LMFDB - Embedded newform 8004.2.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 27 x^{11} + 158 x^{10} + 180 x^{9} - 1652 x^{8} + 65 x^{7} + 7388 x^{6} + \cdots - 5832 $ x^13 - 5x^12 - 27x^11 + 158x^10 + 180x^9 - 1652x^8 + 65x^7 + 7388x^6 - 3259x^5 - 15445x^4 + 7832x^3 + 15102x^2 - 5184x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.91772$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.91772 q^{5} -2.53447 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.91772 q^{5} -2.53447 q^{7} +1.00000 q^{9} -2.79646 q^{11} +2.05656 q^{13} +1.91772 q^{15} +6.58508 q^{17} +0.477899 q^{19} +2.53447 q^{21} -1.00000 q^{23} -1.32234 q^{25} -1.00000 q^{27} -1.00000 q^{29} +2.03218 q^{31} +2.79646 q^{33} +4.86041 q^{35} -11.2922 q^{37} -2.05656 q^{39} -4.45615 q^{41} +6.32597 q^{43} -1.91772 q^{45} -5.49945 q^{47} -0.576470 q^{49} -6.58508 q^{51} +0.163207 q^{53} +5.36284 q^{55} -0.477899 q^{57} +1.28356 q^{59} +3.48409 q^{61} -2.53447 q^{63} -3.94390 q^{65} -9.96086 q^{67} +1.00000 q^{69} -3.03556 q^{71} +5.70799 q^{73} +1.32234 q^{75} +7.08755 q^{77} -13.2090 q^{79} +1.00000 q^{81} -10.7821 q^{83} -12.6284 q^{85} +1.00000 q^{87} +13.7367 q^{89} -5.21228 q^{91} -2.03218 q^{93} -0.916477 q^{95} +10.6793 q^{97} -2.79646 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.91772	−0.857632	−0.428816	−	0.903392i	$-0.641069\pi$
$5$	−1.91772	−0.857632	−0.428816	+	0.903392i	$0.641069\pi$
$6$	0	0
$7$	−2.53447	−0.957939	−0.478970	−	0.877832i	$-0.658990\pi$
$7$	−2.53447	−0.957939	−0.478970	+	0.877832i	$0.658990\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.79646	−0.843166	−0.421583	−	0.906790i	$-0.638525\pi$
$11$	−2.79646	−0.843166	−0.421583	+	0.906790i	$0.638525\pi$
$12$	0	0
$13$	2.05656	0.570386	0.285193	−	0.958470i	$-0.407942\pi$
$13$	2.05656	0.570386	0.285193	+	0.958470i	$0.407942\pi$
$14$	0	0
$15$	1.91772	0.495154
$16$	0	0
$17$	6.58508	1.59712	0.798559	−	0.601917i	$-0.205596\pi$
$17$	6.58508	1.59712	0.798559	+	0.601917i	$0.205596\pi$
$18$	0	0
$19$	0.477899	0.109637	0.0548187	−	0.998496i	$-0.482542\pi$
$19$	0.477899	0.109637	0.0548187	+	0.998496i	$0.482542\pi$
$20$	0	0
$21$	2.53447	0.553066
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.32234	−0.264468
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.03218	0.364990	0.182495	−	0.983207i	$-0.441583\pi$
$31$	2.03218	0.364990	0.182495	+	0.983207i	$0.441583\pi$
$32$	0	0
$33$	2.79646	0.486802
$34$	0	0
$35$	4.86041	0.821559
$36$	0	0
$37$	−11.2922	−1.85642	−0.928209	−	0.372059i	$-0.878652\pi$
$37$	−11.2922	−1.85642	−0.928209	+	0.372059i	$0.878652\pi$
$38$	0	0
$39$	−2.05656	−0.329313
$40$	0	0
$41$	−4.45615	−0.695934	−0.347967	−	0.937507i	$-0.613128\pi$
$41$	−4.45615	−0.695934	−0.347967	+	0.937507i	$0.613128\pi$
$42$	0	0
$43$	6.32597	0.964701	0.482350	−	0.875978i	$-0.339783\pi$
$43$	6.32597	0.964701	0.482350	+	0.875978i	$0.339783\pi$
$44$	0	0
$45$	−1.91772	−0.285877
$46$	0	0
$47$	−5.49945	−0.802177	−0.401088	−	0.916039i	$-0.631368\pi$
$47$	−5.49945	−0.802177	−0.401088	+	0.916039i	$0.631368\pi$
$48$	0	0
$49$	−0.576470	−0.0823529
$50$	0	0
$51$	−6.58508	−0.922096
$52$	0	0
$53$	0.163207	0.0224182	0.0112091	−	0.999937i	$-0.496432\pi$
$53$	0.163207	0.0224182	0.0112091	+	0.999937i	$0.496432\pi$
$54$	0	0
$55$	5.36284	0.723125
$56$	0	0
$57$	−0.477899	−0.0632992
$58$	0	0
$59$	1.28356	0.167105	0.0835527	−	0.996503i	$-0.473373\pi$
$59$	1.28356	0.167105	0.0835527	+	0.996503i	$0.473373\pi$
$60$	0	0
$61$	3.48409	0.446093	0.223046	−	0.974808i	$-0.428400\pi$
$61$	3.48409	0.446093	0.223046	+	0.974808i	$0.428400\pi$
$62$	0	0
$63$	−2.53447	−0.319313
$64$	0	0
$65$	−3.94390	−0.489181
$66$	0	0
$67$	−9.96086	−1.21691	−0.608457	−	0.793587i	$-0.708211\pi$
$67$	−9.96086	−1.21691	−0.608457	+	0.793587i	$0.708211\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−3.03556	−0.360254	−0.180127	−	0.983643i	$-0.557651\pi$
$71$	−3.03556	−0.360254	−0.180127	+	0.983643i	$0.557651\pi$
$72$	0	0
$73$	5.70799	0.668070	0.334035	−	0.942561i	$-0.391590\pi$
$73$	5.70799	0.668070	0.334035	+	0.942561i	$0.391590\pi$
$74$	0	0
$75$	1.32234	0.152691
$76$	0	0
$77$	7.08755	0.807701
$78$	0	0
$79$	−13.2090	−1.48613	−0.743064	−	0.669220i	$-0.766628\pi$
$79$	−13.2090	−1.48613	−0.743064	+	0.669220i	$0.766628\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.7821	−1.18348	−0.591742	−	0.806127i	$-0.701560\pi$
$83$	−10.7821	−1.18348	−0.591742	+	0.806127i	$0.701560\pi$
$84$	0	0
$85$	−12.6284	−1.36974
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	13.7367	1.45609	0.728044	−	0.685531i	$-0.240430\pi$
$89$	13.7367	1.45609	0.728044	+	0.685531i	$0.240430\pi$
$90$	0	0
$91$	−5.21228	−0.546395
$92$	0	0
$93$	−2.03218	−0.210727
$94$	0	0
$95$	−0.916477	−0.0940285
$96$	0	0
$97$	10.6793	1.08432	0.542162	−	0.840274i	$-0.317606\pi$
$97$	10.6793	1.08432	0.542162	+	0.840274i	$0.317606\pi$
$98$	0	0
$99$	−2.79646	−0.281055
$100$	0	0
$101$	−13.6172	−1.35496	−0.677479	−	0.735542i	$-0.736927\pi$
$101$	−13.6172	−1.35496	−0.677479	+	0.735542i	$0.736927\pi$
$102$	0	0
$103$	−13.5731	−1.33739	−0.668697	−	0.743535i	$-0.733148\pi$
$103$	−13.5731	−1.33739	−0.668697	+	0.743535i	$0.733148\pi$
$104$	0	0
$105$	−4.86041	−0.474327
$106$	0	0
$107$	15.1393	1.46358	0.731788	−	0.681533i	$-0.238686\pi$
$107$	15.1393	1.46358	0.731788	+	0.681533i	$0.238686\pi$
$108$	0	0
$109$	1.33221	0.127602	0.0638012	−	0.997963i	$-0.479678\pi$
$109$	1.33221	0.127602	0.0638012	+	0.997963i	$0.479678\pi$
$110$	0	0
$111$	11.2922	1.07180
$112$	0	0
$113$	−12.1281	−1.14092	−0.570459	−	0.821326i	$-0.693235\pi$
$113$	−12.1281	−1.14092	−0.570459	+	0.821326i	$0.693235\pi$
$114$	0	0
$115$	1.91772	0.178829
$116$	0	0
$117$	2.05656	0.190129
$118$	0	0
$119$	−16.6897	−1.52994
$120$	0	0
$121$	−3.17979	−0.289072
$122$	0	0
$123$	4.45615	0.401798
$124$	0	0
$125$	12.1245	1.08445
$126$	0	0
$127$	−16.4511	−1.45980	−0.729898	−	0.683556i	$-0.760432\pi$
$127$	−16.4511	−1.45980	−0.729898	+	0.683556i	$0.760432\pi$
$128$	0	0
$129$	−6.32597	−0.556970
$130$	0	0
$131$	−0.749438	−0.0654787	−0.0327393	−	0.999464i	$-0.510423\pi$
$131$	−0.749438	−0.0654787	−0.0327393	+	0.999464i	$0.510423\pi$
$132$	0	0
$133$	−1.21122	−0.105026
$134$	0	0
$135$	1.91772	0.165051
$136$	0	0
$137$	4.38791	0.374884	0.187442	−	0.982276i	$-0.439980\pi$
$137$	4.38791	0.374884	0.187442	+	0.982276i	$0.439980\pi$
$138$	0	0
$139$	21.8239	1.85108	0.925538	−	0.378655i	$-0.123613\pi$
$139$	21.8239	1.85108	0.925538	+	0.378655i	$0.123613\pi$
$140$	0	0
$141$	5.49945	0.463137
$142$	0	0
$143$	−5.75109	−0.480930
$144$	0	0
$145$	1.91772	0.159258
$146$	0	0
$147$	0.576470	0.0475465
$148$	0	0
$149$	−6.83755	−0.560154	−0.280077	−	0.959978i	$-0.590360\pi$
$149$	−6.83755	−0.560154	−0.280077	+	0.959978i	$0.590360\pi$
$150$	0	0
$151$	−2.41037	−0.196153	−0.0980767	−	0.995179i	$-0.531269\pi$
$151$	−2.41037	−0.196153	−0.0980767	+	0.995179i	$0.531269\pi$
$152$	0	0
$153$	6.58508	0.532372
$154$	0	0
$155$	−3.89716	−0.313027
$156$	0	0
$157$	21.6857	1.73071	0.865355	−	0.501160i	$-0.167093\pi$
$157$	21.6857	1.73071	0.865355	+	0.501160i	$0.167093\pi$
$158$	0	0
$159$	−0.163207	−0.0129432
$160$	0	0
$161$	2.53447	0.199744
$162$	0	0
$163$	−3.37011	−0.263968	−0.131984	−	0.991252i	$-0.542135\pi$
$163$	−3.37011	−0.263968	−0.131984	+	0.991252i	$0.542135\pi$
$164$	0	0
$165$	−5.36284	−0.417497
$166$	0	0
$167$	−4.89843	−0.379052	−0.189526	−	0.981876i	$-0.560695\pi$
$167$	−4.89843	−0.379052	−0.189526	+	0.981876i	$0.560695\pi$
$168$	0	0
$169$	−8.77058	−0.674660
$170$	0	0
$171$	0.477899	0.0365458
$172$	0	0
$173$	24.1859	1.83882	0.919411	−	0.393299i	$-0.128666\pi$
$173$	24.1859	1.83882	0.919411	+	0.393299i	$0.128666\pi$
$174$	0	0
$175$	3.35143	0.253344
$176$	0	0
$177$	−1.28356	−0.0964783
$178$	0	0
$179$	10.0886	0.754058	0.377029	−	0.926201i	$-0.376946\pi$
$179$	10.0886	0.754058	0.377029	+	0.926201i	$0.376946\pi$
$180$	0	0
$181$	15.0355	1.11758	0.558788	−	0.829310i	$-0.311266\pi$
$181$	15.0355	1.11758	0.558788	+	0.829310i	$0.311266\pi$
$182$	0	0
$183$	−3.48409	−0.257552
$184$	0	0
$185$	21.6552	1.59212
$186$	0	0
$187$	−18.4149	−1.34663
$188$	0	0
$189$	2.53447	0.184355
$190$	0	0
$191$	1.66689	0.120612	0.0603060	−	0.998180i	$-0.480792\pi$
$191$	1.66689	0.120612	0.0603060	+	0.998180i	$0.480792\pi$
$192$	0	0
$193$	−10.3313	−0.743664	−0.371832	−	0.928300i	$-0.621270\pi$
$193$	−10.3313	−0.743664	−0.371832	+	0.928300i	$0.621270\pi$
$194$	0	0
$195$	3.94390	0.282429
$196$	0	0
$197$	24.2977	1.73114	0.865570	−	0.500787i	$-0.166956\pi$
$197$	24.2977	1.73114	0.865570	+	0.500787i	$0.166956\pi$
$198$	0	0
$199$	−3.39029	−0.240331	−0.120166	−	0.992754i	$-0.538343\pi$
$199$	−3.39029	−0.240331	−0.120166	+	0.992754i	$0.538343\pi$
$200$	0	0
$201$	9.96086	0.702585
$202$	0	0
$203$	2.53447	0.177885
$204$	0	0
$205$	8.54566	0.596855
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−1.33643	−0.0924425
$210$	0	0
$211$	13.8877	0.956071	0.478036	−	0.878340i	$-0.341349\pi$
$211$	13.8877	0.956071	0.478036	+	0.878340i	$0.341349\pi$
$212$	0	0
$213$	3.03556	0.207993
$214$	0	0
$215$	−12.1314	−0.827358
$216$	0	0
$217$	−5.15050	−0.349639
$218$	0	0
$219$	−5.70799	−0.385710
$220$	0	0
$221$	13.5426	0.910973
$222$	0	0
$223$	5.89758	0.394931	0.197466	−	0.980310i	$-0.436729\pi$
$223$	5.89758	0.394931	0.197466	+	0.980310i	$0.436729\pi$
$224$	0	0
$225$	−1.32234	−0.0881561
$226$	0	0
$227$	4.55343	0.302222	0.151111	−	0.988517i	$-0.451715\pi$
$227$	4.55343	0.302222	0.151111	+	0.988517i	$0.451715\pi$
$228$	0	0
$229$	18.0839	1.19502	0.597509	−	0.801862i	$-0.296157\pi$
$229$	18.0839	1.19502	0.597509	+	0.801862i	$0.296157\pi$
$230$	0	0
$231$	−7.08755	−0.466327
$232$	0	0
$233$	19.2686	1.26233	0.631164	−	0.775650i	$-0.282578\pi$
$233$	19.2686	1.26233	0.631164	+	0.775650i	$0.282578\pi$
$234$	0	0
$235$	10.5464	0.687972
$236$	0	0
$237$	13.2090	0.858017
$238$	0	0
$239$	−4.67472	−0.302382	−0.151191	−	0.988505i	$-0.548311\pi$
$239$	−4.67472	−0.302382	−0.151191	+	0.988505i	$0.548311\pi$
$240$	0	0
$241$	−25.0978	−1.61669	−0.808347	−	0.588707i	$-0.799637\pi$
$241$	−25.0978	−1.61669	−0.808347	+	0.588707i	$0.799637\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.10551	0.0706284
$246$	0	0
$247$	0.982825	0.0625357
$248$	0	0
$249$	10.7821	0.683285
$250$	0	0
$251$	13.8863	0.876495	0.438248	−	0.898854i	$-0.355599\pi$
$251$	13.8863	0.876495	0.438248	+	0.898854i	$0.355599\pi$
$252$	0	0
$253$	2.79646	0.175812
$254$	0	0
$255$	12.6284	0.790819
$256$	0	0
$257$	16.2069	1.01096	0.505479	−	0.862839i	$-0.331316\pi$
$257$	16.2069	1.01096	0.505479	+	0.862839i	$0.331316\pi$
$258$	0	0
$259$	28.6196	1.77834
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	5.73640	0.353721	0.176861	−	0.984236i	$-0.443406\pi$
$263$	5.73640	0.353721	0.176861	+	0.984236i	$0.443406\pi$
$264$	0	0
$265$	−0.312986	−0.0192266
$266$	0	0
$267$	−13.7367	−0.840672
$268$	0	0
$269$	4.81286	0.293445	0.146723	−	0.989178i	$-0.453128\pi$
$269$	4.81286	0.293445	0.146723	+	0.989178i	$0.453128\pi$
$270$	0	0
$271$	−30.7494	−1.86789	−0.933946	−	0.357415i	$-0.883658\pi$
$271$	−30.7494	−1.86789	−0.933946	+	0.357415i	$0.883658\pi$
$272$	0	0
$273$	5.21228	0.315461
$274$	0	0
$275$	3.69788	0.222990
$276$	0	0
$277$	−21.7128	−1.30459	−0.652297	−	0.757964i	$-0.726194\pi$
$277$	−21.7128	−1.30459	−0.652297	+	0.757964i	$0.726194\pi$
$278$	0	0
$279$	2.03218	0.121663
$280$	0	0
$281$	16.1648	0.964313	0.482157	−	0.876085i	$-0.339854\pi$
$281$	16.1648	0.964313	0.482157	+	0.876085i	$0.339854\pi$
$282$	0	0
$283$	0.469095	0.0278848	0.0139424	−	0.999903i	$-0.495562\pi$
$283$	0.469095	0.0278848	0.0139424	+	0.999903i	$0.495562\pi$
$284$	0	0
$285$	0.916477	0.0542874
$286$	0	0
$287$	11.2940	0.666662
$288$	0	0
$289$	26.3633	1.55078
$290$	0	0
$291$	−10.6793	−0.626035
$292$	0	0
$293$	19.0024	1.11013	0.555067	−	0.831805i	$-0.312692\pi$
$293$	19.0024	1.11013	0.555067	+	0.831805i	$0.312692\pi$
$294$	0	0
$295$	−2.46151	−0.143315
$296$	0	0
$297$	2.79646	0.162267
$298$	0	0
$299$	−2.05656	−0.118934
$300$	0	0
$301$	−16.0330	−0.924125
$302$	0	0
$303$	13.6172	0.782285
$304$	0	0
$305$	−6.68153	−0.382583
$306$	0	0
$307$	−30.3246	−1.73071	−0.865357	−	0.501156i	$-0.832908\pi$
$307$	−30.3246	−1.73071	−0.865357	+	0.501156i	$0.832908\pi$
$308$	0	0
$309$	13.5731	0.772145
$310$	0	0
$311$	23.9215	1.35646	0.678232	−	0.734847i	$-0.262746\pi$
$311$	23.9215	1.35646	0.678232	+	0.734847i	$0.262746\pi$
$312$	0	0
$313$	25.7932	1.45792	0.728959	−	0.684557i	$-0.240004\pi$
$313$	25.7932	1.45792	0.728959	+	0.684557i	$0.240004\pi$
$314$	0	0
$315$	4.86041	0.273853
$316$	0	0
$317$	−25.4602	−1.42999	−0.714994	−	0.699130i	$-0.753571\pi$
$317$	−25.4602	−1.42999	−0.714994	+	0.699130i	$0.753571\pi$
$318$	0	0
$319$	2.79646	0.156572
$320$	0	0
$321$	−15.1393	−0.844996
$322$	0	0
$323$	3.14700	0.175104
$324$	0	0
$325$	−2.71947	−0.150849
$326$	0	0
$327$	−1.33221	−0.0736713
$328$	0	0
$329$	13.9382	0.768436
$330$	0	0
$331$	−6.99611	−0.384541	−0.192271	−	0.981342i	$-0.561585\pi$
$331$	−6.99611	−0.384541	−0.192271	+	0.981342i	$0.561585\pi$
$332$	0	0
$333$	−11.2922	−0.618806
$334$	0	0
$335$	19.1022	1.04366
$336$	0	0
$337$	24.4367	1.33115	0.665577	−	0.746329i	$-0.268186\pi$
$337$	24.4367	1.33115	0.665577	+	0.746329i	$0.268186\pi$
$338$	0	0
$339$	12.1281	0.658709
$340$	0	0
$341$	−5.68292	−0.307747
$342$	0	0
$343$	19.2023	1.03683
$344$	0	0
$345$	−1.91772	−0.103247
$346$	0	0
$347$	9.52215	0.511176	0.255588	−	0.966786i	$-0.417731\pi$
$347$	9.52215	0.511176	0.255588	+	0.966786i	$0.417731\pi$
$348$	0	0
$349$	13.1358	0.703144	0.351572	−	0.936161i	$-0.385647\pi$
$349$	13.1358	0.703144	0.351572	+	0.936161i	$0.385647\pi$
$350$	0	0
$351$	−2.05656	−0.109771
$352$	0	0
$353$	−18.6499	−0.992633	−0.496316	−	0.868142i	$-0.665314\pi$
$353$	−18.6499	−0.992633	−0.496316	+	0.868142i	$0.665314\pi$
$354$	0	0
$355$	5.82135	0.308965
$356$	0	0
$357$	16.6897	0.883312
$358$	0	0
$359$	4.60500	0.243043	0.121521	−	0.992589i	$-0.461223\pi$
$359$	4.60500	0.243043	0.121521	+	0.992589i	$0.461223\pi$
$360$	0	0
$361$	−18.7716	−0.987980
$362$	0	0
$363$	3.17979	0.166896
$364$	0	0
$365$	−10.9463	−0.572958
$366$	0	0
$367$	22.8145	1.19091	0.595454	−	0.803390i	$-0.296972\pi$
$367$	22.8145	1.19091	0.595454	+	0.803390i	$0.296972\pi$
$368$	0	0
$369$	−4.45615	−0.231978
$370$	0	0
$371$	−0.413644	−0.0214753
$372$	0	0
$373$	−32.3817	−1.67666	−0.838331	−	0.545162i	$-0.816468\pi$
$373$	−32.3817	−1.67666	−0.838331	+	0.545162i	$0.816468\pi$
$374$	0	0
$375$	−12.1245	−0.626106
$376$	0	0
$377$	−2.05656	−0.105918
$378$	0	0
$379$	24.2970	1.24805	0.624025	−	0.781404i	$-0.285496\pi$
$379$	24.2970	1.24805	0.624025	+	0.781404i	$0.285496\pi$
$380$	0	0
$381$	16.4511	0.842813
$382$	0	0
$383$	−14.1255	−0.721780	−0.360890	−	0.932608i	$-0.617527\pi$
$383$	−14.1255	−0.721780	−0.360890	+	0.932608i	$0.617527\pi$
$384$	0	0
$385$	−13.5920	−0.692710
$386$	0	0
$387$	6.32597	0.321567
$388$	0	0
$389$	−17.5528	−0.889965	−0.444982	−	0.895539i	$-0.646790\pi$
$389$	−17.5528	−0.889965	−0.444982	+	0.895539i	$0.646790\pi$
$390$	0	0
$391$	−6.58508	−0.333022
$392$	0	0
$393$	0.749438	0.0378041
$394$	0	0
$395$	25.3312	1.27455
$396$	0	0
$397$	26.2395	1.31692	0.658462	−	0.752614i	$-0.271207\pi$
$397$	26.2395	1.31692	0.658462	+	0.752614i	$0.271207\pi$
$398$	0	0
$399$	1.21122	0.0606368
$400$	0	0
$401$	−16.3837	−0.818165	−0.409082	−	0.912497i	$-0.634151\pi$
$401$	−16.3837	−0.818165	−0.409082	+	0.912497i	$0.634151\pi$
$402$	0	0
$403$	4.17929	0.208185
$404$	0	0
$405$	−1.91772	−0.0952924
$406$	0	0
$407$	31.5781	1.56527
$408$	0	0
$409$	26.4351	1.30713	0.653565	−	0.756870i	$-0.273272\pi$
$409$	26.4351	1.30713	0.653565	+	0.756870i	$0.273272\pi$
$410$	0	0
$411$	−4.38791	−0.216439
$412$	0	0
$413$	−3.25314	−0.160077
$414$	0	0
$415$	20.6770	1.01499
$416$	0	0
$417$	−21.8239	−1.06872
$418$	0	0
$419$	−30.6261	−1.49618	−0.748091	−	0.663597i	$-0.769029\pi$
$419$	−30.6261	−1.49618	−0.748091	+	0.663597i	$0.769029\pi$
$420$	0	0
$421$	10.2828	0.501155	0.250577	−	0.968097i	$-0.419380\pi$
$421$	10.2828	0.501155	0.250577	+	0.968097i	$0.419380\pi$
$422$	0	0
$423$	−5.49945	−0.267392
$424$	0	0
$425$	−8.70772	−0.422387
$426$	0	0
$427$	−8.83033	−0.427330
$428$	0	0
$429$	5.75109	0.277665
$430$	0	0
$431$	31.0519	1.49572	0.747860	−	0.663857i	$-0.231082\pi$
$431$	31.0519	1.49572	0.747860	+	0.663857i	$0.231082\pi$
$432$	0	0
$433$	−10.3928	−0.499444	−0.249722	−	0.968318i	$-0.580339\pi$
$433$	−10.3928	−0.499444	−0.249722	+	0.968318i	$0.580339\pi$
$434$	0	0
$435$	−1.91772	−0.0919477
$436$	0	0
$437$	−0.477899	−0.0228610
$438$	0	0
$439$	29.0847	1.38814	0.694069	−	0.719909i	$-0.255816\pi$
$439$	29.0847	1.38814	0.694069	+	0.719909i	$0.255816\pi$
$440$	0	0
$441$	−0.576470	−0.0274510
$442$	0	0
$443$	28.7980	1.36824	0.684118	−	0.729371i	$-0.260187\pi$
$443$	28.7980	1.36824	0.684118	+	0.729371i	$0.260187\pi$
$444$	0	0
$445$	−26.3432	−1.24879
$446$	0	0
$447$	6.83755	0.323405
$448$	0	0
$449$	5.03769	0.237743	0.118872	−	0.992910i	$-0.462072\pi$
$449$	5.03769	0.237743	0.118872	+	0.992910i	$0.462072\pi$
$450$	0	0
$451$	12.4615	0.586787
$452$	0	0
$453$	2.41037	0.113249
$454$	0	0
$455$	9.99570	0.468606
$456$	0	0
$457$	6.96542	0.325829	0.162914	−	0.986640i	$-0.447911\pi$
$457$	6.96542	0.325829	0.162914	+	0.986640i	$0.447911\pi$
$458$	0	0
$459$	−6.58508	−0.307365
$460$	0	0
$461$	−25.4541	−1.18551	−0.592757	−	0.805381i	$-0.701961\pi$
$461$	−25.4541	−1.18551	−0.592757	+	0.805381i	$0.701961\pi$
$462$	0	0
$463$	−24.6650	−1.14628	−0.573139	−	0.819458i	$-0.694275\pi$
$463$	−24.6650	−1.14628	−0.573139	+	0.819458i	$0.694275\pi$
$464$	0	0
$465$	3.89716	0.180726
$466$	0	0
$467$	−16.1163	−0.745774	−0.372887	−	0.927877i	$-0.621632\pi$
$467$	−16.1163	−0.745774	−0.372887	+	0.927877i	$0.621632\pi$
$468$	0	0
$469$	25.2455	1.16573
$470$	0	0
$471$	−21.6857	−0.999225
$472$	0	0
$473$	−17.6903	−0.813403
$474$	0	0
$475$	−0.631945	−0.0289956
$476$	0	0
$477$	0.163207	0.00747275
$478$	0	0
$479$	6.76181	0.308955	0.154478	−	0.987996i	$-0.450631\pi$
$479$	6.76181	0.308955	0.154478	+	0.987996i	$0.450631\pi$
$480$	0	0
$481$	−23.2229	−1.05888
$482$	0	0
$483$	−2.53447	−0.115322
$484$	0	0
$485$	−20.4800	−0.929950
$486$	0	0
$487$	−20.7179	−0.938817	−0.469409	−	0.882981i	$-0.655533\pi$
$487$	−20.7179	−0.938817	−0.469409	+	0.882981i	$0.655533\pi$
$488$	0	0
$489$	3.37011	0.152402
$490$	0	0
$491$	32.1180	1.44946	0.724732	−	0.689031i	$-0.241964\pi$
$491$	32.1180	1.44946	0.724732	+	0.689031i	$0.241964\pi$
$492$	0	0
$493$	−6.58508	−0.296577
$494$	0	0
$495$	5.36284	0.241042
$496$	0	0
$497$	7.69352	0.345102
$498$	0	0
$499$	−38.2820	−1.71374	−0.856869	−	0.515535i	$-0.827593\pi$
$499$	−38.2820	−1.71374	−0.856869	+	0.515535i	$0.827593\pi$
$500$	0	0
$501$	4.89843	0.218846
$502$	0	0
$503$	−17.5144	−0.780927	−0.390463	−	0.920618i	$-0.627685\pi$
$503$	−17.5144	−0.780927	−0.390463	+	0.920618i	$0.627685\pi$
$504$	0	0
$505$	26.1139	1.16205
$506$	0	0
$507$	8.77058	0.389515
$508$	0	0
$509$	11.9264	0.528628	0.264314	−	0.964437i	$-0.414854\pi$
$509$	11.9264	0.528628	0.264314	+	0.964437i	$0.414854\pi$
$510$	0	0
$511$	−14.4667	−0.639970
$512$	0	0
$513$	−0.477899	−0.0210997
$514$	0	0
$515$	26.0294	1.14699
$516$	0	0
$517$	15.3790	0.676368
$518$	0	0
$519$	−24.1859	−1.06164
$520$	0	0
$521$	25.4432	1.11469	0.557344	−	0.830282i	$-0.311821\pi$
$521$	25.4432	1.11469	0.557344	+	0.830282i	$0.311821\pi$
$522$	0	0
$523$	3.35016	0.146492	0.0732460	−	0.997314i	$-0.476664\pi$
$523$	3.35016	0.146492	0.0732460	+	0.997314i	$0.476664\pi$
$524$	0	0
$525$	−3.35143	−0.146268
$526$	0	0
$527$	13.3821	0.582932
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	1.28356	0.0557018
$532$	0	0
$533$	−9.16432	−0.396951
$534$	0	0
$535$	−29.0331	−1.25521
$536$	0	0
$537$	−10.0886	−0.435356
$538$	0	0
$539$	1.61208	0.0694371
$540$	0	0
$541$	−39.8733	−1.71429	−0.857144	−	0.515077i	$-0.827763\pi$
$541$	−39.8733	−1.71429	−0.857144	+	0.515077i	$0.827763\pi$
$542$	0	0
$543$	−15.0355	−0.645233
$544$	0	0
$545$	−2.55481	−0.109436
$546$	0	0
$547$	15.0493	0.643463	0.321732	−	0.946831i	$-0.395735\pi$
$547$	15.0493	0.643463	0.321732	+	0.946831i	$0.395735\pi$
$548$	0	0
$549$	3.48409	0.148698
$550$	0	0
$551$	−0.477899	−0.0203592
$552$	0	0
$553$	33.4778	1.42362
$554$	0	0
$555$	−21.6552	−0.919213
$556$	0	0
$557$	10.8139	0.458198	0.229099	−	0.973403i	$-0.426422\pi$
$557$	10.8139	0.458198	0.229099	+	0.973403i	$0.426422\pi$
$558$	0	0
$559$	13.0097	0.550252
$560$	0	0
$561$	18.4149	0.777480
$562$	0	0
$563$	−13.3753	−0.563703	−0.281851	−	0.959458i	$-0.590948\pi$
$563$	−13.3753	−0.563703	−0.281851	+	0.959458i	$0.590948\pi$
$564$	0	0
$565$	23.2584	0.978488
$566$	0	0
$567$	−2.53447	−0.106438
$568$	0	0
$569$	−35.6365	−1.49396	−0.746981	−	0.664846i	$-0.768497\pi$
$569$	−35.6365	−1.49396	−0.746981	+	0.664846i	$0.768497\pi$
$570$	0	0
$571$	−24.3975	−1.02100	−0.510501	−	0.859877i	$-0.670540\pi$
$571$	−24.3975	−1.02100	−0.510501	+	0.859877i	$0.670540\pi$
$572$	0	0
$573$	−1.66689	−0.0696354
$574$	0	0
$575$	1.32234	0.0551454
$576$	0	0
$577$	−45.1035	−1.87768	−0.938842	−	0.344348i	$-0.888100\pi$
$577$	−45.1035	−1.87768	−0.938842	+	0.344348i	$0.888100\pi$
$578$	0	0
$579$	10.3313	0.429355
$580$	0	0
$581$	27.3268	1.13371
$582$	0	0
$583$	−0.456403	−0.0189023
$584$	0	0
$585$	−3.94390	−0.163060
$586$	0	0
$587$	45.6357	1.88359	0.941794	−	0.336190i	$-0.109138\pi$
$587$	45.6357	1.88359	0.941794	+	0.336190i	$0.109138\pi$
$588$	0	0
$589$	0.971176	0.0400166
$590$	0	0
$591$	−24.2977	−0.999475
$592$	0	0
$593$	21.9224	0.900246	0.450123	−	0.892967i	$-0.351380\pi$
$593$	21.9224	0.900246	0.450123	+	0.892967i	$0.351380\pi$
$594$	0	0
$595$	32.0062	1.31213
$596$	0	0
$597$	3.39029	0.138755
$598$	0	0
$599$	39.1293	1.59878	0.799390	−	0.600812i	$-0.205156\pi$
$599$	39.1293	1.59878	0.799390	+	0.600812i	$0.205156\pi$
$600$	0	0
$601$	17.4876	0.713335	0.356668	−	0.934231i	$-0.383913\pi$
$601$	17.4876	0.713335	0.356668	+	0.934231i	$0.383913\pi$
$602$	0	0
$603$	−9.96086	−0.405638
$604$	0	0
$605$	6.09795	0.247917
$606$	0	0
$607$	−6.29183	−0.255377	−0.127689	−	0.991814i	$-0.540756\pi$
$607$	−6.29183	−0.255377	−0.127689	+	0.991814i	$0.540756\pi$
$608$	0	0
$609$	−2.53447	−0.102702
$610$	0	0
$611$	−11.3099	−0.457550
$612$	0	0
$613$	38.9994	1.57517	0.787585	−	0.616206i	$-0.211331\pi$
$613$	38.9994	1.57517	0.787585	+	0.616206i	$0.211331\pi$
$614$	0	0
$615$	−8.54566	−0.344594
$616$	0	0
$617$	18.8487	0.758820	0.379410	−	0.925229i	$-0.376127\pi$
$617$	18.8487	0.758820	0.379410	+	0.925229i	$0.376127\pi$
$618$	0	0
$619$	−0.954896	−0.0383805	−0.0191903	−	0.999816i	$-0.506109\pi$
$619$	−0.954896	−0.0383805	−0.0191903	+	0.999816i	$0.506109\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−34.8152	−1.39484
$624$	0	0
$625$	−16.6397	−0.665588
$626$	0	0
$627$	1.33643	0.0533717
$628$	0	0
$629$	−74.3598	−2.96492
$630$	0	0
$631$	40.1646	1.59893	0.799465	−	0.600713i	$-0.205117\pi$
$631$	40.1646	1.59893	0.799465	+	0.600713i	$0.205117\pi$
$632$	0	0
$633$	−13.8877	−0.551988
$634$	0	0
$635$	31.5486	1.25197
$636$	0	0
$637$	−1.18554	−0.0469729
$638$	0	0
$639$	−3.03556	−0.120085
$640$	0	0
$641$	27.2751	1.07730	0.538652	−	0.842529i	$-0.318934\pi$
$641$	27.2751	1.07730	0.538652	+	0.842529i	$0.318934\pi$
$642$	0	0
$643$	−24.6982	−0.974003	−0.487001	−	0.873401i	$-0.661909\pi$
$643$	−24.6982	−0.974003	−0.487001	+	0.873401i	$0.661909\pi$
$644$	0	0
$645$	12.1314	0.477675
$646$	0	0
$647$	−38.4781	−1.51273	−0.756365	−	0.654150i	$-0.773027\pi$
$647$	−38.4781	−1.51273	−0.756365	+	0.654150i	$0.773027\pi$
$648$	0	0
$649$	−3.58943	−0.140897
$650$	0	0
$651$	5.15050	0.201864
$652$	0	0
$653$	37.0334	1.44923	0.724615	−	0.689154i	$-0.242018\pi$
$653$	37.0334	1.44923	0.724615	+	0.689154i	$0.242018\pi$
$654$	0	0
$655$	1.43721	0.0561566
$656$	0	0
$657$	5.70799	0.222690
$658$	0	0
$659$	16.6957	0.650374	0.325187	−	0.945650i	$-0.394573\pi$
$659$	16.6957	0.650374	0.325187	+	0.945650i	$0.394573\pi$
$660$	0	0
$661$	35.8018	1.39253	0.696264	−	0.717786i	$-0.254844\pi$
$661$	35.8018	1.39253	0.696264	+	0.717786i	$0.254844\pi$
$662$	0	0
$663$	−13.5426	−0.525951
$664$	0	0
$665$	2.32278	0.0900736
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−5.89758	−0.228014
$670$	0	0
$671$	−9.74315	−0.376130
$672$	0	0
$673$	27.8783	1.07463	0.537314	−	0.843382i	$-0.319439\pi$
$673$	27.8783	1.07463	0.537314	+	0.843382i	$0.319439\pi$
$674$	0	0
$675$	1.32234	0.0508969
$676$	0	0
$677$	17.5460	0.674347	0.337174	−	0.941442i	$-0.390529\pi$
$677$	17.5460	0.674347	0.337174	+	0.941442i	$0.390529\pi$
$678$	0	0
$679$	−27.0665	−1.03872
$680$	0	0
$681$	−4.55343	−0.174488
$682$	0	0
$683$	5.52296	0.211330	0.105665	−	0.994402i	$-0.466303\pi$
$683$	5.52296	0.211330	0.105665	+	0.994402i	$0.466303\pi$
$684$	0	0
$685$	−8.41478	−0.321512
$686$	0	0
$687$	−18.0839	−0.689944
$688$	0	0
$689$	0.335645	0.0127871
$690$	0	0
$691$	0.724362	0.0275560	0.0137780	−	0.999905i	$-0.495614\pi$
$691$	0.724362	0.0275560	0.0137780	+	0.999905i	$0.495614\pi$
$692$	0	0
$693$	7.08755	0.269234
$694$	0	0
$695$	−41.8521	−1.58754
$696$	0	0
$697$	−29.3441	−1.11149
$698$	0	0
$699$	−19.2686	−0.728805
$700$	0	0
$701$	45.9821	1.73672	0.868360	−	0.495934i	$-0.165174\pi$
$701$	45.9821	1.73672	0.868360	+	0.495934i	$0.165174\pi$
$702$	0	0
$703$	−5.39650	−0.203533
$704$	0	0
$705$	−10.5464	−0.397201
$706$	0	0
$707$	34.5122	1.29797
$708$	0	0
$709$	−23.0261	−0.864764	−0.432382	−	0.901690i	$-0.642327\pi$
$709$	−23.0261	−0.864764	−0.432382	+	0.901690i	$0.642327\pi$
$710$	0	0
$711$	−13.2090	−0.495376
$712$	0	0
$713$	−2.03218	−0.0761058
$714$	0	0
$715$	11.0290	0.412461
$716$	0	0
$717$	4.67472	0.174581
$718$	0	0
$719$	46.4713	1.73309	0.866544	−	0.499101i	$-0.166336\pi$
$719$	46.4713	1.73309	0.866544	+	0.499101i	$0.166336\pi$
$720$	0	0
$721$	34.4005	1.28114
$722$	0	0
$723$	25.0978	0.933398
$724$	0	0
$725$	1.32234	0.0491105
$726$	0	0
$727$	−19.0051	−0.704859	−0.352429	−	0.935838i	$-0.614644\pi$
$727$	−19.0051	−0.704859	−0.352429	+	0.935838i	$0.614644\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	41.6570	1.54074
$732$	0	0
$733$	−13.5044	−0.498795	−0.249398	−	0.968401i	$-0.580233\pi$
$733$	−13.5044	−0.498795	−0.249398	+	0.968401i	$0.580233\pi$
$734$	0	0
$735$	−1.10551	−0.0407773
$736$	0	0
$737$	27.8552	1.02606
$738$	0	0
$739$	11.6716	0.429349	0.214674	−	0.976686i	$-0.431131\pi$
$739$	11.6716	0.429349	0.214674	+	0.976686i	$0.431131\pi$
$740$	0	0
$741$	−0.982825	−0.0361050
$742$	0	0
$743$	19.8047	0.726565	0.363282	−	0.931679i	$-0.381656\pi$
$743$	19.8047	0.726565	0.363282	+	0.931679i	$0.381656\pi$
$744$	0	0
$745$	13.1125	0.480406
$746$	0	0
$747$	−10.7821	−0.394495
$748$	0	0
$749$	−38.3702	−1.40202
$750$	0	0
$751$	−8.05648	−0.293985	−0.146992	−	0.989138i	$-0.546959\pi$
$751$	−8.05648	−0.293985	−0.146992	+	0.989138i	$0.546959\pi$
$752$	0	0
$753$	−13.8863	−0.506045
$754$	0	0
$755$	4.62243	0.168227
$756$	0	0
$757$	9.64397	0.350516	0.175258	−	0.984523i	$-0.443924\pi$
$757$	9.64397	0.350516	0.175258	+	0.984523i	$0.443924\pi$
$758$	0	0
$759$	−2.79646	−0.101505
$760$	0	0
$761$	2.99588	0.108600	0.0543002	−	0.998525i	$-0.482707\pi$
$761$	2.99588	0.108600	0.0543002	+	0.998525i	$0.482707\pi$
$762$	0	0
$763$	−3.37644	−0.122235
$764$	0	0
$765$	−12.6284	−0.456579
$766$	0	0
$767$	2.63971	0.0953145
$768$	0	0
$769$	36.1077	1.30208	0.651038	−	0.759045i	$-0.274334\pi$
$769$	36.1077	1.30208	0.651038	+	0.759045i	$0.274334\pi$
$770$	0	0
$771$	−16.2069	−0.583677
$772$	0	0
$773$	37.2892	1.34120	0.670600	−	0.741820i	$-0.266037\pi$
$773$	37.2892	1.34120	0.670600	+	0.741820i	$0.266037\pi$
$774$	0	0
$775$	−2.68724	−0.0965284
$776$	0	0
$777$	−28.6196	−1.02672
$778$	0	0
$779$	−2.12959	−0.0763004
$780$	0	0
$781$	8.48882	0.303754
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−41.5872	−1.48431
$786$	0	0
$787$	−5.47927	−0.195315	−0.0976574	−	0.995220i	$-0.531135\pi$
$787$	−5.47927	−0.195315	−0.0976574	+	0.995220i	$0.531135\pi$
$788$	0	0
$789$	−5.73640	−0.204221
$790$	0	0
$791$	30.7384	1.09293
$792$	0	0
$793$	7.16524	0.254445
$794$	0	0
$795$	0.312986	0.0111005
$796$	0	0
$797$	5.66428	0.200639	0.100319	−	0.994955i	$-0.468014\pi$
$797$	5.66428	0.200639	0.100319	+	0.994955i	$0.468014\pi$
$798$	0	0
$799$	−36.2143	−1.28117
$800$	0	0
$801$	13.7367	0.485362
$802$	0	0
$803$	−15.9622	−0.563294
$804$	0	0
$805$	−4.86041	−0.171307
$806$	0	0
$807$	−4.81286	−0.169421
$808$	0	0
$809$	55.0633	1.93592	0.967961	−	0.251102i	$-0.0807930\pi$
$809$	55.0633	1.93592	0.967961	+	0.251102i	$0.0807930\pi$
$810$	0	0
$811$	4.01596	0.141020	0.0705098	−	0.997511i	$-0.477537\pi$
$811$	4.01596	0.141020	0.0705098	+	0.997511i	$0.477537\pi$
$812$	0	0
$813$	30.7494	1.07843
$814$	0	0
$815$	6.46294	0.226387
$816$	0	0
$817$	3.02317	0.105767
$818$	0	0
$819$	−5.21228	−0.182132
$820$	0	0
$821$	25.8082	0.900711	0.450355	−	0.892849i	$-0.351297\pi$
$821$	25.8082	0.900711	0.450355	+	0.892849i	$0.351297\pi$
$822$	0	0
$823$	−18.9256	−0.659703	−0.329852	−	0.944033i	$-0.606999\pi$
$823$	−18.9256	−0.659703	−0.329852	+	0.944033i	$0.606999\pi$
$824$	0	0
$825$	−3.69788	−0.128744
$826$	0	0
$827$	−2.92809	−0.101820	−0.0509098	−	0.998703i	$-0.516212\pi$
$827$	−2.92809	−0.101820	−0.0509098	+	0.998703i	$0.516212\pi$
$828$	0	0
$829$	−6.60528	−0.229411	−0.114705	−	0.993400i	$-0.536592\pi$
$829$	−6.60528	−0.229411	−0.114705	+	0.993400i	$0.536592\pi$
$830$	0	0
$831$	21.7128	0.753207
$832$	0	0
$833$	−3.79610	−0.131527
$834$	0	0
$835$	9.39383	0.325087
$836$	0	0
$837$	−2.03218	−0.0702424
$838$	0	0
$839$	−52.1021	−1.79877	−0.899383	−	0.437162i	$-0.855983\pi$
$839$	−52.1021	−1.79877	−0.899383	+	0.437162i	$0.855983\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−16.1648	−0.556747
$844$	0	0
$845$	16.8195	0.578609
$846$	0	0
$847$	8.05907	0.276913
$848$	0	0
$849$	−0.469095	−0.0160993
$850$	0	0
$851$	11.2922	0.387090
$852$	0	0
$853$	−12.2147	−0.418222	−0.209111	−	0.977892i	$-0.567057\pi$
$853$	−12.2147	−0.418222	−0.209111	+	0.977892i	$0.567057\pi$
$854$	0	0
$855$	−0.916477	−0.0313428
$856$	0	0
$857$	−28.2080	−0.963568	−0.481784	−	0.876290i	$-0.660011\pi$
$857$	−28.2080	−0.963568	−0.481784	+	0.876290i	$0.660011\pi$
$858$	0	0
$859$	−41.0290	−1.39989	−0.699946	−	0.714196i	$-0.746792\pi$
$859$	−41.0290	−1.39989	−0.699946	+	0.714196i	$0.746792\pi$
$860$	0	0
$861$	−11.2940	−0.384898
$862$	0	0
$863$	−6.71078	−0.228438	−0.114219	−	0.993456i	$-0.536436\pi$
$863$	−6.71078	−0.228438	−0.114219	+	0.993456i	$0.536436\pi$
$864$	0	0
$865$	−46.3819	−1.57703
$866$	0	0
$867$	−26.3633	−0.895345
$868$	0	0
$869$	36.9385	1.25305
$870$	0	0
$871$	−20.4851	−0.694110
$872$	0	0
$873$	10.6793	0.361441
$874$	0	0
$875$	−30.7291	−1.03883
$876$	0	0
$877$	3.17210	0.107114	0.0535571	−	0.998565i	$-0.482944\pi$
$877$	3.17210	0.107114	0.0535571	+	0.998565i	$0.482944\pi$
$878$	0	0
$879$	−19.0024	−0.640937
$880$	0	0
$881$	−31.7778	−1.07062	−0.535311	−	0.844655i	$-0.679806\pi$
$881$	−31.7778	−1.07062	−0.535311	+	0.844655i	$0.679806\pi$
$882$	0	0
$883$	32.0802	1.07959	0.539793	−	0.841798i	$-0.318503\pi$
$883$	32.0802	1.07959	0.539793	+	0.841798i	$0.318503\pi$
$884$	0	0
$885$	2.46151	0.0827428
$886$	0	0
$887$	19.5869	0.657664	0.328832	−	0.944388i	$-0.393345\pi$
$887$	19.5869	0.657664	0.328832	+	0.944388i	$0.393345\pi$
$888$	0	0
$889$	41.6947	1.39840
$890$	0	0
$891$	−2.79646	−0.0936851
$892$	0	0
$893$	−2.62818	−0.0879486
$894$	0	0
$895$	−19.3472	−0.646704
$896$	0	0
$897$	2.05656	0.0686664
$898$	0	0
$899$	−2.03218	−0.0677770
$900$	0	0
$901$	1.07473	0.0358046
$902$	0	0
$903$	16.0330	0.533544
$904$	0	0
$905$	−28.8338	−0.958469
$906$	0	0
$907$	−22.6224	−0.751164	−0.375582	−	0.926789i	$-0.622557\pi$
$907$	−22.6224	−0.751164	−0.375582	+	0.926789i	$0.622557\pi$
$908$	0	0
$909$	−13.6172	−0.451652
$910$	0	0
$911$	−24.1958	−0.801641	−0.400821	−	0.916157i	$-0.631275\pi$
$911$	−24.1958	−0.801641	−0.400821	+	0.916157i	$0.631275\pi$
$912$	0	0
$913$	30.1516	0.997874
$914$	0	0
$915$	6.68153	0.220884
$916$	0	0
$917$	1.89943	0.0627246
$918$	0	0
$919$	44.5098	1.46824	0.734121	−	0.679019i	$-0.237595\pi$
$919$	44.5098	1.46824	0.734121	+	0.679019i	$0.237595\pi$
$920$	0	0
$921$	30.3246	0.999228
$922$	0	0
$923$	−6.24279	−0.205484
$924$	0	0
$925$	14.9321	0.490964
$926$	0	0
$927$	−13.5731	−0.445798
$928$	0	0
$929$	−56.2137	−1.84431	−0.922157	−	0.386817i	$-0.873575\pi$
$929$	−56.2137	−1.84431	−0.922157	+	0.386817i	$0.873575\pi$
$930$	0	0
$931$	−0.275494	−0.00902896
$932$	0	0
$933$	−23.9215	−0.783155
$934$	0	0
$935$	35.3148	1.15492
$936$	0	0
$937$	32.9656	1.07694	0.538469	−	0.842645i	$-0.319003\pi$
$937$	32.9656	1.07694	0.538469	+	0.842645i	$0.319003\pi$
$938$	0	0
$939$	−25.7932	−0.841730
$940$	0	0
$941$	−30.0251	−0.978792	−0.489396	−	0.872062i	$-0.662783\pi$
$941$	−30.0251	−0.978792	−0.489396	+	0.872062i	$0.662783\pi$
$942$	0	0
$943$	4.45615	0.145112
$944$	0	0
$945$	−4.86041	−0.158109
$946$	0	0
$947$	13.6276	0.442837	0.221418	−	0.975179i	$-0.428931\pi$
$947$	13.6276	0.442837	0.221418	+	0.975179i	$0.428931\pi$
$948$	0	0
$949$	11.7388	0.381058
$950$	0	0
$951$	25.4602	0.825604
$952$	0	0
$953$	−56.3809	−1.82636	−0.913178	−	0.407561i	$-0.866379\pi$
$953$	−56.3809	−1.82636	−0.913178	+	0.407561i	$0.866379\pi$
$954$	0	0
$955$	−3.19664	−0.103441
$956$	0	0
$957$	−2.79646	−0.0903968
$958$	0	0
$959$	−11.1210	−0.359116
$960$	0	0
$961$	−26.8702	−0.866782
$962$	0	0
$963$	15.1393	0.487858
$964$	0	0
$965$	19.8126	0.637790
$966$	0	0
$967$	−9.25351	−0.297573	−0.148786	−	0.988869i	$-0.547537\pi$
$967$	−9.25351	−0.297573	−0.148786	+	0.988869i	$0.547537\pi$
$968$	0	0
$969$	−3.14700	−0.101096
$970$	0	0
$971$	−33.3188	−1.06925	−0.534625	−	0.845089i	$-0.679547\pi$
$971$	−33.3188	−1.06925	−0.534625	+	0.845089i	$0.679547\pi$
$972$	0	0
$973$	−55.3119	−1.77322
$974$	0	0
$975$	2.71947	0.0870927
$976$	0	0
$977$	−42.5914	−1.36262	−0.681310	−	0.731995i	$-0.738589\pi$
$977$	−42.5914	−1.36262	−0.681310	+	0.731995i	$0.738589\pi$
$978$	0	0
$979$	−38.4142	−1.22772
$980$	0	0
$981$	1.33221	0.0425341
$982$	0	0
$983$	35.4101	1.12941	0.564703	−	0.825294i	$-0.308991\pi$
$983$	35.4101	1.12941	0.564703	+	0.825294i	$0.308991\pi$
$984$	0	0
$985$	−46.5963	−1.48468
$986$	0	0
$987$	−13.9382	−0.443657
$988$	0	0
$989$	−6.32597	−0.201154
$990$	0	0
$991$	−54.1616	−1.72050	−0.860251	−	0.509872i	$-0.829693\pi$
$991$	−54.1616	−1.72050	−0.860251	+	0.509872i	$0.829693\pi$
$992$	0	0
$993$	6.99611	0.222015
$994$	0	0
$995$	6.50163	0.206116
$996$	0	0
$997$	23.6836	0.750067	0.375033	−	0.927011i	$-0.377631\pi$
$997$	23.6836	0.750067	0.375033	+	0.927011i	$0.377631\pi$
$998$	0	0
$999$	11.2922	0.357268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.h.1.3	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.h.1.3	✓	13	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.91772 q^{5} -2.53447 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.91772 q^{5} -2.53447 q^{7} +1.00000 q^{9} -2.79646 q^{11} +2.05656 q^{13} +1.91772 q^{15} +6.58508 q^{17} +0.477899 q^{19} +2.53447 q^{21} -1.00000 q^{23} -1.32234 q^{25} -1.00000 q^{27} -1.00000 q^{29} +2.03218 q^{31} +2.79646 q^{33} +4.86041 q^{35} -11.2922 q^{37} -2.05656 q^{39} -4.45615 q^{41} +6.32597 q^{43} -1.91772 q^{45} -5.49945 q^{47} -0.576470 q^{49} -6.58508 q^{51} +0.163207 q^{53} +5.36284 q^{55} -0.477899 q^{57} +1.28356 q^{59} +3.48409 q^{61} -2.53447 q^{63} -3.94390 q^{65} -9.96086 q^{67} +1.00000 q^{69} -3.03556 q^{71} +5.70799 q^{73} +1.32234 q^{75} +7.08755 q^{77} -13.2090 q^{79} +1.00000 q^{81} -10.7821 q^{83} -12.6284 q^{85} +1.00000 q^{87} +13.7367 q^{89} -5.21228 q^{91} -2.03218 q^{93} -0.916477 q^{95} +10.6793 q^{97} -2.79646 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Introduction

L-functions

Modular forms

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Database

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