LMFDB - Embedded newform 2475.2.c.k.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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, 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("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (of order $2$, degree $1$, not 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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-2.30278i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.k.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} +3.00000i q^{8} +O(q^{10})$	$q-2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} +3.00000i q^{8} +1.00000 q^{11} -5.00000i q^{13} +9.90833 q^{14} +0.302776 q^{16} +3.90833i q^{17} +1.00000 q^{19} -2.30278i q^{22} -3.69722i q^{23} -11.5139 q^{26} -14.2111i q^{28} -9.90833 q^{29} -4.21110 q^{31} +5.30278i q^{32} +9.00000 q^{34} +9.60555i q^{37} -2.30278i q^{38} -1.60555 q^{41} +7.21110i q^{43} -3.30278 q^{44} -8.51388 q^{46} +3.00000i q^{47} -11.5139 q^{49} +16.5139i q^{52} +2.30278i q^{53} -12.9083 q^{56} +22.8167i q^{58} +0.211103 q^{59} +2.90833 q^{61} +9.69722i q^{62} +12.8167 q^{64} -4.00000i q^{67} -12.9083i q^{68} -4.60555 q^{71} -2.90833i q^{73} +22.1194 q^{74} -3.30278 q^{76} +4.30278i q^{77} +0.0916731 q^{79} +3.69722i q^{82} +14.5139i q^{83} +16.6056 q^{86} +3.00000i q^{88} +5.30278 q^{89} +21.5139 q^{91} +12.2111i q^{92} +6.90833 q^{94} +11.6972i q^{97} +26.5139i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4}+O(q^{10}) $ 4 * q - 6 * q^4	$ 4 q - 6 q^{4} + 4 q^{11} + 18 q^{14} - 6 q^{16} + 4 q^{19} - 10 q^{26} - 18 q^{29} + 12 q^{31} + 36 q^{34} + 8 q^{41} - 6 q^{44} + 2 q^{46} - 10 q^{49} - 30 q^{56} - 28 q^{59} - 10 q^{61} + 8 q^{64} - 4 q^{71} + 38 q^{74} - 6 q^{76} + 22 q^{79} + 52 q^{86} + 14 q^{89} + 50 q^{91} + 6 q^{94}+O(q^{100}) $ 4 * q - 6 * q^4 + 4 * q^11 + 18 * q^14 - 6 * q^16 + 4 * q^19 - 10 * q^26 - 18 * q^29 + 12 * q^31 + 36 * q^34 + 8 * q^41 - 6 * q^44 + 2 * q^46 - 10 * q^49 - 30 * q^56 - 28 * q^59 - 10 * q^61 + 8 * q^64 - 4 * q^71 + 38 * q^74 - 6 * q^76 + 22 * q^79 + 52 * q^86 + 14 * q^89 + 50 * q^91 + 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 2.30278i	− 1.62831i	−0.580649	−	0.814154i	$-0.697201\pi$
$2$	− 2.30278i	− 1.62831i	0.580649	−	0.814154i	$-0.302799\pi$
$3$	0	0
$3$	0	0
$4$	−3.30278	−1.65139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.30278i	1.62630i	0.582057	+	0.813148i	$0.302248\pi$
$7$	4.30278i	1.62630i	−0.582057	+	0.813148i	$0.697752\pi$
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 5.00000i	− 1.38675i	−0.720577	−	0.693375i	$-0.756123\pi$
$13$	− 5.00000i	− 1.38675i	0.720577	−	0.693375i	$-0.243877\pi$
$14$	9.90833	2.64811
$15$	0	0
$16$	0.302776	0.0756939
$17$	3.90833i	0.947909i	0.880549	+	0.473954i	$0.157174\pi$
$17$	3.90833i	0.947909i	−0.880549	+	0.473954i	$0.842826\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	− 2.30278i	− 0.490953i
$23$	− 3.69722i	− 0.770925i	−0.922724	−	0.385462i	$-0.874042\pi$
$23$	− 3.69722i	− 0.770925i	0.922724	−	0.385462i	$-0.125958\pi$
$24$	0	0
$25$	0	0
$26$	−11.5139	−2.25806
$27$	0	0
$28$	− 14.2111i	− 2.68565i
$29$	−9.90833	−1.83993	−0.919965	−	0.392000i	$-0.871783\pi$
$29$	−9.90833	−1.83993	−0.919965	+	0.392000i	$0.871783\pi$
$30$	0	0
$31$	−4.21110	−0.756336	−0.378168	−	0.925737i	$-0.623446\pi$
$31$	−4.21110	−0.756336	−0.378168	+	0.925737i	$0.623446\pi$
$32$	5.30278i	0.937407i
$33$	0	0
$34$	9.00000	1.54349
$35$	0	0
$36$	0	0
$37$	9.60555i	1.57914i	0.613659	+	0.789571i	$0.289697\pi$
$37$	9.60555i	1.57914i	−0.613659	+	0.789571i	$0.710303\pi$
$38$	− 2.30278i	− 0.373560i
$39$	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
$43$	7.21110i	1.09968i	0.835269	+	0.549841i	$0.185312\pi$
$43$	7.21110i	1.09968i	−0.835269	+	0.549841i	$0.814688\pi$
$44$	−3.30278	−0.497912
$45$	0	0
$46$	−8.51388	−1.25530
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	0	0
$49$	−11.5139	−1.64484
$50$	0	0
$51$	0	0
$52$	16.5139i	2.29006i
$53$	2.30278i	0.316311i	0.987414	+	0.158155i	$0.0505547\pi$
$53$	2.30278i	0.316311i	−0.987414	+	0.158155i	$0.949445\pi$
$54$	0	0
$55$	0	0
$56$	−12.9083	−1.72495
$57$	0	0
$58$	22.8167i	2.99597i
$59$	0.211103	0.0274832	0.0137416	−	0.999906i	$-0.495626\pi$
$59$	0.211103	0.0274832	0.0137416	+	0.999906i	$0.495626\pi$
$60$	0	0
$61$	2.90833	0.372373	0.186187	−	0.982514i	$-0.440387\pi$
$61$	2.90833	0.372373	0.186187	+	0.982514i	$0.440387\pi$
$62$	9.69722i	1.23155i
$63$	0	0
$64$	12.8167	1.60208
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 12.9083i	− 1.56536i
$69$	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
$73$	− 2.90833i	− 0.340394i	−0.985410	−	0.170197i	$-0.945560\pi$
$73$	− 2.90833i	− 0.340394i	0.985410	−	0.170197i	$-0.0544404\pi$
$74$	22.1194	2.57133
$75$	0	0
$76$	−3.30278	−0.378854
$77$	4.30278i	0.490347i
$78$	0	0
$79$	0.0916731	0.0103140	0.00515701	−	0.999987i	$-0.498358\pi$
$79$	0.0916731	0.0103140	0.00515701	+	0.999987i	$0.498358\pi$
$80$	0	0
$81$	0	0
$82$	3.69722i	0.408290i
$83$	14.5139i	1.59311i	0.604569	+	0.796553i	$0.293345\pi$
$83$	14.5139i	1.59311i	−0.604569	+	0.796553i	$0.706655\pi$
$84$	0	0
$85$	0	0
$86$	16.6056	1.79062
$87$	0	0
$88$	3.00000i	0.319801i
$89$	5.30278	0.562093	0.281047	−	0.959694i	$-0.409318\pi$
$89$	5.30278	0.562093	0.281047	+	0.959694i	$0.409318\pi$
$90$	0	0
$91$	21.5139	2.25527
$92$	12.2111i	1.27310i
$93$	0	0
$94$	6.90833	0.712540
$95$	0	0
$96$	0	0
$97$	11.6972i	1.18767i	0.804586	+	0.593837i	$0.202387\pi$
$97$	11.6972i	1.18767i	−0.804586	+	0.593837i	$0.797613\pi$
$98$	26.5139i	2.67831i
$99$	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
$103$	7.90833i	0.779231i	0.920978	+	0.389615i	$0.127392\pi$
$103$	7.90833i	0.779231i	−0.920978	+	0.389615i	$0.872608\pi$
$104$	15.0000	1.47087
$105$	0	0
$106$	5.30278	0.515051
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	6.51388	0.623916	0.311958	−	0.950096i	$-0.399015\pi$
$109$	6.51388	0.623916	0.311958	+	0.950096i	$0.399015\pi$
$110$	0	0
$111$	0	0
$112$	1.30278i	0.123101i
$113$	10.8167i	1.01755i	0.860901	+	0.508773i	$0.169901\pi$
$113$	10.8167i	1.01755i	−0.860901	+	0.508773i	$0.830099\pi$
$114$	0	0
$115$	0	0
$116$	32.7250	3.03844
$117$	0	0
$118$	− 0.486122i	− 0.0447511i
$119$	−16.8167	−1.54158
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 6.69722i	− 0.606338i
$123$	0	0
$124$	13.9083	1.24900
$125$	0	0
$126$	0	0
$127$	− 17.1194i	− 1.51910i	−0.650447	−	0.759552i	$-0.725418\pi$
$127$	− 17.1194i	− 1.51910i	0.650447	−	0.759552i	$-0.274582\pi$
$128$	− 18.9083i	− 1.67128i
$129$	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
$133$	4.30278i	0.373098i
$134$	−9.21110	−0.795718
$135$	0	0
$136$	−11.7250	−1.00541
$137$	2.09167i	0.178704i	0.996000	+	0.0893518i	$0.0284796\pi$
$137$	2.09167i	0.178704i	−0.996000	+	0.0893518i	$0.971520\pi$
$138$	0	0
$139$	−8.21110	−0.696457	−0.348228	−	0.937410i	$-0.613217\pi$
$139$	−8.21110	−0.696457	−0.348228	+	0.937410i	$0.613217\pi$
$140$	0	0
$141$	0	0
$142$	10.6056i	0.889998i
$143$	− 5.00000i	− 0.418121i
$144$	0	0
$145$	0	0
$146$	−6.69722	−0.554266
$147$	0	0
$148$	− 31.7250i	− 2.60778i
$149$	2.78890	0.228475	0.114238	−	0.993453i	$-0.463557\pi$
$149$	2.78890	0.228475	0.114238	+	0.993453i	$0.463557\pi$
$150$	0	0
$151$	−20.8167	−1.69404	−0.847018	−	0.531565i	$-0.821604\pi$
$151$	−20.8167	−1.69404	−0.847018	+	0.531565i	$0.821604\pi$
$152$	3.00000i	0.243332i
$153$	0	0
$154$	9.90833	0.798436
$155$	0	0
$156$	0	0
$157$	4.78890i	0.382196i	0.981571	+	0.191098i	$0.0612048\pi$
$157$	4.78890i	0.382196i	−0.981571	+	0.191098i	$0.938795\pi$
$158$	− 0.211103i	− 0.0167944i
$159$	0	0
$160$	0	0
$161$	15.9083	1.25375
$162$	0	0
$163$	− 5.69722i	− 0.446241i	−0.974791	−	0.223121i	$-0.928376\pi$
$163$	− 5.69722i	− 0.446241i	0.974791	−	0.223121i	$-0.0716243\pi$
$164$	5.30278	0.414077
$165$	0	0
$166$	33.4222	2.59407
$167$	15.4222i	1.19341i	0.802462	+	0.596703i	$0.203523\pi$
$167$	15.4222i	1.19341i	−0.802462	+	0.596703i	$0.796477\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	− 23.8167i	− 1.81600i
$173$	16.8167i	1.27855i	0.768980	+	0.639273i	$0.220765\pi$
$173$	16.8167i	1.27855i	−0.768980	+	0.639273i	$0.779235\pi$
$174$	0	0
$175$	0	0
$176$	0.302776	0.0228226
$177$	0	0
$178$	− 12.2111i	− 0.915261i
$179$	−5.51388	−0.412127	−0.206063	−	0.978539i	$-0.566065\pi$
$179$	−5.51388	−0.412127	−0.206063	+	0.978539i	$0.566065\pi$
$180$	0	0
$181$	−9.09167	−0.675779	−0.337889	−	0.941186i	$-0.609713\pi$
$181$	−9.09167	−0.675779	−0.337889	+	0.941186i	$0.609713\pi$
$182$	− 49.5416i	− 3.67227i
$183$	0	0
$184$	11.0917	0.817689
$185$	0	0
$186$	0	0
$187$	3.90833i	0.285805i
$188$	− 9.90833i	− 0.722639i
$189$	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
$193$	1.21110i	0.0871771i	0.999050	+	0.0435885i	$0.0138791\pi$
$193$	1.21110i	0.0871771i	−0.999050	+	0.0435885i	$0.986121\pi$
$194$	26.9361	1.93390
$195$	0	0
$196$	38.0278	2.71627
$197$	− 9.69722i	− 0.690899i	−0.938437	−	0.345449i	$-0.887727\pi$
$197$	− 9.69722i	− 0.690899i	0.938437	−	0.345449i	$-0.112273\pi$
$198$	0	0
$199$	24.5139	1.73774	0.868871	−	0.495038i	$-0.164846\pi$
$199$	24.5139	1.73774	0.868871	+	0.495038i	$0.164846\pi$
$200$	0	0
$201$	0	0
$202$	40.3305i	2.83765i
$203$	− 42.6333i	− 2.99227i
$204$	0	0
$205$	0	0
$206$	18.2111	1.26883
$207$	0	0
$208$	− 1.51388i	− 0.104969i
$209$	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	$0.164919\pi$
$212$	− 7.60555i	− 0.522351i
$213$	0	0
$214$	6.90833	0.472244
$215$	0	0
$216$	0	0
$217$	− 18.1194i	− 1.23003i
$218$	− 15.0000i	− 1.01593i
$219$	0	0
$220$	0	0
$221$	19.5416	1.31451
$222$	0	0
$223$	− 20.6333i	− 1.38171i	−0.722994	−	0.690854i	$-0.757235\pi$
$223$	− 20.6333i	− 1.38171i	0.722994	−	0.690854i	$-0.242765\pi$
$224$	−22.8167	−1.52450
$225$	0	0
$226$	24.9083	1.65688
$227$	− 5.30278i	− 0.351958i	−0.984394	−	0.175979i	$-0.943691\pi$
$227$	− 5.30278i	− 0.351958i	0.984394	−	0.175979i	$-0.0563090\pi$
$228$	0	0
$229$	−13.7250	−0.906972	−0.453486	−	0.891263i	$-0.649820\pi$
$229$	−13.7250	−0.906972	−0.453486	+	0.891263i	$0.649820\pi$
$230$	0	0
$231$	0	0
$232$	− 29.7250i	− 1.95154i
$233$	5.09167i	0.333567i	0.985994	+	0.166783i	$0.0533380\pi$
$233$	5.09167i	0.333567i	−0.985994	+	0.166783i	$0.946662\pi$
$234$	0	0
$235$	0	0
$236$	−0.697224	−0.0453854
$237$	0	0
$238$	38.7250i	2.51017i
$239$	−4.11943	−0.266464	−0.133232	−	0.991085i	$-0.542535\pi$
$239$	−4.11943	−0.266464	−0.133232	+	0.991085i	$0.542535\pi$
$240$	0	0
$241$	−24.9361	−1.60627	−0.803137	−	0.595794i	$-0.796837\pi$
$241$	−24.9361	−1.60627	−0.803137	+	0.595794i	$0.796837\pi$
$242$	− 2.30278i	− 0.148028i
$243$	0	0
$244$	−9.60555	−0.614932
$245$	0	0
$246$	0	0
$247$	− 5.00000i	− 0.318142i
$248$	− 12.6333i	− 0.802216i
$249$	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
$253$	− 3.69722i	− 0.232443i
$254$	−39.4222	−2.47357
$255$	0	0
$256$	−17.9083	−1.11927
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	−41.3305	−2.56815
$260$	0	0
$261$	0	0
$262$	2.09167i	0.129224i
$263$	− 1.18335i	− 0.0729683i	−0.999334	−	0.0364841i	$-0.988384\pi$
$263$	− 1.18335i	− 0.0729683i	0.999334	−	0.0364841i	$-0.0116158\pi$
$264$	0	0
$265$	0	0
$266$	9.90833	0.607519
$267$	0	0
$268$	13.2111i	0.806997i
$269$	−23.7250	−1.44654	−0.723269	−	0.690567i	$-0.757361\pi$
$269$	−23.7250	−1.44654	−0.723269	+	0.690567i	$0.757361\pi$
$270$	0	0
$271$	14.2111	0.863263	0.431632	−	0.902050i	$-0.357938\pi$
$271$	14.2111	0.863263	0.431632	+	0.902050i	$0.357938\pi$
$272$	1.18335i	0.0717509i
$273$	0	0
$274$	4.81665	0.290985
$275$	0	0
$276$	0	0
$277$	21.6056i	1.29815i	0.760724	+	0.649076i	$0.224844\pi$
$277$	21.6056i	1.29815i	−0.760724	+	0.649076i	$0.775156\pi$
$278$	18.9083i	1.13405i
$279$	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
$283$	− 2.69722i	− 0.160333i	−0.996781	−	0.0801667i	$-0.974455\pi$
$283$	− 2.69722i	− 0.160333i	0.996781	−	0.0801667i	$-0.0255453\pi$
$284$	15.2111	0.902613
$285$	0	0
$286$	−11.5139	−0.680830
$287$	− 6.90833i	− 0.407786i
$288$	0	0
$289$	1.72498	0.101469
$290$	0	0
$291$	0	0
$292$	9.60555i	0.562122i
$293$	15.2111i	0.888642i	0.895868	+	0.444321i	$0.146555\pi$
$293$	15.2111i	0.888642i	−0.895868	+	0.444321i	$0.853445\pi$
$294$	0	0
$295$	0	0
$296$	−28.8167	−1.67493
$297$	0	0
$298$	− 6.42221i	− 0.372028i
$299$	−18.4861	−1.06908
$300$	0	0
$301$	−31.0278	−1.78841
$302$	47.9361i	2.75841i
$303$	0	0
$304$	0.302776	0.0173654
$305$	0	0
$306$	0	0
$307$	− 6.09167i	− 0.347670i	−0.984775	−	0.173835i	$-0.944384\pi$
$307$	− 6.09167i	− 0.347670i	0.984775	−	0.173835i	$-0.0556160\pi$
$308$	− 14.2111i	− 0.809753i
$309$	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
$313$	− 21.8167i	− 1.23315i	−0.787296	−	0.616575i	$-0.788520\pi$
$313$	− 21.8167i	− 1.23315i	0.787296	−	0.616575i	$-0.211480\pi$
$314$	11.0278	0.622332
$315$	0	0
$316$	−0.302776	−0.0170325
$317$	− 9.90833i	− 0.556507i	−0.960508	−	0.278254i	$-0.910244\pi$
$317$	− 9.90833i	− 0.556507i	0.960508	−	0.278254i	$-0.0897555\pi$
$318$	0	0
$319$	−9.90833	−0.554760
$320$	0	0
$321$	0	0
$322$	− 36.6333i	− 2.04149i
$323$	3.90833i	0.217465i
$324$	0	0
$325$	0	0
$326$	−13.1194	−0.726618
$327$	0	0
$328$	− 4.81665i	− 0.265955i
$329$	−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	$0.629462\pi$
$332$	− 47.9361i	− 2.63083i
$333$	0	0
$334$	35.5139	1.94323
$335$	0	0
$336$	0	0
$337$	26.8444i	1.46231i	0.682212	+	0.731154i	$0.261018\pi$
$337$	26.8444i	1.46231i	−0.682212	+	0.731154i	$0.738982\pi$
$338$	27.6333i	1.50305i
$339$	0	0
$340$	0	0
$341$	−4.21110	−0.228044
$342$	0	0
$343$	− 19.4222i	− 1.04870i
$344$	−21.6333	−1.16639
$345$	0	0
$346$	38.7250	2.08187
$347$	− 5.51388i	− 0.296000i	−0.988987	−	0.148000i	$-0.952716\pi$
$347$	− 5.51388i	− 0.296000i	0.988987	−	0.148000i	$-0.0472836\pi$
$348$	0	0
$349$	26.8167	1.43546	0.717731	−	0.696320i	$-0.245181\pi$
$349$	26.8167	1.43546	0.717731	+	0.696320i	$0.245181\pi$
$350$	0	0
$351$	0	0
$352$	5.30278i	0.282639i
$353$	− 24.6333i	− 1.31110i	−0.755152	−	0.655549i	$-0.772437\pi$
$353$	− 24.6333i	− 1.31110i	0.755152	−	0.655549i	$-0.227563\pi$
$354$	0	0
$355$	0	0
$356$	−17.5139	−0.928234
$357$	0	0
$358$	12.6972i	0.671069i
$359$	15.2111	0.802811	0.401406	−	0.915900i	$-0.368522\pi$
$359$	15.2111	0.802811	0.401406	+	0.915900i	$0.368522\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	20.9361i	1.10038i
$363$	0	0
$364$	−71.0555	−3.72432
$365$	0	0
$366$	0	0
$367$	− 24.3028i	− 1.26859i	−0.773089	−	0.634297i	$-0.781290\pi$
$367$	− 24.3028i	− 1.26859i	0.773089	−	0.634297i	$-0.218710\pi$
$368$	− 1.11943i	− 0.0583543i
$369$	0	0
$370$	0	0
$371$	−9.90833	−0.514415
$372$	0	0
$373$	1.42221i	0.0736390i	0.999322	+	0.0368195i	$0.0117227\pi$
$373$	1.42221i	0.0736390i	−0.999322	+	0.0368195i	$0.988277\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−9.00000	−0.464140
$377$	49.5416i	2.55152i
$378$	0	0
$379$	−24.8167	−1.27475	−0.637373	−	0.770555i	$-0.719979\pi$
$379$	−24.8167	−1.27475	−0.637373	+	0.770555i	$0.719979\pi$
$380$	0	0
$381$	0	0
$382$	15.4222i	0.789069i
$383$	− 21.6333i	− 1.10541i	−0.833377	−	0.552705i	$-0.813596\pi$
$383$	− 21.6333i	− 1.10541i	0.833377	−	0.552705i	$-0.186404\pi$
$384$	0	0
$385$	0	0
$386$	2.78890	0.141951
$387$	0	0
$388$	− 38.6333i	− 1.96131i
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	14.4500	0.730766
$392$	− 34.5416i	− 1.74462i
$393$	0	0
$394$	−22.3305	−1.12500
$395$	0	0
$396$	0	0
$397$	25.3028i	1.26991i	0.772549	+	0.634955i	$0.218981\pi$
$397$	25.3028i	1.26991i	−0.772549	+	0.634955i	$0.781019\pi$
$398$	− 56.4500i	− 2.82958i
$399$	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
$403$	21.0555i	1.04885i
$404$	57.8444	2.87787
$405$	0	0
$406$	−98.1749	−4.87234
$407$	9.60555i	0.476129i
$408$	0	0
$409$	−8.21110	−0.406013	−0.203006	−	0.979177i	$-0.565071\pi$
$409$	−8.21110	−0.406013	−0.203006	+	0.979177i	$0.565071\pi$
$410$	0	0
$411$	0	0
$412$	− 26.1194i	− 1.28681i
$413$	0.908327i	0.0446958i
$414$	0	0
$415$	0	0
$416$	26.5139	1.29995
$417$	0	0
$418$	− 2.30278i	− 0.112632i
$419$	13.6056	0.664675	0.332337	−	0.943161i	$-0.392163\pi$
$419$	13.6056	0.664675	0.332337	+	0.943161i	$0.392163\pi$
$420$	0	0
$421$	4.30278	0.209704	0.104852	−	0.994488i	$-0.466563\pi$
$421$	4.30278	0.209704	0.104852	+	0.994488i	$0.466563\pi$
$422$	− 58.1194i	− 2.82921i
$423$	0	0
$424$	−6.90833	−0.335498
$425$	0	0
$426$	0	0
$427$	12.5139i	0.605589i
$428$	− 9.90833i	− 0.478937i
$429$	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
$433$	− 5.00000i	− 0.240285i	−0.992757	−	0.120142i	$-0.961665\pi$
$433$	− 5.00000i	− 0.240285i	0.992757	−	0.120142i	$-0.0383351\pi$
$434$	−41.7250	−2.00286
$435$	0	0
$436$	−21.5139	−1.03033
$437$	− 3.69722i	− 0.176862i
$438$	0	0
$439$	−20.6972	−0.987825	−0.493912	−	0.869512i	$-0.664434\pi$
$439$	−20.6972	−0.987825	−0.493912	+	0.869512i	$0.664434\pi$
$440$	0	0
$441$	0	0
$442$	− 45.0000i	− 2.14043i
$443$	− 1.39445i	− 0.0662523i	−0.999451	−	0.0331261i	$-0.989454\pi$
$443$	− 1.39445i	− 0.0662523i	0.999451	−	0.0331261i	$-0.0105463\pi$
$444$	0	0
$445$	0	0
$446$	−47.5139	−2.24985
$447$	0	0
$448$	55.1472i	2.60546i
$449$	41.5139	1.95916	0.979581	−	0.201052i	$-0.0644361\pi$
$449$	41.5139	1.95916	0.979581	+	0.201052i	$0.0644361\pi$
$450$	0	0
$451$	−1.60555	−0.0756025
$452$	− 35.7250i	− 1.68036i
$453$	0	0
$454$	−12.2111	−0.573095
$455$	0	0
$456$	0	0
$457$	− 24.3028i	− 1.13684i	−0.822740	−	0.568418i	$-0.807556\pi$
$457$	− 24.3028i	− 1.13684i	0.822740	−	0.568418i	$-0.192444\pi$
$458$	31.6056i	1.47683i
$459$	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
$463$	− 26.2111i	− 1.21813i	−0.793119	−	0.609067i	$-0.791544\pi$
$463$	− 26.2111i	− 1.21813i	0.793119	−	0.609067i	$-0.208456\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	11.7250	0.543149
$467$	− 24.6333i	− 1.13989i	−0.821682	−	0.569947i	$-0.806964\pi$
$467$	− 24.6333i	− 1.13989i	0.821682	−	0.569947i	$-0.193036\pi$
$468$	0	0
$469$	17.2111	0.794735
$470$	0	0
$471$	0	0
$472$	0.633308i	0.0291503i
$473$	7.21110i	0.331567i
$474$	0	0
$475$	0	0
$476$	55.5416	2.54575
$477$	0	0
$478$	9.48612i	0.433885i
$479$	−13.1833	−0.602362	−0.301181	−	0.953567i	$-0.597381\pi$
$479$	−13.1833	−0.602362	−0.301181	+	0.953567i	$0.597381\pi$
$480$	0	0
$481$	48.0278	2.18988
$482$	57.4222i	2.61551i
$483$	0	0
$484$	−3.30278	−0.150126
$485$	0	0
$486$	0	0
$487$	− 10.2111i	− 0.462709i	−0.972869	−	0.231355i	$-0.925684\pi$
$487$	− 10.2111i	− 0.462709i	0.972869	−	0.231355i	$-0.0743158\pi$
$488$	8.72498i	0.394961i
$489$	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
$493$	− 38.7250i	− 1.74409i
$494$	−11.5139	−0.518034
$495$	0	0
$496$	−1.27502	−0.0572501
$497$	− 19.8167i	− 0.888898i
$498$	0	0
$499$	21.5139	0.963093	0.481547	−	0.876420i	$-0.340075\pi$
$499$	21.5139	0.963093	0.481547	+	0.876420i	$0.340075\pi$
$500$	0	0
$501$	0	0
$502$	− 9.00000i	− 0.401690i
$503$	16.6056i	0.740405i	0.928951	+	0.370202i	$0.120712\pi$
$503$	16.6056i	0.740405i	−0.928951	+	0.370202i	$0.879288\pi$
$504$	0	0
$505$	0	0
$506$	−8.51388	−0.378488
$507$	0	0
$508$	56.5416i	2.50863i
$509$	−26.3028	−1.16585	−0.582925	−	0.812526i	$-0.698092\pi$
$509$	−26.3028	−1.16585	−0.582925	+	0.812526i	$0.698092\pi$
$510$	0	0
$511$	12.5139	0.553581
$512$	3.42221i	0.151242i
$513$	0	0
$514$	41.4500	1.82828
$515$	0	0
$516$	0	0
$517$	3.00000i	0.131940i
$518$	95.1749i	4.18175i
$519$	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
$523$	3.57779i	0.156446i	0.996936	+	0.0782230i	$0.0249246\pi$
$523$	3.57779i	0.156446i	−0.996936	+	0.0782230i	$0.975075\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	−2.72498	−0.118815
$527$	− 16.4584i	− 0.716938i
$528$	0	0
$529$	9.33053	0.405675
$530$	0	0
$531$	0	0
$532$	− 14.2111i	− 0.616129i
$533$	8.02776i	0.347721i
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	54.6333i	2.35541i
$539$	−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	$0.546178\pi$
$542$	− 32.7250i	− 1.40566i
$543$	0	0
$544$	−20.7250	−0.888576
$545$	0	0
$546$	0	0
$547$	18.1194i	0.774731i	0.921926	+	0.387365i	$0.126615\pi$
$547$	18.1194i	0.774731i	−0.921926	+	0.387365i	$0.873385\pi$
$548$	− 6.90833i	− 0.295109i
$549$	0	0
$550$	0	0
$551$	−9.90833	−0.422109
$552$	0	0
$553$	0.394449i	0.0167737i
$554$	49.7527	2.11379
$555$	0	0
$556$	27.1194	1.15012
$557$	9.42221i	0.399232i	0.979874	+	0.199616i	$0.0639694\pi$
$557$	9.42221i	0.399232i	−0.979874	+	0.199616i	$0.936031\pi$
$558$	0	0
$559$	36.0555	1.52499
$560$	0	0
$561$	0	0
$562$	− 52.5416i	− 2.21634i
$563$	18.9083i	0.796891i	0.917192	+	0.398445i	$0.130450\pi$
$563$	18.9083i	0.796891i	−0.917192	+	0.398445i	$0.869550\pi$
$564$	0	0
$565$	0	0
$566$	−6.21110	−0.261072
$567$	0	0
$568$	− 13.8167i	− 0.579734i
$569$	15.1472	0.635003	0.317502	−	0.948258i	$-0.397156\pi$
$569$	15.1472	0.635003	0.317502	+	0.948258i	$0.397156\pi$
$570$	0	0
$571$	−17.3305	−0.725260	−0.362630	−	0.931933i	$-0.618121\pi$
$571$	−17.3305	−0.725260	−0.362630	+	0.931933i	$0.618121\pi$
$572$	16.5139i	0.690480i
$573$	0	0
$574$	−15.9083	−0.664001
$575$	0	0
$576$	0	0
$577$	− 31.3583i	− 1.30546i	−0.757589	−	0.652731i	$-0.773623\pi$
$577$	− 31.3583i	− 1.30546i	0.757589	−	0.652731i	$-0.226377\pi$
$578$	− 3.97224i	− 0.165224i
$579$	0	0
$580$	0	0
$581$	−62.4500	−2.59086
$582$	0	0
$583$	2.30278i	0.0953712i
$584$	8.72498	0.361042
$585$	0	0
$586$	35.0278	1.44698
$587$	37.5416i	1.54951i	0.632262	+	0.774755i	$0.282127\pi$
$587$	37.5416i	1.54951i	−0.632262	+	0.774755i	$0.717873\pi$
$588$	0	0
$589$	−4.21110	−0.173515
$590$	0	0
$591$	0	0
$592$	2.90833i	0.119531i
$593$	− 13.6056i	− 0.558713i	−0.960187	−	0.279357i	$-0.909879\pi$
$593$	− 13.6056i	− 0.558713i	0.960187	−	0.279357i	$-0.0901211\pi$
$594$	0	0
$595$	0	0
$596$	−9.21110	−0.377301
$597$	0	0
$598$	42.5694i	1.74079i
$599$	−14.0917	−0.575770	−0.287885	−	0.957665i	$-0.592952\pi$
$599$	−14.0917	−0.575770	−0.287885	+	0.957665i	$0.592952\pi$
$600$	0	0
$601$	8.90833	0.363378	0.181689	−	0.983356i	$-0.441844\pi$
$601$	8.90833	0.363378	0.181689	+	0.983356i	$0.441844\pi$
$602$	71.4500i	2.91208i
$603$	0	0
$604$	68.7527	2.79751
$605$	0	0
$606$	0	0
$607$	− 7.21110i	− 0.292690i	−0.989234	−	0.146345i	$-0.953249\pi$
$607$	− 7.21110i	− 0.292690i	0.989234	−	0.146345i	$-0.0467509\pi$
$608$	5.30278i	0.215056i
$609$	0	0
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	41.1194i	1.66080i	0.557169	+	0.830399i	$0.311888\pi$
$613$	41.1194i	1.66080i	−0.557169	+	0.830399i	$0.688112\pi$
$614$	−14.0278	−0.566114
$615$	0	0
$616$	−12.9083	−0.520091
$617$	− 10.6056i	− 0.426963i	−0.976947	−	0.213482i	$-0.931520\pi$
$617$	− 10.6056i	− 0.426963i	0.976947	−	0.213482i	$-0.0684804\pi$
$618$	0	0
$619$	−17.4222	−0.700258	−0.350129	−	0.936702i	$-0.613862\pi$
$619$	−17.4222	−0.700258	−0.350129	+	0.936702i	$0.613862\pi$
$620$	0	0
$621$	0	0
$622$	− 38.7250i	− 1.55273i
$623$	22.8167i	0.914130i
$624$	0	0
$625$	0	0
$626$	−50.2389	−2.00795
$627$	0	0
$628$	− 15.8167i	− 0.631153i
$629$	−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	$0.792485\pi$
$632$	0.275019i	0.0109397i
$633$	0	0
$634$	−22.8167	−0.906165
$635$	0	0
$636$	0	0
$637$	57.5694i	2.28098i
$638$	22.8167i	0.903320i
$639$	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
$643$	22.0000i	0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$	22.0000i	0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$	−52.5416	−2.07043
$645$	0	0
$646$	9.00000	0.354100
$647$	− 17.2389i	− 0.677729i	−0.940835	−	0.338865i	$-0.889957\pi$
$647$	− 17.2389i	− 0.677729i	0.940835	−	0.338865i	$-0.110043\pi$
$648$	0	0
$649$	0.211103	0.00828650
$650$	0	0
$651$	0	0
$652$	18.8167i	0.736917i
$653$	− 19.1194i	− 0.748201i	−0.927388	−	0.374101i	$-0.877951\pi$
$653$	− 19.1194i	− 0.748201i	0.927388	−	0.374101i	$-0.122049\pi$
$654$	0	0
$655$	0	0
$656$	−0.486122	−0.0189799
$657$	0	0
$658$	29.7250i	1.15880i
$659$	−20.0917	−0.782660	−0.391330	−	0.920250i	$-0.627985\pi$
$659$	−20.0917	−0.782660	−0.391330	+	0.920250i	$0.627985\pi$
$660$	0	0
$661$	12.8167	0.498510	0.249255	−	0.968438i	$-0.419814\pi$
$661$	12.8167	0.498510	0.249255	+	0.968438i	$0.419814\pi$
$662$	33.1472i	1.28830i
$663$	0	0
$664$	−43.5416	−1.68974
$665$	0	0
$666$	0	0
$667$	36.6333i	1.41845i
$668$	− 50.9361i	− 1.97078i
$669$	0	0
$670$	0	0
$671$	2.90833	0.112275
$672$	0	0
$673$	6.02776i	0.232353i	0.993229	+	0.116176i	$0.0370638\pi$
$673$	6.02776i	0.232353i	−0.993229	+	0.116176i	$0.962936\pi$
$674$	61.8167	2.38109
$675$	0	0
$676$	39.6333	1.52436
$677$	26.2389i	1.00844i	0.863575	+	0.504221i	$0.168220\pi$
$677$	26.2389i	1.00844i	−0.863575	+	0.504221i	$0.831780\pi$
$678$	0	0
$679$	−50.3305	−1.93151
$680$	0	0
$681$	0	0
$682$	9.69722i	0.371326i
$683$	− 9.84441i	− 0.376686i	−0.982103	−	0.188343i	$-0.939688\pi$
$683$	− 9.84441i	− 0.376686i	0.982103	−	0.188343i	$-0.0603116\pi$
$684$	0	0
$685$	0	0
$686$	−44.7250	−1.70761
$687$	0	0
$688$	2.18335i	0.0832393i
$689$	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	$0.668451\pi$
$692$	− 55.5416i	− 2.11138i
$693$	0	0
$694$	−12.6972	−0.481980
$695$	0	0
$696$	0	0
$697$	− 6.27502i	− 0.237683i
$698$	− 61.7527i	− 2.33738i
$699$	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
$703$	9.60555i	0.362280i
$704$	12.8167	0.483046
$705$	0	0
$706$	−56.7250	−2.13487
$707$	− 75.3583i	− 2.83414i
$708$	0	0
$709$	−11.6333	−0.436898	−0.218449	−	0.975848i	$-0.570100\pi$
$709$	−11.6333	−0.436898	−0.218449	+	0.975848i	$0.570100\pi$
$710$	0	0
$711$	0	0
$712$	15.9083i	0.596190i
$713$	15.5694i	0.583078i
$714$	0	0
$715$	0	0
$716$	18.2111	0.680581
$717$	0	0
$718$	− 35.0278i	− 1.30722i
$719$	28.8167	1.07468	0.537340	−	0.843366i	$-0.319429\pi$
$719$	28.8167	1.07468	0.537340	+	0.843366i	$0.319429\pi$
$720$	0	0
$721$	−34.0278	−1.26726
$722$	41.4500i	1.54261i
$723$	0	0
$724$	30.0278	1.11597
$725$	0	0
$726$	0	0
$727$	0.330532i	0.0122588i	0.999981	+	0.00612938i	$0.00195105\pi$
$727$	0.330532i	0.0122588i	−0.999981	+	0.00612938i	$0.998049\pi$
$728$	64.5416i	2.39207i
$729$	0	0
$730$	0	0
$731$	−28.1833	−1.04240
$732$	0	0
$733$	− 12.3944i	− 0.457799i	−0.973450	−	0.228900i	$-0.926487\pi$
$733$	− 12.3944i	− 0.457799i	0.973450	−	0.228900i	$-0.0735128\pi$
$734$	−55.9638	−2.06566
$735$	0	0
$736$	19.6056	0.722670
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	−9.88057	−0.363463	−0.181731	−	0.983348i	$-0.558170\pi$
$739$	−9.88057	−0.363463	−0.181731	+	0.983348i	$0.558170\pi$
$740$	0	0
$741$	0	0
$742$	22.8167i	0.837626i
$743$	44.3028i	1.62531i	0.582744	+	0.812656i	$0.301979\pi$
$743$	44.3028i	1.62531i	−0.582744	+	0.812656i	$0.698021\pi$
$744$	0	0
$745$	0	0
$746$	3.27502	0.119907
$747$	0	0
$748$	− 12.9083i	− 0.471975i
$749$	−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	$0.532985\pi$
$752$	0.908327i	0.0331233i
$753$	0	0
$754$	114.083	4.15467
$755$	0	0
$756$	0	0
$757$	− 23.0555i	− 0.837967i	−0.907994	−	0.418983i	$-0.862387\pi$
$757$	− 23.0555i	− 0.837967i	0.907994	−	0.418983i	$-0.137613\pi$
$758$	57.1472i	2.07568i
$759$	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
$763$	28.0278i	1.01467i
$764$	22.1194	0.800253
$765$	0	0
$766$	−49.8167	−1.79995
$767$	− 1.05551i	− 0.0381124i
$768$	0	0
$769$	26.8167	0.967033	0.483517	−	0.875335i	$-0.339359\pi$
$769$	26.8167	0.967033	0.483517	+	0.875335i	$0.339359\pi$
$770$	0	0
$771$	0	0
$772$	− 4.00000i	− 0.143963i
$773$	− 22.1194i	− 0.795581i	−0.917476	−	0.397790i	$-0.869777\pi$
$773$	− 22.1194i	− 0.795581i	0.917476	−	0.397790i	$-0.130223\pi$
$774$	0	0
$775$	0	0
$776$	−35.0917	−1.25972
$777$	0	0
$778$	− 27.6333i	− 0.990702i
$779$	−1.60555	−0.0575248
$780$	0	0
$781$	−4.60555	−0.164800
$782$	− 33.2750i	− 1.18991i
$783$	0	0
$784$	−3.48612	−0.124504
$785$	0	0
$786$	0	0
$787$	− 4.21110i	− 0.150110i	−0.997179	−	0.0750548i	$-0.976087\pi$
$787$	− 4.21110i	− 0.150110i	0.997179	−	0.0750548i	$-0.0239132\pi$
$788$	32.0278i	1.14094i
$789$	0	0
$790$	0	0
$791$	−46.5416	−1.65483
$792$	0	0
$793$	− 14.5416i	− 0.516389i
$794$	58.2666	2.06780
$795$	0	0
$796$	−80.9638	−2.86969
$797$	14.5139i	0.514108i	0.966397	+	0.257054i	$0.0827518\pi$
$797$	14.5139i	0.514108i	−0.966397	+	0.257054i	$0.917248\pi$
$798$	0	0
$799$	−11.7250	−0.414800
$800$	0	0
$801$	0	0
$802$	− 62.6611i	− 2.21264i
$803$	− 2.90833i	− 0.102633i
$804$	0	0
$805$	0	0
$806$	48.4861	1.70785
$807$	0	0
$808$	− 52.5416i	− 1.84841i
$809$	−3.63331	−0.127740	−0.0638701	−	0.997958i	$-0.520344\pi$
$809$	−3.63331	−0.127740	−0.0638701	+	0.997958i	$0.520344\pi$
$810$	0	0
$811$	−54.8722	−1.92682	−0.963411	−	0.268028i	$-0.913628\pi$
$811$	−54.8722	−1.92682	−0.963411	+	0.268028i	$0.913628\pi$
$812$	140.808i	4.94140i
$813$	0	0
$814$	22.1194	0.775286
$815$	0	0
$816$	0	0
$817$	7.21110i	0.252285i
$818$	18.9083i	0.661114i
$819$	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
$823$	10.4222i	0.363295i	0.983364	+	0.181648i	$0.0581430\pi$
$823$	10.4222i	0.363295i	−0.983364	+	0.181648i	$0.941857\pi$
$824$	−23.7250	−0.826499
$825$	0	0
$826$	2.09167	0.0727786
$827$	7.81665i	0.271812i	0.990722	+	0.135906i	$0.0433945\pi$
$827$	7.81665i	0.271812i	−0.990722	+	0.135906i	$0.956606\pi$
$828$	0	0
$829$	38.7527	1.34594	0.672969	−	0.739671i	$-0.265019\pi$
$829$	38.7527	1.34594	0.672969	+	0.739671i	$0.265019\pi$
$830$	0	0
$831$	0	0
$832$	− 64.0833i	− 2.22169i
$833$	− 45.0000i	− 1.55916i
$834$	0	0
$835$	0	0
$836$	−3.30278	−0.114229
$837$	0	0
$838$	− 31.3305i	− 1.08230i
$839$	16.1194	0.556505	0.278252	−	0.960508i	$-0.410245\pi$
$839$	16.1194	0.556505	0.278252	+	0.960508i	$0.410245\pi$
$840$	0	0
$841$	69.1749	2.38534
$842$	− 9.90833i	− 0.341463i
$843$	0	0
$844$	−83.3583	−2.86931
$845$	0	0
$846$	0	0
$847$	4.30278i	0.147845i
$848$	0.697224i	0.0239428i
$849$	0	0
$850$	0	0
$851$	35.5139	1.21740
$852$	0	0
$853$	− 19.7250i	− 0.675370i	−0.941259	−	0.337685i	$-0.890356\pi$
$853$	− 19.7250i	− 0.675370i	0.941259	−	0.337685i	$-0.109644\pi$
$854$	28.8167	0.986086
$855$	0	0
$856$	−9.00000	−0.307614
$857$	3.00000i	0.102478i	0.998686	+	0.0512390i	$0.0163170\pi$
$857$	3.00000i	0.102478i	−0.998686	+	0.0512390i	$0.983683\pi$
$858$	0	0
$859$	−48.6056	−1.65840	−0.829200	−	0.558952i	$-0.811204\pi$
$859$	−48.6056	−1.65840	−0.829200	+	0.558952i	$0.811204\pi$
$860$	0	0
$861$	0	0
$862$	75.9916i	2.58828i
$863$	− 19.6056i	− 0.667381i	−0.942683	−	0.333690i	$-0.891706\pi$
$863$	− 19.6056i	− 0.667381i	0.942683	−	0.333690i	$-0.108294\pi$
$864$	0	0
$865$	0	0
$866$	−11.5139	−0.391258
$867$	0	0
$868$	59.8444i	2.03125i
$869$	0.0916731	0.00310980
$870$	0	0
$871$	−20.0000	−0.677674
$872$	19.5416i	0.661763i
$873$	0	0
$874$	−8.51388	−0.287986
$875$	0	0
$876$	0	0
$877$	20.0000i	0.675352i	0.941262	+	0.337676i	$0.109641\pi$
$877$	20.0000i	0.675352i	−0.941262	+	0.337676i	$0.890359\pi$
$878$	47.6611i	1.60848i
$879$	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
$883$	12.4500i	0.418975i	0.977811	+	0.209487i	$0.0671795\pi$
$883$	12.4500i	0.418975i	−0.977811	+	0.209487i	$0.932821\pi$
$884$	−64.5416	−2.17077
$885$	0	0
$886$	−3.21110	−0.107879
$887$	− 47.2389i	− 1.58613i	−0.609140	−	0.793063i	$-0.708485\pi$
$887$	− 47.2389i	− 1.58613i	0.609140	−	0.793063i	$-0.291515\pi$
$888$	0	0
$889$	73.6611	2.47051
$890$	0	0
$891$	0	0
$892$	68.1472i	2.28174i
$893$	3.00000i	0.100391i
$894$	0	0
$895$	0	0
$896$	81.3583	2.71799
$897$	0	0
$898$	− 95.5971i	− 3.19012i
$899$	41.7250	1.39161
$900$	0	0
$901$	−9.00000	−0.299833
$902$	3.69722i	0.123104i
$903$	0	0
$904$	−32.4500	−1.07927
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	17.5139i	0.581218i
$909$	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
$913$	14.5139i	0.480339i
$914$	−55.9638	−1.85112
$915$	0	0
$916$	45.3305	1.49776
$917$	− 3.90833i	− 0.129064i
$918$	0	0
$919$	−41.2111	−1.35943	−0.679714	−	0.733477i	$-0.737896\pi$
$919$	−41.2111	−1.35943	−0.679714	+	0.733477i	$0.737896\pi$
$920$	0	0
$921$	0	0
$922$	40.9638i	1.34907i
$923$	23.0278i	0.757968i
$924$	0	0
$925$	0	0
$926$	−60.3583	−1.98350
$927$	0	0
$928$	− 52.5416i	− 1.72476i
$929$	46.3944	1.52215	0.761076	−	0.648662i	$-0.224671\pi$
$929$	46.3944	1.52215	0.761076	+	0.648662i	$0.224671\pi$
$930$	0	0
$931$	−11.5139	−0.377352
$932$	− 16.8167i	− 0.550848i
$933$	0	0
$934$	−56.7250	−1.85610
$935$	0	0
$936$	0	0
$937$	5.21110i	0.170239i	0.996371	+	0.0851196i	$0.0271273\pi$
$937$	5.21110i	0.170239i	−0.996371	+	0.0851196i	$0.972873\pi$
$938$	− 39.6333i	− 1.29407i
$939$	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
$943$	5.93608i	0.193305i
$944$	0.0639167	0.00208031
$945$	0	0
$946$	16.6056	0.539893
$947$	36.6333i	1.19042i	0.803569	+	0.595211i	$0.202932\pi$
$947$	36.6333i	1.19042i	−0.803569	+	0.595211i	$0.797068\pi$
$948$	0	0
$949$	−14.5416	−0.472041
$950$	0	0
$951$	0	0
$952$	− 50.4500i	− 1.63509i
$953$	− 49.2666i	− 1.59590i	−0.602722	−	0.797951i	$-0.705917\pi$
$953$	− 49.2666i	− 1.59590i	0.602722	−	0.797951i	$-0.294083\pi$
$954$	0	0
$955$	0	0
$956$	13.6056	0.440035
$957$	0	0
$958$	30.3583i	0.980832i
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−13.2666	−0.427955
$962$	− 110.597i	− 3.56580i
$963$	0	0
$964$	82.3583	2.65258
$965$	0	0
$966$	0	0
$967$	4.09167i	0.131579i	0.997834	+	0.0657897i	$0.0209566\pi$
$967$	4.09167i	0.131579i	−0.997834	+	0.0657897i	$0.979043\pi$
$968$	3.00000i	0.0964237i
$969$	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
$973$	− 35.3305i	− 1.13264i
$974$	−23.5139	−0.753433
$975$	0	0
$976$	0.880571	0.0281864
$977$	15.9722i	0.510997i	0.966809	+	0.255499i	$0.0822396\pi$
$977$	15.9722i	0.510997i	−0.966809	+	0.255499i	$0.917760\pi$
$978$	0	0
$979$	5.30278	0.169477
$980$	0	0
$981$	0	0
$982$	− 55.7527i	− 1.77914i
$983$	− 48.8444i	− 1.55789i	−0.627089	−	0.778947i	$-0.715754\pi$
$983$	− 48.8444i	− 1.55789i	0.627089	−	0.778947i	$-0.284246\pi$
$984$	0	0
$985$	0	0
$986$	−89.1749	−2.83991
$987$	0	0
$988$	16.5139i	0.525376i
$989$	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	$0.530846\pi$
$992$	− 22.3305i	− 0.708995i
$993$	0	0
$994$	−45.6333	−1.44740
$995$	0	0
$996$	0	0
$997$	14.2750i	0.452094i	0.974116	+	0.226047i	$0.0725804\pi$
$997$	14.2750i	0.452094i	−0.974116	+	0.226047i	$0.927420\pi$
$998$	− 49.5416i	− 1.56821i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.1	✓	2	15.2	even	4
275.2.a.f.1.2	yes	2	15.8	even	4
275.2.b.c.199.1		4	15.14	odd	2
275.2.b.c.199.4		4	3.2	odd	2
2475.2.a.o.1.1		2	5.3	odd	4
2475.2.a.t.1.2		2	5.2	odd	4
2475.2.c.k.199.1		4	1.1	even	1	trivial
2475.2.c.k.199.4		4	5.4	even	2	inner
3025.2.a.h.1.1		2	165.98	odd	4
3025.2.a.n.1.2		2	165.32	odd	4
4400.2.a.bh.1.2		2	60.23	odd	4
4400.2.a.bs.1.1		2	60.47	odd	4
4400.2.b.y.4049.2		4	12.11	even	2
4400.2.b.y.4049.3		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q-2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} +3.00000i q^{8} +1.00000 q^{11} -5.00000i q^{13} +9.90833 q^{14} +0.302776 q^{16} +3.90833i q^{17} +1.00000 q^{19} -2.30278i q^{22} -3.69722i q^{23} -11.5139 q^{26} -14.2111i q^{28} -9.90833 q^{29} -4.21110 q^{31} +5.30278i q^{32} +9.00000 q^{34} +9.60555i q^{37} -2.30278i q^{38} -1.60555 q^{41} +7.21110i q^{43} -3.30278 q^{44} -8.51388 q^{46} +3.00000i q^{47} -11.5139 q^{49} +16.5139i q^{52} +2.30278i q^{53} -12.9083 q^{56} +22.8167i q^{58} +0.211103 q^{59} +2.90833 q^{61} +9.69722i q^{62} +12.8167 q^{64} -4.00000i q^{67} -12.9083i q^{68} -4.60555 q^{71} -2.90833i q^{73} +22.1194 q^{74} -3.30278 q^{76} +4.30278i q^{77} +0.0916731 q^{79} +3.69722i q^{82} +14.5139i q^{83} +16.6056 q^{86} +3.00000i q^{88} +5.30278 q^{89} +21.5139 q^{91} +12.2111i q^{92} +6.90833 q^{94} +11.6972i q^{97} +26.5139i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4}+O(q^{10}) \) 4 * q - 6 * q^4	\( 4 q - 6 q^{4} + 4 q^{11} + 18 q^{14} - 6 q^{16} + 4 q^{19} - 10 q^{26} - 18 q^{29} + 12 q^{31} + 36 q^{34} + 8 q^{41} - 6 q^{44} + 2 q^{46} - 10 q^{49} - 30 q^{56} - 28 q^{59} - 10 q^{61} + 8 q^{64} - 4 q^{71} + 38 q^{74} - 6 q^{76} + 22 q^{79} + 52 q^{86} + 14 q^{89} + 50 q^{91} + 6 q^{94}+O(q^{100}) \) 4 * q - 6 * q^4 + 4 * q^11 + 18 * q^14 - 6 * q^16 + 4 * q^19 - 10 * q^26 - 18 * q^29 + 12 * q^31 + 36 * q^34 + 8 * q^41 - 6 * q^44 + 2 * q^46 - 10 * q^49 - 30 * q^56 - 28 * q^59 - 10 * q^61 + 8 * q^64 - 4 * q^71 + 38 * q^74 - 6 * q^76 + 22 * q^79 + 52 * q^86 + 14 * q^89 + 50 * q^91 + 6 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more