LMFDB - Embedded newform 867.6.a.u.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,6,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("867.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	867.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:	$139.052771778$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.803842 q^{2} +9.00000 q^{3} -31.3538 q^{4} +37.1951 q^{5} +7.23458 q^{6} -118.502 q^{7} -50.9265 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+0.803842 q^{2} +9.00000 q^{3} -31.3538 q^{4} +37.1951 q^{5} +7.23458 q^{6} -118.502 q^{7} -50.9265 q^{8} +81.0000 q^{9} +29.8990 q^{10} +452.859 q^{11} -282.185 q^{12} -1075.19 q^{13} -95.2570 q^{14} +334.756 q^{15} +962.386 q^{16} +65.1112 q^{18} +1545.26 q^{19} -1166.21 q^{20} -1066.52 q^{21} +364.027 q^{22} +1285.08 q^{23} -458.338 q^{24} -1741.52 q^{25} -864.283 q^{26} +729.000 q^{27} +3715.49 q^{28} +1528.00 q^{29} +269.091 q^{30} +3594.33 q^{31} +2403.25 q^{32} +4075.73 q^{33} -4407.70 q^{35} -2539.66 q^{36} -1072.53 q^{37} +1242.15 q^{38} -9676.70 q^{39} -1894.22 q^{40} +18275.0 q^{41} -857.313 q^{42} -22141.7 q^{43} -14198.9 q^{44} +3012.80 q^{45} +1033.00 q^{46} -5846.54 q^{47} +8661.47 q^{48} -2764.26 q^{49} -1399.91 q^{50} +33711.3 q^{52} -24750.3 q^{53} +586.001 q^{54} +16844.1 q^{55} +6034.89 q^{56} +13907.4 q^{57} +1228.27 q^{58} +39414.7 q^{59} -10495.9 q^{60} +20506.0 q^{61} +2889.27 q^{62} -9598.67 q^{63} -28864.5 q^{64} -39991.8 q^{65} +3276.25 q^{66} -8233.57 q^{67} +11565.7 q^{69} -3543.09 q^{70} -68036.5 q^{71} -4125.05 q^{72} +26771.3 q^{73} -862.148 q^{74} -15673.7 q^{75} -48450.0 q^{76} -53664.8 q^{77} -7778.54 q^{78} +30920.0 q^{79} +35796.0 q^{80} +6561.00 q^{81} +14690.2 q^{82} -35030.6 q^{83} +33439.5 q^{84} -17798.4 q^{86} +13752.0 q^{87} -23062.5 q^{88} -126996. q^{89} +2421.82 q^{90} +127412. q^{91} -40292.2 q^{92} +32349.0 q^{93} -4699.70 q^{94} +57476.3 q^{95} +21629.3 q^{96} +131033. q^{97} -2222.03 q^{98} +36681.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9}+O(q^{10}) $ 28 * q + 252 * q^3 + 384 * q^4 - 244 * q^5 - 392 * q^7 + 2268 * q^9	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9} - 800 q^{10} - 1132 q^{11} + 3456 q^{12} - 828 q^{13} - 1568 q^{14} - 2196 q^{15} + 4096 q^{16} - 7532 q^{19} - 9216 q^{20} - 3528 q^{21} - 664 q^{22} - 4548 q^{23} + 9904 q^{25} + 20412 q^{27} - 36936 q^{28} - 36680 q^{29} - 7200 q^{30} - 7688 q^{31} - 10188 q^{33} + 13136 q^{35} + 31104 q^{36} - 42360 q^{37} - 1984 q^{38} - 7452 q^{39} - 81000 q^{40} - 6060 q^{41} - 14112 q^{42} - 50044 q^{43} - 56960 q^{44} - 19764 q^{45} - 51488 q^{46} + 16552 q^{47} + 36864 q^{48} + 38084 q^{49} - 48064 q^{50} - 31360 q^{52} - 4328 q^{53} - 71740 q^{55} + 9248 q^{56} - 67788 q^{57} - 95256 q^{58} - 89832 q^{59} - 82944 q^{60} - 154104 q^{61} - 106624 q^{62} - 31752 q^{63} + 93720 q^{64} - 13260 q^{65} - 5976 q^{66} - 182136 q^{67} - 40932 q^{69} - 145464 q^{70} + 24504 q^{71} - 58184 q^{73} - 308992 q^{74} + 89136 q^{75} - 602544 q^{76} - 240576 q^{77} - 136080 q^{79} - 317440 q^{80} + 183708 q^{81} - 46648 q^{82} - 203576 q^{83} - 332424 q^{84} - 421280 q^{86} - 330120 q^{87} - 145152 q^{88} - 120880 q^{89} - 64800 q^{90} - 290152 q^{91} - 491616 q^{92} - 69192 q^{93} - 799168 q^{94} - 543084 q^{95} - 409328 q^{97} + 193072 q^{98} - 91692 q^{99}+O(q^{100}) \) 28 * q + 252 * q^3 + 384 * q^4 - 244 * q^5 - 392 * q^7 + 2268 * q^9 - 800 * q^10 - 1132 * q^11 + 3456 * q^12 - 828 * q^13 - 1568 * q^14 - 2196 * q^15 + 4096 * q^16 - 7532 * q^19 - 9216 * q^20 - 3528 * q^21 - 664 * q^22 - 4548 * q^23 + 9904 * q^25 + 20412 * q^27 - 36936 * q^28 - 36680 * q^29 - 7200 * q^30 - 7688 * q^31 - 10188 * q^33 + 13136 * q^35 + 31104 * q^36 - 42360 * q^37 - 1984 * q^38 - 7452 * q^39 - 81000 * q^40 - 6060 * q^41 - 14112 * q^42 - 50044 * q^43 - 56960 * q^44 - 19764 * q^45 - 51488 * q^46 + 16552 * q^47 + 36864 * q^48 + 38084 * q^49 - 48064 * q^50 - 31360 * q^52 - 4328 * q^53 - 71740 * q^55 + 9248 * q^56 - 67788 * q^57 - 95256 * q^58 - 89832 * q^59 - 82944 * q^60 - 154104 * q^61 - 106624 * q^62 - 31752 * q^63 + 93720 * q^64 - 13260 * q^65 - 5976 * q^66 - 182136 * q^67 - 40932 * q^69 - 145464 * q^70 + 24504 * q^71 - 58184 * q^73 - 308992 * q^74 + 89136 * q^75 - 602544 * q^76 - 240576 * q^77 - 136080 * q^79 - 317440 * q^80 + 183708 * q^81 - 46648 * q^82 - 203576 * q^83 - 332424 * q^84 - 421280 * q^86 - 330120 * q^87 - 145152 * q^88 - 120880 * q^89 - 64800 * q^90 - 290152 * q^91 - 491616 * q^92 - 69192 * q^93 - 799168 * q^94 - 543084 * q^95 - 409328 * q^97 + 193072 * q^98 - 91692 * q^99

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 $

$2$	0.803842	0.142101	0.0710503	−	0.997473i	$-0.477365\pi$
$2$	0.803842	0.142101	0.0710503	+	0.997473i	$0.477365\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−31.3538	−0.979807
$5$	37.1951	0.665366	0.332683	−	0.943039i	$-0.392046\pi$
$5$	37.1951	0.665366	0.332683	+	0.943039i	$0.392046\pi$
$6$	7.23458	0.0820418
$7$	−118.502	−0.914073	−0.457036	−	0.889448i	$-0.651089\pi$
$7$	−118.502	−0.914073	−0.457036	+	0.889448i	$0.651089\pi$
$8$	−50.9265	−0.281332
$9$	81.0000	0.333333
$10$	29.8990	0.0945489
$11$	452.859	1.12845	0.564224	−	0.825622i	$-0.309175\pi$
$11$	452.859	1.12845	0.564224	+	0.825622i	$0.309175\pi$
$12$	−282.185	−0.565692
$13$	−1075.19	−1.76452	−0.882260	−	0.470763i	$-0.843979\pi$
$13$	−1075.19	−1.76452	−0.882260	+	0.470763i	$0.843979\pi$
$14$	−95.2570	−0.129890
$15$	334.756	0.384149
$16$	962.386	0.939830
$17$	0	0
$17$	0	0
$18$	65.1112	0.0473669
$19$	1545.26	0.982016	0.491008	−	0.871155i	$-0.336628\pi$
$19$	1545.26	0.982016	0.491008	+	0.871155i	$0.336628\pi$
$20$	−1166.21	−0.651931
$21$	−1066.52	−0.527740
$22$	364.027	0.160353
$23$	1285.08	0.506536	0.253268	−	0.967396i	$-0.418495\pi$
$23$	1285.08	0.506536	0.253268	+	0.967396i	$0.418495\pi$
$24$	−458.338	−0.162427
$25$	−1741.52	−0.557288
$26$	−864.283	−0.250739
$27$	729.000	0.192450
$28$	3715.49	0.895615
$29$	1528.00	0.337387	0.168693	−	0.985669i	$-0.446045\pi$
$29$	1528.00	0.337387	0.168693	+	0.985669i	$0.446045\pi$
$30$	269.091	0.0545878
$31$	3594.33	0.671759	0.335880	−	0.941905i	$-0.390966\pi$
$31$	3594.33	0.671759	0.335880	+	0.941905i	$0.390966\pi$
$32$	2403.25	0.414882
$33$	4075.73	0.651510
$34$	0	0
$35$	−4407.70	−0.608193
$36$	−2539.66	−0.326602
$37$	−1072.53	−0.128797	−0.0643986	−	0.997924i	$-0.520513\pi$
$37$	−1072.53	−0.128797	−0.0643986	+	0.997924i	$0.520513\pi$
$38$	1242.15	0.139545
$39$	−9676.70	−1.01875
$40$	−1894.22	−0.187189
$41$	18275.0	1.69785	0.848924	−	0.528516i	$-0.177251\pi$
$41$	18275.0	1.69785	0.848924	+	0.528516i	$0.177251\pi$
$42$	−857.313	−0.0749922
$43$	−22141.7	−1.82617	−0.913083	−	0.407774i	$-0.866305\pi$
$43$	−22141.7	−1.82617	−0.913083	+	0.407774i	$0.866305\pi$
$44$	−14198.9	−1.10566
$45$	3012.80	0.221789
$46$	1033.00	0.0719790
$47$	−5846.54	−0.386060	−0.193030	−	0.981193i	$-0.561831\pi$
$47$	−5846.54	−0.386060	−0.193030	+	0.981193i	$0.561831\pi$
$48$	8661.47	0.542611
$49$	−2764.26	−0.164471
$50$	−1399.91	−0.0791909
$51$	0	0
$52$	33711.3	1.72889
$53$	−24750.3	−1.21029	−0.605147	−	0.796114i	$-0.706885\pi$
$53$	−24750.3	−1.21029	−0.605147	+	0.796114i	$0.706885\pi$
$54$	586.001	0.0273473
$55$	16844.1	0.750831
$56$	6034.89	0.257158
$57$	13907.4	0.566967
$58$	1228.27	0.0479429
$59$	39414.7	1.47410	0.737051	−	0.675837i	$-0.236218\pi$
$59$	39414.7	1.47410	0.737051	+	0.675837i	$0.236218\pi$
$60$	−10495.9	−0.376392
$61$	20506.0	0.705596	0.352798	−	0.935700i	$-0.385230\pi$
$61$	20506.0	0.705596	0.352798	+	0.935700i	$0.385230\pi$
$62$	2889.27	0.0954574
$63$	−9598.67	−0.304691
$64$	−28864.5	−0.880875
$65$	−39991.8	−1.17405
$66$	3276.25	0.0925799
$67$	−8233.57	−0.224079	−0.112039	−	0.993704i	$-0.535738\pi$
$67$	−8233.57	−0.224079	−0.112039	+	0.993704i	$0.535738\pi$
$68$	0	0
$69$	11565.7	0.292449
$70$	−3543.09	−0.0864246
$71$	−68036.5	−1.60176	−0.800878	−	0.598828i	$-0.795633\pi$
$71$	−68036.5	−1.60176	−0.800878	+	0.598828i	$0.795633\pi$
$72$	−4125.05	−0.0937773
$73$	26771.3	0.587979	0.293990	−	0.955809i	$-0.405017\pi$
$73$	26771.3	0.587979	0.293990	+	0.955809i	$0.405017\pi$
$74$	−862.148	−0.0183022
$75$	−15673.7	−0.321750
$76$	−48450.0	−0.962187
$77$	−53664.8	−1.03148
$78$	−7778.54	−0.144764
$79$	30920.0	0.557406	0.278703	−	0.960377i	$-0.410095\pi$
$79$	30920.0	0.557406	0.278703	+	0.960377i	$0.410095\pi$
$80$	35796.0	0.625331
$81$	6561.00	0.111111
$82$	14690.2	0.241265
$83$	−35030.6	−0.558151	−0.279076	−	0.960269i	$-0.590028\pi$
$83$	−35030.6	−0.558151	−0.279076	+	0.960269i	$0.590028\pi$
$84$	33439.5	0.517084
$85$	0	0
$86$	−17798.4	−0.259499
$87$	13752.0	0.194790
$88$	−23062.5	−0.317468
$89$	−126996.	−1.69948	−0.849740	−	0.527202i	$-0.823241\pi$
$89$	−126996.	−1.69948	−0.849740	+	0.527202i	$0.823241\pi$
$90$	2421.82	0.0315163
$91$	127412.	1.61290
$92$	−40292.2	−0.496308
$93$	32349.0	0.387840
$94$	−4699.70	−0.0548593
$95$	57476.3	0.653400
$96$	21629.3	0.239532
$97$	131033.	1.41400	0.707002	−	0.707211i	$-0.250047\pi$
$97$	131033.	1.41400	0.707002	+	0.707211i	$0.250047\pi$
$98$	−2222.03	−0.0233714
$99$	36681.6	0.376149
$100$	54603.5	0.546035
$101$	−170219.	−1.66037	−0.830183	−	0.557491i	$-0.811764\pi$
$101$	−170219.	−1.66037	−0.830183	+	0.557491i	$0.811764\pi$
$102$	0	0
$103$	−24377.9	−0.226414	−0.113207	−	0.993571i	$-0.536112\pi$
$103$	−24377.9	−0.226414	−0.113207	+	0.993571i	$0.536112\pi$
$104$	54755.6	0.496415
$105$	−39669.3	−0.351140
$106$	−19895.3	−0.171983
$107$	−38897.8	−0.328448	−0.164224	−	0.986423i	$-0.552512\pi$
$107$	−38897.8	−0.328448	−0.164224	+	0.986423i	$0.552512\pi$
$108$	−22856.9	−0.188564
$109$	−19079.0	−0.153812	−0.0769058	−	0.997038i	$-0.524504\pi$
$109$	−19079.0	−0.153812	−0.0769058	+	0.997038i	$0.524504\pi$
$110$	13540.0	0.106694
$111$	−9652.80	−0.0743611
$112$	−114045.	−0.859073
$113$	−64938.0	−0.478413	−0.239206	−	0.970969i	$-0.576887\pi$
$113$	−64938.0	−0.478413	−0.239206	+	0.970969i	$0.576887\pi$
$114$	11179.3	0.0805664
$115$	47798.6	0.337032
$116$	−47908.7	−0.330574
$117$	−87090.3	−0.588173
$118$	31683.2	0.209471
$119$	0	0
$120$	−17047.9	−0.108073
$121$	44030.6	0.273395
$122$	16483.6	0.100266
$123$	164475.	0.980253
$124$	−112696.	−0.658195
$125$	−181011.	−1.03617
$126$	−7715.81	−0.0432968
$127$	−427.183	−0.00235020	−0.00117510	−	0.999999i	$-0.500374\pi$
$127$	−427.183	−0.00235020	−0.00117510	+	0.999999i	$0.500374\pi$
$128$	−100107.	−0.540055
$129$	−199275.	−1.05434
$130$	−32147.1	−0.166833
$131$	−97788.1	−0.497861	−0.248930	−	0.968521i	$-0.580079\pi$
$131$	−97788.1	−0.497861	−0.248930	+	0.968521i	$0.580079\pi$
$132$	−127790.	−0.638354
$133$	−183117.	−0.897635
$134$	−6618.49	−0.0318418
$135$	27115.2	0.128050
$136$	0	0
$137$	236268.	1.07548	0.537740	−	0.843111i	$-0.319278\pi$
$137$	236268.	1.07548	0.537740	+	0.843111i	$0.319278\pi$
$138$	9297.01	0.0415571
$139$	22280.6	0.0978115	0.0489057	−	0.998803i	$-0.484427\pi$
$139$	22280.6	0.0978115	0.0489057	+	0.998803i	$0.484427\pi$
$140$	138198.	0.595912
$141$	−52618.9	−0.222892
$142$	−54690.6	−0.227610
$143$	−486910.	−1.99117
$144$	77953.3	0.313277
$145$	56834.1	0.224486
$146$	21519.9	0.0835522
$147$	−24878.3	−0.0949572
$148$	33628.0	0.126197
$149$	285298.	1.05277	0.526384	−	0.850247i	$-0.323547\pi$
$149$	285298.	1.05277	0.526384	+	0.850247i	$0.323547\pi$
$150$	−12599.2	−0.0457209
$151$	−527073.	−1.88117	−0.940585	−	0.339558i	$-0.889723\pi$
$151$	−527073.	−1.88117	−0.940585	+	0.339558i	$0.889723\pi$
$152$	−78694.9	−0.276272
$153$	0	0
$154$	−43138.0	−0.146574
$155$	133691.	0.446966
$156$	303402.	0.998175
$157$	−6338.53	−0.0205229	−0.0102615	−	0.999947i	$-0.503266\pi$
$157$	−6338.53	−0.0205229	−0.0102615	+	0.999947i	$0.503266\pi$
$158$	24854.8	0.0792077
$159$	−222753.	−0.698763
$160$	89389.3	0.276049
$161$	−152285.	−0.463011
$162$	5274.01	0.0157890
$163$	−606331.	−1.78748	−0.893739	−	0.448587i	$-0.851927\pi$
$163$	−606331.	−1.78748	−0.893739	+	0.448587i	$0.851927\pi$
$164$	−572993.	−1.66356
$165$	151597.	0.433493
$166$	−28159.1	−0.0793136
$167$	−76561.6	−0.212432	−0.106216	−	0.994343i	$-0.533874\pi$
$167$	−76561.6	−0.212432	−0.106216	+	0.994343i	$0.533874\pi$
$168$	54314.1	0.148470
$169$	784739.	2.11353
$170$	0	0
$171$	125166.	0.327339
$172$	694228.	1.78929
$173$	−281437.	−0.714933	−0.357467	−	0.933926i	$-0.616359\pi$
$173$	−281437.	−0.714933	−0.357467	+	0.933926i	$0.616359\pi$
$174$	11054.4	0.0276798
$175$	206374.	0.509402
$176$	435825.	1.06055
$177$	354732.	0.851073
$178$	−102085.	−0.241497
$179$	137699.	0.321218	0.160609	−	0.987018i	$-0.448654\pi$
$179$	137699.	0.321218	0.160609	+	0.987018i	$0.448654\pi$
$180$	−94462.9	−0.217310
$181$	−24989.9	−0.0566981	−0.0283490	−	0.999598i	$-0.509025\pi$
$181$	−24989.9	−0.0566981	−0.0283490	+	0.999598i	$0.509025\pi$
$182$	102419.	0.229194
$183$	184554.	0.407376
$184$	−65444.6	−0.142505
$185$	−39893.0	−0.0856973
$186$	26003.5	0.0551123
$187$	0	0
$188$	183312.	0.378264
$189$	−86388.0	−0.175913
$190$	46201.8	0.0928486
$191$	−635062.	−1.25960	−0.629800	−	0.776757i	$-0.716863\pi$
$191$	−635062.	−1.25960	−0.629800	+	0.776757i	$0.716863\pi$
$192$	−259781.	−0.508573
$193$	37888.0	0.0732165	0.0366082	−	0.999330i	$-0.488345\pi$
$193$	37888.0	0.0732165	0.0366082	+	0.999330i	$0.488345\pi$
$194$	105330.	0.200931
$195$	−359926.	−0.677839
$196$	86670.1	0.161150
$197$	−362516.	−0.665521	−0.332761	−	0.943011i	$-0.607980\pi$
$197$	−362516.	−0.665521	−0.332761	+	0.943011i	$0.607980\pi$
$198$	29486.2	0.0534510
$199$	−513530.	−0.919249	−0.459624	−	0.888113i	$-0.652016\pi$
$199$	−513530.	−0.919249	−0.459624	+	0.888113i	$0.652016\pi$
$200$	88689.7	0.156783
$201$	−74102.1	−0.129372
$202$	−136829.	−0.235939
$203$	−181071.	−0.308396
$204$	0	0
$205$	679742.	1.12969
$206$	−19596.0	−0.0321736
$207$	104091.	0.168845
$208$	−1.03475e6	−1.65835
$209$	699787.	1.10815
$210$	−31887.8	−0.0498973
$211$	−385027.	−0.595367	−0.297684	−	0.954665i	$-0.596214\pi$
$211$	−385027.	−0.595367	−0.297684	+	0.954665i	$0.596214\pi$
$212$	776016.	1.18585
$213$	−612329.	−0.924774
$214$	−31267.7	−0.0466726
$215$	−823563.	−1.21507
$216$	−37125.4	−0.0541423
$217$	−425935.	−0.614037
$218$	−15336.5	−0.0218567
$219$	240942.	0.339470
$220$	−528129.	−0.735670
$221$	0	0
$222$	−7759.33	−0.0105668
$223$	521847.	0.702718	0.351359	−	0.936241i	$-0.385720\pi$
$223$	521847.	0.702718	0.351359	+	0.936241i	$0.385720\pi$
$224$	−284791.	−0.379233
$225$	−141064.	−0.185763
$226$	−52199.9	−0.0679827
$227$	−1.34039e6	−1.72650	−0.863252	−	0.504773i	$-0.831576\pi$
$227$	−1.34039e6	−1.72650	−0.863252	+	0.504773i	$0.831576\pi$
$228$	−436050.	−0.555519
$229$	334943.	0.422068	0.211034	−	0.977479i	$-0.432317\pi$
$229$	334943.	0.422068	0.211034	+	0.977479i	$0.432317\pi$
$230$	38422.6	0.0478924
$231$	−482983.	−0.595528
$232$	−77815.7	−0.0949177
$233$	710462.	0.857336	0.428668	−	0.903462i	$-0.358983\pi$
$233$	710462.	0.857336	0.428668	+	0.903462i	$0.358983\pi$
$234$	−70006.9	−0.0835798
$235$	−217463.	−0.256871
$236$	−1.23580e6	−1.44434
$237$	278280.	0.321819
$238$	0	0
$239$	62521.3	0.0708000	0.0354000	−	0.999373i	$-0.488729\pi$
$239$	62521.3	0.0708000	0.0354000	+	0.999373i	$0.488729\pi$
$240$	322164.	0.361035
$241$	−468499.	−0.519597	−0.259798	−	0.965663i	$-0.583656\pi$
$241$	−468499.	−0.519597	−0.259798	+	0.965663i	$0.583656\pi$
$242$	35393.6	0.0388496
$243$	59049.0	0.0641500
$244$	−642941.	−0.691348
$245$	−102817.	−0.109433
$246$	132212.	0.139294
$247$	−1.66145e6	−1.73279
$248$	−183047.	−0.188987
$249$	−315275.	−0.322249
$250$	−145504.	−0.147240
$251$	−1.27557e6	−1.27797	−0.638985	−	0.769219i	$-0.720645\pi$
$251$	−1.27557e6	−1.27797	−0.638985	+	0.769219i	$0.720645\pi$
$252$	300955.	0.298538
$253$	581960.	0.571599
$254$	−343.388	−0.000333964 0
$255$	0	0
$256$	843194.	0.804133
$257$	−829345.	−0.783254	−0.391627	−	0.920124i	$-0.628088\pi$
$257$	−829345.	−0.783254	−0.391627	+	0.920124i	$0.628088\pi$
$258$	−160186.	−0.149822
$259$	127097.	0.117730
$260$	1.25390e6	1.15034
$261$	123768.	0.112462
$262$	−78606.2	−0.0707463
$263$	524275.	0.467380	0.233690	−	0.972311i	$-0.424920\pi$
$263$	524275.	0.467380	0.233690	+	0.972311i	$0.424920\pi$
$264$	−207563.	−0.183290
$265$	−920589.	−0.805288
$266$	−147197.	−0.127554
$267$	−1.14297e6	−0.981195
$268$	258154.	0.219554
$269$	−845565.	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$269$	−845565.	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$270$	21796.4	0.0181959
$271$	1.04394e6	0.863476	0.431738	−	0.901999i	$-0.357901\pi$
$271$	1.04394e6	0.863476	0.431738	+	0.901999i	$0.357901\pi$
$272$	0	0
$273$	1.14671e6	0.931208
$274$	189922.	0.152826
$275$	−788666.	−0.628871
$276$	−362629.	−0.286543
$277$	809517.	0.633909	0.316954	−	0.948441i	$-0.397340\pi$
$277$	809517.	0.633909	0.316954	+	0.948441i	$0.397340\pi$
$278$	17910.1	0.0138991
$279$	291141.	0.223920
$280$	224468.	0.171104
$281$	−1.27804e6	−0.965558	−0.482779	−	0.875742i	$-0.660373\pi$
$281$	−1.27804e6	−0.965558	−0.482779	+	0.875742i	$0.660373\pi$
$282$	−42297.3	−0.0316730
$283$	−1.68165e6	−1.24816	−0.624079	−	0.781361i	$-0.714526\pi$
$283$	−1.68165e6	−1.24816	−0.624079	+	0.781361i	$0.714526\pi$
$284$	2.13321e6	1.56941
$285$	517286.	0.377241
$286$	−391398.	−0.282946
$287$	−2.16563e6	−1.55196
$288$	194664.	0.138294
$289$	0	0
$290$	45685.6	0.0318996
$291$	1.17930e6	0.816376
$292$	−839382.	−0.576106
$293$	−1.93317e6	−1.31553	−0.657766	−	0.753222i	$-0.728498\pi$
$293$	−1.93317e6	−1.31553	−0.657766	+	0.753222i	$0.728498\pi$
$294$	−19998.2	−0.0134935
$295$	1.46603e6	0.980818
$296$	54620.4	0.0362348
$297$	330134.	0.217170
$298$	229335.	0.149599
$299$	−1.38170e6	−0.893793
$300$	491431.	0.315253
$301$	2.62384e6	1.66925
$302$	−423683.	−0.267315
$303$	−1.53197e6	−0.958613
$304$	1.48714e6	0.922929
$305$	762722.	0.469480
$306$	0	0
$307$	1.39233e6	0.843133	0.421567	−	0.906797i	$-0.361480\pi$
$307$	1.39233e6	0.843133	0.421567	+	0.906797i	$0.361480\pi$
$308$	1.68260e6	1.01066
$309$	−219401.	−0.130720
$310$	107467.	0.0635141
$311$	−344613.	−0.202037	−0.101018	−	0.994885i	$-0.532210\pi$
$311$	−344613.	−0.202037	−0.101018	+	0.994885i	$0.532210\pi$
$312$	492801.	0.286606
$313$	2.36956e6	1.36712	0.683561	−	0.729893i	$-0.260430\pi$
$313$	2.36956e6	1.36712	0.683561	+	0.729893i	$0.260430\pi$
$314$	−5095.17	−0.00291632
$315$	−357023.	−0.202731
$316$	−969461.	−0.546151
$317$	2.55672e6	1.42901	0.714505	−	0.699630i	$-0.246652\pi$
$317$	2.55672e6	1.42901	0.714505	+	0.699630i	$0.246652\pi$
$318$	−179058.	−0.0992946
$319$	691969.	0.380724
$320$	−1.07362e6	−0.586104
$321$	−350081.	−0.189629
$322$	−122413.	−0.0657941
$323$	0	0
$324$	−205713.	−0.108867
$325$	1.87247e6	0.983346
$326$	−487394.	−0.254002
$327$	−171711.	−0.0888031
$328$	−930683.	−0.477658
$329$	692827.	0.352887
$330$	121860.	0.0615995
$331$	−19952.5	−0.0100098	−0.00500492	−	0.999987i	$-0.501593\pi$
$331$	−19952.5	−0.0100098	−0.00500492	+	0.999987i	$0.501593\pi$
$332$	1.09834e6	0.546881
$333$	−86875.2	−0.0429324
$334$	−61543.5	−0.0301867
$335$	−306248.	−0.149095
$336$	−1.02640e6	−0.495986
$337$	916701.	0.439696	0.219848	−	0.975534i	$-0.429444\pi$
$337$	916701.	0.439696	0.219848	+	0.975534i	$0.429444\pi$
$338$	630806.	0.300334
$339$	−584442.	−0.276212
$340$	0	0
$341$	1.62773e6	0.758045
$342$	100614.	0.0465150
$343$	2.31923e6	1.06441
$344$	1.12760e6	0.513758
$345$	430188.	0.194585
$346$	−226231.	−0.101592
$347$	−3.30936e6	−1.47543	−0.737717	−	0.675110i	$-0.764096\pi$
$347$	−3.30936e6	−1.47543	−0.737717	+	0.675110i	$0.764096\pi$
$348$	−431178.	−0.190857
$349$	−4.10164e6	−1.80258	−0.901288	−	0.433221i	$-0.857377\pi$
$349$	−4.10164e6	−1.80258	−0.901288	+	0.433221i	$0.857377\pi$
$350$	165892.	0.0723863
$351$	−783813.	−0.339582
$352$	1.08834e6	0.468173
$353$	2.30651e6	0.985188	0.492594	−	0.870259i	$-0.336049\pi$
$353$	2.30651e6	0.985188	0.492594	+	0.870259i	$0.336049\pi$
$354$	285148.	0.120938
$355$	−2.53063e6	−1.06575
$356$	3.98182e6	1.66516
$357$	0	0
$358$	110689.	0.0456453
$359$	1.96136e6	0.803194	0.401597	−	0.915817i	$-0.368455\pi$
$359$	1.96136e6	0.803194	0.401597	+	0.915817i	$0.368455\pi$
$360$	−153431.	−0.0623962
$361$	−88257.4	−0.0356437
$362$	−20087.9	−0.00805683
$363$	396275.	0.157845
$364$	−3.99486e6	−1.58033
$365$	995760.	0.391221
$366$	148352.	0.0578884
$367$	−4.72592e6	−1.83156	−0.915780	−	0.401679i	$-0.868427\pi$
$367$	−4.72592e6	−1.83156	−0.915780	+	0.401679i	$0.868427\pi$
$368$	1.23674e6	0.476058
$369$	1.48028e6	0.565949
$370$	−32067.7	−0.0121776
$371$	2.93296e6	1.10630
$372$	−1.01426e6	−0.380009
$373$	−4.61493e6	−1.71748	−0.858742	−	0.512408i	$-0.828754\pi$
$373$	−4.61493e6	−1.71748	−0.858742	+	0.512408i	$0.828754\pi$
$374$	0	0
$375$	−1.62910e6	−0.598231
$376$	297744.	0.108611
$377$	−1.64289e6	−0.595326
$378$	−69442.3	−0.0249974
$379$	1.26132e6	0.451052	0.225526	−	0.974237i	$-0.427590\pi$
$379$	1.26132e6	0.451052	0.225526	+	0.974237i	$0.427590\pi$
$380$	−1.80210e6	−0.640207
$381$	−3844.64	−0.00135689
$382$	−510490.	−0.178990
$383$	−2.73522e6	−0.952786	−0.476393	−	0.879232i	$-0.658056\pi$
$383$	−2.73522e6	−0.952786	−0.476393	+	0.879232i	$0.658056\pi$
$384$	−900960.	−0.311801
$385$	−1.99607e6	−0.686314
$386$	30456.0	0.0104041
$387$	−1.79348e6	−0.608722
$388$	−4.10838e6	−1.38545
$389$	397735.	0.133266	0.0666331	−	0.997778i	$-0.478774\pi$
$389$	397735.	0.133266	0.0666331	+	0.997778i	$0.478774\pi$
$390$	−289324.	−0.0963213
$391$	0	0
$392$	140774.	0.0462708
$393$	−880093.	−0.287440
$394$	−291406.	−0.0945709
$395$	1.15007e6	0.370879
$396$	−1.15011e6	−0.368554
$397$	−5.25000e6	−1.67180	−0.835898	−	0.548885i	$-0.815053\pi$
$397$	−5.25000e6	−1.67180	−0.835898	+	0.548885i	$0.815053\pi$
$398$	−412797.	−0.130626
$399$	−1.64805e6	−0.518250
$400$	−1.67602e6	−0.523756
$401$	1.05176e6	0.326628	0.163314	−	0.986574i	$-0.447782\pi$
$401$	1.05176e6	0.326628	0.163314	+	0.986574i	$0.447782\pi$
$402$	−59566.4	−0.0183838
$403$	−3.86458e6	−1.18533
$404$	5.33701e6	1.62684
$405$	244037.	0.0739296
$406$	−145553.	−0.0438233
$407$	−485707.	−0.145341
$408$	0	0
$409$	−4.17248e6	−1.23335	−0.616675	−	0.787218i	$-0.711521\pi$
$409$	−4.17248e6	−1.23335	−0.616675	+	0.787218i	$0.711521\pi$
$410$	546405.	0.160530
$411$	2.12641e6	0.620929
$412$	764341.	0.221842
$413$	−4.67072e6	−1.34744
$414$	83673.1	0.0239930
$415$	−1.30297e6	−0.371375
$416$	−2.58395e6	−0.732068
$417$	200525.	0.0564715
$418$	562519.	0.157469
$419$	−4.72038e6	−1.31354	−0.656768	−	0.754093i	$-0.728077\pi$
$419$	−4.72038e6	−1.31354	−0.656768	+	0.754093i	$0.728077\pi$
$420$	1.24378e6	0.344050
$421$	−2.49661e6	−0.686508	−0.343254	−	0.939243i	$-0.611529\pi$
$421$	−2.49661e6	−0.686508	−0.343254	+	0.939243i	$0.611529\pi$
$422$	−309501.	−0.0846020
$423$	−473570.	−0.128687
$424$	1.26045e6	0.340494
$425$	0	0
$426$	−492216.	−0.131411
$427$	−2.43000e6	−0.644966
$428$	1.21960e6	0.321815
$429$	−4.38219e6	−1.14960
$430$	−662015.	−0.172662
$431$	1.56568e6	0.405985	0.202993	−	0.979180i	$-0.434933\pi$
$431$	1.56568e6	0.405985	0.202993	+	0.979180i	$0.434933\pi$
$432$	701579.	0.180870
$433$	2.86424e6	0.734159	0.367080	−	0.930189i	$-0.380358\pi$
$433$	2.86424e6	0.734159	0.367080	+	0.930189i	$0.380358\pi$
$434$	−342385.	−0.0872550
$435$	511507.	0.129607
$436$	598199.	0.150706
$437$	1.98579e6	0.497427
$438$	193679.	0.0482389
$439$	7.39389e6	1.83110	0.915549	−	0.402206i	$-0.131756\pi$
$439$	7.39389e6	1.83110	0.915549	+	0.402206i	$0.131756\pi$
$440$	−857813.	−0.211233
$441$	−223905.	−0.0548236
$442$	0	0
$443$	5.15096e6	1.24704	0.623518	−	0.781809i	$-0.285703\pi$
$443$	5.15096e6	1.24704	0.623518	+	0.781809i	$0.285703\pi$
$444$	302652.	0.0728596
$445$	−4.72364e6	−1.13078
$446$	419483.	0.0998566
$447$	2.56768e6	0.607816
$448$	3.42050e6	0.805184
$449$	2.90818e6	0.680777	0.340388	−	0.940285i	$-0.389441\pi$
$449$	2.90818e6	0.680777	0.340388	+	0.940285i	$0.389441\pi$
$450$	−113393.	−0.0263970
$451$	8.27602e6	1.91593
$452$	2.03605e6	0.468752
$453$	−4.74365e6	−1.08609
$454$	−1.07747e6	−0.245337
$455$	4.73911e6	1.07317
$456$	−708254.	−0.159506
$457$	−5.85259e6	−1.31086	−0.655432	−	0.755254i	$-0.727514\pi$
$457$	−5.85259e6	−1.31086	−0.655432	+	0.755254i	$0.727514\pi$
$458$	269241.	0.0599761
$459$	0	0
$460$	−1.49867e6	−0.330226
$461$	−4.70464e6	−1.03104	−0.515519	−	0.856878i	$-0.672401\pi$
$461$	−4.70464e6	−1.03104	−0.515519	+	0.856878i	$0.672401\pi$
$462$	−388242.	−0.0846248
$463$	−1.03289e6	−0.223925	−0.111962	−	0.993712i	$-0.535714\pi$
$463$	−1.03289e6	−0.223925	−0.111962	+	0.993712i	$0.535714\pi$
$464$	1.47053e6	0.317086
$465$	1.20322e6	0.258056
$466$	571100.	0.121828
$467$	−2.25881e6	−0.479277	−0.239639	−	0.970862i	$-0.577029\pi$
$467$	−2.25881e6	−0.479277	−0.239639	+	0.970862i	$0.577029\pi$
$468$	2.73062e6	0.576297
$469$	975695.	0.204825
$470$	−174806.	−0.0365015
$471$	−57046.7	−0.0118489
$472$	−2.00725e6	−0.414712
$473$	−1.00271e7	−2.06073
$474$	223693.	0.0457306
$475$	−2.69112e6	−0.547266
$476$	0	0
$477$	−2.00477e6	−0.403431
$478$	50257.2	0.0100607
$479$	−3.29694e6	−0.656557	−0.328278	−	0.944581i	$-0.606468\pi$
$479$	−3.29694e6	−0.656557	−0.328278	+	0.944581i	$0.606468\pi$
$480$	804503.	0.159377
$481$	1.15318e6	0.227265
$482$	−376600.	−0.0738350
$483$	−1.37056e6	−0.267319
$484$	−1.38053e6	−0.267875
$485$	4.87378e6	0.940831
$486$	47466.1	0.00911576
$487$	−2.95848e6	−0.565258	−0.282629	−	0.959229i	$-0.591206\pi$
$487$	−2.95848e6	−0.565258	−0.282629	+	0.959229i	$0.591206\pi$
$488$	−1.04430e6	−0.198507
$489$	−5.45698e6	−1.03200
$490$	−82648.5	−0.0155505
$491$	621932.	0.116423	0.0582116	−	0.998304i	$-0.481460\pi$
$491$	621932.	0.116423	0.0582116	+	0.998304i	$0.481460\pi$
$492$	−5.15693e6	−0.960459
$493$	0	0
$494$	−1.33554e6	−0.246230
$495$	1.36438e6	0.250277
$496$	3.45913e6	0.631340
$497$	8.06247e6	1.46412
$498$	−253431.	−0.0457917
$499$	8.81811e6	1.58535	0.792673	−	0.609647i	$-0.208689\pi$
$499$	8.81811e6	1.58535	0.792673	+	0.609647i	$0.208689\pi$
$500$	5.67539e6	1.01524
$501$	−689055.	−0.122648
$502$	−1.02536e6	−0.181600
$503$	6.49963e6	1.14543	0.572715	−	0.819755i	$-0.305890\pi$
$503$	6.49963e6	1.14543	0.572715	+	0.819755i	$0.305890\pi$
$504$	488826.	0.0857193
$505$	−6.33130e6	−1.10475
$506$	467804.	0.0812246
$507$	7.06265e6	1.22025
$508$	13393.8	0.00230274
$509$	9.65833e6	1.65237	0.826185	−	0.563398i	$-0.190506\pi$
$509$	9.65833e6	1.65237	0.826185	+	0.563398i	$0.190506\pi$
$510$	0	0
$511$	−3.17245e6	−0.537456
$512$	3.88121e6	0.654323
$513$	1.12650e6	0.188989
$514$	−666663.	−0.111301
$515$	−906739.	−0.150648
$516$	6.24805e6	1.03305
$517$	−2.64766e6	−0.435648
$518$	102166.	0.0167295
$519$	−2.53293e6	−0.412767
$520$	2.03664e6	0.330298
$521$	1.05437e7	1.70176	0.850882	−	0.525357i	$-0.176068\pi$
$521$	1.05437e7	1.70176	0.850882	+	0.525357i	$0.176068\pi$
$522$	99489.9	0.0159810
$523$	−1.01443e6	−0.162168	−0.0810842	−	0.996707i	$-0.525838\pi$
$523$	−1.01443e6	−0.162168	−0.0810842	+	0.996707i	$0.525838\pi$
$524$	3.06603e6	0.487807
$525$	1.85737e6	0.294103
$526$	421434.	0.0664149
$527$	0	0
$528$	3.92243e6	0.612309
$529$	−4.78492e6	−0.743421
$530$	−740009.	−0.114432
$531$	3.19259e6	0.491367
$532$	5.74142e6	0.879509
$533$	−1.96491e7	−2.99589
$534$	−918765.	−0.139428
$535$	−1.44681e6	−0.218538
$536$	419307.	0.0630405
$537$	1.23930e6	0.185455
$538$	−679701.	−0.101242
$539$	−1.25182e6	−0.185597
$540$	−850166.	−0.125464
$541$	6.23439e6	0.915801	0.457900	−	0.889003i	$-0.348602\pi$
$541$	6.23439e6	0.915801	0.457900	+	0.889003i	$0.348602\pi$
$542$	839159.	0.122700
$543$	−224909.	−0.0327346
$544$	0	0
$545$	−709644.	−0.102341
$546$	921773.	0.132325
$547$	5.28866e6	0.755749	0.377875	−	0.925857i	$-0.376655\pi$
$547$	5.28866e6	0.755749	0.377875	+	0.925857i	$0.376655\pi$
$548$	−7.40789e6	−1.05376
$549$	1.66098e6	0.235199
$550$	−633963.	−0.0893629
$551$	2.36116e6	0.331320
$552$	−589001.	−0.0822751
$553$	−3.66408e6	−0.509510
$554$	650724.	0.0900788
$555$	−359037.	−0.0494774
$556$	−698582.	−0.0958364
$557$	−895127.	−0.122249	−0.0611247	−	0.998130i	$-0.519469\pi$
$557$	−895127.	−0.122249	−0.0611247	+	0.998130i	$0.519469\pi$
$558$	234031.	0.0318191
$559$	2.38065e7	3.22231
$560$	−4.24190e6	−0.571598
$561$	0	0
$562$	−1.02734e6	−0.137206
$563$	−1.47961e7	−1.96733	−0.983666	−	0.180006i	$-0.942388\pi$
$563$	−1.47961e7	−1.96733	−0.983666	+	0.180006i	$0.942388\pi$
$564$	1.64980e6	0.218391
$565$	−2.41537e6	−0.318320
$566$	−1.35178e6	−0.177364
$567$	−777492.	−0.101564
$568$	3.46486e6	0.450625
$569$	1.47611e7	1.91134	0.955668	−	0.294446i	$-0.0951350\pi$
$569$	1.47611e7	1.91134	0.955668	+	0.294446i	$0.0951350\pi$
$570$	415817.	0.0536062
$571$	990545.	0.127141	0.0635703	−	0.997977i	$-0.479751\pi$
$571$	990545.	0.127141	0.0635703	+	0.997977i	$0.479751\pi$
$572$	1.52665e7	1.95096
$573$	−5.71556e6	−0.727231
$574$	−1.74082e6	−0.220534
$575$	−2.23800e6	−0.282286
$576$	−2.33803e6	−0.293625
$577$	−967155.	−0.120936	−0.0604681	−	0.998170i	$-0.519259\pi$
$577$	−967155.	−0.120936	−0.0604681	+	0.998170i	$0.519259\pi$
$578$	0	0
$579$	340992.	0.0422715
$580$	−1.78197e6	−0.219953
$581$	4.15120e6	0.510191
$582$	947968.	0.116008
$583$	−1.12084e7	−1.36575
$584$	−1.36337e6	−0.165417
$585$	−3.23933e6	−0.391351
$586$	−1.55397e6	−0.186938
$587$	3.52350e6	0.422065	0.211032	−	0.977479i	$-0.432317\pi$
$587$	3.52350e6	0.422065	0.211032	+	0.977479i	$0.432317\pi$
$588$	780031.	0.0930398
$589$	5.55419e6	0.659679
$590$	1.17846e6	0.139375
$591$	−3.26265e6	−0.384239
$592$	−1.03219e6	−0.121048
$593$	7.39591e6	0.863684	0.431842	−	0.901949i	$-0.357864\pi$
$593$	7.39591e6	0.863684	0.431842	+	0.901949i	$0.357864\pi$
$594$	265376.	0.0308600
$595$	0	0
$596$	−8.94518e6	−1.03151
$597$	−4.62177e6	−0.530729
$598$	−1.11067e6	−0.127008
$599$	1.06373e7	1.21133	0.605667	−	0.795718i	$-0.292906\pi$
$599$	1.06373e7	1.21133	0.605667	+	0.795718i	$0.292906\pi$
$600$	798208.	0.0905186
$601$	−73344.1	−0.00828283	−0.00414142	−	0.999991i	$-0.501318\pi$
$601$	−73344.1	−0.00828283	−0.00414142	+	0.999991i	$0.501318\pi$
$602$	2.10915e6	0.237201
$603$	−666919.	−0.0746930
$604$	1.65258e7	1.84318
$605$	1.63772e6	0.181908
$606$	−1.23146e6	−0.136219
$607$	5.28217e6	0.581890	0.290945	−	0.956740i	$-0.406030\pi$
$607$	5.28217e6	0.581890	0.290945	+	0.956740i	$0.406030\pi$
$608$	3.71366e6	0.407421
$609$	−1.62964e6	−0.178053
$610$	613108.	0.0667133
$611$	6.28614e6	0.681210
$612$	0	0
$613$	6.34089e6	0.681552	0.340776	−	0.940144i	$-0.389310\pi$
$613$	6.34089e6	0.681552	0.340776	+	0.940144i	$0.389310\pi$
$614$	1.11921e6	0.119810
$615$	6.11768e6	0.652227
$616$	2.73296e6	0.290189
$617$	−1.02422e7	−1.08313	−0.541565	−	0.840659i	$-0.682168\pi$
$617$	−1.02422e7	−1.08313	−0.541565	+	0.840659i	$0.682168\pi$
$618$	−176364.	−0.0185754
$619$	1.35555e7	1.42197	0.710983	−	0.703209i	$-0.248250\pi$
$619$	1.35555e7	1.42197	0.710983	+	0.703209i	$0.248250\pi$
$620$	−4.19174e6	−0.437940
$621$	936822.	0.0974829
$622$	−277014.	−0.0287096
$623$	1.50493e7	1.55345
$624$	−9.31272e6	−0.957448
$625$	−1.29045e6	−0.132142
$626$	1.90475e6	0.194269
$627$	6.29809e6	0.639793
$628$	198737.	0.0201085
$629$	0	0
$630$	−286990.	−0.0288082
$631$	6.12044e6	0.611941	0.305970	−	0.952041i	$-0.401019\pi$
$631$	6.12044e6	0.611941	0.305970	+	0.952041i	$0.401019\pi$
$632$	−1.57465e6	−0.156816
$633$	−3.46524e6	−0.343735
$634$	2.05520e6	0.203063
$635$	−15889.1	−0.00156374
$636$	6.98415e6	0.684653
$637$	2.97210e6	0.290212
$638$	556234.	0.0541010
$639$	−5.51096e6	−0.533919
$640$	−3.72348e6	−0.359334
$641$	915737.	0.0880290	0.0440145	−	0.999031i	$-0.485985\pi$
$641$	915737.	0.0880290	0.0440145	+	0.999031i	$0.485985\pi$
$642$	−281410.	−0.0269464
$643$	−7.00352e6	−0.668020	−0.334010	−	0.942570i	$-0.608402\pi$
$643$	−7.00352e6	−0.668020	−0.334010	+	0.942570i	$0.608402\pi$
$644$	4.77470e6	0.453661
$645$	−7.41207e6	−0.701520
$646$	0	0
$647$	1.67308e6	0.157129	0.0785645	−	0.996909i	$-0.474966\pi$
$647$	1.67308e6	0.157129	0.0785645	+	0.996909i	$0.474966\pi$
$648$	−334129.	−0.0312591
$649$	1.78493e7	1.66345
$650$	1.50517e6	0.139734
$651$	−3.83342e6	−0.354514
$652$	1.90108e7	1.75138
$653$	2.04589e6	0.187758	0.0938791	−	0.995584i	$-0.470073\pi$
$653$	2.04589e6	0.187758	0.0938791	+	0.995584i	$0.470073\pi$
$654$	−138028.	−0.0126190
$655$	−3.63724e6	−0.331260
$656$	1.75876e7	1.59569
$657$	2.16847e6	0.195993
$658$	556924.	0.0501454
$659$	1.83509e7	1.64606	0.823028	−	0.568001i	$-0.192283\pi$
$659$	1.83509e7	1.64606	0.823028	+	0.568001i	$0.192283\pi$
$660$	−4.75316e6	−0.424739
$661$	−5.31402e6	−0.473063	−0.236532	−	0.971624i	$-0.576011\pi$
$661$	−5.31402e6	−0.473063	−0.236532	+	0.971624i	$0.576011\pi$
$662$	−16038.6	−0.00142240
$663$	0	0
$664$	1.78398e6	0.157026
$665$	−6.81106e6	−0.597256
$666$	−69834.0	−0.00610072
$667$	1.96360e6	0.170899
$668$	2.40050e6	0.208142
$669$	4.69662e6	0.405714
$670$	−246175.	−0.0211864
$671$	9.28633e6	0.796228
$672$	−2.56312e6	−0.218950
$673$	2.17855e7	1.85409	0.927043	−	0.374956i	$-0.122342\pi$
$673$	2.17855e7	1.85409	0.927043	+	0.374956i	$0.122342\pi$
$674$	736883.	0.0624811
$675$	−1.26957e6	−0.107250
$676$	−2.46046e7	−2.07085
$677$	6.80004e6	0.570217	0.285108	−	0.958495i	$-0.407970\pi$
$677$	6.80004e6	0.570217	0.285108	+	0.958495i	$0.407970\pi$
$678$	−469799.	−0.0392498
$679$	−1.55277e7	−1.29250
$680$	0	0
$681$	−1.20635e7	−0.996798
$682$	1.30843e6	0.107719
$683$	−1.57563e7	−1.29241	−0.646206	−	0.763163i	$-0.723645\pi$
$683$	−1.57563e7	−1.29241	−0.646206	+	0.763163i	$0.723645\pi$
$684$	−3.92445e6	−0.320729
$685$	8.78799e6	0.715588
$686$	1.86430e6	0.151253
$687$	3.01449e6	0.243681
$688$	−2.13089e7	−1.71629
$689$	2.66112e7	2.13559
$690$	345803.	0.0276507
$691$	9.63845e6	0.767913	0.383956	−	0.923351i	$-0.374561\pi$
$691$	9.63845e6	0.767913	0.383956	+	0.923351i	$0.374561\pi$
$692$	8.82412e6	0.700497
$693$	−4.34685e6	−0.343828
$694$	−2.66020e6	−0.209660
$695$	828729.	0.0650804
$696$	−700341.	−0.0548007
$697$	0	0
$698$	−3.29707e6	−0.256147
$699$	6.39416e6	0.494983
$700$	−6.47063e6	−0.499116
$701$	2.31995e7	1.78313	0.891566	−	0.452891i	$-0.149607\pi$
$701$	2.31995e7	1.78313	0.891566	+	0.452891i	$0.149607\pi$
$702$	−630062.	−0.0482548
$703$	−1.65735e6	−0.126481
$704$	−1.30716e7	−0.994022
$705$	−1.95716e6	−0.148305
$706$	1.85407e6	0.139996
$707$	2.01713e7	1.51770
$708$	−1.11222e7	−0.833888
$709$	−8.89519e6	−0.664568	−0.332284	−	0.943179i	$-0.607819\pi$
$709$	−8.89519e6	−0.664568	−0.332284	+	0.943179i	$0.607819\pi$
$710$	−2.03422e6	−0.151444
$711$	2.50452e6	0.185802
$712$	6.46748e6	0.478118
$713$	4.61900e6	0.340270
$714$	0	0
$715$	−1.81106e7	−1.32486
$716$	−4.31741e6	−0.314732
$717$	562691.	0.0408764
$718$	1.57662e6	0.114134
$719$	1.92832e7	1.39109	0.695546	−	0.718481i	$-0.255162\pi$
$719$	1.92832e7	1.39109	0.695546	+	0.718481i	$0.255162\pi$
$720$	2.89948e6	0.208444
$721$	2.88883e6	0.206959
$722$	−70945.0	−0.00506499
$723$	−4.21649e6	−0.299989
$724$	783529.	0.0555532
$725$	−2.66105e6	−0.188022
$726$	318543.	0.0224298
$727$	6.80164e6	0.477285	0.238642	−	0.971108i	$-0.423298\pi$
$727$	6.80164e6	0.477285	0.238642	+	0.971108i	$0.423298\pi$
$728$	−6.48865e6	−0.453760
$729$	531441.	0.0370370
$730$	800434.	0.0555928
$731$	0	0
$732$	−5.78647e6	−0.399150
$733$	4.04803e6	0.278281	0.139141	−	0.990273i	$-0.455566\pi$
$733$	4.04803e6	0.278281	0.139141	+	0.990273i	$0.455566\pi$
$734$	−3.79890e6	−0.260266
$735$	−925352.	−0.0631813
$736$	3.08837e6	0.210153
$737$	−3.72865e6	−0.252862
$738$	1.18991e6	0.0804217
$739$	2.74011e6	0.184569	0.0922843	−	0.995733i	$-0.470583\pi$
$739$	2.74011e6	0.184569	0.0922843	+	0.995733i	$0.470583\pi$
$740$	1.25080e6	0.0839669
$741$	−1.49531e7	−1.00043
$742$	2.35764e6	0.157205
$743$	−466698.	−0.0310145	−0.0155072	−	0.999880i	$-0.504936\pi$
$743$	−466698.	−0.0310145	−0.0155072	+	0.999880i	$0.504936\pi$
$744$	−1.64742e6	−0.109112
$745$	1.06117e7	0.700477
$746$	−3.70967e6	−0.244056
$747$	−2.83748e6	−0.186050
$748$	0	0
$749$	4.60948e6	0.300225
$750$	−1.30954e6	−0.0850090
$751$	2.08868e6	0.135137	0.0675683	−	0.997715i	$-0.478476\pi$
$751$	2.08868e6	0.135137	0.0675683	+	0.997715i	$0.478476\pi$
$752$	−5.62663e6	−0.362830
$753$	−1.14801e7	−0.737836
$754$	−1.32062e6	−0.0845962
$755$	−1.96045e7	−1.25167
$756$	2.70860e6	0.172361
$757$	−899811.	−0.0570705	−0.0285352	−	0.999593i	$-0.509084\pi$
$757$	−899811.	−0.0570705	−0.0285352	+	0.999593i	$0.509084\pi$
$758$	1.01390e6	0.0640948
$759$	5.23764e6	0.330013
$760$	−2.92706e6	−0.183822
$761$	2.84513e7	1.78090	0.890451	−	0.455080i	$-0.150389\pi$
$761$	2.84513e7	1.78090	0.890451	+	0.455080i	$0.150389\pi$
$762$	−3090.49	−0.000192814 0
$763$	2.26090e6	0.140595
$764$	1.99116e7	1.23417
$765$	0	0
$766$	−2.19869e6	−0.135391
$767$	−4.23782e7	−2.60108
$768$	7.58875e6	0.464266
$769$	6.88279e6	0.419710	0.209855	−	0.977733i	$-0.432701\pi$
$769$	6.88279e6	0.419710	0.209855	+	0.977733i	$0.432701\pi$
$770$	−1.60452e6	−0.0975257
$771$	−7.46411e6	−0.452212
$772$	−1.18794e6	−0.0717380
$773$	−7.17307e6	−0.431774	−0.215887	−	0.976418i	$-0.569264\pi$
$773$	−7.17307e6	−0.431774	−0.215887	+	0.976418i	$0.569264\pi$
$774$	−1.44167e6	−0.0864997
$775$	−6.25961e6	−0.374363
$776$	−6.67304e6	−0.397804
$777$	1.14388e6	0.0679715
$778$	319716.	0.0189372
$779$	2.82398e7	1.66731
$780$	1.12851e7	0.664152
$781$	−3.08110e7	−1.80750
$782$	0	0
$783$	1.11391e6	0.0649301
$784$	−2.66028e6	−0.154574
$785$	−235762.	−0.0136553
$786$	−707456.	−0.0408454
$787$	−2.40986e7	−1.38693	−0.693465	−	0.720490i	$-0.743917\pi$
$787$	−2.40986e7	−1.38693	−0.693465	+	0.720490i	$0.743917\pi$
$788$	1.13663e7	0.652082
$789$	4.71847e6	0.269842
$790$	924477.	0.0527021
$791$	7.69528e6	0.437304
$792$	−1.86807e6	−0.105823
$793$	−2.20478e7	−1.24504
$794$	−4.22017e6	−0.237563
$795$	−8.28530e6	−0.464933
$796$	1.61011e7	0.900687
$797$	−2.57405e7	−1.43539	−0.717696	−	0.696356i	$-0.754803\pi$
$797$	−2.57405e7	−1.43539	−0.717696	+	0.696356i	$0.754803\pi$
$798$	−1.32477e6	−0.0736436
$799$	0	0
$800$	−4.18533e6	−0.231209
$801$	−1.02867e7	−0.566493
$802$	845446.	0.0464141
$803$	1.21236e7	0.663504
$804$	2.32339e6	0.126760
$805$	−5.66424e6	−0.308072
$806$	−3.10652e6	−0.168436
$807$	−7.61008e6	−0.411344
$808$	8.66864e6	0.467114
$809$	1.82115e7	0.978307	0.489154	−	0.872198i	$-0.337306\pi$
$809$	1.82115e7	0.978307	0.489154	+	0.872198i	$0.337306\pi$
$810$	196167.	0.0105054
$811$	−1.98604e7	−1.06032	−0.530158	−	0.847899i	$-0.677867\pi$
$811$	−1.98604e7	−1.06032	−0.530158	+	0.847899i	$0.677867\pi$
$812$	5.67727e6	0.302169
$813$	9.39542e6	0.498528
$814$	−390432.	−0.0206530
$815$	−2.25525e7	−1.18933
$816$	0	0
$817$	−3.42148e7	−1.79332
$818$	−3.35402e6	−0.175260
$819$	1.03204e7	0.537633
$820$	−2.13125e7	−1.10688
$821$	−2.96585e7	−1.53565	−0.767823	−	0.640662i	$-0.778660\pi$
$821$	−2.96585e7	−1.53565	−0.767823	+	0.640662i	$0.778660\pi$
$822$	1.70930e6	0.0882344
$823$	1.33764e6	0.0688401	0.0344200	−	0.999407i	$-0.489042\pi$
$823$	1.33764e6	0.0688401	0.0344200	+	0.999407i	$0.489042\pi$
$824$	1.24148e6	0.0636975
$825$	−7.09799e6	−0.363079
$826$	−3.75452e6	−0.191472
$827$	6.58068e6	0.334585	0.167293	−	0.985907i	$-0.446498\pi$
$827$	6.58068e6	0.334585	0.167293	+	0.985907i	$0.446498\pi$
$828$	−3.26366e6	−0.165436
$829$	−8.85598e6	−0.447559	−0.223779	−	0.974640i	$-0.571840\pi$
$829$	−8.85598e6	−0.447559	−0.223779	+	0.974640i	$0.571840\pi$
$830$	−1.04738e6	−0.0527726
$831$	7.28566e6	0.365987
$832$	3.10348e7	1.55432
$833$	0	0
$834$	161191.	0.00802463
$835$	−2.84772e6	−0.141345
$836$	−2.19410e7	−1.08578
$837$	2.62027e6	0.129280
$838$	−3.79444e6	−0.186654
$839$	−2.65981e7	−1.30451	−0.652253	−	0.758002i	$-0.726176\pi$
$839$	−2.65981e7	−1.30451	−0.652253	+	0.758002i	$0.726176\pi$
$840$	2.02022e6	0.0987870
$841$	−1.81764e7	−0.886170
$842$	−2.00688e6	−0.0975531
$843$	−1.15024e7	−0.557465
$844$	1.20721e7	0.583345
$845$	2.91884e7	1.40627
$846$	−380675.	−0.0182864
$847$	−5.21771e6	−0.249903
$848$	−2.38193e7	−1.13747
$849$	−1.51349e7	−0.720625
$850$	0	0
$851$	−1.37829e6	−0.0652404
$852$	1.91989e7	0.906101
$853$	2.51789e7	1.18485	0.592425	−	0.805626i	$-0.298171\pi$
$853$	2.51789e7	1.18485	0.592425	+	0.805626i	$0.298171\pi$
$854$	−1.95334e6	−0.0916500
$855$	4.65558e6	0.217800
$856$	1.98093e6	0.0924027
$857$	−1.33275e7	−0.619864	−0.309932	−	0.950759i	$-0.600306\pi$
$857$	−1.33275e7	−0.619864	−0.309932	+	0.950759i	$0.600306\pi$
$858$	−3.52259e6	−0.163359
$859$	−3.16841e7	−1.46507	−0.732534	−	0.680730i	$-0.761663\pi$
$859$	−3.16841e7	−1.46507	−0.732534	+	0.680730i	$0.761663\pi$
$860$	2.58219e7	1.19053
$861$	−1.94907e7	−0.896022
$862$	1.25856e6	0.0576908
$863$	1.81287e7	0.828591	0.414295	−	0.910143i	$-0.364028\pi$
$863$	1.81287e7	0.828591	0.414295	+	0.910143i	$0.364028\pi$
$864$	1.75197e6	0.0798441
$865$	−1.04681e7	−0.475692
$866$	2.30240e6	0.104324
$867$	0	0
$868$	1.33547e7	0.601638
$869$	1.40024e7	0.629004
$870$	411171.	0.0184172
$871$	8.85265e6	0.395392
$872$	971625.	0.0432721
$873$	1.06137e7	0.471335
$874$	1.59626e6	0.0706846
$875$	2.14502e7	0.947132
$876$	−7.55444e6	−0.332615
$877$	−3.47911e7	−1.52746	−0.763729	−	0.645537i	$-0.776634\pi$
$877$	−3.47911e7	−1.52746	−0.763729	+	0.645537i	$0.776634\pi$
$878$	5.94352e6	0.260200
$879$	−1.73986e7	−0.759523
$880$	1.62106e7	0.705654
$881$	−1.52161e7	−0.660486	−0.330243	−	0.943896i	$-0.607131\pi$
$881$	−1.52161e7	−0.660486	−0.330243	+	0.943896i	$0.607131\pi$
$882$	−179984.	−0.00779046
$883$	3.32712e7	1.43604	0.718021	−	0.696022i	$-0.245048\pi$
$883$	3.32712e7	1.43604	0.718021	+	0.696022i	$0.245048\pi$
$884$	0	0
$885$	1.31943e7	0.566275
$886$	4.14056e6	0.177204
$887$	−1.90177e7	−0.811612	−0.405806	−	0.913959i	$-0.633009\pi$
$887$	−1.90177e7	−0.811612	−0.405806	+	0.913959i	$0.633009\pi$
$888$	491583.	0.0209201
$889$	50622.0	0.00214825
$890$	−3.79706e6	−0.160684
$891$	2.97121e6	0.125383
$892$	−1.63619e7	−0.688528
$893$	−9.03445e6	−0.379117
$894$	2.06401e6	0.0863710
$895$	5.12175e6	0.213728
$896$	1.18628e7	0.493650
$897$	−1.24353e7	−0.516031
$898$	2.33771e6	0.0967388
$899$	5.49213e6	0.226643
$900$	4.42288e6	0.182012
$901$	0	0
$902$	6.65262e6	0.272255
$903$	2.36146e7	0.963741
$904$	3.30706e6	0.134593
$905$	−929502.	−0.0377250
$906$	−3.81315e6	−0.154335
$907$	4.24744e7	1.71439	0.857193	−	0.514995i	$-0.172206\pi$
$907$	4.24744e7	1.71439	0.857193	+	0.514995i	$0.172206\pi$
$908$	4.20265e7	1.69164
$909$	−1.37877e7	−0.553455
$910$	3.80949e6	0.152498
$911$	1.94304e7	0.775686	0.387843	−	0.921726i	$-0.373220\pi$
$911$	1.94304e7	0.775686	0.387843	+	0.921726i	$0.373220\pi$
$912$	1.33843e7	0.532853
$913$	−1.58639e7	−0.629845
$914$	−4.70456e6	−0.186275
$915$	6.86450e6	0.271054
$916$	−1.05017e7	−0.413545
$917$	1.15881e7	0.455081
$918$	0	0
$919$	2.87216e7	1.12181	0.560905	−	0.827880i	$-0.310453\pi$
$919$	2.87216e7	1.12181	0.560905	+	0.827880i	$0.310453\pi$
$920$	−2.43422e6	−0.0948177
$921$	1.25310e7	0.486783
$922$	−3.78179e6	−0.146511
$923$	7.31522e7	2.82633
$924$	1.51434e7	0.583502
$925$	1.86784e6	0.0717772
$926$	−830281.	−0.0318198
$927$	−1.97461e6	−0.0754714
$928$	3.67217e6	0.139976
$929$	−77127.9	−0.00293206	−0.00146603	−	0.999999i	$-0.500467\pi$
$929$	−77127.9	−0.00293206	−0.00146603	+	0.999999i	$0.500467\pi$
$930$	967201.	0.0366699
$931$	−4.27151e6	−0.161513
$932$	−2.22757e7	−0.840024
$933$	−3.10152e6	−0.116646
$934$	−1.81572e6	−0.0681055
$935$	0	0
$936$	4.43520e6	0.165472
$937$	−1.58980e7	−0.591551	−0.295776	−	0.955257i	$-0.595578\pi$
$937$	−1.58980e7	−0.591551	−0.295776	+	0.955257i	$0.595578\pi$
$938$	784305.	0.0291057
$939$	2.13261e7	0.789308
$940$	6.81829e6	0.251684
$941$	−4.78289e7	−1.76082	−0.880412	−	0.474209i	$-0.842734\pi$
$941$	−4.78289e7	−1.76082	−0.880412	+	0.474209i	$0.842734\pi$
$942$	−45856.6	−0.00168374
$943$	2.34849e7	0.860021
$944$	3.79321e7	1.38541
$945$	−3.21321e6	−0.117047
$946$	−8.06019e6	−0.292831
$947$	2.68257e7	0.972021	0.486011	−	0.873953i	$-0.338452\pi$
$947$	2.68257e7	0.972021	0.486011	+	0.873953i	$0.338452\pi$
$948$	−8.72515e6	−0.315320
$949$	−2.87842e7	−1.03750
$950$	−2.16323e6	−0.0777668
$951$	2.30105e7	0.825039
$952$	0	0
$953$	−2.69758e7	−0.962147	−0.481074	−	0.876680i	$-0.659753\pi$
$953$	−2.69758e7	−0.962147	−0.481074	+	0.876680i	$0.659753\pi$
$954$	−1.61152e6	−0.0573278
$955$	−2.36212e7	−0.838096
$956$	−1.96028e6	−0.0693703
$957$	6.22772e6	0.219811
$958$	−2.65022e6	−0.0932971
$959$	−2.79982e7	−0.983068
$960$	−9.66257e6	−0.338388
$961$	−1.57099e7	−0.548740
$962$	926972.	0.0322945
$963$	−3.15073e6	−0.109483
$964$	1.46893e7	0.509105
$965$	1.40925e6	0.0487158
$966$	−1.10171e6	−0.0379862
$967$	2.28176e7	0.784701	0.392351	−	0.919816i	$-0.371662\pi$
$967$	2.28176e7	0.784701	0.392351	+	0.919816i	$0.371662\pi$
$968$	−2.24232e6	−0.0769148
$969$	0	0
$970$	3.91775e6	0.133693
$971$	1.74769e7	0.594862	0.297431	−	0.954743i	$-0.403870\pi$
$971$	1.74769e7	0.594862	0.297431	+	0.954743i	$0.403870\pi$
$972$	−1.85141e6	−0.0628547
$973$	−2.64030e6	−0.0894068
$974$	−2.37815e6	−0.0803234
$975$	1.68522e7	0.567735
$976$	1.97347e7	0.663140
$977$	1.08540e7	0.363793	0.181897	−	0.983318i	$-0.441776\pi$
$977$	1.08540e7	0.363793	0.181897	+	0.983318i	$0.441776\pi$
$978$	−4.38655e6	−0.146648
$979$	−5.75115e7	−1.91777
$980$	3.22370e6	0.107223
$981$	−1.54540e6	−0.0512705
$982$	499936.	0.0165438
$983$	−2.61776e7	−0.864063	−0.432032	−	0.901858i	$-0.642203\pi$
$983$	−2.61776e7	−0.864063	−0.432032	+	0.901858i	$0.642203\pi$
$984$	−8.37615e6	−0.275776
$985$	−1.34838e7	−0.442815
$986$	0	0
$987$	6.23545e6	0.203739
$988$	5.20929e7	1.69780
$989$	−2.84539e7	−0.925018
$990$	1.09674e6	0.0355645
$991$	−3.30777e7	−1.06992	−0.534960	−	0.844877i	$-0.679674\pi$
$991$	−3.30777e7	−1.06992	−0.534960	+	0.844877i	$0.679674\pi$
$992$	8.63809e6	0.278701
$993$	−179572.	−0.00577918
$994$	6.48095e6	0.208053
$995$	−1.91008e7	−0.611637
$996$	9.88509e6	0.315742
$997$	−3.90596e7	−1.24448	−0.622242	−	0.782825i	$-0.713778\pi$
$997$	−3.90596e7	−1.24448	−0.622242	+	0.782825i	$0.713778\pi$
$998$	7.08837e6	0.225279
$999$	−781877.	−0.0247870

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.6.a.u.1.15		28
17.11	odd	16		51.6.h.a.19.8	✓	56
17.14	odd	16		51.6.h.a.43.8	yes	56
17.16	even	2		867.6.a.t.1.15		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.6.h.a.19.8	✓	56	17.11	odd	16
51.6.h.a.43.8	yes	56	17.14	odd	16
867.6.a.t.1.15		28	17.16	even	2
867.6.a.u.1.15		28	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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q+0.803842 q^{2} +9.00000 q^{3} -31.3538 q^{4} +37.1951 q^{5} +7.23458 q^{6} -118.502 q^{7} -50.9265 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+0.803842 q^{2} +9.00000 q^{3} -31.3538 q^{4} +37.1951 q^{5} +7.23458 q^{6} -118.502 q^{7} -50.9265 q^{8} +81.0000 q^{9} +29.8990 q^{10} +452.859 q^{11} -282.185 q^{12} -1075.19 q^{13} -95.2570 q^{14} +334.756 q^{15} +962.386 q^{16} +65.1112 q^{18} +1545.26 q^{19} -1166.21 q^{20} -1066.52 q^{21} +364.027 q^{22} +1285.08 q^{23} -458.338 q^{24} -1741.52 q^{25} -864.283 q^{26} +729.000 q^{27} +3715.49 q^{28} +1528.00 q^{29} +269.091 q^{30} +3594.33 q^{31} +2403.25 q^{32} +4075.73 q^{33} -4407.70 q^{35} -2539.66 q^{36} -1072.53 q^{37} +1242.15 q^{38} -9676.70 q^{39} -1894.22 q^{40} +18275.0 q^{41} -857.313 q^{42} -22141.7 q^{43} -14198.9 q^{44} +3012.80 q^{45} +1033.00 q^{46} -5846.54 q^{47} +8661.47 q^{48} -2764.26 q^{49} -1399.91 q^{50} +33711.3 q^{52} -24750.3 q^{53} +586.001 q^{54} +16844.1 q^{55} +6034.89 q^{56} +13907.4 q^{57} +1228.27 q^{58} +39414.7 q^{59} -10495.9 q^{60} +20506.0 q^{61} +2889.27 q^{62} -9598.67 q^{63} -28864.5 q^{64} -39991.8 q^{65} +3276.25 q^{66} -8233.57 q^{67} +11565.7 q^{69} -3543.09 q^{70} -68036.5 q^{71} -4125.05 q^{72} +26771.3 q^{73} -862.148 q^{74} -15673.7 q^{75} -48450.0 q^{76} -53664.8 q^{77} -7778.54 q^{78} +30920.0 q^{79} +35796.0 q^{80} +6561.00 q^{81} +14690.2 q^{82} -35030.6 q^{83} +33439.5 q^{84} -17798.4 q^{86} +13752.0 q^{87} -23062.5 q^{88} -126996. q^{89} +2421.82 q^{90} +127412. q^{91} -40292.2 q^{92} +32349.0 q^{93} -4699.70 q^{94} +57476.3 q^{95} +21629.3 q^{96} +131033. q^{97} -2222.03 q^{98} +36681.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9}+O(q^{10}) \) 28 * q + 252 * q^3 + 384 * q^4 - 244 * q^5 - 392 * q^7 + 2268 * q^9	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9} - 800 q^{10} - 1132 q^{11} + 3456 q^{12} - 828 q^{13} - 1568 q^{14} - 2196 q^{15} + 4096 q^{16} - 7532 q^{19} - 9216 q^{20} - 3528 q^{21} - 664 q^{22} - 4548 q^{23} + 9904 q^{25} + 20412 q^{27} - 36936 q^{28} - 36680 q^{29} - 7200 q^{30} - 7688 q^{31} - 10188 q^{33} + 13136 q^{35} + 31104 q^{36} - 42360 q^{37} - 1984 q^{38} - 7452 q^{39} - 81000 q^{40} - 6060 q^{41} - 14112 q^{42} - 50044 q^{43} - 56960 q^{44} - 19764 q^{45} - 51488 q^{46} + 16552 q^{47} + 36864 q^{48} + 38084 q^{49} - 48064 q^{50} - 31360 q^{52} - 4328 q^{53} - 71740 q^{55} + 9248 q^{56} - 67788 q^{57} - 95256 q^{58} - 89832 q^{59} - 82944 q^{60} - 154104 q^{61} - 106624 q^{62} - 31752 q^{63} + 93720 q^{64} - 13260 q^{65} - 5976 q^{66} - 182136 q^{67} - 40932 q^{69} - 145464 q^{70} + 24504 q^{71} - 58184 q^{73} - 308992 q^{74} + 89136 q^{75} - 602544 q^{76} - 240576 q^{77} - 136080 q^{79} - 317440 q^{80} + 183708 q^{81} - 46648 q^{82} - 203576 q^{83} - 332424 q^{84} - 421280 q^{86} - 330120 q^{87} - 145152 q^{88} - 120880 q^{89} - 64800 q^{90} - 290152 q^{91} - 491616 q^{92} - 69192 q^{93} - 799168 q^{94} - 543084 q^{95} - 409328 q^{97} + 193072 q^{98} - 91692 q^{99}+O(q^{100}) \) 28 * q + 252 * q^3 + 384 * q^4 - 244 * q^5 - 392 * q^7 + 2268 * q^9 - 800 * q^10 - 1132 * q^11 + 3456 * q^12 - 828 * q^13 - 1568 * q^14 - 2196 * q^15 + 4096 * q^16 - 7532 * q^19 - 9216 * q^20 - 3528 * q^21 - 664 * q^22 - 4548 * q^23 + 9904 * q^25 + 20412 * q^27 - 36936 * q^28 - 36680 * q^29 - 7200 * q^30 - 7688 * q^31 - 10188 * q^33 + 13136 * q^35 + 31104 * q^36 - 42360 * q^37 - 1984 * q^38 - 7452 * q^39 - 81000 * q^40 - 6060 * q^41 - 14112 * q^42 - 50044 * q^43 - 56960 * q^44 - 19764 * q^45 - 51488 * q^46 + 16552 * q^47 + 36864 * q^48 + 38084 * q^49 - 48064 * q^50 - 31360 * q^52 - 4328 * q^53 - 71740 * q^55 + 9248 * q^56 - 67788 * q^57 - 95256 * q^58 - 89832 * q^59 - 82944 * q^60 - 154104 * q^61 - 106624 * q^62 - 31752 * q^63 + 93720 * q^64 - 13260 * q^65 - 5976 * q^66 - 182136 * q^67 - 40932 * q^69 - 145464 * q^70 + 24504 * q^71 - 58184 * q^73 - 308992 * q^74 + 89136 * q^75 - 602544 * q^76 - 240576 * q^77 - 136080 * q^79 - 317440 * q^80 + 183708 * q^81 - 46648 * q^82 - 203576 * q^83 - 332424 * q^84 - 421280 * q^86 - 330120 * q^87 - 145152 * q^88 - 120880 * q^89 - 64800 * q^90 - 290152 * q^91 - 491616 * q^92 - 69192 * q^93 - 799168 * q^94 - 543084 * q^95 - 409328 * q^97 + 193072 * q^98 - 91692 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more