LMFDB - Embedded newform 2646.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(1,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2646.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.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:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	2646.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +1.41421 q^{10} +2.24264 q^{11} +3.00000 q^{13} +1.00000 q^{16} +4.41421 q^{17} +4.58579 q^{19} +1.41421 q^{20} +2.24264 q^{22} -1.00000 q^{23} -3.00000 q^{25} +3.00000 q^{26} -5.24264 q^{29} -1.24264 q^{31} +1.00000 q^{32} +4.41421 q^{34} -10.4853 q^{37} +4.58579 q^{38} +1.41421 q^{40} -2.82843 q^{41} +3.24264 q^{43} +2.24264 q^{44} -1.00000 q^{46} -7.07107 q^{47} -3.00000 q^{50} +3.00000 q^{52} +11.2426 q^{53} +3.17157 q^{55} -5.24264 q^{58} +0.171573 q^{59} +11.6569 q^{61} -1.24264 q^{62} +1.00000 q^{64} +4.24264 q^{65} +13.2426 q^{67} +4.41421 q^{68} -5.48528 q^{71} +9.89949 q^{73} -10.4853 q^{74} +4.58579 q^{76} -10.7279 q^{79} +1.41421 q^{80} -2.82843 q^{82} +17.3137 q^{83} +6.24264 q^{85} +3.24264 q^{86} +2.24264 q^{88} -1.92893 q^{89} -1.00000 q^{92} -7.07107 q^{94} +6.48528 q^{95} +14.8284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 4 q^{11} + 6 q^{13} + 2 q^{16} + 6 q^{17} + 12 q^{19} - 4 q^{22} - 2 q^{23} - 6 q^{25} + 6 q^{26} - 2 q^{29} + 6 q^{31} + 2 q^{32} + 6 q^{34} - 4 q^{37} + 12 q^{38} - 2 q^{43} - 4 q^{44} - 2 q^{46} - 6 q^{50} + 6 q^{52} + 14 q^{53} + 12 q^{55} - 2 q^{58} + 6 q^{59} + 12 q^{61} + 6 q^{62} + 2 q^{64} + 18 q^{67} + 6 q^{68} + 6 q^{71} - 4 q^{74} + 12 q^{76} + 4 q^{79} + 12 q^{83} + 4 q^{85} - 2 q^{86} - 4 q^{88} - 18 q^{89} - 2 q^{92} - 4 q^{95} + 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 4 * q^11 + 6 * q^13 + 2 * q^16 + 6 * q^17 + 12 * q^19 - 4 * q^22 - 2 * q^23 - 6 * q^25 + 6 * q^26 - 2 * q^29 + 6 * q^31 + 2 * q^32 + 6 * q^34 - 4 * q^37 + 12 * q^38 - 2 * q^43 - 4 * q^44 - 2 * q^46 - 6 * q^50 + 6 * q^52 + 14 * q^53 + 12 * q^55 - 2 * q^58 + 6 * q^59 + 12 * q^61 + 6 * q^62 + 2 * q^64 + 18 * q^67 + 6 * q^68 + 6 * q^71 - 4 * q^74 + 12 * q^76 + 4 * q^79 + 12 * q^83 + 4 * q^85 - 2 * q^86 - 4 * q^88 - 18 * q^89 - 2 * q^92 - 4 * q^95 + 24 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	1.41421	0.447214
$11$	2.24264	0.676182	0.338091	−	0.941113i	$-0.390219\pi$
$11$	2.24264	0.676182	0.338091	+	0.941113i	$0.390219\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	4.41421	1.07060	0.535302	−	0.844661i	$-0.320198\pi$
$17$	4.41421	1.07060	0.535302	+	0.844661i	$0.320198\pi$
$18$	0	0
$19$	4.58579	1.05205	0.526026	−	0.850469i	$-0.323682\pi$
$19$	4.58579	1.05205	0.526026	+	0.850469i	$0.323682\pi$
$20$	1.41421	0.316228
$21$	0	0
$22$	2.24264	0.478133
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	3.00000	0.588348
$27$	0	0
$28$	0	0
$29$	−5.24264	−0.973534	−0.486767	−	0.873532i	$-0.661824\pi$
$29$	−5.24264	−0.973534	−0.486767	+	0.873532i	$0.661824\pi$
$30$	0	0
$31$	−1.24264	−0.223185	−0.111592	−	0.993754i	$-0.535595\pi$
$31$	−1.24264	−0.223185	−0.111592	+	0.993754i	$0.535595\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.41421	0.757031
$35$	0	0
$36$	0	0
$37$	−10.4853	−1.72377	−0.861885	−	0.507104i	$-0.830716\pi$
$37$	−10.4853	−1.72377	−0.861885	+	0.507104i	$0.830716\pi$
$38$	4.58579	0.743913
$39$	0	0
$40$	1.41421	0.223607
$41$	−2.82843	−0.441726	−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843	−0.441726	−0.220863	+	0.975305i	$0.570887\pi$
$42$	0	0
$43$	3.24264	0.494498	0.247249	−	0.968952i	$-0.420473\pi$
$43$	3.24264	0.494498	0.247249	+	0.968952i	$0.420473\pi$
$44$	2.24264	0.338091
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−7.07107	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$47$	−7.07107	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$48$	0	0
$49$	0	0
$50$	−3.00000	−0.424264
$51$	0	0
$52$	3.00000	0.416025
$53$	11.2426	1.54430	0.772148	−	0.635443i	$-0.219183\pi$
$53$	11.2426	1.54430	0.772148	+	0.635443i	$0.219183\pi$
$54$	0	0
$55$	3.17157	0.427655
$56$	0	0
$57$	0	0
$58$	−5.24264	−0.688392
$59$	0.171573	0.0223369	0.0111684	−	0.999938i	$-0.496445\pi$
$59$	0.171573	0.0223369	0.0111684	+	0.999938i	$0.496445\pi$
$60$	0	0
$61$	11.6569	1.49251	0.746254	−	0.665662i	$-0.231851\pi$
$61$	11.6569	1.49251	0.746254	+	0.665662i	$0.231851\pi$
$62$	−1.24264	−0.157816
$63$	0	0
$64$	1.00000	0.125000
$65$	4.24264	0.526235
$66$	0	0
$67$	13.2426	1.61785	0.808923	−	0.587915i	$-0.200051\pi$
$67$	13.2426	1.61785	0.808923	+	0.587915i	$0.200051\pi$
$68$	4.41421	0.535302
$69$	0	0
$70$	0	0
$71$	−5.48528	−0.650983	−0.325492	−	0.945545i	$-0.605530\pi$
$71$	−5.48528	−0.650983	−0.325492	+	0.945545i	$0.605530\pi$
$72$	0	0
$73$	9.89949	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$73$	9.89949	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$74$	−10.4853	−1.21889
$75$	0	0
$76$	4.58579	0.526026
$77$	0	0
$78$	0	0
$79$	−10.7279	−1.20699	−0.603493	−	0.797368i	$-0.706225\pi$
$79$	−10.7279	−1.20699	−0.603493	+	0.797368i	$0.706225\pi$
$80$	1.41421	0.158114
$81$	0	0
$82$	−2.82843	−0.312348
$83$	17.3137	1.90043	0.950213	−	0.311601i	$-0.100865\pi$
$83$	17.3137	1.90043	0.950213	+	0.311601i	$0.100865\pi$
$84$	0	0
$85$	6.24264	0.677109
$86$	3.24264	0.349663
$87$	0	0
$88$	2.24264	0.239066
$89$	−1.92893	−0.204466	−0.102233	−	0.994760i	$-0.532599\pi$
$89$	−1.92893	−0.204466	−0.102233	+	0.994760i	$0.532599\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−7.07107	−0.729325
$95$	6.48528	0.665376
$96$	0	0
$97$	14.8284	1.50560	0.752799	−	0.658250i	$-0.228703\pi$
$97$	14.8284	1.50560	0.752799	+	0.658250i	$0.228703\pi$
$98$	0	0
$99$	0	0
$100$	−3.00000	−0.300000
$101$	−10.2426	−1.01918	−0.509590	−	0.860417i	$-0.670203\pi$
$101$	−10.2426	−1.01918	−0.509590	+	0.860417i	$0.670203\pi$
$102$	0	0
$103$	1.92893	0.190063	0.0950317	−	0.995474i	$-0.469705\pi$
$103$	1.92893	0.190063	0.0950317	+	0.995474i	$0.469705\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	11.2426	1.09198
$107$	−0.485281	−0.0469139	−0.0234570	−	0.999725i	$-0.507467\pi$
$107$	−0.485281	−0.0469139	−0.0234570	+	0.999725i	$0.507467\pi$
$108$	0	0
$109$	0.485281	0.0464815	0.0232408	−	0.999730i	$-0.492602\pi$
$109$	0.485281	0.0464815	0.0232408	+	0.999730i	$0.492602\pi$
$110$	3.17157	0.302398
$111$	0	0
$112$	0	0
$113$	−12.7279	−1.19734	−0.598671	−	0.800995i	$-0.704304\pi$
$113$	−12.7279	−1.19734	−0.598671	+	0.800995i	$0.704304\pi$
$114$	0	0
$115$	−1.41421	−0.131876
$116$	−5.24264	−0.486767
$117$	0	0
$118$	0.171573	0.0157946
$119$	0	0
$120$	0	0
$121$	−5.97056	−0.542778
$122$	11.6569	1.05536
$123$	0	0
$124$	−1.24264	−0.111592
$125$	−11.3137	−1.01193
$126$	0	0
$127$	4.72792	0.419535	0.209768	−	0.977751i	$-0.432729\pi$
$127$	4.72792	0.419535	0.209768	+	0.977751i	$0.432729\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.24264	0.372104
$131$	−0.514719	−0.0449712	−0.0224856	−	0.999747i	$-0.507158\pi$
$131$	−0.514719	−0.0449712	−0.0224856	+	0.999747i	$0.507158\pi$
$132$	0	0
$133$	0	0
$134$	13.2426	1.14399
$135$	0	0
$136$	4.41421	0.378516
$137$	−13.7574	−1.17537	−0.587685	−	0.809090i	$-0.699961\pi$
$137$	−13.7574	−1.17537	−0.587685	+	0.809090i	$0.699961\pi$
$138$	0	0
$139$	17.3137	1.46853	0.734265	−	0.678863i	$-0.237527\pi$
$139$	17.3137	1.46853	0.734265	+	0.678863i	$0.237527\pi$
$140$	0	0
$141$	0	0
$142$	−5.48528	−0.460315
$143$	6.72792	0.562617
$144$	0	0
$145$	−7.41421	−0.615717
$146$	9.89949	0.819288
$147$	0	0
$148$	−10.4853	−0.861885
$149$	−9.24264	−0.757187	−0.378593	−	0.925563i	$-0.623592\pi$
$149$	−9.24264	−0.757187	−0.378593	+	0.925563i	$0.623592\pi$
$150$	0	0
$151$	14.7279	1.19854	0.599271	−	0.800546i	$-0.295457\pi$
$151$	14.7279	1.19854	0.599271	+	0.800546i	$0.295457\pi$
$152$	4.58579	0.371956
$153$	0	0
$154$	0	0
$155$	−1.75736	−0.141154
$156$	0	0
$157$	−14.6569	−1.16974	−0.584872	−	0.811125i	$-0.698855\pi$
$157$	−14.6569	−1.16974	−0.584872	+	0.811125i	$0.698855\pi$
$158$	−10.7279	−0.853468
$159$	0	0
$160$	1.41421	0.111803
$161$	0	0
$162$	0	0
$163$	−21.2426	−1.66385	−0.831926	−	0.554887i	$-0.812762\pi$
$163$	−21.2426	−1.66385	−0.831926	+	0.554887i	$0.812762\pi$
$164$	−2.82843	−0.220863
$165$	0	0
$166$	17.3137	1.34380
$167$	−3.89949	−0.301752	−0.150876	−	0.988553i	$-0.548209\pi$
$167$	−3.89949	−0.301752	−0.150876	+	0.988553i	$0.548209\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	6.24264	0.478789
$171$	0	0
$172$	3.24264	0.247249
$173$	−20.4853	−1.55747	−0.778734	−	0.627355i	$-0.784138\pi$
$173$	−20.4853	−1.55747	−0.778734	+	0.627355i	$0.784138\pi$
$174$	0	0
$175$	0	0
$176$	2.24264	0.169045
$177$	0	0
$178$	−1.92893	−0.144580
$179$	22.2426	1.66249	0.831247	−	0.555904i	$-0.187628\pi$
$179$	22.2426	1.66249	0.831247	+	0.555904i	$0.187628\pi$
$180$	0	0
$181$	6.51472	0.484235	0.242118	−	0.970247i	$-0.422158\pi$
$181$	6.51472	0.484235	0.242118	+	0.970247i	$0.422158\pi$
$182$	0	0
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−14.8284	−1.09021
$186$	0	0
$187$	9.89949	0.723923
$188$	−7.07107	−0.515711
$189$	0	0
$190$	6.48528	0.470492
$191$	−26.9706	−1.95152	−0.975761	−	0.218840i	$-0.929773\pi$
$191$	−26.9706	−1.95152	−0.975761	+	0.218840i	$0.929773\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	14.8284	1.06462
$195$	0	0
$196$	0	0
$197$	10.4853	0.747045	0.373523	−	0.927621i	$-0.378150\pi$
$197$	10.4853	0.747045	0.373523	+	0.927621i	$0.378150\pi$
$198$	0	0
$199$	13.2426	0.938746	0.469373	−	0.883000i	$-0.344480\pi$
$199$	13.2426	0.938746	0.469373	+	0.883000i	$0.344480\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−10.2426	−0.720670
$203$	0	0
$204$	0	0
$205$	−4.00000	−0.279372
$206$	1.92893	0.134395
$207$	0	0
$208$	3.00000	0.208013
$209$	10.2843	0.711378
$210$	0	0
$211$	19.2426	1.32472	0.662359	−	0.749187i	$-0.269555\pi$
$211$	19.2426	1.32472	0.662359	+	0.749187i	$0.269555\pi$
$212$	11.2426	0.772148
$213$	0	0
$214$	−0.485281	−0.0331732
$215$	4.58579	0.312748
$216$	0	0
$217$	0	0
$218$	0.485281	0.0328674
$219$	0	0
$220$	3.17157	0.213827
$221$	13.2426	0.890796
$222$	0	0
$223$	−16.9706	−1.13643	−0.568216	−	0.822879i	$-0.692366\pi$
$223$	−16.9706	−1.13643	−0.568216	+	0.822879i	$0.692366\pi$
$224$	0	0
$225$	0	0
$226$	−12.7279	−0.846649
$227$	−19.9706	−1.32549	−0.662746	−	0.748844i	$-0.730609\pi$
$227$	−19.9706	−1.32549	−0.662746	+	0.748844i	$0.730609\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	−1.41421	−0.0932505
$231$	0	0
$232$	−5.24264	−0.344196
$233$	8.48528	0.555889	0.277945	−	0.960597i	$-0.410347\pi$
$233$	8.48528	0.555889	0.277945	+	0.960597i	$0.410347\pi$
$234$	0	0
$235$	−10.0000	−0.652328
$236$	0.171573	0.0111684
$237$	0	0
$238$	0	0
$239$	22.9706	1.48584	0.742921	−	0.669379i	$-0.233440\pi$
$239$	22.9706	1.48584	0.742921	+	0.669379i	$0.233440\pi$
$240$	0	0
$241$	−0.727922	−0.0468896	−0.0234448	−	0.999725i	$-0.507463\pi$
$241$	−0.727922	−0.0468896	−0.0234448	+	0.999725i	$0.507463\pi$
$242$	−5.97056	−0.383802
$243$	0	0
$244$	11.6569	0.746254
$245$	0	0
$246$	0	0
$247$	13.7574	0.875360
$248$	−1.24264	−0.0789078
$249$	0	0
$250$	−11.3137	−0.715542
$251$	−28.6274	−1.80695	−0.903473	−	0.428644i	$-0.858991\pi$
$251$	−28.6274	−1.80695	−0.903473	+	0.428644i	$0.858991\pi$
$252$	0	0
$253$	−2.24264	−0.140994
$254$	4.72792	0.296656
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.9706	−1.43286	−0.716432	−	0.697657i	$-0.754226\pi$
$257$	−22.9706	−1.43286	−0.716432	+	0.697657i	$0.754226\pi$
$258$	0	0
$259$	0	0
$260$	4.24264	0.263117
$261$	0	0
$262$	−0.514719	−0.0317994
$263$	−6.51472	−0.401715	−0.200857	−	0.979620i	$-0.564373\pi$
$263$	−6.51472	−0.401715	−0.200857	+	0.979620i	$0.564373\pi$
$264$	0	0
$265$	15.8995	0.976698
$266$	0	0
$267$	0	0
$268$	13.2426	0.808923
$269$	3.55635	0.216834	0.108417	−	0.994105i	$-0.465422\pi$
$269$	3.55635	0.216834	0.108417	+	0.994105i	$0.465422\pi$
$270$	0	0
$271$	−6.89949	−0.419114	−0.209557	−	0.977796i	$-0.567202\pi$
$271$	−6.89949	−0.419114	−0.209557	+	0.977796i	$0.567202\pi$
$272$	4.41421	0.267651
$273$	0	0
$274$	−13.7574	−0.831112
$275$	−6.72792	−0.405709
$276$	0	0
$277$	−4.48528	−0.269494	−0.134747	−	0.990880i	$-0.543022\pi$
$277$	−4.48528	−0.269494	−0.134747	+	0.990880i	$0.543022\pi$
$278$	17.3137	1.03841
$279$	0	0
$280$	0	0
$281$	−22.4853	−1.34136	−0.670680	−	0.741747i	$-0.733997\pi$
$281$	−22.4853	−1.34136	−0.670680	+	0.741747i	$0.733997\pi$
$282$	0	0
$283$	17.6569	1.04959	0.524796	−	0.851228i	$-0.324142\pi$
$283$	17.6569	1.04959	0.524796	+	0.851228i	$0.324142\pi$
$284$	−5.48528	−0.325492
$285$	0	0
$286$	6.72792	0.397830
$287$	0	0
$288$	0	0
$289$	2.48528	0.146193
$290$	−7.41421	−0.435378
$291$	0	0
$292$	9.89949	0.579324
$293$	−27.1716	−1.58738	−0.793690	−	0.608322i	$-0.791843\pi$
$293$	−27.1716	−1.58738	−0.793690	+	0.608322i	$0.791843\pi$
$294$	0	0
$295$	0.242641	0.0141271
$296$	−10.4853	−0.609445
$297$	0	0
$298$	−9.24264	−0.535412
$299$	−3.00000	−0.173494
$300$	0	0
$301$	0	0
$302$	14.7279	0.847497
$303$	0	0
$304$	4.58579	0.263013
$305$	16.4853	0.943944
$306$	0	0
$307$	−12.3431	−0.704461	−0.352230	−	0.935913i	$-0.614577\pi$
$307$	−12.3431	−0.704461	−0.352230	+	0.935913i	$0.614577\pi$
$308$	0	0
$309$	0	0
$310$	−1.75736	−0.0998113
$311$	−0.686292	−0.0389160	−0.0194580	−	0.999811i	$-0.506194\pi$
$311$	−0.686292	−0.0389160	−0.0194580	+	0.999811i	$0.506194\pi$
$312$	0	0
$313$	−4.24264	−0.239808	−0.119904	−	0.992785i	$-0.538259\pi$
$313$	−4.24264	−0.239808	−0.119904	+	0.992785i	$0.538259\pi$
$314$	−14.6569	−0.827134
$315$	0	0
$316$	−10.7279	−0.603493
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	−11.7574	−0.658286
$320$	1.41421	0.0790569
$321$	0	0
$322$	0	0
$323$	20.2426	1.12633
$324$	0	0
$325$	−9.00000	−0.499230
$326$	−21.2426	−1.17652
$327$	0	0
$328$	−2.82843	−0.156174
$329$	0	0
$330$	0	0
$331$	1.72792	0.0949752	0.0474876	−	0.998872i	$-0.484879\pi$
$331$	1.72792	0.0949752	0.0474876	+	0.998872i	$0.484879\pi$
$332$	17.3137	0.950213
$333$	0	0
$334$	−3.89949	−0.213371
$335$	18.7279	1.02322
$336$	0	0
$337$	25.4853	1.38827	0.694136	−	0.719844i	$-0.255787\pi$
$337$	25.4853	1.38827	0.694136	+	0.719844i	$0.255787\pi$
$338$	−4.00000	−0.217571
$339$	0	0
$340$	6.24264	0.338555
$341$	−2.78680	−0.150913
$342$	0	0
$343$	0	0
$344$	3.24264	0.174831
$345$	0	0
$346$	−20.4853	−1.10130
$347$	2.48528	0.133417	0.0667084	−	0.997773i	$-0.478750\pi$
$347$	2.48528	0.133417	0.0667084	+	0.997773i	$0.478750\pi$
$348$	0	0
$349$	−3.00000	−0.160586	−0.0802932	−	0.996771i	$-0.525586\pi$
$349$	−3.00000	−0.160586	−0.0802932	+	0.996771i	$0.525586\pi$
$350$	0	0
$351$	0	0
$352$	2.24264	0.119533
$353$	−12.5563	−0.668307	−0.334154	−	0.942519i	$-0.608450\pi$
$353$	−12.5563	−0.668307	−0.334154	+	0.942519i	$0.608450\pi$
$354$	0	0
$355$	−7.75736	−0.411718
$356$	−1.92893	−0.102233
$357$	0	0
$358$	22.2426	1.17556
$359$	−21.9706	−1.15956	−0.579781	−	0.814772i	$-0.696862\pi$
$359$	−21.9706	−1.15956	−0.579781	+	0.814772i	$0.696862\pi$
$360$	0	0
$361$	2.02944	0.106812
$362$	6.51472	0.342406
$363$	0	0
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	−13.5858	−0.709172	−0.354586	−	0.935023i	$-0.615378\pi$
$367$	−13.5858	−0.709172	−0.354586	+	0.935023i	$0.615378\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−14.8284	−0.770893
$371$	0	0
$372$	0	0
$373$	33.6985	1.74484	0.872421	−	0.488756i	$-0.162549\pi$
$373$	33.6985	1.74484	0.872421	+	0.488756i	$0.162549\pi$
$374$	9.89949	0.511891
$375$	0	0
$376$	−7.07107	−0.364662
$377$	−15.7279	−0.810029
$378$	0	0
$379$	−18.9706	−0.974452	−0.487226	−	0.873276i	$-0.661991\pi$
$379$	−18.9706	−0.974452	−0.487226	+	0.873276i	$0.661991\pi$
$380$	6.48528	0.332688
$381$	0	0
$382$	−26.9706	−1.37993
$383$	9.55635	0.488307	0.244153	−	0.969737i	$-0.421490\pi$
$383$	9.55635	0.488307	0.244153	+	0.969737i	$0.421490\pi$
$384$	0	0
$385$	0	0
$386$	5.00000	0.254493
$387$	0	0
$388$	14.8284	0.752799
$389$	−38.9706	−1.97589	−0.987943	−	0.154818i	$-0.950521\pi$
$389$	−38.9706	−1.97589	−0.987943	+	0.154818i	$0.950521\pi$
$390$	0	0
$391$	−4.41421	−0.223236
$392$	0	0
$393$	0	0
$394$	10.4853	0.528241
$395$	−15.1716	−0.763365
$396$	0	0
$397$	14.8284	0.744217	0.372109	−	0.928189i	$-0.378635\pi$
$397$	14.8284	0.744217	0.372109	+	0.928189i	$0.378635\pi$
$398$	13.2426	0.663794
$399$	0	0
$400$	−3.00000	−0.150000
$401$	14.4853	0.723360	0.361680	−	0.932302i	$-0.382203\pi$
$401$	14.4853	0.723360	0.361680	+	0.932302i	$0.382203\pi$
$402$	0	0
$403$	−3.72792	−0.185701
$404$	−10.2426	−0.509590
$405$	0	0
$406$	0	0
$407$	−23.5147	−1.16558
$408$	0	0
$409$	14.4853	0.716251	0.358126	−	0.933673i	$-0.383416\pi$
$409$	14.4853	0.716251	0.358126	+	0.933673i	$0.383416\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	1.92893	0.0950317
$413$	0	0
$414$	0	0
$415$	24.4853	1.20194
$416$	3.00000	0.147087
$417$	0	0
$418$	10.2843	0.503020
$419$	26.6569	1.30227	0.651136	−	0.758961i	$-0.274293\pi$
$419$	26.6569	1.30227	0.651136	+	0.758961i	$0.274293\pi$
$420$	0	0
$421$	−40.2426	−1.96131	−0.980653	−	0.195753i	$-0.937285\pi$
$421$	−40.2426	−1.96131	−0.980653	+	0.195753i	$0.937285\pi$
$422$	19.2426	0.936717
$423$	0	0
$424$	11.2426	0.545991
$425$	−13.2426	−0.642362
$426$	0	0
$427$	0	0
$428$	−0.485281	−0.0234570
$429$	0	0
$430$	4.58579	0.221146
$431$	34.9706	1.68447	0.842236	−	0.539108i	$-0.181239\pi$
$431$	34.9706	1.68447	0.842236	+	0.539108i	$0.181239\pi$
$432$	0	0
$433$	−32.4853	−1.56114	−0.780571	−	0.625067i	$-0.785072\pi$
$433$	−32.4853	−1.56114	−0.780571	+	0.625067i	$0.785072\pi$
$434$	0	0
$435$	0	0
$436$	0.485281	0.0232408
$437$	−4.58579	−0.219368
$438$	0	0
$439$	12.2132	0.582904	0.291452	−	0.956585i	$-0.405862\pi$
$439$	12.2132	0.582904	0.291452	+	0.956585i	$0.405862\pi$
$440$	3.17157	0.151199
$441$	0	0
$442$	13.2426	0.629888
$443$	−26.9706	−1.28141	−0.640705	−	0.767787i	$-0.721358\pi$
$443$	−26.9706	−1.28141	−0.640705	+	0.767787i	$0.721358\pi$
$444$	0	0
$445$	−2.72792	−0.129316
$446$	−16.9706	−0.803579
$447$	0	0
$448$	0	0
$449$	−0.727922	−0.0343528	−0.0171764	−	0.999852i	$-0.505468\pi$
$449$	−0.727922	−0.0343528	−0.0171764	+	0.999852i	$0.505468\pi$
$450$	0	0
$451$	−6.34315	−0.298687
$452$	−12.7279	−0.598671
$453$	0	0
$454$	−19.9706	−0.937265
$455$	0	0
$456$	0	0
$457$	−15.0000	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$457$	−15.0000	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$458$	0	0
$459$	0	0
$460$	−1.41421	−0.0659380
$461$	1.07107	0.0498846	0.0249423	−	0.999689i	$-0.492060\pi$
$461$	1.07107	0.0498846	0.0249423	+	0.999689i	$0.492060\pi$
$462$	0	0
$463$	−23.2132	−1.07881	−0.539405	−	0.842047i	$-0.681351\pi$
$463$	−23.2132	−1.07881	−0.539405	+	0.842047i	$0.681351\pi$
$464$	−5.24264	−0.243383
$465$	0	0
$466$	8.48528	0.393073
$467$	23.6569	1.09471	0.547354	−	0.836901i	$-0.315635\pi$
$467$	23.6569	1.09471	0.547354	+	0.836901i	$0.315635\pi$
$468$	0	0
$469$	0	0
$470$	−10.0000	−0.461266
$471$	0	0
$472$	0.171573	0.00789728
$473$	7.27208	0.334370
$474$	0	0
$475$	−13.7574	−0.631231
$476$	0	0
$477$	0	0
$478$	22.9706	1.05065
$479$	−18.0416	−0.824343	−0.412172	−	0.911106i	$-0.635230\pi$
$479$	−18.0416	−0.824343	−0.412172	+	0.911106i	$0.635230\pi$
$480$	0	0
$481$	−31.4558	−1.43426
$482$	−0.727922	−0.0331559
$483$	0	0
$484$	−5.97056	−0.271389
$485$	20.9706	0.952224
$486$	0	0
$487$	−22.2426	−1.00791	−0.503955	−	0.863730i	$-0.668122\pi$
$487$	−22.2426	−1.00791	−0.503955	+	0.863730i	$0.668122\pi$
$488$	11.6569	0.527681
$489$	0	0
$490$	0	0
$491$	−10.2426	−0.462244	−0.231122	−	0.972925i	$-0.574240\pi$
$491$	−10.2426	−0.462244	−0.231122	+	0.972925i	$0.574240\pi$
$492$	0	0
$493$	−23.1421	−1.04227
$494$	13.7574	0.618973
$495$	0	0
$496$	−1.24264	−0.0557962
$497$	0	0
$498$	0	0
$499$	24.9706	1.11784	0.558918	−	0.829223i	$-0.311217\pi$
$499$	24.9706	1.11784	0.558918	+	0.829223i	$0.311217\pi$
$500$	−11.3137	−0.505964
$501$	0	0
$502$	−28.6274	−1.27770
$503$	8.10051	0.361184	0.180592	−	0.983558i	$-0.442199\pi$
$503$	8.10051	0.361184	0.180592	+	0.983558i	$0.442199\pi$
$504$	0	0
$505$	−14.4853	−0.644587
$506$	−2.24264	−0.0996975
$507$	0	0
$508$	4.72792	0.209768
$509$	−8.14214	−0.360894	−0.180447	−	0.983585i	$-0.557754\pi$
$509$	−8.14214	−0.360894	−0.180447	+	0.983585i	$0.557754\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−22.9706	−1.01319
$515$	2.72792	0.120207
$516$	0	0
$517$	−15.8579	−0.697428
$518$	0	0
$519$	0	0
$520$	4.24264	0.186052
$521$	10.4142	0.456255	0.228127	−	0.973631i	$-0.426740\pi$
$521$	10.4142	0.456255	0.228127	+	0.973631i	$0.426740\pi$
$522$	0	0
$523$	−11.2721	−0.492894	−0.246447	−	0.969156i	$-0.579263\pi$
$523$	−11.2721	−0.492894	−0.246447	+	0.969156i	$0.579263\pi$
$524$	−0.514719	−0.0224856
$525$	0	0
$526$	−6.51472	−0.284055
$527$	−5.48528	−0.238943
$528$	0	0
$529$	−22.0000	−0.956522
$530$	15.8995	0.690630
$531$	0	0
$532$	0	0
$533$	−8.48528	−0.367538
$534$	0	0
$535$	−0.686292	−0.0296710
$536$	13.2426	0.571995
$537$	0	0
$538$	3.55635	0.153325
$539$	0	0
$540$	0	0
$541$	10.7279	0.461229	0.230615	−	0.973045i	$-0.425926\pi$
$541$	10.7279	0.461229	0.230615	+	0.973045i	$0.425926\pi$
$542$	−6.89949	−0.296359
$543$	0	0
$544$	4.41421	0.189258
$545$	0.686292	0.0293975
$546$	0	0
$547$	−4.48528	−0.191777	−0.0958884	−	0.995392i	$-0.530569\pi$
$547$	−4.48528	−0.191777	−0.0958884	+	0.995392i	$0.530569\pi$
$548$	−13.7574	−0.587685
$549$	0	0
$550$	−6.72792	−0.286880
$551$	−24.0416	−1.02421
$552$	0	0
$553$	0	0
$554$	−4.48528	−0.190561
$555$	0	0
$556$	17.3137	0.734265
$557$	−14.6985	−0.622795	−0.311397	−	0.950280i	$-0.600797\pi$
$557$	−14.6985	−0.622795	−0.311397	+	0.950280i	$0.600797\pi$
$558$	0	0
$559$	9.72792	0.411447
$560$	0	0
$561$	0	0
$562$	−22.4853	−0.948484
$563$	41.1421	1.73393	0.866967	−	0.498365i	$-0.166066\pi$
$563$	41.1421	1.73393	0.866967	+	0.498365i	$0.166066\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	17.6569	0.742173
$567$	0	0
$568$	−5.48528	−0.230157
$569$	1.21320	0.0508601	0.0254301	−	0.999677i	$-0.491904\pi$
$569$	1.21320	0.0508601	0.0254301	+	0.999677i	$0.491904\pi$
$570$	0	0
$571$	−3.24264	−0.135700	−0.0678501	−	0.997696i	$-0.521614\pi$
$571$	−3.24264	−0.135700	−0.0678501	+	0.997696i	$0.521614\pi$
$572$	6.72792	0.281309
$573$	0	0
$574$	0	0
$575$	3.00000	0.125109
$576$	0	0
$577$	−2.48528	−0.103464	−0.0517318	−	0.998661i	$-0.516474\pi$
$577$	−2.48528	−0.103464	−0.0517318	+	0.998661i	$0.516474\pi$
$578$	2.48528	0.103374
$579$	0	0
$580$	−7.41421	−0.307858
$581$	0	0
$582$	0	0
$583$	25.2132	1.04422
$584$	9.89949	0.409644
$585$	0	0
$586$	−27.1716	−1.12245
$587$	40.4558	1.66979	0.834896	−	0.550408i	$-0.185528\pi$
$587$	40.4558	1.66979	0.834896	+	0.550408i	$0.185528\pi$
$588$	0	0
$589$	−5.69848	−0.234802
$590$	0.242641	0.00998936
$591$	0	0
$592$	−10.4853	−0.430942
$593$	−20.1421	−0.827138	−0.413569	−	0.910473i	$-0.635718\pi$
$593$	−20.1421	−0.827138	−0.413569	+	0.910473i	$0.635718\pi$
$594$	0	0
$595$	0	0
$596$	−9.24264	−0.378593
$597$	0	0
$598$	−3.00000	−0.122679
$599$	−7.00000	−0.286012	−0.143006	−	0.989722i	$-0.545677\pi$
$599$	−7.00000	−0.286012	−0.143006	+	0.989722i	$0.545677\pi$
$600$	0	0
$601$	−34.2426	−1.39679	−0.698393	−	0.715714i	$-0.746101\pi$
$601$	−34.2426	−1.39679	−0.698393	+	0.715714i	$0.746101\pi$
$602$	0	0
$603$	0	0
$604$	14.7279	0.599271
$605$	−8.44365	−0.343283
$606$	0	0
$607$	−32.6985	−1.32719	−0.663595	−	0.748092i	$-0.730970\pi$
$607$	−32.6985	−1.32719	−0.663595	+	0.748092i	$0.730970\pi$
$608$	4.58579	0.185978
$609$	0	0
$610$	16.4853	0.667470
$611$	−21.2132	−0.858194
$612$	0	0
$613$	28.2426	1.14071	0.570355	−	0.821398i	$-0.306806\pi$
$613$	28.2426	1.14071	0.570355	+	0.821398i	$0.306806\pi$
$614$	−12.3431	−0.498129
$615$	0	0
$616$	0	0
$617$	−35.9411	−1.44694	−0.723468	−	0.690358i	$-0.757453\pi$
$617$	−35.9411	−1.44694	−0.723468	+	0.690358i	$0.757453\pi$
$618$	0	0
$619$	8.82843	0.354844	0.177422	−	0.984135i	$-0.443224\pi$
$619$	8.82843	0.354844	0.177422	+	0.984135i	$0.443224\pi$
$620$	−1.75736	−0.0705772
$621$	0	0
$622$	−0.686292	−0.0275178
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	−4.24264	−0.169570
$627$	0	0
$628$	−14.6569	−0.584872
$629$	−46.2843	−1.84547
$630$	0	0
$631$	3.75736	0.149578	0.0747891	−	0.997199i	$-0.476172\pi$
$631$	3.75736	0.149578	0.0747891	+	0.997199i	$0.476172\pi$
$632$	−10.7279	−0.426734
$633$	0	0
$634$	10.0000	0.397151
$635$	6.68629	0.265337
$636$	0	0
$637$	0	0
$638$	−11.7574	−0.465478
$639$	0	0
$640$	1.41421	0.0559017
$641$	9.45584	0.373483	0.186742	−	0.982409i	$-0.440207\pi$
$641$	9.45584	0.373483	0.186742	+	0.982409i	$0.440207\pi$
$642$	0	0
$643$	−16.9289	−0.667612	−0.333806	−	0.942642i	$-0.608333\pi$
$643$	−16.9289	−0.667612	−0.333806	+	0.942642i	$0.608333\pi$
$644$	0	0
$645$	0	0
$646$	20.2426	0.796436
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	0.384776	0.0151038
$650$	−9.00000	−0.353009
$651$	0	0
$652$	−21.2426	−0.831926
$653$	32.6985	1.27959	0.639795	−	0.768545i	$-0.279019\pi$
$653$	32.6985	1.27959	0.639795	+	0.768545i	$0.279019\pi$
$654$	0	0
$655$	−0.727922	−0.0284423
$656$	−2.82843	−0.110432
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	21.9411	0.853411	0.426705	−	0.904391i	$-0.359674\pi$
$661$	21.9411	0.853411	0.426705	+	0.904391i	$0.359674\pi$
$662$	1.72792	0.0671576
$663$	0	0
$664$	17.3137	0.671902
$665$	0	0
$666$	0	0
$667$	5.24264	0.202996
$668$	−3.89949	−0.150876
$669$	0	0
$670$	18.7279	0.723523
$671$	26.1421	1.00921
$672$	0	0
$673$	−28.4558	−1.09689	−0.548446	−	0.836186i	$-0.684780\pi$
$673$	−28.4558	−1.09689	−0.548446	+	0.836186i	$0.684780\pi$
$674$	25.4853	0.981656
$675$	0	0
$676$	−4.00000	−0.153846
$677$	29.6985	1.14141	0.570703	−	0.821157i	$-0.306671\pi$
$677$	29.6985	1.14141	0.570703	+	0.821157i	$0.306671\pi$
$678$	0	0
$679$	0	0
$680$	6.24264	0.239394
$681$	0	0
$682$	−2.78680	−0.106712
$683$	−13.7574	−0.526411	−0.263205	−	0.964740i	$-0.584780\pi$
$683$	−13.7574	−0.526411	−0.263205	+	0.964740i	$0.584780\pi$
$684$	0	0
$685$	−19.4558	−0.743370
$686$	0	0
$687$	0	0
$688$	3.24264	0.123625
$689$	33.7279	1.28493
$690$	0	0
$691$	33.2132	1.26349	0.631745	−	0.775176i	$-0.282339\pi$
$691$	33.2132	1.26349	0.631745	+	0.775176i	$0.282339\pi$
$692$	−20.4853	−0.778734
$693$	0	0
$694$	2.48528	0.0943400
$695$	24.4853	0.928780
$696$	0	0
$697$	−12.4853	−0.472914
$698$	−3.00000	−0.113552
$699$	0	0
$700$	0	0
$701$	1.02944	0.0388813	0.0194407	−	0.999811i	$-0.493811\pi$
$701$	1.02944	0.0388813	0.0194407	+	0.999811i	$0.493811\pi$
$702$	0	0
$703$	−48.0833	−1.81349
$704$	2.24264	0.0845227
$705$	0	0
$706$	−12.5563	−0.472564
$707$	0	0
$708$	0	0
$709$	19.2132	0.721567	0.360784	−	0.932650i	$-0.382509\pi$
$709$	19.2132	0.721567	0.360784	+	0.932650i	$0.382509\pi$
$710$	−7.75736	−0.291129
$711$	0	0
$712$	−1.92893	−0.0722898
$713$	1.24264	0.0465373
$714$	0	0
$715$	9.51472	0.355830
$716$	22.2426	0.831247
$717$	0	0
$718$	−21.9706	−0.819934
$719$	34.6274	1.29138	0.645692	−	0.763598i	$-0.276569\pi$
$719$	34.6274	1.29138	0.645692	+	0.763598i	$0.276569\pi$
$720$	0	0
$721$	0	0
$722$	2.02944	0.0755278
$723$	0	0
$724$	6.51472	0.242118
$725$	15.7279	0.584120
$726$	0	0
$727$	−46.8406	−1.73722	−0.868611	−	0.495494i	$-0.834987\pi$
$727$	−46.8406	−1.73722	−0.868611	+	0.495494i	$0.834987\pi$
$728$	0	0
$729$	0	0
$730$	14.0000	0.518163
$731$	14.3137	0.529412
$732$	0	0
$733$	−36.9411	−1.36445	−0.682226	−	0.731142i	$-0.738988\pi$
$733$	−36.9411	−1.36445	−0.682226	+	0.731142i	$0.738988\pi$
$734$	−13.5858	−0.501461
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	29.6985	1.09396
$738$	0	0
$739$	22.9706	0.844986	0.422493	−	0.906366i	$-0.361155\pi$
$739$	22.9706	0.844986	0.422493	+	0.906366i	$0.361155\pi$
$740$	−14.8284	−0.545104
$741$	0	0
$742$	0	0
$743$	21.4853	0.788219	0.394109	−	0.919064i	$-0.371053\pi$
$743$	21.4853	0.788219	0.394109	+	0.919064i	$0.371053\pi$
$744$	0	0
$745$	−13.0711	−0.478887
$746$	33.6985	1.23379
$747$	0	0
$748$	9.89949	0.361961
$749$	0	0
$750$	0	0
$751$	−22.0000	−0.802791	−0.401396	−	0.915905i	$-0.631475\pi$
$751$	−22.0000	−0.802791	−0.401396	+	0.915905i	$0.631475\pi$
$752$	−7.07107	−0.257855
$753$	0	0
$754$	−15.7279	−0.572777
$755$	20.8284	0.758024
$756$	0	0
$757$	31.6985	1.15210	0.576051	−	0.817414i	$-0.304593\pi$
$757$	31.6985	1.15210	0.576051	+	0.817414i	$0.304593\pi$
$758$	−18.9706	−0.689042
$759$	0	0
$760$	6.48528	0.235246
$761$	11.5269	0.417850	0.208925	−	0.977932i	$-0.433004\pi$
$761$	11.5269	0.417850	0.208925	+	0.977932i	$0.433004\pi$
$762$	0	0
$763$	0	0
$764$	−26.9706	−0.975761
$765$	0	0
$766$	9.55635	0.345285
$767$	0.514719	0.0185854
$768$	0	0
$769$	−27.2132	−0.981333	−0.490667	−	0.871347i	$-0.663247\pi$
$769$	−27.2132	−0.981333	−0.490667	+	0.871347i	$0.663247\pi$
$770$	0	0
$771$	0	0
$772$	5.00000	0.179954
$773$	−18.3848	−0.661254	−0.330627	−	0.943761i	$-0.607260\pi$
$773$	−18.3848	−0.661254	−0.330627	+	0.943761i	$0.607260\pi$
$774$	0	0
$775$	3.72792	0.133911
$776$	14.8284	0.532310
$777$	0	0
$778$	−38.9706	−1.39716
$779$	−12.9706	−0.464719
$780$	0	0
$781$	−12.3015	−0.440183
$782$	−4.41421	−0.157852
$783$	0	0
$784$	0	0
$785$	−20.7279	−0.739811
$786$	0	0
$787$	−32.8284	−1.17021	−0.585104	−	0.810959i	$-0.698946\pi$
$787$	−32.8284	−1.17021	−0.585104	+	0.810959i	$0.698946\pi$
$788$	10.4853	0.373523
$789$	0	0
$790$	−15.1716	−0.539780
$791$	0	0
$792$	0	0
$793$	34.9706	1.24184
$794$	14.8284	0.526241
$795$	0	0
$796$	13.2426	0.469373
$797$	−33.5147	−1.18715	−0.593576	−	0.804778i	$-0.702284\pi$
$797$	−33.5147	−1.18715	−0.593576	+	0.804778i	$0.702284\pi$
$798$	0	0
$799$	−31.2132	−1.10424
$800$	−3.00000	−0.106066
$801$	0	0
$802$	14.4853	0.511493
$803$	22.2010	0.783457
$804$	0	0
$805$	0	0
$806$	−3.72792	−0.131310
$807$	0	0
$808$	−10.2426	−0.360335
$809$	−4.00000	−0.140633	−0.0703163	−	0.997525i	$-0.522401\pi$
$809$	−4.00000	−0.140633	−0.0703163	+	0.997525i	$0.522401\pi$
$810$	0	0
$811$	51.9411	1.82390	0.911950	−	0.410302i	$-0.134577\pi$
$811$	51.9411	1.82390	0.911950	+	0.410302i	$0.134577\pi$
$812$	0	0
$813$	0	0
$814$	−23.5147	−0.824190
$815$	−30.0416	−1.05231
$816$	0	0
$817$	14.8701	0.520237
$818$	14.4853	0.506466
$819$	0	0
$820$	−4.00000	−0.139686
$821$	18.2132	0.635645	0.317823	−	0.948150i	$-0.397048\pi$
$821$	18.2132	0.635645	0.317823	+	0.948150i	$0.397048\pi$
$822$	0	0
$823$	−6.97056	−0.242979	−0.121489	−	0.992593i	$-0.538767\pi$
$823$	−6.97056	−0.242979	−0.121489	+	0.992593i	$0.538767\pi$
$824$	1.92893	0.0671975
$825$	0	0
$826$	0	0
$827$	29.6985	1.03272	0.516359	−	0.856372i	$-0.327287\pi$
$827$	29.6985	1.03272	0.516359	+	0.856372i	$0.327287\pi$
$828$	0	0
$829$	21.1716	0.735319	0.367660	−	0.929960i	$-0.380159\pi$
$829$	21.1716	0.735319	0.367660	+	0.929960i	$0.380159\pi$
$830$	24.4853	0.849897
$831$	0	0
$832$	3.00000	0.104006
$833$	0	0
$834$	0	0
$835$	−5.51472	−0.190845
$836$	10.2843	0.355689
$837$	0	0
$838$	26.6569	0.920846
$839$	−1.79899	−0.0621080	−0.0310540	−	0.999518i	$-0.509886\pi$
$839$	−1.79899	−0.0621080	−0.0310540	+	0.999518i	$0.509886\pi$
$840$	0	0
$841$	−1.51472	−0.0522317
$842$	−40.2426	−1.38685
$843$	0	0
$844$	19.2426	0.662359
$845$	−5.65685	−0.194602
$846$	0	0
$847$	0	0
$848$	11.2426	0.386074
$849$	0	0
$850$	−13.2426	−0.454219
$851$	10.4853	0.359431
$852$	0	0
$853$	45.4264	1.55537	0.777685	−	0.628654i	$-0.216394\pi$
$853$	45.4264	1.55537	0.777685	+	0.628654i	$0.216394\pi$
$854$	0	0
$855$	0	0
$856$	−0.485281	−0.0165866
$857$	43.5858	1.48886	0.744431	−	0.667699i	$-0.232721\pi$
$857$	43.5858	1.48886	0.744431	+	0.667699i	$0.232721\pi$
$858$	0	0
$859$	−42.7696	−1.45928	−0.729639	−	0.683832i	$-0.760312\pi$
$859$	−42.7696	−1.45928	−0.729639	+	0.683832i	$0.760312\pi$
$860$	4.58579	0.156374
$861$	0	0
$862$	34.9706	1.19110
$863$	2.51472	0.0856020	0.0428010	−	0.999084i	$-0.486372\pi$
$863$	2.51472	0.0856020	0.0428010	+	0.999084i	$0.486372\pi$
$864$	0	0
$865$	−28.9706	−0.985029
$866$	−32.4853	−1.10389
$867$	0	0
$868$	0	0
$869$	−24.0589	−0.816141
$870$	0	0
$871$	39.7279	1.34613
$872$	0.485281	0.0164337
$873$	0	0
$874$	−4.58579	−0.155117
$875$	0	0
$876$	0	0
$877$	4.48528	0.151457	0.0757286	−	0.997128i	$-0.475872\pi$
$877$	4.48528	0.151457	0.0757286	+	0.997128i	$0.475872\pi$
$878$	12.2132	0.412176
$879$	0	0
$880$	3.17157	0.106914
$881$	−8.27208	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$881$	−8.27208	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$882$	0	0
$883$	−37.7279	−1.26965	−0.634823	−	0.772658i	$-0.718927\pi$
$883$	−37.7279	−1.26965	−0.634823	+	0.772658i	$0.718927\pi$
$884$	13.2426	0.445398
$885$	0	0
$886$	−26.9706	−0.906094
$887$	38.8284	1.30373	0.651865	−	0.758335i	$-0.273987\pi$
$887$	38.8284	1.30373	0.651865	+	0.758335i	$0.273987\pi$
$888$	0	0
$889$	0	0
$890$	−2.72792	−0.0914402
$891$	0	0
$892$	−16.9706	−0.568216
$893$	−32.4264	−1.08511
$894$	0	0
$895$	31.4558	1.05145
$896$	0	0
$897$	0	0
$898$	−0.727922	−0.0242911
$899$	6.51472	0.217278
$900$	0	0
$901$	49.6274	1.65333
$902$	−6.34315	−0.211204
$903$	0	0
$904$	−12.7279	−0.423324
$905$	9.21320	0.306257
$906$	0	0
$907$	−23.9411	−0.794952	−0.397476	−	0.917613i	$-0.630114\pi$
$907$	−23.9411	−0.794952	−0.397476	+	0.917613i	$0.630114\pi$
$908$	−19.9706	−0.662746
$909$	0	0
$910$	0	0
$911$	34.4853	1.14255	0.571274	−	0.820759i	$-0.306449\pi$
$911$	34.4853	1.14255	0.571274	+	0.820759i	$0.306449\pi$
$912$	0	0
$913$	38.8284	1.28503
$914$	−15.0000	−0.496156
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	14.2426	0.469821	0.234911	−	0.972017i	$-0.424520\pi$
$919$	14.2426	0.469821	0.234911	+	0.972017i	$0.424520\pi$
$920$	−1.41421	−0.0466252
$921$	0	0
$922$	1.07107	0.0352737
$923$	−16.4558	−0.541651
$924$	0	0
$925$	31.4558	1.03426
$926$	−23.2132	−0.762833
$927$	0	0
$928$	−5.24264	−0.172098
$929$	18.6863	0.613077	0.306539	−	0.951858i	$-0.400829\pi$
$929$	18.6863	0.613077	0.306539	+	0.951858i	$0.400829\pi$
$930$	0	0
$931$	0	0
$932$	8.48528	0.277945
$933$	0	0
$934$	23.6569	0.774076
$935$	14.0000	0.457849
$936$	0	0
$937$	−22.9289	−0.749056	−0.374528	−	0.927216i	$-0.622195\pi$
$937$	−22.9289	−0.749056	−0.374528	+	0.927216i	$0.622195\pi$
$938$	0	0
$939$	0	0
$940$	−10.0000	−0.326164
$941$	−34.2843	−1.11764	−0.558818	−	0.829291i	$-0.688745\pi$
$941$	−34.2843	−1.11764	−0.558818	+	0.829291i	$0.688745\pi$
$942$	0	0
$943$	2.82843	0.0921063
$944$	0.171573	0.00558422
$945$	0	0
$946$	7.27208	0.236436
$947$	−11.2721	−0.366293	−0.183147	−	0.983086i	$-0.558628\pi$
$947$	−11.2721	−0.366293	−0.183147	+	0.983086i	$0.558628\pi$
$948$	0	0
$949$	29.6985	0.964054
$950$	−13.7574	−0.446348
$951$	0	0
$952$	0	0
$953$	−49.6985	−1.60989	−0.804946	−	0.593348i	$-0.797806\pi$
$953$	−49.6985	−1.60989	−0.804946	+	0.593348i	$0.797806\pi$
$954$	0	0
$955$	−38.1421	−1.23425
$956$	22.9706	0.742921
$957$	0	0
$958$	−18.0416	−0.582899
$959$	0	0
$960$	0	0
$961$	−29.4558	−0.950189
$962$	−31.4558	−1.01418
$963$	0	0
$964$	−0.727922	−0.0234448
$965$	7.07107	0.227626
$966$	0	0
$967$	55.1543	1.77364	0.886822	−	0.462112i	$-0.152908\pi$
$967$	55.1543	1.77364	0.886822	+	0.462112i	$0.152908\pi$
$968$	−5.97056	−0.191901
$969$	0	0
$970$	20.9706	0.673324
$971$	−45.7696	−1.46881	−0.734407	−	0.678709i	$-0.762540\pi$
$971$	−45.7696	−1.46881	−0.734407	+	0.678709i	$0.762540\pi$
$972$	0	0
$973$	0	0
$974$	−22.2426	−0.712700
$975$	0	0
$976$	11.6569	0.373127
$977$	−40.6690	−1.30112	−0.650559	−	0.759456i	$-0.725465\pi$
$977$	−40.6690	−1.30112	−0.650559	+	0.759456i	$0.725465\pi$
$978$	0	0
$979$	−4.32590	−0.138256
$980$	0	0
$981$	0	0
$982$	−10.2426	−0.326856
$983$	21.5563	0.687541	0.343770	−	0.939054i	$-0.388296\pi$
$983$	21.5563	0.687541	0.343770	+	0.939054i	$0.388296\pi$
$984$	0	0
$985$	14.8284	0.472473
$986$	−23.1421	−0.736996
$987$	0	0
$988$	13.7574	0.437680
$989$	−3.24264	−0.103110
$990$	0	0
$991$	−11.4558	−0.363907	−0.181953	−	0.983307i	$-0.558242\pi$
$991$	−11.4558	−0.363907	−0.181953	+	0.983307i	$0.558242\pi$
$992$	−1.24264	−0.0394539
$993$	0	0
$994$	0	0
$995$	18.7279	0.593715
$996$	0	0
$997$	−18.8579	−0.597235	−0.298617	−	0.954373i	$-0.596525\pi$
$997$	−18.8579	−0.597235	−0.298617	+	0.954373i	$0.596525\pi$
$998$	24.9706	0.790429
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.a.bn.1.2	yes	2
3.2	odd	2		2646.2.a.bh.1.1	yes	2
7.6	odd	2		2646.2.a.bm.1.1	yes	2
21.20	even	2		2646.2.a.bg.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.a.bg.1.2	✓	2	21.20	even	2
2646.2.a.bh.1.1	yes	2	3.2	odd	2
2646.2.a.bm.1.1	yes	2	7.6	odd	2
2646.2.a.bn.1.2	yes	2	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:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.41421 q^{5} +1.00000 q^{8} +1.41421 q^{10} +2.24264 q^{11} +3.00000 q^{13} +1.00000 q^{16} +4.41421 q^{17} +4.58579 q^{19} +1.41421 q^{20} +2.24264 q^{22} -1.00000 q^{23} -3.00000 q^{25} +3.00000 q^{26} -5.24264 q^{29} -1.24264 q^{31} +1.00000 q^{32} +4.41421 q^{34} -10.4853 q^{37} +4.58579 q^{38} +1.41421 q^{40} -2.82843 q^{41} +3.24264 q^{43} +2.24264 q^{44} -1.00000 q^{46} -7.07107 q^{47} -3.00000 q^{50} +3.00000 q^{52} +11.2426 q^{53} +3.17157 q^{55} -5.24264 q^{58} +0.171573 q^{59} +11.6569 q^{61} -1.24264 q^{62} +1.00000 q^{64} +4.24264 q^{65} +13.2426 q^{67} +4.41421 q^{68} -5.48528 q^{71} +9.89949 q^{73} -10.4853 q^{74} +4.58579 q^{76} -10.7279 q^{79} +1.41421 q^{80} -2.82843 q^{82} +17.3137 q^{83} +6.24264 q^{85} +3.24264 q^{86} +2.24264 q^{88} -1.92893 q^{89} -1.00000 q^{92} -7.07107 q^{94} +6.48528 q^{95} +14.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 4 q^{11} + 6 q^{13} + 2 q^{16} + 6 q^{17} + 12 q^{19} - 4 q^{22} - 2 q^{23} - 6 q^{25} + 6 q^{26} - 2 q^{29} + 6 q^{31} + 2 q^{32} + 6 q^{34} - 4 q^{37} + 12 q^{38} - 2 q^{43} - 4 q^{44} - 2 q^{46} - 6 q^{50} + 6 q^{52} + 14 q^{53} + 12 q^{55} - 2 q^{58} + 6 q^{59} + 12 q^{61} + 6 q^{62} + 2 q^{64} + 18 q^{67} + 6 q^{68} + 6 q^{71} - 4 q^{74} + 12 q^{76} + 4 q^{79} + 12 q^{83} + 4 q^{85} - 2 q^{86} - 4 q^{88} - 18 q^{89} - 2 q^{92} - 4 q^{95} + 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 4 * q^11 + 6 * q^13 + 2 * q^16 + 6 * q^17 + 12 * q^19 - 4 * q^22 - 2 * q^23 - 6 * q^25 + 6 * q^26 - 2 * q^29 + 6 * q^31 + 2 * q^32 + 6 * q^34 - 4 * q^37 + 12 * q^38 - 2 * q^43 - 4 * q^44 - 2 * q^46 - 6 * q^50 + 6 * q^52 + 14 * q^53 + 12 * q^55 - 2 * q^58 + 6 * q^59 + 12 * q^61 + 6 * q^62 + 2 * q^64 + 18 * q^67 + 6 * q^68 + 6 * q^71 - 4 * q^74 + 12 * q^76 + 4 * q^79 + 12 * q^83 + 4 * q^85 - 2 * q^86 - 4 * q^88 - 18 * q^89 - 2 * q^92 - 4 * q^95 + 24 * q^97

Introduction

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