LMFDB - Embedded newform 8450.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.16228 q^{3} +1.00000 q^{4} -3.16228 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.16228 q^{3} +1.00000 q^{4} -3.16228 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.00000 q^{9} -2.16228 q^{11} -3.16228 q^{12} -1.00000 q^{14} +1.00000 q^{16} +5.16228 q^{17} +7.00000 q^{18} +1.83772 q^{19} +3.16228 q^{21} -2.16228 q^{22} -6.00000 q^{23} -3.16228 q^{24} -12.6491 q^{27} -1.00000 q^{28} +10.3246 q^{29} -7.16228 q^{31} +1.00000 q^{32} +6.83772 q^{33} +5.16228 q^{34} +7.00000 q^{36} -0.162278 q^{37} +1.83772 q^{38} +10.3246 q^{41} +3.16228 q^{42} -2.00000 q^{43} -2.16228 q^{44} -6.00000 q^{46} -3.00000 q^{47} -3.16228 q^{48} -6.00000 q^{49} -16.3246 q^{51} +2.16228 q^{53} -12.6491 q^{54} -1.00000 q^{56} -5.81139 q^{57} +10.3246 q^{58} +10.3246 q^{59} -7.48683 q^{61} -7.16228 q^{62} -7.00000 q^{63} +1.00000 q^{64} +6.83772 q^{66} -2.32456 q^{67} +5.16228 q^{68} +18.9737 q^{69} +4.32456 q^{71} +7.00000 q^{72} +2.83772 q^{73} -0.162278 q^{74} +1.83772 q^{76} +2.16228 q^{77} -13.4868 q^{79} +19.0000 q^{81} +10.3246 q^{82} +9.48683 q^{83} +3.16228 q^{84} -2.00000 q^{86} -32.6491 q^{87} -2.16228 q^{88} +7.32456 q^{89} -6.00000 q^{92} +22.6491 q^{93} -3.00000 q^{94} -3.16228 q^{96} -7.48683 q^{97} -6.00000 q^{98} -15.1359 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 14 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 14 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 14 q^{9} + 2 q^{11} - 2 q^{14} + 2 q^{16} + 4 q^{17} + 14 q^{18} + 10 q^{19} + 2 q^{22} - 12 q^{23} - 2 q^{28} + 8 q^{29} - 8 q^{31} + 2 q^{32} + 20 q^{33} + 4 q^{34} + 14 q^{36} + 6 q^{37} + 10 q^{38} + 8 q^{41} - 4 q^{43} + 2 q^{44} - 12 q^{46} - 6 q^{47} - 12 q^{49} - 20 q^{51} - 2 q^{53} - 2 q^{56} + 20 q^{57} + 8 q^{58} + 8 q^{59} + 4 q^{61} - 8 q^{62} - 14 q^{63} + 2 q^{64} + 20 q^{66} + 8 q^{67} + 4 q^{68} - 4 q^{71} + 14 q^{72} + 12 q^{73} + 6 q^{74} + 10 q^{76} - 2 q^{77} - 8 q^{79} + 38 q^{81} + 8 q^{82} - 4 q^{86} - 40 q^{87} + 2 q^{88} + 2 q^{89} - 12 q^{92} + 20 q^{93} - 6 q^{94} + 4 q^{97} - 12 q^{98} + 14 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 14 * q^9 + 2 * q^11 - 2 * q^14 + 2 * q^16 + 4 * q^17 + 14 * q^18 + 10 * q^19 + 2 * q^22 - 12 * q^23 - 2 * q^28 + 8 * q^29 - 8 * q^31 + 2 * q^32 + 20 * q^33 + 4 * q^34 + 14 * q^36 + 6 * q^37 + 10 * q^38 + 8 * q^41 - 4 * q^43 + 2 * q^44 - 12 * q^46 - 6 * q^47 - 12 * q^49 - 20 * q^51 - 2 * q^53 - 2 * q^56 + 20 * q^57 + 8 * q^58 + 8 * q^59 + 4 * q^61 - 8 * q^62 - 14 * q^63 + 2 * q^64 + 20 * q^66 + 8 * q^67 + 4 * q^68 - 4 * q^71 + 14 * q^72 + 12 * q^73 + 6 * q^74 + 10 * q^76 - 2 * q^77 - 8 * q^79 + 38 * q^81 + 8 * q^82 - 4 * q^86 - 40 * q^87 + 2 * q^88 + 2 * q^89 - 12 * q^92 + 20 * q^93 - 6 * q^94 + 4 * q^97 - 12 * q^98 + 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.16228	−1.82574	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	−3.16228	−1.82574	−0.912871	+	0.408248i	$0.866140\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.16228	−1.29099
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	7.00000	2.33333
$10$	0	0
$11$	−2.16228	−0.651951	−0.325976	−	0.945378i	$-0.605693\pi$
$11$	−2.16228	−0.651951	−0.325976	+	0.945378i	$0.605693\pi$
$12$	−3.16228	−0.912871
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	5.16228	1.25204	0.626018	−	0.779809i	$-0.284684\pi$
$17$	5.16228	1.25204	0.626018	+	0.779809i	$0.284684\pi$
$18$	7.00000	1.64992
$19$	1.83772	0.421602	0.210801	−	0.977529i	$-0.432393\pi$
$19$	1.83772	0.421602	0.210801	+	0.977529i	$0.432393\pi$
$20$	0	0
$21$	3.16228	0.690066
$22$	−2.16228	−0.460999
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−3.16228	−0.645497
$25$	0	0
$26$	0	0
$27$	−12.6491	−2.43432
$28$	−1.00000	−0.188982
$29$	10.3246	1.91722	0.958611	−	0.284719i	$-0.0919004\pi$
$29$	10.3246	1.91722	0.958611	+	0.284719i	$0.0919004\pi$
$30$	0	0
$31$	−7.16228	−1.28638	−0.643192	−	0.765705i	$-0.722390\pi$
$31$	−7.16228	−1.28638	−0.643192	+	0.765705i	$0.722390\pi$
$32$	1.00000	0.176777
$33$	6.83772	1.19029
$34$	5.16228	0.885323
$35$	0	0
$36$	7.00000	1.16667
$37$	−0.162278	−0.0266783	−0.0133391	−	0.999911i	$-0.504246\pi$
$37$	−0.162278	−0.0266783	−0.0133391	+	0.999911i	$0.504246\pi$
$38$	1.83772	0.298118
$39$	0	0
$40$	0	0
$41$	10.3246	1.61242	0.806212	−	0.591626i	$-0.201514\pi$
$41$	10.3246	1.61242	0.806212	+	0.591626i	$0.201514\pi$
$42$	3.16228	0.487950
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−2.16228	−0.325976
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−3.16228	−0.456435
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−16.3246	−2.28589
$52$	0	0
$53$	2.16228	0.297012	0.148506	−	0.988912i	$-0.452554\pi$
$53$	2.16228	0.297012	0.148506	+	0.988912i	$0.452554\pi$
$54$	−12.6491	−1.72133
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−5.81139	−0.769737
$58$	10.3246	1.35568
$59$	10.3246	1.34414	0.672071	−	0.740486i	$-0.265405\pi$
$59$	10.3246	1.34414	0.672071	+	0.740486i	$0.265405\pi$
$60$	0	0
$61$	−7.48683	−0.958591	−0.479295	−	0.877654i	$-0.659108\pi$
$61$	−7.48683	−0.958591	−0.479295	+	0.877654i	$0.659108\pi$
$62$	−7.16228	−0.909610
$63$	−7.00000	−0.881917
$64$	1.00000	0.125000
$65$	0	0
$66$	6.83772	0.841665
$67$	−2.32456	−0.283990	−0.141995	−	0.989867i	$-0.545352\pi$
$67$	−2.32456	−0.283990	−0.141995	+	0.989867i	$0.545352\pi$
$68$	5.16228	0.626018
$69$	18.9737	2.28416
$70$	0	0
$71$	4.32456	0.513230	0.256615	−	0.966514i	$-0.417393\pi$
$71$	4.32456	0.513230	0.256615	+	0.966514i	$0.417393\pi$
$72$	7.00000	0.824958
$73$	2.83772	0.332130	0.166065	−	0.986115i	$-0.446894\pi$
$73$	2.83772	0.332130	0.166065	+	0.986115i	$0.446894\pi$
$74$	−0.162278	−0.0188644
$75$	0	0
$76$	1.83772	0.210801
$77$	2.16228	0.246414
$78$	0	0
$79$	−13.4868	−1.51739	−0.758694	−	0.651448i	$-0.774162\pi$
$79$	−13.4868	−1.51739	−0.758694	+	0.651448i	$0.774162\pi$
$80$	0	0
$81$	19.0000	2.11111
$82$	10.3246	1.14016
$83$	9.48683	1.04132	0.520658	−	0.853766i	$-0.325687\pi$
$83$	9.48683	1.04132	0.520658	+	0.853766i	$0.325687\pi$
$84$	3.16228	0.345033
$85$	0	0
$86$	−2.00000	−0.215666
$87$	−32.6491	−3.50035
$88$	−2.16228	−0.230500
$89$	7.32456	0.776401	0.388201	−	0.921575i	$-0.373097\pi$
$89$	7.32456	0.776401	0.388201	+	0.921575i	$0.373097\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	22.6491	2.34860
$94$	−3.00000	−0.309426
$95$	0	0
$96$	−3.16228	−0.322749
$97$	−7.48683	−0.760173	−0.380086	−	0.924951i	$-0.624106\pi$
$97$	−7.48683	−0.760173	−0.380086	+	0.924951i	$0.624106\pi$
$98$	−6.00000	−0.606092
$99$	−15.1359	−1.52122
$100$	0	0
$101$	−11.1623	−1.11069	−0.555344	−	0.831621i	$-0.687413\pi$
$101$	−11.1623	−1.11069	−0.555344	+	0.831621i	$0.687413\pi$
$102$	−16.3246	−1.61637
$103$	−0.675445	−0.0665535	−0.0332768	−	0.999446i	$-0.510594\pi$
$103$	−0.675445	−0.0665535	−0.0332768	+	0.999446i	$0.510594\pi$
$104$	0	0
$105$	0	0
$106$	2.16228	0.210019
$107$	3.48683	0.337085	0.168542	−	0.985694i	$-0.446094\pi$
$107$	3.48683	0.337085	0.168542	+	0.985694i	$0.446094\pi$
$108$	−12.6491	−1.21716
$109$	−10.6491	−1.02000	−0.510000	−	0.860174i	$-0.670355\pi$
$109$	−10.6491	−1.02000	−0.510000	+	0.860174i	$0.670355\pi$
$110$	0	0
$111$	0.513167	0.0487077
$112$	−1.00000	−0.0944911
$113$	8.64911	0.813640	0.406820	−	0.913508i	$-0.366638\pi$
$113$	8.64911	0.813640	0.406820	+	0.913508i	$0.366638\pi$
$114$	−5.81139	−0.544286
$115$	0	0
$116$	10.3246	0.958611
$117$	0	0
$118$	10.3246	0.950452
$119$	−5.16228	−0.473225
$120$	0	0
$121$	−6.32456	−0.574960
$122$	−7.48683	−0.677826
$123$	−32.6491	−2.94387
$124$	−7.16228	−0.643192
$125$	0	0
$126$	−7.00000	−0.623610
$127$	−3.32456	−0.295007	−0.147503	−	0.989062i	$-0.547124\pi$
$127$	−3.32456	−0.295007	−0.147503	+	0.989062i	$0.547124\pi$
$128$	1.00000	0.0883883
$129$	6.32456	0.556846
$130$	0	0
$131$	−10.8114	−0.944595	−0.472298	−	0.881439i	$-0.656575\pi$
$131$	−10.8114	−0.944595	−0.472298	+	0.881439i	$0.656575\pi$
$132$	6.83772	0.595147
$133$	−1.83772	−0.159351
$134$	−2.32456	−0.200811
$135$	0	0
$136$	5.16228	0.442662
$137$	15.4868	1.32313	0.661565	−	0.749888i	$-0.269893\pi$
$137$	15.4868	1.32313	0.661565	+	0.749888i	$0.269893\pi$
$138$	18.9737	1.61515
$139$	−12.1623	−1.03159	−0.515795	−	0.856712i	$-0.672504\pi$
$139$	−12.1623	−1.03159	−0.515795	+	0.856712i	$0.672504\pi$
$140$	0	0
$141$	9.48683	0.798935
$142$	4.32456	0.362909
$143$	0	0
$144$	7.00000	0.583333
$145$	0	0
$146$	2.83772	0.234852
$147$	18.9737	1.56492
$148$	−0.162278	−0.0133391
$149$	−16.3246	−1.33736	−0.668680	−	0.743550i	$-0.733140\pi$
$149$	−16.3246	−1.33736	−0.668680	+	0.743550i	$0.733140\pi$
$150$	0	0
$151$	4.83772	0.393688	0.196844	−	0.980435i	$-0.436931\pi$
$151$	4.83772	0.393688	0.196844	+	0.980435i	$0.436931\pi$
$152$	1.83772	0.149059
$153$	36.1359	2.92142
$154$	2.16228	0.174241
$155$	0	0
$156$	0	0
$157$	12.1623	0.970655	0.485328	−	0.874332i	$-0.338700\pi$
$157$	12.1623	0.970655	0.485328	+	0.874332i	$0.338700\pi$
$158$	−13.4868	−1.07295
$159$	−6.83772	−0.542267
$160$	0	0
$161$	6.00000	0.472866
$162$	19.0000	1.49278
$163$	17.4868	1.36967	0.684837	−	0.728696i	$-0.259873\pi$
$163$	17.4868	1.36967	0.684837	+	0.728696i	$0.259873\pi$
$164$	10.3246	0.806212
$165$	0	0
$166$	9.48683	0.736321
$167$	13.3246	1.03109	0.515543	−	0.856864i	$-0.327590\pi$
$167$	13.3246	1.03109	0.515543	+	0.856864i	$0.327590\pi$
$168$	3.16228	0.243975
$169$	0	0
$170$	0	0
$171$	12.8641	0.983739
$172$	−2.00000	−0.152499
$173$	−2.16228	−0.164395	−0.0821975	−	0.996616i	$-0.526194\pi$
$173$	−2.16228	−0.164395	−0.0821975	+	0.996616i	$0.526194\pi$
$174$	−32.6491	−2.47512
$175$	0	0
$176$	−2.16228	−0.162988
$177$	−32.6491	−2.45406
$178$	7.32456	0.548999
$179$	22.3246	1.66862	0.834308	−	0.551299i	$-0.185868\pi$
$179$	22.3246	1.66862	0.834308	+	0.551299i	$0.185868\pi$
$180$	0	0
$181$	9.81139	0.729275	0.364637	−	0.931150i	$-0.381193\pi$
$181$	9.81139	0.729275	0.364637	+	0.931150i	$0.381193\pi$
$182$	0	0
$183$	23.6754	1.75014
$184$	−6.00000	−0.442326
$185$	0	0
$186$	22.6491	1.66071
$187$	−11.1623	−0.816267
$188$	−3.00000	−0.218797
$189$	12.6491	0.920087
$190$	0	0
$191$	9.48683	0.686443	0.343222	−	0.939254i	$-0.388482\pi$
$191$	9.48683	0.686443	0.343222	+	0.939254i	$0.388482\pi$
$192$	−3.16228	−0.228218
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−7.48683	−0.537523
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−18.4868	−1.31713	−0.658566	−	0.752523i	$-0.728837\pi$
$197$	−18.4868	−1.31713	−0.658566	+	0.752523i	$0.728837\pi$
$198$	−15.1359	−1.07566
$199$	−1.35089	−0.0957620	−0.0478810	−	0.998853i	$-0.515247\pi$
$199$	−1.35089	−0.0957620	−0.0478810	+	0.998853i	$0.515247\pi$
$200$	0	0
$201$	7.35089	0.518492
$202$	−11.1623	−0.785375
$203$	−10.3246	−0.724642
$204$	−16.3246	−1.14295
$205$	0	0
$206$	−0.675445	−0.0470605
$207$	−42.0000	−2.91920
$208$	0	0
$209$	−3.97367	−0.274864
$210$	0	0
$211$	11.8377	0.814942	0.407471	−	0.913218i	$-0.366411\pi$
$211$	11.8377	0.814942	0.407471	+	0.913218i	$0.366411\pi$
$212$	2.16228	0.148506
$213$	−13.6754	−0.937026
$214$	3.48683	0.238355
$215$	0	0
$216$	−12.6491	−0.860663
$217$	7.16228	0.486207
$218$	−10.6491	−0.721249
$219$	−8.97367	−0.606384
$220$	0	0
$221$	0	0
$222$	0.513167	0.0344415
$223$	3.32456	0.222629	0.111314	−	0.993785i	$-0.464494\pi$
$223$	3.32456	0.222629	0.111314	+	0.993785i	$0.464494\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	8.64911	0.575330
$227$	10.3246	0.685265	0.342632	−	0.939470i	$-0.388681\pi$
$227$	10.3246	0.685265	0.342632	+	0.939470i	$0.388681\pi$
$228$	−5.81139	−0.384869
$229$	−2.83772	−0.187522	−0.0937610	−	0.995595i	$-0.529889\pi$
$229$	−2.83772	−0.187522	−0.0937610	+	0.995595i	$0.529889\pi$
$230$	0	0
$231$	−6.83772	−0.449889
$232$	10.3246	0.677840
$233$	−9.48683	−0.621503	−0.310752	−	0.950491i	$-0.600581\pi$
$233$	−9.48683	−0.621503	−0.310752	+	0.950491i	$0.600581\pi$
$234$	0	0
$235$	0	0
$236$	10.3246	0.672071
$237$	42.6491	2.77036
$238$	−5.16228	−0.334621
$239$	−11.1623	−0.722028	−0.361014	−	0.932560i	$-0.617569\pi$
$239$	−11.1623	−0.722028	−0.361014	+	0.932560i	$0.617569\pi$
$240$	0	0
$241$	25.9737	1.67311	0.836555	−	0.547882i	$-0.184566\pi$
$241$	25.9737	1.67311	0.836555	+	0.547882i	$0.184566\pi$
$242$	−6.32456	−0.406558
$243$	−22.1359	−1.42002
$244$	−7.48683	−0.479295
$245$	0	0
$246$	−32.6491	−2.08163
$247$	0	0
$248$	−7.16228	−0.454805
$249$	−30.0000	−1.90117
$250$	0	0
$251$	12.4868	0.788162	0.394081	−	0.919076i	$-0.371063\pi$
$251$	12.4868	0.788162	0.394081	+	0.919076i	$0.371063\pi$
$252$	−7.00000	−0.440959
$253$	12.9737	0.815647
$254$	−3.32456	−0.208601
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	6.32456	0.393750
$259$	0.162278	0.0100834
$260$	0	0
$261$	72.2719	4.47352
$262$	−10.8114	−0.667930
$263$	11.6491	0.718315	0.359157	−	0.933277i	$-0.383064\pi$
$263$	11.6491	0.718315	0.359157	+	0.933277i	$0.383064\pi$
$264$	6.83772	0.420833
$265$	0	0
$266$	−1.83772	−0.112678
$267$	−23.1623	−1.41751
$268$	−2.32456	−0.141995
$269$	2.51317	0.153230	0.0766152	−	0.997061i	$-0.475589\pi$
$269$	2.51317	0.153230	0.0766152	+	0.997061i	$0.475589\pi$
$270$	0	0
$271$	−6.32456	−0.384189	−0.192095	−	0.981376i	$-0.561528\pi$
$271$	−6.32456	−0.384189	−0.192095	+	0.981376i	$0.561528\pi$
$272$	5.16228	0.313009
$273$	0	0
$274$	15.4868	0.935594
$275$	0	0
$276$	18.9737	1.14208
$277$	−1.51317	−0.0909174	−0.0454587	−	0.998966i	$-0.514475\pi$
$277$	−1.51317	−0.0909174	−0.0454587	+	0.998966i	$0.514475\pi$
$278$	−12.1623	−0.729445
$279$	−50.1359	−3.00156
$280$	0	0
$281$	18.9737	1.13187	0.565937	−	0.824448i	$-0.308515\pi$
$281$	18.9737	1.13187	0.565937	+	0.824448i	$0.308515\pi$
$282$	9.48683	0.564933
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	4.32456	0.256615
$285$	0	0
$286$	0	0
$287$	−10.3246	−0.609439
$288$	7.00000	0.412479
$289$	9.64911	0.567595
$290$	0	0
$291$	23.6754	1.38788
$292$	2.83772	0.166065
$293$	−21.8377	−1.27577	−0.637887	−	0.770130i	$-0.720191\pi$
$293$	−21.8377	−1.27577	−0.637887	+	0.770130i	$0.720191\pi$
$294$	18.9737	1.10657
$295$	0	0
$296$	−0.162278	−0.00943220
$297$	27.3509	1.58706
$298$	−16.3246	−0.945656
$299$	0	0
$300$	0	0
$301$	2.00000	0.115278
$302$	4.83772	0.278380
$303$	35.2982	2.02783
$304$	1.83772	0.105401
$305$	0	0
$306$	36.1359	2.06575
$307$	15.6754	0.894645	0.447322	−	0.894373i	$-0.352378\pi$
$307$	15.6754	0.894645	0.447322	+	0.894373i	$0.352378\pi$
$308$	2.16228	0.123207
$309$	2.13594	0.121510
$310$	0	0
$311$	−3.48683	−0.197720	−0.0988601	−	0.995101i	$-0.531520\pi$
$311$	−3.48683	−0.197720	−0.0988601	+	0.995101i	$0.531520\pi$
$312$	0	0
$313$	8.32456	0.470532	0.235266	−	0.971931i	$-0.424404\pi$
$313$	8.32456	0.470532	0.235266	+	0.971931i	$0.424404\pi$
$314$	12.1623	0.686357
$315$	0	0
$316$	−13.4868	−0.758694
$317$	12.4868	0.701330	0.350665	−	0.936501i	$-0.385956\pi$
$317$	12.4868	0.701330	0.350665	+	0.936501i	$0.385956\pi$
$318$	−6.83772	−0.383440
$319$	−22.3246	−1.24994
$320$	0	0
$321$	−11.0263	−0.615430
$322$	6.00000	0.334367
$323$	9.48683	0.527862
$324$	19.0000	1.05556
$325$	0	0
$326$	17.4868	0.968506
$327$	33.6754	1.86226
$328$	10.3246	0.570078
$329$	3.00000	0.165395
$330$	0	0
$331$	24.6491	1.35484	0.677419	−	0.735598i	$-0.263099\pi$
$331$	24.6491	1.35484	0.677419	+	0.735598i	$0.263099\pi$
$332$	9.48683	0.520658
$333$	−1.13594	−0.0622493
$334$	13.3246	0.729087
$335$	0	0
$336$	3.16228	0.172516
$337$	−17.4868	−0.952568	−0.476284	−	0.879291i	$-0.658017\pi$
$337$	−17.4868	−0.952568	−0.476284	+	0.879291i	$0.658017\pi$
$338$	0	0
$339$	−27.3509	−1.48550
$340$	0	0
$341$	15.4868	0.838659
$342$	12.8641	0.695609
$343$	13.0000	0.701934
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−2.16228	−0.116245
$347$	−2.51317	−0.134914	−0.0674569	−	0.997722i	$-0.521489\pi$
$347$	−2.51317	−0.134914	−0.0674569	+	0.997722i	$0.521489\pi$
$348$	−32.6491	−1.75018
$349$	−17.4868	−0.936049	−0.468024	−	0.883716i	$-0.655034\pi$
$349$	−17.4868	−0.936049	−0.468024	+	0.883716i	$0.655034\pi$
$350$	0	0
$351$	0	0
$352$	−2.16228	−0.115250
$353$	5.16228	0.274760	0.137380	−	0.990518i	$-0.456132\pi$
$353$	5.16228	0.274760	0.137380	+	0.990518i	$0.456132\pi$
$354$	−32.6491	−1.73528
$355$	0	0
$356$	7.32456	0.388201
$357$	16.3246	0.863987
$358$	22.3246	1.17989
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−15.6228	−0.822251
$362$	9.81139	0.515675
$363$	20.0000	1.04973
$364$	0	0
$365$	0	0
$366$	23.6754	1.23754
$367$	−4.64911	−0.242682	−0.121341	−	0.992611i	$-0.538719\pi$
$367$	−4.64911	−0.242682	−0.121341	+	0.992611i	$0.538719\pi$
$368$	−6.00000	−0.312772
$369$	72.2719	3.76232
$370$	0	0
$371$	−2.16228	−0.112260
$372$	22.6491	1.17430
$373$	30.6491	1.58695	0.793475	−	0.608602i	$-0.208270\pi$
$373$	30.6491	1.58695	0.793475	+	0.608602i	$0.208270\pi$
$374$	−11.1623	−0.577188
$375$	0	0
$376$	−3.00000	−0.154713
$377$	0	0
$378$	12.6491	0.650600
$379$	38.8114	1.99361	0.996804	−	0.0798917i	$-0.0254574\pi$
$379$	38.8114	1.99361	0.996804	+	0.0798917i	$0.0254574\pi$
$380$	0	0
$381$	10.5132	0.538606
$382$	9.48683	0.485389
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	−3.16228	−0.161374
$385$	0	0
$386$	−4.00000	−0.203595
$387$	−14.0000	−0.711660
$388$	−7.48683	−0.380086
$389$	12.8377	0.650898	0.325449	−	0.945560i	$-0.394484\pi$
$389$	12.8377	0.650898	0.325449	+	0.945560i	$0.394484\pi$
$390$	0	0
$391$	−30.9737	−1.56641
$392$	−6.00000	−0.303046
$393$	34.1886	1.72459
$394$	−18.4868	−0.931353
$395$	0	0
$396$	−15.1359	−0.760610
$397$	38.4868	1.93160	0.965799	−	0.259290i	$-0.0834887\pi$
$397$	38.4868	1.93160	0.965799	+	0.259290i	$0.0834887\pi$
$398$	−1.35089	−0.0677140
$399$	5.81139	0.290933
$400$	0	0
$401$	−21.9737	−1.09731	−0.548656	−	0.836048i	$-0.684860\pi$
$401$	−21.9737	−1.09731	−0.548656	+	0.836048i	$0.684860\pi$
$402$	7.35089	0.366629
$403$	0	0
$404$	−11.1623	−0.555344
$405$	0	0
$406$	−10.3246	−0.512399
$407$	0.350889	0.0173929
$408$	−16.3246	−0.808186
$409$	21.6491	1.07048	0.535240	−	0.844700i	$-0.320221\pi$
$409$	21.6491	1.07048	0.535240	+	0.844700i	$0.320221\pi$
$410$	0	0
$411$	−48.9737	−2.41569
$412$	−0.675445	−0.0332768
$413$	−10.3246	−0.508038
$414$	−42.0000	−2.06419
$415$	0	0
$416$	0	0
$417$	38.4605	1.88342
$418$	−3.97367	−0.194358
$419$	14.6491	0.715656	0.357828	−	0.933788i	$-0.383517\pi$
$419$	14.6491	0.715656	0.357828	+	0.933788i	$0.383517\pi$
$420$	0	0
$421$	3.16228	0.154120	0.0770600	−	0.997026i	$-0.475447\pi$
$421$	3.16228	0.154120	0.0770600	+	0.997026i	$0.475447\pi$
$422$	11.8377	0.576251
$423$	−21.0000	−1.02105
$424$	2.16228	0.105009
$425$	0	0
$426$	−13.6754	−0.662577
$427$	7.48683	0.362313
$428$	3.48683	0.168542
$429$	0	0
$430$	0	0
$431$	−15.4868	−0.745974	−0.372987	−	0.927836i	$-0.621666\pi$
$431$	−15.4868	−0.745974	−0.372987	+	0.927836i	$0.621666\pi$
$432$	−12.6491	−0.608581
$433$	−6.32456	−0.303939	−0.151969	−	0.988385i	$-0.548562\pi$
$433$	−6.32456	−0.303939	−0.151969	+	0.988385i	$0.548562\pi$
$434$	7.16228	0.343800
$435$	0	0
$436$	−10.6491	−0.510000
$437$	−11.0263	−0.527461
$438$	−8.97367	−0.428778
$439$	3.81139	0.181908	0.0909538	−	0.995855i	$-0.471008\pi$
$439$	3.81139	0.181908	0.0909538	+	0.995855i	$0.471008\pi$
$440$	0	0
$441$	−42.0000	−2.00000
$442$	0	0
$443$	−26.6491	−1.26614	−0.633069	−	0.774096i	$-0.718205\pi$
$443$	−26.6491	−1.26614	−0.633069	+	0.774096i	$0.718205\pi$
$444$	0.513167	0.0243538
$445$	0	0
$446$	3.32456	0.157422
$447$	51.6228	2.44167
$448$	−1.00000	−0.0472456
$449$	2.02633	0.0956286	0.0478143	−	0.998856i	$-0.484774\pi$
$449$	2.02633	0.0956286	0.0478143	+	0.998856i	$0.484774\pi$
$450$	0	0
$451$	−22.3246	−1.05122
$452$	8.64911	0.406820
$453$	−15.2982	−0.718773
$454$	10.3246	0.484555
$455$	0	0
$456$	−5.81139	−0.272143
$457$	24.4605	1.14421	0.572107	−	0.820179i	$-0.306126\pi$
$457$	24.4605	1.14421	0.572107	+	0.820179i	$0.306126\pi$
$458$	−2.83772	−0.132598
$459$	−65.2982	−3.04786
$460$	0	0
$461$	−21.4868	−1.00074	−0.500371	−	0.865811i	$-0.666803\pi$
$461$	−21.4868	−1.00074	−0.500371	+	0.865811i	$0.666803\pi$
$462$	−6.83772	−0.318120
$463$	12.3246	0.572771	0.286385	−	0.958115i	$-0.407546\pi$
$463$	12.3246	0.572771	0.286385	+	0.958115i	$0.407546\pi$
$464$	10.3246	0.479305
$465$	0	0
$466$	−9.48683	−0.439469
$467$	−32.6491	−1.51082	−0.755410	−	0.655252i	$-0.772562\pi$
$467$	−32.6491	−1.51082	−0.755410	+	0.655252i	$0.772562\pi$
$468$	0	0
$469$	2.32456	0.107338
$470$	0	0
$471$	−38.4605	−1.77217
$472$	10.3246	0.475226
$473$	4.32456	0.198843
$474$	42.6491	1.95894
$475$	0	0
$476$	−5.16228	−0.236613
$477$	15.1359	0.693027
$478$	−11.1623	−0.510551
$479$	12.1359	0.554505	0.277253	−	0.960797i	$-0.410576\pi$
$479$	12.1359	0.554505	0.277253	+	0.960797i	$0.410576\pi$
$480$	0	0
$481$	0	0
$482$	25.9737	1.18307
$483$	−18.9737	−0.863332
$484$	−6.32456	−0.287480
$485$	0	0
$486$	−22.1359	−1.00411
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	−7.48683	−0.338913
$489$	−55.2982	−2.50067
$490$	0	0
$491$	17.5132	0.790358	0.395179	−	0.918604i	$-0.370683\pi$
$491$	17.5132	0.790358	0.395179	+	0.918604i	$0.370683\pi$
$492$	−32.6491	−1.47194
$493$	53.2982	2.40043
$494$	0	0
$495$	0	0
$496$	−7.16228	−0.321596
$497$	−4.32456	−0.193983
$498$	−30.0000	−1.34433
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	0	0
$501$	−42.1359	−1.88250
$502$	12.4868	0.557315
$503$	14.2982	0.637526	0.318763	−	0.947834i	$-0.396733\pi$
$503$	14.2982	0.637526	0.318763	+	0.947834i	$0.396733\pi$
$504$	−7.00000	−0.311805
$505$	0	0
$506$	12.9737	0.576750
$507$	0	0
$508$	−3.32456	−0.147503
$509$	0.973666	0.0431570	0.0215785	−	0.999767i	$-0.493131\pi$
$509$	0.973666	0.0431570	0.0215785	+	0.999767i	$0.493131\pi$
$510$	0	0
$511$	−2.83772	−0.125533
$512$	1.00000	0.0441942
$513$	−23.2456	−1.02632
$514$	0	0
$515$	0	0
$516$	6.32456	0.278423
$517$	6.48683	0.285291
$518$	0.162278	0.00713007
$519$	6.83772	0.300143
$520$	0	0
$521$	22.6754	0.993429	0.496715	−	0.867914i	$-0.334540\pi$
$521$	22.6754	0.993429	0.496715	+	0.867914i	$0.334540\pi$
$522$	72.2719	3.16325
$523$	39.2982	1.71839	0.859196	−	0.511647i	$-0.170965\pi$
$523$	39.2982	1.71839	0.859196	+	0.511647i	$0.170965\pi$
$524$	−10.8114	−0.472298
$525$	0	0
$526$	11.6491	0.507925
$527$	−36.9737	−1.61060
$528$	6.83772	0.297574
$529$	13.0000	0.565217
$530$	0	0
$531$	72.2719	3.13633
$532$	−1.83772	−0.0796754
$533$	0	0
$534$	−23.1623	−1.00233
$535$	0	0
$536$	−2.32456	−0.100405
$537$	−70.5964	−3.04646
$538$	2.51317	0.108350
$539$	12.9737	0.558815
$540$	0	0
$541$	−26.8377	−1.15384	−0.576922	−	0.816799i	$-0.695746\pi$
$541$	−26.8377	−1.15384	−0.576922	+	0.816799i	$0.695746\pi$
$542$	−6.32456	−0.271663
$543$	−31.0263	−1.33147
$544$	5.16228	0.221331
$545$	0	0
$546$	0	0
$547$	6.64911	0.284295	0.142148	−	0.989845i	$-0.454599\pi$
$547$	6.64911	0.284295	0.142148	+	0.989845i	$0.454599\pi$
$548$	15.4868	0.661565
$549$	−52.4078	−2.23671
$550$	0	0
$551$	18.9737	0.808305
$552$	18.9737	0.807573
$553$	13.4868	0.573518
$554$	−1.51317	−0.0642883
$555$	0	0
$556$	−12.1623	−0.515795
$557$	15.8377	0.671066	0.335533	−	0.942028i	$-0.391084\pi$
$557$	15.8377	0.671066	0.335533	+	0.942028i	$0.391084\pi$
$558$	−50.1359	−2.12242
$559$	0	0
$560$	0	0
$561$	35.2982	1.49029
$562$	18.9737	0.800356
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	9.48683	0.399468
$565$	0	0
$566$	22.0000	0.924729
$567$	−19.0000	−0.797925
$568$	4.32456	0.181454
$569$	−33.9737	−1.42425	−0.712125	−	0.702053i	$-0.752267\pi$
$569$	−33.9737	−1.42425	−0.712125	+	0.702053i	$0.752267\pi$
$570$	0	0
$571$	−25.1359	−1.05191	−0.525953	−	0.850513i	$-0.676291\pi$
$571$	−25.1359	−1.05191	−0.525953	+	0.850513i	$0.676291\pi$
$572$	0	0
$573$	−30.0000	−1.25327
$574$	−10.3246	−0.430939
$575$	0	0
$576$	7.00000	0.291667
$577$	7.16228	0.298170	0.149085	−	0.988824i	$-0.452367\pi$
$577$	7.16228	0.298170	0.149085	+	0.988824i	$0.452367\pi$
$578$	9.64911	0.401350
$579$	12.6491	0.525679
$580$	0	0
$581$	−9.48683	−0.393580
$582$	23.6754	0.981379
$583$	−4.67544	−0.193637
$584$	2.83772	0.117426
$585$	0	0
$586$	−21.8377	−0.902108
$587$	12.9737	0.535481	0.267740	−	0.963491i	$-0.413723\pi$
$587$	12.9737	0.535481	0.267740	+	0.963491i	$0.413723\pi$
$588$	18.9737	0.782461
$589$	−13.1623	−0.542342
$590$	0	0
$591$	58.4605	2.40474
$592$	−0.162278	−0.00666957
$593$	−3.35089	−0.137605	−0.0688023	−	0.997630i	$-0.521918\pi$
$593$	−3.35089	−0.137605	−0.0688023	+	0.997630i	$0.521918\pi$
$594$	27.3509	1.12222
$595$	0	0
$596$	−16.3246	−0.668680
$597$	4.27189	0.174837
$598$	0	0
$599$	43.9473	1.79564	0.897820	−	0.440363i	$-0.145150\pi$
$599$	43.9473	1.79564	0.897820	+	0.440363i	$0.145150\pi$
$600$	0	0
$601$	34.2982	1.39905	0.699527	−	0.714606i	$-0.253394\pi$
$601$	34.2982	1.39905	0.699527	+	0.714606i	$0.253394\pi$
$602$	2.00000	0.0815139
$603$	−16.2719	−0.662642
$604$	4.83772	0.196844
$605$	0	0
$606$	35.2982	1.43389
$607$	−5.00000	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$607$	−5.00000	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$608$	1.83772	0.0745295
$609$	32.6491	1.32301
$610$	0	0
$611$	0	0
$612$	36.1359	1.46071
$613$	20.4868	0.827455	0.413728	−	0.910401i	$-0.364227\pi$
$613$	20.4868	0.827455	0.413728	+	0.910401i	$0.364227\pi$
$614$	15.6754	0.632609
$615$	0	0
$616$	2.16228	0.0871206
$617$	−31.9473	−1.28615	−0.643076	−	0.765803i	$-0.722342\pi$
$617$	−31.9473	−1.28615	−0.643076	+	0.765803i	$0.722342\pi$
$618$	2.13594	0.0859203
$619$	29.4605	1.18412	0.592059	−	0.805895i	$-0.298315\pi$
$619$	29.4605	1.18412	0.592059	+	0.805895i	$0.298315\pi$
$620$	0	0
$621$	75.8947	3.04555
$622$	−3.48683	−0.139809
$623$	−7.32456	−0.293452
$624$	0	0
$625$	0	0
$626$	8.32456	0.332716
$627$	12.5658	0.501831
$628$	12.1623	0.485328
$629$	−0.837722	−0.0334022
$630$	0	0
$631$	14.3246	0.570252	0.285126	−	0.958490i	$-0.407965\pi$
$631$	14.3246	0.570252	0.285126	+	0.958490i	$0.407965\pi$
$632$	−13.4868	−0.536477
$633$	−37.4342	−1.48787
$634$	12.4868	0.495915
$635$	0	0
$636$	−6.83772	−0.271133
$637$	0	0
$638$	−22.3246	−0.883838
$639$	30.2719	1.19754
$640$	0	0
$641$	−9.97367	−0.393936	−0.196968	−	0.980410i	$-0.563110\pi$
$641$	−9.97367	−0.393936	−0.196968	+	0.980410i	$0.563110\pi$
$642$	−11.0263	−0.435175
$643$	12.4605	0.491394	0.245697	−	0.969347i	$-0.420983\pi$
$643$	12.4605	0.491394	0.245697	+	0.969347i	$0.420983\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	9.48683	0.373254
$647$	15.9737	0.627990	0.313995	−	0.949425i	$-0.398333\pi$
$647$	15.9737	0.627990	0.313995	+	0.949425i	$0.398333\pi$
$648$	19.0000	0.746390
$649$	−22.3246	−0.876315
$650$	0	0
$651$	−22.6491	−0.887689
$652$	17.4868	0.684837
$653$	−17.5132	−0.685343	−0.342672	−	0.939455i	$-0.611332\pi$
$653$	−17.5132	−0.685343	−0.342672	+	0.939455i	$0.611332\pi$
$654$	33.6754	1.31681
$655$	0	0
$656$	10.3246	0.403106
$657$	19.8641	0.774971
$658$	3.00000	0.116952
$659$	23.2982	0.907570	0.453785	−	0.891111i	$-0.350073\pi$
$659$	23.2982	0.907570	0.453785	+	0.891111i	$0.350073\pi$
$660$	0	0
$661$	10.8377	0.421539	0.210769	−	0.977536i	$-0.432403\pi$
$661$	10.8377	0.421539	0.210769	+	0.977536i	$0.432403\pi$
$662$	24.6491	0.958015
$663$	0	0
$664$	9.48683	0.368161
$665$	0	0
$666$	−1.13594	−0.0440169
$667$	−61.9473	−2.39861
$668$	13.3246	0.515543
$669$	−10.5132	−0.406463
$670$	0	0
$671$	16.1886	0.624954
$672$	3.16228	0.121988
$673$	−14.9737	−0.577192	−0.288596	−	0.957451i	$-0.593189\pi$
$673$	−14.9737	−0.577192	−0.288596	+	0.957451i	$0.593189\pi$
$674$	−17.4868	−0.673568
$675$	0	0
$676$	0	0
$677$	12.9737	0.498618	0.249309	−	0.968424i	$-0.419796\pi$
$677$	12.9737	0.498618	0.249309	+	0.968424i	$0.419796\pi$
$678$	−27.3509	−1.05040
$679$	7.48683	0.287318
$680$	0	0
$681$	−32.6491	−1.25112
$682$	15.4868	0.593021
$683$	19.6754	0.752860	0.376430	−	0.926445i	$-0.377152\pi$
$683$	19.6754	0.752860	0.376430	+	0.926445i	$0.377152\pi$
$684$	12.8641	0.491869
$685$	0	0
$686$	13.0000	0.496342
$687$	8.97367	0.342367
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	0	0
$691$	37.1359	1.41272	0.706359	−	0.707854i	$-0.250336\pi$
$691$	37.1359	1.41272	0.706359	+	0.707854i	$0.250336\pi$
$692$	−2.16228	−0.0821975
$693$	15.1359	0.574967
$694$	−2.51317	−0.0953985
$695$	0	0
$696$	−32.6491	−1.23756
$697$	53.2982	2.01881
$698$	−17.4868	−0.661886
$699$	30.0000	1.13470
$700$	0	0
$701$	−23.1623	−0.874827	−0.437414	−	0.899260i	$-0.644105\pi$
$701$	−23.1623	−0.874827	−0.437414	+	0.899260i	$0.644105\pi$
$702$	0	0
$703$	−0.298221	−0.0112476
$704$	−2.16228	−0.0814939
$705$	0	0
$706$	5.16228	0.194285
$707$	11.1623	0.419801
$708$	−32.6491	−1.22703
$709$	−41.4868	−1.55807	−0.779035	−	0.626980i	$-0.784291\pi$
$709$	−41.4868	−1.55807	−0.779035	+	0.626980i	$0.784291\pi$
$710$	0	0
$711$	−94.4078	−3.54057
$712$	7.32456	0.274499
$713$	42.9737	1.60938
$714$	16.3246	0.610931
$715$	0	0
$716$	22.3246	0.834308
$717$	35.2982	1.31824
$718$	6.00000	0.223918
$719$	12.0000	0.447524	0.223762	−	0.974644i	$-0.428166\pi$
$719$	12.0000	0.447524	0.223762	+	0.974644i	$0.428166\pi$
$720$	0	0
$721$	0.675445	0.0251549
$722$	−15.6228	−0.581420
$723$	−82.1359	−3.05467
$724$	9.81139	0.364637
$725$	0	0
$726$	20.0000	0.742270
$727$	20.6754	0.766810	0.383405	−	0.923580i	$-0.374751\pi$
$727$	20.6754	0.766810	0.383405	+	0.923580i	$0.374751\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−10.3246	−0.381867
$732$	23.6754	0.875070
$733$	8.48683	0.313468	0.156734	−	0.987641i	$-0.449903\pi$
$733$	8.48683	0.313468	0.156734	+	0.987641i	$0.449903\pi$
$734$	−4.64911	−0.171602
$735$	0	0
$736$	−6.00000	−0.221163
$737$	5.02633	0.185147
$738$	72.2719	2.66036
$739$	44.8114	1.64841	0.824207	−	0.566289i	$-0.191621\pi$
$739$	44.8114	1.64841	0.824207	+	0.566289i	$0.191621\pi$
$740$	0	0
$741$	0	0
$742$	−2.16228	−0.0793797
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	22.6491	0.830357
$745$	0	0
$746$	30.6491	1.12214
$747$	66.4078	2.42974
$748$	−11.1623	−0.408133
$749$	−3.48683	−0.127406
$750$	0	0
$751$	−34.9737	−1.27621	−0.638104	−	0.769951i	$-0.720281\pi$
$751$	−34.9737	−1.27621	−0.638104	+	0.769951i	$0.720281\pi$
$752$	−3.00000	−0.109399
$753$	−39.4868	−1.43898
$754$	0	0
$755$	0	0
$756$	12.6491	0.460044
$757$	−34.1623	−1.24165	−0.620825	−	0.783950i	$-0.713202\pi$
$757$	−34.1623	−1.24165	−0.620825	+	0.783950i	$0.713202\pi$
$758$	38.8114	1.40969
$759$	−41.0263	−1.48916
$760$	0	0
$761$	8.29822	0.300810	0.150405	−	0.988624i	$-0.451942\pi$
$761$	8.29822	0.300810	0.150405	+	0.988624i	$0.451942\pi$
$762$	10.5132	0.380852
$763$	10.6491	0.385524
$764$	9.48683	0.343222
$765$	0	0
$766$	30.0000	1.08394
$767$	0	0
$768$	−3.16228	−0.114109
$769$	−6.32456	−0.228069	−0.114035	−	0.993477i	$-0.536377\pi$
$769$	−6.32456	−0.228069	−0.114035	+	0.993477i	$0.536377\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	47.7851	1.71871	0.859354	−	0.511380i	$-0.170866\pi$
$773$	47.7851	1.71871	0.859354	+	0.511380i	$0.170866\pi$
$774$	−14.0000	−0.503220
$775$	0	0
$776$	−7.48683	−0.268762
$777$	−0.513167	−0.0184098
$778$	12.8377	0.460255
$779$	18.9737	0.679802
$780$	0	0
$781$	−9.35089	−0.334601
$782$	−30.9737	−1.10762
$783$	−130.596	−4.66714
$784$	−6.00000	−0.214286
$785$	0	0
$786$	34.1886	1.21947
$787$	−46.1359	−1.64457	−0.822284	−	0.569077i	$-0.807301\pi$
$787$	−46.1359	−1.64457	−0.822284	+	0.569077i	$0.807301\pi$
$788$	−18.4868	−0.658566
$789$	−36.8377	−1.31146
$790$	0	0
$791$	−8.64911	−0.307527
$792$	−15.1359	−0.537832
$793$	0	0
$794$	38.4868	1.36585
$795$	0	0
$796$	−1.35089	−0.0478810
$797$	13.6754	0.484409	0.242205	−	0.970225i	$-0.422129\pi$
$797$	13.6754	0.484409	0.242205	+	0.970225i	$0.422129\pi$
$798$	5.81139	0.205721
$799$	−15.4868	−0.547885
$800$	0	0
$801$	51.2719	1.81160
$802$	−21.9737	−0.775917
$803$	−6.13594	−0.216533
$804$	7.35089	0.259246
$805$	0	0
$806$	0	0
$807$	−7.94733	−0.279759
$808$	−11.1623	−0.392688
$809$	10.3246	0.362992	0.181496	−	0.983392i	$-0.441906\pi$
$809$	10.3246	0.362992	0.181496	+	0.983392i	$0.441906\pi$
$810$	0	0
$811$	6.16228	0.216387	0.108193	−	0.994130i	$-0.465493\pi$
$811$	6.16228	0.216387	0.108193	+	0.994130i	$0.465493\pi$
$812$	−10.3246	−0.362321
$813$	20.0000	0.701431
$814$	0.350889	0.0122987
$815$	0	0
$816$	−16.3246	−0.571474
$817$	−3.67544	−0.128588
$818$	21.6491	0.756943
$819$	0	0
$820$	0	0
$821$	5.16228	0.180165	0.0900824	−	0.995934i	$-0.471287\pi$
$821$	5.16228	0.180165	0.0900824	+	0.995934i	$0.471287\pi$
$822$	−48.9737	−1.70815
$823$	−47.0000	−1.63832	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	−47.0000	−1.63832	−0.819159	+	0.573567i	$0.805559\pi$
$824$	−0.675445	−0.0235302
$825$	0	0
$826$	−10.3246	−0.359237
$827$	−51.4868	−1.79037	−0.895186	−	0.445692i	$-0.852958\pi$
$827$	−51.4868	−1.79037	−0.895186	+	0.445692i	$0.852958\pi$
$828$	−42.0000	−1.45960
$829$	32.0000	1.11141	0.555703	−	0.831381i	$-0.312449\pi$
$829$	32.0000	1.11141	0.555703	+	0.831381i	$0.312449\pi$
$830$	0	0
$831$	4.78505	0.165992
$832$	0	0
$833$	−30.9737	−1.07317
$834$	38.4605	1.33178
$835$	0	0
$836$	−3.97367	−0.137432
$837$	90.5964	3.13147
$838$	14.6491	0.506045
$839$	23.1623	0.799651	0.399825	−	0.916591i	$-0.369071\pi$
$839$	23.1623	0.799651	0.399825	+	0.916591i	$0.369071\pi$
$840$	0	0
$841$	77.5964	2.67574
$842$	3.16228	0.108979
$843$	−60.0000	−2.06651
$844$	11.8377	0.407471
$845$	0	0
$846$	−21.0000	−0.721995
$847$	6.32456	0.217314
$848$	2.16228	0.0742529
$849$	−69.5701	−2.38764
$850$	0	0
$851$	0.973666	0.0333768
$852$	−13.6754	−0.468513
$853$	56.2719	1.92671	0.963356	−	0.268225i	$-0.0864370\pi$
$853$	56.2719	1.92671	0.963356	+	0.268225i	$0.0864370\pi$
$854$	7.48683	0.256194
$855$	0	0
$856$	3.48683	0.119177
$857$	−30.1359	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$857$	−30.1359	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$858$	0	0
$859$	47.1359	1.60826	0.804129	−	0.594455i	$-0.202632\pi$
$859$	47.1359	1.60826	0.804129	+	0.594455i	$0.202632\pi$
$860$	0	0
$861$	32.6491	1.11268
$862$	−15.4868	−0.527484
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	−12.6491	−0.430331
$865$	0	0
$866$	−6.32456	−0.214917
$867$	−30.5132	−1.03628
$868$	7.16228	0.243104
$869$	29.1623	0.989263
$870$	0	0
$871$	0	0
$872$	−10.6491	−0.360624
$873$	−52.4078	−1.77374
$874$	−11.0263	−0.372971
$875$	0	0
$876$	−8.97367	−0.303192
$877$	−23.6754	−0.799463	−0.399731	−	0.916632i	$-0.630897\pi$
$877$	−23.6754	−0.799463	−0.399731	+	0.916632i	$0.630897\pi$
$878$	3.81139	0.128628
$879$	69.0569	2.32923
$880$	0	0
$881$	−57.9737	−1.95318	−0.976591	−	0.215104i	$-0.930991\pi$
$881$	−57.9737	−1.95318	−0.976591	+	0.215104i	$0.930991\pi$
$882$	−42.0000	−1.41421
$883$	−35.4868	−1.19423	−0.597114	−	0.802157i	$-0.703686\pi$
$883$	−35.4868	−1.19423	−0.597114	+	0.802157i	$0.703686\pi$
$884$	0	0
$885$	0	0
$886$	−26.6491	−0.895294
$887$	−36.6228	−1.22967	−0.614836	−	0.788655i	$-0.710778\pi$
$887$	−36.6228	−1.22967	−0.614836	+	0.788655i	$0.710778\pi$
$888$	0.513167	0.0172208
$889$	3.32456	0.111502
$890$	0	0
$891$	−41.0833	−1.37634
$892$	3.32456	0.111314
$893$	−5.51317	−0.184491
$894$	51.6228	1.72652
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	2.02633	0.0676196
$899$	−73.9473	−2.46628
$900$	0	0
$901$	11.1623	0.371869
$902$	−22.3246	−0.743326
$903$	−6.32456	−0.210468
$904$	8.64911	0.287665
$905$	0	0
$906$	−15.2982	−0.508249
$907$	16.1359	0.535785	0.267893	−	0.963449i	$-0.413673\pi$
$907$	16.1359	0.535785	0.267893	+	0.963449i	$0.413673\pi$
$908$	10.3246	0.342632
$909$	−78.1359	−2.59161
$910$	0	0
$911$	−17.2982	−0.573116	−0.286558	−	0.958063i	$-0.592511\pi$
$911$	−17.2982	−0.573116	−0.286558	+	0.958063i	$0.592511\pi$
$912$	−5.81139	−0.192434
$913$	−20.5132	−0.678887
$914$	24.4605	0.809081
$915$	0	0
$916$	−2.83772	−0.0937610
$917$	10.8114	0.357023
$918$	−65.2982	−2.15516
$919$	−11.6754	−0.385137	−0.192569	−	0.981283i	$-0.561682\pi$
$919$	−11.6754	−0.385137	−0.192569	+	0.981283i	$0.561682\pi$
$920$	0	0
$921$	−49.5701	−1.63339
$922$	−21.4868	−0.707631
$923$	0	0
$924$	−6.83772	−0.224945
$925$	0	0
$926$	12.3246	0.405010
$927$	−4.72811	−0.155292
$928$	10.3246	0.338920
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	−11.0263	−0.361374
$932$	−9.48683	−0.310752
$933$	11.0263	0.360986
$934$	−32.6491	−1.06831
$935$	0	0
$936$	0	0
$937$	−27.6754	−0.904117	−0.452059	−	0.891988i	$-0.649310\pi$
$937$	−27.6754	−0.904117	−0.452059	+	0.891988i	$0.649310\pi$
$938$	2.32456	0.0758994
$939$	−26.3246	−0.859069
$940$	0	0
$941$	−22.3246	−0.727760	−0.363880	−	0.931446i	$-0.618548\pi$
$941$	−22.3246	−0.727760	−0.363880	+	0.931446i	$0.618548\pi$
$942$	−38.4605	−1.25311
$943$	−61.9473	−2.01728
$944$	10.3246	0.336036
$945$	0	0
$946$	4.32456	0.140603
$947$	−1.81139	−0.0588622	−0.0294311	−	0.999567i	$-0.509370\pi$
$947$	−1.81139	−0.0588622	−0.0294311	+	0.999567i	$0.509370\pi$
$948$	42.6491	1.38518
$949$	0	0
$950$	0	0
$951$	−39.4868	−1.28045
$952$	−5.16228	−0.167310
$953$	2.51317	0.0814095	0.0407047	−	0.999171i	$-0.487040\pi$
$953$	2.51317	0.0814095	0.0407047	+	0.999171i	$0.487040\pi$
$954$	15.1359	0.490044
$955$	0	0
$956$	−11.1623	−0.361014
$957$	70.5964	2.28206
$958$	12.1359	0.392095
$959$	−15.4868	−0.500096
$960$	0	0
$961$	20.2982	0.654781
$962$	0	0
$963$	24.4078	0.786531
$964$	25.9737	0.836555
$965$	0	0
$966$	−18.9737	−0.610468
$967$	−16.6228	−0.534552	−0.267276	−	0.963620i	$-0.586124\pi$
$967$	−16.6228	−0.534552	−0.267276	+	0.963620i	$0.586124\pi$
$968$	−6.32456	−0.203279
$969$	−30.0000	−0.963739
$970$	0	0
$971$	59.7851	1.91859	0.959297	−	0.282400i	$-0.0911304\pi$
$971$	59.7851	1.91859	0.959297	+	0.282400i	$0.0911304\pi$
$972$	−22.1359	−0.710011
$973$	12.1623	0.389905
$974$	−1.00000	−0.0320421
$975$	0	0
$976$	−7.48683	−0.239648
$977$	18.9737	0.607021	0.303511	−	0.952828i	$-0.401841\pi$
$977$	18.9737	0.607021	0.303511	+	0.952828i	$0.401841\pi$
$978$	−55.2982	−1.76824
$979$	−15.8377	−0.506176
$980$	0	0
$981$	−74.5438	−2.38000
$982$	17.5132	0.558868
$983$	30.3509	0.968043	0.484022	−	0.875056i	$-0.339176\pi$
$983$	30.3509	0.968043	0.484022	+	0.875056i	$0.339176\pi$
$984$	−32.6491	−1.04082
$985$	0	0
$986$	53.2982	1.69736
$987$	−9.48683	−0.301969
$988$	0	0
$989$	12.0000	0.381578
$990$	0	0
$991$	−39.2982	−1.24835	−0.624175	−	0.781285i	$-0.714565\pi$
$991$	−39.2982	−1.24835	−0.624175	+	0.781285i	$0.714565\pi$
$992$	−7.16228	−0.227403
$993$	−77.9473	−2.47358
$994$	−4.32456	−0.137167
$995$	0	0
$996$	−30.0000	−0.950586
$997$	−11.8377	−0.374904	−0.187452	−	0.982274i	$-0.560023\pi$
$997$	−11.8377	−0.374904	−0.187452	+	0.982274i	$0.560023\pi$
$998$	22.0000	0.696398
$999$	2.05267	0.0649435

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.bj.1.1		2
5.4	even	2		1690.2.a.k.1.2		2
13.4	even	6		650.2.e.h.601.2		4
13.10	even	6		650.2.e.h.451.2		4
13.12	even	2		8450.2.a.bc.1.1		2
65.4	even	6		130.2.e.c.81.1	yes	4
65.9	even	6		1690.2.e.m.991.1		4
65.17	odd	12		650.2.o.g.549.2		8
65.19	odd	12		1690.2.l.k.361.3		8
65.23	odd	12		650.2.o.g.399.2		8
65.24	odd	12		1690.2.l.k.1161.3		8
65.29	even	6		1690.2.e.m.191.1		4
65.34	odd	4		1690.2.d.g.1351.2		4
65.43	odd	12		650.2.o.g.549.3		8
65.44	odd	4		1690.2.d.g.1351.4		4
65.49	even	6		130.2.e.c.61.1	✓	4
65.54	odd	12		1690.2.l.k.1161.1		8
65.59	odd	12		1690.2.l.k.361.1		8
65.62	odd	12		650.2.o.g.399.3		8
65.64	even	2		1690.2.a.n.1.2		2
195.134	odd	6		1170.2.i.q.991.2		4
195.179	odd	6		1170.2.i.q.451.2		4
260.179	odd	6		1040.2.q.m.321.2		4
260.199	odd	6		1040.2.q.m.81.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.e.c.61.1	✓	4	65.49	even	6
130.2.e.c.81.1	yes	4	65.4	even	6
650.2.e.h.451.2		4	13.10	even	6
650.2.e.h.601.2		4	13.4	even	6
650.2.o.g.399.2		8	65.23	odd	12
650.2.o.g.399.3		8	65.62	odd	12
650.2.o.g.549.2		8	65.17	odd	12
650.2.o.g.549.3		8	65.43	odd	12
1040.2.q.m.81.2		4	260.199	odd	6
1040.2.q.m.321.2		4	260.179	odd	6
1170.2.i.q.451.2		4	195.179	odd	6
1170.2.i.q.991.2		4	195.134	odd	6
1690.2.a.k.1.2		2	5.4	even	2
1690.2.a.n.1.2		2	65.64	even	2
1690.2.d.g.1351.2		4	65.34	odd	4
1690.2.d.g.1351.4		4	65.44	odd	4
1690.2.e.m.191.1		4	65.29	even	6
1690.2.e.m.991.1		4	65.9	even	6
1690.2.l.k.361.1		8	65.59	odd	12
1690.2.l.k.361.3		8	65.19	odd	12
1690.2.l.k.1161.1		8	65.54	odd	12
1690.2.l.k.1161.3		8	65.24	odd	12
8450.2.a.bc.1.1		2	13.12	even	2
8450.2.a.bj.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.16228 q^{3} +1.00000 q^{4} -3.16228 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.16228 q^{3} +1.00000 q^{4} -3.16228 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.00000 q^{9} -2.16228 q^{11} -3.16228 q^{12} -1.00000 q^{14} +1.00000 q^{16} +5.16228 q^{17} +7.00000 q^{18} +1.83772 q^{19} +3.16228 q^{21} -2.16228 q^{22} -6.00000 q^{23} -3.16228 q^{24} -12.6491 q^{27} -1.00000 q^{28} +10.3246 q^{29} -7.16228 q^{31} +1.00000 q^{32} +6.83772 q^{33} +5.16228 q^{34} +7.00000 q^{36} -0.162278 q^{37} +1.83772 q^{38} +10.3246 q^{41} +3.16228 q^{42} -2.00000 q^{43} -2.16228 q^{44} -6.00000 q^{46} -3.00000 q^{47} -3.16228 q^{48} -6.00000 q^{49} -16.3246 q^{51} +2.16228 q^{53} -12.6491 q^{54} -1.00000 q^{56} -5.81139 q^{57} +10.3246 q^{58} +10.3246 q^{59} -7.48683 q^{61} -7.16228 q^{62} -7.00000 q^{63} +1.00000 q^{64} +6.83772 q^{66} -2.32456 q^{67} +5.16228 q^{68} +18.9737 q^{69} +4.32456 q^{71} +7.00000 q^{72} +2.83772 q^{73} -0.162278 q^{74} +1.83772 q^{76} +2.16228 q^{77} -13.4868 q^{79} +19.0000 q^{81} +10.3246 q^{82} +9.48683 q^{83} +3.16228 q^{84} -2.00000 q^{86} -32.6491 q^{87} -2.16228 q^{88} +7.32456 q^{89} -6.00000 q^{92} +22.6491 q^{93} -3.00000 q^{94} -3.16228 q^{96} -7.48683 q^{97} -6.00000 q^{98} -15.1359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 14 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 14 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} + 14 q^{9} + 2 q^{11} - 2 q^{14} + 2 q^{16} + 4 q^{17} + 14 q^{18} + 10 q^{19} + 2 q^{22} - 12 q^{23} - 2 q^{28} + 8 q^{29} - 8 q^{31} + 2 q^{32} + 20 q^{33} + 4 q^{34} + 14 q^{36} + 6 q^{37} + 10 q^{38} + 8 q^{41} - 4 q^{43} + 2 q^{44} - 12 q^{46} - 6 q^{47} - 12 q^{49} - 20 q^{51} - 2 q^{53} - 2 q^{56} + 20 q^{57} + 8 q^{58} + 8 q^{59} + 4 q^{61} - 8 q^{62} - 14 q^{63} + 2 q^{64} + 20 q^{66} + 8 q^{67} + 4 q^{68} - 4 q^{71} + 14 q^{72} + 12 q^{73} + 6 q^{74} + 10 q^{76} - 2 q^{77} - 8 q^{79} + 38 q^{81} + 8 q^{82} - 4 q^{86} - 40 q^{87} + 2 q^{88} + 2 q^{89} - 12 q^{92} + 20 q^{93} - 6 q^{94} + 4 q^{97} - 12 q^{98} + 14 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 + 14 * q^9 + 2 * q^11 - 2 * q^14 + 2 * q^16 + 4 * q^17 + 14 * q^18 + 10 * q^19 + 2 * q^22 - 12 * q^23 - 2 * q^28 + 8 * q^29 - 8 * q^31 + 2 * q^32 + 20 * q^33 + 4 * q^34 + 14 * q^36 + 6 * q^37 + 10 * q^38 + 8 * q^41 - 4 * q^43 + 2 * q^44 - 12 * q^46 - 6 * q^47 - 12 * q^49 - 20 * q^51 - 2 * q^53 - 2 * q^56 + 20 * q^57 + 8 * q^58 + 8 * q^59 + 4 * q^61 - 8 * q^62 - 14 * q^63 + 2 * q^64 + 20 * q^66 + 8 * q^67 + 4 * q^68 - 4 * q^71 + 14 * q^72 + 12 * q^73 + 6 * q^74 + 10 * q^76 - 2 * q^77 - 8 * q^79 + 38 * q^81 + 8 * q^82 - 4 * q^86 - 40 * q^87 + 2 * q^88 + 2 * q^89 - 12 * q^92 + 20 * q^93 - 6 * q^94 + 4 * q^97 - 12 * q^98 + 14 * q^99

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