LMFDB - Embedded newform 117.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,6,Mod(1,117)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	117.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:	$18.7649069181$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.168897.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 100x + 256 $ x^3 - x^2 - 100*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.3
Root			$-10.6486$ 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)

$f(q)$	$=$	$q+8.64858 q^{2} +42.7979 q^{4} -92.1784 q^{5} +2.86088 q^{7} +93.3863 q^{8} +O(q^{10})$	\(q+8.64858 q^{2} +42.7979 q^{4} -92.1784 q^{5} +2.86088 q^{7} +93.3863 q^{8} -797.212 q^{10} -571.711 q^{11} +169.000 q^{13} +24.7426 q^{14} -561.873 q^{16} +855.571 q^{17} -2355.90 q^{19} -3945.04 q^{20} -4944.49 q^{22} -2509.66 q^{23} +5371.86 q^{25} +1461.61 q^{26} +122.440 q^{28} +5497.50 q^{29} -144.924 q^{31} -7847.77 q^{32} +7399.47 q^{34} -263.712 q^{35} -515.975 q^{37} -20375.2 q^{38} -8608.21 q^{40} +13928.9 q^{41} +7753.58 q^{43} -24468.0 q^{44} -21705.0 q^{46} -8344.77 q^{47} -16798.8 q^{49} +46459.0 q^{50} +7232.84 q^{52} +5976.21 q^{53} +52699.5 q^{55} +267.168 q^{56} +47545.6 q^{58} -2110.84 q^{59} -17394.2 q^{61} -1253.38 q^{62} -49892.1 q^{64} -15578.2 q^{65} -3121.05 q^{67} +36616.6 q^{68} -2280.73 q^{70} -43323.3 q^{71} +6808.31 q^{73} -4462.45 q^{74} -100828. q^{76} -1635.60 q^{77} -1280.80 q^{79} +51792.6 q^{80} +120465. q^{82} -7148.44 q^{83} -78865.2 q^{85} +67057.5 q^{86} -53390.0 q^{88} -107123. q^{89} +483.489 q^{91} -107408. q^{92} -72170.4 q^{94} +217163. q^{95} +42456.2 q^{97} -145286. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 7 q^{2} + 121 q^{4} - 56 q^{5} - 60 q^{7} - 327 q^{8}+O(q^{10}) $ 3 * q - 7 * q^2 + 121 * q^4 - 56 * q^5 - 60 * q^7 - 327 * q^8	$ 3 q - 7 q^{2} + 121 q^{4} - 56 q^{5} - 60 q^{7} - 327 q^{8} - 1291 q^{10} - 556 q^{11} + 507 q^{13} + 793 q^{14} + 2785 q^{16} - 908 q^{17} + 148 q^{19} + 767 q^{20} - 3410 q^{22} - 3624 q^{23} + 2023 q^{25} - 1183 q^{26} - 6661 q^{28} + 8758 q^{29} - 2608 q^{31} - 34871 q^{32} + 15989 q^{34} - 4348 q^{35} - 20632 q^{37} - 33786 q^{38} - 43509 q^{40} + 10998 q^{41} + 2032 q^{43} - 49790 q^{44} - 3408 q^{46} - 34260 q^{47} - 44571 q^{49} + 65038 q^{50} + 20449 q^{52} + 12570 q^{53} + 35312 q^{55} + 49287 q^{56} - 19126 q^{58} - 63948 q^{59} - 12754 q^{61} + 40340 q^{62} + 117393 q^{64} - 9464 q^{65} + 56132 q^{67} + 49175 q^{68} + 41299 q^{70} - 77580 q^{71} - 43026 q^{73} + 131519 q^{74} - 99638 q^{76} + 21052 q^{77} - 61872 q^{79} + 264683 q^{80} + 142260 q^{82} - 98092 q^{83} - 55226 q^{85} + 69897 q^{86} + 158010 q^{88} - 33694 q^{89} - 10140 q^{91} - 300864 q^{92} + 72843 q^{94} + 196560 q^{95} + 76334 q^{97} + 54636 q^{98}+O(q^{100}) $ 3 * q - 7 * q^2 + 121 * q^4 - 56 * q^5 - 60 * q^7 - 327 * q^8 - 1291 * q^10 - 556 * q^11 + 507 * q^13 + 793 * q^14 + 2785 * q^16 - 908 * q^17 + 148 * q^19 + 767 * q^20 - 3410 * q^22 - 3624 * q^23 + 2023 * q^25 - 1183 * q^26 - 6661 * q^28 + 8758 * q^29 - 2608 * q^31 - 34871 * q^32 + 15989 * q^34 - 4348 * q^35 - 20632 * q^37 - 33786 * q^38 - 43509 * q^40 + 10998 * q^41 + 2032 * q^43 - 49790 * q^44 - 3408 * q^46 - 34260 * q^47 - 44571 * q^49 + 65038 * q^50 + 20449 * q^52 + 12570 * q^53 + 35312 * q^55 + 49287 * q^56 - 19126 * q^58 - 63948 * q^59 - 12754 * q^61 + 40340 * q^62 + 117393 * q^64 - 9464 * q^65 + 56132 * q^67 + 49175 * q^68 + 41299 * q^70 - 77580 * q^71 - 43026 * q^73 + 131519 * q^74 - 99638 * q^76 + 21052 * q^77 - 61872 * q^79 + 264683 * q^80 + 142260 * q^82 - 98092 * q^83 - 55226 * q^85 + 69897 * q^86 + 158010 * q^88 - 33694 * q^89 - 10140 * q^91 - 300864 * q^92 + 72843 * q^94 + 196560 * q^95 + 76334 * q^97 + 54636 * 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 $

$2$	8.64858	1.52887	0.764433	−	0.644703i	$-0.223019\pi$
$2$	8.64858	1.52887	0.764433	+	0.644703i	$0.223019\pi$
$3$	0	0
$3$	0	0
$4$	42.7979	1.33743
$5$	−92.1784	−1.64894	−0.824469	−	0.565907i	$-0.808526\pi$
$5$	−92.1784	−1.64894	−0.824469	+	0.565907i	$0.808526\pi$
$6$	0	0
$7$	2.86088	0.0220676	0.0110338	−	0.999939i	$-0.496488\pi$
$7$	2.86088	0.0220676	0.0110338	+	0.999939i	$0.496488\pi$
$8$	93.3863	0.515892
$9$	0	0
$10$	−797.212	−2.52101
$11$	−571.711	−1.42461	−0.712304	−	0.701871i	$-0.752348\pi$
$11$	−571.711	−1.42461	−0.712304	+	0.701871i	$0.752348\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	24.7426	0.0337384
$15$	0	0
$16$	−561.873	−0.548704
$17$	855.571	0.718015	0.359008	−	0.933335i	$-0.383115\pi$
$17$	855.571	0.718015	0.359008	+	0.933335i	$0.383115\pi$
$18$	0	0
$19$	−2355.90	−1.49718	−0.748589	−	0.663034i	$-0.769268\pi$
$19$	−2355.90	−1.49718	−0.748589	+	0.663034i	$0.769268\pi$
$20$	−3945.04	−2.20535
$21$	0	0
$22$	−4944.49	−2.17804
$23$	−2509.66	−0.989227	−0.494613	−	0.869113i	$-0.664690\pi$
$23$	−2509.66	−0.989227	−0.494613	+	0.869113i	$0.664690\pi$
$24$	0	0
$25$	5371.86	1.71900
$26$	1461.61	0.424031
$27$	0	0
$28$	122.440	0.0295140
$29$	5497.50	1.21387	0.606933	−	0.794753i	$-0.292400\pi$
$29$	5497.50	1.21387	0.606933	+	0.794753i	$0.292400\pi$
$30$	0	0
$31$	−144.924	−0.0270854	−0.0135427	−	0.999908i	$-0.504311\pi$
$31$	−144.924	−0.0270854	−0.0135427	+	0.999908i	$0.504311\pi$
$32$	−7847.77	−1.35479
$33$	0	0
$34$	7399.47	1.09775
$35$	−263.712	−0.0363881
$36$	0	0
$37$	−515.975	−0.0619618	−0.0309809	−	0.999520i	$-0.509863\pi$
$37$	−515.975	−0.0619618	−0.0309809	+	0.999520i	$0.509863\pi$
$38$	−20375.2	−2.28899
$39$	0	0
$40$	−8608.21	−0.850673
$41$	13928.9	1.29407	0.647033	−	0.762462i	$-0.276009\pi$
$41$	13928.9	1.29407	0.647033	+	0.762462i	$0.276009\pi$
$42$	0	0
$43$	7753.58	0.639486	0.319743	−	0.947504i	$-0.396403\pi$
$43$	7753.58	0.639486	0.319743	+	0.947504i	$0.396403\pi$
$44$	−24468.0	−1.90532
$45$	0	0
$46$	−21705.0	−1.51240
$47$	−8344.77	−0.551023	−0.275512	−	0.961298i	$-0.588847\pi$
$47$	−8344.77	−0.551023	−0.275512	+	0.961298i	$0.588847\pi$
$48$	0	0
$49$	−16798.8	−0.999513
$50$	46459.0	2.62812
$51$	0	0
$52$	7232.84	0.370937
$53$	5976.21	0.292238	0.146119	−	0.989267i	$-0.453322\pi$
$53$	5976.21	0.292238	0.146119	+	0.989267i	$0.453322\pi$
$54$	0	0
$55$	52699.5	2.34909
$56$	267.168	0.0113845
$57$	0	0
$58$	47545.6	1.85584
$59$	−2110.84	−0.0789451	−0.0394726	−	0.999221i	$-0.512568\pi$
$59$	−2110.84	−0.0789451	−0.0394726	+	0.999221i	$0.512568\pi$
$60$	0	0
$61$	−17394.2	−0.598522	−0.299261	−	0.954171i	$-0.596740\pi$
$61$	−17394.2	−0.598522	−0.299261	+	0.954171i	$0.596740\pi$
$62$	−1253.38	−0.0414099
$63$	0	0
$64$	−49892.1	−1.52259
$65$	−15578.2	−0.457333
$66$	0	0
$67$	−3121.05	−0.0849402	−0.0424701	−	0.999098i	$-0.513523\pi$
$67$	−3121.05	−0.0849402	−0.0424701	+	0.999098i	$0.513523\pi$
$68$	36616.6	0.960298
$69$	0	0
$70$	−2280.73	−0.0556326
$71$	−43323.3	−1.01994	−0.509971	−	0.860191i	$-0.670344\pi$
$71$	−43323.3	−1.01994	−0.509971	+	0.860191i	$0.670344\pi$
$72$	0	0
$73$	6808.31	0.149531	0.0747657	−	0.997201i	$-0.476179\pi$
$73$	6808.31	0.149531	0.0747657	+	0.997201i	$0.476179\pi$
$74$	−4462.45	−0.0947314
$75$	0	0
$76$	−100828.	−2.00238
$77$	−1635.60	−0.0314377
$78$	0	0
$79$	−1280.80	−0.0230895	−0.0115447	−	0.999933i	$-0.503675\pi$
$79$	−1280.80	−0.0230895	−0.0115447	+	0.999933i	$0.503675\pi$
$80$	51792.6	0.904779
$81$	0	0
$82$	120465.	1.97846
$83$	−7148.44	−0.113898	−0.0569490	−	0.998377i	$-0.518137\pi$
$83$	−7148.44	−0.113898	−0.0569490	+	0.998377i	$0.518137\pi$
$84$	0	0
$85$	−78865.2	−1.18396
$86$	67057.5	0.977690
$87$	0	0
$88$	−53390.0	−0.734943
$89$	−107123.	−1.43353	−0.716765	−	0.697315i	$-0.754378\pi$
$89$	−107123.	−1.43353	−0.716765	+	0.697315i	$0.754378\pi$
$90$	0	0
$91$	483.489	0.00612045
$92$	−107408.	−1.32303
$93$	0	0
$94$	−72170.4	−0.842441
$95$	217163.	2.46875
$96$	0	0
$97$	42456.2	0.458155	0.229077	−	0.973408i	$-0.426429\pi$
$97$	42456.2	0.458155	0.229077	+	0.973408i	$0.426429\pi$
$98$	−145286.	−1.52812
$99$	0	0
$100$	229904.	2.29904
$101$	87947.3	0.857866	0.428933	−	0.903336i	$-0.358890\pi$
$101$	87947.3	0.857866	0.428933	+	0.903336i	$0.358890\pi$
$102$	0	0
$103$	96885.8	0.899844	0.449922	−	0.893068i	$-0.351452\pi$
$103$	96885.8	0.899844	0.449922	+	0.893068i	$0.351452\pi$
$104$	15782.3	0.143083
$105$	0	0
$106$	51685.7	0.446792
$107$	−106509.	−0.899346	−0.449673	−	0.893193i	$-0.648460\pi$
$107$	−106509.	−0.899346	−0.449673	+	0.893193i	$0.648460\pi$
$108$	0	0
$109$	−142520.	−1.14897	−0.574485	−	0.818515i	$-0.694798\pi$
$109$	−142520.	−1.14897	−0.574485	+	0.818515i	$0.694798\pi$
$110$	455775.	3.59144
$111$	0	0
$112$	−1607.45	−0.0121086
$113$	209540.	1.54373	0.771864	−	0.635787i	$-0.219324\pi$
$113$	209540.	1.54373	0.771864	+	0.635787i	$0.219324\pi$
$114$	0	0
$115$	231337.	1.63117
$116$	235282.	1.62346
$117$	0	0
$118$	−18255.8	−0.120697
$119$	2447.69	0.0158449
$120$	0	0
$121$	165803.	1.02951
$122$	−150435.	−0.915061
$123$	0	0
$124$	−6202.42	−0.0362249
$125$	−207112.	−1.18558
$126$	0	0
$127$	−54468.2	−0.299663	−0.149832	−	0.988712i	$-0.547873\pi$
$127$	−54468.2	−0.299663	−0.149832	+	0.988712i	$0.547873\pi$
$128$	−180367.	−0.973043
$129$	0	0
$130$	−134729.	−0.699201
$131$	−258252.	−1.31482	−0.657408	−	0.753535i	$-0.728347\pi$
$131$	−258252.	−1.31482	−0.657408	+	0.753535i	$0.728347\pi$
$132$	0	0
$133$	−6739.97	−0.0330391
$134$	−26992.6	−0.129862
$135$	0	0
$136$	79898.6	0.370418
$137$	−293754.	−1.33716	−0.668580	−	0.743641i	$-0.733097\pi$
$137$	−293754.	−1.33716	−0.668580	+	0.743641i	$0.733097\pi$
$138$	0	0
$139$	180400.	0.791955	0.395977	−	0.918260i	$-0.370406\pi$
$139$	180400.	0.791955	0.395977	+	0.918260i	$0.370406\pi$
$140$	−11286.3	−0.0486667
$141$	0	0
$142$	−374685.	−1.55936
$143$	−96619.2	−0.395115
$144$	0	0
$145$	−506751.	−2.00159
$146$	58882.2	0.228614
$147$	0	0
$148$	−22082.6	−0.0828698
$149$	−516009.	−1.90411	−0.952054	−	0.305930i	$-0.901032\pi$
$149$	−516009.	−1.90411	−0.952054	+	0.305930i	$0.901032\pi$
$150$	0	0
$151$	−333770.	−1.19126	−0.595628	−	0.803260i	$-0.703097\pi$
$151$	−333770.	−1.19126	−0.595628	+	0.803260i	$0.703097\pi$
$152$	−220009.	−0.772381
$153$	0	0
$154$	−14145.6	−0.0480640
$155$	13358.8	0.0446621
$156$	0	0
$157$	−265860.	−0.860804	−0.430402	−	0.902637i	$-0.641628\pi$
$157$	−265860.	−0.860804	−0.430402	+	0.902637i	$0.641628\pi$
$158$	−11077.1	−0.0353007
$159$	0	0
$160$	723395.	2.23396
$161$	−7179.86	−0.0218299
$162$	0	0
$163$	−231417.	−0.682223	−0.341112	−	0.940023i	$-0.610803\pi$
$163$	−231417.	−0.682223	−0.341112	+	0.940023i	$0.610803\pi$
$164$	596127.	1.73073
$165$	0	0
$166$	−61823.8	−0.174135
$167$	656226.	1.82080	0.910400	−	0.413730i	$-0.135774\pi$
$167$	656226.	1.82080	0.910400	+	0.413730i	$0.135774\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	−682071.	−1.81012
$171$	0	0
$172$	331837.	0.855271
$173$	−45111.6	−0.114597	−0.0572985	−	0.998357i	$-0.518249\pi$
$173$	−45111.6	−0.114597	−0.0572985	+	0.998357i	$0.518249\pi$
$174$	0	0
$175$	15368.3	0.0379341
$176$	321229.	0.781688
$177$	0	0
$178$	−926460.	−2.19168
$179$	−521846.	−1.21733	−0.608667	−	0.793426i	$-0.708295\pi$
$179$	−521846.	−1.21733	−0.608667	+	0.793426i	$0.708295\pi$
$180$	0	0
$181$	−122780.	−0.278569	−0.139284	−	0.990252i	$-0.544480\pi$
$181$	−122780.	−0.278569	−0.139284	+	0.990252i	$0.544480\pi$
$182$	4181.50	0.00935736
$183$	0	0
$184$	−234368.	−0.510334
$185$	47561.7	0.102171
$186$	0	0
$187$	−489140.	−1.02289
$188$	−357139.	−0.736957
$189$	0	0
$190$	1.87815e6	3.77440
$191$	−350208.	−0.694613	−0.347306	−	0.937752i	$-0.612904\pi$
$191$	−350208.	−0.694613	−0.347306	+	0.937752i	$0.612904\pi$
$192$	0	0
$193$	876316.	1.69343	0.846716	−	0.532046i	$-0.178577\pi$
$193$	876316.	1.69343	0.846716	+	0.532046i	$0.178577\pi$
$194$	367186.	0.700457
$195$	0	0
$196$	−718954.	−1.33678
$197$	−306582.	−0.562835	−0.281418	−	0.959585i	$-0.590805\pi$
$197$	−306582.	−0.562835	−0.281418	+	0.959585i	$0.590805\pi$
$198$	0	0
$199$	−327849.	−0.586869	−0.293434	−	0.955979i	$-0.594798\pi$
$199$	−327849.	−0.586869	−0.293434	+	0.955979i	$0.594798\pi$
$200$	501659.	0.886815
$201$	0	0
$202$	760619.	1.31156
$203$	15727.7	0.0267871
$204$	0	0
$205$	−1.28394e6	−2.13384
$206$	837924.	1.37574
$207$	0	0
$208$	−94956.6	−0.152183
$209$	1.34690e6	2.13289
$210$	0	0
$211$	−509483.	−0.787814	−0.393907	−	0.919150i	$-0.628877\pi$
$211$	−509483.	−0.787814	−0.393907	+	0.919150i	$0.628877\pi$
$212$	255769.	0.390849
$213$	0	0
$214$	−921151.	−1.37498
$215$	−714713.	−1.05447
$216$	0	0
$217$	−414.610	−0.000597709 0
$218$	−1.23259e6	−1.75662
$219$	0	0
$220$	2.25543e6	3.14175
$221$	144591.	0.199142
$222$	0	0
$223$	−15061.4	−0.0202816	−0.0101408	−	0.999949i	$-0.503228\pi$
$223$	−15061.4	−0.0202816	−0.0101408	+	0.999949i	$0.503228\pi$
$224$	−22451.6	−0.0298969
$225$	0	0
$226$	1.81222e6	2.36016
$227$	−273713.	−0.352558	−0.176279	−	0.984340i	$-0.556406\pi$
$227$	−273713.	−0.352558	−0.176279	+	0.984340i	$0.556406\pi$
$228$	0	0
$229$	743056.	0.936339	0.468169	−	0.883639i	$-0.344914\pi$
$229$	743056.	0.936339	0.468169	+	0.883639i	$0.344914\pi$
$230$	2.00073e6	2.49385
$231$	0	0
$232$	513392.	0.626223
$233$	−1.12122e6	−1.35301	−0.676506	−	0.736437i	$-0.736507\pi$
$233$	−1.12122e6	−1.35301	−0.676506	+	0.736437i	$0.736507\pi$
$234$	0	0
$235$	769208.	0.908603
$236$	−90339.5	−0.105584
$237$	0	0
$238$	21169.0	0.0242247
$239$	−279838.	−0.316892	−0.158446	−	0.987368i	$-0.550648\pi$
$239$	−279838.	−0.316892	−0.158446	+	0.987368i	$0.550648\pi$
$240$	0	0
$241$	816889.	0.905983	0.452992	−	0.891515i	$-0.350357\pi$
$241$	816889.	0.905983	0.452992	+	0.891515i	$0.350357\pi$
$242$	1.43396e6	1.57398
$243$	0	0
$244$	−744436.	−0.800484
$245$	1.54849e6	1.64813
$246$	0	0
$247$	−398148.	−0.415242
$248$	−13533.9	−0.0139731
$249$	0	0
$250$	−1.79123e6	−1.81259
$251$	766697.	0.768139	0.384069	−	0.923304i	$-0.374522\pi$
$251$	766697.	0.768139	0.384069	+	0.923304i	$0.374522\pi$
$252$	0	0
$253$	1.43480e6	1.40926
$254$	−471072.	−0.458145
$255$	0	0
$256$	36629.4	0.0349325
$257$	1.82413e6	1.72276	0.861379	−	0.507963i	$-0.169601\pi$
$257$	1.82413e6	1.72276	0.861379	+	0.507963i	$0.169601\pi$
$258$	0	0
$259$	−1476.14	−0.00136735
$260$	−666712.	−0.611653
$261$	0	0
$262$	−2.23351e6	−2.01018
$263$	294257.	0.262323	0.131162	−	0.991361i	$-0.458129\pi$
$263$	294257.	0.262323	0.131162	+	0.991361i	$0.458129\pi$
$264$	0	0
$265$	−550878.	−0.481882
$266$	−58291.1	−0.0505124
$267$	0	0
$268$	−133574.	−0.113602
$269$	−289738.	−0.244132	−0.122066	−	0.992522i	$-0.538952\pi$
$269$	−289738.	−0.244132	−0.122066	+	0.992522i	$0.538952\pi$
$270$	0	0
$271$	1.63179e6	1.34971	0.674855	−	0.737950i	$-0.264206\pi$
$271$	1.63179e6	1.34971	0.674855	+	0.737950i	$0.264206\pi$
$272$	−480722.	−0.393978
$273$	0	0
$274$	−2.54056e6	−2.04434
$275$	−3.07116e6	−2.44889
$276$	0	0
$277$	398868.	0.312341	0.156171	−	0.987730i	$-0.450085\pi$
$277$	398868.	0.312341	0.156171	+	0.987730i	$0.450085\pi$
$278$	1.56021e6	1.21079
$279$	0	0
$280$	−24627.1	−0.0187723
$281$	−274611.	−0.207468	−0.103734	−	0.994605i	$-0.533079\pi$
$281$	−274611.	−0.207468	−0.103734	+	0.994605i	$0.533079\pi$
$282$	0	0
$283$	−794683.	−0.589831	−0.294916	−	0.955523i	$-0.595292\pi$
$283$	−794683.	−0.589831	−0.294916	+	0.955523i	$0.595292\pi$
$284$	−1.85415e6	−1.36411
$285$	0	0
$286$	−835619.	−0.604078
$287$	39848.9	0.0285570
$288$	0	0
$289$	−687856.	−0.484454
$290$	−4.38268e6	−3.06016
$291$	0	0
$292$	291381.	0.199988
$293$	1.55323e6	1.05698	0.528490	−	0.848940i	$-0.322759\pi$
$293$	1.55323e6	1.05698	0.528490	+	0.848940i	$0.322759\pi$
$294$	0	0
$295$	194574.	0.130176
$296$	−48185.0	−0.0319656
$297$	0	0
$298$	−4.46274e6	−2.91113
$299$	−424133.	−0.274362
$300$	0	0
$301$	22182.1	0.0141119
$302$	−2.88664e6	−1.82127
$303$	0	0
$304$	1.32372e6	0.821508
$305$	1.60337e6	0.986926
$306$	0	0
$307$	2.07136e6	1.25433	0.627163	−	0.778888i	$-0.284216\pi$
$307$	2.07136e6	1.25433	0.627163	+	0.778888i	$0.284216\pi$
$308$	−70000.2	−0.0420458
$309$	0	0
$310$	115535.	0.0682824
$311$	2.50827e6	1.47053	0.735263	−	0.677782i	$-0.237059\pi$
$311$	2.50827e6	1.47053	0.735263	+	0.677782i	$0.237059\pi$
$312$	0	0
$313$	832384.	0.480245	0.240123	−	0.970743i	$-0.422812\pi$
$313$	832384.	0.480245	0.240123	+	0.970743i	$0.422812\pi$
$314$	−2.29931e6	−1.31606
$315$	0	0
$316$	−54815.6	−0.0308806
$317$	1.76537e6	0.986708	0.493354	−	0.869829i	$-0.335771\pi$
$317$	1.76537e6	0.986708	0.493354	+	0.869829i	$0.335771\pi$
$318$	0	0
$319$	−3.14299e6	−1.72928
$320$	4.59897e6	2.51065
$321$	0	0
$322$	−62095.5	−0.0333750
$323$	−2.01564e6	−1.07500
$324$	0	0
$325$	907845.	0.476764
$326$	−2.00143e6	−1.04303
$327$	0	0
$328$	1.30077e6	0.667598
$329$	−23873.4	−0.0121598
$330$	0	0
$331$	−285068.	−0.143014	−0.0715070	−	0.997440i	$-0.522781\pi$
$331$	−285068.	−0.143014	−0.0715070	+	0.997440i	$0.522781\pi$
$332$	−305938.	−0.152331
$333$	0	0
$334$	5.67542e6	2.78376
$335$	287693.	0.140061
$336$	0	0
$337$	1.96932e6	0.944584	0.472292	−	0.881442i	$-0.343427\pi$
$337$	1.96932e6	0.944584	0.472292	+	0.881442i	$0.343427\pi$
$338$	247012.	0.117605
$339$	0	0
$340$	−3.37526e6	−1.58347
$341$	82854.5	0.0385860
$342$	0	0
$343$	−96142.4	−0.0441245
$344$	724079.	0.329906
$345$	0	0
$346$	−390151.	−0.175204
$347$	−1.89753e6	−0.845988	−0.422994	−	0.906133i	$-0.639021\pi$
$347$	−1.89753e6	−0.845988	−0.422994	+	0.906133i	$0.639021\pi$
$348$	0	0
$349$	−365808.	−0.160764	−0.0803822	−	0.996764i	$-0.525614\pi$
$349$	−365808.	−0.160764	−0.0803822	+	0.996764i	$0.525614\pi$
$350$	132914.	0.0579962
$351$	0	0
$352$	4.48666e6	1.93004
$353$	−1.56672e6	−0.669196	−0.334598	−	0.942361i	$-0.608601\pi$
$353$	−1.56672e6	−0.669196	−0.334598	+	0.942361i	$0.608601\pi$
$354$	0	0
$355$	3.99348e6	1.68182
$356$	−4.58463e6	−1.91725
$357$	0	0
$358$	−4.51322e6	−1.86114
$359$	4.03307e6	1.65158	0.825789	−	0.563979i	$-0.190730\pi$
$359$	4.03307e6	1.65158	0.825789	+	0.563979i	$0.190730\pi$
$360$	0	0
$361$	3.07418e6	1.24154
$362$	−1.06187e6	−0.425894
$363$	0	0
$364$	20692.3	0.00818570
$365$	−627579.	−0.246568
$366$	0	0
$367$	683844.	0.265028	0.132514	−	0.991181i	$-0.457695\pi$
$367$	683844.	0.265028	0.132514	+	0.991181i	$0.457695\pi$
$368$	1.41011e6	0.542793
$369$	0	0
$370$	411341.	0.156206
$371$	17097.2	0.00644899
$372$	0	0
$373$	−3.09158e6	−1.15056	−0.575279	−	0.817957i	$-0.695107\pi$
$373$	−3.09158e6	−1.15056	−0.575279	+	0.817957i	$0.695107\pi$
$374$	−4.23036e6	−1.56386
$375$	0	0
$376$	−779288.	−0.284268
$377$	929078.	0.336666
$378$	0	0
$379$	−5.24600e6	−1.87599	−0.937995	−	0.346649i	$-0.887319\pi$
$379$	−5.24600e6	−1.87599	−0.937995	+	0.346649i	$0.887319\pi$
$380$	9.29414e6	3.30179
$381$	0	0
$382$	−3.02880e6	−1.06197
$383$	−1.57009e6	−0.546926	−0.273463	−	0.961883i	$-0.588169\pi$
$383$	−1.57009e6	−0.546926	−0.273463	+	0.961883i	$0.588169\pi$
$384$	0	0
$385$	150767.	0.0518388
$386$	7.57889e6	2.58903
$387$	0	0
$388$	1.81704e6	0.612751
$389$	−1.54482e6	−0.517611	−0.258805	−	0.965929i	$-0.583329\pi$
$389$	−1.54482e6	−0.517611	−0.258805	+	0.965929i	$0.583329\pi$
$390$	0	0
$391$	−2.14719e6	−0.710280
$392$	−1.56878e6	−0.515640
$393$	0	0
$394$	−2.65150e6	−0.860500
$395$	118062.	0.0380731
$396$	0	0
$397$	5.06711e6	1.61356	0.806778	−	0.590855i	$-0.201210\pi$
$397$	5.06711e6	1.61356	0.806778	+	0.590855i	$0.201210\pi$
$398$	−2.83543e6	−0.897244
$399$	0	0
$400$	−3.01831e6	−0.943221
$401$	138120.	0.0428939	0.0214469	−	0.999770i	$-0.493173\pi$
$401$	138120.	0.0428939	0.0214469	+	0.999770i	$0.493173\pi$
$402$	0	0
$403$	−24492.1	−0.00751213
$404$	3.76396e6	1.14734
$405$	0	0
$406$	136022.	0.0409539
$407$	294989.	0.0882713
$408$	0	0
$409$	−4.13317e6	−1.22173	−0.610865	−	0.791735i	$-0.709178\pi$
$409$	−4.13317e6	−1.22173	−0.610865	+	0.791735i	$0.709178\pi$
$410$	−1.11043e7	−3.26235
$411$	0	0
$412$	4.14651e6	1.20348
$413$	−6038.87	−0.00174213
$414$	0	0
$415$	658932.	0.187811
$416$	−1.32627e6	−0.375750
$417$	0	0
$418$	1.16487e7	3.26091
$419$	2.96703e6	0.825633	0.412817	−	0.910814i	$-0.364545\pi$
$419$	2.96703e6	0.825633	0.412817	+	0.910814i	$0.364545\pi$
$420$	0	0
$421$	−2.98125e6	−0.819773	−0.409887	−	0.912137i	$-0.634432\pi$
$421$	−2.98125e6	−0.819773	−0.409887	+	0.912137i	$0.634432\pi$
$422$	−4.40630e6	−1.20446
$423$	0	0
$424$	558096.	0.150763
$425$	4.59601e6	1.23426
$426$	0	0
$427$	−49762.8	−0.0132080
$428$	−4.55836e6	−1.20282
$429$	0	0
$430$	−6.18125e6	−1.61215
$431$	3.34790e6	0.868119	0.434059	−	0.900884i	$-0.357081\pi$
$431$	3.34790e6	0.868119	0.434059	+	0.900884i	$0.357081\pi$
$432$	0	0
$433$	−3.76782e6	−0.965764	−0.482882	−	0.875686i	$-0.660410\pi$
$433$	−3.76782e6	−0.965764	−0.482882	+	0.875686i	$0.660410\pi$
$434$	−3585.78	−0.000913818 0
$435$	0	0
$436$	−6.09954e6	−1.53667
$437$	5.91252e6	1.48105
$438$	0	0
$439$	5.63075e6	1.39446	0.697228	−	0.716849i	$-0.254416\pi$
$439$	5.63075e6	1.39446	0.697228	+	0.716849i	$0.254416\pi$
$440$	4.92141e6	1.21188
$441$	0	0
$442$	1.25051e6	0.304461
$443$	−3.69803e6	−0.895284	−0.447642	−	0.894213i	$-0.647736\pi$
$443$	−3.69803e6	−0.895284	−0.447642	+	0.894213i	$0.647736\pi$
$444$	0	0
$445$	9.87441e6	2.36380
$446$	−130260.	−0.0310079
$447$	0	0
$448$	−142735.	−0.0335998
$449$	505514.	0.118336	0.0591681	−	0.998248i	$-0.481155\pi$
$449$	505514.	0.118336	0.0591681	+	0.998248i	$0.481155\pi$
$450$	0	0
$451$	−7.96330e6	−1.84354
$452$	8.96787e6	2.06464
$453$	0	0
$454$	−2.36723e6	−0.539015
$455$	−44567.3	−0.0100922
$456$	0	0
$457$	783627.	0.175517	0.0877584	−	0.996142i	$-0.472030\pi$
$457$	783627.	0.175517	0.0877584	+	0.996142i	$0.472030\pi$
$458$	6.42638e6	1.43154
$459$	0	0
$460$	9.90073e6	2.18159
$461$	−4.64856e6	−1.01875	−0.509373	−	0.860546i	$-0.670123\pi$
$461$	−4.64856e6	−1.01875	−0.509373	+	0.860546i	$0.670123\pi$
$462$	0	0
$463$	2.64872e6	0.574226	0.287113	−	0.957897i	$-0.407304\pi$
$463$	2.64872e6	0.574226	0.287113	+	0.957897i	$0.407304\pi$
$464$	−3.08890e6	−0.666053
$465$	0	0
$466$	−9.69698e6	−2.06858
$467$	2.42172e6	0.513845	0.256923	−	0.966432i	$-0.417291\pi$
$467$	2.42172e6	0.513845	0.256923	+	0.966432i	$0.417291\pi$
$468$	0	0
$469$	−8928.96	−0.00187443
$470$	6.65255e6	1.38913
$471$	0	0
$472$	−197124.	−0.0407271
$473$	−4.43281e6	−0.911017
$474$	0	0
$475$	−1.26556e7	−2.57364
$476$	104756.	0.0211915
$477$	0	0
$478$	−2.42020e6	−0.484486
$479$	−5.58315e6	−1.11184	−0.555918	−	0.831237i	$-0.687633\pi$
$479$	−5.58315e6	−1.11184	−0.555918	+	0.831237i	$0.687633\pi$
$480$	0	0
$481$	−87199.7	−0.0171851
$482$	7.06492e6	1.38513
$483$	0	0
$484$	7.09602e6	1.37690
$485$	−3.91355e6	−0.755468
$486$	0	0
$487$	7.25183e6	1.38556	0.692780	−	0.721149i	$-0.256386\pi$
$487$	7.25183e6	1.38556	0.692780	+	0.721149i	$0.256386\pi$
$488$	−1.62438e6	−0.308773
$489$	0	0
$490$	1.33922e7	2.51978
$491$	−4.17410e6	−0.781373	−0.390687	−	0.920524i	$-0.627762\pi$
$491$	−4.17410e6	−0.781373	−0.390687	+	0.920524i	$0.627762\pi$
$492$	0	0
$493$	4.70350e6	0.871574
$494$	−3.44341e6	−0.634850
$495$	0	0
$496$	81428.7	0.0148619
$497$	−123943.	−0.0225077
$498$	0	0
$499$	−7.83551e6	−1.40869	−0.704345	−	0.709857i	$-0.748759\pi$
$499$	−7.83551e6	−1.40869	−0.704345	+	0.709857i	$0.748759\pi$
$500$	−8.86396e6	−1.58563
$501$	0	0
$502$	6.63084e6	1.17438
$503$	3.72420e6	0.656315	0.328158	−	0.944623i	$-0.393572\pi$
$503$	3.72420e6	0.656315	0.328158	+	0.944623i	$0.393572\pi$
$504$	0	0
$505$	−8.10685e6	−1.41457
$506$	1.24090e7	2.15457
$507$	0	0
$508$	−2.33112e6	−0.400780
$509$	6.76675e6	1.15767	0.578836	−	0.815444i	$-0.303507\pi$
$509$	6.76675e6	1.15767	0.578836	+	0.815444i	$0.303507\pi$
$510$	0	0
$511$	19477.8	0.00329980
$512$	6.08853e6	1.02645
$513$	0	0
$514$	1.57762e7	2.63387
$515$	−8.93078e6	−1.48379
$516$	0	0
$517$	4.77080e6	0.784992
$518$	−12766.5	−0.00209049
$519$	0	0
$520$	−1.45479e6	−0.235934
$521$	−9.08380e6	−1.46613	−0.733066	−	0.680157i	$-0.761912\pi$
$521$	−9.08380e6	−1.46613	−0.733066	+	0.680157i	$0.761912\pi$
$522$	0	0
$523$	−1.09284e7	−1.74703	−0.873515	−	0.486796i	$-0.838165\pi$
$523$	−1.09284e7	−1.74703	−0.873515	+	0.486796i	$0.838165\pi$
$524$	−1.10526e7	−1.75848
$525$	0	0
$526$	2.54490e6	0.401057
$527$	−123992.	−0.0194477
$528$	0	0
$529$	−137934.	−0.0214305
$530$	−4.76431e6	−0.736733
$531$	0	0
$532$	−288456.	−0.0441877
$533$	2.35398e6	0.358910
$534$	0	0
$535$	9.81783e6	1.48297
$536$	−291463.	−0.0438200
$537$	0	0
$538$	−2.50582e6	−0.373245
$539$	9.60408e6	1.42391
$540$	0	0
$541$	1.17853e6	0.173120	0.0865598	−	0.996247i	$-0.472413\pi$
$541$	1.17853e6	0.173120	0.0865598	+	0.996247i	$0.472413\pi$
$542$	1.41126e7	2.06353
$543$	0	0
$544$	−6.71432e6	−0.972758
$545$	1.31372e7	1.89458
$546$	0	0
$547$	9.57702e6	1.36855	0.684277	−	0.729222i	$-0.260118\pi$
$547$	9.57702e6	1.36855	0.684277	+	0.729222i	$0.260118\pi$
$548$	−1.25721e7	−1.78836
$549$	0	0
$550$	−2.65611e7	−3.74403
$551$	−1.29516e7	−1.81737
$552$	0	0
$553$	−3664.22	−0.000509529 0
$554$	3.44964e6	0.477528
$555$	0	0
$556$	7.72075e6	1.05919
$557$	−2.46873e6	−0.337159	−0.168580	−	0.985688i	$-0.553918\pi$
$557$	−2.46873e6	−0.337159	−0.168580	+	0.985688i	$0.553918\pi$
$558$	0	0
$559$	1.31036e6	0.177362
$560$	148173.	0.0199663
$561$	0	0
$562$	−2.37499e6	−0.317191
$563$	−7.81681e6	−1.03934	−0.519671	−	0.854367i	$-0.673945\pi$
$563$	−7.81681e6	−1.03934	−0.519671	+	0.854367i	$0.673945\pi$
$564$	0	0
$565$	−1.93151e7	−2.54551
$566$	−6.87288e6	−0.901774
$567$	0	0
$568$	−4.04581e6	−0.526180
$569$	9.64303e6	1.24863	0.624314	−	0.781174i	$-0.285379\pi$
$569$	9.64303e6	1.24863	0.624314	+	0.781174i	$0.285379\pi$
$570$	0	0
$571$	1.12378e7	1.44242	0.721211	−	0.692715i	$-0.243586\pi$
$571$	1.12378e7	1.44242	0.721211	+	0.692715i	$0.243586\pi$
$572$	−4.13510e6	−0.528440
$573$	0	0
$574$	344636.	0.0436598
$575$	−1.34816e7	−1.70048
$576$	0	0
$577$	−5.44674e6	−0.681078	−0.340539	−	0.940230i	$-0.610610\pi$
$577$	−5.44674e6	−0.681078	−0.340539	+	0.940230i	$0.610610\pi$
$578$	−5.94897e6	−0.740666
$579$	0	0
$580$	−2.16879e7	−2.67699
$581$	−20450.9	−0.00251346
$582$	0	0
$583$	−3.41667e6	−0.416324
$584$	635803.	0.0771420
$585$	0	0
$586$	1.34332e7	1.61598
$587$	−9.96240e6	−1.19335	−0.596676	−	0.802482i	$-0.703512\pi$
$587$	−9.96240e6	−1.19335	−0.596676	+	0.802482i	$0.703512\pi$
$588$	0	0
$589$	341426.	0.0405516
$590$	1.68279e6	0.199021
$591$	0	0
$592$	289912.	0.0339987
$593$	3.17929e6	0.371273	0.185636	−	0.982619i	$-0.440565\pi$
$593$	3.17929e6	0.371273	0.185636	+	0.982619i	$0.440565\pi$
$594$	0	0
$595$	−225624.	−0.0261272
$596$	−2.20841e7	−2.54662
$597$	0	0
$598$	−3.66815e6	−0.419463
$599$	1.66017e7	1.89054	0.945271	−	0.326285i	$-0.105797\pi$
$599$	1.66017e7	1.89054	0.945271	+	0.326285i	$0.105797\pi$
$600$	0	0
$601$	1.05771e6	0.119449	0.0597244	−	0.998215i	$-0.480978\pi$
$601$	1.05771e6	0.119449	0.0597244	+	0.998215i	$0.480978\pi$
$602$	191844.	0.0215753
$603$	0	0
$604$	−1.42847e7	−1.59323
$605$	−1.52835e7	−1.69759
$606$	0	0
$607$	2.42026e6	0.266619	0.133309	−	0.991074i	$-0.457440\pi$
$607$	2.42026e6	0.266619	0.133309	+	0.991074i	$0.457440\pi$
$608$	1.84886e7	2.02836
$609$	0	0
$610$	1.38669e7	1.50888
$611$	−1.41027e6	−0.152826
$612$	0	0
$613$	110410.	0.0118675	0.00593373	−	0.999982i	$-0.498111\pi$
$613$	110410.	0.0118675	0.00593373	+	0.999982i	$0.498111\pi$
$614$	1.79144e7	1.91770
$615$	0	0
$616$	−152743.	−0.0162184
$617$	1.06544e7	1.12672	0.563361	−	0.826211i	$-0.309508\pi$
$617$	1.06544e7	1.12672	0.563361	+	0.826211i	$0.309508\pi$
$618$	0	0
$619$	−1.30456e7	−1.36848	−0.684240	−	0.729256i	$-0.739866\pi$
$619$	−1.30456e7	−1.36848	−0.684240	+	0.729256i	$0.739866\pi$
$620$	571730.	0.0597326
$621$	0	0
$622$	2.16929e7	2.24824
$623$	−306466.	−0.0316346
$624$	0	0
$625$	2.30421e6	0.235951
$626$	7.19894e6	0.734231
$627$	0	0
$628$	−1.13783e7	−1.15127
$629$	−441453.	−0.0444895
$630$	0	0
$631$	−1.53156e6	−0.153130	−0.0765651	−	0.997065i	$-0.524395\pi$
$631$	−1.53156e6	−0.153130	−0.0765651	+	0.997065i	$0.524395\pi$
$632$	−119609.	−0.0119117
$633$	0	0
$634$	1.52680e7	1.50855
$635$	5.02079e6	0.494126
$636$	0	0
$637$	−2.83900e6	−0.277215
$638$	−2.71824e7	−2.64384
$639$	0	0
$640$	1.66259e7	1.60449
$641$	−8.09995e6	−0.778641	−0.389321	−	0.921102i	$-0.627290\pi$
$641$	−8.09995e6	−0.778641	−0.389321	+	0.921102i	$0.627290\pi$
$642$	0	0
$643$	−5.29175e6	−0.504745	−0.252373	−	0.967630i	$-0.581211\pi$
$643$	−5.29175e6	−0.504745	−0.252373	+	0.967630i	$0.581211\pi$
$644$	−307283.	−0.0291960
$645$	0	0
$646$	−1.74324e7	−1.64353
$647$	−1.80626e7	−1.69637	−0.848185	−	0.529700i	$-0.822305\pi$
$647$	−1.80626e7	−1.69637	−0.848185	+	0.529700i	$0.822305\pi$
$648$	0	0
$649$	1.20679e6	0.112466
$650$	7.85156e6	0.728908
$651$	0	0
$652$	−9.90416e6	−0.912428
$653$	7.10212e6	0.651786	0.325893	−	0.945407i	$-0.394335\pi$
$653$	7.10212e6	0.651786	0.325893	+	0.945407i	$0.394335\pi$
$654$	0	0
$655$	2.38052e7	2.16805
$656$	−7.82627e6	−0.710060
$657$	0	0
$658$	−206471.	−0.0185907
$659$	−1.41430e7	−1.26861	−0.634303	−	0.773084i	$-0.718713\pi$
$659$	−1.41430e7	−1.26861	−0.634303	+	0.773084i	$0.718713\pi$
$660$	0	0
$661$	−675807.	−0.0601615	−0.0300808	−	0.999547i	$-0.509576\pi$
$661$	−675807.	−0.0601615	−0.0300808	+	0.999547i	$0.509576\pi$
$662$	−2.46543e6	−0.218649
$663$	0	0
$664$	−667566.	−0.0587590
$665$	621280.	0.0544795
$666$	0	0
$667$	−1.37969e7	−1.20079
$668$	2.80851e7	2.43520
$669$	0	0
$670$	2.48814e6	0.214135
$671$	9.94447e6	0.852659
$672$	0	0
$673$	2.66714e6	0.226991	0.113496	−	0.993539i	$-0.463795\pi$
$673$	2.66714e6	0.226991	0.113496	+	0.993539i	$0.463795\pi$
$674$	1.70318e7	1.44414
$675$	0	0
$676$	1.22235e6	0.102880
$677$	1.20319e7	1.00893	0.504465	−	0.863432i	$-0.331690\pi$
$677$	1.20319e7	1.00893	0.504465	+	0.863432i	$0.331690\pi$
$678$	0	0
$679$	121462.	0.0101104
$680$	−7.36493e6	−0.610796
$681$	0	0
$682$	716573.	0.0589929
$683$	1.83758e7	1.50728	0.753640	−	0.657287i	$-0.228296\pi$
$683$	1.83758e7	1.50728	0.753640	+	0.657287i	$0.228296\pi$
$684$	0	0
$685$	2.70778e7	2.20489
$686$	−831495.	−0.0674604
$687$	0	0
$688$	−4.35653e6	−0.350889
$689$	1.00998e6	0.0810521
$690$	0	0
$691$	1.33412e7	1.06292	0.531459	−	0.847084i	$-0.321644\pi$
$691$	1.33412e7	1.06292	0.531459	+	0.847084i	$0.321644\pi$
$692$	−1.93068e6	−0.153266
$693$	0	0
$694$	−1.64109e7	−1.29340
$695$	−1.66290e7	−1.30588
$696$	0	0
$697$	1.19171e7	0.929159
$698$	−3.16372e6	−0.245787
$699$	0	0
$700$	657730.	0.0507344
$701$	−2.51952e6	−0.193652	−0.0968262	−	0.995301i	$-0.530869\pi$
$701$	−2.51952e6	−0.193652	−0.0968262	+	0.995301i	$0.530869\pi$
$702$	0	0
$703$	1.21559e6	0.0927679
$704$	2.85239e7	2.16909
$705$	0	0
$706$	−1.35499e7	−1.02311
$707$	251607.	0.0189310
$708$	0	0
$709$	2.51441e7	1.87854	0.939272	−	0.343174i	$-0.111502\pi$
$709$	2.51441e7	1.87854	0.939272	+	0.343174i	$0.111502\pi$
$710$	3.45379e7	2.57128
$711$	0	0
$712$	−1.00038e7	−0.739546
$713$	363709.	0.0267936
$714$	0	0
$715$	8.90621e6	0.651520
$716$	−2.23339e7	−1.62810
$717$	0	0
$718$	3.48803e7	2.52504
$719$	−415865.	−0.0300006	−0.0150003	−	0.999887i	$-0.504775\pi$
$719$	−415865.	−0.0300006	−0.0150003	+	0.999887i	$0.504775\pi$
$720$	0	0
$721$	277179.	0.0198574
$722$	2.65873e7	1.89815
$723$	0	0
$724$	−5.25474e6	−0.372567
$725$	2.95318e7	2.08663
$726$	0	0
$727$	−1.53987e7	−1.08056	−0.540278	−	0.841486i	$-0.681681\pi$
$727$	−1.53987e7	−1.08056	−0.540278	+	0.841486i	$0.681681\pi$
$728$	45151.3	0.00315749
$729$	0	0
$730$	−5.42767e6	−0.376970
$731$	6.63374e6	0.459161
$732$	0	0
$733$	−1.82125e7	−1.25202	−0.626008	−	0.779817i	$-0.715312\pi$
$733$	−1.82125e7	−1.25202	−0.626008	+	0.779817i	$0.715312\pi$
$734$	5.91428e6	0.405193
$735$	0	0
$736$	1.96953e7	1.34019
$737$	1.78434e6	0.121007
$738$	0	0
$739$	−1.88107e7	−1.26705	−0.633524	−	0.773723i	$-0.718392\pi$
$739$	−1.88107e7	−1.26705	−0.633524	+	0.773723i	$0.718392\pi$
$740$	2.03554e6	0.136647
$741$	0	0
$742$	147867.	0.00985964
$743$	−2.61292e7	−1.73642	−0.868209	−	0.496199i	$-0.834729\pi$
$743$	−2.61292e7	−1.73642	−0.868209	+	0.496199i	$0.834729\pi$
$744$	0	0
$745$	4.75649e7	3.13976
$746$	−2.67378e7	−1.75905
$747$	0	0
$748$	−2.09341e7	−1.36805
$749$	−304710.	−0.0198464
$750$	0	0
$751$	−1.90282e7	−1.23111	−0.615557	−	0.788092i	$-0.711069\pi$
$751$	−1.90282e7	−1.23111	−0.615557	+	0.788092i	$0.711069\pi$
$752$	4.68871e6	0.302349
$753$	0	0
$754$	8.03520e6	0.514717
$755$	3.07664e7	1.96431
$756$	0	0
$757$	−1.91639e7	−1.21547	−0.607734	−	0.794140i	$-0.707921\pi$
$757$	−1.91639e7	−1.21547	−0.607734	+	0.794140i	$0.707921\pi$
$758$	−4.53705e7	−2.86814
$759$	0	0
$760$	2.02801e7	1.27361
$761$	468091.	0.0293001	0.0146500	−	0.999893i	$-0.495337\pi$
$761$	468091.	0.0293001	0.0146500	+	0.999893i	$0.495337\pi$
$762$	0	0
$763$	−407732.	−0.0253550
$764$	−1.49882e7	−0.928999
$765$	0	0
$766$	−1.35791e7	−0.836177
$767$	−356732.	−0.0218954
$768$	0	0
$769$	−1.92729e7	−1.17525	−0.587627	−	0.809132i	$-0.699938\pi$
$769$	−1.92729e7	−1.17525	−0.587627	+	0.809132i	$0.699938\pi$
$770$	1.30392e6	0.0792546
$771$	0	0
$772$	3.75045e7	2.26485
$773$	−2.42083e7	−1.45719	−0.728593	−	0.684947i	$-0.759825\pi$
$773$	−2.42083e7	−1.45719	−0.728593	+	0.684947i	$0.759825\pi$
$774$	0	0
$775$	−778509.	−0.0465596
$776$	3.96483e6	0.236358
$777$	0	0
$778$	−1.33605e7	−0.791358
$779$	−3.28151e7	−1.93745
$780$	0	0
$781$	2.47684e7	1.45302
$782$	−1.85702e7	−1.08592
$783$	0	0
$784$	9.43881e6	0.548437
$785$	2.45066e7	1.41941
$786$	0	0
$787$	−1.48087e7	−0.852276	−0.426138	−	0.904658i	$-0.640126\pi$
$787$	−1.48087e7	−0.852276	−0.426138	+	0.904658i	$0.640126\pi$
$788$	−1.31211e7	−0.752755
$789$	0	0
$790$	1.02107e6	0.0582087
$791$	599470.	0.0340664
$792$	0	0
$793$	−2.93962e6	−0.166000
$794$	4.38233e7	2.46691
$795$	0	0
$796$	−1.40312e7	−0.784898
$797$	2.90994e7	1.62270	0.811351	−	0.584560i	$-0.198733\pi$
$797$	2.90994e7	1.62270	0.811351	+	0.584560i	$0.198733\pi$
$798$	0	0
$799$	−7.13954e6	−0.395643
$800$	−4.21571e7	−2.32887
$801$	0	0
$802$	1.19454e6	0.0655790
$803$	−3.89239e6	−0.213023
$804$	0	0
$805$	661828.	0.0359961
$806$	−211822.	−0.0114850
$807$	0	0
$808$	8.21308e6	0.442566
$809$	−3.37792e7	−1.81459	−0.907295	−	0.420496i	$-0.861856\pi$
$809$	−3.37792e7	−1.81459	−0.907295	+	0.420496i	$0.861856\pi$
$810$	0	0
$811$	−2.42986e7	−1.29727	−0.648633	−	0.761101i	$-0.724659\pi$
$811$	−2.42986e7	−1.29727	−0.648633	+	0.761101i	$0.724659\pi$
$812$	673113.	0.0358260
$813$	0	0
$814$	2.55123e6	0.134955
$815$	2.13317e7	1.12494
$816$	0	0
$817$	−1.82667e7	−0.957425
$818$	−3.57460e7	−1.86786
$819$	0	0
$820$	−5.49500e7	−2.85386
$821$	7.42442e6	0.384419	0.192209	−	0.981354i	$-0.438435\pi$
$821$	7.42442e6	0.384419	0.192209	+	0.981354i	$0.438435\pi$
$822$	0	0
$823$	−2.98540e7	−1.53640	−0.768198	−	0.640212i	$-0.778847\pi$
$823$	−2.98540e7	−1.53640	−0.768198	+	0.640212i	$0.778847\pi$
$824$	9.04781e6	0.464222
$825$	0	0
$826$	−52227.6	−0.00266348
$827$	−2.47154e7	−1.25662	−0.628311	−	0.777962i	$-0.716253\pi$
$827$	−2.47154e7	−1.25662	−0.628311	+	0.777962i	$0.716253\pi$
$828$	0	0
$829$	−1.14800e7	−0.580169	−0.290084	−	0.957001i	$-0.593683\pi$
$829$	−1.14800e7	−0.580169	−0.290084	+	0.957001i	$0.593683\pi$
$830$	5.69882e6	0.287137
$831$	0	0
$832$	−8.43176e6	−0.422289
$833$	−1.43726e7	−0.717665
$834$	0	0
$835$	−6.04899e7	−3.00239
$836$	5.76443e7	2.85260
$837$	0	0
$838$	2.56606e7	1.26228
$839$	−2.41188e7	−1.18291	−0.591454	−	0.806339i	$-0.701446\pi$
$839$	−2.41188e7	−1.18291	−0.591454	+	0.806339i	$0.701446\pi$
$840$	0	0
$841$	9.71140e6	0.473469
$842$	−2.57836e7	−1.25332
$843$	0	0
$844$	−2.18048e7	−1.05365
$845$	−2.63271e6	−0.126841
$846$	0	0
$847$	474343.	0.0227187
$848$	−3.35787e6	−0.160352
$849$	0	0
$850$	3.97489e7	1.88703
$851$	1.29492e6	0.0612943
$852$	0	0
$853$	−2.02681e6	−0.0953764	−0.0476882	−	0.998862i	$-0.515185\pi$
$853$	−2.02681e6	−0.0953764	−0.0476882	+	0.998862i	$0.515185\pi$
$854$	−430378.	−0.0201932
$855$	0	0
$856$	−9.94649e6	−0.463965
$857$	−1.52859e7	−0.710950	−0.355475	−	0.934686i	$-0.615681\pi$
$857$	−1.52859e7	−0.710950	−0.355475	+	0.934686i	$0.615681\pi$
$858$	0	0
$859$	1.78567e7	0.825693	0.412846	−	0.910801i	$-0.364535\pi$
$859$	1.78567e7	0.825693	0.412846	+	0.910801i	$0.364535\pi$
$860$	−3.05882e7	−1.41029
$861$	0	0
$862$	2.89546e7	1.32724
$863$	1.12315e7	0.513347	0.256673	−	0.966498i	$-0.417374\pi$
$863$	1.12315e7	0.513347	0.256673	+	0.966498i	$0.417374\pi$
$864$	0	0
$865$	4.15832e6	0.188963
$866$	−3.25863e7	−1.47652
$867$	0	0
$868$	−17744.4	−0.000799397 0
$869$	732249.	0.0328934
$870$	0	0
$871$	−527457.	−0.0235582
$872$	−1.33094e7	−0.592744
$873$	0	0
$874$	5.11349e7	2.26433
$875$	−592524.	−0.0261629
$876$	0	0
$877$	1.75233e6	0.0769336	0.0384668	−	0.999260i	$-0.487753\pi$
$877$	1.75233e6	0.0769336	0.0384668	+	0.999260i	$0.487753\pi$
$878$	4.86980e7	2.13194
$879$	0	0
$880$	−2.96104e7	−1.28896
$881$	1.04009e7	0.451474	0.225737	−	0.974188i	$-0.427521\pi$
$881$	1.04009e7	0.451474	0.225737	+	0.974188i	$0.427521\pi$
$882$	0	0
$883$	4.25645e7	1.83715	0.918577	−	0.395241i	$-0.129339\pi$
$883$	4.25645e7	1.83715	0.918577	+	0.395241i	$0.129339\pi$
$884$	6.18821e6	0.266339
$885$	0	0
$886$	−3.19827e7	−1.36877
$887$	−201500.	−0.00859935	−0.00429968	−	0.999991i	$-0.501369\pi$
$887$	−201500.	−0.00859935	−0.00429968	+	0.999991i	$0.501369\pi$
$888$	0	0
$889$	−155827.	−0.00661285
$890$	8.53996e7	3.61394
$891$	0	0
$892$	−644596.	−0.0271253
$893$	1.96595e7	0.824980
$894$	0	0
$895$	4.81029e7	2.00731
$896$	−516009.	−0.0214727
$897$	0	0
$898$	4.37198e6	0.180920
$899$	−796718.	−0.0328780
$900$	0	0
$901$	5.11307e6	0.209831
$902$	−6.88712e7	−2.81852
$903$	0	0
$904$	1.95682e7	0.796397
$905$	1.13177e7	0.459342
$906$	0	0
$907$	1.47930e7	0.597088	0.298544	−	0.954396i	$-0.403499\pi$
$907$	1.47930e7	0.597088	0.298544	+	0.954396i	$0.403499\pi$
$908$	−1.17143e7	−0.471523
$909$	0	0
$910$	−385444.	−0.0154297
$911$	3.50070e7	1.39752	0.698762	−	0.715354i	$-0.253735\pi$
$911$	3.50070e7	1.39752	0.698762	+	0.715354i	$0.253735\pi$
$912$	0	0
$913$	4.08684e6	0.162260
$914$	6.77726e6	0.268342
$915$	0	0
$916$	3.18012e7	1.25229
$917$	−738828.	−0.0290148
$918$	0	0
$919$	4.24526e7	1.65812	0.829060	−	0.559160i	$-0.188876\pi$
$919$	4.24526e7	1.65812	0.829060	+	0.559160i	$0.188876\pi$
$920$	2.16037e7	0.841509
$921$	0	0
$922$	−4.02034e7	−1.55753
$923$	−7.32164e6	−0.282881
$924$	0	0
$925$	−2.77175e6	−0.106512
$926$	2.29076e7	0.877916
$927$	0	0
$928$	−4.31431e7	−1.64453
$929$	3.95487e7	1.50347	0.751733	−	0.659468i	$-0.229218\pi$
$929$	3.95487e7	1.50347	0.751733	+	0.659468i	$0.229218\pi$
$930$	0	0
$931$	3.95764e7	1.49645
$932$	−4.79859e7	−1.80957
$933$	0	0
$934$	2.09445e7	0.785601
$935$	4.50881e7	1.68668
$936$	0	0
$937$	−4.62850e7	−1.72223	−0.861116	−	0.508409i	$-0.830234\pi$
$937$	−4.62850e7	−1.72223	−0.861116	+	0.508409i	$0.830234\pi$
$938$	−77222.8	−0.00286575
$939$	0	0
$940$	3.29205e7	1.21520
$941$	−5.74465e6	−0.211490	−0.105745	−	0.994393i	$-0.533723\pi$
$941$	−5.74465e6	−0.211490	−0.105745	+	0.994393i	$0.533723\pi$
$942$	0	0
$943$	−3.49568e7	−1.28013
$944$	1.18602e6	0.0433175
$945$	0	0
$946$	−3.83375e7	−1.39282
$947$	−5.15966e7	−1.86959	−0.934795	−	0.355187i	$-0.884417\pi$
$947$	−5.15966e7	−1.86959	−0.934795	+	0.355187i	$0.884417\pi$
$948$	0	0
$949$	1.15060e6	0.0414725
$950$	−1.09453e8	−3.93476
$951$	0	0
$952$	228581.	0.00817424
$953$	2.18333e7	0.778729	0.389365	−	0.921084i	$-0.372695\pi$
$953$	2.18333e7	0.778729	0.389365	+	0.921084i	$0.372695\pi$
$954$	0	0
$955$	3.22816e7	1.14537
$956$	−1.19765e7	−0.423822
$957$	0	0
$958$	−4.82863e7	−1.69985
$959$	−840398.	−0.0295079
$960$	0	0
$961$	−2.86081e7	−0.999266
$962$	−754154.	−0.0262738
$963$	0	0
$964$	3.49611e7	1.21169
$965$	−8.07774e7	−2.79236
$966$	0	0
$967$	−7.30509e6	−0.251223	−0.125611	−	0.992080i	$-0.540089\pi$
$967$	−7.30509e6	−0.251223	−0.125611	+	0.992080i	$0.540089\pi$
$968$	1.54837e7	0.531114
$969$	0	0
$970$	−3.38466e7	−1.15501
$971$	1.62933e6	0.0554576	0.0277288	−	0.999615i	$-0.491173\pi$
$971$	1.62933e6	0.0554576	0.0277288	+	0.999615i	$0.491173\pi$
$972$	0	0
$973$	516105.	0.0174765
$974$	6.27180e7	2.11834
$975$	0	0
$976$	9.77335e6	0.328412
$977$	2.55515e7	0.856407	0.428204	−	0.903682i	$-0.359147\pi$
$977$	2.55515e7	0.856407	0.428204	+	0.903682i	$0.359147\pi$
$978$	0	0
$979$	6.12433e7	2.04222
$980$	6.62720e7	2.20427
$981$	0	0
$982$	−3.61000e7	−1.19462
$983$	8.48095e6	0.279937	0.139969	−	0.990156i	$-0.455300\pi$
$983$	8.48095e6	0.279937	0.139969	+	0.990156i	$0.455300\pi$
$984$	0	0
$985$	2.82603e7	0.928080
$986$	4.06786e7	1.33252
$987$	0	0
$988$	−1.70399e7	−0.555359
$989$	−1.94589e7	−0.632597
$990$	0	0
$991$	−2.74483e7	−0.887833	−0.443916	−	0.896068i	$-0.646411\pi$
$991$	−2.74483e7	−0.887833	−0.443916	+	0.896068i	$0.646411\pi$
$992$	1.13733e6	0.0366949
$993$	0	0
$994$	−1.07193e6	−0.0344113
$995$	3.02206e7	0.967710
$996$	0	0
$997$	−2.26765e7	−0.722501	−0.361251	−	0.932469i	$-0.617650\pi$
$997$	−2.26765e7	−0.722501	−0.361251	+	0.932469i	$0.617650\pi$
$998$	−6.77660e7	−2.15370
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.6.a.d.1.3		3
3.2	odd	2		13.6.a.b.1.1	✓	3
12.11	even	2		208.6.a.j.1.2		3
15.2	even	4		325.6.b.c.274.2		6
15.8	even	4		325.6.b.c.274.5		6
15.14	odd	2		325.6.a.c.1.3		3
21.20	even	2		637.6.a.b.1.1		3
24.5	odd	2		832.6.a.s.1.2		3
24.11	even	2		832.6.a.t.1.2		3
39.5	even	4		169.6.b.b.168.5		6
39.8	even	4		169.6.b.b.168.2		6
39.38	odd	2		169.6.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.6.a.b.1.1	✓	3	3.2	odd	2
117.6.a.d.1.3		3	1.1	even	1	trivial
169.6.a.b.1.3		3	39.38	odd	2
169.6.b.b.168.2		6	39.8	even	4
169.6.b.b.168.5		6	39.5	even	4
208.6.a.j.1.2		3	12.11	even	2
325.6.a.c.1.3		3	15.14	odd	2
325.6.b.c.274.2		6	15.2	even	4
325.6.b.c.274.5		6	15.8	even	4
637.6.a.b.1.1		3	21.20	even	2
832.6.a.s.1.2		3	24.5	odd	2
832.6.a.t.1.2		3	24.11	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q+8.64858 q^{2} +42.7979 q^{4} -92.1784 q^{5} +2.86088 q^{7} +93.3863 q^{8} +O(q^{10})\)	\(q+8.64858 q^{2} +42.7979 q^{4} -92.1784 q^{5} +2.86088 q^{7} +93.3863 q^{8} -797.212 q^{10} -571.711 q^{11} +169.000 q^{13} +24.7426 q^{14} -561.873 q^{16} +855.571 q^{17} -2355.90 q^{19} -3945.04 q^{20} -4944.49 q^{22} -2509.66 q^{23} +5371.86 q^{25} +1461.61 q^{26} +122.440 q^{28} +5497.50 q^{29} -144.924 q^{31} -7847.77 q^{32} +7399.47 q^{34} -263.712 q^{35} -515.975 q^{37} -20375.2 q^{38} -8608.21 q^{40} +13928.9 q^{41} +7753.58 q^{43} -24468.0 q^{44} -21705.0 q^{46} -8344.77 q^{47} -16798.8 q^{49} +46459.0 q^{50} +7232.84 q^{52} +5976.21 q^{53} +52699.5 q^{55} +267.168 q^{56} +47545.6 q^{58} -2110.84 q^{59} -17394.2 q^{61} -1253.38 q^{62} -49892.1 q^{64} -15578.2 q^{65} -3121.05 q^{67} +36616.6 q^{68} -2280.73 q^{70} -43323.3 q^{71} +6808.31 q^{73} -4462.45 q^{74} -100828. q^{76} -1635.60 q^{77} -1280.80 q^{79} +51792.6 q^{80} +120465. q^{82} -7148.44 q^{83} -78865.2 q^{85} +67057.5 q^{86} -53390.0 q^{88} -107123. q^{89} +483.489 q^{91} -107408. q^{92} -72170.4 q^{94} +217163. q^{95} +42456.2 q^{97} -145286. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 7 q^{2} + 121 q^{4} - 56 q^{5} - 60 q^{7} - 327 q^{8}+O(q^{10}) \) 3 * q - 7 * q^2 + 121 * q^4 - 56 * q^5 - 60 * q^7 - 327 * q^8	\( 3 q - 7 q^{2} + 121 q^{4} - 56 q^{5} - 60 q^{7} - 327 q^{8} - 1291 q^{10} - 556 q^{11} + 507 q^{13} + 793 q^{14} + 2785 q^{16} - 908 q^{17} + 148 q^{19} + 767 q^{20} - 3410 q^{22} - 3624 q^{23} + 2023 q^{25} - 1183 q^{26} - 6661 q^{28} + 8758 q^{29} - 2608 q^{31} - 34871 q^{32} + 15989 q^{34} - 4348 q^{35} - 20632 q^{37} - 33786 q^{38} - 43509 q^{40} + 10998 q^{41} + 2032 q^{43} - 49790 q^{44} - 3408 q^{46} - 34260 q^{47} - 44571 q^{49} + 65038 q^{50} + 20449 q^{52} + 12570 q^{53} + 35312 q^{55} + 49287 q^{56} - 19126 q^{58} - 63948 q^{59} - 12754 q^{61} + 40340 q^{62} + 117393 q^{64} - 9464 q^{65} + 56132 q^{67} + 49175 q^{68} + 41299 q^{70} - 77580 q^{71} - 43026 q^{73} + 131519 q^{74} - 99638 q^{76} + 21052 q^{77} - 61872 q^{79} + 264683 q^{80} + 142260 q^{82} - 98092 q^{83} - 55226 q^{85} + 69897 q^{86} + 158010 q^{88} - 33694 q^{89} - 10140 q^{91} - 300864 q^{92} + 72843 q^{94} + 196560 q^{95} + 76334 q^{97} + 54636 q^{98}+O(q^{100}) \) 3 * q - 7 * q^2 + 121 * q^4 - 56 * q^5 - 60 * q^7 - 327 * q^8 - 1291 * q^10 - 556 * q^11 + 507 * q^13 + 793 * q^14 + 2785 * q^16 - 908 * q^17 + 148 * q^19 + 767 * q^20 - 3410 * q^22 - 3624 * q^23 + 2023 * q^25 - 1183 * q^26 - 6661 * q^28 + 8758 * q^29 - 2608 * q^31 - 34871 * q^32 + 15989 * q^34 - 4348 * q^35 - 20632 * q^37 - 33786 * q^38 - 43509 * q^40 + 10998 * q^41 + 2032 * q^43 - 49790 * q^44 - 3408 * q^46 - 34260 * q^47 - 44571 * q^49 + 65038 * q^50 + 20449 * q^52 + 12570 * q^53 + 35312 * q^55 + 49287 * q^56 - 19126 * q^58 - 63948 * q^59 - 12754 * q^61 + 40340 * q^62 + 117393 * q^64 - 9464 * q^65 + 56132 * q^67 + 49175 * q^68 + 41299 * q^70 - 77580 * q^71 - 43026 * q^73 + 131519 * q^74 - 99638 * q^76 + 21052 * q^77 - 61872 * q^79 + 264683 * q^80 + 142260 * q^82 - 98092 * q^83 - 55226 * q^85 + 69897 * q^86 + 158010 * q^88 - 33694 * q^89 - 10140 * q^91 - 300864 * q^92 + 72843 * q^94 + 196560 * q^95 + 76334 * q^97 + 54636 * q^98

Introduction

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