LMFDB - Embedded newform 867.6.a.u.1.16

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			1.16
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)

$f(q)$	$=$	$q+1.77098 q^{2} +9.00000 q^{3} -28.8636 q^{4} -66.3668 q^{5} +15.9388 q^{6} +199.014 q^{7} -107.788 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+1.77098 q^{2} +9.00000 q^{3} -28.8636 q^{4} -66.3668 q^{5} +15.9388 q^{6} +199.014 q^{7} -107.788 q^{8} +81.0000 q^{9} -117.534 q^{10} +239.823 q^{11} -259.773 q^{12} -611.300 q^{13} +352.450 q^{14} -597.301 q^{15} +732.745 q^{16} +143.449 q^{18} -2154.17 q^{19} +1915.59 q^{20} +1791.12 q^{21} +424.721 q^{22} +511.526 q^{23} -970.095 q^{24} +1279.56 q^{25} -1082.60 q^{26} +729.000 q^{27} -5744.26 q^{28} +2303.30 q^{29} -1057.81 q^{30} +9878.70 q^{31} +4746.90 q^{32} +2158.40 q^{33} -13207.9 q^{35} -2337.95 q^{36} +7553.81 q^{37} -3814.99 q^{38} -5501.70 q^{39} +7153.57 q^{40} -11444.2 q^{41} +3172.05 q^{42} -12497.8 q^{43} -6922.15 q^{44} -5375.71 q^{45} +905.903 q^{46} +21471.8 q^{47} +6594.70 q^{48} +22799.5 q^{49} +2266.07 q^{50} +17644.3 q^{52} -33887.4 q^{53} +1291.05 q^{54} -15916.3 q^{55} -21451.4 q^{56} -19387.5 q^{57} +4079.10 q^{58} -17413.2 q^{59} +17240.3 q^{60} +20427.5 q^{61} +17495.0 q^{62} +16120.1 q^{63} -15041.2 q^{64} +40570.1 q^{65} +3822.49 q^{66} -12089.7 q^{67} +4603.73 q^{69} -23391.0 q^{70} +59676.8 q^{71} -8730.86 q^{72} -43842.6 q^{73} +13377.7 q^{74} +11516.0 q^{75} +62177.1 q^{76} +47728.0 q^{77} -9743.41 q^{78} +11987.4 q^{79} -48630.0 q^{80} +6561.00 q^{81} -20267.5 q^{82} -390.808 q^{83} -51698.3 q^{84} -22133.3 q^{86} +20729.7 q^{87} -25850.1 q^{88} +12797.7 q^{89} -9520.29 q^{90} -121657. q^{91} -14764.5 q^{92} +88908.3 q^{93} +38026.2 q^{94} +142965. q^{95} +42722.1 q^{96} -65497.6 q^{97} +40377.5 q^{98} +19425.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.77098	0.313068	0.156534	−	0.987673i	$-0.449968\pi$
$2$	1.77098	0.313068	0.156534	+	0.987673i	$0.449968\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−28.8636	−0.901988
$5$	−66.3668	−1.18721	−0.593603	−	0.804758i	$-0.702295\pi$
$5$	−66.3668	−1.18721	−0.593603	+	0.804758i	$0.702295\pi$
$6$	15.9388	0.180750
$7$	199.014	1.53510	0.767552	−	0.640986i	$-0.221474\pi$
$7$	199.014	1.53510	0.767552	+	0.640986i	$0.221474\pi$
$8$	−107.788	−0.595452
$9$	81.0000	0.333333
$10$	−117.534	−0.371676
$11$	239.823	0.597597	0.298799	−	0.954316i	$-0.403414\pi$
$11$	239.823	0.597597	0.298799	+	0.954316i	$0.403414\pi$
$12$	−259.773	−0.520763
$13$	−611.300	−1.00322	−0.501610	−	0.865094i	$-0.667259\pi$
$13$	−611.300	−1.00322	−0.501610	+	0.865094i	$0.667259\pi$
$14$	352.450	0.480593
$15$	−597.301	−0.685434
$16$	732.745	0.715571
$17$	0	0
$17$	0	0
$18$	143.449	0.104356
$19$	−2154.17	−1.36898	−0.684488	−	0.729024i	$-0.739974\pi$
$19$	−2154.17	−1.36898	−0.684488	+	0.729024i	$0.739974\pi$
$20$	1915.59	1.07085
$21$	1791.12	0.886293
$22$	424.721	0.187089
$23$	511.526	0.201627	0.100813	−	0.994905i	$-0.467856\pi$
$23$	511.526	0.201627	0.100813	+	0.994905i	$0.467856\pi$
$24$	−970.095	−0.343784
$25$	1279.56	0.409458
$26$	−1082.60	−0.314076
$27$	729.000	0.192450
$28$	−5744.26	−1.38465
$29$	2303.30	0.508576	0.254288	−	0.967129i	$-0.418159\pi$
$29$	2303.30	0.508576	0.254288	+	0.967129i	$0.418159\pi$
$30$	−1057.81	−0.214587
$31$	9878.70	1.84627	0.923136	−	0.384473i	$-0.125617\pi$
$31$	9878.70	1.84627	0.923136	+	0.384473i	$0.125617\pi$
$32$	4746.90	0.819475
$33$	2158.40	0.345023
$34$	0	0
$35$	−13207.9	−1.82249
$36$	−2337.95	−0.300663
$37$	7553.81	0.907114	0.453557	−	0.891227i	$-0.350155\pi$
$37$	7553.81	0.907114	0.453557	+	0.891227i	$0.350155\pi$
$38$	−3814.99	−0.428583
$39$	−5501.70	−0.579209
$40$	7153.57	0.706924
$41$	−11444.2	−1.06323	−0.531615	−	0.846986i	$-0.678415\pi$
$41$	−11444.2	−1.06323	−0.531615	+	0.846986i	$0.678415\pi$
$42$	3172.05	0.277470
$43$	−12497.8	−1.03077	−0.515385	−	0.856959i	$-0.672351\pi$
$43$	−12497.8	−1.03077	−0.515385	+	0.856959i	$0.672351\pi$
$44$	−6922.15	−0.539026
$45$	−5375.71	−0.395735
$46$	905.903	0.0631229
$47$	21471.8	1.41783	0.708915	−	0.705294i	$-0.249185\pi$
$47$	21471.8	1.41783	0.708915	+	0.705294i	$0.249185\pi$
$48$	6594.70	0.413135
$49$	22799.5	1.35655
$50$	2266.07	0.128188
$51$	0	0
$52$	17644.3	0.904893
$53$	−33887.4	−1.65710	−0.828551	−	0.559914i	$-0.810834\pi$
$53$	−33887.4	−1.65710	−0.828551	+	0.559914i	$0.810834\pi$
$54$	1291.05	0.0602500
$55$	−15916.3	−0.709471
$56$	−21451.4	−0.914081
$57$	−19387.5	−0.790379
$58$	4079.10	0.159219
$59$	−17413.2	−0.651252	−0.325626	−	0.945499i	$-0.605575\pi$
$59$	−17413.2	−0.651252	−0.325626	+	0.945499i	$0.605575\pi$
$60$	17240.3	0.618253
$61$	20427.5	0.702896	0.351448	−	0.936207i	$-0.385689\pi$
$61$	20427.5	0.702896	0.351448	+	0.936207i	$0.385689\pi$
$62$	17495.0	0.578009
$63$	16120.1	0.511702
$64$	−15041.2	−0.459020
$65$	40570.1	1.19103
$66$	3822.49	0.108016
$67$	−12089.7	−0.329024	−0.164512	−	0.986375i	$-0.552605\pi$
$67$	−12089.7	−0.329024	−0.164512	+	0.986375i	$0.552605\pi$
$68$	0	0
$69$	4603.73	0.116409
$70$	−23391.0	−0.570562
$71$	59676.8	1.40495	0.702473	−	0.711710i	$-0.252079\pi$
$71$	59676.8	1.40495	0.702473	+	0.711710i	$0.252079\pi$
$72$	−8730.86	−0.198484
$73$	−43842.6	−0.962917	−0.481459	−	0.876469i	$-0.659893\pi$
$73$	−43842.6	−0.962917	−0.481459	+	0.876469i	$0.659893\pi$
$74$	13377.7	0.283989
$75$	11516.0	0.236401
$76$	62177.1	1.23480
$77$	47728.0	0.917374
$78$	−9743.41	−0.181332
$79$	11987.4	0.216101	0.108050	−	0.994145i	$-0.465539\pi$
$79$	11987.4	0.216101	0.108050	+	0.994145i	$0.465539\pi$
$80$	−48630.0	−0.849530
$81$	6561.00	0.111111
$82$	−20267.5	−0.332863
$83$	−390.808	−0.00622684	−0.00311342	−	0.999995i	$-0.500991\pi$
$83$	−390.808	−0.00622684	−0.00311342	+	0.999995i	$0.500991\pi$
$84$	−51698.3	−0.799426
$85$	0	0
$86$	−22133.3	−0.322701
$87$	20729.7	0.293626
$88$	−25850.1	−0.355840
$89$	12797.7	0.171260	0.0856301	−	0.996327i	$-0.472710\pi$
$89$	12797.7	0.171260	0.0856301	+	0.996327i	$0.472710\pi$
$90$	−9520.29	−0.123892
$91$	−121657.	−1.54005
$92$	−14764.5	−0.181865
$93$	88908.3	1.06595
$94$	38026.2	0.443877
$95$	142965.	1.62526
$96$	42722.1	0.473124
$97$	−65497.6	−0.706799	−0.353400	−	0.935472i	$-0.614974\pi$
$97$	−65497.6	−0.706799	−0.353400	+	0.935472i	$0.614974\pi$
$98$	40377.5	0.424692
$99$	19425.6	0.199199
$100$	−36932.6	−0.369326
$101$	−90356.8	−0.881368	−0.440684	−	0.897662i	$-0.645264\pi$
$101$	−90356.8	−0.881368	−0.440684	+	0.897662i	$0.645264\pi$
$102$	0	0
$103$	59010.1	0.548067	0.274033	−	0.961720i	$-0.411642\pi$
$103$	59010.1	0.548067	0.274033	+	0.961720i	$0.411642\pi$
$104$	65891.1	0.597370
$105$	−118871.	−1.05221
$106$	−60014.0	−0.518786
$107$	187408.	1.58245	0.791223	−	0.611527i	$-0.209445\pi$
$107$	187408.	1.58245	0.791223	+	0.611527i	$0.209445\pi$
$108$	−21041.6	−0.173588
$109$	−75134.9	−0.605725	−0.302862	−	0.953034i	$-0.597942\pi$
$109$	−75134.9	−0.605725	−0.302862	+	0.953034i	$0.597942\pi$
$110$	−28187.4	−0.222113
$111$	67984.3	0.523722
$112$	145826.	1.09848
$113$	−150081.	−1.10568	−0.552841	−	0.833287i	$-0.686456\pi$
$113$	−150081.	−1.10568	−0.552841	+	0.833287i	$0.686456\pi$
$114$	−34334.9	−0.247442
$115$	−33948.4	−0.239372
$116$	−66481.6	−0.458729
$117$	−49515.3	−0.334407
$118$	−30838.5	−0.203886
$119$	0	0
$120$	64382.1	0.408143
$121$	−103536.	−0.642878
$122$	36176.7	0.220054
$123$	−102998.	−0.613856
$124$	−285135.	−1.66532
$125$	122476.	0.701095
$126$	28548.4	0.160198
$127$	−246752.	−1.35753	−0.678767	−	0.734354i	$-0.737485\pi$
$127$	−246752.	−1.35753	−0.678767	+	0.734354i	$0.737485\pi$
$128$	−178539.	−0.963179
$129$	−112480.	−0.595116
$130$	71848.8	0.372873
$131$	−233862.	−1.19064	−0.595320	−	0.803489i	$-0.702975\pi$
$131$	−233862.	−1.19064	−0.595320	+	0.803489i	$0.702975\pi$
$132$	−62299.4	−0.311207
$133$	−428709.	−2.10152
$134$	−21410.6	−0.103007
$135$	−48381.4	−0.228478
$136$	0	0
$137$	−416772.	−1.89713	−0.948566	−	0.316580i	$-0.897465\pi$
$137$	−416772.	−1.89713	−0.948566	+	0.316580i	$0.897465\pi$
$138$	8153.13	0.0364440
$139$	−219543.	−0.963792	−0.481896	−	0.876229i	$-0.660052\pi$
$139$	−219543.	−0.963792	−0.481896	+	0.876229i	$0.660052\pi$
$140$	381228.	1.64386
$141$	193246.	0.818584
$142$	105687.	0.439844
$143$	−146604.	−0.599521
$144$	59352.3	0.238524
$145$	−152863.	−0.603784
$146$	−77644.4	−0.301459
$147$	205195.	0.783203
$148$	−218030.	−0.818206
$149$	148189.	0.546828	0.273414	−	0.961896i	$-0.411847\pi$
$149$	148189.	0.546828	0.273414	+	0.961896i	$0.411847\pi$
$150$	20394.6	0.0740095
$151$	66388.8	0.236948	0.118474	−	0.992957i	$-0.462200\pi$
$151$	66388.8	0.236948	0.118474	+	0.992957i	$0.462200\pi$
$152$	232194.	0.815160
$153$	0	0
$154$	84525.4	0.287201
$155$	−655618.	−2.19191
$156$	158799.	0.522440
$157$	211227.	0.683913	0.341956	−	0.939716i	$-0.388910\pi$
$157$	211227.	0.683913	0.341956	+	0.939716i	$0.388910\pi$
$158$	21229.4	0.0676543
$159$	−304987.	−0.956728
$160$	−315037.	−0.972885
$161$	101801.	0.309518
$162$	11619.4	0.0347854
$163$	−390968.	−1.15258	−0.576291	−	0.817245i	$-0.695500\pi$
$163$	−390968.	−1.15258	−0.576291	+	0.817245i	$0.695500\pi$
$164$	330322.	0.959020
$165$	−143246.	−0.409613
$166$	−692.113	−0.00194943
$167$	288287.	0.799897	0.399949	−	0.916538i	$-0.369028\pi$
$167$	288287.	0.799897	0.399949	+	0.916538i	$0.369028\pi$
$168$	−193062.	−0.527745
$169$	2395.17	0.00645088
$170$	0	0
$171$	−174488.	−0.456325
$172$	360731.	0.929743
$173$	−359215.	−0.912513	−0.456256	−	0.889848i	$-0.650810\pi$
$173$	−359215.	−0.912513	−0.456256	+	0.889848i	$0.650810\pi$
$174$	36711.9	0.0919251
$175$	254649.	0.628561
$176$	175729.	0.427623
$177$	−156719.	−0.376000
$178$	22664.5	0.0536161
$179$	−606085.	−1.41384	−0.706922	−	0.707292i	$-0.749917\pi$
$179$	−606085.	−1.41384	−0.706922	+	0.707292i	$0.749917\pi$
$180$	155163.	0.356949
$181$	−655606.	−1.48746	−0.743732	−	0.668478i	$-0.766946\pi$
$181$	−655606.	−1.48746	−0.743732	+	0.668478i	$0.766946\pi$
$182$	−215453.	−0.482140
$183$	183848.	0.405817
$184$	−55136.5	−0.120059
$185$	−501323.	−1.07693
$186$	157455.	0.333714
$187$	0	0
$188$	−619754.	−1.27887
$189$	145081.	0.295431
$190$	253189.	0.508816
$191$	238034.	0.472124	0.236062	−	0.971738i	$-0.424143\pi$
$191$	238034.	0.472124	0.236062	+	0.971738i	$0.424143\pi$
$192$	−135370.	−0.265015
$193$	−856993.	−1.65609	−0.828045	−	0.560662i	$-0.810547\pi$
$193$	−856993.	−1.65609	−0.828045	+	0.560662i	$0.810547\pi$
$194$	−115995.	−0.221276
$195$	365131.	0.687641
$196$	−658076.	−1.22359
$197$	−398281.	−0.731180	−0.365590	−	0.930776i	$-0.619133\pi$
$197$	−398281.	−0.731180	−0.365590	+	0.930776i	$0.619133\pi$
$198$	34402.4	0.0623629
$199$	−426294.	−0.763092	−0.381546	−	0.924350i	$-0.624608\pi$
$199$	−426294.	−0.763092	−0.381546	+	0.924350i	$0.624608\pi$
$200$	−137921.	−0.243813
$201$	−108807.	−0.189962
$202$	−160020.	−0.275928
$203$	458389.	0.780717
$204$	0	0
$205$	759517.	1.26227
$206$	104506.	0.171582
$207$	41433.6	0.0672089
$208$	−447927.	−0.717875
$209$	−516619.	−0.818096
$210$	−210519.	−0.329414
$211$	−96470.4	−0.149172	−0.0745861	−	0.997215i	$-0.523764\pi$
$211$	−96470.4	−0.149172	−0.0745861	+	0.997215i	$0.523764\pi$
$212$	978115.	1.49469
$213$	537091.	0.811146
$214$	331896.	0.495414
$215$	829438.	1.22374
$216$	−78577.7	−0.114595
$217$	1.96600e6	2.83422
$218$	−133062.	−0.189633
$219$	−394583.	−0.555940
$220$	459401.	0.639934
$221$	0	0
$222$	120399.	0.163961
$223$	−1.22608e6	−1.65104	−0.825520	−	0.564373i	$-0.809118\pi$
$223$	−1.22608e6	−1.65104	−0.825520	+	0.564373i	$0.809118\pi$
$224$	944699.	1.25798
$225$	103644.	0.136486
$226$	−265791.	−0.346154
$227$	−885392.	−1.14044	−0.570218	−	0.821493i	$-0.693141\pi$
$227$	−885392.	−1.14044	−0.570218	+	0.821493i	$0.693141\pi$
$228$	559594.	0.712912
$229$	911021.	1.14799	0.573997	−	0.818858i	$-0.305392\pi$
$229$	911021.	1.14799	0.573997	+	0.818858i	$0.305392\pi$
$230$	−60121.9	−0.0749399
$231$	429552.	0.529646
$232$	−248269.	−0.302832
$233$	341031.	0.411532	0.205766	−	0.978601i	$-0.434031\pi$
$233$	341031.	0.411532	0.205766	+	0.978601i	$0.434031\pi$
$234$	−87690.7	−0.104692
$235$	−1.42502e6	−1.68326
$236$	502609.	0.587422
$237$	107886.	0.124766
$238$	0	0
$239$	1.18300e6	1.33964	0.669822	−	0.742522i	$-0.266370\pi$
$239$	1.18300e6	1.33964	0.669822	+	0.742522i	$0.266370\pi$
$240$	−437670.	−0.490477
$241$	209053.	0.231854	0.115927	−	0.993258i	$-0.463016\pi$
$241$	209053.	0.231854	0.115927	+	0.993258i	$0.463016\pi$
$242$	−183361.	−0.201265
$243$	59049.0	0.0641500
$244$	−589612.	−0.634004
$245$	−1.51313e6	−1.61050
$246$	−182408.	−0.192179
$247$	1.31684e6	1.37338
$248$	−1.06481e6	−1.09937
$249$	−3517.27	−0.00359507
$250$	216903.	0.219491
$251$	637215.	0.638413	0.319206	−	0.947685i	$-0.396584\pi$
$251$	637215.	0.638413	0.319206	+	0.947685i	$0.396584\pi$
$252$	−465285.	−0.461549
$253$	122675.	0.120492
$254$	−436993.	−0.425001
$255$	0	0
$256$	165129.	0.157479
$257$	−351083.	−0.331572	−0.165786	−	0.986162i	$-0.553016\pi$
$257$	−351083.	−0.331572	−0.165786	+	0.986162i	$0.553016\pi$
$258$	−199200.	−0.186312
$259$	1.50331e6	1.39252
$260$	−1.17100e6	−1.07429
$261$	186567.	0.169525
$262$	−414164.	−0.372752
$263$	1.37117e6	1.22237	0.611186	−	0.791487i	$-0.290693\pi$
$263$	1.37117e6	1.22237	0.611186	+	0.791487i	$0.290693\pi$
$264$	−232651.	−0.205445
$265$	2.24900e6	1.96732
$266$	−759236.	−0.657920
$267$	115179.	0.0988771
$268$	348952.	0.296776
$269$	−748362.	−0.630567	−0.315284	−	0.948998i	$-0.602100\pi$
$269$	−748362.	−0.630567	−0.315284	+	0.948998i	$0.602100\pi$
$270$	−85682.6	−0.0715292
$271$	747302.	0.618120	0.309060	−	0.951042i	$-0.399986\pi$
$271$	747302.	0.618120	0.309060	+	0.951042i	$0.399986\pi$
$272$	0	0
$273$	−1.09491e6	−0.889147
$274$	−738096.	−0.593932
$275$	306866.	0.244691
$276$	−132880.	−0.105000
$277$	224153.	0.175527	0.0877636	−	0.996141i	$-0.472028\pi$
$277$	224153.	0.175527	0.0877636	+	0.996141i	$0.472028\pi$
$278$	−388807.	−0.301733
$279$	800175.	0.615424
$280$	1.42366e6	1.08520
$281$	−256514.	−0.193796	−0.0968981	−	0.995294i	$-0.530892\pi$
$281$	−256514.	−0.193796	−0.0968981	+	0.995294i	$0.530892\pi$
$282$	342236.	0.256273
$283$	1.96633e6	1.45946	0.729728	−	0.683738i	$-0.239647\pi$
$283$	1.96633e6	1.45946	0.729728	+	0.683738i	$0.239647\pi$
$284$	−1.72249e6	−1.26725
$285$	1.28669e6	0.938342
$286$	−259632.	−0.187691
$287$	−2.27756e6	−1.63217
$288$	384499.	0.273158
$289$	0	0
$290$	−270717.	−0.189026
$291$	−589478.	−0.408071
$292$	1.26546e6	0.868540
$293$	−1.14872e6	−0.781707	−0.390853	−	0.920453i	$-0.627820\pi$
$293$	−1.14872e6	−0.781707	−0.390853	+	0.920453i	$0.627820\pi$
$294$	363397.	0.245196
$295$	1.15566e6	0.773170
$296$	−814213.	−0.540143
$297$	174831.	0.115008
$298$	262440.	0.171194
$299$	−312696.	−0.202276
$300$	−332394.	−0.213231
$301$	−2.48723e6	−1.58234
$302$	117573.	0.0741808
$303$	−813211.	−0.508858
$304$	−1.57846e6	−0.979600
$305$	−1.35571e6	−0.834482
$306$	0	0
$307$	−1.73043e6	−1.04787	−0.523936	−	0.851758i	$-0.675537\pi$
$307$	−1.73043e6	−1.04787	−0.523936	+	0.851758i	$0.675537\pi$
$308$	−1.37760e6	−0.827461
$309$	531091.	0.316426
$310$	−1.16109e6	−0.686216
$311$	−558249.	−0.327286	−0.163643	−	0.986520i	$-0.552325\pi$
$311$	−558249.	−0.327286	−0.163643	+	0.986520i	$0.552325\pi$
$312$	593019.	0.344891
$313$	−1.35483e6	−0.781671	−0.390836	−	0.920460i	$-0.627814\pi$
$313$	−1.35483e6	−0.781671	−0.390836	+	0.920460i	$0.627814\pi$
$314$	374079.	0.214111
$315$	−1.06984e6	−0.607495
$316$	−345999.	−0.194921
$317$	1.21461e6	0.678874	0.339437	−	0.940629i	$-0.389763\pi$
$317$	1.21461e6	0.678874	0.339437	+	0.940629i	$0.389763\pi$
$318$	−540126.	−0.299521
$319$	552384.	0.303923
$320$	998234.	0.544951
$321$	1.68667e6	0.913626
$322$	180287.	0.0969003
$323$	0	0
$324$	−189374.	−0.100221
$325$	−782193.	−0.410776
$326$	−692396.	−0.360837
$327$	−676214.	−0.349715
$328$	1.23355e6	0.633102
$329$	4.27319e6	2.17652
$330$	−253687.	−0.128237
$331$	−2.79124e6	−1.40032	−0.700161	−	0.713985i	$-0.746888\pi$
$331$	−2.79124e6	−1.40032	−0.700161	+	0.713985i	$0.746888\pi$
$332$	11280.1	0.00561654
$333$	611859.	0.302371
$334$	510551.	0.250422
$335$	802354.	0.390619
$336$	1.31244e6	0.634206
$337$	2.11646e6	1.01516	0.507581	−	0.861604i	$-0.330540\pi$
$337$	2.11646e6	1.01516	0.507581	+	0.861604i	$0.330540\pi$
$338$	4241.80	0.00201957
$339$	−1.35073e6	−0.638366
$340$	0	0
$341$	2.36914e6	1.10333
$342$	−309015.	−0.142861
$343$	1.19259e6	0.547337
$344$	1.34712e6	0.613774
$345$	−305535.	−0.138202
$346$	−636163.	−0.285679
$347$	−2.67239e6	−1.19145	−0.595725	−	0.803188i	$-0.703135\pi$
$347$	−2.67239e6	−1.19145	−0.595725	+	0.803188i	$0.703135\pi$
$348$	−598334.	−0.264848
$349$	137835.	0.0605754	0.0302877	−	0.999541i	$-0.490358\pi$
$349$	137835.	0.0605754	0.0302877	+	0.999541i	$0.490358\pi$
$350$	450979.	0.196782
$351$	−445638.	−0.193070
$352$	1.13842e6	0.489716
$353$	−2.48662e6	−1.06212	−0.531059	−	0.847335i	$-0.678206\pi$
$353$	−2.48662e6	−1.06212	−0.531059	+	0.847335i	$0.678206\pi$
$354$	−277546.	−0.117714
$355$	−3.96056e6	−1.66796
$356$	−369388.	−0.154475
$357$	0	0
$358$	−1.07337e6	−0.442629
$359$	−1.13509e6	−0.464829	−0.232414	−	0.972617i	$-0.574663\pi$
$359$	−1.13509e6	−0.464829	−0.232414	+	0.972617i	$0.574663\pi$
$360$	579439.	0.235641
$361$	2.16435e6	0.874096
$362$	−1.16107e6	−0.465678
$363$	−931825.	−0.371166
$364$	3.51147e6	1.38911
$365$	2.90969e6	1.14318
$366$	325591.	0.127048
$367$	−2.52648e6	−0.979153	−0.489576	−	0.871960i	$-0.662849\pi$
$367$	−2.52648e6	−0.979153	−0.489576	+	0.871960i	$0.662849\pi$
$368$	374818.	0.144278
$369$	−926983.	−0.354410
$370$	−887833.	−0.337153
$371$	−6.74407e6	−2.54383
$372$	−2.56622e6	−0.961471
$373$	−2.28265e6	−0.849506	−0.424753	−	0.905309i	$-0.639639\pi$
$373$	−2.28265e6	−0.849506	−0.424753	+	0.905309i	$0.639639\pi$
$374$	0	0
$375$	1.10229e6	0.404777
$376$	−2.31441e6	−0.844250
$377$	−1.40801e6	−0.510213
$378$	256936.	0.0924901
$379$	697418.	0.249399	0.124700	−	0.992195i	$-0.460203\pi$
$379$	697418.	0.249399	0.124700	+	0.992195i	$0.460203\pi$
$380$	−4.12650e6	−1.46596
$381$	−2.22077e6	−0.783773
$382$	421554.	0.147807
$383$	−3.95512e6	−1.37772	−0.688862	−	0.724892i	$-0.741890\pi$
$383$	−3.95512e6	−1.37772	−0.688862	+	0.724892i	$0.741890\pi$
$384$	−1.60685e6	−0.556092
$385$	−3.16756e6	−1.08911
$386$	−1.51772e6	−0.518469
$387$	−1.01232e6	−0.343590
$388$	1.89050e6	0.637525
$389$	2.57864e6	0.864006	0.432003	−	0.901872i	$-0.357807\pi$
$389$	2.57864e6	0.864006	0.432003	+	0.901872i	$0.357807\pi$
$390$	646639.	0.215279
$391$	0	0
$392$	−2.45752e6	−0.807759
$393$	−2.10475e6	−0.687416
$394$	−705348.	−0.228909
$395$	−795565.	−0.256556
$396$	−560694.	−0.179675
$397$	−805733.	−0.256575	−0.128288	−	0.991737i	$-0.540948\pi$
$397$	−805733.	−0.256575	−0.128288	+	0.991737i	$0.540948\pi$
$398$	−754959.	−0.238900
$399$	−3.85839e6	−1.21331
$400$	937588.	0.292996
$401$	5.06325e6	1.57242	0.786210	−	0.617959i	$-0.212040\pi$
$401$	5.06325e6	1.57242	0.786210	+	0.617959i	$0.212040\pi$
$402$	−192695.	−0.0594711
$403$	−6.03886e6	−1.85222
$404$	2.60803e6	0.794984
$405$	−435433.	−0.131912
$406$	811798.	0.244418
$407$	1.81158e6	0.542089
$408$	0	0
$409$	2.12290e6	0.627511	0.313755	−	0.949504i	$-0.398413\pi$
$409$	2.12290e6	0.627511	0.313755	+	0.949504i	$0.398413\pi$
$410$	1.34509e6	0.395177
$411$	−3.75095e6	−1.09531
$412$	−1.70325e6	−0.494350
$413$	−3.46547e6	−0.999740
$414$	73378.1	0.0210410
$415$	25936.7	0.00739255
$416$	−2.90178e6	−0.822114
$417$	−1.97589e6	−0.556445
$418$	−914922.	−0.256120
$419$	4.23290e6	1.17789	0.588943	−	0.808175i	$-0.299544\pi$
$419$	4.23290e6	1.17789	0.588943	+	0.808175i	$0.299544\pi$
$420$	3.43105e6	0.949083
$421$	−2.71257e6	−0.745891	−0.372945	−	0.927853i	$-0.621652\pi$
$421$	−2.71257e6	−0.745891	−0.372945	+	0.927853i	$0.621652\pi$
$422$	−170847.	−0.0467011
$423$	1.73922e6	0.472610
$424$	3.65267e6	0.986725
$425$	0	0
$426$	951179.	0.253944
$427$	4.06536e6	1.07902
$428$	−5.40928e6	−1.42735
$429$	−1.31943e6	−0.346134
$430$	1.46892e6	0.383113
$431$	2.68529e6	0.696303	0.348151	−	0.937438i	$-0.386810\pi$
$431$	2.68529e6	0.696303	0.348151	+	0.937438i	$0.386810\pi$
$432$	534171.	0.137712
$433$	4.85685e6	1.24490	0.622451	−	0.782659i	$-0.286137\pi$
$433$	4.85685e6	1.24490	0.622451	+	0.782659i	$0.286137\pi$
$434$	3.48175e6	0.887305
$435$	−1.37576e6	−0.348595
$436$	2.16866e6	0.546357
$437$	−1.10191e6	−0.276022
$438$	−698799.	−0.174047
$439$	3.93457e6	0.974398	0.487199	−	0.873291i	$-0.338019\pi$
$439$	3.93457e6	0.974398	0.487199	+	0.873291i	$0.338019\pi$
$440$	1.71559e6	0.422456
$441$	1.84676e6	0.452182
$442$	0	0
$443$	3.77548e6	0.914036	0.457018	−	0.889458i	$-0.348918\pi$
$443$	3.77548e6	0.914036	0.457018	+	0.889458i	$0.348918\pi$
$444$	−1.96227e6	−0.472392
$445$	−849342.	−0.203321
$446$	−2.17137e6	−0.516888
$447$	1.33370e6	0.315711
$448$	−2.99340e6	−0.704643
$449$	7.17372e6	1.67930	0.839650	−	0.543128i	$-0.182760\pi$
$449$	7.17372e6	1.67930	0.839650	+	0.543128i	$0.182760\pi$
$450$	183552.	0.0427294
$451$	−2.74459e6	−0.635383
$452$	4.33189e6	0.997312
$453$	597499.	0.136802
$454$	−1.56801e6	−0.357034
$455$	8.07400e6	1.82835
$456$	2.08975e6	0.470633
$457$	−5.56445e6	−1.24633	−0.623163	−	0.782092i	$-0.714153\pi$
$457$	−5.56445e6	−1.24633	−0.623163	+	0.782092i	$0.714153\pi$
$458$	1.61340e6	0.359400
$459$	0	0
$460$	979873.	0.215911
$461$	7.97101e6	1.74687	0.873436	−	0.486938i	$-0.161886\pi$
$461$	7.97101e6	1.74687	0.873436	+	0.486938i	$0.161886\pi$
$462$	760729.	0.165815
$463$	−5.51092e6	−1.19474	−0.597368	−	0.801967i	$-0.703787\pi$
$463$	−5.51092e6	−1.19474	−0.597368	+	0.801967i	$0.703787\pi$
$464$	1.68773e6	0.363922
$465$	−5.90056e6	−1.26550
$466$	603959.	0.128838
$467$	−1.66102e6	−0.352438	−0.176219	−	0.984351i	$-0.556387\pi$
$467$	−1.66102e6	−0.352438	−0.176219	+	0.984351i	$0.556387\pi$
$468$	1.42919e6	0.301631
$469$	−2.40601e6	−0.505086
$470$	−2.52368e6	−0.526974
$471$	1.90104e6	0.394857
$472$	1.87694e6	0.387789
$473$	−2.99725e6	−0.615985
$474$	191065.	0.0390602
$475$	−2.75638e6	−0.560538
$476$	0	0
$477$	−2.74488e6	−0.552367
$478$	2.09507e6	0.419400
$479$	7.78447e6	1.55021	0.775105	−	0.631833i	$-0.217697\pi$
$479$	7.78447e6	1.55021	0.775105	+	0.631833i	$0.217697\pi$
$480$	−2.83533e6	−0.561696
$481$	−4.61765e6	−0.910035
$482$	370230.	0.0725862
$483$	916206.	0.178700
$484$	2.98843e6	0.579868
$485$	4.34687e6	0.839116
$486$	104575.	0.0200833
$487$	−2.09230e6	−0.399762	−0.199881	−	0.979820i	$-0.564056\pi$
$487$	−2.09230e6	−0.399762	−0.199881	+	0.979820i	$0.564056\pi$
$488$	−2.20185e6	−0.418541
$489$	−3.51871e6	−0.665443
$490$	−2.67972e6	−0.504197
$491$	2.31248e6	0.432887	0.216444	−	0.976295i	$-0.430554\pi$
$491$	2.31248e6	0.432887	0.216444	+	0.976295i	$0.430554\pi$
$492$	2.97290e6	0.553691
$493$	0	0
$494$	2.33211e6	0.429963
$495$	−1.28922e6	−0.236490
$496$	7.23857e6	1.32114
$497$	1.18765e7	2.15674
$498$	−6229.02	−0.00112550
$499$	−6.93956e6	−1.24761	−0.623807	−	0.781578i	$-0.714415\pi$
$499$	−6.93956e6	−1.24761	−0.623807	+	0.781578i	$0.714415\pi$
$500$	−3.53511e6	−0.632380
$501$	2.59459e6	0.461821
$502$	1.12850e6	0.199867
$503$	2.14959e6	0.378822	0.189411	−	0.981898i	$-0.439342\pi$
$503$	2.14959e6	0.378822	0.189411	+	0.981898i	$0.439342\pi$
$504$	−1.73756e6	−0.304694
$505$	5.99670e6	1.04637
$506$	217256.	0.0377221
$507$	21556.5	0.00372442
$508$	7.12215e6	1.22448
$509$	1.46576e6	0.250766	0.125383	−	0.992108i	$-0.459984\pi$
$509$	1.46576e6	0.250766	0.125383	+	0.992108i	$0.459984\pi$
$510$	0	0
$511$	−8.72528e6	−1.47818
$512$	6.00567e6	1.01248
$513$	−1.57039e6	−0.263460
$514$	−621762.	−0.103805
$515$	−3.91631e6	−0.650668
$516$	3.24658e6	0.536787
$517$	5.14943e6	0.847291
$518$	2.66234e6	0.435952
$519$	−3.23293e6	−0.526840
$520$	−4.37298e6	−0.709201
$521$	3.84548e6	0.620663	0.310332	−	0.950628i	$-0.399560\pi$
$521$	3.84548e6	0.620663	0.310332	+	0.950628i	$0.399560\pi$
$522$	330407.	0.0530730
$523$	−2.56252e6	−0.409651	−0.204825	−	0.978799i	$-0.565663\pi$
$523$	−2.56252e6	−0.409651	−0.204825	+	0.978799i	$0.565663\pi$
$524$	6.75009e6	1.07394
$525$	2.29184e6	0.362900
$526$	2.42832e6	0.382686
$527$	0	0
$528$	1.58156e6	0.246888
$529$	−6.17468e6	−0.959347
$530$	3.98294e6	0.615906
$531$	−1.41047e6	−0.217084
$532$	1.23741e7	1.89555
$533$	6.99586e6	1.06665
$534$	203980.	0.0309553
$535$	−1.24377e7	−1.87869
$536$	1.30313e6	0.195918
$537$	−5.45477e6	−0.816283
$538$	−1.32534e6	−0.197411
$539$	5.46783e6	0.810668
$540$	1.39646e6	0.206084
$541$	4.64520e6	0.682357	0.341178	−	0.939999i	$-0.389174\pi$
$541$	4.64520e6	0.682357	0.341178	+	0.939999i	$0.389174\pi$
$542$	1.32346e6	0.193514
$543$	−5.90046e6	−0.858788
$544$	0	0
$545$	4.98646e6	0.719120
$546$	−1.93907e6	−0.278364
$547$	−7.16064e6	−1.02325	−0.511627	−	0.859208i	$-0.670957\pi$
$547$	−7.16064e6	−1.02325	−0.511627	+	0.859208i	$0.670957\pi$
$548$	1.20296e7	1.71119
$549$	1.65463e6	0.234299
$550$	543455.	0.0766049
$551$	−4.96170e6	−0.696228
$552$	−496229.	−0.0693161
$553$	2.38565e6	0.331738
$554$	396970.	0.0549520
$555$	−4.51190e6	−0.621766
$556$	6.33682e6	0.869329
$557$	−3.53070e6	−0.482195	−0.241097	−	0.970501i	$-0.577507\pi$
$557$	−3.53070e6	−0.482195	−0.241097	+	0.970501i	$0.577507\pi$
$558$	1.41710e6	0.192670
$559$	7.63990e6	1.03409
$560$	−9.67803e6	−1.30412
$561$	0	0
$562$	−454282.	−0.0606714
$563$	8.01990e6	1.06635	0.533173	−	0.846006i	$-0.320999\pi$
$563$	8.01990e6	1.06635	0.533173	+	0.846006i	$0.320999\pi$
$564$	−5.57779e6	−0.738354
$565$	9.96042e6	1.31267
$566$	3.48234e6	0.456909
$567$	1.30573e6	0.170567
$568$	−6.43246e6	−0.836578
$569$	−5.92356e6	−0.767011	−0.383506	−	0.923539i	$-0.625283\pi$
$569$	−5.92356e6	−0.767011	−0.383506	+	0.923539i	$0.625283\pi$
$570$	2.27870e6	0.293765
$571$	1.24905e6	0.160321	0.0801603	−	0.996782i	$-0.474457\pi$
$571$	1.24905e6	0.160321	0.0801603	+	0.996782i	$0.474457\pi$
$572$	4.23151e6	0.540761
$573$	2.14231e6	0.272581
$574$	−4.03352e6	−0.510980
$575$	654526.	0.0825577
$576$	−1.21833e6	−0.153007
$577$	7.92318e6	0.990741	0.495370	−	0.868682i	$-0.335032\pi$
$577$	7.92318e6	0.990741	0.495370	+	0.868682i	$0.335032\pi$
$578$	0	0
$579$	−7.71294e6	−0.956144
$580$	4.41217e6	0.544606
$581$	−77776.1	−0.00955886
$582$	−1.04396e6	−0.127754
$583$	−8.12698e6	−0.990279
$584$	4.72572e6	0.573371
$585$	3.28618e6	0.397010
$586$	−2.03436e6	−0.244728
$587$	−1.24650e7	−1.49313	−0.746566	−	0.665311i	$-0.768299\pi$
$587$	−1.24650e7	−1.49313	−0.746566	+	0.665311i	$0.768299\pi$
$588$	−5.92268e6	−0.706440
$589$	−2.12804e7	−2.52750
$590$	2.04665e6	0.242055
$591$	−3.58453e6	−0.422147
$592$	5.53502e6	0.649105
$593$	1.77795e6	0.207627	0.103814	−	0.994597i	$-0.466895\pi$
$593$	1.77795e6	0.207627	0.103814	+	0.994597i	$0.466895\pi$
$594$	309622.	0.0360052
$595$	0	0
$596$	−4.27727e6	−0.493232
$597$	−3.83665e6	−0.440571
$598$	−553779.	−0.0633262
$599$	−1.27300e7	−1.44964	−0.724820	−	0.688938i	$-0.758077\pi$
$599$	−1.27300e7	−1.44964	−0.724820	+	0.688938i	$0.758077\pi$
$600$	−1.24129e6	−0.140765
$601$	1.17061e7	1.32199	0.660994	−	0.750391i	$-0.270135\pi$
$601$	1.17061e7	1.32199	0.660994	+	0.750391i	$0.270135\pi$
$602$	−4.40484e6	−0.495381
$603$	−979264.	−0.109675
$604$	−1.91622e6	−0.213724
$605$	6.87136e6	0.763228
$606$	−1.44018e6	−0.159307
$607$	1.59262e7	1.75445	0.877226	−	0.480077i	$-0.159391\pi$
$607$	1.59262e7	1.75445	0.877226	+	0.480077i	$0.159391\pi$
$608$	−1.02256e7	−1.12184
$609$	4.12550e6	0.450747
$610$	−2.40094e6	−0.261250
$611$	−1.31257e7	−1.42240
$612$	0	0
$613$	7.46048e6	0.801892	0.400946	−	0.916102i	$-0.368682\pi$
$613$	7.46048e6	0.801892	0.400946	+	0.916102i	$0.368682\pi$
$614$	−3.06456e6	−0.328055
$615$	6.83566e6	0.728773
$616$	−5.14452e6	−0.546252
$617$	739639.	0.0782180	0.0391090	−	0.999235i	$-0.487548\pi$
$617$	739639.	0.0782180	0.0391090	+	0.999235i	$0.487548\pi$
$618$	940552.	0.0990631
$619$	2.59004e6	0.271694	0.135847	−	0.990730i	$-0.456624\pi$
$619$	2.59004e6	0.271694	0.135847	+	0.990730i	$0.456624\pi$
$620$	1.89235e7	1.97707
$621$	372902.	0.0388031
$622$	−988649.	−0.102463
$623$	2.54692e6	0.262902
$624$	−4.03135e6	−0.414466
$625$	−1.21270e7	−1.24180
$626$	−2.39938e6	−0.244716
$627$	−4.64957e6	−0.472328
$628$	−6.09678e6	−0.616881
$629$	0	0
$630$	−1.89467e6	−0.190187
$631$	−3.85306e6	−0.385240	−0.192620	−	0.981273i	$-0.561699\pi$
$631$	−3.85306e6	−0.385240	−0.192620	+	0.981273i	$0.561699\pi$
$632$	−1.29210e6	−0.128678
$633$	−868233.	−0.0861246
$634$	2.15105e6	0.212534
$635$	1.63761e7	1.61167
$636$	8.80303e6	0.862958
$637$	−1.39373e7	−1.36092
$638$	978261.	0.0951487
$639$	4.83382e6	0.468315
$640$	1.18490e7	1.14349
$641$	8.33994e6	0.801711	0.400855	−	0.916141i	$-0.368713\pi$
$641$	8.33994e6	0.801711	0.400855	+	0.916141i	$0.368713\pi$
$642$	2.98707e6	0.286027
$643$	−294836.	−0.0281224	−0.0140612	−	0.999901i	$-0.504476\pi$
$643$	−294836.	−0.0281224	−0.0140612	+	0.999901i	$0.504476\pi$
$644$	−2.93834e6	−0.279182
$645$	7.46495e6	0.706525
$646$	0	0
$647$	−8.61072e6	−0.808684	−0.404342	−	0.914608i	$-0.632499\pi$
$647$	−8.61072e6	−0.808684	−0.404342	+	0.914608i	$0.632499\pi$
$648$	−707199.	−0.0661613
$649$	−4.17608e6	−0.389186
$650$	−1.38525e6	−0.128601
$651$	1.76940e7	1.63634
$652$	1.12847e7	1.03962
$653$	5.38173e6	0.493900	0.246950	−	0.969028i	$-0.420572\pi$
$653$	5.38173e6	0.493900	0.246950	+	0.969028i	$0.420572\pi$
$654$	−1.19756e6	−0.109485
$655$	1.55206e7	1.41354
$656$	−8.38570e6	−0.760816
$657$	−3.55125e6	−0.320972
$658$	7.56773e6	0.681398
$659$	−1.50040e7	−1.34584	−0.672921	−	0.739715i	$-0.734960\pi$
$659$	−1.50040e7	−1.34584	−0.672921	+	0.739715i	$0.734960\pi$
$660$	4.13461e6	0.369466
$661$	1.78923e7	1.59281	0.796403	−	0.604766i	$-0.206733\pi$
$661$	1.78923e7	1.59281	0.796403	+	0.604766i	$0.206733\pi$
$662$	−4.94324e6	−0.438396
$663$	0	0
$664$	42124.5	0.00370779
$665$	2.84521e7	2.49494
$666$	1.08359e6	0.0946629
$667$	1.17820e6	0.102542
$668$	−8.32102e6	−0.721498
$669$	−1.10347e7	−0.953228
$670$	1.42095e6	0.122291
$671$	4.89898e6	0.420048
$672$	8.50229e6	0.726295
$673$	−6.60438e6	−0.562075	−0.281038	−	0.959697i	$-0.590679\pi$
$673$	−6.60438e6	−0.562075	−0.281038	+	0.959697i	$0.590679\pi$
$674$	3.74821e6	0.317815
$675$	932796.	0.0788002
$676$	−69133.2	−0.00581862
$677$	−6.81337e6	−0.571334	−0.285667	−	0.958329i	$-0.592215\pi$
$677$	−6.81337e6	−0.571334	−0.285667	+	0.958329i	$0.592215\pi$
$678$	−2.39212e6	−0.199852
$679$	−1.30349e7	−1.08501
$680$	0	0
$681$	−7.96853e6	−0.658431
$682$	4.19570e6	0.345417
$683$	−8.60468e6	−0.705802	−0.352901	−	0.935661i	$-0.614805\pi$
$683$	−8.60468e6	−0.705802	−0.352901	+	0.935661i	$0.614805\pi$
$684$	5.03635e6	0.411600
$685$	2.76599e7	2.25229
$686$	2.11205e6	0.171354
$687$	8.19919e6	0.662794
$688$	−9.15769e6	−0.737590
$689$	2.07154e7	1.66244
$690$	−541097.	−0.0432666
$691$	−7.30847e6	−0.582279	−0.291140	−	0.956681i	$-0.594034\pi$
$691$	−7.30847e6	−0.582279	−0.291140	+	0.956681i	$0.594034\pi$
$692$	1.03682e7	0.823076
$693$	3.86597e6	0.305791
$694$	−4.73275e6	−0.373005
$695$	1.45704e7	1.14422
$696$	−2.23442e6	−0.174840
$697$	0	0
$698$	244103.	0.0189642
$699$	3.06928e6	0.237598
$700$	−7.35010e6	−0.566955
$701$	4.78344e6	0.367659	0.183830	−	0.982958i	$-0.441150\pi$
$701$	4.78344e6	0.367659	0.183830	+	0.982958i	$0.441150\pi$
$702$	−789217.	−0.0604440
$703$	−1.62722e7	−1.24182
$704$	−3.60721e6	−0.274309
$705$	−1.28251e7	−0.971828
$706$	−4.40376e6	−0.332515
$707$	−1.79823e7	−1.35299
$708$	4.52348e6	0.339148
$709$	−1.35741e7	−1.01413	−0.507067	−	0.861906i	$-0.669270\pi$
$709$	−1.35741e7	−1.01413	−0.507067	+	0.861906i	$0.669270\pi$
$710$	−7.01408e6	−0.522185
$711$	970978.	0.0720336
$712$	−1.37944e6	−0.101977
$713$	5.05321e6	0.372258
$714$	0	0
$715$	9.72962e6	0.711755
$716$	1.74938e7	1.27527
$717$	1.06470e7	0.773444
$718$	−2.01022e6	−0.145523
$719$	−2.07106e7	−1.49407	−0.747033	−	0.664787i	$-0.768522\pi$
$719$	−2.07106e7	−1.49407	−0.747033	+	0.664787i	$0.768522\pi$
$720$	−3.93903e6	−0.283177
$721$	1.17438e7	0.841340
$722$	3.83302e6	0.273652
$723$	1.88148e6	0.133861
$724$	1.89232e7	1.34168
$725$	2.94720e6	0.208240
$726$	−1.65024e6	−0.116200
$727$	1.84525e7	1.29485	0.647424	−	0.762130i	$-0.275846\pi$
$727$	1.84525e7	1.29485	0.647424	+	0.762130i	$0.275846\pi$
$728$	1.31132e7	0.917025
$729$	531441.	0.0370370
$730$	5.15301e6	0.357894
$731$	0	0
$732$	−5.30651e6	−0.366042
$733$	−1.79655e7	−1.23503	−0.617517	−	0.786558i	$-0.711861\pi$
$733$	−1.79655e7	−1.23503	−0.617517	+	0.786558i	$0.711861\pi$
$734$	−4.47435e6	−0.306542
$735$	−1.36182e7	−0.929823
$736$	2.42816e6	0.165228
$737$	−2.89938e6	−0.196624
$738$	−1.64167e6	−0.110954
$739$	−2.41487e7	−1.62661	−0.813305	−	0.581838i	$-0.802334\pi$
$739$	−2.41487e7	−1.62661	−0.813305	+	0.581838i	$0.802334\pi$
$740$	1.44700e7	0.971379
$741$	1.18516e7	0.792924
$742$	−1.19436e7	−0.796391
$743$	−8.21203e6	−0.545731	−0.272865	−	0.962052i	$-0.587971\pi$
$743$	−8.21203e6	−0.545731	−0.272865	+	0.962052i	$0.587971\pi$
$744$	−9.58328e6	−0.634720
$745$	−9.83484e6	−0.649197
$746$	−4.04252e6	−0.265953
$747$	−31655.4	−0.00207561
$748$	0	0
$749$	3.72968e7	2.42922
$750$	1.95213e6	0.126723
$751$	−3.30775e6	−0.214009	−0.107005	−	0.994259i	$-0.534126\pi$
$751$	−3.30775e6	−0.214009	−0.107005	+	0.994259i	$0.534126\pi$
$752$	1.57334e7	1.01456
$753$	5.73494e6	0.368588
$754$	−2.49356e6	−0.159732
$755$	−4.40601e6	−0.281306
$756$	−4.18756e6	−0.266475
$757$	−1.76595e7	−1.12005	−0.560027	−	0.828474i	$-0.689209\pi$
$757$	−1.76595e7	−1.12005	−0.560027	+	0.828474i	$0.689209\pi$
$758$	1.23511e6	0.0780790
$759$	1.10408e6	0.0695658
$760$	−1.54100e7	−0.967763
$761$	−1.68591e7	−1.05529	−0.527645	−	0.849465i	$-0.676925\pi$
$761$	−1.68591e7	−1.05529	−0.527645	+	0.849465i	$0.676925\pi$
$762$	−3.93293e6	−0.245374
$763$	−1.49529e7	−0.929851
$764$	−6.87053e6	−0.425850
$765$	0	0
$766$	−7.00444e6	−0.431322
$767$	1.06447e7	0.653349
$768$	1.48616e6	0.0909205
$769$	−2.17990e7	−1.32929	−0.664646	−	0.747158i	$-0.731418\pi$
$769$	−2.17990e7	−1.32929	−0.664646	+	0.747158i	$0.731418\pi$
$770$	−5.60968e6	−0.340966
$771$	−3.15975e6	−0.191433
$772$	2.47359e7	1.49377
$773$	4.48917e6	0.270220	0.135110	−	0.990831i	$-0.456861\pi$
$773$	4.48917e6	0.270220	0.135110	+	0.990831i	$0.456861\pi$
$774$	−1.79280e6	−0.107567
$775$	1.26404e7	0.755971
$776$	7.05988e6	0.420865
$777$	1.35298e7	0.803969
$778$	4.56673e6	0.270493
$779$	2.46528e7	1.45554
$780$	−1.05390e7	−0.620244
$781$	1.43119e7	0.839592
$782$	0	0
$783$	1.67911e6	0.0978754
$784$	1.67062e7	0.970706
$785$	−1.40185e7	−0.811945
$786$	−3.72748e6	−0.215208
$787$	−1.62385e7	−0.934564	−0.467282	−	0.884108i	$-0.654767\pi$
$787$	−1.62385e7	−0.934564	−0.467282	+	0.884108i	$0.654767\pi$
$788$	1.14958e7	0.659516
$789$	1.23406e7	0.705736
$790$	−1.40893e6	−0.0803196
$791$	−2.98682e7	−1.69734
$792$	−2.09386e6	−0.118613
$793$	−1.24873e7	−0.705159
$794$	−1.42694e6	−0.0803256
$795$	2.02410e7	1.13583
$796$	1.23044e7	0.688300
$797$	3.32557e7	1.85447	0.927237	−	0.374475i	$-0.122177\pi$
$797$	3.32557e7	1.85447	0.927237	+	0.374475i	$0.122177\pi$
$798$	−6.83313e6	−0.379850
$799$	0	0
$800$	6.07393e6	0.335540
$801$	1.03661e6	0.0570867
$802$	8.96693e6	0.492275
$803$	−1.05144e7	−0.575436
$804$	3.14057e6	0.171344
$805$	−6.75619e6	−0.367462
$806$	−1.06947e7	−0.579871
$807$	−6.73526e6	−0.364058
$808$	9.73941e6	0.524813
$809$	2.15294e7	1.15654	0.578270	−	0.815846i	$-0.303728\pi$
$809$	2.15294e7	1.15654	0.578270	+	0.815846i	$0.303728\pi$
$810$	−771143.	−0.0412974
$811$	3.16150e7	1.68788	0.843938	−	0.536441i	$-0.180231\pi$
$811$	3.16150e7	1.68788	0.843938	+	0.536441i	$0.180231\pi$
$812$	−1.32308e7	−0.704198
$813$	6.72572e6	0.356872
$814$	3.20827e6	0.169711
$815$	2.59473e7	1.36835
$816$	0	0
$817$	2.69224e7	1.41110
$818$	3.75961e6	0.196454
$819$	−9.85423e6	−0.513349
$820$	−2.19224e7	−1.13855
$821$	−1.29096e7	−0.668431	−0.334215	−	0.942497i	$-0.608471\pi$
$821$	−1.29096e7	−0.668431	−0.334215	+	0.942497i	$0.608471\pi$
$822$	−6.64286e6	−0.342907
$823$	5.87395e6	0.302295	0.151147	−	0.988511i	$-0.451703\pi$
$823$	5.87395e6	0.302295	0.151147	+	0.988511i	$0.451703\pi$
$824$	−6.36060e6	−0.326347
$825$	2.76180e6	0.141272
$826$	−6.13728e6	−0.312987
$827$	1.41574e7	0.719811	0.359905	−	0.932989i	$-0.382809\pi$
$827$	1.41574e7	0.719811	0.359905	+	0.932989i	$0.382809\pi$
$828$	−1.19592e6	−0.0606216
$829$	2.07053e7	1.04639	0.523196	−	0.852212i	$-0.324740\pi$
$829$	2.07053e7	1.04639	0.523196	+	0.852212i	$0.324740\pi$
$830$	45933.4	0.00231437
$831$	2.01737e6	0.101341
$832$	9.19467e6	0.460498
$833$	0	0
$834$	−3.49926e6	−0.174205
$835$	−1.91327e7	−0.949643
$836$	1.49115e7	0.737913
$837$	7.20158e6	0.355315
$838$	7.49639e6	0.368759
$839$	3.71471e7	1.82188	0.910940	−	0.412539i	$-0.135358\pi$
$839$	3.71471e7	1.82188	0.910940	+	0.412539i	$0.135358\pi$
$840$	1.28129e7	0.626542
$841$	−1.52060e7	−0.741351
$842$	−4.80391e6	−0.233515
$843$	−2.30863e6	−0.111888
$844$	2.78448e6	0.134552
$845$	−158960.	−0.00765852
$846$	3.08012e6	0.147959
$847$	−2.06051e7	−0.986885
$848$	−2.48309e7	−1.18577
$849$	1.76970e7	0.842617
$850$	0	0
$851$	3.86397e6	0.182898
$852$	−1.55024e7	−0.731644
$853$	−8.53648e6	−0.401704	−0.200852	−	0.979622i	$-0.564371\pi$
$853$	−8.53648e6	−0.401704	−0.200852	+	0.979622i	$0.564371\pi$
$854$	7.19967e6	0.337806
$855$	1.15802e7	0.541752
$856$	−2.02004e7	−0.942271
$857$	−3.41597e7	−1.58877	−0.794387	−	0.607411i	$-0.792208\pi$
$857$	−3.41597e7	−1.58877	−0.794387	+	0.607411i	$0.792208\pi$
$858$	−2.33669e6	−0.108364
$859$	−2.83315e7	−1.31005	−0.655023	−	0.755609i	$-0.727341\pi$
$859$	−2.83315e7	−1.31005	−0.655023	+	0.755609i	$0.727341\pi$
$860$	−2.39406e7	−1.10380
$861$	−2.04980e7	−0.942333
$862$	4.75560e6	0.217990
$863$	5.30250e6	0.242356	0.121178	−	0.992631i	$-0.461333\pi$
$863$	5.30250e6	0.242356	0.121178	+	0.992631i	$0.461333\pi$
$864$	3.46049e6	0.157708
$865$	2.38399e7	1.08334
$866$	8.60139e6	0.389739
$867$	0	0
$868$	−5.67458e7	−2.55643
$869$	2.87485e6	0.129141
$870$	−2.43645e6	−0.109134
$871$	7.39042e6	0.330084
$872$	8.09866e6	0.360680
$873$	−5.30531e6	−0.235600
$874$	−1.95147e6	−0.0864138
$875$	2.43745e7	1.07625
$876$	1.13891e7	0.501452
$877$	−3.20526e7	−1.40723	−0.703614	−	0.710582i	$-0.748432\pi$
$877$	−3.20526e7	−1.40723	−0.703614	+	0.710582i	$0.748432\pi$
$878$	6.96806e6	0.305053
$879$	−1.03384e7	−0.451319
$880$	−1.16626e7	−0.507677
$881$	2.56582e7	1.11375	0.556873	−	0.830597i	$-0.312001\pi$
$881$	2.56582e7	1.11375	0.556873	+	0.830597i	$0.312001\pi$
$882$	3.27057e6	0.141564
$883$	−5.96594e6	−0.257500	−0.128750	−	0.991677i	$-0.541096\pi$
$883$	−5.96594e6	−0.257500	−0.128750	+	0.991677i	$0.541096\pi$
$884$	0	0
$885$	1.04009e7	0.446390
$886$	6.68631e6	0.286156
$887$	2.66178e7	1.13596	0.567979	−	0.823043i	$-0.307725\pi$
$887$	2.66178e7	1.13596	0.567979	+	0.823043i	$0.307725\pi$
$888$	−7.32792e6	−0.311852
$889$	−4.91070e7	−2.08396
$890$	−1.50417e6	−0.0636534
$891$	1.57348e6	0.0663997
$892$	3.53892e7	1.48922
$893$	−4.62539e7	−1.94098
$894$	2.36196e6	0.0988391
$895$	4.02240e7	1.67852
$896$	−3.55316e7	−1.47858
$897$	−2.81426e6	−0.116784
$898$	1.27045e7	0.525735
$899$	2.27536e7	0.938969
$900$	−2.99154e6	−0.123109
$901$	0	0
$902$	−4.86061e6	−0.198918
$903$	−2.23851e7	−0.913565
$904$	1.61770e7	0.658381
$905$	4.35105e7	1.76593
$906$	1.05816e6	0.0428283
$907$	−8.88692e6	−0.358701	−0.179351	−	0.983785i	$-0.557400\pi$
$907$	−8.88692e6	−0.358701	−0.179351	+	0.983785i	$0.557400\pi$
$908$	2.55556e7	1.02866
$909$	−7.31890e6	−0.293789
$910$	1.42989e7	0.572400
$911$	−1.22973e7	−0.490922	−0.245461	−	0.969407i	$-0.578939\pi$
$911$	−1.22973e7	−0.490922	−0.245461	+	0.969407i	$0.578939\pi$
$912$	−1.42061e7	−0.565572
$913$	−93724.6	−0.00372114
$914$	−9.85453e6	−0.390185
$915$	−1.22014e7	−0.481788
$916$	−2.62954e7	−1.03548
$917$	−4.65417e7	−1.82776
$918$	0	0
$919$	−1.99290e7	−0.778387	−0.389194	−	0.921156i	$-0.627246\pi$
$919$	−1.99290e7	−0.778387	−0.389194	+	0.921156i	$0.627246\pi$
$920$	3.65924e6	0.142535
$921$	−1.55739e7	−0.604989
$922$	1.41165e7	0.546890
$923$	−3.64805e7	−1.40947
$924$	−1.23984e7	−0.477735
$925$	9.66553e6	0.371425
$926$	−9.75974e6	−0.374034
$927$	4.77982e6	0.182689
$928$	1.09335e7	0.416765
$929$	−2.55360e7	−0.970763	−0.485381	−	0.874303i	$-0.661319\pi$
$929$	−2.55360e7	−0.970763	−0.485381	+	0.874303i	$0.661319\pi$
$930$	−1.04498e7	−0.396187
$931$	−4.91140e7	−1.85708
$932$	−9.84338e6	−0.371197
$933$	−5.02425e6	−0.188959
$934$	−2.94164e6	−0.110337
$935$	0	0
$936$	5.33718e6	0.199123
$937$	7.37919e6	0.274574	0.137287	−	0.990531i	$-0.456162\pi$
$937$	7.37919e6	0.274574	0.137287	+	0.990531i	$0.456162\pi$
$938$	−4.26100e6	−0.158127
$939$	−1.21935e7	−0.451298
$940$	4.11311e7	1.51828
$941$	1.42507e7	0.524641	0.262320	−	0.964981i	$-0.415512\pi$
$941$	1.42507e7	0.524641	0.262320	+	0.964981i	$0.415512\pi$
$942$	3.36672e6	0.123617
$943$	−5.85402e6	−0.214375
$944$	−1.27594e7	−0.466017
$945$	−9.62857e6	−0.350738
$946$	−5.30808e6	−0.192845
$947$	4.01454e7	1.45466	0.727328	−	0.686290i	$-0.240762\pi$
$947$	4.01454e7	1.45466	0.727328	+	0.686290i	$0.240762\pi$
$948$	−3.11399e6	−0.112537
$949$	2.68010e7	0.966018
$950$	−4.88150e6	−0.175487
$951$	1.09315e7	0.391948
$952$	0	0
$953$	1.50941e7	0.538364	0.269182	−	0.963089i	$-0.413247\pi$
$953$	1.50941e7	0.538364	0.269182	+	0.963089i	$0.413247\pi$
$954$	−4.86114e6	−0.172929
$955$	−1.57976e7	−0.560508
$956$	−3.41456e7	−1.20834
$957$	4.97145e6	0.175470
$958$	1.37862e7	0.485321
$959$	−8.29434e7	−2.91230
$960$	8.98411e6	0.314628
$961$	6.89597e7	2.40872
$962$	−8.17777e6	−0.284903
$963$	1.51801e7	0.527482
$964$	−6.03404e6	−0.209130
$965$	5.68759e7	1.96612
$966$	1.62258e6	0.0559454
$967$	−2.91776e7	−1.00342	−0.501710	−	0.865036i	$-0.667296\pi$
$967$	−2.91776e7	−1.00342	−0.501710	+	0.865036i	$0.667296\pi$
$968$	1.11600e7	0.382803
$969$	0	0
$970$	7.69822e6	0.262701
$971$	−1.78729e7	−0.608340	−0.304170	−	0.952618i	$-0.598379\pi$
$971$	−1.78729e7	−0.608340	−0.304170	+	0.952618i	$0.598379\pi$
$972$	−1.70437e6	−0.0578626
$973$	−4.36921e7	−1.47952
$974$	−3.70543e6	−0.125153
$975$	−7.03974e6	−0.237162
$976$	1.49682e7	0.502972
$977$	−5.64678e6	−0.189263	−0.0946313	−	0.995512i	$-0.530167\pi$
$977$	−5.64678e6	−0.189263	−0.0946313	+	0.995512i	$0.530167\pi$
$978$	−6.23157e6	−0.208329
$979$	3.06917e6	0.102345
$980$	4.36744e7	1.45265
$981$	−6.08592e6	−0.201908
$982$	4.09537e6	0.135523
$983$	−5.52113e7	−1.82240	−0.911200	−	0.411963i	$-0.864843\pi$
$983$	−5.52113e7	−1.82240	−0.911200	+	0.411963i	$0.864843\pi$
$984$	1.11020e7	0.365522
$985$	2.64327e7	0.868061
$986$	0	0
$987$	3.84587e7	1.25661
$988$	−3.80089e7	−1.23878
$989$	−6.39294e6	−0.207831
$990$	−2.28318e6	−0.0740376
$991$	3.52082e7	1.13883	0.569415	−	0.822050i	$-0.307170\pi$
$991$	3.52082e7	1.13883	0.569415	+	0.822050i	$0.307170\pi$
$992$	4.68933e7	1.51297
$993$	−2.51212e7	−0.808476
$994$	2.10331e7	0.675207
$995$	2.82918e7	0.905947
$996$	101521.	0.00324271
$997$	−3.14382e7	−1.00166	−0.500829	−	0.865546i	$-0.666972\pi$
$997$	−3.14382e7	−1.00166	−0.500829	+	0.865546i	$0.666972\pi$
$998$	−1.22898e7	−0.390589
$999$	5.50673e6	0.174574

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.6.a.u.1.16		28
17.10	odd	16		51.6.h.a.49.7	yes	56
17.12	odd	16		51.6.h.a.25.7	✓	56
17.16	even	2		867.6.a.t.1.16		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.6.h.a.25.7	✓	56	17.12	odd	16
51.6.h.a.49.7	yes	56	17.10	odd	16
867.6.a.t.1.16		28	17.16	even	2
867.6.a.u.1.16		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)

\(f(q)\)	\(=\)	\(q+1.77098 q^{2} +9.00000 q^{3} -28.8636 q^{4} -66.3668 q^{5} +15.9388 q^{6} +199.014 q^{7} -107.788 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+1.77098 q^{2} +9.00000 q^{3} -28.8636 q^{4} -66.3668 q^{5} +15.9388 q^{6} +199.014 q^{7} -107.788 q^{8} +81.0000 q^{9} -117.534 q^{10} +239.823 q^{11} -259.773 q^{12} -611.300 q^{13} +352.450 q^{14} -597.301 q^{15} +732.745 q^{16} +143.449 q^{18} -2154.17 q^{19} +1915.59 q^{20} +1791.12 q^{21} +424.721 q^{22} +511.526 q^{23} -970.095 q^{24} +1279.56 q^{25} -1082.60 q^{26} +729.000 q^{27} -5744.26 q^{28} +2303.30 q^{29} -1057.81 q^{30} +9878.70 q^{31} +4746.90 q^{32} +2158.40 q^{33} -13207.9 q^{35} -2337.95 q^{36} +7553.81 q^{37} -3814.99 q^{38} -5501.70 q^{39} +7153.57 q^{40} -11444.2 q^{41} +3172.05 q^{42} -12497.8 q^{43} -6922.15 q^{44} -5375.71 q^{45} +905.903 q^{46} +21471.8 q^{47} +6594.70 q^{48} +22799.5 q^{49} +2266.07 q^{50} +17644.3 q^{52} -33887.4 q^{53} +1291.05 q^{54} -15916.3 q^{55} -21451.4 q^{56} -19387.5 q^{57} +4079.10 q^{58} -17413.2 q^{59} +17240.3 q^{60} +20427.5 q^{61} +17495.0 q^{62} +16120.1 q^{63} -15041.2 q^{64} +40570.1 q^{65} +3822.49 q^{66} -12089.7 q^{67} +4603.73 q^{69} -23391.0 q^{70} +59676.8 q^{71} -8730.86 q^{72} -43842.6 q^{73} +13377.7 q^{74} +11516.0 q^{75} +62177.1 q^{76} +47728.0 q^{77} -9743.41 q^{78} +11987.4 q^{79} -48630.0 q^{80} +6561.00 q^{81} -20267.5 q^{82} -390.808 q^{83} -51698.3 q^{84} -22133.3 q^{86} +20729.7 q^{87} -25850.1 q^{88} +12797.7 q^{89} -9520.29 q^{90} -121657. q^{91} -14764.5 q^{92} +88908.3 q^{93} +38026.2 q^{94} +142965. q^{95} +42722.1 q^{96} -65497.6 q^{97} +40377.5 q^{98} +19425.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9}+O(q^{10}) \) 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