LMFDB - Embedded newform 1840.2.e.c.369.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			369.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	1840.369
Dual form			1840.2.e.c.369.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803i q^{3} -2.23607 q^{5} -1.85410i q^{7} +0.381966 q^{9} +O(q^{10})$	$q-1.61803i q^{3} -2.23607 q^{5} -1.85410i q^{7} +0.381966 q^{9} +5.61803 q^{11} +2.61803i q^{13} +3.61803i q^{15} -0.854102i q^{17} -0.145898 q^{19} -3.00000 q^{21} -1.00000i q^{23} +5.00000 q^{25} -5.47214i q^{27} +9.70820 q^{29} +2.14590 q^{31} -9.09017i q^{33} +4.14590i q^{35} +9.70820i q^{37} +4.23607 q^{39} -5.61803 q^{41} -11.2361i q^{43} -0.854102 q^{45} +1.70820i q^{47} +3.56231 q^{49} -1.38197 q^{51} -2.00000i q^{53} -12.5623 q^{55} +0.236068i q^{57} -6.00000 q^{59} +2.85410 q^{61} -0.708204i q^{63} -5.85410i q^{65} +5.23607i q^{67} -1.61803 q^{69} -0.381966 q^{71} -16.4721i q^{73} -8.09017i q^{75} -10.4164i q^{77} -7.70820 q^{79} -7.70820 q^{81} -7.70820i q^{83} +1.90983i q^{85} -15.7082i q^{87} -3.70820 q^{89} +4.85410 q^{91} -3.47214i q^{93} +0.326238 q^{95} -13.0344i q^{97} +2.14590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{9}+O(q^{10}) $ 4 * q + 6 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

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
$3$	− 1.61803i	− 0.934172i	−0.884212	−	0.467086i	$-0.845304\pi$
$3$	− 1.61803i	− 0.934172i	0.884212	−	0.467086i	$-0.154696\pi$
$4$	0	0
$5$	−2.23607	−1.00000
$5$	−2.23607	−1.00000
$6$	0	0
$7$	− 1.85410i	− 0.700785i	−0.936603	−	0.350392i	$-0.886048\pi$
$7$	− 1.85410i	− 0.700785i	0.936603	−	0.350392i	$-0.113952\pi$
$8$	0	0
$9$	0.381966	0.127322
$10$	0	0
$11$	5.61803	1.69390	0.846950	−	0.531672i	$-0.178436\pi$
$11$	5.61803	1.69390	0.846950	+	0.531672i	$0.178436\pi$
$12$	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
$15$	3.61803i	0.934172i
$16$	0	0
$17$	− 0.854102i	− 0.207150i	−0.994622	−	0.103575i	$-0.966972\pi$
$17$	− 0.854102i	− 0.207150i	0.994622	−	0.103575i	$-0.0330282\pi$
$18$	0	0
$19$	−0.145898	−0.0334713	−0.0167357	−	0.999860i	$-0.505327\pi$
$19$	−0.145898	−0.0334713	−0.0167357	+	0.999860i	$0.505327\pi$
$20$	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
$27$	− 5.47214i	− 1.05311i
$28$	0	0
$29$	9.70820	1.80277	0.901384	−	0.433020i	$-0.142552\pi$
$29$	9.70820	1.80277	0.901384	+	0.433020i	$0.142552\pi$
$30$	0	0
$31$	2.14590	0.385415	0.192707	−	0.981256i	$-0.438273\pi$
$31$	2.14590	0.385415	0.192707	+	0.981256i	$0.438273\pi$
$32$	0	0
$33$	− 9.09017i	− 1.58240i
$34$	0	0
$35$	4.14590i	0.700785i
$36$	0	0
$37$	9.70820i	1.59602i	0.602645	+	0.798009i	$0.294114\pi$
$37$	9.70820i	1.59602i	−0.602645	+	0.798009i	$0.705886\pi$
$38$	0	0
$39$	4.23607	0.678314
$40$	0	0
$41$	−5.61803	−0.877390	−0.438695	−	0.898636i	$-0.644559\pi$
$41$	−5.61803	−0.877390	−0.438695	+	0.898636i	$0.644559\pi$
$42$	0	0
$43$	− 11.2361i	− 1.71348i	−0.515745	−	0.856742i	$-0.672485\pi$
$43$	− 11.2361i	− 1.71348i	0.515745	−	0.856742i	$-0.327515\pi$
$44$	0	0
$45$	−0.854102	−0.127322
$46$	0	0
$47$	1.70820i	0.249167i	0.992209	+	0.124584i	$0.0397595\pi$
$47$	1.70820i	0.249167i	−0.992209	+	0.124584i	$0.960241\pi$
$48$	0	0
$49$	3.56231	0.508901
$50$	0	0
$51$	−1.38197	−0.193514
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	−12.5623	−1.69390
$56$	0	0
$57$	0.236068i	0.0312680i
$58$	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
$61$	2.85410	0.365430	0.182715	−	0.983166i	$-0.441511\pi$
$61$	2.85410	0.365430	0.182715	+	0.983166i	$0.441511\pi$
$62$	0	0
$63$	− 0.708204i	− 0.0892253i
$64$	0	0
$65$	− 5.85410i	− 0.726112i
$66$	0	0
$67$	5.23607i	0.639688i	0.947470	+	0.319844i	$0.103630\pi$
$67$	5.23607i	0.639688i	−0.947470	+	0.319844i	$0.896370\pi$
$68$	0	0
$69$	−1.61803	−0.194788
$70$	0	0
$71$	−0.381966	−0.0453310	−0.0226655	−	0.999743i	$-0.507215\pi$
$71$	−0.381966	−0.0453310	−0.0226655	+	0.999743i	$0.507215\pi$
$72$	0	0
$73$	− 16.4721i	− 1.92792i	−0.266051	−	0.963959i	$-0.585719\pi$
$73$	− 16.4721i	− 1.92792i	0.266051	−	0.963959i	$-0.414281\pi$
$74$	0	0
$75$	− 8.09017i	− 0.934172i
$76$	0	0
$77$	− 10.4164i	− 1.18706i
$78$	0	0
$79$	−7.70820	−0.867241	−0.433620	−	0.901096i	$-0.642764\pi$
$79$	−7.70820	−0.867241	−0.433620	+	0.901096i	$0.642764\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	− 7.70820i	− 0.846085i	−0.906110	−	0.423043i	$-0.860962\pi$
$83$	− 7.70820i	− 0.846085i	0.906110	−	0.423043i	$-0.139038\pi$
$84$	0	0
$85$	1.90983i	0.207150i
$86$	0	0
$87$	− 15.7082i	− 1.68410i
$88$	0	0
$89$	−3.70820	−0.393069	−0.196534	−	0.980497i	$-0.562969\pi$
$89$	−3.70820	−0.393069	−0.196534	+	0.980497i	$0.562969\pi$
$90$	0	0
$91$	4.85410	0.508848
$92$	0	0
$93$	− 3.47214i	− 0.360044i
$94$	0	0
$95$	0.326238	0.0334713
$96$	0	0
$97$	− 13.0344i	− 1.32345i	−0.749748	−	0.661724i	$-0.769825\pi$
$97$	− 13.0344i	− 1.32345i	0.749748	−	0.661724i	$-0.230175\pi$
$98$	0	0
$99$	2.14590	0.215671
$100$	0	0
$101$	−1.52786	−0.152028	−0.0760141	−	0.997107i	$-0.524219\pi$
$101$	−1.52786	−0.152028	−0.0760141	+	0.997107i	$0.524219\pi$
$102$	0	0
$103$	− 10.8541i	− 1.06949i	−0.845015	−	0.534743i	$-0.820408\pi$
$103$	− 10.8541i	− 1.06949i	0.845015	−	0.534743i	$-0.179592\pi$
$104$	0	0
$105$	6.70820	0.654654
$106$	0	0
$107$	11.7082i	1.13187i	0.824448	+	0.565937i	$0.191486\pi$
$107$	11.7082i	1.13187i	−0.824448	+	0.565937i	$0.808514\pi$
$108$	0	0
$109$	−7.56231	−0.724338	−0.362169	−	0.932113i	$-0.617964\pi$
$109$	−7.56231	−0.724338	−0.362169	+	0.932113i	$0.617964\pi$
$110$	0	0
$111$	15.7082	1.49096
$112$	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
$119$	−1.58359	−0.145168
$120$	0	0
$121$	20.5623	1.86930
$122$	0	0
$123$	9.09017i	0.819633i
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	− 9.70820i	− 0.861464i	−0.902480	−	0.430732i	$-0.858255\pi$
$127$	− 9.70820i	− 0.861464i	0.902480	−	0.430732i	$-0.141745\pi$
$128$	0	0
$129$	−18.1803	−1.60069
$130$	0	0
$131$	14.1803	1.23894	0.619471	−	0.785020i	$-0.287347\pi$
$131$	14.1803	1.23894	0.619471	+	0.785020i	$0.287347\pi$
$132$	0	0
$133$	0.270510i	0.0234562i
$134$	0	0
$135$	12.2361i	1.05311i
$136$	0	0
$137$	13.8541i	1.18364i	0.806072	+	0.591818i	$0.201590\pi$
$137$	13.8541i	1.18364i	−0.806072	+	0.591818i	$0.798410\pi$
$138$	0	0
$139$	4.29180	0.364025	0.182013	−	0.983296i	$-0.441739\pi$
$139$	4.29180	0.364025	0.182013	+	0.983296i	$0.441739\pi$
$140$	0	0
$141$	2.76393	0.232765
$142$	0	0
$143$	14.7082i	1.22996i
$144$	0	0
$145$	−21.7082	−1.80277
$146$	0	0
$147$	− 5.76393i	− 0.475401i
$148$	0	0
$149$	2.61803	0.214478	0.107239	−	0.994233i	$-0.465799\pi$
$149$	2.61803	0.214478	0.107239	+	0.994233i	$0.465799\pi$
$150$	0	0
$151$	−14.2705	−1.16132	−0.580659	−	0.814147i	$-0.697205\pi$
$151$	−14.2705	−1.16132	−0.580659	+	0.814147i	$0.697205\pi$
$152$	0	0
$153$	− 0.326238i	− 0.0263748i
$154$	0	0
$155$	−4.79837	−0.385415
$156$	0	0
$157$	− 2.29180i	− 0.182905i	−0.995809	−	0.0914526i	$-0.970849\pi$
$157$	− 2.29180i	− 0.182905i	0.995809	−	0.0914526i	$-0.0291510\pi$
$158$	0	0
$159$	−3.23607	−0.256637
$160$	0	0
$161$	−1.85410	−0.146124
$162$	0	0
$163$	22.0344i	1.72587i	0.505314	+	0.862935i	$0.331377\pi$
$163$	22.0344i	1.72587i	−0.505314	+	0.862935i	$0.668623\pi$
$164$	0	0
$165$	20.3262i	1.58240i
$166$	0	0
$167$	− 9.70820i	− 0.751243i	−0.926773	−	0.375622i	$-0.877429\pi$
$167$	− 9.70820i	− 0.751243i	0.926773	−	0.375622i	$-0.122571\pi$
$168$	0	0
$169$	6.14590	0.472761
$170$	0	0
$171$	−0.0557281	−0.00426163
$172$	0	0
$173$	16.5623i	1.25921i	0.776916	+	0.629604i	$0.216783\pi$
$173$	16.5623i	1.25921i	−0.776916	+	0.629604i	$0.783217\pi$
$174$	0	0
$175$	− 9.27051i	− 0.700785i
$176$	0	0
$177$	9.70820i	0.729713i
$178$	0	0
$179$	7.52786	0.562659	0.281329	−	0.959611i	$-0.409225\pi$
$179$	7.52786	0.562659	0.281329	+	0.959611i	$0.409225\pi$
$180$	0	0
$181$	15.5623	1.15674	0.578369	−	0.815776i	$-0.303690\pi$
$181$	15.5623	1.15674	0.578369	+	0.815776i	$0.303690\pi$
$182$	0	0
$183$	− 4.61803i	− 0.341375i
$184$	0	0
$185$	− 21.7082i	− 1.59602i
$186$	0	0
$187$	− 4.79837i	− 0.350892i
$188$	0	0
$189$	−10.1459	−0.738005
$190$	0	0
$191$	−25.4164	−1.83907	−0.919533	−	0.393012i	$-0.871433\pi$
$191$	−25.4164	−1.83907	−0.919533	+	0.393012i	$0.871433\pi$
$192$	0	0
$193$	− 15.7082i	− 1.13070i	−0.824851	−	0.565351i	$-0.808741\pi$
$193$	− 15.7082i	− 1.13070i	0.824851	−	0.565351i	$-0.191259\pi$
$194$	0	0
$195$	−9.47214	−0.678314
$196$	0	0
$197$	20.5623i	1.46500i	0.680765	+	0.732502i	$0.261647\pi$
$197$	20.5623i	1.46500i	−0.680765	+	0.732502i	$0.738353\pi$
$198$	0	0
$199$	11.4164	0.809288	0.404644	−	0.914474i	$-0.367395\pi$
$199$	11.4164	0.809288	0.404644	+	0.914474i	$0.367395\pi$
$200$	0	0
$201$	8.47214	0.597578
$202$	0	0
$203$	− 18.0000i	− 1.26335i
$204$	0	0
$205$	12.5623	0.877390
$206$	0	0
$207$	− 0.381966i	− 0.0265485i
$208$	0	0
$209$	−0.819660	−0.0566971
$210$	0	0
$211$	7.70820	0.530655	0.265327	−	0.964158i	$-0.414520\pi$
$211$	7.70820	0.530655	0.265327	+	0.964158i	$0.414520\pi$
$212$	0	0
$213$	0.618034i	0.0423470i
$214$	0	0
$215$	25.1246i	1.71348i
$216$	0	0
$217$	− 3.97871i	− 0.270093i
$218$	0	0
$219$	−26.6525	−1.80101
$220$	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
$225$	1.90983	0.127322
$226$	0	0
$227$	10.2918i	0.683090i	0.939865	+	0.341545i	$0.110950\pi$
$227$	10.2918i	0.683090i	−0.939865	+	0.341545i	$0.889050\pi$
$228$	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
$231$	−16.8541	−1.10892
$232$	0	0
$233$	21.1246i	1.38392i	0.721936	+	0.691960i	$0.243252\pi$
$233$	21.1246i	1.38392i	−0.721936	+	0.691960i	$0.756748\pi$
$234$	0	0
$235$	− 3.81966i	− 0.249167i
$236$	0	0
$237$	12.4721i	0.810152i
$238$	0	0
$239$	−10.4721	−0.677386	−0.338693	−	0.940897i	$-0.609985\pi$
$239$	−10.4721	−0.677386	−0.338693	+	0.940897i	$0.609985\pi$
$240$	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
$243$	− 3.94427i	− 0.253025i
$244$	0	0
$245$	−7.96556	−0.508901
$246$	0	0
$247$	− 0.381966i	− 0.0243039i
$248$	0	0
$249$	−12.4721	−0.790390
$250$	0	0
$251$	16.7984	1.06030	0.530152	−	0.847903i	$-0.322135\pi$
$251$	16.7984	1.06030	0.530152	+	0.847903i	$0.322135\pi$
$252$	0	0
$253$	− 5.61803i	− 0.353203i
$254$	0	0
$255$	3.09017	0.193514
$256$	0	0
$257$	11.4164i	0.712136i	0.934460	+	0.356068i	$0.115883\pi$
$257$	11.4164i	0.712136i	−0.934460	+	0.356068i	$0.884117\pi$
$258$	0	0
$259$	18.0000	1.11847
$260$	0	0
$261$	3.70820	0.229532
$262$	0	0
$263$	18.2705i	1.12661i	0.826250	+	0.563304i	$0.190470\pi$
$263$	18.2705i	1.12661i	−0.826250	+	0.563304i	$0.809530\pi$
$264$	0	0
$265$	4.47214i	0.274721i
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	1.52786	0.0931555	0.0465778	−	0.998915i	$-0.485168\pi$
$269$	1.52786	0.0931555	0.0465778	+	0.998915i	$0.485168\pi$
$270$	0	0
$271$	−25.8541	−1.57052	−0.785262	−	0.619163i	$-0.787472\pi$
$271$	−25.8541	−1.57052	−0.785262	+	0.619163i	$0.787472\pi$
$272$	0	0
$273$	− 7.85410i	− 0.475352i
$274$	0	0
$275$	28.0902	1.69390
$276$	0	0
$277$	− 7.52786i	− 0.452306i	−0.974092	−	0.226153i	$-0.927385\pi$
$277$	− 7.52786i	− 0.452306i	0.974092	−	0.226153i	$-0.0726149\pi$
$278$	0	0
$279$	0.819660	0.0490718
$280$	0	0
$281$	30.6525	1.82857	0.914287	−	0.405068i	$-0.132752\pi$
$281$	30.6525	1.82857	0.914287	+	0.405068i	$0.132752\pi$
$282$	0	0
$283$	− 20.9443i	− 1.24501i	−0.782617	−	0.622504i	$-0.786115\pi$
$283$	− 20.9443i	− 1.24501i	0.782617	−	0.622504i	$-0.213885\pi$
$284$	0	0
$285$	− 0.527864i	− 0.0312680i
$286$	0	0
$287$	10.4164i	0.614861i
$288$	0	0
$289$	16.2705	0.957089
$290$	0	0
$291$	−21.0902	−1.23633
$292$	0	0
$293$	− 14.2918i	− 0.834936i	−0.908692	−	0.417468i	$-0.862918\pi$
$293$	− 14.2918i	− 0.834936i	0.908692	−	0.417468i	$-0.137082\pi$
$294$	0	0
$295$	13.4164	0.781133
$296$	0	0
$297$	− 30.7426i	− 1.78387i
$298$	0	0
$299$	2.61803	0.151405
$300$	0	0
$301$	−20.8328	−1.20078
$302$	0	0
$303$	2.47214i	0.142020i
$304$	0	0
$305$	−6.38197	−0.365430
$306$	0	0
$307$	16.8541i	0.961914i	0.876744	+	0.480957i	$0.159711\pi$
$307$	16.8541i	0.961914i	−0.876744	+	0.480957i	$0.840289\pi$
$308$	0	0
$309$	−17.5623	−0.999085
$310$	0	0
$311$	22.4721	1.27428	0.637139	−	0.770749i	$-0.280118\pi$
$311$	22.4721	1.27428	0.637139	+	0.770749i	$0.280118\pi$
$312$	0	0
$313$	25.8541i	1.46136i	0.682720	+	0.730680i	$0.260797\pi$
$313$	25.8541i	1.46136i	−0.682720	+	0.730680i	$0.739203\pi$
$314$	0	0
$315$	1.58359i	0.0892253i
$316$	0	0
$317$	7.27051i	0.408353i	0.978934	+	0.204176i	$0.0654516\pi$
$317$	7.27051i	0.408353i	−0.978934	+	0.204176i	$0.934548\pi$
$318$	0	0
$319$	54.5410	3.05371
$320$	0	0
$321$	18.9443	1.05737
$322$	0	0
$323$	0.124612i	0.00693359i
$324$	0	0
$325$	13.0902i	0.726112i
$326$	0	0
$327$	12.2361i	0.676656i
$328$	0	0
$329$	3.16718	0.174613
$330$	0	0
$331$	25.1246	1.38097	0.690487	−	0.723345i	$-0.257396\pi$
$331$	25.1246	1.38097	0.690487	+	0.723345i	$0.257396\pi$
$332$	0	0
$333$	3.70820i	0.203208i
$334$	0	0
$335$	− 11.7082i	− 0.639688i
$336$	0	0
$337$	3.38197i	0.184227i	0.995748	+	0.0921137i	$0.0293623\pi$
$337$	3.38197i	0.184227i	−0.995748	+	0.0921137i	$0.970638\pi$
$338$	0	0
$339$	−21.7082	−1.17903
$340$	0	0
$341$	12.0557	0.652854
$342$	0	0
$343$	− 19.5836i	− 1.05741i
$344$	0	0
$345$	3.61803	0.194788
$346$	0	0
$347$	14.5623i	0.781746i	0.920445	+	0.390873i	$0.127827\pi$
$347$	14.5623i	0.781746i	−0.920445	+	0.390873i	$0.872173\pi$
$348$	0	0
$349$	−27.7082	−1.48319	−0.741593	−	0.670850i	$-0.765929\pi$
$349$	−27.7082	−1.48319	−0.741593	+	0.670850i	$0.765929\pi$
$350$	0	0
$351$	14.3262	0.764678
$352$	0	0
$353$	8.00000i	0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$	8.00000i	0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$	0	0
$355$	0.854102	0.0453310
$356$	0	0
$357$	2.56231i	0.135612i
$358$	0	0
$359$	−13.4164	−0.708091	−0.354045	−	0.935228i	$-0.615194\pi$
$359$	−13.4164	−0.708091	−0.354045	+	0.935228i	$0.615194\pi$
$360$	0	0
$361$	−18.9787	−0.998880
$362$	0	0
$363$	− 33.2705i	− 1.74625i
$364$	0	0
$365$	36.8328i	1.92792i
$366$	0	0
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	0	0
$369$	−2.14590	−0.111711
$370$	0	0
$371$	−3.70820	−0.192520
$372$	0	0
$373$	− 20.9443i	− 1.08445i	−0.840232	−	0.542227i	$-0.817581\pi$
$373$	− 20.9443i	− 1.08445i	0.840232	−	0.542227i	$-0.182419\pi$
$374$	0	0
$375$	18.0902i	0.934172i
$376$	0	0
$377$	25.4164i	1.30901i
$378$	0	0
$379$	−24.2705	−1.24669	−0.623346	−	0.781946i	$-0.714227\pi$
$379$	−24.2705	−1.24669	−0.623346	+	0.781946i	$0.714227\pi$
$380$	0	0
$381$	−15.7082	−0.804756
$382$	0	0
$383$	− 0.583592i	− 0.0298202i	−0.999889	−	0.0149101i	$-0.995254\pi$
$383$	− 0.583592i	− 0.0298202i	0.999889	−	0.0149101i	$-0.00474620\pi$
$384$	0	0
$385$	23.2918i	1.18706i
$386$	0	0
$387$	− 4.29180i	− 0.218164i
$388$	0	0
$389$	21.3262	1.08128	0.540642	−	0.841253i	$-0.318182\pi$
$389$	21.3262	1.08128	0.540642	+	0.841253i	$0.318182\pi$
$390$	0	0
$391$	−0.854102	−0.0431938
$392$	0	0
$393$	− 22.9443i	− 1.15739i
$394$	0	0
$395$	17.2361	0.867241
$396$	0	0
$397$	− 15.4377i	− 0.774796i	−0.921913	−	0.387398i	$-0.873374\pi$
$397$	− 15.4377i	− 0.774796i	0.921913	−	0.387398i	$-0.126626\pi$
$398$	0	0
$399$	0.437694	0.0219121
$400$	0	0
$401$	−25.5279	−1.27480	−0.637400	−	0.770533i	$-0.719990\pi$
$401$	−25.5279	−1.27480	−0.637400	+	0.770533i	$0.719990\pi$
$402$	0	0
$403$	5.61803i	0.279854i
$404$	0	0
$405$	17.2361	0.856467
$406$	0	0
$407$	54.5410i	2.70350i
$408$	0	0
$409$	−13.5623	−0.670613	−0.335306	−	0.942109i	$-0.608840\pi$
$409$	−13.5623	−0.670613	−0.335306	+	0.942109i	$0.608840\pi$
$410$	0	0
$411$	22.4164	1.10572
$412$	0	0
$413$	11.1246i	0.547406i
$414$	0	0
$415$	17.2361i	0.846085i
$416$	0	0
$417$	− 6.94427i	− 0.340062i
$418$	0	0
$419$	−29.8885	−1.46015	−0.730075	−	0.683367i	$-0.760515\pi$
$419$	−29.8885	−1.46015	−0.730075	+	0.683367i	$0.760515\pi$
$420$	0	0
$421$	0.145898	0.00711064	0.00355532	−	0.999994i	$-0.498868\pi$
$421$	0.145898	0.00711064	0.00355532	+	0.999994i	$0.498868\pi$
$422$	0	0
$423$	0.652476i	0.0317245i
$424$	0	0
$425$	− 4.27051i	− 0.207150i
$426$	0	0
$427$	− 5.29180i	− 0.256088i
$428$	0	0
$429$	23.7984	1.14900
$430$	0	0
$431$	11.2361	0.541222	0.270611	−	0.962689i	$-0.412774\pi$
$431$	11.2361	0.541222	0.270611	+	0.962689i	$0.412774\pi$
$432$	0	0
$433$	− 27.3820i	− 1.31589i	−0.753065	−	0.657947i	$-0.771425\pi$
$433$	− 27.3820i	− 1.31589i	0.753065	−	0.657947i	$-0.228575\pi$
$434$	0	0
$435$	35.1246i	1.68410i
$436$	0	0
$437$	0.145898i	0.00697925i
$438$	0	0
$439$	33.2705	1.58791	0.793957	−	0.607973i	$-0.208017\pi$
$439$	33.2705	1.58791	0.793957	+	0.607973i	$0.208017\pi$
$440$	0	0
$441$	1.36068	0.0647943
$442$	0	0
$443$	− 0.145898i	− 0.00693182i	−0.999994	−	0.00346591i	$-0.998897\pi$
$443$	− 0.145898i	− 0.00693182i	0.999994	−	0.00346591i	$-0.00110324\pi$
$444$	0	0
$445$	8.29180	0.393069
$446$	0	0
$447$	− 4.23607i	− 0.200359i
$448$	0	0
$449$	17.5623	0.828816	0.414408	−	0.910091i	$-0.363989\pi$
$449$	17.5623	0.828816	0.414408	+	0.910091i	$0.363989\pi$
$450$	0	0
$451$	−31.5623	−1.48621
$452$	0	0
$453$	23.0902i	1.08487i
$454$	0	0
$455$	−10.8541	−0.508848
$456$	0	0
$457$	− 1.41641i	− 0.0662568i	−0.999451	−	0.0331284i	$-0.989453\pi$
$457$	− 1.41641i	− 0.0662568i	0.999451	−	0.0331284i	$-0.0105470\pi$
$458$	0	0
$459$	−4.67376	−0.218153
$460$	0	0
$461$	37.3050	1.73746	0.868732	−	0.495282i	$-0.164935\pi$
$461$	37.3050	1.73746	0.868732	+	0.495282i	$0.164935\pi$
$462$	0	0
$463$	15.7082i	0.730022i	0.931003	+	0.365011i	$0.118935\pi$
$463$	15.7082i	0.730022i	−0.931003	+	0.365011i	$0.881065\pi$
$464$	0	0
$465$	7.76393i	0.360044i
$466$	0	0
$467$	23.1246i	1.07008i	0.844827	+	0.535040i	$0.179703\pi$
$467$	23.1246i	1.07008i	−0.844827	+	0.535040i	$0.820297\pi$
$468$	0	0
$469$	9.70820	0.448283
$470$	0	0
$471$	−3.70820	−0.170865
$472$	0	0
$473$	− 63.1246i	− 2.90247i
$474$	0	0
$475$	−0.729490	−0.0334713
$476$	0	0
$477$	− 0.763932i	− 0.0349780i
$478$	0	0
$479$	−28.4721	−1.30093	−0.650463	−	0.759538i	$-0.725425\pi$
$479$	−28.4721	−1.30093	−0.650463	+	0.759538i	$0.725425\pi$
$480$	0	0
$481$	−25.4164	−1.15889
$482$	0	0
$483$	3.00000i	0.136505i
$484$	0	0
$485$	29.1459i	1.32345i
$486$	0	0
$487$	− 8.29180i	− 0.375737i	−0.982194	−	0.187869i	$-0.939842\pi$
$487$	− 8.29180i	− 0.375737i	0.982194	−	0.187869i	$-0.0601579\pi$
$488$	0	0
$489$	35.6525	1.61226
$490$	0	0
$491$	−26.8328	−1.21095	−0.605474	−	0.795865i	$-0.707016\pi$
$491$	−26.8328	−1.21095	−0.605474	+	0.795865i	$0.707016\pi$
$492$	0	0
$493$	− 8.29180i	− 0.373444i
$494$	0	0
$495$	−4.79837	−0.215671
$496$	0	0
$497$	0.708204i	0.0317673i
$498$	0	0
$499$	−15.4164	−0.690133	−0.345067	−	0.938578i	$-0.612144\pi$
$499$	−15.4164	−0.690133	−0.345067	+	0.938578i	$0.612144\pi$
$500$	0	0
$501$	−15.7082	−0.701791
$502$	0	0
$503$	− 10.1459i	− 0.452383i	−0.974083	−	0.226192i	$-0.927372\pi$
$503$	− 10.1459i	− 0.452383i	0.974083	−	0.226192i	$-0.0726276\pi$
$504$	0	0
$505$	3.41641	0.152028
$506$	0	0
$507$	− 9.94427i	− 0.441641i
$508$	0	0
$509$	−6.65248	−0.294866	−0.147433	−	0.989072i	$-0.547101\pi$
$509$	−6.65248	−0.294866	−0.147433	+	0.989072i	$0.547101\pi$
$510$	0	0
$511$	−30.5410	−1.35106
$512$	0	0
$513$	0.798374i	0.0352491i
$514$	0	0
$515$	24.2705i	1.06949i
$516$	0	0
$517$	9.59675i	0.422064i
$518$	0	0
$519$	26.7984	1.17632
$520$	0	0
$521$	−38.0689	−1.66783	−0.833914	−	0.551894i	$-0.813905\pi$
$521$	−38.0689	−1.66783	−0.833914	+	0.551894i	$0.813905\pi$
$522$	0	0
$523$	− 4.36068i	− 0.190679i	−0.995445	−	0.0953396i	$-0.969606\pi$
$523$	− 4.36068i	− 0.190679i	0.995445	−	0.0953396i	$-0.0303937\pi$
$524$	0	0
$525$	−15.0000	−0.654654
$526$	0	0
$527$	− 1.83282i	− 0.0798387i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−2.29180	−0.0994555
$532$	0	0
$533$	− 14.7082i	− 0.637083i
$534$	0	0
$535$	− 26.1803i	− 1.13187i
$536$	0	0
$537$	− 12.1803i	− 0.525620i
$538$	0	0
$539$	20.0132	0.862028
$540$	0	0
$541$	−0.583592	−0.0250906	−0.0125453	−	0.999921i	$-0.503993\pi$
$541$	−0.583592	−0.0250906	−0.0125453	+	0.999921i	$0.503993\pi$
$542$	0	0
$543$	− 25.1803i	− 1.08059i
$544$	0	0
$545$	16.9098	0.724338
$546$	0	0
$547$	− 7.85410i	− 0.335817i	−0.985803	−	0.167909i	$-0.946299\pi$
$547$	− 7.85410i	− 0.335817i	0.985803	−	0.167909i	$-0.0537013\pi$
$548$	0	0
$549$	1.09017	0.0465273
$550$	0	0
$551$	−1.41641	−0.0603410
$552$	0	0
$553$	14.2918i	0.607749i
$554$	0	0
$555$	−35.1246	−1.49096
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	29.4164	1.24418
$560$	0	0
$561$	−7.76393	−0.327793
$562$	0	0
$563$	15.7082i	0.662022i	0.943627	+	0.331011i	$0.107390\pi$
$563$	15.7082i	0.662022i	−0.943627	+	0.331011i	$0.892610\pi$
$564$	0	0
$565$	30.0000i	1.26211i
$566$	0	0
$567$	14.2918i	0.600199i
$568$	0	0
$569$	16.3607	0.685875	0.342938	−	0.939358i	$-0.388578\pi$
$569$	16.3607	0.685875	0.342938	+	0.939358i	$0.388578\pi$
$570$	0	0
$571$	−8.27051	−0.346110	−0.173055	−	0.984912i	$-0.555364\pi$
$571$	−8.27051	−0.346110	−0.173055	+	0.984912i	$0.555364\pi$
$572$	0	0
$573$	41.1246i	1.71801i
$574$	0	0
$575$	− 5.00000i	− 0.208514i
$576$	0	0
$577$	39.7082i	1.65307i	0.562882	+	0.826537i	$0.309692\pi$
$577$	39.7082i	1.65307i	−0.562882	+	0.826537i	$0.690308\pi$
$578$	0	0
$579$	−25.4164	−1.05627
$580$	0	0
$581$	−14.2918	−0.592924
$582$	0	0
$583$	− 11.2361i	− 0.465350i
$584$	0	0
$585$	− 2.23607i	− 0.0924500i
$586$	0	0
$587$	− 8.85410i	− 0.365448i	−0.983164	−	0.182724i	$-0.941509\pi$
$587$	− 8.85410i	− 0.365448i	0.983164	−	0.182724i	$-0.0584915\pi$
$588$	0	0
$589$	−0.313082	−0.0129003
$590$	0	0
$591$	33.2705	1.36857
$592$	0	0
$593$	6.58359i	0.270356i	0.990821	+	0.135178i	$0.0431606\pi$
$593$	6.58359i	0.270356i	−0.990821	+	0.135178i	$0.956839\pi$
$594$	0	0
$595$	3.54102	0.145168
$596$	0	0
$597$	− 18.4721i	− 0.756014i
$598$	0	0
$599$	23.6180	0.965007	0.482503	−	0.875894i	$-0.339728\pi$
$599$	23.6180	0.965007	0.482503	+	0.875894i	$0.339728\pi$
$600$	0	0
$601$	−9.85410	−0.401957	−0.200979	−	0.979596i	$-0.564412\pi$
$601$	−9.85410	−0.401957	−0.200979	+	0.979596i	$0.564412\pi$
$602$	0	0
$603$	2.00000i	0.0814463i
$604$	0	0
$605$	−45.9787	−1.86930
$606$	0	0
$607$	21.0557i	0.854626i	0.904104	+	0.427313i	$0.140540\pi$
$607$	21.0557i	0.854626i	−0.904104	+	0.427313i	$0.859460\pi$
$608$	0	0
$609$	−29.1246	−1.18019
$610$	0	0
$611$	−4.47214	−0.180923
$612$	0	0
$613$	− 0.763932i	− 0.0308549i	−0.999881	−	0.0154275i	$-0.995089\pi$
$613$	− 0.763932i	− 0.0308549i	0.999881	−	0.0154275i	$-0.00491091\pi$
$614$	0	0
$615$	− 20.3262i	− 0.819633i
$616$	0	0
$617$	2.56231i	0.103155i	0.998669	+	0.0515773i	$0.0164248\pi$
$617$	2.56231i	0.103155i	−0.998669	+	0.0515773i	$0.983575\pi$
$618$	0	0
$619$	−27.8541	−1.11955	−0.559775	−	0.828644i	$-0.689113\pi$
$619$	−27.8541	−1.11955	−0.559775	+	0.828644i	$0.689113\pi$
$620$	0	0
$621$	−5.47214	−0.219589
$622$	0	0
$623$	6.87539i	0.275457i
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	1.32624i	0.0529648i
$628$	0	0
$629$	8.29180	0.330616
$630$	0	0
$631$	23.4164	0.932192	0.466096	−	0.884734i	$-0.345660\pi$
$631$	23.4164	0.932192	0.466096	+	0.884734i	$0.345660\pi$
$632$	0	0
$633$	− 12.4721i	− 0.495723i
$634$	0	0
$635$	21.7082i	0.861464i
$636$	0	0
$637$	9.32624i	0.369519i
$638$	0	0
$639$	−0.145898	−0.00577164
$640$	0	0
$641$	−0.652476	−0.0257712	−0.0128856	−	0.999917i	$-0.504102\pi$
$641$	−0.652476	−0.0257712	−0.0128856	+	0.999917i	$0.504102\pi$
$642$	0	0
$643$	41.0132i	1.61740i	0.588221	+	0.808700i	$0.299829\pi$
$643$	41.0132i	1.61740i	−0.588221	+	0.808700i	$0.700171\pi$
$644$	0	0
$645$	40.6525	1.60069
$646$	0	0
$647$	− 13.7082i	− 0.538925i	−0.963011	−	0.269463i	$-0.913154\pi$
$647$	− 13.7082i	− 0.538925i	0.963011	−	0.269463i	$-0.0868460\pi$
$648$	0	0
$649$	−33.7082	−1.32316
$650$	0	0
$651$	−6.43769	−0.252313
$652$	0	0
$653$	34.9787i	1.36882i	0.729096	+	0.684411i	$0.239941\pi$
$653$	34.9787i	1.36882i	−0.729096	+	0.684411i	$0.760059\pi$
$654$	0	0
$655$	−31.7082	−1.23894
$656$	0	0
$657$	− 6.29180i	− 0.245466i
$658$	0	0
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	−39.3951	−1.53229	−0.766146	−	0.642666i	$-0.777828\pi$
$661$	−39.3951	−1.53229	−0.766146	+	0.642666i	$0.777828\pi$
$662$	0	0
$663$	− 3.61803i	− 0.140513i
$664$	0	0
$665$	− 0.604878i	− 0.0234562i
$666$	0	0
$667$	− 9.70820i	− 0.375903i
$668$	0	0
$669$	−36.1803	−1.39881
$670$	0	0
$671$	16.0344	0.619003
$672$	0	0
$673$	− 27.7082i	− 1.06807i	−0.845461	−	0.534036i	$-0.820675\pi$
$673$	− 27.7082i	− 1.06807i	0.845461	−	0.534036i	$-0.179325\pi$
$674$	0	0
$675$	− 27.3607i	− 1.05311i
$676$	0	0
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	0	0
$679$	−24.1672	−0.927451
$680$	0	0
$681$	16.6525	0.638124
$682$	0	0
$683$	10.9787i	0.420089i	0.977692	+	0.210044i	$0.0673609\pi$
$683$	10.9787i	0.420089i	−0.977692	+	0.210044i	$0.932639\pi$
$684$	0	0
$685$	− 30.9787i	− 1.18364i
$686$	0	0
$687$	− 16.1803i	− 0.617318i
$688$	0	0
$689$	5.23607	0.199478
$690$	0	0
$691$	21.1246	0.803618	0.401809	−	0.915723i	$-0.368382\pi$
$691$	21.1246	0.803618	0.401809	+	0.915723i	$0.368382\pi$
$692$	0	0
$693$	− 3.97871i	− 0.151139i
$694$	0	0
$695$	−9.59675	−0.364025
$696$	0	0
$697$	4.79837i	0.181751i
$698$	0	0
$699$	34.1803	1.29282
$700$	0	0
$701$	−27.9230	−1.05464	−0.527318	−	0.849668i	$-0.676802\pi$
$701$	−27.9230	−1.05464	−0.527318	+	0.849668i	$0.676802\pi$
$702$	0	0
$703$	− 1.41641i	− 0.0534208i
$704$	0	0
$705$	−6.18034	−0.232765
$706$	0	0
$707$	2.83282i	0.106539i
$708$	0	0
$709$	−8.56231	−0.321564	−0.160782	−	0.986990i	$-0.551402\pi$
$709$	−8.56231	−0.321564	−0.160782	+	0.986990i	$0.551402\pi$
$710$	0	0
$711$	−2.94427	−0.110419
$712$	0	0
$713$	− 2.14590i	− 0.0803645i
$714$	0	0
$715$	− 32.8885i	− 1.22996i
$716$	0	0
$717$	16.9443i	0.632795i
$718$	0	0
$719$	23.4508	0.874569	0.437285	−	0.899323i	$-0.355940\pi$
$719$	23.4508	0.874569	0.437285	+	0.899323i	$0.355940\pi$
$720$	0	0
$721$	−20.1246	−0.749480
$722$	0	0
$723$	3.23607i	0.120351i
$724$	0	0
$725$	48.5410	1.80277
$726$	0	0
$727$	40.7426i	1.51106i	0.655113	+	0.755531i	$0.272621\pi$
$727$	40.7426i	1.51106i	−0.655113	+	0.755531i	$0.727379\pi$
$728$	0	0
$729$	−29.5066	−1.09284
$730$	0	0
$731$	−9.59675	−0.354949
$732$	0	0
$733$	0.111456i	0.00411673i	0.999998	+	0.00205836i	$0.000655198\pi$
$733$	0.111456i	0.00411673i	−0.999998	+	0.00205836i	$0.999345\pi$
$734$	0	0
$735$	12.8885i	0.475401i
$736$	0	0
$737$	29.4164i	1.08357i
$738$	0	0
$739$	34.5410	1.27061	0.635306	−	0.772261i	$-0.280874\pi$
$739$	34.5410	1.27061	0.635306	+	0.772261i	$0.280874\pi$
$740$	0	0
$741$	−0.618034	−0.0227040
$742$	0	0
$743$	− 36.9787i	− 1.35662i	−0.734777	−	0.678309i	$-0.762713\pi$
$743$	− 36.9787i	− 1.35662i	0.734777	−	0.678309i	$-0.237287\pi$
$744$	0	0
$745$	−5.85410	−0.214478
$746$	0	0
$747$	− 2.94427i	− 0.107725i
$748$	0	0
$749$	21.7082	0.793201
$750$	0	0
$751$	0.875388	0.0319434	0.0159717	−	0.999872i	$-0.494916\pi$
$751$	0.875388	0.0319434	0.0159717	+	0.999872i	$0.494916\pi$
$752$	0	0
$753$	− 27.1803i	− 0.990507i
$754$	0	0
$755$	31.9098	1.16132
$756$	0	0
$757$	− 29.0132i	− 1.05450i	−0.849710	−	0.527251i	$-0.823223\pi$
$757$	− 29.0132i	− 1.05450i	0.849710	−	0.527251i	$-0.176777\pi$
$758$	0	0
$759$	−9.09017	−0.329952
$760$	0	0
$761$	26.5066	0.960863	0.480431	−	0.877032i	$-0.340480\pi$
$761$	26.5066	0.960863	0.480431	+	0.877032i	$0.340480\pi$
$762$	0	0
$763$	14.0213i	0.507605i
$764$	0	0
$765$	0.729490i	0.0263748i
$766$	0	0
$767$	− 15.7082i	− 0.567190i
$768$	0	0
$769$	47.7082	1.72040	0.860201	−	0.509955i	$-0.170338\pi$
$769$	47.7082	1.72040	0.860201	+	0.509955i	$0.170338\pi$
$770$	0	0
$771$	18.4721	0.665258
$772$	0	0
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	10.7295	0.385415
$776$	0	0
$777$	− 29.1246i	− 1.04484i
$778$	0	0
$779$	0.819660	0.0293674
$780$	0	0
$781$	−2.14590	−0.0767863
$782$	0	0
$783$	− 53.1246i	− 1.89852i
$784$	0	0
$785$	5.12461i	0.182905i
$786$	0	0
$787$	4.58359i	0.163387i	0.996657	+	0.0816937i	$0.0260329\pi$
$787$	4.58359i	0.163387i	−0.996657	+	0.0816937i	$0.973967\pi$
$788$	0	0
$789$	29.5623	1.05245
$790$	0	0
$791$	−24.8754	−0.884467
$792$	0	0
$793$	7.47214i	0.265343i
$794$	0	0
$795$	7.23607	0.256637
$796$	0	0
$797$	− 45.4164i	− 1.60873i	−0.594134	−	0.804366i	$-0.702505\pi$
$797$	− 45.4164i	− 1.60873i	0.594134	−	0.804366i	$-0.297495\pi$
$798$	0	0
$799$	1.45898	0.0516150
$800$	0	0
$801$	−1.41641	−0.0500463
$802$	0	0
$803$	− 92.5410i	− 3.26570i
$804$	0	0
$805$	4.14590	0.146124
$806$	0	0
$807$	− 2.47214i	− 0.0870233i
$808$	0	0
$809$	42.1591	1.48223	0.741117	−	0.671376i	$-0.234297\pi$
$809$	42.1591	1.48223	0.741117	+	0.671376i	$0.234297\pi$
$810$	0	0
$811$	−5.70820	−0.200442	−0.100221	−	0.994965i	$-0.531955\pi$
$811$	−5.70820	−0.200442	−0.100221	+	0.994965i	$0.531955\pi$
$812$	0	0
$813$	41.8328i	1.46714i
$814$	0	0
$815$	− 49.2705i	− 1.72587i
$816$	0	0
$817$	1.63932i	0.0573526i
$818$	0	0
$819$	1.85410	0.0647876
$820$	0	0
$821$	14.9443	0.521559	0.260779	−	0.965398i	$-0.416020\pi$
$821$	14.9443	0.521559	0.260779	+	0.965398i	$0.416020\pi$
$822$	0	0
$823$	32.8328i	1.14448i	0.820086	+	0.572240i	$0.193925\pi$
$823$	32.8328i	1.14448i	−0.820086	+	0.572240i	$0.806075\pi$
$824$	0	0
$825$	− 45.4508i	− 1.58240i
$826$	0	0
$827$	− 32.2492i	− 1.12142i	−0.828014	−	0.560708i	$-0.810529\pi$
$827$	− 32.2492i	− 1.12142i	0.828014	−	0.560708i	$-0.189471\pi$
$828$	0	0
$829$	34.5410	1.19966	0.599830	−	0.800128i	$-0.295235\pi$
$829$	34.5410	1.19966	0.599830	+	0.800128i	$0.295235\pi$
$830$	0	0
$831$	−12.1803	−0.422531
$832$	0	0
$833$	− 3.04257i	− 0.105419i
$834$	0	0
$835$	21.7082i	0.751243i
$836$	0	0
$837$	− 11.7426i	− 0.405885i
$838$	0	0
$839$	−11.2361	−0.387912	−0.193956	−	0.981010i	$-0.562132\pi$
$839$	−11.2361	−0.387912	−0.193956	+	0.981010i	$0.562132\pi$
$840$	0	0
$841$	65.2492	2.24997
$842$	0	0
$843$	− 49.5967i	− 1.70820i
$844$	0	0
$845$	−13.7426	−0.472761
$846$	0	0
$847$	− 38.1246i	− 1.30998i
$848$	0	0
$849$	−33.8885	−1.16305
$850$	0	0
$851$	9.70820	0.332793
$852$	0	0
$853$	0.214782i	0.00735399i	0.999993	+	0.00367699i	$0.00117043\pi$
$853$	0.214782i	0.00735399i	−0.999993	+	0.00367699i	$0.998830\pi$
$854$	0	0
$855$	0.124612	0.00426163
$856$	0	0
$857$	36.0000i	1.22974i	0.788630	+	0.614868i	$0.210791\pi$
$857$	36.0000i	1.22974i	−0.788630	+	0.614868i	$0.789209\pi$
$858$	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
$861$	16.8541	0.574386
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	− 37.0344i	− 1.25921i
$866$	0	0
$867$	− 26.3262i	− 0.894086i
$868$	0	0
$869$	−43.3050	−1.46902
$870$	0	0
$871$	−13.7082	−0.464485
$872$	0	0
$873$	− 4.97871i	− 0.168504i
$874$	0	0
$875$	20.7295i	0.700785i
$876$	0	0
$877$	− 25.6869i	− 0.867386i	−0.901061	−	0.433693i	$-0.857210\pi$
$877$	− 25.6869i	− 0.867386i	0.901061	−	0.433693i	$-0.142790\pi$
$878$	0	0
$879$	−23.1246	−0.779974
$880$	0	0
$881$	−51.7082	−1.74209	−0.871047	−	0.491200i	$-0.836558\pi$
$881$	−51.7082	−1.74209	−0.871047	+	0.491200i	$0.836558\pi$
$882$	0	0
$883$	51.2705i	1.72539i	0.505725	+	0.862695i	$0.331225\pi$
$883$	51.2705i	1.72539i	−0.505725	+	0.862695i	$0.668775\pi$
$884$	0	0
$885$	− 21.7082i	− 0.729713i
$886$	0	0
$887$	30.5410i	1.02547i	0.858548	+	0.512734i	$0.171367\pi$
$887$	30.5410i	1.02547i	−0.858548	+	0.512734i	$0.828633\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−43.3050	−1.45077
$892$	0	0
$893$	− 0.249224i	− 0.00833995i
$894$	0	0
$895$	−16.8328	−0.562659
$896$	0	0
$897$	− 4.23607i	− 0.141438i
$898$	0	0
$899$	20.8328	0.694813
$900$	0	0
$901$	−1.70820	−0.0569085
$902$	0	0
$903$	33.7082i	1.12174i
$904$	0	0
$905$	−34.7984	−1.15674
$906$	0	0
$907$	− 36.5410i	− 1.21332i	−0.794960	−	0.606662i	$-0.792508\pi$
$907$	− 36.5410i	− 1.21332i	0.794960	−	0.606662i	$-0.207492\pi$
$908$	0	0
$909$	−0.583592	−0.0193565
$910$	0	0
$911$	22.3607	0.740842	0.370421	−	0.928864i	$-0.379213\pi$
$911$	22.3607	0.740842	0.370421	+	0.928864i	$0.379213\pi$
$912$	0	0
$913$	− 43.3050i	− 1.43318i
$914$	0	0
$915$	10.3262i	0.341375i
$916$	0	0
$917$	− 26.2918i	− 0.868232i
$918$	0	0
$919$	−41.4164	−1.36620	−0.683101	−	0.730324i	$-0.739369\pi$
$919$	−41.4164	−1.36620	−0.683101	+	0.730324i	$0.739369\pi$
$920$	0	0
$921$	27.2705	0.898594
$922$	0	0
$923$	− 1.00000i	− 0.0329154i
$924$	0	0
$925$	48.5410i	1.59602i
$926$	0	0
$927$	− 4.14590i	− 0.136169i
$928$	0	0
$929$	43.3050	1.42079	0.710395	−	0.703804i	$-0.248516\pi$
$929$	43.3050	1.42079	0.710395	+	0.703804i	$0.248516\pi$
$930$	0	0
$931$	−0.519733	−0.0170336
$932$	0	0
$933$	− 36.3607i	− 1.19040i
$934$	0	0
$935$	10.7295i	0.350892i
$936$	0	0
$937$	24.3262i	0.794704i	0.917666	+	0.397352i	$0.130071\pi$
$937$	24.3262i	0.794704i	−0.917666	+	0.397352i	$0.869929\pi$
$938$	0	0
$939$	41.8328	1.36516
$940$	0	0
$941$	27.2148	0.887177	0.443588	−	0.896231i	$-0.353705\pi$
$941$	27.2148	0.887177	0.443588	+	0.896231i	$0.353705\pi$
$942$	0	0
$943$	5.61803i	0.182948i
$944$	0	0
$945$	22.6869	0.738005
$946$	0	0
$947$	26.3951i	0.857726i	0.903369	+	0.428863i	$0.141086\pi$
$947$	26.3951i	0.857726i	−0.903369	+	0.428863i	$0.858914\pi$
$948$	0	0
$949$	43.1246	1.39988
$950$	0	0
$951$	11.7639	0.381472
$952$	0	0
$953$	37.3951i	1.21135i	0.795713	+	0.605673i	$0.207096\pi$
$953$	37.3951i	1.21135i	−0.795713	+	0.605673i	$0.792904\pi$
$954$	0	0
$955$	56.8328	1.83907
$956$	0	0
$957$	− 88.2492i	− 2.85269i
$958$	0	0
$959$	25.6869	0.829474
$960$	0	0
$961$	−26.3951	−0.851456
$962$	0	0
$963$	4.47214i	0.144113i
$964$	0	0
$965$	35.1246i	1.13070i
$966$	0	0
$967$	29.2361i	0.940169i	0.882622	+	0.470084i	$0.155776\pi$
$967$	29.2361i	0.940169i	−0.882622	+	0.470084i	$0.844224\pi$
$968$	0	0
$969$	0.201626	0.00647716
$970$	0	0
$971$	−19.1459	−0.614421	−0.307211	−	0.951642i	$-0.599396\pi$
$971$	−19.1459	−0.614421	−0.307211	+	0.951642i	$0.599396\pi$
$972$	0	0
$973$	− 7.95743i	− 0.255103i
$974$	0	0
$975$	21.1803	0.678314
$976$	0	0
$977$	− 1.27051i	− 0.0406472i	−0.999793	−	0.0203236i	$-0.993530\pi$
$977$	− 1.27051i	− 0.0406472i	0.999793	−	0.0203236i	$-0.00646965\pi$
$978$	0	0
$979$	−20.8328	−0.665820
$980$	0	0
$981$	−2.88854	−0.0922241
$982$	0	0
$983$	47.3951i	1.51167i	0.654762	+	0.755835i	$0.272769\pi$
$983$	47.3951i	1.51167i	−0.654762	+	0.755835i	$0.727231\pi$
$984$	0	0
$985$	− 45.9787i	− 1.46500i
$986$	0	0
$987$	− 5.12461i	− 0.163118i
$988$	0	0
$989$	−11.2361	−0.357286
$990$	0	0
$991$	−17.7295	−0.563196	−0.281598	−	0.959532i	$-0.590864\pi$
$991$	−17.7295	−0.563196	−0.281598	+	0.959532i	$0.590864\pi$
$992$	0	0
$993$	− 40.6525i	− 1.29007i
$994$	0	0
$995$	−25.5279	−0.809288
$996$	0	0
$997$	62.7214i	1.98641i	0.116398	+	0.993203i	$0.462865\pi$
$997$	62.7214i	1.98641i	−0.116398	+	0.993203i	$0.537135\pi$
$998$	0	0
$999$	53.1246	1.68079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.e.c.369.1		4
4.3	odd	2		230.2.b.a.139.4	yes	4
5.2	odd	4		9200.2.a.bo.1.1		2
5.3	odd	4		9200.2.a.by.1.2		2
5.4	even	2	inner	1840.2.e.c.369.4		4
12.11	even	2		2070.2.d.c.829.2		4
20.3	even	4		1150.2.a.n.1.1		2
20.7	even	4		1150.2.a.l.1.2		2
20.19	odd	2		230.2.b.a.139.1	✓	4
60.59	even	2		2070.2.d.c.829.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.b.a.139.1	✓	4	20.19	odd	2
230.2.b.a.139.4	yes	4	4.3	odd	2
1150.2.a.l.1.2		2	20.7	even	4
1150.2.a.n.1.1		2	20.3	even	4
1840.2.e.c.369.1		4	1.1	even	1	trivial
1840.2.e.c.369.4		4	5.4	even	2	inner
2070.2.d.c.829.2		4	12.11	even	2
2070.2.d.c.829.4		4	60.59	even	2
9200.2.a.bo.1.1		2	5.2	odd	4
9200.2.a.by.1.2		2	5.3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.61803i q^{3} -2.23607 q^{5} -1.85410i q^{7} +0.381966 q^{9} +O(q^{10})\)	\(q-1.61803i q^{3} -2.23607 q^{5} -1.85410i q^{7} +0.381966 q^{9} +5.61803 q^{11} +2.61803i q^{13} +3.61803i q^{15} -0.854102i q^{17} -0.145898 q^{19} -3.00000 q^{21} -1.00000i q^{23} +5.00000 q^{25} -5.47214i q^{27} +9.70820 q^{29} +2.14590 q^{31} -9.09017i q^{33} +4.14590i q^{35} +9.70820i q^{37} +4.23607 q^{39} -5.61803 q^{41} -11.2361i q^{43} -0.854102 q^{45} +1.70820i q^{47} +3.56231 q^{49} -1.38197 q^{51} -2.00000i q^{53} -12.5623 q^{55} +0.236068i q^{57} -6.00000 q^{59} +2.85410 q^{61} -0.708204i q^{63} -5.85410i q^{65} +5.23607i q^{67} -1.61803 q^{69} -0.381966 q^{71} -16.4721i q^{73} -8.09017i q^{75} -10.4164i q^{77} -7.70820 q^{79} -7.70820 q^{81} -7.70820i q^{83} +1.90983i q^{85} -15.7082i q^{87} -3.70820 q^{89} +4.85410 q^{91} -3.47214i q^{93} +0.326238 q^{95} -13.0344i q^{97} +2.14590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{9}+O(q^{10}) \) 4 * q + 6 * q^9	\( 4 q + 6 q^{9} + 18 q^{11} - 14 q^{19} - 12 q^{21} + 20 q^{25} + 12 q^{29} + 22 q^{31} + 8 q^{39} - 18 q^{41} + 10 q^{45} - 26 q^{49} - 10 q^{51} - 10 q^{55} - 24 q^{59} - 2 q^{61} - 2 q^{69} - 6 q^{71} - 4 q^{79} - 4 q^{81} + 12 q^{89} + 6 q^{91} - 30 q^{95} + 22 q^{99}+O(q^{100}) \) 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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