LMFDB - Embedded newform 207.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 8 $ x^3 - x^2 - 11*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.20733$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.49429 q^{2} -19.7900 q^{4} +59.5955 q^{5} +96.8268 q^{7} +180.969 q^{8} +O(q^{10})$	\(q-3.49429 q^{2} -19.7900 q^{4} +59.5955 q^{5} +96.8268 q^{7} +180.969 q^{8} -208.244 q^{10} +731.078 q^{11} -631.879 q^{13} -338.341 q^{14} +0.920920 q^{16} -75.7854 q^{17} +2.35318 q^{19} -1179.39 q^{20} -2554.60 q^{22} -529.000 q^{23} +426.625 q^{25} +2207.97 q^{26} -1916.20 q^{28} +2425.82 q^{29} -349.683 q^{31} -5794.23 q^{32} +264.816 q^{34} +5770.44 q^{35} +8314.43 q^{37} -8.22270 q^{38} +10784.9 q^{40} -4198.56 q^{41} +9372.62 q^{43} -14468.0 q^{44} +1848.48 q^{46} +2436.94 q^{47} -7431.57 q^{49} -1490.75 q^{50} +12504.8 q^{52} +34979.0 q^{53} +43569.0 q^{55} +17522.6 q^{56} -8476.51 q^{58} -15086.9 q^{59} +987.986 q^{61} +1221.89 q^{62} +20217.2 q^{64} -37657.1 q^{65} +44337.8 q^{67} +1499.79 q^{68} -20163.6 q^{70} +33934.4 q^{71} -42483.0 q^{73} -29053.0 q^{74} -46.5694 q^{76} +70788.0 q^{77} +51187.8 q^{79} +54.8827 q^{80} +14671.0 q^{82} +88496.5 q^{83} -4516.47 q^{85} -32750.6 q^{86} +132303. q^{88} +34083.9 q^{89} -61182.8 q^{91} +10468.9 q^{92} -8515.36 q^{94} +140.239 q^{95} -156803. q^{97} +25968.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8}+O(q^{10}) $ 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8	$ 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8} + 282 q^{10} + 376 q^{11} - 858 q^{13} - 588 q^{14} + 2738 q^{16} + 2548 q^{17} - 2846 q^{19} + 4618 q^{20} - 5050 q^{22} - 1587 q^{23} + 753 q^{25} + 7788 q^{26} + 4736 q^{28} + 16370 q^{29} - 14756 q^{31} + 3878 q^{32} + 16520 q^{34} + 18520 q^{35} + 15874 q^{37} - 12438 q^{38} + 38270 q^{40} - 12606 q^{41} + 3154 q^{43} - 27114 q^{44} - 4232 q^{46} - 29928 q^{47} + 4471 q^{49} - 1452 q^{50} + 86856 q^{52} + 44084 q^{53} + 38360 q^{55} + 35704 q^{56} + 73316 q^{58} + 29300 q^{59} + 54010 q^{61} - 99908 q^{62} - 1582 q^{64} + 51216 q^{65} + 43390 q^{67} + 69840 q^{68} - 2476 q^{70} - 23424 q^{71} - 91402 q^{73} + 2294 q^{74} - 14274 q^{76} + 97208 q^{77} - 49398 q^{79} + 52626 q^{80} + 40152 q^{82} + 103936 q^{83} + 5888 q^{85} - 133634 q^{86} + 48898 q^{88} - 96112 q^{89} + 129228 q^{91} - 11638 q^{92} - 133688 q^{94} + 55928 q^{95} - 135318 q^{97} - 108440 q^{98}+O(q^{100}) $ 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8 + 282 * q^10 + 376 * q^11 - 858 * q^13 - 588 * q^14 + 2738 * q^16 + 2548 * q^17 - 2846 * q^19 + 4618 * q^20 - 5050 * q^22 - 1587 * q^23 + 753 * q^25 + 7788 * q^26 + 4736 * q^28 + 16370 * q^29 - 14756 * q^31 + 3878 * q^32 + 16520 * q^34 + 18520 * q^35 + 15874 * q^37 - 12438 * q^38 + 38270 * q^40 - 12606 * q^41 + 3154 * q^43 - 27114 * q^44 - 4232 * q^46 - 29928 * q^47 + 4471 * q^49 - 1452 * q^50 + 86856 * q^52 + 44084 * q^53 + 38360 * q^55 + 35704 * q^56 + 73316 * q^58 + 29300 * q^59 + 54010 * q^61 - 99908 * q^62 - 1582 * q^64 + 51216 * q^65 + 43390 * q^67 + 69840 * q^68 - 2476 * q^70 - 23424 * q^71 - 91402 * q^73 + 2294 * q^74 - 14274 * q^76 + 97208 * q^77 - 49398 * q^79 + 52626 * q^80 + 40152 * q^82 + 103936 * q^83 + 5888 * q^85 - 133634 * q^86 + 48898 * q^88 - 96112 * q^89 + 129228 * q^91 - 11638 * q^92 - 133688 * q^94 + 55928 * q^95 - 135318 * q^97 - 108440 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.49429	−0.617709	−0.308854	−	0.951109i	$-0.599946\pi$
$2$	−3.49429	−0.617709	−0.308854	+	0.951109i	$0.599946\pi$
$3$	0	0
$3$	0	0
$4$	−19.7900	−0.618436
$5$	59.5955	1.06608	0.533038	−	0.846091i	$-0.321050\pi$
$5$	59.5955	1.06608	0.533038	+	0.846091i	$0.321050\pi$
$6$	0	0
$7$	96.8268	0.746879	0.373440	−	0.927654i	$-0.378178\pi$
$7$	96.8268	0.746879	0.373440	+	0.927654i	$0.378178\pi$
$8$	180.969	0.999722
$9$	0	0
$10$	−208.244	−0.658525
$11$	731.078	1.82172	0.910861	−	0.412713i	$-0.135419\pi$
$11$	731.078	1.82172	0.910861	+	0.412713i	$0.135419\pi$
$12$	0	0
$13$	−631.879	−1.03699	−0.518496	−	0.855080i	$-0.673508\pi$
$13$	−631.879	−1.03699	−0.518496	+	0.855080i	$0.673508\pi$
$14$	−338.341	−0.461354
$15$	0	0
$16$	0.920920	0.000899336 0
$17$	−75.7854	−0.0636009	−0.0318005	−	0.999494i	$-0.510124\pi$
$17$	−75.7854	−0.0636009	−0.0318005	+	0.999494i	$0.510124\pi$
$18$	0	0
$19$	2.35318	0.00149545	0.000747725	−	1.00000i	$-0.499762\pi$
$19$	2.35318	0.00149545	0.000747725		1.00000i	$0.499762\pi$
$20$	−1179.39	−0.659300
$21$	0	0
$22$	−2554.60	−1.12529
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	426.625	0.136520
$26$	2207.97	0.640559
$27$	0	0
$28$	−1916.20	−0.461897
$29$	2425.82	0.535628	0.267814	−	0.963471i	$-0.413699\pi$
$29$	2425.82	0.535628	0.267814	+	0.963471i	$0.413699\pi$
$30$	0	0
$31$	−349.683	−0.0653537	−0.0326769	−	0.999466i	$-0.510403\pi$
$31$	−349.683	−0.0653537	−0.0326769	+	0.999466i	$0.510403\pi$
$32$	−5794.23	−1.00028
$33$	0	0
$34$	264.816	0.0392868
$35$	5770.44	0.796231
$36$	0	0
$37$	8314.43	0.998454	0.499227	−	0.866471i	$-0.333617\pi$
$37$	8314.43	0.998454	0.499227	+	0.866471i	$0.333617\pi$
$38$	−8.22270	−0.000923753 0
$39$	0	0
$40$	10784.9	1.06578
$41$	−4198.56	−0.390068	−0.195034	−	0.980796i	$-0.562482\pi$
$41$	−4198.56	−0.390068	−0.195034	+	0.980796i	$0.562482\pi$
$42$	0	0
$43$	9372.62	0.773019	0.386509	−	0.922285i	$-0.373681\pi$
$43$	9372.62	0.773019	0.386509	+	0.922285i	$0.373681\pi$
$44$	−14468.0	−1.12662
$45$	0	0
$46$	1848.48	0.128801
$47$	2436.94	0.160916	0.0804581	−	0.996758i	$-0.474362\pi$
$47$	2436.94	0.160916	0.0804581	+	0.996758i	$0.474362\pi$
$48$	0	0
$49$	−7431.57	−0.442171
$50$	−1490.75	−0.0843296
$51$	0	0
$52$	12504.8	0.641313
$53$	34979.0	1.71048	0.855241	−	0.518231i	$-0.173409\pi$
$53$	34979.0	1.71048	0.855241	+	0.518231i	$0.173409\pi$
$54$	0	0
$55$	43569.0	1.94210
$56$	17522.6	0.746672
$57$	0	0
$58$	−8476.51	−0.330862
$59$	−15086.9	−0.564249	−0.282125	−	0.959378i	$-0.591039\pi$
$59$	−15086.9	−0.564249	−0.282125	+	0.959378i	$0.591039\pi$
$60$	0	0
$61$	987.986	0.0339959	0.0169979	−	0.999856i	$-0.494589\pi$
$61$	987.986	0.0339959	0.0169979	+	0.999856i	$0.494589\pi$
$62$	1221.89	0.0403696
$63$	0	0
$64$	20217.2	0.616981
$65$	−37657.1	−1.10551
$66$	0	0
$67$	44337.8	1.20667	0.603333	−	0.797490i	$-0.293839\pi$
$67$	44337.8	1.20667	0.603333	+	0.797490i	$0.293839\pi$
$68$	1499.79	0.0393331
$69$	0	0
$70$	−20163.6	−0.491839
$71$	33934.4	0.798904	0.399452	−	0.916754i	$-0.369200\pi$
$71$	33934.4	0.798904	0.399452	+	0.916754i	$0.369200\pi$
$72$	0	0
$73$	−42483.0	−0.933058	−0.466529	−	0.884506i	$-0.654496\pi$
$73$	−42483.0	−0.933058	−0.466529	+	0.884506i	$0.654496\pi$
$74$	−29053.0	−0.616753
$75$	0	0
$76$	−46.5694	−0.000924841 0
$77$	70788.0	1.36061
$78$	0	0
$79$	51187.8	0.922782	0.461391	−	0.887197i	$-0.347351\pi$
$79$	51187.8	0.922782	0.461391	+	0.887197i	$0.347351\pi$
$80$	54.8827	0.000958761 0
$81$	0	0
$82$	14671.0	0.240949
$83$	88496.5	1.41004	0.705019	−	0.709188i	$-0.250938\pi$
$83$	88496.5	1.41004	0.705019	+	0.709188i	$0.250938\pi$
$84$	0	0
$85$	−4516.47	−0.0678035
$86$	−32750.6	−0.477500
$87$	0	0
$88$	132303.	1.82122
$89$	34083.9	0.456115	0.228058	−	0.973648i	$-0.426763\pi$
$89$	34083.9	0.456115	0.228058	+	0.973648i	$0.426763\pi$
$90$	0	0
$91$	−61182.8	−0.774508
$92$	10468.9	0.128953
$93$	0	0
$94$	−8515.36	−0.0993993
$95$	140.239	0.00159427
$96$	0	0
$97$	−156803.	−1.69209	−0.846047	−	0.533108i	$-0.821024\pi$
$97$	−156803.	−1.69209	−0.846047	+	0.533108i	$0.821024\pi$
$98$	25968.0	0.273133
$99$	0	0
$100$	−8442.89	−0.0844289
$101$	148400.	1.44754	0.723772	−	0.690039i	$-0.242407\pi$
$101$	148400.	1.44754	0.723772	+	0.690039i	$0.242407\pi$
$102$	0	0
$103$	155116.	1.44067	0.720335	−	0.693626i	$-0.243988\pi$
$103$	155116.	1.44067	0.720335	+	0.693626i	$0.243988\pi$
$104$	−114350.	−1.03670
$105$	0	0
$106$	−122227.	−1.05658
$107$	177527.	1.49901	0.749504	−	0.662000i	$-0.230292\pi$
$107$	177527.	1.49901	0.749504	+	0.662000i	$0.230292\pi$
$108$	0	0
$109$	−169314.	−1.36498	−0.682492	−	0.730893i	$-0.739104\pi$
$109$	−169314.	−1.36498	−0.682492	+	0.730893i	$0.739104\pi$
$110$	−152243.	−1.19965
$111$	0	0
$112$	89.1697	0.000671696 0
$113$	−81710.2	−0.601977	−0.300989	−	0.953628i	$-0.597317\pi$
$113$	−81710.2	−0.601977	−0.300989	+	0.953628i	$0.597317\pi$
$114$	0	0
$115$	−31526.0	−0.222292
$116$	−48006.9	−0.331252
$117$	0	0
$118$	52718.1	0.348542
$119$	−7338.06	−0.0475022
$120$	0	0
$121$	373424.	2.31867
$122$	−3452.31	−0.0209996
$123$	0	0
$124$	6920.21	0.0404171
$125$	−160811.	−0.920536
$126$	0	0
$127$	−317307.	−1.74570	−0.872851	−	0.487986i	$-0.837732\pi$
$127$	−317307.	−1.74570	−0.872851	+	0.487986i	$0.837732\pi$
$128$	114770.	0.619163
$129$	0	0
$130$	131585.	0.682885
$131$	154610.	0.787151	0.393576	−	0.919292i	$-0.371238\pi$
$131$	154610.	0.787151	0.393576	+	0.919292i	$0.371238\pi$
$132$	0	0
$133$	227.851	0.00111692
$134$	−154929.	−0.745368
$135$	0	0
$136$	−13714.8	−0.0635832
$137$	−217676.	−0.990852	−0.495426	−	0.868650i	$-0.664988\pi$
$137$	−217676.	−0.990852	−0.495426	+	0.868650i	$0.664988\pi$
$138$	0	0
$139$	236914.	1.04005	0.520023	−	0.854152i	$-0.325923\pi$
$139$	236914.	1.04005	0.520023	+	0.854152i	$0.325923\pi$
$140$	−114197.	−0.492418
$141$	0	0
$142$	−118577.	−0.493490
$143$	−461953.	−1.88911
$144$	0	0
$145$	144568.	0.571021
$146$	148448.	0.576358
$147$	0	0
$148$	−164542.	−0.617480
$149$	−434197.	−1.60221	−0.801107	−	0.598521i	$-0.795755\pi$
$149$	−434197.	−1.60221	−0.801107	+	0.598521i	$0.795755\pi$
$150$	0	0
$151$	−135760.	−0.484541	−0.242270	−	0.970209i	$-0.577892\pi$
$151$	−135760.	−0.484541	−0.242270	+	0.970209i	$0.577892\pi$
$152$	425.853	0.00149503
$153$	0	0
$154$	−247354.	−0.840459
$155$	−20839.5	−0.0696721
$156$	0	0
$157$	−355399.	−1.15071	−0.575356	−	0.817903i	$-0.695136\pi$
$157$	−355399.	−1.15071	−0.575356	+	0.817903i	$0.695136\pi$
$158$	−178865.	−0.570010
$159$	0	0
$160$	−345310.	−1.06637
$161$	−51221.4	−0.155735
$162$	0	0
$163$	−514344.	−1.51630	−0.758148	−	0.652082i	$-0.773896\pi$
$163$	−514344.	−1.51630	−0.758148	+	0.652082i	$0.773896\pi$
$164$	83089.3	0.241232
$165$	0	0
$166$	−309232.	−0.870993
$167$	−178285.	−0.494680	−0.247340	−	0.968929i	$-0.579556\pi$
$167$	−178285.	−0.494680	−0.247340	+	0.968929i	$0.579556\pi$
$168$	0	0
$169$	27977.5	0.0753515
$170$	15781.8	0.0418828
$171$	0	0
$172$	−185484.	−0.478063
$173$	535921.	1.36140	0.680699	−	0.732563i	$-0.261676\pi$
$173$	535921.	1.36140	0.680699	+	0.732563i	$0.261676\pi$
$174$	0	0
$175$	41308.7	0.101964
$176$	673.265	0.00163834
$177$	0	0
$178$	−119099.	−0.281746
$179$	58891.2	0.137378	0.0686891	−	0.997638i	$-0.478118\pi$
$179$	58891.2	0.137378	0.0686891	+	0.997638i	$0.478118\pi$
$180$	0	0
$181$	829741.	1.88255	0.941274	−	0.337644i	$-0.109630\pi$
$181$	829741.	1.88255	0.941274	+	0.337644i	$0.109630\pi$
$182$	213790.	0.478420
$183$	0	0
$184$	−95732.6	−0.208456
$185$	495502.	1.06443
$186$	0	0
$187$	−55405.1	−0.115863
$188$	−48226.9	−0.0995164
$189$	0	0
$190$	−490.036	−0.000984791 0
$191$	−149335.	−0.296195	−0.148097	−	0.988973i	$-0.547315\pi$
$191$	−149335.	−0.296195	−0.148097	+	0.988973i	$0.547315\pi$
$192$	0	0
$193$	913030.	1.76438	0.882189	−	0.470895i	$-0.156069\pi$
$193$	913030.	1.76438	0.882189	+	0.470895i	$0.156069\pi$
$194$	547914.	1.04522
$195$	0	0
$196$	147070.	0.273455
$197$	919205.	1.68751	0.843756	−	0.536727i	$-0.180339\pi$
$197$	919205.	1.68751	0.843756	+	0.536727i	$0.180339\pi$
$198$	0	0
$199$	493730.	0.883806	0.441903	−	0.897063i	$-0.354304\pi$
$199$	493730.	0.883806	0.441903	+	0.897063i	$0.354304\pi$
$200$	77205.9	0.136482
$201$	0	0
$202$	−518554.	−0.894160
$203$	234884.	0.400050
$204$	0	0
$205$	−250215.	−0.415843
$206$	−542021.	−0.889914
$207$	0	0
$208$	−581.910	−0.000932604 0
$209$	1720.36	0.00272430
$210$	0	0
$211$	−133387.	−0.206257	−0.103129	−	0.994668i	$-0.532885\pi$
$211$	−133387.	−0.206257	−0.103129	+	0.994668i	$0.532885\pi$
$212$	−692234.	−1.05782
$213$	0	0
$214$	−620329.	−0.925950
$215$	558566.	0.824098
$216$	0	0
$217$	−33858.7	−0.0488114
$218$	591633.	0.843162
$219$	0	0
$220$	−862228.	−1.20106
$221$	47887.2	0.0659536
$222$	0	0
$223$	362735.	0.488457	0.244229	−	0.969718i	$-0.421465\pi$
$223$	362735.	0.488457	0.244229	+	0.969718i	$0.421465\pi$
$224$	−561036.	−0.747087
$225$	0	0
$226$	285519.	0.371846
$227$	190962.	0.245970	0.122985	−	0.992409i	$-0.460753\pi$
$227$	190962.	0.245970	0.122985	+	0.992409i	$0.460753\pi$
$228$	0	0
$229$	745968.	0.940008	0.470004	−	0.882664i	$-0.344252\pi$
$229$	745968.	0.940008	0.470004	+	0.882664i	$0.344252\pi$
$230$	110161.	0.137312
$231$	0	0
$232$	438998.	0.535479
$233$	−636804.	−0.768450	−0.384225	−	0.923239i	$-0.625531\pi$
$233$	−636804.	−0.768450	−0.384225	+	0.923239i	$0.625531\pi$
$234$	0	0
$235$	145231.	0.171549
$236$	298570.	0.348952
$237$	0	0
$238$	25641.3	0.0293425
$239$	−1.44028e6	−1.63099	−0.815494	−	0.578765i	$-0.803535\pi$
$239$	−1.44028e6	−1.63099	−0.815494	+	0.578765i	$0.803535\pi$
$240$	0	0
$241$	1.13226e6	1.25575	0.627876	−	0.778313i	$-0.283924\pi$
$241$	1.13226e6	1.25575	0.627876	+	0.778313i	$0.283924\pi$
$242$	−1.30485e6	−1.43226
$243$	0	0
$244$	−19552.2	−0.0210243
$245$	−442888.	−0.471388
$246$	0	0
$247$	−1486.93	−0.00155077
$248$	−63281.8	−0.0653356
$249$	0	0
$250$	561920.	0.568623
$251$	622529.	0.623700	0.311850	−	0.950131i	$-0.399051\pi$
$251$	622529.	0.623700	0.311850	+	0.950131i	$0.399051\pi$
$252$	0	0
$253$	−386740.	−0.379855
$254$	1.10876e6	1.07834
$255$	0	0
$256$	−1.04799e6	−0.999443
$257$	2.10304e6	1.98616	0.993082	−	0.117419i	$-0.0374621\pi$
$257$	2.10304e6	1.98616	0.993082	+	0.117419i	$0.0374621\pi$
$258$	0	0
$259$	805059.	0.745725
$260$	745233.	0.683689
$261$	0	0
$262$	−540250.	−0.486230
$263$	−838283.	−0.747310	−0.373655	−	0.927568i	$-0.621896\pi$
$263$	−838283.	−0.747310	−0.373655	+	0.927568i	$0.621896\pi$
$264$	0	0
$265$	2.08459e6	1.82350
$266$	−796.178	−0.000689932 0
$267$	0	0
$268$	−877443.	−0.746246
$269$	−897532.	−0.756257	−0.378128	−	0.925753i	$-0.623432\pi$
$269$	−897532.	−0.756257	−0.378128	+	0.925753i	$0.623432\pi$
$270$	0	0
$271$	−1.00088e6	−0.827864	−0.413932	−	0.910308i	$-0.635845\pi$
$271$	−1.00088e6	−0.827864	−0.413932	+	0.910308i	$0.635845\pi$
$272$	−69.7923	−5.71986e−5 0
$273$	0	0
$274$	760622.	0.612058
$275$	311896.	0.248701
$276$	0	0
$277$	2.09974e6	1.64424	0.822120	−	0.569314i	$-0.192791\pi$
$277$	2.09974e6	1.64424	0.822120	+	0.569314i	$0.192791\pi$
$278$	−827844.	−0.642446
$279$	0	0
$280$	1.04427e6	0.796009
$281$	−503227.	−0.380188	−0.190094	−	0.981766i	$-0.560879\pi$
$281$	−503227.	−0.380188	−0.190094	+	0.981766i	$0.560879\pi$
$282$	0	0
$283$	−1.61771e6	−1.20070	−0.600349	−	0.799738i	$-0.704972\pi$
$283$	−1.61771e6	−1.20070	−0.600349	+	0.799738i	$0.704972\pi$
$284$	−671561.	−0.494071
$285$	0	0
$286$	1.61420e6	1.16692
$287$	−406533.	−0.291334
$288$	0	0
$289$	−1.41411e6	−0.995955
$290$	−505162.	−0.352725
$291$	0	0
$292$	840738.	0.577037
$293$	−1.04521e6	−0.711270	−0.355635	−	0.934625i	$-0.615735\pi$
$293$	−1.04521e6	−0.711270	−0.355635	+	0.934625i	$0.615735\pi$
$294$	0	0
$295$	−899113.	−0.601533
$296$	1.50465e6	0.998176
$297$	0	0
$298$	1.51721e6	0.989702
$299$	334264.	0.216228
$300$	0	0
$301$	907521.	0.577352
$302$	474385.	0.299305
$303$	0	0
$304$	2.16710	1.34491e−6 0
$305$	58879.6	0.0362422
$306$	0	0
$307$	−697539.	−0.422399	−0.211199	−	0.977443i	$-0.567737\pi$
$307$	−697539.	−0.422399	−0.211199	+	0.977443i	$0.567737\pi$
$308$	−1.40089e6	−0.841448
$309$	0	0
$310$	72819.4	0.0430371
$311$	2.68619e6	1.57484	0.787419	−	0.616419i	$-0.211417\pi$
$311$	2.68619e6	1.57484	0.787419	+	0.616419i	$0.211417\pi$
$312$	0	0
$313$	1.62222e6	0.935944	0.467972	−	0.883743i	$-0.344985\pi$
$313$	1.62222e6	0.935944	0.467972	+	0.883743i	$0.344985\pi$
$314$	1.24186e6	0.710805
$315$	0	0
$316$	−1.01301e6	−0.570682
$317$	−648171.	−0.362277	−0.181139	−	0.983458i	$-0.557978\pi$
$317$	−648171.	−0.362277	−0.181139	+	0.983458i	$0.557978\pi$
$318$	0	0
$319$	1.77346e6	0.975766
$320$	1.20486e6	0.657749
$321$	0	0
$322$	178982.	0.0961989
$323$	−178.337	−9.51120e−5 0
$324$	0	0
$325$	−269575.	−0.141570
$326$	1.79726e6	0.936630
$327$	0	0
$328$	−759809.	−0.389960
$329$	235961.	0.120185
$330$	0	0
$331$	−1.66550e6	−0.835552	−0.417776	−	0.908550i	$-0.637190\pi$
$331$	−1.66550e6	−0.835552	−0.417776	+	0.908550i	$0.637190\pi$
$332$	−1.75134e6	−0.872019
$333$	0	0
$334$	622980.	0.305568
$335$	2.64233e6	1.28640
$336$	0	0
$337$	−2.11943e6	−1.01659	−0.508294	−	0.861184i	$-0.669724\pi$
$337$	−2.11943e6	−1.01659	−0.508294	+	0.861184i	$0.669724\pi$
$338$	−97761.3	−0.0465452
$339$	0	0
$340$	89380.8	0.0419321
$341$	−255646.	−0.119056
$342$	0	0
$343$	−2.34694e6	−1.07713
$344$	1.69615e6	0.772804
$345$	0	0
$346$	−1.87266e6	−0.840948
$347$	791717.	0.352977	0.176488	−	0.984303i	$-0.443526\pi$
$347$	791717.	0.352977	0.176488	+	0.984303i	$0.443526\pi$
$348$	0	0
$349$	743781.	0.326875	0.163437	−	0.986554i	$-0.447742\pi$
$349$	743781.	0.326875	0.163437	+	0.986554i	$0.447742\pi$
$350$	−144345.	−0.0629840
$351$	0	0
$352$	−4.23603e6	−1.82223
$353$	646287.	0.276051	0.138025	−	0.990429i	$-0.455924\pi$
$353$	646287.	0.276051	0.138025	+	0.990429i	$0.455924\pi$
$354$	0	0
$355$	2.02234e6	0.851693
$356$	−674519.	−0.282078
$357$	0	0
$358$	−205783.	−0.0848597
$359$	−1.89862e6	−0.777503	−0.388752	−	0.921343i	$-0.627094\pi$
$359$	−1.89862e6	−0.777503	−0.388752	+	0.921343i	$0.627094\pi$
$360$	0	0
$361$	−2.47609e6	−0.999998
$362$	−2.89935e6	−1.16287
$363$	0	0
$364$	1.21080e6	0.478983
$365$	−2.53180e6	−0.994711
$366$	0	0
$367$	1.19082e6	0.461509	0.230754	−	0.973012i	$-0.425881\pi$
$367$	1.19082e6	0.461509	0.230754	+	0.973012i	$0.425881\pi$
$368$	−487.167	−0.000187525 0
$369$	0	0
$370$	−1.73143e6	−0.657507
$371$	3.38691e6	1.27752
$372$	0	0
$373$	−254301.	−0.0946402	−0.0473201	−	0.998880i	$-0.515068\pi$
$373$	−254301.	−0.0946402	−0.0473201	+	0.998880i	$0.515068\pi$
$374$	193601.	0.0715697
$375$	0	0
$376$	441010.	0.160871
$377$	−1.53282e6	−0.555442
$378$	0	0
$379$	−5.16696e6	−1.84772	−0.923861	−	0.382728i	$-0.874985\pi$
$379$	−5.16696e6	−1.84772	−0.923861	+	0.382728i	$0.874985\pi$
$380$	−2775.33	−0.000985951 0
$381$	0	0
$382$	521818.	0.182962
$383$	−5.10331e6	−1.77769	−0.888843	−	0.458212i	$-0.848490\pi$
$383$	−5.10331e6	−1.77769	−0.888843	+	0.458212i	$0.848490\pi$
$384$	0	0
$385$	4.21865e6	1.45051
$386$	−3.19039e6	−1.08987
$387$	0	0
$388$	3.10312e6	1.04645
$389$	1.00400e6	0.336403	0.168201	−	0.985753i	$-0.446204\pi$
$389$	1.00400e6	0.336403	0.168201	+	0.985753i	$0.446204\pi$
$390$	0	0
$391$	40090.5	0.0132617
$392$	−1.34488e6	−0.442048
$393$	0	0
$394$	−3.21197e6	−1.04239
$395$	3.05057e6	0.983757
$396$	0	0
$397$	3.62003e6	1.15275	0.576377	−	0.817184i	$-0.304466\pi$
$397$	3.62003e6	1.15275	0.576377	+	0.817184i	$0.304466\pi$
$398$	−1.72523e6	−0.545934
$399$	0	0
$400$	392.887	0.000122777 0
$401$	−1.18825e6	−0.369016	−0.184508	−	0.982831i	$-0.559069\pi$
$401$	−1.18825e6	−0.369016	−0.184508	+	0.982831i	$0.559069\pi$
$402$	0	0
$403$	220957.	0.0677713
$404$	−2.93684e6	−0.895213
$405$	0	0
$406$	−820753.	−0.247114
$407$	6.07850e6	1.81891
$408$	0	0
$409$	−64602.3	−0.0190959	−0.00954794	−	0.999954i	$-0.503039\pi$
$409$	−64602.3	−0.0190959	−0.00954794	+	0.999954i	$0.503039\pi$
$410$	874324.	0.256870
$411$	0	0
$412$	−3.06975e6	−0.890962
$413$	−1.46082e6	−0.421426
$414$	0	0
$415$	5.27399e6	1.50321
$416$	3.66125e6	1.03728
$417$	0	0
$418$	−6011.44	−0.00168282
$419$	2.92057e6	0.812704	0.406352	−	0.913717i	$-0.366801\pi$
$419$	2.92057e6	0.812704	0.406352	+	0.913717i	$0.366801\pi$
$420$	0	0
$421$	−2.99592e6	−0.823807	−0.411904	−	0.911227i	$-0.635136\pi$
$421$	−2.99592e6	−0.823807	−0.411904	+	0.911227i	$0.635136\pi$
$422$	466094.	0.127407
$423$	0	0
$424$	6.33012e6	1.71001
$425$	−32331.9	−0.00868280
$426$	0	0
$427$	95663.6	0.0253908
$428$	−3.51324e6	−0.927040
$429$	0	0
$430$	−1.95179e6	−0.509052
$431$	−5.82982e6	−1.51169	−0.755844	−	0.654752i	$-0.772773\pi$
$431$	−5.82982e6	−1.51169	−0.755844	+	0.654752i	$0.772773\pi$
$432$	0	0
$433$	−2.98624e6	−0.765429	−0.382714	−	0.923867i	$-0.625011\pi$
$433$	−2.98624e6	−0.765429	−0.382714	+	0.923867i	$0.625011\pi$
$434$	118312.	0.0301512
$435$	0	0
$436$	3.35072e6	0.844155
$437$	−1244.83	−0.000311823 0
$438$	0	0
$439$	−5.68591e6	−1.40812	−0.704058	−	0.710142i	$-0.748631\pi$
$439$	−5.68591e6	−1.40812	−0.704058	+	0.710142i	$0.748631\pi$
$440$	7.88464e6	1.94156
$441$	0	0
$442$	−167332.	−0.0407401
$443$	68926.8	0.0166870	0.00834352	−	0.999965i	$-0.497344\pi$
$443$	68926.8	0.0166870	0.00834352	+	0.999965i	$0.497344\pi$
$444$	0	0
$445$	2.03125e6	0.486254
$446$	−1.26750e6	−0.301724
$447$	0	0
$448$	1.95757e6	0.460810
$449$	−4.63362e6	−1.08469	−0.542343	−	0.840157i	$-0.682463\pi$
$449$	−4.63362e6	−1.08469	−0.542343	+	0.840157i	$0.682463\pi$
$450$	0	0
$451$	−3.06948e6	−0.710596
$452$	1.61704e6	0.372284
$453$	0	0
$454$	−667275.	−0.151937
$455$	−3.64622e6	−0.825685
$456$	0	0
$457$	−4.33421e6	−0.970777	−0.485388	−	0.874299i	$-0.661322\pi$
$457$	−4.33421e6	−0.970777	−0.485388	+	0.874299i	$0.661322\pi$
$458$	−2.60663e6	−0.580651
$459$	0	0
$460$	623899.	0.137474
$461$	−2.08958e6	−0.457938	−0.228969	−	0.973434i	$-0.573535\pi$
$461$	−2.08958e6	−0.457938	−0.228969	+	0.973434i	$0.573535\pi$
$462$	0	0
$463$	−1.24932e6	−0.270846	−0.135423	−	0.990788i	$-0.543239\pi$
$463$	−1.24932e6	−0.270846	−0.135423	+	0.990788i	$0.543239\pi$
$464$	2233.99	0.000481710 0
$465$	0	0
$466$	2.22517e6	0.474678
$467$	157302.	0.0333765	0.0166883	−	0.999861i	$-0.494688\pi$
$467$	157302.	0.0333765	0.0166883	+	0.999861i	$0.494688\pi$
$468$	0	0
$469$	4.29309e6	0.901234
$470$	−507477.	−0.105967
$471$	0	0
$472$	−2.73027e6	−0.564092
$473$	6.85212e6	1.40823
$474$	0	0
$475$	1003.93	0.000204159 0
$476$	145220.	0.0293771
$477$	0	0
$478$	5.03274e6	1.00748
$479$	−2.74855e6	−0.547350	−0.273675	−	0.961822i	$-0.588239\pi$
$479$	−2.74855e6	−0.547350	−0.273675	+	0.961822i	$0.588239\pi$
$480$	0	0
$481$	−5.25371e6	−1.03539
$482$	−3.95645e6	−0.775689
$483$	0	0
$484$	−7.39005e6	−1.43395
$485$	−9.34475e6	−1.80390
$486$	0	0
$487$	−1.01838e7	−1.94576	−0.972878	−	0.231319i	$-0.925696\pi$
$487$	−1.01838e7	−1.94576	−0.972878	+	0.231319i	$0.925696\pi$
$488$	178795.	0.0339864
$489$	0	0
$490$	1.54758e6	0.291181
$491$	8.81893e6	1.65087	0.825434	−	0.564499i	$-0.190931\pi$
$491$	8.81893e6	1.65087	0.825434	+	0.564499i	$0.190931\pi$
$492$	0	0
$493$	−183842.	−0.0340664
$494$	5195.75	0.000957924 0
$495$	0	0
$496$	−322.030	−5.87750e−5 0
$497$	3.28576e6	0.596685
$498$	0	0
$499$	−7.27705e6	−1.30829	−0.654145	−	0.756369i	$-0.726971\pi$
$499$	−7.27705e6	−1.30829	−0.654145	+	0.756369i	$0.726971\pi$
$500$	3.18244e6	0.569293
$501$	0	0
$502$	−2.17530e6	−0.385265
$503$	2.82184e6	0.497293	0.248646	−	0.968594i	$-0.420014\pi$
$503$	2.82184e6	0.497293	0.248646	+	0.968594i	$0.420014\pi$
$504$	0	0
$505$	8.84400e6	1.54319
$506$	1.35138e6	0.234640
$507$	0	0
$508$	6.27949e6	1.07961
$509$	4.78063e6	0.817881	0.408941	−	0.912561i	$-0.365898\pi$
$509$	4.78063e6	0.817881	0.408941	+	0.912561i	$0.365898\pi$
$510$	0	0
$511$	−4.11350e6	−0.696882
$512$	−10669.1	−0.00179867
$513$	0	0
$514$	−7.34864e6	−1.22687
$515$	9.24424e6	1.53587
$516$	0	0
$517$	1.78159e6	0.293145
$518$	−2.81311e6	−0.460640
$519$	0	0
$520$	−6.81477e6	−1.10521
$521$	−7.11976e6	−1.14914	−0.574568	−	0.818457i	$-0.694830\pi$
$521$	−7.11976e6	−1.14914	−0.574568	+	0.818457i	$0.694830\pi$
$522$	0	0
$523$	4.97980e6	0.796082	0.398041	−	0.917368i	$-0.369690\pi$
$523$	4.97980e6	0.796082	0.398041	+	0.917368i	$0.369690\pi$
$524$	−3.05972e6	−0.486803
$525$	0	0
$526$	2.92920e6	0.461620
$527$	26500.9	0.00415656
$528$	0	0
$529$	279841.	0.0434783
$530$	−7.28417e6	−1.12639
$531$	0	0
$532$	−4509.17	−0.000690744 0
$533$	2.65298e6	0.404498
$534$	0	0
$535$	1.05798e7	1.59806
$536$	8.02376e6	1.20633
$537$	0	0
$538$	3.13624e6	0.467146
$539$	−5.43306e6	−0.805513
$540$	0	0
$541$	2.34995e6	0.345196	0.172598	−	0.984992i	$-0.444784\pi$
$541$	2.34995e6	0.345196	0.172598	+	0.984992i	$0.444784\pi$
$542$	3.49737e6	0.511379
$543$	0	0
$544$	439118.	0.0636186
$545$	−1.00904e7	−1.45518
$546$	0	0
$547$	1.27753e7	1.82559	0.912795	−	0.408419i	$-0.133920\pi$
$547$	1.27753e7	1.82559	0.912795	+	0.408419i	$0.133920\pi$
$548$	4.30779e6	0.612778
$549$	0	0
$550$	−1.08986e6	−0.153625
$551$	5708.40	0.000801005 0
$552$	0	0
$553$	4.95636e6	0.689207
$554$	−7.33708e6	−1.01566
$555$	0	0
$556$	−4.68851e6	−0.643203
$557$	6.13111e6	0.837339	0.418670	−	0.908139i	$-0.362497\pi$
$557$	6.13111e6	0.837339	0.418670	+	0.908139i	$0.362497\pi$
$558$	0	0
$559$	−5.92236e6	−0.801614
$560$	5314.12	0.000716079 0
$561$	0	0
$562$	1.75842e6	0.234845
$563$	−1.30613e7	−1.73667	−0.868334	−	0.495979i	$-0.834809\pi$
$563$	−1.30613e7	−1.73667	−0.868334	+	0.495979i	$0.834809\pi$
$564$	0	0
$565$	−4.86956e6	−0.641754
$566$	5.65274e6	0.741682
$567$	0	0
$568$	6.14108e6	0.798682
$569$	5.03952e6	0.652542	0.326271	−	0.945276i	$-0.394208\pi$
$569$	5.03952e6	0.652542	0.326271	+	0.945276i	$0.394208\pi$
$570$	0	0
$571$	−7.67025e6	−0.984508	−0.492254	−	0.870452i	$-0.663827\pi$
$571$	−7.67025e6	−0.984508	−0.492254	+	0.870452i	$0.663827\pi$
$572$	9.14202e6	1.16829
$573$	0	0
$574$	1.42054e6	0.179960
$575$	−225685.	−0.0284664
$576$	0	0
$577$	1.25985e7	1.57536	0.787681	−	0.616083i	$-0.211281\pi$
$577$	1.25985e7	1.57536	0.787681	+	0.616083i	$0.211281\pi$
$578$	4.94132e6	0.615210
$579$	0	0
$580$	−2.86099e6	−0.353140
$581$	8.56883e6	1.05313
$582$	0	0
$583$	2.55724e7	3.11602
$584$	−7.68811e6	−0.932798
$585$	0	0
$586$	3.65226e6	0.439357
$587$	−5.11295e6	−0.612457	−0.306229	−	0.951958i	$-0.599067\pi$
$587$	−5.11295e6	−0.612457	−0.306229	+	0.951958i	$0.599067\pi$
$588$	0	0
$589$	−822.869	−9.77333e−5 0
$590$	3.14176e6	0.371572
$591$	0	0
$592$	7656.92	0.000897945 0
$593$	−5.45932e6	−0.637532	−0.318766	−	0.947833i	$-0.603268\pi$
$593$	−5.45932e6	−0.637532	−0.318766	+	0.947833i	$0.603268\pi$
$594$	0	0
$595$	−437315.	−0.0506410
$596$	8.59273e6	0.990867
$597$	0	0
$598$	−1.16801e6	−0.133566
$599$	22496.6	0.00256183	0.00128091	−	0.999999i	$-0.499592\pi$
$599$	22496.6	0.00256183	0.00128091	+	0.999999i	$0.499592\pi$
$600$	0	0
$601$	1.00912e6	0.113962	0.0569808	−	0.998375i	$-0.481853\pi$
$601$	1.00912e6	0.113962	0.0569808	+	0.998375i	$0.481853\pi$
$602$	−3.17114e6	−0.356635
$603$	0	0
$604$	2.68669e6	0.299657
$605$	2.22544e7	2.47188
$606$	0	0
$607$	−68503.9	−0.00754646	−0.00377323	−	0.999993i	$-0.501201\pi$
$607$	−68503.9	−0.00754646	−0.00377323	+	0.999993i	$0.501201\pi$
$608$	−13634.9	−0.00149587
$609$	0	0
$610$	−205742.	−0.0223871
$611$	−1.53985e6	−0.166869
$612$	0	0
$613$	7.67884e6	0.825362	0.412681	−	0.910876i	$-0.364593\pi$
$613$	7.67884e6	0.825362	0.412681	+	0.910876i	$0.364593\pi$
$614$	2.43740e6	0.260919
$615$	0	0
$616$	1.28104e7	1.36023
$617$	1.85528e6	0.196199	0.0980995	−	0.995177i	$-0.468724\pi$
$617$	1.85528e6	0.196199	0.0980995	+	0.995177i	$0.468724\pi$
$618$	0	0
$619$	−2.88941e6	−0.303098	−0.151549	−	0.988450i	$-0.548426\pi$
$619$	−2.88941e6	−0.303098	−0.151549	+	0.988450i	$0.548426\pi$
$620$	412414.	0.0430878
$621$	0	0
$622$	−9.38632e6	−0.972791
$623$	3.30024e6	0.340663
$624$	0	0
$625$	−1.09168e7	−1.11788
$626$	−5.66851e6	−0.578140
$627$	0	0
$628$	7.03332e6	0.711642
$629$	−630112.	−0.0635026
$630$	0	0
$631$	−1.22946e7	−1.22925	−0.614625	−	0.788819i	$-0.710693\pi$
$631$	−1.22946e7	−1.22925	−0.614625	+	0.788819i	$0.710693\pi$
$632$	9.26341e6	0.922525
$633$	0	0
$634$	2.26489e6	0.223782
$635$	−1.89101e7	−1.86105
$636$	0	0
$637$	4.69585e6	0.458528
$638$	−6.19699e6	−0.602739
$639$	0	0
$640$	6.83980e6	0.660076
$641$	7.87250e6	0.756776	0.378388	−	0.925647i	$-0.376478\pi$
$641$	7.87250e6	0.756776	0.378388	+	0.925647i	$0.376478\pi$
$642$	0	0
$643$	3.14575e6	0.300052	0.150026	−	0.988682i	$-0.452064\pi$
$643$	3.14575e6	0.300052	0.150026	+	0.988682i	$0.452064\pi$
$644$	1.01367e6	0.0963122
$645$	0	0
$646$	623.161	5.87515e−5 0
$647$	−4.14059e6	−0.388867	−0.194434	−	0.980916i	$-0.562287\pi$
$647$	−4.14059e6	−0.388867	−0.194434	+	0.980916i	$0.562287\pi$
$648$	0	0
$649$	−1.10297e7	−1.02791
$650$	941973.	0.0874490
$651$	0	0
$652$	1.01788e7	0.937733
$653$	−1.09894e7	−1.00854	−0.504268	−	0.863547i	$-0.668238\pi$
$653$	−1.09894e7	−1.00854	−0.504268	+	0.863547i	$0.668238\pi$
$654$	0	0
$655$	9.21404e6	0.839164
$656$	−3866.54	−0.000350803 0
$657$	0	0
$658$	−824515.	−0.0742393
$659$	−1.78286e7	−1.59920	−0.799602	−	0.600530i	$-0.794956\pi$
$659$	−1.78286e7	−1.59920	−0.799602	+	0.600530i	$0.794956\pi$
$660$	0	0
$661$	4.39680e6	0.391411	0.195705	−	0.980663i	$-0.437300\pi$
$661$	4.39680e6	0.391411	0.195705	+	0.980663i	$0.437300\pi$
$662$	5.81972e6	0.516128
$663$	0	0
$664$	1.60151e7	1.40965
$665$	13578.9	0.00119072
$666$	0	0
$667$	−1.28326e6	−0.111686
$668$	3.52826e6	0.305928
$669$	0	0
$670$	−9.23307e6	−0.794619
$671$	722295.	0.0619311
$672$	0	0
$673$	−1.98954e7	−1.69322	−0.846612	−	0.532210i	$-0.821362\pi$
$673$	−1.98954e7	−1.69322	−0.846612	+	0.532210i	$0.821362\pi$
$674$	7.40591e6	0.627955
$675$	0	0
$676$	−553673.	−0.0466001
$677$	796059.	0.0667535	0.0333767	−	0.999443i	$-0.489374\pi$
$677$	796059.	0.0667535	0.0333767	+	0.999443i	$0.489374\pi$
$678$	0	0
$679$	−1.51827e7	−1.26379
$680$	−817341.	−0.0677846
$681$	0	0
$682$	893300.	0.0735421
$683$	9.97449e6	0.818162	0.409081	−	0.912498i	$-0.365849\pi$
$683$	9.97449e6	0.818162	0.409081	+	0.912498i	$0.365849\pi$
$684$	0	0
$685$	−1.29725e7	−1.05632
$686$	8.20089e6	0.665351
$687$	0	0
$688$	8631.44	0.000695204 0
$689$	−2.21025e7	−1.77375
$690$	0	0
$691$	−6.67183e6	−0.531557	−0.265779	−	0.964034i	$-0.585629\pi$
$691$	−6.67183e6	−0.531557	−0.265779	+	0.964034i	$0.585629\pi$
$692$	−1.06058e7	−0.841938
$693$	0	0
$694$	−2.76649e6	−0.218037
$695$	1.41190e7	1.10877
$696$	0	0
$697$	318190.	0.0248087
$698$	−2.59898e6	−0.201913
$699$	0	0
$700$	−817498.	−0.0630582
$701$	−5.64132e6	−0.433597	−0.216798	−	0.976216i	$-0.569561\pi$
$701$	−5.64132e6	−0.433597	−0.216798	+	0.976216i	$0.569561\pi$
$702$	0	0
$703$	19565.4	0.00149314
$704$	1.47804e7	1.12397
$705$	0	0
$706$	−2.25831e6	−0.170519
$707$	1.43691e7	1.08114
$708$	0	0
$709$	−1.86950e6	−0.139672	−0.0698361	−	0.997558i	$-0.522248\pi$
$709$	−1.86950e6	−0.139672	−0.0698361	+	0.997558i	$0.522248\pi$
$710$	−7.06663e6	−0.526098
$711$	0	0
$712$	6.16813e6	0.455988
$713$	184982.	0.0136272
$714$	0	0
$715$	−2.75303e7	−2.01394
$716$	−1.16545e6	−0.0849596
$717$	0	0
$718$	6.63433e6	0.480270
$719$	−1.87348e7	−1.35153	−0.675766	−	0.737116i	$-0.736187\pi$
$719$	−1.87348e7	−1.35153	−0.675766	+	0.737116i	$0.736187\pi$
$720$	0	0
$721$	1.50194e7	1.07601
$722$	8.65218e6	0.617707
$723$	0	0
$724$	−1.64205e7	−1.16424
$725$	1.03492e6	0.0731240
$726$	0	0
$727$	7.88165e6	0.553071	0.276536	−	0.961004i	$-0.410814\pi$
$727$	7.88165e6	0.553071	0.276536	+	0.961004i	$0.410814\pi$
$728$	−1.10722e7	−0.774292
$729$	0	0
$730$	8.84683e6	0.614442
$731$	−710308.	−0.0491647
$732$	0	0
$733$	196627.	0.0135171	0.00675854	−	0.999977i	$-0.497849\pi$
$733$	196627.	0.0135171	0.00675854	+	0.999977i	$0.497849\pi$
$734$	−4.16106e6	−0.285078
$735$	0	0
$736$	3.06515e6	0.208572
$737$	3.24144e7	2.19821
$738$	0	0
$739$	−1.89161e7	−1.27415	−0.637074	−	0.770803i	$-0.719855\pi$
$739$	−1.89161e7	−1.27415	−0.637074	+	0.770803i	$0.719855\pi$
$740$	−9.80597e6	−0.658281
$741$	0	0
$742$	−1.18348e7	−0.789137
$743$	5.29482e6	0.351867	0.175934	−	0.984402i	$-0.443706\pi$
$743$	5.29482e6	0.351867	0.175934	+	0.984402i	$0.443706\pi$
$744$	0	0
$745$	−2.58762e7	−1.70808
$746$	888600.	0.0584601
$747$	0	0
$748$	1.09646e6	0.0716540
$749$	1.71893e7	1.11958
$750$	0	0
$751$	−1.23104e7	−0.796478	−0.398239	−	0.917282i	$-0.630379\pi$
$751$	−1.23104e7	−0.796478	−0.398239	+	0.917282i	$0.630379\pi$
$752$	2244.23	0.000144718 0
$753$	0	0
$754$	5.35612e6	0.343101
$755$	−8.09070e6	−0.516558
$756$	0	0
$757$	−7.33051e6	−0.464938	−0.232469	−	0.972604i	$-0.574680\pi$
$757$	−7.33051e6	−0.464938	−0.232469	+	0.972604i	$0.574680\pi$
$758$	1.80548e7	1.14135
$759$	0	0
$760$	25379.0	0.00159382
$761$	−3.11913e6	−0.195242	−0.0976208	−	0.995224i	$-0.531123\pi$
$761$	−3.11913e6	−0.195242	−0.0976208	+	0.995224i	$0.531123\pi$
$762$	0	0
$763$	−1.63942e7	−1.01948
$764$	2.95533e6	0.183178
$765$	0	0
$766$	1.78324e7	1.09809
$767$	9.53311e6	0.585122
$768$	0	0
$769$	−7.07903e6	−0.431676	−0.215838	−	0.976429i	$-0.569248\pi$
$769$	−7.07903e6	−0.431676	−0.215838	+	0.976429i	$0.569248\pi$
$770$	−1.47412e7	−0.895993
$771$	0	0
$772$	−1.80688e7	−1.09116
$773$	−1.91702e7	−1.15393	−0.576964	−	0.816770i	$-0.695763\pi$
$773$	−1.91702e7	−1.15393	−0.576964	+	0.816770i	$0.695763\pi$
$774$	0	0
$775$	−149184.	−0.00892209
$776$	−2.83765e7	−1.69162
$777$	0	0
$778$	−3.50826e6	−0.207799
$779$	−9879.99	−0.000583328 0
$780$	0	0
$781$	2.48087e7	1.45538
$782$	−140088.	−0.00819187
$783$	0	0
$784$	−6843.88	−0.000397660 0
$785$	−2.11802e7	−1.22675
$786$	0	0
$787$	8.38766e6	0.482730	0.241365	−	0.970434i	$-0.422405\pi$
$787$	8.38766e6	0.482730	0.241365	+	0.970434i	$0.422405\pi$
$788$	−1.81910e7	−1.04362
$789$	0	0
$790$	−1.06596e7	−0.607675
$791$	−7.91173e6	−0.449604
$792$	0	0
$793$	−624287.	−0.0352534
$794$	−1.26494e7	−0.712066
$795$	0	0
$796$	−9.77089e6	−0.546577
$797$	1.36638e7	0.761950	0.380975	−	0.924585i	$-0.375588\pi$
$797$	1.36638e7	0.761950	0.380975	+	0.924585i	$0.375588\pi$
$798$	0	0
$799$	−184684.	−0.0102344
$800$	−2.47196e6	−0.136558
$801$	0	0
$802$	4.15207e6	0.227944
$803$	−3.10584e7	−1.69977
$804$	0	0
$805$	−3.05256e6	−0.166026
$806$	−772088.	−0.0418629
$807$	0	0
$808$	2.68559e7	1.44714
$809$	−5.16198e6	−0.277297	−0.138648	−	0.990342i	$-0.544276\pi$
$809$	−5.16198e6	−0.277297	−0.138648	+	0.990342i	$0.544276\pi$
$810$	0	0
$811$	1.20275e7	0.642131	0.321065	−	0.947057i	$-0.395959\pi$
$811$	1.20275e7	0.642131	0.321065	+	0.947057i	$0.395959\pi$
$812$	−4.64835e6	−0.247405
$813$	0	0
$814$	−2.12400e7	−1.12355
$815$	−3.06526e7	−1.61649
$816$	0	0
$817$	22055.5	0.00115601
$818$	225739.	0.0117957
$819$	0	0
$820$	4.95175e6	0.257172
$821$	4.25894e6	0.220518	0.110259	−	0.993903i	$-0.464832\pi$
$821$	4.25894e6	0.220518	0.110259	+	0.993903i	$0.464832\pi$
$822$	0	0
$823$	1.63473e7	0.841290	0.420645	−	0.907225i	$-0.361804\pi$
$823$	1.63473e7	0.841290	0.420645	+	0.907225i	$0.361804\pi$
$824$	2.80712e7	1.44027
$825$	0	0
$826$	5.10452e6	0.260318
$827$	1.62566e7	0.826542	0.413271	−	0.910608i	$-0.364386\pi$
$827$	1.62566e7	0.826542	0.413271	+	0.910608i	$0.364386\pi$
$828$	0	0
$829$	3.87989e7	1.96080	0.980400	−	0.197018i	$-0.0631256\pi$
$829$	3.87989e7	1.96080	0.980400	+	0.197018i	$0.0631256\pi$
$830$	−1.84289e7	−0.928545
$831$	0	0
$832$	−1.27748e7	−0.639804
$833$	563205.	0.0281225
$834$	0	0
$835$	−1.06250e7	−0.527367
$836$	−34045.9	−0.00168480
$837$	0	0
$838$	−1.02053e7	−0.502014
$839$	2.96779e7	1.45555	0.727776	−	0.685815i	$-0.240554\pi$
$839$	2.96779e7	1.45555	0.727776	+	0.685815i	$0.240554\pi$
$840$	0	0
$841$	−1.46265e7	−0.713102
$842$	1.04686e7	0.508873
$843$	0	0
$844$	2.63973e6	0.127557
$845$	1.66733e6	0.0803305
$846$	0	0
$847$	3.61575e7	1.73177
$848$	32212.9	0.00153830
$849$	0	0
$850$	112977.	0.00536344
$851$	−4.39833e6	−0.208192
$852$	0	0
$853$	−3.20704e7	−1.50915	−0.754573	−	0.656216i	$-0.772156\pi$
$853$	−3.20704e7	−1.50915	−0.754573	+	0.656216i	$0.772156\pi$
$854$	−334276.	−0.0156841
$855$	0	0
$856$	3.21268e7	1.49859
$857$	1.10540e7	0.514123	0.257061	−	0.966395i	$-0.417246\pi$
$857$	1.10540e7	0.514123	0.257061	+	0.966395i	$0.417246\pi$
$858$	0	0
$859$	1.49699e6	0.0692206	0.0346103	−	0.999401i	$-0.488981\pi$
$859$	1.49699e6	0.0692206	0.0346103	+	0.999401i	$0.488981\pi$
$860$	−1.10540e7	−0.509652
$861$	0	0
$862$	2.03711e7	0.933783
$863$	−7.26672e6	−0.332132	−0.166066	−	0.986115i	$-0.553107\pi$
$863$	−7.26672e6	−0.332132	−0.166066	+	0.986115i	$0.553107\pi$
$864$	0	0
$865$	3.19385e7	1.45136
$866$	1.04348e7	0.472812
$867$	0	0
$868$	670062.	0.0301867
$869$	3.74223e7	1.68105
$870$	0	0
$871$	−2.80161e7	−1.25130
$872$	−3.06406e7	−1.36460
$873$	0	0
$874$	4349.81	0.000192616 0
$875$	−1.55708e7	−0.687529
$876$	0	0
$877$	−3.36708e6	−0.147827	−0.0739137	−	0.997265i	$-0.523549\pi$
$877$	−3.36708e6	−0.147827	−0.0739137	+	0.997265i	$0.523549\pi$
$878$	1.98682e7	0.869806
$879$	0	0
$880$	40123.6	0.00174660
$881$	1.40164e7	0.608409	0.304204	−	0.952607i	$-0.401609\pi$
$881$	1.40164e7	0.608409	0.304204	+	0.952607i	$0.401609\pi$
$882$	0	0
$883$	−4.00921e7	−1.73044	−0.865220	−	0.501392i	$-0.832821\pi$
$883$	−4.00921e7	−1.73044	−0.865220	+	0.501392i	$0.832821\pi$
$884$	−947685.	−0.0407881
$885$	0	0
$886$	−240850.	−0.0103077
$887$	−3.57779e7	−1.52688	−0.763442	−	0.645877i	$-0.776492\pi$
$887$	−3.57779e7	−1.52688	−0.763442	+	0.645877i	$0.776492\pi$
$888$	0	0
$889$	−3.07238e7	−1.30383
$890$	−7.09777e6	−0.300363
$891$	0	0
$892$	−7.17850e6	−0.302080
$893$	5734.56	0.000240642 0
$894$	0	0
$895$	3.50965e6	0.146456
$896$	1.11129e7	0.462440
$897$	0	0
$898$	1.61912e7	0.670020
$899$	−848268.	−0.0350053
$900$	0	0
$901$	−2.65090e6	−0.108788
$902$	1.07256e7	0.438941
$903$	0	0
$904$	−1.47870e7	−0.601810
$905$	4.94488e7	2.00694
$906$	0	0
$907$	2.64880e6	0.106913	0.0534565	−	0.998570i	$-0.482976\pi$
$907$	2.64880e6	0.106913	0.0534565	+	0.998570i	$0.482976\pi$
$908$	−3.77912e6	−0.152116
$909$	0	0
$910$	1.27409e7	0.510033
$911$	3.49769e7	1.39632	0.698161	−	0.715941i	$-0.254002\pi$
$911$	3.49769e7	1.39632	0.698161	+	0.715941i	$0.254002\pi$
$912$	0	0
$913$	6.46979e7	2.56870
$914$	1.51450e7	0.599657
$915$	0	0
$916$	−1.47627e7	−0.581335
$917$	1.49704e7	0.587907
$918$	0	0
$919$	−2.60166e7	−1.01616	−0.508079	−	0.861310i	$-0.669644\pi$
$919$	−2.60166e7	−1.01616	−0.508079	+	0.861310i	$0.669644\pi$
$920$	−5.70523e6	−0.222231
$921$	0	0
$922$	7.30160e6	0.282872
$923$	−2.14424e7	−0.828457
$924$	0	0
$925$	3.54714e6	0.136309
$926$	4.36550e6	0.167304
$927$	0	0
$928$	−1.40557e7	−0.535777
$929$	−4.76233e7	−1.81042	−0.905212	−	0.424960i	$-0.860288\pi$
$929$	−4.76233e7	−1.81042	−0.905212	+	0.424960i	$0.860288\pi$
$930$	0	0
$931$	−17487.9	−0.000661245 0
$932$	1.26023e7	0.475237
$933$	0	0
$934$	−549657.	−0.0206170
$935$	−3.30189e6	−0.123519
$936$	0	0
$937$	−5.81079e6	−0.216215	−0.108108	−	0.994139i	$-0.534479\pi$
$937$	−5.81079e6	−0.216215	−0.108108	+	0.994139i	$0.534479\pi$
$938$	−1.50013e7	−0.556700
$939$	0	0
$940$	−2.87411e6	−0.106092
$941$	6.47870e6	0.238514	0.119257	−	0.992863i	$-0.461949\pi$
$941$	6.47870e6	0.238514	0.119257	+	0.992863i	$0.461949\pi$
$942$	0	0
$943$	2.22104e6	0.0813349
$944$	−13893.9	−0.000507450 0
$945$	0	0
$946$	−2.39433e7	−0.869873
$947$	4.16257e6	0.150830	0.0754149	−	0.997152i	$-0.475972\pi$
$947$	4.16257e6	0.150830	0.0754149	+	0.997152i	$0.475972\pi$
$948$	0	0
$949$	2.68441e7	0.967573
$950$	−3508.01	−0.000126111 0
$951$	0	0
$952$	−1.32796e6	−0.0474890
$953$	2.93375e7	1.04638	0.523191	−	0.852216i	$-0.324741\pi$
$953$	2.93375e7	1.04638	0.523191	+	0.852216i	$0.324741\pi$
$954$	0	0
$955$	−8.89968e6	−0.315766
$956$	2.85030e7	1.00866
$957$	0	0
$958$	9.60422e6	0.338103
$959$	−2.10768e7	−0.740047
$960$	0	0
$961$	−2.85069e7	−0.995729
$962$	1.83580e7	0.639568
$963$	0	0
$964$	−2.24074e7	−0.776603
$965$	5.44125e7	1.88096
$966$	0	0
$967$	4.34493e6	0.149423	0.0747113	−	0.997205i	$-0.476196\pi$
$967$	4.34493e6	0.149423	0.0747113	+	0.997205i	$0.476196\pi$
$968$	6.75783e7	2.31803
$969$	0	0
$970$	3.26532e7	1.11429
$971$	1.84656e7	0.628514	0.314257	−	0.949338i	$-0.398245\pi$
$971$	1.84656e7	0.628514	0.314257	+	0.949338i	$0.398245\pi$
$972$	0	0
$973$	2.29396e7	0.776790
$974$	3.55852e7	1.20191
$975$	0	0
$976$	909.857	3.05737e−5 0
$977$	−4.05887e7	−1.36041	−0.680203	−	0.733023i	$-0.738109\pi$
$977$	−4.05887e7	−1.36041	−0.680203	+	0.733023i	$0.738109\pi$
$978$	0	0
$979$	2.49180e7	0.830915
$980$	8.76474e6	0.291524
$981$	0	0
$982$	−3.08159e7	−1.01975
$983$	−5.29637e7	−1.74822	−0.874108	−	0.485732i	$-0.838553\pi$
$983$	−5.29637e7	−1.74822	−0.874108	+	0.485732i	$0.838553\pi$
$984$	0	0
$985$	5.47805e7	1.79902
$986$	642396.	0.0210431
$987$	0	0
$988$	29426.2	0.000959052 0
$989$	−4.95812e6	−0.161186
$990$	0	0
$991$	1.26953e7	0.410636	0.205318	−	0.978695i	$-0.434177\pi$
$991$	1.26953e7	0.410636	0.205318	+	0.978695i	$0.434177\pi$
$992$	2.02614e6	0.0653719
$993$	0	0
$994$	−1.14814e7	−0.368577
$995$	2.94241e7	0.942205
$996$	0	0
$997$	4.65074e7	1.48178	0.740890	−	0.671626i	$-0.234404\pi$
$997$	4.65074e7	1.48178	0.740890	+	0.671626i	$0.234404\pi$
$998$	2.54281e7	0.808142
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.c.1.1		3
3.2	odd	2		69.6.a.b.1.3	✓	3
12.11	even	2		1104.6.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.b.1.3	✓	3	3.2	odd	2
207.6.a.c.1.1		3	1.1	even	1	trivial
1104.6.a.i.1.1		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-3.49429 q^{2} -19.7900 q^{4} +59.5955 q^{5} +96.8268 q^{7} +180.969 q^{8} +O(q^{10})\)	\(q-3.49429 q^{2} -19.7900 q^{4} +59.5955 q^{5} +96.8268 q^{7} +180.969 q^{8} -208.244 q^{10} +731.078 q^{11} -631.879 q^{13} -338.341 q^{14} +0.920920 q^{16} -75.7854 q^{17} +2.35318 q^{19} -1179.39 q^{20} -2554.60 q^{22} -529.000 q^{23} +426.625 q^{25} +2207.97 q^{26} -1916.20 q^{28} +2425.82 q^{29} -349.683 q^{31} -5794.23 q^{32} +264.816 q^{34} +5770.44 q^{35} +8314.43 q^{37} -8.22270 q^{38} +10784.9 q^{40} -4198.56 q^{41} +9372.62 q^{43} -14468.0 q^{44} +1848.48 q^{46} +2436.94 q^{47} -7431.57 q^{49} -1490.75 q^{50} +12504.8 q^{52} +34979.0 q^{53} +43569.0 q^{55} +17522.6 q^{56} -8476.51 q^{58} -15086.9 q^{59} +987.986 q^{61} +1221.89 q^{62} +20217.2 q^{64} -37657.1 q^{65} +44337.8 q^{67} +1499.79 q^{68} -20163.6 q^{70} +33934.4 q^{71} -42483.0 q^{73} -29053.0 q^{74} -46.5694 q^{76} +70788.0 q^{77} +51187.8 q^{79} +54.8827 q^{80} +14671.0 q^{82} +88496.5 q^{83} -4516.47 q^{85} -32750.6 q^{86} +132303. q^{88} +34083.9 q^{89} -61182.8 q^{91} +10468.9 q^{92} -8515.36 q^{94} +140.239 q^{95} -156803. q^{97} +25968.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8}+O(q^{10}) \) 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8	\( 3 q + 8 q^{2} + 22 q^{4} + 56 q^{5} - 114 q^{7} + 510 q^{8} + 282 q^{10} + 376 q^{11} - 858 q^{13} - 588 q^{14} + 2738 q^{16} + 2548 q^{17} - 2846 q^{19} + 4618 q^{20} - 5050 q^{22} - 1587 q^{23} + 753 q^{25} + 7788 q^{26} + 4736 q^{28} + 16370 q^{29} - 14756 q^{31} + 3878 q^{32} + 16520 q^{34} + 18520 q^{35} + 15874 q^{37} - 12438 q^{38} + 38270 q^{40} - 12606 q^{41} + 3154 q^{43} - 27114 q^{44} - 4232 q^{46} - 29928 q^{47} + 4471 q^{49} - 1452 q^{50} + 86856 q^{52} + 44084 q^{53} + 38360 q^{55} + 35704 q^{56} + 73316 q^{58} + 29300 q^{59} + 54010 q^{61} - 99908 q^{62} - 1582 q^{64} + 51216 q^{65} + 43390 q^{67} + 69840 q^{68} - 2476 q^{70} - 23424 q^{71} - 91402 q^{73} + 2294 q^{74} - 14274 q^{76} + 97208 q^{77} - 49398 q^{79} + 52626 q^{80} + 40152 q^{82} + 103936 q^{83} + 5888 q^{85} - 133634 q^{86} + 48898 q^{88} - 96112 q^{89} + 129228 q^{91} - 11638 q^{92} - 133688 q^{94} + 55928 q^{95} - 135318 q^{97} - 108440 q^{98}+O(q^{100}) \) 3 * q + 8 * q^2 + 22 * q^4 + 56 * q^5 - 114 * q^7 + 510 * q^8 + 282 * q^10 + 376 * q^11 - 858 * q^13 - 588 * q^14 + 2738 * q^16 + 2548 * q^17 - 2846 * q^19 + 4618 * q^20 - 5050 * q^22 - 1587 * q^23 + 753 * q^25 + 7788 * q^26 + 4736 * q^28 + 16370 * q^29 - 14756 * q^31 + 3878 * q^32 + 16520 * q^34 + 18520 * q^35 + 15874 * q^37 - 12438 * q^38 + 38270 * q^40 - 12606 * q^41 + 3154 * q^43 - 27114 * q^44 - 4232 * q^46 - 29928 * q^47 + 4471 * q^49 - 1452 * q^50 + 86856 * q^52 + 44084 * q^53 + 38360 * q^55 + 35704 * q^56 + 73316 * q^58 + 29300 * q^59 + 54010 * q^61 - 99908 * q^62 - 1582 * q^64 + 51216 * q^65 + 43390 * q^67 + 69840 * q^68 - 2476 * q^70 - 23424 * q^71 - 91402 * q^73 + 2294 * q^74 - 14274 * q^76 + 97208 * q^77 - 49398 * q^79 + 52626 * q^80 + 40152 * q^82 + 103936 * q^83 + 5888 * q^85 - 133634 * q^86 + 48898 * q^88 - 96112 * q^89 + 129228 * q^91 - 11638 * q^92 - 133688 * q^94 + 55928 * q^95 - 135318 * q^97 - 108440 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more