LMFDB - Embedded newform 1183.2.c.b.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(337,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, 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("1183.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1183 = 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1183.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.44630255912$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1183.337
Dual form			1183.2.c.b.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{5} +1.00000i q^{7} -3.00000 q^{9} +O(q^{10})$	\(q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{5} +1.00000i q^{7} -3.00000 q^{9} -6.00000 q^{10} +6.00000i q^{11} +2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{17} +6.00000i q^{18} +5.00000i q^{19} +6.00000i q^{20} +12.0000 q^{22} -3.00000 q^{23} -4.00000 q^{25} -2.00000i q^{28} -5.00000 q^{29} -3.00000i q^{31} +8.00000i q^{32} +8.00000i q^{34} +3.00000 q^{35} +6.00000 q^{36} +4.00000i q^{37} +10.0000 q^{38} -6.00000i q^{41} +1.00000 q^{43} -12.0000i q^{44} +9.00000i q^{45} +6.00000i q^{46} -7.00000i q^{47} -1.00000 q^{49} +8.00000i q^{50} -9.00000 q^{53} +18.0000 q^{55} +10.0000i q^{58} -8.00000i q^{59} -10.0000 q^{61} -6.00000 q^{62} -3.00000i q^{63} +8.00000 q^{64} -6.00000i q^{67} +8.00000 q^{68} -6.00000i q^{70} -8.00000i q^{71} +13.0000i q^{73} +8.00000 q^{74} -10.0000i q^{76} -6.00000 q^{77} +3.00000 q^{79} +12.0000i q^{80} +9.00000 q^{81} -12.0000 q^{82} +15.0000i q^{83} +12.0000i q^{85} -2.00000i q^{86} -3.00000i q^{89} +18.0000 q^{90} +6.00000 q^{92} -14.0000 q^{94} +15.0000 q^{95} +7.00000i q^{97} +2.00000i q^{98} -18.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 6 q^{9}+O(q^{10}) $ 2 * q - 4 * q^4 - 6 * q^9	$ 2 q - 4 q^{4} - 6 q^{9} - 12 q^{10} + 4 q^{14} - 8 q^{16} - 8 q^{17} + 24 q^{22} - 6 q^{23} - 8 q^{25} - 10 q^{29} + 6 q^{35} + 12 q^{36} + 20 q^{38} + 2 q^{43} - 2 q^{49} - 18 q^{53} + 36 q^{55} - 20 q^{61} - 12 q^{62} + 16 q^{64} + 16 q^{68} + 16 q^{74} - 12 q^{77} + 6 q^{79} + 18 q^{81} - 24 q^{82} + 36 q^{90} + 12 q^{92} - 28 q^{94} + 30 q^{95}+O(q^{100}) $ 2 * q - 4 * q^4 - 6 * q^9 - 12 * q^10 + 4 * q^14 - 8 * q^16 - 8 * q^17 + 24 * q^22 - 6 * q^23 - 8 * q^25 - 10 * q^29 + 6 * q^35 + 12 * q^36 + 20 * q^38 + 2 * q^43 - 2 * q^49 - 18 * q^53 + 36 * q^55 - 20 * q^61 - 12 * q^62 + 16 * q^64 + 16 * q^68 + 16 * q^74 - 12 * q^77 + 6 * q^79 + 18 * q^81 - 24 * q^82 + 36 * q^90 + 12 * q^92 - 28 * q^94 + 30 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$.

$n$	$339$	$1016$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−2.00000	−1.00000
$5$	− 3.00000i	− 1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$	− 3.00000i	− 1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	−3.00000	−1.00000
$10$	−6.00000	−1.89737
$11$	6.00000i	1.80907i	0.426401	+	0.904534i	$0.359781\pi$
$11$	6.00000i	1.80907i	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	6.00000i	1.41421i
$19$	5.00000i	1.14708i	0.819178	+	0.573539i	$0.194430\pi$
$19$	5.00000i	1.14708i	−0.819178	+	0.573539i	$0.805570\pi$
$20$	6.00000i	1.34164i
$21$	0	0
$22$	12.0000	2.55841
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	− 3.00000i	− 0.538816i	−0.963026	−	0.269408i	$-0.913172\pi$
$31$	− 3.00000i	− 0.538816i	0.963026	−	0.269408i	$-0.0868280\pi$
$32$	8.00000i	1.41421i
$33$	0	0
$34$	8.00000i	1.37199i
$35$	3.00000	0.507093
$36$	6.00000	1.00000
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	10.0000	1.62221
$39$	0	0
$40$	0	0
$41$	− 6.00000i	− 0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	− 6.00000i	− 0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	− 12.0000i	− 1.80907i
$45$	9.00000i	1.34164i
$46$	6.00000i	0.884652i
$47$	− 7.00000i	− 1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$	− 7.00000i	− 1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	8.00000i	1.13137i
$51$	0	0
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	18.0000	2.42712
$56$	0	0
$57$	0	0
$58$	10.0000i	1.31306i
$59$	− 8.00000i	− 1.04151i	−0.853706	−	0.520756i	$-0.825650\pi$
$59$	− 8.00000i	− 1.04151i	0.853706	−	0.520756i	$-0.174350\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−6.00000	−0.762001
$63$	− 3.00000i	− 0.377964i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 6.00000i	− 0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$	− 6.00000i	− 0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	8.00000	0.970143
$69$	0	0
$70$	− 6.00000i	− 0.717137i
$71$	− 8.00000i	− 0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$	− 8.00000i	− 0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$	0	0
$73$	13.0000i	1.52153i	0.649025	+	0.760767i	$0.275177\pi$
$73$	13.0000i	1.52153i	−0.649025	+	0.760767i	$0.724823\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	− 10.0000i	− 1.14708i
$77$	−6.00000	−0.683763
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	12.0000i	1.34164i
$81$	9.00000	1.00000
$82$	−12.0000	−1.32518
$83$	15.0000i	1.64646i	0.567705	+	0.823232i	$0.307831\pi$
$83$	15.0000i	1.64646i	−0.567705	+	0.823232i	$0.692169\pi$
$84$	0	0
$85$	12.0000i	1.30158i
$86$	− 2.00000i	− 0.215666i
$87$	0	0
$88$	0	0
$89$	− 3.00000i	− 0.317999i	−0.987279	−	0.159000i	$-0.949173\pi$
$89$	− 3.00000i	− 0.317999i	0.987279	−	0.159000i	$-0.0508269\pi$
$90$	18.0000	1.89737
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	−14.0000	−1.44399
$95$	15.0000	1.53897
$96$	0	0
$97$	7.00000i	0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$	7.00000i	0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$	2.00000i	0.202031i
$99$	− 18.0000i	− 1.80907i
$100$	8.00000	0.800000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	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
$105$	0	0
$106$	18.0000i	1.74831i
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	− 36.0000i	− 3.43247i
$111$	0	0
$112$	− 4.00000i	− 0.377964i
$113$	−3.00000	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$113$	−3.00000	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$114$	0	0
$115$	9.00000i	0.839254i
$116$	10.0000	0.928477
$117$	0	0
$118$	−16.0000	−1.47292
$119$	− 4.00000i	− 0.366679i
$120$	0	0
$121$	−25.0000	−2.27273
$122$	20.0000i	1.81071i
$123$	0	0
$124$	6.00000i	0.538816i
$125$	− 3.00000i	− 0.268328i
$126$	−6.00000	−0.534522
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	−12.0000	−1.03664
$135$	0	0
$136$	0	0
$137$	− 4.00000i	− 0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$	− 4.00000i	− 0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$	0	0
$139$	−18.0000	−1.52674	−0.763370	−	0.645961i	$-0.776457\pi$
$139$	−18.0000	−1.52674	−0.763370	+	0.645961i	$0.776457\pi$
$140$	−6.00000	−0.507093
$141$	0	0
$142$	−16.0000	−1.34269
$143$	0	0
$144$	12.0000	1.00000
$145$	15.0000i	1.24568i
$146$	26.0000	2.15178
$147$	0	0
$148$	− 8.00000i	− 0.657596i
$149$	− 18.0000i	− 1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$	− 18.0000i	− 1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	12.0000	0.970143
$154$	12.0000i	0.966988i
$155$	−9.00000	−0.722897
$156$	0	0
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	− 6.00000i	− 0.477334i
$159$	0	0
$160$	24.0000	1.89737
$161$	− 3.00000i	− 0.236433i
$162$	− 18.0000i	− 1.41421i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	12.0000i	0.937043i
$165$	0	0
$166$	30.0000	2.32845
$167$	− 5.00000i	− 0.386912i	−0.981109	−	0.193456i	$-0.938030\pi$
$167$	− 5.00000i	− 0.386912i	0.981109	−	0.193456i	$-0.0619696\pi$
$168$	0	0
$169$	0	0
$170$	24.0000	1.84072
$171$	− 15.0000i	− 1.14708i
$172$	−2.00000	−0.152499
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	0	0
$175$	− 4.00000i	− 0.302372i
$176$	− 24.0000i	− 1.80907i
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−23.0000	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$180$	− 18.0000i	− 1.34164i
$181$	−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
$183$	0	0
$184$	0	0
$185$	12.0000	0.882258
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	14.0000i	1.02105i
$189$	0	0
$190$	− 30.0000i	− 2.17643i
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	2.00000	0.142857
$197$	2.00000i	0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$	2.00000i	0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$	−36.0000	−2.55841
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	− 28.0000i	− 1.97007i
$203$	− 5.00000i	− 0.350931i
$204$	0	0
$205$	−18.0000	−1.25717
$206$	− 8.00000i	− 0.557386i
$207$	9.00000	0.625543
$208$	0	0
$209$	−30.0000	−2.07514
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	18.0000	1.23625
$213$	0	0
$214$	8.00000i	0.546869i
$215$	− 3.00000i	− 0.204598i
$216$	0	0
$217$	3.00000	0.203653
$218$	−4.00000	−0.270914
$219$	0	0
$220$	−36.0000	−2.42712
$221$	0	0
$222$	0	0
$223$	15.0000i	1.00447i	0.864730	+	0.502237i	$0.167490\pi$
$223$	15.0000i	1.00447i	−0.864730	+	0.502237i	$0.832510\pi$
$224$	−8.00000	−0.534522
$225$	12.0000	0.800000
$226$	6.00000i	0.399114i
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	0	0
$229$	− 14.0000i	− 0.925146i	−0.886581	−	0.462573i	$-0.846926\pi$
$229$	− 14.0000i	− 0.925146i	0.886581	−	0.462573i	$-0.153074\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	0	0
$233$	−15.0000	−0.982683	−0.491341	−	0.870967i	$-0.663493\pi$
$233$	−15.0000	−0.982683	−0.491341	+	0.870967i	$0.663493\pi$
$234$	0	0
$235$	−21.0000	−1.36989
$236$	16.0000i	1.04151i
$237$	0	0
$238$	−8.00000	−0.518563
$239$	− 4.00000i	− 0.258738i	−0.991596	−	0.129369i	$-0.958705\pi$
$239$	− 4.00000i	− 0.258738i	0.991596	−	0.129369i	$-0.0412952\pi$
$240$	0	0
$241$	17.0000i	1.09507i	0.836784	+	0.547533i	$0.184433\pi$
$241$	17.0000i	1.09507i	−0.836784	+	0.547533i	$0.815567\pi$
$242$	50.0000i	3.21412i
$243$	0	0
$244$	20.0000	1.28037
$245$	3.00000i	0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−6.00000	−0.379473
$251$	26.0000	1.64111	0.820553	−	0.571571i	$-0.193666\pi$
$251$	26.0000	1.64111	0.820553	+	0.571571i	$0.193666\pi$
$252$	6.00000i	0.377964i
$253$	− 18.0000i	− 1.13165i
$254$	− 8.00000i	− 0.501965i
$255$	0	0
$256$	16.0000	1.00000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	15.0000	0.928477
$262$	− 16.0000i	− 0.988483i
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	27.0000i	1.65860i
$266$	10.0000i	0.613139i
$267$	0	0
$268$	12.0000i	0.733017i
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	− 8.00000i	− 0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$	− 8.00000i	− 0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$	16.0000	0.970143
$273$	0	0
$274$	−8.00000	−0.483298
$275$	− 24.0000i	− 1.44725i
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	36.0000i	2.15914i
$279$	9.00000i	0.538816i
$280$	0	0
$281$	30.0000i	1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$	30.0000i	1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	16.0000i	0.949425i
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	− 24.0000i	− 1.41421i
$289$	−1.00000	−0.0588235
$290$	30.0000	1.76166
$291$	0	0
$292$	− 26.0000i	− 1.52153i
$293$	19.0000i	1.10999i	0.831853	+	0.554996i	$0.187280\pi$
$293$	19.0000i	1.10999i	−0.831853	+	0.554996i	$0.812720\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	0	0
$297$	0	0
$298$	−36.0000	−2.08542
$299$	0	0
$300$	0	0
$301$	1.00000i	0.0576390i
$302$	0	0
$303$	0	0
$304$	− 20.0000i	− 1.14708i
$305$	30.0000i	1.71780i
$306$	− 24.0000i	− 1.37199i
$307$	33.0000i	1.88341i	0.336440	+	0.941705i	$0.390777\pi$
$307$	33.0000i	1.88341i	−0.336440	+	0.941705i	$0.609223\pi$
$308$	12.0000	0.683763
$309$	0	0
$310$	18.0000i	1.02233i
$311$	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
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	16.0000i	0.902932i
$315$	−9.00000	−0.507093
$316$	−6.00000	−0.337526
$317$	− 24.0000i	− 1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$	− 24.0000i	− 1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$	0	0
$319$	− 30.0000i	− 1.67968i
$320$	− 24.0000i	− 1.34164i
$321$	0	0
$322$	−6.00000	−0.334367
$323$	− 20.0000i	− 1.11283i
$324$	−18.0000	−1.00000
$325$	0	0
$326$	8.00000	0.443079
$327$	0	0
$328$	0	0
$329$	7.00000	0.385922
$330$	0	0
$331$	22.0000i	1.20923i	0.796518	+	0.604615i	$0.206673\pi$
$331$	22.0000i	1.20923i	−0.796518	+	0.604615i	$0.793327\pi$
$332$	− 30.0000i	− 1.64646i
$333$	− 12.0000i	− 0.657596i
$334$	−10.0000	−0.547176
$335$	−18.0000	−0.983445
$336$	0	0
$337$	−17.0000	−0.926049	−0.463025	−	0.886345i	$-0.653236\pi$
$337$	−17.0000	−0.926049	−0.463025	+	0.886345i	$0.653236\pi$
$338$	0	0
$339$	0	0
$340$	− 24.0000i	− 1.30158i
$341$	18.0000	0.974755
$342$	−30.0000	−1.62221
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	− 16.0000i	− 0.860165i
$347$	−32.0000	−1.71785	−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000	−1.71785	−0.858925	+	0.512101i	$0.828867\pi$
$348$	0	0
$349$	− 11.0000i	− 0.588817i	−0.955680	−	0.294408i	$-0.904877\pi$
$349$	− 11.0000i	− 0.588817i	0.955680	−	0.294408i	$-0.0951225\pi$
$350$	−8.00000	−0.427618
$351$	0	0
$352$	−48.0000	−2.55841
$353$	− 10.0000i	− 0.532246i	−0.963939	−	0.266123i	$-0.914257\pi$
$353$	− 10.0000i	− 0.532246i	0.963939	−	0.266123i	$-0.0857428\pi$
$354$	0	0
$355$	−24.0000	−1.27379
$356$	6.00000i	0.317999i
$357$	0	0
$358$	46.0000i	2.43118i
$359$	− 20.0000i	− 1.05556i	−0.849381	−	0.527780i	$-0.823025\pi$
$359$	− 20.0000i	− 1.05556i	0.849381	−	0.527780i	$-0.176975\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	28.0000i	1.47165i
$363$	0	0
$364$	0	0
$365$	39.0000	2.04135
$366$	0	0
$367$	14.0000	0.730794	0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000	0.730794	0.365397	+	0.930852i	$0.380933\pi$
$368$	12.0000	0.625543
$369$	18.0000i	0.937043i
$370$	− 24.0000i	− 1.24770i
$371$	− 9.00000i	− 0.467257i
$372$	0	0
$373$	30.0000	1.55334	0.776671	−	0.629907i	$-0.216907\pi$
$373$	30.0000	1.55334	0.776671	+	0.629907i	$0.216907\pi$
$374$	−48.0000	−2.48202
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 6.00000i	− 0.308199i	−0.988055	−	0.154100i	$-0.950752\pi$
$379$	− 6.00000i	− 0.308199i	0.988055	−	0.154100i	$-0.0492477\pi$
$380$	−30.0000	−1.53897
$381$	0	0
$382$	16.0000i	0.818631i
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	0	0
$385$	18.0000i	0.917365i
$386$	−44.0000	−2.23954
$387$	−3.00000	−0.152499
$388$	− 14.0000i	− 0.710742i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	4.00000	0.201517
$395$	− 9.00000i	− 0.452839i
$396$	36.0000i	1.80907i
$397$	13.0000i	0.652451i	0.945292	+	0.326226i	$0.105777\pi$
$397$	13.0000i	0.652451i	−0.945292	+	0.326226i	$0.894223\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	16.0000	0.800000
$401$	32.0000i	1.59800i	0.601329	+	0.799002i	$0.294638\pi$
$401$	32.0000i	1.59800i	−0.601329	+	0.799002i	$0.705362\pi$
$402$	0	0
$403$	0	0
$404$	−28.0000	−1.39305
$405$	− 27.0000i	− 1.34164i
$406$	−10.0000	−0.496292
$407$	−24.0000	−1.18964
$408$	0	0
$409$	− 13.0000i	− 0.642809i	−0.946942	−	0.321404i	$-0.895845\pi$
$409$	− 13.0000i	− 0.642809i	0.946942	−	0.321404i	$-0.104155\pi$
$410$	36.0000i	1.77791i
$411$	0	0
$412$	−8.00000	−0.394132
$413$	8.00000	0.393654
$414$	− 18.0000i	− 0.884652i
$415$	45.0000	2.20896
$416$	0	0
$417$	0	0
$418$	60.0000i	2.93470i
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	− 12.0000i	− 0.584844i	−0.956289	−	0.292422i	$-0.905539\pi$
$421$	− 12.0000i	− 0.584844i	0.956289	−	0.292422i	$-0.0944612\pi$
$422$	10.0000i	0.486792i
$423$	21.0000i	1.02105i
$424$	0	0
$425$	16.0000	0.776114
$426$	0	0
$427$	− 10.0000i	− 0.483934i
$428$	8.00000	0.386695
$429$	0	0
$430$	−6.00000	−0.289346
$431$	6.00000i	0.289010i	0.989504	+	0.144505i	$0.0461589\pi$
$431$	6.00000i	0.289010i	−0.989504	+	0.144505i	$0.953841\pi$
$432$	0	0
$433$	−12.0000	−0.576683	−0.288342	−	0.957528i	$-0.593104\pi$
$433$	−12.0000	−0.576683	−0.288342	+	0.957528i	$0.593104\pi$
$434$	− 6.00000i	− 0.288009i
$435$	0	0
$436$	4.00000i	0.191565i
$437$	− 15.0000i	− 0.717547i
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	19.0000	0.902717	0.451359	−	0.892343i	$-0.350940\pi$
$443$	19.0000	0.902717	0.451359	+	0.892343i	$0.350940\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	30.0000	1.42054
$447$	0	0
$448$	8.00000i	0.377964i
$449$	− 36.0000i	− 1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$449$	− 36.0000i	− 1.69895i	0.527633	−	0.849473i	$-0.323080\pi$
$450$	− 24.0000i	− 1.13137i
$451$	36.0000	1.69517
$452$	6.00000	0.282216
$453$	0	0
$454$	40.0000	1.87729
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	−28.0000	−1.30835
$459$	0	0
$460$	− 18.0000i	− 0.839254i
$461$	− 22.0000i	− 1.02464i	−0.858794	−	0.512321i	$-0.828786\pi$
$461$	− 22.0000i	− 1.02464i	0.858794	−	0.512321i	$-0.171214\pi$
$462$	0	0
$463$	14.0000i	0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$	14.0000i	0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	20.0000	0.928477
$465$	0	0
$466$	30.0000i	1.38972i
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	42.0000i	1.93732i
$471$	0	0
$472$	0	0
$473$	6.00000i	0.275880i
$474$	0	0
$475$	− 20.0000i	− 0.917663i
$476$	8.00000i	0.366679i
$477$	27.0000	1.23625
$478$	−8.00000	−0.365911
$479$	11.0000i	0.502603i	0.967909	+	0.251301i	$0.0808585\pi$
$479$	11.0000i	0.502603i	−0.967909	+	0.251301i	$0.919141\pi$
$480$	0	0
$481$	0	0
$482$	34.0000	1.54866
$483$	0	0
$484$	50.0000	2.27273
$485$	21.0000	0.953561
$486$	0	0
$487$	− 26.0000i	− 1.17817i	−0.808070	−	0.589086i	$-0.799488\pi$
$487$	− 26.0000i	− 1.17817i	0.808070	−	0.589086i	$-0.200512\pi$
$488$	0	0
$489$	0	0
$490$	6.00000	0.271052
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	−54.0000	−2.42712
$496$	12.0000i	0.538816i
$497$	8.00000	0.358849
$498$	0	0
$499$	− 16.0000i	− 0.716258i	−0.933672	−	0.358129i	$-0.883415\pi$
$499$	− 16.0000i	− 0.716258i	0.933672	−	0.358129i	$-0.116585\pi$
$500$	6.00000i	0.268328i
$501$	0	0
$502$	− 52.0000i	− 2.32087i
$503$	2.00000	0.0891756	0.0445878	−	0.999005i	$-0.485803\pi$
$503$	2.00000	0.0891756	0.0445878	+	0.999005i	$0.485803\pi$
$504$	0	0
$505$	− 42.0000i	− 1.86898i
$506$	−36.0000	−1.60040
$507$	0	0
$508$	−8.00000	−0.354943
$509$	− 19.0000i	− 0.842160i	−0.907023	−	0.421080i	$-0.861651\pi$
$509$	− 19.0000i	− 0.842160i	0.907023	−	0.421080i	$-0.138349\pi$
$510$	0	0
$511$	−13.0000	−0.575086
$512$	− 32.0000i	− 1.41421i
$513$	0	0
$514$	− 4.00000i	− 0.176432i
$515$	− 12.0000i	− 0.528783i
$516$	0	0
$517$	42.0000	1.84716
$518$	8.00000i	0.351500i
$519$	0	0
$520$	0	0
$521$	40.0000	1.75243	0.876216	−	0.481919i	$-0.160060\pi$
$521$	40.0000	1.75243	0.876216	+	0.481919i	$0.160060\pi$
$522$	− 30.0000i	− 1.31306i
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	30.0000i	1.30806i
$527$	12.0000i	0.522728i
$528$	0	0
$529$	−14.0000	−0.608696
$530$	54.0000	2.34561
$531$	24.0000i	1.04151i
$532$	10.0000	0.433555
$533$	0	0
$534$	0	0
$535$	12.0000i	0.518805i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 6.00000i	− 0.258438i
$540$	0	0
$541$	40.0000i	1.71973i	0.510518	+	0.859867i	$0.329454\pi$
$541$	40.0000i	1.71973i	−0.510518	+	0.859867i	$0.670546\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	− 32.0000i	− 1.37199i
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−7.00000	−0.299298	−0.149649	−	0.988739i	$-0.547814\pi$
$547$	−7.00000	−0.299298	−0.149649	+	0.988739i	$0.547814\pi$
$548$	8.00000i	0.341743i
$549$	30.0000	1.28037
$550$	−48.0000	−2.04673
$551$	− 25.0000i	− 1.06504i
$552$	0	0
$553$	3.00000i	0.127573i
$554$	2.00000i	0.0849719i
$555$	0	0
$556$	36.0000	1.52674
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	18.0000	0.762001
$559$	0	0
$560$	−12.0000	−0.507093
$561$	0	0
$562$	60.0000	2.53095
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	9.00000i	0.378633i
$566$	32.0000i	1.34506i
$567$	9.00000i	0.377964i
$568$	0	0
$569$	−7.00000	−0.293455	−0.146728	−	0.989177i	$-0.546874\pi$
$569$	−7.00000	−0.293455	−0.146728	+	0.989177i	$0.546874\pi$
$570$	0	0
$571$	17.0000	0.711428	0.355714	−	0.934595i	$-0.384238\pi$
$571$	17.0000	0.711428	0.355714	+	0.934595i	$0.384238\pi$
$572$	0	0
$573$	0	0
$574$	− 12.0000i	− 0.500870i
$575$	12.0000	0.500435
$576$	−24.0000	−1.00000
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	2.00000i	0.0831890i
$579$	0	0
$580$	− 30.0000i	− 1.24568i
$581$	−15.0000	−0.622305
$582$	0	0
$583$	− 54.0000i	− 2.23645i
$584$	0	0
$585$	0	0
$586$	38.0000	1.56977
$587$	39.0000i	1.60970i	0.593477	+	0.804851i	$0.297755\pi$
$587$	39.0000i	1.60970i	−0.593477	+	0.804851i	$0.702245\pi$
$588$	0	0
$589$	15.0000	0.618064
$590$	48.0000i	1.97613i
$591$	0	0
$592$	− 16.0000i	− 0.657596i
$593$	27.0000i	1.10876i	0.832265	+	0.554379i	$0.187044\pi$
$593$	27.0000i	1.10876i	−0.832265	+	0.554379i	$0.812956\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	36.0000i	1.47462i
$597$	0	0
$598$	0	0
$599$	11.0000	0.449448	0.224724	−	0.974422i	$-0.427852\pi$
$599$	11.0000	0.449448	0.224724	+	0.974422i	$0.427852\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	2.00000	0.0815139
$603$	18.0000i	0.733017i
$604$	0	0
$605$	75.0000i	3.04918i
$606$	0	0
$607$	2.00000	0.0811775	0.0405887	−	0.999176i	$-0.487077\pi$
$607$	2.00000	0.0811775	0.0405887	+	0.999176i	$0.487077\pi$
$608$	−40.0000	−1.62221
$609$	0	0
$610$	60.0000	2.42933
$611$	0	0
$612$	−24.0000	−0.970143
$613$	8.00000i	0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$	8.00000i	0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$	66.0000	2.66354
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	18.0000	0.722897
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	3.00000	0.120192
$624$	0	0
$625$	−29.0000	−1.16000
$626$	− 44.0000i	− 1.75859i
$627$	0	0
$628$	16.0000	0.638470
$629$	− 16.0000i	− 0.637962i
$630$	18.0000i	0.717137i
$631$	− 22.0000i	− 0.875806i	−0.899022	−	0.437903i	$-0.855721\pi$
$631$	− 22.0000i	− 0.875806i	0.899022	−	0.437903i	$-0.144279\pi$
$632$	0	0
$633$	0	0
$634$	−48.0000	−1.90632
$635$	− 12.0000i	− 0.476205i
$636$	0	0
$637$	0	0
$638$	−60.0000	−2.37542
$639$	24.0000i	0.949425i
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	8.00000i	0.315489i	0.987480	+	0.157745i	$0.0504223\pi$
$643$	8.00000i	0.315489i	−0.987480	+	0.157745i	$0.949578\pi$
$644$	6.00000i	0.236433i
$645$	0	0
$646$	−40.0000	−1.57378
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	− 8.00000i	− 0.313304i
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	− 24.0000i	− 0.937758i
$656$	24.0000i	0.937043i
$657$	− 39.0000i	− 1.52153i
$658$	− 14.0000i	− 0.545777i
$659$	17.0000	0.662226	0.331113	−	0.943591i	$-0.392576\pi$
$659$	17.0000	0.662226	0.331113	+	0.943591i	$0.392576\pi$
$660$	0	0
$661$	33.0000i	1.28355i	0.766892	+	0.641776i	$0.221802\pi$
$661$	33.0000i	1.28355i	−0.766892	+	0.641776i	$0.778198\pi$
$662$	44.0000	1.71011
$663$	0	0
$664$	0	0
$665$	15.0000i	0.581675i
$666$	−24.0000	−0.929981
$667$	15.0000	0.580802
$668$	10.0000i	0.386912i
$669$	0	0
$670$	36.0000i	1.39080i
$671$	− 60.0000i	− 2.31627i
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	34.0000i	1.30963i
$675$	0	0
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	0	0
$681$	0	0
$682$	− 36.0000i	− 1.37851i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	30.0000i	1.14708i
$685$	−12.0000	−0.458496
$686$	−2.00000	−0.0763604
$687$	0	0
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	11.0000i	0.418460i	0.977866	+	0.209230i	$0.0670957\pi$
$691$	11.0000i	0.418460i	−0.977866	+	0.209230i	$0.932904\pi$
$692$	−16.0000	−0.608229
$693$	18.0000	0.683763
$694$	64.0000i	2.42941i
$695$	54.0000i	2.04834i
$696$	0	0
$697$	24.0000i	0.909065i
$698$	−22.0000	−0.832712
$699$	0	0
$700$	8.00000i	0.302372i
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	48.0000i	1.80907i
$705$	0	0
$706$	−20.0000	−0.752710
$707$	14.0000i	0.526524i
$708$	0	0
$709$	10.0000i	0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$	10.0000i	0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$	48.0000i	1.80141i
$711$	−9.00000	−0.337526
$712$	0	0
$713$	9.00000i	0.337053i
$714$	0	0
$715$	0	0
$716$	46.0000	1.71910
$717$	0	0
$718$	−40.0000	−1.49279
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	− 36.0000i	− 1.34164i
$721$	4.00000i	0.148968i
$722$	12.0000i	0.446594i
$723$	0	0
$724$	28.0000	1.04061
$725$	20.0000	0.742781
$726$	0	0
$727$	−46.0000	−1.70605	−0.853023	−	0.521874i	$-0.825233\pi$
$727$	−46.0000	−1.70605	−0.853023	+	0.521874i	$0.825233\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	− 78.0000i	− 2.88691i
$731$	−4.00000	−0.147945
$732$	0	0
$733$	51.0000i	1.88373i	0.335994	+	0.941864i	$0.390928\pi$
$733$	51.0000i	1.88373i	−0.335994	+	0.941864i	$0.609072\pi$
$734$	− 28.0000i	− 1.03350i
$735$	0	0
$736$	− 24.0000i	− 0.884652i
$737$	36.0000	1.32608
$738$	36.0000	1.32518
$739$	26.0000i	0.956425i	0.878244	+	0.478213i	$0.158715\pi$
$739$	26.0000i	0.956425i	−0.878244	+	0.478213i	$0.841285\pi$
$740$	−24.0000	−0.882258
$741$	0	0
$742$	−18.0000	−0.660801
$743$	36.0000i	1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$	36.0000i	1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$	0	0
$745$	−54.0000	−1.97841
$746$	− 60.0000i	− 2.19676i
$747$	− 45.0000i	− 1.64646i
$748$	48.0000i	1.75505i
$749$	− 4.00000i	− 0.146157i
$750$	0	0
$751$	17.0000	0.620339	0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000	0.620339	0.310169	+	0.950681i	$0.399614\pi$
$752$	28.0000i	1.02105i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−15.0000	−0.545184	−0.272592	−	0.962130i	$-0.587881\pi$
$757$	−15.0000	−0.545184	−0.272592	+	0.962130i	$0.587881\pi$
$758$	−12.0000	−0.435860
$759$	0	0
$760$	0	0
$761$	− 9.00000i	− 0.326250i	−0.986605	−	0.163125i	$-0.947843\pi$
$761$	− 9.00000i	− 0.326250i	0.986605	−	0.163125i	$-0.0521573\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	16.0000	0.578860
$765$	− 36.0000i	− 1.30158i
$766$	−72.0000	−2.60147
$767$	0	0
$768$	0	0
$769$	− 35.0000i	− 1.26213i	−0.775729	−	0.631066i	$-0.782618\pi$
$769$	− 35.0000i	− 1.26213i	0.775729	−	0.631066i	$-0.217382\pi$
$770$	36.0000	1.29735
$771$	0	0
$772$	44.0000i	1.58359i
$773$	54.0000i	1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$	54.0000i	1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$	6.00000i	0.215666i
$775$	12.0000i	0.431053i
$776$	0	0
$777$	0	0
$778$	60.0000i	2.15110i
$779$	30.0000	1.07486
$780$	0	0
$781$	48.0000	1.71758
$782$	− 24.0000i	− 0.858238i
$783$	0	0
$784$	4.00000	0.142857
$785$	24.0000i	0.856597i
$786$	0	0
$787$	− 37.0000i	− 1.31891i	−0.751745	−	0.659454i	$-0.770788\pi$
$787$	− 37.0000i	− 1.31891i	0.751745	−	0.659454i	$-0.229212\pi$
$788$	− 4.00000i	− 0.142494i
$789$	0	0
$790$	−18.0000	−0.640411
$791$	− 3.00000i	− 0.106668i
$792$	0	0
$793$	0	0
$794$	26.0000	0.922705
$795$	0	0
$796$	8.00000	0.283552
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	28.0000i	0.990569i
$800$	− 32.0000i	− 1.13137i
$801$	9.00000i	0.317999i
$802$	64.0000	2.25992
$803$	−78.0000	−2.75256
$804$	0	0
$805$	−9.00000	−0.317208
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31.0000	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$809$	−31.0000	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$810$	−54.0000	−1.89737
$811$	− 52.0000i	− 1.82597i	−0.407997	−	0.912983i	$-0.633772\pi$
$811$	− 52.0000i	− 1.82597i	0.407997	−	0.912983i	$-0.366228\pi$
$812$	10.0000i	0.350931i
$813$	0	0
$814$	48.0000i	1.68240i
$815$	12.0000	0.420342
$816$	0	0
$817$	5.00000i	0.174928i
$818$	−26.0000	−0.909069
$819$	0	0
$820$	36.0000	1.25717
$821$	− 6.00000i	− 0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$	− 6.00000i	− 0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	− 16.0000i	− 0.556711i
$827$	− 4.00000i	− 0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$	− 4.00000i	− 0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$	−18.0000	−0.625543
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	− 90.0000i	− 3.12395i
$831$	0	0
$832$	0	0
$833$	4.00000	0.138592
$834$	0	0
$835$	−15.0000	−0.519096
$836$	60.0000	2.07514
$837$	0	0
$838$	20.0000i	0.690889i
$839$	8.00000i	0.276191i	0.990419	+	0.138095i	$0.0440980\pi$
$839$	8.00000i	0.276191i	−0.990419	+	0.138095i	$0.955902\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−24.0000	−0.827095
$843$	0	0
$844$	10.0000	0.344214
$845$	0	0
$846$	42.0000	1.44399
$847$	− 25.0000i	− 0.859010i
$848$	36.0000	1.23625
$849$	0	0
$850$	− 32.0000i	− 1.09759i
$851$	− 12.0000i	− 0.411355i
$852$	0	0
$853$	− 45.0000i	− 1.54077i	−0.637579	−	0.770385i	$-0.720064\pi$
$853$	− 45.0000i	− 1.54077i	0.637579	−	0.770385i	$-0.279936\pi$
$854$	−20.0000	−0.684386
$855$	−45.0000	−1.53897
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−2.00000	−0.0682391	−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000	−0.0682391	−0.0341196	+	0.999418i	$0.510863\pi$
$860$	6.00000i	0.204598i
$861$	0	0
$862$	12.0000	0.408722
$863$	− 32.0000i	− 1.08929i	−0.838666	−	0.544646i	$-0.816664\pi$
$863$	− 32.0000i	− 1.08929i	0.838666	−	0.544646i	$-0.183336\pi$
$864$	0	0
$865$	− 24.0000i	− 0.816024i
$866$	24.0000i	0.815553i
$867$	0	0
$868$	−6.00000	−0.203653
$869$	18.0000i	0.610608i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 21.0000i	− 0.710742i
$874$	−30.0000	−1.01477
$875$	3.00000	0.101419
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	− 44.0000i	− 1.48493i
$879$	0	0
$880$	−72.0000	−2.42712
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	− 6.00000i	− 0.202031i
$883$	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
$886$	− 38.0000i	− 1.27663i
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	4.00000i	0.134156i
$890$	18.0000i	0.603361i
$891$	54.0000i	1.80907i
$892$	− 30.0000i	− 1.00447i
$893$	35.0000	1.17123
$894$	0	0
$895$	69.0000i	2.30642i
$896$	0	0
$897$	0	0
$898$	−72.0000	−2.40267
$899$	15.0000i	0.500278i
$900$	−24.0000	−0.800000
$901$	36.0000	1.19933
$902$	− 72.0000i	− 2.39734i
$903$	0	0
$904$	0	0
$905$	42.0000i	1.39613i
$906$	0	0
$907$	−7.00000	−0.232431	−0.116216	−	0.993224i	$-0.537076\pi$
$907$	−7.00000	−0.232431	−0.116216	+	0.993224i	$0.537076\pi$
$908$	− 40.0000i	− 1.32745i
$909$	−42.0000	−1.39305
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	−90.0000	−2.97857
$914$	0	0
$915$	0	0
$916$	28.0000i	0.925146i
$917$	8.00000i	0.264183i
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	0	0
$922$	−44.0000	−1.44906
$923$	0	0
$924$	0	0
$925$	− 16.0000i	− 0.526077i
$926$	28.0000	0.920137
$927$	−12.0000	−0.394132
$928$	− 40.0000i	− 1.31306i
$929$	5.00000i	0.164045i	0.996630	+	0.0820223i	$0.0261379\pi$
$929$	5.00000i	0.164045i	−0.996630	+	0.0820223i	$0.973862\pi$
$930$	0	0
$931$	− 5.00000i	− 0.163868i
$932$	30.0000	0.982683
$933$	0	0
$934$	− 44.0000i	− 1.43972i
$935$	−72.0000	−2.35465
$936$	0	0
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	− 12.0000i	− 0.391814i
$939$	0	0
$940$	42.0000	1.36989
$941$	55.0000i	1.79295i	0.443096	+	0.896474i	$0.353880\pi$
$941$	55.0000i	1.79295i	−0.443096	+	0.896474i	$0.646120\pi$
$942$	0	0
$943$	18.0000i	0.586161i
$944$	32.0000i	1.04151i
$945$	0	0
$946$	12.0000	0.390154
$947$	18.0000i	0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$	18.0000i	0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$	0	0
$949$	0	0
$950$	−40.0000	−1.29777
$951$	0	0
$952$	0	0
$953$	39.0000	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$954$	− 54.0000i	− 1.74831i
$955$	24.0000i	0.776622i
$956$	8.00000i	0.258738i
$957$	0	0
$958$	22.0000	0.710788
$959$	4.00000	0.129167
$960$	0	0
$961$	22.0000	0.709677
$962$	0	0
$963$	12.0000	0.386695
$964$	− 34.0000i	− 1.09507i
$965$	−66.0000	−2.12462
$966$	0	0
$967$	22.0000i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$967$	22.0000i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$968$	0	0
$969$	0	0
$970$	− 42.0000i	− 1.34854i
$971$	38.0000	1.21948	0.609739	−	0.792602i	$-0.291274\pi$
$971$	38.0000	1.21948	0.609739	+	0.792602i	$0.291274\pi$
$972$	0	0
$973$	− 18.0000i	− 0.577054i
$974$	−52.0000	−1.66619
$975$	0	0
$976$	40.0000	1.28037
$977$	10.0000i	0.319928i	0.987123	+	0.159964i	$0.0511379\pi$
$977$	10.0000i	0.319928i	−0.987123	+	0.159964i	$0.948862\pi$
$978$	0	0
$979$	18.0000	0.575282
$980$	− 6.00000i	− 0.191663i
$981$	6.00000i	0.191565i
$982$	− 24.0000i	− 0.765871i
$983$	− 17.0000i	− 0.542216i	−0.962549	−	0.271108i	$-0.912610\pi$
$983$	− 17.0000i	− 0.542216i	0.962549	−	0.271108i	$-0.0873900\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	− 40.0000i	− 1.27386i
$987$	0	0
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	108.000i	3.43247i
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	24.0000	0.762001
$993$	0	0
$994$	− 16.0000i	− 0.507489i
$995$	12.0000i	0.380426i
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1183.2.c.b.337.1		2
13.5	odd	4		91.2.a.a.1.1	✓	1
13.8	odd	4		1183.2.a.b.1.1		1
13.12	even	2	inner	1183.2.c.b.337.2		2
39.5	even	4		819.2.a.f.1.1		1
52.31	even	4		1456.2.a.g.1.1		1
65.44	odd	4		2275.2.a.h.1.1		1
91.5	even	12		637.2.e.d.508.1		2
91.18	odd	12		637.2.e.e.79.1		2
91.31	even	12		637.2.e.d.79.1		2
91.34	even	4		8281.2.a.l.1.1		1
91.44	odd	12		637.2.e.e.508.1		2
91.83	even	4		637.2.a.a.1.1		1
104.5	odd	4		5824.2.a.s.1.1		1
104.83	even	4		5824.2.a.t.1.1		1
273.83	odd	4		5733.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.a.1.1	✓	1	13.5	odd	4
637.2.a.a.1.1		1	91.83	even	4
637.2.e.d.79.1		2	91.31	even	12
637.2.e.d.508.1		2	91.5	even	12
637.2.e.e.79.1		2	91.18	odd	12
637.2.e.e.508.1		2	91.44	odd	12
819.2.a.f.1.1		1	39.5	even	4
1183.2.a.b.1.1		1	13.8	odd	4
1183.2.c.b.337.1		2	1.1	even	1	trivial
1183.2.c.b.337.2		2	13.12	even	2	inner
1456.2.a.g.1.1		1	52.31	even	4
2275.2.a.h.1.1		1	65.44	odd	4
5733.2.a.l.1.1		1	273.83	odd	4
5824.2.a.s.1.1		1	104.5	odd	4
5824.2.a.t.1.1		1	104.83	even	4
8281.2.a.l.1.1		1	91.34	even	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:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1183.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{5} +1.00000i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{2} -2.00000 q^{4} -3.00000i q^{5} +1.00000i q^{7} -3.00000 q^{9} -6.00000 q^{10} +6.00000i q^{11} +2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{17} +6.00000i q^{18} +5.00000i q^{19} +6.00000i q^{20} +12.0000 q^{22} -3.00000 q^{23} -4.00000 q^{25} -2.00000i q^{28} -5.00000 q^{29} -3.00000i q^{31} +8.00000i q^{32} +8.00000i q^{34} +3.00000 q^{35} +6.00000 q^{36} +4.00000i q^{37} +10.0000 q^{38} -6.00000i q^{41} +1.00000 q^{43} -12.0000i q^{44} +9.00000i q^{45} +6.00000i q^{46} -7.00000i q^{47} -1.00000 q^{49} +8.00000i q^{50} -9.00000 q^{53} +18.0000 q^{55} +10.0000i q^{58} -8.00000i q^{59} -10.0000 q^{61} -6.00000 q^{62} -3.00000i q^{63} +8.00000 q^{64} -6.00000i q^{67} +8.00000 q^{68} -6.00000i q^{70} -8.00000i q^{71} +13.0000i q^{73} +8.00000 q^{74} -10.0000i q^{76} -6.00000 q^{77} +3.00000 q^{79} +12.0000i q^{80} +9.00000 q^{81} -12.0000 q^{82} +15.0000i q^{83} +12.0000i q^{85} -2.00000i q^{86} -3.00000i q^{89} +18.0000 q^{90} +6.00000 q^{92} -14.0000 q^{94} +15.0000 q^{95} +7.00000i q^{97} +2.00000i q^{98} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 6 q^{9}+O(q^{10}) \) 2 * q - 4 * q^4 - 6 * q^9	\( 2 q - 4 q^{4} - 6 q^{9} - 12 q^{10} + 4 q^{14} - 8 q^{16} - 8 q^{17} + 24 q^{22} - 6 q^{23} - 8 q^{25} - 10 q^{29} + 6 q^{35} + 12 q^{36} + 20 q^{38} + 2 q^{43} - 2 q^{49} - 18 q^{53} + 36 q^{55} - 20 q^{61} - 12 q^{62} + 16 q^{64} + 16 q^{68} + 16 q^{74} - 12 q^{77} + 6 q^{79} + 18 q^{81} - 24 q^{82} + 36 q^{90} + 12 q^{92} - 28 q^{94} + 30 q^{95}+O(q^{100}) \) 2 * q - 4 * q^4 - 6 * q^9 - 12 * q^10 + 4 * q^14 - 8 * q^16 - 8 * q^17 + 24 * q^22 - 6 * q^23 - 8 * q^25 - 10 * q^29 + 6 * q^35 + 12 * q^36 + 20 * q^38 + 2 * q^43 - 2 * q^49 - 18 * q^53 + 36 * q^55 - 20 * q^61 - 12 * q^62 + 16 * q^64 + 16 * q^68 + 16 * q^74 - 12 * q^77 + 6 * q^79 + 18 * q^81 - 24 * q^82 + 36 * q^90 + 12 * q^92 - 28 * q^94 + 30 * q^95

Introduction

L-functions

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Database

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