LMFDB - Embedded newform 1175.4.a.l.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1175,4,Mod(1,1175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1175, 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("1175.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1175 = 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1175.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:	$69.3272442567$
Analytic rank:	$0$
Dimension:	$35$
Twist minimal:	no (minimal twist has level 235)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	1175.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.91402 q^{2} +8.67089 q^{3} +0.491508 q^{4} -25.2671 q^{6} -6.75637 q^{7} +21.8799 q^{8} +48.1844 q^{9} +O(q^{10})$	\(q-2.91402 q^{2} +8.67089 q^{3} +0.491508 q^{4} -25.2671 q^{6} -6.75637 q^{7} +21.8799 q^{8} +48.1844 q^{9} +13.0885 q^{11} +4.26181 q^{12} +5.65722 q^{13} +19.6882 q^{14} -67.6905 q^{16} +31.9809 q^{17} -140.410 q^{18} +4.86646 q^{19} -58.5837 q^{21} -38.1402 q^{22} +6.08843 q^{23} +189.718 q^{24} -16.4852 q^{26} +183.687 q^{27} -3.32081 q^{28} +152.627 q^{29} +229.694 q^{31} +22.2123 q^{32} +113.489 q^{33} -93.1928 q^{34} +23.6830 q^{36} -265.483 q^{37} -14.1810 q^{38} +49.0531 q^{39} +72.7486 q^{41} +170.714 q^{42} -67.8792 q^{43} +6.43311 q^{44} -17.7418 q^{46} +47.0000 q^{47} -586.937 q^{48} -297.352 q^{49} +277.302 q^{51} +2.78057 q^{52} +217.985 q^{53} -535.268 q^{54} -147.829 q^{56} +42.1966 q^{57} -444.757 q^{58} +493.890 q^{59} -479.736 q^{61} -669.333 q^{62} -325.551 q^{63} +476.797 q^{64} -330.709 q^{66} +783.965 q^{67} +15.7188 q^{68} +52.7921 q^{69} -777.675 q^{71} +1054.27 q^{72} +189.090 q^{73} +773.623 q^{74} +2.39191 q^{76} -88.4308 q^{77} -142.942 q^{78} -59.3826 q^{79} +291.755 q^{81} -211.991 q^{82} +1055.02 q^{83} -28.7944 q^{84} +197.801 q^{86} +1323.41 q^{87} +286.375 q^{88} +445.464 q^{89} -38.2222 q^{91} +2.99251 q^{92} +1991.65 q^{93} -136.959 q^{94} +192.600 q^{96} -112.364 q^{97} +866.488 q^{98} +630.662 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q + 8 q^{2} + 12 q^{3} + 144 q^{4} + 12 q^{6} + 84 q^{7} + 96 q^{8} + 333 q^{9}+O(q^{10}) $ 35 * q + 8 * q^2 + 12 * q^3 + 144 * q^4 + 12 * q^6 + 84 * q^7 + 96 * q^8 + 333 * q^9	\( 35 q + 8 q^{2} + 12 q^{3} + 144 q^{4} + 12 q^{6} + 84 q^{7} + 96 q^{8} + 333 q^{9} + 144 q^{12} + 294 q^{13} + 4 q^{14} + 580 q^{16} + 272 q^{17} + 1230 q^{18} - 96 q^{19} + 12 q^{21} + 484 q^{22} + 304 q^{23} - 54 q^{24} - 218 q^{26} + 432 q^{27} + 1350 q^{28} - 334 q^{29} - 248 q^{31} + 896 q^{32} + 1252 q^{33} + 148 q^{34} + 1836 q^{36} + 1480 q^{37} + 596 q^{38} - 332 q^{39} - 410 q^{41} + 672 q^{42} + 1506 q^{43} + 88 q^{44} - 456 q^{46} + 1645 q^{47} + 1298 q^{48} + 1459 q^{49} - 332 q^{51} + 3144 q^{52} + 1224 q^{53} + 524 q^{54} - 34 q^{56} + 4274 q^{57} + 1774 q^{58} - 86 q^{59} + 528 q^{61} + 2256 q^{62} + 1916 q^{63} + 2668 q^{64} + 1296 q^{66} + 4024 q^{67} + 286 q^{68} - 176 q^{69} + 556 q^{71} + 11280 q^{72} + 4584 q^{73} + 438 q^{74} - 3026 q^{76} + 3624 q^{77} + 446 q^{78} + 48 q^{79} + 4139 q^{81} + 2488 q^{82} + 1228 q^{83} - 3376 q^{84} - 610 q^{86} + 4770 q^{87} + 4168 q^{88} + 794 q^{89} + 1096 q^{91} + 4128 q^{92} + 1420 q^{93} + 376 q^{94} - 1100 q^{96} + 10624 q^{97} + 296 q^{98} + 1000 q^{99}+O(q^{100}) \) 35 * q + 8 * q^2 + 12 * q^3 + 144 * q^4 + 12 * q^6 + 84 * q^7 + 96 * q^8 + 333 * q^9 + 144 * q^12 + 294 * q^13 + 4 * q^14 + 580 * q^16 + 272 * q^17 + 1230 * q^18 - 96 * q^19 + 12 * q^21 + 484 * q^22 + 304 * q^23 - 54 * q^24 - 218 * q^26 + 432 * q^27 + 1350 * q^28 - 334 * q^29 - 248 * q^31 + 896 * q^32 + 1252 * q^33 + 148 * q^34 + 1836 * q^36 + 1480 * q^37 + 596 * q^38 - 332 * q^39 - 410 * q^41 + 672 * q^42 + 1506 * q^43 + 88 * q^44 - 456 * q^46 + 1645 * q^47 + 1298 * q^48 + 1459 * q^49 - 332 * q^51 + 3144 * q^52 + 1224 * q^53 + 524 * q^54 - 34 * q^56 + 4274 * q^57 + 1774 * q^58 - 86 * q^59 + 528 * q^61 + 2256 * q^62 + 1916 * q^63 + 2668 * q^64 + 1296 * q^66 + 4024 * q^67 + 286 * q^68 - 176 * q^69 + 556 * q^71 + 11280 * q^72 + 4584 * q^73 + 438 * q^74 - 3026 * q^76 + 3624 * q^77 + 446 * q^78 + 48 * q^79 + 4139 * q^81 + 2488 * q^82 + 1228 * q^83 - 3376 * q^84 - 610 * q^86 + 4770 * q^87 + 4168 * q^88 + 794 * q^89 + 1096 * q^91 + 4128 * q^92 + 1420 * q^93 + 376 * q^94 - 1100 * q^96 + 10624 * q^97 + 296 * q^98 + 1000 * q^99

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 $

$2$	−2.91402	−1.03026	−0.515131	−	0.857112i	$-0.672257\pi$
$2$	−2.91402	−1.03026	−0.515131	+	0.857112i	$0.672257\pi$
$3$	8.67089	1.66871	0.834357	−	0.551225i	$-0.185839\pi$
$3$	8.67089	1.66871	0.834357	+	0.551225i	$0.185839\pi$
$4$	0.491508	0.0614385
$5$	0	0
$5$	0	0
$6$	−25.2671	−1.71921
$7$	−6.75637	−0.364809	−0.182405	−	0.983224i	$-0.558388\pi$
$7$	−6.75637	−0.364809	−0.182405	+	0.983224i	$0.558388\pi$
$8$	21.8799	0.966964
$9$	48.1844	1.78461
$10$	0	0
$11$	13.0885	0.358758	0.179379	−	0.983780i	$-0.442591\pi$
$11$	13.0885	0.358758	0.179379	+	0.983780i	$0.442591\pi$
$12$	4.26181	0.102523
$13$	5.65722	0.120695	0.0603473	−	0.998177i	$-0.480779\pi$
$13$	5.65722	0.120695	0.0603473	+	0.998177i	$0.480779\pi$
$14$	19.6882	0.375849
$15$	0	0
$16$	−67.6905	−1.05766
$17$	31.9809	0.456264	0.228132	−	0.973630i	$-0.426738\pi$
$17$	31.9809	0.456264	0.228132	+	0.973630i	$0.426738\pi$
$18$	−140.410	−1.83861
$19$	4.86646	0.0587602	0.0293801	−	0.999568i	$-0.490647\pi$
$19$	4.86646	0.0587602	0.0293801	+	0.999568i	$0.490647\pi$
$20$	0	0
$21$	−58.5837	−0.608763
$22$	−38.1402	−0.369614
$23$	6.08843	0.0551967	0.0275984	−	0.999619i	$-0.491214\pi$
$23$	6.08843	0.0551967	0.0275984	+	0.999619i	$0.491214\pi$
$24$	189.718	1.61359
$25$	0	0
$26$	−16.4852	−0.124347
$27$	183.687	1.30928
$28$	−3.32081	−0.0224133
$29$	152.627	0.977312	0.488656	−	0.872476i	$-0.337487\pi$
$29$	152.627	0.977312	0.488656	+	0.872476i	$0.337487\pi$
$30$	0	0
$31$	229.694	1.33078	0.665392	−	0.746494i	$-0.268264\pi$
$31$	229.694	1.33078	0.665392	+	0.746494i	$0.268264\pi$
$32$	22.2123	0.122707
$33$	113.489	0.598664
$34$	−93.1928	−0.470072
$35$	0	0
$36$	23.6830	0.109644
$37$	−265.483	−1.17960	−0.589799	−	0.807550i	$-0.700793\pi$
$37$	−265.483	−1.17960	−0.589799	+	0.807550i	$0.700793\pi$
$38$	−14.1810	−0.0605384
$39$	49.0531	0.201405
$40$	0	0
$41$	72.7486	0.277108	0.138554	−	0.990355i	$-0.455755\pi$
$41$	72.7486	0.277108	0.138554	+	0.990355i	$0.455755\pi$
$42$	170.714	0.627185
$43$	−67.8792	−0.240732	−0.120366	−	0.992730i	$-0.538407\pi$
$43$	−67.8792	−0.240732	−0.120366	+	0.992730i	$0.538407\pi$
$44$	6.43311	0.0220415
$45$	0	0
$46$	−17.7418	−0.0568671
$47$	47.0000	0.145865
$47$	47.0000	0.145865
$48$	−586.937	−1.76494
$49$	−297.352	−0.866914
$50$	0	0
$51$	277.302	0.761375
$52$	2.78057	0.00741530
$53$	217.985	0.564955	0.282477	−	0.959274i	$-0.408844\pi$
$53$	217.985	0.564955	0.282477	+	0.959274i	$0.408844\pi$
$54$	−535.268	−1.34890
$55$	0	0
$56$	−147.829	−0.352757
$57$	42.1966	0.0980539
$58$	−444.757	−1.00689
$59$	493.890	1.08981	0.544907	−	0.838497i	$-0.316565\pi$
$59$	493.890	1.08981	0.544907	+	0.838497i	$0.316565\pi$
$60$	0	0
$61$	−479.736	−1.00695	−0.503474	−	0.864010i	$-0.667945\pi$
$61$	−479.736	−1.00695	−0.503474	+	0.864010i	$0.667945\pi$
$62$	−669.333	−1.37106
$63$	−325.551	−0.651041
$64$	476.797	0.931244
$65$	0	0
$66$	−330.709	−0.616780
$67$	783.965	1.42950	0.714751	−	0.699379i	$-0.246540\pi$
$67$	783.965	1.42950	0.714751	+	0.699379i	$0.246540\pi$
$68$	15.7188	0.0280322
$69$	52.7921	0.0921076
$70$	0	0
$71$	−777.675	−1.29990	−0.649951	−	0.759976i	$-0.725211\pi$
$71$	−777.675	−1.29990	−0.649951	+	0.759976i	$0.725211\pi$
$72$	1054.27	1.72565
$73$	189.090	0.303169	0.151584	−	0.988444i	$-0.451562\pi$
$73$	189.090	0.303169	0.151584	+	0.988444i	$0.451562\pi$
$74$	773.623	1.21530
$75$	0	0
$76$	2.39191	0.00361014
$77$	−88.4308	−0.130878
$78$	−142.942	−0.207500
$79$	−59.3826	−0.0845704	−0.0422852	−	0.999106i	$-0.513464\pi$
$79$	−59.3826	−0.0845704	−0.0422852	+	0.999106i	$0.513464\pi$
$80$	0	0
$81$	291.755	0.400213
$82$	−211.991	−0.285494
$83$	1055.02	1.39522	0.697610	−	0.716478i	$-0.254247\pi$
$83$	1055.02	1.39522	0.697610	+	0.716478i	$0.254247\pi$
$84$	−28.7944	−0.0374015
$85$	0	0
$86$	197.801	0.248017
$87$	1323.41	1.63085
$88$	286.375	0.346906
$89$	445.464	0.530552	0.265276	−	0.964173i	$-0.414537\pi$
$89$	445.464	0.530552	0.265276	+	0.964173i	$0.414537\pi$
$90$	0	0
$91$	−38.2222	−0.0440305
$92$	2.99251	0.00339121
$93$	1991.65	2.22070
$94$	−136.959	−0.150279
$95$	0	0
$96$	192.600	0.204762
$97$	−112.364	−0.117617	−0.0588084	−	0.998269i	$-0.518730\pi$
$97$	−112.364	−0.117617	−0.0588084	+	0.998269i	$0.518730\pi$
$98$	866.488	0.893148
$99$	630.662	0.640241
$100$	0	0
$101$	880.230	0.867190	0.433595	−	0.901108i	$-0.357245\pi$
$101$	880.230	0.867190	0.433595	+	0.901108i	$0.357245\pi$
$102$	−808.065	−0.784415
$103$	1023.79	0.979389	0.489694	−	0.871894i	$-0.337108\pi$
$103$	1023.79	0.979389	0.489694	+	0.871894i	$0.337108\pi$
$104$	123.779	0.116707
$105$	0	0
$106$	−635.214	−0.582051
$107$	1764.44	1.59416	0.797080	−	0.603873i	$-0.206377\pi$
$107$	1764.44	1.59416	0.797080	+	0.603873i	$0.206377\pi$
$108$	90.2838	0.0804404
$109$	−749.210	−0.658361	−0.329180	−	0.944267i	$-0.606772\pi$
$109$	−749.210	−0.658361	−0.329180	+	0.944267i	$0.606772\pi$
$110$	0	0
$111$	−2301.98	−1.96841
$112$	457.342	0.385846
$113$	1770.75	1.47414	0.737071	−	0.675815i	$-0.236208\pi$
$113$	1770.75	1.47414	0.737071	+	0.675815i	$0.236208\pi$
$114$	−122.962	−0.101021
$115$	0	0
$116$	75.0172	0.0600446
$117$	272.589	0.215392
$118$	−1439.20	−1.12279
$119$	−216.074	−0.166450
$120$	0	0
$121$	−1159.69	−0.871293
$122$	1397.96	1.03742
$123$	630.795	0.462414
$124$	112.897	0.0817614
$125$	0	0
$126$	948.662	0.670743
$127$	−293.718	−0.205222	−0.102611	−	0.994722i	$-0.532720\pi$
$127$	−293.718	−0.205222	−0.102611	+	0.994722i	$0.532720\pi$
$128$	−1567.09	−1.08213
$129$	−588.573	−0.401713
$130$	0	0
$131$	−2304.46	−1.53696	−0.768478	−	0.639877i	$-0.778985\pi$
$131$	−2304.46	−1.53696	−0.768478	+	0.639877i	$0.778985\pi$
$132$	55.7808	0.0367810
$133$	−32.8796	−0.0214363
$134$	−2284.49	−1.47276
$135$	0	0
$136$	699.737	0.441191
$137$	−1094.81	−0.682742	−0.341371	−	0.939929i	$-0.610891\pi$
$137$	−1094.81	−0.682742	−0.341371	+	0.939929i	$0.610891\pi$
$138$	−153.837	−0.0948949
$139$	1018.66	0.621592	0.310796	−	0.950477i	$-0.399404\pi$
$139$	1018.66	0.621592	0.310796	+	0.950477i	$0.399404\pi$
$140$	0	0
$141$	407.532	0.243407
$142$	2266.16	1.33924
$143$	74.0446	0.0433001
$144$	−3261.62	−1.88751
$145$	0	0
$146$	−551.012	−0.312343
$147$	−2578.30	−1.44663
$148$	−130.487	−0.0724728
$149$	−441.392	−0.242686	−0.121343	−	0.992611i	$-0.538720\pi$
$149$	−441.392	−0.242686	−0.121343	+	0.992611i	$0.538720\pi$
$150$	0	0
$151$	2675.63	1.44198	0.720992	−	0.692944i	$-0.243687\pi$
$151$	2675.63	1.44198	0.720992	+	0.692944i	$0.243687\pi$
$152$	106.478	0.0568190
$153$	1540.98	0.814252
$154$	257.689	0.134839
$155$	0	0
$156$	24.1100	0.0123740
$157$	795.479	0.404370	0.202185	−	0.979347i	$-0.435196\pi$
$157$	795.479	0.404370	0.202185	+	0.979347i	$0.435196\pi$
$158$	173.042	0.0871297
$159$	1890.13	0.942748
$160$	0	0
$161$	−41.1356	−0.0201363
$162$	−850.180	−0.412324
$163$	391.089	0.187929	0.0939647	−	0.995576i	$-0.470046\pi$
$163$	391.089	0.187929	0.0939647	+	0.995576i	$0.470046\pi$
$164$	35.7565	0.0170251
$165$	0	0
$166$	−3074.34	−1.43744
$167$	−3951.06	−1.83079	−0.915396	−	0.402554i	$-0.868123\pi$
$167$	−3951.06	−1.83079	−0.915396	+	0.402554i	$0.868123\pi$
$168$	−1281.81	−0.588651
$169$	−2165.00	−0.985433
$170$	0	0
$171$	234.487	0.104864
$172$	−33.3632	−0.0147902
$173$	1787.19	0.785421	0.392710	−	0.919662i	$-0.371538\pi$
$173$	1787.19	0.785421	0.392710	+	0.919662i	$0.371538\pi$
$174$	−3856.44	−1.68021
$175$	0	0
$176$	−885.968	−0.379445
$177$	4282.46	1.81859
$178$	−1298.09	−0.546607
$179$	4586.46	1.91513	0.957565	−	0.288216i	$-0.0930620\pi$
$179$	4586.46	1.91513	0.957565	+	0.288216i	$0.0930620\pi$
$180$	0	0
$181$	1916.55	0.787052	0.393526	−	0.919314i	$-0.371255\pi$
$181$	1916.55	0.787052	0.393526	+	0.919314i	$0.371255\pi$
$182$	111.380	0.0453630
$183$	−4159.74	−1.68031
$184$	133.214	0.0533732
$185$	0	0
$186$	−5803.72	−2.28790
$187$	418.582	0.163688
$188$	23.1009	0.00896173
$189$	−1241.06	−0.477639
$190$	0	0
$191$	2980.86	1.12926	0.564628	−	0.825346i	$-0.309020\pi$
$191$	2980.86	1.12926	0.564628	+	0.825346i	$0.309020\pi$
$192$	4134.25	1.55398
$193$	−1834.43	−0.684171	−0.342086	−	0.939669i	$-0.611133\pi$
$193$	−1834.43	−0.684171	−0.342086	+	0.939669i	$0.611133\pi$
$194$	327.431	0.121176
$195$	0	0
$196$	−146.151	−0.0532619
$197$	3883.03	1.40434	0.702168	−	0.712011i	$-0.252215\pi$
$197$	3883.03	1.40434	0.702168	+	0.712011i	$0.252215\pi$
$198$	−1837.76	−0.659616
$199$	−3927.32	−1.39900	−0.699498	−	0.714634i	$-0.746593\pi$
$199$	−3927.32	−1.39900	−0.699498	+	0.714634i	$0.746593\pi$
$200$	0	0
$201$	6797.68	2.38543
$202$	−2565.01	−0.893432
$203$	−1031.20	−0.356533
$204$	136.296	0.0467777
$205$	0	0
$206$	−2983.34	−1.00903
$207$	293.367	0.0985044
$208$	−382.940	−0.127654
$209$	63.6948	0.0210807
$210$	0	0
$211$	425.776	0.138918	0.0694589	−	0.997585i	$-0.477873\pi$
$211$	425.776	0.138918	0.0694589	+	0.997585i	$0.477873\pi$
$212$	107.142	0.0347100
$213$	−6743.14	−2.16916
$214$	−5141.62	−1.64240
$215$	0	0
$216$	4019.06	1.26603
$217$	−1551.90	−0.485483
$218$	2183.21	0.678284
$219$	1639.58	0.505902
$220$	0	0
$221$	180.923	0.0550687
$222$	6708.00	2.02798
$223$	5680.33	1.70575	0.852877	−	0.522113i	$-0.174856\pi$
$223$	5680.33	1.70575	0.852877	+	0.522113i	$0.174856\pi$
$224$	−150.074	−0.0447645
$225$	0	0
$226$	−5160.00	−1.51875
$227$	481.842	0.140885	0.0704427	−	0.997516i	$-0.477559\pi$
$227$	481.842	0.140885	0.0704427	+	0.997516i	$0.477559\pi$
$228$	20.7400	0.00602429
$229$	−1873.22	−0.540549	−0.270274	−	0.962783i	$-0.587114\pi$
$229$	−1873.22	−0.540549	−0.270274	+	0.962783i	$0.587114\pi$
$230$	0	0
$231$	−766.774	−0.218398
$232$	3339.45	0.945026
$233$	4826.07	1.35694	0.678468	−	0.734630i	$-0.262644\pi$
$233$	4826.07	1.35694	0.678468	+	0.734630i	$0.262644\pi$
$234$	−794.331	−0.221910
$235$	0	0
$236$	242.751	0.0669565
$237$	−514.900	−0.141124
$238$	629.645	0.171487
$239$	−5297.44	−1.43374	−0.716869	−	0.697208i	$-0.754425\pi$
$239$	−5297.44	−1.43374	−0.716869	+	0.697208i	$0.754425\pi$
$240$	0	0
$241$	4700.62	1.25640	0.628202	−	0.778050i	$-0.283791\pi$
$241$	4700.62	1.25640	0.628202	+	0.778050i	$0.283791\pi$
$242$	3379.36	0.897659
$243$	−2429.78	−0.641442
$244$	−235.794	−0.0618654
$245$	0	0
$246$	−1838.15	−0.476407
$247$	27.5306	0.00709204
$248$	5025.68	1.28682
$249$	9147.94	2.32822
$250$	0	0
$251$	4029.22	1.01324	0.506618	−	0.862171i	$-0.330895\pi$
$251$	4029.22	1.01324	0.506618	+	0.862171i	$0.330895\pi$
$252$	−160.011	−0.0399990
$253$	79.6885	0.0198023
$254$	855.898	0.211432
$255$	0	0
$256$	752.165	0.183634
$257$	−3209.23	−0.778935	−0.389468	−	0.921040i	$-0.627341\pi$
$257$	−3209.23	−0.778935	−0.389468	+	0.921040i	$0.627341\pi$
$258$	1715.11	0.413869
$259$	1793.70	0.430329
$260$	0	0
$261$	7354.22	1.74412
$262$	6715.23	1.58347
$263$	−2901.28	−0.680231	−0.340116	−	0.940384i	$-0.610466\pi$
$263$	−2901.28	−0.680231	−0.340116	+	0.940384i	$0.610466\pi$
$264$	2483.13	0.578886
$265$	0	0
$266$	95.8118	0.0220850
$267$	3862.57	0.885339
$268$	385.325	0.0878265
$269$	−2479.29	−0.561953	−0.280976	−	0.959715i	$-0.590658\pi$
$269$	−2479.29	−0.561953	−0.280976	+	0.959715i	$0.590658\pi$
$270$	0	0
$271$	−248.503	−0.0557028	−0.0278514	−	0.999612i	$-0.508867\pi$
$271$	−248.503	−0.0557028	−0.0278514	+	0.999612i	$0.508867\pi$
$272$	−2164.80	−0.482574
$273$	−331.421	−0.0734744
$274$	3190.29	0.703403
$275$	0	0
$276$	25.9477	0.00565895
$277$	7045.89	1.52833	0.764164	−	0.645022i	$-0.223152\pi$
$277$	7045.89	1.52833	0.764164	+	0.645022i	$0.223152\pi$
$278$	−2968.39	−0.640403
$279$	11067.7	2.37492
$280$	0	0
$281$	−2176.36	−0.462031	−0.231015	−	0.972950i	$-0.574205\pi$
$281$	−2176.36	−0.462031	−0.231015	+	0.972950i	$0.574205\pi$
$282$	−1187.56	−0.250773
$283$	6926.71	1.45495	0.727473	−	0.686136i	$-0.240694\pi$
$283$	6926.71	1.45495	0.727473	+	0.686136i	$0.240694\pi$
$284$	−382.234	−0.0798640
$285$	0	0
$286$	−215.767	−0.0446104
$287$	−491.516	−0.101092
$288$	1070.28	0.218983
$289$	−3890.23	−0.791823
$290$	0	0
$291$	−974.295	−0.196269
$292$	92.9393	0.0186262
$293$	3880.89	0.773802	0.386901	−	0.922121i	$-0.373546\pi$
$293$	3880.89	0.773802	0.386901	+	0.922121i	$0.373546\pi$
$294$	7513.22	1.49041
$295$	0	0
$296$	−5808.74	−1.14063
$297$	2404.19	0.469715
$298$	1286.23	0.250030
$299$	34.4436	0.00666195
$300$	0	0
$301$	458.616	0.0878213
$302$	−7796.83	−1.48562
$303$	7632.38	1.44709
$304$	−329.413	−0.0621485
$305$	0	0
$306$	−4490.44	−0.838893
$307$	−3018.79	−0.561210	−0.280605	−	0.959823i	$-0.590535\pi$
$307$	−3018.79	−0.561210	−0.280605	+	0.959823i	$0.590535\pi$
$308$	−43.4644	−0.00804096
$309$	8877.17	1.63432
$310$	0	0
$311$	7172.60	1.30778	0.653892	−	0.756588i	$-0.273135\pi$
$311$	7172.60	1.30778	0.653892	+	0.756588i	$0.273135\pi$
$312$	1073.28	0.194751
$313$	8587.23	1.55073	0.775366	−	0.631512i	$-0.217566\pi$
$313$	8587.23	1.55073	0.775366	+	0.631512i	$0.217566\pi$
$314$	−2318.04	−0.416607
$315$	0	0
$316$	−29.1870	−0.00519588
$317$	1674.41	0.296670	0.148335	−	0.988937i	$-0.452609\pi$
$317$	1674.41	0.296670	0.148335	+	0.988937i	$0.452609\pi$
$318$	−5507.87	−0.971277
$319$	1997.66	0.350618
$320$	0	0
$321$	15299.3	2.66020
$322$	119.870	0.0207456
$323$	155.634	0.0268102
$324$	143.400	0.0245885
$325$	0	0
$326$	−1139.64	−0.193616
$327$	−6496.32	−1.09862
$328$	1591.73	0.267953
$329$	−317.549	−0.0532129
$330$	0	0
$331$	2843.72	0.472220	0.236110	−	0.971726i	$-0.424127\pi$
$331$	2843.72	0.472220	0.236110	+	0.971726i	$0.424127\pi$
$332$	518.550	0.0857202
$333$	−12792.1	−2.10512
$334$	11513.5	1.88620
$335$	0	0
$336$	3965.56	0.643866
$337$	3045.53	0.492286	0.246143	−	0.969234i	$-0.420837\pi$
$337$	3045.53	0.492286	0.246143	+	0.969234i	$0.420837\pi$
$338$	6308.84	1.01525
$339$	15354.0	2.45992
$340$	0	0
$341$	3006.36	0.477429
$342$	−683.301	−0.108037
$343$	4326.45	0.681068
$344$	−1485.19	−0.232779
$345$	0	0
$346$	−5207.91	−0.809188
$347$	−9013.91	−1.39450	−0.697251	−	0.716827i	$-0.745594\pi$
$347$	−9013.91	−1.39450	−0.697251	+	0.716827i	$0.745594\pi$
$348$	650.466	0.100197
$349$	1642.45	0.251914	0.125957	−	0.992036i	$-0.459800\pi$
$349$	1642.45	0.251914	0.125957	+	0.992036i	$0.459800\pi$
$350$	0	0
$351$	1039.16	0.158023
$352$	290.725	0.0440219
$353$	11065.8	1.66848	0.834238	−	0.551404i	$-0.185908\pi$
$353$	11065.8	1.66848	0.834238	+	0.551404i	$0.185908\pi$
$354$	−12479.2	−1.87362
$355$	0	0
$356$	218.949	0.0325963
$357$	−1873.56	−0.277757
$358$	−13365.0	−1.97309
$359$	−5643.57	−0.829684	−0.414842	−	0.909894i	$-0.636163\pi$
$359$	−5643.57	−0.829684	−0.414842	+	0.909894i	$0.636163\pi$
$360$	0	0
$361$	−6835.32	−0.996547
$362$	−5584.88	−0.810869
$363$	−10055.6	−1.45394
$364$	−18.7865	−0.00270517
$365$	0	0
$366$	12121.6	1.73116
$367$	5054.38	0.718901	0.359450	−	0.933164i	$-0.382964\pi$
$367$	5054.38	0.718901	0.359450	+	0.933164i	$0.382964\pi$
$368$	−412.129	−0.0583796
$369$	3505.35	0.494528
$370$	0	0
$371$	−1472.79	−0.206101
$372$	978.914	0.136436
$373$	−336.611	−0.0467267	−0.0233633	−	0.999727i	$-0.507437\pi$
$373$	−336.611	−0.0467267	−0.0233633	+	0.999727i	$0.507437\pi$
$374$	−1219.76	−0.168642
$375$	0	0
$376$	1028.35	0.141046
$377$	863.442	0.117956
$378$	3616.47	0.492093
$379$	391.439	0.0530524	0.0265262	−	0.999648i	$-0.491555\pi$
$379$	391.439	0.0530524	0.0265262	+	0.999648i	$0.491555\pi$
$380$	0	0
$381$	−2546.79	−0.342457
$382$	−8686.30	−1.16343
$383$	−10867.1	−1.44982	−0.724910	−	0.688843i	$-0.758119\pi$
$383$	−10867.1	−1.44982	−0.724910	+	0.688843i	$0.758119\pi$
$384$	−13588.1	−1.80577
$385$	0	0
$386$	5345.56	0.704875
$387$	−3270.71	−0.429612
$388$	−55.2278	−0.00722620
$389$	−11505.1	−1.49957	−0.749783	−	0.661684i	$-0.769842\pi$
$389$	−11505.1	−1.49957	−0.749783	+	0.661684i	$0.769842\pi$
$390$	0	0
$391$	194.713	0.0251843
$392$	−6506.02	−0.838274
$393$	−19981.7	−2.56474
$394$	−11315.2	−1.44683
$395$	0	0
$396$	309.975	0.0393355
$397$	−3615.29	−0.457043	−0.228522	−	0.973539i	$-0.573389\pi$
$397$	−3615.29	−0.457043	−0.228522	+	0.973539i	$0.573389\pi$
$398$	11444.3	1.44133
$399$	−285.096	−0.0357710
$400$	0	0
$401$	2805.33	0.349355	0.174678	−	0.984626i	$-0.444112\pi$
$401$	2805.33	0.349355	0.174678	+	0.984626i	$0.444112\pi$
$402$	−19808.6	−2.45762
$403$	1299.43	0.160618
$404$	432.640	0.0532789
$405$	0	0
$406$	3004.94	0.367322
$407$	−3474.78	−0.423190
$408$	6067.35	0.736222
$409$	12145.2	1.46831	0.734156	−	0.678981i	$-0.237578\pi$
$409$	12145.2	1.46831	0.734156	+	0.678981i	$0.237578\pi$
$410$	0	0
$411$	−9492.95	−1.13930
$412$	503.201	0.0601722
$413$	−3336.90	−0.397574
$414$	−854.877	−0.101485
$415$	0	0
$416$	125.660	0.0148100
$417$	8832.66	1.03726
$418$	−185.608	−0.0217186
$419$	5415.56	0.631426	0.315713	−	0.948855i	$-0.397756\pi$
$419$	5415.56	0.631426	0.315713	+	0.948855i	$0.397756\pi$
$420$	0	0
$421$	−7241.59	−0.838322	−0.419161	−	0.907912i	$-0.637676\pi$
$421$	−7241.59	−0.838322	−0.419161	+	0.907912i	$0.637676\pi$
$422$	−1240.72	−0.143122
$423$	2264.67	0.260312
$424$	4769.50	0.546291
$425$	0	0
$426$	19649.6	2.23481
$427$	3241.27	0.367344
$428$	867.238	0.0979428
$429$	642.032	0.0722555
$430$	0	0
$431$	−16495.2	−1.84349	−0.921745	−	0.387797i	$-0.873236\pi$
$431$	−16495.2	−1.84349	−0.921745	+	0.387797i	$0.873236\pi$
$432$	−12433.9	−1.38478
$433$	−17007.0	−1.88754	−0.943771	−	0.330600i	$-0.892749\pi$
$433$	−17007.0	−1.88754	−0.943771	+	0.330600i	$0.892749\pi$
$434$	4522.26	0.500174
$435$	0	0
$436$	−368.243	−0.0404487
$437$	29.6291	0.00324337
$438$	−4777.77	−0.521211
$439$	8469.56	0.920797	0.460398	−	0.887712i	$-0.347707\pi$
$439$	8469.56	0.920797	0.460398	+	0.887712i	$0.347707\pi$
$440$	0	0
$441$	−14327.7	−1.54710
$442$	−527.212	−0.0567351
$443$	−18201.2	−1.95207	−0.976034	−	0.217619i	$-0.930171\pi$
$443$	−18201.2	−1.95207	−0.976034	+	0.217619i	$0.930171\pi$
$444$	−1131.44	−0.120936
$445$	0	0
$446$	−16552.6	−1.75737
$447$	−3827.26	−0.404974
$448$	−3221.41	−0.339727
$449$	5166.30	0.543013	0.271506	−	0.962437i	$-0.412478\pi$
$449$	5166.30	0.543013	0.271506	+	0.962437i	$0.412478\pi$
$450$	0	0
$451$	952.171	0.0994146
$452$	870.337	0.0905691
$453$	23200.1	2.40626
$454$	−1404.10	−0.145149
$455$	0	0
$456$	923.257	0.0948146
$457$	−2791.11	−0.285695	−0.142848	−	0.989745i	$-0.545626\pi$
$457$	−2791.11	−0.285695	−0.142848	+	0.989745i	$0.545626\pi$
$458$	5458.59	0.556907
$459$	5874.48	0.597379
$460$	0	0
$461$	−7045.54	−0.711808	−0.355904	−	0.934522i	$-0.615827\pi$
$461$	−7045.54	−0.711808	−0.355904	+	0.934522i	$0.615827\pi$
$462$	2234.39	0.225007
$463$	−3047.22	−0.305867	−0.152934	−	0.988236i	$-0.548872\pi$
$463$	−3047.22	−0.305867	−0.152934	+	0.988236i	$0.548872\pi$
$464$	−10331.4	−1.03367
$465$	0	0
$466$	−14063.3	−1.39800
$467$	−1937.86	−0.192020	−0.0960100	−	0.995380i	$-0.530608\pi$
$467$	−1937.86	−0.192020	−0.0960100	+	0.995380i	$0.530608\pi$
$468$	133.980	0.0132334
$469$	−5296.76	−0.521496
$470$	0	0
$471$	6897.51	0.674778
$472$	10806.3	1.05381
$473$	−888.437	−0.0863645
$474$	1500.43	0.145394
$475$	0	0
$476$	−106.202	−0.0102264
$477$	10503.5	1.00822
$478$	15436.9	1.47712
$479$	−11078.1	−1.05673	−0.528364	−	0.849018i	$-0.677195\pi$
$479$	−11078.1	−1.05673	−0.528364	+	0.849018i	$0.677195\pi$
$480$	0	0
$481$	−1501.90	−0.142371
$482$	−13697.7	−1.29443
$483$	−356.683	−0.0336017
$484$	−569.997	−0.0535309
$485$	0	0
$486$	7080.43	0.660853
$487$	13043.5	1.21367	0.606834	−	0.794829i	$-0.292439\pi$
$487$	13043.5	1.21367	0.606834	+	0.794829i	$0.292439\pi$
$488$	−10496.6	−0.973683
$489$	3391.09	0.313600
$490$	0	0
$491$	4390.30	0.403526	0.201763	−	0.979434i	$-0.435333\pi$
$491$	4390.30	0.403526	0.201763	+	0.979434i	$0.435333\pi$
$492$	310.041	0.0284100
$493$	4881.13	0.445913
$494$	−80.2248	−0.00730665
$495$	0	0
$496$	−15548.1	−1.40752
$497$	5254.26	0.474217
$498$	−26657.3	−2.39868
$499$	−6824.07	−0.612199	−0.306100	−	0.952000i	$-0.599024\pi$
$499$	−6824.07	−0.612199	−0.306100	+	0.952000i	$0.599024\pi$
$500$	0	0
$501$	−34259.2	−3.05507
$502$	−11741.2	−1.04390
$503$	12309.0	1.09112	0.545559	−	0.838072i	$-0.316317\pi$
$503$	12309.0	1.09112	0.545559	+	0.838072i	$0.316317\pi$
$504$	−7123.02	−0.629533
$505$	0	0
$506$	−232.214	−0.0204015
$507$	−18772.4	−1.64441
$508$	−144.365	−0.0126085
$509$	4148.87	0.361288	0.180644	−	0.983549i	$-0.442182\pi$
$509$	4148.87	0.361288	0.180644	+	0.983549i	$0.442182\pi$
$510$	0	0
$511$	−1277.56	−0.110599
$512$	10344.9	0.892940
$513$	893.908	0.0769337
$514$	9351.76	0.802507
$515$	0	0
$516$	−289.288	−0.0246806
$517$	615.160	0.0523302
$518$	−5226.88	−0.443351
$519$	15496.6	1.31064
$520$	0	0
$521$	−17398.9	−1.46307	−0.731535	−	0.681804i	$-0.761196\pi$
$521$	−17398.9	−1.46307	−0.731535	+	0.681804i	$0.761196\pi$
$522$	−21430.3	−1.79690
$523$	−4631.40	−0.387222	−0.193611	−	0.981078i	$-0.562020\pi$
$523$	−4631.40	−0.387222	−0.193611	+	0.981078i	$0.562020\pi$
$524$	−1132.66	−0.0944283
$525$	0	0
$526$	8454.40	0.700816
$527$	7345.82	0.607189
$528$	−7682.13	−0.633185
$529$	−12129.9	−0.996953
$530$	0	0
$531$	23797.8	1.94489
$532$	−16.1606	−0.00131701
$533$	411.555	0.0334454
$534$	−11255.6	−0.912131
$535$	0	0
$536$	17153.1	1.38228
$537$	39768.7	3.19581
$538$	7224.71	0.578958
$539$	−3891.89	−0.311012
$540$	0	0
$541$	12238.1	0.972560	0.486280	−	0.873803i	$-0.338353\pi$
$541$	12238.1	0.972560	0.486280	+	0.873803i	$0.338353\pi$
$542$	724.141	0.0573884
$543$	16618.2	1.31336
$544$	710.367	0.0559866
$545$	0	0
$546$	965.767	0.0756978
$547$	−15160.8	−1.18507	−0.592533	−	0.805546i	$-0.701872\pi$
$547$	−15160.8	−1.18507	−0.592533	+	0.805546i	$0.701872\pi$
$548$	−538.106	−0.0419466
$549$	−23115.8	−1.79701
$550$	0	0
$551$	742.752	0.0574271
$552$	1155.09	0.0890647
$553$	401.211	0.0308521
$554$	−20531.9	−1.57458
$555$	0	0
$556$	500.678	0.0381897
$557$	3935.83	0.299401	0.149700	−	0.988731i	$-0.452169\pi$
$557$	3935.83	0.299401	0.149700	+	0.988731i	$0.452169\pi$
$558$	−32251.4	−2.44679
$559$	−384.007	−0.0290550
$560$	0	0
$561$	3629.48	0.273149
$562$	6341.95	0.476012
$563$	8960.93	0.670796	0.335398	−	0.942077i	$-0.391129\pi$
$563$	8960.93	0.670796	0.335398	+	0.942077i	$0.391129\pi$
$564$	200.305	0.0149546
$565$	0	0
$566$	−20184.6	−1.49898
$567$	−1971.20	−0.146001
$568$	−17015.4	−1.25696
$569$	−20539.4	−1.51328	−0.756638	−	0.653834i	$-0.773160\pi$
$569$	−20539.4	−1.51328	−0.756638	+	0.653834i	$0.773160\pi$
$570$	0	0
$571$	−13437.9	−0.984868	−0.492434	−	0.870350i	$-0.663893\pi$
$571$	−13437.9	−0.984868	−0.492434	+	0.870350i	$0.663893\pi$
$572$	36.3935	0.00266030
$573$	25846.8	1.88440
$574$	1432.29	0.104151
$575$	0	0
$576$	22974.2	1.66190
$577$	16491.6	1.18987	0.594933	−	0.803775i	$-0.297179\pi$
$577$	16491.6	1.18987	0.594933	+	0.803775i	$0.297179\pi$
$578$	11336.2	0.815784
$579$	−15906.1	−1.14169
$580$	0	0
$581$	−7128.08	−0.508989
$582$	2839.11	0.202208
$583$	2853.11	0.202682
$584$	4137.27	0.293153
$585$	0	0
$586$	−11309.0	−0.797218
$587$	−988.872	−0.0695317	−0.0347659	−	0.999395i	$-0.511069\pi$
$587$	−988.872	−0.0695317	−0.0347659	+	0.999395i	$0.511069\pi$
$588$	−1267.26	−0.0888789
$589$	1117.80	0.0781971
$590$	0	0
$591$	33669.3	2.34344
$592$	17970.7	1.24762
$593$	−19941.1	−1.38092	−0.690459	−	0.723372i	$-0.742591\pi$
$593$	−19941.1	−1.38092	−0.690459	+	0.723372i	$0.742591\pi$
$594$	−7005.87	−0.483930
$595$	0	0
$596$	−216.948	−0.0149103
$597$	−34053.4	−2.33452
$598$	−100.369	−0.00686355
$599$	−20660.7	−1.40931	−0.704653	−	0.709552i	$-0.748897\pi$
$599$	−20660.7	−1.40931	−0.704653	+	0.709552i	$0.748897\pi$
$600$	0	0
$601$	−3289.29	−0.223250	−0.111625	−	0.993750i	$-0.535605\pi$
$601$	−3289.29	−0.223250	−0.111625	+	0.993750i	$0.535605\pi$
$602$	−1336.42	−0.0904789
$603$	37774.9	2.55110
$604$	1315.09	0.0885933
$605$	0	0
$606$	−22240.9	−1.49088
$607$	−13710.5	−0.916792	−0.458396	−	0.888748i	$-0.651576\pi$
$607$	−13710.5	−0.916792	−0.458396	+	0.888748i	$0.651576\pi$
$608$	108.095	0.00721026
$609$	−8941.44	−0.594951
$610$	0	0
$611$	265.889	0.0176051
$612$	757.403	0.0500264
$613$	7176.79	0.472867	0.236434	−	0.971648i	$-0.424021\pi$
$613$	7176.79	0.472867	0.236434	+	0.971648i	$0.424021\pi$
$614$	8796.82	0.578193
$615$	0	0
$616$	−1934.86	−0.126554
$617$	1027.75	0.0670591	0.0335296	−	0.999438i	$-0.489325\pi$
$617$	1027.75	0.0670591	0.0335296	+	0.999438i	$0.489325\pi$
$618$	−25868.3	−1.68378
$619$	16063.5	1.04305	0.521524	−	0.853237i	$-0.325364\pi$
$619$	16063.5	1.04305	0.521524	+	0.853237i	$0.325364\pi$
$620$	0	0
$621$	1118.37	0.0722682
$622$	−20901.1	−1.34736
$623$	−3009.72	−0.193550
$624$	−3320.43	−0.213019
$625$	0	0
$626$	−25023.4	−1.59766
$627$	552.291	0.0351776
$628$	390.984	0.0248439
$629$	−8490.38	−0.538209
$630$	0	0
$631$	13936.3	0.879231	0.439615	−	0.898186i	$-0.355115\pi$
$631$	13936.3	0.879231	0.439615	+	0.898186i	$0.355115\pi$
$632$	−1299.28	−0.0817765
$633$	3691.86	0.231814
$634$	−4879.27	−0.305647
$635$	0	0
$636$	929.013	0.0579210
$637$	−1682.18	−0.104632
$638$	−5821.21	−0.361229
$639$	−37471.8	−2.31981
$640$	0	0
$641$	−17417.9	−1.07327	−0.536634	−	0.843815i	$-0.680304\pi$
$641$	−17417.9	−1.07327	−0.536634	+	0.843815i	$0.680304\pi$
$642$	−44582.4	−2.74070
$643$	6553.24	0.401920	0.200960	−	0.979599i	$-0.435594\pi$
$643$	6553.24	0.401920	0.200960	+	0.979599i	$0.435594\pi$
$644$	−20.2185	−0.00123714
$645$	0	0
$646$	−453.520	−0.0276215
$647$	1158.59	0.0704004	0.0352002	−	0.999380i	$-0.488793\pi$
$647$	1158.59	0.0704004	0.0352002	+	0.999380i	$0.488793\pi$
$648$	6383.57	0.386991
$649$	6464.28	0.390979
$650$	0	0
$651$	−13456.3	−0.810131
$652$	192.224	0.0115461
$653$	−14018.6	−0.840106	−0.420053	−	0.907500i	$-0.637988\pi$
$653$	−14018.6	−0.840106	−0.420053	+	0.907500i	$0.637988\pi$
$654$	18930.4	1.13186
$655$	0	0
$656$	−4924.39	−0.293087
$657$	9111.18	0.541037
$658$	925.344	0.0548232
$659$	1558.81	0.0921434	0.0460717	−	0.998938i	$-0.485330\pi$
$659$	1558.81	0.0921434	0.0460717	+	0.998938i	$0.485330\pi$
$660$	0	0
$661$	−3193.49	−0.187916	−0.0939578	−	0.995576i	$-0.529952\pi$
$661$	−3193.49	−0.187916	−0.0939578	+	0.995576i	$0.529952\pi$
$662$	−8286.65	−0.486510
$663$	1568.76	0.0918938
$664$	23083.7	1.34913
$665$	0	0
$666$	37276.5	2.16882
$667$	929.257	0.0539445
$668$	−1941.98	−0.112481
$669$	49253.5	2.84641
$670$	0	0
$671$	−6279.03	−0.361251
$672$	−1301.28	−0.0746991
$673$	−1761.50	−0.100893	−0.0504463	−	0.998727i	$-0.516064\pi$
$673$	−1761.50	−0.100893	−0.0504463	+	0.998727i	$0.516064\pi$
$674$	−8874.72	−0.507183
$675$	0	0
$676$	−1064.11	−0.0605435
$677$	−8603.43	−0.488415	−0.244207	−	0.969723i	$-0.578528\pi$
$677$	−8603.43	−0.488415	−0.244207	+	0.969723i	$0.578528\pi$
$678$	−44741.8	−2.53436
$679$	759.171	0.0429077
$680$	0	0
$681$	4178.00	0.235097
$682$	−8760.58	−0.491877
$683$	−14744.7	−0.826048	−0.413024	−	0.910720i	$-0.635527\pi$
$683$	−14744.7	−0.826048	−0.413024	+	0.910720i	$0.635527\pi$
$684$	115.253	0.00644267
$685$	0	0
$686$	−12607.4	−0.701678
$687$	−16242.5	−0.902022
$688$	4594.77	0.254613
$689$	1233.19	0.0681870
$690$	0	0
$691$	−20565.2	−1.13218	−0.566090	−	0.824343i	$-0.691545\pi$
$691$	−20565.2	−1.13218	−0.566090	+	0.824343i	$0.691545\pi$
$692$	878.420	0.0482551
$693$	−4260.98	−0.233566
$694$	26266.7	1.43670
$695$	0	0
$696$	28956.0	1.57698
$697$	2326.56	0.126434
$698$	−4786.12	−0.259538
$699$	41846.3	2.26434
$700$	0	0
$701$	−4399.17	−0.237025	−0.118513	−	0.992953i	$-0.537813\pi$
$701$	−4399.17	−0.237025	−0.118513	+	0.992953i	$0.537813\pi$
$702$	−3028.13	−0.162805
$703$	−1291.96	−0.0693134
$704$	6240.56	0.334091
$705$	0	0
$706$	−32245.9	−1.71897
$707$	−5947.16	−0.316359
$708$	2104.87	0.111731
$709$	−21218.1	−1.12393	−0.561963	−	0.827162i	$-0.689954\pi$
$709$	−21218.1	−1.12393	−0.561963	+	0.827162i	$0.689954\pi$
$710$	0	0
$711$	−2861.31	−0.150925
$712$	9746.71	0.513024
$713$	1398.48	0.0734549
$714$	5459.58	0.286162
$715$	0	0
$716$	2254.28	0.117663
$717$	−45933.6	−2.39250
$718$	16445.5	0.854791
$719$	−13905.5	−0.721263	−0.360631	−	0.932708i	$-0.617439\pi$
$719$	−13905.5	−0.721263	−0.360631	+	0.932708i	$0.617439\pi$
$720$	0	0
$721$	−6917.10	−0.357290
$722$	19918.2	1.02670
$723$	40758.6	2.09658
$724$	942.002	0.0483553
$725$	0	0
$726$	29302.1	1.49794
$727$	−21878.3	−1.11612	−0.558060	−	0.829800i	$-0.688454\pi$
$727$	−21878.3	−1.11612	−0.558060	+	0.829800i	$0.688454\pi$
$728$	−836.298	−0.0425759
$729$	−28945.8	−1.47060
$730$	0	0
$731$	−2170.83	−0.109837
$732$	−2044.54	−0.103236
$733$	−3047.48	−0.153563	−0.0767813	−	0.997048i	$-0.524464\pi$
$733$	−3047.48	−0.153563	−0.0767813	+	0.997048i	$0.524464\pi$
$734$	−14728.6	−0.740655
$735$	0	0
$736$	135.238	0.00677300
$737$	10260.9	0.512845
$738$	−10214.6	−0.509494
$739$	32719.2	1.62868	0.814340	−	0.580388i	$-0.197099\pi$
$739$	32719.2	1.62868	0.814340	+	0.580388i	$0.197099\pi$
$740$	0	0
$741$	238.715	0.0118346
$742$	4291.74	0.212338
$743$	569.079	0.0280989	0.0140494	−	0.999901i	$-0.495528\pi$
$743$	569.079	0.0280989	0.0140494	+	0.999901i	$0.495528\pi$
$744$	43577.2	2.14733
$745$	0	0
$746$	980.890	0.0481407
$747$	50835.4	2.48992
$748$	205.736	0.0100568
$749$	−11921.2	−0.581565
$750$	0	0
$751$	12590.1	0.611744	0.305872	−	0.952073i	$-0.401052\pi$
$751$	12590.1	0.611744	0.305872	+	0.952073i	$0.401052\pi$
$752$	−3181.45	−0.154276
$753$	34937.0	1.69080
$754$	−2516.09	−0.121526
$755$	0	0
$756$	−609.990	−0.0293454
$757$	−33733.6	−1.61964	−0.809820	−	0.586678i	$-0.800435\pi$
$757$	−33733.6	−1.61964	−0.809820	+	0.586678i	$0.800435\pi$
$758$	−1140.66	−0.0546579
$759$	690.970	0.0330443
$760$	0	0
$761$	−8792.22	−0.418814	−0.209407	−	0.977829i	$-0.567153\pi$
$761$	−8792.22	−0.418814	−0.209407	+	0.977829i	$0.567153\pi$
$762$	7421.40	0.352820
$763$	5061.94	0.240176
$764$	1465.12	0.0693798
$765$	0	0
$766$	31666.9	1.49369
$767$	2794.04	0.131535
$768$	6521.94	0.306433
$769$	−37423.3	−1.75490	−0.877451	−	0.479666i	$-0.840758\pi$
$769$	−37423.3	−1.75490	−0.877451	+	0.479666i	$0.840758\pi$
$770$	0	0
$771$	−27826.9	−1.29982
$772$	−901.637	−0.0420345
$773$	11625.8	0.540945	0.270473	−	0.962728i	$-0.412820\pi$
$773$	11625.8	0.540945	0.270473	+	0.962728i	$0.412820\pi$
$774$	9530.92	0.442612
$775$	0	0
$776$	−2458.51	−0.113731
$777$	15553.0	0.718096
$778$	33526.0	1.54494
$779$	354.029	0.0162829
$780$	0	0
$781$	−10178.6	−0.466350
$782$	−567.398	−0.0259464
$783$	28035.6	1.27958
$784$	20127.9	0.916904
$785$	0	0
$786$	58227.0	2.64235
$787$	−21047.3	−0.953310	−0.476655	−	0.879090i	$-0.658151\pi$
$787$	−21047.3	−0.953310	−0.476655	+	0.879090i	$0.658151\pi$
$788$	1908.54	0.0862803
$789$	−25156.7	−1.13511
$790$	0	0
$791$	−11963.8	−0.537781
$792$	13798.8	0.619090
$793$	−2713.97	−0.121533
$794$	10535.0	0.470874
$795$	0	0
$796$	−1930.31	−0.0859522
$797$	−34186.9	−1.51940	−0.759701	−	0.650272i	$-0.774655\pi$
$797$	−34186.9	−1.51940	−0.759701	+	0.650272i	$0.774655\pi$
$798$	830.774	0.0368535
$799$	1503.10	0.0665530
$800$	0	0
$801$	21464.4	0.946826
$802$	−8174.79	−0.359927
$803$	2474.91	0.108764
$804$	3341.11	0.146557
$805$	0	0
$806$	−3786.56	−0.165479
$807$	−21497.7	−0.937738
$808$	19259.3	0.838541
$809$	15273.4	0.663763	0.331882	−	0.943321i	$-0.392317\pi$
$809$	15273.4	0.663763	0.331882	+	0.943321i	$0.392317\pi$
$810$	0	0
$811$	35943.4	1.55628	0.778141	−	0.628090i	$-0.216163\pi$
$811$	35943.4	1.55628	0.778141	+	0.628090i	$0.216163\pi$
$812$	−506.844	−0.0219048
$813$	−2154.74	−0.0929520
$814$	10125.6	0.435997
$815$	0	0
$816$	−18770.7	−0.805279
$817$	−330.332	−0.0141455
$818$	−35391.2	−1.51274
$819$	−1841.71	−0.0785771
$820$	0	0
$821$	38543.6	1.63847	0.819234	−	0.573460i	$-0.194399\pi$
$821$	38543.6	1.63847	0.819234	+	0.573460i	$0.194399\pi$
$822$	27662.6	1.17378
$823$	14437.2	0.611480	0.305740	−	0.952115i	$-0.401096\pi$
$823$	14437.2	0.611480	0.305740	+	0.952115i	$0.401096\pi$
$824$	22400.4	0.947033
$825$	0	0
$826$	9723.79	0.409605
$827$	37039.4	1.55742	0.778710	−	0.627384i	$-0.215874\pi$
$827$	37039.4	1.55742	0.778710	+	0.627384i	$0.215874\pi$
$828$	144.192	0.00605197
$829$	−30537.8	−1.27940	−0.639700	−	0.768624i	$-0.720942\pi$
$829$	−30537.8	−1.27940	−0.639700	+	0.768624i	$0.720942\pi$
$830$	0	0
$831$	61094.2	2.55034
$832$	2697.34	0.112396
$833$	−9509.55	−0.395542
$834$	−25738.5	−1.06865
$835$	0	0
$836$	31.3065	0.00129517
$837$	42191.9	1.74237
$838$	−15781.0	−0.650534
$839$	−5657.88	−0.232815	−0.116407	−	0.993202i	$-0.537138\pi$
$839$	−5657.88	−0.232815	−0.116407	+	0.993202i	$0.537138\pi$
$840$	0	0
$841$	−1094.10	−0.0448605
$842$	21102.1	0.863690
$843$	−18871.0	−0.770997
$844$	209.273	0.00853490
$845$	0	0
$846$	−6599.28	−0.268189
$847$	7835.29	0.317856
$848$	−14755.5	−0.597532
$849$	60060.7	2.42789
$850$	0	0
$851$	−1616.37	−0.0651100
$852$	−3314.31	−0.133270
$853$	−43525.7	−1.74712	−0.873558	−	0.486720i	$-0.838193\pi$
$853$	−43525.7	−1.74712	−0.873558	+	0.486720i	$0.838193\pi$
$854$	−9445.13	−0.378461
$855$	0	0
$856$	38605.8	1.54150
$857$	23953.7	0.954775	0.477387	−	0.878693i	$-0.341584\pi$
$857$	23953.7	0.954775	0.477387	+	0.878693i	$0.341584\pi$
$858$	−1870.89	−0.0744421
$859$	8276.60	0.328747	0.164374	−	0.986398i	$-0.447440\pi$
$859$	8276.60	0.328747	0.164374	+	0.986398i	$0.447440\pi$
$860$	0	0
$861$	−4261.88	−0.168693
$862$	48067.2	1.89928
$863$	−27959.3	−1.10284	−0.551418	−	0.834229i	$-0.685913\pi$
$863$	−27959.3	−1.10284	−0.551418	+	0.834229i	$0.685913\pi$
$864$	4080.11	0.160658
$865$	0	0
$866$	49558.8	1.94466
$867$	−33731.7	−1.32133
$868$	−762.770	−0.0298273
$869$	−777.230	−0.0303403
$870$	0	0
$871$	4435.06	0.172533
$872$	−16392.6	−0.636611
$873$	−5414.18	−0.209900
$874$	−86.3398	−0.00334152
$875$	0	0
$876$	805.867	0.0310819
$877$	2139.28	0.0823698	0.0411849	−	0.999152i	$-0.486887\pi$
$877$	2139.28	0.0823698	0.0411849	+	0.999152i	$0.486887\pi$
$878$	−24680.4	−0.948661
$879$	33650.8	1.29125
$880$	0	0
$881$	13318.4	0.509316	0.254658	−	0.967031i	$-0.418037\pi$
$881$	13318.4	0.509316	0.254658	+	0.967031i	$0.418037\pi$
$882$	41751.2	1.59392
$883$	28484.2	1.08558	0.542791	−	0.839868i	$-0.317367\pi$
$883$	28484.2	1.08558	0.542791	+	0.839868i	$0.317367\pi$
$884$	88.9249	0.00338334
$885$	0	0
$886$	53038.7	2.01114
$887$	−30432.3	−1.15199	−0.575995	−	0.817453i	$-0.695385\pi$
$887$	−30432.3	−1.15199	−0.575995	+	0.817453i	$0.695385\pi$
$888$	−50367.0	−1.90338
$889$	1984.46	0.0748670
$890$	0	0
$891$	3818.64	0.143579
$892$	2791.93	0.104799
$893$	228.724	0.00857105
$894$	11152.7	0.417229
$895$	0	0
$896$	10587.9	0.394772
$897$	298.656	0.0111169
$898$	−15054.7	−0.559445
$899$	35057.5	1.30059
$900$	0	0
$901$	6971.36	0.257769
$902$	−2774.65	−0.102423
$903$	3976.61	0.146549
$904$	38743.8	1.42544
$905$	0	0
$906$	−67605.5	−2.47907
$907$	−4214.72	−0.154297	−0.0771486	−	0.997020i	$-0.524582\pi$
$907$	−4214.72	−0.154297	−0.0771486	+	0.997020i	$0.524582\pi$
$908$	236.829	0.00865578
$909$	42413.3	1.54759
$910$	0	0
$911$	−46561.5	−1.69336	−0.846680	−	0.532103i	$-0.821402\pi$
$911$	−46561.5	−1.69336	−0.846680	+	0.532103i	$0.821402\pi$
$912$	−2856.31	−0.103708
$913$	13808.6	0.500546
$914$	8133.35	0.294341
$915$	0	0
$916$	−920.702	−0.0332105
$917$	15569.7	0.560696
$918$	−17118.3	−0.615457
$919$	5870.32	0.210712	0.105356	−	0.994435i	$-0.466402\pi$
$919$	5870.32	0.210712	0.105356	+	0.994435i	$0.466402\pi$
$920$	0	0
$921$	−26175.6	−0.936499
$922$	20530.9	0.733349
$923$	−4399.48	−0.156891
$924$	−376.875	−0.0134181
$925$	0	0
$926$	8879.67	0.315123
$927$	49330.7	1.74782
$928$	3390.18	0.119923
$929$	−47608.2	−1.68135	−0.840675	−	0.541541i	$-0.817841\pi$
$929$	−47608.2	−1.68135	−0.840675	+	0.541541i	$0.817841\pi$
$930$	0	0
$931$	−1447.05	−0.0509400
$932$	2372.05	0.0833682
$933$	62192.9	2.18232
$934$	5646.95	0.197831
$935$	0	0
$936$	5964.23	0.208277
$937$	38132.3	1.32949	0.664743	−	0.747072i	$-0.268541\pi$
$937$	38132.3	1.32949	0.664743	+	0.747072i	$0.268541\pi$
$938$	15434.9	0.537277
$939$	74458.9	2.58773
$940$	0	0
$941$	−27759.5	−0.961673	−0.480836	−	0.876810i	$-0.659667\pi$
$941$	−27759.5	−0.961673	−0.480836	+	0.876810i	$0.659667\pi$
$942$	−20099.5	−0.695198
$943$	442.925	0.0152955
$944$	−33431.6	−1.15266
$945$	0	0
$946$	2588.92	0.0889780
$947$	−2027.42	−0.0695694	−0.0347847	−	0.999395i	$-0.511075\pi$
$947$	−2027.42	−0.0695694	−0.0347847	+	0.999395i	$0.511075\pi$
$948$	−253.078	−0.00867044
$949$	1069.72	0.0365908
$950$	0	0
$951$	14518.6	0.495057
$952$	−4727.68	−0.160951
$953$	−10764.9	−0.365906	−0.182953	−	0.983122i	$-0.558566\pi$
$953$	−10764.9	−0.365906	−0.182953	+	0.983122i	$0.558566\pi$
$954$	−30607.4	−1.03873
$955$	0	0
$956$	−2603.74	−0.0880867
$957$	17321.5	0.585082
$958$	32281.9	1.08871
$959$	7396.91	0.249071
$960$	0	0
$961$	22968.4	0.770986
$962$	4376.55	0.146680
$963$	85018.6	2.84495
$964$	2310.39	0.0771916
$965$	0	0
$966$	1039.38	0.0346185
$967$	8149.21	0.271004	0.135502	−	0.990777i	$-0.456735\pi$
$967$	8149.21	0.271004	0.135502	+	0.990777i	$0.456735\pi$
$968$	−25373.9	−0.842509
$969$	1349.48	0.0447385
$970$	0	0
$971$	−39070.3	−1.29127	−0.645636	−	0.763646i	$-0.723407\pi$
$971$	−39070.3	−1.29127	−0.645636	+	0.763646i	$0.723407\pi$
$972$	−1194.26	−0.0394093
$973$	−6882.42	−0.226763
$974$	−38008.9	−1.25039
$975$	0	0
$976$	32473.6	1.06501
$977$	22544.6	0.738246	0.369123	−	0.929381i	$-0.379658\pi$
$977$	22544.6	0.738246	0.369123	+	0.929381i	$0.379658\pi$
$978$	−9881.71	−0.323090
$979$	5830.46	0.190340
$980$	0	0
$981$	−36100.2	−1.17491
$982$	−12793.4	−0.415737
$983$	−22177.1	−0.719573	−0.359786	−	0.933035i	$-0.617150\pi$
$983$	−22177.1	−0.719573	−0.359786	+	0.933035i	$0.617150\pi$
$984$	13801.7	0.447137
$985$	0	0
$986$	−14223.7	−0.459407
$987$	−2753.43	−0.0887971
$988$	13.5315	0.000435724 0
$989$	−413.277	−0.0132876
$990$	0	0
$991$	−36365.1	−1.16567	−0.582833	−	0.812592i	$-0.698056\pi$
$991$	−36365.1	−1.16567	−0.582833	+	0.812592i	$0.698056\pi$
$992$	5102.03	0.163296
$993$	24657.6	0.788000
$994$	−15311.0	−0.488567
$995$	0	0
$996$	4496.29	0.143043
$997$	22258.6	0.707059	0.353529	−	0.935423i	$-0.384981\pi$
$997$	22258.6	0.707059	0.353529	+	0.935423i	$0.384981\pi$
$998$	19885.5	0.630725
$999$	−48765.9	−1.54443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1175.4.a.l.1.9		35
5.2	odd	4		235.4.c.a.189.19	✓	70
5.3	odd	4		235.4.c.a.189.52	yes	70
5.4	even	2		1175.4.a.k.1.27		35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
235.4.c.a.189.19	✓	70	5.2	odd	4
235.4.c.a.189.52	yes	70	5.3	odd	4
1175.4.a.k.1.27		35	5.4	even	2
1175.4.a.l.1.9		35	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:	\( N \)	\(=\)	\( 1175 = 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1175.a (trivial)

\(f(q)\)	\(=\)	\(q-2.91402 q^{2} +8.67089 q^{3} +0.491508 q^{4} -25.2671 q^{6} -6.75637 q^{7} +21.8799 q^{8} +48.1844 q^{9} +O(q^{10})\)	\(q-2.91402 q^{2} +8.67089 q^{3} +0.491508 q^{4} -25.2671 q^{6} -6.75637 q^{7} +21.8799 q^{8} +48.1844 q^{9} +13.0885 q^{11} +4.26181 q^{12} +5.65722 q^{13} +19.6882 q^{14} -67.6905 q^{16} +31.9809 q^{17} -140.410 q^{18} +4.86646 q^{19} -58.5837 q^{21} -38.1402 q^{22} +6.08843 q^{23} +189.718 q^{24} -16.4852 q^{26} +183.687 q^{27} -3.32081 q^{28} +152.627 q^{29} +229.694 q^{31} +22.2123 q^{32} +113.489 q^{33} -93.1928 q^{34} +23.6830 q^{36} -265.483 q^{37} -14.1810 q^{38} +49.0531 q^{39} +72.7486 q^{41} +170.714 q^{42} -67.8792 q^{43} +6.43311 q^{44} -17.7418 q^{46} +47.0000 q^{47} -586.937 q^{48} -297.352 q^{49} +277.302 q^{51} +2.78057 q^{52} +217.985 q^{53} -535.268 q^{54} -147.829 q^{56} +42.1966 q^{57} -444.757 q^{58} +493.890 q^{59} -479.736 q^{61} -669.333 q^{62} -325.551 q^{63} +476.797 q^{64} -330.709 q^{66} +783.965 q^{67} +15.7188 q^{68} +52.7921 q^{69} -777.675 q^{71} +1054.27 q^{72} +189.090 q^{73} +773.623 q^{74} +2.39191 q^{76} -88.4308 q^{77} -142.942 q^{78} -59.3826 q^{79} +291.755 q^{81} -211.991 q^{82} +1055.02 q^{83} -28.7944 q^{84} +197.801 q^{86} +1323.41 q^{87} +286.375 q^{88} +445.464 q^{89} -38.2222 q^{91} +2.99251 q^{92} +1991.65 q^{93} -136.959 q^{94} +192.600 q^{96} -112.364 q^{97} +866.488 q^{98} +630.662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q + 8 q^{2} + 12 q^{3} + 144 q^{4} + 12 q^{6} + 84 q^{7} + 96 q^{8} + 333 q^{9}+O(q^{10}) \) 35 * q + 8 * q^2 + 12 * q^3 + 144 * q^4 + 12 * q^6 + 84 * q^7 + 96 * q^8 + 333 * q^9	\( 35 q + 8 q^{2} + 12 q^{3} + 144 q^{4} + 12 q^{6} + 84 q^{7} + 96 q^{8} + 333 q^{9} + 144 q^{12} + 294 q^{13} + 4 q^{14} + 580 q^{16} + 272 q^{17} + 1230 q^{18} - 96 q^{19} + 12 q^{21} + 484 q^{22} + 304 q^{23} - 54 q^{24} - 218 q^{26} + 432 q^{27} + 1350 q^{28} - 334 q^{29} - 248 q^{31} + 896 q^{32} + 1252 q^{33} + 148 q^{34} + 1836 q^{36} + 1480 q^{37} + 596 q^{38} - 332 q^{39} - 410 q^{41} + 672 q^{42} + 1506 q^{43} + 88 q^{44} - 456 q^{46} + 1645 q^{47} + 1298 q^{48} + 1459 q^{49} - 332 q^{51} + 3144 q^{52} + 1224 q^{53} + 524 q^{54} - 34 q^{56} + 4274 q^{57} + 1774 q^{58} - 86 q^{59} + 528 q^{61} + 2256 q^{62} + 1916 q^{63} + 2668 q^{64} + 1296 q^{66} + 4024 q^{67} + 286 q^{68} - 176 q^{69} + 556 q^{71} + 11280 q^{72} + 4584 q^{73} + 438 q^{74} - 3026 q^{76} + 3624 q^{77} + 446 q^{78} + 48 q^{79} + 4139 q^{81} + 2488 q^{82} + 1228 q^{83} - 3376 q^{84} - 610 q^{86} + 4770 q^{87} + 4168 q^{88} + 794 q^{89} + 1096 q^{91} + 4128 q^{92} + 1420 q^{93} + 376 q^{94} - 1100 q^{96} + 10624 q^{97} + 296 q^{98} + 1000 q^{99}+O(q^{100}) \) 35 * q + 8 * q^2 + 12 * q^3 + 144 * q^4 + 12 * q^6 + 84 * q^7 + 96 * q^8 + 333 * q^9 + 144 * q^12 + 294 * q^13 + 4 * q^14 + 580 * q^16 + 272 * q^17 + 1230 * q^18 - 96 * q^19 + 12 * q^21 + 484 * q^22 + 304 * q^23 - 54 * q^24 - 218 * q^26 + 432 * q^27 + 1350 * q^28 - 334 * q^29 - 248 * q^31 + 896 * q^32 + 1252 * q^33 + 148 * q^34 + 1836 * q^36 + 1480 * q^37 + 596 * q^38 - 332 * q^39 - 410 * q^41 + 672 * q^42 + 1506 * q^43 + 88 * q^44 - 456 * q^46 + 1645 * q^47 + 1298 * q^48 + 1459 * q^49 - 332 * q^51 + 3144 * q^52 + 1224 * q^53 + 524 * q^54 - 34 * q^56 + 4274 * q^57 + 1774 * q^58 - 86 * q^59 + 528 * q^61 + 2256 * q^62 + 1916 * q^63 + 2668 * q^64 + 1296 * q^66 + 4024 * q^67 + 286 * q^68 - 176 * q^69 + 556 * q^71 + 11280 * q^72 + 4584 * q^73 + 438 * q^74 - 3026 * q^76 + 3624 * q^77 + 446 * q^78 + 48 * q^79 + 4139 * q^81 + 2488 * q^82 + 1228 * q^83 - 3376 * q^84 - 610 * q^86 + 4770 * q^87 + 4168 * q^88 + 794 * q^89 + 1096 * q^91 + 4128 * q^92 + 1420 * q^93 + 376 * q^94 - 1100 * q^96 + 10624 * q^97 + 296 * q^98 + 1000 * q^99

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