LMFDB - Embedded newform 891.6.a.f.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,6,Mod(1,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, 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("891.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	891.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:	$142.901983453$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	891.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.33614 q^{2} -13.1979 q^{4} -59.3925 q^{5} +60.8126 q^{7} -195.984 q^{8} +O(q^{10})$	\(q+4.33614 q^{2} -13.1979 q^{4} -59.3925 q^{5} +60.8126 q^{7} -195.984 q^{8} -257.534 q^{10} -121.000 q^{11} -23.3315 q^{13} +263.692 q^{14} -427.482 q^{16} +520.434 q^{17} +2153.91 q^{19} +783.857 q^{20} -524.673 q^{22} +1650.86 q^{23} +402.472 q^{25} -101.169 q^{26} -802.599 q^{28} +1884.22 q^{29} +1592.58 q^{31} +4417.88 q^{32} +2256.68 q^{34} -3611.81 q^{35} -1682.52 q^{37} +9339.64 q^{38} +11640.0 q^{40} +12648.6 q^{41} -14157.6 q^{43} +1596.95 q^{44} +7158.35 q^{46} +25660.9 q^{47} -13108.8 q^{49} +1745.17 q^{50} +307.927 q^{52} -19169.7 q^{53} +7186.50 q^{55} -11918.3 q^{56} +8170.23 q^{58} -32588.3 q^{59} -39543.6 q^{61} +6905.65 q^{62} +32836.0 q^{64} +1385.72 q^{65} -53124.7 q^{67} -6868.65 q^{68} -15661.3 q^{70} +37842.9 q^{71} +53753.3 q^{73} -7295.64 q^{74} -28427.1 q^{76} -7358.32 q^{77} -3801.71 q^{79} +25389.2 q^{80} +54845.9 q^{82} -15752.5 q^{83} -30909.9 q^{85} -61389.3 q^{86} +23714.1 q^{88} -78068.0 q^{89} -1418.85 q^{91} -21787.9 q^{92} +111269. q^{94} -127926. q^{95} -12141.7 q^{97} -56841.7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) $ 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8	$ 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) $ 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8 - 600 * q^10 - 2783 * q^11 - 1871 * q^13 + 1329 * q^14 + 3584 * q^16 + 267 * q^17 - 3641 * q^19 + 1917 * q^20 + 8292 * q^23 + 10049 * q^25 + 9570 * q^26 + 3793 * q^28 + 5970 * q^29 - 9542 * q^31 + 3831 * q^32 - 2982 * q^34 + 3240 * q^35 - 16007 * q^37 - 1221 * q^38 - 40635 * q^40 - 12030 * q^41 - 25943 * q^43 - 38720 * q^44 - 77004 * q^46 - 9756 * q^47 + 6990 * q^49 - 101805 * q^50 - 144446 * q^52 + 53919 * q^53 - 4356 * q^55 + 16602 * q^56 - 95367 * q^58 + 20310 * q^59 - 100247 * q^61 - 15297 * q^62 - 84577 * q^64 - 20931 * q^65 - 84956 * q^67 - 168471 * q^68 - 212292 * q^70 + 36093 * q^71 - 173444 * q^73 - 86619 * q^74 - 340334 * q^76 + 20207 * q^77 - 123113 * q^79 - 15123 * q^80 - 199983 * q^82 - 30672 * q^83 - 268335 * q^85 + 211260 * q^86 - 25773 * q^88 + 32514 * q^89 - 328021 * q^91 + 196731 * q^92 - 230262 * q^94 + 325926 * q^95 - 357002 * q^97 - 214464 * 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 $

$2$	4.33614	0.766528	0.383264	−	0.923639i	$-0.374800\pi$
$2$	4.33614	0.766528	0.383264	+	0.923639i	$0.374800\pi$
$3$	0	0
$3$	0	0
$4$	−13.1979	−0.412435
$5$	−59.3925	−1.06245	−0.531223	−	0.847232i	$-0.678267\pi$
$5$	−59.3925	−1.06245	−0.531223	+	0.847232i	$0.678267\pi$
$6$	0	0
$7$	60.8126	0.469081	0.234541	−	0.972106i	$-0.424641\pi$
$7$	60.8126	0.469081	0.234541	+	0.972106i	$0.424641\pi$
$8$	−195.984	−1.08267
$9$	0	0
$10$	−257.534	−0.814395
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−23.3315	−0.0382899	−0.0191450	−	0.999817i	$-0.506094\pi$
$13$	−23.3315	−0.0382899	−0.0191450	+	0.999817i	$0.506094\pi$
$14$	263.692	0.359564
$15$	0	0
$16$	−427.482	−0.417463
$17$	520.434	0.436761	0.218380	−	0.975864i	$-0.429923\pi$
$17$	520.434	0.436761	0.218380	+	0.975864i	$0.429923\pi$
$18$	0	0
$19$	2153.91	1.36881	0.684405	−	0.729102i	$-0.260062\pi$
$19$	2153.91	1.36881	0.684405	+	0.729102i	$0.260062\pi$
$20$	783.857	0.438189
$21$	0	0
$22$	−524.673	−0.231117
$23$	1650.86	0.650714	0.325357	−	0.945591i	$-0.394516\pi$
$23$	1650.86	0.650714	0.325357	+	0.945591i	$0.394516\pi$
$24$	0	0
$25$	402.472	0.128791
$26$	−101.169	−0.0293503
$27$	0	0
$28$	−802.599	−0.193465
$29$	1884.22	0.416041	0.208021	−	0.978124i	$-0.433298\pi$
$29$	1884.22	0.416041	0.208021	+	0.978124i	$0.433298\pi$
$30$	0	0
$31$	1592.58	0.297644	0.148822	−	0.988864i	$-0.452452\pi$
$31$	1592.58	0.297644	0.148822	+	0.988864i	$0.452452\pi$
$32$	4417.88	0.762674
$33$	0	0
$34$	2256.68	0.334789
$35$	−3611.81	−0.498374
$36$	0	0
$37$	−1682.52	−0.202049	−0.101024	−	0.994884i	$-0.532212\pi$
$37$	−1682.52	−0.202049	−0.101024	+	0.994884i	$0.532212\pi$
$38$	9339.64	1.04923
$39$	0	0
$40$	11640.0	1.15028
$41$	12648.6	1.17512	0.587559	−	0.809182i	$-0.300089\pi$
$41$	12648.6	1.17512	0.587559	+	0.809182i	$0.300089\pi$
$42$	0	0
$43$	−14157.6	−1.16766	−0.583832	−	0.811874i	$-0.698447\pi$
$43$	−14157.6	−1.16766	−0.583832	+	0.811874i	$0.698447\pi$
$44$	1596.95	0.124354
$45$	0	0
$46$	7158.35	0.498790
$47$	25660.9	1.69444	0.847222	−	0.531239i	$-0.178273\pi$
$47$	25660.9	1.69444	0.847222	+	0.531239i	$0.178273\pi$
$48$	0	0
$49$	−13108.8	−0.779963
$50$	1745.17	0.0987219
$51$	0	0
$52$	307.927	0.0157921
$53$	−19169.7	−0.937401	−0.468700	−	0.883357i	$-0.655278\pi$
$53$	−19169.7	−0.937401	−0.468700	+	0.883357i	$0.655278\pi$
$54$	0	0
$55$	7186.50	0.320339
$56$	−11918.3	−0.507861
$57$	0	0
$58$	8170.23	0.318907
$59$	−32588.3	−1.21880	−0.609398	−	0.792864i	$-0.708589\pi$
$59$	−32588.3	−1.21880	−0.609398	+	0.792864i	$0.708589\pi$
$60$	0	0
$61$	−39543.6	−1.36066	−0.680332	−	0.732904i	$-0.738165\pi$
$61$	−39543.6	−1.36066	−0.680332	+	0.732904i	$0.738165\pi$
$62$	6905.65	0.228153
$63$	0	0
$64$	32836.0	1.00207
$65$	1385.72	0.0406809
$66$	0	0
$67$	−53124.7	−1.44581	−0.722903	−	0.690950i	$-0.757193\pi$
$67$	−53124.7	−1.44581	−0.722903	+	0.690950i	$0.757193\pi$
$68$	−6868.65	−0.180135
$69$	0	0
$70$	−15661.3	−0.382017
$71$	37842.9	0.890919	0.445459	−	0.895302i	$-0.353040\pi$
$71$	37842.9	0.890919	0.445459	+	0.895302i	$0.353040\pi$
$72$	0	0
$73$	53753.3	1.18059	0.590293	−	0.807189i	$-0.299012\pi$
$73$	53753.3	1.18059	0.590293	+	0.807189i	$0.299012\pi$
$74$	−7295.64	−0.154876
$75$	0	0
$76$	−28427.1	−0.564545
$77$	−7358.32	−0.141433
$78$	0	0
$79$	−3801.71	−0.0685348	−0.0342674	−	0.999413i	$-0.510910\pi$
$79$	−3801.71	−0.0685348	−0.0342674	+	0.999413i	$0.510910\pi$
$80$	25389.2	0.443532
$81$	0	0
$82$	54845.9	0.900761
$83$	−15752.5	−0.250988	−0.125494	−	0.992094i	$-0.540052\pi$
$83$	−15752.5	−0.250988	−0.125494	+	0.992094i	$0.540052\pi$
$84$	0	0
$85$	−30909.9	−0.464035
$86$	−61389.3	−0.895048
$87$	0	0
$88$	23714.1	0.326438
$89$	−78068.0	−1.04472	−0.522358	−	0.852726i	$-0.674947\pi$
$89$	−78068.0	−1.04472	−0.522358	+	0.852726i	$0.674947\pi$
$90$	0	0
$91$	−1418.85	−0.0179611
$92$	−21787.9	−0.268377
$93$	0	0
$94$	111269.	1.29884
$95$	−127926.	−1.45429
$96$	0	0
$97$	−12141.7	−0.131024	−0.0655121	−	0.997852i	$-0.520868\pi$
$97$	−12141.7	−0.131024	−0.0655121	+	0.997852i	$0.520868\pi$
$98$	−56841.7	−0.597863
$99$	0	0
$100$	−5311.79	−0.0531179
$101$	146047.	1.42459	0.712294	−	0.701882i	$-0.247656\pi$
$101$	146047.	1.42459	0.712294	+	0.701882i	$0.247656\pi$
$102$	0	0
$103$	89470.0	0.830968	0.415484	−	0.909600i	$-0.363612\pi$
$103$	89470.0	0.830968	0.415484	+	0.909600i	$0.363612\pi$
$104$	4572.61	0.0414554
$105$	0	0
$106$	−83122.4	−0.718544
$107$	−44547.9	−0.376156	−0.188078	−	0.982154i	$-0.560226\pi$
$107$	−44547.9	−0.376156	−0.188078	+	0.982154i	$0.560226\pi$
$108$	0	0
$109$	69893.8	0.563472	0.281736	−	0.959492i	$-0.409090\pi$
$109$	69893.8	0.563472	0.281736	+	0.959492i	$0.409090\pi$
$110$	31161.6	0.245549
$111$	0	0
$112$	−25996.3	−0.195824
$113$	−158867.	−1.17041	−0.585203	−	0.810887i	$-0.698985\pi$
$113$	−158867.	−1.17041	−0.585203	+	0.810887i	$0.698985\pi$
$114$	0	0
$115$	−98048.6	−0.691348
$116$	−24867.8	−0.171590
$117$	0	0
$118$	−141307.	−0.934241
$119$	31649.0	0.204876
$120$	0	0
$121$	14641.0	0.0909091
$122$	−171466.	−1.04299
$123$	0	0
$124$	−21018.7	−0.122759
$125$	161698.	0.925612
$126$	0	0
$127$	−129876.	−0.714526	−0.357263	−	0.934004i	$-0.616290\pi$
$127$	−129876.	−0.714526	−0.357263	+	0.934004i	$0.616290\pi$
$128$	1009.12	0.00544400
$129$	0	0
$130$	6008.66	0.0311831
$131$	−15748.8	−0.0801804	−0.0400902	−	0.999196i	$-0.512765\pi$
$131$	−15748.8	−0.0801804	−0.0400902	+	0.999196i	$0.512765\pi$
$132$	0	0
$133$	130985.	0.642083
$134$	−230356.	−1.10825
$135$	0	0
$136$	−101997.	−0.472868
$137$	−270842.	−1.23286	−0.616432	−	0.787408i	$-0.711422\pi$
$137$	−270842.	−1.23286	−0.616432	+	0.787408i	$0.711422\pi$
$138$	0	0
$139$	−315182.	−1.38364	−0.691822	−	0.722069i	$-0.743192\pi$
$139$	−315182.	−1.38364	−0.691822	+	0.722069i	$0.743192\pi$
$140$	47668.4	0.205547
$141$	0	0
$142$	164092.	0.682914
$143$	2823.11	0.0115448
$144$	0	0
$145$	−111909.	−0.442021
$146$	233082.	0.904953
$147$	0	0
$148$	22205.8	0.0833319
$149$	−113622.	−0.419275	−0.209637	−	0.977779i	$-0.567228\pi$
$149$	−113622.	−0.419275	−0.209637	+	0.977779i	$0.567228\pi$
$150$	0	0
$151$	383.145	0.00136748	0.000683739	−	1.00000i	$-0.499782\pi$
$151$	383.145	0.00136748	0.000683739		1.00000i	$0.499782\pi$
$152$	−422132.	−1.48197
$153$	0	0
$154$	−31906.7	−0.108413
$155$	−94587.4	−0.316231
$156$	0	0
$157$	−360697.	−1.16787	−0.583934	−	0.811801i	$-0.698487\pi$
$157$	−360697.	−1.16787	−0.583934	+	0.811801i	$0.698487\pi$
$158$	−16484.7	−0.0525338
$159$	0	0
$160$	−262389.	−0.810299
$161$	100393.	0.305238
$162$	0	0
$163$	−85429.9	−0.251849	−0.125925	−	0.992040i	$-0.540190\pi$
$163$	−85429.9	−0.251849	−0.125925	+	0.992040i	$0.540190\pi$
$164$	−166934.	−0.484659
$165$	0	0
$166$	−68304.9	−0.192390
$167$	442758.	1.22850	0.614250	−	0.789111i	$-0.289459\pi$
$167$	442758.	1.22850	0.614250	+	0.789111i	$0.289459\pi$
$168$	0	0
$169$	−370749.	−0.998534
$170$	−134030.	−0.355696
$171$	0	0
$172$	186851.	0.481585
$173$	−264730.	−0.672493	−0.336247	−	0.941774i	$-0.609158\pi$
$173$	−264730.	−0.672493	−0.336247	+	0.941774i	$0.609158\pi$
$174$	0	0
$175$	24475.3	0.0604135
$176$	51725.3	0.125870
$177$	0	0
$178$	−338514.	−0.800804
$179$	366684.	0.855381	0.427691	−	0.903925i	$-0.359327\pi$
$179$	366684.	0.855381	0.427691	+	0.903925i	$0.359327\pi$
$180$	0	0
$181$	295830.	0.671190	0.335595	−	0.942006i	$-0.391063\pi$
$181$	295830.	0.671190	0.335595	+	0.942006i	$0.391063\pi$
$182$	−6152.32	−0.0137677
$183$	0	0
$184$	−323542.	−0.704509
$185$	99929.2	0.214666
$186$	0	0
$187$	−62972.6	−0.131688
$188$	−338670.	−0.698848
$189$	0	0
$190$	−554705.	−1.11475
$191$	−770864.	−1.52895	−0.764477	−	0.644651i	$-0.777003\pi$
$191$	−770864.	−1.52895	−0.764477	+	0.644651i	$0.777003\pi$
$192$	0	0
$193$	−30736.7	−0.0593969	−0.0296984	−	0.999559i	$-0.509455\pi$
$193$	−30736.7	−0.0593969	−0.0296984	+	0.999559i	$0.509455\pi$
$194$	−52648.2	−0.100434
$195$	0	0
$196$	173009.	0.321684
$197$	−264627.	−0.485812	−0.242906	−	0.970050i	$-0.578101\pi$
$197$	−264627.	−0.485812	−0.242906	+	0.970050i	$0.578101\pi$
$198$	0	0
$199$	−604685.	−1.08242	−0.541211	−	0.840887i	$-0.682034\pi$
$199$	−604685.	−1.08242	−0.541211	+	0.840887i	$0.682034\pi$
$200$	−78878.2	−0.139438
$201$	0	0
$202$	633280.	1.09199
$203$	114584.	0.195157
$204$	0	0
$205$	−751230.	−1.24850
$206$	387954.	0.636960
$207$	0	0
$208$	9973.80	0.0159846
$209$	−260623.	−0.412712
$210$	0	0
$211$	−1.02219e6	−1.58062	−0.790308	−	0.612709i	$-0.790080\pi$
$211$	−1.02219e6	−1.58062	−0.790308	+	0.612709i	$0.790080\pi$
$212$	253000.	0.386617
$213$	0	0
$214$	−193166.	−0.288334
$215$	840855.	1.24058
$216$	0	0
$217$	96849.0	0.139619
$218$	303069.	0.431917
$219$	0	0
$220$	−94846.7	−0.132119
$221$	−12142.5	−0.0167235
$222$	0	0
$223$	289106.	0.389310	0.194655	−	0.980872i	$-0.437641\pi$
$223$	289106.	0.389310	0.194655	+	0.980872i	$0.437641\pi$
$224$	268663.	0.357756
$225$	0	0
$226$	−688867.	−0.897149
$227$	−923893.	−1.19003	−0.595014	−	0.803716i	$-0.702853\pi$
$227$	−923893.	−1.19003	−0.595014	+	0.803716i	$0.702853\pi$
$228$	0	0
$229$	15085.6	0.0190097	0.00950484	−	0.999955i	$-0.496974\pi$
$229$	15085.6	0.0190097	0.00950484	+	0.999955i	$0.496974\pi$
$230$	−425152.	−0.529938
$231$	0	0
$232$	−369277.	−0.450436
$233$	−204062.	−0.246248	−0.123124	−	0.992391i	$-0.539291\pi$
$233$	−204062.	−0.246248	−0.123124	+	0.992391i	$0.539291\pi$
$234$	0	0
$235$	−1.52407e6	−1.80026
$236$	430097.	0.502674
$237$	0	0
$238$	137234.	0.157044
$239$	855027.	0.968245	0.484122	−	0.875000i	$-0.339139\pi$
$239$	855027.	0.968245	0.484122	+	0.875000i	$0.339139\pi$
$240$	0	0
$241$	1.28850e6	1.42903	0.714517	−	0.699618i	$-0.246646\pi$
$241$	1.28850e6	1.42903	0.714517	+	0.699618i	$0.246646\pi$
$242$	63485.4	0.0696844
$243$	0	0
$244$	521892.	0.561185
$245$	778567.	0.828668
$246$	0	0
$247$	−50253.9	−0.0524116
$248$	−312121.	−0.322251
$249$	0	0
$250$	701144.	0.709508
$251$	−948050.	−0.949833	−0.474916	−	0.880031i	$-0.657522\pi$
$251$	−948050.	−0.949833	−0.474916	+	0.880031i	$0.657522\pi$
$252$	0	0
$253$	−199754.	−0.196198
$254$	−563159.	−0.547705
$255$	0	0
$256$	−1.04637e6	−0.997901
$257$	1.40479e6	1.32672	0.663361	−	0.748300i	$-0.269130\pi$
$257$	1.40479e6	1.32672	0.663361	+	0.748300i	$0.269130\pi$
$258$	0	0
$259$	−102318.	−0.0947773
$260$	−18288.6	−0.0167782
$261$	0	0
$262$	−68288.8	−0.0614605
$263$	1.87583e6	1.67226	0.836130	−	0.548531i	$-0.184813\pi$
$263$	1.87583e6	1.67226	0.836130	+	0.548531i	$0.184813\pi$
$264$	0	0
$265$	1.13854e6	0.995938
$266$	567968.	0.492175
$267$	0	0
$268$	701136.	0.596300
$269$	1.16684e6	0.983174	0.491587	−	0.870828i	$-0.336417\pi$
$269$	1.16684e6	0.983174	0.491587	+	0.870828i	$0.336417\pi$
$270$	0	0
$271$	−861057.	−0.712211	−0.356106	−	0.934446i	$-0.615896\pi$
$271$	−861057.	−0.712211	−0.356106	+	0.934446i	$0.615896\pi$
$272$	−222476.	−0.182331
$273$	0	0
$274$	−1.17441e6	−0.945024
$275$	−48699.1	−0.0388319
$276$	0	0
$277$	−1.37196e6	−1.07434	−0.537170	−	0.843474i	$-0.680507\pi$
$277$	−1.37196e6	−1.07434	−0.537170	+	0.843474i	$0.680507\pi$
$278$	−1.36667e6	−1.06060
$279$	0	0
$280$	707859.	0.539575
$281$	−822695.	−0.621546	−0.310773	−	0.950484i	$-0.600588\pi$
$281$	−822695.	−0.621546	−0.310773	+	0.950484i	$0.600588\pi$
$282$	0	0
$283$	2.44600e6	1.81548	0.907739	−	0.419535i	$-0.137807\pi$
$283$	2.44600e6	1.81548	0.907739	+	0.419535i	$0.137807\pi$
$284$	−499447.	−0.367446
$285$	0	0
$286$	12241.4	0.00884944
$287$	769191.	0.551226
$288$	0	0
$289$	−1.14901e6	−0.809240
$290$	−485251.	−0.338822
$291$	0	0
$292$	−709431.	−0.486915
$293$	1.96116e6	1.33458	0.667289	−	0.744798i	$-0.267454\pi$
$293$	1.96116e6	1.33458	0.667289	+	0.744798i	$0.267454\pi$
$294$	0	0
$295$	1.93550e6	1.29490
$296$	329748.	0.218752
$297$	0	0
$298$	−492683.	−0.321386
$299$	−38517.0	−0.0249158
$300$	0	0
$301$	−860959.	−0.547730
$302$	1661.37	0.00104821
$303$	0	0
$304$	−920757.	−0.571427
$305$	2.34859e6	1.44563
$306$	0	0
$307$	−2.90260e6	−1.75769	−0.878844	−	0.477110i	$-0.841684\pi$
$307$	−2.90260e6	−1.75769	−0.878844	+	0.477110i	$0.841684\pi$
$308$	97114.5	0.0583320
$309$	0	0
$310$	−410144.	−0.242400
$311$	−142496.	−0.0835416	−0.0417708	−	0.999127i	$-0.513300\pi$
$311$	−142496.	−0.0835416	−0.0417708	+	0.999127i	$0.513300\pi$
$312$	0	0
$313$	−924118.	−0.533171	−0.266586	−	0.963811i	$-0.585895\pi$
$313$	−924118.	−0.533171	−0.266586	+	0.963811i	$0.585895\pi$
$314$	−1.56403e6	−0.895203
$315$	0	0
$316$	50174.6	0.0282661
$317$	−1.37996e6	−0.771291	−0.385646	−	0.922647i	$-0.626021\pi$
$317$	−1.37996e6	−0.771291	−0.385646	+	0.922647i	$0.626021\pi$
$318$	0	0
$319$	−227991.	−0.125441
$320$	−1.95021e6	−1.06465
$321$	0	0
$322$	435317.	0.233973
$323$	1.12097e6	0.597843
$324$	0	0
$325$	−9390.27	−0.00493139
$326$	−370436.	−0.193050
$327$	0	0
$328$	−2.47892e6	−1.27227
$329$	1.56051e6	0.794833
$330$	0	0
$331$	−2.75761e6	−1.38345	−0.691724	−	0.722162i	$-0.743149\pi$
$331$	−2.75761e6	−1.38345	−0.691724	+	0.722162i	$0.743149\pi$
$332$	207900.	0.103516
$333$	0	0
$334$	1.91986e6	0.941680
$335$	3.15521e6	1.53609
$336$	0	0
$337$	327782.	0.157221	0.0786105	−	0.996905i	$-0.474952\pi$
$337$	327782.	0.157221	0.0786105	+	0.996905i	$0.474952\pi$
$338$	−1.60762e6	−0.765404
$339$	0	0
$340$	407946.	0.191384
$341$	−192702.	−0.0897431
$342$	0	0
$343$	−1.81926e6	−0.834947
$344$	2.77467e6	1.26420
$345$	0	0
$346$	−1.14791e6	−0.515485
$347$	−4.02530e6	−1.79463	−0.897314	−	0.441394i	$-0.854484\pi$
$347$	−4.02530e6	−1.79463	−0.897314	+	0.441394i	$0.854484\pi$
$348$	0	0
$349$	−3.66143e6	−1.60911	−0.804557	−	0.593875i	$-0.797597\pi$
$349$	−3.66143e6	−1.60911	−0.804557	+	0.593875i	$0.797597\pi$
$350$	106128.	0.0463086
$351$	0	0
$352$	−534563.	−0.229955
$353$	3.65277e6	1.56022	0.780109	−	0.625643i	$-0.215163\pi$
$353$	3.65277e6	1.56022	0.780109	+	0.625643i	$0.215163\pi$
$354$	0	0
$355$	−2.24758e6	−0.946553
$356$	1.03033e6	0.430877
$357$	0	0
$358$	1.58999e6	0.655674
$359$	1.86829e6	0.765082	0.382541	−	0.923939i	$-0.375049\pi$
$359$	1.86829e6	0.765082	0.382541	+	0.923939i	$0.375049\pi$
$360$	0	0
$361$	2.16322e6	0.873641
$362$	1.28276e6	0.514486
$363$	0	0
$364$	18725.8	0.00740777
$365$	−3.19254e6	−1.25431
$366$	0	0
$367$	−3.17647e6	−1.23106	−0.615531	−	0.788113i	$-0.711058\pi$
$367$	−3.17647e6	−1.23106	−0.615531	+	0.788113i	$0.711058\pi$
$368$	−705712.	−0.271649
$369$	0	0
$370$	433307.	0.164547
$371$	−1.16576e6	−0.439717
$372$	0	0
$373$	2.50637e6	0.932766	0.466383	−	0.884583i	$-0.345557\pi$
$373$	2.50637e6	0.932766	0.466383	+	0.884583i	$0.345557\pi$
$374$	−273058.	−0.100943
$375$	0	0
$376$	−5.02914e6	−1.83453
$377$	−43961.7	−0.0159302
$378$	0	0
$379$	2.90232e6	1.03788	0.518939	−	0.854811i	$-0.326327\pi$
$379$	2.90232e6	1.03788	0.518939	+	0.854811i	$0.326327\pi$
$380$	1.68836e6	0.599798
$381$	0	0
$382$	−3.34257e6	−1.17199
$383$	−1.10466e6	−0.384796	−0.192398	−	0.981317i	$-0.561627\pi$
$383$	−1.10466e6	−0.384796	−0.192398	+	0.981317i	$0.561627\pi$
$384$	0	0
$385$	437029.	0.150265
$386$	−133278.	−0.0455294
$387$	0	0
$388$	160245.	0.0540389
$389$	−4.81889e6	−1.61463	−0.807314	−	0.590122i	$-0.799080\pi$
$389$	−4.81889e6	−1.61463	−0.807314	+	0.590122i	$0.799080\pi$
$390$	0	0
$391$	859163.	0.284206
$392$	2.56913e6	0.844443
$393$	0	0
$394$	−1.14746e6	−0.372389
$395$	225793.	0.0728145
$396$	0	0
$397$	4.44015e6	1.41391	0.706954	−	0.707259i	$-0.250069\pi$
$397$	4.44015e6	1.41391	0.706954	+	0.707259i	$0.250069\pi$
$398$	−2.62200e6	−0.829707
$399$	0	0
$400$	−172049.	−0.0537655
$401$	3.23141e6	1.00353	0.501766	−	0.865003i	$-0.332684\pi$
$401$	3.23141e6	1.00353	0.501766	+	0.865003i	$0.332684\pi$
$402$	0	0
$403$	−37157.3	−0.0113968
$404$	−1.92751e6	−0.587549
$405$	0	0
$406$	496853.	0.149594
$407$	203585.	0.0609200
$408$	0	0
$409$	2.50659e6	0.740927	0.370463	−	0.928847i	$-0.379199\pi$
$409$	2.50659e6	0.740927	0.370463	+	0.928847i	$0.379199\pi$
$410$	−3.25744e6	−0.957009
$411$	0	0
$412$	−1.18082e6	−0.342720
$413$	−1.98178e6	−0.571715
$414$	0	0
$415$	935579.	0.266661
$416$	−103076.	−0.0292027
$417$	0	0
$418$	−1.13010e6	−0.316355
$419$	−97029.0	−0.0270002	−0.0135001	−	0.999909i	$-0.504297\pi$
$419$	−97029.0	−0.0270002	−0.0135001	+	0.999909i	$0.504297\pi$
$420$	0	0
$421$	−3.93743e6	−1.08270	−0.541349	−	0.840798i	$-0.682086\pi$
$421$	−3.93743e6	−1.08270	−0.541349	+	0.840798i	$0.682086\pi$
$422$	−4.43237e6	−1.21159
$423$	0	0
$424$	3.75696e6	1.01490
$425$	209460.	0.0562508
$426$	0	0
$427$	−2.40475e6	−0.638263
$428$	587939.	0.155140
$429$	0	0
$430$	3.64606e6	0.950940
$431$	919444.	0.238414	0.119207	−	0.992869i	$-0.461965\pi$
$431$	919444.	0.238414	0.119207	+	0.992869i	$0.461965\pi$
$432$	0	0
$433$	−2.89265e6	−0.741439	−0.370720	−	0.928745i	$-0.620889\pi$
$433$	−2.89265e6	−0.741439	−0.370720	+	0.928745i	$0.620889\pi$
$434$	419950.	0.107022
$435$	0	0
$436$	−922453.	−0.232396
$437$	3.55580e6	0.890704
$438$	0	0
$439$	7.35367e6	1.82114	0.910569	−	0.413357i	$-0.135644\pi$
$439$	7.35367e6	1.82114	0.910569	+	0.413357i	$0.135644\pi$
$440$	−1.40844e6	−0.346822
$441$	0	0
$442$	−52651.6	−0.0128191
$443$	5.70050e6	1.38008	0.690039	−	0.723772i	$-0.257593\pi$
$443$	5.70050e6	1.38008	0.690039	+	0.723772i	$0.257593\pi$
$444$	0	0
$445$	4.63666e6	1.10995
$446$	1.25360e6	0.298417
$447$	0	0
$448$	1.99684e6	0.470054
$449$	5.12012e6	1.19857	0.599286	−	0.800535i	$-0.295451\pi$
$449$	5.12012e6	1.19857	0.599286	+	0.800535i	$0.295451\pi$
$450$	0	0
$451$	−1.53048e6	−0.354311
$452$	2.09671e6	0.482716
$453$	0	0
$454$	−4.00613e6	−0.912189
$455$	84269.0	0.0190827
$456$	0	0
$457$	−4.67041e6	−1.04608	−0.523040	−	0.852308i	$-0.675202\pi$
$457$	−4.67041e6	−1.04608	−0.523040	+	0.852308i	$0.675202\pi$
$458$	65413.4	0.0145715
$459$	0	0
$460$	1.29404e6	0.285136
$461$	−3.01744e6	−0.661282	−0.330641	−	0.943757i	$-0.607265\pi$
$461$	−3.01744e6	−0.661282	−0.330641	+	0.943757i	$0.607265\pi$
$462$	0	0
$463$	−5.29694e6	−1.14835	−0.574173	−	0.818734i	$-0.694676\pi$
$463$	−5.29694e6	−1.14835	−0.574173	+	0.818734i	$0.694676\pi$
$464$	−805470.	−0.173682
$465$	0	0
$466$	−884842.	−0.188756
$467$	−3.84074e6	−0.814934	−0.407467	−	0.913220i	$-0.633588\pi$
$467$	−3.84074e6	−0.814934	−0.407467	+	0.913220i	$0.633588\pi$
$468$	0	0
$469$	−3.23065e6	−0.678201
$470$	−6.60856e6	−1.37995
$471$	0	0
$472$	6.38679e6	1.31955
$473$	1.71307e6	0.352064
$474$	0	0
$475$	866887.	0.176290
$476$	−417700.	−0.0844981
$477$	0	0
$478$	3.70752e6	0.742187
$479$	5.30547e6	1.05654	0.528269	−	0.849077i	$-0.322841\pi$
$479$	5.30547e6	1.05654	0.528269	+	0.849077i	$0.322841\pi$
$480$	0	0
$481$	39255.7	0.00773643
$482$	5.58713e6	1.09540
$483$	0	0
$484$	−193231.	−0.0374941
$485$	721128.	0.139206
$486$	0	0
$487$	−7.82188e6	−1.49447	−0.747237	−	0.664557i	$-0.768620\pi$
$487$	−7.82188e6	−1.49447	−0.747237	+	0.664557i	$0.768620\pi$
$488$	7.74992e6	1.47315
$489$	0	0
$490$	3.37597e6	0.635197
$491$	−5.22307e6	−0.977736	−0.488868	−	0.872358i	$-0.662590\pi$
$491$	−5.22307e6	−0.977736	−0.488868	+	0.872358i	$0.662590\pi$
$492$	0	0
$493$	980612.	0.181711
$494$	−217908.	−0.0401750
$495$	0	0
$496$	−680800.	−0.124255
$497$	2.30132e6	0.417913
$498$	0	0
$499$	−2.90396e6	−0.522083	−0.261041	−	0.965328i	$-0.584066\pi$
$499$	−2.90396e6	−0.522083	−0.261041	+	0.965328i	$0.584066\pi$
$500$	−2.13407e6	−0.381755
$501$	0	0
$502$	−4.11088e6	−0.728073
$503$	−486299.	−0.0857006	−0.0428503	−	0.999082i	$-0.513644\pi$
$503$	−486299.	−0.0857006	−0.0428503	+	0.999082i	$0.513644\pi$
$504$	0	0
$505$	−8.67410e6	−1.51355
$506$	−866160.	−0.150391
$507$	0	0
$508$	1.71409e6	0.294695
$509$	−5.87625e6	−1.00532	−0.502662	−	0.864483i	$-0.667646\pi$
$509$	−5.87625e6	−1.00532	−0.502662	+	0.864483i	$0.667646\pi$
$510$	0	0
$511$	3.26888e6	0.553791
$512$	−4.56952e6	−0.770363
$513$	0	0
$514$	6.09138e6	1.01697
$515$	−5.31385e6	−0.882858
$516$	0	0
$517$	−3.10497e6	−0.510894
$518$	−443667.	−0.0726495
$519$	0	0
$520$	−271579.	−0.0440441
$521$	2.60297e6	0.420122	0.210061	−	0.977688i	$-0.432634\pi$
$521$	2.60297e6	0.420122	0.210061	+	0.977688i	$0.432634\pi$
$522$	0	0
$523$	5.49677e6	0.878726	0.439363	−	0.898310i	$-0.355204\pi$
$523$	5.49677e6	0.878726	0.439363	+	0.898310i	$0.355204\pi$
$524$	207851.	0.0330692
$525$	0	0
$526$	8.13385e6	1.28183
$527$	828834.	0.129999
$528$	0	0
$529$	−3.71101e6	−0.576571
$530$	4.93685e6	0.763414
$531$	0	0
$532$	−1.72872e6	−0.264817
$533$	−295110.	−0.0449951
$534$	0	0
$535$	2.64581e6	0.399645
$536$	1.04116e7	1.56533
$537$	0	0
$538$	5.05958e6	0.753631
$539$	1.58617e6	0.235168
$540$	0	0
$541$	−7.97680e6	−1.17175	−0.585876	−	0.810401i	$-0.699249\pi$
$541$	−7.97680e6	−1.17175	−0.585876	+	0.810401i	$0.699249\pi$
$542$	−3.73366e6	−0.545930
$543$	0	0
$544$	2.29922e6	0.333106
$545$	−4.15117e6	−0.598659
$546$	0	0
$547$	1.18298e7	1.69048	0.845240	−	0.534387i	$-0.179457\pi$
$547$	1.18298e7	1.69048	0.845240	+	0.534387i	$0.179457\pi$
$548$	3.57455e6	0.508476
$549$	0	0
$550$	−211166.	−0.0297658
$551$	4.05843e6	0.569481
$552$	0	0
$553$	−231192.	−0.0321484
$554$	−5.94900e6	−0.823512
$555$	0	0
$556$	4.15974e6	0.570662
$557$	7.30159e6	0.997193	0.498597	−	0.866834i	$-0.333849\pi$
$557$	7.30159e6	0.997193	0.498597	+	0.866834i	$0.333849\pi$
$558$	0	0
$559$	330318.	0.0447098
$560$	1.54398e6	0.208053
$561$	0	0
$562$	−3.56732e6	−0.476432
$563$	3.28061e6	0.436198	0.218099	−	0.975927i	$-0.430015\pi$
$563$	3.28061e6	0.436198	0.218099	+	0.975927i	$0.430015\pi$
$564$	0	0
$565$	9.43549e6	1.24349
$566$	1.06062e7	1.39162
$567$	0	0
$568$	−7.41661e6	−0.964572
$569$	−7.36607e6	−0.953795	−0.476897	−	0.878959i	$-0.658239\pi$
$569$	−7.36607e6	−0.953795	−0.476897	+	0.878959i	$0.658239\pi$
$570$	0	0
$571$	7.83643e6	1.00584	0.502919	−	0.864333i	$-0.332259\pi$
$571$	7.83643e6	1.00584	0.502919	+	0.864333i	$0.332259\pi$
$572$	−37259.2	−0.00476149
$573$	0	0
$574$	3.33532e6	0.422530
$575$	664424.	0.0838061
$576$	0	0
$577$	9.90123e6	1.23808	0.619041	−	0.785358i	$-0.287521\pi$
$577$	9.90123e6	1.23808	0.619041	+	0.785358i	$0.287521\pi$
$578$	−4.98224e6	−0.620305
$579$	0	0
$580$	1.47696e6	0.182305
$581$	−957948.	−0.117734
$582$	0	0
$583$	2.31953e6	0.282637
$584$	−1.05348e7	−1.27819
$585$	0	0
$586$	8.50387e6	1.02299
$587$	−1.44710e7	−1.73342	−0.866709	−	0.498813i	$-0.833769\pi$
$587$	−1.44710e7	−1.73342	−0.866709	+	0.498813i	$0.833769\pi$
$588$	0	0
$589$	3.43027e6	0.407418
$590$	8.39259e6	0.992581
$591$	0	0
$592$	719248.	0.0843479
$593$	2.88455e6	0.336854	0.168427	−	0.985714i	$-0.446131\pi$
$593$	2.88455e6	0.336854	0.168427	+	0.985714i	$0.446131\pi$
$594$	0	0
$595$	−1.87971e6	−0.217670
$596$	1.49958e6	0.172923
$597$	0	0
$598$	−167015.	−0.0190986
$599$	−5.42841e6	−0.618167	−0.309083	−	0.951035i	$-0.600022\pi$
$599$	−5.42841e6	−0.618167	−0.309083	+	0.951035i	$0.600022\pi$
$600$	0	0
$601$	−6.97091e6	−0.787233	−0.393616	−	0.919275i	$-0.628776\pi$
$601$	−6.97091e6	−0.787233	−0.393616	+	0.919275i	$0.628776\pi$
$602$	−3.73324e6	−0.419850
$603$	0	0
$604$	−5056.71	−0.000563996 0
$605$	−869566.	−0.0965860
$606$	0	0
$607$	3.20372e6	0.352925	0.176462	−	0.984307i	$-0.443535\pi$
$607$	3.20372e6	0.352925	0.176462	+	0.984307i	$0.443535\pi$
$608$	9.51570e6	1.04396
$609$	0	0
$610$	1.01838e7	1.10812
$611$	−598707.	−0.0648801
$612$	0	0
$613$	−2.12031e6	−0.227902	−0.113951	−	0.993486i	$-0.536351\pi$
$613$	−2.12031e6	−0.227902	−0.113951	+	0.993486i	$0.536351\pi$
$614$	−1.25861e7	−1.34732
$615$	0	0
$616$	1.44212e6	0.153126
$617$	−1.41804e7	−1.49960	−0.749802	−	0.661662i	$-0.769852\pi$
$617$	−1.41804e7	−1.49960	−0.749802	+	0.661662i	$0.769852\pi$
$618$	0	0
$619$	−8.03399e6	−0.842762	−0.421381	−	0.906884i	$-0.638454\pi$
$619$	−8.03399e6	−0.842762	−0.421381	+	0.906884i	$0.638454\pi$
$620$	1.24836e6	0.130425
$621$	0	0
$622$	−617884.	−0.0640370
$623$	−4.74752e6	−0.490057
$624$	0	0
$625$	−1.08614e7	−1.11220
$626$	−4.00710e6	−0.408691
$627$	0	0
$628$	4.76045e6	0.481669
$629$	−875642.	−0.0882470
$630$	0	0
$631$	−1.81629e6	−0.181598	−0.0907989	−	0.995869i	$-0.528942\pi$
$631$	−1.81629e6	−0.181598	−0.0907989	+	0.995869i	$0.528942\pi$
$632$	745075.	0.0742006
$633$	0	0
$634$	−5.98370e6	−0.591216
$635$	7.71364e6	0.759146
$636$	0	0
$637$	305849.	0.0298647
$638$	−988598.	−0.0961542
$639$	0	0
$640$	−59934.2	−0.00578395
$641$	−1.38588e7	−1.33224	−0.666119	−	0.745845i	$-0.732046\pi$
$641$	−1.38588e7	−1.33224	−0.666119	+	0.745845i	$0.732046\pi$
$642$	0	0
$643$	−8.11589e6	−0.774121	−0.387061	−	0.922054i	$-0.626510\pi$
$643$	−8.11589e6	−0.774121	−0.387061	+	0.922054i	$0.626510\pi$
$644$	−1.32498e6	−0.125891
$645$	0	0
$646$	4.86067e6	0.458263
$647$	−1.76112e7	−1.65397	−0.826984	−	0.562225i	$-0.809946\pi$
$647$	−1.76112e7	−1.65397	−0.826984	+	0.562225i	$0.809946\pi$
$648$	0	0
$649$	3.94318e6	0.367481
$650$	−40717.5	−0.00378005
$651$	0	0
$652$	1.12750e6	0.103871
$653$	6.30309e6	0.578456	0.289228	−	0.957260i	$-0.406601\pi$
$653$	6.30309e6	0.578456	0.289228	+	0.957260i	$0.406601\pi$
$654$	0	0
$655$	935358.	0.0851873
$656$	−5.40703e6	−0.490568
$657$	0	0
$658$	6.76657e6	0.609261
$659$	−7.57883e6	−0.679812	−0.339906	−	0.940459i	$-0.610395\pi$
$659$	−7.57883e6	−0.679812	−0.339906	+	0.940459i	$0.610395\pi$
$660$	0	0
$661$	1.81023e7	1.61150	0.805748	−	0.592259i	$-0.201764\pi$
$661$	1.81023e7	1.61150	0.805748	+	0.592259i	$0.201764\pi$
$662$	−1.19574e7	−1.06045
$663$	0	0
$664$	3.08724e6	0.271738
$665$	−7.77951e6	−0.682179
$666$	0	0
$667$	3.11058e6	0.270724
$668$	−5.84348e6	−0.506676
$669$	0	0
$670$	1.36814e7	1.17746
$671$	4.78477e6	0.410256
$672$	0	0
$673$	1.39989e7	1.19139	0.595696	−	0.803210i	$-0.296876\pi$
$673$	1.39989e7	1.19139	0.595696	+	0.803210i	$0.296876\pi$
$674$	1.42131e6	0.120514
$675$	0	0
$676$	4.89311e6	0.411830
$677$	1.94801e7	1.63350	0.816750	−	0.576991i	$-0.195773\pi$
$677$	1.94801e7	1.63350	0.816750	+	0.576991i	$0.195773\pi$
$678$	0	0
$679$	−738370.	−0.0614610
$680$	6.05786e6	0.502397
$681$	0	0
$682$	−835584.	−0.0687906
$683$	−9.61131e6	−0.788372	−0.394186	−	0.919031i	$-0.628973\pi$
$683$	−9.61131e6	−0.788372	−0.394186	+	0.919031i	$0.628973\pi$
$684$	0	0
$685$	1.60860e7	1.30985
$686$	−7.88856e6	−0.640011
$687$	0	0
$688$	6.05212e6	0.487457
$689$	447258.	0.0358930
$690$	0	0
$691$	3.41088e6	0.271751	0.135875	−	0.990726i	$-0.456615\pi$
$691$	3.41088e6	0.271751	0.135875	+	0.990726i	$0.456615\pi$
$692$	3.49388e6	0.277360
$693$	0	0
$694$	−1.74542e7	−1.37563
$695$	1.87194e7	1.47005
$696$	0	0
$697$	6.58274e6	0.513245
$698$	−1.58765e7	−1.23343
$699$	0	0
$700$	−323023.	−0.0249166
$701$	556273.	0.0427556	0.0213778	−	0.999771i	$-0.493195\pi$
$701$	556273.	0.0427556	0.0213778	+	0.999771i	$0.493195\pi$
$702$	0	0
$703$	−3.62400e6	−0.276566
$704$	−3.97315e6	−0.302137
$705$	0	0
$706$	1.58389e7	1.19595
$707$	8.88149e6	0.668247
$708$	0	0
$709$	−803983.	−0.0600664	−0.0300332	−	0.999549i	$-0.509561\pi$
$709$	−803983.	−0.0600664	−0.0300332	+	0.999549i	$0.509561\pi$
$710$	−9.74583e6	−0.725559
$711$	0	0
$712$	1.53001e7	1.13108
$713$	2.62912e6	0.193681
$714$	0	0
$715$	−167672.	−0.0122658
$716$	−4.83947e6	−0.352789
$717$	0	0
$718$	8.10116e6	0.586457
$719$	−2.19398e7	−1.58275	−0.791373	−	0.611334i	$-0.790633\pi$
$719$	−2.19398e7	−1.58275	−0.791373	+	0.611334i	$0.790633\pi$
$720$	0	0
$721$	5.44090e6	0.389792
$722$	9.38002e6	0.669670
$723$	0	0
$724$	−3.90434e6	−0.276822
$725$	758345.	0.0535824
$726$	0	0
$727$	−2.57342e7	−1.80582	−0.902912	−	0.429826i	$-0.858575\pi$
$727$	−2.57342e7	−1.80582	−0.902912	+	0.429826i	$0.858575\pi$
$728$	278072.	0.0194459
$729$	0	0
$730$	−1.38433e7	−0.961463
$731$	−7.36810e6	−0.509990
$732$	0	0
$733$	−1.74584e7	−1.20017	−0.600086	−	0.799935i	$-0.704867\pi$
$733$	−1.74584e7	−1.20017	−0.600086	+	0.799935i	$0.704867\pi$
$734$	−1.37736e7	−0.943644
$735$	0	0
$736$	7.29329e6	0.496282
$737$	6.42809e6	0.435927
$738$	0	0
$739$	7.89228e6	0.531608	0.265804	−	0.964027i	$-0.414363\pi$
$739$	7.89228e6	0.531608	0.265804	+	0.964027i	$0.414363\pi$
$740$	−1.31886e6	−0.0885357
$741$	0	0
$742$	−5.05489e6	−0.337056
$743$	−3.83136e6	−0.254613	−0.127306	−	0.991863i	$-0.540633\pi$
$743$	−3.83136e6	−0.254613	−0.127306	+	0.991863i	$0.540633\pi$
$744$	0	0
$745$	6.74832e6	0.445456
$746$	1.08679e7	0.714991
$747$	0	0
$748$	831106.	0.0543128
$749$	−2.70907e6	−0.176448
$750$	0	0
$751$	2.68126e6	0.173476	0.0867379	−	0.996231i	$-0.472356\pi$
$751$	2.68126e6	0.173476	0.0867379	+	0.996231i	$0.472356\pi$
$752$	−1.09696e7	−0.707368
$753$	0	0
$754$	−190624.	−0.0122109
$755$	−22755.9	−0.00145287
$756$	0	0
$757$	1.11805e7	0.709121	0.354561	−	0.935033i	$-0.384630\pi$
$757$	1.11805e7	0.709121	0.354561	+	0.935033i	$0.384630\pi$
$758$	1.25848e7	0.795563
$759$	0	0
$760$	2.50715e7	1.57451
$761$	1.58419e7	0.991623	0.495812	−	0.868430i	$-0.334871\pi$
$761$	1.58419e7	0.991623	0.495812	+	0.868430i	$0.334871\pi$
$762$	0	0
$763$	4.25042e6	0.264314
$764$	1.01738e7	0.630594
$765$	0	0
$766$	−4.78995e6	−0.294957
$767$	760333.	0.0466676
$768$	0	0
$769$	1.50314e7	0.916605	0.458303	−	0.888796i	$-0.348458\pi$
$769$	1.50314e7	0.916605	0.458303	+	0.888796i	$0.348458\pi$
$770$	1.89502e6	0.115183
$771$	0	0
$772$	405660.	0.0244973
$773$	4.92955e6	0.296728	0.148364	−	0.988933i	$-0.452599\pi$
$773$	4.92955e6	0.296728	0.148364	+	0.988933i	$0.452599\pi$
$774$	0	0
$775$	640969.	0.0383339
$776$	2.37959e6	0.141856
$777$	0	0
$778$	−2.08954e7	−1.23766
$779$	2.72438e7	1.60851
$780$	0	0
$781$	−4.57899e6	−0.268622
$782$	3.72545e6	0.217852
$783$	0	0
$784$	5.60379e6	0.325605
$785$	2.14227e7	1.24080
$786$	0	0
$787$	1.70314e7	0.980198	0.490099	−	0.871667i	$-0.336961\pi$
$787$	1.70314e7	0.980198	0.490099	+	0.871667i	$0.336961\pi$
$788$	3.49252e6	0.200366
$789$	0	0
$790$	979070.	0.0558143
$791$	−9.66109e6	−0.549016
$792$	0	0
$793$	922610.	0.0520997
$794$	1.92531e7	1.08380
$795$	0	0
$796$	7.98058e6	0.446428
$797$	2.26891e7	1.26524	0.632619	−	0.774463i	$-0.281980\pi$
$797$	2.26891e7	1.26524	0.632619	+	0.774463i	$0.281980\pi$
$798$	0	0
$799$	1.33548e7	0.740067
$800$	1.77807e6	0.0982255
$801$	0	0
$802$	1.40118e7	0.769235
$803$	−6.50415e6	−0.355960
$804$	0	0
$805$	−5.96259e6	−0.324299
$806$	−161119.	−0.00873594
$807$	0	0
$808$	−2.86229e7	−1.54236
$809$	3.25660e7	1.74942	0.874708	−	0.484651i	$-0.161053\pi$
$809$	3.25660e7	1.74942	0.874708	+	0.484651i	$0.161053\pi$
$810$	0	0
$811$	3.60043e7	1.92222	0.961108	−	0.276173i	$-0.0890662\pi$
$811$	3.60043e7	1.92222	0.961108	+	0.276173i	$0.0890662\pi$
$812$	−1.51227e6	−0.0804896
$813$	0	0
$814$	882773.	0.0466969
$815$	5.07390e6	0.267576
$816$	0	0
$817$	−3.04941e7	−1.59831
$818$	1.08689e7	0.567941
$819$	0	0
$820$	9.91466e6	0.514924
$821$	1.41305e7	0.731646	0.365823	−	0.930684i	$-0.380788\pi$
$821$	1.41305e7	0.731646	0.365823	+	0.930684i	$0.380788\pi$
$822$	0	0
$823$	−3.72658e7	−1.91783	−0.958917	−	0.283687i	$-0.908442\pi$
$823$	−3.72658e7	−1.91783	−0.958917	+	0.283687i	$0.908442\pi$
$824$	−1.75347e7	−0.899665
$825$	0	0
$826$	−8.59325e6	−0.438235
$827$	3.00434e6	0.152751	0.0763756	−	0.997079i	$-0.475665\pi$
$827$	3.00434e6	0.152751	0.0763756	+	0.997079i	$0.475665\pi$
$828$	0	0
$829$	−2.71130e7	−1.37022	−0.685111	−	0.728438i	$-0.740246\pi$
$829$	−2.71130e7	−1.37022	−0.685111	+	0.728438i	$0.740246\pi$
$830$	4.05680e6	0.204403
$831$	0	0
$832$	−766112.	−0.0383693
$833$	−6.82229e6	−0.340657
$834$	0	0
$835$	−2.62965e7	−1.30521
$836$	3.43968e6	0.170217
$837$	0	0
$838$	−420731.	−0.0206964
$839$	−7.03727e6	−0.345143	−0.172572	−	0.984997i	$-0.555208\pi$
$839$	−7.03727e6	−0.345143	−0.172572	+	0.984997i	$0.555208\pi$
$840$	0	0
$841$	−1.69609e7	−0.826910
$842$	−1.70732e7	−0.829918
$843$	0	0
$844$	1.34908e7	0.651901
$845$	2.20197e7	1.06089
$846$	0	0
$847$	890357.	0.0426438
$848$	8.19470e6	0.391330
$849$	0	0
$850$	908248.	0.0431179
$851$	−2.77760e6	−0.131476
$852$	0	0
$853$	1.55737e7	0.732859	0.366430	−	0.930446i	$-0.380580\pi$
$853$	1.55737e7	0.732859	0.366430	+	0.930446i	$0.380580\pi$
$854$	−1.04273e7	−0.489246
$855$	0	0
$856$	8.73069e6	0.407253
$857$	2.86475e7	1.33240	0.666201	−	0.745772i	$-0.267919\pi$
$857$	2.86475e7	1.33240	0.666201	+	0.745772i	$0.267919\pi$
$858$	0	0
$859$	−3.25974e7	−1.50730	−0.753651	−	0.657274i	$-0.771709\pi$
$859$	−3.25974e7	−1.50730	−0.753651	+	0.657274i	$0.771709\pi$
$860$	−1.10975e7	−0.511658
$861$	0	0
$862$	3.98683e6	0.182751
$863$	−1.15333e6	−0.0527142	−0.0263571	−	0.999653i	$-0.508391\pi$
$863$	−1.15333e6	−0.0527142	−0.0263571	+	0.999653i	$0.508391\pi$
$864$	0	0
$865$	1.57230e7	0.714488
$866$	−1.25429e7	−0.568334
$867$	0	0
$868$	−1.27820e6	−0.0575839
$869$	460007.	0.0206640
$870$	0	0
$871$	1.23948e6	0.0553597
$872$	−1.36981e7	−0.610055
$873$	0	0
$874$	1.54184e7	0.682749
$875$	9.83326e6	0.434188
$876$	0	0
$877$	1.01823e7	0.447041	0.223520	−	0.974699i	$-0.428245\pi$
$877$	1.01823e7	0.447041	0.223520	+	0.974699i	$0.428245\pi$
$878$	3.18865e7	1.39595
$879$	0	0
$880$	−3.07210e6	−0.133730
$881$	3.09949e7	1.34540	0.672699	−	0.739916i	$-0.265135\pi$
$881$	3.09949e7	1.34540	0.672699	+	0.739916i	$0.265135\pi$
$882$	0	0
$883$	−1.22625e7	−0.529270	−0.264635	−	0.964349i	$-0.585251\pi$
$883$	−1.22625e7	−0.529270	−0.264635	+	0.964349i	$0.585251\pi$
$884$	160256.	0.00689736
$885$	0	0
$886$	2.47181e7	1.05787
$887$	−2.86899e7	−1.22439	−0.612196	−	0.790706i	$-0.709713\pi$
$887$	−2.86899e7	−1.22439	−0.612196	+	0.790706i	$0.709713\pi$
$888$	0	0
$889$	−7.89807e6	−0.335171
$890$	2.01052e7	0.850811
$891$	0	0
$892$	−3.81560e6	−0.160565
$893$	5.52712e7	2.31937
$894$	0	0
$895$	−2.17783e7	−0.908796
$896$	61367.2	0.00255368
$897$	0	0
$898$	2.22016e7	0.918740
$899$	3.00077e6	0.123832
$900$	0	0
$901$	−9.97656e6	−0.409420
$902$	−6.63635e6	−0.271590
$903$	0	0
$904$	3.11354e7	1.26716
$905$	−1.75701e7	−0.713103
$906$	0	0
$907$	3.17887e7	1.28308	0.641542	−	0.767088i	$-0.278295\pi$
$907$	3.17887e7	1.28308	0.641542	+	0.767088i	$0.278295\pi$
$908$	1.21935e7	0.490808
$909$	0	0
$910$	365402.	0.0146274
$911$	−1.75883e7	−0.702147	−0.351073	−	0.936348i	$-0.614183\pi$
$911$	−1.75883e7	−0.702147	−0.351073	+	0.936348i	$0.614183\pi$
$912$	0	0
$913$	1.90605e6	0.0756758
$914$	−2.02515e7	−0.801849
$915$	0	0
$916$	−199099.	−0.00784025
$917$	−957722.	−0.0376111
$918$	0	0
$919$	−3.34743e7	−1.30744	−0.653721	−	0.756736i	$-0.726793\pi$
$919$	−3.34743e7	−1.30744	−0.653721	+	0.756736i	$0.726793\pi$
$920$	1.92160e7	0.748503
$921$	0	0
$922$	−1.30840e7	−0.506891
$923$	−882931.	−0.0341132
$924$	0	0
$925$	−677167.	−0.0260221
$926$	−2.29683e7	−0.880239
$927$	0	0
$928$	8.32425e6	0.317304
$929$	−1.53260e7	−0.582624	−0.291312	−	0.956628i	$-0.594092\pi$
$929$	−1.53260e7	−0.582624	−0.291312	+	0.956628i	$0.594092\pi$
$930$	0	0
$931$	−2.82352e7	−1.06762
$932$	2.69319e6	0.101561
$933$	0	0
$934$	−1.66540e7	−0.624670
$935$	3.74010e6	0.139912
$936$	0	0
$937$	−1.43631e7	−0.534439	−0.267220	−	0.963636i	$-0.586105\pi$
$937$	−1.43631e7	−0.534439	−0.267220	+	0.963636i	$0.586105\pi$
$938$	−1.40086e7	−0.519860
$939$	0	0
$940$	2.01145e7	0.742488
$941$	−2.20728e6	−0.0812611	−0.0406305	−	0.999174i	$-0.512937\pi$
$941$	−2.20728e6	−0.0812611	−0.0406305	+	0.999174i	$0.512937\pi$
$942$	0	0
$943$	2.08810e7	0.764665
$944$	1.39309e7	0.508802
$945$	0	0
$946$	7.42810e6	0.269867
$947$	1.21613e7	0.440661	0.220330	−	0.975425i	$-0.429286\pi$
$947$	1.21613e7	0.440661	0.220330	+	0.975425i	$0.429286\pi$
$948$	0	0
$949$	−1.25414e6	−0.0452046
$950$	3.75894e6	0.135132
$951$	0	0
$952$	−6.20270e6	−0.221814
$953$	−1.44801e7	−0.516462	−0.258231	−	0.966083i	$-0.583140\pi$
$953$	−1.44801e7	−0.516462	−0.258231	+	0.966083i	$0.583140\pi$
$954$	0	0
$955$	4.57836e7	1.62443
$956$	−1.12846e7	−0.399338
$957$	0	0
$958$	2.30052e7	0.809866
$959$	−1.64706e7	−0.578313
$960$	0	0
$961$	−2.60928e7	−0.911408
$962$	170218.	0.00593019
$963$	0	0
$964$	−1.70055e7	−0.589384
$965$	1.82553e6	0.0631059
$966$	0	0
$967$	2.68488e7	0.923333	0.461666	−	0.887054i	$-0.347252\pi$
$967$	2.68488e7	0.923333	0.461666	+	0.887054i	$0.347252\pi$
$968$	−2.86941e6	−0.0984246
$969$	0	0
$970$	3.12691e6	0.106705
$971$	−5.85596e6	−0.199320	−0.0996598	−	0.995022i	$-0.531775\pi$
$971$	−5.85596e6	−0.199320	−0.0996598	+	0.995022i	$0.531775\pi$
$972$	0	0
$973$	−1.91670e7	−0.649041
$974$	−3.39167e7	−1.14556
$975$	0	0
$976$	1.69042e7	0.568027
$977$	4.44242e7	1.48896	0.744481	−	0.667644i	$-0.232697\pi$
$977$	4.44242e7	1.48896	0.744481	+	0.667644i	$0.232697\pi$
$978$	0	0
$979$	9.44623e6	0.314994
$980$	−1.02755e7	−0.341771
$981$	0	0
$982$	−2.26479e7	−0.749462
$983$	1.30751e7	0.431580	0.215790	−	0.976440i	$-0.430767\pi$
$983$	1.30751e7	0.431580	0.215790	+	0.976440i	$0.430767\pi$
$984$	0	0
$985$	1.57169e7	0.516149
$986$	4.25207e6	0.139286
$987$	0	0
$988$	663246.	0.0216164
$989$	−2.33722e7	−0.759816
$990$	0	0
$991$	−3.75007e7	−1.21298	−0.606492	−	0.795090i	$-0.707424\pi$
$991$	−3.75007e7	−1.21298	−0.606492	+	0.795090i	$0.707424\pi$
$992$	7.03583e6	0.227005
$993$	0	0
$994$	9.97885e6	0.320342
$995$	3.59138e7	1.15001
$996$	0	0
$997$	−4.23190e7	−1.34834	−0.674168	−	0.738578i	$-0.735497\pi$
$997$	−4.23190e7	−1.34834	−0.674168	+	0.738578i	$0.735497\pi$
$998$	−1.25920e7	−0.400191
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.6.a.f.1.16		23
3.2	odd	2		891.6.a.e.1.8		23
9.2	odd	6		297.6.e.a.199.16		46
9.4	even	3		99.6.e.a.34.8	✓	46
9.5	odd	6		297.6.e.a.100.16		46
9.7	even	3		99.6.e.a.67.8	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.8	✓	46	9.4	even	3
99.6.e.a.67.8	yes	46	9.7	even	3
297.6.e.a.100.16		46	9.5	odd	6
297.6.e.a.199.16		46	9.2	odd	6
891.6.a.e.1.8		23	3.2	odd	2
891.6.a.f.1.16		23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	891.a (trivial)

\(f(q)\)	\(=\)	\(q+4.33614 q^{2} -13.1979 q^{4} -59.3925 q^{5} +60.8126 q^{7} -195.984 q^{8} +O(q^{10})\)	\(q+4.33614 q^{2} -13.1979 q^{4} -59.3925 q^{5} +60.8126 q^{7} -195.984 q^{8} -257.534 q^{10} -121.000 q^{11} -23.3315 q^{13} +263.692 q^{14} -427.482 q^{16} +520.434 q^{17} +2153.91 q^{19} +783.857 q^{20} -524.673 q^{22} +1650.86 q^{23} +402.472 q^{25} -101.169 q^{26} -802.599 q^{28} +1884.22 q^{29} +1592.58 q^{31} +4417.88 q^{32} +2256.68 q^{34} -3611.81 q^{35} -1682.52 q^{37} +9339.64 q^{38} +11640.0 q^{40} +12648.6 q^{41} -14157.6 q^{43} +1596.95 q^{44} +7158.35 q^{46} +25660.9 q^{47} -13108.8 q^{49} +1745.17 q^{50} +307.927 q^{52} -19169.7 q^{53} +7186.50 q^{55} -11918.3 q^{56} +8170.23 q^{58} -32588.3 q^{59} -39543.6 q^{61} +6905.65 q^{62} +32836.0 q^{64} +1385.72 q^{65} -53124.7 q^{67} -6868.65 q^{68} -15661.3 q^{70} +37842.9 q^{71} +53753.3 q^{73} -7295.64 q^{74} -28427.1 q^{76} -7358.32 q^{77} -3801.71 q^{79} +25389.2 q^{80} +54845.9 q^{82} -15752.5 q^{83} -30909.9 q^{85} -61389.3 q^{86} +23714.1 q^{88} -78068.0 q^{89} -1418.85 q^{91} -21787.9 q^{92} +111269. q^{94} -127926. q^{95} -12141.7 q^{97} -56841.7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) \) 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) \) 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8 - 600 * q^10 - 2783 * q^11 - 1871 * q^13 + 1329 * q^14 + 3584 * q^16 + 267 * q^17 - 3641 * q^19 + 1917 * q^20 + 8292 * q^23 + 10049 * q^25 + 9570 * q^26 + 3793 * q^28 + 5970 * q^29 - 9542 * q^31 + 3831 * q^32 - 2982 * q^34 + 3240 * q^35 - 16007 * q^37 - 1221 * q^38 - 40635 * q^40 - 12030 * q^41 - 25943 * q^43 - 38720 * q^44 - 77004 * q^46 - 9756 * q^47 + 6990 * q^49 - 101805 * q^50 - 144446 * q^52 + 53919 * q^53 - 4356 * q^55 + 16602 * q^56 - 95367 * q^58 + 20310 * q^59 - 100247 * q^61 - 15297 * q^62 - 84577 * q^64 - 20931 * q^65 - 84956 * q^67 - 168471 * q^68 - 212292 * q^70 + 36093 * q^71 - 173444 * q^73 - 86619 * q^74 - 340334 * q^76 + 20207 * q^77 - 123113 * q^79 - 15123 * q^80 - 199983 * q^82 - 30672 * q^83 - 268335 * q^85 + 211260 * q^86 - 25773 * q^88 + 32514 * q^89 - 328021 * q^91 + 196731 * q^92 - 230262 * q^94 + 325926 * q^95 - 357002 * q^97 - 214464 * q^98

Introduction

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