LMFDB - Embedded newform 3450.2.d.z.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(2899,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{57})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 196 $ x^4 + 29*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.2
Root			$-4.27492i$ of defining polynomial
Character	$\chi$	$=$	3450.2899
Dual form			3450.2.d.z.2899.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.27492 q^{11} -1.00000i q^{12} -4.27492i q^{13} -3.00000 q^{14} +1.00000 q^{16} -5.27492i q^{17} +1.00000i q^{18} +4.27492 q^{19} +3.00000 q^{21} -6.27492i q^{22} -1.00000i q^{23} -1.00000 q^{24} -4.27492 q^{26} -1.00000i q^{27} +3.00000i q^{28} -5.54983 q^{29} +6.00000 q^{31} -1.00000i q^{32} +6.27492i q^{33} -5.27492 q^{34} +1.00000 q^{36} +11.8248i q^{37} -4.27492i q^{38} +4.27492 q^{39} -10.2749 q^{41} -3.00000i q^{42} -0.274917i q^{43} -6.27492 q^{44} -1.00000 q^{46} -0.725083i q^{47} +1.00000i q^{48} -2.00000 q^{49} +5.27492 q^{51} +4.27492i q^{52} +4.54983i q^{53} -1.00000 q^{54} +3.00000 q^{56} +4.27492i q^{57} +5.54983i q^{58} +2.54983 q^{59} -14.5498 q^{61} -6.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} +6.27492 q^{66} -10.5498i q^{67} +5.27492i q^{68} +1.00000 q^{69} +13.8248 q^{71} -1.00000i q^{72} -0.450166i q^{73} +11.8248 q^{74} -4.27492 q^{76} -18.8248i q^{77} -4.27492i q^{78} -10.8248 q^{79} +1.00000 q^{81} +10.2749i q^{82} -17.5498i q^{83} -3.00000 q^{84} -0.274917 q^{86} -5.54983i q^{87} +6.27492i q^{88} -0.725083 q^{89} -12.8248 q^{91} +1.00000i q^{92} +6.00000i q^{93} -0.725083 q^{94} +1.00000 q^{96} -14.0000i q^{97} +2.00000i q^{98} -6.27492 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 10 q^{11} - 12 q^{14} + 4 q^{16} + 2 q^{19} + 12 q^{21} - 4 q^{24} - 2 q^{26} + 8 q^{29} + 24 q^{31} - 6 q^{34} + 4 q^{36} + 2 q^{39} - 26 q^{41} - 10 q^{44} - 4 q^{46} - 8 q^{49} + 6 q^{51} - 4 q^{54} + 12 q^{56} - 20 q^{59} - 28 q^{61} - 4 q^{64} + 10 q^{66} + 4 q^{69} + 10 q^{71} + 2 q^{74} - 2 q^{76} + 2 q^{79} + 4 q^{81} - 12 q^{84} + 14 q^{86} - 18 q^{89} - 6 q^{91} - 18 q^{94} + 4 q^{96} - 10 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 10 * q^11 - 12 * q^14 + 4 * q^16 + 2 * q^19 + 12 * q^21 - 4 * q^24 - 2 * q^26 + 8 * q^29 + 24 * q^31 - 6 * q^34 + 4 * q^36 + 2 * q^39 - 26 * q^41 - 10 * q^44 - 4 * q^46 - 8 * q^49 + 6 * q^51 - 4 * q^54 + 12 * q^56 - 20 * q^59 - 28 * q^61 - 4 * q^64 + 10 * q^66 + 4 * q^69 + 10 * q^71 + 2 * q^74 - 2 * q^76 + 2 * q^79 + 4 * q^81 - 12 * q^84 + 14 * q^86 - 18 * q^89 - 6 * q^91 - 18 * q^94 + 4 * q^96 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 3.00000i	− 1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$	− 3.00000i	− 1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	6.27492	1.89196	0.945979	−	0.324227i	$-0.105104\pi$
$11$	6.27492	1.89196	0.945979	+	0.324227i	$0.105104\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 4.27492i	− 1.18565i	−0.805332	−	0.592824i	$-0.798013\pi$
$13$	− 4.27492i	− 1.18565i	0.805332	−	0.592824i	$-0.201987\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	− 5.27492i	− 1.27936i	−0.768643	−	0.639678i	$-0.779068\pi$
$17$	− 5.27492i	− 1.27936i	0.768643	−	0.639678i	$-0.220932\pi$
$18$	1.00000i	0.235702i
$19$	4.27492	0.980733	0.490367	−	0.871516i	$-0.336863\pi$
$19$	4.27492	0.980733	0.490367	+	0.871516i	$0.336863\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	− 6.27492i	− 1.33782i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.27492	−0.838380
$27$	− 1.00000i	− 0.192450i
$28$	3.00000i	0.566947i
$29$	−5.54983	−1.03058	−0.515289	−	0.857016i	$-0.672315\pi$
$29$	−5.54983	−1.03058	−0.515289	+	0.857016i	$0.672315\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	− 1.00000i	− 0.176777i
$33$	6.27492i	1.09232i
$34$	−5.27492	−0.904641
$35$	0	0
$36$	1.00000	0.166667
$37$	11.8248i	1.94398i	0.235028	+	0.971989i	$0.424482\pi$
$37$	11.8248i	1.94398i	−0.235028	+	0.971989i	$0.575518\pi$
$38$	− 4.27492i	− 0.693483i
$39$	4.27492	0.684535
$40$	0	0
$41$	−10.2749	−1.60467	−0.802336	−	0.596872i	$-0.796410\pi$
$41$	−10.2749	−1.60467	−0.802336	+	0.596872i	$0.796410\pi$
$42$	− 3.00000i	− 0.462910i
$43$	− 0.274917i	− 0.0419245i	−0.999780	−	0.0209622i	$-0.993327\pi$
$43$	− 0.274917i	− 0.0419245i	0.999780	−	0.0209622i	$-0.00667298\pi$
$44$	−6.27492	−0.945979
$45$	0	0
$46$	−1.00000	−0.147442
$47$	− 0.725083i	− 0.105764i	−0.998601	−	0.0528821i	$-0.983159\pi$
$47$	− 0.725083i	− 0.105764i	0.998601	−	0.0528821i	$-0.0168407\pi$
$48$	1.00000i	0.144338i
$49$	−2.00000	−0.285714
$50$	0	0
$51$	5.27492	0.738636
$52$	4.27492i	0.592824i
$53$	4.54983i	0.624968i	0.949923	+	0.312484i	$0.101161\pi$
$53$	4.54983i	0.624968i	−0.949923	+	0.312484i	$0.898839\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.00000	0.400892
$57$	4.27492i	0.566227i
$58$	5.54983i	0.728729i
$59$	2.54983	0.331960	0.165980	−	0.986129i	$-0.446921\pi$
$59$	2.54983	0.331960	0.165980	+	0.986129i	$0.446921\pi$
$60$	0	0
$61$	−14.5498	−1.86292	−0.931458	−	0.363850i	$-0.881462\pi$
$61$	−14.5498	−1.86292	−0.931458	+	0.363850i	$0.881462\pi$
$62$	− 6.00000i	− 0.762001i
$63$	3.00000i	0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	6.27492	0.772389
$67$	− 10.5498i	− 1.28887i	−0.764660	−	0.644434i	$-0.777093\pi$
$67$	− 10.5498i	− 1.28887i	0.764660	−	0.644434i	$-0.222907\pi$
$68$	5.27492i	0.639678i
$69$	1.00000	0.120386
$70$	0	0
$71$	13.8248	1.64070	0.820348	−	0.571865i	$-0.193780\pi$
$71$	13.8248	1.64070	0.820348	+	0.571865i	$0.193780\pi$
$72$	− 1.00000i	− 0.117851i
$73$	− 0.450166i	− 0.0526879i	−0.999653	−	0.0263439i	$-0.991613\pi$
$73$	− 0.450166i	− 0.0526879i	0.999653	−	0.0263439i	$-0.00838651\pi$
$74$	11.8248	1.37460
$75$	0	0
$76$	−4.27492	−0.490367
$77$	− 18.8248i	− 2.14528i
$78$	− 4.27492i	− 0.484039i
$79$	−10.8248	−1.21788	−0.608940	−	0.793216i	$-0.708405\pi$
$79$	−10.8248	−1.21788	−0.608940	+	0.793216i	$0.708405\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.2749i	1.13467i
$83$	− 17.5498i	− 1.92634i	−0.268885	−	0.963172i	$-0.586655\pi$
$83$	− 17.5498i	− 1.92634i	0.268885	−	0.963172i	$-0.413345\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−0.274917	−0.0296451
$87$	− 5.54983i	− 0.595005i
$88$	6.27492i	0.668908i
$89$	−0.725083	−0.0768586	−0.0384293	−	0.999261i	$-0.512235\pi$
$89$	−0.725083	−0.0768586	−0.0384293	+	0.999261i	$0.512235\pi$
$90$	0	0
$91$	−12.8248	−1.34440
$92$	1.00000i	0.104257i
$93$	6.00000i	0.622171i
$94$	−0.725083	−0.0747866
$95$	0	0
$96$	1.00000	0.102062
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	2.00000i	0.202031i
$99$	−6.27492	−0.630653
$100$	0	0
$101$	0.725083	0.0721484	0.0360742	−	0.999349i	$-0.488515\pi$
$101$	0.725083	0.0721484	0.0360742	+	0.999349i	$0.488515\pi$
$102$	− 5.27492i	− 0.522295i
$103$	− 3.00000i	− 0.295599i	−0.989017	−	0.147799i	$-0.952781\pi$
$103$	− 3.00000i	− 0.295599i	0.989017	−	0.147799i	$-0.0472190\pi$
$104$	4.27492	0.419190
$105$	0	0
$106$	4.54983	0.441919
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	1.00000i	0.0962250i
$109$	19.8248	1.89887	0.949433	−	0.313968i	$-0.101659\pi$
$109$	19.8248	1.89887	0.949433	+	0.313968i	$0.101659\pi$
$110$	0	0
$111$	−11.8248	−1.12236
$112$	− 3.00000i	− 0.283473i
$113$	− 11.2749i	− 1.06065i	−0.847793	−	0.530327i	$-0.822069\pi$
$113$	− 11.2749i	− 1.06065i	0.847793	−	0.530327i	$-0.177931\pi$
$114$	4.27492	0.400383
$115$	0	0
$116$	5.54983	0.515289
$117$	4.27492i	0.395216i
$118$	− 2.54983i	− 0.234731i
$119$	−15.8248	−1.45065
$120$	0	0
$121$	28.3746	2.57951
$122$	14.5498i	1.31728i
$123$	− 10.2749i	− 0.926458i
$124$	−6.00000	−0.538816
$125$	0	0
$126$	3.00000	0.267261
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	1.00000i	0.0883883i
$129$	0.274917	0.0242051
$130$	0	0
$131$	−8.54983	−0.747003	−0.373501	−	0.927630i	$-0.621843\pi$
$131$	−8.54983	−0.747003	−0.373501	+	0.927630i	$0.621843\pi$
$132$	− 6.27492i	− 0.546161i
$133$	− 12.8248i	− 1.11205i
$134$	−10.5498	−0.911367
$135$	0	0
$136$	5.27492	0.452320
$137$	13.8248i	1.18113i	0.806991	+	0.590564i	$0.201095\pi$
$137$	13.8248i	1.18113i	−0.806991	+	0.590564i	$0.798905\pi$
$138$	− 1.00000i	− 0.0851257i
$139$	1.27492	0.108137	0.0540685	−	0.998537i	$-0.482781\pi$
$139$	1.27492	0.108137	0.0540685	+	0.998537i	$0.482781\pi$
$140$	0	0
$141$	0.725083	0.0610630
$142$	− 13.8248i	− 1.16015i
$143$	− 26.8248i	− 2.24320i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−0.450166	−0.0372560
$147$	− 2.00000i	− 0.164957i
$148$	− 11.8248i	− 0.971989i
$149$	−12.5498	−1.02812	−0.514061	−	0.857753i	$-0.671860\pi$
$149$	−12.5498	−1.02812	−0.514061	+	0.857753i	$0.671860\pi$
$150$	0	0
$151$	12.5498	1.02129	0.510646	−	0.859791i	$-0.329406\pi$
$151$	12.5498	1.02129	0.510646	+	0.859791i	$0.329406\pi$
$152$	4.27492i	0.346742i
$153$	5.27492i	0.426452i
$154$	−18.8248	−1.51694
$155$	0	0
$156$	−4.27492	−0.342267
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	10.8248i	0.861171i
$159$	−4.54983	−0.360825
$160$	0	0
$161$	−3.00000	−0.236433
$162$	− 1.00000i	− 0.0785674i
$163$	21.6495i	1.69572i	0.530220	+	0.847860i	$0.322109\pi$
$163$	21.6495i	1.69572i	−0.530220	+	0.847860i	$0.677891\pi$
$164$	10.2749	0.802336
$165$	0	0
$166$	−17.5498	−1.36213
$167$	4.72508i	0.365638i	0.983147	+	0.182819i	$0.0585222\pi$
$167$	4.72508i	0.365638i	−0.983147	+	0.182819i	$0.941478\pi$
$168$	3.00000i	0.231455i
$169$	−5.27492	−0.405763
$170$	0	0
$171$	−4.27492	−0.326911
$172$	0.274917i	0.0209622i
$173$	− 7.72508i	− 0.587327i	−0.955909	−	0.293664i	$-0.905125\pi$
$173$	− 7.72508i	− 0.587327i	0.955909	−	0.293664i	$-0.0948745\pi$
$174$	−5.54983	−0.420732
$175$	0	0
$176$	6.27492	0.472990
$177$	2.54983i	0.191657i
$178$	0.725083i	0.0543473i
$179$	−11.4502	−0.855826	−0.427913	−	0.903820i	$-0.640751\pi$
$179$	−11.4502	−0.855826	−0.427913	+	0.903820i	$0.640751\pi$
$180$	0	0
$181$	9.27492	0.689399	0.344700	−	0.938713i	$-0.387981\pi$
$181$	9.27492	0.689399	0.344700	+	0.938713i	$0.387981\pi$
$182$	12.8248i	0.950634i
$183$	− 14.5498i	− 1.07555i
$184$	1.00000	0.0737210
$185$	0	0
$186$	6.00000	0.439941
$187$	− 33.0997i	− 2.42049i
$188$	0.725083i	0.0528821i
$189$	−3.00000	−0.218218
$190$	0	0
$191$	15.7251	1.13783	0.568914	−	0.822397i	$-0.307364\pi$
$191$	15.7251	1.13783	0.568914	+	0.822397i	$0.307364\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	− 15.2749i	− 1.09951i	−0.835325	−	0.549756i	$-0.814721\pi$
$193$	− 15.2749i	− 1.09951i	0.835325	−	0.549756i	$-0.185279\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	2.00000	0.142857
$197$	19.5498i	1.39287i	0.717621	+	0.696434i	$0.245231\pi$
$197$	19.5498i	1.39287i	−0.717621	+	0.696434i	$0.754769\pi$
$198$	6.27492i	0.445939i
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	10.5498	0.744128
$202$	− 0.725083i	− 0.0510166i
$203$	16.6495i	1.16857i
$204$	−5.27492	−0.369318
$205$	0	0
$206$	−3.00000	−0.209020
$207$	1.00000i	0.0695048i
$208$	− 4.27492i	− 0.296412i
$209$	26.8248	1.85551
$210$	0	0
$211$	1.82475	0.125621	0.0628105	−	0.998025i	$-0.479994\pi$
$211$	1.82475	0.125621	0.0628105	+	0.998025i	$0.479994\pi$
$212$	− 4.54983i	− 0.312484i
$213$	13.8248i	0.947256i
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	− 18.0000i	− 1.22192i
$218$	− 19.8248i	− 1.34270i
$219$	0.450166	0.0304194
$220$	0	0
$221$	−22.5498	−1.51687
$222$	11.8248i	0.793625i
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	−11.2749	−0.749996
$227$	2.72508i	0.180870i	0.995902	+	0.0904350i	$0.0288258\pi$
$227$	2.72508i	0.180870i	−0.995902	+	0.0904350i	$0.971174\pi$
$228$	− 4.27492i	− 0.283113i
$229$	−11.0997	−0.733487	−0.366743	−	0.930322i	$-0.619527\pi$
$229$	−11.0997	−0.733487	−0.366743	+	0.930322i	$0.619527\pi$
$230$	0	0
$231$	18.8248	1.23858
$232$	− 5.54983i	− 0.364364i
$233$	− 23.3746i	− 1.53132i	−0.643245	−	0.765660i	$-0.722413\pi$
$233$	− 23.3746i	− 1.53132i	0.643245	−	0.765660i	$-0.277587\pi$
$234$	4.27492	0.279460
$235$	0	0
$236$	−2.54983	−0.165980
$237$	− 10.8248i	− 0.703143i
$238$	15.8248i	1.02577i
$239$	−4.72508	−0.305640	−0.152820	−	0.988254i	$-0.548836\pi$
$239$	−4.72508	−0.305640	−0.152820	+	0.988254i	$0.548836\pi$
$240$	0	0
$241$	−4.54983	−0.293081	−0.146540	−	0.989205i	$-0.546814\pi$
$241$	−4.54983	−0.293081	−0.146540	+	0.989205i	$0.546814\pi$
$242$	− 28.3746i	− 1.82399i
$243$	1.00000i	0.0641500i
$244$	14.5498	0.931458
$245$	0	0
$246$	−10.2749	−0.655105
$247$	− 18.2749i	− 1.16281i
$248$	6.00000i	0.381000i
$249$	17.5498	1.11218
$250$	0	0
$251$	4.17525	0.263539	0.131770	−	0.991280i	$-0.457934\pi$
$251$	4.17525	0.263539	0.131770	+	0.991280i	$0.457934\pi$
$252$	− 3.00000i	− 0.188982i
$253$	− 6.27492i	− 0.394501i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	− 0.274917i	− 0.0171156i
$259$	35.4743	2.20426
$260$	0	0
$261$	5.54983	0.343526
$262$	8.54983i	0.528211i
$263$	10.5498i	0.650531i	0.945623	+	0.325265i	$0.105454\pi$
$263$	10.5498i	0.650531i	−0.945623	+	0.325265i	$0.894546\pi$
$264$	−6.27492	−0.386194
$265$	0	0
$266$	−12.8248	−0.786336
$267$	− 0.725083i	− 0.0443743i
$268$	10.5498i	0.644434i
$269$	−4.27492	−0.260646	−0.130323	−	0.991472i	$-0.541601\pi$
$269$	−4.27492	−0.260646	−0.130323	+	0.991472i	$0.541601\pi$
$270$	0	0
$271$	16.5498	1.00533	0.502665	−	0.864481i	$-0.332353\pi$
$271$	16.5498	1.00533	0.502665	+	0.864481i	$0.332353\pi$
$272$	− 5.27492i	− 0.319839i
$273$	− 12.8248i	− 0.776189i
$274$	13.8248	0.835184
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	16.8248i	1.01090i	0.862855	+	0.505451i	$0.168674\pi$
$277$	16.8248i	1.01090i	−0.862855	+	0.505451i	$0.831326\pi$
$278$	− 1.27492i	− 0.0764645i
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−32.3746	−1.93131	−0.965653	−	0.259835i	$-0.916332\pi$
$281$	−32.3746	−1.93131	−0.965653	+	0.259835i	$0.916332\pi$
$282$	− 0.725083i	− 0.0431781i
$283$	− 16.0000i	− 0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$	− 16.0000i	− 0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	−13.8248	−0.820348
$285$	0	0
$286$	−26.8248	−1.58618
$287$	30.8248i	1.81953i
$288$	1.00000i	0.0589256i
$289$	−10.8248	−0.636750
$290$	0	0
$291$	14.0000	0.820695
$292$	0.450166i	0.0263439i
$293$	23.6495i	1.38162i	0.723037	+	0.690809i	$0.242746\pi$
$293$	23.6495i	1.38162i	−0.723037	+	0.690809i	$0.757254\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	−11.8248	−0.687300
$297$	− 6.27492i	− 0.364108i
$298$	12.5498i	0.726992i
$299$	−4.27492	−0.247225
$300$	0	0
$301$	−0.824752	−0.0475379
$302$	− 12.5498i	− 0.722162i
$303$	0.725083i	0.0416549i
$304$	4.27492	0.245183
$305$	0	0
$306$	5.27492	0.301547
$307$	1.27492i	0.0727634i	0.999338	+	0.0363817i	$0.0115832\pi$
$307$	1.27492i	0.0727634i	−0.999338	+	0.0363817i	$0.988417\pi$
$308$	18.8248i	1.07264i
$309$	3.00000	0.170664
$310$	0	0
$311$	11.8248	0.670520	0.335260	−	0.942126i	$-0.391176\pi$
$311$	11.8248	0.670520	0.335260	+	0.942126i	$0.391176\pi$
$312$	4.27492i	0.242020i
$313$	− 32.0000i	− 1.80875i	−0.426742	−	0.904373i	$-0.640339\pi$
$313$	− 32.0000i	− 1.80875i	0.426742	−	0.904373i	$-0.359661\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	10.8248	0.608940
$317$	− 1.54983i	− 0.0870474i	−0.999052	−	0.0435237i	$-0.986142\pi$
$317$	− 1.54983i	− 0.0870474i	0.999052	−	0.0435237i	$-0.0138584\pi$
$318$	4.54983i	0.255142i
$319$	−34.8248	−1.94981
$320$	0	0
$321$	−12.0000	−0.669775
$322$	3.00000i	0.167183i
$323$	− 22.5498i	− 1.25471i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	21.6495	1.19906
$327$	19.8248i	1.09631i
$328$	− 10.2749i	− 0.567337i
$329$	−2.17525	−0.119925
$330$	0	0
$331$	−5.82475	−0.320157	−0.160079	−	0.987104i	$-0.551175\pi$
$331$	−5.82475	−0.320157	−0.160079	+	0.987104i	$0.551175\pi$
$332$	17.5498i	0.963172i
$333$	− 11.8248i	− 0.647992i
$334$	4.72508	0.258545
$335$	0	0
$336$	3.00000	0.163663
$337$	− 24.5498i	− 1.33731i	−0.743571	−	0.668657i	$-0.766869\pi$
$337$	− 24.5498i	− 1.33731i	0.743571	−	0.668657i	$-0.233131\pi$
$338$	5.27492i	0.286918i
$339$	11.2749	0.612369
$340$	0	0
$341$	37.6495	2.03883
$342$	4.27492i	0.231161i
$343$	− 15.0000i	− 0.809924i
$344$	0.274917	0.0148225
$345$	0	0
$346$	−7.72508	−0.415303
$347$	34.5498i	1.85473i	0.374155	+	0.927366i	$0.377933\pi$
$347$	34.5498i	1.85473i	−0.374155	+	0.927366i	$0.622067\pi$
$348$	5.54983i	0.297502i
$349$	19.7251	1.05586	0.527930	−	0.849288i	$-0.322968\pi$
$349$	19.7251	1.05586	0.527930	+	0.849288i	$0.322968\pi$
$350$	0	0
$351$	−4.27492	−0.228178
$352$	− 6.27492i	− 0.334454i
$353$	− 13.3746i	− 0.711857i	−0.934513	−	0.355929i	$-0.884165\pi$
$353$	− 13.3746i	− 0.711857i	0.934513	−	0.355929i	$-0.115835\pi$
$354$	2.54983	0.135522
$355$	0	0
$356$	0.725083	0.0384293
$357$	− 15.8248i	− 0.837535i
$358$	11.4502i	0.605160i
$359$	−23.9244	−1.26268	−0.631341	−	0.775505i	$-0.717495\pi$
$359$	−23.9244	−1.26268	−0.631341	+	0.775505i	$0.717495\pi$
$360$	0	0
$361$	−0.725083	−0.0381623
$362$	− 9.27492i	− 0.487479i
$363$	28.3746i	1.48928i
$364$	12.8248	0.672200
$365$	0	0
$366$	−14.5498	−0.760532
$367$	− 1.72508i	− 0.0900486i	−0.998986	−	0.0450243i	$-0.985663\pi$
$367$	− 1.72508i	− 0.0900486i	0.998986	−	0.0450243i	$-0.0143365\pi$
$368$	− 1.00000i	− 0.0521286i
$369$	10.2749	0.534891
$370$	0	0
$371$	13.6495	0.708647
$372$	− 6.00000i	− 0.311086i
$373$	14.3746i	0.744288i	0.928175	+	0.372144i	$0.121377\pi$
$373$	14.3746i	0.744288i	−0.928175	+	0.372144i	$0.878623\pi$
$374$	−33.0997	−1.71154
$375$	0	0
$376$	0.725083	0.0373933
$377$	23.7251i	1.22190i
$378$	3.00000i	0.154303i
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	− 15.7251i	− 0.804565i
$383$	22.8248i	1.16629i	0.812368	+	0.583145i	$0.198178\pi$
$383$	22.8248i	1.16629i	−0.812368	+	0.583145i	$0.801822\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−15.2749	−0.777473
$387$	0.274917i	0.0139748i
$388$	14.0000i	0.710742i
$389$	7.09967	0.359968	0.179984	−	0.983670i	$-0.442395\pi$
$389$	7.09967	0.359968	0.179984	+	0.983670i	$0.442395\pi$
$390$	0	0
$391$	−5.27492	−0.266764
$392$	− 2.00000i	− 0.101015i
$393$	− 8.54983i	− 0.431282i
$394$	19.5498	0.984906
$395$	0	0
$396$	6.27492	0.315326
$397$	36.5498i	1.83438i	0.398446	+	0.917192i	$0.369550\pi$
$397$	36.5498i	1.83438i	−0.398446	+	0.917192i	$0.630450\pi$
$398$	11.0000i	0.551380i
$399$	12.8248	0.642041
$400$	0	0
$401$	35.6495	1.78025	0.890126	−	0.455715i	$-0.150616\pi$
$401$	35.6495	1.78025	0.890126	+	0.455715i	$0.150616\pi$
$402$	− 10.5498i	− 0.526178i
$403$	− 25.6495i	− 1.27769i
$404$	−0.725083	−0.0360742
$405$	0	0
$406$	16.6495	0.826301
$407$	74.1993i	3.67792i
$408$	5.27492i	0.261147i
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	−13.8248	−0.681925
$412$	3.00000i	0.147799i
$413$	− 7.64950i	− 0.376407i
$414$	1.00000	0.0491473
$415$	0	0
$416$	−4.27492	−0.209595
$417$	1.27492i	0.0624330i
$418$	− 26.8248i	− 1.31204i
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	0.900331	0.0438795	0.0219397	−	0.999759i	$-0.493016\pi$
$421$	0.900331	0.0438795	0.0219397	+	0.999759i	$0.493016\pi$
$422$	− 1.82475i	− 0.0888275i
$423$	0.725083i	0.0352547i
$424$	−4.54983	−0.220959
$425$	0	0
$426$	13.8248	0.669811
$427$	43.6495i	2.11235i
$428$	− 12.0000i	− 0.580042i
$429$	26.8248	1.29511
$430$	0	0
$431$	2.54983	0.122821	0.0614106	−	0.998113i	$-0.480440\pi$
$431$	2.54983	0.122821	0.0614106	+	0.998113i	$0.480440\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	− 27.6495i	− 1.32875i	−0.747399	−	0.664375i	$-0.768698\pi$
$433$	− 27.6495i	− 1.32875i	0.747399	−	0.664375i	$-0.231302\pi$
$434$	−18.0000	−0.864028
$435$	0	0
$436$	−19.8248	−0.949433
$437$	− 4.27492i	− 0.204497i
$438$	− 0.450166i	− 0.0215097i
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	22.5498i	1.07259i
$443$	− 29.6495i	− 1.40869i	−0.709858	−	0.704345i	$-0.751241\pi$
$443$	− 29.6495i	− 1.40869i	0.709858	−	0.704345i	$-0.248759\pi$
$444$	11.8248	0.561178
$445$	0	0
$446$	−14.0000	−0.662919
$447$	− 12.5498i	− 0.593587i
$448$	3.00000i	0.141737i
$449$	15.0997	0.712597	0.356299	−	0.934372i	$-0.384039\pi$
$449$	15.0997	0.712597	0.356299	+	0.934372i	$0.384039\pi$
$450$	0	0
$451$	−64.4743	−3.03597
$452$	11.2749i	0.530327i
$453$	12.5498i	0.589643i
$454$	2.72508	0.127894
$455$	0	0
$456$	−4.27492	−0.200191
$457$	− 5.64950i	− 0.264273i	−0.991232	−	0.132136i	$-0.957816\pi$
$457$	− 5.64950i	− 0.264273i	0.991232	−	0.132136i	$-0.0421837\pi$
$458$	11.0997i	0.518653i
$459$	−5.27492	−0.246212
$460$	0	0
$461$	−30.0997	−1.40188	−0.700941	−	0.713220i	$-0.747236\pi$
$461$	−30.0997	−1.40188	−0.700941	+	0.713220i	$0.747236\pi$
$462$	− 18.8248i	− 0.875807i
$463$	1.09967i	0.0511059i	0.999673	+	0.0255530i	$0.00813465\pi$
$463$	1.09967i	0.0511059i	−0.999673	+	0.0255530i	$0.991865\pi$
$464$	−5.54983	−0.257645
$465$	0	0
$466$	−23.3746	−1.08281
$467$	21.0000i	0.971764i	0.874024	+	0.485882i	$0.161502\pi$
$467$	21.0000i	0.971764i	−0.874024	+	0.485882i	$0.838498\pi$
$468$	− 4.27492i	− 0.197608i
$469$	−31.6495	−1.46144
$470$	0	0
$471$	14.0000	0.645086
$472$	2.54983i	0.117366i
$473$	− 1.72508i	− 0.0793194i
$474$	−10.8248	−0.497197
$475$	0	0
$476$	15.8248	0.725326
$477$	− 4.54983i	− 0.208323i
$478$	4.72508i	0.216120i
$479$	0.274917	0.0125613	0.00628064	−	0.999980i	$-0.498001\pi$
$479$	0.274917	0.0125613	0.00628064	+	0.999980i	$0.498001\pi$
$480$	0	0
$481$	50.5498	2.30487
$482$	4.54983i	0.207239i
$483$	− 3.00000i	− 0.136505i
$484$	−28.3746	−1.28975
$485$	0	0
$486$	1.00000	0.0453609
$487$	− 8.54983i	− 0.387430i	−0.981058	−	0.193715i	$-0.937946\pi$
$487$	− 8.54983i	− 0.387430i	0.981058	−	0.193715i	$-0.0620537\pi$
$488$	− 14.5498i	− 0.658640i
$489$	−21.6495	−0.979024
$490$	0	0
$491$	6.54983	0.295590	0.147795	−	0.989018i	$-0.452782\pi$
$491$	6.54983	0.295590	0.147795	+	0.989018i	$0.452782\pi$
$492$	10.2749i	0.463229i
$493$	29.2749i	1.31848i
$494$	−18.2749	−0.822227
$495$	0	0
$496$	6.00000	0.269408
$497$	− 41.4743i	− 1.86037i
$498$	− 17.5498i	− 0.786427i
$499$	−19.2749	−0.862864	−0.431432	−	0.902146i	$-0.641991\pi$
$499$	−19.2749	−0.862864	−0.431432	+	0.902146i	$0.641991\pi$
$500$	0	0
$501$	−4.72508	−0.211101
$502$	− 4.17525i	− 0.186350i
$503$	19.3746i	0.863870i	0.901905	+	0.431935i	$0.142169\pi$
$503$	19.3746i	0.863870i	−0.901905	+	0.431935i	$0.857831\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−6.27492	−0.278954
$507$	− 5.27492i	− 0.234267i
$508$	8.00000i	0.354943i
$509$	−37.4743	−1.66102	−0.830509	−	0.557006i	$-0.811950\pi$
$509$	−37.4743	−1.66102	−0.830509	+	0.557006i	$0.811950\pi$
$510$	0	0
$511$	−1.35050	−0.0597425
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.27492i	− 0.188742i
$514$	2.00000	0.0882162
$515$	0	0
$516$	−0.274917	−0.0121026
$517$	− 4.54983i	− 0.200101i
$518$	− 35.4743i	− 1.55865i
$519$	7.72508	0.339093
$520$	0	0
$521$	−34.3746	−1.50598	−0.752989	−	0.658033i	$-0.771389\pi$
$521$	−34.3746	−1.50598	−0.752989	+	0.658033i	$0.771389\pi$
$522$	− 5.54983i	− 0.242910i
$523$	− 11.3746i	− 0.497376i	−0.968584	−	0.248688i	$-0.920001\pi$
$523$	− 11.3746i	− 0.497376i	0.968584	−	0.248688i	$-0.0799994\pi$
$524$	8.54983	0.373501
$525$	0	0
$526$	10.5498	0.459995
$527$	− 31.6495i	− 1.37867i
$528$	6.27492i	0.273081i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−2.54983	−0.110653
$532$	12.8248i	0.556024i
$533$	43.9244i	1.90258i
$534$	−0.725083	−0.0313774
$535$	0	0
$536$	10.5498	0.455683
$537$	− 11.4502i	− 0.494111i
$538$	4.27492i	0.184305i
$539$	−12.5498	−0.540560
$540$	0	0
$541$	−9.17525	−0.394475	−0.197237	−	0.980356i	$-0.563197\pi$
$541$	−9.17525	−0.394475	−0.197237	+	0.980356i	$0.563197\pi$
$542$	− 16.5498i	− 0.710876i
$543$	9.27492i	0.398025i
$544$	−5.27492	−0.226160
$545$	0	0
$546$	−12.8248	−0.548849
$547$	22.1752i	0.948145i	0.880486	+	0.474073i	$0.157217\pi$
$547$	22.1752i	0.948145i	−0.880486	+	0.474073i	$0.842783\pi$
$548$	− 13.8248i	− 0.590564i
$549$	14.5498	0.620972
$550$	0	0
$551$	−23.7251	−1.01072
$552$	1.00000i	0.0425628i
$553$	32.4743i	1.38095i
$554$	16.8248	0.714815
$555$	0	0
$556$	−1.27492	−0.0540685
$557$	5.45017i	0.230931i	0.993311	+	0.115465i	$0.0368360\pi$
$557$	5.45017i	0.230931i	−0.993311	+	0.115465i	$0.963164\pi$
$558$	6.00000i	0.254000i
$559$	−1.17525	−0.0497077
$560$	0	0
$561$	33.0997	1.39747
$562$	32.3746i	1.36564i
$563$	14.2749i	0.601616i	0.953685	+	0.300808i	$0.0972563\pi$
$563$	14.2749i	0.601616i	−0.953685	+	0.300808i	$0.902744\pi$
$564$	−0.725083	−0.0305315
$565$	0	0
$566$	−16.0000	−0.672530
$567$	− 3.00000i	− 0.125988i
$568$	13.8248i	0.580074i
$569$	22.5498	0.945338	0.472669	−	0.881240i	$-0.343291\pi$
$569$	22.5498	0.945338	0.472669	+	0.881240i	$0.343291\pi$
$570$	0	0
$571$	−17.4502	−0.730267	−0.365133	−	0.930955i	$-0.618977\pi$
$571$	−17.4502	−0.730267	−0.365133	+	0.930955i	$0.618977\pi$
$572$	26.8248i	1.12160i
$573$	15.7251i	0.656925i
$574$	30.8248	1.28660
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 20.8248i	− 0.866946i	−0.901167	−	0.433473i	$-0.857288\pi$
$577$	− 20.8248i	− 0.866946i	0.901167	−	0.433473i	$-0.142712\pi$
$578$	10.8248i	0.450250i
$579$	15.2749	0.634804
$580$	0	0
$581$	−52.6495	−2.18427
$582$	− 14.0000i	− 0.580319i
$583$	28.5498i	1.18241i
$584$	0.450166	0.0186280
$585$	0	0
$586$	23.6495	0.976952
$587$	10.9003i	0.449905i	0.974370	+	0.224952i	$0.0722227\pi$
$587$	10.9003i	0.449905i	−0.974370	+	0.224952i	$0.927777\pi$
$588$	2.00000i	0.0824786i
$589$	25.6495	1.05687
$590$	0	0
$591$	−19.5498	−0.804173
$592$	11.8248i	0.485994i
$593$	30.8248i	1.26582i	0.774225	+	0.632910i	$0.218140\pi$
$593$	30.8248i	1.26582i	−0.774225	+	0.632910i	$0.781860\pi$
$594$	−6.27492	−0.257463
$595$	0	0
$596$	12.5498	0.514061
$597$	− 11.0000i	− 0.450200i
$598$	4.27492i	0.174814i
$599$	38.7492	1.58325	0.791624	−	0.611008i	$-0.209236\pi$
$599$	38.7492	1.58325	0.791624	+	0.611008i	$0.209236\pi$
$600$	0	0
$601$	27.2749	1.11257	0.556284	−	0.830993i	$-0.312227\pi$
$601$	27.2749	1.11257	0.556284	+	0.830993i	$0.312227\pi$
$602$	0.824752i	0.0336144i
$603$	10.5498i	0.429622i
$604$	−12.5498	−0.510646
$605$	0	0
$606$	0.725083	0.0294545
$607$	− 4.90033i	− 0.198898i	−0.995043	−	0.0994492i	$-0.968292\pi$
$607$	− 4.90033i	− 0.198898i	0.995043	−	0.0994492i	$-0.0317081\pi$
$608$	− 4.27492i	− 0.173371i
$609$	−16.6495	−0.674672
$610$	0	0
$611$	−3.09967	−0.125399
$612$	− 5.27492i	− 0.213226i
$613$	18.7251i	0.756299i	0.925745	+	0.378149i	$0.123439\pi$
$613$	18.7251i	0.756299i	−0.925745	+	0.378149i	$0.876561\pi$
$614$	1.27492	0.0514515
$615$	0	0
$616$	18.8248	0.758471
$617$	7.64950i	0.307957i	0.988074	+	0.153979i	$0.0492087\pi$
$617$	7.64950i	0.307957i	−0.988074	+	0.153979i	$0.950791\pi$
$618$	− 3.00000i	− 0.120678i
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	− 11.8248i	− 0.474129i
$623$	2.17525i	0.0871495i
$624$	4.27492	0.171134
$625$	0	0
$626$	−32.0000	−1.27898
$627$	26.8248i	1.07128i
$628$	14.0000i	0.558661i
$629$	62.3746	2.48704
$630$	0	0
$631$	−32.0997	−1.27787	−0.638934	−	0.769262i	$-0.720624\pi$
$631$	−32.0997	−1.27787	−0.638934	+	0.769262i	$0.720624\pi$
$632$	− 10.8248i	− 0.430586i
$633$	1.82475i	0.0725274i
$634$	−1.54983	−0.0615518
$635$	0	0
$636$	4.54983	0.180413
$637$	8.54983i	0.338757i
$638$	34.8248i	1.37873i
$639$	−13.8248	−0.546899
$640$	0	0
$641$	46.3746	1.83169	0.915843	−	0.401537i	$-0.131524\pi$
$641$	46.3746	1.83169	0.915843	+	0.401537i	$0.131524\pi$
$642$	12.0000i	0.473602i
$643$	− 0.274917i	− 0.0108417i	−0.999985	−	0.00542084i	$-0.998274\pi$
$643$	− 0.274917i	− 0.0108417i	0.999985	−	0.00542084i	$-0.00172551\pi$
$644$	3.00000	0.118217
$645$	0	0
$646$	−22.5498	−0.887211
$647$	37.8248i	1.48704i	0.668711	+	0.743522i	$0.266846\pi$
$647$	37.8248i	1.48704i	−0.668711	+	0.743522i	$0.733154\pi$
$648$	1.00000i	0.0392837i
$649$	16.0000	0.628055
$650$	0	0
$651$	18.0000	0.705476
$652$	− 21.6495i	− 0.847860i
$653$	7.90033i	0.309164i	0.987980	+	0.154582i	$0.0494031\pi$
$653$	7.90033i	0.309164i	−0.987980	+	0.154582i	$0.950597\pi$
$654$	19.8248	0.775209
$655$	0	0
$656$	−10.2749	−0.401168
$657$	0.450166i	0.0175626i
$658$	2.17525i	0.0848000i
$659$	−4.09967	−0.159700	−0.0798502	−	0.996807i	$-0.525444\pi$
$659$	−4.09967	−0.159700	−0.0798502	+	0.996807i	$0.525444\pi$
$660$	0	0
$661$	18.1752	0.706935	0.353468	−	0.935447i	$-0.385002\pi$
$661$	18.1752	0.706935	0.353468	+	0.935447i	$0.385002\pi$
$662$	5.82475i	0.226385i
$663$	− 22.5498i	− 0.875763i
$664$	17.5498	0.681066
$665$	0	0
$666$	−11.8248	−0.458200
$667$	5.54983i	0.214890i
$668$	− 4.72508i	− 0.182819i
$669$	14.0000	0.541271
$670$	0	0
$671$	−91.2990	−3.52456
$672$	− 3.00000i	− 0.115728i
$673$	21.5498i	0.830685i	0.909665	+	0.415343i	$0.136338\pi$
$673$	21.5498i	0.830685i	−0.909665	+	0.415343i	$0.863662\pi$
$674$	−24.5498	−0.945624
$675$	0	0
$676$	5.27492	0.202881
$677$	13.0997i	0.503461i	0.967797	+	0.251731i	$0.0809997\pi$
$677$	13.0997i	0.503461i	−0.967797	+	0.251731i	$0.919000\pi$
$678$	− 11.2749i	− 0.433011i
$679$	−42.0000	−1.61181
$680$	0	0
$681$	−2.72508	−0.104425
$682$	− 37.6495i	− 1.44167i
$683$	− 7.45017i	− 0.285073i	−0.989790	−	0.142536i	$-0.954474\pi$
$683$	− 7.45017i	− 0.285073i	0.989790	−	0.142536i	$-0.0455258\pi$
$684$	4.27492	0.163456
$685$	0	0
$686$	−15.0000	−0.572703
$687$	− 11.0997i	− 0.423479i
$688$	− 0.274917i	− 0.0104811i
$689$	19.4502	0.740992
$690$	0	0
$691$	25.2749	0.961503	0.480752	−	0.876857i	$-0.340364\pi$
$691$	25.2749	0.961503	0.480752	+	0.876857i	$0.340364\pi$
$692$	7.72508i	0.293664i
$693$	18.8248i	0.715093i
$694$	34.5498	1.31149
$695$	0	0
$696$	5.54983	0.210366
$697$	54.1993i	2.05295i
$698$	− 19.7251i	− 0.746605i
$699$	23.3746	0.884108
$700$	0	0
$701$	−27.6495	−1.04431	−0.522154	−	0.852851i	$-0.674871\pi$
$701$	−27.6495	−1.04431	−0.522154	+	0.852851i	$0.674871\pi$
$702$	4.27492i	0.161346i
$703$	50.5498i	1.90652i
$704$	−6.27492	−0.236495
$705$	0	0
$706$	−13.3746	−0.503359
$707$	− 2.17525i	− 0.0818086i
$708$	− 2.54983i	− 0.0958286i
$709$	49.8248	1.87121	0.935604	−	0.353051i	$-0.114856\pi$
$709$	49.8248	1.87121	0.935604	+	0.353051i	$0.114856\pi$
$710$	0	0
$711$	10.8248	0.405960
$712$	− 0.725083i	− 0.0271736i
$713$	− 6.00000i	− 0.224702i
$714$	−15.8248	−0.592226
$715$	0	0
$716$	11.4502	0.427913
$717$	− 4.72508i	− 0.176461i
$718$	23.9244i	0.892851i
$719$	22.3746	0.834431	0.417216	−	0.908808i	$-0.363006\pi$
$719$	22.3746	0.834431	0.417216	+	0.908808i	$0.363006\pi$
$720$	0	0
$721$	−9.00000	−0.335178
$722$	0.725083i	0.0269848i
$723$	− 4.54983i	− 0.169210i
$724$	−9.27492	−0.344700
$725$	0	0
$726$	28.3746	1.05308
$727$	26.1993i	0.971680i	0.874048	+	0.485840i	$0.161486\pi$
$727$	26.1993i	0.971680i	−0.874048	+	0.485840i	$0.838514\pi$
$728$	− 12.8248i	− 0.475317i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−1.45017	−0.0536363
$732$	14.5498i	0.537777i
$733$	− 18.9244i	− 0.698989i	−0.936938	−	0.349495i	$-0.886353\pi$
$733$	− 18.9244i	− 0.698989i	0.936938	−	0.349495i	$-0.113647\pi$
$734$	−1.72508	−0.0636740
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	− 66.1993i	− 2.43848i
$738$	− 10.2749i	− 0.378225i
$739$	−18.9244	−0.696146	−0.348073	−	0.937467i	$-0.613164\pi$
$739$	−18.9244	−0.696146	−0.348073	+	0.937467i	$0.613164\pi$
$740$	0	0
$741$	18.2749	0.671346
$742$	− 13.6495i	− 0.501089i
$743$	− 0.824752i	− 0.0302572i	−0.999886	−	0.0151286i	$-0.995184\pi$
$743$	− 0.824752i	− 0.0302572i	0.999886	−	0.0151286i	$-0.00481577\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	14.3746	0.526291
$747$	17.5498i	0.642115i
$748$	33.0997i	1.21024i
$749$	36.0000	1.31541
$750$	0	0
$751$	36.6495	1.33736	0.668680	−	0.743551i	$-0.266860\pi$
$751$	36.6495	1.33736	0.668680	+	0.743551i	$0.266860\pi$
$752$	− 0.725083i	− 0.0264410i
$753$	4.17525i	0.152155i
$754$	23.7251	0.864017
$755$	0	0
$756$	3.00000	0.109109
$757$	− 29.2749i	− 1.06401i	−0.846740	−	0.532007i	$-0.821438\pi$
$757$	− 29.2749i	− 1.06401i	0.846740	−	0.532007i	$-0.178562\pi$
$758$	12.0000i	0.435860i
$759$	6.27492	0.227765
$760$	0	0
$761$	8.82475	0.319897	0.159948	−	0.987125i	$-0.448867\pi$
$761$	8.82475	0.319897	0.159948	+	0.987125i	$0.448867\pi$
$762$	− 8.00000i	− 0.289809i
$763$	− 59.4743i	− 2.15311i
$764$	−15.7251	−0.568914
$765$	0	0
$766$	22.8248	0.824692
$767$	− 10.9003i	− 0.393588i
$768$	1.00000i	0.0360844i
$769$	6.54983	0.236193	0.118097	−	0.993002i	$-0.462321\pi$
$769$	6.54983	0.236193	0.118097	+	0.993002i	$0.462321\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	15.2749i	0.549756i
$773$	28.7492i	1.03404i	0.855975	+	0.517018i	$0.172958\pi$
$773$	28.7492i	1.03404i	−0.855975	+	0.517018i	$0.827042\pi$
$774$	0.274917	0.00988170
$775$	0	0
$776$	14.0000	0.502571
$777$	35.4743i	1.27263i
$778$	− 7.09967i	− 0.254535i
$779$	−43.9244	−1.57376
$780$	0	0
$781$	86.7492	3.10413
$782$	5.27492i	0.188631i
$783$	5.54983i	0.198335i
$784$	−2.00000	−0.0714286
$785$	0	0
$786$	−8.54983	−0.304962
$787$	1.17525i	0.0418931i	0.999781	+	0.0209465i	$0.00666798\pi$
$787$	1.17525i	0.0418931i	−0.999781	+	0.0209465i	$0.993332\pi$
$788$	− 19.5498i	− 0.696434i
$789$	−10.5498	−0.375584
$790$	0	0
$791$	−33.8248	−1.20267
$792$	− 6.27492i	− 0.222969i
$793$	62.1993i	2.20876i
$794$	36.5498	1.29711
$795$	0	0
$796$	11.0000	0.389885
$797$	2.54983i	0.0903198i	0.998980	+	0.0451599i	$0.0143797\pi$
$797$	2.54983i	0.0903198i	−0.998980	+	0.0451599i	$0.985620\pi$
$798$	− 12.8248i	− 0.453991i
$799$	−3.82475	−0.135310
$800$	0	0
$801$	0.725083	0.0256195
$802$	− 35.6495i	− 1.25883i
$803$	− 2.82475i	− 0.0996833i
$804$	−10.5498	−0.372064
$805$	0	0
$806$	−25.6495	−0.903465
$807$	− 4.27492i	− 0.150484i
$808$	0.725083i	0.0255083i
$809$	54.4743	1.91521	0.957606	−	0.288080i	$-0.0930168\pi$
$809$	54.4743	1.91521	0.957606	+	0.288080i	$0.0930168\pi$
$810$	0	0
$811$	−10.9003	−0.382762	−0.191381	−	0.981516i	$-0.561297\pi$
$811$	−10.9003	−0.382762	−0.191381	+	0.981516i	$0.561297\pi$
$812$	− 16.6495i	− 0.584283i
$813$	16.5498i	0.580428i
$814$	74.1993	2.60069
$815$	0	0
$816$	5.27492	0.184659
$817$	− 1.17525i	− 0.0411167i
$818$	− 7.00000i	− 0.244749i
$819$	12.8248	0.448133
$820$	0	0
$821$	31.0241	1.08275	0.541374	−	0.840782i	$-0.317904\pi$
$821$	31.0241	1.08275	0.541374	+	0.840782i	$0.317904\pi$
$822$	13.8248i	0.482194i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	3.00000	0.104510
$825$	0	0
$826$	−7.64950	−0.266160
$827$	− 17.9003i	− 0.622456i	−0.950335	−	0.311228i	$-0.899260\pi$
$827$	− 17.9003i	− 0.622456i	0.950335	−	0.311228i	$-0.100740\pi$
$828$	− 1.00000i	− 0.0347524i
$829$	−33.3746	−1.15915	−0.579574	−	0.814920i	$-0.696781\pi$
$829$	−33.3746	−1.15915	−0.579574	+	0.814920i	$0.696781\pi$
$830$	0	0
$831$	−16.8248	−0.583644
$832$	4.27492i	0.148206i
$833$	10.5498i	0.365530i
$834$	1.27492	0.0441468
$835$	0	0
$836$	−26.8248	−0.927753
$837$	− 6.00000i	− 0.207390i
$838$	3.00000i	0.103633i
$839$	18.8248	0.649903	0.324951	−	0.945731i	$-0.394652\pi$
$839$	18.8248	0.649903	0.324951	+	0.945731i	$0.394652\pi$
$840$	0	0
$841$	1.80066	0.0620918
$842$	− 0.900331i	− 0.0310275i
$843$	− 32.3746i	− 1.11504i
$844$	−1.82475	−0.0628105
$845$	0	0
$846$	0.725083	0.0249289
$847$	− 85.1238i	− 2.92489i
$848$	4.54983i	0.156242i
$849$	16.0000	0.549119
$850$	0	0
$851$	11.8248	0.405347
$852$	− 13.8248i	− 0.473628i
$853$	26.2749i	0.899636i	0.893120	+	0.449818i	$0.148511\pi$
$853$	26.2749i	0.899636i	−0.893120	+	0.449818i	$0.851489\pi$
$854$	43.6495	1.49366
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 27.0997i	− 0.925707i	−0.886435	−	0.462854i	$-0.846826\pi$
$857$	− 27.0997i	− 0.925707i	0.886435	−	0.462854i	$-0.153174\pi$
$858$	− 26.8248i	− 0.915782i
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	−30.8248	−1.05050
$862$	− 2.54983i	− 0.0868477i
$863$	− 1.07558i	− 0.0366132i	−0.999832	−	0.0183066i	$-0.994173\pi$
$863$	− 1.07558i	− 0.0366132i	0.999832	−	0.0183066i	$-0.00582749\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−27.6495	−0.939568
$867$	− 10.8248i	− 0.367628i
$868$	18.0000i	0.610960i
$869$	−67.9244	−2.30418
$870$	0	0
$871$	−45.0997	−1.52814
$872$	19.8248i	0.671351i
$873$	14.0000i	0.473828i
$874$	−4.27492	−0.144601
$875$	0	0
$876$	−0.450166	−0.0152097
$877$	− 31.0997i	− 1.05016i	−0.851053	−	0.525081i	$-0.824035\pi$
$877$	− 31.0997i	− 1.05016i	0.851053	−	0.525081i	$-0.175965\pi$
$878$	− 20.0000i	− 0.674967i
$879$	−23.6495	−0.797678
$880$	0	0
$881$	44.7492	1.50764	0.753819	−	0.657082i	$-0.228210\pi$
$881$	44.7492	1.50764	0.753819	+	0.657082i	$0.228210\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	− 22.7251i	− 0.764760i	−0.924005	−	0.382380i	$-0.875105\pi$
$883$	− 22.7251i	− 0.764760i	0.924005	−	0.382380i	$-0.124895\pi$
$884$	22.5498	0.758433
$885$	0	0
$886$	−29.6495	−0.996095
$887$	9.62541i	0.323190i	0.986857	+	0.161595i	$0.0516638\pi$
$887$	9.62541i	0.323190i	−0.986857	+	0.161595i	$0.948336\pi$
$888$	− 11.8248i	− 0.396813i
$889$	−24.0000	−0.804934
$890$	0	0
$891$	6.27492	0.210218
$892$	14.0000i	0.468755i
$893$	− 3.09967i	− 0.103726i
$894$	−12.5498	−0.419729
$895$	0	0
$896$	3.00000	0.100223
$897$	− 4.27492i	− 0.142735i
$898$	− 15.0997i	− 0.503882i
$899$	−33.2990	−1.11058
$900$	0	0
$901$	24.0000	0.799556
$902$	64.4743i	2.14676i
$903$	− 0.824752i	− 0.0274460i
$904$	11.2749	0.374998
$905$	0	0
$906$	12.5498	0.416940
$907$	− 37.3746i	− 1.24100i	−0.784205	−	0.620501i	$-0.786929\pi$
$907$	− 37.3746i	− 1.24100i	0.784205	−	0.620501i	$-0.213071\pi$
$908$	− 2.72508i	− 0.0904350i
$909$	−0.725083	−0.0240495
$910$	0	0
$911$	−21.1752	−0.701567	−0.350784	−	0.936457i	$-0.614085\pi$
$911$	−21.1752	−0.701567	−0.350784	+	0.936457i	$0.614085\pi$
$912$	4.27492i	0.141557i
$913$	− 110.124i	− 3.64456i
$914$	−5.64950	−0.186869
$915$	0	0
$916$	11.0997	0.366743
$917$	25.6495i	0.847021i
$918$	5.27492i	0.174098i
$919$	19.4743	0.642396	0.321198	−	0.947012i	$-0.395914\pi$
$919$	19.4743	0.642396	0.321198	+	0.947012i	$0.395914\pi$
$920$	0	0
$921$	−1.27492	−0.0420100
$922$	30.0997i	0.991280i
$923$	− 59.0997i	− 1.94529i
$924$	−18.8248	−0.619289
$925$	0	0
$926$	1.09967	0.0361374
$927$	3.00000i	0.0985329i
$928$	5.54983i	0.182182i
$929$	−29.3746	−0.963749	−0.481874	−	0.876240i	$-0.660044\pi$
$929$	−29.3746	−0.963749	−0.481874	+	0.876240i	$0.660044\pi$
$930$	0	0
$931$	−8.54983	−0.280210
$932$	23.3746i	0.765660i
$933$	11.8248i	0.387125i
$934$	21.0000	0.687141
$935$	0	0
$936$	−4.27492	−0.139730
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	31.6495i	1.03339i
$939$	32.0000	1.04428
$940$	0	0
$941$	44.1993	1.44086	0.720429	−	0.693529i	$-0.243945\pi$
$941$	44.1993	1.44086	0.720429	+	0.693529i	$0.243945\pi$
$942$	− 14.0000i	− 0.456145i
$943$	10.2749i	0.334597i
$944$	2.54983	0.0829900
$945$	0	0
$946$	−1.72508	−0.0560873
$947$	6.35050i	0.206363i	0.994663	+	0.103182i	$0.0329023\pi$
$947$	6.35050i	0.206363i	−0.994663	+	0.103182i	$0.967098\pi$
$948$	10.8248i	0.351572i
$949$	−1.92442	−0.0624693
$950$	0	0
$951$	1.54983	0.0502568
$952$	− 15.8248i	− 0.512883i
$953$	− 16.9244i	− 0.548236i	−0.961696	−	0.274118i	$-0.911614\pi$
$953$	− 16.9244i	− 0.548236i	0.961696	−	0.274118i	$-0.0883858\pi$
$954$	−4.54983	−0.147306
$955$	0	0
$956$	4.72508	0.152820
$957$	− 34.8248i	− 1.12572i
$958$	− 0.274917i	− 0.00888217i
$959$	41.4743	1.33927
$960$	0	0
$961$	5.00000	0.161290
$962$	− 50.5498i	− 1.62979i
$963$	− 12.0000i	− 0.386695i
$964$	4.54983	0.146540
$965$	0	0
$966$	−3.00000	−0.0965234
$967$	19.6495i	0.631885i	0.948778	+	0.315943i	$0.102321\pi$
$967$	19.6495i	0.631885i	−0.948778	+	0.315943i	$0.897679\pi$
$968$	28.3746i	0.911994i
$969$	22.5498	0.724405
$970$	0	0
$971$	−26.0997	−0.837578	−0.418789	−	0.908084i	$-0.637545\pi$
$971$	−26.0997	−0.837578	−0.418789	+	0.908084i	$0.637545\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 3.82475i	− 0.122616i
$974$	−8.54983	−0.273954
$975$	0	0
$976$	−14.5498	−0.465729
$977$	18.9244i	0.605446i	0.953079	+	0.302723i	$0.0978957\pi$
$977$	18.9244i	0.605446i	−0.953079	+	0.302723i	$0.902104\pi$
$978$	21.6495i	0.692275i
$979$	−4.54983	−0.145413
$980$	0	0
$981$	−19.8248	−0.632956
$982$	− 6.54983i	− 0.209014i
$983$	− 15.9244i	− 0.507910i	−0.967216	−	0.253955i	$-0.918268\pi$
$983$	− 15.9244i	− 0.507910i	0.967216	−	0.253955i	$-0.0817315\pi$
$984$	10.2749	0.327552
$985$	0	0
$986$	29.2749	0.932303
$987$	− 2.17525i	− 0.0692389i
$988$	18.2749i	0.581403i
$989$	−0.274917	−0.00874186
$990$	0	0
$991$	−4.54983	−0.144530	−0.0722651	−	0.997385i	$-0.523023\pi$
$991$	−4.54983	−0.144530	−0.0722651	+	0.997385i	$0.523023\pi$
$992$	− 6.00000i	− 0.190500i
$993$	− 5.82475i	− 0.184843i
$994$	−41.4743	−1.31548
$995$	0	0
$996$	−17.5498	−0.556088
$997$	− 16.2749i	− 0.515432i	−0.966221	−	0.257716i	$-0.917030\pi$
$997$	− 16.2749i	− 0.515432i	0.966221	−	0.257716i	$-0.0829699\pi$
$998$	19.2749i	0.610137i
$999$	11.8248	0.374119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.d.z.2899.2		4
5.2	odd	4		3450.2.a.bm.1.2	yes	2
5.3	odd	4		3450.2.a.bd.1.2	✓	2
5.4	even	2	inner	3450.2.d.z.2899.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.bd.1.2	✓	2	5.3	odd	4
3450.2.a.bm.1.2	yes	2	5.2	odd	4
3450.2.d.z.2899.2		4	1.1	even	1	trivial
3450.2.d.z.2899.4		4	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.27492 q^{11} -1.00000i q^{12} -4.27492i q^{13} -3.00000 q^{14} +1.00000 q^{16} -5.27492i q^{17} +1.00000i q^{18} +4.27492 q^{19} +3.00000 q^{21} -6.27492i q^{22} -1.00000i q^{23} -1.00000 q^{24} -4.27492 q^{26} -1.00000i q^{27} +3.00000i q^{28} -5.54983 q^{29} +6.00000 q^{31} -1.00000i q^{32} +6.27492i q^{33} -5.27492 q^{34} +1.00000 q^{36} +11.8248i q^{37} -4.27492i q^{38} +4.27492 q^{39} -10.2749 q^{41} -3.00000i q^{42} -0.274917i q^{43} -6.27492 q^{44} -1.00000 q^{46} -0.725083i q^{47} +1.00000i q^{48} -2.00000 q^{49} +5.27492 q^{51} +4.27492i q^{52} +4.54983i q^{53} -1.00000 q^{54} +3.00000 q^{56} +4.27492i q^{57} +5.54983i q^{58} +2.54983 q^{59} -14.5498 q^{61} -6.00000i q^{62} +3.00000i q^{63} -1.00000 q^{64} +6.27492 q^{66} -10.5498i q^{67} +5.27492i q^{68} +1.00000 q^{69} +13.8248 q^{71} -1.00000i q^{72} -0.450166i q^{73} +11.8248 q^{74} -4.27492 q^{76} -18.8248i q^{77} -4.27492i q^{78} -10.8248 q^{79} +1.00000 q^{81} +10.2749i q^{82} -17.5498i q^{83} -3.00000 q^{84} -0.274917 q^{86} -5.54983i q^{87} +6.27492i q^{88} -0.725083 q^{89} -12.8248 q^{91} +1.00000i q^{92} +6.00000i q^{93} -0.725083 q^{94} +1.00000 q^{96} -14.0000i q^{97} +2.00000i q^{98} -6.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 10 q^{11} - 12 q^{14} + 4 q^{16} + 2 q^{19} + 12 q^{21} - 4 q^{24} - 2 q^{26} + 8 q^{29} + 24 q^{31} - 6 q^{34} + 4 q^{36} + 2 q^{39} - 26 q^{41} - 10 q^{44} - 4 q^{46} - 8 q^{49} + 6 q^{51} - 4 q^{54} + 12 q^{56} - 20 q^{59} - 28 q^{61} - 4 q^{64} + 10 q^{66} + 4 q^{69} + 10 q^{71} + 2 q^{74} - 2 q^{76} + 2 q^{79} + 4 q^{81} - 12 q^{84} + 14 q^{86} - 18 q^{89} - 6 q^{91} - 18 q^{94} + 4 q^{96} - 10 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 10 * q^11 - 12 * q^14 + 4 * q^16 + 2 * q^19 + 12 * q^21 - 4 * q^24 - 2 * q^26 + 8 * q^29 + 24 * q^31 - 6 * q^34 + 4 * q^36 + 2 * q^39 - 26 * q^41 - 10 * q^44 - 4 * q^46 - 8 * q^49 + 6 * q^51 - 4 * q^54 + 12 * q^56 - 20 * q^59 - 28 * q^61 - 4 * q^64 + 10 * q^66 + 4 * q^69 + 10 * q^71 + 2 * q^74 - 2 * q^76 + 2 * q^79 + 4 * q^81 - 12 * q^84 + 14 * q^86 - 18 * q^89 - 6 * q^91 - 18 * q^94 + 4 * q^96 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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