LMFDB - Embedded newform 7920.2.a.cl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7920,2,Mod(1,7920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7920.a (trivial)

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:	yes
Analytic conductor:	$63.2415184009$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3960)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.85577$ of defining polynomial
Character	$\chi$	$=$	7920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.29966 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -1.29966 q^{7} -1.00000 q^{11} -5.01121 q^{13} +3.29966 q^{17} -5.71155 q^{19} +5.71155 q^{23} +1.00000 q^{25} +2.00000 q^{29} -3.71155 q^{31} -1.29966 q^{35} +2.00000 q^{37} +2.00000 q^{41} +4.41188 q^{43} -9.71155 q^{47} -5.31087 q^{49} +4.59933 q^{53} -1.00000 q^{55} +3.71155 q^{59} +15.1346 q^{61} -5.01121 q^{65} +1.40067 q^{67} -16.0224 q^{71} -5.89899 q^{73} +1.29966 q^{77} +14.0224 q^{79} -7.52410 q^{83} +3.29966 q^{85} +16.5353 q^{89} +6.51289 q^{91} -5.71155 q^{95} -6.82376 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5}+O(q^{10}) $ 3 * q + 3 * q^5	$ 3 q + 3 q^{5} - 3 q^{11} + 4 q^{13} + 6 q^{17} - 2 q^{19} + 2 q^{23} + 3 q^{25} + 6 q^{29} + 4 q^{31} + 6 q^{37} + 6 q^{41} + 2 q^{43} - 14 q^{47} + 7 q^{49} + 6 q^{53} - 3 q^{55} - 4 q^{59} + 4 q^{65} + 12 q^{67} - 10 q^{71} - 6 q^{73} + 4 q^{79} - 4 q^{83} + 6 q^{85} + 12 q^{89} + 20 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 3 * q^11 + 4 * q^13 + 6 * q^17 - 2 * q^19 + 2 * q^23 + 3 * q^25 + 6 * q^29 + 4 * q^31 + 6 * q^37 + 6 * q^41 + 2 * q^43 - 14 * q^47 + 7 * q^49 + 6 * q^53 - 3 * q^55 - 4 * q^59 + 4 * q^65 + 12 * q^67 - 10 * q^71 - 6 * q^73 + 4 * q^79 - 4 * q^83 + 6 * q^85 + 12 * q^89 + 20 * q^91 - 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.29966	−0.491227	−0.245613	−	0.969368i	$-0.578989\pi$
$7$	−1.29966	−0.491227	−0.245613	+	0.969368i	$0.578989\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.01121	−1.38986	−0.694930	−	0.719078i	$-0.744565\pi$
$13$	−5.01121	−1.38986	−0.694930	+	0.719078i	$0.744565\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.29966	0.800286	0.400143	−	0.916453i	$-0.368960\pi$
$17$	3.29966	0.800286	0.400143	+	0.916453i	$0.368960\pi$
$18$	0	0
$19$	−5.71155	−1.31032	−0.655159	−	0.755491i	$-0.727398\pi$
$19$	−5.71155	−1.31032	−0.655159	+	0.755491i	$0.727398\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.71155	1.19094	0.595470	−	0.803378i	$-0.296966\pi$
$23$	5.71155	1.19094	0.595470	+	0.803378i	$0.296966\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−3.71155	−0.666613	−0.333307	−	0.942818i	$-0.608164\pi$
$31$	−3.71155	−0.666613	−0.333307	+	0.942818i	$0.608164\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.29966	−0.219683
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.41188	0.672806	0.336403	−	0.941718i	$-0.390790\pi$
$43$	4.41188	0.672806	0.336403	+	0.941718i	$0.390790\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.71155	−1.41657	−0.708287	−	0.705924i	$-0.750532\pi$
$47$	−9.71155	−1.41657	−0.708287	+	0.705924i	$0.750532\pi$
$48$	0	0
$49$	−5.31087	−0.758696
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.59933	0.631766	0.315883	−	0.948798i	$-0.397699\pi$
$53$	4.59933	0.631766	0.315883	+	0.948798i	$0.397699\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.71155	0.483202	0.241601	−	0.970376i	$-0.422327\pi$
$59$	3.71155	0.483202	0.241601	+	0.970376i	$0.422327\pi$
$60$	0	0
$61$	15.1346	1.93779	0.968896	−	0.247469i	$-0.0795990\pi$
$61$	15.1346	1.93779	0.968896	+	0.247469i	$0.0795990\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.01121	−0.621564
$66$	0	0
$67$	1.40067	0.171119	0.0855596	−	0.996333i	$-0.472732\pi$
$67$	1.40067	0.171119	0.0855596	+	0.996333i	$0.472732\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−16.0224	−1.90151	−0.950756	−	0.309942i	$-0.899691\pi$
$71$	−16.0224	−1.90151	−0.950756	+	0.309942i	$0.899691\pi$
$72$	0	0
$73$	−5.89899	−0.690425	−0.345212	−	0.938525i	$-0.612193\pi$
$73$	−5.89899	−0.690425	−0.345212	+	0.938525i	$0.612193\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.29966	0.148110
$78$	0	0
$79$	14.0224	1.57765	0.788823	−	0.614621i	$-0.210691\pi$
$79$	14.0224	1.57765	0.788823	+	0.614621i	$0.210691\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.52410	−0.825877	−0.412939	−	0.910759i	$-0.635498\pi$
$83$	−7.52410	−0.825877	−0.412939	+	0.910759i	$0.635498\pi$
$84$	0	0
$85$	3.29966	0.357899
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.5353	1.75274	0.876370	−	0.481639i	$-0.159958\pi$
$89$	16.5353	1.75274	0.876370	+	0.481639i	$0.159958\pi$
$90$	0	0
$91$	6.51289	0.682736
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.71155	−0.585992
$96$	0	0
$97$	−6.82376	−0.692848	−0.346424	−	0.938078i	$-0.612604\pi$
$97$	−6.82376	−0.692848	−0.346424	+	0.938078i	$0.612604\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.40067	0.338379	0.169190	−	0.985583i	$-0.445885\pi$
$101$	3.40067	0.338379	0.169190	+	0.985583i	$0.445885\pi$
$102$	0	0
$103$	5.19866	0.512239	0.256119	−	0.966645i	$-0.417556\pi$
$103$	5.19866	0.512239	0.256119	+	0.966645i	$0.417556\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.12343	0.205279	0.102640	−	0.994719i	$-0.467271\pi$
$107$	2.12343	0.205279	0.102640	+	0.994719i	$0.467271\pi$
$108$	0	0
$109$	18.3109	1.75386	0.876932	−	0.480615i	$-0.159586\pi$
$109$	18.3109	1.75386	0.876932	+	0.480615i	$0.159586\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.68913	0.158900	0.0794498	−	0.996839i	$-0.474684\pi$
$113$	1.68913	0.158900	0.0794498	+	0.996839i	$0.474684\pi$
$114$	0	0
$115$	5.71155	0.532604
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.28845	−0.393122
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.72275	−0.419077	−0.209538	−	0.977800i	$-0.567196\pi$
$127$	−4.72275	−0.419077	−0.209538	+	0.977800i	$0.567196\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.0224	1.22514	0.612572	−	0.790415i	$-0.290135\pi$
$131$	14.0224	1.22514	0.612572	+	0.790415i	$0.290135\pi$
$132$	0	0
$133$	7.42309	0.643664
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.31087	−0.197431	−0.0987156	−	0.995116i	$-0.531473\pi$
$137$	−2.31087	−0.197431	−0.0987156	+	0.995116i	$0.531473\pi$
$138$	0	0
$139$	−3.11222	−0.263975	−0.131987	−	0.991251i	$-0.542136\pi$
$139$	−3.11222	−0.263975	−0.131987	+	0.991251i	$0.542136\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.01121	0.419058
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.02242	−0.329529	−0.164765	−	0.986333i	$-0.552686\pi$
$149$	−4.02242	−0.329529	−0.164765	+	0.986333i	$0.552686\pi$
$150$	0	0
$151$	6.59933	0.537046	0.268523	−	0.963273i	$-0.413465\pi$
$151$	6.59933	0.537046	0.268523	+	0.963273i	$0.413465\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.71155	−0.298118
$156$	0	0
$157$	21.2211	1.69363	0.846813	−	0.531891i	$-0.178518\pi$
$157$	21.2211	1.69363	0.846813	+	0.531891i	$0.178518\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.42309	−0.585021
$162$	0	0
$163$	19.4231	1.52133	0.760667	−	0.649142i	$-0.224872\pi$
$163$	19.4231	1.52133	0.760667	+	0.649142i	$0.224872\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.49832	0.193326	0.0966629	−	0.995317i	$-0.469183\pi$
$167$	2.49832	0.193326	0.0966629	+	0.995317i	$0.469183\pi$
$168$	0	0
$169$	12.1122	0.931709
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.7228	1.42346	0.711732	−	0.702451i	$-0.247911\pi$
$173$	18.7228	1.42346	0.711732	+	0.702451i	$0.247911\pi$
$174$	0	0
$175$	−1.29966	−0.0982454
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.19866	−0.687540	−0.343770	−	0.939054i	$-0.611704\pi$
$179$	−9.19866	−0.687540	−0.343770	+	0.939054i	$0.611704\pi$
$180$	0	0
$181$	−19.7340	−1.46681	−0.733407	−	0.679790i	$-0.762071\pi$
$181$	−19.7340	−1.46681	−0.733407	+	0.679790i	$0.762071\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	−3.29966	−0.241295
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.68913	−0.266936	−0.133468	−	0.991053i	$-0.542611\pi$
$191$	−3.68913	−0.266936	−0.133468	+	0.991053i	$0.542611\pi$
$192$	0	0
$193$	−13.5241	−0.973486	−0.486743	−	0.873545i	$-0.661815\pi$
$193$	−13.5241	−0.973486	−0.486743	+	0.873545i	$0.661815\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.1458	1.57783	0.788913	−	0.614505i	$-0.210644\pi$
$197$	22.1458	1.57783	0.788913	+	0.614505i	$0.210644\pi$
$198$	0	0
$199$	5.19866	0.368523	0.184261	−	0.982877i	$-0.441011\pi$
$199$	5.19866	0.368523	0.184261	+	0.982877i	$0.441011\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.59933	−0.182437
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.71155	0.395076
$210$	0	0
$211$	18.9102	1.30183	0.650916	−	0.759150i	$-0.274385\pi$
$211$	18.9102	1.30183	0.650916	+	0.759150i	$0.274385\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.41188	0.300888
$216$	0	0
$217$	4.82376	0.327458
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−16.5353	−1.11229
$222$	0	0
$223$	−9.44551	−0.632518	−0.316259	−	0.948673i	$-0.602427\pi$
$223$	−9.44551	−0.632518	−0.316259	+	0.948673i	$0.602427\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.32208	0.485984	0.242992	−	0.970028i	$-0.421871\pi$
$227$	7.32208	0.485984	0.242992	+	0.970028i	$0.421871\pi$
$228$	0	0
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.4983	1.34289	0.671445	−	0.741055i	$-0.265674\pi$
$233$	20.4983	1.34289	0.671445	+	0.741055i	$0.265674\pi$
$234$	0	0
$235$	−9.71155	−0.633511
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.59933	−0.168137	−0.0840683	−	0.996460i	$-0.526791\pi$
$239$	−2.59933	−0.168137	−0.0840683	+	0.996460i	$0.526791\pi$
$240$	0	0
$241$	−2.82376	−0.181894	−0.0909472	−	0.995856i	$-0.528989\pi$
$241$	−2.82376	−0.181894	−0.0909472	+	0.995856i	$0.528989\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.31087	−0.339299
$246$	0	0
$247$	28.6217	1.82116
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−30.5577	−1.92879	−0.964393	−	0.264472i	$-0.914802\pi$
$251$	−30.5577	−1.92879	−0.964393	+	0.264472i	$0.914802\pi$
$252$	0	0
$253$	−5.71155	−0.359082
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.9102	0.805316	0.402658	−	0.915350i	$-0.368086\pi$
$257$	12.9102	0.805316	0.402658	+	0.915350i	$0.368086\pi$
$258$	0	0
$259$	−2.59933	−0.161514
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.72275	0.537868	0.268934	−	0.963159i	$-0.413329\pi$
$263$	8.72275	0.537868	0.268934	+	0.963159i	$0.413329\pi$
$264$	0	0
$265$	4.59933	0.282534
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.7340	−1.20320	−0.601600	−	0.798797i	$-0.705470\pi$
$269$	−19.7340	−1.20320	−0.601600	+	0.798797i	$0.705470\pi$
$270$	0	0
$271$	6.59933	0.400881	0.200440	−	0.979706i	$-0.435763\pi$
$271$	6.59933	0.400881	0.200440	+	0.979706i	$0.435763\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	4.38946	0.263737	0.131869	−	0.991267i	$-0.457902\pi$
$277$	4.38946	0.263737	0.131869	+	0.991267i	$0.457902\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.19866	0.429436	0.214718	−	0.976676i	$-0.431117\pi$
$281$	7.19866	0.429436	0.214718	+	0.976676i	$0.431117\pi$
$282$	0	0
$283$	−13.0336	−0.774769	−0.387384	−	0.921918i	$-0.626621\pi$
$283$	−13.0336	−0.774769	−0.387384	+	0.921918i	$0.626621\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.59933	−0.153433
$288$	0	0
$289$	−6.11222	−0.359542
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.3445	1.36380	0.681900	−	0.731445i	$-0.261154\pi$
$293$	23.3445	1.36380	0.681900	+	0.731445i	$0.261154\pi$
$294$	0	0
$295$	3.71155	0.216095
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−28.6217	−1.65524
$300$	0	0
$301$	−5.73396	−0.330500
$302$	0	0
$303$	0	0
$304$	0	0
$305$	15.1346	0.866607
$306$	0	0
$307$	1.81255	0.103448	0.0517239	−	0.998661i	$-0.483528\pi$
$307$	1.81255	0.103448	0.0517239	+	0.998661i	$0.483528\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−32.6442	−1.85108	−0.925540	−	0.378649i	$-0.876389\pi$
$311$	−32.6442	−1.85108	−0.925540	+	0.378649i	$0.876389\pi$
$312$	0	0
$313$	−2.57691	−0.145656	−0.0728278	−	0.997345i	$-0.523202\pi$
$313$	−2.57691	−0.145656	−0.0728278	+	0.997345i	$0.523202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.4231	−1.20324	−0.601620	−	0.798782i	$-0.705478\pi$
$317$	−21.4231	−1.20324	−0.601620	+	0.798782i	$0.705478\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−18.8462	−1.04863
$324$	0	0
$325$	−5.01121	−0.277972
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.6217	0.695859
$330$	0	0
$331$	12.2885	0.675435	0.337717	−	0.941248i	$-0.390345\pi$
$331$	12.2885	0.675435	0.337717	+	0.941248i	$0.390345\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.40067	0.0765269
$336$	0	0
$337$	20.1234	1.09619	0.548096	−	0.836415i	$-0.315353\pi$
$337$	20.1234	1.09619	0.548096	+	0.836415i	$0.315353\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.71155	0.200991
$342$	0	0
$343$	16.0000	0.863919
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.3221	−0.822532	−0.411266	−	0.911515i	$-0.634913\pi$
$347$	−15.3221	−0.822532	−0.411266	+	0.911515i	$0.634913\pi$
$348$	0	0
$349$	20.2885	1.08602	0.543008	−	0.839727i	$-0.317285\pi$
$349$	20.2885	1.08602	0.543008	+	0.839727i	$0.317285\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.3333	−0.656435	−0.328217	−	0.944602i	$-0.606448\pi$
$353$	−12.3333	−0.656435	−0.328217	+	0.944602i	$0.606448\pi$
$354$	0	0
$355$	−16.0224	−0.850382
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.59933	−0.348299	−0.174150	−	0.984719i	$-0.555718\pi$
$359$	−6.59933	−0.348299	−0.174150	+	0.984719i	$0.555718\pi$
$360$	0	0
$361$	13.6217	0.716934
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.89899	−0.308767
$366$	0	0
$367$	10.0224	0.523166	0.261583	−	0.965181i	$-0.415755\pi$
$367$	10.0224	0.523166	0.261583	+	0.965181i	$0.415755\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.97758	−0.310341
$372$	0	0
$373$	35.0336	1.81397	0.906986	−	0.421160i	$-0.138377\pi$
$373$	35.0336	1.81397	0.906986	+	0.421160i	$0.138377\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.0224	−0.516181
$378$	0	0
$379$	−20.6217	−1.05927	−0.529634	−	0.848226i	$-0.677671\pi$
$379$	−20.6217	−1.05927	−0.529634	+	0.848226i	$0.677671\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	28.9326	1.47839	0.739194	−	0.673493i	$-0.235207\pi$
$383$	28.9326	1.47839	0.739194	+	0.673493i	$0.235207\pi$
$384$	0	0
$385$	1.29966	0.0662370
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.91356	0.0970214	0.0485107	−	0.998823i	$-0.484553\pi$
$389$	1.91356	0.0970214	0.0485107	+	0.998823i	$0.484553\pi$
$390$	0	0
$391$	18.8462	0.953092
$392$	0	0
$393$	0	0
$394$	0	0
$395$	14.0224	0.705544
$396$	0	0
$397$	−14.2469	−0.715029	−0.357514	−	0.933908i	$-0.616376\pi$
$397$	−14.2469	−0.715029	−0.357514	+	0.933908i	$0.616376\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.0864	−0.603568	−0.301784	−	0.953376i	$-0.597582\pi$
$401$	−12.0864	−0.603568	−0.301784	+	0.953376i	$0.597582\pi$
$402$	0	0
$403$	18.5993	0.926499
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	25.4231	1.25709	0.628545	−	0.777773i	$-0.283651\pi$
$409$	25.4231	1.25709	0.628545	+	0.777773i	$0.283651\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.82376	−0.237362
$414$	0	0
$415$	−7.52410	−0.369343
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	−8.31087	−0.405047	−0.202524	−	0.979277i	$-0.564914\pi$
$421$	−8.31087	−0.405047	−0.202524	+	0.979277i	$0.564914\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.29966	0.160057
$426$	0	0
$427$	−19.6699	−0.951895
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.2693	1.26535	0.632673	−	0.774419i	$-0.281958\pi$
$431$	26.2693	1.26535	0.632673	+	0.774419i	$0.281958\pi$
$432$	0	0
$433$	8.02242	0.385533	0.192766	−	0.981245i	$-0.438254\pi$
$433$	8.02242	0.385533	0.192766	+	0.981245i	$0.438254\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−32.6217	−1.56051
$438$	0	0
$439$	−29.1987	−1.39358	−0.696788	−	0.717277i	$-0.745388\pi$
$439$	−29.1987	−1.39358	−0.696788	+	0.717277i	$0.745388\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.0448	0.952359	0.476179	−	0.879348i	$-0.342021\pi$
$443$	20.0448	0.952359	0.476179	+	0.879348i	$0.342021\pi$
$444$	0	0
$445$	16.5353	0.783849
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.3367	−0.723781	−0.361891	−	0.932221i	$-0.617869\pi$
$449$	−15.3367	−0.723781	−0.361891	+	0.932221i	$0.617869\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.51289	0.305329
$456$	0	0
$457$	−33.1201	−1.54929	−0.774646	−	0.632395i	$-0.782072\pi$
$457$	−33.1201	−1.54929	−0.774646	+	0.632395i	$0.782072\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.6442	1.14779	0.573897	−	0.818928i	$-0.305431\pi$
$461$	24.6442	1.14779	0.573897	+	0.818928i	$0.305431\pi$
$462$	0	0
$463$	36.2469	1.68453	0.842267	−	0.539061i	$-0.181221\pi$
$463$	36.2469	1.68453	0.842267	+	0.539061i	$0.181221\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.59933	−0.305380	−0.152690	−	0.988274i	$-0.548794\pi$
$467$	−6.59933	−0.305380	−0.152690	+	0.988274i	$0.548794\pi$
$468$	0	0
$469$	−1.82040	−0.0840584
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.41188	−0.202859
$474$	0	0
$475$	−5.71155	−0.262064
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11.6251	−0.531165	−0.265582	−	0.964088i	$-0.585564\pi$
$479$	−11.6251	−0.531165	−0.265582	+	0.964088i	$0.585564\pi$
$480$	0	0
$481$	−10.0224	−0.456983
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.82376	−0.309851
$486$	0	0
$487$	40.8686	1.85193	0.925966	−	0.377606i	$-0.123253\pi$
$487$	40.8686	1.85193	0.925966	+	0.377606i	$0.123253\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.8204	0.623706	0.311853	−	0.950130i	$-0.399050\pi$
$491$	13.8204	0.623706	0.311853	+	0.950130i	$0.399050\pi$
$492$	0	0
$493$	6.59933	0.297219
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.8238	0.934073
$498$	0	0
$499$	2.84618	0.127413	0.0637063	−	0.997969i	$-0.479708\pi$
$499$	2.84618	0.127413	0.0637063	+	0.997969i	$0.479708\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.70034	0.120402	0.0602010	−	0.998186i	$-0.480826\pi$
$503$	2.70034	0.120402	0.0602010	+	0.998186i	$0.480826\pi$
$504$	0	0
$505$	3.40067	0.151328
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−30.7082	−1.36112	−0.680558	−	0.732694i	$-0.738263\pi$
$509$	−30.7082	−1.36112	−0.680558	+	0.732694i	$0.738263\pi$
$510$	0	0
$511$	7.66671	0.339155
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.19866	0.229080
$516$	0	0
$517$	9.71155	0.427113
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−34.9326	−1.53043	−0.765213	−	0.643777i	$-0.777366\pi$
$521$	−34.9326	−1.53043	−0.765213	+	0.643777i	$0.777366\pi$
$522$	0	0
$523$	33.6105	1.46969	0.734843	−	0.678237i	$-0.237256\pi$
$523$	33.6105	1.46969	0.734843	+	0.678237i	$0.237256\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.2469	−0.533481
$528$	0	0
$529$	9.62175	0.418337
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.0224	−0.434119
$534$	0	0
$535$	2.12343	0.0918037
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.31087	0.228756
$540$	0	0
$541$	17.7340	0.762443	0.381221	−	0.924484i	$-0.375504\pi$
$541$	17.7340	0.762443	0.381221	+	0.924484i	$0.375504\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.3109	0.784352
$546$	0	0
$547$	41.0785	1.75639	0.878194	−	0.478304i	$-0.158748\pi$
$547$	41.0785	1.75639	0.878194	+	0.478304i	$0.158748\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.4231	−0.486640
$552$	0	0
$553$	−18.2244	−0.774982
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−4.49832	−0.190600	−0.0953000	−	0.995449i	$-0.530381\pi$
$557$	−4.49832	−0.190600	−0.0953000	+	0.995449i	$0.530381\pi$
$558$	0	0
$559$	−22.1089	−0.935105
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.6779	0.702890	0.351445	−	0.936208i	$-0.385690\pi$
$563$	16.6779	0.702890	0.351445	+	0.936208i	$0.385690\pi$
$564$	0	0
$565$	1.68913	0.0710621
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.7980	−1.08151	−0.540754	−	0.841181i	$-0.681861\pi$
$569$	−25.7980	−1.08151	−0.540754	+	0.841181i	$0.681861\pi$
$570$	0	0
$571$	−35.1571	−1.47128	−0.735638	−	0.677374i	$-0.763118\pi$
$571$	−35.1571	−1.47128	−0.735638	+	0.677374i	$0.763118\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.71155	0.238188
$576$	0	0
$577$	−15.9776	−0.665155	−0.332578	−	0.943076i	$-0.607918\pi$
$577$	−15.9776	−0.665155	−0.332578	+	0.943076i	$0.607918\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.77880	0.405693
$582$	0	0
$583$	−4.59933	−0.190485
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.57355	−0.230045	−0.115023	−	0.993363i	$-0.536694\pi$
$587$	−5.57355	−0.230045	−0.115023	+	0.993363i	$0.536694\pi$
$588$	0	0
$589$	21.1987	0.873475
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−7.34450	−0.301602	−0.150801	−	0.988564i	$-0.548185\pi$
$593$	−7.34450	−0.301602	−0.150801	+	0.988564i	$0.548185\pi$
$594$	0	0
$595$	−4.28845	−0.175810
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.266037	0.0108700	0.00543498	−	0.999985i	$-0.498270\pi$
$599$	0.266037	0.0108700	0.00543498	+	0.999985i	$0.498270\pi$
$600$	0	0
$601$	14.2469	0.581141	0.290571	−	0.956854i	$-0.406155\pi$
$601$	14.2469	0.581141	0.290571	+	0.956854i	$0.406155\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−19.5689	−0.794279	−0.397139	−	0.917758i	$-0.629997\pi$
$607$	−19.5689	−0.794279	−0.397139	+	0.917758i	$0.629997\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	48.6666	1.96884
$612$	0	0
$613$	−4.80919	−0.194242	−0.0971208	−	0.995273i	$-0.530963\pi$
$613$	−4.80919	−0.194242	−0.0971208	+	0.995273i	$0.530963\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.1571	0.851751	0.425875	−	0.904782i	$-0.359966\pi$
$617$	21.1571	0.851751	0.425875	+	0.904782i	$0.359966\pi$
$618$	0	0
$619$	−5.48711	−0.220546	−0.110273	−	0.993901i	$-0.535172\pi$
$619$	−5.48711	−0.220546	−0.110273	+	0.993901i	$0.535172\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21.4903	−0.860992
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.59933	0.263132
$630$	0	0
$631$	−30.8462	−1.22797	−0.613984	−	0.789319i	$-0.710434\pi$
$631$	−30.8462	−1.22797	−0.613984	+	0.789319i	$0.710434\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.72275	−0.187417
$636$	0	0
$637$	26.6139	1.05448
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.7788	1.49217	0.746086	−	0.665849i	$-0.231931\pi$
$641$	37.7788	1.49217	0.746086	+	0.665849i	$0.231931\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.1313	1.02733	0.513663	−	0.857992i	$-0.328288\pi$
$647$	26.1313	1.02733	0.513663	+	0.857992i	$0.328288\pi$
$648$	0	0
$649$	−3.71155	−0.145691
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.8686	1.52105	0.760523	−	0.649311i	$-0.224943\pi$
$653$	38.8686	1.52105	0.760523	+	0.649311i	$0.224943\pi$
$654$	0	0
$655$	14.0224	0.547901
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−0.621746	−0.0242198	−0.0121099	−	0.999927i	$-0.503855\pi$
$659$	−0.621746	−0.0242198	−0.0121099	+	0.999927i	$0.503855\pi$
$660$	0	0
$661$	−11.1987	−0.435577	−0.217789	−	0.975996i	$-0.569884\pi$
$661$	−11.1987	−0.435577	−0.217789	+	0.975996i	$0.569884\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.42309	0.287855
$666$	0	0
$667$	11.4231	0.442304
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15.1346	−0.584266
$672$	0	0
$673$	6.14584	0.236905	0.118452	−	0.992960i	$-0.462207\pi$
$673$	6.14584	0.236905	0.118452	+	0.992960i	$0.462207\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.27389	0.0873926	0.0436963	−	0.999045i	$-0.486087\pi$
$677$	2.27389	0.0873926	0.0436963	+	0.999045i	$0.486087\pi$
$678$	0	0
$679$	8.86860	0.340346
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.44551	−0.208367	−0.104183	−	0.994558i	$-0.533223\pi$
$683$	−5.44551	−0.208367	−0.104183	+	0.994558i	$0.533223\pi$
$684$	0	0
$685$	−2.31087	−0.0882939
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−23.0482	−0.878066
$690$	0	0
$691$	40.0448	1.52338	0.761689	−	0.647943i	$-0.224371\pi$
$691$	40.0448	1.52338	0.761689	+	0.647943i	$0.224371\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.11222	−0.118053
$696$	0	0
$697$	6.59933	0.249967
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11.7756	−0.444757	−0.222379	−	0.974960i	$-0.571382\pi$
$701$	−11.7756	−0.444757	−0.222379	+	0.974960i	$0.571382\pi$
$702$	0	0
$703$	−11.4231	−0.430830
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.41973	−0.166221
$708$	0	0
$709$	11.2851	0.423821	0.211910	−	0.977289i	$-0.432032\pi$
$709$	11.2851	0.423821	0.211910	+	0.977289i	$0.432032\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.1987	−0.793896
$714$	0	0
$715$	5.01121	0.187409
$716$	0	0
$717$	0	0
$718$	0	0
$719$	42.1313	1.57123	0.785616	−	0.618715i	$-0.212346\pi$
$719$	42.1313	1.57123	0.785616	+	0.618715i	$0.212346\pi$
$720$	0	0
$721$	−6.75651	−0.251625
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−42.8910	−1.59074	−0.795370	−	0.606124i	$-0.792723\pi$
$727$	−42.8910	−1.59074	−0.795370	+	0.606124i	$0.792723\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.5577	0.538437
$732$	0	0
$733$	−32.8832	−1.21457	−0.607284	−	0.794485i	$-0.707741\pi$
$733$	−32.8832	−1.21457	−0.607284	+	0.794485i	$0.707741\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.40067	−0.0515944
$738$	0	0
$739$	−31.1571	−1.14613	−0.573065	−	0.819510i	$-0.694246\pi$
$739$	−31.1571	−1.14613	−0.573065	+	0.819510i	$0.694246\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.7228	1.78746	0.893732	−	0.448601i	$-0.148077\pi$
$743$	48.7228	1.78746	0.893732	+	0.448601i	$0.148077\pi$
$744$	0	0
$745$	−4.02242	−0.147370
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.75974	−0.100839
$750$	0	0
$751$	29.1987	1.06547	0.532737	−	0.846281i	$-0.321163\pi$
$751$	29.1987	1.06547	0.532737	+	0.846281i	$0.321163\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.59933	0.240174
$756$	0	0
$757$	2.20202	0.0800336	0.0400168	−	0.999199i	$-0.487259\pi$
$757$	2.20202	0.0800336	0.0400168	+	0.999199i	$0.487259\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.4713	−1.17708	−0.588542	−	0.808467i	$-0.700298\pi$
$761$	−32.4713	−1.17708	−0.588542	+	0.808467i	$0.700298\pi$
$762$	0	0
$763$	−23.7980	−0.861545
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18.5993	−0.671583
$768$	0	0
$769$	9.75315	0.351708	0.175854	−	0.984416i	$-0.443731\pi$
$769$	9.75315	0.351708	0.175854	+	0.984416i	$0.443731\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12.0673	−0.434029	−0.217014	−	0.976168i	$-0.569632\pi$
$773$	−12.0673	−0.434029	−0.217014	+	0.976168i	$0.569632\pi$
$774$	0	0
$775$	−3.71155	−0.133323
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.4231	−0.409275
$780$	0	0
$781$	16.0224	0.573327
$782$	0	0
$783$	0	0
$784$	0	0
$785$	21.2211	0.757413
$786$	0	0
$787$	10.7644	0.383708	0.191854	−	0.981423i	$-0.438550\pi$
$787$	10.7644	0.383708	0.191854	+	0.981423i	$0.438550\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.19530	−0.0780558
$792$	0	0
$793$	−75.8428	−2.69326
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−51.6475	−1.82945	−0.914725	−	0.404078i	$-0.867592\pi$
$797$	−51.6475	−1.82945	−0.914725	+	0.404078i	$0.867592\pi$
$798$	0	0
$799$	−32.0448	−1.13366
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.89899	0.208171
$804$	0	0
$805$	−7.42309	−0.261630
$806$	0	0
$807$	0	0
$808$	0	0
$809$	43.6924	1.53614	0.768071	−	0.640365i	$-0.221217\pi$
$809$	43.6924	1.53614	0.768071	+	0.640365i	$0.221217\pi$
$810$	0	0
$811$	−23.2851	−0.817650	−0.408825	−	0.912613i	$-0.634061\pi$
$811$	−23.2851	−0.817650	−0.408825	+	0.912613i	$0.634061\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	19.4231	0.680361
$816$	0	0
$817$	−25.1987	−0.881589
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.2693	−1.68461	−0.842305	−	0.539002i	$-0.818802\pi$
$821$	−48.2693	−1.68461	−0.842305	+	0.539002i	$0.818802\pi$
$822$	0	0
$823$	31.4231	1.09534	0.547670	−	0.836694i	$-0.315515\pi$
$823$	31.4231	1.09534	0.547670	+	0.836694i	$0.315515\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.07523	0.106936	0.0534681	−	0.998570i	$-0.482972\pi$
$827$	3.07523	0.106936	0.0534681	+	0.998570i	$0.482972\pi$
$828$	0	0
$829$	23.1122	0.802720	0.401360	−	0.915920i	$-0.368538\pi$
$829$	23.1122	0.802720	0.401360	+	0.915920i	$0.368538\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−17.5241	−0.607174
$834$	0	0
$835$	2.49832	0.0864579
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.62511	0.0561049	0.0280524	−	0.999606i	$-0.491069\pi$
$839$	1.62511	0.0561049	0.0280524	+	0.999606i	$0.491069\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.1122	0.416673
$846$	0	0
$847$	−1.29966	−0.0446570
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.4231	0.391578
$852$	0	0
$853$	−48.4791	−1.65989	−0.829947	−	0.557842i	$-0.811629\pi$
$853$	−48.4791	−1.65989	−0.829947	+	0.557842i	$0.811629\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.9662	−1.36522	−0.682610	−	0.730782i	$-0.739155\pi$
$857$	−39.9662	−1.36522	−0.682610	+	0.730782i	$0.739155\pi$
$858$	0	0
$859$	29.5319	1.00762	0.503809	−	0.863815i	$-0.331932\pi$
$859$	29.5319	1.00762	0.503809	+	0.863815i	$0.331932\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.7340	−0.535590	−0.267795	−	0.963476i	$-0.586295\pi$
$863$	−15.7340	−0.535590	−0.267795	+	0.963476i	$0.586295\pi$
$864$	0	0
$865$	18.7228	0.636593
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.0224	−0.475678
$870$	0	0
$871$	−7.01906	−0.237832
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.29966	−0.0439367
$876$	0	0
$877$	14.2099	0.479833	0.239917	−	0.970794i	$-0.422880\pi$
$877$	14.2099	0.479833	0.239917	+	0.970794i	$0.422880\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−37.8204	−1.27420	−0.637101	−	0.770780i	$-0.719867\pi$
$881$	−37.8204	−1.27420	−0.637101	+	0.770780i	$0.719867\pi$
$882$	0	0
$883$	50.4421	1.69751	0.848757	−	0.528784i	$-0.177352\pi$
$883$	50.4421	1.69751	0.848757	+	0.528784i	$0.177352\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.5207	1.36055	0.680277	−	0.732955i	$-0.261860\pi$
$887$	40.5207	1.36055	0.680277	+	0.732955i	$0.261860\pi$
$888$	0	0
$889$	6.13799	0.205862
$890$	0	0
$891$	0	0
$892$	0	0
$893$	55.4679	1.85616
$894$	0	0
$895$	−9.19866	−0.307477
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.42309	−0.247574
$900$	0	0
$901$	15.1762	0.505594
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.7340	−0.655979
$906$	0	0
$907$	18.0224	0.598425	0.299212	−	0.954187i	$-0.403276\pi$
$907$	18.0224	0.598425	0.299212	+	0.954187i	$0.403276\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−13.4647	−0.446105	−0.223053	−	0.974806i	$-0.571602\pi$
$911$	−13.4647	−0.446105	−0.223053	+	0.974806i	$0.571602\pi$
$912$	0	0
$913$	7.52410	0.249011
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.2244	−0.601824
$918$	0	0
$919$	−48.0448	−1.58485	−0.792426	−	0.609967i	$-0.791182\pi$
$919$	−48.0448	−1.58485	−0.792426	+	0.609967i	$0.791182\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	80.2917	2.64283
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.2693	0.468160	0.234080	−	0.972217i	$-0.424792\pi$
$929$	14.2693	0.468160	0.234080	+	0.972217i	$0.424792\pi$
$930$	0	0
$931$	30.3333	0.994133
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.29966	−0.107911
$936$	0	0
$937$	−46.3479	−1.51412	−0.757059	−	0.653346i	$-0.773365\pi$
$937$	−46.3479	−1.51412	−0.757059	+	0.653346i	$0.773365\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13.1762	−0.429533	−0.214767	−	0.976665i	$-0.568899\pi$
$941$	−13.1762	−0.429533	−0.214767	+	0.976665i	$0.568899\pi$
$942$	0	0
$943$	11.4231	0.371987
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.202015	0.00656462	0.00328231	−	0.999995i	$-0.498955\pi$
$947$	0.202015	0.00656462	0.00328231	+	0.999995i	$0.498955\pi$
$948$	0	0
$949$	29.5611	0.959593
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20.7452	0.672002	0.336001	−	0.941862i	$-0.390925\pi$
$953$	20.7452	0.672002	0.336001	+	0.941862i	$0.390925\pi$
$954$	0	0
$955$	−3.68913	−0.119377
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.00336	0.0969835
$960$	0	0
$961$	−17.2244	−0.555627
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13.5241	−0.435356
$966$	0	0
$967$	4.52074	0.145377	0.0726886	−	0.997355i	$-0.476842\pi$
$967$	4.52074	0.145377	0.0726886	+	0.997355i	$0.476842\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.0706	−0.804554	−0.402277	−	0.915518i	$-0.631781\pi$
$971$	−25.0706	−0.804554	−0.402277	+	0.915518i	$0.631781\pi$
$972$	0	0
$973$	4.04484	0.129672
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31.5095	−1.00808	−0.504040	−	0.863681i	$-0.668153\pi$
$977$	−31.5095	−1.00808	−0.504040	+	0.863681i	$0.668153\pi$
$978$	0	0
$979$	−16.5353	−0.528471
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.5353	0.846345	0.423172	−	0.906049i	$-0.360916\pi$
$983$	26.5353	0.846345	0.423172	+	0.906049i	$0.360916\pi$
$984$	0	0
$985$	22.1458	0.705625
$986$	0	0
$987$	0	0
$988$	0	0
$989$	25.1987	0.801271
$990$	0	0
$991$	21.3591	0.678493	0.339247	−	0.940697i	$-0.389828\pi$
$991$	21.3591	0.678493	0.339247	+	0.940697i	$0.389828\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.19866	0.164808
$996$	0	0
$997$	38.2099	1.21012	0.605059	−	0.796180i	$-0.293149\pi$
$997$	38.2099	1.21012	0.605059	+	0.796180i	$0.293149\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7920.2.a.cl.1.2		3
3.2	odd	2		7920.2.a.ck.1.2		3
4.3	odd	2		3960.2.a.bh.1.2	yes	3
12.11	even	2		3960.2.a.bg.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3960.2.a.bg.1.2	✓	3	12.11	even	2
3960.2.a.bh.1.2	yes	3	4.3	odd	2
7920.2.a.ck.1.2		3	3.2	odd	2
7920.2.a.cl.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7920 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.29966 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -1.29966 q^{7} -1.00000 q^{11} -5.01121 q^{13} +3.29966 q^{17} -5.71155 q^{19} +5.71155 q^{23} +1.00000 q^{25} +2.00000 q^{29} -3.71155 q^{31} -1.29966 q^{35} +2.00000 q^{37} +2.00000 q^{41} +4.41188 q^{43} -9.71155 q^{47} -5.31087 q^{49} +4.59933 q^{53} -1.00000 q^{55} +3.71155 q^{59} +15.1346 q^{61} -5.01121 q^{65} +1.40067 q^{67} -16.0224 q^{71} -5.89899 q^{73} +1.29966 q^{77} +14.0224 q^{79} -7.52410 q^{83} +3.29966 q^{85} +16.5353 q^{89} +6.51289 q^{91} -5.71155 q^{95} -6.82376 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5}+O(q^{10}) \) 3 * q + 3 * q^5	\( 3 q + 3 q^{5} - 3 q^{11} + 4 q^{13} + 6 q^{17} - 2 q^{19} + 2 q^{23} + 3 q^{25} + 6 q^{29} + 4 q^{31} + 6 q^{37} + 6 q^{41} + 2 q^{43} - 14 q^{47} + 7 q^{49} + 6 q^{53} - 3 q^{55} - 4 q^{59} + 4 q^{65} + 12 q^{67} - 10 q^{71} - 6 q^{73} + 4 q^{79} - 4 q^{83} + 6 q^{85} + 12 q^{89} + 20 q^{91} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 3 * q^11 + 4 * q^13 + 6 * q^17 - 2 * q^19 + 2 * q^23 + 3 * q^25 + 6 * q^29 + 4 * q^31 + 6 * q^37 + 6 * q^41 + 2 * q^43 - 14 * q^47 + 7 * q^49 + 6 * q^53 - 3 * q^55 - 4 * q^59 + 4 * q^65 + 12 * q^67 - 10 * q^71 - 6 * q^73 + 4 * q^79 - 4 * q^83 + 6 * q^85 + 12 * q^89 + 20 * q^91 - 2 * q^95 + 2 * q^97

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