LMFDB - Embedded newform 2151.4.a.h.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

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:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$0$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.40394 q^{2} +11.3946 q^{4} +1.03938 q^{5} +0.641944 q^{7} -14.9498 q^{8} +O(q^{10})$	\(q-4.40394 q^{2} +11.3946 q^{4} +1.03938 q^{5} +0.641944 q^{7} -14.9498 q^{8} -4.57738 q^{10} -48.9987 q^{11} -36.7063 q^{13} -2.82708 q^{14} -25.3192 q^{16} +37.8766 q^{17} -135.430 q^{19} +11.8434 q^{20} +215.787 q^{22} -107.109 q^{23} -123.920 q^{25} +161.652 q^{26} +7.31473 q^{28} -164.716 q^{29} -178.114 q^{31} +231.103 q^{32} -166.806 q^{34} +0.667226 q^{35} -312.490 q^{37} +596.427 q^{38} -15.5386 q^{40} -128.393 q^{41} -167.661 q^{43} -558.323 q^{44} +471.700 q^{46} -59.1103 q^{47} -342.588 q^{49} +545.734 q^{50} -418.255 q^{52} +261.864 q^{53} -50.9285 q^{55} -9.59694 q^{56} +725.399 q^{58} -716.904 q^{59} +900.978 q^{61} +784.401 q^{62} -815.207 q^{64} -38.1519 q^{65} -397.047 q^{67} +431.590 q^{68} -2.93842 q^{70} +360.948 q^{71} +145.439 q^{73} +1376.18 q^{74} -1543.18 q^{76} -31.4545 q^{77} -227.148 q^{79} -26.3163 q^{80} +565.437 q^{82} +994.304 q^{83} +39.3683 q^{85} +738.368 q^{86} +732.522 q^{88} +323.348 q^{89} -23.5634 q^{91} -1220.47 q^{92} +260.318 q^{94} -140.764 q^{95} -198.509 q^{97} +1508.74 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) $ 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8	$ 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8} - 36 q^{10} + 132 q^{11} + 104 q^{13} + 280 q^{14} + 822 q^{16} + 408 q^{17} + 20 q^{19} + 800 q^{20} - 2 q^{22} + 276 q^{23} + 1477 q^{25} + 780 q^{26} + 224 q^{28} + 696 q^{29} - 380 q^{31} + 896 q^{32} - 72 q^{34} + 700 q^{35} + 224 q^{37} + 988 q^{38} - 258 q^{40} + 2706 q^{41} - 156 q^{43} + 1584 q^{44} + 428 q^{46} + 1316 q^{47} + 2135 q^{49} + 1400 q^{50} + 1092 q^{52} + 1484 q^{53} - 992 q^{55} + 3360 q^{56} - 120 q^{58} + 3186 q^{59} - 254 q^{61} + 1240 q^{62} + 3054 q^{64} + 5120 q^{65} + 288 q^{67} + 9420 q^{68} + 1108 q^{70} + 4468 q^{71} - 1770 q^{73} + 6214 q^{74} + 720 q^{76} + 6352 q^{77} - 746 q^{79} + 7040 q^{80} + 276 q^{82} + 5484 q^{83} + 588 q^{85} + 10152 q^{86} + 1186 q^{88} + 11570 q^{89} + 1768 q^{91} + 15366 q^{92} - 2142 q^{94} + 5736 q^{95} + 2390 q^{97} + 6912 q^{98}+O(q^{100}) $ 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8 - 36 * q^10 + 132 * q^11 + 104 * q^13 + 280 * q^14 + 822 * q^16 + 408 * q^17 + 20 * q^19 + 800 * q^20 - 2 * q^22 + 276 * q^23 + 1477 * q^25 + 780 * q^26 + 224 * q^28 + 696 * q^29 - 380 * q^31 + 896 * q^32 - 72 * q^34 + 700 * q^35 + 224 * q^37 + 988 * q^38 - 258 * q^40 + 2706 * q^41 - 156 * q^43 + 1584 * q^44 + 428 * q^46 + 1316 * q^47 + 2135 * q^49 + 1400 * q^50 + 1092 * q^52 + 1484 * q^53 - 992 * q^55 + 3360 * q^56 - 120 * q^58 + 3186 * q^59 - 254 * q^61 + 1240 * q^62 + 3054 * q^64 + 5120 * q^65 + 288 * q^67 + 9420 * q^68 + 1108 * q^70 + 4468 * q^71 - 1770 * q^73 + 6214 * q^74 + 720 * q^76 + 6352 * q^77 - 746 * q^79 + 7040 * q^80 + 276 * q^82 + 5484 * q^83 + 588 * q^85 + 10152 * q^86 + 1186 * q^88 + 11570 * q^89 + 1768 * q^91 + 15366 * q^92 - 2142 * q^94 + 5736 * q^95 + 2390 * q^97 + 6912 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−4.40394	−1.55703	−0.778513	−	0.627628i	$-0.784026\pi$
$2$	−4.40394	−1.55703	−0.778513	+	0.627628i	$0.784026\pi$
$3$	0	0
$3$	0	0
$4$	11.3946	1.42433
$5$	1.03938	0.0929653	0.0464826	−	0.998919i	$-0.485199\pi$
$5$	1.03938	0.0929653	0.0464826	+	0.998919i	$0.485199\pi$
$6$	0	0
$7$	0.641944	0.0346617	0.0173309	−	0.999850i	$-0.494483\pi$
$7$	0.641944	0.0346617	0.0173309	+	0.999850i	$0.494483\pi$
$8$	−14.9498	−0.660694
$9$	0	0
$10$	−4.57738	−0.144749
$11$	−48.9987	−1.34306	−0.671531	−	0.740977i	$-0.734363\pi$
$11$	−48.9987	−1.34306	−0.671531	+	0.740977i	$0.734363\pi$
$12$	0	0
$13$	−36.7063	−0.783115	−0.391557	−	0.920154i	$-0.628064\pi$
$13$	−36.7063	−0.783115	−0.391557	+	0.920154i	$0.628064\pi$
$14$	−2.82708	−0.0539692
$15$	0	0
$16$	−25.3192	−0.395612
$17$	37.8766	0.540378	0.270189	−	0.962807i	$-0.412914\pi$
$17$	37.8766	0.540378	0.270189	+	0.962807i	$0.412914\pi$
$18$	0	0
$19$	−135.430	−1.63526	−0.817629	−	0.575746i	$-0.804712\pi$
$19$	−135.430	−1.63526	−0.817629	+	0.575746i	$0.804712\pi$
$20$	11.8434	0.132413
$21$	0	0
$22$	215.787	2.09118
$23$	−107.109	−0.971031	−0.485516	−	0.874228i	$-0.661368\pi$
$23$	−107.109	−0.971031	−0.485516	+	0.874228i	$0.661368\pi$
$24$	0	0
$25$	−123.920	−0.991357
$26$	161.652	1.21933
$27$	0	0
$28$	7.31473	0.0493698
$29$	−164.716	−1.05473	−0.527363	−	0.849640i	$-0.676819\pi$
$29$	−164.716	−1.05473	−0.527363	+	0.849640i	$0.676819\pi$
$30$	0	0
$31$	−178.114	−1.03194	−0.515970	−	0.856606i	$-0.672569\pi$
$31$	−178.114	−1.03194	−0.515970	+	0.856606i	$0.672569\pi$
$32$	231.103	1.27667
$33$	0	0
$34$	−166.806	−0.841382
$35$	0.667226	0.00322234
$36$	0	0
$37$	−312.490	−1.38846	−0.694229	−	0.719754i	$-0.744255\pi$
$37$	−312.490	−1.38846	−0.694229	+	0.719754i	$0.744255\pi$
$38$	596.427	2.54614
$39$	0	0
$40$	−15.5386	−0.0614216
$41$	−128.393	−0.489066	−0.244533	−	0.969641i	$-0.578635\pi$
$41$	−128.393	−0.489066	−0.244533	+	0.969641i	$0.578635\pi$
$42$	0	0
$43$	−167.661	−0.594606	−0.297303	−	0.954783i	$-0.596087\pi$
$43$	−167.661	−0.594606	−0.297303	+	0.954783i	$0.596087\pi$
$44$	−558.323	−1.91296
$45$	0	0
$46$	471.700	1.51192
$47$	−59.1103	−0.183449	−0.0917247	−	0.995784i	$-0.529238\pi$
$47$	−59.1103	−0.183449	−0.0917247	+	0.995784i	$0.529238\pi$
$48$	0	0
$49$	−342.588	−0.998799
$50$	545.734	1.54357
$51$	0	0
$52$	−418.255	−1.11541
$53$	261.864	0.678676	0.339338	−	0.940665i	$-0.389797\pi$
$53$	261.864	0.678676	0.339338	+	0.940665i	$0.389797\pi$
$54$	0	0
$55$	−50.9285	−0.124858
$56$	−9.59694	−0.0229008
$57$	0	0
$58$	725.399	1.64223
$59$	−716.904	−1.58191	−0.790957	−	0.611872i	$-0.790417\pi$
$59$	−716.904	−1.58191	−0.790957	+	0.611872i	$0.790417\pi$
$60$	0	0
$61$	900.978	1.89112	0.945561	−	0.325446i	$-0.105514\pi$
$61$	900.978	1.89112	0.945561	+	0.325446i	$0.105514\pi$
$62$	784.401	1.60676
$63$	0	0
$64$	−815.207	−1.59220
$65$	−38.1519	−0.0728024
$66$	0	0
$67$	−397.047	−0.723986	−0.361993	−	0.932181i	$-0.617904\pi$
$67$	−397.047	−0.723986	−0.361993	+	0.932181i	$0.617904\pi$
$68$	431.590	0.769677
$69$	0	0
$70$	−2.93842	−0.00501726
$71$	360.948	0.603333	0.301666	−	0.953414i	$-0.402457\pi$
$71$	360.948	0.603333	0.301666	+	0.953414i	$0.402457\pi$
$72$	0	0
$73$	145.439	0.233183	0.116592	−	0.993180i	$-0.462803\pi$
$73$	145.439	0.233183	0.116592	+	0.993180i	$0.462803\pi$
$74$	1376.18	2.16187
$75$	0	0
$76$	−1543.18	−2.32915
$77$	−31.4545	−0.0465529
$78$	0	0
$79$	−227.148	−0.323496	−0.161748	−	0.986832i	$-0.551713\pi$
$79$	−227.148	−0.323496	−0.161748	+	0.986832i	$0.551713\pi$
$80$	−26.3163	−0.0367782
$81$	0	0
$82$	565.437	0.761488
$83$	994.304	1.31493	0.657464	−	0.753486i	$-0.271629\pi$
$83$	994.304	1.31493	0.657464	+	0.753486i	$0.271629\pi$
$84$	0	0
$85$	39.3683	0.0502363
$86$	738.368	0.925817
$87$	0	0
$88$	732.522	0.887353
$89$	323.348	0.385111	0.192555	−	0.981286i	$-0.438323\pi$
$89$	323.348	0.385111	0.192555	+	0.981286i	$0.438323\pi$
$90$	0	0
$91$	−23.5634	−0.0271441
$92$	−1220.47	−1.38307
$93$	0	0
$94$	260.318	0.285636
$95$	−140.764	−0.152022
$96$	0	0
$97$	−198.509	−0.207789	−0.103895	−	0.994588i	$-0.533130\pi$
$97$	−198.509	−0.207789	−0.103895	+	0.994588i	$0.533130\pi$
$98$	1508.74	1.55516
$99$	0	0
$100$	−1412.02	−1.41202
$101$	1377.89	1.35748	0.678738	−	0.734380i	$-0.262527\pi$
$101$	1377.89	1.35748	0.678738	+	0.734380i	$0.262527\pi$
$102$	0	0
$103$	−134.668	−0.128828	−0.0644140	−	0.997923i	$-0.520518\pi$
$103$	−134.668	−0.128828	−0.0644140	+	0.997923i	$0.520518\pi$
$104$	548.752	0.517399
$105$	0	0
$106$	−1153.23	−1.05672
$107$	288.843	0.260967	0.130484	−	0.991450i	$-0.458347\pi$
$107$	288.843	0.260967	0.130484	+	0.991450i	$0.458347\pi$
$108$	0	0
$109$	−885.572	−0.778187	−0.389094	−	0.921198i	$-0.627212\pi$
$109$	−885.572	−0.778187	−0.389094	+	0.921198i	$0.627212\pi$
$110$	224.286	0.194407
$111$	0	0
$112$	−16.2535	−0.0137126
$113$	23.9012	0.0198976	0.00994882	−	0.999951i	$-0.496833\pi$
$113$	23.9012	0.0198976	0.00994882	+	0.999951i	$0.496833\pi$
$114$	0	0
$115$	−111.327	−0.0902722
$116$	−1876.88	−1.50228
$117$	0	0
$118$	3157.20	2.46308
$119$	24.3147	0.0187304
$120$	0	0
$121$	1069.88	0.803815
$122$	−3967.85	−2.94453
$123$	0	0
$124$	−2029.54	−1.46983
$125$	−258.723	−0.185127
$126$	0	0
$127$	857.709	0.599286	0.299643	−	0.954051i	$-0.403132\pi$
$127$	857.709	0.599286	0.299643	+	0.954051i	$0.403132\pi$
$128$	1741.30	1.20243
$129$	0	0
$130$	168.018	0.113355
$131$	−1472.96	−0.982392	−0.491196	−	0.871049i	$-0.663440\pi$
$131$	−1472.96	−0.982392	−0.491196	+	0.871049i	$0.663440\pi$
$132$	0	0
$133$	−86.9388	−0.0566809
$134$	1748.57	1.12727
$135$	0	0
$136$	−566.247	−0.357024
$137$	1390.34	0.867041	0.433520	−	0.901144i	$-0.357271\pi$
$137$	1390.34	0.867041	0.433520	+	0.901144i	$0.357271\pi$
$138$	0	0
$139$	−27.7276	−0.0169196	−0.00845981	−	0.999964i	$-0.502693\pi$
$139$	−27.7276	−0.0169196	−0.00845981	+	0.999964i	$0.502693\pi$
$140$	7.60281	0.00458967
$141$	0	0
$142$	−1589.59	−0.939405
$143$	1798.56	1.05177
$144$	0	0
$145$	−171.203	−0.0980528
$146$	−640.505	−0.363072
$147$	0	0
$148$	−3560.71	−1.97762
$149$	−995.083	−0.547116	−0.273558	−	0.961855i	$-0.588201\pi$
$149$	−995.083	−0.547116	−0.273558	+	0.961855i	$0.588201\pi$
$150$	0	0
$151$	−659.208	−0.355269	−0.177634	−	0.984097i	$-0.556844\pi$
$151$	−659.208	−0.355269	−0.177634	+	0.984097i	$0.556844\pi$
$152$	2024.66	1.08041
$153$	0	0
$154$	138.523	0.0724840
$155$	−185.128	−0.0959346
$156$	0	0
$157$	−741.441	−0.376901	−0.188450	−	0.982083i	$-0.560346\pi$
$157$	−741.441	−0.376901	−0.188450	+	0.982083i	$0.560346\pi$
$158$	1000.35	0.503692
$159$	0	0
$160$	240.204	0.118686
$161$	−68.7579	−0.0336576
$162$	0	0
$163$	−181.036	−0.0869926	−0.0434963	−	0.999054i	$-0.513850\pi$
$163$	−181.036	−0.0869926	−0.0434963	+	0.999054i	$0.513850\pi$
$164$	−1463.00	−0.696591
$165$	0	0
$166$	−4378.85	−2.04738
$167$	882.972	0.409140	0.204570	−	0.978852i	$-0.434420\pi$
$167$	882.972	0.409140	0.204570	+	0.978852i	$0.434420\pi$
$168$	0	0
$169$	−849.649	−0.386732
$170$	−173.375	−0.0782193
$171$	0	0
$172$	−1910.44	−0.846916
$173$	589.731	0.259170	0.129585	−	0.991568i	$-0.458635\pi$
$173$	589.731	0.259170	0.129585	+	0.991568i	$0.458635\pi$
$174$	0	0
$175$	−79.5495	−0.0343622
$176$	1240.61	0.531332
$177$	0	0
$178$	−1424.00	−0.599627
$179$	−2060.21	−0.860263	−0.430132	−	0.902766i	$-0.641533\pi$
$179$	−2060.21	−0.860263	−0.430132	+	0.902766i	$0.641533\pi$
$180$	0	0
$181$	−1514.84	−0.622083	−0.311041	−	0.950396i	$-0.600678\pi$
$181$	−1514.84	−0.622083	−0.311041	+	0.950396i	$0.600678\pi$
$182$	103.772	0.0422641
$183$	0	0
$184$	1601.26	0.641555
$185$	−324.796	−0.129078
$186$	0	0
$187$	−1855.90	−0.725760
$188$	−673.541	−0.261293
$189$	0	0
$190$	619.916	0.236702
$191$	1934.29	0.732777	0.366388	−	0.930462i	$-0.380594\pi$
$191$	1934.29	0.732777	0.366388	+	0.930462i	$0.380594\pi$
$192$	0	0
$193$	4952.41	1.84706	0.923530	−	0.383526i	$-0.125290\pi$
$193$	4952.41	1.84706	0.923530	+	0.383526i	$0.125290\pi$
$194$	874.222	0.323534
$195$	0	0
$196$	−3903.67	−1.42262
$197$	−1761.04	−0.636899	−0.318449	−	0.947940i	$-0.603162\pi$
$197$	−1761.04	−0.636899	−0.318449	+	0.947940i	$0.603162\pi$
$198$	0	0
$199$	−3690.29	−1.31456	−0.657281	−	0.753645i	$-0.728294\pi$
$199$	−3690.29	−1.31456	−0.657281	+	0.753645i	$0.728294\pi$
$200$	1852.58	0.654984
$201$	0	0
$202$	−6068.14	−2.11363
$203$	−105.739	−0.0365586
$204$	0	0
$205$	−133.450	−0.0454661
$206$	593.071	0.200588
$207$	0	0
$208$	929.373	0.309810
$209$	6635.92	2.19625
$210$	0	0
$211$	−5426.38	−1.77046	−0.885231	−	0.465151i	$-0.846000\pi$
$211$	−5426.38	−1.77046	−0.885231	+	0.465151i	$0.846000\pi$
$212$	2983.85	0.966659
$213$	0	0
$214$	−1272.04	−0.406333
$215$	−174.264	−0.0552777
$216$	0	0
$217$	−114.339	−0.0357689
$218$	3900.00	1.21166
$219$	0	0
$220$	−580.312	−0.177839
$221$	−1390.31	−0.423178
$222$	0	0
$223$	2368.65	0.711286	0.355643	−	0.934622i	$-0.384262\pi$
$223$	2368.65	0.711286	0.355643	+	0.934622i	$0.384262\pi$
$224$	148.355	0.0442517
$225$	0	0
$226$	−105.259	−0.0309812
$227$	4653.98	1.36077	0.680387	−	0.732853i	$-0.261812\pi$
$227$	4653.98	1.36077	0.680387	+	0.732853i	$0.261812\pi$
$228$	0	0
$229$	−2760.19	−0.796500	−0.398250	−	0.917277i	$-0.630382\pi$
$229$	−2760.19	−0.796500	−0.398250	+	0.917277i	$0.630382\pi$
$230$	490.277	0.140556
$231$	0	0
$232$	2462.48	0.696851
$233$	−4125.32	−1.15991	−0.579955	−	0.814649i	$-0.696930\pi$
$233$	−4125.32	−1.15991	−0.579955	+	0.814649i	$0.696930\pi$
$234$	0	0
$235$	−61.4382	−0.0170544
$236$	−8168.87	−2.25317
$237$	0	0
$238$	−107.080	−0.0291638
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−3229.52	−0.863202	−0.431601	−	0.902065i	$-0.642051\pi$
$241$	−3229.52	−0.863202	−0.431601	+	0.902065i	$0.642051\pi$
$242$	−4711.67	−1.25156
$243$	0	0
$244$	10266.3	2.69358
$245$	−356.080	−0.0928536
$246$	0	0
$247$	4971.15	1.28059
$248$	2662.77	0.681797
$249$	0	0
$250$	1139.40	0.288248
$251$	1164.25	0.292777	0.146389	−	0.989227i	$-0.453235\pi$
$251$	1164.25	0.292777	0.146389	+	0.989227i	$0.453235\pi$
$252$	0	0
$253$	5248.20	1.30416
$254$	−3777.29	−0.933105
$255$	0	0
$256$	−1146.91	−0.280008
$257$	−7690.46	−1.86661	−0.933303	−	0.359090i	$-0.883087\pi$
$257$	−7690.46	−1.86661	−0.933303	+	0.359090i	$0.883087\pi$
$258$	0	0
$259$	−200.601	−0.0481264
$260$	−434.727	−0.103695
$261$	0	0
$262$	6486.83	1.52961
$263$	−3236.93	−0.758927	−0.379463	−	0.925207i	$-0.623891\pi$
$263$	−3236.93	−0.758927	−0.379463	+	0.925207i	$0.623891\pi$
$264$	0	0
$265$	272.177	0.0630933
$266$	382.873	0.0882536
$267$	0	0
$268$	−4524.22	−1.03120
$269$	−709.861	−0.160896	−0.0804479	−	0.996759i	$-0.525635\pi$
$269$	−709.861	−0.160896	−0.0804479	+	0.996759i	$0.525635\pi$
$270$	0	0
$271$	−6243.22	−1.39944	−0.699721	−	0.714416i	$-0.746692\pi$
$271$	−6243.22	−1.39944	−0.699721	+	0.714416i	$0.746692\pi$
$272$	−959.005	−0.213780
$273$	0	0
$274$	−6122.96	−1.35001
$275$	6071.91	1.33145
$276$	0	0
$277$	5525.13	1.19846	0.599229	−	0.800578i	$-0.295474\pi$
$277$	5525.13	1.19846	0.599229	+	0.800578i	$0.295474\pi$
$278$	122.111	0.0263443
$279$	0	0
$280$	−9.97490	−0.00212898
$281$	−148.471	−0.0315197	−0.0157598	−	0.999876i	$-0.505017\pi$
$281$	−148.471	−0.0315197	−0.0157598	+	0.999876i	$0.505017\pi$
$282$	0	0
$283$	−2136.10	−0.448685	−0.224342	−	0.974510i	$-0.572023\pi$
$283$	−2136.10	−0.448685	−0.224342	+	0.974510i	$0.572023\pi$
$284$	4112.87	0.859346
$285$	0	0
$286$	−7920.75	−1.63764
$287$	−82.4215	−0.0169519
$288$	0	0
$289$	−3478.36	−0.707992
$290$	753.968	0.152671
$291$	0	0
$292$	1657.23	0.332130
$293$	5523.29	1.10128	0.550639	−	0.834744i	$-0.314384\pi$
$293$	5523.29	1.10128	0.550639	+	0.834744i	$0.314384\pi$
$294$	0	0
$295$	−745.138	−0.147063
$296$	4671.66	0.917347
$297$	0	0
$298$	4382.28	0.851875
$299$	3931.56	0.760429
$300$	0	0
$301$	−107.629	−0.0206101
$302$	2903.11	0.553163
$303$	0	0
$304$	3428.99	0.646928
$305$	936.461	0.175809
$306$	0	0
$307$	−1415.87	−0.263218	−0.131609	−	0.991302i	$-0.542014\pi$
$307$	−1415.87	−0.263218	−0.131609	+	0.991302i	$0.542014\pi$
$308$	−358.413	−0.0663067
$309$	0	0
$310$	815.293	0.149373
$311$	−6245.37	−1.13872	−0.569361	−	0.822088i	$-0.692809\pi$
$311$	−6245.37	−1.13872	−0.569361	+	0.822088i	$0.692809\pi$
$312$	0	0
$313$	−3634.14	−0.656273	−0.328137	−	0.944630i	$-0.606421\pi$
$313$	−3634.14	−0.656273	−0.328137	+	0.944630i	$0.606421\pi$
$314$	3265.26	0.586844
$315$	0	0
$316$	−2588.27	−0.460765
$317$	1405.97	0.249107	0.124553	−	0.992213i	$-0.460250\pi$
$317$	1405.97	0.249107	0.124553	+	0.992213i	$0.460250\pi$
$318$	0	0
$319$	8070.89	1.41656
$320$	−847.313	−0.148019
$321$	0	0
$322$	302.805	0.0524058
$323$	−5129.64	−0.883656
$324$	0	0
$325$	4548.63	0.776346
$326$	797.269	0.135450
$327$	0	0
$328$	1919.46	0.323123
$329$	−37.9455	−0.00635867
$330$	0	0
$331$	9057.14	1.50400	0.752002	−	0.659161i	$-0.229088\pi$
$331$	9057.14	1.50400	0.752002	+	0.659161i	$0.229088\pi$
$332$	11329.7	1.87289
$333$	0	0
$334$	−3888.55	−0.637042
$335$	−412.684	−0.0673056
$336$	0	0
$337$	1855.81	0.299977	0.149988	−	0.988688i	$-0.452076\pi$
$337$	1855.81	0.299977	0.149988	+	0.988688i	$0.452076\pi$
$338$	3741.80	0.602151
$339$	0	0
$340$	448.588	0.0715532
$341$	8727.35	1.38596
$342$	0	0
$343$	−440.109	−0.0692818
$344$	2506.50	0.392853
$345$	0	0
$346$	−2597.14	−0.403535
$347$	−11986.1	−1.85432	−0.927161	−	0.374663i	$-0.877759\pi$
$347$	−11986.1	−1.85432	−0.927161	+	0.374663i	$0.877759\pi$
$348$	0	0
$349$	6042.16	0.926732	0.463366	−	0.886167i	$-0.346641\pi$
$349$	6042.16	0.926732	0.463366	+	0.886167i	$0.346641\pi$
$350$	350.331	0.0535028
$351$	0	0
$352$	−11323.7	−1.71465
$353$	−5817.21	−0.877107	−0.438554	−	0.898705i	$-0.644509\pi$
$353$	−5817.21	−0.877107	−0.438554	+	0.898705i	$0.644509\pi$
$354$	0	0
$355$	375.163	0.0560890
$356$	3684.44	0.548525
$357$	0	0
$358$	9073.02	1.33945
$359$	3414.05	0.501912	0.250956	−	0.967998i	$-0.419255\pi$
$359$	3414.05	0.501912	0.250956	+	0.967998i	$0.419255\pi$
$360$	0	0
$361$	11482.4	1.67407
$362$	6671.25	0.968599
$363$	0	0
$364$	−268.496	−0.0386622
$365$	151.167	0.0216779
$366$	0	0
$367$	6466.51	0.919753	0.459876	−	0.887983i	$-0.347894\pi$
$367$	6466.51	0.919753	0.459876	+	0.887983i	$0.347894\pi$
$368$	2711.91	0.384152
$369$	0	0
$370$	1430.38	0.200978
$371$	168.102	0.0235241
$372$	0	0
$373$	6467.86	0.897836	0.448918	−	0.893573i	$-0.351810\pi$
$373$	6467.86	0.897836	0.448918	+	0.893573i	$0.351810\pi$
$374$	8173.29	1.13003
$375$	0	0
$376$	883.687	0.121204
$377$	6046.12	0.825971
$378$	0	0
$379$	−4816.72	−0.652818	−0.326409	−	0.945229i	$-0.605839\pi$
$379$	−4816.72	−0.652818	−0.326409	+	0.945229i	$0.605839\pi$
$380$	−1603.96	−0.216530
$381$	0	0
$382$	−8518.49	−1.14095
$383$	−2548.11	−0.339954	−0.169977	−	0.985448i	$-0.554369\pi$
$383$	−2548.11	−0.339954	−0.169977	+	0.985448i	$0.554369\pi$
$384$	0	0
$385$	−32.6932	−0.00432780
$386$	−21810.1	−2.87592
$387$	0	0
$388$	−2261.94	−0.295961
$389$	526.077	0.0685685	0.0342843	−	0.999412i	$-0.489085\pi$
$389$	526.077	0.0685685	0.0342843	+	0.999412i	$0.489085\pi$
$390$	0	0
$391$	−4056.91	−0.524724
$392$	5121.62	0.659901
$393$	0	0
$394$	7755.52	0.991668
$395$	−236.094	−0.0300739
$396$	0	0
$397$	385.566	0.0487431	0.0243715	−	0.999703i	$-0.492242\pi$
$397$	385.566	0.0487431	0.0243715	+	0.999703i	$0.492242\pi$
$398$	16251.8	2.04681
$399$	0	0
$400$	3137.55	0.392193
$401$	−3502.48	−0.436173	−0.218087	−	0.975929i	$-0.569982\pi$
$401$	−3502.48	−0.436173	−0.218087	+	0.975929i	$0.569982\pi$
$402$	0	0
$403$	6537.89	0.808128
$404$	15700.6	1.93350
$405$	0	0
$406$	465.666	0.0569227
$407$	15311.6	1.86479
$408$	0	0
$409$	14696.0	1.77670	0.888352	−	0.459163i	$-0.151851\pi$
$409$	14696.0	1.77670	0.888352	+	0.459163i	$0.151851\pi$
$410$	587.705	0.0707919
$411$	0	0
$412$	−1534.50	−0.183494
$413$	−460.212	−0.0548319
$414$	0	0
$415$	1033.46	0.122243
$416$	−8482.91	−0.999781
$417$	0	0
$418$	−29224.2	−3.41962
$419$	−4073.53	−0.474952	−0.237476	−	0.971393i	$-0.576320\pi$
$419$	−4073.53	−0.474952	−0.237476	+	0.971393i	$0.576320\pi$
$420$	0	0
$421$	16058.5	1.85901	0.929503	−	0.368814i	$-0.120236\pi$
$421$	16058.5	1.85901	0.929503	+	0.368814i	$0.120236\pi$
$422$	23897.4	2.75666
$423$	0	0
$424$	−3914.82	−0.448397
$425$	−4693.65	−0.535707
$426$	0	0
$427$	578.378	0.0655496
$428$	3291.26	0.371703
$429$	0	0
$430$	767.447	0.0860688
$431$	92.4432	0.0103314	0.00516570	−	0.999987i	$-0.498356\pi$
$431$	92.4432	0.0103314	0.00516570	+	0.999987i	$0.498356\pi$
$432$	0	0
$433$	−3712.65	−0.412053	−0.206026	−	0.978546i	$-0.566053\pi$
$433$	−3712.65	−0.412053	−0.206026	+	0.978546i	$0.566053\pi$
$434$	503.542	0.0556931
$435$	0	0
$436$	−10090.8	−1.10840
$437$	14505.8	1.58789
$438$	0	0
$439$	−12513.8	−1.36048	−0.680241	−	0.732989i	$-0.738125\pi$
$439$	−12513.8	−1.36048	−0.680241	+	0.732989i	$0.738125\pi$
$440$	761.371	0.0824930
$441$	0	0
$442$	6122.83	0.658899
$443$	−9236.58	−0.990617	−0.495308	−	0.868717i	$-0.664945\pi$
$443$	−9236.58	−0.990617	−0.495308	+	0.868717i	$0.664945\pi$
$444$	0	0
$445$	336.083	0.0358019
$446$	−10431.4	−1.10749
$447$	0	0
$448$	−523.318	−0.0551885
$449$	−1911.77	−0.200939	−0.100470	−	0.994940i	$-0.532035\pi$
$449$	−1911.77	−0.200939	−0.100470	+	0.994940i	$0.532035\pi$
$450$	0	0
$451$	6291.12	0.656845
$452$	272.345	0.0283408
$453$	0	0
$454$	−20495.8	−2.11876
$455$	−24.4914	−0.00252346
$456$	0	0
$457$	11969.9	1.22523	0.612615	−	0.790381i	$-0.290117\pi$
$457$	11969.9	1.22523	0.612615	+	0.790381i	$0.290117\pi$
$458$	12155.7	1.24017
$459$	0	0
$460$	−1268.53	−0.128577
$461$	1178.25	0.119038	0.0595190	−	0.998227i	$-0.481043\pi$
$461$	1178.25	0.119038	0.0595190	+	0.998227i	$0.481043\pi$
$462$	0	0
$463$	−13223.2	−1.32729	−0.663643	−	0.748050i	$-0.730990\pi$
$463$	−13223.2	−1.32729	−0.663643	+	0.748050i	$0.730990\pi$
$464$	4170.48	0.417262
$465$	0	0
$466$	18167.7	1.80601
$467$	3290.43	0.326045	0.163022	−	0.986622i	$-0.447876\pi$
$467$	3290.43	0.326045	0.163022	+	0.986622i	$0.447876\pi$
$468$	0	0
$469$	−254.882	−0.0250946
$470$	270.570	0.0265542
$471$	0	0
$472$	10717.6	1.04516
$473$	8215.18	0.798592
$474$	0	0
$475$	16782.5	1.62112
$476$	277.057	0.0266783
$477$	0	0
$478$	1052.54	0.100716
$479$	11053.9	1.05441	0.527207	−	0.849737i	$-0.323239\pi$
$479$	11053.9	1.05441	0.527207	+	0.849737i	$0.323239\pi$
$480$	0	0
$481$	11470.3	1.08732
$482$	14222.6	1.34403
$483$	0	0
$484$	12190.9	1.14490
$485$	−206.327	−0.0193172
$486$	0	0
$487$	16349.3	1.52127	0.760635	−	0.649180i	$-0.224888\pi$
$487$	16349.3	1.52127	0.760635	+	0.649180i	$0.224888\pi$
$488$	−13469.4	−1.24945
$489$	0	0
$490$	1568.15	0.144575
$491$	2008.83	0.184638	0.0923189	−	0.995729i	$-0.470572\pi$
$491$	2008.83	0.184638	0.0923189	+	0.995729i	$0.470572\pi$
$492$	0	0
$493$	−6238.89	−0.569950
$494$	−21892.6	−1.99392
$495$	0	0
$496$	4509.70	0.408249
$497$	231.708	0.0209126
$498$	0	0
$499$	15817.0	1.41897	0.709486	−	0.704719i	$-0.248927\pi$
$499$	15817.0	1.41897	0.709486	+	0.704719i	$0.248927\pi$
$500$	−2948.06	−0.263682
$501$	0	0
$502$	−5127.30	−0.455862
$503$	−8353.62	−0.740496	−0.370248	−	0.928933i	$-0.620727\pi$
$503$	−8353.62	−0.740496	−0.370248	+	0.928933i	$0.620727\pi$
$504$	0	0
$505$	1432.15	0.126198
$506$	−23112.7	−2.03060
$507$	0	0
$508$	9773.29	0.853582
$509$	21578.9	1.87911	0.939557	−	0.342394i	$-0.111238\pi$
$509$	21578.9	1.87911	0.939557	+	0.342394i	$0.111238\pi$
$510$	0	0
$511$	93.3639	0.00808253
$512$	−8879.47	−0.766447
$513$	0	0
$514$	33868.3	2.90635
$515$	−139.972	−0.0119765
$516$	0	0
$517$	2896.33	0.246384
$518$	883.433	0.0749340
$519$	0	0
$520$	570.363	0.0481002
$521$	8204.63	0.689926	0.344963	−	0.938616i	$-0.387891\pi$
$521$	8204.63	0.689926	0.344963	+	0.938616i	$0.387891\pi$
$522$	0	0
$523$	−2955.61	−0.247112	−0.123556	−	0.992338i	$-0.539430\pi$
$523$	−2955.61	−0.247112	−0.123556	+	0.992338i	$0.539430\pi$
$524$	−16783.9	−1.39925
$525$	0	0
$526$	14255.2	1.18167
$527$	−6746.34	−0.557638
$528$	0	0
$529$	−694.712	−0.0570980
$530$	−1198.65	−0.0982379
$531$	0	0
$532$	−990.637	−0.0807323
$533$	4712.85	0.382994
$534$	0	0
$535$	300.218	0.0242609
$536$	5935.78	0.478334
$537$	0	0
$538$	3126.18	0.250519
$539$	16786.4	1.34145
$540$	0	0
$541$	6432.08	0.511158	0.255579	−	0.966788i	$-0.417734\pi$
$541$	6432.08	0.511158	0.255579	+	0.966788i	$0.417734\pi$
$542$	27494.8	2.17897
$543$	0	0
$544$	8753.37	0.689886
$545$	−920.449	−0.0723444
$546$	0	0
$547$	−16502.9	−1.28997	−0.644983	−	0.764197i	$-0.723135\pi$
$547$	−16502.9	−1.28997	−0.644983	+	0.764197i	$0.723135\pi$
$548$	15842.4	1.23495
$549$	0	0
$550$	−26740.3	−2.07311
$551$	22307.6	1.72475
$552$	0	0
$553$	−145.817	−0.0112129
$554$	−24332.3	−1.86603
$555$	0	0
$556$	−315.947	−0.0240991
$557$	−1129.33	−0.0859092	−0.0429546	−	0.999077i	$-0.513677\pi$
$557$	−1129.33	−0.0859092	−0.0429546	+	0.999077i	$0.513677\pi$
$558$	0	0
$559$	6154.21	0.465645
$560$	−16.8936	−0.00127480
$561$	0	0
$562$	653.856	0.0490770
$563$	−2308.16	−0.172784	−0.0863920	−	0.996261i	$-0.527534\pi$
$563$	−2308.16	−0.172784	−0.0863920	+	0.996261i	$0.527534\pi$
$564$	0	0
$565$	24.8425	0.00184979
$566$	9407.24	0.698614
$567$	0	0
$568$	−5396.10	−0.398619
$569$	−7530.78	−0.554845	−0.277423	−	0.960748i	$-0.589480\pi$
$569$	−7530.78	−0.554845	−0.277423	+	0.960748i	$0.589480\pi$
$570$	0	0
$571$	−873.313	−0.0640053	−0.0320026	−	0.999488i	$-0.510188\pi$
$571$	−873.313	−0.0640053	−0.0320026	+	0.999488i	$0.510188\pi$
$572$	20494.0	1.49807
$573$	0	0
$574$	362.979	0.0263945
$575$	13272.9	0.962639
$576$	0	0
$577$	−6904.84	−0.498184	−0.249092	−	0.968480i	$-0.580132\pi$
$577$	−6904.84	−0.498184	−0.249092	+	0.968480i	$0.580132\pi$
$578$	15318.5	1.10236
$579$	0	0
$580$	−1950.80	−0.139660
$581$	638.288	0.0455777
$582$	0	0
$583$	−12831.0	−0.911503
$584$	−2174.29	−0.154063
$585$	0	0
$586$	−24324.2	−1.71472
$587$	16415.9	1.15427	0.577134	−	0.816649i	$-0.304171\pi$
$587$	16415.9	1.15427	0.577134	+	0.816649i	$0.304171\pi$
$588$	0	0
$589$	24122.0	1.68749
$590$	3281.54	0.228981
$591$	0	0
$592$	7911.98	0.549291
$593$	15126.7	1.04752	0.523761	−	0.851865i	$-0.324528\pi$
$593$	15126.7	1.04752	0.523761	+	0.851865i	$0.324528\pi$
$594$	0	0
$595$	25.2722	0.00174128
$596$	−11338.6	−0.779275
$597$	0	0
$598$	−17314.4	−1.18401
$599$	−2800.89	−0.191054	−0.0955270	−	0.995427i	$-0.530454\pi$
$599$	−2800.89	−0.191054	−0.0955270	+	0.995427i	$0.530454\pi$
$600$	0	0
$601$	751.290	0.0509913	0.0254956	−	0.999675i	$-0.491884\pi$
$601$	751.290	0.0509913	0.0254956	+	0.999675i	$0.491884\pi$
$602$	473.991	0.0320904
$603$	0	0
$604$	−7511.44	−0.506020
$605$	1112.01	0.0747268
$606$	0	0
$607$	−13057.4	−0.873118	−0.436559	−	0.899676i	$-0.643803\pi$
$607$	−13057.4	−0.873118	−0.436559	+	0.899676i	$0.643803\pi$
$608$	−31298.3	−2.08769
$609$	0	0
$610$	−4124.12	−0.273739
$611$	2169.72	0.143662
$612$	0	0
$613$	−10446.0	−0.688270	−0.344135	−	0.938920i	$-0.611828\pi$
$613$	−10446.0	−0.688270	−0.344135	+	0.938920i	$0.611828\pi$
$614$	6235.39	0.409837
$615$	0	0
$616$	470.238	0.0307572
$617$	−23077.9	−1.50580	−0.752902	−	0.658133i	$-0.771346\pi$
$617$	−23077.9	−1.50580	−0.752902	+	0.658133i	$0.771346\pi$
$618$	0	0
$619$	−15208.7	−0.987541	−0.493771	−	0.869592i	$-0.664382\pi$
$619$	−15208.7	−0.987541	−0.493771	+	0.869592i	$0.664382\pi$
$620$	−2109.47	−0.136643
$621$	0	0
$622$	27504.2	1.77302
$623$	207.572	0.0133486
$624$	0	0
$625$	15221.0	0.974147
$626$	16004.5	1.02184
$627$	0	0
$628$	−8448.46	−0.536831
$629$	−11836.0	−0.750292
$630$	0	0
$631$	12789.5	0.806881	0.403440	−	0.915006i	$-0.367814\pi$
$631$	12789.5	0.806881	0.403440	+	0.915006i	$0.367814\pi$
$632$	3395.82	0.213732
$633$	0	0
$634$	−6191.78	−0.387866
$635$	891.488	0.0557128
$636$	0	0
$637$	12575.1	0.782174
$638$	−35543.7	−2.20562
$639$	0	0
$640$	1809.88	0.111784
$641$	26876.5	1.65610	0.828048	−	0.560657i	$-0.189451\pi$
$641$	26876.5	1.65610	0.828048	+	0.560657i	$0.189451\pi$
$642$	0	0
$643$	−19135.9	−1.17363	−0.586816	−	0.809720i	$-0.699619\pi$
$643$	−19135.9	−1.17363	−0.586816	+	0.809720i	$0.699619\pi$
$644$	−783.472	−0.0479396
$645$	0	0
$646$	22590.6	1.37588
$647$	21262.7	1.29200	0.645999	−	0.763338i	$-0.276441\pi$
$647$	21262.7	1.29200	0.645999	+	0.763338i	$0.276441\pi$
$648$	0	0
$649$	35127.4	2.12461
$650$	−20031.9	−1.20879
$651$	0	0
$652$	−2062.84	−0.123906
$653$	−20411.0	−1.22319	−0.611597	−	0.791169i	$-0.709473\pi$
$653$	−20411.0	−1.22319	−0.611597	+	0.791169i	$0.709473\pi$
$654$	0	0
$655$	−1530.97	−0.0913283
$656$	3250.82	0.193480
$657$	0	0
$658$	167.110	0.00990062
$659$	9829.73	0.581050	0.290525	−	0.956867i	$-0.406170\pi$
$659$	9829.73	0.581050	0.290525	+	0.956867i	$0.406170\pi$
$660$	0	0
$661$	−12394.9	−0.729355	−0.364678	−	0.931134i	$-0.618821\pi$
$661$	−12394.9	−0.729355	−0.364678	+	0.931134i	$0.618821\pi$
$662$	−39887.1	−2.34177
$663$	0	0
$664$	−14864.7	−0.868766
$665$	−90.3628	−0.00526935
$666$	0	0
$667$	17642.5	1.02417
$668$	10061.2	0.582751
$669$	0	0
$670$	1817.44	0.104797
$671$	−44146.8	−2.53989
$672$	0	0
$673$	24772.1	1.41886	0.709430	−	0.704776i	$-0.248952\pi$
$673$	24772.1	1.41886	0.709430	+	0.704776i	$0.248952\pi$
$674$	−8172.85	−0.467072
$675$	0	0
$676$	−9681.46	−0.550834
$677$	−25646.0	−1.45592	−0.727959	−	0.685621i	$-0.759531\pi$
$677$	−25646.0	−1.45592	−0.727959	+	0.685621i	$0.759531\pi$
$678$	0	0
$679$	−127.432	−0.00720234
$680$	−588.548	−0.0331909
$681$	0	0
$682$	−38434.7	−2.15798
$683$	−31682.3	−1.77495	−0.887475	−	0.460857i	$-0.847542\pi$
$683$	−31682.3	−1.77495	−0.887475	+	0.460857i	$0.847542\pi$
$684$	0	0
$685$	1445.09	0.0806047
$686$	1938.21	0.107874
$687$	0	0
$688$	4245.04	0.235233
$689$	−9612.06	−0.531481
$690$	0	0
$691$	−26145.5	−1.43939	−0.719696	−	0.694289i	$-0.755719\pi$
$691$	−26145.5	−1.43939	−0.719696	+	0.694289i	$0.755719\pi$
$692$	6719.78	0.369144
$693$	0	0
$694$	52786.2	2.88723
$695$	−28.8196	−0.00157294
$696$	0	0
$697$	−4863.10	−0.264280
$698$	−26609.3	−1.44295
$699$	0	0
$700$	−906.439	−0.0489431
$701$	−14749.5	−0.794694	−0.397347	−	0.917668i	$-0.630069\pi$
$701$	−14749.5	−0.794694	−0.397347	+	0.917668i	$0.630069\pi$
$702$	0	0
$703$	42320.6	2.27049
$704$	39944.1	2.13842
$705$	0	0
$706$	25618.6	1.36568
$707$	884.528	0.0470525
$708$	0	0
$709$	−17166.5	−0.909313	−0.454657	−	0.890667i	$-0.650238\pi$
$709$	−17166.5	−0.909313	−0.454657	+	0.890667i	$0.650238\pi$
$710$	−1652.19	−0.0873320
$711$	0	0
$712$	−4833.99	−0.254440
$713$	19077.5	1.00205
$714$	0	0
$715$	1869.39	0.0977782
$716$	−23475.3	−1.22530
$717$	0	0
$718$	−15035.2	−0.781491
$719$	−7064.10	−0.366407	−0.183203	−	0.983075i	$-0.558647\pi$
$719$	−7064.10	−0.366407	−0.183203	+	0.983075i	$0.558647\pi$
$720$	0	0
$721$	−86.4496	−0.00446540
$722$	−50567.8	−2.60656
$723$	0	0
$724$	−17261.0	−0.886052
$725$	20411.6	1.04561
$726$	0	0
$727$	5129.92	0.261703	0.130852	−	0.991402i	$-0.458229\pi$
$727$	5129.92	0.261703	0.130852	+	0.991402i	$0.458229\pi$
$728$	352.268	0.0179340
$729$	0	0
$730$	−665.730	−0.0337531
$731$	−6350.42	−0.321312
$732$	0	0
$733$	−36422.5	−1.83533	−0.917664	−	0.397357i	$-0.869928\pi$
$733$	−36422.5	−1.83533	−0.917664	+	0.397357i	$0.869928\pi$
$734$	−28478.1	−1.43208
$735$	0	0
$736$	−24753.1	−1.23969
$737$	19454.8	0.972358
$738$	0	0
$739$	2073.74	0.103226	0.0516129	−	0.998667i	$-0.483564\pi$
$739$	2073.74	0.103226	0.0516129	+	0.998667i	$0.483564\pi$
$740$	−3700.94	−0.183850
$741$	0	0
$742$	−740.311	−0.0366276
$743$	24869.6	1.22797	0.613983	−	0.789319i	$-0.289566\pi$
$743$	24869.6	1.22797	0.613983	+	0.789319i	$0.289566\pi$
$744$	0	0
$745$	−1034.27	−0.0508628
$746$	−28484.0	−1.39795
$747$	0	0
$748$	−21147.4	−1.03372
$749$	185.421	0.00904557
$750$	0	0
$751$	−3087.87	−0.150037	−0.0750187	−	0.997182i	$-0.523902\pi$
$751$	−3087.87	−0.150037	−0.0750187	+	0.997182i	$0.523902\pi$
$752$	1496.62	0.0725749
$753$	0	0
$754$	−26626.7	−1.28606
$755$	−685.170	−0.0330277
$756$	0	0
$757$	536.550	0.0257612	0.0128806	−	0.999917i	$-0.495900\pi$
$757$	536.550	0.0257612	0.0128806	+	0.999917i	$0.495900\pi$
$758$	21212.5	1.01646
$759$	0	0
$760$	2104.40	0.100440
$761$	−35832.5	−1.70687	−0.853435	−	0.521199i	$-0.825485\pi$
$761$	−35832.5	−1.70687	−0.853435	+	0.521199i	$0.825485\pi$
$762$	0	0
$763$	−568.488	−0.0269733
$764$	22040.6	1.04372
$765$	0	0
$766$	11221.7	0.529317
$767$	26314.9	1.23882
$768$	0	0
$769$	−2680.19	−0.125683	−0.0628415	−	0.998024i	$-0.520016\pi$
$769$	−2680.19	−0.125683	−0.0628415	+	0.998024i	$0.520016\pi$
$770$	143.979	0.00673849
$771$	0	0
$772$	56431.0	2.63082
$773$	−29716.1	−1.38268	−0.691342	−	0.722528i	$-0.742980\pi$
$773$	−29716.1	−1.38268	−0.691342	+	0.722528i	$0.742980\pi$
$774$	0	0
$775$	22071.8	1.02302
$776$	2967.68	0.137285
$777$	0	0
$778$	−2316.81	−0.106763
$779$	17388.4	0.799748
$780$	0	0
$781$	−17686.0	−0.810313
$782$	17866.4	0.817008
$783$	0	0
$784$	8674.05	0.395137
$785$	−770.641	−0.0350387
$786$	0	0
$787$	−24465.5	−1.10813	−0.554067	−	0.832472i	$-0.686925\pi$
$787$	−24465.5	−1.10813	−0.554067	+	0.832472i	$0.686925\pi$
$788$	−20066.5	−0.907155
$789$	0	0
$790$	1039.74	0.0468258
$791$	15.3432	0.000689687 0
$792$	0	0
$793$	−33071.5	−1.48096
$794$	−1698.01	−0.0758942
$795$	0	0
$796$	−42049.6	−1.87237
$797$	−40634.3	−1.80595	−0.902974	−	0.429695i	$-0.858621\pi$
$797$	−40634.3	−1.80595	−0.902974	+	0.429695i	$0.858621\pi$
$798$	0	0
$799$	−2238.90	−0.0991319
$800$	−28638.2	−1.26564
$801$	0	0
$802$	15424.7	0.679133
$803$	−7126.34	−0.313179
$804$	0	0
$805$	−71.4658	−0.00312899
$806$	−28792.4	−1.25828
$807$	0	0
$808$	−20599.2	−0.896877
$809$	29160.3	1.26727	0.633636	−	0.773631i	$-0.281562\pi$
$809$	29160.3	1.26727	0.633636	+	0.773631i	$0.281562\pi$
$810$	0	0
$811$	−30266.4	−1.31048	−0.655239	−	0.755422i	$-0.727432\pi$
$811$	−30266.4	−1.31048	−0.655239	+	0.755422i	$0.727432\pi$
$812$	−1204.85	−0.0520715
$813$	0	0
$814$	−67431.3	−2.90352
$815$	−188.165	−0.00808729
$816$	0	0
$817$	22706.4	0.972334
$818$	−64720.4	−2.76637
$819$	0	0
$820$	−1520.62	−0.0647588
$821$	−6700.52	−0.284835	−0.142418	−	0.989807i	$-0.545488\pi$
$821$	−6700.52	−0.284835	−0.142418	+	0.989807i	$0.545488\pi$
$822$	0	0
$823$	26517.6	1.12314	0.561570	−	0.827429i	$-0.310198\pi$
$823$	26517.6	1.12314	0.561570	+	0.827429i	$0.310198\pi$
$824$	2013.27	0.0851159
$825$	0	0
$826$	2026.75	0.0853747
$827$	−30173.6	−1.26873	−0.634364	−	0.773034i	$-0.718738\pi$
$827$	−30173.6	−1.26873	−0.634364	+	0.773034i	$0.718738\pi$
$828$	0	0
$829$	−3774.02	−0.158115	−0.0790573	−	0.996870i	$-0.525191\pi$
$829$	−3774.02	−0.158115	−0.0790573	+	0.996870i	$0.525191\pi$
$830$	−4551.30	−0.190335
$831$	0	0
$832$	29923.2	1.24688
$833$	−12976.1	−0.539728
$834$	0	0
$835$	917.747	0.0380358
$836$	75614.0	3.12819
$837$	0	0
$838$	17939.6	0.739513
$839$	−8724.67	−0.359010	−0.179505	−	0.983757i	$-0.557450\pi$
$839$	−8724.67	−0.359010	−0.179505	+	0.983757i	$0.557450\pi$
$840$	0	0
$841$	2742.42	0.112445
$842$	−70720.4	−2.89452
$843$	0	0
$844$	−61831.7	−2.52172
$845$	−883.111	−0.0359526
$846$	0	0
$847$	686.802	0.0278616
$848$	−6630.19	−0.268493
$849$	0	0
$850$	20670.5	0.834110
$851$	33470.4	1.34824
$852$	0	0
$853$	−26434.7	−1.06109	−0.530543	−	0.847658i	$-0.678012\pi$
$853$	−26434.7	−1.06109	−0.530543	+	0.847658i	$0.678012\pi$
$854$	−2547.14	−0.102062
$855$	0	0
$856$	−4318.14	−0.172419
$857$	−927.595	−0.0369732	−0.0184866	−	0.999829i	$-0.505885\pi$
$857$	−927.595	−0.0369732	−0.0184866	+	0.999829i	$0.505885\pi$
$858$	0	0
$859$	45670.7	1.81405	0.907023	−	0.421081i	$-0.138349\pi$
$859$	45670.7	1.81405	0.907023	+	0.421081i	$0.138349\pi$
$860$	−1985.68	−0.0787337
$861$	0	0
$862$	−407.114	−0.0160863
$863$	4030.68	0.158987	0.0794936	−	0.996835i	$-0.474670\pi$
$863$	4030.68	0.158987	0.0794936	+	0.996835i	$0.474670\pi$
$864$	0	0
$865$	612.957	0.0240938
$866$	16350.3	0.641577
$867$	0	0
$868$	−1302.85	−0.0509467
$869$	11130.0	0.434475
$870$	0	0
$871$	14574.1	0.566964
$872$	13239.1	0.514144
$873$	0	0
$874$	−63882.6	−2.47238
$875$	−166.086	−0.00641682
$876$	0	0
$877$	−28143.7	−1.08363	−0.541816	−	0.840497i	$-0.682263\pi$
$877$	−28143.7	−1.08363	−0.541816	+	0.840497i	$0.682263\pi$
$878$	55110.0	2.11831
$879$	0	0
$880$	1289.47	0.0493954
$881$	−2213.18	−0.0846354	−0.0423177	−	0.999104i	$-0.513474\pi$
$881$	−2213.18	−0.0846354	−0.0423177	+	0.999104i	$0.513474\pi$
$882$	0	0
$883$	35722.9	1.36146	0.680732	−	0.732533i	$-0.261662\pi$
$883$	35722.9	1.36146	0.680732	+	0.732533i	$0.261662\pi$
$884$	−15842.1	−0.602745
$885$	0	0
$886$	40677.3	1.54242
$887$	10710.8	0.405449	0.202725	−	0.979236i	$-0.435020\pi$
$887$	10710.8	0.405449	0.202725	+	0.979236i	$0.435020\pi$
$888$	0	0
$889$	550.601	0.0207723
$890$	−1480.09	−0.0557445
$891$	0	0
$892$	26990.0	1.01311
$893$	8005.33	0.299987
$894$	0	0
$895$	−2141.34	−0.0799746
$896$	1117.82	0.0416782
$897$	0	0
$898$	8419.29	0.312868
$899$	29338.2	1.08841
$900$	0	0
$901$	9918.52	0.366741
$902$	−27705.7	−1.02273
$903$	0	0
$904$	−357.318	−0.0131463
$905$	−1574.50	−0.0578321
$906$	0	0
$907$	1897.49	0.0694655	0.0347327	−	0.999397i	$-0.488942\pi$
$907$	1897.49	0.0694655	0.0347327	+	0.999397i	$0.488942\pi$
$908$	53030.5	1.93819
$909$	0	0
$910$	107.858	0.00392909
$911$	29031.1	1.05581	0.527905	−	0.849303i	$-0.322978\pi$
$911$	29031.1	1.05581	0.527905	+	0.849303i	$0.322978\pi$
$912$	0	0
$913$	−48719.7	−1.76603
$914$	−52714.9	−1.90772
$915$	0	0
$916$	−31451.4	−1.13448
$917$	−945.560	−0.0340514
$918$	0	0
$919$	−28486.3	−1.02250	−0.511249	−	0.859433i	$-0.670817\pi$
$919$	−28486.3	−1.02250	−0.511249	+	0.859433i	$0.670817\pi$
$920$	1664.32	0.0596423
$921$	0	0
$922$	−5188.93	−0.185345
$923$	−13249.1	−0.472479
$924$	0	0
$925$	38723.6	1.37646
$926$	58234.0	2.06662
$927$	0	0
$928$	−38066.3	−1.34654
$929$	−10167.6	−0.359084	−0.179542	−	0.983750i	$-0.557462\pi$
$929$	−10167.6	−0.359084	−0.179542	+	0.983750i	$0.557462\pi$
$930$	0	0
$931$	46396.8	1.63329
$932$	−47006.6	−1.65209
$933$	0	0
$934$	−14490.8	−0.507660
$935$	−1929.00	−0.0674705
$936$	0	0
$937$	−19728.5	−0.687835	−0.343917	−	0.939000i	$-0.611754\pi$
$937$	−19728.5	−0.687835	−0.343917	+	0.939000i	$0.611754\pi$
$938$	1122.49	0.0390730
$939$	0	0
$940$	−700.067	−0.0242911
$941$	7614.04	0.263773	0.131887	−	0.991265i	$-0.457897\pi$
$941$	7614.04	0.263773	0.131887	+	0.991265i	$0.457897\pi$
$942$	0	0
$943$	13752.1	0.474898
$944$	18151.4	0.625825
$945$	0	0
$946$	−36179.1	−1.24343
$947$	−5187.12	−0.177992	−0.0889962	−	0.996032i	$-0.528366\pi$
$947$	−5187.12	−0.177992	−0.0889962	+	0.996032i	$0.528366\pi$
$948$	0	0
$949$	−5338.53	−0.182609
$950$	−73909.1	−2.52413
$951$	0	0
$952$	−363.499	−0.0123751
$953$	4786.50	0.162697	0.0813483	−	0.996686i	$-0.474077\pi$
$953$	4786.50	0.162697	0.0813483	+	0.996686i	$0.474077\pi$
$954$	0	0
$955$	2010.47	0.0681228
$956$	−2723.32	−0.0921323
$957$	0	0
$958$	−48680.5	−1.64175
$959$	892.519	0.0300531
$960$	0	0
$961$	1933.49	0.0649018
$962$	−50514.6	−1.69299
$963$	0	0
$964$	−36799.2	−1.22949
$965$	5147.46	0.171712
$966$	0	0
$967$	−26758.4	−0.889858	−0.444929	−	0.895566i	$-0.646771\pi$
$967$	−26758.4	−0.889858	−0.444929	+	0.895566i	$0.646771\pi$
$968$	−15994.5	−0.531076
$969$	0	0
$970$	908.652	0.0300774
$971$	151.922	0.00502102	0.00251051	−	0.999997i	$-0.499201\pi$
$971$	151.922	0.00502102	0.00251051	+	0.999997i	$0.499201\pi$
$972$	0	0
$973$	−17.7996	−0.000586464 0
$974$	−72001.3	−2.36866
$975$	0	0
$976$	−22812.0	−0.748151
$977$	−49782.9	−1.63019	−0.815096	−	0.579326i	$-0.803316\pi$
$977$	−49782.9	−1.63019	−0.815096	+	0.579326i	$0.803316\pi$
$978$	0	0
$979$	−15843.7	−0.517227
$980$	−4057.41	−0.132254
$981$	0	0
$982$	−8846.75	−0.287486
$983$	14715.5	0.477469	0.238734	−	0.971085i	$-0.423268\pi$
$983$	14715.5	0.477469	0.238734	+	0.971085i	$0.423268\pi$
$984$	0	0
$985$	−1830.40	−0.0592095
$986$	27475.6	0.887427
$987$	0	0
$988$	56644.5	1.82399
$989$	17958.0	0.577381
$990$	0	0
$991$	−30000.8	−0.961662	−0.480831	−	0.876813i	$-0.659665\pi$
$991$	−30000.8	−0.961662	−0.480831	+	0.876813i	$0.659665\pi$
$992$	−41162.5	−1.31745
$993$	0	0
$994$	−1020.43	−0.0325614
$995$	−3835.63	−0.122209
$996$	0	0
$997$	−48254.9	−1.53285	−0.766424	−	0.642335i	$-0.777966\pi$
$997$	−48254.9	−1.53285	−0.766424	+	0.642335i	$0.777966\pi$
$998$	−69657.2	−2.20938
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.h.1.9	yes	59
3.2	odd	2		2151.4.a.g.1.51	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.51	✓	59	3.2	odd	2
2151.4.a.h.1.9	yes	59	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-4.40394 q^{2} +11.3946 q^{4} +1.03938 q^{5} +0.641944 q^{7} -14.9498 q^{8} +O(q^{10})\)	\(q-4.40394 q^{2} +11.3946 q^{4} +1.03938 q^{5} +0.641944 q^{7} -14.9498 q^{8} -4.57738 q^{10} -48.9987 q^{11} -36.7063 q^{13} -2.82708 q^{14} -25.3192 q^{16} +37.8766 q^{17} -135.430 q^{19} +11.8434 q^{20} +215.787 q^{22} -107.109 q^{23} -123.920 q^{25} +161.652 q^{26} +7.31473 q^{28} -164.716 q^{29} -178.114 q^{31} +231.103 q^{32} -166.806 q^{34} +0.667226 q^{35} -312.490 q^{37} +596.427 q^{38} -15.5386 q^{40} -128.393 q^{41} -167.661 q^{43} -558.323 q^{44} +471.700 q^{46} -59.1103 q^{47} -342.588 q^{49} +545.734 q^{50} -418.255 q^{52} +261.864 q^{53} -50.9285 q^{55} -9.59694 q^{56} +725.399 q^{58} -716.904 q^{59} +900.978 q^{61} +784.401 q^{62} -815.207 q^{64} -38.1519 q^{65} -397.047 q^{67} +431.590 q^{68} -2.93842 q^{70} +360.948 q^{71} +145.439 q^{73} +1376.18 q^{74} -1543.18 q^{76} -31.4545 q^{77} -227.148 q^{79} -26.3163 q^{80} +565.437 q^{82} +994.304 q^{83} +39.3683 q^{85} +738.368 q^{86} +732.522 q^{88} +323.348 q^{89} -23.5634 q^{91} -1220.47 q^{92} +260.318 q^{94} -140.764 q^{95} -198.509 q^{97} +1508.74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) \) 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8} - 36 q^{10} + 132 q^{11} + 104 q^{13} + 280 q^{14} + 822 q^{16} + 408 q^{17} + 20 q^{19} + 800 q^{20} - 2 q^{22} + 276 q^{23} + 1477 q^{25} + 780 q^{26} + 224 q^{28} + 696 q^{29} - 380 q^{31} + 896 q^{32} - 72 q^{34} + 700 q^{35} + 224 q^{37} + 988 q^{38} - 258 q^{40} + 2706 q^{41} - 156 q^{43} + 1584 q^{44} + 428 q^{46} + 1316 q^{47} + 2135 q^{49} + 1400 q^{50} + 1092 q^{52} + 1484 q^{53} - 992 q^{55} + 3360 q^{56} - 120 q^{58} + 3186 q^{59} - 254 q^{61} + 1240 q^{62} + 3054 q^{64} + 5120 q^{65} + 288 q^{67} + 9420 q^{68} + 1108 q^{70} + 4468 q^{71} - 1770 q^{73} + 6214 q^{74} + 720 q^{76} + 6352 q^{77} - 746 q^{79} + 7040 q^{80} + 276 q^{82} + 5484 q^{83} + 588 q^{85} + 10152 q^{86} + 1186 q^{88} + 11570 q^{89} + 1768 q^{91} + 15366 q^{92} - 2142 q^{94} + 5736 q^{95} + 2390 q^{97} + 6912 q^{98}+O(q^{100}) \) 59 * q + 8 * q^2 + 238 * q^4 + 80 * q^5 - 10 * q^7 + 96 * q^8 - 36 * q^10 + 132 * q^11 + 104 * q^13 + 280 * q^14 + 822 * q^16 + 408 * q^17 + 20 * q^19 + 800 * q^20 - 2 * q^22 + 276 * q^23 + 1477 * q^25 + 780 * q^26 + 224 * q^28 + 696 * q^29 - 380 * q^31 + 896 * q^32 - 72 * q^34 + 700 * q^35 + 224 * q^37 + 988 * q^38 - 258 * q^40 + 2706 * q^41 - 156 * q^43 + 1584 * q^44 + 428 * q^46 + 1316 * q^47 + 2135 * q^49 + 1400 * q^50 + 1092 * q^52 + 1484 * q^53 - 992 * q^55 + 3360 * q^56 - 120 * q^58 + 3186 * q^59 - 254 * q^61 + 1240 * q^62 + 3054 * q^64 + 5120 * q^65 + 288 * q^67 + 9420 * q^68 + 1108 * q^70 + 4468 * q^71 - 1770 * q^73 + 6214 * q^74 + 720 * q^76 + 6352 * q^77 - 746 * q^79 + 7040 * q^80 + 276 * q^82 + 5484 * q^83 + 588 * q^85 + 10152 * q^86 + 1186 * q^88 + 11570 * q^89 + 1768 * q^91 + 15366 * q^92 - 2142 * q^94 + 5736 * q^95 + 2390 * q^97 + 6912 * q^98

Introduction

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