LMFDB - Embedded newform 588.2.e.a.491.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,2,Mod(491,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("588.491");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	588.e (of order $2$, degree $1$, 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:	$4.69520363885$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			491.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	588.491
Dual form			588.2.e.a.491.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.44949i q^{5} -2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +3.46410i q^{10} -1.41421 q^{11} +3.46410i q^{12} +4.24264 q^{15} +4.00000 q^{16} +7.34847i q^{17} +4.24264 q^{18} +6.92820i q^{19} -4.89898i q^{20} +2.00000 q^{22} +7.07107 q^{23} -4.89898i q^{24} -1.00000 q^{25} -5.19615i q^{27} -6.00000 q^{30} +3.46410i q^{31} -5.65685 q^{32} -2.44949i q^{33} -10.3923i q^{34} -6.00000 q^{36} +8.00000 q^{37} -9.79796i q^{38} +6.92820i q^{40} +12.2474i q^{41} -2.82843 q^{44} +7.34847i q^{45} -10.0000 q^{46} +6.92820i q^{48} +1.41421 q^{50} -12.7279 q^{51} +7.34847i q^{54} +3.46410i q^{55} -12.0000 q^{57} +8.48528 q^{60} -4.89898i q^{62} +8.00000 q^{64} +3.46410i q^{66} +14.6969i q^{68} +12.2474i q^{69} -15.5563 q^{71} +8.48528 q^{72} -11.3137 q^{74} -1.73205i q^{75} +13.8564i q^{76} -9.79796i q^{80} +9.00000 q^{81} -17.3205i q^{82} +18.0000 q^{85} +4.00000 q^{88} -2.44949i q^{89} -10.3923i q^{90} +14.1421 q^{92} -6.00000 q^{93} +16.9706 q^{95} -9.79796i q^{96} +4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 12 q^{9} + 16 q^{16} + 8 q^{22} - 4 q^{25} - 24 q^{30} - 24 q^{36} + 32 q^{37} - 40 q^{46} - 48 q^{57} + 32 q^{64} + 36 q^{81} + 72 q^{85} + 16 q^{88} - 24 q^{93}+O(q^{100}) $ 4 * q + 8 * q^4 - 12 * q^9 + 16 * q^16 + 8 * q^22 - 4 * q^25 - 24 * q^30 - 24 * q^36 + 32 * q^37 - 40 * q^46 - 48 * q^57 + 32 * q^64 + 36 * q^81 + 72 * q^85 + 16 * q^88 - 24 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	−1.41421	−1.00000
$2$	−1.41421	−1.00000
$3$	1.73205i	1.00000i
$3$	1.73205i	1.00000i
$4$	2.00000	1.00000
$5$	− 2.44949i	− 1.09545i	−0.836660	−	0.547723i	$-0.815495\pi$
$5$	− 2.44949i	− 1.09545i	0.836660	−	0.547723i	$-0.184505\pi$
$6$	− 2.44949i	− 1.00000i
$7$	0	0
$7$	0	0
$8$	−2.82843	−1.00000
$9$	−3.00000	−1.00000
$10$	3.46410i	1.09545i
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	3.46410i	1.00000i
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	4.24264	1.09545
$16$	4.00000	1.00000
$17$	7.34847i	1.78227i	0.453743	+	0.891133i	$0.350089\pi$
$17$	7.34847i	1.78227i	−0.453743	+	0.891133i	$0.649911\pi$
$18$	4.24264	1.00000
$19$	6.92820i	1.58944i	0.606977	+	0.794719i	$0.292382\pi$
$19$	6.92820i	1.58944i	−0.606977	+	0.794719i	$0.707618\pi$
$20$	− 4.89898i	− 1.09545i
$21$	0	0
$22$	2.00000	0.426401
$23$	7.07107	1.47442	0.737210	−	0.675664i	$-0.236143\pi$
$23$	7.07107	1.47442	0.737210	+	0.675664i	$0.236143\pi$
$24$	− 4.89898i	− 1.00000i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 5.19615i	− 1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−6.00000	−1.09545
$31$	3.46410i	0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$	3.46410i	0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$	−5.65685	−1.00000
$33$	− 2.44949i	− 0.426401i
$34$	− 10.3923i	− 1.78227i
$35$	0	0
$36$	−6.00000	−1.00000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	− 9.79796i	− 1.58944i
$39$	0	0
$40$	6.92820i	1.09545i
$41$	12.2474i	1.91273i	0.292174	+	0.956365i	$0.405621\pi$
$41$	12.2474i	1.91273i	−0.292174	+	0.956365i	$0.594379\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−2.82843	−0.426401
$45$	7.34847i	1.09545i
$46$	−10.0000	−1.47442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	6.92820i	1.00000i
$49$	0	0
$50$	1.41421	0.200000
$51$	−12.7279	−1.78227
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	7.34847i	1.00000i
$55$	3.46410i	0.467099i
$56$	0	0
$57$	−12.0000	−1.58944
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	8.48528	1.09545
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	− 4.89898i	− 0.622171i
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	3.46410i	0.426401i
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	14.6969i	1.78227i
$69$	12.2474i	1.47442i
$70$	0	0
$71$	−15.5563	−1.84620	−0.923099	−	0.384561i	$-0.874353\pi$
$71$	−15.5563	−1.84620	−0.923099	+	0.384561i	$0.874353\pi$
$72$	8.48528	1.00000
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−11.3137	−1.31519
$75$	− 1.73205i	− 0.200000i
$76$	13.8564i	1.58944i
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	− 9.79796i	− 1.09545i
$81$	9.00000	1.00000
$82$	− 17.3205i	− 1.91273i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	18.0000	1.95237
$86$	0	0
$87$	0	0
$88$	4.00000	0.426401
$89$	− 2.44949i	− 0.259645i	−0.991537	−	0.129823i	$-0.958559\pi$
$89$	− 2.44949i	− 0.259645i	0.991537	−	0.129823i	$-0.0414408\pi$
$90$	− 10.3923i	− 1.09545i
$91$	0	0
$92$	14.1421	1.47442
$93$	−6.00000	−0.622171
$94$	0	0
$95$	16.9706	1.74114
$96$	− 9.79796i	− 1.00000i
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	4.24264	0.426401
$100$	−2.00000	−0.200000
$101$	7.34847i	0.731200i	0.930772	+	0.365600i	$0.119136\pi$
$101$	7.34847i	0.731200i	−0.930772	+	0.365600i	$0.880864\pi$
$102$	18.0000	1.78227
$103$	− 17.3205i	− 1.70664i	−0.521387	−	0.853320i	$-0.674585\pi$
$103$	− 17.3205i	− 1.70664i	0.521387	−	0.853320i	$-0.325415\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.07107	0.683586	0.341793	−	0.939775i	$-0.388966\pi$
$107$	7.07107	0.683586	0.341793	+	0.939775i	$0.388966\pi$
$108$	− 10.3923i	− 1.00000i
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	− 4.89898i	− 0.467099i
$111$	13.8564i	1.31519i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	16.9706	1.58944
$115$	− 17.3205i	− 1.61515i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−12.0000	−1.09545
$121$	−9.00000	−0.818182
$122$	0	0
$123$	−21.2132	−1.91273
$124$	6.92820i	0.622171i
$125$	− 9.79796i	− 0.876356i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 4.89898i	− 0.426401i
$133$	0	0
$134$	0	0
$135$	−12.7279	−1.09545
$136$	− 20.7846i	− 1.78227i
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	− 17.3205i	− 1.47442i
$139$	− 10.3923i	− 0.881464i	−0.897639	−	0.440732i	$-0.854719\pi$
$139$	− 10.3923i	− 0.881464i	0.897639	−	0.440732i	$-0.145281\pi$
$140$	0	0
$141$	0	0
$142$	22.0000	1.84620
$143$	0	0
$144$	−12.0000	−1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	16.0000	1.31519
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	2.44949i	0.200000i
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	− 19.5959i	− 1.58944i
$153$	− 22.0454i	− 1.78227i
$154$	0	0
$155$	8.48528	0.681554
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	13.8564i	1.09545i
$161$	0	0
$162$	−12.7279	−1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	24.4949i	1.91273i
$165$	−6.00000	−0.467099
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−25.4558	−1.95237
$171$	− 20.7846i	− 1.58944i
$172$	0	0
$173$	− 2.44949i	− 0.186231i	−0.995655	−	0.0931156i	$-0.970317\pi$
$173$	− 2.44949i	− 0.186231i	0.995655	−	0.0931156i	$-0.0296826\pi$
$174$	0	0
$175$	0	0
$176$	−5.65685	−0.426401
$177$	0	0
$178$	3.46410i	0.259645i
$179$	18.3848	1.37414	0.687071	−	0.726590i	$-0.258896\pi$
$179$	18.3848	1.37414	0.687071	+	0.726590i	$0.258896\pi$
$180$	14.6969i	1.09545i
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	−20.0000	−1.47442
$185$	− 19.5959i	− 1.44072i
$186$	8.48528	0.622171
$187$	− 10.3923i	− 0.759961i
$188$	0	0
$189$	0	0
$190$	−24.0000	−1.74114
$191$	26.8701	1.94425	0.972125	−	0.234465i	$-0.0753338\pi$
$191$	26.8701	1.94425	0.972125	+	0.234465i	$0.0753338\pi$
$192$	13.8564i	1.00000i
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−6.00000	−0.426401
$199$	27.7128i	1.96451i	0.187552	+	0.982255i	$0.439945\pi$
$199$	27.7128i	1.96451i	−0.187552	+	0.982255i	$0.560055\pi$
$200$	2.82843	0.200000
$201$	0	0
$202$	− 10.3923i	− 0.731200i
$203$	0	0
$204$	−25.4558	−1.78227
$205$	30.0000	2.09529
$206$	24.4949i	1.70664i
$207$	−21.2132	−1.47442
$208$	0	0
$209$	− 9.79796i	− 0.677739i
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	− 26.9444i	− 1.84620i
$214$	−10.0000	−0.683586
$215$	0	0
$216$	14.6969i	1.00000i
$217$	0	0
$218$	14.1421	0.957826
$219$	0	0
$220$	6.92820i	0.467099i
$221$	0	0
$222$	− 19.5959i	− 1.31519i
$223$	13.8564i	0.927894i	0.885863	+	0.463947i	$0.153567\pi$
$223$	13.8564i	0.927894i	−0.885863	+	0.463947i	$0.846433\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	−24.0000	−1.58944
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	24.4949i	1.61515i
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.0416	1.55512	0.777562	−	0.628806i	$-0.216456\pi$
$239$	24.0416	1.55512	0.777562	+	0.628806i	$0.216456\pi$
$240$	16.9706	1.09545
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	12.7279	0.818182
$243$	15.5885i	1.00000i
$244$	0	0
$245$	0	0
$246$	30.0000	1.91273
$247$	0	0
$248$	− 9.79796i	− 0.622171i
$249$	0	0
$250$	13.8564i	0.876356i
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−10.0000	−0.628695
$254$	0	0
$255$	31.1769i	1.95237i
$256$	16.0000	1.00000
$257$	31.8434i	1.98633i	0.116699	+	0.993167i	$0.462769\pi$
$257$	31.8434i	1.98633i	−0.116699	+	0.993167i	$0.537231\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.41421	−0.0872041	−0.0436021	−	0.999049i	$-0.513883\pi$
$263$	−1.41421	−0.0872041	−0.0436021	+	0.999049i	$0.513883\pi$
$264$	6.92820i	0.426401i
$265$	0	0
$266$	0	0
$267$	4.24264	0.259645
$268$	0	0
$269$	− 26.9444i	− 1.64283i	−0.570332	−	0.821414i	$-0.693186\pi$
$269$	− 26.9444i	− 1.64283i	0.570332	−	0.821414i	$-0.306814\pi$
$270$	18.0000	1.09545
$271$	31.1769i	1.89386i	0.321436	+	0.946931i	$0.395835\pi$
$271$	31.1769i	1.89386i	−0.321436	+	0.946931i	$0.604165\pi$
$272$	29.3939i	1.78227i
$273$	0	0
$274$	0	0
$275$	1.41421	0.0852803
$276$	24.4949i	1.47442i
$277$	32.0000	1.92269	0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000	1.92269	0.961347	+	0.275340i	$0.0887905\pi$
$278$	14.6969i	0.881464i
$279$	− 10.3923i	− 0.622171i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 20.7846i	− 1.23552i	−0.786368	−	0.617758i	$-0.788041\pi$
$283$	− 20.7846i	− 1.23552i	0.786368	−	0.617758i	$-0.211959\pi$
$284$	−31.1127	−1.84620
$285$	29.3939i	1.74114i
$286$	0	0
$287$	0	0
$288$	16.9706	1.00000
$289$	−37.0000	−2.17647
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 22.0454i	− 1.28791i	−0.765065	−	0.643953i	$-0.777293\pi$
$293$	− 22.0454i	− 1.28791i	0.765065	−	0.643953i	$-0.222707\pi$
$294$	0	0
$295$	0	0
$296$	−22.6274	−1.31519
$297$	7.34847i	0.426401i
$298$	0	0
$299$	0	0
$300$	− 3.46410i	− 0.200000i
$301$	0	0
$302$	0	0
$303$	−12.7279	−0.731200
$304$	27.7128i	1.58944i
$305$	0	0
$306$	31.1769i	1.78227i
$307$	− 34.6410i	− 1.97707i	−0.151001	−	0.988534i	$-0.548250\pi$
$307$	− 34.6410i	− 1.97707i	0.151001	−	0.988534i	$-0.451750\pi$
$308$	0	0
$309$	30.0000	1.70664
$310$	−12.0000	−0.681554
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	− 19.5959i	− 1.09545i
$321$	12.2474i	0.683586i
$322$	0	0
$323$	−50.9117	−2.83280
$324$	18.0000	1.00000
$325$	0	0
$326$	0	0
$327$	− 17.3205i	− 0.957826i
$328$	− 34.6410i	− 1.91273i
$329$	0	0
$330$	8.48528	0.467099
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−24.0000	−1.31519
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	18.3848	1.00000
$339$	0	0
$340$	36.0000	1.95237
$341$	− 4.89898i	− 0.265295i
$342$	29.3939i	1.58944i
$343$	0	0
$344$	0	0
$345$	30.0000	1.61515
$346$	3.46410i	0.186231i
$347$	18.3848	0.986947	0.493473	−	0.869761i	$-0.335727\pi$
$347$	18.3848	0.986947	0.493473	+	0.869761i	$0.335727\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	8.00000	0.426401
$353$	− 26.9444i	− 1.43411i	−0.697019	−	0.717053i	$-0.745491\pi$
$353$	− 26.9444i	− 1.43411i	0.697019	−	0.717053i	$-0.254509\pi$
$354$	0	0
$355$	38.1051i	2.02241i
$356$	− 4.89898i	− 0.259645i
$357$	0	0
$358$	−26.0000	−1.37414
$359$	−32.5269	−1.71670	−0.858352	−	0.513061i	$-0.828512\pi$
$359$	−32.5269	−1.71670	−0.858352	+	0.513061i	$0.828512\pi$
$360$	− 20.7846i	− 1.09545i
$361$	−29.0000	−1.52632
$362$	0	0
$363$	− 15.5885i	− 0.818182i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	27.7128i	1.44660i	0.690535	+	0.723299i	$0.257375\pi$
$367$	27.7128i	1.44660i	−0.690535	+	0.723299i	$0.742625\pi$
$368$	28.2843	1.47442
$369$	− 36.7423i	− 1.91273i
$370$	27.7128i	1.44072i
$371$	0	0
$372$	−12.0000	−0.622171
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	14.6969i	0.759961i
$375$	16.9706	0.876356
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	33.9411	1.74114
$381$	0	0
$382$	−38.0000	−1.94425
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	− 19.5959i	− 1.00000i
$385$	0	0
$386$	−5.65685	−0.287926
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	51.9615i	2.62781i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	8.48528	0.426401
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	− 39.1918i	− 1.96451i
$399$	0	0
$400$	−4.00000	−0.200000
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	14.6969i	0.731200i
$405$	− 22.0454i	− 1.09545i
$406$	0	0
$407$	−11.3137	−0.560800
$408$	36.0000	1.78227
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	−42.4264	−2.09529
$411$	0	0
$412$	− 34.6410i	− 1.70664i
$413$	0	0
$414$	30.0000	1.47442
$415$	0	0
$416$	0	0
$417$	18.0000	0.881464
$418$	13.8564i	0.677739i
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−40.0000	−1.94948	−0.974740	−	0.223341i	$-0.928304\pi$
$421$	−40.0000	−1.94948	−0.974740	+	0.223341i	$0.928304\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 7.34847i	− 0.356453i
$426$	38.1051i	1.84620i
$427$	0	0
$428$	14.1421	0.683586
$429$	0	0
$430$	0	0
$431$	−41.0122	−1.97549	−0.987744	−	0.156083i	$-0.950113\pi$
$431$	−41.0122	−1.97549	−0.987744	+	0.156083i	$0.950113\pi$
$432$	− 20.7846i	− 1.00000i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−20.0000	−0.957826
$437$	48.9898i	2.34350i
$438$	0	0
$439$	− 41.5692i	− 1.98399i	−0.126275	−	0.991995i	$-0.540302\pi$
$439$	− 41.5692i	− 1.98399i	0.126275	−	0.991995i	$-0.459698\pi$
$440$	− 9.79796i	− 0.467099i
$441$	0	0
$442$	0	0
$443$	26.8701	1.27663	0.638317	−	0.769773i	$-0.279631\pi$
$443$	26.8701	1.27663	0.638317	+	0.769773i	$0.279631\pi$
$444$	27.7128i	1.31519i
$445$	−6.00000	−0.284427
$446$	− 19.5959i	− 0.927894i
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−4.24264	−0.200000
$451$	− 17.3205i	− 0.815591i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	33.9411	1.58944
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	38.1838	1.78227
$460$	− 34.6410i	− 1.61515i
$461$	12.2474i	0.570421i	0.958465	+	0.285210i	$0.0920634\pi$
$461$	12.2474i	0.570421i	−0.958465	+	0.285210i	$0.907937\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	14.6969i	0.681554i
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 6.92820i	− 0.317888i
$476$	0	0
$477$	0	0
$478$	−34.0000	−1.55512
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−24.0000	−1.09545
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−18.0000	−0.818182
$485$	0	0
$486$	− 22.0454i	− 1.00000i
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	43.8406	1.97850	0.989250	−	0.146236i	$-0.0467158\pi$
$491$	43.8406	1.97850	0.989250	+	0.146236i	$0.0467158\pi$
$492$	−42.4264	−1.91273
$493$	0	0
$494$	0	0
$495$	− 10.3923i	− 0.467099i
$496$	13.8564i	0.622171i
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 19.5959i	− 0.876356i
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	18.0000	0.800989
$506$	14.1421	0.628695
$507$	− 22.5167i	− 1.00000i
$508$	0	0
$509$	− 36.7423i	− 1.62858i	−0.580461	−	0.814288i	$-0.697128\pi$
$509$	− 36.7423i	− 1.62858i	0.580461	−	0.814288i	$-0.302872\pi$
$510$	− 44.0908i	− 1.95237i
$511$	0	0
$512$	−22.6274	−1.00000
$513$	36.0000	1.58944
$514$	− 45.0333i	− 1.98633i
$515$	−42.4264	−1.86953
$516$	0	0
$517$	0	0
$518$	0	0
$519$	4.24264	0.186231
$520$	0	0
$521$	41.6413i	1.82434i	0.409812	+	0.912170i	$0.365594\pi$
$521$	41.6413i	1.82434i	−0.409812	+	0.912170i	$0.634406\pi$
$522$	0	0
$523$	− 17.3205i	− 0.757373i	−0.925525	−	0.378686i	$-0.876376\pi$
$523$	− 17.3205i	− 0.757373i	0.925525	−	0.378686i	$-0.123624\pi$
$524$	0	0
$525$	0	0
$526$	2.00000	0.0872041
$527$	−25.4558	−1.10887
$528$	− 9.79796i	− 0.426401i
$529$	27.0000	1.17391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−6.00000	−0.259645
$535$	− 17.3205i	− 0.748831i
$536$	0	0
$537$	31.8434i	1.37414i
$538$	38.1051i	1.64283i
$539$	0	0
$540$	−25.4558	−1.09545
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	− 44.0908i	− 1.89386i
$543$	0	0
$544$	− 41.5692i	− 1.78227i
$545$	24.4949i	1.04925i
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	0	0
$552$	− 34.6410i	− 1.47442i
$553$	0	0
$554$	−45.2548	−1.92269
$555$	33.9411	1.44072
$556$	− 20.7846i	− 0.881464i
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	14.6969i	0.622171i
$559$	0	0
$560$	0	0
$561$	18.0000	0.759961
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	29.3939i	1.23552i
$567$	0	0
$568$	44.0000	1.84620
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	− 41.5692i	− 1.74114i
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	46.5403i	1.94425i
$574$	0	0
$575$	−7.07107	−0.294884
$576$	−24.0000	−1.00000
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	52.3259	2.17647
$579$	6.92820i	0.287926i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	31.1769i	1.28791i
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	0	0
$592$	32.0000	1.31519
$593$	− 2.44949i	− 0.100588i	−0.998734	−	0.0502942i	$-0.983984\pi$
$593$	− 2.44949i	− 0.100588i	0.998734	−	0.0502942i	$-0.0160159\pi$
$594$	− 10.3923i	− 0.426401i
$595$	0	0
$596$	0	0
$597$	−48.0000	−1.96451
$598$	0	0
$599$	18.3848	0.751182	0.375591	−	0.926786i	$-0.377440\pi$
$599$	18.3848	0.751182	0.375591	+	0.926786i	$0.377440\pi$
$600$	4.89898i	0.200000i
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.0454i	0.896273i
$606$	18.0000	0.731200
$607$	− 41.5692i	− 1.68724i	−0.536939	−	0.843621i	$-0.680419\pi$
$607$	− 41.5692i	− 1.68724i	0.536939	−	0.843621i	$-0.319581\pi$
$608$	− 39.1918i	− 1.58944i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	− 44.0908i	− 1.78227i
$613$	46.0000	1.85792	0.928961	−	0.370177i	$-0.120703\pi$
$613$	46.0000	1.85792	0.928961	+	0.370177i	$0.120703\pi$
$614$	48.9898i	1.97707i
$615$	51.9615i	2.09529i
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−42.4264	−1.70664
$619$	− 45.0333i	− 1.81004i	−0.425367	−	0.905021i	$-0.639855\pi$
$619$	− 45.0333i	− 1.81004i	0.425367	−	0.905021i	$-0.360145\pi$
$620$	16.9706	0.681554
$621$	− 36.7423i	− 1.47442i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	16.9706	0.677739
$628$	0	0
$629$	58.7878i	2.34402i
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	46.6690	1.84620
$640$	27.7128i	1.09545i
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	− 17.3205i	− 0.683586i
$643$	− 34.6410i	− 1.36611i	−0.730368	−	0.683054i	$-0.760651\pi$
$643$	− 34.6410i	− 1.36611i	0.730368	−	0.683054i	$-0.239349\pi$
$644$	0	0
$645$	0	0
$646$	72.0000	2.83280
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	−25.4558	−1.00000
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	24.4949i	0.957826i
$655$	0	0
$656$	48.9898i	1.91273i
$657$	0	0
$658$	0	0
$659$	24.0416	0.936529	0.468264	−	0.883588i	$-0.344879\pi$
$659$	24.0416	0.936529	0.468264	+	0.883588i	$0.344879\pi$
$660$	−12.0000	−0.467099
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	33.9411	1.31519
$667$	0	0
$668$	0	0
$669$	−24.0000	−0.927894
$670$	0	0
$671$	0	0
$672$	0	0
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	−2.82843	−0.108947
$675$	5.19615i	0.200000i
$676$	−26.0000	−1.00000
$677$	31.8434i	1.22384i	0.790920	+	0.611920i	$0.209603\pi$
$677$	31.8434i	1.22384i	−0.790920	+	0.611920i	$0.790397\pi$
$678$	0	0
$679$	0	0
$680$	−50.9117	−1.95237
$681$	0	0
$682$	6.92820i	0.265295i
$683$	−41.0122	−1.56929	−0.784644	−	0.619947i	$-0.787154\pi$
$683$	−41.0122	−1.56929	−0.784644	+	0.619947i	$0.787154\pi$
$684$	− 41.5692i	− 1.58944i
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	−42.4264	−1.61515
$691$	31.1769i	1.18603i	0.805193	+	0.593013i	$0.202062\pi$
$691$	31.1769i	1.18603i	−0.805193	+	0.593013i	$0.797938\pi$
$692$	− 4.89898i	− 0.186231i
$693$	0	0
$694$	−26.0000	−0.986947
$695$	−25.4558	−0.965595
$696$	0	0
$697$	−90.0000	−3.40899
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	55.4256i	2.09042i
$704$	−11.3137	−0.426401
$705$	0	0
$706$	38.1051i	1.43411i
$707$	0	0
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	− 53.8888i	− 2.02241i
$711$	0	0
$712$	6.92820i	0.259645i
$713$	24.4949i	0.917341i
$714$	0	0
$715$	0	0
$716$	36.7696	1.37414
$717$	41.6413i	1.55512i
$718$	46.0000	1.71670
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	29.3939i	1.09545i
$721$	0	0
$722$	41.0122	1.52632
$723$	0	0
$724$	0	0
$725$	0	0
$726$	22.0454i	0.818182i
$727$	− 10.3923i	− 0.385429i	−0.981255	−	0.192715i	$-0.938271\pi$
$727$	− 10.3923i	− 0.385429i	0.981255	−	0.192715i	$-0.0617292\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	− 39.1918i	− 1.44660i
$735$	0	0
$736$	−40.0000	−1.47442
$737$	0	0
$738$	51.9615i	1.91273i
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	− 39.1918i	− 1.44072i
$741$	0	0
$742$	0	0
$743$	43.8406	1.60836	0.804178	−	0.594388i	$-0.202606\pi$
$743$	43.8406	1.60836	0.804178	+	0.594388i	$0.202606\pi$
$744$	16.9706	0.622171
$745$	0	0
$746$	48.0833	1.76045
$747$	0	0
$748$	− 20.7846i	− 0.759961i
$749$	0	0
$750$	−24.0000	−0.876356
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	− 17.3205i	− 0.628695i
$760$	−48.0000	−1.74114
$761$	− 36.7423i	− 1.33191i	−0.745992	−	0.665955i	$-0.768024\pi$
$761$	− 36.7423i	− 1.33191i	0.745992	−	0.665955i	$-0.231976\pi$
$762$	0	0
$763$	0	0
$764$	53.7401	1.94425
$765$	−54.0000	−1.95237
$766$	0	0
$767$	0	0
$768$	27.7128i	1.00000i
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	−55.1543	−1.98633
$772$	8.00000	0.287926
$773$	− 26.9444i	− 0.969122i	−0.874757	−	0.484561i	$-0.838979\pi$
$773$	− 26.9444i	− 0.969122i	0.874757	−	0.484561i	$-0.161021\pi$
$774$	0	0
$775$	− 3.46410i	− 0.124434i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−84.8528	−3.04017
$780$	0	0
$781$	22.0000	0.787222
$782$	− 73.4847i	− 2.62781i
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	51.9615i	1.85223i	0.377243	+	0.926114i	$0.376872\pi$
$787$	51.9615i	1.85223i	−0.377243	+	0.926114i	$0.623128\pi$
$788$	0	0
$789$	− 2.44949i	− 0.0872041i
$790$	0	0
$791$	0	0
$792$	−12.0000	−0.426401
$793$	0	0
$794$	0	0
$795$	0	0
$796$	55.4256i	1.96451i
$797$	− 56.3383i	− 1.99560i	−0.0662682	−	0.997802i	$-0.521109\pi$
$797$	− 56.3383i	− 1.99560i	0.0662682	−	0.997802i	$-0.478891\pi$
$798$	0	0
$799$	0	0
$800$	5.65685	0.200000
$801$	7.34847i	0.259645i
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	46.6690	1.64283
$808$	− 20.7846i	− 0.731200i
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	31.1769i	1.09545i
$811$	38.1051i	1.33805i	0.743239	+	0.669026i	$0.233288\pi$
$811$	38.1051i	1.33805i	−0.743239	+	0.669026i	$0.766712\pi$
$812$	0	0
$813$	−54.0000	−1.89386
$814$	16.0000	0.560800
$815$	0	0
$816$	−50.9117	−1.78227
$817$	0	0
$818$	0	0
$819$	0	0
$820$	60.0000	2.09529
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	48.9898i	1.70664i
$825$	2.44949i	0.0852803i
$826$	0	0
$827$	−35.3553	−1.22943	−0.614713	−	0.788751i	$-0.710728\pi$
$827$	−35.3553	−1.22943	−0.614713	+	0.788751i	$0.710728\pi$
$828$	−42.4264	−1.47442
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	55.4256i	1.92269i
$832$	0	0
$833$	0	0
$834$	−25.4558	−0.881464
$835$	0	0
$836$	− 19.5959i	− 0.677739i
$837$	18.0000	0.622171
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	56.5685	1.94948
$843$	0	0
$844$	0	0
$845$	31.8434i	1.09545i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	36.0000	1.23552
$850$	10.3923i	0.356453i
$851$	56.5685	1.93914
$852$	− 53.8888i	− 1.84620i
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	−50.9117	−1.74114
$856$	−20.0000	−0.683586
$857$	41.6413i	1.42244i	0.702969	+	0.711220i	$0.251857\pi$
$857$	41.6413i	1.42244i	−0.702969	+	0.711220i	$0.748143\pi$
$858$	0	0
$859$	6.92820i	0.236387i	0.992991	+	0.118194i	$0.0377103\pi$
$859$	6.92820i	0.236387i	−0.992991	+	0.118194i	$0.962290\pi$
$860$	0	0
$861$	0	0
$862$	58.0000	1.97549
$863$	7.07107	0.240702	0.120351	−	0.992731i	$-0.461598\pi$
$863$	7.07107	0.240702	0.120351	+	0.992731i	$0.461598\pi$
$864$	29.3939i	1.00000i
$865$	−6.00000	−0.204006
$866$	0	0
$867$	− 64.0859i	− 2.17647i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	28.2843	0.957826
$873$	0	0
$874$	− 69.2820i	− 2.34350i
$875$	0	0
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	58.7878i	1.98399i
$879$	38.1838	1.28791
$880$	13.8564i	0.467099i
$881$	− 56.3383i	− 1.89808i	−0.315149	−	0.949042i	$-0.602055\pi$
$881$	− 56.3383i	− 1.89808i	0.315149	−	0.949042i	$-0.397945\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	−38.0000	−1.27663
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	− 39.1918i	− 1.31519i
$889$	0	0
$890$	8.48528	0.284427
$891$	−12.7279	−0.426401
$892$	27.7128i	0.927894i
$893$	0	0
$894$	0	0
$895$	− 45.0333i	− 1.50530i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	6.00000	0.200000
$901$	0	0
$902$	24.4949i	0.815591i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	− 22.0454i	− 0.731200i
$910$	0	0
$911$	−15.5563	−0.515405	−0.257702	−	0.966224i	$-0.582965\pi$
$911$	−15.5563	−0.515405	−0.257702	+	0.966224i	$0.582965\pi$
$912$	−48.0000	−1.58944
$913$	0	0
$914$	−31.1127	−1.02912
$915$	0	0
$916$	0	0
$917$	0	0
$918$	−54.0000	−1.78227
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	48.9898i	1.61515i
$921$	60.0000	1.97707
$922$	− 17.3205i	− 0.570421i
$923$	0	0
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	51.9615i	1.70664i
$928$	0	0
$929$	− 36.7423i	− 1.20548i	−0.797939	−	0.602739i	$-0.794076\pi$
$929$	− 36.7423i	− 1.20548i	0.797939	−	0.602739i	$-0.205924\pi$
$930$	− 20.7846i	− 0.681554i
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−25.4558	−0.832495
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 61.2372i	− 1.99628i	−0.0609873	−	0.998139i	$-0.519425\pi$
$941$	− 61.2372i	− 1.99628i	0.0609873	−	0.998139i	$-0.480575\pi$
$942$	0	0
$943$	86.6025i	2.82017i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.3259	−1.70036	−0.850182	−	0.526489i	$-0.823508\pi$
$947$	−52.3259	−1.70036	−0.850182	+	0.526489i	$0.823508\pi$
$948$	0	0
$949$	0	0
$950$	9.79796i	0.317888i
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	− 65.8179i	− 2.12982i
$956$	48.0833	1.55512
$957$	0	0
$958$	0	0
$959$	0	0
$960$	33.9411	1.09545
$961$	19.0000	0.612903
$962$	0	0
$963$	−21.2132	−0.683586
$964$	0	0
$965$	− 9.79796i	− 0.315407i
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	25.4558	0.818182
$969$	− 88.1816i	− 2.83280i
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	31.1769i	1.00000i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	3.46410i	0.110713i
$980$	0	0
$981$	30.0000	0.957826
$982$	−62.0000	−1.97850
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	60.0000	1.91273
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	14.6969i	0.467099i
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	− 19.5959i	− 0.622171i
$993$	0	0
$994$	0	0
$995$	67.8823	2.15201
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	− 41.5692i	− 1.31519i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.2.e.a.491.2	yes	4
3.2	odd	2	inner	588.2.e.a.491.4	yes	4
4.3	odd	2	inner	588.2.e.a.491.3	yes	4
7.2	even	3		588.2.n.a.263.2		4
7.3	odd	6		588.2.n.a.275.2		4
7.4	even	3		588.2.n.b.275.2		4
7.5	odd	6		588.2.n.b.263.2		4
7.6	odd	2	inner	588.2.e.a.491.1	✓	4
12.11	even	2	inner	588.2.e.a.491.1	✓	4
21.2	odd	6		588.2.n.a.263.1		4
21.5	even	6		588.2.n.b.263.1		4
21.11	odd	6		588.2.n.b.275.1		4
21.17	even	6		588.2.n.a.275.1		4
21.20	even	2	inner	588.2.e.a.491.3	yes	4
28.3	even	6		588.2.n.b.275.1		4
28.11	odd	6		588.2.n.a.275.1		4
28.19	even	6		588.2.n.a.263.1		4
28.23	odd	6		588.2.n.b.263.1		4
28.27	even	2	inner	588.2.e.a.491.4	yes	4
84.11	even	6		588.2.n.a.275.2		4
84.23	even	6		588.2.n.b.263.2		4
84.47	odd	6		588.2.n.a.263.2		4
84.59	odd	6		588.2.n.b.275.2		4
84.83	odd	2	CM	588.2.e.a.491.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.2.e.a.491.1	✓	4	7.6	odd	2	inner
588.2.e.a.491.1	✓	4	12.11	even	2	inner
588.2.e.a.491.2	yes	4	1.1	even	1	trivial
588.2.e.a.491.2	yes	4	84.83	odd	2	CM
588.2.e.a.491.3	yes	4	4.3	odd	2	inner
588.2.e.a.491.3	yes	4	21.20	even	2	inner
588.2.e.a.491.4	yes	4	3.2	odd	2	inner
588.2.e.a.491.4	yes	4	28.27	even	2	inner
588.2.n.a.263.1		4	21.2	odd	6
588.2.n.a.263.1		4	28.19	even	6
588.2.n.a.263.2		4	7.2	even	3
588.2.n.a.263.2		4	84.47	odd	6
588.2.n.a.275.1		4	21.17	even	6
588.2.n.a.275.1		4	28.11	odd	6
588.2.n.a.275.2		4	7.3	odd	6
588.2.n.a.275.2		4	84.11	even	6
588.2.n.b.263.1		4	21.5	even	6
588.2.n.b.263.1		4	28.23	odd	6
588.2.n.b.263.2		4	7.5	odd	6
588.2.n.b.263.2		4	84.23	even	6
588.2.n.b.275.1		4	21.11	odd	6
588.2.n.b.275.1		4	28.3	even	6
588.2.n.b.275.2		4	7.4	even	3
588.2.n.b.275.2		4	84.59	odd	6

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	588.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.44949i q^{5} -2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +3.46410i q^{10} -1.41421 q^{11} +3.46410i q^{12} +4.24264 q^{15} +4.00000 q^{16} +7.34847i q^{17} +4.24264 q^{18} +6.92820i q^{19} -4.89898i q^{20} +2.00000 q^{22} +7.07107 q^{23} -4.89898i q^{24} -1.00000 q^{25} -5.19615i q^{27} -6.00000 q^{30} +3.46410i q^{31} -5.65685 q^{32} -2.44949i q^{33} -10.3923i q^{34} -6.00000 q^{36} +8.00000 q^{37} -9.79796i q^{38} +6.92820i q^{40} +12.2474i q^{41} -2.82843 q^{44} +7.34847i q^{45} -10.0000 q^{46} +6.92820i q^{48} +1.41421 q^{50} -12.7279 q^{51} +7.34847i q^{54} +3.46410i q^{55} -12.0000 q^{57} +8.48528 q^{60} -4.89898i q^{62} +8.00000 q^{64} +3.46410i q^{66} +14.6969i q^{68} +12.2474i q^{69} -15.5563 q^{71} +8.48528 q^{72} -11.3137 q^{74} -1.73205i q^{75} +13.8564i q^{76} -9.79796i q^{80} +9.00000 q^{81} -17.3205i q^{82} +18.0000 q^{85} +4.00000 q^{88} -2.44949i q^{89} -10.3923i q^{90} +14.1421 q^{92} -6.00000 q^{93} +16.9706 q^{95} -9.79796i q^{96} +4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 12 q^{9} + 16 q^{16} + 8 q^{22} - 4 q^{25} - 24 q^{30} - 24 q^{36} + 32 q^{37} - 40 q^{46} - 48 q^{57} + 32 q^{64} + 36 q^{81} + 72 q^{85} + 16 q^{88} - 24 q^{93}+O(q^{100}) \) 4 * q + 8 * q^4 - 12 * q^9 + 16 * q^16 + 8 * q^22 - 4 * q^25 - 24 * q^30 - 24 * q^36 + 32 * q^37 - 40 * q^46 - 48 * q^57 + 32 * q^64 + 36 * q^81 + 72 * q^85 + 16 * q^88 - 24 * q^93

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