LMFDB - Embedded newform 5070.2.b.i.1351.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5070,2,Mod(1351,5070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5070.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5070.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$40.4841538248$
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 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	5070.1351
Dual form			5070.2.b.i.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} -4.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000i q^{18} +4.00000i q^{19} +1.00000i q^{20} -4.00000i q^{21} -8.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +4.00000i q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.00000i q^{31} -1.00000i q^{32} -2.00000i q^{34} -4.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} +4.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} -4.00000 q^{42} -12.0000 q^{43} -1.00000i q^{45} +8.00000i q^{46} +1.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} +2.00000 q^{51} +10.0000 q^{53} -1.00000i q^{54} +4.00000 q^{56} +4.00000i q^{57} -2.00000i q^{58} +1.00000i q^{60} -10.0000 q^{61} -8.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -2.00000 q^{68} -8.00000 q^{69} +4.00000i q^{70} -16.0000i q^{71} +1.00000i q^{72} +6.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000i q^{76} -8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000i q^{83} +4.00000i q^{84} -2.00000i q^{85} +12.0000i q^{86} +2.00000 q^{87} +14.0000i q^{89} -1.00000 q^{90} +8.00000 q^{92} -8.00000i q^{93} +4.00000 q^{95} -1.00000i q^{96} -6.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 8 q^{14} + 2 q^{16} + 4 q^{17} - 16 q^{23} - 2 q^{25} + 2 q^{27} + 4 q^{29} - 2 q^{30} - 8 q^{35} - 2 q^{36} + 8 q^{38} + 2 q^{40} - 8 q^{42} - 24 q^{43} + 2 q^{48} - 18 q^{49} + 4 q^{51} + 20 q^{53} + 8 q^{56} - 20 q^{61} - 16 q^{62} - 2 q^{64} - 4 q^{68} - 16 q^{69} - 4 q^{74} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{82} + 4 q^{87} - 2 q^{90} + 16 q^{92} + 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 - 8 * q^14 + 2 * q^16 + 4 * q^17 - 16 * q^23 - 2 * q^25 + 2 * q^27 + 4 * q^29 - 2 * q^30 - 8 * q^35 - 2 * q^36 + 8 * q^38 + 2 * q^40 - 8 * q^42 - 24 * q^43 + 2 * q^48 - 18 * q^49 + 4 * q^51 + 20 * q^53 + 8 * q^56 - 20 * q^61 - 16 * q^62 - 2 * q^64 - 4 * q^68 - 16 * q^69 - 4 * q^74 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 12 * q^82 + 4 * q^87 - 2 * q^90 + 16 * q^92 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times$.

$n$	$1691$	$1861$	$4057$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	− 1.00000i	− 0.408248i
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−4.00000	−1.06904
$15$	− 1.00000i	− 0.258199i
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	− 1.00000i	− 0.235702i
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	1.00000i	0.223607i
$21$	− 4.00000i	− 0.872872i
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	4.00000i	0.755929i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	− 8.00000i	− 1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$	− 8.00000i	− 1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 2.00000i	− 0.342997i
$35$	−4.00000	−0.676123
$36$	−1.00000	−0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	1.00000	0.158114
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	−4.00000	−0.617213
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	8.00000i	1.17954i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000	0.144338
$49$	−9.00000	−1.28571
$50$	1.00000i	0.141421i
$51$	2.00000	0.280056
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	− 1.00000i	− 0.136083i
$55$	0	0
$56$	4.00000	0.534522
$57$	4.00000i	0.529813i
$58$	− 2.00000i	− 0.262613i
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	1.00000i	0.129099i
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−8.00000	−1.01600
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	−2.00000	−0.242536
$69$	−8.00000	−0.963087
$70$	4.00000i	0.478091i
$71$	− 16.0000i	− 1.89885i	−0.313993	−	0.949425i	$-0.601667\pi$
$71$	− 16.0000i	− 1.89885i	0.313993	−	0.949425i	$-0.398333\pi$
$72$	1.00000i	0.117851i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−2.00000	−0.232495
$75$	−1.00000	−0.115470
$76$	− 4.00000i	− 0.458831i
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	4.00000i	0.436436i
$85$	− 2.00000i	− 0.216930i
$86$	12.0000i	1.29399i
$87$	2.00000	0.214423
$88$	0	0
$89$	14.0000i	1.48400i	0.670402	+	0.741999i	$0.266122\pi$
$89$	14.0000i	1.48400i	−0.670402	+	0.741999i	$0.733878\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	8.00000	0.834058
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	4.00000	0.410391
$96$	− 1.00000i	− 0.102062i
$97$	− 6.00000i	− 0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$	− 6.00000i	− 0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 2.00000i	− 0.198030i
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	− 10.0000i	− 0.971286i
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000i	1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$	14.0000i	1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$	0	0
$111$	− 2.00000i	− 0.189832i
$112$	− 4.00000i	− 0.377964i
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	4.00000	0.374634
$115$	8.00000i	0.746004i
$116$	−2.00000	−0.185695
$117$	0	0
$118$	0	0
$119$	− 8.00000i	− 0.733359i
$120$	1.00000	0.0912871
$121$	11.0000	1.00000
$122$	10.0000i	0.905357i
$123$	− 6.00000i	− 0.541002i
$124$	8.00000i	0.718421i
$125$	1.00000i	0.0894427i
$126$	−4.00000	−0.356348
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000i	0.0883883i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	−4.00000	−0.345547
$135$	− 1.00000i	− 0.0860663i
$136$	2.00000i	0.171499i
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	8.00000i	0.681005i
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−16.0000	−1.34269
$143$	0	0
$144$	1.00000	0.0833333
$145$	− 2.00000i	− 0.166091i
$146$	6.00000	0.496564
$147$	−9.00000	−0.742307
$148$	2.00000i	0.164399i
$149$	10.0000i	0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$	10.0000i	0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$	1.00000i	0.0816497i
$151$	8.00000i	0.651031i	0.945537	+	0.325515i	$0.105538\pi$
$151$	8.00000i	0.651031i	−0.945537	+	0.325515i	$0.894462\pi$
$152$	−4.00000	−0.324443
$153$	2.00000	0.161690
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	8.00000i	0.636446i
$159$	10.0000	0.793052
$160$	−1.00000	−0.0790569
$161$	32.0000i	2.52195i
$162$	− 1.00000i	− 0.0785674i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	6.00000i	0.468521i
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	4.00000	0.308607
$169$	0	0
$170$	−2.00000	−0.153393
$171$	4.00000i	0.305888i
$172$	12.0000	0.914991
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	− 2.00000i	− 0.151620i
$175$	4.00000i	0.302372i
$176$	0	0
$177$	0	0
$178$	14.0000	1.04934
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	1.00000i	0.0745356i
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	− 8.00000i	− 0.589768i
$185$	−2.00000	−0.147043
$186$	−8.00000	−0.586588
$187$	0	0
$188$	0	0
$189$	− 4.00000i	− 0.290957i
$190$	− 4.00000i	− 0.290191i
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	−1.00000	−0.0721688
$193$	− 18.0000i	− 1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$	− 18.0000i	− 1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	9.00000	0.642857
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 4.00000i	− 0.282138i
$202$	10.0000i	0.703598i
$203$	− 8.00000i	− 0.561490i
$204$	−2.00000	−0.140028
$205$	−6.00000	−0.419058
$206$	− 4.00000i	− 0.278693i
$207$	−8.00000	−0.556038
$208$	0	0
$209$	0	0
$210$	4.00000i	0.276026i
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−10.0000	−0.686803
$213$	− 16.0000i	− 1.09630i
$214$	12.0000i	0.820303i
$215$	12.0000i	0.818393i
$216$	1.00000i	0.0680414i
$217$	−32.0000	−2.17230
$218$	14.0000	0.948200
$219$	6.00000i	0.405442i
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	12.0000i	0.803579i	0.915732	+	0.401790i	$0.131612\pi$
$223$	12.0000i	0.803579i	−0.915732	+	0.401790i	$0.868388\pi$
$224$	−4.00000	−0.267261
$225$	−1.00000	−0.0666667
$226$	10.0000i	0.665190i
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 4.00000i	− 0.264906i
$229$	− 14.0000i	− 0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$	− 14.0000i	− 0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	2.00000i	0.131306i
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	−8.00000	−0.518563
$239$	24.0000i	1.55243i	0.630468	+	0.776215i	$0.282863\pi$
$239$	24.0000i	1.55243i	−0.630468	+	0.776215i	$0.717137\pi$
$240$	− 1.00000i	− 0.0645497i
$241$	− 2.00000i	− 0.128831i	−0.997923	−	0.0644157i	$-0.979482\pi$
$241$	− 2.00000i	− 0.128831i	0.997923	−	0.0644157i	$-0.0205183\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	9.00000i	0.574989i
$246$	−6.00000	−0.382546
$247$	0	0
$248$	8.00000	0.508001
$249$	− 4.00000i	− 0.253490i
$250$	1.00000	0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	4.00000i	0.251976i
$253$	0	0
$254$	− 12.0000i	− 0.752947i
$255$	− 2.00000i	− 0.125245i
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	12.0000i	0.747087i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	2.00000	0.123797
$262$	− 8.00000i	− 0.494242i
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	− 10.0000i	− 0.614295i
$266$	− 16.0000i	− 0.981023i
$267$	14.0000i	0.856786i
$268$	4.00000i	0.244339i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	−1.00000	−0.0608581
$271$	32.0000i	1.94386i	0.235267	+	0.971931i	$0.424404\pi$
$271$	32.0000i	1.94386i	−0.235267	+	0.971931i	$0.575596\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	8.00000	0.481543
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	20.0000i	1.19952i
$279$	− 8.00000i	− 0.478947i
$280$	− 4.00000i	− 0.239046i
$281$	− 18.0000i	− 1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$	− 18.0000i	− 1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	16.0000i	0.949425i
$285$	4.00000	0.236940
$286$	0	0
$287$	−24.0000	−1.41668
$288$	− 1.00000i	− 0.0589256i
$289$	−13.0000	−0.764706
$290$	−2.00000	−0.117444
$291$	− 6.00000i	− 0.351726i
$292$	− 6.00000i	− 0.351123i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	9.00000i	0.524891i
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	10.0000	0.579284
$299$	0	0
$300$	1.00000	0.0577350
$301$	48.0000i	2.76667i
$302$	8.00000	0.460348
$303$	−10.0000	−0.574485
$304$	4.00000i	0.229416i
$305$	10.0000i	0.572598i
$306$	− 2.00000i	− 0.114332i
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	8.00000i	0.454369i
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	− 18.0000i	− 1.01580i
$315$	−4.00000	−0.225374
$316$	8.00000	0.450035
$317$	− 14.0000i	− 0.786318i	−0.919470	−	0.393159i	$-0.871382\pi$
$317$	− 14.0000i	− 0.786318i	0.919470	−	0.393159i	$-0.128618\pi$
$318$	− 10.0000i	− 0.560772i
$319$	0	0
$320$	1.00000i	0.0559017i
$321$	−12.0000	−0.669775
$322$	32.0000	1.78329
$323$	8.00000i	0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	14.0000i	0.774202i
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	− 20.0000i	− 1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$	− 20.0000i	− 1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$	4.00000i	0.219529i
$333$	− 2.00000i	− 0.109599i
$334$	−16.0000	−0.875481
$335$	−4.00000	−0.218543
$336$	− 4.00000i	− 0.218218i
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	2.00000i	0.108465i
$341$	0	0
$342$	4.00000	0.216295
$343$	8.00000i	0.431959i
$344$	− 12.0000i	− 0.646997i
$345$	8.00000i	0.430706i
$346$	2.00000i	0.107521i
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	−2.00000	−0.107211
$349$	− 14.0000i	− 0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	− 14.0000i	− 0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	− 14.0000i	− 0.741999i
$357$	− 8.00000i	− 0.423405i
$358$	16.0000i	0.845626i
$359$	− 24.0000i	− 1.26667i	−0.773877	−	0.633336i	$-0.781685\pi$
$359$	− 24.0000i	− 1.26667i	0.773877	−	0.633336i	$-0.218315\pi$
$360$	1.00000	0.0527046
$361$	3.00000	0.157895
$362$	− 2.00000i	− 0.105118i
$363$	11.0000	0.577350
$364$	0	0
$365$	6.00000	0.314054
$366$	10.0000i	0.522708i
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	−8.00000	−0.417029
$369$	− 6.00000i	− 0.312348i
$370$	2.00000i	0.103975i
$371$	− 40.0000i	− 2.07670i
$372$	8.00000i	0.414781i
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	1.00000i	0.0516398i
$376$	0	0
$377$	0	0
$378$	−4.00000	−0.205738
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	−4.00000	−0.205196
$381$	12.0000	0.614779
$382$	− 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	$-0.134046\pi$
$384$	1.00000i	0.0510310i
$385$	0	0
$386$	−18.0000	−0.916176
$387$	−12.0000	−0.609994
$388$	6.00000i	0.304604i
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	− 9.00000i	− 0.454569i
$393$	8.00000	0.403547
$394$	−6.00000	−0.302276
$395$	8.00000i	0.402524i
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	− 8.00000i	− 0.401004i
$399$	16.0000	0.801002
$400$	−1.00000	−0.0500000
$401$	− 26.0000i	− 1.29838i	−0.760627	−	0.649189i	$-0.775108\pi$
$401$	− 26.0000i	− 1.29838i	0.760627	−	0.649189i	$-0.224892\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	10.0000	0.497519
$405$	− 1.00000i	− 0.0496904i
$406$	−8.00000	−0.397033
$407$	0	0
$408$	2.00000i	0.0990148i
$409$	10.0000i	0.494468i	0.968956	+	0.247234i	$0.0795217\pi$
$409$	10.0000i	0.494468i	−0.968956	+	0.247234i	$0.920478\pi$
$410$	6.00000i	0.296319i
$411$	− 6.00000i	− 0.295958i
$412$	−4.00000	−0.197066
$413$	0	0
$414$	8.00000i	0.393179i
$415$	−4.00000	−0.196352
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	4.00000	0.195180
$421$	30.0000i	1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$	30.0000i	1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	10.0000i	0.485643i
$425$	−2.00000	−0.0970143
$426$	−16.0000	−0.775203
$427$	40.0000i	1.93574i
$428$	12.0000	0.580042
$429$	0	0
$430$	12.0000	0.578691
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.00000	0.0481125
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	32.0000i	1.53605i
$435$	− 2.00000i	− 0.0958927i
$436$	− 14.0000i	− 0.670478i
$437$	− 32.0000i	− 1.53077i
$438$	6.00000	0.286691
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	2.00000i	0.0949158i
$445$	14.0000	0.663664
$446$	12.0000	0.568216
$447$	10.0000i	0.472984i
$448$	4.00000i	0.188982i
$449$	6.00000i	0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$	6.00000i	0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$	1.00000i	0.0471405i
$451$	0	0
$452$	10.0000	0.470360
$453$	8.00000i	0.375873i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	−14.0000	−0.654177
$459$	2.00000	0.0933520
$460$	− 8.00000i	− 0.373002i
$461$	2.00000i	0.0931493i	0.998915	+	0.0465746i	$0.0148305\pi$
$461$	2.00000i	0.0931493i	−0.998915	+	0.0465746i	$0.985169\pi$
$462$	0	0
$463$	− 20.0000i	− 0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	2.00000	0.0928477
$465$	−8.00000	−0.370991
$466$	− 26.0000i	− 1.20443i
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	0	0
$474$	8.00000i	0.367452i
$475$	− 4.00000i	− 0.183533i
$476$	8.00000i	0.366679i
$477$	10.0000	0.457869
$478$	24.0000	1.09773
$479$	− 8.00000i	− 0.365529i	−0.983157	−	0.182765i	$-0.941495\pi$
$479$	− 8.00000i	− 0.365529i	0.983157	−	0.182765i	$-0.0585046\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−2.00000	−0.0910975
$483$	32.0000i	1.45605i
$484$	−11.0000	−0.500000
$485$	−6.00000	−0.272446
$486$	− 1.00000i	− 0.0453609i
$487$	− 20.0000i	− 0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$	− 20.0000i	− 0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$	− 10.0000i	− 0.452679i
$489$	20.0000i	0.904431i
$490$	9.00000	0.406579
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	6.00000i	0.270501i
$493$	4.00000	0.180151
$494$	0	0
$495$	0	0
$496$	− 8.00000i	− 0.359211i
$497$	−64.0000	−2.87079
$498$	−4.00000	−0.179244
$499$	− 44.0000i	− 1.96971i	−0.173379	−	0.984855i	$-0.555468\pi$
$499$	− 44.0000i	− 1.96971i	0.173379	−	0.984855i	$-0.444532\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	− 16.0000i	− 0.714827i
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	4.00000	0.178174
$505$	10.0000i	0.444994i
$506$	0	0
$507$	0	0
$508$	−12.0000	−0.532414
$509$	− 38.0000i	− 1.68432i	−0.539227	−	0.842160i	$-0.681284\pi$
$509$	− 38.0000i	− 1.68432i	0.539227	−	0.842160i	$-0.318716\pi$
$510$	−2.00000	−0.0885615
$511$	24.0000	1.06170
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	22.0000i	0.970378i
$515$	− 4.00000i	− 0.176261i
$516$	12.0000	0.528271
$517$	0	0
$518$	8.00000i	0.351500i
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	− 2.00000i	− 0.0875376i
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−8.00000	−0.349482
$525$	4.00000i	0.174574i
$526$	− 24.0000i	− 1.04645i
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	41.0000	1.78261
$530$	−10.0000	−0.434372
$531$	0	0
$532$	−16.0000	−0.693688
$533$	0	0
$534$	14.0000	0.605839
$535$	12.0000i	0.518805i
$536$	4.00000	0.172774
$537$	−16.0000	−0.690451
$538$	14.0000i	0.603583i
$539$	0	0
$540$	1.00000i	0.0430331i
$541$	2.00000i	0.0859867i	0.999075	+	0.0429934i	$0.0136894\pi$
$541$	2.00000i	0.0859867i	−0.999075	+	0.0429934i	$0.986311\pi$
$542$	32.0000	1.37452
$543$	2.00000	0.0858282
$544$	− 2.00000i	− 0.0857493i
$545$	14.0000	0.599694
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	6.00000i	0.256307i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	8.00000i	0.340811i
$552$	− 8.00000i	− 0.340503i
$553$	32.0000i	1.36078i
$554$	18.0000i	0.764747i
$555$	−2.00000	−0.0848953
$556$	20.0000	0.848189
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	−4.00000	−0.169031
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	10.0000i	0.420703i
$566$	20.0000i	0.840663i
$567$	− 4.00000i	− 0.167984i
$568$	16.0000	0.671345
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	− 4.00000i	− 0.167542i
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	24.0000i	1.00174i
$575$	8.00000	0.333623
$576$	−1.00000	−0.0416667
$577$	− 22.0000i	− 0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$	− 22.0000i	− 0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$	13.0000i	0.540729i
$579$	− 18.0000i	− 0.748054i
$580$	2.00000i	0.0830455i
$581$	−16.0000	−0.663792
$582$	−6.00000	−0.248708
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	6.00000	0.247858
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	9.00000	0.371154
$589$	32.0000	1.31854
$590$	0	0
$591$	− 6.00000i	− 0.246807i
$592$	− 2.00000i	− 0.0821995i
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	− 10.0000i	− 0.409616i
$597$	8.00000	0.327418
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	48.0000	1.95633
$603$	− 4.00000i	− 0.162893i
$604$	− 8.00000i	− 0.325515i
$605$	− 11.0000i	− 0.447214i
$606$	10.0000i	0.406222i
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	4.00000	0.162221
$609$	− 8.00000i	− 0.324176i
$610$	10.0000	0.404888
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	12.0000	0.484281
$615$	−6.00000	−0.241943
$616$	0	0
$617$	− 2.00000i	− 0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$	− 2.00000i	− 0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$	− 4.00000i	− 0.160904i
$619$	28.0000i	1.12542i	0.826656	+	0.562708i	$0.190240\pi$
$619$	28.0000i	1.12542i	−0.826656	+	0.562708i	$0.809760\pi$
$620$	8.00000	0.321288
$621$	−8.00000	−0.321029
$622$	0	0
$623$	56.0000	2.24359
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 18.0000i	− 0.719425i
$627$	0	0
$628$	−18.0000	−0.718278
$629$	− 4.00000i	− 0.159490i
$630$	4.00000i	0.159364i
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	− 8.00000i	− 0.318223i
$633$	−12.0000	−0.476957
$634$	−14.0000	−0.556011
$635$	− 12.0000i	− 0.476205i
$636$	−10.0000	−0.396526
$637$	0	0
$638$	0	0
$639$	− 16.0000i	− 0.632950i
$640$	1.00000	0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	12.0000i	0.473602i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	− 32.0000i	− 1.26098i
$645$	12.0000i	0.472500i
$646$	8.00000	0.314756
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	−32.0000	−1.25418
$652$	− 20.0000i	− 0.783260i
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	14.0000	0.547443
$655$	− 8.00000i	− 0.312586i
$656$	− 6.00000i	− 0.234261i
$657$	6.00000i	0.234082i
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	10.0000i	0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$	10.0000i	0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	4.00000	0.155230
$665$	− 16.0000i	− 0.620453i
$666$	−2.00000	−0.0774984
$667$	−16.0000	−0.619522
$668$	16.0000i	0.619059i
$669$	12.0000i	0.463947i
$670$	4.00000i	0.154533i
$671$	0	0
$672$	−4.00000	−0.154303
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	− 22.0000i	− 0.847408i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	26.0000	0.999261	0.499631	−	0.866239i	$-0.333469\pi$
$677$	26.0000	0.999261	0.499631	+	0.866239i	$0.333469\pi$
$678$	10.0000i	0.384048i
$679$	−24.0000	−0.921035
$680$	2.00000	0.0766965
$681$	− 12.0000i	− 0.459841i
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	− 4.00000i	− 0.152944i
$685$	−6.00000	−0.229248
$686$	8.00000	0.305441
$687$	− 14.0000i	− 0.534133i
$688$	−12.0000	−0.457496
$689$	0	0
$690$	8.00000	0.304555
$691$	20.0000i	0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$	20.0000i	0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$	2.00000	0.0760286
$693$	0	0
$694$	4.00000i	0.151838i
$695$	20.0000i	0.758643i
$696$	2.00000i	0.0758098i
$697$	− 12.0000i	− 0.454532i
$698$	−14.0000	−0.529908
$699$	26.0000	0.983410
$700$	− 4.00000i	− 0.151186i
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	−26.0000	−0.978523
$707$	40.0000i	1.50435i
$708$	0	0
$709$	34.0000i	1.27690i	0.769665	+	0.638448i	$0.220423\pi$
$709$	34.0000i	1.27690i	−0.769665	+	0.638448i	$0.779577\pi$
$710$	16.0000i	0.600469i
$711$	−8.00000	−0.300023
$712$	−14.0000	−0.524672
$713$	64.0000i	2.39682i
$714$	−8.00000	−0.299392
$715$	0	0
$716$	16.0000	0.597948
$717$	24.0000i	0.896296i
$718$	−24.0000	−0.895672
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	− 16.0000i	− 0.595871i
$722$	− 3.00000i	− 0.111648i
$723$	− 2.00000i	− 0.0743808i
$724$	−2.00000	−0.0743294
$725$	−2.00000	−0.0742781
$726$	− 11.0000i	− 0.408248i
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 6.00000i	− 0.222070i
$731$	−24.0000	−0.887672
$732$	10.0000	0.369611
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	− 4.00000i	− 0.147643i
$735$	9.00000i	0.331970i
$736$	8.00000i	0.294884i
$737$	0	0
$738$	−6.00000	−0.220863
$739$	− 4.00000i	− 0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$	− 4.00000i	− 0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−40.0000	−1.46845
$743$	− 40.0000i	− 1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	− 40.0000i	− 1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$	8.00000	0.293294
$745$	10.0000	0.366372
$746$	22.0000i	0.805477i
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	48.0000i	1.75388i
$750$	1.00000	0.0365148
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.00000	0.291150
$756$	4.00000i	0.145479i
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	4.00000i	0.145095i
$761$	− 18.0000i	− 0.652499i	−0.945284	−	0.326250i	$-0.894215\pi$
$761$	− 18.0000i	− 0.652499i	0.945284	−	0.326250i	$-0.105785\pi$
$762$	− 12.0000i	− 0.434714i
$763$	56.0000	2.02734
$764$	−8.00000	−0.289430
$765$	− 2.00000i	− 0.0723102i
$766$	−16.0000	−0.578103
$767$	0	0
$768$	1.00000	0.0360844
$769$	2.00000i	0.0721218i	0.999350	+	0.0360609i	$0.0114810\pi$
$769$	2.00000i	0.0721218i	−0.999350	+	0.0360609i	$0.988519\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	18.0000i	0.647834i
$773$	10.0000i	0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$	10.0000i	0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$	12.0000i	0.431331i
$775$	8.00000i	0.287368i
$776$	6.00000	0.215387
$777$	−8.00000	−0.286998
$778$	− 14.0000i	− 0.501924i
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	16.0000i	0.572159i
$783$	2.00000	0.0714742
$784$	−9.00000	−0.321429
$785$	− 18.0000i	− 0.642448i
$786$	− 8.00000i	− 0.285351i
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	6.00000i	0.213741i
$789$	24.0000	0.854423
$790$	8.00000	0.284627
$791$	40.0000i	1.42224i
$792$	0	0
$793$	0	0
$794$	−10.0000	−0.354887
$795$	− 10.0000i	− 0.354663i
$796$	−8.00000	−0.283552
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	− 16.0000i	− 0.566394i
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	14.0000i	0.494666i
$802$	−26.0000	−0.918092
$803$	0	0
$804$	4.00000i	0.141069i
$805$	32.0000	1.12785
$806$	0	0
$807$	−14.0000	−0.492823
$808$	− 10.0000i	− 0.351799i
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	−1.00000	−0.0351364
$811$	4.00000i	0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$	4.00000i	0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$	8.00000i	0.280745i
$813$	32.0000i	1.12229i
$814$	0	0
$815$	20.0000	0.700569
$816$	2.00000	0.0700140
$817$	− 48.0000i	− 1.67931i
$818$	10.0000	0.349642
$819$	0	0
$820$	6.00000	0.209529
$821$	2.00000i	0.0698005i	0.999391	+	0.0349002i	$0.0111113\pi$
$821$	2.00000i	0.0698005i	−0.999391	+	0.0349002i	$0.988889\pi$
$822$	−6.00000	−0.209274
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	4.00000i	0.139347i
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	8.00000	0.278019
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	4.00000i	0.138842i
$831$	−18.0000	−0.624413
$832$	0	0
$833$	−18.0000	−0.623663
$834$	20.0000i	0.692543i
$835$	−16.0000	−0.553703
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	24.0000i	0.829066i
$839$	8.00000i	0.276191i	0.990419	+	0.138095i	$0.0440980\pi$
$839$	8.00000i	0.276191i	−0.990419	+	0.138095i	$0.955902\pi$
$840$	− 4.00000i	− 0.138013i
$841$	−25.0000	−0.862069
$842$	30.0000	1.03387
$843$	− 18.0000i	− 0.619953i
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	10.0000	0.343401
$849$	−20.0000	−0.686398
$850$	2.00000i	0.0685994i
$851$	16.0000i	0.548473i
$852$	16.0000i	0.548151i
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	40.0000	1.36877
$855$	4.00000	0.136797
$856$	− 12.0000i	− 0.410152i
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	− 12.0000i	− 0.409197i
$861$	−24.0000	−0.817918
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	2.00000i	0.0680020i
$866$	10.0000i	0.339814i
$867$	−13.0000	−0.441503
$868$	32.0000	1.08615
$869$	0	0
$870$	−2.00000	−0.0678064
$871$	0	0
$872$	−14.0000	−0.474100
$873$	− 6.00000i	− 0.203069i
$874$	−32.0000	−1.08242
$875$	4.00000	0.135225
$876$	− 6.00000i	− 0.202721i
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	40.0000i	1.34993i
$879$	6.00000i	0.202375i
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	9.00000i	0.303046i
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	0	0
$886$	36.0000i	1.20944i
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	2.00000	0.0671156
$889$	− 48.0000i	− 1.60987i
$890$	− 14.0000i	− 0.469281i
$891$	0	0
$892$	− 12.0000i	− 0.401790i
$893$	0	0
$894$	10.0000	0.334450
$895$	16.0000i	0.534821i
$896$	4.00000	0.133631
$897$	0	0
$898$	6.00000	0.200223
$899$	− 16.0000i	− 0.533630i
$900$	1.00000	0.0333333
$901$	20.0000	0.666297
$902$	0	0
$903$	48.0000i	1.59734i
$904$	− 10.0000i	− 0.332595i
$905$	− 2.00000i	− 0.0664822i
$906$	8.00000	0.265782
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	12.0000i	0.398234i
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	4.00000i	0.132453i
$913$	0	0
$914$	18.0000	0.595387
$915$	10.0000i	0.330590i
$916$	14.0000i	0.462573i
$917$	− 32.0000i	− 1.05673i
$918$	− 2.00000i	− 0.0660098i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−8.00000	−0.263752
$921$	12.0000i	0.395413i
$922$	2.00000	0.0658665
$923$	0	0
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	−20.0000	−0.657241
$927$	4.00000	0.131377
$928$	− 2.00000i	− 0.0656532i
$929$	26.0000i	0.853032i	0.904480	+	0.426516i	$0.140259\pi$
$929$	26.0000i	0.853032i	−0.904480	+	0.426516i	$0.859741\pi$
$930$	8.00000i	0.262330i
$931$	− 36.0000i	− 1.17985i
$932$	−26.0000	−0.851658
$933$	0	0
$934$	− 36.0000i	− 1.17796i
$935$	0	0
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	16.0000i	0.522419i
$939$	18.0000	0.587408
$940$	0	0
$941$	− 30.0000i	− 0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$	− 30.0000i	− 0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$	− 18.0000i	− 0.586472i
$943$	48.0000i	1.56310i
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	−4.00000	−0.129777
$951$	− 14.0000i	− 0.453981i
$952$	8.00000	0.259281
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	− 10.0000i	− 0.323762i
$955$	− 8.00000i	− 0.258874i
$956$	− 24.0000i	− 0.776215i
$957$	0	0
$958$	−8.00000	−0.258468
$959$	−24.0000	−0.775000
$960$	1.00000i	0.0322749i
$961$	−33.0000	−1.06452
$962$	0	0
$963$	−12.0000	−0.386695
$964$	2.00000i	0.0644157i
$965$	−18.0000	−0.579441
$966$	32.0000	1.02958
$967$	− 28.0000i	− 0.900419i	−0.892923	−	0.450210i	$-0.851349\pi$
$967$	− 28.0000i	− 0.900419i	0.892923	−	0.450210i	$-0.148651\pi$
$968$	11.0000i	0.353553i
$969$	8.00000i	0.256997i
$970$	6.00000i	0.192648i
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	−1.00000	−0.0320750
$973$	80.0000i	2.56468i
$974$	−20.0000	−0.640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	38.0000i	1.21573i	0.794041	+	0.607864i	$0.207973\pi$
$977$	38.0000i	1.21573i	−0.794041	+	0.607864i	$0.792027\pi$
$978$	20.0000	0.639529
$979$	0	0
$980$	− 9.00000i	− 0.287494i
$981$	14.0000i	0.446986i
$982$	0	0
$983$	− 56.0000i	− 1.78612i	−0.449935	−	0.893061i	$-0.648553\pi$
$983$	− 56.0000i	− 1.78612i	0.449935	−	0.893061i	$-0.351447\pi$
$984$	6.00000	0.191273
$985$	−6.00000	−0.191176
$986$	− 4.00000i	− 0.127386i
$987$	0	0
$988$	0	0
$989$	96.0000	3.05262
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	−8.00000	−0.254000
$993$	− 20.0000i	− 0.634681i
$994$	64.0000i	2.02996i
$995$	− 8.00000i	− 0.253617i
$996$	4.00000i	0.126745i
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−44.0000	−1.39280
$999$	− 2.00000i	− 0.0632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.i.1351.1		2
13.5	odd	4		390.2.a.c.1.1	✓	1
13.8	odd	4		5070.2.a.u.1.1		1
13.12	even	2	inner	5070.2.b.i.1351.2		2
39.5	even	4		1170.2.a.n.1.1		1
52.31	even	4		3120.2.a.a.1.1		1
65.18	even	4		1950.2.e.e.1249.2		2
65.44	odd	4		1950.2.a.n.1.1		1
65.57	even	4		1950.2.e.e.1249.1		2
156.83	odd	4		9360.2.a.bc.1.1		1
195.44	even	4		5850.2.a.c.1.1		1
195.83	odd	4		5850.2.e.m.5149.1		2
195.122	odd	4		5850.2.e.m.5149.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.c.1.1	✓	1	13.5	odd	4
1170.2.a.n.1.1		1	39.5	even	4
1950.2.a.n.1.1		1	65.44	odd	4
1950.2.e.e.1249.1		2	65.57	even	4
1950.2.e.e.1249.2		2	65.18	even	4
3120.2.a.a.1.1		1	52.31	even	4
5070.2.a.u.1.1		1	13.8	odd	4
5070.2.b.i.1351.1		2	1.1	even	1	trivial
5070.2.b.i.1351.2		2	13.12	even	2	inner
5850.2.a.c.1.1		1	195.44	even	4
5850.2.e.m.5149.1		2	195.83	odd	4
5850.2.e.m.5149.2		2	195.122	odd	4
9360.2.a.bc.1.1		1	156.83	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:	\( N \)	\(=\)	\( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5070.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} -4.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000i q^{18} +4.00000i q^{19} +1.00000i q^{20} -4.00000i q^{21} -8.00000 q^{23} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} +4.00000i q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.00000i q^{31} -1.00000i q^{32} -2.00000i q^{34} -4.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} +4.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} -4.00000 q^{42} -12.0000 q^{43} -1.00000i q^{45} +8.00000i q^{46} +1.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} +2.00000 q^{51} +10.0000 q^{53} -1.00000i q^{54} +4.00000 q^{56} +4.00000i q^{57} -2.00000i q^{58} +1.00000i q^{60} -10.0000 q^{61} -8.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} -2.00000 q^{68} -8.00000 q^{69} +4.00000i q^{70} -16.0000i q^{71} +1.00000i q^{72} +6.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} -4.00000i q^{76} -8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000i q^{83} +4.00000i q^{84} -2.00000i q^{85} +12.0000i q^{86} +2.00000 q^{87} +14.0000i q^{89} -1.00000 q^{90} +8.00000 q^{92} -8.00000i q^{93} +4.00000 q^{95} -1.00000i q^{96} -6.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 8 q^{14} + 2 q^{16} + 4 q^{17} - 16 q^{23} - 2 q^{25} + 2 q^{27} + 4 q^{29} - 2 q^{30} - 8 q^{35} - 2 q^{36} + 8 q^{38} + 2 q^{40} - 8 q^{42} - 24 q^{43} + 2 q^{48} - 18 q^{49} + 4 q^{51} + 20 q^{53} + 8 q^{56} - 20 q^{61} - 16 q^{62} - 2 q^{64} - 4 q^{68} - 16 q^{69} - 4 q^{74} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 12 q^{82} + 4 q^{87} - 2 q^{90} + 16 q^{92} + 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 - 8 * q^14 + 2 * q^16 + 4 * q^17 - 16 * q^23 - 2 * q^25 + 2 * q^27 + 4 * q^29 - 2 * q^30 - 8 * q^35 - 2 * q^36 + 8 * q^38 + 2 * q^40 - 8 * q^42 - 24 * q^43 + 2 * q^48 - 18 * q^49 + 4 * q^51 + 20 * q^53 + 8 * q^56 - 20 * q^61 - 16 * q^62 - 2 * q^64 - 4 * q^68 - 16 * q^69 - 4 * q^74 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 12 * q^82 + 4 * q^87 - 2 * q^90 + 16 * q^92 + 8 * q^95

Introduction

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