LMFDB - Embedded newform 2432.2.c.h.1217.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (of order $2$, degree $1$, 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:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.3
Root			$-0.671462 - 1.24464i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.h.1217.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.14637i q^{3} +3.34292i q^{5} +3.34292 q^{7} +1.68585 q^{9} +O(q^{10})$	$q-1.14637i q^{3} +3.34292i q^{5} +3.34292 q^{7} +1.68585 q^{9} +0.489289i q^{11} +2.29273i q^{13} +3.83221 q^{15} +0.196558 q^{17} +1.00000i q^{19} -3.83221i q^{21} +4.00000 q^{23} -6.17513 q^{25} -5.37169i q^{27} +6.12494i q^{29} +8.22533 q^{31} +0.560904 q^{33} +11.1751i q^{35} -3.83221i q^{37} +2.62831 q^{39} -4.12494 q^{41} -7.17513i q^{43} +5.63565i q^{45} -11.0073 q^{47} +4.17513 q^{49} -0.225327i q^{51} +4.81079i q^{53} -1.63565 q^{55} +1.14637 q^{57} +5.31415i q^{59} -7.73604i q^{61} +5.63565 q^{63} -7.66442 q^{65} +15.4966i q^{67} -4.58546i q^{69} +11.2713 q^{71} -15.4679 q^{73} +7.07896i q^{75} +1.63565i q^{77} -10.5181 q^{79} -1.10038 q^{81} +0.728692i q^{83} +0.657077i q^{85} +7.02142 q^{87} +8.51806 q^{89} +7.66442i q^{91} -9.42923i q^{93} -3.34292 q^{95} -3.31415 q^{97} +0.824865i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 8 q^{7} - 14 q^{9}+O(q^{10}) $ 6 * q + 8 * q^7 - 14 * q^9	$ 6 q + 8 q^{7} - 14 q^{9} - 4 q^{15} - 8 q^{17} + 24 q^{23} + 2 q^{25} + 4 q^{31} + 12 q^{33} + 64 q^{39} + 8 q^{41} - 14 q^{49} + 8 q^{55} + 4 q^{57} + 16 q^{63} + 8 q^{65} + 32 q^{71} - 48 q^{73} - 12 q^{79} + 6 q^{81} + 72 q^{87} - 8 q^{95} - 44 q^{97}+O(q^{100}) $ 6 * q + 8 * q^7 - 14 * q^9 - 4 * q^15 - 8 * q^17 + 24 * q^23 + 2 * q^25 + 4 * q^31 + 12 * q^33 + 64 * q^39 + 8 * q^41 - 14 * q^49 + 8 * q^55 + 4 * q^57 + 16 * q^63 + 8 * q^65 + 32 * q^71 - 48 * q^73 - 12 * q^79 + 6 * q^81 + 72 * q^87 - 8 * q^95 - 44 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.14637i	− 0.661854i	−0.943656	−	0.330927i	$-0.892639\pi$
$3$	− 1.14637i	− 0.661854i	0.943656	−	0.330927i	$-0.107361\pi$
$4$	0	0
$5$	3.34292i	1.49500i	0.664261	+	0.747500i	$0.268746\pi$
$5$	3.34292i	1.49500i	−0.664261	+	0.747500i	$0.731254\pi$
$6$	0	0
$7$	3.34292	1.26351	0.631753	−	0.775170i	$-0.282336\pi$
$7$	3.34292	1.26351	0.631753	+	0.775170i	$0.282336\pi$
$8$	0	0
$9$	1.68585	0.561949
$10$	0	0
$11$	0.489289i	0.147526i	0.997276	+	0.0737630i	$0.0235008\pi$
$11$	0.489289i	0.147526i	−0.997276	+	0.0737630i	$0.976499\pi$
$12$	0	0
$13$	2.29273i	0.635889i	0.948109	+	0.317945i	$0.102993\pi$
$13$	2.29273i	0.635889i	−0.948109	+	0.317945i	$0.897007\pi$
$14$	0	0
$15$	3.83221	0.989473
$16$	0	0
$17$	0.196558	0.0476722	0.0238361	−	0.999716i	$-0.492412\pi$
$17$	0.196558	0.0476722	0.0238361	+	0.999716i	$0.492412\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	− 3.83221i	− 0.836257i
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−6.17513	−1.23503
$26$	0	0
$27$	− 5.37169i	− 1.03378i
$28$	0	0
$29$	6.12494i	1.13737i	0.822554	+	0.568687i	$0.192548\pi$
$29$	6.12494i	1.13737i	−0.822554	+	0.568687i	$0.807452\pi$
$30$	0	0
$31$	8.22533	1.47731	0.738656	−	0.674082i	$-0.235461\pi$
$31$	8.22533	1.47731	0.738656	+	0.674082i	$0.235461\pi$
$32$	0	0
$33$	0.560904	0.0976408
$34$	0	0
$35$	11.1751i	1.88894i
$36$	0	0
$37$	− 3.83221i	− 0.630012i	−0.949090	−	0.315006i	$-0.897994\pi$
$37$	− 3.83221i	− 0.630012i	0.949090	−	0.315006i	$-0.102006\pi$
$38$	0	0
$39$	2.62831	0.420866
$40$	0	0
$41$	−4.12494	−0.644208	−0.322104	−	0.946704i	$-0.604390\pi$
$41$	−4.12494	−0.644208	−0.322104	+	0.946704i	$0.604390\pi$
$42$	0	0
$43$	− 7.17513i	− 1.09420i	−0.837068	−	0.547099i	$-0.815732\pi$
$43$	− 7.17513i	− 1.09420i	0.837068	−	0.547099i	$-0.184268\pi$
$44$	0	0
$45$	5.63565i	0.840114i
$46$	0	0
$47$	−11.0073	−1.60559	−0.802793	−	0.596258i	$-0.796654\pi$
$47$	−11.0073	−1.60559	−0.802793	+	0.596258i	$0.796654\pi$
$48$	0	0
$49$	4.17513	0.596448
$50$	0	0
$51$	− 0.225327i	− 0.0315521i
$52$	0	0
$53$	4.81079i	0.660813i	0.943839	+	0.330406i	$0.107186\pi$
$53$	4.81079i	0.660813i	−0.943839	+	0.330406i	$0.892814\pi$
$54$	0	0
$55$	−1.63565	−0.220552
$56$	0	0
$57$	1.14637	0.151840
$58$	0	0
$59$	5.31415i	0.691844i	0.938263	+	0.345922i	$0.112434\pi$
$59$	5.31415i	0.691844i	−0.938263	+	0.345922i	$0.887566\pi$
$60$	0	0
$61$	− 7.73604i	− 0.990498i	−0.868751	−	0.495249i	$-0.835077\pi$
$61$	− 7.73604i	− 0.990498i	0.868751	−	0.495249i	$-0.164923\pi$
$62$	0	0
$63$	5.63565	0.710026
$64$	0	0
$65$	−7.66442	−0.950655
$66$	0	0
$67$	15.4966i	1.89322i	0.322388	+	0.946608i	$0.395514\pi$
$67$	15.4966i	1.89322i	−0.322388	+	0.946608i	$0.604486\pi$
$68$	0	0
$69$	− 4.58546i	− 0.552025i
$70$	0	0
$71$	11.2713	1.33766	0.668829	−	0.743416i	$-0.266796\pi$
$71$	11.2713	1.33766	0.668829	+	0.743416i	$0.266796\pi$
$72$	0	0
$73$	−15.4679	−1.81038	−0.905188	−	0.425011i	$-0.860270\pi$
$73$	−15.4679	−1.81038	−0.905188	+	0.425011i	$0.860270\pi$
$74$	0	0
$75$	7.07896i	0.817408i
$76$	0	0
$77$	1.63565i	0.186400i
$78$	0	0
$79$	−10.5181	−1.18337	−0.591687	−	0.806168i	$-0.701538\pi$
$79$	−10.5181	−1.18337	−0.591687	+	0.806168i	$0.701538\pi$
$80$	0	0
$81$	−1.10038	−0.122265
$82$	0	0
$83$	0.728692i	0.0799843i	0.999200	+	0.0399922i	$0.0127333\pi$
$83$	0.728692i	0.0799843i	−0.999200	+	0.0399922i	$0.987267\pi$
$84$	0	0
$85$	0.657077i	0.0712700i
$86$	0	0
$87$	7.02142	0.752776
$88$	0	0
$89$	8.51806	0.902912	0.451456	−	0.892293i	$-0.350905\pi$
$89$	8.51806	0.902912	0.451456	+	0.892293i	$0.350905\pi$
$90$	0	0
$91$	7.66442i	0.803450i
$92$	0	0
$93$	− 9.42923i	− 0.977766i
$94$	0	0
$95$	−3.34292	−0.342977
$96$	0	0
$97$	−3.31415	−0.336501	−0.168251	−	0.985744i	$-0.553812\pi$
$97$	−3.31415	−0.336501	−0.168251	+	0.985744i	$0.553812\pi$
$98$	0	0
$99$	0.824865i	0.0829021i
$100$	0	0
$101$	− 10.6858i	− 1.06328i	−0.846970	−	0.531641i	$-0.821576\pi$
$101$	− 10.6858i	− 1.06328i	0.846970	−	0.531641i	$-0.178424\pi$
$102$	0	0
$103$	12.0575	1.18806	0.594032	−	0.804441i	$-0.297535\pi$
$103$	12.0575	1.18806	0.594032	+	0.804441i	$0.297535\pi$
$104$	0	0
$105$	12.8108	1.25020
$106$	0	0
$107$	8.39312i	0.811393i	0.914008	+	0.405697i	$0.132971\pi$
$107$	8.39312i	0.811393i	−0.914008	+	0.405697i	$0.867029\pi$
$108$	0	0
$109$	8.22533i	0.787843i	0.919144	+	0.393922i	$0.128882\pi$
$109$	8.22533i	0.787843i	−0.919144	+	0.393922i	$0.871118\pi$
$110$	0	0
$111$	−4.39312	−0.416976
$112$	0	0
$113$	13.4966	1.26966	0.634828	−	0.772653i	$-0.281071\pi$
$113$	13.4966	1.26966	0.634828	+	0.772653i	$0.281071\pi$
$114$	0	0
$115$	13.3717i	1.24692i
$116$	0	0
$117$	3.86519i	0.357337i
$118$	0	0
$119$	0.657077	0.0602341
$120$	0	0
$121$	10.7606	0.978236
$122$	0	0
$123$	4.72869i	0.426372i
$124$	0	0
$125$	− 3.92839i	− 0.351365i
$126$	0	0
$127$	−1.53948	−0.136607	−0.0683034	−	0.997665i	$-0.521759\pi$
$127$	−1.53948	−0.136607	−0.0683034	+	0.997665i	$0.521759\pi$
$128$	0	0
$129$	−8.22533	−0.724200
$130$	0	0
$131$	3.56825i	0.311759i	0.987776	+	0.155880i	$0.0498212\pi$
$131$	3.56825i	0.311759i	−0.987776	+	0.155880i	$0.950179\pi$
$132$	0	0
$133$	3.34292i	0.289868i
$134$	0	0
$135$	17.9572	1.54551
$136$	0	0
$137$	−10.1537	−0.867490	−0.433745	−	0.901036i	$-0.642808\pi$
$137$	−10.1537	−0.867490	−0.433745	+	0.901036i	$0.642808\pi$
$138$	0	0
$139$	− 3.90383i	− 0.331118i	−0.986200	−	0.165559i	$-0.947057\pi$
$139$	− 3.90383i	− 0.331118i	0.986200	−	0.165559i	$-0.0529429\pi$
$140$	0	0
$141$	12.6184i	1.06266i
$142$	0	0
$143$	−1.12181	−0.0938102
$144$	0	0
$145$	−20.4752	−1.70037
$146$	0	0
$147$	− 4.78623i	− 0.394762i
$148$	0	0
$149$	− 17.3576i	− 1.42199i	−0.703196	−	0.710996i	$-0.748245\pi$
$149$	− 17.3576i	− 1.42199i	0.703196	−	0.710996i	$-0.251755\pi$
$150$	0	0
$151$	4.05754	0.330198	0.165099	−	0.986277i	$-0.447206\pi$
$151$	4.05754	0.330198	0.165099	+	0.986277i	$0.447206\pi$
$152$	0	0
$153$	0.331366	0.0267893
$154$	0	0
$155$	27.4966i	2.20858i
$156$	0	0
$157$	2.35027i	0.187572i	0.995592	+	0.0937860i	$0.0298969\pi$
$157$	2.35027i	0.187572i	−0.995592	+	0.0937860i	$0.970103\pi$
$158$	0	0
$159$	5.51492	0.437362
$160$	0	0
$161$	13.3717	1.05384
$162$	0	0
$163$	− 13.7648i	− 1.07814i	−0.842260	−	0.539071i	$-0.818775\pi$
$163$	− 13.7648i	− 1.07814i	0.842260	−	0.539071i	$-0.181225\pi$
$164$	0	0
$165$	1.87506i	0.145973i
$166$	0	0
$167$	−0.417674	−0.0323206	−0.0161603	−	0.999869i	$-0.505144\pi$
$167$	−0.417674	−0.0323206	−0.0161603	+	0.999869i	$0.505144\pi$
$168$	0	0
$169$	7.74338	0.595645
$170$	0	0
$171$	1.68585i	0.128920i
$172$	0	0
$173$	3.60688i	0.274226i	0.990555	+	0.137113i	$0.0437824\pi$
$173$	3.60688i	0.274226i	−0.990555	+	0.137113i	$0.956218\pi$
$174$	0	0
$175$	−20.6430	−1.56046
$176$	0	0
$177$	6.09196	0.457900
$178$	0	0
$179$	6.51806i	0.487183i	0.969878	+	0.243591i	$0.0783255\pi$
$179$	6.51806i	0.487183i	−0.969878	+	0.243591i	$0.921674\pi$
$180$	0	0
$181$	1.31415i	0.0976803i	0.998807	+	0.0488401i	$0.0155525\pi$
$181$	1.31415i	0.0976803i	−0.998807	+	0.0488401i	$0.984448\pi$
$182$	0	0
$183$	−8.86833	−0.655566
$184$	0	0
$185$	12.8108	0.941868
$186$	0	0
$187$	0.0961734i	0.00703289i
$188$	0	0
$189$	− 17.9572i	− 1.30619i
$190$	0	0
$191$	−9.24254	−0.668767	−0.334383	−	0.942437i	$-0.608528\pi$
$191$	−9.24254	−0.668767	−0.334383	+	0.942437i	$0.608528\pi$
$192$	0	0
$193$	8.85363	0.637299	0.318649	−	0.947873i	$-0.396771\pi$
$193$	8.85363	0.637299	0.318649	+	0.947873i	$0.396771\pi$
$194$	0	0
$195$	8.78623i	0.629195i
$196$	0	0
$197$	27.3288i	1.94710i	0.228475	+	0.973550i	$0.426626\pi$
$197$	27.3288i	1.94710i	−0.228475	+	0.973550i	$0.573374\pi$
$198$	0	0
$199$	18.3643	1.30181	0.650907	−	0.759157i	$-0.274389\pi$
$199$	18.3643	1.30181	0.650907	+	0.759157i	$0.274389\pi$
$200$	0	0
$201$	17.7648	1.25303
$202$	0	0
$203$	20.4752i	1.43708i
$204$	0	0
$205$	− 13.7894i	− 0.963091i
$206$	0	0
$207$	6.74338	0.468698
$208$	0	0
$209$	−0.489289	−0.0338448
$210$	0	0
$211$	− 18.3503i	− 1.26328i	−0.775260	−	0.631642i	$-0.782381\pi$
$211$	− 18.3503i	− 1.26328i	0.775260	−	0.631642i	$-0.217619\pi$
$212$	0	0
$213$	− 12.9210i	− 0.885335i
$214$	0	0
$215$	23.9859	1.63583
$216$	0	0
$217$	27.4966	1.86659
$218$	0	0
$219$	17.7318i	1.19821i
$220$	0	0
$221$	0.450654i	0.0303142i
$222$	0	0
$223$	18.3748	1.23047	0.615235	−	0.788344i	$-0.289061\pi$
$223$	18.3748	1.23047	0.615235	+	0.788344i	$0.289061\pi$
$224$	0	0
$225$	−10.4103	−0.694022
$226$	0	0
$227$	− 18.9933i	− 1.26063i	−0.776340	−	0.630314i	$-0.782926\pi$
$227$	− 18.9933i	− 1.26063i	0.776340	−	0.630314i	$-0.217074\pi$
$228$	0	0
$229$	− 9.05019i	− 0.598054i	−0.954245	−	0.299027i	$-0.903338\pi$
$229$	− 9.05019i	− 0.598054i	0.954245	−	0.299027i	$-0.0966620\pi$
$230$	0	0
$231$	1.87506	0.123370
$232$	0	0
$233$	−9.76060	−0.639438	−0.319719	−	0.947512i	$-0.603589\pi$
$233$	−9.76060	−0.639438	−0.319719	+	0.947512i	$0.603589\pi$
$234$	0	0
$235$	− 36.7967i	− 2.40035i
$236$	0	0
$237$	12.0575i	0.783221i
$238$	0	0
$239$	12.9645	0.838604	0.419302	−	0.907847i	$-0.362275\pi$
$239$	12.9645	0.838604	0.419302	+	0.907847i	$0.362275\pi$
$240$	0	0
$241$	−20.3257	−1.30929	−0.654647	−	0.755935i	$-0.727183\pi$
$241$	−20.3257	−1.30929	−0.654647	+	0.755935i	$0.727183\pi$
$242$	0	0
$243$	− 14.8536i	− 0.952861i
$244$	0	0
$245$	13.9572i	0.891690i
$246$	0	0
$247$	−2.29273	−0.145883
$248$	0	0
$249$	0.835347	0.0529380
$250$	0	0
$251$	− 6.05333i	− 0.382083i	−0.981582	−	0.191041i	$-0.938814\pi$
$251$	− 6.05333i	− 0.382083i	0.981582	−	0.191041i	$-0.0611865\pi$
$252$	0	0
$253$	1.95715i	0.123045i
$254$	0	0
$255$	0.753250	0.0471704
$256$	0	0
$257$	14.0575	0.876885	0.438443	−	0.898759i	$-0.355530\pi$
$257$	14.0575	0.876885	0.438443	+	0.898759i	$0.355530\pi$
$258$	0	0
$259$	− 12.8108i	− 0.796024i
$260$	0	0
$261$	10.3257i	0.639145i
$262$	0	0
$263$	8.26396	0.509578	0.254789	−	0.966997i	$-0.417994\pi$
$263$	8.26396	0.509578	0.254789	+	0.966997i	$0.417994\pi$
$264$	0	0
$265$	−16.0821	−0.987915
$266$	0	0
$267$	− 9.76481i	− 0.597597i
$268$	0	0
$269$	3.18921i	0.194450i	0.995262	+	0.0972248i	$0.0309966\pi$
$269$	3.18921i	0.194450i	−0.995262	+	0.0972248i	$0.969003\pi$
$270$	0	0
$271$	−9.89962	−0.601359	−0.300679	−	0.953725i	$-0.597213\pi$
$271$	−9.89962	−0.601359	−0.300679	+	0.953725i	$0.597213\pi$
$272$	0	0
$273$	8.78623	0.531767
$274$	0	0
$275$	− 3.02142i	− 0.182199i
$276$	0	0
$277$	− 12.7722i	− 0.767404i	−0.923457	−	0.383702i	$-0.874649\pi$
$277$	− 12.7722i	− 0.767404i	0.923457	−	0.383702i	$-0.125351\pi$
$278$	0	0
$279$	13.8666	0.830174
$280$	0	0
$281$	−29.5296	−1.76159	−0.880795	−	0.473499i	$-0.842991\pi$
$281$	−29.5296	−1.76159	−0.880795	+	0.473499i	$0.842991\pi$
$282$	0	0
$283$	− 25.3246i	− 1.50539i	−0.658367	−	0.752697i	$-0.728753\pi$
$283$	− 25.3246i	− 1.50539i	0.658367	−	0.752697i	$-0.271247\pi$
$284$	0	0
$285$	3.83221i	0.227001i
$286$	0	0
$287$	−13.7894	−0.813961
$288$	0	0
$289$	−16.9614	−0.997727
$290$	0	0
$291$	3.79923i	0.222715i
$292$	0	0
$293$	16.8353i	0.983531i	0.870728	+	0.491766i	$0.163648\pi$
$293$	16.8353i	0.983531i	−0.870728	+	0.491766i	$0.836352\pi$
$294$	0	0
$295$	−17.7648	−1.03431
$296$	0	0
$297$	2.62831	0.152510
$298$	0	0
$299$	9.17092i	0.530368i
$300$	0	0
$301$	− 23.9859i	− 1.38253i
$302$	0	0
$303$	−12.2499	−0.703738
$304$	0	0
$305$	25.8610	1.48080
$306$	0	0
$307$	− 17.7894i	− 1.01529i	−0.861566	−	0.507646i	$-0.830516\pi$
$307$	− 17.7894i	− 1.01529i	0.861566	−	0.507646i	$-0.169484\pi$
$308$	0	0
$309$	− 13.8223i	− 0.786326i
$310$	0	0
$311$	20.3215	1.15233	0.576163	−	0.817335i	$-0.304549\pi$
$311$	20.3215	1.15233	0.576163	+	0.817335i	$0.304549\pi$
$312$	0	0
$313$	−34.8929	−1.97226	−0.986131	−	0.165967i	$-0.946925\pi$
$313$	−34.8929	−1.97226	−0.986131	+	0.165967i	$0.946925\pi$
$314$	0	0
$315$	18.8396i	1.06149i
$316$	0	0
$317$	12.7287i	0.714915i	0.933929	+	0.357457i	$0.116356\pi$
$317$	12.7287i	0.714915i	−0.933929	+	0.357457i	$0.883644\pi$
$318$	0	0
$319$	−2.99686	−0.167792
$320$	0	0
$321$	9.62158	0.537024
$322$	0	0
$323$	0.196558i	0.0109368i
$324$	0	0
$325$	− 14.1579i	− 0.785340i
$326$	0	0
$327$	9.42923	0.521438
$328$	0	0
$329$	−36.7967	−2.02867
$330$	0	0
$331$	− 18.4605i	− 1.01468i	−0.861745	−	0.507341i	$-0.830628\pi$
$331$	− 18.4605i	− 1.01468i	0.861745	−	0.507341i	$-0.169372\pi$
$332$	0	0
$333$	− 6.46052i	− 0.354034i
$334$	0	0
$335$	−51.8041	−2.83036
$336$	0	0
$337$	28.5510	1.55527	0.777637	−	0.628713i	$-0.216418\pi$
$337$	28.5510	1.55527	0.777637	+	0.628713i	$0.216418\pi$
$338$	0	0
$339$	− 15.4721i	− 0.840328i
$340$	0	0
$341$	4.02456i	0.217942i
$342$	0	0
$343$	−9.44331	−0.509891
$344$	0	0
$345$	15.3288	0.825277
$346$	0	0
$347$	30.3116i	1.62721i	0.581415	+	0.813607i	$0.302499\pi$
$347$	30.3116i	1.62721i	−0.581415	+	0.813607i	$0.697501\pi$
$348$	0	0
$349$	− 19.6503i	− 1.05186i	−0.850528	−	0.525929i	$-0.823718\pi$
$349$	− 19.6503i	− 1.05186i	0.850528	−	0.525929i	$-0.176282\pi$
$350$	0	0
$351$	12.3158	0.657371
$352$	0	0
$353$	1.80765	0.0962117	0.0481058	−	0.998842i	$-0.484682\pi$
$353$	1.80765	0.0962117	0.0481058	+	0.998842i	$0.484682\pi$
$354$	0	0
$355$	37.6791i	1.99980i
$356$	0	0
$357$	− 0.753250i	− 0.0398662i
$358$	0	0
$359$	−23.4496	−1.23762	−0.618811	−	0.785540i	$-0.712385\pi$
$359$	−23.4496	−1.23762	−0.618811	+	0.785540i	$0.712385\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 12.3356i	− 0.647450i
$364$	0	0
$365$	− 51.7079i	− 2.70651i
$366$	0	0
$367$	16.2499	0.848237	0.424119	−	0.905607i	$-0.360584\pi$
$367$	16.2499	0.848237	0.424119	+	0.905607i	$0.360584\pi$
$368$	0	0
$369$	−6.95402	−0.362012
$370$	0	0
$371$	16.0821i	0.834941i
$372$	0	0
$373$	16.8353i	0.871701i	0.900019	+	0.435851i	$0.143552\pi$
$373$	16.8353i	0.871701i	−0.900019	+	0.435851i	$0.856448\pi$
$374$	0	0
$375$	−4.50337	−0.232553
$376$	0	0
$377$	−14.0428	−0.723243
$378$	0	0
$379$	2.68585i	0.137963i	0.997618	+	0.0689813i	$0.0219749\pi$
$379$	2.68585i	0.137963i	−0.997618	+	0.0689813i	$0.978025\pi$
$380$	0	0
$381$	1.76481i	0.0904138i
$382$	0	0
$383$	−0.110250	−0.00563350	−0.00281675	−	0.999996i	$-0.500897\pi$
$383$	−0.110250	−0.00563350	−0.00281675	+	0.999996i	$0.500897\pi$
$384$	0	0
$385$	−5.46787	−0.278668
$386$	0	0
$387$	− 12.0962i	− 0.614883i
$388$	0	0
$389$	9.24254i	0.468615i	0.972162	+	0.234308i	$0.0752823\pi$
$389$	9.24254i	0.468615i	−0.972162	+	0.234308i	$0.924718\pi$
$390$	0	0
$391$	0.786230	0.0397614
$392$	0	0
$393$	4.09052	0.206339
$394$	0	0
$395$	− 35.1611i	− 1.76914i
$396$	0	0
$397$	32.9070i	1.65155i	0.563997	+	0.825777i	$0.309263\pi$
$397$	32.9070i	1.65155i	−0.563997	+	0.825777i	$0.690737\pi$
$398$	0	0
$399$	3.83221	0.191851
$400$	0	0
$401$	−23.9817	−1.19759	−0.598795	−	0.800902i	$-0.704353\pi$
$401$	−23.9817	−1.19759	−0.598795	+	0.800902i	$0.704353\pi$
$402$	0	0
$403$	18.8585i	0.939407i
$404$	0	0
$405$	− 3.67850i	− 0.182786i
$406$	0	0
$407$	1.87506	0.0929431
$408$	0	0
$409$	−0.182481	−0.00902311	−0.00451156	−	0.999990i	$-0.501436\pi$
$409$	−0.182481	−0.00902311	−0.00451156	+	0.999990i	$0.501436\pi$
$410$	0	0
$411$	11.6399i	0.574152i
$412$	0	0
$413$	17.7648i	0.874149i
$414$	0	0
$415$	−2.43596	−0.119577
$416$	0	0
$417$	−4.47521	−0.219152
$418$	0	0
$419$	13.6497i	0.666833i	0.942780	+	0.333416i	$0.108202\pi$
$419$	13.6497i	0.666833i	−0.942780	+	0.333416i	$0.891798\pi$
$420$	0	0
$421$	14.4324i	0.703390i	0.936115	+	0.351695i	$0.114395\pi$
$421$	14.4324i	0.703390i	−0.936115	+	0.351695i	$0.885605\pi$
$422$	0	0
$423$	−18.5567	−0.902257
$424$	0	0
$425$	−1.21377	−0.0588765
$426$	0	0
$427$	− 25.8610i	− 1.25150i
$428$	0	0
$429$	1.28600i	0.0620887i
$430$	0	0
$431$	−17.1218	−0.824728	−0.412364	−	0.911019i	$-0.635297\pi$
$431$	−17.1218	−0.824728	−0.412364	+	0.911019i	$0.635297\pi$
$432$	0	0
$433$	−28.3748	−1.36361	−0.681804	−	0.731535i	$-0.738804\pi$
$433$	−28.3748	−1.36361	−0.681804	+	0.731535i	$0.738804\pi$
$434$	0	0
$435$	23.4721i	1.12540i
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−15.5823	−0.743704	−0.371852	−	0.928292i	$-0.621277\pi$
$439$	−15.5823	−0.743704	−0.371852	+	0.928292i	$0.621277\pi$
$440$	0	0
$441$	7.03863	0.335173
$442$	0	0
$443$	− 24.7476i	− 1.17579i	−0.808936	−	0.587897i	$-0.799956\pi$
$443$	− 24.7476i	− 1.17579i	0.808936	−	0.587897i	$-0.200044\pi$
$444$	0	0
$445$	28.4752i	1.34985i
$446$	0	0
$447$	−19.8982	−0.941151
$448$	0	0
$449$	−37.7220	−1.78021	−0.890105	−	0.455756i	$-0.849369\pi$
$449$	−37.7220	−1.78021	−0.890105	+	0.455756i	$0.849369\pi$
$450$	0	0
$451$	− 2.01829i	− 0.0950374i
$452$	0	0
$453$	− 4.65142i	− 0.218543i
$454$	0	0
$455$	−25.6216	−1.20116
$456$	0	0
$457$	21.6258	1.01161	0.505806	−	0.862647i	$-0.331195\pi$
$457$	21.6258	1.01161	0.505806	+	0.862647i	$0.331195\pi$
$458$	0	0
$459$	− 1.05585i	− 0.0492827i
$460$	0	0
$461$	− 11.5353i	− 0.537251i	−0.963245	−	0.268626i	$-0.913431\pi$
$461$	− 11.5353i	− 0.537251i	0.963245	−	0.268626i	$-0.0865694\pi$
$462$	0	0
$463$	−40.4282	−1.87886	−0.939428	−	0.342747i	$-0.888643\pi$
$463$	−40.4282	−1.87886	−0.939428	+	0.342747i	$0.888643\pi$
$464$	0	0
$465$	31.5212	1.46176
$466$	0	0
$467$	− 27.7606i	− 1.28461i	−0.766450	−	0.642304i	$-0.777979\pi$
$467$	− 27.7606i	− 1.28461i	0.766450	−	0.642304i	$-0.222021\pi$
$468$	0	0
$469$	51.8041i	2.39209i
$470$	0	0
$471$	2.69427	0.124145
$472$	0	0
$473$	3.51071	0.161423
$474$	0	0
$475$	− 6.17513i	− 0.283335i
$476$	0	0
$477$	8.11025i	0.371343i
$478$	0	0
$479$	10.4935	0.479460	0.239730	−	0.970840i	$-0.422941\pi$
$479$	10.4935	0.479460	0.239730	+	0.970840i	$0.422941\pi$
$480$	0	0
$481$	8.78623	0.400618
$482$	0	0
$483$	− 15.3288i	− 0.697487i
$484$	0	0
$485$	− 11.0790i	− 0.503070i
$486$	0	0
$487$	18.2646	0.827647	0.413824	−	0.910357i	$-0.364193\pi$
$487$	18.2646	0.827647	0.413824	+	0.910357i	$0.364193\pi$
$488$	0	0
$489$	−15.7795	−0.713574
$490$	0	0
$491$	41.6791i	1.88095i	0.339860	+	0.940476i	$0.389620\pi$
$491$	41.6791i	1.88095i	−0.339860	+	0.940476i	$0.610380\pi$
$492$	0	0
$493$	1.20390i	0.0542211i
$494$	0	0
$495$	−2.75746	−0.123939
$496$	0	0
$497$	37.6791	1.69014
$498$	0	0
$499$	− 13.0748i	− 0.585306i	−0.956219	−	0.292653i	$-0.905462\pi$
$499$	− 13.0748i	− 0.585306i	0.956219	−	0.292653i	$-0.0945381\pi$
$500$	0	0
$501$	0.478807i	0.0213915i
$502$	0	0
$503$	−18.4935	−0.824584	−0.412292	−	0.911052i	$-0.635272\pi$
$503$	−18.4935	−0.824584	−0.412292	+	0.911052i	$0.635272\pi$
$504$	0	0
$505$	35.7220	1.58961
$506$	0	0
$507$	− 8.87675i	− 0.394230i
$508$	0	0
$509$	− 11.8652i	− 0.525915i	−0.964807	−	0.262958i	$-0.915302\pi$
$509$	− 11.8652i	− 0.525915i	0.964807	−	0.262958i	$-0.0846980\pi$
$510$	0	0
$511$	−51.7079	−2.28742
$512$	0	0
$513$	5.37169	0.237166
$514$	0	0
$515$	40.3074i	1.77616i
$516$	0	0
$517$	− 5.38577i	− 0.236866i
$518$	0	0
$519$	4.13481	0.181498
$520$	0	0
$521$	24.3832	1.06825	0.534125	−	0.845406i	$-0.320641\pi$
$521$	24.3832	1.06825	0.534125	+	0.845406i	$0.320641\pi$
$522$	0	0
$523$	− 38.5510i	− 1.68572i	−0.538134	−	0.842860i	$-0.680870\pi$
$523$	− 38.5510i	− 1.68572i	0.538134	−	0.842860i	$-0.319130\pi$
$524$	0	0
$525$	23.6644i	1.03280i
$526$	0	0
$527$	1.61675	0.0704268
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	8.95885i	0.388781i
$532$	0	0
$533$	− 9.45738i	− 0.409645i
$534$	0	0
$535$	−28.0575	−1.21303
$536$	0	0
$537$	7.47208	0.322444
$538$	0	0
$539$	2.04285i	0.0879916i
$540$	0	0
$541$	0.857845i	0.0368817i	0.999830	+	0.0184408i	$0.00587023\pi$
$541$	0.857845i	0.0368817i	−0.999830	+	0.0184408i	$0.994130\pi$
$542$	0	0
$543$	1.50650	0.0646501
$544$	0	0
$545$	−27.4966	−1.17783
$546$	0	0
$547$	10.8291i	0.463018i	0.972833	+	0.231509i	$0.0743663\pi$
$547$	10.8291i	0.463018i	−0.972833	+	0.231509i	$0.925634\pi$
$548$	0	0
$549$	− 13.0418i	− 0.556609i
$550$	0	0
$551$	−6.12494	−0.260931
$552$	0	0
$553$	−35.1611	−1.49520
$554$	0	0
$555$	− 14.6858i	− 0.623379i
$556$	0	0
$557$	− 36.7722i	− 1.55809i	−0.626970	−	0.779043i	$-0.715705\pi$
$557$	− 36.7722i	− 1.55809i	0.626970	−	0.779043i	$-0.284295\pi$
$558$	0	0
$559$	16.4507	0.695789
$560$	0	0
$561$	0.110250	0.00465475
$562$	0	0
$563$	− 29.8469i	− 1.25790i	−0.777447	−	0.628949i	$-0.783486\pi$
$563$	− 29.8469i	− 1.25790i	0.777447	−	0.628949i	$-0.216514\pi$
$564$	0	0
$565$	45.1182i	1.89814i
$566$	0	0
$567$	−3.67850	−0.154482
$568$	0	0
$569$	36.2646	1.52029	0.760145	−	0.649753i	$-0.225128\pi$
$569$	36.2646	1.52029	0.760145	+	0.649753i	$0.225128\pi$
$570$	0	0
$571$	− 11.3570i	− 0.475276i	−0.971354	−	0.237638i	$-0.923627\pi$
$571$	− 11.3570i	− 0.475276i	0.971354	−	0.237638i	$-0.0763731\pi$
$572$	0	0
$573$	10.5953i	0.442626i
$574$	0	0
$575$	−24.7005	−1.03008
$576$	0	0
$577$	18.2113	0.758144	0.379072	−	0.925367i	$-0.376243\pi$
$577$	18.2113	0.758144	0.379072	+	0.925367i	$0.376243\pi$
$578$	0	0
$579$	− 10.1495i	− 0.421799i
$580$	0	0
$581$	2.43596i	0.101061i
$582$	0	0
$583$	−2.35386	−0.0974871
$584$	0	0
$585$	−12.9210	−0.534219
$586$	0	0
$587$	− 26.4464i	− 1.09156i	−0.837928	−	0.545781i	$-0.816233\pi$
$587$	− 26.4464i	− 1.09156i	0.837928	−	0.545781i	$-0.183767\pi$
$588$	0	0
$589$	8.22533i	0.338919i
$590$	0	0
$591$	31.3288	1.28870
$592$	0	0
$593$	−26.8929	−1.10436	−0.552179	−	0.833725i	$-0.686204\pi$
$593$	−26.8929	−1.10436	−0.552179	+	0.833725i	$0.686204\pi$
$594$	0	0
$595$	2.19656i	0.0900501i
$596$	0	0
$597$	− 21.0523i	− 0.861611i
$598$	0	0
$599$	34.5181	1.41037	0.705185	−	0.709024i	$-0.250864\pi$
$599$	34.5181	1.41037	0.705185	+	0.709024i	$0.250864\pi$
$600$	0	0
$601$	27.8715	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$601$	27.8715	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$602$	0	0
$603$	26.1249i	1.06389i
$604$	0	0
$605$	35.9718i	1.46246i
$606$	0	0
$607$	−27.5787	−1.11939	−0.559693	−	0.828700i	$-0.689081\pi$
$607$	−27.5787	−1.11939	−0.559693	+	0.828700i	$0.689081\pi$
$608$	0	0
$609$	23.4721	0.951137
$610$	0	0
$611$	− 25.2369i	− 1.02098i
$612$	0	0
$613$	− 2.67177i	− 0.107912i	−0.998543	−	0.0539559i	$-0.982817\pi$
$613$	− 2.67177i	− 0.107912i	0.998543	−	0.0539559i	$-0.0171830\pi$
$614$	0	0
$615$	−15.8077	−0.637426
$616$	0	0
$617$	−15.0832	−0.607226	−0.303613	−	0.952795i	$-0.598193\pi$
$617$	−15.0832	−0.607226	−0.303613	+	0.952795i	$0.598193\pi$
$618$	0	0
$619$	− 31.7220i	− 1.27501i	−0.770445	−	0.637507i	$-0.779966\pi$
$619$	− 31.7220i	− 1.27501i	0.770445	−	0.637507i	$-0.220034\pi$
$620$	0	0
$621$	− 21.4868i	− 0.862234i
$622$	0	0
$623$	28.4752	1.14084
$624$	0	0
$625$	−17.7434	−0.709735
$626$	0	0
$627$	0.560904i	0.0224003i
$628$	0	0
$629$	− 0.753250i	− 0.0300341i
$630$	0	0
$631$	−47.1800	−1.87820	−0.939102	−	0.343638i	$-0.888341\pi$
$631$	−47.1800	−1.87820	−0.939102	+	0.343638i	$0.888341\pi$
$632$	0	0
$633$	−21.0361	−0.836111
$634$	0	0
$635$	− 5.14637i	− 0.204227i
$636$	0	0
$637$	9.57246i	0.379275i
$638$	0	0
$639$	19.0017	0.751695
$640$	0	0
$641$	34.1067	1.34713	0.673566	−	0.739127i	$-0.264762\pi$
$641$	34.1067	1.34713	0.673566	+	0.739127i	$0.264762\pi$
$642$	0	0
$643$	− 21.6686i	− 0.854528i	−0.904127	−	0.427264i	$-0.859478\pi$
$643$	− 21.6686i	− 0.854528i	0.904127	−	0.427264i	$-0.140522\pi$
$644$	0	0
$645$	− 27.4966i	− 1.08268i
$646$	0	0
$647$	35.8427	1.40912	0.704561	−	0.709644i	$-0.251144\pi$
$647$	35.8427	1.40912	0.704561	+	0.709644i	$0.251144\pi$
$648$	0	0
$649$	−2.60015	−0.102065
$650$	0	0
$651$	− 31.5212i	− 1.23541i
$652$	0	0
$653$	21.7507i	0.851172i	0.904918	+	0.425586i	$0.139932\pi$
$653$	21.7507i	0.851172i	−0.904918	+	0.425586i	$0.860068\pi$
$654$	0	0
$655$	−11.9284	−0.466081
$656$	0	0
$657$	−26.0764	−1.01734
$658$	0	0
$659$	− 10.3012i	− 0.401276i	−0.979665	−	0.200638i	$-0.935699\pi$
$659$	− 10.3012i	− 0.401276i	0.979665	−	0.200638i	$-0.0643015\pi$
$660$	0	0
$661$	− 1.95715i	− 0.0761245i	−0.999275	−	0.0380622i	$-0.987881\pi$
$661$	− 1.95715i	− 0.0761245i	0.999275	−	0.0380622i	$-0.0121185\pi$
$662$	0	0
$663$	0.516614	0.0200636
$664$	0	0
$665$	−11.1751	−0.433353
$666$	0	0
$667$	24.4998i	0.948635i
$668$	0	0
$669$	− 21.0643i	− 0.814392i
$670$	0	0
$671$	3.78516	0.146124
$672$	0	0
$673$	−0.0428457	−0.00165158	−0.000825790	−	1.00000i	$-0.500263\pi$
$673$	−0.0428457	−0.00165158	−0.000825790		1.00000i	$0.500263\pi$
$674$	0	0
$675$	33.1709i	1.27675i
$676$	0	0
$677$	− 42.1543i	− 1.62012i	−0.586345	−	0.810061i	$-0.699434\pi$
$677$	− 42.1543i	− 1.62012i	0.586345	−	0.810061i	$-0.300566\pi$
$678$	0	0
$679$	−11.0790	−0.425172
$680$	0	0
$681$	−21.7732	−0.834352
$682$	0	0
$683$	− 28.8353i	− 1.10335i	−0.834058	−	0.551677i	$-0.813988\pi$
$683$	− 28.8353i	− 1.10335i	0.834058	−	0.551677i	$-0.186012\pi$
$684$	0	0
$685$	− 33.9431i	− 1.29690i
$686$	0	0
$687$	−10.3748	−0.395824
$688$	0	0
$689$	−11.0298	−0.420204
$690$	0	0
$691$	− 21.4103i	− 0.814487i	−0.913320	−	0.407244i	$-0.866490\pi$
$691$	− 21.4103i	− 0.814487i	0.913320	−	0.407244i	$-0.133510\pi$
$692$	0	0
$693$	2.75746i	0.104747i
$694$	0	0
$695$	13.0502	0.495022
$696$	0	0
$697$	−0.810789	−0.0307108
$698$	0	0
$699$	11.1892i	0.423215i
$700$	0	0
$701$	− 3.07054i	− 0.115973i	−0.998317	−	0.0579863i	$-0.981532\pi$
$701$	− 3.07054i	− 0.115973i	0.998317	−	0.0579863i	$-0.0184680\pi$
$702$	0	0
$703$	3.83221	0.144535
$704$	0	0
$705$	−42.1825	−1.58868
$706$	0	0
$707$	− 35.7220i	− 1.34346i
$708$	0	0
$709$	− 42.8500i	− 1.60927i	−0.593772	−	0.804634i	$-0.702362\pi$
$709$	− 42.8500i	− 1.60927i	0.593772	−	0.804634i	$-0.297638\pi$
$710$	0	0
$711$	−17.7318	−0.664995
$712$	0	0
$713$	32.9013	1.23216
$714$	0	0
$715$	− 3.75011i	− 0.140246i
$716$	0	0
$717$	− 14.8621i	− 0.555034i
$718$	0	0
$719$	34.5285	1.28770	0.643849	−	0.765153i	$-0.277337\pi$
$719$	34.5285	1.28770	0.643849	+	0.765153i	$0.277337\pi$
$720$	0	0
$721$	40.3074	1.50113
$722$	0	0
$723$	23.3007i	0.866562i
$724$	0	0
$725$	− 37.8223i	− 1.40469i
$726$	0	0
$727$	−44.0350	−1.63317	−0.816585	−	0.577226i	$-0.804135\pi$
$727$	−44.0350	−1.63317	−0.816585	+	0.577226i	$0.804135\pi$
$728$	0	0
$729$	−20.3288	−0.752920
$730$	0	0
$731$	− 1.41033i	− 0.0521628i
$732$	0	0
$733$	− 35.4721i	− 1.31019i	−0.755546	−	0.655096i	$-0.772628\pi$
$733$	− 35.4721i	− 1.31019i	0.755546	−	0.655096i	$-0.227372\pi$
$734$	0	0
$735$	16.0000	0.590169
$736$	0	0
$737$	−7.58233	−0.279299
$738$	0	0
$739$	− 34.4956i	− 1.26894i	−0.772948	−	0.634470i	$-0.781218\pi$
$739$	− 34.4956i	− 1.26894i	0.772948	−	0.634470i	$-0.218782\pi$
$740$	0	0
$741$	2.62831i	0.0965533i
$742$	0	0
$743$	2.40298	0.0881568	0.0440784	−	0.999028i	$-0.485965\pi$
$743$	2.40298	0.0881568	0.0440784	+	0.999028i	$0.485965\pi$
$744$	0	0
$745$	58.0252	2.12588
$746$	0	0
$747$	1.22846i	0.0449471i
$748$	0	0
$749$	28.0575i	1.02520i
$750$	0	0
$751$	46.3221	1.69032	0.845159	−	0.534515i	$-0.179506\pi$
$751$	46.3221	1.69032	0.845159	+	0.534515i	$0.179506\pi$
$752$	0	0
$753$	−6.93933	−0.252883
$754$	0	0
$755$	13.5640i	0.493646i
$756$	0	0
$757$	28.5798i	1.03875i	0.854546	+	0.519375i	$0.173835\pi$
$757$	28.5798i	1.03875i	−0.854546	+	0.519375i	$0.826165\pi$
$758$	0	0
$759$	2.24361	0.0814380
$760$	0	0
$761$	−21.5401	−0.780828	−0.390414	−	0.920639i	$-0.627668\pi$
$761$	−21.5401	−0.780828	−0.390414	+	0.920639i	$0.627668\pi$
$762$	0	0
$763$	27.4966i	0.995445i
$764$	0	0
$765$	1.10773i	0.0400501i
$766$	0	0
$767$	−12.1839	−0.439936
$768$	0	0
$769$	16.6472	0.600314	0.300157	−	0.953890i	$-0.402961\pi$
$769$	16.6472	0.600314	0.300157	+	0.953890i	$0.402961\pi$
$770$	0	0
$771$	− 16.1151i	− 0.580370i
$772$	0	0
$773$	38.8009i	1.39557i	0.716306	+	0.697786i	$0.245831\pi$
$773$	38.8009i	1.39557i	−0.716306	+	0.697786i	$0.754169\pi$
$774$	0	0
$775$	−50.7925	−1.82452
$776$	0	0
$777$	−14.6858	−0.526852
$778$	0	0
$779$	− 4.12494i	− 0.147791i
$780$	0	0
$781$	5.51492i	0.197339i
$782$	0	0
$783$	32.9013	1.17580
$784$	0	0
$785$	−7.85677	−0.280420
$786$	0	0
$787$	− 23.8652i	− 0.850702i	−0.905028	−	0.425351i	$-0.860151\pi$
$787$	− 23.8652i	− 0.850702i	0.905028	−	0.425351i	$-0.139849\pi$
$788$	0	0
$789$	− 9.47352i	− 0.337266i
$790$	0	0
$791$	45.1182	1.60422
$792$	0	0
$793$	17.7367	0.629847
$794$	0	0
$795$	18.4360i	0.653856i
$796$	0	0
$797$	− 2.01829i	− 0.0714914i	−0.999361	−	0.0357457i	$-0.988619\pi$
$797$	− 2.01829i	− 0.0714914i	0.999361	−	0.0357457i	$-0.0113806\pi$
$798$	0	0
$799$	−2.16358	−0.0765419
$800$	0	0
$801$	14.3601	0.507390
$802$	0	0
$803$	− 7.56825i	− 0.267078i
$804$	0	0
$805$	44.7005i	1.57549i
$806$	0	0
$807$	3.65600	0.128697
$808$	0	0
$809$	52.0533	1.83010	0.915049	−	0.403343i	$-0.132152\pi$
$809$	52.0533	1.83010	0.915049	+	0.403343i	$0.132152\pi$
$810$	0	0
$811$	− 34.1579i	− 1.19945i	−0.800207	−	0.599723i	$-0.795277\pi$
$811$	− 34.1579i	− 1.19945i	0.800207	−	0.599723i	$-0.204723\pi$
$812$	0	0
$813$	11.3486i	0.398012i
$814$	0	0
$815$	46.0147	1.61182
$816$	0	0
$817$	7.17513	0.251026
$818$	0	0
$819$	12.9210i	0.451498i
$820$	0	0
$821$	− 30.2787i	− 1.05673i	−0.849017	−	0.528366i	$-0.822805\pi$
$821$	− 30.2787i	− 1.05673i	0.849017	−	0.528366i	$-0.177195\pi$
$822$	0	0
$823$	−39.9284	−1.39182	−0.695908	−	0.718131i	$-0.744998\pi$
$823$	−39.9284	−1.39182	−0.695908	+	0.718131i	$0.744998\pi$
$824$	0	0
$825$	−3.46365	−0.120589
$826$	0	0
$827$	12.0330i	0.418428i	0.977870	+	0.209214i	$0.0670905\pi$
$827$	12.0330i	0.418428i	−0.977870	+	0.209214i	$0.932910\pi$
$828$	0	0
$829$	9.84208i	0.341829i	0.985286	+	0.170915i	$0.0546723\pi$
$829$	9.84208i	0.341829i	−0.985286	+	0.170915i	$0.945328\pi$
$830$	0	0
$831$	−14.6416	−0.507910
$832$	0	0
$833$	0.820654	0.0284340
$834$	0	0
$835$	− 1.39625i	− 0.0483192i
$836$	0	0
$837$	− 44.1839i	− 1.52722i
$838$	0	0
$839$	19.2713	0.665319	0.332660	−	0.943047i	$-0.392054\pi$
$839$	19.2713	0.665319	0.332660	+	0.943047i	$0.392054\pi$
$840$	0	0
$841$	−8.51492	−0.293618
$842$	0	0
$843$	33.8517i	1.16592i
$844$	0	0
$845$	25.8855i	0.890490i
$846$	0	0
$847$	35.9718	1.23601
$848$	0	0
$849$	−29.0313	−0.996351
$850$	0	0
$851$	− 15.3288i	− 0.525466i
$852$	0	0
$853$	− 27.4721i	− 0.940626i	−0.882500	−	0.470313i	$-0.844141\pi$
$853$	− 27.4721i	− 0.940626i	0.882500	−	0.470313i	$-0.155859\pi$
$854$	0	0
$855$	−5.63565	−0.192735
$856$	0	0
$857$	−0.292731	−0.00999950	−0.00499975	−	0.999988i	$-0.501591\pi$
$857$	−0.292731	−0.00999950	−0.00499975	+	0.999988i	$0.501591\pi$
$858$	0	0
$859$	23.9614i	0.817551i	0.912635	+	0.408776i	$0.134044\pi$
$859$	23.9614i	0.817551i	−0.912635	+	0.408776i	$0.865956\pi$
$860$	0	0
$861$	15.8077i	0.538723i
$862$	0	0
$863$	6.12494	0.208495	0.104248	−	0.994551i	$-0.466757\pi$
$863$	6.12494	0.208495	0.104248	+	0.994551i	$0.466757\pi$
$864$	0	0
$865$	−12.0575	−0.409969
$866$	0	0
$867$	19.4439i	0.660350i
$868$	0	0
$869$	− 5.14637i	− 0.174578i
$870$	0	0
$871$	−35.5296	−1.20388
$872$	0	0
$873$	−5.58715	−0.189096
$874$	0	0
$875$	− 13.1323i	− 0.443952i
$876$	0	0
$877$	22.3503i	0.754715i	0.926068	+	0.377357i	$0.123167\pi$
$877$	22.3503i	0.754715i	−0.926068	+	0.377357i	$0.876833\pi$
$878$	0	0
$879$	19.2995	0.650955
$880$	0	0
$881$	5.70306	0.192141	0.0960705	−	0.995375i	$-0.469373\pi$
$881$	5.70306	0.192141	0.0960705	+	0.995375i	$0.469373\pi$
$882$	0	0
$883$	− 8.29694i	− 0.279214i	−0.990207	−	0.139607i	$-0.955416\pi$
$883$	− 8.29694i	− 0.279214i	0.990207	−	0.139607i	$-0.0445840\pi$
$884$	0	0
$885$	20.3650i	0.684561i
$886$	0	0
$887$	25.4783	0.855479	0.427740	−	0.903902i	$-0.359310\pi$
$887$	25.4783	0.855479	0.427740	+	0.903902i	$0.359310\pi$
$888$	0	0
$889$	−5.14637	−0.172604
$890$	0	0
$891$	− 0.538405i	− 0.0180373i
$892$	0	0
$893$	− 11.0073i	− 0.368347i
$894$	0	0
$895$	−21.7894	−0.728338
$896$	0	0
$897$	10.5132	0.351027
$898$	0	0
$899$	50.3797i	1.68026i
$900$	0	0
$901$	0.945597i	0.0315024i
$902$	0	0
$903$	−27.4966	−0.915031
$904$	0	0
$905$	−4.39312	−0.146032
$906$	0	0
$907$	39.9964i	1.32806i	0.747706	+	0.664029i	$0.231155\pi$
$907$	39.9964i	1.32806i	−0.747706	+	0.664029i	$0.768845\pi$
$908$	0	0
$909$	− 18.0147i	− 0.597510i
$910$	0	0
$911$	19.3864	0.642300	0.321150	−	0.947028i	$-0.395931\pi$
$911$	19.3864	0.642300	0.321150	+	0.947028i	$0.395931\pi$
$912$	0	0
$913$	−0.356541	−0.0117998
$914$	0	0
$915$	− 29.6461i	− 0.980071i
$916$	0	0
$917$	11.9284i	0.393910i
$918$	0	0
$919$	38.1213	1.25751	0.628754	−	0.777605i	$-0.283565\pi$
$919$	38.1213	1.25751	0.628754	+	0.777605i	$0.283565\pi$
$920$	0	0
$921$	−20.3931	−0.671976
$922$	0	0
$923$	25.8421i	0.850602i
$924$	0	0
$925$	23.6644i	0.778081i
$926$	0	0
$927$	20.3272	0.667631
$928$	0	0
$929$	−16.8500	−0.552832	−0.276416	−	0.961038i	$-0.589147\pi$
$929$	−16.8500	−0.552832	−0.276416	+	0.961038i	$0.589147\pi$
$930$	0	0
$931$	4.17513i	0.136835i
$932$	0	0
$933$	− 23.2959i	− 0.762672i
$934$	0	0
$935$	−0.321500	−0.0105142
$936$	0	0
$937$	39.4973	1.29032	0.645159	−	0.764048i	$-0.276791\pi$
$937$	39.4973	1.29032	0.645159	+	0.764048i	$0.276791\pi$
$938$	0	0
$939$	40.0000i	1.30535i
$940$	0	0
$941$	− 55.9718i	− 1.82463i	−0.409489	−	0.912315i	$-0.634293\pi$
$941$	− 55.9718i	− 1.82463i	0.409489	−	0.912315i	$-0.365707\pi$
$942$	0	0
$943$	−16.4998	−0.537306
$944$	0	0
$945$	60.0294	1.95276
$946$	0	0
$947$	− 13.5725i	− 0.441046i	−0.975382	−	0.220523i	$-0.929224\pi$
$947$	− 13.5725i	− 0.441046i	0.975382	−	0.220523i	$-0.0707764\pi$
$948$	0	0
$949$	− 35.4637i	− 1.15120i
$950$	0	0
$951$	14.5917	0.473169
$952$	0	0
$953$	−26.0246	−0.843018	−0.421509	−	0.906824i	$-0.638499\pi$
$953$	−26.0246	−0.843018	−0.421509	+	0.906824i	$0.638499\pi$
$954$	0	0
$955$	− 30.8971i	− 0.999807i
$956$	0	0
$957$	3.43550i	0.111054i
$958$	0	0
$959$	−33.9431	−1.09608
$960$	0	0
$961$	36.6560	1.18245
$962$	0	0
$963$	14.1495i	0.455961i
$964$	0	0
$965$	29.5970i	0.952762i
$966$	0	0
$967$	−13.0277	−0.418942	−0.209471	−	0.977815i	$-0.567174\pi$
$967$	−13.0277	−0.418942	−0.209471	+	0.977815i	$0.567174\pi$
$968$	0	0
$969$	0.225327	0.00723854
$970$	0	0
$971$	− 60.6148i	− 1.94522i	−0.232437	−	0.972612i	$-0.574670\pi$
$971$	− 60.6148i	− 1.94522i	0.232437	−	0.972612i	$-0.425330\pi$
$972$	0	0
$973$	− 13.0502i	− 0.418370i
$974$	0	0
$975$	−16.2302	−0.519781
$976$	0	0
$977$	−62.1396	−1.98802	−0.994012	−	0.109275i	$-0.965147\pi$
$977$	−62.1396	−1.98802	−0.994012	+	0.109275i	$0.965147\pi$
$978$	0	0
$979$	4.16779i	0.133203i
$980$	0	0
$981$	13.8666i	0.442728i
$982$	0	0
$983$	6.37842	0.203440	0.101720	−	0.994813i	$-0.467565\pi$
$983$	6.37842	0.203440	0.101720	+	0.994813i	$0.467565\pi$
$984$	0	0
$985$	−91.3582	−2.91092
$986$	0	0
$987$	42.1825i	1.34268i
$988$	0	0
$989$	− 28.7005i	− 0.912624i
$990$	0	0
$991$	14.1249	0.448694	0.224347	−	0.974509i	$-0.427975\pi$
$991$	14.1249	0.448694	0.224347	+	0.974509i	$0.427975\pi$
$992$	0	0
$993$	−21.1625	−0.671572
$994$	0	0
$995$	61.3906i	1.94621i
$996$	0	0
$997$	− 13.1568i	− 0.416682i	−0.978056	−	0.208341i	$-0.933194\pi$
$997$	− 13.1568i	− 0.416682i	0.978056	−	0.208341i	$-0.0668063\pi$
$998$	0	0
$999$	−20.5855	−0.651295

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.h.1217.3	yes	6
4.3	odd	2		2432.2.c.e.1217.4	yes	6
8.3	odd	2		2432.2.c.e.1217.3	✓	6
8.5	even	2	inner	2432.2.c.h.1217.4	yes	6
16.3	odd	4		4864.2.a.bb.1.2		3
16.5	even	4		4864.2.a.ba.1.2		3
16.11	odd	4		4864.2.a.bg.1.2		3
16.13	even	4		4864.2.a.bh.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.e.1217.3	✓	6	8.3	odd	2
2432.2.c.e.1217.4	yes	6	4.3	odd	2
2432.2.c.h.1217.3	yes	6	1.1	even	1	trivial
2432.2.c.h.1217.4	yes	6	8.5	even	2	inner
4864.2.a.ba.1.2		3	16.5	even	4
4864.2.a.bb.1.2		3	16.3	odd	4
4864.2.a.bg.1.2		3	16.11	odd	4
4864.2.a.bh.1.2		3	16.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.14637i q^{3} +3.34292i q^{5} +3.34292 q^{7} +1.68585 q^{9} +O(q^{10})\)	\(q-1.14637i q^{3} +3.34292i q^{5} +3.34292 q^{7} +1.68585 q^{9} +0.489289i q^{11} +2.29273i q^{13} +3.83221 q^{15} +0.196558 q^{17} +1.00000i q^{19} -3.83221i q^{21} +4.00000 q^{23} -6.17513 q^{25} -5.37169i q^{27} +6.12494i q^{29} +8.22533 q^{31} +0.560904 q^{33} +11.1751i q^{35} -3.83221i q^{37} +2.62831 q^{39} -4.12494 q^{41} -7.17513i q^{43} +5.63565i q^{45} -11.0073 q^{47} +4.17513 q^{49} -0.225327i q^{51} +4.81079i q^{53} -1.63565 q^{55} +1.14637 q^{57} +5.31415i q^{59} -7.73604i q^{61} +5.63565 q^{63} -7.66442 q^{65} +15.4966i q^{67} -4.58546i q^{69} +11.2713 q^{71} -15.4679 q^{73} +7.07896i q^{75} +1.63565i q^{77} -10.5181 q^{79} -1.10038 q^{81} +0.728692i q^{83} +0.657077i q^{85} +7.02142 q^{87} +8.51806 q^{89} +7.66442i q^{91} -9.42923i q^{93} -3.34292 q^{95} -3.31415 q^{97} +0.824865i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 8 q^{7} - 14 q^{9}+O(q^{10}) \) 6 * q + 8 * q^7 - 14 * q^9	\( 6 q + 8 q^{7} - 14 q^{9} - 4 q^{15} - 8 q^{17} + 24 q^{23} + 2 q^{25} + 4 q^{31} + 12 q^{33} + 64 q^{39} + 8 q^{41} - 14 q^{49} + 8 q^{55} + 4 q^{57} + 16 q^{63} + 8 q^{65} + 32 q^{71} - 48 q^{73} - 12 q^{79} + 6 q^{81} + 72 q^{87} - 8 q^{95} - 44 q^{97}+O(q^{100}) \) 6 * q + 8 * q^7 - 14 * q^9 - 4 * q^15 - 8 * q^17 + 24 * q^23 + 2 * q^25 + 4 * q^31 + 12 * q^33 + 64 * q^39 + 8 * q^41 - 14 * q^49 + 8 * q^55 + 4 * q^57 + 16 * q^63 + 8 * q^65 + 32 * q^71 - 48 * q^73 - 12 * q^79 + 6 * q^81 + 72 * q^87 - 8 * q^95 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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