LMFDB - Embedded newform 4140.2.f.b.829.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 $ x^12 + 24x^10 + 188x^8 + 530x^6 + 508x^4 + 80*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.7
Root			$-1.26443i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.b.829.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.817027 - 2.08146i) q^{5} +4.41307i q^{7} +O(q^{10})$	$q+(0.817027 - 2.08146i) q^{5} +4.41307i q^{7} -2.29289 q^{11} -6.92936i q^{13} -1.51387i q^{17} -2.89920 q^{19} +1.00000i q^{23} +(-3.66493 - 3.40122i) q^{25} -7.68764 q^{29} +3.85746 q^{31} +(9.18562 + 3.60560i) q^{35} +8.62830i q^{37} +6.44324 q^{41} +3.48497i q^{43} +6.19747i q^{47} -12.4752 q^{49} +2.17710i q^{53} +(-1.87335 + 4.77255i) q^{55} -11.7637 q^{59} -5.11443 q^{61} +(-14.4232 - 5.66148i) q^{65} +9.94597i q^{67} -3.41407 q^{71} +8.95307i q^{73} -10.1187i q^{77} +1.92694 q^{79} +8.04131i q^{83} +(-3.15106 - 1.23687i) q^{85} -1.09273 q^{89} +30.5798 q^{91} +(-2.36872 + 6.03456i) q^{95} -16.9208i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) $ 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.817027	−	2.08146i	0.365386	−	0.930856i
$5$	0.817027	−	2.08146i	0.365386	−	0.930856i
$6$		0			0
$7$			4.41307i			1.66798i	0.551777	+	0.833992i	$0.313950\pi$
$7$			4.41307i			1.66798i	−0.551777	+	0.833992i	$0.686050\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.29289			−0.691332			−0.345666	−	0.938358i	$-0.612347\pi$
$11$	−2.29289			−0.691332			−0.345666	+	0.938358i	$0.612347\pi$
$12$		0			0
$13$		−	6.92936i		−	1.92186i	−0.276792	−	0.960930i	$-0.589271\pi$
$13$		−	6.92936i		−	1.92186i	0.276792	−	0.960930i	$-0.410729\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	1.51387i		−	0.367168i	−0.983004	−	0.183584i	$-0.941230\pi$
$17$		−	1.51387i		−	0.367168i	0.983004	−	0.183584i	$-0.0587699\pi$
$18$		0			0
$19$	−2.89920			−0.665121			−0.332561	−	0.943082i	$-0.607913\pi$
$19$	−2.89920			−0.665121			−0.332561	+	0.943082i	$0.607913\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−3.66493	−	3.40122i	−0.732987	−	0.680243i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−7.68764			−1.42756			−0.713779	−	0.700371i	$-0.753018\pi$
$29$	−7.68764			−1.42756			−0.713779	+	0.700371i	$0.753018\pi$
$30$		0			0
$31$	3.85746			0.692821			0.346410	−	0.938083i	$-0.387401\pi$
$31$	3.85746			0.692821			0.346410	+	0.938083i	$0.387401\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.18562	+	3.60560i	1.55265	+	0.609457i
$36$		0			0
$37$			8.62830i			1.41848i	0.704965	+	0.709242i	$0.250963\pi$
$37$			8.62830i			1.41848i	−0.704965	+	0.709242i	$0.749037\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.44324			1.00626			0.503132	−	0.864209i	$-0.332181\pi$
$41$	6.44324			1.00626			0.503132	+	0.864209i	$0.332181\pi$
$42$		0			0
$43$			3.48497i			0.531453i	0.964048	+	0.265727i	$0.0856119\pi$
$43$			3.48497i			0.531453i	−0.964048	+	0.265727i	$0.914388\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.19747i			0.903994i	0.892019	+	0.451997i	$0.149288\pi$
$47$			6.19747i			0.903994i	−0.892019	+	0.451997i	$0.850712\pi$
$48$		0			0
$49$	−12.4752			−1.78217
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2.17710i			0.299047i	0.988758	+	0.149524i	$0.0477740\pi$
$53$			2.17710i			0.299047i	−0.988758	+	0.149524i	$0.952226\pi$
$54$		0			0
$55$	−1.87335	+	4.77255i	−0.252603	+	0.643530i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.7637			−1.53150			−0.765750	−	0.643138i	$-0.777632\pi$
$59$	−11.7637			−1.53150			−0.765750	+	0.643138i	$0.777632\pi$
$60$		0			0
$61$	−5.11443			−0.654835			−0.327418	−	0.944880i	$-0.606178\pi$
$61$	−5.11443			−0.654835			−0.327418	+	0.944880i	$0.606178\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−14.4232	−	5.66148i	−1.78898	−	0.702220i
$66$		0			0
$67$			9.94597i			1.21509i	0.794284	+	0.607547i	$0.207846\pi$
$67$			9.94597i			1.21509i	−0.794284	+	0.607547i	$0.792154\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.41407			−0.405175			−0.202588	−	0.979264i	$-0.564935\pi$
$71$	−3.41407			−0.405175			−0.202588	+	0.979264i	$0.564935\pi$
$72$		0			0
$73$			8.95307i			1.04788i	0.851756	+	0.523939i	$0.175538\pi$
$73$			8.95307i			1.04788i	−0.851756	+	0.523939i	$0.824462\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	10.1187i		−	1.15313i
$78$		0			0
$79$	1.92694			0.216798			0.108399	−	0.994107i	$-0.465428\pi$
$79$	1.92694			0.216798			0.108399	+	0.994107i	$0.465428\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			8.04131i			0.882648i	0.897348	+	0.441324i	$0.145491\pi$
$83$			8.04131i			0.882648i	−0.897348	+	0.441324i	$0.854509\pi$
$84$		0			0
$85$	−3.15106	−	1.23687i	−0.341781	−	0.134158i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.09273			−0.115829			−0.0579147	−	0.998322i	$-0.518445\pi$
$89$	−1.09273			−0.115829			−0.0579147	+	0.998322i	$0.518445\pi$
$90$		0			0
$91$	30.5798			3.20563
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.36872	+	6.03456i	−0.243026	+	0.619132i
$96$		0			0
$97$		−	16.9208i		−	1.71805i	−0.511933	−	0.859026i	$-0.671070\pi$
$97$		−	16.9208i		−	1.71805i	0.511933	−	0.859026i	$-0.328930\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.9497			−1.28855			−0.644274	−	0.764795i	$-0.722840\pi$
$101$	−12.9497			−1.28855			−0.644274	+	0.764795i	$0.722840\pi$
$102$		0			0
$103$		−	10.9754i		−	1.08144i	−0.841202	−	0.540721i	$-0.818152\pi$
$103$		−	10.9754i		−	1.08144i	0.841202	−	0.540721i	$-0.181848\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19.8821i		−	1.92208i	−0.276411	−	0.961040i	$-0.589145\pi$
$107$		−	19.8821i		−	1.92208i	0.276411	−	0.961040i	$-0.410855\pi$
$108$		0			0
$109$	−0.427249			−0.0409231			−0.0204615	−	0.999791i	$-0.506514\pi$
$109$	−0.427249			−0.0409231			−0.0204615	+	0.999791i	$0.506514\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	5.38533i		−	0.506609i	−0.967387	−	0.253304i	$-0.918483\pi$
$113$		−	5.38533i		−	0.506609i	0.967387	−	0.253304i	$-0.0815174\pi$
$114$		0			0
$115$	2.08146	+	0.817027i	0.194097	+	0.0761882i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.68082			0.612430
$120$		0			0
$121$	−5.74267			−0.522061
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.0738	+	4.84952i	−0.901031	+	0.433754i
$126$		0			0
$127$			6.16014i			0.546624i	0.961925	+	0.273312i	$0.0881191\pi$
$127$			6.16014i			0.546624i	−0.961925	+	0.273312i	$0.911881\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−14.6325			−1.27845			−0.639225	−	0.769020i	$-0.720745\pi$
$131$	−14.6325			−1.27845			−0.639225	+	0.769020i	$0.720745\pi$
$132$		0			0
$133$		−	12.7944i		−	1.10941i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	0.972256i		−	0.0830654i	−0.999137	−	0.0415327i	$-0.986776\pi$
$137$		−	0.972256i		−	0.0830654i	0.999137	−	0.0415327i	$-0.0132241\pi$
$138$		0			0
$139$	−7.74312			−0.656763			−0.328382	−	0.944545i	$-0.606503\pi$
$139$	−7.74312			−0.656763			−0.328382	+	0.944545i	$0.606503\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			15.8883i			1.32864i
$144$		0			0
$145$	−6.28101	+	16.0015i	−0.521609	+	1.32885i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−17.6823			−1.44859			−0.724293	−	0.689492i	$-0.757834\pi$
$149$	−17.6823			−1.44859			−0.724293	+	0.689492i	$0.757834\pi$
$150$		0			0
$151$	−2.01286			−0.163804			−0.0819022	−	0.996640i	$-0.526100\pi$
$151$	−2.01286			−0.163804			−0.0819022	+	0.996640i	$0.526100\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.15165	−	8.02914i	0.253147	−	0.644916i
$156$		0			0
$157$		−	5.82703i		−	0.465048i	−0.972591	−	0.232524i	$-0.925302\pi$
$157$		−	5.82703i		−	0.465048i	0.972591	−	0.232524i	$-0.0746984\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.41307			−0.347799
$162$		0			0
$163$			6.75147i			0.528816i	0.964411	+	0.264408i	$0.0851765\pi$
$163$			6.75147i			0.528816i	−0.964411	+	0.264408i	$0.914823\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			23.4541i			1.81493i	0.420125	+	0.907466i	$0.361986\pi$
$167$			23.4541i			1.81493i	−0.420125	+	0.907466i	$0.638014\pi$
$168$		0			0
$169$	−35.0161			−2.69355
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	8.28850i		−	0.630163i	−0.949065	−	0.315081i	$-0.897968\pi$
$173$		−	8.28850i		−	0.630163i	0.949065	−	0.315081i	$-0.102032\pi$
$174$		0			0
$175$	15.0098	−	16.1736i	1.13463	−	1.22261i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.04002			0.0777345			0.0388673	−	0.999244i	$-0.487625\pi$
$179$	1.04002			0.0777345			0.0388673	+	0.999244i	$0.487625\pi$
$180$		0			0
$181$	2.71850			0.202065			0.101032	−	0.994883i	$-0.467785\pi$
$181$	2.71850			0.202065			0.101032	+	0.994883i	$0.467785\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	17.9594	+	7.04956i	1.32040	+	0.518294i
$186$		0			0
$187$			3.47114i			0.253835i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.04858			−0.582375			−0.291187	−	0.956666i	$-0.594050\pi$
$191$	−8.04858			−0.582375			−0.291187	+	0.956666i	$0.594050\pi$
$192$		0			0
$193$			16.0276i			1.15370i	0.816852	+	0.576848i	$0.195717\pi$
$193$			16.0276i			1.15370i	−0.816852	+	0.576848i	$0.804283\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.06316i		−	0.289488i	−0.989469	−	0.144744i	$-0.953764\pi$
$197$		−	4.06316i		−	0.289488i	0.989469	−	0.144744i	$-0.0462359\pi$
$198$		0			0
$199$	−18.7042			−1.32590			−0.662952	−	0.748662i	$-0.730697\pi$
$199$	−18.7042			−1.32590			−0.662952	+	0.748662i	$0.730697\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	33.9261i		−	2.38114i
$204$		0			0
$205$	5.26430	−	13.4113i	0.367675	−	0.936688i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.64753			0.459819
$210$		0			0
$211$	7.87839			0.542371			0.271185	−	0.962527i	$-0.412584\pi$
$211$	7.87839			0.542371			0.271185	+	0.962527i	$0.412584\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.25382	+	2.84732i	0.494707	+	0.194185i
$216$		0			0
$217$			17.0232i			1.15561i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−10.4902			−0.705645
$222$		0			0
$223$			2.29263i			0.153526i	0.997049	+	0.0767628i	$0.0244584\pi$
$223$			2.29263i			0.153526i	−0.997049	+	0.0767628i	$0.975542\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	12.8325i		−	0.851725i	−0.904788	−	0.425862i	$-0.859971\pi$
$227$		−	12.8325i		−	0.851725i	0.904788	−	0.425862i	$-0.140029\pi$
$228$		0			0
$229$	16.2528			1.07401			0.537007	−	0.843578i	$-0.319555\pi$
$229$	16.2528			1.07401			0.537007	+	0.843578i	$0.319555\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			21.5410i			1.41120i	0.708613	+	0.705598i	$0.249321\pi$
$233$			21.5410i			1.41120i	−0.708613	+	0.705598i	$0.750679\pi$
$234$		0			0
$235$	12.8998	+	5.06350i	0.841489	+	0.330307i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.0622			−0.715555			−0.357778	−	0.933807i	$-0.616465\pi$
$239$	−11.0622			−0.715555			−0.357778	+	0.933807i	$0.616465\pi$
$240$		0			0
$241$	18.2164			1.17342			0.586710	−	0.809797i	$-0.300423\pi$
$241$	18.2164			1.17342			0.586710	+	0.809797i	$0.300423\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.1926	+	25.9666i	−0.651179	+	1.65894i
$246$		0			0
$247$			20.0896i			1.27827i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.0182			0.632345			0.316172	−	0.948702i	$-0.397602\pi$
$251$	10.0182			0.632345			0.316172	+	0.948702i	$0.397602\pi$
$252$		0			0
$253$		−	2.29289i		−	0.144153i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			14.3021i			0.892141i	0.894998	+	0.446070i	$0.147177\pi$
$257$			14.3021i			0.892141i	−0.894998	+	0.446070i	$0.852823\pi$
$258$		0			0
$259$	−38.0773			−2.36601
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	9.44361i		−	0.582318i	−0.956675	−	0.291159i	$-0.905959\pi$
$263$		−	9.44361i		−	0.582318i	0.956675	−	0.291159i	$-0.0940409\pi$
$264$		0			0
$265$	4.53153	+	1.77875i	0.278370	+	0.109268i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	23.7630			1.44886			0.724428	−	0.689350i	$-0.242104\pi$
$269$	23.7630			1.44886			0.724428	+	0.689350i	$0.242104\pi$
$270$		0			0
$271$	−17.5939			−1.06875			−0.534375	−	0.845247i	$-0.679453\pi$
$271$	−17.5939			−1.06875			−0.534375	+	0.845247i	$0.679453\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.40328	+	7.79860i	0.506737	+	0.470273i
$276$		0			0
$277$		−	29.6582i		−	1.78199i	−0.454014	−	0.890995i	$-0.650008\pi$
$277$		−	29.6582i		−	1.78199i	0.454014	−	0.890995i	$-0.349992\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8.25484			0.492442			0.246221	−	0.969214i	$-0.420811\pi$
$281$	8.25484			0.492442			0.246221	+	0.969214i	$0.420811\pi$
$282$		0			0
$283$		−	9.21317i		−	0.547666i	−0.961777	−	0.273833i	$-0.911708\pi$
$283$		−	9.21317i		−	0.547666i	0.961777	−	0.273833i	$-0.0882916\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			28.4344i			1.67843i
$288$		0			0
$289$	14.7082			0.865188
$290$		0			0
$291$		0			0
$292$		0			0
$293$			11.1138i			0.649277i	0.945838	+	0.324639i	$0.105243\pi$
$293$			11.1138i			0.649277i	−0.945838	+	0.324639i	$0.894757\pi$
$294$		0			0
$295$	−9.61125	+	24.4856i	−0.559588	+	1.42561i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.92936			0.400735
$300$		0			0
$301$	−15.3794			−0.886455
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.17863	+	10.6455i	−0.239267	+	0.609558i
$306$		0			0
$307$			18.7800i			1.07183i	0.844272	+	0.535916i	$0.180033\pi$
$307$			18.7800i			1.07183i	−0.844272	+	0.535916i	$0.819967\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.51258			−0.142475			−0.0712377	−	0.997459i	$-0.522695\pi$
$311$	−2.51258			−0.142475			−0.0712377	+	0.997459i	$0.522695\pi$
$312$		0			0
$313$		−	19.9278i		−	1.12638i	−0.826326	−	0.563192i	$-0.809573\pi$
$313$		−	19.9278i		−	1.12638i	0.826326	−	0.563192i	$-0.190427\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			27.8849i			1.56617i	0.621912	+	0.783087i	$0.286356\pi$
$317$			27.8849i			1.56617i	−0.621912	+	0.783087i	$0.713644\pi$
$318$		0			0
$319$	17.6269			0.986916
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4.38901i			0.244211i
$324$		0			0
$325$	−23.5683	+	25.3957i	−1.30733	+	1.40870i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−27.3499			−1.50785
$330$		0			0
$331$	−28.5409			−1.56875			−0.784375	−	0.620287i	$-0.787016\pi$
$331$	−28.5409			−1.56875			−0.784375	+	0.620287i	$0.787016\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	20.7021	+	8.12612i	1.13108	+	0.443978i
$336$		0			0
$337$			3.71371i			0.202298i	0.994871	+	0.101149i	$0.0322520\pi$
$337$			3.71371i			0.202298i	−0.994871	+	0.101149i	$0.967748\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−8.84472			−0.478969
$342$		0			0
$343$		−	24.1624i		−	1.30464i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	14.1459i		−	0.759393i	−0.925111	−	0.379696i	$-0.876028\pi$
$347$		−	14.1459i		−	0.759393i	0.925111	−	0.379696i	$-0.123972\pi$
$348$		0			0
$349$	21.2802			1.13910			0.569550	−	0.821957i	$-0.307117\pi$
$349$	21.2802			1.13910			0.569550	+	0.821957i	$0.307117\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			24.4641i			1.30209i	0.759038	+	0.651046i	$0.225669\pi$
$353$			24.4641i			1.30209i	−0.759038	+	0.651046i	$0.774331\pi$
$354$		0			0
$355$	−2.78939	+	7.10624i	−0.148045	+	0.377160i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−23.0252			−1.21522			−0.607612	−	0.794234i	$-0.707872\pi$
$359$	−23.0252			−1.21522			−0.607612	+	0.794234i	$0.707872\pi$
$360$		0			0
$361$	−10.5947			−0.557613
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.6354	+	7.31490i	0.975424	+	0.382880i
$366$		0			0
$367$			0.478057i			0.0249544i	0.999922	+	0.0124772i	$0.00397171\pi$
$367$			0.478057i			0.0249544i	−0.999922	+	0.0124772i	$0.996028\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.60767			−0.498806
$372$		0			0
$373$		−	27.3508i		−	1.41617i	−0.706127	−	0.708085i	$-0.749559\pi$
$373$		−	27.3508i		−	1.41617i	0.706127	−	0.708085i	$-0.250441\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			53.2704i			2.74357i
$378$		0			0
$379$	7.35358			0.377728			0.188864	−	0.982003i	$-0.439519\pi$
$379$	7.35358			0.377728			0.188864	+	0.982003i	$0.439519\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.34108i		−	0.170721i	−0.996350	−	0.0853606i	$-0.972796\pi$
$383$		−	3.34108i		−	0.170721i	0.996350	−	0.0853606i	$-0.0272042\pi$
$384$		0			0
$385$	−21.0616	−	8.26723i	−1.07340	−	0.421337i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.366568			0.0185858			0.00929288	−	0.999957i	$-0.497042\pi$
$389$	0.366568			0.0185858			0.00929288	+	0.999957i	$0.497042\pi$
$390$		0			0
$391$	1.51387			0.0765598
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.57436	−	4.01085i	0.0792148	−	0.201808i
$396$		0			0
$397$		−	11.3222i		−	0.568245i	−0.958788	−	0.284122i	$-0.908298\pi$
$397$		−	11.3222i		−	0.568245i	0.958788	−	0.284122i	$-0.0917022\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−37.1673			−1.85605			−0.928024	−	0.372520i	$-0.878494\pi$
$401$	−37.1673			−1.85605			−0.928024	+	0.372520i	$0.878494\pi$
$402$		0			0
$403$		−	26.7298i		−	1.33150i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	19.7837i		−	0.980643i
$408$		0			0
$409$	13.7745			0.681107			0.340553	−	0.940225i	$-0.389386\pi$
$409$	13.7745			0.681107			0.340553	+	0.940225i	$0.389386\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	51.9139i		−	2.55452i
$414$		0			0
$415$	16.7376	+	6.56997i	0.821619	+	0.322507i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.50726			0.366753			0.183377	−	0.983043i	$-0.441297\pi$
$419$	7.50726			0.366753			0.183377	+	0.983043i	$0.441297\pi$
$420$		0			0
$421$	−7.65685			−0.373172			−0.186586	−	0.982439i	$-0.559742\pi$
$421$	−7.65685			−0.373172			−0.186586	+	0.982439i	$0.559742\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.14900	+	5.54824i	−0.249763	+	0.269129i
$426$		0			0
$427$		−	22.5703i		−	1.09225i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.22764			−0.299975			−0.149987	−	0.988688i	$-0.547923\pi$
$431$	−6.22764			−0.299975			−0.149987	+	0.988688i	$0.547923\pi$
$432$		0			0
$433$		−	0.704087i		−	0.0338362i	−0.999857	−	0.0169181i	$-0.994615\pi$
$433$		−	0.704087i		−	0.0338362i	0.999857	−	0.0169181i	$-0.00538546\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	2.89920i		−	0.138687i
$438$		0			0
$439$	34.8570			1.66363			0.831816	−	0.555052i	$-0.187301\pi$
$439$	34.8570			1.66363			0.831816	+	0.555052i	$0.187301\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	24.8165i		−	1.17907i	−0.807745	−	0.589533i	$-0.799312\pi$
$443$		−	24.8165i		−	1.17907i	0.807745	−	0.589533i	$-0.200688\pi$
$444$		0			0
$445$	−0.892792	+	2.27448i	−0.0423224	+	0.107820i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.77738			0.319844			0.159922	−	0.987130i	$-0.448876\pi$
$449$	6.77738			0.319844			0.159922	+	0.987130i	$0.448876\pi$
$450$		0			0
$451$	−14.7736			−0.695662
$452$		0			0
$453$		0			0
$454$		0			0
$455$	24.9845	−	63.6505i	1.17129	−	2.98398i
$456$		0			0
$457$			1.47169i			0.0688429i	0.999407	+	0.0344214i	$0.0109588\pi$
$457$			1.47169i			0.0688429i	−0.999407	+	0.0344214i	$0.989041\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.8037			−1.20180			−0.600899	−	0.799325i	$-0.705191\pi$
$461$	−25.8037			−1.20180			−0.600899	+	0.799325i	$0.705191\pi$
$462$		0			0
$463$			15.3469i			0.713231i	0.934251	+	0.356616i	$0.116069\pi$
$463$			15.3469i			0.713231i	−0.934251	+	0.356616i	$0.883931\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	10.2933i		−	0.476315i	−0.971227	−	0.238157i	$-0.923457\pi$
$467$		−	10.2933i		−	0.476315i	0.971227	−	0.238157i	$-0.0765434\pi$
$468$		0			0
$469$	−43.8922			−2.02676
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	7.99065i		−	0.367410i
$474$		0			0
$475$	10.6254	+	9.86079i	0.487525	+	0.452444i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.50453			0.205817			0.102909	−	0.994691i	$-0.467185\pi$
$479$	4.50453			0.205817			0.102909	+	0.994691i	$0.467185\pi$
$480$		0			0
$481$	59.7886			2.72613
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−35.2200	−	13.8248i	−1.59926	−	0.627751i
$486$		0			0
$487$		−	27.2669i		−	1.23558i	−0.786343	−	0.617790i	$-0.788028\pi$
$487$		−	27.2669i		−	1.23558i	0.786343	−	0.617790i	$-0.211972\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−24.4866			−1.10507			−0.552533	−	0.833491i	$-0.686339\pi$
$491$	−24.4866			−1.10507			−0.552533	+	0.833491i	$0.686339\pi$
$492$		0			0
$493$			11.6381i			0.524154i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	15.0665i		−	0.675826i
$498$		0			0
$499$	24.1892			1.08286			0.541429	−	0.840746i	$-0.317883\pi$
$499$	24.1892			1.08286			0.541429	+	0.840746i	$0.317883\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			30.6230i			1.36541i	0.730693	+	0.682706i	$0.239197\pi$
$503$			30.6230i			1.36541i	−0.730693	+	0.682706i	$0.760803\pi$
$504$		0			0
$505$	−10.5803	+	26.9543i	−0.470817	+	1.19945i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.56826			−0.113836			−0.0569181	−	0.998379i	$-0.518127\pi$
$509$	−2.56826			−0.113836			−0.0569181	+	0.998379i	$0.518127\pi$
$510$		0			0
$511$	−39.5105			−1.74784
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−22.8449	−	8.96722i	−1.00667	−	0.395143i
$516$		0			0
$517$		−	14.2101i		−	0.624960i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	44.1355			1.93361			0.966807	−	0.255509i	$-0.0822432\pi$
$521$	44.1355			1.93361			0.966807	+	0.255509i	$0.0822432\pi$
$522$		0			0
$523$		−	42.2884i		−	1.84914i	−0.381008	−	0.924572i	$-0.624423\pi$
$523$		−	42.2884i		−	1.84914i	0.381008	−	0.924572i	$-0.375577\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	5.83970i		−	0.254381i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	44.6475i		−	1.93390i
$534$		0			0
$535$	−41.3838	−	16.2443i	−1.78918	−	0.702300i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	28.6042			1.23207
$540$		0			0
$541$	10.2776			0.441871			0.220935	−	0.975288i	$-0.429089\pi$
$541$	10.2776			0.441871			0.220935	+	0.975288i	$0.429089\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.349074	+	0.889302i	−0.0149527	+	0.0380935i
$546$		0			0
$547$		−	3.87305i		−	0.165600i	−0.996566	−	0.0827998i	$-0.973614\pi$
$547$		−	3.87305i		−	0.165600i	0.996566	−	0.0827998i	$-0.0263862\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.2880			0.949500
$552$		0			0
$553$			8.50373i			0.361615i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			9.62364i			0.407767i	0.978995	+	0.203883i	$0.0653563\pi$
$557$			9.62364i			0.407767i	−0.978995	+	0.203883i	$0.934644\pi$
$558$		0			0
$559$	24.1486			1.02138
$560$		0			0
$561$		0			0
$562$		0			0
$563$			3.35865i			0.141550i	0.997492	+	0.0707751i	$0.0225473\pi$
$563$			3.35865i			0.141550i	−0.997492	+	0.0707751i	$0.977453\pi$
$564$		0			0
$565$	−11.2093	−	4.39996i	−0.471580	−	0.185108i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4.26939			−0.178982			−0.0894910	−	0.995988i	$-0.528524\pi$
$569$	−4.26939			−0.178982			−0.0894910	+	0.995988i	$0.528524\pi$
$570$		0			0
$571$	16.4567			0.688689			0.344345	−	0.938843i	$-0.388101\pi$
$571$	16.4567			0.688689			0.344345	+	0.938843i	$0.388101\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	3.40122	−	3.66493i	0.141840	−	0.152838i
$576$		0			0
$577$			1.98780i			0.0827530i	0.999144	+	0.0413765i	$0.0131743\pi$
$577$			1.98780i			0.0827530i	−0.999144	+	0.0413765i	$0.986826\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−35.4869			−1.47224
$582$		0			0
$583$		−	4.99184i		−	0.206741i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			28.9720i			1.19580i	0.801570	+	0.597900i	$0.203998\pi$
$587$			28.9720i			1.19580i	−0.801570	+	0.597900i	$0.796002\pi$
$588$		0			0
$589$	−11.1835			−0.460810
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	19.4799i		−	0.799944i	−0.916527	−	0.399972i	$-0.869020\pi$
$593$		−	19.4799i		−	0.799944i	0.916527	−	0.399972i	$-0.130980\pi$
$594$		0			0
$595$	5.45841	−	13.9059i	0.223773	−	0.570084i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	37.8442			1.54627			0.773136	−	0.634240i	$-0.218687\pi$
$599$	37.8442			1.54627			0.773136	+	0.634240i	$0.218687\pi$
$600$		0			0
$601$	−20.2347			−0.825389			−0.412695	−	0.910869i	$-0.635412\pi$
$601$	−20.2347			−0.825389			−0.412695	+	0.910869i	$0.635412\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.69191	+	11.9531i	−0.190753	+	0.485963i
$606$		0			0
$607$			24.4075i			0.990670i	0.868702	+	0.495335i	$0.164955\pi$
$607$			24.4075i			0.990670i	−0.868702	+	0.495335i	$0.835045\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	42.9445			1.73735
$612$		0			0
$613$		−	10.6964i		−	0.432025i	−0.976391	−	0.216012i	$-0.930695\pi$
$613$		−	10.6964i		−	0.432025i	0.976391	−	0.216012i	$-0.0693051\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			27.4738i			1.10605i	0.833164	+	0.553026i	$0.186527\pi$
$617$			27.4738i			1.10605i	−0.833164	+	0.553026i	$0.813473\pi$
$618$		0			0
$619$	0.389330			0.0156485			0.00782425	−	0.999969i	$-0.497509\pi$
$619$	0.389330			0.0156485			0.00782425	+	0.999969i	$0.497509\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	4.82230i		−	0.193201i
$624$		0			0
$625$	1.86347	+	24.9305i	0.0745389	+	0.997218i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13.0621			0.520822
$630$		0			0
$631$	−11.9330			−0.475047			−0.237523	−	0.971382i	$-0.576336\pi$
$631$	−11.9330			−0.475047			−0.237523	+	0.971382i	$0.576336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12.8221	+	5.03300i	0.508828	+	0.199729i
$636$		0			0
$637$			86.4451i			3.42508i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−22.8525			−0.902621			−0.451311	−	0.892367i	$-0.649043\pi$
$641$	−22.8525			−0.902621			−0.451311	+	0.892367i	$0.649043\pi$
$642$		0			0
$643$		−	34.9279i		−	1.37742i	−0.725036	−	0.688711i	$-0.758177\pi$
$643$		−	34.9279i		−	1.37742i	0.725036	−	0.688711i	$-0.241823\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	26.3738i		−	1.03686i	−0.855120	−	0.518430i	$-0.826517\pi$
$647$		−	26.3738i		−	1.03686i	0.855120	−	0.518430i	$-0.173483\pi$
$648$		0			0
$649$	26.9728			1.05877
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	2.17022i		−	0.0849274i	−0.999098	−	0.0424637i	$-0.986479\pi$
$653$		−	2.17022i		−	0.0849274i	0.999098	−	0.0424637i	$-0.0135207\pi$
$654$		0			0
$655$	−11.9552	+	30.4570i	−0.467127	+	1.19005i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	19.7820			0.770596			0.385298	−	0.922792i	$-0.374099\pi$
$659$	19.7820			0.770596			0.385298	+	0.922792i	$0.374099\pi$
$660$		0			0
$661$	45.7505			1.77949			0.889743	−	0.456461i	$-0.150883\pi$
$661$	45.7505			1.77949			0.889743	+	0.456461i	$0.150883\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−26.6309	−	10.4533i	−1.03270	−	0.405363i
$666$		0			0
$667$		−	7.68764i		−	0.297666i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.7268			0.452708
$672$		0			0
$673$		−	6.99940i		−	0.269807i	−0.990859	−	0.134904i	$-0.956928\pi$
$673$		−	6.99940i		−	0.269807i	0.990859	−	0.134904i	$-0.0430725\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	26.7178i		−	1.02685i	−0.858134	−	0.513425i	$-0.828376\pi$
$677$		−	26.7178i		−	1.02685i	0.858134	−	0.513425i	$-0.171624\pi$
$678$		0			0
$679$	74.6728			2.86568
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	19.8947i		−	0.761248i	−0.924730	−	0.380624i	$-0.875709\pi$
$683$		−	19.8947i		−	0.761248i	0.924730	−	0.380624i	$-0.124291\pi$
$684$		0			0
$685$	−2.02371	−	0.794359i	−0.0773219	−	0.0303509i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	15.0859			0.574727
$690$		0			0
$691$	1.24438			0.0473385			0.0236693	−	0.999720i	$-0.492465\pi$
$691$	1.24438			0.0473385			0.0236693	+	0.999720i	$0.492465\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.32634	+	16.1170i	−0.239972	+	0.611352i
$696$		0			0
$697$		−	9.75424i		−	0.369468i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29.8454			1.12724			0.563622	−	0.826033i	$-0.309408\pi$
$701$	29.8454			1.12724			0.563622	+	0.826033i	$0.309408\pi$
$702$		0			0
$703$		−	25.0151i		−	0.943464i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	57.1481i		−	2.14928i
$708$		0			0
$709$	−25.2255			−0.947362			−0.473681	−	0.880697i	$-0.657075\pi$
$709$	−25.2255			−0.947362			−0.473681	+	0.880697i	$0.657075\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3.85746i			0.144463i
$714$		0			0
$715$	33.0707	+	12.9811i	1.23678	+	0.485467i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−23.7787			−0.886797			−0.443398	−	0.896325i	$-0.646227\pi$
$719$	−23.7787			−0.886797			−0.443398	+	0.896325i	$0.646227\pi$
$720$		0			0
$721$	48.4353			1.80383
$722$		0			0
$723$		0			0
$724$		0			0
$725$	28.1747	+	26.1473i	1.04638	+	0.971086i
$726$		0			0
$727$			36.5849i			1.35686i	0.734666	+	0.678429i	$0.237339\pi$
$727$			36.5849i			1.35686i	−0.734666	+	0.678429i	$0.762661\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.27580			0.195133
$732$		0			0
$733$			21.7593i			0.803698i	0.915706	+	0.401849i	$0.131632\pi$
$733$			21.7593i			0.803698i	−0.915706	+	0.401849i	$0.868368\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	22.8050i		−	0.840032i
$738$		0			0
$739$	−2.72303			−0.100168			−0.0500842	−	0.998745i	$-0.515949\pi$
$739$	−2.72303			−0.100168			−0.0500842	+	0.998745i	$0.515949\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	32.0161i		−	1.17456i	−0.809385	−	0.587278i	$-0.800200\pi$
$743$		−	32.0161i		−	1.17456i	0.809385	−	0.587278i	$-0.199800\pi$
$744$		0			0
$745$	−14.4469	+	36.8049i	−0.529293	+	1.34843i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	87.7413			3.20600
$750$		0			0
$751$	−36.0949			−1.31712			−0.658561	−	0.752527i	$-0.728835\pi$
$751$	−36.0949			−1.31712			−0.658561	+	0.752527i	$0.728835\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.64456	+	4.18969i	−0.0598517	+	0.152478i
$756$		0			0
$757$		−	46.8188i		−	1.70166i	−0.525442	−	0.850830i	$-0.676100\pi$
$757$		−	46.8188i		−	1.70166i	0.525442	−	0.850830i	$-0.323900\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	28.8182			1.04466			0.522330	−	0.852744i	$-0.325063\pi$
$761$	28.8182			1.04466			0.522330	+	0.852744i	$0.325063\pi$
$762$		0			0
$763$		−	1.88548i		−	0.0682590i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			81.5148i			2.94333i
$768$		0			0
$769$	−51.6243			−1.86162			−0.930811	−	0.365501i	$-0.880898\pi$
$769$	−51.6243			−1.86162			−0.930811	+	0.365501i	$0.880898\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			25.9610i			0.933753i	0.884322	+	0.466877i	$0.154621\pi$
$773$			25.9610i			0.933753i	−0.884322	+	0.466877i	$0.845379\pi$
$774$		0			0
$775$	−14.1373	−	13.1201i	−0.507828	−	0.471286i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−18.6802			−0.669288
$780$		0			0
$781$	7.82807			0.280110
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−12.1287	−	4.76084i	−0.432893	−	0.169922i
$786$		0			0
$787$			53.2258i			1.89730i	0.316333	+	0.948648i	$0.397548\pi$
$787$			53.2258i			1.89730i	−0.316333	+	0.948648i	$0.602452\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	23.7658			0.845015
$792$		0			0
$793$			35.4397i			1.25850i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	31.1874i		−	1.10471i	−0.833608	−	0.552357i	$-0.813729\pi$
$797$		−	31.1874i		−	1.10471i	0.833608	−	0.552357i	$-0.186271\pi$
$798$		0			0
$799$	9.38218			0.331918
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	20.5284i		−	0.724431i
$804$		0			0
$805$	−3.60560	+	9.18562i	−0.127081	+	0.323750i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.60251			−0.0563414			−0.0281707	−	0.999603i	$-0.508968\pi$
$809$	−1.60251			−0.0563414			−0.0281707	+	0.999603i	$0.508968\pi$
$810$		0			0
$811$	−36.9545			−1.29765			−0.648824	−	0.760938i	$-0.724739\pi$
$811$	−36.9545			−1.29765			−0.648824	+	0.760938i	$0.724739\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.0529	+	5.51613i	0.492251	+	0.193222i
$816$		0			0
$817$		−	10.1036i		−	0.353481i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	16.5583			0.577890			0.288945	−	0.957346i	$-0.406696\pi$
$821$	16.5583			0.577890			0.288945	+	0.957346i	$0.406696\pi$
$822$		0			0
$823$			7.63201i			0.266035i	0.991114	+	0.133018i	$0.0424667\pi$
$823$			7.63201i			0.266035i	−0.991114	+	0.133018i	$0.957533\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			19.1969i			0.667543i	0.942654	+	0.333772i	$0.108321\pi$
$827$			19.1969i			0.667543i	−0.942654	+	0.333772i	$0.891679\pi$
$828$		0			0
$829$	6.32413			0.219646			0.109823	−	0.993951i	$-0.464972\pi$
$829$	6.32413			0.219646			0.109823	+	0.993951i	$0.464972\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			18.8858i			0.654355i
$834$		0			0
$835$	48.8187	+	19.1626i	1.68944	+	0.663150i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	23.1158			0.798046			0.399023	−	0.916941i	$-0.369349\pi$
$839$	23.1158			0.798046			0.399023	+	0.916941i	$0.369349\pi$
$840$		0			0
$841$	30.0997			1.03792
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−28.6091	+	72.8845i	−0.984183	+	2.50730i
$846$		0			0
$847$		−	25.3428i		−	0.870789i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.62830			−0.295774
$852$		0			0
$853$			37.7049i			1.29099i	0.763764	+	0.645496i	$0.223349\pi$
$853$			37.7049i			1.29099i	−0.763764	+	0.645496i	$0.776651\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	19.7133i		−	0.673395i	−0.941613	−	0.336697i	$-0.890690\pi$
$857$		−	19.7133i		−	0.673395i	0.941613	−	0.336697i	$-0.109310\pi$
$858$		0			0
$859$	−2.00609			−0.0684471			−0.0342235	−	0.999414i	$-0.510896\pi$
$859$	−2.00609			−0.0684471			−0.0342235	+	0.999414i	$0.510896\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			16.8127i			0.572312i	0.958183	+	0.286156i	$0.0923775\pi$
$863$			16.8127i			0.572312i	−0.958183	+	0.286156i	$0.907623\pi$
$864$		0			0
$865$	−17.2522	−	6.77193i	−0.586591	−	0.230252i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−4.41826			−0.149879
$870$		0			0
$871$	68.9192			2.33524
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−21.4013	−	44.4565i	−0.723495	−	1.50291i
$876$		0			0
$877$			14.8922i			0.502874i	0.967874	+	0.251437i	$0.0809031\pi$
$877$			14.8922i			0.502874i	−0.967874	+	0.251437i	$0.919097\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	16.5218			0.556632			0.278316	−	0.960490i	$-0.410224\pi$
$881$	16.5218			0.556632			0.278316	+	0.960490i	$0.410224\pi$
$882$		0			0
$883$		−	38.7254i		−	1.30321i	−0.758557	−	0.651607i	$-0.774095\pi$
$883$		−	38.7254i		−	1.30321i	0.758557	−	0.651607i	$-0.225905\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1.16364i		−	0.0390711i	−0.999809	−	0.0195355i	$-0.993781\pi$
$887$		−	1.16364i		−	0.0390711i	0.999809	−	0.0195355i	$-0.00621875\pi$
$888$		0			0
$889$	−27.1851			−0.911760
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	17.9677i		−	0.601266i
$894$		0			0
$895$	0.849722	−	2.16475i	0.0284031	−	0.0723597i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−29.6548			−0.989042
$900$		0			0
$901$	3.29584			0.109800
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.22109	−	5.65845i	0.0738316	−	0.188093i
$906$		0			0
$907$			30.2796i			1.00542i	0.864456	+	0.502709i	$0.167663\pi$
$907$			30.2796i			1.00542i	−0.864456	+	0.502709i	$0.832337\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	28.2296			0.935290			0.467645	−	0.883916i	$-0.345103\pi$
$911$	28.2296			0.935290			0.467645	+	0.883916i	$0.345103\pi$
$912$		0			0
$913$		−	18.4378i		−	0.610203i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	64.5744i		−	2.13243i
$918$		0			0
$919$	−3.28150			−0.108247			−0.0541233	−	0.998534i	$-0.517236\pi$
$919$	−3.28150			−0.108247			−0.0541233	+	0.998534i	$0.517236\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			23.6573i			0.778690i
$924$		0			0
$925$	29.3467	−	31.6221i	0.964914	−	1.03973i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.64745			0.250905			0.125452	−	0.992100i	$-0.459962\pi$
$929$	7.64745			0.250905			0.125452	+	0.992100i	$0.459962\pi$
$930$		0			0
$931$	36.1680			1.18536
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.22503	+	2.83601i	0.236284	+	0.0927476i
$936$		0			0
$937$		−	4.95226i		−	0.161783i	−0.996723	−	0.0808917i	$-0.974223\pi$
$937$		−	4.95226i		−	0.161783i	0.996723	−	0.0808917i	$-0.0257768\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.62677			0.281225			0.140612	−	0.990065i	$-0.455093\pi$
$941$	8.62677			0.281225			0.140612	+	0.990065i	$0.455093\pi$
$942$		0			0
$943$			6.44324i			0.209821i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			46.6807i			1.51692i	0.651720	+	0.758460i	$0.274048\pi$
$947$			46.6807i			1.51692i	−0.651720	+	0.758460i	$0.725952\pi$
$948$		0			0
$949$	62.0391			2.01387
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	12.2343i		−	0.396307i	−0.980171	−	0.198154i	$-0.936505\pi$
$953$		−	12.2343i		−	0.396307i	0.980171	−	0.198154i	$-0.0634945\pi$
$954$		0			0
$955$	−6.57591	+	16.7528i	−0.212791	+	0.542107i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.29063			0.138552
$960$		0			0
$961$	−16.1200			−0.520000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	33.3609	+	13.0950i	1.07392	+	0.421544i
$966$		0			0
$967$		−	54.8318i		−	1.76327i	−0.471932	−	0.881635i	$-0.656443\pi$
$967$		−	54.8318i		−	1.76327i	0.471932	−	0.881635i	$-0.343557\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−40.3804			−1.29587			−0.647935	−	0.761696i	$-0.724367\pi$
$971$	−40.3804			−1.29587			−0.647935	+	0.761696i	$0.724367\pi$
$972$		0			0
$973$		−	34.1709i		−	1.09547i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			10.6700i			0.341363i	0.985326	+	0.170681i	$0.0545968\pi$
$977$			10.6700i			0.341363i	−0.985326	+	0.170681i	$0.945403\pi$
$978$		0			0
$979$	2.50551			0.0800765
$980$		0			0
$981$		0			0
$982$		0			0
$983$			6.27812i			0.200241i	0.994975	+	0.100120i	$0.0319228\pi$
$983$			6.27812i			0.200241i	−0.994975	+	0.100120i	$0.968077\pi$
$984$		0			0
$985$	−8.45730	−	3.31971i	−0.269472	−	0.105775i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−3.48497			−0.110816
$990$		0			0
$991$	61.2864			1.94683			0.973413	−	0.229057i	$-0.0735643\pi$
$991$	61.2864			1.94683			0.973413	+	0.229057i	$0.0735643\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−15.2818	+	38.9320i	−0.484466	+	1.23423i
$996$		0			0
$997$			39.9799i			1.26618i	0.774080	+	0.633088i	$0.218213\pi$
$997$			39.9799i			1.26618i	−0.774080	+	0.633088i	$0.781787\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.b.829.7		12
3.2	odd	2		460.2.c.a.369.10	yes	12
5.4	even	2	inner	4140.2.f.b.829.8		12
12.11	even	2		1840.2.e.f.369.3		12
15.2	even	4		2300.2.a.n.1.6		6
15.8	even	4		2300.2.a.o.1.1		6
15.14	odd	2		460.2.c.a.369.3	✓	12
60.23	odd	4		9200.2.a.cx.1.6		6
60.47	odd	4		9200.2.a.cy.1.1		6
60.59	even	2		1840.2.e.f.369.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.3	✓	12	15.14	odd	2
460.2.c.a.369.10	yes	12	3.2	odd	2
1840.2.e.f.369.3		12	12.11	even	2
1840.2.e.f.369.10		12	60.59	even	2
2300.2.a.n.1.6		6	15.2	even	4
2300.2.a.o.1.1		6	15.8	even	4
4140.2.f.b.829.7		12	1.1	even	1	trivial
4140.2.f.b.829.8		12	5.4	even	2	inner
9200.2.a.cx.1.6		6	60.23	odd	4
9200.2.a.cy.1.1		6	60.47	odd	4

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.817027 - 2.08146i) q^{5} +4.41307i q^{7} +O(q^{10})\)	\(q+(0.817027 - 2.08146i) q^{5} +4.41307i q^{7} -2.29289 q^{11} -6.92936i q^{13} -1.51387i q^{17} -2.89920 q^{19} +1.00000i q^{23} +(-3.66493 - 3.40122i) q^{25} -7.68764 q^{29} +3.85746 q^{31} +(9.18562 + 3.60560i) q^{35} +8.62830i q^{37} +6.44324 q^{41} +3.48497i q^{43} +6.19747i q^{47} -12.4752 q^{49} +2.17710i q^{53} +(-1.87335 + 4.77255i) q^{55} -11.7637 q^{59} -5.11443 q^{61} +(-14.4232 - 5.66148i) q^{65} +9.94597i q^{67} -3.41407 q^{71} +8.95307i q^{73} -10.1187i q^{77} +1.92694 q^{79} +8.04131i q^{83} +(-3.15106 - 1.23687i) q^{85} -1.09273 q^{89} +30.5798 q^{91} +(-2.36872 + 6.03456i) q^{95} -16.9208i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) \) 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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