LMFDB - Embedded newform 845.2.c.e.506.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(506,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("845.506");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			506.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	845.506
Dual form			845.2.c.e.506.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{2} -0.732051 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.26795i q^{6} +2.00000i q^{7} -1.73205i q^{8} -2.46410 q^{9} +O(q^{10})$	\(q-1.73205i q^{2} -0.732051 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.26795i q^{6} +2.00000i q^{7} -1.73205i q^{8} -2.46410 q^{9} +1.73205 q^{10} -1.26795i q^{11} +0.732051 q^{12} +3.46410 q^{14} -0.732051i q^{15} -5.00000 q^{16} -3.46410 q^{17} +4.26795i q^{18} -4.19615i q^{19} -1.00000i q^{20} -1.46410i q^{21} -2.19615 q^{22} -4.73205 q^{23} +1.26795i q^{24} -1.00000 q^{25} +4.00000 q^{27} -2.00000i q^{28} -9.46410 q^{29} -1.26795 q^{30} +0.196152i q^{31} +5.19615i q^{32} +0.928203i q^{33} +6.00000i q^{34} -2.00000 q^{35} +2.46410 q^{36} -4.00000i q^{37} -7.26795 q^{38} +1.73205 q^{40} +3.46410i q^{41} -2.53590 q^{42} -10.1962 q^{43} +1.26795i q^{44} -2.46410i q^{45} +8.19615i q^{46} +6.00000i q^{47} +3.66025 q^{48} +3.00000 q^{49} +1.73205i q^{50} +2.53590 q^{51} -10.3923 q^{53} -6.92820i q^{54} +1.26795 q^{55} +3.46410 q^{56} +3.07180i q^{57} +16.3923i q^{58} -15.1244i q^{59} +0.732051i q^{60} +12.3923 q^{61} +0.339746 q^{62} -4.92820i q^{63} -1.00000 q^{64} +1.60770 q^{66} +14.3923i q^{67} +3.46410 q^{68} +3.46410 q^{69} +3.46410i q^{70} -1.26795i q^{71} +4.26795i q^{72} -4.00000i q^{73} -6.92820 q^{74} +0.732051 q^{75} +4.19615i q^{76} +2.53590 q^{77} +12.3923 q^{79} -5.00000i q^{80} +4.46410 q^{81} +6.00000 q^{82} +6.00000i q^{83} +1.46410i q^{84} -3.46410i q^{85} +17.6603i q^{86} +6.92820 q^{87} -2.19615 q^{88} +0.928203i q^{89} -4.26795 q^{90} +4.73205 q^{92} -0.143594i q^{93} +10.3923 q^{94} +4.19615 q^{95} -3.80385i q^{96} -2.00000i q^{97} -5.19615i q^{98} +3.12436i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{12} - 20 q^{16} + 12 q^{22} - 12 q^{23} - 4 q^{25} + 16 q^{27} - 24 q^{29} - 12 q^{30} - 8 q^{35} - 4 q^{36} - 36 q^{38} - 24 q^{42} - 20 q^{43} - 20 q^{48} + 12 q^{49} + 24 q^{51} + 12 q^{55} + 8 q^{61} + 36 q^{62} - 4 q^{64} + 48 q^{66} - 4 q^{75} + 24 q^{77} + 8 q^{79} + 4 q^{81} + 24 q^{82} + 12 q^{88} - 24 q^{90} + 12 q^{92} - 4 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^12 - 20 * q^16 + 12 * q^22 - 12 * q^23 - 4 * q^25 + 16 * q^27 - 24 * q^29 - 12 * q^30 - 8 * q^35 - 4 * q^36 - 36 * q^38 - 24 * q^42 - 20 * q^43 - 20 * q^48 + 12 * q^49 + 24 * q^51 + 12 * q^55 + 8 * q^61 + 36 * q^62 - 4 * q^64 + 48 * q^66 - 4 * q^75 + 24 * q^77 + 8 * q^79 + 4 * q^81 + 24 * q^82 + 12 * q^88 - 24 * q^90 + 12 * q^92 - 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/845\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.73205i	− 1.22474i	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	− 1.73205i	− 1.22474i	0.790569	−	0.612372i	$-0.209785\pi$
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	1.26795i	0.517638i
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 1.73205i	− 0.612372i
$9$	−2.46410	−0.821367
$10$	1.73205	0.547723
$11$	− 1.26795i	− 0.382301i	−0.981561	−	0.191151i	$-0.938778\pi$
$11$	− 1.26795i	− 0.382301i	0.981561	−	0.191151i	$-0.0612219\pi$
$12$	0.732051	0.211325
$13$	0	0
$13$	0	0
$14$	3.46410	0.925820
$15$	− 0.732051i	− 0.189015i
$16$	−5.00000	−1.25000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	4.26795i	1.00597i
$19$	− 4.19615i	− 0.962663i	−0.876539	−	0.481332i	$-0.840153\pi$
$19$	− 4.19615i	− 0.962663i	0.876539	−	0.481332i	$-0.159847\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 1.46410i	− 0.319493i
$22$	−2.19615	−0.468221
$23$	−4.73205	−0.986701	−0.493350	−	0.869831i	$-0.664228\pi$
$23$	−4.73205	−0.986701	−0.493350	+	0.869831i	$0.664228\pi$
$24$	1.26795i	0.258819i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	− 2.00000i	− 0.377964i
$29$	−9.46410	−1.75744	−0.878720	−	0.477338i	$-0.841602\pi$
$29$	−9.46410	−1.75744	−0.878720	+	0.477338i	$0.841602\pi$
$30$	−1.26795	−0.231495
$31$	0.196152i	0.0352300i	0.999845	+	0.0176150i	$0.00560732\pi$
$31$	0.196152i	0.0352300i	−0.999845	+	0.0176150i	$0.994393\pi$
$32$	5.19615i	0.918559i
$33$	0.928203i	0.161579i
$34$	6.00000i	1.02899i
$35$	−2.00000	−0.338062
$36$	2.46410	0.410684
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	−7.26795	−1.17902
$39$	0	0
$40$	1.73205	0.273861
$41$	3.46410i	0.541002i	0.962720	+	0.270501i	$0.0871893\pi$
$41$	3.46410i	0.541002i	−0.962720	+	0.270501i	$0.912811\pi$
$42$	−2.53590	−0.391298
$43$	−10.1962	−1.55490	−0.777449	−	0.628946i	$-0.783487\pi$
$43$	−10.1962	−1.55490	−0.777449	+	0.628946i	$0.783487\pi$
$44$	1.26795i	0.191151i
$45$	− 2.46410i	− 0.367327i
$46$	8.19615i	1.20846i
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	3.66025	0.528312
$49$	3.00000	0.428571
$50$	1.73205i	0.244949i
$51$	2.53590	0.355097
$52$	0	0
$53$	−10.3923	−1.42749	−0.713746	−	0.700404i	$-0.753003\pi$
$53$	−10.3923	−1.42749	−0.713746	+	0.700404i	$0.753003\pi$
$54$	− 6.92820i	− 0.942809i
$55$	1.26795	0.170970
$56$	3.46410	0.462910
$57$	3.07180i	0.406869i
$58$	16.3923i	2.15242i
$59$	− 15.1244i	− 1.96902i	−0.175319	−	0.984512i	$-0.556096\pi$
$59$	− 15.1244i	− 1.96902i	0.175319	−	0.984512i	$-0.443904\pi$
$60$	0.732051i	0.0945074i
$61$	12.3923	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$61$	12.3923	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$62$	0.339746	0.0431478
$63$	− 4.92820i	− 0.620895i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.60770	0.197894
$67$	14.3923i	1.75830i	0.476545	+	0.879150i	$0.341889\pi$
$67$	14.3923i	1.75830i	−0.476545	+	0.879150i	$0.658111\pi$
$68$	3.46410	0.420084
$69$	3.46410	0.417029
$70$	3.46410i	0.414039i
$71$	− 1.26795i	− 0.150478i	−0.997166	−	0.0752389i	$-0.976028\pi$
$71$	− 1.26795i	− 0.150478i	0.997166	−	0.0752389i	$-0.0239720\pi$
$72$	4.26795i	0.502983i
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−6.92820	−0.805387
$75$	0.732051	0.0845299
$76$	4.19615i	0.481332i
$77$	2.53590	0.288992
$78$	0	0
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	− 5.00000i	− 0.559017i
$81$	4.46410	0.496011
$82$	6.00000	0.662589
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	1.46410i	0.159747i
$85$	− 3.46410i	− 0.375735i
$86$	17.6603i	1.90435i
$87$	6.92820	0.742781
$88$	−2.19615	−0.234111
$89$	0.928203i	0.0983893i	0.998789	+	0.0491947i	$0.0156655\pi$
$89$	0.928203i	0.0983893i	−0.998789	+	0.0491947i	$0.984335\pi$
$90$	−4.26795	−0.449881
$91$	0	0
$92$	4.73205	0.493350
$93$	− 0.143594i	− 0.0148900i
$94$	10.3923	1.07188
$95$	4.19615	0.430516
$96$	− 3.80385i	− 0.388229i
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 5.19615i	− 0.524891i
$99$	3.12436i	0.314010i
$100$	1.00000	0.100000
$101$	−12.9282	−1.28640	−0.643202	−	0.765696i	$-0.722395\pi$
$101$	−12.9282	−1.28640	−0.643202	+	0.765696i	$0.722395\pi$
$102$	− 4.39230i	− 0.434903i
$103$	−10.1962	−1.00466	−0.502328	−	0.864677i	$-0.667523\pi$
$103$	−10.1962	−1.00466	−0.502328	+	0.864677i	$0.667523\pi$
$104$	0	0
$105$	1.46410	0.142882
$106$	18.0000i	1.74831i
$107$	0.339746	0.0328445	0.0164222	−	0.999865i	$-0.494772\pi$
$107$	0.339746	0.0328445	0.0164222	+	0.999865i	$0.494772\pi$
$108$	−4.00000	−0.384900
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	− 2.19615i	− 0.209395i
$111$	2.92820i	0.277933i
$112$	− 10.0000i	− 0.944911i
$113$	15.4641	1.45474	0.727370	−	0.686245i	$-0.240742\pi$
$113$	15.4641	1.45474	0.727370	+	0.686245i	$0.240742\pi$
$114$	5.32051	0.498311
$115$	− 4.73205i	− 0.441266i
$116$	9.46410	0.878720
$117$	0	0
$118$	−26.1962	−2.41155
$119$	− 6.92820i	− 0.635107i
$120$	−1.26795	−0.115747
$121$	9.39230	0.853846
$122$	− 21.4641i	− 1.94327i
$123$	− 2.53590i	− 0.228654i
$124$	− 0.196152i	− 0.0176150i
$125$	− 1.00000i	− 0.0894427i
$126$	−8.53590	−0.760438
$127$	−5.80385	−0.515008	−0.257504	−	0.966277i	$-0.582900\pi$
$127$	−5.80385	−0.515008	−0.257504	+	0.966277i	$0.582900\pi$
$128$	12.1244i	1.07165i
$129$	7.46410	0.657178
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 0.928203i	− 0.0807897i
$133$	8.39230	0.727705
$134$	24.9282	2.15347
$135$	4.00000i	0.344265i
$136$	6.00000i	0.514496i
$137$	− 12.9282i	− 1.10453i	−0.833668	−	0.552265i	$-0.813763\pi$
$137$	− 12.9282i	− 1.10453i	0.833668	−	0.552265i	$-0.186237\pi$
$138$	− 6.00000i	− 0.510754i
$139$	−8.39230	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$139$	−8.39230	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$140$	2.00000	0.169031
$141$	− 4.39230i	− 0.369899i
$142$	−2.19615	−0.184297
$143$	0	0
$144$	12.3205	1.02671
$145$	− 9.46410i	− 0.785951i
$146$	−6.92820	−0.573382
$147$	−2.19615	−0.181136
$148$	4.00000i	0.328798i
$149$	− 19.8564i	− 1.62670i	−0.581775	−	0.813350i	$-0.697641\pi$
$149$	− 19.8564i	− 1.62670i	0.581775	−	0.813350i	$-0.302359\pi$
$150$	− 1.26795i	− 0.103528i
$151$	− 12.1962i	− 0.992509i	−0.868177	−	0.496254i	$-0.834708\pi$
$151$	− 12.1962i	− 0.992509i	0.868177	−	0.496254i	$-0.165292\pi$
$152$	−7.26795	−0.589509
$153$	8.53590	0.690086
$154$	− 4.39230i	− 0.353942i
$155$	−0.196152	−0.0157553
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	− 21.4641i	− 1.70759i
$159$	7.60770	0.603329
$160$	−5.19615	−0.410792
$161$	− 9.46410i	− 0.745876i
$162$	− 7.73205i	− 0.607487i
$163$	6.39230i	0.500684i	0.968157	+	0.250342i	$0.0805431\pi$
$163$	6.39230i	0.500684i	−0.968157	+	0.250342i	$0.919457\pi$
$164$	− 3.46410i	− 0.270501i
$165$	−0.928203	−0.0722605
$166$	10.3923	0.806599
$167$	12.9282i	1.00041i	0.865906	+	0.500207i	$0.166743\pi$
$167$	12.9282i	1.00041i	−0.865906	+	0.500207i	$0.833257\pi$
$168$	−2.53590	−0.195649
$169$	0	0
$170$	−6.00000	−0.460179
$171$	10.3397i	0.790700i
$172$	10.1962	0.777449
$173$	−15.4641	−1.17571	−0.587857	−	0.808965i	$-0.700028\pi$
$173$	−15.4641	−1.17571	−0.587857	+	0.808965i	$0.700028\pi$
$174$	− 12.0000i	− 0.909718i
$175$	− 2.00000i	− 0.151186i
$176$	6.33975i	0.477876i
$177$	11.0718i	0.832207i
$178$	1.60770	0.120502
$179$	5.07180	0.379084	0.189542	−	0.981873i	$-0.439300\pi$
$179$	5.07180	0.379084	0.189542	+	0.981873i	$0.439300\pi$
$180$	2.46410i	0.183663i
$181$	20.3923	1.51575	0.757874	−	0.652401i	$-0.226238\pi$
$181$	20.3923	1.51575	0.757874	+	0.652401i	$0.226238\pi$
$182$	0	0
$183$	−9.07180	−0.670607
$184$	8.19615i	0.604228i
$185$	4.00000	0.294086
$186$	−0.248711	−0.0182364
$187$	4.39230i	0.321197i
$188$	− 6.00000i	− 0.437595i
$189$	8.00000i	0.581914i
$190$	− 7.26795i	− 0.527272i
$191$	−18.9282	−1.36960	−0.684798	−	0.728733i	$-0.740110\pi$
$191$	−18.9282	−1.36960	−0.684798	+	0.728733i	$0.740110\pi$
$192$	0.732051	0.0528312
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−3.46410	−0.248708
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 0.928203i	− 0.0661317i	−0.999453	−	0.0330659i	$-0.989473\pi$
$197$	− 0.928203i	− 0.0661317i	0.999453	−	0.0330659i	$-0.0105271\pi$
$198$	5.41154	0.384582
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	1.73205i	0.122474i
$201$	− 10.5359i	− 0.743145i
$202$	22.3923i	1.57552i
$203$	− 18.9282i	− 1.32850i
$204$	−2.53590	−0.177548
$205$	−3.46410	−0.241943
$206$	17.6603i	1.23045i
$207$	11.6603	0.810444
$208$	0	0
$209$	−5.32051	−0.368027
$210$	− 2.53590i	− 0.174994i
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	10.3923	0.713746
$213$	0.928203i	0.0635994i
$214$	− 0.588457i	− 0.0402261i
$215$	− 10.1962i	− 0.695372i
$216$	− 6.92820i	− 0.471405i
$217$	−0.392305	−0.0266314
$218$	−3.46410	−0.234619
$219$	2.92820i	0.197870i
$220$	−1.26795	−0.0854851
$221$	0	0
$222$	5.07180	0.340397
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	−10.3923	−0.694365
$225$	2.46410	0.164273
$226$	− 26.7846i	− 1.78169i
$227$	− 3.46410i	− 0.229920i	−0.993370	−	0.114960i	$-0.963326\pi$
$227$	− 3.46410i	− 0.229920i	0.993370	−	0.114960i	$-0.0366741\pi$
$228$	− 3.07180i	− 0.203435i
$229$	− 14.3923i	− 0.951070i	−0.879697	−	0.475535i	$-0.842254\pi$
$229$	− 14.3923i	− 0.951070i	0.879697	−	0.475535i	$-0.157746\pi$
$230$	−8.19615	−0.540438
$231$	−1.85641	−0.122143
$232$	16.3923i	1.07621i
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−6.00000	−0.391397
$236$	15.1244i	0.984512i
$237$	−9.07180	−0.589277
$238$	−12.0000	−0.777844
$239$	3.80385i	0.246050i	0.992404	+	0.123025i	$0.0392596\pi$
$239$	3.80385i	0.246050i	−0.992404	+	0.123025i	$0.960740\pi$
$240$	3.66025i	0.236268i
$241$	18.3923i	1.18475i	0.805661	+	0.592376i	$0.201810\pi$
$241$	18.3923i	1.18475i	−0.805661	+	0.592376i	$0.798190\pi$
$242$	− 16.2679i	− 1.04574i
$243$	−15.2679	−0.979439
$244$	−12.3923	−0.793336
$245$	3.00000i	0.191663i
$246$	−4.39230	−0.280043
$247$	0	0
$248$	0.339746	0.0215739
$249$	− 4.39230i	− 0.278351i
$250$	−1.73205	−0.109545
$251$	14.5359	0.917498	0.458749	−	0.888566i	$-0.348298\pi$
$251$	14.5359	0.917498	0.458749	+	0.888566i	$0.348298\pi$
$252$	4.92820i	0.310448i
$253$	6.00000i	0.377217i
$254$	10.0526i	0.630754i
$255$	2.53590i	0.158804i
$256$	19.0000	1.18750
$257$	7.85641	0.490069	0.245035	−	0.969514i	$-0.421201\pi$
$257$	7.85641	0.490069	0.245035	+	0.969514i	$0.421201\pi$
$258$	− 12.9282i	− 0.804875i
$259$	8.00000	0.497096
$260$	0	0
$261$	23.3205	1.44350
$262$	0	0
$263$	4.73205	0.291791	0.145895	−	0.989300i	$-0.453394\pi$
$263$	4.73205	0.291791	0.145895	+	0.989300i	$0.453394\pi$
$264$	1.60770	0.0989468
$265$	− 10.3923i	− 0.638394i
$266$	− 14.5359i	− 0.891253i
$267$	− 0.679492i	− 0.0415842i
$268$	− 14.3923i	− 0.879150i
$269$	7.85641	0.479014	0.239507	−	0.970895i	$-0.423014\pi$
$269$	7.85641	0.479014	0.239507	+	0.970895i	$0.423014\pi$
$270$	6.92820	0.421637
$271$	− 20.9808i	− 1.27449i	−0.770661	−	0.637245i	$-0.780074\pi$
$271$	− 20.9808i	− 1.27449i	0.770661	−	0.637245i	$-0.219926\pi$
$272$	17.3205	1.05021
$273$	0	0
$274$	−22.3923	−1.35277
$275$	1.26795i	0.0764602i
$276$	−3.46410	−0.208514
$277$	5.60770	0.336934	0.168467	−	0.985707i	$-0.446118\pi$
$277$	5.60770	0.336934	0.168467	+	0.985707i	$0.446118\pi$
$278$	14.5359i	0.871805i
$279$	− 0.483340i	− 0.0289368i
$280$	3.46410i	0.207020i
$281$	1.60770i	0.0959071i	0.998850	+	0.0479535i	$0.0152699\pi$
$281$	1.60770i	0.0959071i	−0.998850	+	0.0479535i	$0.984730\pi$
$282$	−7.60770	−0.453032
$283$	−1.41154	−0.0839075	−0.0419538	−	0.999120i	$-0.513358\pi$
$283$	−1.41154	−0.0839075	−0.0419538	+	0.999120i	$0.513358\pi$
$284$	1.26795i	0.0752389i
$285$	−3.07180	−0.181958
$286$	0	0
$287$	−6.92820	−0.408959
$288$	− 12.8038i	− 0.754474i
$289$	−5.00000	−0.294118
$290$	−16.3923	−0.962589
$291$	1.46410i	0.0858272i
$292$	4.00000i	0.234082i
$293$	− 18.9282i	− 1.10580i	−0.833248	−	0.552899i	$-0.813522\pi$
$293$	− 18.9282i	− 1.10580i	0.833248	−	0.552899i	$-0.186478\pi$
$294$	3.80385i	0.221845i
$295$	15.1244	0.880574
$296$	−6.92820	−0.402694
$297$	− 5.07180i	− 0.294295i
$298$	−34.3923	−1.99229
$299$	0	0
$300$	−0.732051	−0.0422650
$301$	− 20.3923i	− 1.17539i
$302$	−21.1244	−1.21557
$303$	9.46410	0.543698
$304$	20.9808i	1.20333i
$305$	12.3923i	0.709581i
$306$	− 14.7846i	− 0.845180i
$307$	22.7846i	1.30039i	0.759769	+	0.650193i	$0.225312\pi$
$307$	22.7846i	1.30039i	−0.759769	+	0.650193i	$0.774688\pi$
$308$	−2.53590	−0.144496
$309$	7.46410	0.424618
$310$	0.339746i	0.0192963i
$311$	−4.39230	−0.249065	−0.124532	−	0.992216i	$-0.539743\pi$
$311$	−4.39230	−0.249065	−0.124532	+	0.992216i	$0.539743\pi$
$312$	0	0
$313$	6.39230	0.361314	0.180657	−	0.983546i	$-0.442178\pi$
$313$	6.39230	0.361314	0.180657	+	0.983546i	$0.442178\pi$
$314$	17.3205i	0.977453i
$315$	4.92820	0.277673
$316$	−12.3923	−0.697122
$317$	− 24.0000i	− 1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$	− 24.0000i	− 1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$	− 13.1769i	− 0.738925i
$319$	12.0000i	0.671871i
$320$	− 1.00000i	− 0.0559017i
$321$	−0.248711	−0.0138817
$322$	−16.3923	−0.913507
$323$	14.5359i	0.808799i
$324$	−4.46410	−0.248006
$325$	0	0
$326$	11.0718	0.613210
$327$	1.46410i	0.0809650i
$328$	6.00000	0.331295
$329$	−12.0000	−0.661581
$330$	1.60770i	0.0885007i
$331$	28.5885i	1.57136i	0.618631	+	0.785682i	$0.287688\pi$
$331$	28.5885i	1.57136i	−0.618631	+	0.785682i	$0.712312\pi$
$332$	− 6.00000i	− 0.329293i
$333$	9.85641i	0.540128i
$334$	22.3923	1.22525
$335$	−14.3923	−0.786336
$336$	7.32051i	0.399366i
$337$	5.60770	0.305471	0.152735	−	0.988267i	$-0.451192\pi$
$337$	5.60770	0.305471	0.152735	+	0.988267i	$0.451192\pi$
$338$	0	0
$339$	−11.3205	−0.614846
$340$	3.46410i	0.187867i
$341$	0.248711	0.0134685
$342$	17.9090	0.968406
$343$	20.0000i	1.07990i
$344$	17.6603i	0.952177i
$345$	3.46410i	0.186501i
$346$	26.7846i	1.43995i
$347$	11.6603	0.625955	0.312978	−	0.949761i	$-0.398674\pi$
$347$	11.6603	0.625955	0.312978	+	0.949761i	$0.398674\pi$
$348$	−6.92820	−0.371391
$349$	6.39230i	0.342172i	0.985256	+	0.171086i	$0.0547276\pi$
$349$	6.39230i	0.342172i	−0.985256	+	0.171086i	$0.945272\pi$
$350$	−3.46410	−0.185164
$351$	0	0
$352$	6.58846	0.351166
$353$	− 27.7128i	− 1.47500i	−0.675345	−	0.737502i	$-0.736005\pi$
$353$	− 27.7128i	− 1.47500i	0.675345	−	0.737502i	$-0.263995\pi$
$354$	19.1769	1.01924
$355$	1.26795	0.0672958
$356$	− 0.928203i	− 0.0491947i
$357$	5.07180i	0.268428i
$358$	− 8.78461i	− 0.464281i
$359$	8.19615i	0.432576i	0.976330	+	0.216288i	$0.0693951\pi$
$359$	8.19615i	0.432576i	−0.976330	+	0.216288i	$0.930605\pi$
$360$	−4.26795	−0.224941
$361$	1.39230	0.0732792
$362$	− 35.3205i	− 1.85640i
$363$	−6.87564	−0.360878
$364$	0	0
$365$	4.00000	0.209370
$366$	15.7128i	0.821322i
$367$	22.1962	1.15863	0.579315	−	0.815104i	$-0.303320\pi$
$367$	22.1962	1.15863	0.579315	+	0.815104i	$0.303320\pi$
$368$	23.6603	1.23338
$369$	− 8.53590i	− 0.444361i
$370$	− 6.92820i	− 0.360180i
$371$	− 20.7846i	− 1.07908i
$372$	0.143594i	0.00744498i
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	7.60770	0.393385
$375$	0.732051i	0.0378029i
$376$	10.3923	0.535942
$377$	0	0
$378$	13.8564	0.712697
$379$	32.9808i	1.69411i	0.531507	+	0.847054i	$0.321626\pi$
$379$	32.9808i	1.69411i	−0.531507	+	0.847054i	$0.678374\pi$
$380$	−4.19615	−0.215258
$381$	4.24871	0.217668
$382$	32.7846i	1.67741i
$383$	0.928203i	0.0474290i	0.999719	+	0.0237145i	$0.00754926\pi$
$383$	0.928203i	0.0474290i	−0.999719	+	0.0237145i	$0.992451\pi$
$384$	− 8.87564i	− 0.452933i
$385$	2.53590i	0.129241i
$386$	−17.3205	−0.881591
$387$	25.1244	1.27714
$388$	2.00000i	0.101535i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	16.3923	0.828994
$392$	− 5.19615i	− 0.262445i
$393$	0	0
$394$	−1.60770	−0.0809945
$395$	12.3923i	0.623525i
$396$	− 3.12436i	− 0.157005i
$397$	− 12.7846i	− 0.641641i	−0.947140	−	0.320821i	$-0.896041\pi$
$397$	− 12.7846i	− 0.641641i	0.947140	−	0.320821i	$-0.103959\pi$
$398$	34.6410i	1.73640i
$399$	−6.14359	−0.307564
$400$	5.00000	0.250000
$401$	− 23.0718i	− 1.15215i	−0.817397	−	0.576075i	$-0.804584\pi$
$401$	− 23.0718i	− 1.15215i	0.817397	−	0.576075i	$-0.195416\pi$
$402$	−18.2487	−0.910163
$403$	0	0
$404$	12.9282	0.643202
$405$	4.46410i	0.221823i
$406$	−32.7846	−1.62707
$407$	−5.07180	−0.251400
$408$	− 4.39230i	− 0.217451i
$409$	38.3923i	1.89838i	0.314708	+	0.949189i	$0.398094\pi$
$409$	38.3923i	1.89838i	−0.314708	+	0.949189i	$0.601906\pi$
$410$	6.00000i	0.296319i
$411$	9.46410i	0.466830i
$412$	10.1962	0.502328
$413$	30.2487	1.48844
$414$	− 20.1962i	− 0.992587i
$415$	−6.00000	−0.294528
$416$	0	0
$417$	6.14359	0.300853
$418$	9.21539i	0.450739i
$419$	−9.46410	−0.462352	−0.231176	−	0.972912i	$-0.574257\pi$
$419$	−9.46410	−0.462352	−0.231176	+	0.972912i	$0.574257\pi$
$420$	−1.46410	−0.0714408
$421$	− 10.7846i	− 0.525610i	−0.964849	−	0.262805i	$-0.915352\pi$
$421$	− 10.7846i	− 0.525610i	0.964849	−	0.262805i	$-0.0846475\pi$
$422$	− 13.8564i	− 0.674519i
$423$	− 14.7846i	− 0.718852i
$424$	18.0000i	0.874157i
$425$	3.46410	0.168034
$426$	1.60770	0.0778931
$427$	24.7846i	1.19941i
$428$	−0.339746	−0.0164222
$429$	0	0
$430$	−17.6603	−0.851653
$431$	− 19.5167i	− 0.940084i	−0.882644	−	0.470042i	$-0.844239\pi$
$431$	− 19.5167i	− 0.940084i	0.882644	−	0.470042i	$-0.155761\pi$
$432$	−20.0000	−0.962250
$433$	6.78461	0.326048	0.163024	−	0.986622i	$-0.447875\pi$
$433$	6.78461	0.326048	0.163024	+	0.986622i	$0.447875\pi$
$434$	0.679492i	0.0326167i
$435$	6.92820i	0.332182i
$436$	2.00000i	0.0957826i
$437$	19.8564i	0.949861i
$438$	5.07180	0.242340
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	− 2.19615i	− 0.104697i
$441$	−7.39230	−0.352015
$442$	0	0
$443$	34.9808	1.66199	0.830993	−	0.556283i	$-0.187773\pi$
$443$	34.9808	1.66199	0.830993	+	0.556283i	$0.187773\pi$
$444$	− 2.92820i	− 0.138966i
$445$	−0.928203	−0.0440011
$446$	−3.46410	−0.164030
$447$	14.5359i	0.687524i
$448$	− 2.00000i	− 0.0944911i
$449$	27.4641i	1.29611i	0.761593	+	0.648056i	$0.224418\pi$
$449$	27.4641i	1.29611i	−0.761593	+	0.648056i	$0.775582\pi$
$450$	− 4.26795i	− 0.201193i
$451$	4.39230	0.206826
$452$	−15.4641	−0.727370
$453$	8.92820i	0.419484i
$454$	−6.00000	−0.281594
$455$	0	0
$456$	5.32051	0.249156
$457$	30.7846i	1.44004i	0.693952	+	0.720022i	$0.255868\pi$
$457$	30.7846i	1.44004i	−0.693952	+	0.720022i	$0.744132\pi$
$458$	−24.9282	−1.16482
$459$	−13.8564	−0.646762
$460$	4.73205i	0.220633i
$461$	− 3.46410i	− 0.161339i	−0.996741	−	0.0806696i	$-0.974294\pi$
$461$	− 3.46410i	− 0.161339i	0.996741	−	0.0806696i	$-0.0257059\pi$
$462$	3.21539i	0.149593i
$463$	18.3923i	0.854763i	0.904071	+	0.427381i	$0.140564\pi$
$463$	18.3923i	0.854763i	−0.904071	+	0.427381i	$0.859436\pi$
$464$	47.3205	2.19680
$465$	0.143594	0.00665899
$466$	− 10.3923i	− 0.481414i
$467$	−38.1962	−1.76751	−0.883754	−	0.467953i	$-0.844992\pi$
$467$	−38.1962	−1.76751	−0.883754	+	0.467953i	$0.844992\pi$
$468$	0	0
$469$	−28.7846	−1.32915
$470$	10.3923i	0.479361i
$471$	7.32051	0.337311
$472$	−26.1962	−1.20578
$473$	12.9282i	0.594439i
$474$	15.7128i	0.721713i
$475$	4.19615i	0.192533i
$476$	6.92820i	0.317554i
$477$	25.6077	1.17250
$478$	6.58846	0.301349
$479$	18.3397i	0.837964i	0.907994	+	0.418982i	$0.137613\pi$
$479$	18.3397i	0.837964i	−0.907994	+	0.418982i	$0.862387\pi$
$480$	3.80385	0.173621
$481$	0	0
$482$	31.8564	1.45102
$483$	6.92820i	0.315244i
$484$	−9.39230	−0.426923
$485$	2.00000	0.0908153
$486$	26.4449i	1.19956i
$487$	5.60770i	0.254109i	0.991896	+	0.127054i	$0.0405523\pi$
$487$	5.60770i	0.254109i	−0.991896	+	0.127054i	$0.959448\pi$
$488$	− 21.4641i	− 0.971634i
$489$	− 4.67949i	− 0.211614i
$490$	5.19615	0.234738
$491$	9.46410	0.427109	0.213554	−	0.976931i	$-0.431496\pi$
$491$	9.46410	0.427109	0.213554	+	0.976931i	$0.431496\pi$
$492$	2.53590i	0.114327i
$493$	32.7846	1.47654
$494$	0	0
$495$	−3.12436	−0.140429
$496$	− 0.980762i	− 0.0440375i
$497$	2.53590	0.113751
$498$	−7.60770	−0.340909
$499$	− 12.9808i	− 0.581099i	−0.956860	−	0.290549i	$-0.906162\pi$
$499$	− 12.9808i	− 0.581099i	0.956860	−	0.290549i	$-0.0938380\pi$
$500$	1.00000i	0.0447214i
$501$	− 9.46410i	− 0.422825i
$502$	− 25.1769i	− 1.12370i
$503$	−25.5167	−1.13773	−0.568866	−	0.822430i	$-0.692618\pi$
$503$	−25.5167	−1.13773	−0.568866	+	0.822430i	$0.692618\pi$
$504$	−8.53590	−0.380219
$505$	− 12.9282i	− 0.575297i
$506$	10.3923	0.461994
$507$	0	0
$508$	5.80385	0.257504
$509$	− 32.5359i	− 1.44213i	−0.692868	−	0.721064i	$-0.743653\pi$
$509$	− 32.5359i	− 1.44213i	0.692868	−	0.721064i	$-0.256347\pi$
$510$	4.39230	0.194495
$511$	8.00000	0.353899
$512$	− 8.66025i	− 0.382733i
$513$	− 16.7846i	− 0.741059i
$514$	− 13.6077i	− 0.600210i
$515$	− 10.1962i	− 0.449296i
$516$	−7.46410	−0.328589
$517$	7.60770	0.334586
$518$	− 13.8564i	− 0.608816i
$519$	11.3205	0.496915
$520$	0	0
$521$	−7.60770	−0.333299	−0.166650	−	0.986016i	$-0.553295\pi$
$521$	−7.60770	−0.333299	−0.166650	+	0.986016i	$0.553295\pi$
$522$	− 40.3923i	− 1.76792i
$523$	−13.8038	−0.603600	−0.301800	−	0.953371i	$-0.597588\pi$
$523$	−13.8038	−0.603600	−0.301800	+	0.953371i	$0.597588\pi$
$524$	0	0
$525$	1.46410i	0.0638986i
$526$	− 8.19615i	− 0.357369i
$527$	− 0.679492i	− 0.0295991i
$528$	− 4.64102i	− 0.201974i
$529$	−0.607695	−0.0264215
$530$	−18.0000	−0.781870
$531$	37.2679i	1.61729i
$532$	−8.39230	−0.363853
$533$	0	0
$534$	−1.17691	−0.0509301
$535$	0.339746i	0.0146885i
$536$	24.9282	1.07673
$537$	−3.71281	−0.160220
$538$	− 13.6077i	− 0.586669i
$539$	− 3.80385i	− 0.163843i
$540$	− 4.00000i	− 0.172133i
$541$	− 5.60770i	− 0.241094i	−0.992708	−	0.120547i	$-0.961535\pi$
$541$	− 5.60770i	− 0.241094i	0.992708	−	0.120547i	$-0.0384648\pi$
$542$	−36.3397	−1.56093
$543$	−14.9282	−0.640631
$544$	− 18.0000i	− 0.771744i
$545$	2.00000	0.0856706
$546$	0	0
$547$	−1.80385	−0.0771270	−0.0385635	−	0.999256i	$-0.512278\pi$
$547$	−1.80385	−0.0771270	−0.0385635	+	0.999256i	$0.512278\pi$
$548$	12.9282i	0.552265i
$549$	−30.5359	−1.30324
$550$	2.19615	0.0936443
$551$	39.7128i	1.69182i
$552$	− 6.00000i	− 0.255377i
$553$	24.7846i	1.05395i
$554$	− 9.71281i	− 0.412658i
$555$	−2.92820	−0.124295
$556$	8.39230	0.355913
$557$	− 25.8564i	− 1.09557i	−0.836619	−	0.547786i	$-0.815471\pi$
$557$	− 25.8564i	− 1.09557i	0.836619	−	0.547786i	$-0.184529\pi$
$558$	−0.837169	−0.0354402
$559$	0	0
$560$	10.0000	0.422577
$561$	− 3.21539i	− 0.135754i
$562$	2.78461	0.117462
$563$	−16.0526	−0.676535	−0.338267	−	0.941050i	$-0.609841\pi$
$563$	−16.0526	−0.676535	−0.338267	+	0.941050i	$0.609841\pi$
$564$	4.39230i	0.184949i
$565$	15.4641i	0.650580i
$566$	2.44486i	0.102765i
$567$	8.92820i	0.374949i
$568$	−2.19615	−0.0921485
$569$	9.46410	0.396756	0.198378	−	0.980126i	$-0.436433\pi$
$569$	9.46410	0.396756	0.198378	+	0.980126i	$0.436433\pi$
$570$	5.32051i	0.222852i
$571$	−15.6077	−0.653162	−0.326581	−	0.945169i	$-0.605897\pi$
$571$	−15.6077	−0.653162	−0.326581	+	0.945169i	$0.605897\pi$
$572$	0	0
$573$	13.8564	0.578860
$574$	12.0000i	0.500870i
$575$	4.73205	0.197340
$576$	2.46410	0.102671
$577$	4.00000i	0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$	4.00000i	0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$	8.66025i	0.360219i
$579$	7.32051i	0.304230i
$580$	9.46410i	0.392975i
$581$	−12.0000	−0.497844
$582$	2.53590	0.105116
$583$	13.1769i	0.545732i
$584$	−6.92820	−0.286691
$585$	0	0
$586$	−32.7846	−1.35432
$587$	15.4641i	0.638272i	0.947709	+	0.319136i	$0.103393\pi$
$587$	15.4641i	0.638272i	−0.947709	+	0.319136i	$0.896607\pi$
$588$	2.19615	0.0905678
$589$	0.823085	0.0339146
$590$	− 26.1962i	− 1.07848i
$591$	0.679492i	0.0279506i
$592$	20.0000i	0.821995i
$593$	− 14.7846i	− 0.607131i	−0.952811	−	0.303566i	$-0.901823\pi$
$593$	− 14.7846i	− 0.607131i	0.952811	−	0.303566i	$-0.0981771\pi$
$594$	−8.78461	−0.360437
$595$	6.92820	0.284029
$596$	19.8564i	0.813350i
$597$	14.6410	0.599217
$598$	0	0
$599$	−28.3923	−1.16008	−0.580039	−	0.814589i	$-0.696963\pi$
$599$	−28.3923	−1.16008	−0.580039	+	0.814589i	$0.696963\pi$
$600$	− 1.26795i	− 0.0517638i
$601$	−39.5692	−1.61406	−0.807031	−	0.590509i	$-0.798927\pi$
$601$	−39.5692	−1.61406	−0.807031	+	0.590509i	$0.798927\pi$
$602$	−35.3205	−1.43956
$603$	− 35.4641i	− 1.44421i
$604$	12.1962i	0.496254i
$605$	9.39230i	0.381851i
$606$	− 16.3923i	− 0.665892i
$607$	−26.9808	−1.09512	−0.547558	−	0.836768i	$-0.684442\pi$
$607$	−26.9808	−1.09512	−0.547558	+	0.836768i	$0.684442\pi$
$608$	21.8038	0.884263
$609$	13.8564i	0.561490i
$610$	21.4641	0.869056
$611$	0	0
$612$	−8.53590	−0.345043
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	39.4641	1.59264
$615$	2.53590	0.102257
$616$	− 4.39230i	− 0.176971i
$617$	21.7128i	0.874125i	0.899431	+	0.437062i	$0.143981\pi$
$617$	21.7128i	0.874125i	−0.899431	+	0.437062i	$0.856019\pi$
$618$	− 12.9282i	− 0.520049i
$619$	− 44.9808i	− 1.80793i	−0.427607	−	0.903965i	$-0.640643\pi$
$619$	− 44.9808i	− 1.80793i	0.427607	−	0.903965i	$-0.359357\pi$
$620$	0.196152	0.00787767
$621$	−18.9282	−0.759563
$622$	7.60770i	0.305041i
$623$	−1.85641	−0.0743754
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 11.0718i	− 0.442518i
$627$	3.89488	0.155547
$628$	10.0000	0.399043
$629$	13.8564i	0.552491i
$630$	− 8.53590i	− 0.340078i
$631$	16.1962i	0.644759i	0.946611	+	0.322379i	$0.104483\pi$
$631$	16.1962i	0.644759i	−0.946611	+	0.322379i	$0.895517\pi$
$632$	− 21.4641i	− 0.853796i
$633$	−5.85641	−0.232771
$634$	−41.5692	−1.65092
$635$	− 5.80385i	− 0.230319i
$636$	−7.60770	−0.301665
$637$	0	0
$638$	20.7846	0.822871
$639$	3.12436i	0.123598i
$640$	−12.1244	−0.479257
$641$	0.928203	0.0366618	0.0183309	−	0.999832i	$-0.494165\pi$
$641$	0.928203	0.0366618	0.0183309	+	0.999832i	$0.494165\pi$
$642$	0.430781i	0.0170016i
$643$	− 34.7846i	− 1.37177i	−0.727709	−	0.685886i	$-0.759415\pi$
$643$	− 34.7846i	− 1.37177i	0.727709	−	0.685886i	$-0.240585\pi$
$644$	9.46410i	0.372938i
$645$	7.46410i	0.293899i
$646$	25.1769	0.990572
$647$	16.0526	0.631091	0.315546	−	0.948910i	$-0.397812\pi$
$647$	16.0526	0.631091	0.315546	+	0.948910i	$0.397812\pi$
$648$	− 7.73205i	− 0.303744i
$649$	−19.1769	−0.752760
$650$	0	0
$651$	0.287187	0.0112557
$652$	− 6.39230i	− 0.250342i
$653$	−19.8564	−0.777041	−0.388521	−	0.921440i	$-0.627014\pi$
$653$	−19.8564	−0.777041	−0.388521	+	0.921440i	$0.627014\pi$
$654$	2.53590	0.0991615
$655$	0	0
$656$	− 17.3205i	− 0.676252i
$657$	9.85641i	0.384535i
$658$	20.7846i	0.810268i
$659$	−14.5359	−0.566238	−0.283119	−	0.959085i	$-0.591369\pi$
$659$	−14.5359	−0.566238	−0.283119	+	0.959085i	$0.591369\pi$
$660$	0.928203	0.0361303
$661$	− 30.7846i	− 1.19738i	−0.800980	−	0.598691i	$-0.795688\pi$
$661$	− 30.7846i	− 1.19738i	0.800980	−	0.598691i	$-0.204312\pi$
$662$	49.5167	1.92452
$663$	0	0
$664$	10.3923	0.403300
$665$	8.39230i	0.325440i
$666$	17.0718	0.661519
$667$	44.7846	1.73407
$668$	− 12.9282i	− 0.500207i
$669$	1.46410i	0.0566054i
$670$	24.9282i	0.963061i
$671$	− 15.7128i	− 0.606586i
$672$	7.60770	0.293473
$673$	−6.39230	−0.246405	−0.123203	−	0.992382i	$-0.539317\pi$
$673$	−6.39230	−0.246405	−0.123203	+	0.992382i	$0.539317\pi$
$674$	− 9.71281i	− 0.374124i
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−10.3923	−0.399409	−0.199704	−	0.979856i	$-0.563998\pi$
$677$	−10.3923	−0.399409	−0.199704	+	0.979856i	$0.563998\pi$
$678$	19.6077i	0.753029i
$679$	4.00000	0.153506
$680$	−6.00000	−0.230089
$681$	2.53590i	0.0971758i
$682$	− 0.430781i	− 0.0164954i
$683$	39.4641i	1.51005i	0.655695	+	0.755026i	$0.272376\pi$
$683$	39.4641i	1.51005i	−0.655695	+	0.755026i	$0.727624\pi$
$684$	− 10.3397i	− 0.395350i
$685$	12.9282	0.493961
$686$	34.6410	1.32260
$687$	10.5359i	0.401970i
$688$	50.9808	1.94362
$689$	0	0
$690$	6.00000	0.228416
$691$	− 45.7654i	− 1.74100i	−0.492171	−	0.870498i	$-0.663797\pi$
$691$	− 45.7654i	− 1.74100i	0.492171	−	0.870498i	$-0.336203\pi$
$692$	15.4641	0.587857
$693$	−6.24871	−0.237369
$694$	− 20.1962i	− 0.766635i
$695$	− 8.39230i	− 0.318338i
$696$	− 12.0000i	− 0.454859i
$697$	− 12.0000i	− 0.454532i
$698$	11.0718	0.419074
$699$	−4.39230	−0.166132
$700$	2.00000i	0.0755929i
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	−16.7846	−0.633044
$704$	1.26795i	0.0477876i
$705$	4.39230	0.165424
$706$	−48.0000	−1.80650
$707$	− 25.8564i	− 0.972430i
$708$	− 11.0718i	− 0.416104i
$709$	9.60770i	0.360825i	0.983591	+	0.180412i	$0.0577432\pi$
$709$	9.60770i	0.360825i	−0.983591	+	0.180412i	$0.942257\pi$
$710$	− 2.19615i	− 0.0824201i
$711$	−30.5359	−1.14519
$712$	1.60770	0.0602509
$713$	− 0.928203i	− 0.0347615i
$714$	8.78461	0.328756
$715$	0	0
$716$	−5.07180	−0.189542
$717$	− 2.78461i	− 0.103993i
$718$	14.1962	0.529796
$719$	−1.85641	−0.0692323	−0.0346161	−	0.999401i	$-0.511021\pi$
$719$	−1.85641	−0.0692323	−0.0346161	+	0.999401i	$0.511021\pi$
$720$	12.3205i	0.459158i
$721$	− 20.3923i	− 0.759449i
$722$	− 2.41154i	− 0.0897483i
$723$	− 13.4641i	− 0.500735i
$724$	−20.3923	−0.757874
$725$	9.46410	0.351488
$726$	11.9090i	0.441983i
$727$	−13.4115	−0.497407	−0.248703	−	0.968580i	$-0.580004\pi$
$727$	−13.4115	−0.497407	−0.248703	+	0.968580i	$0.580004\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	− 6.92820i	− 0.256424i
$731$	35.3205	1.30638
$732$	9.07180	0.335303
$733$	− 38.0000i	− 1.40356i	−0.712393	−	0.701781i	$-0.752388\pi$
$733$	− 38.0000i	− 1.40356i	0.712393	−	0.701781i	$-0.247612\pi$
$734$	− 38.4449i	− 1.41903i
$735$	− 2.19615i	− 0.0810063i
$736$	− 24.5885i	− 0.906343i
$737$	18.2487	0.672200
$738$	−14.7846	−0.544229
$739$	− 7.80385i	− 0.287069i	−0.989645	−	0.143535i	$-0.954153\pi$
$739$	− 7.80385i	− 0.287069i	0.989645	−	0.143535i	$-0.0458468\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−36.0000	−1.32160
$743$	43.8564i	1.60894i	0.593996	+	0.804468i	$0.297549\pi$
$743$	43.8564i	1.60894i	−0.593996	+	0.804468i	$0.702451\pi$
$744$	−0.248711	−0.00911820
$745$	19.8564	0.727482
$746$	17.3205i	0.634149i
$747$	− 14.7846i	− 0.540941i
$748$	− 4.39230i	− 0.160599i
$749$	0.679492i	0.0248281i
$750$	1.26795	0.0462990
$751$	−15.6077	−0.569533	−0.284766	−	0.958597i	$-0.591916\pi$
$751$	−15.6077	−0.569533	−0.284766	+	0.958597i	$0.591916\pi$
$752$	− 30.0000i	− 1.09399i
$753$	−10.6410	−0.387780
$754$	0	0
$755$	12.1962	0.443863
$756$	− 8.00000i	− 0.290957i
$757$	18.3923	0.668480	0.334240	−	0.942488i	$-0.391520\pi$
$757$	18.3923	0.668480	0.334240	+	0.942488i	$0.391520\pi$
$758$	57.1244	2.07485
$759$	− 4.39230i	− 0.159431i
$760$	− 7.26795i	− 0.263636i
$761$	7.85641i	0.284795i	0.989810	+	0.142397i	$0.0454810\pi$
$761$	7.85641i	0.284795i	−0.989810	+	0.142397i	$0.954519\pi$
$762$	− 7.35898i	− 0.266588i
$763$	4.00000	0.144810
$764$	18.9282	0.684798
$765$	8.53590i	0.308616i
$766$	1.60770	0.0580884
$767$	0	0
$768$	−13.9090	−0.501897
$769$	6.78461i	0.244659i	0.992490	+	0.122330i	$0.0390365\pi$
$769$	6.78461i	0.244659i	−0.992490	+	0.122330i	$0.960963\pi$
$770$	4.39230	0.158288
$771$	−5.75129	−0.207128
$772$	10.0000i	0.359908i
$773$	6.92820i	0.249190i	0.992208	+	0.124595i	$0.0397632\pi$
$773$	6.92820i	0.249190i	−0.992208	+	0.124595i	$0.960237\pi$
$774$	− 43.5167i	− 1.56417i
$775$	− 0.196152i	− 0.00704600i
$776$	−3.46410	−0.124354
$777$	−5.85641	−0.210097
$778$	10.3923i	0.372582i
$779$	14.5359	0.520803
$780$	0	0
$781$	−1.60770	−0.0575279
$782$	− 28.3923i	− 1.01531i
$783$	−37.8564	−1.35288
$784$	−15.0000	−0.535714
$785$	− 10.0000i	− 0.356915i
$786$	0	0
$787$	− 51.5692i	− 1.83824i	−0.393973	−	0.919122i	$-0.628900\pi$
$787$	− 51.5692i	− 1.83824i	0.393973	−	0.919122i	$-0.371100\pi$
$788$	0.928203i	0.0330659i
$789$	−3.46410	−0.123325
$790$	21.4641	0.763658
$791$	30.9282i	1.09968i
$792$	5.41154	0.192291
$793$	0	0
$794$	−22.1436	−0.785847
$795$	7.60770i	0.269817i
$796$	20.0000	0.708881
$797$	−28.6410	−1.01452	−0.507258	−	0.861794i	$-0.669341\pi$
$797$	−28.6410	−1.01452	−0.507258	+	0.861794i	$0.669341\pi$
$798$	10.6410i	0.376688i
$799$	− 20.7846i	− 0.735307i
$800$	− 5.19615i	− 0.183712i
$801$	− 2.28719i	− 0.0808138i
$802$	−39.9615	−1.41109
$803$	−5.07180	−0.178980
$804$	10.5359i	0.371572i
$805$	9.46410	0.333566
$806$	0	0
$807$	−5.75129	−0.202455
$808$	22.3923i	0.787759i
$809$	−9.46410	−0.332740	−0.166370	−	0.986063i	$-0.553205\pi$
$809$	−9.46410	−0.332740	−0.166370	+	0.986063i	$0.553205\pi$
$810$	7.73205	0.271677
$811$	− 28.1962i	− 0.990101i	−0.868864	−	0.495050i	$-0.835150\pi$
$811$	− 28.1962i	− 0.990101i	0.868864	−	0.495050i	$-0.164850\pi$
$812$	18.9282i	0.664250i
$813$	15.3590i	0.538663i
$814$	8.78461i	0.307900i
$815$	−6.39230	−0.223913
$816$	−12.6795	−0.443871
$817$	42.7846i	1.49684i
$818$	66.4974	2.32503
$819$	0	0
$820$	3.46410	0.120972
$821$	40.6410i	1.41838i	0.705017	+	0.709191i	$0.250939\pi$
$821$	40.6410i	1.41838i	−0.705017	+	0.709191i	$0.749061\pi$
$822$	16.3923	0.571747
$823$	46.5885	1.62397	0.811986	−	0.583677i	$-0.198387\pi$
$823$	46.5885	1.62397	0.811986	+	0.583677i	$0.198387\pi$
$824$	17.6603i	0.615224i
$825$	− 0.928203i	− 0.0323159i
$826$	− 52.3923i	− 1.82296i
$827$	18.0000i	0.625921i	0.949766	+	0.312961i	$0.101321\pi$
$827$	18.0000i	0.625921i	−0.949766	+	0.312961i	$0.898679\pi$
$828$	−11.6603	−0.405222
$829$	20.3923	0.708254	0.354127	−	0.935197i	$-0.384778\pi$
$829$	20.3923	0.708254	0.354127	+	0.935197i	$0.384778\pi$
$830$	10.3923i	0.360722i
$831$	−4.10512	−0.142405
$832$	0	0
$833$	−10.3923	−0.360072
$834$	− 10.6410i	− 0.368468i
$835$	−12.9282	−0.447399
$836$	5.32051	0.184014
$837$	0.784610i	0.0271201i
$838$	16.3923i	0.566263i
$839$	17.6603i	0.609700i	0.952400	+	0.304850i	$0.0986063\pi$
$839$	17.6603i	0.609700i	−0.952400	+	0.304850i	$0.901394\pi$
$840$	− 2.53590i	− 0.0874968i
$841$	60.5692	2.08859
$842$	−18.6795	−0.643738
$843$	− 1.17691i	− 0.0405351i
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−25.6077	−0.880411
$847$	18.7846i	0.645447i
$848$	51.9615	1.78437
$849$	1.03332	0.0354635
$850$	− 6.00000i	− 0.205798i
$851$	18.9282i	0.648850i
$852$	− 0.928203i	− 0.0317997i
$853$	8.00000i	0.273915i	0.990577	+	0.136957i	$0.0437323\pi$
$853$	8.00000i	0.273915i	−0.990577	+	0.136957i	$0.956268\pi$
$854$	42.9282	1.46897
$855$	−10.3397	−0.353612
$856$	− 0.588457i	− 0.0201131i
$857$	−47.5692	−1.62493	−0.812467	−	0.583007i	$-0.801876\pi$
$857$	−47.5692	−1.62493	−0.812467	+	0.583007i	$0.801876\pi$
$858$	0	0
$859$	45.1769	1.54142	0.770708	−	0.637188i	$-0.219903\pi$
$859$	45.1769	1.54142	0.770708	+	0.637188i	$0.219903\pi$
$860$	10.1962i	0.347686i
$861$	5.07180	0.172846
$862$	−33.8038	−1.15136
$863$	− 2.78461i	− 0.0947892i	−0.998876	−	0.0473946i	$-0.984908\pi$
$863$	− 2.78461i	− 0.0947892i	0.998876	−	0.0473946i	$-0.0150918\pi$
$864$	20.7846i	0.707107i
$865$	− 15.4641i	− 0.525795i
$866$	− 11.7513i	− 0.399325i
$867$	3.66025	0.124309
$868$	0.392305	0.0133157
$869$	− 15.7128i	− 0.533021i
$870$	12.0000	0.406838
$871$	0	0
$872$	−3.46410	−0.117309
$873$	4.92820i	0.166794i
$874$	34.3923	1.16334
$875$	2.00000	0.0676123
$876$	− 2.92820i	− 0.0989348i
$877$	− 2.00000i	− 0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$	− 2.00000i	− 0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$	55.4256i	1.87052i
$879$	13.8564i	0.467365i
$880$	−6.33975	−0.213713
$881$	12.6795	0.427183	0.213591	−	0.976923i	$-0.431484\pi$
$881$	12.6795	0.427183	0.213591	+	0.976923i	$0.431484\pi$
$882$	12.8038i	0.431128i
$883$	−34.1962	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$883$	−34.1962	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$884$	0	0
$885$	−11.0718	−0.372174
$886$	− 60.5885i	− 2.03551i
$887$	17.9090	0.601324	0.300662	−	0.953731i	$-0.402792\pi$
$887$	17.9090	0.601324	0.300662	+	0.953731i	$0.402792\pi$
$888$	5.07180	0.170198
$889$	− 11.6077i	− 0.389310i
$890$	1.60770i	0.0538901i
$891$	− 5.66025i	− 0.189626i
$892$	2.00000i	0.0669650i
$893$	25.1769	0.842513
$894$	25.1769	0.842042
$895$	5.07180i	0.169531i
$896$	−24.2487	−0.810093
$897$	0	0
$898$	47.5692	1.58741
$899$	− 1.85641i	− 0.0619146i
$900$	−2.46410	−0.0821367
$901$	36.0000	1.19933
$902$	− 7.60770i	− 0.253309i
$903$	14.9282i	0.496779i
$904$	− 26.7846i	− 0.890843i
$905$	20.3923i	0.677863i
$906$	15.4641	0.513760
$907$	−39.7654	−1.32039	−0.660194	−	0.751095i	$-0.729526\pi$
$907$	−39.7654	−1.32039	−0.660194	+	0.751095i	$0.729526\pi$
$908$	3.46410i	0.114960i
$909$	31.8564	1.05661
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	− 15.3590i	− 0.508587i
$913$	7.60770	0.251778
$914$	53.3205	1.76369
$915$	− 9.07180i	− 0.299904i
$916$	14.3923i	0.475535i
$917$	0	0
$918$	24.0000i	0.792118i
$919$	−53.1769	−1.75414	−0.877072	−	0.480358i	$-0.840507\pi$
$919$	−53.1769	−1.75414	−0.877072	+	0.480358i	$0.840507\pi$
$920$	−8.19615	−0.270219
$921$	− 16.6795i	− 0.549608i
$922$	−6.00000	−0.197599
$923$	0	0
$924$	1.85641	0.0610713
$925$	4.00000i	0.131519i
$926$	31.8564	1.04687
$927$	25.1244	0.825192
$928$	− 49.1769i	− 1.61431i
$929$	− 51.4641i	− 1.68848i	−0.535963	−	0.844241i	$-0.680052\pi$
$929$	− 51.4641i	− 1.68848i	0.535963	−	0.844241i	$-0.319948\pi$
$930$	− 0.248711i	− 0.00815557i
$931$	− 12.5885i	− 0.412570i
$932$	−6.00000	−0.196537
$933$	3.21539	0.105267
$934$	66.1577i	2.16475i
$935$	−4.39230	−0.143644
$936$	0	0
$937$	−6.78461	−0.221644	−0.110822	−	0.993840i	$-0.535348\pi$
$937$	−6.78461	−0.221644	−0.110822	+	0.993840i	$0.535348\pi$
$938$	49.8564i	1.62787i
$939$	−4.67949	−0.152709
$940$	6.00000	0.195698
$941$	31.1769i	1.01634i	0.861257	+	0.508169i	$0.169678\pi$
$941$	31.1769i	1.01634i	−0.861257	+	0.508169i	$0.830322\pi$
$942$	− 12.6795i	− 0.413120i
$943$	− 16.3923i	− 0.533807i
$944$	75.6218i	2.46128i
$945$	−8.00000	−0.260240
$946$	22.3923	0.728037
$947$	− 28.6410i	− 0.930708i	−0.885125	−	0.465354i	$-0.845927\pi$
$947$	− 28.6410i	− 0.930708i	0.885125	−	0.465354i	$-0.154073\pi$
$948$	9.07180	0.294638
$949$	0	0
$950$	7.26795	0.235803
$951$	17.5692i	0.569721i
$952$	−12.0000	−0.388922
$953$	12.9282	0.418786	0.209393	−	0.977832i	$-0.432851\pi$
$953$	12.9282	0.418786	0.209393	+	0.977832i	$0.432851\pi$
$954$	− 44.3538i	− 1.43601i
$955$	− 18.9282i	− 0.612502i
$956$	− 3.80385i	− 0.123025i
$957$	− 8.78461i	− 0.283966i
$958$	31.7654	1.02629
$959$	25.8564	0.834947
$960$	0.732051i	0.0236268i
$961$	30.9615	0.998759
$962$	0	0
$963$	−0.837169	−0.0269774
$964$	− 18.3923i	− 0.592376i
$965$	10.0000	0.321911
$966$	12.0000	0.386094
$967$	29.6077i	0.952119i	0.879413	+	0.476060i	$0.157935\pi$
$967$	29.6077i	0.952119i	−0.879413	+	0.476060i	$0.842065\pi$
$968$	− 16.2679i	− 0.522872i
$969$	− 10.6410i	− 0.341839i
$970$	− 3.46410i	− 0.111226i
$971$	5.07180	0.162762	0.0813809	−	0.996683i	$-0.474067\pi$
$971$	5.07180	0.162762	0.0813809	+	0.996683i	$0.474067\pi$
$972$	15.2679	0.489720
$973$	− 16.7846i	− 0.538090i
$974$	9.71281	0.311219
$975$	0	0
$976$	−61.9615	−1.98334
$977$	− 39.7128i	− 1.27053i	−0.772296	−	0.635263i	$-0.780892\pi$
$977$	− 39.7128i	− 1.27053i	0.772296	−	0.635263i	$-0.219108\pi$
$978$	−8.10512	−0.259173
$979$	1.17691	0.0376144
$980$	− 3.00000i	− 0.0958315i
$981$	4.92820i	0.157345i
$982$	− 16.3923i	− 0.523099i
$983$	− 13.6077i	− 0.434018i	−0.976170	−	0.217009i	$-0.930370\pi$
$983$	− 13.6077i	− 0.434018i	0.976170	−	0.217009i	$-0.0696301\pi$
$984$	−4.39230	−0.140022
$985$	0.928203	0.0295750
$986$	− 56.7846i	− 1.80839i
$987$	8.78461	0.279617
$988$	0	0
$989$	48.2487	1.53422
$990$	5.41154i	0.171990i
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−1.01924	−0.0323608
$993$	− 20.9282i	− 0.664136i
$994$	− 4.39230i	− 0.139315i
$995$	− 20.0000i	− 0.634043i
$996$	4.39230i	0.139176i
$997$	54.3923	1.72262	0.861311	−	0.508078i	$-0.169644\pi$
$997$	54.3923	1.72262	0.861311	+	0.508078i	$0.169644\pi$
$998$	−22.4833	−0.711698
$999$	− 16.0000i	− 0.506218i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.c.e.506.1		4
13.2	odd	12		845.2.e.f.191.2		4
13.3	even	3		845.2.m.c.316.2		4
13.4	even	6		845.2.m.c.361.2		4
13.5	odd	4		845.2.a.d.1.1		2
13.6	odd	12		845.2.e.f.146.2		4
13.7	odd	12		845.2.e.e.146.1		4
13.8	odd	4		65.2.a.c.1.2	✓	2
13.9	even	3		845.2.m.a.361.2		4
13.10	even	6		845.2.m.a.316.2		4
13.11	odd	12		845.2.e.e.191.1		4
13.12	even	2	inner	845.2.c.e.506.3		4
39.5	even	4		7605.2.a.be.1.2		2
39.8	even	4		585.2.a.k.1.1		2
52.47	even	4		1040.2.a.h.1.2		2
65.8	even	4		325.2.b.e.274.2		4
65.34	odd	4		325.2.a.g.1.1		2
65.44	odd	4		4225.2.a.w.1.2		2
65.47	even	4		325.2.b.e.274.3		4
91.34	even	4		3185.2.a.k.1.2		2
104.21	odd	4		4160.2.a.y.1.2		2
104.99	even	4		4160.2.a.bj.1.1		2
143.21	even	4		7865.2.a.h.1.1		2
156.47	odd	4		9360.2.a.cm.1.2		2
195.8	odd	4		2925.2.c.v.2224.3		4
195.47	odd	4		2925.2.c.v.2224.2		4
195.164	even	4		2925.2.a.z.1.2		2
260.99	even	4		5200.2.a.ca.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.c.1.2	✓	2	13.8	odd	4
325.2.a.g.1.1		2	65.34	odd	4
325.2.b.e.274.2		4	65.8	even	4
325.2.b.e.274.3		4	65.47	even	4
585.2.a.k.1.1		2	39.8	even	4
845.2.a.d.1.1		2	13.5	odd	4
845.2.c.e.506.1		4	1.1	even	1	trivial
845.2.c.e.506.3		4	13.12	even	2	inner
845.2.e.e.146.1		4	13.7	odd	12
845.2.e.e.191.1		4	13.11	odd	12
845.2.e.f.146.2		4	13.6	odd	12
845.2.e.f.191.2		4	13.2	odd	12
845.2.m.a.316.2		4	13.10	even	6
845.2.m.a.361.2		4	13.9	even	3
845.2.m.c.316.2		4	13.3	even	3
845.2.m.c.361.2		4	13.4	even	6
1040.2.a.h.1.2		2	52.47	even	4
2925.2.a.z.1.2		2	195.164	even	4
2925.2.c.v.2224.2		4	195.47	odd	4
2925.2.c.v.2224.3		4	195.8	odd	4
3185.2.a.k.1.2		2	91.34	even	4
4160.2.a.y.1.2		2	104.21	odd	4
4160.2.a.bj.1.1		2	104.99	even	4
4225.2.a.w.1.2		2	65.44	odd	4
5200.2.a.ca.1.1		2	260.99	even	4
7605.2.a.be.1.2		2	39.5	even	4
7865.2.a.h.1.1		2	143.21	even	4
9360.2.a.cm.1.2		2	156.47	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{2} -0.732051 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.26795i q^{6} +2.00000i q^{7} -1.73205i q^{8} -2.46410 q^{9} +O(q^{10})\)	\(q-1.73205i q^{2} -0.732051 q^{3} -1.00000 q^{4} +1.00000i q^{5} +1.26795i q^{6} +2.00000i q^{7} -1.73205i q^{8} -2.46410 q^{9} +1.73205 q^{10} -1.26795i q^{11} +0.732051 q^{12} +3.46410 q^{14} -0.732051i q^{15} -5.00000 q^{16} -3.46410 q^{17} +4.26795i q^{18} -4.19615i q^{19} -1.00000i q^{20} -1.46410i q^{21} -2.19615 q^{22} -4.73205 q^{23} +1.26795i q^{24} -1.00000 q^{25} +4.00000 q^{27} -2.00000i q^{28} -9.46410 q^{29} -1.26795 q^{30} +0.196152i q^{31} +5.19615i q^{32} +0.928203i q^{33} +6.00000i q^{34} -2.00000 q^{35} +2.46410 q^{36} -4.00000i q^{37} -7.26795 q^{38} +1.73205 q^{40} +3.46410i q^{41} -2.53590 q^{42} -10.1962 q^{43} +1.26795i q^{44} -2.46410i q^{45} +8.19615i q^{46} +6.00000i q^{47} +3.66025 q^{48} +3.00000 q^{49} +1.73205i q^{50} +2.53590 q^{51} -10.3923 q^{53} -6.92820i q^{54} +1.26795 q^{55} +3.46410 q^{56} +3.07180i q^{57} +16.3923i q^{58} -15.1244i q^{59} +0.732051i q^{60} +12.3923 q^{61} +0.339746 q^{62} -4.92820i q^{63} -1.00000 q^{64} +1.60770 q^{66} +14.3923i q^{67} +3.46410 q^{68} +3.46410 q^{69} +3.46410i q^{70} -1.26795i q^{71} +4.26795i q^{72} -4.00000i q^{73} -6.92820 q^{74} +0.732051 q^{75} +4.19615i q^{76} +2.53590 q^{77} +12.3923 q^{79} -5.00000i q^{80} +4.46410 q^{81} +6.00000 q^{82} +6.00000i q^{83} +1.46410i q^{84} -3.46410i q^{85} +17.6603i q^{86} +6.92820 q^{87} -2.19615 q^{88} +0.928203i q^{89} -4.26795 q^{90} +4.73205 q^{92} -0.143594i q^{93} +10.3923 q^{94} +4.19615 q^{95} -3.80385i q^{96} -2.00000i q^{97} -5.19615i q^{98} +3.12436i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{12} - 20 q^{16} + 12 q^{22} - 12 q^{23} - 4 q^{25} + 16 q^{27} - 24 q^{29} - 12 q^{30} - 8 q^{35} - 4 q^{36} - 36 q^{38} - 24 q^{42} - 20 q^{43} - 20 q^{48} + 12 q^{49} + 24 q^{51} + 12 q^{55} + 8 q^{61} + 36 q^{62} - 4 q^{64} + 48 q^{66} - 4 q^{75} + 24 q^{77} + 8 q^{79} + 4 q^{81} + 24 q^{82} + 12 q^{88} - 24 q^{90} + 12 q^{92} - 4 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^4 + 4 * q^9 - 4 * q^12 - 20 * q^16 + 12 * q^22 - 12 * q^23 - 4 * q^25 + 16 * q^27 - 24 * q^29 - 12 * q^30 - 8 * q^35 - 4 * q^36 - 36 * q^38 - 24 * q^42 - 20 * q^43 - 20 * q^48 + 12 * q^49 + 24 * q^51 + 12 * q^55 + 8 * q^61 + 36 * q^62 - 4 * q^64 + 48 * q^66 - 4 * q^75 + 24 * q^77 + 8 * q^79 + 4 * q^81 + 24 * q^82 + 12 * q^88 - 24 * q^90 + 12 * q^92 - 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more