LMFDB - Embedded newform 1875.2.b.g.1249.10

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			1249.10
Root			$2.59716i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.g.1249.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.770071i q^{2} +1.00000i q^{3} +1.40699 q^{4} -0.770071 q^{6} -3.98808i q^{7} +2.62363i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+0.770071i q^{2} +1.00000i q^{3} +1.40699 q^{4} -0.770071 q^{6} -3.98808i q^{7} +2.62363i q^{8} -1.00000 q^{9} -6.35014 q^{11} +1.40699i q^{12} +5.35433i q^{13} +3.07111 q^{14} +0.793602 q^{16} +3.45746i q^{17} -0.770071i q^{18} -2.32026 q^{19} +3.98808 q^{21} -4.89006i q^{22} +0.955221i q^{23} -2.62363 q^{24} -4.12322 q^{26} -1.00000i q^{27} -5.61119i q^{28} -7.26630 q^{29} +6.40484 q^{31} +5.85838i q^{32} -6.35014i q^{33} -2.66249 q^{34} -1.40699 q^{36} +2.83348i q^{37} -1.78677i q^{38} -5.35433 q^{39} -5.35164 q^{41} +3.07111i q^{42} +3.93593i q^{43} -8.93458 q^{44} -0.735588 q^{46} -2.48348i q^{47} +0.793602i q^{48} -8.90478 q^{49} -3.45746 q^{51} +7.53350i q^{52} +3.95079i q^{53} +0.770071 q^{54} +10.4632 q^{56} -2.32026i q^{57} -5.59557i q^{58} -0.0941232 q^{59} -6.61174 q^{61} +4.93218i q^{62} +3.98808i q^{63} -2.92417 q^{64} +4.89006 q^{66} +10.2811i q^{67} +4.86461i q^{68} -0.955221 q^{69} -5.81587 q^{71} -2.62363i q^{72} +3.79897i q^{73} -2.18198 q^{74} -3.26459 q^{76} +25.3249i q^{77} -4.12322i q^{78} +5.44480 q^{79} +1.00000 q^{81} -4.12114i q^{82} +9.24696i q^{83} +5.61119 q^{84} -3.03095 q^{86} -7.26630i q^{87} -16.6604i q^{88} -10.1278 q^{89} +21.3535 q^{91} +1.34399i q^{92} +6.40484i q^{93} +1.91245 q^{94} -5.85838 q^{96} -7.89936i q^{97} -6.85731i q^{98} +6.35014 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.770071i	0.544523i	0.962223	+	0.272261i	$0.0877715\pi$
$2$	0.770071i	0.544523i	−0.962223	+	0.272261i	$0.912228\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	1.40699	0.703495
$5$	0	0
$5$	0	0
$6$	−0.770071	−0.314380
$7$	− 3.98808i	− 1.50735i	−0.657246	−	0.753676i	$-0.728279\pi$
$7$	− 3.98808i	− 1.50735i	0.657246	−	0.753676i	$-0.271721\pi$
$8$	2.62363i	0.927592i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.35014	−1.91464	−0.957319	−	0.289033i	$-0.906666\pi$
$11$	−6.35014	−1.91464	−0.957319	+	0.289033i	$0.906666\pi$
$12$	1.40699i	0.406163i
$13$	5.35433i	1.48503i	0.669832	+	0.742513i	$0.266366\pi$
$13$	5.35433i	1.48503i	−0.669832	+	0.742513i	$0.733634\pi$
$14$	3.07111	0.820787
$15$	0	0
$16$	0.793602	0.198400
$17$	3.45746i	0.838557i	0.907858	+	0.419279i	$0.137717\pi$
$17$	3.45746i	0.838557i	−0.907858	+	0.419279i	$0.862283\pi$
$18$	− 0.770071i	− 0.181508i
$19$	−2.32026	−0.532305	−0.266152	−	0.963931i	$-0.585752\pi$
$19$	−2.32026	−0.532305	−0.266152	+	0.963931i	$0.585752\pi$
$20$	0	0
$21$	3.98808	0.870270
$22$	− 4.89006i	− 1.04256i
$23$	0.955221i	0.199177i	0.995029	+	0.0995886i	$0.0317527\pi$
$23$	0.955221i	0.199177i	−0.995029	+	0.0995886i	$0.968247\pi$
$24$	−2.62363	−0.535545
$25$	0	0
$26$	−4.12322	−0.808630
$27$	− 1.00000i	− 0.192450i
$28$	− 5.61119i	− 1.06042i
$29$	−7.26630	−1.34932	−0.674659	−	0.738129i	$-0.735709\pi$
$29$	−7.26630	−1.34932	−0.674659	+	0.738129i	$0.735709\pi$
$30$	0	0
$31$	6.40484	1.15034	0.575172	−	0.818033i	$-0.304935\pi$
$31$	6.40484	1.15034	0.575172	+	0.818033i	$0.304935\pi$
$32$	5.85838i	1.03563i
$33$	− 6.35014i	− 1.10542i
$34$	−2.66249	−0.456613
$35$	0	0
$36$	−1.40699	−0.234498
$37$	2.83348i	0.465821i	0.972498	+	0.232911i	$0.0748249\pi$
$37$	2.83348i	0.465821i	−0.972498	+	0.232911i	$0.925175\pi$
$38$	− 1.78677i	− 0.289852i
$39$	−5.35433	−0.857380
$40$	0	0
$41$	−5.35164	−0.835785	−0.417893	−	0.908496i	$-0.637231\pi$
$41$	−5.35164	−0.835785	−0.417893	+	0.908496i	$0.637231\pi$
$42$	3.07111i	0.473882i
$43$	3.93593i	0.600224i	0.953904	+	0.300112i	$0.0970241\pi$
$43$	3.93593i	0.600224i	−0.953904	+	0.300112i	$0.902976\pi$
$44$	−8.93458	−1.34694
$45$	0	0
$46$	−0.735588	−0.108457
$47$	− 2.48348i	− 0.362252i	−0.983460	−	0.181126i	$-0.942026\pi$
$47$	− 2.48348i	− 0.362252i	0.983460	−	0.181126i	$-0.0579742\pi$
$48$	0.793602i	0.114547i
$49$	−8.90478	−1.27211
$50$	0	0
$51$	−3.45746	−0.484141
$52$	7.53350i	1.04471i
$53$	3.95079i	0.542682i	0.962483	+	0.271341i	$0.0874671\pi$
$53$	3.95079i	0.542682i	−0.962483	+	0.271341i	$0.912533\pi$
$54$	0.770071	0.104793
$55$	0	0
$56$	10.4632	1.39821
$57$	− 2.32026i	− 0.307326i
$58$	− 5.59557i	− 0.734735i
$59$	−0.0941232	−0.0122538	−0.00612690	−	0.999981i	$-0.501950\pi$
$59$	−0.0941232	−0.0122538	−0.00612690	+	0.999981i	$0.501950\pi$
$60$	0	0
$61$	−6.61174	−0.846547	−0.423273	−	0.906002i	$-0.639119\pi$
$61$	−6.61174	−0.846547	−0.423273	+	0.906002i	$0.639119\pi$
$62$	4.93218i	0.626388i
$63$	3.98808i	0.502451i
$64$	−2.92417	−0.365521
$65$	0	0
$66$	4.89006	0.601925
$67$	10.2811i	1.25603i	0.778201	+	0.628016i	$0.216133\pi$
$67$	10.2811i	1.25603i	−0.778201	+	0.628016i	$0.783867\pi$
$68$	4.86461i	0.589921i
$69$	−0.955221	−0.114995
$70$	0	0
$71$	−5.81587	−0.690217	−0.345109	−	0.938563i	$-0.612158\pi$
$71$	−5.81587	−0.690217	−0.345109	+	0.938563i	$0.612158\pi$
$72$	− 2.62363i	− 0.309197i
$73$	3.79897i	0.444635i	0.974974	+	0.222318i	$0.0713622\pi$
$73$	3.79897i	0.444635i	−0.974974	+	0.222318i	$0.928638\pi$
$74$	−2.18198	−0.253650
$75$	0	0
$76$	−3.26459	−0.374474
$77$	25.3249i	2.88603i
$78$	− 4.12322i	− 0.466863i
$79$	5.44480	0.612587	0.306294	−	0.951937i	$-0.400911\pi$
$79$	5.44480	0.612587	0.306294	+	0.951937i	$0.400911\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 4.12114i	− 0.455104i
$83$	9.24696i	1.01499i	0.861656	+	0.507493i	$0.169428\pi$
$83$	9.24696i	1.01499i	−0.861656	+	0.507493i	$0.830572\pi$
$84$	5.61119	0.612231
$85$	0	0
$86$	−3.03095	−0.326836
$87$	− 7.26630i	− 0.779030i
$88$	− 16.6604i	− 1.77600i
$89$	−10.1278	−1.07355	−0.536773	−	0.843727i	$-0.680357\pi$
$89$	−10.1278	−1.07355	−0.536773	+	0.843727i	$0.680357\pi$
$90$	0	0
$91$	21.3535	2.23846
$92$	1.34399i	0.140120i
$93$	6.40484i	0.664151i
$94$	1.91245	0.197255
$95$	0	0
$96$	−5.85838	−0.597918
$97$	− 7.89936i	− 0.802058i	−0.916065	−	0.401029i	$-0.868653\pi$
$97$	− 7.89936i	− 0.802058i	0.916065	−	0.401029i	$-0.131347\pi$
$98$	− 6.85731i	− 0.692693i
$99$	6.35014	0.638213
$100$	0	0
$101$	8.72974	0.868641	0.434321	−	0.900758i	$-0.356989\pi$
$101$	8.72974	0.868641	0.434321	+	0.900758i	$0.356989\pi$
$102$	− 2.66249i	− 0.263626i
$103$	− 3.82390i	− 0.376780i	−0.982094	−	0.188390i	$-0.939673\pi$
$103$	− 3.82390i	− 0.376780i	0.982094	−	0.188390i	$-0.0603268\pi$
$104$	−14.0478	−1.37750
$105$	0	0
$106$	−3.04239	−0.295503
$107$	− 14.1662i	− 1.36950i	−0.728780	−	0.684748i	$-0.759912\pi$
$107$	− 14.1662i	− 1.36950i	0.728780	−	0.684748i	$-0.240088\pi$
$108$	− 1.40699i	− 0.135388i
$109$	−14.0557	−1.34629	−0.673145	−	0.739511i	$-0.735057\pi$
$109$	−14.0557	−1.34629	−0.673145	+	0.739511i	$0.735057\pi$
$110$	0	0
$111$	−2.83348	−0.268942
$112$	− 3.16495i	− 0.299059i
$113$	9.26557i	0.871632i	0.900036	+	0.435816i	$0.143540\pi$
$113$	9.26557i	0.871632i	−0.900036	+	0.435816i	$0.856460\pi$
$114$	1.78677	0.167346
$115$	0	0
$116$	−10.2236	−0.949239
$117$	− 5.35433i	− 0.495008i
$118$	− 0.0724815i	− 0.00667247i
$119$	13.7886	1.26400
$120$	0	0
$121$	29.3242	2.66584
$122$	− 5.09151i	− 0.460964i
$123$	− 5.35164i	− 0.482541i
$124$	9.01155	0.809261
$125$	0	0
$126$	−3.07111	−0.273596
$127$	− 6.91919i	− 0.613979i	−0.951713	−	0.306989i	$-0.900678\pi$
$127$	− 6.91919i	− 0.613979i	0.951713	−	0.306989i	$-0.0993216\pi$
$128$	9.46494i	0.836591i
$129$	−3.93593	−0.346540
$130$	0	0
$131$	22.0453	1.92611	0.963055	−	0.269305i	$-0.0867938\pi$
$131$	22.0453	1.92611	0.963055	+	0.269305i	$0.0867938\pi$
$132$	− 8.93458i	− 0.777655i
$133$	9.25339i	0.802371i
$134$	−7.91715	−0.683938
$135$	0	0
$136$	−9.07108	−0.777839
$137$	− 1.29992i	− 0.111060i	−0.998457	−	0.0555299i	$-0.982315\pi$
$137$	− 1.29992i	− 0.111060i	0.998457	−	0.0555299i	$-0.0176848\pi$
$138$	− 0.735588i	− 0.0626174i
$139$	−3.84419	−0.326060	−0.163030	−	0.986621i	$-0.552127\pi$
$139$	−3.84419	−0.326060	−0.163030	+	0.986621i	$0.552127\pi$
$140$	0	0
$141$	2.48348	0.209146
$142$	− 4.47864i	− 0.375839i
$143$	− 34.0008i	− 2.84329i
$144$	−0.793602	−0.0661335
$145$	0	0
$146$	−2.92547	−0.242114
$147$	− 8.90478i	− 0.734454i
$148$	3.98668i	0.327703i
$149$	−16.1035	−1.31925	−0.659623	−	0.751596i	$-0.729284\pi$
$149$	−16.1035	−1.31925	−0.659623	+	0.751596i	$0.729284\pi$
$150$	0	0
$151$	−14.8215	−1.20616	−0.603079	−	0.797682i	$-0.706059\pi$
$151$	−14.8215	−1.20616	−0.603079	+	0.797682i	$0.706059\pi$
$152$	− 6.08750i	− 0.493761i
$153$	− 3.45746i	− 0.279519i
$154$	−19.5019	−1.57151
$155$	0	0
$156$	−7.53350	−0.603162
$157$	5.29762i	0.422796i	0.977400	+	0.211398i	$0.0678017\pi$
$157$	5.29762i	0.422796i	−0.977400	+	0.211398i	$0.932198\pi$
$158$	4.19288i	0.333568i
$159$	−3.95079	−0.313318
$160$	0	0
$161$	3.80950	0.300230
$162$	0.770071i	0.0605025i
$163$	− 2.86408i	− 0.224332i	−0.993689	−	0.112166i	$-0.964221\pi$
$163$	− 2.86408i	− 0.224332i	0.993689	−	0.112166i	$-0.0357788\pi$
$164$	−7.52970	−0.587971
$165$	0	0
$166$	−7.12082	−0.552683
$167$	2.57868i	0.199544i	0.995010	+	0.0997720i	$0.0318113\pi$
$167$	2.57868i	0.199544i	−0.995010	+	0.0997720i	$0.968189\pi$
$168$	10.4632i	0.807255i
$169$	−15.6689	−1.20530
$170$	0	0
$171$	2.32026	0.177435
$172$	5.53782i	0.422255i
$173$	− 20.2272i	− 1.53784i	−0.639343	−	0.768921i	$-0.720794\pi$
$173$	− 20.2272i	− 1.53784i	0.639343	−	0.768921i	$-0.279206\pi$
$174$	5.59557	0.424199
$175$	0	0
$176$	−5.03948	−0.379865
$177$	− 0.0941232i	− 0.00707473i
$178$	− 7.79913i	− 0.584570i
$179$	26.2365	1.96101	0.980506	−	0.196491i	$-0.0629546\pi$
$179$	26.2365	1.96101	0.980506	+	0.196491i	$0.0629546\pi$
$180$	0	0
$181$	2.37843	0.176787	0.0883936	−	0.996086i	$-0.471827\pi$
$181$	2.37843	0.176787	0.0883936	+	0.996086i	$0.471827\pi$
$182$	16.4437i	1.21889i
$183$	− 6.61174i	− 0.488754i
$184$	−2.50614	−0.184755
$185$	0	0
$186$	−4.93218	−0.361645
$187$	− 21.9553i	− 1.60553i
$188$	− 3.49423i	− 0.254843i
$189$	−3.98808	−0.290090
$190$	0	0
$191$	5.71292	0.413372	0.206686	−	0.978407i	$-0.433732\pi$
$191$	5.71292	0.413372	0.206686	+	0.978407i	$0.433732\pi$
$192$	− 2.92417i	− 0.211034i
$193$	4.18017i	0.300895i	0.988618	+	0.150448i	$0.0480715\pi$
$193$	4.18017i	0.300895i	−0.988618	+	0.150448i	$0.951929\pi$
$194$	6.08307	0.436739
$195$	0	0
$196$	−12.5289	−0.894924
$197$	14.6732i	1.04542i	0.852511	+	0.522710i	$0.175079\pi$
$197$	14.6732i	1.04542i	−0.852511	+	0.522710i	$0.824921\pi$
$198$	4.89006i	0.347521i
$199$	−8.77623	−0.622130	−0.311065	−	0.950389i	$-0.600686\pi$
$199$	−8.77623	−0.622130	−0.311065	+	0.950389i	$0.600686\pi$
$200$	0	0
$201$	−10.2811	−0.725170
$202$	6.72252i	0.472995i
$203$	28.9786i	2.03390i
$204$	−4.86461	−0.340591
$205$	0	0
$206$	2.94467	0.205165
$207$	− 0.955221i	− 0.0663924i
$208$	4.24921i	0.294630i
$209$	14.7340	1.01917
$210$	0	0
$211$	18.7411	1.29019	0.645095	−	0.764103i	$-0.276818\pi$
$211$	18.7411	1.29019	0.645095	+	0.764103i	$0.276818\pi$
$212$	5.55872i	0.381774i
$213$	− 5.81587i	− 0.398497i
$214$	10.9090	0.745722
$215$	0	0
$216$	2.62363	0.178515
$217$	− 25.5430i	− 1.73397i
$218$	− 10.8239i	− 0.733085i
$219$	−3.79897	−0.256710
$220$	0	0
$221$	−18.5124	−1.24528
$222$	− 2.18198i	− 0.146445i
$223$	16.5574i	1.10877i	0.832262	+	0.554383i	$0.187046\pi$
$223$	16.5574i	1.10877i	−0.832262	+	0.554383i	$0.812954\pi$
$224$	23.3637	1.56105
$225$	0	0
$226$	−7.13515	−0.474623
$227$	5.07350i	0.336740i	0.985724	+	0.168370i	$0.0538504\pi$
$227$	5.07350i	0.336740i	−0.985724	+	0.168370i	$0.946150\pi$
$228$	− 3.26459i	− 0.216203i
$229$	23.9323	1.58149	0.790744	−	0.612147i	$-0.209694\pi$
$229$	23.9323	1.58149	0.790744	+	0.612147i	$0.209694\pi$
$230$	0	0
$231$	−25.3249	−1.66625
$232$	− 19.0641i	− 1.25162i
$233$	− 18.5301i	− 1.21395i	−0.794722	−	0.606974i	$-0.792383\pi$
$233$	− 18.5301i	− 1.21395i	0.794722	−	0.606974i	$-0.207617\pi$
$234$	4.12322	0.269543
$235$	0	0
$236$	−0.132430	−0.00862048
$237$	5.44480i	0.353678i
$238$	10.6182i	0.688277i
$239$	14.5812	0.943179	0.471590	−	0.881818i	$-0.343680\pi$
$239$	14.5812	0.943179	0.471590	+	0.881818i	$0.343680\pi$
$240$	0	0
$241$	0.0849630	0.00547295	0.00273647	−	0.999996i	$-0.499129\pi$
$241$	0.0849630	0.00547295	0.00273647	+	0.999996i	$0.499129\pi$
$242$	22.5818i	1.45161i
$243$	1.00000i	0.0641500i
$244$	−9.30266	−0.595541
$245$	0	0
$246$	4.12114	0.262754
$247$	− 12.4235i	− 0.790486i
$248$	16.8039i	1.06705i
$249$	−9.24696	−0.586002
$250$	0	0
$251$	20.6190	1.30146	0.650729	−	0.759310i	$-0.274463\pi$
$251$	20.6190	1.30146	0.650729	+	0.759310i	$0.274463\pi$
$252$	5.61119i	0.353472i
$253$	− 6.06578i	− 0.381352i
$254$	5.32827	0.334325
$255$	0	0
$256$	−13.1370	−0.821063
$257$	12.1228i	0.756199i	0.925765	+	0.378100i	$0.123422\pi$
$257$	12.1228i	0.756199i	−0.925765	+	0.378100i	$0.876578\pi$
$258$	− 3.03095i	− 0.188699i
$259$	11.3001	0.702157
$260$	0	0
$261$	7.26630	0.449773
$262$	16.9765i	1.04881i
$263$	− 7.25319i	− 0.447251i	−0.974675	−	0.223625i	$-0.928211\pi$
$263$	− 7.25319i	− 0.447251i	0.974675	−	0.223625i	$-0.0717892\pi$
$264$	16.6604	1.02538
$265$	0	0
$266$	−7.12577	−0.436909
$267$	− 10.1278i	− 0.619812i
$268$	14.4654i	0.883612i
$269$	−10.3778	−0.632744	−0.316372	−	0.948635i	$-0.602465\pi$
$269$	−10.3778	−0.632744	−0.316372	+	0.948635i	$0.602465\pi$
$270$	0	0
$271$	−29.4681	−1.79006	−0.895031	−	0.446004i	$-0.852847\pi$
$271$	−29.4681	−1.79006	−0.895031	+	0.446004i	$0.852847\pi$
$272$	2.74385i	0.166370i
$273$	21.3535i	1.29237i
$274$	1.00103	0.0604745
$275$	0	0
$276$	−1.34399	−0.0808985
$277$	− 17.4446i	− 1.04815i	−0.851673	−	0.524073i	$-0.824412\pi$
$277$	− 17.4446i	− 1.04815i	0.851673	−	0.524073i	$-0.175588\pi$
$278$	− 2.96030i	− 0.177547i
$279$	−6.40484	−0.383448
$280$	0	0
$281$	8.95212	0.534039	0.267019	−	0.963691i	$-0.413961\pi$
$281$	8.95212	0.534039	0.267019	+	0.963691i	$0.413961\pi$
$282$	1.91245i	0.113885i
$283$	− 26.1417i	− 1.55396i	−0.629523	−	0.776982i	$-0.716750\pi$
$283$	− 26.1417i	− 1.55396i	0.629523	−	0.776982i	$-0.283250\pi$
$284$	−8.18288	−0.485564
$285$	0	0
$286$	26.1830	1.54823
$287$	21.3428i	1.25982i
$288$	− 5.85838i	− 0.345208i
$289$	5.04597	0.296822
$290$	0	0
$291$	7.89936	0.463069
$292$	5.34511i	0.312799i
$293$	− 18.9520i	− 1.10719i	−0.832787	−	0.553594i	$-0.813256\pi$
$293$	− 18.9520i	− 1.10719i	0.832787	−	0.553594i	$-0.186744\pi$
$294$	6.85731	0.399927
$295$	0	0
$296$	−7.43399	−0.432092
$297$	6.35014i	0.368472i
$298$	− 12.4008i	− 0.718360i
$299$	−5.11457	−0.295783
$300$	0	0
$301$	15.6968	0.904749
$302$	− 11.4136i	− 0.656780i
$303$	8.72974i	0.501510i
$304$	−1.84137	−0.105610
$305$	0	0
$306$	2.66249	0.152204
$307$	6.75782i	0.385689i	0.981229	+	0.192845i	$0.0617713\pi$
$307$	6.75782i	0.385689i	−0.981229	+	0.192845i	$0.938229\pi$
$308$	35.6318i	2.03031i
$309$	3.82390	0.217534
$310$	0	0
$311$	13.0403	0.739445	0.369723	−	0.929142i	$-0.379453\pi$
$311$	13.0403	0.739445	0.369723	+	0.929142i	$0.379453\pi$
$312$	− 14.0478i	− 0.795298i
$313$	20.1274i	1.13767i	0.822453	+	0.568833i	$0.192605\pi$
$313$	20.1274i	1.13767i	−0.822453	+	0.568833i	$0.807395\pi$
$314$	−4.07955	−0.230222
$315$	0	0
$316$	7.66078	0.430952
$317$	12.7288i	0.714921i	0.933928	+	0.357460i	$0.116357\pi$
$317$	12.7288i	0.714921i	−0.933928	+	0.357460i	$0.883643\pi$
$318$	− 3.04239i	− 0.170609i
$319$	46.1420	2.58346
$320$	0	0
$321$	14.1662	0.790679
$322$	2.93358i	0.163482i
$323$	− 8.02222i	− 0.446368i
$324$	1.40699	0.0781661
$325$	0	0
$326$	2.20554	0.122154
$327$	− 14.0557i	− 0.777281i
$328$	− 14.0407i	− 0.775267i
$329$	−9.90430	−0.546042
$330$	0	0
$331$	−16.3131	−0.896649	−0.448324	−	0.893871i	$-0.647979\pi$
$331$	−16.3131	−0.896649	−0.448324	+	0.893871i	$0.647979\pi$
$332$	13.0104i	0.714038i
$333$	− 2.83348i	− 0.155274i
$334$	−1.98576	−0.108656
$335$	0	0
$336$	3.16495	0.172662
$337$	− 11.0359i	− 0.601166i	−0.953756	−	0.300583i	$-0.902819\pi$
$337$	− 11.0359i	− 0.601166i	0.953756	−	0.300583i	$-0.0971812\pi$
$338$	− 12.0662i	− 0.656313i
$339$	−9.26557	−0.503237
$340$	0	0
$341$	−40.6716	−2.20249
$342$	1.78677i	0.0966173i
$343$	7.59641i	0.410168i
$344$	−10.3264	−0.556763
$345$	0	0
$346$	15.5764	0.837390
$347$	30.3961i	1.63175i	0.578229	+	0.815875i	$0.303744\pi$
$347$	30.3961i	1.63175i	−0.578229	+	0.815875i	$0.696256\pi$
$348$	− 10.2236i	− 0.548044i
$349$	6.45842	0.345711	0.172856	−	0.984947i	$-0.444701\pi$
$349$	6.45842	0.345711	0.172856	+	0.984947i	$0.444701\pi$
$350$	0	0
$351$	5.35433	0.285793
$352$	− 37.2015i	− 1.98285i
$353$	4.25227i	0.226325i	0.993576	+	0.113163i	$0.0360981\pi$
$353$	4.25227i	0.226325i	−0.993576	+	0.113163i	$0.963902\pi$
$354$	0.0724815	0.00385235
$355$	0	0
$356$	−14.2497	−0.755234
$357$	13.7886i	0.729772i
$358$	20.2040i	1.06782i
$359$	8.14893	0.430084	0.215042	−	0.976605i	$-0.431011\pi$
$359$	8.14893	0.430084	0.215042	+	0.976605i	$0.431011\pi$
$360$	0	0
$361$	−13.6164	−0.716652
$362$	1.83156i	0.0962646i
$363$	29.3242i	1.53912i
$364$	30.0442	1.57474
$365$	0	0
$366$	5.09151	0.266138
$367$	− 7.18365i	− 0.374984i	−0.982266	−	0.187492i	$-0.939964\pi$
$367$	− 7.18365i	− 0.374984i	0.982266	−	0.187492i	$-0.0600358\pi$
$368$	0.758065i	0.0395169i
$369$	5.35164	0.278595
$370$	0	0
$371$	15.7561	0.818013
$372$	9.01155i	0.467227i
$373$	− 7.23599i	− 0.374666i	−0.982297	−	0.187333i	$-0.940016\pi$
$373$	− 7.23599i	− 0.374666i	0.982297	−	0.187333i	$-0.0599843\pi$
$374$	16.9072	0.874250
$375$	0	0
$376$	6.51571	0.336022
$377$	− 38.9062i	− 2.00377i
$378$	− 3.07111i	− 0.157961i
$379$	−1.57210	−0.0807535	−0.0403767	−	0.999185i	$-0.512856\pi$
$379$	−1.57210	−0.0807535	−0.0403767	+	0.999185i	$0.512856\pi$
$380$	0	0
$381$	6.91919	0.354481
$382$	4.39935i	0.225090i
$383$	28.4834i	1.45543i	0.685878	+	0.727717i	$0.259419\pi$
$383$	28.4834i	1.45543i	−0.685878	+	0.727717i	$0.740581\pi$
$384$	−9.46494	−0.483006
$385$	0	0
$386$	−3.21903	−0.163844
$387$	− 3.93593i	− 0.200075i
$388$	− 11.1143i	− 0.564244i
$389$	−5.45707	−0.276684	−0.138342	−	0.990385i	$-0.544177\pi$
$389$	−5.45707	−0.276684	−0.138342	+	0.990385i	$0.544177\pi$
$390$	0	0
$391$	−3.30264	−0.167022
$392$	− 23.3628i	− 1.18000i
$393$	22.0453i	1.11204i
$394$	−11.2994	−0.569255
$395$	0	0
$396$	8.93458	0.448980
$397$	13.6918i	0.687172i	0.939121	+	0.343586i	$0.111642\pi$
$397$	13.6918i	0.687172i	−0.939121	+	0.343586i	$0.888358\pi$
$398$	− 6.75832i	− 0.338764i
$399$	−9.25339	−0.463249
$400$	0	0
$401$	10.6255	0.530614	0.265307	−	0.964164i	$-0.414527\pi$
$401$	10.6255	0.530614	0.265307	+	0.964164i	$0.414527\pi$
$402$	− 7.91715i	− 0.394872i
$403$	34.2937i	1.70829i
$404$	12.2827	0.611085
$405$	0	0
$406$	−22.3156	−1.10750
$407$	− 17.9930i	− 0.891879i
$408$	− 9.07108i	− 0.449085i
$409$	20.5777	1.01750	0.508750	−	0.860914i	$-0.330108\pi$
$409$	20.5777	1.01750	0.508750	+	0.860914i	$0.330108\pi$
$410$	0	0
$411$	1.29992	0.0641204
$412$	− 5.38018i	− 0.265063i
$413$	0.375371i	0.0184708i
$414$	0.735588	0.0361522
$415$	0	0
$416$	−31.3677	−1.53793
$417$	− 3.84419i	− 0.188251i
$418$	11.3462i	0.554962i
$419$	−7.44362	−0.363644	−0.181822	−	0.983331i	$-0.558200\pi$
$419$	−7.44362	−0.363644	−0.181822	+	0.983331i	$0.558200\pi$
$420$	0	0
$421$	26.2168	1.27773	0.638863	−	0.769320i	$-0.279405\pi$
$421$	26.2168	1.27773	0.638863	+	0.769320i	$0.279405\pi$
$422$	14.4320i	0.702537i
$423$	2.48348i	0.120751i
$424$	−10.3654	−0.503387
$425$	0	0
$426$	4.47864	0.216991
$427$	26.3682i	1.27604i
$428$	− 19.9317i	− 0.963434i
$429$	34.0008	1.64157
$430$	0	0
$431$	−22.9859	−1.10719	−0.553597	−	0.832785i	$-0.686745\pi$
$431$	−22.9859	−1.10719	−0.553597	+	0.832785i	$0.686745\pi$
$432$	− 0.793602i	− 0.0381822i
$433$	− 29.5347i	− 1.41934i	−0.704532	−	0.709672i	$-0.748843\pi$
$433$	− 29.5347i	− 1.41934i	0.704532	−	0.709672i	$-0.251157\pi$
$434$	19.6699	0.944188
$435$	0	0
$436$	−19.7762	−0.947108
$437$	− 2.21636i	− 0.106023i
$438$	− 2.92547i	− 0.139785i
$439$	−23.4898	−1.12111	−0.560553	−	0.828119i	$-0.689411\pi$
$439$	−23.4898	−1.12111	−0.560553	+	0.828119i	$0.689411\pi$
$440$	0	0
$441$	8.90478	0.424037
$442$	− 14.2559i	− 0.678082i
$443$	19.5349i	0.928130i	0.885801	+	0.464065i	$0.153610\pi$
$443$	19.5349i	0.928130i	−0.885801	+	0.464065i	$0.846390\pi$
$444$	−3.98668	−0.189199
$445$	0	0
$446$	−12.7504	−0.603748
$447$	− 16.1035i	− 0.761667i
$448$	11.6618i	0.550969i
$449$	−1.18687	−0.0560118	−0.0280059	−	0.999608i	$-0.508916\pi$
$449$	−1.18687	−0.0560118	−0.0280059	+	0.999608i	$0.508916\pi$
$450$	0	0
$451$	33.9836	1.60023
$452$	13.0366i	0.613189i
$453$	− 14.8215i	− 0.696375i
$454$	−3.90696	−0.183363
$455$	0	0
$456$	6.08750	0.285073
$457$	23.4359i	1.09629i	0.836384	+	0.548144i	$0.184665\pi$
$457$	23.4359i	1.09629i	−0.836384	+	0.548144i	$0.815335\pi$
$458$	18.4295i	0.861156i
$459$	3.45746	0.161380
$460$	0	0
$461$	35.4249	1.64990	0.824950	−	0.565205i	$-0.191203\pi$
$461$	35.4249	1.64990	0.824950	+	0.565205i	$0.191203\pi$
$462$	− 19.5019i	− 0.907312i
$463$	9.44996i	0.439177i	0.975593	+	0.219588i	$0.0704714\pi$
$463$	9.44996i	0.439177i	−0.975593	+	0.219588i	$0.929529\pi$
$464$	−5.76655	−0.267706
$465$	0	0
$466$	14.2695	0.661022
$467$	0.0277734i	0.00128520i	1.00000		0.000642599i	$0.000204546\pi$
$467$	0.0277734i	0.00128520i	−1.00000		0.000642599i	$0.999795\pi$
$468$	− 7.53350i	− 0.348236i
$469$	41.0017	1.89328
$470$	0	0
$471$	−5.29762	−0.244102
$472$	− 0.246944i	− 0.0113665i
$473$	− 24.9937i	− 1.14921i
$474$	−4.19288	−0.192585
$475$	0	0
$476$	19.4005	0.889219
$477$	− 3.95079i	− 0.180894i
$478$	11.2286i	0.513583i
$479$	−9.30016	−0.424935	−0.212467	−	0.977168i	$-0.568150\pi$
$479$	−9.30016	−0.424935	−0.212467	+	0.977168i	$0.568150\pi$
$480$	0	0
$481$	−15.1714	−0.691756
$482$	0.0654276i	0.00298014i
$483$	3.80950i	0.173338i
$484$	41.2589	1.87541
$485$	0	0
$486$	−0.770071	−0.0349311
$487$	32.5963i	1.47708i	0.674211	+	0.738539i	$0.264484\pi$
$487$	32.5963i	1.47708i	−0.674211	+	0.738539i	$0.735516\pi$
$488$	− 17.3467i	− 0.785250i
$489$	2.86408	0.129518
$490$	0	0
$491$	−1.08119	−0.0487932	−0.0243966	−	0.999702i	$-0.507766\pi$
$491$	−1.08119	−0.0487932	−0.0243966	+	0.999702i	$0.507766\pi$
$492$	− 7.52970i	− 0.339465i
$493$	− 25.1230i	− 1.13148i
$494$	9.56695	0.430437
$495$	0	0
$496$	5.08290	0.228229
$497$	23.1942i	1.04040i
$498$	− 7.12082i	− 0.319092i
$499$	43.9536	1.96763	0.983816	−	0.179181i	$-0.0573449\pi$
$499$	43.9536	1.96763	0.983816	+	0.179181i	$0.0573449\pi$
$500$	0	0
$501$	−2.57868	−0.115207
$502$	15.8781i	0.708673i
$503$	− 29.0142i	− 1.29368i	−0.762625	−	0.646841i	$-0.776090\pi$
$503$	− 29.0142i	− 1.29368i	0.762625	−	0.646841i	$-0.223910\pi$
$504$	−10.4632	−0.466069
$505$	0	0
$506$	4.67108	0.207655
$507$	− 15.6689i	− 0.695880i
$508$	− 9.73523i	− 0.431931i
$509$	−5.13104	−0.227429	−0.113715	−	0.993513i	$-0.536275\pi$
$509$	−5.13104	−0.227429	−0.113715	+	0.993513i	$0.536275\pi$
$510$	0	0
$511$	15.1506	0.670222
$512$	8.81345i	0.389503i
$513$	2.32026i	0.102442i
$514$	−9.33541	−0.411767
$515$	0	0
$516$	−5.53782	−0.243789
$517$	15.7704i	0.693582i
$518$	8.70191i	0.382340i
$519$	20.2272	0.887874
$520$	0	0
$521$	−8.20068	−0.359278	−0.179639	−	0.983733i	$-0.557493\pi$
$521$	−8.20068	−0.359278	−0.179639	+	0.983733i	$0.557493\pi$
$522$	5.59557i	0.244912i
$523$	5.23806i	0.229044i	0.993421	+	0.114522i	$0.0365337\pi$
$523$	5.23806i	0.229044i	−0.993421	+	0.114522i	$0.963466\pi$
$524$	31.0176	1.35501
$525$	0	0
$526$	5.58547	0.243538
$527$	22.1445i	0.964629i
$528$	− 5.03948i	− 0.219315i
$529$	22.0876	0.960328
$530$	0	0
$531$	0.0941232	0.00408460
$532$	13.0194i	0.564464i
$533$	− 28.6545i	− 1.24116i
$534$	7.79913	0.337502
$535$	0	0
$536$	−26.9737	−1.16508
$537$	26.2365i	1.13219i
$538$	− 7.99163i	− 0.344544i
$539$	56.5466	2.43563
$540$	0	0
$541$	13.2888	0.571329	0.285664	−	0.958330i	$-0.407786\pi$
$541$	13.2888	0.571329	0.285664	+	0.958330i	$0.407786\pi$
$542$	− 22.6926i	− 0.974729i
$543$	2.37843i	0.102068i
$544$	−20.2551	−0.868431
$545$	0	0
$546$	−16.4437	−0.703726
$547$	40.1311i	1.71588i	0.513750	+	0.857940i	$0.328256\pi$
$547$	40.1311i	1.71588i	−0.513750	+	0.857940i	$0.671744\pi$
$548$	− 1.82898i	− 0.0781300i
$549$	6.61174	0.282182
$550$	0	0
$551$	16.8597	0.718249
$552$	− 2.50614i	− 0.106668i
$553$	− 21.7143i	− 0.923385i
$554$	13.4336	0.570740
$555$	0	0
$556$	−5.40874	−0.229382
$557$	45.6945i	1.93614i	0.250686	+	0.968069i	$0.419344\pi$
$557$	45.6945i	1.93614i	−0.250686	+	0.968069i	$0.580656\pi$
$558$	− 4.93218i	− 0.208796i
$559$	−21.0743	−0.891348
$560$	0	0
$561$	21.9553	0.926956
$562$	6.89377i	0.290796i
$563$	0.600201i	0.0252955i	0.999920	+	0.0126477i	$0.00402601\pi$
$563$	0.600201i	0.0252955i	−0.999920	+	0.0126477i	$0.995974\pi$
$564$	3.49423	0.147134
$565$	0	0
$566$	20.1310	0.846168
$567$	− 3.98808i	− 0.167484i
$568$	− 15.2587i	− 0.640240i
$569$	11.9725	0.501914	0.250957	−	0.967998i	$-0.419255\pi$
$569$	11.9725	0.501914	0.250957	+	0.967998i	$0.419255\pi$
$570$	0	0
$571$	−19.4133	−0.812422	−0.406211	−	0.913779i	$-0.633150\pi$
$571$	−19.4133	−0.812422	−0.406211	+	0.913779i	$0.633150\pi$
$572$	− 47.8387i	− 2.00024i
$573$	5.71292i	0.238660i
$574$	−16.4354	−0.686002
$575$	0	0
$576$	2.92417	0.121840
$577$	− 26.0641i	− 1.08506i	−0.840035	−	0.542532i	$-0.817466\pi$
$577$	− 26.0641i	− 1.08506i	0.840035	−	0.542532i	$-0.182534\pi$
$578$	3.88575i	0.161626i
$579$	−4.18017	−0.173722
$580$	0	0
$581$	36.8776	1.52994
$582$	6.08307i	0.252151i
$583$	− 25.0880i	− 1.03904i
$584$	−9.96706	−0.412440
$585$	0	0
$586$	14.5944	0.602888
$587$	− 2.53968i	− 0.104824i	−0.998626	−	0.0524120i	$-0.983309\pi$
$587$	− 2.53968i	− 0.104824i	0.998626	−	0.0524120i	$-0.0166909\pi$
$588$	− 12.5289i	− 0.516685i
$589$	−14.8609	−0.612333
$590$	0	0
$591$	−14.6732	−0.603573
$592$	2.24865i	0.0924191i
$593$	4.27995i	0.175757i	0.996131	+	0.0878783i	$0.0280087\pi$
$593$	4.27995i	0.175757i	−0.996131	+	0.0878783i	$0.971991\pi$
$594$	−4.89006	−0.200642
$595$	0	0
$596$	−22.6574	−0.928084
$597$	− 8.77623i	− 0.359187i
$598$	− 3.93858i	− 0.161061i
$599$	−27.1909	−1.11099	−0.555495	−	0.831520i	$-0.687471\pi$
$599$	−27.1909	−1.11099	−0.555495	+	0.831520i	$0.687471\pi$
$600$	0	0
$601$	−44.6772	−1.82242	−0.911211	−	0.411940i	$-0.864851\pi$
$601$	−44.6772	−1.82242	−0.911211	+	0.411940i	$0.864851\pi$
$602$	12.0877i	0.492656i
$603$	− 10.2811i	− 0.418677i
$604$	−20.8537	−0.848526
$605$	0	0
$606$	−6.72252	−0.273084
$607$	− 3.48433i	− 0.141425i	−0.997497	−	0.0707123i	$-0.977473\pi$
$607$	− 3.48433i	− 0.141425i	0.997497	−	0.0707123i	$-0.0225272\pi$
$608$	− 13.5930i	− 0.551268i
$609$	−28.9786	−1.17427
$610$	0	0
$611$	13.2974	0.537954
$612$	− 4.86461i	− 0.196640i
$613$	− 11.1118i	− 0.448803i	−0.974497	−	0.224402i	$-0.927957\pi$
$613$	− 11.1118i	− 0.448803i	0.974497	−	0.224402i	$-0.0720427\pi$
$614$	−5.20400	−0.210016
$615$	0	0
$616$	−66.4429	−2.67706
$617$	33.0355i	1.32996i	0.746861	+	0.664980i	$0.231560\pi$
$617$	33.0355i	1.32996i	−0.746861	+	0.664980i	$0.768440\pi$
$618$	2.94467i	0.118452i
$619$	−17.0417	−0.684964	−0.342482	−	0.939524i	$-0.611268\pi$
$619$	−17.0417	−0.684964	−0.342482	+	0.939524i	$0.611268\pi$
$620$	0	0
$621$	0.955221	0.0383317
$622$	10.0419i	0.402645i
$623$	40.3905i	1.61821i
$624$	−4.24921	−0.170105
$625$	0	0
$626$	−15.4995	−0.619485
$627$	14.7340i	0.588419i
$628$	7.45370i	0.297435i
$629$	−9.79664	−0.390618
$630$	0	0
$631$	7.72790	0.307643	0.153821	−	0.988099i	$-0.450842\pi$
$631$	7.72790	0.307643	0.153821	+	0.988099i	$0.450842\pi$
$632$	14.2851i	0.568231i
$633$	18.7411i	0.744891i
$634$	−9.80208	−0.389290
$635$	0	0
$636$	−5.55872	−0.220417
$637$	− 47.6792i	− 1.88912i
$638$	35.5327i	1.40675i
$639$	5.81587	0.230072
$640$	0	0
$641$	14.4910	0.572361	0.286181	−	0.958176i	$-0.407614\pi$
$641$	14.4910	0.572361	0.286181	+	0.958176i	$0.407614\pi$
$642$	10.9090i	0.430543i
$643$	42.5134i	1.67656i	0.545238	+	0.838282i	$0.316439\pi$
$643$	42.5134i	1.67656i	−0.545238	+	0.838282i	$0.683561\pi$
$644$	5.35992	0.211211
$645$	0	0
$646$	6.17768	0.243058
$647$	26.3343i	1.03531i	0.855590	+	0.517655i	$0.173195\pi$
$647$	26.3343i	1.03531i	−0.855590	+	0.517655i	$0.826805\pi$
$648$	2.62363i	0.103066i
$649$	0.597695	0.0234616
$650$	0	0
$651$	25.5430	1.00111
$652$	− 4.02973i	− 0.157816i
$653$	4.46840i	0.174862i	0.996171	+	0.0874310i	$0.0278657\pi$
$653$	4.46840i	0.174862i	−0.996171	+	0.0874310i	$0.972134\pi$
$654$	10.8239	0.423247
$655$	0	0
$656$	−4.24707	−0.165820
$657$	− 3.79897i	− 0.148212i
$658$	− 7.62702i	− 0.297332i
$659$	−40.2092	−1.56633	−0.783165	−	0.621814i	$-0.786396\pi$
$659$	−40.2092	−1.56633	−0.783165	+	0.621814i	$0.786396\pi$
$660$	0	0
$661$	−16.0012	−0.622374	−0.311187	−	0.950349i	$-0.600727\pi$
$661$	−16.0012	−0.622374	−0.311187	+	0.950349i	$0.600727\pi$
$662$	− 12.5622i	− 0.488246i
$663$	− 18.5124i	− 0.718962i
$664$	−24.2606	−0.941492
$665$	0	0
$666$	2.18198	0.0845500
$667$	− 6.94092i	− 0.268754i
$668$	3.62817i	0.140378i
$669$	−16.5574	−0.640146
$670$	0	0
$671$	41.9855	1.62083
$672$	23.3637i	0.901274i
$673$	− 10.6585i	− 0.410854i	−0.978672	−	0.205427i	$-0.934142\pi$
$673$	− 10.6585i	− 0.410854i	0.978672	−	0.205427i	$-0.0658584\pi$
$674$	8.49846	0.327349
$675$	0	0
$676$	−22.0460	−0.847922
$677$	− 15.9650i	− 0.613585i	−0.951776	−	0.306793i	$-0.900744\pi$
$677$	− 15.9650i	− 0.613585i	0.951776	−	0.306793i	$-0.0992558\pi$
$678$	− 7.13515i	− 0.274024i
$679$	−31.5033	−1.20898
$680$	0	0
$681$	−5.07350	−0.194417
$682$	− 31.3201i	− 1.19931i
$683$	1.13294i	0.0433508i	0.999765	+	0.0216754i	$0.00690004\pi$
$683$	1.13294i	0.0433508i	−0.999765	+	0.0216754i	$0.993100\pi$
$684$	3.26459	0.124825
$685$	0	0
$686$	−5.84978	−0.223346
$687$	23.9323i	0.913072i
$688$	3.12356i	0.119085i
$689$	−21.1538	−0.805897
$690$	0	0
$691$	−11.4786	−0.436668	−0.218334	−	0.975874i	$-0.570062\pi$
$691$	−11.4786	−0.436668	−0.218334	+	0.975874i	$0.570062\pi$
$692$	− 28.4594i	− 1.08186i
$693$	− 25.3249i	− 0.962012i
$694$	−23.4072	−0.888525
$695$	0	0
$696$	19.0641	0.722621
$697$	− 18.5031i	− 0.700854i
$698$	4.97344i	0.188248i
$699$	18.5301	0.700873
$700$	0	0
$701$	−27.9382	−1.05521	−0.527605	−	0.849490i	$-0.676910\pi$
$701$	−27.9382	−1.05521	−0.527605	+	0.849490i	$0.676910\pi$
$702$	4.12322i	0.155621i
$703$	− 6.57442i	− 0.247959i
$704$	18.5689	0.699840
$705$	0	0
$706$	−3.27455	−0.123239
$707$	− 34.8149i	− 1.30935i
$708$	− 0.132430i	− 0.00497704i
$709$	−26.2614	−0.986266	−0.493133	−	0.869954i	$-0.664148\pi$
$709$	−26.2614	−0.986266	−0.493133	+	0.869954i	$0.664148\pi$
$710$	0	0
$711$	−5.44480	−0.204196
$712$	− 26.5716i	− 0.995812i
$713$	6.11804i	0.229122i
$714$	−10.6182	−0.397377
$715$	0	0
$716$	36.9146	1.37956
$717$	14.5812i	0.544545i
$718$	6.27526i	0.234191i
$719$	47.8778	1.78554	0.892771	−	0.450510i	$-0.148758\pi$
$719$	47.8778	1.78554	0.892771	+	0.450510i	$0.148758\pi$
$720$	0	0
$721$	−15.2500	−0.567940
$722$	− 10.4856i	− 0.390233i
$723$	0.0849630i	0.00315981i
$724$	3.34642	0.124369
$725$	0	0
$726$	−22.5818	−0.838088
$727$	6.74514i	0.250163i	0.992146	+	0.125082i	$0.0399193\pi$
$727$	6.74514i	0.250163i	−0.992146	+	0.125082i	$0.960081\pi$
$728$	56.0236i	2.07637i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−13.6083	−0.503322
$732$	− 9.30266i	− 0.343836i
$733$	42.9974i	1.58815i	0.607822	+	0.794073i	$0.292043\pi$
$733$	42.9974i	1.58815i	−0.607822	+	0.794073i	$0.707957\pi$
$734$	5.53192	0.204187
$735$	0	0
$736$	−5.59605	−0.206273
$737$	− 65.2862i	− 2.40485i
$738$	4.12114i	0.151701i
$739$	17.0432	0.626943	0.313472	−	0.949598i	$-0.398508\pi$
$739$	17.0432	0.626943	0.313472	+	0.949598i	$0.398508\pi$
$740$	0	0
$741$	12.4235	0.456387
$742$	12.1333i	0.445427i
$743$	− 50.4068i	− 1.84925i	−0.380884	−	0.924623i	$-0.624380\pi$
$743$	− 50.4068i	− 1.84925i	0.380884	−	0.924623i	$-0.375620\pi$
$744$	−16.8039	−0.616061
$745$	0	0
$746$	5.57223	0.204014
$747$	− 9.24696i	− 0.338329i
$748$	− 30.8910i	− 1.12949i
$749$	−56.4958	−2.06431
$750$	0	0
$751$	−3.00535	−0.109667	−0.0548334	−	0.998496i	$-0.517463\pi$
$751$	−3.00535	−0.109667	−0.0548334	+	0.998496i	$0.517463\pi$
$752$	− 1.97089i	− 0.0718710i
$753$	20.6190i	0.751397i
$754$	29.9606	1.09110
$755$	0	0
$756$	−5.61119	−0.204077
$757$	8.74406i	0.317808i	0.987294	+	0.158904i	$0.0507961\pi$
$757$	8.74406i	0.317808i	−0.987294	+	0.158904i	$0.949204\pi$
$758$	− 1.21063i	− 0.0439721i
$759$	6.06578	0.220174
$760$	0	0
$761$	−31.8086	−1.15306	−0.576530	−	0.817076i	$-0.695594\pi$
$761$	−31.8086	−1.15306	−0.576530	+	0.817076i	$0.695594\pi$
$762$	5.32827i	0.193023i
$763$	56.0551i	2.02933i
$764$	8.03802	0.290805
$765$	0	0
$766$	−21.9342	−0.792517
$767$	− 0.503967i	− 0.0181972i
$768$	− 13.1370i	− 0.474041i
$769$	−28.7776	−1.03775	−0.518874	−	0.854851i	$-0.673649\pi$
$769$	−28.7776	−1.03775	−0.518874	+	0.854851i	$0.673649\pi$
$770$	0	0
$771$	−12.1228	−0.436592
$772$	5.88146i	0.211678i
$773$	34.6320i	1.24563i	0.782370	+	0.622814i	$0.214011\pi$
$773$	34.6320i	1.24563i	−0.782370	+	0.622814i	$0.785989\pi$
$774$	3.03095	0.108945
$775$	0	0
$776$	20.7250	0.743983
$777$	11.3001i	0.405390i
$778$	− 4.20233i	− 0.150661i
$779$	12.4172	0.444893
$780$	0	0
$781$	36.9316	1.32152
$782$	− 2.54327i	− 0.0909470i
$783$	7.26630i	0.259677i
$784$	−7.06685	−0.252388
$785$	0	0
$786$	−16.9765	−0.605531
$787$	52.9281i	1.88668i	0.331821	+	0.943342i	$0.392337\pi$
$787$	52.9281i	1.88668i	−0.331821	+	0.943342i	$0.607663\pi$
$788$	20.6450i	0.735448i
$789$	7.25319	0.258220
$790$	0	0
$791$	36.9518	1.31386
$792$	16.6604i	0.592001i
$793$	− 35.4015i	− 1.25714i
$794$	−10.5437	−0.374181
$795$	0	0
$796$	−12.3481	−0.437666
$797$	− 32.9585i	− 1.16745i	−0.811952	−	0.583724i	$-0.801595\pi$
$797$	− 32.9585i	− 1.16745i	0.811952	−	0.583724i	$-0.198405\pi$
$798$	− 7.12577i	− 0.252250i
$799$	8.58652	0.303769
$800$	0	0
$801$	10.1278	0.357849
$802$	8.18242i	0.288931i
$803$	− 24.1240i	− 0.851316i
$804$	−14.4654	−0.510154
$805$	0	0
$806$	−26.4086	−0.930202
$807$	− 10.3778i	− 0.365315i
$808$	22.9036i	0.805744i
$809$	33.7389	1.18620	0.593098	−	0.805130i	$-0.297905\pi$
$809$	33.7389	1.18620	0.593098	+	0.805130i	$0.297905\pi$
$810$	0	0
$811$	−23.3207	−0.818902	−0.409451	−	0.912332i	$-0.634280\pi$
$811$	−23.3207	−0.818902	−0.409451	+	0.912332i	$0.634280\pi$
$812$	40.7726i	1.43084i
$813$	− 29.4681i	− 1.03349i
$814$	13.8559	0.485648
$815$	0	0
$816$	−2.74385	−0.0960539
$817$	− 9.13240i	− 0.319502i
$818$	15.8463i	0.554052i
$819$	−21.3535	−0.746152
$820$	0	0
$821$	16.9918	0.593018	0.296509	−	0.955030i	$-0.404178\pi$
$821$	16.9918	0.593018	0.296509	+	0.955030i	$0.404178\pi$
$822$	1.00103i	0.0349150i
$823$	− 29.8674i	− 1.04111i	−0.853828	−	0.520555i	$-0.825725\pi$
$823$	− 29.8674i	− 1.04111i	0.853828	−	0.520555i	$-0.174275\pi$
$824$	10.0325	0.349498
$825$	0	0
$826$	−0.289062	−0.0100578
$827$	34.1803i	1.18857i	0.804256	+	0.594283i	$0.202564\pi$
$827$	34.1803i	1.18857i	−0.804256	+	0.594283i	$0.797436\pi$
$828$	− 1.34399i	− 0.0467067i
$829$	−45.5662	−1.58258	−0.791289	−	0.611442i	$-0.790590\pi$
$829$	−45.5662	−1.58258	−0.791289	+	0.611442i	$0.790590\pi$
$830$	0	0
$831$	17.4446	0.605148
$832$	− 15.6570i	− 0.542808i
$833$	− 30.7879i	− 1.06674i
$834$	2.96030	0.102507
$835$	0	0
$836$	20.7306	0.716982
$837$	− 6.40484i	− 0.221384i
$838$	− 5.73212i	− 0.198013i
$839$	5.10480	0.176237	0.0881186	−	0.996110i	$-0.471915\pi$
$839$	5.10480	0.176237	0.0881186	+	0.996110i	$0.471915\pi$
$840$	0	0
$841$	23.7992	0.820661
$842$	20.1888i	0.695751i
$843$	8.95212i	0.308328i
$844$	26.3685	0.907642
$845$	0	0
$846$	−1.91245	−0.0657515
$847$	− 116.947i	− 4.01836i
$848$	3.13535i	0.107668i
$849$	26.1417	0.897181
$850$	0	0
$851$	−2.70660	−0.0927810
$852$	− 8.18288i	− 0.280341i
$853$	57.1027i	1.95516i	0.210564	+	0.977580i	$0.432470\pi$
$853$	57.1027i	1.95516i	−0.210564	+	0.977580i	$0.567530\pi$
$854$	−20.3054	−0.694835
$855$	0	0
$856$	37.1667	1.27033
$857$	− 46.0057i	− 1.57153i	−0.618528	−	0.785763i	$-0.712271\pi$
$857$	− 46.0057i	− 1.57153i	0.618528	−	0.785763i	$-0.287729\pi$
$858$	26.1830i	0.893873i
$859$	39.1766	1.33669	0.668343	−	0.743853i	$-0.267004\pi$
$859$	39.1766	1.33669	0.668343	+	0.743853i	$0.267004\pi$
$860$	0	0
$861$	−21.3428	−0.727359
$862$	− 17.7008i	− 0.602892i
$863$	− 34.8513i	− 1.18635i	−0.805073	−	0.593175i	$-0.797874\pi$
$863$	− 34.8513i	− 1.18635i	0.805073	−	0.593175i	$-0.202126\pi$
$864$	5.85838	0.199306
$865$	0	0
$866$	22.7438	0.772865
$867$	5.04597i	0.171370i
$868$	− 35.9388i	− 1.21984i
$869$	−34.5752	−1.17288
$870$	0	0
$871$	−55.0482	−1.86524
$872$	− 36.8768i	− 1.24881i
$873$	7.89936i	0.267353i
$874$	1.70676	0.0577319
$875$	0	0
$876$	−5.34511	−0.180594
$877$	− 24.0922i	− 0.813537i	−0.913531	−	0.406768i	$-0.866656\pi$
$877$	− 24.0922i	− 0.813537i	0.913531	−	0.406768i	$-0.133344\pi$
$878$	− 18.0888i	− 0.610467i
$879$	18.9520	0.639235
$880$	0	0
$881$	−2.41163	−0.0812499	−0.0406250	−	0.999174i	$-0.512935\pi$
$881$	−2.41163	−0.0812499	−0.0406250	+	0.999174i	$0.512935\pi$
$882$	6.85731i	0.230898i
$883$	− 1.48748i	− 0.0500578i	−0.999687	−	0.0250289i	$-0.992032\pi$
$883$	− 1.48748i	− 0.0500578i	0.999687	−	0.0250289i	$-0.00796778\pi$
$884$	−26.0468	−0.876047
$885$	0	0
$886$	−15.0432	−0.505388
$887$	20.1745i	0.677392i	0.940896	+	0.338696i	$0.109986\pi$
$887$	20.1745i	0.677392i	−0.940896	+	0.338696i	$0.890014\pi$
$888$	− 7.43399i	− 0.249468i
$889$	−27.5943	−0.925482
$890$	0	0
$891$	−6.35014	−0.212738
$892$	23.2961i	0.780011i
$893$	5.76232i	0.192829i
$894$	12.4008	0.414745
$895$	0	0
$896$	37.7470	1.26104
$897$	− 5.11457i	− 0.170771i
$898$	− 0.913974i	− 0.0304997i
$899$	−46.5395	−1.55218
$900$	0	0
$901$	−13.6597	−0.455070
$902$	26.1698i	0.871360i
$903$	15.6968i	0.522357i
$904$	−24.3094	−0.808518
$905$	0	0
$906$	11.4136	0.379192
$907$	− 26.2850i	− 0.872778i	−0.899758	−	0.436389i	$-0.856257\pi$
$907$	− 26.2850i	− 0.872778i	0.899758	−	0.436389i	$-0.143743\pi$
$908$	7.13837i	0.236895i
$909$	−8.72974	−0.289547
$910$	0	0
$911$	−22.8586	−0.757340	−0.378670	−	0.925532i	$-0.623619\pi$
$911$	−22.8586	−0.757340	−0.378670	+	0.925532i	$0.623619\pi$
$912$	− 1.84137i	− 0.0609737i
$913$	− 58.7195i	− 1.94333i
$914$	−18.0473	−0.596953
$915$	0	0
$916$	33.6724	1.11257
$917$	− 87.9186i	− 2.90333i
$918$	2.66249i	0.0878753i
$919$	15.7960	0.521061	0.260530	−	0.965466i	$-0.416103\pi$
$919$	15.7960	0.521061	0.260530	+	0.965466i	$0.416103\pi$
$920$	0	0
$921$	−6.75782	−0.222678
$922$	27.2797i	0.898408i
$923$	− 31.1401i	− 1.02499i
$924$	−35.6318	−1.17220
$925$	0	0
$926$	−7.27714	−0.239142
$927$	3.82390i	0.125593i
$928$	− 42.5688i	− 1.39739i
$929$	−48.5302	−1.59222	−0.796111	−	0.605150i	$-0.793113\pi$
$929$	−48.5302	−1.59222	−0.796111	+	0.605150i	$0.793113\pi$
$930$	0	0
$931$	20.6614	0.677151
$932$	− 26.0717i	− 0.854006i
$933$	13.0403i	0.426919i
$934$	−0.0213875	−0.000699819 0
$935$	0	0
$936$	14.0478	0.459166
$937$	51.4725i	1.68153i	0.541397	+	0.840767i	$0.317896\pi$
$937$	51.4725i	1.68153i	−0.541397	+	0.840767i	$0.682104\pi$
$938$	31.5742i	1.03094i
$939$	−20.1274	−0.656832
$940$	0	0
$941$	−19.4724	−0.634782	−0.317391	−	0.948295i	$-0.602807\pi$
$941$	−19.4724	−0.634782	−0.317391	+	0.948295i	$0.602807\pi$
$942$	− 4.07955i	− 0.132919i
$943$	− 5.11199i	− 0.166469i
$944$	−0.0746963	−0.00243116
$945$	0	0
$946$	19.2469	0.625772
$947$	− 38.9572i	− 1.26594i	−0.774177	−	0.632970i	$-0.781836\pi$
$947$	− 38.9572i	− 1.26594i	0.774177	−	0.632970i	$-0.218164\pi$
$948$	7.66078i	0.248810i
$949$	−20.3409	−0.660295
$950$	0	0
$951$	−12.7288	−0.412760
$952$	36.1762i	1.17248i
$953$	0.532270i	0.0172419i	0.999963	+	0.00862095i	$0.00274417\pi$
$953$	0.532270i	0.0172419i	−0.999963	+	0.00862095i	$0.997256\pi$
$954$	3.04239	0.0985009
$955$	0	0
$956$	20.5156	0.663522
$957$	46.1420i	1.49156i
$958$	− 7.16178i	− 0.231387i
$959$	−5.18419	−0.167406
$960$	0	0
$961$	10.0220	0.323290
$962$	− 11.6831i	− 0.376677i
$963$	14.1662i	0.456499i
$964$	0.119542	0.00385019
$965$	0	0
$966$	−2.93358	−0.0943865
$967$	36.8517i	1.18507i	0.805544	+	0.592536i	$0.201873\pi$
$967$	36.8517i	1.18507i	−0.805544	+	0.592536i	$0.798127\pi$
$968$	76.9358i	2.47281i
$969$	8.02222	0.257711
$970$	0	0
$971$	4.25064	0.136410	0.0682048	−	0.997671i	$-0.478273\pi$
$971$	4.25064	0.136410	0.0682048	+	0.997671i	$0.478273\pi$
$972$	1.40699i	0.0451292i
$973$	15.3310i	0.491488i
$974$	−25.1014	−0.804302
$975$	0	0
$976$	−5.24709	−0.167955
$977$	− 31.6086i	− 1.01125i	−0.862754	−	0.505625i	$-0.831262\pi$
$977$	− 31.6086i	− 1.01125i	0.862754	−	0.505625i	$-0.168738\pi$
$978$	2.20554i	0.0705255i
$979$	64.3130	2.05545
$980$	0	0
$981$	14.0557	0.448763
$982$	− 0.832590i	− 0.0265690i
$983$	− 45.3330i	− 1.44590i	−0.690901	−	0.722949i	$-0.742786\pi$
$983$	− 45.3330i	− 1.44590i	0.690901	−	0.722949i	$-0.257214\pi$
$984$	14.0407	0.447601
$985$	0	0
$986$	19.3465	0.616117
$987$	− 9.90430i	− 0.315257i
$988$	− 17.4797i	− 0.556103i
$989$	−3.75968	−0.119551
$990$	0	0
$991$	9.20124	0.292287	0.146143	−	0.989263i	$-0.453314\pi$
$991$	9.20124	0.292287	0.146143	+	0.989263i	$0.453314\pi$
$992$	37.5220i	1.19132i
$993$	− 16.3131i	− 0.517680i
$994$	−17.8612	−0.566522
$995$	0	0
$996$	−13.0104	−0.412250
$997$	13.1014i	0.414924i	0.978243	+	0.207462i	$0.0665203\pi$
$997$	13.1014i	0.414924i	−0.978243	+	0.207462i	$0.933480\pi$
$998$	33.8474i	1.07142i
$999$	2.83348	0.0896473

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.g.1249.10		16
5.2	odd	4		1875.2.a.o.1.4	yes	8
5.3	odd	4		1875.2.a.n.1.5	✓	8
5.4	even	2	inner	1875.2.b.g.1249.7		16
15.2	even	4		5625.2.a.u.1.5		8
15.8	even	4		5625.2.a.bc.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.n.1.5	✓	8	5.3	odd	4
1875.2.a.o.1.4	yes	8	5.2	odd	4
1875.2.b.g.1249.7		16	5.4	even	2	inner
1875.2.b.g.1249.10		16	1.1	even	1	trivial
5625.2.a.u.1.5		8	15.2	even	4
5625.2.a.bc.1.4		8	15.8	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)

\(f(q)\)	\(=\)	\(q+0.770071i q^{2} +1.00000i q^{3} +1.40699 q^{4} -0.770071 q^{6} -3.98808i q^{7} +2.62363i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+0.770071i q^{2} +1.00000i q^{3} +1.40699 q^{4} -0.770071 q^{6} -3.98808i q^{7} +2.62363i q^{8} -1.00000 q^{9} -6.35014 q^{11} +1.40699i q^{12} +5.35433i q^{13} +3.07111 q^{14} +0.793602 q^{16} +3.45746i q^{17} -0.770071i q^{18} -2.32026 q^{19} +3.98808 q^{21} -4.89006i q^{22} +0.955221i q^{23} -2.62363 q^{24} -4.12322 q^{26} -1.00000i q^{27} -5.61119i q^{28} -7.26630 q^{29} +6.40484 q^{31} +5.85838i q^{32} -6.35014i q^{33} -2.66249 q^{34} -1.40699 q^{36} +2.83348i q^{37} -1.78677i q^{38} -5.35433 q^{39} -5.35164 q^{41} +3.07111i q^{42} +3.93593i q^{43} -8.93458 q^{44} -0.735588 q^{46} -2.48348i q^{47} +0.793602i q^{48} -8.90478 q^{49} -3.45746 q^{51} +7.53350i q^{52} +3.95079i q^{53} +0.770071 q^{54} +10.4632 q^{56} -2.32026i q^{57} -5.59557i q^{58} -0.0941232 q^{59} -6.61174 q^{61} +4.93218i q^{62} +3.98808i q^{63} -2.92417 q^{64} +4.89006 q^{66} +10.2811i q^{67} +4.86461i q^{68} -0.955221 q^{69} -5.81587 q^{71} -2.62363i q^{72} +3.79897i q^{73} -2.18198 q^{74} -3.26459 q^{76} +25.3249i q^{77} -4.12322i q^{78} +5.44480 q^{79} +1.00000 q^{81} -4.12114i q^{82} +9.24696i q^{83} +5.61119 q^{84} -3.03095 q^{86} -7.26630i q^{87} -16.6604i q^{88} -10.1278 q^{89} +21.3535 q^{91} +1.34399i q^{92} +6.40484i q^{93} +1.91245 q^{94} -5.85838 q^{96} -7.89936i q^{97} -6.85731i q^{98} +6.35014 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9}+O(q^{10}) \) 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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