LMFDB - Embedded newform 196.2.e.b.165.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [196,2,Mod(165,196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(196, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("196.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 196 = 2^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	196.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.56506787962$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			165.2
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	196.165
Dual form			196.2.e.b.177.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41421 - 2.44949i) q^{3} +(-0.707107 - 1.22474i) q^{5} +(-2.50000 - 4.33013i) q^{9} +(-2.00000 + 3.46410i) q^{11} +4.24264 q^{13} -4.00000 q^{15} +(-0.707107 + 1.22474i) q^{17} +(-1.41421 - 2.44949i) q^{19} +(2.00000 + 3.46410i) q^{23} +(1.50000 - 2.59808i) q^{25} -5.65685 q^{27} +8.00000 q^{29} +(5.65685 + 9.79796i) q^{33} +(4.00000 + 6.92820i) q^{37} +(6.00000 - 10.3923i) q^{39} -7.07107 q^{41} -4.00000 q^{43} +(-3.53553 + 6.12372i) q^{45} +(-2.82843 - 4.89898i) q^{47} +(2.00000 + 3.46410i) q^{51} +(-5.00000 + 8.66025i) q^{53} +5.65685 q^{55} -8.00000 q^{57} +(-7.07107 + 12.2474i) q^{59} +(3.53553 + 6.12372i) q^{61} +(-3.00000 - 5.19615i) q^{65} +11.3137 q^{69} +(3.53553 - 6.12372i) q^{73} +(-4.24264 - 7.34847i) q^{75} +(-4.00000 - 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} -14.1421 q^{83} +2.00000 q^{85} +(11.3137 - 19.5959i) q^{87} +(-3.53553 - 6.12372i) q^{89} +(-2.00000 + 3.46410i) q^{95} -1.41421 q^{97} +20.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{9} - 8 q^{11} - 16 q^{15} + 8 q^{23} + 6 q^{25} + 32 q^{29} + 16 q^{37} + 24 q^{39} - 16 q^{43} + 8 q^{51} - 20 q^{53} - 32 q^{57} - 12 q^{65} - 16 q^{79} - 2 q^{81} + 8 q^{85} - 8 q^{95} + 80 q^{99}+O(q^{100}) $ 4 * q - 10 * q^9 - 8 * q^11 - 16 * q^15 + 8 * q^23 + 6 * q^25 + 32 * q^29 + 16 * q^37 + 24 * q^39 - 16 * q^43 + 8 * q^51 - 20 * q^53 - 32 * q^57 - 12 * q^65 - 16 * q^79 - 2 * q^81 + 8 * q^85 - 8 * q^95 + 80 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/196\mathbb{Z}\right)^\times$.

$n$	$99$	$101$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.41421	−	2.44949i	0.816497	−	1.41421i	−0.0917517	−	0.995782i	$-0.529247\pi$
$3$	1.41421	−	2.44949i	0.816497	−	1.41421i	0.908248	−	0.418432i	$-0.137420\pi$
$4$		0			0
$5$	−0.707107	−	1.22474i	−0.316228	−	0.547723i	0.663470	−	0.748203i	$-0.269083\pi$
$5$	−0.707107	−	1.22474i	−0.316228	−	0.547723i	−0.979698	+	0.200480i	$0.935750\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.50000	−	4.33013i	−0.833333	−	1.44338i
$10$		0			0
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	0.389338	+	0.921095i	$0.372704\pi$
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	−0.992361	+	0.123371i	$0.960630\pi$
$12$		0			0
$13$	4.24264			1.17670			0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264			1.17670			0.588348	+	0.808608i	$0.299778\pi$
$14$		0			0
$15$	−4.00000			−1.03280
$16$		0			0
$17$	−0.707107	+	1.22474i	−0.171499	+	0.297044i	−0.938944	−	0.344070i	$-0.888194\pi$
$17$	−0.707107	+	1.22474i	−0.171499	+	0.297044i	0.767445	+	0.641114i	$0.221528\pi$
$18$		0			0
$19$	−1.41421	−	2.44949i	−0.324443	−	0.561951i	0.656957	−	0.753928i	$-0.271843\pi$
$19$	−1.41421	−	2.44949i	−0.324443	−	0.561951i	−0.981399	+	0.191977i	$0.938510\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	1.50000	−	2.59808i	0.300000	−	0.519615i
$26$		0			0
$27$	−5.65685			−1.08866
$28$		0			0
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$	5.65685	+	9.79796i	0.984732	+	1.70561i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	0.981236	+	0.192809i	$0.0617599\pi$
$37$	4.00000	+	6.92820i	0.657596	+	1.13899i	−0.323640	+	0.946180i	$0.604907\pi$
$38$		0			0
$39$	6.00000	−	10.3923i	0.960769	−	1.66410i
$40$		0			0
$41$	−7.07107			−1.10432			−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107			−1.10432			−0.552158	+	0.833740i	$0.686195\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$	−3.53553	+	6.12372i	−0.527046	+	0.912871i
$46$		0			0
$47$	−2.82843	−	4.89898i	−0.412568	−	0.714590i	0.582601	−	0.812758i	$-0.302035\pi$
$47$	−2.82843	−	4.89898i	−0.412568	−	0.714590i	−0.995170	+	0.0981685i	$0.968702\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	2.00000	+	3.46410i	0.280056	+	0.485071i
$52$		0			0
$53$	−5.00000	+	8.66025i	−0.686803	+	1.18958i	0.286064	+	0.958211i	$0.407653\pi$
$53$	−5.00000	+	8.66025i	−0.686803	+	1.18958i	−0.972867	+	0.231367i	$0.925680\pi$
$54$		0			0
$55$	5.65685			0.762770
$56$		0			0
$57$	−8.00000			−1.05963
$58$		0			0
$59$	−7.07107	+	12.2474i	−0.920575	+	1.59448i	−0.122047	+	0.992524i	$0.538946\pi$
$59$	−7.07107	+	12.2474i	−0.920575	+	1.59448i	−0.798528	+	0.601958i	$0.794388\pi$
$60$		0			0
$61$	3.53553	+	6.12372i	0.452679	+	0.784063i	0.998551	−	0.0538056i	$-0.0171351\pi$
$61$	3.53553	+	6.12372i	0.452679	+	0.784063i	−0.545873	+	0.837868i	$0.683802\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	−	5.19615i	−0.372104	−	0.644503i
$66$		0			0
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$	11.3137			1.36201
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	3.53553	−	6.12372i	0.413803	−	0.716728i	−0.581499	−	0.813547i	$-0.697534\pi$
$73$	3.53553	−	6.12372i	0.413803	−	0.716728i	0.995302	+	0.0968194i	$0.0308669\pi$
$74$		0			0
$75$	−4.24264	−	7.34847i	−0.489898	−	0.848528i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−14.1421			−1.55230			−0.776151	−	0.630548i	$-0.782830\pi$
$83$	−14.1421			−1.55230			−0.776151	+	0.630548i	$0.782830\pi$
$84$		0			0
$85$	2.00000			0.216930
$86$		0			0
$87$	11.3137	−	19.5959i	1.21296	−	2.10090i
$88$		0			0
$89$	−3.53553	−	6.12372i	−0.374766	−	0.649113i	0.615526	−	0.788116i	$-0.288944\pi$
$89$	−3.53553	−	6.12372i	−0.374766	−	0.649113i	−0.990292	+	0.139003i	$0.955610\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.00000	+	3.46410i	−0.205196	+	0.355409i
$96$		0			0
$97$	−1.41421			−0.143592			−0.0717958	−	0.997419i	$-0.522873\pi$
$97$	−1.41421			−0.143592			−0.0717958	+	0.997419i	$0.522873\pi$
$98$		0			0
$99$	20.0000			2.01008
$100$		0			0
$101$	6.36396	−	11.0227i	0.633238	−	1.09680i	−0.353648	−	0.935379i	$-0.615059\pi$
$101$	6.36396	−	11.0227i	0.633238	−	1.09680i	0.986886	−	0.161421i	$-0.0516078\pi$
$102$		0			0
$103$	5.65685	+	9.79796i	0.557386	+	0.965422i	0.997714	+	0.0675842i	$0.0215291\pi$
$103$	5.65685	+	9.79796i	0.557386	+	0.965422i	−0.440327	+	0.897837i	$0.645138\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.00000	−	6.92820i	−0.386695	−	0.669775i	0.605308	−	0.795991i	$-0.293050\pi$
$107$	−4.00000	−	6.92820i	−0.386695	−	0.669775i	−0.992003	+	0.126217i	$0.959717\pi$
$108$		0			0
$109$	4.00000	−	6.92820i	0.383131	−	0.663602i	−0.608377	−	0.793648i	$-0.708179\pi$
$109$	4.00000	−	6.92820i	0.383131	−	0.663602i	0.991508	+	0.130046i	$0.0415126\pi$
$110$		0			0
$111$	22.6274			2.14770
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$	2.82843	−	4.89898i	0.263752	−	0.456832i
$116$		0			0
$117$	−10.6066	−	18.3712i	−0.980581	−	1.69842i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$		0			0
$123$	−10.0000	+	17.3205i	−0.901670	+	1.56174i
$124$		0			0
$125$	−11.3137			−1.01193
$126$		0			0
$127$	−20.0000			−1.77471			−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000			−1.77471			−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$	−5.65685	+	9.79796i	−0.498058	+	0.862662i
$130$		0			0
$131$	4.24264	+	7.34847i	0.370681	+	0.642039i	0.989671	−	0.143361i	$-0.0457909\pi$
$131$	4.24264	+	7.34847i	0.370681	+	0.642039i	−0.618989	+	0.785399i	$0.712458\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	4.00000	+	6.92820i	0.344265	+	0.596285i
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	2.82843			0.239904			0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843			0.239904			0.119952	+	0.992780i	$0.461726\pi$
$140$		0			0
$141$	−16.0000			−1.34744
$142$		0			0
$143$	−8.48528	+	14.6969i	−0.709575	+	1.22902i
$144$		0			0
$145$	−5.65685	−	9.79796i	−0.469776	−	0.813676i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	0.585231	−	0.810867i	$-0.301004\pi$
$149$	−5.00000	−	8.66025i	−0.409616	−	0.709476i	−0.994847	+	0.101391i	$0.967671\pi$
$150$		0			0
$151$	2.00000	−	3.46410i	0.162758	−	0.281905i	−0.773099	−	0.634285i	$-0.781294\pi$
$151$	2.00000	−	3.46410i	0.162758	−	0.281905i	0.935857	+	0.352381i	$0.114628\pi$
$152$		0			0
$153$	7.07107			0.571662
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−3.53553	+	6.12372i	−0.282166	+	0.488726i	−0.971918	−	0.235320i	$-0.924386\pi$
$157$	−3.53553	+	6.12372i	−0.282166	+	0.488726i	0.689752	+	0.724046i	$0.257720\pi$
$158$		0			0
$159$	14.1421	+	24.4949i	1.12154	+	1.94257i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	0.777007	−	0.629492i	$-0.216737\pi$
$163$	−2.00000	−	3.46410i	−0.156652	−	0.271329i	−0.933659	+	0.358162i	$0.883403\pi$
$164$		0			0
$165$	8.00000	−	13.8564i	0.622799	−	1.07872i
$166$		0			0
$167$	−5.65685			−0.437741			−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685			−0.437741			−0.218870	+	0.975754i	$0.570237\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	−7.07107	+	12.2474i	−0.540738	+	0.936586i
$172$		0			0
$173$	2.12132	+	3.67423i	0.161281	+	0.279347i	0.935328	−	0.353781i	$-0.115104\pi$
$173$	2.12132	+	3.67423i	0.161281	+	0.279347i	−0.774047	+	0.633128i	$0.781771\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	20.0000	+	34.6410i	1.50329	+	2.60378i
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−21.2132			−1.57676			−0.788382	−	0.615185i	$-0.789081\pi$
$181$	−21.2132			−1.57676			−0.788382	+	0.615185i	$0.789081\pi$
$182$		0			0
$183$	20.0000			1.47844
$184$		0			0
$185$	5.65685	−	9.79796i	0.415900	−	0.720360i
$186$		0			0
$187$	−2.82843	−	4.89898i	−0.206835	−	0.358249i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.00000	+	13.8564i	0.578860	+	1.00261i	0.995610	+	0.0935936i	$0.0298354\pi$
$191$	8.00000	+	13.8564i	0.578860	+	1.00261i	−0.416751	+	0.909021i	$0.636831\pi$
$192$		0			0
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	−0.628037	−	0.778183i	$-0.716141\pi$
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	0.987945	+	0.154805i	$0.0494748\pi$
$194$		0			0
$195$	−16.9706			−1.21529
$196$		0			0
$197$	−10.0000			−0.712470			−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000			−0.712470			−0.356235	+	0.934396i	$0.615940\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	5.00000	+	8.66025i	0.349215	+	0.604858i
$206$		0			0
$207$	10.0000	−	17.3205i	0.695048	−	1.20386i
$208$		0			0
$209$	11.3137			0.782586
$210$		0			0
$211$	24.0000			1.65223			0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000			1.65223			0.826114	+	0.563503i	$0.190547\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.82843	+	4.89898i	0.192897	+	0.334108i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−10.0000	−	17.3205i	−0.675737	−	1.17041i
$220$		0			0
$221$	−3.00000	+	5.19615i	−0.201802	+	0.349531i
$222$		0			0
$223$	16.9706			1.13643			0.568216	−	0.822879i	$-0.307634\pi$
$223$	16.9706			1.13643			0.568216	+	0.822879i	$0.307634\pi$
$224$		0			0
$225$	−15.0000			−1.00000
$226$		0			0
$227$	−4.24264	+	7.34847i	−0.281594	+	0.487735i	−0.971778	−	0.235899i	$-0.924196\pi$
$227$	−4.24264	+	7.34847i	−0.281594	+	0.487735i	0.690184	+	0.723634i	$0.257530\pi$
$228$		0			0
$229$	10.6066	+	18.3712i	0.700904	+	1.21400i	0.968149	+	0.250373i	$0.0805532\pi$
$229$	10.6066	+	18.3712i	0.700904	+	1.21400i	−0.267246	+	0.963628i	$0.586113\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$	−4.00000	+	6.92820i	−0.260931	+	0.451946i
$236$		0			0
$237$	−22.6274			−1.46981
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−6.36396	+	11.0227i	−0.409939	+	0.710035i	−0.994882	−	0.101039i	$-0.967783\pi$
$241$	−6.36396	+	11.0227i	−0.409939	+	0.710035i	0.584944	+	0.811074i	$0.301117\pi$
$242$		0			0
$243$	−7.07107	−	12.2474i	−0.453609	−	0.785674i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.00000	−	10.3923i	−0.381771	−	0.661247i
$248$		0			0
$249$	−20.0000	+	34.6410i	−1.26745	+	2.19529i
$250$		0			0
$251$	19.7990			1.24970			0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990			1.24970			0.624851	+	0.780744i	$0.285160\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$		0			0
$255$	2.82843	−	4.89898i	0.177123	−	0.306786i
$256$		0			0
$257$	−10.6066	−	18.3712i	−0.661622	−	1.14596i	−0.980189	−	0.198062i	$-0.936535\pi$
$257$	−10.6066	−	18.3712i	−0.661622	−	1.14596i	0.318568	−	0.947900i	$-0.396798\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−20.0000	−	34.6410i	−1.23797	−	2.14423i
$262$		0			0
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	−0.212565	−	0.977147i	$-0.568182\pi$
$263$	12.0000	−	20.7846i	0.739952	−	1.28163i	0.952517	−	0.304487i	$-0.0984850\pi$
$264$		0			0
$265$	14.1421			0.868744
$266$		0			0
$267$	−20.0000			−1.22398
$268$		0			0
$269$	9.19239	−	15.9217i	0.560470	−	0.970762i	−0.436986	−	0.899469i	$-0.643954\pi$
$269$	9.19239	−	15.9217i	0.560470	−	0.970762i	0.997455	−	0.0712937i	$-0.0227127\pi$
$270$		0			0
$271$	−14.1421	−	24.4949i	−0.859074	−	1.48796i	−0.872814	−	0.488053i	$-0.837707\pi$
$271$	−14.1421	−	24.4949i	−0.859074	−	1.48796i	0.0137402	−	0.999906i	$-0.495626\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	+	10.3923i	0.361814	+	0.626680i
$276$		0			0
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	0.319447	+	0.947604i	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.0000			0.954480			0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000			0.954480			0.477240	+	0.878773i	$0.341637\pi$
$282$		0			0
$283$	1.41421	−	2.44949i	0.0840663	−	0.145607i	−0.820927	−	0.571034i	$-0.806543\pi$
$283$	1.41421	−	2.44949i	0.0840663	−	0.145607i	0.904993	+	0.425427i	$0.139876\pi$
$284$		0			0
$285$	5.65685	+	9.79796i	0.335083	+	0.580381i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	7.50000	+	12.9904i	0.441176	+	0.764140i
$290$		0			0
$291$	−2.00000	+	3.46410i	−0.117242	+	0.203069i
$292$		0			0
$293$	32.5269			1.90024			0.950121	−	0.311881i	$-0.100959\pi$
$293$	32.5269			1.90024			0.950121	+	0.311881i	$0.100959\pi$
$294$		0			0
$295$	20.0000			1.16445
$296$		0			0
$297$	11.3137	−	19.5959i	0.656488	−	1.13707i
$298$		0			0
$299$	8.48528	+	14.6969i	0.490716	+	0.849946i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−18.0000	−	31.1769i	−1.03407	−	1.79107i
$304$		0			0
$305$	5.00000	−	8.66025i	0.286299	−	0.495885i
$306$		0			0
$307$	−19.7990			−1.12999			−0.564994	−	0.825095i	$-0.691122\pi$
$307$	−19.7990			−1.12999			−0.564994	+	0.825095i	$0.691122\pi$
$308$		0			0
$309$	32.0000			1.82042
$310$		0			0
$311$	11.3137	−	19.5959i	0.641542	−	1.11118i	−0.343547	−	0.939135i	$-0.611629\pi$
$311$	11.3137	−	19.5959i	0.641542	−	1.11118i	0.985089	−	0.172047i	$-0.0550381\pi$
$312$		0			0
$313$	−2.12132	−	3.67423i	−0.119904	−	0.207680i	0.799825	−	0.600233i	$-0.204925\pi$
$313$	−2.12132	−	3.67423i	−0.119904	−	0.207680i	−0.919730	+	0.392553i	$0.871592\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.00000	+	1.73205i	0.0561656	+	0.0972817i	0.892741	−	0.450570i	$-0.148779\pi$
$317$	1.00000	+	1.73205i	0.0561656	+	0.0972817i	−0.836576	+	0.547852i	$0.815446\pi$
$318$		0			0
$319$	−16.0000	+	27.7128i	−0.895828	+	1.55162i
$320$		0			0
$321$	−22.6274			−1.26294
$322$		0			0
$323$	4.00000			0.222566
$324$		0			0
$325$	6.36396	−	11.0227i	0.353009	−	0.611430i
$326$		0			0
$327$	−11.3137	−	19.5959i	−0.625650	−	1.08366i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	0.998298	+	0.0583130i	$0.0185721\pi$
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	−0.448649	+	0.893708i	$0.648095\pi$
$332$		0			0
$333$	20.0000	−	34.6410i	1.09599	−	1.89832i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$		0			0
$339$	8.48528	−	14.6969i	0.460857	−	0.798228i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.00000	−	13.8564i	−0.430706	−	0.746004i
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	−4.24264			−0.227103			−0.113552	−	0.993532i	$-0.536223\pi$
$349$	−4.24264			−0.227103			−0.113552	+	0.993532i	$0.536223\pi$
$350$		0			0
$351$	−24.0000			−1.28103
$352$		0			0
$353$	−4.94975	+	8.57321i	−0.263448	+	0.456306i	−0.967156	−	0.254184i	$-0.918193\pi$
$353$	−4.94975	+	8.57321i	−0.263448	+	0.456306i	0.703707	+	0.710490i	$0.251527\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	0.999476	+	0.0323801i	$0.0103087\pi$
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	−0.471696	+	0.881761i	$0.656358\pi$
$360$		0			0
$361$	5.50000	−	9.52628i	0.289474	−	0.501383i
$362$		0			0
$363$	−14.1421			−0.742270
$364$		0			0
$365$	−10.0000			−0.523424
$366$		0			0
$367$	−2.82843	+	4.89898i	−0.147643	+	0.255725i	−0.930356	−	0.366658i	$-0.880502\pi$
$367$	−2.82843	+	4.89898i	−0.147643	+	0.255725i	0.782713	+	0.622383i	$0.213835\pi$
$368$		0			0
$369$	17.6777	+	30.6186i	0.920263	+	1.59394i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	0.707055	−	0.707159i	$-0.250023\pi$
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	−0.965945	+	0.258748i	$0.916690\pi$
$374$		0			0
$375$	−16.0000	+	27.7128i	−0.826236	+	1.43108i
$376$		0			0
$377$	33.9411			1.74806
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	−28.2843	+	48.9898i	−1.44905	+	2.50982i
$382$		0			0
$383$	2.82843	+	4.89898i	0.144526	+	0.250326i	0.929196	−	0.369587i	$-0.120501\pi$
$383$	2.82843	+	4.89898i	0.144526	+	0.250326i	−0.784670	+	0.619914i	$0.787168\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	10.0000	+	17.3205i	0.508329	+	0.880451i
$388$		0			0
$389$	−4.00000	+	6.92820i	−0.202808	+	0.351274i	−0.949432	−	0.313972i	$-0.898340\pi$
$389$	−4.00000	+	6.92820i	−0.202808	+	0.351274i	0.746624	+	0.665246i	$0.231673\pi$
$390$		0			0
$391$	−5.65685			−0.286079
$392$		0			0
$393$	24.0000			1.21064
$394$		0			0
$395$	−5.65685	+	9.79796i	−0.284627	+	0.492989i
$396$		0			0
$397$	−7.77817	−	13.4722i	−0.390375	−	0.676150i	0.602124	−	0.798403i	$-0.294321\pi$
$397$	−7.77817	−	13.4722i	−0.390375	−	0.676150i	−0.992499	+	0.122253i	$0.960988\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	−0.992932	−	0.118686i	$-0.962132\pi$
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	0.393680	−	0.919247i	$-0.371202\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.41421			0.0702728
$406$		0			0
$407$	−32.0000			−1.58618
$408$		0			0
$409$	−19.0919	+	33.0681i	−0.944033	+	1.63511i	−0.186357	+	0.982482i	$0.559668\pi$
$409$	−19.0919	+	33.0681i	−0.944033	+	1.63511i	−0.757676	+	0.652631i	$0.773665\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	10.0000	+	17.3205i	0.490881	+	0.850230i
$416$		0			0
$417$	4.00000	−	6.92820i	0.195881	−	0.339276i
$418$		0			0
$419$	14.1421			0.690889			0.345444	−	0.938439i	$-0.387728\pi$
$419$	14.1421			0.690889			0.345444	+	0.938439i	$0.387728\pi$
$420$		0			0
$421$	6.00000			0.292422			0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000			0.292422			0.146211	+	0.989253i	$0.453292\pi$
$422$		0			0
$423$	−14.1421	+	24.4949i	−0.687614	+	1.19098i
$424$		0			0
$425$	2.12132	+	3.67423i	0.102899	+	0.178227i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	24.0000	+	41.5692i	1.15873	+	2.00698i
$430$		0			0
$431$	18.0000	−	31.1769i	0.867029	−	1.50174i	0.00201168	−	0.999998i	$-0.499360\pi$
$431$	18.0000	−	31.1769i	0.867029	−	1.50174i	0.865018	−	0.501741i	$-0.167307\pi$
$432$		0			0
$433$	−21.2132			−1.01944			−0.509721	−	0.860340i	$-0.670251\pi$
$433$	−21.2132			−1.01944			−0.509721	+	0.860340i	$0.670251\pi$
$434$		0			0
$435$	−32.0000			−1.53428
$436$		0			0
$437$	5.65685	−	9.79796i	0.270604	−	0.468700i
$438$		0			0
$439$	8.48528	+	14.6969i	0.404980	+	0.701447i	0.994319	−	0.106439i	$-0.0339450\pi$
$439$	8.48528	+	14.6969i	0.404980	+	0.701447i	−0.589339	+	0.807886i	$0.700612\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.00000	+	13.8564i	0.380091	+	0.658338i	0.991075	−	0.133306i	$-0.0425592\pi$
$443$	8.00000	+	13.8564i	0.380091	+	0.658338i	−0.610984	+	0.791643i	$0.709226\pi$
$444$		0			0
$445$	−5.00000	+	8.66025i	−0.237023	+	0.410535i
$446$		0			0
$447$	−28.2843			−1.33780
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	14.1421	−	24.4949i	0.665927	−	1.15342i
$452$		0			0
$453$	−5.65685	−	9.79796i	−0.265782	−	0.460348i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−15.0000	−	25.9808i	−0.701670	−	1.21533i	−0.967880	−	0.251414i	$-0.919105\pi$
$457$	−15.0000	−	25.9808i	−0.701670	−	1.21533i	0.266209	−	0.963915i	$-0.414229\pi$
$458$		0			0
$459$	4.00000	−	6.92820i	0.186704	−	0.323381i
$460$		0			0
$461$	7.07107			0.329332			0.164666	−	0.986349i	$-0.447345\pi$
$461$	7.07107			0.329332			0.164666	+	0.986349i	$0.447345\pi$
$462$		0			0
$463$	40.0000			1.85896			0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000			1.85896			0.929479	+	0.368875i	$0.120257\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9.89949	+	17.1464i	0.458094	+	0.793442i	0.998860	−	0.0477308i	$-0.0151990\pi$
$467$	9.89949	+	17.1464i	0.458094	+	0.793442i	−0.540766	+	0.841173i	$0.681866\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	10.0000	+	17.3205i	0.460776	+	0.798087i
$472$		0			0
$473$	8.00000	−	13.8564i	0.367840	−	0.637118i
$474$		0			0
$475$	−8.48528			−0.389331
$476$		0			0
$477$	50.0000			2.28934
$478$		0			0
$479$	5.65685	−	9.79796i	0.258468	−	0.447680i	−0.707364	−	0.706850i	$-0.750116\pi$
$479$	5.65685	−	9.79796i	0.258468	−	0.447680i	0.965832	+	0.259170i	$0.0834489\pi$
$480$		0			0
$481$	16.9706	+	29.3939i	0.773791	+	1.34025i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.00000	+	1.73205i	0.0454077	+	0.0786484i
$486$		0			0
$487$	6.00000	−	10.3923i	0.271886	−	0.470920i	−0.697459	−	0.716625i	$-0.745686\pi$
$487$	6.00000	−	10.3923i	0.271886	−	0.470920i	0.969345	+	0.245705i	$0.0790193\pi$
$488$		0			0
$489$	−11.3137			−0.511624
$490$		0			0
$491$	24.0000			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$492$		0			0
$493$	−5.65685	+	9.79796i	−0.254772	+	0.441278i
$494$		0			0
$495$	−14.1421	−	24.4949i	−0.635642	−	1.10096i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−16.0000	−	27.7128i	−0.716258	−	1.24060i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−16.0000	−	27.7128i	−0.716258	−	1.24060i	0.246214	−	0.969216i	$-0.420813\pi$
$500$		0			0
$501$	−8.00000	+	13.8564i	−0.357414	+	0.619059i
$502$		0			0
$503$	−39.5980			−1.76559			−0.882793	−	0.469762i	$-0.844340\pi$
$503$	−39.5980			−1.76559			−0.882793	+	0.469762i	$0.844340\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$	7.07107	−	12.2474i	0.314037	−	0.543928i
$508$		0			0
$509$	9.19239	+	15.9217i	0.407445	+	0.705716i	0.994603	−	0.103757i	$-0.0330863\pi$
$509$	9.19239	+	15.9217i	0.407445	+	0.705716i	−0.587157	+	0.809473i	$0.699753\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	8.00000	+	13.8564i	0.353209	+	0.611775i
$514$		0			0
$515$	8.00000	−	13.8564i	0.352522	−	0.610586i
$516$		0			0
$517$	22.6274			0.995153
$518$		0			0
$519$	12.0000			0.526742
$520$		0			0
$521$	20.5061	−	35.5176i	0.898388	−	1.55605i	0.0688342	−	0.997628i	$-0.478072\pi$
$521$	20.5061	−	35.5176i	0.898388	−	1.55605i	0.829554	−	0.558426i	$-0.188595\pi$
$522$		0			0
$523$	−21.2132	−	36.7423i	−0.927589	−	1.60663i	−0.787344	−	0.616514i	$-0.788544\pi$
$523$	−21.2132	−	36.7423i	−0.927589	−	1.60663i	−0.140244	−	0.990117i	$-0.544789\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	70.7107			3.06858
$532$		0			0
$533$	−30.0000			−1.29944
$534$		0			0
$535$	−5.65685	+	9.79796i	−0.244567	+	0.423603i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.00000	+	12.1244i	0.300954	+	0.521267i	0.976352	−	0.216186i	$-0.0693618\pi$
$541$	7.00000	+	12.1244i	0.300954	+	0.521267i	−0.675399	+	0.737453i	$0.736028\pi$
$542$		0			0
$543$	−30.0000	+	51.9615i	−1.28742	+	2.22988i
$544$		0			0
$545$	−11.3137			−0.484626
$546$		0			0
$547$	−20.0000			−0.855138			−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000			−0.855138			−0.427569	+	0.903983i	$0.640630\pi$
$548$		0			0
$549$	17.6777	−	30.6186i	0.754465	−	1.30677i
$550$		0			0
$551$	−11.3137	−	19.5959i	−0.481980	−	0.834814i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−16.0000	−	27.7128i	−0.679162	−	1.17634i
$556$		0			0
$557$	−15.0000	+	25.9808i	−0.635570	+	1.10084i	0.350824	+	0.936442i	$0.385902\pi$
$557$	−15.0000	+	25.9808i	−0.635570	+	1.10084i	−0.986394	+	0.164399i	$0.947432\pi$
$558$		0			0
$559$	−16.9706			−0.717778
$560$		0			0
$561$	−16.0000			−0.675521
$562$		0			0
$563$	−7.07107	+	12.2474i	−0.298010	+	0.516168i	−0.975681	−	0.219197i	$-0.929656\pi$
$563$	−7.07107	+	12.2474i	−0.298010	+	0.516168i	0.677671	+	0.735366i	$0.262990\pi$
$564$		0			0
$565$	−4.24264	−	7.34847i	−0.178489	−	0.309152i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20.0000	−	34.6410i	−0.838444	−	1.45223i	−0.891196	−	0.453619i	$-0.850133\pi$
$569$	−20.0000	−	34.6410i	−0.838444	−	1.45223i	0.0527519	−	0.998608i	$-0.483201\pi$
$570$		0			0
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	−0.577301	−	0.816532i	$-0.695894\pi$
$571$	10.0000	−	17.3205i	0.418487	−	0.724841i	0.995788	+	0.0916910i	$0.0292272\pi$
$572$		0			0
$573$	45.2548			1.89055
$574$		0			0
$575$	12.0000			0.500435
$576$		0			0
$577$	−6.36396	+	11.0227i	−0.264935	+	0.458881i	−0.967547	−	0.252693i	$-0.918684\pi$
$577$	−6.36396	+	11.0227i	−0.264935	+	0.458881i	0.702611	+	0.711574i	$0.252017\pi$
$578$		0			0
$579$	−14.1421	−	24.4949i	−0.587727	−	1.01797i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−20.0000	−	34.6410i	−0.828315	−	1.43468i
$584$		0			0
$585$	−15.0000	+	25.9808i	−0.620174	+	1.07417i
$586$		0			0
$587$	25.4558			1.05068			0.525338	−	0.850894i	$-0.323939\pi$
$587$	25.4558			1.05068			0.525338	+	0.850894i	$0.323939\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−14.1421	+	24.4949i	−0.581730	+	1.00759i
$592$		0			0
$593$	4.94975	+	8.57321i	0.203262	+	0.352060i	0.949578	−	0.313532i	$-0.101512\pi$
$593$	4.94975	+	8.57321i	0.203262	+	0.352060i	−0.746316	+	0.665592i	$0.768179\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.00000	+	6.92820i	−0.163436	+	0.283079i	−0.936099	−	0.351738i	$-0.885591\pi$
$599$	−4.00000	+	6.92820i	−0.163436	+	0.283079i	0.772663	+	0.634816i	$0.218924\pi$
$600$		0			0
$601$	−29.6985			−1.21143			−0.605713	−	0.795683i	$-0.707112\pi$
$601$	−29.6985			−1.21143			−0.605713	+	0.795683i	$0.707112\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.53553	+	6.12372i	−0.143740	+	0.248965i
$606$		0			0
$607$	16.9706	+	29.3939i	0.688814	+	1.19306i	0.972222	+	0.234061i	$0.0752016\pi$
$607$	16.9706	+	29.3939i	0.688814	+	1.19306i	−0.283408	+	0.958999i	$0.591465\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000	−	20.7846i	−0.485468	−	0.840855i
$612$		0			0
$613$	−12.0000	+	20.7846i	−0.484675	+	0.839482i	−0.999845	−	0.0176058i	$-0.994396\pi$
$613$	−12.0000	+	20.7846i	−0.484675	+	0.839482i	0.515170	+	0.857088i	$0.327729\pi$
$614$		0			0
$615$	28.2843			1.14053
$616$		0			0
$617$	−8.00000			−0.322068			−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000			−0.322068			−0.161034	+	0.986949i	$0.551483\pi$
$618$		0			0
$619$	15.5563	−	26.9444i	0.625262	−	1.08299i	−0.363228	−	0.931700i	$-0.618325\pi$
$619$	15.5563	−	26.9444i	0.625262	−	1.08299i	0.988490	−	0.151286i	$-0.0483414\pi$
$620$		0			0
$621$	−11.3137	−	19.5959i	−0.454003	−	0.786357i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	+	0.866025i	0.0200000	+	0.0346410i
$626$		0			0
$627$	16.0000	−	27.7128i	0.638978	−	1.10674i
$628$		0			0
$629$	−11.3137			−0.451107
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$	33.9411	−	58.7878i	1.34904	−	2.33660i
$634$		0			0
$635$	14.1421	+	24.4949i	0.561214	+	0.972050i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.00000	+	6.92820i	−0.157991	+	0.273648i	−0.934144	−	0.356897i	$-0.883835\pi$
$641$	−4.00000	+	6.92820i	−0.157991	+	0.273648i	0.776153	+	0.630544i	$0.217168\pi$
$642$		0			0
$643$	2.82843			0.111542			0.0557711	−	0.998444i	$-0.482238\pi$
$643$	2.82843			0.111542			0.0557711	+	0.998444i	$0.482238\pi$
$644$		0			0
$645$	16.0000			0.629999
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$	−28.2843	−	48.9898i	−1.11025	−	1.92302i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.0000	+	20.7846i	0.469596	+	0.813365i	0.999396	−	0.0347583i	$-0.0110661\pi$
$653$	12.0000	+	20.7846i	0.469596	+	0.813365i	−0.529799	+	0.848123i	$0.677733\pi$
$654$		0			0
$655$	6.00000	−	10.3923i	0.234439	−	0.406061i
$656$		0			0
$657$	−35.3553			−1.37934
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	10.6066	−	18.3712i	0.412549	−	0.714556i	−0.582619	−	0.812746i	$-0.697972\pi$
$661$	10.6066	−	18.3712i	0.412549	−	0.714556i	0.995168	+	0.0981898i	$0.0313052\pi$
$662$		0			0
$663$	8.48528	+	14.6969i	0.329541	+	0.570782i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16.0000	+	27.7128i	0.619522	+	1.07304i
$668$		0			0
$669$	24.0000	−	41.5692i	0.927894	−	1.60716i
$670$		0			0
$671$	−28.2843			−1.09190
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$	−8.48528	+	14.6969i	−0.326599	+	0.565685i
$676$		0			0
$677$	−6.36396	−	11.0227i	−0.244587	−	0.423637i	0.717428	−	0.696632i	$-0.245319\pi$
$677$	−6.36396	−	11.0227i	−0.244587	−	0.423637i	−0.962015	+	0.272995i	$0.911986\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	12.0000	+	20.7846i	0.459841	+	0.796468i
$682$		0			0
$683$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$683$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	60.0000			2.28914
$688$		0			0
$689$	−21.2132	+	36.7423i	−0.808159	+	1.39977i
$690$		0			0
$691$	21.2132	+	36.7423i	0.806988	+	1.39774i	0.914941	+	0.403589i	$0.132237\pi$
$691$	21.2132	+	36.7423i	0.806988	+	1.39774i	−0.107952	+	0.994156i	$0.534429\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.00000	−	3.46410i	−0.0758643	−	0.131401i
$696$		0			0
$697$	5.00000	−	8.66025i	0.189389	−	0.328031i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	24.0000			0.906467			0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000			0.906467			0.453234	+	0.891392i	$0.350270\pi$
$702$		0			0
$703$	11.3137	−	19.5959i	0.426705	−	0.739074i
$704$		0			0
$705$	11.3137	+	19.5959i	0.426099	+	0.738025i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4.00000	+	6.92820i	0.150223	+	0.260194i	0.931309	−	0.364229i	$-0.118667\pi$
$709$	4.00000	+	6.92820i	0.150223	+	0.260194i	−0.781086	+	0.624423i	$0.785334\pi$
$710$		0			0
$711$	−20.0000	+	34.6410i	−0.750059	+	1.29914i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	24.0000			0.897549
$716$		0			0
$717$	−16.9706	+	29.3939i	−0.633777	+	1.09773i
$718$		0			0
$719$	−19.7990	−	34.2929i	−0.738378	−	1.27891i	−0.953225	−	0.302260i	$-0.902259\pi$
$719$	−19.7990	−	34.2929i	−0.738378	−	1.27891i	0.214848	−	0.976648i	$-0.431074\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	18.0000	+	31.1769i	0.669427	+	1.15948i
$724$		0			0
$725$	12.0000	−	20.7846i	0.445669	−	0.771921i
$726$		0			0
$727$	28.2843			1.04901			0.524503	−	0.851409i	$-0.324251\pi$
$727$	28.2843			1.04901			0.524503	+	0.851409i	$0.324251\pi$
$728$		0			0
$729$	−43.0000			−1.59259
$730$		0			0
$731$	2.82843	−	4.89898i	0.104613	−	0.181195i
$732$		0			0
$733$	19.0919	+	33.0681i	0.705175	+	1.22140i	0.966628	+	0.256183i	$0.0824648\pi$
$733$	19.0919	+	33.0681i	0.705175	+	1.22140i	−0.261454	+	0.965216i	$0.584202\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	6.00000	−	10.3923i	0.220714	−	0.382287i	−0.734311	−	0.678813i	$-0.762495\pi$
$739$	6.00000	−	10.3923i	0.220714	−	0.382287i	0.955025	+	0.296526i	$0.0958281\pi$
$740$		0			0
$741$	−33.9411			−1.24686
$742$		0			0
$743$	−20.0000			−0.733729			−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000			−0.733729			−0.366864	+	0.930274i	$0.619569\pi$
$744$		0			0
$745$	−7.07107	+	12.2474i	−0.259064	+	0.448712i
$746$		0			0
$747$	35.3553	+	61.2372i	1.29358	+	2.24055i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.00000	+	3.46410i	0.0729810	+	0.126407i	0.900207	−	0.435463i	$-0.143415\pi$
$751$	2.00000	+	3.46410i	0.0729810	+	0.126407i	−0.827225	+	0.561870i	$0.810082\pi$
$752$		0			0
$753$	28.0000	−	48.4974i	1.02038	−	1.76734i
$754$		0			0
$755$	−5.65685			−0.205874
$756$		0			0
$757$	−40.0000			−1.45382			−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000			−1.45382			−0.726912	+	0.686730i	$0.759045\pi$
$758$		0			0
$759$	−22.6274	+	39.1918i	−0.821323	+	1.42257i
$760$		0			0
$761$	−20.5061	−	35.5176i	−0.743345	−	1.28751i	−0.950964	−	0.309302i	$-0.899905\pi$
$761$	−20.5061	−	35.5176i	−0.743345	−	1.28751i	0.207618	−	0.978210i	$-0.433429\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−5.00000	−	8.66025i	−0.180775	−	0.313112i
$766$		0			0
$767$	−30.0000	+	51.9615i	−1.08324	+	1.87622i
$768$		0			0
$769$	46.6690			1.68293			0.841464	−	0.540312i	$-0.181694\pi$
$769$	46.6690			1.68293			0.841464	+	0.540312i	$0.181694\pi$
$770$		0			0
$771$	−60.0000			−2.16085
$772$		0			0
$773$	−12.0208	+	20.8207i	−0.432359	+	0.748867i	−0.997076	−	0.0764173i	$-0.975652\pi$
$773$	−12.0208	+	20.8207i	−0.432359	+	0.748867i	0.564717	+	0.825284i	$0.308985\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.0000	+	17.3205i	0.358287	+	0.620572i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−45.2548			−1.61728
$784$		0			0
$785$	10.0000			0.356915
$786$		0			0
$787$	−24.0416	+	41.6413i	−0.856992	+	1.48435i	0.0177934	+	0.999842i	$0.494336\pi$
$787$	−24.0416	+	41.6413i	−0.856992	+	1.48435i	−0.874785	+	0.484511i	$0.838997\pi$
$788$		0			0
$789$	−33.9411	−	58.7878i	−1.20834	−	2.09290i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	15.0000	+	25.9808i	0.532666	+	0.922604i
$794$		0			0
$795$	20.0000	−	34.6410i	0.709327	−	1.22859i
$796$		0			0
$797$	−29.6985			−1.05197			−0.525987	−	0.850493i	$-0.676304\pi$
$797$	−29.6985			−1.05197			−0.525987	+	0.850493i	$0.676304\pi$
$798$		0			0
$799$	8.00000			0.283020
$800$		0			0
$801$	−17.6777	+	30.6186i	−0.624610	+	1.08186i
$802$		0			0
$803$	14.1421	+	24.4949i	0.499065	+	0.864406i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−26.0000	−	45.0333i	−0.915243	−	1.58525i
$808$		0			0
$809$	−1.00000	+	1.73205i	−0.0351581	+	0.0608957i	−0.883069	−	0.469243i	$-0.844527\pi$
$809$	−1.00000	+	1.73205i	−0.0351581	+	0.0608957i	0.847911	+	0.530139i	$0.177860\pi$
$810$		0			0
$811$	8.48528			0.297959			0.148979	−	0.988840i	$-0.452401\pi$
$811$	8.48528			0.297959			0.148979	+	0.988840i	$0.452401\pi$
$812$		0			0
$813$	−80.0000			−2.80572
$814$		0			0
$815$	−2.82843	+	4.89898i	−0.0990755	+	0.171604i
$816$		0			0
$817$	5.65685	+	9.79796i	0.197908	+	0.342787i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.00000	−	5.19615i	−0.104701	−	0.181347i	0.808915	−	0.587925i	$-0.200055\pi$
$821$	−3.00000	−	5.19615i	−0.104701	−	0.181347i	−0.913616	+	0.406578i	$0.866722\pi$
$822$		0			0
$823$	−20.0000	+	34.6410i	−0.697156	+	1.20751i	0.272292	+	0.962215i	$0.412218\pi$
$823$	−20.0000	+	34.6410i	−0.697156	+	1.20751i	−0.969448	+	0.245295i	$0.921115\pi$
$824$		0			0
$825$	33.9411			1.18168
$826$		0			0
$827$	48.0000			1.66912			0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000			1.66912			0.834562	+	0.550914i	$0.185721\pi$
$828$		0			0
$829$	3.53553	−	6.12372i	0.122794	−	0.212686i	−0.798074	−	0.602559i	$-0.794148\pi$
$829$	3.53553	−	6.12372i	0.122794	−	0.212686i	0.920869	+	0.389873i	$0.127481\pi$
$830$		0			0
$831$	31.1127	+	53.8888i	1.07929	+	1.86938i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	4.00000	+	6.92820i	0.138426	+	0.239760i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.3137			−0.390593			−0.195296	−	0.980744i	$-0.562567\pi$
$839$	−11.3137			−0.390593			−0.195296	+	0.980744i	$0.562567\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$	22.6274	−	39.1918i	0.779330	−	1.34984i
$844$		0			0
$845$	−3.53553	−	6.12372i	−0.121626	−	0.210663i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−4.00000	−	6.92820i	−0.137280	−	0.237775i
$850$		0			0
$851$	−16.0000	+	27.7128i	−0.548473	+	0.949983i
$852$		0			0
$853$	12.7279			0.435796			0.217898	−	0.975972i	$-0.430080\pi$
$853$	12.7279			0.435796			0.217898	+	0.975972i	$0.430080\pi$
$854$		0			0
$855$	20.0000			0.683986
$856$		0			0
$857$	−3.53553	+	6.12372i	−0.120772	+	0.209182i	−0.920072	−	0.391749i	$-0.871870\pi$
$857$	−3.53553	+	6.12372i	−0.120772	+	0.209182i	0.799301	+	0.600931i	$0.205204\pi$
$858$		0			0
$859$	−7.07107	−	12.2474i	−0.241262	−	0.417878i	0.719812	−	0.694169i	$-0.244228\pi$
$859$	−7.07107	−	12.2474i	−0.241262	−	0.417878i	−0.961074	+	0.276291i	$0.910895\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.0000	+	34.6410i	0.680808	+	1.17919i	0.974735	+	0.223366i	$0.0717045\pi$
$863$	20.0000	+	34.6410i	0.680808	+	1.17919i	−0.293927	+	0.955828i	$0.594962\pi$
$864$		0			0
$865$	3.00000	−	5.19615i	0.102003	−	0.176674i
$866$		0			0
$867$	42.4264			1.44088
$868$		0			0
$869$	32.0000			1.08553
$870$		0			0
$871$		0			0
$872$		0			0
$873$	3.53553	+	6.12372i	0.119660	+	0.207257i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.00000	−	6.92820i	−0.135070	−	0.233949i	0.790554	−	0.612392i	$-0.209793\pi$
$877$	−4.00000	−	6.92820i	−0.135070	−	0.233949i	−0.925624	+	0.378444i	$0.876459\pi$
$878$		0			0
$879$	46.0000	−	79.6743i	1.55154	−	2.68735i
$880$		0			0
$881$	−21.2132			−0.714691			−0.357345	−	0.933972i	$-0.616318\pi$
$881$	−21.2132			−0.714691			−0.357345	+	0.933972i	$0.616318\pi$
$882$		0			0
$883$	−40.0000			−1.34611			−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000			−1.34611			−0.673054	+	0.739594i	$0.735018\pi$
$884$		0			0
$885$	28.2843	−	48.9898i	0.950765	−	1.64677i
$886$		0			0
$887$	−14.1421	−	24.4949i	−0.474846	−	0.822458i	0.524739	−	0.851263i	$-0.324163\pi$
$887$	−14.1421	−	24.4949i	−0.474846	−	0.822458i	−0.999585	+	0.0288053i	$0.990830\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−2.00000	−	3.46410i	−0.0670025	−	0.116052i
$892$		0			0
$893$	−8.00000	+	13.8564i	−0.267710	+	0.463687i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	48.0000			1.60267
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−7.07107	−	12.2474i	−0.235571	−	0.408022i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	15.0000	+	25.9808i	0.498617	+	0.863630i
$906$		0			0
$907$	4.00000	−	6.92820i	0.132818	−	0.230047i	−0.791944	−	0.610594i	$-0.790931\pi$
$907$	4.00000	−	6.92820i	0.132818	−	0.230047i	0.924762	+	0.380547i	$0.124264\pi$
$908$		0			0
$909$	−63.6396			−2.11079
$910$		0			0
$911$	20.0000			0.662630			0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000			0.662630			0.331315	+	0.943520i	$0.392508\pi$
$912$		0			0
$913$	28.2843	−	48.9898i	0.936073	−	1.62133i
$914$		0			0
$915$	−14.1421	−	24.4949i	−0.467525	−	0.809776i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	4.00000	+	6.92820i	0.131948	+	0.228540i	0.924427	−	0.381358i	$-0.124544\pi$
$919$	4.00000	+	6.92820i	0.131948	+	0.228540i	−0.792480	+	0.609898i	$0.791210\pi$
$920$		0			0
$921$	−28.0000	+	48.4974i	−0.922631	+	1.59804i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	24.0000			0.789115
$926$		0			0
$927$	28.2843	−	48.9898i	0.928977	−	1.60904i
$928$		0			0
$929$	−17.6777	−	30.6186i	−0.579986	−	1.00456i	−0.995480	−	0.0949688i	$-0.969725\pi$
$929$	−17.6777	−	30.6186i	−0.579986	−	1.00456i	0.415495	−	0.909596i	$-0.363608\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−32.0000	−	55.4256i	−1.04763	−	1.81455i
$934$		0			0
$935$	−4.00000	+	6.92820i	−0.130814	+	0.226576i
$936$		0			0
$937$	15.5563			0.508204			0.254102	−	0.967177i	$-0.418220\pi$
$937$	15.5563			0.508204			0.254102	+	0.967177i	$0.418220\pi$
$938$		0			0
$939$	−12.0000			−0.391605
$940$		0			0
$941$	3.53553	−	6.12372i	0.115255	−	0.199628i	−0.802627	−	0.596482i	$-0.796565\pi$
$941$	3.53553	−	6.12372i	0.115255	−	0.199628i	0.917882	+	0.396854i	$0.129898\pi$
$942$		0			0
$943$	−14.1421	−	24.4949i	−0.460531	−	0.797664i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30.0000	−	51.9615i	−0.974869	−	1.68852i	−0.680367	−	0.732872i	$-0.738179\pi$
$947$	−30.0000	−	51.9615i	−0.974869	−	1.68852i	−0.294502	−	0.955651i	$-0.595154\pi$
$948$		0			0
$949$	15.0000	−	25.9808i	0.486921	−	0.843371i
$950$		0			0
$951$	5.65685			0.183436
$952$		0			0
$953$	10.0000			0.323932			0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000			0.323932			0.161966	+	0.986796i	$0.448217\pi$
$954$		0			0
$955$	11.3137	−	19.5959i	0.366103	−	0.634109i
$956$		0			0
$957$	45.2548	+	78.3837i	1.46288	+	2.53378i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$	−20.0000	+	34.6410i	−0.644491	+	1.11629i
$964$		0			0
$965$	−14.1421			−0.455251
$966$		0			0
$967$	12.0000			0.385894			0.192947	−	0.981209i	$-0.438195\pi$
$967$	12.0000			0.385894			0.192947	+	0.981209i	$0.438195\pi$
$968$		0			0
$969$	5.65685	−	9.79796i	0.181724	−	0.314756i
$970$		0			0
$971$	−7.07107	−	12.2474i	−0.226921	−	0.393039i	0.729973	−	0.683476i	$-0.239533\pi$
$971$	−7.07107	−	12.2474i	−0.226921	−	0.393039i	−0.956894	+	0.290437i	$0.906199\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−18.0000	−	31.1769i	−0.576461	−	0.998460i
$976$		0			0
$977$	−24.0000	+	41.5692i	−0.767828	+	1.32992i	0.170910	+	0.985287i	$0.445329\pi$
$977$	−24.0000	+	41.5692i	−0.767828	+	1.32992i	−0.938738	+	0.344631i	$0.888004\pi$
$978$		0			0
$979$	28.2843			0.903969
$980$		0			0
$981$	−40.0000			−1.27710
$982$		0			0
$983$	−22.6274	+	39.1918i	−0.721703	+	1.25003i	0.238614	+	0.971114i	$0.423307\pi$
$983$	−22.6274	+	39.1918i	−0.721703	+	1.25003i	−0.960317	+	0.278911i	$0.910027\pi$
$984$		0			0
$985$	7.07107	+	12.2474i	0.225303	+	0.390236i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−8.00000	−	13.8564i	−0.254385	−	0.440608i
$990$		0			0
$991$	20.0000	−	34.6410i	0.635321	−	1.10041i	−0.351126	−	0.936328i	$-0.614201\pi$
$991$	20.0000	−	34.6410i	0.635321	−	1.10041i	0.986447	−	0.164080i	$-0.0524655\pi$
$992$		0			0
$993$	56.5685			1.79515
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.707107	+	1.22474i	−0.0223943	+	0.0387881i	−0.877005	−	0.480481i	$-0.840462\pi$
$997$	−0.707107	+	1.22474i	−0.0223943	+	0.0387881i	0.854611	+	0.519269i	$0.173796\pi$
$998$		0			0
$999$	−22.6274	−	39.1918i	−0.715900	−	1.23997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	196.2.e.b.165.2		4
3.2	odd	2		1764.2.k.l.361.2		4
4.3	odd	2		784.2.i.l.753.1		4
7.2	even	3	inner	196.2.e.b.177.2		4
7.3	odd	6		196.2.a.c.1.2	yes	2
7.4	even	3		196.2.a.c.1.1	✓	2
7.5	odd	6	inner	196.2.e.b.177.1		4
7.6	odd	2	inner	196.2.e.b.165.1		4
21.2	odd	6		1764.2.k.l.1549.2		4
21.5	even	6		1764.2.k.l.1549.1		4
21.11	odd	6		1764.2.a.l.1.1		2
21.17	even	6		1764.2.a.l.1.2		2
21.20	even	2		1764.2.k.l.361.1		4
28.3	even	6		784.2.a.m.1.1		2
28.11	odd	6		784.2.a.m.1.2		2
28.19	even	6		784.2.i.l.177.2		4
28.23	odd	6		784.2.i.l.177.1		4
28.27	even	2		784.2.i.l.753.2		4
35.3	even	12		4900.2.e.p.2549.4		4
35.4	even	6		4900.2.a.y.1.2		2
35.17	even	12		4900.2.e.p.2549.2		4
35.18	odd	12		4900.2.e.p.2549.1		4
35.24	odd	6		4900.2.a.y.1.1		2
35.32	odd	12		4900.2.e.p.2549.3		4
56.3	even	6		3136.2.a.bs.1.2		2
56.11	odd	6		3136.2.a.bs.1.1		2
56.45	odd	6		3136.2.a.br.1.1		2
56.53	even	6		3136.2.a.br.1.2		2
84.11	even	6		7056.2.a.cr.1.1		2
84.59	odd	6		7056.2.a.cr.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.2.a.c.1.1	✓	2	7.4	even	3
196.2.a.c.1.2	yes	2	7.3	odd	6
196.2.e.b.165.1		4	7.6	odd	2	inner
196.2.e.b.165.2		4	1.1	even	1	trivial
196.2.e.b.177.1		4	7.5	odd	6	inner
196.2.e.b.177.2		4	7.2	even	3	inner
784.2.a.m.1.1		2	28.3	even	6
784.2.a.m.1.2		2	28.11	odd	6
784.2.i.l.177.1		4	28.23	odd	6
784.2.i.l.177.2		4	28.19	even	6
784.2.i.l.753.1		4	4.3	odd	2
784.2.i.l.753.2		4	28.27	even	2
1764.2.a.l.1.1		2	21.11	odd	6
1764.2.a.l.1.2		2	21.17	even	6
1764.2.k.l.361.1		4	21.20	even	2
1764.2.k.l.361.2		4	3.2	odd	2
1764.2.k.l.1549.1		4	21.5	even	6
1764.2.k.l.1549.2		4	21.2	odd	6
3136.2.a.br.1.1		2	56.45	odd	6
3136.2.a.br.1.2		2	56.53	even	6
3136.2.a.bs.1.1		2	56.11	odd	6
3136.2.a.bs.1.2		2	56.3	even	6
4900.2.a.y.1.1		2	35.24	odd	6
4900.2.a.y.1.2		2	35.4	even	6
4900.2.e.p.2549.1		4	35.18	odd	12
4900.2.e.p.2549.2		4	35.17	even	12
4900.2.e.p.2549.3		4	35.32	odd	12
4900.2.e.p.2549.4		4	35.3	even	12
7056.2.a.cr.1.1		2	84.11	even	6
7056.2.a.cr.1.2		2	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:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.41421 - 2.44949i) q^{3} +(-0.707107 - 1.22474i) q^{5} +(-2.50000 - 4.33013i) q^{9} +(-2.00000 + 3.46410i) q^{11} +4.24264 q^{13} -4.00000 q^{15} +(-0.707107 + 1.22474i) q^{17} +(-1.41421 - 2.44949i) q^{19} +(2.00000 + 3.46410i) q^{23} +(1.50000 - 2.59808i) q^{25} -5.65685 q^{27} +8.00000 q^{29} +(5.65685 + 9.79796i) q^{33} +(4.00000 + 6.92820i) q^{37} +(6.00000 - 10.3923i) q^{39} -7.07107 q^{41} -4.00000 q^{43} +(-3.53553 + 6.12372i) q^{45} +(-2.82843 - 4.89898i) q^{47} +(2.00000 + 3.46410i) q^{51} +(-5.00000 + 8.66025i) q^{53} +5.65685 q^{55} -8.00000 q^{57} +(-7.07107 + 12.2474i) q^{59} +(3.53553 + 6.12372i) q^{61} +(-3.00000 - 5.19615i) q^{65} +11.3137 q^{69} +(3.53553 - 6.12372i) q^{73} +(-4.24264 - 7.34847i) q^{75} +(-4.00000 - 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} -14.1421 q^{83} +2.00000 q^{85} +(11.3137 - 19.5959i) q^{87} +(-3.53553 - 6.12372i) q^{89} +(-2.00000 + 3.46410i) q^{95} -1.41421 q^{97} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{9} - 8 q^{11} - 16 q^{15} + 8 q^{23} + 6 q^{25} + 32 q^{29} + 16 q^{37} + 24 q^{39} - 16 q^{43} + 8 q^{51} - 20 q^{53} - 32 q^{57} - 12 q^{65} - 16 q^{79} - 2 q^{81} + 8 q^{85} - 8 q^{95} + 80 q^{99}+O(q^{100}) \) 4 * q - 10 * q^9 - 8 * q^11 - 16 * q^15 + 8 * q^23 + 6 * q^25 + 32 * q^29 + 16 * q^37 + 24 * q^39 - 16 * q^43 + 8 * q^51 - 20 * q^53 - 32 * q^57 - 12 * q^65 - 16 * q^79 - 2 * q^81 + 8 * q^85 - 8 * q^95 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more