LMFDB - Embedded newform 92.2.b.b.91.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [92,2,Mod(91,92)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(92, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("92.91");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 92 = 2^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	92.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.734623698596$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			91.4
Root			$-0.261988 - 1.38973i$ of defining polynomial
Character	$\chi$	$=$	92.91
Dual form			92.2.b.b.91.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.261988 + 1.38973i) q^{2} +1.32309i q^{3} +(-1.86272 + 0.728188i) q^{4} +(-1.83875 + 0.346635i) q^{6} +(-1.50000 - 2.39792i) q^{8} +1.24943 q^{9} +O(q^{10})$	\(q+(0.261988 + 1.38973i) q^{2} +1.32309i q^{3} +(-1.86272 + 0.728188i) q^{4} +(-1.83875 + 0.346635i) q^{6} +(-1.50000 - 2.39792i) q^{8} +1.24943 q^{9} +(-0.963461 - 2.46456i) q^{12} +2.15352 q^{13} +(2.93948 - 2.71283i) q^{16} +(0.327335 + 1.73637i) q^{18} -4.79583i q^{23} +(3.17267 - 1.98464i) q^{24} +5.00000 q^{25} +(0.564197 + 2.99282i) q^{26} +5.62238i q^{27} -10.6524 q^{29} -9.79478i q^{31} +(4.54022 + 3.37437i) q^{32} +(-2.32733 + 0.909817i) q^{36} +2.84931i q^{39} -8.55646 q^{41} +(6.66493 - 1.25645i) q^{46} +7.14860i q^{47} +(3.58932 + 3.88921i) q^{48} -7.00000 q^{49} +(1.30994 + 6.94867i) q^{50} +(-4.01141 + 1.56817i) q^{52} +(-7.81362 + 1.47300i) q^{54} +(-2.79080 - 14.8040i) q^{58} +9.59166i q^{59} +(13.6121 - 2.56612i) q^{62} +(-3.50000 + 7.19375i) q^{64} +6.34533 q^{69} -15.0872i q^{71} +(-1.87414 - 2.99602i) q^{72} +17.0553 q^{73} +6.61546i q^{75} +(-3.95978 + 0.746485i) q^{78} -3.69066 q^{81} +(-2.24169 - 11.8912i) q^{82} -14.0941i q^{87} +(3.49227 + 8.93331i) q^{92} +12.9594 q^{93} +(-9.93465 + 1.87285i) q^{94} +(-4.46461 + 6.00713i) q^{96} +(-1.83392 - 9.72814i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{6} - 9 q^{8} - 18 q^{9}+O(q^{10}) $ 6 * q - 3 * q^6 - 9 * q^8 - 18 * q^9	$ 6 q - 3 q^{6} - 9 q^{8} - 18 q^{9} + 15 q^{12} + 21 q^{18} + 30 q^{25} - 27 q^{26} - 33 q^{36} + 39 q^{48} - 42 q^{49} + 3 q^{52} + 9 q^{54} - 15 q^{58} + 45 q^{62} - 21 q^{64} + 27 q^{72} - 51 q^{78} + 54 q^{81} + 33 q^{82} - 12 q^{93} - 39 q^{94} - 57 q^{96}+O(q^{100}) $ 6 * q - 3 * q^6 - 9 * q^8 - 18 * q^9 + 15 * q^12 + 21 * q^18 + 30 * q^25 - 27 * q^26 - 33 * q^36 + 39 * q^48 - 42 * q^49 + 3 * q^52 + 9 * q^54 - 15 * q^58 + 45 * q^62 - 21 * q^64 + 27 * q^72 - 51 * q^78 + 54 * q^81 + 33 * q^82 - 12 * q^93 - 39 * q^94 - 57 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/92\mathbb{Z}\right)^\times$.

$n$	$5$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.261988	+	1.38973i	0.185254	+	0.982691i
$2$	0.261988	+	1.38973i	0.185254	+	0.982691i
$3$			1.32309i			0.763888i	0.924185	+	0.381944i	$0.124745\pi$
$3$			1.32309i			0.763888i	−0.924185	+	0.381944i	$0.875255\pi$
$4$	−1.86272	+	0.728188i	−0.931362	+	0.364094i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	−1.83875	+	0.346635i	−0.750666	+	0.141513i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−1.50000	−	2.39792i	−0.530330	−	0.847791i
$9$	1.24943			0.416475
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.963461	−	2.46456i	−0.278127	−	0.711456i
$13$	2.15352			0.597279			0.298639	−	0.954366i	$-0.403467\pi$
$13$	2.15352			0.597279			0.298639	+	0.954366i	$0.403467\pi$
$14$		0			0
$15$		0			0
$16$	2.93948	−	2.71283i	0.734871	−	0.678207i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	0.327335	+	1.73637i	0.0771535	+	0.409266i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	4.79583i		−	1.00000i
$23$		−	4.79583i		−	1.00000i
$24$	3.17267	−	1.98464i	0.647618	−	0.405113i
$25$	5.00000			1.00000
$26$	0.564197	+	2.99282i	0.110648	+	0.586940i
$27$			5.62238i			1.08203i
$28$		0			0
$29$	−10.6524			−1.97810			−0.989048	−	0.147596i	$-0.952846\pi$
$29$	−10.6524			−1.97810			−0.989048	+	0.147596i	$0.952846\pi$
$30$		0			0
$31$		−	9.79478i		−	1.75920i	−0.475719	−	0.879598i	$-0.657812\pi$
$31$		−	9.79478i		−	1.75920i	0.475719	−	0.879598i	$-0.342188\pi$
$32$	4.54022	+	3.37437i	0.802605	+	0.596511i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.32733	+	0.909817i	−0.387889	+	0.151636i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$			2.84931i			0.456254i
$40$		0			0
$41$	−8.55646			−1.33630			−0.668148	−	0.744029i	$-0.732913\pi$
$41$	−8.55646			−1.33630			−0.668148	+	0.744029i	$0.732913\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	6.66493	−	1.25645i	0.982691	−	0.185254i
$47$			7.14860i			1.04273i	0.853334	+	0.521365i	$0.174577\pi$
$47$			7.14860i			1.04273i	−0.853334	+	0.521365i	$0.825423\pi$
$48$	3.58932	+	3.88921i	0.518074	+	0.561359i
$49$	−7.00000			−1.00000
$50$	1.30994	+	6.94867i	0.185254	+	0.982691i
$51$		0			0
$52$	−4.01141	+	1.56817i	−0.556283	+	0.217466i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−7.81362	+	1.47300i	−1.06330	+	0.200450i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−2.79080	−	14.8040i	−0.366449	−	1.94386i
$59$			9.59166i			1.24873i	0.781133	+	0.624364i	$0.214642\pi$
$59$			9.59166i			1.24873i	−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	13.6121	−	2.56612i	1.72874	−	0.325897i
$63$		0			0
$64$	−3.50000	+	7.19375i	−0.437500	+	0.899218i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	6.34533			0.763888
$70$		0			0
$71$		−	15.0872i		−	1.79052i	−0.445548	−	0.895258i	$-0.646991\pi$
$71$		−	15.0872i		−	1.79052i	0.445548	−	0.895258i	$-0.353009\pi$
$72$	−1.87414	−	2.99602i	−0.220869	−	0.353084i
$73$	17.0553			1.99617			0.998087	−	0.0618285i	$-0.0196932\pi$
$73$	17.0553			1.99617			0.998087	+	0.0618285i	$0.0196932\pi$
$74$		0			0
$75$			6.61546i			0.763888i
$76$		0			0
$77$		0			0
$78$	−3.95978	+	0.746485i	−0.448357	+	0.0845227i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−3.69066			−0.410073
$82$	−2.24169	−	11.8912i	−0.247554	−	1.31316i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	14.0941i		−	1.51104i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	3.49227	+	8.93331i	0.364094	+	0.931362i
$93$	12.9594			1.34383
$94$	−9.93465	+	1.87285i	−1.02468	+	0.193170i
$95$		0			0
$96$	−4.46461	+	6.00713i	−0.455667	+	0.613100i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−1.83392	−	9.72814i	−0.185254	−	0.982691i
$99$		0			0
$100$	−9.31362	+	3.64094i	−0.931362	+	0.364094i
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	−3.23028	−	5.16396i	−0.316755	−	0.506368i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−4.09415	−	10.4730i	−0.393960	−	1.00776i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	19.8424	−	7.75693i	1.84232	−	0.720213i
$117$	2.69066			0.248752
$118$	−13.3299	+	2.51290i	−1.22711	+	0.231331i
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	11.3210i		−	1.02078i
$124$	7.13245	+	18.2450i	0.640512	+	1.63845i
$125$		0			0
$126$		0			0
$127$			17.7333i			1.57358i	0.617221	+	0.786790i	$0.288258\pi$
$127$			17.7333i			1.57358i	−0.617221	+	0.786790i	$0.711742\pi$
$128$	−10.9144	−	2.97939i	−0.964702	−	0.263344i
$129$		0			0
$130$		0			0
$131$			18.2665i			1.59595i	0.602691	+	0.797975i	$0.294095\pi$
$131$			18.2665i			1.59595i	−0.602691	+	0.797975i	$0.705905\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	1.66240	+	8.81833i	0.141513	+	0.750666i
$139$			23.5588i			1.99824i	0.0419997	+	0.999118i	$0.486627\pi$
$139$			23.5588i			1.99824i	−0.0419997	+	0.999118i	$0.513373\pi$
$140$		0			0
$141$	−9.45826			−0.796529
$142$	20.9671	−	3.95266i	1.75952	−	0.331700i
$143$		0			0
$144$	3.67267	−	3.38947i	0.306055	−	0.282456i
$145$		0			0
$146$	4.46829	+	23.7024i	0.369798	+	1.96162i
$147$		−	9.26165i		−	0.763888i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−9.19374	+	1.73317i	−0.750666	+	0.141513i
$151$			1.85623i			0.151058i	0.997144	+	0.0755288i	$0.0240645\pi$
$151$			1.85623i			0.151058i	−0.997144	+	0.0755288i	$0.975936\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−2.07483	−	5.30747i	−0.166119	−	0.424938i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.966910	−	5.12904i	−0.0759676	−	0.402975i
$163$		−	15.6203i		−	1.22348i	−0.791061	−	0.611738i	$-0.790471\pi$
$163$		−	15.6203i		−	1.22348i	0.791061	−	0.611738i	$-0.209529\pi$
$164$	15.9383	−	6.23072i	1.24457	−	0.486537i
$165$		0			0
$166$		0			0
$167$		−	9.59166i		−	0.742225i	−0.928588	−	0.371113i	$-0.878976\pi$
$167$		−	9.59166i		−	0.742225i	0.928588	−	0.371113i	$-0.121024\pi$
$168$		0			0
$169$	−8.36235			−0.643258
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.0000			−1.36851			−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000			−1.36851			−0.684257	+	0.729241i	$0.739873\pi$
$174$	19.5870	−	3.69248i	1.48489	−	0.279926i
$175$		0			0
$176$		0			0
$177$	−12.6907			−0.953889
$178$		0			0
$179$		−	26.2050i		−	1.95866i	−0.202279	−	0.979328i	$-0.564835\pi$
$179$		−	26.2050i		−	1.95866i	0.202279	−	0.979328i	$-0.435165\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−11.5000	+	7.19375i	−0.847791	+	0.530330i
$185$		0			0
$186$	3.39521	+	18.0101i	0.248949	+	1.32057i
$187$		0			0
$188$	−5.20552	−	13.3159i	−0.379652	−	0.971159i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−9.51800	−	4.63083i	−0.686902	−	0.334201i
$193$	8.44124			0.607613			0.303807	−	0.952734i	$-0.401742\pi$
$193$	8.44124			0.607613			0.303807	+	0.952734i	$0.401742\pi$
$194$		0			0
$195$		0			0
$196$	13.0391	−	5.09732i	0.931362	−	0.364094i
$197$	−14.8442			−1.05760			−0.528802	−	0.848745i	$-0.677359\pi$
$197$	−14.8442			−1.05760			−0.528802	+	0.848745i	$0.677359\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−7.50000	−	11.9896i	−0.530330	−	0.847791i
$201$		0			0
$202$	1.57193	+	8.33841i	0.110601	+	0.586688i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	5.99203i		−	0.416475i
$208$	6.33024	−	5.84213i	0.438923	−	0.405079i
$209$		0			0
$210$		0			0
$211$		−	28.7750i		−	1.98095i	−0.137686	−	0.990476i	$-0.543966\pi$
$211$		−	28.7750i		−	1.98095i	0.137686	−	0.990476i	$-0.456034\pi$
$212$		0			0
$213$	19.9617			1.36775
$214$		0			0
$215$		0			0
$216$	13.4820	−	8.43358i	0.917334	−	0.573832i
$217$		0			0
$218$		0			0
$219$			22.5658i			1.52485i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			28.7750i			1.92692i	0.267860	+	0.963458i	$0.413684\pi$
$223$			28.7750i			1.92692i	−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$	6.24713			0.416475
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	15.9786	+	25.5435i	1.04904	+	1.67701i
$233$	29.8612			1.95627			0.978136	−	0.207965i	$-0.0666840\pi$
$233$	29.8612			1.95627			0.978136	+	0.207965i	$0.0666840\pi$
$234$	0.704922	+	3.73931i	0.0460822	+	0.244446i
$235$		0			0
$236$	−6.98454	−	17.8666i	−0.454655	−	1.16302i
$237$		0			0
$238$		0			0
$239$			0.789960i			0.0510982i	0.999674	+	0.0255491i	$0.00813342\pi$
$239$			0.789960i			0.0510982i	−0.999674	+	0.0255491i	$0.991867\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−2.88187	−	15.2871i	−0.185254	−	0.982691i
$243$			11.9841i			0.768778i
$244$		0			0
$245$		0			0
$246$	15.7332	−	2.96597i	1.00311	−	0.189103i
$247$		0			0
$248$	−23.4871	+	14.6922i	−1.49143	+	0.932954i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	−24.6446	+	4.64593i	−1.54634	+	0.291511i
$255$		0			0
$256$	1.28113	−	15.9486i	0.0800709	−	0.996789i
$257$	−4.36465			−0.272260			−0.136130	−	0.990691i	$-0.543466\pi$
$257$	−4.36465			−0.272260			−0.136130	+	0.990691i	$0.543466\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−13.3093			−0.823827
$262$	−25.3856	+	4.78560i	−1.56832	+	0.295655i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	27.7653			1.69288			0.846440	−	0.532484i	$-0.178741\pi$
$269$	27.7653			1.69288			0.846440	+	0.532484i	$0.178741\pi$
$270$		0			0
$271$			28.7750i			1.74796i	0.485965	+	0.873978i	$0.338468\pi$
$271$			28.7750i			1.74796i	−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−11.8196	+	4.62059i	−0.711456	+	0.278127i
$277$	−27.6501			−1.66133			−0.830666	−	0.556771i	$-0.812040\pi$
$277$	−27.6501			−1.66133			−0.830666	+	0.556771i	$0.812040\pi$
$278$	−32.7405	+	6.17214i	−1.96365	+	0.370180i
$279$		−	12.2378i		−	0.732661i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−2.47795	−	13.1445i	−0.147560	−	0.782742i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	10.9863	+	28.1032i	0.651916	+	1.66762i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	5.67267	+	4.21603i	0.334265	+	0.248432i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	−31.7693	+	12.4195i	−1.85916	+	0.726795i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	12.8712	−	2.42644i	0.750666	−	0.141513i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	10.3279i		−	0.597279i
$300$	−4.81730	−	12.3228i	−0.278127	−	0.711456i
$301$		0			0
$302$	−2.57966	+	0.486309i	−0.148443	+	0.0279840i
$303$			7.93856i			0.456058i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	28.7750i		−	1.64228i	−0.570730	−	0.821138i	$-0.693340\pi$
$307$		−	28.7750i		−	1.64228i	0.570730	−	0.821138i	$-0.306660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			23.0257i			1.30567i	0.757501	+	0.652834i	$0.226420\pi$
$311$			23.0257i			1.30567i	−0.757501	+	0.652834i	$0.773580\pi$
$312$	6.83240	−	4.27396i	0.386808	−	0.241965i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	30.0000			1.68497			0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000			1.68497			0.842484	+	0.538721i	$0.181092\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	6.87468	−	2.68750i	0.381927	−	0.149305i
$325$	10.7676			0.597279
$326$	21.7081	−	4.09233i	1.20230	−	0.226653i
$327$		0			0
$328$	12.8347	+	20.5177i	0.708678	+	1.13290i
$329$		0			0
$330$		0			0
$331$		−	31.4974i		−	1.73125i	−0.500690	−	0.865627i	$-0.666920\pi$
$331$		−	31.4974i		−	1.73125i	0.500690	−	0.865627i	$-0.333080\pi$
$332$		0			0
$333$		0			0
$334$	13.3299	−	2.51290i	0.729378	−	0.137500i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−2.19084	−	11.6215i	−0.119166	−	0.632124i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−4.71579	−	25.0152i	−0.253522	−	1.34483i
$347$			9.59166i			0.514907i	0.966291	+	0.257454i	$0.0828835\pi$
$347$			9.59166i			0.514907i	−0.966291	+	0.257454i	$0.917117\pi$
$348$	10.2631	+	26.2534i	0.550162	+	1.40733i
$349$	−36.2641			−1.94118			−0.970588	−	0.240748i	$-0.922607\pi$
$349$	−36.2641			−1.94118			−0.970588	+	0.240748i	$0.922607\pi$
$350$		0			0
$351$			12.1079i			0.646273i
$352$		0			0
$353$	34.0530			1.81246			0.906230	−	0.422786i	$-0.138948\pi$
$353$	34.0530			1.81246			0.906230	+	0.422786i	$0.138948\pi$
$354$	−3.32480	−	17.6367i	−0.176711	−	0.937377i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	36.4180	−	6.86541i	1.92475	−	0.362848i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	14.5540i		−	0.763888i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−13.0103	−	14.0973i	−0.678207	−	0.734871i
$369$	−10.6907			−0.556534
$370$		0			0
$371$		0			0
$372$	−24.1398	+	9.43689i	−1.25159	+	0.489280i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	17.1417	−	10.7229i	0.884018	−	0.552991i
$377$	−22.9401			−1.18147
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−23.4629			−1.20204
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	3.94202	−	14.4407i	0.201165	−	0.736924i
$385$		0			0
$386$	2.21150	+	11.7311i	0.112563	+	0.597096i
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	10.5000	+	16.7854i	0.530330	+	0.847791i
$393$	−24.1682			−1.21913
$394$	−3.88900	−	20.6295i	−0.195925	−	1.03930i
$395$		0			0
$396$		0			0
$397$	23.3430			1.17155			0.585777	−	0.810473i	$-0.300790\pi$
$397$	23.3430			1.17155			0.585777	+	0.810473i	$0.300790\pi$
$398$		0			0
$399$		0			0
$400$	14.6974	−	13.5641i	0.734871	−	0.678207i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		−	21.0933i		−	1.05073i
$404$	−11.1763	+	4.36913i	−0.556044	+	0.217372i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4.13420			−0.204423			−0.102211	−	0.994763i	$-0.532592\pi$
$409$	−4.13420			−0.204423			−0.102211	+	0.994763i	$0.532592\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	8.32733	−	1.56984i	0.409266	−	0.0771535i
$415$		0			0
$416$	9.77745	+	7.26678i	0.479379	+	0.356283i
$417$	−31.1705			−1.52643
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	39.9896	−	7.53871i	1.94666	−	0.366979i
$423$			8.93164i			0.434271i
$424$		0			0
$425$		0			0
$426$	5.22973	+	27.7415i	0.253381	+	1.34408i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	15.2526	+	16.5269i	0.733839	+	0.795151i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−31.3604	+	5.91196i	−1.49846	+	0.282485i
$439$		−	25.6719i		−	1.22525i	−0.790373	−	0.612626i	$-0.790113\pi$
$439$		−	25.6719i		−	1.22525i	0.790373	−	0.612626i	$-0.209887\pi$
$440$		0			0
$441$	−8.74598			−0.416475
$442$		0			0
$443$			34.1436i			1.62221i	0.584900	+	0.811105i	$0.301134\pi$
$443$			34.1436i			1.62221i	−0.584900	+	0.811105i	$0.698866\pi$
$444$		0			0
$445$		0			0
$446$	−39.9896	+	7.53871i	−1.89356	+	0.356968i
$447$		0			0
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$	1.63667	+	8.68185i	0.0771535	+	0.409266i
$451$		0			0
$452$		0			0
$453$	−2.45596			−0.115391
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.26875			−0.105666			−0.0528331	−	0.998603i	$-0.516825\pi$
$461$	−2.26875			−0.105666			−0.0528331	+	0.998603i	$0.516825\pi$
$462$		0			0
$463$			28.7750i			1.33729i	0.743583	+	0.668644i	$0.233125\pi$
$463$			28.7750i			1.33729i	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−31.3125	+	28.8980i	−1.45365	+	1.34156i
$465$		0			0
$466$	7.82328	+	41.4991i	0.362407	+	1.92241i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	−5.01196	+	1.95931i	−0.231678	+	0.0905690i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	23.0000	−	14.3875i	1.05866	−	0.662238i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−1.09783	+	0.206960i	−0.0502138	+	0.00946614i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	20.4900	−	8.01007i	0.931362	−	0.364094i
$485$		0			0
$486$	−16.6547	+	3.13968i	−0.755471	+	0.142419i
$487$			6.08233i			0.275617i	0.990459	+	0.137808i	$0.0440058\pi$
$487$			6.08233i			0.275617i	−0.990459	+	0.137808i	$0.955994\pi$
$488$		0			0
$489$	20.6671			0.934598
$490$		0			0
$491$			40.5022i			1.82784i	0.405894	+	0.913920i	$0.366960\pi$
$491$			40.5022i			1.82784i	−0.405894	+	0.913920i	$0.633040\pi$
$492$	8.24382	+	21.0879i	0.371660	+	0.950716i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−26.5716	−	28.7916i	−1.19310	−	1.29278i
$497$		0			0
$498$		0			0
$499$		−	43.1484i		−	1.93159i	−0.259310	−	0.965794i	$-0.583495\pi$
$499$		−	43.1484i		−	1.93159i	0.259310	−	0.965794i	$-0.416505\pi$
$500$		0			0
$501$	12.6907			0.566977
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	11.0642i		−	0.491377i
$508$	−12.9132	−	33.0323i	−0.572931	−	1.46557i
$509$	36.1489			1.60227			0.801136	−	0.598482i	$-0.204229\pi$
$509$	36.1489			1.60227			0.801136	+	0.598482i	$0.204229\pi$
$510$		0			0
$511$		0			0
$512$	22.5000	−	2.39792i	0.994369	−	0.105974i
$513$		0			0
$514$	−1.14349	−	6.06571i	−0.0504371	−	0.267547i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	23.8157i		−	1.04539i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−3.48689	−	18.4964i	−0.152617	−	0.809568i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−13.3014	−	34.0254i	−0.581076	−	1.48641i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$			11.9841i			0.520064i
$532$		0			0
$533$	−18.4265			−0.798141
$534$		0			0
$535$		0			0
$536$		0			0
$537$	34.6717			1.49619
$538$	7.27418	+	38.5864i	0.313612	+	1.66358i
$539$		0			0
$540$		0			0
$541$	40.5712			1.74429			0.872146	−	0.489246i	$-0.162728\pi$
$541$	40.5712			1.74429			0.872146	+	0.489246i	$0.162728\pi$
$542$	−39.9896	+	7.53871i	−1.71770	+	0.323815i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			35.2099i			1.50546i	0.658327	+	0.752732i	$0.271265\pi$
$547$			35.2099i			1.50546i	−0.658327	+	0.752732i	$0.728735\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−9.51800	−	15.2156i	−0.405113	−	0.647618i
$553$		0			0
$554$	−7.24399	−	38.4263i	−0.307768	−	1.63258i
$555$		0			0
$556$	−17.1553	−	43.8836i	−0.727546	−	1.86108i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	17.0074	−	3.20617i	0.719979	−	0.135728i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	17.6181	−	6.88739i	0.741857	−	0.290012i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−36.1777	+	22.6307i	−1.51798	+	0.949564i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	23.9792i		−	1.00000i
$576$	−4.37299	+	8.98805i	−0.182208	+	0.374502i
$577$	46.8589			1.95076			0.975381	−	0.220527i	$-0.0707777\pi$
$577$	46.8589			1.95076			0.975381	+	0.220527i	$0.0707777\pi$
$578$	4.45380	+	23.6255i	0.185254	+	0.982691i
$579$			11.1685i			0.464149i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−25.5830	−	40.8972i	−1.05863	−	1.69234i
$585$		0			0
$586$		0			0
$587$		−	48.4408i		−	1.99937i	−0.0251938	−	0.999683i	$-0.508020\pi$
$587$		−	48.4408i		−	1.99937i	0.0251938	−	0.999683i	$-0.491980\pi$
$588$	6.74422	+	17.2519i	0.278127	+	0.711456i
$589$		0			0
$590$		0			0
$591$		−	19.6402i		−	0.807891i
$592$		0			0
$593$	−30.0000			−1.23195			−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000			−1.23195			−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	14.3531	−	2.70579i	0.586940	−	0.110648i
$599$		−	9.59166i		−	0.391905i	−0.980613	−	0.195952i	$-0.937220\pi$
$599$		−	9.59166i		−	0.391905i	0.980613	−	0.195952i	$-0.0627798\pi$
$600$	15.8633	−	9.92320i	0.647618	−	0.405113i
$601$	−42.5519			−1.73573			−0.867863	−	0.496803i	$-0.834507\pi$
$601$	−42.5519			−1.73573			−0.867863	+	0.496803i	$0.834507\pi$
$602$		0			0
$603$		0			0
$604$	−1.35168	−	3.45764i	−0.0549992	−	0.140689i
$605$		0			0
$606$	−11.0325	+	2.07981i	−0.448164	+	0.0844864i
$607$			28.7750i			1.16794i	0.811775	+	0.583970i	$0.198502\pi$
$607$			28.7750i			1.16794i	−0.811775	+	0.583970i	$0.801498\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			15.3946i			0.622801i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	39.9896	−	7.53871i	1.61385	−	0.304237i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	26.9640			1.08203
$622$	−31.9996	+	6.03247i	−1.28307	+	0.241880i
$623$		0			0
$624$	7.72968	+	8.37549i	0.309435	+	0.335288i
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	38.0720			1.51323
$634$	7.85965	+	41.6920i	0.312146	+	1.65580i
$635$		0			0
$636$		0			0
$637$	−15.0746			−0.597279
$638$		0			0
$639$		−	18.8503i		−	0.745705i
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	8.72852i		−	0.343153i	−0.985171	−	0.171577i	$-0.945114\pi$
$647$		−	8.72852i		−	0.343153i	0.985171	−	0.171577i	$-0.0548861\pi$
$648$	5.53599	+	8.84990i	0.217474	+	0.347657i
$649$		0			0
$650$	2.82098	+	14.9641i	0.110648	+	0.586940i
$651$		0			0
$652$	11.3745	+	29.0963i	0.445460	+	1.13950i
$653$	−49.0700			−1.92026			−0.960129	−	0.279556i	$-0.909813\pi$
$653$	−49.0700			−1.92026			−0.960129	+	0.279556i	$0.909813\pi$
$654$		0			0
$655$		0			0
$656$	−25.1516	+	23.2122i	−0.982005	+	0.906285i
$657$	21.3093			0.831356
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	43.7730	−	8.25195i	1.70129	−	0.320721i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			51.0870i			1.97810i
$668$	6.98454	+	17.8666i	0.270240	+	0.691280i
$669$	−38.0720			−1.47195
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−29.9764			−1.15551			−0.577753	−	0.816211i	$-0.696070\pi$
$673$	−29.9764			−1.15551			−0.577753	+	0.816211i	$0.696070\pi$
$674$		0			0
$675$			28.1119i			1.08203i
$676$	15.5768	−	6.08937i	0.599106	−	0.234206i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			2.38936i			0.0914263i	0.998955	+	0.0457131i	$0.0145560\pi$
$683$			2.38936i			0.0914263i	−0.998955	+	0.0457131i	$0.985444\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	28.7750i		−	1.09465i	−0.836919	−	0.547326i	$-0.815646\pi$
$691$		−	28.7750i		−	1.09465i	0.836919	−	0.547326i	$-0.184354\pi$
$692$	33.5290	−	13.1074i	1.27458	−	0.498268i
$693$		0			0
$694$	−13.3299	+	2.51290i	−0.505995	+	0.0953885i
$695$		0			0
$696$	−33.7964	+	21.1411i	−1.28105	+	0.801352i
$697$		0			0
$698$	−9.50078	−	50.3975i	−0.359610	−	1.90758i
$699$			39.5091i			1.49437i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−16.8268	+	3.17213i	−0.635086	+	0.119724i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	8.92149	+	47.3247i	0.335765	+	1.78109i
$707$		0			0
$708$	23.6392	−	9.24119i	0.888416	−	0.347305i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−46.9741			−1.75920
$714$		0			0
$715$		0			0
$716$	19.0822	+	48.8128i	0.713135	+	1.82422i
$717$	−1.04519			−0.0390333
$718$		0			0
$719$		−	47.9583i		−	1.78854i	−0.447524	−	0.894272i	$-0.647694\pi$
$719$		−	47.9583i		−	1.78854i	0.447524	−	0.894272i	$-0.352306\pi$
$720$		0			0
$721$		0			0
$722$	−4.97778	−	26.4050i	−0.185254	−	0.982691i
$723$		0			0
$724$		0			0
$725$	−53.2618			−1.97810
$726$	20.2262	−	3.81298i	0.750666	−	0.141513i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−26.9280			−0.997334
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	16.1829	−	21.7741i	0.596511	−	0.802605i
$737$		0			0
$738$	−2.80083	−	14.8572i	−0.103100	−	0.546900i
$739$			39.4360i			1.45068i	0.688393	+	0.725338i	$0.258316\pi$
$739$			39.4360i			1.45068i	−0.688393	+	0.725338i	$0.741684\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	−19.4391	−	31.0756i	−0.712672	−	1.13929i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	19.3929	+	21.0132i	0.707187	+	0.766272i
$753$		0			0
$754$	−6.01003	−	31.8806i	−0.218872	−	1.16102i
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	21.4776			0.778562			0.389281	−	0.921119i	$-0.372724\pi$
$761$	21.4776			0.778562			0.389281	+	0.921119i	$0.372724\pi$
$762$	−6.14699	−	32.6071i	−0.222682	−	1.18123i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			20.6558i			0.745839i
$768$	21.1015	+	1.69506i	0.761435	+	0.0611652i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		−	5.77484i		−	0.207976i
$772$	−15.7237	+	6.14681i	−0.565908	+	0.221228i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		−	48.9739i		−	1.75920i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	59.8917i		−	2.14036i
$784$	−20.5764	+	18.9898i	−0.734871	+	0.678207i
$785$		0			0
$786$	−6.33179	−	33.5874i	−0.225848	−	1.19802i
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	27.6506	−	10.8094i	0.985012	−	0.385067i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	6.11560	+	32.4406i	0.217034	+	1.15127i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	22.7011	+	16.8719i	0.802605	+	0.596511i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	29.3140	−	5.52618i	1.03254	−	0.194652i
$807$			36.7361i			1.29317i
$808$	−9.00000	−	14.3875i	−0.316619	−	0.506150i
$809$	42.0000			1.47664			0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000			1.47664			0.738321	+	0.674450i	$0.235619\pi$
$810$		0			0
$811$		−	27.2713i		−	0.957625i	−0.877917	−	0.478812i	$-0.841067\pi$
$811$		−	27.2713i		−	0.957625i	0.877917	−	0.478812i	$-0.158933\pi$
$812$		0			0
$813$	−38.0720			−1.33524
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.08311	−	5.74544i	−0.0378701	−	0.200884i
$819$		0			0
$820$		0			0
$821$	54.0000			1.88461			0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000			1.88461			0.942306	+	0.334751i	$0.108652\pi$
$822$		0			0
$823$			56.9125i			1.98384i	0.126849	+	0.991922i	$0.459514\pi$
$823$			56.9125i			1.98384i	−0.126849	+	0.991922i	$0.540486\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	4.36333	+	11.1615i	0.151636	+	0.387889i
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$		−	36.5836i		−	1.26907i
$832$	−7.53732	+	15.4919i	−0.261309	+	0.537084i
$833$		0			0
$834$	−8.16631	−	43.3188i	−0.282776	−	1.50001i
$835$		0			0
$836$		0			0
$837$	55.0700			1.90350
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	84.4730			2.91286
$842$		0			0
$843$		0			0
$844$	20.9536	+	53.5999i	0.721253	+	1.84498i
$845$		0			0
$846$	−12.4126	+	2.33998i	−0.426754	+	0.0804503i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−37.1832	+	14.5359i	−1.27387	+	0.497991i
$853$	−10.0000			−0.342393			−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000			−0.342393			−0.171197	+	0.985237i	$0.554763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−55.3578			−1.89098			−0.945492	−	0.325644i	$-0.894419\pi$
$857$	−55.3578			−1.89098			−0.945492	+	0.325644i	$0.894419\pi$
$858$		0			0
$859$			0.256826i			0.00876280i	0.999990	+	0.00438140i	$0.00139465\pi$
$859$			0.256826i			0.00876280i	−0.999990	+	0.00438140i	$0.998605\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	43.6815i		−	1.48694i	−0.668771	−	0.743469i	$-0.733179\pi$
$863$		−	43.6815i		−	1.48694i	0.668771	−	0.743469i	$-0.266821\pi$
$864$	−18.9720	+	25.5269i	−0.645441	+	0.868442i
$865$		0			0
$866$		0			0
$867$			22.4926i			0.763888i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−16.4321	−	42.0338i	−0.555190	−	1.42019i
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$	35.6771	−	6.72573i	1.20404	−	0.226983i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−2.29134	−	12.1546i	−0.0771535	−	0.409266i
$883$		−	28.7750i		−	0.968355i	−0.874970	−	0.484178i	$-0.839119\pi$
$883$		−	28.7750i		−	0.968355i	0.874970	−	0.484178i	$-0.160881\pi$
$884$		0			0
$885$		0			0
$886$	−47.4505	+	8.94522i	−1.59413	+	0.300520i
$887$		−	59.5587i		−	1.99978i	−0.0146917	−	0.999892i	$-0.504677\pi$
$887$		−	59.5587i		−	1.99978i	0.0146917	−	0.999892i	$-0.495323\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−20.9536	−	53.5999i	−0.701579	−	1.79466i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	13.6648			0.456254
$898$	4.71579	+	25.0152i	0.157368	+	0.834769i
$899$			104.338i			3.47986i
$900$	−11.6367	+	4.54908i	−0.387889	+	0.151636i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−0.643432	−	3.41313i	−0.0213766	−	0.113394i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$	7.49655			0.248645
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	38.0720			1.25451
$922$	−0.594385	−	3.15296i	−0.0195750	−	0.103837i
$923$		−	32.4905i		−	1.06944i
$924$		0			0
$925$		0			0
$926$	−39.9896	+	7.53871i	−1.31414	+	0.247737i
$927$		0			0
$928$	−48.3641	−	35.9451i	−1.58763	−	1.17996i
$929$	−21.1319			−0.693315			−0.346658	−	0.937992i	$-0.612683\pi$
$929$	−21.1319			−0.693315			−0.346658	+	0.937992i	$0.612683\pi$
$930$		0			0
$931$		0			0
$932$	−55.6232	+	21.7446i	−1.82200	+	0.712267i
$933$	−30.4652			−0.997384
$934$		0			0
$935$		0			0
$936$	−4.03599	−	6.45198i	−0.131921	−	0.210890i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$			41.0354i			1.33630i
$944$	26.0205	+	28.1945i	0.846896	+	0.917654i
$945$		0			0
$946$		0			0
$947$			24.6251i			0.800209i	0.916470	+	0.400104i	$0.131026\pi$
$947$			24.6251i			0.800209i	−0.916470	+	0.400104i	$0.868974\pi$
$948$		0			0
$949$	36.7290			1.19227
$950$		0			0
$951$			39.6928i			1.28713i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.575239	−	1.47148i	−0.0186046	−	0.0475910i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−64.9378			−2.09477
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			33.6105i			1.08084i	0.841396	+	0.540420i	$0.181735\pi$
$967$			33.6105i			1.08084i	−0.841396	+	0.540420i	$0.818265\pi$
$968$	16.5000	+	26.3771i	0.530330	+	0.847791i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−8.72665	−	22.3230i	−0.279908	−	0.716011i
$973$		0			0
$974$	−8.45283	+	1.59350i	−0.270846	+	0.0510590i
$975$			14.2465i			0.456254i
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$	5.41453	+	28.7218i	0.173138	+	0.918421i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−56.2873	+	10.6111i	−1.79620	+	0.338614i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−27.1468	+	16.9815i	−0.865408	+	0.541350i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			28.7750i			0.914068i	0.889449	+	0.457034i	$0.151088\pi$
$991$			28.7750i			0.914068i	−0.889449	+	0.457034i	$0.848912\pi$
$992$	33.0513	−	44.4705i	1.04938	−	1.41194i
$993$	41.6740			1.32248
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−26.0000			−0.823428			−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000			−0.823428			−0.411714	+	0.911313i	$0.635070\pi$
$998$	59.9648	−	11.3044i	1.89815	−	0.357834i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	92.2.b.b.91.4	yes	6
3.2	odd	2		828.2.e.b.91.3		6
4.3	odd	2	inner	92.2.b.b.91.3	✓	6
8.3	odd	2		1472.2.c.c.1471.4		6
8.5	even	2		1472.2.c.c.1471.3		6
12.11	even	2		828.2.e.b.91.4		6
23.22	odd	2	CM	92.2.b.b.91.4	yes	6
69.68	even	2		828.2.e.b.91.3		6
92.91	even	2	inner	92.2.b.b.91.3	✓	6
184.45	odd	2		1472.2.c.c.1471.3		6
184.91	even	2		1472.2.c.c.1471.4		6
276.275	odd	2		828.2.e.b.91.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.2.b.b.91.3	✓	6	4.3	odd	2	inner
92.2.b.b.91.3	✓	6	92.91	even	2	inner
92.2.b.b.91.4	yes	6	1.1	even	1	trivial
92.2.b.b.91.4	yes	6	23.22	odd	2	CM
828.2.e.b.91.3		6	3.2	odd	2
828.2.e.b.91.3		6	69.68	even	2
828.2.e.b.91.4		6	12.11	even	2
828.2.e.b.91.4		6	276.275	odd	2
1472.2.c.c.1471.3		6	8.5	even	2
1472.2.c.c.1471.3		6	184.45	odd	2
1472.2.c.c.1471.4		6	8.3	odd	2
1472.2.c.c.1471.4		6	184.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 92 = 2^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	92.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.261988 + 1.38973i) q^{2} +1.32309i q^{3} +(-1.86272 + 0.728188i) q^{4} +(-1.83875 + 0.346635i) q^{6} +(-1.50000 - 2.39792i) q^{8} +1.24943 q^{9} +O(q^{10})\)	\(q+(0.261988 + 1.38973i) q^{2} +1.32309i q^{3} +(-1.86272 + 0.728188i) q^{4} +(-1.83875 + 0.346635i) q^{6} +(-1.50000 - 2.39792i) q^{8} +1.24943 q^{9} +(-0.963461 - 2.46456i) q^{12} +2.15352 q^{13} +(2.93948 - 2.71283i) q^{16} +(0.327335 + 1.73637i) q^{18} -4.79583i q^{23} +(3.17267 - 1.98464i) q^{24} +5.00000 q^{25} +(0.564197 + 2.99282i) q^{26} +5.62238i q^{27} -10.6524 q^{29} -9.79478i q^{31} +(4.54022 + 3.37437i) q^{32} +(-2.32733 + 0.909817i) q^{36} +2.84931i q^{39} -8.55646 q^{41} +(6.66493 - 1.25645i) q^{46} +7.14860i q^{47} +(3.58932 + 3.88921i) q^{48} -7.00000 q^{49} +(1.30994 + 6.94867i) q^{50} +(-4.01141 + 1.56817i) q^{52} +(-7.81362 + 1.47300i) q^{54} +(-2.79080 - 14.8040i) q^{58} +9.59166i q^{59} +(13.6121 - 2.56612i) q^{62} +(-3.50000 + 7.19375i) q^{64} +6.34533 q^{69} -15.0872i q^{71} +(-1.87414 - 2.99602i) q^{72} +17.0553 q^{73} +6.61546i q^{75} +(-3.95978 + 0.746485i) q^{78} -3.69066 q^{81} +(-2.24169 - 11.8912i) q^{82} -14.0941i q^{87} +(3.49227 + 8.93331i) q^{92} +12.9594 q^{93} +(-9.93465 + 1.87285i) q^{94} +(-4.46461 + 6.00713i) q^{96} +(-1.83392 - 9.72814i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{6} - 9 q^{8} - 18 q^{9}+O(q^{10}) \) 6 * q - 3 * q^6 - 9 * q^8 - 18 * q^9	\( 6 q - 3 q^{6} - 9 q^{8} - 18 q^{9} + 15 q^{12} + 21 q^{18} + 30 q^{25} - 27 q^{26} - 33 q^{36} + 39 q^{48} - 42 q^{49} + 3 q^{52} + 9 q^{54} - 15 q^{58} + 45 q^{62} - 21 q^{64} + 27 q^{72} - 51 q^{78} + 54 q^{81} + 33 q^{82} - 12 q^{93} - 39 q^{94} - 57 q^{96}+O(q^{100}) \) 6 * q - 3 * q^6 - 9 * q^8 - 18 * q^9 + 15 * q^12 + 21 * q^18 + 30 * q^25 - 27 * q^26 - 33 * q^36 + 39 * q^48 - 42 * q^49 + 3 * q^52 + 9 * q^54 - 15 * q^58 + 45 * q^62 - 21 * q^64 + 27 * q^72 - 51 * q^78 + 54 * q^81 + 33 * q^82 - 12 * q^93 - 39 * q^94 - 57 * q^96

Introduction

L-functions

Modular forms

Varieties

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Database

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