LMFDB - Embedded newform 735.4.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,4,Mod(1,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, 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("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	735.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:	$43.3664038542$
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} - 2x^{4} - 30x^{3} + 22x^{2} + 153x - 84 $ x^5 - 2x^4 - 30x^3 + 22x^2 + 153x - 84
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.80647$ of defining polynomial
Character	$\chi$	$=$	735.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.80647 q^{2} +3.00000 q^{3} +15.1022 q^{4} +5.00000 q^{5} -14.4194 q^{6} -34.1365 q^{8} +9.00000 q^{9} -24.0324 q^{10} +53.2563 q^{11} +45.3066 q^{12} +60.9548 q^{13} +15.0000 q^{15} +43.2586 q^{16} +31.1198 q^{17} -43.2583 q^{18} -17.9133 q^{19} +75.5109 q^{20} -255.975 q^{22} -34.8223 q^{23} -102.409 q^{24} +25.0000 q^{25} -292.978 q^{26} +27.0000 q^{27} +141.423 q^{29} -72.0971 q^{30} +117.739 q^{31} +65.1706 q^{32} +159.769 q^{33} -149.576 q^{34} +135.920 q^{36} +175.319 q^{37} +86.0996 q^{38} +182.864 q^{39} -170.682 q^{40} +411.157 q^{41} -498.509 q^{43} +804.286 q^{44} +45.0000 q^{45} +167.373 q^{46} -290.262 q^{47} +129.776 q^{48} -120.162 q^{50} +93.3594 q^{51} +920.551 q^{52} +582.633 q^{53} -129.775 q^{54} +266.281 q^{55} -53.7398 q^{57} -679.745 q^{58} -657.710 q^{59} +226.533 q^{60} -417.167 q^{61} -565.911 q^{62} -659.310 q^{64} +304.774 q^{65} -767.925 q^{66} -567.911 q^{67} +469.977 q^{68} -104.467 q^{69} +887.933 q^{71} -307.228 q^{72} -1092.51 q^{73} -842.666 q^{74} +75.0000 q^{75} -270.529 q^{76} -878.933 q^{78} +135.425 q^{79} +216.293 q^{80} +81.0000 q^{81} -1976.22 q^{82} -464.523 q^{83} +155.599 q^{85} +2396.07 q^{86} +424.269 q^{87} -1817.98 q^{88} +31.8799 q^{89} -216.291 q^{90} -525.894 q^{92} +353.218 q^{93} +1395.14 q^{94} -89.5663 q^{95} +195.512 q^{96} -254.781 q^{97} +479.306 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 15 q^{3} + 25 q^{4} + 25 q^{5} + 9 q^{6} + 21 q^{8} + 45 q^{9} + 15 q^{10} + 43 q^{11} + 75 q^{12} + 123 q^{13} + 75 q^{15} + 161 q^{16} + 124 q^{17} + 27 q^{18} + 37 q^{19} + 125 q^{20}+ \cdots + 387 q^{99}+O(q^{100}) $ 5 * q + 3 * q^2 + 15 * q^3 + 25 * q^4 + 25 * q^5 + 9 * q^6 + 21 * q^8 + 45 * q^9 + 15 * q^10 + 43 * q^11 + 75 * q^12 + 123 * q^13 + 75 * q^15 + 161 * q^16 + 124 * q^17 + 27 * q^18 + 37 * q^19 + 125 * q^20 - 221 * q^22 + 77 * q^23 + 63 * q^24 + 125 * q^25 - 79 * q^26 + 135 * q^27 + 360 * q^29 + 45 * q^30 + 314 * q^31 - 59 * q^32 + 129 * q^33 + 176 * q^34 + 225 * q^36 + 225 * q^37 + 759 * q^38 + 369 * q^39 + 105 * q^40 + 341 * q^41 + 32 * q^43 + 679 * q^44 + 225 * q^45 - 331 * q^46 + 25 * q^47 + 483 * q^48 + 75 * q^50 + 372 * q^51 + 2299 * q^52 - 317 * q^53 + 81 * q^54 + 215 * q^55 + 111 * q^57 + 8 * q^58 + 676 * q^59 + 375 * q^60 - 188 * q^61 - 348 * q^62 - 1103 * q^64 + 615 * q^65 - 663 * q^66 - 1776 * q^67 + 1280 * q^68 + 231 * q^69 - 6 * q^71 + 189 * q^72 + 2006 * q^73 - 2729 * q^74 + 375 * q^75 + 1417 * q^76 - 237 * q^78 + 200 * q^79 + 805 * q^80 + 405 * q^81 - 539 * q^82 - 332 * q^83 + 620 * q^85 + 4262 * q^86 + 1080 * q^87 - 4529 * q^88 + 894 * q^89 + 135 * q^90 - 3687 * q^92 + 942 * q^93 + 4233 * q^94 + 185 * q^95 - 177 * q^96 - 576 * q^97 + 387 * 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 $

$2$	−4.80647	−1.69935	−0.849673	−	0.527311i	$-0.823200\pi$
$2$	−4.80647	−1.69935	−0.849673	+	0.527311i	$0.823200\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	15.1022	1.88777
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−14.4194	−0.981117
$7$	0	0
$7$	0	0
$8$	−34.1365	−1.50863
$9$	9.00000	0.333333
$10$	−24.0324	−0.759970
$11$	53.2563	1.45976	0.729881	−	0.683575i	$-0.239576\pi$
$11$	53.2563	1.45976	0.729881	+	0.683575i	$0.239576\pi$
$12$	45.3066	1.08991
$13$	60.9548	1.30045	0.650224	−	0.759743i	$-0.274675\pi$
$13$	60.9548	1.30045	0.650224	+	0.759743i	$0.274675\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	43.2586	0.675915
$17$	31.1198	0.443980	0.221990	−	0.975049i	$-0.428745\pi$
$17$	31.1198	0.443980	0.221990	+	0.975049i	$0.428745\pi$
$18$	−43.2583	−0.566448
$19$	−17.9133	−0.216294	−0.108147	−	0.994135i	$-0.534492\pi$
$19$	−17.9133	−0.216294	−0.108147	+	0.994135i	$0.534492\pi$
$20$	75.5109	0.844238
$21$	0	0
$22$	−255.975	−2.48064
$23$	−34.8223	−0.315694	−0.157847	−	0.987464i	$-0.550455\pi$
$23$	−34.8223	−0.315694	−0.157847	+	0.987464i	$0.550455\pi$
$24$	−102.409	−0.871010
$25$	25.0000	0.200000
$26$	−292.978	−2.20991
$27$	27.0000	0.192450
$28$	0	0
$29$	141.423	0.905571	0.452786	−	0.891619i	$-0.350430\pi$
$29$	141.423	0.905571	0.452786	+	0.891619i	$0.350430\pi$
$30$	−72.0971	−0.438769
$31$	117.739	0.682149	0.341075	−	0.940036i	$-0.389209\pi$
$31$	117.739	0.682149	0.341075	+	0.940036i	$0.389209\pi$
$32$	65.1706	0.360020
$33$	159.769	0.842793
$34$	−149.576	−0.754475
$35$	0	0
$36$	135.920	0.629258
$37$	175.319	0.778980	0.389490	−	0.921031i	$-0.372651\pi$
$37$	175.319	0.778980	0.389490	+	0.921031i	$0.372651\pi$
$38$	86.0996	0.367558
$39$	182.864	0.750814
$40$	−170.682	−0.674681
$41$	411.157	1.56615	0.783073	−	0.621930i	$-0.213651\pi$
$41$	411.157	1.56615	0.783073	+	0.621930i	$0.213651\pi$
$42$	0	0
$43$	−498.509	−1.76795	−0.883976	−	0.467532i	$-0.845143\pi$
$43$	−498.509	−1.76795	−0.883976	+	0.467532i	$0.845143\pi$
$44$	804.286	2.75570
$45$	45.0000	0.149071
$46$	167.373	0.536473
$47$	−290.262	−0.900832	−0.450416	−	0.892819i	$-0.648724\pi$
$47$	−290.262	−0.900832	−0.450416	+	0.892819i	$0.648724\pi$
$48$	129.776	0.390240
$49$	0	0
$50$	−120.162	−0.339869
$51$	93.3594	0.256332
$52$	920.551	2.45495
$53$	582.633	1.51001	0.755007	−	0.655717i	$-0.227633\pi$
$53$	582.633	1.51001	0.755007	+	0.655717i	$0.227633\pi$
$54$	−129.775	−0.327039
$55$	266.281	0.652825
$56$	0	0
$57$	−53.7398	−0.124877
$58$	−679.745	−1.53888
$59$	−657.710	−1.45130	−0.725649	−	0.688065i	$-0.758460\pi$
$59$	−657.710	−1.45130	−0.725649	+	0.688065i	$0.758460\pi$
$60$	226.533	0.487421
$61$	−417.167	−0.875618	−0.437809	−	0.899068i	$-0.644245\pi$
$61$	−417.167	−0.875618	−0.437809	+	0.899068i	$0.644245\pi$
$62$	−565.911	−1.15921
$63$	0	0
$64$	−659.310	−1.28771
$65$	304.774	0.581578
$66$	−767.925	−1.43220
$67$	−567.911	−1.03554	−0.517771	−	0.855519i	$-0.673238\pi$
$67$	−567.911	−1.03554	−0.517771	+	0.855519i	$0.673238\pi$
$68$	469.977	0.838134
$69$	−104.467	−0.182266
$70$	0	0
$71$	887.933	1.48420	0.742100	−	0.670289i	$-0.233830\pi$
$71$	887.933	1.48420	0.742100	+	0.670289i	$0.233830\pi$
$72$	−307.228	−0.502878
$73$	−1092.51	−1.75162	−0.875809	−	0.482657i	$-0.839672\pi$
$73$	−1092.51	−1.75162	−0.875809	+	0.482657i	$0.839672\pi$
$74$	−842.666	−1.32376
$75$	75.0000	0.115470
$76$	−270.529	−0.408314
$77$	0	0
$78$	−878.933	−1.27589
$79$	135.425	0.192867	0.0964337	−	0.995339i	$-0.469256\pi$
$79$	135.425	0.192867	0.0964337	+	0.995339i	$0.469256\pi$
$80$	216.293	0.302278
$81$	81.0000	0.111111
$82$	−1976.22	−2.66142
$83$	−464.523	−0.614313	−0.307157	−	0.951659i	$-0.599378\pi$
$83$	−464.523	−0.614313	−0.307157	+	0.951659i	$0.599378\pi$
$84$	0	0
$85$	155.599	0.198554
$86$	2396.07	3.00436
$87$	424.269	0.522832
$88$	−1817.98	−2.20224
$89$	31.8799	0.0379692	0.0189846	−	0.999820i	$-0.493957\pi$
$89$	31.8799	0.0379692	0.0189846	+	0.999820i	$0.493957\pi$
$90$	−216.291	−0.253323
$91$	0	0
$92$	−525.894	−0.595959
$93$	353.218	0.393839
$94$	1395.14	1.53082
$95$	−89.5663	−0.0967295
$96$	195.512	0.207858
$97$	−254.781	−0.266691	−0.133346	−	0.991070i	$-0.542572\pi$
$97$	−254.781	−0.266691	−0.133346	+	0.991070i	$0.542572\pi$
$98$	0	0
$99$	479.306	0.486587
$100$	377.555	0.377555
$101$	1359.53	1.33939	0.669694	−	0.742637i	$-0.266425\pi$
$101$	1359.53	1.33939	0.669694	+	0.742637i	$0.266425\pi$
$102$	−448.729	−0.435596
$103$	1448.06	1.38526	0.692629	−	0.721294i	$-0.256453\pi$
$103$	1448.06	1.38526	0.692629	+	0.721294i	$0.256453\pi$
$104$	−2080.78	−1.96190
$105$	0	0
$106$	−2800.41	−2.56603
$107$	−1666.45	−1.50563	−0.752814	−	0.658234i	$-0.771304\pi$
$107$	−1666.45	−1.50563	−0.752814	+	0.658234i	$0.771304\pi$
$108$	407.759	0.363302
$109$	−1118.40	−0.982780	−0.491390	−	0.870940i	$-0.663511\pi$
$109$	−1118.40	−0.982780	−0.491390	+	0.870940i	$0.663511\pi$
$110$	−1279.87	−1.10937
$111$	525.957	0.449745
$112$	0	0
$113$	806.732	0.671602	0.335801	−	0.941933i	$-0.390993\pi$
$113$	806.732	0.671602	0.335801	+	0.941933i	$0.390993\pi$
$114$	258.299	0.212210
$115$	−174.112	−0.141183
$116$	2135.80	1.70951
$117$	548.593	0.433483
$118$	3161.27	2.46625
$119$	0	0
$120$	−512.047	−0.389527
$121$	1505.23	1.13090
$122$	2005.10	1.48798
$123$	1233.47	0.904214
$124$	1778.12	1.28774
$125$	125.000	0.0894427
$126$	0	0
$127$	−1433.92	−1.00189	−0.500945	−	0.865479i	$-0.667014\pi$
$127$	−1433.92	−1.00189	−0.500945	+	0.865479i	$0.667014\pi$
$128$	2647.59	1.82825
$129$	−1495.53	−1.02073
$130$	−1464.89	−0.988301
$131$	650.933	0.434140	0.217070	−	0.976156i	$-0.430350\pi$
$131$	650.933	0.434140	0.217070	+	0.976156i	$0.430350\pi$
$132$	2412.86	1.59100
$133$	0	0
$134$	2729.65	1.75974
$135$	135.000	0.0860663
$136$	−1062.32	−0.669803
$137$	−617.782	−0.385260	−0.192630	−	0.981271i	$-0.561702\pi$
$137$	−617.782	−0.385260	−0.192630	+	0.981271i	$0.561702\pi$
$138$	502.118	0.309733
$139$	1525.08	0.930614	0.465307	−	0.885149i	$-0.345944\pi$
$139$	1525.08	0.930614	0.465307	+	0.885149i	$0.345944\pi$
$140$	0	0
$141$	−870.787	−0.520096
$142$	−4267.82	−2.52217
$143$	3246.23	1.89834
$144$	389.327	0.225305
$145$	707.114	0.404984
$146$	5251.10	2.97660
$147$	0	0
$148$	2647.70	1.47054
$149$	−1387.82	−0.763051	−0.381525	−	0.924358i	$-0.624601\pi$
$149$	−1387.82	−0.763051	−0.381525	+	0.924358i	$0.624601\pi$
$150$	−360.486	−0.196223
$151$	2284.34	1.23111	0.615554	−	0.788095i	$-0.288932\pi$
$151$	2284.34	1.23111	0.615554	+	0.788095i	$0.288932\pi$
$152$	611.495	0.326308
$153$	280.078	0.147993
$154$	0	0
$155$	588.697	0.305066
$156$	2761.65	1.41737
$157$	268.308	0.136391	0.0681953	−	0.997672i	$-0.478276\pi$
$157$	268.308	0.136391	0.0681953	+	0.997672i	$0.478276\pi$
$158$	−650.918	−0.327748
$159$	1747.90	0.871807
$160$	325.853	0.161006
$161$	0	0
$162$	−389.324	−0.188816
$163$	−2727.58	−1.31068	−0.655340	−	0.755334i	$-0.727475\pi$
$163$	−2727.58	−1.31068	−0.655340	+	0.755334i	$0.727475\pi$
$164$	6209.37	2.95653
$165$	798.844	0.376909
$166$	2232.72	1.04393
$167$	2234.41	1.03535	0.517675	−	0.855577i	$-0.326798\pi$
$167$	2234.41	1.03535	0.517675	+	0.855577i	$0.326798\pi$
$168$	0	0
$169$	1518.49	0.691164
$170$	−747.882	−0.337412
$171$	−161.219	−0.0720979
$172$	−7528.58	−3.33749
$173$	2339.66	1.02822	0.514108	−	0.857725i	$-0.328123\pi$
$173$	2339.66	1.02822	0.514108	+	0.857725i	$0.328123\pi$
$174$	−2039.24	−0.888472
$175$	0	0
$176$	2303.79	0.986675
$177$	−1973.13	−0.837907
$178$	−153.230	−0.0645228
$179$	2445.75	1.02125	0.510626	−	0.859803i	$-0.329414\pi$
$179$	2445.75	1.02125	0.510626	+	0.859803i	$0.329414\pi$
$180$	679.598	0.281413
$181$	1178.45	0.483944	0.241972	−	0.970283i	$-0.422206\pi$
$181$	1178.45	0.483944	0.241972	+	0.970283i	$0.422206\pi$
$182$	0	0
$183$	−1251.50	−0.505539
$184$	1188.71	0.476266
$185$	876.595	0.348371
$186$	−1697.73	−0.669268
$187$	1657.32	0.648105
$188$	−4383.60	−1.70057
$189$	0	0
$190$	430.498	0.164377
$191$	−2275.29	−0.861958	−0.430979	−	0.902362i	$-0.641832\pi$
$191$	−2275.29	−0.861958	−0.430979	+	0.902362i	$0.641832\pi$
$192$	−1977.93	−0.743462
$193$	4411.04	1.64515	0.822574	−	0.568658i	$-0.192537\pi$
$193$	4411.04	1.64515	0.822574	+	0.568658i	$0.192537\pi$
$194$	1224.60	0.453200
$195$	914.322	0.335774
$196$	0	0
$197$	−2011.41	−0.727447	−0.363723	−	0.931507i	$-0.618495\pi$
$197$	−2011.41	−0.727447	−0.363723	+	0.931507i	$0.618495\pi$
$198$	−2303.77	−0.826879
$199$	1111.24	0.395846	0.197923	−	0.980218i	$-0.436580\pi$
$199$	1111.24	0.395846	0.197923	+	0.980218i	$0.436580\pi$
$200$	−853.412	−0.301727
$201$	−1703.73	−0.597871
$202$	−6534.54	−2.27608
$203$	0	0
$204$	1409.93	0.483897
$205$	2055.79	0.700402
$206$	−6960.06	−2.35403
$207$	−313.401	−0.105231
$208$	2636.82	0.878992
$209$	−953.993	−0.315737
$210$	0	0
$211$	3031.04	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$211$	3031.04	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$212$	8799.03	2.85056
$213$	2663.80	0.856903
$214$	8009.77	2.55858
$215$	−2492.55	−0.790652
$216$	−921.685	−0.290337
$217$	0	0
$218$	5375.55	1.67008
$219$	−3277.52	−1.01130
$220$	4021.43	1.23239
$221$	1896.90	0.577373
$222$	−2528.00	−0.764271
$223$	−3789.93	−1.13808	−0.569042	−	0.822309i	$-0.692686\pi$
$223$	−3789.93	−1.13808	−0.569042	+	0.822309i	$0.692686\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	−3877.54	−1.14128
$227$	3876.31	1.13339	0.566696	−	0.823927i	$-0.308222\pi$
$227$	3876.31	1.13339	0.566696	+	0.823927i	$0.308222\pi$
$228$	−811.588	−0.235740
$229$	5926.44	1.71018	0.855088	−	0.518483i	$-0.173503\pi$
$229$	5926.44	1.71018	0.855088	+	0.518483i	$0.173503\pi$
$230$	836.863	0.239918
$231$	0	0
$232$	−4827.68	−1.36618
$233$	4643.03	1.30547	0.652737	−	0.757585i	$-0.273621\pi$
$233$	4643.03	1.30547	0.652737	+	0.757585i	$0.273621\pi$
$234$	−2636.80	−0.736636
$235$	−1451.31	−0.402864
$236$	−9932.86	−2.73972
$237$	406.276	0.111352
$238$	0	0
$239$	944.025	0.255497	0.127749	−	0.991807i	$-0.459225\pi$
$239$	944.025	0.255497	0.127749	+	0.991807i	$0.459225\pi$
$240$	648.879	0.174521
$241$	1515.01	0.404939	0.202470	−	0.979289i	$-0.435103\pi$
$241$	1515.01	0.404939	0.202470	+	0.979289i	$0.435103\pi$
$242$	−7234.85	−1.92179
$243$	243.000	0.0641500
$244$	−6300.13	−1.65297
$245$	0	0
$246$	−5928.65	−1.53657
$247$	−1091.90	−0.281279
$248$	−4019.21	−1.02911
$249$	−1393.57	−0.354674
$250$	−600.809	−0.151994
$251$	3219.82	0.809695	0.404847	−	0.914384i	$-0.367325\pi$
$251$	3219.82	0.809695	0.404847	+	0.914384i	$0.367325\pi$
$252$	0	0
$253$	−1854.51	−0.460838
$254$	6892.11	1.70256
$255$	466.797	0.114635
$256$	−7451.09	−1.81911
$257$	2214.55	0.537509	0.268754	−	0.963209i	$-0.413388\pi$
$257$	2214.55	0.537509	0.268754	+	0.963209i	$0.413388\pi$
$258$	7188.21	1.73457
$259$	0	0
$260$	4602.75	1.09789
$261$	1272.81	0.301857
$262$	−3128.69	−0.737753
$263$	−5457.02	−1.27945	−0.639723	−	0.768605i	$-0.720951\pi$
$263$	−5457.02	−1.27945	−0.639723	+	0.768605i	$0.720951\pi$
$264$	−5453.94	−1.27147
$265$	2913.16	0.675299
$266$	0	0
$267$	95.6396	0.0219215
$268$	−8576.69	−1.95487
$269$	−5820.76	−1.31932	−0.659661	−	0.751563i	$-0.729300\pi$
$269$	−5820.76	−1.31932	−0.659661	+	0.751563i	$0.729300\pi$
$270$	−648.874	−0.146256
$271$	3300.74	0.739874	0.369937	−	0.929057i	$-0.379379\pi$
$271$	3300.74	0.739874	0.369937	+	0.929057i	$0.379379\pi$
$272$	1346.20	0.300093
$273$	0	0
$274$	2969.35	0.654690
$275$	1331.41	0.291952
$276$	−1577.68	−0.344077
$277$	−3871.25	−0.839714	−0.419857	−	0.907590i	$-0.637920\pi$
$277$	−3871.25	−0.839714	−0.419857	+	0.907590i	$0.637920\pi$
$278$	−7330.24	−1.58143
$279$	1059.65	0.227383
$280$	0	0
$281$	−1411.40	−0.299634	−0.149817	−	0.988714i	$-0.547869\pi$
$281$	−1411.40	−0.299634	−0.149817	+	0.988714i	$0.547869\pi$
$282$	4185.41	0.883822
$283$	−4752.51	−0.998259	−0.499129	−	0.866528i	$-0.666347\pi$
$283$	−4752.51	−0.998259	−0.499129	+	0.866528i	$0.666347\pi$
$284$	13409.7	2.80183
$285$	−268.699	−0.0558468
$286$	−15602.9	−3.22594
$287$	0	0
$288$	586.536	0.120007
$289$	−3944.56	−0.802882
$290$	−3398.73	−0.688207
$291$	−764.342	−0.153974
$292$	−16499.2	−3.30666
$293$	−1301.32	−0.259468	−0.129734	−	0.991549i	$-0.541412\pi$
$293$	−1301.32	−0.259468	−0.129734	+	0.991549i	$0.541412\pi$
$294$	0	0
$295$	−3288.55	−0.649040
$296$	−5984.77	−1.17520
$297$	1437.92	0.280931
$298$	6670.51	1.29669
$299$	−2122.59	−0.410543
$300$	1132.66	0.217981
$301$	0	0
$302$	−10979.6	−2.09208
$303$	4078.59	0.773296
$304$	−774.902	−0.146196
$305$	−2085.83	−0.391588
$306$	−1346.19	−0.251492
$307$	6138.12	1.14111	0.570555	−	0.821259i	$-0.306728\pi$
$307$	6138.12	1.14111	0.570555	+	0.821259i	$0.306728\pi$
$308$	0	0
$309$	4344.18	0.799779
$310$	−2829.56	−0.518413
$311$	4082.04	0.744281	0.372140	−	0.928176i	$-0.378624\pi$
$311$	4082.04	0.744281	0.372140	+	0.928176i	$0.378624\pi$
$312$	−6242.35	−1.13270
$313$	7143.19	1.28996	0.644979	−	0.764200i	$-0.276866\pi$
$313$	7143.19	1.28996	0.644979	+	0.764200i	$0.276866\pi$
$314$	−1289.62	−0.231775
$315$	0	0
$316$	2045.22	0.364090
$317$	−7481.62	−1.32558	−0.662791	−	0.748804i	$-0.730628\pi$
$317$	−7481.62	−1.32558	−0.662791	+	0.748804i	$0.730628\pi$
$318$	−8401.22	−1.48150
$319$	7531.66	1.32192
$320$	−3296.55	−0.575883
$321$	−4999.36	−0.869275
$322$	0	0
$323$	−557.457	−0.0960301
$324$	1223.28	0.209753
$325$	1523.87	0.260090
$326$	13110.1	2.22730
$327$	−3355.19	−0.567408
$328$	−14035.5	−2.36274
$329$	0	0
$330$	−3839.62	−0.640498
$331$	−1944.12	−0.322835	−0.161417	−	0.986886i	$-0.551607\pi$
$331$	−1944.12	−0.322835	−0.161417	+	0.986886i	$0.551607\pi$
$332$	−7015.31	−1.15968
$333$	1577.87	0.259660
$334$	−10739.6	−1.75942
$335$	−2839.55	−0.463109
$336$	0	0
$337$	−1920.36	−0.310411	−0.155205	−	0.987882i	$-0.549604\pi$
$337$	−1920.36	−0.310411	−0.155205	+	0.987882i	$0.549604\pi$
$338$	−7298.57	−1.17453
$339$	2420.20	0.387749
$340$	2349.89	0.374825
$341$	6270.36	0.995775
$342$	774.896	0.122519
$343$	0	0
$344$	17017.3	2.66719
$345$	−522.335	−0.0815118
$346$	−11245.5	−1.74729
$347$	2093.45	0.323868	0.161934	−	0.986802i	$-0.448227\pi$
$347$	2093.45	0.323868	0.161934	+	0.986802i	$0.448227\pi$
$348$	6407.39	0.986988
$349$	2199.00	0.337277	0.168638	−	0.985678i	$-0.446063\pi$
$349$	2199.00	0.337277	0.168638	+	0.985678i	$0.446063\pi$
$350$	0	0
$351$	1645.78	0.250271
$352$	3470.75	0.525544
$353$	−5861.17	−0.883735	−0.441867	−	0.897080i	$-0.645684\pi$
$353$	−5861.17	−0.883735	−0.441867	+	0.897080i	$0.645684\pi$
$354$	9483.80	1.42389
$355$	4439.66	0.663755
$356$	481.456	0.0716772
$357$	0	0
$358$	−11755.4	−1.73546
$359$	7271.44	1.06900	0.534502	−	0.845167i	$-0.320499\pi$
$359$	7271.44	1.06900	0.534502	+	0.845167i	$0.320499\pi$
$360$	−1536.14	−0.224894
$361$	−6538.12	−0.953217
$362$	−5664.21	−0.822387
$363$	4515.69	0.652927
$364$	0	0
$365$	−5462.53	−0.783348
$366$	6015.30	0.859084
$367$	−12638.0	−1.79754	−0.898768	−	0.438425i	$-0.855537\pi$
$367$	−12638.0	−1.79754	−0.898768	+	0.438425i	$0.855537\pi$
$368$	−1506.36	−0.213382
$369$	3700.41	0.522048
$370$	−4213.33	−0.592002
$371$	0	0
$372$	5334.37	0.743479
$373$	−6212.81	−0.862431	−0.431216	−	0.902249i	$-0.641915\pi$
$373$	−6212.81	−0.862431	−0.431216	+	0.902249i	$0.641915\pi$
$374$	−7965.89	−1.10135
$375$	375.000	0.0516398
$376$	9908.53	1.35903
$377$	8620.40	1.17765
$378$	0	0
$379$	5075.53	0.687896	0.343948	−	0.938989i	$-0.388236\pi$
$379$	5075.53	0.687896	0.343948	+	0.938989i	$0.388236\pi$
$380$	−1352.65	−0.182603
$381$	−4301.76	−0.578441
$382$	10936.1	1.46476
$383$	−8413.71	−1.12251	−0.561254	−	0.827644i	$-0.689681\pi$
$383$	−8413.71	−1.12251	−0.561254	+	0.827644i	$0.689681\pi$
$384$	7942.77	1.05554
$385$	0	0
$386$	−21201.5	−2.79567
$387$	−4486.58	−0.589317
$388$	−3847.74	−0.503453
$389$	3708.69	0.483389	0.241694	−	0.970352i	$-0.422297\pi$
$389$	3708.69	0.483389	0.241694	+	0.970352i	$0.422297\pi$
$390$	−4394.66	−0.570596
$391$	−1083.66	−0.140162
$392$	0	0
$393$	1952.80	0.250651
$394$	9667.79	1.23618
$395$	677.126	0.0862529
$396$	7238.58	0.918566
$397$	−4520.05	−0.571422	−0.285711	−	0.958316i	$-0.592230\pi$
$397$	−4520.05	−0.571422	−0.285711	+	0.958316i	$0.592230\pi$
$398$	−5341.12	−0.672679
$399$	0	0
$400$	1081.46	0.135183
$401$	−12591.8	−1.56809	−0.784043	−	0.620707i	$-0.786846\pi$
$401$	−12591.8	−1.56809	−0.784043	+	0.620707i	$0.786846\pi$
$402$	8188.94	1.01599
$403$	7176.78	0.887099
$404$	20531.9	2.52846
$405$	405.000	0.0496904
$406$	0	0
$407$	9336.84	1.13713
$408$	−3186.96	−0.386711
$409$	9017.62	1.09020	0.545101	−	0.838370i	$-0.316491\pi$
$409$	9017.62	1.09020	0.545101	+	0.838370i	$0.316491\pi$
$410$	−9881.08	−1.19022
$411$	−1853.35	−0.222430
$412$	21868.9	2.61505
$413$	0	0
$414$	1506.35	0.178824
$415$	−2322.61	−0.274729
$416$	3972.46	0.468188
$417$	4575.23	0.537290
$418$	4585.34	0.536547
$419$	3.80670	0.000443842 0	0.000221921	−	1.00000i	$-0.499929\pi$
$419$	3.80670	0.000443842 0	0.000221921		1.00000i	$0.499929\pi$
$420$	0	0
$421$	12076.0	1.39798	0.698989	−	0.715132i	$-0.253633\pi$
$421$	12076.0	1.39798	0.698989	+	0.715132i	$0.253633\pi$
$422$	−14568.6	−1.68054
$423$	−2612.36	−0.300277
$424$	−19889.0	−2.27806
$425$	777.995	0.0887960
$426$	−12803.5	−1.45617
$427$	0	0
$428$	−25167.1	−2.84228
$429$	9738.68	1.09601
$430$	11980.4	1.34359
$431$	−10317.7	−1.15310	−0.576551	−	0.817061i	$-0.695602\pi$
$431$	−10317.7	−1.15310	−0.576551	+	0.817061i	$0.695602\pi$
$432$	1167.98	0.130080
$433$	12728.4	1.41267	0.706334	−	0.707878i	$-0.250347\pi$
$433$	12728.4	1.41267	0.706334	+	0.707878i	$0.250347\pi$
$434$	0	0
$435$	2121.34	0.233818
$436$	−16890.2	−1.85527
$437$	623.781	0.0682826
$438$	15753.3	1.71854
$439$	−9972.58	−1.08420	−0.542102	−	0.840313i	$-0.682371\pi$
$439$	−9972.58	−1.08420	−0.542102	+	0.840313i	$0.682371\pi$
$440$	−9089.91	−0.984874
$441$	0	0
$442$	−9117.40	−0.981155
$443$	10462.8	1.12213	0.561063	−	0.827773i	$-0.310393\pi$
$443$	10462.8	1.12213	0.561063	+	0.827773i	$0.310393\pi$
$444$	7943.10	0.849016
$445$	159.399	0.0169803
$446$	18216.2	1.93400
$447$	−4163.46	−0.440547
$448$	0	0
$449$	−11664.7	−1.22604	−0.613019	−	0.790068i	$-0.710045\pi$
$449$	−11664.7	−1.22604	−0.613019	+	0.790068i	$0.710045\pi$
$450$	−1081.46	−0.113290
$451$	21896.7	2.28620
$452$	12183.4	1.26783
$453$	6853.03	0.710780
$454$	−18631.4	−1.92602
$455$	0	0
$456$	1834.49	0.188394
$457$	4542.53	0.464968	0.232484	−	0.972600i	$-0.425315\pi$
$457$	4542.53	0.464968	0.232484	+	0.972600i	$0.425315\pi$
$458$	−28485.3	−2.90618
$459$	840.235	0.0854440
$460$	−2629.47	−0.266521
$461$	7887.54	0.796875	0.398437	−	0.917195i	$-0.369553\pi$
$461$	7887.54	0.796875	0.398437	+	0.917195i	$0.369553\pi$
$462$	0	0
$463$	−5444.15	−0.546460	−0.273230	−	0.961949i	$-0.588092\pi$
$463$	−5444.15	−0.546460	−0.273230	+	0.961949i	$0.588092\pi$
$464$	6117.75	0.612089
$465$	1766.09	0.176130
$466$	−22316.6	−2.21845
$467$	−10578.0	−1.04816	−0.524082	−	0.851668i	$-0.675591\pi$
$467$	−10578.0	−1.04816	−0.524082	+	0.851668i	$0.675591\pi$
$468$	8284.96	0.818317
$469$	0	0
$470$	6975.69	0.684606
$471$	804.925	0.0787452
$472$	22451.9	2.18948
$473$	−26548.7	−2.58079
$474$	−1952.75	−0.189226
$475$	−447.831	−0.0432588
$476$	0	0
$477$	5243.69	0.503338
$478$	−4537.43	−0.434178
$479$	11644.7	1.11077	0.555386	−	0.831592i	$-0.312570\pi$
$479$	11644.7	1.11077	0.555386	+	0.831592i	$0.312570\pi$
$480$	977.560	0.0929568
$481$	10686.5	1.01302
$482$	−7281.85	−0.688131
$483$	0	0
$484$	22732.3	2.13489
$485$	−1273.90	−0.119268
$486$	−1167.97	−0.109013
$487$	−9237.23	−0.859504	−0.429752	−	0.902947i	$-0.641399\pi$
$487$	−9237.23	−0.859504	−0.429752	+	0.902947i	$0.641399\pi$
$488$	14240.6	1.32099
$489$	−8182.75	−0.756722
$490$	0	0
$491$	−2704.47	−0.248576	−0.124288	−	0.992246i	$-0.539665\pi$
$491$	−2704.47	−0.248576	−0.124288	+	0.992246i	$0.539665\pi$
$492$	18628.1	1.70695
$493$	4401.05	0.402056
$494$	5248.18	0.477990
$495$	2396.53	0.217608
$496$	5093.24	0.461075
$497$	0	0
$498$	6698.15	0.602713
$499$	−763.351	−0.0684815	−0.0342408	−	0.999414i	$-0.510901\pi$
$499$	−763.351	−0.0684815	−0.0342408	+	0.999414i	$0.510901\pi$
$500$	1887.77	0.168848
$501$	6703.22	0.597760
$502$	−15476.0	−1.37595
$503$	5902.36	0.523207	0.261604	−	0.965175i	$-0.415749\pi$
$503$	5902.36	0.523207	0.261604	+	0.965175i	$0.415749\pi$
$504$	0	0
$505$	6797.64	0.598993
$506$	8913.64	0.783122
$507$	4555.46	0.399044
$508$	−21655.4	−1.89134
$509$	2200.30	0.191604	0.0958020	−	0.995400i	$-0.469458\pi$
$509$	2200.30	0.191604	0.0958020	+	0.995400i	$0.469458\pi$
$510$	−2243.65	−0.194805
$511$	0	0
$512$	14632.8	1.26305
$513$	−483.658	−0.0416258
$514$	−10644.2	−0.913413
$515$	7240.29	0.619506
$516$	−22585.7	−1.92690
$517$	−15458.3	−1.31500
$518$	0	0
$519$	7018.99	0.593641
$520$	−10403.9	−0.877388
$521$	−1213.69	−0.102059	−0.0510296	−	0.998697i	$-0.516250\pi$
$521$	−1213.69	−0.102059	−0.0510296	+	0.998697i	$0.516250\pi$
$522$	−6117.71	−0.512959
$523$	−8864.18	−0.741116	−0.370558	−	0.928809i	$-0.620833\pi$
$523$	−8864.18	−0.741116	−0.370558	+	0.928809i	$0.620833\pi$
$524$	9830.52	0.819558
$525$	0	0
$526$	26229.0	2.17422
$527$	3664.03	0.302861
$528$	6911.37	0.569657
$529$	−10954.4	−0.900337
$530$	−14002.0	−1.14757
$531$	−5919.39	−0.483766
$532$	0	0
$533$	25062.0	2.03669
$534$	−459.689	−0.0372522
$535$	−8332.27	−0.673337
$536$	19386.5	1.56225
$537$	7337.25	0.589620
$538$	27977.3	2.24198
$539$	0	0
$540$	2038.80	0.162474
$541$	−11585.7	−0.920714	−0.460357	−	0.887734i	$-0.652279\pi$
$541$	−11585.7	−0.920714	−0.460357	+	0.887734i	$0.652279\pi$
$542$	−15864.9	−1.25730
$543$	3535.36	0.279405
$544$	2028.10	0.159842
$545$	−5591.98	−0.439513
$546$	0	0
$547$	4233.45	0.330913	0.165456	−	0.986217i	$-0.447090\pi$
$547$	4233.45	0.330913	0.165456	+	0.986217i	$0.447090\pi$
$548$	−9329.86	−0.727284
$549$	−3754.50	−0.291873
$550$	−6399.37	−0.496128
$551$	−2533.34	−0.195869
$552$	3566.14	0.274973
$553$	0	0
$554$	18607.1	1.42696
$555$	2629.79	0.201132
$556$	23032.0	1.75679
$557$	2198.23	0.167221	0.0836106	−	0.996499i	$-0.473355\pi$
$557$	2198.23	0.167221	0.0836106	+	0.996499i	$0.473355\pi$
$558$	−5093.20	−0.386402
$559$	−30386.5	−2.29913
$560$	0	0
$561$	4971.97	0.374183
$562$	6783.87	0.509182
$563$	−5918.67	−0.443059	−0.221529	−	0.975154i	$-0.571105\pi$
$563$	−5918.67	−0.443059	−0.221529	+	0.975154i	$0.571105\pi$
$564$	−13150.8	−0.981823
$565$	4033.66	0.300349
$566$	22842.8	1.69639
$567$	0	0
$568$	−30310.9	−2.23911
$569$	−2560.97	−0.188684	−0.0943422	−	0.995540i	$-0.530075\pi$
$569$	−2560.97	−0.188684	−0.0943422	+	0.995540i	$0.530075\pi$
$570$	1291.49	0.0949030
$571$	−24683.8	−1.80908	−0.904540	−	0.426389i	$-0.859786\pi$
$571$	−24683.8	−1.80908	−0.904540	+	0.426389i	$0.859786\pi$
$572$	49025.1	3.58364
$573$	−6825.86	−0.497652
$574$	0	0
$575$	−870.559	−0.0631388
$576$	−5933.79	−0.429238
$577$	10144.6	0.731932	0.365966	−	0.930628i	$-0.380739\pi$
$577$	10144.6	0.731932	0.365966	+	0.930628i	$0.380739\pi$
$578$	18959.4	1.36437
$579$	13233.1	0.949826
$580$	10679.0	0.764518
$581$	0	0
$582$	3673.79	0.261655
$583$	31028.8	2.20426
$584$	37294.3	2.64255
$585$	2742.97	0.193859
$586$	6254.78	0.440926
$587$	12204.6	0.858158	0.429079	−	0.903267i	$-0.358838\pi$
$587$	12204.6	0.858158	0.429079	+	0.903267i	$0.358838\pi$
$588$	0	0
$589$	−2109.10	−0.147545
$590$	15806.3	1.10294
$591$	−6034.23	−0.419992
$592$	7584.05	0.526525
$593$	10372.0	0.718255	0.359128	−	0.933288i	$-0.383074\pi$
$593$	10372.0	0.718255	0.359128	+	0.933288i	$0.383074\pi$
$594$	−6911.32	−0.477399
$595$	0	0
$596$	−20959.1	−1.44047
$597$	3333.71	0.228542
$598$	10202.2	0.697655
$599$	−20908.1	−1.42618	−0.713091	−	0.701072i	$-0.752705\pi$
$599$	−20908.1	−1.42618	−0.713091	+	0.701072i	$0.752705\pi$
$600$	−2560.24	−0.174202
$601$	−758.785	−0.0515000	−0.0257500	−	0.999668i	$-0.508197\pi$
$601$	−758.785	−0.0515000	−0.0257500	+	0.999668i	$0.508197\pi$
$602$	0	0
$603$	−5111.20	−0.345181
$604$	34498.6	2.32405
$605$	7526.15	0.505755
$606$	−19603.6	−1.31410
$607$	18780.8	1.25583	0.627917	−	0.778281i	$-0.283908\pi$
$607$	18780.8	1.25583	0.627917	+	0.778281i	$0.283908\pi$
$608$	−1167.42	−0.0778702
$609$	0	0
$610$	10025.5	0.665444
$611$	−17692.9	−1.17149
$612$	4229.79	0.279378
$613$	27346.0	1.80179	0.900893	−	0.434040i	$-0.142912\pi$
$613$	27346.0	1.80179	0.900893	+	0.434040i	$0.142912\pi$
$614$	−29502.7	−1.93914
$615$	6167.36	0.404377
$616$	0	0
$617$	−16384.9	−1.06910	−0.534548	−	0.845138i	$-0.679518\pi$
$617$	−16384.9	−1.06910	−0.534548	+	0.845138i	$0.679518\pi$
$618$	−20880.2	−1.35910
$619$	−9536.46	−0.619229	−0.309614	−	0.950862i	$-0.600200\pi$
$619$	−9536.46	−0.619229	−0.309614	+	0.950862i	$0.600200\pi$
$620$	8890.62	0.575896
$621$	−940.203	−0.0607553
$622$	−19620.2	−1.26479
$623$	0	0
$624$	7910.45	0.507486
$625$	625.000	0.0400000
$626$	−34333.6	−2.19209
$627$	−2861.98	−0.182291
$628$	4052.04	0.257475
$629$	5455.89	0.345852
$630$	0	0
$631$	−20656.2	−1.30318	−0.651592	−	0.758570i	$-0.725899\pi$
$631$	−20656.2	−1.30318	−0.651592	+	0.758570i	$0.725899\pi$
$632$	−4622.94	−0.290966
$633$	9093.12	0.570962
$634$	35960.2	2.25262
$635$	−7169.61	−0.448059
$636$	26397.1	1.64577
$637$	0	0
$638$	−36200.7	−2.24639
$639$	7991.39	0.494733
$640$	13237.9	0.817618
$641$	−17904.5	−1.10326	−0.551628	−	0.834091i	$-0.685993\pi$
$641$	−17904.5	−1.10326	−0.551628	+	0.834091i	$0.685993\pi$
$642$	24029.3	1.47720
$643$	−4497.56	−0.275842	−0.137921	−	0.990443i	$-0.544042\pi$
$643$	−4497.56	−0.275842	−0.137921	+	0.990443i	$0.544042\pi$
$644$	0	0
$645$	−7477.64	−0.456483
$646$	2679.40	0.163188
$647$	261.315	0.0158784	0.00793922	−	0.999968i	$-0.497473\pi$
$647$	261.315	0.0158784	0.00793922	+	0.999968i	$0.497473\pi$
$648$	−2765.05	−0.167626
$649$	−35027.2	−2.11855
$650$	−7324.44	−0.441982
$651$	0	0
$652$	−41192.5	−2.47427
$653$	3791.14	0.227196	0.113598	−	0.993527i	$-0.463762\pi$
$653$	3791.14	0.227196	0.113598	+	0.993527i	$0.463762\pi$
$654$	16126.6	0.964222
$655$	3254.67	0.194153
$656$	17786.1	1.05858
$657$	−9832.55	−0.583873
$658$	0	0
$659$	1149.18	0.0679298	0.0339649	−	0.999423i	$-0.489187\pi$
$659$	1149.18	0.0679298	0.0339649	+	0.999423i	$0.489187\pi$
$660$	12064.3	0.711518
$661$	1943.10	0.114338	0.0571692	−	0.998365i	$-0.481793\pi$
$661$	1943.10	0.114338	0.0571692	+	0.998365i	$0.481793\pi$
$662$	9344.35	0.548608
$663$	5690.70	0.333346
$664$	15857.2	0.926773
$665$	0	0
$666$	−7584.00	−0.441252
$667$	−4924.68	−0.285883
$668$	33744.4	1.95451
$669$	−11369.8	−0.657073
$670$	13648.2	0.786981
$671$	−22216.7	−1.27819
$672$	0	0
$673$	31546.2	1.80686	0.903431	−	0.428734i	$-0.141040\pi$
$673$	31546.2	1.80686	0.903431	+	0.428734i	$0.141040\pi$
$674$	9230.14	0.527495
$675$	675.000	0.0384900
$676$	22932.5	1.30476
$677$	−3409.55	−0.193560	−0.0967798	−	0.995306i	$-0.530854\pi$
$677$	−3409.55	−0.193560	−0.0967798	+	0.995306i	$0.530854\pi$
$678$	−11632.6	−0.658920
$679$	0	0
$680$	−5311.60	−0.299545
$681$	11628.9	0.654364
$682$	−30138.3	−1.69217
$683$	2632.86	0.147502	0.0737508	−	0.997277i	$-0.476503\pi$
$683$	2632.86	0.147502	0.0737508	+	0.997277i	$0.476503\pi$
$684$	−2434.76	−0.136105
$685$	−3088.91	−0.172294
$686$	0	0
$687$	17779.3	0.987370
$688$	−21564.8	−1.19499
$689$	35514.2	1.96369
$690$	2510.59	0.138517
$691$	25251.1	1.39015	0.695076	−	0.718936i	$-0.255371\pi$
$691$	25251.1	1.39015	0.695076	+	0.718936i	$0.255371\pi$
$692$	35334.1	1.94104
$693$	0	0
$694$	−10062.1	−0.550363
$695$	7625.39	0.416183
$696$	−14483.0	−0.788762
$697$	12795.1	0.695337
$698$	−10569.4	−0.573149
$699$	13929.1	0.753715
$700$	0	0
$701$	−18046.3	−0.972325	−0.486162	−	0.873869i	$-0.661604\pi$
$701$	−18046.3	−0.972325	−0.486162	+	0.873869i	$0.661604\pi$
$702$	−7910.40	−0.425297
$703$	−3140.53	−0.168489
$704$	−35112.4	−1.87975
$705$	−4353.94	−0.232594
$706$	28171.5	1.50177
$707$	0	0
$708$	−29798.6	−1.58178
$709$	−21736.6	−1.15139	−0.575693	−	0.817666i	$-0.695268\pi$
$709$	−21736.6	−1.15139	−0.575693	+	0.817666i	$0.695268\pi$
$710$	−21339.1	−1.12795
$711$	1218.83	0.0642891
$712$	−1088.27	−0.0572816
$713$	−4099.96	−0.215350
$714$	0	0
$715$	16231.1	0.848965
$716$	36936.2	1.92789
$717$	2832.07	0.147511
$718$	−34950.0	−1.81661
$719$	10921.1	0.566467	0.283233	−	0.959051i	$-0.408593\pi$
$719$	10921.1	0.566467	0.283233	+	0.959051i	$0.408593\pi$
$720$	1946.64	0.100759
$721$	0	0
$722$	31425.3	1.61984
$723$	4545.03	0.233792
$724$	17797.2	0.913576
$725$	3535.57	0.181114
$726$	−21704.6	−1.10955
$727$	−15670.3	−0.799420	−0.399710	−	0.916642i	$-0.630889\pi$
$727$	−15670.3	−0.799420	−0.399710	+	0.916642i	$0.630889\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	26255.5	1.33118
$731$	−15513.5	−0.784935
$732$	−18900.4	−0.954342
$733$	−8032.15	−0.404740	−0.202370	−	0.979309i	$-0.564864\pi$
$733$	−8032.15	−0.404740	−0.202370	+	0.979309i	$0.564864\pi$
$734$	60744.0	3.05463
$735$	0	0
$736$	−2269.39	−0.113656
$737$	−30244.8	−1.51164
$738$	−17785.9	−0.887140
$739$	9602.50	0.477989	0.238994	−	0.971021i	$-0.423182\pi$
$739$	9602.50	0.477989	0.238994	+	0.971021i	$0.423182\pi$
$740$	13238.5	0.657645
$741$	−3275.70	−0.162396
$742$	0	0
$743$	−17641.7	−0.871078	−0.435539	−	0.900170i	$-0.643442\pi$
$743$	−17641.7	−0.871078	−0.435539	+	0.900170i	$0.643442\pi$
$744$	−12057.6	−0.594159
$745$	−6939.09	−0.341247
$746$	29861.7	1.46557
$747$	−4180.70	−0.204771
$748$	25029.2	1.22347
$749$	0	0
$750$	−1802.43	−0.0877538
$751$	5688.94	0.276421	0.138211	−	0.990403i	$-0.455865\pi$
$751$	5688.94	0.276421	0.138211	+	0.990403i	$0.455865\pi$
$752$	−12556.3	−0.608886
$753$	9659.47	0.467478
$754$	−41433.7	−2.00123
$755$	11421.7	0.550568
$756$	0	0
$757$	−18939.2	−0.909324	−0.454662	−	0.890664i	$-0.650240\pi$
$757$	−18939.2	−0.909324	−0.454662	+	0.890664i	$0.650240\pi$
$758$	−24395.4	−1.16897
$759$	−5563.52	−0.266065
$760$	3057.48	0.145929
$761$	−16101.6	−0.766992	−0.383496	−	0.923542i	$-0.625280\pi$
$761$	−16101.6	−0.766992	−0.383496	+	0.923542i	$0.625280\pi$
$762$	20676.3	0.982971
$763$	0	0
$764$	−34361.8	−1.62718
$765$	1400.39	0.0661846
$766$	40440.3	1.90753
$767$	−40090.6	−1.88734
$768$	−22353.3	−1.05027
$769$	10781.2	0.505568	0.252784	−	0.967523i	$-0.418654\pi$
$769$	10781.2	0.505568	0.252784	+	0.967523i	$0.418654\pi$
$770$	0	0
$771$	6643.64	0.310331
$772$	66616.3	3.10567
$773$	−10224.8	−0.475755	−0.237878	−	0.971295i	$-0.576452\pi$
$773$	−10224.8	−0.475755	−0.237878	+	0.971295i	$0.576452\pi$
$774$	21564.6	1.00145
$775$	2943.49	0.136430
$776$	8697.31	0.402339
$777$	0	0
$778$	−17825.7	−0.821444
$779$	−7365.16	−0.338748
$780$	13808.3	0.633865
$781$	47288.0	2.16658
$782$	5208.60	0.238183
$783$	3818.42	0.174277
$784$	0	0
$785$	1341.54	0.0609958
$786$	−9386.08	−0.425942
$787$	−12823.4	−0.580817	−0.290409	−	0.956903i	$-0.593791\pi$
$787$	−12823.4	−0.580817	−0.290409	+	0.956903i	$0.593791\pi$
$788$	−30376.7	−1.37326
$789$	−16371.1	−0.738689
$790$	−3254.59	−0.146573
$791$	0	0
$792$	−16361.8	−0.734081
$793$	−25428.3	−1.13870
$794$	21725.5	0.971044
$795$	8739.49	0.389884
$796$	16782.1	0.747268
$797$	−28745.8	−1.27758	−0.638789	−	0.769382i	$-0.720564\pi$
$797$	−28745.8	−1.27758	−0.638789	+	0.769382i	$0.720564\pi$
$798$	0	0
$799$	−9032.91	−0.399951
$800$	1629.27	0.0720041
$801$	286.919	0.0126564
$802$	60522.0	2.66472
$803$	−58182.8	−2.55694
$804$	−25730.1	−1.12864
$805$	0	0
$806$	−34495.0	−1.50749
$807$	−17462.3	−0.761711
$808$	−46409.5	−2.02065
$809$	−31758.8	−1.38020	−0.690099	−	0.723715i	$-0.742433\pi$
$809$	−31758.8	−1.38020	−0.690099	+	0.723715i	$0.742433\pi$
$810$	−1946.62	−0.0844411
$811$	−3269.28	−0.141554	−0.0707769	−	0.997492i	$-0.522548\pi$
$811$	−3269.28	−0.141554	−0.0707769	+	0.997492i	$0.522548\pi$
$812$	0	0
$813$	9902.23	0.427167
$814$	−44877.3	−1.93237
$815$	−13637.9	−0.586154
$816$	4038.59	0.173259
$817$	8929.92	0.382397
$818$	−43343.0	−1.85263
$819$	0	0
$820$	31046.9	1.32220
$821$	17959.3	0.763441	0.381720	−	0.924278i	$-0.375332\pi$
$821$	17959.3	0.763441	0.381720	+	0.924278i	$0.375332\pi$
$822$	8908.06	0.377986
$823$	−22816.3	−0.966373	−0.483187	−	0.875517i	$-0.660521\pi$
$823$	−22816.3	−0.966373	−0.483187	+	0.875517i	$0.660521\pi$
$824$	−49431.6	−2.08985
$825$	3994.22	0.168559
$826$	0	0
$827$	38776.6	1.63047	0.815233	−	0.579133i	$-0.196609\pi$
$827$	38776.6	1.63047	0.815233	+	0.579133i	$0.196609\pi$
$828$	−4733.04	−0.198653
$829$	−43239.0	−1.81152	−0.905761	−	0.423789i	$-0.860700\pi$
$829$	−43239.0	−1.81152	−0.905761	+	0.423789i	$0.860700\pi$
$830$	11163.6	0.466860
$831$	−11613.8	−0.484809
$832$	−40188.1	−1.67460
$833$	0	0
$834$	−21990.7	−0.913042
$835$	11172.0	0.463023
$836$	−14407.4	−0.596040
$837$	3178.96	0.131280
$838$	−18.2968	−0.000754240 0
$839$	−38639.2	−1.58996	−0.794978	−	0.606638i	$-0.792518\pi$
$839$	−38639.2	−1.58996	−0.794978	+	0.606638i	$0.792518\pi$
$840$	0	0
$841$	−4388.57	−0.179940
$842$	−58043.1	−2.37565
$843$	−4234.21	−0.172994
$844$	45775.3	1.86689
$845$	7592.44	0.309098
$846$	12556.2	0.510275
$847$	0	0
$848$	25203.9	1.02064
$849$	−14257.5	−0.576345
$850$	−3739.41	−0.150895
$851$	−6105.02	−0.245919
$852$	40229.2	1.61764
$853$	21024.3	0.843914	0.421957	−	0.906616i	$-0.361343\pi$
$853$	21024.3	0.843914	0.421957	+	0.906616i	$0.361343\pi$
$854$	0	0
$855$	−806.096	−0.0322432
$856$	56886.9	2.27144
$857$	40140.9	1.59998	0.799991	−	0.600011i	$-0.204837\pi$
$857$	40140.9	1.59998	0.799991	+	0.600011i	$0.204837\pi$
$858$	−46808.7	−1.86250
$859$	16970.0	0.674050	0.337025	−	0.941496i	$-0.390579\pi$
$859$	16970.0	0.674050	0.337025	+	0.941496i	$0.390579\pi$
$860$	−37642.9	−1.49257
$861$	0	0
$862$	49591.8	1.95952
$863$	28638.9	1.12964	0.564819	−	0.825215i	$-0.308946\pi$
$863$	28638.9	1.12964	0.564819	+	0.825215i	$0.308946\pi$
$864$	1759.61	0.0692859
$865$	11698.3	0.459832
$866$	−61178.5	−2.40061
$867$	−11833.7	−0.463544
$868$	0	0
$869$	7212.24	0.281540
$870$	−10196.2	−0.397337
$871$	−34616.9	−1.34667
$872$	38178.1	1.48265
$873$	−2293.03	−0.0888971
$874$	−2998.19	−0.116036
$875$	0	0
$876$	−49497.7	−1.90910
$877$	1459.67	0.0562026	0.0281013	−	0.999605i	$-0.491054\pi$
$877$	1459.67	0.0562026	0.0281013	+	0.999605i	$0.491054\pi$
$878$	47933.0	1.84244
$879$	−3903.97	−0.149804
$880$	11519.0	0.441254
$881$	−34268.7	−1.31049	−0.655246	−	0.755416i	$-0.727435\pi$
$881$	−34268.7	−1.31049	−0.655246	+	0.755416i	$0.727435\pi$
$882$	0	0
$883$	−23106.9	−0.880645	−0.440322	−	0.897840i	$-0.645136\pi$
$883$	−23106.9	−0.880645	−0.440322	+	0.897840i	$0.645136\pi$
$884$	28647.4	1.08995
$885$	−9865.65	−0.374723
$886$	−50289.1	−1.90688
$887$	22858.2	0.865280	0.432640	−	0.901567i	$-0.357582\pi$
$887$	22858.2	0.865280	0.432640	+	0.901567i	$0.357582\pi$
$888$	−17954.3	−0.678500
$889$	0	0
$890$	−766.148	−0.0288555
$891$	4313.76	0.162196
$892$	−57236.3	−2.14844
$893$	5199.54	0.194844
$894$	20011.5	0.748642
$895$	12228.8	0.456718
$896$	0	0
$897$	−6367.77	−0.237027
$898$	56066.0	2.08346
$899$	16651.1	0.617735
$900$	3397.99	0.125852
$901$	18131.4	0.670416
$902$	−105246.	−3.88504
$903$	0	0
$904$	−27539.0	−1.01320
$905$	5892.27	0.216426
$906$	−32938.9	−1.20786
$907$	−4420.76	−0.161840	−0.0809199	−	0.996721i	$-0.525786\pi$
$907$	−4420.76	−0.161840	−0.0809199	+	0.996721i	$0.525786\pi$
$908$	58540.8	2.13959
$909$	12235.8	0.446463
$910$	0	0
$911$	31813.3	1.15699	0.578496	−	0.815685i	$-0.303640\pi$
$911$	31813.3	1.15699	0.578496	+	0.815685i	$0.303640\pi$
$912$	−2324.71	−0.0844065
$913$	−24738.7	−0.896750
$914$	−21833.5	−0.790142
$915$	−6257.50	−0.226084
$916$	89502.2	3.22842
$917$	0	0
$918$	−4038.57	−0.145199
$919$	22036.5	0.790988	0.395494	−	0.918469i	$-0.370573\pi$
$919$	22036.5	0.790988	0.395494	+	0.918469i	$0.370573\pi$
$920$	5943.56	0.212993
$921$	18414.3	0.658820
$922$	−37911.3	−1.35417
$923$	54123.8	1.93012
$924$	0	0
$925$	4382.98	0.155796
$926$	26167.1	0.928624
$927$	13032.5	0.461752
$928$	9216.62	0.326024
$929$	−27015.4	−0.954086	−0.477043	−	0.878880i	$-0.658291\pi$
$929$	−27015.4	−0.954086	−0.477043	+	0.878880i	$0.658291\pi$
$930$	−8488.67	−0.299306
$931$	0	0
$932$	70120.0	2.46444
$933$	12246.1	0.429711
$934$	50843.0	1.78119
$935$	8286.62	0.289841
$936$	−18727.0	−0.653966
$937$	33711.1	1.17534	0.587670	−	0.809101i	$-0.300046\pi$
$937$	33711.1	1.17534	0.587670	+	0.809101i	$0.300046\pi$
$938$	0	0
$939$	21429.6	0.744758
$940$	−21918.0	−0.760517
$941$	23168.8	0.802638	0.401319	−	0.915938i	$-0.368552\pi$
$941$	23168.8	0.802638	0.401319	+	0.915938i	$0.368552\pi$
$942$	−3868.85	−0.133815
$943$	−14317.5	−0.494423
$944$	−28451.6	−0.980954
$945$	0	0
$946$	127606.	4.38565
$947$	39932.2	1.37024	0.685122	−	0.728428i	$-0.259749\pi$
$947$	39932.2	1.37024	0.685122	+	0.728428i	$0.259749\pi$
$948$	6135.65	0.210207
$949$	−66593.5	−2.27789
$950$	2152.49	0.0735116
$951$	−22444.9	−0.765325
$952$	0	0
$953$	−31125.8	−1.05799	−0.528995	−	0.848625i	$-0.677431\pi$
$953$	−31125.8	−1.05799	−0.528995	+	0.848625i	$0.677431\pi$
$954$	−25203.7	−0.855345
$955$	−11376.4	−0.385479
$956$	14256.8	0.482321
$957$	22595.0	0.763210
$958$	−55970.0	−1.88759
$959$	0	0
$960$	−9889.64	−0.332486
$961$	−15928.4	−0.534672
$962$	−51364.6	−1.72148
$963$	−14998.1	−0.501876
$964$	22880.0	0.764433
$965$	22055.2	0.735732
$966$	0	0
$967$	−35679.1	−1.18652	−0.593259	−	0.805011i	$-0.702159\pi$
$967$	−35679.1	−1.18652	−0.593259	+	0.805011i	$0.702159\pi$
$968$	−51383.3	−1.70612
$969$	−1672.37	−0.0554430
$970$	6122.98	0.202677
$971$	−4137.64	−0.136749	−0.0683744	−	0.997660i	$-0.521781\pi$
$971$	−4137.64	−0.136749	−0.0683744	+	0.997660i	$0.521781\pi$
$972$	3669.83	0.121101
$973$	0	0
$974$	44398.5	1.46059
$975$	4571.61	0.150163
$976$	−18046.0	−0.591844
$977$	30405.7	0.995664	0.497832	−	0.867273i	$-0.334130\pi$
$977$	30405.7	0.995664	0.497832	+	0.867273i	$0.334130\pi$
$978$	39330.2	1.28593
$979$	1697.80	0.0554259
$980$	0	0
$981$	−10065.6	−0.327593
$982$	12998.9	0.422416
$983$	10011.3	0.324832	0.162416	−	0.986722i	$-0.448071\pi$
$983$	10011.3	0.324832	0.162416	+	0.986722i	$0.448071\pi$
$984$	−42106.4	−1.36413
$985$	−10057.1	−0.325324
$986$	−21153.5	−0.683231
$987$	0	0
$988$	−16490.1	−0.530991
$989$	17359.3	0.558132
$990$	−11518.9	−0.369792
$991$	−2889.51	−0.0926219	−0.0463110	−	0.998927i	$-0.514747\pi$
$991$	−2889.51	−0.0926219	−0.0463110	+	0.998927i	$0.514747\pi$
$992$	7673.15	0.245588
$993$	−5832.35	−0.186389
$994$	0	0
$995$	5556.18	0.177028
$996$	−21045.9	−0.669544
$997$	−6005.63	−0.190772	−0.0953862	−	0.995440i	$-0.530409\pi$
$997$	−6005.63	−0.190772	−0.0953862	+	0.995440i	$0.530409\pi$
$998$	3669.03	0.116374
$999$	4733.61	0.149915

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.4.a.ba.1.1		5
3.2	odd	2		2205.4.a.br.1.5		5
7.2	even	3		105.4.i.d.46.5	yes	10
7.4	even	3		105.4.i.d.16.5	✓	10
7.6	odd	2		735.4.a.z.1.1		5
21.2	odd	6		315.4.j.h.46.1		10
21.11	odd	6		315.4.j.h.226.1		10
21.20	even	2		2205.4.a.bs.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.i.d.16.5	✓	10	7.4	even	3
105.4.i.d.46.5	yes	10	7.2	even	3
315.4.j.h.46.1		10	21.2	odd	6
315.4.j.h.226.1		10	21.11	odd	6
735.4.a.z.1.1		5	7.6	odd	2
735.4.a.ba.1.1		5	1.1	even	1	trivial
2205.4.a.br.1.5		5	3.2	odd	2
2205.4.a.bs.1.5		5	21.20	even	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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

\(f(q)\)	\(=\)	\(q-4.80647 q^{2} +3.00000 q^{3} +15.1022 q^{4} +5.00000 q^{5} -14.4194 q^{6} -34.1365 q^{8} +9.00000 q^{9} -24.0324 q^{10} +53.2563 q^{11} +45.3066 q^{12} +60.9548 q^{13} +15.0000 q^{15} +43.2586 q^{16} +31.1198 q^{17} -43.2583 q^{18} -17.9133 q^{19} +75.5109 q^{20} -255.975 q^{22} -34.8223 q^{23} -102.409 q^{24} +25.0000 q^{25} -292.978 q^{26} +27.0000 q^{27} +141.423 q^{29} -72.0971 q^{30} +117.739 q^{31} +65.1706 q^{32} +159.769 q^{33} -149.576 q^{34} +135.920 q^{36} +175.319 q^{37} +86.0996 q^{38} +182.864 q^{39} -170.682 q^{40} +411.157 q^{41} -498.509 q^{43} +804.286 q^{44} +45.0000 q^{45} +167.373 q^{46} -290.262 q^{47} +129.776 q^{48} -120.162 q^{50} +93.3594 q^{51} +920.551 q^{52} +582.633 q^{53} -129.775 q^{54} +266.281 q^{55} -53.7398 q^{57} -679.745 q^{58} -657.710 q^{59} +226.533 q^{60} -417.167 q^{61} -565.911 q^{62} -659.310 q^{64} +304.774 q^{65} -767.925 q^{66} -567.911 q^{67} +469.977 q^{68} -104.467 q^{69} +887.933 q^{71} -307.228 q^{72} -1092.51 q^{73} -842.666 q^{74} +75.0000 q^{75} -270.529 q^{76} -878.933 q^{78} +135.425 q^{79} +216.293 q^{80} +81.0000 q^{81} -1976.22 q^{82} -464.523 q^{83} +155.599 q^{85} +2396.07 q^{86} +424.269 q^{87} -1817.98 q^{88} +31.8799 q^{89} -216.291 q^{90} -525.894 q^{92} +353.218 q^{93} +1395.14 q^{94} -89.5663 q^{95} +195.512 q^{96} -254.781 q^{97} +479.306 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 15 q^{3} + 25 q^{4} + 25 q^{5} + 9 q^{6} + 21 q^{8} + 45 q^{9} + 15 q^{10} + 43 q^{11} + 75 q^{12} + 123 q^{13} + 75 q^{15} + 161 q^{16} + 124 q^{17} + 27 q^{18} + 37 q^{19} + 125 q^{20}+ \cdots + 387 q^{99}+O(q^{100}) \) 5 * q + 3 * q^2 + 15 * q^3 + 25 * q^4 + 25 * q^5 + 9 * q^6 + 21 * q^8 + 45 * q^9 + 15 * q^10 + 43 * q^11 + 75 * q^12 + 123 * q^13 + 75 * q^15 + 161 * q^16 + 124 * q^17 + 27 * q^18 + 37 * q^19 + 125 * q^20 - 221 * q^22 + 77 * q^23 + 63 * q^24 + 125 * q^25 - 79 * q^26 + 135 * q^27 + 360 * q^29 + 45 * q^30 + 314 * q^31 - 59 * q^32 + 129 * q^33 + 176 * q^34 + 225 * q^36 + 225 * q^37 + 759 * q^38 + 369 * q^39 + 105 * q^40 + 341 * q^41 + 32 * q^43 + 679 * q^44 + 225 * q^45 - 331 * q^46 + 25 * q^47 + 483 * q^48 + 75 * q^50 + 372 * q^51 + 2299 * q^52 - 317 * q^53 + 81 * q^54 + 215 * q^55 + 111 * q^57 + 8 * q^58 + 676 * q^59 + 375 * q^60 - 188 * q^61 - 348 * q^62 - 1103 * q^64 + 615 * q^65 - 663 * q^66 - 1776 * q^67 + 1280 * q^68 + 231 * q^69 - 6 * q^71 + 189 * q^72 + 2006 * q^73 - 2729 * q^74 + 375 * q^75 + 1417 * q^76 - 237 * q^78 + 200 * q^79 + 805 * q^80 + 405 * q^81 - 539 * q^82 - 332 * q^83 + 620 * q^85 + 4262 * q^86 + 1080 * q^87 - 4529 * q^88 + 894 * q^89 + 135 * q^90 - 3687 * q^92 + 942 * q^93 + 4233 * q^94 + 185 * q^95 - 177 * q^96 - 576 * q^97 + 387 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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