LMFDB - Embedded newform 3087.2.c.c.3086.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3087,2,Mod(3086,3087)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3087, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3087.3086");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3087 = 3^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3087.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:	$24.6498191040$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3086.6
Character	$\chi$	$=$	3087.3086
Dual form			3087.2.c.c.3086.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.660558i q^{2} +1.56366 q^{4} +3.37724 q^{5} +2.35401i q^{8} +O(q^{10})$	$q+0.660558i q^{2} +1.56366 q^{4} +3.37724 q^{5} +2.35401i q^{8} +2.23086i q^{10} -0.0311739i q^{11} -1.36274i q^{13} +1.57237 q^{16} +0.499555 q^{17} +8.07475i q^{19} +5.28087 q^{20} +0.0205922 q^{22} -4.60641i q^{23} +6.40575 q^{25} +0.900167 q^{26} +1.74045i q^{29} -5.76897i q^{31} +5.74665i q^{32} +0.329985i q^{34} +6.29093 q^{37} -5.33384 q^{38} +7.95005i q^{40} -10.5192 q^{41} +6.83001 q^{43} -0.0487454i q^{44} +3.04280 q^{46} +1.86114 q^{47} +4.23137i q^{50} -2.13086i q^{52} +11.7475i q^{53} -0.105282i q^{55} -1.14967 q^{58} +8.79391 q^{59} -4.53664i q^{61} +3.81074 q^{62} -0.651263 q^{64} -4.60229i q^{65} +2.05552 q^{67} +0.781136 q^{68} +6.15204i q^{71} +13.0459i q^{73} +4.15552i q^{74} +12.6262i q^{76} -2.05331 q^{79} +5.31026 q^{80} -6.94858i q^{82} -3.34932 q^{83} +1.68712 q^{85} +4.51162i q^{86} +0.0733835 q^{88} +0.198747 q^{89} -7.20287i q^{92} +1.22939i q^{94} +27.2704i q^{95} -14.0334i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) $ 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3087\mathbb{Z}\right)^\times$.

$n$	$344$	$2404$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.660558i	0.467085i	0.972347	+	0.233543i	$0.0750318\pi$
$2$	0.660558i	0.467085i	−0.972347	+	0.233543i	$0.924968\pi$
$3$	0	0
$3$	0	0
$4$	1.56366	0.781831
$5$	3.37724	1.51035	0.755174	−	0.655525i	$-0.227552\pi$
$5$	3.37724	1.51035	0.755174	+	0.655525i	$0.227552\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.35401i	0.832267i
$9$	0	0
$10$	2.23086i	0.705461i
$11$	− 0.0311739i	− 0.00939928i	−0.999989	−	0.00469964i	$-0.998504\pi$
$11$	− 0.0311739i	− 0.00939928i	0.999989	−	0.00469964i	$-0.00149595\pi$
$12$	0	0
$13$	− 1.36274i	− 0.377955i	−0.981981	−	0.188978i	$-0.939483\pi$
$13$	− 1.36274i	− 0.377955i	0.981981	−	0.188978i	$-0.0605174\pi$
$14$	0	0
$15$	0	0
$16$	1.57237	0.393092
$17$	0.499555	0.121160	0.0605799	−	0.998163i	$-0.480705\pi$
$17$	0.499555	0.121160	0.0605799	+	0.998163i	$0.480705\pi$
$18$	0	0
$19$	8.07475i	1.85247i	0.376942	+	0.926237i	$0.376976\pi$
$19$	8.07475i	1.85247i	−0.376942	+	0.926237i	$0.623024\pi$
$20$	5.28087	1.18084
$21$	0	0
$22$	0.0205922	0.00439026
$23$	− 4.60641i	− 0.960502i	−0.877131	−	0.480251i	$-0.840546\pi$
$23$	− 4.60641i	− 0.960502i	0.877131	−	0.480251i	$-0.159454\pi$
$24$	0	0
$25$	6.40575	1.28115
$26$	0.900167	0.176537
$27$	0	0
$28$	0	0
$29$	1.74045i	0.323193i	0.986857	+	0.161596i	$0.0516642\pi$
$29$	1.74045i	0.323193i	−0.986857	+	0.161596i	$0.948336\pi$
$30$	0	0
$31$	− 5.76897i	− 1.03614i	−0.855339	−	0.518069i	$-0.826651\pi$
$31$	− 5.76897i	− 1.03614i	0.855339	−	0.518069i	$-0.173349\pi$
$32$	5.74665i	1.01587i
$33$	0	0
$34$	0.329985i	0.0565920i
$35$	0	0
$36$	0	0
$37$	6.29093	1.03422	0.517111	−	0.855918i	$-0.327007\pi$
$37$	6.29093	1.03422	0.517111	+	0.855918i	$0.327007\pi$
$38$	−5.33384	−0.865263
$39$	0	0
$40$	7.95005i	1.25701i
$41$	−10.5192	−1.64283	−0.821415	−	0.570330i	$-0.806815\pi$
$41$	−10.5192	−1.64283	−0.821415	+	0.570330i	$0.806815\pi$
$42$	0	0
$43$	6.83001	1.04157	0.520783	−	0.853689i	$-0.325640\pi$
$43$	6.83001	1.04157	0.520783	+	0.853689i	$0.325640\pi$
$44$	− 0.0487454i	− 0.00734865i
$45$	0	0
$46$	3.04280	0.448636
$47$	1.86114	0.271476	0.135738	−	0.990745i	$-0.456660\pi$
$47$	1.86114	0.271476	0.135738	+	0.990745i	$0.456660\pi$
$48$	0	0
$49$	0	0
$50$	4.23137i	0.598406i
$51$	0	0
$52$	− 2.13086i	− 0.295497i
$53$	11.7475i	1.61365i	0.590792	+	0.806824i	$0.298815\pi$
$53$	11.7475i	1.61365i	−0.590792	+	0.806824i	$0.701185\pi$
$54$	0	0
$55$	− 0.105282i	− 0.0141962i
$56$	0	0
$57$	0	0
$58$	−1.14967	−0.150958
$59$	8.79391	1.14487	0.572435	−	0.819950i	$-0.305999\pi$
$59$	8.79391	1.14487	0.572435	+	0.819950i	$0.305999\pi$
$60$	0	0
$61$	− 4.53664i	− 0.580858i	−0.956897	−	0.290429i	$-0.906202\pi$
$61$	− 4.53664i	− 0.580858i	0.956897	−	0.290429i	$-0.0937980\pi$
$62$	3.81074	0.483964
$63$	0	0
$64$	−0.651263	−0.0814079
$65$	− 4.60229i	− 0.570844i
$66$	0	0
$67$	2.05552	0.251122	0.125561	−	0.992086i	$-0.459927\pi$
$67$	2.05552	0.251122	0.125561	+	0.992086i	$0.459927\pi$
$68$	0.781136	0.0947266
$69$	0	0
$70$	0	0
$71$	6.15204i	0.730113i	0.930985	+	0.365057i	$0.118950\pi$
$71$	6.15204i	0.730113i	−0.930985	+	0.365057i	$0.881050\pi$
$72$	0	0
$73$	13.0459i	1.52690i	0.645864	+	0.763452i	$0.276497\pi$
$73$	13.0459i	1.52690i	−0.645864	+	0.763452i	$0.723503\pi$
$74$	4.15552i	0.483070i
$75$	0	0
$76$	12.6262i	1.44832i
$77$	0	0
$78$	0	0
$79$	−2.05331	−0.231016	−0.115508	−	0.993307i	$-0.536850\pi$
$79$	−2.05331	−0.231016	−0.115508	+	0.993307i	$0.536850\pi$
$80$	5.31026	0.593706
$81$	0	0
$82$	− 6.94858i	− 0.767342i
$83$	−3.34932	−0.367635	−0.183818	−	0.982960i	$-0.558846\pi$
$83$	−3.34932	−0.367635	−0.183818	+	0.982960i	$0.558846\pi$
$84$	0	0
$85$	1.68712	0.182994
$86$	4.51162i	0.486500i
$87$	0	0
$88$	0.0733835	0.00782271
$89$	0.198747	0.0210672	0.0105336	−	0.999945i	$-0.496647\pi$
$89$	0.198747	0.0210672	0.0105336	+	0.999945i	$0.496647\pi$
$90$	0	0
$91$	0	0
$92$	− 7.20287i	− 0.750951i
$93$	0	0
$94$	1.22939i	0.126802i
$95$	27.2704i	2.79788i
$96$	0	0
$97$	− 14.0334i	− 1.42488i	−0.701734	−	0.712439i	$-0.747590\pi$
$97$	− 14.0334i	− 1.42488i	0.701734	−	0.712439i	$-0.252410\pi$
$98$	0	0
$99$	0	0
$100$	10.0164	1.00164
$101$	−13.9514	−1.38822	−0.694108	−	0.719871i	$-0.744201\pi$
$101$	−13.9514	−1.38822	−0.694108	+	0.719871i	$0.744201\pi$
$102$	0	0
$103$	− 18.6819i	− 1.84079i	−0.390995	−	0.920393i	$-0.627869\pi$
$103$	− 18.6819i	− 1.84079i	0.390995	−	0.920393i	$-0.372131\pi$
$104$	3.20789	0.314560
$105$	0	0
$106$	−7.75993	−0.753711
$107$	− 10.4874i	− 1.01386i	−0.861988	−	0.506928i	$-0.830781\pi$
$107$	− 10.4874i	− 1.01386i	0.861988	−	0.506928i	$-0.169219\pi$
$108$	0	0
$109$	12.7854	1.22462	0.612308	−	0.790619i	$-0.290241\pi$
$109$	12.7854	1.22462	0.612308	+	0.790619i	$0.290241\pi$
$110$	0.0695447	0.00663082
$111$	0	0
$112$	0	0
$113$	− 10.9530i	− 1.03037i	−0.857079	−	0.515185i	$-0.827723\pi$
$113$	− 10.9530i	− 1.03037i	0.857079	−	0.515185i	$-0.172277\pi$
$114$	0	0
$115$	− 15.5569i	− 1.45069i
$116$	2.72147i	0.252682i
$117$	0	0
$118$	5.80889i	0.534752i
$119$	0	0
$120$	0	0
$121$	10.9990	0.999912
$122$	2.99672	0.271310
$123$	0	0
$124$	− 9.02072i	− 0.810085i
$125$	4.74756	0.424635
$126$	0	0
$127$	−16.3743	−1.45298	−0.726490	−	0.687177i	$-0.758850\pi$
$127$	−16.3743	−1.45298	−0.726490	+	0.687177i	$0.758850\pi$
$128$	11.0631i	0.977850i
$129$	0	0
$130$	3.04008	0.266633
$131$	−20.1932	−1.76429	−0.882146	−	0.470976i	$-0.843902\pi$
$131$	−20.1932	−1.76429	−0.882146	+	0.470976i	$0.843902\pi$
$132$	0	0
$133$	0	0
$134$	1.35779i	0.117295i
$135$	0	0
$136$	1.17596i	0.100837i
$137$	12.9961i	1.11033i	0.831739	+	0.555167i	$0.187346\pi$
$137$	12.9961i	1.11033i	−0.831739	+	0.555167i	$0.812654\pi$
$138$	0	0
$139$	− 15.4249i	− 1.30832i	−0.756355	−	0.654162i	$-0.773022\pi$
$139$	− 15.4249i	− 1.30832i	0.756355	−	0.654162i	$-0.226978\pi$
$140$	0	0
$141$	0	0
$142$	−4.06378	−0.341025
$143$	−0.0424818	−0.00355251
$144$	0	0
$145$	5.87790i	0.488133i
$146$	−8.61756	−0.713194
$147$	0	0
$148$	9.83689	0.808588
$149$	− 8.77095i	− 0.718544i	−0.933233	−	0.359272i	$-0.883025\pi$
$149$	− 8.77095i	− 0.718544i	0.933233	−	0.359272i	$-0.116975\pi$
$150$	0	0
$151$	11.3597	0.924436	0.462218	−	0.886766i	$-0.347054\pi$
$151$	11.3597	0.924436	0.462218	+	0.886766i	$0.347054\pi$
$152$	−19.0080	−1.54175
$153$	0	0
$154$	0	0
$155$	− 19.4832i	− 1.56493i
$156$	0	0
$157$	10.0937i	0.805568i	0.915295	+	0.402784i	$0.131957\pi$
$157$	10.0937i	0.805568i	−0.915295	+	0.402784i	$0.868043\pi$
$158$	− 1.35633i	− 0.107904i
$159$	0	0
$160$	19.4078i	1.53432i
$161$	0	0
$162$	0	0
$163$	−11.6522	−0.912667	−0.456334	−	0.889809i	$-0.650838\pi$
$163$	−11.6522	−0.912667	−0.456334	+	0.889809i	$0.650838\pi$
$164$	−16.4486	−1.28442
$165$	0	0
$166$	− 2.21242i	− 0.171717i
$167$	11.3993	0.882106	0.441053	−	0.897481i	$-0.354605\pi$
$167$	11.3993	0.882106	0.441053	+	0.897481i	$0.354605\pi$
$168$	0	0
$169$	11.1429	0.857150
$170$	1.11444i	0.0854736i
$171$	0	0
$172$	10.6798	0.814329
$173$	−11.1602	−0.848497	−0.424248	−	0.905546i	$-0.639462\pi$
$173$	−11.1602	−0.848497	−0.424248	+	0.905546i	$0.639462\pi$
$174$	0	0
$175$	0	0
$176$	− 0.0490168i	− 0.00369478i
$177$	0	0
$178$	0.131284i	0.00984016i
$179$	1.87137i	0.139873i	0.997551	+	0.0699365i	$0.0222797\pi$
$179$	1.87137i	0.139873i	−0.997551	+	0.0699365i	$0.977720\pi$
$180$	0	0
$181$	− 5.04242i	− 0.374800i	−0.982284	−	0.187400i	$-0.939994\pi$
$181$	− 5.04242i	− 0.374800i	0.982284	−	0.187400i	$-0.0600061\pi$
$182$	0	0
$183$	0	0
$184$	10.8435	0.799394
$185$	21.2460	1.56204
$186$	0	0
$187$	− 0.0155731i	− 0.00113882i
$188$	2.91020	0.212248
$189$	0	0
$190$	−18.0137	−1.30685
$191$	− 12.2020i	− 0.882904i	−0.897285	−	0.441452i	$-0.854464\pi$
$191$	− 12.2020i	− 0.882904i	0.897285	−	0.441452i	$-0.145536\pi$
$192$	0	0
$193$	19.0113	1.36847	0.684233	−	0.729264i	$-0.260137\pi$
$193$	19.0113	1.36847	0.684233	+	0.729264i	$0.260137\pi$
$194$	9.26989	0.665540
$195$	0	0
$196$	0	0
$197$	14.6438i	1.04333i	0.853151	+	0.521663i	$0.174688\pi$
$197$	14.6438i	1.04333i	−0.853151	+	0.521663i	$0.825312\pi$
$198$	0	0
$199$	− 14.7868i	− 1.04821i	−0.851655	−	0.524103i	$-0.824401\pi$
$199$	− 14.7868i	− 1.04821i	0.851655	−	0.524103i	$-0.175599\pi$
$200$	15.0792i	1.06626i
$201$	0	0
$202$	− 9.21571i	− 0.648415i
$203$	0	0
$204$	0	0
$205$	−35.5260	−2.48125
$206$	12.3405	0.859804
$207$	0	0
$208$	− 2.14272i	− 0.148571i
$209$	0.251721	0.0174119
$210$	0	0
$211$	−3.76788	−0.259392	−0.129696	−	0.991554i	$-0.541400\pi$
$211$	−3.76788	−0.259392	−0.129696	+	0.991554i	$0.541400\pi$
$212$	18.3692i	1.26160i
$213$	0	0
$214$	6.92755	0.473557
$215$	23.0666	1.57313
$216$	0	0
$217$	0	0
$218$	8.44548i	0.572000i
$219$	0	0
$220$	− 0.164625i	− 0.0110990i
$221$	− 0.680762i	− 0.0457930i
$222$	0	0
$223$	− 9.49956i	− 0.636138i	−0.948068	−	0.318069i	$-0.896966\pi$
$223$	− 9.49956i	− 0.636138i	0.948068	−	0.318069i	$-0.103034\pi$
$224$	0	0
$225$	0	0
$226$	7.23508	0.481270
$227$	−25.5234	−1.69405	−0.847023	−	0.531556i	$-0.821608\pi$
$227$	−25.5234	−1.69405	−0.847023	+	0.531556i	$0.821608\pi$
$228$	0	0
$229$	− 9.63613i	− 0.636773i	−0.947961	−	0.318387i	$-0.896859\pi$
$229$	− 9.63613i	− 0.636773i	0.947961	−	0.318387i	$-0.103141\pi$
$230$	10.2763	0.677597
$231$	0	0
$232$	−4.09702	−0.268983
$233$	18.2445i	1.19524i	0.801779	+	0.597620i	$0.203887\pi$
$233$	18.2445i	1.19524i	−0.801779	+	0.597620i	$0.796113\pi$
$234$	0	0
$235$	6.28553	0.410023
$236$	13.7507	0.895095
$237$	0	0
$238$	0	0
$239$	− 24.0211i	− 1.55380i	−0.629627	−	0.776898i	$-0.716792\pi$
$239$	− 24.0211i	− 1.55380i	0.629627	−	0.776898i	$-0.283208\pi$
$240$	0	0
$241$	28.1749i	1.81491i	0.420154	+	0.907453i	$0.361976\pi$
$241$	28.1749i	1.81491i	−0.420154	+	0.907453i	$0.638024\pi$
$242$	7.26550i	0.467044i
$243$	0	0
$244$	− 7.09378i	− 0.454133i
$245$	0	0
$246$	0	0
$247$	11.0038	0.700152
$248$	13.5802	0.862343
$249$	0	0
$250$	3.13604i	0.198341i
$251$	−24.4849	−1.54547	−0.772737	−	0.634726i	$-0.781113\pi$
$251$	−24.4849	−1.54547	−0.772737	+	0.634726i	$0.781113\pi$
$252$	0	0
$253$	−0.143600	−0.00902803
$254$	− 10.8161i	− 0.678666i
$255$	0	0
$256$	−8.61035	−0.538147
$257$	11.6606	0.727371	0.363686	−	0.931522i	$-0.381518\pi$
$257$	11.6606	0.727371	0.363686	+	0.931522i	$0.381518\pi$
$258$	0	0
$259$	0	0
$260$	− 7.19643i	− 0.446304i
$261$	0	0
$262$	− 13.3388i	− 0.824074i
$263$	− 2.55316i	− 0.157434i	−0.996897	−	0.0787172i	$-0.974918\pi$
$263$	− 2.55316i	− 0.157434i	0.996897	−	0.0787172i	$-0.0250824\pi$
$264$	0	0
$265$	39.6742i	2.43717i
$266$	0	0
$267$	0	0
$268$	3.21414	0.196335
$269$	15.2731	0.931217	0.465609	−	0.884991i	$-0.345835\pi$
$269$	15.2731	0.931217	0.465609	+	0.884991i	$0.345835\pi$
$270$	0	0
$271$	13.8771i	0.842971i	0.906835	+	0.421486i	$0.138491\pi$
$271$	13.8771i	0.842971i	−0.906835	+	0.421486i	$0.861509\pi$
$272$	0.785484	0.0476270
$273$	0	0
$274$	−8.58470	−0.518621
$275$	− 0.199692i	− 0.0120419i
$276$	0	0
$277$	−8.58054	−0.515554	−0.257777	−	0.966204i	$-0.582990\pi$
$277$	−8.58054	−0.515554	−0.257777	+	0.966204i	$0.582990\pi$
$278$	10.1890	0.611098
$279$	0	0
$280$	0	0
$281$	− 2.80960i	− 0.167607i	−0.996482	−	0.0838034i	$-0.973293\pi$
$281$	− 2.80960i	− 0.167607i	0.996482	−	0.0838034i	$-0.0267068\pi$
$282$	0	0
$283$	− 7.17856i	− 0.426721i	−0.976974	−	0.213361i	$-0.931559\pi$
$283$	− 7.17856i	− 0.426721i	0.976974	−	0.213361i	$-0.0684409\pi$
$284$	9.61972i	0.570826i
$285$	0	0
$286$	− 0.0280617i	− 0.00165932i
$287$	0	0
$288$	0	0
$289$	−16.7504	−0.985320
$290$	−3.88270	−0.228000
$291$	0	0
$292$	20.3994i	1.19378i
$293$	−8.72223	−0.509558	−0.254779	−	0.966999i	$-0.582003\pi$
$293$	−8.72223	−0.509558	−0.254779	+	0.966999i	$0.582003\pi$
$294$	0	0
$295$	29.6992	1.72915
$296$	14.8089i	0.860749i
$297$	0	0
$298$	5.79372	0.335621
$299$	−6.27732	−0.363027
$300$	0	0
$301$	0	0
$302$	7.50372i	0.431790i
$303$	0	0
$304$	12.6965i	0.728193i
$305$	− 15.3213i	− 0.877297i
$306$	0	0
$307$	6.71201i	0.383075i	0.981485	+	0.191537i	$0.0613473\pi$
$307$	6.71201i	0.383075i	−0.981485	+	0.191537i	$0.938653\pi$
$308$	0	0
$309$	0	0
$310$	12.8698	0.730954
$311$	−10.4899	−0.594825	−0.297412	−	0.954749i	$-0.596124\pi$
$311$	−10.4899	−0.594825	−0.297412	+	0.954749i	$0.596124\pi$
$312$	0	0
$313$	31.3431i	1.77162i	0.464051	+	0.885809i	$0.346396\pi$
$313$	31.3431i	1.77162i	−0.464051	+	0.885809i	$0.653604\pi$
$314$	−6.66750	−0.376269
$315$	0	0
$316$	−3.21069	−0.180615
$317$	− 14.1819i	− 0.796532i	−0.917270	−	0.398266i	$-0.869612\pi$
$317$	− 14.1819i	− 0.796532i	0.917270	−	0.398266i	$-0.130388\pi$
$318$	0	0
$319$	0.0542564	0.00303778
$320$	−2.19947	−0.122954
$321$	0	0
$322$	0	0
$323$	4.03378i	0.224446i
$324$	0	0
$325$	− 8.72935i	− 0.484217i
$326$	− 7.69693i	− 0.426293i
$327$	0	0
$328$	− 24.7624i	− 1.36727i
$329$	0	0
$330$	0	0
$331$	−19.5517	−1.07466	−0.537330	−	0.843372i	$-0.680567\pi$
$331$	−19.5517	−1.07466	−0.537330	+	0.843372i	$0.680567\pi$
$332$	−5.23720	−0.287429
$333$	0	0
$334$	7.52992i	0.412019i
$335$	6.94198	0.379281
$336$	0	0
$337$	17.0678	0.929743	0.464871	−	0.885378i	$-0.346101\pi$
$337$	17.0678	0.929743	0.464871	+	0.885378i	$0.346101\pi$
$338$	7.36057i	0.400362i
$339$	0	0
$340$	2.63808	0.143070
$341$	−0.179841	−0.00973894
$342$	0	0
$343$	0	0
$344$	16.0779i	0.866861i
$345$	0	0
$346$	− 7.37198i	− 0.396320i
$347$	27.9914i	1.50265i	0.659930	+	0.751327i	$0.270586\pi$
$347$	27.9914i	1.50265i	−0.659930	+	0.751327i	$0.729414\pi$
$348$	0	0
$349$	7.67849i	0.411020i	0.978655	+	0.205510i	$0.0658853\pi$
$349$	7.67849i	0.411020i	−0.978655	+	0.205510i	$0.934115\pi$
$350$	0	0
$351$	0	0
$352$	0.179145	0.00954849
$353$	−20.2636	−1.07853	−0.539263	−	0.842138i	$-0.681297\pi$
$353$	−20.2636	−1.07853	−0.539263	+	0.842138i	$0.681297\pi$
$354$	0	0
$355$	20.7769i	1.10273i
$356$	0.310774	0.0164710
$357$	0	0
$358$	−1.23615	−0.0653326
$359$	− 6.12495i	− 0.323263i	−0.986851	−	0.161631i	$-0.948324\pi$
$359$	− 6.12495i	− 0.323263i	0.986851	−	0.161631i	$-0.0516755\pi$
$360$	0	0
$361$	−46.2015	−2.43166
$362$	3.33081	0.175063
$363$	0	0
$364$	0	0
$365$	44.0591i	2.30616i
$366$	0	0
$367$	− 2.77656i	− 0.144935i	−0.997371	−	0.0724677i	$-0.976913\pi$
$367$	− 2.77656i	− 0.144935i	0.997371	−	0.0724677i	$-0.0230874\pi$
$368$	− 7.24296i	− 0.377566i
$369$	0	0
$370$	14.0342i	0.729604i
$371$	0	0
$372$	0	0
$373$	−17.8679	−0.925164	−0.462582	−	0.886577i	$-0.653077\pi$
$373$	−17.8679	−0.925164	−0.462582	+	0.886577i	$0.653077\pi$
$374$	0.0102869	0.000531924 0
$375$	0	0
$376$	4.38114i	0.225940i
$377$	2.37177	0.122152
$378$	0	0
$379$	−22.6890	−1.16546	−0.582728	−	0.812668i	$-0.698015\pi$
$379$	−22.6890	−1.16546	−0.582728	+	0.812668i	$0.698015\pi$
$380$	42.6417i	2.18747i
$381$	0	0
$382$	8.06012	0.412391
$383$	16.8751	0.862275	0.431137	−	0.902286i	$-0.358112\pi$
$383$	16.8751	0.862275	0.431137	+	0.902286i	$0.358112\pi$
$384$	0	0
$385$	0	0
$386$	12.5581i	0.639190i
$387$	0	0
$388$	− 21.9435i	− 1.11401i
$389$	14.3380i	0.726967i	0.931601	+	0.363484i	$0.118413\pi$
$389$	14.3380i	0.726967i	−0.931601	+	0.363484i	$0.881587\pi$
$390$	0	0
$391$	− 2.30115i	− 0.116374i
$392$	0	0
$393$	0	0
$394$	−9.67308	−0.487323
$395$	−6.93453	−0.348914
$396$	0	0
$397$	− 3.58702i	− 0.180027i	−0.995941	−	0.0900136i	$-0.971309\pi$
$397$	− 3.58702i	− 0.180027i	0.995941	−	0.0900136i	$-0.0286911\pi$
$398$	9.76751	0.489601
$399$	0	0
$400$	10.0722	0.503610
$401$	− 27.6593i	− 1.38124i	−0.723219	−	0.690619i	$-0.757338\pi$
$401$	− 27.6593i	− 1.38124i	0.723219	−	0.690619i	$-0.242662\pi$
$402$	0	0
$403$	−7.86158	−0.391613
$404$	−21.8153	−1.08535
$405$	0	0
$406$	0	0
$407$	− 0.196113i	− 0.00972094i
$408$	0	0
$409$	23.3964i	1.15688i	0.815725	+	0.578439i	$0.196338\pi$
$409$	23.3964i	1.15688i	−0.815725	+	0.578439i	$0.803662\pi$
$410$	− 23.4670i	− 1.15895i
$411$	0	0
$412$	− 29.2122i	− 1.43918i
$413$	0	0
$414$	0	0
$415$	−11.3114	−0.555257
$416$	7.83118	0.383955
$417$	0	0
$418$	0.166276i	0.00813285i
$419$	9.91640	0.484448	0.242224	−	0.970220i	$-0.422123\pi$
$419$	9.91640	0.484448	0.242224	+	0.970220i	$0.422123\pi$
$420$	0	0
$421$	30.9183	1.50686	0.753432	−	0.657525i	$-0.228397\pi$
$421$	30.9183	1.50686	0.753432	+	0.657525i	$0.228397\pi$
$422$	− 2.48890i	− 0.121158i
$423$	0	0
$424$	−27.6538	−1.34299
$425$	3.20003	0.155224
$426$	0	0
$427$	0	0
$428$	− 16.3988i	− 0.792665i
$429$	0	0
$430$	15.2368i	0.734784i
$431$	− 34.2411i	− 1.64933i	−0.565618	−	0.824667i	$-0.691362\pi$
$431$	− 34.2411i	− 1.64933i	0.565618	−	0.824667i	$-0.308638\pi$
$432$	0	0
$433$	− 14.1597i	− 0.680471i	−0.940340	−	0.340236i	$-0.889493\pi$
$433$	− 14.1597i	− 0.680471i	0.940340	−	0.340236i	$-0.110507\pi$
$434$	0	0
$435$	0	0
$436$	19.9920	0.957444
$437$	37.1956	1.77931
$438$	0	0
$439$	− 18.5502i	− 0.885353i	−0.896681	−	0.442677i	$-0.854029\pi$
$439$	− 18.5502i	− 0.885353i	0.896681	−	0.442677i	$-0.145971\pi$
$440$	0.247834	0.0118150
$441$	0	0
$442$	0.449683	0.0213892
$443$	34.2330i	1.62646i	0.581944	+	0.813229i	$0.302292\pi$
$443$	34.2330i	1.62646i	−0.581944	+	0.813229i	$0.697708\pi$
$444$	0	0
$445$	0.671217	0.0318187
$446$	6.27501	0.297131
$447$	0	0
$448$	0	0
$449$	− 22.1046i	− 1.04318i	−0.853197	−	0.521590i	$-0.825339\pi$
$449$	− 22.1046i	− 1.04318i	0.853197	−	0.521590i	$-0.174661\pi$
$450$	0	0
$451$	0.327926i	0.0154414i
$452$	− 17.1268i	− 0.805575i
$453$	0	0
$454$	− 16.8597i	− 0.791264i
$455$	0	0
$456$	0	0
$457$	−10.1624	−0.475376	−0.237688	−	0.971342i	$-0.576390\pi$
$457$	−10.1624	−0.475376	−0.237688	+	0.971342i	$0.576390\pi$
$458$	6.36522	0.297427
$459$	0	0
$460$	− 24.3258i	− 1.13420i
$461$	7.67757	0.357580	0.178790	−	0.983887i	$-0.442782\pi$
$461$	7.67757	0.357580	0.178790	+	0.983887i	$0.442782\pi$
$462$	0	0
$463$	5.57314	0.259006	0.129503	−	0.991579i	$-0.458662\pi$
$463$	5.57314	0.259006	0.129503	+	0.991579i	$0.458662\pi$
$464$	2.73662i	0.127044i
$465$	0	0
$466$	−12.0516	−0.558279
$467$	15.2947	0.707756	0.353878	−	0.935292i	$-0.384863\pi$
$467$	15.2947	0.707756	0.353878	+	0.935292i	$0.384863\pi$
$468$	0	0
$469$	0	0
$470$	4.15196i	0.191516i
$471$	0	0
$472$	20.7009i	0.952838i
$473$	− 0.212918i	− 0.00978997i
$474$	0	0
$475$	51.7248i	2.37330i
$476$	0	0
$477$	0	0
$478$	15.8673	0.725755
$479$	−21.6957	−0.991301	−0.495650	−	0.868522i	$-0.665070\pi$
$479$	−21.6957	−0.991301	−0.495650	+	0.868522i	$0.665070\pi$
$480$	0	0
$481$	− 8.57288i	− 0.390890i
$482$	−18.6112	−0.847715
$483$	0	0
$484$	17.1988	0.781762
$485$	− 47.3942i	− 2.15206i
$486$	0	0
$487$	24.5225	1.11122	0.555611	−	0.831442i	$-0.312484\pi$
$487$	24.5225	1.11122	0.555611	+	0.831442i	$0.312484\pi$
$488$	10.6793	0.483429
$489$	0	0
$490$	0	0
$491$	− 0.156928i	− 0.00708206i	−0.999994	−	0.00354103i	$-0.998873\pi$
$491$	− 0.156928i	− 0.00708206i	0.999994	−	0.00354103i	$-0.00112715\pi$
$492$	0	0
$493$	0.869448i	0.0391580i
$494$	7.26862i	0.327031i
$495$	0	0
$496$	− 9.07094i	− 0.407297i
$497$	0	0
$498$	0	0
$499$	−32.5224	−1.45590	−0.727952	−	0.685629i	$-0.759527\pi$
$499$	−32.5224	−1.45590	−0.727952	+	0.685629i	$0.759527\pi$
$500$	7.42359	0.331993
$501$	0	0
$502$	− 16.1737i	− 0.721868i
$503$	16.9850	0.757323	0.378661	−	0.925535i	$-0.376384\pi$
$503$	16.9850	0.757323	0.378661	+	0.925535i	$0.376384\pi$
$504$	0	0
$505$	−47.1172	−2.09669
$506$	− 0.0948558i	− 0.00421686i
$507$	0	0
$508$	−25.6038	−1.13599
$509$	−23.4891	−1.04114	−0.520569	−	0.853820i	$-0.674280\pi$
$509$	−23.4891	−1.04114	−0.520569	+	0.853820i	$0.674280\pi$
$510$	0	0
$511$	0	0
$512$	16.4386i	0.726489i
$513$	0	0
$514$	7.70253i	0.339744i
$515$	− 63.0934i	− 2.78023i
$516$	0	0
$517$	− 0.0580191i	− 0.00255168i
$518$	0	0
$519$	0	0
$520$	10.8338	0.475094
$521$	26.7941	1.17387	0.586936	−	0.809634i	$-0.300334\pi$
$521$	26.7941	1.17387	0.586936	+	0.809634i	$0.300334\pi$
$522$	0	0
$523$	− 21.8819i	− 0.956829i	−0.878134	−	0.478414i	$-0.841212\pi$
$523$	− 21.8819i	− 0.956829i	0.878134	−	0.478414i	$-0.158788\pi$
$524$	−31.5754	−1.37938
$525$	0	0
$526$	1.68651	0.0735353
$527$	− 2.88192i	− 0.125538i
$528$	0	0
$529$	1.78102	0.0774358
$530$	−26.2071	−1.13837
$531$	0	0
$532$	0	0
$533$	14.3350i	0.620916i
$534$	0	0
$535$	− 35.4185i	− 1.53128i
$536$	4.83871i	0.209000i
$537$	0	0
$538$	10.0888i	0.434958i
$539$	0	0
$540$	0	0
$541$	36.9385	1.58811	0.794055	−	0.607845i	$-0.207966\pi$
$541$	36.9385	1.58811	0.794055	+	0.607845i	$0.207966\pi$
$542$	−9.16660	−0.393739
$543$	0	0
$544$	2.87077i	0.123083i
$545$	43.1793	1.84960
$546$	0	0
$547$	−4.43337	−0.189557	−0.0947785	−	0.995498i	$-0.530214\pi$
$547$	−4.43337	−0.189557	−0.0947785	+	0.995498i	$0.530214\pi$
$548$	20.3216i	0.868094i
$549$	0	0
$550$	0.131908	0.00562459
$551$	−14.0537	−0.598706
$552$	0	0
$553$	0	0
$554$	− 5.66794i	− 0.240808i
$555$	0	0
$556$	− 24.1193i	− 1.02289i
$557$	− 25.3212i	− 1.07289i	−0.843934	−	0.536447i	$-0.819766\pi$
$557$	− 25.3212i	− 1.07289i	0.843934	−	0.536447i	$-0.180234\pi$
$558$	0	0
$559$	− 9.30750i	− 0.393665i
$560$	0	0
$561$	0	0
$562$	1.85591	0.0782866
$563$	−27.0100	−1.13834	−0.569168	−	0.822221i	$-0.692735\pi$
$563$	−27.0100	−1.13834	−0.569168	+	0.822221i	$0.692735\pi$
$564$	0	0
$565$	− 36.9908i	− 1.55622i
$566$	4.74186	0.199315
$567$	0	0
$568$	−14.4820	−0.607649
$569$	− 22.1564i	− 0.928843i	−0.885614	−	0.464422i	$-0.846262\pi$
$569$	− 22.1564i	− 0.928843i	0.885614	−	0.464422i	$-0.153738\pi$
$570$	0	0
$571$	−9.87253	−0.413153	−0.206576	−	0.978430i	$-0.566232\pi$
$571$	−9.87253	−0.413153	−0.206576	+	0.978430i	$0.566232\pi$
$572$	−0.0664272	−0.00277746
$573$	0	0
$574$	0	0
$575$	− 29.5075i	− 1.23055i
$576$	0	0
$577$	− 1.96927i	− 0.0819817i	−0.999160	−	0.0409909i	$-0.986949\pi$
$577$	− 1.96927i	− 0.0819817i	0.999160	−	0.0409909i	$-0.0130515\pi$
$578$	− 11.0646i	− 0.460228i
$579$	0	0
$580$	9.19106i	0.381638i
$581$	0	0
$582$	0	0
$583$	0.366216	0.0151671
$584$	−30.7101	−1.27079
$585$	0	0
$586$	− 5.76154i	− 0.238007i
$587$	3.46687	0.143093	0.0715466	−	0.997437i	$-0.477207\pi$
$587$	3.46687	0.143093	0.0715466	+	0.997437i	$0.477207\pi$
$588$	0	0
$589$	46.5829	1.91942
$590$	19.6180i	0.807661i
$591$	0	0
$592$	9.89166	0.406545
$593$	5.29165	0.217302	0.108651	−	0.994080i	$-0.465347\pi$
$593$	5.29165	0.217302	0.108651	+	0.994080i	$0.465347\pi$
$594$	0	0
$595$	0	0
$596$	− 13.7148i	− 0.561781i
$597$	0	0
$598$	− 4.14653i	− 0.169564i
$599$	− 14.6362i	− 0.598017i	−0.954250	−	0.299009i	$-0.903344\pi$
$599$	− 14.6362i	− 0.598017i	0.954250	−	0.299009i	$-0.0966559\pi$
$600$	0	0
$601$	− 31.3558i	− 1.27903i	−0.768778	−	0.639516i	$-0.779135\pi$
$601$	− 31.3558i	− 1.27903i	0.768778	−	0.639516i	$-0.220865\pi$
$602$	0	0
$603$	0	0
$604$	17.7627	0.722753
$605$	37.1464	1.51021
$606$	0	0
$607$	− 15.6192i	− 0.633965i	−0.948431	−	0.316983i	$-0.897330\pi$
$607$	− 15.6192i	− 0.633965i	0.948431	−	0.316983i	$-0.102670\pi$
$608$	−46.4028	−1.88188
$609$	0	0
$610$	10.1206	0.409772
$611$	− 2.53625i	− 0.102606i
$612$	0	0
$613$	−19.4069	−0.783838	−0.391919	−	0.920000i	$-0.628189\pi$
$613$	−19.4069	−0.783838	−0.391919	+	0.920000i	$0.628189\pi$
$614$	−4.43367	−0.178928
$615$	0	0
$616$	0	0
$617$	28.2345i	1.13668i	0.822795	+	0.568339i	$0.192414\pi$
$617$	28.2345i	1.13668i	−0.822795	+	0.568339i	$0.807586\pi$
$618$	0	0
$619$	− 20.6347i	− 0.829378i	−0.909963	−	0.414689i	$-0.863890\pi$
$619$	− 20.6347i	− 0.829378i	0.909963	−	0.414689i	$-0.136110\pi$
$620$	− 30.4651i	− 1.22351i
$621$	0	0
$622$	− 6.92916i	− 0.277834i
$623$	0	0
$624$	0	0
$625$	−15.9951	−0.639804
$626$	−20.7039	−0.827496
$627$	0	0
$628$	15.7832i	0.629819i
$629$	3.14266	0.125306
$630$	0	0
$631$	−34.3938	−1.36919	−0.684597	−	0.728922i	$-0.740022\pi$
$631$	−34.3938	−1.36919	−0.684597	+	0.728922i	$0.740022\pi$
$632$	− 4.83351i	− 0.192267i
$633$	0	0
$634$	9.36794	0.372048
$635$	−55.2998	−2.19451
$636$	0	0
$637$	0	0
$638$	0.0358395i	0.00141890i
$639$	0	0
$640$	37.3628i	1.47689i
$641$	− 47.1929i	− 1.86401i	−0.362451	−	0.932003i	$-0.618060\pi$
$641$	− 47.1929i	− 1.86401i	0.362451	−	0.932003i	$-0.381940\pi$
$642$	0	0
$643$	15.5559i	0.613463i	0.951796	+	0.306732i	$0.0992354\pi$
$643$	15.5559i	0.613463i	−0.951796	+	0.306732i	$0.900765\pi$
$644$	0	0
$645$	0	0
$646$	−2.66455	−0.104835
$647$	−10.5267	−0.413848	−0.206924	−	0.978357i	$-0.566345\pi$
$647$	−10.5267	−0.413848	−0.206924	+	0.978357i	$0.566345\pi$
$648$	0	0
$649$	− 0.274140i	− 0.0107610i
$650$	5.76625	0.226171
$651$	0	0
$652$	−18.2200	−0.713552
$653$	25.6663i	1.00440i	0.864752	+	0.502200i	$0.167476\pi$
$653$	25.6663i	1.00440i	−0.864752	+	0.502200i	$0.832524\pi$
$654$	0	0
$655$	−68.1974	−2.66469
$656$	−16.5401	−0.645784
$657$	0	0
$658$	0	0
$659$	2.15471i	0.0839356i	0.999119	+	0.0419678i	$0.0133627\pi$
$659$	2.15471i	0.0839356i	−0.999119	+	0.0419678i	$0.986637\pi$
$660$	0	0
$661$	− 0.0829326i	− 0.00322570i	−0.999999	−	0.00161285i	$-0.999487\pi$
$661$	− 0.0829326i	− 0.00322570i	0.999999	−	0.00161285i	$-0.000513387\pi$
$662$	− 12.9151i	− 0.501958i
$663$	0	0
$664$	− 7.88431i	− 0.305971i
$665$	0	0
$666$	0	0
$667$	8.01720	0.310427
$668$	17.8247	0.689659
$669$	0	0
$670$	4.58558i	0.177157i
$671$	−0.141425	−0.00545964
$672$	0	0
$673$	−0.176566	−0.00680614	−0.00340307	−	0.999994i	$-0.501083\pi$
$673$	−0.176566	−0.00680614	−0.00340307	+	0.999994i	$0.501083\pi$
$674$	11.2743i	0.434269i
$675$	0	0
$676$	17.4238	0.670147
$677$	−44.8239	−1.72272	−0.861362	−	0.507992i	$-0.830388\pi$
$677$	−44.8239	−1.72272	−0.861362	+	0.507992i	$0.830388\pi$
$678$	0	0
$679$	0	0
$680$	3.97148i	0.152299i
$681$	0	0
$682$	− 0.118795i	− 0.00454891i
$683$	− 4.27014i	− 0.163392i	−0.996657	−	0.0816962i	$-0.973966\pi$
$683$	− 4.27014i	− 0.163392i	0.996657	−	0.0816962i	$-0.0260337\pi$
$684$	0	0
$685$	43.8911i	1.67699i
$686$	0	0
$687$	0	0
$688$	10.7393	0.409431
$689$	16.0088	0.609886
$690$	0	0
$691$	27.2454i	1.03646i	0.855240	+	0.518232i	$0.173409\pi$
$691$	27.2454i	1.03646i	−0.855240	+	0.518232i	$0.826591\pi$
$692$	−17.4508	−0.663381
$693$	0	0
$694$	−18.4899	−0.701868
$695$	− 52.0936i	− 1.97602i
$696$	0	0
$697$	−5.25494	−0.199045
$698$	−5.07209	−0.191981
$699$	0	0
$700$	0	0
$701$	− 40.6804i	− 1.53648i	−0.640162	−	0.768240i	$-0.721133\pi$
$701$	− 40.6804i	− 1.53648i	0.640162	−	0.768240i	$-0.278867\pi$
$702$	0	0
$703$	50.7977i	1.91587i
$704$	0.0203024i	0 0.000765175i
$705$	0	0
$706$	− 13.3853i	− 0.503763i
$707$	0	0
$708$	0	0
$709$	−8.00099	−0.300484	−0.150242	−	0.988649i	$-0.548005\pi$
$709$	−8.00099	−0.300484	−0.150242	+	0.988649i	$0.548005\pi$
$710$	−13.7244	−0.515066
$711$	0	0
$712$	0.467852i	0.0175335i
$713$	−26.5742	−0.995212
$714$	0	0
$715$	−0.143471	−0.00536552
$716$	2.92620i	0.109357i
$717$	0	0
$718$	4.04589	0.150991
$719$	36.9061	1.37637	0.688183	−	0.725537i	$-0.258408\pi$
$719$	36.9061	1.37637	0.688183	+	0.725537i	$0.258408\pi$
$720$	0	0
$721$	0	0
$722$	− 30.5188i	− 1.13579i
$723$	0	0
$724$	− 7.88464i	− 0.293030i
$725$	11.1489i	0.414058i
$726$	0	0
$727$	31.3119i	1.16129i	0.814155	+	0.580647i	$0.197200\pi$
$727$	31.3119i	1.16129i	−0.814155	+	0.580647i	$0.802800\pi$
$728$	0	0
$729$	0	0
$730$	−29.1036	−1.07717
$731$	3.41196	0.126196
$732$	0	0
$733$	− 18.6695i	− 0.689573i	−0.938681	−	0.344787i	$-0.887951\pi$
$733$	− 18.6695i	− 0.689573i	0.938681	−	0.344787i	$-0.112049\pi$
$734$	1.83408	0.0676972
$735$	0	0
$736$	26.4714	0.975749
$737$	− 0.0640785i	− 0.00236036i
$738$	0	0
$739$	−17.0849	−0.628477	−0.314239	−	0.949344i	$-0.601749\pi$
$739$	−17.0849	−0.628477	−0.314239	+	0.949344i	$0.601749\pi$
$740$	33.2216	1.22125
$741$	0	0
$742$	0	0
$743$	− 36.8935i	− 1.35349i	−0.736216	−	0.676746i	$-0.763389\pi$
$743$	− 36.8935i	− 1.35349i	0.736216	−	0.676746i	$-0.236611\pi$
$744$	0	0
$745$	− 29.6216i	− 1.08525i
$746$	− 11.8028i	− 0.432130i
$747$	0	0
$748$	− 0.0243510i	0 0.000890362i
$749$	0	0
$750$	0	0
$751$	29.1079	1.06216	0.531081	−	0.847321i	$-0.321786\pi$
$751$	29.1079	1.06216	0.531081	+	0.847321i	$0.321786\pi$
$752$	2.92640	0.106715
$753$	0	0
$754$	1.56669i	0.0570555i
$755$	38.3643	1.39622
$756$	0	0
$757$	−25.7085	−0.934390	−0.467195	−	0.884154i	$-0.654735\pi$
$757$	−25.7085	−0.934390	−0.467195	+	0.884154i	$0.654735\pi$
$758$	− 14.9874i	− 0.544367i
$759$	0	0
$760$	−64.1946	−2.32858
$761$	46.8615	1.69873	0.849364	−	0.527807i	$-0.176986\pi$
$761$	46.8615	1.69873	0.849364	+	0.527807i	$0.176986\pi$
$762$	0	0
$763$	0	0
$764$	− 19.0798i	− 0.690282i
$765$	0	0
$766$	11.1470i	0.402756i
$767$	− 11.9838i	− 0.432710i
$768$	0	0
$769$	0.935222i	0.0337250i	0.999858	+	0.0168625i	$0.00536775\pi$
$769$	0.935222i	0.0337250i	−0.999858	+	0.0168625i	$0.994632\pi$
$770$	0	0
$771$	0	0
$772$	29.7273	1.06991
$773$	10.4185	0.374726	0.187363	−	0.982291i	$-0.440006\pi$
$773$	10.4185	0.374726	0.187363	+	0.982291i	$0.440006\pi$
$774$	0	0
$775$	− 36.9546i	− 1.32745i
$776$	33.0348	1.18588
$777$	0	0
$778$	−9.47111	−0.339556
$779$	− 84.9403i	− 3.04330i
$780$	0	0
$781$	0.191783	0.00686254
$782$	1.52005	0.0543567
$783$	0	0
$784$	0	0
$785$	34.0890i	1.21669i
$786$	0	0
$787$	9.68700i	0.345304i	0.984983	+	0.172652i	$0.0552336\pi$
$787$	9.68700i	0.345304i	−0.984983	+	0.172652i	$0.944766\pi$
$788$	22.8980i	0.815706i
$789$	0	0
$790$	− 4.58066i	− 0.162973i
$791$	0	0
$792$	0	0
$793$	−6.18225	−0.219538
$794$	2.36943	0.0840880
$795$	0	0
$796$	− 23.1215i	− 0.819520i
$797$	−53.8730	−1.90828	−0.954139	−	0.299363i	$-0.903226\pi$
$797$	−53.8730	−1.90828	−0.954139	+	0.299363i	$0.903226\pi$
$798$	0	0
$799$	0.929743	0.0328920
$800$	36.8116i	1.30149i
$801$	0	0
$802$	18.2705	0.645155
$803$	0.406691	0.0143518
$804$	0	0
$805$	0	0
$806$	− 5.19303i	− 0.182917i
$807$	0	0
$808$	− 32.8417i	− 1.15537i
$809$	− 48.4520i	− 1.70348i	−0.523964	−	0.851740i	$-0.675548\pi$
$809$	− 48.4520i	− 1.70348i	0.523964	−	0.851740i	$-0.324452\pi$
$810$	0	0
$811$	− 42.1420i	− 1.47981i	−0.672713	−	0.739903i	$-0.734871\pi$
$811$	− 42.1420i	− 1.47981i	0.672713	−	0.739903i	$-0.265129\pi$
$812$	0	0
$813$	0	0
$814$	0.129544	0.00454051
$815$	−39.3521	−1.37845
$816$	0	0
$817$	55.1506i	1.92947i
$818$	−15.4547	−0.540361
$819$	0	0
$820$	−55.5507	−1.93992
$821$	20.0758i	0.700651i	0.936628	+	0.350325i	$0.113929\pi$
$821$	20.0758i	0.700651i	−0.936628	+	0.350325i	$0.886071\pi$
$822$	0	0
$823$	−7.37710	−0.257150	−0.128575	−	0.991700i	$-0.541040\pi$
$823$	−7.37710	−0.257150	−0.128575	+	0.991700i	$0.541040\pi$
$824$	43.9774	1.53203
$825$	0	0
$826$	0	0
$827$	− 23.3830i	− 0.813105i	−0.913627	−	0.406552i	$-0.866731\pi$
$827$	− 23.3830i	− 0.813105i	0.913627	−	0.406552i	$-0.133269\pi$
$828$	0	0
$829$	31.2165i	1.08419i	0.840317	+	0.542096i	$0.182369\pi$
$829$	31.2165i	1.08419i	−0.840317	+	0.542096i	$0.817631\pi$
$830$	− 7.47187i	− 0.259352i
$831$	0	0
$832$	0.887500i	0.0307685i
$833$	0	0
$834$	0	0
$835$	38.4983	1.33229
$836$	0.393607	0.0136132
$837$	0	0
$838$	6.55036i	0.226278i
$839$	−18.4041	−0.635379	−0.317689	−	0.948195i	$-0.602907\pi$
$839$	−18.4041	−0.635379	−0.317689	+	0.948195i	$0.602907\pi$
$840$	0	0
$841$	25.9709	0.895547
$842$	20.4233i	0.703834i
$843$	0	0
$844$	−5.89170	−0.202801
$845$	37.6324	1.29459
$846$	0	0
$847$	0	0
$848$	18.4714i	0.634312i
$849$	0	0
$850$	2.11380i	0.0725028i
$851$	− 28.9786i	− 0.993373i
$852$	0	0
$853$	− 31.8130i	− 1.08926i	−0.838678	−	0.544628i	$-0.816671\pi$
$853$	− 31.8130i	− 1.08926i	0.838678	−	0.544628i	$-0.183329\pi$
$854$	0	0
$855$	0	0
$856$	24.6874	0.843800
$857$	16.6649	0.569262	0.284631	−	0.958637i	$-0.408129\pi$
$857$	16.6649	0.569262	0.284631	+	0.958637i	$0.408129\pi$
$858$	0	0
$859$	56.7639i	1.93676i	0.249481	+	0.968380i	$0.419740\pi$
$859$	56.7639i	1.93676i	−0.249481	+	0.968380i	$0.580260\pi$
$860$	36.0684	1.22992
$861$	0	0
$862$	22.6182	0.770380
$863$	− 17.8765i	− 0.608524i	−0.952588	−	0.304262i	$-0.901590\pi$
$863$	− 17.8765i	− 0.608524i	0.952588	−	0.304262i	$-0.0984099\pi$
$864$	0	0
$865$	−37.6908	−1.28152
$866$	9.35330	0.317838
$867$	0	0
$868$	0	0
$869$	0.0640097i	0.00217138i
$870$	0	0
$871$	− 2.80113i	− 0.0949127i
$872$	30.0969i	1.01921i
$873$	0	0
$874$	24.5698i	0.831087i
$875$	0	0
$876$	0	0
$877$	−4.33543	−0.146397	−0.0731985	−	0.997317i	$-0.523321\pi$
$877$	−4.33543	−0.146397	−0.0731985	+	0.997317i	$0.523321\pi$
$878$	12.2535	0.413535
$879$	0	0
$880$	− 0.165542i	− 0.00558040i
$881$	42.5687	1.43417	0.717087	−	0.696983i	$-0.245475\pi$
$881$	42.5687	1.43417	0.717087	+	0.696983i	$0.245475\pi$
$882$	0	0
$883$	−32.5186	−1.09434	−0.547169	−	0.837022i	$-0.684295\pi$
$883$	−32.5186	−1.09434	−0.547169	+	0.837022i	$0.684295\pi$
$884$	− 1.06448i	− 0.0358024i
$885$	0	0
$886$	−22.6129	−0.759694
$887$	10.8190	0.363268	0.181634	−	0.983366i	$-0.441861\pi$
$887$	10.8190	0.363268	0.181634	+	0.983366i	$0.441861\pi$
$888$	0	0
$889$	0	0
$890$	0.443378i	0.0148621i
$891$	0	0
$892$	− 14.8541i	− 0.497353i
$893$	15.0283i	0.502902i
$894$	0	0
$895$	6.32007i	0.211257i
$896$	0	0
$897$	0	0
$898$	14.6014	0.487254
$899$	10.0406	0.334872
$900$	0	0
$901$	5.86854i	0.195509i
$902$	−0.216614	−0.00721246
$903$	0	0
$904$	25.7834	0.857542
$905$	− 17.0295i	− 0.566078i
$906$	0	0
$907$	−20.3926	−0.677124	−0.338562	−	0.940944i	$-0.609941\pi$
$907$	−20.3926	−0.677124	−0.338562	+	0.940944i	$0.609941\pi$
$908$	−39.9100	−1.32446
$909$	0	0
$910$	0	0
$911$	23.6479i	0.783490i	0.920074	+	0.391745i	$0.128128\pi$
$911$	23.6479i	0.783490i	−0.920074	+	0.391745i	$0.871872\pi$
$912$	0	0
$913$	0.104411i	0.00345551i
$914$	− 6.71284i	− 0.222041i
$915$	0	0
$916$	− 15.0677i	− 0.497849i
$917$	0	0
$918$	0	0
$919$	−31.5979	−1.04232	−0.521160	−	0.853459i	$-0.674500\pi$
$919$	−31.5979	−1.04232	−0.521160	+	0.853459i	$0.674500\pi$
$920$	36.6211	1.20736
$921$	0	0
$922$	5.07148i	0.167020i
$923$	8.38362	0.275950
$924$	0	0
$925$	40.2981	1.32499
$926$	3.68139i	0.120978i
$927$	0	0
$928$	−10.0017	−0.328323
$929$	25.6468	0.841443	0.420722	−	0.907190i	$-0.361777\pi$
$929$	25.6468	0.841443	0.420722	+	0.907190i	$0.361777\pi$
$930$	0	0
$931$	0	0
$932$	28.5283i	0.934476i
$933$	0	0
$934$	10.1031i	0.330582i
$935$	− 0.0525940i	− 0.00172001i
$936$	0	0
$937$	4.78137i	0.156201i	0.996946	+	0.0781003i	$0.0248854\pi$
$937$	4.78137i	0.156201i	−0.996946	+	0.0781003i	$0.975115\pi$
$938$	0	0
$939$	0	0
$940$	9.82845	0.320569
$941$	−47.1928	−1.53844	−0.769221	−	0.638983i	$-0.779356\pi$
$941$	−47.1928	−1.53844	−0.769221	+	0.638983i	$0.779356\pi$
$942$	0	0
$943$	48.4559i	1.57794i
$944$	13.8273	0.450039
$945$	0	0
$946$	0.140645	0.00457275
$947$	20.7021i	0.672727i	0.941732	+	0.336363i	$0.109197\pi$
$947$	20.7021i	0.672727i	−0.941732	+	0.336363i	$0.890803\pi$
$948$	0	0
$949$	17.7781	0.577101
$950$	−34.1673	−1.10853
$951$	0	0
$952$	0	0
$953$	50.3100i	1.62970i	0.579671	+	0.814851i	$0.303181\pi$
$953$	50.3100i	1.62970i	−0.579671	+	0.814851i	$0.696819\pi$
$954$	0	0
$955$	− 41.2090i	− 1.33349i
$956$	− 37.5609i	− 1.21481i
$957$	0	0
$958$	− 14.3313i	− 0.463022i
$959$	0	0
$960$	0	0
$961$	−2.28098	−0.0735799
$962$	5.66289	0.182579
$963$	0	0
$964$	44.0561i	1.41895i
$965$	64.2058	2.06686
$966$	0	0
$967$	−48.7686	−1.56829	−0.784146	−	0.620577i	$-0.786899\pi$
$967$	−48.7686	−1.56829	−0.784146	+	0.620577i	$0.786899\pi$
$968$	25.8918i	0.832193i
$969$	0	0
$970$	31.3067	1.00520
$971$	−19.5162	−0.626303	−0.313152	−	0.949703i	$-0.601385\pi$
$971$	−19.5162	−0.626303	−0.313152	+	0.949703i	$0.601385\pi$
$972$	0	0
$973$	0	0
$974$	16.1986i	0.519035i
$975$	0	0
$976$	− 7.13327i	− 0.228330i
$977$	39.4057i	1.26070i	0.776310	+	0.630351i	$0.217089\pi$
$977$	39.4057i	1.26070i	−0.776310	+	0.630351i	$0.782911\pi$
$978$	0	0
$979$	− 0.00619572i	0 0.000198016i
$980$	0	0
$981$	0	0
$982$	0.103660	0.00330793
$983$	37.8359	1.20678	0.603389	−	0.797447i	$-0.293817\pi$
$983$	37.8359	1.20678	0.603389	+	0.797447i	$0.293817\pi$
$984$	0	0
$985$	49.4556i	1.57579i
$986$	−0.574321	−0.0182901
$987$	0	0
$988$	17.2062	0.547401
$989$	− 31.4618i	− 1.00043i
$990$	0	0
$991$	−0.440691	−0.0139990	−0.00699951	−	0.999976i	$-0.502228\pi$
$991$	−0.440691	−0.0139990	−0.00699951	+	0.999976i	$0.502228\pi$
$992$	33.1523	1.05259
$993$	0	0
$994$	0	0
$995$	− 49.9384i	− 1.58315i
$996$	0	0
$997$	16.4582i	0.521236i	0.965442	+	0.260618i	$0.0839263\pi$
$997$	16.4582i	0.521236i	−0.965442	+	0.260618i	$0.916074\pi$
$998$	− 21.4829i	− 0.680031i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3087.2.c.c.3086.6	yes	24
3.2	odd	2	inner	3087.2.c.c.3086.19	yes	24
7.6	odd	2	inner	3087.2.c.c.3086.20	yes	24
21.20	even	2	inner	3087.2.c.c.3086.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3087.2.c.c.3086.5	✓	24	21.20	even	2	inner
3087.2.c.c.3086.6	yes	24	1.1	even	1	trivial
3087.2.c.c.3086.19	yes	24	3.2	odd	2	inner
3087.2.c.c.3086.20	yes	24	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 3087 = 3^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3087.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.660558i q^{2} +1.56366 q^{4} +3.37724 q^{5} +2.35401i q^{8} +O(q^{10})\)	\(q+0.660558i q^{2} +1.56366 q^{4} +3.37724 q^{5} +2.35401i q^{8} +2.23086i q^{10} -0.0311739i q^{11} -1.36274i q^{13} +1.57237 q^{16} +0.499555 q^{17} +8.07475i q^{19} +5.28087 q^{20} +0.0205922 q^{22} -4.60641i q^{23} +6.40575 q^{25} +0.900167 q^{26} +1.74045i q^{29} -5.76897i q^{31} +5.74665i q^{32} +0.329985i q^{34} +6.29093 q^{37} -5.33384 q^{38} +7.95005i q^{40} -10.5192 q^{41} +6.83001 q^{43} -0.0487454i q^{44} +3.04280 q^{46} +1.86114 q^{47} +4.23137i q^{50} -2.13086i q^{52} +11.7475i q^{53} -0.105282i q^{55} -1.14967 q^{58} +8.79391 q^{59} -4.53664i q^{61} +3.81074 q^{62} -0.651263 q^{64} -4.60229i q^{65} +2.05552 q^{67} +0.781136 q^{68} +6.15204i q^{71} +13.0459i q^{73} +4.15552i q^{74} +12.6262i q^{76} -2.05331 q^{79} +5.31026 q^{80} -6.94858i q^{82} -3.34932 q^{83} +1.68712 q^{85} +4.51162i q^{86} +0.0733835 q^{88} +0.198747 q^{89} -7.20287i q^{92} +1.22939i q^{94} +27.2704i q^{95} -14.0334i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 40 q^{16} + 64 q^{22} + 136 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{46} - 128 q^{58} + 32 q^{64} + 40 q^{67} + 8 q^{79} + 64 q^{85} + 16 q^{88}+O(q^{100}) \) 24 * q - 40 * q^16 + 64 * q^22 + 136 * q^25 + 32 * q^37 + 32 * q^43 + 32 * q^46 - 128 * q^58 + 32 * q^64 + 40 * q^67 + 8 * q^79 + 64 * q^85 + 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more