LMFDB - Embedded newform 1617.4.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, 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("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.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:	$95.4060884793$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.54138$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.54138 q^{2} -3.00000 q^{3} +12.6241 q^{4} -13.1655 q^{5} -13.6241 q^{6} +21.0000 q^{8} +9.00000 q^{9} -59.7897 q^{10} -11.0000 q^{11} -37.8724 q^{12} +54.8345 q^{13} +39.4966 q^{15} -5.62414 q^{16} +79.9104 q^{17} +40.8724 q^{18} -21.2551 q^{19} -166.203 q^{20} -49.9552 q^{22} +119.414 q^{23} -63.0000 q^{24} +48.3311 q^{25} +249.024 q^{26} -27.0000 q^{27} -87.4829 q^{29} +179.369 q^{30} -191.248 q^{31} -193.541 q^{32} +33.0000 q^{33} +362.904 q^{34} +113.617 q^{36} +91.6484 q^{37} -96.5277 q^{38} -164.503 q^{39} -276.476 q^{40} +60.4829 q^{41} -213.552 q^{43} -138.866 q^{44} -118.490 q^{45} +542.304 q^{46} -417.421 q^{47} +16.8724 q^{48} +219.490 q^{50} -239.731 q^{51} +692.238 q^{52} -414.904 q^{53} -122.617 q^{54} +144.821 q^{55} +63.7654 q^{57} -397.293 q^{58} +358.069 q^{59} +498.610 q^{60} -515.297 q^{61} -868.531 q^{62} -833.952 q^{64} -721.925 q^{65} +149.866 q^{66} -107.572 q^{67} +1008.80 q^{68} -358.241 q^{69} -711.793 q^{71} +189.000 q^{72} -131.773 q^{73} +416.210 q^{74} -144.993 q^{75} -268.328 q^{76} -747.073 q^{78} +10.4966 q^{79} +74.0448 q^{80} +81.0000 q^{81} +274.676 q^{82} +618.352 q^{83} -1052.06 q^{85} -969.821 q^{86} +262.449 q^{87} -231.000 q^{88} -1376.44 q^{89} -538.107 q^{90} +1507.50 q^{92} +573.745 q^{93} -1895.67 q^{94} +279.835 q^{95} +580.624 q^{96} +534.752 q^{97} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} - 6 q^{3} + 7 q^{4} - 2 q^{5} - 9 q^{6} + 42 q^{8} + 18 q^{9} - 77 q^{10} - 22 q^{11} - 21 q^{12} + 134 q^{13} + 6 q^{15} + 7 q^{16} + 26 q^{17} + 27 q^{18} - 152 q^{19} - 229 q^{20} - 33 q^{22}+ \cdots - 198 q^{99}+O(q^{100}) $ 2 * q + 3 * q^2 - 6 * q^3 + 7 * q^4 - 2 * q^5 - 9 * q^6 + 42 * q^8 + 18 * q^9 - 77 * q^10 - 22 * q^11 - 21 * q^12 + 134 * q^13 + 6 * q^15 + 7 * q^16 + 26 * q^17 + 27 * q^18 - 152 * q^19 - 229 * q^20 - 33 * q^22 + 178 * q^23 - 126 * q^24 + 48 * q^25 + 127 * q^26 - 54 * q^27 + 190 * q^29 + 231 * q^30 - 346 * q^31 - 381 * q^32 + 66 * q^33 + 446 * q^34 + 63 * q^36 - 206 * q^37 + 105 * q^38 - 402 * q^39 - 42 * q^40 - 244 * q^41 + 242 * q^43 - 77 * q^44 - 18 * q^45 + 452 * q^46 - 920 * q^47 - 21 * q^48 + 220 * q^50 - 78 * q^51 + 247 * q^52 - 550 * q^53 - 81 * q^54 + 22 * q^55 + 456 * q^57 - 825 * q^58 + 412 * q^59 + 687 * q^60 - 252 * q^61 - 630 * q^62 - 646 * q^64 + 162 * q^65 + 99 * q^66 + 16 * q^67 + 1312 * q^68 - 534 * q^69 - 572 * q^71 + 378 * q^72 + 1026 * q^73 + 875 * q^74 - 144 * q^75 + 467 * q^76 - 381 * q^78 - 52 * q^79 + 215 * q^80 + 162 * q^81 + 744 * q^82 - 138 * q^83 - 1654 * q^85 - 1672 * q^86 - 570 * q^87 - 462 * q^88 - 1512 * q^89 - 693 * q^90 + 1178 * q^92 + 1038 * q^93 - 1121 * q^94 - 1180 * q^95 + 1143 * q^96 + 1106 * q^97 - 198 * 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 $

$2$	4.54138	1.60562	0.802810	−	0.596234i	$-0.203337\pi$
$2$	4.54138	1.60562	0.802810	+	0.596234i	$0.203337\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	12.6241	1.57802
$5$	−13.1655	−1.17756	−0.588780	−	0.808293i	$-0.700392\pi$
$5$	−13.1655	−1.17756	−0.588780	+	0.808293i	$0.700392\pi$
$6$	−13.6241	−0.927006
$7$	0	0
$7$	0	0
$8$	21.0000	0.928078
$9$	9.00000	0.333333
$10$	−59.7897	−1.89072
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−37.8724	−0.911069
$13$	54.8345	1.16987	0.584936	−	0.811079i	$-0.301119\pi$
$13$	54.8345	1.16987	0.584936	+	0.811079i	$0.301119\pi$
$14$	0	0
$15$	39.4966	0.679865
$16$	−5.62414	−0.0878772
$17$	79.9104	1.14007	0.570033	−	0.821622i	$-0.306930\pi$
$17$	79.9104	1.14007	0.570033	+	0.821622i	$0.306930\pi$
$18$	40.8724	0.535207
$19$	−21.2551	−0.256645	−0.128323	−	0.991732i	$-0.540959\pi$
$19$	−21.2551	−0.256645	−0.128323	+	0.991732i	$0.540959\pi$
$20$	−166.203	−1.85821
$21$	0	0
$22$	−49.9552	−0.484113
$23$	119.414	1.08259	0.541294	−	0.840834i	$-0.317935\pi$
$23$	119.414	1.08259	0.541294	+	0.840834i	$0.317935\pi$
$24$	−63.0000	−0.535826
$25$	48.3311	0.386648
$26$	249.024	1.87837
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−87.4829	−0.560178	−0.280089	−	0.959974i	$-0.590364\pi$
$29$	−87.4829	−0.560178	−0.280089	+	0.959974i	$0.590364\pi$
$30$	179.369	1.09161
$31$	−191.248	−1.10804	−0.554019	−	0.832504i	$-0.686907\pi$
$31$	−191.248	−1.10804	−0.554019	+	0.832504i	$0.686907\pi$
$32$	−193.541	−1.06918
$33$	33.0000	0.174078
$34$	362.904	1.83051
$35$	0	0
$36$	113.617	0.526006
$37$	91.6484	0.407214	0.203607	−	0.979053i	$-0.434734\pi$
$37$	91.6484	0.407214	0.203607	+	0.979053i	$0.434734\pi$
$38$	−96.5277	−0.412075
$39$	−164.503	−0.675426
$40$	−276.476	−1.09287
$41$	60.4829	0.230386	0.115193	−	0.993343i	$-0.463251\pi$
$41$	60.4829	0.230386	0.115193	+	0.993343i	$0.463251\pi$
$42$	0	0
$43$	−213.552	−0.757357	−0.378679	−	0.925528i	$-0.623621\pi$
$43$	−213.552	−0.757357	−0.378679	+	0.925528i	$0.623621\pi$
$44$	−138.866	−0.475790
$45$	−118.490	−0.392520
$46$	542.304	1.73822
$47$	−417.421	−1.29547	−0.647735	−	0.761866i	$-0.724283\pi$
$47$	−417.421	−1.29547	−0.647735	+	0.761866i	$0.724283\pi$
$48$	16.8724	0.0507360
$49$	0	0
$50$	219.490	0.620811
$51$	−239.731	−0.658217
$52$	692.238	1.84608
$53$	−414.904	−1.07531	−0.537655	−	0.843165i	$-0.680690\pi$
$53$	−414.904	−1.07531	−0.537655	+	0.843165i	$0.680690\pi$
$54$	−122.617	−0.309002
$55$	144.821	0.355048
$56$	0	0
$57$	63.7654	0.148174
$58$	−397.293	−0.899433
$59$	358.069	0.790112	0.395056	−	0.918657i	$-0.370725\pi$
$59$	358.069	0.790112	0.395056	+	0.918657i	$0.370725\pi$
$60$	498.610	1.07284
$61$	−515.297	−1.08159	−0.540795	−	0.841154i	$-0.681877\pi$
$61$	−515.297	−1.08159	−0.540795	+	0.841154i	$0.681877\pi$
$62$	−868.531	−1.77909
$63$	0	0
$64$	−833.952	−1.62881
$65$	−721.925	−1.37760
$66$	149.866	0.279503
$67$	−107.572	−0.196150	−0.0980752	−	0.995179i	$-0.531269\pi$
$67$	−107.572	−0.196150	−0.0980752	+	0.995179i	$0.531269\pi$
$68$	1008.80	1.79904
$69$	−358.241	−0.625032
$70$	0	0
$71$	−711.793	−1.18978	−0.594890	−	0.803807i	$-0.702804\pi$
$71$	−711.793	−1.18978	−0.594890	+	0.803807i	$0.702804\pi$
$72$	189.000	0.309359
$73$	−131.773	−0.211272	−0.105636	−	0.994405i	$-0.533688\pi$
$73$	−131.773	−0.211272	−0.105636	+	0.994405i	$0.533688\pi$
$74$	416.210	0.653831
$75$	−144.993	−0.223232
$76$	−268.328	−0.404991
$77$	0	0
$78$	−747.073	−1.08448
$79$	10.4966	0.0149488	0.00747441	−	0.999972i	$-0.497621\pi$
$79$	10.4966	0.0149488	0.00747441	+	0.999972i	$0.497621\pi$
$80$	74.0448	0.103481
$81$	81.0000	0.111111
$82$	274.676	0.369913
$83$	618.352	0.817747	0.408873	−	0.912591i	$-0.365922\pi$
$83$	618.352	0.817747	0.408873	+	0.912591i	$0.365922\pi$
$84$	0	0
$85$	−1052.06	−1.34250
$86$	−969.821	−1.21603
$87$	262.449	0.323419
$88$	−231.000	−0.279826
$89$	−1376.44	−1.63935	−0.819677	−	0.572826i	$-0.805847\pi$
$89$	−1376.44	−1.63935	−0.819677	+	0.572826i	$0.805847\pi$
$90$	−538.107	−0.630238
$91$	0	0
$92$	1507.50	1.70834
$93$	573.745	0.639727
$94$	−1895.67	−2.08003
$95$	279.835	0.302215
$96$	580.624	0.617289
$97$	534.752	0.559751	0.279875	−	0.960036i	$-0.409707\pi$
$97$	534.752	0.559751	0.279875	+	0.960036i	$0.409707\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	610.138	0.610138
$101$	−1620.01	−1.59601	−0.798004	−	0.602652i	$-0.794111\pi$
$101$	−1620.01	−1.59601	−0.798004	+	0.602652i	$0.794111\pi$
$102$	−1088.71	−1.05685
$103$	−822.814	−0.787129	−0.393564	−	0.919297i	$-0.628758\pi$
$103$	−822.814	−0.787129	−0.393564	+	0.919297i	$0.628758\pi$
$104$	1151.52	1.08573
$105$	0	0
$106$	−1884.24	−1.72654
$107$	−1749.78	−1.58091	−0.790456	−	0.612518i	$-0.790157\pi$
$107$	−1749.78	−1.58091	−0.790456	+	0.612518i	$0.790157\pi$
$108$	−340.852	−0.303690
$109$	−647.607	−0.569078	−0.284539	−	0.958665i	$-0.591840\pi$
$109$	−647.607	−0.569078	−0.284539	+	0.958665i	$0.591840\pi$
$110$	657.686	0.570072
$111$	−274.945	−0.235105
$112$	0	0
$113$	1151.50	0.958622	0.479311	−	0.877645i	$-0.340887\pi$
$113$	1151.50	0.958622	0.479311	+	0.877645i	$0.340887\pi$
$114$	289.583	0.237912
$115$	−1572.15	−1.27481
$116$	−1104.40	−0.883971
$117$	493.510	0.389958
$118$	1626.13	1.26862
$119$	0	0
$120$	829.428	0.630967
$121$	121.000	0.0909091
$122$	−2340.16	−1.73662
$123$	−181.449	−0.133014
$124$	−2414.35	−1.74851
$125$	1009.39	0.722259
$126$	0	0
$127$	278.146	0.194342	0.0971710	−	0.995268i	$-0.469021\pi$
$127$	278.146	0.194342	0.0971710	+	0.995268i	$0.469021\pi$
$128$	−2238.96	−1.54608
$129$	640.656	0.437260
$130$	−3278.54	−2.21190
$131$	−2332.22	−1.55547	−0.777737	−	0.628590i	$-0.783633\pi$
$131$	−2332.22	−1.55547	−0.777737	+	0.628590i	$0.783633\pi$
$132$	416.597	0.274698
$133$	0	0
$134$	−488.528	−0.314943
$135$	355.469	0.226622
$136$	1678.12	1.05807
$137$	2612.23	1.62903	0.814517	−	0.580139i	$-0.197002\pi$
$137$	2612.23	1.62903	0.814517	+	0.580139i	$0.197002\pi$
$138$	−1626.91	−1.00356
$139$	2884.68	1.76025	0.880126	−	0.474740i	$-0.157458\pi$
$139$	2884.68	1.76025	0.880126	+	0.474740i	$0.157458\pi$
$140$	0	0
$141$	1252.26	0.747940
$142$	−3232.53	−1.91033
$143$	−603.179	−0.352730
$144$	−50.6173	−0.0292924
$145$	1151.76	0.659643
$146$	−598.431	−0.339222
$147$	0	0
$148$	1156.98	0.642590
$149$	2909.54	1.59972	0.799862	−	0.600184i	$-0.204906\pi$
$149$	2909.54	1.59972	0.799862	+	0.600184i	$0.204906\pi$
$150$	−658.469	−0.358425
$151$	−2327.06	−1.25413	−0.627063	−	0.778968i	$-0.715743\pi$
$151$	−2327.06	−1.25413	−0.627063	+	0.778968i	$0.715743\pi$
$152$	−446.358	−0.238187
$153$	719.193	0.380022
$154$	0	0
$155$	2517.88	1.30478
$156$	−2076.71	−1.06584
$157$	927.572	0.471518	0.235759	−	0.971812i	$-0.424242\pi$
$157$	927.572	0.471518	0.235759	+	0.971812i	$0.424242\pi$
$158$	47.6689	0.0240021
$159$	1244.71	0.620830
$160$	2548.07	1.25902
$161$	0	0
$162$	367.852	0.178402
$163$	−2966.31	−1.42539	−0.712696	−	0.701473i	$-0.752526\pi$
$163$	−2966.31	−1.42539	−0.712696	+	0.701473i	$0.752526\pi$
$164$	763.545	0.363554
$165$	−434.462	−0.204987
$166$	2808.17	1.31299
$167$	−3856.35	−1.78691	−0.893454	−	0.449156i	$-0.851725\pi$
$167$	−3856.35	−1.78691	−0.893454	+	0.449156i	$0.851725\pi$
$168$	0	0
$169$	809.820	0.368602
$170$	−4777.82	−2.15554
$171$	−191.296	−0.0855485
$172$	−2695.91	−1.19512
$173$	2699.52	1.18636	0.593181	−	0.805069i	$-0.297872\pi$
$173$	2699.52	1.18636	0.593181	+	0.805069i	$0.297872\pi$
$174$	1191.88	0.519288
$175$	0	0
$176$	61.8656	0.0264960
$177$	−1074.21	−0.456172
$178$	−6250.95	−2.63218
$179$	1328.58	0.554764	0.277382	−	0.960760i	$-0.410533\pi$
$179$	1328.58	0.554764	0.277382	+	0.960760i	$0.410533\pi$
$180$	−1495.83	−0.619404
$181$	−534.944	−0.219680	−0.109840	−	0.993949i	$-0.535034\pi$
$181$	−534.944	−0.219680	−0.109840	+	0.993949i	$0.535034\pi$
$182$	0	0
$183$	1545.89	0.624456
$184$	2507.69	1.00472
$185$	−1206.60	−0.479519
$186$	2605.59	1.02716
$187$	−879.014	−0.343743
$188$	−5269.58	−2.04427
$189$	0	0
$190$	1270.84	0.485243
$191$	−3456.20	−1.30933	−0.654665	−	0.755919i	$-0.727190\pi$
$191$	−3456.20	−1.30933	−0.654665	+	0.755919i	$0.727190\pi$
$192$	2501.86	0.940395
$193$	−3106.02	−1.15843	−0.579213	−	0.815176i	$-0.696640\pi$
$193$	−3106.02	−1.15843	−0.579213	+	0.815176i	$0.696640\pi$
$194$	2428.51	0.898747
$195$	2165.77	0.795355
$196$	0	0
$197$	4886.49	1.76725	0.883623	−	0.468198i	$-0.155097\pi$
$197$	4886.49	1.76725	0.883623	+	0.468198i	$0.155097\pi$
$198$	−449.597	−0.161371
$199$	−3319.74	−1.18257	−0.591283	−	0.806464i	$-0.701378\pi$
$199$	−3319.74	−1.18257	−0.591283	+	0.806464i	$0.701378\pi$
$200$	1014.95	0.358840
$201$	322.717	0.113247
$202$	−7357.07	−2.56258
$203$	0	0
$204$	−3026.40	−1.03868
$205$	−796.289	−0.271294
$206$	−3736.71	−1.26383
$207$	1074.72	0.360862
$208$	−308.397	−0.102805
$209$	233.807	0.0773815
$210$	0	0
$211$	4262.51	1.39073	0.695364	−	0.718658i	$-0.255243\pi$
$211$	4262.51	1.39073	0.695364	+	0.718658i	$0.255243\pi$
$212$	−5237.80	−1.69686
$213$	2135.38	0.686919
$214$	−7946.42	−2.53835
$215$	2811.52	0.891834
$216$	−567.000	−0.178609
$217$	0	0
$218$	−2941.03	−0.913723
$219$	395.318	0.121978
$220$	1828.24	0.560272
$221$	4381.84	1.33373
$222$	−1248.63	−0.377489
$223$	−2766.16	−0.830655	−0.415328	−	0.909672i	$-0.636333\pi$
$223$	−2766.16	−0.830655	−0.415328	+	0.909672i	$0.636333\pi$
$224$	0	0
$225$	434.979	0.128883
$226$	5229.42	1.53918
$227$	−545.724	−0.159564	−0.0797818	−	0.996812i	$-0.525422\pi$
$227$	−545.724	−0.159564	−0.0797818	+	0.996812i	$0.525422\pi$
$228$	804.984	0.233822
$229$	−3197.48	−0.922688	−0.461344	−	0.887221i	$-0.652633\pi$
$229$	−3197.48	−0.922688	−0.461344	+	0.887221i	$0.652633\pi$
$230$	−7139.71	−2.04686
$231$	0	0
$232$	−1837.14	−0.519889
$233$	4203.76	1.18196	0.590982	−	0.806684i	$-0.298740\pi$
$233$	4203.76	1.18196	0.590982	+	0.806684i	$0.298740\pi$
$234$	2241.22	0.626124
$235$	5495.56	1.52549
$236$	4520.32	1.24681
$237$	−31.4897	−0.00863071
$238$	0	0
$239$	5720.73	1.54830	0.774150	−	0.633003i	$-0.218178\pi$
$239$	5720.73	1.54830	0.774150	+	0.633003i	$0.218178\pi$
$240$	−222.134	−0.0597446
$241$	−5191.77	−1.38768	−0.693841	−	0.720128i	$-0.744083\pi$
$241$	−5191.77	−1.38768	−0.693841	+	0.720128i	$0.744083\pi$
$242$	549.507	0.145966
$243$	−243.000	−0.0641500
$244$	−6505.18	−1.70677
$245$	0	0
$246$	−824.027	−0.213569
$247$	−1165.51	−0.300243
$248$	−4016.21	−1.02835
$249$	−1855.06	−0.472126
$250$	4584.01	1.15967
$251$	−847.944	−0.213234	−0.106617	−	0.994300i	$-0.534002\pi$
$251$	−847.944	−0.213234	−0.106617	+	0.994300i	$0.534002\pi$
$252$	0	0
$253$	−1313.55	−0.326412
$254$	1263.16	0.312039
$255$	3156.19	0.775090
$256$	−3496.37	−0.853606
$257$	1493.87	0.362588	0.181294	−	0.983429i	$-0.441971\pi$
$257$	1493.87	0.362588	0.181294	+	0.983429i	$0.441971\pi$
$258$	2909.46	0.702074
$259$	0	0
$260$	−9113.68	−2.17387
$261$	−787.346	−0.186726
$262$	−10591.5	−2.49750
$263$	135.848	0.0318507	0.0159253	−	0.999873i	$-0.494931\pi$
$263$	135.848	0.0318507	0.0159253	+	0.999873i	$0.494931\pi$
$264$	693.000	0.161558
$265$	5462.42	1.26624
$266$	0	0
$267$	4129.33	0.946482
$268$	−1358.01	−0.309529
$269$	739.406	0.167593	0.0837963	−	0.996483i	$-0.473295\pi$
$269$	739.406	0.167593	0.0837963	+	0.996483i	$0.473295\pi$
$270$	1614.32	0.363868
$271$	−4360.29	−0.977375	−0.488688	−	0.872459i	$-0.662524\pi$
$271$	−4360.29	−0.977375	−0.488688	+	0.872459i	$0.662524\pi$
$272$	−449.428	−0.100186
$273$	0	0
$274$	11863.1	2.61561
$275$	−531.642	−0.116579
$276$	−4522.49	−0.986312
$277$	−5779.74	−1.25369	−0.626843	−	0.779146i	$-0.715653\pi$
$277$	−5779.74	−1.25369	−0.626843	+	0.779146i	$0.715653\pi$
$278$	13100.4	2.82630
$279$	−1721.23	−0.369346
$280$	0	0
$281$	−4366.84	−0.927059	−0.463530	−	0.886081i	$-0.653417\pi$
$281$	−4366.84	−0.927059	−0.463530	+	0.886081i	$0.653417\pi$
$282$	5687.00	1.20091
$283$	8294.14	1.74218	0.871088	−	0.491128i	$-0.163415\pi$
$283$	8294.14	1.74218	0.871088	+	0.491128i	$0.163415\pi$
$284$	−8985.78	−1.87749
$285$	−839.505	−0.174484
$286$	−2739.27	−0.566351
$287$	0	0
$288$	−1741.87	−0.356392
$289$	1472.67	0.299750
$290$	5230.57	1.05914
$291$	−1604.26	−0.323172
$292$	−1663.52	−0.333391
$293$	−2434.82	−0.485473	−0.242736	−	0.970092i	$-0.578045\pi$
$293$	−2434.82	−0.485473	−0.242736	+	0.970092i	$0.578045\pi$
$294$	0	0
$295$	−4714.17	−0.930405
$296$	1924.62	0.377926
$297$	297.000	0.0580259
$298$	13213.3	2.56855
$299$	6547.99	1.26649
$300$	−1830.41	−0.352263
$301$	0	0
$302$	−10568.1	−2.01365
$303$	4860.02	0.921455
$304$	119.542	0.0225533
$305$	6784.15	1.27364
$306$	3266.13	0.610171
$307$	5972.33	1.11029	0.555145	−	0.831754i	$-0.312663\pi$
$307$	5972.33	1.11029	0.555145	+	0.831754i	$0.312663\pi$
$308$	0	0
$309$	2468.44	0.454449
$310$	11434.7	2.09499
$311$	8278.65	1.50945	0.754725	−	0.656041i	$-0.227770\pi$
$311$	8278.65	1.50945	0.754725	+	0.656041i	$0.227770\pi$
$312$	−3454.57	−0.626848
$313$	1386.60	0.250400	0.125200	−	0.992131i	$-0.460043\pi$
$313$	1386.60	0.250400	0.125200	+	0.992131i	$0.460043\pi$
$314$	4212.46	0.757079
$315$	0	0
$316$	132.510	0.0235895
$317$	11.2346	0.00199053	0.000995264	−	1.00000i	$-0.499683\pi$
$317$	11.2346	0.00199053	0.000995264		1.00000i	$0.499683\pi$
$318$	5652.71	0.996818
$319$	962.312	0.168900
$320$	10979.4	1.91803
$321$	5249.34	0.912740
$322$	0	0
$323$	−1698.51	−0.292593
$324$	1022.56	0.175335
$325$	2650.21	0.452329
$326$	−13471.1	−2.28864
$327$	1942.82	0.328557
$328$	1270.14	0.213816
$329$	0	0
$330$	−1973.06	−0.329131
$331$	−4516.29	−0.749963	−0.374981	−	0.927032i	$-0.622351\pi$
$331$	−4516.29	−0.749963	−0.374981	+	0.927032i	$0.622351\pi$
$332$	7806.17	1.29042
$333$	824.836	0.135738
$334$	−17513.2	−2.86910
$335$	1416.25	0.230979
$336$	0	0
$337$	−676.780	−0.109396	−0.0546982	−	0.998503i	$-0.517420\pi$
$337$	−676.780	−0.109396	−0.0546982	+	0.998503i	$0.517420\pi$
$338$	3677.70	0.591836
$339$	−3454.51	−0.553461
$340$	−13281.4	−2.11848
$341$	2103.73	0.334086
$342$	−868.749	−0.137358
$343$	0	0
$344$	−4484.59	−0.702886
$345$	4716.44	0.736013
$346$	12259.5	1.90485
$347$	7619.43	1.17877	0.589384	−	0.807853i	$-0.299371\pi$
$347$	7619.43	1.17877	0.589384	+	0.807853i	$0.299371\pi$
$348$	3313.19	0.510361
$349$	−3178.35	−0.487488	−0.243744	−	0.969840i	$-0.578376\pi$
$349$	−3178.35	−0.487488	−0.243744	+	0.969840i	$0.578376\pi$
$350$	0	0
$351$	−1480.53	−0.225142
$352$	2128.96	0.322368
$353$	−1776.12	−0.267800	−0.133900	−	0.990995i	$-0.542750\pi$
$353$	−1776.12	−0.267800	−0.133900	+	0.990995i	$0.542750\pi$
$354$	−4878.38	−0.732438
$355$	9371.13	1.40104
$356$	−17376.4	−2.58693
$357$	0	0
$358$	6033.59	0.890740
$359$	−11425.3	−1.67968	−0.839838	−	0.542838i	$-0.817350\pi$
$359$	−11425.3	−1.67968	−0.839838	+	0.542838i	$0.817350\pi$
$360$	−2488.28	−0.364289
$361$	−6407.22	−0.934133
$362$	−2429.38	−0.352722
$363$	−363.000	−0.0524864
$364$	0	0
$365$	1734.86	0.248785
$366$	7020.48	1.00264
$367$	1321.34	0.187938	0.0939689	−	0.995575i	$-0.470045\pi$
$367$	1321.34	0.187938	0.0939689	+	0.995575i	$0.470045\pi$
$368$	−671.600	−0.0951348
$369$	544.346	0.0767954
$370$	−5479.63	−0.769925
$371$	0	0
$372$	7243.04	1.00950
$373$	−2011.16	−0.279180	−0.139590	−	0.990209i	$-0.544578\pi$
$373$	−2011.16	−0.279180	−0.139590	+	0.990209i	$0.544578\pi$
$374$	−3991.94	−0.551920
$375$	−3028.16	−0.416996
$376$	−8765.83	−1.20230
$377$	−4797.08	−0.655337
$378$	0	0
$379$	−1376.20	−0.186519	−0.0932597	−	0.995642i	$-0.529729\pi$
$379$	−1376.20	−0.186519	−0.0932597	+	0.995642i	$0.529729\pi$
$380$	3532.68	0.476901
$381$	−834.437	−0.112203
$382$	−15695.9	−2.10229
$383$	−68.4429	−0.00913125	−0.00456563	−	0.999990i	$-0.501453\pi$
$383$	−68.4429	−0.00913125	−0.00456563	+	0.999990i	$0.501453\pi$
$384$	6716.89	0.892630
$385$	0	0
$386$	−14105.6	−1.85999
$387$	−1921.97	−0.252452
$388$	6750.78	0.883297
$389$	−1796.33	−0.234133	−0.117066	−	0.993124i	$-0.537349\pi$
$389$	−1796.33	−0.234133	−0.117066	+	0.993124i	$0.537349\pi$
$390$	9835.61	1.27704
$391$	9542.40	1.23422
$392$	0	0
$393$	6996.66	0.898054
$394$	22191.4	2.83753
$395$	−138.193	−0.0176031
$396$	−1249.79	−0.158597
$397$	−4175.59	−0.527876	−0.263938	−	0.964540i	$-0.585022\pi$
$397$	−4175.59	−0.527876	−0.263938	+	0.964540i	$0.585022\pi$
$398$	−15076.2	−1.89875
$399$	0	0
$400$	−271.821	−0.0339776
$401$	−5735.25	−0.714226	−0.357113	−	0.934061i	$-0.616239\pi$
$401$	−5735.25	−0.714226	−0.357113	+	0.934061i	$0.616239\pi$
$402$	1465.58	0.181832
$403$	−10487.0	−1.29626
$404$	−20451.2	−2.51853
$405$	−1066.41	−0.130840
$406$	0	0
$407$	−1008.13	−0.122780
$408$	−5034.35	−0.610877
$409$	−4552.26	−0.550355	−0.275177	−	0.961394i	$-0.588737\pi$
$409$	−4552.26	−0.550355	−0.275177	+	0.961394i	$0.588737\pi$
$410$	−3616.25	−0.435595
$411$	−7836.69	−0.940524
$412$	−10387.3	−1.24210
$413$	0	0
$414$	4880.73	0.579408
$415$	−8140.93	−0.962946
$416$	−10612.7	−1.25080
$417$	−8654.03	−1.01628
$418$	1061.80	0.124245
$419$	−12761.1	−1.48788	−0.743940	−	0.668247i	$-0.767045\pi$
$419$	−12761.1	−1.48788	−0.743940	+	0.668247i	$0.767045\pi$
$420$	0	0
$421$	−2838.93	−0.328649	−0.164324	−	0.986406i	$-0.552544\pi$
$421$	−2838.93	−0.328649	−0.164324	+	0.986406i	$0.552544\pi$
$422$	19357.7	2.23298
$423$	−3756.79	−0.431823
$424$	−8712.97	−0.997970
$425$	3862.15	0.440805
$426$	9697.58	1.10293
$427$	0	0
$428$	−22089.5	−2.49471
$429$	1809.54	0.203649
$430$	12768.2	1.43195
$431$	−12820.2	−1.43277	−0.716387	−	0.697704i	$-0.754205\pi$
$431$	−12820.2	−1.43277	−0.716387	+	0.697704i	$0.754205\pi$
$432$	151.852	0.0169120
$433$	12570.9	1.39519	0.697595	−	0.716493i	$-0.254254\pi$
$433$	12570.9	1.39519	0.697595	+	0.716493i	$0.254254\pi$
$434$	0	0
$435$	−3455.27	−0.380845
$436$	−8175.48	−0.898015
$437$	−2538.16	−0.277841
$438$	1795.29	0.195850
$439$	11589.8	1.26003	0.630013	−	0.776585i	$-0.283050\pi$
$439$	11589.8	1.26003	0.630013	+	0.776585i	$0.283050\pi$
$440$	3041.24	0.329512
$441$	0	0
$442$	19899.6	2.14147
$443$	15298.1	1.64071	0.820355	−	0.571855i	$-0.193776\pi$
$443$	15298.1	1.64071	0.820355	+	0.571855i	$0.193776\pi$
$444$	−3470.95	−0.371000
$445$	18121.6	1.93044
$446$	−12562.2	−1.33372
$447$	−8728.62	−0.923601
$448$	0	0
$449$	3370.37	0.354249	0.177125	−	0.984188i	$-0.443320\pi$
$449$	3370.37	0.354249	0.177125	+	0.984188i	$0.443320\pi$
$450$	1975.41	0.206937
$451$	−665.312	−0.0694641
$452$	14536.7	1.51272
$453$	6981.17	0.724071
$454$	−2478.34	−0.256199
$455$	0	0
$456$	1339.07	0.137517
$457$	−2682.12	−0.274539	−0.137269	−	0.990534i	$-0.543833\pi$
$457$	−2682.12	−0.274539	−0.137269	+	0.990534i	$0.543833\pi$
$458$	−14521.0	−1.48149
$459$	−2157.58	−0.219406
$460$	−19847.0	−2.01168
$461$	−1313.76	−0.132729	−0.0663645	−	0.997795i	$-0.521140\pi$
$461$	−1313.76	−0.132729	−0.0663645	+	0.997795i	$0.521140\pi$
$462$	0	0
$463$	17339.2	1.74043	0.870215	−	0.492672i	$-0.163980\pi$
$463$	17339.2	1.74043	0.870215	+	0.492672i	$0.163980\pi$
$464$	492.016	0.0492269
$465$	−7553.65	−0.753317
$466$	19090.9	1.89779
$467$	8810.22	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$467$	8810.22	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$468$	6230.14	0.615360
$469$	0	0
$470$	24957.4	2.44936
$471$	−2782.72	−0.272231
$472$	7519.45	0.733285
$473$	2349.07	0.228352
$474$	−143.007	−0.0138576
$475$	−1027.28	−0.0992315
$476$	0	0
$477$	−3734.13	−0.358436
$478$	25980.0	2.48598
$479$	708.948	0.0676256	0.0338128	−	0.999428i	$-0.489235\pi$
$479$	708.948	0.0676256	0.0338128	+	0.999428i	$0.489235\pi$
$480$	−7644.22	−0.726895
$481$	5025.49	0.476388
$482$	−23577.8	−2.22809
$483$	0	0
$484$	1527.52	0.143456
$485$	−7040.29	−0.659140
$486$	−1103.56	−0.103001
$487$	12253.6	1.14017	0.570085	−	0.821586i	$-0.306910\pi$
$487$	12253.6	1.14017	0.570085	+	0.821586i	$0.306910\pi$
$488$	−10821.2	−1.00380
$489$	8898.92	0.822951
$490$	0	0
$491$	−5429.37	−0.499031	−0.249515	−	0.968371i	$-0.580271\pi$
$491$	−5429.37	−0.499031	−0.249515	+	0.968371i	$0.580271\pi$
$492$	−2290.63	−0.209898
$493$	−6990.79	−0.638640
$494$	−5293.04	−0.482076
$495$	1303.39	0.118349
$496$	1075.61	0.0973714
$497$	0	0
$498$	−8424.52	−0.758056
$499$	−14531.6	−1.30366	−0.651829	−	0.758366i	$-0.725998\pi$
$499$	−14531.6	−1.30366	−0.651829	+	0.758366i	$0.725998\pi$
$500$	12742.6	1.13974
$501$	11569.1	1.03167
$502$	−3850.83	−0.342373
$503$	6349.27	0.562823	0.281412	−	0.959587i	$-0.409197\pi$
$503$	6349.27	0.562823	0.281412	+	0.959587i	$0.409197\pi$
$504$	0	0
$505$	21328.2	1.87940
$506$	−5965.34	−0.524094
$507$	−2429.46	−0.212813
$508$	3511.35	0.306675
$509$	2082.16	0.181317	0.0906585	−	0.995882i	$-0.471103\pi$
$509$	2082.16	0.181317	0.0906585	+	0.995882i	$0.471103\pi$
$510$	14333.4	1.24450
$511$	0	0
$512$	2033.36	0.175513
$513$	573.889	0.0493914
$514$	6784.24	0.582179
$515$	10832.8	0.926892
$516$	8087.73	0.690005
$517$	4591.63	0.390599
$518$	0	0
$519$	−8098.55	−0.684946
$520$	−15160.4	−1.27852
$521$	17339.3	1.45806	0.729030	−	0.684481i	$-0.239971\pi$
$521$	17339.3	1.45806	0.729030	+	0.684481i	$0.239971\pi$
$522$	−3575.64	−0.299811
$523$	11310.6	0.945660	0.472830	−	0.881154i	$-0.343232\pi$
$523$	11310.6	0.945660	0.472830	+	0.881154i	$0.343232\pi$
$524$	−29442.3	−2.45457
$525$	0	0
$526$	616.936	0.0511401
$527$	−15282.7	−1.26324
$528$	−185.597	−0.0152975
$529$	2092.66	0.171995
$530$	24806.9	2.03310
$531$	3222.62	0.263371
$532$	0	0
$533$	3316.55	0.269523
$534$	18752.8	1.51969
$535$	23036.8	1.86162
$536$	−2259.02	−0.182043
$537$	−3985.74	−0.320293
$538$	3357.93	0.269090
$539$	0	0
$540$	4487.49	0.357613
$541$	−1483.01	−0.117855	−0.0589275	−	0.998262i	$-0.518768\pi$
$541$	−1483.01	−0.117855	−0.0589275	+	0.998262i	$0.518768\pi$
$542$	−19801.7	−1.56929
$543$	1604.83	0.126832
$544$	−15466.0	−1.21893
$545$	8526.08	0.670123
$546$	0	0
$547$	3512.31	0.274544	0.137272	−	0.990533i	$-0.456167\pi$
$547$	3512.31	0.274544	0.137272	+	0.990533i	$0.456167\pi$
$548$	32977.2	2.57065
$549$	−4637.67	−0.360530
$550$	−2414.39	−0.187181
$551$	1859.46	0.143767
$552$	−7523.07	−0.580078
$553$	0	0
$554$	−26248.0	−2.01294
$555$	3619.80	0.276850
$556$	36416.6	2.77771
$557$	14069.1	1.07025	0.535123	−	0.844774i	$-0.320265\pi$
$557$	14069.1	1.07025	0.535123	+	0.844774i	$0.320265\pi$
$558$	−7816.78	−0.593030
$559$	−11710.0	−0.886012
$560$	0	0
$561$	2637.04	0.198460
$562$	−19831.5	−1.48851
$563$	−12982.7	−0.971860	−0.485930	−	0.873998i	$-0.661519\pi$
$563$	−12982.7	−0.971860	−0.485930	+	0.873998i	$0.661519\pi$
$564$	15808.7	1.18026
$565$	−15160.1	−1.12884
$566$	37666.9	2.79727
$567$	0	0
$568$	−14947.7	−1.10421
$569$	−1668.21	−0.122909	−0.0614545	−	0.998110i	$-0.519574\pi$
$569$	−1668.21	−0.122909	−0.0614545	+	0.998110i	$0.519574\pi$
$570$	−3812.51	−0.280155
$571$	−22037.1	−1.61510	−0.807552	−	0.589797i	$-0.799208\pi$
$571$	−22037.1	−1.61510	−0.807552	+	0.589797i	$0.799208\pi$
$572$	−7614.62	−0.556614
$573$	10368.6	0.755942
$574$	0	0
$575$	5771.39	0.418581
$576$	−7505.57	−0.542938
$577$	−22415.8	−1.61730	−0.808649	−	0.588292i	$-0.799801\pi$
$577$	−22415.8	−1.61730	−0.808649	+	0.588292i	$0.799801\pi$
$578$	6687.96	0.481284
$579$	9318.07	0.668818
$580$	14540.0	1.04093
$581$	0	0
$582$	−7285.53	−0.518892
$583$	4563.94	0.324218
$584$	−2767.23	−0.196077
$585$	−6497.32	−0.459199
$586$	−11057.4	−0.779485
$587$	24252.2	1.70527	0.852636	−	0.522506i	$-0.175003\pi$
$587$	24252.2	1.70527	0.852636	+	0.522506i	$0.175003\pi$
$588$	0	0
$589$	4065.01	0.284373
$590$	−21408.8	−1.49388
$591$	−14659.5	−1.02032
$592$	−515.444	−0.0357848
$593$	−18666.9	−1.29268	−0.646340	−	0.763050i	$-0.723701\pi$
$593$	−18666.9	−1.29268	−0.646340	+	0.763050i	$0.723701\pi$
$594$	1348.79	0.0931676
$595$	0	0
$596$	36730.5	2.52439
$597$	9959.23	0.682754
$598$	29736.9	2.03350
$599$	−11400.1	−0.777620	−0.388810	−	0.921318i	$-0.627114\pi$
$599$	−11400.1	−0.777620	−0.388810	+	0.921318i	$0.627114\pi$
$600$	−3044.86	−0.207176
$601$	9722.11	0.659856	0.329928	−	0.944006i	$-0.392976\pi$
$601$	9722.11	0.659856	0.329928	+	0.944006i	$0.392976\pi$
$602$	0	0
$603$	−968.152	−0.0653834
$604$	−29377.1	−1.97903
$605$	−1593.03	−0.107051
$606$	22071.2	1.47951
$607$	20616.5	1.37858	0.689291	−	0.724484i	$-0.257922\pi$
$607$	20616.5	1.37858	0.689291	+	0.724484i	$0.257922\pi$
$608$	4113.75	0.274399
$609$	0	0
$610$	30809.4	2.04498
$611$	−22889.0	−1.51553
$612$	9079.20	0.599681
$613$	−363.351	−0.0239406	−0.0119703	−	0.999928i	$-0.503810\pi$
$613$	−363.351	−0.0239406	−0.0119703	+	0.999928i	$0.503810\pi$
$614$	27122.6	1.78270
$615$	2388.87	0.156632
$616$	0	0
$617$	−19599.7	−1.27886	−0.639429	−	0.768850i	$-0.720829\pi$
$617$	−19599.7	−1.27886	−0.639429	+	0.768850i	$0.720829\pi$
$618$	11210.1	0.729673
$619$	−10178.6	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$619$	−10178.6	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$620$	31786.1	2.05897
$621$	−3224.17	−0.208344
$622$	37596.5	2.42361
$623$	0	0
$624$	925.191	0.0593546
$625$	−19330.5	−1.23715
$626$	6297.08	0.402048
$627$	−701.420	−0.0446762
$628$	11709.8	0.744064
$629$	7323.66	0.464250
$630$	0	0
$631$	−13503.9	−0.851949	−0.425975	−	0.904735i	$-0.640069\pi$
$631$	−13503.9	−0.851949	−0.425975	+	0.904735i	$0.640069\pi$
$632$	220.428	0.0138737
$633$	−12787.5	−0.802937
$634$	51.0205	0.00319603
$635$	−3661.93	−0.228849
$636$	15713.4	0.979681
$637$	0	0
$638$	4370.22	0.271189
$639$	−6406.14	−0.396593
$640$	29477.1	1.82060
$641$	29389.5	1.81095	0.905473	−	0.424404i	$-0.139517\pi$
$641$	29389.5	1.81095	0.905473	+	0.424404i	$0.139517\pi$
$642$	23839.3	1.46551
$643$	11632.9	0.713461	0.356730	−	0.934207i	$-0.383892\pi$
$643$	11632.9	0.713461	0.356730	+	0.934207i	$0.383892\pi$
$644$	0	0
$645$	−8434.57	−0.514901
$646$	−7713.56	−0.469793
$647$	7522.70	0.457106	0.228553	−	0.973531i	$-0.426601\pi$
$647$	7522.70	0.457106	0.228553	+	0.973531i	$0.426601\pi$
$648$	1701.00	0.103120
$649$	−3938.76	−0.238228
$650$	12035.6	0.726270
$651$	0	0
$652$	−37447.1	−2.24929
$653$	31890.8	1.91115	0.955577	−	0.294743i	$-0.0952341\pi$
$653$	31890.8	1.91115	0.955577	+	0.294743i	$0.0952341\pi$
$654$	8823.09	0.527538
$655$	30704.9	1.83166
$656$	−340.164	−0.0202457
$657$	−1185.96	−0.0704239
$658$	0	0
$659$	20234.0	1.19606	0.598029	−	0.801474i	$-0.295951\pi$
$659$	20234.0	1.19606	0.598029	+	0.801474i	$0.295951\pi$
$660$	−5484.71	−0.323473
$661$	−16493.0	−0.970505	−0.485252	−	0.874374i	$-0.661272\pi$
$661$	−16493.0	−0.970505	−0.485252	+	0.874374i	$0.661272\pi$
$662$	−20510.2	−1.20416
$663$	−13145.5	−0.770030
$664$	12985.4	0.758932
$665$	0	0
$666$	3745.89	0.217944
$667$	−10446.7	−0.606441
$668$	−48683.1	−2.81977
$669$	8298.49	0.479579
$670$	6431.72	0.370864
$671$	5668.26	0.326112
$672$	0	0
$673$	7338.87	0.420346	0.210173	−	0.977664i	$-0.432597\pi$
$673$	7338.87	0.420346	0.210173	+	0.977664i	$0.432597\pi$
$674$	−3073.52	−0.175649
$675$	−1304.94	−0.0744105
$676$	10223.3	0.581661
$677$	−10296.8	−0.584549	−0.292274	−	0.956334i	$-0.594412\pi$
$677$	−10296.8	−0.584549	−0.292274	+	0.956334i	$0.594412\pi$
$678$	−15688.2	−0.888648
$679$	0	0
$680$	−22093.3	−1.24594
$681$	1637.17	0.0921241
$682$	9553.85	0.536416
$683$	19884.5	1.11400	0.556998	−	0.830514i	$-0.311953\pi$
$683$	19884.5	1.11400	0.556998	+	0.830514i	$0.311953\pi$
$684$	−2414.95	−0.134997
$685$	−34391.4	−1.91829
$686$	0	0
$687$	9592.45	0.532714
$688$	1201.05	0.0665545
$689$	−22751.0	−1.25797
$690$	21419.1	1.18176
$691$	−26756.6	−1.47304	−0.736519	−	0.676417i	$-0.763532\pi$
$691$	−26756.6	−1.47304	−0.736519	+	0.676417i	$0.763532\pi$
$692$	34079.1	1.87210
$693$	0	0
$694$	34602.7	1.89265
$695$	−37978.3	−2.07280
$696$	5511.42	0.300158
$697$	4833.21	0.262656
$698$	−14434.1	−0.782720
$699$	−12611.3	−0.682408
$700$	0	0
$701$	−7556.29	−0.407128	−0.203564	−	0.979062i	$-0.565253\pi$
$701$	−7556.29	−0.407128	−0.203564	+	0.979062i	$0.565253\pi$
$702$	−6723.65	−0.361493
$703$	−1948.00	−0.104510
$704$	9173.47	0.491105
$705$	−16486.7	−0.880744
$706$	−8066.03	−0.429985
$707$	0	0
$708$	−13560.9	−0.719847
$709$	18371.1	0.973117	0.486558	−	0.873648i	$-0.338252\pi$
$709$	18371.1	0.973117	0.486558	+	0.873648i	$0.338252\pi$
$710$	42557.9	2.24953
$711$	94.4692	0.00498294
$712$	−28905.3	−1.52145
$713$	−22837.7	−1.19955
$714$	0	0
$715$	7941.17	0.415361
$716$	16772.2	0.875427
$717$	−17162.2	−0.893911
$718$	−51886.6	−2.69692
$719$	−24627.6	−1.27740	−0.638702	−	0.769454i	$-0.720528\pi$
$719$	−24627.6	−1.27740	−0.638702	+	0.769454i	$0.720528\pi$
$720$	666.403	0.0344936
$721$	0	0
$722$	−29097.6	−1.49986
$723$	15575.3	0.801179
$724$	−6753.20	−0.346659
$725$	−4228.14	−0.216592
$726$	−1648.52	−0.0842732
$727$	−19040.6	−0.971360	−0.485680	−	0.874137i	$-0.661428\pi$
$727$	−19040.6	−0.971360	−0.485680	+	0.874137i	$0.661428\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	7878.65	0.399455
$731$	−17065.0	−0.863437
$732$	19515.5	0.985403
$733$	35623.2	1.79505	0.897526	−	0.440961i	$-0.145362\pi$
$733$	35623.2	1.79505	0.897526	+	0.440961i	$0.145362\pi$
$734$	6000.69	0.301757
$735$	0	0
$736$	−23111.5	−1.15748
$737$	1183.30	0.0591415
$738$	2472.08	0.123304
$739$	7248.54	0.360814	0.180407	−	0.983592i	$-0.442258\pi$
$739$	7248.54	0.360814	0.180407	+	0.983592i	$0.442258\pi$
$740$	−15232.3	−0.756689
$741$	3496.54	0.173345
$742$	0	0
$743$	−29033.2	−1.43354	−0.716772	−	0.697307i	$-0.754381\pi$
$743$	−29033.2	−1.43354	−0.716772	+	0.697307i	$0.754381\pi$
$744$	12048.6	0.593716
$745$	−38305.6	−1.88377
$746$	−9133.46	−0.448257
$747$	5565.17	0.272582
$748$	−11096.8	−0.542432
$749$	0	0
$750$	−13752.0	−0.669538
$751$	−32062.0	−1.55787	−0.778933	−	0.627107i	$-0.784239\pi$
$751$	−32062.0	−1.55787	−0.778933	+	0.627107i	$0.784239\pi$
$752$	2347.63	0.113842
$753$	2543.83	0.123111
$754$	−21785.4	−1.05222
$755$	30636.9	1.47681
$756$	0	0
$757$	16132.9	0.774584	0.387292	−	0.921957i	$-0.373411\pi$
$757$	16132.9	0.774584	0.387292	+	0.921957i	$0.373411\pi$
$758$	−6249.87	−0.299479
$759$	3940.66	0.188454
$760$	5876.54	0.280479
$761$	37232.9	1.77358	0.886789	−	0.462175i	$-0.152931\pi$
$761$	37232.9	1.77358	0.886789	+	0.462175i	$0.152931\pi$
$762$	−3789.49	−0.180156
$763$	0	0
$764$	−43631.6	−2.06615
$765$	−9468.56	−0.447499
$766$	−310.825	−0.0146613
$767$	19634.5	0.924331
$768$	10489.1	0.492829
$769$	−36009.7	−1.68861	−0.844305	−	0.535862i	$-0.819987\pi$
$769$	−36009.7	−1.68861	−0.844305	+	0.535862i	$0.819987\pi$
$770$	0	0
$771$	−4481.61	−0.209340
$772$	−39210.9	−1.82802
$773$	−20156.0	−0.937856	−0.468928	−	0.883236i	$-0.655360\pi$
$773$	−20156.0	−0.937856	−0.468928	+	0.883236i	$0.655360\pi$
$774$	−8728.39	−0.405343
$775$	−9243.23	−0.428421
$776$	11229.8	0.519492
$777$	0	0
$778$	−8157.83	−0.375929
$779$	−1285.57	−0.0591276
$780$	27341.0	1.25509
$781$	7829.73	0.358732
$782$	43335.7	1.98169
$783$	2362.04	0.107806
$784$	0	0
$785$	−12212.0	−0.555241
$786$	31774.5	1.44193
$787$	19711.9	0.892823	0.446411	−	0.894828i	$-0.352702\pi$
$787$	19711.9	0.892823	0.446411	+	0.894828i	$0.352702\pi$
$788$	61687.7	2.78875
$789$	−407.543	−0.0183890
$790$	−627.587	−0.0282640
$791$	0	0
$792$	−2079.00	−0.0932753
$793$	−28256.0	−1.26532
$794$	−18963.0	−0.847569
$795$	−16387.3	−0.731065
$796$	−41908.9	−1.86611
$797$	8274.02	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$797$	8274.02	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$798$	0	0
$799$	−33356.2	−1.47692
$800$	−9354.06	−0.413395
$801$	−12388.0	−0.546451
$802$	−26045.9	−1.14678
$803$	1449.50	0.0637009
$804$	4074.03	0.178706
$805$	0	0
$806$	−47625.5	−2.08131
$807$	−2218.22	−0.0967596
$808$	−34020.2	−1.48122
$809$	15039.4	0.653593	0.326797	−	0.945095i	$-0.394031\pi$
$809$	15039.4	0.653593	0.326797	+	0.945095i	$0.394031\pi$
$810$	−4842.96	−0.210079
$811$	31217.1	1.35164	0.675820	−	0.737067i	$-0.263790\pi$
$811$	31217.1	1.35164	0.675820	+	0.737067i	$0.263790\pi$
$812$	0	0
$813$	13080.9	0.564288
$814$	−4578.31	−0.197137
$815$	39053.0	1.67849
$816$	1348.28	0.0578423
$817$	4539.08	0.194372
$818$	−20673.6	−0.883661
$819$	0	0
$820$	−10052.5	−0.428106
$821$	6151.17	0.261483	0.130741	−	0.991417i	$-0.458264\pi$
$821$	6151.17	0.261483	0.130741	+	0.991417i	$0.458264\pi$
$822$	−35589.4	−1.51012
$823$	−34084.1	−1.44362	−0.721808	−	0.692093i	$-0.756689\pi$
$823$	−34084.1	−1.44362	−0.721808	+	0.692093i	$0.756689\pi$
$824$	−17279.1	−0.730517
$825$	1594.92	0.0673068
$826$	0	0
$827$	19451.9	0.817907	0.408953	−	0.912555i	$-0.365894\pi$
$827$	19451.9	0.817907	0.408953	+	0.912555i	$0.365894\pi$
$828$	13567.5	0.569447
$829$	−26492.7	−1.10993	−0.554963	−	0.831875i	$-0.687268\pi$
$829$	−26492.7	−1.10993	−0.554963	+	0.831875i	$0.687268\pi$
$830$	−36971.1	−1.54613
$831$	17339.2	0.723816
$832$	−45729.3	−1.90550
$833$	0	0
$834$	−39301.3	−1.63176
$835$	50770.9	2.10419
$836$	2951.61	0.122109
$837$	5163.70	0.213242
$838$	−57953.1	−2.38897
$839$	8978.20	0.369442	0.184721	−	0.982791i	$-0.440862\pi$
$839$	8978.20	0.369442	0.184721	+	0.982791i	$0.440862\pi$
$840$	0	0
$841$	−16735.7	−0.686201
$842$	−12892.7	−0.527685
$843$	13100.5	0.535238
$844$	53810.6	2.19459
$845$	−10661.7	−0.434052
$846$	−17061.0	−0.693344
$847$	0	0
$848$	2333.48	0.0944952
$849$	−24882.4	−1.00585
$850$	17539.5	0.707765
$851$	10944.1	0.440844
$852$	26957.3	1.08397
$853$	27980.8	1.12315	0.561574	−	0.827427i	$-0.310196\pi$
$853$	27980.8	1.12315	0.561574	+	0.827427i	$0.310196\pi$
$854$	0	0
$855$	2518.52	0.100738
$856$	−36745.4	−1.46721
$857$	4175.08	0.166416	0.0832078	−	0.996532i	$-0.473483\pi$
$857$	4175.08	0.166416	0.0832078	+	0.996532i	$0.473483\pi$
$858$	8217.80	0.326983
$859$	−38793.4	−1.54088	−0.770439	−	0.637514i	$-0.779963\pi$
$859$	−38793.4	−1.54088	−0.770439	+	0.637514i	$0.779963\pi$
$860$	35493.1	1.40733
$861$	0	0
$862$	−58221.2	−2.30049
$863$	−19429.6	−0.766385	−0.383193	−	0.923668i	$-0.625175\pi$
$863$	−19429.6	−0.766385	−0.383193	+	0.923668i	$0.625175\pi$
$864$	5225.62	0.205763
$865$	−35540.6	−1.39701
$866$	57089.1	2.24015
$867$	−4418.01	−0.173061
$868$	0	0
$869$	−115.462	−0.00450724
$870$	−15691.7	−0.611493
$871$	−5898.68	−0.229471
$872$	−13599.7	−0.528148
$873$	4812.77	0.186584
$874$	−11526.7	−0.446107
$875$	0	0
$876$	4990.56	0.192483
$877$	−17547.9	−0.675656	−0.337828	−	0.941208i	$-0.609692\pi$
$877$	−17547.9	−0.675656	−0.337828	+	0.941208i	$0.609692\pi$
$878$	52633.8	2.02312
$879$	7304.45	0.280288
$880$	−814.493	−0.0312006
$881$	44948.7	1.71891	0.859455	−	0.511212i	$-0.170803\pi$
$881$	44948.7	1.71891	0.859455	+	0.511212i	$0.170803\pi$
$882$	0	0
$883$	−37819.3	−1.44136	−0.720680	−	0.693268i	$-0.756170\pi$
$883$	−37819.3	−1.44136	−0.720680	+	0.693268i	$0.756170\pi$
$884$	55317.0	2.10465
$885$	14142.5	0.537169
$886$	69474.4	2.63436
$887$	7203.82	0.272695	0.136348	−	0.990661i	$-0.456464\pi$
$887$	7203.82	0.272695	0.136348	+	0.990661i	$0.456464\pi$
$888$	−5773.85	−0.218196
$889$	0	0
$890$	82297.0	3.09955
$891$	−891.000	−0.0335013
$892$	−34920.5	−1.31079
$893$	8872.33	0.332476
$894$	−39640.0	−1.48295
$895$	−17491.5	−0.653268
$896$	0	0
$897$	−19644.0	−0.731208
$898$	15306.2	0.568790
$899$	16731.0	0.620699
$900$	5491.24	0.203379
$901$	−33155.1	−1.22592
$902$	−3021.43	−0.111533
$903$	0	0
$904$	24181.6	0.889676
$905$	7042.81	0.258686
$906$	31704.2	1.16258
$907$	9420.96	0.344893	0.172446	−	0.985019i	$-0.444833\pi$
$907$	9420.96	0.344893	0.172446	+	0.985019i	$0.444833\pi$
$908$	−6889.30	−0.251794
$909$	−14580.1	−0.532003
$910$	0	0
$911$	40045.0	1.45637	0.728184	−	0.685382i	$-0.240365\pi$
$911$	40045.0	1.45637	0.728184	+	0.685382i	$0.240365\pi$
$912$	−358.626	−0.0130211
$913$	−6801.87	−0.246560
$914$	−12180.5	−0.440805
$915$	−20352.5	−0.735335
$916$	−40365.5	−1.45602
$917$	0	0
$918$	−9798.40	−0.352282
$919$	−8881.69	−0.318803	−0.159402	−	0.987214i	$-0.550956\pi$
$919$	−8881.69	−0.318803	−0.159402	+	0.987214i	$0.550956\pi$
$920$	−33015.1	−1.18312
$921$	−17917.0	−0.641026
$922$	−5966.30	−0.213112
$923$	−39030.8	−1.39189
$924$	0	0
$925$	4429.46	0.157448
$926$	78743.7	2.79447
$927$	−7405.33	−0.262376
$928$	16931.6	0.598928
$929$	−39037.4	−1.37866	−0.689331	−	0.724447i	$-0.742095\pi$
$929$	−39037.4	−1.37866	−0.689331	+	0.724447i	$0.742095\pi$
$930$	−34304.0	−1.20954
$931$	0	0
$932$	53068.9	1.86516
$933$	−24836.0	−0.871482
$934$	40010.6	1.40170
$935$	11572.7	0.404778
$936$	10363.7	0.361911
$937$	−17549.7	−0.611870	−0.305935	−	0.952052i	$-0.598969\pi$
$937$	−17549.7	−0.611870	−0.305935	+	0.952052i	$0.598969\pi$
$938$	0	0
$939$	−4159.80	−0.144569
$940$	69376.8	2.40726
$941$	28176.8	0.976128	0.488064	−	0.872808i	$-0.337703\pi$
$941$	28176.8	0.976128	0.488064	+	0.872808i	$0.337703\pi$
$942$	−12637.4	−0.437100
$943$	7222.49	0.249413
$944$	−2013.83	−0.0694329
$945$	0	0
$946$	10668.0	0.366646
$947$	−11092.6	−0.380636	−0.190318	−	0.981722i	$-0.560952\pi$
$947$	−11092.6	−0.380636	−0.190318	+	0.981722i	$0.560952\pi$
$948$	−397.531	−0.0136194
$949$	−7225.69	−0.247161
$950$	−4665.28	−0.159328
$951$	−33.7038	−0.00114923
$952$	0	0
$953$	21883.0	0.743820	0.371910	−	0.928269i	$-0.378703\pi$
$953$	21883.0	0.743820	0.371910	+	0.928269i	$0.378703\pi$
$954$	−16958.1	−0.575513
$955$	45502.7	1.54181
$956$	72219.4	2.44324
$957$	−2886.93	−0.0975145
$958$	3219.60	0.108581
$959$	0	0
$960$	−32938.2	−1.10737
$961$	6784.91	0.227750
$962$	22822.7	0.764899
$963$	−15748.0	−0.526971
$964$	−65541.7	−2.18979
$965$	40892.4	1.36412
$966$	0	0
$967$	40235.4	1.33804	0.669019	−	0.743245i	$-0.266714\pi$
$967$	40235.4	1.33804	0.669019	+	0.743245i	$0.266714\pi$
$968$	2541.00	0.0843707
$969$	5095.52	0.168928
$970$	−31972.6	−1.05833
$971$	−44470.2	−1.46974	−0.734870	−	0.678208i	$-0.762757\pi$
$971$	−44470.2	−1.46974	−0.734870	+	0.678208i	$0.762757\pi$
$972$	−3067.67	−0.101230
$973$	0	0
$974$	55648.2	1.83068
$975$	−7950.62	−0.261153
$976$	2898.10	0.0950472
$977$	−55989.4	−1.83343	−0.916715	−	0.399542i	$-0.869169\pi$
$977$	−55989.4	−1.83343	−0.916715	+	0.399542i	$0.869169\pi$
$978$	40413.4	1.32135
$979$	15140.9	0.494284
$980$	0	0
$981$	−5828.46	−0.189693
$982$	−24656.8	−0.801254
$983$	5987.22	0.194265	0.0971326	−	0.995271i	$-0.469033\pi$
$983$	5987.22	0.194265	0.0971326	+	0.995271i	$0.469033\pi$
$984$	−3810.42	−0.123447
$985$	−64333.1	−2.08104
$986$	−31747.8	−1.02541
$987$	0	0
$988$	−14713.6	−0.473788
$989$	−25501.1	−0.819905
$990$	5919.18	0.190024
$991$	−1965.53	−0.0630043	−0.0315022	−	0.999504i	$-0.510029\pi$
$991$	−1965.53	−0.0630043	−0.0315022	+	0.999504i	$0.510029\pi$
$992$	37014.5	1.18469
$993$	13548.9	0.432991
$994$	0	0
$995$	43706.2	1.39254
$996$	−23418.5	−0.745024
$997$	10802.9	0.343161	0.171581	−	0.985170i	$-0.445113\pi$
$997$	10802.9	0.343161	0.171581	+	0.985170i	$0.445113\pi$
$998$	−65993.7	−2.09318
$999$	−2474.51	−0.0783683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.4.a.l.1.2		2
7.6	odd	2		231.4.a.i.1.2	✓	2
21.20	even	2		693.4.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.i.1.2	✓	2	7.6	odd	2
693.4.a.h.1.1		2	21.20	even	2
1617.4.a.l.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q+4.54138 q^{2} -3.00000 q^{3} +12.6241 q^{4} -13.1655 q^{5} -13.6241 q^{6} +21.0000 q^{8} +9.00000 q^{9} -59.7897 q^{10} -11.0000 q^{11} -37.8724 q^{12} +54.8345 q^{13} +39.4966 q^{15} -5.62414 q^{16} +79.9104 q^{17} +40.8724 q^{18} -21.2551 q^{19} -166.203 q^{20} -49.9552 q^{22} +119.414 q^{23} -63.0000 q^{24} +48.3311 q^{25} +249.024 q^{26} -27.0000 q^{27} -87.4829 q^{29} +179.369 q^{30} -191.248 q^{31} -193.541 q^{32} +33.0000 q^{33} +362.904 q^{34} +113.617 q^{36} +91.6484 q^{37} -96.5277 q^{38} -164.503 q^{39} -276.476 q^{40} +60.4829 q^{41} -213.552 q^{43} -138.866 q^{44} -118.490 q^{45} +542.304 q^{46} -417.421 q^{47} +16.8724 q^{48} +219.490 q^{50} -239.731 q^{51} +692.238 q^{52} -414.904 q^{53} -122.617 q^{54} +144.821 q^{55} +63.7654 q^{57} -397.293 q^{58} +358.069 q^{59} +498.610 q^{60} -515.297 q^{61} -868.531 q^{62} -833.952 q^{64} -721.925 q^{65} +149.866 q^{66} -107.572 q^{67} +1008.80 q^{68} -358.241 q^{69} -711.793 q^{71} +189.000 q^{72} -131.773 q^{73} +416.210 q^{74} -144.993 q^{75} -268.328 q^{76} -747.073 q^{78} +10.4966 q^{79} +74.0448 q^{80} +81.0000 q^{81} +274.676 q^{82} +618.352 q^{83} -1052.06 q^{85} -969.821 q^{86} +262.449 q^{87} -231.000 q^{88} -1376.44 q^{89} -538.107 q^{90} +1507.50 q^{92} +573.745 q^{93} -1895.67 q^{94} +279.835 q^{95} +580.624 q^{96} +534.752 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 6 q^{3} + 7 q^{4} - 2 q^{5} - 9 q^{6} + 42 q^{8} + 18 q^{9} - 77 q^{10} - 22 q^{11} - 21 q^{12} + 134 q^{13} + 6 q^{15} + 7 q^{16} + 26 q^{17} + 27 q^{18} - 152 q^{19} - 229 q^{20} - 33 q^{22}+ \cdots - 198 q^{99}+O(q^{100}) \) 2 * q + 3 * q^2 - 6 * q^3 + 7 * q^4 - 2 * q^5 - 9 * q^6 + 42 * q^8 + 18 * q^9 - 77 * q^10 - 22 * q^11 - 21 * q^12 + 134 * q^13 + 6 * q^15 + 7 * q^16 + 26 * q^17 + 27 * q^18 - 152 * q^19 - 229 * q^20 - 33 * q^22 + 178 * q^23 - 126 * q^24 + 48 * q^25 + 127 * q^26 - 54 * q^27 + 190 * q^29 + 231 * q^30 - 346 * q^31 - 381 * q^32 + 66 * q^33 + 446 * q^34 + 63 * q^36 - 206 * q^37 + 105 * q^38 - 402 * q^39 - 42 * q^40 - 244 * q^41 + 242 * q^43 - 77 * q^44 - 18 * q^45 + 452 * q^46 - 920 * q^47 - 21 * q^48 + 220 * q^50 - 78 * q^51 + 247 * q^52 - 550 * q^53 - 81 * q^54 + 22 * q^55 + 456 * q^57 - 825 * q^58 + 412 * q^59 + 687 * q^60 - 252 * q^61 - 630 * q^62 - 646 * q^64 + 162 * q^65 + 99 * q^66 + 16 * q^67 + 1312 * q^68 - 534 * q^69 - 572 * q^71 + 378 * q^72 + 1026 * q^73 + 875 * q^74 - 144 * q^75 + 467 * q^76 - 381 * q^78 - 52 * q^79 + 215 * q^80 + 162 * q^81 + 744 * q^82 - 138 * q^83 - 1654 * q^85 - 1672 * q^86 - 570 * q^87 - 462 * q^88 - 1512 * q^89 - 693 * q^90 + 1178 * q^92 + 1038 * q^93 - 1121 * q^94 - 1180 * q^95 + 1143 * q^96 + 1106 * q^97 - 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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