LMFDB - Embedded newform 2320.2.d.d.929.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2320,2,Mod(929,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2320.929");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.d (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:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 290)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			929.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	2320.929
Dual form			2320.2.d.d.929.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803i q^{3} +2.23607i q^{5} +0.381966i q^{7} -3.85410 q^{9} +O(q^{10})$	$q-2.61803i q^{3} +2.23607i q^{5} +0.381966i q^{7} -3.85410 q^{9} -2.00000 q^{11} +2.61803i q^{13} +5.85410 q^{15} -5.61803i q^{17} -2.00000 q^{19} +1.00000 q^{21} +1.85410i q^{23} -5.00000 q^{25} +2.23607i q^{27} +1.00000 q^{29} -2.85410 q^{31} +5.23607i q^{33} -0.854102 q^{35} +8.94427i q^{37} +6.85410 q^{39} -5.70820 q^{41} -5.61803i q^{43} -8.61803i q^{45} +10.4721i q^{47} +6.85410 q^{49} -14.7082 q^{51} +11.6180i q^{53} -4.47214i q^{55} +5.23607i q^{57} -3.85410 q^{59} -4.14590 q^{61} -1.47214i q^{63} -5.85410 q^{65} +8.94427i q^{67} +4.85410 q^{69} -7.70820 q^{71} +14.5623i q^{73} +13.0902i q^{75} -0.763932i q^{77} -13.5623 q^{79} -5.70820 q^{81} +2.94427i q^{83} +12.5623 q^{85} -2.61803i q^{87} -2.00000 q^{89} -1.00000 q^{91} +7.47214i q^{93} -4.47214i q^{95} -13.8541i q^{97} +7.70820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^9	$ 4 q - 2 q^{9} - 8 q^{11} + 10 q^{15} - 8 q^{19} + 4 q^{21} - 20 q^{25} + 4 q^{29} + 2 q^{31} + 10 q^{35} + 14 q^{39} + 4 q^{41} + 14 q^{49} - 32 q^{51} - 2 q^{59} - 30 q^{61} - 10 q^{65} + 6 q^{69} - 4 q^{71} - 14 q^{79} + 4 q^{81} + 10 q^{85} - 8 q^{89} - 4 q^{91} + 4 q^{99}+O(q^{100}) $ 4 * q - 2 * q^9 - 8 * q^11 + 10 * q^15 - 8 * q^19 + 4 * q^21 - 20 * q^25 + 4 * q^29 + 2 * q^31 + 10 * q^35 + 14 * q^39 + 4 * q^41 + 14 * q^49 - 32 * q^51 - 2 * q^59 - 30 * q^61 - 10 * q^65 + 6 * q^69 - 4 * q^71 - 14 * q^79 + 4 * q^81 + 10 * q^85 - 8 * q^89 - 4 * q^91 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 2.61803i	− 1.51152i	−0.654847	−	0.755761i	$-0.727267\pi$
$3$	− 2.61803i	− 1.51152i	0.654847	−	0.755761i	$-0.272733\pi$
$4$	0	0
$5$	2.23607i	1.00000i
$5$	2.23607i	1.00000i
$6$	0	0
$7$	0.381966i	0.144370i	0.997391	+	0.0721848i	$0.0229971\pi$
$7$	0.381966i	0.144370i	−0.997391	+	0.0721848i	$0.977003\pi$
$8$	0	0
$9$	−3.85410	−1.28470
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.61803i	0.726112i	0.931767	+	0.363056i	$0.118267\pi$
$13$	2.61803i	0.726112i	−0.931767	+	0.363056i	$0.881733\pi$
$14$	0	0
$15$	5.85410	1.51152
$16$	0	0
$17$	− 5.61803i	− 1.36257i	−0.732017	−	0.681287i	$-0.761421\pi$
$17$	− 5.61803i	− 1.36257i	0.732017	−	0.681287i	$-0.238579\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.85410i	0.386607i	0.981139	+	0.193303i	$0.0619202\pi$
$23$	1.85410i	0.386607i	−0.981139	+	0.193303i	$0.938080\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	2.23607i	0.430331i
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.85410	−0.512612	−0.256306	−	0.966596i	$-0.582505\pi$
$31$	−2.85410	−0.512612	−0.256306	+	0.966596i	$0.582505\pi$
$32$	0	0
$33$	5.23607i	0.911482i
$34$	0	0
$35$	−0.854102	−0.144370
$36$	0	0
$37$	8.94427i	1.47043i	0.677834	+	0.735215i	$0.262919\pi$
$37$	8.94427i	1.47043i	−0.677834	+	0.735215i	$0.737081\pi$
$38$	0	0
$39$	6.85410	1.09753
$40$	0	0
$41$	−5.70820	−0.891472	−0.445736	−	0.895165i	$-0.647058\pi$
$41$	−5.70820	−0.891472	−0.445736	+	0.895165i	$0.647058\pi$
$42$	0	0
$43$	− 5.61803i	− 0.856742i	−0.903603	−	0.428371i	$-0.859088\pi$
$43$	− 5.61803i	− 0.856742i	0.903603	−	0.428371i	$-0.140912\pi$
$44$	0	0
$45$	− 8.61803i	− 1.28470i
$46$	0	0
$47$	10.4721i	1.52752i	0.645501	+	0.763759i	$0.276648\pi$
$47$	10.4721i	1.52752i	−0.645501	+	0.763759i	$0.723352\pi$
$48$	0	0
$49$	6.85410	0.979157
$50$	0	0
$51$	−14.7082	−2.05956
$52$	0	0
$53$	11.6180i	1.59586i	0.602750	+	0.797930i	$0.294071\pi$
$53$	11.6180i	1.59586i	−0.602750	+	0.797930i	$0.705929\pi$
$54$	0	0
$55$	− 4.47214i	− 0.603023i
$56$	0	0
$57$	5.23607i	0.693534i
$58$	0	0
$59$	−3.85410	−0.501761	−0.250881	−	0.968018i	$-0.580720\pi$
$59$	−3.85410	−0.501761	−0.250881	+	0.968018i	$0.580720\pi$
$60$	0	0
$61$	−4.14590	−0.530828	−0.265414	−	0.964135i	$-0.585509\pi$
$61$	−4.14590	−0.530828	−0.265414	+	0.964135i	$0.585509\pi$
$62$	0	0
$63$	− 1.47214i	− 0.185472i
$64$	0	0
$65$	−5.85410	−0.726112
$66$	0	0
$67$	8.94427i	1.09272i	0.837552	+	0.546358i	$0.183986\pi$
$67$	8.94427i	1.09272i	−0.837552	+	0.546358i	$0.816014\pi$
$68$	0	0
$69$	4.85410	0.584365
$70$	0	0
$71$	−7.70820	−0.914796	−0.457398	−	0.889262i	$-0.651218\pi$
$71$	−7.70820	−0.914796	−0.457398	+	0.889262i	$0.651218\pi$
$72$	0	0
$73$	14.5623i	1.70439i	0.523225	+	0.852194i	$0.324729\pi$
$73$	14.5623i	1.70439i	−0.523225	+	0.852194i	$0.675271\pi$
$74$	0	0
$75$	13.0902i	1.51152i
$76$	0	0
$77$	− 0.763932i	− 0.0870581i
$78$	0	0
$79$	−13.5623	−1.52588	−0.762939	−	0.646470i	$-0.776245\pi$
$79$	−13.5623	−1.52588	−0.762939	+	0.646470i	$0.776245\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	2.94427i	0.323176i	0.986858	+	0.161588i	$0.0516615\pi$
$83$	2.94427i	0.323176i	−0.986858	+	0.161588i	$0.948338\pi$
$84$	0	0
$85$	12.5623	1.36257
$86$	0	0
$87$	− 2.61803i	− 0.280683i
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	7.47214i	0.774824i
$94$	0	0
$95$	− 4.47214i	− 0.458831i
$96$	0	0
$97$	− 13.8541i	− 1.40667i	−0.710858	−	0.703335i	$-0.751693\pi$
$97$	− 13.8541i	− 1.40667i	0.710858	−	0.703335i	$-0.248307\pi$
$98$	0	0
$99$	7.70820	0.774704
$100$	0	0
$101$	4.14590	0.412532	0.206266	−	0.978496i	$-0.433869\pi$
$101$	4.14590	0.412532	0.206266	+	0.978496i	$0.433869\pi$
$102$	0	0
$103$	19.4164i	1.91316i	0.291477	+	0.956578i	$0.405853\pi$
$103$	19.4164i	1.91316i	−0.291477	+	0.956578i	$0.594147\pi$
$104$	0	0
$105$	2.23607i	0.218218i
$106$	0	0
$107$	8.18034i	0.790823i	0.918504	+	0.395412i	$0.129398\pi$
$107$	8.18034i	0.790823i	−0.918504	+	0.395412i	$0.870602\pi$
$108$	0	0
$109$	7.41641	0.710363	0.355182	−	0.934797i	$-0.384419\pi$
$109$	7.41641	0.710363	0.355182	+	0.934797i	$0.384419\pi$
$110$	0	0
$111$	23.4164	2.22259
$112$	0	0
$113$	− 7.09017i	− 0.666987i	−0.942752	−	0.333494i	$-0.891772\pi$
$113$	− 7.09017i	− 0.666987i	0.942752	−	0.333494i	$-0.108228\pi$
$114$	0	0
$115$	−4.14590	−0.386607
$116$	0	0
$117$	− 10.0902i	− 0.932837i
$118$	0	0
$119$	2.14590	0.196714
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	14.9443i	1.34748i
$124$	0	0
$125$	− 11.1803i	− 1.00000i
$126$	0	0
$127$	− 21.7082i	− 1.92629i	−0.268980	−	0.963146i	$-0.586687\pi$
$127$	− 21.7082i	− 1.92629i	0.268980	−	0.963146i	$-0.413313\pi$
$128$	0	0
$129$	−14.7082	−1.29499
$130$	0	0
$131$	5.41641	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$131$	5.41641	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$132$	0	0
$133$	− 0.763932i	− 0.0662413i
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	− 1.85410i	− 0.158407i	−0.996858	−	0.0792033i	$-0.974762\pi$
$137$	− 1.85410i	− 0.158407i	0.996858	−	0.0792033i	$-0.0252376\pi$
$138$	0	0
$139$	14.5623	1.23516	0.617579	−	0.786509i	$-0.288113\pi$
$139$	14.5623	1.23516	0.617579	+	0.786509i	$0.288113\pi$
$140$	0	0
$141$	27.4164	2.30888
$142$	0	0
$143$	− 5.23607i	− 0.437862i
$144$	0	0
$145$	2.23607i	0.185695i
$146$	0	0
$147$	− 17.9443i	− 1.48002i
$148$	0	0
$149$	−5.41641	−0.443729	−0.221865	−	0.975077i	$-0.571214\pi$
$149$	−5.41641	−0.443729	−0.221865	+	0.975077i	$0.571214\pi$
$150$	0	0
$151$	−17.7082	−1.44107	−0.720537	−	0.693417i	$-0.756104\pi$
$151$	−17.7082	−1.44107	−0.720537	+	0.693417i	$0.756104\pi$
$152$	0	0
$153$	21.6525i	1.75050i
$154$	0	0
$155$	− 6.38197i	− 0.512612i
$156$	0	0
$157$	3.70820i	0.295947i	0.988991	+	0.147973i	$0.0472750\pi$
$157$	3.70820i	0.295947i	−0.988991	+	0.147973i	$0.952725\pi$
$158$	0	0
$159$	30.4164	2.41218
$160$	0	0
$161$	−0.708204	−0.0558143
$162$	0	0
$163$	19.4164i	1.52081i	0.649449	+	0.760405i	$0.275000\pi$
$163$	19.4164i	1.52081i	−0.649449	+	0.760405i	$0.725000\pi$
$164$	0	0
$165$	−11.7082	−0.911482
$166$	0	0
$167$	0.381966i	0.0295574i	0.999891	+	0.0147787i	$0.00470438\pi$
$167$	0.381966i	0.0295574i	−0.999891	+	0.0147787i	$0.995296\pi$
$168$	0	0
$169$	6.14590	0.472761
$170$	0	0
$171$	7.70820	0.589461
$172$	0	0
$173$	− 20.5066i	− 1.55909i	−0.626349	−	0.779543i	$-0.715451\pi$
$173$	− 20.5066i	− 1.55909i	0.626349	−	0.779543i	$-0.284549\pi$
$174$	0	0
$175$	− 1.90983i	− 0.144370i
$176$	0	0
$177$	10.0902i	0.758424i
$178$	0	0
$179$	15.8541	1.18499	0.592496	−	0.805574i	$-0.298143\pi$
$179$	15.8541	1.18499	0.592496	+	0.805574i	$0.298143\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	10.8541i	0.802358i
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	11.2361i	0.821663i
$188$	0	0
$189$	−0.854102	−0.0621268
$190$	0	0
$191$	10.2705	0.743148	0.371574	−	0.928403i	$-0.378818\pi$
$191$	10.2705	0.743148	0.371574	+	0.928403i	$0.378818\pi$
$192$	0	0
$193$	17.5623i	1.26416i	0.774902	+	0.632081i	$0.217799\pi$
$193$	17.5623i	1.26416i	−0.774902	+	0.632081i	$0.782201\pi$
$194$	0	0
$195$	15.3262i	1.09753i
$196$	0	0
$197$	− 11.6180i	− 0.827751i	−0.910334	−	0.413875i	$-0.864175\pi$
$197$	− 11.6180i	− 0.827751i	0.910334	−	0.413875i	$-0.135825\pi$
$198$	0	0
$199$	−21.1246	−1.49748	−0.748742	−	0.662862i	$-0.769342\pi$
$199$	−21.1246	−1.49748	−0.748742	+	0.662862i	$0.769342\pi$
$200$	0	0
$201$	23.4164	1.65167
$202$	0	0
$203$	0.381966i	0.0268088i
$204$	0	0
$205$	− 12.7639i	− 0.891472i
$206$	0	0
$207$	− 7.14590i	− 0.496674i
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−15.4164	−1.06131	−0.530655	−	0.847588i	$-0.678054\pi$
$211$	−15.4164	−1.06131	−0.530655	+	0.847588i	$0.678054\pi$
$212$	0	0
$213$	20.1803i	1.38273i
$214$	0	0
$215$	12.5623	0.856742
$216$	0	0
$217$	− 1.09017i	− 0.0740056i
$218$	0	0
$219$	38.1246	2.57622
$220$	0	0
$221$	14.7082	0.989381
$222$	0	0
$223$	6.43769i	0.431100i	0.976493	+	0.215550i	$0.0691544\pi$
$223$	6.43769i	0.431100i	−0.976493	+	0.215550i	$0.930846\pi$
$224$	0	0
$225$	19.2705	1.28470
$226$	0	0
$227$	7.52786i	0.499642i	0.968292	+	0.249821i	$0.0803718\pi$
$227$	7.52786i	0.499642i	−0.968292	+	0.249821i	$0.919628\pi$
$228$	0	0
$229$	14.8541	0.981587	0.490793	−	0.871276i	$-0.336707\pi$
$229$	14.8541	0.981587	0.490793	+	0.871276i	$0.336707\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	− 6.76393i	− 0.443120i	−0.975147	−	0.221560i	$-0.928885\pi$
$233$	− 6.76393i	− 0.443120i	0.975147	−	0.221560i	$-0.0711149\pi$
$234$	0	0
$235$	−23.4164	−1.52752
$236$	0	0
$237$	35.5066i	2.30640i
$238$	0	0
$239$	−7.70820	−0.498602	−0.249301	−	0.968426i	$-0.580201\pi$
$239$	−7.70820	−0.498602	−0.249301	+	0.968426i	$0.580201\pi$
$240$	0	0
$241$	−2.14590	−0.138229	−0.0691147	−	0.997609i	$-0.522017\pi$
$241$	−2.14590	−0.138229	−0.0691147	+	0.997609i	$0.522017\pi$
$242$	0	0
$243$	21.6525i	1.38901i
$244$	0	0
$245$	15.3262i	0.979157i
$246$	0	0
$247$	− 5.23607i	− 0.333163i
$248$	0	0
$249$	7.70820	0.488488
$250$	0	0
$251$	−11.1246	−0.702179	−0.351090	−	0.936342i	$-0.614189\pi$
$251$	−11.1246	−0.702179	−0.351090	+	0.936342i	$0.614189\pi$
$252$	0	0
$253$	− 3.70820i	− 0.233133i
$254$	0	0
$255$	− 32.8885i	− 2.05956i
$256$	0	0
$257$	− 15.7082i	− 0.979851i	−0.871764	−	0.489925i	$-0.837024\pi$
$257$	− 15.7082i	− 0.979851i	0.871764	−	0.489925i	$-0.162976\pi$
$258$	0	0
$259$	−3.41641	−0.212285
$260$	0	0
$261$	−3.85410	−0.238563
$262$	0	0
$263$	− 27.5967i	− 1.70169i	−0.525418	−	0.850844i	$-0.676091\pi$
$263$	− 27.5967i	− 1.70169i	0.525418	−	0.850844i	$-0.323909\pi$
$264$	0	0
$265$	−25.9787	−1.59586
$266$	0	0
$267$	5.23607i	0.320442i
$268$	0	0
$269$	30.2705	1.84563	0.922813	−	0.385249i	$-0.125884\pi$
$269$	30.2705	1.84563	0.922813	+	0.385249i	$0.125884\pi$
$270$	0	0
$271$	3.41641	0.207532	0.103766	−	0.994602i	$-0.466911\pi$
$271$	3.41641	0.207532	0.103766	+	0.994602i	$0.466911\pi$
$272$	0	0
$273$	2.61803i	0.158451i
$274$	0	0
$275$	10.0000	0.603023
$276$	0	0
$277$	30.0000i	1.80253i	0.433273	+	0.901263i	$0.357359\pi$
$277$	30.0000i	1.80253i	−0.433273	+	0.901263i	$0.642641\pi$
$278$	0	0
$279$	11.0000	0.658553
$280$	0	0
$281$	2.56231	0.152854	0.0764272	−	0.997075i	$-0.475649\pi$
$281$	2.56231	0.152854	0.0764272	+	0.997075i	$0.475649\pi$
$282$	0	0
$283$	11.2361i	0.667915i	0.942588	+	0.333957i	$0.108384\pi$
$283$	11.2361i	0.667915i	−0.942588	+	0.333957i	$0.891616\pi$
$284$	0	0
$285$	−11.7082	−0.693534
$286$	0	0
$287$	− 2.18034i	− 0.128701i
$288$	0	0
$289$	−14.5623	−0.856606
$290$	0	0
$291$	−36.2705	−2.12621
$292$	0	0
$293$	7.41641i	0.433271i	0.976253	+	0.216636i	$0.0695083\pi$
$293$	7.41641i	0.433271i	−0.976253	+	0.216636i	$0.930492\pi$
$294$	0	0
$295$	− 8.61803i	− 0.501761i
$296$	0	0
$297$	− 4.47214i	− 0.259500i
$298$	0	0
$299$	−4.85410	−0.280720
$300$	0	0
$301$	2.14590	0.123688
$302$	0	0
$303$	− 10.8541i	− 0.623552i
$304$	0	0
$305$	− 9.27051i	− 0.530828i
$306$	0	0
$307$	− 24.0000i	− 1.36975i	−0.728659	−	0.684876i	$-0.759856\pi$
$307$	− 24.0000i	− 1.36975i	0.728659	−	0.684876i	$-0.240144\pi$
$308$	0	0
$309$	50.8328	2.89178
$310$	0	0
$311$	14.4377	0.818687	0.409343	−	0.912380i	$-0.365758\pi$
$311$	14.4377	0.818687	0.409343	+	0.912380i	$0.365758\pi$
$312$	0	0
$313$	− 17.2361i	− 0.974240i	−0.873335	−	0.487120i	$-0.838047\pi$
$313$	− 17.2361i	− 0.974240i	0.873335	−	0.487120i	$-0.161953\pi$
$314$	0	0
$315$	3.29180	0.185472
$316$	0	0
$317$	− 2.18034i	− 0.122460i	−0.998124	−	0.0612300i	$-0.980498\pi$
$317$	− 2.18034i	− 0.122460i	0.998124	−	0.0612300i	$-0.0195023\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	21.4164	1.19535
$322$	0	0
$323$	11.2361i	0.625192i
$324$	0	0
$325$	− 13.0902i	− 0.726112i
$326$	0	0
$327$	− 19.4164i	− 1.07373i
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	13.1246	0.721394	0.360697	−	0.932683i	$-0.382539\pi$
$331$	13.1246	0.721394	0.360697	+	0.932683i	$0.382539\pi$
$332$	0	0
$333$	− 34.4721i	− 1.88906i
$334$	0	0
$335$	−20.0000	−1.09272
$336$	0	0
$337$	− 6.38197i	− 0.347648i	−0.984777	−	0.173824i	$-0.944388\pi$
$337$	− 6.38197i	− 0.347648i	0.984777	−	0.173824i	$-0.0556124\pi$
$338$	0	0
$339$	−18.5623	−1.00817
$340$	0	0
$341$	5.70820	0.309117
$342$	0	0
$343$	5.29180i	0.285730i
$344$	0	0
$345$	10.8541i	0.584365i
$346$	0	0
$347$	− 6.00000i	− 0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$	− 6.00000i	− 0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$	0	0
$349$	−28.8328	−1.54339	−0.771693	−	0.635996i	$-0.780590\pi$
$349$	−28.8328	−1.54339	−0.771693	+	0.635996i	$0.780590\pi$
$350$	0	0
$351$	−5.85410	−0.312469
$352$	0	0
$353$	− 20.9443i	− 1.11475i	−0.830260	−	0.557376i	$-0.811808\pi$
$353$	− 20.9443i	− 1.11475i	0.830260	−	0.557376i	$-0.188192\pi$
$354$	0	0
$355$	− 17.2361i	− 0.914796i
$356$	0	0
$357$	− 5.61803i	− 0.297338i
$358$	0	0
$359$	−33.2705	−1.75595	−0.877975	−	0.478706i	$-0.841106\pi$
$359$	−33.2705	−1.75595	−0.877975	+	0.478706i	$0.841106\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	18.3262i	0.961878i
$364$	0	0
$365$	−32.5623	−1.70439
$366$	0	0
$367$	− 5.12461i	− 0.267503i	−0.991015	−	0.133751i	$-0.957298\pi$
$367$	− 5.12461i	− 0.267503i	0.991015	−	0.133751i	$-0.0427023\pi$
$368$	0	0
$369$	22.0000	1.14527
$370$	0	0
$371$	−4.43769	−0.230394
$372$	0	0
$373$	− 18.2705i	− 0.946011i	−0.881059	−	0.473006i	$-0.843169\pi$
$373$	− 18.2705i	− 0.946011i	0.881059	−	0.473006i	$-0.156831\pi$
$374$	0	0
$375$	−29.2705	−1.51152
$376$	0	0
$377$	2.61803i	0.134836i
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	−56.8328	−2.91163
$382$	0	0
$383$	1.14590i	0.0585527i	0.999571	+	0.0292763i	$0.00932028\pi$
$383$	1.14590i	0.0585527i	−0.999571	+	0.0292763i	$0.990680\pi$
$384$	0	0
$385$	1.70820	0.0870581
$386$	0	0
$387$	21.6525i	1.10066i
$388$	0	0
$389$	−32.8328	−1.66469	−0.832345	−	0.554258i	$-0.813002\pi$
$389$	−32.8328	−1.66469	−0.832345	+	0.554258i	$0.813002\pi$
$390$	0	0
$391$	10.4164	0.526780
$392$	0	0
$393$	− 14.1803i	− 0.715304i
$394$	0	0
$395$	− 30.3262i	− 1.52588i
$396$	0	0
$397$	31.0344i	1.55757i	0.627288	+	0.778787i	$0.284165\pi$
$397$	31.0344i	1.55757i	−0.627288	+	0.778787i	$0.715835\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−24.9787	−1.24738	−0.623689	−	0.781673i	$-0.714367\pi$
$401$	−24.9787	−1.24738	−0.623689	+	0.781673i	$0.714367\pi$
$402$	0	0
$403$	− 7.47214i	− 0.372214i
$404$	0	0
$405$	− 12.7639i	− 0.634245i
$406$	0	0
$407$	− 17.8885i	− 0.886702i
$408$	0	0
$409$	−33.1246	−1.63791	−0.818953	−	0.573860i	$-0.805445\pi$
$409$	−33.1246	−1.63791	−0.818953	+	0.573860i	$0.805445\pi$
$410$	0	0
$411$	−4.85410	−0.239435
$412$	0	0
$413$	− 1.47214i	− 0.0724391i
$414$	0	0
$415$	−6.58359	−0.323176
$416$	0	0
$417$	− 38.1246i	− 1.86697i
$418$	0	0
$419$	−15.1459	−0.739926	−0.369963	−	0.929047i	$-0.620630\pi$
$419$	−15.1459	−0.739926	−0.369963	+	0.929047i	$0.620630\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	− 40.3607i	− 1.96240i
$424$	0	0
$425$	28.0902i	1.36257i
$426$	0	0
$427$	− 1.58359i	− 0.0766354i
$428$	0	0
$429$	−13.7082	−0.661838
$430$	0	0
$431$	6.58359	0.317120	0.158560	−	0.987349i	$-0.449315\pi$
$431$	6.58359	0.317120	0.158560	+	0.987349i	$0.449315\pi$
$432$	0	0
$433$	− 7.52786i	− 0.361766i	−0.983505	−	0.180883i	$-0.942104\pi$
$433$	− 7.52786i	− 0.361766i	0.983505	−	0.180883i	$-0.0578955\pi$
$434$	0	0
$435$	5.85410	0.280683
$436$	0	0
$437$	− 3.70820i	− 0.177387i
$438$	0	0
$439$	3.12461	0.149130	0.0745648	−	0.997216i	$-0.476243\pi$
$439$	3.12461	0.149130	0.0745648	+	0.997216i	$0.476243\pi$
$440$	0	0
$441$	−26.4164	−1.25792
$442$	0	0
$443$	13.0902i	0.621933i	0.950421	+	0.310966i	$0.100653\pi$
$443$	13.0902i	0.621933i	−0.950421	+	0.310966i	$0.899347\pi$
$444$	0	0
$445$	− 4.47214i	− 0.212000i
$446$	0	0
$447$	14.1803i	0.670707i
$448$	0	0
$449$	31.4164	1.48263	0.741316	−	0.671156i	$-0.234202\pi$
$449$	31.4164	1.48263	0.741316	+	0.671156i	$0.234202\pi$
$450$	0	0
$451$	11.4164	0.537578
$452$	0	0
$453$	46.3607i	2.17821i
$454$	0	0
$455$	− 2.23607i	− 0.104828i
$456$	0	0
$457$	27.7082i	1.29614i	0.761583	+	0.648068i	$0.224423\pi$
$457$	27.7082i	1.29614i	−0.761583	+	0.648068i	$0.775577\pi$
$458$	0	0
$459$	12.5623	0.586358
$460$	0	0
$461$	−30.8541	−1.43702	−0.718509	−	0.695517i	$-0.755175\pi$
$461$	−30.8541	−1.43702	−0.718509	+	0.695517i	$0.755175\pi$
$462$	0	0
$463$	− 17.8885i	− 0.831351i	−0.909513	−	0.415676i	$-0.863545\pi$
$463$	− 17.8885i	− 0.831351i	0.909513	−	0.415676i	$-0.136455\pi$
$464$	0	0
$465$	−16.7082	−0.774824
$466$	0	0
$467$	10.2016i	0.472075i	0.971744	+	0.236037i	$0.0758488\pi$
$467$	10.2016i	0.472075i	−0.971744	+	0.236037i	$0.924151\pi$
$468$	0	0
$469$	−3.41641	−0.157755
$470$	0	0
$471$	9.70820	0.447330
$472$	0	0
$473$	11.2361i	0.516635i
$474$	0	0
$475$	10.0000	0.458831
$476$	0	0
$477$	− 44.7771i	− 2.05020i
$478$	0	0
$479$	27.9787	1.27838	0.639190	−	0.769049i	$-0.279270\pi$
$479$	27.9787	1.27838	0.639190	+	0.769049i	$0.279270\pi$
$480$	0	0
$481$	−23.4164	−1.06770
$482$	0	0
$483$	1.85410i	0.0843646i
$484$	0	0
$485$	30.9787	1.40667
$486$	0	0
$487$	18.4377i	0.835492i	0.908564	+	0.417746i	$0.137180\pi$
$487$	18.4377i	0.835492i	−0.908564	+	0.417746i	$0.862820\pi$
$488$	0	0
$489$	50.8328	2.29874
$490$	0	0
$491$	19.7082	0.889419	0.444709	−	0.895675i	$-0.353307\pi$
$491$	19.7082	0.889419	0.444709	+	0.895675i	$0.353307\pi$
$492$	0	0
$493$	− 5.61803i	− 0.253024i
$494$	0	0
$495$	17.2361i	0.774704i
$496$	0	0
$497$	− 2.94427i	− 0.132069i
$498$	0	0
$499$	8.43769	0.377723	0.188862	−	0.982004i	$-0.439520\pi$
$499$	8.43769	0.377723	0.188862	+	0.982004i	$0.439520\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	− 15.0557i	− 0.671302i	−0.941986	−	0.335651i	$-0.891044\pi$
$503$	− 15.0557i	− 0.671302i	0.941986	−	0.335651i	$-0.108956\pi$
$504$	0	0
$505$	9.27051i	0.412532i
$506$	0	0
$507$	− 16.0902i	− 0.714590i
$508$	0	0
$509$	−19.1246	−0.847684	−0.423842	−	0.905736i	$-0.639319\pi$
$509$	−19.1246	−0.847684	−0.423842	+	0.905736i	$0.639319\pi$
$510$	0	0
$511$	−5.56231	−0.246062
$512$	0	0
$513$	− 4.47214i	− 0.197450i
$514$	0	0
$515$	−43.4164	−1.91316
$516$	0	0
$517$	− 20.9443i	− 0.921128i
$518$	0	0
$519$	−53.6869	−2.35659
$520$	0	0
$521$	37.3951	1.63831	0.819155	−	0.573572i	$-0.194443\pi$
$521$	37.3951	1.63831	0.819155	+	0.573572i	$0.194443\pi$
$522$	0	0
$523$	− 29.1246i	− 1.27353i	−0.771058	−	0.636765i	$-0.780272\pi$
$523$	− 29.1246i	− 1.27353i	0.771058	−	0.636765i	$-0.219728\pi$
$524$	0	0
$525$	−5.00000	−0.218218
$526$	0	0
$527$	16.0344i	0.698471i
$528$	0	0
$529$	19.5623	0.850535
$530$	0	0
$531$	14.8541	0.644613
$532$	0	0
$533$	− 14.9443i	− 0.647308i
$534$	0	0
$535$	−18.2918	−0.790823
$536$	0	0
$537$	− 41.5066i	− 1.79114i
$538$	0	0
$539$	−13.7082	−0.590454
$540$	0	0
$541$	−7.43769	−0.319771	−0.159886	−	0.987136i	$-0.551113\pi$
$541$	−7.43769	−0.319771	−0.159886	+	0.987136i	$0.551113\pi$
$542$	0	0
$543$	26.1803i	1.12351i
$544$	0	0
$545$	16.5836i	0.710363i
$546$	0	0
$547$	38.8328i	1.66037i	0.557487	+	0.830186i	$0.311766\pi$
$547$	38.8328i	1.66037i	−0.557487	+	0.830186i	$0.688234\pi$
$548$	0	0
$549$	15.9787	0.681955
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	− 5.18034i	− 0.220290i
$554$	0	0
$555$	52.3607i	2.22259i
$556$	0	0
$557$	− 16.7984i	− 0.711770i	−0.934530	−	0.355885i	$-0.884179\pi$
$557$	− 16.7984i	− 0.711770i	0.934530	−	0.355885i	$-0.115821\pi$
$558$	0	0
$559$	14.7082	0.622091
$560$	0	0
$561$	29.4164	1.24196
$562$	0	0
$563$	− 17.6738i	− 0.744860i	−0.928060	−	0.372430i	$-0.878525\pi$
$563$	− 17.6738i	− 0.744860i	0.928060	−	0.372430i	$-0.121475\pi$
$564$	0	0
$565$	15.8541	0.666987
$566$	0	0
$567$	− 2.18034i	− 0.0915657i
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	0.145898	0.00610564	0.00305282	−	0.999995i	$-0.499028\pi$
$571$	0.145898	0.00610564	0.00305282	+	0.999995i	$0.499028\pi$
$572$	0	0
$573$	− 26.8885i	− 1.12329i
$574$	0	0
$575$	− 9.27051i	− 0.386607i
$576$	0	0
$577$	− 1.03444i	− 0.0430644i	−0.999768	−	0.0215322i	$-0.993146\pi$
$577$	− 1.03444i	− 0.0430644i	0.999768	−	0.0215322i	$-0.00685444\pi$
$578$	0	0
$579$	45.9787	1.91081
$580$	0	0
$581$	−1.12461	−0.0466568
$582$	0	0
$583$	− 23.2361i	− 0.962340i
$584$	0	0
$585$	22.5623	0.932837
$586$	0	0
$587$	− 18.6525i	− 0.769870i	−0.922944	−	0.384935i	$-0.874224\pi$
$587$	− 18.6525i	− 0.769870i	0.922944	−	0.384935i	$-0.125776\pi$
$588$	0	0
$589$	5.70820	0.235202
$590$	0	0
$591$	−30.4164	−1.25116
$592$	0	0
$593$	− 19.3050i	− 0.792759i	−0.918087	−	0.396380i	$-0.870266\pi$
$593$	− 19.3050i	− 0.792759i	0.918087	−	0.396380i	$-0.129734\pi$
$594$	0	0
$595$	4.79837i	0.196714i
$596$	0	0
$597$	55.3050i	2.26348i
$598$	0	0
$599$	3.43769	0.140460	0.0702302	−	0.997531i	$-0.477627\pi$
$599$	3.43769	0.140460	0.0702302	+	0.997531i	$0.477627\pi$
$600$	0	0
$601$	−7.70820	−0.314424	−0.157212	−	0.987565i	$-0.550251\pi$
$601$	−7.70820	−0.314424	−0.157212	+	0.987565i	$0.550251\pi$
$602$	0	0
$603$	− 34.4721i	− 1.40381i
$604$	0	0
$605$	− 15.6525i	− 0.636364i
$606$	0	0
$607$	5.23607i	0.212525i	0.994338	+	0.106263i	$0.0338885\pi$
$607$	5.23607i	0.212525i	−0.994338	+	0.106263i	$0.966112\pi$
$608$	0	0
$609$	1.00000	0.0405220
$610$	0	0
$611$	−27.4164	−1.10915
$612$	0	0
$613$	− 28.7426i	− 1.16090i	−0.814294	−	0.580452i	$-0.802876\pi$
$613$	− 28.7426i	− 1.16090i	0.814294	−	0.580452i	$-0.197124\pi$
$614$	0	0
$615$	−33.4164	−1.34748
$616$	0	0
$617$	− 30.3820i	− 1.22313i	−0.791193	−	0.611566i	$-0.790540\pi$
$617$	− 30.3820i	− 1.22313i	0.791193	−	0.611566i	$-0.209460\pi$
$618$	0	0
$619$	19.4164	0.780411	0.390206	−	0.920728i	$-0.372404\pi$
$619$	19.4164	0.780411	0.390206	+	0.920728i	$0.372404\pi$
$620$	0	0
$621$	−4.14590	−0.166369
$622$	0	0
$623$	− 0.763932i	− 0.0306063i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	− 10.4721i	− 0.418217i
$628$	0	0
$629$	50.2492	2.00357
$630$	0	0
$631$	−18.8328	−0.749723	−0.374861	−	0.927081i	$-0.622310\pi$
$631$	−18.8328	−0.749723	−0.374861	+	0.927081i	$0.622310\pi$
$632$	0	0
$633$	40.3607i	1.60419i
$634$	0	0
$635$	48.5410	1.92629
$636$	0	0
$637$	17.9443i	0.710978i
$638$	0	0
$639$	29.7082	1.17524
$640$	0	0
$641$	13.7082	0.541442	0.270721	−	0.962658i	$-0.412738\pi$
$641$	13.7082	0.541442	0.270721	+	0.962658i	$0.412738\pi$
$642$	0	0
$643$	43.4164i	1.71218i	0.516830	+	0.856088i	$0.327112\pi$
$643$	43.4164i	1.71218i	−0.516830	+	0.856088i	$0.672888\pi$
$644$	0	0
$645$	− 32.8885i	− 1.29499i
$646$	0	0
$647$	22.4721i	0.883471i	0.897145	+	0.441735i	$0.145637\pi$
$647$	22.4721i	0.883471i	−0.897145	+	0.441735i	$0.854363\pi$
$648$	0	0
$649$	7.70820	0.302573
$650$	0	0
$651$	−2.85410	−0.111861
$652$	0	0
$653$	− 25.3050i	− 0.990259i	−0.868819	−	0.495130i	$-0.835121\pi$
$653$	− 25.3050i	− 0.990259i	0.868819	−	0.495130i	$-0.164879\pi$
$654$	0	0
$655$	12.1115i	0.473234i
$656$	0	0
$657$	− 56.1246i	− 2.18963i
$658$	0	0
$659$	−26.5410	−1.03389	−0.516946	−	0.856018i	$-0.672931\pi$
$659$	−26.5410	−1.03389	−0.516946	+	0.856018i	$0.672931\pi$
$660$	0	0
$661$	−24.8328	−0.965885	−0.482942	−	0.875652i	$-0.660432\pi$
$661$	−24.8328	−0.965885	−0.482942	+	0.875652i	$0.660432\pi$
$662$	0	0
$663$	− 38.5066i	− 1.49547i
$664$	0	0
$665$	1.70820	0.0662413
$666$	0	0
$667$	1.85410i	0.0717911i
$668$	0	0
$669$	16.8541	0.651617
$670$	0	0
$671$	8.29180	0.320101
$672$	0	0
$673$	9.05573i	0.349073i	0.984651	+	0.174536i	$0.0558426\pi$
$673$	9.05573i	0.349073i	−0.984651	+	0.174536i	$0.944157\pi$
$674$	0	0
$675$	− 11.1803i	− 0.430331i
$676$	0	0
$677$	− 15.0557i	− 0.578639i	−0.957233	−	0.289319i	$-0.906571\pi$
$677$	− 15.0557i	− 0.578639i	0.957233	−	0.289319i	$-0.0934289\pi$
$678$	0	0
$679$	5.29180	0.203080
$680$	0	0
$681$	19.7082	0.755220
$682$	0	0
$683$	− 47.7771i	− 1.82814i	−0.405557	−	0.914070i	$-0.632922\pi$
$683$	− 47.7771i	− 1.82814i	0.405557	−	0.914070i	$-0.367078\pi$
$684$	0	0
$685$	4.14590	0.158407
$686$	0	0
$687$	− 38.8885i	− 1.48369i
$688$	0	0
$689$	−30.4164	−1.15877
$690$	0	0
$691$	−6.56231	−0.249642	−0.124821	−	0.992179i	$-0.539836\pi$
$691$	−6.56231	−0.249642	−0.124821	+	0.992179i	$0.539836\pi$
$692$	0	0
$693$	2.94427i	0.111844i
$694$	0	0
$695$	32.5623i	1.23516i
$696$	0	0
$697$	32.0689i	1.21470i
$698$	0	0
$699$	−17.7082	−0.669786
$700$	0	0
$701$	31.1246	1.17556	0.587780	−	0.809021i	$-0.300002\pi$
$701$	31.1246	1.17556	0.587780	+	0.809021i	$0.300002\pi$
$702$	0	0
$703$	− 17.8885i	− 0.674679i
$704$	0	0
$705$	61.3050i	2.30888i
$706$	0	0
$707$	1.58359i	0.0595571i
$708$	0	0
$709$	−25.7082	−0.965492	−0.482746	−	0.875760i	$-0.660361\pi$
$709$	−25.7082	−0.965492	−0.482746	+	0.875760i	$0.660361\pi$
$710$	0	0
$711$	52.2705	1.96030
$712$	0	0
$713$	− 5.29180i	− 0.198179i
$714$	0	0
$715$	11.7082	0.437862
$716$	0	0
$717$	20.1803i	0.753649i
$718$	0	0
$719$	21.7082	0.809579	0.404790	−	0.914410i	$-0.367345\pi$
$719$	21.7082	0.809579	0.404790	+	0.914410i	$0.367345\pi$
$720$	0	0
$721$	−7.41641	−0.276201
$722$	0	0
$723$	5.61803i	0.208937i
$724$	0	0
$725$	−5.00000	−0.185695
$726$	0	0
$727$	− 15.0557i	− 0.558386i	−0.960235	−	0.279193i	$-0.909933\pi$
$727$	− 15.0557i	− 0.558386i	0.960235	−	0.279193i	$-0.0900669\pi$
$728$	0	0
$729$	39.5623	1.46527
$730$	0	0
$731$	−31.5623	−1.16737
$732$	0	0
$733$	12.0000i	0.443230i	0.975134	+	0.221615i	$0.0711328\pi$
$733$	12.0000i	0.443230i	−0.975134	+	0.221615i	$0.928867\pi$
$734$	0	0
$735$	40.1246	1.48002
$736$	0	0
$737$	− 17.8885i	− 0.658933i
$738$	0	0
$739$	41.7082	1.53426	0.767131	−	0.641491i	$-0.221684\pi$
$739$	41.7082	1.53426	0.767131	+	0.641491i	$0.221684\pi$
$740$	0	0
$741$	−13.7082	−0.503583
$742$	0	0
$743$	− 1.41641i	− 0.0519630i	−0.999662	−	0.0259815i	$-0.991729\pi$
$743$	− 1.41641i	− 0.0519630i	0.999662	−	0.0259815i	$-0.00827109\pi$
$744$	0	0
$745$	− 12.1115i	− 0.443729i
$746$	0	0
$747$	− 11.3475i	− 0.415184i
$748$	0	0
$749$	−3.12461	−0.114171
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	29.1246i	1.06136i
$754$	0	0
$755$	− 39.5967i	− 1.44107i
$756$	0	0
$757$	26.0689i	0.947490i	0.880662	+	0.473745i	$0.157098\pi$
$757$	26.0689i	0.947490i	−0.880662	+	0.473745i	$0.842902\pi$
$758$	0	0
$759$	−9.70820	−0.352385
$760$	0	0
$761$	−3.14590	−0.114039	−0.0570194	−	0.998373i	$-0.518160\pi$
$761$	−3.14590	−0.114039	−0.0570194	+	0.998373i	$0.518160\pi$
$762$	0	0
$763$	2.83282i	0.102555i
$764$	0	0
$765$	−48.4164	−1.75050
$766$	0	0
$767$	− 10.0902i	− 0.364335i
$768$	0	0
$769$	45.4164	1.63776	0.818879	−	0.573967i	$-0.194596\pi$
$769$	45.4164	1.63776	0.818879	+	0.573967i	$0.194596\pi$
$770$	0	0
$771$	−41.1246	−1.48107
$772$	0	0
$773$	11.2361i	0.404133i	0.979372	+	0.202067i	$0.0647658\pi$
$773$	11.2361i	0.404133i	−0.979372	+	0.202067i	$0.935234\pi$
$774$	0	0
$775$	14.2705	0.512612
$776$	0	0
$777$	8.94427i	0.320874i
$778$	0	0
$779$	11.4164	0.409035
$780$	0	0
$781$	15.4164	0.551642
$782$	0	0
$783$	2.23607i	0.0799106i
$784$	0	0
$785$	−8.29180	−0.295947
$786$	0	0
$787$	29.7771i	1.06144i	0.847548	+	0.530719i	$0.178078\pi$
$787$	29.7771i	1.06144i	−0.847548	+	0.530719i	$0.821922\pi$
$788$	0	0
$789$	−72.2492	−2.57214
$790$	0	0
$791$	2.70820	0.0962926
$792$	0	0
$793$	− 10.8541i	− 0.385440i
$794$	0	0
$795$	68.0132i	2.41218i
$796$	0	0
$797$	30.7639i	1.08971i	0.838529	+	0.544857i	$0.183416\pi$
$797$	30.7639i	1.08971i	−0.838529	+	0.544857i	$0.816584\pi$
$798$	0	0
$799$	58.8328	2.08136
$800$	0	0
$801$	7.70820	0.272356
$802$	0	0
$803$	− 29.1246i	− 1.02779i
$804$	0	0
$805$	− 1.58359i	− 0.0558143i
$806$	0	0
$807$	− 79.2492i	− 2.78970i
$808$	0	0
$809$	12.5836	0.442416	0.221208	−	0.975227i	$-0.429000\pi$
$809$	12.5836	0.442416	0.221208	+	0.975227i	$0.429000\pi$
$810$	0	0
$811$	25.3951	0.891743	0.445872	−	0.895097i	$-0.352894\pi$
$811$	25.3951	0.891743	0.445872	+	0.895097i	$0.352894\pi$
$812$	0	0
$813$	− 8.94427i	− 0.313689i
$814$	0	0
$815$	−43.4164	−1.52081
$816$	0	0
$817$	11.2361i	0.393100i
$818$	0	0
$819$	3.85410	0.134673
$820$	0	0
$821$	−8.00000	−0.279202	−0.139601	−	0.990208i	$-0.544582\pi$
$821$	−8.00000	−0.279202	−0.139601	+	0.990208i	$0.544582\pi$
$822$	0	0
$823$	− 0.763932i	− 0.0266290i	−0.999911	−	0.0133145i	$-0.995762\pi$
$823$	− 0.763932i	− 0.0266290i	0.999911	−	0.0133145i	$-0.00423826\pi$
$824$	0	0
$825$	− 26.1803i	− 0.911482i
$826$	0	0
$827$	49.7984i	1.73166i	0.500339	+	0.865830i	$0.333209\pi$
$827$	49.7984i	1.73166i	−0.500339	+	0.865830i	$0.666791\pi$
$828$	0	0
$829$	34.2705	1.19026	0.595132	−	0.803628i	$-0.297100\pi$
$829$	34.2705	1.19026	0.595132	+	0.803628i	$0.297100\pi$
$830$	0	0
$831$	78.5410	2.72456
$832$	0	0
$833$	− 38.5066i	− 1.33417i
$834$	0	0
$835$	−0.854102	−0.0295574
$836$	0	0
$837$	− 6.38197i	− 0.220593i
$838$	0	0
$839$	14.8328	0.512086	0.256043	−	0.966665i	$-0.417581\pi$
$839$	14.8328	0.512086	0.256043	+	0.966665i	$0.417581\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	− 6.70820i	− 0.231043i
$844$	0	0
$845$	13.7426i	0.472761i
$846$	0	0
$847$	− 2.67376i	− 0.0918716i
$848$	0	0
$849$	29.4164	1.00957
$850$	0	0
$851$	−16.5836	−0.568478
$852$	0	0
$853$	44.8328i	1.53505i	0.641021	+	0.767523i	$0.278511\pi$
$853$	44.8328i	1.53505i	−0.641021	+	0.767523i	$0.721489\pi$
$854$	0	0
$855$	17.2361i	0.589461i
$856$	0	0
$857$	− 44.1803i	− 1.50917i	−0.656201	−	0.754586i	$-0.727838\pi$
$857$	− 44.1803i	− 1.50917i	0.656201	−	0.754586i	$-0.272162\pi$
$858$	0	0
$859$	14.2918	0.487630	0.243815	−	0.969822i	$-0.421601\pi$
$859$	14.2918	0.487630	0.243815	+	0.969822i	$0.421601\pi$
$860$	0	0
$861$	−5.70820	−0.194535
$862$	0	0
$863$	33.9230i	1.15475i	0.816478	+	0.577376i	$0.195923\pi$
$863$	33.9230i	1.15475i	−0.816478	+	0.577376i	$0.804077\pi$
$864$	0	0
$865$	45.8541	1.55909
$866$	0	0
$867$	38.1246i	1.29478i
$868$	0	0
$869$	27.1246	0.920139
$870$	0	0
$871$	−23.4164	−0.793435
$872$	0	0
$873$	53.3951i	1.80715i
$874$	0	0
$875$	4.27051	0.144370
$876$	0	0
$877$	8.72949i	0.294774i	0.989079	+	0.147387i	$0.0470863\pi$
$877$	8.72949i	0.294774i	−0.989079	+	0.147387i	$0.952914\pi$
$878$	0	0
$879$	19.4164	0.654899
$880$	0	0
$881$	−41.4164	−1.39535	−0.697677	−	0.716412i	$-0.745783\pi$
$881$	−41.4164	−1.39535	−0.697677	+	0.716412i	$0.745783\pi$
$882$	0	0
$883$	20.2918i	0.682873i	0.939905	+	0.341437i	$0.110913\pi$
$883$	20.2918i	0.682873i	−0.939905	+	0.341437i	$0.889087\pi$
$884$	0	0
$885$	−22.5623	−0.758424
$886$	0	0
$887$	24.0000i	0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$887$	24.0000i	0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$888$	0	0
$889$	8.29180	0.278098
$890$	0	0
$891$	11.4164	0.382464
$892$	0	0
$893$	− 20.9443i	− 0.700873i
$894$	0	0
$895$	35.4508i	1.18499i
$896$	0	0
$897$	12.7082i	0.424315i
$898$	0	0
$899$	−2.85410	−0.0951896
$900$	0	0
$901$	65.2705	2.17448
$902$	0	0
$903$	− 5.61803i	− 0.186956i
$904$	0	0
$905$	− 22.3607i	− 0.743294i
$906$	0	0
$907$	34.7426i	1.15361i	0.816882	+	0.576805i	$0.195701\pi$
$907$	34.7426i	1.15361i	−0.816882	+	0.576805i	$0.804299\pi$
$908$	0	0
$909$	−15.9787	−0.529980
$910$	0	0
$911$	−6.14590	−0.203623	−0.101811	−	0.994804i	$-0.532464\pi$
$911$	−6.14590	−0.203623	−0.101811	+	0.994804i	$0.532464\pi$
$912$	0	0
$913$	− 5.88854i	− 0.194882i
$914$	0	0
$915$	−24.2705	−0.802358
$916$	0	0
$917$	2.06888i	0.0683206i
$918$	0	0
$919$	14.5410	0.479664	0.239832	−	0.970814i	$-0.422908\pi$
$919$	14.5410	0.479664	0.239832	+	0.970814i	$0.422908\pi$
$920$	0	0
$921$	−62.8328	−2.07041
$922$	0	0
$923$	− 20.1803i	− 0.664244i
$924$	0	0
$925$	− 44.7214i	− 1.47043i
$926$	0	0
$927$	− 74.8328i	− 2.45783i
$928$	0	0
$929$	−40.8541	−1.34038	−0.670190	−	0.742190i	$-0.733787\pi$
$929$	−40.8541	−1.34038	−0.670190	+	0.742190i	$0.733787\pi$
$930$	0	0
$931$	−13.7082	−0.449268
$932$	0	0
$933$	− 37.7984i	− 1.23746i
$934$	0	0
$935$	−25.1246	−0.821663
$936$	0	0
$937$	14.0689i	0.459610i	0.973237	+	0.229805i	$0.0738089\pi$
$937$	14.0689i	0.459610i	−0.973237	+	0.229805i	$0.926191\pi$
$938$	0	0
$939$	−45.1246	−1.47259
$940$	0	0
$941$	−8.29180	−0.270305	−0.135152	−	0.990825i	$-0.543152\pi$
$941$	−8.29180	−0.270305	−0.135152	+	0.990825i	$0.543152\pi$
$942$	0	0
$943$	− 10.5836i	− 0.344649i
$944$	0	0
$945$	− 1.90983i	− 0.0621268i
$946$	0	0
$947$	− 9.15905i	− 0.297629i	−0.988865	−	0.148815i	$-0.952454\pi$
$947$	− 9.15905i	− 0.297629i	0.988865	−	0.148815i	$-0.0475458\pi$
$948$	0	0
$949$	−38.1246	−1.23758
$950$	0	0
$951$	−5.70820	−0.185101
$952$	0	0
$953$	50.8328i	1.64664i	0.567580	+	0.823318i	$0.307880\pi$
$953$	50.8328i	1.64664i	−0.567580	+	0.823318i	$0.692120\pi$
$954$	0	0
$955$	22.9656i	0.743148i
$956$	0	0
$957$	5.23607i	0.169258i
$958$	0	0
$959$	0.708204	0.0228691
$960$	0	0
$961$	−22.8541	−0.737229
$962$	0	0
$963$	− 31.5279i	− 1.01597i
$964$	0	0
$965$	−39.2705	−1.26416
$966$	0	0
$967$	20.2918i	0.652540i	0.945277	+	0.326270i	$0.105792\pi$
$967$	20.2918i	0.652540i	−0.945277	+	0.326270i	$0.894208\pi$
$968$	0	0
$969$	29.4164	0.944991
$970$	0	0
$971$	11.7082	0.375734	0.187867	−	0.982194i	$-0.439843\pi$
$971$	11.7082	0.375734	0.187867	+	0.982194i	$0.439843\pi$
$972$	0	0
$973$	5.56231i	0.178319i
$974$	0	0
$975$	−34.2705	−1.09753
$976$	0	0
$977$	50.9443i	1.62985i	0.579565	+	0.814926i	$0.303222\pi$
$977$	50.9443i	1.62985i	−0.579565	+	0.814926i	$0.696778\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	0	0
$981$	−28.5836	−0.912604
$982$	0	0
$983$	− 45.5967i	− 1.45431i	−0.686473	−	0.727155i	$-0.740842\pi$
$983$	− 45.5967i	− 1.45431i	0.686473	−	0.727155i	$-0.259158\pi$
$984$	0	0
$985$	25.9787	0.827751
$986$	0	0
$987$	10.4721i	0.333332i
$988$	0	0
$989$	10.4164	0.331223
$990$	0	0
$991$	14.5836	0.463263	0.231632	−	0.972804i	$-0.425594\pi$
$991$	14.5836	0.463263	0.231632	+	0.972804i	$0.425594\pi$
$992$	0	0
$993$	− 34.3607i	− 1.09040i
$994$	0	0
$995$	− 47.2361i	− 1.49748i
$996$	0	0
$997$	23.1246i	0.732364i	0.930543	+	0.366182i	$0.119335\pi$
$997$	23.1246i	0.732364i	−0.930543	+	0.366182i	$0.880665\pi$
$998$	0	0
$999$	−20.0000	−0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.d.d.929.1		4
4.3	odd	2		290.2.b.a.59.4	yes	4
5.4	even	2	inner	2320.2.d.d.929.4		4
12.11	even	2		2610.2.e.e.2089.1		4
20.3	even	4		1450.2.a.l.1.1		2
20.7	even	4		1450.2.a.k.1.2		2
20.19	odd	2		290.2.b.a.59.1	✓	4
60.59	even	2		2610.2.e.e.2089.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
290.2.b.a.59.1	✓	4	20.19	odd	2
290.2.b.a.59.4	yes	4	4.3	odd	2
1450.2.a.k.1.2		2	20.7	even	4
1450.2.a.l.1.1		2	20.3	even	4
2320.2.d.d.929.1		4	1.1	even	1	trivial
2320.2.d.d.929.4		4	5.4	even	2	inner
2610.2.e.e.2089.1		4	12.11	even	2
2610.2.e.e.2089.4		4	60.59	even	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:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.61803i q^{3} +2.23607i q^{5} +0.381966i q^{7} -3.85410 q^{9} +O(q^{10})\)	\(q-2.61803i q^{3} +2.23607i q^{5} +0.381966i q^{7} -3.85410 q^{9} -2.00000 q^{11} +2.61803i q^{13} +5.85410 q^{15} -5.61803i q^{17} -2.00000 q^{19} +1.00000 q^{21} +1.85410i q^{23} -5.00000 q^{25} +2.23607i q^{27} +1.00000 q^{29} -2.85410 q^{31} +5.23607i q^{33} -0.854102 q^{35} +8.94427i q^{37} +6.85410 q^{39} -5.70820 q^{41} -5.61803i q^{43} -8.61803i q^{45} +10.4721i q^{47} +6.85410 q^{49} -14.7082 q^{51} +11.6180i q^{53} -4.47214i q^{55} +5.23607i q^{57} -3.85410 q^{59} -4.14590 q^{61} -1.47214i q^{63} -5.85410 q^{65} +8.94427i q^{67} +4.85410 q^{69} -7.70820 q^{71} +14.5623i q^{73} +13.0902i q^{75} -0.763932i q^{77} -13.5623 q^{79} -5.70820 q^{81} +2.94427i q^{83} +12.5623 q^{85} -2.61803i q^{87} -2.00000 q^{89} -1.00000 q^{91} +7.47214i q^{93} -4.47214i q^{95} -13.8541i q^{97} +7.70820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^9	\( 4 q - 2 q^{9} - 8 q^{11} + 10 q^{15} - 8 q^{19} + 4 q^{21} - 20 q^{25} + 4 q^{29} + 2 q^{31} + 10 q^{35} + 14 q^{39} + 4 q^{41} + 14 q^{49} - 32 q^{51} - 2 q^{59} - 30 q^{61} - 10 q^{65} + 6 q^{69} - 4 q^{71} - 14 q^{79} + 4 q^{81} + 10 q^{85} - 8 q^{89} - 4 q^{91} + 4 q^{99}+O(q^{100}) \) 4 * q - 2 * q^9 - 8 * q^11 + 10 * q^15 - 8 * q^19 + 4 * q^21 - 20 * q^25 + 4 * q^29 + 2 * q^31 + 10 * q^35 + 14 * q^39 + 4 * q^41 + 14 * q^49 - 32 * q^51 - 2 * q^59 - 30 * q^61 - 10 * q^65 + 6 * q^69 - 4 * q^71 - 14 * q^79 + 4 * q^81 + 10 * q^85 - 8 * q^89 - 4 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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