LMFDB - Embedded newform 5070.2.b.k.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:	$1$
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 30)
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.k.1351.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +4.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000i q^{18} -4.00000i q^{19} +1.00000i q^{20} +4.00000i q^{21} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -4.00000i q^{28} -6.00000 q^{29} -1.00000 q^{30} +8.00000i q^{31} -1.00000i q^{32} +6.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} -4.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} +4.00000 q^{42} +4.00000 q^{43} -1.00000i q^{45} +1.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} -1.00000i q^{54} -4.00000 q^{56} -4.00000i q^{57} +6.00000i q^{58} +1.00000i q^{60} -10.0000 q^{61} +8.00000 q^{62} +4.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{68} -4.00000i q^{70} +1.00000i q^{72} -2.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} +4.00000i q^{76} +8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000i q^{83} -4.00000i q^{84} +6.00000i q^{85} -4.00000i q^{86} -6.00000 q^{87} -18.0000i q^{89} -1.00000 q^{90} +8.00000i q^{93} -4.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 12 q^{17} - 2 q^{25} + 2 q^{27} - 12 q^{29} - 2 q^{30} + 8 q^{35} - 2 q^{36} - 8 q^{38} + 2 q^{40} + 8 q^{42} + 8 q^{43} + 2 q^{48} - 18 q^{49} - 12 q^{51} - 12 q^{53} - 8 q^{56} - 20 q^{61} + 16 q^{62} - 2 q^{64} + 12 q^{68} - 4 q^{74} - 2 q^{75} + 16 q^{79} + 2 q^{81} - 12 q^{82} - 12 q^{87} - 2 q^{90} - 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 - 12 * q^17 - 2 * q^25 + 2 * q^27 - 12 * q^29 - 2 * q^30 + 8 * q^35 - 2 * q^36 - 8 * q^38 + 2 * q^40 + 8 * q^42 + 8 * q^43 + 2 * q^48 - 18 * q^49 - 12 * q^51 - 12 * q^53 - 8 * q^56 - 20 * q^61 + 16 * q^62 - 2 * q^64 + 12 * q^68 - 4 * q^74 - 2 * q^75 + 16 * q^79 + 2 * q^81 - 12 * q^82 - 12 * q^87 - 2 * q^90 - 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	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	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
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	1.00000i	0.223607i
$21$	4.00000i	0.872872i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000i	0.204124i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	− 4.00000i	− 0.755929i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	8.00000i	1.43684i	0.695608	+	0.718421i	$0.255135\pi$
$31$	8.00000i	1.43684i	−0.695608	+	0.718421i	$0.744865\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	6.00000i	1.02899i
$35$	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
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	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
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	− 1.00000i	− 0.136083i
$55$	0	0
$56$	−4.00000	−0.534522
$57$	− 4.00000i	− 0.529813i
$58$	6.00000i	0.787839i
$59$	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	$-0.0785709\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	− 4.00000i	− 0.478091i
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−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	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	− 1.00000i	− 0.111803i
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	− 4.00000i	− 0.436436i
$85$	6.00000i	0.650791i
$86$	− 4.00000i	− 0.431331i
$87$	−6.00000	−0.643268
$88$	0	0
$89$	− 18.0000i	− 1.90800i	−0.299813	−	0.953998i	$-0.596924\pi$
$89$	− 18.0000i	− 1.90800i	0.299813	−	0.953998i	$-0.403076\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	− 1.00000i	− 0.102062i
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	6.00000i	0.594089i
$103$	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
$106$	6.00000i	0.582772i
$107$	−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
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	0	0
$111$	− 2.00000i	− 0.189832i
$112$	4.00000i	0.377964i
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	0	0
$119$	− 24.0000i	− 2.20008i
$120$	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
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	−4.00000	−0.345547
$135$	− 1.00000i	− 0.0860663i
$136$	− 6.00000i	− 0.514496i
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	6.00000i	0.498273i
$146$	−2.00000	−0.165521
$147$	−9.00000	−0.742307
$148$	2.00000i	0.164399i
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	1.00000i	0.0816497i
$151$	− 8.00000i	− 0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$	− 8.00000i	− 0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$	4.00000	0.324443
$153$	−6.00000	−0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	− 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	$0.949930\pi$
$164$	6.00000i	0.468521i
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−4.00000	−0.308607
$169$	0	0
$170$	6.00000	0.460179
$171$	− 4.00000i	− 0.305888i
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	6.00000i	0.454859i
$175$	− 4.00000i	− 0.302372i
$176$	0	0
$177$	0	0
$178$	−18.0000	−1.34916
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	1.00000i	0.0745356i
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−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
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	−1.00000	−0.0721688
$193$	22.0000i	1.58359i	0.610784	+	0.791797i	$0.290854\pi$
$193$	22.0000i	1.58359i	−0.610784	+	0.791797i	$0.709146\pi$
$194$	2.00000	0.143592
$195$	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	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	− 1.00000i	− 0.0707107i
$201$	− 4.00000i	− 0.282138i
$202$	18.0000i	1.26648i
$203$	− 24.0000i	− 1.68447i
$204$	6.00000	0.420084
$205$	−6.00000	−0.419058
$206$	− 4.00000i	− 0.278693i
$207$	0	0
$208$	0	0
$209$	0	0
$210$	− 4.00000i	− 0.276026i
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	12.0000i	0.820303i
$215$	− 4.00000i	− 0.272798i
$216$	1.00000i	0.0680414i
$217$	−32.0000	−2.17230
$218$	−10.0000	−0.677285
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	20.0000i	1.33930i	0.742677	+	0.669650i	$0.233556\pi$
$223$	20.0000i	1.33930i	−0.742677	+	0.669650i	$0.766444\pi$
$224$	4.00000	0.267261
$225$	−1.00000	−0.0666667
$226$	18.0000i	1.19734i
$227$	− 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
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	−24.0000	−1.55569
$239$	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
$249$	12.0000i	0.760469i
$250$	1.00000	0.0632456
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	− 4.00000i	− 0.251976i
$253$	0	0
$254$	20.0000i	1.25491i
$255$	6.00000i	0.375735i
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	− 4.00000i	− 0.249029i
$259$	8.00000	0.497096
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	6.00000i	0.368577i
$266$	− 16.0000i	− 0.981023i
$267$	− 18.0000i	− 1.10158i
$268$	4.00000i	0.244339i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−1.00000	−0.0608581
$271$	16.0000i	0.971931i	0.873978	+	0.485965i	$0.161532\pi$
$271$	16.0000i	0.971931i	−0.873978	+	0.485965i	$0.838468\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	4.00000i	0.239904i
$279$	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
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	24.0000	1.41668
$288$	− 1.00000i	− 0.0589256i
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	2.00000i	0.117242i
$292$	2.00000i	0.117041i
$293$	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
$298$	−6.00000	−0.347571
$299$	0	0
$300$	1.00000	0.0577350
$301$	16.0000i	0.922225i
$302$	−8.00000	−0.460348
$303$	−18.0000	−1.03407
$304$	− 4.00000i	− 0.229416i
$305$	10.0000i	0.572598i
$306$	6.00000i	0.342997i
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	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
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	− 2.00000i	− 0.112867i
$315$	4.00000	0.225374
$316$	−8.00000	−0.450035
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	6.00000i	0.336463i
$319$	0	0
$320$	1.00000i	0.0559017i
$321$	−12.0000	−0.669775
$322$	0	0
$323$	24.0000i	1.33540i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	− 10.0000i	− 0.553001i
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	− 28.0000i	− 1.53902i	−0.638635	−	0.769510i	$-0.720501\pi$
$331$	− 28.0000i	− 1.53902i	0.638635	−	0.769510i	$-0.279499\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	−4.00000	−0.218543
$336$	4.00000i	0.218218i
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	− 6.00000i	− 0.325396i
$341$	0	0
$342$	−4.00000	−0.216295
$343$	− 8.00000i	− 0.431959i
$344$	4.00000i	0.215666i
$345$	0	0
$346$	18.0000i	0.967686i
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	6.00000	0.321634
$349$	10.0000i	0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$	10.0000i	0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	18.0000i	0.953998i
$357$	− 24.0000i	− 1.27021i
$358$	24.0000i	1.26844i
$359$	24.0000i	1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$	24.0000i	1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	1.00000	0.0527046
$361$	3.00000	0.157895
$362$	14.0000i	0.735824i
$363$	11.0000	0.577350
$364$	0	0
$365$	−2.00000	−0.104685
$366$	10.0000i	0.522708i
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	− 6.00000i	− 0.312348i
$370$	2.00000i	0.103975i
$371$	− 24.0000i	− 1.24602i
$372$	− 8.00000i	− 0.414781i
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	1.00000i	0.0516398i
$376$	0	0
$377$	0	0
$378$	4.00000	0.205738
$379$	− 4.00000i	− 0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$	− 4.00000i	− 0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	4.00000	0.205196
$381$	−20.0000	−1.02463
$382$	24.0000i	1.22795i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	1.00000i	0.0510310i
$385$	0	0
$386$	22.0000	1.11977
$387$	4.00000	0.203331
$388$	− 2.00000i	− 0.101535i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	8.00000i	0.401004i
$399$	16.0000	0.801002
$400$	−1.00000	−0.0500000
$401$	6.00000i	0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$	6.00000i	0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$	−4.00000	−0.199502
$403$	0	0
$404$	18.0000	0.895533
$405$	− 1.00000i	− 0.0496904i
$406$	−24.0000	−1.19110
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	26.0000i	1.28562i	0.766027	+	0.642809i	$0.222231\pi$
$409$	26.0000i	1.28562i	−0.766027	+	0.642809i	$0.777769\pi$
$410$	6.00000i	0.296319i
$411$	− 6.00000i	− 0.295958i
$412$	−4.00000	−0.197066
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	−4.00000	−0.195180
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	− 20.0000i	− 0.973585i
$423$	0	0
$424$	− 6.00000i	− 0.291386i
$425$	6.00000	0.291043
$426$	0	0
$427$	− 40.0000i	− 1.93574i
$428$	12.0000	0.580042
$429$	0	0
$430$	−4.00000	−0.192897
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	32.0000i	1.53605i
$435$	6.00000i	0.287678i
$436$	10.0000i	0.478913i
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	−9.00000	−0.428571
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	2.00000i	0.0949158i
$445$	−18.0000	−0.853282
$446$	20.0000	0.947027
$447$	− 6.00000i	− 0.283790i
$448$	− 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
$452$	18.0000	0.846649
$453$	− 8.00000i	− 0.375873i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	26.0000i	1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$	26.0000i	1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$	10.0000	0.467269
$459$	−6.00000	−0.280056
$460$	0	0
$461$	− 30.0000i	− 1.39724i	−0.715493	−	0.698620i	$-0.753798\pi$
$461$	− 30.0000i	− 1.39724i	0.715493	−	0.698620i	$-0.246202\pi$
$462$	0	0
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−6.00000	−0.278543
$465$	8.00000	0.370991
$466$	− 18.0000i	− 0.833834i
$467$	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
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	− 8.00000i	− 0.367452i
$475$	4.00000i	0.183533i
$476$	24.0000i	1.10004i
$477$	−6.00000	−0.274721
$478$	24.0000	1.09773
$479$	24.0000i	1.09659i	0.836286	+	0.548294i	$0.184723\pi$
$479$	24.0000i	1.09659i	−0.836286	+	0.548294i	$0.815277\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	−11.0000	−0.500000
$485$	2.00000	0.0908153
$486$	− 1.00000i	− 0.0453609i
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	− 10.0000i	− 0.452679i
$489$	4.00000i	0.180886i
$490$	9.00000	0.406579
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	6.00000i	0.270501i
$493$	36.0000	1.62136
$494$	0	0
$495$	0	0
$496$	8.00000i	0.359211i
$497$	0	0
$498$	12.0000	0.537733
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	− 1.00000i	− 0.0447214i
$501$	0	0
$502$	− 24.0000i	− 1.07117i
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−4.00000	−0.178174
$505$	18.0000i	0.800989i
$506$	0	0
$507$	0	0
$508$	20.0000	0.887357
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	6.00000	0.265684
$511$	8.00000	0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	− 4.00000i	− 0.176604i
$514$	− 18.0000i	− 0.793946i
$515$	− 4.00000i	− 0.176261i
$516$	−4.00000	−0.176090
$517$	0	0
$518$	− 8.00000i	− 0.351500i
$519$	−18.0000	−0.790112
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	6.00000i	0.262613i
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	− 4.00000i	− 0.174574i
$526$	0	0
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	−23.0000	−1.00000
$530$	6.00000	0.260623
$531$	0	0
$532$	−16.0000	−0.693688
$533$	0	0
$534$	−18.0000	−0.778936
$535$	12.0000i	0.518805i
$536$	4.00000	0.172774
$537$	−24.0000	−1.03568
$538$	6.00000i	0.258678i
$539$	0	0
$540$	1.00000i	0.0430331i
$541$	10.0000i	0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$	10.0000i	0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$	16.0000	0.687259
$543$	−14.0000	−0.600798
$544$	6.00000i	0.257248i
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	6.00000i	0.256307i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	24.0000i	1.02243i
$552$	0	0
$553$	32.0000i	1.36078i
$554$	2.00000i	0.0849719i
$555$	−2.00000	−0.0848953
$556$	4.00000	0.169638
$557$	− 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
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	18.0000i	0.757266i
$566$	− 28.0000i	− 1.17693i
$567$	4.00000i	0.167984i
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	4.00000i	0.167542i
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	− 24.0000i	− 1.00174i
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 19.0000i	− 0.790296i
$579$	22.0000i	0.914289i
$580$	− 6.00000i	− 0.249136i
$581$	−48.0000	−1.99138
$582$	2.00000	0.0829027
$583$	0	0
$584$	2.00000	0.0827606
$585$	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
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	0	0
$595$	−24.0000	−0.983904
$596$	6.00000i	0.245770i
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	16.0000	0.652111
$603$	− 4.00000i	− 0.162893i
$604$	8.00000i	0.325515i
$605$	− 11.0000i	− 0.447214i
$606$	18.0000i	0.731200i
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	−4.00000	−0.162221
$609$	− 24.0000i	− 0.972529i
$610$	10.0000	0.404888
$611$	0	0
$612$	6.00000	0.242536
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	−20.0000	−0.807134
$615$	−6.00000	−0.241943
$616$	0	0
$617$	30.0000i	1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	− 4.00000i	− 0.160904i
$619$	− 44.0000i	− 1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$	− 44.0000i	− 1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	0	0
$623$	72.0000	2.88462
$624$	0	0
$625$	1.00000	0.0400000
$626$	− 2.00000i	− 0.0799361i
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	12.0000i	0.478471i
$630$	− 4.00000i	− 0.159364i
$631$	− 32.0000i	− 1.27390i	−0.770905	−	0.636950i	$-0.780196\pi$
$631$	− 32.0000i	− 1.27390i	0.770905	−	0.636950i	$-0.219804\pi$
$632$	8.00000i	0.318223i
$633$	20.0000	0.794929
$634$	18.0000	0.714871
$635$	20.0000i	0.793676i
$636$	6.00000	0.237915
$637$	0	0
$638$	0	0
$639$	0	0
$640$	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
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	0	0
$645$	− 4.00000i	− 0.157500i
$646$	24.0000	0.944267
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	−32.0000	−1.25418
$652$	− 4.00000i	− 0.156652i
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	− 6.00000i	− 0.234261i
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	48.0000	1.86981	0.934907	−	0.354892i	$-0.115482\pi$
$659$	48.0000	1.86981	0.934907	+	0.354892i	$0.115482\pi$
$660$	0	0
$661$	− 14.0000i	− 0.544537i	−0.962221	−	0.272268i	$-0.912226\pi$
$661$	− 14.0000i	− 0.544537i	0.962221	−	0.272268i	$-0.0877739\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	−12.0000	−0.465690
$665$	− 16.0000i	− 0.620453i
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	0	0
$669$	20.0000i	0.773245i
$670$	4.00000i	0.154533i
$671$	0	0
$672$	4.00000	0.154303
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	26.0000i	1.00148i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	18.0000i	0.691286i
$679$	−8.00000	−0.307012
$680$	−6.00000	−0.230089
$681$	− 12.0000i	− 0.459841i
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	4.00000i	0.152944i
$685$	−6.00000	−0.229248
$686$	−8.00000	−0.305441
$687$	10.0000i	0.381524i
$688$	4.00000	0.152499
$689$	0	0
$690$	0	0
$691$	44.0000i	1.67384i	0.547326	+	0.836919i	$0.315646\pi$
$691$	44.0000i	1.67384i	−0.547326	+	0.836919i	$0.684354\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	− 12.0000i	− 0.455514i
$695$	4.00000i	0.151729i
$696$	− 6.00000i	− 0.227429i
$697$	36.0000i	1.36360i
$698$	10.0000	0.378506
$699$	18.0000	0.680823
$700$	4.00000i	0.151186i
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	0	0
$705$	0	0
$706$	6.00000	0.225813
$707$	− 72.0000i	− 2.70784i
$708$	0	0
$709$	− 38.0000i	− 1.42712i	−0.700594	−	0.713560i	$-0.747082\pi$
$709$	− 38.0000i	− 1.42712i	0.700594	−	0.713560i	$-0.252918\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	18.0000	0.674579
$713$	0	0
$714$	−24.0000	−0.898177
$715$	0	0
$716$	24.0000	0.896922
$717$	24.0000i	0.896296i
$718$	24.0000	0.895672
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	− 1.00000i	− 0.0372678i
$721$	16.0000i	0.595871i
$722$	− 3.00000i	− 0.111648i
$723$	− 2.00000i	− 0.0743808i
$724$	14.0000	0.520306
$725$	6.00000	0.222834
$726$	− 11.0000i	− 0.408248i
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	2.00000i	0.0740233i
$731$	−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	$-0.133179\pi$
$734$	28.0000i	1.03350i
$735$	9.00000i	0.331970i
$736$	0	0
$737$	0	0
$738$	−6.00000	−0.220863
$739$	52.0000i	1.91285i	0.291977	+	0.956425i	$0.405687\pi$
$739$	52.0000i	1.91285i	−0.291977	+	0.956425i	$0.594313\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	−24.0000	−0.881068
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	−8.00000	−0.293294
$745$	−6.00000	−0.219823
$746$	− 26.0000i	− 0.951928i
$747$	12.0000i	0.439057i
$748$	0	0
$749$	− 48.0000i	− 1.75388i
$750$	1.00000	0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	−8.00000	−0.291150
$756$	− 4.00000i	− 0.145479i
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−4.00000	−0.145287
$759$	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	$-0.105785\pi$
$762$	20.0000i	0.724524i
$763$	40.0000	1.44810
$764$	24.0000	0.868290
$765$	6.00000i	0.216930i
$766$	0	0
$767$	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
$771$	18.0000	0.648254
$772$	− 22.0000i	− 0.791797i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	− 4.00000i	− 0.143777i
$775$	− 8.00000i	− 0.287368i
$776$	−2.00000	−0.0717958
$777$	8.00000	0.286998
$778$	− 6.00000i	− 0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−6.00000	−0.214423
$784$	−9.00000	−0.321429
$785$	− 2.00000i	− 0.0713831i
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	−8.00000	−0.284627
$791$	− 72.0000i	− 2.56003i
$792$	0	0
$793$	0	0
$794$	22.0000	0.780751
$795$	6.00000i	0.212798i
$796$	8.00000	0.283552
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	− 16.0000i	− 0.566394i
$799$	0	0
$800$	1.00000i	0.0353553i
$801$	− 18.0000i	− 0.635999i
$802$	6.00000	0.211867
$803$	0	0
$804$	4.00000i	0.141069i
$805$	0	0
$806$	0	0
$807$	−6.00000	−0.211210
$808$	− 18.0000i	− 0.633238i
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	−1.00000	−0.0351364
$811$	− 4.00000i	− 0.140459i	−0.997531	−	0.0702295i	$-0.977627\pi$
$811$	− 4.00000i	− 0.140459i	0.997531	−	0.0702295i	$-0.0223732\pi$
$812$	24.0000i	0.842235i
$813$	16.0000i	0.561144i
$814$	0	0
$815$	4.00000	0.140114
$816$	−6.00000	−0.210042
$817$	− 16.0000i	− 0.559769i
$818$	26.0000	0.909069
$819$	0	0
$820$	6.00000	0.209529
$821$	18.0000i	0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$	18.0000i	0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$	−6.00000	−0.209274
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	4.00000i	0.139347i
$825$	0	0
$826$	0	0
$827$	− 12.0000i	− 0.417281i	−0.977992	−	0.208640i	$-0.933096\pi$
$827$	− 12.0000i	− 0.417281i	0.977992	−	0.208640i	$-0.0669038\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	− 12.0000i	− 0.416526i
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	54.0000	1.87099
$834$	4.00000i	0.138509i
$835$	0	0
$836$	0	0
$837$	8.00000i	0.276520i
$838$	0	0
$839$	− 24.0000i	− 0.828572i	−0.910147	−	0.414286i	$-0.864031\pi$
$839$	− 24.0000i	− 0.828572i	0.910147	−	0.414286i	$-0.135969\pi$
$840$	4.00000i	0.138013i
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	− 18.0000i	− 0.619953i
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	44.0000i	1.51186i
$848$	−6.00000	−0.206041
$849$	28.0000	0.960958
$850$	− 6.00000i	− 0.205798i
$851$	0	0
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	−40.0000	−1.36877
$855$	−4.00000	−0.136797
$856$	− 12.0000i	− 0.410152i
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	4.00000i	0.136399i
$861$	24.0000	0.817918
$862$	0	0
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	− 1.00000i	− 0.0340207i
$865$	18.0000i	0.612018i
$866$	26.0000i	0.883516i
$867$	19.0000	0.645274
$868$	32.0000	1.08615
$869$	0	0
$870$	6.00000	0.203419
$871$	0	0
$872$	10.0000	0.338643
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	−4.00000	−0.135225
$876$	2.00000i	0.0675737i
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	8.00000i	0.269987i
$879$	6.00000i	0.202375i
$880$	0	0
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	9.00000i	0.303046i
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	0	0
$885$	0	0
$886$	− 12.0000i	− 0.403148i
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	2.00000	0.0671156
$889$	− 80.0000i	− 2.68311i
$890$	18.0000i	0.603361i
$891$	0	0
$892$	− 20.0000i	− 0.669650i
$893$	0	0
$894$	−6.00000	−0.200670
$895$	24.0000i	0.802232i
$896$	−4.00000	−0.133631
$897$	0	0
$898$	6.00000	0.200223
$899$	− 48.0000i	− 1.60089i
$900$	1.00000	0.0333333
$901$	36.0000	1.19933
$902$	0	0
$903$	16.0000i	0.532447i
$904$	− 18.0000i	− 0.598671i
$905$	14.0000i	0.465376i
$906$	−8.00000	−0.265782
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	12.0000i	0.398234i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	− 4.00000i	− 0.132453i
$913$	0	0
$914$	26.0000	0.860004
$915$	10.0000i	0.330590i
$916$	− 10.0000i	− 0.330409i
$917$	0	0
$918$	6.00000i	0.198030i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	− 20.0000i	− 0.659022i
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	4.00000	0.131448
$927$	4.00000	0.131377
$928$	6.00000i	0.196960i
$929$	− 6.00000i	− 0.196854i	−0.995144	−	0.0984268i	$-0.968619\pi$
$929$	− 6.00000i	− 0.196854i	0.995144	−	0.0984268i	$-0.0313810\pi$
$930$	− 8.00000i	− 0.262330i
$931$	36.0000i	1.17985i
$932$	−18.0000	−0.589610
$933$	0	0
$934$	− 36.0000i	− 1.17796i
$935$	0	0
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	− 16.0000i	− 0.522419i
$939$	2.00000	0.0652675
$940$	0	0
$941$	18.0000i	0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$	18.0000i	0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	0	0
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	4.00000	0.129777
$951$	18.0000i	0.583690i
$952$	24.0000	0.777844
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	6.00000i	0.194257i
$955$	24.0000i	0.776622i
$956$	− 24.0000i	− 0.776215i
$957$	0	0
$958$	24.0000	0.775405
$959$	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
$965$	22.0000	0.708205
$966$	0	0
$967$	− 4.00000i	− 0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$	− 4.00000i	− 0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$	11.0000i	0.353553i
$969$	24.0000i	0.770991i
$970$	− 2.00000i	− 0.0642161i
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	−1.00000	−0.0320750
$973$	− 16.0000i	− 0.512936i
$974$	−28.0000	−0.897178
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	4.00000	0.127906
$979$	0	0
$980$	− 9.00000i	− 0.287494i
$981$	− 10.0000i	− 0.319275i
$982$	24.0000i	0.765871i
$983$	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
$985$	−6.00000	−0.191176
$986$	− 36.0000i	− 1.14647i
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	8.00000	0.254000
$993$	− 28.0000i	− 0.888553i
$994$	0	0
$995$	8.00000i	0.253617i
$996$	− 12.0000i	− 0.380235i
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−4.00000	−0.126618
$999$	− 2.00000i	− 0.0632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5070.2.b.k.1351.1		2
13.5	odd	4		30.2.a.a.1.1	✓	1
13.8	odd	4		5070.2.a.w.1.1		1
13.12	even	2	inner	5070.2.b.k.1351.2		2
39.5	even	4		90.2.a.c.1.1		1
52.31	even	4		240.2.a.b.1.1		1
65.18	even	4		150.2.c.a.49.2		2
65.44	odd	4		150.2.a.b.1.1		1
65.57	even	4		150.2.c.a.49.1		2
91.5	even	12		1470.2.i.q.361.1		2
91.18	odd	12		1470.2.i.o.961.1		2
91.31	even	12		1470.2.i.q.961.1		2
91.44	odd	12		1470.2.i.o.361.1		2
91.83	even	4		1470.2.a.d.1.1		1
104.5	odd	4		960.2.a.e.1.1		1
104.83	even	4		960.2.a.p.1.1		1
117.5	even	12		810.2.e.b.541.1		2
117.31	odd	12		810.2.e.l.541.1		2
117.70	odd	12		810.2.e.l.271.1		2
117.83	even	12		810.2.e.b.271.1		2
143.109	even	4		3630.2.a.w.1.1		1
156.83	odd	4		720.2.a.j.1.1		1
195.44	even	4		450.2.a.d.1.1		1
195.83	odd	4		450.2.c.b.199.1		2
195.122	odd	4		450.2.c.b.199.2		2
208.5	odd	4		3840.2.k.y.1921.1		2
208.83	even	4		3840.2.k.f.1921.1		2
208.109	odd	4		3840.2.k.y.1921.2		2
208.187	even	4		3840.2.k.f.1921.2		2
221.135	odd	4		8670.2.a.g.1.1		1
260.83	odd	4		1200.2.f.e.49.1		2
260.187	odd	4		1200.2.f.e.49.2		2
260.239	even	4		1200.2.a.k.1.1		1
273.83	odd	4		4410.2.a.z.1.1		1
312.5	even	4		2880.2.a.a.1.1		1
312.83	odd	4		2880.2.a.q.1.1		1
455.174	even	4		7350.2.a.ct.1.1		1
520.83	odd	4		4800.2.f.w.3649.2		2
520.109	odd	4		4800.2.a.cq.1.1		1
520.187	odd	4		4800.2.f.w.3649.1		2
520.213	even	4		4800.2.f.p.3649.1		2
520.317	even	4		4800.2.f.p.3649.2		2
520.499	even	4		4800.2.a.d.1.1		1
780.83	even	4		3600.2.f.i.2449.1		2
780.239	odd	4		3600.2.a.f.1.1		1
780.707	even	4		3600.2.f.i.2449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.a.a.1.1	✓	1	13.5	odd	4
90.2.a.c.1.1		1	39.5	even	4
150.2.a.b.1.1		1	65.44	odd	4
150.2.c.a.49.1		2	65.57	even	4
150.2.c.a.49.2		2	65.18	even	4
240.2.a.b.1.1		1	52.31	even	4
450.2.a.d.1.1		1	195.44	even	4
450.2.c.b.199.1		2	195.83	odd	4
450.2.c.b.199.2		2	195.122	odd	4
720.2.a.j.1.1		1	156.83	odd	4
810.2.e.b.271.1		2	117.83	even	12
810.2.e.b.541.1		2	117.5	even	12
810.2.e.l.271.1		2	117.70	odd	12
810.2.e.l.541.1		2	117.31	odd	12
960.2.a.e.1.1		1	104.5	odd	4
960.2.a.p.1.1		1	104.83	even	4
1200.2.a.k.1.1		1	260.239	even	4
1200.2.f.e.49.1		2	260.83	odd	4
1200.2.f.e.49.2		2	260.187	odd	4
1470.2.a.d.1.1		1	91.83	even	4
1470.2.i.o.361.1		2	91.44	odd	12
1470.2.i.o.961.1		2	91.18	odd	12
1470.2.i.q.361.1		2	91.5	even	12
1470.2.i.q.961.1		2	91.31	even	12
2880.2.a.a.1.1		1	312.5	even	4
2880.2.a.q.1.1		1	312.83	odd	4
3600.2.a.f.1.1		1	780.239	odd	4
3600.2.f.i.2449.1		2	780.83	even	4
3600.2.f.i.2449.2		2	780.707	even	4
3630.2.a.w.1.1		1	143.109	even	4
3840.2.k.f.1921.1		2	208.83	even	4
3840.2.k.f.1921.2		2	208.187	even	4
3840.2.k.y.1921.1		2	208.5	odd	4
3840.2.k.y.1921.2		2	208.109	odd	4
4410.2.a.z.1.1		1	273.83	odd	4
4800.2.a.d.1.1		1	520.499	even	4
4800.2.a.cq.1.1		1	520.109	odd	4
4800.2.f.p.3649.1		2	520.213	even	4
4800.2.f.p.3649.2		2	520.317	even	4
4800.2.f.w.3649.1		2	520.187	odd	4
4800.2.f.w.3649.2		2	520.83	odd	4
5070.2.a.w.1.1		1	13.8	odd	4
5070.2.b.k.1351.1		2	1.1	even	1	trivial
5070.2.b.k.1351.2		2	13.12	even	2	inner
7350.2.a.ct.1.1		1	455.174	even	4
8670.2.a.g.1.1		1	221.135	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{5} -1.00000i q^{6} +4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +4.00000 q^{14} -1.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000i q^{18} -4.00000i q^{19} +1.00000i q^{20} +4.00000i q^{21} +1.00000i q^{24} -1.00000 q^{25} +1.00000 q^{27} -4.00000i q^{28} -6.00000 q^{29} -1.00000 q^{30} +8.00000i q^{31} -1.00000i q^{32} +6.00000i q^{34} +4.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} -4.00000 q^{38} +1.00000 q^{40} -6.00000i q^{41} +4.00000 q^{42} +4.00000 q^{43} -1.00000i q^{45} +1.00000 q^{48} -9.00000 q^{49} +1.00000i q^{50} -6.00000 q^{51} -6.00000 q^{53} -1.00000i q^{54} -4.00000 q^{56} -4.00000i q^{57} +6.00000i q^{58} +1.00000i q^{60} -10.0000 q^{61} +8.00000 q^{62} +4.00000i q^{63} -1.00000 q^{64} -4.00000i q^{67} +6.00000 q^{68} -4.00000i q^{70} +1.00000i q^{72} -2.00000i q^{73} -2.00000 q^{74} -1.00000 q^{75} +4.00000i q^{76} +8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000i q^{83} -4.00000i q^{84} +6.00000i q^{85} -4.00000i q^{86} -6.00000 q^{87} -18.0000i q^{89} -1.00000 q^{90} +8.00000i q^{93} -4.00000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 8 q^{14} + 2 q^{16} - 12 q^{17} - 2 q^{25} + 2 q^{27} - 12 q^{29} - 2 q^{30} + 8 q^{35} - 2 q^{36} - 8 q^{38} + 2 q^{40} + 8 q^{42} + 8 q^{43} + 2 q^{48} - 18 q^{49} - 12 q^{51} - 12 q^{53} - 8 q^{56} - 20 q^{61} + 16 q^{62} - 2 q^{64} + 12 q^{68} - 4 q^{74} - 2 q^{75} + 16 q^{79} + 2 q^{81} - 12 q^{82} - 12 q^{87} - 2 q^{90} - 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 + 8 * q^14 + 2 * q^16 - 12 * q^17 - 2 * q^25 + 2 * q^27 - 12 * q^29 - 2 * q^30 + 8 * q^35 - 2 * q^36 - 8 * q^38 + 2 * q^40 + 8 * q^42 + 8 * q^43 + 2 * q^48 - 18 * q^49 - 12 * q^51 - 12 * q^53 - 8 * q^56 - 20 * q^61 + 16 * q^62 - 2 * q^64 + 12 * q^68 - 4 * q^74 - 2 * q^75 + 16 * q^79 + 2 * q^81 - 12 * q^82 - 12 * q^87 - 2 * q^90 - 8 * q^95

Introduction

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