LMFDB - Embedded newform 2205.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.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:	$17.6070136457$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.210756$ of defining polynomial
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21076 q^{2} -0.534070 q^{4} +1.00000 q^{5} +3.06814 q^{8} +O(q^{10})$	$q-1.21076 q^{2} -0.534070 q^{4} +1.00000 q^{5} +3.06814 q^{8} -1.21076 q^{10} -0.255174 q^{11} +0.744826 q^{13} -2.64663 q^{16} -1.21076 q^{17} -1.11256 q^{19} -0.534070 q^{20} +0.308953 q^{22} -7.85738 q^{23} +1.00000 q^{25} -0.901803 q^{26} -3.32331 q^{29} -6.91116 q^{31} -2.93186 q^{32} +1.46593 q^{34} +9.27890 q^{37} +1.34704 q^{38} +3.06814 q^{40} +8.81297 q^{41} +5.70041 q^{43} +0.136281 q^{44} +9.51337 q^{46} -7.91116 q^{47} -1.21076 q^{50} -0.397789 q^{52} -2.42151 q^{53} -0.255174 q^{55} +4.02372 q^{58} -11.8811 q^{59} +10.9556 q^{61} +8.36773 q^{62} +8.84302 q^{64} +0.744826 q^{65} -9.16634 q^{67} +0.646629 q^{68} -10.5878 q^{71} +0.435873 q^{73} -11.2345 q^{74} +0.594184 q^{76} +2.37709 q^{79} -2.64663 q^{80} -10.6704 q^{82} +10.2789 q^{83} -1.21076 q^{85} -6.90180 q^{86} -0.782909 q^{88} -14.1663 q^{89} +4.19639 q^{92} +9.57849 q^{94} -1.11256 q^{95} -8.44523 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 6 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 6 * q^4 + 3 * q^5 - 6 * q^8	$ 3 q - 2 q^{2} + 6 q^{4} + 3 q^{5} - 6 q^{8} - 2 q^{10} - 10 q^{11} - 7 q^{13} + 4 q^{16} - 2 q^{17} + q^{19} + 6 q^{20} + 2 q^{22} - 10 q^{23} + 3 q^{25} - 4 q^{29} + q^{31} - 24 q^{32} + 12 q^{34} + 11 q^{37} - 28 q^{38} - 6 q^{40} + 2 q^{41} - 3 q^{43} - 30 q^{44} - 16 q^{46} - 2 q^{47} - 2 q^{50} - 24 q^{52} - 4 q^{53} - 10 q^{55} - 14 q^{58} + 4 q^{59} + 22 q^{61} + 30 q^{62} + 20 q^{64} - 7 q^{65} - 15 q^{67} - 10 q^{68} - 16 q^{71} - 9 q^{73} - 6 q^{74} + 30 q^{76} - 7 q^{79} + 4 q^{80} + 6 q^{82} + 14 q^{83} - 2 q^{85} - 18 q^{86} + 40 q^{88} - 30 q^{89} + 18 q^{92} + 32 q^{94} + q^{95} + 4 q^{97}+O(q^{100}) $ 3 * q - 2 * q^2 + 6 * q^4 + 3 * q^5 - 6 * q^8 - 2 * q^10 - 10 * q^11 - 7 * q^13 + 4 * q^16 - 2 * q^17 + q^19 + 6 * q^20 + 2 * q^22 - 10 * q^23 + 3 * q^25 - 4 * q^29 + q^31 - 24 * q^32 + 12 * q^34 + 11 * q^37 - 28 * q^38 - 6 * q^40 + 2 * q^41 - 3 * q^43 - 30 * q^44 - 16 * q^46 - 2 * q^47 - 2 * q^50 - 24 * q^52 - 4 * q^53 - 10 * q^55 - 14 * q^58 + 4 * q^59 + 22 * q^61 + 30 * q^62 + 20 * q^64 - 7 * q^65 - 15 * q^67 - 10 * q^68 - 16 * q^71 - 9 * q^73 - 6 * q^74 + 30 * q^76 - 7 * q^79 + 4 * q^80 + 6 * q^82 + 14 * q^83 - 2 * q^85 - 18 * q^86 + 40 * q^88 - 30 * q^89 + 18 * q^92 + 32 * q^94 + q^95 + 4 * q^97

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 $

$2$	−1.21076	−0.856134	−0.428067	−	0.903747i	$-0.640805\pi$
$2$	−1.21076	−0.856134	−0.428067	+	0.903747i	$0.640805\pi$
$3$	0	0
$3$	0	0
$4$	−0.534070	−0.267035
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.06814	1.08475
$9$	0	0
$10$	−1.21076	−0.382875
$11$	−0.255174	−0.0769378	−0.0384689	−	0.999260i	$-0.512248\pi$
$11$	−0.255174	−0.0769378	−0.0384689	+	0.999260i	$0.512248\pi$
$12$	0	0
$13$	0.744826	0.206578	0.103289	−	0.994651i	$-0.467063\pi$
$13$	0.744826	0.206578	0.103289	+	0.994651i	$0.467063\pi$
$14$	0	0
$15$	0	0
$16$	−2.64663	−0.661657
$17$	−1.21076	−0.293651	−0.146826	−	0.989162i	$-0.546906\pi$
$17$	−1.21076	−0.293651	−0.146826	+	0.989162i	$0.546906\pi$
$18$	0	0
$19$	−1.11256	−0.255238	−0.127619	−	0.991823i	$-0.540734\pi$
$19$	−1.11256	−0.255238	−0.127619	+	0.991823i	$0.540734\pi$
$20$	−0.534070	−0.119422
$21$	0	0
$22$	0.308953	0.0658691
$23$	−7.85738	−1.63838	−0.819189	−	0.573524i	$-0.805576\pi$
$23$	−7.85738	−1.63838	−0.819189	+	0.573524i	$0.805576\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.901803	−0.176858
$27$	0	0
$28$	0	0
$29$	−3.32331	−0.617124	−0.308562	−	0.951204i	$-0.599848\pi$
$29$	−3.32331	−0.617124	−0.308562	+	0.951204i	$0.599848\pi$
$30$	0	0
$31$	−6.91116	−1.24128	−0.620641	−	0.784095i	$-0.713127\pi$
$31$	−6.91116	−1.24128	−0.620641	+	0.784095i	$0.713127\pi$
$32$	−2.93186	−0.518284
$33$	0	0
$34$	1.46593	0.251405
$35$	0	0
$36$	0	0
$37$	9.27890	1.52544	0.762721	−	0.646728i	$-0.223863\pi$
$37$	9.27890	1.52544	0.762721	+	0.646728i	$0.223863\pi$
$38$	1.34704	0.218518
$39$	0	0
$40$	3.06814	0.485116
$41$	8.81297	1.37635	0.688177	−	0.725543i	$-0.258411\pi$
$41$	8.81297	1.37635	0.688177	+	0.725543i	$0.258411\pi$
$42$	0	0
$43$	5.70041	0.869304	0.434652	−	0.900598i	$-0.356871\pi$
$43$	5.70041	0.869304	0.434652	+	0.900598i	$0.356871\pi$
$44$	0.136281	0.0205451
$45$	0	0
$46$	9.51337	1.40267
$47$	−7.91116	−1.15396	−0.576981	−	0.816758i	$-0.695769\pi$
$47$	−7.91116	−1.15396	−0.576981	+	0.816758i	$0.695769\pi$
$48$	0	0
$49$	0	0
$50$	−1.21076	−0.171227
$51$	0	0
$52$	−0.397789	−0.0551635
$53$	−2.42151	−0.332620	−0.166310	−	0.986073i	$-0.553185\pi$
$53$	−2.42151	−0.332620	−0.166310	+	0.986073i	$0.553185\pi$
$54$	0	0
$55$	−0.255174	−0.0344076
$56$	0	0
$57$	0	0
$58$	4.02372	0.528341
$59$	−11.8811	−1.54679	−0.773394	−	0.633925i	$-0.781443\pi$
$59$	−11.8811	−1.54679	−0.773394	+	0.633925i	$0.781443\pi$
$60$	0	0
$61$	10.9556	1.40272	0.701359	−	0.712808i	$-0.252577\pi$
$61$	10.9556	1.40272	0.701359	+	0.712808i	$0.252577\pi$
$62$	8.36773	1.06270
$63$	0	0
$64$	8.84302	1.10538
$65$	0.744826	0.0923843
$66$	0	0
$67$	−9.16634	−1.11985	−0.559923	−	0.828545i	$-0.689169\pi$
$67$	−9.16634	−1.11985	−0.559923	+	0.828545i	$0.689169\pi$
$68$	0.646629	0.0784152
$69$	0	0
$70$	0	0
$71$	−10.5878	−1.25655	−0.628273	−	0.777993i	$-0.716238\pi$
$71$	−10.5878	−1.25655	−0.628273	+	0.777993i	$0.716238\pi$
$72$	0	0
$73$	0.435873	0.0510150	0.0255075	−	0.999675i	$-0.491880\pi$
$73$	0.435873	0.0510150	0.0255075	+	0.999675i	$0.491880\pi$
$74$	−11.2345	−1.30598
$75$	0	0
$76$	0.594184	0.0681576
$77$	0	0
$78$	0	0
$79$	2.37709	0.267444	0.133722	−	0.991019i	$-0.457307\pi$
$79$	2.37709	0.267444	0.133722	+	0.991019i	$0.457307\pi$
$80$	−2.64663	−0.295902
$81$	0	0
$82$	−10.6704	−1.17834
$83$	10.2789	1.12826	0.564128	−	0.825688i	$-0.309213\pi$
$83$	10.2789	1.12826	0.564128	+	0.825688i	$0.309213\pi$
$84$	0	0
$85$	−1.21076	−0.131325
$86$	−6.90180	−0.744241
$87$	0	0
$88$	−0.782909	−0.0834584
$89$	−14.1663	−1.50163	−0.750814	−	0.660513i	$-0.770339\pi$
$89$	−14.1663	−1.50163	−0.750814	+	0.660513i	$0.770339\pi$
$90$	0	0
$91$	0	0
$92$	4.19639	0.437504
$93$	0	0
$94$	9.57849	0.987946
$95$	−1.11256	−0.114146
$96$	0	0
$97$	−8.44523	−0.857484	−0.428742	−	0.903427i	$-0.641043\pi$
$97$	−8.44523	−0.857484	−0.428742	+	0.903427i	$0.641043\pi$
$98$	0	0
$99$	0	0
$100$	−0.534070	−0.0534070
$101$	10.0775	1.00275	0.501374	−	0.865230i	$-0.332828\pi$
$101$	10.0775	1.00275	0.501374	+	0.865230i	$0.332828\pi$
$102$	0	0
$103$	4.76855	0.469859	0.234930	−	0.972012i	$-0.424514\pi$
$103$	4.76855	0.469859	0.234930	+	0.972012i	$0.424514\pi$
$104$	2.28523	0.224085
$105$	0	0
$106$	2.93186	0.284767
$107$	−16.2789	−1.57374	−0.786870	−	0.617119i	$-0.788300\pi$
$107$	−16.2789	−1.57374	−0.786870	+	0.617119i	$0.788300\pi$
$108$	0	0
$109$	−1.30895	−0.125375	−0.0626875	−	0.998033i	$-0.519967\pi$
$109$	−1.30895	−0.125375	−0.0626875	+	0.998033i	$0.519967\pi$
$110$	0.308953	0.0294575
$111$	0	0
$112$	0	0
$113$	−6.64663	−0.625262	−0.312631	−	0.949875i	$-0.601210\pi$
$113$	−6.64663	−0.625262	−0.312631	+	0.949875i	$0.601210\pi$
$114$	0	0
$115$	−7.85738	−0.732705
$116$	1.77488	0.164794
$117$	0	0
$118$	14.3851	1.32426
$119$	0	0
$120$	0	0
$121$	−10.9349	−0.994081
$122$	−13.2645	−1.20091
$123$	0	0
$124$	3.69105	0.331466
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.09820	−0.541128	−0.270564	−	0.962702i	$-0.587210\pi$
$127$	−6.09820	−0.541128	−0.270564	+	0.962702i	$0.587210\pi$
$128$	−4.84302	−0.428067
$129$	0	0
$130$	−0.901803	−0.0790933
$131$	21.3120	1.86204	0.931018	−	0.364973i	$-0.118922\pi$
$131$	21.3120	1.86204	0.931018	+	0.364973i	$0.118922\pi$
$132$	0	0
$133$	0	0
$134$	11.0982	0.958738
$135$	0	0
$136$	−3.71477	−0.318539
$137$	−15.2582	−1.30360	−0.651798	−	0.758393i	$-0.725985\pi$
$137$	−15.2582	−1.30360	−0.651798	+	0.758393i	$0.725985\pi$
$138$	0	0
$139$	12.1807	1.03315	0.516577	−	0.856241i	$-0.327206\pi$
$139$	12.1807	1.03315	0.516577	+	0.856241i	$0.327206\pi$
$140$	0	0
$141$	0	0
$142$	12.8193	1.07577
$143$	−0.190060	−0.0158936
$144$	0	0
$145$	−3.32331	−0.275986
$146$	−0.527735	−0.0436757
$147$	0	0
$148$	−4.95558	−0.407346
$149$	−16.9793	−1.39100	−0.695499	−	0.718527i	$-0.744817\pi$
$149$	−16.9793	−1.39100	−0.695499	+	0.718527i	$0.744817\pi$
$150$	0	0
$151$	1.18070	0.0960839	0.0480420	−	0.998845i	$-0.484702\pi$
$151$	1.18070	0.0960839	0.0480420	+	0.998845i	$0.484702\pi$
$152$	−3.41349	−0.276870
$153$	0	0
$154$	0	0
$155$	−6.91116	−0.555118
$156$	0	0
$157$	−17.7148	−1.41379	−0.706896	−	0.707317i	$-0.749905\pi$
$157$	−17.7148	−1.41379	−0.706896	+	0.707317i	$0.749905\pi$
$158$	−2.87808	−0.228968
$159$	0	0
$160$	−2.93186	−0.231784
$161$	0	0
$162$	0	0
$163$	−15.4897	−1.21324	−0.606622	−	0.794991i	$-0.707476\pi$
$163$	−15.4897	−1.21324	−0.606622	+	0.794991i	$0.707476\pi$
$164$	−4.70674	−0.367535
$165$	0	0
$166$	−12.4452	−0.965938
$167$	−11.4359	−0.884934	−0.442467	−	0.896785i	$-0.645897\pi$
$167$	−11.4359	−0.884934	−0.442467	+	0.896785i	$0.645897\pi$
$168$	0	0
$169$	−12.4452	−0.957326
$170$	1.46593	0.112432
$171$	0	0
$172$	−3.04442	−0.232135
$173$	−10.2251	−0.777401	−0.388701	−	0.921364i	$-0.627076\pi$
$173$	−10.2251	−0.777401	−0.388701	+	0.921364i	$0.627076\pi$
$174$	0	0
$175$	0	0
$176$	0.675351	0.0509065
$177$	0	0
$178$	17.1520	1.28560
$179$	1.91116	0.142847	0.0714235	−	0.997446i	$-0.477246\pi$
$179$	1.91116	0.142847	0.0714235	+	0.997446i	$0.477246\pi$
$180$	0	0
$181$	−5.44523	−0.404741	−0.202371	−	0.979309i	$-0.564865\pi$
$181$	−5.44523	−0.404741	−0.202371	+	0.979309i	$0.564865\pi$
$182$	0	0
$183$	0	0
$184$	−24.1076	−1.77723
$185$	9.27890	0.682198
$186$	0	0
$187$	0.308953	0.0225929
$188$	4.22512	0.308148
$189$	0	0
$190$	1.34704	0.0977243
$191$	−4.22512	−0.305719	−0.152859	−	0.988248i	$-0.548848\pi$
$191$	−4.22512	−0.305719	−0.152859	+	0.988248i	$0.548848\pi$
$192$	0	0
$193$	−6.54343	−0.471007	−0.235503	−	0.971874i	$-0.575674\pi$
$193$	−6.54343	−0.471007	−0.235503	+	0.971874i	$0.575674\pi$
$194$	10.2251	0.734121
$195$	0	0
$196$	0	0
$197$	17.1694	1.22327	0.611633	−	0.791141i	$-0.290513\pi$
$197$	17.1694	1.22327	0.611633	+	0.791141i	$0.290513\pi$
$198$	0	0
$199$	25.0030	1.77242	0.886209	−	0.463286i	$-0.153330\pi$
$199$	25.0030	1.77242	0.886209	+	0.463286i	$0.153330\pi$
$200$	3.06814	0.216950
$201$	0	0
$202$	−12.2014	−0.858487
$203$	0	0
$204$	0	0
$205$	8.81297	0.615524
$206$	−5.77355	−0.402262
$207$	0	0
$208$	−1.97128	−0.136684
$209$	0.283896	0.0196375
$210$	0	0
$211$	−12.2201	−0.841268	−0.420634	−	0.907231i	$-0.638192\pi$
$211$	−12.2201	−0.841268	−0.420634	+	0.907231i	$0.638192\pi$
$212$	1.29326	0.0888213
$213$	0	0
$214$	19.7098	1.34733
$215$	5.70041	0.388765
$216$	0	0
$217$	0	0
$218$	1.58482	0.107338
$219$	0	0
$220$	0.136281	0.00918805
$221$	−0.901803	−0.0606618
$222$	0	0
$223$	−18.9793	−1.27095	−0.635474	−	0.772122i	$-0.719195\pi$
$223$	−18.9793	−1.27095	−0.635474	+	0.772122i	$0.719195\pi$
$224$	0	0
$225$	0	0
$226$	8.04744	0.535308
$227$	−2.98564	−0.198164	−0.0990819	−	0.995079i	$-0.531591\pi$
$227$	−2.98564	−0.198164	−0.0990819	+	0.995079i	$0.531591\pi$
$228$	0	0
$229$	−3.97930	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$229$	−3.97930	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$230$	9.51337	0.627293
$231$	0	0
$232$	−10.1964	−0.669426
$233$	11.6259	0.761640	0.380820	−	0.924649i	$-0.375642\pi$
$233$	11.6259	0.761640	0.380820	+	0.924649i	$0.375642\pi$
$234$	0	0
$235$	−7.91116	−0.516067
$236$	6.34535	0.413047
$237$	0	0
$238$	0	0
$239$	−7.02070	−0.454131	−0.227066	−	0.973879i	$-0.572913\pi$
$239$	−7.02070	−0.454131	−0.227066	+	0.973879i	$0.572913\pi$
$240$	0	0
$241$	28.5815	1.84110	0.920549	−	0.390628i	$-0.127742\pi$
$241$	28.5815	1.84110	0.920549	+	0.390628i	$0.127742\pi$
$242$	13.2395	0.851066
$243$	0	0
$244$	−5.85105	−0.374575
$245$	0	0
$246$	0	0
$247$	−0.828663	−0.0527265
$248$	−21.2044	−1.34648
$249$	0	0
$250$	−1.21076	−0.0765749
$251$	−4.48029	−0.282793	−0.141397	−	0.989953i	$-0.545159\pi$
$251$	−4.48029	−0.282793	−0.141397	+	0.989953i	$0.545159\pi$
$252$	0	0
$253$	2.00500	0.126053
$254$	7.38343	0.463278
$255$	0	0
$256$	−11.8223	−0.738895
$257$	−26.2375	−1.63665	−0.818325	−	0.574755i	$-0.805097\pi$
$257$	−26.2375	−1.63665	−0.818325	+	0.574755i	$0.805097\pi$
$258$	0	0
$259$	0	0
$260$	−0.397789	−0.0246699
$261$	0	0
$262$	−25.8036	−1.59415
$263$	−10.8430	−0.668609	−0.334305	−	0.942465i	$-0.608501\pi$
$263$	−10.8430	−0.668609	−0.334305	+	0.942465i	$0.608501\pi$
$264$	0	0
$265$	−2.42151	−0.148752
$266$	0	0
$267$	0	0
$268$	4.89547	0.299038
$269$	27.8510	1.69811	0.849054	−	0.528306i	$-0.177172\pi$
$269$	27.8510	1.69811	0.849054	+	0.528306i	$0.177172\pi$
$270$	0	0
$271$	−3.60221	−0.218819	−0.109409	−	0.993997i	$-0.534896\pi$
$271$	−3.60221	−0.218819	−0.109409	+	0.993997i	$0.534896\pi$
$272$	3.20442	0.194297
$273$	0	0
$274$	18.4740	1.11605
$275$	−0.255174	−0.0153876
$276$	0	0
$277$	2.76855	0.166346	0.0831730	−	0.996535i	$-0.473495\pi$
$277$	2.76855	0.166346	0.0831730	+	0.996535i	$0.473495\pi$
$278$	−14.7479	−0.884517
$279$	0	0
$280$	0	0
$281$	−26.0474	−1.55386	−0.776930	−	0.629587i	$-0.783224\pi$
$281$	−26.0474	−1.55386	−0.776930	+	0.629587i	$0.783224\pi$
$282$	0	0
$283$	3.36773	0.200191	0.100095	−	0.994978i	$-0.468085\pi$
$283$	3.36773	0.200191	0.100095	+	0.994978i	$0.468085\pi$
$284$	5.65465	0.335542
$285$	0	0
$286$	0.230116	0.0136071
$287$	0	0
$288$	0	0
$289$	−15.5341	−0.913769
$290$	4.02372	0.236281
$291$	0	0
$292$	−0.232787	−0.0136228
$293$	−15.8510	−0.926028	−0.463014	−	0.886351i	$-0.653232\pi$
$293$	−15.8510	−0.926028	−0.463014	+	0.886351i	$0.653232\pi$
$294$	0	0
$295$	−11.8811	−0.691745
$296$	28.4690	1.65472
$297$	0	0
$298$	20.5578	1.19088
$299$	−5.85238	−0.338452
$300$	0	0
$301$	0	0
$302$	−1.42954	−0.0822607
$303$	0	0
$304$	2.94453	0.168880
$305$	10.9556	0.627315
$306$	0	0
$307$	−22.8811	−1.30589	−0.652947	−	0.757404i	$-0.726468\pi$
$307$	−22.8811	−1.30589	−0.652947	+	0.757404i	$0.726468\pi$
$308$	0	0
$309$	0	0
$310$	8.36773	0.475255
$311$	19.6560	1.11459	0.557294	−	0.830315i	$-0.311839\pi$
$311$	19.6560	1.11459	0.557294	+	0.830315i	$0.311839\pi$
$312$	0	0
$313$	2.31831	0.131039	0.0655194	−	0.997851i	$-0.479130\pi$
$313$	2.31831	0.131039	0.0655194	+	0.997851i	$0.479130\pi$
$314$	21.4483	1.21040
$315$	0	0
$316$	−1.26953	−0.0714169
$317$	5.43587	0.305309	0.152655	−	0.988280i	$-0.451218\pi$
$317$	5.43587	0.305309	0.152655	+	0.988280i	$0.451218\pi$
$318$	0	0
$319$	0.848023	0.0474802
$320$	8.84302	0.494340
$321$	0	0
$322$	0	0
$323$	1.34704	0.0749511
$324$	0	0
$325$	0.744826	0.0413155
$326$	18.7542	1.03870
$327$	0	0
$328$	27.0394	1.49300
$329$	0	0
$330$	0	0
$331$	12.9112	0.709662	0.354831	−	0.934931i	$-0.384538\pi$
$331$	12.9112	0.709662	0.354831	+	0.934931i	$0.384538\pi$
$332$	−5.48965	−0.301284
$333$	0	0
$334$	13.8461	0.757622
$335$	−9.16634	−0.500811
$336$	0	0
$337$	−29.8604	−1.62660	−0.813300	−	0.581844i	$-0.802331\pi$
$337$	−29.8604	−1.62660	−0.813300	+	0.581844i	$0.802331\pi$
$338$	15.0681	0.819599
$339$	0	0
$340$	0.646629	0.0350684
$341$	1.76355	0.0955015
$342$	0	0
$343$	0	0
$344$	17.4897	0.942979
$345$	0	0
$346$	12.3801	0.665559
$347$	−14.9319	−0.801584	−0.400792	−	0.916169i	$-0.631265\pi$
$347$	−14.9319	−0.801584	−0.400792	+	0.916169i	$0.631265\pi$
$348$	0	0
$349$	30.8304	1.65031	0.825156	−	0.564906i	$-0.191087\pi$
$349$	30.8304	1.65031	0.825156	+	0.564906i	$0.191087\pi$
$350$	0	0
$351$	0	0
$352$	0.748134	0.0398757
$353$	7.82866	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$353$	7.82866	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$354$	0	0
$355$	−10.5878	−0.561945
$356$	7.56582	0.400988
$357$	0	0
$358$	−2.31395	−0.122296
$359$	−5.85238	−0.308877	−0.154439	−	0.988002i	$-0.549357\pi$
$359$	−5.85238	−0.308877	−0.154439	+	0.988002i	$0.549357\pi$
$360$	0	0
$361$	−17.7622	−0.934853
$362$	6.59285	0.346512
$363$	0	0
$364$	0	0
$365$	0.435873	0.0228146
$366$	0	0
$367$	−23.4990	−1.22664	−0.613319	−	0.789835i	$-0.710166\pi$
$367$	−23.4990	−1.22664	−0.613319	+	0.789835i	$0.710166\pi$
$368$	20.7956	1.08404
$369$	0	0
$370$	−11.2345	−0.584053
$371$	0	0
$372$	0	0
$373$	−28.7034	−1.48621	−0.743104	−	0.669176i	$-0.766647\pi$
$373$	−28.7034	−1.48621	−0.743104	+	0.669176i	$0.766647\pi$
$374$	−0.374067	−0.0193425
$375$	0	0
$376$	−24.2726	−1.25176
$377$	−2.47529	−0.127484
$378$	0	0
$379$	11.5103	0.591247	0.295623	−	0.955305i	$-0.404473\pi$
$379$	11.5103	0.591247	0.295623	+	0.955305i	$0.404473\pi$
$380$	0.594184	0.0304810
$381$	0	0
$382$	5.11559	0.261736
$383$	27.7148	1.41616	0.708079	−	0.706133i	$-0.249562\pi$
$383$	27.7148	1.41616	0.708079	+	0.706133i	$0.249562\pi$
$384$	0	0
$385$	0	0
$386$	7.92250	0.403245
$387$	0	0
$388$	4.51035	0.228978
$389$	11.0982	0.562701	0.281350	−	0.959605i	$-0.409218\pi$
$389$	11.0982	0.562701	0.281350	+	0.959605i	$0.409218\pi$
$390$	0	0
$391$	9.51337	0.481112
$392$	0	0
$393$	0	0
$394$	−20.7879	−1.04728
$395$	2.37709	0.119605
$396$	0	0
$397$	20.8574	1.04680	0.523401	−	0.852086i	$-0.324663\pi$
$397$	20.8574	1.04680	0.523401	+	0.852086i	$0.324663\pi$
$398$	−30.2726	−1.51743
$399$	0	0
$400$	−2.64663	−0.132331
$401$	26.6941	1.33304	0.666519	−	0.745488i	$-0.267783\pi$
$401$	26.6941	1.33304	0.666519	+	0.745488i	$0.267783\pi$
$402$	0	0
$403$	−5.14762	−0.256421
$404$	−5.38209	−0.267769
$405$	0	0
$406$	0	0
$407$	−2.36773	−0.117364
$408$	0	0
$409$	5.59918	0.276862	0.138431	−	0.990372i	$-0.455794\pi$
$409$	5.59918	0.276862	0.138431	+	0.990372i	$0.455794\pi$
$410$	−10.6704	−0.526971
$411$	0	0
$412$	−2.54674	−0.125469
$413$	0	0
$414$	0	0
$415$	10.2789	0.504571
$416$	−2.18373	−0.107066
$417$	0	0
$418$	−0.343729	−0.0168123
$419$	−37.4008	−1.82715	−0.913575	−	0.406671i	$-0.866690\pi$
$419$	−37.4008	−1.82715	−0.913575	+	0.406671i	$0.866690\pi$
$420$	0	0
$421$	18.3357	0.893627	0.446814	−	0.894627i	$-0.352559\pi$
$421$	18.3357	0.893627	0.446814	+	0.894627i	$0.352559\pi$
$422$	14.7956	0.720238
$423$	0	0
$424$	−7.42954	−0.360810
$425$	−1.21076	−0.0587303
$426$	0	0
$427$	0	0
$428$	8.69407	0.420244
$429$	0	0
$430$	−6.90180	−0.332834
$431$	20.6941	0.996798	0.498399	−	0.866948i	$-0.333921\pi$
$431$	20.6941	0.996798	0.498399	+	0.866948i	$0.333921\pi$
$432$	0	0
$433$	7.25517	0.348661	0.174331	−	0.984687i	$-0.444224\pi$
$433$	7.25517	0.348661	0.174331	+	0.984687i	$0.444224\pi$
$434$	0	0
$435$	0	0
$436$	0.699073	0.0334795
$437$	8.74180	0.418177
$438$	0	0
$439$	9.69105	0.462528	0.231264	−	0.972891i	$-0.425714\pi$
$439$	9.69105	0.462528	0.231264	+	0.972891i	$0.425714\pi$
$440$	−0.782909	−0.0373237
$441$	0	0
$442$	1.09186	0.0519346
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	−14.1663	−0.671549
$446$	22.9793	1.08810
$447$	0	0
$448$	0	0
$449$	16.2251	0.765711	0.382855	−	0.923808i	$-0.374941\pi$
$449$	16.2251	0.765711	0.382855	+	0.923808i	$0.374941\pi$
$450$	0	0
$451$	−2.24884	−0.105894
$452$	3.54977	0.166967
$453$	0	0
$454$	3.61488	0.169655
$455$	0	0
$456$	0	0
$457$	33.8317	1.58258	0.791290	−	0.611441i	$-0.209410\pi$
$457$	33.8317	1.58258	0.791290	+	0.611441i	$0.209410\pi$
$458$	4.81797	0.225129
$459$	0	0
$460$	4.19639	0.195658
$461$	−14.8417	−0.691246	−0.345623	−	0.938373i	$-0.612332\pi$
$461$	−14.8417	−0.691246	−0.345623	+	0.938373i	$0.612332\pi$
$462$	0	0
$463$	34.3708	1.59734	0.798672	−	0.601766i	$-0.205536\pi$
$463$	34.3708	1.59734	0.798672	+	0.601766i	$0.205536\pi$
$464$	8.79558	0.408324
$465$	0	0
$466$	−14.0762	−0.652066
$467$	2.69407	0.124667	0.0623334	−	0.998055i	$-0.480146\pi$
$467$	2.69407	0.124667	0.0623334	+	0.998055i	$0.480146\pi$
$468$	0	0
$469$	0	0
$470$	9.57849	0.441823
$471$	0	0
$472$	−36.4529	−1.67788
$473$	−1.45460	−0.0668824
$474$	0	0
$475$	−1.11256	−0.0510477
$476$	0	0
$477$	0	0
$478$	8.50035	0.388797
$479$	−14.6654	−0.670077	−0.335039	−	0.942204i	$-0.608749\pi$
$479$	−14.6654	−0.670077	−0.335039	+	0.942204i	$0.608749\pi$
$480$	0	0
$481$	6.91116	0.315122
$482$	−34.6052	−1.57623
$483$	0	0
$484$	5.84000	0.265454
$485$	−8.44523	−0.383478
$486$	0	0
$487$	−7.01739	−0.317988	−0.158994	−	0.987280i	$-0.550825\pi$
$487$	−7.01739	−0.317988	−0.158994	+	0.987280i	$0.550825\pi$
$488$	33.6133	1.52160
$489$	0	0
$490$	0	0
$491$	−21.7148	−0.979974	−0.489987	−	0.871730i	$-0.662998\pi$
$491$	−21.7148	−0.979974	−0.489987	+	0.871730i	$0.662998\pi$
$492$	0	0
$493$	4.02372	0.181219
$494$	1.00331	0.0451410
$495$	0	0
$496$	18.2913	0.821303
$497$	0	0
$498$	0	0
$499$	−11.3327	−0.507320	−0.253660	−	0.967293i	$-0.581634\pi$
$499$	−11.3327	−0.507320	−0.253660	+	0.967293i	$0.581634\pi$
$500$	−0.534070	−0.0238843
$501$	0	0
$502$	5.42454	0.242109
$503$	−2.31395	−0.103174	−0.0515870	−	0.998669i	$-0.516428\pi$
$503$	−2.31395	−0.103174	−0.0515870	+	0.998669i	$0.516428\pi$
$504$	0	0
$505$	10.0775	0.448443
$506$	−2.42757	−0.107918
$507$	0	0
$508$	3.25687	0.144500
$509$	−8.28523	−0.367236	−0.183618	−	0.982998i	$-0.558781\pi$
$509$	−8.28523	−0.367236	−0.183618	+	0.982998i	$0.558781\pi$
$510$	0	0
$511$	0	0
$512$	24.0000	1.06066
$513$	0	0
$514$	31.7672	1.40119
$515$	4.76855	0.210127
$516$	0	0
$517$	2.01872	0.0887833
$518$	0	0
$519$	0	0
$520$	2.28523	0.100214
$521$	18.6466	0.816924	0.408462	−	0.912775i	$-0.366065\pi$
$521$	18.6466	0.816924	0.408462	+	0.912775i	$0.366065\pi$
$522$	0	0
$523$	1.39145	0.0608441	0.0304220	−	0.999537i	$-0.490315\pi$
$523$	1.39145	0.0608441	0.0304220	+	0.999537i	$0.490315\pi$
$524$	−11.3821	−0.497229
$525$	0	0
$526$	13.1283	0.572419
$527$	8.36773	0.364504
$528$	0	0
$529$	38.7385	1.68428
$530$	2.93186	0.127352
$531$	0	0
$532$	0	0
$533$	6.56413	0.284324
$534$	0	0
$535$	−16.2789	−0.703798
$536$	−28.1236	−1.21475
$537$	0	0
$538$	−33.7208	−1.45381
$539$	0	0
$540$	0	0
$541$	10.6259	0.456845	0.228422	−	0.973562i	$-0.426643\pi$
$541$	10.6259	0.456845	0.228422	+	0.973562i	$0.426643\pi$
$542$	4.36140	0.187338
$543$	0	0
$544$	3.54977	0.152195
$545$	−1.30895	−0.0560694
$546$	0	0
$547$	18.5765	0.794274	0.397137	−	0.917759i	$-0.370004\pi$
$547$	18.5765	0.794274	0.397137	+	0.917759i	$0.370004\pi$
$548$	8.14895	0.348106
$549$	0	0
$550$	0.308953	0.0131738
$551$	3.69738	0.157514
$552$	0	0
$553$	0	0
$554$	−3.35204	−0.142414
$555$	0	0
$556$	−6.50535	−0.275888
$557$	36.1363	1.53114	0.765572	−	0.643351i	$-0.222456\pi$
$557$	36.1363	1.53114	0.765572	+	0.643351i	$0.222456\pi$
$558$	0	0
$559$	4.24581	0.179579
$560$	0	0
$561$	0	0
$562$	31.5371	1.33031
$563$	23.7622	1.00146	0.500729	−	0.865604i	$-0.333065\pi$
$563$	23.7622	1.00146	0.500729	+	0.865604i	$0.333065\pi$
$564$	0	0
$565$	−6.64663	−0.279626
$566$	−4.07750	−0.171390
$567$	0	0
$568$	−32.4850	−1.36304
$569$	20.6827	0.867066	0.433533	−	0.901138i	$-0.357267\pi$
$569$	20.6827	0.867066	0.433533	+	0.901138i	$0.357267\pi$
$570$	0	0
$571$	37.3851	1.56452	0.782259	−	0.622953i	$-0.214067\pi$
$571$	37.3851	1.56452	0.782259	+	0.622953i	$0.214067\pi$
$572$	0.101505	0.00424416
$573$	0	0
$574$	0	0
$575$	−7.85738	−0.327676
$576$	0	0
$577$	−34.6433	−1.44222	−0.721110	−	0.692820i	$-0.756368\pi$
$577$	−34.6433	−1.44222	−0.721110	+	0.692820i	$0.756368\pi$
$578$	18.8080	0.782308
$579$	0	0
$580$	1.77488	0.0736980
$581$	0	0
$582$	0	0
$583$	0.617907	0.0255911
$584$	1.33732	0.0553386
$585$	0	0
$586$	19.1918	0.792804
$587$	15.7685	0.650838	0.325419	−	0.945570i	$-0.394495\pi$
$587$	15.7685	0.650838	0.325419	+	0.945570i	$0.394495\pi$
$588$	0	0
$589$	7.68907	0.316823
$590$	14.3851	0.592226
$591$	0	0
$592$	−24.5578	−1.00932
$593$	−23.4359	−0.962396	−0.481198	−	0.876612i	$-0.659798\pi$
$593$	−23.4359	−0.962396	−0.481198	+	0.876612i	$0.659798\pi$
$594$	0	0
$595$	0	0
$596$	9.06814	0.371446
$597$	0	0
$598$	7.08581	0.289760
$599$	−11.0080	−0.449776	−0.224888	−	0.974385i	$-0.572202\pi$
$599$	−11.0080	−0.449776	−0.224888	+	0.974385i	$0.572202\pi$
$600$	0	0
$601$	−12.7385	−0.519614	−0.259807	−	0.965661i	$-0.583659\pi$
$601$	−12.7385	−0.519614	−0.259807	+	0.965661i	$0.583659\pi$
$602$	0	0
$603$	0	0
$604$	−0.630576	−0.0256578
$605$	−10.9349	−0.444566
$606$	0	0
$607$	−25.7054	−1.04335	−0.521675	−	0.853144i	$-0.674693\pi$
$607$	−25.7054	−1.04335	−0.521675	+	0.853144i	$0.674693\pi$
$608$	3.26187	0.132286
$609$	0	0
$610$	−13.2645	−0.537065
$611$	−5.89244	−0.238383
$612$	0	0
$613$	22.6290	0.913975	0.456988	−	0.889473i	$-0.348928\pi$
$613$	22.6290	0.913975	0.456988	+	0.889473i	$0.348928\pi$
$614$	27.7034	1.11802
$615$	0	0
$616$	0	0
$617$	−24.2726	−0.977177	−0.488588	−	0.872514i	$-0.662488\pi$
$617$	−24.2726	−0.977177	−0.488588	+	0.872514i	$0.662488\pi$
$618$	0	0
$619$	10.8510	0.436141	0.218070	−	0.975933i	$-0.430024\pi$
$619$	10.8510	0.436141	0.218070	+	0.975933i	$0.430024\pi$
$620$	3.69105	0.148236
$621$	0	0
$622$	−23.7986	−0.954237
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−2.80691	−0.112187
$627$	0	0
$628$	9.46093	0.377532
$629$	−11.2345	−0.447948
$630$	0	0
$631$	−15.4897	−0.616633	−0.308317	−	0.951284i	$-0.599766\pi$
$631$	−15.4897	−0.616633	−0.308317	+	0.951284i	$0.599766\pi$
$632$	7.29326	0.290110
$633$	0	0
$634$	−6.58151	−0.261385
$635$	−6.09820	−0.242000
$636$	0	0
$637$	0	0
$638$	−1.02675	−0.0406494
$639$	0	0
$640$	−4.84302	−0.191437
$641$	15.3520	0.606369	0.303184	−	0.952932i	$-0.401950\pi$
$641$	15.3520	0.606369	0.303184	+	0.952932i	$0.401950\pi$
$642$	0	0
$643$	41.0775	1.61994	0.809969	−	0.586472i	$-0.199484\pi$
$643$	41.0775	1.61994	0.809969	+	0.586472i	$0.199484\pi$
$644$	0	0
$645$	0	0
$646$	−1.63093	−0.0641682
$647$	−6.08250	−0.239128	−0.119564	−	0.992827i	$-0.538150\pi$
$647$	−6.08250	−0.239128	−0.119564	+	0.992827i	$0.538150\pi$
$648$	0	0
$649$	3.03175	0.119007
$650$	−0.901803	−0.0353716
$651$	0	0
$652$	8.27256	0.323979
$653$	5.27087	0.206265	0.103133	−	0.994668i	$-0.467113\pi$
$653$	5.27087	0.206265	0.103133	+	0.994668i	$0.467113\pi$
$654$	0	0
$655$	21.3120	0.832728
$656$	−23.3246	−0.910675
$657$	0	0
$658$	0	0
$659$	−44.6654	−1.73992	−0.869958	−	0.493127i	$-0.835854\pi$
$659$	−44.6654	−1.73992	−0.869958	+	0.493127i	$0.835854\pi$
$660$	0	0
$661$	−2.99500	−0.116492	−0.0582460	−	0.998302i	$-0.518551\pi$
$661$	−2.99500	−0.116492	−0.0582460	+	0.998302i	$0.518551\pi$
$662$	−15.6323	−0.607565
$663$	0	0
$664$	31.5371	1.22388
$665$	0	0
$666$	0	0
$667$	26.1126	1.01108
$668$	6.10756	0.236309
$669$	0	0
$670$	11.0982	0.428761
$671$	−2.79558	−0.107922
$672$	0	0
$673$	40.3420	1.55507	0.777536	−	0.628839i	$-0.216470\pi$
$673$	40.3420	1.55507	0.777536	+	0.628839i	$0.216470\pi$
$674$	36.1537	1.39259
$675$	0	0
$676$	6.64663	0.255640
$677$	22.5515	0.866723	0.433361	−	0.901220i	$-0.357327\pi$
$677$	22.5515	0.866723	0.433361	+	0.901220i	$0.357327\pi$
$678$	0	0
$679$	0	0
$680$	−3.71477	−0.142455
$681$	0	0
$682$	−2.13523	−0.0817621
$683$	−6.42785	−0.245955	−0.122977	−	0.992409i	$-0.539244\pi$
$683$	−6.42785	−0.245955	−0.122977	+	0.992409i	$0.539244\pi$
$684$	0	0
$685$	−15.2582	−0.582986
$686$	0	0
$687$	0	0
$688$	−15.0869	−0.575181
$689$	−1.80361	−0.0687119
$690$	0	0
$691$	48.2519	1.83559	0.917794	−	0.397057i	$-0.129969\pi$
$691$	48.2519	1.83559	0.917794	+	0.397057i	$0.129969\pi$
$692$	5.46093	0.207593
$693$	0	0
$694$	18.0788	0.686263
$695$	12.1807	0.462040
$696$	0	0
$697$	−10.6704	−0.404168
$698$	−37.3280	−1.41289
$699$	0	0
$700$	0	0
$701$	42.6640	1.61140	0.805699	−	0.592325i	$-0.201790\pi$
$701$	42.6640	1.61140	0.805699	+	0.592325i	$0.201790\pi$
$702$	0	0
$703$	−10.3233	−0.389351
$704$	−2.25651	−0.0850454
$705$	0	0
$706$	−9.47860	−0.356732
$707$	0	0
$708$	0	0
$709$	4.97430	0.186814	0.0934070	−	0.995628i	$-0.470224\pi$
$709$	4.97430	0.186814	0.0934070	+	0.995628i	$0.470224\pi$
$710$	12.8193	0.481100
$711$	0	0
$712$	−43.4643	−1.62889
$713$	54.3037	2.03369
$714$	0	0
$715$	−0.190060	−0.00710785
$716$	−1.02070	−0.0381452
$717$	0	0
$718$	7.08581	0.264440
$719$	−3.44826	−0.128598	−0.0642992	−	0.997931i	$-0.520481\pi$
$719$	−3.44826	−0.128598	−0.0642992	+	0.997931i	$0.520481\pi$
$720$	0	0
$721$	0	0
$722$	21.5057	0.800359
$723$	0	0
$724$	2.90814	0.108080
$725$	−3.32331	−0.123425
$726$	0	0
$727$	−34.6797	−1.28620	−0.643100	−	0.765782i	$-0.722352\pi$
$727$	−34.6797	−1.28620	−0.643100	+	0.765782i	$0.722352\pi$
$728$	0	0
$729$	0	0
$730$	−0.527735	−0.0195324
$731$	−6.90180	−0.255272
$732$	0	0
$733$	−24.7635	−0.914663	−0.457331	−	0.889296i	$-0.651195\pi$
$733$	−24.7635	−0.914663	−0.457331	+	0.889296i	$0.651195\pi$
$734$	28.4516	1.05017
$735$	0	0
$736$	23.0367	0.849146
$737$	2.33901	0.0861586
$738$	0	0
$739$	−16.9162	−0.622271	−0.311136	−	0.950366i	$-0.600709\pi$
$739$	−16.9162	−0.622271	−0.311136	+	0.950366i	$0.600709\pi$
$740$	−4.95558	−0.182171
$741$	0	0
$742$	0	0
$743$	−17.2996	−0.634660	−0.317330	−	0.948315i	$-0.602786\pi$
$743$	−17.2996	−0.634660	−0.317330	+	0.948315i	$0.602786\pi$
$744$	0	0
$745$	−16.9793	−0.622074
$746$	34.7529	1.27239
$747$	0	0
$748$	−0.165003	−0.00603310
$749$	0	0
$750$	0	0
$751$	−34.0969	−1.24421	−0.622106	−	0.782933i	$-0.713723\pi$
$751$	−34.0969	−1.24421	−0.622106	+	0.782933i	$0.713723\pi$
$752$	20.9379	0.763527
$753$	0	0
$754$	2.99697	0.109143
$755$	1.18070	0.0429700
$756$	0	0
$757$	−30.5578	−1.11064	−0.555321	−	0.831636i	$-0.687405\pi$
$757$	−30.5578	−1.11064	−0.555321	+	0.831636i	$0.687405\pi$
$758$	−13.9362	−0.506186
$759$	0	0
$760$	−3.41349	−0.123820
$761$	−4.04878	−0.146768	−0.0733841	−	0.997304i	$-0.523380\pi$
$761$	−4.04878	−0.146768	−0.0733841	+	0.997304i	$0.523380\pi$
$762$	0	0
$763$	0	0
$764$	2.25651	0.0816376
$765$	0	0
$766$	−33.5558	−1.21242
$767$	−8.84936	−0.319532
$768$	0	0
$769$	3.97430	0.143317	0.0716585	−	0.997429i	$-0.477171\pi$
$769$	3.97430	0.143317	0.0716585	+	0.997429i	$0.477171\pi$
$770$	0	0
$771$	0	0
$772$	3.49465	0.125775
$773$	−1.82866	−0.0657724	−0.0328862	−	0.999459i	$-0.510470\pi$
$773$	−1.82866	−0.0657724	−0.0328862	+	0.999459i	$0.510470\pi$
$774$	0	0
$775$	−6.91116	−0.248256
$776$	−25.9112	−0.930157
$777$	0	0
$778$	−13.4372	−0.481747
$779$	−9.80494	−0.351298
$780$	0	0
$781$	2.70174	0.0966760
$782$	−11.5184	−0.411896
$783$	0	0
$784$	0	0
$785$	−17.7148	−0.632267
$786$	0	0
$787$	23.1343	0.824649	0.412325	−	0.911037i	$-0.364717\pi$
$787$	23.1343	0.824649	0.412325	+	0.911037i	$0.364717\pi$
$788$	−9.16965	−0.326655
$789$	0	0
$790$	−2.87808	−0.102397
$791$	0	0
$792$	0	0
$793$	8.16000	0.289770
$794$	−25.2532	−0.896203
$795$	0	0
$796$	−13.3534	−0.473298
$797$	1.07448	0.0380599	0.0190299	−	0.999819i	$-0.493942\pi$
$797$	1.07448	0.0380599	0.0190299	+	0.999819i	$0.493942\pi$
$798$	0	0
$799$	9.57849	0.338863
$800$	−2.93186	−0.103657
$801$	0	0
$802$	−32.3200	−1.14126
$803$	−0.111223	−0.00392499
$804$	0	0
$805$	0	0
$806$	6.23251	0.219531
$807$	0	0
$808$	30.9192	1.08773
$809$	45.4195	1.59687	0.798433	−	0.602084i	$-0.205663\pi$
$809$	45.4195	1.59687	0.798433	+	0.602084i	$0.205663\pi$
$810$	0	0
$811$	40.2074	1.41187	0.705937	−	0.708274i	$-0.250526\pi$
$811$	40.2074	1.41187	0.705937	+	0.708274i	$0.250526\pi$
$812$	0	0
$813$	0	0
$814$	2.86675	0.100479
$815$	−15.4897	−0.542579
$816$	0	0
$817$	−6.34204	−0.221880
$818$	−6.77924	−0.237031
$819$	0	0
$820$	−4.70674	−0.164367
$821$	−43.7923	−1.52836	−0.764180	−	0.645003i	$-0.776856\pi$
$821$	−43.7923	−1.52836	−0.764180	+	0.645003i	$0.776856\pi$
$822$	0	0
$823$	−30.3851	−1.05916	−0.529579	−	0.848260i	$-0.677650\pi$
$823$	−30.3851	−1.05916	−0.529579	+	0.848260i	$0.677650\pi$
$824$	14.6306	0.509680
$825$	0	0
$826$	0	0
$827$	−30.3801	−1.05642	−0.528210	−	0.849114i	$-0.677137\pi$
$827$	−30.3801	−1.05642	−0.528210	+	0.849114i	$0.677137\pi$
$828$	0	0
$829$	40.6209	1.41082	0.705412	−	0.708798i	$-0.250762\pi$
$829$	40.6209	1.41082	0.705412	+	0.708798i	$0.250762\pi$
$830$	−12.4452	−0.431980
$831$	0	0
$832$	6.58651	0.228346
$833$	0	0
$834$	0	0
$835$	−11.4359	−0.395755
$836$	−0.151620	−0.00524390
$837$	0	0
$838$	45.2833	1.56428
$839$	−36.0962	−1.24618	−0.623090	−	0.782150i	$-0.714123\pi$
$839$	−36.0962	−1.24618	−0.623090	+	0.782150i	$0.714123\pi$
$840$	0	0
$841$	−17.9556	−0.619158
$842$	−22.2001	−0.765065
$843$	0	0
$844$	6.52640	0.224648
$845$	−12.4452	−0.428129
$846$	0	0
$847$	0	0
$848$	6.40884	0.220081
$849$	0	0
$850$	1.46593	0.0502810
$851$	−72.9079	−2.49925
$852$	0	0
$853$	−30.4546	−1.04275	−0.521373	−	0.853329i	$-0.674580\pi$
$853$	−30.4546	−1.04275	−0.521373	+	0.853329i	$0.674580\pi$
$854$	0	0
$855$	0	0
$856$	−49.9459	−1.70712
$857$	10.7605	0.367572	0.183786	−	0.982966i	$-0.441165\pi$
$857$	10.7605	0.367572	0.183786	+	0.982966i	$0.441165\pi$
$858$	0	0
$859$	−17.7859	−0.606848	−0.303424	−	0.952856i	$-0.598130\pi$
$859$	−17.7859	−0.606848	−0.303424	+	0.952856i	$0.598130\pi$
$860$	−3.04442	−0.103814
$861$	0	0
$862$	−25.0555	−0.853393
$863$	18.1076	0.616388	0.308194	−	0.951323i	$-0.400275\pi$
$863$	18.1076	0.616388	0.308194	+	0.951323i	$0.400275\pi$
$864$	0	0
$865$	−10.2251	−0.347664
$866$	−8.78424	−0.298501
$867$	0	0
$868$	0	0
$869$	−0.606572	−0.0205766
$870$	0	0
$871$	−6.82733	−0.231335
$872$	−4.01605	−0.136001
$873$	0	0
$874$	−10.5842	−0.358015
$875$	0	0
$876$	0	0
$877$	10.9606	0.370113	0.185056	−	0.982728i	$-0.440753\pi$
$877$	10.9606	0.370113	0.185056	+	0.982728i	$0.440753\pi$
$878$	−11.7335	−0.395986
$879$	0	0
$880$	0.675351	0.0227661
$881$	37.1570	1.25185	0.625925	−	0.779883i	$-0.284722\pi$
$881$	37.1570	1.25185	0.625925	+	0.779883i	$0.284722\pi$
$882$	0	0
$883$	−13.9730	−0.470228	−0.235114	−	0.971968i	$-0.575546\pi$
$883$	−13.9730	−0.470228	−0.235114	+	0.971968i	$0.575546\pi$
$884$	0.481626	0.0161988
$885$	0	0
$886$	7.26454	0.244057
$887$	39.4733	1.32538	0.662692	−	0.748892i	$-0.269414\pi$
$887$	39.4733	1.32538	0.662692	+	0.748892i	$0.269414\pi$
$888$	0	0
$889$	0	0
$890$	17.1520	0.574936
$891$	0	0
$892$	10.1363	0.339388
$893$	8.80163	0.294535
$894$	0	0
$895$	1.91116	0.0638832
$896$	0	0
$897$	0	0
$898$	−19.6447	−0.655551
$899$	22.9680	0.766025
$900$	0	0
$901$	2.93186	0.0976744
$902$	2.72280	0.0906592
$903$	0	0
$904$	−20.3928	−0.678254
$905$	−5.44523	−0.181006
$906$	0	0
$907$	−14.4833	−0.480911	−0.240455	−	0.970660i	$-0.577297\pi$
$907$	−14.4833	−0.480911	−0.240455	+	0.970660i	$0.577297\pi$
$908$	1.59454	0.0529167
$909$	0	0
$910$	0	0
$911$	35.1043	1.16306	0.581528	−	0.813526i	$-0.302455\pi$
$911$	35.1043	1.16306	0.581528	+	0.813526i	$0.302455\pi$
$912$	0	0
$913$	−2.62291	−0.0868055
$914$	−40.9619	−1.35490
$915$	0	0
$916$	2.12523	0.0702195
$917$	0	0
$918$	0	0
$919$	−21.4276	−0.706830	−0.353415	−	0.935467i	$-0.614980\pi$
$919$	−21.4276	−0.706830	−0.353415	+	0.935467i	$0.614980\pi$
$920$	−24.1076	−0.794803
$921$	0	0
$922$	17.9697	0.591799
$923$	−7.88611	−0.259574
$924$	0	0
$925$	9.27890	0.305088
$926$	−41.6146	−1.36754
$927$	0	0
$928$	9.74349	0.319846
$929$	−20.0301	−0.657165	−0.328582	−	0.944475i	$-0.606571\pi$
$929$	−20.0301	−0.657165	−0.328582	+	0.944475i	$0.606571\pi$
$930$	0	0
$931$	0	0
$932$	−6.20906	−0.203385
$933$	0	0
$934$	−3.26187	−0.106731
$935$	0.308953	0.0101039
$936$	0	0
$937$	49.4626	1.61587	0.807937	−	0.589269i	$-0.200584\pi$
$937$	49.4626	1.61587	0.807937	+	0.589269i	$0.200584\pi$
$938$	0	0
$939$	0	0
$940$	4.22512	0.137808
$941$	−42.5051	−1.38563	−0.692813	−	0.721117i	$-0.743629\pi$
$941$	−42.5051	−1.38563	−0.692813	+	0.721117i	$0.743629\pi$
$942$	0	0
$943$	−69.2469	−2.25499
$944$	31.4449	1.02344
$945$	0	0
$946$	1.76116	0.0572603
$947$	−17.8099	−0.578745	−0.289373	−	0.957217i	$-0.593447\pi$
$947$	−17.8099	−0.578745	−0.289373	+	0.957217i	$0.593447\pi$
$948$	0	0
$949$	0.324649	0.0105386
$950$	1.34704	0.0437036
$951$	0	0
$952$	0	0
$953$	23.4897	0.760904	0.380452	−	0.924801i	$-0.375768\pi$
$953$	23.4897	0.760904	0.380452	+	0.924801i	$0.375768\pi$
$954$	0	0
$955$	−4.22512	−0.136722
$956$	3.74954	0.121269
$957$	0	0
$958$	17.7562	0.573676
$959$	0	0
$960$	0	0
$961$	16.7642	0.540780
$962$	−8.36773	−0.269787
$963$	0	0
$964$	−15.2645	−0.491638
$965$	−6.54343	−0.210641
$966$	0	0
$967$	−15.5040	−0.498575	−0.249288	−	0.968429i	$-0.580196\pi$
$967$	−15.5040	−0.498575	−0.249288	+	0.968429i	$0.580196\pi$
$968$	−33.5498	−1.07833
$969$	0	0
$970$	10.2251	0.328309
$971$	−26.9379	−0.864479	−0.432239	−	0.901759i	$-0.642276\pi$
$971$	−26.9379	−0.864479	−0.432239	+	0.901759i	$0.642276\pi$
$972$	0	0
$973$	0	0
$974$	8.49634	0.272240
$975$	0	0
$976$	−28.9954	−0.928119
$977$	43.7272	1.39896	0.699478	−	0.714654i	$-0.253416\pi$
$977$	43.7272	1.39896	0.699478	+	0.714654i	$0.253416\pi$
$978$	0	0
$979$	3.61488	0.115532
$980$	0	0
$981$	0	0
$982$	26.2913	0.838989
$983$	17.2769	0.551048	0.275524	−	0.961294i	$-0.411149\pi$
$983$	17.2769	0.551048	0.275524	+	0.961294i	$0.411149\pi$
$984$	0	0
$985$	17.1694	0.547061
$986$	−4.87175	−0.155148
$987$	0	0
$988$	0.442564	0.0140798
$989$	−44.7903	−1.42425
$990$	0	0
$991$	41.2074	1.30900	0.654499	−	0.756063i	$-0.272880\pi$
$991$	41.2074	1.30900	0.654499	+	0.756063i	$0.272880\pi$
$992$	20.2626	0.643337
$993$	0	0
$994$	0	0
$995$	25.0030	0.792649
$996$	0	0
$997$	9.36773	0.296679	0.148340	−	0.988936i	$-0.452607\pi$
$997$	9.36773	0.296679	0.148340	+	0.988936i	$0.452607\pi$
$998$	13.7211	0.434334
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.a.bc.1.2		3
3.2	odd	2		2205.2.a.bd.1.2		3
7.3	odd	6		315.2.j.g.226.2	yes	6
7.5	odd	6		315.2.j.g.46.2	yes	6
7.6	odd	2		2205.2.a.bb.1.2		3
21.5	even	6		315.2.j.f.46.2	✓	6
21.17	even	6		315.2.j.f.226.2	yes	6
21.20	even	2		2205.2.a.be.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.j.f.46.2	✓	6	21.5	even	6
315.2.j.f.226.2	yes	6	21.17	even	6
315.2.j.g.46.2	yes	6	7.5	odd	6
315.2.j.g.226.2	yes	6	7.3	odd	6
2205.2.a.bb.1.2		3	7.6	odd	2
2205.2.a.bc.1.2		3	1.1	even	1	trivial
2205.2.a.bd.1.2		3	3.2	odd	2
2205.2.a.be.1.2		3	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21076 q^{2} -0.534070 q^{4} +1.00000 q^{5} +3.06814 q^{8} +O(q^{10})\)	\(q-1.21076 q^{2} -0.534070 q^{4} +1.00000 q^{5} +3.06814 q^{8} -1.21076 q^{10} -0.255174 q^{11} +0.744826 q^{13} -2.64663 q^{16} -1.21076 q^{17} -1.11256 q^{19} -0.534070 q^{20} +0.308953 q^{22} -7.85738 q^{23} +1.00000 q^{25} -0.901803 q^{26} -3.32331 q^{29} -6.91116 q^{31} -2.93186 q^{32} +1.46593 q^{34} +9.27890 q^{37} +1.34704 q^{38} +3.06814 q^{40} +8.81297 q^{41} +5.70041 q^{43} +0.136281 q^{44} +9.51337 q^{46} -7.91116 q^{47} -1.21076 q^{50} -0.397789 q^{52} -2.42151 q^{53} -0.255174 q^{55} +4.02372 q^{58} -11.8811 q^{59} +10.9556 q^{61} +8.36773 q^{62} +8.84302 q^{64} +0.744826 q^{65} -9.16634 q^{67} +0.646629 q^{68} -10.5878 q^{71} +0.435873 q^{73} -11.2345 q^{74} +0.594184 q^{76} +2.37709 q^{79} -2.64663 q^{80} -10.6704 q^{82} +10.2789 q^{83} -1.21076 q^{85} -6.90180 q^{86} -0.782909 q^{88} -14.1663 q^{89} +4.19639 q^{92} +9.57849 q^{94} -1.11256 q^{95} -8.44523 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 6 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 6 * q^4 + 3 * q^5 - 6 * q^8	\( 3 q - 2 q^{2} + 6 q^{4} + 3 q^{5} - 6 q^{8} - 2 q^{10} - 10 q^{11} - 7 q^{13} + 4 q^{16} - 2 q^{17} + q^{19} + 6 q^{20} + 2 q^{22} - 10 q^{23} + 3 q^{25} - 4 q^{29} + q^{31} - 24 q^{32} + 12 q^{34} + 11 q^{37} - 28 q^{38} - 6 q^{40} + 2 q^{41} - 3 q^{43} - 30 q^{44} - 16 q^{46} - 2 q^{47} - 2 q^{50} - 24 q^{52} - 4 q^{53} - 10 q^{55} - 14 q^{58} + 4 q^{59} + 22 q^{61} + 30 q^{62} + 20 q^{64} - 7 q^{65} - 15 q^{67} - 10 q^{68} - 16 q^{71} - 9 q^{73} - 6 q^{74} + 30 q^{76} - 7 q^{79} + 4 q^{80} + 6 q^{82} + 14 q^{83} - 2 q^{85} - 18 q^{86} + 40 q^{88} - 30 q^{89} + 18 q^{92} + 32 q^{94} + q^{95} + 4 q^{97}+O(q^{100}) \) 3 * q - 2 * q^2 + 6 * q^4 + 3 * q^5 - 6 * q^8 - 2 * q^10 - 10 * q^11 - 7 * q^13 + 4 * q^16 - 2 * q^17 + q^19 + 6 * q^20 + 2 * q^22 - 10 * q^23 + 3 * q^25 - 4 * q^29 + q^31 - 24 * q^32 + 12 * q^34 + 11 * q^37 - 28 * q^38 - 6 * q^40 + 2 * q^41 - 3 * q^43 - 30 * q^44 - 16 * q^46 - 2 * q^47 - 2 * q^50 - 24 * q^52 - 4 * q^53 - 10 * q^55 - 14 * q^58 + 4 * q^59 + 22 * q^61 + 30 * q^62 + 20 * q^64 - 7 * q^65 - 15 * q^67 - 10 * q^68 - 16 * q^71 - 9 * q^73 - 6 * q^74 + 30 * q^76 - 7 * q^79 + 4 * q^80 + 6 * q^82 + 14 * q^83 - 2 * q^85 - 18 * q^86 + 40 * q^88 - 30 * q^89 + 18 * q^92 + 32 * q^94 + q^95 + 4 * q^97

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