LMFDB - Embedded newform 4140.2.f.b.829.10

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			829.10
Root			$3.08006i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.b.829.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.89824 + 1.18182i) q^{5} +0.992530i q^{7} +O(q^{10})$	$q+(1.89824 + 1.18182i) q^{5} +0.992530i q^{7} -1.83236 q^{11} +3.28666i q^{13} -6.63631i q^{17} +5.64378 q^{19} +1.00000i q^{23} +(2.20661 + 4.48675i) q^{25} +2.01596 q^{29} -0.315080 q^{31} +(-1.17299 + 1.88406i) q^{35} +3.07470i q^{37} +1.34964 q^{41} -5.97905i q^{43} +0.306285i q^{47} +6.01489 q^{49} +6.98500i q^{53} +(-3.47826 - 2.16552i) q^{55} +9.49533 q^{59} +5.56160 q^{61} +(-3.88424 + 6.23887i) q^{65} -0.853521i q^{67} -0.797419 q^{71} +7.67189i q^{73} -1.81867i q^{77} +3.62884 q^{79} +17.1966i q^{83} +(7.84291 - 12.5973i) q^{85} -7.01565 q^{89} -3.26211 q^{91} +(10.7132 + 6.66992i) q^{95} +18.5807i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.89824	+	1.18182i	0.848917	+	0.528526i
$5$	1.89824	+	1.18182i	0.848917	+	0.528526i
$6$		0			0
$7$			0.992530i			0.375141i	0.982251	+	0.187570i	$0.0600613\pi$
$7$			0.992530i			0.375141i	−0.982251	+	0.187570i	$0.939939\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.83236			−0.552478			−0.276239	−	0.961089i	$-0.589088\pi$
$11$	−1.83236			−0.552478			−0.276239	+	0.961089i	$0.589088\pi$
$12$		0			0
$13$			3.28666i			0.911556i	0.890093	+	0.455778i	$0.150639\pi$
$13$			3.28666i			0.911556i	−0.890093	+	0.455778i	$0.849361\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	6.63631i		−	1.60954i	−0.593586	−	0.804770i	$-0.702288\pi$
$17$		−	6.63631i		−	1.60954i	0.593586	−	0.804770i	$-0.297712\pi$
$18$		0			0
$19$	5.64378			1.29477			0.647386	−	0.762163i	$-0.275862\pi$
$19$	5.64378			1.29477			0.647386	+	0.762163i	$0.275862\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	2.20661	+	4.48675i	0.441321	+	0.897349i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.01596			0.374354			0.187177	−	0.982326i	$-0.440066\pi$
$29$	2.01596			0.374354			0.187177	+	0.982326i	$0.440066\pi$
$30$		0			0
$31$	−0.315080			−0.0565901			−0.0282951	−	0.999600i	$-0.509008\pi$
$31$	−0.315080			−0.0565901			−0.0282951	+	0.999600i	$0.509008\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.17299	+	1.88406i	−0.198272	+	0.318464i
$36$		0			0
$37$			3.07470i			0.505478i	0.967534	+	0.252739i	$0.0813314\pi$
$37$			3.07470i			0.505478i	−0.967534	+	0.252739i	$0.918669\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.34964			0.210779			0.105389	−	0.994431i	$-0.466391\pi$
$41$	1.34964			0.210779			0.105389	+	0.994431i	$0.466391\pi$
$42$		0			0
$43$		−	5.97905i		−	0.911797i	−0.890032	−	0.455899i	$-0.849318\pi$
$43$		−	5.97905i		−	0.911797i	0.890032	−	0.455899i	$-0.150682\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			0.306285i			0.0446762i	0.999750	+	0.0223381i	$0.00711103\pi$
$47$			0.306285i			0.0446762i	−0.999750	+	0.0223381i	$0.992889\pi$
$48$		0			0
$49$	6.01489			0.859269
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.98500i			0.959464i	0.877415	+	0.479732i	$0.159266\pi$
$53$			6.98500i			0.959464i	−0.877415	+	0.479732i	$0.840734\pi$
$54$		0			0
$55$	−3.47826	−	2.16552i	−0.469008	−	0.291999i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.49533			1.23619			0.618094	−	0.786105i	$-0.287905\pi$
$59$	9.49533			1.23619			0.618094	+	0.786105i	$0.287905\pi$
$60$		0			0
$61$	5.56160			0.712090			0.356045	−	0.934469i	$-0.384125\pi$
$61$	5.56160			0.712090			0.356045	+	0.934469i	$0.384125\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.88424	+	6.23887i	−0.481781	+	0.773836i
$66$		0			0
$67$		−	0.853521i		−	0.104274i	−0.998640	−	0.0521371i	$-0.983397\pi$
$67$		−	0.853521i		−	0.104274i	0.998640	−	0.0521371i	$-0.0166033\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.797419			−0.0946363			−0.0473181	−	0.998880i	$-0.515067\pi$
$71$	−0.797419			−0.0946363			−0.0473181	+	0.998880i	$0.515067\pi$
$72$		0			0
$73$			7.67189i			0.897926i	0.893550	+	0.448963i	$0.148207\pi$
$73$			7.67189i			0.897926i	−0.893550	+	0.448963i	$0.851793\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1.81867i		−	0.207257i
$78$		0			0
$79$	3.62884			0.408276			0.204138	−	0.978942i	$-0.434561\pi$
$79$	3.62884			0.408276			0.204138	+	0.978942i	$0.434561\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			17.1966i			1.88757i	0.330560	+	0.943785i	$0.392762\pi$
$83$			17.1966i			1.88757i	−0.330560	+	0.943785i	$0.607238\pi$
$84$		0			0
$85$	7.84291	−	12.5973i	0.850683	−	1.36637i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−7.01565			−0.743658			−0.371829	−	0.928301i	$-0.621269\pi$
$89$	−7.01565			−0.743658			−0.371829	+	0.928301i	$0.621269\pi$
$90$		0			0
$91$	−3.26211			−0.341962
$92$		0			0
$93$		0			0
$94$		0			0
$95$	10.7132	+	6.66992i	1.09915	+	0.684320i
$96$		0			0
$97$			18.5807i			1.88659i	0.331958	+	0.943294i	$0.392291\pi$
$97$			18.5807i			1.88659i	−0.331958	+	0.943294i	$0.607709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.96538			0.792585			0.396293	−	0.918124i	$-0.370297\pi$
$101$	7.96538			0.792585			0.396293	+	0.918124i	$0.370297\pi$
$102$		0			0
$103$		−	16.8268i		−	1.65799i	−0.559253	−	0.828997i	$-0.688912\pi$
$103$		−	16.8268i		−	1.65799i	0.559253	−	0.828997i	$-0.311088\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			10.7429i			1.03855i	0.854606	+	0.519276i	$0.173798\pi$
$107$			10.7429i			1.03855i	−0.854606	+	0.519276i	$0.826202\pi$
$108$		0			0
$109$	−8.34117			−0.798939			−0.399470	−	0.916746i	$-0.630806\pi$
$109$	−8.34117			−0.798939			−0.399470	+	0.916746i	$0.630806\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.28008i			0.778925i	0.921042	+	0.389462i	$0.127339\pi$
$113$			8.28008i			0.778925i	−0.921042	+	0.389462i	$0.872661\pi$
$114$		0			0
$115$	−1.18182	+	1.89824i	−0.110205	+	0.177012i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.58673			0.603805
$120$		0			0
$121$	−7.64245			−0.694768
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.11386	+	11.1247i	−0.0996265	+	0.995025i
$126$		0			0
$127$			10.5531i			0.936437i	0.883613	+	0.468219i	$0.155104\pi$
$127$			10.5531i			0.936437i	−0.883613	+	0.468219i	$0.844896\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.84966			−0.685828			−0.342914	−	0.939367i	$-0.611414\pi$
$131$	−7.84966			−0.685828			−0.342914	+	0.939367i	$0.611414\pi$
$132$		0			0
$133$			5.60161i			0.485722i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			9.27261i			0.792213i	0.918205	+	0.396106i	$0.129639\pi$
$137$			9.27261i			0.792213i	−0.918205	+	0.396106i	$0.870361\pi$
$138$		0			0
$139$	−18.5293			−1.57163			−0.785816	−	0.618461i	$-0.787757\pi$
$139$	−18.5293			−1.57163			−0.785816	+	0.618461i	$0.787757\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	6.02236i		−	0.503615i
$144$		0			0
$145$	3.82677	+	2.38250i	0.317796	+	0.197856i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	23.6227			1.93525			0.967624	−	0.252395i	$-0.0812183\pi$
$149$	23.6227			1.93525			0.967624	+	0.252395i	$0.0812183\pi$
$150$		0			0
$151$	−19.2472			−1.56631			−0.783157	−	0.621825i	$-0.786392\pi$
$151$	−19.2472			−1.56631			−0.783157	+	0.621825i	$0.786392\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.598097	−	0.372368i	−0.0480403	−	0.0299093i
$156$		0			0
$157$		−	6.61180i		−	0.527679i	−0.964567	−	0.263840i	$-0.915011\pi$
$157$		−	6.61180i		−	0.527679i	0.964567	−	0.263840i	$-0.0849890\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.992530			−0.0782223
$162$		0			0
$163$		−	11.6157i		−	0.909809i	−0.890540	−	0.454905i	$-0.849673\pi$
$163$		−	11.6157i		−	0.909809i	0.890540	−	0.454905i	$-0.150327\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.4280i		−	1.42600i	−0.701162	−	0.713002i	$-0.747335\pi$
$167$		−	18.4280i		−	1.42600i	0.701162	−	0.713002i	$-0.252665\pi$
$168$		0			0
$169$	2.19784			0.169065
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	5.51955i		−	0.419644i	−0.977740	−	0.209822i	$-0.932712\pi$
$173$		−	5.51955i		−	0.419644i	0.977740	−	0.209822i	$-0.0672884\pi$
$174$		0			0
$175$	−4.45323	+	2.19012i	−0.336632	+	0.165558i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.73640			−0.727733			−0.363866	−	0.931451i	$-0.618544\pi$
$179$	−9.73640			−0.727733			−0.363866	+	0.931451i	$0.618544\pi$
$180$		0			0
$181$	21.8277			1.62244			0.811220	−	0.584741i	$-0.198804\pi$
$181$	21.8277			1.62244			0.811220	+	0.584741i	$0.198804\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.63374	+	5.83652i	−0.267158	+	0.429109i
$186$		0			0
$187$			12.1601i			0.889235i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.4807			−1.19250			−0.596251	−	0.802798i	$-0.703344\pi$
$191$	−16.4807			−1.19250			−0.596251	+	0.802798i	$0.703344\pi$
$192$		0			0
$193$			4.60541i			0.331504i	0.986167	+	0.165752i	$0.0530052\pi$
$193$			4.60541i			0.331504i	−0.986167	+	0.165752i	$0.946995\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	0.157186i		−	0.0111990i	−0.999984	−	0.00559951i	$-0.998218\pi$
$197$		−	0.157186i		−	0.0111990i	0.999984	−	0.00559951i	$-0.00178239\pi$
$198$		0			0
$199$	1.94253			0.137702			0.0688511	−	0.997627i	$-0.478067\pi$
$199$	1.94253			0.137702			0.0688511	+	0.997627i	$0.478067\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			2.00090i			0.140436i
$204$		0			0
$205$	2.56194	+	1.59503i	0.178934	+	0.111402i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.3414			−0.715332
$210$		0			0
$211$	23.5602			1.62195			0.810976	−	0.585079i	$-0.198937\pi$
$211$	23.5602			1.62195			0.810976	+	0.585079i	$0.198937\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.06616	−	11.3497i	0.481908	−	0.774040i
$216$		0			0
$217$		−	0.312727i		−	0.0212293i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	21.8113			1.46719
$222$		0			0
$223$		−	9.59356i		−	0.642432i	−0.947006	−	0.321216i	$-0.895908\pi$
$223$		−	9.59356i		−	0.642432i	0.947006	−	0.321216i	$-0.104092\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	9.90765i		−	0.657594i	−0.944401	−	0.328797i	$-0.893357\pi$
$227$		−	9.90765i		−	0.657594i	0.944401	−	0.328797i	$-0.106643\pi$
$228$		0			0
$229$	4.90156			0.323904			0.161952	−	0.986799i	$-0.448221\pi$
$229$	4.90156			0.323904			0.161952	+	0.986799i	$0.448221\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	15.3123i		−	1.00314i	−0.865116	−	0.501571i	$-0.832755\pi$
$233$		−	15.3123i		−	1.00314i	0.865116	−	0.501571i	$-0.167245\pi$
$234$		0			0
$235$	−0.361973	+	0.581401i	−0.0236125	+	0.0379264i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.88065			−0.251018			−0.125509	−	0.992092i	$-0.540056\pi$
$239$	−3.88065			−0.251018			−0.125509	+	0.992092i	$0.540056\pi$
$240$		0			0
$241$	6.64578			0.428092			0.214046	−	0.976824i	$-0.431336\pi$
$241$	6.64578			0.428092			0.214046	+	0.976824i	$0.431336\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.4177	+	7.10851i	0.729449	+	0.454146i
$246$		0			0
$247$			18.5492i			1.18026i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.82636			−0.367756			−0.183878	−	0.982949i	$-0.558865\pi$
$251$	−5.82636			−0.367756			−0.183878	+	0.982949i	$0.558865\pi$
$252$		0			0
$253$		−	1.83236i		−	0.115200i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			17.1979i			1.07278i	0.843971	+	0.536388i	$0.180212\pi$
$257$			17.1979i			1.07278i	−0.843971	+	0.536388i	$0.819788\pi$
$258$		0			0
$259$	−3.05173			−0.189626
$260$		0			0
$261$		0			0
$262$		0			0
$263$			19.5353i			1.20460i	0.798269	+	0.602301i	$0.205749\pi$
$263$			19.5353i			1.20460i	−0.798269	+	0.602301i	$0.794251\pi$
$264$		0			0
$265$	−8.25501	+	13.2592i	−0.507101	+	0.814506i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.2275			−1.29426			−0.647132	−	0.762378i	$-0.724032\pi$
$269$	−21.2275			−1.29426			−0.647132	+	0.762378i	$0.724032\pi$
$270$		0			0
$271$	9.19621			0.558630			0.279315	−	0.960200i	$-0.409893\pi$
$271$	9.19621			0.558630			0.279315	+	0.960200i	$0.409893\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.04330	−	8.22134i	−0.243820	−	0.495765i
$276$		0			0
$277$			2.77325i			0.166628i	0.996523	+	0.0833141i	$0.0265505\pi$
$277$			2.77325i			0.166628i	−0.996523	+	0.0833141i	$0.973450\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	32.7971			1.95651			0.978255	−	0.207407i	$-0.0665025\pi$
$281$	32.7971			1.95651			0.978255	+	0.207407i	$0.0665025\pi$
$282$		0			0
$283$		−	23.6017i		−	1.40298i	−0.712680	−	0.701489i	$-0.752519\pi$
$283$		−	23.6017i		−	1.40298i	0.712680	−	0.701489i	$-0.247481\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			1.33956i			0.0790717i
$288$		0			0
$289$	−27.0406			−1.59062
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.83939i		−	0.107458i	−0.998556	−	0.0537291i	$-0.982889\pi$
$293$		−	1.83939i		−	0.107458i	0.998556	−	0.0537291i	$-0.0171107\pi$
$294$		0			0
$295$	18.0244	+	11.2218i	1.04942	+	0.653356i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.28666			−0.190073
$300$		0			0
$301$	5.93439			0.342052
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.5572	+	6.57281i	0.604506	+	0.376358i
$306$		0			0
$307$			10.3956i			0.593310i	0.954985	+	0.296655i	$0.0958711\pi$
$307$			10.3956i			0.593310i	−0.954985	+	0.296655i	$0.904129\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.2967			0.697281			0.348640	−	0.937257i	$-0.386643\pi$
$311$	12.2967			0.697281			0.348640	+	0.937257i	$0.386643\pi$
$312$		0			0
$313$			17.2954i			0.977592i	0.872398	+	0.488796i	$0.162564\pi$
$313$			17.2954i			0.977592i	−0.872398	+	0.488796i	$0.837436\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			33.6362i			1.88920i	0.328227	+	0.944599i	$0.393549\pi$
$317$			33.6362i			1.88920i	−0.328227	+	0.944599i	$0.606451\pi$
$318$		0			0
$319$	−3.69397			−0.206822
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	37.4538i		−	2.08399i
$324$		0			0
$325$	−14.7464	+	7.25237i	−0.817984	+	0.402289i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.303997			−0.0167599
$330$		0			0
$331$	−9.81704			−0.539593			−0.269797	−	0.962917i	$-0.586956\pi$
$331$	−9.81704			−0.539593			−0.269797	+	0.962917i	$0.586956\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.00871	−	1.62018i	0.0551115	−	0.0885201i
$336$		0			0
$337$			9.15644i			0.498783i	0.968403	+	0.249392i	$0.0802306\pi$
$337$			9.15644i			0.498783i	−0.968403	+	0.249392i	$0.919769\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.577341			0.0312648
$342$		0			0
$343$			12.9177i			0.697488i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			32.5921i			1.74963i	0.484453	+	0.874817i	$0.339019\pi$
$347$			32.5921i			1.74963i	−0.484453	+	0.874817i	$0.660981\pi$
$348$		0			0
$349$	15.5701			0.833448			0.416724	−	0.909033i	$-0.363178\pi$
$349$	15.5701			0.833448			0.416724	+	0.909033i	$0.363178\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	10.9313i		−	0.581816i	−0.956751	−	0.290908i	$-0.906043\pi$
$353$		−	10.9313i		−	0.581816i	0.956751	−	0.290908i	$-0.0939573\pi$
$354$		0			0
$355$	−1.51369	−	0.942405i	−0.0803384	−	0.0500177i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.405120			0.0213814			0.0106907	−	0.999943i	$-0.496597\pi$
$359$	0.405120			0.0213814			0.0106907	+	0.999943i	$0.496597\pi$
$360$		0			0
$361$	12.8522			0.676432
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.06678	+	14.5631i	−0.474577	+	0.762265i
$366$		0			0
$367$		−	7.26443i		−	0.379200i	−0.981861	−	0.189600i	$-0.939281\pi$
$367$		−	7.26443i		−	0.379200i	0.981861	−	0.189600i	$-0.0607191\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.93282			−0.359934
$372$		0			0
$373$		−	3.36770i		−	0.174373i	−0.996192	−	0.0871866i	$-0.972212\pi$
$373$		−	3.36770i		−	0.174373i	0.996192	−	0.0871866i	$-0.0277876\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.62578i			0.341245i
$378$		0			0
$379$	4.54534			0.233478			0.116739	−	0.993163i	$-0.462756\pi$
$379$	4.54534			0.233478			0.116739	+	0.993163i	$0.462756\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			12.0978i			0.618167i	0.951035	+	0.309084i	$0.100022\pi$
$383$			12.0978i			0.618167i	−0.951035	+	0.309084i	$0.899978\pi$
$384$		0			0
$385$	2.14934	−	3.45227i	0.109541	−	0.175944i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.8934			1.16074			0.580371	−	0.814352i	$-0.302908\pi$
$389$	22.8934			1.16074			0.580371	+	0.814352i	$0.302908\pi$
$390$		0			0
$391$	6.63631			0.335612
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.88839	+	4.28863i	0.346593	+	0.215784i
$396$		0			0
$397$		−	23.9748i		−	1.20326i	−0.798774	−	0.601631i	$-0.794518\pi$
$397$		−	23.9748i		−	1.20326i	0.798774	−	0.601631i	$-0.205482\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−26.8722			−1.34193			−0.670967	−	0.741487i	$-0.734121\pi$
$401$	−26.8722			−1.34193			−0.670967	+	0.741487i	$0.734121\pi$
$402$		0			0
$403$		−	1.03556i		−	0.0515851i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5.63397i		−	0.279266i
$408$		0			0
$409$	−15.7840			−0.780467			−0.390233	−	0.920716i	$-0.627606\pi$
$409$	−15.7840			−0.780467			−0.390233	+	0.920716i	$0.627606\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			9.42440i			0.463744i
$414$		0			0
$415$	−20.3232	+	32.6432i	−0.997629	+	1.60239i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	19.6999			0.962402			0.481201	−	0.876610i	$-0.340201\pi$
$419$	19.6999			0.962402			0.481201	+	0.876610i	$0.340201\pi$
$420$		0			0
$421$	−15.6913			−0.764748			−0.382374	−	0.924008i	$-0.624893\pi$
$421$	−15.6913			−0.764748			−0.382374	+	0.924008i	$0.624893\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	29.7754	−	14.6437i	1.44432	−	0.710325i
$426$		0			0
$427$			5.52005i			0.267134i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.33660			0.0643819			0.0321909	−	0.999482i	$-0.489752\pi$
$431$	1.33660			0.0643819			0.0321909	+	0.999482i	$0.489752\pi$
$432$		0			0
$433$		−	4.26379i		−	0.204905i	−0.994738	−	0.102452i	$-0.967331\pi$
$433$		−	4.26379i		−	0.204905i	0.994738	−	0.102452i	$-0.0326689\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5.64378i			0.269978i
$438$		0			0
$439$	32.6899			1.56020			0.780101	−	0.625653i	$-0.215167\pi$
$439$	32.6899			1.56020			0.780101	+	0.625653i	$0.215167\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	16.8373i		−	0.799964i	−0.916523	−	0.399982i	$-0.869016\pi$
$443$		−	16.8373i		−	0.799964i	0.916523	−	0.399982i	$-0.130984\pi$
$444$		0			0
$445$	−13.3174	−	8.29123i	−0.631304	−	0.393042i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.8502			0.606440			0.303220	−	0.952921i	$-0.401938\pi$
$449$	12.8502			0.606440			0.303220	+	0.952921i	$0.401938\pi$
$450$		0			0
$451$	−2.47303			−0.116451
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.19226	−	3.85522i	−0.290298	−	0.180736i
$456$		0			0
$457$		−	40.1900i		−	1.88001i	−0.341161	−	0.940005i	$-0.610820\pi$
$457$		−	40.1900i		−	1.88001i	0.341161	−	0.940005i	$-0.389180\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	34.7208			1.61711			0.808554	−	0.588422i	$-0.200250\pi$
$461$	34.7208			1.61711			0.808554	+	0.588422i	$0.200250\pi$
$462$		0			0
$463$		−	26.7292i		−	1.24221i	−0.783727	−	0.621106i	$-0.786684\pi$
$463$		−	26.7292i		−	1.24221i	0.783727	−	0.621106i	$-0.213316\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			17.6579i			0.817112i	0.912733	+	0.408556i	$0.133968\pi$
$467$			17.6579i			0.817112i	−0.912733	+	0.408556i	$0.866032\pi$
$468$		0			0
$469$	0.847144			0.0391175
$470$		0			0
$471$		0			0
$472$		0			0
$473$			10.9558i			0.503748i
$474$		0			0
$475$	12.4536	+	25.3222i	0.571410	+	1.16186i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−37.8085			−1.72752			−0.863758	−	0.503907i	$-0.831896\pi$
$479$	−37.8085			−1.72752			−0.863758	+	0.503907i	$0.831896\pi$
$480$		0			0
$481$	−10.1055			−0.460772
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−21.9591	+	35.2706i	−0.997110	+	1.60156i
$486$		0			0
$487$		−	27.9951i		−	1.26858i	−0.773096	−	0.634289i	$-0.781293\pi$
$487$		−	27.9951i		−	1.26858i	0.773096	−	0.634289i	$-0.218707\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10.1212			−0.456762			−0.228381	−	0.973572i	$-0.573343\pi$
$491$	−10.1212			−0.456762			−0.228381	+	0.973572i	$0.573343\pi$
$492$		0			0
$493$		−	13.3785i		−	0.602538i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	0.791462i		−	0.0355019i
$498$		0			0
$499$	37.0240			1.65742			0.828712	−	0.559676i	$-0.189074\pi$
$499$	37.0240			1.65742			0.828712	+	0.559676i	$0.189074\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	22.5700i		−	1.00635i	−0.864185	−	0.503174i	$-0.832166\pi$
$503$		−	22.5700i		−	1.00635i	0.864185	−	0.503174i	$-0.167834\pi$
$504$		0			0
$505$	15.1202	+	9.41364i	0.672839	+	0.418902i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	29.3212			1.29964			0.649819	−	0.760089i	$-0.274845\pi$
$509$	29.3212			1.29964			0.649819	+	0.760089i	$0.274845\pi$
$510$		0			0
$511$	−7.61457			−0.336849
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.8862	−	31.9413i	0.876292	−	1.40750i
$516$		0			0
$517$		−	0.561224i		−	0.0246826i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.9057			−0.609220			−0.304610	−	0.952477i	$-0.598526\pi$
$521$	−13.9057			−0.609220			−0.304610	+	0.952477i	$0.598526\pi$
$522$		0			0
$523$		−	36.6591i		−	1.60299i	−0.598001	−	0.801496i	$-0.704038\pi$
$523$		−	36.6591i		−	1.60299i	0.598001	−	0.801496i	$-0.295962\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			2.09097i			0.0910841i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			4.43582i			0.192137i
$534$		0			0
$535$	−12.6961	+	20.3925i	−0.548902	+	0.881645i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−11.0214			−0.474727
$540$		0			0
$541$	15.8082			0.679647			0.339823	−	0.940489i	$-0.389633\pi$
$541$	15.8082			0.679647			0.339823	+	0.940489i	$0.389633\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.8335	−	9.85776i	−0.678233	−	0.422260i
$546$		0			0
$547$			25.6129i			1.09513i	0.836764	+	0.547563i	$0.184444\pi$
$547$			25.6129i			1.09513i	−0.836764	+	0.547563i	$0.815556\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	11.3776			0.484703
$552$		0			0
$553$			3.60173i			0.153161i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	21.9292i		−	0.929172i	−0.885528	−	0.464586i	$-0.846203\pi$
$557$		−	21.9292i		−	0.929172i	0.885528	−	0.464586i	$-0.153797\pi$
$558$		0			0
$559$	19.6511			0.831154
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	34.2718i		−	1.44439i	−0.691692	−	0.722193i	$-0.743134\pi$
$563$		−	34.2718i		−	1.44439i	0.691692	−	0.722193i	$-0.256866\pi$
$564$		0			0
$565$	−9.78556	+	15.7176i	−0.411682	+	0.661243i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	4.05998			0.170203			0.0851015	−	0.996372i	$-0.472879\pi$
$569$	4.05998			0.170203			0.0851015	+	0.996372i	$0.472879\pi$
$570$		0			0
$571$	−21.7533			−0.910347			−0.455174	−	0.890403i	$-0.650423\pi$
$571$	−21.7533			−0.910347			−0.455174	+	0.890403i	$0.650423\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.48675	+	2.20661i	−0.187110	+	0.0920219i
$576$		0			0
$577$		−	23.6094i		−	0.982872i	−0.870914	−	0.491436i	$-0.836472\pi$
$577$		−	23.6094i		−	0.982872i	0.870914	−	0.491436i	$-0.163528\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−17.0681			−0.708105
$582$		0			0
$583$		−	12.7991i		−	0.530083i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	13.2321i		−	0.546146i	−0.961993	−	0.273073i	$-0.911960\pi$
$587$		−	13.2321i		−	0.546146i	0.961993	−	0.273073i	$-0.0880400\pi$
$588$		0			0
$589$	−1.77824			−0.0732713
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.85971i		−	0.363825i	−0.983315	−	0.181912i	$-0.941771\pi$
$593$		−	8.85971i		−	0.363825i	0.983315	−	0.181912i	$-0.0582287\pi$
$594$		0			0
$595$	12.5032	+	7.78432i	0.512580	+	0.319126i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.01552			0.327505			0.163753	−	0.986501i	$-0.447640\pi$
$599$	8.01552			0.327505			0.163753	+	0.986501i	$0.447640\pi$
$600$		0			0
$601$	24.3621			0.993752			0.496876	−	0.867821i	$-0.334480\pi$
$601$	24.3621			0.993752			0.496876	+	0.867821i	$0.334480\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−14.5072	−	9.03199i	−0.589801	−	0.367203i
$606$		0			0
$607$		−	36.5642i		−	1.48409i	−0.670348	−	0.742047i	$-0.733855\pi$
$607$		−	36.5642i		−	1.48409i	0.670348	−	0.742047i	$-0.266145\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.00665			−0.0407249
$612$		0			0
$613$		−	9.21498i		−	0.372190i	−0.982532	−	0.186095i	$-0.940417\pi$
$613$		−	9.21498i		−	0.372190i	0.982532	−	0.186095i	$-0.0595832\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			36.4154i			1.46603i	0.680213	+	0.733015i	$0.261887\pi$
$617$			36.4154i			1.46603i	−0.680213	+	0.733015i	$0.738113\pi$
$618$		0			0
$619$	−34.3781			−1.38177			−0.690886	−	0.722963i	$-0.742780\pi$
$619$	−34.3781			−1.38177			−0.690886	+	0.722963i	$0.742780\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	6.96324i		−	0.278976i
$624$		0			0
$625$	−15.2618	+	19.8010i	−0.610471	+	0.792039i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	20.4047			0.813588
$630$		0			0
$631$	42.3309			1.68517			0.842584	−	0.538565i	$-0.181033\pi$
$631$	42.3309			1.68517			0.842584	+	0.538565i	$0.181033\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.4719	+	20.0323i	−0.494931	+	0.794958i
$636$		0			0
$637$			19.7689i			0.783272i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−43.2608			−1.70870			−0.854349	−	0.519699i	$-0.826044\pi$
$641$	−43.2608			−1.70870			−0.854349	+	0.519699i	$0.826044\pi$
$642$		0			0
$643$			2.36283i			0.0931809i	0.998914	+	0.0465905i	$0.0148356\pi$
$643$			2.36283i			0.0931809i	−0.998914	+	0.0465905i	$0.985164\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	21.9473i		−	0.862837i	−0.902152	−	0.431418i	$-0.858013\pi$
$647$		−	21.9473i		−	0.862837i	0.902152	−	0.431418i	$-0.141987\pi$
$648$		0			0
$649$	−17.3989			−0.682966
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	34.7170i		−	1.35858i	−0.733869	−	0.679291i	$-0.762287\pi$
$653$		−	34.7170i		−	1.35858i	0.733869	−	0.679291i	$-0.237713\pi$
$654$		0			0
$655$	−14.9005	−	9.27687i	−0.582211	−	0.362477i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23.1858			−0.903192			−0.451596	−	0.892223i	$-0.649145\pi$
$659$	−23.1858			−0.903192			−0.451596	+	0.892223i	$0.649145\pi$
$660$		0			0
$661$	−39.6148			−1.54084			−0.770418	−	0.637539i	$-0.779953\pi$
$661$	−39.6148			−1.54084			−0.770418	+	0.637539i	$0.779953\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.62010	+	10.6332i	−0.256716	+	0.412338i
$666$		0			0
$667$			2.01596i			0.0780583i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.1909			−0.393414
$672$		0			0
$673$			30.3084i			1.16830i	0.811645	+	0.584152i	$0.198573\pi$
$673$			30.3084i			1.16830i	−0.811645	+	0.584152i	$0.801427\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	9.26419i		−	0.356052i	−0.984026	−	0.178026i	$-0.943029\pi$
$677$		−	9.26419i		−	0.356052i	0.984026	−	0.178026i	$-0.0569711\pi$
$678$		0			0
$679$	−18.4419			−0.707736
$680$		0			0
$681$		0			0
$682$		0			0
$683$			22.9835i			0.879439i	0.898135	+	0.439720i	$0.144922\pi$
$683$			22.9835i			0.879439i	−0.898135	+	0.439720i	$0.855078\pi$
$684$		0			0
$685$	−10.9585	+	17.6016i	−0.418705	+	0.672523i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−22.9574			−0.874606
$690$		0			0
$691$	−27.2916			−1.03822			−0.519111	−	0.854707i	$-0.673737\pi$
$691$	−27.2916			−1.03822			−0.519111	+	0.854707i	$0.673737\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−35.1729	−	21.8982i	−1.33419	−	0.830648i
$696$		0			0
$697$		−	8.95664i		−	0.339257i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−24.1308			−0.911407			−0.455704	−	0.890132i	$-0.650612\pi$
$701$	−24.1308			−0.911407			−0.455704	+	0.890132i	$0.650612\pi$
$702$		0			0
$703$			17.3529i			0.654479i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			7.90588i			0.297331i
$708$		0			0
$709$	15.0156			0.563922			0.281961	−	0.959426i	$-0.409015\pi$
$709$	15.0156			0.563922			0.281961	+	0.959426i	$0.409015\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	0.315080i		−	0.0117999i
$714$		0			0
$715$	7.11733	−	11.4319i	0.266173	−	0.427527i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	29.4211			1.09722			0.548611	−	0.836078i	$-0.315157\pi$
$719$	29.4211			1.09722			0.548611	+	0.836078i	$0.315157\pi$
$720$		0			0
$721$	16.7011			0.621981
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4.44843	+	9.04510i	0.165211	+	0.335926i
$726$		0			0
$727$			47.8025i			1.77290i	0.462829	+	0.886448i	$0.346834\pi$
$727$			47.8025i			1.77290i	−0.462829	+	0.886448i	$0.653166\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−39.6788			−1.46757
$732$		0			0
$733$		−	44.0763i		−	1.62800i	−0.580867	−	0.813998i	$-0.697286\pi$
$733$		−	44.0763i		−	1.62800i	0.580867	−	0.813998i	$-0.302714\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.56396i			0.0576091i
$738$		0			0
$739$	−3.75235			−0.138032			−0.0690162	−	0.997616i	$-0.521986\pi$
$739$	−3.75235			−0.138032			−0.0690162	+	0.997616i	$0.521986\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	19.6859i		−	0.722205i	−0.932526	−	0.361103i	$-0.882400\pi$
$743$		−	19.6859i		−	0.722205i	0.932526	−	0.361103i	$-0.117600\pi$
$744$		0			0
$745$	44.8415	+	27.9178i	1.64287	+	1.02283i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−10.6626			−0.389604
$750$		0			0
$751$	−26.3594			−0.961868			−0.480934	−	0.876757i	$-0.659702\pi$
$751$	−26.3594			−0.961868			−0.480934	+	0.876757i	$0.659702\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−36.5357	−	22.7467i	−1.32967	−	0.827837i
$756$		0			0
$757$			11.9270i			0.433493i	0.976228	+	0.216747i	$0.0695446\pi$
$757$			11.9270i			0.433493i	−0.976228	+	0.216747i	$0.930455\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.03123			−0.0373821			−0.0186911	−	0.999825i	$-0.505950\pi$
$761$	−1.03123			−0.0373821			−0.0186911	+	0.999825i	$0.505950\pi$
$762$		0			0
$763$		−	8.27886i		−	0.299715i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			31.2080i			1.12685i
$768$		0			0
$769$	3.37170			0.121587			0.0607933	−	0.998150i	$-0.480637\pi$
$769$	3.37170			0.121587			0.0607933	+	0.998150i	$0.480637\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	33.0031i		−	1.18704i	−0.804819	−	0.593520i	$-0.797738\pi$
$773$		−	33.0031i		−	1.18704i	0.804819	−	0.593520i	$-0.202262\pi$
$774$		0			0
$775$	−0.695259	−	1.41369i	−0.0249744	−	0.0507811i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.61708			0.272910
$780$		0			0
$781$	1.46116			0.0522844
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.81395	−	12.5508i	0.278892	−	0.447956i
$786$		0			0
$787$		−	7.29525i		−	0.260048i	−0.991511	−	0.130024i	$-0.958495\pi$
$787$		−	7.29525i		−	0.260048i	0.991511	−	0.130024i	$-0.0415054\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8.21823			−0.292206
$792$		0			0
$793$			18.2791i			0.649110i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	32.1730i		−	1.13963i	−0.821775	−	0.569813i	$-0.807016\pi$
$797$		−	32.1730i		−	1.13963i	0.821775	−	0.569813i	$-0.192984\pi$
$798$		0			0
$799$	2.03260			0.0719082
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	14.0577i		−	0.496084i
$804$		0			0
$805$	−1.88406	−	1.17299i	−0.0664043	−	0.0413425i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15.6881			−0.551564			−0.275782	−	0.961220i	$-0.588937\pi$
$809$	−15.6881			−0.551564			−0.275782	+	0.961220i	$0.588937\pi$
$810$		0			0
$811$	−52.8629			−1.85627			−0.928133	−	0.372249i	$-0.878587\pi$
$811$	−52.8629			−1.85627			−0.928133	+	0.372249i	$0.878587\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13.7276	−	22.0493i	0.480857	−	0.772353i
$816$		0			0
$817$		−	33.7444i		−	1.18057i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	10.7411			0.374866			0.187433	−	0.982277i	$-0.439983\pi$
$821$	10.7411			0.374866			0.187433	+	0.982277i	$0.439983\pi$
$822$		0			0
$823$		−	50.9828i		−	1.77715i	−0.458733	−	0.888574i	$-0.651697\pi$
$823$		−	50.9828i		−	1.77715i	0.458733	−	0.888574i	$-0.348303\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	29.8663i		−	1.03855i	−0.854607	−	0.519276i	$-0.826202\pi$
$827$		−	29.8663i		−	1.03855i	0.854607	−	0.519276i	$-0.173798\pi$
$828$		0			0
$829$	25.5010			0.885685			0.442842	−	0.896599i	$-0.353970\pi$
$829$	25.5010			0.885685			0.442842	+	0.896599i	$0.353970\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	39.9166i		−	1.38303i
$834$		0			0
$835$	21.7786	−	34.9808i	0.753680	−	1.21056i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10.6417			−0.367392			−0.183696	−	0.982983i	$-0.558806\pi$
$839$	−10.6417			−0.367392			−0.183696	+	0.982983i	$0.558806\pi$
$840$		0			0
$841$	−24.9359			−0.859859
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.17203	+	2.59745i	0.143522	+	0.0893552i
$846$		0			0
$847$		−	7.58536i		−	0.260636i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.07470			−0.105399
$852$		0			0
$853$		−	10.9480i		−	0.374853i	−0.982279	−	0.187427i	$-0.939985\pi$
$853$		−	10.9480i		−	0.374853i	0.982279	−	0.187427i	$-0.0600147\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			20.2019i			0.690084i	0.938587	+	0.345042i	$0.112135\pi$
$857$			20.2019i			0.690084i	−0.938587	+	0.345042i	$0.887865\pi$
$858$		0			0
$859$	−18.8889			−0.644480			−0.322240	−	0.946658i	$-0.604436\pi$
$859$	−18.8889			−0.644480			−0.322240	+	0.946658i	$0.604436\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			12.8722i			0.438174i	0.975705	+	0.219087i	$0.0703079\pi$
$863$			12.8722i			0.438174i	−0.975705	+	0.219087i	$0.929692\pi$
$864$		0			0
$865$	6.52311	−	10.4774i	0.221792	−	0.356243i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.64934			−0.225563
$870$		0			0
$871$	2.80523			0.0950518
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−11.0416	−	1.10554i	−0.373275	−	0.0373740i
$876$		0			0
$877$			2.27417i			0.0767933i	0.999263	+	0.0383966i	$0.0122250\pi$
$877$			2.27417i			0.0767933i	−0.999263	+	0.0383966i	$0.987775\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	33.3451			1.12343			0.561713	−	0.827332i	$-0.310143\pi$
$881$	33.3451			1.12343			0.561713	+	0.827332i	$0.310143\pi$
$882$		0			0
$883$		−	27.6240i		−	0.929622i	−0.885410	−	0.464811i	$-0.846122\pi$
$883$		−	27.6240i		−	0.929622i	0.885410	−	0.464811i	$-0.153878\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	40.9934i		−	1.37643i	−0.725509	−	0.688213i	$-0.758396\pi$
$887$		−	40.9934i		−	1.37643i	0.725509	−	0.688213i	$-0.241604\pi$
$888$		0			0
$889$	−10.4743			−0.351296
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1.72860i			0.0578455i
$894$		0			0
$895$	−18.4820	−	11.5067i	−0.617785	−	0.384625i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.635189			−0.0211848
$900$		0			0
$901$	46.3546			1.54430
$902$		0			0
$903$		0			0
$904$		0			0
$905$	41.4342	+	25.7964i	1.37732	+	0.857501i
$906$		0			0
$907$			12.3313i			0.409455i	0.978819	+	0.204727i	$0.0656308\pi$
$907$			12.3313i			0.409455i	−0.978819	+	0.204727i	$0.934369\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	22.2732			0.737943			0.368972	−	0.929441i	$-0.379710\pi$
$911$	22.2732			0.737943			0.368972	+	0.929441i	$0.379710\pi$
$912$		0			0
$913$		−	31.5104i		−	1.04284i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	7.79102i		−	0.257282i
$918$		0			0
$919$	15.8277			0.522108			0.261054	−	0.965324i	$-0.415930\pi$
$919$	15.8277			0.522108			0.261054	+	0.965324i	$0.415930\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	2.62085i		−	0.0862663i
$924$		0			0
$925$	−13.7954	+	6.78466i	−0.453590	+	0.223078i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	33.6716			1.10473			0.552365	−	0.833602i	$-0.313725\pi$
$929$	33.6716			1.10473			0.552365	+	0.833602i	$0.313725\pi$
$930$		0			0
$931$	33.9467			1.11256
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14.3711	+	23.0828i	−0.469984	+	0.754887i
$936$		0			0
$937$			46.9679i			1.53438i	0.641422	+	0.767188i	$0.278345\pi$
$937$			46.9679i			1.53438i	−0.641422	+	0.767188i	$0.721655\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−40.9027			−1.33339			−0.666696	−	0.745330i	$-0.732292\pi$
$941$	−40.9027			−1.33339			−0.666696	+	0.745330i	$0.732292\pi$
$942$		0			0
$943$			1.34964i			0.0439504i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	27.4956i		−	0.893486i	−0.894662	−	0.446743i	$-0.852584\pi$
$947$		−	27.4956i		−	0.893486i	0.894662	−	0.446743i	$-0.147416\pi$
$948$		0			0
$949$	−25.2149			−0.818511
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	5.30199i		−	0.171748i	−0.996306	−	0.0858742i	$-0.972632\pi$
$953$		−	5.30199i		−	0.171748i	0.996306	−	0.0858742i	$-0.0273683\pi$
$954$		0			0
$955$	−31.2843	−	19.4772i	−1.01234	−	0.630268i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.20334			−0.297191
$960$		0			0
$961$	−30.9007			−0.996798
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−5.44276	+	8.74215i	−0.175209	+	0.281420i
$966$		0			0
$967$		−	38.8814i		−	1.25034i	−0.780489	−	0.625170i	$-0.785030\pi$
$967$		−	38.8814i		−	1.25034i	0.780489	−	0.625170i	$-0.214970\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	54.4921			1.74873			0.874367	−	0.485265i	$-0.161277\pi$
$971$	54.4921			1.74873			0.874367	+	0.485265i	$0.161277\pi$
$972$		0			0
$973$		−	18.3908i		−	0.589583i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.49089i			0.0476979i	0.999716	+	0.0238490i	$0.00759208\pi$
$977$			1.49089i			0.0476979i	−0.999716	+	0.0238490i	$0.992408\pi$
$978$		0			0
$979$	12.8552			0.410854
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	29.0626i		−	0.926952i	−0.886110	−	0.463476i	$-0.846602\pi$
$983$		−	29.0626i		−	0.926952i	0.886110	−	0.463476i	$-0.153398\pi$
$984$		0			0
$985$	0.185765	−	0.298376i	0.00591896	−	0.00950704i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.97905			0.190123
$990$		0			0
$991$	−50.9937			−1.61987			−0.809933	−	0.586522i	$-0.800497\pi$
$991$	−50.9937			−1.61987			−0.809933	+	0.586522i	$0.800497\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.68738	+	2.29572i	0.116898	+	0.0727791i
$996$		0			0
$997$			52.4224i			1.66023i	0.557590	+	0.830116i	$0.311726\pi$
$997$			52.4224i			1.66023i	−0.557590	+	0.830116i	$0.688274\pi$
$998$		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.829.10		12
3.2	odd	2		460.2.c.a.369.8	yes	12
5.4	even	2	inner	4140.2.f.b.829.9		12
12.11	even	2		1840.2.e.f.369.5		12
15.2	even	4		2300.2.a.n.1.5		6
15.8	even	4		2300.2.a.o.1.2		6
15.14	odd	2		460.2.c.a.369.5	✓	12
60.23	odd	4		9200.2.a.cx.1.5		6
60.47	odd	4		9200.2.a.cy.1.2		6
60.59	even	2		1840.2.e.f.369.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.5	✓	12	15.14	odd	2
460.2.c.a.369.8	yes	12	3.2	odd	2
1840.2.e.f.369.5		12	12.11	even	2
1840.2.e.f.369.8		12	60.59	even	2
2300.2.a.n.1.5		6	15.2	even	4
2300.2.a.o.1.2		6	15.8	even	4
4140.2.f.b.829.9		12	5.4	even	2	inner
4140.2.f.b.829.10		12	1.1	even	1	trivial
9200.2.a.cx.1.5		6	60.23	odd	4
9200.2.a.cy.1.2		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)

\(f(q)\)	\(=\)	\(q+(1.89824 + 1.18182i) q^{5} +0.992530i q^{7} +O(q^{10})\)	\(q+(1.89824 + 1.18182i) q^{5} +0.992530i q^{7} -1.83236 q^{11} +3.28666i q^{13} -6.63631i q^{17} +5.64378 q^{19} +1.00000i q^{23} +(2.20661 + 4.48675i) q^{25} +2.01596 q^{29} -0.315080 q^{31} +(-1.17299 + 1.88406i) q^{35} +3.07470i q^{37} +1.34964 q^{41} -5.97905i q^{43} +0.306285i q^{47} +6.01489 q^{49} +6.98500i q^{53} +(-3.47826 - 2.16552i) q^{55} +9.49533 q^{59} +5.56160 q^{61} +(-3.88424 + 6.23887i) q^{65} -0.853521i q^{67} -0.797419 q^{71} +7.67189i q^{73} -1.81867i q^{77} +3.62884 q^{79} +17.1966i q^{83} +(7.84291 - 12.5973i) q^{85} -7.01565 q^{89} -3.26211 q^{91} +(10.7132 + 6.66992i) q^{95} +18.5807i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 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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