LMFDB - Embedded newform 3584.2.b.g.1793.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} + 12x^{8} - 12x^{7} + 3x^{6} - 48x^{5} + 158x^{4} - 80x^{3} - 66x^{2} + 72x + 36 $ x^10 - 4x^9 + 12x^8 - 12x^7 + 3x^6 - 48x^5 + 158x^4 - 80x^3 - 66x^2 + 72*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.4
Root			$-0.484731 - 0.120245i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.g.1793.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.783186i q^{3} +3.56843i q^{5} -1.00000 q^{7} +2.38662 q^{9} +O(q^{10})$	$q-0.783186i q^{3} +3.56843i q^{5} -1.00000 q^{7} +2.38662 q^{9} +2.51888i q^{11} -6.52509i q^{13} +2.79474 q^{15} -1.93892 q^{17} -6.56828i q^{19} +0.783186i q^{21} -2.76749 q^{23} -7.73367 q^{25} -4.21872i q^{27} -2.34745i q^{29} -7.50115 q^{31} +1.97275 q^{33} -3.56843i q^{35} +6.05146i q^{37} -5.11036 q^{39} +7.71634 q^{41} -10.9178i q^{43} +8.51648i q^{45} -8.18136 q^{47} +1.00000 q^{49} +1.51854i q^{51} -2.04735i q^{53} -8.98842 q^{55} -5.14418 q^{57} +0.911427i q^{59} -2.08843i q^{61} -2.38662 q^{63} +23.2843 q^{65} -1.87592i q^{67} +2.16746i q^{69} +12.3355 q^{71} -14.1203 q^{73} +6.05690i q^{75} -2.51888i q^{77} -8.33547 q^{79} +3.85582 q^{81} -6.44004i q^{83} -6.91891i q^{85} -1.83849 q^{87} +1.56223 q^{89} +6.52509i q^{91} +5.87480i q^{93} +23.4384 q^{95} +18.9273 q^{97} +6.01160i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{7} - 22 q^{9}+O(q^{10}) $ 10 * q - 10 * q^7 - 22 * q^9	$ 10 q - 10 q^{7} - 22 q^{9} + 16 q^{15} + 16 q^{17} - 8 q^{23} - 30 q^{25} - 8 q^{31} + 12 q^{33} - 20 q^{41} - 24 q^{47} + 10 q^{49} + 32 q^{55} - 28 q^{57} + 22 q^{63} + 32 q^{65} - 48 q^{73} + 40 q^{79} + 62 q^{81} + 8 q^{87} - 16 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) $ 10 * q - 10 * q^7 - 22 * q^9 + 16 * q^15 + 16 * q^17 - 8 * q^23 - 30 * q^25 - 8 * q^31 + 12 * q^33 - 20 * q^41 - 24 * q^47 + 10 * q^49 + 32 * q^55 - 28 * q^57 + 22 * q^63 + 32 * q^65 - 48 * q^73 + 40 * q^79 + 62 * q^81 + 8 * q^87 - 16 * q^89 - 32 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 0.783186i	− 0.452173i	−0.974107	−	0.226086i	$-0.927407\pi$
$3$	− 0.783186i	− 0.452173i	0.974107	−	0.226086i	$-0.0725931\pi$
$4$	0	0
$5$	3.56843i	1.59585i	0.602758	+	0.797924i	$0.294069\pi$
$5$	3.56843i	1.59585i	−0.602758	+	0.797924i	$0.705931\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.38662	0.795540
$10$	0	0
$11$	2.51888i	0.759470i	0.925095	+	0.379735i	$0.123985\pi$
$11$	2.51888i	0.759470i	−0.925095	+	0.379735i	$0.876015\pi$
$12$	0	0
$13$	− 6.52509i	− 1.80974i	−0.425693	−	0.904868i	$-0.639970\pi$
$13$	− 6.52509i	− 1.80974i	0.425693	−	0.904868i	$-0.360030\pi$
$14$	0	0
$15$	2.79474	0.721599
$16$	0	0
$17$	−1.93892	−0.470258	−0.235129	−	0.971964i	$-0.575551\pi$
$17$	−1.93892	−0.470258	−0.235129	+	0.971964i	$0.575551\pi$
$18$	0	0
$19$	− 6.56828i	− 1.50687i	−0.657524	−	0.753434i	$-0.728396\pi$
$19$	− 6.56828i	− 1.50687i	0.657524	−	0.753434i	$-0.271604\pi$
$20$	0	0
$21$	0.783186i	0.170905i
$22$	0	0
$23$	−2.76749	−0.577061	−0.288531	−	0.957471i	$-0.593167\pi$
$23$	−2.76749	−0.577061	−0.288531	+	0.957471i	$0.593167\pi$
$24$	0	0
$25$	−7.73367	−1.54673
$26$	0	0
$27$	− 4.21872i	− 0.811894i
$28$	0	0
$29$	− 2.34745i	− 0.435910i	−0.975959	−	0.217955i	$-0.930061\pi$
$29$	− 2.34745i	− 0.435910i	0.975959	−	0.217955i	$-0.0699386\pi$
$30$	0	0
$31$	−7.50115	−1.34725	−0.673623	−	0.739075i	$-0.735263\pi$
$31$	−7.50115	−1.34725	−0.673623	+	0.739075i	$0.735263\pi$
$32$	0	0
$33$	1.97275	0.343411
$34$	0	0
$35$	− 3.56843i	− 0.603174i
$36$	0	0
$37$	6.05146i	0.994854i	0.867506	+	0.497427i	$0.165722\pi$
$37$	6.05146i	0.994854i	−0.867506	+	0.497427i	$0.834278\pi$
$38$	0	0
$39$	−5.11036	−0.818313
$40$	0	0
$41$	7.71634	1.20509	0.602545	−	0.798085i	$-0.294154\pi$
$41$	7.71634	1.20509	0.602545	+	0.798085i	$0.294154\pi$
$42$	0	0
$43$	− 10.9178i	− 1.66495i	−0.554065	−	0.832473i	$-0.686924\pi$
$43$	− 10.9178i	− 1.66495i	0.554065	−	0.832473i	$-0.313076\pi$
$44$	0	0
$45$	8.51648i	1.26956i
$46$	0	0
$47$	−8.18136	−1.19337	−0.596687	−	0.802474i	$-0.703517\pi$
$47$	−8.18136	−1.19337	−0.596687	+	0.802474i	$0.703517\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.51854i	0.212638i
$52$	0	0
$53$	− 2.04735i	− 0.281225i	−0.990065	−	0.140613i	$-0.955093\pi$
$53$	− 2.04735i	− 0.281225i	0.990065	−	0.140613i	$-0.0449072\pi$
$54$	0	0
$55$	−8.98842	−1.21200
$56$	0	0
$57$	−5.14418	−0.681364
$58$	0	0
$59$	0.911427i	0.118658i	0.998238	+	0.0593288i	$0.0188961\pi$
$59$	0.911427i	0.118658i	−0.998238	+	0.0593288i	$0.981104\pi$
$60$	0	0
$61$	− 2.08843i	− 0.267396i	−0.991022	−	0.133698i	$-0.957315\pi$
$61$	− 2.08843i	− 0.267396i	0.991022	−	0.133698i	$-0.0426852\pi$
$62$	0	0
$63$	−2.38662	−0.300686
$64$	0	0
$65$	23.2843	2.88806
$66$	0	0
$67$	− 1.87592i	− 0.229180i	−0.993413	−	0.114590i	$-0.963445\pi$
$67$	− 1.87592i	− 0.229180i	0.993413	−	0.114590i	$-0.0365555\pi$
$68$	0	0
$69$	2.16746i	0.260931i
$70$	0	0
$71$	12.3355	1.46395	0.731975	−	0.681331i	$-0.238599\pi$
$71$	12.3355	1.46395	0.731975	+	0.681331i	$0.238599\pi$
$72$	0	0
$73$	−14.1203	−1.65265	−0.826327	−	0.563190i	$-0.809574\pi$
$73$	−14.1203	−1.65265	−0.826327	+	0.563190i	$0.809574\pi$
$74$	0	0
$75$	6.05690i	0.699390i
$76$	0	0
$77$	− 2.51888i	− 0.287053i
$78$	0	0
$79$	−8.33547	−0.937814	−0.468907	−	0.883248i	$-0.655352\pi$
$79$	−8.33547	−0.937814	−0.468907	+	0.883248i	$0.655352\pi$
$80$	0	0
$81$	3.85582	0.428424
$82$	0	0
$83$	− 6.44004i	− 0.706886i	−0.935456	−	0.353443i	$-0.885011\pi$
$83$	− 6.44004i	− 0.706886i	0.935456	−	0.353443i	$-0.114989\pi$
$84$	0	0
$85$	− 6.91891i	− 0.750461i
$86$	0	0
$87$	−1.83849	−0.197107
$88$	0	0
$89$	1.56223	0.165596	0.0827980	−	0.996566i	$-0.473614\pi$
$89$	1.56223	0.165596	0.0827980	+	0.996566i	$0.473614\pi$
$90$	0	0
$91$	6.52509i	0.684016i
$92$	0	0
$93$	5.87480i	0.609188i
$94$	0	0
$95$	23.4384	2.40473
$96$	0	0
$97$	18.9273	1.92178	0.960891	−	0.276928i	$-0.0893165\pi$
$97$	18.9273	1.92178	0.960891	+	0.276928i	$0.0893165\pi$
$98$	0	0
$99$	6.01160i	0.604189i
$100$	0	0
$101$	− 4.33962i	− 0.431808i	−0.976415	−	0.215904i	$-0.930730\pi$
$101$	− 4.33962i	− 0.431808i	0.976415	−	0.215904i	$-0.0692699\pi$
$102$	0	0
$103$	−1.28597	−0.126710	−0.0633552	−	0.997991i	$-0.520180\pi$
$103$	−1.28597	−0.126710	−0.0633552	+	0.997991i	$0.520180\pi$
$104$	0	0
$105$	−2.79474	−0.272739
$106$	0	0
$107$	− 16.1882i	− 1.56497i	−0.622670	−	0.782485i	$-0.713952\pi$
$107$	− 16.1882i	− 1.56497i	0.622670	−	0.782485i	$-0.286048\pi$
$108$	0	0
$109$	3.27087i	0.313292i	0.987655	+	0.156646i	$0.0500682\pi$
$109$	3.27087i	0.313292i	−0.987655	+	0.156646i	$0.949932\pi$
$110$	0	0
$111$	4.73942	0.449846
$112$	0	0
$113$	−10.6115	−0.998247	−0.499124	−	0.866531i	$-0.666345\pi$
$113$	−10.6115	−0.998247	−0.499124	+	0.866531i	$0.666345\pi$
$114$	0	0
$115$	− 9.87558i	− 0.920902i
$116$	0	0
$117$	− 15.5729i	− 1.43972i
$118$	0	0
$119$	1.93892	0.177741
$120$	0	0
$121$	4.65526	0.423206
$122$	0	0
$123$	− 6.04333i	− 0.544908i
$124$	0	0
$125$	− 9.75488i	− 0.872503i
$126$	0	0
$127$	−4.00575	−0.355453	−0.177726	−	0.984080i	$-0.556874\pi$
$127$	−4.00575	−0.355453	−0.177726	+	0.984080i	$0.556874\pi$
$128$	0	0
$129$	−8.55065	−0.752843
$130$	0	0
$131$	− 22.6188i	− 1.97621i	−0.153782	−	0.988105i	$-0.549145\pi$
$131$	− 22.6188i	− 1.97621i	0.153782	−	0.988105i	$-0.450855\pi$
$132$	0	0
$133$	6.56828i	0.569542i
$134$	0	0
$135$	15.0542	1.29566
$136$	0	0
$137$	−2.65295	−0.226657	−0.113329	−	0.993558i	$-0.536151\pi$
$137$	−2.65295	−0.226657	−0.113329	+	0.993558i	$0.536151\pi$
$138$	0	0
$139$	12.0524i	1.02227i	0.859500	+	0.511135i	$0.170775\pi$
$139$	12.0524i	1.02227i	−0.859500	+	0.511135i	$0.829225\pi$
$140$	0	0
$141$	6.40752i	0.539611i
$142$	0	0
$143$	16.4359	1.37444
$144$	0	0
$145$	8.37670	0.695647
$146$	0	0
$147$	− 0.783186i	− 0.0645961i
$148$	0	0
$149$	− 0.867450i	− 0.0710643i	−0.999369	−	0.0355321i	$-0.988687\pi$
$149$	− 0.867450i	− 0.0710643i	0.999369	−	0.0355321i	$-0.0113126\pi$
$150$	0	0
$151$	16.6652	1.35619	0.678097	−	0.734973i	$-0.262805\pi$
$151$	16.6652	1.35619	0.678097	+	0.734973i	$0.262805\pi$
$152$	0	0
$153$	−4.62748	−0.374109
$154$	0	0
$155$	− 26.7673i	− 2.15000i
$156$	0	0
$157$	− 10.9584i	− 0.874578i	−0.899321	−	0.437289i	$-0.855939\pi$
$157$	− 10.9584i	− 0.874578i	0.899321	−	0.437289i	$-0.144061\pi$
$158$	0	0
$159$	−1.60346	−0.127162
$160$	0	0
$161$	2.76749	0.218109
$162$	0	0
$163$	− 3.05160i	− 0.239020i	−0.992833	−	0.119510i	$-0.961868\pi$
$163$	− 3.05160i	− 0.239020i	0.992833	−	0.119510i	$-0.0381324\pi$
$164$	0	0
$165$	7.03961i	0.548033i
$166$	0	0
$167$	14.8209	1.14688	0.573440	−	0.819248i	$-0.305609\pi$
$167$	14.8209	1.14688	0.573440	+	0.819248i	$0.305609\pi$
$168$	0	0
$169$	−29.5769	−2.27514
$170$	0	0
$171$	− 15.6760i	− 1.19877i
$172$	0	0
$173$	4.00098i	0.304189i	0.988366	+	0.152095i	$0.0486018\pi$
$173$	4.00098i	0.304189i	−0.988366	+	0.152095i	$0.951398\pi$
$174$	0	0
$175$	7.73367	0.584610
$176$	0	0
$177$	0.713817	0.0536537
$178$	0	0
$179$	− 3.78093i	− 0.282600i	−0.989967	−	0.141300i	$-0.954872\pi$
$179$	− 3.78093i	− 0.282600i	0.989967	−	0.141300i	$-0.0451282\pi$
$180$	0	0
$181$	− 21.3282i	− 1.58531i	−0.609669	−	0.792656i	$-0.708698\pi$
$181$	− 21.3282i	− 1.58531i	0.609669	−	0.792656i	$-0.291302\pi$
$182$	0	0
$183$	−1.63563	−0.120909
$184$	0	0
$185$	−21.5942	−1.58764
$186$	0	0
$187$	− 4.88391i	− 0.357147i
$188$	0	0
$189$	4.21872i	0.306867i
$190$	0	0
$191$	20.3082	1.46945	0.734726	−	0.678365i	$-0.237311\pi$
$191$	20.3082	1.46945	0.734726	+	0.678365i	$0.237311\pi$
$192$	0	0
$193$	−11.2843	−0.812263	−0.406132	−	0.913815i	$-0.633123\pi$
$193$	−11.2843	−0.812263	−0.406132	+	0.913815i	$0.633123\pi$
$194$	0	0
$195$	− 18.2359i	− 1.30590i
$196$	0	0
$197$	21.4452i	1.52791i	0.645272	+	0.763953i	$0.276744\pi$
$197$	21.4452i	1.52791i	−0.645272	+	0.763953i	$0.723256\pi$
$198$	0	0
$199$	−23.8094	−1.68780	−0.843901	−	0.536499i	$-0.819746\pi$
$199$	−23.8094	−1.68780	−0.843901	+	0.536499i	$0.819746\pi$
$200$	0	0
$201$	−1.46920	−0.103629
$202$	0	0
$203$	2.34745i	0.164759i
$204$	0	0
$205$	27.5352i	1.92314i
$206$	0	0
$207$	−6.60494	−0.459075
$208$	0	0
$209$	16.5447	1.14442
$210$	0	0
$211$	21.2738i	1.46455i	0.681011	+	0.732273i	$0.261540\pi$
$211$	21.2738i	1.46455i	−0.681011	+	0.732273i	$0.738460\pi$
$212$	0	0
$213$	− 9.66096i	− 0.661958i
$214$	0	0
$215$	38.9593	2.65700
$216$	0	0
$217$	7.50115	0.509211
$218$	0	0
$219$	11.0588i	0.747285i
$220$	0	0
$221$	12.6517i	0.851043i
$222$	0	0
$223$	−11.8047	−0.790499	−0.395249	−	0.918574i	$-0.629342\pi$
$223$	−11.8047	−0.790499	−0.395249	+	0.918574i	$0.629342\pi$
$224$	0	0
$225$	−18.4573	−1.23049
$226$	0	0
$227$	18.4000i	1.22125i	0.791919	+	0.610626i	$0.209082\pi$
$227$	18.4000i	1.22125i	−0.791919	+	0.610626i	$0.790918\pi$
$228$	0	0
$229$	− 9.65784i	− 0.638208i	−0.947720	−	0.319104i	$-0.896618\pi$
$229$	− 9.65784i	− 0.638208i	0.947720	−	0.319104i	$-0.103382\pi$
$230$	0	0
$231$	−1.97275	−0.129797
$232$	0	0
$233$	4.53080	0.296823	0.148411	−	0.988926i	$-0.452584\pi$
$233$	4.53080	0.296823	0.148411	+	0.988926i	$0.452584\pi$
$234$	0	0
$235$	− 29.1946i	− 1.90444i
$236$	0	0
$237$	6.52822i	0.424053i
$238$	0	0
$239$	−19.2524	−1.24533	−0.622666	−	0.782488i	$-0.713950\pi$
$239$	−19.2524	−1.24533	−0.622666	+	0.782488i	$0.713950\pi$
$240$	0	0
$241$	19.9588	1.28566	0.642829	−	0.766010i	$-0.277761\pi$
$241$	19.9588	1.28566	0.642829	+	0.766010i	$0.277761\pi$
$242$	0	0
$243$	− 15.6760i	− 1.00562i
$244$	0	0
$245$	3.56843i	0.227978i
$246$	0	0
$247$	−42.8587	−2.72703
$248$	0	0
$249$	−5.04375	−0.319635
$250$	0	0
$251$	− 9.57278i	− 0.604229i	−0.953272	−	0.302114i	$-0.902308\pi$
$251$	− 9.57278i	− 0.604229i	0.953272	−	0.302114i	$-0.0976924\pi$
$252$	0	0
$253$	− 6.97096i	− 0.438261i
$254$	0	0
$255$	−5.41879	−0.339338
$256$	0	0
$257$	12.5779	0.784588	0.392294	−	0.919840i	$-0.371682\pi$
$257$	12.5779	0.784588	0.392294	+	0.919840i	$0.371682\pi$
$258$	0	0
$259$	− 6.05146i	− 0.376019i
$260$	0	0
$261$	− 5.60247i	− 0.346784i
$262$	0	0
$263$	−15.9051	−0.980751	−0.490375	−	0.871511i	$-0.663140\pi$
$263$	−15.9051	−0.980751	−0.490375	+	0.871511i	$0.663140\pi$
$264$	0	0
$265$	7.30582	0.448793
$266$	0	0
$267$	− 1.22352i	− 0.0748779i
$268$	0	0
$269$	− 4.00098i	− 0.243944i	−0.992534	−	0.121972i	$-0.961078\pi$
$269$	− 4.00098i	− 0.243944i	0.992534	−	0.121972i	$-0.0389219\pi$
$270$	0	0
$271$	3.80466	0.231117	0.115558	−	0.993301i	$-0.463134\pi$
$271$	3.80466	0.231117	0.115558	+	0.993301i	$0.463134\pi$
$272$	0	0
$273$	5.11036	0.309293
$274$	0	0
$275$	− 19.4801i	− 1.17470i
$276$	0	0
$277$	− 12.6170i	− 0.758085i	−0.925379	−	0.379042i	$-0.876253\pi$
$277$	− 12.6170i	− 0.758085i	0.925379	−	0.379042i	$-0.123747\pi$
$278$	0	0
$279$	−17.9024	−1.07179
$280$	0	0
$281$	−13.1161	−0.782442	−0.391221	−	0.920297i	$-0.627947\pi$
$281$	−13.1161	−0.782442	−0.391221	+	0.920297i	$0.627947\pi$
$282$	0	0
$283$	− 12.4251i	− 0.738593i	−0.929312	−	0.369297i	$-0.879599\pi$
$283$	− 12.4251i	− 0.738593i	0.929312	−	0.369297i	$-0.120401\pi$
$284$	0	0
$285$	− 18.3566i	− 1.08735i
$286$	0	0
$287$	−7.71634	−0.455481
$288$	0	0
$289$	−13.2406	−0.778857
$290$	0	0
$291$	− 14.8236i	− 0.868977i
$292$	0	0
$293$	19.8515i	1.15974i	0.814709	+	0.579869i	$0.196896\pi$
$293$	19.8515i	1.15974i	−0.814709	+	0.579869i	$0.803104\pi$
$294$	0	0
$295$	−3.25236	−0.189360
$296$	0	0
$297$	10.6264	0.616609
$298$	0	0
$299$	18.0581i	1.04433i
$300$	0	0
$301$	10.9178i	0.629291i
$302$	0	0
$303$	−3.39873	−0.195252
$304$	0	0
$305$	7.45240	0.426723
$306$	0	0
$307$	− 2.17681i	− 0.124237i	−0.998069	−	0.0621186i	$-0.980214\pi$
$307$	− 2.17681i	− 0.124237i	0.998069	−	0.0621186i	$-0.0197857\pi$
$308$	0	0
$309$	1.00715i	0.0572950i
$310$	0	0
$311$	−3.14523	−0.178350	−0.0891748	−	0.996016i	$-0.528423\pi$
$311$	−3.14523	−0.178350	−0.0891748	+	0.996016i	$0.528423\pi$
$312$	0	0
$313$	12.3476	0.697926	0.348963	−	0.937136i	$-0.386534\pi$
$313$	12.3476	0.697926	0.348963	+	0.937136i	$0.386534\pi$
$314$	0	0
$315$	− 8.51648i	− 0.479849i
$316$	0	0
$317$	− 17.6070i	− 0.988906i	−0.869204	−	0.494453i	$-0.835368\pi$
$317$	− 17.6070i	− 0.988906i	0.869204	−	0.494453i	$-0.164632\pi$
$318$	0	0
$319$	5.91293	0.331061
$320$	0	0
$321$	−12.6783	−0.707636
$322$	0	0
$323$	12.7354i	0.708617i
$324$	0	0
$325$	50.4629i	2.79918i
$326$	0	0
$327$	2.56170	0.141662
$328$	0	0
$329$	8.18136	0.451053
$330$	0	0
$331$	22.0587i	1.21246i	0.795290	+	0.606229i	$0.207319\pi$
$331$	22.0587i	1.21246i	−0.795290	+	0.606229i	$0.792681\pi$
$332$	0	0
$333$	14.4425i	0.791446i
$334$	0	0
$335$	6.69409	0.365737
$336$	0	0
$337$	14.0594	0.765866	0.382933	−	0.923776i	$-0.374914\pi$
$337$	14.0594	0.765866	0.382933	+	0.923776i	$0.374914\pi$
$338$	0	0
$339$	8.31079i	0.451380i
$340$	0	0
$341$	− 18.8945i	− 1.02319i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−7.73441	−0.416407
$346$	0	0
$347$	8.30730i	0.445959i	0.974823	+	0.222980i	$0.0715783\pi$
$347$	8.30730i	0.445959i	−0.974823	+	0.222980i	$0.928422\pi$
$348$	0	0
$349$	− 5.30333i	− 0.283881i	−0.989875	−	0.141940i	$-0.954666\pi$
$349$	− 5.30333i	− 0.283881i	0.989875	−	0.141940i	$-0.0453341\pi$
$350$	0	0
$351$	−27.5276	−1.46931
$352$	0	0
$353$	−21.6280	−1.15114	−0.575571	−	0.817752i	$-0.695220\pi$
$353$	−21.6280	−1.15114	−0.575571	+	0.817752i	$0.695220\pi$
$354$	0	0
$355$	44.0182i	2.33624i
$356$	0	0
$357$	− 1.51854i	− 0.0803696i
$358$	0	0
$359$	26.8859	1.41898	0.709492	−	0.704713i	$-0.248924\pi$
$359$	26.8859	1.41898	0.709492	+	0.704713i	$0.248924\pi$
$360$	0	0
$361$	−24.1423	−1.27065
$362$	0	0
$363$	− 3.64594i	− 0.191362i
$364$	0	0
$365$	− 50.3872i	− 2.63739i
$366$	0	0
$367$	−2.41052	−0.125828	−0.0629140	−	0.998019i	$-0.520039\pi$
$367$	−2.41052	−0.125828	−0.0629140	+	0.998019i	$0.520039\pi$
$368$	0	0
$369$	18.4160	0.958697
$370$	0	0
$371$	2.04735i	0.106293i
$372$	0	0
$373$	− 11.0549i	− 0.572401i	−0.958170	−	0.286200i	$-0.907608\pi$
$373$	− 11.0549i	− 0.572401i	0.958170	−	0.286200i	$-0.0923923\pi$
$374$	0	0
$375$	−7.63989	−0.394522
$376$	0	0
$377$	−15.3173	−0.788882
$378$	0	0
$379$	− 7.96444i	− 0.409106i	−0.978855	−	0.204553i	$-0.934426\pi$
$379$	− 7.96444i	− 0.409106i	0.978855	−	0.204553i	$-0.0655741\pi$
$380$	0	0
$381$	3.13725i	0.160726i
$382$	0	0
$383$	−21.8440	−1.11618	−0.558089	−	0.829781i	$-0.688465\pi$
$383$	−21.8440	−1.11618	−0.558089	+	0.829781i	$0.688465\pi$
$384$	0	0
$385$	8.98842	0.458093
$386$	0	0
$387$	− 26.0566i	− 1.32453i
$388$	0	0
$389$	26.6875i	1.35311i	0.736392	+	0.676555i	$0.236528\pi$
$389$	26.6875i	1.35311i	−0.736392	+	0.676555i	$0.763472\pi$
$390$	0	0
$391$	5.36595	0.271368
$392$	0	0
$393$	−17.7147	−0.893588
$394$	0	0
$395$	− 29.7445i	− 1.49661i
$396$	0	0
$397$	− 22.8082i	− 1.14471i	−0.820006	−	0.572355i	$-0.806030\pi$
$397$	− 22.8082i	− 1.14471i	0.820006	−	0.572355i	$-0.193970\pi$
$398$	0	0
$399$	5.14418	0.257531
$400$	0	0
$401$	−0.202861	−0.0101304	−0.00506520	−	0.999987i	$-0.501612\pi$
$401$	−0.202861	−0.0101304	−0.00506520	+	0.999987i	$0.501612\pi$
$402$	0	0
$403$	48.9457i	2.43816i
$404$	0	0
$405$	13.7592i	0.683700i
$406$	0	0
$407$	−15.2429	−0.755561
$408$	0	0
$409$	20.5826	1.01774	0.508872	−	0.860842i	$-0.330063\pi$
$409$	20.5826	1.01774	0.508872	+	0.860842i	$0.330063\pi$
$410$	0	0
$411$	2.07776i	0.102488i
$412$	0	0
$413$	− 0.911427i	− 0.0448484i
$414$	0	0
$415$	22.9808	1.12808
$416$	0	0
$417$	9.43926	0.462243
$418$	0	0
$419$	− 11.4778i	− 0.560727i	−0.959894	−	0.280363i	$-0.909545\pi$
$419$	− 11.4778i	− 0.560727i	0.959894	−	0.280363i	$-0.0904550\pi$
$420$	0	0
$421$	− 12.6170i	− 0.614917i	−0.951561	−	0.307459i	$-0.900521\pi$
$421$	− 12.6170i	− 0.614917i	0.951561	−	0.307459i	$-0.0994785\pi$
$422$	0	0
$423$	−19.5258	−0.949377
$424$	0	0
$425$	14.9950	0.727364
$426$	0	0
$427$	2.08843i	0.101066i
$428$	0	0
$429$	− 12.8724i	− 0.621484i
$430$	0	0
$431$	16.7873	0.808618	0.404309	−	0.914623i	$-0.367512\pi$
$431$	16.7873	0.808618	0.404309	+	0.914623i	$0.367512\pi$
$432$	0	0
$433$	−2.91585	−0.140127	−0.0700633	−	0.997543i	$-0.522320\pi$
$433$	−2.91585	−0.140127	−0.0700633	+	0.997543i	$0.522320\pi$
$434$	0	0
$435$	− 6.56051i	− 0.314552i
$436$	0	0
$437$	18.1776i	0.869555i
$438$	0	0
$439$	34.3281	1.63839	0.819195	−	0.573516i	$-0.194421\pi$
$439$	34.3281	1.63839	0.819195	+	0.573516i	$0.194421\pi$
$440$	0	0
$441$	2.38662	0.113649
$442$	0	0
$443$	7.70289i	0.365975i	0.983115	+	0.182988i	$0.0585768\pi$
$443$	7.70289i	0.365975i	−0.983115	+	0.182988i	$0.941423\pi$
$444$	0	0
$445$	5.57470i	0.264266i
$446$	0	0
$447$	−0.679375	−0.0321333
$448$	0	0
$449$	−32.5705	−1.53710	−0.768549	−	0.639792i	$-0.779021\pi$
$449$	−32.5705	−1.53710	−0.768549	+	0.639792i	$0.779021\pi$
$450$	0	0
$451$	19.4365i	0.915229i
$452$	0	0
$453$	− 13.0519i	− 0.613233i
$454$	0	0
$455$	−23.2843	−1.09159
$456$	0	0
$457$	−35.2825	−1.65044	−0.825222	−	0.564809i	$-0.808950\pi$
$457$	−35.2825	−1.65044	−0.825222	+	0.564809i	$0.808950\pi$
$458$	0	0
$459$	8.17979i	0.381800i
$460$	0	0
$461$	1.63944i	0.0763561i	0.999271	+	0.0381781i	$0.0121554\pi$
$461$	1.63944i	0.0763561i	−0.999271	+	0.0381781i	$0.987845\pi$
$462$	0	0
$463$	4.05450	0.188429	0.0942144	−	0.995552i	$-0.469966\pi$
$463$	4.05450	0.188429	0.0942144	+	0.995552i	$0.469966\pi$
$464$	0	0
$465$	−20.9638	−0.972172
$466$	0	0
$467$	− 5.80041i	− 0.268411i	−0.990954	−	0.134206i	$-0.957152\pi$
$467$	− 5.80041i	− 0.268411i	0.990954	−	0.134206i	$-0.0428482\pi$
$468$	0	0
$469$	1.87592i	0.0866220i
$470$	0	0
$471$	−8.58249	−0.395460
$472$	0	0
$473$	27.5005	1.26448
$474$	0	0
$475$	50.7969i	2.33072i
$476$	0	0
$477$	− 4.88625i	− 0.223726i
$478$	0	0
$479$	−7.83014	−0.357768	−0.178884	−	0.983870i	$-0.557249\pi$
$479$	−7.83014	−0.357768	−0.178884	+	0.983870i	$0.557249\pi$
$480$	0	0
$481$	39.4863	1.80042
$482$	0	0
$483$	− 2.16746i	− 0.0986227i
$484$	0	0
$485$	67.5408i	3.06687i
$486$	0	0
$487$	−11.8721	−0.537976	−0.268988	−	0.963144i	$-0.586689\pi$
$487$	−11.8721	−0.537976	−0.268988	+	0.963144i	$0.586689\pi$
$488$	0	0
$489$	−2.38997	−0.108078
$490$	0	0
$491$	10.3134i	0.465436i	0.972544	+	0.232718i	$0.0747619\pi$
$491$	10.3134i	0.465436i	−0.972544	+	0.232718i	$0.925238\pi$
$492$	0	0
$493$	4.55153i	0.204990i
$494$	0	0
$495$	−21.4520	−0.964194
$496$	0	0
$497$	−12.3355	−0.553321
$498$	0	0
$499$	− 18.2726i	− 0.817993i	−0.912536	−	0.408996i	$-0.865879\pi$
$499$	− 18.2726i	− 0.817993i	0.912536	−	0.408996i	$-0.134121\pi$
$500$	0	0
$501$	− 11.6076i	− 0.518587i
$502$	0	0
$503$	3.13596	0.139826	0.0699128	−	0.997553i	$-0.477728\pi$
$503$	3.13596	0.139826	0.0699128	+	0.997553i	$0.477728\pi$
$504$	0	0
$505$	15.4856	0.689101
$506$	0	0
$507$	23.1642i	1.02876i
$508$	0	0
$509$	− 18.9768i	− 0.841133i	−0.907262	−	0.420567i	$-0.861831\pi$
$509$	− 18.9768i	− 0.841133i	0.907262	−	0.420567i	$-0.138169\pi$
$510$	0	0
$511$	14.1203	0.624645
$512$	0	0
$513$	−27.7098	−1.22342
$514$	0	0
$515$	− 4.58889i	− 0.202211i
$516$	0	0
$517$	− 20.6078i	− 0.906332i
$518$	0	0
$519$	3.13351	0.137546
$520$	0	0
$521$	3.60942	0.158132	0.0790658	−	0.996869i	$-0.474806\pi$
$521$	3.60942	0.158132	0.0790658	+	0.996869i	$0.474806\pi$
$522$	0	0
$523$	− 3.26279i	− 0.142672i	−0.997452	−	0.0713360i	$-0.977274\pi$
$523$	− 3.26279i	− 0.142672i	0.997452	−	0.0713360i	$-0.0227262\pi$
$524$	0	0
$525$	− 6.05690i	− 0.264345i
$526$	0	0
$527$	14.5442	0.633554
$528$	0	0
$529$	−15.3410	−0.667000
$530$	0	0
$531$	2.17523i	0.0943969i
$532$	0	0
$533$	− 50.3498i	− 2.18089i
$534$	0	0
$535$	57.7663	2.49745
$536$	0	0
$537$	−2.96117	−0.127784
$538$	0	0
$539$	2.51888i	0.108496i
$540$	0	0
$541$	40.8816i	1.75764i	0.477158	+	0.878818i	$0.341667\pi$
$541$	40.8816i	1.75764i	−0.477158	+	0.878818i	$0.658333\pi$
$542$	0	0
$543$	−16.7039	−0.716835
$544$	0	0
$545$	−11.6718	−0.499967
$546$	0	0
$547$	− 39.4933i	− 1.68861i	−0.535863	−	0.844305i	$-0.680014\pi$
$547$	− 39.4933i	− 1.68861i	0.535863	−	0.844305i	$-0.319986\pi$
$548$	0	0
$549$	− 4.98428i	− 0.212724i
$550$	0	0
$551$	−15.4187	−0.656859
$552$	0	0
$553$	8.33547	0.354460
$554$	0	0
$555$	16.9123i	0.717885i
$556$	0	0
$557$	38.9287i	1.64946i	0.565525	+	0.824731i	$0.308674\pi$
$557$	38.9287i	1.64946i	−0.565525	+	0.824731i	$0.691326\pi$
$558$	0	0
$559$	−71.2396	−3.01311
$560$	0	0
$561$	−3.82501	−0.161492
$562$	0	0
$563$	− 34.1618i	− 1.43975i	−0.694105	−	0.719873i	$-0.744200\pi$
$563$	− 34.1618i	− 1.43975i	0.694105	−	0.719873i	$-0.255800\pi$
$564$	0	0
$565$	− 37.8664i	− 1.59305i
$566$	0	0
$567$	−3.85582	−0.161929
$568$	0	0
$569$	21.5861	0.904935	0.452467	−	0.891781i	$-0.350544\pi$
$569$	21.5861	0.904935	0.452467	+	0.891781i	$0.350544\pi$
$570$	0	0
$571$	− 8.03105i	− 0.336089i	−0.985779	−	0.168045i	$-0.946255\pi$
$571$	− 8.03105i	− 0.336089i	0.985779	−	0.168045i	$-0.0537453\pi$
$572$	0	0
$573$	− 15.9051i	− 0.664445i
$574$	0	0
$575$	21.4028	0.892560
$576$	0	0
$577$	45.8009	1.90672	0.953359	−	0.301839i	$-0.0976005\pi$
$577$	45.8009	1.90672	0.953359	+	0.301839i	$0.0976005\pi$
$578$	0	0
$579$	8.83772i	0.367283i
$580$	0	0
$581$	6.44004i	0.267178i
$582$	0	0
$583$	5.15702	0.213582
$584$	0	0
$585$	55.5708	2.29757
$586$	0	0
$587$	− 3.87142i	− 0.159791i	−0.996803	−	0.0798954i	$-0.974541\pi$
$587$	− 3.87142i	− 0.159791i	0.996803	−	0.0798954i	$-0.0254586\pi$
$588$	0	0
$589$	49.2697i	2.03012i
$590$	0	0
$591$	16.7956	0.690877
$592$	0	0
$593$	−40.4013	−1.65908	−0.829540	−	0.558447i	$-0.811397\pi$
$593$	−40.4013	−1.65908	−0.829540	+	0.558447i	$0.811397\pi$
$594$	0	0
$595$	6.91891i	0.283648i
$596$	0	0
$597$	18.6472i	0.763177i
$598$	0	0
$599$	16.8607	0.688911	0.344455	−	0.938803i	$-0.388064\pi$
$599$	16.8607	0.688911	0.344455	+	0.938803i	$0.388064\pi$
$600$	0	0
$601$	31.7796	1.29632	0.648159	−	0.761505i	$-0.275539\pi$
$601$	31.7796	1.29632	0.648159	+	0.761505i	$0.275539\pi$
$602$	0	0
$603$	− 4.47711i	− 0.182322i
$604$	0	0
$605$	16.6120i	0.675372i
$606$	0	0
$607$	−35.3989	−1.43680	−0.718399	−	0.695631i	$-0.755125\pi$
$607$	−35.3989	−1.43680	−0.718399	+	0.695631i	$0.755125\pi$
$608$	0	0
$609$	1.83849	0.0744993
$610$	0	0
$611$	53.3842i	2.15969i
$612$	0	0
$613$	− 13.6414i	− 0.550971i	−0.961305	−	0.275485i	$-0.911161\pi$
$613$	− 13.6414i	− 0.550971i	0.961305	−	0.275485i	$-0.0888386\pi$
$614$	0	0
$615$	21.5652	0.869591
$616$	0	0
$617$	40.2668	1.62108	0.810540	−	0.585684i	$-0.199174\pi$
$617$	40.2668	1.62108	0.810540	+	0.585684i	$0.199174\pi$
$618$	0	0
$619$	16.1769i	0.650205i	0.945679	+	0.325102i	$0.105399\pi$
$619$	16.1769i	0.650205i	−0.945679	+	0.325102i	$0.894601\pi$
$620$	0	0
$621$	11.6753i	0.468512i
$622$	0	0
$623$	−1.56223	−0.0625894
$624$	0	0
$625$	−3.85875	−0.154350
$626$	0	0
$627$	− 12.9576i	− 0.517475i
$628$	0	0
$629$	− 11.7333i	− 0.467838i
$630$	0	0
$631$	4.79309	0.190810	0.0954049	−	0.995439i	$-0.469585\pi$
$631$	4.79309	0.190810	0.0954049	+	0.995439i	$0.469585\pi$
$632$	0	0
$633$	16.6613	0.662227
$634$	0	0
$635$	− 14.2942i	− 0.567249i
$636$	0	0
$637$	− 6.52509i	− 0.258534i
$638$	0	0
$639$	29.4401	1.16463
$640$	0	0
$641$	4.63323	0.183002	0.0915008	−	0.995805i	$-0.470834\pi$
$641$	4.63323	0.183002	0.0915008	+	0.995805i	$0.470834\pi$
$642$	0	0
$643$	5.95425i	0.234813i	0.993084	+	0.117406i	$0.0374580\pi$
$643$	5.95425i	0.234813i	−0.993084	+	0.117406i	$0.962542\pi$
$644$	0	0
$645$	− 30.5124i	− 1.20142i
$646$	0	0
$647$	−16.8071	−0.660754	−0.330377	−	0.943849i	$-0.607176\pi$
$647$	−16.8071	−0.660754	−0.330377	+	0.943849i	$0.607176\pi$
$648$	0	0
$649$	−2.29577	−0.0901169
$650$	0	0
$651$	− 5.87480i	− 0.230251i
$652$	0	0
$653$	6.80213i	0.266188i	0.991103	+	0.133094i	$0.0424912\pi$
$653$	6.80213i	0.266188i	−0.991103	+	0.133094i	$0.957509\pi$
$654$	0	0
$655$	80.7134	3.15373
$656$	0	0
$657$	−33.6998	−1.31475
$658$	0	0
$659$	45.8568i	1.78633i	0.449733	+	0.893163i	$0.351519\pi$
$659$	45.8568i	1.78633i	−0.449733	+	0.893163i	$0.648481\pi$
$660$	0	0
$661$	3.22646i	0.125495i	0.998029	+	0.0627474i	$0.0199862\pi$
$661$	3.22646i	0.125495i	−0.998029	+	0.0627474i	$0.980014\pi$
$662$	0	0
$663$	9.90861	0.384818
$664$	0	0
$665$	−23.4384	−0.908903
$666$	0	0
$667$	6.49654i	0.251547i
$668$	0	0
$669$	9.24525i	0.357442i
$670$	0	0
$671$	5.26049	0.203079
$672$	0	0
$673$	10.0871	0.388828	0.194414	−	0.980920i	$-0.437719\pi$
$673$	10.0871	0.388828	0.194414	+	0.980920i	$0.437719\pi$
$674$	0	0
$675$	32.6262i	1.25578i
$676$	0	0
$677$	− 32.3792i	− 1.24443i	−0.782845	−	0.622216i	$-0.786232\pi$
$677$	− 32.3792i	− 1.24443i	0.782845	−	0.622216i	$-0.213768\pi$
$678$	0	0
$679$	−18.9273	−0.726365
$680$	0	0
$681$	14.4106	0.552217
$682$	0	0
$683$	35.7666i	1.36857i	0.729215	+	0.684285i	$0.239886\pi$
$683$	35.7666i	1.36857i	−0.729215	+	0.684285i	$0.760114\pi$
$684$	0	0
$685$	− 9.46687i	− 0.361711i
$686$	0	0
$687$	−7.56388	−0.288580
$688$	0	0
$689$	−13.3592	−0.508943
$690$	0	0
$691$	20.9942i	0.798657i	0.916808	+	0.399329i	$0.130757\pi$
$691$	20.9942i	0.798657i	−0.916808	+	0.399329i	$0.869243\pi$
$692$	0	0
$693$	− 6.01160i	− 0.228362i
$694$	0	0
$695$	−43.0081	−1.63139
$696$	0	0
$697$	−14.9614	−0.566703
$698$	0	0
$699$	− 3.54846i	− 0.134215i
$700$	0	0
$701$	− 25.0388i	− 0.945704i	−0.881142	−	0.472852i	$-0.843225\pi$
$701$	− 25.0388i	− 0.945704i	0.881142	−	0.472852i	$-0.156775\pi$
$702$	0	0
$703$	39.7477	1.49911
$704$	0	0
$705$	−22.8648	−0.861137
$706$	0	0
$707$	4.33962i	0.163208i
$708$	0	0
$709$	− 32.5877i	− 1.22386i	−0.790913	−	0.611929i	$-0.790394\pi$
$709$	− 32.5877i	− 1.22386i	0.790913	−	0.611929i	$-0.209606\pi$
$710$	0	0
$711$	−19.8936	−0.746068
$712$	0	0
$713$	20.7594	0.777444
$714$	0	0
$715$	58.6503i	2.19340i
$716$	0	0
$717$	15.0782i	0.563105i
$718$	0	0
$719$	9.02299	0.336501	0.168250	−	0.985744i	$-0.446188\pi$
$719$	9.02299	0.336501	0.168250	+	0.985744i	$0.446188\pi$
$720$	0	0
$721$	1.28597	0.0478920
$722$	0	0
$723$	− 15.6314i	− 0.581339i
$724$	0	0
$725$	18.1544i	0.674237i
$726$	0	0
$727$	16.8802	0.626054	0.313027	−	0.949744i	$-0.398657\pi$
$727$	16.8802	0.626054	0.313027	+	0.949744i	$0.398657\pi$
$728$	0	0
$729$	−0.709766	−0.0262876
$730$	0	0
$731$	21.1688i	0.782955i
$732$	0	0
$733$	− 27.9470i	− 1.03225i	−0.856514	−	0.516123i	$-0.827375\pi$
$733$	− 27.9470i	− 1.03225i	0.856514	−	0.516123i	$-0.172625\pi$
$734$	0	0
$735$	2.79474	0.103086
$736$	0	0
$737$	4.72522	0.174056
$738$	0	0
$739$	10.6219i	0.390733i	0.980730	+	0.195367i	$0.0625896\pi$
$739$	10.6219i	0.390733i	−0.980730	+	0.195367i	$0.937410\pi$
$740$	0	0
$741$	33.5663i	1.23309i
$742$	0	0
$743$	43.1417	1.58272	0.791358	−	0.611353i	$-0.209374\pi$
$743$	43.1417	1.58272	0.791358	+	0.611353i	$0.209374\pi$
$744$	0	0
$745$	3.09543	0.113408
$746$	0	0
$747$	− 15.3699i	− 0.562356i
$748$	0	0
$749$	16.1882i	0.591503i
$750$	0	0
$751$	4.11267	0.150073	0.0750367	−	0.997181i	$-0.476093\pi$
$751$	4.11267	0.150073	0.0750367	+	0.997181i	$0.476093\pi$
$752$	0	0
$753$	−7.49727	−0.273216
$754$	0	0
$755$	59.4685i	2.16428i
$756$	0	0
$757$	4.08900i	0.148617i	0.997235	+	0.0743087i	$0.0236750\pi$
$757$	4.08900i	0.148617i	−0.997235	+	0.0743087i	$0.976325\pi$
$758$	0	0
$759$	−5.45956	−0.198169
$760$	0	0
$761$	−1.32450	−0.0480131	−0.0240065	−	0.999712i	$-0.507642\pi$
$761$	−1.32450	−0.0480131	−0.0240065	+	0.999712i	$0.507642\pi$
$762$	0	0
$763$	− 3.27087i	− 0.118413i
$764$	0	0
$765$	− 16.5128i	− 0.597022i
$766$	0	0
$767$	5.94715	0.214739
$768$	0	0
$769$	17.8450	0.643506	0.321753	−	0.946824i	$-0.395728\pi$
$769$	17.8450	0.643506	0.321753	+	0.946824i	$0.395728\pi$
$770$	0	0
$771$	− 9.85084i	− 0.354769i
$772$	0	0
$773$	13.2378i	0.476131i	0.971249	+	0.238066i	$0.0765134\pi$
$773$	13.2378i	0.476131i	−0.971249	+	0.238066i	$0.923487\pi$
$774$	0	0
$775$	58.0114	2.08383
$776$	0	0
$777$	−4.73942	−0.170026
$778$	0	0
$779$	− 50.6831i	− 1.81591i
$780$	0	0
$781$	31.0715i	1.11183i
$782$	0	0
$783$	−9.90324	−0.353913
$784$	0	0
$785$	39.1044	1.39569
$786$	0	0
$787$	− 25.2795i	− 0.901118i	−0.892747	−	0.450559i	$-0.851225\pi$
$787$	− 25.2795i	− 0.901118i	0.892747	−	0.450559i	$-0.148775\pi$
$788$	0	0
$789$	12.4566i	0.443468i
$790$	0	0
$791$	10.6115	0.377302
$792$	0	0
$793$	−13.6272	−0.483916
$794$	0	0
$795$	− 5.72181i	− 0.202932i
$796$	0	0
$797$	36.5312i	1.29400i	0.762490	+	0.647000i	$0.223977\pi$
$797$	36.5312i	1.29400i	−0.762490	+	0.647000i	$0.776023\pi$
$798$	0	0
$799$	15.8630	0.561194
$800$	0	0
$801$	3.72845	0.131738
$802$	0	0
$803$	− 35.5673i	− 1.25514i
$804$	0	0
$805$	9.87558i	0.348068i
$806$	0	0
$807$	−3.13351	−0.110305
$808$	0	0
$809$	−8.75351	−0.307757	−0.153879	−	0.988090i	$-0.549176\pi$
$809$	−8.75351	−0.307757	−0.153879	+	0.988090i	$0.549176\pi$
$810$	0	0
$811$	23.4106i	0.822057i	0.911622	+	0.411029i	$0.134830\pi$
$811$	23.4106i	0.822057i	−0.911622	+	0.411029i	$0.865170\pi$
$812$	0	0
$813$	− 2.97976i	− 0.104505i
$814$	0	0
$815$	10.8894	0.381440
$816$	0	0
$817$	−71.7111	−2.50885
$818$	0	0
$819$	15.5729i	0.544162i
$820$	0	0
$821$	− 23.1296i	− 0.807229i	−0.914929	−	0.403614i	$-0.867754\pi$
$821$	− 23.1296i	− 0.807229i	0.914929	−	0.403614i	$-0.132246\pi$
$822$	0	0
$823$	−40.2133	−1.40175	−0.700874	−	0.713285i	$-0.747207\pi$
$823$	−40.2133	−1.40175	−0.700874	+	0.713285i	$0.747207\pi$
$824$	0	0
$825$	−15.2566	−0.531166
$826$	0	0
$827$	34.4702i	1.19865i	0.800507	+	0.599323i	$0.204564\pi$
$827$	34.4702i	1.19865i	−0.800507	+	0.599323i	$0.795436\pi$
$828$	0	0
$829$	23.4776i	0.815412i	0.913113	+	0.407706i	$0.133671\pi$
$829$	23.4776i	0.815412i	−0.913113	+	0.407706i	$0.866329\pi$
$830$	0	0
$831$	−9.88149	−0.342785
$832$	0	0
$833$	−1.93892	−0.0671798
$834$	0	0
$835$	52.8875i	1.83025i
$836$	0	0
$837$	31.6453i	1.09382i
$838$	0	0
$839$	−2.16748	−0.0748296	−0.0374148	−	0.999300i	$-0.511912\pi$
$839$	−2.16748	−0.0748296	−0.0374148	+	0.999300i	$0.511912\pi$
$840$	0	0
$841$	23.4895	0.809982
$842$	0	0
$843$	10.2724i	0.353799i
$844$	0	0
$845$	− 105.543i	− 3.63078i
$846$	0	0
$847$	−4.65526	−0.159957
$848$	0	0
$849$	−9.73113	−0.333972
$850$	0	0
$851$	− 16.7473i	− 0.574092i
$852$	0	0
$853$	19.1247i	0.654818i	0.944883	+	0.327409i	$0.106175\pi$
$853$	19.1247i	0.654818i	−0.944883	+	0.327409i	$0.893825\pi$
$854$	0	0
$855$	55.9386	1.91306
$856$	0	0
$857$	−9.73619	−0.332582	−0.166291	−	0.986077i	$-0.553179\pi$
$857$	−9.73619	−0.332582	−0.166291	+	0.986077i	$0.553179\pi$
$858$	0	0
$859$	37.3577i	1.27463i	0.770604	+	0.637314i	$0.219955\pi$
$859$	37.3577i	1.27463i	−0.770604	+	0.637314i	$0.780045\pi$
$860$	0	0
$861$	6.04333i	0.205956i
$862$	0	0
$863$	−17.6964	−0.602392	−0.301196	−	0.953562i	$-0.597386\pi$
$863$	−17.6964	−0.602392	−0.301196	+	0.953562i	$0.597386\pi$
$864$	0	0
$865$	−14.2772	−0.485440
$866$	0	0
$867$	10.3698i	0.352178i
$868$	0	0
$869$	− 20.9960i	− 0.712241i
$870$	0	0
$871$	−12.2406	−0.414756
$872$	0	0
$873$	45.1724	1.52885
$874$	0	0
$875$	9.75488i	0.329775i
$876$	0	0
$877$	14.9379i	0.504417i	0.967673	+	0.252209i	$0.0811569\pi$
$877$	14.9379i	0.504417i	−0.967673	+	0.252209i	$0.918843\pi$
$878$	0	0
$879$	15.5474	0.524402
$880$	0	0
$881$	−12.1360	−0.408873	−0.204437	−	0.978880i	$-0.565536\pi$
$881$	−12.1360	−0.408873	−0.204437	+	0.978880i	$0.565536\pi$
$882$	0	0
$883$	10.5340i	0.354496i	0.984166	+	0.177248i	$0.0567194\pi$
$883$	10.5340i	0.354496i	−0.984166	+	0.177248i	$0.943281\pi$
$884$	0	0
$885$	2.54720i	0.0856233i
$886$	0	0
$887$	15.9916	0.536944	0.268472	−	0.963287i	$-0.413481\pi$
$887$	15.9916	0.536944	0.268472	+	0.963287i	$0.413481\pi$
$888$	0	0
$889$	4.00575	0.134349
$890$	0	0
$891$	9.71232i	0.325375i
$892$	0	0
$893$	53.7375i	1.79826i
$894$	0	0
$895$	13.4920	0.450987
$896$	0	0
$897$	14.1429	0.472217
$898$	0	0
$899$	17.6086i	0.587279i
$900$	0	0
$901$	3.96966i	0.132248i
$902$	0	0
$903$	8.55065	0.284548
$904$	0	0
$905$	76.1081	2.52992
$906$	0	0
$907$	43.0247i	1.42861i	0.699833	+	0.714306i	$0.253258\pi$
$907$	43.0247i	1.42861i	−0.699833	+	0.714306i	$0.746742\pi$
$908$	0	0
$909$	− 10.3570i	− 0.343521i
$910$	0	0
$911$	−35.3708	−1.17189	−0.585943	−	0.810352i	$-0.699276\pi$
$911$	−35.3708	−1.17189	−0.585943	+	0.810352i	$0.699276\pi$
$912$	0	0
$913$	16.2217	0.536859
$914$	0	0
$915$	− 5.83661i	− 0.192953i
$916$	0	0
$917$	22.6188i	0.746937i
$918$	0	0
$919$	−44.0087	−1.45171	−0.725857	−	0.687846i	$-0.758556\pi$
$919$	−44.0087	−1.45171	−0.725857	+	0.687846i	$0.758556\pi$
$920$	0	0
$921$	−1.70485	−0.0561766
$922$	0	0
$923$	− 80.4901i	− 2.64936i
$924$	0	0
$925$	− 46.8000i	− 1.53877i
$926$	0	0
$927$	−3.06912	−0.100803
$928$	0	0
$929$	−46.4669	−1.52453	−0.762265	−	0.647265i	$-0.775913\pi$
$929$	−46.4669	−1.52453	−0.762265	+	0.647265i	$0.775913\pi$
$930$	0	0
$931$	− 6.56828i	− 0.215267i
$932$	0	0
$933$	2.46330i	0.0806448i
$934$	0	0
$935$	17.4279	0.569953
$936$	0	0
$937$	−46.4077	−1.51607	−0.758037	−	0.652211i	$-0.773841\pi$
$937$	−46.4077	−1.51607	−0.758037	+	0.652211i	$0.773841\pi$
$938$	0	0
$939$	− 9.67045i	− 0.315583i
$940$	0	0
$941$	− 21.9398i	− 0.715216i	−0.933872	−	0.357608i	$-0.883592\pi$
$941$	− 21.9398i	− 0.715216i	0.933872	−	0.357608i	$-0.116408\pi$
$942$	0	0
$943$	−21.3549	−0.695410
$944$	0	0
$945$	−15.0542	−0.489713
$946$	0	0
$947$	52.5392i	1.70730i	0.520851	+	0.853648i	$0.325615\pi$
$947$	52.5392i	1.70730i	−0.520851	+	0.853648i	$0.674385\pi$
$948$	0	0
$949$	92.1362i	2.99087i
$950$	0	0
$951$	−13.7895	−0.447156
$952$	0	0
$953$	59.7325	1.93493	0.967463	−	0.253011i	$-0.0814208\pi$
$953$	59.7325	1.93493	0.967463	+	0.253011i	$0.0814208\pi$
$954$	0	0
$955$	72.4684i	2.34502i
$956$	0	0
$957$	− 4.63092i	− 0.149697i
$958$	0	0
$959$	2.65295	0.0856684
$960$	0	0
$961$	25.2673	0.815074
$962$	0	0
$963$	− 38.6350i	− 1.24500i
$964$	0	0
$965$	− 40.2673i	− 1.29625i
$966$	0	0
$967$	−9.91826	−0.318950	−0.159475	−	0.987202i	$-0.550980\pi$
$967$	−9.91826	−0.318950	−0.159475	+	0.987202i	$0.550980\pi$
$968$	0	0
$969$	9.97419	0.320417
$970$	0	0
$971$	− 16.4899i	− 0.529188i	−0.964360	−	0.264594i	$-0.914762\pi$
$971$	− 16.4899i	− 0.529188i	0.964360	−	0.264594i	$-0.0852379\pi$
$972$	0	0
$973$	− 12.0524i	− 0.386382i
$974$	0	0
$975$	39.5218	1.26571
$976$	0	0
$977$	−20.8390	−0.666699	−0.333350	−	0.942803i	$-0.608179\pi$
$977$	−20.8390	−0.666699	−0.333350	+	0.942803i	$0.608179\pi$
$978$	0	0
$979$	3.93506i	0.125765i
$980$	0	0
$981$	7.80631i	0.249236i
$982$	0	0
$983$	−43.8056	−1.39718	−0.698591	−	0.715521i	$-0.746189\pi$
$983$	−43.8056	−1.39718	−0.698591	+	0.715521i	$0.746189\pi$
$984$	0	0
$985$	−76.5255	−2.43831
$986$	0	0
$987$	− 6.40752i	− 0.203954i
$988$	0	0
$989$	30.2148i	0.960776i
$990$	0	0
$991$	−52.6400	−1.67216	−0.836082	−	0.548605i	$-0.815159\pi$
$991$	−52.6400	−1.67216	−0.836082	+	0.548605i	$0.815159\pi$
$992$	0	0
$993$	17.2761	0.548240
$994$	0	0
$995$	− 84.9620i	− 2.69348i
$996$	0	0
$997$	− 22.2608i	− 0.705006i	−0.935811	−	0.352503i	$-0.885331\pi$
$997$	− 22.2608i	− 0.705006i	0.935811	−	0.352503i	$-0.114669\pi$
$998$	0	0
$999$	25.5294	0.807716

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.g.1793.4		10
4.3	odd	2		3584.2.b.h.1793.7		10
8.3	odd	2		3584.2.b.h.1793.4		10
8.5	even	2	inner	3584.2.b.g.1793.7		10
16.3	odd	4		3584.2.a.o.1.4	✓	10
16.5	even	4		3584.2.a.p.1.4	yes	10
16.11	odd	4		3584.2.a.o.1.7	yes	10
16.13	even	4		3584.2.a.p.1.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.o.1.4	✓	10	16.3	odd	4
3584.2.a.o.1.7	yes	10	16.11	odd	4
3584.2.a.p.1.4	yes	10	16.5	even	4
3584.2.a.p.1.7	yes	10	16.13	even	4
3584.2.b.g.1793.4		10	1.1	even	1	trivial
3584.2.b.g.1793.7		10	8.5	even	2	inner
3584.2.b.h.1793.4		10	8.3	odd	2
3584.2.b.h.1793.7		10	4.3	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:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.783186i q^{3} +3.56843i q^{5} -1.00000 q^{7} +2.38662 q^{9} +O(q^{10})\)	\(q-0.783186i q^{3} +3.56843i q^{5} -1.00000 q^{7} +2.38662 q^{9} +2.51888i q^{11} -6.52509i q^{13} +2.79474 q^{15} -1.93892 q^{17} -6.56828i q^{19} +0.783186i q^{21} -2.76749 q^{23} -7.73367 q^{25} -4.21872i q^{27} -2.34745i q^{29} -7.50115 q^{31} +1.97275 q^{33} -3.56843i q^{35} +6.05146i q^{37} -5.11036 q^{39} +7.71634 q^{41} -10.9178i q^{43} +8.51648i q^{45} -8.18136 q^{47} +1.00000 q^{49} +1.51854i q^{51} -2.04735i q^{53} -8.98842 q^{55} -5.14418 q^{57} +0.911427i q^{59} -2.08843i q^{61} -2.38662 q^{63} +23.2843 q^{65} -1.87592i q^{67} +2.16746i q^{69} +12.3355 q^{71} -14.1203 q^{73} +6.05690i q^{75} -2.51888i q^{77} -8.33547 q^{79} +3.85582 q^{81} -6.44004i q^{83} -6.91891i q^{85} -1.83849 q^{87} +1.56223 q^{89} +6.52509i q^{91} +5.87480i q^{93} +23.4384 q^{95} +18.9273 q^{97} +6.01160i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{7} - 22 q^{9}+O(q^{10}) \) 10 * q - 10 * q^7 - 22 * q^9	\( 10 q - 10 q^{7} - 22 q^{9} + 16 q^{15} + 16 q^{17} - 8 q^{23} - 30 q^{25} - 8 q^{31} + 12 q^{33} - 20 q^{41} - 24 q^{47} + 10 q^{49} + 32 q^{55} - 28 q^{57} + 22 q^{63} + 32 q^{65} - 48 q^{73} + 40 q^{79} + 62 q^{81} + 8 q^{87} - 16 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100}) \) 10 * q - 10 * q^7 - 22 * q^9 + 16 * q^15 + 16 * q^17 - 8 * q^23 - 30 * q^25 - 8 * q^31 + 12 * q^33 - 20 * q^41 - 24 * q^47 + 10 * q^49 + 32 * q^55 - 28 * q^57 + 22 * q^63 + 32 * q^65 - 48 * q^73 + 40 * q^79 + 62 * q^81 + 8 * q^87 - 16 * q^89 - 32 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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