LMFDB - Embedded newform 525.2.d.c.274.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.d (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:	$4.19214610612$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.2.d.c.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{2} -1.00000i q^{3} -3.00000 q^{4} +2.23607 q^{6} -1.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.23607i q^{2} -1.00000i q^{3} -3.00000 q^{4} +2.23607 q^{6} -1.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +6.47214 q^{11} +3.00000i q^{12} +4.47214i q^{13} +2.23607 q^{14} -1.00000 q^{16} +2.00000i q^{17} -2.23607i q^{18} +2.47214 q^{19} -1.00000 q^{21} +14.4721i q^{22} +4.00000i q^{23} -2.23607 q^{24} -10.0000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +2.00000 q^{29} +1.52786 q^{31} -6.70820i q^{32} -6.47214i q^{33} -4.47214 q^{34} +3.00000 q^{36} +6.94427i q^{37} +5.52786i q^{38} +4.47214 q^{39} -2.00000 q^{41} -2.23607i q^{42} +8.94427i q^{43} -19.4164 q^{44} -8.94427 q^{46} -12.9443i q^{47} +1.00000i q^{48} -1.00000 q^{49} +2.00000 q^{51} -13.4164i q^{52} -3.52786i q^{53} -2.23607 q^{54} -2.23607 q^{56} -2.47214i q^{57} +4.47214i q^{58} +8.94427 q^{59} -2.00000 q^{61} +3.41641i q^{62} +1.00000i q^{63} +13.0000 q^{64} +14.4721 q^{66} +4.00000i q^{67} -6.00000i q^{68} +4.00000 q^{69} +5.52786 q^{71} +2.23607i q^{72} -12.4721i q^{73} -15.5279 q^{74} -7.41641 q^{76} -6.47214i q^{77} +10.0000i q^{78} -12.9443 q^{79} +1.00000 q^{81} -4.47214i q^{82} -16.9443i q^{83} +3.00000 q^{84} -20.0000 q^{86} -2.00000i q^{87} -14.4721i q^{88} +2.00000 q^{89} +4.47214 q^{91} -12.0000i q^{92} -1.52786i q^{93} +28.9443 q^{94} -6.70820 q^{96} -8.47214i q^{97} -2.23607i q^{98} -6.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{4} - 4 q^{9}+O(q^{10}) $ 4 * q - 12 * q^4 - 4 * q^9	$ 4 q - 12 q^{4} - 4 q^{9} + 8 q^{11} - 4 q^{16} - 8 q^{19} - 4 q^{21} - 40 q^{26} + 8 q^{29} + 24 q^{31} + 12 q^{36} - 8 q^{41} - 24 q^{44} - 4 q^{49} + 8 q^{51} - 8 q^{61} + 52 q^{64} + 40 q^{66} + 16 q^{69} + 40 q^{71} - 80 q^{74} + 24 q^{76} - 16 q^{79} + 4 q^{81} + 12 q^{84} - 80 q^{86} + 8 q^{89} + 80 q^{94} - 8 q^{99}+O(q^{100}) $ 4 * q - 12 * q^4 - 4 * q^9 + 8 * q^11 - 4 * q^16 - 8 * q^19 - 4 * q^21 - 40 * q^26 + 8 * q^29 + 24 * q^31 + 12 * q^36 - 8 * q^41 - 24 * q^44 - 4 * q^49 + 8 * q^51 - 8 * q^61 + 52 * q^64 + 40 * q^66 + 16 * q^69 + 40 * q^71 - 80 * q^74 + 24 * q^76 - 16 * q^79 + 4 * q^81 + 12 * q^84 - 80 * q^86 + 8 * q^89 + 80 * q^94 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	2.23607i	1.58114i	0.612372	+	0.790569i	$0.290215\pi$
$2$	2.23607i	1.58114i	−0.612372	+	0.790569i	$0.709785\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−3.00000	−1.50000
$5$	0	0
$5$	0	0
$6$	2.23607	0.912871
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	− 2.23607i	− 0.790569i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	6.47214	1.95142	0.975711	−	0.219061i	$-0.0702993\pi$
$11$	6.47214	1.95142	0.975711	+	0.219061i	$0.0702993\pi$
$12$	3.00000i	0.866025i
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	2.23607	0.597614
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	− 2.23607i	− 0.527046i
$19$	2.47214	0.567147	0.283573	−	0.958951i	$-0.408480\pi$
$19$	2.47214	0.567147	0.283573	+	0.958951i	$0.408480\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	14.4721i	3.08547i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−10.0000	−1.96116
$27$	1.00000i	0.192450i
$28$	3.00000i	0.566947i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	− 6.70820i	− 1.18585i
$33$	− 6.47214i	− 1.12665i
$34$	−4.47214	−0.766965
$35$	0	0
$36$	3.00000	0.500000
$37$	6.94427i	1.14163i	0.821078	+	0.570816i	$0.193373\pi$
$37$	6.94427i	1.14163i	−0.821078	+	0.570816i	$0.806627\pi$
$38$	5.52786i	0.896738i
$39$	4.47214	0.716115
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	− 2.23607i	− 0.345033i
$43$	8.94427i	1.36399i	0.731357	+	0.681994i	$0.238887\pi$
$43$	8.94427i	1.36399i	−0.731357	+	0.681994i	$0.761113\pi$
$44$	−19.4164	−2.92713
$45$	0	0
$46$	−8.94427	−1.31876
$47$	− 12.9443i	− 1.88812i	−0.329779	−	0.944058i	$-0.606974\pi$
$47$	− 12.9443i	− 1.88812i	0.329779	−	0.944058i	$-0.393026\pi$
$48$	1.00000i	0.144338i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	− 13.4164i	− 1.86052i
$53$	− 3.52786i	− 0.484589i	−0.970203	−	0.242295i	$-0.922100\pi$
$53$	− 3.52786i	− 0.484589i	0.970203	−	0.242295i	$-0.0779001\pi$
$54$	−2.23607	−0.304290
$55$	0	0
$56$	−2.23607	−0.298807
$57$	− 2.47214i	− 0.327442i
$58$	4.47214i	0.587220i
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.41641i	0.433884i
$63$	1.00000i	0.125988i
$64$	13.0000	1.62500
$65$	0	0
$66$	14.4721	1.78140
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 6.00000i	− 0.727607i
$69$	4.00000	0.481543
$70$	0	0
$71$	5.52786	0.656037	0.328018	−	0.944671i	$-0.393619\pi$
$71$	5.52786	0.656037	0.328018	+	0.944671i	$0.393619\pi$
$72$	2.23607i	0.263523i
$73$	− 12.4721i	− 1.45975i	−0.683579	−	0.729877i	$-0.739578\pi$
$73$	− 12.4721i	− 1.45975i	0.683579	−	0.729877i	$-0.260422\pi$
$74$	−15.5279	−1.80508
$75$	0	0
$76$	−7.41641	−0.850720
$77$	− 6.47214i	− 0.737568i
$78$	10.0000i	1.13228i
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 4.47214i	− 0.493865i
$83$	− 16.9443i	− 1.85988i	−0.367717	−	0.929938i	$-0.619860\pi$
$83$	− 16.9443i	− 1.85988i	0.367717	−	0.929938i	$-0.380140\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−20.0000	−2.15666
$87$	− 2.00000i	− 0.214423i
$88$	− 14.4721i	− 1.54273i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	− 12.0000i	− 1.25109i
$93$	− 1.52786i	− 0.158432i
$94$	28.9443	2.98537
$95$	0	0
$96$	−6.70820	−0.684653
$97$	− 8.47214i	− 0.860215i	−0.902778	−	0.430108i	$-0.858476\pi$
$97$	− 8.47214i	− 0.860215i	0.902778	−	0.430108i	$-0.141524\pi$
$98$	− 2.23607i	− 0.225877i
$99$	−6.47214	−0.650474
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	4.47214i	0.442807i
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	10.0000	0.980581
$105$	0	0
$106$	7.88854	0.766203
$107$	12.9443i	1.25137i	0.780076	+	0.625685i	$0.215180\pi$
$107$	12.9443i	1.25137i	−0.780076	+	0.625685i	$0.784820\pi$
$108$	− 3.00000i	− 0.288675i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	6.94427	0.659121
$112$	1.00000i	0.0944911i
$113$	0.472136i	0.0444148i	0.999753	+	0.0222074i	$0.00706942\pi$
$113$	0.472136i	0.0444148i	−0.999753	+	0.0222074i	$0.992931\pi$
$114$	5.52786	0.517732
$115$	0	0
$116$	−6.00000	−0.557086
$117$	− 4.47214i	− 0.413449i
$118$	20.0000i	1.84115i
$119$	2.00000	0.183340
$120$	0	0
$121$	30.8885	2.80805
$122$	− 4.47214i	− 0.404888i
$123$	2.00000i	0.180334i
$124$	−4.58359	−0.411619
$125$	0	0
$126$	−2.23607	−0.199205
$127$	4.94427i	0.438733i	0.975643	+	0.219367i	$0.0703991\pi$
$127$	4.94427i	0.438733i	−0.975643	+	0.219367i	$0.929601\pi$
$128$	15.6525i	1.38350i
$129$	8.94427	0.787499
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	19.4164i	1.68998i
$133$	− 2.47214i	− 0.214361i
$134$	−8.94427	−0.772667
$135$	0	0
$136$	4.47214	0.383482
$137$	− 3.52786i	− 0.301406i	−0.988579	−	0.150703i	$-0.951846\pi$
$137$	− 3.52786i	− 0.301406i	0.988579	−	0.150703i	$-0.0481537\pi$
$138$	8.94427i	0.761387i
$139$	−7.41641	−0.629052	−0.314526	−	0.949249i	$-0.601845\pi$
$139$	−7.41641	−0.629052	−0.314526	+	0.949249i	$0.601845\pi$
$140$	0	0
$141$	−12.9443	−1.09010
$142$	12.3607i	1.03729i
$143$	28.9443i	2.42044i
$144$	1.00000	0.0833333
$145$	0	0
$146$	27.8885	2.30807
$147$	1.00000i	0.0824786i
$148$	− 20.8328i	− 1.71245i
$149$	14.9443	1.22428	0.612141	−	0.790748i	$-0.290308\pi$
$149$	14.9443	1.22428	0.612141	+	0.790748i	$0.290308\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 5.52786i	− 0.448369i
$153$	− 2.00000i	− 0.161690i
$154$	14.4721	1.16620
$155$	0	0
$156$	−13.4164	−1.07417
$157$	0.472136i	0.0376806i	0.999823	+	0.0188403i	$0.00599740\pi$
$157$	0.472136i	0.0376806i	−0.999823	+	0.0188403i	$0.994003\pi$
$158$	− 28.9443i	− 2.30268i
$159$	−3.52786	−0.279778
$160$	0	0
$161$	4.00000	0.315244
$162$	2.23607i	0.175682i
$163$	− 16.9443i	− 1.32718i	−0.748097	−	0.663589i	$-0.769032\pi$
$163$	− 16.9443i	− 1.32718i	0.748097	−	0.663589i	$-0.230968\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	37.8885	2.94072
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	2.23607i	0.172516i
$169$	−7.00000	−0.538462
$170$	0	0
$171$	−2.47214	−0.189049
$172$	− 26.8328i	− 2.04598i
$173$	− 2.94427i	− 0.223849i	−0.993717	−	0.111924i	$-0.964299\pi$
$173$	− 2.94427i	− 0.223849i	0.993717	−	0.111924i	$-0.0357015\pi$
$174$	4.47214	0.339032
$175$	0	0
$176$	−6.47214	−0.487856
$177$	− 8.94427i	− 0.672293i
$178$	4.47214i	0.335201i
$179$	−6.47214	−0.483750	−0.241875	−	0.970307i	$-0.577762\pi$
$179$	−6.47214	−0.483750	−0.241875	+	0.970307i	$0.577762\pi$
$180$	0	0
$181$	1.05573	0.0784717	0.0392358	−	0.999230i	$-0.487508\pi$
$181$	1.05573	0.0784717	0.0392358	+	0.999230i	$0.487508\pi$
$182$	10.0000i	0.741249i
$183$	2.00000i	0.147844i
$184$	8.94427	0.659380
$185$	0	0
$186$	3.41641	0.250503
$187$	12.9443i	0.946579i
$188$	38.8328i	2.83217i
$189$	1.00000	0.0727393
$190$	0	0
$191$	0.583592	0.0422272	0.0211136	−	0.999777i	$-0.493279\pi$
$191$	0.583592	0.0422272	0.0211136	+	0.999777i	$0.493279\pi$
$192$	− 13.0000i	− 0.938194i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	18.9443	1.36012
$195$	0	0
$196$	3.00000	0.214286
$197$	− 15.5279i	− 1.10631i	−0.833077	−	0.553157i	$-0.813423\pi$
$197$	− 15.5279i	− 1.10631i	0.833077	−	0.553157i	$-0.186577\pi$
$198$	− 14.4721i	− 1.02849i
$199$	−27.4164	−1.94350	−0.971749	−	0.236017i	$-0.924158\pi$
$199$	−27.4164	−1.94350	−0.971749	+	0.236017i	$0.924158\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	− 31.3050i	− 2.20261i
$203$	− 2.00000i	− 0.140372i
$204$	−6.00000	−0.420084
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	− 4.47214i	− 0.310087i
$209$	16.0000	1.10674
$210$	0	0
$211$	−16.9443	−1.16649	−0.583246	−	0.812296i	$-0.698218\pi$
$211$	−16.9443	−1.16649	−0.583246	+	0.812296i	$0.698218\pi$
$212$	10.5836i	0.726884i
$213$	− 5.52786i	− 0.378763i
$214$	−28.9443	−1.97859
$215$	0	0
$216$	2.23607	0.152145
$217$	− 1.52786i	− 0.103718i
$218$	4.47214i	0.302891i
$219$	−12.4721	−0.842789
$220$	0	0
$221$	−8.94427	−0.601657
$222$	15.5279i	1.04216i
$223$	− 12.9443i	− 0.866813i	−0.901199	−	0.433406i	$-0.857312\pi$
$223$	− 12.9443i	− 0.866813i	0.901199	−	0.433406i	$-0.142688\pi$
$224$	−6.70820	−0.448211
$225$	0	0
$226$	−1.05573	−0.0702260
$227$	− 0.944272i	− 0.0626735i	−0.999509	−	0.0313368i	$-0.990024\pi$
$227$	− 0.944272i	− 0.0626735i	0.999509	−	0.0313368i	$-0.00997644\pi$
$228$	7.41641i	0.491164i
$229$	−23.8885	−1.57860	−0.789300	−	0.614008i	$-0.789556\pi$
$229$	−23.8885	−1.57860	−0.789300	+	0.614008i	$0.789556\pi$
$230$	0	0
$231$	−6.47214	−0.425835
$232$	− 4.47214i	− 0.293610i
$233$	− 9.41641i	− 0.616889i	−0.951242	−	0.308445i	$-0.900192\pi$
$233$	− 9.41641i	− 0.616889i	0.951242	−	0.308445i	$-0.0998085\pi$
$234$	10.0000	0.653720
$235$	0	0
$236$	−26.8328	−1.74667
$237$	12.9443i	0.840821i
$238$	4.47214i	0.289886i
$239$	10.4721	0.677386	0.338693	−	0.940897i	$-0.390015\pi$
$239$	10.4721	0.677386	0.338693	+	0.940897i	$0.390015\pi$
$240$	0	0
$241$	−18.9443	−1.22031	−0.610154	−	0.792283i	$-0.708892\pi$
$241$	−18.9443	−1.22031	−0.610154	+	0.792283i	$0.708892\pi$
$242$	69.0689i	4.43992i
$243$	− 1.00000i	− 0.0641500i
$244$	6.00000	0.384111
$245$	0	0
$246$	−4.47214	−0.285133
$247$	11.0557i	0.703459i
$248$	− 3.41641i	− 0.216942i
$249$	−16.9443	−1.07380
$250$	0	0
$251$	16.9443	1.06951	0.534756	−	0.845006i	$-0.320403\pi$
$251$	16.9443	1.06951	0.534756	+	0.845006i	$0.320403\pi$
$252$	− 3.00000i	− 0.188982i
$253$	25.8885i	1.62760i
$254$	−11.0557	−0.693698
$255$	0	0
$256$	−9.00000	−0.562500
$257$	− 18.9443i	− 1.18171i	−0.806777	−	0.590856i	$-0.798790\pi$
$257$	− 18.9443i	− 1.18171i	0.806777	−	0.590856i	$-0.201210\pi$
$258$	20.0000i	1.24515i
$259$	6.94427	0.431496
$260$	0	0
$261$	−2.00000	−0.123797
$262$	8.94427i	0.552579i
$263$	7.05573i	0.435075i	0.976052	+	0.217537i	$0.0698024\pi$
$263$	7.05573i	0.435075i	−0.976052	+	0.217537i	$0.930198\pi$
$264$	−14.4721	−0.890698
$265$	0	0
$266$	5.52786	0.338935
$267$	− 2.00000i	− 0.122398i
$268$	− 12.0000i	− 0.733017i
$269$	−11.8885	−0.724857	−0.362429	−	0.932012i	$-0.618052\pi$
$269$	−11.8885	−0.724857	−0.362429	+	0.932012i	$0.618052\pi$
$270$	0	0
$271$	−1.52786	−0.0928111	−0.0464056	−	0.998923i	$-0.514777\pi$
$271$	−1.52786	−0.0928111	−0.0464056	+	0.998923i	$0.514777\pi$
$272$	− 2.00000i	− 0.121268i
$273$	− 4.47214i	− 0.270666i
$274$	7.88854	0.476564
$275$	0	0
$276$	−12.0000	−0.722315
$277$	− 18.9443i	− 1.13825i	−0.822251	−	0.569125i	$-0.807282\pi$
$277$	− 18.9443i	− 1.13825i	0.822251	−	0.569125i	$-0.192718\pi$
$278$	− 16.5836i	− 0.994618i
$279$	−1.52786	−0.0914708
$280$	0	0
$281$	−10.9443	−0.652881	−0.326440	−	0.945218i	$-0.605849\pi$
$281$	−10.9443	−0.652881	−0.326440	+	0.945218i	$0.605849\pi$
$282$	− 28.9443i	− 1.72361i
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	−16.5836	−0.984055
$285$	0	0
$286$	−64.7214	−3.82705
$287$	2.00000i	0.118056i
$288$	6.70820i	0.395285i
$289$	13.0000	0.764706
$290$	0	0
$291$	−8.47214	−0.496645
$292$	37.4164i	2.18963i
$293$	5.05573i	0.295359i	0.989035	+	0.147679i	$0.0471804\pi$
$293$	5.05573i	0.295359i	−0.989035	+	0.147679i	$0.952820\pi$
$294$	−2.23607	−0.130410
$295$	0	0
$296$	15.5279	0.902539
$297$	6.47214i	0.375551i
$298$	33.4164i	1.93576i
$299$	−17.8885	−1.03452
$300$	0	0
$301$	8.94427	0.515539
$302$	− 35.7771i	− 2.05874i
$303$	14.0000i	0.804279i
$304$	−2.47214	−0.141787
$305$	0	0
$306$	4.47214	0.255655
$307$	15.0557i	0.859276i	0.903001	+	0.429638i	$0.141359\pi$
$307$	15.0557i	0.859276i	−0.903001	+	0.429638i	$0.858641\pi$
$308$	19.4164i	1.10635i
$309$	0	0
$310$	0	0
$311$	25.8885	1.46800	0.734002	−	0.679147i	$-0.237650\pi$
$311$	25.8885	1.46800	0.734002	+	0.679147i	$0.237650\pi$
$312$	− 10.0000i	− 0.566139i
$313$	− 17.4164i	− 0.984434i	−0.870473	−	0.492217i	$-0.836187\pi$
$313$	− 17.4164i	− 0.984434i	0.870473	−	0.492217i	$-0.163813\pi$
$314$	−1.05573	−0.0595782
$315$	0	0
$316$	38.8328	2.18452
$317$	− 14.3607i	− 0.806576i	−0.915073	−	0.403288i	$-0.867867\pi$
$317$	− 14.3607i	− 0.806576i	0.915073	−	0.403288i	$-0.132133\pi$
$318$	− 7.88854i	− 0.442368i
$319$	12.9443	0.724740
$320$	0	0
$321$	12.9443	0.722479
$322$	8.94427i	0.498445i
$323$	4.94427i	0.275107i
$324$	−3.00000	−0.166667
$325$	0	0
$326$	37.8885	2.09845
$327$	− 2.00000i	− 0.110600i
$328$	4.47214i	0.246932i
$329$	−12.9443	−0.713641
$330$	0	0
$331$	0.944272	0.0519019	0.0259509	−	0.999663i	$-0.491739\pi$
$331$	0.944272	0.0519019	0.0259509	+	0.999663i	$0.491739\pi$
$332$	50.8328i	2.78981i
$333$	− 6.94427i	− 0.380544i
$334$	−17.8885	−0.978818
$335$	0	0
$336$	1.00000	0.0545545
$337$	23.8885i	1.30129i	0.759381	+	0.650646i	$0.225502\pi$
$337$	23.8885i	1.30129i	−0.759381	+	0.650646i	$0.774498\pi$
$338$	− 15.6525i	− 0.851382i
$339$	0.472136	0.0256429
$340$	0	0
$341$	9.88854	0.535495
$342$	− 5.52786i	− 0.298913i
$343$	1.00000i	0.0539949i
$344$	20.0000	1.07833
$345$	0	0
$346$	6.58359	0.353936
$347$	8.00000i	0.429463i	0.976673	+	0.214731i	$0.0688876\pi$
$347$	8.00000i	0.429463i	−0.976673	+	0.214731i	$0.931112\pi$
$348$	6.00000i	0.321634i
$349$	11.8885	0.636379	0.318190	−	0.948027i	$-0.396925\pi$
$349$	11.8885	0.636379	0.318190	+	0.948027i	$0.396925\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	− 43.4164i	− 2.31410i
$353$	7.88854i	0.419865i	0.977716	+	0.209932i	$0.0673244\pi$
$353$	7.88854i	0.419865i	−0.977716	+	0.209932i	$0.932676\pi$
$354$	20.0000	1.06299
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 2.00000i	− 0.105851i
$358$	− 14.4721i	− 0.764876i
$359$	−18.4721	−0.974922	−0.487461	−	0.873145i	$-0.662077\pi$
$359$	−18.4721	−0.974922	−0.487461	+	0.873145i	$0.662077\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	2.36068i	0.124075i
$363$	− 30.8885i	− 1.62123i
$364$	−13.4164	−0.703211
$365$	0	0
$366$	−4.47214	−0.233762
$367$	− 3.05573i	− 0.159508i	−0.996815	−	0.0797539i	$-0.974587\pi$
$367$	− 3.05573i	− 0.159508i	0.996815	−	0.0797539i	$-0.0254134\pi$
$368$	− 4.00000i	− 0.208514i
$369$	2.00000	0.104116
$370$	0	0
$371$	−3.52786	−0.183158
$372$	4.58359i	0.237648i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	−28.9443	−1.49667
$375$	0	0
$376$	−28.9443	−1.49269
$377$	8.94427i	0.460653i
$378$	2.23607i	0.115011i
$379$	37.8885	1.94620	0.973102	−	0.230375i	$-0.0739953\pi$
$379$	37.8885	1.94620	0.973102	+	0.230375i	$0.0739953\pi$
$380$	0	0
$381$	4.94427	0.253303
$382$	1.30495i	0.0667671i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	15.6525	0.798762
$385$	0	0
$386$	31.3050	1.59338
$387$	− 8.94427i	− 0.454663i
$388$	25.4164i	1.29032i
$389$	6.94427	0.352089	0.176044	−	0.984382i	$-0.443670\pi$
$389$	6.94427	0.352089	0.176044	+	0.984382i	$0.443670\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	2.23607i	0.112938i
$393$	− 4.00000i	− 0.201773i
$394$	34.7214	1.74924
$395$	0	0
$396$	19.4164	0.975711
$397$	13.4164i	0.673350i	0.941621	+	0.336675i	$0.109302\pi$
$397$	13.4164i	0.673350i	−0.941621	+	0.336675i	$0.890698\pi$
$398$	− 61.3050i	− 3.07294i
$399$	−2.47214	−0.123762
$400$	0	0
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	8.94427i	0.446100i
$403$	6.83282i	0.340367i
$404$	42.0000	2.08958
$405$	0	0
$406$	4.47214	0.221948
$407$	44.9443i	2.22780i
$408$	− 4.47214i	− 0.221404i
$409$	−11.8885	−0.587851	−0.293925	−	0.955828i	$-0.594962\pi$
$409$	−11.8885	−0.587851	−0.293925	+	0.955828i	$0.594962\pi$
$410$	0	0
$411$	−3.52786	−0.174017
$412$	0	0
$413$	− 8.94427i	− 0.440119i
$414$	8.94427	0.439587
$415$	0	0
$416$	30.0000	1.47087
$417$	7.41641i	0.363183i
$418$	35.7771i	1.74991i
$419$	29.8885	1.46015	0.730075	−	0.683367i	$-0.239485\pi$
$419$	29.8885	1.46015	0.730075	+	0.683367i	$0.239485\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 37.8885i	− 1.84439i
$423$	12.9443i	0.629372i
$424$	−7.88854	−0.383102
$425$	0	0
$426$	12.3607	0.598877
$427$	2.00000i	0.0967868i
$428$	− 38.8328i	− 1.87705i
$429$	28.9443	1.39744
$430$	0	0
$431$	−18.4721	−0.889771	−0.444886	−	0.895587i	$-0.646756\pi$
$431$	−18.4721	−0.889771	−0.444886	+	0.895587i	$0.646756\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	16.4721i	0.791600i	0.918337	+	0.395800i	$0.129533\pi$
$433$	16.4721i	0.791600i	−0.918337	+	0.395800i	$0.870467\pi$
$434$	3.41641	0.163993
$435$	0	0
$436$	−6.00000	−0.287348
$437$	9.88854i	0.473033i
$438$	− 27.8885i	− 1.33257i
$439$	−1.52786	−0.0729210	−0.0364605	−	0.999335i	$-0.511608\pi$
$439$	−1.52786	−0.0729210	−0.0364605	+	0.999335i	$0.511608\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	− 20.0000i	− 0.951303i
$443$	− 8.00000i	− 0.380091i	−0.981775	−	0.190046i	$-0.939136\pi$
$443$	− 8.00000i	− 0.380091i	0.981775	−	0.190046i	$-0.0608636\pi$
$444$	−20.8328	−0.988682
$445$	0	0
$446$	28.9443	1.37055
$447$	− 14.9443i	− 0.706840i
$448$	− 13.0000i	− 0.614192i
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	−12.9443	−0.609522
$452$	− 1.41641i	− 0.0666222i
$453$	16.0000i	0.751746i
$454$	2.11146	0.0990955
$455$	0	0
$456$	−5.52786	−0.258866
$457$	− 6.94427i	− 0.324839i	−0.986722	−	0.162420i	$-0.948070\pi$
$457$	− 6.94427i	− 0.324839i	0.986722	−	0.162420i	$-0.0519298\pi$
$458$	− 53.4164i	− 2.49598i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	3.88854	0.181108	0.0905538	−	0.995892i	$-0.471136\pi$
$461$	3.88854	0.181108	0.0905538	+	0.995892i	$0.471136\pi$
$462$	− 14.4721i	− 0.673305i
$463$	20.9443i	0.973363i	0.873580	+	0.486681i	$0.161793\pi$
$463$	20.9443i	0.973363i	−0.873580	+	0.486681i	$0.838207\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	21.0557	0.975388
$467$	8.94427i	0.413892i	0.978352	+	0.206946i	$0.0663524\pi$
$467$	8.94427i	0.413892i	−0.978352	+	0.206946i	$0.933648\pi$
$468$	13.4164i	0.620174i
$469$	4.00000	0.184703
$470$	0	0
$471$	0.472136	0.0217549
$472$	− 20.0000i	− 0.920575i
$473$	57.8885i	2.66172i
$474$	−28.9443	−1.32945
$475$	0	0
$476$	−6.00000	−0.275010
$477$	3.52786i	0.161530i
$478$	23.4164i	1.07104i
$479$	17.8885	0.817348	0.408674	−	0.912680i	$-0.365991\pi$
$479$	17.8885	0.817348	0.408674	+	0.912680i	$0.365991\pi$
$480$	0	0
$481$	−31.0557	−1.41602
$482$	− 42.3607i	− 1.92948i
$483$	− 4.00000i	− 0.182006i
$484$	−92.6656	−4.21207
$485$	0	0
$486$	2.23607	0.101430
$487$	20.9443i	0.949076i	0.880235	+	0.474538i	$0.157385\pi$
$487$	20.9443i	0.949076i	−0.880235	+	0.474538i	$0.842615\pi$
$488$	4.47214i	0.202444i
$489$	−16.9443	−0.766246
$490$	0	0
$491$	−21.3050	−0.961479	−0.480740	−	0.876863i	$-0.659632\pi$
$491$	−21.3050	−0.961479	−0.480740	+	0.876863i	$0.659632\pi$
$492$	− 6.00000i	− 0.270501i
$493$	4.00000i	0.180151i
$494$	−24.7214	−1.11227
$495$	0	0
$496$	−1.52786	−0.0686031
$497$	− 5.52786i	− 0.247959i
$498$	− 37.8885i	− 1.69783i
$499$	13.8885	0.621737	0.310868	−	0.950453i	$-0.399380\pi$
$499$	13.8885	0.621737	0.310868	+	0.950453i	$0.399380\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	37.8885i	1.69105i
$503$	32.0000i	1.42681i	0.700752	+	0.713405i	$0.252848\pi$
$503$	32.0000i	1.42681i	−0.700752	+	0.713405i	$0.747152\pi$
$504$	2.23607	0.0996024
$505$	0	0
$506$	−57.8885	−2.57346
$507$	7.00000i	0.310881i
$508$	− 14.8328i	− 0.658100i
$509$	23.8885	1.05884	0.529421	−	0.848360i	$-0.322409\pi$
$509$	23.8885	1.05884	0.529421	+	0.848360i	$0.322409\pi$
$510$	0	0
$511$	−12.4721	−0.551735
$512$	11.1803i	0.494106i
$513$	2.47214i	0.109147i
$514$	42.3607	1.86845
$515$	0	0
$516$	−26.8328	−1.18125
$517$	− 83.7771i	− 3.68451i
$518$	15.5279i	0.682255i
$519$	−2.94427	−0.129239
$520$	0	0
$521$	−19.8885	−0.871333	−0.435666	−	0.900108i	$-0.643487\pi$
$521$	−19.8885	−0.871333	−0.435666	+	0.900108i	$0.643487\pi$
$522$	− 4.47214i	− 0.195740i
$523$	− 8.94427i	− 0.391106i	−0.980693	−	0.195553i	$-0.937350\pi$
$523$	− 8.94427i	− 0.391106i	0.980693	−	0.195553i	$-0.0626501\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−15.7771	−0.687914
$527$	3.05573i	0.133110i
$528$	6.47214i	0.281664i
$529$	7.00000	0.304348
$530$	0	0
$531$	−8.94427	−0.388148
$532$	7.41641i	0.321542i
$533$	− 8.94427i	− 0.387419i
$534$	4.47214	0.193528
$535$	0	0
$536$	8.94427	0.386334
$537$	6.47214i	0.279293i
$538$	− 26.5836i	− 1.14610i
$539$	−6.47214	−0.278775
$540$	0	0
$541$	−11.8885	−0.511128	−0.255564	−	0.966792i	$-0.582261\pi$
$541$	−11.8885	−0.511128	−0.255564	+	0.966792i	$0.582261\pi$
$542$	− 3.41641i	− 0.146747i
$543$	− 1.05573i	− 0.0453056i
$544$	13.4164	0.575224
$545$	0	0
$546$	10.0000	0.427960
$547$	− 5.88854i	− 0.251776i	−0.992044	−	0.125888i	$-0.959822\pi$
$547$	− 5.88854i	− 0.251776i	0.992044	−	0.125888i	$-0.0401780\pi$
$548$	10.5836i	0.452109i
$549$	2.00000	0.0853579
$550$	0	0
$551$	4.94427	0.210633
$552$	− 8.94427i	− 0.380693i
$553$	12.9443i	0.550446i
$554$	42.3607	1.79973
$555$	0	0
$556$	22.2492	0.943577
$557$	− 20.4721i	− 0.867432i	−0.901050	−	0.433716i	$-0.857202\pi$
$557$	− 20.4721i	− 0.867432i	0.901050	−	0.433716i	$-0.142798\pi$
$558$	− 3.41641i	− 0.144628i
$559$	−40.0000	−1.69182
$560$	0	0
$561$	12.9443	0.546508
$562$	− 24.4721i	− 1.03229i
$563$	13.8885i	0.585332i	0.956215	+	0.292666i	$0.0945425\pi$
$563$	13.8885i	0.585332i	−0.956215	+	0.292666i	$0.905458\pi$
$564$	38.8328	1.63516
$565$	0	0
$566$	−26.8328	−1.12787
$567$	− 1.00000i	− 0.0419961i
$568$	− 12.3607i	− 0.518643i
$569$	39.8885	1.67221	0.836107	−	0.548566i	$-0.184826\pi$
$569$	39.8885	1.67221	0.836107	+	0.548566i	$0.184826\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 86.8328i	− 3.63066i
$573$	− 0.583592i	− 0.0243799i
$574$	−4.47214	−0.186663
$575$	0	0
$576$	−13.0000	−0.541667
$577$	− 10.3607i	− 0.431321i	−0.976468	−	0.215660i	$-0.930810\pi$
$577$	− 10.3607i	− 0.431321i	0.976468	−	0.215660i	$-0.0691904\pi$
$578$	29.0689i	1.20911i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−16.9443	−0.702967
$582$	− 18.9443i	− 0.785265i
$583$	− 22.8328i	− 0.945639i
$584$	−27.8885	−1.15404
$585$	0	0
$586$	−11.3050	−0.467003
$587$	4.00000i	0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$	4.00000i	0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	− 3.00000i	− 0.123718i
$589$	3.77709	0.155632
$590$	0	0
$591$	−15.5279	−0.638731
$592$	− 6.94427i	− 0.285408i
$593$	23.8885i	0.980985i	0.871445	+	0.490492i	$0.163183\pi$
$593$	23.8885i	0.980985i	−0.871445	+	0.490492i	$0.836817\pi$
$594$	−14.4721	−0.593799
$595$	0	0
$596$	−44.8328	−1.83642
$597$	27.4164i	1.12208i
$598$	− 40.0000i	− 1.63572i
$599$	−12.3607	−0.505044	−0.252522	−	0.967591i	$-0.581260\pi$
$599$	−12.3607	−0.505044	−0.252522	+	0.967591i	$0.581260\pi$
$600$	0	0
$601$	38.9443	1.58857	0.794285	−	0.607545i	$-0.207846\pi$
$601$	38.9443	1.58857	0.794285	+	0.607545i	$0.207846\pi$
$602$	20.0000i	0.815139i
$603$	− 4.00000i	− 0.162893i
$604$	48.0000	1.95309
$605$	0	0
$606$	−31.3050	−1.27168
$607$	− 38.8328i	− 1.57618i	−0.615563	−	0.788088i	$-0.711071\pi$
$607$	− 38.8328i	− 1.57618i	0.615563	−	0.788088i	$-0.288929\pi$
$608$	− 16.5836i	− 0.672553i
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	57.8885	2.34192
$612$	6.00000i	0.242536i
$613$	− 6.94427i	− 0.280477i	−0.990118	−	0.140238i	$-0.955213\pi$
$613$	− 6.94427i	− 0.280477i	0.990118	−	0.140238i	$-0.0447868\pi$
$614$	−33.6656	−1.35863
$615$	0	0
$616$	−14.4721	−0.583099
$617$	− 16.4721i	− 0.663143i	−0.943430	−	0.331572i	$-0.892421\pi$
$617$	− 16.4721i	− 0.663143i	0.943430	−	0.331572i	$-0.107579\pi$
$618$	0	0
$619$	−39.4164	−1.58428	−0.792140	−	0.610340i	$-0.791033\pi$
$619$	−39.4164	−1.58428	−0.792140	+	0.610340i	$0.791033\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	57.8885i	2.32112i
$623$	− 2.00000i	− 0.0801283i
$624$	−4.47214	−0.179029
$625$	0	0
$626$	38.9443	1.55653
$627$	− 16.0000i	− 0.638978i
$628$	− 1.41641i	− 0.0565208i
$629$	−13.8885	−0.553773
$630$	0	0
$631$	30.8328	1.22744	0.613718	−	0.789526i	$-0.289673\pi$
$631$	30.8328	1.22744	0.613718	+	0.789526i	$0.289673\pi$
$632$	28.9443i	1.15134i
$633$	16.9443i	0.673474i
$634$	32.1115	1.27531
$635$	0	0
$636$	10.5836	0.419667
$637$	− 4.47214i	− 0.177192i
$638$	28.9443i	1.14591i
$639$	−5.52786	−0.218679
$640$	0	0
$641$	16.8328	0.664856	0.332428	−	0.943129i	$-0.392132\pi$
$641$	16.8328	0.664856	0.332428	+	0.943129i	$0.392132\pi$
$642$	28.9443i	1.14234i
$643$	− 15.0557i	− 0.593740i	−0.954918	−	0.296870i	$-0.904057\pi$
$643$	− 15.0557i	− 0.593740i	0.954918	−	0.296870i	$-0.0959428\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−11.0557	−0.434982
$647$	1.88854i	0.0742463i	0.999311	+	0.0371232i	$0.0118194\pi$
$647$	1.88854i	0.0742463i	−0.999311	+	0.0371232i	$0.988181\pi$
$648$	− 2.23607i	− 0.0878410i
$649$	57.8885	2.27232
$650$	0	0
$651$	−1.52786	−0.0598817
$652$	50.8328i	1.99077i
$653$	− 22.5836i	− 0.883764i	−0.897073	−	0.441882i	$-0.854311\pi$
$653$	− 22.5836i	− 0.883764i	0.897073	−	0.441882i	$-0.145689\pi$
$654$	4.47214	0.174874
$655$	0	0
$656$	2.00000	0.0780869
$657$	12.4721i	0.486584i
$658$	− 28.9443i	− 1.12837i
$659$	−21.3050	−0.829923	−0.414962	−	0.909839i	$-0.636205\pi$
$659$	−21.3050	−0.829923	−0.414962	+	0.909839i	$0.636205\pi$
$660$	0	0
$661$	−35.8885	−1.39590	−0.697951	−	0.716145i	$-0.745905\pi$
$661$	−35.8885	−1.39590	−0.697951	+	0.716145i	$0.745905\pi$
$662$	2.11146i	0.0820641i
$663$	8.94427i	0.347367i
$664$	−37.8885	−1.47036
$665$	0	0
$666$	15.5279	0.601693
$667$	8.00000i	0.309761i
$668$	− 24.0000i	− 0.928588i
$669$	−12.9443	−0.500454
$670$	0	0
$671$	−12.9443	−0.499708
$672$	6.70820i	0.258775i
$673$	8.83282i	0.340480i	0.985403	+	0.170240i	$0.0544543\pi$
$673$	8.83282i	0.340480i	−0.985403	+	0.170240i	$0.945546\pi$
$674$	−53.4164	−2.05752
$675$	0	0
$676$	21.0000	0.807692
$677$	− 21.0557i	− 0.809237i	−0.914485	−	0.404619i	$-0.867404\pi$
$677$	− 21.0557i	− 0.809237i	0.914485	−	0.404619i	$-0.132596\pi$
$678$	1.05573i	0.0405450i
$679$	−8.47214	−0.325131
$680$	0	0
$681$	−0.944272	−0.0361846
$682$	22.1115i	0.846691i
$683$	− 1.88854i	− 0.0722631i	−0.999347	−	0.0361316i	$-0.988496\pi$
$683$	− 1.88854i	− 0.0722631i	0.999347	−	0.0361316i	$-0.0115035\pi$
$684$	7.41641	0.283573
$685$	0	0
$686$	−2.23607	−0.0853735
$687$	23.8885i	0.911405i
$688$	− 8.94427i	− 0.340997i
$689$	15.7771	0.601059
$690$	0	0
$691$	44.3607	1.68756	0.843780	−	0.536689i	$-0.180325\pi$
$691$	44.3607	1.68756	0.843780	+	0.536689i	$0.180325\pi$
$692$	8.83282i	0.335773i
$693$	6.47214i	0.245856i
$694$	−17.8885	−0.679040
$695$	0	0
$696$	−4.47214	−0.169516
$697$	− 4.00000i	− 0.151511i
$698$	26.5836i	1.00620i
$699$	−9.41641	−0.356161
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	− 10.0000i	− 0.377426i
$703$	17.1672i	0.647473i
$704$	84.1378	3.17106
$705$	0	0
$706$	−17.6393	−0.663865
$707$	14.0000i	0.526524i
$708$	26.8328i	1.00844i
$709$	−25.7771	−0.968079	−0.484039	−	0.875046i	$-0.660831\pi$
$709$	−25.7771	−0.968079	−0.484039	+	0.875046i	$0.660831\pi$
$710$	0	0
$711$	12.9443	0.485448
$712$	− 4.47214i	− 0.167600i
$713$	6.11146i	0.228876i
$714$	4.47214	0.167365
$715$	0	0
$716$	19.4164	0.725625
$717$	− 10.4721i	− 0.391089i
$718$	− 41.3050i	− 1.54149i
$719$	6.83282	0.254821	0.127411	−	0.991850i	$-0.459333\pi$
$719$	6.83282	0.254821	0.127411	+	0.991850i	$0.459333\pi$
$720$	0	0
$721$	0	0
$722$	− 28.8197i	− 1.07256i
$723$	18.9443i	0.704545i
$724$	−3.16718	−0.117707
$725$	0	0
$726$	69.0689	2.56339
$727$	38.8328i	1.44023i	0.693855	+	0.720115i	$0.255911\pi$
$727$	38.8328i	1.44023i	−0.693855	+	0.720115i	$0.744089\pi$
$728$	− 10.0000i	− 0.370625i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−17.8885	−0.661632
$732$	− 6.00000i	− 0.221766i
$733$	10.5836i	0.390914i	0.980712	+	0.195457i	$0.0626190\pi$
$733$	10.5836i	0.390914i	−0.980712	+	0.195457i	$0.937381\pi$
$734$	6.83282	0.252204
$735$	0	0
$736$	26.8328	0.989071
$737$	25.8885i	0.953617i
$738$	4.47214i	0.164622i
$739$	5.88854	0.216614	0.108307	−	0.994118i	$-0.465457\pi$
$739$	5.88854	0.216614	0.108307	+	0.994118i	$0.465457\pi$
$740$	0	0
$741$	11.0557	0.406142
$742$	− 7.88854i	− 0.289598i
$743$	34.8328i	1.27789i	0.769252	+	0.638946i	$0.220629\pi$
$743$	34.8328i	1.27789i	−0.769252	+	0.638946i	$0.779371\pi$
$744$	−3.41641	−0.125252
$745$	0	0
$746$	−13.4164	−0.491210
$747$	16.9443i	0.619958i
$748$	− 38.8328i	− 1.41987i
$749$	12.9443	0.472973
$750$	0	0
$751$	−20.9443	−0.764267	−0.382134	−	0.924107i	$-0.624811\pi$
$751$	−20.9443	−0.764267	−0.382134	+	0.924107i	$0.624811\pi$
$752$	12.9443i	0.472029i
$753$	− 16.9443i	− 0.617484i
$754$	−20.0000	−0.728357
$755$	0	0
$756$	−3.00000	−0.109109
$757$	− 31.8885i	− 1.15901i	−0.814969	−	0.579504i	$-0.803246\pi$
$757$	− 31.8885i	− 1.15901i	0.814969	−	0.579504i	$-0.196754\pi$
$758$	84.7214i	3.07722i
$759$	25.8885	0.939695
$760$	0	0
$761$	−27.8885	−1.01096	−0.505479	−	0.862839i	$-0.668684\pi$
$761$	−27.8885	−1.01096	−0.505479	+	0.862839i	$0.668684\pi$
$762$	11.0557i	0.400507i
$763$	− 2.00000i	− 0.0724049i
$764$	−1.75078	−0.0633409
$765$	0	0
$766$	17.8885	0.646339
$767$	40.0000i	1.44432i
$768$	9.00000i	0.324760i
$769$	52.8328	1.90520	0.952600	−	0.304226i	$-0.0983976\pi$
$769$	52.8328	1.90520	0.952600	+	0.304226i	$0.0983976\pi$
$770$	0	0
$771$	−18.9443	−0.682261
$772$	42.0000i	1.51161i
$773$	− 42.9443i	− 1.54460i	−0.635259	−	0.772299i	$-0.719107\pi$
$773$	− 42.9443i	− 1.54460i	0.635259	−	0.772299i	$-0.280893\pi$
$774$	20.0000	0.718885
$775$	0	0
$776$	−18.9443	−0.680060
$777$	− 6.94427i	− 0.249124i
$778$	15.5279i	0.556701i
$779$	−4.94427	−0.177147
$780$	0	0
$781$	35.7771	1.28020
$782$	− 17.8885i	− 0.639693i
$783$	2.00000i	0.0714742i
$784$	1.00000	0.0357143
$785$	0	0
$786$	8.94427	0.319032
$787$	− 31.0557i	− 1.10702i	−0.832843	−	0.553509i	$-0.813289\pi$
$787$	− 31.0557i	− 1.10702i	0.832843	−	0.553509i	$-0.186711\pi$
$788$	46.5836i	1.65947i
$789$	7.05573	0.251191
$790$	0	0
$791$	0.472136	0.0167872
$792$	14.4721i	0.514245i
$793$	− 8.94427i	− 0.317620i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	82.2492	2.91525
$797$	18.9443i	0.671041i	0.942033	+	0.335520i	$0.108912\pi$
$797$	18.9443i	0.671041i	−0.942033	+	0.335520i	$0.891088\pi$
$798$	− 5.52786i	− 0.195684i
$799$	25.8885	0.915871
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	22.3607i	0.789583i
$803$	− 80.7214i	− 2.84859i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−15.2786	−0.538167
$807$	11.8885i	0.418497i
$808$	31.3050i	1.10130i
$809$	−38.9443	−1.36921	−0.684604	−	0.728915i	$-0.740025\pi$
$809$	−38.9443	−1.36921	−0.684604	+	0.728915i	$0.740025\pi$
$810$	0	0
$811$	55.4164	1.94593	0.972967	−	0.230946i	$-0.0741820\pi$
$811$	55.4164	1.94593	0.972967	+	0.230946i	$0.0741820\pi$
$812$	6.00000i	0.210559i
$813$	1.52786i	0.0535845i
$814$	−100.498	−3.52247
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	22.1115i	0.773582i
$818$	− 26.5836i	− 0.929474i
$819$	−4.47214	−0.156269
$820$	0	0
$821$	33.7771	1.17883	0.589414	−	0.807831i	$-0.299359\pi$
$821$	33.7771	1.17883	0.589414	+	0.807831i	$0.299359\pi$
$822$	− 7.88854i	− 0.275145i
$823$	44.9443i	1.56666i	0.621607	+	0.783329i	$0.286480\pi$
$823$	44.9443i	1.56666i	−0.621607	+	0.783329i	$0.713520\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	− 12.9443i	− 0.450116i	−0.974345	−	0.225058i	$-0.927743\pi$
$827$	− 12.9443i	− 0.450116i	0.974345	−	0.225058i	$-0.0722572\pi$
$828$	12.0000i	0.417029i
$829$	13.0557	0.453444	0.226722	−	0.973959i	$-0.427199\pi$
$829$	13.0557	0.453444	0.226722	+	0.973959i	$0.427199\pi$
$830$	0	0
$831$	−18.9443	−0.657170
$832$	58.1378i	2.01556i
$833$	− 2.00000i	− 0.0692959i
$834$	−16.5836	−0.574243
$835$	0	0
$836$	−48.0000	−1.66011
$837$	1.52786i	0.0528107i
$838$	66.8328i	2.30870i
$839$	−54.8328	−1.89304	−0.946520	−	0.322647i	$-0.895427\pi$
$839$	−54.8328	−1.89304	−0.946520	+	0.322647i	$0.895427\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	49.1935i	1.69532i
$843$	10.9443i	0.376941i
$844$	50.8328	1.74974
$845$	0	0
$846$	−28.9443	−0.995125
$847$	− 30.8885i	− 1.06134i
$848$	3.52786i	0.121147i
$849$	12.0000	0.411839
$850$	0	0
$851$	−27.7771	−0.952186
$852$	16.5836i	0.568145i
$853$	− 31.3050i	− 1.07186i	−0.844262	−	0.535931i	$-0.819961\pi$
$853$	− 31.3050i	− 1.07186i	0.844262	−	0.535931i	$-0.180039\pi$
$854$	−4.47214	−0.153033
$855$	0	0
$856$	28.9443	0.989295
$857$	− 36.8328i	− 1.25819i	−0.777330	−	0.629093i	$-0.783427\pi$
$857$	− 36.8328i	− 1.25819i	0.777330	−	0.629093i	$-0.216573\pi$
$858$	64.7214i	2.20955i
$859$	−50.4721	−1.72209	−0.861044	−	0.508531i	$-0.830189\pi$
$859$	−50.4721	−1.72209	−0.861044	+	0.508531i	$0.830189\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	− 41.3050i	− 1.40685i
$863$	21.8885i	0.745095i	0.928013	+	0.372547i	$0.121516\pi$
$863$	21.8885i	0.745095i	−0.928013	+	0.372547i	$0.878484\pi$
$864$	6.70820	0.228218
$865$	0	0
$866$	−36.8328	−1.25163
$867$	− 13.0000i	− 0.441503i
$868$	4.58359i	0.155577i
$869$	−83.7771	−2.84194
$870$	0	0
$871$	−17.8885	−0.606130
$872$	− 4.47214i	− 0.151446i
$873$	8.47214i	0.286738i
$874$	−22.1115	−0.747931
$875$	0	0
$876$	37.4164	1.26418
$877$	56.8328i	1.91911i	0.281525	+	0.959554i	$0.409160\pi$
$877$	56.8328i	1.91911i	−0.281525	+	0.959554i	$0.590840\pi$
$878$	− 3.41641i	− 0.115298i
$879$	5.05573	0.170525
$880$	0	0
$881$	−27.8885	−0.939589	−0.469794	−	0.882776i	$-0.655672\pi$
$881$	−27.8885	−0.939589	−0.469794	+	0.882776i	$0.655672\pi$
$882$	2.23607i	0.0752923i
$883$	− 37.8885i	− 1.27505i	−0.770429	−	0.637526i	$-0.779958\pi$
$883$	− 37.8885i	− 1.27505i	0.770429	−	0.637526i	$-0.220042\pi$
$884$	26.8328	0.902485
$885$	0	0
$886$	17.8885	0.600977
$887$	30.8328i	1.03526i	0.855603	+	0.517632i	$0.173186\pi$
$887$	30.8328i	1.03526i	−0.855603	+	0.517632i	$0.826814\pi$
$888$	− 15.5279i	− 0.521081i
$889$	4.94427	0.165826
$890$	0	0
$891$	6.47214	0.216825
$892$	38.8328i	1.30022i
$893$	− 32.0000i	− 1.07084i
$894$	33.4164	1.11761
$895$	0	0
$896$	15.6525	0.522913
$897$	17.8885i	0.597281i
$898$	31.3050i	1.04466i
$899$	3.05573	0.101914
$900$	0	0
$901$	7.05573	0.235060
$902$	− 28.9443i	− 0.963739i
$903$	− 8.94427i	− 0.297647i
$904$	1.05573	0.0351130
$905$	0	0
$906$	−35.7771	−1.18861
$907$	− 53.8885i	− 1.78934i	−0.446728	−	0.894670i	$-0.647411\pi$
$907$	− 53.8885i	− 1.78934i	0.446728	−	0.894670i	$-0.352589\pi$
$908$	2.83282i	0.0940103i
$909$	14.0000	0.464351
$910$	0	0
$911$	46.2492	1.53231	0.766153	−	0.642659i	$-0.222169\pi$
$911$	46.2492	1.53231	0.766153	+	0.642659i	$0.222169\pi$
$912$	2.47214i	0.0818606i
$913$	− 109.666i	− 3.62940i
$914$	15.5279	0.513616
$915$	0	0
$916$	71.6656	2.36790
$917$	− 4.00000i	− 0.132092i
$918$	− 4.47214i	− 0.147602i
$919$	35.0557	1.15638	0.578191	−	0.815902i	$-0.303759\pi$
$919$	35.0557	1.15638	0.578191	+	0.815902i	$0.303759\pi$
$920$	0	0
$921$	15.0557	0.496103
$922$	8.69505i	0.286356i
$923$	24.7214i	0.813713i
$924$	19.4164	0.638753
$925$	0	0
$926$	−46.8328	−1.53902
$927$	0	0
$928$	− 13.4164i	− 0.440415i
$929$	16.1115	0.528600	0.264300	−	0.964441i	$-0.414859\pi$
$929$	16.1115	0.528600	0.264300	+	0.964441i	$0.414859\pi$
$930$	0	0
$931$	−2.47214	−0.0810210
$932$	28.2492i	0.925334i
$933$	− 25.8885i	− 0.847553i
$934$	−20.0000	−0.654420
$935$	0	0
$936$	−10.0000	−0.326860
$937$	52.4721i	1.71419i	0.515158	+	0.857095i	$0.327733\pi$
$937$	52.4721i	1.71419i	−0.515158	+	0.857095i	$0.672267\pi$
$938$	8.94427i	0.292041i
$939$	−17.4164	−0.568363
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	1.05573i	0.0343975i
$943$	− 8.00000i	− 0.260516i
$944$	−8.94427	−0.291111
$945$	0	0
$946$	−129.443	−4.20855
$947$	17.8885i	0.581300i	0.956830	+	0.290650i	$0.0938715\pi$
$947$	17.8885i	0.581300i	−0.956830	+	0.290650i	$0.906129\pi$
$948$	− 38.8328i	− 1.26123i
$949$	55.7771	1.81060
$950$	0	0
$951$	−14.3607	−0.465677
$952$	− 4.47214i	− 0.144943i
$953$	− 33.4164i	− 1.08246i	−0.840873	−	0.541232i	$-0.817958\pi$
$953$	− 33.4164i	− 1.08246i	0.840873	−	0.541232i	$-0.182042\pi$
$954$	−7.88854	−0.255401
$955$	0	0
$956$	−31.4164	−1.01608
$957$	− 12.9443i	− 0.418429i
$958$	40.0000i	1.29234i
$959$	−3.52786	−0.113921
$960$	0	0
$961$	−28.6656	−0.924698
$962$	− 69.4427i	− 2.23892i
$963$	− 12.9443i	− 0.417123i
$964$	56.8328	1.83046
$965$	0	0
$966$	8.94427	0.287777
$967$	− 25.8885i	− 0.832519i	−0.909246	−	0.416260i	$-0.863341\pi$
$967$	− 25.8885i	− 0.832519i	0.909246	−	0.416260i	$-0.136659\pi$
$968$	− 69.0689i	− 2.21996i
$969$	4.94427	0.158833
$970$	0	0
$971$	40.9443	1.31396	0.656982	−	0.753906i	$-0.271833\pi$
$971$	40.9443	1.31396	0.656982	+	0.753906i	$0.271833\pi$
$972$	3.00000i	0.0962250i
$973$	7.41641i	0.237759i
$974$	−46.8328	−1.50062
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 30.5836i	− 0.978456i	−0.872156	−	0.489228i	$-0.837279\pi$
$977$	− 30.5836i	− 0.978456i	0.872156	−	0.489228i	$-0.162721\pi$
$978$	− 37.8885i	− 1.21154i
$979$	12.9443	0.413701
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	− 47.6393i	− 1.52023i
$983$	− 22.8328i	− 0.728254i	−0.931349	−	0.364127i	$-0.881367\pi$
$983$	− 22.8328i	− 0.728254i	0.931349	−	0.364127i	$-0.118633\pi$
$984$	4.47214	0.142566
$985$	0	0
$986$	−8.94427	−0.284844
$987$	12.9443i	0.412021i
$988$	− 33.1672i	− 1.05519i
$989$	−35.7771	−1.13765
$990$	0	0
$991$	4.94427	0.157060	0.0785300	−	0.996912i	$-0.474977\pi$
$991$	4.94427	0.157060	0.0785300	+	0.996912i	$0.474977\pi$
$992$	− 10.2492i	− 0.325413i
$993$	− 0.944272i	− 0.0299656i
$994$	12.3607	0.392057
$995$	0	0
$996$	50.8328	1.61070
$997$	5.41641i	0.171539i	0.996315	+	0.0857697i	$0.0273349\pi$
$997$	5.41641i	0.171539i	−0.996315	+	0.0857697i	$0.972665\pi$
$998$	31.0557i	0.983052i
$999$	−6.94427	−0.219707

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.d.c.274.3		4
3.2	odd	2		1575.2.d.d.1324.1		4
5.2	odd	4		105.2.a.b.1.1	✓	2
5.3	odd	4		525.2.a.g.1.2		2
5.4	even	2	inner	525.2.d.c.274.2		4
15.2	even	4		315.2.a.d.1.2		2
15.8	even	4		1575.2.a.r.1.1		2
15.14	odd	2		1575.2.d.d.1324.4		4
20.3	even	4		8400.2.a.cx.1.1		2
20.7	even	4		1680.2.a.v.1.1		2
35.2	odd	12		735.2.i.k.361.2		4
35.12	even	12		735.2.i.i.361.2		4
35.13	even	4		3675.2.a.y.1.2		2
35.17	even	12		735.2.i.i.226.2		4
35.27	even	4		735.2.a.k.1.1		2
35.32	odd	12		735.2.i.k.226.2		4
40.27	even	4		6720.2.a.cs.1.2		2
40.37	odd	4		6720.2.a.cx.1.1		2
60.47	odd	4		5040.2.a.bw.1.2		2
105.62	odd	4		2205.2.a.w.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.1	✓	2	5.2	odd	4
315.2.a.d.1.2		2	15.2	even	4
525.2.a.g.1.2		2	5.3	odd	4
525.2.d.c.274.2		4	5.4	even	2	inner
525.2.d.c.274.3		4	1.1	even	1	trivial
735.2.a.k.1.1		2	35.27	even	4
735.2.i.i.226.2		4	35.17	even	12
735.2.i.i.361.2		4	35.12	even	12
735.2.i.k.226.2		4	35.32	odd	12
735.2.i.k.361.2		4	35.2	odd	12
1575.2.a.r.1.1		2	15.8	even	4
1575.2.d.d.1324.1		4	3.2	odd	2
1575.2.d.d.1324.4		4	15.14	odd	2
1680.2.a.v.1.1		2	20.7	even	4
2205.2.a.w.1.2		2	105.62	odd	4
3675.2.a.y.1.2		2	35.13	even	4
5040.2.a.bw.1.2		2	60.47	odd	4
6720.2.a.cs.1.2		2	40.27	even	4
6720.2.a.cx.1.1		2	40.37	odd	4
8400.2.a.cx.1.1		2	20.3	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{2} -1.00000i q^{3} -3.00000 q^{4} +2.23607 q^{6} -1.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.23607i q^{2} -1.00000i q^{3} -3.00000 q^{4} +2.23607 q^{6} -1.00000i q^{7} -2.23607i q^{8} -1.00000 q^{9} +6.47214 q^{11} +3.00000i q^{12} +4.47214i q^{13} +2.23607 q^{14} -1.00000 q^{16} +2.00000i q^{17} -2.23607i q^{18} +2.47214 q^{19} -1.00000 q^{21} +14.4721i q^{22} +4.00000i q^{23} -2.23607 q^{24} -10.0000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +2.00000 q^{29} +1.52786 q^{31} -6.70820i q^{32} -6.47214i q^{33} -4.47214 q^{34} +3.00000 q^{36} +6.94427i q^{37} +5.52786i q^{38} +4.47214 q^{39} -2.00000 q^{41} -2.23607i q^{42} +8.94427i q^{43} -19.4164 q^{44} -8.94427 q^{46} -12.9443i q^{47} +1.00000i q^{48} -1.00000 q^{49} +2.00000 q^{51} -13.4164i q^{52} -3.52786i q^{53} -2.23607 q^{54} -2.23607 q^{56} -2.47214i q^{57} +4.47214i q^{58} +8.94427 q^{59} -2.00000 q^{61} +3.41641i q^{62} +1.00000i q^{63} +13.0000 q^{64} +14.4721 q^{66} +4.00000i q^{67} -6.00000i q^{68} +4.00000 q^{69} +5.52786 q^{71} +2.23607i q^{72} -12.4721i q^{73} -15.5279 q^{74} -7.41641 q^{76} -6.47214i q^{77} +10.0000i q^{78} -12.9443 q^{79} +1.00000 q^{81} -4.47214i q^{82} -16.9443i q^{83} +3.00000 q^{84} -20.0000 q^{86} -2.00000i q^{87} -14.4721i q^{88} +2.00000 q^{89} +4.47214 q^{91} -12.0000i q^{92} -1.52786i q^{93} +28.9443 q^{94} -6.70820 q^{96} -8.47214i q^{97} -2.23607i q^{98} -6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4} - 4 q^{9}+O(q^{10}) \) 4 * q - 12 * q^4 - 4 * q^9	\( 4 q - 12 q^{4} - 4 q^{9} + 8 q^{11} - 4 q^{16} - 8 q^{19} - 4 q^{21} - 40 q^{26} + 8 q^{29} + 24 q^{31} + 12 q^{36} - 8 q^{41} - 24 q^{44} - 4 q^{49} + 8 q^{51} - 8 q^{61} + 52 q^{64} + 40 q^{66} + 16 q^{69} + 40 q^{71} - 80 q^{74} + 24 q^{76} - 16 q^{79} + 4 q^{81} + 12 q^{84} - 80 q^{86} + 8 q^{89} + 80 q^{94} - 8 q^{99}+O(q^{100}) \) 4 * q - 12 * q^4 - 4 * q^9 + 8 * q^11 - 4 * q^16 - 8 * q^19 - 4 * q^21 - 40 * q^26 + 8 * q^29 + 24 * q^31 + 12 * q^36 - 8 * q^41 - 24 * q^44 - 4 * q^49 + 8 * q^51 - 8 * q^61 + 52 * q^64 + 40 * q^66 + 16 * q^69 + 40 * q^71 - 80 * q^74 + 24 * q^76 - 16 * q^79 + 4 * q^81 + 12 * q^84 - 80 * q^86 + 8 * q^89 + 80 * q^94 - 8 * q^99

Introduction

L-functions

Modular forms

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Database

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