LMFDB - Embedded newform 1875.2.b.g.1249.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 $ x^16 + 25x^14 + 246x^12 + 1220x^10 + 3281x^8 + 4880x^6 + 3936x^4 + 1600*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.6
Root			$-2.23365i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.g.1249.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.895394i q^{2} -1.00000i q^{3} +1.19827 q^{4} -0.895394 q^{6} +5.08992i q^{7} -2.86371i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-0.895394i q^{2} -1.00000i q^{3} +1.19827 q^{4} -0.895394 q^{6} +5.08992i q^{7} -2.86371i q^{8} -1.00000 q^{9} +2.64310 q^{11} -1.19827i q^{12} -2.13295i q^{13} +4.55748 q^{14} -0.167607 q^{16} +7.75001i q^{17} +0.895394i q^{18} -3.08652 q^{19} +5.08992 q^{21} -2.36661i q^{22} +6.14107i q^{23} -2.86371 q^{24} -1.90983 q^{26} +1.00000i q^{27} +6.09909i q^{28} +4.13435 q^{29} -2.74277 q^{31} -5.57735i q^{32} -2.64310i q^{33} +6.93931 q^{34} -1.19827 q^{36} -0.0157706i q^{37} +2.76365i q^{38} -2.13295 q^{39} +3.72829 q^{41} -4.55748i q^{42} +3.81468i q^{43} +3.16715 q^{44} +5.49868 q^{46} -0.897385i q^{47} +0.167607i q^{48} -18.9072 q^{49} +7.75001 q^{51} -2.55585i q^{52} +9.26724i q^{53} +0.895394 q^{54} +14.5760 q^{56} +3.08652i q^{57} -3.70187i q^{58} +11.0693 q^{59} +6.38696 q^{61} +2.45585i q^{62} -5.08992i q^{63} -5.32913 q^{64} -2.36661 q^{66} -5.54154i q^{67} +9.28660i q^{68} +6.14107 q^{69} -0.0828976 q^{71} +2.86371i q^{72} +9.92024i q^{73} -0.0141209 q^{74} -3.69848 q^{76} +13.4532i q^{77} +1.90983i q^{78} -5.30049 q^{79} +1.00000 q^{81} -3.33829i q^{82} -0.723557i q^{83} +6.09909 q^{84} +3.41564 q^{86} -4.13435i q^{87} -7.56907i q^{88} +13.2548 q^{89} +10.8565 q^{91} +7.35867i q^{92} +2.74277i q^{93} -0.803513 q^{94} -5.57735 q^{96} -2.22836i q^{97} +16.9294i q^{98} -2.64310 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9	$ 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44} + 66 q^{46} - 60 q^{49} - 2 q^{51} - 2 q^{54} + 120 q^{56} - 28 q^{59} + 20 q^{61} - 82 q^{64} + 36 q^{66} + 8 q^{69} + 42 q^{71} + 18 q^{74} - 2 q^{76} - 20 q^{79} + 16 q^{81} + 42 q^{84} + 84 q^{86} + 18 q^{89} - 24 q^{91} - 28 q^{94} - 36 q^{96} - 24 q^{99}+O(q^{100}) $ 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9 + 24 * q^11 - 32 * q^14 + 30 * q^16 - 32 * q^19 + 24 * q^21 - 6 * q^24 - 68 * q^26 - 4 * q^29 + 26 * q^31 + 74 * q^34 + 18 * q^36 - 28 * q^39 - 24 * q^41 - 94 * q^44 + 66 * q^46 - 60 * q^49 - 2 * q^51 - 2 * q^54 + 120 * q^56 - 28 * q^59 + 20 * q^61 - 82 * q^64 + 36 * q^66 + 8 * q^69 + 42 * q^71 + 18 * q^74 - 2 * q^76 - 20 * q^79 + 16 * q^81 + 42 * q^84 + 84 * q^86 + 18 * q^89 - 24 * q^91 - 28 * q^94 - 36 * q^96 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 0.895394i	− 0.633139i	−0.948569	−	0.316569i	$-0.897469\pi$
$2$	− 0.895394i	− 0.633139i	0.948569	−	0.316569i	$-0.102531\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.19827	0.599135
$5$	0	0
$5$	0	0
$6$	−0.895394	−0.365543
$7$	5.08992i	1.92381i	0.273389	+	0.961904i	$0.411855\pi$
$7$	5.08992i	1.92381i	−0.273389	+	0.961904i	$0.588145\pi$
$8$	− 2.86371i	− 1.01247i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.64310	0.796925	0.398462	−	0.917185i	$-0.369544\pi$
$11$	2.64310	0.796925	0.398462	+	0.917185i	$0.369544\pi$
$12$	− 1.19827i	− 0.345911i
$13$	− 2.13295i	− 0.591575i	−0.955254	−	0.295787i	$-0.904418\pi$
$13$	− 2.13295i	− 0.591575i	0.955254	−	0.295787i	$-0.0955820\pi$
$14$	4.55748	1.21804
$15$	0	0
$16$	−0.167607	−0.0419018
$17$	7.75001i	1.87965i	0.341653	+	0.939826i	$0.389013\pi$
$17$	7.75001i	1.87965i	−0.341653	+	0.939826i	$0.610987\pi$
$18$	0.895394i	0.211046i
$19$	−3.08652	−0.708096	−0.354048	−	0.935227i	$-0.615195\pi$
$19$	−3.08652	−0.708096	−0.354048	+	0.935227i	$0.615195\pi$
$20$	0	0
$21$	5.08992	1.11071
$22$	− 2.36661i	− 0.504564i
$23$	6.14107i	1.28050i	0.768166	+	0.640251i	$0.221170\pi$
$23$	6.14107i	1.28050i	−0.768166	+	0.640251i	$0.778830\pi$
$24$	−2.86371	−0.584552
$25$	0	0
$26$	−1.90983	−0.374549
$27$	1.00000i	0.192450i
$28$	6.09909i	1.15262i
$29$	4.13435	0.767730	0.383865	−	0.923389i	$-0.374593\pi$
$29$	4.13435	0.767730	0.383865	+	0.923389i	$0.374593\pi$
$30$	0	0
$31$	−2.74277	−0.492615	−0.246308	−	0.969192i	$-0.579217\pi$
$31$	−2.74277	−0.492615	−0.246308	+	0.969192i	$0.579217\pi$
$32$	− 5.57735i	− 0.985945i
$33$	− 2.64310i	− 0.460105i
$34$	6.93931	1.19008
$35$	0	0
$36$	−1.19827	−0.199712
$37$	− 0.0157706i	− 0.00259267i	−0.999999	−	0.00129634i	$-0.999587\pi$
$37$	− 0.0157706i	− 0.00259267i	0.999999	−	0.00129634i	$-0.000412636\pi$
$38$	2.76365i	0.448323i
$39$	−2.13295	−0.341546
$40$	0	0
$41$	3.72829	0.582262	0.291131	−	0.956683i	$-0.405969\pi$
$41$	3.72829	0.582262	0.291131	+	0.956683i	$0.405969\pi$
$42$	− 4.55748i	− 0.703234i
$43$	3.81468i	0.581733i	0.956764	+	0.290866i	$0.0939435\pi$
$43$	3.81468i	0.581733i	−0.956764	+	0.290866i	$0.906056\pi$
$44$	3.16715	0.477466
$45$	0	0
$46$	5.49868	0.810736
$47$	− 0.897385i	− 0.130897i	−0.997856	−	0.0654486i	$-0.979152\pi$
$47$	− 0.897385i	− 0.130897i	0.997856	−	0.0654486i	$-0.0208478\pi$
$48$	0.167607i	0.0241920i
$49$	−18.9072	−2.70103
$50$	0	0
$51$	7.75001	1.08522
$52$	− 2.55585i	− 0.354433i
$53$	9.26724i	1.27295i	0.771296	+	0.636477i	$0.219609\pi$
$53$	9.26724i	1.27295i	−0.771296	+	0.636477i	$0.780391\pi$
$54$	0.895394	0.121848
$55$	0	0
$56$	14.5760	1.94781
$57$	3.08652i	0.408819i
$58$	− 3.70187i	− 0.486080i
$59$	11.0693	1.44111	0.720553	−	0.693400i	$-0.243888\pi$
$59$	11.0693	1.44111	0.720553	+	0.693400i	$0.243888\pi$
$60$	0	0
$61$	6.38696	0.817766	0.408883	−	0.912587i	$-0.365918\pi$
$61$	6.38696	0.817766	0.408883	+	0.912587i	$0.365918\pi$
$62$	2.45585i	0.311894i
$63$	− 5.08992i	− 0.641269i
$64$	−5.32913	−0.666142
$65$	0	0
$66$	−2.36661	−0.291310
$67$	− 5.54154i	− 0.677007i	−0.940965	−	0.338504i	$-0.890079\pi$
$67$	− 5.54154i	− 0.677007i	0.940965	−	0.338504i	$-0.109921\pi$
$68$	9.28660i	1.12617i
$69$	6.14107	0.739298
$70$	0	0
$71$	−0.0828976	−0.00983814	−0.00491907	−	0.999988i	$-0.501566\pi$
$71$	−0.0828976	−0.00983814	−0.00491907	+	0.999988i	$0.501566\pi$
$72$	2.86371i	0.337492i
$73$	9.92024i	1.16108i	0.814233	+	0.580538i	$0.197158\pi$
$73$	9.92024i	1.16108i	−0.814233	+	0.580538i	$0.802842\pi$
$74$	−0.0141209	−0.00164152
$75$	0	0
$76$	−3.69848	−0.424245
$77$	13.4532i	1.53313i
$78$	1.90983i	0.216246i
$79$	−5.30049	−0.596352	−0.298176	−	0.954511i	$-0.596378\pi$
$79$	−5.30049	−0.596352	−0.298176	+	0.954511i	$0.596378\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 3.33829i	− 0.368652i
$83$	− 0.723557i	− 0.0794207i	−0.999211	−	0.0397104i	$-0.987356\pi$
$83$	− 0.723557i	− 0.0794207i	0.999211	−	0.0397104i	$-0.0126435\pi$
$84$	6.09909	0.665466
$85$	0	0
$86$	3.41564	0.368318
$87$	− 4.13435i	− 0.443249i
$88$	− 7.56907i	− 0.806866i
$89$	13.2548	1.40500	0.702500	−	0.711683i	$-0.252067\pi$
$89$	13.2548	1.40500	0.702500	+	0.711683i	$0.252067\pi$
$90$	0	0
$91$	10.8565	1.13808
$92$	7.35867i	0.767194i
$93$	2.74277i	0.284412i
$94$	−0.803513	−0.0828760
$95$	0	0
$96$	−5.57735	−0.569236
$97$	− 2.22836i	− 0.226256i	−0.993580	−	0.113128i	$-0.963913\pi$
$97$	− 2.22836i	− 0.226256i	0.993580	−	0.113128i	$-0.0360870\pi$
$98$	16.9294i	1.71013i
$99$	−2.64310	−0.265642
$100$	0	0
$101$	15.4908	1.54140	0.770698	−	0.637201i	$-0.219908\pi$
$101$	15.4908	1.54140	0.770698	+	0.637201i	$0.219908\pi$
$102$	− 6.93931i	− 0.687094i
$103$	− 11.5680i	− 1.13983i	−0.821705	−	0.569913i	$-0.806977\pi$
$103$	− 11.5680i	− 1.13983i	0.821705	−	0.569913i	$-0.193023\pi$
$104$	−6.10816	−0.598954
$105$	0	0
$106$	8.29783	0.805956
$107$	1.10562i	0.106884i	0.998571	+	0.0534419i	$0.0170192\pi$
$107$	1.10562i	0.106884i	−0.998571	+	0.0534419i	$0.982981\pi$
$108$	1.19827i	0.115304i
$109$	−10.2626	−0.982976	−0.491488	−	0.870884i	$-0.663547\pi$
$109$	−10.2626	−0.982976	−0.491488	+	0.870884i	$0.663547\pi$
$110$	0	0
$111$	−0.0157706	−0.00149688
$112$	− 0.853106i	− 0.0806109i
$113$	− 1.74010i	− 0.163695i	−0.996645	−	0.0818474i	$-0.973918\pi$
$113$	− 1.74010i	− 0.163695i	0.996645	−	0.0818474i	$-0.0260820\pi$
$114$	2.76365	0.258839
$115$	0	0
$116$	4.95407	0.459974
$117$	2.13295i	0.197192i
$118$	− 9.91142i	− 0.912421i
$119$	−39.4469	−3.61609
$120$	0	0
$121$	−4.01402	−0.364911
$122$	− 5.71884i	− 0.517760i
$123$	− 3.72829i	− 0.336169i
$124$	−3.28657	−0.295143
$125$	0	0
$126$	−4.55748	−0.406012
$127$	− 9.77368i	− 0.867274i	−0.901088	−	0.433637i	$-0.857230\pi$
$127$	− 9.77368i	− 0.867274i	0.901088	−	0.433637i	$-0.142770\pi$
$128$	− 6.38302i	− 0.564185i
$129$	3.81468	0.335864
$130$	0	0
$131$	−10.9407	−0.955893	−0.477947	−	0.878389i	$-0.658619\pi$
$131$	−10.9407	−0.955893	−0.477947	+	0.878389i	$0.658619\pi$
$132$	− 3.16715i	− 0.275665i
$133$	− 15.7101i	− 1.36224i
$134$	−4.96186	−0.428640
$135$	0	0
$136$	22.1938	1.90310
$137$	− 16.7888i	− 1.43436i	−0.696887	−	0.717181i	$-0.745432\pi$
$137$	− 16.7888i	− 1.43436i	0.696887	−	0.717181i	$-0.254568\pi$
$138$	− 5.49868i	− 0.468078i
$139$	5.48342	0.465098	0.232549	−	0.972585i	$-0.425293\pi$
$139$	5.48342	0.465098	0.232549	+	0.972585i	$0.425293\pi$
$140$	0	0
$141$	−0.897385	−0.0755735
$142$	0.0742260i	0.00622891i
$143$	− 5.63761i	− 0.471440i
$144$	0.167607	0.0139673
$145$	0	0
$146$	8.88251	0.735122
$147$	18.9072i	1.55944i
$148$	− 0.0188974i	− 0.00155336i
$149$	0.273699	0.0224223	0.0112112	−	0.999937i	$-0.496431\pi$
$149$	0.273699	0.0224223	0.0112112	+	0.999937i	$0.496431\pi$
$150$	0	0
$151$	1.60377	0.130513	0.0652564	−	0.997869i	$-0.479213\pi$
$151$	1.60377	0.130513	0.0652564	+	0.997869i	$0.479213\pi$
$152$	8.83890i	0.716929i
$153$	− 7.75001i	− 0.626551i
$154$	12.0459	0.970684
$155$	0	0
$156$	−2.55585	−0.204632
$157$	16.5512i	1.32093i	0.750857	+	0.660465i	$0.229641\pi$
$157$	16.5512i	1.32093i	−0.750857	+	0.660465i	$0.770359\pi$
$158$	4.74603i	0.377574i
$159$	9.26724	0.734940
$160$	0	0
$161$	−31.2575	−2.46344
$162$	− 0.895394i	− 0.0703488i
$163$	− 22.9917i	− 1.80085i	−0.435014	−	0.900424i	$-0.643257\pi$
$163$	− 22.9917i	− 1.80085i	0.435014	−	0.900424i	$-0.356743\pi$
$164$	4.46750	0.348853
$165$	0	0
$166$	−0.647868	−0.0502843
$167$	− 17.1191i	− 1.32472i	−0.749187	−	0.662359i	$-0.769555\pi$
$167$	− 17.1191i	− 1.32472i	0.749187	−	0.662359i	$-0.230445\pi$
$168$	− 14.5760i	− 1.12457i
$169$	8.45051	0.650039
$170$	0	0
$171$	3.08652	0.236032
$172$	4.57101i	0.348537i
$173$	− 3.30634i	− 0.251376i	−0.992070	−	0.125688i	$-0.959886\pi$
$173$	− 3.30634i	− 0.251376i	0.992070	−	0.125688i	$-0.0401139\pi$
$174$	−3.70187	−0.280638
$175$	0	0
$176$	−0.443002	−0.0333926
$177$	− 11.0693i	− 0.832023i
$178$	− 11.8682i	− 0.889561i
$179$	−12.1351	−0.907020	−0.453510	−	0.891251i	$-0.649828\pi$
$179$	−12.1351	−0.907020	−0.453510	+	0.891251i	$0.649828\pi$
$180$	0	0
$181$	15.5920	1.15894	0.579472	−	0.814992i	$-0.303259\pi$
$181$	15.5920	1.15894	0.579472	+	0.814992i	$0.303259\pi$
$182$	− 9.72088i	− 0.720560i
$183$	− 6.38696i	− 0.472138i
$184$	17.5863	1.29648
$185$	0	0
$186$	2.45585	0.180072
$187$	20.4840i	1.49794i
$188$	− 1.07531i	− 0.0784251i
$189$	−5.08992	−0.370237
$190$	0	0
$191$	−4.82025	−0.348781	−0.174391	−	0.984677i	$-0.555796\pi$
$191$	−4.82025	−0.348781	−0.174391	+	0.984677i	$0.555796\pi$
$192$	5.32913i	0.384597i
$193$	17.7887i	1.28046i	0.768183	+	0.640230i	$0.221161\pi$
$193$	17.7887i	1.28046i	−0.768183	+	0.640230i	$0.778839\pi$
$194$	−1.99526	−0.143251
$195$	0	0
$196$	−22.6560	−1.61828
$197$	− 22.2222i	− 1.58327i	−0.610997	−	0.791633i	$-0.709231\pi$
$197$	− 22.2222i	− 1.58327i	0.610997	−	0.791633i	$-0.290769\pi$
$198$	2.36661i	0.168188i
$199$	16.3687	1.16035	0.580175	−	0.814492i	$-0.302984\pi$
$199$	16.3687	1.16035	0.580175	+	0.814492i	$0.302984\pi$
$200$	0	0
$201$	−5.54154	−0.390870
$202$	− 13.8704i	− 0.975918i
$203$	21.0435i	1.47696i
$204$	9.28660	0.650192
$205$	0	0
$206$	−10.3579	−0.721668
$207$	− 6.14107i	− 0.426834i
$208$	0.357498i	0.0247880i
$209$	−8.15798	−0.564299
$210$	0	0
$211$	−14.7224	−1.01353	−0.506765	−	0.862084i	$-0.669159\pi$
$211$	−14.7224	−1.01353	−0.506765	+	0.862084i	$0.669159\pi$
$212$	11.1047i	0.762671i
$213$	0.0828976i	0.00568005i
$214$	0.989961	0.0676723
$215$	0	0
$216$	2.86371	0.194851
$217$	− 13.9604i	− 0.947697i
$218$	9.18904i	0.622360i
$219$	9.92024	0.670347
$220$	0	0
$221$	16.5304	1.11196
$222$	0.0141209i	0 0.000947732i
$223$	− 21.7310i	− 1.45521i	−0.685994	−	0.727607i	$-0.740632\pi$
$223$	− 21.7310i	− 1.45521i	0.685994	−	0.727607i	$-0.259368\pi$
$224$	28.3882	1.89677
$225$	0	0
$226$	−1.55807	−0.103642
$227$	27.4127i	1.81944i	0.415218	+	0.909722i	$0.363705\pi$
$227$	27.4127i	1.81944i	−0.415218	+	0.909722i	$0.636295\pi$
$228$	3.69848i	0.244938i
$229$	−19.4643	−1.28624	−0.643118	−	0.765767i	$-0.722360\pi$
$229$	−19.4643	−1.28624	−0.643118	+	0.765767i	$0.722360\pi$
$230$	0	0
$231$	13.4532	0.885152
$232$	− 11.8396i	− 0.777307i
$233$	10.2000i	0.668224i	0.942533	+	0.334112i	$0.108436\pi$
$233$	10.2000i	0.668224i	−0.942533	+	0.334112i	$0.891564\pi$
$234$	1.90983	0.124850
$235$	0	0
$236$	13.2641	0.863418
$237$	5.30049i	0.344304i
$238$	35.3205i	2.28949i
$239$	3.18653	0.206120	0.103060	−	0.994675i	$-0.467137\pi$
$239$	3.18653	0.206120	0.103060	+	0.994675i	$0.467137\pi$
$240$	0	0
$241$	12.4630	0.802811	0.401405	−	0.915901i	$-0.368522\pi$
$241$	12.4630	0.802811	0.401405	+	0.915901i	$0.368522\pi$
$242$	3.59413i	0.231040i
$243$	− 1.00000i	− 0.0641500i
$244$	7.65330	0.489953
$245$	0	0
$246$	−3.33829	−0.212842
$247$	6.58340i	0.418892i
$248$	7.85449i	0.498760i
$249$	−0.723557	−0.0458536
$250$	0	0
$251$	1.49139	0.0941357	0.0470679	−	0.998892i	$-0.485012\pi$
$251$	1.49139	0.0941357	0.0470679	+	0.998892i	$0.485012\pi$
$252$	− 6.09909i	− 0.384207i
$253$	16.2315i	1.02046i
$254$	−8.75129	−0.549105
$255$	0	0
$256$	−16.3736	−1.02335
$257$	− 19.3647i	− 1.20794i	−0.797008	−	0.603969i	$-0.793585\pi$
$257$	− 19.3647i	− 1.20794i	0.797008	−	0.603969i	$-0.206415\pi$
$258$	− 3.41564i	− 0.212648i
$259$	0.0802710	0.00498780
$260$	0	0
$261$	−4.13435	−0.255910
$262$	9.79623i	0.605213i
$263$	13.7684i	0.848996i	0.905429	+	0.424498i	$0.139549\pi$
$263$	13.7684i	0.848996i	−0.905429	+	0.424498i	$0.860451\pi$
$264$	−7.56907	−0.465844
$265$	0	0
$266$	−14.0667	−0.862487
$267$	− 13.2548i	− 0.811178i
$268$	− 6.64027i	− 0.405619i
$269$	−2.85430	−0.174030	−0.0870149	−	0.996207i	$-0.527733\pi$
$269$	−2.85430	−0.174030	−0.0870149	+	0.996207i	$0.527733\pi$
$270$	0	0
$271$	21.9647	1.33426	0.667131	−	0.744941i	$-0.267522\pi$
$271$	21.9647	1.33426	0.667131	+	0.744941i	$0.267522\pi$
$272$	− 1.29896i	− 0.0787608i
$273$	− 10.8565i	− 0.657068i
$274$	−15.0326	−0.908150
$275$	0	0
$276$	7.35867	0.442940
$277$	2.63773i	0.158486i	0.996855	+	0.0792428i	$0.0252502\pi$
$277$	2.63773i	0.158486i	−0.996855	+	0.0792428i	$0.974750\pi$
$278$	− 4.90982i	− 0.294472i
$279$	2.74277	0.164205
$280$	0	0
$281$	−8.16358	−0.486998	−0.243499	−	0.969901i	$-0.578295\pi$
$281$	−8.16358	−0.486998	−0.243499	+	0.969901i	$0.578295\pi$
$282$	0.803513i	0.0478485i
$283$	6.06194i	0.360345i	0.983635	+	0.180173i	$0.0576656\pi$
$283$	6.06194i	0.360345i	−0.983635	+	0.180173i	$0.942334\pi$
$284$	−0.0993338	−0.00589437
$285$	0	0
$286$	−5.04788	−0.298487
$287$	18.9767i	1.12016i
$288$	5.57735i	0.328648i
$289$	−43.0626	−2.53309
$290$	0	0
$291$	−2.22836	−0.130629
$292$	11.8871i	0.695641i
$293$	16.5940i	0.969432i	0.874672	+	0.484716i	$0.161077\pi$
$293$	16.5940i	0.969432i	−0.874672	+	0.484716i	$0.838923\pi$
$294$	16.9294	0.987344
$295$	0	0
$296$	−0.0451624	−0.00262501
$297$	2.64310i	0.153368i
$298$	− 0.245069i	− 0.0141964i
$299$	13.0986	0.757513
$300$	0	0
$301$	−19.4164	−1.11914
$302$	− 1.43600i	− 0.0826327i
$303$	− 15.4908i	− 0.889925i
$304$	0.517323	0.0296705
$305$	0	0
$306$	−6.93931	−0.396694
$307$	14.8424i	0.847098i	0.905873	+	0.423549i	$0.139216\pi$
$307$	14.8424i	0.847098i	−0.905873	+	0.423549i	$0.860784\pi$
$308$	16.1205i	0.918552i
$309$	−11.5680	−0.658079
$310$	0	0
$311$	13.6460	0.773796	0.386898	−	0.922123i	$-0.373547\pi$
$311$	13.6460	0.773796	0.386898	+	0.922123i	$0.373547\pi$
$312$	6.10816i	0.345806i
$313$	− 3.18194i	− 0.179854i	−0.995948	−	0.0899270i	$-0.971337\pi$
$313$	− 3.18194i	− 0.179854i	0.995948	−	0.0899270i	$-0.0286634\pi$
$314$	14.8198	0.836332
$315$	0	0
$316$	−6.35142	−0.357295
$317$	3.70586i	0.208142i	0.994570	+	0.104071i	$0.0331869\pi$
$317$	3.70586i	0.208142i	−0.994570	+	0.104071i	$0.966813\pi$
$318$	− 8.29783i	− 0.465319i
$319$	10.9275	0.611823
$320$	0	0
$321$	1.10562	0.0617094
$322$	27.9878i	1.55970i
$323$	− 23.9205i	− 1.33097i
$324$	1.19827	0.0665706
$325$	0	0
$326$	−20.5866	−1.14019
$327$	10.2626i	0.567522i
$328$	− 10.6768i	− 0.589525i
$329$	4.56762	0.251821
$330$	0	0
$331$	−32.3878	−1.78020	−0.890098	−	0.455769i	$-0.849364\pi$
$331$	−32.3878	−1.78020	−0.890098	+	0.455769i	$0.849364\pi$
$332$	− 0.867017i	− 0.0475838i
$333$	0.0157706i	0 0.000864223i
$334$	−15.3284	−0.838730
$335$	0	0
$336$	−0.853106	−0.0465408
$337$	− 21.9294i	− 1.19457i	−0.802029	−	0.597285i	$-0.796246\pi$
$337$	− 21.9294i	− 1.19457i	0.802029	−	0.597285i	$-0.203754\pi$
$338$	− 7.56653i	− 0.411565i
$339$	−1.74010	−0.0945092
$340$	0	0
$341$	−7.24940	−0.392577
$342$	− 2.76365i	− 0.149441i
$343$	− 60.6068i	− 3.27246i
$344$	10.9241	0.588990
$345$	0	0
$346$	−2.96047	−0.159156
$347$	− 24.8312i	− 1.33301i	−0.745502	−	0.666503i	$-0.767790\pi$
$347$	− 24.8312i	− 1.33301i	0.745502	−	0.666503i	$-0.232210\pi$
$348$	− 4.95407i	− 0.265566i
$349$	1.99222	0.106641	0.0533204	−	0.998577i	$-0.483020\pi$
$349$	1.99222	0.106641	0.0533204	+	0.998577i	$0.483020\pi$
$350$	0	0
$351$	2.13295	0.113849
$352$	− 14.7415i	− 0.785724i
$353$	− 2.00997i	− 0.106980i	−0.998568	−	0.0534900i	$-0.982965\pi$
$353$	− 2.00997i	− 0.106980i	0.998568	−	0.0534900i	$-0.0170345\pi$
$354$	−9.91142	−0.526786
$355$	0	0
$356$	15.8828	0.841785
$357$	39.4469i	2.08775i
$358$	10.8657i	0.574270i
$359$	−24.2078	−1.27764	−0.638818	−	0.769358i	$-0.720576\pi$
$359$	−24.2078	−1.27764	−0.638818	+	0.769358i	$0.720576\pi$
$360$	0	0
$361$	−9.47340	−0.498600
$362$	− 13.9610i	− 0.733772i
$363$	4.01402i	0.210682i
$364$	13.0091	0.681861
$365$	0	0
$366$	−5.71884	−0.298929
$367$	− 0.592824i	− 0.0309452i	−0.999880	−	0.0154726i	$-0.995075\pi$
$367$	− 0.592824i	− 0.0309452i	0.999880	−	0.0154726i	$-0.00492528\pi$
$368$	− 1.02929i	− 0.0536553i
$369$	−3.72829	−0.194087
$370$	0	0
$371$	−47.1695	−2.44892
$372$	3.28657i	0.170401i
$373$	− 8.42516i	− 0.436238i	−0.975922	−	0.218119i	$-0.930008\pi$
$373$	− 8.42516i	− 0.436238i	0.975922	−	0.218119i	$-0.0699921\pi$
$374$	18.3413	0.948405
$375$	0	0
$376$	−2.56985	−0.132530
$377$	− 8.81838i	− 0.454170i
$378$	4.55748i	0.234411i
$379$	18.6130	0.956087	0.478043	−	0.878336i	$-0.341346\pi$
$379$	18.6130	0.956087	0.478043	+	0.878336i	$0.341346\pi$
$380$	0	0
$381$	−9.77368	−0.500721
$382$	4.31602i	0.220827i
$383$	9.94283i	0.508055i	0.967197	+	0.254027i	$0.0817553\pi$
$383$	9.94283i	0.508055i	−0.967197	+	0.254027i	$0.918245\pi$
$384$	−6.38302	−0.325732
$385$	0	0
$386$	15.9279	0.810709
$387$	− 3.81468i	− 0.193911i
$388$	− 2.67018i	− 0.135558i
$389$	5.93571	0.300952	0.150476	−	0.988614i	$-0.451919\pi$
$389$	5.93571	0.300952	0.150476	+	0.988614i	$0.451919\pi$
$390$	0	0
$391$	−47.5934	−2.40690
$392$	54.1449i	2.73473i
$393$	10.9407i	0.551885i
$394$	−19.8976	−1.00243
$395$	0	0
$396$	−3.16715	−0.159155
$397$	− 3.07681i	− 0.154421i	−0.997015	−	0.0772105i	$-0.975399\pi$
$397$	− 3.07681i	− 0.154421i	0.997015	−	0.0772105i	$-0.0246013\pi$
$398$	− 14.6565i	− 0.734662i
$399$	−15.7101	−0.786490
$400$	0	0
$401$	20.6370	1.03056	0.515281	−	0.857021i	$-0.327687\pi$
$401$	20.6370	1.03056	0.515281	+	0.857021i	$0.327687\pi$
$402$	4.96186i	0.247475i
$403$	5.85019i	0.291419i
$404$	18.5622	0.923505
$405$	0	0
$406$	18.8422	0.935124
$407$	− 0.0416833i	− 0.00206616i
$408$	− 22.1938i	− 1.09876i
$409$	−20.8240	−1.02968	−0.514840	−	0.857286i	$-0.672149\pi$
$409$	−20.8240	−1.02968	−0.514840	+	0.857286i	$0.672149\pi$
$410$	0	0
$411$	−16.7888	−0.828129
$412$	− 13.8616i	− 0.682910i
$413$	56.3421i	2.77241i
$414$	−5.49868	−0.270245
$415$	0	0
$416$	−11.8962	−0.583260
$417$	− 5.48342i	− 0.268524i
$418$	7.30460i	0.357280i
$419$	−11.3349	−0.553748	−0.276874	−	0.960906i	$-0.589298\pi$
$419$	−11.3349	−0.553748	−0.276874	+	0.960906i	$0.589298\pi$
$420$	0	0
$421$	−2.03813	−0.0993323	−0.0496662	−	0.998766i	$-0.515816\pi$
$421$	−2.03813	−0.0993323	−0.0496662	+	0.998766i	$0.515816\pi$
$422$	13.1823i	0.641705i
$423$	0.897385i	0.0436324i
$424$	26.5387	1.28883
$425$	0	0
$426$	0.0742260	0.00359626
$427$	32.5091i	1.57322i
$428$	1.32483i	0.0640379i
$429$	−5.63761	−0.272186
$430$	0	0
$431$	−17.9230	−0.863319	−0.431660	−	0.902037i	$-0.642072\pi$
$431$	−17.9230	−0.863319	−0.431660	+	0.902037i	$0.642072\pi$
$432$	− 0.167607i	− 0.00806400i
$433$	14.0213i	0.673821i	0.941537	+	0.336911i	$0.109382\pi$
$433$	14.0213i	0.673821i	−0.941537	+	0.336911i	$0.890618\pi$
$434$	−12.5001	−0.600024
$435$	0	0
$436$	−12.2973	−0.588936
$437$	− 18.9545i	− 0.906718i
$438$	− 8.88251i	− 0.424423i
$439$	1.35014	0.0644385	0.0322192	−	0.999481i	$-0.489743\pi$
$439$	1.35014	0.0644385	0.0322192	+	0.999481i	$0.489743\pi$
$440$	0	0
$441$	18.9072	0.900345
$442$	− 14.8012i	− 0.704022i
$443$	8.53290i	0.405410i	0.979240	+	0.202705i	$0.0649733\pi$
$443$	8.53290i	0.405410i	−0.979240	+	0.202705i	$0.935027\pi$
$444$	−0.0188974	−0.000896833 0
$445$	0	0
$446$	−19.4578	−0.921353
$447$	− 0.273699i	− 0.0129455i
$448$	− 27.1248i	− 1.28153i
$449$	33.6080	1.58606	0.793030	−	0.609183i	$-0.208503\pi$
$449$	33.6080	1.58606	0.793030	+	0.609183i	$0.208503\pi$
$450$	0	0
$451$	9.85425	0.464019
$452$	− 2.08511i	− 0.0980753i
$453$	− 1.60377i	− 0.0753516i
$454$	24.5451	1.15196
$455$	0	0
$456$	8.83890	0.413919
$457$	15.8701i	0.742372i	0.928559	+	0.371186i	$0.121049\pi$
$457$	15.8701i	0.742372i	−0.928559	+	0.371186i	$0.878951\pi$
$458$	17.4282i	0.814367i
$459$	−7.75001	−0.361739
$460$	0	0
$461$	12.1978	0.568110	0.284055	−	0.958808i	$-0.408320\pi$
$461$	12.1978	0.568110	0.284055	+	0.958808i	$0.408320\pi$
$462$	− 12.0459i	− 0.560424i
$463$	12.6538i	0.588070i	0.955795	+	0.294035i	$0.0949983\pi$
$463$	12.6538i	0.588070i	−0.955795	+	0.294035i	$0.905002\pi$
$464$	−0.692947	−0.0321693
$465$	0	0
$466$	9.13301	0.423078
$467$	11.1891i	0.517769i	0.965908	+	0.258884i	$0.0833549\pi$
$467$	11.1891i	0.517769i	−0.965908	+	0.258884i	$0.916645\pi$
$468$	2.55585i	0.118144i
$469$	28.2060	1.30243
$470$	0	0
$471$	16.5512	0.762639
$472$	− 31.6994i	− 1.45908i
$473$	10.0826i	0.463597i
$474$	4.74603	0.217992
$475$	0	0
$476$	−47.2680	−2.16653
$477$	− 9.26724i	− 0.424318i
$478$	− 2.85320i	− 0.130502i
$479$	−12.5593	−0.573848	−0.286924	−	0.957953i	$-0.592633\pi$
$479$	−12.5593	−0.573848	−0.286924	+	0.957953i	$0.592633\pi$
$480$	0	0
$481$	−0.0336379	−0.00153376
$482$	− 11.1593i	− 0.508291i
$483$	31.2575i	1.42227i
$484$	−4.80989	−0.218631
$485$	0	0
$486$	−0.895394	−0.0406159
$487$	− 25.3662i	− 1.14945i	−0.818346	−	0.574726i	$-0.805109\pi$
$487$	− 25.3662i	− 1.14945i	0.818346	−	0.574726i	$-0.194891\pi$
$488$	− 18.2904i	− 0.827968i
$489$	−22.9917	−1.03972
$490$	0	0
$491$	20.6927	0.933847	0.466924	−	0.884298i	$-0.345362\pi$
$491$	20.6927	0.933847	0.466924	+	0.884298i	$0.345362\pi$
$492$	− 4.46750i	− 0.201411i
$493$	32.0413i	1.44307i
$494$	5.89473	0.265217
$495$	0	0
$496$	0.459707	0.0206415
$497$	− 0.421942i	− 0.0189267i
$498$	0.647868i	0.0290317i
$499$	−40.3649	−1.80698	−0.903490	−	0.428608i	$-0.859004\pi$
$499$	−40.3649	−1.80698	−0.903490	+	0.428608i	$0.859004\pi$
$500$	0	0
$501$	−17.1191	−0.764826
$502$	− 1.33538i	− 0.0596010i
$503$	− 4.91088i	− 0.218966i	−0.993989	−	0.109483i	$-0.965081\pi$
$503$	− 4.91088i	− 0.218966i	0.993989	−	0.109483i	$-0.0349194\pi$
$504$	−14.5760	−0.649269
$505$	0	0
$506$	14.5335	0.646095
$507$	− 8.45051i	− 0.375300i
$508$	− 11.7115i	− 0.519614i
$509$	−16.5254	−0.732476	−0.366238	−	0.930521i	$-0.619354\pi$
$509$	−16.5254	−0.732476	−0.366238	+	0.930521i	$0.619354\pi$
$510$	0	0
$511$	−50.4932	−2.23369
$512$	1.89476i	0.0837373i
$513$	− 3.08652i	− 0.136273i
$514$	−17.3390	−0.764792
$515$	0	0
$516$	4.57101	0.201228
$517$	− 2.37188i	− 0.104315i
$518$	− 0.0718741i	− 0.00315797i
$519$	−3.30634	−0.145132
$520$	0	0
$521$	10.2405	0.448646	0.224323	−	0.974515i	$-0.427983\pi$
$521$	10.2405	0.448646	0.224323	+	0.974515i	$0.427983\pi$
$522$	3.70187i	0.162027i
$523$	18.5451i	0.810919i	0.914113	+	0.405459i	$0.132888\pi$
$523$	18.5451i	0.810919i	−0.914113	+	0.405459i	$0.867112\pi$
$524$	−13.1099	−0.572709
$525$	0	0
$526$	12.3281	0.537532
$527$	− 21.2564i	− 0.925945i
$528$	0.443002i	0.0192792i
$529$	−14.7128	−0.639686
$530$	0	0
$531$	−11.0693	−0.480369
$532$	− 18.8250i	− 0.816166i
$533$	− 7.95228i	− 0.344451i
$534$	−11.8682	−0.513588
$535$	0	0
$536$	−15.8694	−0.685453
$537$	12.1351i	0.523669i
$538$	2.55572i	0.110185i
$539$	−49.9737	−2.15252
$540$	0	0
$541$	−22.9520	−0.986782	−0.493391	−	0.869808i	$-0.664243\pi$
$541$	−22.9520	−0.986782	−0.493391	+	0.869808i	$0.664243\pi$
$542$	− 19.6671i	− 0.844773i
$543$	− 15.5920i	− 0.669116i
$544$	43.2245	1.85323
$545$	0	0
$546$	−9.72088	−0.416015
$547$	31.3724i	1.34139i	0.741735	+	0.670693i	$0.234003\pi$
$547$	31.3724i	1.34139i	−0.741735	+	0.670693i	$0.765997\pi$
$548$	− 20.1175i	− 0.859377i
$549$	−6.38696	−0.272589
$550$	0	0
$551$	−12.7608	−0.543627
$552$	− 17.5863i	− 0.748521i
$553$	− 26.9791i	− 1.14727i
$554$	2.36180	0.100343
$555$	0	0
$556$	6.57063	0.278657
$557$	− 19.4590i	− 0.824506i	−0.911069	−	0.412253i	$-0.864742\pi$
$557$	− 19.4590i	− 0.824506i	0.911069	−	0.412253i	$-0.135258\pi$
$558$	− 2.45585i	− 0.103965i
$559$	8.13653	0.344138
$560$	0	0
$561$	20.4840	0.864837
$562$	7.30962i	0.308338i
$563$	14.0217i	0.590944i	0.955351	+	0.295472i	$0.0954768\pi$
$563$	14.0217i	0.590944i	−0.955351	+	0.295472i	$0.904523\pi$
$564$	−1.07531	−0.0452787
$565$	0	0
$566$	5.42783	0.228149
$567$	5.08992i	0.213756i
$568$	0.237395i	0.00996086i
$569$	40.4550	1.69596	0.847982	−	0.530025i	$-0.177818\pi$
$569$	40.4550	1.69596	0.847982	+	0.530025i	$0.177818\pi$
$570$	0	0
$571$	−10.9176	−0.456886	−0.228443	−	0.973557i	$-0.573363\pi$
$571$	−10.9176	−0.456886	−0.228443	+	0.973557i	$0.573363\pi$
$572$	− 6.75538i	− 0.282457i
$573$	4.82025i	0.201369i
$574$	16.9916	0.709216
$575$	0	0
$576$	5.32913	0.222047
$577$	18.0389i	0.750968i	0.926829	+	0.375484i	$0.122524\pi$
$577$	18.0389i	0.750968i	−0.926829	+	0.375484i	$0.877476\pi$
$578$	38.5580i	1.60380i
$579$	17.7887	0.739274
$580$	0	0
$581$	3.68285	0.152790
$582$	1.99526i	0.0827062i
$583$	24.4942i	1.01445i
$584$	28.4087	1.17556
$585$	0	0
$586$	14.8582	0.613785
$587$	10.1243i	0.417874i	0.977929	+	0.208937i	$0.0670003\pi$
$587$	10.1243i	0.417874i	−0.977929	+	0.208937i	$0.933000\pi$
$588$	22.6560i	0.934317i
$589$	8.46560	0.348819
$590$	0	0
$591$	−22.2222	−0.914099
$592$	0.00264326i	0 0.000108637i
$593$	− 8.24833i	− 0.338718i	−0.985554	−	0.169359i	$-0.945830\pi$
$593$	− 8.24833i	− 0.338718i	0.985554	−	0.169359i	$-0.0541698\pi$
$594$	2.36661	0.0971034
$595$	0	0
$596$	0.327966	0.0134340
$597$	− 16.3687i	− 0.669928i
$598$	− 11.7284i	− 0.479611i
$599$	−33.3982	−1.36461	−0.682307	−	0.731066i	$-0.739023\pi$
$599$	−33.3982	−1.36461	−0.682307	+	0.731066i	$0.739023\pi$
$600$	0	0
$601$	23.7251	0.967767	0.483884	−	0.875132i	$-0.339226\pi$
$601$	23.7251	0.967767	0.483884	+	0.875132i	$0.339226\pi$
$602$	17.3853i	0.708572i
$603$	5.54154i	0.225669i
$604$	1.92175	0.0781948
$605$	0	0
$606$	−13.8704	−0.563446
$607$	− 30.5545i	− 1.24017i	−0.784535	−	0.620084i	$-0.787098\pi$
$607$	− 30.5545i	− 1.24017i	0.784535	−	0.620084i	$-0.212902\pi$
$608$	17.2146i	0.698144i
$609$	21.0435	0.852726
$610$	0	0
$611$	−1.91408	−0.0774354
$612$	− 9.28660i	− 0.375389i
$613$	9.49041i	0.383314i	0.981462	+	0.191657i	$0.0613861\pi$
$613$	9.49041i	0.383314i	−0.981462	+	0.191657i	$0.938614\pi$
$614$	13.2898	0.536331
$615$	0	0
$616$	38.5259	1.55225
$617$	− 28.4765i	− 1.14642i	−0.819408	−	0.573210i	$-0.805698\pi$
$617$	− 28.4765i	− 1.14642i	0.819408	−	0.573210i	$-0.194302\pi$
$618$	10.3579i	0.416655i
$619$	6.76342	0.271845	0.135922	−	0.990719i	$-0.456600\pi$
$619$	6.76342	0.271845	0.135922	+	0.990719i	$0.456600\pi$
$620$	0	0
$621$	−6.14107	−0.246433
$622$	− 12.2186i	− 0.489920i
$623$	67.4656i	2.70295i
$624$	0.357498	0.0143114
$625$	0	0
$626$	−2.84909	−0.113873
$627$	8.15798i	0.325798i
$628$	19.8328i	0.791415i
$629$	0.122222	0.00487332
$630$	0	0
$631$	−11.2740	−0.448812	−0.224406	−	0.974496i	$-0.572044\pi$
$631$	−11.2740	−0.448812	−0.224406	+	0.974496i	$0.572044\pi$
$632$	15.1791i	0.603791i
$633$	14.7224i	0.585162i
$634$	3.31821	0.131783
$635$	0	0
$636$	11.1047	0.440328
$637$	40.3282i	1.59786i
$638$	− 9.78442i	− 0.387369i
$639$	0.0828976	0.00327938
$640$	0	0
$641$	12.4989	0.493676	0.246838	−	0.969057i	$-0.420609\pi$
$641$	12.4989	0.493676	0.246838	+	0.969057i	$0.420609\pi$
$642$	− 0.989961i	− 0.0390706i
$643$	15.6325i	0.616487i	0.951307	+	0.308244i	$0.0997412\pi$
$643$	15.6325i	0.616487i	−0.951307	+	0.308244i	$0.900259\pi$
$644$	−37.4550	−1.47593
$645$	0	0
$646$	−21.4183	−0.842692
$647$	− 34.4383i	− 1.35391i	−0.736025	−	0.676955i	$-0.763299\pi$
$647$	− 34.4383i	− 1.35391i	0.736025	−	0.676955i	$-0.236701\pi$
$648$	− 2.86371i	− 0.112497i
$649$	29.2574	1.14845
$650$	0	0
$651$	−13.9604	−0.547153
$652$	− 27.5503i	− 1.07895i
$653$	− 34.6831i	− 1.35725i	−0.734483	−	0.678627i	$-0.762575\pi$
$653$	− 34.6831i	− 1.35725i	0.734483	−	0.678627i	$-0.237425\pi$
$654$	9.18904	0.359320
$655$	0	0
$656$	−0.624889	−0.0243978
$657$	− 9.92024i	− 0.387025i
$658$	− 4.08981i	− 0.159438i
$659$	4.40271	0.171505	0.0857526	−	0.996316i	$-0.472671\pi$
$659$	4.40271	0.171505	0.0857526	+	0.996316i	$0.472671\pi$
$660$	0	0
$661$	14.4318	0.561333	0.280666	−	0.959805i	$-0.409445\pi$
$661$	14.4318	0.561333	0.280666	+	0.959805i	$0.409445\pi$
$662$	28.9999i	1.12711i
$663$	− 16.5304i	− 0.641988i
$664$	−2.07206	−0.0804115
$665$	0	0
$666$	0.0141209	0.000547173 0
$667$	25.3894i	0.983080i
$668$	− 20.5133i	− 0.793685i
$669$	−21.7310	−0.840169
$670$	0	0
$671$	16.8814	0.651698
$672$	− 28.3882i	− 1.09510i
$673$	− 2.75839i	− 0.106328i	−0.998586	−	0.0531640i	$-0.983069\pi$
$673$	− 2.75839i	− 0.106328i	0.998586	−	0.0531640i	$-0.0169306\pi$
$674$	−19.6354	−0.756328
$675$	0	0
$676$	10.1260	0.389461
$677$	25.9431i	0.997073i	0.866869	+	0.498537i	$0.166129\pi$
$677$	25.9431i	0.997073i	−0.866869	+	0.498537i	$0.833871\pi$
$678$	1.55807i	0.0598375i
$679$	11.3422	0.435273
$680$	0	0
$681$	27.4127	1.05046
$682$	6.49107i	0.248556i
$683$	16.5958i	0.635019i	0.948255	+	0.317510i	$0.102847\pi$
$683$	16.5958i	0.635019i	−0.948255	+	0.317510i	$0.897153\pi$
$684$	3.69848	0.141415
$685$	0	0
$686$	−54.2670	−2.07192
$687$	19.4643i	0.742609i
$688$	− 0.639367i	− 0.0243756i
$689$	19.7666	0.753047
$690$	0	0
$691$	12.4942	0.475300	0.237650	−	0.971351i	$-0.423623\pi$
$691$	12.4942	0.475300	0.237650	+	0.971351i	$0.423623\pi$
$692$	− 3.96189i	− 0.150608i
$693$	− 13.4532i	− 0.511043i
$694$	−22.2337	−0.843978
$695$	0	0
$696$	−11.8396	−0.448779
$697$	28.8943i	1.09445i
$698$	− 1.78382i	− 0.0675185i
$699$	10.2000	0.385799
$700$	0	0
$701$	50.7235	1.91580	0.957900	−	0.287101i	$-0.0926917\pi$
$701$	50.7235	1.91580	0.957900	+	0.287101i	$0.0926917\pi$
$702$	− 1.90983i	− 0.0720820i
$703$	0.0486762i	0.00183586i
$704$	−14.0854	−0.530865
$705$	0	0
$706$	−1.79971	−0.0677331
$707$	78.8470i	2.96535i
$708$	− 13.2641i	− 0.498494i
$709$	11.7870	0.442671	0.221335	−	0.975198i	$-0.428958\pi$
$709$	11.7870	0.442671	0.221335	+	0.975198i	$0.428958\pi$
$710$	0	0
$711$	5.30049	0.198784
$712$	− 37.9578i	− 1.42253i
$713$	− 16.8435i	− 0.630795i
$714$	35.3205	1.32184
$715$	0	0
$716$	−14.5411	−0.543428
$717$	− 3.18653i	− 0.119003i
$718$	21.6755i	0.808921i
$719$	−7.52191	−0.280520	−0.140260	−	0.990115i	$-0.544794\pi$
$719$	−7.52191	−0.280520	−0.140260	+	0.990115i	$0.544794\pi$
$720$	0	0
$721$	58.8800	2.19281
$722$	8.48242i	0.315683i
$723$	− 12.4630i	− 0.463503i
$724$	18.6834	0.694364
$725$	0	0
$726$	3.59413	0.133391
$727$	− 7.51316i	− 0.278648i	−0.990247	−	0.139324i	$-0.955507\pi$
$727$	− 7.51316i	− 0.278648i	0.990247	−	0.139324i	$-0.0444929\pi$
$728$	− 31.0900i	− 1.15227i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−29.5638	−1.09346
$732$	− 7.65330i	− 0.282874i
$733$	4.39758i	0.162428i	0.996697	+	0.0812141i	$0.0258798\pi$
$733$	4.39758i	0.162428i	−0.996697	+	0.0812141i	$0.974120\pi$
$734$	−0.530811	−0.0195926
$735$	0	0
$736$	34.2509	1.26250
$737$	− 14.6469i	− 0.539524i
$738$	3.33829i	0.122884i
$739$	45.4541	1.67206	0.836028	−	0.548686i	$-0.184872\pi$
$739$	45.4541	1.67206	0.836028	+	0.548686i	$0.184872\pi$
$740$	0	0
$741$	6.58340	0.241847
$742$	42.2352i	1.55050i
$743$	− 45.0800i	− 1.65382i	−0.562331	−	0.826912i	$-0.690095\pi$
$743$	− 45.0800i	− 1.65382i	0.562331	−	0.826912i	$-0.309905\pi$
$744$	7.85449	0.287959
$745$	0	0
$746$	−7.54384	−0.276199
$747$	0.723557i	0.0264736i
$748$	24.5454i	0.897469i
$749$	−5.62749	−0.205624
$750$	0	0
$751$	−14.7555	−0.538435	−0.269217	−	0.963079i	$-0.586765\pi$
$751$	−14.7555	−0.538435	−0.269217	+	0.963079i	$0.586765\pi$
$752$	0.150408i	0.00548482i
$753$	− 1.49139i	− 0.0543493i
$754$	−7.89592	−0.287552
$755$	0	0
$756$	−6.09909	−0.221822
$757$	− 41.8730i	− 1.52190i	−0.648811	−	0.760950i	$-0.724733\pi$
$757$	− 41.8730i	− 1.52190i	0.648811	−	0.760950i	$-0.275267\pi$
$758$	− 16.6660i	− 0.605336i
$759$	16.2315	0.589165
$760$	0	0
$761$	−1.58549	−0.0574738	−0.0287369	−	0.999587i	$-0.509149\pi$
$761$	−1.58549	−0.0574738	−0.0287369	+	0.999587i	$0.509149\pi$
$762$	8.75129i	0.317026i
$763$	− 52.2356i	− 1.89106i
$764$	−5.77596	−0.208967
$765$	0	0
$766$	8.90275	0.321669
$767$	− 23.6104i	− 0.852522i
$768$	16.3736i	0.590831i
$769$	−17.0820	−0.615993	−0.307996	−	0.951388i	$-0.599658\pi$
$769$	−17.0820	−0.615993	−0.307996	+	0.951388i	$0.599658\pi$
$770$	0	0
$771$	−19.3647	−0.697403
$772$	21.3157i	0.767169i
$773$	− 51.0507i	− 1.83617i	−0.396388	−	0.918083i	$-0.629737\pi$
$773$	− 51.0507i	− 1.83617i	0.396388	−	0.918083i	$-0.370263\pi$
$774$	−3.41564	−0.122773
$775$	0	0
$776$	−6.38138	−0.229078
$777$	− 0.0802710i	− 0.00287971i
$778$	− 5.31479i	− 0.190545i
$779$	−11.5075	−0.412297
$780$	0	0
$781$	−0.219107	−0.00784025
$782$	42.6148i	1.52390i
$783$	4.13435i	0.147750i
$784$	3.16899	0.113178
$785$	0	0
$786$	9.79623	0.349420
$787$	− 41.2363i	− 1.46992i	−0.678113	−	0.734958i	$-0.737202\pi$
$787$	− 41.2363i	− 1.46992i	0.678113	−	0.734958i	$-0.262798\pi$
$788$	− 26.6282i	− 0.948590i
$789$	13.7684	0.490168
$790$	0	0
$791$	8.85696	0.314917
$792$	7.56907i	0.268955i
$793$	− 13.6231i	− 0.483770i
$794$	−2.75496	−0.0977699
$795$	0	0
$796$	19.6142	0.695206
$797$	38.0725i	1.34860i	0.738458	+	0.674299i	$0.235554\pi$
$797$	38.0725i	1.34860i	−0.738458	+	0.674299i	$0.764446\pi$
$798$	14.0667i	0.497957i
$799$	6.95474	0.246041
$800$	0	0
$801$	−13.2548	−0.468334
$802$	− 18.4782i	− 0.652489i
$803$	26.2202i	0.925290i
$804$	−6.64027	−0.234184
$805$	0	0
$806$	5.23822	0.184508
$807$	2.85430i	0.100476i
$808$	− 44.3613i	− 1.56062i
$809$	−33.8845	−1.19131	−0.595657	−	0.803239i	$-0.703108\pi$
$809$	−33.8845	−1.19131	−0.595657	+	0.803239i	$0.703108\pi$
$810$	0	0
$811$	9.04529	0.317623	0.158812	−	0.987309i	$-0.449234\pi$
$811$	9.04529	0.317623	0.158812	+	0.987309i	$0.449234\pi$
$812$	25.2158i	0.884902i
$813$	− 21.9647i	− 0.770336i
$814$	−0.0373229	−0.00130817
$815$	0	0
$816$	−1.29896	−0.0454726
$817$	− 11.7741i	− 0.411923i
$818$	18.6457i	0.651930i
$819$	−10.8565	−0.379359
$820$	0	0
$821$	42.7195	1.49092	0.745461	−	0.666549i	$-0.232229\pi$
$821$	42.7195	1.49092	0.745461	+	0.666549i	$0.232229\pi$
$822$	15.0326i	0.524321i
$823$	47.3852i	1.65174i	0.563858	+	0.825872i	$0.309317\pi$
$823$	47.3852i	1.65174i	−0.563858	+	0.825872i	$0.690683\pi$
$824$	−33.1273	−1.15404
$825$	0	0
$826$	50.4483	1.75532
$827$	− 39.0928i	− 1.35939i	−0.733495	−	0.679695i	$-0.762112\pi$
$827$	− 39.0928i	− 1.35939i	0.733495	−	0.679695i	$-0.237888\pi$
$828$	− 7.35867i	− 0.255731i
$829$	−30.5779	−1.06202	−0.531008	−	0.847367i	$-0.678186\pi$
$829$	−30.5779	−1.06202	−0.531008	+	0.847367i	$0.678186\pi$
$830$	0	0
$831$	2.63773	0.0915017
$832$	11.3668i	0.394073i
$833$	− 146.531i	− 5.07701i
$834$	−4.90982	−0.170013
$835$	0	0
$836$	−9.77546	−0.338091
$837$	− 2.74277i	− 0.0948038i
$838$	10.1492i	0.350599i
$839$	−30.3660	−1.04835	−0.524176	−	0.851610i	$-0.675627\pi$
$839$	−30.3660	−1.04835	−0.524176	+	0.851610i	$0.675627\pi$
$840$	0	0
$841$	−11.9071	−0.410590
$842$	1.82493i	0.0628911i
$843$	8.16358i	0.281169i
$844$	−17.6414	−0.607241
$845$	0	0
$846$	0.803513	0.0276253
$847$	− 20.4310i	− 0.702019i
$848$	− 1.55326i	− 0.0533390i
$849$	6.06194	0.208045
$850$	0	0
$851$	0.0968484	0.00331992
$852$	0.0993338i	0.00340312i
$853$	3.41673i	0.116987i	0.998288	+	0.0584933i	$0.0186296\pi$
$853$	3.41673i	0.116987i	−0.998288	+	0.0584933i	$0.981370\pi$
$854$	29.1084	0.996070
$855$	0	0
$856$	3.16616	0.108217
$857$	− 42.0948i	− 1.43793i	−0.695045	−	0.718966i	$-0.744616\pi$
$857$	− 42.0948i	− 1.43793i	0.695045	−	0.718966i	$-0.255384\pi$
$858$	5.04788i	0.172332i
$859$	33.1680	1.13168	0.565840	−	0.824515i	$-0.308552\pi$
$859$	33.1680	1.13168	0.565840	+	0.824515i	$0.308552\pi$
$860$	0	0
$861$	18.9767	0.646724
$862$	16.0481i	0.546601i
$863$	15.5446i	0.529143i	0.964366	+	0.264572i	$0.0852306\pi$
$863$	15.5446i	0.529143i	−0.964366	+	0.264572i	$0.914769\pi$
$864$	5.57735	0.189745
$865$	0	0
$866$	12.5546	0.426622
$867$	43.0626i	1.46248i
$868$	− 16.7284i	− 0.567798i
$869$	−14.0097	−0.475248
$870$	0	0
$871$	−11.8199	−0.400500
$872$	29.3890i	0.995238i
$873$	2.22836i	0.0754186i
$874$	−16.9718	−0.574079
$875$	0	0
$876$	11.8871	0.401629
$877$	− 35.7948i	− 1.20871i	−0.796717	−	0.604353i	$-0.793432\pi$
$877$	− 35.7948i	− 1.20871i	0.796717	−	0.604353i	$-0.206568\pi$
$878$	− 1.20890i	− 0.0407985i
$879$	16.5940	0.559702
$880$	0	0
$881$	32.9640	1.11059	0.555293	−	0.831655i	$-0.312606\pi$
$881$	32.9640	1.11059	0.555293	+	0.831655i	$0.312606\pi$
$882$	− 16.9294i	− 0.570043i
$883$	− 19.2788i	− 0.648784i	−0.945923	−	0.324392i	$-0.894840\pi$
$883$	− 19.2788i	− 0.648784i	0.945923	−	0.324392i	$-0.105160\pi$
$884$	19.8079	0.666211
$885$	0	0
$886$	7.64030	0.256681
$887$	− 26.4213i	− 0.887141i	−0.896239	−	0.443571i	$-0.853712\pi$
$887$	− 26.4213i	− 0.887141i	0.896239	−	0.443571i	$-0.146288\pi$
$888$	0.0451624i	0.00151555i
$889$	49.7472	1.66847
$890$	0	0
$891$	2.64310	0.0885472
$892$	− 26.0396i	− 0.871870i
$893$	2.76980i	0.0926877i
$894$	−0.245069	−0.00819632
$895$	0	0
$896$	32.4890	1.08538
$897$	− 13.0986i	− 0.437350i
$898$	− 30.0924i	− 1.00420i
$899$	−11.3396	−0.378196
$900$	0	0
$901$	−71.8212	−2.39271
$902$	− 8.82343i	− 0.293788i
$903$	19.4164i	0.646137i
$904$	−4.98314	−0.165737
$905$	0	0
$906$	−1.43600	−0.0477080
$907$	4.32841i	0.143722i	0.997415	+	0.0718612i	$0.0228939\pi$
$907$	4.32841i	0.143722i	−0.997415	+	0.0718612i	$0.977106\pi$
$908$	32.8478i	1.09009i
$909$	−15.4908	−0.513799
$910$	0	0
$911$	32.8983	1.08997	0.544985	−	0.838446i	$-0.316535\pi$
$911$	32.8983	1.08997	0.544985	+	0.838446i	$0.316535\pi$
$912$	− 0.517323i	− 0.0171303i
$913$	− 1.91243i	− 0.0632923i
$914$	14.2100	0.470024
$915$	0	0
$916$	−23.3235	−0.770630
$917$	− 55.6872i	− 1.83895i
$918$	6.93931i	0.229031i
$919$	−49.8351	−1.64391	−0.821954	−	0.569554i	$-0.807116\pi$
$919$	−49.8351	−1.64391	−0.821954	+	0.569554i	$0.807116\pi$
$920$	0	0
$921$	14.8424	0.489072
$922$	− 10.9219i	− 0.359692i
$923$	0.176817i	0.00581999i
$924$	16.1205	0.530326
$925$	0	0
$926$	11.3301	0.372330
$927$	11.5680i	0.379942i
$928$	− 23.0587i	− 0.756940i
$929$	−16.1832	−0.530955	−0.265477	−	0.964117i	$-0.585530\pi$
$929$	−16.1832	−0.530955	−0.265477	+	0.964117i	$0.585530\pi$
$930$	0	0
$931$	58.3575	1.91259
$932$	12.2224i	0.400356i
$933$	− 13.6460i	− 0.446751i
$934$	10.0186	0.327820
$935$	0	0
$936$	6.10816	0.199651
$937$	34.9825i	1.14283i	0.820662	+	0.571413i	$0.193605\pi$
$937$	34.9825i	1.14283i	−0.820662	+	0.571413i	$0.806395\pi$
$938$	− 25.2555i	− 0.824620i
$939$	−3.18194	−0.103839
$940$	0	0
$941$	51.6450	1.68358	0.841790	−	0.539805i	$-0.181502\pi$
$941$	51.6450	1.68358	0.841790	+	0.539805i	$0.181502\pi$
$942$	− 14.8198i	− 0.482856i
$943$	22.8957i	0.745587i
$944$	−1.85530	−0.0603849
$945$	0	0
$946$	9.02787	0.293521
$947$	− 1.85002i	− 0.0601176i	−0.999548	−	0.0300588i	$-0.990431\pi$
$947$	− 1.85002i	− 0.0601176i	0.999548	−	0.0300588i	$-0.00956946\pi$
$948$	6.35142i	0.206285i
$949$	21.1594	0.686863
$950$	0	0
$951$	3.70586	0.120171
$952$	112.964i	3.66120i
$953$	2.18860i	0.0708956i	0.999372	+	0.0354478i	$0.0112857\pi$
$953$	2.18860i	0.0708956i	−0.999372	+	0.0354478i	$0.988714\pi$
$954$	−8.29783	−0.268652
$955$	0	0
$956$	3.81833	0.123494
$957$	− 10.9275i	− 0.353236i
$958$	11.2455i	0.363326i
$959$	85.4534	2.75944
$960$	0	0
$961$	−23.4772	−0.757330
$962$	0.0301192i	0 0.000971082i
$963$	− 1.10562i	− 0.0356280i
$964$	14.9340	0.480992
$965$	0	0
$966$	27.9878	0.900493
$967$	− 13.2573i	− 0.426326i	−0.977017	−	0.213163i	$-0.931623\pi$
$967$	− 13.2573i	− 0.426326i	0.977017	−	0.213163i	$-0.0683766\pi$
$968$	11.4950i	0.369463i
$969$	−23.9205	−0.768439
$970$	0	0
$971$	−35.2892	−1.13248	−0.566242	−	0.824239i	$-0.691603\pi$
$971$	−35.2892	−1.13248	−0.566242	+	0.824239i	$0.691603\pi$
$972$	− 1.19827i	− 0.0384345i
$973$	27.9102i	0.894759i
$974$	−22.7127	−0.727763
$975$	0	0
$976$	−1.07050	−0.0342659
$977$	12.6542i	0.404844i	0.979298	+	0.202422i	$0.0648813\pi$
$977$	12.6542i	0.404844i	−0.979298	+	0.202422i	$0.935119\pi$
$978$	20.5866i	0.658287i
$979$	35.0336	1.11968
$980$	0	0
$981$	10.2626	0.327659
$982$	− 18.5281i	− 0.591255i
$983$	− 22.3964i	− 0.714335i	−0.934040	−	0.357167i	$-0.883743\pi$
$983$	− 22.3964i	− 0.714335i	0.934040	−	0.357167i	$-0.116257\pi$
$984$	−10.6768	−0.340363
$985$	0	0
$986$	28.6895	0.913661
$987$	− 4.56762i	− 0.145389i
$988$	7.88869i	0.250973i
$989$	−23.4262	−0.744910
$990$	0	0
$991$	−60.7912	−1.93110	−0.965548	−	0.260224i	$-0.916204\pi$
$991$	−60.7912	−1.93110	−0.965548	+	0.260224i	$0.916204\pi$
$992$	15.2974i	0.485691i
$993$	32.3878i	1.02780i
$994$	−0.377804	−0.0119832
$995$	0	0
$996$	−0.867017	−0.0274725
$997$	− 42.5995i	− 1.34914i	−0.738211	−	0.674570i	$-0.764329\pi$
$997$	− 42.5995i	− 1.34914i	0.738211	−	0.674570i	$-0.235671\pi$
$998$	36.1425i	1.14407i
$999$	0.0157706	0.000498960 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.g.1249.6		16
5.2	odd	4		1875.2.a.n.1.6	✓	8
5.3	odd	4		1875.2.a.o.1.3	yes	8
5.4	even	2	inner	1875.2.b.g.1249.11		16
15.2	even	4		5625.2.a.bc.1.3		8
15.8	even	4		5625.2.a.u.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.6	✓	8	5.2	odd	4
1875.2.a.o.1.3	yes	8	5.3	odd	4
1875.2.b.g.1249.6		16	1.1	even	1	trivial
1875.2.b.g.1249.11		16	5.4	even	2	inner
5625.2.a.u.1.6		8	15.8	even	4
5625.2.a.bc.1.3		8	15.2	even	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.895394i q^{2} -1.00000i q^{3} +1.19827 q^{4} -0.895394 q^{6} +5.08992i q^{7} -2.86371i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-0.895394i q^{2} -1.00000i q^{3} +1.19827 q^{4} -0.895394 q^{6} +5.08992i q^{7} -2.86371i q^{8} -1.00000 q^{9} +2.64310 q^{11} -1.19827i q^{12} -2.13295i q^{13} +4.55748 q^{14} -0.167607 q^{16} +7.75001i q^{17} +0.895394i q^{18} -3.08652 q^{19} +5.08992 q^{21} -2.36661i q^{22} +6.14107i q^{23} -2.86371 q^{24} -1.90983 q^{26} +1.00000i q^{27} +6.09909i q^{28} +4.13435 q^{29} -2.74277 q^{31} -5.57735i q^{32} -2.64310i q^{33} +6.93931 q^{34} -1.19827 q^{36} -0.0157706i q^{37} +2.76365i q^{38} -2.13295 q^{39} +3.72829 q^{41} -4.55748i q^{42} +3.81468i q^{43} +3.16715 q^{44} +5.49868 q^{46} -0.897385i q^{47} +0.167607i q^{48} -18.9072 q^{49} +7.75001 q^{51} -2.55585i q^{52} +9.26724i q^{53} +0.895394 q^{54} +14.5760 q^{56} +3.08652i q^{57} -3.70187i q^{58} +11.0693 q^{59} +6.38696 q^{61} +2.45585i q^{62} -5.08992i q^{63} -5.32913 q^{64} -2.36661 q^{66} -5.54154i q^{67} +9.28660i q^{68} +6.14107 q^{69} -0.0828976 q^{71} +2.86371i q^{72} +9.92024i q^{73} -0.0141209 q^{74} -3.69848 q^{76} +13.4532i q^{77} +1.90983i q^{78} -5.30049 q^{79} +1.00000 q^{81} -3.33829i q^{82} -0.723557i q^{83} +6.09909 q^{84} +3.41564 q^{86} -4.13435i q^{87} -7.56907i q^{88} +13.2548 q^{89} +10.8565 q^{91} +7.35867i q^{92} +2.74277i q^{93} -0.803513 q^{94} -5.57735 q^{96} -2.22836i q^{97} +16.9294i q^{98} -2.64310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44} + 66 q^{46} - 60 q^{49} - 2 q^{51} - 2 q^{54} + 120 q^{56} - 28 q^{59} + 20 q^{61} - 82 q^{64} + 36 q^{66} + 8 q^{69} + 42 q^{71} + 18 q^{74} - 2 q^{76} - 20 q^{79} + 16 q^{81} + 42 q^{84} + 84 q^{86} + 18 q^{89} - 24 q^{91} - 28 q^{94} - 36 q^{96} - 24 q^{99}+O(q^{100}) \) 16 * q - 18 * q^4 + 2 * q^6 - 16 * q^9 + 24 * q^11 - 32 * q^14 + 30 * q^16 - 32 * q^19 + 24 * q^21 - 6 * q^24 - 68 * q^26 - 4 * q^29 + 26 * q^31 + 74 * q^34 + 18 * q^36 - 28 * q^39 - 24 * q^41 - 94 * q^44 + 66 * q^46 - 60 * q^49 - 2 * q^51 - 2 * q^54 + 120 * q^56 - 28 * q^59 + 20 * q^61 - 82 * q^64 + 36 * q^66 + 8 * q^69 + 42 * q^71 + 18 * q^74 - 2 * q^76 - 20 * q^79 + 16 * q^81 + 42 * q^84 + 84 * q^86 + 18 * q^89 - 24 * q^91 - 28 * q^94 - 36 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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