LMFDB - Embedded newform 1323.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.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:	$10.5642081874$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 7 $ x^4 - 6*x^2 + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.10100$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.10100 q^{2} +2.41421 q^{4} +2.10100 q^{5} -0.870264 q^{8} -4.41421 q^{10} +3.84153 q^{11} -6.82843 q^{13} -3.00000 q^{16} +1.23074 q^{17} -4.41421 q^{19} +5.07227 q^{20} -8.07107 q^{22} -6.81280 q^{23} -0.585786 q^{25} +14.3465 q^{26} -8.40401 q^{29} +0.656854 q^{31} +8.04354 q^{32} -2.58579 q^{34} +3.24264 q^{37} +9.27428 q^{38} -1.82843 q^{40} -5.07227 q^{41} -7.07107 q^{43} +9.27428 q^{44} +14.3137 q^{46} -10.6543 q^{47} +1.23074 q^{50} -16.4853 q^{52} +10.1445 q^{53} +8.07107 q^{55} +17.6569 q^{58} +2.97127 q^{59} +6.48528 q^{61} -1.38005 q^{62} -10.8995 q^{64} -14.3465 q^{65} -5.65685 q^{67} +2.97127 q^{68} +14.7070 q^{71} -11.4142 q^{73} -6.81280 q^{74} -10.6569 q^{76} +5.89949 q^{79} -6.30301 q^{80} +10.6569 q^{82} -0.720950 q^{83} +2.58579 q^{85} +14.8563 q^{86} -3.34315 q^{88} +0.149314 q^{89} -16.4475 q^{92} +22.3848 q^{94} -9.27428 q^{95} +1.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 12 q^{10} - 16 q^{13} - 12 q^{16} - 12 q^{19} - 4 q^{22} - 8 q^{25} - 20 q^{31} - 16 q^{34} - 4 q^{37} + 4 q^{40} + 12 q^{46} - 32 q^{52} + 4 q^{55} + 48 q^{58} - 8 q^{61} - 4 q^{64} - 40 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 12 * q^10 - 16 * q^13 - 12 * q^16 - 12 * q^19 - 4 * q^22 - 8 * q^25 - 20 * q^31 - 16 * q^34 - 4 * q^37 + 4 * q^40 + 12 * q^46 - 32 * q^52 + 4 * q^55 + 48 * q^58 - 8 * q^61 - 4 * q^64 - 40 * q^73 - 20 * q^76 - 16 * q^79 + 20 * q^82 + 16 * q^85 - 36 * q^88 + 16 * q^94 - 16 * q^97

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 $

$2$	−2.10100	−1.48563	−0.742817	−	0.669495i	$-0.766511\pi$
$2$	−2.10100	−1.48563	−0.742817	+	0.669495i	$0.766511\pi$
$3$	0	0
$3$	0	0
$4$	2.41421	1.20711
$5$	2.10100	0.939597	0.469799	−	0.882774i	$-0.344327\pi$
$5$	2.10100	0.939597	0.469799	+	0.882774i	$0.344327\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−0.870264	−0.307685
$9$	0	0
$10$	−4.41421	−1.39590
$11$	3.84153	1.15827	0.579133	−	0.815233i	$-0.303391\pi$
$11$	3.84153	1.15827	0.579133	+	0.815233i	$0.303391\pi$
$12$	0	0
$13$	−6.82843	−1.89386	−0.946932	−	0.321433i	$-0.895836\pi$
$13$	−6.82843	−1.89386	−0.946932	+	0.321433i	$0.895836\pi$
$14$	0	0
$15$	0	0
$16$	−3.00000	−0.750000
$17$	1.23074	0.298498	0.149249	−	0.988800i	$-0.452314\pi$
$17$	1.23074	0.298498	0.149249	+	0.988800i	$0.452314\pi$
$18$	0	0
$19$	−4.41421	−1.01269	−0.506345	−	0.862331i	$-0.669004\pi$
$19$	−4.41421	−1.01269	−0.506345	+	0.862331i	$0.669004\pi$
$20$	5.07227	1.13419
$21$	0	0
$22$	−8.07107	−1.72076
$23$	−6.81280	−1.42057	−0.710283	−	0.703916i	$-0.751433\pi$
$23$	−6.81280	−1.42057	−0.710283	+	0.703916i	$0.751433\pi$
$24$	0	0
$25$	−0.585786	−0.117157
$26$	14.3465	2.81359
$27$	0	0
$28$	0	0
$29$	−8.40401	−1.56059	−0.780293	−	0.625414i	$-0.784930\pi$
$29$	−8.40401	−1.56059	−0.780293	+	0.625414i	$0.784930\pi$
$30$	0	0
$31$	0.656854	0.117975	0.0589873	−	0.998259i	$-0.481213\pi$
$31$	0.656854	0.117975	0.0589873	+	0.998259i	$0.481213\pi$
$32$	8.04354	1.42191
$33$	0	0
$34$	−2.58579	−0.443459
$35$	0	0
$36$	0	0
$37$	3.24264	0.533087	0.266543	−	0.963823i	$-0.414118\pi$
$37$	3.24264	0.533087	0.266543	+	0.963823i	$0.414118\pi$
$38$	9.27428	1.50449
$39$	0	0
$40$	−1.82843	−0.289100
$41$	−5.07227	−0.792155	−0.396078	−	0.918217i	$-0.629629\pi$
$41$	−5.07227	−0.792155	−0.396078	+	0.918217i	$0.629629\pi$
$42$	0	0
$43$	−7.07107	−1.07833	−0.539164	−	0.842201i	$-0.681260\pi$
$43$	−7.07107	−1.07833	−0.539164	+	0.842201i	$0.681260\pi$
$44$	9.27428	1.39815
$45$	0	0
$46$	14.3137	2.11044
$47$	−10.6543	−1.55409	−0.777047	−	0.629443i	$-0.783283\pi$
$47$	−10.6543	−1.55409	−0.777047	+	0.629443i	$0.783283\pi$
$48$	0	0
$49$	0	0
$50$	1.23074	0.174053
$51$	0	0
$52$	−16.4853	−2.28610
$53$	10.1445	1.39346	0.696730	−	0.717334i	$-0.254638\pi$
$53$	10.1445	1.39346	0.696730	+	0.717334i	$0.254638\pi$
$54$	0	0
$55$	8.07107	1.08830
$56$	0	0
$57$	0	0
$58$	17.6569	2.31846
$59$	2.97127	0.386826	0.193413	−	0.981117i	$-0.438044\pi$
$59$	2.97127	0.386826	0.193413	+	0.981117i	$0.438044\pi$
$60$	0	0
$61$	6.48528	0.830355	0.415178	−	0.909740i	$-0.363719\pi$
$61$	6.48528	0.830355	0.415178	+	0.909740i	$0.363719\pi$
$62$	−1.38005	−0.175267
$63$	0	0
$64$	−10.8995	−1.36244
$65$	−14.3465	−1.77947
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	2.97127	0.360319
$69$	0	0
$70$	0	0
$71$	14.7070	1.74540	0.872701	−	0.488255i	$-0.162366\pi$
$71$	14.7070	1.74540	0.872701	+	0.488255i	$0.162366\pi$
$72$	0	0
$73$	−11.4142	−1.33593	−0.667966	−	0.744192i	$-0.732835\pi$
$73$	−11.4142	−1.33593	−0.667966	+	0.744192i	$0.732835\pi$
$74$	−6.81280	−0.791972
$75$	0	0
$76$	−10.6569	−1.22243
$77$	0	0
$78$	0	0
$79$	5.89949	0.663745	0.331873	−	0.943324i	$-0.392320\pi$
$79$	5.89949	0.663745	0.331873	+	0.943324i	$0.392320\pi$
$80$	−6.30301	−0.704698
$81$	0	0
$82$	10.6569	1.17685
$83$	−0.720950	−0.0791346	−0.0395673	−	0.999217i	$-0.512598\pi$
$83$	−0.720950	−0.0791346	−0.0395673	+	0.999217i	$0.512598\pi$
$84$	0	0
$85$	2.58579	0.280468
$86$	14.8563	1.60200
$87$	0	0
$88$	−3.34315	−0.356381
$89$	0.149314	0.0158272	0.00791361	−	0.999969i	$-0.497481\pi$
$89$	0.149314	0.0158272	0.00791361	+	0.999969i	$0.497481\pi$
$90$	0	0
$91$	0	0
$92$	−16.4475	−1.71478
$93$	0	0
$94$	22.3848	2.30881
$95$	−9.27428	−0.951521
$96$	0	0
$97$	1.65685	0.168228	0.0841140	−	0.996456i	$-0.473194\pi$
$97$	1.65685	0.168228	0.0841140	+	0.996456i	$0.473194\pi$
$98$	0	0
$99$	0	0
$100$	−1.41421	−0.141421
$101$	2.25032	0.223915	0.111957	−	0.993713i	$-0.464288\pi$
$101$	2.25032	0.223915	0.111957	+	0.993713i	$0.464288\pi$
$102$	0	0
$103$	−13.4853	−1.32874	−0.664372	−	0.747402i	$-0.731301\pi$
$103$	−13.4853	−1.32874	−0.664372	+	0.747402i	$0.731301\pi$
$104$	5.94253	0.582713
$105$	0	0
$106$	−21.3137	−2.07017
$107$	5.43275	0.525203	0.262602	−	0.964904i	$-0.415419\pi$
$107$	5.43275	0.525203	0.262602	+	0.964904i	$0.415419\pi$
$108$	0	0
$109$	−8.65685	−0.829176	−0.414588	−	0.910009i	$-0.636074\pi$
$109$	−8.65685	−0.829176	−0.414588	+	0.910009i	$0.636074\pi$
$110$	−16.9573	−1.61682
$111$	0	0
$112$	0	0
$113$	10.6543	1.00227	0.501137	−	0.865368i	$-0.332915\pi$
$113$	10.6543	1.00227	0.501137	+	0.865368i	$0.332915\pi$
$114$	0	0
$115$	−14.3137	−1.33476
$116$	−20.2891	−1.88379
$117$	0	0
$118$	−6.24264	−0.574682
$119$	0	0
$120$	0	0
$121$	3.75736	0.341578
$122$	−13.6256	−1.23360
$123$	0	0
$124$	1.58579	0.142408
$125$	−11.7358	−1.04968
$126$	0	0
$127$	0.828427	0.0735110	0.0367555	−	0.999324i	$-0.488298\pi$
$127$	0.828427	0.0735110	0.0367555	+	0.999324i	$0.488298\pi$
$128$	6.81280	0.602172
$129$	0	0
$130$	30.1421	2.64364
$131$	−7.89422	−0.689721	−0.344861	−	0.938654i	$-0.612074\pi$
$131$	−7.89422	−0.689721	−0.344861	+	0.938654i	$0.612074\pi$
$132$	0	0
$133$	0	0
$134$	11.8851	1.02671
$135$	0	0
$136$	−1.07107	−0.0918433
$137$	−16.0871	−1.37441	−0.687206	−	0.726463i	$-0.741163\pi$
$137$	−16.0871	−1.37441	−0.687206	+	0.726463i	$0.741163\pi$
$138$	0	0
$139$	3.17157	0.269009	0.134505	−	0.990913i	$-0.457056\pi$
$139$	3.17157	0.269009	0.134505	+	0.990913i	$0.457056\pi$
$140$	0	0
$141$	0	0
$142$	−30.8995	−2.59303
$143$	−26.2316	−2.19360
$144$	0	0
$145$	−17.6569	−1.46632
$146$	23.9813	1.98471
$147$	0	0
$148$	7.82843	0.643493
$149$	−5.43275	−0.445068	−0.222534	−	0.974925i	$-0.571433\pi$
$149$	−5.43275	−0.445068	−0.222534	+	0.974925i	$0.571433\pi$
$150$	0	0
$151$	−4.24264	−0.345261	−0.172631	−	0.984987i	$-0.555227\pi$
$151$	−4.24264	−0.345261	−0.172631	+	0.984987i	$0.555227\pi$
$152$	3.84153	0.311589
$153$	0	0
$154$	0	0
$155$	1.38005	0.110849
$156$	0	0
$157$	−1.51472	−0.120888	−0.0604439	−	0.998172i	$-0.519252\pi$
$157$	−1.51472	−0.120888	−0.0604439	+	0.998172i	$0.519252\pi$
$158$	−12.3949	−0.986082
$159$	0	0
$160$	16.8995	1.33602
$161$	0	0
$162$	0	0
$163$	−0.343146	−0.0268772	−0.0134386	−	0.999910i	$-0.504278\pi$
$163$	−0.343146	−0.0268772	−0.0134386	+	0.999910i	$0.504278\pi$
$164$	−12.2455	−0.956216
$165$	0	0
$166$	1.51472	0.117565
$167$	−10.6543	−0.824457	−0.412228	−	0.911081i	$-0.635249\pi$
$167$	−10.6543	−0.824457	−0.412228	+	0.911081i	$0.635249\pi$
$168$	0	0
$169$	33.6274	2.58672
$170$	−5.43275	−0.416673
$171$	0	0
$172$	−17.0711	−1.30166
$173$	−15.9378	−1.21173	−0.605863	−	0.795569i	$-0.707172\pi$
$173$	−15.9378	−1.21173	−0.605863	+	0.795569i	$0.707172\pi$
$174$	0	0
$175$	0	0
$176$	−11.5246	−0.868699
$177$	0	0
$178$	−0.313708	−0.0235134
$179$	12.0962	0.904115	0.452057	−	0.891989i	$-0.350690\pi$
$179$	12.0962	0.904115	0.452057	+	0.891989i	$0.350690\pi$
$180$	0	0
$181$	−23.8995	−1.77644	−0.888218	−	0.459423i	$-0.848056\pi$
$181$	−23.8995	−1.77644	−0.888218	+	0.459423i	$0.848056\pi$
$182$	0	0
$183$	0	0
$184$	5.92893	0.437087
$185$	6.81280	0.500887
$186$	0	0
$187$	4.72792	0.345740
$188$	−25.7218	−1.87596
$189$	0	0
$190$	19.4853	1.41361
$191$	2.61079	0.188910	0.0944551	−	0.995529i	$-0.469889\pi$
$191$	2.61079	0.188910	0.0944551	+	0.995529i	$0.469889\pi$
$192$	0	0
$193$	12.3431	0.888479	0.444240	−	0.895908i	$-0.353474\pi$
$193$	12.3431	0.888479	0.444240	+	0.895908i	$0.353474\pi$
$194$	−3.48106	−0.249925
$195$	0	0
$196$	0	0
$197$	11.3753	0.810455	0.405228	−	0.914216i	$-0.367192\pi$
$197$	11.3753	0.810455	0.405228	+	0.914216i	$0.367192\pi$
$198$	0	0
$199$	18.0711	1.28102	0.640512	−	0.767948i	$-0.278722\pi$
$199$	18.0711	1.28102	0.640512	+	0.767948i	$0.278722\pi$
$200$	0.509789	0.0360475
$201$	0	0
$202$	−4.72792	−0.332655
$203$	0	0
$204$	0	0
$205$	−10.6569	−0.744307
$206$	28.3326	1.97403
$207$	0	0
$208$	20.4853	1.42040
$209$	−16.9573	−1.17296
$210$	0	0
$211$	0.242641	0.0167041	0.00835204	−	0.999965i	$-0.497341\pi$
$211$	0.242641	0.0167041	0.00835204	+	0.999965i	$0.497341\pi$
$212$	24.4911	1.68205
$213$	0	0
$214$	−11.4142	−0.780260
$215$	−14.8563	−1.01319
$216$	0	0
$217$	0	0
$218$	18.1881	1.23185
$219$	0	0
$220$	19.4853	1.31370
$221$	−8.40401	−0.565315
$222$	0	0
$223$	−11.7279	−0.785360	−0.392680	−	0.919675i	$-0.628452\pi$
$223$	−11.7279	−0.785360	−0.392680	+	0.919675i	$0.628452\pi$
$224$	0	0
$225$	0	0
$226$	−22.3848	−1.48901
$227$	−4.71179	−0.312733	−0.156366	−	0.987699i	$-0.549978\pi$
$227$	−4.71179	−0.312733	−0.156366	+	0.987699i	$0.549978\pi$
$228$	0	0
$229$	−18.4853	−1.22154	−0.610771	−	0.791807i	$-0.709140\pi$
$229$	−18.4853	−1.22154	−0.610771	+	0.791807i	$0.709140\pi$
$230$	30.0731	1.98296
$231$	0	0
$232$	7.31371	0.480168
$233$	−8.19285	−0.536731	−0.268366	−	0.963317i	$-0.586484\pi$
$233$	−8.19285	−0.536731	−0.268366	+	0.963317i	$0.586484\pi$
$234$	0	0
$235$	−22.3848	−1.46022
$236$	7.17327	0.466940
$237$	0	0
$238$	0	0
$239$	−2.97127	−0.192195	−0.0960976	−	0.995372i	$-0.530636\pi$
$239$	−2.97127	−0.192195	−0.0960976	+	0.995372i	$0.530636\pi$
$240$	0	0
$241$	−0.727922	−0.0468896	−0.0234448	−	0.999725i	$-0.507463\pi$
$241$	−0.727922	−0.0468896	−0.0234448	+	0.999725i	$0.507463\pi$
$242$	−7.89422	−0.507460
$243$	0	0
$244$	15.6569	1.00233
$245$	0	0
$246$	0	0
$247$	30.1421	1.91790
$248$	−0.571637	−0.0362990
$249$	0	0
$250$	24.6569	1.55944
$251$	27.9721	1.76559	0.882793	−	0.469762i	$-0.155660\pi$
$251$	27.9721	1.76559	0.882793	+	0.469762i	$0.155660\pi$
$252$	0	0
$253$	−26.1716	−1.64539
$254$	−1.74053	−0.109210
$255$	0	0
$256$	7.48528	0.467830
$257$	29.0536	1.81231	0.906156	−	0.422944i	$-0.139003\pi$
$257$	29.0536	1.81231	0.906156	+	0.422944i	$0.139003\pi$
$258$	0	0
$259$	0	0
$260$	−34.6356	−2.14801
$261$	0	0
$262$	16.5858	1.02467
$263$	31.3039	1.93028	0.965140	−	0.261734i	$-0.0842943\pi$
$263$	31.3039	1.93028	0.965140	+	0.261734i	$0.0842943\pi$
$264$	0	0
$265$	21.3137	1.30929
$266$	0	0
$267$	0	0
$268$	−13.6569	−0.834225
$269$	5.07227	0.309262	0.154631	−	0.987972i	$-0.450581\pi$
$269$	5.07227	0.309262	0.154631	+	0.987972i	$0.450581\pi$
$270$	0	0
$271$	11.1716	0.678625	0.339312	−	0.940674i	$-0.389806\pi$
$271$	11.1716	0.678625	0.339312	+	0.940674i	$0.389806\pi$
$272$	−3.69222	−0.223874
$273$	0	0
$274$	33.7990	2.04187
$275$	−2.25032	−0.135699
$276$	0	0
$277$	−22.5563	−1.35528	−0.677640	−	0.735394i	$-0.736997\pi$
$277$	−22.5563	−1.35528	−0.677640	+	0.735394i	$0.736997\pi$
$278$	−6.66348	−0.399649
$279$	0	0
$280$	0	0
$281$	−3.99084	−0.238074	−0.119037	−	0.992890i	$-0.537981\pi$
$281$	−3.99084	−0.238074	−0.119037	+	0.992890i	$0.537981\pi$
$282$	0	0
$283$	8.48528	0.504398	0.252199	−	0.967675i	$-0.418846\pi$
$283$	8.48528	0.504398	0.252199	+	0.967675i	$0.418846\pi$
$284$	35.5059	2.10689
$285$	0	0
$286$	55.1127	3.25888
$287$	0	0
$288$	0	0
$289$	−15.4853	−0.910899
$290$	37.0971	2.17842
$291$	0	0
$292$	−27.5563	−1.61261
$293$	−25.0009	−1.46057	−0.730283	−	0.683144i	$-0.760612\pi$
$293$	−25.0009	−1.46057	−0.730283	+	0.683144i	$0.760612\pi$
$294$	0	0
$295$	6.24264	0.363461
$296$	−2.82195	−0.164023
$297$	0	0
$298$	11.4142	0.661208
$299$	46.5207	2.69036
$300$	0	0
$301$	0	0
$302$	8.91380	0.512932
$303$	0	0
$304$	13.2426	0.759518
$305$	13.6256	0.780199
$306$	0	0
$307$	24.0711	1.37381	0.686904	−	0.726748i	$-0.258969\pi$
$307$	24.0711	1.37381	0.686904	+	0.726748i	$0.258969\pi$
$308$	0	0
$309$	0	0
$310$	−2.89949	−0.164680
$311$	13.3270	0.755703	0.377852	−	0.925866i	$-0.376663\pi$
$311$	13.3270	0.755703	0.377852	+	0.925866i	$0.376663\pi$
$312$	0	0
$313$	−19.1716	−1.08364	−0.541821	−	0.840494i	$-0.682265\pi$
$313$	−19.1716	−1.08364	−0.541821	+	0.840494i	$0.682265\pi$
$314$	3.18243	0.179595
$315$	0	0
$316$	14.2426	0.801211
$317$	26.4428	1.48517	0.742587	−	0.669750i	$-0.233599\pi$
$317$	26.4428	1.48517	0.742587	+	0.669750i	$0.233599\pi$
$318$	0	0
$319$	−32.2843	−1.80757
$320$	−22.8999	−1.28014
$321$	0	0
$322$	0	0
$323$	−5.43275	−0.302286
$324$	0	0
$325$	4.00000	0.221880
$326$	0.720950	0.0399297
$327$	0	0
$328$	4.41421	0.243734
$329$	0	0
$330$	0	0
$331$	30.9706	1.70230	0.851148	−	0.524926i	$-0.175907\pi$
$331$	30.9706	1.70230	0.851148	+	0.524926i	$0.175907\pi$
$332$	−1.74053	−0.0955239
$333$	0	0
$334$	22.3848	1.22484
$335$	−11.8851	−0.649351
$336$	0	0
$337$	10.7990	0.588258	0.294129	−	0.955766i	$-0.404970\pi$
$337$	10.7990	0.588258	0.294129	+	0.955766i	$0.404970\pi$
$338$	−70.6513	−3.84292
$339$	0	0
$340$	6.24264	0.338555
$341$	2.52333	0.136646
$342$	0	0
$343$	0	0
$344$	6.15370	0.331785
$345$	0	0
$346$	33.4853	1.80018
$347$	−1.59121	−0.0854209	−0.0427104	−	0.999087i	$-0.513599\pi$
$347$	−1.59121	−0.0854209	−0.0427104	+	0.999087i	$0.513599\pi$
$348$	0	0
$349$	−13.0711	−0.699678	−0.349839	−	0.936810i	$-0.613764\pi$
$349$	−13.0711	−0.699678	−0.349839	+	0.936810i	$0.613764\pi$
$350$	0	0
$351$	0	0
$352$	30.8995	1.64695
$353$	−17.9769	−0.956815	−0.478407	−	0.878138i	$-0.658786\pi$
$353$	−17.9769	−0.956815	−0.478407	+	0.878138i	$0.658786\pi$
$354$	0	0
$355$	30.8995	1.63997
$356$	0.360475	0.0191051
$357$	0	0
$358$	−25.4142	−1.34318
$359$	−14.5577	−0.768326	−0.384163	−	0.923265i	$-0.625510\pi$
$359$	−14.5577	−0.768326	−0.384163	+	0.923265i	$0.625510\pi$
$360$	0	0
$361$	0.485281	0.0255411
$362$	50.2129	2.63913
$363$	0	0
$364$	0	0
$365$	−23.9813	−1.25524
$366$	0	0
$367$	−29.3848	−1.53387	−0.766936	−	0.641723i	$-0.778220\pi$
$367$	−29.3848	−1.53387	−0.766936	+	0.641723i	$0.778220\pi$
$368$	20.4384	1.06542
$369$	0	0
$370$	−14.3137	−0.744134
$371$	0	0
$372$	0	0
$373$	−15.1421	−0.784030	−0.392015	−	0.919959i	$-0.628222\pi$
$373$	−15.1421	−0.784030	−0.392015	+	0.919959i	$0.628222\pi$
$374$	−9.93338	−0.513643
$375$	0	0
$376$	9.27208	0.478171
$377$	57.3862	2.95554
$378$	0	0
$379$	17.8995	0.919435	0.459718	−	0.888065i	$-0.347951\pi$
$379$	17.8995	0.919435	0.459718	+	0.888065i	$0.347951\pi$
$380$	−22.3901	−1.14859
$381$	0	0
$382$	−5.48528	−0.280651
$383$	−5.43275	−0.277600	−0.138800	−	0.990320i	$-0.544325\pi$
$383$	−5.43275	−0.277600	−0.138800	+	0.990320i	$0.544325\pi$
$384$	0	0
$385$	0	0
$386$	−25.9330	−1.31995
$387$	0	0
$388$	4.00000	0.203069
$389$	−6.66348	−0.337852	−0.168926	−	0.985629i	$-0.554030\pi$
$389$	−6.66348	−0.337852	−0.168926	+	0.985629i	$0.554030\pi$
$390$	0	0
$391$	−8.38478	−0.424036
$392$	0	0
$393$	0	0
$394$	−23.8995	−1.20404
$395$	12.3949	0.623653
$396$	0	0
$397$	−23.0711	−1.15790	−0.578952	−	0.815362i	$-0.696538\pi$
$397$	−23.0711	−1.15790	−0.578952	+	0.815362i	$0.696538\pi$
$398$	−37.9674	−1.90313
$399$	0	0
$400$	1.75736	0.0878680
$401$	11.1641	0.557509	0.278755	−	0.960362i	$-0.410078\pi$
$401$	11.1641	0.557509	0.278755	+	0.960362i	$0.410078\pi$
$402$	0	0
$403$	−4.48528	−0.223428
$404$	5.43275	0.270289
$405$	0	0
$406$	0	0
$407$	12.4567	0.617456
$408$	0	0
$409$	−5.07107	−0.250748	−0.125374	−	0.992110i	$-0.540013\pi$
$409$	−5.07107	−0.250748	−0.125374	+	0.992110i	$0.540013\pi$
$410$	22.3901	1.10577
$411$	0	0
$412$	−32.5563	−1.60394
$413$	0	0
$414$	0	0
$415$	−1.51472	−0.0743546
$416$	−54.9247	−2.69291
$417$	0	0
$418$	35.6274	1.74259
$419$	12.3949	0.605528	0.302764	−	0.953066i	$-0.402091\pi$
$419$	12.3949	0.605528	0.302764	+	0.953066i	$0.402091\pi$
$420$	0	0
$421$	7.92893	0.386433	0.193216	−	0.981156i	$-0.438108\pi$
$421$	7.92893	0.386433	0.193216	+	0.981156i	$0.438108\pi$
$422$	−0.509789	−0.0248161
$423$	0	0
$424$	−8.82843	−0.428746
$425$	−0.720950	−0.0349712
$426$	0	0
$427$	0	0
$428$	13.1158	0.633976
$429$	0	0
$430$	31.2132	1.50523
$431$	−0.149314	−0.00719219	−0.00359609	−	0.999994i	$-0.501145\pi$
$431$	−0.149314	−0.00719219	−0.00359609	+	0.999994i	$0.501145\pi$
$432$	0	0
$433$	8.97056	0.431098	0.215549	−	0.976493i	$-0.430846\pi$
$433$	8.97056	0.431098	0.215549	+	0.976493i	$0.430846\pi$
$434$	0	0
$435$	0	0
$436$	−20.8995	−1.00090
$437$	30.0731	1.43859
$438$	0	0
$439$	19.7990	0.944954	0.472477	−	0.881343i	$-0.343360\pi$
$439$	19.7990	0.944954	0.472477	+	0.881343i	$0.343360\pi$
$440$	−7.02396	−0.334854
$441$	0	0
$442$	17.6569	0.839851
$443$	−11.5246	−0.547550	−0.273775	−	0.961794i	$-0.588272\pi$
$443$	−11.5246	−0.547550	−0.273775	+	0.961794i	$0.588272\pi$
$444$	0	0
$445$	0.313708	0.0148712
$446$	24.6404	1.16676
$447$	0	0
$448$	0	0
$449$	−25.0009	−1.17986	−0.589932	−	0.807453i	$-0.700846\pi$
$449$	−25.0009	−1.17986	−0.589932	+	0.807453i	$0.700846\pi$
$450$	0	0
$451$	−19.4853	−0.917526
$452$	25.7218	1.20985
$453$	0	0
$454$	9.89949	0.464606
$455$	0	0
$456$	0	0
$457$	−13.9706	−0.653515	−0.326758	−	0.945108i	$-0.605956\pi$
$457$	−13.9706	−0.653515	−0.326758	+	0.945108i	$0.605956\pi$
$458$	38.8376	1.81476
$459$	0	0
$460$	−34.5563	−1.61120
$461$	20.8607	0.971580	0.485790	−	0.874075i	$-0.338532\pi$
$461$	20.8607	0.971580	0.485790	+	0.874075i	$0.338532\pi$
$462$	0	0
$463$	−7.89949	−0.367121	−0.183560	−	0.983008i	$-0.558762\pi$
$463$	−7.89949	−0.367121	−0.183560	+	0.983008i	$0.558762\pi$
$464$	25.2120	1.17044
$465$	0	0
$466$	17.2132	0.797386
$467$	−13.1158	−0.606927	−0.303464	−	0.952843i	$-0.598143\pi$
$467$	−13.1158	−0.606927	−0.303464	+	0.952843i	$0.598143\pi$
$468$	0	0
$469$	0	0
$470$	47.0305	2.16935
$471$	0	0
$472$	−2.58579	−0.119020
$473$	−27.1637	−1.24899
$474$	0	0
$475$	2.58579	0.118644
$476$	0	0
$477$	0	0
$478$	6.24264	0.285532
$479$	27.2512	1.24514	0.622569	−	0.782565i	$-0.286089\pi$
$479$	27.2512	1.24514	0.622569	+	0.782565i	$0.286089\pi$
$480$	0	0
$481$	−22.1421	−1.00959
$482$	1.52937	0.0696607
$483$	0	0
$484$	9.07107	0.412321
$485$	3.48106	0.158067
$486$	0	0
$487$	35.4142	1.60477	0.802386	−	0.596806i	$-0.203564\pi$
$487$	35.4142	1.60477	0.802386	+	0.596806i	$0.203564\pi$
$488$	−5.64391	−0.255488
$489$	0	0
$490$	0	0
$491$	−22.8999	−1.03346	−0.516728	−	0.856149i	$-0.672850\pi$
$491$	−22.8999	−1.03346	−0.516728	+	0.856149i	$0.672850\pi$
$492$	0	0
$493$	−10.3431	−0.465832
$494$	−63.3287	−2.84929
$495$	0	0
$496$	−1.97056	−0.0884809
$497$	0	0
$498$	0	0
$499$	−10.4437	−0.467522	−0.233761	−	0.972294i	$-0.575103\pi$
$499$	−10.4437	−0.467522	−0.233761	+	0.972294i	$0.575103\pi$
$500$	−28.3326	−1.26707
$501$	0	0
$502$	−58.7696	−2.62301
$503$	5.22158	0.232819	0.116409	−	0.993201i	$-0.462862\pi$
$503$	5.22158	0.232819	0.116409	+	0.993201i	$0.462862\pi$
$504$	0	0
$505$	4.72792	0.210390
$506$	54.9866	2.44445
$507$	0	0
$508$	2.00000	0.0887357
$509$	−14.8563	−0.658495	−0.329248	−	0.944244i	$-0.606795\pi$
$509$	−14.8563	−0.658495	−0.329248	+	0.944244i	$0.606795\pi$
$510$	0	0
$511$	0	0
$512$	−29.3522	−1.29720
$513$	0	0
$514$	−61.0416	−2.69243
$515$	−28.3326	−1.24848
$516$	0	0
$517$	−40.9289	−1.80005
$518$	0	0
$519$	0	0
$520$	12.4853	0.547516
$521$	−30.0731	−1.31753	−0.658764	−	0.752349i	$-0.728921\pi$
$521$	−30.0731	−1.31753	−0.658764	+	0.752349i	$0.728921\pi$
$522$	0	0
$523$	−43.7696	−1.91391	−0.956954	−	0.290238i	$-0.906265\pi$
$523$	−43.7696	−1.91391	−0.956954	+	0.290238i	$0.906265\pi$
$524$	−19.0583	−0.832567
$525$	0	0
$526$	−65.7696	−2.86769
$527$	0.808416	0.0352152
$528$	0	0
$529$	23.4142	1.01801
$530$	−44.7802	−1.94513
$531$	0	0
$532$	0	0
$533$	34.6356	1.50024
$534$	0	0
$535$	11.4142	0.493479
$536$	4.92296	0.212639
$537$	0	0
$538$	−10.6569	−0.459450
$539$	0	0
$540$	0	0
$541$	−21.3848	−0.919403	−0.459702	−	0.888073i	$-0.652044\pi$
$541$	−21.3848	−0.919403	−0.459702	+	0.888073i	$0.652044\pi$
$542$	−23.4715	−1.00819
$543$	0	0
$544$	9.89949	0.424437
$545$	−18.1881	−0.779092
$546$	0	0
$547$	7.51472	0.321306	0.160653	−	0.987011i	$-0.448640\pi$
$547$	7.51472	0.321306	0.160653	+	0.987011i	$0.448640\pi$
$548$	−38.8376	−1.65906
$549$	0	0
$550$	4.72792	0.201599
$551$	37.0971	1.58039
$552$	0	0
$553$	0	0
$554$	47.3910	2.01345
$555$	0	0
$556$	7.65685	0.324723
$557$	0.932112	0.0394948	0.0197474	−	0.999805i	$-0.493714\pi$
$557$	0.932112	0.0394948	0.0197474	+	0.999805i	$0.493714\pi$
$558$	0	0
$559$	48.2843	2.04221
$560$	0	0
$561$	0	0
$562$	8.38478	0.353690
$563$	−8.10538	−0.341601	−0.170801	−	0.985306i	$-0.554635\pi$
$563$	−8.10538	−0.341601	−0.170801	+	0.985306i	$0.554635\pi$
$564$	0	0
$565$	22.3848	0.941735
$566$	−17.8276	−0.749350
$567$	0	0
$568$	−12.7990	−0.537034
$569$	24.2799	1.01787	0.508934	−	0.860806i	$-0.330040\pi$
$569$	24.2799	1.01787	0.508934	+	0.860806i	$0.330040\pi$
$570$	0	0
$571$	16.0416	0.671321	0.335661	−	0.941983i	$-0.391040\pi$
$571$	16.0416	0.671321	0.335661	+	0.941983i	$0.391040\pi$
$572$	−63.3287	−2.64791
$573$	0	0
$574$	0	0
$575$	3.99084	0.166430
$576$	0	0
$577$	29.3553	1.22208	0.611039	−	0.791600i	$-0.290752\pi$
$577$	29.3553	1.22208	0.611039	+	0.791600i	$0.290752\pi$
$578$	32.5346	1.35326
$579$	0	0
$580$	−42.6274	−1.77001
$581$	0	0
$582$	0	0
$583$	38.9706	1.61400
$584$	9.93338	0.411046
$585$	0	0
$586$	52.5269	2.16987
$587$	17.0192	0.702457	0.351228	−	0.936290i	$-0.385764\pi$
$587$	17.0192	0.702457	0.351228	+	0.936290i	$0.385764\pi$
$588$	0	0
$589$	−2.89949	−0.119472
$590$	−13.1158	−0.539969
$591$	0	0
$592$	−9.72792	−0.399815
$593$	−32.5346	−1.33604	−0.668018	−	0.744145i	$-0.732857\pi$
$593$	−32.5346	−1.33604	−0.668018	+	0.744145i	$0.732857\pi$
$594$	0	0
$595$	0	0
$596$	−13.1158	−0.537244
$597$	0	0
$598$	−97.7401	−3.99689
$599$	−2.90942	−0.118876	−0.0594378	−	0.998232i	$-0.518931\pi$
$599$	−2.90942	−0.118876	−0.0594378	+	0.998232i	$0.518931\pi$
$600$	0	0
$601$	−26.6274	−1.08615	−0.543077	−	0.839683i	$-0.682741\pi$
$601$	−26.6274	−1.08615	−0.543077	+	0.839683i	$0.682741\pi$
$602$	0	0
$603$	0	0
$604$	−10.2426	−0.416767
$605$	7.89422	0.320946
$606$	0	0
$607$	−6.68629	−0.271388	−0.135694	−	0.990751i	$-0.543326\pi$
$607$	−6.68629	−0.271388	−0.135694	+	0.990751i	$0.543326\pi$
$608$	−35.5059	−1.43995
$609$	0	0
$610$	−28.6274	−1.15909
$611$	72.7523	2.94324
$612$	0	0
$613$	5.62742	0.227289	0.113645	−	0.993521i	$-0.463747\pi$
$613$	5.62742	0.227289	0.113645	+	0.993521i	$0.463747\pi$
$614$	−50.5734	−2.04098
$615$	0	0
$616$	0	0
$617$	−13.8368	−0.557047	−0.278523	−	0.960429i	$-0.589845\pi$
$617$	−13.8368	−0.557047	−0.278523	+	0.960429i	$0.589845\pi$
$618$	0	0
$619$	34.4558	1.38490	0.692449	−	0.721467i	$-0.256532\pi$
$619$	34.4558	1.38490	0.692449	+	0.721467i	$0.256532\pi$
$620$	3.33174	0.133806
$621$	0	0
$622$	−28.0000	−1.12270
$623$	0	0
$624$	0	0
$625$	−21.7279	−0.869117
$626$	40.2795	1.60989
$627$	0	0
$628$	−3.65685	−0.145924
$629$	3.99084	0.159125
$630$	0	0
$631$	−41.3553	−1.64633	−0.823165	−	0.567802i	$-0.807794\pi$
$631$	−41.3553	−1.64633	−0.823165	+	0.567802i	$0.807794\pi$
$632$	−5.13412	−0.204224
$633$	0	0
$634$	−55.5563	−2.20642
$635$	1.74053	0.0690707
$636$	0	0
$637$	0	0
$638$	67.8294	2.68539
$639$	0	0
$640$	14.3137	0.565799
$641$	−25.2995	−0.999270	−0.499635	−	0.866236i	$-0.666533\pi$
$641$	−25.2995	−0.999270	−0.499635	+	0.866236i	$0.666533\pi$
$642$	0	0
$643$	11.2426	0.443366	0.221683	−	0.975119i	$-0.428845\pi$
$643$	11.2426	0.443366	0.221683	+	0.975119i	$0.428845\pi$
$644$	0	0
$645$	0	0
$646$	11.4142	0.449086
$647$	5.43275	0.213583	0.106792	−	0.994281i	$-0.465942\pi$
$647$	5.43275	0.213583	0.106792	+	0.994281i	$0.465942\pi$
$648$	0	0
$649$	11.4142	0.448047
$650$	−8.40401	−0.329632
$651$	0	0
$652$	−0.828427	−0.0324437
$653$	21.2212	0.830449	0.415225	−	0.909719i	$-0.363703\pi$
$653$	21.2212	0.830449	0.415225	+	0.909719i	$0.363703\pi$
$654$	0	0
$655$	−16.5858	−0.648060
$656$	15.2168	0.594117
$657$	0	0
$658$	0	0
$659$	−23.1110	−0.900278	−0.450139	−	0.892958i	$-0.648626\pi$
$659$	−23.1110	−0.900278	−0.450139	+	0.892958i	$0.648626\pi$
$660$	0	0
$661$	−30.1421	−1.17239	−0.586197	−	0.810169i	$-0.699375\pi$
$661$	−30.1421	−1.17239	−0.586197	+	0.810169i	$0.699375\pi$
$662$	−65.0692	−2.52899
$663$	0	0
$664$	0.627417	0.0243485
$665$	0	0
$666$	0	0
$667$	57.2548	2.21692
$668$	−25.7218	−0.995207
$669$	0	0
$670$	24.9706	0.964697
$671$	24.9134	0.961771
$672$	0	0
$673$	20.8284	0.802877	0.401438	−	0.915886i	$-0.368510\pi$
$673$	20.8284	0.802877	0.401438	+	0.915886i	$0.368510\pi$
$674$	−22.6887	−0.873936
$675$	0	0
$676$	81.1838	3.12245
$677$	−15.4280	−0.592945	−0.296473	−	0.955041i	$-0.595810\pi$
$677$	−15.4280	−0.592945	−0.296473	+	0.955041i	$0.595810\pi$
$678$	0	0
$679$	0	0
$680$	−2.25032	−0.0862957
$681$	0	0
$682$	−5.30152	−0.203006
$683$	−23.1110	−0.884319	−0.442160	−	0.896936i	$-0.645788\pi$
$683$	−23.1110	−0.884319	−0.442160	+	0.896936i	$0.645788\pi$
$684$	0	0
$685$	−33.7990	−1.29139
$686$	0	0
$687$	0	0
$688$	21.2132	0.808746
$689$	−69.2713	−2.63902
$690$	0	0
$691$	39.1716	1.49016	0.745078	−	0.666977i	$-0.232412\pi$
$691$	39.1716	1.49016	0.745078	+	0.666977i	$0.232412\pi$
$692$	−38.4772	−1.46268
$693$	0	0
$694$	3.34315	0.126904
$695$	6.66348	0.252760
$696$	0	0
$697$	−6.24264	−0.236457
$698$	27.4624	1.03947
$699$	0	0
$700$	0	0
$701$	−4.20201	−0.158708	−0.0793538	−	0.996847i	$-0.525286\pi$
$701$	−4.20201	−0.158708	−0.0793538	+	0.996847i	$0.525286\pi$
$702$	0	0
$703$	−14.3137	−0.539852
$704$	−41.8707	−1.57806
$705$	0	0
$706$	37.7696	1.42148
$707$	0	0
$708$	0	0
$709$	42.6569	1.60201	0.801006	−	0.598656i	$-0.204299\pi$
$709$	42.6569	1.60201	0.801006	+	0.598656i	$0.204299\pi$
$710$	−64.9199	−2.43640
$711$	0	0
$712$	−0.129942	−0.00486979
$713$	−4.47502	−0.167591
$714$	0	0
$715$	−55.1127	−2.06110
$716$	29.2029	1.09136
$717$	0	0
$718$	30.5858	1.14145
$719$	−18.7597	−0.699619	−0.349810	−	0.936821i	$-0.613754\pi$
$719$	−18.7597	−0.699619	−0.349810	+	0.936821i	$0.613754\pi$
$720$	0	0
$721$	0	0
$722$	−1.01958	−0.0379447
$723$	0	0
$724$	−57.6985	−2.14435
$725$	4.92296	0.182834
$726$	0	0
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	0	0
$729$	0	0
$730$	50.3848	1.86482
$731$	−8.70264	−0.321879
$732$	0	0
$733$	−15.7574	−0.582011	−0.291006	−	0.956721i	$-0.593990\pi$
$733$	−15.7574	−0.582011	−0.291006	+	0.956721i	$0.593990\pi$
$734$	61.7375	2.27877
$735$	0	0
$736$	−54.7990	−2.01992
$737$	−21.7310	−0.800471
$738$	0	0
$739$	−29.5563	−1.08725	−0.543624	−	0.839329i	$-0.682948\pi$
$739$	−29.5563	−1.08725	−0.543624	+	0.839329i	$0.682948\pi$
$740$	16.4475	0.604624
$741$	0	0
$742$	0	0
$743$	−29.0536	−1.06587	−0.532936	−	0.846156i	$-0.678911\pi$
$743$	−29.0536	−1.06587	−0.532936	+	0.846156i	$0.678911\pi$
$744$	0	0
$745$	−11.4142	−0.418184
$746$	31.8137	1.16478
$747$	0	0
$748$	11.4142	0.417345
$749$	0	0
$750$	0	0
$751$	−12.1005	−0.441554	−0.220777	−	0.975324i	$-0.570859\pi$
$751$	−12.1005	−0.441554	−0.220777	+	0.975324i	$0.570859\pi$
$752$	31.9630	1.16557
$753$	0	0
$754$	−120.569	−4.39085
$755$	−8.91380	−0.324406
$756$	0	0
$757$	−9.65685	−0.350984	−0.175492	−	0.984481i	$-0.556152\pi$
$757$	−9.65685	−0.350984	−0.175492	+	0.984481i	$0.556152\pi$
$758$	−37.6069	−1.36594
$759$	0	0
$760$	8.07107	0.292768
$761$	17.3178	0.627770	0.313885	−	0.949461i	$-0.398369\pi$
$761$	17.3178	0.627770	0.313885	+	0.949461i	$0.398369\pi$
$762$	0	0
$763$	0	0
$764$	6.30301	0.228035
$765$	0	0
$766$	11.4142	0.412412
$767$	−20.2891	−0.732596
$768$	0	0
$769$	10.9706	0.395609	0.197804	−	0.980242i	$-0.436619\pi$
$769$	10.9706	0.395609	0.197804	+	0.980242i	$0.436619\pi$
$770$	0	0
$771$	0	0
$772$	29.7990	1.07249
$773$	34.9961	1.25872	0.629361	−	0.777113i	$-0.283317\pi$
$773$	34.9961	1.25872	0.629361	+	0.777113i	$0.283317\pi$
$774$	0	0
$775$	−0.384776	−0.0138216
$776$	−1.44190	−0.0517612
$777$	0	0
$778$	14.0000	0.501924
$779$	22.3901	0.802208
$780$	0	0
$781$	56.4975	2.02164
$782$	17.6164	0.629963
$783$	0	0
$784$	0	0
$785$	−3.18243	−0.113586
$786$	0	0
$787$	6.34315	0.226109	0.113054	−	0.993589i	$-0.463937\pi$
$787$	6.34315	0.226109	0.113054	+	0.993589i	$0.463937\pi$
$788$	27.4624	0.978306
$789$	0	0
$790$	−26.0416	−0.926520
$791$	0	0
$792$	0	0
$793$	−44.2843	−1.57258
$794$	48.4724	1.72022
$795$	0	0
$796$	43.6274	1.54633
$797$	40.9386	1.45012	0.725060	−	0.688685i	$-0.241812\pi$
$797$	40.9386	1.45012	0.725060	+	0.688685i	$0.241812\pi$
$798$	0	0
$799$	−13.1127	−0.463894
$800$	−4.71179	−0.166587
$801$	0	0
$802$	−23.4558	−0.828255
$803$	−43.8481	−1.54736
$804$	0	0
$805$	0	0
$806$	9.42359	0.331932
$807$	0	0
$808$	−1.95837	−0.0688952
$809$	33.1937	1.16703	0.583515	−	0.812103i	$-0.301677\pi$
$809$	33.1937	1.16703	0.583515	+	0.812103i	$0.301677\pi$
$810$	0	0
$811$	7.62742	0.267835	0.133917	−	0.990992i	$-0.457244\pi$
$811$	7.62742	0.267835	0.133917	+	0.990992i	$0.457244\pi$
$812$	0	0
$813$	0	0
$814$	−26.1716	−0.917313
$815$	−0.720950	−0.0252538
$816$	0	0
$817$	31.2132	1.09201
$818$	10.6543	0.372520
$819$	0	0
$820$	−25.7279	−0.898458
$821$	−5.13412	−0.179182	−0.0895910	−	0.995979i	$-0.528556\pi$
$821$	−5.13412	−0.179182	−0.0895910	+	0.995979i	$0.528556\pi$
$822$	0	0
$823$	−25.2721	−0.880929	−0.440465	−	0.897770i	$-0.645186\pi$
$823$	−25.2721	−0.880929	−0.440465	+	0.897770i	$0.645186\pi$
$824$	11.7358	0.408834
$825$	0	0
$826$	0	0
$827$	24.3418	0.846446	0.423223	−	0.906025i	$-0.360899\pi$
$827$	24.3418	0.846446	0.423223	+	0.906025i	$0.360899\pi$
$828$	0	0
$829$	3.21320	0.111599	0.0557996	−	0.998442i	$-0.482229\pi$
$829$	3.21320	0.111599	0.0557996	+	0.998442i	$0.482229\pi$
$830$	3.18243	0.110464
$831$	0	0
$832$	74.4264	2.58027
$833$	0	0
$834$	0	0
$835$	−22.3848	−0.774657
$836$	−40.9386	−1.41589
$837$	0	0
$838$	−26.0416	−0.899593
$839$	−3.48106	−0.120179	−0.0600897	−	0.998193i	$-0.519139\pi$
$839$	−3.48106	−0.120179	−0.0600897	+	0.998193i	$0.519139\pi$
$840$	0	0
$841$	41.6274	1.43543
$842$	−16.6587	−0.574097
$843$	0	0
$844$	0.585786	0.0201636
$845$	70.6513	2.43048
$846$	0	0
$847$	0	0
$848$	−30.4336	−1.04509
$849$	0	0
$850$	1.51472	0.0519544
$851$	−22.0915	−0.757285
$852$	0	0
$853$	−33.1716	−1.13577	−0.567887	−	0.823107i	$-0.692239\pi$
$853$	−33.1716	−1.13577	−0.567887	+	0.823107i	$0.692239\pi$
$854$	0	0
$855$	0	0
$856$	−4.72792	−0.161597
$857$	34.0640	1.16360	0.581802	−	0.813331i	$-0.302348\pi$
$857$	34.0640	1.16360	0.581802	+	0.813331i	$0.302348\pi$
$858$	0	0
$859$	31.7279	1.08254	0.541271	−	0.840848i	$-0.317943\pi$
$859$	31.7279	1.08254	0.541271	+	0.840848i	$0.317943\pi$
$860$	−35.8664	−1.22303
$861$	0	0
$862$	0.313708	0.0106850
$863$	−33.7035	−1.14728	−0.573640	−	0.819107i	$-0.694469\pi$
$863$	−33.7035	−1.14728	−0.573640	+	0.819107i	$0.694469\pi$
$864$	0	0
$865$	−33.4853	−1.13853
$866$	−18.8472	−0.640453
$867$	0	0
$868$	0	0
$869$	22.6631	0.768793
$870$	0	0
$871$	38.6274	1.30884
$872$	7.53375	0.255125
$873$	0	0
$874$	−63.1838	−2.13722
$875$	0	0
$876$	0	0
$877$	−26.8284	−0.905932	−0.452966	−	0.891528i	$-0.649634\pi$
$877$	−26.8284	−0.905932	−0.452966	+	0.891528i	$0.649634\pi$
$878$	−41.5977	−1.40386
$879$	0	0
$880$	−24.2132	−0.816227
$881$	9.69660	0.326687	0.163343	−	0.986569i	$-0.447772\pi$
$881$	9.69660	0.326687	0.163343	+	0.986569i	$0.447772\pi$
$882$	0	0
$883$	−44.6274	−1.50183	−0.750916	−	0.660398i	$-0.770388\pi$
$883$	−44.6274	−1.50183	−0.750916	+	0.660398i	$0.770388\pi$
$884$	−20.2891	−0.682396
$885$	0	0
$886$	24.2132	0.813458
$887$	48.6835	1.63463	0.817317	−	0.576189i	$-0.195461\pi$
$887$	48.6835	1.63463	0.817317	+	0.576189i	$0.195461\pi$
$888$	0	0
$889$	0	0
$890$	−0.659102	−0.0220932
$891$	0	0
$892$	−28.3137	−0.948013
$893$	47.0305	1.57382
$894$	0	0
$895$	25.4142	0.849503
$896$	0	0
$897$	0	0
$898$	52.5269	1.75285
$899$	−5.52021	−0.184109
$900$	0	0
$901$	12.4853	0.415945
$902$	40.9386	1.36311
$903$	0	0
$904$	−9.27208	−0.308385
$905$	−50.2129	−1.66913
$906$	0	0
$907$	25.2721	0.839146	0.419573	−	0.907722i	$-0.362180\pi$
$907$	25.2721	0.839146	0.419573	+	0.907722i	$0.362180\pi$
$908$	−11.3753	−0.377502
$909$	0	0
$910$	0	0
$911$	−51.9534	−1.72129	−0.860647	−	0.509202i	$-0.829941\pi$
$911$	−51.9534	−1.72129	−0.860647	+	0.509202i	$0.829941\pi$
$912$	0	0
$913$	−2.76955	−0.0916588
$914$	29.3522	0.970884
$915$	0	0
$916$	−44.6274	−1.47453
$917$	0	0
$918$	0	0
$919$	−18.8284	−0.621093	−0.310546	−	0.950558i	$-0.600512\pi$
$919$	−18.8284	−0.621093	−0.310546	+	0.950558i	$0.600512\pi$
$920$	12.4567	0.410685
$921$	0	0
$922$	−43.8284	−1.44341
$923$	−100.426	−3.30556
$924$	0	0
$925$	−1.89949	−0.0624550
$926$	16.5969	0.545407
$927$	0	0
$928$	−67.5980	−2.21901
$929$	11.0767	0.363413	0.181707	−	0.983353i	$-0.441838\pi$
$929$	11.0767	0.363413	0.181707	+	0.983353i	$0.441838\pi$
$930$	0	0
$931$	0	0
$932$	−19.7793	−0.647892
$933$	0	0
$934$	27.5563	0.901671
$935$	9.93338	0.324856
$936$	0	0
$937$	−59.2548	−1.93577	−0.967886	−	0.251391i	$-0.919112\pi$
$937$	−59.2548	−1.93577	−0.967886	+	0.251391i	$0.919112\pi$
$938$	0	0
$939$	0	0
$940$	−54.0416	−1.76264
$941$	29.0536	0.947119	0.473560	−	0.880762i	$-0.342969\pi$
$941$	29.0536	0.947119	0.473560	+	0.880762i	$0.342969\pi$
$942$	0	0
$943$	34.5563	1.12531
$944$	−8.91380	−0.290120
$945$	0	0
$946$	57.0711	1.85554
$947$	−31.3039	−1.01724	−0.508620	−	0.860991i	$-0.669844\pi$
$947$	−31.3039	−1.01724	−0.508620	+	0.860991i	$0.669844\pi$
$948$	0	0
$949$	77.9411	2.53008
$950$	−5.43275	−0.176262
$951$	0	0
$952$	0	0
$953$	40.2795	1.30478	0.652391	−	0.757883i	$-0.273766\pi$
$953$	40.2795	1.30478	0.652391	+	0.757883i	$0.273766\pi$
$954$	0	0
$955$	5.48528	0.177500
$956$	−7.17327	−0.232000
$957$	0	0
$958$	−57.2548	−1.84982
$959$	0	0
$960$	0	0
$961$	−30.5685	−0.986082
$962$	46.5207	1.49989
$963$	0	0
$964$	−1.75736	−0.0566007
$965$	25.9330	0.834812
$966$	0	0
$967$	1.89949	0.0610836	0.0305418	−	0.999533i	$-0.490277\pi$
$967$	1.89949	0.0610836	0.0305418	+	0.999533i	$0.490277\pi$
$968$	−3.26989	−0.105098
$969$	0	0
$970$	−7.31371	−0.234829
$971$	24.4911	0.785956	0.392978	−	0.919548i	$-0.371445\pi$
$971$	24.4911	0.785956	0.392978	+	0.919548i	$0.371445\pi$
$972$	0	0
$973$	0	0
$974$	−74.4054	−2.38410
$975$	0	0
$976$	−19.4558	−0.622766
$977$	53.9051	1.72458	0.862289	−	0.506417i	$-0.169030\pi$
$977$	53.9051	1.72458	0.862289	+	0.506417i	$0.169030\pi$
$978$	0	0
$979$	0.573593	0.0183321
$980$	0	0
$981$	0	0
$982$	48.1127	1.53534
$983$	−15.5773	−0.496838	−0.248419	−	0.968653i	$-0.579911\pi$
$983$	−15.5773	−0.496838	−0.248419	+	0.968653i	$0.579911\pi$
$984$	0	0
$985$	23.8995	0.761501
$986$	21.7310	0.692055
$987$	0	0
$988$	72.7696	2.31511
$989$	48.1738	1.53184
$990$	0	0
$991$	26.9706	0.856748	0.428374	−	0.903601i	$-0.359087\pi$
$991$	26.9706	0.856748	0.428374	+	0.903601i	$0.359087\pi$
$992$	5.28343	0.167749
$993$	0	0
$994$	0	0
$995$	37.9674	1.20365
$996$	0	0
$997$	−52.8701	−1.67441	−0.837206	−	0.546888i	$-0.815812\pi$
$997$	−52.8701	−1.67441	−0.837206	+	0.546888i	$0.815812\pi$
$998$	21.9421	0.694566
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.bc.1.1	✓	4
3.2	odd	2	inner	1323.2.a.bc.1.4	yes	4
7.6	odd	2		1323.2.a.bd.1.1	yes	4
21.20	even	2		1323.2.a.bd.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.a.bc.1.1	✓	4	1.1	even	1	trivial
1323.2.a.bc.1.4	yes	4	3.2	odd	2	inner
1323.2.a.bd.1.1	yes	4	7.6	odd	2
1323.2.a.bd.1.4	yes	4	21.20	even	2

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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10100 q^{2} +2.41421 q^{4} +2.10100 q^{5} -0.870264 q^{8} -4.41421 q^{10} +3.84153 q^{11} -6.82843 q^{13} -3.00000 q^{16} +1.23074 q^{17} -4.41421 q^{19} +5.07227 q^{20} -8.07107 q^{22} -6.81280 q^{23} -0.585786 q^{25} +14.3465 q^{26} -8.40401 q^{29} +0.656854 q^{31} +8.04354 q^{32} -2.58579 q^{34} +3.24264 q^{37} +9.27428 q^{38} -1.82843 q^{40} -5.07227 q^{41} -7.07107 q^{43} +9.27428 q^{44} +14.3137 q^{46} -10.6543 q^{47} +1.23074 q^{50} -16.4853 q^{52} +10.1445 q^{53} +8.07107 q^{55} +17.6569 q^{58} +2.97127 q^{59} +6.48528 q^{61} -1.38005 q^{62} -10.8995 q^{64} -14.3465 q^{65} -5.65685 q^{67} +2.97127 q^{68} +14.7070 q^{71} -11.4142 q^{73} -6.81280 q^{74} -10.6569 q^{76} +5.89949 q^{79} -6.30301 q^{80} +10.6569 q^{82} -0.720950 q^{83} +2.58579 q^{85} +14.8563 q^{86} -3.34315 q^{88} +0.149314 q^{89} -16.4475 q^{92} +22.3848 q^{94} -9.27428 q^{95} +1.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 12 q^{10} - 16 q^{13} - 12 q^{16} - 12 q^{19} - 4 q^{22} - 8 q^{25} - 20 q^{31} - 16 q^{34} - 4 q^{37} + 4 q^{40} + 12 q^{46} - 32 q^{52} + 4 q^{55} + 48 q^{58} - 8 q^{61} - 4 q^{64} - 40 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 12 * q^10 - 16 * q^13 - 12 * q^16 - 12 * q^19 - 4 * q^22 - 8 * q^25 - 20 * q^31 - 16 * q^34 - 4 * q^37 + 4 * q^40 + 12 * q^46 - 32 * q^52 + 4 * q^55 + 48 * q^58 - 8 * q^61 - 4 * q^64 - 40 * q^73 - 20 * q^76 - 16 * q^79 + 20 * q^82 + 16 * q^85 - 36 * q^88 + 16 * q^94 - 16 * q^97

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L-functions

Modular forms

Varieties

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