LMFDB - Embedded newform 2366.2.d.f.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2366,2,Mod(337,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2366.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2366.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.8926051182$
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 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2366.337
Dual form			2366.2.d.f.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{5} -1.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{5} -1.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{9} +3.00000 q^{10} -1.00000 q^{12} -1.00000 q^{14} +3.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} +2.00000i q^{18} -4.00000i q^{19} -3.00000i q^{20} -1.00000i q^{21} -3.00000 q^{23} +1.00000i q^{24} -4.00000 q^{25} -5.00000 q^{27} +1.00000i q^{28} +6.00000 q^{29} +3.00000 q^{30} -10.0000i q^{31} -1.00000i q^{32} +6.00000i q^{34} +3.00000 q^{35} +2.00000 q^{36} -8.00000i q^{37} -4.00000 q^{38} -3.00000 q^{40} -1.00000 q^{42} -8.00000 q^{43} -6.00000i q^{45} +3.00000i q^{46} -6.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +4.00000i q^{50} -6.00000 q^{51} +12.0000 q^{53} +5.00000i q^{54} +1.00000 q^{56} -4.00000i q^{57} -6.00000i q^{58} -3.00000i q^{59} -3.00000i q^{60} +11.0000 q^{61} -10.0000 q^{62} +2.00000i q^{63} -1.00000 q^{64} +2.00000i q^{67} +6.00000 q^{68} -3.00000 q^{69} -3.00000i q^{70} -3.00000i q^{71} -2.00000i q^{72} -2.00000i q^{73} -8.00000 q^{74} -4.00000 q^{75} +4.00000i q^{76} -4.00000 q^{79} +3.00000i q^{80} +1.00000 q^{81} +1.00000i q^{84} -18.0000i q^{85} +8.00000i q^{86} +6.00000 q^{87} +6.00000i q^{89} -6.00000 q^{90} +3.00000 q^{92} -10.0000i q^{93} -6.00000 q^{94} +12.0000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9	$ 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9} + 6 q^{10} - 2 q^{12} - 2 q^{14} + 2 q^{16} - 12 q^{17} - 6 q^{23} - 8 q^{25} - 10 q^{27} + 12 q^{29} + 6 q^{30} + 6 q^{35} + 4 q^{36} - 8 q^{38} - 6 q^{40} - 2 q^{42} - 16 q^{43} + 2 q^{48} - 2 q^{49} - 12 q^{51} + 24 q^{53} + 2 q^{56} + 22 q^{61} - 20 q^{62} - 2 q^{64} + 12 q^{68} - 6 q^{69} - 16 q^{74} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 12 q^{87} - 12 q^{90} + 6 q^{92} - 12 q^{94} + 24 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9 + 6 * q^10 - 2 * q^12 - 2 * q^14 + 2 * q^16 - 12 * q^17 - 6 * q^23 - 8 * q^25 - 10 * q^27 + 12 * q^29 + 6 * q^30 + 6 * q^35 + 4 * q^36 - 8 * q^38 - 6 * q^40 - 2 * q^42 - 16 * q^43 + 2 * q^48 - 2 * q^49 - 12 * q^51 + 24 * q^53 + 2 * q^56 + 22 * q^61 - 20 * q^62 - 2 * q^64 + 12 * q^68 - 6 * q^69 - 16 * q^74 - 8 * q^75 - 8 * q^79 + 2 * q^81 + 12 * q^87 - 12 * q^90 + 6 * q^92 - 12 * q^94 + 24 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times$.

$n$	$339$	$2199$
$\chi(n)$	$1$	$-1$

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−1.00000	−0.500000
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	− 1.00000i	− 0.408248i
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.00000i	0.353553i
$9$	−2.00000	−0.666667
$10$	3.00000	0.948683
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	3.00000i	0.774597i
$16$	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	$0.759367\pi$
$18$	2.00000i	0.471405i
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	− 3.00000i	− 0.670820i
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.00000i	0.204124i
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	1.00000i	0.188982i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	3.00000	0.547723
$31$	− 10.0000i	− 1.79605i	−0.439941	−	0.898027i	$-0.645001\pi$
$31$	− 10.0000i	− 1.79605i	0.439941	−	0.898027i	$-0.354999\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	6.00000i	1.02899i
$35$	3.00000	0.507093
$36$	2.00000	0.333333
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−3.00000	−0.474342
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	−1.00000	−0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	− 6.00000i	− 0.894427i
$46$	3.00000i	0.442326i
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	1.00000	0.144338
$49$	−1.00000	−0.142857
$50$	4.00000i	0.565685i
$51$	−6.00000	−0.840168
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	5.00000i	0.680414i
$55$	0	0
$56$	1.00000	0.133631
$57$	− 4.00000i	− 0.529813i
$58$	− 6.00000i	− 0.787839i
$59$	− 3.00000i	− 0.390567i	−0.980747	−	0.195283i	$-0.937437\pi$
$59$	− 3.00000i	− 0.390567i	0.980747	−	0.195283i	$-0.0625627\pi$
$60$	− 3.00000i	− 0.387298i
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	−10.0000	−1.27000
$63$	2.00000i	0.251976i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	6.00000	0.727607
$69$	−3.00000	−0.361158
$70$	− 3.00000i	− 0.358569i
$71$	− 3.00000i	− 0.356034i	−0.984027	−	0.178017i	$-0.943032\pi$
$71$	− 3.00000i	− 0.356034i	0.984027	−	0.178017i	$-0.0569683\pi$
$72$	− 2.00000i	− 0.235702i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	−8.00000	−0.929981
$75$	−4.00000	−0.461880
$76$	4.00000i	0.458831i
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	3.00000i	0.335410i
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	1.00000i	0.109109i
$85$	− 18.0000i	− 1.95237i
$86$	8.00000i	0.862662i
$87$	6.00000	0.643268
$88$	0	0
$89$	6.00000i	0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$	6.00000i	0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$	−6.00000	−0.632456
$91$	0	0
$92$	3.00000	0.312772
$93$	− 10.0000i	− 1.03695i
$94$	−6.00000	−0.618853
$95$	12.0000	1.23117
$96$	− 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	$0.967625\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	4.00000	0.400000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	6.00000i	0.594089i
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	− 12.0000i	− 1.16554i
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	5.00000	0.481125
$109$	− 16.0000i	− 1.53252i	−0.642529	−	0.766261i	$-0.722115\pi$
$109$	− 16.0000i	− 1.53252i	0.642529	−	0.766261i	$-0.277885\pi$
$110$	0	0
$111$	− 8.00000i	− 0.759326i
$112$	− 1.00000i	− 0.0944911i
$113$	−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
$115$	− 9.00000i	− 0.839254i
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−3.00000	−0.276172
$119$	6.00000i	0.550019i
$120$	−3.00000	−0.273861
$121$	11.0000	1.00000
$122$	− 11.0000i	− 0.995893i
$123$	0	0
$124$	10.0000i	0.898027i
$125$	3.00000i	0.268328i
$126$	2.00000	0.178174
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	1.00000i	0.0883883i
$129$	−8.00000	−0.704361
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	2.00000	0.172774
$135$	− 15.0000i	− 1.29099i
$136$	− 6.00000i	− 0.514496i
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	3.00000i	0.255377i
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−3.00000	−0.253546
$141$	− 6.00000i	− 0.505291i
$142$	−3.00000	−0.251754
$143$	0	0
$144$	−2.00000	−0.166667
$145$	18.0000i	1.49482i
$146$	−2.00000	−0.165521
$147$	−1.00000	−0.0824786
$148$	8.00000i	0.657596i
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	4.00000i	0.326599i
$151$	1.00000i	0.0813788i	0.999172	+	0.0406894i	$0.0129554\pi$
$151$	1.00000i	0.0813788i	−0.999172	+	0.0406894i	$0.987045\pi$
$152$	4.00000	0.324443
$153$	12.0000	0.970143
$154$	0	0
$155$	30.0000	2.40966
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	4.00000i	0.318223i
$159$	12.0000	0.951662
$160$	3.00000	0.237171
$161$	3.00000i	0.236433i
$162$	− 1.00000i	− 0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	−18.0000	−1.38054
$171$	8.00000i	0.611775i
$172$	8.00000	0.609994
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	− 6.00000i	− 0.454859i
$175$	4.00000i	0.302372i
$176$	0	0
$177$	− 3.00000i	− 0.225494i
$178$	6.00000	0.449719
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	6.00000i	0.447214i
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	11.0000	0.813143
$184$	− 3.00000i	− 0.221163i
$185$	24.0000	1.76452
$186$	−10.0000	−0.733236
$187$	0	0
$188$	6.00000i	0.437595i
$189$	5.00000i	0.363696i
$190$	− 12.0000i	− 0.870572i
$191$	−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
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	− 4.00000i	− 0.282843i
$201$	2.00000i	0.141069i
$202$	− 6.00000i	− 0.422159i
$203$	− 6.00000i	− 0.421117i
$204$	6.00000	0.420084
$205$	0	0
$206$	14.0000i	0.975426i
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	− 3.00000i	− 0.207020i
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−12.0000	−0.824163
$213$	− 3.00000i	− 0.205557i
$214$	18.0000i	1.23045i
$215$	− 24.0000i	− 1.63679i
$216$	− 5.00000i	− 0.340207i
$217$	−10.0000	−0.678844
$218$	−16.0000	−1.08366
$219$	− 2.00000i	− 0.135147i
$220$	0	0
$221$	0	0
$222$	−8.00000	−0.536925
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	−1.00000	−0.0668153
$225$	8.00000	0.533333
$226$	18.0000i	1.19734i
$227$	− 15.0000i	− 0.995585i	−0.867296	−	0.497792i	$-0.834144\pi$
$227$	− 15.0000i	− 0.995585i	0.867296	−	0.497792i	$-0.165856\pi$
$228$	4.00000i	0.264906i
$229$	22.0000i	1.45380i	0.686743	+	0.726900i	$0.259040\pi$
$229$	22.0000i	1.45380i	−0.686743	+	0.726900i	$0.740960\pi$
$230$	−9.00000	−0.593442
$231$	0	0
$232$	6.00000i	0.393919i
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	3.00000i	0.195283i
$237$	−4.00000	−0.259828
$238$	6.00000	0.388922
$239$	21.0000i	1.35838i	0.733964	+	0.679189i	$0.237668\pi$
$239$	21.0000i	1.35838i	−0.733964	+	0.679189i	$0.762332\pi$
$240$	3.00000i	0.193649i
$241$	28.0000i	1.80364i	0.432113	+	0.901819i	$0.357768\pi$
$241$	28.0000i	1.80364i	−0.432113	+	0.901819i	$0.642232\pi$
$242$	− 11.0000i	− 0.707107i
$243$	16.0000	1.02640
$244$	−11.0000	−0.704203
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	0	0
$248$	10.0000	0.635001
$249$	0	0
$250$	3.00000	0.189737
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	− 2.00000i	− 0.125988i
$253$	0	0
$254$	11.0000i	0.690201i
$255$	− 18.0000i	− 1.12720i
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	8.00000i	0.498058i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	−12.0000	−0.742781
$262$	− 15.0000i	− 0.926703i
$263$	−27.0000	−1.66489	−0.832446	−	0.554107i	$-0.813060\pi$
$263$	−27.0000	−1.66489	−0.832446	+	0.554107i	$0.813060\pi$
$264$	0	0
$265$	36.0000i	2.21146i
$266$	4.00000i	0.245256i
$267$	6.00000i	0.367194i
$268$	− 2.00000i	− 0.122169i
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−15.0000	−0.912871
$271$	− 2.00000i	− 0.121491i	−0.998153	−	0.0607457i	$-0.980652\pi$
$271$	− 2.00000i	− 0.121491i	0.998153	−	0.0607457i	$-0.0193479\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	3.00000	0.180579
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	16.0000i	0.959616i
$279$	20.0000i	1.19737i
$280$	3.00000i	0.179284i
$281$	− 18.0000i	− 1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$	− 18.0000i	− 1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$	−6.00000	−0.357295
$283$	31.0000	1.84276	0.921379	−	0.388664i	$-0.127063\pi$
$283$	31.0000	1.84276	0.921379	+	0.388664i	$0.127063\pi$
$284$	3.00000i	0.178017i
$285$	12.0000	0.710819
$286$	0	0
$287$	0	0
$288$	2.00000i	0.117851i
$289$	19.0000	1.11765
$290$	18.0000	1.05700
$291$	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	$0.943923\pi$
$294$	1.00000i	0.0583212i
$295$	9.00000	0.524000
$296$	8.00000	0.464991
$297$	0	0
$298$	0	0
$299$	0	0
$300$	4.00000	0.230940
$301$	8.00000i	0.461112i
$302$	1.00000	0.0575435
$303$	6.00000	0.344691
$304$	− 4.00000i	− 0.229416i
$305$	33.0000i	1.88957i
$306$	− 12.0000i	− 0.685994i
$307$	13.0000i	0.741949i	0.928643	+	0.370975i	$0.120976\pi$
$307$	13.0000i	0.741949i	−0.928643	+	0.370975i	$0.879024\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	− 30.0000i	− 1.70389i
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	− 2.00000i	− 0.112867i
$315$	−6.00000	−0.338062
$316$	4.00000	0.225018
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	− 12.0000i	− 0.672927i
$319$	0	0
$320$	− 3.00000i	− 0.167705i
$321$	−18.0000	−1.00466
$322$	3.00000	0.167183
$323$	24.0000i	1.33540i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	− 16.0000i	− 0.884802i
$328$	0	0
$329$	−6.00000	−0.330791
$330$	0	0
$331$	− 10.0000i	− 0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$	− 10.0000i	− 0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$	0	0
$333$	16.0000i	0.876795i
$334$	12.0000	0.656611
$335$	−6.00000	−0.327815
$336$	− 1.00000i	− 0.0545545i
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	18.0000i	0.976187i
$341$	0	0
$342$	8.00000	0.432590
$343$	1.00000i	0.0539949i
$344$	− 8.00000i	− 0.431331i
$345$	− 9.00000i	− 0.484544i
$346$	9.00000i	0.483843i
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	−6.00000	−0.321634
$349$	− 29.0000i	− 1.55233i	−0.630527	−	0.776167i	$-0.717161\pi$
$349$	− 29.0000i	− 1.55233i	0.630527	−	0.776167i	$-0.282839\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	−3.00000	−0.159448
$355$	9.00000	0.477670
$356$	− 6.00000i	− 0.317999i
$357$	6.00000i	0.317554i
$358$	6.00000i	0.317110i
$359$	9.00000i	0.475002i	0.971387	+	0.237501i	$0.0763283\pi$
$359$	9.00000i	0.475002i	−0.971387	+	0.237501i	$0.923672\pi$
$360$	6.00000	0.316228
$361$	3.00000	0.157895
$362$	5.00000i	0.262794i
$363$	11.0000	0.577350
$364$	0	0
$365$	6.00000	0.314054
$366$	− 11.0000i	− 0.574979i
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	− 24.0000i	− 1.24770i
$371$	− 12.0000i	− 0.623009i
$372$	10.0000i	0.518476i
$373$	32.0000	1.65690	0.828449	−	0.560065i	$-0.189224\pi$
$373$	32.0000	1.65690	0.828449	+	0.560065i	$0.189224\pi$
$374$	0	0
$375$	3.00000i	0.154919i
$376$	6.00000	0.309426
$377$	0	0
$378$	5.00000	0.257172
$379$	− 28.0000i	− 1.43826i	−0.694874	−	0.719132i	$-0.744540\pi$
$379$	− 28.0000i	− 1.43826i	0.694874	−	0.719132i	$-0.255460\pi$
$380$	−12.0000	−0.615587
$381$	−11.0000	−0.563547
$382$	24.0000i	1.22795i
$383$	− 6.00000i	− 0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$	− 6.00000i	− 0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$	1.00000i	0.0510310i
$385$	0	0
$386$	−5.00000	−0.254493
$387$	16.0000	0.813326
$388$	− 2.00000i	− 0.101535i
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	− 1.00000i	− 0.0505076i
$393$	15.0000	0.756650
$394$	12.0000	0.604551
$395$	− 12.0000i	− 0.603786i
$396$	0	0
$397$	7.00000i	0.351320i	0.984451	+	0.175660i	$0.0562059\pi$
$397$	7.00000i	0.351320i	−0.984451	+	0.175660i	$0.943794\pi$
$398$	14.0000i	0.701757i
$399$	−4.00000	−0.200250
$400$	−4.00000	−0.200000
$401$	− 6.00000i	− 0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$	− 6.00000i	− 0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$	2.00000	0.0997509
$403$	0	0
$404$	−6.00000	−0.298511
$405$	3.00000i	0.149071i
$406$	−6.00000	−0.297775
$407$	0	0
$408$	− 6.00000i	− 0.297044i
$409$	− 22.0000i	− 1.08783i	−0.839140	−	0.543915i	$-0.816941\pi$
$409$	− 22.0000i	− 1.08783i	0.839140	−	0.543915i	$-0.183059\pi$
$410$	0	0
$411$	− 3.00000i	− 0.147979i
$412$	14.0000	0.689730
$413$	−3.00000	−0.147620
$414$	− 6.00000i	− 0.294884i
$415$	0	0
$416$	0	0
$417$	−16.0000	−0.783523
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	−3.00000	−0.146385
$421$	26.0000i	1.26716i	0.773676	+	0.633581i	$0.218416\pi$
$421$	26.0000i	1.26716i	−0.773676	+	0.633581i	$0.781584\pi$
$422$	4.00000i	0.194717i
$423$	12.0000i	0.583460i
$424$	12.0000i	0.582772i
$425$	24.0000	1.16417
$426$	−3.00000	−0.145350
$427$	− 11.0000i	− 0.532327i
$428$	18.0000	0.870063
$429$	0	0
$430$	−24.0000	−1.15738
$431$	− 33.0000i	− 1.58955i	−0.606902	−	0.794777i	$-0.707588\pi$
$431$	− 33.0000i	− 1.58955i	0.606902	−	0.794777i	$-0.292412\pi$
$432$	−5.00000	−0.240563
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	10.0000i	0.480015i
$435$	18.0000i	0.863034i
$436$	16.0000i	0.766261i
$437$	12.0000i	0.574038i
$438$	−2.00000	−0.0955637
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	8.00000i	0.379663i
$445$	−18.0000	−0.853282
$446$	8.00000	0.378811
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	− 9.00000i	− 0.424736i	−0.977190	−	0.212368i	$-0.931882\pi$
$449$	− 9.00000i	− 0.424736i	0.977190	−	0.212368i	$-0.0681176\pi$
$450$	− 8.00000i	− 0.377124i
$451$	0	0
$452$	18.0000	0.846649
$453$	1.00000i	0.0469841i
$454$	−15.0000	−0.703985
$455$	0	0
$456$	4.00000	0.187317
$457$	29.0000i	1.35656i	0.734802	+	0.678281i	$0.237275\pi$
$457$	29.0000i	1.35656i	−0.734802	+	0.678281i	$0.762725\pi$
$458$	22.0000	1.02799
$459$	30.0000	1.40028
$460$	9.00000i	0.419627i
$461$	21.0000i	0.978068i	0.872265	+	0.489034i	$0.162651\pi$
$461$	21.0000i	0.978068i	−0.872265	+	0.489034i	$0.837349\pi$
$462$	0	0
$463$	31.0000i	1.44069i	0.693615	+	0.720346i	$0.256017\pi$
$463$	31.0000i	1.44069i	−0.693615	+	0.720346i	$0.743983\pi$
$464$	6.00000	0.278543
$465$	30.0000	1.39122
$466$	− 3.00000i	− 0.138972i
$467$	3.00000	0.138823	0.0694117	−	0.997588i	$-0.477888\pi$
$467$	3.00000	0.138823	0.0694117	+	0.997588i	$0.477888\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	− 18.0000i	− 0.830278i
$471$	2.00000	0.0921551
$472$	3.00000	0.138086
$473$	0	0
$474$	4.00000i	0.183726i
$475$	16.0000i	0.734130i
$476$	− 6.00000i	− 0.275010i
$477$	−24.0000	−1.09888
$478$	21.0000	0.960518
$479$	− 30.0000i	− 1.37073i	−0.728197	−	0.685367i	$-0.759642\pi$
$479$	− 30.0000i	− 1.37073i	0.728197	−	0.685367i	$-0.240358\pi$
$480$	3.00000	0.136931
$481$	0	0
$482$	28.0000	1.27537
$483$	3.00000i	0.136505i
$484$	−11.0000	−0.500000
$485$	−6.00000	−0.272446
$486$	− 16.0000i	− 0.725775i
$487$	− 13.0000i	− 0.589086i	−0.955638	−	0.294543i	$-0.904833\pi$
$487$	− 13.0000i	− 0.589086i	0.955638	−	0.294543i	$-0.0951675\pi$
$488$	11.0000i	0.497947i
$489$	− 20.0000i	− 0.904431i
$490$	−3.00000	−0.135526
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	0	0
$495$	0	0
$496$	− 10.0000i	− 0.449013i
$497$	−3.00000	−0.134568
$498$	0	0
$499$	− 22.0000i	− 0.984855i	−0.870353	−	0.492428i	$-0.836110\pi$
$499$	− 22.0000i	− 0.984855i	0.870353	−	0.492428i	$-0.163890\pi$
$500$	− 3.00000i	− 0.134164i
$501$	12.0000i	0.536120i
$502$	3.00000i	0.133897i
$503$	42.0000	1.87269	0.936344	−	0.351085i	$-0.114187\pi$
$503$	42.0000	1.87269	0.936344	+	0.351085i	$0.114187\pi$
$504$	−2.00000	−0.0890871
$505$	18.0000i	0.800989i
$506$	0	0
$507$	0	0
$508$	11.0000	0.488046
$509$	− 15.0000i	− 0.664863i	−0.943127	−	0.332432i	$-0.892131\pi$
$509$	− 15.0000i	− 0.664863i	0.943127	−	0.332432i	$-0.107869\pi$
$510$	−18.0000	−0.797053
$511$	−2.00000	−0.0884748
$512$	− 1.00000i	− 0.0441942i
$513$	20.0000i	0.883022i
$514$	− 6.00000i	− 0.264649i
$515$	− 42.0000i	− 1.85074i
$516$	8.00000	0.352180
$517$	0	0
$518$	8.00000i	0.351500i
$519$	−9.00000	−0.395056
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	12.0000i	0.525226i
$523$	−1.00000	−0.0437269	−0.0218635	−	0.999761i	$-0.506960\pi$
$523$	−1.00000	−0.0437269	−0.0218635	+	0.999761i	$0.506960\pi$
$524$	−15.0000	−0.655278
$525$	4.00000i	0.174574i
$526$	27.0000i	1.17726i
$527$	60.0000i	2.61364i
$528$	0	0
$529$	−14.0000	−0.608696
$530$	36.0000	1.56374
$531$	6.00000i	0.260378i
$532$	4.00000	0.173422
$533$	0	0
$534$	6.00000	0.259645
$535$	− 54.0000i	− 2.33462i
$536$	−2.00000	−0.0863868
$537$	−6.00000	−0.258919
$538$	− 9.00000i	− 0.388018i
$539$	0	0
$540$	15.0000i	0.645497i
$541$	− 38.0000i	− 1.63375i	−0.576816	−	0.816874i	$-0.695705\pi$
$541$	− 38.0000i	− 1.63375i	0.576816	−	0.816874i	$-0.304295\pi$
$542$	−2.00000	−0.0859074
$543$	−5.00000	−0.214571
$544$	6.00000i	0.257248i
$545$	48.0000	2.05609
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	3.00000i	0.128154i
$549$	−22.0000	−0.938937
$550$	0	0
$551$	− 24.0000i	− 1.02243i
$552$	− 3.00000i	− 0.127688i
$553$	4.00000i	0.170097i
$554$	26.0000i	1.10463i
$555$	24.0000	1.01874
$556$	16.0000	0.678551
$557$	− 24.0000i	− 1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$	− 24.0000i	− 1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	20.0000	0.846668
$559$	0	0
$560$	3.00000	0.126773
$561$	0	0
$562$	−18.0000	−0.759284
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	6.00000i	0.252646i
$565$	− 54.0000i	− 2.27180i
$566$	− 31.0000i	− 1.30303i
$567$	− 1.00000i	− 0.0419961i
$568$	3.00000	0.125877
$569$	45.0000	1.88650	0.943249	−	0.332086i	$-0.107752\pi$
$569$	45.0000	1.88650	0.943249	+	0.332086i	$0.107752\pi$
$570$	− 12.0000i	− 0.502625i
$571$	−8.00000	−0.334790	−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000	−0.334790	−0.167395	+	0.985890i	$0.553535\pi$
$572$	0	0
$573$	−24.0000	−1.00261
$574$	0	0
$575$	12.0000	0.500435
$576$	2.00000	0.0833333
$577$	− 28.0000i	− 1.16566i	−0.812596	−	0.582828i	$-0.801946\pi$
$577$	− 28.0000i	− 1.16566i	0.812596	−	0.582828i	$-0.198054\pi$
$578$	− 19.0000i	− 0.790296i
$579$	− 5.00000i	− 0.207793i
$580$	− 18.0000i	− 0.747409i
$581$	0	0
$582$	2.00000	0.0829027
$583$	0	0
$584$	2.00000	0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	3.00000i	0.123823i	0.998082	+	0.0619116i	$0.0197197\pi$
$587$	3.00000i	0.123823i	−0.998082	+	0.0619116i	$0.980280\pi$
$588$	1.00000	0.0412393
$589$	−40.0000	−1.64817
$590$	− 9.00000i	− 0.370524i
$591$	12.0000i	0.493614i
$592$	− 8.00000i	− 0.328798i
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	−18.0000	−0.737928
$596$	0	0
$597$	−14.0000	−0.572982
$598$	0	0
$599$	3.00000	0.122577	0.0612883	−	0.998120i	$-0.480479\pi$
$599$	3.00000	0.122577	0.0612883	+	0.998120i	$0.480479\pi$
$600$	− 4.00000i	− 0.163299i
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	8.00000	0.326056
$603$	− 4.00000i	− 0.162893i
$604$	− 1.00000i	− 0.0406894i
$605$	33.0000i	1.34164i
$606$	− 6.00000i	− 0.243733i
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	−4.00000	−0.162221
$609$	− 6.00000i	− 0.243132i
$610$	33.0000	1.33613
$611$	0	0
$612$	−12.0000	−0.485071
$613$	− 22.0000i	− 0.888572i	−0.895885	−	0.444286i	$-0.853457\pi$
$613$	− 22.0000i	− 0.888572i	0.895885	−	0.444286i	$-0.146543\pi$
$614$	13.0000	0.524637
$615$	0	0
$616$	0	0
$617$	− 3.00000i	− 0.120775i	−0.998175	−	0.0603877i	$-0.980766\pi$
$617$	− 3.00000i	− 0.120775i	0.998175	−	0.0603877i	$-0.0192337\pi$
$618$	14.0000i	0.563163i
$619$	37.0000i	1.48716i	0.668649	+	0.743578i	$0.266873\pi$
$619$	37.0000i	1.48716i	−0.668649	+	0.743578i	$0.733127\pi$
$620$	−30.0000	−1.20483
$621$	15.0000	0.601929
$622$	− 6.00000i	− 0.240578i
$623$	6.00000	0.240385
$624$	0	0
$625$	−29.0000	−1.16000
$626$	− 8.00000i	− 0.319744i
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	48.0000i	1.91389i
$630$	6.00000i	0.239046i
$631$	− 5.00000i	− 0.199047i	−0.995035	−	0.0995234i	$-0.968268\pi$
$631$	− 5.00000i	− 0.199047i	0.995035	−	0.0995234i	$-0.0317318\pi$
$632$	− 4.00000i	− 0.159111i
$633$	−4.00000	−0.158986
$634$	−12.0000	−0.476581
$635$	− 33.0000i	− 1.30957i
$636$	−12.0000	−0.475831
$637$	0	0
$638$	0	0
$639$	6.00000i	0.237356i
$640$	−3.00000	−0.118585
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	18.0000i	0.710403i
$643$	41.0000i	1.61688i	0.588577	+	0.808441i	$0.299688\pi$
$643$	41.0000i	1.61688i	−0.588577	+	0.808441i	$0.700312\pi$
$644$	− 3.00000i	− 0.118217i
$645$	− 24.0000i	− 0.944999i
$646$	24.0000	0.944267
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	1.00000i	0.0392837i
$649$	0	0
$650$	0	0
$651$	−10.0000	−0.391931
$652$	20.0000i	0.783260i
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	−16.0000	−0.625650
$655$	45.0000i	1.75830i
$656$	0	0
$657$	4.00000i	0.156055i
$658$	6.00000i	0.233904i
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	− 23.0000i	− 0.894596i	−0.894385	−	0.447298i	$-0.852386\pi$
$661$	− 23.0000i	− 0.894596i	0.894385	−	0.447298i	$-0.147614\pi$
$662$	−10.0000	−0.388661
$663$	0	0
$664$	0	0
$665$	− 12.0000i	− 0.465340i
$666$	16.0000	0.619987
$667$	−18.0000	−0.696963
$668$	− 12.0000i	− 0.464294i
$669$	8.00000i	0.309298i
$670$	6.00000i	0.231800i
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	14.0000i	0.539260i
$675$	20.0000	0.769800
$676$	0	0
$677$	−51.0000	−1.96009	−0.980045	−	0.198778i	$-0.936303\pi$
$677$	−51.0000	−1.96009	−0.980045	+	0.198778i	$0.936303\pi$
$678$	18.0000i	0.691286i
$679$	2.00000	0.0767530
$680$	18.0000	0.690268
$681$	− 15.0000i	− 0.574801i
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	− 8.00000i	− 0.305888i
$685$	9.00000	0.343872
$686$	1.00000	0.0381802
$687$	22.0000i	0.839352i
$688$	−8.00000	−0.304997
$689$	0	0
$690$	−9.00000	−0.342624
$691$	− 1.00000i	− 0.0380418i	−0.999819	−	0.0190209i	$-0.993945\pi$
$691$	− 1.00000i	− 0.0380418i	0.999819	−	0.0190209i	$-0.00605490\pi$
$692$	9.00000	0.342129
$693$	0	0
$694$	24.0000i	0.911028i
$695$	− 48.0000i	− 1.82074i
$696$	6.00000i	0.227429i
$697$	0	0
$698$	−29.0000	−1.09767
$699$	3.00000	0.113470
$700$	− 4.00000i	− 0.151186i
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	0	0
$705$	18.0000	0.677919
$706$	−6.00000	−0.225813
$707$	− 6.00000i	− 0.225653i
$708$	3.00000i	0.112747i
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	− 9.00000i	− 0.337764i
$711$	8.00000	0.300023
$712$	−6.00000	−0.224860
$713$	30.0000i	1.12351i
$714$	6.00000	0.224544
$715$	0	0
$716$	6.00000	0.224231
$717$	21.0000i	0.784259i
$718$	9.00000	0.335877
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	− 6.00000i	− 0.223607i
$721$	14.0000i	0.521387i
$722$	− 3.00000i	− 0.111648i
$723$	28.0000i	1.04133i
$724$	5.00000	0.185824
$725$	−24.0000	−0.891338
$726$	− 11.0000i	− 0.408248i
$727$	52.0000	1.92857	0.964287	−	0.264861i	$-0.0853260\pi$
$727$	52.0000	1.92857	0.964287	+	0.264861i	$0.0853260\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	− 6.00000i	− 0.222070i
$731$	48.0000	1.77534
$732$	−11.0000	−0.406572
$733$	35.0000i	1.29275i	0.763018	+	0.646377i	$0.223717\pi$
$733$	35.0000i	1.29275i	−0.763018	+	0.646377i	$0.776283\pi$
$734$	10.0000i	0.369107i
$735$	− 3.00000i	− 0.110657i
$736$	3.00000i	0.110581i
$737$	0	0
$738$	0	0
$739$	16.0000i	0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$	16.0000i	0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$	−24.0000	−0.882258
$741$	0	0
$742$	−12.0000	−0.440534
$743$	48.0000i	1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$	48.0000i	1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	− 32.0000i	− 1.17160i
$747$	0	0
$748$	0	0
$749$	18.0000i	0.657706i
$750$	3.00000	0.109545
$751$	−17.0000	−0.620339	−0.310169	−	0.950681i	$-0.600386\pi$
$751$	−17.0000	−0.620339	−0.310169	+	0.950681i	$0.600386\pi$
$752$	− 6.00000i	− 0.218797i
$753$	−3.00000	−0.109326
$754$	0	0
$755$	−3.00000	−0.109181
$756$	− 5.00000i	− 0.181848i
$757$	−52.0000	−1.88997	−0.944986	−	0.327111i	$-0.893925\pi$
$757$	−52.0000	−1.88997	−0.944986	+	0.327111i	$0.893925\pi$
$758$	−28.0000	−1.01701
$759$	0	0
$760$	12.0000i	0.435286i
$761$	− 12.0000i	− 0.435000i	−0.976060	−	0.217500i	$-0.930210\pi$
$761$	− 12.0000i	− 0.435000i	0.976060	−	0.217500i	$-0.0697902\pi$
$762$	11.0000i	0.398488i
$763$	−16.0000	−0.579239
$764$	24.0000	0.868290
$765$	36.0000i	1.30158i
$766$	−6.00000	−0.216789
$767$	0	0
$768$	1.00000	0.0360844
$769$	− 10.0000i	− 0.360609i	−0.983611	−	0.180305i	$-0.942292\pi$
$769$	− 10.0000i	− 0.360609i	0.983611	−	0.180305i	$-0.0577084\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	5.00000i	0.179954i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	− 16.0000i	− 0.575108i
$775$	40.0000i	1.43684i
$776$	−2.00000	−0.0717958
$777$	−8.00000	−0.286998
$778$	− 30.0000i	− 1.07555i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	− 18.0000i	− 0.643679i
$783$	−30.0000	−1.07211
$784$	−1.00000	−0.0357143
$785$	6.00000i	0.214149i
$786$	− 15.0000i	− 0.535032i
$787$	− 53.0000i	− 1.88925i	−0.328158	−	0.944623i	$-0.606428\pi$
$787$	− 53.0000i	− 1.88925i	0.328158	−	0.944623i	$-0.393572\pi$
$788$	− 12.0000i	− 0.427482i
$789$	−27.0000	−0.961225
$790$	−12.0000	−0.426941
$791$	18.0000i	0.640006i
$792$	0	0
$793$	0	0
$794$	7.00000	0.248421
$795$	36.0000i	1.27679i
$796$	14.0000	0.496217
$797$	15.0000	0.531327	0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000	0.531327	0.265664	+	0.964066i	$0.414409\pi$
$798$	4.00000i	0.141598i
$799$	36.0000i	1.27359i
$800$	4.00000i	0.141421i
$801$	− 12.0000i	− 0.423999i
$802$	−6.00000	−0.211867
$803$	0	0
$804$	− 2.00000i	− 0.0705346i
$805$	−9.00000	−0.317208
$806$	0	0
$807$	9.00000	0.316815
$808$	6.00000i	0.211079i
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	3.00000	0.105409
$811$	− 19.0000i	− 0.667180i	−0.942718	−	0.333590i	$-0.891740\pi$
$811$	− 19.0000i	− 0.667180i	0.942718	−	0.333590i	$-0.108260\pi$
$812$	6.00000i	0.210559i
$813$	− 2.00000i	− 0.0701431i
$814$	0	0
$815$	60.0000	2.10171
$816$	−6.00000	−0.210042
$817$	32.0000i	1.11954i
$818$	−22.0000	−0.769212
$819$	0	0
$820$	0	0
$821$	12.0000i	0.418803i	0.977830	+	0.209401i	$0.0671515\pi$
$821$	12.0000i	0.418803i	−0.977830	+	0.209401i	$0.932848\pi$
$822$	−3.00000	−0.104637
$823$	13.0000	0.453152	0.226576	−	0.973994i	$-0.427247\pi$
$823$	13.0000	0.453152	0.226576	+	0.973994i	$0.427247\pi$
$824$	− 14.0000i	− 0.487713i
$825$	0	0
$826$	3.00000i	0.104383i
$827$	− 24.0000i	− 0.834562i	−0.908778	−	0.417281i	$-0.862983\pi$
$827$	− 24.0000i	− 0.834562i	0.908778	−	0.417281i	$-0.137017\pi$
$828$	−6.00000	−0.208514
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	0	0
$833$	6.00000	0.207888
$834$	16.0000i	0.554035i
$835$	−36.0000	−1.24583
$836$	0	0
$837$	50.0000i	1.72825i
$838$	0	0
$839$	54.0000i	1.86429i	0.362089	+	0.932144i	$0.382064\pi$
$839$	54.0000i	1.86429i	−0.362089	+	0.932144i	$0.617936\pi$
$840$	3.00000i	0.103510i
$841$	7.00000	0.241379
$842$	26.0000	0.896019
$843$	− 18.0000i	− 0.619953i
$844$	4.00000	0.137686
$845$	0	0
$846$	12.0000	0.412568
$847$	− 11.0000i	− 0.377964i
$848$	12.0000	0.412082
$849$	31.0000	1.06392
$850$	− 24.0000i	− 0.823193i
$851$	24.0000i	0.822709i
$852$	3.00000i	0.102778i
$853$	− 17.0000i	− 0.582069i	−0.956713	−	0.291034i	$-0.906001\pi$
$853$	− 17.0000i	− 0.582069i	0.956713	−	0.291034i	$-0.0939994\pi$
$854$	−11.0000	−0.376412
$855$	−24.0000	−0.820783
$856$	− 18.0000i	− 0.615227i
$857$	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	$0.521738\pi$
$860$	24.0000i	0.818393i
$861$	0	0
$862$	−33.0000	−1.12398
$863$	− 57.0000i	− 1.94030i	−0.242500	−	0.970151i	$-0.577968\pi$
$863$	− 57.0000i	− 1.94030i	0.242500	−	0.970151i	$-0.422032\pi$
$864$	5.00000i	0.170103i
$865$	− 27.0000i	− 0.918028i
$866$	− 4.00000i	− 0.135926i
$867$	19.0000	0.645274
$868$	10.0000	0.339422
$869$	0	0
$870$	18.0000	0.610257
$871$	0	0
$872$	16.0000	0.541828
$873$	− 4.00000i	− 0.135379i
$874$	12.0000	0.405906
$875$	3.00000	0.101419
$876$	2.00000i	0.0675737i
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	14.0000i	0.472477i
$879$	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	$0.467773\pi$
$882$	− 2.00000i	− 0.0673435i
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	9.00000	0.302532
$886$	− 18.0000i	− 0.604722i
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	8.00000	0.268462
$889$	11.0000i	0.368928i
$890$	18.0000i	0.603361i
$891$	0	0
$892$	− 8.00000i	− 0.267860i
$893$	−24.0000	−0.803129
$894$	0	0
$895$	− 18.0000i	− 0.601674i
$896$	1.00000	0.0334077
$897$	0	0
$898$	−9.00000	−0.300334
$899$	− 60.0000i	− 2.00111i
$900$	−8.00000	−0.266667
$901$	−72.0000	−2.39867
$902$	0	0
$903$	8.00000i	0.266223i
$904$	− 18.0000i	− 0.598671i
$905$	− 15.0000i	− 0.498617i
$906$	1.00000	0.0332228
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	15.0000i	0.497792i
$909$	−12.0000	−0.398015
$910$	0	0
$911$	21.0000	0.695761	0.347881	−	0.937539i	$-0.386901\pi$
$911$	21.0000	0.695761	0.347881	+	0.937539i	$0.386901\pi$
$912$	− 4.00000i	− 0.132453i
$913$	0	0
$914$	29.0000	0.959235
$915$	33.0000i	1.09095i
$916$	− 22.0000i	− 0.726900i
$917$	− 15.0000i	− 0.495344i
$918$	− 30.0000i	− 0.990148i
$919$	−7.00000	−0.230909	−0.115454	−	0.993313i	$-0.536832\pi$
$919$	−7.00000	−0.230909	−0.115454	+	0.993313i	$0.536832\pi$
$920$	9.00000	0.296721
$921$	13.0000i	0.428365i
$922$	21.0000	0.691598
$923$	0	0
$924$	0	0
$925$	32.0000i	1.05215i
$926$	31.0000	1.01872
$927$	28.0000	0.919641
$928$	− 6.00000i	− 0.196960i
$929$	18.0000i	0.590561i	0.955411	+	0.295280i	$0.0954131\pi$
$929$	18.0000i	0.590561i	−0.955411	+	0.295280i	$0.904587\pi$
$930$	− 30.0000i	− 0.983739i
$931$	4.00000i	0.131095i
$932$	−3.00000	−0.0982683
$933$	6.00000	0.196431
$934$	− 3.00000i	− 0.0981630i
$935$	0	0
$936$	0	0
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	− 2.00000i	− 0.0653023i
$939$	8.00000	0.261070
$940$	−18.0000	−0.587095
$941$	15.0000i	0.488986i	0.969651	+	0.244493i	$0.0786215\pi$
$941$	15.0000i	0.488986i	−0.969651	+	0.244493i	$0.921378\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	0	0
$944$	− 3.00000i	− 0.0976417i
$945$	−15.0000	−0.487950
$946$	0	0
$947$	42.0000i	1.36482i	0.730971	+	0.682408i	$0.239067\pi$
$947$	42.0000i	1.36482i	−0.730971	+	0.682408i	$0.760933\pi$
$948$	4.00000	0.129914
$949$	0	0
$950$	16.0000	0.519109
$951$	− 12.0000i	− 0.389127i
$952$	−6.00000	−0.194461
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	24.0000i	0.777029i
$955$	− 72.0000i	− 2.32987i
$956$	− 21.0000i	− 0.679189i
$957$	0	0
$958$	−30.0000	−0.969256
$959$	−3.00000	−0.0968751
$960$	− 3.00000i	− 0.0968246i
$961$	−69.0000	−2.22581
$962$	0	0
$963$	36.0000	1.16008
$964$	− 28.0000i	− 0.901819i
$965$	15.0000	0.482867
$966$	3.00000	0.0965234
$967$	− 16.0000i	− 0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$	− 16.0000i	− 0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$	11.0000i	0.353553i
$969$	24.0000i	0.770991i
$970$	6.00000i	0.192648i
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	−16.0000	−0.513200
$973$	16.0000i	0.512936i
$974$	−13.0000	−0.416547
$975$	0	0
$976$	11.0000	0.352101
$977$	39.0000i	1.24772i	0.781536	+	0.623860i	$0.214437\pi$
$977$	39.0000i	1.24772i	−0.781536	+	0.623860i	$0.785563\pi$
$978$	−20.0000	−0.639529
$979$	0	0
$980$	3.00000i	0.0958315i
$981$	32.0000i	1.02168i
$982$	12.0000i	0.382935i
$983$	30.0000i	0.956851i	0.878128	+	0.478426i	$0.158792\pi$
$983$	30.0000i	0.956851i	−0.878128	+	0.478426i	$0.841208\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	36.0000i	1.14647i
$987$	−6.00000	−0.190982
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	17.0000	0.540023	0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000	0.540023	0.270011	+	0.962857i	$0.412973\pi$
$992$	−10.0000	−0.317500
$993$	− 10.0000i	− 0.317340i
$994$	3.00000i	0.0951542i
$995$	− 42.0000i	− 1.33149i
$996$	0	0
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	−22.0000	−0.696398
$999$	40.0000i	1.26554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.d.f.337.1		2
13.2	odd	12		182.2.g.b.113.1	yes	2
13.5	odd	4		2366.2.a.f.1.1		1
13.6	odd	12		182.2.g.b.29.1	✓	2
13.8	odd	4		2366.2.a.n.1.1		1
13.12	even	2	inner	2366.2.d.f.337.2		2
39.2	even	12		1638.2.r.c.1387.1		2
39.32	even	12		1638.2.r.c.757.1		2
52.15	even	12		1456.2.s.e.113.1		2
52.19	even	12		1456.2.s.e.1121.1		2
91.2	odd	12		1274.2.e.d.165.1		2
91.6	even	12		1274.2.g.g.393.1		2
91.19	even	12		1274.2.h.i.263.1		2
91.32	odd	12		1274.2.e.d.471.1		2
91.41	even	12		1274.2.g.g.295.1		2
91.45	even	12		1274.2.e.i.471.1		2
91.54	even	12		1274.2.e.i.165.1		2
91.58	odd	12		1274.2.h.j.263.1		2
91.67	odd	12		1274.2.h.j.373.1		2
91.80	even	12		1274.2.h.i.373.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.g.b.29.1	✓	2	13.6	odd	12
182.2.g.b.113.1	yes	2	13.2	odd	12
1274.2.e.d.165.1		2	91.2	odd	12
1274.2.e.d.471.1		2	91.32	odd	12
1274.2.e.i.165.1		2	91.54	even	12
1274.2.e.i.471.1		2	91.45	even	12
1274.2.g.g.295.1		2	91.41	even	12
1274.2.g.g.393.1		2	91.6	even	12
1274.2.h.i.263.1		2	91.19	even	12
1274.2.h.i.373.1		2	91.80	even	12
1274.2.h.j.263.1		2	91.58	odd	12
1274.2.h.j.373.1		2	91.67	odd	12
1456.2.s.e.113.1		2	52.15	even	12
1456.2.s.e.1121.1		2	52.19	even	12
1638.2.r.c.757.1		2	39.32	even	12
1638.2.r.c.1387.1		2	39.2	even	12
2366.2.a.f.1.1		1	13.5	odd	4
2366.2.a.n.1.1		1	13.8	odd	4
2366.2.d.f.337.1		2	1.1	even	1	trivial
2366.2.d.f.337.2		2	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{5} -1.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{5} -1.00000i q^{6} -1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{9} +3.00000 q^{10} -1.00000 q^{12} -1.00000 q^{14} +3.00000i q^{15} +1.00000 q^{16} -6.00000 q^{17} +2.00000i q^{18} -4.00000i q^{19} -3.00000i q^{20} -1.00000i q^{21} -3.00000 q^{23} +1.00000i q^{24} -4.00000 q^{25} -5.00000 q^{27} +1.00000i q^{28} +6.00000 q^{29} +3.00000 q^{30} -10.0000i q^{31} -1.00000i q^{32} +6.00000i q^{34} +3.00000 q^{35} +2.00000 q^{36} -8.00000i q^{37} -4.00000 q^{38} -3.00000 q^{40} -1.00000 q^{42} -8.00000 q^{43} -6.00000i q^{45} +3.00000i q^{46} -6.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +4.00000i q^{50} -6.00000 q^{51} +12.0000 q^{53} +5.00000i q^{54} +1.00000 q^{56} -4.00000i q^{57} -6.00000i q^{58} -3.00000i q^{59} -3.00000i q^{60} +11.0000 q^{61} -10.0000 q^{62} +2.00000i q^{63} -1.00000 q^{64} +2.00000i q^{67} +6.00000 q^{68} -3.00000 q^{69} -3.00000i q^{70} -3.00000i q^{71} -2.00000i q^{72} -2.00000i q^{73} -8.00000 q^{74} -4.00000 q^{75} +4.00000i q^{76} -4.00000 q^{79} +3.00000i q^{80} +1.00000 q^{81} +1.00000i q^{84} -18.0000i q^{85} +8.00000i q^{86} +6.00000 q^{87} +6.00000i q^{89} -6.00000 q^{90} +3.00000 q^{92} -10.0000i q^{93} -6.00000 q^{94} +12.0000 q^{95} -1.00000i q^{96} +2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9	\( 2 q + 2 q^{3} - 2 q^{4} - 4 q^{9} + 6 q^{10} - 2 q^{12} - 2 q^{14} + 2 q^{16} - 12 q^{17} - 6 q^{23} - 8 q^{25} - 10 q^{27} + 12 q^{29} + 6 q^{30} + 6 q^{35} + 4 q^{36} - 8 q^{38} - 6 q^{40} - 2 q^{42} - 16 q^{43} + 2 q^{48} - 2 q^{49} - 12 q^{51} + 24 q^{53} + 2 q^{56} + 22 q^{61} - 20 q^{62} - 2 q^{64} + 12 q^{68} - 6 q^{69} - 16 q^{74} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 12 q^{87} - 12 q^{90} + 6 q^{92} - 12 q^{94} + 24 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^4 - 4 * q^9 + 6 * q^10 - 2 * q^12 - 2 * q^14 + 2 * q^16 - 12 * q^17 - 6 * q^23 - 8 * q^25 - 10 * q^27 + 12 * q^29 + 6 * q^30 + 6 * q^35 + 4 * q^36 - 8 * q^38 - 6 * q^40 - 2 * q^42 - 16 * q^43 + 2 * q^48 - 2 * q^49 - 12 * q^51 + 24 * q^53 + 2 * q^56 + 22 * q^61 - 20 * q^62 - 2 * q^64 + 12 * q^68 - 6 * q^69 - 16 * q^74 - 8 * q^75 - 8 * q^79 + 2 * q^81 + 12 * q^87 - 12 * q^90 + 6 * q^92 - 12 * q^94 + 24 * q^95

Introduction

L-functions

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