LMFDB - Embedded newform 1161.4.a.d.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1161,4,Mod(1,1161)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1161, 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("1161.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1161 = 3^{3} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1161.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:	$68.5012175167$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	1161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.88873 q^{2} -4.43270 q^{4} +7.93489 q^{5} +11.0434 q^{7} -23.4820 q^{8} +O(q^{10})$	\(q+1.88873 q^{2} -4.43270 q^{4} +7.93489 q^{5} +11.0434 q^{7} -23.4820 q^{8} +14.9869 q^{10} +38.0646 q^{11} -40.0082 q^{13} +20.8580 q^{14} -8.88951 q^{16} -66.3621 q^{17} -75.9953 q^{19} -35.1730 q^{20} +71.8937 q^{22} +98.2887 q^{23} -62.0375 q^{25} -75.5646 q^{26} -48.9522 q^{28} -255.137 q^{29} +62.2094 q^{31} +171.066 q^{32} -125.340 q^{34} +87.6284 q^{35} +79.8729 q^{37} -143.535 q^{38} -186.327 q^{40} +140.746 q^{41} -43.0000 q^{43} -168.729 q^{44} +185.641 q^{46} -40.6624 q^{47} -221.043 q^{49} -117.172 q^{50} +177.344 q^{52} +529.124 q^{53} +302.038 q^{55} -259.322 q^{56} -481.885 q^{58} +57.8648 q^{59} -722.339 q^{61} +117.497 q^{62} +394.214 q^{64} -317.461 q^{65} -924.625 q^{67} +294.163 q^{68} +165.506 q^{70} +337.796 q^{71} -504.963 q^{73} +150.858 q^{74} +336.865 q^{76} +420.363 q^{77} +419.819 q^{79} -70.5373 q^{80} +265.832 q^{82} -522.575 q^{83} -526.576 q^{85} -81.2153 q^{86} -893.833 q^{88} -1572.00 q^{89} -441.827 q^{91} -435.684 q^{92} -76.8003 q^{94} -603.015 q^{95} -646.667 q^{97} -417.490 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - q^{2} + 87 q^{4} - 3 q^{5} + 3 q^{7} - 69 q^{8}+O(q^{10}) $ 21 * q - q^2 + 87 * q^4 - 3 * q^5 + 3 * q^7 - 69 * q^8	$ 21 q - q^{2} + 87 q^{4} - 3 q^{5} + 3 q^{7} - 69 q^{8} - 9 q^{10} - 129 q^{11} + 54 q^{13} - 184 q^{14} + 111 q^{16} - 68 q^{17} + 78 q^{19} - 123 q^{20} - 51 q^{22} - 586 q^{23} + 402 q^{25} - 125 q^{26} - 6 q^{28} - 556 q^{29} - 111 q^{31} - 758 q^{32} - 420 q^{34} - 1409 q^{35} + 330 q^{37} - 1067 q^{38} + 180 q^{40} - 678 q^{41} - 903 q^{43} - 1510 q^{44} - 1062 q^{46} - 1580 q^{47} + 1218 q^{49} - 2054 q^{50} + 717 q^{52} - 1069 q^{53} - 2259 q^{55} - 2823 q^{56} + 666 q^{58} - 2248 q^{59} - 162 q^{61} - 3211 q^{62} - 2025 q^{64} - 910 q^{65} - 216 q^{67} - 1868 q^{68} + 3636 q^{70} - 3088 q^{71} - 1605 q^{73} - 2516 q^{74} + 1263 q^{76} - 891 q^{77} + 2094 q^{79} - 5251 q^{80} - 966 q^{82} - 1993 q^{83} - 114 q^{85} + 43 q^{86} + 846 q^{88} - 2494 q^{89} - 2382 q^{91} - 8034 q^{92} - 1062 q^{94} - 4798 q^{95} + 2565 q^{97} - 3103 q^{98}+O(q^{100}) $ 21 * q - q^2 + 87 * q^4 - 3 * q^5 + 3 * q^7 - 69 * q^8 - 9 * q^10 - 129 * q^11 + 54 * q^13 - 184 * q^14 + 111 * q^16 - 68 * q^17 + 78 * q^19 - 123 * q^20 - 51 * q^22 - 586 * q^23 + 402 * q^25 - 125 * q^26 - 6 * q^28 - 556 * q^29 - 111 * q^31 - 758 * q^32 - 420 * q^34 - 1409 * q^35 + 330 * q^37 - 1067 * q^38 + 180 * q^40 - 678 * q^41 - 903 * q^43 - 1510 * q^44 - 1062 * q^46 - 1580 * q^47 + 1218 * q^49 - 2054 * q^50 + 717 * q^52 - 1069 * q^53 - 2259 * q^55 - 2823 * q^56 + 666 * q^58 - 2248 * q^59 - 162 * q^61 - 3211 * q^62 - 2025 * q^64 - 910 * q^65 - 216 * q^67 - 1868 * q^68 + 3636 * q^70 - 3088 * q^71 - 1605 * q^73 - 2516 * q^74 + 1263 * q^76 - 891 * q^77 + 2094 * q^79 - 5251 * q^80 - 966 * q^82 - 1993 * q^83 - 114 * q^85 + 43 * q^86 + 846 * q^88 - 2494 * q^89 - 2382 * q^91 - 8034 * q^92 - 1062 * q^94 - 4798 * q^95 + 2565 * q^97 - 3103 * 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 $

$2$	1.88873	0.667766	0.333883	−	0.942614i	$-0.391641\pi$
$2$	1.88873	0.667766	0.333883	+	0.942614i	$0.391641\pi$
$3$	0	0
$3$	0	0
$4$	−4.43270	−0.554088
$5$	7.93489	0.709719	0.354859	−	0.934920i	$-0.384529\pi$
$5$	7.93489	0.709719	0.354859	+	0.934920i	$0.384529\pi$
$6$	0	0
$7$	11.0434	0.596289	0.298144	−	0.954521i	$-0.403632\pi$
$7$	11.0434	0.596289	0.298144	+	0.954521i	$0.403632\pi$
$8$	−23.4820	−1.03777
$9$	0	0
$10$	14.9869	0.473926
$11$	38.0646	1.04336	0.521678	−	0.853143i	$-0.325306\pi$
$11$	38.0646	1.04336	0.521678	+	0.853143i	$0.325306\pi$
$12$	0	0
$13$	−40.0082	−0.853560	−0.426780	−	0.904356i	$-0.640352\pi$
$13$	−40.0082	−0.853560	−0.426780	+	0.904356i	$0.640352\pi$
$14$	20.8580	0.398182
$15$	0	0
$16$	−8.88951	−0.138899
$17$	−66.3621	−0.946774	−0.473387	−	0.880854i	$-0.656969\pi$
$17$	−66.3621	−0.946774	−0.473387	+	0.880854i	$0.656969\pi$
$18$	0	0
$19$	−75.9953	−0.917606	−0.458803	−	0.888538i	$-0.651722\pi$
$19$	−75.9953	−0.917606	−0.458803	+	0.888538i	$0.651722\pi$
$20$	−35.1730	−0.393247
$21$	0	0
$22$	71.8937	0.696717
$23$	98.2887	0.891070	0.445535	−	0.895265i	$-0.353014\pi$
$23$	98.2887	0.891070	0.445535	+	0.895265i	$0.353014\pi$
$24$	0	0
$25$	−62.0375	−0.496300
$26$	−75.5646	−0.569978
$27$	0	0
$28$	−48.9522	−0.330397
$29$	−255.137	−1.63372	−0.816858	−	0.576838i	$-0.804286\pi$
$29$	−255.137	−1.63372	−0.816858	+	0.576838i	$0.804286\pi$
$30$	0	0
$31$	62.2094	0.360424	0.180212	−	0.983628i	$-0.442322\pi$
$31$	62.2094	0.360424	0.180212	+	0.983628i	$0.442322\pi$
$32$	171.066	0.945016
$33$	0	0
$34$	−125.340	−0.632224
$35$	87.6284	0.423197
$36$	0	0
$37$	79.8729	0.354892	0.177446	−	0.984131i	$-0.443216\pi$
$37$	79.8729	0.354892	0.177446	+	0.984131i	$0.443216\pi$
$38$	−143.535	−0.612747
$39$	0	0
$40$	−186.327	−0.736523
$41$	140.746	0.536120	0.268060	−	0.963402i	$-0.413618\pi$
$41$	140.746	0.536120	0.268060	+	0.963402i	$0.413618\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	−168.729	−0.578110
$45$	0	0
$46$	185.641	0.595026
$47$	−40.6624	−0.126196	−0.0630981	−	0.998007i	$-0.520098\pi$
$47$	−40.6624	−0.126196	−0.0630981	+	0.998007i	$0.520098\pi$
$48$	0	0
$49$	−221.043	−0.644439
$50$	−117.172	−0.331412
$51$	0	0
$52$	177.344	0.472947
$53$	529.124	1.37133	0.685667	−	0.727915i	$-0.259511\pi$
$53$	529.124	1.37133	0.685667	+	0.727915i	$0.259511\pi$
$54$	0	0
$55$	302.038	0.740488
$56$	−259.322	−0.618810
$57$	0	0
$58$	−481.885	−1.09094
$59$	57.8648	0.127684	0.0638419	−	0.997960i	$-0.479665\pi$
$59$	57.8648	0.127684	0.0638419	+	0.997960i	$0.479665\pi$
$60$	0	0
$61$	−722.339	−1.51616	−0.758082	−	0.652159i	$-0.773863\pi$
$61$	−722.339	−1.51616	−0.758082	+	0.652159i	$0.773863\pi$
$62$	117.497	0.240679
$63$	0	0
$64$	394.214	0.769949
$65$	−317.461	−0.605787
$66$	0	0
$67$	−924.625	−1.68598	−0.842992	−	0.537926i	$-0.819208\pi$
$67$	−924.625	−1.68598	−0.842992	+	0.537926i	$0.819208\pi$
$68$	294.163	0.524596
$69$	0	0
$70$	165.506	0.282597
$71$	337.796	0.564633	0.282317	−	0.959321i	$-0.408897\pi$
$71$	337.796	0.564633	0.282317	+	0.959321i	$0.408897\pi$
$72$	0	0
$73$	−504.963	−0.809610	−0.404805	−	0.914403i	$-0.632661\pi$
$73$	−504.963	−0.809610	−0.404805	+	0.914403i	$0.632661\pi$
$74$	150.858	0.236985
$75$	0	0
$76$	336.865	0.508435
$77$	420.363	0.622141
$78$	0	0
$79$	419.819	0.597890	0.298945	−	0.954270i	$-0.403365\pi$
$79$	419.819	0.597890	0.298945	+	0.954270i	$0.403365\pi$
$80$	−70.5373	−0.0985789
$81$	0	0
$82$	265.832	0.358003
$83$	−522.575	−0.691086	−0.345543	−	0.938403i	$-0.612305\pi$
$83$	−522.575	−0.691086	−0.345543	+	0.938403i	$0.612305\pi$
$84$	0	0
$85$	−526.576	−0.671943
$86$	−81.2153	−0.101833
$87$	0	0
$88$	−893.833	−1.08276
$89$	−1572.00	−1.87227	−0.936135	−	0.351642i	$-0.885624\pi$
$89$	−1572.00	−1.87227	−0.936135	+	0.351642i	$0.885624\pi$
$90$	0	0
$91$	−441.827	−0.508968
$92$	−435.684	−0.493731
$93$	0	0
$94$	−76.8003	−0.0842696
$95$	−603.015	−0.651242
$96$	0	0
$97$	−646.667	−0.676898	−0.338449	−	0.940985i	$-0.609902\pi$
$97$	−646.667	−0.676898	−0.338449	+	0.940985i	$0.609902\pi$
$98$	−417.490	−0.430335
$99$	0	0
$100$	274.994	0.274994
$101$	226.564	0.223208	0.111604	−	0.993753i	$-0.464401\pi$
$101$	226.564	0.223208	0.111604	+	0.993753i	$0.464401\pi$
$102$	0	0
$103$	308.947	0.295548	0.147774	−	0.989021i	$-0.452789\pi$
$103$	308.947	0.295548	0.147774	+	0.989021i	$0.452789\pi$
$104$	939.472	0.885797
$105$	0	0
$106$	999.371	0.915731
$107$	−1549.85	−1.40028	−0.700140	−	0.714006i	$-0.746879\pi$
$107$	−1549.85	−1.40028	−0.700140	+	0.714006i	$0.746879\pi$
$108$	0	0
$109$	−528.865	−0.464735	−0.232368	−	0.972628i	$-0.574647\pi$
$109$	−528.865	−0.464735	−0.232368	+	0.972628i	$0.574647\pi$
$110$	570.469	0.494473
$111$	0	0
$112$	−98.1706	−0.0828237
$113$	−1788.43	−1.48886	−0.744432	−	0.667698i	$-0.767280\pi$
$113$	−1788.43	−1.48886	−0.744432	+	0.667698i	$0.767280\pi$
$114$	0	0
$115$	779.910	0.632409
$116$	1130.95	0.905223
$117$	0	0
$118$	109.291	0.0852630
$119$	−732.865	−0.564551
$120$	0	0
$121$	117.913	0.0885898
$122$	−1364.30	−1.01244
$123$	0	0
$124$	−275.756	−0.199707
$125$	−1484.12	−1.06195
$126$	0	0
$127$	419.056	0.292797	0.146398	−	0.989226i	$-0.453232\pi$
$127$	419.056	0.292797	0.146398	+	0.989226i	$0.453232\pi$
$128$	−623.967	−0.430870
$129$	0	0
$130$	−599.597	−0.404524
$131$	355.849	0.237333	0.118667	−	0.992934i	$-0.462138\pi$
$131$	355.849	0.237333	0.118667	+	0.992934i	$0.462138\pi$
$132$	0	0
$133$	−839.249	−0.547159
$134$	−1746.37	−1.12584
$135$	0	0
$136$	1558.31	0.982532
$137$	−1215.23	−0.757838	−0.378919	−	0.925430i	$-0.623704\pi$
$137$	−1215.23	−0.757838	−0.378919	+	0.925430i	$0.623704\pi$
$138$	0	0
$139$	2208.69	1.34776	0.673880	−	0.738840i	$-0.264626\pi$
$139$	2208.69	1.34776	0.673880	+	0.738840i	$0.264626\pi$
$140$	−388.431	−0.234489
$141$	0	0
$142$	638.004	0.377043
$143$	−1522.90	−0.890566
$144$	0	0
$145$	−2024.49	−1.15948
$146$	−953.739	−0.540630
$147$	0	0
$148$	−354.053	−0.196642
$149$	−828.291	−0.455411	−0.227706	−	0.973730i	$-0.573122\pi$
$149$	−828.291	−0.455411	−0.227706	+	0.973730i	$0.573122\pi$
$150$	0	0
$151$	2036.38	1.09747	0.548735	−	0.835997i	$-0.315110\pi$
$151$	2036.38	1.09747	0.548735	+	0.835997i	$0.315110\pi$
$152$	1784.52	0.952262
$153$	0	0
$154$	793.953	0.415445
$155$	493.625	0.255800
$156$	0	0
$157$	1835.03	0.932809	0.466405	−	0.884571i	$-0.345549\pi$
$157$	1835.03	0.932809	0.466405	+	0.884571i	$0.345549\pi$
$158$	792.924	0.399251
$159$	0	0
$160$	1357.39	0.670695
$161$	1085.44	0.531335
$162$	0	0
$163$	2558.86	1.22960	0.614802	−	0.788681i	$-0.289236\pi$
$163$	2558.86	1.22960	0.614802	+	0.788681i	$0.289236\pi$
$164$	−623.887	−0.297057
$165$	0	0
$166$	−987.003	−0.461484
$167$	−889.825	−0.412316	−0.206158	−	0.978519i	$-0.566096\pi$
$167$	−889.825	−0.412316	−0.206158	+	0.978519i	$0.566096\pi$
$168$	0	0
$169$	−596.345	−0.271436
$170$	−994.559	−0.448701
$171$	0	0
$172$	190.606	0.0844976
$173$	−2286.70	−1.00494	−0.502470	−	0.864595i	$-0.667575\pi$
$173$	−2286.70	−1.00494	−0.502470	+	0.864595i	$0.667575\pi$
$174$	0	0
$175$	−685.106	−0.295938
$176$	−338.375	−0.144920
$177$	0	0
$178$	−2969.09	−1.25024
$179$	−1868.63	−0.780269	−0.390134	−	0.920758i	$-0.627571\pi$
$179$	−1868.63	−0.780269	−0.390134	+	0.920758i	$0.627571\pi$
$180$	0	0
$181$	−1885.45	−0.774280	−0.387140	−	0.922021i	$-0.626537\pi$
$181$	−1885.45	−0.774280	−0.387140	+	0.922021i	$0.626537\pi$
$182$	−834.492	−0.339872
$183$	0	0
$184$	−2308.01	−0.924723
$185$	633.783	0.251874
$186$	0	0
$187$	−2526.04	−0.987822
$188$	180.244	0.0699238
$189$	0	0
$190$	−1138.93	−0.434878
$191$	330.285	0.125124	0.0625618	−	0.998041i	$-0.480073\pi$
$191$	330.285	0.125124	0.0625618	+	0.998041i	$0.480073\pi$
$192$	0	0
$193$	−3151.60	−1.17543	−0.587713	−	0.809070i	$-0.699972\pi$
$193$	−3151.60	−1.17543	−0.587713	+	0.809070i	$0.699972\pi$
$194$	−1221.38	−0.452010
$195$	0	0
$196$	979.817	0.357076
$197$	2300.70	0.832072	0.416036	−	0.909348i	$-0.363419\pi$
$197$	2300.70	0.832072	0.416036	+	0.909348i	$0.363419\pi$
$198$	0	0
$199$	−341.966	−0.121816	−0.0609078	−	0.998143i	$-0.519400\pi$
$199$	−341.966	−0.121816	−0.0609078	+	0.998143i	$0.519400\pi$
$200$	1456.76	0.515044
$201$	0	0
$202$	427.919	0.149051
$203$	−2817.59	−0.974167
$204$	0	0
$205$	1116.81	0.380494
$206$	583.518	0.197357
$207$	0	0
$208$	355.653	0.118558
$209$	−2892.73	−0.957389
$210$	0	0
$211$	−5106.25	−1.66601	−0.833007	−	0.553263i	$-0.813382\pi$
$211$	−5106.25	−1.66601	−0.833007	+	0.553263i	$0.813382\pi$
$212$	−2345.45	−0.759840
$213$	0	0
$214$	−2927.25	−0.935060
$215$	−341.200	−0.108231
$216$	0	0
$217$	687.005	0.214917
$218$	−998.883	−0.310334
$219$	0	0
$220$	−1338.85	−0.410296
$221$	2655.03	0.808128
$222$	0	0
$223$	4201.79	1.26176	0.630880	−	0.775880i	$-0.282694\pi$
$223$	4201.79	1.26176	0.630880	+	0.775880i	$0.282694\pi$
$224$	1889.16	0.563503
$225$	0	0
$226$	−3377.86	−0.994213
$227$	1423.58	0.416240	0.208120	−	0.978103i	$-0.433266\pi$
$227$	1423.58	0.416240	0.208120	+	0.978103i	$0.433266\pi$
$228$	0	0
$229$	−1854.94	−0.535273	−0.267637	−	0.963520i	$-0.586243\pi$
$229$	−1854.94	−0.535273	−0.267637	+	0.963520i	$0.586243\pi$
$230$	1473.04	0.422301
$231$	0	0
$232$	5991.13	1.69542
$233$	−4816.74	−1.35432	−0.677158	−	0.735838i	$-0.736788\pi$
$233$	−4816.74	−1.35432	−0.677158	+	0.735838i	$0.736788\pi$
$234$	0	0
$235$	−322.652	−0.0895638
$236$	−256.497	−0.0707481
$237$	0	0
$238$	−1384.18	−0.376988
$239$	−927.680	−0.251074	−0.125537	−	0.992089i	$-0.540065\pi$
$239$	−927.680	−0.251074	−0.125537	+	0.992089i	$0.540065\pi$
$240$	0	0
$241$	6009.33	1.60620	0.803101	−	0.595843i	$-0.203182\pi$
$241$	6009.33	1.60620	0.803101	+	0.595843i	$0.203182\pi$
$242$	222.706	0.0591573
$243$	0	0
$244$	3201.91	0.840088
$245$	−1753.95	−0.457371
$246$	0	0
$247$	3040.43	0.783232
$248$	−1460.80	−0.374036
$249$	0	0
$250$	−2803.10	−0.709136
$251$	7377.94	1.85534	0.927672	−	0.373396i	$-0.121807\pi$
$251$	7377.94	1.85534	0.927672	+	0.373396i	$0.121807\pi$
$252$	0	0
$253$	3741.32	0.929702
$254$	791.483	0.195520
$255$	0	0
$256$	−4332.21	−1.05767
$257$	−4465.62	−1.08388	−0.541941	−	0.840417i	$-0.682310\pi$
$257$	−4465.62	−1.08388	−0.541941	+	0.840417i	$0.682310\pi$
$258$	0	0
$259$	882.070	0.211618
$260$	1407.21	0.335659
$261$	0	0
$262$	672.103	0.158483
$263$	1018.00	0.238679	0.119340	−	0.992853i	$-0.461922\pi$
$263$	1018.00	0.238679	0.119340	+	0.992853i	$0.461922\pi$
$264$	0	0
$265$	4198.54	0.973261
$266$	−1585.11	−0.365374
$267$	0	0
$268$	4098.59	0.934183
$269$	2025.49	0.459093	0.229547	−	0.973298i	$-0.426276\pi$
$269$	2025.49	0.459093	0.229547	+	0.973298i	$0.426276\pi$
$270$	0	0
$271$	7040.79	1.57822	0.789110	−	0.614252i	$-0.210542\pi$
$271$	7040.79	1.57822	0.789110	+	0.614252i	$0.210542\pi$
$272$	589.926	0.131506
$273$	0	0
$274$	−2295.23	−0.506059
$275$	−2361.43	−0.517817
$276$	0	0
$277$	1720.14	0.373116	0.186558	−	0.982444i	$-0.440267\pi$
$277$	1720.14	0.373116	0.186558	+	0.982444i	$0.440267\pi$
$278$	4171.62	0.899989
$279$	0	0
$280$	−2057.69	−0.439181
$281$	−6373.12	−1.35298	−0.676491	−	0.736450i	$-0.736500\pi$
$281$	−6373.12	−1.35298	−0.676491	+	0.736450i	$0.736500\pi$
$282$	0	0
$283$	4132.72	0.868073	0.434036	−	0.900895i	$-0.357089\pi$
$283$	4132.72	0.868073	0.434036	+	0.900895i	$0.357089\pi$
$284$	−1497.35	−0.312856
$285$	0	0
$286$	−2876.34	−0.594690
$287$	1554.32	0.319682
$288$	0	0
$289$	−509.077	−0.103618
$290$	−3823.71	−0.774261
$291$	0	0
$292$	2238.35	0.448595
$293$	7150.20	1.42566	0.712831	−	0.701335i	$-0.247412\pi$
$293$	7150.20	1.42566	0.712831	+	0.701335i	$0.247412\pi$
$294$	0	0
$295$	459.151	0.0906196
$296$	−1875.57	−0.368296
$297$	0	0
$298$	−1564.42	−0.304108
$299$	−3932.35	−0.760581
$300$	0	0
$301$	−474.867	−0.0909332
$302$	3846.16	0.732853
$303$	0	0
$304$	675.561	0.127454
$305$	−5731.68	−1.07605
$306$	0	0
$307$	2192.83	0.407660	0.203830	−	0.979006i	$-0.434661\pi$
$307$	2192.83	0.407660	0.203830	+	0.979006i	$0.434661\pi$
$308$	−1863.35	−0.344721
$309$	0	0
$310$	932.324	0.170814
$311$	−265.247	−0.0483627	−0.0241813	−	0.999708i	$-0.507698\pi$
$311$	−265.247	−0.0483627	−0.0241813	+	0.999708i	$0.507698\pi$
$312$	0	0
$313$	7859.36	1.41929	0.709644	−	0.704561i	$-0.248856\pi$
$313$	7859.36	1.41929	0.709644	+	0.704561i	$0.248856\pi$
$314$	3465.87	0.622899
$315$	0	0
$316$	−1860.93	−0.331284
$317$	−3930.67	−0.696430	−0.348215	−	0.937415i	$-0.613212\pi$
$317$	−3930.67	−0.696430	−0.348215	+	0.937415i	$0.613212\pi$
$318$	0	0
$319$	−9711.69	−1.70455
$320$	3128.04	0.546447
$321$	0	0
$322$	2050.11	0.354808
$323$	5043.21	0.868766
$324$	0	0
$325$	2482.01	0.423621
$326$	4833.00	0.821089
$327$	0	0
$328$	−3305.01	−0.556368
$329$	−449.052	−0.0752494
$330$	0	0
$331$	8899.38	1.47781	0.738904	−	0.673811i	$-0.235344\pi$
$331$	8899.38	1.47781	0.738904	+	0.673811i	$0.235344\pi$
$332$	2316.42	0.382922
$333$	0	0
$334$	−1680.64	−0.275331
$335$	−7336.80	−1.19657
$336$	0	0
$337$	1263.01	0.204155	0.102078	−	0.994776i	$-0.467451\pi$
$337$	1263.01	0.204155	0.102078	+	0.994776i	$0.467451\pi$
$338$	−1126.33	−0.181256
$339$	0	0
$340$	2334.16	0.372316
$341$	2367.98	0.376050
$342$	0	0
$343$	−6228.96	−0.980561
$344$	1009.73	0.158258
$345$	0	0
$346$	−4318.96	−0.671066
$347$	−7883.19	−1.21957	−0.609786	−	0.792566i	$-0.708745\pi$
$347$	−7883.19	−1.21957	−0.609786	+	0.792566i	$0.708745\pi$
$348$	0	0
$349$	10939.6	1.67789	0.838946	−	0.544215i	$-0.183172\pi$
$349$	10939.6	1.67789	0.838946	+	0.544215i	$0.183172\pi$
$350$	−1293.98	−0.197617
$351$	0	0
$352$	6511.56	0.985987
$353$	739.154	0.111448	0.0557241	−	0.998446i	$-0.482253\pi$
$353$	739.154	0.111448	0.0557241	+	0.998446i	$0.482253\pi$
$354$	0	0
$355$	2680.37	0.400731
$356$	6968.22	1.03740
$357$	0	0
$358$	−3529.34	−0.521037
$359$	3580.29	0.526352	0.263176	−	0.964748i	$-0.415230\pi$
$359$	3580.29	0.526352	0.263176	+	0.964748i	$0.415230\pi$
$360$	0	0
$361$	−1083.71	−0.157999
$362$	−3561.11	−0.517038
$363$	0	0
$364$	1958.49	0.282013
$365$	−4006.83	−0.574595
$366$	0	0
$367$	−11833.1	−1.68306	−0.841531	−	0.540209i	$-0.818345\pi$
$367$	−11833.1	−1.68306	−0.841531	+	0.540209i	$0.818345\pi$
$368$	−873.737	−0.123768
$369$	0	0
$370$	1197.04	0.168193
$371$	5843.34	0.817712
$372$	0	0
$373$	12398.6	1.72112	0.860558	−	0.509352i	$-0.170115\pi$
$373$	12398.6	1.72112	0.860558	+	0.509352i	$0.170115\pi$
$374$	−4771.01	−0.659634
$375$	0	0
$376$	954.835	0.130962
$377$	10207.6	1.39447
$378$	0	0
$379$	1280.13	0.173498	0.0867489	−	0.996230i	$-0.472352\pi$
$379$	1280.13	0.173498	0.0867489	+	0.996230i	$0.472352\pi$
$380$	2672.99	0.360846
$381$	0	0
$382$	623.820	0.0835534
$383$	13328.0	1.77814	0.889071	−	0.457770i	$-0.151352\pi$
$383$	13328.0	1.77814	0.889071	+	0.457770i	$0.151352\pi$
$384$	0	0
$385$	3335.54	0.441545
$386$	−5952.52	−0.784910
$387$	0	0
$388$	2866.48	0.375061
$389$	−327.389	−0.0426717	−0.0213359	−	0.999772i	$-0.506792\pi$
$389$	−327.389	−0.0426717	−0.0213359	+	0.999772i	$0.506792\pi$
$390$	0	0
$391$	−6522.64	−0.843642
$392$	5190.53	0.668779
$393$	0	0
$394$	4345.40	0.555630
$395$	3331.22	0.424334
$396$	0	0
$397$	−8133.35	−1.02821	−0.514107	−	0.857726i	$-0.671877\pi$
$397$	−8133.35	−1.02821	−0.514107	+	0.857726i	$0.671877\pi$
$398$	−645.880	−0.0813443
$399$	0	0
$400$	551.482	0.0689353
$401$	−3916.96	−0.487790	−0.243895	−	0.969802i	$-0.578425\pi$
$401$	−3916.96	−0.487790	−0.243895	+	0.969802i	$0.578425\pi$
$402$	0	0
$403$	−2488.89	−0.307643
$404$	−1004.29	−0.123677
$405$	0	0
$406$	−5321.66	−0.650516
$407$	3040.33	0.370279
$408$	0	0
$409$	−2446.43	−0.295765	−0.147883	−	0.989005i	$-0.547246\pi$
$409$	−2446.43	−0.295765	−0.147883	+	0.989005i	$0.547246\pi$
$410$	2109.35	0.254081
$411$	0	0
$412$	−1369.47	−0.163760
$413$	639.025	0.0761365
$414$	0	0
$415$	−4146.58	−0.490476
$416$	−6844.05	−0.806627
$417$	0	0
$418$	−5463.58	−0.639312
$419$	−8091.59	−0.943437	−0.471718	−	0.881749i	$-0.656366\pi$
$419$	−8091.59	−0.943437	−0.471718	+	0.881749i	$0.656366\pi$
$420$	0	0
$421$	−11243.0	−1.30155	−0.650774	−	0.759272i	$-0.725555\pi$
$421$	−11243.0	−1.30155	−0.650774	+	0.759272i	$0.725555\pi$
$422$	−9644.32	−1.11251
$423$	0	0
$424$	−12424.9	−1.42313
$425$	4116.93	0.469884
$426$	0	0
$427$	−7977.10	−0.904072
$428$	6870.04	0.775878
$429$	0	0
$430$	−644.435	−0.0722731
$431$	1809.82	0.202264	0.101132	−	0.994873i	$-0.467753\pi$
$431$	1809.82	0.202264	0.101132	+	0.994873i	$0.467753\pi$
$432$	0	0
$433$	2217.54	0.246116	0.123058	−	0.992399i	$-0.460730\pi$
$433$	2217.54	0.246116	0.123058	+	0.992399i	$0.460730\pi$
$434$	1297.57	0.143514
$435$	0	0
$436$	2344.30	0.257504
$437$	−7469.48	−0.817651
$438$	0	0
$439$	−10800.1	−1.17417	−0.587086	−	0.809524i	$-0.699725\pi$
$439$	−10800.1	−1.17417	−0.587086	+	0.809524i	$0.699725\pi$
$440$	−7092.47	−0.768455
$441$	0	0
$442$	5014.62	0.539641
$443$	−4702.08	−0.504294	−0.252147	−	0.967689i	$-0.581137\pi$
$443$	−4702.08	−0.504294	−0.252147	+	0.967689i	$0.581137\pi$
$444$	0	0
$445$	−12473.7	−1.32878
$446$	7936.04	0.842561
$447$	0	0
$448$	4353.47	0.459112
$449$	12921.3	1.35812	0.679059	−	0.734084i	$-0.262388\pi$
$449$	12921.3	1.35812	0.679059	+	0.734084i	$0.262388\pi$
$450$	0	0
$451$	5357.46	0.559363
$452$	7927.59	0.824962
$453$	0	0
$454$	2688.76	0.277951
$455$	−3505.85	−0.361224
$456$	0	0
$457$	2502.04	0.256107	0.128053	−	0.991767i	$-0.459127\pi$
$457$	2502.04	0.256107	0.128053	+	0.991767i	$0.459127\pi$
$458$	−3503.47	−0.357438
$459$	0	0
$460$	−3457.11	−0.350410
$461$	−8149.77	−0.823368	−0.411684	−	0.911327i	$-0.635059\pi$
$461$	−8149.77	−0.823368	−0.411684	+	0.911327i	$0.635059\pi$
$462$	0	0
$463$	−6960.66	−0.698681	−0.349340	−	0.936996i	$-0.613594\pi$
$463$	−6960.66	−0.698681	−0.349340	+	0.936996i	$0.613594\pi$
$464$	2268.04	0.226921
$465$	0	0
$466$	−9097.52	−0.904366
$467$	−7857.29	−0.778570	−0.389285	−	0.921117i	$-0.627278\pi$
$467$	−7857.29	−0.778570	−0.389285	+	0.921117i	$0.627278\pi$
$468$	0	0
$469$	−10211.0	−1.00533
$470$	−609.402	−0.0598077
$471$	0	0
$472$	−1358.78	−0.132506
$473$	−1636.78	−0.159110
$474$	0	0
$475$	4714.56	0.455408
$476$	3248.57	0.312811
$477$	0	0
$478$	−1752.14	−0.167659
$479$	−1836.93	−0.175222	−0.0876112	−	0.996155i	$-0.527923\pi$
$479$	−1836.93	−0.175222	−0.0876112	+	0.996155i	$0.527923\pi$
$480$	0	0
$481$	−3195.57	−0.302922
$482$	11350.0	1.07257
$483$	0	0
$484$	−522.673	−0.0490865
$485$	−5131.24	−0.480407
$486$	0	0
$487$	17357.0	1.61503	0.807517	−	0.589844i	$-0.200811\pi$
$487$	17357.0	1.61503	0.807517	+	0.589844i	$0.200811\pi$
$488$	16962.0	1.57343
$489$	0	0
$490$	−3312.74	−0.305417
$491$	14658.2	1.34728	0.673642	−	0.739057i	$-0.264729\pi$
$491$	14658.2	1.34728	0.673642	+	0.739057i	$0.264729\pi$
$492$	0	0
$493$	16931.4	1.54676
$494$	5742.56	0.523016
$495$	0	0
$496$	−553.011	−0.0500624
$497$	3730.42	0.336685
$498$	0	0
$499$	−10778.5	−0.966960	−0.483480	−	0.875355i	$-0.660627\pi$
$499$	−10778.5	−0.966960	−0.483480	+	0.875355i	$0.660627\pi$
$500$	6578.68	0.588415
$501$	0	0
$502$	13934.9	1.23894
$503$	1061.13	0.0940626	0.0470313	−	0.998893i	$-0.485024\pi$
$503$	1061.13	0.0940626	0.0470313	+	0.998893i	$0.485024\pi$
$504$	0	0
$505$	1797.76	0.158415
$506$	7066.33	0.620824
$507$	0	0
$508$	−1857.55	−0.162235
$509$	2890.46	0.251704	0.125852	−	0.992049i	$-0.459834\pi$
$509$	2890.46	0.251704	0.125852	+	0.992049i	$0.459834\pi$
$510$	0	0
$511$	−5576.53	−0.482761
$512$	−3190.64	−0.275406
$513$	0	0
$514$	−8434.34	−0.723780
$515$	2451.46	0.209756
$516$	0	0
$517$	−1547.80	−0.131668
$518$	1665.99	0.141312
$519$	0	0
$520$	7454.61	0.628666
$521$	−3200.61	−0.269139	−0.134570	−	0.990904i	$-0.542965\pi$
$521$	−3200.61	−0.269139	−0.134570	+	0.990904i	$0.542965\pi$
$522$	0	0
$523$	11380.2	0.951472	0.475736	−	0.879588i	$-0.342182\pi$
$523$	11380.2	0.951472	0.475736	+	0.879588i	$0.342182\pi$
$524$	−1577.37	−0.131504
$525$	0	0
$526$	1922.73	0.159382
$527$	−4128.35	−0.341240
$528$	0	0
$529$	−2506.34	−0.205995
$530$	7929.90	0.649911
$531$	0	0
$532$	3720.14	0.303174
$533$	−5631.01	−0.457610
$534$	0	0
$535$	−12297.9	−0.993804
$536$	21712.0	1.74966
$537$	0	0
$538$	3825.59	0.306567
$539$	−8413.90	−0.672379
$540$	0	0
$541$	4469.09	0.355159	0.177580	−	0.984106i	$-0.443173\pi$
$541$	4469.09	0.355159	0.177580	+	0.984106i	$0.443173\pi$
$542$	13298.1	1.05388
$543$	0	0
$544$	−11352.3	−0.894717
$545$	−4196.49	−0.329831
$546$	0	0
$547$	5828.71	0.455608	0.227804	−	0.973707i	$-0.426845\pi$
$547$	5828.71	0.455608	0.227804	+	0.973707i	$0.426845\pi$
$548$	5386.74	0.419909
$549$	0	0
$550$	−4460.10	−0.345781
$551$	19389.2	1.49911
$552$	0	0
$553$	4636.24	0.356515
$554$	3248.87	0.249154
$555$	0	0
$556$	−9790.47	−0.746778
$557$	19254.2	1.46468	0.732339	−	0.680940i	$-0.238429\pi$
$557$	19254.2	1.46468	0.732339	+	0.680940i	$0.238429\pi$
$558$	0	0
$559$	1720.35	0.130167
$560$	−778.973	−0.0587815
$561$	0	0
$562$	−12037.1	−0.903477
$563$	−2834.56	−0.212189	−0.106095	−	0.994356i	$-0.533835\pi$
$563$	−2834.56	−0.212189	−0.106095	+	0.994356i	$0.533835\pi$
$564$	0	0
$565$	−14191.0	−1.05667
$566$	7805.59	0.579670
$567$	0	0
$568$	−7932.12	−0.585958
$569$	23171.0	1.70717	0.853583	−	0.520957i	$-0.174425\pi$
$569$	23171.0	1.70717	0.853583	+	0.520957i	$0.174425\pi$
$570$	0	0
$571$	−5652.30	−0.414258	−0.207129	−	0.978314i	$-0.566412\pi$
$571$	−5652.30	−0.414258	−0.207129	+	0.978314i	$0.566412\pi$
$572$	6750.54	0.493452
$573$	0	0
$574$	2935.70	0.213473
$575$	−6097.58	−0.442238
$576$	0	0
$577$	−2118.52	−0.152852	−0.0764258	−	0.997075i	$-0.524351\pi$
$577$	−2118.52	−0.152852	−0.0764258	+	0.997075i	$0.524351\pi$
$578$	−961.508	−0.0691929
$579$	0	0
$580$	8973.95	0.642453
$581$	−5771.02	−0.412087
$582$	0	0
$583$	20140.9	1.43079
$584$	11857.6	0.840187
$585$	0	0
$586$	13504.8	0.952010
$587$	16355.2	1.15000	0.575000	−	0.818154i	$-0.305002\pi$
$587$	16355.2	1.15000	0.575000	+	0.818154i	$0.305002\pi$
$588$	0	0
$589$	−4727.63	−0.330727
$590$	867.211	0.0605127
$591$	0	0
$592$	−710.030	−0.0492940
$593$	872.023	0.0603873	0.0301937	−	0.999544i	$-0.490388\pi$
$593$	872.023	0.0603873	0.0301937	+	0.999544i	$0.490388\pi$
$594$	0	0
$595$	−5815.20	−0.400672
$596$	3671.57	0.252338
$597$	0	0
$598$	−7427.14	−0.507890
$599$	−18981.3	−1.29475	−0.647374	−	0.762173i	$-0.724133\pi$
$599$	−18981.3	−1.29475	−0.647374	+	0.762173i	$0.724133\pi$
$600$	0	0
$601$	20699.4	1.40490	0.702450	−	0.711733i	$-0.252089\pi$
$601$	20699.4	1.40490	0.702450	+	0.711733i	$0.252089\pi$
$602$	−896.896	−0.0607222
$603$	0	0
$604$	−9026.65	−0.608095
$605$	935.627	0.0628738
$606$	0	0
$607$	17340.9	1.15955	0.579773	−	0.814778i	$-0.303141\pi$
$607$	17340.9	1.15955	0.579773	+	0.814778i	$0.303141\pi$
$608$	−13000.2	−0.867153
$609$	0	0
$610$	−10825.6	−0.718550
$611$	1626.83	0.107716
$612$	0	0
$613$	17208.1	1.13382	0.566908	−	0.823781i	$-0.308140\pi$
$613$	17208.1	1.13382	0.566908	+	0.823781i	$0.308140\pi$
$614$	4141.67	0.272222
$615$	0	0
$616$	−9870.98	−0.645638
$617$	−17559.2	−1.14572	−0.572858	−	0.819655i	$-0.694165\pi$
$617$	−17559.2	−1.14572	−0.572858	+	0.819655i	$0.694165\pi$
$618$	0	0
$619$	5211.57	0.338402	0.169201	−	0.985582i	$-0.445881\pi$
$619$	5211.57	0.338402	0.169201	+	0.985582i	$0.445881\pi$
$620$	−2188.09	−0.141735
$621$	0	0
$622$	−500.980	−0.0322950
$623$	−17360.3	−1.11641
$624$	0	0
$625$	−4021.67	−0.257387
$626$	14844.2	0.947753
$627$	0	0
$628$	−8134.13	−0.516859
$629$	−5300.53	−0.336003
$630$	0	0
$631$	24188.9	1.52606	0.763030	−	0.646363i	$-0.223711\pi$
$631$	24188.9	1.52606	0.763030	+	0.646363i	$0.223711\pi$
$632$	−9858.19	−0.620471
$633$	0	0
$634$	−7423.97	−0.465053
$635$	3325.17	0.207803
$636$	0	0
$637$	8843.52	0.550067
$638$	−18342.8	−1.13824
$639$	0	0
$640$	−4951.11	−0.305797
$641$	−17252.2	−1.06306	−0.531530	−	0.847039i	$-0.678383\pi$
$641$	−17252.2	−1.06306	−0.531530	+	0.847039i	$0.678383\pi$
$642$	0	0
$643$	−16300.9	−0.999756	−0.499878	−	0.866096i	$-0.666622\pi$
$643$	−16300.9	−0.999756	−0.499878	+	0.866096i	$0.666622\pi$
$644$	−4811.45	−0.294406
$645$	0	0
$646$	9525.25	0.580133
$647$	−4467.83	−0.271482	−0.135741	−	0.990744i	$-0.543341\pi$
$647$	−4467.83	−0.271482	−0.135741	+	0.990744i	$0.543341\pi$
$648$	0	0
$649$	2202.60	0.133220
$650$	4687.84	0.282880
$651$	0	0
$652$	−11342.7	−0.681309
$653$	18596.2	1.11443	0.557216	−	0.830367i	$-0.311869\pi$
$653$	18596.2	1.11443	0.557216	+	0.830367i	$0.311869\pi$
$654$	0	0
$655$	2823.63	0.168440
$656$	−1251.17	−0.0744662
$657$	0	0
$658$	−848.138	−0.0502490
$659$	16863.2	0.996811	0.498406	−	0.866944i	$-0.333919\pi$
$659$	16863.2	0.996811	0.498406	+	0.866944i	$0.333919\pi$
$660$	0	0
$661$	18006.1	1.05954	0.529772	−	0.848140i	$-0.322278\pi$
$661$	18006.1	1.05954	0.529772	+	0.848140i	$0.322278\pi$
$662$	16808.5	0.986830
$663$	0	0
$664$	12271.1	0.717186
$665$	−6659.35	−0.388329
$666$	0	0
$667$	−25077.1	−1.45576
$668$	3944.33	0.228459
$669$	0	0
$670$	−13857.2	−0.799032
$671$	−27495.5	−1.58190
$672$	0	0
$673$	7425.14	0.425287	0.212644	−	0.977130i	$-0.431793\pi$
$673$	7425.14	0.425287	0.212644	+	0.977130i	$0.431793\pi$
$674$	2385.48	0.136328
$675$	0	0
$676$	2643.42	0.150399
$677$	8749.04	0.496681	0.248340	−	0.968673i	$-0.420115\pi$
$677$	8749.04	0.496681	0.248340	+	0.968673i	$0.420115\pi$
$678$	0	0
$679$	−7141.42	−0.403627
$680$	12365.1	0.697321
$681$	0	0
$682$	4472.46	0.251114
$683$	−24690.6	−1.38325	−0.691625	−	0.722257i	$-0.743105\pi$
$683$	−24690.6	−1.38325	−0.691625	+	0.722257i	$0.743105\pi$
$684$	0	0
$685$	−9642.69	−0.537852
$686$	−11764.8	−0.654786
$687$	0	0
$688$	382.249	0.0211818
$689$	−21169.3	−1.17052
$690$	0	0
$691$	10777.5	0.593335	0.296667	−	0.954981i	$-0.404125\pi$
$691$	10777.5	0.593335	0.296667	+	0.954981i	$0.404125\pi$
$692$	10136.3	0.556825
$693$	0	0
$694$	−14889.2	−0.814389
$695$	17525.7	0.956531
$696$	0	0
$697$	−9340.23	−0.507584
$698$	20662.0	1.12044
$699$	0	0
$700$	3036.87	0.163976
$701$	24779.3	1.33509	0.667547	−	0.744568i	$-0.267344\pi$
$701$	24779.3	1.33509	0.667547	+	0.744568i	$0.267344\pi$
$702$	0	0
$703$	−6069.96	−0.325651
$704$	15005.6	0.803330
$705$	0	0
$706$	1396.06	0.0744214
$707$	2502.05	0.133096
$708$	0	0
$709$	−28762.6	−1.52356	−0.761778	−	0.647838i	$-0.775673\pi$
$709$	−28762.6	−1.52356	−0.761778	+	0.647838i	$0.775673\pi$
$710$	5062.50	0.267594
$711$	0	0
$712$	36913.8	1.94298
$713$	6114.48	0.321163
$714$	0	0
$715$	−12084.0	−0.632051
$716$	8283.09	0.432338
$717$	0	0
$718$	6762.19	0.351480
$719$	−11366.3	−0.589555	−0.294778	−	0.955566i	$-0.595246\pi$
$719$	−11366.3	−0.589555	−0.294778	+	0.955566i	$0.595246\pi$
$720$	0	0
$721$	3411.84	0.176232
$722$	−2046.84	−0.105506
$723$	0	0
$724$	8357.65	0.429019
$725$	15828.1	0.810813
$726$	0	0
$727$	−19684.1	−1.00418	−0.502092	−	0.864814i	$-0.667436\pi$
$727$	−19684.1	−1.00418	−0.502092	+	0.864814i	$0.667436\pi$
$728$	10375.0	0.528191
$729$	0	0
$730$	−7567.82	−0.383695
$731$	2853.57	0.144382
$732$	0	0
$733$	11084.2	0.558534	0.279267	−	0.960213i	$-0.409909\pi$
$733$	11084.2	0.558534	0.279267	+	0.960213i	$0.409909\pi$
$734$	−22349.6	−1.12389
$735$	0	0
$736$	16813.9	0.842075
$737$	−35195.5	−1.75908
$738$	0	0
$739$	−32551.5	−1.62033	−0.810166	−	0.586200i	$-0.800623\pi$
$739$	−32551.5	−1.62033	−0.810166	+	0.586200i	$0.800623\pi$
$740$	−2809.37	−0.139560
$741$	0	0
$742$	11036.5	0.546040
$743$	−7712.05	−0.380791	−0.190395	−	0.981707i	$-0.560977\pi$
$743$	−7712.05	−0.380791	−0.190395	+	0.981707i	$0.560977\pi$
$744$	0	0
$745$	−6572.40	−0.323214
$746$	23417.6	1.14930
$747$	0	0
$748$	11197.2	0.547340
$749$	−17115.7	−0.834971
$750$	0	0
$751$	19253.4	0.935508	0.467754	−	0.883859i	$-0.345063\pi$
$751$	19253.4	0.935508	0.467754	+	0.883859i	$0.345063\pi$
$752$	361.469	0.0175285
$753$	0	0
$754$	19279.3	0.931183
$755$	16158.4	0.778894
$756$	0	0
$757$	12426.6	0.596636	0.298318	−	0.954467i	$-0.403574\pi$
$757$	12426.6	0.596636	0.298318	+	0.954467i	$0.403574\pi$
$758$	2417.81	0.115856
$759$	0	0
$760$	14160.0	0.675838
$761$	−19431.6	−0.925618	−0.462809	−	0.886458i	$-0.653159\pi$
$761$	−19431.6	−0.925618	−0.462809	+	0.886458i	$0.653159\pi$
$762$	0	0
$763$	−5840.49	−0.277116
$764$	−1464.06	−0.0693295
$765$	0	0
$766$	25172.9	1.18738
$767$	−2315.06	−0.108986
$768$	0	0
$769$	−38343.4	−1.79805	−0.899024	−	0.437900i	$-0.855723\pi$
$769$	−38343.4	−1.79805	−0.899024	+	0.437900i	$0.855723\pi$
$770$	6299.93	0.294849
$771$	0	0
$772$	13970.1	0.651289
$773$	−18166.0	−0.845259	−0.422630	−	0.906302i	$-0.638893\pi$
$773$	−18166.0	−0.845259	−0.422630	+	0.906302i	$0.638893\pi$
$774$	0	0
$775$	−3859.31	−0.178878
$776$	15185.0	0.702463
$777$	0	0
$778$	−618.349	−0.0284947
$779$	−10696.1	−0.491947
$780$	0	0
$781$	12858.0	0.589113
$782$	−12319.5	−0.563356
$783$	0	0
$784$	1964.96	0.0895117
$785$	14560.7	0.662032
$786$	0	0
$787$	−24389.3	−1.10468	−0.552341	−	0.833618i	$-0.686266\pi$
$787$	−24389.3	−1.10468	−0.552341	+	0.833618i	$0.686266\pi$
$788$	−10198.3	−0.461041
$789$	0	0
$790$	6291.77	0.283356
$791$	−19750.4	−0.887793
$792$	0	0
$793$	28899.5	1.29414
$794$	−15361.7	−0.686607
$795$	0	0
$796$	1515.83	0.0674965
$797$	27341.1	1.21514	0.607572	−	0.794264i	$-0.292144\pi$
$797$	27341.1	1.21514	0.607572	+	0.794264i	$0.292144\pi$
$798$	0	0
$799$	2698.44	0.119479
$800$	−10612.5	−0.469011
$801$	0	0
$802$	−7398.08	−0.325730
$803$	−19221.2	−0.844710
$804$	0	0
$805$	8612.88	0.377098
$806$	−4700.83	−0.205434
$807$	0	0
$808$	−5320.18	−0.231638
$809$	28610.4	1.24337	0.621685	−	0.783267i	$-0.286448\pi$
$809$	28610.4	1.24337	0.621685	+	0.783267i	$0.286448\pi$
$810$	0	0
$811$	20977.5	0.908284	0.454142	−	0.890929i	$-0.349946\pi$
$811$	20977.5	0.908284	0.454142	+	0.890929i	$0.349946\pi$
$812$	12489.5	0.539774
$813$	0	0
$814$	5742.35	0.247260
$815$	20304.3	0.872673
$816$	0	0
$817$	3267.80	0.139934
$818$	−4620.64	−0.197502
$819$	0	0
$820$	−4950.48	−0.210827
$821$	−20078.3	−0.853515	−0.426758	−	0.904366i	$-0.640344\pi$
$821$	−20078.3	−0.853515	−0.426758	+	0.904366i	$0.640344\pi$
$822$	0	0
$823$	−27473.5	−1.16363	−0.581813	−	0.813322i	$-0.697657\pi$
$823$	−27473.5	−1.16363	−0.581813	+	0.813322i	$0.697657\pi$
$824$	−7254.70	−0.306711
$825$	0	0
$826$	1206.95	0.0508414
$827$	−15097.0	−0.634792	−0.317396	−	0.948293i	$-0.602808\pi$
$827$	−15097.0	−0.634792	−0.317396	+	0.948293i	$0.602808\pi$
$828$	0	0
$829$	30521.9	1.27873	0.639366	−	0.768902i	$-0.279197\pi$
$829$	30521.9	1.27873	0.639366	+	0.768902i	$0.279197\pi$
$830$	−7831.77	−0.327524
$831$	0	0
$832$	−15771.8	−0.657197
$833$	14668.9	0.610139
$834$	0	0
$835$	−7060.67	−0.292628
$836$	12822.6	0.530478
$837$	0	0
$838$	−15282.8	−0.629995
$839$	9892.69	0.407072	0.203536	−	0.979067i	$-0.434757\pi$
$839$	9892.69	0.407072	0.203536	+	0.979067i	$0.434757\pi$
$840$	0	0
$841$	40706.0	1.66903
$842$	−21235.0	−0.869129
$843$	0	0
$844$	22634.5	0.923118
$845$	−4731.93	−0.192643
$846$	0	0
$847$	1302.16	0.0528251
$848$	−4703.65	−0.190476
$849$	0	0
$850$	7775.77	0.313773
$851$	7850.60	0.316234
$852$	0	0
$853$	11141.5	0.447220	0.223610	−	0.974679i	$-0.428216\pi$
$853$	11141.5	0.447220	0.223610	+	0.974679i	$0.428216\pi$
$854$	−15066.6	−0.603709
$855$	0	0
$856$	36393.6	1.45317
$857$	15957.8	0.636064	0.318032	−	0.948080i	$-0.396978\pi$
$857$	15957.8	0.636064	0.318032	+	0.948080i	$0.396978\pi$
$858$	0	0
$859$	−11813.6	−0.469236	−0.234618	−	0.972088i	$-0.575384\pi$
$859$	−11813.6	−0.469236	−0.234618	+	0.972088i	$0.575384\pi$
$860$	1512.44	0.0599695
$861$	0	0
$862$	3418.26	0.135065
$863$	20352.9	0.802804	0.401402	−	0.915902i	$-0.368523\pi$
$863$	20352.9	0.802804	0.401402	+	0.915902i	$0.368523\pi$
$864$	0	0
$865$	−18144.7	−0.713225
$866$	4188.33	0.164348
$867$	0	0
$868$	−3045.29	−0.119083
$869$	15980.2	0.623812
$870$	0	0
$871$	36992.6	1.43909
$872$	12418.8	0.482287
$873$	0	0
$874$	−14107.8	−0.546000
$875$	−16389.8	−0.633230
$876$	0	0
$877$	−23229.7	−0.894427	−0.447213	−	0.894427i	$-0.647584\pi$
$877$	−23229.7	−0.894427	−0.447213	+	0.894427i	$0.647584\pi$
$878$	−20398.5	−0.784073
$879$	0	0
$880$	−2684.97	−0.102853
$881$	−29443.9	−1.12598	−0.562992	−	0.826462i	$-0.690350\pi$
$881$	−29443.9	−1.12598	−0.562992	+	0.826462i	$0.690350\pi$
$882$	0	0
$883$	−1339.17	−0.0510381	−0.0255191	−	0.999674i	$-0.508124\pi$
$883$	−1339.17	−0.0510381	−0.0255191	+	0.999674i	$0.508124\pi$
$884$	−11768.9	−0.447774
$885$	0	0
$886$	−8880.95	−0.336751
$887$	−46970.2	−1.77802	−0.889011	−	0.457887i	$-0.848607\pi$
$887$	−46970.2	−1.77802	−0.889011	+	0.457887i	$0.848607\pi$
$888$	0	0
$889$	4627.82	0.174592
$890$	−23559.4	−0.887318
$891$	0	0
$892$	−18625.3	−0.699126
$893$	3090.15	0.115798
$894$	0	0
$895$	−14827.4	−0.553771
$896$	−6890.73	−0.256923
$897$	0	0
$898$	24404.9	0.906906
$899$	−15871.9	−0.588831
$900$	0	0
$901$	−35113.7	−1.29834
$902$	10118.8	0.373524
$903$	0	0
$904$	41996.0	1.54510
$905$	−14960.9	−0.549521
$906$	0	0
$907$	−28436.0	−1.04102	−0.520509	−	0.853856i	$-0.674258\pi$
$907$	−28436.0	−1.04102	−0.520509	+	0.853856i	$0.674258\pi$
$908$	−6310.32	−0.230634
$909$	0	0
$910$	−6621.61	−0.241213
$911$	−53330.0	−1.93952	−0.969759	−	0.244065i	$-0.921519\pi$
$911$	−53330.0	−1.93952	−0.969759	+	0.244065i	$0.921519\pi$
$912$	0	0
$913$	−19891.6	−0.721048
$914$	4725.68	0.171019
$915$	0	0
$916$	8222.38	0.296588
$917$	3929.79	0.141519
$918$	0	0
$919$	38701.2	1.38916	0.694579	−	0.719416i	$-0.255591\pi$
$919$	38701.2	1.38916	0.694579	+	0.719416i	$0.255591\pi$
$920$	−18313.9	−0.656293
$921$	0	0
$922$	−15392.7	−0.549818
$923$	−13514.6	−0.481948
$924$	0	0
$925$	−4955.11	−0.176133
$926$	−13146.8	−0.466556
$927$	0	0
$928$	−43645.3	−1.54389
$929$	14086.7	0.497493	0.248746	−	0.968569i	$-0.419981\pi$
$929$	14086.7	0.497493	0.248746	+	0.968569i	$0.419981\pi$
$930$	0	0
$931$	16798.2	0.591342
$932$	21351.2	0.750410
$933$	0	0
$934$	−14840.3	−0.519903
$935$	−20043.9	−0.701075
$936$	0	0
$937$	5700.03	0.198732	0.0993660	−	0.995051i	$-0.468319\pi$
$937$	5700.03	0.198732	0.0993660	+	0.995051i	$0.468319\pi$
$938$	−19285.9	−0.671328
$939$	0	0
$940$	1430.22	0.0496262
$941$	37751.7	1.30783	0.653917	−	0.756567i	$-0.273125\pi$
$941$	37751.7	1.30783	0.653917	+	0.756567i	$0.273125\pi$
$942$	0	0
$943$	13833.8	0.477720
$944$	−514.389	−0.0177351
$945$	0	0
$946$	−3091.43	−0.106248
$947$	1348.84	0.0462844	0.0231422	−	0.999732i	$-0.492633\pi$
$947$	1348.84	0.0462844	0.0231422	+	0.999732i	$0.492633\pi$
$948$	0	0
$949$	20202.7	0.691050
$950$	8904.52	0.304106
$951$	0	0
$952$	17209.1	0.585873
$953$	1896.25	0.0644548	0.0322274	−	0.999481i	$-0.489740\pi$
$953$	1896.25	0.0644548	0.0322274	+	0.999481i	$0.489740\pi$
$954$	0	0
$955$	2620.78	0.0888026
$956$	4112.13	0.139117
$957$	0	0
$958$	−3469.47	−0.117008
$959$	−13420.3	−0.451890
$960$	0	0
$961$	−25921.0	−0.870095
$962$	−6035.56	−0.202281
$963$	0	0
$964$	−26637.6	−0.889977
$965$	−25007.6	−0.834221
$966$	0	0
$967$	4853.86	0.161416	0.0807082	−	0.996738i	$-0.474282\pi$
$967$	4853.86	0.161416	0.0807082	+	0.996738i	$0.474282\pi$
$968$	−2768.83	−0.0919356
$969$	0	0
$970$	−9691.51	−0.320800
$971$	−16566.8	−0.547534	−0.273767	−	0.961796i	$-0.588270\pi$
$971$	−16566.8	−0.547534	−0.273767	+	0.961796i	$0.588270\pi$
$972$	0	0
$973$	24391.5	0.803655
$974$	32782.7	1.07847
$975$	0	0
$976$	6421.23	0.210593
$977$	−5209.92	−0.170604	−0.0853020	−	0.996355i	$-0.527186\pi$
$977$	−5209.92	−0.170604	−0.0853020	+	0.996355i	$0.527186\pi$
$978$	0	0
$979$	−59837.6	−1.95344
$980$	7774.74	0.253424
$981$	0	0
$982$	27685.4	0.899672
$983$	15460.1	0.501628	0.250814	−	0.968035i	$-0.419302\pi$
$983$	15460.1	0.501628	0.250814	+	0.968035i	$0.419302\pi$
$984$	0	0
$985$	18255.8	0.590537
$986$	31978.9	1.03288
$987$	0	0
$988$	−13477.3	−0.433979
$989$	−4226.41	−0.135887
$990$	0	0
$991$	−32935.0	−1.05572	−0.527859	−	0.849332i	$-0.677005\pi$
$991$	−32935.0	−1.05572	−0.527859	+	0.849332i	$0.677005\pi$
$992$	10641.9	0.340606
$993$	0	0
$994$	7045.75	0.224827
$995$	−2713.46	−0.0864547
$996$	0	0
$997$	−33495.5	−1.06401	−0.532003	−	0.846743i	$-0.678560\pi$
$997$	−33495.5	−1.06401	−0.532003	+	0.846743i	$0.678560\pi$
$998$	−20357.7	−0.645704
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1161.4.a.d.1.14	✓	21
3.2	odd	2		1161.4.a.e.1.8	yes	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1161.4.a.d.1.14	✓	21	1.1	even	1	trivial
1161.4.a.e.1.8	yes	21	3.2	odd	2

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:	\( N \)	\(=\)	\( 1161 = 3^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1161.a (trivial)

\(f(q)\)	\(=\)	\(q+1.88873 q^{2} -4.43270 q^{4} +7.93489 q^{5} +11.0434 q^{7} -23.4820 q^{8} +O(q^{10})\)	\(q+1.88873 q^{2} -4.43270 q^{4} +7.93489 q^{5} +11.0434 q^{7} -23.4820 q^{8} +14.9869 q^{10} +38.0646 q^{11} -40.0082 q^{13} +20.8580 q^{14} -8.88951 q^{16} -66.3621 q^{17} -75.9953 q^{19} -35.1730 q^{20} +71.8937 q^{22} +98.2887 q^{23} -62.0375 q^{25} -75.5646 q^{26} -48.9522 q^{28} -255.137 q^{29} +62.2094 q^{31} +171.066 q^{32} -125.340 q^{34} +87.6284 q^{35} +79.8729 q^{37} -143.535 q^{38} -186.327 q^{40} +140.746 q^{41} -43.0000 q^{43} -168.729 q^{44} +185.641 q^{46} -40.6624 q^{47} -221.043 q^{49} -117.172 q^{50} +177.344 q^{52} +529.124 q^{53} +302.038 q^{55} -259.322 q^{56} -481.885 q^{58} +57.8648 q^{59} -722.339 q^{61} +117.497 q^{62} +394.214 q^{64} -317.461 q^{65} -924.625 q^{67} +294.163 q^{68} +165.506 q^{70} +337.796 q^{71} -504.963 q^{73} +150.858 q^{74} +336.865 q^{76} +420.363 q^{77} +419.819 q^{79} -70.5373 q^{80} +265.832 q^{82} -522.575 q^{83} -526.576 q^{85} -81.2153 q^{86} -893.833 q^{88} -1572.00 q^{89} -441.827 q^{91} -435.684 q^{92} -76.8003 q^{94} -603.015 q^{95} -646.667 q^{97} -417.490 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - q^{2} + 87 q^{4} - 3 q^{5} + 3 q^{7} - 69 q^{8}+O(q^{10}) \) 21 * q - q^2 + 87 * q^4 - 3 * q^5 + 3 * q^7 - 69 * q^8	\( 21 q - q^{2} + 87 q^{4} - 3 q^{5} + 3 q^{7} - 69 q^{8} - 9 q^{10} - 129 q^{11} + 54 q^{13} - 184 q^{14} + 111 q^{16} - 68 q^{17} + 78 q^{19} - 123 q^{20} - 51 q^{22} - 586 q^{23} + 402 q^{25} - 125 q^{26} - 6 q^{28} - 556 q^{29} - 111 q^{31} - 758 q^{32} - 420 q^{34} - 1409 q^{35} + 330 q^{37} - 1067 q^{38} + 180 q^{40} - 678 q^{41} - 903 q^{43} - 1510 q^{44} - 1062 q^{46} - 1580 q^{47} + 1218 q^{49} - 2054 q^{50} + 717 q^{52} - 1069 q^{53} - 2259 q^{55} - 2823 q^{56} + 666 q^{58} - 2248 q^{59} - 162 q^{61} - 3211 q^{62} - 2025 q^{64} - 910 q^{65} - 216 q^{67} - 1868 q^{68} + 3636 q^{70} - 3088 q^{71} - 1605 q^{73} - 2516 q^{74} + 1263 q^{76} - 891 q^{77} + 2094 q^{79} - 5251 q^{80} - 966 q^{82} - 1993 q^{83} - 114 q^{85} + 43 q^{86} + 846 q^{88} - 2494 q^{89} - 2382 q^{91} - 8034 q^{92} - 1062 q^{94} - 4798 q^{95} + 2565 q^{97} - 3103 q^{98}+O(q^{100}) \) 21 * q - q^2 + 87 * q^4 - 3 * q^5 + 3 * q^7 - 69 * q^8 - 9 * q^10 - 129 * q^11 + 54 * q^13 - 184 * q^14 + 111 * q^16 - 68 * q^17 + 78 * q^19 - 123 * q^20 - 51 * q^22 - 586 * q^23 + 402 * q^25 - 125 * q^26 - 6 * q^28 - 556 * q^29 - 111 * q^31 - 758 * q^32 - 420 * q^34 - 1409 * q^35 + 330 * q^37 - 1067 * q^38 + 180 * q^40 - 678 * q^41 - 903 * q^43 - 1510 * q^44 - 1062 * q^46 - 1580 * q^47 + 1218 * q^49 - 2054 * q^50 + 717 * q^52 - 1069 * q^53 - 2259 * q^55 - 2823 * q^56 + 666 * q^58 - 2248 * q^59 - 162 * q^61 - 3211 * q^62 - 2025 * q^64 - 910 * q^65 - 216 * q^67 - 1868 * q^68 + 3636 * q^70 - 3088 * q^71 - 1605 * q^73 - 2516 * q^74 + 1263 * q^76 - 891 * q^77 + 2094 * q^79 - 5251 * q^80 - 966 * q^82 - 1993 * q^83 - 114 * q^85 + 43 * q^86 + 846 * q^88 - 2494 * q^89 - 2382 * q^91 - 8034 * q^92 - 1062 * q^94 - 4798 * q^95 + 2565 * q^97 - 3103 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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