LMFDB - Embedded newform 1521.2.b.m.1351.4

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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.4
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.m.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445042i q^{2} +1.80194 q^{4} -0.246980i q^{5} +1.75302i q^{7} +1.69202i q^{8} +O(q^{10})$	$q+0.445042i q^{2} +1.80194 q^{4} -0.246980i q^{5} +1.75302i q^{7} +1.69202i q^{8} +0.109916 q^{10} -5.65279i q^{11} -0.780167 q^{14} +2.85086 q^{16} -3.80194 q^{17} -5.58211i q^{19} -0.445042i q^{20} +2.51573 q^{22} +8.34481 q^{23} +4.93900 q^{25} +3.15883i q^{28} +5.93900 q^{29} -5.26875i q^{31} +4.65279i q^{32} -1.69202i q^{34} +0.432960 q^{35} +3.19806i q^{37} +2.48427 q^{38} +0.417895 q^{40} +0.445042i q^{41} -1.71379 q^{43} -10.1860i q^{44} +3.71379i q^{46} +6.73556i q^{47} +3.92692 q^{49} +2.19806i q^{50} +1.06100 q^{53} -1.39612 q^{55} -2.96615 q^{56} +2.64310i q^{58} +13.7017i q^{59} -8.51573 q^{61} +2.34481 q^{62} +3.63102 q^{64} -5.96077i q^{67} -6.85086 q^{68} +0.192685i q^{70} -5.71917i q^{71} +7.35690i q^{73} -1.42327 q^{74} -10.0586i q^{76} +9.90946 q^{77} +4.45473 q^{79} -0.704103i q^{80} -0.198062 q^{82} -10.1860i q^{83} +0.939001i q^{85} -0.762709i q^{86} +9.56465 q^{88} -0.137063i q^{89} +15.0368 q^{92} -2.99761 q^{94} -1.37867 q^{95} +13.6896i q^{97} +1.74764i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{4}+O(q^{10}) $ 6 * q + 2 * q^4	$ 6 q + 2 q^{4} + 2 q^{10} - 2 q^{14} - 10 q^{16} - 14 q^{17} - 10 q^{22} + 4 q^{23} + 10 q^{25} + 16 q^{29} - 36 q^{35} + 40 q^{38} + 14 q^{40} + 6 q^{43} - 34 q^{49} + 26 q^{53} - 26 q^{55} + 14 q^{56} - 26 q^{61} - 32 q^{62} - 8 q^{64} - 14 q^{68} - 14 q^{74} - 30 q^{77} - 18 q^{79} - 10 q^{82} + 14 q^{88} + 34 q^{92} + 64 q^{94} + 6 q^{95}+O(q^{100}) $ 6 * q + 2 * q^4 + 2 * q^10 - 2 * q^14 - 10 * q^16 - 14 * q^17 - 10 * q^22 + 4 * q^23 + 10 * q^25 + 16 * q^29 - 36 * q^35 + 40 * q^38 + 14 * q^40 + 6 * q^43 - 34 * q^49 + 26 * q^53 - 26 * q^55 + 14 * q^56 - 26 * q^61 - 32 * q^62 - 8 * q^64 - 14 * q^68 - 14 * q^74 - 30 * q^77 - 18 * q^79 - 10 * q^82 + 14 * q^88 + 34 * q^92 + 64 * q^94 + 6 * 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 $

$2$	0.445042i	0.314692i	0.987544	+	0.157346i	$0.0502938\pi$
$2$	0.445042i	0.314692i	−0.987544	+	0.157346i	$0.949706\pi$
$3$	0	0
$3$	0	0
$4$	1.80194	0.900969
$5$	− 0.246980i	− 0.110453i	−0.998474	−	0.0552263i	$-0.982412\pi$
$5$	− 0.246980i	− 0.110453i	0.998474	−	0.0552263i	$-0.0175880\pi$
$6$	0	0
$7$	1.75302i	0.662579i	0.943529	+	0.331290i	$0.107484\pi$
$7$	1.75302i	0.662579i	−0.943529	+	0.331290i	$0.892516\pi$
$8$	1.69202i	0.598220i
$9$	0	0
$10$	0.109916	0.0347586
$11$	− 5.65279i	− 1.70438i	−0.523232	−	0.852191i	$-0.675274\pi$
$11$	− 5.65279i	− 1.70438i	0.523232	−	0.852191i	$-0.324726\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−0.780167	−0.208509
$15$	0	0
$16$	2.85086	0.712714
$17$	−3.80194	−0.922105	−0.461053	−	0.887373i	$-0.652528\pi$
$17$	−3.80194	−0.922105	−0.461053	+	0.887373i	$0.652528\pi$
$18$	0	0
$19$	− 5.58211i	− 1.28062i	−0.768115	−	0.640311i	$-0.778805\pi$
$19$	− 5.58211i	− 1.28062i	0.768115	−	0.640311i	$-0.221195\pi$
$20$	− 0.445042i	− 0.0995144i
$21$	0	0
$22$	2.51573	0.536355
$23$	8.34481	1.74001	0.870007	−	0.493039i	$-0.164114\pi$
$23$	8.34481	1.74001	0.870007	+	0.493039i	$0.164114\pi$
$24$	0	0
$25$	4.93900	0.987800
$26$	0	0
$27$	0	0
$28$	3.15883i	0.596963i
$29$	5.93900	1.10284	0.551422	−	0.834226i	$-0.314085\pi$
$29$	5.93900	1.10284	0.551422	+	0.834226i	$0.314085\pi$
$30$	0	0
$31$	− 5.26875i	− 0.946295i	−0.880983	−	0.473148i	$-0.843118\pi$
$31$	− 5.26875i	− 0.946295i	0.880983	−	0.473148i	$-0.156882\pi$
$32$	4.65279i	0.822505i
$33$	0	0
$34$	− 1.69202i	− 0.290179i
$35$	0.432960	0.0731836
$36$	0	0
$37$	3.19806i	0.525758i	0.964829	+	0.262879i	$0.0846720\pi$
$37$	3.19806i	0.525758i	−0.964829	+	0.262879i	$0.915328\pi$
$38$	2.48427	0.403002
$39$	0	0
$40$	0.417895	0.0660750
$41$	0.445042i	0.0695039i	0.999396	+	0.0347519i	$0.0110641\pi$
$41$	0.445042i	0.0695039i	−0.999396	+	0.0347519i	$0.988936\pi$
$42$	0	0
$43$	−1.71379	−0.261351	−0.130675	−	0.991425i	$-0.541715\pi$
$43$	−1.71379	−0.261351	−0.130675	+	0.991425i	$0.541715\pi$
$44$	− 10.1860i	− 1.53559i
$45$	0	0
$46$	3.71379i	0.547569i
$47$	6.73556i	0.982483i	0.871024	+	0.491241i	$0.163457\pi$
$47$	6.73556i	0.982483i	−0.871024	+	0.491241i	$0.836543\pi$
$48$	0	0
$49$	3.92692	0.560988
$50$	2.19806i	0.310853i
$51$	0	0
$52$	0	0
$53$	1.06100	0.145739	0.0728697	−	0.997341i	$-0.476784\pi$
$53$	1.06100	0.145739	0.0728697	+	0.997341i	$0.476784\pi$
$54$	0	0
$55$	−1.39612	−0.188253
$56$	−2.96615	−0.396368
$57$	0	0
$58$	2.64310i	0.347057i
$59$	13.7017i	1.78381i	0.452222	+	0.891905i	$0.350631\pi$
$59$	13.7017i	1.78381i	−0.452222	+	0.891905i	$0.649369\pi$
$60$	0	0
$61$	−8.51573	−1.09033	−0.545164	−	0.838330i	$-0.683533\pi$
$61$	−8.51573	−1.09033	−0.545164	+	0.838330i	$0.683533\pi$
$62$	2.34481	0.297792
$63$	0	0
$64$	3.63102	0.453878
$65$	0	0
$66$	0	0
$67$	− 5.96077i	− 0.728224i	−0.931355	−	0.364112i	$-0.881373\pi$
$67$	− 5.96077i	− 0.728224i	0.931355	−	0.364112i	$-0.118627\pi$
$68$	−6.85086	−0.830788
$69$	0	0
$70$	0.192685i	0.0230303i
$71$	− 5.71917i	− 0.678740i	−0.940653	−	0.339370i	$-0.889786\pi$
$71$	− 5.71917i	− 0.678740i	0.940653	−	0.339370i	$-0.110214\pi$
$72$	0	0
$73$	7.35690i	0.861060i	0.902576	+	0.430530i	$0.141673\pi$
$73$	7.35690i	0.861060i	−0.902576	+	0.430530i	$0.858327\pi$
$74$	−1.42327	−0.165452
$75$	0	0
$76$	− 10.0586i	− 1.15380i
$77$	9.90946	1.12929
$78$	0	0
$79$	4.45473	0.501196	0.250598	−	0.968091i	$-0.419373\pi$
$79$	4.45473	0.501196	0.250598	+	0.968091i	$0.419373\pi$
$80$	− 0.704103i	− 0.0787211i
$81$	0	0
$82$	−0.198062	−0.0218723
$83$	− 10.1860i	− 1.11806i	−0.829149	−	0.559028i	$-0.811174\pi$
$83$	− 10.1860i	− 1.11806i	0.829149	−	0.559028i	$-0.188826\pi$
$84$	0	0
$85$	0.939001i	0.101849i
$86$	− 0.762709i	− 0.0822450i
$87$	0	0
$88$	9.56465	1.01959
$89$	− 0.137063i	− 0.0145287i	−0.999974	−	0.00726434i	$-0.997688\pi$
$89$	− 0.137063i	− 0.0145287i	0.999974	−	0.00726434i	$-0.00231233\pi$
$90$	0	0
$91$	0	0
$92$	15.0368	1.56770
$93$	0	0
$94$	−2.99761	−0.309180
$95$	−1.37867	−0.141448
$96$	0	0
$97$	13.6896i	1.38997i	0.719024	+	0.694986i	$0.244589\pi$
$97$	13.6896i	1.38997i	−0.719024	+	0.694986i	$0.755411\pi$
$98$	1.74764i	0.176539i
$99$	0	0
$100$	8.89977	0.889977
$101$	−5.41119	−0.538434	−0.269217	−	0.963080i	$-0.586765\pi$
$101$	−5.41119	−0.538434	−0.269217	+	0.963080i	$0.586765\pi$
$102$	0	0
$103$	−13.7560	−1.35542	−0.677710	−	0.735330i	$-0.737027\pi$
$103$	−13.7560	−1.35542	−0.677710	+	0.735330i	$0.737027\pi$
$104$	0	0
$105$	0	0
$106$	0.472189i	0.0458630i
$107$	12.8170	1.23907	0.619533	−	0.784970i	$-0.287322\pi$
$107$	12.8170	1.23907	0.619533	+	0.784970i	$0.287322\pi$
$108$	0	0
$109$	− 12.1468i	− 1.16345i	−0.813386	−	0.581724i	$-0.802378\pi$
$109$	− 12.1468i	− 1.16345i	0.813386	−	0.581724i	$-0.197622\pi$
$110$	− 0.621334i	− 0.0592419i
$111$	0	0
$112$	4.99761i	0.472229i
$113$	1.63773	0.154064	0.0770322	−	0.997029i	$-0.475456\pi$
$113$	1.63773	0.154064	0.0770322	+	0.997029i	$0.475456\pi$
$114$	0	0
$115$	− 2.06100i	− 0.192189i
$116$	10.7017	0.993629
$117$	0	0
$118$	−6.09783	−0.561351
$119$	− 6.66487i	− 0.610968i
$120$	0	0
$121$	−20.9541	−1.90492
$122$	− 3.78986i	− 0.343117i
$123$	0	0
$124$	− 9.49396i	− 0.852583i
$125$	− 2.45473i	− 0.219558i
$126$	0	0
$127$	−10.7995	−0.958305	−0.479152	−	0.877732i	$-0.659056\pi$
$127$	−10.7995	−0.958305	−0.479152	+	0.877732i	$0.659056\pi$
$128$	10.9215i	0.965337i
$129$	0	0
$130$	0	0
$131$	−0.907542	−0.0792923	−0.0396462	−	0.999214i	$-0.512623\pi$
$131$	−0.907542	−0.0792923	−0.0396462	+	0.999214i	$0.512623\pi$
$132$	0	0
$133$	9.78554	0.848514
$134$	2.65279	0.229166
$135$	0	0
$136$	− 6.43296i	− 0.551622i
$137$	9.54825i	0.815762i	0.913035	+	0.407881i	$0.133732\pi$
$137$	9.54825i	0.815762i	−0.913035	+	0.407881i	$0.866268\pi$
$138$	0	0
$139$	−4.09246	−0.347118	−0.173559	−	0.984823i	$-0.555527\pi$
$139$	−4.09246	−0.347118	−0.173559	+	0.984823i	$0.555527\pi$
$140$	0.780167	0.0659362
$141$	0	0
$142$	2.54527	0.213594
$143$	0	0
$144$	0	0
$145$	− 1.46681i	− 0.121812i
$146$	−3.27413	−0.270969
$147$	0	0
$148$	5.76271i	0.473692i
$149$	− 15.3884i	− 1.26066i	−0.776326	−	0.630332i	$-0.782919\pi$
$149$	− 15.3884i	− 1.26066i	0.776326	−	0.630332i	$-0.217081\pi$
$150$	0	0
$151$	− 3.67456i	− 0.299032i	−0.988759	−	0.149516i	$-0.952228\pi$
$151$	− 3.67456i	− 0.299032i	0.988759	−	0.149516i	$-0.0477715\pi$
$152$	9.44504	0.766094
$153$	0	0
$154$	4.41013i	0.355378i
$155$	−1.30127	−0.104521
$156$	0	0
$157$	−4.87800	−0.389307	−0.194653	−	0.980872i	$-0.562358\pi$
$157$	−4.87800	−0.389307	−0.194653	+	0.980872i	$0.562358\pi$
$158$	1.98254i	0.157723i
$159$	0	0
$160$	1.14914	0.0908479
$161$	14.6286i	1.15290i
$162$	0	0
$163$	8.63102i	0.676034i	0.941140	+	0.338017i	$0.109756\pi$
$163$	8.63102i	0.676034i	−0.941140	+	0.338017i	$0.890244\pi$
$164$	0.801938i	0.0626208i
$165$	0	0
$166$	4.53319	0.351844
$167$	− 9.46980i	− 0.732795i	−0.930458	−	0.366397i	$-0.880591\pi$
$167$	− 9.46980i	− 0.732795i	0.930458	−	0.366397i	$-0.119409\pi$
$168$	0	0
$169$	0	0
$170$	−0.417895	−0.0320511
$171$	0	0
$172$	−3.08815	−0.235469
$173$	4.77479	0.363021	0.181510	−	0.983389i	$-0.441901\pi$
$173$	4.77479	0.363021	0.181510	+	0.983389i	$0.441901\pi$
$174$	0	0
$175$	8.65817i	0.654496i
$176$	− 16.1153i	− 1.21474i
$177$	0	0
$178$	0.0609989	0.00457206
$179$	3.43535	0.256770	0.128385	−	0.991724i	$-0.459021\pi$
$179$	3.43535	0.256770	0.128385	+	0.991724i	$0.459021\pi$
$180$	0	0
$181$	13.4862	1.00242	0.501210	−	0.865326i	$-0.332888\pi$
$181$	13.4862	1.00242	0.501210	+	0.865326i	$0.332888\pi$
$182$	0	0
$183$	0	0
$184$	14.1196i	1.04091i
$185$	0.789856	0.0580714
$186$	0	0
$187$	21.4916i	1.57162i
$188$	12.1371i	0.885186i
$189$	0	0
$190$	− 0.613564i	− 0.0445126i
$191$	1.30127	0.0941569	0.0470784	−	0.998891i	$-0.485009\pi$
$191$	1.30127	0.0941569	0.0470784	+	0.998891i	$0.485009\pi$
$192$	0	0
$193$	− 9.19567i	− 0.661919i	−0.943645	−	0.330959i	$-0.892628\pi$
$193$	− 9.19567i	− 0.661919i	0.943645	−	0.330959i	$-0.107372\pi$
$194$	−6.09246	−0.437413
$195$	0	0
$196$	7.07606	0.505433
$197$	4.11231i	0.292990i	0.989211	+	0.146495i	$0.0467992\pi$
$197$	4.11231i	0.292990i	−0.989211	+	0.146495i	$0.953201\pi$
$198$	0	0
$199$	24.7724	1.75607	0.878034	−	0.478598i	$-0.158855\pi$
$199$	24.7724	1.75607	0.878034	+	0.478598i	$0.158855\pi$
$200$	8.35690i	0.590922i
$201$	0	0
$202$	− 2.40821i	− 0.169441i
$203$	10.4112i	0.730722i
$204$	0	0
$205$	0.109916	0.00767688
$206$	− 6.12200i	− 0.426540i
$207$	0	0
$208$	0	0
$209$	−31.5545	−2.18267
$210$	0	0
$211$	−5.93900	−0.408858	−0.204429	−	0.978881i	$-0.565534\pi$
$211$	−5.93900	−0.408858	−0.204429	+	0.978881i	$0.565534\pi$
$212$	1.91185	0.131307
$213$	0	0
$214$	5.70410i	0.389924i
$215$	0.423272i	0.0288669i
$216$	0	0
$217$	9.23623	0.626996
$218$	5.40581	0.366128
$219$	0	0
$220$	−2.51573	−0.169610
$221$	0	0
$222$	0	0
$223$	− 14.2010i	− 0.950972i	−0.879723	−	0.475486i	$-0.842272\pi$
$223$	− 14.2010i	− 0.950972i	0.879723	−	0.475486i	$-0.157728\pi$
$224$	−8.15644	−0.544975
$225$	0	0
$226$	0.728857i	0.0484829i
$227$	16.0073i	1.06244i	0.847233	+	0.531221i	$0.178267\pi$
$227$	16.0073i	1.06244i	−0.847233	+	0.531221i	$0.821733\pi$
$228$	0	0
$229$	1.84117i	0.121668i	0.998148	+	0.0608339i	$0.0193760\pi$
$229$	1.84117i	0.121668i	−0.998148	+	0.0608339i	$0.980624\pi$
$230$	0.917231	0.0604804
$231$	0	0
$232$	10.0489i	0.659744i
$233$	−23.4252	−1.53464	−0.767318	−	0.641267i	$-0.778409\pi$
$233$	−23.4252	−1.53464	−0.767318	+	0.641267i	$0.778409\pi$
$234$	0	0
$235$	1.66355	0.108518
$236$	24.6896i	1.60716i
$237$	0	0
$238$	2.96615	0.192267
$239$	14.6015i	0.944491i	0.881467	+	0.472246i	$0.156556\pi$
$239$	14.6015i	0.944491i	−0.881467	+	0.472246i	$0.843444\pi$
$240$	0	0
$241$	8.63102i	0.555973i	0.960585	+	0.277987i	$0.0896671\pi$
$241$	8.63102i	0.555973i	−0.960585	+	0.277987i	$0.910333\pi$
$242$	− 9.32544i	− 0.599462i
$243$	0	0
$244$	−15.3448	−0.982351
$245$	− 0.969869i	− 0.0619627i
$246$	0	0
$247$	0	0
$248$	8.91484	0.566093
$249$	0	0
$250$	1.09246	0.0690931
$251$	−3.80194	−0.239976	−0.119988	−	0.992775i	$-0.538286\pi$
$251$	−3.80194	−0.239976	−0.119988	+	0.992775i	$0.538286\pi$
$252$	0	0
$253$	− 47.1715i	− 2.96565i
$254$	− 4.80625i	− 0.301571i
$255$	0	0
$256$	2.40150	0.150094
$257$	−20.5961	−1.28475	−0.642375	−	0.766391i	$-0.722051\pi$
$257$	−20.5961	−1.28475	−0.642375	+	0.766391i	$0.722051\pi$
$258$	0	0
$259$	−5.60627	−0.348357
$260$	0	0
$261$	0	0
$262$	− 0.403894i	− 0.0249527i
$263$	−0.332733	−0.0205172	−0.0102586	−	0.999947i	$-0.503265\pi$
$263$	−0.332733	−0.0205172	−0.0102586	+	0.999947i	$0.503265\pi$
$264$	0	0
$265$	− 0.262045i	− 0.0160973i
$266$	4.35498i	0.267021i
$267$	0	0
$268$	− 10.7409i	− 0.656107i
$269$	−27.3032	−1.66471	−0.832353	−	0.554247i	$-0.813006\pi$
$269$	−27.3032	−1.66471	−0.832353	+	0.554247i	$0.813006\pi$
$270$	0	0
$271$	27.9855i	1.70000i	0.526783	+	0.850000i	$0.323398\pi$
$271$	27.9855i	1.70000i	−0.526783	+	0.850000i	$0.676602\pi$
$272$	−10.8388	−0.657197
$273$	0	0
$274$	−4.24937	−0.256714
$275$	− 27.9191i	− 1.68359i
$276$	0	0
$277$	2.10321	0.126370	0.0631849	−	0.998002i	$-0.479874\pi$
$277$	2.10321	0.126370	0.0631849	+	0.998002i	$0.479874\pi$
$278$	− 1.82132i	− 0.109235i
$279$	0	0
$280$	0.732578i	0.0437799i
$281$	− 27.2349i	− 1.62470i	−0.583172	−	0.812349i	$-0.698189\pi$
$281$	− 27.2349i	− 1.62470i	0.583172	−	0.812349i	$-0.301811\pi$
$282$	0	0
$283$	−5.28382	−0.314090	−0.157045	−	0.987591i	$-0.550197\pi$
$283$	−5.28382	−0.314090	−0.157045	+	0.987591i	$0.550197\pi$
$284$	− 10.3056i	− 0.611524i
$285$	0	0
$286$	0	0
$287$	−0.780167	−0.0460518
$288$	0	0
$289$	−2.54527	−0.149722
$290$	0.652793	0.0383333
$291$	0	0
$292$	13.2567i	0.775788i
$293$	− 32.6625i	− 1.90816i	−0.299548	−	0.954081i	$-0.596836\pi$
$293$	− 32.6625i	− 1.90816i	0.299548	−	0.954081i	$-0.403164\pi$
$294$	0	0
$295$	3.38404	0.197027
$296$	−5.41119	−0.314519
$297$	0	0
$298$	6.84846	0.396721
$299$	0	0
$300$	0	0
$301$	− 3.00431i	− 0.173166i
$302$	1.63533	0.0941029
$303$	0	0
$304$	− 15.9138i	− 0.912717i
$305$	2.10321i	0.120430i
$306$	0	0
$307$	− 20.7614i	− 1.18491i	−0.805602	−	0.592457i	$-0.798158\pi$
$307$	− 20.7614i	− 1.18491i	0.805602	−	0.592457i	$-0.201842\pi$
$308$	17.8562	1.01745
$309$	0	0
$310$	− 0.579121i	− 0.0328919i
$311$	−11.3013	−0.640836	−0.320418	−	0.947276i	$-0.603823\pi$
$311$	−11.3013	−0.640836	−0.320418	+	0.947276i	$0.603823\pi$
$312$	0	0
$313$	−4.27173	−0.241453	−0.120726	−	0.992686i	$-0.538522\pi$
$313$	−4.27173	−0.241453	−0.120726	+	0.992686i	$0.538522\pi$
$314$	− 2.17092i	− 0.122512i
$315$	0	0
$316$	8.02715	0.451562
$317$	15.4776i	0.869307i	0.900598	+	0.434653i	$0.143129\pi$
$317$	15.4776i	0.869307i	−0.900598	+	0.434653i	$0.856871\pi$
$318$	0	0
$319$	− 33.5719i	− 1.87967i
$320$	− 0.896789i	− 0.0501320i
$321$	0	0
$322$	−6.51035	−0.362808
$323$	21.2228i	1.18087i
$324$	0	0
$325$	0	0
$326$	−3.84117	−0.212743
$327$	0	0
$328$	−0.753020	−0.0415786
$329$	−11.8076	−0.650973
$330$	0	0
$331$	− 6.06829i	− 0.333544i	−0.985996	−	0.166772i	$-0.946666\pi$
$331$	− 6.06829i	− 0.333544i	0.985996	−	0.166772i	$-0.0533343\pi$
$332$	− 18.3545i	− 1.00733i
$333$	0	0
$334$	4.21446	0.230605
$335$	−1.47219	−0.0804343
$336$	0	0
$337$	−12.1239	−0.660432	−0.330216	−	0.943905i	$-0.607122\pi$
$337$	−12.1239	−0.660432	−0.330216	+	0.943905i	$0.607122\pi$
$338$	0	0
$339$	0	0
$340$	1.69202i	0.0917627i
$341$	−29.7832	−1.61285
$342$	0	0
$343$	19.1551i	1.03428i
$344$	− 2.89977i	− 0.156345i
$345$	0	0
$346$	2.12498i	0.114240i
$347$	−23.1497	−1.24274	−0.621371	−	0.783516i	$-0.713424\pi$
$347$	−23.1497	−1.24274	−0.621371	+	0.783516i	$0.713424\pi$
$348$	0	0
$349$	− 22.1957i	− 1.18811i	−0.804426	−	0.594053i	$-0.797527\pi$
$349$	− 22.1957i	− 1.18811i	0.804426	−	0.594053i	$-0.202473\pi$
$350$	−3.85325	−0.205965
$351$	0	0
$352$	26.3013	1.40186
$353$	− 5.07069i	− 0.269885i	−0.990853	−	0.134943i	$-0.956915\pi$
$353$	− 5.07069i	− 0.269885i	0.990853	−	0.134943i	$-0.0430850\pi$
$354$	0	0
$355$	−1.41252	−0.0749687
$356$	− 0.246980i	− 0.0130899i
$357$	0	0
$358$	1.52888i	0.0808036i
$359$	16.6746i	0.880050i	0.897986	+	0.440025i	$0.145030\pi$
$359$	16.6746i	0.880050i	−0.897986	+	0.440025i	$0.854970\pi$
$360$	0	0
$361$	−12.1599	−0.639995
$362$	6.00192i	0.315454i
$363$	0	0
$364$	0	0
$365$	1.81700	0.0951063
$366$	0	0
$367$	−1.17928	−0.0615577	−0.0307789	−	0.999526i	$-0.509799\pi$
$367$	−1.17928	−0.0615577	−0.0307789	+	0.999526i	$0.509799\pi$
$368$	23.7899	1.24013
$369$	0	0
$370$	0.351519i	0.0182746i
$371$	1.85995i	0.0965639i
$372$	0	0
$373$	−30.0925	−1.55813	−0.779064	−	0.626944i	$-0.784305\pi$
$373$	−30.0925	−1.55813	−0.779064	+	0.626944i	$0.784305\pi$
$374$	−9.56465	−0.494576
$375$	0	0
$376$	−11.3967	−0.587741
$377$	0	0
$378$	0	0
$379$	19.1631i	0.984345i	0.870498	+	0.492172i	$0.163797\pi$
$379$	19.1631i	0.984345i	−0.870498	+	0.492172i	$0.836203\pi$
$380$	−2.48427	−0.127440
$381$	0	0
$382$	0.579121i	0.0296304i
$383$	− 15.3884i	− 0.786308i	−0.919473	−	0.393154i	$-0.871384\pi$
$383$	− 15.3884i	− 0.786308i	0.919473	−	0.393154i	$-0.128616\pi$
$384$	0	0
$385$	− 2.44743i	− 0.124733i
$386$	4.09246	0.208301
$387$	0	0
$388$	24.6679i	1.25232i
$389$	−24.0315	−1.21844	−0.609222	−	0.793000i	$-0.708518\pi$
$389$	−24.0315	−1.21844	−0.609222	+	0.793000i	$0.708518\pi$
$390$	0	0
$391$	−31.7265	−1.60448
$392$	6.64443i	0.335594i
$393$	0	0
$394$	−1.83015	−0.0922016
$395$	− 1.10023i	− 0.0553585i
$396$	0	0
$397$	29.6015i	1.48566i	0.669482	+	0.742828i	$0.266516\pi$
$397$	29.6015i	1.48566i	−0.669482	+	0.742828i	$0.733484\pi$
$398$	11.0248i	0.552621i
$399$	0	0
$400$	14.0804	0.704019
$401$	21.1032i	1.05384i	0.849914	+	0.526922i	$0.176654\pi$
$401$	21.1032i	1.05384i	−0.849914	+	0.526922i	$0.823346\pi$
$402$	0	0
$403$	0	0
$404$	−9.75063	−0.485112
$405$	0	0
$406$	−4.63342	−0.229953
$407$	18.0780	0.896092
$408$	0	0
$409$	36.0224i	1.78119i	0.454796	+	0.890596i	$0.349712\pi$
$409$	36.0224i	1.78119i	−0.454796	+	0.890596i	$0.650288\pi$
$410$	0.0489173i	0.00241586i
$411$	0	0
$412$	−24.7875	−1.22119
$413$	−24.0194	−1.18192
$414$	0	0
$415$	−2.51573	−0.123492
$416$	0	0
$417$	0	0
$418$	− 14.0431i	− 0.686869i
$419$	−5.96854	−0.291582	−0.145791	−	0.989315i	$-0.546573\pi$
$419$	−5.96854	−0.291582	−0.145791	+	0.989315i	$0.546573\pi$
$420$	0	0
$421$	2.09544i	0.102126i	0.998695	+	0.0510628i	$0.0162609\pi$
$421$	2.09544i	0.102126i	−0.998695	+	0.0510628i	$0.983739\pi$
$422$	− 2.64310i	− 0.128664i
$423$	0	0
$424$	1.79523i	0.0871842i
$425$	−18.7778	−0.910856
$426$	0	0
$427$	− 14.9282i	− 0.722429i
$428$	23.0954	1.11636
$429$	0	0
$430$	−0.188374	−0.00908418
$431$	− 2.88577i	− 0.139003i	−0.997582	−	0.0695014i	$-0.977859\pi$
$431$	− 2.88577i	− 0.139003i	0.997582	−	0.0695014i	$-0.0221408\pi$
$432$	0	0
$433$	12.6485	0.607847	0.303924	−	0.952696i	$-0.401703\pi$
$433$	12.6485	0.607847	0.303924	+	0.952696i	$0.401703\pi$
$434$	4.11051i	0.197311i
$435$	0	0
$436$	− 21.8877i	− 1.04823i
$437$	− 46.5816i	− 2.22830i
$438$	0	0
$439$	10.9269	0.521513	0.260757	−	0.965405i	$-0.416028\pi$
$439$	10.9269	0.521513	0.260757	+	0.965405i	$0.416028\pi$
$440$	− 2.36227i	− 0.112617i
$441$	0	0
$442$	0	0
$443$	19.2403	0.914133	0.457067	−	0.889433i	$-0.348900\pi$
$443$	19.2403	0.914133	0.457067	+	0.889433i	$0.348900\pi$
$444$	0	0
$445$	−0.0338518	−0.00160473
$446$	6.32006	0.299264
$447$	0	0
$448$	6.36526i	0.300730i
$449$	28.8200i	1.36010i	0.733166	+	0.680050i	$0.238042\pi$
$449$	28.8200i	1.36010i	−0.733166	+	0.680050i	$0.761958\pi$
$450$	0	0
$451$	2.51573	0.118461
$452$	2.95108	0.138807
$453$	0	0
$454$	−7.12392	−0.334342
$455$	0	0
$456$	0	0
$457$	18.0707i	0.845311i	0.906290	+	0.422656i	$0.138902\pi$
$457$	18.0707i	0.845311i	−0.906290	+	0.422656i	$0.861098\pi$
$458$	−0.819396	−0.0382879
$459$	0	0
$460$	− 3.71379i	− 0.173156i
$461$	− 7.56763i	− 0.352460i	−0.984349	−	0.176230i	$-0.943610\pi$
$461$	− 7.56763i	− 0.352460i	0.984349	−	0.176230i	$-0.0563902\pi$
$462$	0	0
$463$	35.3551i	1.64309i	0.570143	+	0.821545i	$0.306888\pi$
$463$	35.3551i	1.64309i	−0.570143	+	0.821545i	$0.693112\pi$
$464$	16.9312	0.786013
$465$	0	0
$466$	− 10.4252i	− 0.482938i
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	0	0
$469$	10.4494	0.482506
$470$	0.740348i	0.0341497i
$471$	0	0
$472$	−23.1836	−1.06711
$473$	9.68771i	0.445441i
$474$	0	0
$475$	− 27.5700i	− 1.26500i
$476$	− 12.0097i	− 0.550463i
$477$	0	0
$478$	−6.49827	−0.297224
$479$	25.5265i	1.16633i	0.812352	+	0.583167i	$0.198187\pi$
$479$	25.5265i	1.16633i	−0.812352	+	0.583167i	$0.801813\pi$
$480$	0	0
$481$	0	0
$482$	−3.84117	−0.174960
$483$	0	0
$484$	−37.7579	−1.71627
$485$	3.38106	0.153526
$486$	0	0
$487$	− 16.0073i	− 0.725360i	−0.931914	−	0.362680i	$-0.881862\pi$
$487$	− 16.0073i	− 0.725360i	0.931914	−	0.362680i	$-0.118138\pi$
$488$	− 14.4088i	− 0.652256i
$489$	0	0
$490$	0.431632	0.0194992
$491$	20.7385	0.935917	0.467959	−	0.883750i	$-0.344990\pi$
$491$	20.7385	0.935917	0.467959	+	0.883750i	$0.344990\pi$
$492$	0	0
$493$	−22.5797	−1.01694
$494$	0	0
$495$	0	0
$496$	− 15.0204i	− 0.674438i
$497$	10.0258	0.449719
$498$	0	0
$499$	8.06770i	0.361160i	0.983560	+	0.180580i	$0.0577975\pi$
$499$	8.06770i	0.361160i	−0.983560	+	0.180580i	$0.942203\pi$
$500$	− 4.42327i	− 0.197815i
$501$	0	0
$502$	− 1.69202i	− 0.0755186i
$503$	30.2422	1.34843	0.674216	−	0.738534i	$-0.264482\pi$
$503$	30.2422	1.34843	0.674216	+	0.738534i	$0.264482\pi$
$504$	0	0
$505$	1.33645i	0.0594714i
$506$	20.9933	0.933266
$507$	0	0
$508$	−19.4601	−0.863403
$509$	16.0495i	0.711382i	0.934604	+	0.355691i	$0.115754\pi$
$509$	16.0495i	0.711382i	−0.934604	+	0.355691i	$0.884246\pi$
$510$	0	0
$511$	−12.8968	−0.570520
$512$	22.9119i	1.01257i
$513$	0	0
$514$	− 9.16613i	− 0.404301i
$515$	3.39745i	0.149710i
$516$	0	0
$517$	38.0747	1.67452
$518$	− 2.49502i	− 0.109625i
$519$	0	0
$520$	0	0
$521$	2.69309	0.117986	0.0589931	−	0.998258i	$-0.481211\pi$
$521$	2.69309	0.117986	0.0589931	+	0.998258i	$0.481211\pi$
$522$	0	0
$523$	35.3957	1.54774	0.773872	−	0.633342i	$-0.218317\pi$
$523$	35.3957	1.54774	0.773872	+	0.633342i	$0.218317\pi$
$524$	−1.63533	−0.0714399
$525$	0	0
$526$	− 0.148080i	− 0.00645659i
$527$	20.0315i	0.872584i
$528$	0	0
$529$	46.6359	2.02765
$530$	0.116621	0.00506569
$531$	0	0
$532$	17.6329	0.764485
$533$	0	0
$534$	0	0
$535$	− 3.16554i	− 0.136858i
$536$	10.0858	0.435638
$537$	0	0
$538$	− 12.1511i	− 0.523870i
$539$	− 22.1981i	− 0.956138i
$540$	0	0
$541$	− 34.7338i	− 1.49332i	−0.665205	−	0.746660i	$-0.731656\pi$
$541$	− 34.7338i	− 1.49332i	0.665205	−	0.746660i	$-0.268344\pi$
$542$	−12.4547	−0.534976
$543$	0	0
$544$	− 17.6896i	− 0.758437i
$545$	−3.00000	−0.128506
$546$	0	0
$547$	−26.1183	−1.11674	−0.558368	−	0.829593i	$-0.688572\pi$
$547$	−26.1183	−1.11674	−0.558368	+	0.829593i	$0.688572\pi$
$548$	17.2054i	0.734976i
$549$	0	0
$550$	12.4252	0.529812
$551$	− 33.1521i	− 1.41233i
$552$	0	0
$553$	7.80923i	0.332082i
$554$	0.936017i	0.0397676i
$555$	0	0
$556$	−7.37435	−0.312742
$557$	− 24.7748i	− 1.04974i	−0.851182	−	0.524871i	$-0.824114\pi$
$557$	− 24.7748i	− 1.04974i	0.851182	−	0.524871i	$-0.175886\pi$
$558$	0	0
$559$	0	0
$560$	1.23431	0.0521590
$561$	0	0
$562$	12.1207	0.511280
$563$	5.26098	0.221724	0.110862	−	0.993836i	$-0.464639\pi$
$563$	5.26098	0.221724	0.110862	+	0.993836i	$0.464639\pi$
$564$	0	0
$565$	− 0.404485i	− 0.0170168i
$566$	− 2.35152i	− 0.0988417i
$567$	0	0
$568$	9.67696	0.406036
$569$	−33.7458	−1.41470	−0.707350	−	0.706864i	$-0.750109\pi$
$569$	−33.7458	−1.41470	−0.707350	+	0.706864i	$0.750109\pi$
$570$	0	0
$571$	−23.0887	−0.966234	−0.483117	−	0.875556i	$-0.660495\pi$
$571$	−23.0887	−0.966234	−0.483117	+	0.875556i	$0.660495\pi$
$572$	0	0
$573$	0	0
$574$	− 0.347207i	− 0.0144921i
$575$	41.2150	1.71879
$576$	0	0
$577$	− 3.57002i	− 0.148622i	−0.997235	−	0.0743110i	$-0.976324\pi$
$577$	− 3.57002i	− 0.148622i	0.997235	−	0.0743110i	$-0.0236758\pi$
$578$	− 1.13275i	− 0.0471162i
$579$	0	0
$580$	− 2.64310i	− 0.109749i
$581$	17.8562	0.740801
$582$	0	0
$583$	− 5.99761i	− 0.248396i
$584$	−12.4480	−0.515103
$585$	0	0
$586$	14.5362	0.600484
$587$	− 11.4625i	− 0.473108i	−0.971618	−	0.236554i	$-0.923982\pi$
$587$	− 11.4625i	− 0.473108i	0.971618	−	0.236554i	$-0.0760180\pi$
$588$	0	0
$589$	−29.4107	−1.21185
$590$	1.50604i	0.0620027i
$591$	0	0
$592$	9.11721i	0.374715i
$593$	21.8538i	0.897430i	0.893675	+	0.448715i	$0.148118\pi$
$593$	21.8538i	0.897430i	−0.893675	+	0.448715i	$0.851882\pi$
$594$	0	0
$595$	−1.64609	−0.0674830
$596$	− 27.7289i	− 1.13582i
$597$	0	0
$598$	0	0
$599$	−27.0573	−1.10553	−0.552765	−	0.833337i	$-0.686427\pi$
$599$	−27.0573	−1.10553	−0.552765	+	0.833337i	$0.686427\pi$
$600$	0	0
$601$	10.8780	0.443723	0.221861	−	0.975078i	$-0.428787\pi$
$601$	10.8780	0.443723	0.221861	+	0.975078i	$0.428787\pi$
$602$	1.33704	0.0544939
$603$	0	0
$604$	− 6.62133i	− 0.269418i
$605$	5.17523i	0.210403i
$606$	0	0
$607$	−29.6359	−1.20289	−0.601443	−	0.798916i	$-0.705407\pi$
$607$	−29.6359	−1.20289	−0.601443	+	0.798916i	$0.705407\pi$
$608$	25.9724	1.05332
$609$	0	0
$610$	−0.936017	−0.0378982
$611$	0	0
$612$	0	0
$613$	− 10.2343i	− 0.413360i	−0.978409	−	0.206680i	$-0.933734\pi$
$613$	− 10.2343i	− 0.413360i	0.978409	−	0.206680i	$-0.0662659\pi$
$614$	9.23968	0.372883
$615$	0	0
$616$	16.7670i	0.675563i
$617$	26.2828i	1.05810i	0.848590	+	0.529052i	$0.177452\pi$
$617$	26.2828i	1.05810i	−0.848590	+	0.529052i	$0.822548\pi$
$618$	0	0
$619$	29.0834i	1.16896i	0.811408	+	0.584479i	$0.198701\pi$
$619$	29.0834i	1.16896i	−0.811408	+	0.584479i	$0.801299\pi$
$620$	−2.34481	−0.0941700
$621$	0	0
$622$	− 5.02954i	− 0.201666i
$623$	0.240275	0.00962641
$624$	0	0
$625$	24.0887	0.963549
$626$	− 1.90110i	− 0.0759833i
$627$	0	0
$628$	−8.78986	−0.350753
$629$	− 12.1588i	− 0.484804i
$630$	0	0
$631$	25.4480i	1.01307i	0.862219	+	0.506535i	$0.169074\pi$
$631$	25.4480i	1.01307i	−0.862219	+	0.506535i	$0.830926\pi$
$632$	7.53750i	0.299826i
$633$	0	0
$634$	−6.88816	−0.273564
$635$	2.66727i	0.105847i
$636$	0	0
$637$	0	0
$638$	14.9409	0.591517
$639$	0	0
$640$	2.69740	0.106624
$641$	−26.7409	−1.05620	−0.528102	−	0.849181i	$-0.677096\pi$
$641$	−26.7409	−1.05620	−0.528102	+	0.849181i	$0.677096\pi$
$642$	0	0
$643$	32.9614i	1.29987i	0.759990	+	0.649935i	$0.225204\pi$
$643$	32.9614i	1.29987i	−0.759990	+	0.649935i	$0.774796\pi$
$644$	26.3599i	1.03872i
$645$	0	0
$646$	−9.44504	−0.371610
$647$	34.4946	1.35612	0.678060	−	0.735006i	$-0.262821\pi$
$647$	34.4946	1.35612	0.678060	+	0.735006i	$0.262821\pi$
$648$	0	0
$649$	77.4529	3.04029
$650$	0	0
$651$	0	0
$652$	15.5526i	0.609085i
$653$	−36.1517	−1.41472	−0.707362	−	0.706852i	$-0.750115\pi$
$653$	−36.1517	−1.41472	−0.707362	+	0.706852i	$0.750115\pi$
$654$	0	0
$655$	0.224144i	0.00875805i
$656$	1.26875i	0.0495364i
$657$	0	0
$658$	− 5.25487i	− 0.204856i
$659$	6.81700	0.265553	0.132776	−	0.991146i	$-0.457611\pi$
$659$	6.81700	0.265553	0.132776	+	0.991146i	$0.457611\pi$
$660$	0	0
$661$	− 10.8944i	− 0.423743i	−0.977298	−	0.211871i	$-0.932044\pi$
$661$	− 10.8944i	− 0.423743i	0.977298	−	0.211871i	$-0.0679558\pi$
$662$	2.70065	0.104964
$663$	0	0
$664$	17.2349	0.668844
$665$	− 2.41683i	− 0.0937206i
$666$	0	0
$667$	49.5599	1.91897
$668$	− 17.0640i	− 0.660225i
$669$	0	0
$670$	− 0.655186i	− 0.0253120i
$671$	48.1377i	1.85833i
$672$	0	0
$673$	−20.7385	−0.799412	−0.399706	−	0.916643i	$-0.630888\pi$
$673$	−20.7385	−0.799412	−0.399706	+	0.916643i	$0.630888\pi$
$674$	− 5.39565i	− 0.207833i
$675$	0	0
$676$	0	0
$677$	25.5786	0.983067	0.491534	−	0.870859i	$-0.336436\pi$
$677$	25.5786	0.983067	0.491534	+	0.870859i	$0.336436\pi$
$678$	0	0
$679$	−23.9982	−0.920966
$680$	−1.58881	−0.0609281
$681$	0	0
$682$	− 13.2547i	− 0.507551i
$683$	− 21.6310i	− 0.827688i	−0.910348	−	0.413844i	$-0.864186\pi$
$683$	− 21.6310i	− 0.827688i	0.910348	−	0.413844i	$-0.135814\pi$
$684$	0	0
$685$	2.35822	0.0901031
$686$	−8.52483	−0.325479
$687$	0	0
$688$	−4.88577	−0.186268
$689$	0	0
$690$	0	0
$691$	− 2.62996i	− 0.100048i	−0.998748	−	0.0500242i	$-0.984070\pi$
$691$	− 2.62996i	− 0.100048i	0.998748	−	0.0500242i	$-0.0159298\pi$
$692$	8.60388	0.327070
$693$	0	0
$694$	− 10.3026i	− 0.391081i
$695$	1.01075i	0.0383401i
$696$	0	0
$697$	− 1.69202i	− 0.0640899i
$698$	9.87800	0.373888
$699$	0	0
$700$	15.6015i	0.589681i
$701$	40.0925	1.51427	0.757136	−	0.653258i	$-0.226598\pi$
$701$	40.0925	1.51427	0.757136	+	0.653258i	$0.226598\pi$
$702$	0	0
$703$	17.8519	0.673298
$704$	− 20.5254i	− 0.773581i
$705$	0	0
$706$	2.25667	0.0849308
$707$	− 9.48593i	− 0.356755i
$708$	0	0
$709$	23.2097i	0.871657i	0.900030	+	0.435829i	$0.143545\pi$
$709$	23.2097i	0.871657i	−0.900030	+	0.435829i	$0.856455\pi$
$710$	− 0.628630i	− 0.0235921i
$711$	0	0
$712$	0.231914	0.00869135
$713$	− 43.9667i	− 1.64657i
$714$	0	0
$715$	0	0
$716$	6.19029	0.231342
$717$	0	0
$718$	−7.42088	−0.276945
$719$	−26.0146	−0.970181	−0.485090	−	0.874464i	$-0.661213\pi$
$719$	−26.0146	−0.970181	−0.485090	+	0.874464i	$0.661213\pi$
$720$	0	0
$721$	− 24.1146i	− 0.898073i
$722$	− 5.41166i	− 0.201401i
$723$	0	0
$724$	24.3013	0.903150
$725$	29.3327	1.08939
$726$	0	0
$727$	16.5472	0.613701	0.306851	−	0.951758i	$-0.400725\pi$
$727$	16.5472	0.613701	0.306851	+	0.951758i	$0.400725\pi$
$728$	0	0
$729$	0	0
$730$	0.808643i	0.0299292i
$731$	6.51573	0.240993
$732$	0	0
$733$	− 18.8750i	− 0.697165i	−0.937278	−	0.348582i	$-0.886663\pi$
$733$	− 18.8750i	− 0.697165i	0.937278	−	0.348582i	$-0.113337\pi$
$734$	− 0.524827i	− 0.0193717i
$735$	0	0
$736$	38.8267i	1.43117i
$737$	−33.6950	−1.24117
$738$	0	0
$739$	47.3239i	1.74084i	0.492312	+	0.870419i	$0.336152\pi$
$739$	47.3239i	1.74084i	−0.492312	+	0.870419i	$0.663848\pi$
$740$	1.42327	0.0523205
$741$	0	0
$742$	−0.827757	−0.0303879
$743$	− 8.88769i	− 0.326058i	−0.986621	−	0.163029i	$-0.947874\pi$
$743$	− 8.88769i	− 0.326058i	0.986621	−	0.163029i	$-0.0521264\pi$
$744$	0	0
$745$	−3.80061	−0.139244
$746$	− 13.3924i	− 0.490331i
$747$	0	0
$748$	38.7265i	1.41598i
$749$	22.4685i	0.820980i
$750$	0	0
$751$	−0.710808	−0.0259377	−0.0129689	−	0.999916i	$-0.504128\pi$
$751$	−0.710808	−0.0259377	−0.0129689	+	0.999916i	$0.504128\pi$
$752$	19.2021i	0.700229i
$753$	0	0
$754$	0	0
$755$	−0.907542	−0.0330288
$756$	0	0
$757$	9.78554	0.355662	0.177831	−	0.984061i	$-0.443092\pi$
$757$	9.78554	0.355662	0.177831	+	0.984061i	$0.443092\pi$
$758$	−8.52840	−0.309766
$759$	0	0
$760$	− 2.33273i	− 0.0846171i
$761$	18.8810i	0.684435i	0.939621	+	0.342218i	$0.111178\pi$
$761$	18.8810i	0.684435i	−0.939621	+	0.342218i	$0.888822\pi$
$762$	0	0
$763$	21.2935	0.770877
$764$	2.34481	0.0848324
$765$	0	0
$766$	6.84846	0.247445
$767$	0	0
$768$	0	0
$769$	12.4349i	0.448413i	0.974542	+	0.224207i	$0.0719790\pi$
$769$	12.4349i	0.448413i	−0.974542	+	0.224207i	$0.928021\pi$
$770$	1.08921	0.0392524
$771$	0	0
$772$	− 16.5700i	− 0.596368i
$773$	45.6746i	1.64280i	0.570353	+	0.821400i	$0.306807\pi$
$773$	45.6746i	1.64280i	−0.570353	+	0.821400i	$0.693193\pi$
$774$	0	0
$775$	− 26.0224i	− 0.934751i
$776$	−23.1631	−0.831508
$777$	0	0
$778$	− 10.6950i	− 0.383435i
$779$	2.48427	0.0890082
$780$	0	0
$781$	−32.3293	−1.15683
$782$	− 14.1196i	− 0.504916i
$783$	0	0
$784$	11.1951	0.399824
$785$	1.20477i	0.0430000i
$786$	0	0
$787$	− 4.51871i	− 0.161075i	−0.996752	−	0.0805374i	$-0.974336\pi$
$787$	− 4.51871i	− 0.161075i	0.996752	−	0.0805374i	$-0.0256636\pi$
$788$	7.41013i	0.263975i
$789$	0	0
$790$	0.489647	0.0174209
$791$	2.87097i	0.102080i
$792$	0	0
$793$	0	0
$794$	−13.1739	−0.467524
$795$	0	0
$796$	44.6383	1.58216
$797$	−28.7391	−1.01799	−0.508996	−	0.860769i	$-0.669983\pi$
$797$	−28.7391	−1.01799	−0.508996	+	0.860769i	$0.669983\pi$
$798$	0	0
$799$	− 25.6082i	− 0.905953i
$800$	22.9801i	0.812471i
$801$	0	0
$802$	−9.39181	−0.331636
$803$	41.5870	1.46757
$804$	0	0
$805$	3.61297	0.127341
$806$	0	0
$807$	0	0
$808$	− 9.15585i	− 0.322102i
$809$	−5.42891	−0.190870	−0.0954352	−	0.995436i	$-0.530424\pi$
$809$	−5.42891	−0.190870	−0.0954352	+	0.995436i	$0.530424\pi$
$810$	0	0
$811$	− 0.629104i	− 0.0220908i	−0.999939	−	0.0110454i	$-0.996484\pi$
$811$	− 0.629104i	− 0.0220908i	0.999939	−	0.0110454i	$-0.00351593\pi$
$812$	18.7603i	0.658358i
$813$	0	0
$814$	8.04546i	0.281993i
$815$	2.13169	0.0746697
$816$	0	0
$817$	9.56657i	0.334692i
$818$	−16.0315	−0.560527
$819$	0	0
$820$	0.198062	0.00691663
$821$	36.2640i	1.26562i	0.774307	+	0.632811i	$0.218099\pi$
$821$	36.2640i	1.26562i	−0.774307	+	0.632811i	$0.781901\pi$
$822$	0	0
$823$	−41.7396	−1.45495	−0.727476	−	0.686133i	$-0.759307\pi$
$823$	−41.7396	−1.45495	−0.727476	+	0.686133i	$0.759307\pi$
$824$	− 23.2755i	− 0.810839i
$825$	0	0
$826$	− 10.6896i	− 0.371940i
$827$	− 38.1997i	− 1.32833i	−0.747584	−	0.664167i	$-0.768786\pi$
$827$	− 38.1997i	− 1.32833i	0.747584	−	0.664167i	$-0.231214\pi$
$828$	0	0
$829$	−15.9788	−0.554967	−0.277484	−	0.960730i	$-0.589500\pi$
$829$	−15.9788	−0.554967	−0.277484	+	0.960730i	$0.589500\pi$
$830$	− 1.11960i	− 0.0388621i
$831$	0	0
$832$	0	0
$833$	−14.9299	−0.517290
$834$	0	0
$835$	−2.33885	−0.0809391
$836$	−56.8592	−1.96652
$837$	0	0
$838$	− 2.65625i	− 0.0917587i
$839$	4.63879i	0.160149i	0.996789	+	0.0800744i	$0.0255158\pi$
$839$	4.63879i	0.160149i	−0.996789	+	0.0800744i	$0.974484\pi$
$840$	0	0
$841$	6.27173	0.216267
$842$	−0.932559	−0.0321381
$843$	0	0
$844$	−10.7017	−0.368368
$845$	0	0
$846$	0	0
$847$	− 36.7329i	− 1.26216i
$848$	3.02475	0.103870
$849$	0	0
$850$	− 8.35690i	− 0.286639i
$851$	26.6872i	0.914827i
$852$	0	0
$853$	18.3884i	0.629605i	0.949157	+	0.314803i	$0.101938\pi$
$853$	18.3884i	0.629605i	−0.949157	+	0.314803i	$0.898062\pi$
$854$	6.64370	0.227343
$855$	0	0
$856$	21.6866i	0.741234i
$857$	28.1849	0.962778	0.481389	−	0.876507i	$-0.340132\pi$
$857$	28.1849	0.962778	0.481389	+	0.876507i	$0.340132\pi$
$858$	0	0
$859$	33.3957	1.13944	0.569722	−	0.821837i	$-0.307051\pi$
$859$	33.3957	1.13944	0.569722	+	0.821837i	$0.307051\pi$
$860$	0.762709i	0.0260082i
$861$	0	0
$862$	1.28429	0.0437431
$863$	− 43.0640i	− 1.46592i	−0.680274	−	0.732958i	$-0.738139\pi$
$863$	− 43.0640i	− 1.46592i	0.680274	−	0.732958i	$-0.261861\pi$
$864$	0	0
$865$	− 1.17928i	− 0.0400966i
$866$	5.62910i	0.191285i
$867$	0	0
$868$	16.6431	0.564904
$869$	− 25.1817i	− 0.854230i
$870$	0	0
$871$	0	0
$872$	20.5526	0.695998
$873$	0	0
$874$	20.7308	0.701229
$875$	4.30319	0.145474
$876$	0	0
$877$	30.1702i	1.01877i	0.860537	+	0.509387i	$0.170128\pi$
$877$	30.1702i	1.01877i	−0.860537	+	0.509387i	$0.829872\pi$
$878$	4.86294i	0.164116i
$879$	0	0
$880$	−3.98015	−0.134171
$881$	−3.56273	−0.120031	−0.0600157	−	0.998197i	$-0.519115\pi$
$881$	−3.56273	−0.120031	−0.0600157	+	0.998197i	$0.519115\pi$
$882$	0	0
$883$	10.2088	0.343554	0.171777	−	0.985136i	$-0.445049\pi$
$883$	10.2088	0.343554	0.171777	+	0.985136i	$0.445049\pi$
$884$	0	0
$885$	0	0
$886$	8.56273i	0.287670i
$887$	9.33645	0.313487	0.156744	−	0.987639i	$-0.449900\pi$
$887$	9.33645	0.313487	0.156744	+	0.987639i	$0.449900\pi$
$888$	0	0
$889$	− 18.9318i	− 0.634953i
$890$	− 0.0150655i	0 0.000504996i
$891$	0	0
$892$	− 25.5894i	− 0.856797i
$893$	37.5986	1.25819
$894$	0	0
$895$	− 0.848462i	− 0.0283610i
$896$	−19.1457	−0.639613
$897$	0	0
$898$	−12.8261	−0.428013
$899$	− 31.2911i	− 1.04362i
$900$	0	0
$901$	−4.03385	−0.134387
$902$	1.11960i	0.0372788i
$903$	0	0
$904$	2.77107i	0.0921644i
$905$	− 3.33081i	− 0.110720i
$906$	0	0
$907$	41.0804	1.36405	0.682026	−	0.731328i	$-0.261099\pi$
$907$	41.0804	1.36405	0.682026	+	0.731328i	$0.261099\pi$
$908$	28.8442i	0.957227i
$909$	0	0
$910$	0	0
$911$	18.9705	0.628519	0.314260	−	0.949337i	$-0.398244\pi$
$911$	18.9705	0.628519	0.314260	+	0.949337i	$0.398244\pi$
$912$	0	0
$913$	−57.5792	−1.90559
$914$	−8.04221	−0.266013
$915$	0	0
$916$	3.31767i	0.109619i
$917$	− 1.59094i	− 0.0525375i
$918$	0	0
$919$	29.0019	0.956685	0.478343	−	0.878173i	$-0.341238\pi$
$919$	29.0019	0.956685	0.478343	+	0.878173i	$0.341238\pi$
$920$	3.48725	0.114971
$921$	0	0
$922$	3.36791	0.110916
$923$	0	0
$924$	0	0
$925$	15.7952i	0.519344i
$926$	−15.7345	−0.517068
$927$	0	0
$928$	27.6329i	0.907096i
$929$	− 50.7211i	− 1.66410i	−0.554697	−	0.832052i	$-0.687166\pi$
$929$	− 50.7211i	− 1.66410i	0.554697	−	0.832052i	$-0.312834\pi$
$930$	0	0
$931$	− 21.9205i	− 0.718415i
$932$	−42.2107	−1.38266
$933$	0	0
$934$	5.78554i	0.189309i
$935$	5.30798	0.173589
$936$	0	0
$937$	51.3051	1.67606	0.838032	−	0.545620i	$-0.183706\pi$
$937$	51.3051	1.67606	0.838032	+	0.545620i	$0.183706\pi$
$938$	4.65040i	0.151841i
$939$	0	0
$940$	2.99761	0.0977712
$941$	− 34.7036i	− 1.13131i	−0.824643	−	0.565653i	$-0.808624\pi$
$941$	− 34.7036i	− 1.13131i	0.824643	−	0.565653i	$-0.191376\pi$
$942$	0	0
$943$	3.71379i	0.120938i
$944$	39.0616i	1.27135i
$945$	0	0
$946$	−4.31144	−0.140177
$947$	13.0127i	0.422855i	0.977394	+	0.211428i	$0.0678112\pi$
$947$	13.0127i	0.422855i	−0.977394	+	0.211428i	$0.932189\pi$
$948$	0	0
$949$	0	0
$950$	12.2698	0.398085
$951$	0	0
$952$	11.2771	0.365493
$953$	26.0151	0.842711	0.421355	−	0.906896i	$-0.361555\pi$
$953$	26.0151	0.842711	0.421355	+	0.906896i	$0.361555\pi$
$954$	0	0
$955$	− 0.321388i	− 0.0103999i
$956$	26.3110i	0.850957i
$957$	0	0
$958$	−11.3604	−0.367036
$959$	−16.7383	−0.540507
$960$	0	0
$961$	3.24027	0.104525
$962$	0	0
$963$	0	0
$964$	15.5526i	0.500914i
$965$	−2.27114	−0.0731107
$966$	0	0
$967$	− 36.6644i	− 1.17905i	−0.807751	−	0.589524i	$-0.799315\pi$
$967$	− 36.6644i	− 1.17905i	0.807751	−	0.589524i	$-0.200685\pi$
$968$	− 35.4547i	− 1.13956i
$969$	0	0
$970$	1.50471i	0.0483134i
$971$	37.8465	1.21455	0.607277	−	0.794490i	$-0.292262\pi$
$971$	37.8465	1.21455	0.607277	+	0.794490i	$0.292262\pi$
$972$	0	0
$973$	− 7.17416i	− 0.229993i
$974$	7.12392	0.228265
$975$	0	0
$976$	−24.2771	−0.777091
$977$	28.8998i	0.924586i	0.886727	+	0.462293i	$0.152973\pi$
$977$	28.8998i	0.924586i	−0.886727	+	0.462293i	$0.847027\pi$
$978$	0	0
$979$	−0.774791	−0.0247624
$980$	− 1.74764i	− 0.0558264i
$981$	0	0
$982$	9.22952i	0.294526i
$983$	19.3991i	0.618735i	0.950942	+	0.309368i	$0.100117\pi$
$983$	19.3991i	0.618735i	−0.950942	+	0.309368i	$0.899883\pi$
$984$	0	0
$985$	1.01566	0.0323615
$986$	− 10.0489i	− 0.320023i
$987$	0	0
$988$	0	0
$989$	−14.3013	−0.454754
$990$	0	0
$991$	−5.18300	−0.164643	−0.0823217	−	0.996606i	$-0.526233\pi$
$991$	−5.18300	−0.164643	−0.0823217	+	0.996606i	$0.526233\pi$
$992$	24.5144	0.778333
$993$	0	0
$994$	4.46191i	0.141523i
$995$	− 6.11828i	− 0.193962i
$996$	0	0
$997$	−49.3642	−1.56338	−0.781690	−	0.623667i	$-0.785642\pi$
$997$	−49.3642	−1.56338	−0.781690	+	0.623667i	$0.785642\pi$
$998$	−3.59047	−0.113654
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.m.1351.4		6
3.2	odd	2		507.2.b.g.337.3		6
13.5	odd	4		1521.2.a.q.1.2		3
13.8	odd	4		1521.2.a.p.1.2		3
13.12	even	2	inner	1521.2.b.m.1351.3		6
39.2	even	12		507.2.e.k.22.2		6
39.5	even	4		507.2.a.j.1.2	✓	3
39.8	even	4		507.2.a.k.1.2	yes	3
39.11	even	12		507.2.e.j.22.2		6
39.17	odd	6		507.2.j.h.361.4		12
39.20	even	12		507.2.e.j.484.2		6
39.23	odd	6		507.2.j.h.316.3		12
39.29	odd	6		507.2.j.h.316.4		12
39.32	even	12		507.2.e.k.484.2		6
39.35	odd	6		507.2.j.h.361.3		12
39.38	odd	2		507.2.b.g.337.4		6
156.47	odd	4		8112.2.a.cf.1.1		3
156.83	odd	4		8112.2.a.by.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.2	✓	3	39.5	even	4
507.2.a.k.1.2	yes	3	39.8	even	4
507.2.b.g.337.3		6	3.2	odd	2
507.2.b.g.337.4		6	39.38	odd	2
507.2.e.j.22.2		6	39.11	even	12
507.2.e.j.484.2		6	39.20	even	12
507.2.e.k.22.2		6	39.2	even	12
507.2.e.k.484.2		6	39.32	even	12
507.2.j.h.316.3		12	39.23	odd	6
507.2.j.h.316.4		12	39.29	odd	6
507.2.j.h.361.3		12	39.35	odd	6
507.2.j.h.361.4		12	39.17	odd	6
1521.2.a.p.1.2		3	13.8	odd	4
1521.2.a.q.1.2		3	13.5	odd	4
1521.2.b.m.1351.3		6	13.12	even	2	inner
1521.2.b.m.1351.4		6	1.1	even	1	trivial
8112.2.a.by.1.3		3	156.83	odd	4
8112.2.a.cf.1.1		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)

\(f(q)\)	\(=\)	\(q+0.445042i q^{2} +1.80194 q^{4} -0.246980i q^{5} +1.75302i q^{7} +1.69202i q^{8} +O(q^{10})\)	\(q+0.445042i q^{2} +1.80194 q^{4} -0.246980i q^{5} +1.75302i q^{7} +1.69202i q^{8} +0.109916 q^{10} -5.65279i q^{11} -0.780167 q^{14} +2.85086 q^{16} -3.80194 q^{17} -5.58211i q^{19} -0.445042i q^{20} +2.51573 q^{22} +8.34481 q^{23} +4.93900 q^{25} +3.15883i q^{28} +5.93900 q^{29} -5.26875i q^{31} +4.65279i q^{32} -1.69202i q^{34} +0.432960 q^{35} +3.19806i q^{37} +2.48427 q^{38} +0.417895 q^{40} +0.445042i q^{41} -1.71379 q^{43} -10.1860i q^{44} +3.71379i q^{46} +6.73556i q^{47} +3.92692 q^{49} +2.19806i q^{50} +1.06100 q^{53} -1.39612 q^{55} -2.96615 q^{56} +2.64310i q^{58} +13.7017i q^{59} -8.51573 q^{61} +2.34481 q^{62} +3.63102 q^{64} -5.96077i q^{67} -6.85086 q^{68} +0.192685i q^{70} -5.71917i q^{71} +7.35690i q^{73} -1.42327 q^{74} -10.0586i q^{76} +9.90946 q^{77} +4.45473 q^{79} -0.704103i q^{80} -0.198062 q^{82} -10.1860i q^{83} +0.939001i q^{85} -0.762709i q^{86} +9.56465 q^{88} -0.137063i q^{89} +15.0368 q^{92} -2.99761 q^{94} -1.37867 q^{95} +13.6896i q^{97} +1.74764i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{4}+O(q^{10}) \) 6 * q + 2 * q^4	\( 6 q + 2 q^{4} + 2 q^{10} - 2 q^{14} - 10 q^{16} - 14 q^{17} - 10 q^{22} + 4 q^{23} + 10 q^{25} + 16 q^{29} - 36 q^{35} + 40 q^{38} + 14 q^{40} + 6 q^{43} - 34 q^{49} + 26 q^{53} - 26 q^{55} + 14 q^{56} - 26 q^{61} - 32 q^{62} - 8 q^{64} - 14 q^{68} - 14 q^{74} - 30 q^{77} - 18 q^{79} - 10 q^{82} + 14 q^{88} + 34 q^{92} + 64 q^{94} + 6 q^{95}+O(q^{100}) \) 6 * q + 2 * q^4 + 2 * q^10 - 2 * q^14 - 10 * q^16 - 14 * q^17 - 10 * q^22 + 4 * q^23 + 10 * q^25 + 16 * q^29 - 36 * q^35 + 40 * q^38 + 14 * q^40 + 6 * q^43 - 34 * q^49 + 26 * q^53 - 26 * q^55 + 14 * q^56 - 26 * q^61 - 32 * q^62 - 8 * q^64 - 14 * q^68 - 14 * q^74 - 30 * q^77 - 18 * q^79 - 10 * q^82 + 14 * q^88 + 34 * q^92 + 64 * q^94 + 6 * q^95

Introduction

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