LMFDB - Embedded newform 2175.2.c.j.349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(349,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.c (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:	$17.3674624396$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.j.349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +3.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +3.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} -5.47214 q^{11} +1.61803i q^{12} -6.23607i q^{13} +1.85410 q^{14} +1.85410 q^{16} -3.47214i q^{17} +0.618034i q^{18} -7.70820 q^{19} -3.00000 q^{21} +3.38197i q^{22} +2.23607 q^{24} -3.85410 q^{26} -1.00000i q^{27} +4.85410i q^{28} -1.00000 q^{29} -8.00000 q^{31} -5.61803i q^{32} -5.47214i q^{33} -2.14590 q^{34} -1.61803 q^{36} +8.00000i q^{37} +4.76393i q^{38} +6.23607 q^{39} -4.47214 q^{41} +1.85410i q^{42} +3.23607i q^{43} -8.85410 q^{44} -6.70820i q^{47} +1.85410i q^{48} -2.00000 q^{49} +3.47214 q^{51} -10.0902i q^{52} -6.76393i q^{53} -0.618034 q^{54} +6.70820 q^{56} -7.70820i q^{57} +0.618034i q^{58} -5.23607 q^{59} -5.70820 q^{61} +4.94427i q^{62} -3.00000i q^{63} +0.236068 q^{64} -3.38197 q^{66} +11.4721i q^{67} -5.61803i q^{68} +7.23607 q^{71} +2.23607i q^{72} -8.00000i q^{73} +4.94427 q^{74} -12.4721 q^{76} -16.4164i q^{77} -3.85410i q^{78} +6.18034 q^{79} +1.00000 q^{81} +2.76393i q^{82} -3.70820i q^{83} -4.85410 q^{84} +2.00000 q^{86} -1.00000i q^{87} +12.2361i q^{88} -11.1803 q^{89} +18.7082 q^{91} -8.00000i q^{93} -4.14590 q^{94} +5.61803 q^{96} -2.76393i q^{97} +1.23607i q^{98} +5.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9	$ 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 4 q^{11} - 6 q^{14} - 6 q^{16} - 4 q^{19} - 12 q^{21} - 2 q^{26} - 4 q^{29} - 32 q^{31} - 22 q^{34} - 2 q^{36} + 16 q^{39} - 22 q^{44} - 8 q^{49} - 4 q^{51} + 2 q^{54} - 12 q^{59} + 4 q^{61} - 8 q^{64} - 18 q^{66} + 20 q^{71} - 16 q^{74} - 32 q^{76} - 20 q^{79} + 4 q^{81} - 6 q^{84} + 8 q^{86} + 48 q^{91} - 30 q^{94} + 18 q^{96} + 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 4 * q^11 - 6 * q^14 - 6 * q^16 - 4 * q^19 - 12 * q^21 - 2 * q^26 - 4 * q^29 - 32 * q^31 - 22 * q^34 - 2 * q^36 + 16 * q^39 - 22 * q^44 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 12 * q^59 + 4 * q^61 - 8 * q^64 - 18 * q^66 + 20 * q^71 - 16 * q^74 - 32 * q^76 - 20 * q^79 + 4 * q^81 - 6 * q^84 + 8 * q^86 + 48 * q^91 - 30 * q^94 + 18 * q^96 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 0.618034i	− 0.437016i	−0.975835	−	0.218508i	$-0.929881\pi$
$2$	− 0.618034i	− 0.437016i	0.975835	−	0.218508i	$-0.0701190\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	0.618034	0.252311
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	− 2.23607i	− 0.790569i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.47214	−1.64991	−0.824956	−	0.565198i	$-0.808800\pi$
$11$	−5.47214	−1.64991	−0.824956	+	0.565198i	$0.808800\pi$
$12$	1.61803i	0.467086i
$13$	− 6.23607i	− 1.72957i	−0.502139	−	0.864787i	$-0.667453\pi$
$13$	− 6.23607i	− 1.72957i	0.502139	−	0.864787i	$-0.332547\pi$
$14$	1.85410	0.495530
$15$	0	0
$16$	1.85410	0.463525
$17$	− 3.47214i	− 0.842117i	−0.907034	−	0.421058i	$-0.861659\pi$
$17$	− 3.47214i	− 0.842117i	0.907034	−	0.421058i	$-0.138341\pi$
$18$	0.618034i	0.145672i
$19$	−7.70820	−1.76838	−0.884192	−	0.467124i	$-0.845290\pi$
$19$	−7.70820	−1.76838	−0.884192	+	0.467124i	$0.845290\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	3.38197i	0.721038i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−3.85410	−0.755852
$27$	− 1.00000i	− 0.192450i
$28$	4.85410i	0.917339i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 5.61803i	− 0.993137i
$33$	− 5.47214i	− 0.952577i
$34$	−2.14590	−0.368018
$35$	0	0
$36$	−1.61803	−0.269672
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	4.76393i	0.772812i
$39$	6.23607	0.998570
$40$	0	0
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	1.85410i	0.286094i
$43$	3.23607i	0.493496i	0.969080	+	0.246748i	$0.0793619\pi$
$43$	3.23607i	0.493496i	−0.969080	+	0.246748i	$0.920638\pi$
$44$	−8.85410	−1.33481
$45$	0	0
$46$	0	0
$47$	− 6.70820i	− 0.978492i	−0.872146	−	0.489246i	$-0.837272\pi$
$47$	− 6.70820i	− 0.978492i	0.872146	−	0.489246i	$-0.162728\pi$
$48$	1.85410i	0.267617i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	3.47214	0.486196
$52$	− 10.0902i	− 1.39925i
$53$	− 6.76393i	− 0.929098i	−0.885548	−	0.464549i	$-0.846217\pi$
$53$	− 6.76393i	− 0.929098i	0.885548	−	0.464549i	$-0.153783\pi$
$54$	−0.618034	−0.0841038
$55$	0	0
$56$	6.70820	0.896421
$57$	− 7.70820i	− 1.02098i
$58$	0.618034i	0.0811518i
$59$	−5.23607	−0.681678	−0.340839	−	0.940122i	$-0.610711\pi$
$59$	−5.23607	−0.681678	−0.340839	+	0.940122i	$0.610711\pi$
$60$	0	0
$61$	−5.70820	−0.730861	−0.365430	−	0.930839i	$-0.619078\pi$
$61$	−5.70820	−0.730861	−0.365430	+	0.930839i	$0.619078\pi$
$62$	4.94427i	0.627923i
$63$	− 3.00000i	− 0.377964i
$64$	0.236068	0.0295085
$65$	0	0
$66$	−3.38197	−0.416291
$67$	11.4721i	1.40154i	0.713385	+	0.700772i	$0.247161\pi$
$67$	11.4721i	1.40154i	−0.713385	+	0.700772i	$0.752839\pi$
$68$	− 5.61803i	− 0.681287i
$69$	0	0
$70$	0	0
$71$	7.23607	0.858763	0.429382	−	0.903123i	$-0.358732\pi$
$71$	7.23607	0.858763	0.429382	+	0.903123i	$0.358732\pi$
$72$	2.23607i	0.263523i
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	4.94427	0.574760
$75$	0	0
$76$	−12.4721	−1.43065
$77$	− 16.4164i	− 1.87082i
$78$	− 3.85410i	− 0.436391i
$79$	6.18034	0.695343	0.347671	−	0.937616i	$-0.386973\pi$
$79$	6.18034	0.695343	0.347671	+	0.937616i	$0.386973\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.76393i	0.305225i
$83$	− 3.70820i	− 0.407028i	−0.979072	−	0.203514i	$-0.934764\pi$
$83$	− 3.70820i	− 0.407028i	0.979072	−	0.203514i	$-0.0652363\pi$
$84$	−4.85410	−0.529626
$85$	0	0
$86$	2.00000	0.215666
$87$	− 1.00000i	− 0.107211i
$88$	12.2361i	1.30437i
$89$	−11.1803	−1.18511	−0.592557	−	0.805529i	$-0.701881\pi$
$89$	−11.1803	−1.18511	−0.592557	+	0.805529i	$0.701881\pi$
$90$	0	0
$91$	18.7082	1.96115
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	−4.14590	−0.427617
$95$	0	0
$96$	5.61803	0.573388
$97$	− 2.76393i	− 0.280635i	−0.990107	−	0.140317i	$-0.955188\pi$
$97$	− 2.76393i	− 0.280635i	0.990107	−	0.140317i	$-0.0448123\pi$
$98$	1.23607i	0.124862i
$99$	5.47214	0.549970
$100$	0	0
$101$	4.23607	0.421505	0.210752	−	0.977540i	$-0.432409\pi$
$101$	4.23607	0.421505	0.210752	+	0.977540i	$0.432409\pi$
$102$	− 2.14590i	− 0.212476i
$103$	7.41641i	0.730760i	0.930858	+	0.365380i	$0.119061\pi$
$103$	7.41641i	0.730760i	−0.930858	+	0.365380i	$0.880939\pi$
$104$	−13.9443	−1.36735
$105$	0	0
$106$	−4.18034	−0.406031
$107$	− 7.52786i	− 0.727746i	−0.931449	−	0.363873i	$-0.881454\pi$
$107$	− 7.52786i	− 0.727746i	0.931449	−	0.363873i	$-0.118546\pi$
$108$	− 1.61803i	− 0.155695i
$109$	16.4164	1.57241	0.786203	−	0.617968i	$-0.212044\pi$
$109$	16.4164	1.57241	0.786203	+	0.617968i	$0.212044\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	5.56231i	0.525589i
$113$	− 7.94427i	− 0.747334i	−0.927563	−	0.373667i	$-0.878100\pi$
$113$	− 7.94427i	− 0.747334i	0.927563	−	0.373667i	$-0.121900\pi$
$114$	−4.76393	−0.446183
$115$	0	0
$116$	−1.61803	−0.150231
$117$	6.23607i	0.576525i
$118$	3.23607i	0.297904i
$119$	10.4164	0.954871
$120$	0	0
$121$	18.9443	1.72221
$122$	3.52786i	0.319398i
$123$	− 4.47214i	− 0.403239i
$124$	−12.9443	−1.16243
$125$	0	0
$126$	−1.85410	−0.165177
$127$	6.00000i	0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$	6.00000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	− 11.3820i	− 1.00603i
$129$	−3.23607	−0.284920
$130$	0	0
$131$	3.47214	0.303362	0.151681	−	0.988430i	$-0.451531\pi$
$131$	3.47214	0.303362	0.151681	+	0.988430i	$0.451531\pi$
$132$	− 8.85410i	− 0.770651i
$133$	− 23.1246i	− 2.00516i
$134$	7.09017	0.612497
$135$	0	0
$136$	−7.76393	−0.665752
$137$	10.9443i	0.935032i	0.883985	+	0.467516i	$0.154851\pi$
$137$	10.9443i	0.935032i	−0.883985	+	0.467516i	$0.845149\pi$
$138$	0	0
$139$	0.708204	0.0600691	0.0300345	−	0.999549i	$-0.490438\pi$
$139$	0.708204	0.0600691	0.0300345	+	0.999549i	$0.490438\pi$
$140$	0	0
$141$	6.70820	0.564933
$142$	− 4.47214i	− 0.375293i
$143$	34.1246i	2.85364i
$144$	−1.85410	−0.154508
$145$	0	0
$146$	−4.94427	−0.409191
$147$	− 2.00000i	− 0.164957i
$148$	12.9443i	1.06401i
$149$	−20.1803	−1.65324	−0.826619	−	0.562762i	$-0.809739\pi$
$149$	−20.1803	−1.65324	−0.826619	+	0.562762i	$0.809739\pi$
$150$	0	0
$151$	2.47214	0.201180	0.100590	−	0.994928i	$-0.467927\pi$
$151$	2.47214	0.201180	0.100590	+	0.994928i	$0.467927\pi$
$152$	17.2361i	1.39803i
$153$	3.47214i	0.280706i
$154$	−10.1459	−0.817580
$155$	0	0
$156$	10.0902	0.807860
$157$	11.7082i	0.934416i	0.884147	+	0.467208i	$0.154740\pi$
$157$	11.7082i	0.934416i	−0.884147	+	0.467208i	$0.845260\pi$
$158$	− 3.81966i	− 0.303876i
$159$	6.76393	0.536415
$160$	0	0
$161$	0	0
$162$	− 0.618034i	− 0.0485573i
$163$	− 15.1246i	− 1.18465i	−0.805699	−	0.592326i	$-0.798210\pi$
$163$	− 15.1246i	− 1.18465i	0.805699	−	0.592326i	$-0.201790\pi$
$164$	−7.23607	−0.565042
$165$	0	0
$166$	−2.29180	−0.177878
$167$	17.8885i	1.38426i	0.721774	+	0.692129i	$0.243327\pi$
$167$	17.8885i	1.38426i	−0.721774	+	0.692129i	$0.756673\pi$
$168$	6.70820i	0.517549i
$169$	−25.8885	−1.99143
$170$	0	0
$171$	7.70820	0.589461
$172$	5.23607i	0.399246i
$173$	− 19.4164i	− 1.47620i	−0.674690	−	0.738101i	$-0.735723\pi$
$173$	− 19.4164i	− 1.47620i	0.674690	−	0.738101i	$-0.264277\pi$
$174$	−0.618034	−0.0468530
$175$	0	0
$176$	−10.1459	−0.764776
$177$	− 5.23607i	− 0.393567i
$178$	6.90983i	0.517914i
$179$	−4.18034	−0.312453	−0.156227	−	0.987721i	$-0.549933\pi$
$179$	−4.18034	−0.312453	−0.156227	+	0.987721i	$0.549933\pi$
$180$	0	0
$181$	8.41641	0.625587	0.312793	−	0.949821i	$-0.398735\pi$
$181$	8.41641	0.625587	0.312793	+	0.949821i	$0.398735\pi$
$182$	− 11.5623i	− 0.857055i
$183$	− 5.70820i	− 0.421963i
$184$	0	0
$185$	0	0
$186$	−4.94427	−0.362532
$187$	19.0000i	1.38942i
$188$	− 10.8541i	− 0.791617i
$189$	3.00000	0.218218
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0.236068i	0.0170367i
$193$	− 6.00000i	− 0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$	− 6.00000i	− 0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$	−1.70820	−0.122642
$195$	0	0
$196$	−3.23607	−0.231148
$197$	− 14.9443i	− 1.06474i	−0.846513	−	0.532368i	$-0.821302\pi$
$197$	− 14.9443i	− 1.06474i	0.846513	−	0.532368i	$-0.178698\pi$
$198$	− 3.38197i	− 0.240346i
$199$	−20.7082	−1.46797	−0.733983	−	0.679168i	$-0.762341\pi$
$199$	−20.7082	−1.46797	−0.733983	+	0.679168i	$0.762341\pi$
$200$	0	0
$201$	−11.4721	−0.809182
$202$	− 2.61803i	− 0.184204i
$203$	− 3.00000i	− 0.210559i
$204$	5.61803	0.393341
$205$	0	0
$206$	4.58359	0.319354
$207$	0	0
$208$	− 11.5623i	− 0.801702i
$209$	42.1803	2.91768
$210$	0	0
$211$	−27.8885	−1.91993	−0.959963	−	0.280126i	$-0.909624\pi$
$211$	−27.8885	−1.91993	−0.959963	+	0.280126i	$0.909624\pi$
$212$	− 10.9443i	− 0.751656i
$213$	7.23607i	0.495807i
$214$	−4.65248	−0.318037
$215$	0	0
$216$	−2.23607	−0.152145
$217$	− 24.0000i	− 1.62923i
$218$	− 10.1459i	− 0.687167i
$219$	8.00000	0.540590
$220$	0	0
$221$	−21.6525	−1.45650
$222$	4.94427i	0.331838i
$223$	− 13.0000i	− 0.870544i	−0.900299	−	0.435272i	$-0.856652\pi$
$223$	− 13.0000i	− 0.870544i	0.900299	−	0.435272i	$-0.143348\pi$
$224$	16.8541	1.12611
$225$	0	0
$226$	−4.90983	−0.326597
$227$	5.81966i	0.386264i	0.981173	+	0.193132i	$0.0618646\pi$
$227$	5.81966i	0.386264i	−0.981173	+	0.193132i	$0.938135\pi$
$228$	− 12.4721i	− 0.825987i
$229$	−9.41641	−0.622254	−0.311127	−	0.950368i	$-0.600706\pi$
$229$	−9.41641	−0.622254	−0.311127	+	0.950368i	$0.600706\pi$
$230$	0	0
$231$	16.4164	1.08012
$232$	2.23607i	0.146805i
$233$	1.41641i	0.0927920i	0.998923	+	0.0463960i	$0.0147736\pi$
$233$	1.41641i	0.0927920i	−0.998923	+	0.0463960i	$0.985226\pi$
$234$	3.85410	0.251951
$235$	0	0
$236$	−8.47214	−0.551489
$237$	6.18034i	0.401456i
$238$	− 6.43769i	− 0.417294i
$239$	−11.8885	−0.769006	−0.384503	−	0.923124i	$-0.625627\pi$
$239$	−11.8885	−0.769006	−0.384503	+	0.923124i	$0.625627\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	− 11.7082i	− 0.752632i
$243$	1.00000i	0.0641500i
$244$	−9.23607	−0.591279
$245$	0	0
$246$	−2.76393	−0.176222
$247$	48.0689i	3.05855i
$248$	17.8885i	1.13592i
$249$	3.70820	0.234998
$250$	0	0
$251$	−24.8885	−1.57095	−0.785475	−	0.618893i	$-0.787582\pi$
$251$	−24.8885	−1.57095	−0.785475	+	0.618893i	$0.787582\pi$
$252$	− 4.85410i	− 0.305780i
$253$	0	0
$254$	3.70820	0.232673
$255$	0	0
$256$	−6.56231	−0.410144
$257$	9.70820i	0.605581i	0.953057	+	0.302791i	$0.0979183\pi$
$257$	9.70820i	0.605581i	−0.953057	+	0.302791i	$0.902082\pi$
$258$	2.00000i	0.124515i
$259$	−24.0000	−1.49129
$260$	0	0
$261$	1.00000	0.0618984
$262$	− 2.14590i	− 0.132574i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	−12.2361	−0.753078
$265$	0	0
$266$	−14.2918	−0.876286
$267$	− 11.1803i	− 0.684226i
$268$	18.5623i	1.13387i
$269$	30.2361	1.84353	0.921763	−	0.387754i	$-0.126749\pi$
$269$	30.2361	1.84353	0.921763	+	0.387754i	$0.126749\pi$
$270$	0	0
$271$	14.3607	0.872349	0.436175	−	0.899862i	$-0.356333\pi$
$271$	14.3607	0.872349	0.436175	+	0.899862i	$0.356333\pi$
$272$	− 6.43769i	− 0.390343i
$273$	18.7082i	1.13227i
$274$	6.76393	0.408624
$275$	0	0
$276$	0	0
$277$	− 8.70820i	− 0.523225i	−0.965173	−	0.261613i	$-0.915746\pi$
$277$	− 8.70820i	− 0.523225i	0.965173	−	0.261613i	$-0.0842543\pi$
$278$	− 0.437694i	− 0.0262511i
$279$	8.00000	0.478947
$280$	0	0
$281$	−23.1246	−1.37950	−0.689749	−	0.724048i	$-0.742279\pi$
$281$	−23.1246	−1.37950	−0.689749	+	0.724048i	$0.742279\pi$
$282$	− 4.14590i	− 0.246885i
$283$	18.4721i	1.09805i	0.835804	+	0.549027i	$0.185002\pi$
$283$	18.4721i	1.09805i	−0.835804	+	0.549027i	$0.814998\pi$
$284$	11.7082	0.694754
$285$	0	0
$286$	21.0902	1.24709
$287$	− 13.4164i	− 0.791946i
$288$	5.61803i	0.331046i
$289$	4.94427	0.290840
$290$	0	0
$291$	2.76393	0.162025
$292$	− 12.9443i	− 0.757506i
$293$	− 17.9443i	− 1.04832i	−0.851621	−	0.524158i	$-0.824380\pi$
$293$	− 17.9443i	− 1.04832i	0.851621	−	0.524158i	$-0.175620\pi$
$294$	−1.23607	−0.0720889
$295$	0	0
$296$	17.8885	1.03975
$297$	5.47214i	0.317526i
$298$	12.4721i	0.722491i
$299$	0	0
$300$	0	0
$301$	−9.70820	−0.559572
$302$	− 1.52786i	− 0.0879187i
$303$	4.23607i	0.243356i
$304$	−14.2918	−0.819691
$305$	0	0
$306$	2.14590	0.122673
$307$	− 24.9443i	− 1.42364i	−0.702360	−	0.711822i	$-0.747870\pi$
$307$	− 24.9443i	− 1.42364i	0.702360	−	0.711822i	$-0.252130\pi$
$308$	− 26.5623i	− 1.51353i
$309$	−7.41641	−0.421905
$310$	0	0
$311$	23.4721	1.33098	0.665491	−	0.746406i	$-0.268222\pi$
$311$	23.4721	1.33098	0.665491	+	0.746406i	$0.268222\pi$
$312$	− 13.9443i	− 0.789439i
$313$	1.18034i	0.0667168i	0.999443	+	0.0333584i	$0.0106203\pi$
$313$	1.18034i	0.0667168i	−0.999443	+	0.0333584i	$0.989380\pi$
$314$	7.23607	0.408355
$315$	0	0
$316$	10.0000	0.562544
$317$	28.4164i	1.59602i	0.602641	+	0.798012i	$0.294115\pi$
$317$	28.4164i	1.59602i	−0.602641	+	0.798012i	$0.705885\pi$
$318$	− 4.18034i	− 0.234422i
$319$	5.47214	0.306381
$320$	0	0
$321$	7.52786	0.420164
$322$	0	0
$323$	26.7639i	1.48919i
$324$	1.61803	0.0898908
$325$	0	0
$326$	−9.34752	−0.517711
$327$	16.4164i	0.907829i
$328$	10.0000i	0.552158i
$329$	20.1246	1.10951
$330$	0	0
$331$	−15.8885	−0.873313	−0.436657	−	0.899628i	$-0.643838\pi$
$331$	−15.8885	−0.873313	−0.436657	+	0.899628i	$0.643838\pi$
$332$	− 6.00000i	− 0.329293i
$333$	− 8.00000i	− 0.438397i
$334$	11.0557	0.604943
$335$	0	0
$336$	−5.56231	−0.303449
$337$	20.4721i	1.11519i	0.830114	+	0.557594i	$0.188275\pi$
$337$	20.4721i	1.11519i	−0.830114	+	0.557594i	$0.811725\pi$
$338$	16.0000i	0.870285i
$339$	7.94427	0.431474
$340$	0	0
$341$	43.7771	2.37066
$342$	− 4.76393i	− 0.257604i
$343$	15.0000i	0.809924i
$344$	7.23607	0.390143
$345$	0	0
$346$	−12.0000	−0.645124
$347$	− 17.5279i	− 0.940945i	−0.882415	−	0.470473i	$-0.844083\pi$
$347$	− 17.5279i	− 0.940945i	0.882415	−	0.470473i	$-0.155917\pi$
$348$	− 1.61803i	− 0.0867357i
$349$	18.9443	1.01406	0.507032	−	0.861927i	$-0.330743\pi$
$349$	18.9443	1.01406	0.507032	+	0.861927i	$0.330743\pi$
$350$	0	0
$351$	−6.23607	−0.332857
$352$	30.7426i	1.63859i
$353$	3.52786i	0.187769i	0.995583	+	0.0938846i	$0.0299285\pi$
$353$	3.52786i	0.187769i	−0.995583	+	0.0938846i	$0.970072\pi$
$354$	−3.23607	−0.171995
$355$	0	0
$356$	−18.0902	−0.958777
$357$	10.4164i	0.551295i
$358$	2.58359i	0.136547i
$359$	4.58359	0.241913	0.120956	−	0.992658i	$-0.461404\pi$
$359$	4.58359	0.241913	0.120956	+	0.992658i	$0.461404\pi$
$360$	0	0
$361$	40.4164	2.12718
$362$	− 5.20163i	− 0.273391i
$363$	18.9443i	0.994316i
$364$	30.2705	1.58661
$365$	0	0
$366$	−3.52786	−0.184404
$367$	− 15.4164i	− 0.804730i	−0.915479	−	0.402365i	$-0.868188\pi$
$367$	− 15.4164i	− 0.804730i	0.915479	−	0.402365i	$-0.131812\pi$
$368$	0	0
$369$	4.47214	0.232810
$370$	0	0
$371$	20.2918	1.05350
$372$	− 12.9443i	− 0.671129i
$373$	15.8885i	0.822678i	0.911483	+	0.411339i	$0.134939\pi$
$373$	15.8885i	0.822678i	−0.911483	+	0.411339i	$0.865061\pi$
$374$	11.7426	0.607198
$375$	0	0
$376$	−15.0000	−0.773566
$377$	6.23607i	0.321174i
$378$	− 1.85410i	− 0.0953647i
$379$	−6.94427	−0.356703	−0.178352	−	0.983967i	$-0.557076\pi$
$379$	−6.94427	−0.356703	−0.178352	+	0.983967i	$0.557076\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	− 7.41641i	− 0.379456i
$383$	− 20.0689i	− 1.02547i	−0.858546	−	0.512736i	$-0.828632\pi$
$383$	− 20.0689i	− 1.02547i	0.858546	−	0.512736i	$-0.171368\pi$
$384$	11.3820	0.580834
$385$	0	0
$386$	−3.70820	−0.188743
$387$	− 3.23607i	− 0.164499i
$388$	− 4.47214i	− 0.227038i
$389$	−22.2361	−1.12741	−0.563707	−	0.825975i	$-0.690625\pi$
$389$	−22.2361	−1.12741	−0.563707	+	0.825975i	$0.690625\pi$
$390$	0	0
$391$	0	0
$392$	4.47214i	0.225877i
$393$	3.47214i	0.175146i
$394$	−9.23607	−0.465306
$395$	0	0
$396$	8.85410	0.444935
$397$	6.58359i	0.330421i	0.986258	+	0.165211i	$0.0528304\pi$
$397$	6.58359i	0.330421i	−0.986258	+	0.165211i	$0.947170\pi$
$398$	12.7984i	0.641525i
$399$	23.1246	1.15768
$400$	0	0
$401$	−36.6525	−1.83034	−0.915169	−	0.403071i	$-0.867943\pi$
$401$	−36.6525	−1.83034	−0.915169	+	0.403071i	$0.867943\pi$
$402$	7.09017i	0.353626i
$403$	49.8885i	2.48513i
$404$	6.85410	0.341004
$405$	0	0
$406$	−1.85410	−0.0920175
$407$	− 43.7771i	− 2.16995i
$408$	− 7.76393i	− 0.384372i
$409$	−8.65248	−0.427837	−0.213919	−	0.976851i	$-0.568623\pi$
$409$	−8.65248	−0.427837	−0.213919	+	0.976851i	$0.568623\pi$
$410$	0	0
$411$	−10.9443	−0.539841
$412$	12.0000i	0.591198i
$413$	− 15.7082i	− 0.772950i
$414$	0	0
$415$	0	0
$416$	−35.0344	−1.71770
$417$	0.708204i	0.0346809i
$418$	− 26.0689i	− 1.27507i
$419$	−34.3607	−1.67863	−0.839315	−	0.543646i	$-0.817043\pi$
$419$	−34.3607	−1.67863	−0.839315	+	0.543646i	$0.817043\pi$
$420$	0	0
$421$	−1.81966	−0.0886848	−0.0443424	−	0.999016i	$-0.514119\pi$
$421$	−1.81966	−0.0886848	−0.0443424	+	0.999016i	$0.514119\pi$
$422$	17.2361i	0.839039i
$423$	6.70820i	0.326164i
$424$	−15.1246	−0.734516
$425$	0	0
$426$	4.47214	0.216676
$427$	− 17.1246i	− 0.828718i
$428$	− 12.1803i	− 0.588759i
$429$	−34.1246	−1.64755
$430$	0	0
$431$	34.4721	1.66046	0.830232	−	0.557418i	$-0.188208\pi$
$431$	34.4721	1.66046	0.830232	+	0.557418i	$0.188208\pi$
$432$	− 1.85410i	− 0.0892055i
$433$	− 4.65248i	− 0.223584i	−0.993732	−	0.111792i	$-0.964341\pi$
$433$	− 4.65248i	− 0.223584i	0.993732	−	0.111792i	$-0.0356590\pi$
$434$	−14.8328	−0.711998
$435$	0	0
$436$	26.5623	1.27210
$437$	0	0
$438$	− 4.94427i	− 0.236246i
$439$	10.1246	0.483221	0.241611	−	0.970373i	$-0.422324\pi$
$439$	10.1246	0.483221	0.241611	+	0.970373i	$0.422324\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	13.3820i	0.636515i
$443$	23.7639i	1.12906i	0.825413	+	0.564529i	$0.190942\pi$
$443$	23.7639i	1.12906i	−0.825413	+	0.564529i	$0.809058\pi$
$444$	−12.9443	−0.614308
$445$	0	0
$446$	−8.03444	−0.380442
$447$	− 20.1803i	− 0.954497i
$448$	0.708204i	0.0334595i
$449$	−1.76393	−0.0832451	−0.0416225	−	0.999133i	$-0.513253\pi$
$449$	−1.76393	−0.0832451	−0.0416225	+	0.999133i	$0.513253\pi$
$450$	0	0
$451$	24.4721	1.15235
$452$	− 12.8541i	− 0.604606i
$453$	2.47214i	0.116151i
$454$	3.59675	0.168804
$455$	0	0
$456$	−17.2361	−0.807153
$457$	− 6.12461i	− 0.286497i	−0.989687	−	0.143249i	$-0.954245\pi$
$457$	− 6.12461i	− 0.286497i	0.989687	−	0.143249i	$-0.0457549\pi$
$458$	5.81966i	0.271935i
$459$	−3.47214	−0.162065
$460$	0	0
$461$	2.36068	0.109948	0.0549739	−	0.998488i	$-0.482492\pi$
$461$	2.36068	0.109948	0.0549739	+	0.998488i	$0.482492\pi$
$462$	− 10.1459i	− 0.472030i
$463$	− 2.52786i	− 0.117480i	−0.998273	−	0.0587399i	$-0.981292\pi$
$463$	− 2.52786i	− 0.117480i	0.998273	−	0.0587399i	$-0.0187083\pi$
$464$	−1.85410	−0.0860745
$465$	0	0
$466$	0.875388	0.0405516
$467$	12.9443i	0.598989i	0.954098	+	0.299495i	$0.0968181\pi$
$467$	12.9443i	0.598989i	−0.954098	+	0.299495i	$0.903182\pi$
$468$	10.0902i	0.466418i
$469$	−34.4164	−1.58920
$470$	0	0
$471$	−11.7082	−0.539486
$472$	11.7082i	0.538914i
$473$	− 17.7082i	− 0.814224i
$474$	3.81966	0.175443
$475$	0	0
$476$	16.8541	0.772506
$477$	6.76393i	0.309699i
$478$	7.34752i	0.336068i
$479$	6.47214	0.295719	0.147860	−	0.989008i	$-0.452762\pi$
$479$	6.47214	0.295719	0.147860	+	0.989008i	$0.452762\pi$
$480$	0	0
$481$	49.8885	2.27472
$482$	− 4.32624i	− 0.197055i
$483$	0	0
$484$	30.6525	1.39329
$485$	0	0
$486$	0.618034	0.0280346
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	12.7639i	0.577796i
$489$	15.1246	0.683959
$490$	0	0
$491$	25.8885	1.16833	0.584167	−	0.811634i	$-0.301421\pi$
$491$	25.8885	1.16833	0.584167	+	0.811634i	$0.301421\pi$
$492$	− 7.23607i	− 0.326227i
$493$	3.47214i	0.156377i
$494$	29.7082	1.33664
$495$	0	0
$496$	−14.8328	−0.666013
$497$	21.7082i	0.973746i
$498$	− 2.29180i	− 0.102698i
$499$	23.7639	1.06382	0.531910	−	0.846801i	$-0.321475\pi$
$499$	23.7639	1.06382	0.531910	+	0.846801i	$0.321475\pi$
$500$	0	0
$501$	−17.8885	−0.799201
$502$	15.3820i	0.686531i
$503$	− 30.5967i	− 1.36424i	−0.731240	−	0.682121i	$-0.761058\pi$
$503$	− 30.5967i	− 1.36424i	0.731240	−	0.682121i	$-0.238942\pi$
$504$	−6.70820	−0.298807
$505$	0	0
$506$	0	0
$507$	− 25.8885i	− 1.14975i
$508$	9.70820i	0.430732i
$509$	−6.18034	−0.273939	−0.136969	−	0.990575i	$-0.543736\pi$
$509$	−6.18034	−0.273939	−0.136969	+	0.990575i	$0.543736\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	− 18.7082i	− 0.826794i
$513$	7.70820i	0.340326i
$514$	6.00000	0.264649
$515$	0	0
$516$	−5.23607	−0.230505
$517$	36.7082i	1.61442i
$518$	14.8328i	0.651717i
$519$	19.4164	0.852286
$520$	0	0
$521$	−20.7639	−0.909684	−0.454842	−	0.890572i	$-0.650304\pi$
$521$	−20.7639	−0.909684	−0.454842	+	0.890572i	$0.650304\pi$
$522$	− 0.618034i	− 0.0270506i
$523$	− 29.8328i	− 1.30450i	−0.758005	−	0.652249i	$-0.773826\pi$
$523$	− 29.8328i	− 1.30450i	0.758005	−	0.652249i	$-0.226174\pi$
$524$	5.61803	0.245425
$525$	0	0
$526$	14.8328	0.646741
$527$	27.7771i	1.20999i
$528$	− 10.1459i	− 0.441544i
$529$	23.0000	1.00000
$530$	0	0
$531$	5.23607	0.227226
$532$	− 37.4164i	− 1.62221i
$533$	27.8885i	1.20799i
$534$	−6.90983	−0.299018
$535$	0	0
$536$	25.6525	1.10802
$537$	− 4.18034i	− 0.180395i
$538$	− 18.6869i	− 0.805650i
$539$	10.9443	0.471403
$540$	0	0
$541$	38.3607	1.64925	0.824627	−	0.565677i	$-0.191385\pi$
$541$	38.3607	1.64925	0.824627	+	0.565677i	$0.191385\pi$
$542$	− 8.87539i	− 0.381231i
$543$	8.41641i	0.361183i
$544$	−19.5066	−0.836338
$545$	0	0
$546$	11.5623	0.494821
$547$	− 27.4721i	− 1.17462i	−0.809361	−	0.587312i	$-0.800186\pi$
$547$	− 27.4721i	− 1.17462i	0.809361	−	0.587312i	$-0.199814\pi$
$548$	17.7082i	0.756457i
$549$	5.70820	0.243620
$550$	0	0
$551$	7.70820	0.328381
$552$	0	0
$553$	18.5410i	0.788444i
$554$	−5.38197	−0.228658
$555$	0	0
$556$	1.14590	0.0485969
$557$	− 2.76393i	− 0.117112i	−0.998284	−	0.0585558i	$-0.981350\pi$
$557$	− 2.76393i	− 0.117112i	0.998284	−	0.0585558i	$-0.0186496\pi$
$558$	− 4.94427i	− 0.209308i
$559$	20.1803	0.853537
$560$	0	0
$561$	−19.0000	−0.802181
$562$	14.2918i	0.602863i
$563$	44.1246i	1.85963i	0.368026	+	0.929815i	$0.380034\pi$
$563$	44.1246i	1.85963i	−0.368026	+	0.929815i	$0.619966\pi$
$564$	10.8541	0.457040
$565$	0	0
$566$	11.4164	0.479867
$567$	3.00000i	0.125988i
$568$	− 16.1803i	− 0.678912i
$569$	5.18034	0.217171	0.108586	−	0.994087i	$-0.465368\pi$
$569$	5.18034	0.217171	0.108586	+	0.994087i	$0.465368\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	55.2148i	2.30865i
$573$	12.0000i	0.501307i
$574$	−8.29180	−0.346093
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	29.3050i	1.21998i	0.792409	+	0.609991i	$0.208827\pi$
$577$	29.3050i	1.21998i	−0.792409	+	0.609991i	$0.791173\pi$
$578$	− 3.05573i	− 0.127102i
$579$	6.00000	0.249351
$580$	0	0
$581$	11.1246	0.461527
$582$	− 1.70820i	− 0.0708073i
$583$	37.0132i	1.53293i
$584$	−17.8885	−0.740233
$585$	0	0
$586$	−11.0902	−0.458131
$587$	1.81966i	0.0751054i	0.999295	+	0.0375527i	$0.0119562\pi$
$587$	1.81966i	0.0751054i	−0.999295	+	0.0375527i	$0.988044\pi$
$588$	− 3.23607i	− 0.133453i
$589$	61.6656	2.54089
$590$	0	0
$591$	14.9443	0.614725
$592$	14.8328i	0.609625i
$593$	− 24.6525i	− 1.01236i	−0.862429	−	0.506178i	$-0.831058\pi$
$593$	− 24.6525i	− 1.01236i	0.862429	−	0.506178i	$-0.168942\pi$
$594$	3.38197	0.138764
$595$	0	0
$596$	−32.6525	−1.33750
$597$	− 20.7082i	− 0.847530i
$598$	0	0
$599$	8.88854	0.363176	0.181588	−	0.983375i	$-0.441876\pi$
$599$	8.88854	0.363176	0.181588	+	0.983375i	$0.441876\pi$
$600$	0	0
$601$	8.11146	0.330873	0.165437	−	0.986220i	$-0.447097\pi$
$601$	8.11146	0.330873	0.165437	+	0.986220i	$0.447097\pi$
$602$	6.00000i	0.244542i
$603$	− 11.4721i	− 0.467181i
$604$	4.00000	0.162758
$605$	0	0
$606$	2.61803	0.106350
$607$	− 40.6525i	− 1.65003i	−0.565109	−	0.825017i	$-0.691166\pi$
$607$	− 40.6525i	− 1.65003i	0.565109	−	0.825017i	$-0.308834\pi$
$608$	43.3050i	1.75625i
$609$	3.00000	0.121566
$610$	0	0
$611$	−41.8328	−1.69237
$612$	5.61803i	0.227096i
$613$	7.29180i	0.294513i	0.989098	+	0.147256i	$0.0470443\pi$
$613$	7.29180i	0.294513i	−0.989098	+	0.147256i	$0.952956\pi$
$614$	−15.4164	−0.622156
$615$	0	0
$616$	−36.7082	−1.47902
$617$	− 43.3050i	− 1.74339i	−0.490047	−	0.871696i	$-0.663020\pi$
$617$	− 43.3050i	− 1.74339i	0.490047	−	0.871696i	$-0.336980\pi$
$618$	4.58359i	0.184379i
$619$	3.52786	0.141797	0.0708984	−	0.997484i	$-0.477413\pi$
$619$	3.52786	0.141797	0.0708984	+	0.997484i	$0.477413\pi$
$620$	0	0
$621$	0	0
$622$	− 14.5066i	− 0.581661i
$623$	− 33.5410i	− 1.34379i
$624$	11.5623	0.462863
$625$	0	0
$626$	0.729490	0.0291563
$627$	42.1803i	1.68452i
$628$	18.9443i	0.755959i
$629$	27.7771	1.10755
$630$	0	0
$631$	−38.4853	−1.53208	−0.766038	−	0.642796i	$-0.777774\pi$
$631$	−38.4853	−1.53208	−0.766038	+	0.642796i	$0.777774\pi$
$632$	− 13.8197i	− 0.549717i
$633$	− 27.8885i	− 1.10847i
$634$	17.5623	0.697488
$635$	0	0
$636$	10.9443	0.433969
$637$	12.4721i	0.494164i
$638$	− 3.38197i	− 0.133893i
$639$	−7.23607	−0.286254
$640$	0	0
$641$	46.2361	1.82621	0.913107	−	0.407719i	$-0.133676\pi$
$641$	46.2361	1.82621	0.913107	+	0.407719i	$0.133676\pi$
$642$	− 4.65248i	− 0.183619i
$643$	− 35.8328i	− 1.41311i	−0.707659	−	0.706554i	$-0.750249\pi$
$643$	− 35.8328i	− 1.41311i	0.707659	−	0.706554i	$-0.249751\pi$
$644$	0	0
$645$	0	0
$646$	16.5410	0.650798
$647$	− 11.2361i	− 0.441735i	−0.975304	−	0.220868i	$-0.929111\pi$
$647$	− 11.2361i	− 0.441735i	0.975304	−	0.220868i	$-0.0708889\pi$
$648$	− 2.23607i	− 0.0878410i
$649$	28.6525	1.12471
$650$	0	0
$651$	24.0000	0.940634
$652$	− 24.4721i	− 0.958403i
$653$	8.88854i	0.347836i	0.984760	+	0.173918i	$0.0556427\pi$
$653$	8.88854i	0.347836i	−0.984760	+	0.173918i	$0.944357\pi$
$654$	10.1459	0.396736
$655$	0	0
$656$	−8.29180	−0.323740
$657$	8.00000i	0.312110i
$658$	− 12.4377i	− 0.484872i
$659$	21.0000	0.818044	0.409022	−	0.912525i	$-0.365870\pi$
$659$	21.0000	0.818044	0.409022	+	0.912525i	$0.365870\pi$
$660$	0	0
$661$	−37.8328	−1.47153	−0.735763	−	0.677239i	$-0.763176\pi$
$661$	−37.8328	−1.47153	−0.735763	+	0.677239i	$0.763176\pi$
$662$	9.81966i	0.381652i
$663$	− 21.6525i	− 0.840912i
$664$	−8.29180	−0.321784
$665$	0	0
$666$	−4.94427	−0.191587
$667$	0	0
$668$	28.9443i	1.11989i
$669$	13.0000	0.502609
$670$	0	0
$671$	31.2361	1.20586
$672$	16.8541i	0.650161i
$673$	− 33.2918i	− 1.28330i	−0.766996	−	0.641652i	$-0.778249\pi$
$673$	− 33.2918i	− 1.28330i	0.766996	−	0.641652i	$-0.221751\pi$
$674$	12.6525	0.487355
$675$	0	0
$676$	−41.8885	−1.61110
$677$	20.8885i	0.802812i	0.915900	+	0.401406i	$0.131478\pi$
$677$	20.8885i	0.802812i	−0.915900	+	0.401406i	$0.868522\pi$
$678$	− 4.90983i	− 0.188561i
$679$	8.29180	0.318210
$680$	0	0
$681$	−5.81966	−0.223010
$682$	− 27.0557i	− 1.03602i
$683$	7.41641i	0.283781i	0.989882	+	0.141890i	$0.0453181\pi$
$683$	7.41641i	0.283781i	−0.989882	+	0.141890i	$0.954682\pi$
$684$	12.4721	0.476884
$685$	0	0
$686$	9.27051	0.353950
$687$	− 9.41641i	− 0.359258i
$688$	6.00000i	0.228748i
$689$	−42.1803	−1.60694
$690$	0	0
$691$	−34.7082	−1.32036	−0.660181	−	0.751106i	$-0.729520\pi$
$691$	−34.7082	−1.32036	−0.660181	+	0.751106i	$0.729520\pi$
$692$	− 31.4164i	− 1.19427i
$693$	16.4164i	0.623608i
$694$	−10.8328	−0.411208
$695$	0	0
$696$	−2.23607	−0.0847579
$697$	15.5279i	0.588160i
$698$	− 11.7082i	− 0.443162i
$699$	−1.41641	−0.0535735
$700$	0	0
$701$	−5.12461	−0.193554	−0.0967770	−	0.995306i	$-0.530853\pi$
$701$	−5.12461	−0.193554	−0.0967770	+	0.995306i	$0.530853\pi$
$702$	3.85410i	0.145464i
$703$	− 61.6656i	− 2.32576i
$704$	−1.29180	−0.0486864
$705$	0	0
$706$	2.18034	0.0820582
$707$	12.7082i	0.477941i
$708$	− 8.47214i	− 0.318402i
$709$	−19.8885	−0.746930	−0.373465	−	0.927644i	$-0.621830\pi$
$709$	−19.8885	−0.746930	−0.373465	+	0.927644i	$0.621830\pi$
$710$	0	0
$711$	−6.18034	−0.231781
$712$	25.0000i	0.936915i
$713$	0	0
$714$	6.43769	0.240925
$715$	0	0
$716$	−6.76393	−0.252780
$717$	− 11.8885i	− 0.443986i
$718$	− 2.83282i	− 0.105720i
$719$	11.2361	0.419035	0.209517	−	0.977805i	$-0.432811\pi$
$719$	11.2361	0.419035	0.209517	+	0.977805i	$0.432811\pi$
$720$	0	0
$721$	−22.2492	−0.828604
$722$	− 24.9787i	− 0.929611i
$723$	7.00000i	0.260333i
$724$	13.6180	0.506110
$725$	0	0
$726$	11.7082	0.434532
$727$	− 2.11146i	− 0.0783096i	−0.999233	−	0.0391548i	$-0.987533\pi$
$727$	− 2.11146i	− 0.0783096i	0.999233	−	0.0391548i	$-0.0124665\pi$
$728$	− 41.8328i	− 1.55043i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	11.2361	0.415581
$732$	− 9.23607i	− 0.341375i
$733$	− 14.2918i	− 0.527880i	−0.964539	−	0.263940i	$-0.914978\pi$
$733$	− 14.2918i	− 0.527880i	0.964539	−	0.263940i	$-0.0850220\pi$
$734$	−9.52786	−0.351680
$735$	0	0
$736$	0	0
$737$	− 62.7771i	− 2.31242i
$738$	− 2.76393i	− 0.101742i
$739$	−41.4853	−1.52606	−0.763031	−	0.646362i	$-0.776289\pi$
$739$	−41.4853	−1.52606	−0.763031	+	0.646362i	$0.776289\pi$
$740$	0	0
$741$	−48.0689	−1.76585
$742$	− 12.5410i	− 0.460395i
$743$	− 10.8197i	− 0.396935i	−0.980107	−	0.198467i	$-0.936404\pi$
$743$	− 10.8197i	− 0.396935i	0.980107	−	0.198467i	$-0.0635964\pi$
$744$	−17.8885	−0.655826
$745$	0	0
$746$	9.81966	0.359523
$747$	3.70820i	0.135676i
$748$	30.7426i	1.12406i
$749$	22.5836	0.825186
$750$	0	0
$751$	43.4164	1.58429	0.792144	−	0.610335i	$-0.208965\pi$
$751$	43.4164	1.58429	0.792144	+	0.610335i	$0.208965\pi$
$752$	− 12.4377i	− 0.453556i
$753$	− 24.8885i	− 0.906989i
$754$	3.85410	0.140358
$755$	0	0
$756$	4.85410	0.176542
$757$	34.8328i	1.26602i	0.774144	+	0.633010i	$0.218181\pi$
$757$	34.8328i	1.26602i	−0.774144	+	0.633010i	$0.781819\pi$
$758$	4.29180i	0.155885i
$759$	0	0
$760$	0	0
$761$	33.0132	1.19673	0.598363	−	0.801225i	$-0.295818\pi$
$761$	33.0132	1.19673	0.598363	+	0.801225i	$0.295818\pi$
$762$	3.70820i	0.134334i
$763$	49.2492i	1.78294i
$764$	19.4164	0.702461
$765$	0	0
$766$	−12.4033	−0.448148
$767$	32.6525i	1.17901i
$768$	− 6.56231i	− 0.236797i
$769$	−34.6525	−1.24960	−0.624800	−	0.780785i	$-0.714820\pi$
$769$	−34.6525	−1.24960	−0.624800	+	0.780785i	$0.714820\pi$
$770$	0	0
$771$	−9.70820	−0.349632
$772$	− 9.70820i	− 0.349406i
$773$	− 52.2492i	− 1.87927i	−0.342173	−	0.939637i	$-0.611163\pi$
$773$	− 52.2492i	− 1.87927i	0.342173	−	0.939637i	$-0.388837\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	−6.18034	−0.221861
$777$	− 24.0000i	− 0.860995i
$778$	13.7426i	0.492698i
$779$	34.4721	1.23509
$780$	0	0
$781$	−39.5967	−1.41688
$782$	0	0
$783$	1.00000i	0.0357371i
$784$	−3.70820	−0.132436
$785$	0	0
$786$	2.14590	0.0765416
$787$	− 1.88854i	− 0.0673193i	−0.999433	−	0.0336597i	$-0.989284\pi$
$787$	− 1.88854i	− 0.0673193i	0.999433	−	0.0336597i	$-0.0107162\pi$
$788$	− 24.1803i	− 0.861389i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	23.8328	0.847397
$792$	− 12.2361i	− 0.434790i
$793$	35.5967i	1.26408i
$794$	4.06888	0.144399
$795$	0	0
$796$	−33.5066	−1.18761
$797$	− 6.94427i	− 0.245979i	−0.992408	−	0.122989i	$-0.960752\pi$
$797$	− 6.94427i	− 0.245979i	0.992408	−	0.122989i	$-0.0392481\pi$
$798$	− 14.2918i	− 0.505924i
$799$	−23.2918	−0.824005
$800$	0	0
$801$	11.1803	0.395038
$802$	22.6525i	0.799887i
$803$	43.7771i	1.54486i
$804$	−18.5623	−0.654642
$805$	0	0
$806$	30.8328	1.08604
$807$	30.2361i	1.06436i
$808$	− 9.47214i	− 0.333229i
$809$	−44.2361	−1.55526	−0.777629	−	0.628724i	$-0.783578\pi$
$809$	−44.2361	−1.55526	−0.777629	+	0.628724i	$0.783578\pi$
$810$	0	0
$811$	−22.7082	−0.797393	−0.398696	−	0.917083i	$-0.630537\pi$
$811$	−22.7082	−0.797393	−0.398696	+	0.917083i	$0.630537\pi$
$812$	− 4.85410i	− 0.170346i
$813$	14.3607i	0.503651i
$814$	−27.0557	−0.948303
$815$	0	0
$816$	6.43769	0.225364
$817$	− 24.9443i	− 0.872690i
$818$	5.34752i	0.186972i
$819$	−18.7082	−0.653718
$820$	0	0
$821$	−10.5836	−0.369370	−0.184685	−	0.982798i	$-0.559126\pi$
$821$	−10.5836	−0.369370	−0.184685	+	0.982798i	$0.559126\pi$
$822$	6.76393i	0.235919i
$823$	− 28.7639i	− 1.00265i	−0.865260	−	0.501324i	$-0.832847\pi$
$823$	− 28.7639i	− 1.00265i	0.865260	−	0.501324i	$-0.167153\pi$
$824$	16.5836	0.577717
$825$	0	0
$826$	−9.70820	−0.337792
$827$	7.41641i	0.257894i	0.991652	+	0.128947i	$0.0411597\pi$
$827$	7.41641i	0.257894i	−0.991652	+	0.128947i	$0.958840\pi$
$828$	0	0
$829$	20.0000	0.694629	0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000	0.694629	0.347314	+	0.937749i	$0.387094\pi$
$830$	0	0
$831$	8.70820	0.302084
$832$	− 1.47214i	− 0.0510371i
$833$	6.94427i	0.240605i
$834$	0.437694	0.0151561
$835$	0	0
$836$	68.2492	2.36045
$837$	8.00000i	0.276520i
$838$	21.2361i	0.733588i
$839$	−14.8885	−0.514010	−0.257005	−	0.966410i	$-0.582736\pi$
$839$	−14.8885	−0.514010	−0.257005	+	0.966410i	$0.582736\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	1.12461i	0.0387567i
$843$	− 23.1246i	− 0.796454i
$844$	−45.1246	−1.55325
$845$	0	0
$846$	4.14590	0.142539
$847$	56.8328i	1.95280i
$848$	− 12.5410i	− 0.430660i
$849$	−18.4721	−0.633962
$850$	0	0
$851$	0	0
$852$	11.7082i	0.401116i
$853$	− 28.7639i	− 0.984858i	−0.870353	−	0.492429i	$-0.836109\pi$
$853$	− 28.7639i	− 0.984858i	0.870353	−	0.492429i	$-0.163891\pi$
$854$	−10.5836	−0.362163
$855$	0	0
$856$	−16.8328	−0.575334
$857$	23.8885i	0.816017i	0.912978	+	0.408009i	$0.133777\pi$
$857$	23.8885i	0.816017i	−0.912978	+	0.408009i	$0.866223\pi$
$858$	21.0902i	0.720007i
$859$	15.7082	0.535957	0.267979	−	0.963425i	$-0.413644\pi$
$859$	15.7082	0.535957	0.267979	+	0.963425i	$0.413644\pi$
$860$	0	0
$861$	13.4164	0.457230
$862$	− 21.3050i	− 0.725650i
$863$	15.5967i	0.530919i	0.964122	+	0.265460i	$0.0855237\pi$
$863$	15.5967i	0.530919i	−0.964122	+	0.265460i	$0.914476\pi$
$864$	−5.61803	−0.191129
$865$	0	0
$866$	−2.87539	−0.0977097
$867$	4.94427i	0.167916i
$868$	− 38.8328i	− 1.31807i
$869$	−33.8197	−1.14725
$870$	0	0
$871$	71.5410	2.42407
$872$	− 36.7082i	− 1.24310i
$873$	2.76393i	0.0935449i
$874$	0	0
$875$	0	0
$876$	12.9443	0.437346
$877$	− 48.8328i	− 1.64897i	−0.565886	−	0.824484i	$-0.691466\pi$
$877$	− 48.8328i	− 1.64897i	0.565886	−	0.824484i	$-0.308534\pi$
$878$	− 6.25735i	− 0.211175i
$879$	17.9443	0.605245
$880$	0	0
$881$	12.7082	0.428150	0.214075	−	0.976817i	$-0.431326\pi$
$881$	12.7082	0.428150	0.214075	+	0.976817i	$0.431326\pi$
$882$	− 1.23607i	− 0.0416206i
$883$	− 32.0000i	− 1.07689i	−0.842662	−	0.538443i	$-0.819013\pi$
$883$	− 32.0000i	− 1.07689i	0.842662	−	0.538443i	$-0.180987\pi$
$884$	−35.0344	−1.17834
$885$	0	0
$886$	14.6869	0.493417
$887$	44.5967i	1.49741i	0.662902	+	0.748706i	$0.269325\pi$
$887$	44.5967i	1.49741i	−0.662902	+	0.748706i	$0.730675\pi$
$888$	17.8885i	0.600300i
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−5.47214	−0.183323
$892$	− 21.0344i	− 0.704285i
$893$	51.7082i	1.73035i
$894$	−12.4721	−0.417131
$895$	0	0
$896$	34.1459	1.14073
$897$	0	0
$898$	1.09017i	0.0363794i
$899$	8.00000	0.266815
$900$	0	0
$901$	−23.4853	−0.782409
$902$	− 15.1246i	− 0.503594i
$903$	− 9.70820i	− 0.323069i
$904$	−17.7639	−0.590820
$905$	0	0
$906$	1.52786	0.0507599
$907$	− 26.7639i	− 0.888682i	−0.895858	−	0.444341i	$-0.853438\pi$
$907$	− 26.7639i	− 0.888682i	0.895858	−	0.444341i	$-0.146562\pi$
$908$	9.41641i	0.312494i
$909$	−4.23607	−0.140502
$910$	0	0
$911$	−18.0557	−0.598213	−0.299106	−	0.954220i	$-0.596689\pi$
$911$	−18.0557	−0.598213	−0.299106	+	0.954220i	$0.596689\pi$
$912$	− 14.2918i	− 0.473249i
$913$	20.2918i	0.671560i
$914$	−3.78522	−0.125204
$915$	0	0
$916$	−15.2361	−0.503414
$917$	10.4164i	0.343980i
$918$	2.14590i	0.0708252i
$919$	−20.1246	−0.663850	−0.331925	−	0.943306i	$-0.607698\pi$
$919$	−20.1246	−0.663850	−0.331925	+	0.943306i	$0.607698\pi$
$920$	0	0
$921$	24.9443	0.821942
$922$	− 1.45898i	− 0.0480490i
$923$	− 45.1246i	− 1.48529i
$924$	26.5623	0.873836
$925$	0	0
$926$	−1.56231	−0.0513406
$927$	− 7.41641i	− 0.243587i
$928$	5.61803i	0.184421i
$929$	56.8328	1.86462	0.932312	−	0.361655i	$-0.117788\pi$
$929$	56.8328	1.86462	0.932312	+	0.361655i	$0.117788\pi$
$930$	0	0
$931$	15.4164	0.505252
$932$	2.29180i	0.0750703i
$933$	23.4721i	0.768443i
$934$	8.00000	0.261768
$935$	0	0
$936$	13.9443	0.455783
$937$	− 19.7639i	− 0.645660i	−0.946457	−	0.322830i	$-0.895366\pi$
$937$	− 19.7639i	− 0.645660i	0.946457	−	0.322830i	$-0.104634\pi$
$938$	21.2705i	0.694507i
$939$	−1.18034	−0.0385189
$940$	0	0
$941$	−46.0689	−1.50180	−0.750901	−	0.660414i	$-0.770381\pi$
$941$	−46.0689	−1.50180	−0.750901	+	0.660414i	$0.770381\pi$
$942$	7.23607i	0.235764i
$943$	0	0
$944$	−9.70820	−0.315975
$945$	0	0
$946$	−10.9443	−0.355829
$947$	12.7082i	0.412961i	0.978451	+	0.206481i	$0.0662010\pi$
$947$	12.7082i	0.412961i	−0.978451	+	0.206481i	$0.933799\pi$
$948$	10.0000i	0.324785i
$949$	−49.8885	−1.61945
$950$	0	0
$951$	−28.4164	−0.921465
$952$	− 23.2918i	− 0.754891i
$953$	19.0132i	0.615897i	0.951403	+	0.307948i	$0.0996424\pi$
$953$	19.0132i	0.615897i	−0.951403	+	0.307948i	$0.900358\pi$
$954$	4.18034	0.135344
$955$	0	0
$956$	−19.2361	−0.622139
$957$	5.47214i	0.176889i
$958$	− 4.00000i	− 0.129234i
$959$	−32.8328	−1.06023
$960$	0	0
$961$	33.0000	1.06452
$962$	− 30.8328i	− 0.994090i
$963$	7.52786i	0.242582i
$964$	11.3262	0.364794
$965$	0	0
$966$	0	0
$967$	35.1246i	1.12953i	0.825251	+	0.564766i	$0.191033\pi$
$967$	35.1246i	1.12953i	−0.825251	+	0.564766i	$0.808967\pi$
$968$	− 42.3607i	− 1.36152i
$969$	−26.7639	−0.859781
$970$	0	0
$971$	−23.0557	−0.739894	−0.369947	−	0.929053i	$-0.620624\pi$
$971$	−23.0557	−0.739894	−0.369947	+	0.929053i	$0.620624\pi$
$972$	1.61803i	0.0518985i
$973$	2.12461i	0.0681119i
$974$	7.41641	0.237637
$975$	0	0
$976$	−10.5836	−0.338773
$977$	22.4721i	0.718947i	0.933155	+	0.359474i	$0.117044\pi$
$977$	22.4721i	0.718947i	−0.933155	+	0.359474i	$0.882956\pi$
$978$	− 9.34752i	− 0.298901i
$979$	61.1803	1.95533
$980$	0	0
$981$	−16.4164	−0.524136
$982$	− 16.0000i	− 0.510581i
$983$	37.5279i	1.19695i	0.801140	+	0.598476i	$0.204227\pi$
$983$	37.5279i	1.19695i	−0.801140	+	0.598476i	$0.795773\pi$
$984$	−10.0000	−0.318788
$985$	0	0
$986$	2.14590	0.0683393
$987$	20.1246i	0.640573i
$988$	77.7771i	2.47442i
$989$	0	0
$990$	0	0
$991$	22.4853	0.714269	0.357134	−	0.934053i	$-0.383754\pi$
$991$	22.4853	0.714269	0.357134	+	0.934053i	$0.383754\pi$
$992$	44.9443i	1.42698i
$993$	− 15.8885i	− 0.504208i
$994$	13.4164	0.425543
$995$	0	0
$996$	6.00000	0.190117
$997$	− 7.41641i	− 0.234880i	−0.993080	−	0.117440i	$-0.962531\pi$
$997$	− 7.41641i	− 0.234880i	0.993080	−	0.117440i	$-0.0374688\pi$
$998$	− 14.6869i	− 0.464906i
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.j.349.2		4
5.2	odd	4		435.2.a.e.1.2	✓	2
5.3	odd	4		2175.2.a.q.1.1		2
5.4	even	2	inner	2175.2.c.j.349.3		4
15.2	even	4		1305.2.a.k.1.1		2
15.8	even	4		6525.2.a.s.1.2		2
20.7	even	4		6960.2.a.bu.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.e.1.2	✓	2	5.2	odd	4
1305.2.a.k.1.1		2	15.2	even	4
2175.2.a.q.1.1		2	5.3	odd	4
2175.2.c.j.349.2		4	1.1	even	1	trivial
2175.2.c.j.349.3		4	5.4	even	2	inner
6525.2.a.s.1.2		2	15.8	even	4
6960.2.a.bu.1.2		2	20.7	even	4

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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +3.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +3.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} -5.47214 q^{11} +1.61803i q^{12} -6.23607i q^{13} +1.85410 q^{14} +1.85410 q^{16} -3.47214i q^{17} +0.618034i q^{18} -7.70820 q^{19} -3.00000 q^{21} +3.38197i q^{22} +2.23607 q^{24} -3.85410 q^{26} -1.00000i q^{27} +4.85410i q^{28} -1.00000 q^{29} -8.00000 q^{31} -5.61803i q^{32} -5.47214i q^{33} -2.14590 q^{34} -1.61803 q^{36} +8.00000i q^{37} +4.76393i q^{38} +6.23607 q^{39} -4.47214 q^{41} +1.85410i q^{42} +3.23607i q^{43} -8.85410 q^{44} -6.70820i q^{47} +1.85410i q^{48} -2.00000 q^{49} +3.47214 q^{51} -10.0902i q^{52} -6.76393i q^{53} -0.618034 q^{54} +6.70820 q^{56} -7.70820i q^{57} +0.618034i q^{58} -5.23607 q^{59} -5.70820 q^{61} +4.94427i q^{62} -3.00000i q^{63} +0.236068 q^{64} -3.38197 q^{66} +11.4721i q^{67} -5.61803i q^{68} +7.23607 q^{71} +2.23607i q^{72} -8.00000i q^{73} +4.94427 q^{74} -12.4721 q^{76} -16.4164i q^{77} -3.85410i q^{78} +6.18034 q^{79} +1.00000 q^{81} +2.76393i q^{82} -3.70820i q^{83} -4.85410 q^{84} +2.00000 q^{86} -1.00000i q^{87} +12.2361i q^{88} -11.1803 q^{89} +18.7082 q^{91} -8.00000i q^{93} -4.14590 q^{94} +5.61803 q^{96} -2.76393i q^{97} +1.23607i q^{98} +5.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9	\( 4 q + 2 q^{4} - 2 q^{6} - 4 q^{9} - 4 q^{11} - 6 q^{14} - 6 q^{16} - 4 q^{19} - 12 q^{21} - 2 q^{26} - 4 q^{29} - 32 q^{31} - 22 q^{34} - 2 q^{36} + 16 q^{39} - 22 q^{44} - 8 q^{49} - 4 q^{51} + 2 q^{54} - 12 q^{59} + 4 q^{61} - 8 q^{64} - 18 q^{66} + 20 q^{71} - 16 q^{74} - 32 q^{76} - 20 q^{79} + 4 q^{81} - 6 q^{84} + 8 q^{86} + 48 q^{91} - 30 q^{94} + 18 q^{96} + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^6 - 4 * q^9 - 4 * q^11 - 6 * q^14 - 6 * q^16 - 4 * q^19 - 12 * q^21 - 2 * q^26 - 4 * q^29 - 32 * q^31 - 22 * q^34 - 2 * q^36 + 16 * q^39 - 22 * q^44 - 8 * q^49 - 4 * q^51 + 2 * q^54 - 12 * q^59 + 4 * q^61 - 8 * q^64 - 18 * q^66 + 20 * q^71 - 16 * q^74 - 32 * q^76 - 20 * q^79 + 4 * q^81 - 6 * q^84 + 8 * q^86 + 48 * q^91 - 30 * q^94 + 18 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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