LMFDB - Embedded newform 92.2.b.b.91.6

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			91.6
Root			$-1.07255 - 0.921756i$ of defining polynomial
Character	$\chi$	$=$	92.91
Dual form			92.2.b.b.91.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.07255 + 0.921756i) q^{2} -2.11101i q^{3} +(0.300733 + 1.97726i) q^{4} +(1.94584 - 2.26417i) q^{6} +(-1.50000 + 2.39792i) q^{8} -1.45636 q^{9} +O(q^{10})$	\(q+(1.07255 + 0.921756i) q^{2} -2.11101i q^{3} +(0.300733 + 1.97726i) q^{4} +(1.94584 - 2.26417i) q^{6} +(-1.50000 + 2.39792i) q^{8} -1.45636 q^{9} +(4.17402 - 0.634850i) q^{12} -7.03677 q^{13} +(-3.81912 + 1.18925i) q^{16} +(-1.56202 - 1.34241i) q^{18} +4.79583i q^{23} +(5.06202 + 3.16651i) q^{24} +5.00000 q^{25} +(-7.54730 - 6.48619i) q^{26} -3.25863i q^{27} +3.94950 q^{29} -9.48506i q^{31} +(-5.19240 - 2.24476i) q^{32} +(-0.437976 - 2.87961i) q^{36} +14.8547i q^{39} +12.5299 q^{41} +(-4.42059 + 5.14378i) q^{46} +13.7071i q^{47} +(2.51053 + 8.06220i) q^{48} -7.00000 q^{49} +(5.36276 + 4.60878i) q^{50} +(-2.11619 - 13.9135i) q^{52} +(3.00366 - 3.49505i) q^{54} +(4.23604 + 3.64047i) q^{58} -9.59166i q^{59} +(8.74290 - 10.1732i) q^{62} +(-3.50000 - 7.19375i) q^{64} +10.1240 q^{69} -1.04102i q^{71} +(2.18454 - 3.49224i) q^{72} -9.44264 q^{73} -10.5550i q^{75} +(-13.6924 + 15.9324i) q^{78} -11.2481 q^{81} +(13.4390 + 11.5495i) q^{82} -8.33743i q^{87} +(-9.48261 + 1.44226i) q^{92} -20.0230 q^{93} +(-12.6346 + 14.7015i) q^{94} +(-4.73871 + 10.9612i) q^{96} +(-7.50786 - 6.45229i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.07255	+	0.921756i	0.758408	+	0.651780i
$2$	1.07255	+	0.921756i	0.758408	+	0.651780i
$3$		−	2.11101i		−	1.21879i	−0.792866	−	0.609396i	$-0.791412\pi$
$3$		−	2.11101i		−	1.21879i	0.792866	−	0.609396i	$-0.208588\pi$
$4$	0.300733	+	1.97726i	0.150366	+	0.988630i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	1.94584	−	2.26417i	0.794384	−	0.924342i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−1.50000	+	2.39792i	−0.530330	+	0.847791i
$9$	−1.45636			−0.485454
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	4.17402	−	0.634850i	1.20493	−	0.183265i
$13$	−7.03677			−1.95165			−0.975825	−	0.218554i	$-0.929866\pi$
$13$	−7.03677			−1.95165			−0.975825	+	0.218554i	$0.929866\pi$
$14$		0			0
$15$		0			0
$16$	−3.81912	+	1.18925i	−0.954780	+	0.297314i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−1.56202	−	1.34241i	−0.368173	−	0.316409i
$19$		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
$24$	5.06202	+	3.16651i	1.03328	+	0.646362i
$25$	5.00000			1.00000
$26$	−7.54730	−	6.48619i	−1.48015	−	1.27205i
$27$		−	3.25863i		−	0.627124i
$28$		0			0
$29$	3.94950			0.733404			0.366702	−	0.930339i	$-0.380487\pi$
$29$	3.94950			0.733404			0.366702	+	0.930339i	$0.380487\pi$
$30$		0			0
$31$		−	9.48506i		−	1.70357i	−0.523895	−	0.851783i	$-0.675521\pi$
$31$		−	9.48506i		−	1.70357i	0.523895	−	0.851783i	$-0.324479\pi$
$32$	−5.19240	−	2.24476i	−0.917896	−	0.396821i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.437976	−	2.87961i	−0.0729960	−	0.479935i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$			14.8547i			2.37866i
$40$		0			0
$41$	12.5299			1.95684			0.978422	−	0.206618i	$-0.0662459\pi$
$41$	12.5299			1.95684			0.978422	+	0.206618i	$0.0662459\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$	−4.42059	+	5.14378i	−0.651780	+	0.758408i
$47$			13.7071i			1.99938i	0.0248485	+	0.999691i	$0.492090\pi$
$47$			13.7071i			1.99938i	−0.0248485	+	0.999691i	$0.507910\pi$
$48$	2.51053	+	8.06220i	0.362363	+	1.16368i
$49$	−7.00000			−1.00000
$50$	5.36276	+	4.60878i	0.758408	+	0.651780i
$51$		0			0
$52$	−2.11619	−	13.9135i	−0.293463	−	1.92946i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	3.00366	−	3.49505i	0.408747	−	0.475616i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	4.23604	+	3.64047i	0.556219	+	0.478018i
$59$		−	9.59166i		−	1.24873i	−0.781133	−	0.624364i	$-0.785358\pi$
$59$		−	9.59166i		−	1.24873i	0.781133	−	0.624364i	$-0.214642\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	8.74290	−	10.1732i	1.11035	−	1.29200i
$63$		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
$69$	10.1240			1.21879
$70$		0			0
$71$		−	1.04102i		−	0.123546i	−0.998090	−	0.0617729i	$-0.980325\pi$
$71$		−	1.04102i		−	0.123546i	0.998090	−	0.0617729i	$-0.0196755\pi$
$72$	2.18454	−	3.49224i	0.257451	−	0.411564i
$73$	−9.44264			−1.10518			−0.552588	−	0.833454i	$-0.686360\pi$
$73$	−9.44264			−1.10518			−0.552588	+	0.833454i	$0.686360\pi$
$74$		0			0
$75$		−	10.5550i		−	1.21879i
$76$		0			0
$77$		0			0
$78$	−13.6924	+	15.9324i	−1.55036	+	1.80399i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−11.2481			−1.24979
$82$	13.4390	+	11.5495i	1.48409	+	1.27543i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	8.33743i		−	0.893867i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−9.48261	+	1.44226i	−0.988630	+	0.150366i
$93$	−20.0230			−2.07629
$94$	−12.6346	+	14.7015i	−1.30316	+	1.51635i
$95$		0			0
$96$	−4.73871	+	10.9612i	−0.483642	+	1.11872i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−7.50786	−	6.45229i	−0.758408	−	0.651780i
$99$		0			0
$100$	1.50366	+	9.88630i	0.150366	+	0.988630i
$101$	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	$0.5\pi$
$104$	10.5552	−	16.8736i	1.03502	−	1.65459i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	6.44317	−	0.979978i	0.619994	−	0.0942984i
$109$		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
$116$	1.18774	+	7.80919i	0.110279	+	0.725065i
$117$	10.2481			0.947437
$118$	8.84117	−	10.2876i	0.813896	−	0.947046i
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	26.4508i		−	2.38499i
$124$	18.7544	−	2.85247i	1.68420	−	0.256159i
$125$		0			0
$126$		0			0
$127$		−	3.18100i		−	0.282268i	−0.989990	−	0.141134i	$-0.954925\pi$
$127$		−	3.18100i		−	0.282268i	0.989990	−	0.141134i	$-0.0450749\pi$
$128$	2.87695	−	10.9418i	0.254289	−	0.967128i
$129$		0			0
$130$		0			0
$131$			21.0811i			1.84187i	0.389721	+	0.920933i	$0.372572\pi$
$131$			21.0811i			1.84187i	−0.389721	+	0.920933i	$0.627428\pi$
$132$		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			$\pi$
$138$	10.8586	+	9.33190i	0.924342	+	0.794384i
$139$			12.6371i			1.07186i	0.844261	+	0.535932i	$0.180040\pi$
$139$			12.6371i			1.07186i	−0.844261	+	0.535932i	$0.819960\pi$
$140$		0			0
$141$	28.9358			2.43683
$142$	0.959562	−	1.11654i	0.0805247	−	0.0936982i
$143$		0			0
$144$	5.56202	−	1.73199i	0.463502	−	0.144332i
$145$		0			0
$146$	−10.1277	−	8.70380i	−0.838175	−	0.720332i
$147$			14.7771i			1.21879i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	9.72918	−	11.3208i	0.794384	−	0.924342i
$151$			22.1511i			1.80263i	0.433162	+	0.901316i	$0.357398\pi$
$151$			22.1511i			1.80263i	−0.433162	+	0.901316i	$0.642602\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−29.3716	+	4.46729i	−2.35161	+	0.357670i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−12.0642	−	10.3680i	−0.947850	−	0.814587i
$163$		−	25.3031i		−	1.98189i	−0.134250	−	0.990947i	$-0.542863\pi$
$163$		−	25.3031i		−	1.98189i	0.134250	−	0.990947i	$-0.457137\pi$
$164$	3.76815	+	24.7749i	0.294243	+	1.93459i
$165$		0			0
$166$		0			0
$167$			9.59166i			0.742225i	0.928588	+	0.371113i	$0.121024\pi$
$167$			9.59166i			0.742225i	−0.928588	+	0.371113i	$0.878976\pi$
$168$		0			0
$169$	36.5162			2.80894
$170$		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	$0.739873\pi$
$174$	7.68508	−	8.94232i	0.582604	−	0.677916i
$175$		0			0
$176$		0			0
$177$	−20.2481			−1.52194
$178$		0			0
$179$		−	8.41506i		−	0.628971i	−0.949262	−	0.314486i	$-0.898168\pi$
$179$		−	8.41506i		−	0.628971i	0.949262	−	0.314486i	$-0.101832\pi$
$180$		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
$186$	−21.4757	−	18.4564i	−1.57468	−	1.35329i
$187$		0			0
$188$	−27.1025	+	4.12217i	−1.97665	+	0.300640i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−15.1861	+	7.38853i	−1.09596	+	0.533222i
$193$	18.7045			1.34638			0.673188	−	0.739471i	$-0.264924\pi$
$193$	18.7045			1.34638			0.673188	+	0.739471i	$0.264924\pi$
$194$		0			0
$195$		0			0
$196$	−2.10513	−	13.8408i	−0.150366	−	0.988630i
$197$	−13.2113			−0.941268			−0.470634	−	0.882329i	$-0.655975\pi$
$197$	−13.2113			−0.941268			−0.470634	+	0.882329i	$0.655975\pi$
$198$		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
$202$	6.43531	+	5.53053i	0.452787	+	0.389127i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	6.98447i		−	0.485454i
$208$	26.8743	−	8.36851i	1.86340	−	0.580252i
$209$		0			0
$210$		0			0
$211$			28.7750i			1.98095i	0.137686	+	0.990476i	$0.456034\pi$
$211$			28.7750i			1.98095i	−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$	−2.19760			−0.150577
$214$		0			0
$215$		0			0
$216$	7.81393	+	4.88795i	0.531670	+	0.332583i
$217$		0			0
$218$		0			0
$219$			19.9335i			1.34698i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	28.7750i		−	1.92692i	−0.267860	−	0.963458i	$-0.586316\pi$
$223$		−	28.7750i		−	1.92692i	0.267860	−	0.963458i	$-0.413684\pi$
$224$		0			0
$225$	−7.28181			−0.485454
$226$		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
$232$	−5.92425	+	9.47057i	−0.388946	+	0.621773i
$233$	−20.4289			−1.33834			−0.669171	−	0.743108i	$-0.733351\pi$
$233$	−20.4289			−1.33834			−0.669171	+	0.743108i	$0.733351\pi$
$234$	10.9916	+	9.44624i	0.718544	+	0.617520i
$235$		0			0
$236$	18.9652	−	2.88453i	1.23453	−	0.187767i
$237$		0			0
$238$		0			0
$239$		−	26.3731i		−	1.70594i	−0.521963	−	0.852968i	$-0.674800\pi$
$239$		−	26.3731i		−	1.70594i	0.521963	−	0.852968i	$-0.325200\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−11.7981	−	10.1393i	−0.758408	−	0.651780i
$243$			13.9689i			0.896108i
$244$		0			0
$245$		0			0
$246$	24.3811	−	28.3698i	1.55448	−	1.80879i
$247$		0			0
$248$	22.7444	+	14.2276i	1.44427	+	0.903452i
$249$		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
$254$	2.93211	−	3.41179i	0.183977	−	0.214075i
$255$		0			0
$256$	13.1713	−	9.08381i	0.823209	−	0.567738i
$257$	29.6907			1.85206			0.926028	−	0.377454i	$-0.123200\pi$
$257$	29.6907			1.85206			0.926028	+	0.377454i	$0.123200\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−5.75190			−0.356034
$262$	−19.4316	+	22.6106i	−1.20049	+	1.39689i
$263$		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
$269$	−29.0093			−1.76873			−0.884365	−	0.466797i	$-0.845408\pi$
$269$	−29.0093			−1.76873			−0.884365	+	0.466797i	$0.845408\pi$
$270$		0			0
$271$		−	28.7750i		−	1.74796i	−0.485965	−	0.873978i	$-0.661532\pi$
$271$		−	28.7750i		−	1.74796i	0.485965	−	0.873978i	$-0.338468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	3.04463	+	20.0179i	0.183265	+	1.20493i
$277$	−2.22505			−0.133690			−0.0668451	−	0.997763i	$-0.521293\pi$
$277$	−2.22505			−0.133690			−0.0668451	+	0.997763i	$0.521293\pi$
$278$	−11.6483	+	13.5539i	−0.698619	+	0.812910i
$279$			13.8137i			0.827004i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	31.0351	+	26.6717i	1.84811	+	1.58828i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	2.05836	−	0.313068i	0.122141	−	0.0185771i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	7.56202	+	3.26918i	0.445597	+	0.192639i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	−2.83971	−	18.6706i	−0.166181	−	1.09261i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	−13.6208	+	15.8492i	−0.794384	+	0.924342i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	33.7472i		−	1.95165i
$300$	20.8701	−	3.17425i	1.20493	−	0.183265i
$301$		0			0
$302$	−20.4179	+	23.7582i	−1.17492	+	1.36713i
$303$		−	12.6661i		−	0.727646i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			28.7750i			1.64228i	0.570730	+	0.821138i	$0.306660\pi$
$307$			28.7750i			1.64228i	−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	11.6250i		−	0.659196i	−0.944121	−	0.329598i	$-0.893087\pi$
$311$		−	11.6250i		−	0.659196i	0.944121	−	0.329598i	$-0.106913\pi$
$312$	−35.6203	−	22.2820i	−2.01660	−	1.26147i
$313$		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
$324$	−3.38267	−	22.2404i	−0.187926	−	1.23558i
$325$	−35.1839			−1.95165
$326$	23.3233	−	27.1389i	1.29176	−	1.50309i
$327$		0			0
$328$	−18.7949	+	30.0457i	−1.03777	+	1.65899i
$329$		0			0
$330$		0			0
$331$			0.0289779i			0.00159277i	1.00000		0.000796384i	$0.000253497\pi$
$331$			0.0289779i			0.00159277i	−1.00000		0.000796384i	$0.999747\pi$
$332$		0			0
$333$		0			0
$334$	−8.84117	+	10.2876i	−0.483767	+	0.562910i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	39.1655	+	33.6590i	2.13032	+	1.83081i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−19.3059	−	16.5916i	−1.03789	−	0.891970i
$347$		−	9.59166i		−	0.514907i	−0.966291	−	0.257454i	$-0.917117\pi$
$347$		−	9.59166i		−	0.514907i	0.966291	−	0.257454i	$-0.0828835\pi$
$348$	16.4853	−	2.50734i	0.883704	−	0.134407i
$349$	25.9220			1.38758			0.693788	−	0.720180i	$-0.255941\pi$
$349$	25.9220			1.38758			0.693788	+	0.720180i	$0.255941\pi$
$350$		0			0
$351$			22.9303i			1.22393i
$352$		0			0
$353$	−3.26809			−0.173943			−0.0869714	−	0.996211i	$-0.527719\pi$
$353$	−3.26809			−0.173943			−0.0869714	+	0.996211i	$0.527719\pi$
$354$	−21.7171	−	18.6638i	−1.15425	−	0.991970i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	7.75663	−	9.02559i	0.409951	−	0.477017i
$359$		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
$363$			23.2211i			1.21879i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−5.70346	−	18.3159i	−0.297314	−	0.954780i
$369$	−18.2481			−0.949958
$370$		0			0
$371$		0			0
$372$	−6.02159	−	39.5908i	−0.312205	−	2.05269i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−32.8684	−	20.5606i	−1.69506	−	1.06033i
$377$	−27.7917			−1.43135
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−6.71513			−0.344027
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−23.0983	−	6.07327i	−1.17873	−	0.309925i
$385$		0			0
$386$	20.0615	+	17.2409i	1.02110	+	0.877541i
$387$		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
$393$	44.5025			2.24485
$394$	−14.1698	−	12.1776i	−0.713865	−	0.613499i
$395$		0			0
$396$		0			0
$397$	16.2986			0.818003			0.409002	−	0.912534i	$-0.365877\pi$
$397$	16.2986			0.818003			0.409002	+	0.912534i	$0.365877\pi$
$398$		0			0
$399$		0			0
$400$	−19.0956	+	5.94627i	−0.954780	+	0.297314i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$			66.7442i			3.32476i
$404$	1.80440	+	11.8636i	0.0897721	+	0.590234i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−32.7780			−1.62077			−0.810384	−	0.585899i	$-0.800742\pi$
$409$	−32.7780			−1.62077			−0.810384	+	0.585899i	$0.800742\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	6.43798	−	7.49120i	0.316409	−	0.368173i
$415$		0			0
$416$	36.5378	+	15.7959i	1.79141	+	0.774456i
$417$	26.6770			1.30638
$418$		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			$\pi$
$422$	−26.5235	+	30.8627i	−1.29114	+	1.50237i
$423$		−	19.9625i		−	0.970609i
$424$		0			0
$425$		0			0
$426$	−2.35703	−	2.02565i	−0.114199	−	0.0981429i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	3.87534	+	12.4451i	0.186453	+	0.598766i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−18.3738	+	21.3797i	−0.877935	+	1.02156i
$439$			15.8471i			0.756339i	0.925736	+	0.378170i	$0.123446\pi$
$439$			15.8471i			0.756339i	−0.925736	+	0.378170i	$0.876554\pi$
$440$		0			0
$441$	10.1945			0.485454
$442$		0			0
$443$		−	4.25100i		−	0.201971i	−0.994888	−	0.100985i	$-0.967800\pi$
$443$		−	4.25100i		−	0.201971i	0.994888	−	0.100985i	$-0.0321996\pi$
$444$		0			0
$445$		0			0
$446$	26.5235	−	30.8627i	1.25592	−	1.46139i
$447$		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	$0.360367\pi$
$450$	−7.81012	−	6.71205i	−0.368173	−	0.316409i
$451$		0			0
$452$		0			0
$453$	46.7612			2.19703
$454$		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
$461$	38.2711			1.78246			0.891232	−	0.453547i	$-0.149842\pi$
$461$	38.2711			1.78246			0.891232	+	0.453547i	$0.149842\pi$
$462$		0			0
$463$		−	28.7750i		−	1.33729i	−0.743583	−	0.668644i	$-0.766875\pi$
$463$		−	28.7750i		−	1.33729i	0.743583	−	0.668644i	$-0.233125\pi$
$464$	−15.0836	+	4.69696i	−0.700239	+	0.218051i
$465$		0			0
$466$	−21.9111	−	18.8305i	−1.01501	−	0.872304i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	3.08194	+	20.2632i	0.142463	+	0.936665i
$469$		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
$478$	24.3096	−	28.2865i	1.11189	−	1.29380i
$479$		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
$484$	−3.30806	−	21.7499i	−0.150366	−	0.988630i
$485$		0			0
$486$	−12.8760	+	14.9824i	−0.584065	+	0.679616i
$487$		−	34.8172i		−	1.57772i	−0.614575	−	0.788858i	$-0.710672\pi$
$487$		−	34.8172i		−	1.57772i	0.614575	−	0.788858i	$-0.289328\pi$
$488$		0			0
$489$	−53.4152			−2.41552
$490$		0			0
$491$			35.8292i			1.61695i	0.588531	+	0.808475i	$0.299707\pi$
$491$			35.8292i			1.61695i	−0.588531	+	0.808475i	$0.700293\pi$
$492$	52.3001	−	7.95461i	2.35787	−	0.358622i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	11.2801	+	36.2246i	0.506493	+	1.62653i
$497$		0			0
$498$		0			0
$499$		−	31.6072i		−	1.41493i	−0.706747	−	0.707466i	$-0.749838\pi$
$499$		−	31.6072i		−	1.41493i	0.706747	−	0.707466i	$-0.250162\pi$
$500$		0			0
$501$	20.2481			0.904618
$502$		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
$507$		−	77.0860i		−	3.42351i
$508$	6.28967	−	0.956632i	0.279059	−	0.0424437i
$509$	5.31232			0.235465			0.117732	−	0.993045i	$-0.462438\pi$
$509$	5.31232			0.235465			0.117732	+	0.993045i	$0.462438\pi$
$510$		0			0
$511$		0			0
$512$	22.5000	+	2.39792i	0.994369	+	0.105974i
$513$		0			0
$514$	31.8448	+	27.3676i	1.40462	+	1.20713i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			37.9982i			1.66793i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−6.16921	−	5.30185i	−0.270019	−	0.232056i
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−41.6829	+	6.33978i	−1.82092	+	0.276955i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$			13.9689i			0.606200i
$532$		0			0
$533$	−88.1701			−3.81907
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−17.7643			−0.766585
$538$	−31.1140	−	26.7395i	−1.34142	−	1.15282i
$539$		0			0
$540$		0			0
$541$	−39.9956			−1.71954			−0.859772	−	0.510677i	$-0.829395\pi$
$541$	−39.9956			−1.71954			−0.859772	+	0.510677i	$0.829395\pi$
$542$	26.5235	−	30.8627i	1.13928	−	1.32566i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			44.2733i			1.89299i	0.322722	+	0.946494i	$0.395402\pi$
$547$			44.2733i			1.89299i	−0.322722	+	0.946494i	$0.604598\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−15.1861	+	24.2766i	−0.646362	+	1.03328i
$553$		0			0
$554$	−2.38648	−	2.05095i	−0.101392	−	0.0871366i
$555$		0			0
$556$	−24.9868	+	3.80038i	−1.05968	+	0.161172i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	−12.7328	+	14.8159i	−0.539024	+	0.627206i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	8.70194	+	57.2136i	0.366418	+	2.40913i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	2.49627	+	1.56152i	0.104741	+	0.0655201i
$569$		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
$576$	5.09727	+	10.4767i	0.212386	+	0.436529i
$577$	−14.2544			−0.593417			−0.296708	−	0.954968i	$-0.595889\pi$
$577$	−14.2544			−0.593417			−0.296708	+	0.954968i	$0.595889\pi$
$578$	18.2334	+	15.6698i	0.758408	+	0.651780i
$579$		−	39.4853i		−	1.64095i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	14.1640	−	22.6426i	0.586109	−	0.936959i
$585$		0			0
$586$		0			0
$587$		−	23.1632i		−	0.956046i	−0.878347	−	0.478023i	$-0.841354\pi$
$587$		−	23.1632i		−	0.956046i	0.878347	−	0.478023i	$-0.158646\pi$
$588$	−29.2181	+	4.44395i	−1.20493	+	0.183265i
$589$		0			0
$590$		0			0
$591$			27.8892i			1.14721i
$592$		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
$598$	31.1067	−	36.1956i	1.27205	−	1.48015i
$599$			9.59166i			0.391905i	0.980613	+	0.195952i	$0.0627798\pi$
$599$			9.59166i			0.391905i	−0.980613	+	0.195952i	$0.937220\pi$
$600$	25.3101	+	15.8326i	1.03328	+	0.646362i
$601$	0.180813			0.00737552			0.00368776	−	0.999993i	$-0.498826\pi$
$601$	0.180813			0.00737552			0.00368776	+	0.999993i	$0.498826\pi$
$602$		0			0
$603$		0			0
$604$	−43.7985	+	6.66157i	−1.78214	+	0.271055i
$605$		0			0
$606$	11.6750	−	13.5850i	0.474265	−	0.551853i
$607$		−	28.7750i		−	1.16794i	−0.811775	−	0.583970i	$-0.801498\pi$
$607$		−	28.7750i		−	1.16794i	0.811775	−	0.583970i	$-0.198502\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	96.4536i		−	3.90209i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−26.5235	+	30.8627i	−1.07040	+	1.24552i
$615$		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
$621$	15.6279			0.627124
$622$	10.7155	−	12.4685i	0.429650	−	0.499940i
$623$		0			0
$624$	−17.6660	−	56.7319i	−0.707207	−	2.27109i
$625$	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
$633$	60.7443			2.41437
$634$	32.1765	+	27.6527i	1.27789	+	1.09823i
$635$		0			0
$636$		0			0
$637$	49.2574			1.95165
$638$		0			0
$639$			1.51610i			0.0599759i
$640$		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
$647$			39.0392i			1.53479i	0.641175	+	0.767395i	$0.278447\pi$
$647$			39.0392i			1.53479i	−0.641175	+	0.767395i	$0.721553\pi$
$648$	16.8721	−	26.9720i	0.662800	−	1.05956i
$649$		0			0
$650$	−37.7365	−	32.4309i	−1.48015	−	1.27205i
$651$		0			0
$652$	50.0309	−	7.60948i	1.95936	−	0.298010i
$653$	36.9083			1.44433			0.722167	−	0.691719i	$-0.243146\pi$
$653$	36.9083			1.44433			0.722167	+	0.691719i	$0.243146\pi$
$654$		0			0
$655$		0			0
$656$	−47.8532	+	14.9012i	−1.86835	+	0.581796i
$657$	13.7519			0.536513
$658$		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			$\pi$
$662$	−0.0267105	+	0.0310802i	−0.00103813	+	0.00120797i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			18.9411i			0.733404i
$668$	−18.9652	+	2.88453i	−0.733786	+	0.111606i
$669$	−60.7443			−2.34851
$670$		0			0
$671$		0			0
$672$		0			0
$673$	51.6633			1.99147			0.995737	−	0.0922433i	$-0.0294037\pi$
$673$	51.6633			1.99147			0.995737	+	0.0922433i	$0.0294037\pi$
$674$		0			0
$675$		−	16.2932i		−	0.627124i
$676$	10.9816	+	72.2020i	0.422370	+	2.77700i
$677$		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
$683$			46.4132i			1.77595i	0.459889	+	0.887977i	$0.347889\pi$
$683$			46.4132i			1.77595i	−0.459889	+	0.887977i	$0.652111\pi$
$684$		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	$0.184354\pi$
$691$			28.7750i			1.09465i	−0.836919	+	0.547326i	$0.815646\pi$
$692$	−5.41319	−	35.5907i	−0.205779	−	1.35296i
$693$		0			0
$694$	8.84117	−	10.2876i	0.335606	−	0.390510i
$695$		0			0
$696$	19.9925	+	12.5061i	0.757812	+	0.474044i
$697$		0			0
$698$	27.8027	+	23.8938i	1.05235	+	0.904393i
$699$			43.1256i			1.63116i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−21.1361	+	24.5939i	−0.797731	+	0.928236i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−3.50519	−	3.01238i	−0.131920	−	0.113372i
$707$		0			0
$708$	−6.08927	−	40.0358i	−0.228849	−	1.50464i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	45.4887			1.70357
$714$		0			0
$715$		0			0
$716$	16.6388	−	2.53068i	0.621820	−	0.0945761i
$717$	−55.6739			−2.07918
$718$		0			0
$719$			47.9583i			1.78854i	0.447524	+	0.894272i	$0.352306\pi$
$719$			47.9583i			1.78854i	−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$		0			0
$722$	−20.3785	−	17.5134i	−0.758408	−	0.651780i
$723$		0			0
$724$		0			0
$725$	19.7475			0.733404
$726$	−21.4042	+	24.9058i	−0.794384	+	0.924342i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−4.25571			−0.157619
$730$		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
$736$	10.7655	−	24.9019i	0.396821	−	0.917896i
$737$		0			0
$738$	−19.5720	−	16.8203i	−0.720456	−	0.619163i
$739$		−	12.6950i		−	0.466994i	−0.972357	−	0.233497i	$-0.924983\pi$
$739$		−	12.6950i		−	0.466994i	0.972357	−	0.233497i	$-0.0750170\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	30.0346	−	48.0136i	1.10112	−	1.76026i
$745$		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	$0.5\pi$
$752$	−16.3012	−	52.3490i	−0.594443	−	1.90897i
$753$		0			0
$754$	−29.8081	−	25.6172i	−1.08555	−	0.932923i
$755$		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
$761$	−54.7506			−1.98471			−0.992353	−	0.123432i	$-0.960610\pi$
$761$	−54.7506			−1.98471			−0.992353	+	0.123432i	$0.960610\pi$
$762$	−7.20232	−	6.18971i	−0.260913	−	0.224230i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			67.4944i			2.43708i
$768$	−19.1760	−	27.8048i	−0.691955	−	1.00332i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		−	62.6774i		−	2.25727i
$772$	5.62504	+	36.9836i	0.202450	+	1.33107i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		−	47.4253i		−	1.70357i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	12.8700i		−	0.459935i
$784$	26.7338	−	8.32478i	0.954780	−	0.297314i
$785$		0			0
$786$	47.7312	+	41.0204i	1.70251	+	1.46315i
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−3.97308	−	26.1222i	−0.141535	−	0.930566i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	17.4811	+	15.0233i	0.620380	+	0.533158i
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−25.9620	−	11.2238i	−0.917896	−	0.396821i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−61.5218	+	71.5866i	−2.16701	+	2.52153i
$807$			61.2390i			2.15571i
$808$	−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
$811$		−	56.9393i		−	1.99941i	−0.0242949	−	0.999705i	$-0.507734\pi$
$811$		−	56.9393i		−	1.99941i	0.0242949	−	0.999705i	$-0.492266\pi$
$812$		0			0
$813$	−60.7443			−2.13040
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−35.1561	−	30.2133i	−1.22920	−	1.05638i
$819$		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
$823$			34.7592i			1.21163i	0.795605	+	0.605815i	$0.207153\pi$
$823$			34.7592i			1.21163i	−0.795605	+	0.605815i	$0.792847\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	13.8101	−	2.10046i	0.479935	−	0.0729960i
$829$	−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
$831$			4.69710i			0.162941i
$832$	24.6287	+	50.6208i	0.853847	+	1.75496i
$833$		0			0
$834$	28.6125	+	24.5897i	0.990768	+	0.851471i
$835$		0			0
$836$		0			0
$837$	−30.9083			−1.06835
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−13.4015			−0.462119
$842$		0			0
$843$		0			0
$844$	−56.8957	+	8.65358i	−1.95843	+	0.297869i
$845$		0			0
$846$	18.4005	−	21.4108i	0.632623	−	0.736118i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−0.660889	−	4.34522i	−0.0226417	−	0.148865i
$853$	−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
$857$	11.1671			0.381460			0.190730	−	0.981642i	$-0.438914\pi$
$857$	11.1671			0.381460			0.190730	+	0.981642i	$0.438914\pi$
$858$		0			0
$859$		−	50.6353i		−	1.72765i	−0.503790	−	0.863826i	$-0.668061\pi$
$859$		−	50.6353i		−	1.72765i	0.503790	−	0.863826i	$-0.331939\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	55.8693i		−	1.90181i	−0.309477	−	0.950907i	$-0.600154\pi$
$863$		−	55.8693i		−	1.90181i	0.309477	−	0.950907i	$-0.399846\pi$
$864$	−7.31485	+	16.9201i	−0.248856	+	0.575635i
$865$		0			0
$866$		0			0
$867$		−	35.8872i		−	1.21879i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−39.4137	+	5.99466i	−1.33167	+	0.202541i
$877$	14.0000			0.472746			0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000			0.472746			0.236373	+	0.971662i	$0.424041\pi$
$878$	−14.6071	+	16.9968i	−0.492966	+	0.573614i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	10.9342	+	9.39688i	0.368173	+	0.316409i
$883$			28.7750i			0.968355i	0.874970	+	0.484178i	$0.160881\pi$
$883$			28.7750i			0.968355i	−0.874970	+	0.484178i	$0.839119\pi$
$884$		0			0
$885$		0			0
$886$	3.91838	−	4.55941i	0.131641	−	0.153176i
$887$		−	30.5372i		−	1.02534i	−0.858586	−	0.512669i	$-0.828657\pi$
$887$		−	30.5372i		−	1.02534i	0.858586	−	0.512669i	$-0.171343\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	56.8957	−	8.65358i	1.90501	−	0.289743i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−71.2406			−2.37866
$898$	19.3059	+	16.5916i	0.644247	+	0.553669i
$899$		−	37.4612i		−	1.24940i
$900$	−2.18988	−	14.3980i	−0.0729960	−	0.479935i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	50.1538	+	43.1024i	1.66625	+	1.43198i
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$	−8.73818			−0.289827
$910$		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
$921$	60.7443			2.00159
$922$	41.0478	+	35.2766i	1.35184	+	1.16177i
$923$			7.32539i			0.241118i
$924$		0			0
$925$		0			0
$926$	26.5235	−	30.8627i	0.871617	−	1.01421i
$927$		0			0
$928$	−20.5074	−	8.86567i	−0.673188	−	0.291030i
$929$	−38.9526			−1.27799			−0.638996	−	0.769210i	$-0.720650\pi$
$929$	−38.9526			−1.27799			−0.638996	+	0.769210i	$0.720650\pi$
$930$		0			0
$931$		0			0
$932$	−6.14364	−	40.3933i	−0.201242	−	1.32313i
$933$	−24.5406			−0.803423
$934$		0			0
$935$		0			0
$936$	−15.3721	+	24.5741i	−0.502454	+	0.803229i
$937$		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
$943$			60.0913i			1.95684i
$944$	11.4069	+	36.6317i	0.371264	+	1.19226i
$945$		0			0
$946$		0			0
$947$			61.1613i			1.98748i	0.111734	+	0.993738i	$0.464359\pi$
$947$			61.1613i			1.98748i	−0.111734	+	0.993738i	$0.535641\pi$
$948$		0			0
$949$	66.4457			2.15692
$950$		0			0
$951$		−	63.3303i		−	2.05363i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	52.1466	−	7.93127i	1.68654	−	0.256515i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−58.9663			−1.90214
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	28.5131i		−	0.916920i	−0.888715	−	0.458460i	$-0.848401\pi$
$967$		−	28.5131i		−	0.916920i	0.888715	−	0.458460i	$-0.151599\pi$
$968$	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	$0.5\pi$
$972$	−27.6202	+	4.20092i	−0.885920	+	0.134745i
$973$		0			0
$974$	32.0929	−	37.3432i	1.02832	−	1.19655i
$975$			74.2735i			2.37866i
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$	−57.2905	−	49.2357i	−1.83195	−	1.57439i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−33.0258	+	38.4287i	−1.05390	+	1.22631i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	63.4267	+	39.6761i	2.02197	+	1.26483i
$985$		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	$-0.848912\pi$
$991$		−	28.7750i		−	0.914068i	0.889449	−	0.457034i	$-0.151088\pi$
$992$	−21.2917	+	49.2502i	−0.676011	+	1.56370i
$993$	0.0611725			0.00194125
$994$		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	$0.635070\pi$
$998$	29.1341	−	33.9003i	0.922224	−	1.07310i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	92.2.b.b.91.6	yes	6
3.2	odd	2		828.2.e.b.91.1		6
4.3	odd	2	inner	92.2.b.b.91.5	✓	6
8.3	odd	2		1472.2.c.c.1471.2		6
8.5	even	2		1472.2.c.c.1471.5		6
12.11	even	2		828.2.e.b.91.2		6
23.22	odd	2	CM	92.2.b.b.91.6	yes	6
69.68	even	2		828.2.e.b.91.1		6
92.91	even	2	inner	92.2.b.b.91.5	✓	6
184.45	odd	2		1472.2.c.c.1471.5		6
184.91	even	2		1472.2.c.c.1471.2		6
276.275	odd	2		828.2.e.b.91.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.2.b.b.91.5	✓	6	4.3	odd	2	inner
92.2.b.b.91.5	✓	6	92.91	even	2	inner
92.2.b.b.91.6	yes	6	1.1	even	1	trivial
92.2.b.b.91.6	yes	6	23.22	odd	2	CM
828.2.e.b.91.1		6	3.2	odd	2
828.2.e.b.91.1		6	69.68	even	2
828.2.e.b.91.2		6	12.11	even	2
828.2.e.b.91.2		6	276.275	odd	2
1472.2.c.c.1471.2		6	8.3	odd	2
1472.2.c.c.1471.2		6	184.91	even	2
1472.2.c.c.1471.5		6	8.5	even	2
1472.2.c.c.1471.5		6	184.45	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(1.07255 + 0.921756i) q^{2} -2.11101i q^{3} +(0.300733 + 1.97726i) q^{4} +(1.94584 - 2.26417i) q^{6} +(-1.50000 + 2.39792i) q^{8} -1.45636 q^{9} +O(q^{10})\)	\(q+(1.07255 + 0.921756i) q^{2} -2.11101i q^{3} +(0.300733 + 1.97726i) q^{4} +(1.94584 - 2.26417i) q^{6} +(-1.50000 + 2.39792i) q^{8} -1.45636 q^{9} +(4.17402 - 0.634850i) q^{12} -7.03677 q^{13} +(-3.81912 + 1.18925i) q^{16} +(-1.56202 - 1.34241i) q^{18} +4.79583i q^{23} +(5.06202 + 3.16651i) q^{24} +5.00000 q^{25} +(-7.54730 - 6.48619i) q^{26} -3.25863i q^{27} +3.94950 q^{29} -9.48506i q^{31} +(-5.19240 - 2.24476i) q^{32} +(-0.437976 - 2.87961i) q^{36} +14.8547i q^{39} +12.5299 q^{41} +(-4.42059 + 5.14378i) q^{46} +13.7071i q^{47} +(2.51053 + 8.06220i) q^{48} -7.00000 q^{49} +(5.36276 + 4.60878i) q^{50} +(-2.11619 - 13.9135i) q^{52} +(3.00366 - 3.49505i) q^{54} +(4.23604 + 3.64047i) q^{58} -9.59166i q^{59} +(8.74290 - 10.1732i) q^{62} +(-3.50000 - 7.19375i) q^{64} +10.1240 q^{69} -1.04102i q^{71} +(2.18454 - 3.49224i) q^{72} -9.44264 q^{73} -10.5550i q^{75} +(-13.6924 + 15.9324i) q^{78} -11.2481 q^{81} +(13.4390 + 11.5495i) q^{82} -8.33743i q^{87} +(-9.48261 + 1.44226i) q^{92} -20.0230 q^{93} +(-12.6346 + 14.7015i) q^{94} +(-4.73871 + 10.9612i) q^{96} +(-7.50786 - 6.45229i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{6} - 9 q^{8} - 18 q^{9}+O(q^{10}) \) 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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