LMFDB - Embedded newform 2646.2.a.bn.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} -1.41421 q^{10} -6.24264 q^{11} +3.00000 q^{13} +1.00000 q^{16} +1.58579 q^{17} +7.41421 q^{19} -1.41421 q^{20} -6.24264 q^{22} -1.00000 q^{23} -3.00000 q^{25} +3.00000 q^{26} +3.24264 q^{29} +7.24264 q^{31} +1.00000 q^{32} +1.58579 q^{34} +6.48528 q^{37} +7.41421 q^{38} -1.41421 q^{40} +2.82843 q^{41} -5.24264 q^{43} -6.24264 q^{44} -1.00000 q^{46} +7.07107 q^{47} -3.00000 q^{50} +3.00000 q^{52} +2.75736 q^{53} +8.82843 q^{55} +3.24264 q^{58} +5.82843 q^{59} +0.343146 q^{61} +7.24264 q^{62} +1.00000 q^{64} -4.24264 q^{65} +4.75736 q^{67} +1.58579 q^{68} +11.4853 q^{71} -9.89949 q^{73} +6.48528 q^{74} +7.41421 q^{76} +14.7279 q^{79} -1.41421 q^{80} +2.82843 q^{82} -5.31371 q^{83} -2.24264 q^{85} -5.24264 q^{86} -6.24264 q^{88} -16.0711 q^{89} -1.00000 q^{92} +7.07107 q^{94} -10.4853 q^{95} +9.17157 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.41421	−0.447214
$11$	−6.24264	−1.88223	−0.941113	−	0.338091i	$-0.890219\pi$
$11$	−6.24264	−1.88223	−0.941113	+	0.338091i	$0.890219\pi$
$12$	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
$17$	1.58579	0.384610	0.192305	−	0.981335i	$-0.438404\pi$
$17$	1.58579	0.384610	0.192305	+	0.981335i	$0.438404\pi$
$18$	0	0
$19$	7.41421	1.70094	0.850469	−	0.526026i	$-0.176318\pi$
$19$	7.41421	1.70094	0.850469	+	0.526026i	$0.176318\pi$
$20$	−1.41421	−0.316228
$21$	0	0
$22$	−6.24264	−1.33094
$23$	−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
$29$	3.24264	0.602143	0.301072	−	0.953602i	$-0.402656\pi$
$29$	3.24264	0.602143	0.301072	+	0.953602i	$0.402656\pi$
$30$	0	0
$31$	7.24264	1.30082	0.650408	−	0.759585i	$-0.274598\pi$
$31$	7.24264	1.30082	0.650408	+	0.759585i	$0.274598\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.58579	0.271960
$35$	0	0
$36$	0	0
$37$	6.48528	1.06617	0.533087	−	0.846061i	$-0.321032\pi$
$37$	6.48528	1.06617	0.533087	+	0.846061i	$0.321032\pi$
$38$	7.41421	1.20274
$39$	0	0
$40$	−1.41421	−0.223607
$41$	2.82843	0.441726	0.220863	−	0.975305i	$-0.429113\pi$
$41$	2.82843	0.441726	0.220863	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	−5.24264	−0.799495	−0.399748	−	0.916625i	$-0.630902\pi$
$43$	−5.24264	−0.799495	−0.399748	+	0.916625i	$0.630902\pi$
$44$	−6.24264	−0.941113
$45$	0	0
$46$	−1.00000	−0.147442
$47$	7.07107	1.03142	0.515711	−	0.856763i	$-0.327528\pi$
$47$	7.07107	1.03142	0.515711	+	0.856763i	$0.327528\pi$
$48$	0	0
$49$	0	0
$50$	−3.00000	−0.424264
$51$	0	0
$52$	3.00000	0.416025
$53$	2.75736	0.378752	0.189376	−	0.981905i	$-0.439353\pi$
$53$	2.75736	0.378752	0.189376	+	0.981905i	$0.439353\pi$
$54$	0	0
$55$	8.82843	1.19042
$56$	0	0
$57$	0	0
$58$	3.24264	0.425780
$59$	5.82843	0.758797	0.379398	−	0.925233i	$-0.376131\pi$
$59$	5.82843	0.758797	0.379398	+	0.925233i	$0.376131\pi$
$60$	0	0
$61$	0.343146	0.0439353	0.0219677	−	0.999759i	$-0.493007\pi$
$61$	0.343146	0.0439353	0.0219677	+	0.999759i	$0.493007\pi$
$62$	7.24264	0.919816
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.24264	−0.526235
$66$	0	0
$67$	4.75736	0.581204	0.290602	−	0.956844i	$-0.406144\pi$
$67$	4.75736	0.581204	0.290602	+	0.956844i	$0.406144\pi$
$68$	1.58579	0.192305
$69$	0	0
$70$	0	0
$71$	11.4853	1.36305	0.681526	−	0.731794i	$-0.261317\pi$
$71$	11.4853	1.36305	0.681526	+	0.731794i	$0.261317\pi$
$72$	0	0
$73$	−9.89949	−1.15865	−0.579324	−	0.815097i	$-0.696683\pi$
$73$	−9.89949	−1.15865	−0.579324	+	0.815097i	$0.696683\pi$
$74$	6.48528	0.753899
$75$	0	0
$76$	7.41421	0.850469
$77$	0	0
$78$	0	0
$79$	14.7279	1.65702	0.828510	−	0.559974i	$-0.189189\pi$
$79$	14.7279	1.65702	0.828510	+	0.559974i	$0.189189\pi$
$80$	−1.41421	−0.158114
$81$	0	0
$82$	2.82843	0.312348
$83$	−5.31371	−0.583255	−0.291628	−	0.956532i	$-0.594197\pi$
$83$	−5.31371	−0.583255	−0.291628	+	0.956532i	$0.594197\pi$
$84$	0	0
$85$	−2.24264	−0.243249
$86$	−5.24264	−0.565328
$87$	0	0
$88$	−6.24264	−0.665468
$89$	−16.0711	−1.70353	−0.851765	−	0.523924i	$-0.824468\pi$
$89$	−16.0711	−1.70353	−0.851765	+	0.523924i	$0.824468\pi$
$90$	0	0
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	7.07107	0.729325
$95$	−10.4853	−1.07577
$96$	0	0
$97$	9.17157	0.931232	0.465616	−	0.884987i	$-0.345833\pi$
$97$	9.17157	0.931232	0.465616	+	0.884987i	$0.345833\pi$
$98$	0	0
$99$	0	0
$100$	−3.00000	−0.300000
$101$	−1.75736	−0.174864	−0.0874319	−	0.996170i	$-0.527866\pi$
$101$	−1.75736	−0.174864	−0.0874319	+	0.996170i	$0.527866\pi$
$102$	0	0
$103$	16.0711	1.58353	0.791765	−	0.610826i	$-0.209163\pi$
$103$	16.0711	1.58353	0.791765	+	0.610826i	$0.209163\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	2.75736	0.267818
$107$	16.4853	1.59369	0.796846	−	0.604182i	$-0.206500\pi$
$107$	16.4853	1.59369	0.796846	+	0.604182i	$0.206500\pi$
$108$	0	0
$109$	−16.4853	−1.57900	−0.789502	−	0.613748i	$-0.789661\pi$
$109$	−16.4853	−1.57900	−0.789502	+	0.613748i	$0.789661\pi$
$110$	8.82843	0.841757
$111$	0	0
$112$	0	0
$113$	12.7279	1.19734	0.598671	−	0.800995i	$-0.295696\pi$
$113$	12.7279	1.19734	0.598671	+	0.800995i	$0.295696\pi$
$114$	0	0
$115$	1.41421	0.131876
$116$	3.24264	0.301072
$117$	0	0
$118$	5.82843	0.536550
$119$	0	0
$120$	0	0
$121$	27.9706	2.54278
$122$	0.343146	0.0310670
$123$	0	0
$124$	7.24264	0.650408
$125$	11.3137	1.01193
$126$	0	0
$127$	−20.7279	−1.83931	−0.919653	−	0.392732i	$-0.871530\pi$
$127$	−20.7279	−1.83931	−0.919653	+	0.392732i	$0.871530\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.24264	−0.372104
$131$	−17.4853	−1.52770	−0.763848	−	0.645396i	$-0.776692\pi$
$131$	−17.4853	−1.52770	−0.763848	+	0.645396i	$0.776692\pi$
$132$	0	0
$133$	0	0
$134$	4.75736	0.410973
$135$	0	0
$136$	1.58579	0.135980
$137$	−22.2426	−1.90032	−0.950159	−	0.311767i	$-0.899079\pi$
$137$	−22.2426	−1.90032	−0.950159	+	0.311767i	$0.899079\pi$
$138$	0	0
$139$	−5.31371	−0.450703	−0.225351	−	0.974278i	$-0.572353\pi$
$139$	−5.31371	−0.450703	−0.225351	+	0.974278i	$0.572353\pi$
$140$	0	0
$141$	0	0
$142$	11.4853	0.963823
$143$	−18.7279	−1.56611
$144$	0	0
$145$	−4.58579	−0.380829
$146$	−9.89949	−0.819288
$147$	0	0
$148$	6.48528	0.533087
$149$	−0.757359	−0.0620453	−0.0310226	−	0.999519i	$-0.509876\pi$
$149$	−0.757359	−0.0620453	−0.0310226	+	0.999519i	$0.509876\pi$
$150$	0	0
$151$	−10.7279	−0.873026	−0.436513	−	0.899698i	$-0.643787\pi$
$151$	−10.7279	−0.873026	−0.436513	+	0.899698i	$0.643787\pi$
$152$	7.41421	0.601372
$153$	0	0
$154$	0	0
$155$	−10.2426	−0.822709
$156$	0	0
$157$	−3.34315	−0.266812	−0.133406	−	0.991061i	$-0.542591\pi$
$157$	−3.34315	−0.266812	−0.133406	+	0.991061i	$0.542591\pi$
$158$	14.7279	1.17169
$159$	0	0
$160$	−1.41421	−0.111803
$161$	0	0
$162$	0	0
$163$	−12.7574	−0.999233	−0.499617	−	0.866247i	$-0.666526\pi$
$163$	−12.7574	−0.999233	−0.499617	+	0.866247i	$0.666526\pi$
$164$	2.82843	0.220863
$165$	0	0
$166$	−5.31371	−0.412424
$167$	15.8995	1.23034	0.615170	−	0.788395i	$-0.289087\pi$
$167$	15.8995	1.23034	0.615170	+	0.788395i	$0.289087\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	−2.24264	−0.172003
$171$	0	0
$172$	−5.24264	−0.399748
$173$	−3.51472	−0.267219	−0.133610	−	0.991034i	$-0.542657\pi$
$173$	−3.51472	−0.267219	−0.133610	+	0.991034i	$0.542657\pi$
$174$	0	0
$175$	0	0
$176$	−6.24264	−0.470557
$177$	0	0
$178$	−16.0711	−1.20458
$179$	13.7574	1.02827	0.514137	−	0.857708i	$-0.328112\pi$
$179$	13.7574	1.02827	0.514137	+	0.857708i	$0.328112\pi$
$180$	0	0
$181$	23.4853	1.74565	0.872824	−	0.488036i	$-0.162286\pi$
$181$	23.4853	1.74565	0.872824	+	0.488036i	$0.162286\pi$
$182$	0	0
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−9.17157	−0.674307
$186$	0	0
$187$	−9.89949	−0.723923
$188$	7.07107	0.515711
$189$	0	0
$190$	−10.4853	−0.760682
$191$	6.97056	0.504372	0.252186	−	0.967679i	$-0.418850\pi$
$191$	6.97056	0.504372	0.252186	+	0.967679i	$0.418850\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	9.17157	0.658481
$195$	0	0
$196$	0	0
$197$	−6.48528	−0.462057	−0.231029	−	0.972947i	$-0.574209\pi$
$197$	−6.48528	−0.462057	−0.231029	+	0.972947i	$0.574209\pi$
$198$	0	0
$199$	4.75736	0.337240	0.168620	−	0.985681i	$-0.446069\pi$
$199$	4.75736	0.337240	0.168620	+	0.985681i	$0.446069\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	−1.75736	−0.123647
$203$	0	0
$204$	0	0
$205$	−4.00000	−0.279372
$206$	16.0711	1.11972
$207$	0	0
$208$	3.00000	0.208013
$209$	−46.2843	−3.20155
$210$	0	0
$211$	10.7574	0.740567	0.370284	−	0.928919i	$-0.379261\pi$
$211$	10.7574	0.740567	0.370284	+	0.928919i	$0.379261\pi$
$212$	2.75736	0.189376
$213$	0	0
$214$	16.4853	1.12691
$215$	7.41421	0.505645
$216$	0	0
$217$	0	0
$218$	−16.4853	−1.11652
$219$	0	0
$220$	8.82843	0.595212
$221$	4.75736	0.320015
$222$	0	0
$223$	16.9706	1.13643	0.568216	−	0.822879i	$-0.307634\pi$
$223$	16.9706	1.13643	0.568216	+	0.822879i	$0.307634\pi$
$224$	0	0
$225$	0	0
$226$	12.7279	0.846649
$227$	13.9706	0.927259	0.463629	−	0.886029i	$-0.346547\pi$
$227$	13.9706	0.927259	0.463629	+	0.886029i	$0.346547\pi$
$228$	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
$232$	3.24264	0.212890
$233$	−8.48528	−0.555889	−0.277945	−	0.960597i	$-0.589653\pi$
$233$	−8.48528	−0.555889	−0.277945	+	0.960597i	$0.589653\pi$
$234$	0	0
$235$	−10.0000	−0.652328
$236$	5.82843	0.379398
$237$	0	0
$238$	0	0
$239$	−10.9706	−0.709627	−0.354813	−	0.934937i	$-0.615456\pi$
$239$	−10.9706	−0.709627	−0.354813	+	0.934937i	$0.615456\pi$
$240$	0	0
$241$	24.7279	1.59287	0.796433	−	0.604727i	$-0.206718\pi$
$241$	24.7279	1.59287	0.796433	+	0.604727i	$0.206718\pi$
$242$	27.9706	1.79802
$243$	0	0
$244$	0.343146	0.0219677
$245$	0	0
$246$	0	0
$247$	22.2426	1.41527
$248$	7.24264	0.459908
$249$	0	0
$250$	11.3137	0.715542
$251$	16.6274	1.04951	0.524757	−	0.851252i	$-0.324156\pi$
$251$	16.6274	1.04951	0.524757	+	0.851252i	$0.324156\pi$
$252$	0	0
$253$	6.24264	0.392471
$254$	−20.7279	−1.30059
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.9706	0.684325	0.342162	−	0.939641i	$-0.388841\pi$
$257$	10.9706	0.684325	0.342162	+	0.939641i	$0.388841\pi$
$258$	0	0
$259$	0	0
$260$	−4.24264	−0.263117
$261$	0	0
$262$	−17.4853	−1.08024
$263$	−23.4853	−1.44816	−0.724082	−	0.689714i	$-0.757736\pi$
$263$	−23.4853	−1.44816	−0.724082	+	0.689714i	$0.757736\pi$
$264$	0	0
$265$	−3.89949	−0.239544
$266$	0	0
$267$	0	0
$268$	4.75736	0.290602
$269$	−27.5563	−1.68014	−0.840070	−	0.542478i	$-0.817486\pi$
$269$	−27.5563	−1.68014	−0.840070	+	0.542478i	$0.817486\pi$
$270$	0	0
$271$	12.8995	0.783589	0.391794	−	0.920053i	$-0.371855\pi$
$271$	12.8995	0.783589	0.391794	+	0.920053i	$0.371855\pi$
$272$	1.58579	0.0961524
$273$	0	0
$274$	−22.2426	−1.34373
$275$	18.7279	1.12934
$276$	0	0
$277$	12.4853	0.750168	0.375084	−	0.926991i	$-0.377614\pi$
$277$	12.4853	0.750168	0.375084	+	0.926991i	$0.377614\pi$
$278$	−5.31371	−0.318695
$279$	0	0
$280$	0	0
$281$	−5.51472	−0.328981	−0.164490	−	0.986379i	$-0.552598\pi$
$281$	−5.51472	−0.328981	−0.164490	+	0.986379i	$0.552598\pi$
$282$	0	0
$283$	6.34315	0.377061	0.188530	−	0.982067i	$-0.439628\pi$
$283$	6.34315	0.377061	0.188530	+	0.982067i	$0.439628\pi$
$284$	11.4853	0.681526
$285$	0	0
$286$	−18.7279	−1.10741
$287$	0	0
$288$	0	0
$289$	−14.4853	−0.852075
$290$	−4.58579	−0.269287
$291$	0	0
$292$	−9.89949	−0.579324
$293$	−32.8284	−1.91786	−0.958929	−	0.283648i	$-0.908455\pi$
$293$	−32.8284	−1.91786	−0.958929	+	0.283648i	$0.908455\pi$
$294$	0	0
$295$	−8.24264	−0.479905
$296$	6.48528	0.376949
$297$	0	0
$298$	−0.757359	−0.0438726
$299$	−3.00000	−0.173494
$300$	0	0
$301$	0	0
$302$	−10.7279	−0.617323
$303$	0	0
$304$	7.41421	0.425234
$305$	−0.485281	−0.0277871
$306$	0	0
$307$	−23.6569	−1.35017	−0.675084	−	0.737741i	$-0.735893\pi$
$307$	−23.6569	−1.35017	−0.675084	+	0.737741i	$0.735893\pi$
$308$	0	0
$309$	0	0
$310$	−10.2426	−0.581743
$311$	−23.3137	−1.32200	−0.661000	−	0.750386i	$-0.729867\pi$
$311$	−23.3137	−1.32200	−0.661000	+	0.750386i	$0.729867\pi$
$312$	0	0
$313$	4.24264	0.239808	0.119904	−	0.992785i	$-0.461741\pi$
$313$	4.24264	0.239808	0.119904	+	0.992785i	$0.461741\pi$
$314$	−3.34315	−0.188665
$315$	0	0
$316$	14.7279	0.828510
$317$	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
$319$	−20.2426	−1.13337
$320$	−1.41421	−0.0790569
$321$	0	0
$322$	0	0
$323$	11.7574	0.654197
$324$	0	0
$325$	−9.00000	−0.499230
$326$	−12.7574	−0.706565
$327$	0	0
$328$	2.82843	0.156174
$329$	0	0
$330$	0	0
$331$	−23.7279	−1.30420	−0.652102	−	0.758131i	$-0.726113\pi$
$331$	−23.7279	−1.30420	−0.652102	+	0.758131i	$0.726113\pi$
$332$	−5.31371	−0.291628
$333$	0	0
$334$	15.8995	0.869982
$335$	−6.72792	−0.367586
$336$	0	0
$337$	8.51472	0.463826	0.231913	−	0.972736i	$-0.425501\pi$
$337$	8.51472	0.463826	0.231913	+	0.972736i	$0.425501\pi$
$338$	−4.00000	−0.217571
$339$	0	0
$340$	−2.24264	−0.121624
$341$	−45.2132	−2.44843
$342$	0	0
$343$	0	0
$344$	−5.24264	−0.282664
$345$	0	0
$346$	−3.51472	−0.188952
$347$	−14.4853	−0.777611	−0.388805	−	0.921320i	$-0.627112\pi$
$347$	−14.4853	−0.777611	−0.388805	+	0.921320i	$0.627112\pi$
$348$	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
$352$	−6.24264	−0.332734
$353$	18.5563	0.987655	0.493827	−	0.869560i	$-0.335597\pi$
$353$	18.5563	0.987655	0.493827	+	0.869560i	$0.335597\pi$
$354$	0	0
$355$	−16.2426	−0.862070
$356$	−16.0711	−0.851765
$357$	0	0
$358$	13.7574	0.727099
$359$	11.9706	0.631782	0.315891	−	0.948795i	$-0.397697\pi$
$359$	11.9706	0.631782	0.315891	+	0.948795i	$0.397697\pi$
$360$	0	0
$361$	35.9706	1.89319
$362$	23.4853	1.23436
$363$	0	0
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	−16.4142	−0.856815	−0.428407	−	0.903586i	$-0.640925\pi$
$367$	−16.4142	−0.856815	−0.428407	+	0.903586i	$0.640925\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−9.17157	−0.476807
$371$	0	0
$372$	0	0
$373$	−25.6985	−1.33062	−0.665309	−	0.746569i	$-0.731700\pi$
$373$	−25.6985	−1.33062	−0.665309	+	0.746569i	$0.731700\pi$
$374$	−9.89949	−0.511891
$375$	0	0
$376$	7.07107	0.364662
$377$	9.72792	0.501013
$378$	0	0
$379$	14.9706	0.768986	0.384493	−	0.923128i	$-0.374376\pi$
$379$	14.9706	0.768986	0.384493	+	0.923128i	$0.374376\pi$
$380$	−10.4853	−0.537884
$381$	0	0
$382$	6.97056	0.356645
$383$	−21.5563	−1.10148	−0.550739	−	0.834678i	$-0.685654\pi$
$383$	−21.5563	−1.10148	−0.550739	+	0.834678i	$0.685654\pi$
$384$	0	0
$385$	0	0
$386$	5.00000	0.254493
$387$	0	0
$388$	9.17157	0.465616
$389$	−5.02944	−0.255003	−0.127501	−	0.991838i	$-0.540696\pi$
$389$	−5.02944	−0.255003	−0.127501	+	0.991838i	$0.540696\pi$
$390$	0	0
$391$	−1.58579	−0.0801967
$392$	0	0
$393$	0	0
$394$	−6.48528	−0.326724
$395$	−20.8284	−1.04799
$396$	0	0
$397$	9.17157	0.460308	0.230154	−	0.973154i	$-0.426077\pi$
$397$	9.17157	0.460308	0.230154	+	0.973154i	$0.426077\pi$
$398$	4.75736	0.238465
$399$	0	0
$400$	−3.00000	−0.150000
$401$	−2.48528	−0.124109	−0.0620545	−	0.998073i	$-0.519765\pi$
$401$	−2.48528	−0.124109	−0.0620545	+	0.998073i	$0.519765\pi$
$402$	0	0
$403$	21.7279	1.08234
$404$	−1.75736	−0.0874319
$405$	0	0
$406$	0	0
$407$	−40.4853	−2.00678
$408$	0	0
$409$	−2.48528	−0.122889	−0.0614446	−	0.998110i	$-0.519571\pi$
$409$	−2.48528	−0.122889	−0.0614446	+	0.998110i	$0.519571\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	16.0711	0.791765
$413$	0	0
$414$	0	0
$415$	7.51472	0.368883
$416$	3.00000	0.147087
$417$	0	0
$418$	−46.2843	−2.26384
$419$	15.3431	0.749562	0.374781	−	0.927113i	$-0.377718\pi$
$419$	15.3431	0.749562	0.374781	+	0.927113i	$0.377718\pi$
$420$	0	0
$421$	−31.7574	−1.54776	−0.773879	−	0.633333i	$-0.781686\pi$
$421$	−31.7574	−1.54776	−0.773879	+	0.633333i	$0.781686\pi$
$422$	10.7574	0.523660
$423$	0	0
$424$	2.75736	0.133909
$425$	−4.75736	−0.230766
$426$	0	0
$427$	0	0
$428$	16.4853	0.796846
$429$	0	0
$430$	7.41421	0.357545
$431$	1.02944	0.0495862	0.0247931	−	0.999693i	$-0.492107\pi$
$431$	1.02944	0.0495862	0.0247931	+	0.999693i	$0.492107\pi$
$432$	0	0
$433$	−15.5147	−0.745590	−0.372795	−	0.927914i	$-0.621600\pi$
$433$	−15.5147	−0.745590	−0.372795	+	0.927914i	$0.621600\pi$
$434$	0	0
$435$	0	0
$436$	−16.4853	−0.789502
$437$	−7.41421	−0.354670
$438$	0	0
$439$	−30.2132	−1.44200	−0.720999	−	0.692936i	$-0.756317\pi$
$439$	−30.2132	−1.44200	−0.720999	+	0.692936i	$0.756317\pi$
$440$	8.82843	0.420879
$441$	0	0
$442$	4.75736	0.226285
$443$	6.97056	0.331181	0.165591	−	0.986195i	$-0.447047\pi$
$443$	6.97056	0.331181	0.165591	+	0.986195i	$0.447047\pi$
$444$	0	0
$445$	22.7279	1.07741
$446$	16.9706	0.803579
$447$	0	0
$448$	0	0
$449$	24.7279	1.16698	0.583491	−	0.812119i	$-0.301686\pi$
$449$	24.7279	1.16698	0.583491	+	0.812119i	$0.301686\pi$
$450$	0	0
$451$	−17.6569	−0.831429
$452$	12.7279	0.598671
$453$	0	0
$454$	13.9706	0.655671
$455$	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
$461$	−13.0711	−0.608780	−0.304390	−	0.952547i	$-0.598453\pi$
$461$	−13.0711	−0.608780	−0.304390	+	0.952547i	$0.598453\pi$
$462$	0	0
$463$	19.2132	0.892913	0.446457	−	0.894805i	$-0.352686\pi$
$463$	19.2132	0.892913	0.446457	+	0.894805i	$0.352686\pi$
$464$	3.24264	0.150536
$465$	0	0
$466$	−8.48528	−0.393073
$467$	12.3431	0.571173	0.285586	−	0.958353i	$-0.407812\pi$
$467$	12.3431	0.571173	0.285586	+	0.958353i	$0.407812\pi$
$468$	0	0
$469$	0	0
$470$	−10.0000	−0.461266
$471$	0	0
$472$	5.82843	0.268275
$473$	32.7279	1.50483
$474$	0	0
$475$	−22.2426	−1.02056
$476$	0	0
$477$	0	0
$478$	−10.9706	−0.501782
$479$	30.0416	1.37264	0.686319	−	0.727301i	$-0.259226\pi$
$479$	30.0416	1.37264	0.686319	+	0.727301i	$0.259226\pi$
$480$	0	0
$481$	19.4558	0.887110
$482$	24.7279	1.12633
$483$	0	0
$484$	27.9706	1.27139
$485$	−12.9706	−0.588963
$486$	0	0
$487$	−13.7574	−0.623405	−0.311703	−	0.950180i	$-0.600899\pi$
$487$	−13.7574	−0.623405	−0.311703	+	0.950180i	$0.600899\pi$
$488$	0.343146	0.0155335
$489$	0	0
$490$	0	0
$491$	−1.75736	−0.0793085	−0.0396543	−	0.999213i	$-0.512626\pi$
$491$	−1.75736	−0.0793085	−0.0396543	+	0.999213i	$0.512626\pi$
$492$	0	0
$493$	5.14214	0.231590
$494$	22.2426	1.00074
$495$	0	0
$496$	7.24264	0.325204
$497$	0	0
$498$	0	0
$499$	−8.97056	−0.401578	−0.200789	−	0.979635i	$-0.564350\pi$
$499$	−8.97056	−0.401578	−0.200789	+	0.979635i	$0.564350\pi$
$500$	11.3137	0.505964
$501$	0	0
$502$	16.6274	0.742118
$503$	27.8995	1.24398	0.621988	−	0.783026i	$-0.286325\pi$
$503$	27.8995	1.24398	0.621988	+	0.783026i	$0.286325\pi$
$504$	0	0
$505$	2.48528	0.110594
$506$	6.24264	0.277519
$507$	0	0
$508$	−20.7279	−0.919653
$509$	20.1421	0.892784	0.446392	−	0.894837i	$-0.352709\pi$
$509$	20.1421	0.892784	0.446392	+	0.894837i	$0.352709\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	10.9706	0.483891
$515$	−22.7279	−1.00151
$516$	0	0
$517$	−44.1421	−1.94137
$518$	0	0
$519$	0	0
$520$	−4.24264	−0.186052
$521$	7.58579	0.332339	0.166170	−	0.986097i	$-0.446860\pi$
$521$	7.58579	0.332339	0.166170	+	0.986097i	$0.446860\pi$
$522$	0	0
$523$	−36.7279	−1.60600	−0.803000	−	0.595979i	$-0.796764\pi$
$523$	−36.7279	−1.60600	−0.803000	+	0.595979i	$0.796764\pi$
$524$	−17.4853	−0.763848
$525$	0	0
$526$	−23.4853	−1.02401
$527$	11.4853	0.500307
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−3.89949	−0.169383
$531$	0	0
$532$	0	0
$533$	8.48528	0.367538
$534$	0	0
$535$	−23.3137	−1.00794
$536$	4.75736	0.205487
$537$	0	0
$538$	−27.5563	−1.18804
$539$	0	0
$540$	0	0
$541$	−14.7279	−0.633203	−0.316601	−	0.948559i	$-0.602542\pi$
$541$	−14.7279	−0.633203	−0.316601	+	0.948559i	$0.602542\pi$
$542$	12.8995	0.554081
$543$	0	0
$544$	1.58579	0.0679900
$545$	23.3137	0.998650
$546$	0	0
$547$	12.4853	0.533832	0.266916	−	0.963720i	$-0.413995\pi$
$547$	12.4853	0.533832	0.266916	+	0.963720i	$0.413995\pi$
$548$	−22.2426	−0.950159
$549$	0	0
$550$	18.7279	0.798561
$551$	24.0416	1.02421
$552$	0	0
$553$	0	0
$554$	12.4853	0.530449
$555$	0	0
$556$	−5.31371	−0.225351
$557$	44.6985	1.89394	0.946968	−	0.321328i	$-0.104129\pi$
$557$	44.6985	1.89394	0.946968	+	0.321328i	$0.104129\pi$
$558$	0	0
$559$	−15.7279	−0.665220
$560$	0	0
$561$	0	0
$562$	−5.51472	−0.232624
$563$	12.8579	0.541894	0.270947	−	0.962594i	$-0.412663\pi$
$563$	12.8579	0.541894	0.270947	+	0.962594i	$0.412663\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	6.34315	0.266622
$567$	0	0
$568$	11.4853	0.481912
$569$	−41.2132	−1.72775	−0.863874	−	0.503709i	$-0.831969\pi$
$569$	−41.2132	−1.72775	−0.863874	+	0.503709i	$0.831969\pi$
$570$	0	0
$571$	5.24264	0.219398	0.109699	−	0.993965i	$-0.465011\pi$
$571$	5.24264	0.219398	0.109699	+	0.993965i	$0.465011\pi$
$572$	−18.7279	−0.783054
$573$	0	0
$574$	0	0
$575$	3.00000	0.125109
$576$	0	0
$577$	14.4853	0.603030	0.301515	−	0.953461i	$-0.402508\pi$
$577$	14.4853	0.603030	0.301515	+	0.953461i	$0.402508\pi$
$578$	−14.4853	−0.602508
$579$	0	0
$580$	−4.58579	−0.190414
$581$	0	0
$582$	0	0
$583$	−17.2132	−0.712898
$584$	−9.89949	−0.409644
$585$	0	0
$586$	−32.8284	−1.35613
$587$	−10.4558	−0.431559	−0.215779	−	0.976442i	$-0.569229\pi$
$587$	−10.4558	−0.431559	−0.215779	+	0.976442i	$0.569229\pi$
$588$	0	0
$589$	53.6985	2.21261
$590$	−8.24264	−0.339344
$591$	0	0
$592$	6.48528	0.266543
$593$	8.14214	0.334357	0.167179	−	0.985927i	$-0.446534\pi$
$593$	8.14214	0.334357	0.167179	+	0.985927i	$0.446534\pi$
$594$	0	0
$595$	0	0
$596$	−0.757359	−0.0310226
$597$	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
$601$	−25.7574	−1.05066	−0.525332	−	0.850897i	$-0.676059\pi$
$601$	−25.7574	−1.05066	−0.525332	+	0.850897i	$0.676059\pi$
$602$	0	0
$603$	0	0
$604$	−10.7279	−0.436513
$605$	−39.5563	−1.60819
$606$	0	0
$607$	26.6985	1.08366	0.541829	−	0.840489i	$-0.317732\pi$
$607$	26.6985	1.08366	0.541829	+	0.840489i	$0.317732\pi$
$608$	7.41421	0.300686
$609$	0	0
$610$	−0.485281	−0.0196485
$611$	21.2132	0.858194
$612$	0	0
$613$	19.7574	0.797992	0.398996	−	0.916953i	$-0.369359\pi$
$613$	19.7574	0.797992	0.398996	+	0.916953i	$0.369359\pi$
$614$	−23.6569	−0.954713
$615$	0	0
$616$	0	0
$617$	31.9411	1.28590	0.642951	−	0.765908i	$-0.277710\pi$
$617$	31.9411	1.28590	0.642951	+	0.765908i	$0.277710\pi$
$618$	0	0
$619$	3.17157	0.127476	0.0637381	−	0.997967i	$-0.479698\pi$
$619$	3.17157	0.127476	0.0637381	+	0.997967i	$0.479698\pi$
$620$	−10.2426	−0.411354
$621$	0	0
$622$	−23.3137	−0.934795
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	4.24264	0.169570
$627$	0	0
$628$	−3.34315	−0.133406
$629$	10.2843	0.410061
$630$	0	0
$631$	12.2426	0.487372	0.243686	−	0.969854i	$-0.421643\pi$
$631$	12.2426	0.487372	0.243686	+	0.969854i	$0.421643\pi$
$632$	14.7279	0.585845
$633$	0	0
$634$	10.0000	0.397151
$635$	29.3137	1.16328
$636$	0	0
$637$	0	0
$638$	−20.2426	−0.801414
$639$	0	0
$640$	−1.41421	−0.0559017
$641$	−41.4558	−1.63741	−0.818704	−	0.574216i	$-0.805307\pi$
$641$	−41.4558	−1.63741	−0.818704	+	0.574216i	$0.805307\pi$
$642$	0	0
$643$	−31.0711	−1.22532	−0.612662	−	0.790345i	$-0.709901\pi$
$643$	−31.0711	−1.22532	−0.612662	+	0.790345i	$0.709901\pi$
$644$	0	0
$645$	0	0
$646$	11.7574	0.462587
$647$	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
$649$	−36.3848	−1.42823
$650$	−9.00000	−0.353009
$651$	0	0
$652$	−12.7574	−0.499617
$653$	−26.6985	−1.04479	−0.522396	−	0.852703i	$-0.674962\pi$
$653$	−26.6985	−1.04479	−0.522396	+	0.852703i	$0.674962\pi$
$654$	0	0
$655$	24.7279	0.966200
$656$	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
$661$	−45.9411	−1.78690	−0.893451	−	0.449160i	$-0.851723\pi$
$661$	−45.9411	−1.78690	−0.893451	+	0.449160i	$0.851723\pi$
$662$	−23.7279	−0.922212
$663$	0	0
$664$	−5.31371	−0.206212
$665$	0	0
$666$	0	0
$667$	−3.24264	−0.125556
$668$	15.8995	0.615170
$669$	0	0
$670$	−6.72792	−0.259922
$671$	−2.14214	−0.0826962
$672$	0	0
$673$	22.4558	0.865609	0.432805	−	0.901488i	$-0.357524\pi$
$673$	22.4558	0.865609	0.432805	+	0.901488i	$0.357524\pi$
$674$	8.51472	0.327975
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−29.6985	−1.14141	−0.570703	−	0.821157i	$-0.693329\pi$
$677$	−29.6985	−1.14141	−0.570703	+	0.821157i	$0.693329\pi$
$678$	0	0
$679$	0	0
$680$	−2.24264	−0.0860013
$681$	0	0
$682$	−45.2132	−1.73130
$683$	−22.2426	−0.851091	−0.425545	−	0.904937i	$-0.639918\pi$
$683$	−22.2426	−0.851091	−0.425545	+	0.904937i	$0.639918\pi$
$684$	0	0
$685$	31.4558	1.20187
$686$	0	0
$687$	0	0
$688$	−5.24264	−0.199874
$689$	8.27208	0.315141
$690$	0	0
$691$	−9.21320	−0.350487	−0.175243	−	0.984525i	$-0.556071\pi$
$691$	−9.21320	−0.350487	−0.175243	+	0.984525i	$0.556071\pi$
$692$	−3.51472	−0.133610
$693$	0	0
$694$	−14.4853	−0.549854
$695$	7.51472	0.285050
$696$	0	0
$697$	4.48528	0.169892
$698$	−3.00000	−0.113552
$699$	0	0
$700$	0	0
$701$	34.9706	1.32082	0.660410	−	0.750905i	$-0.270383\pi$
$701$	34.9706	1.32082	0.660410	+	0.750905i	$0.270383\pi$
$702$	0	0
$703$	48.0833	1.81349
$704$	−6.24264	−0.235278
$705$	0	0
$706$	18.5563	0.698377
$707$	0	0
$708$	0	0
$709$	−23.2132	−0.871790	−0.435895	−	0.899997i	$-0.643568\pi$
$709$	−23.2132	−0.871790	−0.435895	+	0.899997i	$0.643568\pi$
$710$	−16.2426	−0.609575
$711$	0	0
$712$	−16.0711	−0.602289
$713$	−7.24264	−0.271239
$714$	0	0
$715$	26.4853	0.990493
$716$	13.7574	0.514137
$717$	0	0
$718$	11.9706	0.446737
$719$	−10.6274	−0.396336	−0.198168	−	0.980168i	$-0.563499\pi$
$719$	−10.6274	−0.396336	−0.198168	+	0.980168i	$0.563499\pi$
$720$	0	0
$721$	0	0
$722$	35.9706	1.33869
$723$	0	0
$724$	23.4853	0.872824
$725$	−9.72792	−0.361286
$726$	0	0
$727$	40.8406	1.51469	0.757347	−	0.653012i	$-0.226495\pi$
$727$	40.8406	1.51469	0.757347	+	0.653012i	$0.226495\pi$
$728$	0	0
$729$	0	0
$730$	14.0000	0.518163
$731$	−8.31371	−0.307494
$732$	0	0
$733$	30.9411	1.14284	0.571418	−	0.820659i	$-0.306393\pi$
$733$	30.9411	1.14284	0.571418	+	0.820659i	$0.306393\pi$
$734$	−16.4142	−0.605860
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−29.6985	−1.09396
$738$	0	0
$739$	−10.9706	−0.403559	−0.201779	−	0.979431i	$-0.564672\pi$
$739$	−10.9706	−0.403559	−0.201779	+	0.979431i	$0.564672\pi$
$740$	−9.17157	−0.337154
$741$	0	0
$742$	0	0
$743$	4.51472	0.165629	0.0828145	−	0.996565i	$-0.473609\pi$
$743$	4.51472	0.165629	0.0828145	+	0.996565i	$0.473609\pi$
$744$	0	0
$745$	1.07107	0.0392409
$746$	−25.6985	−0.940888
$747$	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
$754$	9.72792	0.354270
$755$	15.1716	0.552150
$756$	0	0
$757$	−27.6985	−1.00672	−0.503359	−	0.864077i	$-0.667903\pi$
$757$	−27.6985	−1.00672	−0.503359	+	0.864077i	$0.667903\pi$
$758$	14.9706	0.543755
$759$	0	0
$760$	−10.4853	−0.380341
$761$	−53.5269	−1.94035	−0.970175	−	0.242408i	$-0.922063\pi$
$761$	−53.5269	−1.94035	−0.970175	+	0.242408i	$0.922063\pi$
$762$	0	0
$763$	0	0
$764$	6.97056	0.252186
$765$	0	0
$766$	−21.5563	−0.778863
$767$	17.4853	0.631357
$768$	0	0
$769$	15.2132	0.548602	0.274301	−	0.961644i	$-0.411554\pi$
$769$	15.2132	0.548602	0.274301	+	0.961644i	$0.411554\pi$
$770$	0	0
$771$	0	0
$772$	5.00000	0.179954
$773$	18.3848	0.661254	0.330627	−	0.943761i	$-0.392740\pi$
$773$	18.3848	0.661254	0.330627	+	0.943761i	$0.392740\pi$
$774$	0	0
$775$	−21.7279	−0.780490
$776$	9.17157	0.329240
$777$	0	0
$778$	−5.02944	−0.180314
$779$	20.9706	0.751348
$780$	0	0
$781$	−71.6985	−2.56557
$782$	−1.58579	−0.0567076
$783$	0	0
$784$	0	0
$785$	4.72792	0.168747
$786$	0	0
$787$	−27.1716	−0.968562	−0.484281	−	0.874913i	$-0.660919\pi$
$787$	−27.1716	−0.968562	−0.484281	+	0.874913i	$0.660919\pi$
$788$	−6.48528	−0.231029
$789$	0	0
$790$	−20.8284	−0.741042
$791$	0	0
$792$	0	0
$793$	1.02944	0.0365564
$794$	9.17157	0.325487
$795$	0	0
$796$	4.75736	0.168620
$797$	−50.4853	−1.78828	−0.894140	−	0.447787i	$-0.852212\pi$
$797$	−50.4853	−1.78828	−0.894140	+	0.447787i	$0.852212\pi$
$798$	0	0
$799$	11.2132	0.396695
$800$	−3.00000	−0.106066
$801$	0	0
$802$	−2.48528	−0.0877583
$803$	61.7990	2.18084
$804$	0	0
$805$	0	0
$806$	21.7279	0.765333
$807$	0	0
$808$	−1.75736	−0.0618237
$809$	−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
$811$	−15.9411	−0.559769	−0.279884	−	0.960034i	$-0.590296\pi$
$811$	−15.9411	−0.559769	−0.279884	+	0.960034i	$0.590296\pi$
$812$	0	0
$813$	0	0
$814$	−40.4853	−1.41901
$815$	18.0416	0.631971
$816$	0	0
$817$	−38.8701	−1.35989
$818$	−2.48528	−0.0868958
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−24.2132	−0.845047	−0.422523	−	0.906352i	$-0.638856\pi$
$821$	−24.2132	−0.845047	−0.422523	+	0.906352i	$0.638856\pi$
$822$	0	0
$823$	26.9706	0.940135	0.470067	−	0.882631i	$-0.344230\pi$
$823$	26.9706	0.940135	0.470067	+	0.882631i	$0.344230\pi$
$824$	16.0711	0.559862
$825$	0	0
$826$	0	0
$827$	−29.6985	−1.03272	−0.516359	−	0.856372i	$-0.672713\pi$
$827$	−29.6985	−1.03272	−0.516359	+	0.856372i	$0.672713\pi$
$828$	0	0
$829$	26.8284	0.931790	0.465895	−	0.884840i	$-0.345732\pi$
$829$	26.8284	0.931790	0.465895	+	0.884840i	$0.345732\pi$
$830$	7.51472	0.260840
$831$	0	0
$832$	3.00000	0.104006
$833$	0	0
$834$	0	0
$835$	−22.4853	−0.778135
$836$	−46.2843	−1.60077
$837$	0	0
$838$	15.3431	0.530020
$839$	37.7990	1.30497	0.652483	−	0.757803i	$-0.273727\pi$
$839$	37.7990	1.30497	0.652483	+	0.757803i	$0.273727\pi$
$840$	0	0
$841$	−18.4853	−0.637423
$842$	−31.7574	−1.09443
$843$	0	0
$844$	10.7574	0.370284
$845$	5.65685	0.194602
$846$	0	0
$847$	0	0
$848$	2.75736	0.0946881
$849$	0	0
$850$	−4.75736	−0.163176
$851$	−6.48528	−0.222313
$852$	0	0
$853$	−39.4264	−1.34993	−0.674967	−	0.737848i	$-0.735842\pi$
$853$	−39.4264	−1.34993	−0.674967	+	0.737848i	$0.735842\pi$
$854$	0	0
$855$	0	0
$856$	16.4853	0.563455
$857$	46.4142	1.58548	0.792740	−	0.609560i	$-0.208654\pi$
$857$	46.4142	1.58548	0.792740	+	0.609560i	$0.208654\pi$
$858$	0	0
$859$	30.7696	1.04984	0.524922	−	0.851150i	$-0.324095\pi$
$859$	30.7696	1.04984	0.524922	+	0.851150i	$0.324095\pi$
$860$	7.41421	0.252823
$861$	0	0
$862$	1.02944	0.0350628
$863$	19.4853	0.663287	0.331643	−	0.943405i	$-0.392397\pi$
$863$	19.4853	0.663287	0.331643	+	0.943405i	$0.392397\pi$
$864$	0	0
$865$	4.97056	0.169004
$866$	−15.5147	−0.527212
$867$	0	0
$868$	0	0
$869$	−91.9411	−3.11889
$870$	0	0
$871$	14.2721	0.483591
$872$	−16.4853	−0.558262
$873$	0	0
$874$	−7.41421	−0.250790
$875$	0	0
$876$	0	0
$877$	−12.4853	−0.421598	−0.210799	−	0.977529i	$-0.567607\pi$
$877$	−12.4853	−0.421598	−0.210799	+	0.977529i	$0.567607\pi$
$878$	−30.2132	−1.01965
$879$	0	0
$880$	8.82843	0.297606
$881$	−33.7279	−1.13632	−0.568161	−	0.822917i	$-0.692345\pi$
$881$	−33.7279	−1.13632	−0.568161	+	0.822917i	$0.692345\pi$
$882$	0	0
$883$	−12.2721	−0.412988	−0.206494	−	0.978448i	$-0.566205\pi$
$883$	−12.2721	−0.412988	−0.206494	+	0.978448i	$0.566205\pi$
$884$	4.75736	0.160007
$885$	0	0
$886$	6.97056	0.234181
$887$	33.1716	1.11379	0.556896	−	0.830582i	$-0.311992\pi$
$887$	33.1716	1.11379	0.556896	+	0.830582i	$0.311992\pi$
$888$	0	0
$889$	0	0
$890$	22.7279	0.761842
$891$	0	0
$892$	16.9706	0.568216
$893$	52.4264	1.75438
$894$	0	0
$895$	−19.4558	−0.650337
$896$	0	0
$897$	0	0
$898$	24.7279	0.825181
$899$	23.4853	0.783278
$900$	0	0
$901$	4.37258	0.145672
$902$	−17.6569	−0.587909
$903$	0	0
$904$	12.7279	0.423324
$905$	−33.2132	−1.10404
$906$	0	0
$907$	43.9411	1.45904	0.729521	−	0.683959i	$-0.239743\pi$
$907$	43.9411	1.45904	0.729521	+	0.683959i	$0.239743\pi$
$908$	13.9706	0.463629
$909$	0	0
$910$	0	0
$911$	17.5147	0.580289	0.290144	−	0.956983i	$-0.406297\pi$
$911$	17.5147	0.580289	0.290144	+	0.956983i	$0.406297\pi$
$912$	0	0
$913$	33.1716	1.09782
$914$	−15.0000	−0.496156
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.75736	0.189918	0.0949589	−	0.995481i	$-0.469728\pi$
$919$	5.75736	0.189918	0.0949589	+	0.995481i	$0.469728\pi$
$920$	1.41421	0.0466252
$921$	0	0
$922$	−13.0711	−0.430473
$923$	34.4558	1.13413
$924$	0	0
$925$	−19.4558	−0.639704
$926$	19.2132	0.631385
$927$	0	0
$928$	3.24264	0.106445
$929$	41.3137	1.35546	0.677729	−	0.735311i	$-0.262964\pi$
$929$	41.3137	1.35546	0.677729	+	0.735311i	$0.262964\pi$
$930$	0	0
$931$	0	0
$932$	−8.48528	−0.277945
$933$	0	0
$934$	12.3431	0.403880
$935$	14.0000	0.457849
$936$	0	0
$937$	−37.0711	−1.21106	−0.605529	−	0.795823i	$-0.707039\pi$
$937$	−37.0711	−1.21106	−0.605529	+	0.795823i	$0.707039\pi$
$938$	0	0
$939$	0	0
$940$	−10.0000	−0.326164
$941$	22.2843	0.726446	0.363223	−	0.931702i	$-0.381676\pi$
$941$	22.2843	0.726446	0.363223	+	0.931702i	$0.381676\pi$
$942$	0	0
$943$	−2.82843	−0.0921063
$944$	5.82843	0.189699
$945$	0	0
$946$	32.7279	1.06408
$947$	−36.7279	−1.19350	−0.596749	−	0.802428i	$-0.703541\pi$
$947$	−36.7279	−1.19350	−0.596749	+	0.802428i	$0.703541\pi$
$948$	0	0
$949$	−29.6985	−0.964054
$950$	−22.2426	−0.721647
$951$	0	0
$952$	0	0
$953$	9.69848	0.314165	0.157082	−	0.987586i	$-0.449791\pi$
$953$	9.69848	0.314165	0.157082	+	0.987586i	$0.449791\pi$
$954$	0	0
$955$	−9.85786	−0.318993
$956$	−10.9706	−0.354813
$957$	0	0
$958$	30.0416	0.970601
$959$	0	0
$960$	0	0
$961$	21.4558	0.692124
$962$	19.4558	0.627282
$963$	0	0
$964$	24.7279	0.796433
$965$	−7.07107	−0.227626
$966$	0	0
$967$	−55.1543	−1.77364	−0.886822	−	0.462112i	$-0.847092\pi$
$967$	−55.1543	−1.77364	−0.886822	+	0.462112i	$0.847092\pi$
$968$	27.9706	0.899008
$969$	0	0
$970$	−12.9706	−0.416460
$971$	27.7696	0.891167	0.445584	−	0.895240i	$-0.352996\pi$
$971$	27.7696	0.891167	0.445584	+	0.895240i	$0.352996\pi$
$972$	0	0
$973$	0	0
$974$	−13.7574	−0.440814
$975$	0	0
$976$	0.343146	0.0109838
$977$	52.6690	1.68503	0.842516	−	0.538671i	$-0.181073\pi$
$977$	52.6690	1.68503	0.842516	+	0.538671i	$0.181073\pi$
$978$	0	0
$979$	100.326	3.20643
$980$	0	0
$981$	0	0
$982$	−1.75736	−0.0560796
$983$	−9.55635	−0.304800	−0.152400	−	0.988319i	$-0.548700\pi$
$983$	−9.55635	−0.304800	−0.152400	+	0.988319i	$0.548700\pi$
$984$	0	0
$985$	9.17157	0.292231
$986$	5.14214	0.163759
$987$	0	0
$988$	22.2426	0.707633
$989$	5.24264	0.166706
$990$	0	0
$991$	39.4558	1.25336	0.626678	−	0.779278i	$-0.284414\pi$
$991$	39.4558	1.25336	0.626678	+	0.779278i	$0.284414\pi$
$992$	7.24264	0.229954
$993$	0	0
$994$	0	0
$995$	−6.72792	−0.213289
$996$	0	0
$997$	−47.1421	−1.49301	−0.746503	−	0.665382i	$-0.768269\pi$
$997$	−47.1421	−1.49301	−0.746503	+	0.665382i	$0.768269\pi$
$998$	−8.97056	−0.283958
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.a.bg.1.1	✓	2	21.20	even	2
2646.2.a.bh.1.2	yes	2	3.2	odd	2
2646.2.a.bm.1.2	yes	2	7.6	odd	2
2646.2.a.bn.1.1	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} -1.41421 q^{10} -6.24264 q^{11} +3.00000 q^{13} +1.00000 q^{16} +1.58579 q^{17} +7.41421 q^{19} -1.41421 q^{20} -6.24264 q^{22} -1.00000 q^{23} -3.00000 q^{25} +3.00000 q^{26} +3.24264 q^{29} +7.24264 q^{31} +1.00000 q^{32} +1.58579 q^{34} +6.48528 q^{37} +7.41421 q^{38} -1.41421 q^{40} +2.82843 q^{41} -5.24264 q^{43} -6.24264 q^{44} -1.00000 q^{46} +7.07107 q^{47} -3.00000 q^{50} +3.00000 q^{52} +2.75736 q^{53} +8.82843 q^{55} +3.24264 q^{58} +5.82843 q^{59} +0.343146 q^{61} +7.24264 q^{62} +1.00000 q^{64} -4.24264 q^{65} +4.75736 q^{67} +1.58579 q^{68} +11.4853 q^{71} -9.89949 q^{73} +6.48528 q^{74} +7.41421 q^{76} +14.7279 q^{79} -1.41421 q^{80} +2.82843 q^{82} -5.31371 q^{83} -2.24264 q^{85} -5.24264 q^{86} -6.24264 q^{88} -16.0711 q^{89} -1.00000 q^{92} +7.07107 q^{94} -10.4853 q^{95} +9.17157 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 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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