LMFDB - Embedded newform 495.6.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.21865.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 30x + 40 $ x^3 - x^2 - 30*x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.61356$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.94924 q^{2} -7.50499 q^{4} -25.0000 q^{5} -1.89401 q^{7} -195.520 q^{8} +O(q^{10})$	\(q+4.94924 q^{2} -7.50499 q^{4} -25.0000 q^{5} -1.89401 q^{7} -195.520 q^{8} -123.731 q^{10} -121.000 q^{11} -378.388 q^{13} -9.37392 q^{14} -727.515 q^{16} +1083.22 q^{17} -3117.36 q^{19} +187.625 q^{20} -598.858 q^{22} +3605.50 q^{23} +625.000 q^{25} -1872.73 q^{26} +14.2145 q^{28} -2834.03 q^{29} +3702.72 q^{31} +2655.98 q^{32} +5361.12 q^{34} +47.3503 q^{35} -1867.45 q^{37} -15428.6 q^{38} +4888.00 q^{40} -11772.7 q^{41} +18783.0 q^{43} +908.104 q^{44} +17844.5 q^{46} +19172.7 q^{47} -16803.4 q^{49} +3093.28 q^{50} +2839.80 q^{52} +36095.8 q^{53} +3025.00 q^{55} +370.317 q^{56} -14026.3 q^{58} -32752.8 q^{59} +11854.9 q^{61} +18325.7 q^{62} +36425.6 q^{64} +9459.70 q^{65} -26101.1 q^{67} -8129.56 q^{68} +234.348 q^{70} +51549.5 q^{71} +37342.3 q^{73} -9242.49 q^{74} +23395.8 q^{76} +229.175 q^{77} +44628.8 q^{79} +18187.9 q^{80} -58265.7 q^{82} +41233.4 q^{83} -27080.5 q^{85} +92961.8 q^{86} +23657.9 q^{88} -43519.3 q^{89} +716.671 q^{91} -27059.2 q^{92} +94890.3 q^{94} +77934.0 q^{95} -85975.4 q^{97} -83164.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 7 q^{2} + 41 q^{4} - 75 q^{5} - 102 q^{7} + 15 q^{8}+O(q^{10}) $ 3 * q + 7 * q^2 + 41 * q^4 - 75 * q^5 - 102 * q^7 + 15 * q^8	$ 3 q + 7 q^{2} + 41 q^{4} - 75 q^{5} - 102 q^{7} + 15 q^{8} - 175 q^{10} - 363 q^{11} - 1646 q^{13} + 963 q^{14} - 2687 q^{16} + 1742 q^{17} - 10 q^{19} - 1025 q^{20} - 847 q^{22} + 3876 q^{23} + 1875 q^{25} + 2894 q^{26} - 209 q^{28} - 1970 q^{29} + 6596 q^{31} - 2353 q^{32} + 16887 q^{34} + 2550 q^{35} - 22032 q^{37} + 2205 q^{38} - 375 q^{40} + 7214 q^{41} + 29644 q^{43} - 4961 q^{44} + 21126 q^{46} + 8432 q^{47} - 34939 q^{49} + 4375 q^{50} - 15462 q^{52} + 31656 q^{53} + 9075 q^{55} - 16235 q^{56} + 94035 q^{58} + 31440 q^{59} - 49374 q^{61} + 100169 q^{62} + 9601 q^{64} + 41150 q^{65} - 111892 q^{67} + 30099 q^{68} - 24075 q^{70} + 58444 q^{71} + 5554 q^{73} + 35743 q^{74} + 128385 q^{76} + 12342 q^{77} + 76080 q^{79} + 67175 q^{80} + 18026 q^{82} - 44844 q^{83} - 43550 q^{85} + 157244 q^{86} - 1815 q^{88} - 25460 q^{89} + 123236 q^{91} - 14338 q^{92} + 411002 q^{94} + 250 q^{95} - 150482 q^{97} - 209376 q^{98}+O(q^{100}) $ 3 * q + 7 * q^2 + 41 * q^4 - 75 * q^5 - 102 * q^7 + 15 * q^8 - 175 * q^10 - 363 * q^11 - 1646 * q^13 + 963 * q^14 - 2687 * q^16 + 1742 * q^17 - 10 * q^19 - 1025 * q^20 - 847 * q^22 + 3876 * q^23 + 1875 * q^25 + 2894 * q^26 - 209 * q^28 - 1970 * q^29 + 6596 * q^31 - 2353 * q^32 + 16887 * q^34 + 2550 * q^35 - 22032 * q^37 + 2205 * q^38 - 375 * q^40 + 7214 * q^41 + 29644 * q^43 - 4961 * q^44 + 21126 * q^46 + 8432 * q^47 - 34939 * q^49 + 4375 * q^50 - 15462 * q^52 + 31656 * q^53 + 9075 * q^55 - 16235 * q^56 + 94035 * q^58 + 31440 * q^59 - 49374 * q^61 + 100169 * q^62 + 9601 * q^64 + 41150 * q^65 - 111892 * q^67 + 30099 * q^68 - 24075 * q^70 + 58444 * q^71 + 5554 * q^73 + 35743 * q^74 + 128385 * q^76 + 12342 * q^77 + 76080 * q^79 + 67175 * q^80 + 18026 * q^82 - 44844 * q^83 - 43550 * q^85 + 157244 * q^86 - 1815 * q^88 - 25460 * q^89 + 123236 * q^91 - 14338 * q^92 + 411002 * q^94 + 250 * q^95 - 150482 * q^97 - 209376 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	4.94924	0.874911	0.437455	−	0.899240i	$-0.355880\pi$
$2$	4.94924	0.874911	0.437455	+	0.899240i	$0.355880\pi$
$3$	0	0
$3$	0	0
$4$	−7.50499	−0.234531
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−1.89401	−0.0146096	−0.00730478	−	0.999973i	$-0.502325\pi$
$7$	−1.89401	−0.0146096	−0.00730478	+	0.999973i	$0.502325\pi$
$8$	−195.520	−1.08010
$9$	0	0
$10$	−123.731	−0.391272
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−378.388	−0.620982	−0.310491	−	0.950576i	$-0.600493\pi$
$13$	−378.388	−0.620982	−0.310491	+	0.950576i	$0.600493\pi$
$14$	−9.37392	−0.0127821
$15$	0	0
$16$	−727.515	−0.710464
$17$	1083.22	0.909064	0.454532	−	0.890730i	$-0.349807\pi$
$17$	1083.22	0.909064	0.454532	+	0.890730i	$0.349807\pi$
$18$	0	0
$19$	−3117.36	−1.98108	−0.990542	−	0.137209i	$-0.956187\pi$
$19$	−3117.36	−1.98108	−0.990542	+	0.137209i	$0.956187\pi$
$20$	187.625	0.104885
$21$	0	0
$22$	−598.858	−0.263796
$23$	3605.50	1.42117	0.710584	−	0.703613i	$-0.248431\pi$
$23$	3605.50	1.42117	0.710584	+	0.703613i	$0.248431\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	−1872.73	−0.543304
$27$	0	0
$28$	14.2145	0.00342640
$29$	−2834.03	−0.625763	−0.312882	−	0.949792i	$-0.601294\pi$
$29$	−2834.03	−0.625763	−0.312882	+	0.949792i	$0.601294\pi$
$30$	0	0
$31$	3702.72	0.692018	0.346009	−	0.938231i	$-0.387537\pi$
$31$	3702.72	0.692018	0.346009	+	0.938231i	$0.387537\pi$
$32$	2655.98	0.458512
$33$	0	0
$34$	5361.12	0.795350
$35$	47.3503	0.00653360
$36$	0	0
$37$	−1867.45	−0.224257	−0.112128	−	0.993694i	$-0.535767\pi$
$37$	−1867.45	−0.224257	−0.112128	+	0.993694i	$0.535767\pi$
$38$	−15428.6	−1.73327
$39$	0	0
$40$	4888.00	0.483037
$41$	−11772.7	−1.09374	−0.546871	−	0.837217i	$-0.684181\pi$
$41$	−11772.7	−1.09374	−0.546871	+	0.837217i	$0.684181\pi$
$42$	0	0
$43$	18783.0	1.54915	0.774577	−	0.632480i	$-0.217963\pi$
$43$	18783.0	1.54915	0.774577	+	0.632480i	$0.217963\pi$
$44$	908.104	0.0707138
$45$	0	0
$46$	17844.5	1.24340
$47$	19172.7	1.26601	0.633007	−	0.774146i	$-0.281820\pi$
$47$	19172.7	1.26601	0.633007	+	0.774146i	$0.281820\pi$
$48$	0	0
$49$	−16803.4	−0.999787
$50$	3093.28	0.174982
$51$	0	0
$52$	2839.80	0.145640
$53$	36095.8	1.76509	0.882545	−	0.470228i	$-0.155828\pi$
$53$	36095.8	1.76509	0.882545	+	0.470228i	$0.155828\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	370.317	0.0157799
$57$	0	0
$58$	−14026.3	−0.547487
$59$	−32752.8	−1.22495	−0.612475	−	0.790490i	$-0.709826\pi$
$59$	−32752.8	−1.22495	−0.612475	+	0.790490i	$0.709826\pi$
$60$	0	0
$61$	11854.9	0.407917	0.203959	−	0.978980i	$-0.434619\pi$
$61$	11854.9	0.407917	0.203959	+	0.978980i	$0.434619\pi$
$62$	18325.7	0.605454
$63$	0	0
$64$	36425.6	1.11162
$65$	9459.70	0.277712
$66$	0	0
$67$	−26101.1	−0.710348	−0.355174	−	0.934800i	$-0.615578\pi$
$67$	−26101.1	−0.710348	−0.355174	+	0.934800i	$0.615578\pi$
$68$	−8129.56	−0.213204
$69$	0	0
$70$	234.348	0.00571631
$71$	51549.5	1.21361	0.606805	−	0.794851i	$-0.292451\pi$
$71$	51549.5	1.21361	0.606805	+	0.794851i	$0.292451\pi$
$72$	0	0
$73$	37342.3	0.820151	0.410075	−	0.912052i	$-0.365502\pi$
$73$	37342.3	0.820151	0.410075	+	0.912052i	$0.365502\pi$
$74$	−9242.49	−0.196205
$75$	0	0
$76$	23395.8	0.464626
$77$	229.175	0.00440495
$78$	0	0
$79$	44628.8	0.804540	0.402270	−	0.915521i	$-0.368221\pi$
$79$	44628.8	0.804540	0.402270	+	0.915521i	$0.368221\pi$
$80$	18187.9	0.317729
$81$	0	0
$82$	−58265.7	−0.956926
$83$	41233.4	0.656983	0.328491	−	0.944507i	$-0.393460\pi$
$83$	41233.4	0.656983	0.328491	+	0.944507i	$0.393460\pi$
$84$	0	0
$85$	−27080.5	−0.406546
$86$	92961.8	1.35537
$87$	0	0
$88$	23657.9	0.325664
$89$	−43519.3	−0.582381	−0.291190	−	0.956665i	$-0.594051\pi$
$89$	−43519.3	−0.582381	−0.291190	+	0.956665i	$0.594051\pi$
$90$	0	0
$91$	716.671	0.00907228
$92$	−27059.2	−0.333308
$93$	0	0
$94$	94890.3	1.10765
$95$	77934.0	0.885968
$96$	0	0
$97$	−85975.4	−0.927779	−0.463890	−	0.885893i	$-0.653547\pi$
$97$	−85975.4	−0.927779	−0.463890	+	0.885893i	$0.653547\pi$
$98$	−83164.2	−0.874724
$99$	0	0
$100$	−4690.62	−0.0469062
$101$	129916.	1.26724	0.633620	−	0.773645i	$-0.281568\pi$
$101$	129916.	1.26724	0.633620	+	0.773645i	$0.281568\pi$
$102$	0	0
$103$	150586.	1.39859	0.699295	−	0.714834i	$-0.253498\pi$
$103$	150586.	1.39859	0.699295	+	0.714834i	$0.253498\pi$
$104$	73982.4	0.670726
$105$	0	0
$106$	178647.	1.54430
$107$	18690.7	0.157822	0.0789108	−	0.996882i	$-0.474856\pi$
$107$	18690.7	0.157822	0.0789108	+	0.996882i	$0.474856\pi$
$108$	0	0
$109$	151235.	1.21923	0.609617	−	0.792696i	$-0.291323\pi$
$109$	151235.	1.21923	0.609617	+	0.792696i	$0.291323\pi$
$110$	14971.5	0.117973
$111$	0	0
$112$	1377.92	0.0103796
$113$	5683.54	0.0418719	0.0209359	−	0.999781i	$-0.493335\pi$
$113$	5683.54	0.0418719	0.0209359	+	0.999781i	$0.493335\pi$
$114$	0	0
$115$	−90137.4	−0.635566
$116$	21269.4	0.146761
$117$	0	0
$118$	−162101.	−1.07172
$119$	−2051.63	−0.0132810
$120$	0	0
$121$	14641.0	0.0909091
$122$	58672.6	0.356891
$123$	0	0
$124$	−27788.9	−0.162300
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	190536.	1.04826	0.524128	−	0.851639i	$-0.324391\pi$
$127$	190536.	1.04826	0.524128	+	0.851639i	$0.324391\pi$
$128$	95287.7	0.514058
$129$	0	0
$130$	46818.4	0.242973
$131$	−259228.	−1.31978	−0.659892	−	0.751360i	$-0.729398\pi$
$131$	−259228.	−1.31978	−0.659892	+	0.751360i	$0.729398\pi$
$132$	0	0
$133$	5904.31	0.0289428
$134$	−129181.	−0.621491
$135$	0	0
$136$	−211791.	−0.981884
$137$	146907.	0.668715	0.334358	−	0.942446i	$-0.391481\pi$
$137$	146907.	0.668715	0.334358	+	0.942446i	$0.391481\pi$
$138$	0	0
$139$	−269665.	−1.18382	−0.591912	−	0.806003i	$-0.701627\pi$
$139$	−269665.	−1.18382	−0.591912	+	0.806003i	$0.701627\pi$
$140$	−355.363	−0.00153233
$141$	0	0
$142$	255131.	1.06180
$143$	45785.0	0.187233
$144$	0	0
$145$	70850.8	0.279850
$146$	184816.	0.717559
$147$	0	0
$148$	14015.2	0.0525952
$149$	407608.	1.50410	0.752050	−	0.659106i	$-0.229065\pi$
$149$	407608.	1.50410	0.752050	+	0.659106i	$0.229065\pi$
$150$	0	0
$151$	−61604.8	−0.219873	−0.109937	−	0.993939i	$-0.535065\pi$
$151$	−61604.8	−0.219873	−0.109937	+	0.993939i	$0.535065\pi$
$152$	609506.	2.13978
$153$	0	0
$154$	1134.24	0.00385394
$155$	−92568.1	−0.309480
$156$	0	0
$157$	80442.5	0.260457	0.130229	−	0.991484i	$-0.458429\pi$
$157$	80442.5	0.260457	0.130229	+	0.991484i	$0.458429\pi$
$158$	220879.	0.703901
$159$	0	0
$160$	−66399.6	−0.205053
$161$	−6828.85	−0.0207626
$162$	0	0
$163$	−341306.	−1.00618	−0.503090	−	0.864234i	$-0.667803\pi$
$163$	−341306.	−1.00618	−0.503090	+	0.864234i	$0.667803\pi$
$164$	88353.7	0.256516
$165$	0	0
$166$	204074.	0.574801
$167$	−504106.	−1.39872	−0.699360	−	0.714769i	$-0.746532\pi$
$167$	−504106.	−1.39872	−0.699360	+	0.714769i	$0.746532\pi$
$168$	0	0
$169$	−228115.	−0.614381
$170$	−134028.	−0.355691
$171$	0	0
$172$	−140966.	−0.363325
$173$	346912.	0.881260	0.440630	−	0.897689i	$-0.354755\pi$
$173$	346912.	0.881260	0.440630	+	0.897689i	$0.354755\pi$
$174$	0	0
$175$	−1183.76	−0.00292191
$176$	88029.4	0.214213
$177$	0	0
$178$	−215388.	−0.509531
$179$	−264949.	−0.618058	−0.309029	−	0.951053i	$-0.600004\pi$
$179$	−264949.	−0.618058	−0.309029	+	0.951053i	$0.600004\pi$
$180$	0	0
$181$	−133409.	−0.302684	−0.151342	−	0.988481i	$-0.548360\pi$
$181$	−133409.	−0.302684	−0.151342	+	0.988481i	$0.548360\pi$
$182$	3546.98	0.00793744
$183$	0	0
$184$	−704946.	−1.53501
$185$	46686.4	0.100291
$186$	0	0
$187$	−131070.	−0.274093
$188$	−143891.	−0.296920
$189$	0	0
$190$	385714.	0.775143
$191$	−431716.	−0.856277	−0.428139	−	0.903713i	$-0.640830\pi$
$191$	−431716.	−0.856277	−0.428139	+	0.903713i	$0.640830\pi$
$192$	0	0
$193$	298294.	0.576437	0.288218	−	0.957565i	$-0.406937\pi$
$193$	298294.	0.576437	0.288218	+	0.957565i	$0.406937\pi$
$194$	−425513.	−0.811724
$195$	0	0
$196$	126109.	0.234481
$197$	2845.11	0.00522315	0.00261158	−	0.999997i	$-0.499169\pi$
$197$	2845.11	0.00522315	0.00261158	+	0.999997i	$0.499169\pi$
$198$	0	0
$199$	405609.	0.726063	0.363032	−	0.931777i	$-0.381742\pi$
$199$	405609.	0.726063	0.363032	+	0.931777i	$0.381742\pi$
$200$	−122200.	−0.216021
$201$	0	0
$202$	642985.	1.10872
$203$	5367.69	0.00914213
$204$	0	0
$205$	294316.	0.489136
$206$	745284.	1.22364
$207$	0	0
$208$	275283.	0.441186
$209$	377201.	0.597319
$210$	0	0
$211$	1.12008e6	1.73199	0.865993	−	0.500057i	$-0.166687\pi$
$211$	1.12008e6	1.73199	0.865993	+	0.500057i	$0.166687\pi$
$212$	−270899.	−0.413968
$213$	0	0
$214$	92504.9	0.138080
$215$	−469576.	−0.692802
$216$	0	0
$217$	−7013.00	−0.0101101
$218$	748501.	1.06672
$219$	0	0
$220$	−22702.6	−0.0316242
$221$	−409878.	−0.564512
$222$	0	0
$223$	−107784.	−0.145141	−0.0725705	−	0.997363i	$-0.523120\pi$
$223$	−107784.	−0.145141	−0.0725705	+	0.997363i	$0.523120\pi$
$224$	−5030.46	−0.00669866
$225$	0	0
$226$	28129.2	0.0366342
$227$	423724.	0.545781	0.272891	−	0.962045i	$-0.412020\pi$
$227$	423724.	0.545781	0.272891	+	0.962045i	$0.412020\pi$
$228$	0	0
$229$	−1.45283e6	−1.83073	−0.915366	−	0.402622i	$-0.868099\pi$
$229$	−1.45283e6	−1.83073	−0.915366	+	0.402622i	$0.868099\pi$
$230$	−446112.	−0.556063
$231$	0	0
$232$	554110.	0.675890
$233$	367393.	0.443344	0.221672	−	0.975121i	$-0.428849\pi$
$233$	367393.	0.443344	0.221672	+	0.975121i	$0.428849\pi$
$234$	0	0
$235$	−479317.	−0.566179
$236$	245809.	0.287289
$237$	0	0
$238$	−10154.0	−0.0116197
$239$	546441.	0.618798	0.309399	−	0.950932i	$-0.399872\pi$
$239$	546441.	0.618798	0.309399	+	0.950932i	$0.399872\pi$
$240$	0	0
$241$	−676946.	−0.750778	−0.375389	−	0.926867i	$-0.622491\pi$
$241$	−676946.	−0.750778	−0.375389	+	0.926867i	$0.622491\pi$
$242$	72461.9	0.0795373
$243$	0	0
$244$	−88970.7	−0.0956692
$245$	420085.	0.447118
$246$	0	0
$247$	1.17957e6	1.23022
$248$	−723956.	−0.747451
$249$	0	0
$250$	−77331.9	−0.0782544
$251$	246453.	0.246917	0.123458	−	0.992350i	$-0.460601\pi$
$251$	246453.	0.246917	0.123458	+	0.992350i	$0.460601\pi$
$252$	0	0
$253$	−436265.	−0.428498
$254$	943008.	0.917131
$255$	0	0
$256$	−694017.	−0.661867
$257$	69607.2	0.0657388	0.0328694	−	0.999460i	$-0.489535\pi$
$257$	69607.2	0.0657388	0.0328694	+	0.999460i	$0.489535\pi$
$258$	0	0
$259$	3536.98	0.00327629
$260$	−70995.0	−0.0651320
$261$	0	0
$262$	−1.28298e6	−1.15469
$263$	1.34607e6	1.19999	0.599996	−	0.800003i	$-0.295169\pi$
$263$	1.34607e6	1.19999	0.599996	+	0.800003i	$0.295169\pi$
$264$	0	0
$265$	−902395.	−0.789372
$266$	29221.9	0.0253223
$267$	0	0
$268$	195888.	0.166599
$269$	−1.16307e6	−0.980002	−0.490001	−	0.871722i	$-0.663004\pi$
$269$	−1.16307e6	−0.980002	−0.490001	+	0.871722i	$0.663004\pi$
$270$	0	0
$271$	283879.	0.234806	0.117403	−	0.993084i	$-0.462543\pi$
$271$	283879.	0.234806	0.117403	+	0.993084i	$0.462543\pi$
$272$	−788059.	−0.645857
$273$	0	0
$274$	727079.	0.585066
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	−201333.	−0.157658	−0.0788288	−	0.996888i	$-0.525118\pi$
$277$	−201333.	−0.157658	−0.0788288	+	0.996888i	$0.525118\pi$
$278$	−1.33464e6	−1.03574
$279$	0	0
$280$	−9257.91	−0.00705697
$281$	−1.06994e6	−0.808340	−0.404170	−	0.914684i	$-0.632440\pi$
$281$	−1.06994e6	−0.808340	−0.404170	+	0.914684i	$0.632440\pi$
$282$	0	0
$283$	−328502.	−0.243822	−0.121911	−	0.992541i	$-0.538902\pi$
$283$	−328502.	−0.243822	−0.121911	+	0.992541i	$0.538902\pi$
$284$	−386879.	−0.284629
$285$	0	0
$286$	226601.	0.163812
$287$	22297.5	0.0159791
$288$	0	0
$289$	−246492.	−0.173603
$290$	350658.	0.244844
$291$	0	0
$292$	−280254.	−0.192351
$293$	1.74357e6	1.18651	0.593254	−	0.805015i	$-0.297843\pi$
$293$	1.74357e6	1.18651	0.593254	+	0.805015i	$0.297843\pi$
$294$	0	0
$295$	818820.	0.547814
$296$	365124.	0.242221
$297$	0	0
$298$	2.01735e6	1.31595
$299$	−1.36428e6	−0.882520
$300$	0	0
$301$	−35575.2	−0.0226325
$302$	−304897.	−0.192370
$303$	0	0
$304$	2.26793e6	1.40749
$305$	−296372.	−0.182426
$306$	0	0
$307$	579249.	0.350768	0.175384	−	0.984500i	$-0.443883\pi$
$307$	579249.	0.350768	0.175384	+	0.984500i	$0.443883\pi$
$308$	−1719.96	−0.00103310
$309$	0	0
$310$	−458142.	−0.270767
$311$	1.71450e6	1.00516	0.502581	−	0.864530i	$-0.332384\pi$
$311$	1.71450e6	1.00516	0.502581	+	0.864530i	$0.332384\pi$
$312$	0	0
$313$	−2.76316e6	−1.59421	−0.797105	−	0.603841i	$-0.793636\pi$
$313$	−2.76316e6	−1.59421	−0.797105	+	0.603841i	$0.793636\pi$
$314$	398129.	0.227877
$315$	0	0
$316$	−334939.	−0.188690
$317$	1.32752e6	0.741978	0.370989	−	0.928637i	$-0.379019\pi$
$317$	1.32752e6	0.741978	0.370989	+	0.928637i	$0.379019\pi$
$318$	0	0
$319$	342918.	0.188675
$320$	−910640.	−0.497132
$321$	0	0
$322$	−33797.6	−0.0181655
$323$	−3.37679e6	−1.80093
$324$	0	0
$325$	−236493.	−0.124196
$326$	−1.68921e6	−0.880317
$327$	0	0
$328$	2.30179e6	1.18136
$329$	−36313.3	−0.0184959
$330$	0	0
$331$	−330933.	−0.166024	−0.0830119	−	0.996549i	$-0.526454\pi$
$331$	−330933.	−0.166024	−0.0830119	+	0.996549i	$0.526454\pi$
$332$	−309456.	−0.154083
$333$	0	0
$334$	−2.49494e6	−1.22376
$335$	652527.	0.317677
$336$	0	0
$337$	−2.17940e6	−1.04535	−0.522675	−	0.852532i	$-0.675066\pi$
$337$	−2.17940e6	−1.04535	−0.522675	+	0.852532i	$0.675066\pi$
$338$	−1.12900e6	−0.537529
$339$	0	0
$340$	203239.	0.0953476
$341$	−448030.	−0.208651
$342$	0	0
$343$	63658.5	0.0292160
$344$	−3.67245e6	−1.67325
$345$	0	0
$346$	1.71695e6	0.771024
$347$	2.89823e6	1.29214	0.646069	−	0.763279i	$-0.276412\pi$
$347$	2.89823e6	1.29214	0.646069	+	0.763279i	$0.276412\pi$
$348$	0	0
$349$	1.43020e6	0.628540	0.314270	−	0.949334i	$-0.398240\pi$
$349$	1.43020e6	0.628540	0.314270	+	0.949334i	$0.398240\pi$
$350$	−5858.70	−0.00255641
$351$	0	0
$352$	−321374.	−0.138247
$353$	3.02584e6	1.29244	0.646218	−	0.763153i	$-0.276350\pi$
$353$	3.02584e6	1.29244	0.646218	+	0.763153i	$0.276350\pi$
$354$	0	0
$355$	−1.28874e6	−0.542742
$356$	326612.	0.136586
$357$	0	0
$358$	−1.31130e6	−0.540746
$359$	−2.57569e6	−1.05477	−0.527384	−	0.849627i	$-0.676827\pi$
$359$	−2.57569e6	−1.05477	−0.527384	+	0.849627i	$0.676827\pi$
$360$	0	0
$361$	7.24183e6	2.92469
$362$	−660275.	−0.264822
$363$	0	0
$364$	−5378.61	−0.00212773
$365$	−933557.	−0.366783
$366$	0	0
$367$	3.37921e6	1.30963	0.654817	−	0.755787i	$-0.272746\pi$
$367$	3.37921e6	1.30963	0.654817	+	0.755787i	$0.272746\pi$
$368$	−2.62305e6	−1.00969
$369$	0	0
$370$	231062.	0.0877454
$371$	−68365.8	−0.0257872
$372$	0	0
$373$	−1.57502e6	−0.586157	−0.293078	−	0.956088i	$-0.594680\pi$
$373$	−1.57502e6	−0.586157	−0.293078	+	0.956088i	$0.594680\pi$
$374$	−648695.	−0.239807
$375$	0	0
$376$	−3.74864e6	−1.36743
$377$	1.07236e6	0.388588
$378$	0	0
$379$	−1.46645e6	−0.524406	−0.262203	−	0.965013i	$-0.584449\pi$
$379$	−1.46645e6	−0.524406	−0.262203	+	0.965013i	$0.584449\pi$
$380$	−584894.	−0.207787
$381$	0	0
$382$	−2.13667e6	−0.749166
$383$	243072.	0.0846716	0.0423358	−	0.999103i	$-0.486520\pi$
$383$	243072.	0.0846716	0.0423358	+	0.999103i	$0.486520\pi$
$384$	0	0
$385$	−5729.38	−0.00196995
$386$	1.47633e6	0.504331
$387$	0	0
$388$	645245.	0.217593
$389$	−5.45247e6	−1.82692	−0.913459	−	0.406930i	$-0.866599\pi$
$389$	−5.45247e6	−1.82692	−0.913459	+	0.406930i	$0.866599\pi$
$390$	0	0
$391$	3.90554e6	1.29193
$392$	3.28540e6	1.07987
$393$	0	0
$394$	14081.1	0.00456979
$395$	−1.11572e6	−0.359801
$396$	0	0
$397$	−2.74224e6	−0.873232	−0.436616	−	0.899648i	$-0.643823\pi$
$397$	−2.74224e6	−0.873232	−0.436616	+	0.899648i	$0.643823\pi$
$398$	2.00746e6	0.635240
$399$	0	0
$400$	−454697.	−0.142093
$401$	5.54146e6	1.72093	0.860465	−	0.509510i	$-0.170173\pi$
$401$	5.54146e6	1.72093	0.860465	+	0.509510i	$0.170173\pi$
$402$	0	0
$403$	−1.40107e6	−0.429731
$404$	−975017.	−0.297207
$405$	0	0
$406$	26566.0	0.00799855
$407$	225962.	0.0676160
$408$	0	0
$409$	2.04392e6	0.604165	0.302083	−	0.953282i	$-0.402318\pi$
$409$	2.04392e6	0.604165	0.302083	+	0.953282i	$0.402318\pi$
$410$	1.45664e6	0.427950
$411$	0	0
$412$	−1.13014e6	−0.328013
$413$	62034.1	0.0178960
$414$	0	0
$415$	−1.03084e6	−0.293812
$416$	−1.00499e6	−0.284728
$417$	0	0
$418$	1.86686e6	0.522601
$419$	340963.	0.0948796	0.0474398	−	0.998874i	$-0.484894\pi$
$419$	340963.	0.0948796	0.0474398	+	0.998874i	$0.484894\pi$
$420$	0	0
$421$	−1.89163e6	−0.520152	−0.260076	−	0.965588i	$-0.583748\pi$
$421$	−1.89163e6	−0.520152	−0.260076	+	0.965588i	$0.583748\pi$
$422$	5.54357e6	1.51533
$423$	0	0
$424$	−7.05744e6	−1.90648
$425$	677012.	0.181813
$426$	0	0
$427$	−22453.2	−0.00595949
$428$	−140274.	−0.0370141
$429$	0	0
$430$	−2.32404e6	−0.606140
$431$	−4.45748e6	−1.15584	−0.577918	−	0.816095i	$-0.696135\pi$
$431$	−4.45748e6	−1.15584	−0.577918	+	0.816095i	$0.696135\pi$
$432$	0	0
$433$	−2.92437e6	−0.749571	−0.374786	−	0.927111i	$-0.622284\pi$
$433$	−2.92437e6	−0.749571	−0.374786	+	0.927111i	$0.622284\pi$
$434$	−34709.0	−0.00884541
$435$	0	0
$436$	−1.13502e6	−0.285948
$437$	−1.12396e7	−2.81545
$438$	0	0
$439$	−2.03890e6	−0.504934	−0.252467	−	0.967605i	$-0.581242\pi$
$439$	−2.03890e6	−0.504934	−0.252467	+	0.967605i	$0.581242\pi$
$440$	−591447.	−0.145641
$441$	0	0
$442$	−2.02858e6	−0.493898
$443$	3.95910e6	0.958488	0.479244	−	0.877682i	$-0.340911\pi$
$443$	3.95910e6	0.958488	0.479244	+	0.877682i	$0.340911\pi$
$444$	0	0
$445$	1.08798e6	0.260449
$446$	−533447.	−0.126985
$447$	0	0
$448$	−68990.5	−0.0162403
$449$	−5.90201e6	−1.38161	−0.690803	−	0.723043i	$-0.742743\pi$
$449$	−5.90201e6	−1.38161	−0.690803	+	0.723043i	$0.742743\pi$
$450$	0	0
$451$	1.42449e6	0.329775
$452$	−42654.9	−0.00982026
$453$	0	0
$454$	2.09711e6	0.477510
$455$	−17916.8	−0.00405725
$456$	0	0
$457$	−5.94885e6	−1.33242	−0.666212	−	0.745762i	$-0.732086\pi$
$457$	−5.94885e6	−1.33242	−0.666212	+	0.745762i	$0.732086\pi$
$458$	−7.19039e6	−1.60173
$459$	0	0
$460$	676480.	0.149060
$461$	4.89147e6	1.07198	0.535991	−	0.844224i	$-0.319938\pi$
$461$	4.89147e6	1.07198	0.535991	+	0.844224i	$0.319938\pi$
$462$	0	0
$463$	1.38567e6	0.300406	0.150203	−	0.988655i	$-0.452007\pi$
$463$	1.38567e6	0.300406	0.150203	+	0.988655i	$0.452007\pi$
$464$	2.06180e6	0.444582
$465$	0	0
$466$	1.81832e6	0.387886
$467$	−8.55531e6	−1.81528	−0.907640	−	0.419750i	$-0.862118\pi$
$467$	−8.55531e6	−1.81528	−0.907640	+	0.419750i	$0.862118\pi$
$468$	0	0
$469$	49435.7	0.0103779
$470$	−2.37226e6	−0.495356
$471$	0	0
$472$	6.40382e6	1.32307
$473$	−2.27275e6	−0.467087
$474$	0	0
$475$	−1.94835e6	−0.396217
$476$	15397.5	0.00311481
$477$	0	0
$478$	2.70447e6	0.541393
$479$	1.76253e6	0.350993	0.175497	−	0.984480i	$-0.443847\pi$
$479$	1.76253e6	0.350993	0.175497	+	0.984480i	$0.443847\pi$
$480$	0	0
$481$	706623.	0.139259
$482$	−3.35037e6	−0.656864
$483$	0	0
$484$	−109881.	−0.0213210
$485$	2.14938e6	0.414916
$486$	0	0
$487$	4.94747e6	0.945280	0.472640	−	0.881256i	$-0.343301\pi$
$487$	4.94747e6	0.945280	0.472640	+	0.881256i	$0.343301\pi$
$488$	−2.31786e6	−0.440593
$489$	0	0
$490$	2.07910e6	0.391189
$491$	−3.17558e6	−0.594456	−0.297228	−	0.954807i	$-0.596062\pi$
$491$	−3.17558e6	−0.594456	−0.297228	+	0.954807i	$0.596062\pi$
$492$	0	0
$493$	−3.06988e6	−0.568859
$494$	5.83799e6	1.07633
$495$	0	0
$496$	−2.69379e6	−0.491654
$497$	−97635.3	−0.0177303
$498$	0	0
$499$	616758.	0.110883	0.0554413	−	0.998462i	$-0.482343\pi$
$499$	616758.	0.110883	0.0554413	+	0.998462i	$0.482343\pi$
$500$	117266.	0.0209771
$501$	0	0
$502$	1.21976e6	0.216030
$503$	6.22348e6	1.09677	0.548383	−	0.836228i	$-0.315244\pi$
$503$	6.22348e6	1.09677	0.548383	+	0.836228i	$0.315244\pi$
$504$	0	0
$505$	−3.24790e6	−0.566727
$506$	−2.15918e6	−0.374898
$507$	0	0
$508$	−1.42997e6	−0.245849
$509$	−3.93927e6	−0.673941	−0.336970	−	0.941515i	$-0.609402\pi$
$509$	−3.93927e6	−0.673941	−0.336970	+	0.941515i	$0.609402\pi$
$510$	0	0
$511$	−70726.7	−0.0119820
$512$	−6.48407e6	−1.09313
$513$	0	0
$514$	344503.	0.0575156
$515$	−3.76464e6	−0.625468
$516$	0	0
$517$	−2.31990e6	−0.381718
$518$	17505.4	0.00286647
$519$	0	0
$520$	−1.84956e6	−0.299958
$521$	1.15068e7	1.85721	0.928605	−	0.371071i	$-0.121009\pi$
$521$	1.15068e7	1.85721	0.928605	+	0.371071i	$0.121009\pi$
$522$	0	0
$523$	−9.12252e6	−1.45835	−0.729173	−	0.684329i	$-0.760095\pi$
$523$	−9.12252e6	−1.45835	−0.729173	+	0.684329i	$0.760095\pi$
$524$	1.94550e6	0.309530
$525$	0	0
$526$	6.66203e6	1.04989
$527$	4.01086e6	0.629088
$528$	0	0
$529$	6.56325e6	1.01972
$530$	−4.46617e6	−0.690630
$531$	0	0
$532$	−44311.8	−0.00678798
$533$	4.45463e6	0.679194
$534$	0	0
$535$	−467268.	−0.0705799
$536$	5.10328e6	0.767250
$537$	0	0
$538$	−5.75634e6	−0.857414
$539$	2.03321e6	0.301447
$540$	0	0
$541$	−2.88055e6	−0.423138	−0.211569	−	0.977363i	$-0.567857\pi$
$541$	−2.88055e6	−0.423138	−0.211569	+	0.977363i	$0.567857\pi$
$542$	1.40498e6	0.205434
$543$	0	0
$544$	2.87701e6	0.416816
$545$	−3.78088e6	−0.545258
$546$	0	0
$547$	1.22189e7	1.74608	0.873038	−	0.487653i	$-0.162147\pi$
$547$	1.22189e7	1.74608	0.873038	+	0.487653i	$0.162147\pi$
$548$	−1.10254e6	−0.156834
$549$	0	0
$550$	−374287.	−0.0527591
$551$	8.83470e6	1.23969
$552$	0	0
$553$	−84527.5	−0.0117540
$554$	−996445.	−0.137936
$555$	0	0
$556$	2.02383e6	0.277644
$557$	5.30627e6	0.724688	0.362344	−	0.932044i	$-0.381976\pi$
$557$	5.30627e6	0.724688	0.362344	+	0.932044i	$0.381976\pi$
$558$	0	0
$559$	−7.10727e6	−0.961997
$560$	−34448.0	−0.00464189
$561$	0	0
$562$	−5.29540e6	−0.707225
$563$	−1.17446e6	−0.156160	−0.0780799	−	0.996947i	$-0.524879\pi$
$563$	−1.17446e6	−0.156160	−0.0780799	+	0.996947i	$0.524879\pi$
$564$	0	0
$565$	−142088.	−0.0187257
$566$	−1.62584e6	−0.213322
$567$	0	0
$568$	−1.00790e7	−1.31082
$569$	1.06868e7	1.38379	0.691893	−	0.722000i	$-0.256777\pi$
$569$	1.06868e7	1.38379	0.691893	+	0.722000i	$0.256777\pi$
$570$	0	0
$571$	−8.65573e6	−1.11100	−0.555499	−	0.831517i	$-0.687473\pi$
$571$	−8.65573e6	−1.11100	−0.555499	+	0.831517i	$0.687473\pi$
$572$	−343616.	−0.0439120
$573$	0	0
$574$	110356.	0.0139803
$575$	2.25343e6	0.284234
$576$	0	0
$577$	1.22736e7	1.53473	0.767363	−	0.641213i	$-0.221568\pi$
$577$	1.22736e7	1.53473	0.767363	+	0.641213i	$0.221568\pi$
$578$	−1.21995e6	−0.151887
$579$	0	0
$580$	−531735.	−0.0656335
$581$	−78096.5	−0.00959823
$582$	0	0
$583$	−4.36759e6	−0.532195
$584$	−7.30116e6	−0.885849
$585$	0	0
$586$	8.62936e6	1.03809
$587$	2.68106e6	0.321152	0.160576	−	0.987023i	$-0.448665\pi$
$587$	2.68106e6	0.321152	0.160576	+	0.987023i	$0.448665\pi$
$588$	0	0
$589$	−1.15427e7	−1.37095
$590$	4.05254e6	0.479288
$591$	0	0
$592$	1.35860e6	0.159326
$593$	1.22632e7	1.43208	0.716041	−	0.698058i	$-0.245952\pi$
$593$	1.22632e7	1.43208	0.716041	+	0.698058i	$0.245952\pi$
$594$	0	0
$595$	51290.7	0.00593945
$596$	−3.05909e6	−0.352758
$597$	0	0
$598$	−6.75214e6	−0.772126
$599$	1.10008e7	1.25273	0.626367	−	0.779529i	$-0.284541\pi$
$599$	1.10008e7	1.25273	0.626367	+	0.779529i	$0.284541\pi$
$600$	0	0
$601$	1.43987e7	1.62607	0.813033	−	0.582217i	$-0.197815\pi$
$601$	1.43987e7	1.62607	0.813033	+	0.582217i	$0.197815\pi$
$602$	−176071.	−0.0198014
$603$	0	0
$604$	462344.	0.0515671
$605$	−366025.	−0.0406558
$606$	0	0
$607$	1.24380e7	1.37019	0.685093	−	0.728456i	$-0.259762\pi$
$607$	1.24380e7	1.37019	0.685093	+	0.728456i	$0.259762\pi$
$608$	−8.27966e6	−0.908350
$609$	0	0
$610$	−1.46682e6	−0.159607
$611$	−7.25472e6	−0.786172
$612$	0	0
$613$	8.31062e6	0.893269	0.446635	−	0.894716i	$-0.352622\pi$
$613$	8.31062e6	0.893269	0.446635	+	0.894716i	$0.352622\pi$
$614$	2.86685e6	0.306891
$615$	0	0
$616$	−44808.3	−0.00475781
$617$	1.70807e7	1.80631	0.903156	−	0.429312i	$-0.141244\pi$
$617$	1.70807e7	1.80631	0.903156	+	0.429312i	$0.141244\pi$
$618$	0	0
$619$	−4.29657e6	−0.450708	−0.225354	−	0.974277i	$-0.572354\pi$
$619$	−4.29657e6	−0.450708	−0.225354	+	0.974277i	$0.572354\pi$
$620$	694723.	0.0725826
$621$	0	0
$622$	8.48547e6	0.879427
$623$	82426.0	0.00850833
$624$	0	0
$625$	390625.	0.0400000
$626$	−1.36756e7	−1.39479
$627$	0	0
$628$	−603720.	−0.0610853
$629$	−2.02286e6	−0.203864
$630$	0	0
$631$	9.11607e6	0.911453	0.455727	−	0.890120i	$-0.349379\pi$
$631$	9.11607e6	0.911453	0.455727	+	0.890120i	$0.349379\pi$
$632$	−8.72582e6	−0.868987
$633$	0	0
$634$	6.57019e6	0.649165
$635$	−4.76340e6	−0.468794
$636$	0	0
$637$	6.35821e6	0.620850
$638$	1.69719e6	0.165074
$639$	0	0
$640$	−2.38219e6	−0.229894
$641$	1.38475e7	1.33114	0.665572	−	0.746333i	$-0.268187\pi$
$641$	1.38475e7	1.33114	0.665572	+	0.746333i	$0.268187\pi$
$642$	0	0
$643$	−4.72967e6	−0.451132	−0.225566	−	0.974228i	$-0.572423\pi$
$643$	−4.72967e6	−0.451132	−0.225566	+	0.974228i	$0.572423\pi$
$644$	51250.4	0.00486948
$645$	0	0
$646$	−1.67125e7	−1.57565
$647$	1.72220e7	1.61742	0.808708	−	0.588210i	$-0.200167\pi$
$647$	1.72220e7	1.61742	0.808708	+	0.588210i	$0.200167\pi$
$648$	0	0
$649$	3.96309e6	0.369336
$650$	−1.17046e6	−0.108661
$651$	0	0
$652$	2.56150e6	0.235980
$653$	−8.17825e6	−0.750546	−0.375273	−	0.926914i	$-0.622451\pi$
$653$	−8.17825e6	−0.750546	−0.375273	+	0.926914i	$0.622451\pi$
$654$	0	0
$655$	6.48069e6	0.590225
$656$	8.56478e6	0.777064
$657$	0	0
$658$	−179723.	−0.0161823
$659$	1.92561e7	1.72725	0.863623	−	0.504138i	$-0.168190\pi$
$659$	1.92561e7	1.72725	0.863623	+	0.504138i	$0.168190\pi$
$660$	0	0
$661$	2.14935e7	1.91339	0.956694	−	0.291094i	$-0.0940193\pi$
$661$	2.14935e7	1.91339	0.956694	+	0.291094i	$0.0940193\pi$
$662$	−1.63787e6	−0.145256
$663$	0	0
$664$	−8.06195e6	−0.709610
$665$	−147608.	−0.0129436
$666$	0	0
$667$	−1.02181e7	−0.889315
$668$	3.78331e6	0.328043
$669$	0	0
$670$	3.22951e6	0.277939
$671$	−1.43444e6	−0.122992
$672$	0	0
$673$	1.27765e7	1.08736	0.543680	−	0.839293i	$-0.317031\pi$
$673$	1.27765e7	1.08736	0.543680	+	0.839293i	$0.317031\pi$
$674$	−1.07864e7	−0.914588
$675$	0	0
$676$	1.71200e6	0.144091
$677$	−5.84159e6	−0.489846	−0.244923	−	0.969543i	$-0.578763\pi$
$677$	−5.84159e6	−0.489846	−0.244923	+	0.969543i	$0.578763\pi$
$678$	0	0
$679$	162838.	0.0135545
$680$	5.29477e6	0.439112
$681$	0	0
$682$	−2.21741e6	−0.182551
$683$	−1.20216e7	−0.986077	−0.493038	−	0.870008i	$-0.664114\pi$
$683$	−1.20216e7	−0.986077	−0.493038	+	0.870008i	$0.664114\pi$
$684$	0	0
$685$	−3.67268e6	−0.299059
$686$	315061.	0.0255614
$687$	0	0
$688$	−1.36649e7	−1.10062
$689$	−1.36582e7	−1.09609
$690$	0	0
$691$	−546946.	−0.0435762	−0.0217881	−	0.999763i	$-0.506936\pi$
$691$	−546946.	−0.0435762	−0.0217881	+	0.999763i	$0.506936\pi$
$692$	−2.60357e6	−0.206683
$693$	0	0
$694$	1.43440e7	1.13051
$695$	6.74162e6	0.529422
$696$	0	0
$697$	−1.27524e7	−0.994281
$698$	7.07841e6	0.549917
$699$	0	0
$700$	8884.08	0.000685279 0
$701$	29732.9	0.00228529	0.00114265	−	0.999999i	$-0.499636\pi$
$701$	29732.9	0.00228529	0.00114265	+	0.999999i	$0.499636\pi$
$702$	0	0
$703$	5.82153e6	0.444272
$704$	−4.40750e6	−0.335166
$705$	0	0
$706$	1.49756e7	1.13077
$707$	−246062.	−0.0185138
$708$	0	0
$709$	1.98410e7	1.48234	0.741170	−	0.671317i	$-0.234271\pi$
$709$	1.98410e7	1.48234	0.741170	+	0.671317i	$0.234271\pi$
$710$	−6.37828e6	−0.474851
$711$	0	0
$712$	8.50889e6	0.629032
$713$	1.33502e7	0.983473
$714$	0	0
$715$	−1.14462e6	−0.0837332
$716$	1.98844e6	0.144954
$717$	0	0
$718$	−1.27477e7	−0.922828
$719$	−6.54460e6	−0.472129	−0.236065	−	0.971737i	$-0.575858\pi$
$719$	−6.54460e6	−0.472129	−0.236065	+	0.971737i	$0.575858\pi$
$720$	0	0
$721$	−285211.	−0.0204328
$722$	3.58416e7	2.55885
$723$	0	0
$724$	1.00124e6	0.0709888
$725$	−1.77127e6	−0.125153
$726$	0	0
$727$	−1.01284e7	−0.710733	−0.355367	−	0.934727i	$-0.615644\pi$
$727$	−1.01284e7	−0.710733	−0.355367	+	0.934727i	$0.615644\pi$
$728$	−140123.	−0.00979901
$729$	0	0
$730$	−4.62040e6	−0.320902
$731$	2.03461e7	1.40828
$732$	0	0
$733$	2.60835e7	1.79311	0.896553	−	0.442937i	$-0.146064\pi$
$733$	2.60835e7	1.79311	0.896553	+	0.442937i	$0.146064\pi$
$734$	1.67245e7	1.14581
$735$	0	0
$736$	9.57614e6	0.651622
$737$	3.15823e6	0.214178
$738$	0	0
$739$	−1.19682e7	−0.806155	−0.403077	−	0.915166i	$-0.632059\pi$
$739$	−1.19682e7	−0.806155	−0.403077	+	0.915166i	$0.632059\pi$
$740$	−350381.	−0.0235213
$741$	0	0
$742$	−338359.	−0.0225615
$743$	−2.13249e6	−0.141715	−0.0708573	−	0.997486i	$-0.522574\pi$
$743$	−2.13249e6	−0.141715	−0.0708573	+	0.997486i	$0.522574\pi$
$744$	0	0
$745$	−1.01902e7	−0.672654
$746$	−7.79515e6	−0.512835
$747$	0	0
$748$	983677.	0.0642833
$749$	−35400.4	−0.00230570
$750$	0	0
$751$	−2.81351e6	−0.182032	−0.0910162	−	0.995849i	$-0.529012\pi$
$751$	−2.81351e6	−0.182032	−0.0910162	+	0.995849i	$0.529012\pi$
$752$	−1.39484e7	−0.899458
$753$	0	0
$754$	5.30739e6	0.339980
$755$	1.54012e6	0.0983303
$756$	0	0
$757$	1.86775e6	0.118462	0.0592311	−	0.998244i	$-0.481135\pi$
$757$	1.86775e6	0.118462	0.0592311	+	0.998244i	$0.481135\pi$
$758$	−7.25780e6	−0.458809
$759$	0	0
$760$	−1.52376e7	−0.956938
$761$	−1.07993e6	−0.0675978	−0.0337989	−	0.999429i	$-0.510761\pi$
$761$	−1.07993e6	−0.0675978	−0.0337989	+	0.999429i	$0.510761\pi$
$762$	0	0
$763$	−286441.	−0.0178125
$764$	3.24002e6	0.200824
$765$	0	0
$766$	1.20302e6	0.0740801
$767$	1.23933e7	0.760672
$768$	0	0
$769$	2.29875e7	1.40177	0.700883	−	0.713276i	$-0.252789\pi$
$769$	2.29875e7	1.40177	0.700883	+	0.713276i	$0.252789\pi$
$770$	−28356.1	−0.00172353
$771$	0	0
$772$	−2.23870e6	−0.135192
$773$	−2.92098e7	−1.75824	−0.879122	−	0.476597i	$-0.841870\pi$
$773$	−2.92098e7	−1.75824	−0.879122	+	0.476597i	$0.841870\pi$
$774$	0	0
$775$	2.31420e6	0.138404
$776$	1.68099e7	1.00210
$777$	0	0
$778$	−2.69856e7	−1.59839
$779$	3.66996e7	2.16679
$780$	0	0
$781$	−6.23749e6	−0.365917
$782$	1.93295e7	1.13033
$783$	0	0
$784$	1.22247e7	0.710313
$785$	−2.01106e6	−0.116480
$786$	0	0
$787$	3.07230e7	1.76818	0.884090	−	0.467317i	$-0.154779\pi$
$787$	3.07230e7	1.76818	0.884090	+	0.467317i	$0.154779\pi$
$788$	−21352.5	−0.00122499
$789$	0	0
$790$	−5.52197e6	−0.314794
$791$	−10764.7	−0.000611730 0
$792$	0	0
$793$	−4.48574e6	−0.253309
$794$	−1.35720e7	−0.764000
$795$	0	0
$796$	−3.04409e6	−0.170284
$797$	5.67459e6	0.316438	0.158219	−	0.987404i	$-0.449425\pi$
$797$	5.67459e6	0.316438	0.158219	+	0.987404i	$0.449425\pi$
$798$	0	0
$799$	2.07682e7	1.15089
$800$	1.65999e6	0.0917024
$801$	0	0
$802$	2.74260e7	1.50566
$803$	−4.51842e6	−0.247285
$804$	0	0
$805$	170721.	0.00928533
$806$	−6.93422e6	−0.375976
$807$	0	0
$808$	−2.54011e7	−1.36875
$809$	5.87280e6	0.315482	0.157741	−	0.987481i	$-0.449579\pi$
$809$	5.87280e6	0.315482	0.157741	+	0.987481i	$0.449579\pi$
$810$	0	0
$811$	−6.49092e6	−0.346541	−0.173270	−	0.984874i	$-0.555433\pi$
$811$	−6.49092e6	−0.346541	−0.173270	+	0.984874i	$0.555433\pi$
$812$	−40284.5	−0.00214411
$813$	0	0
$814$	1.11834e6	0.0591580
$815$	8.53266e6	0.449977
$816$	0	0
$817$	−5.85535e7	−3.06900
$818$	1.01159e7	0.528591
$819$	0	0
$820$	−2.20884e6	−0.114718
$821$	1.07277e7	0.555455	0.277728	−	0.960660i	$-0.410419\pi$
$821$	1.07277e7	0.555455	0.277728	+	0.960660i	$0.410419\pi$
$822$	0	0
$823$	9.30076e6	0.478651	0.239325	−	0.970939i	$-0.423074\pi$
$823$	9.30076e6	0.478651	0.239325	+	0.970939i	$0.423074\pi$
$824$	−2.94425e7	−1.51062
$825$	0	0
$826$	307022.	0.0156574
$827$	−2.15608e7	−1.09623	−0.548115	−	0.836403i	$-0.684654\pi$
$827$	−2.15608e7	−1.09623	−0.548115	+	0.836403i	$0.684654\pi$
$828$	0	0
$829$	1.88178e7	0.951003	0.475502	−	0.879715i	$-0.342267\pi$
$829$	1.88178e7	0.951003	0.475502	+	0.879715i	$0.342267\pi$
$830$	−5.10185e6	−0.257059
$831$	0	0
$832$	−1.37830e7	−0.690297
$833$	−1.82018e7	−0.908870
$834$	0	0
$835$	1.26027e7	0.625527
$836$	−2.83089e6	−0.140090
$837$	0	0
$838$	1.68751e6	0.0830112
$839$	−3.49101e6	−0.171217	−0.0856084	−	0.996329i	$-0.527283\pi$
$839$	−3.49101e6	−0.171217	−0.0856084	+	0.996329i	$0.527283\pi$
$840$	0	0
$841$	−1.24794e7	−0.608420
$842$	−9.36212e6	−0.455087
$843$	0	0
$844$	−8.40622e6	−0.406204
$845$	5.70289e6	0.274760
$846$	0	0
$847$	−27730.2	−0.00132814
$848$	−2.62602e7	−1.25403
$849$	0	0
$850$	3.35070e6	0.159070
$851$	−6.73310e6	−0.318707
$852$	0	0
$853$	−2.22413e7	−1.04661	−0.523307	−	0.852144i	$-0.675302\pi$
$853$	−2.22413e7	−1.04661	−0.523307	+	0.852144i	$0.675302\pi$
$854$	−111127.	−0.00521402
$855$	0	0
$856$	−3.65440e6	−0.170464
$857$	−1.58585e7	−0.737582	−0.368791	−	0.929512i	$-0.620228\pi$
$857$	−1.58585e7	−0.737582	−0.368791	+	0.929512i	$0.620228\pi$
$858$	0	0
$859$	3.38437e7	1.56493	0.782464	−	0.622696i	$-0.213962\pi$
$859$	3.38437e7	1.56493	0.782464	+	0.622696i	$0.213962\pi$
$860$	3.52416e6	0.162484
$861$	0	0
$862$	−2.20611e7	−1.01125
$863$	−6.72008e6	−0.307148	−0.153574	−	0.988137i	$-0.549078\pi$
$863$	−6.72008e6	−0.307148	−0.153574	+	0.988137i	$0.549078\pi$
$864$	0	0
$865$	−8.67280e6	−0.394112
$866$	−1.44734e7	−0.655808
$867$	0	0
$868$	52632.5	0.00237113
$869$	−5.40009e6	−0.242578
$870$	0	0
$871$	9.87633e6	0.441114
$872$	−2.95695e7	−1.31690
$873$	0	0
$874$	−5.56276e7	−2.46327
$875$	29593.9	0.00130672
$876$	0	0
$877$	2.59286e7	1.13836	0.569180	−	0.822213i	$-0.307261\pi$
$877$	2.59286e7	1.13836	0.569180	+	0.822213i	$0.307261\pi$
$878$	−1.00910e7	−0.441772
$879$	0	0
$880$	−2.20073e6	−0.0957990
$881$	2.56439e6	0.111313	0.0556564	−	0.998450i	$-0.482275\pi$
$881$	2.56439e6	0.111313	0.0556564	+	0.998450i	$0.482275\pi$
$882$	0	0
$883$	−2.80262e7	−1.20966	−0.604830	−	0.796355i	$-0.706759\pi$
$883$	−2.80262e7	−1.20966	−0.604830	+	0.796355i	$0.706759\pi$
$884$	3.07613e6	0.132396
$885$	0	0
$886$	1.95945e7	0.838592
$887$	−1.80878e7	−0.771928	−0.385964	−	0.922514i	$-0.626131\pi$
$887$	−1.80878e7	−0.771928	−0.385964	+	0.922514i	$0.626131\pi$
$888$	0	0
$889$	−360877.	−0.0153146
$890$	5.38469e6	0.227869
$891$	0	0
$892$	808915.	0.0340401
$893$	−5.97682e7	−2.50808
$894$	0	0
$895$	6.62372e6	0.276404
$896$	−180476.	−0.00751016
$897$	0	0
$898$	−2.92105e7	−1.20878
$899$	−1.04936e7	−0.433039
$900$	0	0
$901$	3.90997e7	1.60458
$902$	7.05015e6	0.288524
$903$	0	0
$904$	−1.11124e6	−0.0452260
$905$	3.33523e6	0.135364
$906$	0	0
$907$	2.90377e7	1.17204	0.586022	−	0.810295i	$-0.300693\pi$
$907$	2.90377e7	1.17204	0.586022	+	0.810295i	$0.300693\pi$
$908$	−3.18005e6	−0.128003
$909$	0	0
$910$	−88674.5	−0.00354973
$911$	4.95469e7	1.97798	0.988988	−	0.147999i	$-0.0472831\pi$
$911$	4.95469e7	1.97798	0.988988	+	0.147999i	$0.0472831\pi$
$912$	0	0
$913$	−4.98924e6	−0.198088
$914$	−2.94423e7	−1.16575
$915$	0	0
$916$	1.09035e7	0.429364
$917$	490980.	0.0192815
$918$	0	0
$919$	−4.28254e7	−1.67268	−0.836339	−	0.548212i	$-0.815309\pi$
$919$	−4.28254e7	−1.67268	−0.836339	+	0.548212i	$0.815309\pi$
$920$	1.76236e7	0.686477
$921$	0	0
$922$	2.42091e7	0.937888
$923$	−1.95057e7	−0.753630
$924$	0	0
$925$	−1.16716e6	−0.0448514
$926$	6.85803e6	0.262828
$927$	0	0
$928$	−7.52715e6	−0.286920
$929$	1.80864e7	0.687564	0.343782	−	0.939049i	$-0.388292\pi$
$929$	1.80864e7	0.687564	0.343782	+	0.939049i	$0.388292\pi$
$930$	0	0
$931$	5.23823e7	1.98066
$932$	−2.75728e6	−0.103978
$933$	0	0
$934$	−4.23423e7	−1.58821
$935$	3.27674e6	0.122578
$936$	0	0
$937$	−1.51452e7	−0.563540	−0.281770	−	0.959482i	$-0.590922\pi$
$937$	−1.51452e7	−0.563540	−0.281770	+	0.959482i	$0.590922\pi$
$938$	244669.	0.00907972
$939$	0	0
$940$	3.59727e6	0.132786
$941$	2.49831e7	0.919756	0.459878	−	0.887982i	$-0.347893\pi$
$941$	2.49831e7	0.919756	0.459878	+	0.887982i	$0.347893\pi$
$942$	0	0
$943$	−4.24462e7	−1.55439
$944$	2.38282e7	0.870283
$945$	0	0
$946$	−1.12484e7	−0.408660
$947$	4.54274e7	1.64605	0.823025	−	0.568005i	$-0.192285\pi$
$947$	4.54274e7	1.64605	0.823025	+	0.568005i	$0.192285\pi$
$948$	0	0
$949$	−1.41299e7	−0.509299
$950$	−9.64286e6	−0.346654
$951$	0	0
$952$	401134.	0.0143449
$953$	−3.72825e6	−0.132976	−0.0664880	−	0.997787i	$-0.521179\pi$
$953$	−3.72825e6	−0.132976	−0.0664880	+	0.997787i	$0.521179\pi$
$954$	0	0
$955$	1.07929e7	0.382939
$956$	−4.10104e6	−0.145127
$957$	0	0
$958$	8.72321e6	0.307088
$959$	−278243.	−0.00976964
$960$	0	0
$961$	−1.49190e7	−0.521112
$962$	3.49725e6	0.121840
$963$	0	0
$964$	5.08048e6	0.176081
$965$	−7.45736e6	−0.257790
$966$	0	0
$967$	−2.83238e6	−0.0974060	−0.0487030	−	0.998813i	$-0.515509\pi$
$967$	−2.83238e6	−0.0974060	−0.0487030	+	0.998813i	$0.515509\pi$
$968$	−2.86261e6	−0.0981913
$969$	0	0
$970$	1.06378e7	0.363014
$971$	−2.56419e7	−0.872775	−0.436387	−	0.899759i	$-0.643742\pi$
$971$	−2.56419e7	−0.872775	−0.436387	+	0.899759i	$0.643742\pi$
$972$	0	0
$973$	510748.	0.0172952
$974$	2.44862e7	0.827036
$975$	0	0
$976$	−8.62460e6	−0.289811
$977$	−6.53289e6	−0.218962	−0.109481	−	0.993989i	$-0.534919\pi$
$977$	−6.53289e6	−0.218962	−0.109481	+	0.993989i	$0.534919\pi$
$978$	0	0
$979$	5.26584e6	0.175594
$980$	−3.15274e6	−0.104863
$981$	0	0
$982$	−1.57167e7	−0.520096
$983$	−1.60792e7	−0.530739	−0.265370	−	0.964147i	$-0.585494\pi$
$983$	−1.60792e7	−0.530739	−0.265370	+	0.964147i	$0.585494\pi$
$984$	0	0
$985$	−71127.7	−0.00233587
$986$	−1.51936e7	−0.497701
$987$	0	0
$988$	−8.85268e6	−0.288524
$989$	6.77221e7	2.20161
$990$	0	0
$991$	−5.58796e7	−1.80746	−0.903732	−	0.428099i	$-0.859183\pi$
$991$	−5.58796e7	−1.80746	−0.903732	+	0.428099i	$0.859183\pi$
$992$	9.83438e6	0.317298
$993$	0	0
$994$	−483221.	−0.0155124
$995$	−1.01402e7	−0.324705
$996$	0	0
$997$	−2.90161e7	−0.924486	−0.462243	−	0.886753i	$-0.652955\pi$
$997$	−2.90161e7	−0.924486	−0.462243	+	0.886753i	$0.652955\pi$
$998$	3.05249e6	0.0970124
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.f.1.2		3
3.2	odd	2		55.6.a.a.1.2	✓	3
12.11	even	2		880.6.a.l.1.1		3
15.2	even	4		275.6.b.c.199.3		6
15.8	even	4		275.6.b.c.199.4		6
15.14	odd	2		275.6.a.c.1.2		3
33.32	even	2		605.6.a.b.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.a.a.1.2	✓	3	3.2	odd	2
275.6.a.c.1.2		3	15.14	odd	2
275.6.b.c.199.3		6	15.2	even	4
275.6.b.c.199.4		6	15.8	even	4
495.6.a.f.1.2		3	1.1	even	1	trivial
605.6.a.b.1.2		3	33.32	even	2
880.6.a.l.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+4.94924 q^{2} -7.50499 q^{4} -25.0000 q^{5} -1.89401 q^{7} -195.520 q^{8} +O(q^{10})\)	\(q+4.94924 q^{2} -7.50499 q^{4} -25.0000 q^{5} -1.89401 q^{7} -195.520 q^{8} -123.731 q^{10} -121.000 q^{11} -378.388 q^{13} -9.37392 q^{14} -727.515 q^{16} +1083.22 q^{17} -3117.36 q^{19} +187.625 q^{20} -598.858 q^{22} +3605.50 q^{23} +625.000 q^{25} -1872.73 q^{26} +14.2145 q^{28} -2834.03 q^{29} +3702.72 q^{31} +2655.98 q^{32} +5361.12 q^{34} +47.3503 q^{35} -1867.45 q^{37} -15428.6 q^{38} +4888.00 q^{40} -11772.7 q^{41} +18783.0 q^{43} +908.104 q^{44} +17844.5 q^{46} +19172.7 q^{47} -16803.4 q^{49} +3093.28 q^{50} +2839.80 q^{52} +36095.8 q^{53} +3025.00 q^{55} +370.317 q^{56} -14026.3 q^{58} -32752.8 q^{59} +11854.9 q^{61} +18325.7 q^{62} +36425.6 q^{64} +9459.70 q^{65} -26101.1 q^{67} -8129.56 q^{68} +234.348 q^{70} +51549.5 q^{71} +37342.3 q^{73} -9242.49 q^{74} +23395.8 q^{76} +229.175 q^{77} +44628.8 q^{79} +18187.9 q^{80} -58265.7 q^{82} +41233.4 q^{83} -27080.5 q^{85} +92961.8 q^{86} +23657.9 q^{88} -43519.3 q^{89} +716.671 q^{91} -27059.2 q^{92} +94890.3 q^{94} +77934.0 q^{95} -85975.4 q^{97} -83164.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 7 q^{2} + 41 q^{4} - 75 q^{5} - 102 q^{7} + 15 q^{8}+O(q^{10}) \) 3 * q + 7 * q^2 + 41 * q^4 - 75 * q^5 - 102 * q^7 + 15 * q^8	\( 3 q + 7 q^{2} + 41 q^{4} - 75 q^{5} - 102 q^{7} + 15 q^{8} - 175 q^{10} - 363 q^{11} - 1646 q^{13} + 963 q^{14} - 2687 q^{16} + 1742 q^{17} - 10 q^{19} - 1025 q^{20} - 847 q^{22} + 3876 q^{23} + 1875 q^{25} + 2894 q^{26} - 209 q^{28} - 1970 q^{29} + 6596 q^{31} - 2353 q^{32} + 16887 q^{34} + 2550 q^{35} - 22032 q^{37} + 2205 q^{38} - 375 q^{40} + 7214 q^{41} + 29644 q^{43} - 4961 q^{44} + 21126 q^{46} + 8432 q^{47} - 34939 q^{49} + 4375 q^{50} - 15462 q^{52} + 31656 q^{53} + 9075 q^{55} - 16235 q^{56} + 94035 q^{58} + 31440 q^{59} - 49374 q^{61} + 100169 q^{62} + 9601 q^{64} + 41150 q^{65} - 111892 q^{67} + 30099 q^{68} - 24075 q^{70} + 58444 q^{71} + 5554 q^{73} + 35743 q^{74} + 128385 q^{76} + 12342 q^{77} + 76080 q^{79} + 67175 q^{80} + 18026 q^{82} - 44844 q^{83} - 43550 q^{85} + 157244 q^{86} - 1815 q^{88} - 25460 q^{89} + 123236 q^{91} - 14338 q^{92} + 411002 q^{94} + 250 q^{95} - 150482 q^{97} - 209376 q^{98}+O(q^{100}) \) 3 * q + 7 * q^2 + 41 * q^4 - 75 * q^5 - 102 * q^7 + 15 * q^8 - 175 * q^10 - 363 * q^11 - 1646 * q^13 + 963 * q^14 - 2687 * q^16 + 1742 * q^17 - 10 * q^19 - 1025 * q^20 - 847 * q^22 + 3876 * q^23 + 1875 * q^25 + 2894 * q^26 - 209 * q^28 - 1970 * q^29 + 6596 * q^31 - 2353 * q^32 + 16887 * q^34 + 2550 * q^35 - 22032 * q^37 + 2205 * q^38 - 375 * q^40 + 7214 * q^41 + 29644 * q^43 - 4961 * q^44 + 21126 * q^46 + 8432 * q^47 - 34939 * q^49 + 4375 * q^50 - 15462 * q^52 + 31656 * q^53 + 9075 * q^55 - 16235 * q^56 + 94035 * q^58 + 31440 * q^59 - 49374 * q^61 + 100169 * q^62 + 9601 * q^64 + 41150 * q^65 - 111892 * q^67 + 30099 * q^68 - 24075 * q^70 + 58444 * q^71 + 5554 * q^73 + 35743 * q^74 + 128385 * q^76 + 12342 * q^77 + 76080 * q^79 + 67175 * q^80 + 18026 * q^82 - 44844 * q^83 - 43550 * q^85 + 157244 * q^86 - 1815 * q^88 - 25460 * q^89 + 123236 * q^91 - 14338 * q^92 + 411002 * q^94 + 250 * q^95 - 150482 * q^97 - 209376 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more