LMFDB - Embedded newform 1183.2.c.j.337.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1183,2,Mod(337,1183)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [24,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$9.44630255912$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.6
Character	$\chi$	$=$	1183.337
Dual form			1183.2.c.j.337.19

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.06743i q^{2} +2.11889 q^{3} -2.27425 q^{4} -2.43928i q^{5} -4.38065i q^{6} +1.00000i q^{7} +0.566992i q^{8} +1.48970 q^{9} -5.04303 q^{10} -3.64779i q^{11} -4.81889 q^{12} +2.06743 q^{14} -5.16857i q^{15} -3.37629 q^{16} -7.04993 q^{17} -3.07985i q^{18} +2.76410i q^{19} +5.54753i q^{20} +2.11889i q^{21} -7.54154 q^{22} +7.75601 q^{23} +1.20139i q^{24} -0.950076 q^{25} -3.20015 q^{27} -2.27425i q^{28} +2.31600 q^{29} -10.6856 q^{30} -7.00738i q^{31} +8.11421i q^{32} -7.72927i q^{33} +14.5752i q^{34} +2.43928 q^{35} -3.38796 q^{36} -7.28412i q^{37} +5.71456 q^{38} +1.38305 q^{40} +3.49162i q^{41} +4.38065 q^{42} -3.44630 q^{43} +8.29599i q^{44} -3.63380i q^{45} -16.0350i q^{46} +2.66883i q^{47} -7.15399 q^{48} -1.00000 q^{49} +1.96421i q^{50} -14.9380 q^{51} +13.3378 q^{53} +6.61608i q^{54} -8.89797 q^{55} -0.566992 q^{56} +5.85682i q^{57} -4.78816i q^{58} -8.24978i q^{59} +11.7546i q^{60} +13.9465 q^{61} -14.4872 q^{62} +1.48970i q^{63} +10.0229 q^{64} -15.9797 q^{66} +3.20893i q^{67} +16.0333 q^{68} +16.4341 q^{69} -5.04303i q^{70} +9.74080i q^{71} +0.844650i q^{72} -3.75697i q^{73} -15.0594 q^{74} -2.01311 q^{75} -6.28624i q^{76} +3.64779 q^{77} -12.8682 q^{79} +8.23570i q^{80} -11.2499 q^{81} +7.21867 q^{82} +5.42451i q^{83} -4.81889i q^{84} +17.1967i q^{85} +7.12496i q^{86} +4.90736 q^{87} +2.06827 q^{88} -0.335808i q^{89} -7.51262 q^{90} -17.6391 q^{92} -14.8479i q^{93} +5.51762 q^{94} +6.74240 q^{95} +17.1931i q^{96} -10.9424i q^{97} +2.06743i q^{98} -5.43413i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q + 16 q^{3} - 30 q^{4} + 52 q^{9} + 12 q^{10} - 26 q^{12} + 6 q^{14} + 26 q^{16} - 62 q^{17} - 8 q^{22} - 36 q^{23} - 64 q^{25} + 64 q^{27} + 30 q^{29} + 20 q^{30} - 8 q^{35} - 98 q^{36} - 90 q^{38}+ \cdots - 58 q^{95}+O(q^{100})$ 24 * q + 16 * q^3 - 30 * q^4 + 52 * q^9 + 12 * q^10 - 26 * q^12 + 6 * q^14 + 26 * q^16 - 62 * q^17 - 8 * q^22 - 36 * q^23 - 64 * q^25 + 64 * q^27 + 30 * q^29 + 20 * q^30 - 8 * q^35 - 98 * q^36 - 90 * q^38 - 40 * q^40 + 4 * q^42 + 44 * q^43 + 22 * q^48 - 24 * q^49 - 36 * q^51 + 106 * q^53 - 52 * q^55 - 24 * q^56 + 44 * q^61 - 38 * q^62 - 4 * q^64 - 68 * q^66 + 68 * q^68 - 6 * q^69 - 80 * q^74 - 30 * q^75 - 24 * q^77 + 4 * q^79 + 72 * q^81 + 64 * q^82 + 54 * q^87 + 96 * q^88 + 52 * q^90 + 104 * q^92 - 88 * q^94 - 58 * q^95

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	− 2.06743i	− 1.46189i	−0.682436	−	0.730945i	$-0.739079\pi$
$2$	− 2.06743i	− 1.46189i	0.682436	−	0.730945i	$-0.260921\pi$
$3$	2.11889	1.22334	0.611671	−	0.791112i	$-0.290497\pi$
$3$	2.11889	1.22334	0.611671	+	0.791112i	$0.290497\pi$
$4$	−2.27425	−1.13713
$5$	− 2.43928i	− 1.09088i	−0.838150	−	0.545439i	$-0.816363\pi$
$5$	− 2.43928i	− 1.09088i	0.838150	−	0.545439i	$-0.183637\pi$
$6$	− 4.38065i	− 1.78839i
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0.566992i	0.200462i
$9$	1.48970	0.496568
$10$	−5.04303	−1.59474
$11$	− 3.64779i	− 1.09985i	−0.835214	−	0.549925i	$-0.814656\pi$
$11$	− 3.64779i	− 1.09985i	0.835214	−	0.549925i	$-0.185344\pi$
$12$	−4.81889	−1.39109
$13$	0	0
$13$	0	0
$14$	2.06743	0.552543
$15$	− 5.16857i	− 1.33452i
$16$	−3.37629	−0.844072
$17$	−7.04993	−1.70986	−0.854929	−	0.518745i	$-0.826400\pi$
$17$	−7.04993	−1.70986	−0.854929	+	0.518745i	$0.826400\pi$
$18$	− 3.07985i	− 0.725928i
$19$	2.76410i	0.634127i	0.948404	+	0.317063i	$0.102697\pi$
$19$	2.76410i	0.634127i	−0.948404	+	0.317063i	$0.897303\pi$
$20$	5.54753i	1.24046i
$21$	2.11889i	0.462380i
$22$	−7.54154	−1.60786
$23$	7.75601	1.61724	0.808620	−	0.588332i	$-0.200215\pi$
$23$	7.75601	1.61724	0.808620	+	0.588332i	$0.200215\pi$
$24$	1.20139i	0.245234i
$25$	−0.950076	−0.190015
$26$	0	0
$27$	−3.20015	−0.615870
$28$	− 2.27425i	− 0.429793i
$29$	2.31600	0.430071	0.215035	−	0.976606i	$-0.431013\pi$
$29$	2.31600	0.430071	0.215035	+	0.976606i	$0.431013\pi$
$30$	−10.6856	−1.95092
$31$	− 7.00738i	− 1.25856i	−0.777178	−	0.629281i	$-0.783349\pi$
$31$	− 7.00738i	− 1.25856i	0.777178	−	0.629281i	$-0.216651\pi$
$32$	8.11421i	1.43440i
$33$	− 7.72927i	− 1.34549i
$34$	14.5752i	2.49963i
$35$	2.43928	0.412313
$36$	−3.38796	−0.564660
$37$	− 7.28412i	− 1.19750i	−0.800935	−	0.598751i	$-0.795664\pi$
$37$	− 7.28412i	− 1.19750i	0.800935	−	0.598751i	$-0.204336\pi$
$38$	5.71456	0.927024
$39$	0	0
$40$	1.38305	0.218679
$41$	3.49162i	0.545300i	0.962113	+	0.272650i	$0.0879000\pi$
$41$	3.49162i	0.545300i	−0.962113	+	0.272650i	$0.912100\pi$
$42$	4.38065	0.675949
$43$	−3.44630	−0.525555	−0.262778	−	0.964856i	$-0.584639\pi$
$43$	−3.44630	−0.525555	−0.262778	+	0.964856i	$0.584639\pi$
$44$	8.29599i	1.25067i
$45$	− 3.63380i	− 0.541695i
$46$	− 16.0350i	− 2.36423i
$47$	2.66883i	0.389289i	0.980874	+	0.194645i	$0.0623554\pi$
$47$	2.66883i	0.389289i	−0.980874	+	0.194645i	$0.937645\pi$
$48$	−7.15399	−1.03259
$49$	−1.00000	−0.142857
$50$	1.96421i	0.277782i
$51$	−14.9380	−2.09174
$52$	0	0
$53$	13.3378	1.83208	0.916041	−	0.401086i	$-0.131367\pi$
$53$	13.3378	1.83208	0.916041	+	0.401086i	$0.131367\pi$
$54$	6.61608i	0.900335i
$55$	−8.89797	−1.19980
$56$	−0.566992	−0.0757675
$57$	5.85682i	0.775755i
$58$	− 4.78816i	− 0.628716i
$59$	− 8.24978i	− 1.07403i	−0.843573	−	0.537015i	$-0.819552\pi$
$59$	− 8.24978i	− 1.07403i	0.843573	−	0.537015i	$-0.180448\pi$
$60$	11.7546i	1.51751i
$61$	13.9465	1.78566	0.892831	−	0.450392i	$-0.148716\pi$
$61$	13.9465	1.78566	0.892831	+	0.450392i	$0.148716\pi$
$62$	−14.4872	−1.83988
$63$	1.48970i	0.187685i
$64$	10.0229	1.25287
$65$	0	0
$66$	−15.9797	−1.96697
$67$	3.20893i	0.392033i	0.980601	+	0.196017i	$0.0628007\pi$
$67$	3.20893i	0.392033i	−0.980601	+	0.196017i	$0.937199\pi$
$68$	16.0333	1.94432
$69$	16.4341	1.97844
$70$	− 5.04303i	− 0.602757i
$71$	9.74080i	1.15602i	0.816029	+	0.578010i	$0.196171\pi$
$71$	9.74080i	1.15602i	−0.816029	+	0.578010i	$0.803829\pi$
$72$	0.844650i	0.0995429i
$73$	− 3.75697i	− 0.439720i	−0.975531	−	0.219860i	$-0.929440\pi$
$73$	− 3.75697i	− 0.439720i	0.975531	−	0.219860i	$-0.0705601\pi$
$74$	−15.0594	−1.75062
$75$	−2.01311	−0.232454
$76$	− 6.28624i	− 0.721082i
$77$	3.64779	0.415704
$78$	0	0
$79$	−12.8682	−1.44778	−0.723890	−	0.689915i	$-0.757648\pi$
$79$	−12.8682	−1.44778	−0.723890	+	0.689915i	$0.757648\pi$
$80$	8.23570i	0.920779i
$81$	−11.2499	−1.24999
$82$	7.21867	0.797168
$83$	5.42451i	0.595417i	0.954657	+	0.297709i	$0.0962224\pi$
$83$	5.42451i	0.595417i	−0.954657	+	0.297709i	$0.903778\pi$
$84$	− 4.81889i	− 0.525784i
$85$	17.1967i	1.86525i
$86$	7.12496i	0.768304i
$87$	4.90736	0.526124
$88$	2.06827	0.220478
$89$	− 0.335808i	− 0.0355955i	−0.999842	−	0.0177978i	$-0.994334\pi$
$89$	− 0.335808i	− 0.0355955i	0.999842	−	0.0177978i	$-0.00566551\pi$
$90$	−7.51262	−0.791899
$91$	0	0
$92$	−17.6391	−1.83900
$93$	− 14.8479i	− 1.53965i
$94$	5.51762	0.569099
$95$	6.74240	0.691755
$96$	17.1931i	1.75477i
$97$	− 10.9424i	− 1.11103i	−0.831505	−	0.555517i	$-0.812520\pi$
$97$	− 10.9424i	− 1.11103i	0.831505	−	0.555517i	$-0.187480\pi$
$98$	2.06743i	0.208842i
$99$	− 5.43413i	− 0.546150i
$100$	2.16071	0.216071
$101$	−8.92269	−0.887841	−0.443920	−	0.896066i	$-0.646413\pi$
$101$	−8.92269	−0.887841	−0.443920	+	0.896066i	$0.646413\pi$
$102$	30.8833i	3.05790i
$103$	2.66966	0.263049	0.131525	−	0.991313i	$-0.458013\pi$
$103$	2.66966	0.263049	0.131525	+	0.991313i	$0.458013\pi$
$104$	0	0
$105$	5.16857	0.504400
$106$	− 27.5748i	− 2.67830i
$107$	14.9485	1.44513	0.722563	−	0.691305i	$-0.242964\pi$
$107$	14.9485	1.44513	0.722563	+	0.691305i	$0.242964\pi$
$108$	7.27795	0.700321
$109$	1.45686i	0.139542i	0.997563	+	0.0697710i	$0.0222269\pi$
$109$	1.45686i	0.139542i	−0.997563	+	0.0697710i	$0.977773\pi$
$110$	18.3959i	1.75398i
$111$	− 15.4343i	− 1.46496i
$112$	− 3.37629i	− 0.319029i
$113$	14.8440	1.39641	0.698204	−	0.715899i	$-0.253983\pi$
$113$	14.8440	1.39641	0.698204	+	0.715899i	$0.253983\pi$
$114$	12.1085	1.13407
$115$	− 18.9191i	− 1.76421i
$116$	−5.26716	−0.489044
$117$	0	0
$118$	−17.0558	−1.57011
$119$	− 7.04993i	− 0.646266i
$120$	2.93053	0.267520
$121$	−2.30638	−0.209670
$122$	− 28.8333i	− 2.61044i
$123$	7.39837i	0.667088i
$124$	15.9365i	1.43114i
$125$	− 9.87889i	− 0.883595i
$126$	3.07985	0.274375
$127$	−5.83060	−0.517382	−0.258691	−	0.965960i	$-0.583291\pi$
$127$	−5.83060	−0.517382	−0.258691	+	0.965960i	$0.583291\pi$
$128$	− 4.49329i	− 0.397154i
$129$	−7.30233	−0.642934
$130$	0	0
$131$	15.0558	1.31543	0.657714	−	0.753268i	$-0.271523\pi$
$131$	15.0558	1.31543	0.657714	+	0.753268i	$0.271523\pi$
$132$	17.5783i	1.52999i
$133$	−2.76410	−0.239677
$134$	6.63423	0.573110
$135$	7.80607i	0.671839i
$136$	− 3.99725i	− 0.342761i
$137$	4.23709i	0.361999i	0.983483	+	0.181000i	$0.0579333\pi$
$137$	4.23709i	0.361999i	−0.983483	+	0.181000i	$0.942067\pi$
$138$	− 33.9764i	− 2.89226i
$139$	−2.70802	−0.229691	−0.114846	−	0.993383i	$-0.536637\pi$
$139$	−2.70802	−0.229691	−0.114846	+	0.993383i	$0.536637\pi$
$140$	−5.54753	−0.468852
$141$	5.65497i	0.476235i
$142$	20.1384	1.68998
$143$	0	0
$144$	−5.02967	−0.419139
$145$	− 5.64937i	− 0.469155i
$146$	−7.76726	−0.642823
$147$	−2.11889	−0.174763
$148$	16.5659i	1.36171i
$149$	− 0.891186i	− 0.0730088i	−0.999333	−	0.0365044i	$-0.988378\pi$
$149$	− 0.891186i	− 0.0730088i	0.999333	−	0.0365044i	$-0.0116223\pi$
$150$	4.16195i	0.339822i
$151$	16.2557i	1.32287i	0.750001	+	0.661436i	$0.230053\pi$
$151$	16.2557i	1.32287i	−0.750001	+	0.661436i	$0.769947\pi$
$152$	−1.56722	−0.127118
$153$	−10.5023	−0.849061
$154$	− 7.54154i	− 0.607714i
$155$	−17.0929	−1.37294
$156$	0	0
$157$	−3.94721	−0.315022	−0.157511	−	0.987517i	$-0.550347\pi$
$157$	−3.94721	−0.315022	−0.157511	+	0.987517i	$0.550347\pi$
$158$	26.6040i	2.11650i
$159$	28.2613	2.24126
$160$	19.7928	1.56476
$161$	7.75601i	0.611259i
$162$	23.2583i	1.82735i
$163$	− 4.48439i	− 0.351245i	−0.984458	−	0.175622i	$-0.943806\pi$
$163$	− 4.48439i	− 0.351245i	0.984458	−	0.175622i	$-0.0561937\pi$
$164$	− 7.94082i	− 0.620074i
$165$	−18.8538	−1.46777
$166$	11.2148	0.870435
$167$	8.06225i	0.623875i	0.950103	+	0.311938i	$0.100978\pi$
$167$	8.06225i	0.623875i	−0.950103	+	0.311938i	$0.899022\pi$
$168$	−1.20139	−0.0926896
$169$	0	0
$170$	35.5530	2.72679
$171$	4.11768i	0.314887i
$172$	7.83774	0.597622
$173$	7.74395	0.588761	0.294381	−	0.955688i	$-0.404887\pi$
$173$	7.74395	0.588761	0.294381	+	0.955688i	$0.404887\pi$
$174$	− 10.1456i	− 0.769136i
$175$	− 0.950076i	− 0.0718190i
$176$	12.3160i	0.928352i
$177$	− 17.4804i	− 1.31391i
$178$	−0.694258	−0.0520368
$179$	16.3124	1.21924	0.609621	−	0.792693i	$-0.291321\pi$
$179$	16.3124	1.21924	0.609621	+	0.792693i	$0.291321\pi$
$180$	8.26417i	0.615975i
$181$	−8.28425	−0.615763	−0.307882	−	0.951425i	$-0.599620\pi$
$181$	−8.28425	−0.615763	−0.307882	+	0.951425i	$0.599620\pi$
$182$	0	0
$183$	29.5510	2.18448
$184$	4.39759i	0.324195i
$185$	−17.7680	−1.30633
$186$	−30.6969	−2.25081
$187$	25.7167i	1.88059i
$188$	− 6.06960i	− 0.442671i
$189$	− 3.20015i	− 0.232777i
$190$	− 13.9394i	− 1.01127i
$191$	10.7356	0.776803	0.388402	−	0.921490i	$-0.373027\pi$
$191$	10.7356	0.776803	0.388402	+	0.921490i	$0.373027\pi$
$192$	21.2375	1.53269
$193$	0.762728i	0.0549024i	0.999623	+	0.0274512i	$0.00873908\pi$
$193$	0.762728i	0.0549024i	−0.999623	+	0.0274512i	$0.991261\pi$
$194$	−22.6226	−1.62421
$195$	0	0
$196$	2.27425	0.162446
$197$	12.1127i	0.862997i	0.902114	+	0.431498i	$0.142015\pi$
$197$	12.1127i	0.862997i	−0.902114	+	0.431498i	$0.857985\pi$
$198$	−11.2347	−0.798412
$199$	4.27032	0.302715	0.151358	−	0.988479i	$-0.451636\pi$
$199$	4.27032	0.302715	0.151358	+	0.988479i	$0.451636\pi$
$200$	− 0.538685i	− 0.0380908i
$201$	6.79938i	0.479591i
$202$	18.4470i	1.29793i
$203$	2.31600i	0.162551i
$204$	33.9728	2.37857
$205$	8.51703	0.594855
$206$	− 5.51932i	− 0.384549i
$207$	11.5542	0.803069
$208$	0	0
$209$	10.0828	0.697445
$210$	− 10.6856i	− 0.737378i
$211$	−7.26849	−0.500384	−0.250192	−	0.968196i	$-0.580494\pi$
$211$	−7.26849	−0.500384	−0.250192	+	0.968196i	$0.580494\pi$
$212$	−30.3334	−2.08331
$213$	20.6397i	1.41421i
$214$	− 30.9049i	− 2.11262i
$215$	8.40647i	0.573317i
$216$	− 1.81446i	− 0.123458i
$217$	7.00738	0.475692
$218$	3.01195	0.203995
$219$	− 7.96062i	− 0.537929i
$220$	20.2362	1.36433
$221$	0	0
$222$	−31.9092	−2.14161
$223$	11.5634i	0.774343i	0.922008	+	0.387172i	$0.126548\pi$
$223$	11.5634i	0.774343i	−0.922008	+	0.387172i	$0.873452\pi$
$224$	−8.11421	−0.542153
$225$	−1.41533	−0.0943555
$226$	− 30.6889i	− 2.04140i
$227$	3.02967i	0.201086i	0.994933	+	0.100543i	$0.0320580\pi$
$227$	3.02967i	0.201086i	−0.994933	+	0.100543i	$0.967942\pi$
$228$	− 13.3199i	− 0.882130i
$229$	3.56377i	0.235500i	0.993043	+	0.117750i	$0.0375682\pi$
$229$	3.56377i	0.235500i	−0.993043	+	0.117750i	$0.962432\pi$
$230$	−39.1137	−2.57908
$231$	7.72927	0.508549
$232$	1.31315i	0.0862127i
$233$	12.8151	0.839547	0.419774	−	0.907629i	$-0.362109\pi$
$233$	12.8151	0.839547	0.419774	+	0.907629i	$0.362109\pi$
$234$	0	0
$235$	6.51003	0.424667
$236$	18.7621i	1.22131i
$237$	−27.2662	−1.77113
$238$	−14.5752	−0.944770
$239$	− 25.3757i	− 1.64142i	−0.571347	−	0.820708i	$-0.693579\pi$
$239$	− 25.3757i	− 1.64142i	0.571347	−	0.820708i	$-0.306421\pi$
$240$	17.4506i	1.12643i
$241$	6.80302i	0.438221i	0.975700	+	0.219111i	$0.0703156\pi$
$241$	6.80302i	0.438221i	−0.975700	+	0.219111i	$0.929684\pi$
$242$	4.76826i	0.306515i
$243$	−14.2368	−0.913294
$244$	−31.7177	−2.03052
$245$	2.43928i	0.155840i
$246$	15.2956	0.975210
$247$	0	0
$248$	3.97313	0.252294
$249$	11.4940i	0.728400i
$250$	−20.4239	−1.29172
$251$	22.8556	1.44263	0.721317	−	0.692605i	$-0.243537\pi$
$251$	22.8556	1.44263	0.721317	+	0.692605i	$0.243537\pi$
$252$	− 3.38796i	− 0.213421i
$253$	− 28.2923i	− 1.77872i
$254$	12.0543i	0.756356i
$255$	36.4380i	2.28184i
$256$	10.7564	0.672272
$257$	3.85941	0.240743	0.120372	−	0.992729i	$-0.461591\pi$
$257$	3.85941	0.240743	0.120372	+	0.992729i	$0.461591\pi$
$258$	15.0970i	0.939899i
$259$	7.28412	0.452613
$260$	0	0
$261$	3.45016	0.213559
$262$	− 31.1267i	− 1.92301i
$263$	−16.2197	−1.00015	−0.500075	−	0.865982i	$-0.666694\pi$
$263$	−16.2197	−1.00015	−0.500075	+	0.865982i	$0.666694\pi$
$264$	4.38243	0.269720
$265$	− 32.5345i	− 1.99858i
$266$	5.71456i	0.350382i
$267$	− 0.711540i	− 0.0435456i
$268$	− 7.29791i	− 0.445791i
$269$	−23.0815	−1.40731	−0.703653	−	0.710544i	$-0.748449\pi$
$269$	−23.0815	−1.40731	−0.703653	+	0.710544i	$0.748449\pi$
$270$	16.1385	0.982156
$271$	− 10.7673i	− 0.654067i	−0.945013	−	0.327034i	$-0.893951\pi$
$271$	− 10.7673i	− 0.654067i	0.945013	−	0.327034i	$-0.106049\pi$
$272$	23.8026	1.44324
$273$	0	0
$274$	8.75987	0.529203
$275$	3.46568i	0.208988i
$276$	−37.3753	−2.24973
$277$	11.7786	0.707708	0.353854	−	0.935301i	$-0.384871\pi$
$277$	11.7786	0.707708	0.353854	+	0.935301i	$0.384871\pi$
$278$	5.59863i	0.335784i
$279$	− 10.4389i	− 0.624962i
$280$	1.38305i	0.0826531i
$281$	− 27.3072i	− 1.62901i	−0.580154	−	0.814507i	$-0.697008\pi$
$281$	− 27.3072i	− 1.62901i	0.580154	−	0.814507i	$-0.302992\pi$
$282$	11.6912	0.696203
$283$	−22.8247	−1.35679	−0.678393	−	0.734699i	$-0.737323\pi$
$283$	−22.8247	−1.35679	−0.678393	+	0.734699i	$0.737323\pi$
$284$	− 22.1530i	− 1.31454i
$285$	14.2864	0.846254
$286$	0	0
$287$	−3.49162	−0.206104
$288$	12.0878i	0.712278i
$289$	32.7015	1.92362
$290$	−11.6797	−0.685853
$291$	− 23.1858i	− 1.35918i
$292$	8.54430i	0.500017i
$293$	− 19.8388i	− 1.15899i	−0.814974	−	0.579497i	$-0.803249\pi$
$293$	− 19.8388i	− 1.15899i	0.814974	−	0.579497i	$-0.196751\pi$
$294$	4.38065i	0.255485i
$295$	−20.1235	−1.17164
$296$	4.13004	0.240054
$297$	11.6735i	0.677365i
$298$	−1.84246	−0.106731
$299$	0	0
$300$	4.57831	0.264329
$301$	− 3.44630i	− 0.198641i
$302$	33.6075	1.93390
$303$	−18.9062	−1.08613
$304$	− 9.33238i	− 0.535249i
$305$	− 34.0193i	− 1.94794i
$306$	21.7127i	1.24123i
$307$	10.3240i	0.589219i	0.955618	+	0.294610i	$0.0951896\pi$
$307$	10.3240i	0.589219i	−0.955618	+	0.294610i	$0.904810\pi$
$308$	−8.29599	−0.472708
$309$	5.65672	0.321800
$310$	35.3384i	2.00709i
$311$	−22.2930	−1.26412	−0.632060	−	0.774919i	$-0.717790\pi$
$311$	−22.2930	−1.26412	−0.632060	+	0.774919i	$0.717790\pi$
$312$	0	0
$313$	−19.4600	−1.09995	−0.549973	−	0.835182i	$-0.685362\pi$
$313$	−19.4600	−1.09995	−0.549973	+	0.835182i	$0.685362\pi$
$314$	8.16057i	0.460528i
$315$	3.63380	0.204742
$316$	29.2654	1.64631
$317$	− 27.4413i	− 1.54126i	−0.637284	−	0.770629i	$-0.719942\pi$
$317$	− 27.4413i	− 1.54126i	0.637284	−	0.770629i	$-0.280058\pi$
$318$	− 58.4281i	− 3.27648i
$319$	− 8.44829i	− 0.473013i
$320$	− 24.4488i	− 1.36673i
$321$	31.6743	1.76789
$322$	16.0350	0.893594
$323$	− 19.4867i	− 1.08427i
$324$	25.5851	1.42139
$325$	0	0
$326$	−9.27115	−0.513481
$327$	3.08693i	0.170708i
$328$	−1.97972	−0.109312
$329$	−2.66883	−0.147138
$330$	38.9789i	2.14572i
$331$	− 20.3710i	− 1.11969i	−0.828596	−	0.559846i	$-0.810860\pi$
$331$	− 20.3710i	− 1.11969i	0.828596	−	0.559846i	$-0.189140\pi$
$332$	− 12.3367i	− 0.677064i
$333$	− 10.8512i	− 0.594641i
$334$	16.6681	0.912038
$335$	7.82748	0.427661
$336$	− 7.15399i	− 0.390282i
$337$	2.67053	0.145473	0.0727366	−	0.997351i	$-0.476827\pi$
$337$	2.67053	0.145473	0.0727366	+	0.997351i	$0.476827\pi$
$338$	0	0
$339$	31.4529	1.70829
$340$	− 39.1097i	− 2.12102i
$341$	−25.5615	−1.38423
$342$	8.51301	0.460331
$343$	− 1.00000i	− 0.0539949i
$344$	− 1.95402i	− 0.105354i
$345$	− 40.0874i	− 2.15823i
$346$	− 16.0100i	− 0.860705i
$347$	−0.317819	−0.0170614	−0.00853071	−	0.999964i	$-0.502715\pi$
$347$	−0.317819	−0.0170614	−0.00853071	+	0.999964i	$0.502715\pi$
$348$	−11.1606	−0.598268
$349$	0.514513i	0.0275413i	0.999905	+	0.0137706i	$0.00438347\pi$
$349$	0.514513i	0.0275413i	−0.999905	+	0.0137706i	$0.995617\pi$
$350$	−1.96421	−0.104992
$351$	0	0
$352$	29.5989	1.57763
$353$	− 0.627678i	− 0.0334080i	−0.999860	−	0.0167040i	$-0.994683\pi$
$353$	− 0.627678i	− 0.0334080i	0.999860	−	0.0167040i	$-0.00531729\pi$
$354$	−36.1394	−1.92079
$355$	23.7605	1.26108
$356$	0.763711i	0.0404766i
$357$	− 14.9380i	− 0.790605i
$358$	− 33.7246i	− 1.78240i
$359$	31.9224i	1.68480i	0.538854	+	0.842399i	$0.318857\pi$
$359$	31.9224i	1.68480i	−0.538854	+	0.842399i	$0.681143\pi$
$360$	2.06034	0.108589
$361$	11.3598	0.597883
$362$	17.1271i	0.900179i
$363$	−4.88696	−0.256499
$364$	0	0
$365$	−9.16430	−0.479681
$366$	− 61.0946i	− 3.19347i
$367$	10.6064	0.553651	0.276826	−	0.960920i	$-0.410718\pi$
$367$	10.6064	0.553651	0.276826	+	0.960920i	$0.410718\pi$
$368$	−26.1865	−1.36507
$369$	5.20148i	0.270778i
$370$	36.7340i	1.90971i
$371$	13.3378i	0.692462i
$372$	33.7678i	1.75078i
$373$	10.6129	0.549514	0.274757	−	0.961514i	$-0.411403\pi$
$373$	10.6129	0.549514	0.274757	+	0.961514i	$0.411403\pi$
$374$	53.1673	2.74921
$375$	− 20.9323i	− 1.08094i
$376$	−1.51321	−0.0780377
$377$	0	0
$378$	−6.61608	−0.340295
$379$	13.5259i	0.694779i	0.937721	+	0.347390i	$0.112932\pi$
$379$	13.5259i	0.694779i	−0.937721	+	0.347390i	$0.887068\pi$
$380$	−15.3339	−0.786612
$381$	−12.3544	−0.632935
$382$	− 22.1951i	− 1.13560i
$383$	14.4264i	0.737155i	0.929597	+	0.368577i	$0.120155\pi$
$383$	14.4264i	0.737155i	−0.929597	+	0.368577i	$0.879845\pi$
$384$	− 9.52079i	− 0.485856i
$385$	− 8.89797i	− 0.453483i
$386$	1.57688	0.0802613
$387$	−5.13396	−0.260974
$388$	24.8858i	1.26338i
$389$	−15.1018	−0.765692	−0.382846	−	0.923812i	$-0.625056\pi$
$389$	−15.1018	−0.765692	−0.382846	+	0.923812i	$0.625056\pi$
$390$	0	0
$391$	−54.6793	−2.76525
$392$	− 0.566992i	− 0.0286374i
$393$	31.9015	1.60922
$394$	25.0422	1.26161
$395$	31.3890i	1.57935i
$396$	12.3586i	0.621041i
$397$	15.6944i	0.787679i	0.919179	+	0.393840i	$0.128853\pi$
$397$	15.6944i	0.787679i	−0.919179	+	0.393840i	$0.871147\pi$
$398$	− 8.82858i	− 0.442536i
$399$	−5.85682	−0.293208
$400$	3.20773	0.160386
$401$	18.9270i	0.945168i	0.881286	+	0.472584i	$0.156679\pi$
$401$	18.9270i	0.945168i	−0.881286	+	0.472584i	$0.843321\pi$
$402$	14.0572	0.701110
$403$	0	0
$404$	20.2924	1.00959
$405$	27.4416i	1.36358i
$406$	4.78816	0.237632
$407$	−26.5710	−1.31707
$408$	− 8.46974i	− 0.419315i
$409$	2.37052i	0.117215i	0.998281	+	0.0586074i	$0.0186660\pi$
$409$	2.37052i	0.117215i	−0.998281	+	0.0586074i	$0.981334\pi$
$410$	− 17.6083i	− 0.869614i
$411$	8.97794i	0.442849i
$412$	−6.07147	−0.299120
$413$	8.24978	0.405945
$414$	− 23.8874i	− 1.17400i
$415$	13.2319	0.649528
$416$	0	0
$417$	−5.73800	−0.280991
$418$	− 20.8455i	− 1.01959i
$419$	21.4796	1.04935	0.524673	−	0.851304i	$-0.324188\pi$
$419$	21.4796	1.04935	0.524673	+	0.851304i	$0.324188\pi$
$420$	−11.7546	−0.573566
$421$	20.1413i	0.981626i	0.871265	+	0.490813i	$0.163300\pi$
$421$	20.1413i	0.981626i	−0.871265	+	0.490813i	$0.836700\pi$
$422$	15.0271i	0.731506i
$423$	3.97577i	0.193309i
$424$	7.56239i	0.367262i
$425$	6.69797	0.324899
$426$	42.6711	2.06742
$427$	13.9465i	0.674917i
$428$	−33.9966	−1.64329
$429$	0	0
$430$	17.3798	0.838126
$431$	− 18.0221i	− 0.868092i	−0.900891	−	0.434046i	$-0.857085\pi$
$431$	− 18.0221i	− 0.868092i	0.900891	−	0.434046i	$-0.142915\pi$
$432$	10.8046	0.519838
$433$	−35.8293	−1.72184	−0.860922	−	0.508737i	$-0.830113\pi$
$433$	−35.8293	−1.72184	−0.860922	+	0.508737i	$0.830113\pi$
$434$	− 14.4872i	− 0.695410i
$435$	− 11.9704i	− 0.573937i
$436$	− 3.31327i	− 0.158677i
$437$	21.4383i	1.02553i
$438$	−16.4580	−0.786393
$439$	2.24154	0.106983	0.0534914	−	0.998568i	$-0.482965\pi$
$439$	2.24154	0.106983	0.0534914	+	0.998568i	$0.482965\pi$
$440$	− 5.04508i	− 0.240515i
$441$	−1.48970	−0.0709383
$442$	0	0
$443$	−5.70925	−0.271254	−0.135627	−	0.990760i	$-0.543305\pi$
$443$	−5.70925	−0.271254	−0.135627	+	0.990760i	$0.543305\pi$
$444$	35.1014i	1.66584i
$445$	−0.819128	−0.0388304
$446$	23.9065	1.13201
$447$	− 1.88833i	− 0.0893148i
$448$	10.0229i	0.473540i
$449$	32.3594i	1.52714i	0.645728	+	0.763568i	$0.276554\pi$
$449$	32.3594i	1.52714i	−0.645728	+	0.763568i	$0.723446\pi$
$450$	2.92609i	0.137937i
$451$	12.7367	0.599748
$452$	−33.7590	−1.58789
$453$	34.4441i	1.61833i
$454$	6.26362	0.293966
$455$	0	0
$456$	−3.32077	−0.155509
$457$	24.8614i	1.16297i	0.813559	+	0.581483i	$0.197527\pi$
$457$	24.8614i	1.16297i	−0.813559	+	0.581483i	$0.802473\pi$
$458$	7.36782	0.344276
$459$	22.5609	1.05305
$460$	43.0267i	2.00613i
$461$	17.5977i	0.819607i	0.912174	+	0.409804i	$0.134403\pi$
$461$	17.5977i	0.819607i	−0.912174	+	0.409804i	$0.865597\pi$
$462$	− 15.9797i	− 0.743443i
$463$	− 31.9863i	− 1.48653i	−0.668997	−	0.743265i	$-0.733276\pi$
$463$	− 31.9863i	− 1.48653i	0.668997	−	0.743265i	$-0.266724\pi$
$464$	−7.81948	−0.363010
$465$	−36.2181	−1.67957
$466$	− 26.4943i	− 1.22733i
$467$	−2.98855	−0.138294	−0.0691468	−	0.997606i	$-0.522028\pi$
$467$	−2.98855	−0.138294	−0.0691468	+	0.997606i	$0.522028\pi$
$468$	0	0
$469$	−3.20893	−0.148175
$470$	− 13.4590i	− 0.620817i
$471$	−8.36372	−0.385380
$472$	4.67756	0.215302
$473$	12.5714i	0.578032i
$474$	56.3709i	2.58920i
$475$	− 2.62610i	− 0.120494i
$476$	16.0333i	0.734885i
$477$	19.8693	0.909753
$478$	−52.4624	−2.39957
$479$	19.2718i	0.880553i	0.897862	+	0.440277i	$0.145120\pi$
$479$	19.2718i	0.880553i	−0.897862	+	0.440277i	$0.854880\pi$
$480$	41.9388	1.91424
$481$	0	0
$482$	14.0647	0.640632
$483$	16.4341i	0.747779i
$484$	5.24527	0.238422
$485$	−26.6916	−1.21200
$486$	29.4336i	1.33514i
$487$	18.2603i	0.827451i	0.910402	+	0.413726i	$0.135773\pi$
$487$	18.2603i	0.827451i	−0.910402	+	0.413726i	$0.864227\pi$
$488$	7.90753i	0.357957i
$489$	− 9.50194i	− 0.429693i
$490$	5.04303	0.227821
$491$	32.6604	1.47394	0.736971	−	0.675924i	$-0.236255\pi$
$491$	32.6604	1.47394	0.736971	+	0.675924i	$0.236255\pi$
$492$	− 16.8257i	− 0.758563i
$493$	−16.3276	−0.735360
$494$	0	0
$495$	−13.2553	−0.595784
$496$	23.6589i	1.06232i
$497$	−9.74080	−0.436935
$498$	23.7629	1.06484
$499$	− 5.35967i	− 0.239932i	−0.992778	−	0.119966i	$-0.961721\pi$
$499$	− 5.35967i	− 0.239932i	0.992778	−	0.119966i	$-0.0382785\pi$
$500$	22.4671i	1.00476i
$501$	17.0830i	0.763214i
$502$	− 47.2523i	− 2.10897i
$503$	−20.9055	−0.932129	−0.466065	−	0.884751i	$-0.654329\pi$
$503$	−20.9055	−0.932129	−0.466065	+	0.884751i	$0.654329\pi$
$504$	−0.844650	−0.0376237
$505$	21.7649i	0.968526i
$506$	−58.4922	−2.60030
$507$	0	0
$508$	13.2602	0.588328
$509$	− 25.0852i	− 1.11188i	−0.831222	−	0.555941i	$-0.812358\pi$
$509$	− 25.0852i	− 1.11188i	0.831222	−	0.555941i	$-0.187642\pi$
$510$	75.3329	3.33580
$511$	3.75697	0.166199
$512$	− 31.2245i	− 1.37994i
$513$	− 8.84553i	− 0.390540i
$514$	− 7.97905i	− 0.351941i
$515$	− 6.51204i	− 0.286955i
$516$	16.6073	0.731096
$517$	9.73535	0.428160
$518$	− 15.0594i	− 0.661671i
$519$	16.4086	0.720257
$520$	0	0
$521$	−8.13575	−0.356434	−0.178217	−	0.983991i	$-0.557033\pi$
$521$	−8.13575	−0.356434	−0.178217	+	0.983991i	$0.557033\pi$
$522$	− 7.13294i	− 0.312200i
$523$	36.6634	1.60318	0.801589	−	0.597875i	$-0.203988\pi$
$523$	36.6634	1.60318	0.801589	+	0.597875i	$0.203988\pi$
$524$	−34.2406	−1.49581
$525$	− 2.01311i	− 0.0878593i
$526$	33.5330i	1.46211i
$527$	49.4015i	2.15196i
$528$	26.0962i	1.13569i
$529$	37.1556	1.61546
$530$	−67.2626	−2.92170
$531$	− 12.2897i	− 0.533329i
$532$	6.28624	0.272543
$533$	0	0
$534$	−1.47106	−0.0636589
$535$	− 36.4636i	− 1.57646i
$536$	−1.81944	−0.0785877
$537$	34.5641	1.49155
$538$	47.7194i	2.05733i
$539$	3.64779i	0.157121i
$540$	− 17.7529i	− 0.763965i
$541$	16.6320i	0.715063i	0.933901	+	0.357532i	$0.116382\pi$
$541$	16.6320i	0.715063i	−0.933901	+	0.357532i	$0.883618\pi$
$542$	−22.2606	−0.956175
$543$	−17.5534	−0.753290
$544$	− 57.2046i	− 2.45263i
$545$	3.55369	0.152223
$546$	0	0
$547$	−36.0139	−1.53984	−0.769921	−	0.638139i	$-0.779704\pi$
$547$	−36.0139	−1.53984	−0.769921	+	0.638139i	$0.779704\pi$
$548$	− 9.63620i	− 0.411638i
$549$	20.7761	0.886702
$550$	7.16503	0.305518
$551$	6.40165i	0.272719i
$552$	9.31802i	0.396601i
$553$	− 12.8682i	− 0.547210i
$554$	− 24.3514i	− 1.03459i
$555$	−37.6485	−1.59809
$556$	6.15872	0.261188
$557$	− 11.2328i	− 0.475948i	−0.971272	−	0.237974i	$-0.923517\pi$
$557$	− 11.2328i	− 0.475948i	0.971272	−	0.237974i	$-0.0764833\pi$
$558$	−21.5817	−0.913626
$559$	0	0
$560$	−8.23570	−0.348022
$561$	54.4908i	2.30060i
$562$	−56.4557	−2.38144
$563$	−11.1199	−0.468649	−0.234324	−	0.972158i	$-0.575288\pi$
$563$	−11.1199	−0.468649	−0.234324	+	0.972158i	$0.575288\pi$
$564$	− 12.8608i	− 0.541538i
$565$	− 36.2087i	− 1.52331i
$566$	47.1883i	1.98347i
$567$	− 11.2499i	− 0.472451i
$568$	−5.52296	−0.231738
$569$	2.48545	0.104196	0.0520978	−	0.998642i	$-0.483409\pi$
$569$	2.48545	0.104196	0.0520978	+	0.998642i	$0.483409\pi$
$570$	− 29.5361i	− 1.23713i
$571$	−0.537621	−0.0224987	−0.0112494	−	0.999937i	$-0.503581\pi$
$571$	−0.537621	−0.0224987	−0.0112494	+	0.999937i	$0.503581\pi$
$572$	0	0
$573$	22.7476	0.950297
$574$	7.21867i	0.301301i
$575$	−7.36880	−0.307300
$576$	14.9312	0.622134
$577$	− 27.0563i	− 1.12637i	−0.826331	−	0.563184i	$-0.809576\pi$
$577$	− 27.0563i	− 1.12637i	0.826331	−	0.563184i	$-0.190424\pi$
$578$	− 67.6079i	− 2.81212i
$579$	1.61614i	0.0671644i
$580$	12.8481i	0.533487i
$581$	−5.42451	−0.225047
$582$	−47.9349	−1.98697
$583$	− 48.6533i	− 2.01501i
$584$	2.13017	0.0881472
$585$	0	0
$586$	−41.0152	−1.69432
$587$	− 20.7196i	− 0.855189i	−0.903971	−	0.427594i	$-0.859361\pi$
$587$	− 20.7196i	− 0.855189i	0.903971	−	0.427594i	$-0.140639\pi$
$588$	4.81889	0.198728
$589$	19.3691	0.798089
$590$	41.6039i	1.71280i
$591$	25.6656i	1.05574i
$592$	24.5933i	1.01078i
$593$	23.2747i	0.955776i	0.878421	+	0.477888i	$0.158598\pi$
$593$	23.2747i	0.955776i	−0.878421	+	0.477888i	$0.841402\pi$
$594$	24.1341	0.990233
$595$	−17.1967	−0.704997
$596$	2.02678i	0.0830202i
$597$	9.04835	0.370324
$598$	0	0
$599$	9.28114	0.379217	0.189609	−	0.981860i	$-0.439278\pi$
$599$	9.28114	0.379217	0.189609	+	0.981860i	$0.439278\pi$
$600$	− 1.14142i	− 0.0465981i
$601$	−23.6534	−0.964843	−0.482421	−	0.875939i	$-0.660243\pi$
$601$	−23.6534	−0.964843	−0.482421	+	0.875939i	$0.660243\pi$
$602$	−7.12496	−0.290392
$603$	4.78036i	0.194671i
$604$	− 36.9696i	− 1.50427i
$605$	5.62589i	0.228725i
$606$	39.0872i	1.58781i
$607$	−24.1600	−0.980623	−0.490312	−	0.871547i	$-0.663117\pi$
$607$	−24.1600	−0.980623	−0.490312	+	0.871547i	$0.663117\pi$
$608$	−22.4284	−0.909593
$609$	4.90736i	0.198856i
$610$	−70.3324	−2.84767
$611$	0	0
$612$	23.8849	0.965488
$613$	27.6780i	1.11790i	0.829200	+	0.558952i	$0.188796\pi$
$613$	27.6780i	1.11790i	−0.829200	+	0.558952i	$0.811204\pi$
$614$	21.3440	0.861374
$615$	18.0467	0.727712
$616$	2.06827i	0.0833328i
$617$	10.5982i	0.426669i	0.976979	+	0.213335i	$0.0684325\pi$
$617$	10.5982i	0.426669i	−0.976979	+	0.213335i	$0.931568\pi$
$618$	− 11.6948i	− 0.470436i
$619$	42.9657i	1.72694i	0.504403	+	0.863468i	$0.331712\pi$
$619$	42.9657i	1.72694i	−0.504403	+	0.863468i	$0.668288\pi$
$620$	38.8736	1.56120
$621$	−24.8204	−0.996009
$622$	46.0891i	1.84801i
$623$	0.335808	0.0134539
$624$	0	0
$625$	−28.8477	−1.15391
$626$	40.2322i	1.60800i
$627$	21.3645	0.853214
$628$	8.97695	0.358219
$629$	51.3525i	2.04756i
$630$	− 7.51262i	− 0.299310i
$631$	15.4963i	0.616899i	0.951241	+	0.308449i	$0.0998100\pi$
$631$	15.4963i	0.616899i	−0.951241	+	0.308449i	$0.900190\pi$
$632$	− 7.29614i	− 0.290225i
$633$	−15.4012	−0.612141
$634$	−56.7329	−2.25315
$635$	14.2224i	0.564400i
$636$	−64.2732	−2.54860
$637$	0	0
$638$	−17.4662	−0.691494
$639$	14.5109i	0.574043i
$640$	−10.9604	−0.433247
$641$	−22.2240	−0.877796	−0.438898	−	0.898537i	$-0.644631\pi$
$641$	−22.2240	−0.877796	−0.438898	+	0.898537i	$0.644631\pi$
$642$	− 65.4842i	− 2.58446i
$643$	26.6384i	1.05052i	0.850943	+	0.525259i	$0.176031\pi$
$643$	26.6384i	1.05052i	−0.850943	+	0.525259i	$0.823969\pi$
$644$	− 17.6391i	− 0.695078i
$645$	17.8124i	0.701363i
$646$	−40.2872	−1.58508
$647$	−15.7389	−0.618758	−0.309379	−	0.950939i	$-0.600121\pi$
$647$	−15.7389	−0.618758	−0.309379	+	0.950939i	$0.600121\pi$
$648$	− 6.37860i	− 0.250575i
$649$	−30.0935	−1.18127
$650$	0	0
$651$	14.8479	0.581934
$652$	10.1986i	0.399409i
$653$	−9.13364	−0.357427	−0.178714	−	0.983901i	$-0.557194\pi$
$653$	−9.13364	−0.357427	−0.178714	+	0.983901i	$0.557194\pi$
$654$	6.38201	0.249556
$655$	− 36.7252i	− 1.43497i
$656$	− 11.7887i	− 0.460272i
$657$	− 5.59678i	− 0.218351i
$658$	5.51762i	0.215099i
$659$	0.447782	0.0174431	0.00872156	−	0.999962i	$-0.497224\pi$
$659$	0.447782	0.0174431	0.00872156	+	0.999962i	$0.497224\pi$
$660$	42.8784	1.66904
$661$	46.9277i	1.82528i	0.408769	+	0.912638i	$0.365958\pi$
$661$	46.9277i	1.82528i	−0.408769	+	0.912638i	$0.634042\pi$
$662$	−42.1156	−1.63687
$663$	0	0
$664$	−3.07565	−0.119358
$665$	6.74240i	0.261459i
$666$	−22.4340	−0.869301
$667$	17.9629	0.695527
$668$	− 18.3356i	− 0.709424i
$669$	24.5016i	0.947287i
$670$	− 16.1827i	− 0.625193i
$671$	− 50.8738i	− 1.96396i
$672$	−17.1931	−0.663239
$673$	16.5829	0.639225	0.319612	−	0.947548i	$-0.396447\pi$
$673$	16.5829	0.639225	0.319612	+	0.947548i	$0.396447\pi$
$674$	− 5.52113i	− 0.212666i
$675$	3.04039	0.117025
$676$	0	0
$677$	16.1027	0.618878	0.309439	−	0.950919i	$-0.399859\pi$
$677$	16.1027	0.618878	0.309439	+	0.950919i	$0.399859\pi$
$678$	− 65.0265i	− 2.49733i
$679$	10.9424	0.419931
$680$	−9.75040	−0.373911
$681$	6.41954i	0.245997i
$682$	52.8464i	2.02359i
$683$	10.7304i	0.410589i	0.978700	+	0.205294i	$0.0658152\pi$
$683$	10.7304i	0.410589i	−0.978700	+	0.205294i	$0.934185\pi$
$684$	− 9.36464i	− 0.358066i
$685$	10.3354	0.394897
$686$	−2.06743	−0.0789347
$687$	7.55123i	0.288098i
$688$	11.6357	0.443606
$689$	0	0
$690$	−82.8778	−3.15510
$691$	24.1742i	0.919631i	0.888015	+	0.459815i	$0.152084\pi$
$691$	24.1742i	0.919631i	−0.888015	+	0.459815i	$0.847916\pi$
$692$	−17.6117	−0.669495
$693$	5.43413	0.206425
$694$	0.657068i	0.0249419i
$695$	6.60562i	0.250565i
$696$	2.78243i	0.105468i
$697$	− 24.6157i	− 0.932385i
$698$	1.06372	0.0402623
$699$	27.1539	1.02705
$700$	2.16071i	0.0816672i
$701$	14.4328	0.545120	0.272560	−	0.962139i	$-0.412130\pi$
$701$	14.4328	0.545120	0.272560	+	0.962139i	$0.412130\pi$
$702$	0	0
$703$	20.1340	0.759369
$704$	− 36.5616i	− 1.37797i
$705$	13.7940	0.519514
$706$	−1.29768	−0.0488388
$707$	− 8.92269i	− 0.335572i
$708$	39.7548i	1.49408i
$709$	− 15.1940i	− 0.570623i	−0.958435	−	0.285311i	$-0.907903\pi$
$709$	− 15.1940i	− 0.570623i	0.958435	−	0.285311i	$-0.0920970\pi$
$710$	− 49.1231i	− 1.84356i
$711$	−19.1697	−0.718921
$712$	0.190400	0.00713555
$713$	− 54.3493i	− 2.03540i
$714$	−30.8833	−1.15578
$715$	0	0
$716$	−37.0984	−1.38643
$717$	− 53.7683i	− 2.00802i
$718$	65.9971	2.46299
$719$	−4.99848	−0.186412	−0.0932060	−	0.995647i	$-0.529712\pi$
$719$	−4.99848	−0.186412	−0.0932060	+	0.995647i	$0.529712\pi$
$720$	12.2688i	0.457230i
$721$	2.66966i	0.0994233i
$722$	− 23.4855i	− 0.874040i
$723$	14.4149i	0.536095i
$724$	18.8405	0.700200
$725$	−2.20038	−0.0817199
$726$	10.1034i	0.374973i
$727$	7.22850	0.268090	0.134045	−	0.990975i	$-0.457203\pi$
$727$	7.22850	0.268090	0.134045	+	0.990975i	$0.457203\pi$
$728$	0	0
$729$	3.58334	0.132716
$730$	18.9465i	0.701242i
$731$	24.2961	0.898625
$732$	−67.2065	−2.48402
$733$	34.3382i	1.26831i	0.773206	+	0.634154i	$0.218652\pi$
$733$	34.3382i	1.26831i	−0.773206	+	0.634154i	$0.781348\pi$
$734$	− 21.9280i	− 0.809377i
$735$	5.16857i	0.190645i
$736$	62.9338i	2.31977i
$737$	11.7055	0.431178
$738$	10.7537	0.395848
$739$	40.3619i	1.48474i	0.669992	+	0.742369i	$0.266298\pi$
$739$	40.3619i	1.48474i	−0.669992	+	0.742369i	$0.733702\pi$
$740$	40.4089	1.48546
$741$	0	0
$742$	27.5748	1.01230
$743$	− 15.9115i	− 0.583736i	−0.956459	−	0.291868i	$-0.905723\pi$
$743$	− 15.9115i	− 0.583736i	0.956459	−	0.291868i	$-0.0942768\pi$
$744$	8.41863	0.308642
$745$	−2.17385	−0.0796437
$746$	− 21.9413i	− 0.803329i
$747$	8.08091i	0.295665i
$748$	− 58.4861i	− 2.13846i
$749$	14.9485i	0.546207i
$750$	−43.2760	−1.58022
$751$	−1.10100	−0.0401760	−0.0200880	−	0.999798i	$-0.506395\pi$
$751$	−1.10100	−0.0401760	−0.0200880	+	0.999798i	$0.506395\pi$
$752$	− 9.01075i	− 0.328588i
$753$	48.4286	1.76484
$754$	0	0
$755$	39.6522	1.44309
$756$	7.27795i	0.264697i
$757$	−28.9466	−1.05208	−0.526041	−	0.850459i	$-0.676324\pi$
$757$	−28.9466	−1.05208	−0.526041	+	0.850459i	$0.676324\pi$
$758$	27.9638	1.01569
$759$	− 59.9483i	− 2.17599i
$760$	3.82288i	0.138671i
$761$	12.6820i	0.459721i	0.973224	+	0.229860i	$0.0738270\pi$
$761$	12.6820i	0.459721i	−0.973224	+	0.229860i	$0.926173\pi$
$762$	25.5418i	0.925282i
$763$	−1.45686	−0.0527419
$764$	−24.4155	−0.883322
$765$	25.6180i	0.926222i
$766$	29.8255	1.07764
$767$	0	0
$768$	22.7916	0.822419
$769$	− 40.2042i	− 1.44980i	−0.688855	−	0.724899i	$-0.741886\pi$
$769$	− 40.2042i	− 1.44980i	0.688855	−	0.724899i	$-0.258114\pi$
$770$	−18.3959	−0.662942
$771$	8.17768	0.294512
$772$	− 1.73463i	− 0.0624309i
$773$	42.3905i	1.52468i	0.647177	+	0.762340i	$0.275949\pi$
$773$	42.3905i	1.52468i	−0.647177	+	0.762340i	$0.724051\pi$
$774$	10.6141i	0.381515i
$775$	6.65754i	0.239146i
$776$	6.20426	0.222720
$777$	15.4343	0.553701
$778$	31.2218i	1.11936i
$779$	−9.65117	−0.345789
$780$	0	0
$781$	35.5324	1.27145
$782$	113.045i	4.04249i
$783$	−7.41156	−0.264868
$784$	3.37629	0.120582
$785$	9.62835i	0.343650i
$786$	− 65.9541i	− 2.35250i
$787$	− 43.9833i	− 1.56784i	−0.620865	−	0.783918i	$-0.713218\pi$
$787$	− 43.9833i	− 1.56784i	0.620865	−	0.783918i	$-0.286782\pi$
$788$	− 27.5474i	− 0.981335i
$789$	−34.3678	−1.22353
$790$	64.8944	2.30884
$791$	14.8440i	0.527793i
$792$	3.08111	0.109482
$793$	0	0
$794$	32.4470	1.15150
$795$	− 68.9371i	− 2.44495i
$796$	−9.71178	−0.344225
$797$	−48.0486	−1.70197	−0.850984	−	0.525192i	$-0.823993\pi$
$797$	−48.0486	−1.70197	−0.850984	+	0.525192i	$0.823993\pi$
$798$	12.1085i	0.428638i
$799$	− 18.8151i	− 0.665630i
$800$	− 7.70911i	− 0.272558i
$801$	− 0.500254i	− 0.0176756i
$802$	39.1301	1.38173
$803$	−13.7046	−0.483627
$804$	− 15.4635i	− 0.545355i
$805$	18.9191	0.666809
$806$	0	0
$807$	−48.9073	−1.72162
$808$	− 5.05909i	− 0.177978i
$809$	0.460055	0.0161747	0.00808733	−	0.999967i	$-0.497426\pi$
$809$	0.460055	0.0161747	0.00808733	+	0.999967i	$0.497426\pi$
$810$	56.7335	1.99341
$811$	29.7828i	1.04582i	0.852389	+	0.522908i	$0.175153\pi$
$811$	29.7828i	1.04582i	−0.852389	+	0.522908i	$0.824847\pi$
$812$	− 5.26716i	− 0.184841i
$813$	− 22.8148i	− 0.800148i
$814$	54.9335i	1.92542i
$815$	−10.9387	−0.383165
$816$	50.4351	1.76558
$817$	− 9.52589i	− 0.333269i
$818$	4.90088	0.171355
$819$	0	0
$820$	−19.3699	−0.676425
$821$	− 3.59326i	− 0.125406i	−0.998032	−	0.0627028i	$-0.980028\pi$
$821$	− 3.59326i	− 0.125406i	0.998032	−	0.0627028i	$-0.0199720\pi$
$822$	18.5612	0.647397
$823$	−21.8141	−0.760391	−0.380196	−	0.924906i	$-0.624143\pi$
$823$	−21.8141	−0.760391	−0.380196	+	0.924906i	$0.624143\pi$
$824$	1.51367i	0.0527313i
$825$	7.34340i	0.255664i
$826$	− 17.0558i	− 0.593447i
$827$	42.7217i	1.48558i	0.669525	+	0.742790i	$0.266498\pi$
$827$	42.7217i	1.48558i	−0.669525	+	0.742790i	$0.733502\pi$
$828$	−26.2770	−0.913190
$829$	36.5078	1.26797	0.633985	−	0.773345i	$-0.281418\pi$
$829$	36.5078	1.26797	0.633985	+	0.773345i	$0.281418\pi$
$830$	− 27.3559i	− 0.949539i
$831$	24.9576	0.865770
$832$	0	0
$833$	7.04993	0.244265
$834$	11.8629i	0.410779i
$835$	19.6661	0.680572
$836$	−22.9309	−0.793082
$837$	22.4247i	0.775111i
$838$	− 44.4075i	− 1.53403i
$839$	− 6.70588i	− 0.231513i	−0.993278	−	0.115756i	$-0.963071\pi$
$839$	− 6.70588i	− 0.231513i	0.993278	−	0.115756i	$-0.0369292\pi$
$840$	2.93053i	0.101113i
$841$	−23.6361	−0.815039
$842$	41.6406	1.43503
$843$	− 57.8611i	− 1.99284i
$844$	16.5304	0.568999
$845$	0	0
$846$	8.21962	0.282596
$847$	− 2.30638i	− 0.0792480i
$848$	−45.0321	−1.54641
$849$	−48.3630	−1.65981
$850$	− 13.8475i	− 0.474967i
$851$	− 56.4957i	− 1.93665i
$852$	− 46.9399i	− 1.60813i
$853$	− 27.2554i	− 0.933207i	−0.884467	−	0.466603i	$-0.845478\pi$
$853$	− 27.2554i	− 0.933207i	0.884467	−	0.466603i	$-0.154522\pi$
$854$	28.8333	0.986654
$855$	10.0442	0.343504
$856$	8.47568i	0.289693i
$857$	−46.1474	−1.57637	−0.788183	−	0.615441i	$-0.788978\pi$
$857$	−46.1474	−1.57637	−0.788183	+	0.615441i	$0.788978\pi$
$858$	0	0
$859$	−9.45682	−0.322662	−0.161331	−	0.986900i	$-0.551579\pi$
$859$	−9.45682	−0.322662	−0.161331	+	0.986900i	$0.551579\pi$
$860$	− 19.1184i	− 0.651933i
$861$	−7.39837	−0.252136
$862$	−37.2593	−1.26906
$863$	28.8550i	0.982237i	0.871093	+	0.491119i	$0.163412\pi$
$863$	28.8550i	0.982237i	−0.871093	+	0.491119i	$0.836588\pi$
$864$	− 25.9667i	− 0.883406i
$865$	− 18.8896i	− 0.642267i
$866$	74.0743i	2.51715i
$867$	69.2909	2.35324
$868$	−15.9365	−0.540921
$869$	46.9403i	1.59234i
$870$	−24.7479	−0.839033
$871$	0	0
$872$	−0.826029	−0.0279729
$873$	− 16.3010i	− 0.551704i
$874$	44.3222	1.49922
$875$	9.87889	0.333967
$876$	18.1044i	0.611692i
$877$	8.55393i	0.288846i	0.989516	+	0.144423i	$0.0461326\pi$
$877$	8.55393i	0.288846i	−0.989516	+	0.144423i	$0.953867\pi$
$878$	− 4.63422i	− 0.156397i
$879$	− 42.0363i	− 1.41785i
$880$	30.0421	1.01272
$881$	−27.9911	−0.943046	−0.471523	−	0.881854i	$-0.656296\pi$
$881$	−27.9911	−0.943046	−0.471523	+	0.881854i	$0.656296\pi$
$882$	3.07985i	0.103704i
$883$	44.7564	1.50617	0.753086	−	0.657922i	$-0.228564\pi$
$883$	44.7564	1.50617	0.753086	+	0.657922i	$0.228564\pi$
$884$	0	0
$885$	−42.6395	−1.43331
$886$	11.8034i	0.396544i
$887$	−9.56744	−0.321243	−0.160622	−	0.987016i	$-0.551350\pi$
$887$	−9.56744	−0.321243	−0.160622	+	0.987016i	$0.551350\pi$
$888$	8.75110	0.293668
$889$	− 5.83060i	− 0.195552i
$890$	1.69349i	0.0567658i
$891$	41.0373i	1.37480i
$892$	− 26.2981i	− 0.880525i
$893$	−7.37691	−0.246859
$894$	−3.90398	−0.130569
$895$	− 39.7904i	− 1.33005i
$896$	4.49329	0.150110
$897$	0	0
$898$	66.9007	2.23251
$899$	− 16.2291i	− 0.541271i
$900$	3.21882	0.107294
$901$	−94.0302	−3.13260
$902$	− 26.3322i	− 0.876766i
$903$	− 7.30233i	− 0.243006i
$904$	8.41644i	0.279926i
$905$	20.2076i	0.671723i
$906$	71.2107	2.36582
$907$	18.8652	0.626408	0.313204	−	0.949686i	$-0.398598\pi$
$907$	18.8652	0.626408	0.313204	+	0.949686i	$0.398598\pi$
$908$	− 6.89023i	− 0.228660i
$909$	−13.2922	−0.440873
$910$	0	0
$911$	−0.358293	−0.0118708	−0.00593539	−	0.999982i	$-0.501889\pi$
$911$	−0.358293	−0.0118708	−0.00593539	+	0.999982i	$0.501889\pi$
$912$	− 19.7743i	− 0.654793i
$913$	19.7875	0.654870
$914$	51.3990	1.70013
$915$	− 72.0832i	− 2.38300i
$916$	− 8.10489i	− 0.267793i
$917$	15.0558i	0.497185i
$918$	− 46.6429i	− 1.53944i
$919$	−12.0902	−0.398818	−0.199409	−	0.979916i	$-0.563902\pi$
$919$	−12.0902	−0.398818	−0.199409	+	0.979916i	$0.563902\pi$
$920$	10.7269	0.353657
$921$	21.8753i	0.720817i
$922$	36.3820	1.19818
$923$	0	0
$924$	−17.5783	−0.578284
$925$	6.92047i	0.227544i
$926$	−66.1294	−2.17315
$927$	3.97700	0.130622
$928$	18.7925i	0.616894i
$929$	4.34400i	0.142522i	0.997458	+	0.0712610i	$0.0227023\pi$
$929$	4.34400i	0.142522i	−0.997458	+	0.0712610i	$0.977298\pi$
$930$	74.8783i	2.45535i
$931$	− 2.76410i	− 0.0905896i
$932$	−29.1448	−0.954670
$933$	−47.2365	−1.54645
$934$	6.17860i	0.202170i
$935$	62.7301	2.05149
$936$	0	0
$937$	44.7092	1.46059	0.730294	−	0.683133i	$-0.239383\pi$
$937$	44.7092	1.46059	0.730294	+	0.683133i	$0.239383\pi$
$938$	6.63423i	0.216615i
$939$	−41.2337	−1.34561
$940$	−14.8054	−0.482900
$941$	57.3375i	1.86915i	0.355768	+	0.934574i	$0.384219\pi$
$941$	57.3375i	1.86915i	−0.355768	+	0.934574i	$0.615781\pi$
$942$	17.2914i	0.563383i
$943$	27.0810i	0.881880i
$944$	27.8536i	0.906558i
$945$	−7.80607	−0.253931
$946$	25.9904	0.845020
$947$	− 46.9376i	− 1.52527i	−0.646830	−	0.762634i	$-0.723906\pi$
$947$	− 46.9376i	− 1.52527i	0.646830	−	0.762634i	$-0.276094\pi$
$948$	62.0102	2.01400
$949$	0	0
$950$	−5.42927	−0.176149
$951$	− 58.1452i	− 1.88549i
$952$	3.99725	0.129552
$953$	35.5773	1.15246	0.576231	−	0.817287i	$-0.304523\pi$
$953$	35.5773	1.15246	0.576231	+	0.817287i	$0.304523\pi$
$954$	− 41.0783i	− 1.32996i
$955$	− 26.1872i	− 0.847398i
$956$	57.7107i	1.86650i
$957$	− 17.9010i	− 0.578657i
$958$	39.8431	1.28727
$959$	−4.23709	−0.136823
$960$	− 51.8043i	− 1.67198i
$961$	−18.1034	−0.583980
$962$	0	0
$963$	22.2689	0.717604
$964$	− 15.4718i	− 0.498312i
$965$	1.86051	0.0598918
$966$	33.9764	1.09317
$967$	− 18.5576i	− 0.596772i	−0.954445	−	0.298386i	$-0.903552\pi$
$967$	− 18.5576i	− 0.596772i	0.954445	−	0.298386i	$-0.0964482\pi$
$968$	− 1.30770i	− 0.0420309i
$969$	− 41.2902i	− 1.32643i
$970$	55.1829i	1.77182i
$971$	47.2286	1.51564	0.757819	−	0.652465i	$-0.226265\pi$
$971$	47.2286	1.51564	0.757819	+	0.652465i	$0.226265\pi$
$972$	32.3782	1.03853
$973$	− 2.70802i	− 0.0868152i
$974$	37.7517	1.20964
$975$	0	0
$976$	−47.0873	−1.50723
$977$	− 47.9846i	− 1.53516i	−0.640951	−	0.767582i	$-0.721460\pi$
$977$	− 47.9846i	− 1.53516i	0.640951	−	0.767582i	$-0.278540\pi$
$978$	−19.6446	−0.628164
$979$	−1.22496	−0.0391498
$980$	− 5.54753i	− 0.177209i
$981$	2.17029i	0.0692921i
$982$	− 67.5229i	− 2.15474i
$983$	− 27.9789i	− 0.892387i	−0.894937	−	0.446193i	$-0.852779\pi$
$983$	− 27.9789i	− 0.892387i	0.894937	−	0.446193i	$-0.147221\pi$
$984$	−4.19481	−0.133726
$985$	29.5463	0.941425
$986$	33.7562i	1.07502i
$987$	−5.65497	−0.180000
$988$	0	0
$989$	−26.7295	−0.849948
$990$	27.4044i	0.870971i
$991$	−19.4925	−0.619199	−0.309600	−	0.950867i	$-0.600195\pi$
$991$	−19.4925	−0.619199	−0.309600	+	0.950867i	$0.600195\pi$
$992$	56.8593	1.80529
$993$	− 43.1640i	− 1.36977i
$994$	20.1384i	0.638751i
$995$	− 10.4165i	− 0.330225i
$996$	− 26.1401i	− 0.828281i
$997$	17.2637	0.546748	0.273374	−	0.961908i	$-0.411860\pi$
$997$	17.2637	0.546748	0.273374	+	0.961908i	$0.411860\pi$
$998$	−11.0807	−0.350754
$999$	23.3103i	0.737506i

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a_n

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a_n

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1183.2.c.j.337.6		24
13.5	odd	4		1183.2.a.q.1.4	✓	12
13.8	odd	4		1183.2.a.r.1.9	yes	12
13.12	even	2	inner	1183.2.c.j.337.19		24
91.34	even	4		8281.2.a.cq.1.9		12
91.83	even	4		8281.2.a.cn.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.a.q.1.4	✓	12	13.5	odd	4
1183.2.a.r.1.9	yes	12	13.8	odd	4
1183.2.c.j.337.6		24	1.1	even	1	trivial
1183.2.c.j.337.19		24	13.12	even	2	inner
8281.2.a.cn.1.4		12	91.83	even	4
8281.2.a.cq.1.9		12	91.34	even	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1183.2.c.j.337.6

Level	$1183$
Weight	$2$
Character	1183.337
Analytic conductor	$9.446$
Analytic rank	$0$
Dimension	$24$
Inner twists	$2$