LMFDB - Embedded newform 117.6.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,6,Mod(1,117)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("117.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	117.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.7649069181$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	117.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.56155 q^{2} -11.1922 q^{4} +103.462 q^{5} +126.309 q^{7} -197.024 q^{8} +471.948 q^{10} +14.8296 q^{11} -169.000 q^{13} +576.164 q^{14} -540.582 q^{16} +1051.12 q^{17} -213.723 q^{19} -1157.97 q^{20} +67.6458 q^{22} +4231.16 q^{23} +7579.41 q^{25} -770.902 q^{26} -1413.68 q^{28} +504.955 q^{29} +4783.58 q^{31} +3838.86 q^{32} +4794.75 q^{34} +13068.2 q^{35} -4635.74 q^{37} -974.911 q^{38} -20384.5 q^{40} -7944.15 q^{41} -8516.41 q^{43} -165.976 q^{44} +19300.7 q^{46} -24921.2 q^{47} -853.113 q^{49} +34573.9 q^{50} +1891.49 q^{52} +7808.46 q^{53} +1534.30 q^{55} -24885.8 q^{56} +2303.38 q^{58} +37337.5 q^{59} -18172.2 q^{61} +21820.5 q^{62} +34809.8 q^{64} -17485.1 q^{65} -34559.9 q^{67} -11764.4 q^{68} +59611.1 q^{70} -41255.7 q^{71} -1056.42 q^{73} -21146.2 q^{74} +2392.04 q^{76} +1873.10 q^{77} -47719.3 q^{79} -55929.8 q^{80} -36237.7 q^{82} +74799.0 q^{83} +108751. q^{85} -38848.0 q^{86} -2921.78 q^{88} -9799.26 q^{89} -21346.2 q^{91} -47356.1 q^{92} -113679. q^{94} -22112.3 q^{95} -138432. q^{97} -3891.52 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 5 q^{2} - 43 q^{4} + 42 q^{5} - 36 q^{7} - 225 q^{8} + 445 q^{10} + 376 q^{11} - 338 q^{13} + 505 q^{14} + 465 q^{16} + 2630 q^{17} - 312 q^{19} + 797 q^{20} + 226 q^{22} + 2624 q^{23} + 8232 q^{25}+ \cdots + 290 q^{98}+O(q^{100}) $ 2 * q + 5 * q^2 - 43 * q^4 + 42 * q^5 - 36 * q^7 - 225 * q^8 + 445 * q^10 + 376 * q^11 - 338 * q^13 + 505 * q^14 + 465 * q^16 + 2630 * q^17 - 312 * q^19 + 797 * q^20 + 226 * q^22 + 2624 * q^23 + 8232 * q^25 - 845 * q^26 + 3749 * q^28 + 812 * q^29 + 7720 * q^31 + 5175 * q^32 + 5487 * q^34 + 23044 * q^35 - 16858 * q^37 - 1018 * q^38 - 18665 * q^40 - 7840 * q^41 + 2420 * q^43 - 11654 * q^44 + 18596 * q^46 - 9972 * q^47 + 8684 * q^49 + 34860 * q^50 + 7267 * q^52 + 43720 * q^53 - 20664 * q^55 - 20345 * q^56 + 2438 * q^58 + 38936 * q^59 + 1984 * q^61 + 23108 * q^62 + 3217 * q^64 - 7098 * q^65 - 69928 * q^67 - 61985 * q^68 + 63985 * q^70 - 67396 * q^71 + 74412 * q^73 - 26505 * q^74 + 5518 * q^76 - 56748 * q^77 - 55296 * q^79 - 117735 * q^80 - 36192 * q^82 + 75712 * q^83 + 11710 * q^85 - 34053 * q^86 - 13026 * q^88 - 116508 * q^89 + 6084 * q^91 + 3764 * q^92 - 107125 * q^94 - 16072 * q^95 - 34756 * q^97 + 290 * q^98

Coefficient data

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

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

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

$2$	4.56155	0.806376	0.403188	−	0.915117i	$-0.367902\pi$
$2$	4.56155	0.806376	0.403188	+	0.915117i	$0.367902\pi$
$3$	0	0
$3$	0	0
$4$	−11.1922	−0.349757
$5$	103.462	1.85079	0.925393	−	0.379008i	$-0.123735\pi$
$5$	103.462	1.85079	0.925393	+	0.379008i	$0.123735\pi$
$6$	0	0
$7$	126.309	0.974290	0.487145	−	0.873321i	$-0.338038\pi$
$7$	126.309	0.974290	0.487145	+	0.873321i	$0.338038\pi$
$8$	−197.024	−1.08841
$9$	0	0
$10$	471.948	1.49243
$11$	14.8296	0.0369527	0.0184764	−	0.999829i	$-0.494118\pi$
$11$	14.8296	0.0369527	0.0184764	+	0.999829i	$0.494118\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	576.164	0.785644
$15$	0	0
$16$	−540.582	−0.527912
$17$	1051.12	0.882126	0.441063	−	0.897476i	$-0.354602\pi$
$17$	1051.12	0.882126	0.441063	+	0.897476i	$0.354602\pi$
$18$	0	0
$19$	−213.723	−0.135821	−0.0679107	−	0.997691i	$-0.521633\pi$
$19$	−213.723	−0.135821	−0.0679107	+	0.997691i	$0.521633\pi$
$20$	−1157.97	−0.647326
$21$	0	0
$22$	67.6458	0.0297978
$23$	4231.16	1.66778	0.833892	−	0.551928i	$-0.186108\pi$
$23$	4231.16	1.66778	0.833892	+	0.551928i	$0.186108\pi$
$24$	0	0
$25$	7579.41	2.42541
$26$	−770.902	−0.223649
$27$	0	0
$28$	−1413.68	−0.340765
$29$	504.955	0.111495	0.0557477	−	0.998445i	$-0.482246\pi$
$29$	504.955	0.111495	0.0557477	+	0.998445i	$0.482246\pi$
$30$	0	0
$31$	4783.58	0.894022	0.447011	−	0.894528i	$-0.352488\pi$
$31$	4783.58	0.894022	0.447011	+	0.894528i	$0.352488\pi$
$32$	3838.86	0.662716
$33$	0	0
$34$	4794.75	0.711325
$35$	13068.2	1.80320
$36$	0	0
$37$	−4635.74	−0.556692	−0.278346	−	0.960481i	$-0.589786\pi$
$37$	−4635.74	−0.556692	−0.278346	+	0.960481i	$0.589786\pi$
$38$	−974.911	−0.109523
$39$	0	0
$40$	−20384.5	−2.01442
$41$	−7944.15	−0.738054	−0.369027	−	0.929419i	$-0.620309\pi$
$41$	−7944.15	−0.738054	−0.369027	+	0.929419i	$0.620309\pi$
$42$	0	0
$43$	−8516.41	−0.702401	−0.351201	−	0.936300i	$-0.614226\pi$
$43$	−8516.41	−0.702401	−0.351201	+	0.936300i	$0.614226\pi$
$44$	−165.976	−0.0129245
$45$	0	0
$46$	19300.7	1.34486
$47$	−24921.2	−1.64560	−0.822801	−	0.568330i	$-0.807590\pi$
$47$	−24921.2	−1.64560	−0.822801	+	0.568330i	$0.807590\pi$
$48$	0	0
$49$	−853.113	−0.0507594
$50$	34573.9	1.95579
$51$	0	0
$52$	1891.49	0.0970052
$53$	7808.46	0.381835	0.190917	−	0.981606i	$-0.438854\pi$
$53$	7808.46	0.381835	0.190917	+	0.981606i	$0.438854\pi$
$54$	0	0
$55$	1534.30	0.0683916
$56$	−24885.8	−1.06043
$57$	0	0
$58$	2303.38	0.0899073
$59$	37337.5	1.39642	0.698209	−	0.715894i	$-0.253980\pi$
$59$	37337.5	1.39642	0.698209	+	0.715894i	$0.253980\pi$
$60$	0	0
$61$	−18172.2	−0.625292	−0.312646	−	0.949870i	$-0.601215\pi$
$61$	−18172.2	−0.625292	−0.312646	+	0.949870i	$0.601215\pi$
$62$	21820.5	0.720918
$63$	0	0
$64$	34809.8	1.06231
$65$	−17485.1	−0.513316
$66$	0	0
$67$	−34559.9	−0.940559	−0.470279	−	0.882518i	$-0.655847\pi$
$67$	−34559.9	−0.940559	−0.470279	+	0.882518i	$0.655847\pi$
$68$	−11764.4	−0.308530
$69$	0	0
$70$	59611.1	1.45406
$71$	−41255.7	−0.971265	−0.485632	−	0.874163i	$-0.661411\pi$
$71$	−41255.7	−0.971265	−0.485632	+	0.874163i	$0.661411\pi$
$72$	0	0
$73$	−1056.42	−0.0232022	−0.0116011	−	0.999933i	$-0.503693\pi$
$73$	−1056.42	−0.0232022	−0.0116011	+	0.999933i	$0.503693\pi$
$74$	−21146.2	−0.448903
$75$	0	0
$76$	2392.04	0.0475045
$77$	1873.10	0.0360027
$78$	0	0
$79$	−47719.3	−0.860253	−0.430126	−	0.902769i	$-0.641531\pi$
$79$	−47719.3	−0.860253	−0.430126	+	0.902769i	$0.641531\pi$
$80$	−55929.8	−0.977053
$81$	0	0
$82$	−36237.7	−0.595149
$83$	74799.0	1.19179	0.595896	−	0.803061i	$-0.296797\pi$
$83$	74799.0	1.19179	0.595896	+	0.803061i	$0.296797\pi$
$84$	0	0
$85$	108751.	1.63263
$86$	−38848.0	−0.566400
$87$	0	0
$88$	−2921.78	−0.0402198
$89$	−9799.26	−0.131135	−0.0655675	−	0.997848i	$-0.520886\pi$
$89$	−9799.26	−0.131135	−0.0655675	+	0.997848i	$0.520886\pi$
$90$	0	0
$91$	−21346.2	−0.270219
$92$	−47356.1	−0.583320
$93$	0	0
$94$	−113679.	−1.32697
$95$	−22112.3	−0.251376
$96$	0	0
$97$	−138432.	−1.49385	−0.746927	−	0.664906i	$-0.768472\pi$
$97$	−138432.	−1.49385	−0.746927	+	0.664906i	$0.768472\pi$
$98$	−3891.52	−0.0409312
$99$	0	0
$100$	−84830.5	−0.848305
$101$	−139151.	−1.35733	−0.678663	−	0.734450i	$-0.737440\pi$
$101$	−139151.	−1.35733	−0.678663	+	0.734450i	$0.737440\pi$
$102$	0	0
$103$	98512.2	0.914950	0.457475	−	0.889223i	$-0.348754\pi$
$103$	98512.2	0.914950	0.457475	+	0.889223i	$0.348754\pi$
$104$	33297.0	0.301871
$105$	0	0
$106$	35618.7	0.307902
$107$	26848.4	0.226704	0.113352	−	0.993555i	$-0.463841\pi$
$107$	26848.4	0.226704	0.113352	+	0.993555i	$0.463841\pi$
$108$	0	0
$109$	−63220.9	−0.509676	−0.254838	−	0.966984i	$-0.582022\pi$
$109$	−63220.9	−0.509676	−0.254838	+	0.966984i	$0.582022\pi$
$110$	6998.78	0.0551494
$111$	0	0
$112$	−68280.2	−0.514340
$113$	−114434.	−0.843058	−0.421529	−	0.906815i	$-0.638506\pi$
$113$	−114434.	−0.843058	−0.421529	+	0.906815i	$0.638506\pi$
$114$	0	0
$115$	437765.	3.08671
$116$	−5651.57	−0.0389964
$117$	0	0
$118$	170317.	1.12604
$119$	132766.	0.859446
$120$	0	0
$121$	−160831.	−0.998634
$122$	−82893.4	−0.504221
$123$	0	0
$124$	−53538.9	−0.312691
$125$	460863.	2.63813
$126$	0	0
$127$	−248871.	−1.36919	−0.684596	−	0.728922i	$-0.740022\pi$
$127$	−248871.	−1.36919	−0.684596	+	0.728922i	$0.740022\pi$
$128$	35943.2	0.193906
$129$	0	0
$130$	−79759.2	−0.413926
$131$	−102963.	−0.524205	−0.262102	−	0.965040i	$-0.584416\pi$
$131$	−102963.	−0.524205	−0.262102	+	0.965040i	$0.584416\pi$
$132$	0	0
$133$	−26995.1	−0.132329
$134$	−157647.	−0.758444
$135$	0	0
$136$	−207096.	−0.960117
$137$	36037.4	0.164041	0.0820204	−	0.996631i	$-0.473863\pi$
$137$	36037.4	0.164041	0.0820204	+	0.996631i	$0.473863\pi$
$138$	0	0
$139$	152655.	0.670151	0.335076	−	0.942191i	$-0.391238\pi$
$139$	152655.	0.670151	0.335076	+	0.942191i	$0.391238\pi$
$140$	−146262.	−0.630683
$141$	0	0
$142$	−188190.	−0.783205
$143$	−2506.20	−0.0102488
$144$	0	0
$145$	52243.7	0.206354
$146$	−4818.92	−0.0187097
$147$	0	0
$148$	51884.3	0.194707
$149$	−72547.1	−0.267704	−0.133852	−	0.991001i	$-0.542735\pi$
$149$	−72547.1	−0.267704	−0.133852	+	0.991001i	$0.542735\pi$
$150$	0	0
$151$	489021.	1.74536	0.872681	−	0.488291i	$-0.162379\pi$
$151$	489021.	1.74536	0.872681	+	0.488291i	$0.162379\pi$
$152$	42108.6	0.147830
$153$	0	0
$154$	8544.26	0.0290317
$155$	494919.	1.65464
$156$	0	0
$157$	89467.9	0.289680	0.144840	−	0.989455i	$-0.453733\pi$
$157$	89467.9	0.289680	0.144840	+	0.989455i	$0.453733\pi$
$158$	−217674.	−0.693687
$159$	0	0
$160$	397177.	1.22655
$161$	534432.	1.62490
$162$	0	0
$163$	−225668.	−0.665275	−0.332637	−	0.943055i	$-0.607939\pi$
$163$	−225668.	−0.665275	−0.332637	+	0.943055i	$0.607939\pi$
$164$	88912.8	0.258140
$165$	0	0
$166$	341200.	0.961033
$167$	−209528.	−0.581367	−0.290683	−	0.956819i	$-0.593883\pi$
$167$	−209528.	−0.581367	−0.290683	+	0.956819i	$0.593883\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	496074.	1.31651
$171$	0	0
$172$	95317.6	0.245670
$173$	465184.	1.18171	0.590853	−	0.806779i	$-0.298791\pi$
$173$	465184.	1.18171	0.590853	+	0.806779i	$0.298791\pi$
$174$	0	0
$175$	957345.	2.36305
$176$	−8016.60	−0.0195078
$177$	0	0
$178$	−44699.9	−0.105744
$179$	472573.	1.10239	0.551197	−	0.834375i	$-0.314171\pi$
$179$	472573.	1.10239	0.551197	+	0.834375i	$0.314171\pi$
$180$	0	0
$181$	−74099.4	−0.168120	−0.0840598	−	0.996461i	$-0.526789\pi$
$181$	−74099.4	−0.168120	−0.0840598	+	0.996461i	$0.526789\pi$
$182$	−97371.7	−0.217898
$183$	0	0
$184$	−833638.	−1.81524
$185$	−479624.	−1.03032
$186$	0	0
$187$	15587.7	0.0325970
$188$	278924.	0.575561
$189$	0	0
$190$	−100866.	−0.202704
$191$	224128.	0.444543	0.222271	−	0.974985i	$-0.428653\pi$
$191$	224128.	0.444543	0.222271	+	0.974985i	$0.428653\pi$
$192$	0	0
$193$	−662265.	−1.27979	−0.639895	−	0.768463i	$-0.721022\pi$
$193$	−662265.	−1.27979	−0.639895	+	0.768463i	$0.721022\pi$
$194$	−631467.	−1.20461
$195$	0	0
$196$	9548.24	0.0177535
$197$	−666464.	−1.22352	−0.611760	−	0.791043i	$-0.709538\pi$
$197$	−666464.	−1.22352	−0.611760	+	0.791043i	$0.709538\pi$
$198$	0	0
$199$	645720.	1.15588	0.577938	−	0.816081i	$-0.303858\pi$
$199$	645720.	1.15588	0.577938	+	0.816081i	$0.303858\pi$
$200$	−1.49332e6	−2.63985
$201$	0	0
$202$	−634746.	−1.09451
$203$	63780.1	0.108629
$204$	0	0
$205$	−821919.	−1.36598
$206$	449369.	0.737794
$207$	0	0
$208$	91358.4	0.146417
$209$	−3169.43	−0.00501897
$210$	0	0
$211$	−868021.	−1.34222	−0.671110	−	0.741357i	$-0.734182\pi$
$211$	−868021.	−1.34222	−0.671110	+	0.741357i	$0.734182\pi$
$212$	−87394.1	−0.133550
$213$	0	0
$214$	122470.	0.182808
$215$	−881125.	−1.29999
$216$	0	0
$217$	604207.	0.871037
$218$	−288385.	−0.410991
$219$	0	0
$220$	−17172.2	−0.0239205
$221$	−177639.	−0.244658
$222$	0	0
$223$	−58342.9	−0.0785644	−0.0392822	−	0.999228i	$-0.512507\pi$
$223$	−58342.9	−0.0785644	−0.0392822	+	0.999228i	$0.512507\pi$
$224$	484882.	0.645678
$225$	0	0
$226$	−521995.	−0.679822
$227$	−111768.	−0.143964	−0.0719820	−	0.997406i	$-0.522932\pi$
$227$	−111768.	−0.143964	−0.0719820	+	0.997406i	$0.522932\pi$
$228$	0	0
$229$	1.14984e6	1.44893	0.724467	−	0.689309i	$-0.242086\pi$
$229$	1.14984e6	1.44893	0.724467	+	0.689309i	$0.242086\pi$
$230$	1.99689e6	2.48905
$231$	0	0
$232$	−99488.0	−0.121353
$233$	−630470.	−0.760807	−0.380404	−	0.924821i	$-0.624215\pi$
$233$	−630470.	−0.760807	−0.380404	+	0.924821i	$0.624215\pi$
$234$	0	0
$235$	−2.57840e6	−3.04566
$236$	−417891.	−0.488408
$237$	0	0
$238$	605618.	0.693037
$239$	165628.	0.187559	0.0937797	−	0.995593i	$-0.470105\pi$
$239$	165628.	0.187559	0.0937797	+	0.995593i	$0.470105\pi$
$240$	0	0
$241$	690968.	0.766329	0.383165	−	0.923680i	$-0.374834\pi$
$241$	690968.	0.766329	0.383165	+	0.923680i	$0.374834\pi$
$242$	−733639.	−0.805275
$243$	0	0
$244$	203387.	0.218700
$245$	−88264.9	−0.0939448
$246$	0	0
$247$	36119.3	0.0376701
$248$	−942478.	−0.973065
$249$	0	0
$250$	2.10225e6	2.12733
$251$	386887.	0.387615	0.193807	−	0.981040i	$-0.437916\pi$
$251$	386887.	0.387615	0.193807	+	0.981040i	$0.437916\pi$
$252$	0	0
$253$	62746.2	0.0616292
$254$	−1.13524e6	−1.10408
$255$	0	0
$256$	−949957.	−0.905950
$257$	−258260.	−0.243907	−0.121953	−	0.992536i	$-0.538916\pi$
$257$	−258260.	−0.243907	−0.121953	+	0.992536i	$0.538916\pi$
$258$	0	0
$259$	−585535.	−0.542379
$260$	195697.	0.179536
$261$	0	0
$262$	−469669.	−0.422706
$263$	1.19053e6	1.06133	0.530665	−	0.847582i	$-0.321942\pi$
$263$	1.19053e6	1.06133	0.530665	+	0.847582i	$0.321942\pi$
$264$	0	0
$265$	807879.	0.706695
$266$	−123140.	−0.106707
$267$	0	0
$268$	386803.	0.328967
$269$	−850968.	−0.717022	−0.358511	−	0.933525i	$-0.616715\pi$
$269$	−850968.	−0.717022	−0.358511	+	0.933525i	$0.616715\pi$
$270$	0	0
$271$	−40926.1	−0.0338514	−0.0169257	−	0.999857i	$-0.505388\pi$
$271$	−40926.1	−0.0338514	−0.0169257	+	0.999857i	$0.505388\pi$
$272$	−568218.	−0.465685
$273$	0	0
$274$	164387.	0.132279
$275$	112399.	0.0896256
$276$	0	0
$277$	1.00054e6	0.783490	0.391745	−	0.920074i	$-0.371872\pi$
$277$	1.00054e6	0.783490	0.391745	+	0.920074i	$0.371872\pi$
$278$	696342.	0.540394
$279$	0	0
$280$	−2.57474e6	−1.96263
$281$	1.73596e6	1.31151	0.655757	−	0.754972i	$-0.272350\pi$
$281$	1.73596e6	1.31151	0.655757	+	0.754972i	$0.272350\pi$
$282$	0	0
$283$	−1.27363e6	−0.945319	−0.472660	−	0.881245i	$-0.656706\pi$
$283$	−1.27363e6	−0.945319	−0.472660	+	0.881245i	$0.656706\pi$
$284$	461743.	0.339707
$285$	0	0
$286$	−11432.1	−0.00826443
$287$	−1.00342e6	−0.719078
$288$	0	0
$289$	−315001.	−0.221854
$290$	238312.	0.166399
$291$	0	0
$292$	11823.7	0.00811515
$293$	2043.70	0.00139075	0.000695374	−	1.00000i	$-0.499779\pi$
$293$	2043.70	0.00139075	0.000695374		1.00000i	$0.499779\pi$
$294$	0	0
$295$	3.86302e6	2.58447
$296$	913351.	0.605910
$297$	0	0
$298$	−330927.	−0.215870
$299$	−715066.	−0.462560
$300$	0	0
$301$	−1.07570e6	−0.684342
$302$	2.23070e6	1.40742
$303$	0	0
$304$	115535.	0.0717018
$305$	−1.88013e6	−1.15728
$306$	0	0
$307$	−401308.	−0.243014	−0.121507	−	0.992591i	$-0.538773\pi$
$307$	−401308.	−0.243014	−0.121507	+	0.992591i	$0.538773\pi$
$308$	−20964.2	−0.0125922
$309$	0	0
$310$	2.25760e6	1.33427
$311$	1.92628e6	1.12933	0.564663	−	0.825322i	$-0.309006\pi$
$311$	1.92628e6	1.12933	0.564663	+	0.825322i	$0.309006\pi$
$312$	0	0
$313$	1.64519e6	0.949196	0.474598	−	0.880203i	$-0.342593\pi$
$313$	1.64519e6	0.949196	0.474598	+	0.880203i	$0.342593\pi$
$314$	408113.	0.233591
$315$	0	0
$316$	534085.	0.300880
$317$	1.99476e6	1.11492	0.557459	−	0.830205i	$-0.311777\pi$
$317$	1.99476e6	1.11492	0.557459	+	0.830205i	$0.311777\pi$
$318$	0	0
$319$	7488.26	0.00412006
$320$	3.60150e6	1.96611
$321$	0	0
$322$	2.43784e6	1.31028
$323$	−224649.	−0.119812
$324$	0	0
$325$	−1.28092e6	−0.672688
$326$	−1.02940e6	−0.536462
$327$	0	0
$328$	1.56519e6	0.803306
$329$	−3.14777e6	−1.60329
$330$	0	0
$331$	−675924.	−0.339100	−0.169550	−	0.985522i	$-0.554231\pi$
$331$	−675924.	−0.339100	−0.169550	+	0.985522i	$0.554231\pi$
$332$	−837168.	−0.416838
$333$	0	0
$334$	−955772.	−0.468800
$335$	−3.57564e6	−1.74077
$336$	0	0
$337$	−2.13552e6	−1.02430	−0.512152	−	0.858895i	$-0.671152\pi$
$337$	−2.13552e6	−1.02430	−0.512152	+	0.858895i	$0.671152\pi$
$338$	130283.	0.0620289
$339$	0	0
$340$	−1.21717e6	−0.571023
$341$	70938.3	0.0330366
$342$	0	0
$343$	−2.23063e6	−1.02374
$344$	1.67793e6	0.764502
$345$	0	0
$346$	2.12196e6	0.952899
$347$	2.57257e6	1.14695	0.573473	−	0.819225i	$-0.305596\pi$
$347$	2.57257e6	1.14695	0.573473	+	0.819225i	$0.305596\pi$
$348$	0	0
$349$	2.02363e6	0.889339	0.444670	−	0.895695i	$-0.353321\pi$
$349$	2.02363e6	0.889339	0.444670	+	0.895695i	$0.353321\pi$
$350$	4.36698e6	1.90551
$351$	0	0
$352$	56928.7	0.0244892
$353$	−2.04810e6	−0.874810	−0.437405	−	0.899265i	$-0.644102\pi$
$353$	−2.04810e6	−0.874810	−0.437405	+	0.899265i	$0.644102\pi$
$354$	0	0
$355$	−4.26840e6	−1.79760
$356$	109676.	0.0458654
$357$	0	0
$358$	2.15567e6	0.888944
$359$	1.59901e6	0.654808	0.327404	−	0.944885i	$-0.393826\pi$
$359$	1.59901e6	0.654808	0.327404	+	0.944885i	$0.393826\pi$
$360$	0	0
$361$	−2.43042e6	−0.981553
$362$	−338008.	−0.135568
$363$	0	0
$364$	238911.	0.0945112
$365$	−109299.	−0.0429424
$366$	0	0
$367$	−3.86389e6	−1.49747	−0.748737	−	0.662867i	$-0.769339\pi$
$367$	−3.86389e6	−1.49747	−0.748737	+	0.662867i	$0.769339\pi$
$368$	−2.28729e6	−0.880444
$369$	0	0
$370$	−2.18783e6	−0.830824
$371$	986276.	0.372018
$372$	0	0
$373$	−1.56702e6	−0.583179	−0.291589	−	0.956544i	$-0.594184\pi$
$373$	−1.56702e6	−0.583179	−0.291589	+	0.956544i	$0.594184\pi$
$374$	71104.0	0.0262854
$375$	0	0
$376$	4.91007e6	1.79109
$377$	−85337.3	−0.0309233
$378$	0	0
$379$	3.19239e6	1.14161	0.570805	−	0.821086i	$-0.306631\pi$
$379$	3.19239e6	1.14161	0.570805	+	0.821086i	$0.306631\pi$
$380$	247486.	0.0879208
$381$	0	0
$382$	1.02237e6	0.358469
$383$	−400432.	−0.139486	−0.0697432	−	0.997565i	$-0.522218\pi$
$383$	−400432.	−0.139486	−0.0697432	+	0.997565i	$0.522218\pi$
$384$	0	0
$385$	193795.	0.0666333
$386$	−3.02096e6	−1.03199
$387$	0	0
$388$	1.54937e6	0.522487
$389$	−413440.	−0.138528	−0.0692642	−	0.997598i	$-0.522065\pi$
$389$	−413440.	−0.138528	−0.0692642	+	0.997598i	$0.522065\pi$
$390$	0	0
$391$	4.44746e6	1.47120
$392$	168083.	0.0552471
$393$	0	0
$394$	−3.04011e6	−0.986618
$395$	−4.93714e6	−1.59214
$396$	0	0
$397$	−102926.	−0.0327753	−0.0163877	−	0.999866i	$-0.505217\pi$
$397$	−102926.	−0.0327753	−0.0163877	+	0.999866i	$0.505217\pi$
$398$	2.94548e6	0.932071
$399$	0	0
$400$	−4.09729e6	−1.28040
$401$	−2.23365e6	−0.693671	−0.346836	−	0.937926i	$-0.612744\pi$
$401$	−2.23365e6	−0.693671	−0.346836	+	0.937926i	$0.612744\pi$
$402$	0	0
$403$	−808424.	−0.247957
$404$	1.55741e6	0.474734
$405$	0	0
$406$	290937.	0.0875958
$407$	−68746.0	−0.0205713
$408$	0	0
$409$	−4.46150e6	−1.31878	−0.659390	−	0.751801i	$-0.729185\pi$
$409$	−4.46150e6	−1.31878	−0.659390	+	0.751801i	$0.729185\pi$
$410$	−3.74923e6	−1.10149
$411$	0	0
$412$	−1.10257e6	−0.320010
$413$	4.71606e6	1.36052
$414$	0	0
$415$	7.73887e6	2.20575
$416$	−648768.	−0.183804
$417$	0	0
$418$	−14457.5	−0.00404718
$419$	−4.22792e6	−1.17650	−0.588250	−	0.808679i	$-0.700183\pi$
$419$	−4.22792e6	−1.17650	−0.588250	+	0.808679i	$0.700183\pi$
$420$	0	0
$421$	4.11791e6	1.13233	0.566163	−	0.824293i	$-0.308427\pi$
$421$	4.11791e6	1.13233	0.566163	+	0.824293i	$0.308427\pi$
$422$	−3.95952e6	−1.08233
$423$	0	0
$424$	−1.53845e6	−0.415594
$425$	7.96688e6	2.13952
$426$	0	0
$427$	−2.29531e6	−0.609216
$428$	−300493.	−0.0792912
$429$	0	0
$430$	−4.01930e6	−1.04828
$431$	−1.15324e6	−0.299038	−0.149519	−	0.988759i	$-0.547773\pi$
$431$	−1.15324e6	−0.299038	−0.149519	+	0.988759i	$0.547773\pi$
$432$	0	0
$433$	−33734.3	−0.00864673	−0.00432337	−	0.999991i	$-0.501376\pi$
$433$	−33734.3	−0.00864673	−0.00432337	+	0.999991i	$0.501376\pi$
$434$	2.75612e6	0.702383
$435$	0	0
$436$	707583.	0.178263
$437$	−904298.	−0.226521
$438$	0	0
$439$	7.48363e6	1.85332	0.926661	−	0.375898i	$-0.122666\pi$
$439$	7.48363e6	1.85332	0.926661	+	0.375898i	$0.122666\pi$
$440$	−302293.	−0.0744383
$441$	0	0
$442$	−810312.	−0.197286
$443$	3.28028e6	0.794148	0.397074	−	0.917786i	$-0.370026\pi$
$443$	3.28028e6	0.794148	0.397074	+	0.917786i	$0.370026\pi$
$444$	0	0
$445$	−1.01385e6	−0.242703
$446$	−266134.	−0.0633525
$447$	0	0
$448$	4.39678e6	1.03500
$449$	−7.95356e6	−1.86185	−0.930927	−	0.365205i	$-0.880999\pi$
$449$	−7.95356e6	−1.86185	−0.930927	+	0.365205i	$0.880999\pi$
$450$	0	0
$451$	−117808.	−0.0272731
$452$	1.28077e6	0.294866
$453$	0	0
$454$	−509837.	−0.116089
$455$	−2.20852e6	−0.500118
$456$	0	0
$457$	−3.35187e6	−0.750753	−0.375377	−	0.926872i	$-0.622487\pi$
$457$	−3.35187e6	−0.750753	−0.375377	+	0.926872i	$0.622487\pi$
$458$	5.24506e6	1.16839
$459$	0	0
$460$	−4.89956e6	−1.07960
$461$	−4.68627e6	−1.02701	−0.513505	−	0.858086i	$-0.671653\pi$
$461$	−4.68627e6	−1.02701	−0.513505	+	0.858086i	$0.671653\pi$
$462$	0	0
$463$	6.64697e6	1.44102	0.720512	−	0.693442i	$-0.243907\pi$
$463$	6.64697e6	1.44102	0.720512	+	0.693442i	$0.243907\pi$
$464$	−272969.	−0.0588599
$465$	0	0
$466$	−2.87592e6	−0.613497
$467$	3.14141e6	0.666549	0.333275	−	0.942830i	$-0.391846\pi$
$467$	3.14141e6	0.666549	0.333275	+	0.942830i	$0.391846\pi$
$468$	0	0
$469$	−4.36522e6	−0.916377
$470$	−1.17615e7	−2.45594
$471$	0	0
$472$	−7.35638e6	−1.51988
$473$	−126295.	−0.0259556
$474$	0	0
$475$	−1.61990e6	−0.329423
$476$	−1.48595e6	−0.300598
$477$	0	0
$478$	755521.	0.151243
$479$	6.68286e6	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$479$	6.68286e6	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$480$	0	0
$481$	783441.	0.154399
$482$	3.15189e6	0.617950
$483$	0	0
$484$	1.80006e6	0.349280
$485$	−1.43225e7	−2.76481
$486$	0	0
$487$	−4.06478e6	−0.776631	−0.388316	−	0.921526i	$-0.626943\pi$
$487$	−4.06478e6	−0.776631	−0.388316	+	0.921526i	$0.626943\pi$
$488$	3.58035e6	0.680575
$489$	0	0
$490$	−402625.	−0.0757549
$491$	2.10434e6	0.393923	0.196962	−	0.980411i	$-0.436893\pi$
$491$	2.10434e6	0.393923	0.196962	+	0.980411i	$0.436893\pi$
$492$	0	0
$493$	530768.	0.0983530
$494$	164760.	0.0303763
$495$	0	0
$496$	−2.58592e6	−0.471966
$497$	−5.21095e6	−0.946293
$498$	0	0
$499$	5.96715e6	1.07279	0.536396	−	0.843966i	$-0.319785\pi$
$499$	5.96715e6	1.07279	0.536396	+	0.843966i	$0.319785\pi$
$500$	−5.15808e6	−0.922706
$501$	0	0
$502$	1.76481e6	0.312563
$503$	1.00144e7	1.76483	0.882417	−	0.470467i	$-0.155915\pi$
$503$	1.00144e7	1.76483	0.882417	+	0.470467i	$0.155915\pi$
$504$	0	0
$505$	−1.43969e7	−2.51212
$506$	286220.	0.0496963
$507$	0	0
$508$	2.78542e6	0.478885
$509$	8.47321e6	1.44962	0.724809	−	0.688950i	$-0.241928\pi$
$509$	8.47321e6	1.44962	0.724809	+	0.688950i	$0.241928\pi$
$510$	0	0
$511$	−133435.	−0.0226057
$512$	−5.48346e6	−0.924442
$513$	0	0
$514$	−1.17807e6	−0.196681
$515$	1.01923e7	1.69338
$516$	0	0
$517$	−369571.	−0.0608095
$518$	−2.67095e6	−0.437362
$519$	0	0
$520$	3.44498e6	0.558699
$521$	197614.	0.0318951	0.0159476	−	0.999873i	$-0.494924\pi$
$521$	197614.	0.0318951	0.0159476	+	0.999873i	$0.494924\pi$
$522$	0	0
$523$	8.27263e6	1.32248	0.661240	−	0.750174i	$-0.270030\pi$
$523$	8.27263e6	1.32248	0.661240	+	0.750174i	$0.270030\pi$
$524$	1.15238e6	0.183345
$525$	0	0
$526$	5.43066e6	0.855831
$527$	5.02812e6	0.788640
$528$	0	0
$529$	1.14664e7	1.78150
$530$	3.68518e6	0.569862
$531$	0	0
$532$	302136.	0.0462832
$533$	1.34256e6	0.204699
$534$	0	0
$535$	2.77779e6	0.419580
$536$	6.80913e6	1.02372
$537$	0	0
$538$	−3.88174e6	−0.578190
$539$	−12651.3	−0.00187570
$540$	0	0
$541$	363216.	0.0533546	0.0266773	−	0.999644i	$-0.491507\pi$
$541$	363216.	0.0533546	0.0266773	+	0.999644i	$0.491507\pi$
$542$	−186686.	−0.0272970
$543$	0	0
$544$	4.03511e6	0.584599
$545$	−6.54097e6	−0.943302
$546$	0	0
$547$	−620452.	−0.0886624	−0.0443312	−	0.999017i	$-0.514116\pi$
$547$	−620452.	−0.0886624	−0.0443312	+	0.999017i	$0.514116\pi$
$548$	−403339.	−0.0573745
$549$	0	0
$550$	512715.	0.0722719
$551$	−107921.	−0.0151435
$552$	0	0
$553$	−6.02736e6	−0.838136
$554$	4.56400e6	0.631788
$555$	0	0
$556$	−1.70855e6	−0.234390
$557$	−3.89737e6	−0.532272	−0.266136	−	0.963935i	$-0.585747\pi$
$557$	−3.89737e6	−0.532272	−0.266136	+	0.963935i	$0.585747\pi$
$558$	0	0
$559$	1.43927e6	0.194811
$560$	−7.06442e6	−0.951933
$561$	0	0
$562$	7.91865e6	1.05757
$563$	−510725.	−0.0679073	−0.0339536	−	0.999423i	$-0.510810\pi$
$563$	−510725.	−0.0679073	−0.0339536	+	0.999423i	$0.510810\pi$
$564$	0	0
$565$	−1.18395e7	−1.56032
$566$	−5.80975e6	−0.762283
$567$	0	0
$568$	8.12834e6	1.05714
$569$	−9.75625e6	−1.26329	−0.631644	−	0.775259i	$-0.717619\pi$
$569$	−9.75625e6	−1.26329	−0.631644	+	0.775259i	$0.717619\pi$
$570$	0	0
$571$	−1.41952e7	−1.82201	−0.911006	−	0.412393i	$-0.864693\pi$
$571$	−1.41952e7	−1.82201	−0.911006	+	0.412393i	$0.864693\pi$
$572$	28049.9	0.00358461
$573$	0	0
$574$	−4.57713e6	−0.579847
$575$	3.20697e7	4.04506
$576$	0	0
$577$	−1.16423e6	−0.145579	−0.0727896	−	0.997347i	$-0.523190\pi$
$577$	−1.16423e6	−0.145579	−0.0727896	+	0.997347i	$0.523190\pi$
$578$	−1.43689e6	−0.178898
$579$	0	0
$580$	−584723.	−0.0721740
$581$	9.44777e6	1.16115
$582$	0	0
$583$	115796.	0.0141098
$584$	208140.	0.0252536
$585$	0	0
$586$	9322.45	0.00112147
$587$	−6.58038e6	−0.788234	−0.394117	−	0.919060i	$-0.628950\pi$
$587$	−6.58038e6	−0.788234	−0.394117	+	0.919060i	$0.628950\pi$
$588$	0	0
$589$	−1.02236e6	−0.121427
$590$	1.76214e7	2.08406
$591$	0	0
$592$	2.50600e6	0.293885
$593$	1.91423e6	0.223541	0.111771	−	0.993734i	$-0.464348\pi$
$593$	1.91423e6	0.223541	0.111771	+	0.993734i	$0.464348\pi$
$594$	0	0
$595$	1.37362e7	1.59065
$596$	811964.	0.0936313
$597$	0	0
$598$	−3.26181e6	−0.372997
$599$	2.33678e6	0.266104	0.133052	−	0.991109i	$-0.457522\pi$
$599$	2.33678e6	0.266104	0.133052	+	0.991109i	$0.457522\pi$
$600$	0	0
$601$	1.04273e7	1.17757	0.588786	−	0.808289i	$-0.299606\pi$
$601$	1.04273e7	1.17757	0.588786	+	0.808289i	$0.299606\pi$
$602$	−4.90684e6	−0.551837
$603$	0	0
$604$	−5.47324e6	−0.610453
$605$	−1.66399e7	−1.84826
$606$	0	0
$607$	−120274.	−0.0132495	−0.00662474	−	0.999978i	$-0.502109\pi$
$607$	−120274.	−0.0132495	−0.00662474	+	0.999978i	$0.502109\pi$
$608$	−820455.	−0.0900111
$609$	0	0
$610$	−8.57633e6	−0.933205
$611$	4.21169e6	0.456408
$612$	0	0
$613$	1.34576e7	1.44649	0.723245	−	0.690592i	$-0.242650\pi$
$613$	1.34576e7	1.44649	0.723245	+	0.690592i	$0.242650\pi$
$614$	−1.83059e6	−0.195961
$615$	0	0
$616$	−369046.	−0.0391858
$617$	6.84879e6	0.724270	0.362135	−	0.932126i	$-0.382048\pi$
$617$	6.84879e6	0.724270	0.362135	+	0.932126i	$0.382048\pi$
$618$	0	0
$619$	−5.40663e6	−0.567153	−0.283577	−	0.958950i	$-0.591521\pi$
$619$	−5.40663e6	−0.567153	−0.283577	+	0.958950i	$0.591521\pi$
$620$	−5.53925e6	−0.578724
$621$	0	0
$622$	8.78684e6	0.910661
$623$	−1.23773e6	−0.127763
$624$	0	0
$625$	2.39962e7	2.45721
$626$	7.50463e6	0.765409
$627$	0	0
$628$	−1.00135e6	−0.101318
$629$	−4.87273e6	−0.491072
$630$	0	0
$631$	−9.62552e6	−0.962390	−0.481195	−	0.876614i	$-0.659797\pi$
$631$	−9.62552e6	−0.962390	−0.481195	+	0.876614i	$0.659797\pi$
$632$	9.40183e6	0.936310
$633$	0	0
$634$	9.09920e6	0.899043
$635$	−2.57487e7	−2.53408
$636$	0	0
$637$	144176.	0.0140781
$638$	34158.1	0.00332232
$639$	0	0
$640$	3.71876e6	0.358879
$641$	1.58752e7	1.52607	0.763037	−	0.646355i	$-0.223707\pi$
$641$	1.58752e7	1.52607	0.763037	+	0.646355i	$0.223707\pi$
$642$	0	0
$643$	1.57235e7	1.49976	0.749880	−	0.661574i	$-0.230111\pi$
$643$	1.57235e7	1.49976	0.749880	+	0.661574i	$0.230111\pi$
$644$	−5.98149e6	−0.568322
$645$	0	0
$646$	−1.02475e6	−0.0966132
$647$	1.58173e7	1.48549	0.742747	−	0.669572i	$-0.233523\pi$
$647$	1.58173e7	1.48549	0.742747	+	0.669572i	$0.233523\pi$
$648$	0	0
$649$	553699.	0.0516015
$650$	−5.84298e6	−0.542440
$651$	0	0
$652$	2.52573e6	0.232685
$653$	−5.31229e6	−0.487527	−0.243763	−	0.969835i	$-0.578382\pi$
$653$	−5.31229e6	−0.487527	−0.243763	+	0.969835i	$0.578382\pi$
$654$	0	0
$655$	−1.06527e7	−0.970192
$656$	4.29447e6	0.389628
$657$	0	0
$658$	−1.43587e7	−1.29286
$659$	−9.90554e6	−0.888514	−0.444257	−	0.895899i	$-0.646532\pi$
$659$	−9.90554e6	−0.888514	−0.444257	+	0.895899i	$0.646532\pi$
$660$	0	0
$661$	−1.29988e7	−1.15717	−0.578587	−	0.815621i	$-0.696396\pi$
$661$	−1.29988e7	−1.15717	−0.578587	+	0.815621i	$0.696396\pi$
$662$	−3.08326e6	−0.273442
$663$	0	0
$664$	−1.47372e7	−1.29716
$665$	−2.79297e6	−0.244913
$666$	0	0
$667$	2.13654e6	0.185950
$668$	2.34508e6	0.203337
$669$	0	0
$670$	−1.63105e7	−1.40372
$671$	−269486.	−0.0231062
$672$	0	0
$673$	−1.32503e7	−1.12769	−0.563844	−	0.825881i	$-0.690678\pi$
$673$	−1.32503e7	−1.12769	−0.563844	+	0.825881i	$0.690678\pi$
$674$	−9.74129e6	−0.825975
$675$	0	0
$676$	−319661.	−0.0269044
$677$	−2.23310e7	−1.87257	−0.936284	−	0.351245i	$-0.885758\pi$
$677$	−2.23310e7	−1.87257	−0.936284	+	0.351245i	$0.885758\pi$
$678$	0	0
$679$	−1.74852e7	−1.45545
$680$	−2.14266e7	−1.77697
$681$	0	0
$682$	323589.	0.0266399
$683$	1.40049e7	1.14876	0.574380	−	0.818588i	$-0.305243\pi$
$683$	1.40049e7	1.14876	0.574380	+	0.818588i	$0.305243\pi$
$684$	0	0
$685$	3.72851e6	0.303605
$686$	−1.01751e7	−0.825523
$687$	0	0
$688$	4.60382e6	0.370806
$689$	−1.31963e6	−0.105902
$690$	0	0
$691$	−5.24817e6	−0.418132	−0.209066	−	0.977902i	$-0.567042\pi$
$691$	−5.24817e6	−0.418132	−0.209066	+	0.977902i	$0.567042\pi$
$692$	−5.20645e6	−0.413310
$693$	0	0
$694$	1.17349e7	0.924870
$695$	1.57940e7	1.24031
$696$	0	0
$697$	−8.35027e6	−0.651056
$698$	9.23090e6	0.717142
$699$	0	0
$700$	−1.07148e7	−0.826495
$701$	2.13994e7	1.64477	0.822386	−	0.568930i	$-0.192642\pi$
$701$	2.13994e7	1.64477	0.822386	+	0.568930i	$0.192642\pi$
$702$	0	0
$703$	990767.	0.0756107
$704$	516214.	0.0392553
$705$	0	0
$706$	−9.34250e6	−0.705426
$707$	−1.75760e7	−1.32243
$708$	0	0
$709$	4.35333e6	0.325242	0.162621	−	0.986689i	$-0.448005\pi$
$709$	4.35333e6	0.325242	0.162621	+	0.986689i	$0.448005\pi$
$710$	−1.94705e7	−1.44954
$711$	0	0
$712$	1.93069e6	0.142729
$713$	2.02401e7	1.49104
$714$	0	0
$715$	−259296.	−0.0189684
$716$	−5.28915e6	−0.385570
$717$	0	0
$718$	7.29395e6	0.528021
$719$	−2.21389e7	−1.59710	−0.798552	−	0.601926i	$-0.794400\pi$
$719$	−2.21389e7	−1.59710	−0.798552	+	0.601926i	$0.794400\pi$
$720$	0	0
$721$	1.24430e7	0.891426
$722$	−1.10865e7	−0.791501
$723$	0	0
$724$	829338.	0.0588010
$725$	3.82726e6	0.270422
$726$	0	0
$727$	−4.83218e6	−0.339084	−0.169542	−	0.985523i	$-0.554229\pi$
$727$	−4.83218e6	−0.339084	−0.169542	+	0.985523i	$0.554229\pi$
$728$	4.20570e6	0.294110
$729$	0	0
$730$	−498575.	−0.0346277
$731$	−8.95178e6	−0.619606
$732$	0	0
$733$	−2.47827e7	−1.70368	−0.851840	−	0.523803i	$-0.824513\pi$
$733$	−2.47827e7	−1.70368	−0.851840	+	0.523803i	$0.824513\pi$
$734$	−1.76253e7	−1.20753
$735$	0	0
$736$	1.62428e7	1.10527
$737$	−512509.	−0.0347562
$738$	0	0
$739$	7.09289e6	0.477762	0.238881	−	0.971049i	$-0.423219\pi$
$739$	7.09289e6	0.477762	0.238881	+	0.971049i	$0.423219\pi$
$740$	5.36806e6	0.360361
$741$	0	0
$742$	4.49895e6	0.299986
$743$	−1.95117e7	−1.29665	−0.648327	−	0.761362i	$-0.724531\pi$
$743$	−1.95117e7	−1.29665	−0.648327	+	0.761362i	$0.724531\pi$
$744$	0	0
$745$	−7.50588e6	−0.495462
$746$	−7.14803e6	−0.470262
$747$	0	0
$748$	−174461.	−0.0114010
$749$	3.39118e6	0.220875
$750$	0	0
$751$	1.66103e7	1.07468	0.537339	−	0.843366i	$-0.319430\pi$
$751$	1.66103e7	1.07468	0.537339	+	0.843366i	$0.319430\pi$
$752$	1.34720e7	0.868733
$753$	0	0
$754$	−389271.	−0.0249358
$755$	5.05952e7	3.23029
$756$	0	0
$757$	1.22902e6	0.0779508	0.0389754	−	0.999240i	$-0.487591\pi$
$757$	1.22902e6	0.0779508	0.0389754	+	0.999240i	$0.487591\pi$
$758$	1.45622e7	0.920567
$759$	0	0
$760$	4.35664e6	0.273601
$761$	−1.37482e7	−0.860569	−0.430284	−	0.902693i	$-0.641587\pi$
$761$	−1.37482e7	−0.860569	−0.430284	+	0.902693i	$0.641587\pi$
$762$	0	0
$763$	−7.98535e6	−0.496572
$764$	−2.50850e6	−0.155482
$765$	0	0
$766$	−1.82659e6	−0.112478
$767$	−6.31004e6	−0.387297
$768$	0	0
$769$	−1.01549e7	−0.619240	−0.309620	−	0.950860i	$-0.600202\pi$
$769$	−1.01549e7	−0.619240	−0.309620	+	0.950860i	$0.600202\pi$
$770$	884007.	0.0537315
$771$	0	0
$772$	7.41222e6	0.447616
$773$	−1.41511e7	−0.851807	−0.425903	−	0.904769i	$-0.640044\pi$
$773$	−1.41511e7	−0.851807	−0.425903	+	0.904769i	$0.640044\pi$
$774$	0	0
$775$	3.62567e7	2.16837
$776$	2.72745e7	1.62593
$777$	0	0
$778$	−1.88593e6	−0.111706
$779$	1.69785e6	0.100243
$780$	0	0
$781$	−611803.	−0.0358909
$782$	2.02873e7	1.18634
$783$	0	0
$784$	461178.	0.0267965
$785$	9.25654e6	0.536136
$786$	0	0
$787$	−1.03095e6	−0.0593338	−0.0296669	−	0.999560i	$-0.509445\pi$
$787$	−1.03095e6	−0.0593338	−0.0296669	+	0.999560i	$0.509445\pi$
$788$	7.45923e6	0.427935
$789$	0	0
$790$	−2.25210e7	−1.28387
$791$	−1.44540e7	−0.821383
$792$	0	0
$793$	3.07110e6	0.173425
$794$	−469500.	−0.0264292
$795$	0	0
$796$	−7.22705e6	−0.404276
$797$	1.43337e7	0.799303	0.399651	−	0.916667i	$-0.369131\pi$
$797$	1.43337e7	0.799303	0.399651	+	0.916667i	$0.369131\pi$
$798$	0	0
$799$	−2.61952e7	−1.45163
$800$	2.90963e7	1.60736
$801$	0	0
$802$	−1.01889e7	−0.559360
$803$	−15666.3	−0.000857386 0
$804$	0	0
$805$	5.52935e7	3.00735
$806$	−3.68767e6	−0.199947
$807$	0	0
$808$	2.74161e7	1.47733
$809$	−1.55020e7	−0.832751	−0.416376	−	0.909193i	$-0.636700\pi$
$809$	−1.55020e7	−0.832751	−0.416376	+	0.909193i	$0.636700\pi$
$810$	0	0
$811$	2.45861e7	1.31261	0.656307	−	0.754494i	$-0.272118\pi$
$811$	2.45861e7	1.31261	0.656307	+	0.754494i	$0.272118\pi$
$812$	−713842.	−0.0379938
$813$	0	0
$814$	−313589.	−0.0165882
$815$	−2.33481e7	−1.23128
$816$	0	0
$817$	1.82016e6	0.0954011
$818$	−2.03513e7	−1.06343
$819$	0	0
$820$	9.19911e6	0.477761
$821$	3.33396e6	0.172624	0.0863122	−	0.996268i	$-0.472492\pi$
$821$	3.33396e6	0.172624	0.0863122	+	0.996268i	$0.472492\pi$
$822$	0	0
$823$	3.08787e6	0.158913	0.0794564	−	0.996838i	$-0.474682\pi$
$823$	3.08787e6	0.158913	0.0794564	+	0.996838i	$0.474682\pi$
$824$	−1.94092e7	−0.995842
$825$	0	0
$826$	2.15125e7	1.09709
$827$	−6.89555e6	−0.350595	−0.175297	−	0.984516i	$-0.556089\pi$
$827$	−6.89555e6	−0.350595	−0.175297	+	0.984516i	$0.556089\pi$
$828$	0	0
$829$	−2.00007e7	−1.01078	−0.505391	−	0.862890i	$-0.668652\pi$
$829$	−2.00007e7	−1.01078	−0.505391	+	0.862890i	$0.668652\pi$
$830$	3.53012e7	1.77867
$831$	0	0
$832$	−5.88286e6	−0.294632
$833$	−896725.	−0.0447762
$834$	0	0
$835$	−2.16782e7	−1.07599
$836$	35473.0	0.00175542
$837$	0	0
$838$	−1.92859e7	−0.948701
$839$	−5.23988e6	−0.256990	−0.128495	−	0.991710i	$-0.541015\pi$
$839$	−5.23988e6	−0.256990	−0.128495	+	0.991710i	$0.541015\pi$
$840$	0	0
$841$	−2.02562e7	−0.987569
$842$	1.87841e7	0.913081
$843$	0	0
$844$	9.71509e6	0.469452
$845$	2.95498e6	0.142368
$846$	0	0
$847$	−2.03144e7	−0.972959
$848$	−4.22111e6	−0.201575
$849$	0	0
$850$	3.63413e7	1.72526
$851$	−1.96146e7	−0.928442
$852$	0	0
$853$	−1.32853e7	−0.625170	−0.312585	−	0.949890i	$-0.601195\pi$
$853$	−1.32853e7	−0.625170	−0.312585	+	0.949890i	$0.601195\pi$
$854$	−1.04702e7	−0.491257
$855$	0	0
$856$	−5.28976e6	−0.246747
$857$	8.34473e6	0.388115	0.194057	−	0.980990i	$-0.437835\pi$
$857$	8.34473e6	0.388115	0.194057	+	0.980990i	$0.437835\pi$
$858$	0	0
$859$	−4.07521e7	−1.88437	−0.942187	−	0.335088i	$-0.891234\pi$
$859$	−4.07521e7	−1.88437	−0.942187	+	0.335088i	$0.891234\pi$
$860$	9.86176e6	0.454683
$861$	0	0
$862$	−5.26056e6	−0.241137
$863$	1.73853e7	0.794614	0.397307	−	0.917686i	$-0.369945\pi$
$863$	1.73853e7	0.794614	0.397307	+	0.917686i	$0.369945\pi$
$864$	0	0
$865$	4.81289e7	2.18708
$866$	−153881.	−0.00697252
$867$	0	0
$868$	−6.76243e6	−0.304652
$869$	−707656.	−0.0317887
$870$	0	0
$871$	5.84063e6	0.260864
$872$	1.24560e7	0.554738
$873$	0	0
$874$	−4.12500e6	−0.182661
$875$	5.82109e7	2.57030
$876$	0	0
$877$	2.58279e6	0.113394	0.0566971	−	0.998391i	$-0.481943\pi$
$877$	2.58279e6	0.113394	0.0566971	+	0.998391i	$0.481943\pi$
$878$	3.41370e7	1.49447
$879$	0	0
$880$	−829414.	−0.0361048
$881$	1.66814e7	0.724090	0.362045	−	0.932161i	$-0.382079\pi$
$881$	1.66814e7	0.724090	0.362045	+	0.932161i	$0.382079\pi$
$882$	0	0
$883$	2.36384e7	1.02027	0.510137	−	0.860093i	$-0.329595\pi$
$883$	2.36384e7	1.02027	0.510137	+	0.860093i	$0.329595\pi$
$884$	1.98818e6	0.0855708
$885$	0	0
$886$	1.49632e7	0.640382
$887$	5.25660e6	0.224334	0.112167	−	0.993689i	$-0.464221\pi$
$887$	5.25660e6	0.224334	0.112167	+	0.993689i	$0.464221\pi$
$888$	0	0
$889$	−3.14345e7	−1.33399
$890$	−4.62474e6	−0.195710
$891$	0	0
$892$	652988.	0.0274785
$893$	5.32625e6	0.223508
$894$	0	0
$895$	4.88935e7	2.04030
$896$	4.53994e6	0.188921
$897$	0	0
$898$	−3.62806e7	−1.50135
$899$	2.41549e6	0.0996795
$900$	0	0
$901$	8.20763e6	0.336826
$902$	−537389.	−0.0219924
$903$	0	0
$904$	2.25461e7	0.917595
$905$	−7.66648e6	−0.311153
$906$	0	0
$907$	3.22789e7	1.30287	0.651435	−	0.758705i	$-0.274167\pi$
$907$	3.22789e7	1.30287	0.651435	+	0.758705i	$0.274167\pi$
$908$	1.25094e6	0.0503525
$909$	0	0
$910$	−1.00743e7	−0.403284
$911$	−4.20975e7	−1.68058	−0.840292	−	0.542134i	$-0.817617\pi$
$911$	−4.20975e7	−1.68058	−0.840292	+	0.542134i	$0.817617\pi$
$912$	0	0
$913$	1.10924e6	0.0440400
$914$	−1.52897e7	−0.605389
$915$	0	0
$916$	−1.28693e7	−0.506775
$917$	−1.30051e7	−0.510728
$918$	0	0
$919$	−2.19460e7	−0.857168	−0.428584	−	0.903502i	$-0.640987\pi$
$919$	−2.19460e7	−0.857168	−0.428584	+	0.903502i	$0.640987\pi$
$920$	−8.62500e7	−3.35961
$921$	0	0
$922$	−2.13767e7	−0.828157
$923$	6.97221e6	0.269380
$924$	0	0
$925$	−3.51362e7	−1.35021
$926$	3.03205e7	1.16201
$927$	0	0
$928$	1.93845e6	0.0738899
$929$	1.35921e7	0.516710	0.258355	−	0.966050i	$-0.416820\pi$
$929$	1.35921e7	0.516710	0.258355	+	0.966050i	$0.416820\pi$
$930$	0	0
$931$	182330.	0.00689421
$932$	7.05637e6	0.266098
$933$	0	0
$934$	1.43297e7	0.537489
$935$	1.61273e6	0.0603300
$936$	0	0
$937$	3.01018e7	1.12007	0.560033	−	0.828470i	$-0.310788\pi$
$937$	3.01018e7	1.12007	0.560033	+	0.828470i	$0.310788\pi$
$938$	−1.99122e7	−0.738944
$939$	0	0
$940$	2.88581e7	1.06524
$941$	1.06275e7	0.391252	0.195626	−	0.980679i	$-0.437326\pi$
$941$	1.06275e7	0.391252	0.195626	+	0.980679i	$0.437326\pi$
$942$	0	0
$943$	−3.36130e7	−1.23091
$944$	−2.01840e7	−0.737187
$945$	0	0
$946$	−576099.	−0.0209300
$947$	2.41709e7	0.875826	0.437913	−	0.899017i	$-0.355718\pi$
$947$	2.41709e7	0.875826	0.437913	+	0.899017i	$0.355718\pi$
$948$	0	0
$949$	178535.	0.00643514
$950$	−7.38925e6	−0.265639
$951$	0	0
$952$	−2.61580e7	−0.935432
$953$	−4.27043e7	−1.52314	−0.761569	−	0.648084i	$-0.775570\pi$
$953$	−4.27043e7	−1.52314	−0.761569	+	0.648084i	$0.775570\pi$
$954$	0	0
$955$	2.31888e7	0.822754
$956$	−1.85375e6	−0.0656003
$957$	0	0
$958$	3.04842e7	1.07315
$959$	4.55184e6	0.159823
$960$	0	0
$961$	−5.74655e6	−0.200724
$962$	3.57371e6	0.124503
$963$	0	0
$964$	−7.73348e6	−0.268029
$965$	−6.85193e7	−2.36862
$966$	0	0
$967$	−4.42692e7	−1.52242	−0.761211	−	0.648504i	$-0.775395\pi$
$967$	−4.42692e7	−1.52242	−0.761211	+	0.648504i	$0.775395\pi$
$968$	3.16875e7	1.08693
$969$	0	0
$970$	−6.53329e7	−2.22947
$971$	−3.88962e7	−1.32391	−0.661956	−	0.749542i	$-0.730274\pi$
$971$	−3.88962e7	−1.32391	−0.661956	+	0.749542i	$0.730274\pi$
$972$	0	0
$973$	1.92816e7	0.652922
$974$	−1.85417e7	−0.626257
$975$	0	0
$976$	9.82357e6	0.330099
$977$	2.71611e7	0.910354	0.455177	−	0.890401i	$-0.349576\pi$
$977$	2.71611e7	0.910354	0.455177	+	0.890401i	$0.349576\pi$
$978$	0	0
$979$	−145319.	−0.00484580
$980$	987881.	0.0328579
$981$	0	0
$982$	9.59904e6	0.317650
$983$	1.98048e7	0.653714	0.326857	−	0.945074i	$-0.394011\pi$
$983$	1.98048e7	0.653714	0.326857	+	0.945074i	$0.394011\pi$
$984$	0	0
$985$	−6.89538e7	−2.26448
$986$	2.42113e6	0.0793096
$987$	0	0
$988$	−404255.	−0.0131754
$989$	−3.60343e7	−1.17145
$990$	0	0
$991$	−1.44104e7	−0.466115	−0.233057	−	0.972463i	$-0.574873\pi$
$991$	−1.44104e7	−0.466115	−0.233057	+	0.972463i	$0.574873\pi$
$992$	1.83635e7	0.592483
$993$	0	0
$994$	−2.37700e7	−0.763068
$995$	6.68075e7	2.13928
$996$	0	0
$997$	3.16635e7	1.00884	0.504418	−	0.863459i	$-0.331707\pi$
$997$	3.16635e7	1.00884	0.504418	+	0.863459i	$0.331707\pi$
$998$	2.72195e7	0.865074
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.6.a.c.1.2		2
3.2	odd	2		13.6.a.a.1.1	✓	2
12.11	even	2		208.6.a.h.1.1		2
15.2	even	4		325.6.b.b.274.1		4
15.8	even	4		325.6.b.b.274.4		4
15.14	odd	2		325.6.a.b.1.2		2
21.20	even	2		637.6.a.a.1.1		2
24.5	odd	2		832.6.a.p.1.1		2
24.11	even	2		832.6.a.i.1.2		2
39.5	even	4		169.6.b.a.168.4		4
39.8	even	4		169.6.b.a.168.1		4
39.38	odd	2		169.6.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.a.1.1	✓	2	3.2	odd	2
117.6.a.c.1.2		2	1.1	even	1	trivial
169.6.a.a.1.2		2	39.38	odd	2
169.6.b.a.168.1		4	39.8	even	4
169.6.b.a.168.4		4	39.5	even	4
208.6.a.h.1.1		2	12.11	even	2
325.6.a.b.1.2		2	15.14	odd	2
325.6.b.b.274.1		4	15.2	even	4
325.6.b.b.274.4		4	15.8	even	4
637.6.a.a.1.1		2	21.20	even	2
832.6.a.i.1.2		2	24.11	even	2
832.6.a.p.1.1		2	24.5	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q+4.56155 q^{2} -11.1922 q^{4} +103.462 q^{5} +126.309 q^{7} -197.024 q^{8} +471.948 q^{10} +14.8296 q^{11} -169.000 q^{13} +576.164 q^{14} -540.582 q^{16} +1051.12 q^{17} -213.723 q^{19} -1157.97 q^{20} +67.6458 q^{22} +4231.16 q^{23} +7579.41 q^{25} -770.902 q^{26} -1413.68 q^{28} +504.955 q^{29} +4783.58 q^{31} +3838.86 q^{32} +4794.75 q^{34} +13068.2 q^{35} -4635.74 q^{37} -974.911 q^{38} -20384.5 q^{40} -7944.15 q^{41} -8516.41 q^{43} -165.976 q^{44} +19300.7 q^{46} -24921.2 q^{47} -853.113 q^{49} +34573.9 q^{50} +1891.49 q^{52} +7808.46 q^{53} +1534.30 q^{55} -24885.8 q^{56} +2303.38 q^{58} +37337.5 q^{59} -18172.2 q^{61} +21820.5 q^{62} +34809.8 q^{64} -17485.1 q^{65} -34559.9 q^{67} -11764.4 q^{68} +59611.1 q^{70} -41255.7 q^{71} -1056.42 q^{73} -21146.2 q^{74} +2392.04 q^{76} +1873.10 q^{77} -47719.3 q^{79} -55929.8 q^{80} -36237.7 q^{82} +74799.0 q^{83} +108751. q^{85} -38848.0 q^{86} -2921.78 q^{88} -9799.26 q^{89} -21346.2 q^{91} -47356.1 q^{92} -113679. q^{94} -22112.3 q^{95} -138432. q^{97} -3891.52 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} - 43 q^{4} + 42 q^{5} - 36 q^{7} - 225 q^{8} + 445 q^{10} + 376 q^{11} - 338 q^{13} + 505 q^{14} + 465 q^{16} + 2630 q^{17} - 312 q^{19} + 797 q^{20} + 226 q^{22} + 2624 q^{23} + 8232 q^{25}+ \cdots + 290 q^{98}+O(q^{100}) \) 2 * q + 5 * q^2 - 43 * q^4 + 42 * q^5 - 36 * q^7 - 225 * q^8 + 445 * q^10 + 376 * q^11 - 338 * q^13 + 505 * q^14 + 465 * q^16 + 2630 * q^17 - 312 * q^19 + 797 * q^20 + 226 * q^22 + 2624 * q^23 + 8232 * q^25 - 845 * q^26 + 3749 * q^28 + 812 * q^29 + 7720 * q^31 + 5175 * q^32 + 5487 * q^34 + 23044 * q^35 - 16858 * q^37 - 1018 * q^38 - 18665 * q^40 - 7840 * q^41 + 2420 * q^43 - 11654 * q^44 + 18596 * q^46 - 9972 * q^47 + 8684 * q^49 + 34860 * q^50 + 7267 * q^52 + 43720 * q^53 - 20664 * q^55 - 20345 * q^56 + 2438 * q^58 + 38936 * q^59 + 1984 * q^61 + 23108 * q^62 + 3217 * q^64 - 7098 * q^65 - 69928 * q^67 - 61985 * q^68 + 63985 * q^70 - 67396 * q^71 + 74412 * q^73 - 26505 * q^74 + 5518 * q^76 - 56748 * q^77 - 55296 * q^79 - 117735 * q^80 - 36192 * q^82 + 75712 * q^83 + 11710 * q^85 - 34053 * q^86 - 13026 * q^88 - 116508 * q^89 + 6084 * q^91 + 3764 * q^92 - 107125 * q^94 - 16072 * q^95 - 34756 * q^97 + 290 * q^98

Introduction

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