LMFDB - Embedded newform 975.6.a.u.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,6,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("975.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	975.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:	$156.374224318$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 286 x^{9} + 442 x^{8} + 28715 x^{7} - 29138 x^{6} - 1208172 x^{5} + 509768 x^{4} + \cdots + 55036800 $ x^11 - 2x^10 - 286x^9 + 442x^8 + 28715x^7 - 29138x^6 - 1208172x^5 + 509768x^4 + 20144528x^3 - 3093984x^2 - 98068800x + 55036800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3\cdot 5^{3}\cdot 11 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.592612$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.592612 q^{2} -9.00000 q^{3} -31.6488 q^{4} -5.33351 q^{6} +209.544 q^{7} -37.7190 q^{8} +81.0000 q^{9} -237.848 q^{11} +284.839 q^{12} +169.000 q^{13} +124.178 q^{14} +990.409 q^{16} -395.630 q^{17} +48.0016 q^{18} +76.9152 q^{19} -1885.89 q^{21} -140.951 q^{22} -1225.63 q^{23} +339.471 q^{24} +100.151 q^{26} -729.000 q^{27} -6631.81 q^{28} -7424.29 q^{29} -7897.86 q^{31} +1793.94 q^{32} +2140.63 q^{33} -234.455 q^{34} -2563.55 q^{36} +14219.0 q^{37} +45.5808 q^{38} -1521.00 q^{39} -6518.30 q^{41} -1117.60 q^{42} +16867.4 q^{43} +7527.59 q^{44} -726.323 q^{46} +28838.8 q^{47} -8913.68 q^{48} +27101.6 q^{49} +3560.67 q^{51} -5348.65 q^{52} -27895.5 q^{53} -432.014 q^{54} -7903.79 q^{56} -692.237 q^{57} -4399.72 q^{58} +9602.62 q^{59} +1884.63 q^{61} -4680.36 q^{62} +16973.1 q^{63} -30630.0 q^{64} +1268.56 q^{66} +1281.38 q^{67} +12521.2 q^{68} +11030.7 q^{69} -65168.1 q^{71} -3055.24 q^{72} +80657.5 q^{73} +8426.34 q^{74} -2434.27 q^{76} -49839.5 q^{77} -901.362 q^{78} +48672.5 q^{79} +6561.00 q^{81} -3862.82 q^{82} -77761.1 q^{83} +59686.3 q^{84} +9995.84 q^{86} +66818.6 q^{87} +8971.38 q^{88} +103091. q^{89} +35412.9 q^{91} +38789.8 q^{92} +71080.7 q^{93} +17090.2 q^{94} -16145.4 q^{96} -101727. q^{97} +16060.7 q^{98} -19265.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} - 99 q^{3} + 224 q^{4} - 18 q^{6} - 55 q^{7} + 270 q^{8} + 891 q^{9} - 125 q^{11} - 2016 q^{12} + 1859 q^{13} - 1311 q^{14} + 5756 q^{16} - 4507 q^{17} + 162 q^{18} + 142 q^{19} + 495 q^{21}+ \cdots - 10125 q^{99}+O(q^{100}) $ 11 * q + 2 * q^2 - 99 * q^3 + 224 * q^4 - 18 * q^6 - 55 * q^7 + 270 * q^8 + 891 * q^9 - 125 * q^11 - 2016 * q^12 + 1859 * q^13 - 1311 * q^14 + 5756 * q^16 - 4507 * q^17 + 162 * q^18 + 142 * q^19 + 495 * q^21 - 5975 * q^22 - 4672 * q^23 - 2430 * q^24 + 338 * q^26 - 8019 * q^27 - 4149 * q^28 + 5197 * q^29 - 277 * q^31 + 2272 * q^32 + 1125 * q^33 - 21225 * q^34 + 18144 * q^36 - 1696 * q^37 - 27485 * q^38 - 16731 * q^39 + 2040 * q^41 + 11799 * q^42 - 31436 * q^43 - 39883 * q^44 + 67530 * q^46 + 15499 * q^47 - 51804 * q^48 + 29212 * q^49 + 40563 * q^51 + 37856 * q^52 - 119577 * q^53 - 1458 * q^54 + 1677 * q^56 - 1278 * q^57 - 91154 * q^58 + 17115 * q^59 + 26273 * q^61 - 45211 * q^62 - 4455 * q^63 + 137480 * q^64 + 53775 * q^66 + 26313 * q^67 - 191297 * q^68 + 42048 * q^69 - 105756 * q^71 + 21870 * q^72 - 103230 * q^73 - 6501 * q^74 + 124495 * q^76 - 104415 * q^77 - 3042 * q^78 + 210732 * q^79 + 72171 * q^81 - 60283 * q^82 - 144247 * q^83 + 37341 * q^84 - 42260 * q^86 - 46773 * q^87 - 415403 * q^88 + 117088 * q^89 - 9295 * q^91 - 70700 * q^92 + 2493 * q^93 + 383414 * q^94 - 20448 * q^96 - 112934 * q^97 - 159305 * q^98 - 10125 * 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 $

$2$	0.592612	0.104760	0.0523800	−	0.998627i	$-0.483319\pi$
$2$	0.592612	0.104760	0.0523800	+	0.998627i	$0.483319\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−31.6488	−0.989025
$5$	0	0
$5$	0	0
$6$	−5.33351	−0.0604832
$7$	209.544	1.61633	0.808165	−	0.588957i	$-0.200461\pi$
$7$	209.544	1.61633	0.808165	+	0.588957i	$0.200461\pi$
$8$	−37.7190	−0.208370
$9$	81.0000	0.333333
$10$	0	0
$11$	−237.848	−0.592675	−0.296338	−	0.955083i	$-0.595765\pi$
$11$	−237.848	−0.592675	−0.296338	+	0.955083i	$0.595765\pi$
$12$	284.839	0.571014
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	124.178	0.169327
$15$	0	0
$16$	990.409	0.967197
$17$	−395.630	−0.332022	−0.166011	−	0.986124i	$-0.553089\pi$
$17$	−395.630	−0.332022	−0.166011	+	0.986124i	$0.553089\pi$
$18$	48.0016	0.0349200
$19$	76.9152	0.0488797	0.0244398	−	0.999701i	$-0.492220\pi$
$19$	76.9152	0.0488797	0.0244398	+	0.999701i	$0.492220\pi$
$20$	0	0
$21$	−1885.89	−0.933188
$22$	−140.951	−0.0620886
$23$	−1225.63	−0.483103	−0.241552	−	0.970388i	$-0.577656\pi$
$23$	−1225.63	−0.483103	−0.241552	+	0.970388i	$0.577656\pi$
$24$	339.471	0.120303
$25$	0	0
$26$	100.151	0.0290552
$27$	−729.000	−0.192450
$28$	−6631.81	−1.59859
$29$	−7424.29	−1.63930	−0.819652	−	0.572861i	$-0.805833\pi$
$29$	−7424.29	−1.63930	−0.819652	+	0.572861i	$0.805833\pi$
$30$	0	0
$31$	−7897.86	−1.47606	−0.738032	−	0.674766i	$-0.764245\pi$
$31$	−7897.86	−1.47606	−0.738032	+	0.674766i	$0.764245\pi$
$32$	1793.94	0.309694
$33$	2140.63	0.342181
$34$	−234.455	−0.0347826
$35$	0	0
$36$	−2563.55	−0.329675
$37$	14219.0	1.70751	0.853757	−	0.520672i	$-0.174318\pi$
$37$	14219.0	1.70751	0.853757	+	0.520672i	$0.174318\pi$
$38$	45.5808	0.00512063
$39$	−1521.00	−0.160128
$40$	0	0
$41$	−6518.30	−0.605584	−0.302792	−	0.953057i	$-0.597919\pi$
$41$	−6518.30	−0.605584	−0.302792	+	0.953057i	$0.597919\pi$
$42$	−1117.60	−0.0977607
$43$	16867.4	1.39116	0.695581	−	0.718447i	$-0.255147\pi$
$43$	16867.4	1.39116	0.695581	+	0.718447i	$0.255147\pi$
$44$	7527.59	0.586171
$45$	0	0
$46$	−726.323	−0.0506099
$47$	28838.8	1.90429	0.952144	−	0.305649i	$-0.0988735\pi$
$47$	28838.8	1.90429	0.952144	+	0.305649i	$0.0988735\pi$
$48$	−8913.68	−0.558411
$49$	27101.6	1.61252
$50$	0	0
$51$	3560.67	0.191693
$52$	−5348.65	−0.274306
$53$	−27895.5	−1.36409	−0.682046	−	0.731309i	$-0.738910\pi$
$53$	−27895.5	−1.36409	−0.682046	+	0.731309i	$0.738910\pi$
$54$	−432.014	−0.0201611
$55$	0	0
$56$	−7903.79	−0.336795
$57$	−692.237	−0.0282207
$58$	−4399.72	−0.171734
$59$	9602.62	0.359137	0.179568	−	0.983745i	$-0.442530\pi$
$59$	9602.62	0.359137	0.179568	+	0.983745i	$0.442530\pi$
$60$	0	0
$61$	1884.63	0.0648488	0.0324244	−	0.999474i	$-0.489677\pi$
$61$	1884.63	0.0648488	0.0324244	+	0.999474i	$0.489677\pi$
$62$	−4680.36	−0.154632
$63$	16973.1	0.538776
$64$	−30630.0	−0.934753
$65$	0	0
$66$	1268.56	0.0358469
$67$	1281.38	0.0348731	0.0174366	−	0.999848i	$-0.494449\pi$
$67$	1281.38	0.0348731	0.0174366	+	0.999848i	$0.494449\pi$
$68$	12521.2	0.328378
$69$	11030.7	0.278920
$70$	0	0
$71$	−65168.1	−1.53423	−0.767113	−	0.641512i	$-0.778307\pi$
$71$	−65168.1	−1.53423	−0.767113	+	0.641512i	$0.778307\pi$
$72$	−3055.24	−0.0694567
$73$	80657.5	1.77148	0.885742	−	0.464177i	$-0.153650\pi$
$73$	80657.5	1.77148	0.885742	+	0.464177i	$0.153650\pi$
$74$	8426.34	0.178879
$75$	0	0
$76$	−2434.27	−0.0483432
$77$	−49839.5	−0.957958
$78$	−901.362	−0.0167750
$79$	48672.5	0.877437	0.438719	−	0.898624i	$-0.355432\pi$
$79$	48672.5	0.877437	0.438719	+	0.898624i	$0.355432\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−3862.82	−0.0634410
$83$	−77761.1	−1.23899	−0.619494	−	0.785001i	$-0.712662\pi$
$83$	−77761.1	−1.23899	−0.619494	+	0.785001i	$0.712662\pi$
$84$	59686.3	0.922947
$85$	0	0
$86$	9995.84	0.145738
$87$	66818.6	0.946453
$88$	8971.38	0.123496
$89$	103091.	1.37957	0.689787	−	0.724013i	$-0.257704\pi$
$89$	103091.	1.37957	0.689787	+	0.724013i	$0.257704\pi$
$90$	0	0
$91$	35412.9	0.448289
$92$	38789.8	0.477801
$93$	71080.7	0.852206
$94$	17090.2	0.199493
$95$	0	0
$96$	−16145.4	−0.178802
$97$	−101727.	−1.09776	−0.548878	−	0.835902i	$-0.684945\pi$
$97$	−101727.	−1.09776	−0.548878	+	0.835902i	$0.684945\pi$
$98$	16060.7	0.168927
$99$	−19265.6	−0.197558
$100$	0	0
$101$	49647.1	0.484273	0.242137	−	0.970242i	$-0.422152\pi$
$101$	49647.1	0.484273	0.242137	+	0.970242i	$0.422152\pi$
$102$	2110.09	0.0200817
$103$	111189.	1.03269	0.516344	−	0.856381i	$-0.327293\pi$
$103$	111189.	1.03269	0.516344	+	0.856381i	$0.327293\pi$
$104$	−6374.52	−0.0577915
$105$	0	0
$106$	−16531.2	−0.142902
$107$	−140837.	−1.18920	−0.594602	−	0.804020i	$-0.702690\pi$
$107$	−140837.	−1.18920	−0.594602	+	0.804020i	$0.702690\pi$
$108$	23072.0	0.190338
$109$	65953.1	0.531702	0.265851	−	0.964014i	$-0.414347\pi$
$109$	65953.1	0.531702	0.265851	+	0.964014i	$0.414347\pi$
$110$	0	0
$111$	−127971.	−0.985834
$112$	207534.	1.56331
$113$	−187956.	−1.38471	−0.692357	−	0.721555i	$-0.743428\pi$
$113$	−187956.	−1.38471	−0.692357	+	0.721555i	$0.743428\pi$
$114$	−410.228	−0.00295640
$115$	0	0
$116$	234970.	1.62131
$117$	13689.0	0.0924500
$118$	5690.63	0.0376231
$119$	−82901.7	−0.536656
$120$	0	0
$121$	−104480.	−0.648736
$122$	1116.86	0.00679356
$123$	58664.7	0.349634
$124$	249958.	1.45986
$125$	0	0
$126$	10058.4	0.0564422
$127$	−119369.	−0.656720	−0.328360	−	0.944553i	$-0.606496\pi$
$127$	−119369.	−0.656720	−0.328360	+	0.944553i	$0.606496\pi$
$128$	−75557.7	−0.407618
$129$	−151807.	−0.803188
$130$	0	0
$131$	322045.	1.63960	0.819801	−	0.572648i	$-0.194084\pi$
$131$	322045.	1.63960	0.819801	+	0.572648i	$0.194084\pi$
$132$	−67748.3	−0.338426
$133$	16117.1	0.0790056
$134$	759.361	0.00365331
$135$	0	0
$136$	14922.8	0.0691834
$137$	264585.	1.20438	0.602191	−	0.798352i	$-0.294295\pi$
$137$	264585.	1.20438	0.602191	+	0.798352i	$0.294295\pi$
$138$	6536.91	0.0292196
$139$	−195552.	−0.858472	−0.429236	−	0.903192i	$-0.641217\pi$
$139$	−195552.	−0.858472	−0.429236	+	0.903192i	$0.641217\pi$
$140$	0	0
$141$	−259549.	−1.09944
$142$	−38619.4	−0.160725
$143$	−40196.2	−0.164379
$144$	80223.1	0.322399
$145$	0	0
$146$	47798.6	0.185581
$147$	−243915.	−0.930989
$148$	−450014.	−1.68877
$149$	59840.9	0.220817	0.110409	−	0.993886i	$-0.464784\pi$
$149$	59840.9	0.220817	0.110409	+	0.993886i	$0.464784\pi$
$150$	0	0
$151$	−147696.	−0.527139	−0.263570	−	0.964640i	$-0.584900\pi$
$151$	−147696.	−0.527139	−0.263570	+	0.964640i	$0.584900\pi$
$152$	−2901.17	−0.0101851
$153$	−32046.0	−0.110674
$154$	−29535.5	−0.100356
$155$	0	0
$156$	48137.8	0.158371
$157$	164713.	0.533308	0.266654	−	0.963792i	$-0.414082\pi$
$157$	164713.	0.533308	0.266654	+	0.963792i	$0.414082\pi$
$158$	28843.9	0.0919203
$159$	251059.	0.787559
$160$	0	0
$161$	−256823.	−0.780854
$162$	3888.13	0.0116400
$163$	−123359.	−0.363666	−0.181833	−	0.983329i	$-0.558203\pi$
$163$	−123359.	−0.363666	−0.181833	+	0.983329i	$0.558203\pi$
$164$	206296.	0.598938
$165$	0	0
$166$	−46082.1	−0.129796
$167$	−630892.	−1.75051	−0.875254	−	0.483663i	$-0.839306\pi$
$167$	−630892.	−1.75051	−0.875254	+	0.483663i	$0.839306\pi$
$168$	71134.1	0.194449
$169$	28561.0	0.0769231
$170$	0	0
$171$	6230.13	0.0162932
$172$	−533834.	−1.37589
$173$	−310137.	−0.787841	−0.393920	−	0.919145i	$-0.628881\pi$
$173$	−310137.	−0.787841	−0.393920	+	0.919145i	$0.628881\pi$
$174$	39597.5	0.0991504
$175$	0	0
$176$	−235566.	−0.573234
$177$	−86423.6	−0.207348
$178$	61092.8	0.144524
$179$	173768.	0.405357	0.202678	−	0.979245i	$-0.435035\pi$
$179$	173768.	0.405357	0.202678	+	0.979245i	$0.435035\pi$
$180$	0	0
$181$	40809.7	0.0925907	0.0462953	−	0.998928i	$-0.485258\pi$
$181$	40809.7	0.0925907	0.0462953	+	0.998928i	$0.485258\pi$
$182$	20986.1	0.0469627
$183$	−16961.7	−0.0374405
$184$	46229.6	0.100664
$185$	0	0
$186$	42123.3	0.0892771
$187$	94099.5	0.196781
$188$	−912714.	−1.88339
$189$	−152757.	−0.311063
$190$	0	0
$191$	−524380.	−1.04007	−0.520035	−	0.854145i	$-0.674081\pi$
$191$	−524380.	−1.04007	−0.520035	+	0.854145i	$0.674081\pi$
$192$	275670.	0.539680
$193$	−187971.	−0.363243	−0.181621	−	0.983369i	$-0.558135\pi$
$193$	−187971.	−0.363243	−0.181621	+	0.983369i	$0.558135\pi$
$194$	−60284.5	−0.115001
$195$	0	0
$196$	−857734.	−1.59482
$197$	−731510.	−1.34293	−0.671467	−	0.741034i	$-0.734336\pi$
$197$	−731510.	−1.34293	−0.671467	+	0.741034i	$0.734336\pi$
$198$	−11417.0	−0.0206962
$199$	−555001.	−0.993484	−0.496742	−	0.867898i	$-0.665470\pi$
$199$	−555001.	−0.993484	−0.496742	+	0.867898i	$0.665470\pi$
$200$	0	0
$201$	−11532.4	−0.0201340
$202$	29421.4	0.0507324
$203$	−1.55571e6	−2.64966
$204$	−112691.	−0.189589
$205$	0	0
$206$	65891.9	0.108184
$207$	−99276.1	−0.161034
$208$	167379.	0.268252
$209$	−18294.1	−0.0289698
$210$	0	0
$211$	−116540.	−0.180205	−0.0901026	−	0.995932i	$-0.528720\pi$
$211$	−116540.	−0.180205	−0.0901026	+	0.995932i	$0.528720\pi$
$212$	882858.	1.34912
$213$	586513.	0.885786
$214$	−83461.4	−0.124581
$215$	0	0
$216$	27497.2	0.0401009
$217$	−1.65495e6	−2.38581
$218$	39084.6	0.0557011
$219$	−725917.	−1.02277
$220$	0	0
$221$	−66861.4	−0.0920863
$222$	−75837.0	−0.103276
$223$	710939.	0.957349	0.478674	−	0.877992i	$-0.341117\pi$
$223$	710939.	0.957349	0.478674	+	0.877992i	$0.341117\pi$
$224$	375908.	0.500567
$225$	0	0
$226$	−111385.	−0.145063
$227$	666892.	0.858996	0.429498	−	0.903068i	$-0.358691\pi$
$227$	666892.	0.858996	0.429498	+	0.903068i	$0.358691\pi$
$228$	21908.5	0.0279110
$229$	−1.15830e6	−1.45960	−0.729799	−	0.683662i	$-0.760386\pi$
$229$	−1.15830e6	−1.45960	−0.729799	+	0.683662i	$0.760386\pi$
$230$	0	0
$231$	448555.	0.553078
$232$	280037.	0.341582
$233$	1.10558e6	1.33414	0.667071	−	0.744994i	$-0.267548\pi$
$233$	1.10558e6	1.33414	0.667071	+	0.744994i	$0.267548\pi$
$234$	8112.26	0.00968506
$235$	0	0
$236$	−303912.	−0.355195
$237$	−438053.	−0.506589
$238$	−49128.6	−0.0562201
$239$	417955.	0.473298	0.236649	−	0.971595i	$-0.423951\pi$
$239$	417955.	0.473298	0.236649	+	0.971595i	$0.423951\pi$
$240$	0	0
$241$	−812480.	−0.901094	−0.450547	−	0.892753i	$-0.648771\pi$
$241$	−812480.	−0.901094	−0.450547	+	0.892753i	$0.648771\pi$
$242$	−61915.8	−0.0679615
$243$	−59049.0	−0.0641500
$244$	−59646.4	−0.0641371
$245$	0	0
$246$	34765.4	0.0366277
$247$	12998.7	0.0135568
$248$	297900.	0.307568
$249$	699850.	0.715330
$250$	0	0
$251$	955016.	0.956812	0.478406	−	0.878139i	$-0.341215\pi$
$251$	955016.	0.956812	0.478406	+	0.878139i	$0.341215\pi$
$252$	−537177.	−0.532863
$253$	291513.	0.286323
$254$	−70739.2	−0.0687980
$255$	0	0
$256$	935383.	0.892051
$257$	−1.07522e6	−1.01546	−0.507730	−	0.861516i	$-0.669515\pi$
$257$	−1.07522e6	−1.01546	−0.507730	+	0.861516i	$0.669515\pi$
$258$	−89962.6	−0.0841419
$259$	2.97950e6	2.75990
$260$	0	0
$261$	−601367.	−0.546435
$262$	190848.	0.171765
$263$	179082.	0.159647	0.0798237	−	0.996809i	$-0.474564\pi$
$263$	179082.	0.159647	0.0798237	+	0.996809i	$0.474564\pi$
$264$	−80742.4	−0.0713004
$265$	0	0
$266$	9551.19	0.00827662
$267$	−927817.	−0.796497
$268$	−40554.2	−0.0344904
$269$	840441.	0.708152	0.354076	−	0.935217i	$-0.384795\pi$
$269$	840441.	0.708152	0.354076	+	0.935217i	$0.384795\pi$
$270$	0	0
$271$	−874424.	−0.723267	−0.361634	−	0.932320i	$-0.617781\pi$
$271$	−874424.	−0.723267	−0.361634	+	0.932320i	$0.617781\pi$
$272$	−391835.	−0.321130
$273$	−318716.	−0.258820
$274$	156796.	0.126171
$275$	0	0
$276$	−349108.	−0.275859
$277$	−2.13281e6	−1.67014	−0.835070	−	0.550143i	$-0.814573\pi$
$277$	−2.13281e6	−1.67014	−0.835070	+	0.550143i	$0.814573\pi$
$278$	−115887.	−0.0899335
$279$	−639727.	−0.492021
$280$	0	0
$281$	−2.21279e6	−1.67177	−0.835883	−	0.548908i	$-0.815044\pi$
$281$	−2.21279e6	−1.67177	−0.835883	+	0.548908i	$0.815044\pi$
$282$	−153812.	−0.115177
$283$	−2.37960e6	−1.76619	−0.883095	−	0.469194i	$-0.844544\pi$
$283$	−2.37960e6	−1.76619	−0.883095	+	0.469194i	$0.844544\pi$
$284$	2.06249e6	1.51739
$285$	0	0
$286$	−23820.8	−0.0172203
$287$	−1.36587e6	−0.978823
$288$	145309.	0.103231
$289$	−1.26333e6	−0.889762
$290$	0	0
$291$	915541.	0.633790
$292$	−2.55271e6	−1.75204
$293$	2.57230e6	1.75046	0.875232	−	0.483704i	$-0.160709\pi$
$293$	2.57230e6	1.75046	0.875232	+	0.483704i	$0.160709\pi$
$294$	−144547.	−0.0975303
$295$	0	0
$296$	−536326.	−0.355795
$297$	173391.	0.114060
$298$	35462.4	0.0231328
$299$	−207132.	−0.133989
$300$	0	0
$301$	3.53447e6	2.24858
$302$	−87526.2	−0.0552231
$303$	−446824.	−0.279595
$304$	76177.5	0.0472762
$305$	0	0
$306$	−18990.8	−0.0115942
$307$	−1.55047e6	−0.938897	−0.469449	−	0.882960i	$-0.655547\pi$
$307$	−1.55047e6	−0.938897	−0.469449	+	0.882960i	$0.655547\pi$
$308$	1.57736e6	0.947445
$309$	−1.00070e6	−0.596222
$310$	0	0
$311$	−2.27644e6	−1.33461	−0.667305	−	0.744784i	$-0.732552\pi$
$311$	−2.27644e6	−1.33461	−0.667305	+	0.744784i	$0.732552\pi$
$312$	57370.7	0.0333659
$313$	84961.1	0.0490184	0.0245092	−	0.999700i	$-0.492198\pi$
$313$	84961.1	0.0490184	0.0245092	+	0.999700i	$0.492198\pi$
$314$	97610.7	0.0558693
$315$	0	0
$316$	−1.54043e6	−0.867808
$317$	277900.	0.155325	0.0776623	−	0.996980i	$-0.475254\pi$
$317$	277900.	0.155325	0.0776623	+	0.996980i	$0.475254\pi$
$318$	148781.	0.0825047
$319$	1.76585e6	0.971576
$320$	0	0
$321$	1.26753e6	0.686587
$322$	−152197.	−0.0818022
$323$	−30429.9	−0.0162291
$324$	−207648.	−0.109892
$325$	0	0
$326$	−73104.0	−0.0380976
$327$	−593577.	−0.306979
$328$	245864.	0.126186
$329$	6.04300e6	3.07796
$330$	0	0
$331$	−544837.	−0.273336	−0.136668	−	0.990617i	$-0.543639\pi$
$331$	−544837.	−0.273336	−0.136668	+	0.990617i	$0.543639\pi$
$332$	2.46105e6	1.22539
$333$	1.15174e6	0.569171
$334$	−373874.	−0.183383
$335$	0	0
$336$	−1.86781e6	−0.902576
$337$	−1.62448e6	−0.779185	−0.389592	−	0.920987i	$-0.627384\pi$
$337$	−1.62448e6	−0.779185	−0.389592	+	0.920987i	$0.627384\pi$
$338$	16925.6	0.00805846
$339$	1.69160e6	0.799465
$340$	0	0
$341$	1.87849e6	0.874827
$342$	3692.05	0.00170688
$343$	2.15717e6	0.990033
$344$	−636223.	−0.289877
$345$	0	0
$346$	−183791.	−0.0825342
$347$	865578.	0.385907	0.192953	−	0.981208i	$-0.438193\pi$
$347$	865578.	0.385907	0.192953	+	0.981208i	$0.438193\pi$
$348$	−2.11473e6	−0.936066
$349$	−131105.	−0.0576178	−0.0288089	−	0.999585i	$-0.509171\pi$
$349$	−131105.	−0.0576178	−0.0288089	+	0.999585i	$0.509171\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	−426684.	−0.183548
$353$	−4.16672e6	−1.77974	−0.889871	−	0.456211i	$-0.849206\pi$
$353$	−4.16672e6	−1.77974	−0.889871	+	0.456211i	$0.849206\pi$
$354$	−51215.6	−0.0217217
$355$	0	0
$356$	−3.26270e6	−1.36443
$357$	746116.	0.309839
$358$	102977.	0.0424651
$359$	−827904.	−0.339034	−0.169517	−	0.985527i	$-0.554221\pi$
$359$	−827904.	−0.339034	−0.169517	+	0.985527i	$0.554221\pi$
$360$	0	0
$361$	−2.47018e6	−0.997611
$362$	24184.3	0.00969979
$363$	940316.	0.374548
$364$	−1.12078e6	−0.443369
$365$	0	0
$366$	−10051.7	−0.00392226
$367$	4.51043e6	1.74805	0.874023	−	0.485884i	$-0.161502\pi$
$367$	4.51043e6	1.74805	0.874023	+	0.485884i	$0.161502\pi$
$368$	−1.21388e6	−0.467256
$369$	−527982.	−0.201861
$370$	0	0
$371$	−5.84532e6	−2.20482
$372$	−2.24962e6	−0.842853
$373$	−2.89506e6	−1.07742	−0.538711	−	0.842490i	$-0.681089\pi$
$373$	−2.89506e6	−1.07742	−0.538711	+	0.842490i	$0.681089\pi$
$374$	55764.5	0.0206148
$375$	0	0
$376$	−1.08777e6	−0.396797
$377$	−1.25470e6	−0.454661
$378$	−90525.9	−0.0325869
$379$	1.90622e6	0.681673	0.340836	−	0.940123i	$-0.389290\pi$
$379$	1.90622e6	0.681673	0.340836	+	0.940123i	$0.389290\pi$
$380$	0	0
$381$	1.07432e6	0.379158
$382$	−310754.	−0.108958
$383$	−1.96221e6	−0.683514	−0.341757	−	0.939788i	$-0.611022\pi$
$383$	−1.96221e6	−0.683514	−0.341757	+	0.939788i	$0.611022\pi$
$384$	680019.	0.235339
$385$	0	0
$386$	−111394.	−0.0380533
$387$	1.36626e6	0.463721
$388$	3.21953e6	1.08571
$389$	3.01996e6	1.01188	0.505938	−	0.862570i	$-0.331147\pi$
$389$	3.01996e6	1.01188	0.505938	+	0.862570i	$0.331147\pi$
$390$	0	0
$391$	484896.	0.160401
$392$	−1.02225e6	−0.336001
$393$	−2.89841e6	−0.946625
$394$	−433501.	−0.140686
$395$	0	0
$396$	609735.	0.195390
$397$	−346453.	−0.110323	−0.0551617	−	0.998477i	$-0.517567\pi$
$397$	−346453.	−0.110323	−0.0551617	+	0.998477i	$0.517567\pi$
$398$	−328900.	−0.104077
$399$	−145054.	−0.0456139
$400$	0	0
$401$	1.22322e6	0.379878	0.189939	−	0.981796i	$-0.439171\pi$
$401$	1.22322e6	0.379878	0.189939	+	0.981796i	$0.439171\pi$
$402$	−6834.25	−0.00210924
$403$	−1.33474e6	−0.409387
$404$	−1.57127e6	−0.478958
$405$	0	0
$406$	−921934.	−0.277578
$407$	−3.38195e6	−1.01200
$408$	−134305.	−0.0399431
$409$	835795.	0.247054	0.123527	−	0.992341i	$-0.460579\pi$
$409$	835795.	0.247054	0.123527	+	0.992341i	$0.460579\pi$
$410$	0	0
$411$	−2.38127e6	−0.695350
$412$	−3.51900e6	−1.02135
$413$	2.01217e6	0.580483
$414$	−58832.2	−0.0168700
$415$	0	0
$416$	303175.	0.0858936
$417$	1.75997e6	0.495639
$418$	−10841.3	−0.00303487
$419$	−4.82020e6	−1.34131	−0.670656	−	0.741768i	$-0.733987\pi$
$419$	−4.82020e6	−1.34131	−0.670656	+	0.741768i	$0.733987\pi$
$420$	0	0
$421$	5.66795e6	1.55855	0.779276	−	0.626681i	$-0.215587\pi$
$421$	5.66795e6	1.55855	0.779276	+	0.626681i	$0.215587\pi$
$422$	−69062.8	−0.0188783
$423$	2.33594e6	0.634763
$424$	1.05219e6	0.284236
$425$	0	0
$426$	347575.	0.0927949
$427$	394913.	0.104817
$428$	4.45731e6	1.17615
$429$	361766.	0.0949040
$430$	0	0
$431$	−6.02344e6	−1.56189	−0.780947	−	0.624597i	$-0.785263\pi$
$431$	−6.02344e6	−1.56189	−0.780947	+	0.624597i	$0.785263\pi$
$432$	−722008.	−0.186137
$433$	−6.78059e6	−1.73799	−0.868996	−	0.494820i	$-0.835234\pi$
$433$	−6.78059e6	−1.73799	−0.868996	+	0.494820i	$0.835234\pi$
$434$	−980742.	−0.249937
$435$	0	0
$436$	−2.08734e6	−0.525867
$437$	−94269.6	−0.0236139
$438$	−430187.	−0.107145
$439$	1.06487e6	0.263716	0.131858	−	0.991269i	$-0.457906\pi$
$439$	1.06487e6	0.263716	0.131858	+	0.991269i	$0.457906\pi$
$440$	0	0
$441$	2.19523e6	0.537507
$442$	−39622.9	−0.00964695
$443$	−3.68799e6	−0.892854	−0.446427	−	0.894820i	$-0.647304\pi$
$443$	−3.68799e6	−0.892854	−0.446427	+	0.894820i	$0.647304\pi$
$444$	4.05013e6	0.975014
$445$	0	0
$446$	421311.	0.100292
$447$	−538568.	−0.127489
$448$	−6.41832e6	−1.51087
$449$	6.69503e6	1.56724	0.783622	−	0.621238i	$-0.213370\pi$
$449$	6.69503e6	1.56724	0.783622	+	0.621238i	$0.213370\pi$
$450$	0	0
$451$	1.55036e6	0.358915
$452$	5.94859e6	1.36952
$453$	1.32926e6	0.304344
$454$	395208.	0.0899884
$455$	0	0
$456$	26110.5	0.00588035
$457$	2.68543e6	0.601483	0.300741	−	0.953706i	$-0.402766\pi$
$457$	2.68543e6	0.601483	0.300741	+	0.953706i	$0.402766\pi$
$458$	−686423.	−0.152907
$459$	288414.	0.0638976
$460$	0	0
$461$	5.41249e6	1.18616	0.593082	−	0.805142i	$-0.297911\pi$
$461$	5.41249e6	1.18616	0.593082	+	0.805142i	$0.297911\pi$
$462$	265819.	0.0579404
$463$	2.42560e6	0.525856	0.262928	−	0.964815i	$-0.415312\pi$
$463$	2.42560e6	0.525856	0.262928	+	0.964815i	$0.415312\pi$
$464$	−7.35308e6	−1.58553
$465$	0	0
$466$	655182.	0.139765
$467$	2.26854e6	0.481343	0.240672	−	0.970607i	$-0.422632\pi$
$467$	2.26854e6	0.481343	0.240672	+	0.970607i	$0.422632\pi$
$468$	−433241.	−0.0914354
$469$	268505.	0.0563665
$470$	0	0
$471$	−1.48241e6	−0.307905
$472$	−362202.	−0.0748334
$473$	−4.01188e6	−0.824508
$474$	−259595.	−0.0530702
$475$	0	0
$476$	2.62374e6	0.530767
$477$	−2.25953e6	−0.454698
$478$	247685.	0.0495827
$479$	879181.	0.175081	0.0875406	−	0.996161i	$-0.472099\pi$
$479$	879181.	0.175081	0.0875406	+	0.996161i	$0.472099\pi$
$480$	0	0
$481$	2.40301e6	0.473579
$482$	−481485.	−0.0943985
$483$	2.31141e6	0.450826
$484$	3.30665e6	0.641616
$485$	0	0
$486$	−34993.1	−0.00672035
$487$	−3.00595e6	−0.574327	−0.287163	−	0.957882i	$-0.592712\pi$
$487$	−3.00595e6	−0.574327	−0.287163	+	0.957882i	$0.592712\pi$
$488$	−71086.5	−0.0135126
$489$	1.11023e6	0.209962
$490$	0	0
$491$	−884317.	−0.165540	−0.0827702	−	0.996569i	$-0.526377\pi$
$491$	−884317.	−0.165540	−0.0827702	+	0.996569i	$0.526377\pi$
$492$	−1.85667e6	−0.345797
$493$	2.93727e6	0.544285
$494$	7703.16	0.00142021
$495$	0	0
$496$	−7.82211e6	−1.42764
$497$	−1.36556e7	−2.47981
$498$	414739.	0.0749380
$499$	8.24181e6	1.48174	0.740869	−	0.671650i	$-0.234414\pi$
$499$	8.24181e6	1.48174	0.740869	+	0.671650i	$0.234414\pi$
$500$	0	0
$501$	5.67803e6	1.01066
$502$	565954.	0.100236
$503$	−3.40773e6	−0.600545	−0.300273	−	0.953853i	$-0.597078\pi$
$503$	−3.40773e6	−0.600545	−0.300273	+	0.953853i	$0.597078\pi$
$504$	−640207.	−0.112265
$505$	0	0
$506$	172754.	0.0299952
$507$	−257049.	−0.0444116
$508$	3.77787e6	0.649513
$509$	−8.95621e6	−1.53225	−0.766125	−	0.642691i	$-0.777818\pi$
$509$	−8.95621e6	−1.53225	−0.766125	+	0.642691i	$0.777818\pi$
$510$	0	0
$511$	1.69013e7	2.86330
$512$	2.97216e6	0.501070
$513$	−56071.2	−0.00940689
$514$	−637186.	−0.106380
$515$	0	0
$516$	4.80451e6	0.794373
$517$	−6.85924e6	−1.12862
$518$	1.76569e6	0.289127
$519$	2.79123e6	0.454860
$520$	0	0
$521$	2.70744e6	0.436982	0.218491	−	0.975839i	$-0.429887\pi$
$521$	2.70744e6	0.436982	0.218491	+	0.975839i	$0.429887\pi$
$522$	−356377.	−0.0572445
$523$	−2.72225e6	−0.435185	−0.217593	−	0.976040i	$-0.569820\pi$
$523$	−2.72225e6	−0.435185	−0.217593	+	0.976040i	$0.569820\pi$
$524$	−1.01923e7	−1.62161
$525$	0	0
$526$	106126.	0.0167247
$527$	3.12463e6	0.490085
$528$	2.12010e6	0.330957
$529$	−4.93417e6	−0.766611
$530$	0	0
$531$	777812.	0.119712
$532$	−510087.	−0.0781385
$533$	−1.10159e6	−0.167959
$534$	−549835.	−0.0834410
$535$	0	0
$536$	−48332.4	−0.00726652
$537$	−1.56391e6	−0.234033
$538$	498055.	0.0741860
$539$	−6.44605e6	−0.955701
$540$	0	0
$541$	2.49859e6	0.367030	0.183515	−	0.983017i	$-0.441252\pi$
$541$	2.49859e6	0.367030	0.183515	+	0.983017i	$0.441252\pi$
$542$	−518194.	−0.0757695
$543$	−367287.	−0.0534572
$544$	−709735.	−0.102825
$545$	0	0
$546$	−188875.	−0.0271139
$547$	6.43873e6	0.920093	0.460046	−	0.887895i	$-0.347833\pi$
$547$	6.43873e6	0.920093	0.460046	+	0.887895i	$0.347833\pi$
$548$	−8.37381e6	−1.19116
$549$	152655.	0.0216163
$550$	0	0
$551$	−571040.	−0.0801287
$552$	−416066.	−0.0581186
$553$	1.01990e7	1.41823
$554$	−1.26393e6	−0.174964
$555$	0	0
$556$	6.18900e6	0.849051
$557$	−2.84713e6	−0.388838	−0.194419	−	0.980919i	$-0.562282\pi$
$557$	−2.84713e6	−0.388838	−0.194419	+	0.980919i	$0.562282\pi$
$558$	−379110.	−0.0515441
$559$	2.85060e6	0.385839
$560$	0	0
$561$	−846896.	−0.113612
$562$	−1.31133e6	−0.175134
$563$	−1.29070e7	−1.71615	−0.858075	−	0.513524i	$-0.828340\pi$
$563$	−1.29070e7	−1.71615	−0.858075	+	0.513524i	$0.828340\pi$
$564$	8.21443e6	1.08738
$565$	0	0
$566$	−1.41018e6	−0.185026
$567$	1.37482e6	0.179592
$568$	2.45808e6	0.319687
$569$	−2.35627e6	−0.305102	−0.152551	−	0.988296i	$-0.548749\pi$
$569$	−2.35627e6	−0.305102	−0.152551	+	0.988296i	$0.548749\pi$
$570$	0	0
$571$	−1.23131e7	−1.58044	−0.790221	−	0.612823i	$-0.790034\pi$
$571$	−1.23131e7	−1.58044	−0.790221	+	0.612823i	$0.790034\pi$
$572$	1.27216e6	0.162575
$573$	4.71942e6	0.600484
$574$	−809430.	−0.102541
$575$	0	0
$576$	−2.48103e6	−0.311584
$577$	1.55443e6	0.194371	0.0971855	−	0.995266i	$-0.469016\pi$
$577$	1.55443e6	0.194371	0.0971855	+	0.995266i	$0.469016\pi$
$578$	−748667.	−0.0932114
$579$	1.69174e6	0.209718
$580$	0	0
$581$	−1.62944e7	−2.00261
$582$	542560.	0.0663958
$583$	6.63487e6	0.808464
$584$	−3.04232e6	−0.369125
$585$	0	0
$586$	1.52438e6	0.183378
$587$	1.03799e7	1.24336	0.621680	−	0.783271i	$-0.286450\pi$
$587$	1.03799e7	1.24336	0.621680	+	0.783271i	$0.286450\pi$
$588$	7.71961e6	0.920771
$589$	−607465.	−0.0721495
$590$	0	0
$591$	6.58359e6	0.775343
$592$	1.40826e7	1.65150
$593$	−4.16373e6	−0.486234	−0.243117	−	0.969997i	$-0.578170\pi$
$593$	−4.16373e6	−0.486234	−0.243117	+	0.969997i	$0.578170\pi$
$594$	102753.	0.0119490
$595$	0	0
$596$	−1.89389e6	−0.218394
$597$	4.99501e6	0.573588
$598$	−122749.	−0.0140367
$599$	−2.96108e6	−0.337196	−0.168598	−	0.985685i	$-0.553924\pi$
$599$	−2.96108e6	−0.337196	−0.168598	+	0.985685i	$0.553924\pi$
$600$	0	0
$601$	619340.	0.0699428	0.0349714	−	0.999388i	$-0.488866\pi$
$601$	619340.	0.0699428	0.0349714	+	0.999388i	$0.488866\pi$
$602$	2.09457e6	0.235561
$603$	103792.	0.0116244
$604$	4.67439e6	0.521354
$605$	0	0
$606$	−264793.	−0.0292904
$607$	−2.93963e6	−0.323833	−0.161916	−	0.986805i	$-0.551767\pi$
$607$	−2.93963e6	−0.323833	−0.161916	+	0.986805i	$0.551767\pi$
$608$	137981.	0.0151377
$609$	1.40014e7	1.52978
$610$	0	0
$611$	4.87376e6	0.528155
$612$	1.01422e6	0.109459
$613$	9.30961e6	1.00065	0.500323	−	0.865839i	$-0.333215\pi$
$613$	9.30961e6	1.00065	0.500323	+	0.865839i	$0.333215\pi$
$614$	−918828.	−0.0983588
$615$	0	0
$616$	1.87990e6	0.199610
$617$	−7.00275e6	−0.740552	−0.370276	−	0.928922i	$-0.620737\pi$
$617$	−7.00275e6	−0.740552	−0.370276	+	0.928922i	$0.620737\pi$
$618$	−593027.	−0.0624602
$619$	−1.30205e7	−1.36585	−0.682923	−	0.730490i	$-0.739292\pi$
$619$	−1.30205e7	−1.36585	−0.682923	+	0.730490i	$0.739292\pi$
$620$	0	0
$621$	893485.	0.0929733
$622$	−1.34904e6	−0.139814
$623$	2.16020e7	2.22984
$624$	−1.50641e6	−0.154875
$625$	0	0
$626$	50349.0	0.00513517
$627$	164647.	0.0167257
$628$	−5.21296e6	−0.527455
$629$	−5.62545e6	−0.566932
$630$	0	0
$631$	1.89038e6	0.189006	0.0945031	−	0.995525i	$-0.469874\pi$
$631$	1.89038e6	0.189006	0.0945031	+	0.995525i	$0.469874\pi$
$632$	−1.83588e6	−0.182832
$633$	1.04886e6	0.104042
$634$	164687.	0.0162718
$635$	0	0
$636$	−7.94572e6	−0.778916
$637$	4.58017e6	0.447232
$638$	1.04646e6	0.101782
$639$	−5.27862e6	−0.511409
$640$	0	0
$641$	−9.94745e6	−0.956240	−0.478120	−	0.878295i	$-0.658682\pi$
$641$	−9.94745e6	−0.956240	−0.478120	+	0.878295i	$0.658682\pi$
$642$	751153.	0.0719268
$643$	−1.86174e7	−1.77579	−0.887894	−	0.460048i	$-0.847832\pi$
$643$	−1.86174e7	−1.77579	−0.887894	+	0.460048i	$0.847832\pi$
$644$	8.12815e6	0.772284
$645$	0	0
$646$	−18033.1	−0.00170016
$647$	1.08663e7	1.02052	0.510259	−	0.860021i	$-0.329550\pi$
$647$	1.08663e7	1.02052	0.510259	+	0.860021i	$0.329550\pi$
$648$	−247475.	−0.0231522
$649$	−2.28396e6	−0.212852
$650$	0	0
$651$	1.48945e7	1.37745
$652$	3.90417e6	0.359674
$653$	−1.06902e7	−0.981075	−0.490537	−	0.871420i	$-0.663199\pi$
$653$	−1.06902e7	−0.981075	−0.490537	+	0.871420i	$0.663199\pi$
$654$	−351761.	−0.0321591
$655$	0	0
$656$	−6.45578e6	−0.585719
$657$	6.53325e6	0.590495
$658$	3.58115e6	0.322447
$659$	−1.47827e7	−1.32599	−0.662993	−	0.748626i	$-0.730714\pi$
$659$	−1.47827e7	−1.32599	−0.662993	+	0.748626i	$0.730714\pi$
$660$	0	0
$661$	−5.48442e6	−0.488233	−0.244116	−	0.969746i	$-0.578498\pi$
$661$	−5.48442e6	−0.488233	−0.244116	+	0.969746i	$0.578498\pi$
$662$	−322877.	−0.0286347
$663$	601753.	0.0531660
$664$	2.93307e6	0.258168
$665$	0	0
$666$	682533.	0.0596264
$667$	9.09943e6	0.791954
$668$	1.99670e7	1.73130
$669$	−6.39845e6	−0.552726
$670$	0	0
$671$	−448255.	−0.0384343
$672$	−3.38318e6	−0.289002
$673$	1.69559e6	0.144305	0.0721526	−	0.997394i	$-0.477013\pi$
$673$	1.69559e6	0.144305	0.0721526	+	0.997394i	$0.477013\pi$
$674$	−962688.	−0.0816273
$675$	0	0
$676$	−903922.	−0.0760789
$677$	1.29586e7	1.08664	0.543320	−	0.839525i	$-0.317167\pi$
$677$	1.29586e7	1.08664	0.543320	+	0.839525i	$0.317167\pi$
$678$	1.00246e6	0.0837519
$679$	−2.13162e7	−1.77434
$680$	0	0
$681$	−6.00203e6	−0.495942
$682$	1.11321e6	0.0916468
$683$	−2.14578e6	−0.176008	−0.0880041	−	0.996120i	$-0.528049\pi$
$683$	−2.14578e6	−0.176008	−0.0880041	+	0.996120i	$0.528049\pi$
$684$	−197176.	−0.0161144
$685$	0	0
$686$	1.27837e6	0.103716
$687$	1.04247e7	0.842699
$688$	1.67057e7	1.34553
$689$	−4.71433e6	−0.378331
$690$	0	0
$691$	2.52870e6	0.201466	0.100733	−	0.994913i	$-0.467881\pi$
$691$	2.52870e6	0.201466	0.100733	+	0.994913i	$0.467881\pi$
$692$	9.81547e6	0.779194
$693$	−4.03700e6	−0.319319
$694$	512951.	0.0404276
$695$	0	0
$696$	−2.52033e6	−0.197213
$697$	2.57883e6	0.201067
$698$	−77694.5	−0.00603603
$699$	−9.95026e6	−0.770267
$700$	0	0
$701$	−2.06075e7	−1.58391	−0.791953	−	0.610581i	$-0.790936\pi$
$701$	−2.06075e7	−1.58391	−0.791953	+	0.610581i	$0.790936\pi$
$702$	−73010.4	−0.00559167
$703$	1.09366e6	0.0834627
$704$	7.28527e6	0.554005
$705$	0	0
$706$	−2.46925e6	−0.186446
$707$	1.04032e7	0.782745
$708$	2.73520e6	0.205072
$709$	4.44682e6	0.332227	0.166113	−	0.986107i	$-0.446878\pi$
$709$	4.44682e6	0.332227	0.166113	+	0.986107i	$0.446878\pi$
$710$	0	0
$711$	3.94247e6	0.292479
$712$	−3.88849e6	−0.287462
$713$	9.67986e6	0.713092
$714$	442157.	0.0324587
$715$	0	0
$716$	−5.49955e6	−0.400908
$717$	−3.76159e6	−0.273259
$718$	−490625.	−0.0355172
$719$	−6.90228e6	−0.497932	−0.248966	−	0.968512i	$-0.580091\pi$
$719$	−6.90228e6	−0.497932	−0.248966	+	0.968512i	$0.580091\pi$
$720$	0	0
$721$	2.32990e7	1.66916
$722$	−1.46386e6	−0.104510
$723$	7.31232e6	0.520247
$724$	−1.29158e6	−0.0915745
$725$	0	0
$726$	557242.	0.0392376
$727$	8.09243e6	0.567862	0.283931	−	0.958845i	$-0.408361\pi$
$727$	8.09243e6	0.567862	0.283931	+	0.958845i	$0.408361\pi$
$728$	−1.33574e6	−0.0934101
$729$	531441.	0.0370370
$730$	0	0
$731$	−6.67326e6	−0.461896
$732$	536818.	0.0370296
$733$	1.45926e7	1.00316	0.501582	−	0.865110i	$-0.332751\pi$
$733$	1.45926e7	1.00316	0.501582	+	0.865110i	$0.332751\pi$
$734$	2.67294e6	0.183125
$735$	0	0
$736$	−2.19870e6	−0.149614
$737$	−304773.	−0.0206685
$738$	−312888.	−0.0211470
$739$	7.53378e6	0.507460	0.253730	−	0.967275i	$-0.418343\pi$
$739$	7.53378e6	0.507460	0.253730	+	0.967275i	$0.418343\pi$
$740$	0	0
$741$	−116988.	−0.00782701
$742$	−3.46401e6	−0.230977
$743$	−1.37801e7	−0.915759	−0.457880	−	0.889014i	$-0.651391\pi$
$743$	−1.37801e7	−0.915759	−0.457880	+	0.889014i	$0.651391\pi$
$744$	−2.68110e6	−0.177574
$745$	0	0
$746$	−1.71565e6	−0.112871
$747$	−6.29865e6	−0.412996
$748$	−2.97814e6	−0.194622
$749$	−2.95114e7	−1.92214
$750$	0	0
$751$	1.52138e7	0.984321	0.492160	−	0.870505i	$-0.336207\pi$
$751$	1.52138e7	0.984321	0.492160	+	0.870505i	$0.336207\pi$
$752$	2.85622e7	1.84182
$753$	−8.59515e6	−0.552415
$754$	−743553.	−0.0476303
$755$	0	0
$756$	4.83459e6	0.307649
$757$	−1.39902e7	−0.887329	−0.443664	−	0.896193i	$-0.646322\pi$
$757$	−1.39902e7	−0.887329	−0.443664	+	0.896193i	$0.646322\pi$
$758$	1.12965e6	0.0714120
$759$	−2.62362e6	−0.165309
$760$	0	0
$761$	2.81683e7	1.76319	0.881594	−	0.472009i	$-0.156471\pi$
$761$	2.81683e7	1.76319	0.881594	+	0.472009i	$0.156471\pi$
$762$	636653.	0.0397205
$763$	1.38201e7	0.859406
$764$	1.65960e7	1.02866
$765$	0	0
$766$	−1.16283e6	−0.0716049
$767$	1.62284e6	0.0996066
$768$	−8.41845e6	−0.515026
$769$	2.07358e7	1.26446	0.632231	−	0.774780i	$-0.282139\pi$
$769$	2.07358e7	1.26446	0.632231	+	0.774780i	$0.282139\pi$
$770$	0	0
$771$	9.67694e6	0.586276
$772$	5.94905e6	0.359256
$773$	1.26313e7	0.760325	0.380162	−	0.924920i	$-0.375868\pi$
$773$	1.26313e7	0.760325	0.380162	+	0.924920i	$0.375868\pi$
$774$	809663.	0.0485794
$775$	0	0
$776$	3.83704e6	0.228740
$777$	−2.68155e7	−1.59343
$778$	1.78966e6	0.106004
$779$	−501356.	−0.0296007
$780$	0	0
$781$	1.55001e7	0.909298
$782$	287355.	0.0168036
$783$	5.41231e6	0.315484
$784$	2.68417e7	1.55962
$785$	0	0
$786$	−1.71763e6	−0.0991684
$787$	−5.87368e6	−0.338044	−0.169022	−	0.985612i	$-0.554061\pi$
$787$	−5.87368e6	−0.338044	−0.169022	+	0.985612i	$0.554061\pi$
$788$	2.31514e7	1.32820
$789$	−1.61174e6	−0.0921725
$790$	0	0
$791$	−3.93850e7	−2.23815
$792$	726682.	0.0411653
$793$	318503.	0.0179858
$794$	−205312.	−0.0115575
$795$	0	0
$796$	1.75651e7	0.982581
$797$	4.96813e6	0.277043	0.138522	−	0.990359i	$-0.455765\pi$
$797$	4.96813e6	0.277043	0.138522	+	0.990359i	$0.455765\pi$
$798$	−85960.7	−0.00477851
$799$	−1.14095e7	−0.632265
$800$	0	0
$801$	8.35035e6	0.459858
$802$	724896.	0.0397961
$803$	−1.91842e7	−1.04992
$804$	364987.	0.0199131
$805$	0	0
$806$	−790982.	−0.0428873
$807$	−7.56397e6	−0.408852
$808$	−1.87264e6	−0.100908
$809$	−2.25969e7	−1.21389	−0.606943	−	0.794745i	$-0.707604\pi$
$809$	−2.25969e7	−1.21389	−0.606943	+	0.794745i	$0.707604\pi$
$810$	0	0
$811$	−7.31663e6	−0.390624	−0.195312	−	0.980741i	$-0.562572\pi$
$811$	−7.31663e6	−0.390624	−0.195312	+	0.980741i	$0.562572\pi$
$812$	4.92365e7	2.62058
$813$	7.86982e6	0.417579
$814$	−2.00418e6	−0.106017
$815$	0	0
$816$	3.52652e6	0.185405
$817$	1.29736e6	0.0679995
$818$	495302.	0.0258813
$819$	2.86845e6	0.149430
$820$	0	0
$821$	2.23939e7	1.15950	0.579751	−	0.814794i	$-0.303150\pi$
$821$	2.23939e7	1.15950	0.579751	+	0.814794i	$0.303150\pi$
$822$	−1.41117e6	−0.0728449
$823$	−3.22046e7	−1.65737	−0.828683	−	0.559719i	$-0.810909\pi$
$823$	−3.22046e7	−1.65737	−0.828683	+	0.559719i	$0.810909\pi$
$824$	−4.19394e6	−0.215181
$825$	0	0
$826$	1.19244e6	0.0608114
$827$	8.56536e6	0.435494	0.217747	−	0.976005i	$-0.430129\pi$
$827$	8.56536e6	0.435494	0.217747	+	0.976005i	$0.430129\pi$
$828$	3.14197e6	0.159267
$829$	−2.46321e7	−1.24485	−0.622423	−	0.782681i	$-0.713852\pi$
$829$	−2.46321e7	−1.24485	−0.622423	+	0.782681i	$0.713852\pi$
$830$	0	0
$831$	1.91953e7	0.964256
$832$	−5.17647e6	−0.259254
$833$	−1.07222e7	−0.535392
$834$	1.04298e6	0.0519231
$835$	0	0
$836$	578986.	0.0286518
$837$	5.75754e6	0.284069
$838$	−2.85651e6	−0.140516
$839$	1.05438e7	0.517119	0.258560	−	0.965995i	$-0.416752\pi$
$839$	1.05438e7	0.517119	0.258560	+	0.965995i	$0.416752\pi$
$840$	0	0
$841$	3.46089e7	1.68732
$842$	3.35890e6	0.163274
$843$	1.99152e7	0.965194
$844$	3.68834e6	0.178228
$845$	0	0
$846$	1.38431e6	0.0664977
$847$	−2.18930e7	−1.04857
$848$	−2.76279e7	−1.31935
$849$	2.14164e7	1.01971
$850$	0	0
$851$	−1.74272e7	−0.824906
$852$	−1.85624e7	−0.876064
$853$	1.18131e7	0.555895	0.277948	−	0.960596i	$-0.410346\pi$
$853$	1.18131e7	0.555895	0.277948	+	0.960596i	$0.410346\pi$
$854$	234030.	0.0109806
$855$	0	0
$856$	5.31222e6	0.247795
$857$	−3.45083e6	−0.160499	−0.0802494	−	0.996775i	$-0.525572\pi$
$857$	−3.45083e6	−0.160499	−0.0802494	+	0.996775i	$0.525572\pi$
$858$	214387.	0.00994214
$859$	−2.37296e7	−1.09725	−0.548627	−	0.836068i	$-0.684849\pi$
$859$	−2.37296e7	−1.09725	−0.548627	+	0.836068i	$0.684849\pi$
$860$	0	0
$861$	1.22928e7	0.565124
$862$	−3.56956e6	−0.163624
$863$	−3.66091e6	−0.167325	−0.0836626	−	0.996494i	$-0.526662\pi$
$863$	−3.66091e6	−0.167325	−0.0836626	+	0.996494i	$0.526662\pi$
$864$	−1.30778e6	−0.0596006
$865$	0	0
$866$	−4.01826e6	−0.182072
$867$	1.13700e7	0.513704
$868$	5.23771e7	2.35962
$869$	−1.15766e7	−0.520035
$870$	0	0
$871$	216553.	0.00967207
$872$	−2.48769e6	−0.110791
$873$	−8.23987e6	−0.365919
$874$	−55865.3	−0.00247379
$875$	0	0
$876$	2.29744e7	1.01154
$877$	5.00580e6	0.219773	0.109887	−	0.993944i	$-0.464951\pi$
$877$	5.00580e6	0.219773	0.109887	+	0.993944i	$0.464951\pi$
$878$	631057.	0.0276269
$879$	−2.31507e7	−1.01063
$880$	0	0
$881$	−246502.	−0.0106999	−0.00534997	−	0.999986i	$-0.501703\pi$
$881$	−246502.	−0.0106999	−0.00534997	+	0.999986i	$0.501703\pi$
$882$	1.30092e6	0.0563092
$883$	−3.11271e7	−1.34350	−0.671749	−	0.740779i	$-0.734457\pi$
$883$	−3.11271e7	−1.34350	−0.671749	+	0.740779i	$0.734457\pi$
$884$	2.11608e6	0.0910756
$885$	0	0
$886$	−2.18555e6	−0.0935354
$887$	1.82476e7	0.778746	0.389373	−	0.921080i	$-0.372692\pi$
$887$	1.82476e7	0.778746	0.389373	+	0.921080i	$0.372692\pi$
$888$	4.82694e6	0.205418
$889$	−2.50129e7	−1.06148
$890$	0	0
$891$	−1.56052e6	−0.0658528
$892$	−2.25004e7	−0.946842
$893$	2.21814e6	0.0930810
$894$	−319162.	−0.0133557
$895$	0	0
$896$	−1.58326e7	−0.658845
$897$	1.86418e6	0.0773585
$898$	3.96755e6	0.164184
$899$	5.86360e7	2.41972
$900$	0	0
$901$	1.10363e7	0.452908
$902$	918762.	0.0375999
$903$	−3.18102e7	−1.29822
$904$	7.08952e6	0.288533
$905$	0	0
$906$	787736.	0.0318831
$907$	2.12243e7	0.856675	0.428337	−	0.903619i	$-0.359100\pi$
$907$	2.12243e7	0.856675	0.428337	+	0.903619i	$0.359100\pi$
$908$	−2.11064e7	−0.849569
$909$	4.02141e6	0.161424
$910$	0	0
$911$	−3.18110e7	−1.26993	−0.634967	−	0.772540i	$-0.718986\pi$
$911$	−3.18110e7	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$912$	−685598.	−0.0272949
$913$	1.84953e7	0.734318
$914$	1.59142e6	0.0630113
$915$	0	0
$916$	3.66589e7	1.44358
$917$	6.74826e7	2.65014
$918$	170918.	0.00669391
$919$	−2.08307e7	−0.813606	−0.406803	−	0.913516i	$-0.633357\pi$
$919$	−2.08307e7	−0.813606	−0.406803	+	0.913516i	$0.633357\pi$
$920$	0	0
$921$	1.39542e7	0.542073
$922$	3.20750e6	0.124262
$923$	−1.10134e7	−0.425518
$924$	−1.41962e7	−0.547008
$925$	0	0
$926$	1.43744e6	0.0550887
$927$	9.00631e6	0.344229
$928$	−1.33187e7	−0.507682
$929$	2.46999e7	0.938980	0.469490	−	0.882938i	$-0.344438\pi$
$929$	2.46999e7	0.938980	0.469490	+	0.882938i	$0.344438\pi$
$930$	0	0
$931$	2.08453e6	0.0788194
$932$	−3.49904e7	−1.31950
$933$	2.04879e7	0.770538
$934$	1.34436e6	0.0504255
$935$	0	0
$936$	−516336.	−0.0192638
$937$	−8.66681e6	−0.322486	−0.161243	−	0.986915i	$-0.551550\pi$
$937$	−8.66681e6	−0.322486	−0.161243	+	0.986915i	$0.551550\pi$
$938$	159119.	0.00590495
$939$	−764650.	−0.0283008
$940$	0	0
$941$	6.11509e6	0.225127	0.112564	−	0.993645i	$-0.464094\pi$
$941$	6.11509e6	0.225127	0.112564	+	0.993645i	$0.464094\pi$
$942$	−878496.	−0.0322561
$943$	7.98902e6	0.292560
$944$	9.51053e6	0.347356
$945$	0	0
$946$	−2.37749e6	−0.0863754
$947$	−3.15310e6	−0.114252	−0.0571259	−	0.998367i	$-0.518194\pi$
$947$	−3.15310e6	−0.114252	−0.0571259	+	0.998367i	$0.518194\pi$
$948$	1.38638e7	0.501029
$949$	1.36311e7	0.491321
$950$	0	0
$951$	−2.50110e6	−0.0896767
$952$	3.12697e6	0.111823
$953$	−2.87392e7	−1.02504	−0.512522	−	0.858674i	$-0.671289\pi$
$953$	−2.87392e7	−1.02504	−0.512522	+	0.858674i	$0.671289\pi$
$954$	−1.33903e6	−0.0476341
$955$	0	0
$956$	−1.32278e7	−0.468104
$957$	−1.58926e7	−0.560939
$958$	521013.	0.0183415
$959$	5.54422e7	1.94668
$960$	0	0
$961$	3.37470e7	1.17877
$962$	1.42405e6	0.0496121
$963$	−1.14078e7	−0.396401
$964$	2.57140e7	0.891204
$965$	0	0
$966$	1.36977e6	0.0472285
$967$	5.04557e7	1.73518	0.867590	−	0.497281i	$-0.165668\pi$
$967$	5.04557e7	1.73518	0.867590	+	0.497281i	$0.165668\pi$
$968$	3.94087e6	0.135177
$969$	273869.	0.00936988
$970$	0	0
$971$	6.26481e6	0.213236	0.106618	−	0.994300i	$-0.465998\pi$
$971$	6.26481e6	0.213236	0.106618	+	0.994300i	$0.465998\pi$
$972$	1.86883e6	0.0634460
$973$	−4.09768e7	−1.38757
$974$	−1.78136e6	−0.0601665
$975$	0	0
$976$	1.86656e6	0.0627216
$977$	2.54287e7	0.852292	0.426146	−	0.904655i	$-0.359871\pi$
$977$	2.54287e7	0.852292	0.426146	+	0.904655i	$0.359871\pi$
$978$	657936.	0.0219956
$979$	−2.45199e7	−0.817639
$980$	0	0
$981$	5.34220e6	0.177234
$982$	−524056.	−0.0173420
$983$	−2.73152e7	−0.901614	−0.450807	−	0.892621i	$-0.648864\pi$
$983$	−2.73152e7	−0.901614	−0.450807	+	0.892621i	$0.648864\pi$
$984$	−2.21277e6	−0.0728533
$985$	0	0
$986$	1.74066e6	0.0570193
$987$	−5.43870e7	−1.77706
$988$	−411392.	−0.0134080
$989$	−2.06732e7	−0.672075
$990$	0	0
$991$	−5.79527e7	−1.87452	−0.937259	−	0.348633i	$-0.886646\pi$
$991$	−5.79527e7	−1.87452	−0.937259	+	0.348633i	$0.886646\pi$
$992$	−1.41683e7	−0.457128
$993$	4.90353e6	0.157811
$994$	−8.09245e6	−0.259785
$995$	0	0
$996$	−2.21494e7	−0.707480
$997$	−6.87536e6	−0.219057	−0.109529	−	0.993984i	$-0.534934\pi$
$997$	−6.87536e6	−0.219057	−0.109529	+	0.993984i	$0.534934\pi$
$998$	4.88420e6	0.155227
$999$	−1.03656e7	−0.328611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.6.a.u.1.6	yes	11
5.4	even	2		975.6.a.r.1.6	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.6.a.r.1.6	✓	11	5.4	even	2
975.6.a.u.1.6	yes	11	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q+0.592612 q^{2} -9.00000 q^{3} -31.6488 q^{4} -5.33351 q^{6} +209.544 q^{7} -37.7190 q^{8} +81.0000 q^{9} -237.848 q^{11} +284.839 q^{12} +169.000 q^{13} +124.178 q^{14} +990.409 q^{16} -395.630 q^{17} +48.0016 q^{18} +76.9152 q^{19} -1885.89 q^{21} -140.951 q^{22} -1225.63 q^{23} +339.471 q^{24} +100.151 q^{26} -729.000 q^{27} -6631.81 q^{28} -7424.29 q^{29} -7897.86 q^{31} +1793.94 q^{32} +2140.63 q^{33} -234.455 q^{34} -2563.55 q^{36} +14219.0 q^{37} +45.5808 q^{38} -1521.00 q^{39} -6518.30 q^{41} -1117.60 q^{42} +16867.4 q^{43} +7527.59 q^{44} -726.323 q^{46} +28838.8 q^{47} -8913.68 q^{48} +27101.6 q^{49} +3560.67 q^{51} -5348.65 q^{52} -27895.5 q^{53} -432.014 q^{54} -7903.79 q^{56} -692.237 q^{57} -4399.72 q^{58} +9602.62 q^{59} +1884.63 q^{61} -4680.36 q^{62} +16973.1 q^{63} -30630.0 q^{64} +1268.56 q^{66} +1281.38 q^{67} +12521.2 q^{68} +11030.7 q^{69} -65168.1 q^{71} -3055.24 q^{72} +80657.5 q^{73} +8426.34 q^{74} -2434.27 q^{76} -49839.5 q^{77} -901.362 q^{78} +48672.5 q^{79} +6561.00 q^{81} -3862.82 q^{82} -77761.1 q^{83} +59686.3 q^{84} +9995.84 q^{86} +66818.6 q^{87} +8971.38 q^{88} +103091. q^{89} +35412.9 q^{91} +38789.8 q^{92} +71080.7 q^{93} +17090.2 q^{94} -16145.4 q^{96} -101727. q^{97} +16060.7 q^{98} -19265.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} - 99 q^{3} + 224 q^{4} - 18 q^{6} - 55 q^{7} + 270 q^{8} + 891 q^{9} - 125 q^{11} - 2016 q^{12} + 1859 q^{13} - 1311 q^{14} + 5756 q^{16} - 4507 q^{17} + 162 q^{18} + 142 q^{19} + 495 q^{21}+ \cdots - 10125 q^{99}+O(q^{100}) \) 11 * q + 2 * q^2 - 99 * q^3 + 224 * q^4 - 18 * q^6 - 55 * q^7 + 270 * q^8 + 891 * q^9 - 125 * q^11 - 2016 * q^12 + 1859 * q^13 - 1311 * q^14 + 5756 * q^16 - 4507 * q^17 + 162 * q^18 + 142 * q^19 + 495 * q^21 - 5975 * q^22 - 4672 * q^23 - 2430 * q^24 + 338 * q^26 - 8019 * q^27 - 4149 * q^28 + 5197 * q^29 - 277 * q^31 + 2272 * q^32 + 1125 * q^33 - 21225 * q^34 + 18144 * q^36 - 1696 * q^37 - 27485 * q^38 - 16731 * q^39 + 2040 * q^41 + 11799 * q^42 - 31436 * q^43 - 39883 * q^44 + 67530 * q^46 + 15499 * q^47 - 51804 * q^48 + 29212 * q^49 + 40563 * q^51 + 37856 * q^52 - 119577 * q^53 - 1458 * q^54 + 1677 * q^56 - 1278 * q^57 - 91154 * q^58 + 17115 * q^59 + 26273 * q^61 - 45211 * q^62 - 4455 * q^63 + 137480 * q^64 + 53775 * q^66 + 26313 * q^67 - 191297 * q^68 + 42048 * q^69 - 105756 * q^71 + 21870 * q^72 - 103230 * q^73 - 6501 * q^74 + 124495 * q^76 - 104415 * q^77 - 3042 * q^78 + 210732 * q^79 + 72171 * q^81 - 60283 * q^82 - 144247 * q^83 + 37341 * q^84 - 42260 * q^86 - 46773 * q^87 - 415403 * q^88 + 117088 * q^89 - 9295 * q^91 - 70700 * q^92 + 2493 * q^93 + 383414 * q^94 - 20448 * q^96 - 112934 * q^97 - 159305 * q^98 - 10125 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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