LMFDB - Embedded newform 2475.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

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:	yes
Analytic conductor:	$19.7629745003$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{2} +3.30278 q^{4} -4.30278 q^{7} +3.00000 q^{8} +O(q^{10})$	$q+2.30278 q^{2} +3.30278 q^{4} -4.30278 q^{7} +3.00000 q^{8} +1.00000 q^{11} -5.00000 q^{13} -9.90833 q^{14} +0.302776 q^{16} -3.90833 q^{17} -1.00000 q^{19} +2.30278 q^{22} -3.69722 q^{23} -11.5139 q^{26} -14.2111 q^{28} +9.90833 q^{29} -4.21110 q^{31} -5.30278 q^{32} -9.00000 q^{34} -9.60555 q^{37} -2.30278 q^{38} -1.60555 q^{41} +7.21110 q^{43} +3.30278 q^{44} -8.51388 q^{46} -3.00000 q^{47} +11.5139 q^{49} -16.5139 q^{52} +2.30278 q^{53} -12.9083 q^{56} +22.8167 q^{58} -0.211103 q^{59} +2.90833 q^{61} -9.69722 q^{62} -12.8167 q^{64} +4.00000 q^{67} -12.9083 q^{68} -4.60555 q^{71} -2.90833 q^{73} -22.1194 q^{74} -3.30278 q^{76} -4.30278 q^{77} -0.0916731 q^{79} -3.69722 q^{82} +14.5139 q^{83} +16.6056 q^{86} +3.00000 q^{88} -5.30278 q^{89} +21.5139 q^{91} -12.2111 q^{92} -6.90833 q^{94} -11.6972 q^{97} +26.5139 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + 3 * q^4 - 5 * q^7 + 6 * q^8	$ 2 q + q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8} + 2 q^{11} - 10 q^{13} - 9 q^{14} - 3 q^{16} + 3 q^{17} - 2 q^{19} + q^{22} - 11 q^{23} - 5 q^{26} - 14 q^{28} + 9 q^{29} + 6 q^{31} - 7 q^{32} - 18 q^{34} - 12 q^{37} - q^{38} + 4 q^{41} + 3 q^{44} + q^{46} - 6 q^{47} + 5 q^{49} - 15 q^{52} + q^{53} - 15 q^{56} + 24 q^{58} + 14 q^{59} - 5 q^{61} - 23 q^{62} - 4 q^{64} + 8 q^{67} - 15 q^{68} - 2 q^{71} + 5 q^{73} - 19 q^{74} - 3 q^{76} - 5 q^{77} - 11 q^{79} - 11 q^{82} + 11 q^{83} + 26 q^{86} + 6 q^{88} - 7 q^{89} + 25 q^{91} - 10 q^{92} - 3 q^{94} - 27 q^{97} + 35 q^{98}+O(q^{100}) $ 2 * q + q^2 + 3 * q^4 - 5 * q^7 + 6 * q^8 + 2 * q^11 - 10 * q^13 - 9 * q^14 - 3 * q^16 + 3 * q^17 - 2 * q^19 + q^22 - 11 * q^23 - 5 * q^26 - 14 * q^28 + 9 * q^29 + 6 * q^31 - 7 * q^32 - 18 * q^34 - 12 * q^37 - q^38 + 4 * q^41 + 3 * q^44 + q^46 - 6 * q^47 + 5 * q^49 - 15 * q^52 + q^53 - 15 * q^56 + 24 * q^58 + 14 * q^59 - 5 * q^61 - 23 * q^62 - 4 * q^64 + 8 * q^67 - 15 * q^68 - 2 * q^71 + 5 * q^73 - 19 * q^74 - 3 * q^76 - 5 * q^77 - 11 * q^79 - 11 * q^82 + 11 * q^83 + 26 * q^86 + 6 * q^88 - 7 * q^89 + 25 * q^91 - 10 * q^92 - 3 * q^94 - 27 * q^97 + 35 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.30278	1.62831	0.814154	−	0.580649i	$-0.197201\pi$
$2$	2.30278	1.62831	0.814154	+	0.580649i	$0.197201\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.30278	−1.62630	−0.813148	−	0.582057i	$-0.802248\pi$
$7$	−4.30278	−1.62630	−0.813148	+	0.582057i	$0.802248\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−9.90833	−2.64811
$15$	0	0
$16$	0.302776	0.0756939
$17$	−3.90833	−0.947909	−0.473954	−	0.880549i	$-0.657174\pi$
$17$	−3.90833	−0.947909	−0.473954	+	0.880549i	$0.657174\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	2.30278	0.490953
$23$	−3.69722	−0.770925	−0.385462	−	0.922724i	$-0.625958\pi$
$23$	−3.69722	−0.770925	−0.385462	+	0.922724i	$0.625958\pi$
$24$	0	0
$25$	0	0
$26$	−11.5139	−2.25806
$27$	0	0
$28$	−14.2111	−2.68565
$29$	9.90833	1.83993	0.919965	−	0.392000i	$-0.128217\pi$
$29$	9.90833	1.83993	0.919965	+	0.392000i	$0.128217\pi$
$30$	0	0
$31$	−4.21110	−0.756336	−0.378168	−	0.925737i	$-0.623446\pi$
$31$	−4.21110	−0.756336	−0.378168	+	0.925737i	$0.623446\pi$
$32$	−5.30278	−0.937407
$33$	0	0
$34$	−9.00000	−1.54349
$35$	0	0
$36$	0	0
$37$	−9.60555	−1.57914	−0.789571	−	0.613659i	$-0.789697\pi$
$37$	−9.60555	−1.57914	−0.789571	+	0.613659i	$0.789697\pi$
$38$	−2.30278	−0.373560
$39$	0	0
$40$	0	0
$41$	−1.60555	−0.250745	−0.125372	−	0.992110i	$-0.540013\pi$
$41$	−1.60555	−0.250745	−0.125372	+	0.992110i	$0.540013\pi$
$42$	0	0
$43$	7.21110	1.09968	0.549841	−	0.835269i	$-0.314688\pi$
$43$	7.21110	1.09968	0.549841	+	0.835269i	$0.314688\pi$
$44$	3.30278	0.497912
$45$	0	0
$46$	−8.51388	−1.25530
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	11.5139	1.64484
$50$	0	0
$51$	0	0
$52$	−16.5139	−2.29006
$53$	2.30278	0.316311	0.158155	−	0.987414i	$-0.449445\pi$
$53$	2.30278	0.316311	0.158155	+	0.987414i	$0.449445\pi$
$54$	0	0
$55$	0	0
$56$	−12.9083	−1.72495
$57$	0	0
$58$	22.8167	2.99597
$59$	−0.211103	−0.0274832	−0.0137416	−	0.999906i	$-0.504374\pi$
$59$	−0.211103	−0.0274832	−0.0137416	+	0.999906i	$0.504374\pi$
$60$	0	0
$61$	2.90833	0.372373	0.186187	−	0.982514i	$-0.440387\pi$
$61$	2.90833	0.372373	0.186187	+	0.982514i	$0.440387\pi$
$62$	−9.69722	−1.23155
$63$	0	0
$64$	−12.8167	−1.60208
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−12.9083	−1.56536
$69$	0	0
$70$	0	0
$71$	−4.60555	−0.546578	−0.273289	−	0.961932i	$-0.588112\pi$
$71$	−4.60555	−0.546578	−0.273289	+	0.961932i	$0.588112\pi$
$72$	0	0
$73$	−2.90833	−0.340394	−0.170197	−	0.985410i	$-0.554440\pi$
$73$	−2.90833	−0.340394	−0.170197	+	0.985410i	$0.554440\pi$
$74$	−22.1194	−2.57133
$75$	0	0
$76$	−3.30278	−0.378854
$77$	−4.30278	−0.490347
$78$	0	0
$79$	−0.0916731	−0.0103140	−0.00515701	−	0.999987i	$-0.501642\pi$
$79$	−0.0916731	−0.0103140	−0.00515701	+	0.999987i	$0.501642\pi$
$80$	0	0
$81$	0	0
$82$	−3.69722	−0.408290
$83$	14.5139	1.59311	0.796553	−	0.604569i	$-0.206655\pi$
$83$	14.5139	1.59311	0.796553	+	0.604569i	$0.206655\pi$
$84$	0	0
$85$	0	0
$86$	16.6056	1.79062
$87$	0	0
$88$	3.00000	0.319801
$89$	−5.30278	−0.562093	−0.281047	−	0.959694i	$-0.590682\pi$
$89$	−5.30278	−0.562093	−0.281047	+	0.959694i	$0.590682\pi$
$90$	0	0
$91$	21.5139	2.25527
$92$	−12.2111	−1.27310
$93$	0	0
$94$	−6.90833	−0.712540
$95$	0	0
$96$	0	0
$97$	−11.6972	−1.18767	−0.593837	−	0.804586i	$-0.702387\pi$
$97$	−11.6972	−1.18767	−0.593837	+	0.804586i	$0.702387\pi$
$98$	26.5139	2.67831
$99$	0	0
$100$	0	0
$101$	−17.5139	−1.74270	−0.871348	−	0.490666i	$-0.836754\pi$
$101$	−17.5139	−1.74270	−0.871348	+	0.490666i	$0.836754\pi$
$102$	0	0
$103$	7.90833	0.779231	0.389615	−	0.920978i	$-0.372608\pi$
$103$	7.90833	0.779231	0.389615	+	0.920978i	$0.372608\pi$
$104$	−15.0000	−1.47087
$105$	0	0
$106$	5.30278	0.515051
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−6.51388	−0.623916	−0.311958	−	0.950096i	$-0.600985\pi$
$109$	−6.51388	−0.623916	−0.311958	+	0.950096i	$0.600985\pi$
$110$	0	0
$111$	0	0
$112$	−1.30278	−0.123101
$113$	10.8167	1.01755	0.508773	−	0.860901i	$-0.330099\pi$
$113$	10.8167	1.01755	0.508773	+	0.860901i	$0.330099\pi$
$114$	0	0
$115$	0	0
$116$	32.7250	3.03844
$117$	0	0
$118$	−0.486122	−0.0447511
$119$	16.8167	1.54158
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.69722	0.606338
$123$	0	0
$124$	−13.9083	−1.24900
$125$	0	0
$126$	0	0
$127$	17.1194	1.51910	0.759552	−	0.650447i	$-0.225418\pi$
$127$	17.1194	1.51910	0.759552	+	0.650447i	$0.225418\pi$
$128$	−18.9083	−1.67128
$129$	0	0
$130$	0	0
$131$	−0.908327	−0.0793609	−0.0396804	−	0.999212i	$-0.512634\pi$
$131$	−0.908327	−0.0793609	−0.0396804	+	0.999212i	$0.512634\pi$
$132$	0	0
$133$	4.30278	0.373098
$134$	9.21110	0.795718
$135$	0	0
$136$	−11.7250	−1.00541
$137$	−2.09167	−0.178704	−0.0893518	−	0.996000i	$-0.528480\pi$
$137$	−2.09167	−0.178704	−0.0893518	+	0.996000i	$0.528480\pi$
$138$	0	0
$139$	8.21110	0.696457	0.348228	−	0.937410i	$-0.386783\pi$
$139$	8.21110	0.696457	0.348228	+	0.937410i	$0.386783\pi$
$140$	0	0
$141$	0	0
$142$	−10.6056	−0.889998
$143$	−5.00000	−0.418121
$144$	0	0
$145$	0	0
$146$	−6.69722	−0.554266
$147$	0	0
$148$	−31.7250	−2.60778
$149$	−2.78890	−0.228475	−0.114238	−	0.993453i	$-0.536443\pi$
$149$	−2.78890	−0.228475	−0.114238	+	0.993453i	$0.536443\pi$
$150$	0	0
$151$	−20.8167	−1.69404	−0.847018	−	0.531565i	$-0.821604\pi$
$151$	−20.8167	−1.69404	−0.847018	+	0.531565i	$0.821604\pi$
$152$	−3.00000	−0.243332
$153$	0	0
$154$	−9.90833	−0.798436
$155$	0	0
$156$	0	0
$157$	−4.78890	−0.382196	−0.191098	−	0.981571i	$-0.561205\pi$
$157$	−4.78890	−0.382196	−0.191098	+	0.981571i	$0.561205\pi$
$158$	−0.211103	−0.0167944
$159$	0	0
$160$	0	0
$161$	15.9083	1.25375
$162$	0	0
$163$	−5.69722	−0.446241	−0.223121	−	0.974791i	$-0.571624\pi$
$163$	−5.69722	−0.446241	−0.223121	+	0.974791i	$0.571624\pi$
$164$	−5.30278	−0.414077
$165$	0	0
$166$	33.4222	2.59407
$167$	−15.4222	−1.19341	−0.596703	−	0.802462i	$-0.703523\pi$
$167$	−15.4222	−1.19341	−0.596703	+	0.802462i	$0.703523\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	23.8167	1.81600
$173$	16.8167	1.27855	0.639273	−	0.768980i	$-0.279235\pi$
$173$	16.8167	1.27855	0.639273	+	0.768980i	$0.279235\pi$
$174$	0	0
$175$	0	0
$176$	0.302776	0.0228226
$177$	0	0
$178$	−12.2111	−0.915261
$179$	5.51388	0.412127	0.206063	−	0.978539i	$-0.433935\pi$
$179$	5.51388	0.412127	0.206063	+	0.978539i	$0.433935\pi$
$180$	0	0
$181$	−9.09167	−0.675779	−0.337889	−	0.941186i	$-0.609713\pi$
$181$	−9.09167	−0.675779	−0.337889	+	0.941186i	$0.609713\pi$
$182$	49.5416	3.67227
$183$	0	0
$184$	−11.0917	−0.817689
$185$	0	0
$186$	0	0
$187$	−3.90833	−0.285805
$188$	−9.90833	−0.722639
$189$	0	0
$190$	0	0
$191$	−6.69722	−0.484594	−0.242297	−	0.970202i	$-0.577901\pi$
$191$	−6.69722	−0.484594	−0.242297	+	0.970202i	$0.577901\pi$
$192$	0	0
$193$	1.21110	0.0871771	0.0435885	−	0.999050i	$-0.486121\pi$
$193$	1.21110	0.0871771	0.0435885	+	0.999050i	$0.486121\pi$
$194$	−26.9361	−1.93390
$195$	0	0
$196$	38.0278	2.71627
$197$	9.69722	0.690899	0.345449	−	0.938437i	$-0.387727\pi$
$197$	9.69722	0.690899	0.345449	+	0.938437i	$0.387727\pi$
$198$	0	0
$199$	−24.5139	−1.73774	−0.868871	−	0.495038i	$-0.835154\pi$
$199$	−24.5139	−1.73774	−0.868871	+	0.495038i	$0.835154\pi$
$200$	0	0
$201$	0	0
$202$	−40.3305	−2.83765
$203$	−42.6333	−2.99227
$204$	0	0
$205$	0	0
$206$	18.2111	1.26883
$207$	0	0
$208$	−1.51388	−0.104969
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	25.2389	1.73751	0.868757	−	0.495238i	$-0.164919\pi$
$211$	25.2389	1.73751	0.868757	+	0.495238i	$0.164919\pi$
$212$	7.60555	0.522351
$213$	0	0
$214$	−6.90833	−0.472244
$215$	0	0
$216$	0	0
$217$	18.1194	1.23003
$218$	−15.0000	−1.01593
$219$	0	0
$220$	0	0
$221$	19.5416	1.31451
$222$	0	0
$223$	−20.6333	−1.38171	−0.690854	−	0.722994i	$-0.742765\pi$
$223$	−20.6333	−1.38171	−0.690854	+	0.722994i	$0.742765\pi$
$224$	22.8167	1.52450
$225$	0	0
$226$	24.9083	1.65688
$227$	5.30278	0.351958	0.175979	−	0.984394i	$-0.443691\pi$
$227$	5.30278	0.351958	0.175979	+	0.984394i	$0.443691\pi$
$228$	0	0
$229$	13.7250	0.906972	0.453486	−	0.891263i	$-0.350180\pi$
$229$	13.7250	0.906972	0.453486	+	0.891263i	$0.350180\pi$
$230$	0	0
$231$	0	0
$232$	29.7250	1.95154
$233$	5.09167	0.333567	0.166783	−	0.985994i	$-0.446662\pi$
$233$	5.09167	0.333567	0.166783	+	0.985994i	$0.446662\pi$
$234$	0	0
$235$	0	0
$236$	−0.697224	−0.0453854
$237$	0	0
$238$	38.7250	2.51017
$239$	4.11943	0.266464	0.133232	−	0.991085i	$-0.457465\pi$
$239$	4.11943	0.266464	0.133232	+	0.991085i	$0.457465\pi$
$240$	0	0
$241$	−24.9361	−1.60627	−0.803137	−	0.595794i	$-0.796837\pi$
$241$	−24.9361	−1.60627	−0.803137	+	0.595794i	$0.796837\pi$
$242$	2.30278	0.148028
$243$	0	0
$244$	9.60555	0.614932
$245$	0	0
$246$	0	0
$247$	5.00000	0.318142
$248$	−12.6333	−0.802216
$249$	0	0
$250$	0	0
$251$	3.90833	0.246691	0.123346	−	0.992364i	$-0.460638\pi$
$251$	3.90833	0.246691	0.123346	+	0.992364i	$0.460638\pi$
$252$	0	0
$253$	−3.69722	−0.232443
$254$	39.4222	2.47357
$255$	0	0
$256$	−17.9083	−1.11927
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	41.3305	2.56815
$260$	0	0
$261$	0	0
$262$	−2.09167	−0.129224
$263$	−1.18335	−0.0729683	−0.0364841	−	0.999334i	$-0.511616\pi$
$263$	−1.18335	−0.0729683	−0.0364841	+	0.999334i	$0.511616\pi$
$264$	0	0
$265$	0	0
$266$	9.90833	0.607519
$267$	0	0
$268$	13.2111	0.806997
$269$	23.7250	1.44654	0.723269	−	0.690567i	$-0.242639\pi$
$269$	23.7250	1.44654	0.723269	+	0.690567i	$0.242639\pi$
$270$	0	0
$271$	14.2111	0.863263	0.431632	−	0.902050i	$-0.357938\pi$
$271$	14.2111	0.863263	0.431632	+	0.902050i	$0.357938\pi$
$272$	−1.18335	−0.0717509
$273$	0	0
$274$	−4.81665	−0.290985
$275$	0	0
$276$	0	0
$277$	−21.6056	−1.29815	−0.649076	−	0.760724i	$-0.724844\pi$
$277$	−21.6056	−1.29815	−0.649076	+	0.760724i	$0.724844\pi$
$278$	18.9083	1.13405
$279$	0	0
$280$	0	0
$281$	22.8167	1.36113	0.680564	−	0.732689i	$-0.261735\pi$
$281$	22.8167	1.36113	0.680564	+	0.732689i	$0.261735\pi$
$282$	0	0
$283$	−2.69722	−0.160333	−0.0801667	−	0.996781i	$-0.525545\pi$
$283$	−2.69722	−0.160333	−0.0801667	+	0.996781i	$0.525545\pi$
$284$	−15.2111	−0.902613
$285$	0	0
$286$	−11.5139	−0.680830
$287$	6.90833	0.407786
$288$	0	0
$289$	−1.72498	−0.101469
$290$	0	0
$291$	0	0
$292$	−9.60555	−0.562122
$293$	15.2111	0.888642	0.444321	−	0.895868i	$-0.353445\pi$
$293$	15.2111	0.888642	0.444321	+	0.895868i	$0.353445\pi$
$294$	0	0
$295$	0	0
$296$	−28.8167	−1.67493
$297$	0	0
$298$	−6.42221	−0.372028
$299$	18.4861	1.06908
$300$	0	0
$301$	−31.0278	−1.78841
$302$	−47.9361	−2.75841
$303$	0	0
$304$	−0.302776	−0.0173654
$305$	0	0
$306$	0	0
$307$	6.09167	0.347670	0.173835	−	0.984775i	$-0.444384\pi$
$307$	6.09167	0.347670	0.173835	+	0.984775i	$0.444384\pi$
$308$	−14.2111	−0.809753
$309$	0	0
$310$	0	0
$311$	16.8167	0.953585	0.476792	−	0.879016i	$-0.341799\pi$
$311$	16.8167	0.953585	0.476792	+	0.879016i	$0.341799\pi$
$312$	0	0
$313$	−21.8167	−1.23315	−0.616575	−	0.787296i	$-0.711480\pi$
$313$	−21.8167	−1.23315	−0.616575	+	0.787296i	$0.711480\pi$
$314$	−11.0278	−0.622332
$315$	0	0
$316$	−0.302776	−0.0170325
$317$	9.90833	0.556507	0.278254	−	0.960508i	$-0.410244\pi$
$317$	9.90833	0.556507	0.278254	+	0.960508i	$0.410244\pi$
$318$	0	0
$319$	9.90833	0.554760
$320$	0	0
$321$	0	0
$322$	36.6333	2.04149
$323$	3.90833	0.217465
$324$	0	0
$325$	0	0
$326$	−13.1194	−0.726618
$327$	0	0
$328$	−4.81665	−0.265955
$329$	12.9083	0.711659
$330$	0	0
$331$	−14.3944	−0.791190	−0.395595	−	0.918425i	$-0.629462\pi$
$331$	−14.3944	−0.791190	−0.395595	+	0.918425i	$0.629462\pi$
$332$	47.9361	2.63083
$333$	0	0
$334$	−35.5139	−1.94323
$335$	0	0
$336$	0	0
$337$	−26.8444	−1.46231	−0.731154	−	0.682212i	$-0.761018\pi$
$337$	−26.8444	−1.46231	−0.731154	+	0.682212i	$0.761018\pi$
$338$	27.6333	1.50305
$339$	0	0
$340$	0	0
$341$	−4.21110	−0.228044
$342$	0	0
$343$	−19.4222	−1.04870
$344$	21.6333	1.16639
$345$	0	0
$346$	38.7250	2.08187
$347$	5.51388	0.296000	0.148000	−	0.988987i	$-0.452716\pi$
$347$	5.51388	0.296000	0.148000	+	0.988987i	$0.452716\pi$
$348$	0	0
$349$	−26.8167	−1.43546	−0.717731	−	0.696320i	$-0.754819\pi$
$349$	−26.8167	−1.43546	−0.717731	+	0.696320i	$0.754819\pi$
$350$	0	0
$351$	0	0
$352$	−5.30278	−0.282639
$353$	−24.6333	−1.31110	−0.655549	−	0.755152i	$-0.727563\pi$
$353$	−24.6333	−1.31110	−0.655549	+	0.755152i	$0.727563\pi$
$354$	0	0
$355$	0	0
$356$	−17.5139	−0.928234
$357$	0	0
$358$	12.6972	0.671069
$359$	−15.2111	−0.802811	−0.401406	−	0.915900i	$-0.631478\pi$
$359$	−15.2111	−0.802811	−0.401406	+	0.915900i	$0.631478\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−20.9361	−1.10038
$363$	0	0
$364$	71.0555	3.72432
$365$	0	0
$366$	0	0
$367$	24.3028	1.26859	0.634297	−	0.773089i	$-0.281290\pi$
$367$	24.3028	1.26859	0.634297	+	0.773089i	$0.281290\pi$
$368$	−1.11943	−0.0583543
$369$	0	0
$370$	0	0
$371$	−9.90833	−0.514415
$372$	0	0
$373$	1.42221	0.0736390	0.0368195	−	0.999322i	$-0.488277\pi$
$373$	1.42221	0.0736390	0.0368195	+	0.999322i	$0.488277\pi$
$374$	−9.00000	−0.465379
$375$	0	0
$376$	−9.00000	−0.464140
$377$	−49.5416	−2.55152
$378$	0	0
$379$	24.8167	1.27475	0.637373	−	0.770555i	$-0.280021\pi$
$379$	24.8167	1.27475	0.637373	+	0.770555i	$0.280021\pi$
$380$	0	0
$381$	0	0
$382$	−15.4222	−0.789069
$383$	−21.6333	−1.10541	−0.552705	−	0.833377i	$-0.686404\pi$
$383$	−21.6333	−1.10541	−0.552705	+	0.833377i	$0.686404\pi$
$384$	0	0
$385$	0	0
$386$	2.78890	0.141951
$387$	0	0
$388$	−38.6333	−1.96131
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	14.4500	0.730766
$392$	34.5416	1.74462
$393$	0	0
$394$	22.3305	1.12500
$395$	0	0
$396$	0	0
$397$	−25.3028	−1.26991	−0.634955	−	0.772549i	$-0.718981\pi$
$397$	−25.3028	−1.26991	−0.634955	+	0.772549i	$0.718981\pi$
$398$	−56.4500	−2.82958
$399$	0	0
$400$	0	0
$401$	27.2111	1.35886	0.679429	−	0.733741i	$-0.262228\pi$
$401$	27.2111	1.35886	0.679429	+	0.733741i	$0.262228\pi$
$402$	0	0
$403$	21.0555	1.04885
$404$	−57.8444	−2.87787
$405$	0	0
$406$	−98.1749	−4.87234
$407$	−9.60555	−0.476129
$408$	0	0
$409$	8.21110	0.406013	0.203006	−	0.979177i	$-0.434929\pi$
$409$	8.21110	0.406013	0.203006	+	0.979177i	$0.434929\pi$
$410$	0	0
$411$	0	0
$412$	26.1194	1.28681
$413$	0.908327	0.0446958
$414$	0	0
$415$	0	0
$416$	26.5139	1.29995
$417$	0	0
$418$	−2.30278	−0.112632
$419$	−13.6056	−0.664675	−0.332337	−	0.943161i	$-0.607837\pi$
$419$	−13.6056	−0.664675	−0.332337	+	0.943161i	$0.607837\pi$
$420$	0	0
$421$	4.30278	0.209704	0.104852	−	0.994488i	$-0.466563\pi$
$421$	4.30278	0.209704	0.104852	+	0.994488i	$0.466563\pi$
$422$	58.1194	2.82921
$423$	0	0
$424$	6.90833	0.335498
$425$	0	0
$426$	0	0
$427$	−12.5139	−0.605589
$428$	−9.90833	−0.478937
$429$	0	0
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	41.7250	2.00286
$435$	0	0
$436$	−21.5139	−1.03033
$437$	3.69722	0.176862
$438$	0	0
$439$	20.6972	0.987825	0.493912	−	0.869512i	$-0.335566\pi$
$439$	20.6972	0.987825	0.493912	+	0.869512i	$0.335566\pi$
$440$	0	0
$441$	0	0
$442$	45.0000	2.14043
$443$	−1.39445	−0.0662523	−0.0331261	−	0.999451i	$-0.510546\pi$
$443$	−1.39445	−0.0662523	−0.0331261	+	0.999451i	$0.510546\pi$
$444$	0	0
$445$	0	0
$446$	−47.5139	−2.24985
$447$	0	0
$448$	55.1472	2.60546
$449$	−41.5139	−1.95916	−0.979581	−	0.201052i	$-0.935564\pi$
$449$	−41.5139	−1.95916	−0.979581	+	0.201052i	$0.935564\pi$
$450$	0	0
$451$	−1.60555	−0.0756025
$452$	35.7250	1.68036
$453$	0	0
$454$	12.2111	0.573095
$455$	0	0
$456$	0	0
$457$	24.3028	1.13684	0.568418	−	0.822740i	$-0.307556\pi$
$457$	24.3028	1.13684	0.568418	+	0.822740i	$0.307556\pi$
$458$	31.6056	1.47683
$459$	0	0
$460$	0	0
$461$	−17.7889	−0.828512	−0.414256	−	0.910161i	$-0.635958\pi$
$461$	−17.7889	−0.828512	−0.414256	+	0.910161i	$0.635958\pi$
$462$	0	0
$463$	−26.2111	−1.21813	−0.609067	−	0.793119i	$-0.708456\pi$
$463$	−26.2111	−1.21813	−0.609067	+	0.793119i	$0.708456\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	11.7250	0.543149
$467$	24.6333	1.13989	0.569947	−	0.821682i	$-0.306964\pi$
$467$	24.6333	1.13989	0.569947	+	0.821682i	$0.306964\pi$
$468$	0	0
$469$	−17.2111	−0.794735
$470$	0	0
$471$	0	0
$472$	−0.633308	−0.0291503
$473$	7.21110	0.331567
$474$	0	0
$475$	0	0
$476$	55.5416	2.54575
$477$	0	0
$478$	9.48612	0.433885
$479$	13.1833	0.602362	0.301181	−	0.953567i	$-0.402619\pi$
$479$	13.1833	0.602362	0.301181	+	0.953567i	$0.402619\pi$
$480$	0	0
$481$	48.0278	2.18988
$482$	−57.4222	−2.61551
$483$	0	0
$484$	3.30278	0.150126
$485$	0	0
$486$	0	0
$487$	10.2111	0.462709	0.231355	−	0.972869i	$-0.425684\pi$
$487$	10.2111	0.462709	0.231355	+	0.972869i	$0.425684\pi$
$488$	8.72498	0.394961
$489$	0	0
$490$	0	0
$491$	24.2111	1.09263	0.546316	−	0.837579i	$-0.316030\pi$
$491$	24.2111	1.09263	0.546316	+	0.837579i	$0.316030\pi$
$492$	0	0
$493$	−38.7250	−1.74409
$494$	11.5139	0.518034
$495$	0	0
$496$	−1.27502	−0.0572501
$497$	19.8167	0.888898
$498$	0	0
$499$	−21.5139	−0.963093	−0.481547	−	0.876420i	$-0.659925\pi$
$499$	−21.5139	−0.963093	−0.481547	+	0.876420i	$0.659925\pi$
$500$	0	0
$501$	0	0
$502$	9.00000	0.401690
$503$	16.6056	0.740405	0.370202	−	0.928951i	$-0.379288\pi$
$503$	16.6056	0.740405	0.370202	+	0.928951i	$0.379288\pi$
$504$	0	0
$505$	0	0
$506$	−8.51388	−0.378488
$507$	0	0
$508$	56.5416	2.50863
$509$	26.3028	1.16585	0.582925	−	0.812526i	$-0.301908\pi$
$509$	26.3028	1.16585	0.582925	+	0.812526i	$0.301908\pi$
$510$	0	0
$511$	12.5139	0.553581
$512$	−3.42221	−0.151242
$513$	0	0
$514$	−41.4500	−1.82828
$515$	0	0
$516$	0	0
$517$	−3.00000	−0.131940
$518$	95.1749	4.18175
$519$	0	0
$520$	0	0
$521$	−23.4500	−1.02736	−0.513681	−	0.857981i	$-0.671718\pi$
$521$	−23.4500	−1.02736	−0.513681	+	0.857981i	$0.671718\pi$
$522$	0	0
$523$	3.57779	0.156446	0.0782230	−	0.996936i	$-0.475075\pi$
$523$	3.57779	0.156446	0.0782230	+	0.996936i	$0.475075\pi$
$524$	−3.00000	−0.131056
$525$	0	0
$526$	−2.72498	−0.118815
$527$	16.4584	0.716938
$528$	0	0
$529$	−9.33053	−0.405675
$530$	0	0
$531$	0	0
$532$	14.2111	0.616129
$533$	8.02776	0.347721
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	54.6333	2.35541
$539$	11.5139	0.495938
$540$	0	0
$541$	−6.72498	−0.289130	−0.144565	−	0.989495i	$-0.546178\pi$
$541$	−6.72498	−0.289130	−0.144565	+	0.989495i	$0.546178\pi$
$542$	32.7250	1.40566
$543$	0	0
$544$	20.7250	0.888576
$545$	0	0
$546$	0	0
$547$	−18.1194	−0.774731	−0.387365	−	0.921926i	$-0.626615\pi$
$547$	−18.1194	−0.774731	−0.387365	+	0.921926i	$0.626615\pi$
$548$	−6.90833	−0.295109
$549$	0	0
$550$	0	0
$551$	−9.90833	−0.422109
$552$	0	0
$553$	0.394449	0.0167737
$554$	−49.7527	−2.11379
$555$	0	0
$556$	27.1194	1.15012
$557$	−9.42221	−0.399232	−0.199616	−	0.979874i	$-0.563969\pi$
$557$	−9.42221	−0.399232	−0.199616	+	0.979874i	$0.563969\pi$
$558$	0	0
$559$	−36.0555	−1.52499
$560$	0	0
$561$	0	0
$562$	52.5416	2.21634
$563$	18.9083	0.796891	0.398445	−	0.917192i	$-0.369550\pi$
$563$	18.9083	0.796891	0.398445	+	0.917192i	$0.369550\pi$
$564$	0	0
$565$	0	0
$566$	−6.21110	−0.261072
$567$	0	0
$568$	−13.8167	−0.579734
$569$	−15.1472	−0.635003	−0.317502	−	0.948258i	$-0.602844\pi$
$569$	−15.1472	−0.635003	−0.317502	+	0.948258i	$0.602844\pi$
$570$	0	0
$571$	−17.3305	−0.725260	−0.362630	−	0.931933i	$-0.618121\pi$
$571$	−17.3305	−0.725260	−0.362630	+	0.931933i	$0.618121\pi$
$572$	−16.5139	−0.690480
$573$	0	0
$574$	15.9083	0.664001
$575$	0	0
$576$	0	0
$577$	31.3583	1.30546	0.652731	−	0.757589i	$-0.273623\pi$
$577$	31.3583	1.30546	0.652731	+	0.757589i	$0.273623\pi$
$578$	−3.97224	−0.165224
$579$	0	0
$580$	0	0
$581$	−62.4500	−2.59086
$582$	0	0
$583$	2.30278	0.0953712
$584$	−8.72498	−0.361042
$585$	0	0
$586$	35.0278	1.44698
$587$	−37.5416	−1.54951	−0.774755	−	0.632262i	$-0.782127\pi$
$587$	−37.5416	−1.54951	−0.774755	+	0.632262i	$0.782127\pi$
$588$	0	0
$589$	4.21110	0.173515
$590$	0	0
$591$	0	0
$592$	−2.90833	−0.119531
$593$	−13.6056	−0.558713	−0.279357	−	0.960187i	$-0.590121\pi$
$593$	−13.6056	−0.558713	−0.279357	+	0.960187i	$0.590121\pi$
$594$	0	0
$595$	0	0
$596$	−9.21110	−0.377301
$597$	0	0
$598$	42.5694	1.74079
$599$	14.0917	0.575770	0.287885	−	0.957665i	$-0.407048\pi$
$599$	14.0917	0.575770	0.287885	+	0.957665i	$0.407048\pi$
$600$	0	0
$601$	8.90833	0.363378	0.181689	−	0.983356i	$-0.441844\pi$
$601$	8.90833	0.363378	0.181689	+	0.983356i	$0.441844\pi$
$602$	−71.4500	−2.91208
$603$	0	0
$604$	−68.7527	−2.79751
$605$	0	0
$606$	0	0
$607$	7.21110	0.292690	0.146345	−	0.989234i	$-0.453249\pi$
$607$	7.21110	0.292690	0.146345	+	0.989234i	$0.453249\pi$
$608$	5.30278	0.215056
$609$	0	0
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	41.1194	1.66080	0.830399	−	0.557169i	$-0.188112\pi$
$613$	41.1194	1.66080	0.830399	+	0.557169i	$0.188112\pi$
$614$	14.0278	0.566114
$615$	0	0
$616$	−12.9083	−0.520091
$617$	10.6056	0.426963	0.213482	−	0.976947i	$-0.431520\pi$
$617$	10.6056	0.426963	0.213482	+	0.976947i	$0.431520\pi$
$618$	0	0
$619$	17.4222	0.700258	0.350129	−	0.936702i	$-0.386138\pi$
$619$	17.4222	0.700258	0.350129	+	0.936702i	$0.386138\pi$
$620$	0	0
$621$	0	0
$622$	38.7250	1.55273
$623$	22.8167	0.914130
$624$	0	0
$625$	0	0
$626$	−50.2389	−2.00795
$627$	0	0
$628$	−15.8167	−0.631153
$629$	37.5416	1.49688
$630$	0	0
$631$	−39.9361	−1.58983	−0.794915	−	0.606721i	$-0.792485\pi$
$631$	−39.9361	−1.58983	−0.794915	+	0.606721i	$0.792485\pi$
$632$	−0.275019	−0.0109397
$633$	0	0
$634$	22.8167	0.906165
$635$	0	0
$636$	0	0
$637$	−57.5694	−2.28098
$638$	22.8167	0.903320
$639$	0	0
$640$	0	0
$641$	−42.2111	−1.66724	−0.833619	−	0.552340i	$-0.813735\pi$
$641$	−42.2111	−1.66724	−0.833619	+	0.552340i	$0.813735\pi$
$642$	0	0
$643$	22.0000	0.867595	0.433798	−	0.901010i	$-0.357173\pi$
$643$	22.0000	0.867595	0.433798	+	0.901010i	$0.357173\pi$
$644$	52.5416	2.07043
$645$	0	0
$646$	9.00000	0.354100
$647$	17.2389	0.677729	0.338865	−	0.940835i	$-0.389957\pi$
$647$	17.2389	0.677729	0.338865	+	0.940835i	$0.389957\pi$
$648$	0	0
$649$	−0.211103	−0.00828650
$650$	0	0
$651$	0	0
$652$	−18.8167	−0.736917
$653$	−19.1194	−0.748201	−0.374101	−	0.927388i	$-0.622049\pi$
$653$	−19.1194	−0.748201	−0.374101	+	0.927388i	$0.622049\pi$
$654$	0	0
$655$	0	0
$656$	−0.486122	−0.0189799
$657$	0	0
$658$	29.7250	1.15880
$659$	20.0917	0.782660	0.391330	−	0.920250i	$-0.372015\pi$
$659$	20.0917	0.782660	0.391330	+	0.920250i	$0.372015\pi$
$660$	0	0
$661$	12.8167	0.498510	0.249255	−	0.968438i	$-0.419814\pi$
$661$	12.8167	0.498510	0.249255	+	0.968438i	$0.419814\pi$
$662$	−33.1472	−1.28830
$663$	0	0
$664$	43.5416	1.68974
$665$	0	0
$666$	0	0
$667$	−36.6333	−1.41845
$668$	−50.9361	−1.97078
$669$	0	0
$670$	0	0
$671$	2.90833	0.112275
$672$	0	0
$673$	6.02776	0.232353	0.116176	−	0.993229i	$-0.462936\pi$
$673$	6.02776	0.232353	0.116176	+	0.993229i	$0.462936\pi$
$674$	−61.8167	−2.38109
$675$	0	0
$676$	39.6333	1.52436
$677$	−26.2389	−1.00844	−0.504221	−	0.863575i	$-0.668220\pi$
$677$	−26.2389	−1.00844	−0.504221	+	0.863575i	$0.668220\pi$
$678$	0	0
$679$	50.3305	1.93151
$680$	0	0
$681$	0	0
$682$	−9.69722	−0.371326
$683$	−9.84441	−0.376686	−0.188343	−	0.982103i	$-0.560312\pi$
$683$	−9.84441	−0.376686	−0.188343	+	0.982103i	$0.560312\pi$
$684$	0	0
$685$	0	0
$686$	−44.7250	−1.70761
$687$	0	0
$688$	2.18335	0.0832393
$689$	−11.5139	−0.438644
$690$	0	0
$691$	−26.5416	−1.00969	−0.504846	−	0.863210i	$-0.668451\pi$
$691$	−26.5416	−1.00969	−0.504846	+	0.863210i	$0.668451\pi$
$692$	55.5416	2.11138
$693$	0	0
$694$	12.6972	0.481980
$695$	0	0
$696$	0	0
$697$	6.27502	0.237683
$698$	−61.7527	−2.33738
$699$	0	0
$700$	0	0
$701$	26.7889	1.01180	0.505901	−	0.862591i	$-0.331160\pi$
$701$	26.7889	1.01180	0.505901	+	0.862591i	$0.331160\pi$
$702$	0	0
$703$	9.60555	0.362280
$704$	−12.8167	−0.483046
$705$	0	0
$706$	−56.7250	−2.13487
$707$	75.3583	2.83414
$708$	0	0
$709$	11.6333	0.436898	0.218449	−	0.975848i	$-0.429900\pi$
$709$	11.6333	0.436898	0.218449	+	0.975848i	$0.429900\pi$
$710$	0	0
$711$	0	0
$712$	−15.9083	−0.596190
$713$	15.5694	0.583078
$714$	0	0
$715$	0	0
$716$	18.2111	0.680581
$717$	0	0
$718$	−35.0278	−1.30722
$719$	−28.8167	−1.07468	−0.537340	−	0.843366i	$-0.680571\pi$
$719$	−28.8167	−1.07468	−0.537340	+	0.843366i	$0.680571\pi$
$720$	0	0
$721$	−34.0278	−1.26726
$722$	−41.4500	−1.54261
$723$	0	0
$724$	−30.0278	−1.11597
$725$	0	0
$726$	0	0
$727$	−0.330532	−0.0122588	−0.00612938	−	0.999981i	$-0.501951\pi$
$727$	−0.330532	−0.0122588	−0.00612938	+	0.999981i	$0.501951\pi$
$728$	64.5416	2.39207
$729$	0	0
$730$	0	0
$731$	−28.1833	−1.04240
$732$	0	0
$733$	−12.3944	−0.457799	−0.228900	−	0.973450i	$-0.573513\pi$
$733$	−12.3944	−0.457799	−0.228900	+	0.973450i	$0.573513\pi$
$734$	55.9638	2.06566
$735$	0	0
$736$	19.6056	0.722670
$737$	4.00000	0.147342
$738$	0	0
$739$	9.88057	0.363463	0.181731	−	0.983348i	$-0.441830\pi$
$739$	9.88057	0.363463	0.181731	+	0.983348i	$0.441830\pi$
$740$	0	0
$741$	0	0
$742$	−22.8167	−0.837626
$743$	44.3028	1.62531	0.812656	−	0.582744i	$-0.198021\pi$
$743$	44.3028	1.62531	0.812656	+	0.582744i	$0.198021\pi$
$744$	0	0
$745$	0	0
$746$	3.27502	0.119907
$747$	0	0
$748$	−12.9083	−0.471975
$749$	12.9083	0.471660
$750$	0	0
$751$	−5.66947	−0.206882	−0.103441	−	0.994636i	$-0.532985\pi$
$751$	−5.66947	−0.206882	−0.103441	+	0.994636i	$0.532985\pi$
$752$	−0.908327	−0.0331233
$753$	0	0
$754$	−114.083	−4.15467
$755$	0	0
$756$	0	0
$757$	23.0555	0.837967	0.418983	−	0.907994i	$-0.362387\pi$
$757$	23.0555	0.837967	0.418983	+	0.907994i	$0.362387\pi$
$758$	57.1472	2.07568
$759$	0	0
$760$	0	0
$761$	42.4222	1.53780	0.768902	−	0.639367i	$-0.220803\pi$
$761$	42.4222	1.53780	0.768902	+	0.639367i	$0.220803\pi$
$762$	0	0
$763$	28.0278	1.01467
$764$	−22.1194	−0.800253
$765$	0	0
$766$	−49.8167	−1.79995
$767$	1.05551	0.0381124
$768$	0	0
$769$	−26.8167	−0.967033	−0.483517	−	0.875335i	$-0.660641\pi$
$769$	−26.8167	−0.967033	−0.483517	+	0.875335i	$0.660641\pi$
$770$	0	0
$771$	0	0
$772$	4.00000	0.143963
$773$	−22.1194	−0.795581	−0.397790	−	0.917476i	$-0.630223\pi$
$773$	−22.1194	−0.795581	−0.397790	+	0.917476i	$0.630223\pi$
$774$	0	0
$775$	0	0
$776$	−35.0917	−1.25972
$777$	0	0
$778$	−27.6333	−0.990702
$779$	1.60555	0.0575248
$780$	0	0
$781$	−4.60555	−0.164800
$782$	33.2750	1.18991
$783$	0	0
$784$	3.48612	0.124504
$785$	0	0
$786$	0	0
$787$	4.21110	0.150110	0.0750548	−	0.997179i	$-0.476087\pi$
$787$	4.21110	0.150110	0.0750548	+	0.997179i	$0.476087\pi$
$788$	32.0278	1.14094
$789$	0	0
$790$	0	0
$791$	−46.5416	−1.65483
$792$	0	0
$793$	−14.5416	−0.516389
$794$	−58.2666	−2.06780
$795$	0	0
$796$	−80.9638	−2.86969
$797$	−14.5139	−0.514108	−0.257054	−	0.966397i	$-0.582752\pi$
$797$	−14.5139	−0.514108	−0.257054	+	0.966397i	$0.582752\pi$
$798$	0	0
$799$	11.7250	0.414800
$800$	0	0
$801$	0	0
$802$	62.6611	2.21264
$803$	−2.90833	−0.102633
$804$	0	0
$805$	0	0
$806$	48.4861	1.70785
$807$	0	0
$808$	−52.5416	−1.84841
$809$	3.63331	0.127740	0.0638701	−	0.997958i	$-0.479656\pi$
$809$	3.63331	0.127740	0.0638701	+	0.997958i	$0.479656\pi$
$810$	0	0
$811$	−54.8722	−1.92682	−0.963411	−	0.268028i	$-0.913628\pi$
$811$	−54.8722	−1.92682	−0.963411	+	0.268028i	$0.913628\pi$
$812$	−140.808	−4.94140
$813$	0	0
$814$	−22.1194	−0.775286
$815$	0	0
$816$	0	0
$817$	−7.21110	−0.252285
$818$	18.9083	0.661114
$819$	0	0
$820$	0	0
$821$	−12.0000	−0.418803	−0.209401	−	0.977830i	$-0.567152\pi$
$821$	−12.0000	−0.418803	−0.209401	+	0.977830i	$0.567152\pi$
$822$	0	0
$823$	10.4222	0.363295	0.181648	−	0.983364i	$-0.441857\pi$
$823$	10.4222	0.363295	0.181648	+	0.983364i	$0.441857\pi$
$824$	23.7250	0.826499
$825$	0	0
$826$	2.09167	0.0727786
$827$	−7.81665	−0.271812	−0.135906	−	0.990722i	$-0.543394\pi$
$827$	−7.81665	−0.271812	−0.135906	+	0.990722i	$0.543394\pi$
$828$	0	0
$829$	−38.7527	−1.34594	−0.672969	−	0.739671i	$-0.734981\pi$
$829$	−38.7527	−1.34594	−0.672969	+	0.739671i	$0.734981\pi$
$830$	0	0
$831$	0	0
$832$	64.0833	2.22169
$833$	−45.0000	−1.55916
$834$	0	0
$835$	0	0
$836$	−3.30278	−0.114229
$837$	0	0
$838$	−31.3305	−1.08230
$839$	−16.1194	−0.556505	−0.278252	−	0.960508i	$-0.589755\pi$
$839$	−16.1194	−0.556505	−0.278252	+	0.960508i	$0.589755\pi$
$840$	0	0
$841$	69.1749	2.38534
$842$	9.90833	0.341463
$843$	0	0
$844$	83.3583	2.86931
$845$	0	0
$846$	0	0
$847$	−4.30278	−0.147845
$848$	0.697224	0.0239428
$849$	0	0
$850$	0	0
$851$	35.5139	1.21740
$852$	0	0
$853$	−19.7250	−0.675370	−0.337685	−	0.941259i	$-0.609644\pi$
$853$	−19.7250	−0.675370	−0.337685	+	0.941259i	$0.609644\pi$
$854$	−28.8167	−0.986086
$855$	0	0
$856$	−9.00000	−0.307614
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	48.6056	1.65840	0.829200	−	0.558952i	$-0.188796\pi$
$859$	48.6056	1.65840	0.829200	+	0.558952i	$0.188796\pi$
$860$	0	0
$861$	0	0
$862$	−75.9916	−2.58828
$863$	−19.6056	−0.667381	−0.333690	−	0.942683i	$-0.608294\pi$
$863$	−19.6056	−0.667381	−0.333690	+	0.942683i	$0.608294\pi$
$864$	0	0
$865$	0	0
$866$	−11.5139	−0.391258
$867$	0	0
$868$	59.8444	2.03125
$869$	−0.0916731	−0.00310980
$870$	0	0
$871$	−20.0000	−0.677674
$872$	−19.5416	−0.661763
$873$	0	0
$874$	8.51388	0.287986
$875$	0	0
$876$	0	0
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	47.6611	1.60848
$879$	0	0
$880$	0	0
$881$	34.5416	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$881$	34.5416	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$882$	0	0
$883$	12.4500	0.418975	0.209487	−	0.977811i	$-0.432821\pi$
$883$	12.4500	0.418975	0.209487	+	0.977811i	$0.432821\pi$
$884$	64.5416	2.17077
$885$	0	0
$886$	−3.21110	−0.107879
$887$	47.2389	1.58613	0.793063	−	0.609140i	$-0.208485\pi$
$887$	47.2389	1.58613	0.793063	+	0.609140i	$0.208485\pi$
$888$	0	0
$889$	−73.6611	−2.47051
$890$	0	0
$891$	0	0
$892$	−68.1472	−2.28174
$893$	3.00000	0.100391
$894$	0	0
$895$	0	0
$896$	81.3583	2.71799
$897$	0	0
$898$	−95.5971	−3.19012
$899$	−41.7250	−1.39161
$900$	0	0
$901$	−9.00000	−0.299833
$902$	−3.69722	−0.123104
$903$	0	0
$904$	32.4500	1.07927
$905$	0	0
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	17.5139	0.581218
$909$	0	0
$910$	0	0
$911$	39.2111	1.29912	0.649561	−	0.760310i	$-0.274953\pi$
$911$	39.2111	1.29912	0.649561	+	0.760310i	$0.274953\pi$
$912$	0	0
$913$	14.5139	0.480339
$914$	55.9638	1.85112
$915$	0	0
$916$	45.3305	1.49776
$917$	3.90833	0.129064
$918$	0	0
$919$	41.2111	1.35943	0.679714	−	0.733477i	$-0.262104\pi$
$919$	41.2111	1.35943	0.679714	+	0.733477i	$0.262104\pi$
$920$	0	0
$921$	0	0
$922$	−40.9638	−1.34907
$923$	23.0278	0.757968
$924$	0	0
$925$	0	0
$926$	−60.3583	−1.98350
$927$	0	0
$928$	−52.5416	−1.72476
$929$	−46.3944	−1.52215	−0.761076	−	0.648662i	$-0.775329\pi$
$929$	−46.3944	−1.52215	−0.761076	+	0.648662i	$0.775329\pi$
$930$	0	0
$931$	−11.5139	−0.377352
$932$	16.8167	0.550848
$933$	0	0
$934$	56.7250	1.85610
$935$	0	0
$936$	0	0
$937$	−5.21110	−0.170239	−0.0851196	−	0.996371i	$-0.527127\pi$
$937$	−5.21110	−0.170239	−0.0851196	+	0.996371i	$0.527127\pi$
$938$	−39.6333	−1.29407
$939$	0	0
$940$	0	0
$941$	−52.3944	−1.70801	−0.854005	−	0.520265i	$-0.825833\pi$
$941$	−52.3944	−1.70801	−0.854005	+	0.520265i	$0.825833\pi$
$942$	0	0
$943$	5.93608	0.193305
$944$	−0.0639167	−0.00208031
$945$	0	0
$946$	16.6056	0.539893
$947$	−36.6333	−1.19042	−0.595211	−	0.803569i	$-0.702932\pi$
$947$	−36.6333	−1.19042	−0.595211	+	0.803569i	$0.702932\pi$
$948$	0	0
$949$	14.5416	0.472041
$950$	0	0
$951$	0	0
$952$	50.4500	1.63509
$953$	−49.2666	−1.59590	−0.797951	−	0.602722i	$-0.794083\pi$
$953$	−49.2666	−1.59590	−0.797951	+	0.602722i	$0.794083\pi$
$954$	0	0
$955$	0	0
$956$	13.6056	0.440035
$957$	0	0
$958$	30.3583	0.980832
$959$	9.00000	0.290625
$960$	0	0
$961$	−13.2666	−0.427955
$962$	110.597	3.56580
$963$	0	0
$964$	−82.3583	−2.65258
$965$	0	0
$966$	0	0
$967$	−4.09167	−0.131579	−0.0657897	−	0.997834i	$-0.520957\pi$
$967$	−4.09167	−0.131579	−0.0657897	+	0.997834i	$0.520957\pi$
$968$	3.00000	0.0964237
$969$	0	0
$970$	0	0
$971$	30.3583	0.974244	0.487122	−	0.873334i	$-0.338047\pi$
$971$	30.3583	0.974244	0.487122	+	0.873334i	$0.338047\pi$
$972$	0	0
$973$	−35.3305	−1.13264
$974$	23.5139	0.753433
$975$	0	0
$976$	0.880571	0.0281864
$977$	−15.9722	−0.510997	−0.255499	−	0.966809i	$-0.582240\pi$
$977$	−15.9722	−0.510997	−0.255499	+	0.966809i	$0.582240\pi$
$978$	0	0
$979$	−5.30278	−0.169477
$980$	0	0
$981$	0	0
$982$	55.7527	1.77914
$983$	−48.8444	−1.55789	−0.778947	−	0.627089i	$-0.784246\pi$
$983$	−48.8444	−1.55789	−0.778947	+	0.627089i	$0.784246\pi$
$984$	0	0
$985$	0	0
$986$	−89.1749	−2.83991
$987$	0	0
$988$	16.5139	0.525376
$989$	−26.6611	−0.847773
$990$	0	0
$991$	−6.09167	−0.193508	−0.0967542	−	0.995308i	$-0.530846\pi$
$991$	−6.09167	−0.193508	−0.0967542	+	0.995308i	$0.530846\pi$
$992$	22.3305	0.708995
$993$	0	0
$994$	45.6333	1.44740
$995$	0	0
$996$	0	0
$997$	−14.2750	−0.452094	−0.226047	−	0.974116i	$-0.572580\pi$
$997$	−14.2750	−0.452094	−0.226047	+	0.974116i	$0.572580\pi$
$998$	−49.5416	−1.56821
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.t.1.2		2
3.2	odd	2		275.2.a.e.1.1	✓	2
5.2	odd	4		2475.2.c.k.199.4		4
5.3	odd	4		2475.2.c.k.199.1		4
5.4	even	2		2475.2.a.o.1.1		2
12.11	even	2		4400.2.a.bs.1.1		2
15.2	even	4		275.2.b.c.199.1		4
15.8	even	4		275.2.b.c.199.4		4
15.14	odd	2		275.2.a.f.1.2	yes	2
33.32	even	2		3025.2.a.n.1.2		2
60.23	odd	4		4400.2.b.y.4049.2		4
60.47	odd	4		4400.2.b.y.4049.3		4
60.59	even	2		4400.2.a.bh.1.2		2
165.164	even	2		3025.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.1	✓	2	3.2	odd	2
275.2.a.f.1.2	yes	2	15.14	odd	2
275.2.b.c.199.1		4	15.2	even	4
275.2.b.c.199.4		4	15.8	even	4
2475.2.a.o.1.1		2	5.4	even	2
2475.2.a.t.1.2		2	1.1	even	1	trivial
2475.2.c.k.199.1		4	5.3	odd	4
2475.2.c.k.199.4		4	5.2	odd	4
3025.2.a.h.1.1		2	165.164	even	2
3025.2.a.n.1.2		2	33.32	even	2
4400.2.a.bh.1.2		2	60.59	even	2
4400.2.a.bs.1.1		2	12.11	even	2
4400.2.b.y.4049.2		4	60.23	odd	4
4400.2.b.y.4049.3		4	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{2} +3.30278 q^{4} -4.30278 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q+2.30278 q^{2} +3.30278 q^{4} -4.30278 q^{7} +3.00000 q^{8} +1.00000 q^{11} -5.00000 q^{13} -9.90833 q^{14} +0.302776 q^{16} -3.90833 q^{17} -1.00000 q^{19} +2.30278 q^{22} -3.69722 q^{23} -11.5139 q^{26} -14.2111 q^{28} +9.90833 q^{29} -4.21110 q^{31} -5.30278 q^{32} -9.00000 q^{34} -9.60555 q^{37} -2.30278 q^{38} -1.60555 q^{41} +7.21110 q^{43} +3.30278 q^{44} -8.51388 q^{46} -3.00000 q^{47} +11.5139 q^{49} -16.5139 q^{52} +2.30278 q^{53} -12.9083 q^{56} +22.8167 q^{58} -0.211103 q^{59} +2.90833 q^{61} -9.69722 q^{62} -12.8167 q^{64} +4.00000 q^{67} -12.9083 q^{68} -4.60555 q^{71} -2.90833 q^{73} -22.1194 q^{74} -3.30278 q^{76} -4.30278 q^{77} -0.0916731 q^{79} -3.69722 q^{82} +14.5139 q^{83} +16.6056 q^{86} +3.00000 q^{88} -5.30278 q^{89} +21.5139 q^{91} -12.2111 q^{92} -6.90833 q^{94} -11.6972 q^{97} +26.5139 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + 3 * q^4 - 5 * q^7 + 6 * q^8	\( 2 q + q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8} + 2 q^{11} - 10 q^{13} - 9 q^{14} - 3 q^{16} + 3 q^{17} - 2 q^{19} + q^{22} - 11 q^{23} - 5 q^{26} - 14 q^{28} + 9 q^{29} + 6 q^{31} - 7 q^{32} - 18 q^{34} - 12 q^{37} - q^{38} + 4 q^{41} + 3 q^{44} + q^{46} - 6 q^{47} + 5 q^{49} - 15 q^{52} + q^{53} - 15 q^{56} + 24 q^{58} + 14 q^{59} - 5 q^{61} - 23 q^{62} - 4 q^{64} + 8 q^{67} - 15 q^{68} - 2 q^{71} + 5 q^{73} - 19 q^{74} - 3 q^{76} - 5 q^{77} - 11 q^{79} - 11 q^{82} + 11 q^{83} + 26 q^{86} + 6 q^{88} - 7 q^{89} + 25 q^{91} - 10 q^{92} - 3 q^{94} - 27 q^{97} + 35 q^{98}+O(q^{100}) \) 2 * q + q^2 + 3 * q^4 - 5 * q^7 + 6 * q^8 + 2 * q^11 - 10 * q^13 - 9 * q^14 - 3 * q^16 + 3 * q^17 - 2 * q^19 + q^22 - 11 * q^23 - 5 * q^26 - 14 * q^28 + 9 * q^29 + 6 * q^31 - 7 * q^32 - 18 * q^34 - 12 * q^37 - q^38 + 4 * q^41 + 3 * q^44 + q^46 - 6 * q^47 + 5 * q^49 - 15 * q^52 + q^53 - 15 * q^56 + 24 * q^58 + 14 * q^59 - 5 * q^61 - 23 * q^62 - 4 * q^64 + 8 * q^67 - 15 * q^68 - 2 * q^71 + 5 * q^73 - 19 * q^74 - 3 * q^76 - 5 * q^77 - 11 * q^79 - 11 * q^82 + 11 * q^83 + 26 * q^86 + 6 * q^88 - 7 * q^89 + 25 * q^91 - 10 * q^92 - 3 * q^94 - 27 * q^97 + 35 * q^98

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