LMFDB - Embedded newform 525.2.d.c.274.1

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			274.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.2.d.c.274.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{2} -1.00000i q^{3} -3.00000 q^{4} -2.23607 q^{6} -1.00000i q^{7} +2.23607i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.23607i q^{2} -1.00000i q^{3} -3.00000 q^{4} -2.23607 q^{6} -1.00000i q^{7} +2.23607i q^{8} -1.00000 q^{9} -2.47214 q^{11} +3.00000i q^{12} -4.47214i q^{13} -2.23607 q^{14} -1.00000 q^{16} +2.00000i q^{17} +2.23607i q^{18} -6.47214 q^{19} -1.00000 q^{21} +5.52786i q^{22} +4.00000i q^{23} +2.23607 q^{24} -10.0000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +2.00000 q^{29} +10.4721 q^{31} +6.70820i q^{32} +2.47214i q^{33} +4.47214 q^{34} +3.00000 q^{36} -10.9443i q^{37} +14.4721i q^{38} -4.47214 q^{39} -2.00000 q^{41} +2.23607i q^{42} -8.94427i q^{43} +7.41641 q^{44} +8.94427 q^{46} +4.94427i q^{47} +1.00000i q^{48} -1.00000 q^{49} +2.00000 q^{51} +13.4164i q^{52} -12.4721i q^{53} +2.23607 q^{54} +2.23607 q^{56} +6.47214i q^{57} -4.47214i q^{58} -8.94427 q^{59} -2.00000 q^{61} -23.4164i q^{62} +1.00000i q^{63} +13.0000 q^{64} +5.52786 q^{66} +4.00000i q^{67} -6.00000i q^{68} +4.00000 q^{69} +14.4721 q^{71} -2.23607i q^{72} -3.52786i q^{73} -24.4721 q^{74} +19.4164 q^{76} +2.47214i q^{77} +10.0000i q^{78} +4.94427 q^{79} +1.00000 q^{81} +4.47214i q^{82} +0.944272i q^{83} +3.00000 q^{84} -20.0000 q^{86} -2.00000i q^{87} -5.52786i q^{88} +2.00000 q^{89} -4.47214 q^{91} -12.0000i q^{92} -10.4721i q^{93} +11.0557 q^{94} +6.70820 q^{96} +0.472136i q^{97} +2.23607i q^{98} +2.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.709785\pi$
$2$	− 2.23607i	− 1.58114i	0.612372	−	0.790569i	$-0.290215\pi$
$3$	− 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
$11$	−2.47214	−0.745377	−0.372689	−	0.927957i	$-0.621564\pi$
$11$	−2.47214	−0.745377	−0.372689	+	0.927957i	$0.621564\pi$
$12$	3.00000i	0.866025i
$13$	− 4.47214i	− 1.24035i	−0.784465	−	0.620174i	$-0.787062\pi$
$13$	− 4.47214i	− 1.24035i	0.784465	−	0.620174i	$-0.212938\pi$
$14$	−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
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	5.52786i	1.17854i
$23$	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
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	6.70820i	1.18585i
$33$	2.47214i	0.430344i
$34$	4.47214	0.766965
$35$	0	0
$36$	3.00000	0.500000
$37$	− 10.9443i	− 1.79923i	−0.436687	−	0.899614i	$-0.643848\pi$
$37$	− 10.9443i	− 1.79923i	0.436687	−	0.899614i	$-0.356152\pi$
$38$	14.4721i	2.34769i
$39$	−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	$-0.761113\pi$
$43$	− 8.94427i	− 1.36399i	0.731357	−	0.681994i	$-0.238887\pi$
$44$	7.41641	1.11807
$45$	0	0
$46$	8.94427	1.31876
$47$	4.94427i	0.721196i	0.932721	+	0.360598i	$0.117427\pi$
$47$	4.94427i	0.721196i	−0.932721	+	0.360598i	$0.882573\pi$
$48$	1.00000i	0.144338i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	13.4164i	1.86052i
$53$	− 12.4721i	− 1.71318i	−0.515998	−	0.856590i	$-0.672579\pi$
$53$	− 12.4721i	− 1.71318i	0.515998	−	0.856590i	$-0.327421\pi$
$54$	2.23607	0.304290
$55$	0	0
$56$	2.23607	0.298807
$57$	6.47214i	0.857255i
$58$	− 4.47214i	− 0.587220i
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	− 23.4164i	− 2.97389i
$63$	1.00000i	0.125988i
$64$	13.0000	1.62500
$65$	0	0
$66$	5.52786	0.680433
$67$	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
$71$	14.4721	1.71753	0.858763	−	0.512373i	$-0.171233\pi$
$71$	14.4721	1.71753	0.858763	+	0.512373i	$0.171233\pi$
$72$	− 2.23607i	− 0.263523i
$73$	− 3.52786i	− 0.412905i	−0.978457	−	0.206453i	$-0.933808\pi$
$73$	− 3.52786i	− 0.412905i	0.978457	−	0.206453i	$-0.0661919\pi$
$74$	−24.4721	−2.84483
$75$	0	0
$76$	19.4164	2.22721
$77$	2.47214i	0.281726i
$78$	10.0000i	1.13228i
$79$	4.94427	0.556274	0.278137	−	0.960541i	$-0.410283\pi$
$79$	4.94427	0.556274	0.278137	+	0.960541i	$0.410283\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.47214i	0.493865i
$83$	0.944272i	0.103647i	0.998656	+	0.0518237i	$0.0165034\pi$
$83$	0.944272i	0.103647i	−0.998656	+	0.0518237i	$0.983497\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	−20.0000	−2.15666
$87$	− 2.00000i	− 0.214423i
$88$	− 5.52786i	− 0.589272i
$89$	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
$93$	− 10.4721i	− 1.08591i
$94$	11.0557	1.14031
$95$	0	0
$96$	6.70820	0.684653
$97$	0.472136i	0.0479381i	0.999713	+	0.0239691i	$0.00763032\pi$
$97$	0.472136i	0.0479381i	−0.999713	+	0.0239691i	$0.992370\pi$
$98$	2.23607i	0.225877i
$99$	2.47214	0.248459
$100$	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
$106$	−27.8885	−2.70877
$107$	− 4.94427i	− 0.477981i	−0.971022	−	0.238990i	$-0.923184\pi$
$107$	− 4.94427i	− 0.477981i	0.971022	−	0.238990i	$-0.0768164\pi$
$108$	− 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
$111$	−10.9443	−1.03878
$112$	1.00000i	0.0944911i
$113$	− 8.47214i	− 0.796992i	−0.917170	−	0.398496i	$-0.869532\pi$
$113$	− 8.47214i	− 0.796992i	0.917170	−	0.398496i	$-0.130468\pi$
$114$	14.4721	1.35544
$115$	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
$121$	−4.88854	−0.444413
$122$	4.47214i	0.404888i
$123$	2.00000i	0.180334i
$124$	−31.4164	−2.82128
$125$	0	0
$126$	2.23607	0.199205
$127$	− 12.9443i	− 1.14862i	−0.818638	−	0.574309i	$-0.805271\pi$
$127$	− 12.9443i	− 1.14862i	0.818638	−	0.574309i	$-0.194729\pi$
$128$	− 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	$0.444091\pi$
$132$	− 7.41641i	− 0.645515i
$133$	6.47214i	0.561205i
$134$	8.94427	0.772667
$135$	0	0
$136$	−4.47214	−0.383482
$137$	− 12.4721i	− 1.06557i	−0.846252	−	0.532783i	$-0.821146\pi$
$137$	− 12.4721i	− 1.06557i	0.846252	−	0.532783i	$-0.178854\pi$
$138$	− 8.94427i	− 0.761387i
$139$	19.4164	1.64688	0.823439	−	0.567405i	$-0.192052\pi$
$139$	19.4164	1.64688	0.823439	+	0.567405i	$0.192052\pi$
$140$	0	0
$141$	4.94427	0.416383
$142$	− 32.3607i	− 2.71565i
$143$	11.0557i	0.924526i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−7.88854	−0.652861
$147$	1.00000i	0.0824786i
$148$	32.8328i	2.69884i
$149$	−2.94427	−0.241204	−0.120602	−	0.992701i	$-0.538483\pi$
$149$	−2.94427	−0.241204	−0.120602	+	0.992701i	$0.538483\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 14.4721i	− 1.17385i
$153$	− 2.00000i	− 0.161690i
$154$	5.52786	0.445448
$155$	0	0
$156$	13.4164	1.07417
$157$	− 8.47214i	− 0.676150i	−0.941119	−	0.338075i	$-0.890224\pi$
$157$	− 8.47214i	− 0.676150i	0.941119	−	0.338075i	$-0.109776\pi$
$158$	− 11.0557i	− 0.879547i
$159$	−12.4721	−0.989105
$160$	0	0
$161$	4.00000	0.315244
$162$	− 2.23607i	− 0.175682i
$163$	0.944272i	0.0739611i	0.999316	+	0.0369805i	$0.0117740\pi$
$163$	0.944272i	0.0739611i	−0.999316	+	0.0369805i	$0.988226\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	2.11146	0.163881
$167$	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
$171$	6.47214	0.494937
$172$	26.8328i	2.04598i
$173$	14.9443i	1.13619i	0.822962	+	0.568096i	$0.192320\pi$
$173$	14.9443i	1.13619i	−0.822962	+	0.568096i	$0.807680\pi$
$174$	−4.47214	−0.339032
$175$	0	0
$176$	2.47214	0.186344
$177$	8.94427i	0.672293i
$178$	− 4.47214i	− 0.335201i
$179$	2.47214	0.184776	0.0923881	−	0.995723i	$-0.470550\pi$
$179$	2.47214	0.184776	0.0923881	+	0.995723i	$0.470550\pi$
$180$	0	0
$181$	18.9443	1.40812	0.704058	−	0.710142i	$-0.251369\pi$
$181$	18.9443	1.40812	0.704058	+	0.710142i	$0.251369\pi$
$182$	10.0000i	0.741249i
$183$	2.00000i	0.147844i
$184$	−8.94427	−0.659380
$185$	0	0
$186$	−23.4164	−1.71697
$187$	− 4.94427i	− 0.361561i
$188$	− 14.8328i	− 1.08179i
$189$	1.00000	0.0727393
$190$	0	0
$191$	27.4164	1.98378	0.991891	−	0.127093i	$-0.0405646\pi$
$191$	27.4164	1.98378	0.991891	+	0.127093i	$0.0405646\pi$
$192$	− 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	$-0.168091\pi$
$194$	1.05573	0.0757969
$195$	0	0
$196$	3.00000	0.214286
$197$	− 24.4721i	− 1.74357i	−0.489891	−	0.871784i	$-0.662963\pi$
$197$	− 24.4721i	− 1.74357i	0.489891	−	0.871784i	$-0.337037\pi$
$198$	− 5.52786i	− 0.392848i
$199$	−0.583592	−0.0413697	−0.0206849	−	0.999786i	$-0.506585\pi$
$199$	−0.583592	−0.0413697	−0.0206849	+	0.999786i	$0.506585\pi$
$200$	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
$211$	0.944272	0.0650064	0.0325032	−	0.999472i	$-0.489652\pi$
$211$	0.944272	0.0650064	0.0325032	+	0.999472i	$0.489652\pi$
$212$	37.4164i	2.56977i
$213$	− 14.4721i	− 0.991614i
$214$	−11.0557	−0.755754
$215$	0	0
$216$	−2.23607	−0.152145
$217$	− 10.4721i	− 0.710895i
$218$	− 4.47214i	− 0.302891i
$219$	−3.52786	−0.238391
$220$	0	0
$221$	8.94427	0.601657
$222$	24.4721i	1.64246i
$223$	4.94427i	0.331093i	0.986202	+	0.165546i	$0.0529388\pi$
$223$	4.94427i	0.331093i	−0.986202	+	0.165546i	$0.947061\pi$
$224$	6.70820	0.448211
$225$	0	0
$226$	−18.9443	−1.26015
$227$	16.9443i	1.12463i	0.826923	+	0.562315i	$0.190089\pi$
$227$	16.9443i	1.12463i	−0.826923	+	0.562315i	$0.809911\pi$
$228$	− 19.4164i	− 1.28588i
$229$	11.8885	0.785617	0.392809	−	0.919620i	$-0.371504\pi$
$229$	11.8885	0.785617	0.392809	+	0.919620i	$0.371504\pi$
$230$	0	0
$231$	2.47214	0.162655
$232$	4.47214i	0.293610i
$233$	17.4164i	1.14099i	0.821302	+	0.570493i	$0.193248\pi$
$233$	17.4164i	1.14099i	−0.821302	+	0.570493i	$0.806752\pi$
$234$	10.0000	0.653720
$235$	0	0
$236$	26.8328	1.74667
$237$	− 4.94427i	− 0.321165i
$238$	− 4.47214i	− 0.289886i
$239$	1.52786	0.0988293	0.0494147	−	0.998778i	$-0.484264\pi$
$239$	1.52786	0.0988293	0.0494147	+	0.998778i	$0.484264\pi$
$240$	0	0
$241$	−1.05573	−0.0680054	−0.0340027	−	0.999422i	$-0.510825\pi$
$241$	−1.05573	−0.0680054	−0.0340027	+	0.999422i	$0.510825\pi$
$242$	10.9311i	0.702679i
$243$	− 1.00000i	− 0.0641500i
$244$	6.00000	0.384111
$245$	0	0
$246$	4.47214	0.285133
$247$	28.9443i	1.84168i
$248$	23.4164i	1.48694i
$249$	0.944272	0.0598408
$250$	0	0
$251$	−0.944272	−0.0596019	−0.0298010	−	0.999556i	$-0.509487\pi$
$251$	−0.944272	−0.0596019	−0.0298010	+	0.999556i	$0.509487\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 9.88854i	− 0.621687i
$254$	−28.9443	−1.81613
$255$	0	0
$256$	−9.00000	−0.562500
$257$	− 1.05573i	− 0.0658545i	−0.999458	−	0.0329273i	$-0.989517\pi$
$257$	− 1.05573i	− 0.0658545i	0.999458	−	0.0329273i	$-0.0104830\pi$
$258$	20.0000i	1.24515i
$259$	−10.9443	−0.680044
$260$	0	0
$261$	−2.00000	−0.123797
$262$	− 8.94427i	− 0.552579i
$263$	24.9443i	1.53813i	0.639171	+	0.769065i	$0.279278\pi$
$263$	24.9443i	1.53813i	−0.639171	+	0.769065i	$0.720722\pi$
$264$	−5.52786	−0.340217
$265$	0	0
$266$	14.4721	0.887344
$267$	− 2.00000i	− 0.122398i
$268$	− 12.0000i	− 0.733017i
$269$	23.8885	1.45651	0.728255	−	0.685306i	$-0.240332\pi$
$269$	23.8885	1.45651	0.728255	+	0.685306i	$0.240332\pi$
$270$	0	0
$271$	−10.4721	−0.636137	−0.318068	−	0.948068i	$-0.603034\pi$
$271$	−10.4721	−0.636137	−0.318068	+	0.948068i	$0.603034\pi$
$272$	− 2.00000i	− 0.121268i
$273$	4.47214i	0.270666i
$274$	−27.8885	−1.68481
$275$	0	0
$276$	−12.0000	−0.722315
$277$	− 1.05573i	− 0.0634326i	−0.999497	−	0.0317163i	$-0.989903\pi$
$277$	− 1.05573i	− 0.0634326i	0.999497	−	0.0317163i	$-0.0100973\pi$
$278$	− 43.4164i	− 2.60394i
$279$	−10.4721	−0.626950
$280$	0	0
$281$	6.94427	0.414261	0.207130	−	0.978313i	$-0.433588\pi$
$281$	6.94427	0.414261	0.207130	+	0.978313i	$0.433588\pi$
$282$	− 11.0557i	− 0.658359i
$283$	12.0000i	0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$	12.0000i	0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$	−43.4164	−2.57629
$285$	0	0
$286$	24.7214	1.46180
$287$	2.00000i	0.118056i
$288$	− 6.70820i	− 0.395285i
$289$	13.0000	0.764706
$290$	0	0
$291$	0.472136	0.0276771
$292$	10.5836i	0.619358i
$293$	22.9443i	1.34042i	0.742172	+	0.670209i	$0.233796\pi$
$293$	22.9443i	1.34042i	−0.742172	+	0.670209i	$0.766204\pi$
$294$	2.23607	0.130410
$295$	0	0
$296$	24.4721	1.42241
$297$	− 2.47214i	− 0.143448i
$298$	6.58359i	0.381377i
$299$	17.8885	1.03452
$300$	0	0
$301$	−8.94427	−0.515539
$302$	35.7771i	2.05874i
$303$	14.0000i	0.804279i
$304$	6.47214	0.371202
$305$	0	0
$306$	−4.47214	−0.255655
$307$	32.9443i	1.88023i	0.340859	+	0.940114i	$0.389282\pi$
$307$	32.9443i	1.88023i	−0.340859	+	0.940114i	$0.610718\pi$
$308$	− 7.41641i	− 0.422589i
$309$	0	0
$310$	0	0
$311$	−9.88854	−0.560728	−0.280364	−	0.959894i	$-0.590455\pi$
$311$	−9.88854	−0.560728	−0.280364	+	0.959894i	$0.590455\pi$
$312$	− 10.0000i	− 0.566139i
$313$	9.41641i	0.532247i	0.963939	+	0.266123i	$0.0857429\pi$
$313$	9.41641i	0.532247i	−0.963939	+	0.266123i	$0.914257\pi$
$314$	−18.9443	−1.06909
$315$	0	0
$316$	−14.8328	−0.834411
$317$	30.3607i	1.70523i	0.522543	+	0.852613i	$0.324983\pi$
$317$	30.3607i	1.70523i	−0.522543	+	0.852613i	$0.675017\pi$
$318$	27.8885i	1.56391i
$319$	−4.94427	−0.276826
$320$	0	0
$321$	−4.94427	−0.275962
$322$	− 8.94427i	− 0.498445i
$323$	− 12.9443i	− 0.720239i
$324$	−3.00000	−0.166667
$325$	0	0
$326$	2.11146	0.116943
$327$	− 2.00000i	− 0.110600i
$328$	− 4.47214i	− 0.246932i
$329$	4.94427	0.272587
$330$	0	0
$331$	−16.9443	−0.931341	−0.465671	−	0.884958i	$-0.654187\pi$
$331$	−16.9443	−0.931341	−0.465671	+	0.884958i	$0.654187\pi$
$332$	− 2.83282i	− 0.155471i
$333$	10.9443i	0.599742i
$334$	17.8885	0.978818
$335$	0	0
$336$	1.00000	0.0545545
$337$	− 11.8885i	− 0.647610i	−0.946124	−	0.323805i	$-0.895038\pi$
$337$	− 11.8885i	− 0.647610i	0.946124	−	0.323805i	$-0.104962\pi$
$338$	15.6525i	0.851382i
$339$	−8.47214	−0.460143
$340$	0	0
$341$	−25.8885	−1.40194
$342$	− 14.4721i	− 0.782563i
$343$	1.00000i	0.0539949i
$344$	20.0000	1.07833
$345$	0	0
$346$	33.4164	1.79648
$347$	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
$349$	−23.8885	−1.27872	−0.639362	−	0.768906i	$-0.720802\pi$
$349$	−23.8885	−1.27872	−0.639362	+	0.768906i	$0.720802\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	− 16.5836i	− 0.883908i
$353$	− 27.8885i	− 1.48436i	−0.670202	−	0.742179i	$-0.733793\pi$
$353$	− 27.8885i	− 1.48436i	0.670202	−	0.742179i	$-0.266207\pi$
$354$	20.0000	1.06299
$355$	0	0
$356$	−6.00000	−0.317999
$357$	− 2.00000i	− 0.105851i
$358$	− 5.52786i	− 0.292157i
$359$	−9.52786	−0.502861	−0.251431	−	0.967875i	$-0.580901\pi$
$359$	−9.52786	−0.502861	−0.251431	+	0.967875i	$0.580901\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	− 42.3607i	− 2.22643i
$363$	4.88854i	0.256582i
$364$	13.4164	0.703211
$365$	0	0
$366$	4.47214	0.233762
$367$	− 20.9443i	− 1.09328i	−0.837367	−	0.546641i	$-0.815906\pi$
$367$	− 20.9443i	− 1.09328i	0.837367	−	0.546641i	$-0.184094\pi$
$368$	− 4.00000i	− 0.208514i
$369$	2.00000	0.104116
$370$	0	0
$371$	−12.4721	−0.647521
$372$	31.4164i	1.62886i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	−11.0557	−0.571678
$375$	0	0
$376$	−11.0557	−0.570156
$377$	− 8.94427i	− 0.460653i
$378$	− 2.23607i	− 0.115011i
$379$	2.11146	0.108458	0.0542291	−	0.998529i	$-0.482730\pi$
$379$	2.11146	0.108458	0.0542291	+	0.998529i	$0.482730\pi$
$380$	0	0
$381$	−12.9443	−0.663155
$382$	− 61.3050i	− 3.13663i
$383$	− 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
$388$	− 1.41641i	− 0.0719072i
$389$	−10.9443	−0.554897	−0.277448	−	0.960741i	$-0.589489\pi$
$389$	−10.9443	−0.554897	−0.277448	+	0.960741i	$0.589489\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	− 2.23607i	− 0.112938i
$393$	− 4.00000i	− 0.201773i
$394$	−54.7214	−2.75682
$395$	0	0
$396$	−7.41641	−0.372689
$397$	− 13.4164i	− 0.673350i	−0.941621	−	0.336675i	$-0.890698\pi$
$397$	− 13.4164i	− 0.673350i	0.941621	−	0.336675i	$-0.109302\pi$
$398$	1.30495i	0.0654113i
$399$	6.47214	0.324012
$400$	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
$403$	− 46.8328i	− 2.33291i
$404$	42.0000	2.08958
$405$	0	0
$406$	−4.47214	−0.221948
$407$	27.0557i	1.34110i
$408$	4.47214i	0.221404i
$409$	23.8885	1.18121	0.590606	−	0.806960i	$-0.298889\pi$
$409$	23.8885	1.18121	0.590606	+	0.806960i	$0.298889\pi$
$410$	0	0
$411$	−12.4721	−0.615205
$412$	0	0
$413$	8.94427i	0.440119i
$414$	−8.94427	−0.439587
$415$	0	0
$416$	30.0000	1.47087
$417$	− 19.4164i	− 0.950826i
$418$	− 35.7771i	− 1.74991i
$419$	−5.88854	−0.287674	−0.143837	−	0.989601i	$-0.545944\pi$
$419$	−5.88854	−0.287674	−0.143837	+	0.989601i	$0.545944\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 2.11146i	− 0.102784i
$423$	− 4.94427i	− 0.240399i
$424$	27.8885	1.35439
$425$	0	0
$426$	−32.3607	−1.56788
$427$	2.00000i	0.0967868i
$428$	14.8328i	0.716971i
$429$	11.0557	0.533776
$430$	0	0
$431$	−9.52786	−0.458941	−0.229471	−	0.973316i	$-0.573699\pi$
$431$	−9.52786	−0.458941	−0.229471	+	0.973316i	$0.573699\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	7.52786i	0.361766i	0.983505	+	0.180883i	$0.0578955\pi$
$433$	7.52786i	0.361766i	−0.983505	+	0.180883i	$0.942104\pi$
$434$	−23.4164	−1.12402
$435$	0	0
$436$	−6.00000	−0.287348
$437$	− 25.8885i	− 1.23842i
$438$	7.88854i	0.376929i
$439$	−10.4721	−0.499808	−0.249904	−	0.968271i	$-0.580399\pi$
$439$	−10.4721	−0.499808	−0.249904	+	0.968271i	$0.580399\pi$
$440$	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	$-0.0608636\pi$
$444$	32.8328	1.55818
$445$	0	0
$446$	11.0557	0.523504
$447$	2.94427i	0.139259i
$448$	− 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
$451$	4.94427	0.232817
$452$	25.4164i	1.19549i
$453$	16.0000i	0.751746i
$454$	37.8885	1.77820
$455$	0	0
$456$	−14.4721	−0.677720
$457$	10.9443i	0.511951i	0.966683	+	0.255976i	$0.0823967\pi$
$457$	10.9443i	0.511951i	−0.966683	+	0.255976i	$0.917603\pi$
$458$	− 26.5836i	− 1.24217i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−31.8885	−1.48520	−0.742599	−	0.669737i	$-0.766407\pi$
$461$	−31.8885	−1.48520	−0.742599	+	0.669737i	$0.766407\pi$
$462$	− 5.52786i	− 0.257180i
$463$	3.05573i	0.142012i	0.997476	+	0.0710059i	$0.0226209\pi$
$463$	3.05573i	0.142012i	−0.997476	+	0.0710059i	$0.977379\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	38.9443	1.80406
$467$	− 8.94427i	− 0.413892i	−0.978352	−	0.206946i	$-0.933648\pi$
$467$	− 8.94427i	− 0.413892i	0.978352	−	0.206946i	$-0.0663524\pi$
$468$	− 13.4164i	− 0.620174i
$469$	4.00000	0.184703
$470$	0	0
$471$	−8.47214	−0.390375
$472$	− 20.0000i	− 0.920575i
$473$	22.1115i	1.01669i
$474$	−11.0557	−0.507806
$475$	0	0
$476$	−6.00000	−0.275010
$477$	12.4721i	0.571060i
$478$	− 3.41641i	− 0.156263i
$479$	−17.8885	−0.817348	−0.408674	−	0.912680i	$-0.634009\pi$
$479$	−17.8885	−0.817348	−0.408674	+	0.912680i	$0.634009\pi$
$480$	0	0
$481$	−48.9443	−2.23167
$482$	2.36068i	0.107526i
$483$	− 4.00000i	− 0.182006i
$484$	14.6656	0.666620
$485$	0	0
$486$	−2.23607	−0.101430
$487$	3.05573i	0.138468i	0.997600	+	0.0692341i	$0.0220556\pi$
$487$	3.05573i	0.138468i	−0.997600	+	0.0692341i	$0.977944\pi$
$488$	− 4.47214i	− 0.202444i
$489$	0.944272	0.0427015
$490$	0	0
$491$	41.3050	1.86407	0.932033	−	0.362373i	$-0.118033\pi$
$491$	41.3050	1.86407	0.932033	+	0.362373i	$0.118033\pi$
$492$	− 6.00000i	− 0.270501i
$493$	4.00000i	0.180151i
$494$	64.7214	2.91195
$495$	0	0
$496$	−10.4721	−0.470213
$497$	− 14.4721i	− 0.649164i
$498$	− 2.11146i	− 0.0946166i
$499$	−21.8885	−0.979866	−0.489933	−	0.871760i	$-0.662979\pi$
$499$	−21.8885	−0.979866	−0.489933	+	0.871760i	$0.662979\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	2.11146i	0.0942389i
$503$	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
$506$	−22.1115	−0.982974
$507$	7.00000i	0.310881i
$508$	38.8328i	1.72293i
$509$	−11.8885	−0.526950	−0.263475	−	0.964666i	$-0.584869\pi$
$509$	−11.8885	−0.526950	−0.263475	+	0.964666i	$0.584869\pi$
$510$	0	0
$511$	−3.52786	−0.156064
$512$	− 11.1803i	− 0.494106i
$513$	− 6.47214i	− 0.285752i
$514$	−2.36068	−0.104125
$515$	0	0
$516$	26.8328	1.18125
$517$	− 12.2229i	− 0.537563i
$518$	24.4721i	1.07524i
$519$	14.9443	0.655981
$520$	0	0
$521$	15.8885	0.696090	0.348045	−	0.937478i	$-0.386846\pi$
$521$	15.8885	0.696090	0.348045	+	0.937478i	$0.386846\pi$
$522$	4.47214i	0.195740i
$523$	8.94427i	0.391106i	0.980693	+	0.195553i	$0.0626501\pi$
$523$	8.94427i	0.391106i	−0.980693	+	0.195553i	$0.937350\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	55.7771	2.43200
$527$	20.9443i	0.912347i
$528$	− 2.47214i	− 0.107586i
$529$	7.00000	0.304348
$530$	0	0
$531$	8.94427	0.388148
$532$	− 19.4164i	− 0.841808i
$533$	8.94427i	0.387419i
$534$	−4.47214	−0.193528
$535$	0	0
$536$	−8.94427	−0.386334
$537$	− 2.47214i	− 0.106681i
$538$	− 53.4164i	− 2.30294i
$539$	2.47214	0.106482
$540$	0	0
$541$	23.8885	1.02705	0.513524	−	0.858075i	$-0.328340\pi$
$541$	23.8885	1.02705	0.513524	+	0.858075i	$0.328340\pi$
$542$	23.4164i	1.00582i
$543$	− 18.9443i	− 0.812977i
$544$	−13.4164	−0.575224
$545$	0	0
$546$	10.0000	0.427960
$547$	29.8885i	1.27794i	0.769231	+	0.638971i	$0.220640\pi$
$547$	29.8885i	1.27794i	−0.769231	+	0.638971i	$0.779360\pi$
$548$	37.4164i	1.59835i
$549$	2.00000	0.0853579
$550$	0	0
$551$	−12.9443	−0.551445
$552$	8.94427i	0.380693i
$553$	− 4.94427i	− 0.210252i
$554$	−2.36068	−0.100296
$555$	0	0
$556$	−58.2492	−2.47032
$557$	− 11.5279i	− 0.488451i	−0.969718	−	0.244226i	$-0.921466\pi$
$557$	− 11.5279i	− 0.488451i	0.969718	−	0.244226i	$-0.0785338\pi$
$558$	23.4164i	0.991296i
$559$	−40.0000	−1.69182
$560$	0	0
$561$	−4.94427	−0.208747
$562$	− 15.5279i	− 0.655003i
$563$	− 21.8885i	− 0.922492i	−0.887272	−	0.461246i	$-0.847403\pi$
$563$	− 21.8885i	− 0.922492i	0.887272	−	0.461246i	$-0.152597\pi$
$564$	−14.8328	−0.624574
$565$	0	0
$566$	26.8328	1.12787
$567$	− 1.00000i	− 0.0419961i
$568$	32.3607i	1.35782i
$569$	4.11146	0.172361	0.0861806	−	0.996280i	$-0.472534\pi$
$569$	4.11146	0.172361	0.0861806	+	0.996280i	$0.472534\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 33.1672i	− 1.38679i
$573$	− 27.4164i	− 1.14534i
$574$	4.47214	0.186663
$575$	0	0
$576$	−13.0000	−0.541667
$577$	34.3607i	1.43045i	0.698892	+	0.715227i	$0.253677\pi$
$577$	34.3607i	1.43045i	−0.698892	+	0.715227i	$0.746323\pi$
$578$	− 29.0689i	− 1.20911i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0.944272	0.0391750
$582$	− 1.05573i	− 0.0437613i
$583$	30.8328i	1.27696i
$584$	7.88854	0.326430
$585$	0	0
$586$	51.3050	2.11939
$587$	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
$589$	−67.7771	−2.79271
$590$	0	0
$591$	−24.4721	−1.00665
$592$	10.9443i	0.449807i
$593$	− 11.8885i	− 0.488204i	−0.969750	−	0.244102i	$-0.921507\pi$
$593$	− 11.8885i	− 0.488204i	0.969750	−	0.244102i	$-0.0784932\pi$
$594$	−5.52786	−0.226811
$595$	0	0
$596$	8.83282	0.361806
$597$	0.583592i	0.0238848i
$598$	− 40.0000i	− 1.63572i
$599$	32.3607	1.32222	0.661111	−	0.750288i	$-0.270085\pi$
$599$	32.3607	1.32222	0.661111	+	0.750288i	$0.270085\pi$
$600$	0	0
$601$	21.0557	0.858881	0.429441	−	0.903095i	$-0.358711\pi$
$601$	21.0557	0.858881	0.429441	+	0.903095i	$0.358711\pi$
$602$	20.0000i	0.815139i
$603$	− 4.00000i	− 0.162893i
$604$	48.0000	1.95309
$605$	0	0
$606$	31.3050	1.27168
$607$	14.8328i	0.602045i	0.953617	+	0.301023i	$0.0973280\pi$
$607$	14.8328i	0.602045i	−0.953617	+	0.301023i	$0.902672\pi$
$608$	− 43.4164i	− 1.76077i
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	22.1115	0.894534
$612$	6.00000i	0.242536i
$613$	10.9443i	0.442035i	0.975270	+	0.221017i	$0.0709378\pi$
$613$	10.9443i	0.442035i	−0.975270	+	0.221017i	$0.929062\pi$
$614$	73.6656	2.97290
$615$	0	0
$616$	−5.52786	−0.222724
$617$	− 7.52786i	− 0.303060i	−0.988453	−	0.151530i	$-0.951580\pi$
$617$	− 7.52786i	− 0.303060i	0.988453	−	0.151530i	$-0.0484201\pi$
$618$	0	0
$619$	−12.5836	−0.505777	−0.252889	−	0.967495i	$-0.581381\pi$
$619$	−12.5836	−0.505777	−0.252889	+	0.967495i	$0.581381\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	22.1115i	0.886589i
$623$	− 2.00000i	− 0.0801283i
$624$	4.47214	0.179029
$625$	0	0
$626$	21.0557	0.841556
$627$	− 16.0000i	− 0.638978i
$628$	25.4164i	1.01423i
$629$	21.8885	0.872753
$630$	0	0
$631$	−22.8328	−0.908960	−0.454480	−	0.890757i	$-0.650175\pi$
$631$	−22.8328	−0.908960	−0.454480	+	0.890757i	$0.650175\pi$
$632$	11.0557i	0.439773i
$633$	− 0.944272i	− 0.0375314i
$634$	67.8885	2.69620
$635$	0	0
$636$	37.4164	1.48366
$637$	4.47214i	0.177192i
$638$	11.0557i	0.437700i
$639$	−14.4721	−0.572509
$640$	0	0
$641$	−36.8328	−1.45481	−0.727404	−	0.686209i	$-0.759274\pi$
$641$	−36.8328	−1.45481	−0.727404	+	0.686209i	$0.759274\pi$
$642$	11.0557i	0.436335i
$643$	− 32.9443i	− 1.29920i	−0.760278	−	0.649598i	$-0.774937\pi$
$643$	− 32.9443i	− 1.29920i	0.760278	−	0.649598i	$-0.225063\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−28.9443	−1.13880
$647$	− 33.8885i	− 1.33230i	−0.745820	−	0.666148i	$-0.767942\pi$
$647$	− 33.8885i	− 1.33230i	0.745820	−	0.666148i	$-0.232058\pi$
$648$	2.23607i	0.0878410i
$649$	22.1115	0.867951
$650$	0	0
$651$	−10.4721	−0.410435
$652$	− 2.83282i	− 0.110942i
$653$	− 49.4164i	− 1.93381i	−0.255130	−	0.966907i	$-0.582118\pi$
$653$	− 49.4164i	− 1.93381i	0.255130	−	0.966907i	$-0.417882\pi$
$654$	−4.47214	−0.174874
$655$	0	0
$656$	2.00000	0.0780869
$657$	3.52786i	0.137635i
$658$	− 11.0557i	− 0.430997i
$659$	41.3050	1.60901	0.804506	−	0.593944i	$-0.202430\pi$
$659$	41.3050	1.60901	0.804506	+	0.593944i	$0.202430\pi$
$660$	0	0
$661$	−0.111456	−0.00433514	−0.00216757	−	0.999998i	$-0.500690\pi$
$661$	−0.111456	−0.00433514	−0.00216757	+	0.999998i	$0.500690\pi$
$662$	37.8885i	1.47258i
$663$	− 8.94427i	− 0.347367i
$664$	−2.11146	−0.0819404
$665$	0	0
$666$	24.4721	0.948276
$667$	8.00000i	0.309761i
$668$	− 24.0000i	− 0.928588i
$669$	4.94427	0.191157
$670$	0	0
$671$	4.94427	0.190872
$672$	− 6.70820i	− 0.258775i
$673$	− 44.8328i	− 1.72818i	−0.503339	−	0.864089i	$-0.667895\pi$
$673$	− 44.8328i	− 1.72818i	0.503339	−	0.864089i	$-0.332105\pi$
$674$	−26.5836	−1.02396
$675$	0	0
$676$	21.0000	0.807692
$677$	− 38.9443i	− 1.49675i	−0.663276	−	0.748375i	$-0.730834\pi$
$677$	− 38.9443i	− 1.49675i	0.663276	−	0.748375i	$-0.269166\pi$
$678$	18.9443i	0.727550i
$679$	0.472136	0.0181189
$680$	0	0
$681$	16.9443	0.649306
$682$	57.8885i	2.21667i
$683$	33.8885i	1.29671i	0.761339	+	0.648355i	$0.224543\pi$
$683$	33.8885i	1.29671i	−0.761339	+	0.648355i	$0.775457\pi$
$684$	−19.4164	−0.742405
$685$	0	0
$686$	2.23607	0.0853735
$687$	− 11.8885i	− 0.453576i
$688$	8.94427i	0.340997i
$689$	−55.7771	−2.12494
$690$	0	0
$691$	−0.360680	−0.0137209	−0.00686045	−	0.999976i	$-0.502184\pi$
$691$	−0.360680	−0.0137209	−0.00686045	+	0.999976i	$0.502184\pi$
$692$	− 44.8328i	− 1.70429i
$693$	− 2.47214i	− 0.0939087i
$694$	17.8885	0.679040
$695$	0	0
$696$	4.47214	0.169516
$697$	− 4.00000i	− 0.151511i
$698$	53.4164i	2.02184i
$699$	17.4164	0.658749
$700$	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
$703$	70.8328i	2.67151i
$704$	−32.1378	−1.21124
$705$	0	0
$706$	−62.3607	−2.34698
$707$	14.0000i	0.526524i
$708$	− 26.8328i	− 1.00844i
$709$	45.7771	1.71919	0.859597	−	0.510972i	$-0.170714\pi$
$709$	45.7771	1.71919	0.859597	+	0.510972i	$0.170714\pi$
$710$	0	0
$711$	−4.94427	−0.185425
$712$	4.47214i	0.167600i
$713$	41.8885i	1.56874i
$714$	−4.47214	−0.167365
$715$	0	0
$716$	−7.41641	−0.277164
$717$	− 1.52786i	− 0.0570591i
$718$	21.3050i	0.795094i
$719$	−46.8328	−1.74657	−0.873285	−	0.487210i	$-0.838015\pi$
$719$	−46.8328	−1.74657	−0.873285	+	0.487210i	$0.838015\pi$
$720$	0	0
$721$	0	0
$722$	− 51.1803i	− 1.90474i
$723$	1.05573i	0.0392630i
$724$	−56.8328	−2.11217
$725$	0	0
$726$	10.9311	0.405692
$727$	− 14.8328i	− 0.550119i	−0.961427	−	0.275059i	$-0.911302\pi$
$727$	− 14.8328i	− 0.550119i	0.961427	−	0.275059i	$-0.0886975\pi$
$728$	− 10.0000i	− 0.370625i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	17.8885	0.661632
$732$	− 6.00000i	− 0.221766i
$733$	37.4164i	1.38201i	0.722852	+	0.691003i	$0.242831\pi$
$733$	37.4164i	1.38201i	−0.722852	+	0.691003i	$0.757169\pi$
$734$	−46.8328	−1.72863
$735$	0	0
$736$	−26.8328	−0.989071
$737$	− 9.88854i	− 0.364249i
$738$	− 4.47214i	− 0.164622i
$739$	−29.8885	−1.09947	−0.549734	−	0.835340i	$-0.685271\pi$
$739$	−29.8885	−1.09947	−0.549734	+	0.835340i	$0.685271\pi$
$740$	0	0
$741$	28.9443	1.06329
$742$	27.8885i	1.02382i
$743$	− 18.8328i	− 0.690909i	−0.938436	−	0.345455i	$-0.887725\pi$
$743$	− 18.8328i	− 0.690909i	0.938436	−	0.345455i	$-0.112275\pi$
$744$	23.4164	0.858487
$745$	0	0
$746$	13.4164	0.491210
$747$	− 0.944272i	− 0.0345491i
$748$	14.8328i	0.542341i
$749$	−4.94427	−0.180660
$750$	0	0
$751$	−3.05573	−0.111505	−0.0557526	−	0.998445i	$-0.517756\pi$
$751$	−3.05573	−0.111505	−0.0557526	+	0.998445i	$0.517756\pi$
$752$	− 4.94427i	− 0.180299i
$753$	0.944272i	0.0344112i
$754$	−20.0000	−0.728357
$755$	0	0
$756$	−3.00000	−0.109109
$757$	3.88854i	0.141332i	0.997500	+	0.0706658i	$0.0225124\pi$
$757$	3.88854i	0.141332i	−0.997500	+	0.0706658i	$0.977488\pi$
$758$	− 4.72136i	− 0.171488i
$759$	−9.88854	−0.358931
$760$	0	0
$761$	7.88854	0.285959	0.142980	−	0.989726i	$-0.454332\pi$
$761$	7.88854	0.285959	0.142980	+	0.989726i	$0.454332\pi$
$762$	28.9443i	1.04854i
$763$	− 2.00000i	− 0.0724049i
$764$	−82.2492	−2.97567
$765$	0	0
$766$	−17.8885	−0.646339
$767$	40.0000i	1.44432i
$768$	9.00000i	0.324760i
$769$	−0.832816	−0.0300321	−0.0150161	−	0.999887i	$-0.504780\pi$
$769$	−0.832816	−0.0300321	−0.0150161	+	0.999887i	$0.504780\pi$
$770$	0	0
$771$	−1.05573	−0.0380211
$772$	42.0000i	1.51161i
$773$	− 25.0557i	− 0.901192i	−0.892728	−	0.450596i	$-0.851212\pi$
$773$	− 25.0557i	− 0.901192i	0.892728	−	0.450596i	$-0.148788\pi$
$774$	20.0000	0.718885
$775$	0	0
$776$	−1.05573	−0.0378984
$777$	10.9443i	0.392624i
$778$	24.4721i	0.877369i
$779$	12.9443	0.463777
$780$	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
$787$	− 48.9443i	− 1.74467i	−0.488904	−	0.872337i	$-0.662603\pi$
$787$	− 48.9443i	− 1.74467i	0.488904	−	0.872337i	$-0.337397\pi$
$788$	73.4164i	2.61535i
$789$	24.9443	0.888040
$790$	0	0
$791$	−8.47214	−0.301234
$792$	5.52786i	0.196424i
$793$	8.94427i	0.317620i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	1.75078	0.0620546
$797$	1.05573i	0.0373958i	0.999825	+	0.0186979i	$0.00595207\pi$
$797$	1.05573i	0.0373958i	−0.999825	+	0.0186979i	$0.994048\pi$
$798$	− 14.4721i	− 0.512308i
$799$	−9.88854	−0.349832
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	− 22.3607i	− 0.789583i
$803$	8.72136i	0.307770i
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−104.721	−3.68865
$807$	− 23.8885i	− 0.840917i
$808$	− 31.3050i	− 1.10130i
$809$	−21.0557	−0.740280	−0.370140	−	0.928976i	$-0.620690\pi$
$809$	−21.0557	−0.740280	−0.370140	+	0.928976i	$0.620690\pi$
$810$	0	0
$811$	28.5836	1.00371	0.501853	−	0.864953i	$-0.332652\pi$
$811$	28.5836	1.00371	0.501853	+	0.864953i	$0.332652\pi$
$812$	6.00000i	0.210559i
$813$	10.4721i	0.367274i
$814$	60.4984	2.12047
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	57.8885i	2.02526i
$818$	− 53.4164i	− 1.86766i
$819$	4.47214	0.156269
$820$	0	0
$821$	−37.7771	−1.31843	−0.659215	−	0.751955i	$-0.729111\pi$
$821$	−37.7771	−1.31843	−0.659215	+	0.751955i	$0.729111\pi$
$822$	27.8885i	0.972725i
$823$	27.0557i	0.943103i	0.881838	+	0.471552i	$0.156306\pi$
$823$	27.0557i	0.943103i	−0.881838	+	0.471552i	$0.843694\pi$
$824$	0	0
$825$	0	0
$826$	20.0000	0.695889
$827$	4.94427i	0.171929i	0.996298	+	0.0859646i	$0.0273972\pi$
$827$	4.94427i	0.171929i	−0.996298	+	0.0859646i	$0.972603\pi$
$828$	12.0000i	0.417029i
$829$	30.9443	1.07474	0.537369	−	0.843347i	$-0.319418\pi$
$829$	30.9443	1.07474	0.537369	+	0.843347i	$0.319418\pi$
$830$	0	0
$831$	−1.05573	−0.0366228
$832$	− 58.1378i	− 2.01556i
$833$	− 2.00000i	− 0.0692959i
$834$	−43.4164	−1.50339
$835$	0	0
$836$	−48.0000	−1.66011
$837$	10.4721i	0.361970i
$838$	13.1672i	0.454853i
$839$	−1.16718	−0.0402957	−0.0201478	−	0.999797i	$-0.506414\pi$
$839$	−1.16718	−0.0402957	−0.0201478	+	0.999797i	$0.506414\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	− 49.1935i	− 1.69532i
$843$	− 6.94427i	− 0.239173i
$844$	−2.83282	−0.0975095
$845$	0	0
$846$	−11.0557	−0.380104
$847$	4.88854i	0.167972i
$848$	12.4721i	0.428295i
$849$	12.0000	0.411839
$850$	0	0
$851$	43.7771	1.50066
$852$	43.4164i	1.48742i
$853$	31.3050i	1.07186i	0.844262	+	0.535931i	$0.180039\pi$
$853$	31.3050i	1.07186i	−0.844262	+	0.535931i	$0.819961\pi$
$854$	4.47214	0.153033
$855$	0	0
$856$	11.0557	0.377877
$857$	16.8328i	0.574998i	0.957781	+	0.287499i	$0.0928238\pi$
$857$	16.8328i	0.574998i	−0.957781	+	0.287499i	$0.907176\pi$
$858$	− 24.7214i	− 0.843973i
$859$	−41.5279	−1.41691	−0.708456	−	0.705755i	$-0.750608\pi$
$859$	−41.5279	−1.41691	−0.708456	+	0.705755i	$0.750608\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	21.3050i	0.725650i
$863$	− 13.8885i	− 0.472772i	−0.971659	−	0.236386i	$-0.924037\pi$
$863$	− 13.8885i	− 0.472772i	0.971659	−	0.236386i	$-0.0759629\pi$
$864$	−6.70820	−0.228218
$865$	0	0
$866$	16.8328	0.572002
$867$	− 13.0000i	− 0.441503i
$868$	31.4164i	1.06634i
$869$	−12.2229	−0.414634
$870$	0	0
$871$	17.8885	0.606130
$872$	4.47214i	0.151446i
$873$	− 0.472136i	− 0.0159794i
$874$	−57.8885	−1.95811
$875$	0	0
$876$	10.5836	0.357586
$877$	3.16718i	0.106948i	0.998569	+	0.0534741i	$0.0170295\pi$
$877$	3.16718i	0.106948i	−0.998569	+	0.0534741i	$0.982971\pi$
$878$	23.4164i	0.790265i
$879$	22.9443	0.773891
$880$	0	0
$881$	7.88854	0.265772	0.132886	−	0.991131i	$-0.457576\pi$
$881$	7.88854	0.265772	0.132886	+	0.991131i	$0.457576\pi$
$882$	− 2.23607i	− 0.0752923i
$883$	− 2.11146i	− 0.0710562i	−0.999369	−	0.0355281i	$-0.988689\pi$
$883$	− 2.11146i	− 0.0710562i	0.999369	−	0.0355281i	$-0.0113113\pi$
$884$	−26.8328	−0.902485
$885$	0	0
$886$	−17.8885	−0.600977
$887$	− 22.8328i	− 0.766651i	−0.923613	−	0.383325i	$-0.874779\pi$
$887$	− 22.8328i	− 0.766651i	0.923613	−	0.383325i	$-0.125221\pi$
$888$	− 24.4721i	− 0.821231i
$889$	−12.9443	−0.434137
$890$	0	0
$891$	−2.47214	−0.0828197
$892$	− 14.8328i	− 0.496639i
$893$	− 32.0000i	− 1.07084i
$894$	6.58359	0.220188
$895$	0	0
$896$	−15.6525	−0.522913
$897$	− 17.8885i	− 0.597281i
$898$	− 31.3050i	− 1.04466i
$899$	20.9443	0.698531
$900$	0	0
$901$	24.9443	0.831014
$902$	− 11.0557i	− 0.368115i
$903$	8.94427i	0.297647i
$904$	18.9443	0.630077
$905$	0	0
$906$	35.7771	1.18861
$907$	− 18.1115i	− 0.601381i	−0.953722	−	0.300691i	$-0.902783\pi$
$907$	− 18.1115i	− 0.601381i	0.953722	−	0.300691i	$-0.0972171\pi$
$908$	− 50.8328i	− 1.68695i
$909$	14.0000	0.464351
$910$	0	0
$911$	−34.2492	−1.13473	−0.567364	−	0.823467i	$-0.692037\pi$
$911$	−34.2492	−1.13473	−0.567364	+	0.823467i	$0.692037\pi$
$912$	− 6.47214i	− 0.214314i
$913$	− 2.33437i	− 0.0772563i
$914$	24.4721	0.809466
$915$	0	0
$916$	−35.6656	−1.17843
$917$	− 4.00000i	− 0.132092i
$918$	4.47214i	0.147602i
$919$	52.9443	1.74647	0.873235	−	0.487299i	$-0.162018\pi$
$919$	52.9443	1.74647	0.873235	+	0.487299i	$0.162018\pi$
$920$	0	0
$921$	32.9443	1.08555
$922$	71.3050i	2.34830i
$923$	− 64.7214i	− 2.13033i
$924$	−7.41641	−0.243982
$925$	0	0
$926$	6.83282	0.224540
$927$	0	0
$928$	13.4164i	0.440415i
$929$	51.8885	1.70241	0.851204	−	0.524835i	$-0.175873\pi$
$929$	51.8885	1.70241	0.851204	+	0.524835i	$0.175873\pi$
$930$	0	0
$931$	6.47214	0.212116
$932$	− 52.2492i	− 1.71148i
$933$	9.88854i	0.323736i
$934$	−20.0000	−0.654420
$935$	0	0
$936$	−10.0000	−0.326860
$937$	43.5279i	1.42199i	0.703195	+	0.710997i	$0.251756\pi$
$937$	43.5279i	1.42199i	−0.703195	+	0.710997i	$0.748244\pi$
$938$	− 8.94427i	− 0.292041i
$939$	9.41641	0.307293
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	18.9443i	0.617238i
$943$	− 8.00000i	− 0.260516i
$944$	8.94427	0.291111
$945$	0	0
$946$	49.4427	1.60752
$947$	− 17.8885i	− 0.581300i	−0.956830	−	0.290650i	$-0.906129\pi$
$947$	− 17.8885i	− 0.581300i	0.956830	−	0.290650i	$-0.0938715\pi$
$948$	14.8328i	0.481747i
$949$	−15.7771	−0.512146
$950$	0	0
$951$	30.3607	0.984512
$952$	4.47214i	0.144943i
$953$	− 6.58359i	− 0.213263i	−0.994299	−	0.106632i	$-0.965993\pi$
$953$	− 6.58359i	− 0.213263i	0.994299	−	0.106632i	$-0.0340066\pi$
$954$	27.8885	0.902925
$955$	0	0
$956$	−4.58359	−0.148244
$957$	4.94427i	0.159826i
$958$	40.0000i	1.29234i
$959$	−12.4721	−0.402746
$960$	0	0
$961$	78.6656	2.53760
$962$	109.443i	3.52857i
$963$	4.94427i	0.159327i
$964$	3.16718	0.102008
$965$	0	0
$966$	−8.94427	−0.287777
$967$	9.88854i	0.317994i	0.987279	+	0.158997i	$0.0508260\pi$
$967$	9.88854i	0.317994i	−0.987279	+	0.158997i	$0.949174\pi$
$968$	− 10.9311i	− 0.351339i
$969$	−12.9443	−0.415830
$970$	0	0
$971$	23.0557	0.739894	0.369947	−	0.929053i	$-0.379376\pi$
$971$	23.0557	0.739894	0.369947	+	0.929053i	$0.379376\pi$
$972$	3.00000i	0.0962250i
$973$	− 19.4164i	− 0.622461i
$974$	6.83282	0.218938
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 57.4164i	− 1.83691i	−0.395521	−	0.918457i	$-0.629436\pi$
$977$	− 57.4164i	− 1.83691i	0.395521	−	0.918457i	$-0.370564\pi$
$978$	− 2.11146i	− 0.0675169i
$979$	−4.94427	−0.158020
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	− 92.3607i	− 2.94735i
$983$	30.8328i	0.983414i	0.870761	+	0.491707i	$0.163627\pi$
$983$	30.8328i	0.983414i	−0.870761	+	0.491707i	$0.836373\pi$
$984$	−4.47214	−0.142566
$985$	0	0
$986$	8.94427	0.284844
$987$	− 4.94427i	− 0.157378i
$988$	− 86.8328i	− 2.76252i
$989$	35.7771	1.13765
$990$	0	0
$991$	−12.9443	−0.411188	−0.205594	−	0.978637i	$-0.565913\pi$
$991$	−12.9443	−0.411188	−0.205594	+	0.978637i	$0.565913\pi$
$992$	70.2492i	2.23042i
$993$	16.9443i	0.537710i
$994$	−32.3607	−1.02642
$995$	0	0
$996$	−2.83282	−0.0897612
$997$	− 21.4164i	− 0.678264i	−0.940739	−	0.339132i	$-0.889867\pi$
$997$	− 21.4164i	− 0.678264i	0.940739	−	0.339132i	$-0.110133\pi$
$998$	48.9443i	1.54930i
$999$	10.9443	0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.d.c.274.1		4
3.2	odd	2		1575.2.d.d.1324.3		4
5.2	odd	4		105.2.a.b.1.2	✓	2
5.3	odd	4		525.2.a.g.1.1		2
5.4	even	2	inner	525.2.d.c.274.4		4
15.2	even	4		315.2.a.d.1.1		2
15.8	even	4		1575.2.a.r.1.2		2
15.14	odd	2		1575.2.d.d.1324.2		4
20.3	even	4		8400.2.a.cx.1.2		2
20.7	even	4		1680.2.a.v.1.2		2
35.2	odd	12		735.2.i.k.361.1		4
35.12	even	12		735.2.i.i.361.1		4
35.13	even	4		3675.2.a.y.1.1		2
35.17	even	12		735.2.i.i.226.1		4
35.27	even	4		735.2.a.k.1.2		2
35.32	odd	12		735.2.i.k.226.1		4
40.27	even	4		6720.2.a.cs.1.1		2
40.37	odd	4		6720.2.a.cx.1.2		2
60.47	odd	4		5040.2.a.bw.1.1		2
105.62	odd	4		2205.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.a.b.1.2	✓	2	5.2	odd	4
315.2.a.d.1.1		2	15.2	even	4
525.2.a.g.1.1		2	5.3	odd	4
525.2.d.c.274.1		4	1.1	even	1	trivial
525.2.d.c.274.4		4	5.4	even	2	inner
735.2.a.k.1.2		2	35.27	even	4
735.2.i.i.226.1		4	35.17	even	12
735.2.i.i.361.1		4	35.12	even	12
735.2.i.k.226.1		4	35.32	odd	12
735.2.i.k.361.1		4	35.2	odd	12
1575.2.a.r.1.2		2	15.8	even	4
1575.2.d.d.1324.2		4	15.14	odd	2
1575.2.d.d.1324.3		4	3.2	odd	2
1680.2.a.v.1.2		2	20.7	even	4
2205.2.a.w.1.1		2	105.62	odd	4
3675.2.a.y.1.1		2	35.13	even	4
5040.2.a.bw.1.1		2	60.47	odd	4
6720.2.a.cs.1.1		2	40.27	even	4
6720.2.a.cx.1.2		2	40.37	odd	4
8400.2.a.cx.1.2		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)

\(f(q)\)	\(=\)	\(q-2.23607i q^{2} -1.00000i q^{3} -3.00000 q^{4} -2.23607 q^{6} -1.00000i q^{7} +2.23607i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.23607i q^{2} -1.00000i q^{3} -3.00000 q^{4} -2.23607 q^{6} -1.00000i q^{7} +2.23607i q^{8} -1.00000 q^{9} -2.47214 q^{11} +3.00000i q^{12} -4.47214i q^{13} -2.23607 q^{14} -1.00000 q^{16} +2.00000i q^{17} +2.23607i q^{18} -6.47214 q^{19} -1.00000 q^{21} +5.52786i q^{22} +4.00000i q^{23} +2.23607 q^{24} -10.0000 q^{26} +1.00000i q^{27} +3.00000i q^{28} +2.00000 q^{29} +10.4721 q^{31} +6.70820i q^{32} +2.47214i q^{33} +4.47214 q^{34} +3.00000 q^{36} -10.9443i q^{37} +14.4721i q^{38} -4.47214 q^{39} -2.00000 q^{41} +2.23607i q^{42} -8.94427i q^{43} +7.41641 q^{44} +8.94427 q^{46} +4.94427i q^{47} +1.00000i q^{48} -1.00000 q^{49} +2.00000 q^{51} +13.4164i q^{52} -12.4721i q^{53} +2.23607 q^{54} +2.23607 q^{56} +6.47214i q^{57} -4.47214i q^{58} -8.94427 q^{59} -2.00000 q^{61} -23.4164i q^{62} +1.00000i q^{63} +13.0000 q^{64} +5.52786 q^{66} +4.00000i q^{67} -6.00000i q^{68} +4.00000 q^{69} +14.4721 q^{71} -2.23607i q^{72} -3.52786i q^{73} -24.4721 q^{74} +19.4164 q^{76} +2.47214i q^{77} +10.0000i q^{78} +4.94427 q^{79} +1.00000 q^{81} +4.47214i q^{82} +0.944272i q^{83} +3.00000 q^{84} -20.0000 q^{86} -2.00000i q^{87} -5.52786i q^{88} +2.00000 q^{89} -4.47214 q^{91} -12.0000i q^{92} -10.4721i q^{93} +11.0557 q^{94} +6.70820 q^{96} +0.472136i q^{97} +2.23607i q^{98} +2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4} - 4 q^{9}+O(q^{10}) \) 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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