LMFDB - Embedded newform 1275.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1275,4,Mod(1,1275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1275 = 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1275.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:	$75.2274352573$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.24264 q^{2} +3.00000 q^{3} +10.0000 q^{4} -12.7279 q^{6} +20.9706 q^{7} -8.48528 q^{8} +9.00000 q^{9} +16.0294 q^{11} +30.0000 q^{12} +34.9411 q^{13} -88.9706 q^{14} -44.0000 q^{16} +17.0000 q^{17} -38.1838 q^{18} -80.8823 q^{19} +62.9117 q^{21} -68.0071 q^{22} +115.971 q^{23} -25.4558 q^{24} -148.243 q^{26} +27.0000 q^{27} +209.706 q^{28} +154.118 q^{29} +299.941 q^{31} +254.558 q^{32} +48.0883 q^{33} -72.1249 q^{34} +90.0000 q^{36} -315.529 q^{37} +343.154 q^{38} +104.823 q^{39} +132.265 q^{41} -266.912 q^{42} +23.1177 q^{43} +160.294 q^{44} -492.021 q^{46} -260.912 q^{47} -132.000 q^{48} +96.7645 q^{49} +51.0000 q^{51} +349.411 q^{52} +676.087 q^{53} -114.551 q^{54} -177.941 q^{56} -242.647 q^{57} -653.866 q^{58} +629.294 q^{59} -461.852 q^{61} -1272.54 q^{62} +188.735 q^{63} -728.000 q^{64} -204.021 q^{66} +789.470 q^{67} +170.000 q^{68} +347.912 q^{69} -686.412 q^{71} -76.3675 q^{72} -484.912 q^{73} +1338.68 q^{74} -808.823 q^{76} +336.146 q^{77} -444.728 q^{78} +254.000 q^{79} +81.0000 q^{81} -561.153 q^{82} -548.912 q^{83} +629.117 q^{84} -98.0803 q^{86} +462.353 q^{87} -136.014 q^{88} +925.145 q^{89} +732.735 q^{91} +1159.71 q^{92} +899.823 q^{93} +1106.95 q^{94} +763.675 q^{96} -732.617 q^{97} -410.537 q^{98} +144.265 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 20 q^{4} + 8 q^{7} + 18 q^{9} + 66 q^{11} + 60 q^{12} + 2 q^{13} - 144 q^{14} - 88 q^{16} + 34 q^{17} - 26 q^{19} + 24 q^{21} + 144 q^{22} + 198 q^{23} - 288 q^{26} + 54 q^{27} + 80 q^{28}+ \cdots + 594 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 20 * q^4 + 8 * q^7 + 18 * q^9 + 66 * q^11 + 60 * q^12 + 2 * q^13 - 144 * q^14 - 88 * q^16 + 34 * q^17 - 26 * q^19 + 24 * q^21 + 144 * q^22 + 198 * q^23 - 288 * q^26 + 54 * q^27 + 80 * q^28 + 444 * q^29 + 532 * q^31 + 198 * q^33 + 180 * q^36 - 88 * q^37 + 576 * q^38 + 6 * q^39 + 570 * q^41 - 432 * q^42 + 182 * q^43 + 660 * q^44 - 144 * q^46 - 420 * q^47 - 264 * q^48 - 78 * q^49 + 102 * q^51 + 20 * q^52 + 300 * q^53 - 288 * q^56 - 78 * q^57 + 576 * q^58 + 444 * q^59 + 400 * q^61 - 288 * q^62 + 72 * q^63 - 1456 * q^64 + 432 * q^66 + 968 * q^67 + 340 * q^68 + 594 * q^69 - 1848 * q^71 - 868 * q^73 + 2304 * q^74 - 260 * q^76 - 312 * q^77 - 864 * q^78 + 508 * q^79 + 162 * q^81 + 1296 * q^82 - 996 * q^83 + 240 * q^84 + 576 * q^86 + 1332 * q^87 + 288 * q^88 - 288 * q^89 + 1160 * q^91 + 1980 * q^92 + 1596 * q^93 + 432 * q^94 - 1024 * q^97 - 1152 * q^98 + 594 * q^99

Coefficient data

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

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

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

$2$	−4.24264	−1.50000	−0.750000	−	0.661438i	$-0.769947\pi$
$2$	−4.24264	−1.50000	−0.750000	+	0.661438i	$0.769947\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	10.0000	1.25000
$5$	0	0
$5$	0	0
$6$	−12.7279	−0.866025
$7$	20.9706	1.13230	0.566152	−	0.824301i	$-0.308432\pi$
$7$	20.9706	1.13230	0.566152	+	0.824301i	$0.308432\pi$
$8$	−8.48528	−0.375000
$9$	9.00000	0.333333
$10$	0	0
$11$	16.0294	0.439369	0.219684	−	0.975571i	$-0.429497\pi$
$11$	16.0294	0.439369	0.219684	+	0.975571i	$0.429497\pi$
$12$	30.0000	0.721688
$13$	34.9411	0.745456	0.372728	−	0.927941i	$-0.378422\pi$
$13$	34.9411	0.745456	0.372728	+	0.927941i	$0.378422\pi$
$14$	−88.9706	−1.69846
$15$	0	0
$16$	−44.0000	−0.687500
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−38.1838	−0.500000
$19$	−80.8823	−0.976614	−0.488307	−	0.872672i	$-0.662385\pi$
$19$	−80.8823	−0.976614	−0.488307	+	0.872672i	$0.662385\pi$
$20$	0	0
$21$	62.9117	0.653736
$22$	−68.0071	−0.659053
$23$	115.971	1.05137	0.525686	−	0.850679i	$-0.323809\pi$
$23$	115.971	1.05137	0.525686	+	0.850679i	$0.323809\pi$
$24$	−25.4558	−0.216506
$25$	0	0
$26$	−148.243	−1.11818
$27$	27.0000	0.192450
$28$	209.706	1.41538
$29$	154.118	0.986860	0.493430	−	0.869785i	$-0.335743\pi$
$29$	154.118	0.986860	0.493430	+	0.869785i	$0.335743\pi$
$30$	0	0
$31$	299.941	1.73777	0.868887	−	0.495010i	$-0.164836\pi$
$31$	299.941	1.73777	0.868887	+	0.495010i	$0.164836\pi$
$32$	254.558	1.40625
$33$	48.0883	0.253670
$34$	−72.1249	−0.363803
$35$	0	0
$36$	90.0000	0.416667
$37$	−315.529	−1.40196	−0.700982	−	0.713179i	$-0.747255\pi$
$37$	−315.529	−1.40196	−0.700982	+	0.713179i	$0.747255\pi$
$38$	343.154	1.46492
$39$	104.823	0.430389
$40$	0	0
$41$	132.265	0.503813	0.251906	−	0.967752i	$-0.418943\pi$
$41$	132.265	0.503813	0.251906	+	0.967752i	$0.418943\pi$
$42$	−266.912	−0.980604
$43$	23.1177	0.0819866	0.0409933	−	0.999159i	$-0.486948\pi$
$43$	23.1177	0.0819866	0.0409933	+	0.999159i	$0.486948\pi$
$44$	160.294	0.549211
$45$	0	0
$46$	−492.021	−1.57706
$47$	−260.912	−0.809742	−0.404871	−	0.914374i	$-0.632684\pi$
$47$	−260.912	−0.809742	−0.404871	+	0.914374i	$0.632684\pi$
$48$	−132.000	−0.396928
$49$	96.7645	0.282112
$50$	0	0
$51$	51.0000	0.140028
$52$	349.411	0.931820
$53$	676.087	1.75222	0.876111	−	0.482110i	$-0.160129\pi$
$53$	676.087	1.75222	0.876111	+	0.482110i	$0.160129\pi$
$54$	−114.551	−0.288675
$55$	0	0
$56$	−177.941	−0.424614
$57$	−242.647	−0.563848
$58$	−653.866	−1.48029
$59$	629.294	1.38859	0.694297	−	0.719689i	$-0.255715\pi$
$59$	629.294	1.38859	0.694297	+	0.719689i	$0.255715\pi$
$60$	0	0
$61$	−461.852	−0.969411	−0.484706	−	0.874677i	$-0.661073\pi$
$61$	−461.852	−0.969411	−0.484706	+	0.874677i	$0.661073\pi$
$62$	−1272.54	−2.60666
$63$	188.735	0.377435
$64$	−728.000	−1.42188
$65$	0	0
$66$	−204.021	−0.380505
$67$	789.470	1.43954	0.719770	−	0.694213i	$-0.244247\pi$
$67$	789.470	1.43954	0.719770	+	0.694213i	$0.244247\pi$
$68$	170.000	0.303170
$69$	347.912	0.607009
$70$	0	0
$71$	−686.412	−1.14735	−0.573677	−	0.819082i	$-0.694484\pi$
$71$	−686.412	−1.14735	−0.573677	+	0.819082i	$0.694484\pi$
$72$	−76.3675	−0.125000
$73$	−484.912	−0.777461	−0.388730	−	0.921352i	$-0.627086\pi$
$73$	−484.912	−0.777461	−0.388730	+	0.921352i	$0.627086\pi$
$74$	1338.68	2.10295
$75$	0	0
$76$	−808.823	−1.22077
$77$	336.146	0.497499
$78$	−444.728	−0.645584
$79$	254.000	0.361737	0.180869	−	0.983507i	$-0.442109\pi$
$79$	254.000	0.361737	0.180869	+	0.983507i	$0.442109\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−561.153	−0.755719
$83$	−548.912	−0.725914	−0.362957	−	0.931806i	$-0.618233\pi$
$83$	−548.912	−0.725914	−0.362957	+	0.931806i	$0.618233\pi$
$84$	629.117	0.817170
$85$	0	0
$86$	−98.0803	−0.122980
$87$	462.353	0.569764
$88$	−136.014	−0.164763
$89$	925.145	1.10186	0.550928	−	0.834553i	$-0.314274\pi$
$89$	925.145	1.10186	0.550928	+	0.834553i	$0.314274\pi$
$90$	0	0
$91$	732.735	0.844082
$92$	1159.71	1.31421
$93$	899.823	1.00330
$94$	1106.95	1.21461
$95$	0	0
$96$	763.675	0.811899
$97$	−732.617	−0.766866	−0.383433	−	0.923569i	$-0.625258\pi$
$97$	−732.617	−0.766866	−0.383433	+	0.923569i	$0.625258\pi$
$98$	−410.537	−0.423168
$99$	144.265	0.146456
$100$	0	0
$101$	−128.528	−0.126624	−0.0633120	−	0.997994i	$-0.520166\pi$
$101$	−128.528	−0.126624	−0.0633120	+	0.997994i	$0.520166\pi$
$102$	−216.375	−0.210042
$103$	−5.35325	−0.00512108	−0.00256054	−	0.999997i	$-0.500815\pi$
$103$	−5.35325	−0.00512108	−0.00256054	+	0.999997i	$0.500815\pi$
$104$	−296.485	−0.279546
$105$	0	0
$106$	−2868.40	−2.62833
$107$	−473.382	−0.427697	−0.213848	−	0.976867i	$-0.568600\pi$
$107$	−473.382	−0.427697	−0.213848	+	0.976867i	$0.568600\pi$
$108$	270.000	0.240563
$109$	−352.353	−0.309627	−0.154813	−	0.987944i	$-0.549478\pi$
$109$	−352.353	−0.309627	−0.154813	+	0.987944i	$0.549478\pi$
$110$	0	0
$111$	−946.587	−0.809424
$112$	−922.705	−0.778459
$113$	733.617	0.610734	0.305367	−	0.952235i	$-0.401221\pi$
$113$	733.617	0.610734	0.305367	+	0.952235i	$0.401221\pi$
$114$	1029.46	0.845772
$115$	0	0
$116$	1541.18	1.23358
$117$	314.470	0.248485
$118$	−2669.87	−2.08289
$119$	356.500	0.274624
$120$	0	0
$121$	−1074.06	−0.806955
$122$	1959.47	1.45412
$123$	396.795	0.290876
$124$	2999.41	2.17222
$125$	0	0
$126$	−800.735	−0.566152
$127$	−340.410	−0.237846	−0.118923	−	0.992903i	$-0.537944\pi$
$127$	−340.410	−0.237846	−0.118923	+	0.992903i	$0.537944\pi$
$128$	1052.17	0.726562
$129$	69.3532	0.0473350
$130$	0	0
$131$	−2133.85	−1.42317	−0.711586	−	0.702599i	$-0.752023\pi$
$131$	−2133.85	−1.42317	−0.711586	+	0.702599i	$0.752023\pi$
$132$	480.883	0.317087
$133$	−1696.15	−1.10582
$134$	−3349.44	−2.15931
$135$	0	0
$136$	−144.250	−0.0909509
$137$	−55.1455	−0.0343897	−0.0171949	−	0.999852i	$-0.505474\pi$
$137$	−55.1455	−0.0343897	−0.0171949	+	0.999852i	$0.505474\pi$
$138$	−1476.06	−0.910514
$139$	256.266	0.156375	0.0781877	−	0.996939i	$-0.475087\pi$
$139$	256.266	0.156375	0.0781877	+	0.996939i	$0.475087\pi$
$140$	0	0
$141$	−782.735	−0.467505
$142$	2912.20	1.72103
$143$	560.087	0.327530
$144$	−396.000	−0.229167
$145$	0	0
$146$	2057.31	1.16619
$147$	290.294	0.162878
$148$	−3155.29	−1.75245
$149$	−84.3836	−0.0463958	−0.0231979	−	0.999731i	$-0.507385\pi$
$149$	−84.3836	−0.0463958	−0.0231979	+	0.999731i	$0.507385\pi$
$150$	0	0
$151$	1051.65	0.566767	0.283383	−	0.959007i	$-0.408543\pi$
$151$	1051.65	0.566767	0.283383	+	0.959007i	$0.408543\pi$
$152$	686.309	0.366230
$153$	153.000	0.0808452
$154$	−1426.15	−0.746249
$155$	0	0
$156$	1048.23	0.537986
$157$	−1587.47	−0.806968	−0.403484	−	0.914987i	$-0.632201\pi$
$157$	−1587.47	−0.806968	−0.403484	+	0.914987i	$0.632201\pi$
$158$	−1077.63	−0.542606
$159$	2028.26	1.01165
$160$	0	0
$161$	2431.97	1.19047
$162$	−343.654	−0.166667
$163$	−1749.59	−0.840727	−0.420363	−	0.907356i	$-0.638097\pi$
$163$	−1749.59	−0.840727	−0.420363	+	0.907356i	$0.638097\pi$
$164$	1322.65	0.629766
$165$	0	0
$166$	2328.84	1.08887
$167$	1624.21	0.752604	0.376302	−	0.926497i	$-0.377196\pi$
$167$	1624.21	0.752604	0.376302	+	0.926497i	$0.377196\pi$
$168$	−533.823	−0.245151
$169$	−976.118	−0.444296
$170$	0	0
$171$	−727.940	−0.325538
$172$	231.177	0.102483
$173$	3734.50	1.64120	0.820602	−	0.571500i	$-0.193638\pi$
$173$	3734.50	1.64120	0.820602	+	0.571500i	$0.193638\pi$
$174$	−1961.60	−0.854646
$175$	0	0
$176$	−705.295	−0.302066
$177$	1887.88	0.801705
$178$	−3925.06	−1.65278
$179$	−1434.65	−0.599053	−0.299526	−	0.954088i	$-0.596829\pi$
$179$	−1434.65	−0.599053	−0.299526	+	0.954088i	$0.596829\pi$
$180$	0	0
$181$	−219.263	−0.0900426	−0.0450213	−	0.998986i	$-0.514336\pi$
$181$	−219.263	−0.0900426	−0.0450213	+	0.998986i	$0.514336\pi$
$182$	−3108.73	−1.26612
$183$	−1385.56	−0.559690
$184$	−984.043	−0.394264
$185$	0	0
$186$	−3817.63	−1.50496
$187$	272.500	0.106563
$188$	−2609.12	−1.01218
$189$	566.205	0.217912
$190$	0	0
$191$	−3116.44	−1.18062	−0.590308	−	0.807178i	$-0.700994\pi$
$191$	−3116.44	−1.18062	−0.590308	+	0.807178i	$0.700994\pi$
$192$	−2184.00	−0.820920
$193$	3921.82	1.46269	0.731344	−	0.682009i	$-0.238894\pi$
$193$	3921.82	1.46269	0.731344	+	0.682009i	$0.238894\pi$
$194$	3108.23	1.15030
$195$	0	0
$196$	967.645	0.352640
$197$	−3141.68	−1.13622	−0.568110	−	0.822953i	$-0.692325\pi$
$197$	−3141.68	−1.13622	−0.568110	+	0.822953i	$0.692325\pi$
$198$	−612.064	−0.219684
$199$	1832.79	0.652880	0.326440	−	0.945218i	$-0.394151\pi$
$199$	1832.79	0.652880	0.326440	+	0.945218i	$0.394151\pi$
$200$	0	0
$201$	2368.41	0.831118
$202$	545.299	0.189936
$203$	3231.94	1.11743
$204$	510.000	0.175035
$205$	0	0
$206$	22.7119	0.00768162
$207$	1043.74	0.350457
$208$	−1537.41	−0.512501
$209$	−1296.50	−0.429094
$210$	0	0
$211$	4928.41	1.60799	0.803994	−	0.594637i	$-0.202704\pi$
$211$	4928.41	1.60799	0.803994	+	0.594637i	$0.202704\pi$
$212$	6760.87	2.19028
$213$	−2059.24	−0.662425
$214$	2008.39	0.641545
$215$	0	0
$216$	−229.103	−0.0721688
$217$	6289.93	1.96769
$218$	1494.91	0.464440
$219$	−1454.74	−0.448867
$220$	0	0
$221$	593.999	0.180800
$222$	4016.03	1.21414
$223$	75.4727	0.0226638	0.0113319	−	0.999936i	$-0.496393\pi$
$223$	75.4727	0.0226638	0.0113319	+	0.999936i	$0.496393\pi$
$224$	5338.23	1.59230
$225$	0	0
$226$	−3112.47	−0.916101
$227$	1629.97	0.476587	0.238293	−	0.971193i	$-0.423412\pi$
$227$	1629.97	0.476587	0.238293	+	0.971193i	$0.423412\pi$
$228$	−2426.47	−0.704810
$229$	2000.35	0.577235	0.288617	−	0.957445i	$-0.406804\pi$
$229$	2000.35	0.577235	0.288617	+	0.957445i	$0.406804\pi$
$230$	0	0
$231$	1008.44	0.287231
$232$	−1307.73	−0.370073
$233$	4225.26	1.18801	0.594004	−	0.804462i	$-0.297546\pi$
$233$	4225.26	1.18801	0.594004	+	0.804462i	$0.297546\pi$
$234$	−1334.18	−0.372728
$235$	0	0
$236$	6292.94	1.73574
$237$	762.000	0.208849
$238$	−1512.50	−0.411936
$239$	−1614.97	−0.437086	−0.218543	−	0.975827i	$-0.570130\pi$
$239$	−1614.97	−0.437086	−0.218543	+	0.975827i	$0.570130\pi$
$240$	0	0
$241$	−6761.67	−1.80729	−0.903646	−	0.428280i	$-0.859120\pi$
$241$	−6761.67	−1.80729	−0.903646	+	0.428280i	$0.859120\pi$
$242$	4556.84	1.21043
$243$	243.000	0.0641500
$244$	−4618.52	−1.21176
$245$	0	0
$246$	−1683.46	−0.436314
$247$	−2826.12	−0.728022
$248$	−2545.08	−0.651666
$249$	−1646.74	−0.419107
$250$	0	0
$251$	2880.29	0.724313	0.362156	−	0.932117i	$-0.382041\pi$
$251$	2880.29	0.724313	0.362156	+	0.932117i	$0.382041\pi$
$252$	1887.35	0.471793
$253$	1858.94	0.461940
$254$	1444.24	0.356769
$255$	0	0
$256$	1360.00	0.332031
$257$	1460.44	0.354474	0.177237	−	0.984168i	$-0.443284\pi$
$257$	1460.44	0.354474	0.177237	+	0.984168i	$0.443284\pi$
$258$	−294.241	−0.0710025
$259$	−6616.82	−1.58745
$260$	0	0
$261$	1387.06	0.328953
$262$	9053.17	2.13476
$263$	7601.29	1.78219	0.891094	−	0.453819i	$-0.149939\pi$
$263$	7601.29	1.78219	0.891094	+	0.453819i	$0.149939\pi$
$264$	−408.043	−0.0951261
$265$	0	0
$266$	7196.14	1.65874
$267$	2775.44	0.636157
$268$	7894.70	1.79942
$269$	6896.26	1.56309	0.781547	−	0.623846i	$-0.214431\pi$
$269$	6896.26	1.56309	0.781547	+	0.623846i	$0.214431\pi$
$270$	0	0
$271$	849.234	0.190359	0.0951795	−	0.995460i	$-0.469658\pi$
$271$	849.234	0.190359	0.0951795	+	0.995460i	$0.469658\pi$
$272$	−748.000	−0.166743
$273$	2198.21	0.487331
$274$	233.962	0.0515846
$275$	0	0
$276$	3479.12	0.758762
$277$	−4080.09	−0.885013	−0.442507	−	0.896765i	$-0.645911\pi$
$277$	−4080.09	−0.885013	−0.442507	+	0.896765i	$0.645911\pi$
$278$	−1087.24	−0.234563
$279$	2699.47	0.579258
$280$	0	0
$281$	4967.90	1.05466	0.527331	−	0.849660i	$-0.323193\pi$
$281$	4967.90	1.05466	0.527331	+	0.849660i	$0.323193\pi$
$282$	3320.86	0.701257
$283$	1475.83	0.309996	0.154998	−	0.987915i	$-0.450463\pi$
$283$	1475.83	0.309996	0.154998	+	0.987915i	$0.450463\pi$
$284$	−6864.12	−1.43419
$285$	0	0
$286$	−2376.25	−0.491295
$287$	2773.67	0.570469
$288$	2291.03	0.468750
$289$	289.000	0.0588235
$290$	0	0
$291$	−2197.85	−0.442750
$292$	−4849.12	−0.971826
$293$	−521.056	−0.103892	−0.0519461	−	0.998650i	$-0.516542\pi$
$293$	−521.056	−0.103892	−0.0519461	+	0.998650i	$0.516542\pi$
$294$	−1231.61	−0.244316
$295$	0	0
$296$	2677.35	0.525736
$297$	432.795	0.0845566
$298$	358.009	0.0695937
$299$	4052.14	0.783751
$300$	0	0
$301$	484.792	0.0928337
$302$	−4461.76	−0.850150
$303$	−385.584	−0.0731064
$304$	3558.82	0.671422
$305$	0	0
$306$	−649.124	−0.121268
$307$	1718.23	0.319429	0.159715	−	0.987163i	$-0.448943\pi$
$307$	1718.23	0.319429	0.159715	+	0.987163i	$0.448943\pi$
$308$	3361.46	0.621874
$309$	−16.0597	−0.00295666
$310$	0	0
$311$	4916.05	0.896347	0.448173	−	0.893947i	$-0.352075\pi$
$311$	4916.05	0.896347	0.448173	+	0.893947i	$0.352075\pi$
$312$	−889.456	−0.161396
$313$	−7375.84	−1.33197	−0.665986	−	0.745964i	$-0.731989\pi$
$313$	−7375.84	−1.33197	−0.665986	+	0.745964i	$0.731989\pi$
$314$	6735.07	1.21045
$315$	0	0
$316$	2540.00	0.452171
$317$	−522.706	−0.0926124	−0.0463062	−	0.998927i	$-0.514745\pi$
$317$	−522.706	−0.0926124	−0.0463062	+	0.998927i	$0.514745\pi$
$318$	−8605.19	−1.51747
$319$	2470.42	0.433596
$320$	0	0
$321$	−1420.15	−0.246931
$322$	−10318.0	−1.78571
$323$	−1375.00	−0.236864
$324$	810.000	0.138889
$325$	0	0
$326$	7422.88	1.26109
$327$	−1057.06	−0.178763
$328$	−1122.31	−0.188930
$329$	−5471.46	−0.916874
$330$	0	0
$331$	5016.46	0.833020	0.416510	−	0.909131i	$-0.363253\pi$
$331$	5016.46	0.833020	0.416510	+	0.909131i	$0.363253\pi$
$332$	−5489.12	−0.907393
$333$	−2839.76	−0.467321
$334$	−6890.92	−1.12891
$335$	0	0
$336$	−2768.11	−0.449443
$337$	10810.8	1.74748	0.873741	−	0.486392i	$-0.161687\pi$
$337$	10810.8	1.74748	0.873741	+	0.486392i	$0.161687\pi$
$338$	4141.32	0.666444
$339$	2200.85	0.352607
$340$	0	0
$341$	4807.89	0.763524
$342$	3088.39	0.488307
$343$	−5163.70	−0.812867
$344$	−196.161	−0.0307450
$345$	0	0
$346$	−15844.1	−2.46181
$347$	4137.35	0.640070	0.320035	−	0.947406i	$-0.396305\pi$
$347$	4137.35	0.640070	0.320035	+	0.947406i	$0.396305\pi$
$348$	4623.53	0.712205
$349$	7531.11	1.15510	0.577552	−	0.816354i	$-0.304008\pi$
$349$	7531.11	1.15510	0.577552	+	0.816354i	$0.304008\pi$
$350$	0	0
$351$	943.410	0.143463
$352$	4080.43	0.617862
$353$	4872.29	0.734634	0.367317	−	0.930096i	$-0.380276\pi$
$353$	4872.29	0.734634	0.367317	+	0.930096i	$0.380276\pi$
$354$	−8009.60	−1.20256
$355$	0	0
$356$	9251.45	1.37732
$357$	1069.50	0.158554
$358$	6086.69	0.898579
$359$	−31.9429	−0.00469604	−0.00234802	−	0.999997i	$-0.500747\pi$
$359$	−31.9429	−0.00469604	−0.00234802	+	0.999997i	$0.500747\pi$
$360$	0	0
$361$	−317.061	−0.0462256
$362$	930.255	0.135064
$363$	−3222.17	−0.465896
$364$	7327.35	1.05510
$365$	0	0
$366$	5878.42	0.839535
$367$	−2262.42	−0.321791	−0.160895	−	0.986971i	$-0.551438\pi$
$367$	−2262.42	−0.321791	−0.160895	+	0.986971i	$0.551438\pi$
$368$	−5102.70	−0.722818
$369$	1190.38	0.167938
$370$	0	0
$371$	14177.9	1.98405
$372$	8998.23	1.25413
$373$	5788.70	0.803559	0.401780	−	0.915736i	$-0.368392\pi$
$373$	5788.70	0.803559	0.401780	+	0.915736i	$0.368392\pi$
$374$	−1156.12	−0.159844
$375$	0	0
$376$	2213.91	0.303653
$377$	5385.05	0.735661
$378$	−2402.21	−0.326868
$379$	10940.9	1.48284	0.741420	−	0.671041i	$-0.234153\pi$
$379$	10940.9	1.48284	0.741420	+	0.671041i	$0.234153\pi$
$380$	0	0
$381$	−1021.23	−0.137321
$382$	13221.9	1.77092
$383$	−2414.03	−0.322066	−0.161033	−	0.986949i	$-0.551483\pi$
$383$	−2414.03	−0.322066	−0.161033	+	0.986949i	$0.551483\pi$
$384$	3156.52	0.419481
$385$	0	0
$386$	−16638.9	−2.19403
$387$	208.060	0.0273289
$388$	−7326.17	−0.958583
$389$	−4479.41	−0.583844	−0.291922	−	0.956442i	$-0.594295\pi$
$389$	−4479.41	−0.583844	−0.291922	+	0.956442i	$0.594295\pi$
$390$	0	0
$391$	1971.50	0.254995
$392$	−821.074	−0.105792
$393$	−6401.56	−0.821669
$394$	13329.0	1.70433
$395$	0	0
$396$	1442.65	0.183070
$397$	1780.62	0.225105	0.112553	−	0.993646i	$-0.464097\pi$
$397$	1780.62	0.225105	0.112553	+	0.993646i	$0.464097\pi$
$398$	−7775.88	−0.979321
$399$	−5088.44	−0.638448
$400$	0	0
$401$	−4067.97	−0.506595	−0.253297	−	0.967388i	$-0.581515\pi$
$401$	−4067.97	−0.506595	−0.253297	+	0.967388i	$0.581515\pi$
$402$	−10048.3	−1.24668
$403$	10480.3	1.29543
$404$	−1285.28	−0.158280
$405$	0	0
$406$	−13711.9	−1.67614
$407$	−5057.75	−0.615979
$408$	−432.749	−0.0525105
$409$	−632.302	−0.0764434	−0.0382217	−	0.999269i	$-0.512169\pi$
$409$	−632.302	−0.0764434	−0.0382217	+	0.999269i	$0.512169\pi$
$410$	0	0
$411$	−165.436	−0.0198549
$412$	−53.5325	−0.00640135
$413$	13196.6	1.57231
$414$	−4428.19	−0.525686
$415$	0	0
$416$	8894.56	1.04830
$417$	768.797	0.0902834
$418$	5500.57	0.643640
$419$	−1107.06	−0.129077	−0.0645386	−	0.997915i	$-0.520558\pi$
$419$	−1107.06	−0.129077	−0.0645386	+	0.997915i	$0.520558\pi$
$420$	0	0
$421$	−15977.3	−1.84961	−0.924804	−	0.380444i	$-0.875771\pi$
$421$	−15977.3	−1.84961	−0.924804	+	0.380444i	$0.875771\pi$
$422$	−20909.5	−2.41198
$423$	−2348.21	−0.269914
$424$	−5736.79	−0.657083
$425$	0	0
$426$	8736.60	0.993638
$427$	−9685.30	−1.09767
$428$	−4733.82	−0.534621
$429$	1680.26	0.189100
$430$	0	0
$431$	10283.0	1.14922	0.574610	−	0.818427i	$-0.305154\pi$
$431$	10283.0	1.14922	0.574610	+	0.818427i	$0.305154\pi$
$432$	−1188.00	−0.132309
$433$	7084.29	0.786257	0.393128	−	0.919484i	$-0.371393\pi$
$433$	7084.29	0.786257	0.393128	+	0.919484i	$0.371393\pi$
$434$	−26685.9	−2.95153
$435$	0	0
$436$	−3523.53	−0.387033
$437$	−9379.96	−1.02678
$438$	6171.92	0.673301
$439$	−5308.51	−0.577133	−0.288567	−	0.957460i	$-0.593179\pi$
$439$	−5308.51	−0.577133	−0.288567	+	0.957460i	$0.593179\pi$
$440$	0	0
$441$	870.881	0.0940374
$442$	−2520.12	−0.271199
$443$	−3533.26	−0.378939	−0.189470	−	0.981887i	$-0.560677\pi$
$443$	−3533.26	−0.378939	−0.189470	+	0.981887i	$0.560677\pi$
$444$	−9465.87	−1.01178
$445$	0	0
$446$	−320.204	−0.0339957
$447$	−253.151	−0.0267866
$448$	−15266.6	−1.60999
$449$	4787.18	0.503165	0.251582	−	0.967836i	$-0.419049\pi$
$449$	4787.18	0.503165	0.251582	+	0.967836i	$0.419049\pi$
$450$	0	0
$451$	2120.13	0.221360
$452$	7336.17	0.763417
$453$	3154.94	0.327223
$454$	−6915.39	−0.714880
$455$	0	0
$456$	2058.93	0.211443
$457$	−13168.8	−1.34795	−0.673973	−	0.738756i	$-0.735414\pi$
$457$	−13168.8	−1.34795	−0.673973	+	0.738756i	$0.735414\pi$
$458$	−8486.76	−0.865852
$459$	459.000	0.0466760
$460$	0	0
$461$	−7145.81	−0.721938	−0.360969	−	0.932578i	$-0.617554\pi$
$461$	−7145.81	−0.721938	−0.360969	+	0.932578i	$0.617554\pi$
$462$	−4278.44	−0.430847
$463$	−9462.71	−0.949826	−0.474913	−	0.880033i	$-0.657520\pi$
$463$	−9462.71	−0.949826	−0.474913	+	0.880033i	$0.657520\pi$
$464$	−6781.18	−0.678466
$465$	0	0
$466$	−17926.3	−1.78201
$467$	5306.09	0.525774	0.262887	−	0.964827i	$-0.415325\pi$
$467$	5306.09	0.525774	0.262887	+	0.964827i	$0.415325\pi$
$468$	3144.70	0.310607
$469$	16555.6	1.63000
$470$	0	0
$471$	−4762.41	−0.465903
$472$	−5339.73	−0.520723
$473$	370.565	0.0360224
$474$	−3232.89	−0.313274
$475$	0	0
$476$	3565.00	0.343280
$477$	6084.79	0.584074
$478$	6851.73	0.655630
$479$	11731.4	1.11904	0.559522	−	0.828815i	$-0.310985\pi$
$479$	11731.4	1.11904	0.559522	+	0.828815i	$0.310985\pi$
$480$	0	0
$481$	−11024.9	−1.04510
$482$	28687.3	2.71094
$483$	7295.90	0.687319
$484$	−10740.6	−1.00869
$485$	0	0
$486$	−1030.96	−0.0962250
$487$	−11075.5	−1.03055	−0.515274	−	0.857026i	$-0.672310\pi$
$487$	−11075.5	−1.03055	−0.515274	+	0.857026i	$0.672310\pi$
$488$	3918.94	0.363529
$489$	−5248.77	−0.485394
$490$	0	0
$491$	2009.14	0.184666	0.0923331	−	0.995728i	$-0.470568\pi$
$491$	2009.14	0.184666	0.0923331	+	0.995728i	$0.470568\pi$
$492$	3967.95	0.363595
$493$	2620.00	0.239349
$494$	11990.2	1.09203
$495$	0	0
$496$	−13197.4	−1.19472
$497$	−14394.4	−1.29915
$498$	6986.51	0.628660
$499$	9956.24	0.893192	0.446596	−	0.894736i	$-0.352636\pi$
$499$	9956.24	0.893192	0.446596	+	0.894736i	$0.352636\pi$
$500$	0	0
$501$	4872.62	0.434516
$502$	−12220.0	−1.08647
$503$	−11760.9	−1.04253	−0.521266	−	0.853394i	$-0.674540\pi$
$503$	−11760.9	−1.04253	−0.521266	+	0.853394i	$0.674540\pi$
$504$	−1601.47	−0.141538
$505$	0	0
$506$	−7886.83	−0.692910
$507$	−2928.35	−0.256514
$508$	−3404.10	−0.297308
$509$	−948.293	−0.0825783	−0.0412891	−	0.999147i	$-0.513146\pi$
$509$	−948.293	−0.0825783	−0.0412891	+	0.999147i	$0.513146\pi$
$510$	0	0
$511$	−10168.9	−0.880322
$512$	−14187.4	−1.22461
$513$	−2183.82	−0.187949
$514$	−6196.13	−0.531711
$515$	0	0
$516$	693.532	0.0591687
$517$	−4182.27	−0.355775
$518$	28072.8	2.38117
$519$	11203.5	0.947550
$520$	0	0
$521$	−9796.08	−0.823751	−0.411875	−	0.911240i	$-0.635126\pi$
$521$	−9796.08	−0.823751	−0.411875	+	0.911240i	$0.635126\pi$
$522$	−5884.80	−0.493430
$523$	4501.89	0.376394	0.188197	−	0.982131i	$-0.439736\pi$
$523$	4501.89	0.376394	0.188197	+	0.982131i	$0.439736\pi$
$524$	−21338.5	−1.77897
$525$	0	0
$526$	−32249.5	−2.67328
$527$	5099.00	0.421472
$528$	−2115.89	−0.174398
$529$	1282.17	0.105381
$530$	0	0
$531$	5663.64	0.462865
$532$	−16961.5	−1.38228
$533$	4621.49	0.375570
$534$	−11775.2	−0.954236
$535$	0	0
$536$	−6698.88	−0.539827
$537$	−4303.94	−0.345863
$538$	−29258.3	−2.34464
$539$	1551.08	0.123951
$540$	0	0
$541$	−8419.68	−0.669114	−0.334557	−	0.942376i	$-0.608587\pi$
$541$	−8419.68	−0.669114	−0.334557	+	0.942376i	$0.608587\pi$
$542$	−3602.99	−0.285538
$543$	−657.790	−0.0519861
$544$	4327.49	0.341066
$545$	0	0
$546$	−9326.19	−0.730997
$547$	−4655.26	−0.363884	−0.181942	−	0.983309i	$-0.558238\pi$
$547$	−4655.26	−0.363884	−0.181942	+	0.983309i	$0.558238\pi$
$548$	−551.455	−0.0429872
$549$	−4156.67	−0.323137
$550$	0	0
$551$	−12465.4	−0.963781
$552$	−2952.13	−0.227629
$553$	5326.52	0.409596
$554$	17310.3	1.32752
$555$	0	0
$556$	2562.66	0.195469
$557$	−17855.9	−1.35831	−0.679157	−	0.733993i	$-0.737654\pi$
$557$	−17855.9	−1.35831	−0.679157	+	0.733993i	$0.737654\pi$
$558$	−11452.9	−0.868887
$559$	807.760	0.0611174
$560$	0	0
$561$	817.501	0.0615239
$562$	−21077.0	−1.58199
$563$	21434.5	1.60454	0.802269	−	0.596962i	$-0.203626\pi$
$563$	21434.5	1.60454	0.802269	+	0.596962i	$0.203626\pi$
$564$	−7827.35	−0.584381
$565$	0	0
$566$	−6261.40	−0.464994
$567$	1698.62	0.125812
$568$	5824.40	0.430258
$569$	−14412.8	−1.06189	−0.530945	−	0.847406i	$-0.678163\pi$
$569$	−14412.8	−1.06189	−0.530945	+	0.847406i	$0.678163\pi$
$570$	0	0
$571$	4492.48	0.329255	0.164627	−	0.986356i	$-0.447358\pi$
$571$	4492.48	0.329255	0.164627	+	0.986356i	$0.447358\pi$
$572$	5600.87	0.409413
$573$	−9349.31	−0.681628
$574$	−11767.7	−0.855703
$575$	0	0
$576$	−6552.00	−0.473958
$577$	−9544.29	−0.688621	−0.344310	−	0.938856i	$-0.611887\pi$
$577$	−9544.29	−0.688621	−0.344310	+	0.938856i	$0.611887\pi$
$578$	−1226.12	−0.0882353
$579$	11765.5	0.844483
$580$	0	0
$581$	−11511.0	−0.821956
$582$	9324.70	0.664126
$583$	10837.3	0.769872
$584$	4114.61	0.291548
$585$	0	0
$586$	2210.65	0.155838
$587$	15671.9	1.10195	0.550977	−	0.834520i	$-0.314255\pi$
$587$	15671.9	1.10195	0.550977	+	0.834520i	$0.314255\pi$
$588$	2902.94	0.203597
$589$	−24259.9	−1.69713
$590$	0	0
$591$	−9425.03	−0.655996
$592$	13883.3	0.963850
$593$	−6737.03	−0.466537	−0.233269	−	0.972412i	$-0.574942\pi$
$593$	−6737.03	−0.466537	−0.233269	+	0.972412i	$0.574942\pi$
$594$	−1836.19	−0.126835
$595$	0	0
$596$	−843.836	−0.0579947
$597$	5498.38	0.376941
$598$	−17191.8	−1.17563
$599$	−25077.1	−1.71056	−0.855278	−	0.518169i	$-0.826614\pi$
$599$	−25077.1	−1.71056	−0.855278	+	0.518169i	$0.826614\pi$
$600$	0	0
$601$	18123.5	1.23007	0.615037	−	0.788499i	$-0.289141\pi$
$601$	18123.5	1.23007	0.615037	+	0.788499i	$0.289141\pi$
$602$	−2056.80	−0.139251
$603$	7105.23	0.479846
$604$	10516.5	0.708459
$605$	0	0
$606$	1635.90	0.109660
$607$	−18377.1	−1.22884	−0.614419	−	0.788980i	$-0.710610\pi$
$607$	−18377.1	−1.22884	−0.614419	+	0.788980i	$0.710610\pi$
$608$	−20589.3	−1.37336
$609$	9695.81	0.645146
$610$	0	0
$611$	−9116.55	−0.603627
$612$	1530.00	0.101057
$613$	−19642.4	−1.29421	−0.647105	−	0.762401i	$-0.724021\pi$
$613$	−19642.4	−1.29421	−0.647105	+	0.762401i	$0.724021\pi$
$614$	−7289.85	−0.479144
$615$	0	0
$616$	−2852.30	−0.186562
$617$	−8738.58	−0.570181	−0.285091	−	0.958501i	$-0.592024\pi$
$617$	−8738.58	−0.570181	−0.285091	+	0.958501i	$0.592024\pi$
$618$	68.1357	0.00443498
$619$	18996.2	1.23348	0.616739	−	0.787168i	$-0.288453\pi$
$619$	18996.2	1.23348	0.616739	+	0.787168i	$0.288453\pi$
$620$	0	0
$621$	3131.21	0.202336
$622$	−20857.0	−1.34452
$623$	19400.8	1.24764
$624$	−4612.23	−0.295892
$625$	0	0
$626$	31293.1	1.99796
$627$	−3889.49	−0.247737
$628$	−15874.7	−1.00871
$629$	−5363.99	−0.340026
$630$	0	0
$631$	18454.2	1.16426	0.582130	−	0.813096i	$-0.302219\pi$
$631$	18454.2	1.16426	0.582130	+	0.813096i	$0.302219\pi$
$632$	−2155.26	−0.135651
$633$	14785.2	0.928373
$634$	2217.66	0.138919
$635$	0	0
$636$	20282.6	1.26456
$637$	3381.06	0.210302
$638$	−10481.1	−0.650393
$639$	−6177.71	−0.382451
$640$	0	0
$641$	−2208.09	−0.136060	−0.0680300	−	0.997683i	$-0.521671\pi$
$641$	−2208.09	−0.136060	−0.0680300	+	0.997683i	$0.521671\pi$
$642$	6025.17	0.370396
$643$	14048.7	0.861625	0.430813	−	0.902441i	$-0.358227\pi$
$643$	14048.7	0.861625	0.430813	+	0.902441i	$0.358227\pi$
$644$	24319.7	1.48809
$645$	0	0
$646$	5833.62	0.355295
$647$	−7959.60	−0.483654	−0.241827	−	0.970319i	$-0.577747\pi$
$647$	−7959.60	−0.483654	−0.241827	+	0.970319i	$0.577747\pi$
$648$	−687.308	−0.0416667
$649$	10087.2	0.610105
$650$	0	0
$651$	18869.8	1.13605
$652$	−17495.9	−1.05091
$653$	−8533.37	−0.511388	−0.255694	−	0.966758i	$-0.582304\pi$
$653$	−8533.37	−0.511388	−0.255694	+	0.966758i	$0.582304\pi$
$654$	4484.72	0.268145
$655$	0	0
$656$	−5819.66	−0.346371
$657$	−4364.21	−0.259154
$658$	23213.5	1.37531
$659$	8532.32	0.504358	0.252179	−	0.967681i	$-0.418853\pi$
$659$	8532.32	0.504358	0.252179	+	0.967681i	$0.418853\pi$
$660$	0	0
$661$	24194.0	1.42366	0.711828	−	0.702353i	$-0.247867\pi$
$661$	24194.0	1.42366	0.711828	+	0.702353i	$0.247867\pi$
$662$	−21283.1	−1.24953
$663$	1782.00	0.104385
$664$	4657.67	0.272218
$665$	0	0
$666$	12048.1	0.700982
$667$	17873.1	1.03756
$668$	16242.1	0.940755
$669$	226.418	0.0130850
$670$	0	0
$671$	−7403.23	−0.425929
$672$	16014.7	0.919316
$673$	10069.6	0.576750	0.288375	−	0.957518i	$-0.406885\pi$
$673$	10069.6	0.576750	0.288375	+	0.957518i	$0.406885\pi$
$674$	−45866.3	−2.62122
$675$	0	0
$676$	−9761.18	−0.555370
$677$	−13155.9	−0.746856	−0.373428	−	0.927659i	$-0.621818\pi$
$677$	−13155.9	−0.746856	−0.373428	+	0.927659i	$0.621818\pi$
$678$	−9337.42	−0.528911
$679$	−15363.4	−0.868326
$680$	0	0
$681$	4889.92	0.275157
$682$	−20398.1	−1.14529
$683$	21814.1	1.22210	0.611049	−	0.791593i	$-0.290748\pi$
$683$	21814.1	1.22210	0.611049	+	0.791593i	$0.290748\pi$
$684$	−7279.40	−0.406922
$685$	0	0
$686$	21907.7	1.21930
$687$	6001.04	0.333267
$688$	−1017.18	−0.0563658
$689$	23623.3	1.30620
$690$	0	0
$691$	−8237.48	−0.453500	−0.226750	−	0.973953i	$-0.572810\pi$
$691$	−8237.48	−0.453500	−0.226750	+	0.973953i	$0.572810\pi$
$692$	37345.0	2.05151
$693$	3025.32	0.165833
$694$	−17553.3	−0.960105
$695$	0	0
$696$	−3923.20	−0.213662
$697$	2248.50	0.122192
$698$	−31951.8	−1.73265
$699$	12675.8	0.685897
$700$	0	0
$701$	3684.13	0.198499	0.0992495	−	0.995063i	$-0.468356\pi$
$701$	3684.13	0.198499	0.0992495	+	0.995063i	$0.468356\pi$
$702$	−4002.55	−0.215195
$703$	25520.7	1.36918
$704$	−11669.4	−0.624728
$705$	0	0
$706$	−20671.4	−1.10195
$707$	−2695.31	−0.143377
$708$	18878.8	1.00213
$709$	26765.8	1.41779	0.708894	−	0.705315i	$-0.249194\pi$
$709$	26765.8	1.41779	0.708894	+	0.705315i	$0.249194\pi$
$710$	0	0
$711$	2286.00	0.120579
$712$	−7850.12	−0.413196
$713$	34784.3	1.82705
$714$	−4537.50	−0.237831
$715$	0	0
$716$	−14346.5	−0.748816
$717$	−4844.91	−0.252352
$718$	135.522	0.00704407
$719$	20163.3	1.04585	0.522924	−	0.852379i	$-0.324841\pi$
$719$	20163.3	1.04585	0.522924	+	0.852379i	$0.324841\pi$
$720$	0	0
$721$	−112.261	−0.00579862
$722$	1345.18	0.0693384
$723$	−20285.0	−1.04344
$724$	−2192.63	−0.112553
$725$	0	0
$726$	13670.5	0.698844
$727$	−23230.6	−1.18511	−0.592556	−	0.805529i	$-0.701881\pi$
$727$	−23230.6	−1.18511	−0.592556	+	0.805529i	$0.701881\pi$
$728$	−6217.46	−0.316531
$729$	729.000	0.0370370
$730$	0	0
$731$	393.002	0.0198847
$732$	−13855.6	−0.699612
$733$	28594.4	1.44087	0.720436	−	0.693521i	$-0.243942\pi$
$733$	28594.4	1.44087	0.720436	+	0.693521i	$0.243942\pi$
$734$	9598.62	0.482686
$735$	0	0
$736$	29521.3	1.47849
$737$	12654.8	0.632489
$738$	−5050.37	−0.251906
$739$	8294.73	0.412891	0.206446	−	0.978458i	$-0.433810\pi$
$739$	8294.73	0.412891	0.206446	+	0.978458i	$0.433810\pi$
$740$	0	0
$741$	−8478.35	−0.420324
$742$	−60151.9	−2.97607
$743$	5881.61	0.290411	0.145205	−	0.989402i	$-0.453616\pi$
$743$	5881.61	0.290411	0.145205	+	0.989402i	$0.453616\pi$
$744$	−7635.25	−0.376239
$745$	0	0
$746$	−24559.4	−1.20534
$747$	−4940.21	−0.241971
$748$	2725.00	0.133203
$749$	−9927.08	−0.484283
$750$	0	0
$751$	1255.39	0.0609984	0.0304992	−	0.999535i	$-0.490290\pi$
$751$	1255.39	0.0609984	0.0304992	+	0.999535i	$0.490290\pi$
$752$	11480.1	0.556698
$753$	8640.88	0.418182
$754$	−22846.8	−1.10349
$755$	0	0
$756$	5662.05	0.272390
$757$	38652.8	1.85582	0.927912	−	0.372799i	$-0.121602\pi$
$757$	38652.8	1.85582	0.927912	+	0.372799i	$0.121602\pi$
$758$	−46418.3	−2.22426
$759$	5576.83	0.266701
$760$	0	0
$761$	−3286.48	−0.156550	−0.0782752	−	0.996932i	$-0.524941\pi$
$761$	−3286.48	−0.156550	−0.0782752	+	0.996932i	$0.524941\pi$
$762$	4332.71	0.205981
$763$	−7389.05	−0.350592
$764$	−31164.4	−1.47577
$765$	0	0
$766$	10241.9	0.483099
$767$	21988.2	1.03514
$768$	4080.00	0.191698
$769$	−29939.6	−1.40396	−0.701982	−	0.712195i	$-0.747701\pi$
$769$	−29939.6	−1.40396	−0.701982	+	0.712195i	$0.747701\pi$
$770$	0	0
$771$	4381.33	0.204656
$772$	39218.2	1.82836
$773$	−40700.7	−1.89379	−0.946896	−	0.321541i	$-0.895799\pi$
$773$	−40700.7	−1.89379	−0.946896	+	0.321541i	$0.895799\pi$
$774$	−882.723	−0.0409933
$775$	0	0
$776$	6216.46	0.287575
$777$	−19850.5	−0.916514
$778$	19004.5	0.875765
$779$	−10697.9	−0.492030
$780$	0	0
$781$	−11002.8	−0.504112
$782$	−8364.36	−0.382492
$783$	4161.18	0.189921
$784$	−4257.64	−0.193952
$785$	0	0
$786$	27159.5	1.23250
$787$	8126.31	0.368071	0.184035	−	0.982920i	$-0.441084\pi$
$787$	8126.31	0.368071	0.184035	+	0.982920i	$0.441084\pi$
$788$	−31416.8	−1.42027
$789$	22803.9	1.02895
$790$	0	0
$791$	15384.4	0.691536
$792$	−1224.13	−0.0549211
$793$	−16137.6	−0.722653
$794$	−7554.53	−0.337658
$795$	0	0
$796$	18327.9	0.816100
$797$	−11987.8	−0.532787	−0.266393	−	0.963864i	$-0.585832\pi$
$797$	−11987.8	−0.532787	−0.266393	+	0.963864i	$0.585832\pi$
$798$	21588.4	0.957671
$799$	−4435.50	−0.196391
$800$	0	0
$801$	8326.31	0.367285
$802$	17258.9	0.759892
$803$	−7772.86	−0.341592
$804$	23684.1	1.03890
$805$	0	0
$806$	−44464.1	−1.94315
$807$	20688.8	0.902453
$808$	1090.60	0.0474840
$809$	−36146.3	−1.57087	−0.785436	−	0.618943i	$-0.787561\pi$
$809$	−36146.3	−1.57087	−0.785436	+	0.618943i	$0.787561\pi$
$810$	0	0
$811$	−6244.38	−0.270370	−0.135185	−	0.990820i	$-0.543163\pi$
$811$	−6244.38	−0.270370	−0.135185	+	0.990820i	$0.543163\pi$
$812$	32319.4	1.39678
$813$	2547.70	0.109904
$814$	21458.2	0.923969
$815$	0	0
$816$	−2244.00	−0.0962693
$817$	−1869.82	−0.0800692
$818$	2682.63	0.114665
$819$	6594.62	0.281361
$820$	0	0
$821$	28413.7	1.20785	0.603924	−	0.797042i	$-0.293603\pi$
$821$	28413.7	1.20785	0.603924	+	0.797042i	$0.293603\pi$
$822$	701.887	0.0297824
$823$	18803.1	0.796399	0.398199	−	0.917299i	$-0.369635\pi$
$823$	18803.1	0.796399	0.398199	+	0.917299i	$0.369635\pi$
$824$	45.4238	0.00192040
$825$	0	0
$826$	−55988.6	−2.35847
$827$	−33860.3	−1.42375	−0.711873	−	0.702309i	$-0.752153\pi$
$827$	−33860.3	−1.42375	−0.711873	+	0.702309i	$0.752153\pi$
$828$	10437.4	0.438071
$829$	−19746.0	−0.827270	−0.413635	−	0.910443i	$-0.635741\pi$
$829$	−19746.0	−0.827270	−0.413635	+	0.910443i	$0.635741\pi$
$830$	0	0
$831$	−12240.3	−0.510963
$832$	−25437.1	−1.05994
$833$	1645.00	0.0684223
$834$	−3261.73	−0.135425
$835$	0	0
$836$	−12965.0	−0.536367
$837$	8098.41	0.334435
$838$	4696.85	0.193616
$839$	−31045.8	−1.27750	−0.638748	−	0.769416i	$-0.720547\pi$
$839$	−31045.8	−1.27750	−0.638748	+	0.769416i	$0.720547\pi$
$840$	0	0
$841$	−636.719	−0.0261068
$842$	67785.9	2.77441
$843$	14903.7	0.608910
$844$	49284.1	2.00999
$845$	0	0
$846$	9962.59	0.404871
$847$	−22523.6	−0.913718
$848$	−29747.8	−1.20465
$849$	4427.48	0.178976
$850$	0	0
$851$	−36592.1	−1.47398
$852$	−20592.4	−0.828031
$853$	−36020.9	−1.44587	−0.722937	−	0.690914i	$-0.757208\pi$
$853$	−36020.9	−1.44587	−0.722937	+	0.690914i	$0.757208\pi$
$854$	41091.2	1.64650
$855$	0	0
$856$	4016.78	0.160386
$857$	35927.0	1.43202	0.716010	−	0.698090i	$-0.245966\pi$
$857$	35927.0	1.43202	0.716010	+	0.698090i	$0.245966\pi$
$858$	−7128.74	−0.283649
$859$	−20914.3	−0.830718	−0.415359	−	0.909658i	$-0.636344\pi$
$859$	−20914.3	−0.830718	−0.415359	+	0.909658i	$0.636344\pi$
$860$	0	0
$861$	8321.01	0.329360
$862$	−43627.0	−1.72383
$863$	5395.57	0.212824	0.106412	−	0.994322i	$-0.466064\pi$
$863$	5395.57	0.212824	0.106412	+	0.994322i	$0.466064\pi$
$864$	6873.08	0.270633
$865$	0	0
$866$	−30056.1	−1.17939
$867$	867.000	0.0339618
$868$	62899.3	2.45961
$869$	4071.48	0.158936
$870$	0	0
$871$	27585.0	1.07311
$872$	2989.82	0.116110
$873$	−6593.56	−0.255622
$874$	39795.8	1.54018
$875$	0	0
$876$	−14547.4	−0.561084
$877$	31305.2	1.20536	0.602681	−	0.797983i	$-0.294099\pi$
$877$	31305.2	1.20536	0.602681	+	0.797983i	$0.294099\pi$
$878$	22522.1	0.865700
$879$	−1563.17	−0.0599822
$880$	0	0
$881$	19975.4	0.763891	0.381945	−	0.924185i	$-0.375254\pi$
$881$	19975.4	0.763891	0.381945	+	0.924185i	$0.375254\pi$
$882$	−3694.83	−0.141056
$883$	−32032.5	−1.22081	−0.610407	−	0.792088i	$-0.708994\pi$
$883$	−32032.5	−1.22081	−0.610407	+	0.792088i	$0.708994\pi$
$884$	5939.99	0.225999
$885$	0	0
$886$	14990.3	0.568409
$887$	−32668.4	−1.23664	−0.618319	−	0.785927i	$-0.712186\pi$
$887$	−32668.4	−1.23664	−0.618319	+	0.785927i	$0.712186\pi$
$888$	8032.06	0.303534
$889$	−7138.58	−0.269314
$890$	0	0
$891$	1298.38	0.0488188
$892$	754.727	0.0283298
$893$	21103.1	0.790805
$894$	1074.03	0.0401799
$895$	0	0
$896$	22064.7	0.822690
$897$	12156.4	0.452499
$898$	−20310.3	−0.754747
$899$	46226.3	1.71494
$900$	0	0
$901$	11493.5	0.424976
$902$	−8994.96	−0.332039
$903$	1454.38	0.0535976
$904$	−6224.95	−0.229025
$905$	0	0
$906$	−13385.3	−0.490835
$907$	39521.1	1.44683	0.723417	−	0.690411i	$-0.242571\pi$
$907$	39521.1	1.44683	0.723417	+	0.690411i	$0.242571\pi$
$908$	16299.7	0.595733
$909$	−1156.75	−0.0422080
$910$	0	0
$911$	47835.9	1.73971	0.869854	−	0.493310i	$-0.164213\pi$
$911$	47835.9	1.73971	0.869854	+	0.493310i	$0.164213\pi$
$912$	10676.5	0.387646
$913$	−8798.75	−0.318944
$914$	55870.6	2.02192
$915$	0	0
$916$	20003.5	0.721543
$917$	−44748.1	−1.61146
$918$	−1947.37	−0.0700140
$919$	−39689.7	−1.42464	−0.712319	−	0.701856i	$-0.752355\pi$
$919$	−39689.7	−1.42464	−0.712319	+	0.701856i	$0.752355\pi$
$920$	0	0
$921$	5154.70	0.184423
$922$	30317.1	1.08291
$923$	−23984.0	−0.855302
$924$	10084.4	0.359039
$925$	0	0
$926$	40146.9	1.42474
$927$	−48.1792	−0.00170703
$928$	39232.0	1.38777
$929$	−4579.27	−0.161723	−0.0808617	−	0.996725i	$-0.525767\pi$
$929$	−4579.27	−0.161723	−0.0808617	+	0.996725i	$0.525767\pi$
$930$	0	0
$931$	−7826.53	−0.275515
$932$	42252.6	1.48501
$933$	14748.2	0.517506
$934$	−22511.8	−0.788661
$935$	0	0
$936$	−2668.37	−0.0931820
$937$	38046.5	1.32649	0.663246	−	0.748401i	$-0.269178\pi$
$937$	38046.5	1.32649	0.663246	+	0.748401i	$0.269178\pi$
$938$	−70239.6	−2.44499
$939$	−22127.5	−0.769015
$940$	0	0
$941$	−9081.31	−0.314604	−0.157302	−	0.987551i	$-0.550280\pi$
$941$	−9081.31	−0.314604	−0.157302	+	0.987551i	$0.550280\pi$
$942$	20205.2	0.698855
$943$	15338.8	0.529694
$944$	−27688.9	−0.954658
$945$	0	0
$946$	−1572.17	−0.0540335
$947$	−39575.9	−1.35802	−0.679009	−	0.734130i	$-0.737590\pi$
$947$	−39575.9	−1.35802	−0.679009	+	0.734130i	$0.737590\pi$
$948$	7620.00	0.261061
$949$	−16943.4	−0.579562
$950$	0	0
$951$	−1568.12	−0.0534698
$952$	−3025.00	−0.102984
$953$	28601.6	0.972189	0.486095	−	0.873906i	$-0.338421\pi$
$953$	28601.6	0.972189	0.486095	+	0.873906i	$0.338421\pi$
$954$	−25815.6	−0.876111
$955$	0	0
$956$	−16149.7	−0.546358
$957$	7411.26	0.250337
$958$	−49772.2	−1.67857
$959$	−1156.43	−0.0389396
$960$	0	0
$961$	60173.7	2.01986
$962$	46774.9	1.56765
$963$	−4260.44	−0.142566
$964$	−67616.7	−2.25912
$965$	0	0
$966$	−30953.9	−1.03098
$967$	−2591.39	−0.0861773	−0.0430887	−	0.999071i	$-0.513720\pi$
$967$	−2591.39	−0.0861773	−0.0430887	+	0.999071i	$0.513720\pi$
$968$	9113.68	0.302608
$969$	−4124.99	−0.136753
$970$	0	0
$971$	−49692.4	−1.64233	−0.821166	−	0.570689i	$-0.806676\pi$
$971$	−49692.4	−1.64233	−0.821166	+	0.570689i	$0.806676\pi$
$972$	2430.00	0.0801875
$973$	5374.04	0.177064
$974$	46989.2	1.54582
$975$	0	0
$976$	20321.5	0.666470
$977$	44489.8	1.45686	0.728432	−	0.685119i	$-0.240250\pi$
$977$	44489.8	1.45686	0.728432	+	0.685119i	$0.240250\pi$
$978$	22268.6	0.728091
$979$	14829.6	0.484121
$980$	0	0
$981$	−3171.18	−0.103209
$982$	−8524.05	−0.276999
$983$	37606.9	1.22022	0.610109	−	0.792318i	$-0.291126\pi$
$983$	37606.9	1.22022	0.610109	+	0.792318i	$0.291126\pi$
$984$	−3366.92	−0.109079
$985$	0	0
$986$	−11115.7	−0.359023
$987$	−16414.4	−0.529358
$988$	−28261.2	−0.910028
$989$	2680.98	0.0861983
$990$	0	0
$991$	−55446.9	−1.77732	−0.888662	−	0.458563i	$-0.848364\pi$
$991$	−55446.9	−1.77732	−0.888662	+	0.458563i	$0.848364\pi$
$992$	76352.5	2.44375
$993$	15049.4	0.480945
$994$	61070.5	1.94873
$995$	0	0
$996$	−16467.4	−0.523884
$997$	39314.6	1.24885	0.624426	−	0.781084i	$-0.285333\pi$
$997$	39314.6	1.24885	0.624426	+	0.781084i	$0.285333\pi$
$998$	−42240.8	−1.33979
$999$	−8519.28	−0.269808

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1275.4.a.m.1.1		2
5.4	even	2		51.4.a.d.1.2	✓	2
15.14	odd	2		153.4.a.e.1.1		2
20.19	odd	2		816.4.a.o.1.2		2
35.34	odd	2		2499.4.a.l.1.2		2
60.59	even	2		2448.4.a.v.1.1		2
85.84	even	2		867.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.d.1.2	✓	2	5.4	even	2
153.4.a.e.1.1		2	15.14	odd	2
816.4.a.o.1.2		2	20.19	odd	2
867.4.a.j.1.2		2	85.84	even	2
1275.4.a.m.1.1		2	1.1	even	1	trivial
2448.4.a.v.1.1		2	60.59	even	2
2499.4.a.l.1.2		2	35.34	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:	\( N \)	\(=\)	\( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1275.a (trivial)

\(f(q)\)	\(=\)	\(q-4.24264 q^{2} +3.00000 q^{3} +10.0000 q^{4} -12.7279 q^{6} +20.9706 q^{7} -8.48528 q^{8} +9.00000 q^{9} +16.0294 q^{11} +30.0000 q^{12} +34.9411 q^{13} -88.9706 q^{14} -44.0000 q^{16} +17.0000 q^{17} -38.1838 q^{18} -80.8823 q^{19} +62.9117 q^{21} -68.0071 q^{22} +115.971 q^{23} -25.4558 q^{24} -148.243 q^{26} +27.0000 q^{27} +209.706 q^{28} +154.118 q^{29} +299.941 q^{31} +254.558 q^{32} +48.0883 q^{33} -72.1249 q^{34} +90.0000 q^{36} -315.529 q^{37} +343.154 q^{38} +104.823 q^{39} +132.265 q^{41} -266.912 q^{42} +23.1177 q^{43} +160.294 q^{44} -492.021 q^{46} -260.912 q^{47} -132.000 q^{48} +96.7645 q^{49} +51.0000 q^{51} +349.411 q^{52} +676.087 q^{53} -114.551 q^{54} -177.941 q^{56} -242.647 q^{57} -653.866 q^{58} +629.294 q^{59} -461.852 q^{61} -1272.54 q^{62} +188.735 q^{63} -728.000 q^{64} -204.021 q^{66} +789.470 q^{67} +170.000 q^{68} +347.912 q^{69} -686.412 q^{71} -76.3675 q^{72} -484.912 q^{73} +1338.68 q^{74} -808.823 q^{76} +336.146 q^{77} -444.728 q^{78} +254.000 q^{79} +81.0000 q^{81} -561.153 q^{82} -548.912 q^{83} +629.117 q^{84} -98.0803 q^{86} +462.353 q^{87} -136.014 q^{88} +925.145 q^{89} +732.735 q^{91} +1159.71 q^{92} +899.823 q^{93} +1106.95 q^{94} +763.675 q^{96} -732.617 q^{97} -410.537 q^{98} +144.265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 20 q^{4} + 8 q^{7} + 18 q^{9} + 66 q^{11} + 60 q^{12} + 2 q^{13} - 144 q^{14} - 88 q^{16} + 34 q^{17} - 26 q^{19} + 24 q^{21} + 144 q^{22} + 198 q^{23} - 288 q^{26} + 54 q^{27} + 80 q^{28}+ \cdots + 594 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 20 * q^4 + 8 * q^7 + 18 * q^9 + 66 * q^11 + 60 * q^12 + 2 * q^13 - 144 * q^14 - 88 * q^16 + 34 * q^17 - 26 * q^19 + 24 * q^21 + 144 * q^22 + 198 * q^23 - 288 * q^26 + 54 * q^27 + 80 * q^28 + 444 * q^29 + 532 * q^31 + 198 * q^33 + 180 * q^36 - 88 * q^37 + 576 * q^38 + 6 * q^39 + 570 * q^41 - 432 * q^42 + 182 * q^43 + 660 * q^44 - 144 * q^46 - 420 * q^47 - 264 * q^48 - 78 * q^49 + 102 * q^51 + 20 * q^52 + 300 * q^53 - 288 * q^56 - 78 * q^57 + 576 * q^58 + 444 * q^59 + 400 * q^61 - 288 * q^62 + 72 * q^63 - 1456 * q^64 + 432 * q^66 + 968 * q^67 + 340 * q^68 + 594 * q^69 - 1848 * q^71 - 868 * q^73 + 2304 * q^74 - 260 * q^76 - 312 * q^77 - 864 * q^78 + 508 * q^79 + 162 * q^81 + 1296 * q^82 - 996 * q^83 + 240 * q^84 + 576 * q^86 + 1332 * q^87 + 288 * q^88 - 288 * q^89 + 1160 * q^91 + 1980 * q^92 + 1596 * q^93 + 432 * q^94 - 1024 * q^97 - 1152 * q^98 + 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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