LMFDB - Embedded newform 1150.4.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(1,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, 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("1150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.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:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 38x^{3} + 38x^{2} + 202x + 101 $ x^5 - x^4 - 38x^3 + 38x^2 + 202*x + 101
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.33268$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -2.43578 q^{3} +4.00000 q^{4} +4.87156 q^{6} +3.42292 q^{7} -8.00000 q^{8} -21.0670 q^{9} -53.2677 q^{11} -9.74311 q^{12} -28.9262 q^{13} -6.84584 q^{14} +16.0000 q^{16} -116.357 q^{17} +42.1340 q^{18} -57.7956 q^{19} -8.33748 q^{21} +106.535 q^{22} -23.0000 q^{23} +19.4862 q^{24} +57.8524 q^{26} +117.081 q^{27} +13.6917 q^{28} -54.5385 q^{29} -263.480 q^{31} -32.0000 q^{32} +129.748 q^{33} +232.714 q^{34} -84.2679 q^{36} -142.376 q^{37} +115.591 q^{38} +70.4578 q^{39} +316.236 q^{41} +16.6750 q^{42} +235.809 q^{43} -213.071 q^{44} +46.0000 q^{46} +504.756 q^{47} -38.9725 q^{48} -331.284 q^{49} +283.420 q^{51} -115.705 q^{52} +380.218 q^{53} -234.161 q^{54} -27.3834 q^{56} +140.777 q^{57} +109.077 q^{58} -653.064 q^{59} -507.741 q^{61} +526.961 q^{62} -72.1106 q^{63} +64.0000 q^{64} -259.497 q^{66} +740.813 q^{67} -465.427 q^{68} +56.0229 q^{69} -302.446 q^{71} +168.536 q^{72} -265.457 q^{73} +284.752 q^{74} -231.183 q^{76} -182.331 q^{77} -140.916 q^{78} -1182.67 q^{79} +283.626 q^{81} -632.473 q^{82} -489.827 q^{83} -33.3499 q^{84} -471.619 q^{86} +132.844 q^{87} +426.142 q^{88} +484.871 q^{89} -99.0121 q^{91} -92.0000 q^{92} +641.780 q^{93} -1009.51 q^{94} +77.9449 q^{96} -1192.07 q^{97} +662.567 q^{98} +1122.19 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 + 12 * q^3 + 20 * q^4 - 24 * q^6 - 24 * q^7 - 40 * q^8 + 11 * q^9 - 54 * q^11 + 48 * q^12 + 36 * q^13 + 48 * q^14 + 80 * q^16 + 132 * q^17 - 22 * q^18 - 50 * q^19 - 158 * q^21 + 108 * q^22 - 115 * q^23 - 96 * q^24 - 72 * q^26 + 489 * q^27 - 96 * q^28 - 104 * q^29 - 342 * q^31 - 160 * q^32 - 348 * q^33 - 264 * q^34 + 44 * q^36 + 380 * q^37 + 100 * q^38 + 193 * q^39 + 2 * q^41 + 316 * q^42 - 114 * q^43 - 216 * q^44 + 230 * q^46 + 966 * q^47 + 192 * q^48 - 441 * q^49 + 1200 * q^51 + 144 * q^52 + 308 * q^53 - 978 * q^54 + 192 * q^56 + 1410 * q^57 + 208 * q^58 - 277 * q^59 + 482 * q^61 + 684 * q^62 - 222 * q^63 + 320 * q^64 + 696 * q^66 + 2044 * q^67 + 528 * q^68 - 276 * q^69 + 208 * q^71 - 88 * q^72 + 646 * q^73 - 760 * q^74 - 200 * q^76 + 2124 * q^77 - 386 * q^78 - 3444 * q^79 + 2933 * q^81 - 4 * q^82 + 1218 * q^83 - 632 * q^84 + 228 * q^86 + 103 * q^87 + 432 * q^88 + 140 * q^89 + 1976 * q^91 - 460 * q^92 + 473 * q^93 - 1932 * q^94 - 384 * q^96 + 2816 * q^97 + 882 * q^98 - 1740 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−2.43578	−0.468766	−0.234383	−	0.972144i	$-0.575307\pi$
$3$	−2.43578	−0.468766	−0.234383	+	0.972144i	$0.575307\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	4.87156	0.331467
$7$	3.42292	0.184820	0.0924102	−	0.995721i	$-0.470543\pi$
$7$	3.42292	0.184820	0.0924102	+	0.995721i	$0.470543\pi$
$8$	−8.00000	−0.353553
$9$	−21.0670	−0.780259
$10$	0	0
$11$	−53.2677	−1.46007	−0.730037	−	0.683408i	$-0.760497\pi$
$11$	−53.2677	−1.46007	−0.730037	+	0.683408i	$0.760497\pi$
$12$	−9.74311	−0.234383
$13$	−28.9262	−0.617130	−0.308565	−	0.951203i	$-0.599849\pi$
$13$	−28.9262	−0.617130	−0.308565	+	0.951203i	$0.599849\pi$
$14$	−6.84584	−0.130688
$15$	0	0
$16$	16.0000	0.250000
$17$	−116.357	−1.66004	−0.830020	−	0.557733i	$-0.811671\pi$
$17$	−116.357	−1.66004	−0.830020	+	0.557733i	$0.811671\pi$
$18$	42.1340	0.551726
$19$	−57.7956	−0.697854	−0.348927	−	0.937150i	$-0.613454\pi$
$19$	−57.7956	−0.697854	−0.348927	+	0.937150i	$0.613454\pi$
$20$	0	0
$21$	−8.33748	−0.0866375
$22$	106.535	1.03243
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	19.4862	0.165734
$25$	0	0
$26$	57.8524	0.436377
$27$	117.081	0.834524
$28$	13.6917	0.0924102
$29$	−54.5385	−0.349225	−0.174613	−	0.984637i	$-0.555867\pi$
$29$	−54.5385	−0.349225	−0.174613	+	0.984637i	$0.555867\pi$
$30$	0	0
$31$	−263.480	−1.52653	−0.763266	−	0.646085i	$-0.776405\pi$
$31$	−263.480	−1.52653	−0.763266	+	0.646085i	$0.776405\pi$
$32$	−32.0000	−0.176777
$33$	129.748	0.684433
$34$	232.714	1.17383
$35$	0	0
$36$	−84.2679	−0.390129
$37$	−142.376	−0.632608	−0.316304	−	0.948658i	$-0.602442\pi$
$37$	−142.376	−0.632608	−0.316304	+	0.948658i	$0.602442\pi$
$38$	115.591	0.493457
$39$	70.4578	0.289289
$40$	0	0
$41$	316.236	1.20458	0.602290	−	0.798277i	$-0.294255\pi$
$41$	316.236	1.20458	0.602290	+	0.798277i	$0.294255\pi$
$42$	16.6750	0.0612619
$43$	235.809	0.836293	0.418146	−	0.908380i	$-0.362680\pi$
$43$	235.809	0.836293	0.418146	+	0.908380i	$0.362680\pi$
$44$	−213.071	−0.730037
$45$	0	0
$46$	46.0000	0.147442
$47$	504.756	1.56652	0.783258	−	0.621697i	$-0.213556\pi$
$47$	504.756	1.56652	0.783258	+	0.621697i	$0.213556\pi$
$48$	−38.9725	−0.117191
$49$	−331.284	−0.965841
$50$	0	0
$51$	283.420	0.778170
$52$	−115.705	−0.308565
$53$	380.218	0.985415	0.492707	−	0.870195i	$-0.336007\pi$
$53$	380.218	0.985415	0.492707	+	0.870195i	$0.336007\pi$
$54$	−234.161	−0.590098
$55$	0	0
$56$	−27.3834	−0.0653439
$57$	140.777	0.327130
$58$	109.077	0.246940
$59$	−653.064	−1.44104	−0.720522	−	0.693432i	$-0.756098\pi$
$59$	−653.064	−1.44104	−0.720522	+	0.693432i	$0.756098\pi$
$60$	0	0
$61$	−507.741	−1.06573	−0.532865	−	0.846200i	$-0.678885\pi$
$61$	−507.741	−1.06573	−0.532865	+	0.846200i	$0.678885\pi$
$62$	526.961	1.07942
$63$	−72.1106	−0.144208
$64$	64.0000	0.125000
$65$	0	0
$66$	−259.497	−0.483967
$67$	740.813	1.35082	0.675408	−	0.737444i	$-0.263968\pi$
$67$	740.813	1.35082	0.675408	+	0.737444i	$0.263968\pi$
$68$	−465.427	−0.830020
$69$	56.0229	0.0977444
$70$	0	0
$71$	−302.446	−0.505546	−0.252773	−	0.967526i	$-0.581343\pi$
$71$	−302.446	−0.505546	−0.252773	+	0.967526i	$0.581343\pi$
$72$	168.536	0.275863
$73$	−265.457	−0.425608	−0.212804	−	0.977095i	$-0.568260\pi$
$73$	−265.457	−0.425608	−0.212804	+	0.977095i	$0.568260\pi$
$74$	284.752	0.447322
$75$	0	0
$76$	−231.183	−0.348927
$77$	−182.331	−0.269851
$78$	−140.916	−0.204558
$79$	−1182.67	−1.68431	−0.842156	−	0.539233i	$-0.818714\pi$
$79$	−1182.67	−1.68431	−0.842156	+	0.539233i	$0.818714\pi$
$80$	0	0
$81$	283.626	0.389062
$82$	−632.473	−0.851767
$83$	−489.827	−0.647778	−0.323889	−	0.946095i	$-0.604990\pi$
$83$	−489.827	−0.647778	−0.323889	+	0.946095i	$0.604990\pi$
$84$	−33.3499	−0.0433187
$85$	0	0
$86$	−471.619	−0.591348
$87$	132.844	0.163705
$88$	426.142	0.516214
$89$	484.871	0.577486	0.288743	−	0.957407i	$-0.406763\pi$
$89$	484.871	0.577486	0.288743	+	0.957407i	$0.406763\pi$
$90$	0	0
$91$	−99.0121	−0.114058
$92$	−92.0000	−0.104257
$93$	641.780	0.715586
$94$	−1009.51	−1.10769
$95$	0	0
$96$	77.9449	0.0828669
$97$	−1192.07	−1.24779	−0.623896	−	0.781507i	$-0.714451\pi$
$97$	−1192.07	−1.24779	−0.623896	+	0.781507i	$0.714451\pi$
$98$	662.567	0.682953
$99$	1122.19	1.13924
$100$	0	0
$101$	1695.78	1.67066	0.835329	−	0.549751i	$-0.185277\pi$
$101$	1695.78	1.67066	0.835329	+	0.549751i	$0.185277\pi$
$102$	−566.839	−0.550249
$103$	768.895	0.735548	0.367774	−	0.929915i	$-0.380120\pi$
$103$	768.895	0.735548	0.367774	+	0.929915i	$0.380120\pi$
$104$	231.410	0.218188
$105$	0	0
$106$	−760.436	−0.696794
$107$	854.991	0.772478	0.386239	−	0.922399i	$-0.373774\pi$
$107$	854.991	0.772478	0.386239	+	0.922399i	$0.373774\pi$
$108$	468.322	0.417262
$109$	−1312.71	−1.15353	−0.576765	−	0.816910i	$-0.695685\pi$
$109$	−1312.71	−1.15353	−0.576765	+	0.816910i	$0.695685\pi$
$110$	0	0
$111$	346.797	0.296545
$112$	54.7667	0.0462051
$113$	−1804.62	−1.50234	−0.751170	−	0.660109i	$-0.770510\pi$
$113$	−1804.62	−1.50234	−0.751170	+	0.660109i	$0.770510\pi$
$114$	−281.555	−0.231316
$115$	0	0
$116$	−218.154	−0.174613
$117$	609.388	0.481521
$118$	1306.13	1.01897
$119$	−398.280	−0.306809
$120$	0	0
$121$	1506.45	1.13182
$122$	1015.48	0.753585
$123$	−770.282	−0.564666
$124$	−1053.92	−0.763266
$125$	0	0
$126$	144.221	0.101970
$127$	−2104.31	−1.47029	−0.735146	−	0.677908i	$-0.762887\pi$
$127$	−2104.31	−1.47029	−0.735146	+	0.677908i	$0.762887\pi$
$128$	−128.000	−0.0883883
$129$	−574.379	−0.392025
$130$	0	0
$131$	1161.59	0.774721	0.387360	−	0.921928i	$-0.373387\pi$
$131$	1161.59	0.774721	0.387360	+	0.921928i	$0.373387\pi$
$132$	518.993	0.342216
$133$	−197.830	−0.128978
$134$	−1481.63	−0.955171
$135$	0	0
$136$	930.855	0.586913
$137$	2754.88	1.71799	0.858996	−	0.511982i	$-0.171089\pi$
$137$	2754.88	1.71799	0.858996	+	0.511982i	$0.171089\pi$
$138$	−112.046	−0.0691157
$139$	−1479.61	−0.902867	−0.451433	−	0.892305i	$-0.649087\pi$
$139$	−1479.61	−0.902867	−0.451433	+	0.892305i	$0.649087\pi$
$140$	0	0
$141$	−1229.47	−0.734329
$142$	604.893	0.357475
$143$	1540.83	0.901055
$144$	−337.072	−0.195065
$145$	0	0
$146$	530.913	0.300950
$147$	806.934	0.452753
$148$	−569.505	−0.316304
$149$	64.8634	0.0356632	0.0178316	−	0.999841i	$-0.494324\pi$
$149$	64.8634	0.0356632	0.0178316	+	0.999841i	$0.494324\pi$
$150$	0	0
$151$	3234.72	1.74330	0.871648	−	0.490133i	$-0.163052\pi$
$151$	3234.72	1.74330	0.871648	+	0.490133i	$0.163052\pi$
$152$	462.365	0.246729
$153$	2451.29	1.29526
$154$	364.662	0.190814
$155$	0	0
$156$	281.831	0.144645
$157$	−3027.52	−1.53899	−0.769497	−	0.638650i	$-0.779493\pi$
$157$	−3027.52	−1.53899	−0.769497	+	0.638650i	$0.779493\pi$
$158$	2365.34	1.19099
$159$	−926.127	−0.461929
$160$	0	0
$161$	−78.7272	−0.0385377
$162$	−567.253	−0.275108
$163$	412.823	0.198373	0.0991865	−	0.995069i	$-0.468376\pi$
$163$	412.823	0.198373	0.0991865	+	0.995069i	$0.468376\pi$
$164$	1264.95	0.602290
$165$	0	0
$166$	979.655	0.458048
$167$	1133.68	0.525308	0.262654	−	0.964890i	$-0.415402\pi$
$167$	1133.68	0.525308	0.262654	+	0.964890i	$0.415402\pi$
$168$	66.6998	0.0306310
$169$	−1360.27	−0.619151
$170$	0	0
$171$	1217.58	0.544507
$172$	943.237	0.418146
$173$	3572.44	1.56999	0.784994	−	0.619504i	$-0.212666\pi$
$173$	3572.44	1.56999	0.784994	+	0.619504i	$0.212666\pi$
$174$	−265.687	−0.115757
$175$	0	0
$176$	−852.283	−0.365019
$177$	1590.72	0.675512
$178$	−969.742	−0.408344
$179$	−3535.07	−1.47611	−0.738055	−	0.674740i	$-0.764256\pi$
$179$	−3535.07	−1.47611	−0.738055	+	0.674740i	$0.764256\pi$
$180$	0	0
$181$	2213.79	0.909116	0.454558	−	0.890717i	$-0.349797\pi$
$181$	2213.79	0.909116	0.454558	+	0.890717i	$0.349797\pi$
$182$	198.024	0.0806513
$183$	1236.74	0.499578
$184$	184.000	0.0737210
$185$	0	0
$186$	−1283.56	−0.505996
$187$	6198.06	2.42378
$188$	2019.02	0.783258
$189$	400.757	0.154237
$190$	0	0
$191$	2675.85	1.01370	0.506852	−	0.862033i	$-0.330809\pi$
$191$	2675.85	1.01370	0.506852	+	0.862033i	$0.330809\pi$
$192$	−155.890	−0.0585957
$193$	3662.81	1.36609	0.683043	−	0.730378i	$-0.260656\pi$
$193$	3662.81	1.36609	0.683043	+	0.730378i	$0.260656\pi$
$194$	2384.13	0.882322
$195$	0	0
$196$	−1325.13	−0.482921
$197$	−2962.85	−1.07154	−0.535772	−	0.844362i	$-0.679980\pi$
$197$	−2962.85	−1.07154	−0.535772	+	0.844362i	$0.679980\pi$
$198$	−2244.38	−0.805561
$199$	5331.22	1.89909	0.949547	−	0.313624i	$-0.101543\pi$
$199$	5331.22	1.89909	0.949547	+	0.313624i	$0.101543\pi$
$200$	0	0
$201$	−1804.46	−0.633216
$202$	−3391.56	−1.18133
$203$	−186.681	−0.0645440
$204$	1133.68	0.389085
$205$	0	0
$206$	−1537.79	−0.520111
$207$	484.541	0.162695
$208$	−462.819	−0.154282
$209$	3078.64	1.01892
$210$	0	0
$211$	−461.010	−0.150414	−0.0752068	−	0.997168i	$-0.523962\pi$
$211$	−461.010	−0.150414	−0.0752068	+	0.997168i	$0.523962\pi$
$212$	1520.87	0.492707
$213$	736.693	0.236983
$214$	−1709.98	−0.546224
$215$	0	0
$216$	−936.644	−0.295049
$217$	−901.872	−0.282134
$218$	2625.42	0.815669
$219$	646.594	0.199510
$220$	0	0
$221$	3365.76	1.02446
$222$	−693.594	−0.209689
$223$	−5375.30	−1.61416	−0.807078	−	0.590445i	$-0.798952\pi$
$223$	−5375.30	−1.61416	−0.807078	+	0.590445i	$0.798952\pi$
$224$	−109.533	−0.0326719
$225$	0	0
$226$	3609.24	1.06231
$227$	−2426.32	−0.709430	−0.354715	−	0.934975i	$-0.615422\pi$
$227$	−2426.32	−0.709430	−0.354715	+	0.934975i	$0.615422\pi$
$228$	563.109	0.163565
$229$	2811.80	0.811394	0.405697	−	0.914008i	$-0.367029\pi$
$229$	2811.80	0.811394	0.405697	+	0.914008i	$0.367029\pi$
$230$	0	0
$231$	444.118	0.126497
$232$	436.308	0.123470
$233$	−1953.94	−0.549386	−0.274693	−	0.961532i	$-0.588576\pi$
$233$	−1953.94	−0.549386	−0.274693	+	0.961532i	$0.588576\pi$
$234$	−1218.78	−0.340487
$235$	0	0
$236$	−2612.25	−0.720522
$237$	2880.72	0.789548
$238$	796.561	0.216947
$239$	−5743.81	−1.55454	−0.777272	−	0.629165i	$-0.783397\pi$
$239$	−5743.81	−1.55454	−0.777272	+	0.629165i	$0.783397\pi$
$240$	0	0
$241$	2650.58	0.708461	0.354230	−	0.935158i	$-0.384743\pi$
$241$	2650.58	0.708461	0.354230	+	0.935158i	$0.384743\pi$
$242$	−3012.90	−0.800315
$243$	−3852.03	−1.01690
$244$	−2030.96	−0.532865
$245$	0	0
$246$	1540.56	0.399279
$247$	1671.81	0.430666
$248$	2107.84	0.539710
$249$	1193.11	0.303656
$250$	0	0
$251$	1044.43	0.262646	0.131323	−	0.991340i	$-0.458078\pi$
$251$	1044.43	0.262646	0.131323	+	0.991340i	$0.458078\pi$
$252$	−288.442	−0.0721038
$253$	1225.16	0.304446
$254$	4208.62	1.03965
$255$	0	0
$256$	256.000	0.0625000
$257$	6368.51	1.54575	0.772873	−	0.634560i	$-0.218819\pi$
$257$	6368.51	1.54575	0.772873	+	0.634560i	$0.218819\pi$
$258$	1148.76	0.277204
$259$	−487.342	−0.116919
$260$	0	0
$261$	1148.96	0.272486
$262$	−2323.18	−0.547810
$263$	1142.77	0.267931	0.133966	−	0.990986i	$-0.457229\pi$
$263$	1142.77	0.267931	0.133966	+	0.990986i	$0.457229\pi$
$264$	−1037.99	−0.241984
$265$	0	0
$266$	395.660	0.0912010
$267$	−1181.04	−0.270706
$268$	2963.25	0.675408
$269$	−6164.14	−1.39715	−0.698577	−	0.715535i	$-0.746183\pi$
$269$	−6164.14	−1.39715	−0.698577	+	0.715535i	$0.746183\pi$
$270$	0	0
$271$	−1921.34	−0.430676	−0.215338	−	0.976540i	$-0.569085\pi$
$271$	−1921.34	−0.430676	−0.215338	+	0.976540i	$0.569085\pi$
$272$	−1861.71	−0.415010
$273$	241.172	0.0534665
$274$	−5509.75	−1.21480
$275$	0	0
$276$	224.092	0.0488722
$277$	5499.03	1.19280	0.596398	−	0.802689i	$-0.296598\pi$
$277$	5499.03	1.19280	0.596398	+	0.802689i	$0.296598\pi$
$278$	2959.21	0.638423
$279$	5550.74	1.19109
$280$	0	0
$281$	−1313.72	−0.278896	−0.139448	−	0.990229i	$-0.544533\pi$
$281$	−1313.72	−0.278896	−0.139448	+	0.990229i	$0.544533\pi$
$282$	2458.95	0.519249
$283$	4002.49	0.840718	0.420359	−	0.907358i	$-0.361904\pi$
$283$	4002.49	0.840718	0.420359	+	0.907358i	$0.361904\pi$
$284$	−1209.79	−0.252773
$285$	0	0
$286$	−3081.66	−0.637142
$287$	1082.45	0.222631
$288$	674.143	0.137932
$289$	8625.92	1.75573
$290$	0	0
$291$	2903.61	0.584922
$292$	−1061.83	−0.212804
$293$	6904.82	1.37674	0.688368	−	0.725362i	$-0.258327\pi$
$293$	6904.82	1.37674	0.688368	+	0.725362i	$0.258327\pi$
$294$	−1613.87	−0.320145
$295$	0	0
$296$	1139.01	0.223661
$297$	−6236.61	−1.21847
$298$	−129.727	−0.0252177
$299$	665.303	0.128680
$300$	0	0
$301$	807.157	0.154564
$302$	−6469.43	−1.23270
$303$	−4130.54	−0.783147
$304$	−924.730	−0.174464
$305$	0	0
$306$	−4902.58	−0.915888
$307$	−5126.59	−0.953061	−0.476531	−	0.879158i	$-0.658106\pi$
$307$	−5126.59	−0.953061	−0.476531	+	0.879158i	$0.658106\pi$
$308$	−729.324	−0.134926
$309$	−1872.86	−0.344800
$310$	0	0
$311$	4637.42	0.845543	0.422772	−	0.906236i	$-0.361057\pi$
$311$	4637.42	0.845543	0.422772	+	0.906236i	$0.361057\pi$
$312$	−563.663	−0.102279
$313$	459.272	0.0829380	0.0414690	−	0.999140i	$-0.486796\pi$
$313$	459.272	0.0829380	0.0414690	+	0.999140i	$0.486796\pi$
$314$	6055.03	1.08823
$315$	0	0
$316$	−4730.68	−0.842156
$317$	3397.75	0.602008	0.301004	−	0.953623i	$-0.402678\pi$
$317$	3397.75	0.602008	0.301004	+	0.953623i	$0.402678\pi$
$318$	1852.25	0.326633
$319$	2905.14	0.509895
$320$	0	0
$321$	−2082.57	−0.362111
$322$	157.454	0.0272503
$323$	6724.92	1.15847
$324$	1134.51	0.194531
$325$	0	0
$326$	−825.646	−0.140271
$327$	3197.47	0.540735
$328$	−2529.89	−0.425884
$329$	1727.74	0.289524
$330$	0	0
$331$	−4112.38	−0.682891	−0.341445	−	0.939902i	$-0.610916\pi$
$331$	−4112.38	−0.682891	−0.341445	+	0.939902i	$0.610916\pi$
$332$	−1959.31	−0.323889
$333$	2999.44	0.493598
$334$	−2267.35	−0.371449
$335$	0	0
$336$	−133.400	−0.0216594
$337$	−1791.60	−0.289598	−0.144799	−	0.989461i	$-0.546254\pi$
$337$	−1791.60	−0.289598	−0.144799	+	0.989461i	$0.546254\pi$
$338$	2720.55	0.437806
$339$	4395.65	0.704245
$340$	0	0
$341$	14035.0	2.22885
$342$	−2435.16	−0.385024
$343$	−2308.02	−0.363327
$344$	−1886.47	−0.295674
$345$	0	0
$346$	−7144.89	−1.11015
$347$	−5134.49	−0.794335	−0.397167	−	0.917746i	$-0.630007\pi$
$347$	−5134.49	−0.794335	−0.397167	+	0.917746i	$0.630007\pi$
$348$	531.374	0.0818525
$349$	−5278.46	−0.809597	−0.404799	−	0.914406i	$-0.632658\pi$
$349$	−5278.46	−0.809597	−0.404799	+	0.914406i	$0.632658\pi$
$350$	0	0
$351$	−3386.69	−0.515010
$352$	1704.57	0.258107
$353$	−7027.23	−1.05955	−0.529775	−	0.848138i	$-0.677724\pi$
$353$	−7027.23	−1.05955	−0.529775	+	0.848138i	$0.677724\pi$
$354$	−3181.44	−0.477659
$355$	0	0
$356$	1939.48	0.288743
$357$	970.123	0.143822
$358$	7070.15	1.04377
$359$	7592.72	1.11623	0.558117	−	0.829762i	$-0.311524\pi$
$359$	7592.72	1.11623	0.558117	+	0.829762i	$0.311524\pi$
$360$	0	0
$361$	−3518.67	−0.513000
$362$	−4427.59	−0.642842
$363$	−3669.37	−0.530557
$364$	−396.048	−0.0570291
$365$	0	0
$366$	−2473.49	−0.353255
$367$	−1964.86	−0.279468	−0.139734	−	0.990189i	$-0.544625\pi$
$367$	−1964.86	−0.279468	−0.139734	+	0.990189i	$0.544625\pi$
$368$	−368.000	−0.0521286
$369$	−6662.14	−0.939885
$370$	0	0
$371$	1301.46	0.182125
$372$	2567.12	0.357793
$373$	−5152.92	−0.715303	−0.357652	−	0.933855i	$-0.616422\pi$
$373$	−5152.92	−0.715303	−0.357652	+	0.933855i	$0.616422\pi$
$374$	−12396.1	−1.71387
$375$	0	0
$376$	−4038.05	−0.553847
$377$	1577.59	0.215517
$378$	−801.515	−0.109062
$379$	12293.3	1.66613	0.833066	−	0.553174i	$-0.186584\pi$
$379$	12293.3	1.66613	0.833066	+	0.553174i	$0.186584\pi$
$380$	0	0
$381$	5125.63	0.689223
$382$	−5351.69	−0.716797
$383$	−5120.93	−0.683204	−0.341602	−	0.939845i	$-0.610969\pi$
$383$	−5120.93	−0.683204	−0.341602	+	0.939845i	$0.610969\pi$
$384$	311.780	0.0414334
$385$	0	0
$386$	−7325.62	−0.965969
$387$	−4967.79	−0.652525
$388$	−4768.26	−0.623896
$389$	5810.57	0.757346	0.378673	−	0.925530i	$-0.376380\pi$
$389$	5810.57	0.757346	0.378673	+	0.925530i	$0.376380\pi$
$390$	0	0
$391$	2676.21	0.346142
$392$	2650.27	0.341477
$393$	−2829.37	−0.363163
$394$	5925.70	0.757697
$395$	0	0
$396$	4488.76	0.569618
$397$	5594.61	0.707268	0.353634	−	0.935384i	$-0.384946\pi$
$397$	5594.61	0.707268	0.353634	+	0.935384i	$0.384946\pi$
$398$	−10662.4	−1.34286
$399$	481.870	0.0604603
$400$	0	0
$401$	−11832.5	−1.47354	−0.736769	−	0.676145i	$-0.763649\pi$
$401$	−11832.5	−1.47354	−0.736769	+	0.676145i	$0.763649\pi$
$402$	3608.91	0.447752
$403$	7621.49	0.942068
$404$	6783.12	0.835329
$405$	0	0
$406$	373.362	0.0456395
$407$	7584.05	0.923655
$408$	−2267.36	−0.275125
$409$	−9352.34	−1.13067	−0.565334	−	0.824862i	$-0.691253\pi$
$409$	−9352.34	−1.13067	−0.565334	+	0.824862i	$0.691253\pi$
$410$	0	0
$411$	−6710.27	−0.805336
$412$	3075.58	0.367774
$413$	−2235.38	−0.266334
$414$	−969.081	−0.115043
$415$	0	0
$416$	925.638	0.109094
$417$	3603.99	0.423233
$418$	−6157.28	−0.720484
$419$	−11362.2	−1.32477	−0.662386	−	0.749163i	$-0.730456\pi$
$419$	−11362.2	−1.32477	−0.662386	+	0.749163i	$0.730456\pi$
$420$	0	0
$421$	7193.77	0.832786	0.416393	−	0.909185i	$-0.363294\pi$
$421$	7193.77	0.832786	0.416393	+	0.909185i	$0.363294\pi$
$422$	922.020	0.106358
$423$	−10633.7	−1.22229
$424$	−3041.75	−0.348397
$425$	0	0
$426$	−1473.39	−0.167572
$427$	−1737.96	−0.196969
$428$	3419.96	0.386239
$429$	−3753.13	−0.422384
$430$	0	0
$431$	−1955.83	−0.218583	−0.109291	−	0.994010i	$-0.534858\pi$
$431$	−1955.83	−0.218583	−0.109291	+	0.994010i	$0.534858\pi$
$432$	1873.29	0.208631
$433$	−5924.76	−0.657566	−0.328783	−	0.944406i	$-0.606638\pi$
$433$	−5924.76	−0.657566	−0.328783	+	0.944406i	$0.606638\pi$
$434$	1803.74	0.199499
$435$	0	0
$436$	−5250.84	−0.576765
$437$	1329.30	0.145513
$438$	−1293.19	−0.141075
$439$	835.218	0.0908036	0.0454018	−	0.998969i	$-0.485543\pi$
$439$	835.218	0.0908036	0.0454018	+	0.998969i	$0.485543\pi$
$440$	0	0
$441$	6979.15	0.753606
$442$	−6731.52	−0.724403
$443$	−6203.12	−0.665280	−0.332640	−	0.943054i	$-0.607939\pi$
$443$	−6203.12	−0.665280	−0.332640	+	0.943054i	$0.607939\pi$
$444$	1387.19	0.148273
$445$	0	0
$446$	10750.6	1.14138
$447$	−157.993	−0.0167177
$448$	219.067	0.0231025
$449$	14091.7	1.48113	0.740567	−	0.671983i	$-0.234557\pi$
$449$	14091.7	1.48113	0.740567	+	0.671983i	$0.234557\pi$
$450$	0	0
$451$	−16845.2	−1.75878
$452$	−7218.48	−0.751170
$453$	−7879.06	−0.817197
$454$	4852.64	0.501642
$455$	0	0
$456$	−1126.22	−0.115658
$457$	−604.386	−0.0618643	−0.0309321	−	0.999521i	$-0.509848\pi$
$457$	−604.386	−0.0618643	−0.0309321	+	0.999521i	$0.509848\pi$
$458$	−5623.61	−0.573742
$459$	−13623.1	−1.38534
$460$	0	0
$461$	−6531.56	−0.659881	−0.329940	−	0.944002i	$-0.607029\pi$
$461$	−6531.56	−0.659881	−0.329940	+	0.944002i	$0.607029\pi$
$462$	−888.236	−0.0894470
$463$	856.811	0.0860030	0.0430015	−	0.999075i	$-0.486308\pi$
$463$	856.811	0.0860030	0.0430015	+	0.999075i	$0.486308\pi$
$464$	−872.615	−0.0873064
$465$	0	0
$466$	3907.88	0.388474
$467$	−314.136	−0.0311273	−0.0155637	−	0.999879i	$-0.504954\pi$
$467$	−314.136	−0.0311273	−0.0155637	+	0.999879i	$0.504954\pi$
$468$	2437.55	0.240760
$469$	2535.74	0.249658
$470$	0	0
$471$	7374.36	0.721428
$472$	5224.51	0.509486
$473$	−12561.0	−1.22105
$474$	−5761.44	−0.558295
$475$	0	0
$476$	−1593.12	−0.153405
$477$	−8010.05	−0.768878
$478$	11487.6	1.09923
$479$	17059.1	1.62724	0.813621	−	0.581396i	$-0.197493\pi$
$479$	17059.1	1.62724	0.813621	+	0.581396i	$0.197493\pi$
$480$	0	0
$481$	4118.40	0.390401
$482$	−5301.16	−0.500957
$483$	191.762	0.0180652
$484$	6025.79	0.565908
$485$	0	0
$486$	7704.05	0.719059
$487$	−8170.42	−0.760241	−0.380120	−	0.924937i	$-0.624117\pi$
$487$	−8170.42	−0.760241	−0.380120	+	0.924937i	$0.624117\pi$
$488$	4061.93	0.376793
$489$	−1005.55	−0.0929905
$490$	0	0
$491$	−12815.1	−1.17788	−0.588940	−	0.808177i	$-0.700455\pi$
$491$	−12815.1	−1.17788	−0.588940	+	0.808177i	$0.700455\pi$
$492$	−3081.13	−0.282333
$493$	6345.92	0.579728
$494$	−3343.62	−0.304527
$495$	0	0
$496$	−4215.69	−0.381633
$497$	−1035.25	−0.0934352
$498$	−2386.22	−0.214717
$499$	−12617.1	−1.13190	−0.565951	−	0.824439i	$-0.691491\pi$
$499$	−12617.1	−1.13190	−0.565951	+	0.824439i	$0.691491\pi$
$500$	0	0
$501$	−2761.38	−0.246246
$502$	−2088.87	−0.185718
$503$	−442.974	−0.0392669	−0.0196334	−	0.999807i	$-0.506250\pi$
$503$	−442.974	−0.0392669	−0.0196334	+	0.999807i	$0.506250\pi$
$504$	576.885	0.0509851
$505$	0	0
$506$	−2450.31	−0.215276
$507$	3313.33	0.290237
$508$	−8417.23	−0.735146
$509$	6420.52	0.559106	0.279553	−	0.960130i	$-0.409814\pi$
$509$	6420.52	0.559106	0.279553	+	0.960130i	$0.409814\pi$
$510$	0	0
$511$	−908.637	−0.0786609
$512$	−512.000	−0.0441942
$513$	−6766.74	−0.582376
$514$	−12737.0	−1.09301
$515$	0	0
$516$	−2297.52	−0.196013
$517$	−26887.2	−2.28723
$518$	974.685	0.0826741
$519$	−8701.68	−0.735956
$520$	0	0
$521$	17068.2	1.43527	0.717633	−	0.696422i	$-0.245226\pi$
$521$	17068.2	1.43527	0.717633	+	0.696422i	$0.245226\pi$
$522$	−2297.92	−0.192677
$523$	−15522.6	−1.29781	−0.648907	−	0.760867i	$-0.724774\pi$
$523$	−15522.6	−1.29781	−0.648907	+	0.760867i	$0.724774\pi$
$524$	4646.35	0.387360
$525$	0	0
$526$	−2285.53	−0.189456
$527$	30657.8	2.53410
$528$	2075.97	0.171108
$529$	529.000	0.0434783
$530$	0	0
$531$	13758.1	1.12439
$532$	−791.319	−0.0644888
$533$	−9147.51	−0.743382
$534$	2362.08	0.191418
$535$	0	0
$536$	−5926.50	−0.477586
$537$	8610.66	0.691950
$538$	12328.3	0.987937
$539$	17646.7	1.41020
$540$	0	0
$541$	−1483.28	−0.117877	−0.0589384	−	0.998262i	$-0.518772\pi$
$541$	−1483.28	−0.117877	−0.0589384	+	0.998262i	$0.518772\pi$
$542$	3842.69	0.304534
$543$	−5392.31	−0.426163
$544$	3723.42	0.293456
$545$	0	0
$546$	−482.343	−0.0378066
$547$	12652.0	0.988958	0.494479	−	0.869190i	$-0.335359\pi$
$547$	12652.0	0.988958	0.494479	+	0.869190i	$0.335359\pi$
$548$	11019.5	0.858996
$549$	10696.6	0.831545
$550$	0	0
$551$	3152.08	0.243708
$552$	−448.183	−0.0345579
$553$	−4048.18	−0.311295
$554$	−10998.1	−0.843435
$555$	0	0
$556$	−5918.42	−0.451433
$557$	12077.5	0.918741	0.459370	−	0.888245i	$-0.348075\pi$
$557$	12077.5	0.918741	0.459370	+	0.888245i	$0.348075\pi$
$558$	−11101.5	−0.842227
$559$	−6821.07	−0.516101
$560$	0	0
$561$	−15097.1	−1.13619
$562$	2627.44	0.197209
$563$	−3687.15	−0.276012	−0.138006	−	0.990431i	$-0.544069\pi$
$563$	−3687.15	−0.276012	−0.138006	+	0.990431i	$0.544069\pi$
$564$	−4917.90	−0.367165
$565$	0	0
$566$	−8004.98	−0.594478
$567$	970.830	0.0719066
$568$	2419.57	0.178738
$569$	−16302.1	−1.20109	−0.600544	−	0.799592i	$-0.705049\pi$
$569$	−16302.1	−1.20109	−0.600544	+	0.799592i	$0.705049\pi$
$570$	0	0
$571$	−26612.6	−1.95044	−0.975221	−	0.221232i	$-0.928992\pi$
$571$	−26612.6	−1.95044	−0.975221	+	0.221232i	$0.928992\pi$
$572$	6163.33	0.450527
$573$	−6517.77	−0.475190
$574$	−2164.90	−0.157424
$575$	0	0
$576$	−1348.29	−0.0975323
$577$	8796.67	0.634680	0.317340	−	0.948312i	$-0.397210\pi$
$577$	8796.67	0.634680	0.317340	+	0.948312i	$0.397210\pi$
$578$	−17251.8	−1.24149
$579$	−8921.79	−0.640375
$580$	0	0
$581$	−1676.64	−0.119722
$582$	−5807.21	−0.413603
$583$	−20253.3	−1.43878
$584$	2123.65	0.150475
$585$	0	0
$586$	−13809.6	−0.973499
$587$	−18951.5	−1.33256	−0.666279	−	0.745703i	$-0.732114\pi$
$587$	−18951.5	−1.33256	−0.666279	+	0.745703i	$0.732114\pi$
$588$	3227.73	0.226377
$589$	15228.0	1.06530
$590$	0	0
$591$	7216.85	0.502304
$592$	−2278.02	−0.158152
$593$	−15133.0	−1.04795	−0.523976	−	0.851733i	$-0.675552\pi$
$593$	−15133.0	−1.04795	−0.523976	+	0.851733i	$0.675552\pi$
$594$	12473.2	0.861587
$595$	0	0
$596$	259.454	0.0178316
$597$	−12985.7	−0.890231
$598$	−1330.61	−0.0909908
$599$	−3606.66	−0.246017	−0.123008	−	0.992406i	$-0.539254\pi$
$599$	−3606.66	−0.246017	−0.123008	+	0.992406i	$0.539254\pi$
$600$	0	0
$601$	9249.64	0.627788	0.313894	−	0.949458i	$-0.398366\pi$
$601$	9249.64	0.627788	0.313894	+	0.949458i	$0.398366\pi$
$602$	−1614.31	−0.109293
$603$	−15606.7	−1.05399
$604$	12938.9	0.871648
$605$	0	0
$606$	8261.09	0.553769
$607$	−5624.31	−0.376085	−0.188043	−	0.982161i	$-0.560214\pi$
$607$	−5624.31	−0.376085	−0.188043	+	0.982161i	$0.560214\pi$
$608$	1849.46	0.123364
$609$	454.713	0.0302560
$610$	0	0
$611$	−14600.7	−0.966743
$612$	9805.15	0.647630
$613$	22217.5	1.46388	0.731938	−	0.681371i	$-0.238616\pi$
$613$	22217.5	1.46388	0.731938	+	0.681371i	$0.238616\pi$
$614$	10253.2	0.673916
$615$	0	0
$616$	1458.65	0.0954069
$617$	18489.6	1.20642	0.603211	−	0.797582i	$-0.293888\pi$
$617$	18489.6	1.20642	0.603211	+	0.797582i	$0.293888\pi$
$618$	3745.72	0.243810
$619$	4607.09	0.299151	0.149575	−	0.988750i	$-0.452209\pi$
$619$	4607.09	0.299151	0.149575	+	0.988750i	$0.452209\pi$
$620$	0	0
$621$	−2692.85	−0.174010
$622$	−9274.84	−0.597889
$623$	1659.68	0.106731
$624$	1127.33	0.0723223
$625$	0	0
$626$	−918.544	−0.0586460
$627$	−7498.89	−0.477634
$628$	−12110.1	−0.769497
$629$	16566.4	1.05016
$630$	0	0
$631$	−22988.7	−1.45034	−0.725170	−	0.688570i	$-0.758239\pi$
$631$	−22988.7	−1.45034	−0.725170	+	0.688570i	$0.758239\pi$
$632$	9461.36	0.595495
$633$	1122.92	0.0705087
$634$	−6795.50	−0.425684
$635$	0	0
$636$	−3704.51	−0.230964
$637$	9582.78	0.596049
$638$	−5810.28	−0.360550
$639$	6371.63	0.394457
$640$	0	0
$641$	1602.02	0.0987144	0.0493572	−	0.998781i	$-0.484283\pi$
$641$	1602.02	0.0987144	0.0493572	+	0.998781i	$0.484283\pi$
$642$	4165.14	0.256051
$643$	−23798.8	−1.45962	−0.729809	−	0.683651i	$-0.760391\pi$
$643$	−23798.8	−1.45962	−0.729809	+	0.683651i	$0.760391\pi$
$644$	−314.909	−0.0192689
$645$	0	0
$646$	−13449.8	−0.819159
$647$	8096.76	0.491988	0.245994	−	0.969271i	$-0.420886\pi$
$647$	8096.76	0.491988	0.245994	+	0.969271i	$0.420886\pi$
$648$	−2269.01	−0.137554
$649$	34787.2	2.10403
$650$	0	0
$651$	2196.76	0.132255
$652$	1651.29	0.0991865
$653$	21008.0	1.25897	0.629483	−	0.777014i	$-0.283267\pi$
$653$	21008.0	1.25897	0.629483	+	0.777014i	$0.283267\pi$
$654$	−6394.94	−0.382358
$655$	0	0
$656$	5059.78	0.301145
$657$	5592.37	0.332084
$658$	−3455.48	−0.204724
$659$	−15473.7	−0.914672	−0.457336	−	0.889294i	$-0.651196\pi$
$659$	−15473.7	−0.914672	−0.457336	+	0.889294i	$0.651196\pi$
$660$	0	0
$661$	10836.2	0.637638	0.318819	−	0.947816i	$-0.396714\pi$
$661$	10836.2	0.637638	0.318819	+	0.947816i	$0.396714\pi$
$662$	8224.76	0.482877
$663$	−8198.25	−0.480232
$664$	3918.62	0.229024
$665$	0	0
$666$	−5998.87	−0.349027
$667$	1254.38	0.0728185
$668$	4534.70	0.262654
$669$	13093.0	0.756661
$670$	0	0
$671$	27046.2	1.55605
$672$	266.799	0.0153155
$673$	25480.4	1.45943	0.729715	−	0.683752i	$-0.239653\pi$
$673$	25480.4	1.45943	0.729715	+	0.683752i	$0.239653\pi$
$674$	3583.20	0.204777
$675$	0	0
$676$	−5441.10	−0.309576
$677$	−10514.6	−0.596913	−0.298457	−	0.954423i	$-0.596472\pi$
$677$	−10514.6	−0.596913	−0.298457	+	0.954423i	$0.596472\pi$
$678$	−8791.31	−0.497977
$679$	−4080.34	−0.230617
$680$	0	0
$681$	5909.98	0.332556
$682$	−28070.0	−1.57603
$683$	13868.7	0.776973	0.388487	−	0.921454i	$-0.372998\pi$
$683$	13868.7	0.776973	0.388487	+	0.921454i	$0.372998\pi$
$684$	4870.32	0.272253
$685$	0	0
$686$	4616.04	0.256911
$687$	−6848.93	−0.380354
$688$	3772.95	0.209073
$689$	−10998.3	−0.608129
$690$	0	0
$691$	12877.1	0.708928	0.354464	−	0.935070i	$-0.384663\pi$
$691$	12877.1	0.708928	0.354464	+	0.935070i	$0.384663\pi$
$692$	14289.8	0.784994
$693$	3841.17	0.210554
$694$	10269.0	0.561679
$695$	0	0
$696$	−1062.75	−0.0578784
$697$	−36796.3	−1.99965
$698$	10556.9	0.572472
$699$	4759.37	0.257533
$700$	0	0
$701$	−9869.22	−0.531748	−0.265874	−	0.964008i	$-0.585661\pi$
$701$	−9869.22	−0.531748	−0.265874	+	0.964008i	$0.585661\pi$
$702$	6773.39	0.364167
$703$	8228.72	0.441468
$704$	−3409.13	−0.182509
$705$	0	0
$706$	14054.5	0.749216
$707$	5804.52	0.308771
$708$	6362.87	0.337756
$709$	−3479.51	−0.184310	−0.0921551	−	0.995745i	$-0.529376\pi$
$709$	−3479.51	−0.184310	−0.0921551	+	0.995745i	$0.529376\pi$
$710$	0	0
$711$	24915.3	1.31420
$712$	−3878.97	−0.204172
$713$	6060.05	0.318304
$714$	−1940.25	−0.101697
$715$	0	0
$716$	−14140.3	−0.738055
$717$	13990.6	0.728717
$718$	−15185.4	−0.789297
$719$	−2812.17	−0.145864	−0.0729319	−	0.997337i	$-0.523236\pi$
$719$	−2812.17	−0.145864	−0.0729319	+	0.997337i	$0.523236\pi$
$720$	0	0
$721$	2631.87	0.135944
$722$	7037.33	0.362746
$723$	−6456.23	−0.332102
$724$	8855.18	0.454558
$725$	0	0
$726$	7338.75	0.375160
$727$	−3326.25	−0.169689	−0.0848445	−	0.996394i	$-0.527039\pi$
$727$	−3326.25	−0.169689	−0.0848445	+	0.996394i	$0.527039\pi$
$728$	792.097	0.0403256
$729$	1724.77	0.0876274
$730$	0	0
$731$	−27438.0	−1.38828
$732$	4946.98	0.249789
$733$	−6522.98	−0.328693	−0.164346	−	0.986403i	$-0.552552\pi$
$733$	−6522.98	−0.328693	−0.164346	+	0.986403i	$0.552552\pi$
$734$	3929.71	0.197613
$735$	0	0
$736$	736.000	0.0368605
$737$	−39461.4	−1.97229
$738$	13324.3	0.664599
$739$	−17729.7	−0.882541	−0.441271	−	0.897374i	$-0.645472\pi$
$739$	−17729.7	−0.882541	−0.441271	+	0.897374i	$0.645472\pi$
$740$	0	0
$741$	−4072.15	−0.201882
$742$	−2602.91	−0.128782
$743$	12465.1	0.615480	0.307740	−	0.951470i	$-0.400427\pi$
$743$	12465.1	0.615480	0.307740	+	0.951470i	$0.400427\pi$
$744$	−5134.24	−0.252998
$745$	0	0
$746$	10305.8	0.505796
$747$	10319.2	0.505434
$748$	24792.2	1.21189
$749$	2926.57	0.142770
$750$	0	0
$751$	14658.7	0.712253	0.356127	−	0.934438i	$-0.384097\pi$
$751$	14658.7	0.712253	0.356127	+	0.934438i	$0.384097\pi$
$752$	8076.10	0.391629
$753$	−2544.01	−0.123119
$754$	−3155.18	−0.152394
$755$	0	0
$756$	1603.03	0.0771185
$757$	−35441.4	−1.70164	−0.850819	−	0.525459i	$-0.823894\pi$
$757$	−35441.4	−1.70164	−0.850819	+	0.525459i	$0.823894\pi$
$758$	−24586.6	−1.17813
$759$	−2984.21	−0.142714
$760$	0	0
$761$	−17542.8	−0.835647	−0.417824	−	0.908528i	$-0.637207\pi$
$761$	−17542.8	−0.835647	−0.417824	+	0.908528i	$0.637207\pi$
$762$	−10251.3	−0.487354
$763$	−4493.30	−0.213196
$764$	10703.4	0.506852
$765$	0	0
$766$	10241.9	0.483098
$767$	18890.6	0.889311
$768$	−623.559	−0.0292979
$769$	−23071.0	−1.08187	−0.540937	−	0.841063i	$-0.681930\pi$
$769$	−23071.0	−1.08187	−0.540937	+	0.841063i	$0.681930\pi$
$770$	0	0
$771$	−15512.3	−0.724593
$772$	14651.2	0.683043
$773$	21356.1	0.993693	0.496846	−	0.867838i	$-0.334491\pi$
$773$	21356.1	0.993693	0.496846	+	0.867838i	$0.334491\pi$
$774$	9935.58	0.461405
$775$	0	0
$776$	9536.52	0.441161
$777$	1187.06	0.0548076
$778$	−11621.1	−0.535525
$779$	−18277.1	−0.840622
$780$	0	0
$781$	16110.6	0.738135
$782$	−5352.42	−0.244760
$783$	−6385.39	−0.291437
$784$	−5300.54	−0.241460
$785$	0	0
$786$	5658.74	0.256795
$787$	−23753.3	−1.07588	−0.537938	−	0.842984i	$-0.680797\pi$
$787$	−23753.3	−1.07588	−0.537938	+	0.842984i	$0.680797\pi$
$788$	−11851.4	−0.535772
$789$	−2783.52	−0.125597
$790$	0	0
$791$	−6177.07	−0.277663
$792$	−8977.52	−0.402781
$793$	14687.0	0.657694
$794$	−11189.2	−0.500114
$795$	0	0
$796$	21324.9	0.949547
$797$	−10129.9	−0.450213	−0.225106	−	0.974334i	$-0.572273\pi$
$797$	−10129.9	−0.450213	−0.225106	+	0.974334i	$0.572273\pi$
$798$	−963.739	−0.0427519
$799$	−58731.8	−2.60048
$800$	0	0
$801$	−10214.8	−0.450588
$802$	23665.1	1.04195
$803$	14140.3	0.621419
$804$	−7217.82	−0.316608
$805$	0	0
$806$	−15243.0	−0.666142
$807$	15014.5	0.654938
$808$	−13566.2	−0.590667
$809$	37955.8	1.64951	0.824757	−	0.565488i	$-0.191312\pi$
$809$	37955.8	1.64951	0.824757	+	0.565488i	$0.191312\pi$
$810$	0	0
$811$	37134.1	1.60784	0.803918	−	0.594740i	$-0.202745\pi$
$811$	37134.1	1.60784	0.803918	+	0.594740i	$0.202745\pi$
$812$	−746.723	−0.0322720
$813$	4679.96	0.201886
$814$	−15168.1	−0.653123
$815$	0	0
$816$	4534.71	0.194543
$817$	−13628.7	−0.583610
$818$	18704.7	0.799503
$819$	2085.89	0.0889948
$820$	0	0
$821$	−15633.1	−0.664553	−0.332277	−	0.943182i	$-0.607817\pi$
$821$	−15633.1	−0.664553	−0.332277	+	0.943182i	$0.607817\pi$
$822$	13420.5	0.569459
$823$	31827.6	1.34805	0.674023	−	0.738711i	$-0.264565\pi$
$823$	31827.6	1.34805	0.674023	+	0.738711i	$0.264565\pi$
$824$	−6151.16	−0.260056
$825$	0	0
$826$	4470.77	0.188327
$827$	1635.92	0.0687866	0.0343933	−	0.999408i	$-0.489050\pi$
$827$	1635.92	0.0687866	0.0343933	+	0.999408i	$0.489050\pi$
$828$	1938.16	0.0813476
$829$	12715.0	0.532703	0.266351	−	0.963876i	$-0.414182\pi$
$829$	12715.0	0.532703	0.266351	+	0.963876i	$0.414182\pi$
$830$	0	0
$831$	−13394.4	−0.559142
$832$	−1851.28	−0.0771412
$833$	38547.1	1.60334
$834$	−7207.98	−0.299271
$835$	0	0
$836$	12314.6	0.509459
$837$	−30848.4	−1.27393
$838$	22724.4	0.936756
$839$	−9126.14	−0.375529	−0.187765	−	0.982214i	$-0.560124\pi$
$839$	−9126.14	−0.375529	−0.187765	+	0.982214i	$0.560124\pi$
$840$	0	0
$841$	−21414.6	−0.878042
$842$	−14387.5	−0.588869
$843$	3199.93	0.130737
$844$	−1844.04	−0.0752068
$845$	0	0
$846$	21267.4	0.864288
$847$	5156.45	0.209183
$848$	6083.49	0.246354
$849$	−9749.18	−0.394100
$850$	0	0
$851$	3274.65	0.131908
$852$	2946.77	0.118491
$853$	26436.9	1.06117	0.530587	−	0.847631i	$-0.321972\pi$
$853$	26436.9	1.06117	0.530587	+	0.847631i	$0.321972\pi$
$854$	3475.91	0.139278
$855$	0	0
$856$	−6839.93	−0.273112
$857$	32204.7	1.28366	0.641828	−	0.766849i	$-0.278176\pi$
$857$	32204.7	1.28366	0.641828	+	0.766849i	$0.278176\pi$
$858$	7506.25	0.298670
$859$	32712.2	1.29933	0.649666	−	0.760220i	$-0.274909\pi$
$859$	32712.2	1.29933	0.649666	+	0.760220i	$0.274909\pi$
$860$	0	0
$861$	−2636.61	−0.104362
$862$	3911.67	0.154561
$863$	−16384.9	−0.646291	−0.323145	−	0.946349i	$-0.604740\pi$
$863$	−16384.9	−0.646291	−0.323145	+	0.946349i	$0.604740\pi$
$864$	−3746.58	−0.147524
$865$	0	0
$866$	11849.5	0.464969
$867$	−21010.8	−0.823028
$868$	−3607.49	−0.141067
$869$	62998.1	2.45922
$870$	0	0
$871$	−21428.9	−0.833629
$872$	10501.7	0.407834
$873$	25113.2	0.973601
$874$	−2658.60	−0.102893
$875$	0	0
$876$	2586.37	0.0997551
$877$	52.2430	0.00201154	0.00100577	−	0.999999i	$-0.499680\pi$
$877$	52.2430	0.00201154	0.00100577	+	0.999999i	$0.499680\pi$
$878$	−1670.44	−0.0642078
$879$	−16818.6	−0.645367
$880$	0	0
$881$	27812.9	1.06361	0.531806	−	0.846866i	$-0.321514\pi$
$881$	27812.9	1.06361	0.531806	+	0.846866i	$0.321514\pi$
$882$	−13958.3	−0.532880
$883$	−16718.1	−0.637157	−0.318578	−	0.947897i	$-0.603205\pi$
$883$	−16718.1	−0.637157	−0.318578	+	0.947897i	$0.603205\pi$
$884$	13463.0	0.512230
$885$	0	0
$886$	12406.2	0.470424
$887$	16171.1	0.612144	0.306072	−	0.952008i	$-0.400985\pi$
$887$	16171.1	0.612144	0.306072	+	0.952008i	$0.400985\pi$
$888$	−2774.38	−0.104845
$889$	−7202.88	−0.271740
$890$	0	0
$891$	−15108.1	−0.568060
$892$	−21501.2	−0.807078
$893$	−29172.7	−1.09320
$894$	315.986	0.0118212
$895$	0	0
$896$	−438.134	−0.0163360
$897$	−1620.53	−0.0603210
$898$	−28183.4	−1.04732
$899$	14369.8	0.533104
$900$	0	0
$901$	−44241.0	−1.63583
$902$	33690.4	1.24364
$903$	−1966.06	−0.0724543
$904$	14437.0	0.531157
$905$	0	0
$906$	15758.1	0.577846
$907$	−26499.5	−0.970124	−0.485062	−	0.874480i	$-0.661203\pi$
$907$	−26499.5	−0.970124	−0.485062	+	0.874480i	$0.661203\pi$
$908$	−9705.28	−0.354715
$909$	−35725.0	−1.30354
$910$	0	0
$911$	−26021.5	−0.946357	−0.473178	−	0.880967i	$-0.656893\pi$
$911$	−26021.5	−0.946357	−0.473178	+	0.880967i	$0.656893\pi$
$912$	2252.44	0.0817825
$913$	26092.0	0.945803
$914$	1208.77	0.0437447
$915$	0	0
$916$	11247.2	0.405697
$917$	3976.02	0.143184
$918$	27246.2	0.979586
$919$	−12637.1	−0.453599	−0.226800	−	0.973941i	$-0.572826\pi$
$919$	−12637.1	−0.453599	−0.226800	+	0.973941i	$0.572826\pi$
$920$	0	0
$921$	12487.2	0.446763
$922$	13063.1	0.466606
$923$	8748.63	0.311988
$924$	1776.47	0.0632486
$925$	0	0
$926$	−1713.62	−0.0608133
$927$	−16198.3	−0.573918
$928$	1745.23	0.0617349
$929$	−3755.28	−0.132623	−0.0663115	−	0.997799i	$-0.521123\pi$
$929$	−3755.28	−0.132623	−0.0663115	+	0.997799i	$0.521123\pi$
$930$	0	0
$931$	19146.7	0.674016
$932$	−7815.76	−0.274693
$933$	−11295.7	−0.396362
$934$	628.271	0.0220103
$935$	0	0
$936$	−4875.10	−0.170243
$937$	−5107.90	−0.178087	−0.0890437	−	0.996028i	$-0.528381\pi$
$937$	−5107.90	−0.178087	−0.0890437	+	0.996028i	$0.528381\pi$
$938$	−5071.49	−0.176535
$939$	−1118.69	−0.0388785
$940$	0	0
$941$	15383.2	0.532919	0.266460	−	0.963846i	$-0.414146\pi$
$941$	15383.2	0.532919	0.266460	+	0.963846i	$0.414146\pi$
$942$	−14748.7	−0.510127
$943$	−7273.43	−0.251172
$944$	−10449.0	−0.360261
$945$	0	0
$946$	25122.0	0.863412
$947$	−22066.7	−0.757203	−0.378601	−	0.925560i	$-0.623595\pi$
$947$	−22066.7	−0.757203	−0.378601	+	0.925560i	$0.623595\pi$
$948$	11522.9	0.394774
$949$	7678.65	0.262655
$950$	0	0
$951$	−8276.17	−0.282201
$952$	3186.24	0.108473
$953$	−6299.93	−0.214139	−0.107070	−	0.994252i	$-0.534147\pi$
$953$	−6299.93	−0.214139	−0.107070	+	0.994252i	$0.534147\pi$
$954$	16020.1	0.543679
$955$	0	0
$956$	−22975.2	−0.777272
$957$	−7076.27	−0.239021
$958$	−34118.1	−1.15063
$959$	9429.72	0.317520
$960$	0	0
$961$	39630.9	1.33030
$962$	−8236.81	−0.276055
$963$	−18012.1	−0.602732
$964$	10602.3	0.354230
$965$	0	0
$966$	−383.524	−0.0127740
$967$	40723.0	1.35425	0.677127	−	0.735866i	$-0.263225\pi$
$967$	40723.0	1.35425	0.677127	+	0.735866i	$0.263225\pi$
$968$	−12051.6	−0.400158
$969$	−16380.4	−0.543049
$970$	0	0
$971$	−15502.8	−0.512368	−0.256184	−	0.966628i	$-0.582465\pi$
$971$	−15502.8	−0.512368	−0.256184	+	0.966628i	$0.582465\pi$
$972$	−15408.1	−0.508452
$973$	−5064.57	−0.166868
$974$	16340.8	0.537571
$975$	0	0
$976$	−8123.85	−0.266433
$977$	52622.6	1.72318	0.861589	−	0.507606i	$-0.169469\pi$
$977$	52622.6	1.72318	0.861589	+	0.507606i	$0.169469\pi$
$978$	2011.09	0.0657542
$979$	−25828.0	−0.843172
$980$	0	0
$981$	27654.8	0.900052
$982$	25630.3	0.832887
$983$	52619.3	1.70732	0.853659	−	0.520832i	$-0.174378\pi$
$983$	52619.3	1.70732	0.853659	+	0.520832i	$0.174378\pi$
$984$	6162.25	0.199640
$985$	0	0
$986$	−12691.8	−0.409930
$987$	−4208.39	−0.135719
$988$	6687.23	0.215333
$989$	−5423.62	−0.174379
$990$	0	0
$991$	−396.488	−0.0127092	−0.00635461	−	0.999980i	$-0.502023\pi$
$991$	−396.488	−0.0127092	−0.00635461	+	0.999980i	$0.502023\pi$
$992$	8431.37	0.269855
$993$	10016.8	0.320116
$994$	2070.50	0.0660687
$995$	0	0
$996$	4772.44	0.151828
$997$	−40506.0	−1.28670	−0.643349	−	0.765573i	$-0.722456\pi$
$997$	−40506.0	−1.28670	−0.643349	+	0.765573i	$0.722456\pi$
$998$	25234.2	0.800375
$999$	−16669.5	−0.527927

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.s.1.2	✓	5
5.2	odd	4		1150.4.b.p.599.4		10
5.3	odd	4		1150.4.b.p.599.7		10
5.4	even	2		1150.4.a.t.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.s.1.2	✓	5	1.1	even	1	trivial
1150.4.a.t.1.4	yes	5	5.4	even	2
1150.4.b.p.599.4		10	5.2	odd	4
1150.4.b.p.599.7		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -2.43578 q^{3} +4.00000 q^{4} +4.87156 q^{6} +3.42292 q^{7} -8.00000 q^{8} -21.0670 q^{9} -53.2677 q^{11} -9.74311 q^{12} -28.9262 q^{13} -6.84584 q^{14} +16.0000 q^{16} -116.357 q^{17} +42.1340 q^{18} -57.7956 q^{19} -8.33748 q^{21} +106.535 q^{22} -23.0000 q^{23} +19.4862 q^{24} +57.8524 q^{26} +117.081 q^{27} +13.6917 q^{28} -54.5385 q^{29} -263.480 q^{31} -32.0000 q^{32} +129.748 q^{33} +232.714 q^{34} -84.2679 q^{36} -142.376 q^{37} +115.591 q^{38} +70.4578 q^{39} +316.236 q^{41} +16.6750 q^{42} +235.809 q^{43} -213.071 q^{44} +46.0000 q^{46} +504.756 q^{47} -38.9725 q^{48} -331.284 q^{49} +283.420 q^{51} -115.705 q^{52} +380.218 q^{53} -234.161 q^{54} -27.3834 q^{56} +140.777 q^{57} +109.077 q^{58} -653.064 q^{59} -507.741 q^{61} +526.961 q^{62} -72.1106 q^{63} +64.0000 q^{64} -259.497 q^{66} +740.813 q^{67} -465.427 q^{68} +56.0229 q^{69} -302.446 q^{71} +168.536 q^{72} -265.457 q^{73} +284.752 q^{74} -231.183 q^{76} -182.331 q^{77} -140.916 q^{78} -1182.67 q^{79} +283.626 q^{81} -632.473 q^{82} -489.827 q^{83} -33.3499 q^{84} -471.619 q^{86} +132.844 q^{87} +426.142 q^{88} +484.871 q^{89} -99.0121 q^{91} -92.0000 q^{92} +641.780 q^{93} -1009.51 q^{94} +77.9449 q^{96} -1192.07 q^{97} +662.567 q^{98} +1122.19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + 12 * q^3 + 20 * q^4 - 24 * q^6 - 24 * q^7 - 40 * q^8 + 11 * q^9 - 54 * q^11 + 48 * q^12 + 36 * q^13 + 48 * q^14 + 80 * q^16 + 132 * q^17 - 22 * q^18 - 50 * q^19 - 158 * q^21 + 108 * q^22 - 115 * q^23 - 96 * q^24 - 72 * q^26 + 489 * q^27 - 96 * q^28 - 104 * q^29 - 342 * q^31 - 160 * q^32 - 348 * q^33 - 264 * q^34 + 44 * q^36 + 380 * q^37 + 100 * q^38 + 193 * q^39 + 2 * q^41 + 316 * q^42 - 114 * q^43 - 216 * q^44 + 230 * q^46 + 966 * q^47 + 192 * q^48 - 441 * q^49 + 1200 * q^51 + 144 * q^52 + 308 * q^53 - 978 * q^54 + 192 * q^56 + 1410 * q^57 + 208 * q^58 - 277 * q^59 + 482 * q^61 + 684 * q^62 - 222 * q^63 + 320 * q^64 + 696 * q^66 + 2044 * q^67 + 528 * q^68 - 276 * q^69 + 208 * q^71 - 88 * q^72 + 646 * q^73 - 760 * q^74 - 200 * q^76 + 2124 * q^77 - 386 * q^78 - 3444 * q^79 + 2933 * q^81 - 4 * q^82 + 1218 * q^83 - 632 * q^84 + 228 * q^86 + 103 * q^87 + 432 * q^88 + 140 * q^89 + 1976 * q^91 - 460 * q^92 + 473 * q^93 - 1932 * q^94 - 384 * q^96 + 2816 * q^97 + 882 * q^98 - 1740 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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