
$473$	−9.18152	−0.422167
$474$	0	0
$475$	−20.6038	−0.945367
$476$	0	0
$477$	0	0
$478$	24.1123	1.10287
$479$	−20.4180	−0.932922	−0.466461	−	0.884542i	$-0.654471\pi$
$479$	−20.4180	−0.932922	−0.466461	+	0.884542i	$0.654471\pi$
$480$	0	0
$481$	−41.0160	−1.87017
$482$	49.2743	2.24438
$483$	0	0
$484$	−25.3818	−1.15372
$485$	16.1327	0.732549
$486$	0	0
$487$	−4.49697	−0.203777	−0.101889	−	0.994796i	$-0.532489\pi$
$487$	−4.49697	−0.203777	−0.101889	+	0.994796i	$0.532489\pi$
$488$	−11.5110	−0.521077
$489$	0	0
$490$	19.3842	0.875687
$491$	−29.7834	−1.34411	−0.672054	−	0.740502i	$-0.734588\pi$
$491$	−29.7834	−1.34411	−0.672054	+	0.740502i	$0.734588\pi$
$492$	0	0
$493$	0	0
$494$	−73.9750	−3.32829
$495$	0	0
$496$	62.2473	2.79499
$497$	6.13892	0.275368
$498$	0	0
$499$	−5.64495	−0.252703	−0.126351	−	0.991986i	$-0.540327\pi$
$499$	−5.64495	−0.252703	−0.126351	+	0.991986i	$0.540327\pi$
$500$	−46.4242	−2.07615
$501$	0	0
$502$	5.66954	0.253044
$503$	16.6016	0.740230	0.370115	−	0.928986i	$-0.379318\pi$
$503$	16.6016	0.740230	0.370115	+	0.928986i	$0.379318\pi$
$504$	0	0
$505$	18.0899	0.804991
$506$	43.2573	1.92302
$507$	0	0
$508$	57.8058	2.56472
$509$	42.2496	1.87268	0.936340	−	0.351094i	$-0.114190\pi$
$509$	42.2496	1.87268	0.936340	+	0.351094i	$0.114190\pi$
$510$	0	0
$511$	6.69961	0.296373
$512$	50.8469	2.24713
$513$	0	0
$514$	4.03652	0.178043
$515$	5.61073	0.247238
$516$	0	0
$517$	−9.60617	−0.422479
$518$	−17.8012	−0.782139
$519$	0	0
$520$	−38.2815	−1.67876
$521$	−31.3124	−1.37182	−0.685910	−	0.727686i	$-0.740596\pi$
$521$	−31.3124	−1.37182	−0.685910	+	0.727686i	$0.740596\pi$
$522$	0	0
$523$	−17.8178	−0.779116	−0.389558	−	0.921002i	$-0.627372\pi$
$523$	−17.8178	−0.779116	−0.389558	+	0.921002i	$0.627372\pi$
$524$	−18.7892	−0.820810
$525$	0	0
$526$	−9.39804	−0.409774
$527$	0	0
$528$	0	0
$529$	30.8868	1.34290
$530$	39.8746	1.73204
$531$	0	0
$532$	−22.2142	−0.963109
$533$	−28.9329	−1.25322
$534$	0	0
$535$	−19.7516	−0.853936
$536$	67.3027	2.90703
$537$	0	0
$538$	−9.86533	−0.425325
$539$	−14.5266	−0.625703
$540$	0	0
$541$	3.31264	0.142421	0.0712107	−	0.997461i	$-0.477314\pi$
$541$	3.31264	0.142421	0.0712107	+	0.997461i	$0.477314\pi$
$542$	76.6539	3.29257
$543$	0	0
$544$	0	0
$545$	5.94728	0.254753
$546$	0	0
$547$	−19.6443	−0.839928	−0.419964	−	0.907541i	$-0.637957\pi$
$547$	−19.6443	−0.839928	−0.419964	+	0.907541i	$0.637957\pi$
$548$	−38.7780	−1.65651
$549$	0	0
$550$	20.8178	0.887675
$551$	33.0877	1.40958
$552$	0	0
$553$	−0.309314	−0.0131534
$554$	15.6548	0.665108
$555$	0	0
$556$	−19.7514	−0.837648
$557$	−6.60616	−0.279912	−0.139956	−	0.990158i	$-0.544696\pi$
$557$	−6.60616	−0.279912	−0.139956	+	0.990158i	$0.544696\pi$
$558$	0	0
$559$	−19.7629	−0.835881
$560$	−7.38703	−0.312159
$561$	0	0
$562$	−0.984365	−0.0415229
$563$	27.7911	1.17125	0.585627	−	0.810581i	$-0.300848\pi$
$563$	27.7911	1.17125	0.585627	+	0.810581i	$0.300848\pi$
$564$	0	0
$565$	−9.69417	−0.407837
$566$	13.3767	0.562263
$567$	0	0
$568$	45.9581	1.92836
$569$	−22.6455	−0.949347	−0.474674	−	0.880162i	$-0.657434\pi$
$569$	−22.6455	−0.949347	−0.474674	+	0.880162i	$0.657434\pi$
$570$	0	0
$571$	45.3836	1.89925	0.949623	−	0.313395i	$-0.101466\pi$
$571$	45.3836	1.89925	0.949623	+	0.313395i	$0.101466\pi$
$572$	51.7159	2.16235
$573$	0	0
$574$	−12.5570	−0.524121
$575$	25.9333	1.08149
$576$	0	0
$577$	−31.0789	−1.29383	−0.646916	−	0.762561i	$-0.723942\pi$
$577$	−31.0789	−1.29383	−0.646916	+	0.762561i	$0.723942\pi$
$578$	0	0
$579$	0	0
$580$	30.8667	1.28167
$581$	5.16650	0.214343
$582$	0	0
$583$	−29.8822	−1.23759
$584$	50.1557	2.07546
$585$	0	0
$586$	−5.64012	−0.232991
$587$	6.64021	0.274071	0.137035	−	0.990566i	$-0.456243\pi$
$587$	6.64021	0.274071	0.137035	+	0.990566i	$0.456243\pi$
$588$	0	0
$589$	50.4805	2.08001
$590$	29.4592	1.21282
$591$	0	0
$592$	−59.2522	−2.43525
$593$	−22.2981	−0.915671	−0.457836	−	0.889037i	$-0.651375\pi$
$593$	−22.2981	−0.915671	−0.457836	+	0.889037i	$0.651375\pi$
$594$	0	0
$595$	0	0
$596$	−27.2144	−1.11474
$597$	0	0
$598$	93.1099	3.80754
$599$	13.2415	0.541033	0.270516	−	0.962715i	$-0.412806\pi$
$599$	13.2415	0.541033	0.270516	+	0.962715i	$0.412806\pi$
$600$	0	0
$601$	−0.113522	−0.00463068	−0.00231534	−	0.999997i	$-0.500737\pi$
$601$	−0.113522	−0.00463068	−0.00231534	+	0.999997i	$0.500737\pi$
$602$	−8.57721	−0.349581
$603$	0	0
$604$	−24.9180	−1.01390
$605$	−6.84485	−0.278283
$606$	0	0
$607$	27.6974	1.12420	0.562101	−	0.827069i	$-0.309993\pi$
$607$	27.6974	1.12420	0.562101	+	0.827069i	$0.309993\pi$
$608$	−32.8150	−1.33082
$609$	0	0
$610$	−5.59593	−0.226572
$611$	−20.6769	−0.836499
$612$	0	0
$613$	−7.74254	−0.312718	−0.156359	−	0.987700i	$-0.549976\pi$
$613$	−7.74254	−0.312718	−0.156359	+	0.987700i	$0.549976\pi$
$614$	−31.4517	−1.26929
$615$	0	0
$616$	12.4509	0.501661
$617$	−36.6963	−1.47734	−0.738668	−	0.674070i	$-0.764545\pi$
$617$	−36.6963	−1.47734	−0.738668	+	0.674070i	$0.764545\pi$
$618$	0	0
$619$	30.4273	1.22298	0.611488	−	0.791253i	$-0.290571\pi$
$619$	30.4273	1.22298	0.611488	+	0.791253i	$0.290571\pi$
$620$	47.0919	1.89126
$621$	0	0
$622$	−35.5321	−1.42471
$623$	7.98838	0.320048
$624$	0	0
$625$	5.14445	0.205778
$626$	−28.0560	−1.12134
$627$	0	0
$628$	−66.6071	−2.65791
$629$	0	0
$630$	0	0
$631$	19.1334	0.761691	0.380845	−	0.924639i	$-0.375633\pi$
$631$	19.1334	0.761691	0.380845	+	0.924639i	$0.375633\pi$
$632$	−2.31564	−0.0921111
$633$	0	0
$634$	−59.4696	−2.36184
$635$	15.5888	0.618623
$636$	0	0
$637$	−31.2679	−1.23888
$638$	−33.4314	−1.32356
$639$	0	0
$640$	13.7775	0.544603
$641$	9.84916	0.389019	0.194509	−	0.980901i	$-0.437689\pi$
$641$	9.84916	0.389019	0.194509	+	0.980901i	$0.437689\pi$
$642$	0	0
$643$	−30.6319	−1.20801	−0.604003	−	0.796982i	$-0.706428\pi$
$643$	−30.6319	−1.20801	−0.604003	+	0.796982i	$0.706428\pi$
$644$	27.9603	1.10179
$645$	0	0
$646$	0	0
$647$	−2.27404	−0.0894017	−0.0447009	−	0.999000i	$-0.514233\pi$
$647$	−2.27404	−0.0894017	−0.0447009	+	0.999000i	$0.514233\pi$
$648$	0	0
$649$	−22.0768	−0.866592
$650$	44.8096	1.75758
$651$	0	0
$652$	92.1538	3.60902
$653$	−11.4496	−0.448057	−0.224029	−	0.974583i	$-0.571921\pi$
$653$	−11.4496	−0.448057	−0.224029	+	0.974583i	$0.571921\pi$
$654$	0	0
$655$	−5.06698	−0.197983
$656$	−41.7968	−1.63189
$657$	0	0
$658$	−8.97391	−0.349839
$659$	−11.1427	−0.434058	−0.217029	−	0.976165i	$-0.569637\pi$
$659$	−11.1427	−0.434058	−0.217029	+	0.976165i	$0.569637\pi$
$660$	0	0
$661$	−30.7222	−1.19495	−0.597477	−	0.801886i	$-0.703830\pi$
$661$	−30.7222	−1.19495	−0.597477	+	0.801886i	$0.703830\pi$
$662$	9.32787	0.362538
$663$	0	0
$664$	38.6783	1.50101
$665$	−5.99063	−0.232307
$666$	0	0
$667$	−41.6464	−1.61255
$668$	−35.7632	−1.38372
$669$	0	0
$670$	32.7184	1.26402
$671$	4.19361	0.161892
$672$	0	0
$673$	23.1713	0.893188	0.446594	−	0.894737i	$-0.352637\pi$
$673$	23.1713	0.893188	0.446594	+	0.894737i	$0.352637\pi$
$674$	−12.2670	−0.472506
$675$	0	0
$676$	52.9251	2.03558
$677$	−11.5127	−0.442470	−0.221235	−	0.975221i	$-0.571009\pi$
$677$	−11.5127	−0.442470	−0.221235	+	0.975221i	$0.571009\pi$
$678$	0	0
$679$	−11.2942	−0.433431
$680$	0	0
$681$	0	0
$682$	−51.0049	−1.95308
$683$	12.8418	0.491377	0.245689	−	0.969349i	$-0.420986\pi$
$683$	12.8418	0.491377	0.245689	+	0.969349i	$0.420986\pi$
$684$	0	0
$685$	−10.4575	−0.399559
$686$	−28.6946	−1.09556
$687$	0	0
$688$	−28.5497	−1.08845
$689$	−64.3203	−2.45041
$690$	0	0
$691$	31.5223	1.19916	0.599582	−	0.800314i	$-0.295334\pi$
$691$	31.5223	1.19916	0.599582	+	0.800314i	$0.295334\pi$
$692$	80.3404	3.05409
$693$	0	0
$694$	−61.5357	−2.33586
$695$	−5.32648	−0.202045
$696$	0	0
$697$	0	0
$698$	−59.8709	−2.26615
$699$	0	0
$700$	13.4560	0.508591
$701$	0.971208	0.0366820	0.0183410	−	0.999832i	$-0.494162\pi$
$701$	0.971208	0.0366820	0.0183410	+	0.999832i	$0.494162\pi$
$702$	0	0
$703$	−48.0515	−1.81230
$704$	−0.109967	−0.00414453
$705$	0	0
$706$	18.1951	0.684782
$707$	−12.6644	−0.476293
$708$	0	0
$709$	−20.7932	−0.780905	−0.390452	−	0.920623i	$-0.627681\pi$
$709$	−20.7932	−0.780905	−0.390452	+	0.920623i	$0.627681\pi$
$710$	22.3420	0.838480
$711$	0	0
$712$	59.8038	2.24124
$713$	−63.5381	−2.37952
$714$	0	0
$715$	13.9465	0.521570
$716$	−88.6333	−3.31238
$717$	0	0
$718$	−94.0144	−3.50859
$719$	44.4467	1.65758	0.828790	−	0.559560i	$-0.189030\pi$
$719$	44.4467	1.65758	0.828790	+	0.559560i	$0.189030\pi$
$720$	0	0
$721$	−3.92795	−0.146285
$722$	−38.2544	−1.42368
$723$	0	0
$724$	−13.9968	−0.520186
$725$	−20.0425	−0.744361
$726$	0	0
$727$	3.78404	0.140342	0.0701711	−	0.997535i	$-0.477645\pi$
$727$	3.78404	0.140342	0.0701711	+	0.997535i	$0.477645\pi$
$728$	26.8001	0.993278
$729$	0	0
$730$	24.3826	0.902440
$731$	0	0
$732$	0	0
$733$	−14.4391	−0.533321	−0.266660	−	0.963791i	$-0.585920\pi$
$733$	−14.4391	−0.533321	−0.266660	+	0.963791i	$0.585920\pi$
$734$	−4.52702	−0.167095
$735$	0	0
$736$	41.3032	1.52245
$737$	−24.5193	−0.903181
$738$	0	0
$739$	32.8229	1.20741	0.603704	−	0.797208i	$-0.293691\pi$
$739$	32.8229	1.20741	0.603704	+	0.797208i	$0.293691\pi$
$740$	−44.8260	−1.64784
$741$	0	0
$742$	−27.9154	−1.02481
$743$	−31.5527	−1.15756	−0.578779	−	0.815485i	$-0.696471\pi$
$743$	−31.5527	−1.15756	−0.578779	+	0.815485i	$0.696471\pi$
$744$	0	0
$745$	−7.33905	−0.268882
$746$	3.90990	0.143152
$747$	0	0
$748$	0	0
$749$	13.8277	0.505253
$750$	0	0
$751$	−13.1301	−0.479123	−0.239562	−	0.970881i	$-0.577004\pi$
$751$	−13.1301	−0.479123	−0.239562	+	0.970881i	$0.577004\pi$
$752$	−29.8701	−1.08925
$753$	0	0
$754$	−71.9599	−2.62063
$755$	−6.71978	−0.244558
$756$	0	0
$757$	20.6308	0.749840	0.374920	−	0.927057i	$-0.377670\pi$
$757$	20.6308	0.749840	0.374920	+	0.927057i	$0.377670\pi$
$758$	64.5463	2.34443
$759$	0	0
$760$	−44.8480	−1.62681
$761$	−0.498800	−0.0180815	−0.00904074	−	0.999959i	$-0.502878\pi$
$761$	−0.498800	−0.0180815	−0.00904074	+	0.999959i	$0.502878\pi$
$762$	0	0
$763$	−4.16356	−0.150731
$764$	23.5843	0.853249
$765$	0	0
$766$	−80.5870	−2.91173
$767$	−47.5196	−1.71583
$768$	0	0
$769$	−7.46086	−0.269046	−0.134523	−	0.990911i	$-0.542950\pi$
$769$	−7.46086	−0.269046	−0.134523	+	0.990911i	$0.542950\pi$
$770$	6.05286	0.218130
$771$	0	0
$772$	−120.080	−4.32179
$773$	4.10366	0.147598	0.0737991	−	0.997273i	$-0.476488\pi$
$773$	4.10366	0.147598	0.0737991	+	0.997273i	$0.476488\pi$
$774$	0	0
$775$	−30.5780	−1.09840
$776$	−84.5522	−3.03525
$777$	0	0
$778$	−58.5332	−2.09852
$779$	−33.8958	−1.21444
$780$	0	0
$781$	−16.7432	−0.599118
$782$	0	0
$783$	0	0
$784$	−45.1700	−1.61321
$785$	−17.9623	−0.641102
$786$	0	0
$787$	38.1810	1.36101	0.680503	−	0.732745i	$-0.261761\pi$
$787$	38.1810	1.36101	0.680503	+	0.732745i	$0.261761\pi$
$788$	−49.7765	−1.77322
$789$	0	0
$790$	−1.12572	−0.0400513
$791$	6.78669	0.241307
$792$	0	0
$793$	9.02659	0.320544
$794$	−80.5983	−2.86033
$795$	0	0
$796$	−34.6189	−1.22704
$797$	−53.9035	−1.90936	−0.954681	−	0.297632i	$-0.903803\pi$
$797$	−53.9035	−1.90936	−0.954681	+	0.297632i	$0.903803\pi$
$798$	0	0
$799$	0	0
$800$	19.8774	0.702771
$801$	0	0
$802$	6.80082	0.240145
$803$	−18.2724	−0.644819
$804$	0	0
$805$	7.54021	0.265757
$806$	−109.786	−3.86705
$807$	0	0
$808$	−94.8100	−3.33541
$809$	−5.00636	−0.176014	−0.0880071	−	0.996120i	$-0.528050\pi$
$809$	−5.00636	−0.176014	−0.0880071	+	0.996120i	$0.528050\pi$
$810$	0	0
$811$	2.24418	0.0788038	0.0394019	−	0.999223i	$-0.487455\pi$
$811$	2.24418	0.0788038	0.0394019	+	0.999223i	$0.487455\pi$
$812$	−21.6091	−0.758331
$813$	0	0
$814$	48.5507	1.70170
$815$	24.8516	0.870514
$816$	0	0
$817$	−23.1528	−0.810015
$818$	−1.76049	−0.0615540
$819$	0	0
$820$	−31.6205	−1.10424
$821$	−40.5090	−1.41377	−0.706887	−	0.707326i	$-0.749901\pi$
$821$	−40.5090	−1.41377	−0.706887	+	0.707326i	$0.749901\pi$
$822$	0	0
$823$	21.6820	0.755785	0.377893	−	0.925849i	$-0.376649\pi$
$823$	21.6820	0.755785	0.377893	+	0.925849i	$0.376649\pi$
$824$	−29.4061	−1.02441
$825$	0	0
$826$	−20.6238	−0.717593
$827$	9.86865	0.343167	0.171583	−	0.985170i	$-0.445112\pi$
$827$	9.86865	0.343167	0.171583	+	0.985170i	$0.445112\pi$
$828$	0	0
$829$	−56.6233	−1.96661	−0.983304	−	0.181973i	$-0.941752\pi$
$829$	−56.6233	−1.96661	−0.983304	+	0.181973i	$0.941752\pi$
$830$	18.8030	0.652661
$831$	0	0
$832$	−0.236700	−0.00820609
$833$	0	0
$834$	0	0
$835$	−9.64446	−0.333760
$836$	60.5867	2.09544
$837$	0	0
$838$	−5.09446	−0.175985
$839$	−22.9292	−0.791602	−0.395801	−	0.918336i	$-0.629533\pi$
$839$	−22.9292	−0.791602	−0.395801	+	0.918336i	$0.629533\pi$
$840$	0	0
$841$	3.18640	0.109876
$842$	47.4044	1.63366
$843$	0	0
$844$	39.2012	1.34936
$845$	14.2726	0.490992
$846$	0	0
$847$	4.79194	0.164653
$848$	−92.9178	−3.19081
$849$	0	0
$850$	0	0
$851$	60.4808	2.07326
$852$	0	0
$853$	35.0514	1.20014	0.600068	−	0.799949i	$-0.295140\pi$
$853$	35.0514	1.20014	0.600068	+	0.799949i	$0.295140\pi$
$854$	3.91759	0.134057
$855$	0	0
$856$	103.519	3.53821
$857$	−1.20586	−0.0411913	−0.0205956	−	0.999788i	$-0.506556\pi$
$857$	−1.20586	−0.0411913	−0.0205956	+	0.999788i	$0.506556\pi$
$858$	0	0
$859$	−13.6732	−0.466522	−0.233261	−	0.972414i	$-0.574940\pi$
$859$	−13.6732	−0.466522	−0.233261	+	0.972414i	$0.574940\pi$
$860$	−21.5987	−0.736509
$861$	0	0
$862$	−76.3306	−2.59983
$863$	−44.1853	−1.50408	−0.752042	−	0.659115i	$-0.770931\pi$
$863$	−44.1853	−1.50408	−0.752042	+	0.659115i	$0.770931\pi$
$864$	0	0
$865$	21.6658	0.736661
$866$	104.838	3.56255
$867$	0	0
$868$	−32.9681	−1.11901
$869$	0.843620	0.0286178
$870$	0	0
$871$	−52.7769	−1.78828
$872$	−31.1699	−1.05555
$873$	0	0
$874$	109.081	3.68972
$875$	8.76461	0.296298
$876$	0	0
$877$	23.6095	0.797235	0.398617	−	0.917117i	$-0.369490\pi$
$877$	23.6095	0.797235	0.398617	+	0.917117i	$0.369490\pi$
$878$	54.9458	1.85433
$879$	0	0
$880$	20.1473	0.679165
$881$	27.0910	0.912718	0.456359	−	0.889796i	$-0.349153\pi$
$881$	27.0910	0.912718	0.456359	+	0.889796i	$0.349153\pi$
$882$	0	0
$883$	17.3387	0.583495	0.291747	−	0.956495i	$-0.405763\pi$
$883$	17.3387	0.583495	0.291747	+	0.956495i	$0.405763\pi$
$884$	0	0
$885$	0	0
$886$	−100.342	−3.37107
$887$	22.1688	0.744354	0.372177	−	0.928162i	$-0.378611\pi$
$887$	22.1688	0.744354	0.372177	+	0.928162i	$0.378611\pi$
$888$	0	0
$889$	−10.9134	−0.366024
$890$	29.0729	0.974527
$891$	0	0
$892$	−94.2473	−3.15563
$893$	−24.2236	−0.810614
$894$	0	0
$895$	−23.9022	−0.798963
$896$	−9.64533	−0.322228
$897$	0	0
$898$	−62.1157	−2.07283
$899$	49.1054	1.63776
$900$	0	0
$901$	0	0
$902$	34.2479	1.14033
$903$	0	0
$904$	50.8076	1.68983
$905$	−3.77458	−0.125471
$906$	0	0
$907$	54.1335	1.79747	0.898736	−	0.438490i	$-0.144487\pi$
$907$	54.1335	1.79747	0.898736	+	0.438490i	$0.144487\pi$
$908$	24.0275	0.797379
$909$	0	0
$910$	13.0286	0.431893
$911$	−4.61843	−0.153015	−0.0765076	−	0.997069i	$-0.524377\pi$
$911$	−4.61843	−0.153015	−0.0765076	+	0.997069i	$0.524377\pi$
$912$	0	0
$913$	−14.0910	−0.466345
$914$	16.6169	0.549639
$915$	0	0
$916$	119.171	3.93752
$917$	3.54729	0.117142
$918$	0	0
$919$	−44.5640	−1.47003	−0.735015	−	0.678051i	$-0.762825\pi$
$919$	−44.5640	−1.47003	−0.735015	+	0.678051i	$0.762825\pi$
$920$	56.4487	1.86106
$921$	0	0
$922$	43.8232	1.44324
$923$	−36.0391	−1.18624
$924$	0	0
$925$	29.1067	0.957023
$926$	46.5211	1.52878
$927$	0	0
$928$	−31.9211	−1.04786
$929$	0.305784	0.0100324	0.00501622	−	0.999987i	$-0.498403\pi$
$929$	0.305784	0.0100324	0.00501622	+	0.999987i	$0.498403\pi$
$930$	0	0
$931$	−36.6313	−1.20054
$932$	80.0993	2.62374
$933$	0	0
$934$	−18.1542	−0.594023
$935$	0	0
$936$	0	0
$937$	−40.1111	−1.31037	−0.655187	−	0.755467i	$-0.727410\pi$
$937$	−40.1111	−1.31037	−0.655187	+	0.755467i	$0.727410\pi$
$938$	−22.9055	−0.747891
$939$	0	0
$940$	−22.5976	−0.737053
$941$	−42.4502	−1.38384	−0.691919	−	0.721975i	$-0.743234\pi$
$941$	−42.4502	−1.38384	−0.691919	+	0.721975i	$0.743234\pi$
$942$	0	0
$943$	42.6635	1.38931
$944$	−68.6474	−2.23428
$945$	0	0
$946$	23.3933	0.760583
$947$	54.6640	1.77634	0.888171	−	0.459512i	$-0.151976\pi$
$947$	54.6640	1.77634	0.888171	+	0.459512i	$0.151976\pi$
$948$	0	0
$949$	−39.3307	−1.27673
$950$	52.4958	1.70319
$951$	0	0
$952$	0	0
$953$	−5.69466	−0.184468	−0.0922341	−	0.995737i	$-0.529401\pi$
$953$	−5.69466	−0.184468	−0.0922341	+	0.995737i	$0.529401\pi$
$954$	0	0
$955$	6.36010	0.205808
$956$	−42.5077	−1.37480
$957$	0	0
$958$	52.0224	1.68077
$959$	7.32105	0.236409
$960$	0	0
$961$	43.9180	1.41671
$962$	104.504	3.36933
$963$	0	0
$964$	−86.8658	−2.79776
$965$	−32.3827	−1.04244
$966$	0	0
$967$	3.55944	0.114464	0.0572320	−	0.998361i	$-0.481773\pi$
$967$	3.55944	0.114464	0.0572320	+	0.998361i	$0.481773\pi$
$968$	35.8741	1.15304
$969$	0	0
$970$	−41.1041	−1.31977
$971$	−21.5588	−0.691856	−0.345928	−	0.938261i	$-0.612436\pi$
$971$	−21.5588	−0.691856	−0.345928	+	0.938261i	$0.612436\pi$
$972$	0	0
$973$	3.72896	0.119545
$974$	11.4577	0.367129
$975$	0	0
$976$	13.0399	0.417397
$977$	−40.8344	−1.30641	−0.653204	−	0.757182i	$-0.726576\pi$
$977$	−40.8344	−1.30641	−0.653204	+	0.757182i	$0.726576\pi$
$978$	0	0
$979$	−21.7874	−0.696327
$980$	−34.1724	−1.09160
$981$	0	0
$982$	75.8844	2.42157
$983$	56.9723	1.81714	0.908568	−	0.417738i	$-0.137177\pi$
$983$	56.9723	1.81714	0.908568	+	0.417738i	$0.137177\pi$
$984$	0	0
$985$	−13.4235	−0.427708
$986$	0	0
$987$	0	0
$988$	130.411	4.14892
$989$	29.1417	0.926652
$990$	0	0
$991$	−29.7036	−0.943566	−0.471783	−	0.881715i	$-0.656389\pi$
$991$	−29.7036	−0.943566	−0.471783	+	0.881715i	$0.656389\pi$
$992$	−48.7007	−1.54625
$993$	0	0
$994$	−15.6412	−0.496108
$995$	−9.33587	−0.295967
$996$	0	0
$997$	−9.85427	−0.312088	−0.156044	−	0.987750i	$-0.549874\pi$
$997$	−9.85427	−0.312088	−0.156044	+	0.987750i	$0.549874\pi$
$998$	14.3826	0.455274
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.by.1.2	yes	15
3.2	odd	2		7803.2.a.bz.1.14	yes	15
17.16	even	2		7803.2.a.bx.1.2	✓	15
51.50	odd	2		7803.2.a.ca.1.14	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7803.2.a.bx.1.2	✓	15	17.16	even	2
7803.2.a.by.1.2	yes	15	1.1	even	1	trivial
7803.2.a.bz.1.14	yes	15	3.2	odd	2
7803.2.a.ca.1.14	yes	15	51.50	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54787 q^{2} +4.49165 q^{4} +1.21129 q^{5} -0.847998 q^{7} -6.34841 q^{8} +O(q^{10})\)	\(q-2.54787 q^{2} +4.49165 q^{4} +1.21129 q^{5} -0.847998 q^{7} -6.34841 q^{8} -3.08621 q^{10} +2.31282 q^{11} +4.97825 q^{13} +2.16059 q^{14} +7.19164 q^{16} +5.83217 q^{19} +5.44069 q^{20} -5.89276 q^{22} -7.34076 q^{23} -3.53278 q^{25} -12.6839 q^{26} -3.80891 q^{28} +5.67331 q^{29} +8.65552 q^{31} -5.62655 q^{32} -1.02717 q^{35} -8.23904 q^{37} -14.8596 q^{38} -7.68976 q^{40} -5.81186 q^{41} -3.96985 q^{43} +10.3884 q^{44} +18.7033 q^{46} -4.15345 q^{47} -6.28090 q^{49} +9.00107 q^{50} +22.3606 q^{52} -12.9203 q^{53} +2.80149 q^{55} +5.38344 q^{56} -14.4549 q^{58} -9.54544 q^{59} +1.81320 q^{61} -22.0531 q^{62} -0.0475468 q^{64} +6.03010 q^{65} -10.6015 q^{67} +2.61710 q^{70} -7.23931 q^{71} -7.90050 q^{73} +20.9920 q^{74} +26.1961 q^{76} -1.96126 q^{77} +0.364759 q^{79} +8.71115 q^{80} +14.8079 q^{82} -6.09259 q^{83} +10.1147 q^{86} -14.6827 q^{88} -9.42028 q^{89} -4.22155 q^{91} -32.9722 q^{92} +10.5825 q^{94} +7.06444 q^{95} +13.3186 q^{97} +16.0029 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 6 q^{2} + 12 q^{4} + 3 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) 15 * q - 6 * q^2 + 12 * q^4 + 3 * q^5 + 3 * q^7 - 18 * q^8	\( 15 q - 6 q^{2} + 12 q^{4} + 3 q^{5} + 3 q^{7} - 18 q^{8} - 6 q^{11} - 3 q^{13} - 3 q^{14} + 6 q^{16} - 3 q^{19} - 6 q^{20} - 12 q^{22} + 3 q^{23} + 6 q^{25} + 24 q^{26} + 9 q^{28} + 6 q^{29} - 42 q^{32} - 33 q^{35} - 36 q^{38} + 15 q^{40} - 3 q^{43} + 18 q^{44} + 12 q^{46} - 24 q^{47} + 18 q^{49} - 42 q^{50} - 12 q^{52} - 48 q^{53} - 3 q^{55} + 15 q^{56} - 12 q^{58} - 18 q^{59} - 63 q^{62} + 30 q^{64} - 24 q^{65} + 12 q^{67} + 51 q^{70} + 21 q^{71} - 18 q^{73} + 48 q^{74} + 60 q^{76} - 54 q^{77} - 39 q^{79} - 6 q^{80} + 39 q^{82} - 51 q^{83} - 48 q^{86} - 48 q^{88} - 36 q^{89} + 9 q^{91} + 3 q^{92} + 36 q^{94} - 39 q^{95} + 3 q^{97} - 15 q^{98}+O(q^{100}) \) 15 * q - 6 * q^2 + 12 * q^4 + 3 * q^5 + 3 * q^7 - 18 * q^8 - 6 * q^11 - 3 * q^13 - 3 * q^14 + 6 * q^16 - 3 * q^19 - 6 * q^20 - 12 * q^22 + 3 * q^23 + 6 * q^25 + 24 * q^26 + 9 * q^28 + 6 * q^29 - 42 * q^32 - 33 * q^35 - 36 * q^38 + 15 * q^40 - 3 * q^43 + 18 * q^44 + 12 * q^46 - 24 * q^47 + 18 * q^49 - 42 * q^50 - 12 * q^52 - 48 * q^53 - 3 * q^55 + 15 * q^56 - 12 * q^58 - 18 * q^59 - 63 * q^62 + 30 * q^64 - 24 * q^65 + 12 * q^67 + 51 * q^70 + 21 * q^71 - 18 * q^73 + 48 * q^74 + 60 * q^76 - 54 * q^77 - 39 * q^79 - 6 * q^80 + 39 * q^82 - 51 * q^83 - 48 * q^86 - 48 * q^88 - 36 * q^89 + 9 * q^91 + 3 * q^92 + 36 * q^94 - 39 * q^95 + 3 * q^97 - 15 * q^98

Introduction

L-functions

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