LMFDB - Embedded newform 294.2.d.a.293.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,2,Mod(293,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("294.293"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	294.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$2.34760181943$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			293.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	294.293
Dual form			294.2.d.a.293.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} -1.73205 q^{5} -1.73205i q^{6} -1.00000i q^{8} +3.00000 q^{9} -1.73205i q^{10} -3.00000i q^{11} +1.73205 q^{12} -3.46410i q^{13} +3.00000 q^{15} +1.00000 q^{16} +3.46410 q^{17} +3.00000i q^{18} -3.46410i q^{19} +1.73205 q^{20} +3.00000 q^{22} -6.00000i q^{23} +1.73205i q^{24} -2.00000 q^{25} +3.46410 q^{26} -5.19615 q^{27} +3.00000i q^{29} +3.00000i q^{30} -1.73205i q^{31} +1.00000i q^{32} +5.19615i q^{33} +3.46410i q^{34} -3.00000 q^{36} -2.00000 q^{37} +3.46410 q^{38} +6.00000i q^{39} +1.73205i q^{40} +6.92820 q^{41} -8.00000 q^{43} +3.00000i q^{44} -5.19615 q^{45} +6.00000 q^{46} -6.92820 q^{47} -1.73205 q^{48} -2.00000i q^{50} -6.00000 q^{51} +3.46410i q^{52} -9.00000i q^{53} -5.19615i q^{54} +5.19615i q^{55} +6.00000i q^{57} -3.00000 q^{58} +1.73205 q^{59} -3.00000 q^{60} +1.73205 q^{62} -1.00000 q^{64} +6.00000i q^{65} -5.19615 q^{66} +2.00000 q^{67} -3.46410 q^{68} +10.3923i q^{69} -12.0000i q^{71} -3.00000i q^{72} +6.92820i q^{73} -2.00000i q^{74} +3.46410 q^{75} +3.46410i q^{76} -6.00000 q^{78} -1.00000 q^{79} -1.73205 q^{80} +9.00000 q^{81} +6.92820i q^{82} -8.66025 q^{83} -6.00000 q^{85} -8.00000i q^{86} -5.19615i q^{87} -3.00000 q^{88} +10.3923 q^{89} -5.19615i q^{90} +6.00000i q^{92} +3.00000i q^{93} -6.92820i q^{94} +6.00000i q^{95} -1.73205i q^{96} -5.19615i q^{97} -9.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 12 q^{9} + 12 q^{15} + 4 q^{16} + 12 q^{22} - 8 q^{25} - 12 q^{36} - 8 q^{37} - 32 q^{43} + 24 q^{46} - 24 q^{51} - 12 q^{58} - 12 q^{60} - 4 q^{64} + 8 q^{67} - 24 q^{78} - 4 q^{79} + 36 q^{81}+ \cdots - 12 q^{88}+O(q^{100}) $ 4 * q - 4 * q^4 + 12 * q^9 + 12 * q^15 + 4 * q^16 + 12 * q^22 - 8 * q^25 - 12 * q^36 - 8 * q^37 - 32 * q^43 + 24 * q^46 - 24 * q^51 - 12 * q^58 - 12 * q^60 - 4 * q^64 + 8 * q^67 - 24 * q^78 - 4 * q^79 + 36 * q^81 - 24 * q^85 - 12 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$.

$n$	$197$	$199$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	−1.73205	−1.00000
$3$	−1.73205	−1.00000
$4$	−1.00000	−0.500000
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	− 1.73205i	− 0.707107i
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	3.00000	1.00000
$10$	− 1.73205i	− 0.547723i
$11$	− 3.00000i	− 0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$	− 3.00000i	− 0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$	1.73205	0.500000
$13$	− 3.46410i	− 0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$	− 3.46410i	− 0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	3.00000i	0.707107i
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	1.73205	0.387298
$21$	0	0
$22$	3.00000	0.639602
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	1.73205i	0.353553i
$25$	−2.00000	−0.400000
$26$	3.46410	0.679366
$27$	−5.19615	−1.00000
$28$	0	0
$29$	3.00000i	0.557086i	0.960424	+	0.278543i	$0.0898515\pi$
$29$	3.00000i	0.557086i	−0.960424	+	0.278543i	$0.910149\pi$
$30$	3.00000i	0.547723i
$31$	− 1.73205i	− 0.311086i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	− 1.73205i	− 0.311086i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	1.00000i	0.176777i
$33$	5.19615i	0.904534i
$34$	3.46410i	0.594089i
$35$	0	0
$36$	−3.00000	−0.500000
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	3.46410	0.561951
$39$	6.00000i	0.960769i
$40$	1.73205i	0.273861i
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	3.00000i	0.452267i
$45$	−5.19615	−0.774597
$46$	6.00000	0.884652
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	−1.73205	−0.250000
$49$	0	0
$50$	− 2.00000i	− 0.282843i
$51$	−6.00000	−0.840168
$52$	3.46410i	0.480384i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	− 5.19615i	− 0.707107i
$55$	5.19615i	0.700649i
$56$	0	0
$57$	6.00000i	0.794719i
$58$	−3.00000	−0.393919
$59$	1.73205	0.225494	0.112747	−	0.993624i	$-0.464035\pi$
$59$	1.73205	0.225494	0.112747	+	0.993624i	$0.464035\pi$
$60$	−3.00000	−0.387298
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	1.73205	0.219971
$63$	0	0
$64$	−1.00000	−0.125000
$65$	6.00000i	0.744208i
$66$	−5.19615	−0.639602
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−3.46410	−0.420084
$69$	10.3923i	1.25109i
$70$	0	0
$71$	− 12.0000i	− 1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$	− 12.0000i	− 1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$	− 3.00000i	− 0.353553i
$73$	6.92820i	0.810885i	0.914121	+	0.405442i	$0.132883\pi$
$73$	6.92820i	0.810885i	−0.914121	+	0.405442i	$0.867117\pi$
$74$	− 2.00000i	− 0.232495i
$75$	3.46410	0.400000
$76$	3.46410i	0.397360i
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	−1.73205	−0.193649
$81$	9.00000	1.00000
$82$	6.92820i	0.765092i
$83$	−8.66025	−0.950586	−0.475293	−	0.879827i	$-0.657658\pi$
$83$	−8.66025	−0.950586	−0.475293	+	0.879827i	$0.657658\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	− 8.00000i	− 0.862662i
$87$	− 5.19615i	− 0.557086i
$88$	−3.00000	−0.319801
$89$	10.3923	1.10158	0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923	1.10158	0.550791	+	0.834643i	$0.314326\pi$
$90$	− 5.19615i	− 0.547723i
$91$	0	0
$92$	6.00000i	0.625543i
$93$	3.00000i	0.311086i
$94$	− 6.92820i	− 0.714590i
$95$	6.00000i	0.615587i
$96$	− 1.73205i	− 0.176777i
$97$	− 5.19615i	− 0.527589i	−0.964579	−	0.263795i	$-0.915026\pi$
$97$	− 5.19615i	− 0.527589i	0.964579	−	0.263795i	$-0.0849741\pi$
$98$	0	0
$99$	− 9.00000i	− 0.904534i
$100$	2.00000	0.200000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	− 6.00000i	− 0.594089i
$103$	3.46410i	0.341328i	0.985329	+	0.170664i	$0.0545913\pi$
$103$	3.46410i	0.341328i	−0.985329	+	0.170664i	$0.945409\pi$
$104$	−3.46410	−0.339683
$105$	0	0
$106$	9.00000	0.874157
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	5.19615	0.500000
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−5.19615	−0.495434
$111$	3.46410	0.328798
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	−6.00000	−0.561951
$115$	10.3923i	0.969087i
$116$	− 3.00000i	− 0.278543i
$117$	− 10.3923i	− 0.960769i
$118$	1.73205i	0.159448i
$119$	0	0
$120$	− 3.00000i	− 0.273861i
$121$	2.00000	0.181818
$122$	0	0
$123$	−12.0000	−1.08200
$124$	1.73205i	0.155543i
$125$	12.1244	1.08444
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	13.8564	1.21999
$130$	−6.00000	−0.526235
$131$	5.19615	0.453990	0.226995	−	0.973896i	$-0.427110\pi$
$131$	5.19615	0.453990	0.226995	+	0.973896i	$0.427110\pi$
$132$	− 5.19615i	− 0.452267i
$133$	0	0
$134$	2.00000i	0.172774i
$135$	9.00000	0.774597
$136$	− 3.46410i	− 0.297044i
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	−10.3923	−0.884652
$139$	− 17.3205i	− 1.46911i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	− 17.3205i	− 1.46911i	0.678551	−	0.734553i	$-0.262608\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	12.0000	1.00702
$143$	−10.3923	−0.869048
$144$	3.00000	0.250000
$145$	− 5.19615i	− 0.431517i
$146$	−6.92820	−0.573382
$147$	0	0
$148$	2.00000	0.164399
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	3.46410i	0.282843i
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	−3.46410	−0.280976
$153$	10.3923	0.840168
$154$	0	0
$155$	3.00000i	0.240966i
$156$	− 6.00000i	− 0.480384i
$157$	− 20.7846i	− 1.65879i	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	− 20.7846i	− 1.65879i	0.558661	−	0.829396i	$-0.311315\pi$
$158$	− 1.00000i	− 0.0795557i
$159$	15.5885i	1.23625i
$160$	− 1.73205i	− 0.136931i
$161$	0	0
$162$	9.00000i	0.707107i
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	−6.92820	−0.541002
$165$	− 9.00000i	− 0.700649i
$166$	− 8.66025i	− 0.672166i
$167$	−17.3205	−1.34030	−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205	−1.34030	−0.670151	+	0.742225i	$0.733770\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	− 6.00000i	− 0.460179i
$171$	− 10.3923i	− 0.794719i
$172$	8.00000	0.609994
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	5.19615	0.393919
$175$	0	0
$176$	− 3.00000i	− 0.226134i
$177$	−3.00000	−0.225494
$178$	10.3923i	0.778936i
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	5.19615	0.387298
$181$	− 6.92820i	− 0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	− 6.92820i	− 0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	3.46410	0.254686
$186$	−3.00000	−0.219971
$187$	− 10.3923i	− 0.759961i
$188$	6.92820	0.505291
$189$	0	0
$190$	−6.00000	−0.435286
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	1.73205	0.125000
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	5.19615	0.373062
$195$	− 10.3923i	− 0.744208i
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	9.00000	0.639602
$199$	10.3923i	0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$	10.3923i	0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$	2.00000i	0.141421i
$201$	−3.46410	−0.244339
$202$	0	0
$203$	0	0
$204$	6.00000	0.420084
$205$	−12.0000	−0.838116
$206$	−3.46410	−0.241355
$207$	− 18.0000i	− 1.25109i
$208$	− 3.46410i	− 0.240192i
$209$	−10.3923	−0.718851
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	9.00000i	0.618123i
$213$	20.7846i	1.42414i
$214$	3.00000	0.205076
$215$	13.8564	0.944999
$216$	5.19615i	0.353553i
$217$	0	0
$218$	− 2.00000i	− 0.135457i
$219$	− 12.0000i	− 0.810885i
$220$	− 5.19615i	− 0.350325i
$221$	− 12.0000i	− 0.807207i
$222$	3.46410i	0.232495i
$223$	25.9808i	1.73980i	0.493228	+	0.869900i	$0.335817\pi$
$223$	25.9808i	1.73980i	−0.493228	+	0.869900i	$0.664183\pi$
$224$	0	0
$225$	−6.00000	−0.400000
$226$	−12.0000	−0.798228
$227$	−5.19615	−0.344881	−0.172440	−	0.985020i	$-0.555165\pi$
$227$	−5.19615	−0.344881	−0.172440	+	0.985020i	$0.555165\pi$
$228$	− 6.00000i	− 0.397360i
$229$	13.8564i	0.915657i	0.889041	+	0.457829i	$0.151373\pi$
$229$	13.8564i	0.915657i	−0.889041	+	0.457829i	$0.848627\pi$
$230$	−10.3923	−0.685248
$231$	0	0
$232$	3.00000	0.196960
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	10.3923	0.679366
$235$	12.0000	0.782794
$236$	−1.73205	−0.112747
$237$	1.73205	0.112509
$238$	0	0
$239$	6.00000i	0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$	6.00000i	0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$	3.00000	0.193649
$241$	− 25.9808i	− 1.67357i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	− 25.9808i	− 1.67357i	0.547533	−	0.836784i	$-0.315567\pi$
$242$	2.00000i	0.128565i
$243$	−15.5885	−1.00000
$244$	0	0
$245$	0	0
$246$	− 12.0000i	− 0.765092i
$247$	−12.0000	−0.763542
$248$	−1.73205	−0.109985
$249$	15.0000	0.950586
$250$	12.1244i	0.766812i
$251$	19.0526	1.20259	0.601293	−	0.799028i	$-0.294652\pi$
$251$	19.0526	1.20259	0.601293	+	0.799028i	$0.294652\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	11.0000i	0.690201i
$255$	10.3923	0.650791
$256$	1.00000	0.0625000
$257$	−10.3923	−0.648254	−0.324127	−	0.946014i	$-0.605071\pi$
$257$	−10.3923	−0.648254	−0.324127	+	0.946014i	$0.605071\pi$
$258$	13.8564i	0.862662i
$259$	0	0
$260$	− 6.00000i	− 0.372104i
$261$	9.00000i	0.557086i
$262$	5.19615i	0.321019i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	5.19615	0.319801
$265$	15.5885i	0.957591i
$266$	0	0
$267$	−18.0000	−1.10158
$268$	−2.00000	−0.122169
$269$	−29.4449	−1.79529	−0.897643	−	0.440724i	$-0.854722\pi$
$269$	−29.4449	−1.79529	−0.897643	+	0.440724i	$0.854722\pi$
$270$	9.00000i	0.547723i
$271$	− 5.19615i	− 0.315644i	−0.987468	−	0.157822i	$-0.949553\pi$
$271$	− 5.19615i	− 0.315644i	0.987468	−	0.157822i	$-0.0504472\pi$
$272$	3.46410	0.210042
$273$	0	0
$274$	−18.0000	−1.08742
$275$	6.00000i	0.361814i
$276$	− 10.3923i	− 0.625543i
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	17.3205	1.03882
$279$	− 5.19615i	− 0.311086i
$280$	0	0
$281$	30.0000i	1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$	30.0000i	1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	12.0000i	0.714590i
$283$	27.7128i	1.64736i	0.567058	+	0.823678i	$0.308082\pi$
$283$	27.7128i	1.64736i	−0.567058	+	0.823678i	$0.691918\pi$
$284$	12.0000i	0.712069i
$285$	− 10.3923i	− 0.615587i
$286$	− 10.3923i	− 0.614510i
$287$	0	0
$288$	3.00000i	0.176777i
$289$	−5.00000	−0.294118
$290$	5.19615	0.305129
$291$	9.00000i	0.527589i
$292$	− 6.92820i	− 0.405442i
$293$	−19.0526	−1.11306	−0.556531	−	0.830827i	$-0.687868\pi$
$293$	−19.0526	−1.11306	−0.556531	+	0.830827i	$0.687868\pi$
$294$	0	0
$295$	−3.00000	−0.174667
$296$	2.00000i	0.116248i
$297$	15.5885i	0.904534i
$298$	−18.0000	−1.04271
$299$	−20.7846	−1.20201
$300$	−3.46410	−0.200000
$301$	0	0
$302$	7.00000i	0.402805i
$303$	0	0
$304$	− 3.46410i	− 0.198680i
$305$	0	0
$306$	10.3923i	0.594089i
$307$	24.2487i	1.38395i	0.721923	+	0.691974i	$0.243259\pi$
$307$	24.2487i	1.38395i	−0.721923	+	0.691974i	$0.756741\pi$
$308$	0	0
$309$	− 6.00000i	− 0.341328i
$310$	−3.00000	−0.170389
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	6.00000	0.339683
$313$	− 1.73205i	− 0.0979013i	−0.998801	−	0.0489506i	$-0.984412\pi$
$313$	− 1.73205i	− 0.0979013i	0.998801	−	0.0489506i	$-0.0155877\pi$
$314$	20.7846	1.17294
$315$	0	0
$316$	1.00000	0.0562544
$317$	− 15.0000i	− 0.842484i	−0.906948	−	0.421242i	$-0.861594\pi$
$317$	− 15.0000i	− 0.842484i	0.906948	−	0.421242i	$-0.138406\pi$
$318$	−15.5885	−0.874157
$319$	9.00000	0.503903
$320$	1.73205	0.0968246
$321$	5.19615i	0.290021i
$322$	0	0
$323$	− 12.0000i	− 0.667698i
$324$	−9.00000	−0.500000
$325$	6.92820i	0.384308i
$326$	14.0000i	0.775388i
$327$	3.46410	0.191565
$328$	− 6.92820i	− 0.382546i
$329$	0	0
$330$	9.00000	0.495434
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	8.66025	0.475293
$333$	−6.00000	−0.328798
$334$	− 17.3205i	− 0.947736i
$335$	−3.46410	−0.189264
$336$	0	0
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	1.00000i	0.0543928i
$339$	− 20.7846i	− 1.12887i
$340$	6.00000	0.325396
$341$	−5.19615	−0.281387
$342$	10.3923	0.561951
$343$	0	0
$344$	8.00000i	0.431331i
$345$	− 18.0000i	− 0.969087i
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	5.19615i	0.278543i
$349$	− 10.3923i	− 0.556287i	−0.960539	−	0.278144i	$-0.910281\pi$
$349$	− 10.3923i	− 0.556287i	0.960539	−	0.278144i	$-0.0897191\pi$
$350$	0	0
$351$	18.0000i	0.960769i
$352$	3.00000	0.159901
$353$	34.6410	1.84376	0.921878	−	0.387481i	$-0.126655\pi$
$353$	34.6410	1.84376	0.921878	+	0.387481i	$0.126655\pi$
$354$	− 3.00000i	− 0.159448i
$355$	20.7846i	1.10313i
$356$	−10.3923	−0.550791
$357$	0	0
$358$	−12.0000	−0.634220
$359$	6.00000i	0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$	6.00000i	0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$	5.19615i	0.273861i
$361$	7.00000	0.368421
$362$	6.92820	0.364138
$363$	−3.46410	−0.181818
$364$	0	0
$365$	− 12.0000i	− 0.628109i
$366$	0	0
$367$	− 22.5167i	− 1.17536i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	− 22.5167i	− 1.17536i	0.809093	−	0.587680i	$-0.199959\pi$
$368$	− 6.00000i	− 0.312772i
$369$	20.7846	1.08200
$370$	3.46410i	0.180090i
$371$	0	0
$372$	− 3.00000i	− 0.155543i
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	10.3923	0.537373
$375$	−21.0000	−1.08444
$376$	6.92820i	0.357295i
$377$	10.3923	0.535231
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	− 6.00000i	− 0.307794i
$381$	−19.0526	−0.976092
$382$	0	0
$383$	−3.46410	−0.177007	−0.0885037	−	0.996076i	$-0.528208\pi$
$383$	−3.46410	−0.177007	−0.0885037	+	0.996076i	$0.528208\pi$
$384$	1.73205i	0.0883883i
$385$	0	0
$386$	− 23.0000i	− 1.17067i
$387$	−24.0000	−1.21999
$388$	5.19615i	0.263795i
$389$	− 18.0000i	− 0.912636i	−0.889817	−	0.456318i	$-0.849168\pi$
$389$	− 18.0000i	− 0.912636i	0.889817	−	0.456318i	$-0.150832\pi$
$390$	10.3923	0.526235
$391$	− 20.7846i	− 1.05112i
$392$	0	0
$393$	−9.00000	−0.453990
$394$	18.0000	0.906827
$395$	1.73205	0.0871489
$396$	9.00000i	0.452267i
$397$	27.7128i	1.39087i	0.718591	+	0.695433i	$0.244787\pi$
$397$	27.7128i	1.39087i	−0.718591	+	0.695433i	$0.755213\pi$
$398$	−10.3923	−0.520919
$399$	0	0
$400$	−2.00000	−0.100000
$401$	− 12.0000i	− 0.599251i	−0.954057	−	0.299626i	$-0.903138\pi$
$401$	− 12.0000i	− 0.599251i	0.954057	−	0.299626i	$-0.0968618\pi$
$402$	− 3.46410i	− 0.172774i
$403$	−6.00000	−0.298881
$404$	0	0
$405$	−15.5885	−0.774597
$406$	0	0
$407$	6.00000i	0.297409i
$408$	6.00000i	0.297044i
$409$	8.66025i	0.428222i	0.976809	+	0.214111i	$0.0686854\pi$
$409$	8.66025i	0.428222i	−0.976809	+	0.214111i	$0.931315\pi$
$410$	− 12.0000i	− 0.592638i
$411$	− 31.1769i	− 1.53784i
$412$	− 3.46410i	− 0.170664i
$413$	0	0
$414$	18.0000	0.884652
$415$	15.0000	0.736321
$416$	3.46410	0.169842
$417$	30.0000i	1.46911i
$418$	− 10.3923i	− 0.508304i
$419$	24.2487	1.18463	0.592314	−	0.805708i	$-0.298215\pi$
$419$	24.2487	1.18463	0.592314	+	0.805708i	$0.298215\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	4.00000i	0.194717i
$423$	−20.7846	−1.01058
$424$	−9.00000	−0.437079
$425$	−6.92820	−0.336067
$426$	−20.7846	−1.00702
$427$	0	0
$428$	3.00000i	0.145010i
$429$	18.0000	0.869048
$430$	13.8564i	0.668215i
$431$	− 24.0000i	− 1.15604i	−0.816023	−	0.578020i	$-0.803826\pi$
$431$	− 24.0000i	− 1.15604i	0.816023	−	0.578020i	$-0.196174\pi$
$432$	−5.19615	−0.250000
$433$	− 34.6410i	− 1.66474i	−0.554220	−	0.832370i	$-0.686983\pi$
$433$	− 34.6410i	− 1.66474i	0.554220	−	0.832370i	$-0.313017\pi$
$434$	0	0
$435$	9.00000i	0.431517i
$436$	2.00000	0.0957826
$437$	−20.7846	−0.994263
$438$	12.0000	0.573382
$439$	− 36.3731i	− 1.73599i	−0.496571	−	0.867996i	$-0.665408\pi$
$439$	− 36.3731i	− 1.73599i	0.496571	−	0.867996i	$-0.334592\pi$
$440$	5.19615	0.247717
$441$	0	0
$442$	12.0000	0.570782
$443$	9.00000i	0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$	9.00000i	0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	−3.46410	−0.164399
$445$	−18.0000	−0.853282
$446$	−25.9808	−1.23022
$447$	− 31.1769i	− 1.47462i
$448$	0	0
$449$	− 30.0000i	− 1.41579i	−0.706319	−	0.707894i	$-0.749646\pi$
$449$	− 30.0000i	− 1.41579i	0.706319	−	0.707894i	$-0.250354\pi$
$450$	− 6.00000i	− 0.282843i
$451$	− 20.7846i	− 0.978709i
$452$	− 12.0000i	− 0.564433i
$453$	−12.1244	−0.569652
$454$	− 5.19615i	− 0.243868i
$455$	0	0
$456$	6.00000	0.280976
$457$	−5.00000	−0.233890	−0.116945	−	0.993138i	$-0.537310\pi$
$457$	−5.00000	−0.233890	−0.116945	+	0.993138i	$0.537310\pi$
$458$	−13.8564	−0.647467
$459$	−18.0000	−0.840168
$460$	− 10.3923i	− 0.484544i
$461$	−13.8564	−0.645357	−0.322679	−	0.946509i	$-0.604583\pi$
$461$	−13.8564	−0.645357	−0.322679	+	0.946509i	$0.604583\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	3.00000i	0.139272i
$465$	− 5.19615i	− 0.240966i
$466$	18.0000	0.833834
$467$	31.1769	1.44270	0.721348	−	0.692573i	$-0.243523\pi$
$467$	31.1769	1.44270	0.721348	+	0.692573i	$0.243523\pi$
$468$	10.3923i	0.480384i
$469$	0	0
$470$	12.0000i	0.553519i
$471$	36.0000i	1.65879i
$472$	− 1.73205i	− 0.0797241i
$473$	24.0000i	1.10352i
$474$	1.73205i	0.0795557i
$475$	6.92820i	0.317888i
$476$	0	0
$477$	− 27.0000i	− 1.23625i
$478$	−6.00000	−0.274434
$479$	6.92820	0.316558	0.158279	−	0.987394i	$-0.449406\pi$
$479$	6.92820	0.316558	0.158279	+	0.987394i	$0.449406\pi$
$480$	3.00000i	0.136931i
$481$	6.92820i	0.315899i
$482$	25.9808	1.18339
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	9.00000i	0.408669i
$486$	− 15.5885i	− 0.707107i
$487$	1.00000	0.0453143	0.0226572	−	0.999743i	$-0.492787\pi$
$487$	1.00000	0.0453143	0.0226572	+	0.999743i	$0.492787\pi$
$488$	0	0
$489$	−24.2487	−1.09656
$490$	0	0
$491$	33.0000i	1.48927i	0.667472	+	0.744635i	$0.267376\pi$
$491$	33.0000i	1.48927i	−0.667472	+	0.744635i	$0.732624\pi$
$492$	12.0000	0.541002
$493$	10.3923i	0.468046i
$494$	− 12.0000i	− 0.539906i
$495$	15.5885i	0.700649i
$496$	− 1.73205i	− 0.0777714i
$497$	0	0
$498$	15.0000i	0.672166i
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	−12.1244	−0.542218
$501$	30.0000	1.34030
$502$	19.0526i	0.850357i
$503$	38.1051	1.69902	0.849512	−	0.527570i	$-0.176897\pi$
$503$	38.1051	1.69902	0.849512	+	0.527570i	$0.176897\pi$
$504$	0	0
$505$	0	0
$506$	− 18.0000i	− 0.800198i
$507$	−1.73205	−0.0769231
$508$	−11.0000	−0.488046
$509$	−19.0526	−0.844490	−0.422245	−	0.906482i	$-0.638758\pi$
$509$	−19.0526	−0.844490	−0.422245	+	0.906482i	$0.638758\pi$
$510$	10.3923i	0.460179i
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	18.0000i	0.794719i
$514$	− 10.3923i	− 0.458385i
$515$	− 6.00000i	− 0.264392i
$516$	−13.8564	−0.609994
$517$	20.7846i	0.914106i
$518$	0	0
$519$	0	0
$520$	6.00000	0.263117
$521$	27.7128	1.21412	0.607060	−	0.794656i	$-0.292349\pi$
$521$	27.7128	1.21412	0.607060	+	0.794656i	$0.292349\pi$
$522$	−9.00000	−0.393919
$523$	38.1051i	1.66622i	0.553107	+	0.833110i	$0.313442\pi$
$523$	38.1051i	1.66622i	−0.553107	+	0.833110i	$0.686558\pi$
$524$	−5.19615	−0.226995
$525$	0	0
$526$	0	0
$527$	− 6.00000i	− 0.261364i
$528$	5.19615i	0.226134i
$529$	−13.0000	−0.565217
$530$	−15.5885	−0.677119
$531$	5.19615	0.225494
$532$	0	0
$533$	− 24.0000i	− 1.03956i
$534$	− 18.0000i	− 0.778936i
$535$	5.19615i	0.224649i
$536$	− 2.00000i	− 0.0863868i
$537$	− 20.7846i	− 0.896922i
$538$	− 29.4449i	− 1.26946i
$539$	0	0
$540$	−9.00000	−0.387298
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	5.19615	0.223194
$543$	12.0000i	0.514969i
$544$	3.46410i	0.148522i
$545$	3.46410	0.148386
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	− 18.0000i	− 0.768922i
$549$	0	0
$550$	−6.00000	−0.255841
$551$	10.3923	0.442727
$552$	10.3923	0.442326
$553$	0	0
$554$	8.00000i	0.339887i
$555$	−6.00000	−0.254686
$556$	17.3205i	0.734553i
$557$	− 3.00000i	− 0.127114i	−0.997978	−	0.0635570i	$-0.979756\pi$
$557$	− 3.00000i	− 0.127114i	0.997978	−	0.0635570i	$-0.0202445\pi$
$558$	5.19615	0.219971
$559$	27.7128i	1.17213i
$560$	0	0
$561$	18.0000i	0.759961i
$562$	−30.0000	−1.26547
$563$	25.9808	1.09496	0.547479	−	0.836819i	$-0.315587\pi$
$563$	25.9808	1.09496	0.547479	+	0.836819i	$0.315587\pi$
$564$	−12.0000	−0.505291
$565$	− 20.7846i	− 0.874415i
$566$	−27.7128	−1.16486
$567$	0	0
$568$	−12.0000	−0.503509
$569$	− 6.00000i	− 0.251533i	−0.992060	−	0.125767i	$-0.959861\pi$
$569$	− 6.00000i	− 0.251533i	0.992060	−	0.125767i	$-0.0401390\pi$
$570$	10.3923	0.435286
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	10.3923	0.434524
$573$	0	0
$574$	0	0
$575$	12.0000i	0.500435i
$576$	−3.00000	−0.125000
$577$	− 1.73205i	− 0.0721062i	−0.999350	−	0.0360531i	$-0.988521\pi$
$577$	− 1.73205i	− 0.0721062i	0.999350	−	0.0360531i	$-0.0114785\pi$
$578$	− 5.00000i	− 0.207973i
$579$	39.8372	1.65558
$580$	5.19615i	0.215758i
$581$	0	0
$582$	−9.00000	−0.373062
$583$	−27.0000	−1.11823
$584$	6.92820	0.286691
$585$	18.0000i	0.744208i
$586$	− 19.0526i	− 0.787054i
$587$	−15.5885	−0.643404	−0.321702	−	0.946841i	$-0.604255\pi$
$587$	−15.5885	−0.643404	−0.321702	+	0.946841i	$0.604255\pi$
$588$	0	0
$589$	−6.00000	−0.247226
$590$	− 3.00000i	− 0.123508i
$591$	31.1769i	1.28245i
$592$	−2.00000	−0.0821995
$593$	38.1051	1.56479	0.782395	−	0.622783i	$-0.213998\pi$
$593$	38.1051	1.56479	0.782395	+	0.622783i	$0.213998\pi$
$594$	−15.5885	−0.639602
$595$	0	0
$596$	− 18.0000i	− 0.737309i
$597$	− 18.0000i	− 0.736691i
$598$	− 20.7846i	− 0.849946i
$599$	30.0000i	1.22577i	0.790173	+	0.612883i	$0.209990\pi$
$599$	30.0000i	1.22577i	−0.790173	+	0.612883i	$0.790010\pi$
$600$	− 3.46410i	− 0.141421i
$601$	29.4449i	1.20108i	0.799594	+	0.600541i	$0.205048\pi$
$601$	29.4449i	1.20108i	−0.799594	+	0.600541i	$0.794952\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	−7.00000	−0.284826
$605$	−3.46410	−0.140836
$606$	0	0
$607$	− 22.5167i	− 0.913923i	−0.889486	−	0.456962i	$-0.848938\pi$
$607$	− 22.5167i	− 0.913923i	0.889486	−	0.456962i	$-0.151062\pi$
$608$	3.46410	0.140488
$609$	0	0
$610$	0	0
$611$	24.0000i	0.970936i
$612$	−10.3923	−0.420084
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−24.2487	−0.978598
$615$	20.7846	0.838116
$616$	0	0
$617$	− 12.0000i	− 0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$	− 12.0000i	− 0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$	6.00000	0.241355
$619$	6.92820i	0.278468i	0.990260	+	0.139234i	$0.0444640\pi$
$619$	6.92820i	0.278468i	−0.990260	+	0.139234i	$0.955536\pi$
$620$	− 3.00000i	− 0.120483i
$621$	31.1769i	1.25109i
$622$	13.8564i	0.555591i
$623$	0	0
$624$	6.00000i	0.240192i
$625$	−11.0000	−0.440000
$626$	1.73205	0.0692267
$627$	18.0000	0.718851
$628$	20.7846i	0.829396i
$629$	−6.92820	−0.276246
$630$	0	0
$631$	−31.0000	−1.23409	−0.617045	−	0.786928i	$-0.711670\pi$
$631$	−31.0000	−1.23409	−0.617045	+	0.786928i	$0.711670\pi$
$632$	1.00000i	0.0397779i
$633$	−6.92820	−0.275371
$634$	15.0000	0.595726
$635$	−19.0526	−0.756078
$636$	− 15.5885i	− 0.618123i
$637$	0	0
$638$	9.00000i	0.356313i
$639$	− 36.0000i	− 1.42414i
$640$	1.73205i	0.0684653i
$641$	24.0000i	0.947943i	0.880540	+	0.473972i	$0.157180\pi$
$641$	24.0000i	0.947943i	−0.880540	+	0.473972i	$0.842820\pi$
$642$	−5.19615	−0.205076
$643$	17.3205i	0.683054i	0.939872	+	0.341527i	$0.110944\pi$
$643$	17.3205i	0.683054i	−0.939872	+	0.341527i	$0.889056\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	12.0000	0.472134
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	− 9.00000i	− 0.353553i
$649$	− 5.19615i	− 0.203967i
$650$	−6.92820	−0.271746
$651$	0	0
$652$	−14.0000	−0.548282
$653$	− 3.00000i	− 0.117399i	−0.998276	−	0.0586995i	$-0.981305\pi$
$653$	− 3.00000i	− 0.117399i	0.998276	−	0.0586995i	$-0.0186954\pi$
$654$	3.46410i	0.135457i
$655$	−9.00000	−0.351659
$656$	6.92820	0.270501
$657$	20.7846i	0.810885i
$658$	0	0
$659$	12.0000i	0.467454i	0.972302	+	0.233727i	$0.0750921\pi$
$659$	12.0000i	0.467454i	−0.972302	+	0.233727i	$0.924908\pi$
$660$	9.00000i	0.350325i
$661$	− 6.92820i	− 0.269476i	−0.990881	−	0.134738i	$-0.956981\pi$
$661$	− 6.92820i	− 0.269476i	0.990881	−	0.134738i	$-0.0430193\pi$
$662$	− 8.00000i	− 0.310929i
$663$	20.7846i	0.807207i
$664$	8.66025i	0.336083i
$665$	0	0
$666$	− 6.00000i	− 0.232495i
$667$	18.0000	0.696963
$668$	17.3205	0.670151
$669$	− 45.0000i	− 1.73980i
$670$	− 3.46410i	− 0.133830i
$671$	0	0
$672$	0	0
$673$	−41.0000	−1.58043	−0.790217	−	0.612827i	$-0.790032\pi$
$673$	−41.0000	−1.58043	−0.790217	+	0.612827i	$0.790032\pi$
$674$	13.0000i	0.500741i
$675$	10.3923	0.400000
$676$	−1.00000	−0.0384615
$677$	5.19615	0.199704	0.0998522	−	0.995002i	$-0.468163\pi$
$677$	5.19615	0.199704	0.0998522	+	0.995002i	$0.468163\pi$
$678$	20.7846	0.798228
$679$	0	0
$680$	6.00000i	0.230089i
$681$	9.00000	0.344881
$682$	− 5.19615i	− 0.198971i
$683$	− 21.0000i	− 0.803543i	−0.915740	−	0.401771i	$-0.868395\pi$
$683$	− 21.0000i	− 0.803543i	0.915740	−	0.401771i	$-0.131605\pi$
$684$	10.3923i	0.397360i
$685$	− 31.1769i	− 1.19121i
$686$	0	0
$687$	− 24.0000i	− 0.915657i
$688$	−8.00000	−0.304997
$689$	−31.1769	−1.18775
$690$	18.0000	0.685248
$691$	6.92820i	0.263561i	0.991279	+	0.131781i	$0.0420694\pi$
$691$	6.92820i	0.263561i	−0.991279	+	0.131781i	$0.957931\pi$
$692$	0	0
$693$	0	0
$694$	12.0000	0.455514
$695$	30.0000i	1.13796i
$696$	−5.19615	−0.196960
$697$	24.0000	0.909065
$698$	10.3923	0.393355
$699$	31.1769i	1.17922i
$700$	0	0
$701$	− 3.00000i	− 0.113308i	−0.998394	−	0.0566542i	$-0.981957\pi$
$701$	− 3.00000i	− 0.113308i	0.998394	−	0.0566542i	$-0.0180433\pi$
$702$	−18.0000	−0.679366
$703$	6.92820i	0.261302i
$704$	3.00000i	0.113067i
$705$	−20.7846	−0.782794
$706$	34.6410i	1.30373i
$707$	0	0
$708$	3.00000	0.112747
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	−20.7846	−0.780033
$711$	−3.00000	−0.112509
$712$	− 10.3923i	− 0.389468i
$713$	−10.3923	−0.389195
$714$	0	0
$715$	18.0000	0.673162
$716$	− 12.0000i	− 0.448461i
$717$	− 10.3923i	− 0.388108i
$718$	−6.00000	−0.223918
$719$	−45.0333	−1.67946	−0.839730	−	0.543005i	$-0.817287\pi$
$719$	−45.0333	−1.67946	−0.839730	+	0.543005i	$0.817287\pi$
$720$	−5.19615	−0.193649
$721$	0	0
$722$	7.00000i	0.260513i
$723$	45.0000i	1.67357i
$724$	6.92820i	0.257485i
$725$	− 6.00000i	− 0.222834i
$726$	− 3.46410i	− 0.128565i
$727$	− 29.4449i	− 1.09205i	−0.837769	−	0.546025i	$-0.816140\pi$
$727$	− 29.4449i	− 1.09205i	0.837769	−	0.546025i	$-0.183860\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	12.0000	0.444140
$731$	−27.7128	−1.02500
$732$	0	0
$733$	− 34.6410i	− 1.27950i	−0.768585	−	0.639748i	$-0.779039\pi$
$733$	− 34.6410i	− 1.27950i	0.768585	−	0.639748i	$-0.220961\pi$
$734$	22.5167	0.831105
$735$	0	0
$736$	6.00000	0.221163
$737$	− 6.00000i	− 0.221013i
$738$	20.7846i	0.765092i
$739$	−10.0000	−0.367856	−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000	−0.367856	−0.183928	+	0.982940i	$0.558881\pi$
$740$	−3.46410	−0.127343
$741$	20.7846	0.763542
$742$	0	0
$743$	− 18.0000i	− 0.660356i	−0.943919	−	0.330178i	$-0.892891\pi$
$743$	− 18.0000i	− 0.660356i	0.943919	−	0.330178i	$-0.107109\pi$
$744$	3.00000	0.109985
$745$	− 31.1769i	− 1.14223i
$746$	32.0000i	1.17160i
$747$	−25.9808	−0.950586
$748$	10.3923i	0.379980i
$749$	0	0
$750$	− 21.0000i	− 0.766812i
$751$	−11.0000	−0.401396	−0.200698	−	0.979653i	$-0.564321\pi$
$751$	−11.0000	−0.401396	−0.200698	+	0.979653i	$0.564321\pi$
$752$	−6.92820	−0.252646
$753$	−33.0000	−1.20259
$754$	10.3923i	0.378465i
$755$	−12.1244	−0.441250
$756$	0	0
$757$	4.00000	0.145382	0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000	0.145382	0.0726912	+	0.997354i	$0.476841\pi$
$758$	16.0000i	0.581146i
$759$	31.1769	1.13165
$760$	6.00000	0.217643
$761$	17.3205	0.627868	0.313934	−	0.949445i	$-0.398353\pi$
$761$	17.3205	0.627868	0.313934	+	0.949445i	$0.398353\pi$
$762$	− 19.0526i	− 0.690201i
$763$	0	0
$764$	0	0
$765$	−18.0000	−0.650791
$766$	− 3.46410i	− 0.125163i
$767$	− 6.00000i	− 0.216647i
$768$	−1.73205	−0.0625000
$769$	− 19.0526i	− 0.687053i	−0.939143	−	0.343526i	$-0.888379\pi$
$769$	− 19.0526i	− 0.687053i	0.939143	−	0.343526i	$-0.111621\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	23.0000	0.827788
$773$	−27.7128	−0.996761	−0.498380	−	0.866959i	$-0.666072\pi$
$773$	−27.7128	−0.996761	−0.498380	+	0.866959i	$0.666072\pi$
$774$	− 24.0000i	− 0.862662i
$775$	3.46410i	0.124434i
$776$	−5.19615	−0.186531
$777$	0	0
$778$	18.0000	0.645331
$779$	− 24.0000i	− 0.859889i
$780$	10.3923i	0.372104i
$781$	−36.0000	−1.28818
$782$	20.7846	0.743256
$783$	− 15.5885i	− 0.557086i
$784$	0	0
$785$	36.0000i	1.28490i
$786$	− 9.00000i	− 0.321019i
$787$	41.5692i	1.48178i	0.671625	+	0.740891i	$0.265597\pi$
$787$	41.5692i	1.48178i	−0.671625	+	0.740891i	$0.734403\pi$
$788$	18.0000i	0.641223i
$789$	0	0
$790$	1.73205i	0.0616236i
$791$	0	0
$792$	−9.00000	−0.319801
$793$	0	0
$794$	−27.7128	−0.983491
$795$	− 27.0000i	− 0.957591i
$796$	− 10.3923i	− 0.368345i
$797$	−25.9808	−0.920286	−0.460143	−	0.887845i	$-0.652202\pi$
$797$	−25.9808	−0.920286	−0.460143	+	0.887845i	$0.652202\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	− 2.00000i	− 0.0707107i
$801$	31.1769	1.10158
$802$	12.0000	0.423735
$803$	20.7846	0.733473
$804$	3.46410	0.122169
$805$	0	0
$806$	− 6.00000i	− 0.211341i
$807$	51.0000	1.79529
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	− 15.5885i	− 0.547723i
$811$	− 31.1769i	− 1.09477i	−0.836881	−	0.547385i	$-0.815623\pi$
$811$	− 31.1769i	− 1.09477i	0.836881	−	0.547385i	$-0.184377\pi$
$812$	0	0
$813$	9.00000i	0.315644i
$814$	−6.00000	−0.210300
$815$	−24.2487	−0.849395
$816$	−6.00000	−0.210042
$817$	27.7128i	0.969549i
$818$	−8.66025	−0.302799
$819$	0	0
$820$	12.0000	0.419058
$821$	3.00000i	0.104701i	0.998629	+	0.0523504i	$0.0166713\pi$
$821$	3.00000i	0.104701i	−0.998629	+	0.0523504i	$0.983329\pi$
$822$	31.1769	1.08742
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	3.46410	0.120678
$825$	− 10.3923i	− 0.361814i
$826$	0	0
$827$	− 9.00000i	− 0.312961i	−0.987681	−	0.156480i	$-0.949985\pi$
$827$	− 9.00000i	− 0.312961i	0.987681	−	0.156480i	$-0.0500148\pi$
$828$	18.0000i	0.625543i
$829$	− 17.3205i	− 0.601566i	−0.953693	−	0.300783i	$-0.902752\pi$
$829$	− 17.3205i	− 0.601566i	0.953693	−	0.300783i	$-0.0972480\pi$
$830$	15.0000i	0.520658i
$831$	−13.8564	−0.480673
$832$	3.46410i	0.120096i
$833$	0	0
$834$	−30.0000	−1.03882
$835$	30.0000	1.03819
$836$	10.3923	0.359425
$837$	9.00000i	0.311086i
$838$	24.2487i	0.837658i
$839$	3.46410	0.119594	0.0597970	−	0.998211i	$-0.480955\pi$
$839$	3.46410	0.119594	0.0597970	+	0.998211i	$0.480955\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	32.0000i	1.10279i
$843$	− 51.9615i	− 1.78965i
$844$	−4.00000	−0.137686
$845$	−1.73205	−0.0595844
$846$	− 20.7846i	− 0.714590i
$847$	0	0
$848$	− 9.00000i	− 0.309061i
$849$	− 48.0000i	− 1.64736i
$850$	− 6.92820i	− 0.237635i
$851$	12.0000i	0.411355i
$852$	− 20.7846i	− 0.712069i
$853$	24.2487i	0.830260i	0.909762	+	0.415130i	$0.136264\pi$
$853$	24.2487i	0.830260i	−0.909762	+	0.415130i	$0.863736\pi$
$854$	0	0
$855$	18.0000i	0.615587i
$856$	−3.00000	−0.102538
$857$	13.8564	0.473326	0.236663	−	0.971592i	$-0.423946\pi$
$857$	13.8564	0.473326	0.236663	+	0.971592i	$0.423946\pi$
$858$	18.0000i	0.614510i
$859$	20.7846i	0.709162i	0.935025	+	0.354581i	$0.115376\pi$
$859$	20.7846i	0.709162i	−0.935025	+	0.354581i	$0.884624\pi$
$860$	−13.8564	−0.472500
$861$	0	0
$862$	24.0000	0.817443
$863$	− 54.0000i	− 1.83818i	−0.394046	−	0.919091i	$-0.628925\pi$
$863$	− 54.0000i	− 1.83818i	0.394046	−	0.919091i	$-0.371075\pi$
$864$	− 5.19615i	− 0.176777i
$865$	0	0
$866$	34.6410	1.17715
$867$	8.66025	0.294118
$868$	0	0
$869$	3.00000i	0.101768i
$870$	−9.00000	−0.305129
$871$	− 6.92820i	− 0.234753i
$872$	2.00000i	0.0677285i
$873$	− 15.5885i	− 0.527589i
$874$	− 20.7846i	− 0.703050i
$875$	0	0
$876$	12.0000i	0.405442i
$877$	44.0000	1.48577	0.742887	−	0.669417i	$-0.233456\pi$
$877$	44.0000	1.48577	0.742887	+	0.669417i	$0.233456\pi$
$878$	36.3731	1.22753
$879$	33.0000	1.11306
$880$	5.19615i	0.175162i
$881$	10.3923	0.350126	0.175063	−	0.984557i	$-0.443987\pi$
$881$	10.3923	0.350126	0.175063	+	0.984557i	$0.443987\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	12.0000i	0.403604i
$885$	5.19615	0.174667
$886$	−9.00000	−0.302361
$887$	24.2487	0.814192	0.407096	−	0.913385i	$-0.366541\pi$
$887$	24.2487	0.814192	0.407096	+	0.913385i	$0.366541\pi$
$888$	− 3.46410i	− 0.116248i
$889$	0	0
$890$	− 18.0000i	− 0.603361i
$891$	− 27.0000i	− 0.904534i
$892$	− 25.9808i	− 0.869900i
$893$	24.0000i	0.803129i
$894$	31.1769	1.04271
$895$	− 20.7846i	− 0.694753i
$896$	0	0
$897$	36.0000	1.20201
$898$	30.0000	1.00111
$899$	5.19615	0.173301
$900$	6.00000	0.200000
$901$	− 31.1769i	− 1.03865i
$902$	20.7846	0.692052
$903$	0	0
$904$	12.0000	0.399114
$905$	12.0000i	0.398893i
$906$	− 12.1244i	− 0.402805i
$907$	−26.0000	−0.863316	−0.431658	−	0.902037i	$-0.642071\pi$
$907$	−26.0000	−0.863316	−0.431658	+	0.902037i	$0.642071\pi$
$908$	5.19615	0.172440
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	6.00000i	0.198680i
$913$	25.9808i	0.859838i
$914$	− 5.00000i	− 0.165385i
$915$	0	0
$916$	− 13.8564i	− 0.457829i
$917$	0	0
$918$	− 18.0000i	− 0.594089i
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	10.3923	0.342624
$921$	− 42.0000i	− 1.38395i
$922$	− 13.8564i	− 0.456336i
$923$	−41.5692	−1.36827
$924$	0	0
$925$	4.00000	0.131519
$926$	− 4.00000i	− 0.131448i
$927$	10.3923i	0.341328i
$928$	−3.00000	−0.0984798
$929$	−48.4974	−1.59115	−0.795574	−	0.605856i	$-0.792831\pi$
$929$	−48.4974	−1.59115	−0.795574	+	0.605856i	$0.792831\pi$
$930$	5.19615	0.170389
$931$	0	0
$932$	18.0000i	0.589610i
$933$	−24.0000	−0.785725
$934$	31.1769i	1.02014i
$935$	18.0000i	0.588663i
$936$	−10.3923	−0.339683
$937$	− 22.5167i	− 0.735587i	−0.929907	−	0.367794i	$-0.880113\pi$
$937$	− 22.5167i	− 0.735587i	0.929907	−	0.367794i	$-0.119887\pi$
$938$	0	0
$939$	3.00000i	0.0979013i
$940$	−12.0000	−0.391397
$941$	32.9090	1.07280	0.536401	−	0.843963i	$-0.319784\pi$
$941$	32.9090	1.07280	0.536401	+	0.843963i	$0.319784\pi$
$942$	−36.0000	−1.17294
$943$	− 41.5692i	− 1.35368i
$944$	1.73205	0.0563735
$945$	0	0
$946$	−24.0000	−0.780307
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	−1.73205	−0.0562544
$949$	24.0000	0.779073
$950$	−6.92820	−0.224781
$951$	25.9808i	0.842484i
$952$	0	0
$953$	24.0000i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$	24.0000i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$	27.0000	0.874157
$955$	0	0
$956$	− 6.00000i	− 0.194054i
$957$	−15.5885	−0.503903
$958$	6.92820i	0.223840i
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	28.0000	0.903226
$962$	−6.92820	−0.223374
$963$	− 9.00000i	− 0.290021i
$964$	25.9808i	0.836784i
$965$	39.8372	1.28240
$966$	0	0
$967$	−7.00000	−0.225105	−0.112552	−	0.993646i	$-0.535903\pi$
$967$	−7.00000	−0.225105	−0.112552	+	0.993646i	$0.535903\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	20.7846i	0.667698i
$970$	−9.00000	−0.288973
$971$	−8.66025	−0.277921	−0.138960	−	0.990298i	$-0.544376\pi$
$971$	−8.66025	−0.277921	−0.138960	+	0.990298i	$0.544376\pi$
$972$	15.5885	0.500000
$973$	0	0
$974$	1.00000i	0.0320421i
$975$	− 12.0000i	− 0.384308i
$976$	0	0
$977$	− 24.0000i	− 0.767828i	−0.923369	−	0.383914i	$-0.874576\pi$
$977$	− 24.0000i	− 0.767828i	0.923369	−	0.383914i	$-0.125424\pi$
$978$	− 24.2487i	− 0.775388i
$979$	− 31.1769i	− 0.996419i
$980$	0	0
$981$	−6.00000	−0.191565
$982$	−33.0000	−1.05307
$983$	−13.8564	−0.441951	−0.220975	−	0.975279i	$-0.570924\pi$
$983$	−13.8564	−0.441951	−0.220975	+	0.975279i	$0.570924\pi$
$984$	12.0000i	0.382546i
$985$	31.1769i	0.993379i
$986$	−10.3923	−0.330958
$987$	0	0
$988$	12.0000	0.381771
$989$	48.0000i	1.52631i
$990$	−15.5885	−0.495434
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	1.73205	0.0549927
$993$	13.8564	0.439720
$994$	0	0
$995$	− 18.0000i	− 0.570638i
$996$	−15.0000	−0.475293
$997$	17.3205i	0.548546i	0.961652	+	0.274273i	$0.0884372\pi$
$997$	17.3205i	0.548546i	−0.961652	+	0.274273i	$0.911563\pi$
$998$	22.0000i	0.696398i
$999$	10.3923	0.328798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.2.d.a.293.3		4
3.2	odd	2	inner	294.2.d.a.293.2		4
4.3	odd	2		2352.2.k.e.881.4		4
7.2	even	3		294.2.f.a.227.2		4
7.3	odd	6		294.2.f.a.215.1		4
7.4	even	3		42.2.f.a.5.1	✓	4
7.5	odd	6		42.2.f.a.17.2	yes	4
7.6	odd	2	inner	294.2.d.a.293.4		4
12.11	even	2		2352.2.k.e.881.1		4
21.2	odd	6		294.2.f.a.227.1		4
21.5	even	6		42.2.f.a.17.1	yes	4
21.11	odd	6		42.2.f.a.5.2	yes	4
21.17	even	6		294.2.f.a.215.2		4
21.20	even	2	inner	294.2.d.a.293.1		4
28.11	odd	6		336.2.bc.e.257.1		4
28.19	even	6		336.2.bc.e.17.2		4
28.27	even	2		2352.2.k.e.881.2		4
35.4	even	6		1050.2.s.b.551.2		4
35.12	even	12		1050.2.u.d.899.1		4
35.18	odd	12		1050.2.u.a.299.1		4
35.19	odd	6		1050.2.s.b.101.1		4
35.32	odd	12		1050.2.u.d.299.2		4
35.33	even	12		1050.2.u.a.899.2		4
63.4	even	3		1134.2.t.d.593.2		4
63.5	even	6		1134.2.l.c.269.1		4
63.11	odd	6		1134.2.l.c.215.1		4
63.25	even	3		1134.2.l.c.215.2		4
63.32	odd	6		1134.2.t.d.593.1		4
63.40	odd	6		1134.2.l.c.269.2		4
63.47	even	6		1134.2.t.d.1025.2		4
63.61	odd	6		1134.2.t.d.1025.1		4
84.11	even	6		336.2.bc.e.257.2		4
84.47	odd	6		336.2.bc.e.17.1		4
84.83	odd	2		2352.2.k.e.881.3		4
105.32	even	12		1050.2.u.a.299.2		4
105.47	odd	12		1050.2.u.a.899.1		4
105.53	even	12		1050.2.u.d.299.1		4
105.68	odd	12		1050.2.u.d.899.2		4
105.74	odd	6		1050.2.s.b.551.1		4
105.89	even	6		1050.2.s.b.101.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.f.a.5.1	✓	4	7.4	even	3
42.2.f.a.5.2	yes	4	21.11	odd	6
42.2.f.a.17.1	yes	4	21.5	even	6
42.2.f.a.17.2	yes	4	7.5	odd	6
294.2.d.a.293.1		4	21.20	even	2	inner
294.2.d.a.293.2		4	3.2	odd	2	inner
294.2.d.a.293.3		4	1.1	even	1	trivial
294.2.d.a.293.4		4	7.6	odd	2	inner
294.2.f.a.215.1		4	7.3	odd	6
294.2.f.a.215.2		4	21.17	even	6
294.2.f.a.227.1		4	21.2	odd	6
294.2.f.a.227.2		4	7.2	even	3
336.2.bc.e.17.1		4	84.47	odd	6
336.2.bc.e.17.2		4	28.19	even	6
336.2.bc.e.257.1		4	28.11	odd	6
336.2.bc.e.257.2		4	84.11	even	6
1050.2.s.b.101.1		4	35.19	odd	6
1050.2.s.b.101.2		4	105.89	even	6
1050.2.s.b.551.1		4	105.74	odd	6
1050.2.s.b.551.2		4	35.4	even	6
1050.2.u.a.299.1		4	35.18	odd	12
1050.2.u.a.299.2		4	105.32	even	12
1050.2.u.a.899.1		4	105.47	odd	12
1050.2.u.a.899.2		4	35.33	even	12
1050.2.u.d.299.1		4	105.53	even	12
1050.2.u.d.299.2		4	35.32	odd	12
1050.2.u.d.899.1		4	35.12	even	12
1050.2.u.d.899.2		4	105.68	odd	12
1134.2.l.c.215.1		4	63.11	odd	6
1134.2.l.c.215.2		4	63.25	even	3
1134.2.l.c.269.1		4	63.5	even	6
1134.2.l.c.269.2		4	63.40	odd	6
1134.2.t.d.593.1		4	63.32	odd	6
1134.2.t.d.593.2		4	63.4	even	3
1134.2.t.d.1025.1		4	63.61	odd	6
1134.2.t.d.1025.2		4	63.47	even	6
2352.2.k.e.881.1		4	12.11	even	2
2352.2.k.e.881.2		4	28.27	even	2
2352.2.k.e.881.3		4	84.83	odd	2
2352.2.k.e.881.4		4	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	294.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.73205 q^{3} -1.00000 q^{4} -1.73205 q^{5} -1.73205i q^{6} -1.00000i q^{8} +3.00000 q^{9} -1.73205i q^{10} -3.00000i q^{11} +1.73205 q^{12} -3.46410i q^{13} +3.00000 q^{15} +1.00000 q^{16} +3.46410 q^{17} +3.00000i q^{18} -3.46410i q^{19} +1.73205 q^{20} +3.00000 q^{22} -6.00000i q^{23} +1.73205i q^{24} -2.00000 q^{25} +3.46410 q^{26} -5.19615 q^{27} +3.00000i q^{29} +3.00000i q^{30} -1.73205i q^{31} +1.00000i q^{32} +5.19615i q^{33} +3.46410i q^{34} -3.00000 q^{36} -2.00000 q^{37} +3.46410 q^{38} +6.00000i q^{39} +1.73205i q^{40} +6.92820 q^{41} -8.00000 q^{43} +3.00000i q^{44} -5.19615 q^{45} +6.00000 q^{46} -6.92820 q^{47} -1.73205 q^{48} -2.00000i q^{50} -6.00000 q^{51} +3.46410i q^{52} -9.00000i q^{53} -5.19615i q^{54} +5.19615i q^{55} +6.00000i q^{57} -3.00000 q^{58} +1.73205 q^{59} -3.00000 q^{60} +1.73205 q^{62} -1.00000 q^{64} +6.00000i q^{65} -5.19615 q^{66} +2.00000 q^{67} -3.46410 q^{68} +10.3923i q^{69} -12.0000i q^{71} -3.00000i q^{72} +6.92820i q^{73} -2.00000i q^{74} +3.46410 q^{75} +3.46410i q^{76} -6.00000 q^{78} -1.00000 q^{79} -1.73205 q^{80} +9.00000 q^{81} +6.92820i q^{82} -8.66025 q^{83} -6.00000 q^{85} -8.00000i q^{86} -5.19615i q^{87} -3.00000 q^{88} +10.3923 q^{89} -5.19615i q^{90} +6.00000i q^{92} +3.00000i q^{93} -6.92820i q^{94} +6.00000i q^{95} -1.73205i q^{96} -5.19615i q^{97} -9.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 12 q^{9} + 12 q^{15} + 4 q^{16} + 12 q^{22} - 8 q^{25} - 12 q^{36} - 8 q^{37} - 32 q^{43} + 24 q^{46} - 24 q^{51} - 12 q^{58} - 12 q^{60} - 4 q^{64} + 8 q^{67} - 24 q^{78} - 4 q^{79} + 36 q^{81}+ \cdots - 12 q^{88}+O(q^{100}) \) 4 * q - 4 * q^4 + 12 * q^9 + 12 * q^15 + 4 * q^16 + 12 * q^22 - 8 * q^25 - 12 * q^36 - 8 * q^37 - 32 * q^43 + 24 * q^46 - 24 * q^51 - 12 * q^58 - 12 * q^60 - 4 * q^64 + 8 * q^67 - 24 * q^78 - 4 * q^79 + 36 * q^81 - 24 * q^85 - 12 * q^88

Introduction

L-functions

Modular forms

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Database

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