LMFDB - Embedded newform 1183.2.c.a.337.2

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:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} +2.00000 q^{4} +3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} +2.00000 q^{4} +3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} -4.00000 q^{12} -6.00000i q^{15} +4.00000 q^{16} +6.00000 q^{17} +7.00000i q^{19} +6.00000i q^{20} -2.00000i q^{21} -3.00000 q^{23} -4.00000 q^{25} +4.00000 q^{27} +2.00000i q^{28} -9.00000 q^{29} -5.00000i q^{31} -3.00000 q^{35} +2.00000 q^{36} +2.00000i q^{37} +6.00000i q^{41} +1.00000 q^{43} +3.00000i q^{45} +3.00000i q^{47} -8.00000 q^{48} -1.00000 q^{49} -12.0000 q^{51} -9.00000 q^{53} -14.0000i q^{57} -12.0000i q^{60} -10.0000 q^{61} +1.00000i q^{63} +8.00000 q^{64} -14.0000i q^{67} +12.0000 q^{68} +6.00000 q^{69} +6.00000i q^{71} +11.0000i q^{73} +8.00000 q^{75} +14.0000i q^{76} -1.00000 q^{79} +12.0000i q^{80} -11.0000 q^{81} -3.00000i q^{83} -4.00000i q^{84} +18.0000i q^{85} +18.0000 q^{87} +15.0000i q^{89} -6.00000 q^{92} +10.0000i q^{93} -21.0000 q^{95} +1.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 4 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 4 * q^4 + 2 * q^9	$ 2 q - 4 q^{3} + 4 q^{4} + 2 q^{9} - 8 q^{12} + 8 q^{16} + 12 q^{17} - 6 q^{23} - 8 q^{25} + 8 q^{27} - 18 q^{29} - 6 q^{35} + 4 q^{36} + 2 q^{43} - 16 q^{48} - 2 q^{49} - 24 q^{51} - 18 q^{53} - 20 q^{61} + 16 q^{64} + 24 q^{68} + 12 q^{69} + 16 q^{75} - 2 q^{79} - 22 q^{81} + 36 q^{87} - 12 q^{92} - 42 q^{95}+O(q^{100}) $ 2 * q - 4 * q^3 + 4 * q^4 + 2 * q^9 - 8 * q^12 + 8 * q^16 + 12 * q^17 - 6 * q^23 - 8 * q^25 + 8 * q^27 - 18 * q^29 - 6 * q^35 + 4 * q^36 + 2 * q^43 - 16 * q^48 - 2 * q^49 - 24 * q^51 - 18 * q^53 - 20 * q^61 + 16 * q^64 + 24 * q^68 + 12 * q^69 + 16 * q^75 - 2 * q^79 - 22 * q^81 + 36 * q^87 - 12 * q^92 - 42 * 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 $

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	2.00000	1.00000
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−4.00000	−1.15470
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 6.00000i	− 1.54919i
$16$	4.00000	1.00000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	7.00000i	1.60591i	0.596040	+	0.802955i	$0.296740\pi$
$19$	7.00000i	1.60591i	−0.596040	+	0.802955i	$0.703260\pi$
$20$	6.00000i	1.34164i
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	4.00000	0.769800
$28$	2.00000i	0.377964i
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	− 5.00000i	− 0.898027i	−0.893525	−	0.449013i	$-0.851776\pi$
$31$	− 5.00000i	− 0.898027i	0.893525	−	0.449013i	$-0.148224\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.00000	−0.507093
$36$	2.00000	0.333333
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000i	0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$	6.00000i	0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	3.00000i	0.447214i
$46$	0	0
$47$	3.00000i	0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$	3.00000i	0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$	−8.00000	−1.15470
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−12.0000	−1.68034
$52$	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
$55$	0	0
$56$	0	0
$57$	− 14.0000i	− 1.85435i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	− 12.0000i	− 1.54919i
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	12.0000	1.45521
$69$	6.00000	0.722315
$70$	0	0
$71$	6.00000i	0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$	6.00000i	0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$	0	0
$73$	11.0000i	1.28745i	0.765256	+	0.643726i	$0.222612\pi$
$73$	11.0000i	1.28745i	−0.765256	+	0.643726i	$0.777388\pi$
$74$	0	0
$75$	8.00000	0.923760
$76$	14.0000i	1.60591i
$77$	0	0
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	12.0000i	1.34164i
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 3.00000i	− 0.329293i	−0.986353	−	0.164646i	$-0.947352\pi$
$83$	− 3.00000i	− 0.329293i	0.986353	−	0.164646i	$-0.0526483\pi$
$84$	− 4.00000i	− 0.436436i
$85$	18.0000i	1.95237i
$86$	0	0
$87$	18.0000	1.92980
$88$	0	0
$89$	15.0000i	1.59000i	0.606612	+	0.794998i	$0.292528\pi$
$89$	15.0000i	1.59000i	−0.606612	+	0.794998i	$0.707472\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	10.0000i	1.03695i
$94$	0	0
$95$	−21.0000	−2.15455
$96$	0	0
$97$	1.00000i	0.101535i	0.998711	+	0.0507673i	$0.0161667\pi$
$97$	1.00000i	0.101535i	−0.998711	+	0.0507673i	$0.983833\pi$
$98$	0	0
$99$	0	0
$100$	−8.00000	−0.800000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	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
$105$	6.00000	0.585540
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	8.00000	0.769800
$109$	16.0000i	1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$	16.0000i	1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	− 4.00000i	− 0.379663i
$112$	4.00000i	0.377964i
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	0	0
$115$	− 9.00000i	− 0.839254i
$116$	−18.0000	−1.67126
$117$	0	0
$118$	0	0
$119$	6.00000i	0.550019i
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	− 10.0000i	− 0.898027i
$125$	3.00000i	0.268328i
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	−7.00000	−0.606977
$134$	0	0
$135$	12.0000i	1.03280i
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−6.00000	−0.507093
$141$	− 6.00000i	− 0.505291i
$142$	0	0
$143$	0	0
$144$	4.00000	0.333333
$145$	− 27.0000i	− 2.24223i
$146$	0	0
$147$	2.00000	0.164957
$148$	4.00000i	0.328798i
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	− 10.0000i	− 0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$	− 10.0000i	− 0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	15.0000	1.20483
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	18.0000	1.42749
$160$	0	0
$161$	− 3.00000i	− 0.236433i
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	12.0000i	0.937043i
$165$	0	0
$166$	0	0
$167$	− 15.0000i	− 1.16073i	−0.814355	−	0.580367i	$-0.802909\pi$
$167$	− 15.0000i	− 1.16073i	0.814355	−	0.580367i	$-0.197091\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7.00000i	0.535303i
$172$	2.00000	0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	− 4.00000i	− 0.302372i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	6.00000i	0.447214i
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	20.0000	1.47844
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	6.00000i	0.437595i
$189$	4.00000i	0.290957i
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	−16.0000	−1.15470
$193$	− 22.0000i	− 1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$	− 22.0000i	− 1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	0	0
$195$	0	0
$196$	−2.00000	−0.142857
$197$	− 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
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	28.0000i	1.97497i
$202$	0	0
$203$	− 9.00000i	− 0.631676i
$204$	−24.0000	−1.68034
$205$	−18.0000	−1.25717
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	0	0
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	−18.0000	−1.23625
$213$	− 12.0000i	− 0.822226i
$214$	0	0
$215$	3.00000i	0.204598i
$216$	0	0
$217$	5.00000	0.339422
$218$	0	0
$219$	− 22.0000i	− 1.48662i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.00000i	0.0669650i	0.999439	+	0.0334825i	$0.0106598\pi$
$223$	1.00000i	0.0669650i	−0.999439	+	0.0334825i	$0.989340\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	− 28.0000i	− 1.85435i
$229$	14.0000i	0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$	14.0000i	0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	0	0
$237$	2.00000	0.129914
$238$	0	0
$239$	12.0000i	0.776215i	0.921614	+	0.388108i	$0.126871\pi$
$239$	12.0000i	0.776215i	−0.921614	+	0.388108i	$0.873129\pi$
$240$	− 24.0000i	− 1.54919i
$241$	− 1.00000i	− 0.0644157i	−0.999481	−	0.0322078i	$-0.989746\pi$
$241$	− 1.00000i	− 0.0644157i	0.999481	−	0.0322078i	$-0.0102538\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	−20.0000	−1.28037
$245$	− 3.00000i	− 0.191663i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.00000i	0.380235i
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	0	0
$255$	− 36.0000i	− 2.25441i
$256$	16.0000	1.00000
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−9.00000	−0.557086
$262$	0	0
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	− 27.0000i	− 1.65860i
$266$	0	0
$267$	− 30.0000i	− 1.83597i
$268$	− 28.0000i	− 1.71037i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	− 16.0000i	− 0.971931i	−0.873978	−	0.485965i	$-0.838468\pi$
$271$	− 16.0000i	− 0.971931i	0.873978	−	0.485965i	$-0.161532\pi$
$272$	24.0000	1.45521
$273$	0	0
$274$	0	0
$275$	0	0
$276$	12.0000	0.722315
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	− 5.00000i	− 0.299342i
$280$	0	0
$281$	12.0000i	0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$	12.0000i	0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	12.0000i	0.712069i
$285$	42.0000	2.48787
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	22.0000i	1.28745i
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	16.0000	0.923760
$301$	1.00000i	0.0576390i
$302$	0	0
$303$	0	0
$304$	28.0000i	1.60591i
$305$	− 30.0000i	− 1.71780i
$306$	0	0
$307$	11.0000i	0.627803i	0.949456	+	0.313902i	$0.101636\pi$
$307$	11.0000i	0.627803i	−0.949456	+	0.313902i	$0.898364\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	−2.00000	−0.112509
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	0	0
$319$	0	0
$320$	24.0000i	1.34164i
$321$	24.0000	1.33955
$322$	0	0
$323$	42.0000i	2.33694i
$324$	−22.0000	−1.22222
$325$	0	0
$326$	0	0
$327$	− 32.0000i	− 1.76960i
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	− 26.0000i	− 1.42909i	−0.699590	−	0.714545i	$-0.746634\pi$
$331$	− 26.0000i	− 1.42909i	0.699590	−	0.714545i	$-0.253366\pi$
$332$	− 6.00000i	− 0.329293i
$333$	2.00000i	0.109599i
$334$	0	0
$335$	42.0000	2.29471
$336$	− 8.00000i	− 0.436436i
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	36.0000i	1.95237i
$341$	0	0
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	18.0000i	0.969087i
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	36.0000	1.92980
$349$	− 1.00000i	− 0.0535288i	−0.999642	−	0.0267644i	$-0.991480\pi$
$349$	− 1.00000i	− 0.0535288i	0.999642	−	0.0267644i	$-0.00852039\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	−18.0000	−0.955341
$356$	30.0000i	1.59000i
$357$	− 12.0000i	− 0.635107i
$358$	0	0
$359$	18.0000i	0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$	18.0000i	0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$	0	0
$361$	−30.0000	−1.57895
$362$	0	0
$363$	−22.0000	−1.15470
$364$	0	0
$365$	−33.0000	−1.72730
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	−12.0000	−0.625543
$369$	6.00000i	0.312348i
$370$	0	0
$371$	− 9.00000i	− 0.467257i
$372$	20.0000i	1.03695i
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	− 6.00000i	− 0.309839i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.0000i	0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$	16.0000i	0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$	−42.0000	−2.15455
$381$	−32.0000	−1.63941
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.00000	0.0508329
$388$	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
$391$	−18.0000	−0.910299
$392$	0	0
$393$	24.0000	1.21064
$394$	0	0
$395$	− 3.00000i	− 0.150946i
$396$	0	0
$397$	11.0000i	0.552074i	0.961147	+	0.276037i	$0.0890213\pi$
$397$	11.0000i	0.552074i	−0.961147	+	0.276037i	$0.910979\pi$
$398$	0	0
$399$	14.0000	0.700877
$400$	−16.0000	−0.800000
$401$	12.0000i	0.599251i	0.954057	+	0.299626i	$0.0968618\pi$
$401$	12.0000i	0.599251i	−0.954057	+	0.299626i	$0.903138\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 33.0000i	− 1.63978i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 23.0000i	− 1.13728i	−0.822588	−	0.568638i	$-0.807470\pi$
$409$	− 23.0000i	− 1.13728i	0.822588	−	0.568638i	$-0.192530\pi$
$410$	0	0
$411$	− 12.0000i	− 0.591916i
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	9.00000	0.441793
$416$	0	0
$417$	8.00000	0.391762
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	12.0000	0.585540
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	3.00000i	0.145865i
$424$	0	0
$425$	−24.0000	−1.16417
$426$	0	0
$427$	− 10.0000i	− 0.483934i
$428$	−24.0000	−1.16008
$429$	0	0
$430$	0	0
$431$	30.0000i	1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$	30.0000i	1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$	16.0000	0.769800
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	54.0000i	2.58910i
$436$	32.0000i	1.53252i
$437$	− 21.0000i	− 1.00457i
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	39.0000	1.85295	0.926473	−	0.376361i	$-0.122825\pi$
$443$	39.0000	1.85295	0.926473	+	0.376361i	$0.122825\pi$
$444$	− 8.00000i	− 0.379663i
$445$	−45.0000	−2.13320
$446$	0	0
$447$	0	0
$448$	8.00000i	0.377964i
$449$	− 24.0000i	− 1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$	− 24.0000i	− 1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$	0	0
$451$	0	0
$452$	18.0000	0.846649
$453$	20.0000i	0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	0	0
$459$	24.0000	1.12022
$460$	− 18.0000i	− 0.839254i
$461$	30.0000i	1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$	30.0000i	1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−36.0000	−1.67126
$465$	−30.0000	−1.39122
$466$	0	0
$467$	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
$469$	14.0000	0.646460
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 28.0000i	− 1.28473i
$476$	12.0000i	0.550019i
$477$	−9.00000	−0.412082
$478$	0	0
$479$	9.00000i	0.411220i	0.978634	+	0.205610i	$0.0659179\pi$
$479$	9.00000i	0.411220i	−0.978634	+	0.205610i	$0.934082\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	6.00000i	0.273009i
$484$	22.0000	1.00000
$485$	−3.00000	−0.136223
$486$	0	0
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	0	0
$489$	32.0000i	1.44709i
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	− 24.0000i	− 1.08200i
$493$	−54.0000	−2.43204
$494$	0	0
$495$	0	0
$496$	− 20.0000i	− 0.898027i
$497$	−6.00000	−0.269137
$498$	0	0
$499$	− 14.0000i	− 0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$	− 14.0000i	− 0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$	6.00000i	0.268328i
$501$	30.0000i	1.34030i
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	32.0000	1.41977
$509$	− 21.0000i	− 0.930809i	−0.885098	−	0.465404i	$-0.845909\pi$
$509$	− 21.0000i	− 0.930809i	0.885098	−	0.465404i	$-0.154091\pi$
$510$	0	0
$511$	−11.0000	−0.486611
$512$	0	0
$513$	28.0000i	1.23623i
$514$	0	0
$515$	12.0000i	0.528783i
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	−24.0000	−1.04844
$525$	8.00000i	0.349149i
$526$	0	0
$527$	− 30.0000i	− 1.30682i
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	−14.0000	−0.606977
$533$	0	0
$534$	0	0
$535$	− 36.0000i	− 1.55642i
$536$	0	0
$537$	30.0000	1.29460
$538$	0	0
$539$	0	0
$540$	24.0000i	1.03280i
$541$	− 34.0000i	− 1.46177i	−0.682498	−	0.730887i	$-0.739107\pi$
$541$	− 34.0000i	− 1.46177i	0.682498	−	0.730887i	$-0.260893\pi$
$542$	0	0
$543$	−32.0000	−1.37325
$544$	0	0
$545$	−48.0000	−2.05609
$546$	0	0
$547$	17.0000	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$547$	17.0000	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$548$	12.0000i	0.512615i
$549$	−10.0000	−0.426790
$550$	0	0
$551$	− 63.0000i	− 2.68389i
$552$	0	0
$553$	− 1.00000i	− 0.0425243i
$554$	0	0
$555$	12.0000	0.509372
$556$	−8.00000	−0.339276
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	0	0
$559$	0	0
$560$	−12.0000	−0.507093
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	− 12.0000i	− 0.505291i
$565$	27.0000i	1.13590i
$566$	0	0
$567$	− 11.0000i	− 0.461957i
$568$	0	0
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	0	0
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	0	0
$573$	−48.0000	−2.00523
$574$	0	0
$575$	12.0000	0.500435
$576$	8.00000	0.333333
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	0	0
$579$	44.0000i	1.82858i
$580$	− 54.0000i	− 2.24223i
$581$	3.00000	0.124461
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.00000i	0.371470i	0.982600	+	0.185735i	$0.0594666\pi$
$587$	9.00000i	0.371470i	−0.982600	+	0.185735i	$0.940533\pi$
$588$	4.00000	0.164957
$589$	35.0000	1.44215
$590$	0	0
$591$	12.0000i	0.493614i
$592$	8.00000i	0.328798i
$593$	9.00000i	0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$	9.00000i	0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$	0	0
$595$	−18.0000	−0.737928
$596$	0	0
$597$	4.00000	0.163709
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	− 14.0000i	− 0.570124i
$604$	− 20.0000i	− 0.813788i
$605$	33.0000i	1.34164i
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	18.0000i	0.729397i
$610$	0	0
$611$	0	0
$612$	12.0000	0.485071
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	36.0000	1.45166
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	− 28.0000i	− 1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$	− 28.0000i	− 1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$	30.0000	1.20483
$621$	−12.0000	−0.481543
$622$	0	0
$623$	−15.0000	−0.600962
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	28.0000	1.11732
$629$	12.0000i	0.478471i
$630$	0	0
$631$	2.00000i	0.0796187i	0.999207	+	0.0398094i	$0.0126751\pi$
$631$	2.00000i	0.0796187i	−0.999207	+	0.0398094i	$0.987325\pi$
$632$	0	0
$633$	−46.0000	−1.82834
$634$	0	0
$635$	48.0000i	1.90482i
$636$	36.0000	1.42749
$637$	0	0
$638$	0	0
$639$	6.00000i	0.237356i
$640$	0	0
$641$	39.0000	1.54041	0.770204	−	0.637798i	$-0.220155\pi$
$641$	39.0000	1.54041	0.770204	+	0.637798i	$0.220155\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	− 6.00000i	− 0.236433i
$645$	− 6.00000i	− 0.236250i
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−10.0000	−0.391931
$652$	− 32.0000i	− 1.25322i
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	− 36.0000i	− 1.40664i
$656$	24.0000i	0.937043i
$657$	11.0000i	0.429151i
$658$	0	0
$659$	−3.00000	−0.116863	−0.0584317	−	0.998291i	$-0.518610\pi$
$659$	−3.00000	−0.116863	−0.0584317	+	0.998291i	$0.518610\pi$
$660$	0	0
$661$	− 13.0000i	− 0.505641i	−0.967513	−	0.252821i	$-0.918642\pi$
$661$	− 13.0000i	− 0.505641i	0.967513	−	0.252821i	$-0.0813583\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 21.0000i	− 0.814345i
$666$	0	0
$667$	27.0000	1.04544
$668$	− 30.0000i	− 1.16073i
$669$	− 2.00000i	− 0.0773245i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	0	0
$675$	−16.0000	−0.615840
$676$	0	0
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	0	0
$679$	−1.00000	−0.0383765
$680$	0	0
$681$	48.0000i	1.83936i
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	14.0000i	0.535303i
$685$	−18.0000	−0.687745
$686$	0	0
$687$	− 28.0000i	− 1.06827i
$688$	4.00000	0.152499
$689$	0	0
$690$	0	0
$691$	37.0000i	1.40755i	0.710425	+	0.703773i	$0.248503\pi$
$691$	37.0000i	1.40755i	−0.710425	+	0.703773i	$0.751497\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	0	0
$695$	− 12.0000i	− 0.455186i
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	−18.0000	−0.680823
$700$	− 8.00000i	− 0.302372i
$701$	−9.00000	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	0	0
$705$	18.0000	0.677919
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8.00000i	0.300446i	0.988652	+	0.150223i	$0.0479992\pi$
$709$	8.00000i	0.300446i	−0.988652	+	0.150223i	$0.952001\pi$
$710$	0	0
$711$	−1.00000	−0.0375029
$712$	0	0
$713$	15.0000i	0.561754i
$714$	0	0
$715$	0	0
$716$	−30.0000	−1.12115
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	12.0000i	0.447214i
$721$	4.00000i	0.148968i
$722$	0	0
$723$	2.00000i	0.0743808i
$724$	32.0000	1.18927
$725$	36.0000	1.33701
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	6.00000	0.221918
$732$	40.0000	1.47844
$733$	49.0000i	1.80986i	0.425564	+	0.904928i	$0.360076\pi$
$733$	49.0000i	1.80986i	−0.425564	+	0.904928i	$0.639924\pi$
$734$	0	0
$735$	6.00000i	0.221313i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	0	0
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.00000i	− 0.109764i
$748$	0	0
$749$	− 12.0000i	− 0.438470i
$750$	0	0
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	12.0000i	0.437595i
$753$	24.0000	0.874609
$754$	0	0
$755$	30.0000	1.09181
$756$	8.00000i	0.290957i
$757$	−43.0000	−1.56286	−0.781431	−	0.623992i	$-0.785510\pi$
$757$	−43.0000	−1.56286	−0.781431	+	0.623992i	$0.785510\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 27.0000i	− 0.978749i	−0.872074	−	0.489375i	$-0.837225\pi$
$761$	− 27.0000i	− 0.978749i	0.872074	−	0.489375i	$-0.162775\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	48.0000	1.73658
$765$	18.0000i	0.650791i
$766$	0	0
$767$	0	0
$768$	−32.0000	−1.15470
$769$	− 5.00000i	− 0.180305i	−0.995928	−	0.0901523i	$-0.971265\pi$
$769$	− 5.00000i	− 0.180305i	0.995928	−	0.0901523i	$-0.0287354\pi$
$770$	0	0
$771$	−48.0000	−1.72868
$772$	− 44.0000i	− 1.58359i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	0	0
$775$	20.0000i	0.718421i
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−42.0000	−1.50481
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−36.0000	−1.28654
$784$	−4.00000	−0.142857
$785$	42.0000i	1.49904i
$786$	0	0
$787$	5.00000i	0.178231i	0.996021	+	0.0891154i	$0.0284040\pi$
$787$	5.00000i	0.178231i	−0.996021	+	0.0891154i	$0.971596\pi$
$788$	− 12.0000i	− 0.427482i
$789$	−18.0000	−0.640817
$790$	0	0
$791$	9.00000i	0.320003i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	54.0000i	1.91518i
$796$	−4.00000	−0.141776
$797$	−54.0000	−1.91278	−0.956389	−	0.292096i	$-0.905647\pi$
$797$	−54.0000	−1.91278	−0.956389	+	0.292096i	$0.905647\pi$
$798$	0	0
$799$	18.0000i	0.636794i
$800$	0	0
$801$	15.0000i	0.529999i
$802$	0	0
$803$	0	0
$804$	56.0000i	1.97497i
$805$	9.00000	0.317208
$806$	0	0
$807$	48.0000	1.68968
$808$	0	0
$809$	−3.00000	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$809$	−3.00000	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$810$	0	0
$811$	− 20.0000i	− 0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$	− 20.0000i	− 0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$	− 18.0000i	− 0.631676i
$813$	32.0000i	1.12229i
$814$	0	0
$815$	48.0000	1.68137
$816$	−48.0000	−1.68034
$817$	7.00000i	0.244899i
$818$	0	0
$819$	0	0
$820$	−36.0000	−1.25717
$821$	54.0000i	1.88461i	0.334751	+	0.942306i	$0.391348\pi$
$821$	54.0000i	1.88461i	−0.334751	+	0.942306i	$0.608652\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−6.00000	−0.208514
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	0	0
$831$	−38.0000	−1.31821
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	45.0000	1.55729
$836$	0	0
$837$	− 20.0000i	− 0.691301i
$838$	0	0
$839$	− 12.0000i	− 0.414286i	−0.978311	−	0.207143i	$-0.933583\pi$
$839$	− 12.0000i	− 0.414286i	0.978311	−	0.207143i	$-0.0664165\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	− 24.0000i	− 0.826604i
$844$	46.0000	1.58339
$845$	0	0
$846$	0	0
$847$	11.0000i	0.377964i
$848$	−36.0000	−1.23625
$849$	−8.00000	−0.274559
$850$	0	0
$851$	− 6.00000i	− 0.205677i
$852$	− 24.0000i	− 0.822226i
$853$	17.0000i	0.582069i	0.956713	+	0.291034i	$0.0939994\pi$
$853$	17.0000i	0.582069i	−0.956713	+	0.291034i	$0.906001\pi$
$854$	0	0
$855$	−21.0000	−0.718185
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	50.0000	1.70598	0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000	1.70598	0.852989	+	0.521929i	$0.174787\pi$
$860$	6.00000i	0.204598i
$861$	12.0000	0.408959
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	− 18.0000i	− 0.612018i
$866$	0	0
$867$	−38.0000	−1.29055
$868$	10.0000	0.339422
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.00000i	0.0338449i
$874$	0	0
$875$	−3.00000	−0.101419
$876$	− 44.0000i	− 1.48662i
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	0	0
$879$	− 18.0000i	− 0.607125i
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	16.0000i	0.536623i
$890$	0	0
$891$	0	0
$892$	2.00000i	0.0669650i
$893$	−21.0000	−0.702738
$894$	0	0
$895$	− 45.0000i	− 1.50418i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	45.0000i	1.50083i
$900$	−8.00000	−0.266667
$901$	−54.0000	−1.79900
$902$	0	0
$903$	− 2.00000i	− 0.0665558i
$904$	0	0
$905$	48.0000i	1.59557i
$906$	0	0
$907$	37.0000	1.22856	0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000	1.22856	0.614282	+	0.789086i	$0.289446\pi$
$908$	− 48.0000i	− 1.59294i
$909$	0	0
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	− 56.0000i	− 1.85435i
$913$	0	0
$914$	0	0
$915$	60.0000i	1.98354i
$916$	28.0000i	0.925146i
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−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
$921$	− 22.0000i	− 0.724925i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 8.00000i	− 0.263038i
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	3.00000i	0.0984268i	0.998788	+	0.0492134i	$0.0156714\pi$
$929$	3.00000i	0.0984268i	−0.998788	+	0.0492134i	$0.984329\pi$
$930$	0	0
$931$	− 7.00000i	− 0.229416i
$932$	18.0000	0.589610
$933$	−24.0000	−0.785725
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−16.0000	−0.522140
$940$	−18.0000	−0.587095
$941$	− 15.0000i	− 0.488986i	−0.969651	−	0.244493i	$-0.921378\pi$
$941$	− 15.0000i	− 0.488986i	0.969651	−	0.244493i	$-0.0786215\pi$
$942$	0	0
$943$	− 18.0000i	− 0.586161i
$944$	0	0
$945$	−12.0000	−0.390360
$946$	0	0
$947$	54.0000i	1.75476i	0.479792	+	0.877382i	$0.340712\pi$
$947$	54.0000i	1.75476i	−0.479792	+	0.877382i	$0.659288\pi$
$948$	4.00000	0.129914
$949$	0	0
$950$	0	0
$951$	24.0000i	0.778253i
$952$	0	0
$953$	−21.0000	−0.680257	−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000	−0.680257	−0.340128	+	0.940379i	$0.610471\pi$
$954$	0	0
$955$	72.0000i	2.32987i
$956$	24.0000i	0.776215i
$957$	0	0
$958$	0	0
$959$	−6.00000	−0.193750
$960$	− 48.0000i	− 1.54919i
$961$	6.00000	0.193548
$962$	0	0
$963$	−12.0000	−0.386695
$964$	− 2.00000i	− 0.0644157i
$965$	66.0000	2.12462
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	0	0
$969$	− 84.0000i	− 2.69847i
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	20.0000	0.641500
$973$	− 4.00000i	− 0.128234i
$974$	0	0
$975$	0	0
$976$	−40.0000	−1.28037
$977$	− 48.0000i	− 1.53566i	−0.640656	−	0.767828i	$-0.721338\pi$
$977$	− 48.0000i	− 1.53566i	0.640656	−	0.767828i	$-0.278662\pi$
$978$	0	0
$979$	0	0
$980$	− 6.00000i	− 0.191663i
$981$	16.0000i	0.510841i
$982$	0	0
$983$	57.0000i	1.81802i	0.416777	+	0.909009i	$0.363160\pi$
$983$	57.0000i	1.81802i	−0.416777	+	0.909009i	$0.636840\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	52.0000i	1.65017i
$994$	0	0
$995$	− 6.00000i	− 0.190213i
$996$	12.0000i	0.380235i
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	0	0
$999$	8.00000i	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1183.2.c.a.337.2		2
13.5	odd	4		1183.2.a.a.1.1		1
13.8	odd	4		91.2.a.b.1.1	✓	1
13.12	even	2	inner	1183.2.c.a.337.1		2
39.8	even	4		819.2.a.c.1.1		1
52.47	even	4		1456.2.a.k.1.1		1
65.34	odd	4		2275.2.a.d.1.1		1
91.34	even	4		637.2.a.b.1.1		1
91.47	even	12		637.2.e.b.508.1		2
91.60	odd	12		637.2.e.c.79.1		2
91.73	even	12		637.2.e.b.79.1		2
91.83	even	4		8281.2.a.h.1.1		1
91.86	odd	12		637.2.e.c.508.1		2
104.21	odd	4		5824.2.a.bd.1.1		1
104.99	even	4		5824.2.a.f.1.1		1
273.125	odd	4		5733.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.b.1.1	✓	1	13.8	odd	4
637.2.a.b.1.1		1	91.34	even	4
637.2.e.b.79.1		2	91.73	even	12
637.2.e.b.508.1		2	91.47	even	12
637.2.e.c.79.1		2	91.60	odd	12
637.2.e.c.508.1		2	91.86	odd	12
819.2.a.c.1.1		1	39.8	even	4
1183.2.a.a.1.1		1	13.5	odd	4
1183.2.c.a.337.1		2	13.12	even	2	inner
1183.2.c.a.337.2		2	1.1	even	1	trivial
1456.2.a.k.1.1		1	52.47	even	4
2275.2.a.d.1.1		1	65.34	odd	4
5733.2.a.f.1.1		1	273.125	odd	4
5824.2.a.f.1.1		1	104.99	even	4
5824.2.a.bd.1.1		1	104.21	odd	4
8281.2.a.h.1.1		1	91.83	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +2.00000 q^{4} +3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} +2.00000 q^{4} +3.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} -4.00000 q^{12} -6.00000i q^{15} +4.00000 q^{16} +6.00000 q^{17} +7.00000i q^{19} +6.00000i q^{20} -2.00000i q^{21} -3.00000 q^{23} -4.00000 q^{25} +4.00000 q^{27} +2.00000i q^{28} -9.00000 q^{29} -5.00000i q^{31} -3.00000 q^{35} +2.00000 q^{36} +2.00000i q^{37} +6.00000i q^{41} +1.00000 q^{43} +3.00000i q^{45} +3.00000i q^{47} -8.00000 q^{48} -1.00000 q^{49} -12.0000 q^{51} -9.00000 q^{53} -14.0000i q^{57} -12.0000i q^{60} -10.0000 q^{61} +1.00000i q^{63} +8.00000 q^{64} -14.0000i q^{67} +12.0000 q^{68} +6.00000 q^{69} +6.00000i q^{71} +11.0000i q^{73} +8.00000 q^{75} +14.0000i q^{76} -1.00000 q^{79} +12.0000i q^{80} -11.0000 q^{81} -3.00000i q^{83} -4.00000i q^{84} +18.0000i q^{85} +18.0000 q^{87} +15.0000i q^{89} -6.00000 q^{92} +10.0000i q^{93} -21.0000 q^{95} +1.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 4 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 4 * q^4 + 2 * q^9	\( 2 q - 4 q^{3} + 4 q^{4} + 2 q^{9} - 8 q^{12} + 8 q^{16} + 12 q^{17} - 6 q^{23} - 8 q^{25} + 8 q^{27} - 18 q^{29} - 6 q^{35} + 4 q^{36} + 2 q^{43} - 16 q^{48} - 2 q^{49} - 24 q^{51} - 18 q^{53} - 20 q^{61} + 16 q^{64} + 24 q^{68} + 12 q^{69} + 16 q^{75} - 2 q^{79} - 22 q^{81} + 36 q^{87} - 12 q^{92} - 42 q^{95}+O(q^{100}) \) 2 * q - 4 * q^3 + 4 * q^4 + 2 * q^9 - 8 * q^12 + 8 * q^16 + 12 * q^17 - 6 * q^23 - 8 * q^25 + 8 * q^27 - 18 * q^29 - 6 * q^35 + 4 * q^36 + 2 * q^43 - 16 * q^48 - 2 * q^49 - 24 * q^51 - 18 * q^53 - 20 * q^61 + 16 * q^64 + 24 * q^68 + 12 * q^69 + 16 * q^75 - 2 * q^79 - 22 * q^81 + 36 * q^87 - 12 * q^92 - 42 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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