LMFDB - Embedded newform 75.4.b.c.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Root			$-2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.4.b.c.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.35890i q^{2} +3.00000i q^{3} -20.7178 q^{4} -16.0767 q^{6} +4.43560i q^{7} -68.1534i q^{8} -9.00000 q^{9} -3.43560 q^{11} -62.1534i q^{12} +78.7424i q^{13} -23.7699 q^{14} +199.485 q^{16} -53.1780i q^{17} -48.2301i q^{18} -20.4356 q^{19} -13.3068 q^{21} -18.4110i q^{22} +118.307i q^{23} +204.460 q^{24} -421.972 q^{26} -27.0000i q^{27} -91.8958i q^{28} -168.049 q^{29} -61.0492 q^{31} +523.792i q^{32} -10.3068i q^{33} +284.975 q^{34} +186.460 q^{36} +246.614i q^{37} -109.512i q^{38} -236.227 q^{39} +422.663 q^{41} -71.3097i q^{42} +362.436i q^{43} +71.1780 q^{44} -633.994 q^{46} +170.515i q^{47} +598.454i q^{48} +323.325 q^{49} +159.534 q^{51} -1631.37i q^{52} -546.049i q^{53} +144.690 q^{54} +302.301 q^{56} -61.3068i q^{57} -900.559i q^{58} +216.970 q^{59} +130.902 q^{61} -327.156i q^{62} -39.9204i q^{63} -1211.07 q^{64} +55.2330 q^{66} +614.890i q^{67} +1101.73i q^{68} -354.920 q^{69} +324.822 q^{71} +613.381i q^{72} -88.8712i q^{73} -1321.58 q^{74} +423.381 q^{76} -15.2389i q^{77} -1265.92i q^{78} +1137.42 q^{79} +81.0000 q^{81} +2265.01i q^{82} -758.909i q^{83} +275.687 q^{84} -1942.26 q^{86} -504.148i q^{87} +234.148i q^{88} -195.681 q^{89} -349.269 q^{91} -2451.06i q^{92} -183.148i q^{93} -913.774 q^{94} -1571.37 q^{96} -521.000i q^{97} +1732.67i q^{98} +30.9204 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 48 q^{4} - 12 q^{6} - 36 q^{9} + 56 q^{11} - 252 q^{14} + 240 q^{16} - 12 q^{19} + 156 q^{21} + 504 q^{24} - 1252 q^{26} - 184 q^{29} + 244 q^{31} + 1384 q^{34} + 432 q^{36} - 108 q^{39} + 784 q^{41}+ \cdots - 504 q^{99}+O(q^{100}) $ 4 * q - 48 * q^4 - 12 * q^6 - 36 * q^9 + 56 * q^11 - 252 * q^14 + 240 * q^16 - 12 * q^19 + 156 * q^21 + 504 * q^24 - 1252 * q^26 - 184 * q^29 + 244 * q^31 + 1384 * q^34 + 432 * q^36 - 108 * q^39 + 784 * q^41 - 64 * q^44 - 1176 * q^46 - 520 * q^49 - 408 * q^51 + 108 * q^54 - 360 * q^56 - 248 * q^59 + 1500 * q^61 - 2752 * q^64 + 744 * q^66 - 792 * q^69 + 1648 * q^71 - 2392 * q^74 + 752 * q^76 + 1760 * q^79 + 324 * q^81 - 48 * q^84 - 1684 * q^86 + 1728 * q^89 - 4396 * q^91 + 1192 * q^94 - 3984 * q^96 - 504 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	5.35890i	1.89466i	0.320264	+	0.947328i	$0.396228\pi$
$2$	5.35890i	1.89466i	−0.320264	+	0.947328i	$0.603772\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	−20.7178	−2.58972
$5$	0	0
$5$	0	0
$6$	−16.0767	−1.09388
$7$	4.43560i	0.239500i	0.992804	+	0.119750i	$0.0382092\pi$
$7$	4.43560i	0.239500i	−0.992804	+	0.119750i	$0.961791\pi$
$8$	− 68.1534i	− 3.01198i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−3.43560	−0.0941701	−0.0470851	−	0.998891i	$-0.514993\pi$
$11$	−3.43560	−0.0941701	−0.0470851	+	0.998891i	$0.514993\pi$
$12$	− 62.1534i	− 1.49518i
$13$	78.7424i	1.67994i	0.542634	+	0.839970i	$0.317427\pi$
$13$	78.7424i	1.67994i	−0.542634	+	0.839970i	$0.682573\pi$
$14$	−23.7699	−0.453770
$15$	0	0
$16$	199.485	3.11695
$17$	− 53.1780i	− 0.758680i	−0.925257	−	0.379340i	$-0.876151\pi$
$17$	− 53.1780i	− 0.758680i	0.925257	−	0.379340i	$-0.123849\pi$
$18$	− 48.2301i	− 0.631552i
$19$	−20.4356	−0.246750	−0.123375	−	0.992360i	$-0.539372\pi$
$19$	−20.4356	−0.246750	−0.123375	+	0.992360i	$0.539372\pi$
$20$	0	0
$21$	−13.3068	−0.138275
$22$	− 18.4110i	− 0.178420i
$23$	118.307i	1.07255i	0.844043	+	0.536275i	$0.180169\pi$
$23$	118.307i	1.07255i	−0.844043	+	0.536275i	$0.819831\pi$
$24$	204.460	1.73897
$25$	0	0
$26$	−421.972	−3.18291
$27$	− 27.0000i	− 0.192450i
$28$	− 91.8958i	− 0.620238i
$29$	−168.049	−1.07607	−0.538034	−	0.842923i	$-0.680833\pi$
$29$	−168.049	−1.07607	−0.538034	+	0.842923i	$0.680833\pi$
$30$	0	0
$31$	−61.0492	−0.353702	−0.176851	−	0.984238i	$-0.556591\pi$
$31$	−61.0492	−0.353702	−0.176851	+	0.984238i	$0.556591\pi$
$32$	523.792i	2.89357i
$33$	− 10.3068i	− 0.0543691i
$34$	284.975	1.43744
$35$	0	0
$36$	186.460	0.863242
$37$	246.614i	1.09576i	0.836558	+	0.547879i	$0.184564\pi$
$37$	246.614i	1.09576i	−0.836558	+	0.547879i	$0.815436\pi$
$38$	− 109.512i	− 0.467506i
$39$	−236.227	−0.969913
$40$	0	0
$41$	422.663	1.60997	0.804986	−	0.593294i	$-0.202173\pi$
$41$	422.663	1.60997	0.804986	+	0.593294i	$0.202173\pi$
$42$	− 71.3097i	− 0.261984i
$43$	362.436i	1.28537i	0.766131	+	0.642685i	$0.222180\pi$
$43$	362.436i	1.28537i	−0.766131	+	0.642685i	$0.777820\pi$
$44$	71.1780	0.243875
$45$	0	0
$46$	−633.994	−2.03212
$47$	170.515i	0.529196i	0.964359	+	0.264598i	$0.0852392\pi$
$47$	170.515i	0.529196i	−0.964359	+	0.264598i	$0.914761\pi$
$48$	598.454i	1.79957i
$49$	323.325	0.942640
$50$	0	0
$51$	159.534	0.438024
$52$	− 1631.37i	− 4.35058i
$53$	− 546.049i	− 1.41520i	−0.706613	−	0.707600i	$-0.749778\pi$
$53$	− 546.049i	− 1.41520i	0.706613	−	0.707600i	$-0.250222\pi$
$54$	144.690	0.364627
$55$	0	0
$56$	302.301	0.721369
$57$	− 61.3068i	− 0.142461i
$58$	− 900.559i	− 2.03878i
$59$	216.970	0.478763	0.239382	−	0.970926i	$-0.423055\pi$
$59$	216.970	0.478763	0.239382	+	0.970926i	$0.423055\pi$
$60$	0	0
$61$	130.902	0.274758	0.137379	−	0.990519i	$-0.456132\pi$
$61$	130.902	0.274758	0.137379	+	0.990519i	$0.456132\pi$
$62$	− 327.156i	− 0.670143i
$63$	− 39.9204i	− 0.0798332i
$64$	−1211.07	−2.36537
$65$	0	0
$66$	55.2330	0.103011
$67$	614.890i	1.12121i	0.828085	+	0.560603i	$0.189430\pi$
$67$	614.890i	1.12121i	−0.828085	+	0.560603i	$0.810570\pi$
$68$	1101.73i	1.96477i
$69$	−354.920	−0.619238
$70$	0	0
$71$	324.822	0.542948	0.271474	−	0.962446i	$-0.412489\pi$
$71$	324.822	0.542948	0.271474	+	0.962446i	$0.412489\pi$
$72$	613.381i	1.00399i
$73$	− 88.8712i	− 0.142487i	−0.997459	−	0.0712437i	$-0.977303\pi$
$73$	− 88.8712i	− 0.142487i	0.997459	−	0.0712437i	$-0.0226968\pi$
$74$	−1321.58	−2.07608
$75$	0	0
$76$	423.381	0.639014
$77$	− 15.2389i	− 0.0225537i
$78$	− 1265.92i	− 1.83765i
$79$	1137.42	1.61988	0.809938	−	0.586516i	$-0.199501\pi$
$79$	1137.42	1.61988	0.809938	+	0.586516i	$0.199501\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	2265.01i	3.05034i
$83$	− 758.909i	− 1.00363i	−0.864976	−	0.501813i	$-0.832666\pi$
$83$	− 758.909i	− 1.00363i	0.864976	−	0.501813i	$-0.167334\pi$
$84$	275.687	0.358095
$85$	0	0
$86$	−1942.26	−2.43534
$87$	− 504.148i	− 0.621268i
$88$	234.148i	0.283639i
$89$	−195.681	−0.233058	−0.116529	−	0.993187i	$-0.537177\pi$
$89$	−195.681	−0.233058	−0.116529	+	0.993187i	$0.537177\pi$
$90$	0	0
$91$	−349.269	−0.402345
$92$	− 2451.06i	− 2.77761i
$93$	− 183.148i	− 0.204210i
$94$	−913.774	−1.00264
$95$	0	0
$96$	−1571.37	−1.67060
$97$	− 521.000i	− 0.545356i	−0.962105	−	0.272678i	$-0.912091\pi$
$97$	− 521.000i	− 0.545356i	0.962105	−	0.272678i	$-0.0879094\pi$
$98$	1732.67i	1.78598i
$99$	30.9204	0.0313900
$100$	0	0
$101$	660.920	0.651129	0.325565	−	0.945520i	$-0.394446\pi$
$101$	660.920	0.651129	0.325565	+	0.945520i	$0.394446\pi$
$102$	854.926i	0.829905i
$103$	1530.75i	1.46436i	0.681110	+	0.732181i	$0.261497\pi$
$103$	1530.75i	1.46436i	−0.681110	+	0.732181i	$0.738503\pi$
$104$	5366.56	5.05995
$105$	0	0
$106$	2926.22	2.68132
$107$	264.625i	0.239087i	0.992829	+	0.119543i	$0.0381431\pi$
$107$	264.625i	0.239087i	−0.992829	+	0.119543i	$0.961857\pi$
$108$	559.381i	0.498393i
$109$	−1117.61	−0.982091	−0.491046	−	0.871134i	$-0.663385\pi$
$109$	−1117.61	−0.982091	−0.491046	+	0.871134i	$0.663385\pi$
$110$	0	0
$111$	−739.841	−0.632636
$112$	884.834i	0.746508i
$113$	− 934.061i	− 0.777602i	−0.921322	−	0.388801i	$-0.872889\pi$
$113$	− 934.061i	− 0.777602i	0.921322	−	0.388801i	$-0.127111\pi$
$114$	328.537	0.269915
$115$	0	0
$116$	3481.61	2.78672
$117$	− 708.681i	− 0.559980i
$118$	1162.72i	0.907092i
$119$	235.876	0.181704
$120$	0	0
$121$	−1319.20	−0.991132
$122$	701.489i	0.520572i
$123$	1267.99i	0.929517i
$124$	1264.80	0.915990
$125$	0	0
$126$	213.929	0.151257
$127$	− 630.356i	− 0.440433i	−0.975451	−	0.220217i	$-0.929324\pi$
$127$	− 630.356i	− 0.440433i	0.975451	−	0.220217i	$-0.0706765\pi$
$128$	− 2299.66i	− 1.58799i
$129$	−1087.31	−0.742109
$130$	0	0
$131$	−2163.06	−1.44265	−0.721325	−	0.692597i	$-0.756466\pi$
$131$	−2163.06	−1.44265	−0.721325	+	0.692597i	$0.756466\pi$
$132$	213.534i	0.140801i
$133$	− 90.6440i	− 0.0590965i
$134$	−3295.13	−2.12430
$135$	0	0
$136$	−3624.26	−2.28513
$137$	− 1118.61i	− 0.697588i	−0.937199	−	0.348794i	$-0.886591\pi$
$137$	− 1118.61i	− 0.697588i	0.937199	−	0.348794i	$-0.113409\pi$
$138$	− 1901.98i	− 1.17324i
$139$	166.478	0.101586	0.0507930	−	0.998709i	$-0.483825\pi$
$139$	166.478	0.101586	0.0507930	+	0.998709i	$0.483825\pi$
$140$	0	0
$141$	−511.546	−0.305531
$142$	1740.69i	1.02870i
$143$	− 270.527i	− 0.158200i
$144$	−1795.36	−1.03898
$145$	0	0
$146$	476.252	0.269965
$147$	969.976i	0.544233i
$148$	− 5109.29i	− 2.83771i
$149$	653.143	0.359111	0.179555	−	0.983748i	$-0.442534\pi$
$149$	653.143	0.359111	0.179555	+	0.983748i	$0.442534\pi$
$150$	0	0
$151$	−1929.38	−1.03981	−0.519903	−	0.854225i	$-0.674032\pi$
$151$	−1929.38	−1.03981	−0.519903	+	0.854225i	$0.674032\pi$
$152$	1392.76i	0.743206i
$153$	478.602i	0.252893i
$154$	81.6638	0.0427315
$155$	0	0
$156$	4894.11	2.51181
$157$	− 2169.75i	− 1.10296i	−0.834188	−	0.551480i	$-0.814063\pi$
$157$	− 2169.75i	− 1.10296i	0.834188	−	0.551480i	$-0.185937\pi$
$158$	6095.34i	3.06911i
$159$	1638.15	0.817066
$160$	0	0
$161$	−524.761	−0.256876
$162$	434.071i	0.210517i
$163$	− 763.738i	− 0.366997i	−0.983020	−	0.183499i	$-0.941258\pi$
$163$	− 763.738i	− 0.366997i	0.983020	−	0.183499i	$-0.0587423\pi$
$164$	−8756.64	−4.16938
$165$	0	0
$166$	4066.91	1.90153
$167$	2564.28i	1.18820i	0.804389	+	0.594102i	$0.202493\pi$
$167$	2564.28i	1.18820i	−0.804389	+	0.594102i	$0.797507\pi$
$168$	906.903i	0.416483i
$169$	−4003.36	−1.82220
$170$	0	0
$171$	183.920	0.0822500
$172$	− 7508.87i	− 3.32875i
$173$	51.8290i	0.0227774i	0.999935	+	0.0113887i	$0.00362521\pi$
$173$	51.8290i	0.0227774i	−0.999935	+	0.0113887i	$0.996375\pi$
$174$	2701.68	1.17709
$175$	0	0
$176$	−685.349	−0.293523
$177$	650.909i	0.276414i
$178$	− 1048.64i	− 0.441566i
$179$	3956.63	1.65214	0.826068	−	0.563571i	$-0.190573\pi$
$179$	3956.63	1.65214	0.826068	+	0.563571i	$0.190573\pi$
$180$	0	0
$181$	1804.04	0.740848	0.370424	−	0.928863i	$-0.379212\pi$
$181$	1804.04	0.740848	0.370424	+	0.928863i	$0.379212\pi$
$182$	− 1871.70i	− 0.762305i
$183$	392.705i	0.158632i
$184$	8063.01	3.23050
$185$	0	0
$186$	981.469	0.386908
$187$	182.698i	0.0714449i
$188$	− 3532.70i	− 1.37047i
$189$	119.761	0.0460917
$190$	0	0
$191$	3666.75	1.38909	0.694547	−	0.719448i	$-0.255605\pi$
$191$	3666.75	1.38909	0.694547	+	0.719448i	$0.255605\pi$
$192$	− 3633.20i	− 1.36565i
$193$	− 2716.98i	− 1.01333i	−0.862144	−	0.506664i	$-0.830879\pi$
$193$	− 2716.98i	− 1.01333i	0.862144	−	0.506664i	$-0.169121\pi$
$194$	2791.99	1.03326
$195$	0	0
$196$	−6698.59	−2.44118
$197$	− 2034.30i	− 0.735723i	−0.929881	−	0.367862i	$-0.880090\pi$
$197$	− 2034.30i	− 0.735723i	0.929881	−	0.367862i	$-0.119910\pi$
$198$	165.699i	0.0594733i
$199$	1551.27	0.552596	0.276298	−	0.961072i	$-0.410892\pi$
$199$	1551.27	0.552596	0.276298	+	0.961072i	$0.410892\pi$
$200$	0	0
$201$	−1844.67	−0.647328
$202$	3541.81i	1.23367i
$203$	− 745.398i	− 0.257718i
$204$	−3305.19	−1.13436
$205$	0	0
$206$	−8203.13	−2.77446
$207$	− 1064.76i	− 0.357517i
$208$	15707.9i	5.23629i
$209$	70.2084	0.0232365
$210$	0	0
$211$	3192.51	1.04162	0.520809	−	0.853673i	$-0.325630\pi$
$211$	3192.51	1.04162	0.520809	+	0.853673i	$0.325630\pi$
$212$	11312.9i	3.66498i
$213$	974.466i	0.313471i
$214$	−1418.10	−0.452988
$215$	0	0
$216$	−1840.14	−0.579656
$217$	− 270.789i	− 0.0847115i
$218$	− 5989.18i	− 1.86073i
$219$	266.614	0.0822652
$220$	0	0
$221$	4187.36	1.27454
$222$	− 3964.73i	− 1.19863i
$223$	− 1555.55i	− 0.467120i	−0.972342	−	0.233560i	$-0.924963\pi$
$223$	− 1555.55i	− 0.467120i	0.972342	−	0.233560i	$-0.0750374\pi$
$224$	−2323.33	−0.693008
$225$	0	0
$226$	5005.54	1.47329
$227$	6206.86i	1.81482i	0.420248	+	0.907409i	$0.361943\pi$
$227$	6206.86i	1.81482i	−0.420248	+	0.907409i	$0.638057\pi$
$228$	1270.14i	0.368935i
$229$	−4679.51	−1.35035	−0.675176	−	0.737657i	$-0.735932\pi$
$229$	−4679.51	−1.35035	−0.675176	+	0.737657i	$0.735932\pi$
$230$	0	0
$231$	45.7167	0.0130214
$232$	11453.1i	3.24110i
$233$	− 3244.53i	− 0.912259i	−0.889913	−	0.456129i	$-0.849235\pi$
$233$	− 3244.53i	− 0.912259i	0.889913	−	0.456129i	$-0.150765\pi$
$234$	3797.75	1.06097
$235$	0	0
$236$	−4495.13	−1.23986
$237$	3412.27i	0.935236i
$238$	1264.04i	0.344266i
$239$	3658.62	0.990193	0.495097	−	0.868838i	$-0.335133\pi$
$239$	3658.62	0.990193	0.495097	+	0.868838i	$0.335133\pi$
$240$	0	0
$241$	1931.01	0.516131	0.258065	−	0.966127i	$-0.416915\pi$
$241$	1931.01	0.516131	0.258065	+	0.966127i	$0.416915\pi$
$242$	− 7069.44i	− 1.87786i
$243$	243.000i	0.0641500i
$244$	−2711.99	−0.711548
$245$	0	0
$246$	−6795.02	−1.76112
$247$	− 1609.15i	− 0.414525i
$248$	4160.71i	1.06534i
$249$	2276.73	0.579444
$250$	0	0
$251$	−5843.34	−1.46944	−0.734718	−	0.678373i	$-0.762685\pi$
$251$	−5843.34	−1.46944	−0.734718	+	0.678373i	$0.762685\pi$
$252$	827.062i	0.206746i
$253$	− 406.454i	− 0.101002i
$254$	3378.01	0.834470
$255$	0	0
$256$	2635.09	0.643333
$257$	− 4506.11i	− 1.09371i	−0.837227	−	0.546855i	$-0.815825\pi$
$257$	− 4506.11i	− 1.09371i	0.837227	−	0.546855i	$-0.184175\pi$
$258$	− 5826.77i	− 1.40604i
$259$	−1093.88	−0.262434
$260$	0	0
$261$	1512.44	0.358689
$262$	− 11591.6i	− 2.73333i
$263$	5340.16i	1.25205i	0.779804	+	0.626024i	$0.215319\pi$
$263$	5340.16i	1.25205i	−0.779804	+	0.626024i	$0.784681\pi$
$264$	−702.443	−0.163759
$265$	0	0
$266$	485.752	0.111968
$267$	− 587.044i	− 0.134556i
$268$	− 12739.2i	− 2.90361i
$269$	−2809.79	−0.636863	−0.318431	−	0.947946i	$-0.603156\pi$
$269$	−2809.79	−0.636863	−0.318431	+	0.947946i	$0.603156\pi$
$270$	0	0
$271$	3102.95	0.695537	0.347769	−	0.937580i	$-0.386940\pi$
$271$	3102.95	0.695537	0.347769	+	0.937580i	$0.386940\pi$
$272$	− 10608.2i	− 2.36477i
$273$	− 1047.81i	− 0.232294i
$274$	5994.54	1.32169
$275$	0	0
$276$	7353.17	1.60365
$277$	4598.93i	0.997555i	0.866730	+	0.498777i	$0.166217\pi$
$277$	4598.93i	0.997555i	−0.866730	+	0.498777i	$0.833783\pi$
$278$	892.138i	0.192471i
$279$	549.443	0.117901
$280$	0	0
$281$	2571.83	0.545987	0.272994	−	0.962016i	$-0.411986\pi$
$281$	2571.83	0.545987	0.272994	+	0.962016i	$0.411986\pi$
$282$	− 2741.32i	− 0.578877i
$283$	5575.31i	1.17109i	0.810641	+	0.585544i	$0.199119\pi$
$283$	5575.31i	1.17109i	−0.810641	+	0.585544i	$0.800881\pi$
$284$	−6729.60	−1.40608
$285$	0	0
$286$	1449.73	0.299735
$287$	1874.76i	0.385588i
$288$	− 4714.12i	− 0.964522i
$289$	2085.10	0.424405
$290$	0	0
$291$	1563.00	0.314861
$292$	1841.22i	0.369003i
$293$	5794.27i	1.15531i	0.816282	+	0.577654i	$0.196032\pi$
$293$	5794.27i	1.15531i	−0.816282	+	0.577654i	$0.803968\pi$
$294$	−5198.01	−1.03114
$295$	0	0
$296$	16807.6	3.30040
$297$	92.7611i	0.0181230i
$298$	3500.13i	0.680392i
$299$	−9315.76	−1.80182
$300$	0	0
$301$	−1607.62	−0.307846
$302$	− 10339.4i	− 1.97008i
$303$	1982.76i	0.375930i
$304$	−4076.59	−0.769107
$305$	0	0
$306$	−2564.78	−0.479146
$307$	1404.47i	0.261099i	0.991442	+	0.130550i	$0.0416742\pi$
$307$	1404.47i	0.261099i	−0.991442	+	0.130550i	$0.958326\pi$
$308$	315.717i	0.0584079i
$309$	−4592.25	−0.845449
$310$	0	0
$311$	−4096.75	−0.746963	−0.373481	−	0.927638i	$-0.621836\pi$
$311$	−4096.75	−0.746963	−0.373481	+	0.927638i	$0.621836\pi$
$312$	16099.7i	2.92136i
$313$	974.611i	0.176001i	0.996120	+	0.0880004i	$0.0280477\pi$
$313$	974.611i	0.176001i	−0.996120	+	0.0880004i	$0.971952\pi$
$314$	11627.5	2.08973
$315$	0	0
$316$	−23564.9	−4.19503
$317$	2071.69i	0.367058i	0.983014	+	0.183529i	$0.0587522\pi$
$317$	2071.69i	0.367058i	−0.983014	+	0.183529i	$0.941248\pi$
$318$	8778.67i	1.54806i
$319$	577.349	0.101333
$320$	0	0
$321$	−793.876	−0.138037
$322$	− 2812.14i	− 0.486691i
$323$	1086.72i	0.187204i
$324$	−1678.14	−0.287747
$325$	0	0
$326$	4092.79	0.695334
$327$	− 3352.84i	− 0.567011i
$328$	− 28805.9i	− 4.84921i
$329$	−756.337	−0.126742
$330$	0	0
$331$	−6159.17	−1.02278	−0.511388	−	0.859350i	$-0.670868\pi$
$331$	−6159.17	−1.02278	−0.511388	+	0.859350i	$0.670868\pi$
$332$	15722.9i	2.59912i
$333$	− 2219.52i	− 0.365252i
$334$	−13741.7	−2.25124
$335$	0	0
$336$	−2654.50	−0.430997
$337$	− 2791.26i	− 0.451186i	−0.974222	−	0.225593i	$-0.927568\pi$
$337$	− 2791.26i	− 0.451186i	0.974222	−	0.225593i	$-0.0724320\pi$
$338$	− 21453.6i	− 3.45243i
$339$	2802.18	0.448949
$340$	0	0
$341$	209.740	0.0333081
$342$	985.611i	0.155835i
$343$	2955.55i	0.465262i
$344$	24701.2	3.87151
$345$	0	0
$346$	−277.746	−0.0431553
$347$	− 940.848i	− 0.145554i	−0.997348	−	0.0727772i	$-0.976814\pi$
$347$	− 940.848i	− 0.145554i	0.997348	−	0.0727772i	$-0.0231862\pi$
$348$	10444.8i	1.60891i
$349$	3519.62	0.539831	0.269915	−	0.962884i	$-0.413004\pi$
$349$	3519.62	0.539831	0.269915	+	0.962884i	$0.413004\pi$
$350$	0	0
$351$	2126.04	0.323304
$352$	− 1799.54i	− 0.272487i
$353$	5021.60i	0.757147i	0.925571	+	0.378573i	$0.123585\pi$
$353$	5021.60i	0.757147i	−0.925571	+	0.378573i	$0.876415\pi$
$354$	−3488.15	−0.523710
$355$	0	0
$356$	4054.09	0.603557
$357$	707.628i	0.104907i
$358$	21203.2i	3.13023i
$359$	−6811.99	−1.00146	−0.500728	−	0.865604i	$-0.666934\pi$
$359$	−6811.99	−1.00146	−0.500728	+	0.865604i	$0.666934\pi$
$360$	0	0
$361$	−6441.39	−0.939115
$362$	9667.69i	1.40365i
$363$	− 3957.59i	− 0.572230i
$364$	7236.09	1.04196
$365$	0	0
$366$	−2104.47	−0.300553
$367$	− 3748.07i	− 0.533099i	−0.963821	−	0.266550i	$-0.914116\pi$
$367$	− 3748.07i	− 0.533099i	0.963821	−	0.266550i	$-0.0858836\pi$
$368$	23600.4i	3.34309i
$369$	−3803.96	−0.536657
$370$	0	0
$371$	2422.05	0.338940
$372$	3794.41i	0.528847i
$373$	− 898.302i	− 0.124698i	−0.998054	−	0.0623489i	$-0.980141\pi$
$373$	− 898.302i	− 0.124698i	0.998054	−	0.0623489i	$-0.0198592\pi$
$374$	−979.060	−0.135364
$375$	0	0
$376$	11621.2	1.59393
$377$	− 13232.6i	− 1.80773i
$378$	641.788i	0.0873280i
$379$	9378.99	1.27115	0.635576	−	0.772038i	$-0.280763\pi$
$379$	9378.99	1.27115	0.635576	+	0.772038i	$0.280763\pi$
$380$	0	0
$381$	1891.07	0.254284
$382$	19649.8i	2.63186i
$383$	− 9446.29i	− 1.26027i	−0.776486	−	0.630134i	$-0.783000\pi$
$383$	− 9446.29i	− 1.26027i	0.776486	−	0.630134i	$-0.217000\pi$
$384$	6898.97	0.916828
$385$	0	0
$386$	14560.0	1.91991
$387$	− 3261.92i	− 0.428457i
$388$	10794.0i	1.41232i
$389$	7643.23	0.996214	0.498107	−	0.867116i	$-0.334029\pi$
$389$	7643.23	0.996214	0.498107	+	0.867116i	$0.334029\pi$
$390$	0	0
$391$	6291.32	0.813723
$392$	− 22035.7i	− 2.83922i
$393$	− 6489.17i	− 0.832914i
$394$	10901.6	1.39394
$395$	0	0
$396$	−640.602	−0.0812915
$397$	12013.6i	1.51876i	0.650650	+	0.759378i	$0.274497\pi$
$397$	12013.6i	1.51876i	−0.650650	+	0.759378i	$0.725503\pi$
$398$	8313.10i	1.04698i
$399$	271.932	0.0341194
$400$	0	0
$401$	−8538.51	−1.06332	−0.531662	−	0.846957i	$-0.678432\pi$
$401$	−8538.51	−1.06332	−0.531662	+	0.846957i	$0.678432\pi$
$402$	− 9885.40i	− 1.22646i
$403$	− 4807.16i	− 0.594197i
$404$	−13692.8	−1.68625
$405$	0	0
$406$	3994.51	0.488287
$407$	− 847.265i	− 0.103188i
$408$	− 10872.8i	− 1.31932i
$409$	12267.6	1.48312	0.741558	−	0.670889i	$-0.234087\pi$
$409$	12267.6	1.48312	0.741558	+	0.670889i	$0.234087\pi$
$410$	0	0
$411$	3355.84	0.402753
$412$	− 31713.8i	− 3.79229i
$413$	962.389i	0.114664i
$414$	5705.95	0.677372
$415$	0	0
$416$	−41244.6	−4.86102
$417$	499.433i	0.0586508i
$418$	376.240i	0.0440251i
$419$	−15493.0	−1.80641	−0.903204	−	0.429212i	$-0.858791\pi$
$419$	−15493.0	−1.80641	−0.903204	+	0.429212i	$0.858791\pi$
$420$	0	0
$421$	7510.67	0.869472	0.434736	−	0.900558i	$-0.356842\pi$
$421$	7510.67	0.869472	0.434736	+	0.900558i	$0.356842\pi$
$422$	17108.3i	1.97351i
$423$	− 1534.64i	− 0.176399i
$424$	−37215.1	−4.26256
$425$	0	0
$426$	−5222.07	−0.593920
$427$	580.627i	0.0658045i
$428$	− 5482.45i	− 0.619169i
$429$	811.581	0.0913368
$430$	0	0
$431$	−15675.3	−1.75186	−0.875932	−	0.482434i	$-0.839753\pi$
$431$	−15675.3	−1.75186	−0.875932	+	0.482434i	$0.839753\pi$
$432$	− 5386.09i	− 0.599857i
$433$	− 9604.78i	− 1.06600i	−0.846117	−	0.532998i	$-0.821065\pi$
$433$	− 9604.78i	− 1.06600i	0.846117	−	0.532998i	$-0.178935\pi$
$434$	1451.13	0.160499
$435$	0	0
$436$	23154.5	2.54335
$437$	− 2417.67i	− 0.264652i
$438$	1428.76i	0.155864i
$439$	6362.06	0.691673	0.345837	−	0.938295i	$-0.387595\pi$
$439$	6362.06	0.691673	0.345837	+	0.938295i	$0.387595\pi$
$440$	0	0
$441$	−2909.93	−0.314213
$442$	22439.6i	2.41481i
$443$	931.658i	0.0999196i	0.998751	+	0.0499598i	$0.0159093\pi$
$443$	931.658i	0.0999196i	−0.998751	+	0.0499598i	$0.984091\pi$
$444$	15327.9	1.63835
$445$	0	0
$446$	8336.06	0.885031
$447$	1959.43i	0.207333i
$448$	− 5371.81i	− 0.566505i
$449$	18684.1	1.96383	0.981914	−	0.189329i	$-0.0606314\pi$
$449$	18684.1	1.96383	0.981914	+	0.189329i	$0.0606314\pi$
$450$	0	0
$451$	−1452.10	−0.151611
$452$	19351.7i	2.01378i
$453$	− 5788.14i	− 0.600333i
$454$	−33261.9	−3.43846
$455$	0	0
$456$	−4178.27	−0.429090
$457$	11565.9i	1.18387i	0.805985	+	0.591936i	$0.201636\pi$
$457$	11565.9i	1.18387i	−0.805985	+	0.591936i	$0.798364\pi$
$458$	− 25077.0i	− 2.55845i
$459$	−1435.81	−0.146008
$460$	0	0
$461$	19401.0	1.96008	0.980039	−	0.198806i	$-0.0637062\pi$
$461$	19401.0	1.96008	0.980039	+	0.198806i	$0.0637062\pi$
$462$	244.991i	0.0246711i
$463$	1576.28i	0.158220i	0.996866	+	0.0791099i	$0.0252078\pi$
$463$	1576.28i	0.158220i	−0.996866	+	0.0791099i	$0.974792\pi$
$464$	−33523.2	−3.35405
$465$	0	0
$466$	17387.1	1.72842
$467$	− 3256.55i	− 0.322687i	−0.986898	−	0.161344i	$-0.948417\pi$
$467$	− 3256.55i	− 0.322687i	0.986898	−	0.161344i	$-0.0515827\pi$
$468$	14682.3i	1.45019i
$469$	−2727.40	−0.268528
$470$	0	0
$471$	6509.25	0.636795
$472$	− 14787.2i	− 1.44203i
$473$	− 1245.18i	− 0.121043i
$474$	−18286.0	−1.77195
$475$	0	0
$476$	−4886.83	−0.470562
$477$	4914.44i	0.471733i
$478$	19606.2i	1.87608i
$479$	−8291.59	−0.790924	−0.395462	−	0.918482i	$-0.629415\pi$
$479$	−8291.59	−0.790924	−0.395462	+	0.918482i	$0.629415\pi$
$480$	0	0
$481$	−19418.9	−1.84081
$482$	10348.1i	0.977891i
$483$	− 1574.28i	− 0.148307i
$484$	27330.9	2.56676
$485$	0	0
$486$	−1302.21	−0.121542
$487$	− 4758.55i	− 0.442773i	−0.975186	−	0.221387i	$-0.928942\pi$
$487$	− 4758.55i	− 0.442773i	0.975186	−	0.221387i	$-0.0710583\pi$
$488$	− 8921.39i	− 0.827567i
$489$	2291.21	0.211886
$490$	0	0
$491$	3906.46	0.359055	0.179528	−	0.983753i	$-0.442543\pi$
$491$	3906.46	0.359055	0.179528	+	0.983753i	$0.442543\pi$
$492$	− 26269.9i	− 2.40719i
$493$	8936.52i	0.816390i
$494$	8623.26	0.785382
$495$	0	0
$496$	−12178.4	−1.10247
$497$	1440.78i	0.130036i
$498$	12200.7i	1.09785i
$499$	3093.31	0.277506	0.138753	−	0.990327i	$-0.455691\pi$
$499$	3093.31	0.277506	0.138753	+	0.990327i	$0.455691\pi$
$500$	0	0
$501$	−7692.85	−0.686010
$502$	− 31313.9i	− 2.78408i
$503$	− 18153.9i	− 1.60923i	−0.593796	−	0.804616i	$-0.702371\pi$
$503$	− 18153.9i	− 1.60923i	0.593796	−	0.804616i	$-0.297629\pi$
$504$	−2720.71	−0.240456
$505$	0	0
$506$	2178.15	0.191365
$507$	− 12010.1i	− 1.05204i
$508$	13059.6i	1.14060i
$509$	−2281.32	−0.198660	−0.0993298	−	0.995055i	$-0.531670\pi$
$509$	−2281.32	−0.198660	−0.0993298	+	0.995055i	$0.531670\pi$
$510$	0	0
$511$	394.197	0.0341257
$512$	− 4276.07i	− 0.369097i
$513$	551.761i	0.0474870i
$514$	24147.8	2.07221
$515$	0	0
$516$	22526.6	1.92186
$517$	− 585.821i	− 0.0498344i
$518$	− 5861.98i	− 0.497221i
$519$	−155.487	−0.0131505
$520$	0	0
$521$	−16691.9	−1.40362	−0.701809	−	0.712366i	$-0.747624\pi$
$521$	−16691.9	−1.40362	−0.701809	+	0.712366i	$0.747624\pi$
$522$	8105.03i	0.679593i
$523$	− 17090.4i	− 1.42889i	−0.699690	−	0.714446i	$-0.746679\pi$
$523$	− 17090.4i	− 1.42889i	0.699690	−	0.714446i	$-0.253321\pi$
$524$	44813.8	3.73607
$525$	0	0
$526$	−28617.4	−2.37220
$527$	3246.47i	0.268346i
$528$	− 2056.05i	− 0.169466i
$529$	−1829.50	−0.150365
$530$	0	0
$531$	−1952.73	−0.159588
$532$	1877.94i	0.153044i
$533$	33281.5i	2.70465i
$534$	3145.91	0.254938
$535$	0	0
$536$	41906.8	3.37705
$537$	11869.9i	0.953861i
$538$	− 15057.4i	− 1.20664i
$539$	−1110.82	−0.0887685
$540$	0	0
$541$	5271.40	0.418919	0.209459	−	0.977817i	$-0.432830\pi$
$541$	5271.40	0.418919	0.209459	+	0.977817i	$0.432830\pi$
$542$	16628.4i	1.31780i
$543$	5412.13i	0.427729i
$544$	27854.2	2.19529
$545$	0	0
$546$	5615.10	0.440117
$547$	− 15182.2i	− 1.18673i	−0.804933	−	0.593366i	$-0.797799\pi$
$547$	− 15182.2i	− 1.18673i	0.804933	−	0.593366i	$-0.202201\pi$
$548$	23175.2i	1.80656i
$549$	−1178.11	−0.0915860
$550$	0	0
$551$	3434.18	0.265519
$552$	24189.0i	1.86513i
$553$	5045.15i	0.387960i
$554$	−24645.2	−1.89002
$555$	0	0
$556$	−3449.05	−0.263080
$557$	− 12241.2i	− 0.931198i	−0.884996	−	0.465599i	$-0.845839\pi$
$557$	− 12241.2i	− 0.931198i	0.884996	−	0.465599i	$-0.154161\pi$
$558$	2944.41i	0.223381i
$559$	−28539.0	−2.15934
$560$	0	0
$561$	−548.094	−0.0412488
$562$	13782.2i	1.03446i
$563$	14196.4i	1.06271i	0.847149	+	0.531355i	$0.178317\pi$
$563$	14196.4i	1.06271i	−0.847149	+	0.531355i	$0.821683\pi$
$564$	10598.1	0.791242
$565$	0	0
$566$	−29877.5	−2.21881
$567$	359.283i	0.0266111i
$568$	− 22137.7i	− 1.63535i
$569$	−9150.05	−0.674148	−0.337074	−	0.941478i	$-0.609437\pi$
$569$	−9150.05	−0.674148	−0.337074	+	0.941478i	$0.609437\pi$
$570$	0	0
$571$	23582.1	1.72833	0.864167	−	0.503206i	$-0.167846\pi$
$571$	23582.1	1.72833	0.864167	+	0.503206i	$0.167846\pi$
$572$	5604.72i	0.409695i
$573$	11000.3i	0.801993i
$574$	−10046.7	−0.730556
$575$	0	0
$576$	10899.6	0.788456
$577$	− 3906.22i	− 0.281834i	−0.990021	−	0.140917i	$-0.954995\pi$
$577$	− 3906.22i	− 0.281834i	0.990021	−	0.140917i	$-0.0450050\pi$
$578$	11173.9i	0.804102i
$579$	8150.93	0.585045
$580$	0	0
$581$	3366.21	0.240368
$582$	8375.96i	0.596554i
$583$	1876.00i	0.133270i
$584$	−6056.87	−0.429170
$585$	0	0
$586$	−31050.9	−2.18891
$587$	25938.0i	1.82381i	0.410401	+	0.911905i	$0.365389\pi$
$587$	25938.0i	1.82381i	−0.410401	+	0.911905i	$0.634611\pi$
$588$	− 20095.8i	− 1.40941i
$589$	1247.58	0.0872759
$590$	0	0
$591$	6102.89	0.424770
$592$	49195.7i	3.41542i
$593$	1908.23i	0.132145i	0.997815	+	0.0660723i	$0.0210468\pi$
$593$	1908.23i	0.132145i	−0.997815	+	0.0660723i	$0.978953\pi$
$594$	−497.097	−0.0343370
$595$	0	0
$596$	−13531.7	−0.929999
$597$	4653.81i	0.319041i
$598$	− 49922.2i	− 3.41383i
$599$	3495.41	0.238429	0.119214	−	0.992869i	$-0.461962\pi$
$599$	3495.41	0.238429	0.119214	+	0.992869i	$0.461962\pi$
$600$	0	0
$601$	−18267.2	−1.23983	−0.619913	−	0.784671i	$-0.712832\pi$
$601$	−18267.2	−1.23983	−0.619913	+	0.784671i	$0.712832\pi$
$602$	− 8615.06i	− 0.583262i
$603$	− 5534.01i	− 0.373735i
$604$	39972.5	2.69281
$605$	0	0
$606$	−10625.4	−0.712257
$607$	− 11538.2i	− 0.771534i	−0.922596	−	0.385767i	$-0.873937\pi$
$607$	− 11538.2i	− 0.771534i	0.922596	−	0.385767i	$-0.126063\pi$
$608$	− 10704.0i	− 0.713987i
$609$	2236.19	0.148793
$610$	0	0
$611$	−13426.8	−0.889017
$612$	− 9915.58i	− 0.654924i
$613$	21136.9i	1.39268i	0.717713	+	0.696340i	$0.245189\pi$
$613$	21136.9i	1.39268i	−0.717713	+	0.696340i	$0.754811\pi$
$614$	−7526.43	−0.494694
$615$	0	0
$616$	−1038.58	−0.0679314
$617$	− 15673.2i	− 1.02266i	−0.859385	−	0.511329i	$-0.829153\pi$
$617$	− 15673.2i	− 1.02266i	0.859385	−	0.511329i	$-0.170847\pi$
$618$	− 24609.4i	− 1.60184i
$619$	−22923.7	−1.48850	−0.744249	−	0.667902i	$-0.767193\pi$
$619$	−22923.7	−1.48850	−0.744249	+	0.667902i	$0.767193\pi$
$620$	0	0
$621$	3194.28	0.206413
$622$	− 21954.1i	− 1.41524i
$623$	− 867.964i	− 0.0558174i
$624$	−47123.7	−3.02317
$625$	0	0
$626$	−5222.84	−0.333461
$627$	210.625i	0.0134156i
$628$	44952.4i	2.85636i
$629$	13114.4	0.831329
$630$	0	0
$631$	9108.23	0.574632	0.287316	−	0.957836i	$-0.407237\pi$
$631$	9108.23	0.574632	0.287316	+	0.957836i	$0.407237\pi$
$632$	− 77519.3i	− 4.87904i
$633$	9577.53i	0.601379i
$634$	−11102.0	−0.695449
$635$	0	0
$636$	−33938.8	−2.11598
$637$	25459.4i	1.58358i
$638$	3093.96i	0.191992i
$639$	−2923.40	−0.180983
$640$	0	0
$641$	20103.5	1.23875	0.619375	−	0.785095i	$-0.287386\pi$
$641$	20103.5	1.23875	0.619375	+	0.785095i	$0.287386\pi$
$642$	− 4254.30i	− 0.261533i
$643$	5934.92i	0.363997i	0.983299	+	0.181999i	$0.0582567\pi$
$643$	5934.92i	0.363997i	−0.983299	+	0.181999i	$0.941743\pi$
$644$	10871.9	0.665237
$645$	0	0
$646$	−5823.64	−0.354688
$647$	14193.7i	0.862460i	0.902242	+	0.431230i	$0.141920\pi$
$647$	14193.7i	0.862460i	−0.902242	+	0.431230i	$0.858080\pi$
$648$	− 5520.42i	− 0.334665i
$649$	−745.420	−0.0450852
$650$	0	0
$651$	812.368	0.0489082
$652$	15823.0i	0.950422i
$653$	− 4795.80i	− 0.287403i	−0.989621	−	0.143701i	$-0.954100\pi$
$653$	− 4795.80i	− 0.287403i	0.989621	−	0.143701i	$-0.0459005\pi$
$654$	17967.5	1.07429
$655$	0	0
$656$	84314.8	5.01820
$657$	799.841i	0.0474958i
$658$	− 4053.13i	− 0.240133i
$659$	−4399.57	−0.260065	−0.130032	−	0.991510i	$-0.541508\pi$
$659$	−4399.57	−0.260065	−0.130032	+	0.991510i	$0.541508\pi$
$660$	0	0
$661$	24096.0	1.41789	0.708945	−	0.705263i	$-0.249171\pi$
$661$	24096.0	1.41789	0.708945	+	0.705263i	$0.249171\pi$
$662$	− 33006.4i	− 1.93781i
$663$	12562.1i	0.735853i
$664$	−51722.2	−3.02291
$665$	0	0
$666$	11894.2	0.692028
$667$	− 19881.4i	− 1.15414i
$668$	− 53126.3i	− 3.07712i
$669$	4666.66	0.269692
$670$	0	0
$671$	−449.725	−0.0258740
$672$	− 6969.98i	− 0.400109i
$673$	− 27648.3i	− 1.58360i	−0.610781	−	0.791800i	$-0.709144\pi$
$673$	− 27648.3i	− 1.58360i	0.610781	−	0.791800i	$-0.290856\pi$
$674$	14958.1	0.854843
$675$	0	0
$676$	82940.9	4.71898
$677$	27605.5i	1.56716i	0.621292	+	0.783580i	$0.286608\pi$
$677$	27605.5i	1.56716i	−0.621292	+	0.783580i	$0.713392\pi$
$678$	15016.6i	0.850604i
$679$	2310.95	0.130613
$680$	0	0
$681$	−18620.6	−1.04779
$682$	1123.98i	0.0631075i
$683$	− 14949.4i	− 0.837513i	−0.908099	−	0.418756i	$-0.862466\pi$
$683$	− 14949.4i	− 0.837513i	0.908099	−	0.418756i	$-0.137534\pi$
$684$	−3810.42	−0.213005
$685$	0	0
$686$	−15838.5	−0.881511
$687$	− 14038.5i	− 0.779626i
$688$	72300.4i	4.00643i
$689$	42997.2	2.37745
$690$	0	0
$691$	8884.30	0.489110	0.244555	−	0.969635i	$-0.421358\pi$
$691$	8884.30	0.489110	0.244555	+	0.969635i	$0.421358\pi$
$692$	− 1073.78i	− 0.0589871i
$693$	137.150i	0.00751790i
$694$	5041.91	0.275775
$695$	0	0
$696$	−34359.4	−1.87125
$697$	− 22476.4i	− 1.22145i
$698$	18861.3i	1.02279i
$699$	9733.59	0.526693
$700$	0	0
$701$	10556.9	0.568798	0.284399	−	0.958706i	$-0.408206\pi$
$701$	10556.9	0.568798	0.284399	+	0.958706i	$0.408206\pi$
$702$	11393.3i	0.612551i
$703$	− 5039.70i	− 0.270378i
$704$	4160.74	0.222747
$705$	0	0
$706$	−26910.2	−1.43453
$707$	2931.58i	0.155945i
$708$	− 13485.4i	− 0.715836i
$709$	−25351.9	−1.34289	−0.671445	−	0.741055i	$-0.734326\pi$
$709$	−25351.9	−1.34289	−0.671445	+	0.741055i	$0.734326\pi$
$710$	0	0
$711$	−10236.8	−0.539959
$712$	13336.4i	0.701968i
$713$	− 7222.53i	− 0.379363i
$714$	−3792.11	−0.198762
$715$	0	0
$716$	−81972.6	−4.27858
$717$	10975.8i	0.571688i
$718$	− 36504.8i	− 1.89742i
$719$	−9719.94	−0.504162	−0.252081	−	0.967706i	$-0.581115\pi$
$719$	−9719.94	−0.504162	−0.252081	+	0.967706i	$0.581115\pi$
$720$	0	0
$721$	−6789.79	−0.350714
$722$	− 34518.7i	− 1.77930i
$723$	5793.04i	0.297988i
$724$	−37375.8	−1.91859
$725$	0	0
$726$	21208.3	1.08418
$727$	27509.3i	1.40339i	0.712479	+	0.701694i	$0.247572\pi$
$727$	27509.3i	1.40339i	−0.712479	+	0.701694i	$0.752428\pi$
$728$	23803.9i	1.21186i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	19273.6	0.975184
$732$	− 8135.98i	− 0.410812i
$733$	− 7240.49i	− 0.364848i	−0.983220	−	0.182424i	$-0.941606\pi$
$733$	− 7240.49i	− 0.364848i	0.983220	−	0.182424i	$-0.0583944\pi$
$734$	20085.5	1.01004
$735$	0	0
$736$	−61968.1	−3.10350
$737$	− 2112.51i	− 0.105584i
$738$	− 20385.1i	− 1.01678i
$739$	15875.3	0.790234	0.395117	−	0.918631i	$-0.370704\pi$
$739$	15875.3	0.790234	0.395117	+	0.918631i	$0.370704\pi$
$740$	0	0
$741$	4827.44	0.239326
$742$	12979.5i	0.642175i
$743$	− 25714.3i	− 1.26967i	−0.772647	−	0.634836i	$-0.781068\pi$
$743$	− 25714.3i	− 1.26967i	0.772647	−	0.634836i	$-0.218932\pi$
$744$	−12482.1	−0.615076
$745$	0	0
$746$	4813.91	0.236260
$747$	6830.18i	0.334542i
$748$	− 3785.10i	− 0.185023i
$749$	−1173.77	−0.0572612
$750$	0	0
$751$	−9709.09	−0.471757	−0.235879	−	0.971783i	$-0.575797\pi$
$751$	−9709.09	−0.471757	−0.235879	+	0.971783i	$0.575797\pi$
$752$	34015.2i	1.64948i
$753$	− 17530.0i	− 0.848379i
$754$	70912.1	3.42502
$755$	0	0
$756$	−2481.19	−0.119365
$757$	9567.13i	0.459344i	0.973268	+	0.229672i	$0.0737653\pi$
$757$	9567.13i	0.459344i	−0.973268	+	0.229672i	$0.926235\pi$
$758$	50261.1i	2.40840i
$759$	1219.36	0.0583137
$760$	0	0
$761$	−12322.5	−0.586980	−0.293490	−	0.955962i	$-0.594817\pi$
$761$	−12322.5	−0.586980	−0.293490	+	0.955962i	$0.594817\pi$
$762$	10134.0i	0.481782i
$763$	− 4957.28i	− 0.235211i
$764$	−75967.0	−3.59737
$765$	0	0
$766$	50621.7	2.38778
$767$	17084.7i	0.804293i
$768$	7905.27i	0.371428i
$769$	−2575.56	−0.120776	−0.0603881	−	0.998175i	$-0.519234\pi$
$769$	−2575.56	−0.120776	−0.0603881	+	0.998175i	$0.519234\pi$
$770$	0	0
$771$	13518.3	0.631454
$772$	56289.8i	2.62424i
$773$	− 6606.23i	− 0.307386i	−0.988119	−	0.153693i	$-0.950883\pi$
$773$	− 6606.23i	− 0.307386i	0.988119	−	0.153693i	$-0.0491167\pi$
$774$	17480.3	0.811778
$775$	0	0
$776$	−35507.9	−1.64260
$777$	− 3281.63i	− 0.151516i
$778$	40959.3i	1.88748i
$779$	−8637.37	−0.397260
$780$	0	0
$781$	−1115.96	−0.0511294
$782$	33714.5i	1.54173i
$783$	4537.33i	0.207089i
$784$	64498.5	2.93816
$785$	0	0
$786$	34774.8	1.57809
$787$	− 16417.0i	− 0.743587i	−0.928315	−	0.371793i	$-0.878743\pi$
$787$	− 16417.0i	− 0.743587i	0.928315	−	0.371793i	$-0.121257\pi$
$788$	42146.1i	1.90532i
$789$	−16020.5	−0.722870
$790$	0	0
$791$	4143.12	0.186235
$792$	− 2107.33i	− 0.0945462i
$793$	10307.5i	0.461577i
$794$	−64379.8	−2.87752
$795$	0	0
$796$	−32138.9	−1.43107
$797$	3944.19i	0.175295i	0.996152	+	0.0876477i	$0.0279350\pi$
$797$	3944.19i	0.175295i	−0.996152	+	0.0876477i	$0.972065\pi$
$798$	1457.26i	0.0646445i
$799$	9067.66	0.401490
$800$	0	0
$801$	1761.13	0.0776861
$802$	− 45757.0i	− 2.01463i
$803$	305.325i	0.0134181i
$804$	38217.5	1.67640
$805$	0	0
$806$	25761.1	1.12580
$807$	− 8429.37i	− 0.367693i
$808$	− 45044.0i	− 1.96119i
$809$	17960.7	0.780549	0.390275	−	0.920699i	$-0.372380\pi$
$809$	17960.7	0.780549	0.390275	+	0.920699i	$0.372380\pi$
$810$	0	0
$811$	−13162.5	−0.569912	−0.284956	−	0.958541i	$-0.591979\pi$
$811$	−13162.5	−0.569912	−0.284956	+	0.958541i	$0.591979\pi$
$812$	15443.0i	0.667418i
$813$	9308.84i	0.401569i
$814$	4540.41	0.195505
$815$	0	0
$816$	31824.6	1.36530
$817$	− 7406.59i	− 0.317165i
$818$	65740.9i	2.81000i
$819$	3143.42	0.134115
$820$	0	0
$821$	26502.4	1.12660	0.563302	−	0.826251i	$-0.309531\pi$
$821$	26502.4	1.12660	0.563302	+	0.826251i	$0.309531\pi$
$822$	17983.6i	0.763078i
$823$	6937.86i	0.293850i	0.989148	+	0.146925i	$0.0469376\pi$
$823$	6937.86i	0.293850i	−0.989148	+	0.146925i	$0.953062\pi$
$824$	104326.	4.41063
$825$	0	0
$826$	−5157.35	−0.217248
$827$	− 41197.9i	− 1.73228i	−0.499805	−	0.866138i	$-0.666595\pi$
$827$	− 41197.9i	− 1.73228i	0.499805	−	0.866138i	$-0.333405\pi$
$828$	22059.5i	0.925871i
$829$	693.324	0.0290472	0.0145236	−	0.999895i	$-0.495377\pi$
$829$	693.324	0.0290472	0.0145236	+	0.999895i	$0.495377\pi$
$830$	0	0
$831$	−13796.8	−0.575938
$832$	− 95362.4i	− 3.97367i
$833$	− 17193.8i	− 0.715162i
$834$	−2676.41	−0.111123
$835$	0	0
$836$	−1454.56	−0.0601760
$837$	1648.33i	0.0680699i
$838$	− 83025.7i	− 3.42252i
$839$	6491.28	0.267108	0.133554	−	0.991042i	$-0.457361\pi$
$839$	6491.28	0.267108	0.133554	+	0.991042i	$0.457361\pi$
$840$	0	0
$841$	3851.52	0.157921
$842$	40248.9i	1.64735i
$843$	7715.49i	0.315226i
$844$	−66141.8	−2.69750
$845$	0	0
$846$	8223.97	0.334215
$847$	− 5851.42i	− 0.237376i
$848$	− 108928.i	− 4.41111i
$849$	−16725.9	−0.676128
$850$	0	0
$851$	−29176.1	−1.17526
$852$	− 20188.8i	− 0.811803i
$853$	− 1116.68i	− 0.0448233i	−0.999749	−	0.0224117i	$-0.992866\pi$
$853$	− 1116.68i	− 0.0448233i	0.999749	−	0.0224117i	$-0.00713445\pi$
$854$	−3111.52	−0.124677
$855$	0	0
$856$	18035.1	0.720126
$857$	− 44383.9i	− 1.76911i	−0.466438	−	0.884554i	$-0.654463\pi$
$857$	− 44383.9i	− 1.76911i	0.466438	−	0.884554i	$-0.345537\pi$
$858$	4349.18i	0.173052i
$859$	25579.3	1.01601	0.508006	−	0.861354i	$-0.330383\pi$
$859$	25579.3	1.01601	0.508006	+	0.861354i	$0.330383\pi$
$860$	0	0
$861$	−5624.28	−0.222619
$862$	− 84002.5i	− 3.31918i
$863$	− 11194.8i	− 0.441570i	−0.975323	−	0.220785i	$-0.929138\pi$
$863$	− 11194.8i	− 0.441570i	0.975323	−	0.220785i	$-0.0708619\pi$
$864$	14142.4	0.556867
$865$	0	0
$866$	51471.0	2.01970
$867$	6255.31i	0.245030i
$868$	5610.16i	0.219379i
$869$	−3907.73	−0.152544
$870$	0	0
$871$	−48417.9	−1.88356
$872$	76169.2i	2.95804i
$873$	4689.00i	0.181785i
$874$	12956.0	0.501424
$875$	0	0
$876$	−5523.65	−0.213044
$877$	5721.75i	0.220308i	0.993915	+	0.110154i	$0.0351344\pi$
$877$	5721.75i	0.220308i	−0.993915	+	0.110154i	$0.964866\pi$
$878$	34093.6i	1.31048i
$879$	−17382.8	−0.667017
$880$	0	0
$881$	−34682.8	−1.32633	−0.663163	−	0.748475i	$-0.730786\pi$
$881$	−34682.8	−1.32633	−0.663163	+	0.748475i	$0.730786\pi$
$882$	− 15594.0i	− 0.595326i
$883$	37990.4i	1.44788i	0.689862	+	0.723941i	$0.257671\pi$
$883$	37990.4i	1.44788i	−0.689862	+	0.723941i	$0.742329\pi$
$884$	−86752.9	−3.30070
$885$	0	0
$886$	−4992.66	−0.189313
$887$	28299.0i	1.07124i	0.844460	+	0.535618i	$0.179921\pi$
$887$	28299.0i	1.07124i	−0.844460	+	0.535618i	$0.820079\pi$
$888$	50422.7i	1.90549i
$889$	2796.00	0.105484
$890$	0	0
$891$	−278.283	−0.0104633
$892$	32227.7i	1.20971i
$893$	− 3484.58i	− 0.130579i
$894$	−10500.4	−0.392825
$895$	0	0
$896$	10200.4	0.380324
$897$	− 27947.3i	− 1.04028i
$898$	100126.i	3.72078i
$899$	10259.3	0.380607
$900$	0	0
$901$	−29037.8	−1.07368
$902$	− 7781.65i	− 0.287251i
$903$	− 4822.85i	− 0.177735i
$904$	−63659.4	−2.34212
$905$	0	0
$906$	31018.1	1.13742
$907$	17388.0i	0.636559i	0.947997	+	0.318280i	$0.103105\pi$
$907$	17388.0i	0.636559i	−0.947997	+	0.318280i	$0.896895\pi$
$908$	− 128592.i	− 4.69988i
$909$	−5948.28	−0.217043
$910$	0	0
$911$	23555.3	0.856663	0.428332	−	0.903622i	$-0.359101\pi$
$911$	23555.3	0.856663	0.428332	+	0.903622i	$0.359101\pi$
$912$	− 12229.8i	− 0.444044i
$913$	2607.30i	0.0945117i
$914$	−61980.4	−2.24303
$915$	0	0
$916$	96949.1	3.49704
$917$	− 9594.44i	− 0.345514i
$918$	− 7694.34i	− 0.276635i
$919$	5983.09	0.214760	0.107380	−	0.994218i	$-0.465754\pi$
$919$	5983.09	0.214760	0.107380	+	0.994218i	$0.465754\pi$
$920$	0	0
$921$	−4213.42	−0.150746
$922$	103968.i	3.71368i
$923$	25577.3i	0.912119i
$924$	−947.150	−0.0337218
$925$	0	0
$926$	−8447.10	−0.299772
$927$	− 13776.7i	− 0.488120i
$928$	− 88022.7i	− 3.11367i
$929$	−20576.7	−0.726694	−0.363347	−	0.931654i	$-0.618366\pi$
$929$	−20576.7	−0.726694	−0.363347	+	0.931654i	$0.618366\pi$
$930$	0	0
$931$	−6607.35	−0.232596
$932$	67219.5i	2.36250i
$933$	− 12290.2i	− 0.431259i
$934$	17451.5	0.611382
$935$	0	0
$936$	−48299.0	−1.68665
$937$	11228.6i	0.391485i	0.980655	+	0.195743i	$0.0627117\pi$
$937$	11228.6i	0.391485i	−0.980655	+	0.195743i	$0.937288\pi$
$938$	− 14615.9i	− 0.508769i
$939$	−2923.83	−0.101614
$940$	0	0
$941$	38567.6	1.33610	0.668049	−	0.744118i	$-0.267130\pi$
$941$	38567.6	1.33610	0.668049	+	0.744118i	$0.267130\pi$
$942$	34882.4i	1.20651i
$943$	50003.9i	1.72678i
$944$	43282.1	1.49228
$945$	0	0
$946$	6672.81	0.229336
$947$	− 4606.17i	− 0.158057i	−0.996872	−	0.0790287i	$-0.974818\pi$
$947$	− 4606.17i	− 0.158057i	0.996872	−	0.0790287i	$-0.0251819\pi$
$948$	− 70694.8i	− 2.42200i
$949$	6997.93	0.239370
$950$	0	0
$951$	−6215.06	−0.211921
$952$	− 16075.8i	− 0.547288i
$953$	− 25559.7i	− 0.868795i	−0.900721	−	0.434397i	$-0.856961\pi$
$953$	− 25559.7i	− 0.868795i	0.900721	−	0.434397i	$-0.143039\pi$
$954$	−26336.0	−0.893773
$955$	0	0
$956$	−75798.5	−2.56433
$957$	1732.05i	0.0585048i
$958$	− 44433.8i	− 1.49853i
$959$	4961.72	0.167072
$960$	0	0
$961$	−26064.0	−0.874895
$962$	− 104064.i	− 3.48769i
$963$	− 2381.63i	− 0.0796956i
$964$	−40006.4	−1.33664
$965$	0	0
$966$	8436.42	0.280991
$967$	− 37895.8i	− 1.26023i	−0.776500	−	0.630117i	$-0.783007\pi$
$967$	− 37895.8i	− 1.26023i	0.776500	−	0.630117i	$-0.216993\pi$
$968$	89907.7i	2.98527i
$969$	−3260.17	−0.108082
$970$	0	0
$971$	−46761.0	−1.54545	−0.772726	−	0.634740i	$-0.781107\pi$
$971$	−46761.0	−1.54545	−0.772726	+	0.634740i	$0.781107\pi$
$972$	− 5034.42i	− 0.166131i
$973$	738.428i	0.0243298i
$974$	25500.6	0.838903
$975$	0	0
$976$	26112.9	0.856407
$977$	− 3070.29i	− 0.100540i	−0.998736	−	0.0502698i	$-0.983992\pi$
$977$	− 3070.29i	− 0.100540i	0.998736	−	0.0502698i	$-0.0160081\pi$
$978$	12278.4i	0.401451i
$979$	672.282	0.0219471
$980$	0	0
$981$	10058.5	0.327364
$982$	20934.3i	0.680287i
$983$	16319.0i	0.529498i	0.964317	+	0.264749i	$0.0852891\pi$
$983$	16319.0i	0.529498i	−0.964317	+	0.264749i	$0.914711\pi$
$984$	86417.7	2.79969
$985$	0	0
$986$	−47889.9	−1.54678
$987$	− 2269.01i	− 0.0731747i
$988$	33338.0i	1.07350i
$989$	−42878.6	−1.37862
$990$	0	0
$991$	5105.79	0.163664	0.0818319	−	0.996646i	$-0.473923\pi$
$991$	5105.79	0.163664	0.0818319	+	0.996646i	$0.473923\pi$
$992$	− 31977.0i	− 1.02346i
$993$	− 18477.5i	− 0.590500i
$994$	−7720.99	−0.246373
$995$	0	0
$996$	−47168.7	−1.50060
$997$	− 7206.97i	− 0.228934i	−0.993427	−	0.114467i	$-0.963484\pi$
$997$	− 7206.97i	− 0.228934i	0.993427	−	0.114467i	$-0.0365160\pi$
$998$	16576.7i	0.525779i
$999$	6658.57	0.210879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.b.c.49.4		4
3.2	odd	2		225.4.b.h.199.1		4
4.3	odd	2		1200.4.f.v.49.1		4
5.2	odd	4		75.4.a.d.1.1	✓	2
5.3	odd	4		75.4.a.e.1.2	yes	2
5.4	even	2	inner	75.4.b.c.49.1		4
15.2	even	4		225.4.a.n.1.2		2
15.8	even	4		225.4.a.j.1.1		2
15.14	odd	2		225.4.b.h.199.4		4
20.3	even	4		1200.4.a.bu.1.1		2
20.7	even	4		1200.4.a.bl.1.2		2
20.19	odd	2		1200.4.f.v.49.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.1	✓	2	5.2	odd	4
75.4.a.e.1.2	yes	2	5.3	odd	4
75.4.b.c.49.1		4	5.4	even	2	inner
75.4.b.c.49.4		4	1.1	even	1	trivial
225.4.a.j.1.1		2	15.8	even	4
225.4.a.n.1.2		2	15.2	even	4
225.4.b.h.199.1		4	3.2	odd	2
225.4.b.h.199.4		4	15.14	odd	2
1200.4.a.bl.1.2		2	20.7	even	4
1200.4.a.bu.1.1		2	20.3	even	4
1200.4.f.v.49.1		4	4.3	odd	2
1200.4.f.v.49.4		4	20.19	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.35890i q^{2} +3.00000i q^{3} -20.7178 q^{4} -16.0767 q^{6} +4.43560i q^{7} -68.1534i q^{8} -9.00000 q^{9} -3.43560 q^{11} -62.1534i q^{12} +78.7424i q^{13} -23.7699 q^{14} +199.485 q^{16} -53.1780i q^{17} -48.2301i q^{18} -20.4356 q^{19} -13.3068 q^{21} -18.4110i q^{22} +118.307i q^{23} +204.460 q^{24} -421.972 q^{26} -27.0000i q^{27} -91.8958i q^{28} -168.049 q^{29} -61.0492 q^{31} +523.792i q^{32} -10.3068i q^{33} +284.975 q^{34} +186.460 q^{36} +246.614i q^{37} -109.512i q^{38} -236.227 q^{39} +422.663 q^{41} -71.3097i q^{42} +362.436i q^{43} +71.1780 q^{44} -633.994 q^{46} +170.515i q^{47} +598.454i q^{48} +323.325 q^{49} +159.534 q^{51} -1631.37i q^{52} -546.049i q^{53} +144.690 q^{54} +302.301 q^{56} -61.3068i q^{57} -900.559i q^{58} +216.970 q^{59} +130.902 q^{61} -327.156i q^{62} -39.9204i q^{63} -1211.07 q^{64} +55.2330 q^{66} +614.890i q^{67} +1101.73i q^{68} -354.920 q^{69} +324.822 q^{71} +613.381i q^{72} -88.8712i q^{73} -1321.58 q^{74} +423.381 q^{76} -15.2389i q^{77} -1265.92i q^{78} +1137.42 q^{79} +81.0000 q^{81} +2265.01i q^{82} -758.909i q^{83} +275.687 q^{84} -1942.26 q^{86} -504.148i q^{87} +234.148i q^{88} -195.681 q^{89} -349.269 q^{91} -2451.06i q^{92} -183.148i q^{93} -913.774 q^{94} -1571.37 q^{96} -521.000i q^{97} +1732.67i q^{98} +30.9204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 48 q^{4} - 12 q^{6} - 36 q^{9} + 56 q^{11} - 252 q^{14} + 240 q^{16} - 12 q^{19} + 156 q^{21} + 504 q^{24} - 1252 q^{26} - 184 q^{29} + 244 q^{31} + 1384 q^{34} + 432 q^{36} - 108 q^{39} + 784 q^{41}+ \cdots - 504 q^{99}+O(q^{100}) \) 4 * q - 48 * q^4 - 12 * q^6 - 36 * q^9 + 56 * q^11 - 252 * q^14 + 240 * q^16 - 12 * q^19 + 156 * q^21 + 504 * q^24 - 1252 * q^26 - 184 * q^29 + 244 * q^31 + 1384 * q^34 + 432 * q^36 - 108 * q^39 + 784 * q^41 - 64 * q^44 - 1176 * q^46 - 520 * q^49 - 408 * q^51 + 108 * q^54 - 360 * q^56 - 248 * q^59 + 1500 * q^61 - 2752 * q^64 + 744 * q^66 - 792 * q^69 + 1648 * q^71 - 2392 * q^74 + 752 * q^76 + 1760 * q^79 + 324 * q^81 - 48 * q^84 - 1684 * q^86 + 1728 * q^89 - 4396 * q^91 + 1192 * q^94 - 3984 * q^96 - 504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more