LMFDB - Embedded newform 1521.2.b.k.1351.3

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:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.3
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.k.1351.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.04892i q^{2} -2.19806 q^{4} +3.35690i q^{5} +2.24698i q^{7} +0.405813i q^{8} +O(q^{10})$	$q-2.04892i q^{2} -2.19806 q^{4} +3.35690i q^{5} +2.24698i q^{7} +0.405813i q^{8} +6.87800 q^{10} -4.93900i q^{11} +4.60388 q^{14} -3.56465 q^{16} +0.911854 q^{17} +3.80194i q^{19} -7.37867i q^{20} -10.1196 q^{22} +2.02715 q^{23} -6.26875 q^{25} -4.93900i q^{28} +3.93900 q^{29} +8.82908i q^{31} +8.11529i q^{32} -1.86831i q^{34} -7.54288 q^{35} +8.80194i q^{37} +7.78986 q^{38} -1.36227 q^{40} +6.93900i q^{41} +2.28621 q^{43} +10.8562i q^{44} -4.15346i q^{46} -3.80194i q^{47} +1.95108 q^{49} +12.8442i q^{50} -0.542877 q^{53} +16.5797 q^{55} -0.911854 q^{56} -8.07069i q^{58} +4.71379i q^{59} +3.67994 q^{61} +18.0901 q^{62} +9.49827 q^{64} +1.52111i q^{67} -2.00431 q^{68} +15.4547i q^{70} +2.37867i q^{71} +7.41119i q^{73} +18.0344 q^{74} -8.35690i q^{76} +11.0978 q^{77} -3.74094 q^{79} -11.9661i q^{80} +14.2174 q^{82} +2.30798i q^{83} +3.06100i q^{85} -4.68425i q^{86} +2.00431 q^{88} +10.0586i q^{89} -4.45580 q^{92} -7.78986 q^{94} -12.7627 q^{95} +16.1293i q^{97} -3.99761i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 22 q^{4}+O(q^{10}) $ 6 * q - 22 * q^4	$ 6 q - 22 q^{4} + 2 q^{10} + 10 q^{14} + 22 q^{16} - 2 q^{17} - 18 q^{22} - 22 q^{25} + 4 q^{29} - 8 q^{35} + 6 q^{40} + 30 q^{43} + 30 q^{49} + 34 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 4 q^{62} + 26 q^{68} - 30 q^{74} + 30 q^{77} + 6 q^{79} + 6 q^{82} - 26 q^{88} - 14 q^{92} - 42 q^{95}+O(q^{100}) $ 6 * q - 22 * q^4 + 2 * q^10 + 10 * q^14 + 22 * q^16 - 2 * q^17 - 18 * q^22 - 22 * q^25 + 4 * q^29 - 8 * q^35 + 6 * q^40 + 30 * q^43 + 30 * q^49 + 34 * q^53 + 6 * q^55 + 2 * q^56 - 26 * q^61 - 4 * q^62 + 26 * q^68 - 30 * q^74 + 30 * q^77 + 6 * q^79 + 6 * q^82 - 26 * q^88 - 14 * q^92 - 42 * 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 $

$2$	− 2.04892i	− 1.44880i	−0.689378	−	0.724402i	$-0.742116\pi$
$2$	− 2.04892i	− 1.44880i	0.689378	−	0.724402i	$-0.257884\pi$
$3$	0	0
$3$	0	0
$4$	−2.19806	−1.09903
$5$	3.35690i	1.50125i	0.660729	+	0.750625i	$0.270247\pi$
$5$	3.35690i	1.50125i	−0.660729	+	0.750625i	$0.729753\pi$
$6$	0	0
$7$	2.24698i	0.849278i	0.905363	+	0.424639i	$0.139599\pi$
$7$	2.24698i	0.849278i	−0.905363	+	0.424639i	$0.860401\pi$
$8$	0.405813i	0.143477i
$9$	0	0
$10$	6.87800	2.17502
$11$	− 4.93900i	− 1.48916i	−0.667531	−	0.744582i	$-0.732649\pi$
$11$	− 4.93900i	− 1.48916i	0.667531	−	0.744582i	$-0.267351\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	4.60388	1.23044
$15$	0	0
$16$	−3.56465	−0.891162
$17$	0.911854	0.221157	0.110579	−	0.993867i	$-0.464730\pi$
$17$	0.911854	0.221157	0.110579	+	0.993867i	$0.464730\pi$
$18$	0	0
$19$	3.80194i	0.872224i	0.899892	+	0.436112i	$0.143645\pi$
$19$	3.80194i	0.872224i	−0.899892	+	0.436112i	$0.856355\pi$
$20$	− 7.37867i	− 1.64992i
$21$	0	0
$22$	−10.1196	−2.15751
$23$	2.02715	0.422689	0.211345	−	0.977412i	$-0.432216\pi$
$23$	2.02715	0.422689	0.211345	+	0.977412i	$0.432216\pi$
$24$	0	0
$25$	−6.26875	−1.25375
$26$	0	0
$27$	0	0
$28$	− 4.93900i	− 0.933383i
$29$	3.93900	0.731454	0.365727	−	0.930722i	$-0.380820\pi$
$29$	3.93900	0.731454	0.365727	+	0.930722i	$0.380820\pi$
$30$	0	0
$31$	8.82908i	1.58575i	0.609384	+	0.792875i	$0.291417\pi$
$31$	8.82908i	1.58575i	−0.609384	+	0.792875i	$0.708583\pi$
$32$	8.11529i	1.43459i
$33$	0	0
$34$	− 1.86831i	− 0.320413i
$35$	−7.54288	−1.27498
$36$	0	0
$37$	8.80194i	1.44703i	0.690309	+	0.723515i	$0.257475\pi$
$37$	8.80194i	1.44703i	−0.690309	+	0.723515i	$0.742525\pi$
$38$	7.78986	1.26368
$39$	0	0
$40$	−1.36227	−0.215394
$41$	6.93900i	1.08369i	0.840478	+	0.541845i	$0.182274\pi$
$41$	6.93900i	1.08369i	−0.840478	+	0.541845i	$0.817726\pi$
$42$	0	0
$43$	2.28621	0.348643	0.174322	−	0.984689i	$-0.444227\pi$
$43$	2.28621	0.348643	0.174322	+	0.984689i	$0.444227\pi$
$44$	10.8562i	1.63664i
$45$	0	0
$46$	− 4.15346i	− 0.612394i
$47$	− 3.80194i	− 0.554570i	−0.960788	−	0.277285i	$-0.910565\pi$
$47$	− 3.80194i	− 0.554570i	0.960788	−	0.277285i	$-0.0894346\pi$
$48$	0	0
$49$	1.95108	0.278726
$50$	12.8442i	1.81644i
$51$	0	0
$52$	0	0
$53$	−0.542877	−0.0745698	−0.0372849	−	0.999305i	$-0.511871\pi$
$53$	−0.542877	−0.0745698	−0.0372849	+	0.999305i	$0.511871\pi$
$54$	0	0
$55$	16.5797	2.23561
$56$	−0.911854	−0.121852
$57$	0	0
$58$	− 8.07069i	− 1.05973i
$59$	4.71379i	0.613683i	0.951761	+	0.306842i	$0.0992722\pi$
$59$	4.71379i	0.613683i	−0.951761	+	0.306842i	$0.900728\pi$
$60$	0	0
$61$	3.67994	0.471168	0.235584	−	0.971854i	$-0.424300\pi$
$61$	3.67994	0.471168	0.235584	+	0.971854i	$0.424300\pi$
$62$	18.0901	2.29744
$63$	0	0
$64$	9.49827	1.18728
$65$	0	0
$66$	0	0
$67$	1.52111i	0.185833i	0.995674	+	0.0929164i	$0.0296189\pi$
$67$	1.52111i	0.185833i	−0.995674	+	0.0929164i	$0.970381\pi$
$68$	−2.00431	−0.243059
$69$	0	0
$70$	15.4547i	1.84719i
$71$	2.37867i	0.282296i	0.989989	+	0.141148i	$0.0450793\pi$
$71$	2.37867i	0.282296i	−0.989989	+	0.141148i	$0.954921\pi$
$72$	0	0
$73$	7.41119i	0.867414i	0.901054	+	0.433707i	$0.142795\pi$
$73$	7.41119i	0.867414i	−0.901054	+	0.433707i	$0.857205\pi$
$74$	18.0344	2.09646
$75$	0	0
$76$	− 8.35690i	− 0.958602i
$77$	11.0978	1.26472
$78$	0	0
$79$	−3.74094	−0.420888	−0.210444	−	0.977606i	$-0.567491\pi$
$79$	−3.74094	−0.420888	−0.210444	+	0.977606i	$0.567491\pi$
$80$	− 11.9661i	− 1.33786i
$81$	0	0
$82$	14.2174	1.57005
$83$	2.30798i	0.253334i	0.991945	+	0.126667i	$0.0404279\pi$
$83$	2.30798i	0.253334i	−0.991945	+	0.126667i	$0.959572\pi$
$84$	0	0
$85$	3.06100i	0.332012i
$86$	− 4.68425i	− 0.505116i
$87$	0	0
$88$	2.00431	0.213660
$89$	10.0586i	1.06621i	0.846049	+	0.533105i	$0.178975\pi$
$89$	10.0586i	1.06621i	−0.846049	+	0.533105i	$0.821025\pi$
$90$	0	0
$91$	0	0
$92$	−4.45580	−0.464549
$93$	0	0
$94$	−7.78986	−0.803462
$95$	−12.7627	−1.30943
$96$	0	0
$97$	16.1293i	1.63768i	0.574021	+	0.818841i	$0.305383\pi$
$97$	16.1293i	1.63768i	−0.574021	+	0.818841i	$0.694617\pi$
$98$	− 3.99761i	− 0.403819i
$99$	0	0
$100$	13.7791	1.37791
$101$	−9.94869	−0.989932	−0.494966	−	0.868912i	$-0.664819\pi$
$101$	−9.94869	−0.989932	−0.494966	+	0.868912i	$0.664819\pi$
$102$	0	0
$103$	10.9879	1.08267	0.541336	−	0.840806i	$-0.317919\pi$
$103$	10.9879	1.08267	0.541336	+	0.840806i	$0.317919\pi$
$104$	0	0
$105$	0	0
$106$	1.11231i	0.108037i
$107$	9.87263	0.954423	0.477211	−	0.878789i	$-0.341648\pi$
$107$	9.87263	0.954423	0.477211	+	0.878789i	$0.341648\pi$
$108$	0	0
$109$	− 20.2446i	− 1.93908i	−0.244934	−	0.969540i	$-0.578766\pi$
$109$	− 20.2446i	− 1.93908i	0.244934	−	0.969540i	$-0.421234\pi$
$110$	− 33.9705i	− 3.23896i
$111$	0	0
$112$	− 8.00969i	− 0.756844i
$113$	−9.69202	−0.911749	−0.455874	−	0.890044i	$-0.650673\pi$
$113$	−9.69202	−0.911749	−0.455874	+	0.890044i	$0.650673\pi$
$114$	0	0
$115$	6.80492i	0.634562i
$116$	−8.65817	−0.803891
$117$	0	0
$118$	9.65817	0.889107
$119$	2.04892i	0.187824i
$120$	0	0
$121$	−13.3937	−1.21761
$122$	− 7.53989i	− 0.682630i
$123$	0	0
$124$	− 19.4069i	− 1.74279i
$125$	− 4.25906i	− 0.380942i
$126$	0	0
$127$	−13.8116	−1.22558	−0.612792	−	0.790244i	$-0.709954\pi$
$127$	−13.8116	−1.22558	−0.612792	+	0.790244i	$0.709954\pi$
$128$	− 3.23059i	− 0.285546i
$129$	0	0
$130$	0	0
$131$	−2.99462	−0.261641	−0.130821	−	0.991406i	$-0.541761\pi$
$131$	−2.99462	−0.261641	−0.130821	+	0.991406i	$0.541761\pi$
$132$	0	0
$133$	−8.54288	−0.740761
$134$	3.11662	0.269235
$135$	0	0
$136$	0.370042i	0.0317309i
$137$	− 23.0194i	− 1.96668i	−0.181781	−	0.983339i	$-0.558186\pi$
$137$	− 23.0194i	− 1.96668i	0.181781	−	0.983339i	$-0.441814\pi$
$138$	0	0
$139$	0.982542	0.0833381	0.0416690	−	0.999131i	$-0.486732\pi$
$139$	0.982542	0.0833381	0.0416690	+	0.999131i	$0.486732\pi$
$140$	16.5797	1.40124
$141$	0	0
$142$	4.87369	0.408991
$143$	0	0
$144$	0	0
$145$	13.2228i	1.09810i
$146$	15.1849	1.25671
$147$	0	0
$148$	− 19.3472i	− 1.59033i
$149$	10.2591i	0.840455i	0.907419	+	0.420228i	$0.138050\pi$
$149$	10.2591i	0.840455i	−0.907419	+	0.420228i	$0.861950\pi$
$150$	0	0
$151$	− 20.1685i	− 1.64129i	−0.571438	−	0.820646i	$-0.693614\pi$
$151$	− 20.1685i	− 1.64129i	0.571438	−	0.820646i	$-0.306386\pi$
$152$	−1.54288	−0.125144
$153$	0	0
$154$	− 22.7385i	− 1.83232i
$155$	−29.6383	−2.38061
$156$	0	0
$157$	10.4383	0.833070	0.416535	−	0.909120i	$-0.363244\pi$
$157$	10.4383	0.833070	0.416535	+	0.909120i	$0.363244\pi$
$158$	7.66487i	0.609785i
$159$	0	0
$160$	−27.2422	−2.15368
$161$	4.55496i	0.358981i
$162$	0	0
$163$	− 11.0465i	− 0.865231i	−0.901579	−	0.432615i	$-0.857591\pi$
$163$	− 11.0465i	− 0.865231i	0.901579	−	0.432615i	$-0.142409\pi$
$164$	− 15.2524i	− 1.19101i
$165$	0	0
$166$	4.72886	0.367031
$167$	− 8.10992i	− 0.627564i	−0.949495	−	0.313782i	$-0.898404\pi$
$167$	− 8.10992i	− 0.627564i	0.949495	−	0.313782i	$-0.101596\pi$
$168$	0	0
$169$	0	0
$170$	6.27173	0.481020
$171$	0	0
$172$	−5.02523	−0.383170
$173$	−18.0562	−1.37279	−0.686394	−	0.727230i	$-0.740808\pi$
$173$	−18.0562	−1.37279	−0.686394	+	0.727230i	$0.740808\pi$
$174$	0	0
$175$	− 14.0858i	− 1.06478i
$176$	17.6058i	1.32709i
$177$	0	0
$178$	20.6093	1.54473
$179$	−19.8702	−1.48517	−0.742585	−	0.669751i	$-0.766401\pi$
$179$	−19.8702	−1.48517	−0.742585	+	0.669751i	$0.766401\pi$
$180$	0	0
$181$	10.0828	0.749446	0.374723	−	0.927137i	$-0.377738\pi$
$181$	10.0828	0.749446	0.374723	+	0.927137i	$0.377738\pi$
$182$	0	0
$183$	0	0
$184$	0.822643i	0.0606461i
$185$	−29.5472	−2.17235
$186$	0	0
$187$	− 4.50365i	− 0.329339i
$188$	8.35690i	0.609489i
$189$	0	0
$190$	26.1497i	1.89710i
$191$	6.58748	0.476653	0.238327	−	0.971185i	$-0.423401\pi$
$191$	6.58748	0.476653	0.238327	+	0.971185i	$0.423401\pi$
$192$	0	0
$193$	10.8672i	0.782242i	0.920339	+	0.391121i	$0.127913\pi$
$193$	10.8672i	0.782242i	−0.920339	+	0.391121i	$0.872087\pi$
$194$	33.0476	2.37268
$195$	0	0
$196$	−4.28860	−0.306329
$197$	9.24160i	0.658437i	0.944254	+	0.329218i	$0.106785\pi$
$197$	9.24160i	0.658437i	−0.944254	+	0.329218i	$0.893215\pi$
$198$	0	0
$199$	1.56465	0.110915	0.0554574	−	0.998461i	$-0.482338\pi$
$199$	1.56465	0.110915	0.0554574	+	0.998461i	$0.482338\pi$
$200$	− 2.54394i	− 0.179884i
$201$	0	0
$202$	20.3840i	1.43422i
$203$	8.85086i	0.621208i
$204$	0	0
$205$	−23.2935	−1.62689
$206$	− 22.5133i	− 1.56858i
$207$	0	0
$208$	0	0
$209$	18.7778	1.29889
$210$	0	0
$211$	23.2446	1.60022	0.800112	−	0.599851i	$-0.204774\pi$
$211$	23.2446	1.60022	0.800112	+	0.599851i	$0.204774\pi$
$212$	1.19328	0.0819546
$213$	0	0
$214$	− 20.2282i	− 1.38277i
$215$	7.67456i	0.523401i
$216$	0	0
$217$	−19.8388	−1.34674
$218$	−41.4795	−2.80935
$219$	0	0
$220$	−36.4432	−2.45700
$221$	0	0
$222$	0	0
$223$	− 10.7385i	− 0.719106i	−0.933125	−	0.359553i	$-0.882929\pi$
$223$	− 10.7385i	− 0.719106i	0.933125	−	0.359553i	$-0.117071\pi$
$224$	−18.2349	−1.21837
$225$	0	0
$226$	19.8582i	1.32094i
$227$	− 6.97584i	− 0.463003i	−0.972835	−	0.231501i	$-0.925636\pi$
$227$	− 6.97584i	− 0.463003i	0.972835	−	0.231501i	$-0.0743637\pi$
$228$	0	0
$229$	− 16.8049i	− 1.11050i	−0.831683	−	0.555250i	$-0.812622\pi$
$229$	− 16.8049i	− 1.11050i	0.831683	−	0.555250i	$-0.187378\pi$
$230$	13.9427	0.919356
$231$	0	0
$232$	1.59850i	0.104947i
$233$	8.69202	0.569433	0.284717	−	0.958612i	$-0.408100\pi$
$233$	8.69202	0.569433	0.284717	+	0.958612i	$0.408100\pi$
$234$	0	0
$235$	12.7627	0.832547
$236$	− 10.3612i	− 0.674457i
$237$	0	0
$238$	4.19806	0.272120
$239$	− 22.9191i	− 1.48252i	−0.671220	−	0.741258i	$-0.734229\pi$
$239$	− 22.9191i	− 1.48252i	0.671220	−	0.741258i	$-0.265771\pi$
$240$	0	0
$241$	21.9801i	1.41587i	0.706280	+	0.707933i	$0.250372\pi$
$241$	21.9801i	1.41587i	−0.706280	+	0.707933i	$0.749628\pi$
$242$	27.4426i	1.76408i
$243$	0	0
$244$	−8.08874	−0.517828
$245$	6.54958i	0.418437i
$246$	0	0
$247$	0	0
$248$	−3.58296	−0.227518
$249$	0	0
$250$	−8.72646	−0.551910
$251$	26.6437	1.68174	0.840868	−	0.541241i	$-0.182045\pi$
$251$	26.6437	1.68174	0.840868	+	0.541241i	$0.182045\pi$
$252$	0	0
$253$	− 10.0121i	− 0.629454i
$254$	28.2989i	1.77563i
$255$	0	0
$256$	12.3773	0.773584
$257$	23.1444	1.44371	0.721853	−	0.692047i	$-0.243291\pi$
$257$	23.1444	1.44371	0.721853	+	0.692047i	$0.243291\pi$
$258$	0	0
$259$	−19.7778	−1.22893
$260$	0	0
$261$	0	0
$262$	6.13574i	0.379067i
$263$	18.5284	1.14251	0.571255	−	0.820773i	$-0.306457\pi$
$263$	18.5284	1.14251	0.571255	+	0.820773i	$0.306457\pi$
$264$	0	0
$265$	− 1.82238i	− 0.111948i
$266$	17.5036i	1.07322i
$267$	0	0
$268$	− 3.34349i	− 0.204236i
$269$	−9.75840	−0.594980	−0.297490	−	0.954725i	$-0.596149\pi$
$269$	−9.75840	−0.594980	−0.297490	+	0.954725i	$0.596149\pi$
$270$	0	0
$271$	− 22.8019i	− 1.38512i	−0.721361	−	0.692560i	$-0.756483\pi$
$271$	− 22.8019i	− 1.38512i	0.721361	−	0.692560i	$-0.243517\pi$
$272$	−3.25044	−0.197087
$273$	0	0
$274$	−47.1648	−2.84933
$275$	30.9614i	1.86704i
$276$	0	0
$277$	23.9705	1.44025	0.720123	−	0.693847i	$-0.244086\pi$
$277$	23.9705	1.44025	0.720123	+	0.693847i	$0.244086\pi$
$278$	− 2.01315i	− 0.120741i
$279$	0	0
$280$	− 3.06100i	− 0.182930i
$281$	4.12498i	0.246076i	0.992402	+	0.123038i	$0.0392637\pi$
$281$	4.12498i	0.246076i	−0.992402	+	0.123038i	$0.960736\pi$
$282$	0	0
$283$	15.6558	0.930639	0.465320	−	0.885143i	$-0.345939\pi$
$283$	15.6558	0.930639	0.465320	+	0.885143i	$0.345939\pi$
$284$	− 5.22846i	− 0.310252i
$285$	0	0
$286$	0	0
$287$	−15.5918	−0.920354
$288$	0	0
$289$	−16.1685	−0.951090
$290$	27.0925	1.59092
$291$	0	0
$292$	− 16.2903i	− 0.953315i
$293$	− 22.5948i	− 1.32000i	−0.751265	−	0.660001i	$-0.770556\pi$
$293$	− 22.5948i	− 1.32000i	0.751265	−	0.660001i	$-0.229444\pi$
$294$	0	0
$295$	−15.8237	−0.921292
$296$	−3.57194	−0.207615
$297$	0	0
$298$	21.0200	1.21765
$299$	0	0
$300$	0	0
$301$	5.13706i	0.296095i
$302$	−41.3236	−2.37791
$303$	0	0
$304$	− 13.5526i	− 0.777293i
$305$	12.3532i	0.707341i
$306$	0	0
$307$	− 6.55496i	− 0.374111i	−0.982349	−	0.187056i	$-0.940106\pi$
$307$	− 6.55496i	− 0.374111i	0.982349	−	0.187056i	$-0.0598945\pi$
$308$	−24.3937	−1.38996
$309$	0	0
$310$	60.7265i	3.44903i
$311$	−12.0392	−0.682682	−0.341341	−	0.939940i	$-0.610881\pi$
$311$	−12.0392	−0.682682	−0.341341	+	0.939940i	$0.610881\pi$
$312$	0	0
$313$	−33.8950	−1.91586	−0.957929	−	0.287005i	$-0.907340\pi$
$313$	−33.8950	−1.91586	−0.957929	+	0.287005i	$0.907340\pi$
$314$	− 21.3873i	− 1.20695i
$315$	0	0
$316$	8.22282	0.462570
$317$	− 4.49827i	− 0.252648i	−0.991989	−	0.126324i	$-0.959682\pi$
$317$	− 4.49827i	− 0.252648i	0.991989	−	0.126324i	$-0.0403179\pi$
$318$	0	0
$319$	− 19.4547i	− 1.08926i
$320$	31.8847i	1.78241i
$321$	0	0
$322$	9.33273	0.520093
$323$	3.46681i	0.192899i
$324$	0	0
$325$	0	0
$326$	−22.6334	−1.25355
$327$	0	0
$328$	−2.81594	−0.155484
$329$	8.54288	0.470984
$330$	0	0
$331$	11.2131i	0.616329i	0.951333	+	0.308165i	$0.0997148\pi$
$331$	11.2131i	0.616329i	−0.951333	+	0.308165i	$0.900285\pi$
$332$	− 5.07308i	− 0.278421i
$333$	0	0
$334$	−16.6165	−0.909217
$335$	−5.10620	−0.278981
$336$	0	0
$337$	−7.04892	−0.383979	−0.191989	−	0.981397i	$-0.561494\pi$
$337$	−7.04892	−0.383979	−0.191989	+	0.981397i	$0.561494\pi$
$338$	0	0
$339$	0	0
$340$	− 6.72827i	− 0.364891i
$341$	43.6069	2.36144
$342$	0	0
$343$	20.1129i	1.08599i
$344$	0.927774i	0.0500222i
$345$	0	0
$346$	36.9957i	1.98890i
$347$	2.70410	0.145164	0.0725819	−	0.997362i	$-0.476876\pi$
$347$	2.70410	0.145164	0.0725819	+	0.997362i	$0.476876\pi$
$348$	0	0
$349$	− 0.415502i	− 0.0222413i	−0.999938	−	0.0111207i	$-0.996460\pi$
$349$	− 0.415502i	− 0.0222413i	0.999938	−	0.0111207i	$-0.00353989\pi$
$350$	−28.8605	−1.54266
$351$	0	0
$352$	40.0814	2.13635
$353$	− 32.1672i	− 1.71209i	−0.516904	−	0.856044i	$-0.672916\pi$
$353$	− 32.1672i	− 1.71209i	0.516904	−	0.856044i	$-0.327084\pi$
$354$	0	0
$355$	−7.98493	−0.423796
$356$	− 22.1094i	− 1.17180i
$357$	0	0
$358$	40.7125i	2.15172i
$359$	− 22.3521i	− 1.17970i	−0.807513	−	0.589850i	$-0.799187\pi$
$359$	− 22.3521i	− 1.17970i	0.807513	−	0.589850i	$-0.200813\pi$
$360$	0	0
$361$	4.54527	0.239225
$362$	− 20.6588i	− 1.08580i
$363$	0	0
$364$	0	0
$365$	−24.8786	−1.30221
$366$	0	0
$367$	−2.30260	−0.120195	−0.0600974	−	0.998193i	$-0.519141\pi$
$367$	−2.30260	−0.120195	−0.0600974	+	0.998193i	$0.519141\pi$
$368$	−7.22606	−0.376685
$369$	0	0
$370$	60.5397i	3.14731i
$371$	− 1.21983i	− 0.0633305i
$372$	0	0
$373$	−19.2760	−0.998076	−0.499038	−	0.866580i	$-0.666313\pi$
$373$	−19.2760	−0.998076	−0.499038	+	0.866580i	$0.666313\pi$
$374$	−9.22760	−0.477148
$375$	0	0
$376$	1.54288	0.0795678
$377$	0	0
$378$	0	0
$379$	7.33944i	0.377002i	0.982073	+	0.188501i	$0.0603628\pi$
$379$	7.33944i	0.377002i	−0.982073	+	0.188501i	$0.939637\pi$
$380$	28.0532	1.43910
$381$	0	0
$382$	− 13.4972i	− 0.690577i
$383$	19.0901i	0.975457i	0.872995	+	0.487728i	$0.162174\pi$
$383$	19.0901i	0.975457i	−0.872995	+	0.487728i	$0.837826\pi$
$384$	0	0
$385$	37.2543i	1.89865i
$386$	22.2661	1.13331
$387$	0	0
$388$	− 35.4532i	− 1.79986i
$389$	−23.9879	−1.21624	−0.608118	−	0.793847i	$-0.708075\pi$
$389$	−23.9879	−1.21624	−0.608118	+	0.793847i	$0.708075\pi$
$390$	0	0
$391$	1.84846	0.0934807
$392$	0.791775i	0.0399907i
$393$	0	0
$394$	18.9353	0.953946
$395$	− 12.5579i	− 0.631859i
$396$	0	0
$397$	− 4.41789i	− 0.221728i	−0.993836	−	0.110864i	$-0.964638\pi$
$397$	− 4.41789i	− 0.221728i	0.993836	−	0.110864i	$-0.0353618\pi$
$398$	− 3.20583i	− 0.160694i
$399$	0	0
$400$	22.3459	1.11729
$401$	20.4088i	1.01917i	0.860421	+	0.509583i	$0.170200\pi$
$401$	20.4088i	1.01917i	−0.860421	+	0.509583i	$0.829800\pi$
$402$	0	0
$403$	0	0
$404$	21.8678	1.08797
$405$	0	0
$406$	18.1347	0.900009
$407$	43.4728	2.15487
$408$	0	0
$409$	22.1491i	1.09520i	0.836739	+	0.547602i	$0.184459\pi$
$409$	22.1491i	1.09520i	−0.836739	+	0.547602i	$0.815541\pi$
$410$	47.7265i	2.35704i
$411$	0	0
$412$	−24.1521	−1.18989
$413$	−10.5918	−0.521188
$414$	0	0
$415$	−7.74764	−0.380317
$416$	0	0
$417$	0	0
$418$	− 38.4741i	− 1.88183i
$419$	−14.7560	−0.720878	−0.360439	−	0.932783i	$-0.617373\pi$
$419$	−14.7560	−0.720878	−0.360439	+	0.932783i	$0.617373\pi$
$420$	0	0
$421$	− 8.47219i	− 0.412909i	−0.978456	−	0.206455i	$-0.933807\pi$
$421$	− 8.47219i	− 0.412909i	0.978456	−	0.206455i	$-0.0661926\pi$
$422$	− 47.6262i	− 2.31841i
$423$	0	0
$424$	− 0.220306i	− 0.0106990i
$425$	−5.71618	−0.277276
$426$	0	0
$427$	8.26875i	0.400153i
$428$	−21.7006	−1.04894
$429$	0	0
$430$	15.7245	0.758305
$431$	2.39181i	0.115210i	0.998339	+	0.0576048i	$0.0183463\pi$
$431$	2.39181i	0.115210i	−0.998339	+	0.0576048i	$0.981654\pi$
$432$	0	0
$433$	10.4286	0.501169	0.250584	−	0.968095i	$-0.419377\pi$
$433$	10.4286	0.501169	0.250584	+	0.968095i	$0.419377\pi$
$434$	40.6480i	1.95117i
$435$	0	0
$436$	44.4989i	2.13111i
$437$	7.70709i	0.368680i
$438$	0	0
$439$	32.5502	1.55353	0.776767	−	0.629787i	$-0.216858\pi$
$439$	32.5502	1.55353	0.776767	+	0.629787i	$0.216858\pi$
$440$	6.72827i	0.320758i
$441$	0	0
$442$	0	0
$443$	−9.58211	−0.455260	−0.227630	−	0.973748i	$-0.573098\pi$
$443$	−9.58211	−0.455260	−0.227630	+	0.973748i	$0.573098\pi$
$444$	0	0
$445$	−33.7657	−1.60065
$446$	−22.0024	−1.04184
$447$	0	0
$448$	21.3424i	1.00833i
$449$	28.3937i	1.33998i	0.742369	+	0.669992i	$0.233702\pi$
$449$	28.3937i	1.33998i	−0.742369	+	0.669992i	$0.766298\pi$
$450$	0	0
$451$	34.2717	1.61379
$452$	21.3037	1.00204
$453$	0	0
$454$	−14.2929	−0.670800
$455$	0	0
$456$	0	0
$457$	− 4.99569i	− 0.233688i	−0.993150	−	0.116844i	$-0.962722\pi$
$457$	− 4.99569i	− 0.233688i	0.993150	−	0.116844i	$-0.0372778\pi$
$458$	−34.4319	−1.60890
$459$	0	0
$460$	− 14.9576i	− 0.697404i
$461$	1.35258i	0.0629961i	0.999504	+	0.0314981i	$0.0100278\pi$
$461$	1.35258i	0.0629961i	−0.999504	+	0.0314981i	$0.989972\pi$
$462$	0	0
$463$	− 3.36467i	− 0.156369i	−0.996939	−	0.0781846i	$-0.975088\pi$
$463$	− 3.36467i	− 0.156369i	0.996939	−	0.0781846i	$-0.0249124\pi$
$464$	−14.0411	−0.651844
$465$	0	0
$466$	− 17.8092i	− 0.824997i
$467$	6.91079	0.319793	0.159897	−	0.987134i	$-0.448884\pi$
$467$	6.91079	0.319793	0.159897	+	0.987134i	$0.448884\pi$
$468$	0	0
$469$	−3.41789	−0.157824
$470$	− 26.1497i	− 1.20620i
$471$	0	0
$472$	−1.91292	−0.0880492
$473$	− 11.2916i	− 0.519188i
$474$	0	0
$475$	− 23.8334i	− 1.09355i
$476$	− 4.50365i	− 0.206424i
$477$	0	0
$478$	−46.9594	−2.14787
$479$	− 3.51573i	− 0.160638i	−0.996769	−	0.0803189i	$-0.974406\pi$
$479$	− 3.51573i	− 0.160638i	0.996769	−	0.0803189i	$-0.0255939\pi$
$480$	0	0
$481$	0	0
$482$	45.0355	2.05131
$483$	0	0
$484$	29.4403	1.33819
$485$	−54.1444	−2.45857
$486$	0	0
$487$	12.8479i	0.582193i	0.956694	+	0.291096i	$0.0940200\pi$
$487$	12.8479i	0.582193i	−0.956694	+	0.291096i	$0.905980\pi$
$488$	1.49337i	0.0676016i
$489$	0	0
$490$	13.4196	0.606234
$491$	28.6708	1.29390	0.646948	−	0.762534i	$-0.276045\pi$
$491$	28.6708	1.29390	0.646948	+	0.762534i	$0.276045\pi$
$492$	0	0
$493$	3.59179	0.161766
$494$	0	0
$495$	0	0
$496$	− 31.4726i	− 1.41316i
$497$	−5.34481	−0.239748
$498$	0	0
$499$	33.5555i	1.50215i	0.660215	+	0.751076i	$0.270465\pi$
$499$	33.5555i	1.50215i	−0.660215	+	0.751076i	$0.729535\pi$
$500$	9.36168i	0.418667i
$501$	0	0
$502$	− 54.5907i	− 2.43650i
$503$	−21.5633	−0.961461	−0.480730	−	0.876868i	$-0.659628\pi$
$503$	−21.5633	−0.961461	−0.480730	+	0.876868i	$0.659628\pi$
$504$	0	0
$505$	− 33.3967i	− 1.48613i
$506$	−20.5139	−0.911955
$507$	0	0
$508$	30.3588	1.34695
$509$	− 8.89008i	− 0.394046i	−0.980399	−	0.197023i	$-0.936873\pi$
$509$	− 8.89008i	− 0.394046i	0.980399	−	0.197023i	$-0.0631274\pi$
$510$	0	0
$511$	−16.6528	−0.736676
$512$	− 31.8213i	− 1.40632i
$513$	0	0
$514$	− 47.4209i	− 2.09165i
$515$	36.8853i	1.62536i
$516$	0	0
$517$	−18.7778	−0.825846
$518$	40.5230i	1.78048i
$519$	0	0
$520$	0	0
$521$	−19.3478	−0.847642	−0.423821	−	0.905746i	$-0.639312\pi$
$521$	−19.3478	−0.847642	−0.423821	+	0.905746i	$0.639312\pi$
$522$	0	0
$523$	−12.5948	−0.550731	−0.275366	−	0.961340i	$-0.588799\pi$
$523$	−12.5948	−0.550731	−0.275366	+	0.961340i	$0.588799\pi$
$524$	6.58237	0.287552
$525$	0	0
$526$	− 37.9632i	− 1.65527i
$527$	8.05084i	0.350700i
$528$	0	0
$529$	−18.8907	−0.821334
$530$	−3.73391	−0.162191
$531$	0	0
$532$	18.7778	0.814120
$533$	0	0
$534$	0	0
$535$	33.1414i	1.43283i
$536$	−0.617285	−0.0266627
$537$	0	0
$538$	19.9941i	0.862009i
$539$	− 9.63640i	− 0.415069i
$540$	0	0
$541$	29.0019i	1.24689i	0.781867	+	0.623445i	$0.214267\pi$
$541$	29.0019i	1.24689i	−0.781867	+	0.623445i	$0.785733\pi$
$542$	−46.7193	−2.00677
$543$	0	0
$544$	7.39996i	0.317271i
$545$	67.9590	2.91104
$546$	0	0
$547$	27.7006	1.18439	0.592197	−	0.805793i	$-0.298261\pi$
$547$	27.7006	1.18439	0.592197	+	0.805793i	$0.298261\pi$
$548$	50.5980i	2.16144i
$549$	0	0
$550$	63.4373	2.70497
$551$	14.9758i	0.637992i
$552$	0	0
$553$	− 8.40581i	− 0.357452i
$554$	− 49.1135i	− 2.08663i
$555$	0	0
$556$	−2.15969	−0.0915912
$557$	− 6.80971i	− 0.288537i	−0.989539	−	0.144268i	$-0.953917\pi$
$557$	− 6.80971i	− 0.288537i	0.989539	−	0.144268i	$-0.0460828\pi$
$558$	0	0
$559$	0	0
$560$	26.8877	1.13621
$561$	0	0
$562$	8.45175	0.356515
$563$	−19.2524	−0.811390	−0.405695	−	0.914008i	$-0.632970\pi$
$563$	−19.2524	−0.811390	−0.405695	+	0.914008i	$0.632970\pi$
$564$	0	0
$565$	− 32.5351i	− 1.36876i
$566$	− 32.0774i	− 1.34831i
$567$	0	0
$568$	−0.965294	−0.0405028
$569$	7.84846	0.329025	0.164512	−	0.986375i	$-0.447395\pi$
$569$	7.84846	0.329025	0.164512	+	0.986375i	$0.447395\pi$
$570$	0	0
$571$	−29.8568	−1.24947	−0.624735	−	0.780837i	$-0.714793\pi$
$571$	−29.8568	−1.24947	−0.624735	+	0.780837i	$0.714793\pi$
$572$	0	0
$573$	0	0
$574$	31.9463i	1.33341i
$575$	−12.7077	−0.529947
$576$	0	0
$577$	− 8.97823i	− 0.373769i	−0.982382	−	0.186884i	$-0.940161\pi$
$577$	− 8.97823i	− 0.373769i	0.982382	−	0.186884i	$-0.0598389\pi$
$578$	33.1280i	1.37794i
$579$	0	0
$580$	− 29.0646i	− 1.20684i
$581$	−5.18598	−0.215151
$582$	0	0
$583$	2.68127i	0.111047i
$584$	−3.00756	−0.124454
$585$	0	0
$586$	−46.2948	−1.91242
$587$	23.8538i	0.984553i	0.870439	+	0.492277i	$0.163835\pi$
$587$	23.8538i	0.984553i	−0.870439	+	0.492277i	$0.836165\pi$
$588$	0	0
$589$	−33.5676	−1.38313
$590$	32.4215i	1.33477i
$591$	0	0
$592$	− 31.3758i	− 1.28954i
$593$	− 11.9866i	− 0.492230i	−0.969241	−	0.246115i	$-0.920846\pi$
$593$	− 11.9866i	− 0.492230i	0.969241	−	0.246115i	$-0.0791541\pi$
$594$	0	0
$595$	−6.87800	−0.281971
$596$	− 22.5501i	− 0.923686i
$597$	0	0
$598$	0	0
$599$	−29.1142	−1.18958	−0.594788	−	0.803883i	$-0.702764\pi$
$599$	−29.1142	−1.18958	−0.594788	+	0.803883i	$0.702764\pi$
$600$	0	0
$601$	37.1366	1.51483	0.757417	−	0.652932i	$-0.226461\pi$
$601$	37.1366	1.51483	0.757417	+	0.652932i	$0.226461\pi$
$602$	10.5254	0.428984
$603$	0	0
$604$	44.3317i	1.80383i
$605$	− 44.9614i	− 1.82794i
$606$	0	0
$607$	−14.2325	−0.577680	−0.288840	−	0.957377i	$-0.593269\pi$
$607$	−14.2325	−0.577680	−0.288840	+	0.957377i	$0.593269\pi$
$608$	−30.8538	−1.25129
$609$	0	0
$610$	25.3106	1.02480
$611$	0	0
$612$	0	0
$613$	− 30.3139i	− 1.22437i	−0.790715	−	0.612184i	$-0.790291\pi$
$613$	− 30.3139i	− 1.22437i	0.790715	−	0.612184i	$-0.209709\pi$
$614$	−13.4306	−0.542014
$615$	0	0
$616$	4.50365i	0.181457i
$617$	− 24.9638i	− 1.00500i	−0.864576	−	0.502501i	$-0.832413\pi$
$617$	− 24.9638i	− 1.00500i	0.864576	−	0.502501i	$-0.167587\pi$
$618$	0	0
$619$	35.6122i	1.43138i	0.698420	+	0.715688i	$0.253887\pi$
$619$	35.6122i	1.43138i	−0.698420	+	0.715688i	$0.746113\pi$
$620$	65.1469	2.61636
$621$	0	0
$622$	24.6674i	0.989072i
$623$	−22.6015	−0.905509
$624$	0	0
$625$	−17.0465	−0.681861
$626$	69.4480i	2.77570i
$627$	0	0
$628$	−22.9441	−0.915570
$629$	8.02608i	0.320021i
$630$	0	0
$631$	23.8829i	0.950763i	0.879780	+	0.475382i	$0.157690\pi$
$631$	23.8829i	0.950763i	−0.879780	+	0.475382i	$0.842310\pi$
$632$	− 1.51812i	− 0.0603877i
$633$	0	0
$634$	−9.21659	−0.366037
$635$	− 46.3642i	− 1.83991i
$636$	0	0
$637$	0	0
$638$	−39.8611	−1.57812
$639$	0	0
$640$	10.8447	0.428676
$641$	−24.6577	−0.973920	−0.486960	−	0.873424i	$-0.661894\pi$
$641$	−24.6577	−0.973920	−0.486960	+	0.873424i	$0.661894\pi$
$642$	0	0
$643$	− 47.8165i	− 1.88570i	−0.333218	−	0.942850i	$-0.608134\pi$
$643$	− 47.8165i	− 1.88570i	0.333218	−	0.942850i	$-0.391866\pi$
$644$	− 10.0121i	− 0.394531i
$645$	0	0
$646$	7.10321	0.279472
$647$	−5.38942	−0.211880	−0.105940	−	0.994373i	$-0.533785\pi$
$647$	−5.38942	−0.211880	−0.105940	+	0.994373i	$0.533785\pi$
$648$	0	0
$649$	23.2814	0.913876
$650$	0	0
$651$	0	0
$652$	24.2809i	0.950915i
$653$	5.03790	0.197148	0.0985741	−	0.995130i	$-0.468572\pi$
$653$	5.03790	0.197148	0.0985741	+	0.995130i	$0.468572\pi$
$654$	0	0
$655$	− 10.0526i	− 0.392789i
$656$	− 24.7351i	− 0.965743i
$657$	0	0
$658$	− 17.5036i	− 0.682363i
$659$	43.9812	1.71326	0.856632	−	0.515927i	$-0.172553\pi$
$659$	43.9812	1.71326	0.856632	+	0.515927i	$0.172553\pi$
$660$	0	0
$661$	− 38.2194i	− 1.48656i	−0.668980	−	0.743280i	$-0.733269\pi$
$661$	− 38.2194i	− 1.48656i	0.668980	−	0.743280i	$-0.266731\pi$
$662$	22.9748	0.892940
$663$	0	0
$664$	−0.936608	−0.0363474
$665$	− 28.6775i	− 1.11207i
$666$	0	0
$667$	7.98493	0.309178
$668$	17.8261i	0.689713i
$669$	0	0
$670$	10.4622i	0.404189i
$671$	− 18.1752i	− 0.701647i
$672$	0	0
$673$	6.66487	0.256912	0.128456	−	0.991715i	$-0.458998\pi$
$673$	6.66487	0.256912	0.128456	+	0.991715i	$0.458998\pi$
$674$	14.4426i	0.556310i
$675$	0	0
$676$	0	0
$677$	−4.80194	−0.184553	−0.0922767	−	0.995733i	$-0.529414\pi$
$677$	−4.80194	−0.184553	−0.0922767	+	0.995733i	$0.529414\pi$
$678$	0	0
$679$	−36.2422	−1.39085
$680$	−1.24219	−0.0476360
$681$	0	0
$682$	− 89.3469i	− 3.42127i
$683$	− 11.2591i	− 0.430816i	−0.976524	−	0.215408i	$-0.930892\pi$
$683$	− 11.2591i	− 0.430816i	0.976524	−	0.215408i	$-0.0691081\pi$
$684$	0	0
$685$	77.2737	2.95247
$686$	41.2097	1.57339
$687$	0	0
$688$	−8.14952	−0.310698
$689$	0	0
$690$	0	0
$691$	24.7144i	0.940179i	0.882619	+	0.470090i	$0.155778\pi$
$691$	24.7144i	0.940179i	−0.882619	+	0.470090i	$0.844222\pi$
$692$	39.6887	1.50874
$693$	0	0
$694$	− 5.54048i	− 0.210314i
$695$	3.29829i	0.125111i
$696$	0	0
$697$	6.32736i	0.239666i
$698$	−0.851329	−0.0322233
$699$	0	0
$700$	30.9614i	1.17023i
$701$	25.8920	0.977927	0.488964	−	0.872304i	$-0.337375\pi$
$701$	25.8920	0.977927	0.488964	+	0.872304i	$0.337375\pi$
$702$	0	0
$703$	−33.4644	−1.26213
$704$	− 46.9120i	− 1.76806i
$705$	0	0
$706$	−65.9079	−2.48048
$707$	− 22.3545i	− 0.840728i
$708$	0	0
$709$	0.0851621i	0.00319833i	0.999999	+	0.00159916i	$0.000509030\pi$
$709$	0.0851621i	0.00319833i	−0.999999	+	0.00159916i	$0.999491\pi$
$710$	16.3605i	0.613998i
$711$	0	0
$712$	−4.08192	−0.152976
$713$	17.8979i	0.670280i
$714$	0	0
$715$	0	0
$716$	43.6760	1.63225
$717$	0	0
$718$	−45.7976	−1.70915
$719$	27.0508	1.00883	0.504413	−	0.863463i	$-0.331709\pi$
$719$	27.0508	1.00883	0.504413	+	0.863463i	$0.331709\pi$
$720$	0	0
$721$	24.6896i	0.919490i
$722$	− 9.31288i	− 0.346590i
$723$	0	0
$724$	−22.1626	−0.823665
$725$	−24.6926	−0.917061
$726$	0	0
$727$	47.1584	1.74901	0.874503	−	0.485019i	$-0.161187\pi$
$727$	47.1584	1.74901	0.874503	+	0.485019i	$0.161187\pi$
$728$	0	0
$729$	0	0
$730$	50.9742i	1.88664i
$731$	2.08469	0.0771050
$732$	0	0
$733$	9.68186i	0.357608i	0.983885	+	0.178804i	$0.0572227\pi$
$733$	9.68186i	0.357608i	−0.983885	+	0.178804i	$0.942777\pi$
$734$	4.71784i	0.174139i
$735$	0	0
$736$	16.4509i	0.606388i
$737$	7.51275	0.276736
$738$	0	0
$739$	36.3139i	1.33583i	0.744238	+	0.667915i	$0.232813\pi$
$739$	36.3139i	1.33583i	−0.744238	+	0.667915i	$0.767187\pi$
$740$	64.9466	2.38748
$741$	0	0
$742$	−2.49934	−0.0917535
$743$	− 41.7536i	− 1.53179i	−0.642965	−	0.765896i	$-0.722296\pi$
$743$	− 41.7536i	− 1.53179i	0.642965	−	0.765896i	$-0.277704\pi$
$744$	0	0
$745$	−34.4386	−1.26173
$746$	39.4950i	1.44602i
$747$	0	0
$748$	9.89930i	0.361954i
$749$	22.1836i	0.810571i
$750$	0	0
$751$	22.1927	0.809823	0.404911	−	0.914356i	$-0.367302\pi$
$751$	22.1927	0.809823	0.404911	+	0.914356i	$0.367302\pi$
$752$	13.5526i	0.494211i
$753$	0	0
$754$	0	0
$755$	67.7036	2.46399
$756$	0	0
$757$	9.23729	0.335735	0.167868	−	0.985810i	$-0.446312\pi$
$757$	9.23729	0.335735	0.167868	+	0.985810i	$0.446312\pi$
$758$	15.0379	0.546201
$759$	0	0
$760$	− 5.17928i	− 0.187872i
$761$	7.50173i	0.271937i	0.990713	+	0.135969i	$0.0434147\pi$
$761$	7.50173i	0.271937i	−0.990713	+	0.135969i	$0.956585\pi$
$762$	0	0
$763$	45.4892	1.64682
$764$	−14.4797	−0.523857
$765$	0	0
$766$	39.1140	1.41325
$767$	0	0
$768$	0	0
$769$	− 5.14005i	− 0.185355i	−0.995696	−	0.0926774i	$-0.970457\pi$
$769$	− 5.14005i	− 0.185355i	0.995696	−	0.0926774i	$-0.0295425\pi$
$770$	76.3309	2.75078
$771$	0	0
$772$	− 23.8869i	− 0.859708i
$773$	− 8.50173i	− 0.305786i	−0.988243	−	0.152893i	$-0.951141\pi$
$773$	− 8.50173i	− 0.305786i	0.988243	−	0.152893i	$-0.0488590\pi$
$774$	0	0
$775$	− 55.3473i	− 1.98813i
$776$	−6.54548	−0.234969
$777$	0	0
$778$	49.1493i	1.76209i
$779$	−26.3817	−0.945221
$780$	0	0
$781$	11.7482	0.420385
$782$	− 3.78735i	− 0.135435i
$783$	0	0
$784$	−6.95492	−0.248390
$785$	35.0404i	1.25065i
$786$	0	0
$787$	− 7.78554i	− 0.277525i	−0.990326	−	0.138762i	$-0.955688\pi$
$787$	− 7.78554i	− 0.277525i	0.990326	−	0.138762i	$-0.0443124\pi$
$788$	− 20.3136i	− 0.723643i
$789$	0	0
$790$	−25.7302	−0.915439
$791$	− 21.7778i	− 0.774329i
$792$	0	0
$793$	0	0
$794$	−9.05190	−0.321240
$795$	0	0
$796$	−3.43919	−0.121899
$797$	21.3840	0.757462	0.378731	−	0.925507i	$-0.376361\pi$
$797$	21.3840	0.757462	0.378731	+	0.925507i	$0.376361\pi$
$798$	0	0
$799$	− 3.46681i	− 0.122647i
$800$	− 50.8727i	− 1.79862i
$801$	0	0
$802$	41.8159	1.47657
$803$	36.6039	1.29172
$804$	0	0
$805$	−15.2905	−0.538920
$806$	0	0
$807$	0	0
$808$	− 4.03731i	− 0.142032i
$809$	−2.28621	−0.0803788	−0.0401894	−	0.999192i	$-0.512796\pi$
$809$	−2.28621	−0.0803788	−0.0401894	+	0.999192i	$0.512796\pi$
$810$	0	0
$811$	− 17.8079i	− 0.625320i	−0.949865	−	0.312660i	$-0.898780\pi$
$811$	− 17.8079i	− 0.625320i	0.949865	−	0.312660i	$-0.101220\pi$
$812$	− 19.4547i	− 0.682727i
$813$	0	0
$814$	− 89.0721i	− 3.12198i
$815$	37.0820	1.29893
$816$	0	0
$817$	8.69202i	0.304095i
$818$	45.3818	1.58674
$819$	0	0
$820$	51.2006	1.78800
$821$	13.9054i	0.485302i	0.970114	+	0.242651i	$0.0780170\pi$
$821$	13.9054i	0.485302i	−0.970114	+	0.242651i	$0.921983\pi$
$822$	0	0
$823$	7.24831	0.252660	0.126330	−	0.991988i	$-0.459680\pi$
$823$	7.24831	0.252660	0.126330	+	0.991988i	$0.459680\pi$
$824$	4.45904i	0.155338i
$825$	0	0
$826$	21.7017i	0.755099i
$827$	− 54.1191i	− 1.88191i	−0.338536	−	0.940953i	$-0.609932\pi$
$827$	− 54.1191i	− 1.88191i	0.338536	−	0.940953i	$-0.390068\pi$
$828$	0	0
$829$	5.79178	0.201157	0.100578	−	0.994929i	$-0.467931\pi$
$829$	5.79178	0.201157	0.100578	+	0.994929i	$0.467931\pi$
$830$	15.8743i	0.551004i
$831$	0	0
$832$	0	0
$833$	1.77910	0.0616422
$834$	0	0
$835$	27.2241	0.942130
$836$	−41.2747	−1.42752
$837$	0	0
$838$	30.2338i	1.04441i
$839$	5.29350i	0.182752i	0.995816	+	0.0913760i	$0.0291265\pi$
$839$	5.29350i	0.182752i	−0.995816	+	0.0913760i	$0.970873\pi$
$840$	0	0
$841$	−13.4843	−0.464975
$842$	−17.3588	−0.598224
$843$	0	0
$844$	−51.0930	−1.75870
$845$	0	0
$846$	0	0
$847$	− 30.0954i	− 1.03409i
$848$	1.93516	0.0664538
$849$	0	0
$850$	11.7120i	0.401718i
$851$	17.8428i	0.611644i
$852$	0	0
$853$	− 13.5961i	− 0.465522i	−0.972534	−	0.232761i	$-0.925224\pi$
$853$	− 13.5961i	− 0.465522i	0.972534	−	0.232761i	$-0.0747760\pi$
$854$	16.9420	0.579743
$855$	0	0
$856$	4.00644i	0.136937i
$857$	−23.8323	−0.814097	−0.407048	−	0.913407i	$-0.633442\pi$
$857$	−23.8323	−0.814097	−0.407048	+	0.913407i	$0.633442\pi$
$858$	0	0
$859$	−26.9861	−0.920754	−0.460377	−	0.887723i	$-0.652286\pi$
$859$	−26.9861	−0.920754	−0.460377	+	0.887723i	$0.652286\pi$
$860$	− 16.8692i	− 0.575234i
$861$	0	0
$862$	4.90063	0.166916
$863$	− 27.0291i	− 0.920080i	−0.887898	−	0.460040i	$-0.847835\pi$
$863$	− 27.0291i	− 0.920080i	0.887898	−	0.460040i	$-0.152165\pi$
$864$	0	0
$865$	− 60.6128i	− 2.06090i
$866$	− 21.3674i	− 0.726095i
$867$	0	0
$868$	43.6069	1.48011
$869$	18.4765i	0.626772i
$870$	0	0
$871$	0	0
$872$	8.21552	0.278213
$873$	0	0
$874$	15.7912	0.534145
$875$	9.57002	0.323526
$876$	0	0
$877$	− 40.8780i	− 1.38035i	−0.723642	−	0.690176i	$-0.757533\pi$
$877$	− 40.8780i	− 1.38035i	0.723642	−	0.690176i	$-0.242467\pi$
$878$	− 66.6926i	− 2.25077i
$879$	0	0
$880$	−59.1008	−1.99229
$881$	−22.4964	−0.757921	−0.378961	−	0.925413i	$-0.623718\pi$
$881$	−22.4964	−0.757921	−0.378961	+	0.925413i	$0.623718\pi$
$882$	0	0
$883$	4.16315	0.140101	0.0700505	−	0.997543i	$-0.477684\pi$
$883$	4.16315	0.140101	0.0700505	+	0.997543i	$0.477684\pi$
$884$	0	0
$885$	0	0
$886$	19.6329i	0.659582i
$887$	26.3002	0.883075	0.441537	−	0.897243i	$-0.354433\pi$
$887$	26.3002	0.883075	0.441537	+	0.897243i	$0.354433\pi$
$888$	0	0
$889$	− 31.0344i	− 1.04086i
$890$	69.1831i	2.31902i
$891$	0	0
$892$	23.6040i	0.790320i
$893$	14.4547	0.483709
$894$	0	0
$895$	− 66.7023i	− 2.22961i
$896$	7.25906	0.242508
$897$	0	0
$898$	58.1764	1.94137
$899$	34.7778i	1.15990i
$900$	0	0
$901$	−0.495024	−0.0164916
$902$	− 70.2199i	− 2.33807i
$903$	0	0
$904$	− 3.93315i	− 0.130815i
$905$	33.8468i	1.12511i
$906$	0	0
$907$	57.9114	1.92292	0.961458	−	0.274952i	$-0.0886620\pi$
$907$	57.9114	1.92292	0.961458	+	0.274952i	$0.0886620\pi$
$908$	15.3333i	0.508854i
$909$	0	0
$910$	0	0
$911$	0.286799	0.00950208	0.00475104	−	0.999989i	$-0.498488\pi$
$911$	0.286799	0.00950208	0.00475104	+	0.999989i	$0.498488\pi$
$912$	0	0
$913$	11.3991	0.377255
$914$	−10.2358	−0.338569
$915$	0	0
$916$	36.9383i	1.22047i
$917$	− 6.72886i	− 0.222206i
$918$	0	0
$919$	−31.1239	−1.02668	−0.513342	−	0.858184i	$-0.671593\pi$
$919$	−31.1239	−1.02668	−0.513342	+	0.858184i	$0.671593\pi$
$920$	−2.76153	−0.0910449
$921$	0	0
$922$	2.77133	0.0912690
$923$	0	0
$924$	0	0
$925$	− 55.1771i	− 1.81421i
$926$	−6.89392	−0.226548
$927$	0	0
$928$	31.9661i	1.04934i
$929$	7.62671i	0.250224i	0.992143	+	0.125112i	$0.0399291\pi$
$929$	7.62671i	0.250224i	−0.992143	+	0.125112i	$0.960071\pi$
$930$	0	0
$931$	7.41789i	0.243112i
$932$	−19.1056	−0.625825
$933$	0	0
$934$	− 14.1596i	− 0.463317i
$935$	15.1183	0.494421
$936$	0	0
$937$	−5.67324	−0.185337	−0.0926683	−	0.995697i	$-0.529540\pi$
$937$	−5.67324	−0.185337	−0.0926683	+	0.995697i	$0.529540\pi$
$938$	7.00298i	0.228656i
$939$	0	0
$940$	−28.0532	−0.914995
$941$	41.5394i	1.35415i	0.735916	+	0.677073i	$0.236752\pi$
$941$	41.5394i	1.35415i	−0.735916	+	0.677073i	$0.763248\pi$
$942$	0	0
$943$	14.0664i	0.458064i
$944$	− 16.8030i	− 0.546891i
$945$	0	0
$946$	−23.1355	−0.752201
$947$	47.3110i	1.53740i	0.639610	+	0.768700i	$0.279096\pi$
$947$	47.3110i	1.53740i	−0.639610	+	0.768700i	$0.720904\pi$
$948$	0	0
$949$	0	0
$950$	−48.8327	−1.58434
$951$	0	0
$952$	−0.831478	−0.0269483
$953$	−34.3435	−1.11249	−0.556247	−	0.831017i	$-0.687759\pi$
$953$	−34.3435	−1.11249	−0.556247	+	0.831017i	$0.687759\pi$
$954$	0	0
$955$	22.1135i	0.715576i
$956$	50.3777i	1.62933i
$957$	0	0
$958$	−7.20344	−0.232733
$959$	51.7241	1.67026
$960$	0	0
$961$	−46.9527	−1.51460
$962$	0	0
$963$	0	0
$964$	− 48.3137i	− 1.55608i
$965$	−36.4802	−1.17434
$966$	0	0
$967$	− 48.5096i	− 1.55996i	−0.625802	−	0.779982i	$-0.715228\pi$
$967$	− 48.5096i	− 1.55996i	0.625802	−	0.779982i	$-0.284772\pi$
$968$	− 5.43535i	− 0.174699i
$969$	0	0
$970$	110.937i	3.56198i
$971$	−41.4650	−1.33068	−0.665338	−	0.746542i	$-0.731712\pi$
$971$	−41.4650	−1.33068	−0.665338	+	0.746542i	$0.731712\pi$
$972$	0	0
$973$	2.20775i	0.0707772i
$974$	26.3242	0.843483
$975$	0	0
$976$	−13.1177	−0.419887
$977$	− 2.09677i	− 0.0670816i	−0.999437	−	0.0335408i	$-0.989322\pi$
$977$	− 2.09677i	− 0.0670816i	0.999437	−	0.0335408i	$-0.0106784\pi$
$978$	0	0
$979$	49.6795	1.58776
$980$	− 14.3964i	− 0.459876i
$981$	0	0
$982$	− 58.7442i	− 1.87460i
$983$	− 25.2336i	− 0.804826i	−0.915458	−	0.402413i	$-0.868172\pi$
$983$	− 25.2336i	− 0.804826i	0.915458	−	0.402413i	$-0.131828\pi$
$984$	0	0
$985$	−31.0231	−0.988478
$986$	− 7.35929i	− 0.234367i
$987$	0	0
$988$	0	0
$989$	4.63448	0.147368
$990$	0	0
$991$	12.0489	0.382746	0.191373	−	0.981517i	$-0.438706\pi$
$991$	12.0489	0.382746	0.191373	+	0.981517i	$0.438706\pi$
$992$	−71.6506	−2.27491
$993$	0	0
$994$	10.9511i	0.347347i
$995$	5.25236i	0.166511i
$996$	0	0
$997$	−1.43403	−0.0454160	−0.0227080	−	0.999742i	$-0.507229\pi$
$997$	−1.43403	−0.0454160	−0.0227080	+	0.999742i	$0.507229\pi$
$998$	68.7525	2.17632
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.k.1351.3		6
3.2	odd	2		507.2.b.f.337.4		6
13.5	odd	4		1521.2.a.s.1.1		3
13.8	odd	4		1521.2.a.n.1.3		3
13.12	even	2	inner	1521.2.b.k.1351.4		6
39.2	even	12		507.2.e.l.22.1		6
39.5	even	4		507.2.a.i.1.3	✓	3
39.8	even	4		507.2.a.l.1.1	yes	3
39.11	even	12		507.2.e.i.22.3		6
39.17	odd	6		507.2.j.i.361.3		12
39.20	even	12		507.2.e.i.484.3		6
39.23	odd	6		507.2.j.i.316.4		12
39.29	odd	6		507.2.j.i.316.3		12
39.32	even	12		507.2.e.l.484.1		6
39.35	odd	6		507.2.j.i.361.4		12
39.38	odd	2		507.2.b.f.337.3		6
156.47	odd	4		8112.2.a.cp.1.2		3
156.83	odd	4		8112.2.a.cg.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.3	✓	3	39.5	even	4
507.2.a.l.1.1	yes	3	39.8	even	4
507.2.b.f.337.3		6	39.38	odd	2
507.2.b.f.337.4		6	3.2	odd	2
507.2.e.i.22.3		6	39.11	even	12
507.2.e.i.484.3		6	39.20	even	12
507.2.e.l.22.1		6	39.2	even	12
507.2.e.l.484.1		6	39.32	even	12
507.2.j.i.316.3		12	39.29	odd	6
507.2.j.i.316.4		12	39.23	odd	6
507.2.j.i.361.3		12	39.17	odd	6
507.2.j.i.361.4		12	39.35	odd	6
1521.2.a.n.1.3		3	13.8	odd	4
1521.2.a.s.1.1		3	13.5	odd	4
1521.2.b.k.1351.3		6	1.1	even	1	trivial
1521.2.b.k.1351.4		6	13.12	even	2	inner
8112.2.a.cg.1.2		3	156.83	odd	4
8112.2.a.cp.1.2		3	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)

\(f(q)\)	\(=\)	\(q-2.04892i q^{2} -2.19806 q^{4} +3.35690i q^{5} +2.24698i q^{7} +0.405813i q^{8} +O(q^{10})\)	\(q-2.04892i q^{2} -2.19806 q^{4} +3.35690i q^{5} +2.24698i q^{7} +0.405813i q^{8} +6.87800 q^{10} -4.93900i q^{11} +4.60388 q^{14} -3.56465 q^{16} +0.911854 q^{17} +3.80194i q^{19} -7.37867i q^{20} -10.1196 q^{22} +2.02715 q^{23} -6.26875 q^{25} -4.93900i q^{28} +3.93900 q^{29} +8.82908i q^{31} +8.11529i q^{32} -1.86831i q^{34} -7.54288 q^{35} +8.80194i q^{37} +7.78986 q^{38} -1.36227 q^{40} +6.93900i q^{41} +2.28621 q^{43} +10.8562i q^{44} -4.15346i q^{46} -3.80194i q^{47} +1.95108 q^{49} +12.8442i q^{50} -0.542877 q^{53} +16.5797 q^{55} -0.911854 q^{56} -8.07069i q^{58} +4.71379i q^{59} +3.67994 q^{61} +18.0901 q^{62} +9.49827 q^{64} +1.52111i q^{67} -2.00431 q^{68} +15.4547i q^{70} +2.37867i q^{71} +7.41119i q^{73} +18.0344 q^{74} -8.35690i q^{76} +11.0978 q^{77} -3.74094 q^{79} -11.9661i q^{80} +14.2174 q^{82} +2.30798i q^{83} +3.06100i q^{85} -4.68425i q^{86} +2.00431 q^{88} +10.0586i q^{89} -4.45580 q^{92} -7.78986 q^{94} -12.7627 q^{95} +16.1293i q^{97} -3.99761i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 22 q^{4}+O(q^{10}) \) 6 * q - 22 * q^4	\( 6 q - 22 q^{4} + 2 q^{10} + 10 q^{14} + 22 q^{16} - 2 q^{17} - 18 q^{22} - 22 q^{25} + 4 q^{29} - 8 q^{35} + 6 q^{40} + 30 q^{43} + 30 q^{49} + 34 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 4 q^{62} + 26 q^{68} - 30 q^{74} + 30 q^{77} + 6 q^{79} + 6 q^{82} - 26 q^{88} - 14 q^{92} - 42 q^{95}+O(q^{100}) \) 6 * q - 22 * q^4 + 2 * q^10 + 10 * q^14 + 22 * q^16 - 2 * q^17 - 18 * q^22 - 22 * q^25 + 4 * q^29 - 8 * q^35 + 6 * q^40 + 30 * q^43 + 30 * q^49 + 34 * q^53 + 6 * q^55 + 2 * q^56 - 26 * q^61 - 4 * q^62 + 26 * q^68 - 30 * q^74 + 30 * q^77 + 6 * q^79 + 6 * q^82 - 26 * q^88 - 14 * q^92 - 42 * q^95

Introduction

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