LMFDB - Embedded newform 1840.2.e.c.369.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1840,2,Mod(369,1840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1840.369"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,6,0,18,0,0,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.6924739719$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.2
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.c.369.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034i q^{3} +2.23607 q^{5} -4.85410i q^{7} +2.61803 q^{9} +3.38197 q^{11} -0.381966i q^{13} -1.38197i q^{15} -5.85410i q^{17} -6.85410 q^{19} -3.00000 q^{21} +1.00000i q^{23} +5.00000 q^{25} -3.47214i q^{27} -3.70820 q^{29} +8.85410 q^{31} -2.09017i q^{33} -10.8541i q^{35} +3.70820i q^{37} -0.236068 q^{39} -3.38197 q^{41} +6.76393i q^{43} +5.85410 q^{45} +11.7082i q^{47} -16.5623 q^{49} -3.61803 q^{51} +2.00000i q^{53} +7.56231 q^{55} +4.23607i q^{57} -6.00000 q^{59} -3.85410 q^{61} -12.7082i q^{63} -0.854102i q^{65} -0.763932i q^{67} +0.618034 q^{69} -2.61803 q^{71} +7.52786i q^{73} -3.09017i q^{75} -16.4164i q^{77} +5.70820 q^{79} +5.70820 q^{81} -5.70820i q^{83} -13.0902i q^{85} +2.29180i q^{87} +9.70820 q^{89} -1.85410 q^{91} -5.47214i q^{93} -15.3262 q^{95} -16.0344i q^{97} +8.85410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{9} + 18 q^{11} - 14 q^{19} - 12 q^{21} + 20 q^{25} + 12 q^{29} + 22 q^{31} + 8 q^{39} - 18 q^{41} + 10 q^{45} - 26 q^{49} - 10 q^{51} - 10 q^{55} - 24 q^{59} - 2 q^{61} - 2 q^{69} - 6 q^{71}+ \cdots + 22 q^{99}+O(q^{100}) $ 4 * q + 6 * q^9 + 18 * q^11 - 14 * q^19 - 12 * q^21 + 20 * q^25 + 12 * q^29 + 22 * q^31 + 8 * q^39 - 18 * q^41 + 10 * q^45 - 26 * q^49 - 10 * q^51 - 10 * q^55 - 24 * q^59 - 2 * q^61 - 2 * q^69 - 6 * q^71 - 4 * q^79 - 4 * q^81 + 12 * q^89 + 6 * q^91 - 30 * q^95 + 22 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 0.618034i	− 0.356822i	−0.983956	−	0.178411i	$-0.942904\pi$
$3$	− 0.618034i	− 0.356822i	0.983956	−	0.178411i	$-0.0570957\pi$
$4$	0	0
$5$	2.23607	1.00000
$5$	2.23607	1.00000
$6$	0	0
$7$	− 4.85410i	− 1.83468i	−0.398107	−	0.917339i	$-0.630333\pi$
$7$	− 4.85410i	− 1.83468i	0.398107	−	0.917339i	$-0.369667\pi$
$8$	0	0
$9$	2.61803	0.872678
$10$	0	0
$11$	3.38197	1.01970	0.509851	−	0.860263i	$-0.329701\pi$
$11$	3.38197	1.01970	0.509851	+	0.860263i	$0.329701\pi$
$12$	0	0
$13$	− 0.381966i	− 0.105938i	−0.998596	−	0.0529692i	$-0.983131\pi$
$13$	− 0.381966i	− 0.105938i	0.998596	−	0.0529692i	$-0.0168685\pi$
$14$	0	0
$15$	− 1.38197i	− 0.356822i
$16$	0	0
$17$	− 5.85410i	− 1.41983i	−0.704288	−	0.709914i	$-0.748734\pi$
$17$	− 5.85410i	− 1.41983i	0.704288	−	0.709914i	$-0.251266\pi$
$18$	0	0
$19$	−6.85410	−1.57244	−0.786219	−	0.617947i	$-0.787964\pi$
$19$	−6.85410	−1.57244	−0.786219	+	0.617947i	$0.787964\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	5.00000	1.00000
$26$	0	0
$27$	− 3.47214i	− 0.668213i
$28$	0	0
$29$	−3.70820	−0.688596	−0.344298	−	0.938860i	$-0.611883\pi$
$29$	−3.70820	−0.688596	−0.344298	+	0.938860i	$0.611883\pi$
$30$	0	0
$31$	8.85410	1.59024	0.795122	−	0.606450i	$-0.207407\pi$
$31$	8.85410	1.59024	0.795122	+	0.606450i	$0.207407\pi$
$32$	0	0
$33$	− 2.09017i	− 0.363852i
$34$	0	0
$35$	− 10.8541i	− 1.83468i
$36$	0	0
$37$	3.70820i	0.609625i	0.952412	+	0.304812i	$0.0985938\pi$
$37$	3.70820i	0.609625i	−0.952412	+	0.304812i	$0.901406\pi$
$38$	0	0
$39$	−0.236068	−0.0378011
$40$	0	0
$41$	−3.38197	−0.528174	−0.264087	−	0.964499i	$-0.585071\pi$
$41$	−3.38197	−0.528174	−0.264087	+	0.964499i	$0.585071\pi$
$42$	0	0
$43$	6.76393i	1.03149i	0.856742	+	0.515745i	$0.172485\pi$
$43$	6.76393i	1.03149i	−0.856742	+	0.515745i	$0.827515\pi$
$44$	0	0
$45$	5.85410	0.872678
$46$	0	0
$47$	11.7082i	1.70782i	0.520423	+	0.853909i	$0.325774\pi$
$47$	11.7082i	1.70782i	−0.520423	+	0.853909i	$0.674226\pi$
$48$	0	0
$49$	−16.5623	−2.36604
$50$	0	0
$51$	−3.61803	−0.506626
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	7.56231	1.01970
$56$	0	0
$57$	4.23607i	0.561081i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−3.85410	−0.493467	−0.246734	−	0.969083i	$-0.579357\pi$
$61$	−3.85410	−0.493467	−0.246734	+	0.969083i	$0.579357\pi$
$62$	0	0
$63$	− 12.7082i	− 1.60108i
$64$	0	0
$65$	− 0.854102i	− 0.105938i
$66$	0	0
$67$	− 0.763932i	− 0.0933292i	−0.998911	−	0.0466646i	$-0.985141\pi$
$67$	− 0.763932i	− 0.0933292i	0.998911	−	0.0466646i	$-0.0148592\pi$
$68$	0	0
$69$	0.618034	0.0744025
$70$	0	0
$71$	−2.61803	−0.310703	−0.155352	−	0.987859i	$-0.549651\pi$
$71$	−2.61803	−0.310703	−0.155352	+	0.987859i	$0.549651\pi$
$72$	0	0
$73$	7.52786i	0.881070i	0.897735	+	0.440535i	$0.145211\pi$
$73$	7.52786i	0.881070i	−0.897735	+	0.440535i	$0.854789\pi$
$74$	0	0
$75$	− 3.09017i	− 0.356822i
$76$	0	0
$77$	− 16.4164i	− 1.87082i
$78$	0	0
$79$	5.70820	0.642223	0.321112	−	0.947041i	$-0.395944\pi$
$79$	5.70820	0.642223	0.321112	+	0.947041i	$0.395944\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	− 5.70820i	− 0.626557i	−0.949661	−	0.313278i	$-0.898573\pi$
$83$	− 5.70820i	− 0.626557i	0.949661	−	0.313278i	$-0.101427\pi$
$84$	0	0
$85$	− 13.0902i	− 1.41983i
$86$	0	0
$87$	2.29180i	0.245706i
$88$	0	0
$89$	9.70820	1.02907	0.514534	−	0.857470i	$-0.327965\pi$
$89$	9.70820	1.02907	0.514534	+	0.857470i	$0.327965\pi$
$90$	0	0
$91$	−1.85410	−0.194363
$92$	0	0
$93$	− 5.47214i	− 0.567434i
$94$	0	0
$95$	−15.3262	−1.57244
$96$	0	0
$97$	− 16.0344i	− 1.62805i	−0.580829	−	0.814025i	$-0.697272\pi$
$97$	− 16.0344i	− 1.62805i	0.580829	−	0.814025i	$-0.302728\pi$
$98$	0	0
$99$	8.85410	0.889871
$100$	0	0
$101$	−10.4721	−1.04202	−0.521008	−	0.853552i	$-0.674444\pi$
$101$	−10.4721	−1.04202	−0.521008	+	0.853552i	$0.674444\pi$
$102$	0	0
$103$	4.14590i	0.408507i	0.978918	+	0.204254i	$0.0654768\pi$
$103$	4.14590i	0.408507i	−0.978918	+	0.204254i	$0.934523\pi$
$104$	0	0
$105$	−6.70820	−0.654654
$106$	0	0
$107$	1.70820i	0.165138i	0.996585	+	0.0825692i	$0.0263125\pi$
$107$	1.70820i	0.165138i	−0.996585	+	0.0825692i	$0.973687\pi$
$108$	0	0
$109$	12.5623	1.20325	0.601625	−	0.798778i	$-0.294520\pi$
$109$	12.5623	1.20325	0.601625	+	0.798778i	$0.294520\pi$
$110$	0	0
$111$	2.29180	0.217528
$112$	0	0
$113$	− 13.4164i	− 1.26211i	−0.775738	−	0.631055i	$-0.782622\pi$
$113$	− 13.4164i	− 1.26211i	0.775738	−	0.631055i	$-0.217378\pi$
$114$	0	0
$115$	2.23607i	0.208514i
$116$	0	0
$117$	− 1.00000i	− 0.0924500i
$118$	0	0
$119$	−28.4164	−2.60493
$120$	0	0
$121$	0.437694	0.0397904
$122$	0	0
$123$	2.09017i	0.188464i
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	− 3.70820i	− 0.329050i	−0.986373	−	0.164525i	$-0.947391\pi$
$127$	− 3.70820i	− 0.329050i	0.986373	−	0.164525i	$-0.0526091\pi$
$128$	0	0
$129$	4.18034	0.368058
$130$	0	0
$131$	−8.18034	−0.714720	−0.357360	−	0.933967i	$-0.616323\pi$
$131$	−8.18034	−0.714720	−0.357360	+	0.933967i	$0.616323\pi$
$132$	0	0
$133$	33.2705i	2.88492i
$134$	0	0
$135$	− 7.76393i	− 0.668213i
$136$	0	0
$137$	− 7.14590i	− 0.610515i	−0.952270	−	0.305258i	$-0.901257\pi$
$137$	− 7.14590i	− 0.610515i	0.952270	−	0.305258i	$-0.0987426\pi$
$138$	0	0
$139$	17.7082	1.50199	0.750995	−	0.660308i	$-0.229574\pi$
$139$	17.7082	1.50199	0.750995	+	0.660308i	$0.229574\pi$
$140$	0	0
$141$	7.23607	0.609387
$142$	0	0
$143$	− 1.29180i	− 0.108025i
$144$	0	0
$145$	−8.29180	−0.688596
$146$	0	0
$147$	10.2361i	0.844257i
$148$	0	0
$149$	0.381966	0.0312919	0.0156459	−	0.999878i	$-0.495020\pi$
$149$	0.381966	0.0312919	0.0156459	+	0.999878i	$0.495020\pi$
$150$	0	0
$151$	19.2705	1.56821	0.784106	−	0.620627i	$-0.213122\pi$
$151$	19.2705	1.56821	0.784106	+	0.620627i	$0.213122\pi$
$152$	0	0
$153$	− 15.3262i	− 1.23905i
$154$	0	0
$155$	19.7984	1.59024
$156$	0	0
$157$	15.7082i	1.25365i	0.779160	+	0.626826i	$0.215646\pi$
$157$	15.7082i	1.25365i	−0.779160	+	0.626826i	$0.784354\pi$
$158$	0	0
$159$	1.23607	0.0980266
$160$	0	0
$161$	4.85410	0.382557
$162$	0	0
$163$	7.03444i	0.550980i	0.961304	+	0.275490i	$0.0888401\pi$
$163$	7.03444i	0.550980i	−0.961304	+	0.275490i	$0.911160\pi$
$164$	0	0
$165$	− 4.67376i	− 0.363852i
$166$	0	0
$167$	− 3.70820i	− 0.286949i	−0.989654	−	0.143475i	$-0.954172\pi$
$167$	− 3.70820i	− 0.286949i	0.989654	−	0.143475i	$-0.0458276\pi$
$168$	0	0
$169$	12.8541	0.988777
$170$	0	0
$171$	−17.9443	−1.37223
$172$	0	0
$173$	3.56231i	0.270837i	0.990788	+	0.135419i	$0.0432379\pi$
$173$	3.56231i	0.270837i	−0.990788	+	0.135419i	$0.956762\pi$
$174$	0	0
$175$	− 24.2705i	− 1.83468i
$176$	0	0
$177$	3.70820i	0.278726i
$178$	0	0
$179$	16.4721	1.23119	0.615593	−	0.788065i	$-0.288917\pi$
$179$	16.4721	1.23119	0.615593	+	0.788065i	$0.288917\pi$
$180$	0	0
$181$	−4.56231	−0.339114	−0.169557	−	0.985520i	$-0.554234\pi$
$181$	−4.56231	−0.339114	−0.169557	+	0.985520i	$0.554234\pi$
$182$	0	0
$183$	2.38197i	0.176080i
$184$	0	0
$185$	8.29180i	0.609625i
$186$	0	0
$187$	− 19.7984i	− 1.44780i
$188$	0	0
$189$	−16.8541	−1.22596
$190$	0	0
$191$	1.41641	0.102488	0.0512438	−	0.998686i	$-0.483681\pi$
$191$	1.41641	0.102488	0.0512438	+	0.998686i	$0.483681\pi$
$192$	0	0
$193$	2.29180i	0.164967i	0.996592	+	0.0824835i	$0.0262852\pi$
$193$	2.29180i	0.164967i	−0.996592	+	0.0824835i	$0.973715\pi$
$194$	0	0
$195$	−0.527864	−0.0378011
$196$	0	0
$197$	− 0.437694i	− 0.0311844i	−0.999878	−	0.0155922i	$-0.995037\pi$
$197$	− 0.437694i	− 0.0311844i	0.999878	−	0.0155922i	$-0.00496335\pi$
$198$	0	0
$199$	−15.4164	−1.09284	−0.546420	−	0.837511i	$-0.684010\pi$
$199$	−15.4164	−1.09284	−0.546420	+	0.837511i	$0.684010\pi$
$200$	0	0
$201$	−0.472136	−0.0333019
$202$	0	0
$203$	18.0000i	1.26335i
$204$	0	0
$205$	−7.56231	−0.528174
$206$	0	0
$207$	2.61803i	0.181966i
$208$	0	0
$209$	−23.1803	−1.60342
$210$	0	0
$211$	−5.70820	−0.392969	−0.196484	−	0.980507i	$-0.562953\pi$
$211$	−5.70820	−0.392969	−0.196484	+	0.980507i	$0.562953\pi$
$212$	0	0
$213$	1.61803i	0.110866i
$214$	0	0
$215$	15.1246i	1.03149i
$216$	0	0
$217$	− 42.9787i	− 2.91759i
$218$	0	0
$219$	4.65248	0.314385
$220$	0	0
$221$	−2.23607	−0.150414
$222$	0	0
$223$	− 22.3607i	− 1.49738i	−0.662919	−	0.748691i	$-0.730683\pi$
$223$	− 22.3607i	− 1.49738i	0.662919	−	0.748691i	$-0.269317\pi$
$224$	0	0
$225$	13.0902	0.872678
$226$	0	0
$227$	− 23.7082i	− 1.57357i	−0.617228	−	0.786784i	$-0.711744\pi$
$227$	− 23.7082i	− 1.57357i	0.617228	−	0.786784i	$-0.288256\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−10.1459	−0.667551
$232$	0	0
$233$	19.1246i	1.25289i	0.779464	+	0.626447i	$0.215492\pi$
$233$	19.1246i	1.25289i	−0.779464	+	0.626447i	$0.784508\pi$
$234$	0	0
$235$	26.1803i	1.70782i
$236$	0	0
$237$	− 3.52786i	− 0.229159i
$238$	0	0
$239$	−1.52786	−0.0988293	−0.0494147	−	0.998778i	$-0.515736\pi$
$239$	−1.52786	−0.0988293	−0.0494147	+	0.998778i	$0.515736\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	− 13.9443i	− 0.894525i
$244$	0	0
$245$	−37.0344	−2.36604
$246$	0	0
$247$	2.61803i	0.166582i
$248$	0	0
$249$	−3.52786	−0.223569
$250$	0	0
$251$	−7.79837	−0.492229	−0.246114	−	0.969241i	$-0.579154\pi$
$251$	−7.79837	−0.492229	−0.246114	+	0.969241i	$0.579154\pi$
$252$	0	0
$253$	3.38197i	0.212622i
$254$	0	0
$255$	−8.09017	−0.506626
$256$	0	0
$257$	15.4164i	0.961649i	0.876817	+	0.480825i	$0.159663\pi$
$257$	15.4164i	0.961649i	−0.876817	+	0.480825i	$0.840337\pi$
$258$	0	0
$259$	18.0000	1.11847
$260$	0	0
$261$	−9.70820	−0.600923
$262$	0	0
$263$	15.2705i	0.941620i	0.882235	+	0.470810i	$0.156038\pi$
$263$	15.2705i	0.941620i	−0.882235	+	0.470810i	$0.843962\pi$
$264$	0	0
$265$	4.47214i	0.274721i
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	0	0
$269$	10.4721	0.638497	0.319249	−	0.947671i	$-0.396569\pi$
$269$	10.4721	0.638497	0.319249	+	0.947671i	$0.396569\pi$
$270$	0	0
$271$	−19.1459	−1.16303	−0.581515	−	0.813536i	$-0.697540\pi$
$271$	−19.1459	−1.16303	−0.581515	+	0.813536i	$0.697540\pi$
$272$	0	0
$273$	1.14590i	0.0693529i
$274$	0	0
$275$	16.9098	1.01970
$276$	0	0
$277$	16.4721i	0.989715i	0.868974	+	0.494857i	$0.164780\pi$
$277$	16.4721i	0.989715i	−0.868974	+	0.494857i	$0.835220\pi$
$278$	0	0
$279$	23.1803	1.38777
$280$	0	0
$281$	−0.652476	−0.0389234	−0.0194617	−	0.999811i	$-0.506195\pi$
$281$	−0.652476	−0.0389234	−0.0194617	+	0.999811i	$0.506195\pi$
$282$	0	0
$283$	3.05573i	0.181644i	0.995867	+	0.0908221i	$0.0289495\pi$
$283$	3.05573i	0.181644i	−0.995867	+	0.0908221i	$0.971051\pi$
$284$	0	0
$285$	9.47214i	0.561081i
$286$	0	0
$287$	16.4164i	0.969030i
$288$	0	0
$289$	−17.2705	−1.01591
$290$	0	0
$291$	−9.90983	−0.580925
$292$	0	0
$293$	27.7082i	1.61873i	0.587306	+	0.809365i	$0.300189\pi$
$293$	27.7082i	1.61873i	−0.587306	+	0.809365i	$0.699811\pi$
$294$	0	0
$295$	−13.4164	−0.781133
$296$	0	0
$297$	− 11.7426i	− 0.681377i
$298$	0	0
$299$	0.381966	0.0220897
$300$	0	0
$301$	32.8328	1.89245
$302$	0	0
$303$	6.47214i	0.371814i
$304$	0	0
$305$	−8.61803	−0.493467
$306$	0	0
$307$	− 10.1459i	− 0.579057i	−0.957169	−	0.289528i	$-0.906502\pi$
$307$	− 10.1459i	− 0.579057i	0.957169	−	0.289528i	$-0.0934985\pi$
$308$	0	0
$309$	2.56231	0.145764
$310$	0	0
$311$	13.5279	0.767095	0.383547	−	0.923521i	$-0.374702\pi$
$311$	13.5279	0.767095	0.383547	+	0.923521i	$0.374702\pi$
$312$	0	0
$313$	− 19.1459i	− 1.08219i	−0.840961	−	0.541095i	$-0.818010\pi$
$313$	− 19.1459i	− 1.08219i	0.840961	−	0.541095i	$-0.181990\pi$
$314$	0	0
$315$	− 28.4164i	− 1.60108i
$316$	0	0
$317$	26.2705i	1.47550i	0.675075	+	0.737749i	$0.264111\pi$
$317$	26.2705i	1.47550i	−0.675075	+	0.737749i	$0.735889\pi$
$318$	0	0
$319$	−12.5410	−0.702162
$320$	0	0
$321$	1.05573	0.0589250
$322$	0	0
$323$	40.1246i	2.23259i
$324$	0	0
$325$	− 1.90983i	− 0.105938i
$326$	0	0
$327$	− 7.76393i	− 0.429346i
$328$	0	0
$329$	56.8328	3.13329
$330$	0	0
$331$	−15.1246	−0.831324	−0.415662	−	0.909519i	$-0.636450\pi$
$331$	−15.1246	−0.831324	−0.415662	+	0.909519i	$0.636450\pi$
$332$	0	0
$333$	9.70820i	0.532006i
$334$	0	0
$335$	− 1.70820i	− 0.0933292i
$336$	0	0
$337$	− 5.61803i	− 0.306034i	−0.988224	−	0.153017i	$-0.951101\pi$
$337$	− 5.61803i	− 0.306034i	0.988224	−	0.153017i	$-0.0488989\pi$
$338$	0	0
$339$	−8.29180	−0.450349
$340$	0	0
$341$	29.9443	1.62157
$342$	0	0
$343$	46.4164i	2.50625i
$344$	0	0
$345$	1.38197	0.0744025
$346$	0	0
$347$	5.56231i	0.298600i	0.988792	+	0.149300i	$0.0477021\pi$
$347$	5.56231i	0.298600i	−0.988792	+	0.149300i	$0.952298\pi$
$348$	0	0
$349$	−14.2918	−0.765022	−0.382511	−	0.923951i	$-0.624941\pi$
$349$	−14.2918	−0.765022	−0.382511	+	0.923951i	$0.624941\pi$
$350$	0	0
$351$	−1.32624	−0.0707893
$352$	0	0
$353$	− 8.00000i	− 0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$	− 8.00000i	− 0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$	0	0
$355$	−5.85410	−0.310703
$356$	0	0
$357$	17.5623i	0.929496i
$358$	0	0
$359$	13.4164	0.708091	0.354045	−	0.935228i	$-0.384806\pi$
$359$	13.4164	0.708091	0.354045	+	0.935228i	$0.384806\pi$
$360$	0	0
$361$	27.9787	1.47256
$362$	0	0
$363$	− 0.270510i	− 0.0141981i
$364$	0	0
$365$	16.8328i	0.881070i
$366$	0	0
$367$	− 12.0000i	− 0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$	− 12.0000i	− 0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$	0	0
$369$	−8.85410	−0.460926
$370$	0	0
$371$	9.70820	0.504025
$372$	0	0
$373$	3.05573i	0.158220i	0.996866	+	0.0791098i	$0.0252078\pi$
$373$	3.05573i	0.158220i	−0.996866	+	0.0791098i	$0.974792\pi$
$374$	0	0
$375$	− 6.90983i	− 0.356822i
$376$	0	0
$377$	1.41641i	0.0729487i
$378$	0	0
$379$	9.27051	0.476194	0.238097	−	0.971241i	$-0.423476\pi$
$379$	9.27051	0.476194	0.238097	+	0.971241i	$0.423476\pi$
$380$	0	0
$381$	−2.29180	−0.117412
$382$	0	0
$383$	27.4164i	1.40091i	0.713695	+	0.700456i	$0.247020\pi$
$383$	27.4164i	1.40091i	−0.713695	+	0.700456i	$0.752980\pi$
$384$	0	0
$385$	− 36.7082i	− 1.87082i
$386$	0	0
$387$	17.7082i	0.900159i
$388$	0	0
$389$	5.67376	0.287671	0.143836	−	0.989602i	$-0.454056\pi$
$389$	5.67376	0.287671	0.143836	+	0.989602i	$0.454056\pi$
$390$	0	0
$391$	5.85410	0.296055
$392$	0	0
$393$	5.05573i	0.255028i
$394$	0	0
$395$	12.7639	0.642223
$396$	0	0
$397$	35.5623i	1.78482i	0.451224	+	0.892410i	$0.350987\pi$
$397$	35.5623i	1.78482i	−0.451224	+	0.892410i	$0.649013\pi$
$398$	0	0
$399$	20.5623	1.02940
$400$	0	0
$401$	−34.4721	−1.72146	−0.860728	−	0.509065i	$-0.829991\pi$
$401$	−34.4721	−1.72146	−0.860728	+	0.509065i	$0.829991\pi$
$402$	0	0
$403$	− 3.38197i	− 0.168468i
$404$	0	0
$405$	12.7639	0.634245
$406$	0	0
$407$	12.5410i	0.621635i
$408$	0	0
$409$	6.56231	0.324485	0.162243	−	0.986751i	$-0.448127\pi$
$409$	6.56231	0.324485	0.162243	+	0.986751i	$0.448127\pi$
$410$	0	0
$411$	−4.41641	−0.217845
$412$	0	0
$413$	29.1246i	1.43313i
$414$	0	0
$415$	− 12.7639i	− 0.626557i
$416$	0	0
$417$	− 10.9443i	− 0.535943i
$418$	0	0
$419$	5.88854	0.287674	0.143837	−	0.989601i	$-0.454056\pi$
$419$	5.88854	0.287674	0.143837	+	0.989601i	$0.454056\pi$
$420$	0	0
$421$	6.85410	0.334048	0.167024	−	0.985953i	$-0.446584\pi$
$421$	6.85410	0.334048	0.167024	+	0.985953i	$0.446584\pi$
$422$	0	0
$423$	30.6525i	1.49037i
$424$	0	0
$425$	− 29.2705i	− 1.41983i
$426$	0	0
$427$	18.7082i	0.905353i
$428$	0	0
$429$	−0.798374	−0.0385459
$430$	0	0
$431$	6.76393	0.325807	0.162904	−	0.986642i	$-0.447914\pi$
$431$	6.76393	0.325807	0.162904	+	0.986642i	$0.447914\pi$
$432$	0	0
$433$	29.6180i	1.42335i	0.702508	+	0.711676i	$0.252064\pi$
$433$	29.6180i	1.42335i	−0.702508	+	0.711676i	$0.747936\pi$
$434$	0	0
$435$	5.12461i	0.245706i
$436$	0	0
$437$	− 6.85410i	− 0.327876i
$438$	0	0
$439$	−0.270510	−0.0129107	−0.00645536	−	0.999979i	$-0.502055\pi$
$439$	−0.270510	−0.0129107	−0.00645536	+	0.999979i	$0.502055\pi$
$440$	0	0
$441$	−43.3607	−2.06479
$442$	0	0
$443$	6.85410i	0.325648i	0.986655	+	0.162824i	$0.0520603\pi$
$443$	6.85410i	0.325648i	−0.986655	+	0.162824i	$0.947940\pi$
$444$	0	0
$445$	21.7082	1.02907
$446$	0	0
$447$	− 0.236068i	− 0.0111656i
$448$	0	0
$449$	−2.56231	−0.120923	−0.0604613	−	0.998171i	$-0.519257\pi$
$449$	−2.56231	−0.120923	−0.0604613	+	0.998171i	$0.519257\pi$
$450$	0	0
$451$	−11.4377	−0.538580
$452$	0	0
$453$	− 11.9098i	− 0.559573i
$454$	0	0
$455$	−4.14590	−0.194363
$456$	0	0
$457$	− 25.4164i	− 1.18893i	−0.804122	−	0.594465i	$-0.797364\pi$
$457$	− 25.4164i	− 1.18893i	0.804122	−	0.594465i	$-0.202636\pi$
$458$	0	0
$459$	−20.3262	−0.948748
$460$	0	0
$461$	−25.3050	−1.17857	−0.589285	−	0.807926i	$-0.700590\pi$
$461$	−25.3050	−1.17857	−0.589285	+	0.807926i	$0.700590\pi$
$462$	0	0
$463$	− 2.29180i	− 0.106509i	−0.998581	−	0.0532544i	$-0.983041\pi$
$463$	− 2.29180i	− 0.106509i	0.998581	−	0.0532544i	$-0.0169594\pi$
$464$	0	0
$465$	− 12.2361i	− 0.567434i
$466$	0	0
$467$	17.1246i	0.792433i	0.918157	+	0.396216i	$0.129677\pi$
$467$	17.1246i	0.792433i	−0.918157	+	0.396216i	$0.870323\pi$
$468$	0	0
$469$	−3.70820	−0.171229
$470$	0	0
$471$	9.70820	0.447330
$472$	0	0
$473$	22.8754i	1.05181i
$474$	0	0
$475$	−34.2705	−1.57244
$476$	0	0
$477$	5.23607i	0.239743i
$478$	0	0
$479$	−19.5279	−0.892251	−0.446125	−	0.894970i	$-0.647196\pi$
$479$	−19.5279	−0.892251	−0.446125	+	0.894970i	$0.647196\pi$
$480$	0	0
$481$	1.41641	0.0645826
$482$	0	0
$483$	− 3.00000i	− 0.136505i
$484$	0	0
$485$	− 35.8541i	− 1.62805i
$486$	0	0
$487$	21.7082i	0.983693i	0.870682	+	0.491846i	$0.163678\pi$
$487$	21.7082i	0.983693i	−0.870682	+	0.491846i	$0.836322\pi$
$488$	0	0
$489$	4.34752	0.196602
$490$	0	0
$491$	26.8328	1.21095	0.605474	−	0.795865i	$-0.292984\pi$
$491$	26.8328	1.21095	0.605474	+	0.795865i	$0.292984\pi$
$492$	0	0
$493$	21.7082i	0.977688i
$494$	0	0
$495$	19.7984	0.889871
$496$	0	0
$497$	12.7082i	0.570041i
$498$	0	0
$499$	11.4164	0.511069	0.255534	−	0.966800i	$-0.417749\pi$
$499$	11.4164	0.511069	0.255534	+	0.966800i	$0.417749\pi$
$500$	0	0
$501$	−2.29180	−0.102390
$502$	0	0
$503$	16.8541i	0.751487i	0.926724	+	0.375744i	$0.122613\pi$
$503$	16.8541i	0.751487i	−0.926724	+	0.375744i	$0.877387\pi$
$504$	0	0
$505$	−23.4164	−1.04202
$506$	0	0
$507$	− 7.94427i	− 0.352818i
$508$	0	0
$509$	24.6525	1.09270	0.546351	−	0.837556i	$-0.316017\pi$
$509$	24.6525	1.09270	0.546351	+	0.837556i	$0.316017\pi$
$510$	0	0
$511$	36.5410	1.61648
$512$	0	0
$513$	23.7984i	1.05072i
$514$	0	0
$515$	9.27051i	0.408507i
$516$	0	0
$517$	39.5967i	1.74146i
$518$	0	0
$519$	2.20163	0.0966407
$520$	0	0
$521$	20.0689	0.879234	0.439617	−	0.898185i	$-0.355114\pi$
$521$	20.0689	0.879234	0.439617	+	0.898185i	$0.355114\pi$
$522$	0	0
$523$	− 40.3607i	− 1.76485i	−0.470454	−	0.882425i	$-0.655910\pi$
$523$	− 40.3607i	− 1.76485i	0.470454	−	0.882425i	$-0.344090\pi$
$524$	0	0
$525$	−15.0000	−0.654654
$526$	0	0
$527$	− 51.8328i	− 2.25787i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−15.7082	−0.681678
$532$	0	0
$533$	1.29180i	0.0559539i
$534$	0	0
$535$	3.81966i	0.165138i
$536$	0	0
$537$	− 10.1803i	− 0.439314i
$538$	0	0
$539$	−56.0132	−2.41266
$540$	0	0
$541$	−27.4164	−1.17872	−0.589362	−	0.807869i	$-0.700621\pi$
$541$	−27.4164	−1.17872	−0.589362	+	0.807869i	$0.700621\pi$
$542$	0	0
$543$	2.81966i	0.121003i
$544$	0	0
$545$	28.0902	1.20325
$546$	0	0
$547$	1.14590i	0.0489951i	0.999700	+	0.0244975i	$0.00779859\pi$
$547$	1.14590i	0.0489951i	−0.999700	+	0.0244975i	$0.992201\pi$
$548$	0	0
$549$	−10.0902	−0.430638
$550$	0	0
$551$	25.4164	1.08278
$552$	0	0
$553$	− 27.7082i	− 1.17827i
$554$	0	0
$555$	5.12461	0.217528
$556$	0	0
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	2.58359	0.109274
$560$	0	0
$561$	−12.2361	−0.516607
$562$	0	0
$563$	− 2.29180i	− 0.0965877i	−0.998833	−	0.0482938i	$-0.984622\pi$
$563$	− 2.29180i	− 0.0965877i	0.998833	−	0.0482938i	$-0.0153784\pi$
$564$	0	0
$565$	− 30.0000i	− 1.26211i
$566$	0	0
$567$	− 27.7082i	− 1.16364i
$568$	0	0
$569$	−28.3607	−1.18894	−0.594471	−	0.804117i	$-0.702638\pi$
$569$	−28.3607	−1.18894	−0.594471	+	0.804117i	$0.702638\pi$
$570$	0	0
$571$	25.2705	1.05754	0.528769	−	0.848766i	$-0.322654\pi$
$571$	25.2705	1.05754	0.528769	+	0.848766i	$0.322654\pi$
$572$	0	0
$573$	− 0.875388i	− 0.0365699i
$574$	0	0
$575$	5.00000i	0.208514i
$576$	0	0
$577$	− 26.2918i	− 1.09454i	−0.836956	−	0.547271i	$-0.815667\pi$
$577$	− 26.2918i	− 1.09454i	0.836956	−	0.547271i	$-0.184333\pi$
$578$	0	0
$579$	1.41641	0.0588639
$580$	0	0
$581$	−27.7082	−1.14953
$582$	0	0
$583$	6.76393i	0.280133i
$584$	0	0
$585$	− 2.23607i	− 0.0924500i
$586$	0	0
$587$	2.14590i	0.0885707i	0.999019	+	0.0442853i	$0.0141011\pi$
$587$	2.14590i	0.0885707i	−0.999019	+	0.0442853i	$0.985899\pi$
$588$	0	0
$589$	−60.6869	−2.50056
$590$	0	0
$591$	−0.270510	−0.0111273
$592$	0	0
$593$	− 33.4164i	− 1.37225i	−0.727485	−	0.686124i	$-0.759311\pi$
$593$	− 33.4164i	− 1.37225i	0.727485	−	0.686124i	$-0.240689\pi$
$594$	0	0
$595$	−63.5410	−2.60493
$596$	0	0
$597$	9.52786i	0.389950i
$598$	0	0
$599$	21.3820	0.873643	0.436822	−	0.899548i	$-0.356104\pi$
$599$	21.3820	0.873643	0.436822	+	0.899548i	$0.356104\pi$
$600$	0	0
$601$	−3.14590	−0.128324	−0.0641619	−	0.997940i	$-0.520437\pi$
$601$	−3.14590	−0.128324	−0.0641619	+	0.997940i	$0.520437\pi$
$602$	0	0
$603$	− 2.00000i	− 0.0814463i
$604$	0	0
$605$	0.978714	0.0397904
$606$	0	0
$607$	− 38.9443i	− 1.58070i	−0.612656	−	0.790350i	$-0.709899\pi$
$607$	− 38.9443i	− 1.58070i	0.612656	−	0.790350i	$-0.290101\pi$
$608$	0	0
$609$	11.1246	0.450792
$610$	0	0
$611$	4.47214	0.180923
$612$	0	0
$613$	5.23607i	0.211483i	0.994394	+	0.105741i	$0.0337216\pi$
$613$	5.23607i	0.211483i	−0.994394	+	0.105741i	$0.966278\pi$
$614$	0	0
$615$	4.67376i	0.188464i
$616$	0	0
$617$	17.5623i	0.707032i	0.935429	+	0.353516i	$0.115014\pi$
$617$	17.5623i	0.707032i	−0.935429	+	0.353516i	$0.884986\pi$
$618$	0	0
$619$	−21.1459	−0.849925	−0.424963	−	0.905211i	$-0.639713\pi$
$619$	−21.1459	−0.849925	−0.424963	+	0.905211i	$0.639713\pi$
$620$	0	0
$621$	3.47214	0.139332
$622$	0	0
$623$	− 47.1246i	− 1.88801i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	14.3262i	0.572135i
$628$	0	0
$629$	21.7082	0.865563
$630$	0	0
$631$	−3.41641	−0.136005	−0.0680025	−	0.997685i	$-0.521663\pi$
$631$	−3.41641	−0.136005	−0.0680025	+	0.997685i	$0.521663\pi$
$632$	0	0
$633$	3.52786i	0.140220i
$634$	0	0
$635$	− 8.29180i	− 0.329050i
$636$	0	0
$637$	6.32624i	0.250655i
$638$	0	0
$639$	−6.85410	−0.271144
$640$	0	0
$641$	30.6525	1.21070	0.605350	−	0.795959i	$-0.293033\pi$
$641$	30.6525	1.21070	0.605350	+	0.795959i	$0.293033\pi$
$642$	0	0
$643$	35.0132i	1.38078i	0.723435	+	0.690392i	$0.242562\pi$
$643$	35.0132i	1.38078i	−0.723435	+	0.690392i	$0.757438\pi$
$644$	0	0
$645$	9.34752	0.368058
$646$	0	0
$647$	0.291796i	0.0114717i	0.999984	+	0.00573584i	$0.00182579\pi$
$647$	0.291796i	0.0114717i	−0.999984	+	0.00573584i	$0.998174\pi$
$648$	0	0
$649$	−20.2918	−0.796523
$650$	0	0
$651$	−26.5623	−1.04106
$652$	0	0
$653$	11.9787i	0.468763i	0.972145	+	0.234382i	$0.0753065\pi$
$653$	11.9787i	0.468763i	−0.972145	+	0.234382i	$0.924693\pi$
$654$	0	0
$655$	−18.2918	−0.714720
$656$	0	0
$657$	19.7082i	0.768890i
$658$	0	0
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	34.3951	1.33782	0.668908	−	0.743346i	$-0.266762\pi$
$661$	34.3951	1.33782	0.668908	+	0.743346i	$0.266762\pi$
$662$	0	0
$663$	1.38197i	0.0536711i
$664$	0	0
$665$	74.3951i	2.88492i
$666$	0	0
$667$	− 3.70820i	− 0.143582i
$668$	0	0
$669$	−13.8197	−0.534299
$670$	0	0
$671$	−13.0344	−0.503189
$672$	0	0
$673$	14.2918i	0.550908i	0.961314	+	0.275454i	$0.0888282\pi$
$673$	14.2918i	0.550908i	−0.961314	+	0.275454i	$0.911172\pi$
$674$	0	0
$675$	− 17.3607i	− 0.668213i
$676$	0	0
$677$	42.0000i	1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$	42.0000i	1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$	0	0
$679$	−77.8328	−2.98695
$680$	0	0
$681$	−14.6525	−0.561484
$682$	0	0
$683$	35.9787i	1.37669i	0.725385	+	0.688344i	$0.241662\pi$
$683$	35.9787i	1.37669i	−0.725385	+	0.688344i	$0.758338\pi$
$684$	0	0
$685$	− 15.9787i	− 0.610515i
$686$	0	0
$687$	− 6.18034i	− 0.235795i
$688$	0	0
$689$	0.763932	0.0291035
$690$	0	0
$691$	−19.1246	−0.727535	−0.363767	−	0.931490i	$-0.618510\pi$
$691$	−19.1246	−0.727535	−0.363767	+	0.931490i	$0.618510\pi$
$692$	0	0
$693$	− 42.9787i	− 1.63263i
$694$	0	0
$695$	39.5967	1.50199
$696$	0	0
$697$	19.7984i	0.749917i
$698$	0	0
$699$	11.8197	0.447061
$700$	0	0
$701$	36.9230	1.39456	0.697281	−	0.716798i	$-0.254393\pi$
$701$	36.9230	1.39456	0.697281	+	0.716798i	$0.254393\pi$
$702$	0	0
$703$	− 25.4164i	− 0.958598i
$704$	0	0
$705$	16.1803	0.609387
$706$	0	0
$707$	50.8328i	1.91176i
$708$	0	0
$709$	11.5623	0.434232	0.217116	−	0.976146i	$-0.430335\pi$
$709$	11.5623	0.434232	0.217116	+	0.976146i	$0.430335\pi$
$710$	0	0
$711$	14.9443	0.560454
$712$	0	0
$713$	8.85410i	0.331589i
$714$	0	0
$715$	− 2.88854i	− 0.108025i
$716$	0	0
$717$	0.944272i	0.0352645i
$718$	0	0
$719$	−32.4508	−1.21021	−0.605106	−	0.796145i	$-0.706869\pi$
$719$	−32.4508	−1.21021	−0.605106	+	0.796145i	$0.706869\pi$
$720$	0	0
$721$	20.1246	0.749480
$722$	0	0
$723$	1.23607i	0.0459699i
$724$	0	0
$725$	−18.5410	−0.688596
$726$	0	0
$727$	1.74265i	0.0646312i	0.999478	+	0.0323156i	$0.0102882\pi$
$727$	1.74265i	0.0646312i	−0.999478	+	0.0323156i	$0.989712\pi$
$728$	0	0
$729$	8.50658	0.315058
$730$	0	0
$731$	39.5967	1.46454
$732$	0	0
$733$	− 35.8885i	− 1.32557i	−0.748808	−	0.662787i	$-0.769374\pi$
$733$	− 35.8885i	− 1.32557i	0.748808	−	0.662787i	$-0.230626\pi$
$734$	0	0
$735$	22.8885i	0.844257i
$736$	0	0
$737$	− 2.58359i	− 0.0951678i
$738$	0	0
$739$	−32.5410	−1.19704	−0.598520	−	0.801108i	$-0.704244\pi$
$739$	−32.5410	−1.19704	−0.598520	+	0.801108i	$0.704244\pi$
$740$	0	0
$741$	1.61803	0.0594400
$742$	0	0
$743$	− 9.97871i	− 0.366084i	−0.983105	−	0.183042i	$-0.941406\pi$
$743$	− 9.97871i	− 0.366084i	0.983105	−	0.183042i	$-0.0585944\pi$
$744$	0	0
$745$	0.854102	0.0312919
$746$	0	0
$747$	− 14.9443i	− 0.546782i
$748$	0	0
$749$	8.29180	0.302976
$750$	0	0
$751$	41.1246	1.50066	0.750329	−	0.661064i	$-0.229895\pi$
$751$	41.1246	1.50066	0.750329	+	0.661064i	$0.229895\pi$
$752$	0	0
$753$	4.81966i	0.175638i
$754$	0	0
$755$	43.0902	1.56821
$756$	0	0
$757$	− 47.0132i	− 1.70872i	−0.519680	−	0.854361i	$-0.673949\pi$
$757$	− 47.0132i	− 1.70872i	0.519680	−	0.854361i	$-0.326051\pi$
$758$	0	0
$759$	2.09017	0.0758684
$760$	0	0
$761$	−11.5066	−0.417113	−0.208557	−	0.978010i	$-0.566877\pi$
$761$	−11.5066	−0.417113	−0.208557	+	0.978010i	$0.566877\pi$
$762$	0	0
$763$	− 60.9787i	− 2.20758i
$764$	0	0
$765$	− 34.2705i	− 1.23905i
$766$	0	0
$767$	2.29180i	0.0827520i
$768$	0	0
$769$	34.2918	1.23659	0.618297	−	0.785945i	$-0.287823\pi$
$769$	34.2918	1.23659	0.618297	+	0.785945i	$0.287823\pi$
$770$	0	0
$771$	9.52786	0.343138
$772$	0	0
$773$	− 14.0000i	− 0.503545i	−0.967786	−	0.251773i	$-0.918987\pi$
$773$	− 14.0000i	− 0.503545i	0.967786	−	0.251773i	$-0.0810135\pi$
$774$	0	0
$775$	44.2705	1.59024
$776$	0	0
$777$	− 11.1246i	− 0.399093i
$778$	0	0
$779$	23.1803	0.830522
$780$	0	0
$781$	−8.85410	−0.316825
$782$	0	0
$783$	12.8754i	0.460129i
$784$	0	0
$785$	35.1246i	1.25365i
$786$	0	0
$787$	− 31.4164i	− 1.11987i	−0.828535	−	0.559937i	$-0.810825\pi$
$787$	− 31.4164i	− 1.11987i	0.828535	−	0.559937i	$-0.189175\pi$
$788$	0	0
$789$	9.43769	0.335991
$790$	0	0
$791$	−65.1246	−2.31556
$792$	0	0
$793$	1.47214i	0.0522771i
$794$	0	0
$795$	2.76393	0.0980266
$796$	0	0
$797$	18.5836i	0.658265i	0.944284	+	0.329132i	$0.106756\pi$
$797$	18.5836i	0.658265i	−0.944284	+	0.329132i	$0.893244\pi$
$798$	0	0
$799$	68.5410	2.42481
$800$	0	0
$801$	25.4164	0.898045
$802$	0	0
$803$	25.4590i	0.898428i
$804$	0	0
$805$	10.8541	0.382557
$806$	0	0
$807$	− 6.47214i	− 0.227830i
$808$	0	0
$809$	−27.1591	−0.954861	−0.477431	−	0.878669i	$-0.658432\pi$
$809$	−27.1591	−0.954861	−0.477431	+	0.878669i	$0.658432\pi$
$810$	0	0
$811$	7.70820	0.270672	0.135336	−	0.990800i	$-0.456789\pi$
$811$	7.70820	0.270672	0.135336	+	0.990800i	$0.456789\pi$
$812$	0	0
$813$	11.8328i	0.414995i
$814$	0	0
$815$	15.7295i	0.550980i
$816$	0	0
$817$	− 46.3607i	− 1.62195i
$818$	0	0
$819$	−4.85410	−0.169616
$820$	0	0
$821$	−2.94427	−0.102756	−0.0513779	−	0.998679i	$-0.516361\pi$
$821$	−2.94427	−0.102756	−0.0513779	+	0.998679i	$0.516361\pi$
$822$	0	0
$823$	20.8328i	0.726186i	0.931753	+	0.363093i	$0.118279\pi$
$823$	20.8328i	0.726186i	−0.931753	+	0.363093i	$0.881721\pi$
$824$	0	0
$825$	− 10.4508i	− 0.363852i
$826$	0	0
$827$	− 48.2492i	− 1.67779i	−0.544293	−	0.838895i	$-0.683202\pi$
$827$	− 48.2492i	− 1.67779i	0.544293	−	0.838895i	$-0.316798\pi$
$828$	0	0
$829$	−32.5410	−1.13020	−0.565098	−	0.825024i	$-0.691162\pi$
$829$	−32.5410	−1.13020	−0.565098	+	0.825024i	$0.691162\pi$
$830$	0	0
$831$	10.1803	0.353152
$832$	0	0
$833$	96.9574i	3.35938i
$834$	0	0
$835$	− 8.29180i	− 0.286949i
$836$	0	0
$837$	− 30.7426i	− 1.06262i
$838$	0	0
$839$	−6.76393	−0.233517	−0.116758	−	0.993160i	$-0.537250\pi$
$839$	−6.76393	−0.233517	−0.116758	+	0.993160i	$0.537250\pi$
$840$	0	0
$841$	−15.2492	−0.525835
$842$	0	0
$843$	0.403252i	0.0138887i
$844$	0	0
$845$	28.7426	0.988777
$846$	0	0
$847$	− 2.12461i	− 0.0730025i
$848$	0	0
$849$	1.88854	0.0648147
$850$	0	0
$851$	−3.70820	−0.127116
$852$	0	0
$853$	51.2148i	1.75356i	0.480891	+	0.876780i	$0.340313\pi$
$853$	51.2148i	1.75356i	−0.480891	+	0.876780i	$0.659687\pi$
$854$	0	0
$855$	−40.1246	−1.37223
$856$	0	0
$857$	− 36.0000i	− 1.22974i	−0.788630	−	0.614868i	$-0.789209\pi$
$857$	− 36.0000i	− 1.22974i	0.788630	−	0.614868i	$-0.210791\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	10.1459	0.345771
$862$	0	0
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	0	0
$865$	7.96556i	0.270837i
$866$	0	0
$867$	10.6738i	0.362500i
$868$	0	0
$869$	19.3050	0.654876
$870$	0	0
$871$	−0.291796	−0.00988713
$872$	0	0
$873$	− 41.9787i	− 1.42076i
$874$	0	0
$875$	− 54.2705i	− 1.83468i
$876$	0	0
$877$	− 34.6869i	− 1.17129i	−0.810566	−	0.585647i	$-0.800840\pi$
$877$	− 34.6869i	− 1.17129i	0.810566	−	0.585647i	$-0.199160\pi$
$878$	0	0
$879$	17.1246	0.577599
$880$	0	0
$881$	−38.2918	−1.29008	−0.645042	−	0.764147i	$-0.723160\pi$
$881$	−38.2918	−1.29008	−0.645042	+	0.764147i	$0.723160\pi$
$882$	0	0
$883$	− 17.7295i	− 0.596645i	−0.954465	−	0.298322i	$-0.903573\pi$
$883$	− 17.7295i	− 0.596645i	0.954465	−	0.298322i	$-0.0964271\pi$
$884$	0	0
$885$	8.29180i	0.278726i
$886$	0	0
$887$	36.5410i	1.22693i	0.789723	+	0.613464i	$0.210224\pi$
$887$	36.5410i	1.22693i	−0.789723	+	0.613464i	$0.789776\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	19.3050	0.646740
$892$	0	0
$893$	− 80.2492i	− 2.68544i
$894$	0	0
$895$	36.8328	1.23119
$896$	0	0
$897$	− 0.236068i	− 0.00788208i
$898$	0	0
$899$	−32.8328	−1.09504
$900$	0	0
$901$	11.7082	0.390057
$902$	0	0
$903$	− 20.2918i	− 0.675269i
$904$	0	0
$905$	−10.2016	−0.339114
$906$	0	0
$907$	− 30.5410i	− 1.01410i	−0.861917	−	0.507049i	$-0.830736\pi$
$907$	− 30.5410i	− 1.01410i	0.861917	−	0.507049i	$-0.169264\pi$
$908$	0	0
$909$	−27.4164	−0.909345
$910$	0	0
$911$	−22.3607	−0.740842	−0.370421	−	0.928864i	$-0.620787\pi$
$911$	−22.3607	−0.740842	−0.370421	+	0.928864i	$0.620787\pi$
$912$	0	0
$913$	− 19.3050i	− 0.638901i
$914$	0	0
$915$	5.32624i	0.176080i
$916$	0	0
$917$	39.7082i	1.31128i
$918$	0	0
$919$	−14.5836	−0.481068	−0.240534	−	0.970641i	$-0.577323\pi$
$919$	−14.5836	−0.481068	−0.240534	+	0.970641i	$0.577323\pi$
$920$	0	0
$921$	−6.27051	−0.206620
$922$	0	0
$923$	1.00000i	0.0329154i
$924$	0	0
$925$	18.5410i	0.609625i
$926$	0	0
$927$	10.8541i	0.356495i
$928$	0	0
$929$	−19.3050	−0.633375	−0.316687	−	0.948530i	$-0.602571\pi$
$929$	−19.3050	−0.633375	−0.316687	+	0.948530i	$0.602571\pi$
$930$	0	0
$931$	113.520	3.72046
$932$	0	0
$933$	− 8.36068i	− 0.273716i
$934$	0	0
$935$	− 44.2705i	− 1.44780i
$936$	0	0
$937$	− 8.67376i	− 0.283359i	−0.989913	−	0.141680i	$-0.954750\pi$
$937$	− 8.67376i	− 0.283359i	0.989913	−	0.141680i	$-0.0452503\pi$
$938$	0	0
$939$	−11.8328	−0.386149
$940$	0	0
$941$	−24.2148	−0.789379	−0.394690	−	0.918814i	$-0.629148\pi$
$941$	−24.2148	−0.789379	−0.394690	+	0.918814i	$0.629148\pi$
$942$	0	0
$943$	− 3.38197i	− 0.110132i
$944$	0	0
$945$	−37.6869	−1.22596
$946$	0	0
$947$	47.3951i	1.54013i	0.637963	+	0.770067i	$0.279777\pi$
$947$	47.3951i	1.54013i	−0.637963	+	0.770067i	$0.720223\pi$
$948$	0	0
$949$	2.87539	0.0933391
$950$	0	0
$951$	16.2361	0.526491
$952$	0	0
$953$	36.3951i	1.17895i	0.807785	+	0.589477i	$0.200666\pi$
$953$	36.3951i	1.17895i	−0.807785	+	0.589477i	$0.799334\pi$
$954$	0	0
$955$	3.16718	0.102488
$956$	0	0
$957$	7.75078i	0.250547i
$958$	0	0
$959$	−34.6869	−1.12010
$960$	0	0
$961$	47.3951	1.52887
$962$	0	0
$963$	4.47214i	0.144113i
$964$	0	0
$965$	5.12461i	0.164967i
$966$	0	0
$967$	− 24.7639i	− 0.796354i	−0.917309	−	0.398177i	$-0.869643\pi$
$967$	− 24.7639i	− 0.796354i	0.917309	−	0.398177i	$-0.130357\pi$
$968$	0	0
$969$	24.7984	0.796639
$970$	0	0
$971$	−25.8541	−0.829698	−0.414849	−	0.909890i	$-0.636166\pi$
$971$	−25.8541	−0.829698	−0.414849	+	0.909890i	$0.636166\pi$
$972$	0	0
$973$	− 85.9574i	− 2.75567i
$974$	0	0
$975$	−1.18034	−0.0378011
$976$	0	0
$977$	− 32.2705i	− 1.03243i	−0.856461	−	0.516213i	$-0.827341\pi$
$977$	− 32.2705i	− 1.03243i	0.856461	−	0.516213i	$-0.172659\pi$
$978$	0	0
$979$	32.8328	1.04934
$980$	0	0
$981$	32.8885	1.05005
$982$	0	0
$983$	26.3951i	0.841874i	0.907090	+	0.420937i	$0.138299\pi$
$983$	26.3951i	0.841874i	−0.907090	+	0.420937i	$0.861701\pi$
$984$	0	0
$985$	− 0.978714i	− 0.0311844i
$986$	0	0
$987$	− 35.1246i	− 1.11803i
$988$	0	0
$989$	−6.76393	−0.215081
$990$	0	0
$991$	−51.2705	−1.62866	−0.814331	−	0.580401i	$-0.802896\pi$
$991$	−51.2705	−1.62866	−0.814331	+	0.580401i	$0.802896\pi$
$992$	0	0
$993$	9.34752i	0.296635i
$994$	0	0
$995$	−34.4721	−1.09284
$996$	0	0
$997$	26.7214i	0.846274i	0.906066	+	0.423137i	$0.139071\pi$
$997$	26.7214i	0.846274i	−0.906066	+	0.423137i	$0.860929\pi$
$998$	0	0
$999$	12.8754	0.407359

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.c.369.2		4
4.3	odd	2		230.2.b.a.139.2	✓	4
5.2	odd	4		9200.2.a.by.1.1		2
5.3	odd	4		9200.2.a.bo.1.2		2
5.4	even	2	inner	1840.2.e.c.369.3		4
12.11	even	2		2070.2.d.c.829.3		4
20.3	even	4		1150.2.a.l.1.1		2
20.7	even	4		1150.2.a.n.1.2		2
20.19	odd	2		230.2.b.a.139.3	yes	4
60.59	even	2		2070.2.d.c.829.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.b.a.139.2	✓	4	4.3	odd	2
230.2.b.a.139.3	yes	4	20.19	odd	2
1150.2.a.l.1.1		2	20.3	even	4
1150.2.a.n.1.2		2	20.7	even	4
1840.2.e.c.369.2		4	1.1	even	1	trivial
1840.2.e.c.369.3		4	5.4	even	2	inner
2070.2.d.c.829.1		4	60.59	even	2
2070.2.d.c.829.3		4	12.11	even	2
9200.2.a.bo.1.2		2	5.3	odd	4
9200.2.a.by.1.1		2	5.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.618034i q^{3} +2.23607 q^{5} -4.85410i q^{7} +2.61803 q^{9} +3.38197 q^{11} -0.381966i q^{13} -1.38197i q^{15} -5.85410i q^{17} -6.85410 q^{19} -3.00000 q^{21} +1.00000i q^{23} +5.00000 q^{25} -3.47214i q^{27} -3.70820 q^{29} +8.85410 q^{31} -2.09017i q^{33} -10.8541i q^{35} +3.70820i q^{37} -0.236068 q^{39} -3.38197 q^{41} +6.76393i q^{43} +5.85410 q^{45} +11.7082i q^{47} -16.5623 q^{49} -3.61803 q^{51} +2.00000i q^{53} +7.56231 q^{55} +4.23607i q^{57} -6.00000 q^{59} -3.85410 q^{61} -12.7082i q^{63} -0.854102i q^{65} -0.763932i q^{67} +0.618034 q^{69} -2.61803 q^{71} +7.52786i q^{73} -3.09017i q^{75} -16.4164i q^{77} +5.70820 q^{79} +5.70820 q^{81} -5.70820i q^{83} -13.0902i q^{85} +2.29180i q^{87} +9.70820 q^{89} -1.85410 q^{91} -5.47214i q^{93} -15.3262 q^{95} -16.0344i q^{97} +8.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{9} + 18 q^{11} - 14 q^{19} - 12 q^{21} + 20 q^{25} + 12 q^{29} + 22 q^{31} + 8 q^{39} - 18 q^{41} + 10 q^{45} - 26 q^{49} - 10 q^{51} - 10 q^{55} - 24 q^{59} - 2 q^{61} - 2 q^{69} - 6 q^{71}+ \cdots + 22 q^{99}+O(q^{100}) \) 4 * q + 6 * q^9 + 18 * q^11 - 14 * q^19 - 12 * q^21 + 20 * q^25 + 12 * q^29 + 22 * q^31 + 8 * q^39 - 18 * q^41 + 10 * q^45 - 26 * q^49 - 10 * q^51 - 10 * q^55 - 24 * q^59 - 2 * q^61 - 2 * q^69 - 6 * q^71 - 4 * q^79 - 4 * q^81 + 12 * q^89 + 6 * q^91 - 30 * q^95 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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