LMFDB - Embedded newform 825.2.c.a.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, 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("825.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.c (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:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.2.c.a.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} -4.00000 q^{21} -1.00000i q^{22} +8.00000i q^{23} -3.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} -8.00000 q^{31} -5.00000i q^{32} -1.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{42} +1.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -12.0000 q^{56} -6.00000i q^{58} +4.00000 q^{59} +6.00000 q^{61} +8.00000i q^{62} +4.00000i q^{63} -7.00000 q^{64} -1.00000 q^{66} +4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} +3.00000i q^{72} -14.0000i q^{73} -6.00000 q^{74} -4.00000i q^{77} +2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +12.0000i q^{83} -4.00000 q^{84} -6.00000i q^{87} -3.00000i q^{88} +6.00000 q^{89} -8.00000 q^{91} +8.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -5.00000 q^{96} -2.00000i q^{97} +9.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{11} - 8 q^{14} - 2 q^{16} - 8 q^{21} - 6 q^{24} - 4 q^{26} + 12 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 4 q^{39} - 4 q^{41} + 2 q^{44} + 16 q^{46} - 18 q^{49} + 4 q^{51} + 2 q^{54} - 24 q^{56} + 8 q^{59} + 12 q^{61} - 14 q^{64} - 2 q^{66} + 16 q^{69} - 12 q^{74} + 8 q^{79} + 2 q^{81} - 8 q^{84} + 12 q^{89} - 16 q^{91} - 16 q^{94} - 10 q^{96} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^11 - 8 * q^14 - 2 * q^16 - 8 * q^21 - 6 * q^24 - 4 * q^26 + 12 * q^29 - 16 * q^31 + 4 * q^34 - 2 * q^36 - 4 * q^39 - 4 * q^41 + 2 * q^44 + 16 * q^46 - 18 * q^49 + 4 * q^51 + 2 * q^54 - 24 * q^56 + 8 * q^59 + 12 * q^61 - 14 * q^64 - 2 * q^66 + 16 * q^69 - 12 * q^74 + 8 * q^79 + 2 * q^81 - 8 * q^84 + 12 * q^89 - 16 * q^91 - 16 * q^94 - 10 * q^96 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	− 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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 3.00000i	− 1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	− 1.00000i	− 0.288675i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	−4.00000	−1.06904
$15$	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	$0.922021\pi$
$18$	1.00000i	0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	− 1.00000i	− 0.213201i
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	−3.00000	−0.612372
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	− 4.00000i	− 0.755929i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	− 5.00000i	− 0.883883i
$33$	− 1.00000i	− 0.174078i
$34$	2.00000	0.342997
$35$	0	0
$36$	−1.00000	−0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	4.00000i	0.617213i
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	8.00000	1.17954
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	2.00000	0.280056
$52$	− 2.00000i	− 0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−12.0000	−1.60357
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	8.00000i	1.01600i
$63$	4.00000i	0.503953i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	2.00000i	0.242536i
$69$	8.00000	0.963087
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000i	0.353553i
$73$	− 14.0000i	− 1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$	− 14.0000i	− 1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	2.00000i	0.226455i
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.00000i	0.220863i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	0	0
$87$	− 6.00000i	− 0.643268i
$88$	− 3.00000i	− 0.319801i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	8.00000i	0.834058i
$93$	8.00000i	0.829561i
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−5.00000	−0.510310
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	9.00000i	0.909137i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	− 2.00000i	− 0.198030i
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	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
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	4.00000i	0.377964i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000i	0.184900i
$118$	− 4.00000i	− 0.368230i
$119$	8.00000	0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 6.00000i	− 0.543214i
$123$	2.00000i	0.180334i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	4.00000	0.356348
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	− 3.00000i	− 0.265165i
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	− 2.00000i	− 0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$	− 2.00000i	− 0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$	− 8.00000i	− 0.681005i
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.0000	−1.15865
$147$	9.00000i	0.742307i
$148$	− 6.00000i	− 0.493197i
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	− 2.00000i	− 0.161690i
$154$	−4.00000	−0.322329
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	− 4.00000i	− 0.318223i
$159$	6.00000	0.475831
$160$	0	0
$161$	32.0000	2.52195
$162$	− 1.00000i	− 0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	12.0000i	0.925820i
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	− 4.00000i	− 0.300658i
$178$	− 6.00000i	− 0.449719i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	8.00000i	0.592999i
$183$	− 6.00000i	− 0.443533i
$184$	24.0000	1.76930
$185$	0	0
$186$	8.00000	0.586588
$187$	2.00000i	0.146254i
$188$	− 8.00000i	− 0.583460i
$189$	4.00000	0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	7.00000i	0.505181i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	14.0000i	0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$	14.0000i	0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$	1.00000i	0.0710669i
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	− 2.00000i	− 0.140720i
$203$	− 24.0000i	− 1.68447i
$204$	2.00000	0.140028
$205$	0	0
$206$	8.00000	0.557386
$207$	− 8.00000i	− 0.556038i
$208$	2.00000i	0.138675i
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	3.00000	0.204124
$217$	32.0000i	2.17230i
$218$	− 2.00000i	− 0.135457i
$219$	−14.0000	−0.946032
$220$	0	0
$221$	4.00000	0.269069
$222$	6.00000i	0.402694i
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−20.0000	−1.33631
$225$	0	0
$226$	−6.00000	−0.399114
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	− 18.0000i	− 1.18176i
$233$	30.0000i	1.96537i	0.185296	+	0.982683i	$0.440675\pi$
$233$	30.0000i	1.96537i	−0.185296	+	0.982683i	$0.559325\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	4.00000	0.260378
$237$	− 4.00000i	− 0.259828i
$238$	− 8.00000i	− 0.518563i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	− 1.00000i	− 0.0641500i
$244$	6.00000	0.384111
$245$	0	0
$246$	2.00000	0.127515
$247$	0	0
$248$	24.0000i	1.52400i
$249$	12.0000	0.760469
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	4.00000i	0.251976i
$253$	8.00000i	0.502956i
$254$	4.00000	0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	−6.00000	−0.371391
$262$	12.0000i	0.741362i
$263$	− 16.0000i	− 0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$	− 16.0000i	− 0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	4.00000i	0.244339i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	− 2.00000i	− 0.121268i
$273$	8.00000i	0.484182i
$274$	−2.00000	−0.120824
$275$	0	0
$276$	8.00000	0.481543
$277$	26.0000i	1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$	26.0000i	1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$	− 8.00000i	− 0.479808i
$279$	8.00000	0.478947
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	8.00000i	0.476393i
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	−2.00000	−0.118262
$287$	8.00000i	0.472225i
$288$	5.00000i	0.294628i
$289$	13.0000	0.764706
$290$	0	0
$291$	−2.00000	−0.117242
$292$	− 14.0000i	− 0.819288i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−18.0000	−1.04623
$297$	1.00000i	0.0580259i
$298$	− 22.0000i	− 1.27443i
$299$	16.0000	0.925304
$300$	0	0
$301$	0	0
$302$	− 20.0000i	− 1.15087i
$303$	− 2.00000i	− 0.114897i
$304$	0	0
$305$	0	0
$306$	−2.00000	−0.114332
$307$	− 32.0000i	− 1.82634i	−0.407583	−	0.913168i	$-0.633628\pi$
$307$	− 32.0000i	− 1.82634i	0.407583	−	0.913168i	$-0.366372\pi$
$308$	− 4.00000i	− 0.227921i
$309$	8.00000	0.455104
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	6.00000i	0.339683i
$313$	− 22.0000i	− 1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$	− 22.0000i	− 1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	4.00000	0.225018
$317$	− 22.0000i	− 1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	− 22.0000i	− 1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$	− 6.00000i	− 0.336463i
$319$	6.00000	0.335936
$320$	0	0
$321$	12.0000	0.669775
$322$	− 32.0000i	− 1.78329i
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	− 2.00000i	− 0.110600i
$328$	6.00000i	0.331295i
$329$	−32.0000	−1.76422
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	12.0000i	0.658586i
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	4.00000	0.218218
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	− 9.00000i	− 0.489535i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	−6.00000	−0.322562
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	− 5.00000i	− 0.266501i
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	− 8.00000i	− 0.423405i
$358$	12.0000i	0.634220i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 22.0000i	− 1.15629i
$363$	− 1.00000i	− 0.0524864i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	−6.00000	−0.313625
$367$	32.0000i	1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$	32.0000i	1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$	− 8.00000i	− 0.417029i
$369$	2.00000	0.104116
$370$	0	0
$371$	24.0000	1.24602
$372$	8.00000i	0.414781i
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−24.0000	−1.23771
$377$	− 12.0000i	− 0.618031i
$378$	− 4.00000i	− 0.205738i
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	− 8.00000i	− 0.409316i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	−14.0000	−0.712581
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	27.0000i	1.36371i
$393$	12.0000i	0.605320i
$394$	14.0000	0.705310
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	− 4.00000i	− 0.199502i
$403$	16.0000i	0.797017i
$404$	2.00000	0.0995037
$405$	0	0
$406$	−24.0000	−1.19110
$407$	− 6.00000i	− 0.297409i
$408$	− 6.00000i	− 0.297044i
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	8.00000i	0.394132i
$413$	− 16.0000i	− 0.787309i
$414$	−8.00000	−0.393179
$415$	0	0
$416$	−10.0000	−0.490290
$417$	− 8.00000i	− 0.391762i
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	8.00000i	0.388973i
$424$	18.0000	0.874157
$425$	0	0
$426$	0	0
$427$	− 24.0000i	− 1.16144i
$428$	12.0000i	0.580042i
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	32.0000	1.53605
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	14.0000i	0.668946i
$439$	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
$441$	9.00000	0.428571
$442$	− 4.00000i	− 0.190261i
$443$	28.0000i	1.33032i	0.746701	+	0.665160i	$0.231637\pi$
$443$	28.0000i	1.33032i	−0.746701	+	0.665160i	$0.768363\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	16.0000	0.757622
$447$	− 22.0000i	− 1.04056i
$448$	28.0000i	1.32288i
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	− 6.00000i	− 0.282216i
$453$	− 20.0000i	− 0.939682i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	6.00000i	0.280362i
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	4.00000i	0.186097i
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	30.0000	1.38972
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	2.00000i	0.0924500i
$469$	16.0000	0.738811
$470$	0	0
$471$	−14.0000	−0.645086
$472$	− 12.0000i	− 0.552345i
$473$	0	0
$474$	−4.00000	−0.183726
$475$	0	0
$476$	8.00000	0.366679
$477$	− 6.00000i	− 0.274721i
$478$	24.0000i	1.09773i
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	− 10.0000i	− 0.455488i
$483$	− 32.0000i	− 1.45605i
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	− 18.0000i	− 0.814822i
$489$	4.00000	0.180886
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	2.00000i	0.0901670i
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	0	0
$498$	− 12.0000i	− 0.537733i
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	− 4.00000i	− 0.178529i
$503$	− 32.0000i	− 1.42681i	−0.700752	−	0.713405i	$-0.747152\pi$
$503$	− 32.0000i	− 1.42681i	0.700752	−	0.713405i	$-0.252848\pi$
$504$	12.0000	0.534522
$505$	0	0
$506$	8.00000	0.355643
$507$	− 9.00000i	− 0.399704i
$508$	4.00000i	0.177471i
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	11.0000i	0.486136i
$513$	0	0
$514$	14.0000	0.617514
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	24.0000i	1.05450i
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	6.00000i	0.262613i
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−16.0000	−0.697633
$527$	− 16.0000i	− 0.696971i
$528$	1.00000i	0.0435194i
$529$	−41.0000	−1.78261
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	4.00000i	0.173259i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	12.0000	0.518321
$537$	12.0000i	0.517838i
$538$	− 2.00000i	− 0.0862261i
$539$	−9.00000	−0.387657
$540$	0	0
$541$	46.0000	1.97769	0.988847	−	0.148933i	$-0.0475840\pi$
$541$	46.0000	1.97769	0.988847	+	0.148933i	$0.0475840\pi$
$542$	− 20.0000i	− 0.859074i
$543$	− 22.0000i	− 0.944110i
$544$	10.0000	0.428746
$545$	0	0
$546$	8.00000	0.342368
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	− 2.00000i	− 0.0854358i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	0	0
$552$	− 24.0000i	− 1.02151i
$553$	− 16.0000i	− 0.680389i
$554$	26.0000	1.10463
$555$	0	0
$556$	8.00000	0.339276
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	− 8.00000i	− 0.338667i
$559$	0	0
$560$	0	0
$561$	2.00000	0.0844401
$562$	18.0000i	0.759284i
$563$	− 44.0000i	− 1.85438i	−0.374593	−	0.927189i	$-0.622217\pi$
$563$	− 44.0000i	− 1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	16.0000	0.672530
$567$	− 4.00000i	− 0.167984i
$568$	0	0
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	− 8.00000i	− 0.334205i
$574$	8.00000	0.333914
$575$	0	0
$576$	7.00000	0.291667
$577$	30.0000i	1.24892i	0.781058	+	0.624458i	$0.214680\pi$
$577$	30.0000i	1.24892i	−0.781058	+	0.624458i	$0.785320\pi$
$578$	− 13.0000i	− 0.540729i
$579$	−14.0000	−0.581820
$580$	0	0
$581$	48.0000	1.99138
$582$	2.00000i	0.0829027i
$583$	6.00000i	0.248495i
$584$	−42.0000	−1.73797
$585$	0	0
$586$	−6.00000	−0.247858
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	9.00000i	0.371154i
$589$	0	0
$590$	0	0
$591$	14.0000	0.575883
$592$	6.00000i	0.246598i
$593$	38.0000i	1.56047i	0.625485	+	0.780236i	$0.284901\pi$
$593$	38.0000i	1.56047i	−0.625485	+	0.780236i	$0.715099\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	22.0000	0.901155
$597$	0	0
$598$	− 16.0000i	− 0.654289i
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	20.0000	0.813788
$605$	0	0
$606$	−2.00000	−0.0812444
$607$	4.00000i	0.162355i	0.996700	+	0.0811775i	$0.0258681\pi$
$607$	4.00000i	0.162355i	−0.996700	+	0.0811775i	$0.974132\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	−16.0000	−0.647291
$612$	− 2.00000i	− 0.0808452i
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	−32.0000	−1.29141
$615$	0	0
$616$	−12.0000	−0.483494
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	− 8.00000i	− 0.321807i
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	24.0000i	0.962312i
$623$	− 24.0000i	− 0.961540i
$624$	2.00000	0.0800641
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	12.0000	0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	− 12.0000i	− 0.477334i
$633$	0	0
$634$	−22.0000	−0.873732
$635$	0	0
$636$	6.00000	0.237915
$637$	18.0000i	0.713186i
$638$	− 6.00000i	− 0.237542i
$639$	0	0
$640$	0	0
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	− 12.0000i	− 0.473602i
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	32.0000	1.26098
$645$	0	0
$646$	0	0
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	− 3.00000i	− 0.117851i
$649$	4.00000	0.157014
$650$	0	0
$651$	32.0000	1.25418
$652$	4.00000i	0.156652i
$653$	− 2.00000i	− 0.0782660i	−0.999234	−	0.0391330i	$-0.987540\pi$
$653$	− 2.00000i	− 0.0782660i	0.999234	−	0.0391330i	$-0.0124596\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	2.00000	0.0780869
$657$	14.0000i	0.546192i
$658$	32.0000i	1.24749i
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	20.0000i	0.777322i
$663$	− 4.00000i	− 0.155347i
$664$	36.0000	1.39707
$665$	0	0
$666$	6.00000	0.232495
$667$	48.0000i	1.85857i
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	6.00000	0.231627
$672$	20.0000i	0.771517i
$673$	− 46.0000i	− 1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$	− 46.0000i	− 1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	9.00000	0.346154
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	6.00000i	0.230429i
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−12.0000	−0.459841
$682$	8.00000i	0.306336i
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	0	0
$685$	0	0
$686$	8.00000	0.305441
$687$	6.00000i	0.228914i
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 6.00000i	− 0.228086i
$693$	4.00000i	0.151947i
$694$	−4.00000	−0.151838
$695$	0	0
$696$	−18.0000	−0.682288
$697$	− 4.00000i	− 0.151511i
$698$	6.00000i	0.227103i
$699$	30.0000	1.13470
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	0	0
$704$	−7.00000	−0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	− 8.00000i	− 0.300871i
$708$	− 4.00000i	− 0.150329i
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	− 18.0000i	− 0.674579i
$713$	− 64.0000i	− 2.39682i
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	24.0000i	0.896296i
$718$	− 8.00000i	− 0.298557i
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	19.0000i	0.707107i
$723$	− 10.0000i	− 0.371904i
$724$	22.0000	0.817624
$725$	0	0
$726$	−1.00000	−0.0371135
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	24.0000i	0.889499i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	− 6.00000i	− 0.221766i
$733$	30.0000i	1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$	30.0000i	1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	40.0000	1.47442
$737$	4.00000i	0.147342i
$738$	− 2.00000i	− 0.0736210i
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.881068i
$743$	40.0000i	1.46746i	0.679442	+	0.733729i	$0.262222\pi$
$743$	40.0000i	1.46746i	−0.679442	+	0.733729i	$0.737778\pi$
$744$	24.0000	0.879883
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	− 12.0000i	− 0.439057i
$748$	2.00000i	0.0731272i
$749$	48.0000	1.75388
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	8.00000i	0.291730i
$753$	− 4.00000i	− 0.145768i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	4.00000	0.145479
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	28.0000i	1.01701i
$759$	8.00000	0.290382
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	− 4.00000i	− 0.144905i
$763$	− 8.00000i	− 0.289619i
$764$	8.00000	0.289430
$765$	0	0
$766$	−16.0000	−0.578103
$767$	− 8.00000i	− 0.288863i
$768$	17.0000i	0.613435i
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	− 14.0000i	− 0.503871i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	−6.00000	−0.215387
$777$	24.0000i	0.860995i
$778$	− 18.0000i	− 0.645331i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	16.0000i	0.572159i
$783$	6.00000i	0.214423i
$784$	9.00000	0.321429
$785$	0	0
$786$	12.0000	0.428026
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	14.0000i	0.498729i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	−24.0000	−0.853342
$792$	3.00000i	0.106600i
$793$	− 12.0000i	− 0.426132i
$794$	2.00000	0.0709773
$795$	0	0
$796$	0	0
$797$	10.0000i	0.354218i	0.984191	+	0.177109i	$0.0566745\pi$
$797$	10.0000i	0.354218i	−0.984191	+	0.177109i	$0.943325\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 26.0000i	− 0.918092i
$803$	− 14.0000i	− 0.494049i
$804$	4.00000	0.141069
$805$	0	0
$806$	16.0000	0.563576
$807$	− 2.00000i	− 0.0704033i
$808$	− 6.00000i	− 0.211079i
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−56.0000	−1.96643	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000	−1.96643	−0.983213	+	0.182462i	$0.941593\pi$
$812$	− 24.0000i	− 0.842235i
$813$	− 20.0000i	− 0.701431i
$814$	−6.00000	−0.210300
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	18.0000i	0.629355i
$819$	8.00000	0.279543
$820$	0	0
$821$	−14.0000	−0.488603	−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000	−0.488603	−0.244302	+	0.969699i	$0.578559\pi$
$822$	2.00000i	0.0697580i
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	−16.0000	−0.556711
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	− 8.00000i	− 0.278019i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	14.0000i	0.485363i
$833$	− 18.0000i	− 0.623663i
$834$	−8.00000	−0.277017
$835$	0	0
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	− 4.00000i	− 0.138178i
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	26.0000i	0.896019i
$843$	18.0000i	0.619953i
$844$	0	0
$845$	0	0
$846$	8.00000	0.275046
$847$	− 4.00000i	− 0.137442i
$848$	− 6.00000i	− 0.206041i
$849$	16.0000	0.549119
$850$	0	0
$851$	48.0000	1.64542
$852$	0	0
$853$	− 34.0000i	− 1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$	− 34.0000i	− 1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$	−24.0000	−0.821263
$855$	0	0
$856$	36.0000	1.23045
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	2.00000i	0.0682789i
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	24.0000i	0.817443i
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	34.0000	1.15537
$867$	− 13.0000i	− 0.441503i
$868$	32.0000i	1.08615i
$869$	4.00000	0.135691
$870$	0	0
$871$	8.00000	0.271070
$872$	− 6.00000i	− 0.203186i
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	−14.0000	−0.473016
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	− 20.0000i	− 0.674967i
$879$	−6.00000	−0.202375
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	− 9.00000i	− 0.303046i
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	28.0000	0.940678
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	18.0000i	0.604040i
$889$	16.0000	0.536623
$890$	0	0
$891$	1.00000	0.0335013
$892$	16.0000i	0.535720i
$893$	0	0
$894$	−22.0000	−0.735790
$895$	0	0
$896$	−12.0000	−0.400892
$897$	− 16.0000i	− 0.534224i
$898$	2.00000i	0.0667409i
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−12.0000	−0.399778
$902$	2.00000i	0.0665927i
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−20.0000	−0.664455
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	12.0000i	0.397142i
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−6.00000	−0.198246
$917$	48.0000i	1.58510i
$918$	2.00000i	0.0660098i
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	30.0000i	0.987997i
$923$	0	0
$924$	−4.00000	−0.131590
$925$	0	0
$926$	16.0000	0.525793
$927$	− 8.00000i	− 0.262754i
$928$	− 30.0000i	− 0.984798i
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	0	0
$932$	30.0000i	0.982683i
$933$	24.0000i	0.785725i
$934$	12.0000	0.392652
$935$	0	0
$936$	6.00000	0.196116
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	− 16.0000i	− 0.522419i
$939$	−22.0000	−0.717943
$940$	0	0
$941$	−54.0000	−1.76035	−0.880175	−	0.474650i	$-0.842575\pi$
$941$	−54.0000	−1.76035	−0.880175	+	0.474650i	$0.842575\pi$
$942$	14.0000i	0.456145i
$943$	− 16.0000i	− 0.521032i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	0	0
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	− 4.00000i	− 0.129914i
$949$	−28.0000	−0.908918
$950$	0	0
$951$	−22.0000	−0.713399
$952$	− 24.0000i	− 0.777844i
$953$	22.0000i	0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$	22.0000i	0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−24.0000	−0.776215
$957$	− 6.00000i	− 0.193952i
$958$	8.00000i	0.258468i
$959$	−8.00000	−0.258333
$960$	0	0
$961$	33.0000	1.06452
$962$	12.0000i	0.386896i
$963$	− 12.0000i	− 0.386695i
$964$	10.0000	0.322078
$965$	0	0
$966$	−32.0000	−1.02958
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	− 3.00000i	− 0.0964237i
$969$	0	0
$970$	0	0
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 32.0000i	− 1.02587i
$974$	16.0000	0.512673
$975$	0	0
$976$	−6.00000	−0.192055
$977$	6.00000i	0.191957i	0.995383	+	0.0959785i	$0.0305980\pi$
$977$	6.00000i	0.191957i	−0.995383	+	0.0959785i	$0.969402\pi$
$978$	− 4.00000i	− 0.127906i
$979$	6.00000	0.191761
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	− 4.00000i	− 0.127645i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	12.0000	0.382158
$987$	32.0000i	1.01857i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	40.0000i	1.27000i
$993$	20.0000i	0.634681i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	− 4.00000i	− 0.126618i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.c.a.199.1		2
3.2	odd	2		2475.2.c.d.199.2		2
5.2	odd	4		33.2.a.a.1.1	✓	1
5.3	odd	4		825.2.a.a.1.1		1
5.4	even	2	inner	825.2.c.a.199.2		2
15.2	even	4		99.2.a.b.1.1		1
15.8	even	4		2475.2.a.g.1.1		1
15.14	odd	2		2475.2.c.d.199.1		2
20.7	even	4		528.2.a.g.1.1		1
35.27	even	4		1617.2.a.j.1.1		1
40.27	even	4		2112.2.a.j.1.1		1
40.37	odd	4		2112.2.a.bb.1.1		1
45.2	even	12		891.2.e.g.595.1		2
45.7	odd	12		891.2.e.e.595.1		2
45.22	odd	12		891.2.e.e.298.1		2
45.32	even	12		891.2.e.g.298.1		2
55.2	even	20		363.2.e.g.202.1		4
55.7	even	20		363.2.e.g.148.1		4
55.17	even	20		363.2.e.g.124.1		4
55.27	odd	20		363.2.e.e.124.1		4
55.32	even	4		363.2.a.b.1.1		1
55.37	odd	20		363.2.e.e.148.1		4
55.42	odd	20		363.2.e.e.202.1		4
55.43	even	4		9075.2.a.q.1.1		1
55.47	odd	20		363.2.e.e.130.1		4
55.52	even	20		363.2.e.g.130.1		4
60.47	odd	4		1584.2.a.o.1.1		1
65.12	odd	4		5577.2.a.a.1.1		1
85.67	odd	4		9537.2.a.m.1.1		1
105.62	odd	4		4851.2.a.b.1.1		1
120.77	even	4		6336.2.a.x.1.1		1
120.107	odd	4		6336.2.a.n.1.1		1
165.32	odd	4		1089.2.a.j.1.1		1
220.87	odd	4		5808.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.a.a.1.1	✓	1	5.2	odd	4
99.2.a.b.1.1		1	15.2	even	4
363.2.a.b.1.1		1	55.32	even	4
363.2.e.e.124.1		4	55.27	odd	20
363.2.e.e.130.1		4	55.47	odd	20
363.2.e.e.148.1		4	55.37	odd	20
363.2.e.e.202.1		4	55.42	odd	20
363.2.e.g.124.1		4	55.17	even	20
363.2.e.g.130.1		4	55.52	even	20
363.2.e.g.148.1		4	55.7	even	20
363.2.e.g.202.1		4	55.2	even	20
528.2.a.g.1.1		1	20.7	even	4
825.2.a.a.1.1		1	5.3	odd	4
825.2.c.a.199.1		2	1.1	even	1	trivial
825.2.c.a.199.2		2	5.4	even	2	inner
891.2.e.e.298.1		2	45.22	odd	12
891.2.e.e.595.1		2	45.7	odd	12
891.2.e.g.298.1		2	45.32	even	12
891.2.e.g.595.1		2	45.2	even	12
1089.2.a.j.1.1		1	165.32	odd	4
1584.2.a.o.1.1		1	60.47	odd	4
1617.2.a.j.1.1		1	35.27	even	4
2112.2.a.j.1.1		1	40.27	even	4
2112.2.a.bb.1.1		1	40.37	odd	4
2475.2.a.g.1.1		1	15.8	even	4
2475.2.c.d.199.1		2	15.14	odd	2
2475.2.c.d.199.2		2	3.2	odd	2
4851.2.a.b.1.1		1	105.62	odd	4
5577.2.a.a.1.1		1	65.12	odd	4
5808.2.a.t.1.1		1	220.87	odd	4
6336.2.a.n.1.1		1	120.107	odd	4
6336.2.a.x.1.1		1	120.77	even	4
9075.2.a.q.1.1		1	55.43	even	4
9537.2.a.m.1.1		1	85.67	odd	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} -1.00000i q^{12} -2.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +1.00000i q^{18} -4.00000 q^{21} -1.00000i q^{22} +8.00000i q^{23} -3.00000 q^{24} -2.00000 q^{26} +1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} -8.00000 q^{31} -5.00000i q^{32} -1.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} -2.00000 q^{39} -2.00000 q^{41} +4.00000i q^{42} +1.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} -12.0000 q^{56} -6.00000i q^{58} +4.00000 q^{59} +6.00000 q^{61} +8.00000i q^{62} +4.00000i q^{63} -7.00000 q^{64} -1.00000 q^{66} +4.00000i q^{67} +2.00000i q^{68} +8.00000 q^{69} +3.00000i q^{72} -14.0000i q^{73} -6.00000 q^{74} -4.00000i q^{77} +2.00000i q^{78} +4.00000 q^{79} +1.00000 q^{81} +2.00000i q^{82} +12.0000i q^{83} -4.00000 q^{84} -6.00000i q^{87} -3.00000i q^{88} +6.00000 q^{89} -8.00000 q^{91} +8.00000i q^{92} +8.00000i q^{93} -8.00000 q^{94} -5.00000 q^{96} -2.00000i q^{97} +9.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{11} - 8 q^{14} - 2 q^{16} - 8 q^{21} - 6 q^{24} - 4 q^{26} + 12 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 4 q^{39} - 4 q^{41} + 2 q^{44} + 16 q^{46} - 18 q^{49} + 4 q^{51} + 2 q^{54} - 24 q^{56} + 8 q^{59} + 12 q^{61} - 14 q^{64} - 2 q^{66} + 16 q^{69} - 12 q^{74} + 8 q^{79} + 2 q^{81} - 8 q^{84} + 12 q^{89} - 16 q^{91} - 16 q^{94} - 10 q^{96} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^11 - 8 * q^14 - 2 * q^16 - 8 * q^21 - 6 * q^24 - 4 * q^26 + 12 * q^29 - 16 * q^31 + 4 * q^34 - 2 * q^36 - 4 * q^39 - 4 * q^41 + 2 * q^44 + 16 * q^46 - 18 * q^49 + 4 * q^51 + 2 * q^54 - 24 * q^56 + 8 * q^59 + 12 * q^61 - 14 * q^64 - 2 * q^66 + 16 * q^69 - 12 * q^74 + 8 * q^79 + 2 * q^81 - 8 * q^84 + 12 * q^89 - 16 * q^91 - 16 * q^94 - 10 * q^96 - 2 * q^99

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