LMFDB - Embedded newform 1375.2.b.e.749.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1375,2,Mod(749,1375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1375, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1375.749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1375 = 5^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1375.b (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:	$10.9794302779$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 31 x^{18} + 393 x^{16} + 2674 x^{14} + 10768 x^{12} + 26658 x^{10} + 40807 x^{8} + 37898 x^{6} + \cdots + 625 $ x^20 + 31x^18 + 393x^16 + 2674x^14 + 10768x^12 + 26658x^10 + 40807x^8 + 37898x^6 + 20331x^4 + 5675*x^2 + 625
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.5
Root			$-0.852114i$ of defining polynomial
Character	$\chi$	$=$	1375.749
Dual form			1375.2.b.e.749.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85211i q^{2} +2.68354i q^{3} -1.43033 q^{4} +4.97021 q^{6} -0.339766i q^{7} -1.05510i q^{8} -4.20136 q^{9} +O(q^{10})$	\(q-1.85211i q^{2} +2.68354i q^{3} -1.43033 q^{4} +4.97021 q^{6} -0.339766i q^{7} -1.05510i q^{8} -4.20136 q^{9} -1.00000 q^{11} -3.83833i q^{12} +4.86425i q^{13} -0.629285 q^{14} -4.81482 q^{16} +1.31796i q^{17} +7.78140i q^{18} -2.55970 q^{19} +0.911773 q^{21} +1.85211i q^{22} +4.94321i q^{23} +2.83140 q^{24} +9.00914 q^{26} -3.22390i q^{27} +0.485976i q^{28} -6.14267 q^{29} +5.17997 q^{31} +6.80740i q^{32} -2.68354i q^{33} +2.44101 q^{34} +6.00932 q^{36} +1.49416i q^{37} +4.74085i q^{38} -13.0534 q^{39} -6.80393 q^{41} -1.68871i q^{42} -10.6907i q^{43} +1.43033 q^{44} +9.15538 q^{46} +12.5559i q^{47} -12.9207i q^{48} +6.88456 q^{49} -3.53679 q^{51} -6.95746i q^{52} +2.49993i q^{53} -5.97103 q^{54} -0.358486 q^{56} -6.86903i q^{57} +11.3769i q^{58} -3.44252 q^{59} -4.59459 q^{61} -9.59389i q^{62} +1.42748i q^{63} +2.97844 q^{64} -4.97021 q^{66} +10.3237i q^{67} -1.88511i q^{68} -13.2653 q^{69} -11.1357 q^{71} +4.43285i q^{72} -5.71155i q^{73} +2.76736 q^{74} +3.66120 q^{76} +0.339766i q^{77} +24.1763i q^{78} +1.02578 q^{79} -3.95264 q^{81} +12.6017i q^{82} +14.7736i q^{83} -1.30413 q^{84} -19.8004 q^{86} -16.4841i q^{87} +1.05510i q^{88} +2.13669 q^{89} +1.65270 q^{91} -7.07040i q^{92} +13.9006i q^{93} +23.2549 q^{94} -18.2679 q^{96} +7.02963i q^{97} -12.7510i q^{98} +4.20136 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 30 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) $ 20 * q - 30 * q^4 + 2 * q^6 - 16 * q^9	$ 20 q - 30 q^{4} + 2 q^{6} - 16 q^{9} - 20 q^{11} - 12 q^{14} + 42 q^{16} + 2 q^{19} - 42 q^{21} + 36 q^{24} - 12 q^{26} - 2 q^{29} + 26 q^{31} - 50 q^{34} + 52 q^{36} - 4 q^{41} + 30 q^{44} - 36 q^{46} - 2 q^{49} - 42 q^{51} - 30 q^{54} + 40 q^{56} + 16 q^{59} + 50 q^{61} - 116 q^{64} - 2 q^{66} + 36 q^{69} - 52 q^{71} + 76 q^{74} + 22 q^{76} + 26 q^{79} + 4 q^{81} - 48 q^{84} + 98 q^{86} - 26 q^{89} - 60 q^{91} + 52 q^{94} - 106 q^{96} + 16 q^{99}+O(q^{100}) $ 20 * q - 30 * q^4 + 2 * q^6 - 16 * q^9 - 20 * q^11 - 12 * q^14 + 42 * q^16 + 2 * q^19 - 42 * q^21 + 36 * q^24 - 12 * q^26 - 2 * q^29 + 26 * q^31 - 50 * q^34 + 52 * q^36 - 4 * q^41 + 30 * q^44 - 36 * q^46 - 2 * q^49 - 42 * q^51 - 30 * q^54 + 40 * q^56 + 16 * q^59 + 50 * q^61 - 116 * q^64 - 2 * q^66 + 36 * q^69 - 52 * q^71 + 76 * q^74 + 22 * q^76 + 26 * q^79 + 4 * q^81 - 48 * q^84 + 98 * q^86 - 26 * q^89 - 60 * q^91 + 52 * q^94 - 106 * q^96 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times$.

$n$	$376$	$1002$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 1.85211i	− 1.30964i	−0.755784	−	0.654821i	$-0.772744\pi$
$2$	− 1.85211i	− 1.30964i	0.755784	−	0.654821i	$-0.227256\pi$
$3$	2.68354i	1.54934i	0.632366	+	0.774670i	$0.282084\pi$
$3$	2.68354i	1.54934i	−0.632366	+	0.774670i	$0.717916\pi$
$4$	−1.43033	−0.715164
$5$	0	0
$5$	0	0
$6$	4.97021	2.02908
$7$	− 0.339766i	− 0.128419i	−0.997936	−	0.0642097i	$-0.979547\pi$
$7$	− 0.339766i	− 0.128419i	0.997936	−	0.0642097i	$-0.0204526\pi$
$8$	− 1.05510i	− 0.373034i
$9$	−4.20136	−1.40045
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	− 3.83833i	− 1.10803i
$13$	4.86425i	1.34910i	0.738230	+	0.674550i	$0.235662\pi$
$13$	4.86425i	1.34910i	−0.738230	+	0.674550i	$0.764338\pi$
$14$	−0.629285	−0.168183
$15$	0	0
$16$	−4.81482	−1.20370
$17$	1.31796i	0.319652i	0.987145	+	0.159826i	$0.0510933\pi$
$17$	1.31796i	0.319652i	−0.987145	+	0.159826i	$0.948907\pi$
$18$	7.78140i	1.83409i
$19$	−2.55970	−0.587234	−0.293617	−	0.955923i	$-0.594859\pi$
$19$	−2.55970	−0.587234	−0.293617	+	0.955923i	$0.594859\pi$
$20$	0	0
$21$	0.911773	0.198965
$22$	1.85211i	0.394872i
$23$	4.94321i	1.03073i	0.856971	+	0.515365i	$0.172344\pi$
$23$	4.94321i	1.03073i	−0.856971	+	0.515365i	$0.827656\pi$
$24$	2.83140	0.577956
$25$	0	0
$26$	9.00914	1.76684
$27$	− 3.22390i	− 0.620440i
$28$	0.485976i	0.0918408i
$29$	−6.14267	−1.14067	−0.570333	−	0.821414i	$-0.693186\pi$
$29$	−6.14267	−1.14067	−0.570333	+	0.821414i	$0.693186\pi$
$30$	0	0
$31$	5.17997	0.930350	0.465175	−	0.885219i	$-0.345991\pi$
$31$	5.17997	0.930350	0.465175	+	0.885219i	$0.345991\pi$
$32$	6.80740i	1.20339i
$33$	− 2.68354i	− 0.467144i
$34$	2.44101	0.418630
$35$	0	0
$36$	6.00932	1.00155
$37$	1.49416i	0.245639i	0.992429	+	0.122820i	$0.0391937\pi$
$37$	1.49416i	0.245639i	−0.992429	+	0.122820i	$0.960806\pi$
$38$	4.74085i	0.769067i
$39$	−13.0534	−2.09021
$40$	0	0
$41$	−6.80393	−1.06260	−0.531298	−	0.847185i	$-0.678296\pi$
$41$	−6.80393	−1.06260	−0.531298	+	0.847185i	$0.678296\pi$
$42$	− 1.68871i	− 0.260573i
$43$	− 10.6907i	− 1.63031i	−0.579240	−	0.815157i	$-0.696651\pi$
$43$	− 10.6907i	− 1.63031i	0.579240	−	0.815157i	$-0.303349\pi$
$44$	1.43033	0.215630
$45$	0	0
$46$	9.15538	1.34989
$47$	12.5559i	1.83146i	0.401794	+	0.915730i	$0.368387\pi$
$47$	12.5559i	1.83146i	−0.401794	+	0.915730i	$0.631613\pi$
$48$	− 12.9207i	− 1.86495i
$49$	6.88456	0.983508
$50$	0	0
$51$	−3.53679	−0.495249
$52$	− 6.95746i	− 0.964827i
$53$	2.49993i	0.343392i	0.985150	+	0.171696i	$0.0549247\pi$
$53$	2.49993i	0.343392i	−0.985150	+	0.171696i	$0.945075\pi$
$54$	−5.97103	−0.812554
$55$	0	0
$56$	−0.358486	−0.0479048
$57$	− 6.86903i	− 0.909826i
$58$	11.3769i	1.49386i
$59$	−3.44252	−0.448177	−0.224089	−	0.974569i	$-0.571941\pi$
$59$	−3.44252	−0.448177	−0.224089	+	0.974569i	$0.571941\pi$
$60$	0	0
$61$	−4.59459	−0.588277	−0.294138	−	0.955763i	$-0.595033\pi$
$61$	−4.59459	−0.588277	−0.294138	+	0.955763i	$0.595033\pi$
$62$	− 9.59389i	− 1.21843i
$63$	1.42748i	0.179845i
$64$	2.97844	0.372305
$65$	0	0
$66$	−4.97021	−0.611791
$67$	10.3237i	1.26125i	0.776089	+	0.630623i	$0.217201\pi$
$67$	10.3237i	1.26125i	−0.776089	+	0.630623i	$0.782799\pi$
$68$	− 1.88511i	− 0.228603i
$69$	−13.2653	−1.59695
$70$	0	0
$71$	−11.1357	−1.32156	−0.660781	−	0.750579i	$-0.729775\pi$
$71$	−11.1357	−1.32156	−0.660781	+	0.750579i	$0.729775\pi$
$72$	4.43285i	0.522417i
$73$	− 5.71155i	− 0.668487i	−0.942487	−	0.334243i	$-0.891519\pi$
$73$	− 5.71155i	− 0.668487i	0.942487	−	0.334243i	$-0.108481\pi$
$74$	2.76736	0.321700
$75$	0	0
$76$	3.66120	0.419969
$77$	0.339766i	0.0387199i
$78$	24.1763i	2.73743i
$79$	1.02578	0.115409	0.0577044	−	0.998334i	$-0.481622\pi$
$79$	1.02578	0.115409	0.0577044	+	0.998334i	$0.481622\pi$
$80$	0	0
$81$	−3.95264	−0.439182
$82$	12.6017i	1.39162i
$83$	14.7736i	1.62162i	0.585311	+	0.810809i	$0.300973\pi$
$83$	14.7736i	1.62162i	−0.585311	+	0.810809i	$0.699027\pi$
$84$	−1.30413	−0.142293
$85$	0	0
$86$	−19.8004	−2.13513
$87$	− 16.4841i	− 1.76728i
$88$	1.05510i	0.112474i
$89$	2.13669	0.226488	0.113244	−	0.993567i	$-0.463876\pi$
$89$	2.13669	0.226488	0.113244	+	0.993567i	$0.463876\pi$
$90$	0	0
$91$	1.65270	0.173250
$92$	− 7.07040i	− 0.737140i
$93$	13.9006i	1.44143i
$94$	23.2549	2.39856
$95$	0	0
$96$	−18.2679	−1.86446
$97$	7.02963i	0.713751i	0.934152	+	0.356875i	$0.116158\pi$
$97$	7.02963i	0.713751i	−0.934152	+	0.356875i	$0.883842\pi$
$98$	− 12.7510i	− 1.28804i
$99$	4.20136	0.422253
$100$	0	0
$101$	−2.76977	−0.275602	−0.137801	−	0.990460i	$-0.544003\pi$
$101$	−2.76977	−0.275602	−0.137801	+	0.990460i	$0.544003\pi$
$102$	6.55053i	0.648599i
$103$	− 3.31605i	− 0.326740i	−0.986565	−	0.163370i	$-0.947764\pi$
$103$	− 3.31605i	− 0.326740i	0.986565	−	0.163370i	$-0.0522364\pi$
$104$	5.13226	0.503260
$105$	0	0
$106$	4.63016	0.449721
$107$	− 0.363749i	− 0.0351650i	−0.999845	−	0.0175825i	$-0.994403\pi$
$107$	− 0.363749i	− 0.0351650i	0.999845	−	0.0175825i	$-0.00559697\pi$
$108$	4.61123i	0.443716i
$109$	11.1303	1.06608	0.533042	−	0.846089i	$-0.321049\pi$
$109$	11.1303	1.06608	0.533042	+	0.846089i	$0.321049\pi$
$110$	0	0
$111$	−4.00964	−0.380579
$112$	1.63591i	0.154579i
$113$	18.8736i	1.77548i	0.460345	+	0.887740i	$0.347726\pi$
$113$	18.8736i	1.77548i	−0.460345	+	0.887740i	$0.652274\pi$
$114$	−12.7222	−1.19155
$115$	0	0
$116$	8.78603	0.815762
$117$	− 20.4365i	− 1.88935i
$118$	6.37593i	0.586952i
$119$	0.447797	0.0410495
$120$	0	0
$121$	1.00000	0.0909091
$122$	8.50970i	0.770432i
$123$	− 18.2586i	− 1.64632i
$124$	−7.40905	−0.665352
$125$	0	0
$126$	2.64385	0.235533
$127$	7.65574i	0.679337i	0.940545	+	0.339669i	$0.110315\pi$
$127$	7.65574i	0.679337i	−0.940545	+	0.339669i	$0.889685\pi$
$128$	8.09838i	0.715803i
$129$	28.6888	2.52591
$130$	0	0
$131$	14.3588	1.25454	0.627268	−	0.778803i	$-0.284173\pi$
$131$	14.3588	1.25454	0.627268	+	0.778803i	$0.284173\pi$
$132$	3.83833i	0.334084i
$133$	0.869696i	0.0754122i
$134$	19.1208	1.65178
$135$	0	0
$136$	1.39058	0.119241
$137$	− 16.8937i	− 1.44332i	−0.692246	−	0.721662i	$-0.743379\pi$
$137$	− 16.8937i	− 1.44332i	0.692246	−	0.721662i	$-0.256621\pi$
$138$	24.5688i	2.09143i
$139$	−8.30932	−0.704788	−0.352394	−	0.935852i	$-0.614632\pi$
$139$	−8.30932	−0.704788	−0.352394	+	0.935852i	$0.614632\pi$
$140$	0	0
$141$	−33.6941	−2.83755
$142$	20.6246i	1.73077i
$143$	− 4.86425i	− 0.406769i
$144$	20.2288	1.68573
$145$	0	0
$146$	−10.5785	−0.875479
$147$	18.4750i	1.52379i
$148$	− 2.13714i	− 0.175672i
$149$	−13.6972	−1.12212	−0.561059	−	0.827776i	$-0.689606\pi$
$149$	−13.6972	−1.12212	−0.561059	+	0.827776i	$0.689606\pi$
$150$	0	0
$151$	−15.7961	−1.28547	−0.642736	−	0.766088i	$-0.722201\pi$
$151$	−15.7961	−1.28547	−0.642736	+	0.766088i	$0.722201\pi$
$152$	2.70073i	0.219058i
$153$	− 5.53722i	− 0.447658i
$154$	0.629285	0.0507092
$155$	0	0
$156$	18.6706	1.49484
$157$	− 22.3255i	− 1.78177i	−0.454227	−	0.890886i	$-0.650084\pi$
$157$	− 22.3255i	− 1.78177i	0.454227	−	0.890886i	$-0.349916\pi$
$158$	− 1.89985i	− 0.151144i
$159$	−6.70866	−0.532031
$160$	0	0
$161$	1.67953	0.132366
$162$	7.32074i	0.575172i
$163$	− 7.81687i	− 0.612265i	−0.951989	−	0.306132i	$-0.900965\pi$
$163$	− 7.81687i	− 0.612265i	0.951989	−	0.306132i	$-0.0990350\pi$
$164$	9.73185	0.759930
$165$	0	0
$166$	27.3625	2.12374
$167$	− 24.5079i	− 1.89648i	−0.317559	−	0.948239i	$-0.602863\pi$
$167$	− 24.5079i	− 1.89648i	0.317559	−	0.948239i	$-0.397137\pi$
$168$	− 0.962011i	− 0.0742207i
$169$	−10.6609	−0.820068
$170$	0	0
$171$	10.7542	0.822395
$172$	15.2912i	1.16594i
$173$	− 7.21378i	− 0.548454i	−0.961665	−	0.274227i	$-0.911578\pi$
$173$	− 7.21378i	− 0.548454i	0.961665	−	0.274227i	$-0.0884219\pi$
$174$	−30.5304	−2.31450
$175$	0	0
$176$	4.81482	0.362931
$177$	− 9.23811i	− 0.694379i
$178$	− 3.95739i	− 0.296619i
$179$	22.4159	1.67544	0.837720	−	0.546099i	$-0.183888\pi$
$179$	22.4159	1.67544	0.837720	+	0.546099i	$0.183888\pi$
$180$	0	0
$181$	9.89420	0.735430	0.367715	−	0.929938i	$-0.380140\pi$
$181$	9.89420	0.735430	0.367715	+	0.929938i	$0.380140\pi$
$182$	− 3.06100i	− 0.226896i
$183$	− 12.3297i	− 0.911440i
$184$	5.21557	0.384497
$185$	0	0
$186$	25.7456	1.88776
$187$	− 1.31796i	− 0.0963786i
$188$	− 17.9590i	− 1.30979i
$189$	−1.09537	−0.0796764
$190$	0	0
$191$	8.48049	0.613627	0.306813	−	0.951770i	$-0.400737\pi$
$191$	8.48049	0.613627	0.306813	+	0.951770i	$0.400737\pi$
$192$	7.99274i	0.576827i
$193$	17.7545i	1.27800i	0.769207	+	0.639000i	$0.220651\pi$
$193$	17.7545i	1.27800i	−0.769207	+	0.639000i	$0.779349\pi$
$194$	13.0197	0.934758
$195$	0	0
$196$	−9.84717	−0.703369
$197$	− 3.33103i	− 0.237326i	−0.992935	−	0.118663i	$-0.962139\pi$
$197$	− 3.33103i	− 0.237326i	0.992935	−	0.118663i	$-0.0378608\pi$
$198$	− 7.78140i	− 0.553000i
$199$	11.1591	0.791051	0.395526	−	0.918455i	$-0.370562\pi$
$199$	11.1591	0.791051	0.395526	+	0.918455i	$0.370562\pi$
$200$	0	0
$201$	−27.7041	−1.95410
$202$	5.12992i	0.360940i
$203$	2.08707i	0.146483i
$204$	5.05876	0.354184
$205$	0	0
$206$	−6.14170	−0.427912
$207$	− 20.7682i	− 1.44349i
$208$	− 23.4205i	− 1.62392i
$209$	2.55970	0.177058
$210$	0	0
$211$	10.6078	0.730271	0.365136	−	0.930954i	$-0.381023\pi$
$211$	10.6078	0.730271	0.365136	+	0.930954i	$0.381023\pi$
$212$	− 3.57572i	− 0.245582i
$213$	− 29.8830i	− 2.04755i
$214$	−0.673706	−0.0460536
$215$	0	0
$216$	−3.40153	−0.231445
$217$	− 1.75997i	− 0.119475i
$218$	− 20.6145i	− 1.39619i
$219$	15.3272	1.03571
$220$	0	0
$221$	−6.41087	−0.431242
$222$	7.42632i	0.498422i
$223$	− 11.5301i	− 0.772115i	−0.922475	−	0.386058i	$-0.873837\pi$
$223$	− 11.5301i	− 0.772115i	0.922475	−	0.386058i	$-0.126163\pi$
$224$	2.31292	0.154538
$225$	0	0
$226$	34.9561	2.32525
$227$	− 25.8807i	− 1.71776i	−0.512174	−	0.858882i	$-0.671160\pi$
$227$	− 25.8807i	− 1.71776i	0.512174	−	0.858882i	$-0.328840\pi$
$228$	9.82497i	0.650674i
$229$	6.41967	0.424224	0.212112	−	0.977245i	$-0.431966\pi$
$229$	6.41967	0.424224	0.212112	+	0.977245i	$0.431966\pi$
$230$	0	0
$231$	−0.911773	−0.0599903
$232$	6.48113i	0.425507i
$233$	8.90631i	0.583472i	0.956499	+	0.291736i	$0.0942328\pi$
$233$	8.90631i	0.583472i	−0.956499	+	0.291736i	$0.905767\pi$
$234$	−37.8507	−2.47438
$235$	0	0
$236$	4.92392	0.320520
$237$	2.75271i	0.178807i
$238$	− 0.829371i	− 0.0537601i
$239$	27.8843	1.80368	0.901842	−	0.432067i	$-0.142216\pi$
$239$	27.8843	1.80368	0.901842	+	0.432067i	$0.142216\pi$
$240$	0	0
$241$	19.2209	1.23813	0.619064	−	0.785341i	$-0.287512\pi$
$241$	19.2209	1.23813	0.619064	+	0.785341i	$0.287512\pi$
$242$	− 1.85211i	− 0.119058i
$243$	− 20.2787i	− 1.30088i
$244$	6.57176	0.420714
$245$	0	0
$246$	−33.8170	−2.15609
$247$	− 12.4510i	− 0.792237i
$248$	− 5.46538i	− 0.347052i
$249$	−39.6456	−2.51244
$250$	0	0
$251$	−11.8980	−0.750997	−0.375498	−	0.926823i	$-0.622528\pi$
$251$	−11.8980	−0.750997	−0.375498	+	0.926823i	$0.622528\pi$
$252$	− 2.04176i	− 0.128619i
$253$	− 4.94321i	− 0.310777i
$254$	14.1793	0.889689
$255$	0	0
$256$	20.9560	1.30975
$257$	− 0.574789i	− 0.0358544i	−0.999839	−	0.0179272i	$-0.994293\pi$
$257$	− 0.574789i	− 0.0358544i	0.999839	−	0.0179272i	$-0.00570670\pi$
$258$	− 53.1350i	− 3.30804i
$259$	0.507666	0.0315448
$260$	0	0
$261$	25.8076	1.59745
$262$	− 26.5942i	− 1.64299i
$263$	18.9872i	1.17080i	0.810745	+	0.585399i	$0.199062\pi$
$263$	18.9872i	1.17080i	−0.810745	+	0.585399i	$0.800938\pi$
$264$	−2.83140	−0.174260
$265$	0	0
$266$	1.61078	0.0987631
$267$	5.73387i	0.350907i
$268$	− 14.7663i	− 0.901998i
$269$	−18.5134	−1.12878	−0.564391	−	0.825508i	$-0.690889\pi$
$269$	−18.5134	−1.12878	−0.564391	+	0.825508i	$0.690889\pi$
$270$	0	0
$271$	20.3081	1.23363	0.616815	−	0.787108i	$-0.288423\pi$
$271$	20.3081	1.23363	0.616815	+	0.787108i	$0.288423\pi$
$272$	− 6.34573i	− 0.384766i
$273$	4.43509i	0.268424i
$274$	−31.2890	−1.89024
$275$	0	0
$276$	18.9737	1.14208
$277$	5.79699i	0.348307i	0.984719	+	0.174154i	$0.0557189\pi$
$277$	5.79699i	0.348307i	−0.984719	+	0.174154i	$0.944281\pi$
$278$	15.3898i	0.923020i
$279$	−21.7629	−1.30291
$280$	0	0
$281$	−22.3380	−1.33257	−0.666287	−	0.745695i	$-0.732117\pi$
$281$	−22.3380	−1.33257	−0.666287	+	0.745695i	$0.732117\pi$
$282$	62.4053i	3.71618i
$283$	17.4380i	1.03658i	0.855205	+	0.518290i	$0.173431\pi$
$283$	17.4380i	1.03658i	−0.855205	+	0.518290i	$0.826569\pi$
$284$	15.9277	0.945133
$285$	0	0
$286$	−9.00914	−0.532722
$287$	2.31174i	0.136458i
$288$	− 28.6003i	− 1.68529i
$289$	15.2630	0.897823
$290$	0	0
$291$	−18.8643	−1.10584
$292$	8.16939i	0.478077i
$293$	8.16679i	0.477109i	0.971129	+	0.238554i	$0.0766735\pi$
$293$	8.16679i	0.477109i	−0.971129	+	0.238554i	$0.923326\pi$
$294$	34.2177	1.99562
$295$	0	0
$296$	1.57649	0.0916317
$297$	3.22390i	0.187070i
$298$	25.3688i	1.46957i
$299$	−24.0450	−1.39056
$300$	0	0
$301$	−3.63232	−0.209364
$302$	29.2562i	1.68351i
$303$	− 7.43277i	− 0.427001i
$304$	12.3245	0.706857
$305$	0	0
$306$	−10.2556	−0.586272
$307$	− 24.3574i	− 1.39015i	−0.718936	−	0.695076i	$-0.755371\pi$
$307$	− 24.3574i	− 1.39015i	0.718936	−	0.695076i	$-0.244629\pi$
$308$	− 0.485976i	− 0.0276910i
$309$	8.89873	0.506231
$310$	0	0
$311$	24.6088	1.39544	0.697719	−	0.716371i	$-0.254198\pi$
$311$	24.6088	1.39544	0.697719	+	0.716371i	$0.254198\pi$
$312$	13.7726i	0.779720i
$313$	0.908596i	0.0513569i	0.999670	+	0.0256784i	$0.00817460\pi$
$313$	0.908596i	0.0513569i	−0.999670	+	0.0256784i	$0.991825\pi$
$314$	−41.3495	−2.33348
$315$	0	0
$316$	−1.46719	−0.0825362
$317$	35.3028i	1.98280i	0.130861	+	0.991401i	$0.458226\pi$
$317$	35.3028i	1.98280i	−0.130861	+	0.991401i	$0.541774\pi$
$318$	12.4252i	0.696771i
$319$	6.14267	0.343923
$320$	0	0
$321$	0.976135	0.0544825
$322$	− 3.11068i	− 0.173352i
$323$	− 3.37357i	− 0.187711i
$324$	5.65357	0.314087
$325$	0	0
$326$	−14.4777	−0.801848
$327$	29.8684i	1.65173i
$328$	7.17882i	0.396384i
$329$	4.26605	0.235195
$330$	0	0
$331$	28.5160	1.56738	0.783692	−	0.621150i	$-0.213334\pi$
$331$	28.5160	1.56738	0.783692	+	0.621150i	$0.213334\pi$
$332$	− 21.1311i	− 1.15972i
$333$	− 6.27753i	− 0.344006i
$334$	−45.3914	−2.48371
$335$	0	0
$336$	−4.39002	−0.239495
$337$	6.40649i	0.348984i	0.984659	+	0.174492i	$0.0558283\pi$
$337$	6.40649i	0.348984i	−0.984659	+	0.174492i	$0.944172\pi$
$338$	19.7452i	1.07400i
$339$	−50.6480	−2.75082
$340$	0	0
$341$	−5.17997	−0.280511
$342$	− 19.9180i	− 1.07704i
$343$	− 4.71749i	− 0.254721i
$344$	−11.2797	−0.608162
$345$	0	0
$346$	−13.3608	−0.718278
$347$	− 19.1352i	− 1.02723i	−0.858020	−	0.513616i	$-0.828306\pi$
$347$	− 19.1352i	− 1.02723i	0.858020	−	0.513616i	$-0.171694\pi$
$348$	23.5776i	1.26389i
$349$	8.12440	0.434889	0.217445	−	0.976073i	$-0.430228\pi$
$349$	8.12440	0.434889	0.217445	+	0.976073i	$0.430228\pi$
$350$	0	0
$351$	15.6818	0.837035
$352$	− 6.80740i	− 0.362835i
$353$	− 10.6944i	− 0.569208i	−0.958645	−	0.284604i	$-0.908138\pi$
$353$	− 10.6944i	− 0.569208i	0.958645	−	0.284604i	$-0.0918621\pi$
$354$	−17.1100	−0.909388
$355$	0	0
$356$	−3.05616	−0.161976
$357$	1.20168i	0.0635996i
$358$	− 41.5167i	− 2.19423i
$359$	5.55846	0.293364	0.146682	−	0.989184i	$-0.453141\pi$
$359$	5.55846	0.293364	0.146682	+	0.989184i	$0.453141\pi$
$360$	0	0
$361$	−12.4480	−0.655156
$362$	− 18.3252i	− 0.963151i
$363$	2.68354i	0.140849i
$364$	−2.36391	−0.123902
$365$	0	0
$366$	−22.8361	−1.19366
$367$	23.8208i	1.24343i	0.783242	+	0.621717i	$0.213565\pi$
$367$	23.8208i	1.24343i	−0.783242	+	0.621717i	$0.786435\pi$
$368$	− 23.8006i	− 1.24069i
$369$	28.5858	1.48812
$370$	0	0
$371$	0.849391	0.0440982
$372$	− 19.8824i	− 1.03086i
$373$	16.7597i	0.867782i	0.900965	+	0.433891i	$0.142860\pi$
$373$	16.7597i	0.867782i	−0.900965	+	0.433891i	$0.857140\pi$
$374$	−2.44101	−0.126222
$375$	0	0
$376$	13.2477	0.683197
$377$	− 29.8795i	− 1.53887i
$378$	2.02875i	0.104348i
$379$	−20.3771	−1.04670	−0.523349	−	0.852118i	$-0.675318\pi$
$379$	−20.3771	−1.04670	−0.523349	+	0.852118i	$0.675318\pi$
$380$	0	0
$381$	−20.5445	−1.05252
$382$	− 15.7068i	− 0.803632i
$383$	15.5428i	0.794201i	0.917775	+	0.397100i	$0.129984\pi$
$383$	15.5428i	0.794201i	−0.917775	+	0.397100i	$0.870016\pi$
$384$	−21.7323	−1.10902
$385$	0	0
$386$	32.8834	1.67372
$387$	44.9154i	2.28318i
$388$	− 10.0547i	− 0.510449i
$389$	−15.8468	−0.803464	−0.401732	−	0.915757i	$-0.631592\pi$
$389$	−15.8468	−0.803464	−0.401732	+	0.915757i	$0.631592\pi$
$390$	0	0
$391$	−6.51494	−0.329475
$392$	− 7.26389i	− 0.366882i
$393$	38.5324i	1.94370i
$394$	−6.16944	−0.310812
$395$	0	0
$396$	−6.00932	−0.301980
$397$	29.7140i	1.49130i	0.666335	+	0.745652i	$0.267862\pi$
$397$	29.7140i	1.49130i	−0.666335	+	0.745652i	$0.732138\pi$
$398$	− 20.6680i	− 1.03599i
$399$	−2.33386	−0.116839
$400$	0	0
$401$	−32.8019	−1.63805	−0.819023	−	0.573760i	$-0.805484\pi$
$401$	−32.8019	−1.63805	−0.819023	+	0.573760i	$0.805484\pi$
$402$	51.3112i	2.55917i
$403$	25.1966i	1.25513i
$404$	3.96167	0.197101
$405$	0	0
$406$	3.86549	0.191841
$407$	− 1.49416i	− 0.0740630i
$408$	3.73166i	0.184745i
$409$	34.1901	1.69059	0.845295	−	0.534300i	$-0.179425\pi$
$409$	34.1901	1.69059	0.845295	+	0.534300i	$0.179425\pi$
$410$	0	0
$411$	45.3348	2.23620
$412$	4.74303i	0.233672i
$413$	1.16965i	0.0575546i
$414$	−38.4651	−1.89046
$415$	0	0
$416$	−33.1128	−1.62349
$417$	− 22.2984i	− 1.09196i
$418$	− 4.74085i	− 0.231883i
$419$	33.1308	1.61855	0.809273	−	0.587433i	$-0.199861\pi$
$419$	33.1308	1.61855	0.809273	+	0.587433i	$0.199861\pi$
$420$	0	0
$421$	−34.6983	−1.69109	−0.845547	−	0.533901i	$-0.820726\pi$
$421$	−34.6983	−1.69109	−0.845547	+	0.533901i	$0.820726\pi$
$422$	− 19.6469i	− 0.956394i
$423$	− 52.7517i	− 2.56488i
$424$	2.63768	0.128097
$425$	0	0
$426$	−55.3467	−2.68156
$427$	1.56108i	0.0755461i
$428$	0.520281i	0.0251487i
$429$	13.0534	0.630223
$430$	0	0
$431$	11.1172	0.535495	0.267747	−	0.963489i	$-0.413721\pi$
$431$	11.1172	0.535495	0.267747	+	0.963489i	$0.413721\pi$
$432$	15.5225i	0.746826i
$433$	24.9459i	1.19882i	0.800441	+	0.599412i	$0.204599\pi$
$433$	24.9459i	1.19882i	−0.800441	+	0.599412i	$0.795401\pi$
$434$	−3.25967	−0.156469
$435$	0	0
$436$	−15.9199	−0.762425
$437$	− 12.6531i	− 0.605280i
$438$	− 28.3876i	− 1.35641i
$439$	37.8011	1.80415	0.902074	−	0.431581i	$-0.142044\pi$
$439$	37.8011	1.80415	0.902074	+	0.431581i	$0.142044\pi$
$440$	0	0
$441$	−28.9245	−1.37736
$442$	11.8737i	0.564773i
$443$	− 9.70226i	− 0.460968i	−0.973076	−	0.230484i	$-0.925969\pi$
$443$	− 9.70226i	− 0.460968i	0.973076	−	0.230484i	$-0.0740310\pi$
$444$	5.73510	0.272176
$445$	0	0
$446$	−21.3551	−1.01119
$447$	− 36.7569i	− 1.73854i
$448$	− 1.01197i	− 0.0478111i
$449$	−6.59428	−0.311203	−0.155602	−	0.987820i	$-0.549732\pi$
$449$	−6.59428	−0.311203	−0.155602	+	0.987820i	$0.549732\pi$
$450$	0	0
$451$	6.80393	0.320385
$452$	− 26.9954i	− 1.26976i
$453$	− 42.3895i	− 1.99163i
$454$	−47.9340	−2.24966
$455$	0	0
$456$	−7.24751	−0.339396
$457$	− 21.1958i	− 0.991496i	−0.868466	−	0.495748i	$-0.834894\pi$
$457$	− 21.1958i	− 0.991496i	0.868466	−	0.495748i	$-0.165106\pi$
$458$	− 11.8900i	− 0.555582i
$459$	4.24896	0.198325
$460$	0	0
$461$	−17.6602	−0.822515	−0.411258	−	0.911519i	$-0.634910\pi$
$461$	−17.6602	−0.822515	−0.411258	+	0.911519i	$0.634910\pi$
$462$	1.68871i	0.0785658i
$463$	− 38.8863i	− 1.80720i	−0.428376	−	0.903600i	$-0.640914\pi$
$463$	− 38.8863i	− 1.80720i	0.428376	−	0.903600i	$-0.359086\pi$
$464$	29.5758	1.37302
$465$	0	0
$466$	16.4955	0.764140
$467$	− 3.20247i	− 0.148193i	−0.997251	−	0.0740964i	$-0.976393\pi$
$467$	− 3.20247i	− 0.148193i	0.997251	−	0.0740964i	$-0.0236073\pi$
$468$	29.2308i	1.35120i
$469$	3.50765	0.161968
$470$	0	0
$471$	59.9114	2.76057
$472$	3.63219i	0.167185i
$473$	10.6907i	0.491558i
$474$	5.09832	0.234174
$475$	0	0
$476$	−0.640496	−0.0293571
$477$	− 10.5031i	− 0.480905i
$478$	− 51.6448i	− 2.36218i
$479$	9.92200	0.453348	0.226674	−	0.973971i	$-0.427215\pi$
$479$	9.92200	0.453348	0.226674	+	0.973971i	$0.427215\pi$
$480$	0	0
$481$	−7.26799	−0.331392
$482$	− 35.5993i	− 1.62150i
$483$	4.50708i	0.205079i
$484$	−1.43033	−0.0650149
$485$	0	0
$486$	−37.5586	−1.70369
$487$	− 6.17014i	− 0.279596i	−0.990180	−	0.139798i	$-0.955355\pi$
$487$	− 6.17014i	− 0.279596i	0.990180	−	0.139798i	$-0.0446453\pi$
$488$	4.84774i	0.219447i
$489$	20.9769	0.948606
$490$	0	0
$491$	−18.1539	−0.819274	−0.409637	−	0.912249i	$-0.634345\pi$
$491$	−18.1539	−0.819274	−0.409637	+	0.912249i	$0.634345\pi$
$492$	26.1158i	1.17739i
$493$	− 8.09578i	− 0.364616i
$494$	−23.0607	−1.03755
$495$	0	0
$496$	−24.9406	−1.11987
$497$	3.78352i	0.169714i
$498$	73.4282i	3.29040i
$499$	−10.7665	−0.481974	−0.240987	−	0.970528i	$-0.577471\pi$
$499$	−10.7665	−0.481974	−0.240987	+	0.970528i	$0.577471\pi$
$500$	0	0
$501$	65.7678	2.93829
$502$	22.0365i	0.983538i
$503$	− 19.4300i	− 0.866343i	−0.901312	−	0.433171i	$-0.857394\pi$
$503$	− 19.4300i	− 0.866343i	0.901312	−	0.433171i	$-0.142606\pi$
$504$	1.50613	0.0670884
$505$	0	0
$506$	−9.15538	−0.407006
$507$	− 28.6089i	− 1.27056i
$508$	− 10.9502i	− 0.485837i
$509$	−21.1187	−0.936068	−0.468034	−	0.883710i	$-0.655038\pi$
$509$	−21.1187	−0.936068	−0.468034	+	0.883710i	$0.655038\pi$
$510$	0	0
$511$	−1.94059	−0.0858466
$512$	− 22.6162i	− 0.999502i
$513$	8.25220i	0.364344i
$514$	−1.06457	−0.0469564
$515$	0	0
$516$	−41.0344	−1.80644
$517$	− 12.5559i	− 0.552206i
$518$	− 0.940255i	− 0.0413124i
$519$	19.3584	0.849741
$520$	0	0
$521$	−9.78021	−0.428479	−0.214239	−	0.976781i	$-0.568727\pi$
$521$	−9.78021	−0.428479	−0.214239	+	0.976781i	$0.568727\pi$
$522$	− 47.7986i	− 2.09209i
$523$	4.62407i	0.202197i	0.994876	+	0.101098i	$0.0322357\pi$
$523$	4.62407i	0.202197i	−0.994876	+	0.101098i	$0.967764\pi$
$524$	−20.5378	−0.897199
$525$	0	0
$526$	35.1664	1.53333
$527$	6.82698i	0.297388i
$528$	12.9207i	0.562303i
$529$	−1.43529	−0.0624040
$530$	0	0
$531$	14.4633	0.627652
$532$	− 1.24395i	− 0.0539321i
$533$	− 33.0960i	− 1.43355i
$534$	10.6198	0.459563
$535$	0	0
$536$	10.8926	0.470488
$537$	60.1538i	2.59583i
$538$	34.2889i	1.47830i
$539$	−6.88456	−0.296539
$540$	0	0
$541$	−14.9104	−0.641046	−0.320523	−	0.947241i	$-0.603859\pi$
$541$	−14.9104	−0.641046	−0.320523	+	0.947241i	$0.603859\pi$
$542$	− 37.6129i	− 1.61561i
$543$	26.5514i	1.13943i
$544$	−8.97186	−0.384665
$545$	0	0
$546$	8.21429	0.351539
$547$	− 1.58429i	− 0.0677395i	−0.999426	−	0.0338697i	$-0.989217\pi$
$547$	− 1.58429i	− 0.0677395i	0.999426	−	0.0338697i	$-0.0107831\pi$
$548$	24.1635i	1.03221i
$549$	19.3035	0.823854
$550$	0	0
$551$	15.7234	0.669838
$552$	13.9962i	0.595717i
$553$	− 0.348523i	− 0.0148207i
$554$	10.7367	0.456158
$555$	0	0
$556$	11.8851	0.504039
$557$	11.3462i	0.480754i	0.970680	+	0.240377i	$0.0772710\pi$
$557$	11.3462i	0.480754i	−0.970680	+	0.240377i	$0.922729\pi$
$558$	40.3074i	1.70635i
$559$	52.0021	2.19945
$560$	0	0
$561$	3.53679	0.149323
$562$	41.3726i	1.74520i
$563$	− 32.1619i	− 1.35546i	−0.735310	−	0.677731i	$-0.762963\pi$
$563$	− 32.1619i	− 1.35546i	0.735310	−	0.677731i	$-0.237037\pi$
$564$	48.1936	2.02932
$565$	0	0
$566$	32.2971	1.35755
$567$	1.34297i	0.0563995i
$568$	11.7492i	0.492988i
$569$	−37.8707	−1.58762	−0.793812	−	0.608163i	$-0.791907\pi$
$569$	−37.8707	−1.58762	−0.793812	+	0.608163i	$0.791907\pi$
$570$	0	0
$571$	1.97617	0.0827000	0.0413500	−	0.999145i	$-0.486834\pi$
$571$	1.97617	0.0827000	0.0413500	+	0.999145i	$0.486834\pi$
$572$	6.95746i	0.290906i
$573$	22.7577i	0.950716i
$574$	4.28161	0.178711
$575$	0	0
$576$	−12.5135	−0.521396
$577$	7.74133i	0.322276i	0.986932	+	0.161138i	$0.0515164\pi$
$577$	7.74133i	0.322276i	−0.986932	+	0.161138i	$0.948484\pi$
$578$	− 28.2688i	− 1.17583i
$579$	−47.6449	−1.98006
$580$	0	0
$581$	5.01958	0.208247
$582$	34.9388i	1.44826i
$583$	− 2.49993i	− 0.103537i
$584$	−6.02626	−0.249368
$585$	0	0
$586$	15.1258	0.624842
$587$	9.00907i	0.371844i	0.982564	+	0.185922i	$0.0595272\pi$
$587$	9.00907i	0.371844i	−0.982564	+	0.185922i	$0.940473\pi$
$588$	− 26.4252i	− 1.08976i
$589$	−13.2591	−0.546333
$590$	0	0
$591$	8.93893	0.367699
$592$	− 7.19413i	− 0.295677i
$593$	27.5610i	1.13179i	0.824476	+	0.565897i	$0.191470\pi$
$593$	27.5610i	1.13179i	−0.824476	+	0.565897i	$0.808530\pi$
$594$	5.97103	0.244994
$595$	0	0
$596$	19.5915	0.802498
$597$	29.9460i	1.22561i
$598$	44.5340i	1.82113i
$599$	29.2334	1.19444	0.597221	−	0.802076i	$-0.296271\pi$
$599$	29.2334	1.19444	0.597221	+	0.802076i	$0.296271\pi$
$600$	0	0
$601$	31.9920	1.30498	0.652490	−	0.757797i	$-0.273724\pi$
$601$	31.9920	1.30498	0.652490	+	0.757797i	$0.273724\pi$
$602$	6.72748i	0.274192i
$603$	− 43.3738i	− 1.76632i
$604$	22.5936	0.919322
$605$	0	0
$606$	−13.7663	−0.559219
$607$	− 15.1893i	− 0.616515i	−0.951303	−	0.308258i	$-0.900254\pi$
$607$	− 15.1893i	− 0.616515i	0.951303	−	0.308258i	$-0.0997459\pi$
$608$	− 17.4249i	− 0.706671i
$609$	−5.60072	−0.226953
$610$	0	0
$611$	−61.0748	−2.47082
$612$	7.92004i	0.320148i
$613$	37.0573i	1.49673i	0.663287	+	0.748365i	$0.269161\pi$
$613$	37.0573i	1.49673i	−0.663287	+	0.748365i	$0.730839\pi$
$614$	−45.1128	−1.82060
$615$	0	0
$616$	0.358486	0.0144438
$617$	− 25.0196i	− 1.00725i	−0.863922	−	0.503626i	$-0.831999\pi$
$617$	− 25.0196i	− 1.00725i	0.863922	−	0.503626i	$-0.168001\pi$
$618$	− 16.4815i	− 0.662982i
$619$	−30.4104	−1.22230	−0.611148	−	0.791516i	$-0.709292\pi$
$619$	−30.4104	−1.22230	−0.611148	+	0.791516i	$0.709292\pi$
$620$	0	0
$621$	15.9364	0.639506
$622$	− 45.5784i	− 1.82753i
$623$	− 0.725972i	− 0.0290855i
$624$	62.8496	2.51600
$625$	0	0
$626$	1.68282	0.0672592
$627$	6.86903i	0.274323i
$628$	31.9328i	1.27426i
$629$	−1.96925	−0.0785190
$630$	0	0
$631$	20.9541	0.834171	0.417085	−	0.908867i	$-0.363052\pi$
$631$	20.9541	0.834171	0.417085	+	0.908867i	$0.363052\pi$
$632$	− 1.08230i	− 0.0430514i
$633$	28.4664i	1.13144i
$634$	65.3848	2.59676
$635$	0	0
$636$	9.59558	0.380490
$637$	33.4882i	1.32685i
$638$	− 11.3769i	− 0.450417i
$639$	46.7850	1.85079
$640$	0	0
$641$	24.0572	0.950201	0.475101	−	0.879931i	$-0.342412\pi$
$641$	24.0572	0.950201	0.475101	+	0.879931i	$0.342412\pi$
$642$	− 1.80791i	− 0.0713526i
$643$	− 16.8597i	− 0.664884i	−0.943124	−	0.332442i	$-0.892128\pi$
$643$	− 16.8597i	− 0.664884i	0.943124	−	0.332442i	$-0.107872\pi$
$644$	−2.40228	−0.0946631
$645$	0	0
$646$	−6.24824	−0.245834
$647$	28.3088i	1.11294i	0.830869	+	0.556468i	$0.187844\pi$
$647$	28.3088i	1.11294i	−0.830869	+	0.556468i	$0.812156\pi$
$648$	4.17043i	0.163830i
$649$	3.44252	0.135131
$650$	0	0
$651$	4.72295	0.185107
$652$	11.1807i	0.437870i
$653$	0.690331i	0.0270148i	0.999909	+	0.0135074i	$0.00429966\pi$
$653$	0.690331i	0.0270148i	−0.999909	+	0.0135074i	$0.995700\pi$
$654$	55.3197	2.16317
$655$	0	0
$656$	32.7597	1.27905
$657$	23.9963i	0.936185i
$658$	− 7.90121i	− 0.308021i
$659$	−14.0952	−0.549073	−0.274536	−	0.961577i	$-0.588524\pi$
$659$	−14.0952	−0.549073	−0.274536	+	0.961577i	$0.588524\pi$
$660$	0	0
$661$	−37.8923	−1.47384	−0.736920	−	0.675980i	$-0.763720\pi$
$661$	−37.8923	−1.47384	−0.736920	+	0.675980i	$0.763720\pi$
$662$	− 52.8150i	− 2.05271i
$663$	− 17.2038i	− 0.668140i
$664$	15.5877	0.604919
$665$	0	0
$666$	−11.6267	−0.450525
$667$	− 30.3645i	− 1.17572i
$668$	35.0543i	1.35629i
$669$	30.9415	1.19627
$670$	0	0
$671$	4.59459	0.177372
$672$	6.20680i	0.239432i
$673$	49.3462i	1.90216i	0.308952	+	0.951078i	$0.400022\pi$
$673$	49.3462i	1.90216i	−0.308952	+	0.951078i	$0.599978\pi$
$674$	11.8655	0.457044
$675$	0	0
$676$	15.2486	0.586483
$677$	− 26.8343i	− 1.03133i	−0.856791	−	0.515664i	$-0.827545\pi$
$677$	− 26.8343i	− 1.03133i	0.856791	−	0.515664i	$-0.172455\pi$
$678$	93.8059i	3.60259i
$679$	2.38843	0.0916594
$680$	0	0
$681$	69.4518	2.66140
$682$	9.59389i	0.367369i
$683$	36.6861i	1.40375i	0.712299	+	0.701876i	$0.247654\pi$
$683$	36.6861i	1.40375i	−0.712299	+	0.701876i	$0.752346\pi$
$684$	−15.3820	−0.588147
$685$	0	0
$686$	−8.73734	−0.333593
$687$	17.2274i	0.657267i
$688$	51.4737i	1.96242i
$689$	−12.1603	−0.463270
$690$	0	0
$691$	−6.81029	−0.259076	−0.129538	−	0.991574i	$-0.541349\pi$
$691$	−6.81029	−0.259076	−0.129538	+	0.991574i	$0.541349\pi$
$692$	10.3181i	0.392234i
$693$	− 1.42748i	− 0.0542254i
$694$	−35.4406	−1.34531
$695$	0	0
$696$	−17.3923	−0.659255
$697$	− 8.96730i	− 0.339661i
$698$	− 15.0473i	− 0.569550i
$699$	−23.9004	−0.903996
$700$	0	0
$701$	1.76773	0.0667661	0.0333831	−	0.999443i	$-0.489372\pi$
$701$	1.76773	0.0667661	0.0333831	+	0.999443i	$0.489372\pi$
$702$	− 29.0446i	− 1.09622i
$703$	− 3.82461i	− 0.144248i
$704$	−2.97844	−0.112254
$705$	0	0
$706$	−19.8073	−0.745459
$707$	0.941071i	0.0353926i
$708$	13.2135i	0.496595i
$709$	6.47447	0.243154	0.121577	−	0.992582i	$-0.461205\pi$
$709$	6.47447	0.243154	0.121577	+	0.992582i	$0.461205\pi$
$710$	0	0
$711$	−4.30966	−0.161625
$712$	− 2.25442i	− 0.0844878i
$713$	25.6057i	0.958939i
$714$	2.22565	0.0832927
$715$	0	0
$716$	−32.0620	−1.19821
$717$	74.8284i	2.79452i
$718$	− 10.2949i	− 0.384202i
$719$	−27.0016	−1.00699	−0.503496	−	0.863998i	$-0.667953\pi$
$719$	−27.0016	−1.00699	−0.503496	+	0.863998i	$0.667953\pi$
$720$	0	0
$721$	−1.12668	−0.0419597
$722$	23.0550i	0.858020i
$723$	51.5800i	1.91828i
$724$	−14.1519	−0.525953
$725$	0	0
$726$	4.97021	0.184462
$727$	6.10099i	0.226273i	0.993579	+	0.113137i	$0.0360898\pi$
$727$	6.10099i	0.226273i	−0.993579	+	0.113137i	$0.963910\pi$
$728$	− 1.74377i	− 0.0646283i
$729$	42.5608	1.57633
$730$	0	0
$731$	14.0899	0.521133
$732$	17.6356i	0.651829i
$733$	− 1.77481i	− 0.0655543i	−0.999463	−	0.0327771i	$-0.989565\pi$
$733$	− 1.77481i	− 0.0655543i	0.999463	−	0.0327771i	$-0.0104352\pi$
$734$	44.1188	1.62845
$735$	0	0
$736$	−33.6504	−1.24037
$737$	− 10.3237i	− 0.380280i
$738$	− 52.9442i	− 1.94890i
$739$	6.90773	0.254105	0.127052	−	0.991896i	$-0.459448\pi$
$739$	6.90773	0.254105	0.127052	+	0.991896i	$0.459448\pi$
$740$	0	0
$741$	33.4127	1.22745
$742$	− 1.57317i	− 0.0577529i
$743$	− 10.5928i	− 0.388611i	−0.980941	−	0.194305i	$-0.937755\pi$
$743$	− 10.5928i	− 0.388611i	0.980941	−	0.194305i	$-0.0622453\pi$
$744$	14.6665	0.537701
$745$	0	0
$746$	31.0408	1.13648
$747$	− 62.0694i	− 2.27100i
$748$	1.88511i	0.0689265i
$749$	−0.123590	−0.00451586
$750$	0	0
$751$	5.47689	0.199855	0.0999273	−	0.994995i	$-0.468139\pi$
$751$	5.47689	0.199855	0.0999273	+	0.994995i	$0.468139\pi$
$752$	− 60.4542i	− 2.20454i
$753$	− 31.9288i	− 1.16355i
$754$	−55.3402	−2.01537
$755$	0	0
$756$	1.56674	0.0569817
$757$	13.6035i	0.494427i	0.968961	+	0.247213i	$0.0795149\pi$
$757$	13.6035i	0.494427i	−0.968961	+	0.247213i	$0.920485\pi$
$758$	37.7406i	1.37080i
$759$	13.2653	0.481499
$760$	0	0
$761$	9.54198	0.345896	0.172948	−	0.984931i	$-0.444671\pi$
$761$	9.54198	0.345896	0.172948	+	0.984931i	$0.444671\pi$
$762$	38.0507i	1.37843i
$763$	− 3.78168i	− 0.136906i
$764$	−12.1299	−0.438843
$765$	0	0
$766$	28.7871	1.04012
$767$	− 16.7452i	− 0.604636i
$768$	56.2362i	2.02925i
$769$	2.23015	0.0804213	0.0402107	−	0.999191i	$-0.487197\pi$
$769$	2.23015	0.0804213	0.0402107	+	0.999191i	$0.487197\pi$
$770$	0	0
$771$	1.54247	0.0555506
$772$	− 25.3948i	− 0.913978i
$773$	18.3610i	0.660401i	0.943911	+	0.330200i	$0.107116\pi$
$773$	18.3610i	0.660401i	−0.943911	+	0.330200i	$0.892884\pi$
$774$	83.1885	2.99015
$775$	0	0
$776$	7.41696	0.266253
$777$	1.36234i	0.0488736i
$778$	29.3500i	1.05225i
$779$	17.4160	0.623993
$780$	0	0
$781$	11.1357	0.398466
$782$	12.0664i	0.431494i
$783$	19.8033i	0.707714i
$784$	−33.1479	−1.18385
$785$	0	0
$786$	71.3665	2.54556
$787$	24.4774i	0.872524i	0.899820	+	0.436262i	$0.143698\pi$
$787$	24.4774i	0.872524i	−0.899820	+	0.436262i	$0.856302\pi$
$788$	4.76446i	0.169727i
$789$	−50.9527	−1.81397
$790$	0	0
$791$	6.41261	0.228006
$792$	− 4.43285i	− 0.157515i
$793$	− 22.3492i	− 0.793643i
$794$	55.0338	1.95308
$795$	0	0
$796$	−15.9612	−0.565731
$797$	− 3.83074i	− 0.135692i	−0.997696	−	0.0678460i	$-0.978387\pi$
$797$	− 3.83074i	− 0.135692i	0.997696	−	0.0678460i	$-0.0216126\pi$
$798$	4.32258i	0.153018i
$799$	−16.5481	−0.585430
$800$	0	0
$801$	−8.97699	−0.317186
$802$	60.7528i	2.14526i
$803$	5.71155i	0.201556i
$804$	39.6260	1.39750
$805$	0	0
$806$	46.6671	1.64378
$807$	− 49.6814i	− 1.74887i
$808$	2.92238i	0.102809i
$809$	14.4683	0.508678	0.254339	−	0.967115i	$-0.418142\pi$
$809$	14.4683	0.508678	0.254339	+	0.967115i	$0.418142\pi$
$810$	0	0
$811$	2.70562	0.0950072	0.0475036	−	0.998871i	$-0.484873\pi$
$811$	2.70562	0.0950072	0.0475036	+	0.998871i	$0.484873\pi$
$812$	− 2.98519i	− 0.104760i
$813$	54.4975i	1.91131i
$814$	−2.76736	−0.0969961
$815$	0	0
$816$	17.0290	0.596134
$817$	27.3649i	0.957376i
$818$	− 63.3239i	− 2.21407i
$819$	−6.94361	−0.242629
$820$	0	0
$821$	19.2422	0.671557	0.335778	−	0.941941i	$-0.391001\pi$
$821$	19.2422	0.671557	0.335778	+	0.941941i	$0.391001\pi$
$822$	− 83.9652i	− 2.92862i
$823$	− 0.696512i	− 0.0242789i	−0.999926	−	0.0121394i	$-0.996136\pi$
$823$	− 0.696512i	− 0.0242789i	0.999926	−	0.0121394i	$-0.00386420\pi$
$824$	−3.49876	−0.121885
$825$	0	0
$826$	2.16632	0.0753760
$827$	− 48.7156i	− 1.69401i	−0.531587	−	0.847004i	$-0.678404\pi$
$827$	− 48.7156i	− 1.69401i	0.531587	−	0.847004i	$-0.321596\pi$
$828$	29.7053i	1.03233i
$829$	−35.6887	−1.23952	−0.619760	−	0.784791i	$-0.712770\pi$
$829$	−35.6887	−1.23952	−0.619760	+	0.784791i	$0.712770\pi$
$830$	0	0
$831$	−15.5564	−0.539646
$832$	14.4879i	0.502276i
$833$	9.07356i	0.314380i
$834$	−41.2991	−1.43007
$835$	0	0
$836$	−3.66120	−0.126625
$837$	− 16.6997i	− 0.577226i
$838$	− 61.3620i	− 2.11972i
$839$	17.9672	0.620296	0.310148	−	0.950688i	$-0.399621\pi$
$839$	17.9672	0.620296	0.310148	+	0.950688i	$0.399621\pi$
$840$	0	0
$841$	8.73239	0.301117
$842$	64.2653i	2.21473i
$843$	− 59.9449i	− 2.06461i
$844$	−15.1726	−0.522263
$845$	0	0
$846$	−97.7022	−3.35907
$847$	− 0.339766i	− 0.0116745i
$848$	− 12.0367i	− 0.413343i
$849$	−46.7954	−1.60601
$850$	0	0
$851$	−7.38597	−0.253188
$852$	42.7425i	1.46433i
$853$	− 16.0716i	− 0.550281i	−0.961404	−	0.275141i	$-0.911276\pi$
$853$	− 16.0716i	− 0.550281i	0.961404	−	0.275141i	$-0.0887244\pi$
$854$	2.89130	0.0989383
$855$	0	0
$856$	−0.383792	−0.0131177
$857$	18.9322i	0.646710i	0.946278	+	0.323355i	$0.104811\pi$
$857$	18.9322i	0.646710i	−0.946278	+	0.323355i	$0.895189\pi$
$858$	− 24.1763i	− 0.825367i
$859$	−34.0751	−1.16263	−0.581313	−	0.813680i	$-0.697461\pi$
$859$	−34.0751	−1.16263	−0.581313	+	0.813680i	$0.697461\pi$
$860$	0	0
$861$	−6.20364	−0.211420
$862$	− 20.5903i	− 0.701307i
$863$	33.3267i	1.13445i	0.823562	+	0.567226i	$0.191983\pi$
$863$	33.3267i	1.13445i	−0.823562	+	0.567226i	$0.808017\pi$
$864$	21.9464	0.746630
$865$	0	0
$866$	46.2026	1.57003
$867$	40.9588i	1.39103i
$868$	2.51734i	0.0854441i
$869$	−1.02578	−0.0347971
$870$	0	0
$871$	−50.2173	−1.70155
$872$	− 11.7435i	− 0.397686i
$873$	− 29.5340i	− 0.999575i
$874$	−23.4350	−0.792701
$875$	0	0
$876$	−21.9228	−0.740704
$877$	− 7.72385i	− 0.260816i	−0.991460	−	0.130408i	$-0.958371\pi$
$877$	− 7.72385i	− 0.260816i	0.991460	−	0.130408i	$-0.0416287\pi$
$878$	− 70.0120i	− 2.36279i
$879$	−21.9159	−0.739204
$880$	0	0
$881$	−28.3698	−0.955802	−0.477901	−	0.878414i	$-0.658602\pi$
$881$	−28.3698	−0.955802	−0.477901	+	0.878414i	$0.658602\pi$
$882$	53.5715i	1.80385i
$883$	− 10.0458i	− 0.338070i	−0.985610	−	0.169035i	$-0.945935\pi$
$883$	− 10.0458i	− 0.338070i	0.985610	−	0.169035i	$-0.0540650\pi$
$884$	9.16964	0.308409
$885$	0	0
$886$	−17.9697	−0.603703
$887$	19.9372i	0.669425i	0.942320	+	0.334713i	$0.108639\pi$
$887$	19.9372i	0.669425i	−0.942320	+	0.334713i	$0.891361\pi$
$888$	4.23057i	0.141969i
$889$	2.60116	0.0872400
$890$	0	0
$891$	3.95264	0.132418
$892$	16.4919i	0.552189i
$893$	− 32.1392i	− 1.07550i
$894$	−68.0780	−2.27687
$895$	0	0
$896$	2.75155	0.0919229
$897$	− 64.5255i	− 2.15445i
$898$	12.2134i	0.407565i
$899$	−31.8188	−1.06122
$900$	0	0
$901$	−3.29481	−0.109766
$902$	− 12.6017i	− 0.419589i
$903$	− 9.74747i	− 0.324376i
$904$	19.9135	0.662314
$905$	0	0
$906$	−78.5102	−2.60833
$907$	15.4497i	0.512998i	0.966544	+	0.256499i	$0.0825690\pi$
$907$	15.4497i	0.512998i	−0.966544	+	0.256499i	$0.917431\pi$
$908$	37.0179i	1.22848i
$909$	11.6368	0.385968
$910$	0	0
$911$	−29.9758	−0.993143	−0.496571	−	0.867996i	$-0.665408\pi$
$911$	−29.9758	−0.993143	−0.496571	+	0.867996i	$0.665408\pi$
$912$	33.0732i	1.09516i
$913$	− 14.7736i	− 0.488936i
$914$	−39.2570	−1.29851
$915$	0	0
$916$	−9.18223	−0.303389
$917$	− 4.87864i	− 0.161107i
$918$	− 7.86957i	− 0.259734i
$919$	49.7408	1.64080	0.820400	−	0.571791i	$-0.193751\pi$
$919$	49.7408	1.64080	0.820400	+	0.571791i	$0.193751\pi$
$920$	0	0
$921$	65.3640	2.15382
$922$	32.7086i	1.07720i
$923$	− 54.1667i	− 1.78292i
$924$	1.30413	0.0429028
$925$	0	0
$926$	−72.0219	−2.36679
$927$	13.9319i	0.457584i
$928$	− 41.8156i	− 1.37266i
$929$	−22.9162	−0.751857	−0.375928	−	0.926649i	$-0.622676\pi$
$929$	−22.9162	−0.751857	−0.375928	+	0.926649i	$0.622676\pi$
$930$	0	0
$931$	−17.6224	−0.577550
$932$	− 12.7389i	− 0.417278i
$933$	66.0387i	2.16201i
$934$	−5.93135	−0.194080
$935$	0	0
$936$	−21.5625	−0.704792
$937$	− 36.1322i	− 1.18039i	−0.807261	−	0.590194i	$-0.799051\pi$
$937$	− 36.1322i	− 1.18039i	0.807261	−	0.590194i	$-0.200949\pi$
$938$	− 6.49658i	− 0.212121i
$939$	−2.43825	−0.0795693
$940$	0	0
$941$	−18.7912	−0.612577	−0.306288	−	0.951939i	$-0.599087\pi$
$941$	−18.7912	−0.612577	−0.306288	+	0.951939i	$0.599087\pi$
$942$	− 110.963i	− 3.61536i
$943$	− 33.6332i	− 1.09525i
$944$	16.5751	0.539473
$945$	0	0
$946$	19.8004	0.643765
$947$	49.4615i	1.60728i	0.595114	+	0.803642i	$0.297107\pi$
$947$	49.4615i	1.60728i	−0.595114	+	0.803642i	$0.702893\pi$
$948$	− 3.93727i	− 0.127877i
$949$	27.7824	0.901855
$950$	0	0
$951$	−94.7363	−3.07203
$952$	− 0.472470i	− 0.0153128i
$953$	44.6278i	1.44564i	0.691039	+	0.722818i	$0.257153\pi$
$953$	44.6278i	1.44564i	−0.691039	+	0.722818i	$0.742847\pi$
$954$	−19.4530	−0.629814
$955$	0	0
$956$	−39.8836	−1.28993
$957$	16.4841i	0.532854i
$958$	− 18.3767i	− 0.593724i
$959$	−5.73989	−0.185351
$960$	0	0
$961$	−4.16793	−0.134449
$962$	13.4611i	0.434005i
$963$	1.52824i	0.0492470i
$964$	−27.4922	−0.885464
$965$	0	0
$966$	8.34763	0.268581
$967$	− 4.40647i	− 0.141703i	−0.997487	−	0.0708513i	$-0.977428\pi$
$967$	− 4.40647i	− 0.141703i	0.997487	−	0.0708513i	$-0.0225716\pi$
$968$	− 1.05510i	− 0.0339122i
$969$	9.05310	0.290827
$970$	0	0
$971$	−5.35924	−0.171986	−0.0859930	−	0.996296i	$-0.527406\pi$
$971$	−5.35924	−0.171986	−0.0859930	+	0.996296i	$0.527406\pi$
$972$	29.0052i	0.930344i
$973$	2.82322i	0.0905084i
$974$	−11.4278	−0.366170
$975$	0	0
$976$	22.1221	0.708111
$977$	− 24.8480i	− 0.794958i	−0.917611	−	0.397479i	$-0.869885\pi$
$977$	− 24.8480i	− 0.794958i	0.917611	−	0.397479i	$-0.130115\pi$
$978$	− 38.8515i	− 1.24234i
$979$	−2.13669	−0.0682888
$980$	0	0
$981$	−46.7622	−1.49300
$982$	33.6231i	1.07296i
$983$	8.49524i	0.270956i	0.990780	+	0.135478i	$0.0432570\pi$
$983$	8.49524i	0.270956i	−0.990780	+	0.135478i	$0.956743\pi$
$984$	−19.2646	−0.614134
$985$	0	0
$986$	−14.9943	−0.477516
$987$	11.4481i	0.364397i
$988$	17.8090i	0.566579i
$989$	52.8462	1.68041
$990$	0	0
$991$	36.9086	1.17244	0.586220	−	0.810152i	$-0.300615\pi$
$991$	36.9086	1.17244	0.586220	+	0.810152i	$0.300615\pi$
$992$	35.2621i	1.11957i
$993$	76.5238i	2.42841i
$994$	7.00751	0.222265
$995$	0	0
$996$	56.7062	1.79680
$997$	41.7386i	1.32187i	0.750442	+	0.660937i	$0.229841\pi$
$997$	41.7386i	1.32187i	−0.750442	+	0.660937i	$0.770159\pi$
$998$	19.9408i	0.631214i
$999$	4.81704	0.152404

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1375.2.b.e.749.5		20
5.2	odd	4		1375.2.a.g.1.7	yes	10
5.3	odd	4		1375.2.a.d.1.4	✓	10
5.4	even	2	inner	1375.2.b.e.749.16		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1375.2.a.d.1.4	✓	10	5.3	odd	4
1375.2.a.g.1.7	yes	10	5.2	odd	4
1375.2.b.e.749.5		20	1.1	even	1	trivial
1375.2.b.e.749.16		20	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1375 = 5^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1375.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.85211i q^{2} +2.68354i q^{3} -1.43033 q^{4} +4.97021 q^{6} -0.339766i q^{7} -1.05510i q^{8} -4.20136 q^{9} +O(q^{10})\)	\(q-1.85211i q^{2} +2.68354i q^{3} -1.43033 q^{4} +4.97021 q^{6} -0.339766i q^{7} -1.05510i q^{8} -4.20136 q^{9} -1.00000 q^{11} -3.83833i q^{12} +4.86425i q^{13} -0.629285 q^{14} -4.81482 q^{16} +1.31796i q^{17} +7.78140i q^{18} -2.55970 q^{19} +0.911773 q^{21} +1.85211i q^{22} +4.94321i q^{23} +2.83140 q^{24} +9.00914 q^{26} -3.22390i q^{27} +0.485976i q^{28} -6.14267 q^{29} +5.17997 q^{31} +6.80740i q^{32} -2.68354i q^{33} +2.44101 q^{34} +6.00932 q^{36} +1.49416i q^{37} +4.74085i q^{38} -13.0534 q^{39} -6.80393 q^{41} -1.68871i q^{42} -10.6907i q^{43} +1.43033 q^{44} +9.15538 q^{46} +12.5559i q^{47} -12.9207i q^{48} +6.88456 q^{49} -3.53679 q^{51} -6.95746i q^{52} +2.49993i q^{53} -5.97103 q^{54} -0.358486 q^{56} -6.86903i q^{57} +11.3769i q^{58} -3.44252 q^{59} -4.59459 q^{61} -9.59389i q^{62} +1.42748i q^{63} +2.97844 q^{64} -4.97021 q^{66} +10.3237i q^{67} -1.88511i q^{68} -13.2653 q^{69} -11.1357 q^{71} +4.43285i q^{72} -5.71155i q^{73} +2.76736 q^{74} +3.66120 q^{76} +0.339766i q^{77} +24.1763i q^{78} +1.02578 q^{79} -3.95264 q^{81} +12.6017i q^{82} +14.7736i q^{83} -1.30413 q^{84} -19.8004 q^{86} -16.4841i q^{87} +1.05510i q^{88} +2.13669 q^{89} +1.65270 q^{91} -7.07040i q^{92} +13.9006i q^{93} +23.2549 q^{94} -18.2679 q^{96} +7.02963i q^{97} -12.7510i q^{98} +4.20136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 30 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) 20 * q - 30 * q^4 + 2 * q^6 - 16 * q^9	\( 20 q - 30 q^{4} + 2 q^{6} - 16 q^{9} - 20 q^{11} - 12 q^{14} + 42 q^{16} + 2 q^{19} - 42 q^{21} + 36 q^{24} - 12 q^{26} - 2 q^{29} + 26 q^{31} - 50 q^{34} + 52 q^{36} - 4 q^{41} + 30 q^{44} - 36 q^{46} - 2 q^{49} - 42 q^{51} - 30 q^{54} + 40 q^{56} + 16 q^{59} + 50 q^{61} - 116 q^{64} - 2 q^{66} + 36 q^{69} - 52 q^{71} + 76 q^{74} + 22 q^{76} + 26 q^{79} + 4 q^{81} - 48 q^{84} + 98 q^{86} - 26 q^{89} - 60 q^{91} + 52 q^{94} - 106 q^{96} + 16 q^{99}+O(q^{100}) \) 20 * q - 30 * q^4 + 2 * q^6 - 16 * q^9 - 20 * q^11 - 12 * q^14 + 42 * q^16 + 2 * q^19 - 42 * q^21 + 36 * q^24 - 12 * q^26 - 2 * q^29 + 26 * q^31 - 50 * q^34 + 52 * q^36 - 4 * q^41 + 30 * q^44 - 36 * q^46 - 2 * q^49 - 42 * q^51 - 30 * q^54 + 40 * q^56 + 16 * q^59 + 50 * q^61 - 116 * q^64 - 2 * q^66 + 36 * q^69 - 52 * q^71 + 76 * q^74 + 22 * q^76 + 26 * q^79 + 4 * q^81 - 48 * q^84 + 98 * q^86 - 26 * q^89 - 60 * q^91 + 52 * q^94 - 106 * q^96 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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