LMFDB - Embedded newform 1521.2.b.h.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.h.1351.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155i q^{2} -4.56155 q^{4} -0.561553i q^{5} +3.56155i q^{7} +6.56155i q^{8} +O(q^{10})$	$q-2.56155i q^{2} -4.56155 q^{4} -0.561553i q^{5} +3.56155i q^{7} +6.56155i q^{8} -1.43845 q^{10} -2.00000i q^{11} +9.12311 q^{14} +7.68466 q^{16} +2.56155 q^{17} -1.12311i q^{19} +2.56155i q^{20} -5.12311 q^{22} +2.00000 q^{23} +4.68466 q^{25} -16.2462i q^{28} +5.68466 q^{29} -1.56155i q^{31} -6.56155i q^{32} -6.56155i q^{34} +2.00000 q^{35} -3.43845i q^{37} -2.87689 q^{38} +3.68466 q^{40} -2.56155i q^{41} -0.438447 q^{43} +9.12311i q^{44} -5.12311i q^{46} -8.24621i q^{47} -5.68466 q^{49} -12.0000i q^{50} -11.6847 q^{53} -1.12311 q^{55} -23.3693 q^{56} -14.5616i q^{58} -11.1231i q^{59} +12.1231 q^{61} -4.00000 q^{62} -1.43845 q^{64} +0.438447i q^{67} -11.6847 q^{68} -5.12311i q^{70} -14.0000i q^{71} +1.87689i q^{73} -8.80776 q^{74} +5.12311i q^{76} +7.12311 q^{77} +9.56155 q^{79} -4.31534i q^{80} -6.56155 q^{82} +9.12311i q^{83} -1.43845i q^{85} +1.12311i q^{86} +13.1231 q^{88} +13.1231i q^{89} -9.12311 q^{92} -21.1231 q^{94} -0.630683 q^{95} -4.43845i q^{97} +14.5616i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4}+O(q^{10}) $ 4 * q - 10 * q^4	$ 4 q - 10 q^{4} - 14 q^{10} + 20 q^{14} + 6 q^{16} + 2 q^{17} - 4 q^{22} + 8 q^{23} - 6 q^{25} - 2 q^{29} + 8 q^{35} - 28 q^{38} - 10 q^{40} - 10 q^{43} + 2 q^{49} - 22 q^{53} + 12 q^{55} - 44 q^{56} + 32 q^{61} - 16 q^{62} - 14 q^{64} - 22 q^{68} + 6 q^{74} + 12 q^{77} + 30 q^{79} - 18 q^{82} + 36 q^{88} - 20 q^{92} - 68 q^{94} - 52 q^{95}+O(q^{100}) $ 4 * q - 10 * q^4 - 14 * q^10 + 20 * q^14 + 6 * q^16 + 2 * q^17 - 4 * q^22 + 8 * q^23 - 6 * q^25 - 2 * q^29 + 8 * q^35 - 28 * q^38 - 10 * q^40 - 10 * q^43 + 2 * q^49 - 22 * q^53 + 12 * q^55 - 44 * q^56 + 32 * q^61 - 16 * q^62 - 14 * q^64 - 22 * q^68 + 6 * q^74 + 12 * q^77 + 30 * q^79 - 18 * q^82 + 36 * q^88 - 20 * q^92 - 68 * q^94 - 52 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 2.56155i	− 1.81129i	−0.424035	−	0.905646i	$-0.639387\pi$
$2$	− 2.56155i	− 1.81129i	0.424035	−	0.905646i	$-0.360613\pi$
$3$	0	0
$3$	0	0
$4$	−4.56155	−2.28078
$5$	− 0.561553i	− 0.251134i	−0.992085	−	0.125567i	$-0.959925\pi$
$5$	− 0.561553i	− 0.251134i	0.992085	−	0.125567i	$-0.0400750\pi$
$6$	0	0
$7$	3.56155i	1.34614i	0.739579	+	0.673070i	$0.235025\pi$
$7$	3.56155i	1.34614i	−0.739579	+	0.673070i	$0.764975\pi$
$8$	6.56155i	2.31986i
$9$	0	0
$10$	−1.43845	−0.454877
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	9.12311	2.43825
$15$	0	0
$16$	7.68466	1.92116
$17$	2.56155	0.621268	0.310634	−	0.950530i	$-0.399459\pi$
$17$	2.56155	0.621268	0.310634	+	0.950530i	$0.399459\pi$
$18$	0	0
$19$	− 1.12311i	− 0.257658i	−0.991667	−	0.128829i	$-0.958878\pi$
$19$	− 1.12311i	− 0.257658i	0.991667	−	0.128829i	$-0.0411218\pi$
$20$	2.56155i	0.572781i
$21$	0	0
$22$	−5.12311	−1.09225
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	4.68466	0.936932
$26$	0	0
$27$	0	0
$28$	− 16.2462i	− 3.07025i
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	− 1.56155i	− 0.280463i	−0.990119	−	0.140232i	$-0.955215\pi$
$31$	− 1.56155i	− 0.280463i	0.990119	−	0.140232i	$-0.0447847\pi$
$32$	− 6.56155i	− 1.15993i
$33$	0	0
$34$	− 6.56155i	− 1.12530i
$35$	2.00000	0.338062
$36$	0	0
$37$	− 3.43845i	− 0.565277i	−0.959226	−	0.282639i	$-0.908790\pi$
$37$	− 3.43845i	− 0.565277i	0.959226	−	0.282639i	$-0.0912097\pi$
$38$	−2.87689	−0.466694
$39$	0	0
$40$	3.68466	0.582596
$41$	− 2.56155i	− 0.400047i	−0.979791	−	0.200024i	$-0.935898\pi$
$41$	− 2.56155i	− 0.400047i	0.979791	−	0.200024i	$-0.0641019\pi$
$42$	0	0
$43$	−0.438447	−0.0668626	−0.0334313	−	0.999441i	$-0.510643\pi$
$43$	−0.438447	−0.0668626	−0.0334313	+	0.999441i	$0.510643\pi$
$44$	9.12311i	1.37536i
$45$	0	0
$46$	− 5.12311i	− 0.755361i
$47$	− 8.24621i	− 1.20283i	−0.798935	−	0.601417i	$-0.794603\pi$
$47$	− 8.24621i	− 1.20283i	0.798935	−	0.601417i	$-0.205397\pi$
$48$	0	0
$49$	−5.68466	−0.812094
$50$	− 12.0000i	− 1.69706i
$51$	0	0
$52$	0	0
$53$	−11.6847	−1.60501	−0.802506	−	0.596645i	$-0.796500\pi$
$53$	−11.6847	−1.60501	−0.802506	+	0.596645i	$0.796500\pi$
$54$	0	0
$55$	−1.12311	−0.151440
$56$	−23.3693	−3.12286
$57$	0	0
$58$	− 14.5616i	− 1.91203i
$59$	− 11.1231i	− 1.44811i	−0.689745	−	0.724053i	$-0.742277\pi$
$59$	− 11.1231i	− 1.44811i	0.689745	−	0.724053i	$-0.257723\pi$
$60$	0	0
$61$	12.1231	1.55220	0.776102	−	0.630607i	$-0.217194\pi$
$61$	12.1231	1.55220	0.776102	+	0.630607i	$0.217194\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	−1.43845	−0.179806
$65$	0	0
$66$	0	0
$67$	0.438447i	0.0535648i	0.999641	+	0.0267824i	$0.00852613\pi$
$67$	0.438447i	0.0535648i	−0.999641	+	0.0267824i	$0.991474\pi$
$68$	−11.6847	−1.41697
$69$	0	0
$70$	− 5.12311i	− 0.612328i
$71$	− 14.0000i	− 1.66149i	−0.556650	−	0.830747i	$-0.687914\pi$
$71$	− 14.0000i	− 1.66149i	0.556650	−	0.830747i	$-0.312086\pi$
$72$	0	0
$73$	1.87689i	0.219674i	0.993950	+	0.109837i	$0.0350329\pi$
$73$	1.87689i	0.219674i	−0.993950	+	0.109837i	$0.964967\pi$
$74$	−8.80776	−1.02388
$75$	0	0
$76$	5.12311i	0.587661i
$77$	7.12311	0.811753
$78$	0	0
$79$	9.56155	1.07576	0.537879	−	0.843022i	$-0.319226\pi$
$79$	9.56155	1.07576	0.537879	+	0.843022i	$0.319226\pi$
$80$	− 4.31534i	− 0.482470i
$81$	0	0
$82$	−6.56155	−0.724602
$83$	9.12311i	1.00139i	0.865624	+	0.500695i	$0.166922\pi$
$83$	9.12311i	1.00139i	−0.865624	+	0.500695i	$0.833078\pi$
$84$	0	0
$85$	− 1.43845i	− 0.156022i
$86$	1.12311i	0.121108i
$87$	0	0
$88$	13.1231	1.39893
$89$	13.1231i	1.39105i	0.718504	+	0.695523i	$0.244827\pi$
$89$	13.1231i	1.39105i	−0.718504	+	0.695523i	$0.755173\pi$
$90$	0	0
$91$	0	0
$92$	−9.12311	−0.951150
$93$	0	0
$94$	−21.1231	−2.17868
$95$	−0.630683	−0.0647067
$96$	0	0
$97$	− 4.43845i	− 0.450656i	−0.974283	−	0.225328i	$-0.927655\pi$
$97$	− 4.43845i	− 0.450656i	0.974283	−	0.225328i	$-0.0723454\pi$
$98$	14.5616i	1.47094i
$99$	0	0
$100$	−21.3693	−2.13693
$101$	−3.43845	−0.342138	−0.171069	−	0.985259i	$-0.554722\pi$
$101$	−3.43845	−0.342138	−0.171069	+	0.985259i	$0.554722\pi$
$102$	0	0
$103$	7.56155	0.745062	0.372531	−	0.928020i	$-0.378490\pi$
$103$	7.56155	0.745062	0.372531	+	0.928020i	$0.378490\pi$
$104$	0	0
$105$	0	0
$106$	29.9309i	2.90714i
$107$	−8.24621	−0.797191	−0.398596	−	0.917127i	$-0.630502\pi$
$107$	−8.24621	−0.797191	−0.398596	+	0.917127i	$0.630502\pi$
$108$	0	0
$109$	17.8078i	1.70567i	0.522177	+	0.852837i	$0.325120\pi$
$109$	17.8078i	1.70567i	−0.522177	+	0.852837i	$0.674880\pi$
$110$	2.87689i	0.274301i
$111$	0	0
$112$	27.3693i	2.58616i
$113$	14.8078	1.39300	0.696499	−	0.717558i	$-0.254740\pi$
$113$	14.8078	1.39300	0.696499	+	0.717558i	$0.254740\pi$
$114$	0	0
$115$	− 1.12311i	− 0.104730i
$116$	−25.9309	−2.40762
$117$	0	0
$118$	−28.4924	−2.62294
$119$	9.12311i	0.836314i
$120$	0	0
$121$	7.00000	0.636364
$122$	− 31.0540i	− 2.81149i
$123$	0	0
$124$	7.12311i	0.639674i
$125$	− 5.43845i	− 0.486430i
$126$	0	0
$127$	−9.56155	−0.848451	−0.424225	−	0.905557i	$-0.639454\pi$
$127$	−9.56155	−0.848451	−0.424225	+	0.905557i	$0.639454\pi$
$128$	− 9.43845i	− 0.834249i
$129$	0	0
$130$	0	0
$131$	17.3693	1.51756	0.758782	−	0.651345i	$-0.225795\pi$
$131$	17.3693	1.51756	0.758782	+	0.651345i	$0.225795\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	1.12311	0.0970215
$135$	0	0
$136$	16.8078i	1.44125i
$137$	− 1.43845i	− 0.122895i	−0.998110	−	0.0614474i	$-0.980428\pi$
$137$	− 1.43845i	− 0.122895i	0.998110	−	0.0614474i	$-0.0195717\pi$
$138$	0	0
$139$	10.9309	0.927144	0.463572	−	0.886059i	$-0.346567\pi$
$139$	10.9309	0.927144	0.463572	+	0.886059i	$0.346567\pi$
$140$	−9.12311	−0.771043
$141$	0	0
$142$	−35.8617	−3.00945
$143$	0	0
$144$	0	0
$145$	− 3.19224i	− 0.265101i
$146$	4.80776	0.397893
$147$	0	0
$148$	15.6847i	1.28927i
$149$	6.56155i	0.537543i	0.963204	+	0.268772i	$0.0866177\pi$
$149$	6.56155i	0.537543i	−0.963204	+	0.268772i	$0.913382\pi$
$150$	0	0
$151$	− 15.3693i	− 1.25074i	−0.780329	−	0.625369i	$-0.784949\pi$
$151$	− 15.3693i	− 1.25074i	0.780329	−	0.625369i	$-0.215051\pi$
$152$	7.36932	0.597731
$153$	0	0
$154$	− 18.2462i	− 1.47032i
$155$	−0.876894	−0.0704339
$156$	0	0
$157$	−4.36932	−0.348709	−0.174355	−	0.984683i	$-0.555784\pi$
$157$	−4.36932	−0.348709	−0.174355	+	0.984683i	$0.555784\pi$
$158$	− 24.4924i	− 1.94851i
$159$	0	0
$160$	−3.68466	−0.291298
$161$	7.12311i	0.561379i
$162$	0	0
$163$	− 15.8078i	− 1.23816i	−0.785328	−	0.619080i	$-0.787506\pi$
$163$	− 15.8078i	− 1.23816i	0.785328	−	0.619080i	$-0.212494\pi$
$164$	11.6847i	0.912419i
$165$	0	0
$166$	23.3693	1.81381
$167$	− 6.24621i	− 0.483346i	−0.970358	−	0.241673i	$-0.922304\pi$
$167$	− 6.24621i	− 0.483346i	0.970358	−	0.241673i	$-0.0776962\pi$
$168$	0	0
$169$	0	0
$170$	−3.68466	−0.282600
$171$	0	0
$172$	2.00000	0.152499
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	0	0
$175$	16.6847i	1.26124i
$176$	− 15.3693i	− 1.15851i
$177$	0	0
$178$	33.6155	2.51959
$179$	−13.1231	−0.980867	−0.490433	−	0.871479i	$-0.663162\pi$
$179$	−13.1231	−0.980867	−0.490433	+	0.871479i	$0.663162\pi$
$180$	0	0
$181$	−9.68466	−0.719855	−0.359927	−	0.932980i	$-0.617199\pi$
$181$	−9.68466	−0.719855	−0.359927	+	0.932980i	$0.617199\pi$
$182$	0	0
$183$	0	0
$184$	13.1231i	0.967448i
$185$	−1.93087	−0.141960
$186$	0	0
$187$	− 5.12311i	− 0.374639i
$188$	37.6155i	2.74339i
$189$	0	0
$190$	1.61553i	0.117203i
$191$	0.876894	0.0634499	0.0317249	−	0.999497i	$-0.489900\pi$
$191$	0.876894	0.0634499	0.0317249	+	0.999497i	$0.489900\pi$
$192$	0	0
$193$	− 19.4924i	− 1.40310i	−0.712623	−	0.701548i	$-0.752493\pi$
$193$	− 19.4924i	− 1.40310i	0.712623	−	0.701548i	$-0.247507\pi$
$194$	−11.3693	−0.816269
$195$	0	0
$196$	25.9309	1.85220
$197$	11.3693i	0.810030i	0.914310	+	0.405015i	$0.132734\pi$
$197$	11.3693i	0.810030i	−0.914310	+	0.405015i	$0.867266\pi$
$198$	0	0
$199$	23.1771	1.64298	0.821490	−	0.570223i	$-0.193143\pi$
$199$	23.1771	1.64298	0.821490	+	0.570223i	$0.193143\pi$
$200$	30.7386i	2.17355i
$201$	0	0
$202$	8.80776i	0.619712i
$203$	20.2462i	1.42101i
$204$	0	0
$205$	−1.43845	−0.100466
$206$	− 19.3693i	− 1.34952i
$207$	0	0
$208$	0	0
$209$	−2.24621	−0.155374
$210$	0	0
$211$	7.31534	0.503609	0.251804	−	0.967778i	$-0.418976\pi$
$211$	7.31534	0.503609	0.251804	+	0.967778i	$0.418976\pi$
$212$	53.3002	3.66067
$213$	0	0
$214$	21.1231i	1.44395i
$215$	0.246211i	0.0167915i
$216$	0	0
$217$	5.56155	0.377543
$218$	45.6155	3.08947
$219$	0	0
$220$	5.12311	0.345400
$221$	0	0
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	23.3693	1.56143
$225$	0	0
$226$	− 37.9309i	− 2.52312i
$227$	1.12311i	0.0745431i	0.999305	+	0.0372716i	$0.0118667\pi$
$227$	1.12311i	0.0745431i	−0.999305	+	0.0372716i	$0.988133\pi$
$228$	0	0
$229$	− 0.246211i	− 0.0162701i	−0.999967	−	0.00813505i	$-0.997411\pi$
$229$	− 0.246211i	− 0.0162701i	0.999967	−	0.00813505i	$-0.00258949\pi$
$230$	−2.87689	−0.189697
$231$	0	0
$232$	37.3002i	2.44888i
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−4.63068	−0.302072
$236$	50.7386i	3.30280i
$237$	0	0
$238$	23.3693	1.51481
$239$	0.630683i	0.0407955i	0.999792	+	0.0203977i	$0.00649326\pi$
$239$	0.630683i	0.0407955i	−0.999792	+	0.0203977i	$0.993507\pi$
$240$	0	0
$241$	− 2.80776i	− 0.180864i	−0.995903	−	0.0904320i	$-0.971175\pi$
$241$	− 2.80776i	− 0.180864i	0.995903	−	0.0904320i	$-0.0288248\pi$
$242$	− 17.9309i	− 1.15264i
$243$	0	0
$244$	−55.3002	−3.54023
$245$	3.19224i	0.203944i
$246$	0	0
$247$	0	0
$248$	10.2462	0.650635
$249$	0	0
$250$	−13.9309	−0.881066
$251$	−30.7386	−1.94021	−0.970103	−	0.242695i	$-0.921969\pi$
$251$	−30.7386	−1.94021	−0.970103	+	0.242695i	$0.921969\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	24.4924i	1.53679i
$255$	0	0
$256$	−27.0540	−1.69087
$257$	−16.1771	−1.00910	−0.504549	−	0.863383i	$-0.668341\pi$
$257$	−16.1771	−1.00910	−0.504549	+	0.863383i	$0.668341\pi$
$258$	0	0
$259$	12.2462	0.760943
$260$	0	0
$261$	0	0
$262$	− 44.4924i	− 2.74875i
$263$	15.3693	0.947713	0.473856	−	0.880602i	$-0.342862\pi$
$263$	15.3693	0.947713	0.473856	+	0.880602i	$0.342862\pi$
$264$	0	0
$265$	6.56155i	0.403073i
$266$	− 10.2462i	− 0.628236i
$267$	0	0
$268$	− 2.00000i	− 0.122169i
$269$	−3.36932	−0.205431	−0.102715	−	0.994711i	$-0.532753\pi$
$269$	−3.36932	−0.205431	−0.102715	+	0.994711i	$0.532753\pi$
$270$	0	0
$271$	1.06913i	0.0649450i	0.999473	+	0.0324725i	$0.0103381\pi$
$271$	1.06913i	0.0649450i	−0.999473	+	0.0324725i	$0.989662\pi$
$272$	19.6847	1.19356
$273$	0	0
$274$	−3.68466	−0.222598
$275$	− 9.36932i	− 0.564991i
$276$	0	0
$277$	−17.6847	−1.06257	−0.531284	−	0.847194i	$-0.678290\pi$
$277$	−17.6847	−1.06257	−0.531284	+	0.847194i	$0.678290\pi$
$278$	− 28.0000i	− 1.67933i
$279$	0	0
$280$	13.1231i	0.784256i
$281$	− 2.80776i	− 0.167497i	−0.996487	−	0.0837486i	$-0.973311\pi$
$281$	− 2.80776i	− 0.167497i	0.996487	−	0.0837486i	$-0.0266893\pi$
$282$	0	0
$283$	1.31534	0.0781889	0.0390945	−	0.999236i	$-0.487553\pi$
$283$	1.31534	0.0781889	0.0390945	+	0.999236i	$0.487553\pi$
$284$	63.8617i	3.78950i
$285$	0	0
$286$	0	0
$287$	9.12311	0.538520
$288$	0	0
$289$	−10.4384	−0.614026
$290$	−8.17708	−0.480175
$291$	0	0
$292$	− 8.56155i	− 0.501027i
$293$	24.5616i	1.43490i	0.696609	+	0.717451i	$0.254691\pi$
$293$	24.5616i	1.43490i	−0.696609	+	0.717451i	$0.745309\pi$
$294$	0	0
$295$	−6.24621	−0.363668
$296$	22.5616	1.31136
$297$	0	0
$298$	16.8078	0.973648
$299$	0	0
$300$	0	0
$301$	− 1.56155i	− 0.0900064i
$302$	−39.3693	−2.26545
$303$	0	0
$304$	− 8.63068i	− 0.495004i
$305$	− 6.80776i	− 0.389811i
$306$	0	0
$307$	− 10.1922i	− 0.581702i	−0.956768	−	0.290851i	$-0.906062\pi$
$307$	− 10.1922i	− 0.581702i	0.956768	−	0.290851i	$-0.0939383\pi$
$308$	−32.4924	−1.85143
$309$	0	0
$310$	2.24621i	0.127576i
$311$	−10.8769	−0.616772	−0.308386	−	0.951261i	$-0.599789\pi$
$311$	−10.8769	−0.616772	−0.308386	+	0.951261i	$0.599789\pi$
$312$	0	0
$313$	−1.31534	−0.0743475	−0.0371738	−	0.999309i	$-0.511835\pi$
$313$	−1.31534	−0.0743475	−0.0371738	+	0.999309i	$0.511835\pi$
$314$	11.1922i	0.631614i
$315$	0	0
$316$	−43.6155	−2.45357
$317$	23.0540i	1.29484i	0.762133	+	0.647420i	$0.224152\pi$
$317$	23.0540i	1.29484i	−0.762133	+	0.647420i	$0.775848\pi$
$318$	0	0
$319$	− 11.3693i	− 0.636560i
$320$	0.807764i	0.0451554i
$321$	0	0
$322$	18.2462	1.01682
$323$	− 2.87689i	− 0.160075i
$324$	0	0
$325$	0	0
$326$	−40.4924	−2.24267
$327$	0	0
$328$	16.8078	0.928054
$329$	29.3693	1.61918
$330$	0	0
$331$	− 23.8078i	− 1.30859i	−0.756238	−	0.654297i	$-0.772965\pi$
$331$	− 23.8078i	− 1.30859i	0.756238	−	0.654297i	$-0.227035\pi$
$332$	− 41.6155i	− 2.28395i
$333$	0	0
$334$	−16.0000	−0.875481
$335$	0.246211	0.0134520
$336$	0	0
$337$	−2.12311	−0.115653	−0.0578265	−	0.998327i	$-0.518417\pi$
$337$	−2.12311	−0.115653	−0.0578265	+	0.998327i	$0.518417\pi$
$338$	0	0
$339$	0	0
$340$	6.56155i	0.355850i
$341$	−3.12311	−0.169126
$342$	0	0
$343$	4.68466i	0.252948i
$344$	− 2.87689i	− 0.155112i
$345$	0	0
$346$	− 9.61553i	− 0.516934i
$347$	13.6155	0.730920	0.365460	−	0.930827i	$-0.380912\pi$
$347$	13.6155	0.730920	0.365460	+	0.930827i	$0.380912\pi$
$348$	0	0
$349$	13.8078i	0.739113i	0.929208	+	0.369556i	$0.120490\pi$
$349$	13.8078i	0.739113i	−0.929208	+	0.369556i	$0.879510\pi$
$350$	42.7386	2.28448
$351$	0	0
$352$	−13.1231	−0.699464
$353$	− 17.6847i	− 0.941259i	−0.882331	−	0.470630i	$-0.844027\pi$
$353$	− 17.6847i	− 0.941259i	0.882331	−	0.470630i	$-0.155973\pi$
$354$	0	0
$355$	−7.86174	−0.417258
$356$	− 59.8617i	− 3.17267i
$357$	0	0
$358$	33.6155i	1.77664i
$359$	15.3693i	0.811162i	0.914059	+	0.405581i	$0.132931\pi$
$359$	15.3693i	0.811162i	−0.914059	+	0.405581i	$0.867069\pi$
$360$	0	0
$361$	17.7386	0.933612
$362$	24.8078i	1.30387i
$363$	0	0
$364$	0	0
$365$	1.05398	0.0551676
$366$	0	0
$367$	20.0540	1.04681	0.523404	−	0.852084i	$-0.324662\pi$
$367$	20.0540	1.04681	0.523404	+	0.852084i	$0.324662\pi$
$368$	15.3693	0.801181
$369$	0	0
$370$	4.94602i	0.257132i
$371$	− 41.6155i	− 2.16057i
$372$	0	0
$373$	3.63068	0.187990	0.0939948	−	0.995573i	$-0.470036\pi$
$373$	3.63068	0.187990	0.0939948	+	0.995573i	$0.470036\pi$
$374$	−13.1231	−0.678580
$375$	0	0
$376$	54.1080	2.79040
$377$	0	0
$378$	0	0
$379$	11.3153i	0.581230i	0.956840	+	0.290615i	$0.0938599\pi$
$379$	11.3153i	0.581230i	−0.956840	+	0.290615i	$0.906140\pi$
$380$	2.87689	0.147582
$381$	0	0
$382$	− 2.24621i	− 0.114926i
$383$	− 26.7386i	− 1.36628i	−0.730287	−	0.683140i	$-0.760614\pi$
$383$	− 26.7386i	− 1.36628i	0.730287	−	0.683140i	$-0.239386\pi$
$384$	0	0
$385$	− 4.00000i	− 0.203859i
$386$	−49.9309	−2.54141
$387$	0	0
$388$	20.2462i	1.02785i
$389$	−3.05398	−0.154843	−0.0774213	−	0.996998i	$-0.524669\pi$
$389$	−3.05398	−0.154843	−0.0774213	+	0.996998i	$0.524669\pi$
$390$	0	0
$391$	5.12311	0.259087
$392$	− 37.3002i	− 1.88394i
$393$	0	0
$394$	29.1231	1.46720
$395$	− 5.36932i	− 0.270160i
$396$	0	0
$397$	12.0540i	0.604972i	0.953154	+	0.302486i	$0.0978165\pi$
$397$	12.0540i	0.604972i	−0.953154	+	0.302486i	$0.902184\pi$
$398$	− 59.3693i	− 2.97591i
$399$	0	0
$400$	36.0000	1.80000
$401$	18.5616i	0.926920i	0.886118	+	0.463460i	$0.153392\pi$
$401$	18.5616i	0.926920i	−0.886118	+	0.463460i	$0.846608\pi$
$402$	0	0
$403$	0	0
$404$	15.6847	0.780341
$405$	0	0
$406$	51.8617	2.57385
$407$	−6.87689	−0.340875
$408$	0	0
$409$	− 18.3693i	− 0.908304i	−0.890924	−	0.454152i	$-0.849942\pi$
$409$	− 18.3693i	− 0.908304i	0.890924	−	0.454152i	$-0.150058\pi$
$410$	3.68466i	0.181972i
$411$	0	0
$412$	−34.4924	−1.69932
$413$	39.6155	1.94935
$414$	0	0
$415$	5.12311	0.251483
$416$	0	0
$417$	0	0
$418$	5.75379i	0.281427i
$419$	17.7538	0.867329	0.433665	−	0.901074i	$-0.357220\pi$
$419$	17.7538	0.867329	0.433665	+	0.901074i	$0.357220\pi$
$420$	0	0
$421$	14.7538i	0.719056i	0.933134	+	0.359528i	$0.117062\pi$
$421$	14.7538i	0.719056i	−0.933134	+	0.359528i	$0.882938\pi$
$422$	− 18.7386i	− 0.912182i
$423$	0	0
$424$	− 76.6695i	− 3.72340i
$425$	12.0000	0.582086
$426$	0	0
$427$	43.1771i	2.08949i
$428$	37.6155	1.81822
$429$	0	0
$430$	0.630683	0.0304142
$431$	2.87689i	0.138575i	0.997597	+	0.0692876i	$0.0220726\pi$
$431$	2.87689i	0.138575i	−0.997597	+	0.0692876i	$0.977927\pi$
$432$	0	0
$433$	−25.2462	−1.21326	−0.606628	−	0.794986i	$-0.707478\pi$
$433$	−25.2462	−1.21326	−0.606628	+	0.794986i	$0.707478\pi$
$434$	− 14.2462i	− 0.683840i
$435$	0	0
$436$	− 81.2311i	− 3.89026i
$437$	− 2.24621i	− 0.107451i
$438$	0	0
$439$	1.31534	0.0627778	0.0313889	−	0.999507i	$-0.490007\pi$
$439$	1.31534	0.0627778	0.0313889	+	0.999507i	$0.490007\pi$
$440$	− 7.36932i	− 0.351318i
$441$	0	0
$442$	0	0
$443$	−14.7386	−0.700254	−0.350127	−	0.936702i	$-0.613862\pi$
$443$	−14.7386	−0.700254	−0.350127	+	0.936702i	$0.613862\pi$
$444$	0	0
$445$	7.36932	0.349339
$446$	20.4924	0.970344
$447$	0	0
$448$	− 5.12311i	− 0.242044i
$449$	8.24621i	0.389163i	0.980886	+	0.194581i	$0.0623348\pi$
$449$	8.24621i	0.389163i	−0.980886	+	0.194581i	$0.937665\pi$
$450$	0	0
$451$	−5.12311	−0.241238
$452$	−67.5464	−3.17712
$453$	0	0
$454$	2.87689	0.135019
$455$	0	0
$456$	0	0
$457$	− 28.6155i	− 1.33858i	−0.743002	−	0.669289i	$-0.766599\pi$
$457$	− 28.6155i	− 1.33858i	0.743002	−	0.669289i	$-0.233401\pi$
$458$	−0.630683	−0.0294699
$459$	0	0
$460$	5.12311i	0.238866i
$461$	36.8078i	1.71431i	0.515060	+	0.857154i	$0.327770\pi$
$461$	36.8078i	1.71431i	−0.515060	+	0.857154i	$0.672230\pi$
$462$	0	0
$463$	26.6847i	1.24014i	0.784546	+	0.620071i	$0.212896\pi$
$463$	26.6847i	1.24014i	−0.784546	+	0.620071i	$0.787104\pi$
$464$	43.6847	2.02801
$465$	0	0
$466$	− 66.6004i	− 3.08520i
$467$	−26.0000	−1.20314	−0.601568	−	0.798821i	$-0.705457\pi$
$467$	−26.0000	−1.20314	−0.601568	+	0.798821i	$0.705457\pi$
$468$	0	0
$469$	−1.56155	−0.0721058
$470$	11.8617i	0.547141i
$471$	0	0
$472$	72.9848	3.35940
$473$	0.876894i	0.0403196i
$474$	0	0
$475$	− 5.26137i	− 0.241408i
$476$	− 41.6155i	− 1.90744i
$477$	0	0
$478$	1.61553	0.0738925
$479$	− 6.24621i	− 0.285397i	−0.989766	−	0.142698i	$-0.954422\pi$
$479$	− 6.24621i	− 0.285397i	0.989766	−	0.142698i	$-0.0455779\pi$
$480$	0	0
$481$	0	0
$482$	−7.19224	−0.327597
$483$	0	0
$484$	−31.9309	−1.45140
$485$	−2.49242	−0.113175
$486$	0	0
$487$	− 1.12311i	− 0.0508928i	−0.999676	−	0.0254464i	$-0.991899\pi$
$487$	− 1.12311i	− 0.0508928i	0.999676	−	0.0254464i	$-0.00810071\pi$
$488$	79.5464i	3.60090i
$489$	0	0
$490$	8.17708	0.369403
$491$	−19.7538	−0.891476	−0.445738	−	0.895163i	$-0.647059\pi$
$491$	−19.7538	−0.891476	−0.445738	+	0.895163i	$0.647059\pi$
$492$	0	0
$493$	14.5616	0.655819
$494$	0	0
$495$	0	0
$496$	− 12.0000i	− 0.538816i
$497$	49.8617	2.23660
$498$	0	0
$499$	− 28.4924i	− 1.27550i	−0.770245	−	0.637748i	$-0.779866\pi$
$499$	− 28.4924i	− 1.27550i	0.770245	−	0.637748i	$-0.220134\pi$
$500$	24.8078i	1.10944i
$501$	0	0
$502$	78.7386i	3.51428i
$503$	11.7538	0.524076	0.262038	−	0.965058i	$-0.415605\pi$
$503$	11.7538	0.524076	0.262038	+	0.965058i	$0.415605\pi$
$504$	0	0
$505$	1.93087i	0.0859226i
$506$	−10.2462	−0.455500
$507$	0	0
$508$	43.6155	1.93513
$509$	− 6.80776i	− 0.301749i	−0.988553	−	0.150874i	$-0.951791\pi$
$509$	− 6.80776i	− 0.301749i	0.988553	−	0.150874i	$-0.0482089\pi$
$510$	0	0
$511$	−6.68466	−0.295712
$512$	50.4233i	2.22842i
$513$	0	0
$514$	41.4384i	1.82777i
$515$	− 4.24621i	− 0.187110i
$516$	0	0
$517$	−16.4924	−0.725336
$518$	− 31.3693i	− 1.37829i
$519$	0	0
$520$	0	0
$521$	37.9309	1.66178	0.830891	−	0.556436i	$-0.187831\pi$
$521$	37.9309	1.66178	0.830891	+	0.556436i	$0.187831\pi$
$522$	0	0
$523$	−23.8617	−1.04340	−0.521701	−	0.853129i	$-0.674702\pi$
$523$	−23.8617	−1.04340	−0.521701	+	0.853129i	$0.674702\pi$
$524$	−79.2311	−3.46122
$525$	0	0
$526$	− 39.3693i	− 1.71658i
$527$	− 4.00000i	− 0.174243i
$528$	0	0
$529$	−19.0000	−0.826087
$530$	16.8078	0.730083
$531$	0	0
$532$	−18.2462	−0.791074
$533$	0	0
$534$	0	0
$535$	4.63068i	0.200202i
$536$	−2.87689	−0.124263
$537$	0	0
$538$	8.63068i	0.372095i
$539$	11.3693i	0.489711i
$540$	0	0
$541$	29.7386i	1.27856i	0.768972	+	0.639282i	$0.220768\pi$
$541$	29.7386i	1.27856i	−0.768972	+	0.639282i	$0.779232\pi$
$542$	2.73863	0.117634
$543$	0	0
$544$	− 16.8078i	− 0.720627i
$545$	10.0000	0.428353
$546$	0	0
$547$	−24.9309	−1.06597	−0.532984	−	0.846126i	$-0.678929\pi$
$547$	−24.9309	−1.06597	−0.532984	+	0.846126i	$0.678929\pi$
$548$	6.56155i	0.280296i
$549$	0	0
$550$	−24.0000	−1.02336
$551$	− 6.38447i	− 0.271988i
$552$	0	0
$553$	34.0540i	1.44812i
$554$	45.3002i	1.92462i
$555$	0	0
$556$	−49.8617	−2.11461
$557$	− 14.0691i	− 0.596128i	−0.954546	−	0.298064i	$-0.903659\pi$
$557$	− 14.0691i	− 0.596128i	0.954546	−	0.298064i	$-0.0963409\pi$
$558$	0	0
$559$	0	0
$560$	15.3693	0.649472
$561$	0	0
$562$	−7.19224	−0.303386
$563$	1.36932	0.0577098	0.0288549	−	0.999584i	$-0.490814\pi$
$563$	1.36932	0.0577098	0.0288549	+	0.999584i	$0.490814\pi$
$564$	0	0
$565$	− 8.31534i	− 0.349829i
$566$	− 3.36932i	− 0.141623i
$567$	0	0
$568$	91.8617	3.85443
$569$	40.7386	1.70785	0.853926	−	0.520394i	$-0.174215\pi$
$569$	40.7386	1.70785	0.853926	+	0.520394i	$0.174215\pi$
$570$	0	0
$571$	−19.3693	−0.810581	−0.405290	−	0.914188i	$-0.632830\pi$
$571$	−19.3693	−0.810581	−0.405290	+	0.914188i	$0.632830\pi$
$572$	0	0
$573$	0	0
$574$	− 23.3693i	− 0.975416i
$575$	9.36932	0.390728
$576$	0	0
$577$	− 29.6847i	− 1.23579i	−0.786261	−	0.617894i	$-0.787986\pi$
$577$	− 29.6847i	− 1.23579i	0.786261	−	0.617894i	$-0.212014\pi$
$578$	26.7386i	1.11218i
$579$	0	0
$580$	14.5616i	0.604636i
$581$	−32.4924	−1.34801
$582$	0	0
$583$	23.3693i	0.967858i
$584$	−12.3153	−0.509612
$585$	0	0
$586$	62.9157	2.59902
$587$	− 14.6307i	− 0.603873i	−0.953328	−	0.301936i	$-0.902367\pi$
$587$	− 14.6307i	− 0.603873i	0.953328	−	0.301936i	$-0.0976330\pi$
$588$	0	0
$589$	−1.75379	−0.0722636
$590$	16.0000i	0.658710i
$591$	0	0
$592$	− 26.4233i	− 1.08599i
$593$	44.4233i	1.82425i	0.409917	+	0.912123i	$0.365558\pi$
$593$	44.4233i	1.82425i	−0.409917	+	0.912123i	$0.634442\pi$
$594$	0	0
$595$	5.12311	0.210027
$596$	− 29.9309i	− 1.22602i
$597$	0	0
$598$	0	0
$599$	0.384472	0.0157091	0.00785455	−	0.999969i	$-0.497500\pi$
$599$	0.384472	0.0157091	0.00785455	+	0.999969i	$0.497500\pi$
$600$	0	0
$601$	−35.9309	−1.46565	−0.732825	−	0.680417i	$-0.761799\pi$
$601$	−35.9309	−1.46565	−0.732825	+	0.680417i	$0.761799\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	70.1080i	2.85265i
$605$	− 3.93087i	− 0.159813i
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−7.36932	−0.298865
$609$	0	0
$610$	−17.4384	−0.706062
$611$	0	0
$612$	0	0
$613$	22.8617i	0.923377i	0.887042	+	0.461688i	$0.152756\pi$
$613$	22.8617i	0.923377i	−0.887042	+	0.461688i	$0.847244\pi$
$614$	−26.1080	−1.05363
$615$	0	0
$616$	46.7386i	1.88315i
$617$	10.8078i	0.435104i	0.976049	+	0.217552i	$0.0698072\pi$
$617$	10.8078i	0.435104i	−0.976049	+	0.217552i	$0.930193\pi$
$618$	0	0
$619$	24.3002i	0.976707i	0.872646	+	0.488353i	$0.162402\pi$
$619$	24.3002i	0.976707i	−0.872646	+	0.488353i	$0.837598\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	27.8617i	1.11715i
$623$	−46.7386	−1.87254
$624$	0	0
$625$	20.3693	0.814773
$626$	3.36932i	0.134665i
$627$	0	0
$628$	19.9309	0.795328
$629$	− 8.80776i	− 0.351189i
$630$	0	0
$631$	14.4384i	0.574786i	0.957813	+	0.287393i	$0.0927885\pi$
$631$	14.4384i	0.574786i	−0.957813	+	0.287393i	$0.907212\pi$
$632$	62.7386i	2.49561i
$633$	0	0
$634$	59.0540	2.34533
$635$	5.36932i	0.213075i
$636$	0	0
$637$	0	0
$638$	−29.1231	−1.15299
$639$	0	0
$640$	−5.30019	−0.209508
$641$	26.1771	1.03393	0.516966	−	0.856006i	$-0.327061\pi$
$641$	26.1771	1.03393	0.516966	+	0.856006i	$0.327061\pi$
$642$	0	0
$643$	38.5464i	1.52012i	0.649852	+	0.760061i	$0.274831\pi$
$643$	38.5464i	1.52012i	−0.649852	+	0.760061i	$0.725169\pi$
$644$	− 32.4924i	− 1.28038i
$645$	0	0
$646$	−7.36932	−0.289942
$647$	−47.6155	−1.87196	−0.935980	−	0.352054i	$-0.885483\pi$
$647$	−47.6155	−1.87196	−0.935980	+	0.352054i	$0.885483\pi$
$648$	0	0
$649$	−22.2462	−0.873240
$650$	0	0
$651$	0	0
$652$	72.1080i	2.82397i
$653$	−14.8769	−0.582178	−0.291089	−	0.956696i	$-0.594018\pi$
$653$	−14.8769	−0.582178	−0.291089	+	0.956696i	$0.594018\pi$
$654$	0	0
$655$	− 9.75379i	− 0.381112i
$656$	− 19.6847i	− 0.768557i
$657$	0	0
$658$	− 75.2311i	− 2.93281i
$659$	−14.2462	−0.554954	−0.277477	−	0.960732i	$-0.589498\pi$
$659$	−14.2462	−0.554954	−0.277477	+	0.960732i	$0.589498\pi$
$660$	0	0
$661$	− 30.3693i	− 1.18123i	−0.806954	−	0.590615i	$-0.798885\pi$
$661$	− 30.3693i	− 1.18123i	0.806954	−	0.590615i	$-0.201115\pi$
$662$	−60.9848	−2.37024
$663$	0	0
$664$	−59.8617	−2.32309
$665$	− 2.24621i	− 0.0871043i
$666$	0	0
$667$	11.3693	0.440222
$668$	28.4924i	1.10240i
$669$	0	0
$670$	− 0.630683i	− 0.0243654i
$671$	− 24.2462i	− 0.936015i
$672$	0	0
$673$	6.75379	0.260339	0.130170	−	0.991492i	$-0.458448\pi$
$673$	6.75379	0.260339	0.130170	+	0.991492i	$0.458448\pi$
$674$	5.43845i	0.209481i
$675$	0	0
$676$	0	0
$677$	−25.6155	−0.984485	−0.492242	−	0.870458i	$-0.663823\pi$
$677$	−25.6155	−0.984485	−0.492242	+	0.870458i	$0.663823\pi$
$678$	0	0
$679$	15.8078	0.606646
$680$	9.43845	0.361948
$681$	0	0
$682$	8.00000i	0.306336i
$683$	36.1080i	1.38163i	0.723030	+	0.690816i	$0.242749\pi$
$683$	36.1080i	1.38163i	−0.723030	+	0.690816i	$0.757251\pi$
$684$	0	0
$685$	−0.807764	−0.0308631
$686$	12.0000	0.458162
$687$	0	0
$688$	−3.36932	−0.128454
$689$	0	0
$690$	0	0
$691$	2.30019i	0.0875032i	0.999042	+	0.0437516i	$0.0139310\pi$
$691$	2.30019i	0.0875032i	−0.999042	+	0.0437516i	$0.986069\pi$
$692$	−17.1231	−0.650923
$693$	0	0
$694$	− 34.8769i	− 1.32391i
$695$	− 6.13826i	− 0.232837i
$696$	0	0
$697$	− 6.56155i	− 0.248537i
$698$	35.3693	1.33875
$699$	0	0
$700$	− 76.1080i	− 2.87661i
$701$	19.3693	0.731569	0.365785	−	0.930700i	$-0.380801\pi$
$701$	19.3693	0.731569	0.365785	+	0.930700i	$0.380801\pi$
$702$	0	0
$703$	−3.86174	−0.145648
$704$	2.87689i	0.108427i
$705$	0	0
$706$	−45.3002	−1.70490
$707$	− 12.2462i	− 0.460566i
$708$	0	0
$709$	− 25.4924i	− 0.957388i	−0.877982	−	0.478694i	$-0.841110\pi$
$709$	− 25.4924i	− 0.957388i	0.877982	−	0.478694i	$-0.158890\pi$
$710$	20.1383i	0.755775i
$711$	0	0
$712$	−86.1080	−3.22703
$713$	− 3.12311i	− 0.116961i
$714$	0	0
$715$	0	0
$716$	59.8617	2.23714
$717$	0	0
$718$	39.3693	1.46925
$719$	1.36932	0.0510669	0.0255335	−	0.999674i	$-0.491872\pi$
$719$	1.36932	0.0510669	0.0255335	+	0.999674i	$0.491872\pi$
$720$	0	0
$721$	26.9309i	1.00296i
$722$	− 45.4384i	− 1.69104i
$723$	0	0
$724$	44.1771	1.64183
$725$	26.6307	0.989039
$726$	0	0
$727$	−39.6695	−1.47126	−0.735630	−	0.677383i	$-0.763114\pi$
$727$	−39.6695	−1.47126	−0.735630	+	0.677383i	$0.763114\pi$
$728$	0	0
$729$	0	0
$730$	− 2.69981i	− 0.0999246i
$731$	−1.12311	−0.0415396
$732$	0	0
$733$	53.4924i	1.97579i	0.155131	+	0.987894i	$0.450420\pi$
$733$	53.4924i	1.97579i	−0.155131	+	0.987894i	$0.549580\pi$
$734$	− 51.3693i	− 1.89608i
$735$	0	0
$736$	− 13.1231i	− 0.483724i
$737$	0.876894	0.0323008
$738$	0	0
$739$	6.24621i	0.229771i	0.993379	+	0.114885i	$0.0366501\pi$
$739$	6.24621i	0.229771i	−0.993379	+	0.114885i	$0.963350\pi$
$740$	8.80776	0.323780
$741$	0	0
$742$	−106.600	−3.91342
$743$	37.3693i	1.37095i	0.728097	+	0.685474i	$0.240405\pi$
$743$	37.3693i	1.37095i	−0.728097	+	0.685474i	$0.759595\pi$
$744$	0	0
$745$	3.68466	0.134995
$746$	− 9.30019i	− 0.340504i
$747$	0	0
$748$	23.3693i	0.854467i
$749$	− 29.3693i	− 1.07313i
$750$	0	0
$751$	30.1080	1.09865	0.549327	−	0.835607i	$-0.314884\pi$
$751$	30.1080	1.09865	0.549327	+	0.835607i	$0.314884\pi$
$752$	− 63.3693i	− 2.31084i
$753$	0	0
$754$	0	0
$755$	−8.63068	−0.314103
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	28.9848	1.05278
$759$	0	0
$760$	− 4.13826i	− 0.150110i
$761$	− 15.3693i	− 0.557137i	−0.960416	−	0.278569i	$-0.910140\pi$
$761$	− 15.3693i	− 0.557137i	0.960416	−	0.278569i	$-0.0898600\pi$
$762$	0	0
$763$	−63.4233	−2.29608
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−68.4924	−2.47473
$767$	0	0
$768$	0	0
$769$	− 18.0000i	− 0.649097i	−0.945869	−	0.324548i	$-0.894788\pi$
$769$	− 18.0000i	− 0.649097i	0.945869	−	0.324548i	$-0.105212\pi$
$770$	−10.2462	−0.369248
$771$	0	0
$772$	88.9157i	3.20015i
$773$	7.75379i	0.278884i	0.990230	+	0.139442i	$0.0445309\pi$
$773$	7.75379i	0.278884i	−0.990230	+	0.139442i	$0.955469\pi$
$774$	0	0
$775$	− 7.31534i	− 0.262775i
$776$	29.1231	1.04546
$777$	0	0
$778$	7.82292i	0.280465i
$779$	−2.87689	−0.103075
$780$	0	0
$781$	−28.0000	−1.00192
$782$	− 13.1231i	− 0.469281i
$783$	0	0
$784$	−43.6847	−1.56017
$785$	2.45360i	0.0875728i
$786$	0	0
$787$	1.17708i	0.0419584i	0.999780	+	0.0209792i	$0.00667838\pi$
$787$	1.17708i	0.0419584i	−0.999780	+	0.0209792i	$0.993322\pi$
$788$	− 51.8617i	− 1.84750i
$789$	0	0
$790$	−13.7538	−0.489338
$791$	52.7386i	1.87517i
$792$	0	0
$793$	0	0
$794$	30.8769	1.09578
$795$	0	0
$796$	−105.723	−3.74727
$797$	−41.6155	−1.47410	−0.737049	−	0.675840i	$-0.763781\pi$
$797$	−41.6155	−1.47410	−0.737049	+	0.675840i	$0.763781\pi$
$798$	0	0
$799$	− 21.1231i	− 0.747282i
$800$	− 30.7386i	− 1.08677i
$801$	0	0
$802$	47.5464	1.67892
$803$	3.75379	0.132468
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	0	0
$808$	− 22.5616i	− 0.793713i
$809$	−37.3002	−1.31140	−0.655702	−	0.755019i	$-0.727627\pi$
$809$	−37.3002	−1.31140	−0.655702	+	0.755019i	$0.727627\pi$
$810$	0	0
$811$	− 1.56155i	− 0.0548335i	−0.999624	−	0.0274168i	$-0.991272\pi$
$811$	− 1.56155i	− 0.0548335i	0.999624	−	0.0274168i	$-0.00872812\pi$
$812$	− 92.3542i	− 3.24100i
$813$	0	0
$814$	17.6155i	0.617424i
$815$	−8.87689	−0.310944
$816$	0	0
$817$	0.492423i	0.0172277i
$818$	−47.0540	−1.64520
$819$	0	0
$820$	6.56155	0.229139
$821$	26.4924i	0.924592i	0.886726	+	0.462296i	$0.152974\pi$
$821$	26.4924i	0.924592i	−0.886726	+	0.462296i	$0.847026\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	49.6155i	1.72844i
$825$	0	0
$826$	− 101.477i	− 3.53085i
$827$	34.7386i	1.20798i	0.796992	+	0.603990i	$0.206423\pi$
$827$	34.7386i	1.20798i	−0.796992	+	0.603990i	$0.793577\pi$
$828$	0	0
$829$	19.4924	0.677000	0.338500	−	0.940966i	$-0.390081\pi$
$829$	19.4924	0.677000	0.338500	+	0.940966i	$0.390081\pi$
$830$	− 13.1231i	− 0.455510i
$831$	0	0
$832$	0	0
$833$	−14.5616	−0.504528
$834$	0	0
$835$	−3.50758	−0.121385
$836$	10.2462	0.354373
$837$	0	0
$838$	− 45.4773i	− 1.57099i
$839$	− 19.6155i	− 0.677203i	−0.940930	−	0.338602i	$-0.890046\pi$
$839$	− 19.6155i	− 0.677203i	0.940930	−	0.338602i	$-0.109954\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	37.7926	1.30242
$843$	0	0
$844$	−33.3693	−1.14862
$845$	0	0
$846$	0	0
$847$	24.9309i	0.856635i
$848$	−89.7926	−3.08349
$849$	0	0
$850$	− 30.7386i	− 1.05433i
$851$	− 6.87689i	− 0.235737i
$852$	0	0
$853$	6.12311i	0.209651i	0.994491	+	0.104826i	$0.0334284\pi$
$853$	6.12311i	0.209651i	−0.994491	+	0.104826i	$0.966572\pi$
$854$	110.600	3.78467
$855$	0	0
$856$	− 54.1080i	− 1.84937i
$857$	31.4384	1.07392	0.536958	−	0.843609i	$-0.319573\pi$
$857$	31.4384	1.07392	0.536958	+	0.843609i	$0.319573\pi$
$858$	0	0
$859$	20.4384	0.697351	0.348675	−	0.937244i	$-0.386632\pi$
$859$	20.4384	0.697351	0.348675	+	0.937244i	$0.386632\pi$
$860$	− 1.12311i	− 0.0382976i
$861$	0	0
$862$	7.36932	0.251000
$863$	2.49242i	0.0848430i	0.999100	+	0.0424215i	$0.0135072\pi$
$863$	2.49242i	0.0848430i	−0.999100	+	0.0424215i	$0.986493\pi$
$864$	0	0
$865$	− 2.10795i	− 0.0716725i
$866$	64.6695i	2.19756i
$867$	0	0
$868$	−25.3693	−0.861091
$869$	− 19.1231i	− 0.648707i
$870$	0	0
$871$	0	0
$872$	−116.847	−3.95692
$873$	0	0
$874$	−5.75379	−0.194625
$875$	19.3693	0.654802
$876$	0	0
$877$	19.4384i	0.656390i	0.944610	+	0.328195i	$0.106440\pi$
$877$	19.4384i	0.656390i	−0.944610	+	0.328195i	$0.893560\pi$
$878$	− 3.36932i	− 0.113709i
$879$	0	0
$880$	−8.63068	−0.290940
$881$	−37.9309	−1.27792	−0.638962	−	0.769239i	$-0.720636\pi$
$881$	−37.9309	−1.27792	−0.638962	+	0.769239i	$0.720636\pi$
$882$	0	0
$883$	11.8078	0.397363	0.198681	−	0.980064i	$-0.436334\pi$
$883$	11.8078	0.397363	0.198681	+	0.980064i	$0.436334\pi$
$884$	0	0
$885$	0	0
$886$	37.7538i	1.26836i
$887$	−49.3693	−1.65766	−0.828830	−	0.559501i	$-0.810993\pi$
$887$	−49.3693	−1.65766	−0.828830	+	0.559501i	$0.810993\pi$
$888$	0	0
$889$	− 34.0540i	− 1.14213i
$890$	− 18.8769i	− 0.632755i
$891$	0	0
$892$	− 36.4924i	− 1.22186i
$893$	−9.26137	−0.309920
$894$	0	0
$895$	7.36932i	0.246329i
$896$	33.6155	1.12302
$897$	0	0
$898$	21.1231	0.704887
$899$	− 8.87689i	− 0.296061i
$900$	0	0
$901$	−29.9309	−0.997142
$902$	13.1231i	0.436952i
$903$	0	0
$904$	97.1619i	3.23156i
$905$	5.43845i	0.180780i
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	− 5.12311i	− 0.170016i
$909$	0	0
$910$	0	0
$911$	10.7386	0.355787	0.177893	−	0.984050i	$-0.443072\pi$
$911$	10.7386	0.355787	0.177893	+	0.984050i	$0.443072\pi$
$912$	0	0
$913$	18.2462	0.603861
$914$	−73.3002	−2.42455
$915$	0	0
$916$	1.12311i	0.0371085i
$917$	61.8617i	2.04285i
$918$	0	0
$919$	44.4924	1.46767	0.733835	−	0.679328i	$-0.237729\pi$
$919$	44.4924	1.46767	0.733835	+	0.679328i	$0.237729\pi$
$920$	7.36932	0.242959
$921$	0	0
$922$	94.2850	3.10511
$923$	0	0
$924$	0	0
$925$	− 16.1080i	− 0.529626i
$926$	68.3542	2.24626
$927$	0	0
$928$	− 37.3002i	− 1.22444i
$929$	− 12.8078i	− 0.420209i	−0.977679	−	0.210105i	$-0.932620\pi$
$929$	− 12.8078i	− 0.420209i	0.977679	−	0.210105i	$-0.0673804\pi$
$930$	0	0
$931$	6.38447i	0.209243i
$932$	−118.600	−3.88488
$933$	0	0
$934$	66.6004i	2.17923i
$935$	−2.87689	−0.0940845
$936$	0	0
$937$	−3.43845	−0.112329	−0.0561646	−	0.998422i	$-0.517887\pi$
$937$	−3.43845	−0.112329	−0.0561646	+	0.998422i	$0.517887\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	21.1231	0.688960
$941$	2.49242i	0.0812507i	0.999174	+	0.0406253i	$0.0129350\pi$
$941$	2.49242i	0.0812507i	−0.999174	+	0.0406253i	$0.987065\pi$
$942$	0	0
$943$	− 5.12311i	− 0.166831i
$944$	− 85.4773i	− 2.78205i
$945$	0	0
$946$	2.24621	0.0730306
$947$	10.7386i	0.348959i	0.984661	+	0.174479i	$0.0558242\pi$
$947$	10.7386i	0.348959i	−0.984661	+	0.174479i	$0.944176\pi$
$948$	0	0
$949$	0	0
$950$	−13.4773	−0.437260
$951$	0	0
$952$	−59.8617	−1.94013
$953$	−34.9848	−1.13327	−0.566635	−	0.823969i	$-0.691755\pi$
$953$	−34.9848	−1.13327	−0.566635	+	0.823969i	$0.691755\pi$
$954$	0	0
$955$	− 0.492423i	− 0.0159344i
$956$	− 2.87689i	− 0.0930454i
$957$	0	0
$958$	−16.0000	−0.516937
$959$	5.12311	0.165434
$960$	0	0
$961$	28.5616	0.921340
$962$	0	0
$963$	0	0
$964$	12.8078i	0.412510i
$965$	−10.9460	−0.352365
$966$	0	0
$967$	− 9.12311i	− 0.293379i	−0.989183	−	0.146690i	$-0.953138\pi$
$967$	− 9.12311i	− 0.293379i	0.989183	−	0.146690i	$-0.0468619\pi$
$968$	45.9309i	1.47627i
$969$	0	0
$970$	6.38447i	0.204993i
$971$	−52.9848	−1.70036	−0.850182	−	0.526488i	$-0.823508\pi$
$971$	−52.9848	−1.70036	−0.850182	+	0.526488i	$0.823508\pi$
$972$	0	0
$973$	38.9309i	1.24807i
$974$	−2.87689	−0.0921816
$975$	0	0
$976$	93.1619	2.98204
$977$	− 15.8229i	− 0.506220i	−0.967438	−	0.253110i	$-0.918547\pi$
$977$	− 15.8229i	− 0.506220i	0.967438	−	0.253110i	$-0.0814535\pi$
$978$	0	0
$979$	26.2462	0.838833
$980$	− 14.5616i	− 0.465152i
$981$	0	0
$982$	50.6004i	1.61472i
$983$	− 27.6155i	− 0.880799i	−0.897802	−	0.440399i	$-0.854837\pi$
$983$	− 27.6155i	− 0.880799i	0.897802	−	0.440399i	$-0.145163\pi$
$984$	0	0
$985$	6.38447	0.203426
$986$	− 37.3002i	− 1.18788i
$987$	0	0
$988$	0	0
$989$	−0.876894	−0.0278836
$990$	0	0
$991$	40.3542	1.28189	0.640946	−	0.767586i	$-0.278542\pi$
$991$	40.3542	1.28189	0.640946	+	0.767586i	$0.278542\pi$
$992$	−10.2462	−0.325318
$993$	0	0
$994$	− 127.723i	− 4.05114i
$995$	− 13.0152i	− 0.412608i
$996$	0	0
$997$	20.6155	0.652900	0.326450	−	0.945214i	$-0.394147\pi$
$997$	20.6155	0.652900	0.326450	+	0.945214i	$0.394147\pi$
$998$	−72.9848	−2.31029
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.h.1351.1		4
3.2	odd	2		507.2.b.d.337.4		4
13.2	odd	12		117.2.g.c.100.2		4
13.5	odd	4		1521.2.a.g.1.1		2
13.6	odd	12		117.2.g.c.55.2		4
13.8	odd	4		1521.2.a.m.1.2		2
13.12	even	2	inner	1521.2.b.h.1351.4		4
39.2	even	12		39.2.e.b.22.1	yes	4
39.5	even	4		507.2.a.g.1.2		2
39.8	even	4		507.2.a.d.1.1		2
39.11	even	12		507.2.e.g.22.2		4
39.17	odd	6		507.2.j.g.361.1		8
39.20	even	12		507.2.e.g.484.2		4
39.23	odd	6		507.2.j.g.316.4		8
39.29	odd	6		507.2.j.g.316.1		8
39.32	even	12		39.2.e.b.16.1	✓	4
39.35	odd	6		507.2.j.g.361.4		8
39.38	odd	2		507.2.b.d.337.1		4
52.15	even	12		1872.2.t.r.1153.1		4
52.19	even	12		1872.2.t.r.289.1		4
156.47	odd	4		8112.2.a.bo.1.1		2
156.71	odd	12		624.2.q.h.289.2		4
156.83	odd	4		8112.2.a.bk.1.2		2
156.119	odd	12		624.2.q.h.529.2		4
195.2	odd	12		975.2.bb.i.724.1		8
195.32	odd	12		975.2.bb.i.874.4		8
195.119	even	12		975.2.i.k.451.2		4
195.149	even	12		975.2.i.k.601.2		4
195.158	odd	12		975.2.bb.i.724.4		8
195.188	odd	12		975.2.bb.i.874.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.1	✓	4	39.32	even	12
39.2.e.b.22.1	yes	4	39.2	even	12
117.2.g.c.55.2		4	13.6	odd	12
117.2.g.c.100.2		4	13.2	odd	12
507.2.a.d.1.1		2	39.8	even	4
507.2.a.g.1.2		2	39.5	even	4
507.2.b.d.337.1		4	39.38	odd	2
507.2.b.d.337.4		4	3.2	odd	2
507.2.e.g.22.2		4	39.11	even	12
507.2.e.g.484.2		4	39.20	even	12
507.2.j.g.316.1		8	39.29	odd	6
507.2.j.g.316.4		8	39.23	odd	6
507.2.j.g.361.1		8	39.17	odd	6
507.2.j.g.361.4		8	39.35	odd	6
624.2.q.h.289.2		4	156.71	odd	12
624.2.q.h.529.2		4	156.119	odd	12
975.2.i.k.451.2		4	195.119	even	12
975.2.i.k.601.2		4	195.149	even	12
975.2.bb.i.724.1		8	195.2	odd	12
975.2.bb.i.724.4		8	195.158	odd	12
975.2.bb.i.874.1		8	195.188	odd	12
975.2.bb.i.874.4		8	195.32	odd	12
1521.2.a.g.1.1		2	13.5	odd	4
1521.2.a.m.1.2		2	13.8	odd	4
1521.2.b.h.1351.1		4	1.1	even	1	trivial
1521.2.b.h.1351.4		4	13.12	even	2	inner
1872.2.t.r.289.1		4	52.19	even	12
1872.2.t.r.1153.1		4	52.15	even	12
8112.2.a.bk.1.2		2	156.83	odd	4
8112.2.a.bo.1.1		2	156.47	odd	4

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.56155i q^{2} -4.56155 q^{4} -0.561553i q^{5} +3.56155i q^{7} +6.56155i q^{8} +O(q^{10})\)	\(q-2.56155i q^{2} -4.56155 q^{4} -0.561553i q^{5} +3.56155i q^{7} +6.56155i q^{8} -1.43845 q^{10} -2.00000i q^{11} +9.12311 q^{14} +7.68466 q^{16} +2.56155 q^{17} -1.12311i q^{19} +2.56155i q^{20} -5.12311 q^{22} +2.00000 q^{23} +4.68466 q^{25} -16.2462i q^{28} +5.68466 q^{29} -1.56155i q^{31} -6.56155i q^{32} -6.56155i q^{34} +2.00000 q^{35} -3.43845i q^{37} -2.87689 q^{38} +3.68466 q^{40} -2.56155i q^{41} -0.438447 q^{43} +9.12311i q^{44} -5.12311i q^{46} -8.24621i q^{47} -5.68466 q^{49} -12.0000i q^{50} -11.6847 q^{53} -1.12311 q^{55} -23.3693 q^{56} -14.5616i q^{58} -11.1231i q^{59} +12.1231 q^{61} -4.00000 q^{62} -1.43845 q^{64} +0.438447i q^{67} -11.6847 q^{68} -5.12311i q^{70} -14.0000i q^{71} +1.87689i q^{73} -8.80776 q^{74} +5.12311i q^{76} +7.12311 q^{77} +9.56155 q^{79} -4.31534i q^{80} -6.56155 q^{82} +9.12311i q^{83} -1.43845i q^{85} +1.12311i q^{86} +13.1231 q^{88} +13.1231i q^{89} -9.12311 q^{92} -21.1231 q^{94} -0.630683 q^{95} -4.43845i q^{97} +14.5616i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4}+O(q^{10}) \) 4 * q - 10 * q^4	\( 4 q - 10 q^{4} - 14 q^{10} + 20 q^{14} + 6 q^{16} + 2 q^{17} - 4 q^{22} + 8 q^{23} - 6 q^{25} - 2 q^{29} + 8 q^{35} - 28 q^{38} - 10 q^{40} - 10 q^{43} + 2 q^{49} - 22 q^{53} + 12 q^{55} - 44 q^{56} + 32 q^{61} - 16 q^{62} - 14 q^{64} - 22 q^{68} + 6 q^{74} + 12 q^{77} + 30 q^{79} - 18 q^{82} + 36 q^{88} - 20 q^{92} - 68 q^{94} - 52 q^{95}+O(q^{100}) \) 4 * q - 10 * q^4 - 14 * q^10 + 20 * q^14 + 6 * q^16 + 2 * q^17 - 4 * q^22 + 8 * q^23 - 6 * q^25 - 2 * q^29 + 8 * q^35 - 28 * q^38 - 10 * q^40 - 10 * q^43 + 2 * q^49 - 22 * q^53 + 12 * q^55 - 44 * q^56 + 32 * q^61 - 16 * q^62 - 14 * q^64 - 22 * q^68 + 6 * q^74 + 12 * q^77 + 30 * q^79 - 18 * q^82 + 36 * q^88 - 20 * q^92 - 68 * q^94 - 52 * q^95

Introduction

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Database

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