LMFDB - Embedded newform 1104.6.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 374x^{3} + 1565x^{2} + 19136x - 84640 $ x^5 - 2x^4 - 374x^3 + 1565x^2 + 19136x - 84640
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$15.8959$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -46.0012 q^{5} +118.619 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -46.0012 q^{5} +118.619 q^{7} +81.0000 q^{9} -77.1271 q^{11} +209.197 q^{13} +414.011 q^{15} +859.506 q^{17} -1190.01 q^{19} -1067.57 q^{21} -529.000 q^{23} -1008.89 q^{25} -729.000 q^{27} +16.3162 q^{29} -6629.17 q^{31} +694.144 q^{33} -5456.63 q^{35} +4035.42 q^{37} -1882.77 q^{39} -4637.03 q^{41} +18684.6 q^{43} -3726.10 q^{45} +12450.0 q^{47} -2736.47 q^{49} -7735.56 q^{51} -12777.1 q^{53} +3547.94 q^{55} +10710.1 q^{57} +9292.92 q^{59} +25711.5 q^{61} +9608.16 q^{63} -9623.31 q^{65} +47990.2 q^{67} +4761.00 q^{69} +28908.2 q^{71} -85092.6 q^{73} +9080.00 q^{75} -9148.77 q^{77} -13510.2 q^{79} +6561.00 q^{81} +66196.8 q^{83} -39538.3 q^{85} -146.846 q^{87} +16461.8 q^{89} +24814.8 q^{91} +59662.5 q^{93} +54741.8 q^{95} -64655.6 q^{97} -6247.30 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 45 q^{3} + 16 q^{5} + 134 q^{7} + 405 q^{9}+O(q^{10}) $ 5 * q - 45 * q^3 + 16 * q^5 + 134 * q^7 + 405 * q^9	$ 5 q - 45 q^{3} + 16 q^{5} + 134 q^{7} + 405 q^{9} + 632 q^{11} + 326 q^{13} - 144 q^{15} - 1044 q^{17} + 722 q^{19} - 1206 q^{21} - 2645 q^{23} - 4525 q^{25} - 3645 q^{27} - 7822 q^{29} + 2228 q^{31} - 5688 q^{33} + 3020 q^{35} - 18818 q^{37} - 2934 q^{39} - 4550 q^{41} + 2226 q^{43} + 1296 q^{45} + 16164 q^{47} - 24563 q^{49} + 9396 q^{51} - 8972 q^{53} + 37496 q^{55} - 6498 q^{57} + 56168 q^{59} - 61474 q^{61} + 10854 q^{63} - 32312 q^{65} + 58270 q^{67} + 23805 q^{69} + 75920 q^{71} + 7970 q^{73} + 40725 q^{75} - 86424 q^{77} + 64818 q^{79} + 32805 q^{81} + 92680 q^{83} - 18556 q^{85} + 70398 q^{87} - 52256 q^{89} + 80636 q^{91} - 20052 q^{93} + 132324 q^{95} + 42230 q^{97} + 51192 q^{99}+O(q^{100}) $ 5 * q - 45 * q^3 + 16 * q^5 + 134 * q^7 + 405 * q^9 + 632 * q^11 + 326 * q^13 - 144 * q^15 - 1044 * q^17 + 722 * q^19 - 1206 * q^21 - 2645 * q^23 - 4525 * q^25 - 3645 * q^27 - 7822 * q^29 + 2228 * q^31 - 5688 * q^33 + 3020 * q^35 - 18818 * q^37 - 2934 * q^39 - 4550 * q^41 + 2226 * q^43 + 1296 * q^45 + 16164 * q^47 - 24563 * q^49 + 9396 * q^51 - 8972 * q^53 + 37496 * q^55 - 6498 * q^57 + 56168 * q^59 - 61474 * q^61 + 10854 * q^63 - 32312 * q^65 + 58270 * q^67 + 23805 * q^69 + 75920 * q^71 + 7970 * q^73 + 40725 * q^75 - 86424 * q^77 + 64818 * q^79 + 32805 * q^81 + 92680 * q^83 - 18556 * q^85 + 70398 * q^87 - 52256 * q^89 + 80636 * q^91 - 20052 * q^93 + 132324 * q^95 + 42230 * q^97 + 51192 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−46.0012	−0.822895	−0.411447	−	0.911434i	$-0.634977\pi$
$5$	−46.0012	−0.822895	−0.411447	+	0.911434i	$0.634977\pi$
$6$	0	0
$7$	118.619	0.914977	0.457489	−	0.889216i	$-0.348749\pi$
$7$	118.619	0.914977	0.457489	+	0.889216i	$0.348749\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−77.1271	−0.192188	−0.0960938	−	0.995372i	$-0.530635\pi$
$11$	−77.1271	−0.192188	−0.0960938	+	0.995372i	$0.530635\pi$
$12$	0	0
$13$	209.197	0.343318	0.171659	−	0.985156i	$-0.445087\pi$
$13$	209.197	0.343318	0.171659	+	0.985156i	$0.445087\pi$
$14$	0	0
$15$	414.011	0.475098
$16$	0	0
$17$	859.506	0.721318	0.360659	−	0.932698i	$-0.382552\pi$
$17$	859.506	0.721318	0.360659	+	0.932698i	$0.382552\pi$
$18$	0	0
$19$	−1190.01	−0.756250	−0.378125	−	0.925754i	$-0.623431\pi$
$19$	−1190.01	−0.756250	−0.378125	+	0.925754i	$0.623431\pi$
$20$	0	0
$21$	−1067.57	−0.528262
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−1008.89	−0.322844
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	16.3162	0.00360266	0.00180133	−	0.999998i	$-0.499427\pi$
$29$	16.3162	0.00360266	0.00180133	+	0.999998i	$0.499427\pi$
$30$	0	0
$31$	−6629.17	−1.23895	−0.619476	−	0.785015i	$-0.712655\pi$
$31$	−6629.17	−1.23895	−0.619476	+	0.785015i	$0.712655\pi$
$32$	0	0
$33$	694.144	0.110960
$34$	0	0
$35$	−5456.63	−0.752930
$36$	0	0
$37$	4035.42	0.484602	0.242301	−	0.970201i	$-0.422098\pi$
$37$	4035.42	0.484602	0.242301	+	0.970201i	$0.422098\pi$
$38$	0	0
$39$	−1882.77	−0.198215
$40$	0	0
$41$	−4637.03	−0.430805	−0.215402	−	0.976525i	$-0.569106\pi$
$41$	−4637.03	−0.430805	−0.215402	+	0.976525i	$0.569106\pi$
$42$	0	0
$43$	18684.6	1.54104	0.770518	−	0.637418i	$-0.219998\pi$
$43$	18684.6	1.54104	0.770518	+	0.637418i	$0.219998\pi$
$44$	0	0
$45$	−3726.10	−0.274298
$46$	0	0
$47$	12450.0	0.822100	0.411050	−	0.911613i	$-0.365162\pi$
$47$	12450.0	0.822100	0.411050	+	0.911613i	$0.365162\pi$
$48$	0	0
$49$	−2736.47	−0.162817
$50$	0	0
$51$	−7735.56	−0.416453
$52$	0	0
$53$	−12777.1	−0.624805	−0.312402	−	0.949950i	$-0.601134\pi$
$53$	−12777.1	−0.624805	−0.312402	+	0.949950i	$0.601134\pi$
$54$	0	0
$55$	3547.94	0.158150
$56$	0	0
$57$	10710.1	0.436621
$58$	0	0
$59$	9292.92	0.347554	0.173777	−	0.984785i	$-0.444403\pi$
$59$	9292.92	0.347554	0.173777	+	0.984785i	$0.444403\pi$
$60$	0	0
$61$	25711.5	0.884715	0.442358	−	0.896839i	$-0.354142\pi$
$61$	25711.5	0.884715	0.442358	+	0.896839i	$0.354142\pi$
$62$	0	0
$63$	9608.16	0.304992
$64$	0	0
$65$	−9623.31	−0.282515
$66$	0	0
$67$	47990.2	1.30607	0.653033	−	0.757329i	$-0.273496\pi$
$67$	47990.2	1.30607	0.653033	+	0.757329i	$0.273496\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	28908.2	0.680573	0.340286	−	0.940322i	$-0.389476\pi$
$71$	28908.2	0.680573	0.340286	+	0.940322i	$0.389476\pi$
$72$	0	0
$73$	−85092.6	−1.86889	−0.934447	−	0.356103i	$-0.884105\pi$
$73$	−85092.6	−1.86889	−0.934447	+	0.356103i	$0.884105\pi$
$74$	0	0
$75$	9080.00	0.186394
$76$	0	0
$77$	−9148.77	−0.175847
$78$	0	0
$79$	−13510.2	−0.243553	−0.121776	−	0.992558i	$-0.538859\pi$
$79$	−13510.2	−0.243553	−0.121776	+	0.992558i	$0.538859\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	66196.8	1.05473	0.527366	−	0.849638i	$-0.323180\pi$
$83$	66196.8	1.05473	0.527366	+	0.849638i	$0.323180\pi$
$84$	0	0
$85$	−39538.3	−0.593569
$86$	0	0
$87$	−146.846	−0.00208000
$88$	0	0
$89$	16461.8	0.220294	0.110147	−	0.993915i	$-0.464868\pi$
$89$	16461.8	0.220294	0.110147	+	0.993915i	$0.464868\pi$
$90$	0	0
$91$	24814.8	0.314128
$92$	0	0
$93$	59662.5	0.715310
$94$	0	0
$95$	54741.8	0.622314
$96$	0	0
$97$	−64655.6	−0.697713	−0.348856	−	0.937176i	$-0.613430\pi$
$97$	−64655.6	−0.697713	−0.348856	+	0.937176i	$0.613430\pi$
$98$	0	0
$99$	−6247.30	−0.0640625
$100$	0	0
$101$	55038.0	0.536858	0.268429	−	0.963300i	$-0.413496\pi$
$101$	55038.0	0.536858	0.268429	+	0.963300i	$0.413496\pi$
$102$	0	0
$103$	−12138.6	−0.112740	−0.0563699	−	0.998410i	$-0.517953\pi$
$103$	−12138.6	−0.112740	−0.0563699	+	0.998410i	$0.517953\pi$
$104$	0	0
$105$	49109.7	0.434704
$106$	0	0
$107$	21332.6	0.180129	0.0900645	−	0.995936i	$-0.471293\pi$
$107$	21332.6	0.180129	0.0900645	+	0.995936i	$0.471293\pi$
$108$	0	0
$109$	82452.2	0.664716	0.332358	−	0.943153i	$-0.392156\pi$
$109$	82452.2	0.664716	0.332358	+	0.943153i	$0.392156\pi$
$110$	0	0
$111$	−36318.8	−0.279785
$112$	0	0
$113$	−52494.7	−0.386740	−0.193370	−	0.981126i	$-0.561942\pi$
$113$	−52494.7	−0.386740	−0.193370	+	0.981126i	$0.561942\pi$
$114$	0	0
$115$	24334.6	0.171585
$116$	0	0
$117$	16945.0	0.114439
$118$	0	0
$119$	101954.	0.659989
$120$	0	0
$121$	−155102.	−0.963064
$122$	0	0
$123$	41733.3	0.248725
$124$	0	0
$125$	190164.	1.08856
$126$	0	0
$127$	−257438.	−1.41632	−0.708162	−	0.706050i	$-0.750476\pi$
$127$	−257438.	−1.41632	−0.708162	+	0.706050i	$0.750476\pi$
$128$	0	0
$129$	−168161.	−0.889717
$130$	0	0
$131$	−72177.5	−0.367471	−0.183736	−	0.982976i	$-0.558819\pi$
$131$	−72177.5	−0.367471	−0.183736	+	0.982976i	$0.558819\pi$
$132$	0	0
$133$	−141158.	−0.691952
$134$	0	0
$135$	33534.9	0.158366
$136$	0	0
$137$	136579.	0.621701	0.310850	−	0.950459i	$-0.399386\pi$
$137$	136579.	0.621701	0.310850	+	0.950459i	$0.399386\pi$
$138$	0	0
$139$	340312.	1.49397	0.746983	−	0.664843i	$-0.231502\pi$
$139$	340312.	1.49397	0.746983	+	0.664843i	$0.231502\pi$
$140$	0	0
$141$	−112050.	−0.474640
$142$	0	0
$143$	−16134.8	−0.0659815
$144$	0	0
$145$	−750.564	−0.00296461
$146$	0	0
$147$	24628.2	0.0940025
$148$	0	0
$149$	201154.	0.742272	0.371136	−	0.928579i	$-0.378968\pi$
$149$	201154.	0.742272	0.371136	+	0.928579i	$0.378968\pi$
$150$	0	0
$151$	302966.	1.08131	0.540656	−	0.841244i	$-0.318176\pi$
$151$	302966.	1.08131	0.540656	+	0.841244i	$0.318176\pi$
$152$	0	0
$153$	69620.0	0.240439
$154$	0	0
$155$	304950.	1.01953
$156$	0	0
$157$	−380158.	−1.23088	−0.615438	−	0.788185i	$-0.711021\pi$
$157$	−380158.	−1.23088	−0.615438	+	0.788185i	$0.711021\pi$
$158$	0	0
$159$	114994.	0.360731
$160$	0	0
$161$	−62749.6	−0.190786
$162$	0	0
$163$	−435705.	−1.28447	−0.642234	−	0.766508i	$-0.721992\pi$
$163$	−435705.	−1.28447	−0.642234	+	0.766508i	$0.721992\pi$
$164$	0	0
$165$	−31931.5	−0.0913081
$166$	0	0
$167$	17733.6	0.0492046	0.0246023	−	0.999697i	$-0.492168\pi$
$167$	17733.6	0.0492046	0.0246023	+	0.999697i	$0.492168\pi$
$168$	0	0
$169$	−327530.	−0.882133
$170$	0	0
$171$	−96390.6	−0.252083
$172$	0	0
$173$	−317339.	−0.806136	−0.403068	−	0.915170i	$-0.632056\pi$
$173$	−317339.	−0.806136	−0.403068	+	0.915170i	$0.632056\pi$
$174$	0	0
$175$	−119674.	−0.295395
$176$	0	0
$177$	−83636.3	−0.200660
$178$	0	0
$179$	−328750.	−0.766889	−0.383445	−	0.923564i	$-0.625262\pi$
$179$	−328750.	−0.766889	−0.383445	+	0.923564i	$0.625262\pi$
$180$	0	0
$181$	11928.3	0.0270633	0.0135317	−	0.999908i	$-0.495693\pi$
$181$	11928.3	0.0270633	0.0135317	+	0.999908i	$0.495693\pi$
$182$	0	0
$183$	−231404.	−0.510791
$184$	0	0
$185$	−185634.	−0.398776
$186$	0	0
$187$	−66291.2	−0.138628
$188$	0	0
$189$	−86473.5	−0.176087
$190$	0	0
$191$	−102595.	−0.203491	−0.101745	−	0.994810i	$-0.532443\pi$
$191$	−102595.	−0.203491	−0.101745	+	0.994810i	$0.532443\pi$
$192$	0	0
$193$	−814643.	−1.57425	−0.787126	−	0.616793i	$-0.788432\pi$
$193$	−814643.	−1.57425	−0.787126	+	0.616793i	$0.788432\pi$
$194$	0	0
$195$	86609.8	0.163110
$196$	0	0
$197$	−779450.	−1.43094	−0.715472	−	0.698641i	$-0.753789\pi$
$197$	−779450.	−1.43094	−0.715472	+	0.698641i	$0.753789\pi$
$198$	0	0
$199$	403572.	0.722418	0.361209	−	0.932485i	$-0.382364\pi$
$199$	403572.	0.722418	0.361209	+	0.932485i	$0.382364\pi$
$200$	0	0
$201$	−431912.	−0.754058
$202$	0	0
$203$	1935.41	0.00329635
$204$	0	0
$205$	213309.	0.354507
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	91781.8	0.145342
$210$	0	0
$211$	−230170.	−0.355913	−0.177956	−	0.984038i	$-0.556949\pi$
$211$	−230170.	−0.355913	−0.177956	+	0.984038i	$0.556949\pi$
$212$	0	0
$213$	−260173.	−0.392929
$214$	0	0
$215$	−859514.	−1.26811
$216$	0	0
$217$	−786347.	−1.13361
$218$	0	0
$219$	765833.	1.07901
$220$	0	0
$221$	179806.	0.247642
$222$	0	0
$223$	−464523.	−0.625525	−0.312763	−	0.949831i	$-0.601254\pi$
$223$	−464523.	−0.625525	−0.312763	+	0.949831i	$0.601254\pi$
$224$	0	0
$225$	−81720.0	−0.107615
$226$	0	0
$227$	1.22678e6	1.58017	0.790083	−	0.612999i	$-0.210037\pi$
$227$	1.22678e6	1.58017	0.790083	+	0.612999i	$0.210037\pi$
$228$	0	0
$229$	−893156.	−1.12548	−0.562741	−	0.826633i	$-0.690253\pi$
$229$	−893156.	−1.12548	−0.562741	+	0.826633i	$0.690253\pi$
$230$	0	0
$231$	82338.9	0.101525
$232$	0	0
$233$	−299915.	−0.361916	−0.180958	−	0.983491i	$-0.557920\pi$
$233$	−299915.	−0.361916	−0.180958	+	0.983491i	$0.557920\pi$
$234$	0	0
$235$	−572715.	−0.676502
$236$	0	0
$237$	121591.	0.140615
$238$	0	0
$239$	−295159.	−0.334242	−0.167121	−	0.985936i	$-0.553447\pi$
$239$	−295159.	−0.334242	−0.167121	+	0.985936i	$0.553447\pi$
$240$	0	0
$241$	−471614.	−0.523051	−0.261526	−	0.965197i	$-0.584226\pi$
$241$	−471614.	−0.523051	−0.261526	+	0.965197i	$0.584226\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	125881.	0.133981
$246$	0	0
$247$	−248946.	−0.259635
$248$	0	0
$249$	−595771.	−0.608949
$250$	0	0
$251$	596681.	0.597802	0.298901	−	0.954284i	$-0.403380\pi$
$251$	596681.	0.597802	0.298901	+	0.954284i	$0.403380\pi$
$252$	0	0
$253$	40800.3	0.0400739
$254$	0	0
$255$	355845.	0.342697
$256$	0	0
$257$	−341221.	−0.322258	−0.161129	−	0.986933i	$-0.551513\pi$
$257$	−341221.	−0.322258	−0.161129	+	0.986933i	$0.551513\pi$
$258$	0	0
$259$	478679.	0.443399
$260$	0	0
$261$	1321.61	0.00120089
$262$	0	0
$263$	−911380.	−0.812475	−0.406238	−	0.913767i	$-0.633159\pi$
$263$	−911380.	−0.812475	−0.406238	+	0.913767i	$0.633159\pi$
$264$	0	0
$265$	587764.	0.514148
$266$	0	0
$267$	−148156.	−0.127187
$268$	0	0
$269$	306750.	0.258466	0.129233	−	0.991614i	$-0.458748\pi$
$269$	306750.	0.258466	0.129233	+	0.991614i	$0.458748\pi$
$270$	0	0
$271$	577521.	0.477688	0.238844	−	0.971058i	$-0.423231\pi$
$271$	577521.	0.477688	0.238844	+	0.971058i	$0.423231\pi$
$272$	0	0
$273$	−223333.	−0.181362
$274$	0	0
$275$	77812.7	0.0620467
$276$	0	0
$277$	556011.	0.435395	0.217698	−	0.976016i	$-0.430145\pi$
$277$	556011.	0.435395	0.217698	+	0.976016i	$0.430145\pi$
$278$	0	0
$279$	−536962.	−0.412984
$280$	0	0
$281$	582519.	0.440093	0.220046	−	0.975489i	$-0.429379\pi$
$281$	582519.	0.440093	0.220046	+	0.975489i	$0.429379\pi$
$282$	0	0
$283$	−241537.	−0.179274	−0.0896372	−	0.995974i	$-0.528571\pi$
$283$	−241537.	−0.179274	−0.0896372	+	0.995974i	$0.528571\pi$
$284$	0	0
$285$	−492676.	−0.359293
$286$	0	0
$287$	−550041.	−0.394176
$288$	0	0
$289$	−681106.	−0.479701
$290$	0	0
$291$	581900.	0.402825
$292$	0	0
$293$	−644354.	−0.438486	−0.219243	−	0.975670i	$-0.570359\pi$
$293$	−644354.	−0.438486	−0.219243	+	0.975670i	$0.570359\pi$
$294$	0	0
$295$	−427486.	−0.286000
$296$	0	0
$297$	56225.7	0.0369865
$298$	0	0
$299$	−110665.	−0.0715868
$300$	0	0
$301$	2.21635e6	1.41001
$302$	0	0
$303$	−495342.	−0.309955
$304$	0	0
$305$	−1.18276e6	−0.728027
$306$	0	0
$307$	−230054.	−0.139310	−0.0696552	−	0.997571i	$-0.522190\pi$
$307$	−230054.	−0.139310	−0.0696552	+	0.997571i	$0.522190\pi$
$308$	0	0
$309$	109248.	0.0650904
$310$	0	0
$311$	846281.	0.496151	0.248075	−	0.968741i	$-0.420202\pi$
$311$	846281.	0.496151	0.248075	+	0.968741i	$0.420202\pi$
$312$	0	0
$313$	−2.01329e6	−1.16157	−0.580785	−	0.814057i	$-0.697254\pi$
$313$	−2.01329e6	−1.16157	−0.580785	+	0.814057i	$0.697254\pi$
$314$	0	0
$315$	−441987.	−0.250977
$316$	0	0
$317$	−1.67374e6	−0.935489	−0.467745	−	0.883864i	$-0.654933\pi$
$317$	−1.67374e6	−0.935489	−0.467745	+	0.883864i	$0.654933\pi$
$318$	0	0
$319$	−1258.42	−0.000692387 0
$320$	0	0
$321$	−191993.	−0.103998
$322$	0	0
$323$	−1.02282e6	−0.545497
$324$	0	0
$325$	−211056.	−0.110838
$326$	0	0
$327$	−742070.	−0.383774
$328$	0	0
$329$	1.47681e6	0.752202
$330$	0	0
$331$	−575399.	−0.288668	−0.144334	−	0.989529i	$-0.546104\pi$
$331$	−575399.	−0.288668	−0.144334	+	0.989529i	$0.546104\pi$
$332$	0	0
$333$	326869.	0.161534
$334$	0	0
$335$	−2.20761e6	−1.07476
$336$	0	0
$337$	785407.	0.376721	0.188361	−	0.982100i	$-0.439683\pi$
$337$	785407.	0.376721	0.188361	+	0.982100i	$0.439683\pi$
$338$	0	0
$339$	472452.	0.223284
$340$	0	0
$341$	511289.	0.238111
$342$	0	0
$343$	−2.31823e6	−1.06395
$344$	0	0
$345$	−219012.	−0.0990649
$346$	0	0
$347$	−2.10976e6	−0.940608	−0.470304	−	0.882504i	$-0.655856\pi$
$347$	−2.10976e6	−0.940608	−0.470304	+	0.882504i	$0.655856\pi$
$348$	0	0
$349$	551980.	0.242583	0.121291	−	0.992617i	$-0.461296\pi$
$349$	551980.	0.242583	0.121291	+	0.992617i	$0.461296\pi$
$350$	0	0
$351$	−152505.	−0.0660716
$352$	0	0
$353$	1.11837e6	0.477695	0.238847	−	0.971057i	$-0.423230\pi$
$353$	1.11837e6	0.477695	0.238847	+	0.971057i	$0.423230\pi$
$354$	0	0
$355$	−1.32981e6	−0.560040
$356$	0	0
$357$	−917586.	−0.381045
$358$	0	0
$359$	−3.62546e6	−1.48466	−0.742331	−	0.670034i	$-0.766280\pi$
$359$	−3.62546e6	−1.48466	−0.742331	+	0.670034i	$0.766280\pi$
$360$	0	0
$361$	−1.05998e6	−0.428086
$362$	0	0
$363$	1.39592e6	0.556025
$364$	0	0
$365$	3.91436e6	1.53790
$366$	0	0
$367$	764832.	0.296415	0.148208	−	0.988956i	$-0.452650\pi$
$367$	764832.	0.296415	0.148208	+	0.988956i	$0.452650\pi$
$368$	0	0
$369$	−375600.	−0.143602
$370$	0	0
$371$	−1.51562e6	−0.571682
$372$	0	0
$373$	−1.55651e6	−0.579270	−0.289635	−	0.957137i	$-0.593534\pi$
$373$	−1.55651e6	−0.579270	−0.289635	+	0.957137i	$0.593534\pi$
$374$	0	0
$375$	−1.71147e6	−0.628481
$376$	0	0
$377$	3413.30	0.00123686
$378$	0	0
$379$	−2.25695e6	−0.807094	−0.403547	−	0.914959i	$-0.632223\pi$
$379$	−2.25695e6	−0.807094	−0.403547	+	0.914959i	$0.632223\pi$
$380$	0	0
$381$	2.31694e6	0.817715
$382$	0	0
$383$	−3.39523e6	−1.18269	−0.591347	−	0.806418i	$-0.701403\pi$
$383$	−3.39523e6	−1.18269	−0.591347	+	0.806418i	$0.701403\pi$
$384$	0	0
$385$	420854.	0.144704
$386$	0	0
$387$	1.51345e6	0.513679
$388$	0	0
$389$	2.21257e6	0.741351	0.370675	−	0.928763i	$-0.379126\pi$
$389$	2.21257e6	0.741351	0.370675	+	0.928763i	$0.379126\pi$
$390$	0	0
$391$	−454679.	−0.150405
$392$	0	0
$393$	649597.	0.212160
$394$	0	0
$395$	621484.	0.200418
$396$	0	0
$397$	−1.81867e6	−0.579132	−0.289566	−	0.957158i	$-0.593511\pi$
$397$	−1.81867e6	−0.579132	−0.289566	+	0.957158i	$0.593511\pi$
$398$	0	0
$399$	1.27042e6	0.399498
$400$	0	0
$401$	2.83101e6	0.879186	0.439593	−	0.898197i	$-0.355123\pi$
$401$	2.83101e6	0.879186	0.439593	+	0.898197i	$0.355123\pi$
$402$	0	0
$403$	−1.38680e6	−0.425355
$404$	0	0
$405$	−301814.	−0.0914327
$406$	0	0
$407$	−311241.	−0.0931344
$408$	0	0
$409$	−5.84175e6	−1.72677	−0.863385	−	0.504546i	$-0.831660\pi$
$409$	−5.84175e6	−1.72677	−0.863385	+	0.504546i	$0.831660\pi$
$410$	0	0
$411$	−1.22921e6	−0.358939
$412$	0	0
$413$	1.10232e6	0.318004
$414$	0	0
$415$	−3.04513e6	−0.867933
$416$	0	0
$417$	−3.06281e6	−0.862541
$418$	0	0
$419$	−1.09412e6	−0.304459	−0.152230	−	0.988345i	$-0.548645\pi$
$419$	−1.09412e6	−0.304459	−0.152230	+	0.988345i	$0.548645\pi$
$420$	0	0
$421$	−5.67090e6	−1.55936	−0.779680	−	0.626178i	$-0.784618\pi$
$421$	−5.67090e6	−1.55936	−0.779680	+	0.626178i	$0.784618\pi$
$422$	0	0
$423$	1.00845e6	0.274033
$424$	0	0
$425$	−867146.	−0.232873
$426$	0	0
$427$	3.04988e6	0.809494
$428$	0	0
$429$	145213.	0.0380945
$430$	0	0
$431$	−2.79903e6	−0.725795	−0.362897	−	0.931829i	$-0.618212\pi$
$431$	−2.79903e6	−0.725795	−0.362897	+	0.931829i	$0.618212\pi$
$432$	0	0
$433$	4.76910e6	1.22241	0.611204	−	0.791473i	$-0.290685\pi$
$433$	4.76910e6	1.22241	0.611204	+	0.791473i	$0.290685\pi$
$434$	0	0
$435$	6755.08	0.00171162
$436$	0	0
$437$	629514.	0.157689
$438$	0	0
$439$	120078.	0.0297373	0.0148687	−	0.999889i	$-0.495267\pi$
$439$	120078.	0.0297373	0.0148687	+	0.999889i	$0.495267\pi$
$440$	0	0
$441$	−221654.	−0.0542723
$442$	0	0
$443$	−4.88890e6	−1.18359	−0.591796	−	0.806088i	$-0.701581\pi$
$443$	−4.88890e6	−1.18359	−0.591796	+	0.806088i	$0.701581\pi$
$444$	0	0
$445$	−757262.	−0.181278
$446$	0	0
$447$	−1.81039e6	−0.428551
$448$	0	0
$449$	2.70612e6	0.633477	0.316739	−	0.948513i	$-0.397412\pi$
$449$	2.70612e6	0.633477	0.316739	+	0.948513i	$0.397412\pi$
$450$	0	0
$451$	357641.	0.0827953
$452$	0	0
$453$	−2.72669e6	−0.624296
$454$	0	0
$455$	−1.14151e6	−0.258495
$456$	0	0
$457$	−2.37586e6	−0.532146	−0.266073	−	0.963953i	$-0.585726\pi$
$457$	−2.37586e6	−0.532146	−0.266073	+	0.963953i	$0.585726\pi$
$458$	0	0
$459$	−626580.	−0.138818
$460$	0	0
$461$	−4.66685e6	−1.02276	−0.511378	−	0.859356i	$-0.670865\pi$
$461$	−4.66685e6	−1.02276	−0.511378	+	0.859356i	$0.670865\pi$
$462$	0	0
$463$	6.23544e6	1.35181	0.675904	−	0.736990i	$-0.263753\pi$
$463$	6.23544e6	1.35181	0.675904	+	0.736990i	$0.263753\pi$
$464$	0	0
$465$	−2.74455e6	−0.588624
$466$	0	0
$467$	−5.46380e6	−1.15932	−0.579659	−	0.814859i	$-0.696814\pi$
$467$	−5.46380e6	−1.15932	−0.579659	+	0.814859i	$0.696814\pi$
$468$	0	0
$469$	5.69256e6	1.19502
$470$	0	0
$471$	3.42142e6	0.710647
$472$	0	0
$473$	−1.44109e6	−0.296168
$474$	0	0
$475$	1.20058e6	0.244151
$476$	0	0
$477$	−1.03495e6	−0.208268
$478$	0	0
$479$	−1.26648e6	−0.252209	−0.126105	−	0.992017i	$-0.540247\pi$
$479$	−1.26648e6	−0.252209	−0.126105	+	0.992017i	$0.540247\pi$
$480$	0	0
$481$	844198.	0.166373
$482$	0	0
$483$	564746.	0.110150
$484$	0	0
$485$	2.97424e6	0.574144
$486$	0	0
$487$	1.66203e6	0.317553	0.158776	−	0.987315i	$-0.449245\pi$
$487$	1.66203e6	0.317553	0.158776	+	0.987315i	$0.449245\pi$
$488$	0	0
$489$	3.92135e6	0.741588
$490$	0	0
$491$	−8.06724e6	−1.51015	−0.755077	−	0.655636i	$-0.772401\pi$
$491$	−8.06724e6	−1.51015	−0.755077	+	0.655636i	$0.772401\pi$
$492$	0	0
$493$	14023.9	0.00259866
$494$	0	0
$495$	287383.	0.0527167
$496$	0	0
$497$	3.42906e6	0.622708
$498$	0	0
$499$	847177.	0.152308	0.0761540	−	0.997096i	$-0.475736\pi$
$499$	847177.	0.152308	0.0761540	+	0.997096i	$0.475736\pi$
$500$	0	0
$501$	−159602.	−0.0284083
$502$	0	0
$503$	5.42104e6	0.955350	0.477675	−	0.878537i	$-0.341480\pi$
$503$	5.42104e6	0.955350	0.477675	+	0.878537i	$0.341480\pi$
$504$	0	0
$505$	−2.53181e6	−0.441777
$506$	0	0
$507$	2.94777e6	0.509299
$508$	0	0
$509$	−1.49808e6	−0.256296	−0.128148	−	0.991755i	$-0.540903\pi$
$509$	−1.49808e6	−0.256296	−0.128148	+	0.991755i	$0.540903\pi$
$510$	0	0
$511$	−1.00936e7	−1.70999
$512$	0	0
$513$	867515.	0.145540
$514$	0	0
$515$	558393.	0.0927730
$516$	0	0
$517$	−960232.	−0.157997
$518$	0	0
$519$	2.85605e6	0.465423
$520$	0	0
$521$	4.89185e6	0.789549	0.394774	−	0.918778i	$-0.370823\pi$
$521$	4.89185e6	0.789549	0.394774	+	0.918778i	$0.370823\pi$
$522$	0	0
$523$	−1.43346e6	−0.229157	−0.114578	−	0.993414i	$-0.536552\pi$
$523$	−1.43346e6	−0.229157	−0.114578	+	0.993414i	$0.536552\pi$
$524$	0	0
$525$	1.07706e6	0.170546
$526$	0	0
$527$	−5.69781e6	−0.893679
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	752727.	0.115851
$532$	0	0
$533$	−970053.	−0.147903
$534$	0	0
$535$	−981324.	−0.148227
$536$	0	0
$537$	2.95875e6	0.442764
$538$	0	0
$539$	211056.	0.0312914
$540$	0	0
$541$	−7.01584e6	−1.03059	−0.515296	−	0.857012i	$-0.672318\pi$
$541$	−7.01584e6	−1.03059	−0.515296	+	0.857012i	$0.672318\pi$
$542$	0	0
$543$	−107354.	−0.0156250
$544$	0	0
$545$	−3.79290e6	−0.546991
$546$	0	0
$547$	2.46166e6	0.351770	0.175885	−	0.984411i	$-0.443721\pi$
$547$	2.46166e6	0.351770	0.175885	+	0.984411i	$0.443721\pi$
$548$	0	0
$549$	2.08263e6	0.294905
$550$	0	0
$551$	−19416.4	−0.00272451
$552$	0	0
$553$	−1.60257e6	−0.222845
$554$	0	0
$555$	1.67071e6	0.230233
$556$	0	0
$557$	2.92803e6	0.399887	0.199943	−	0.979807i	$-0.435924\pi$
$557$	2.92803e6	0.399887	0.199943	+	0.979807i	$0.435924\pi$
$558$	0	0
$559$	3.90876e6	0.529066
$560$	0	0
$561$	596621.	0.0800371
$562$	0	0
$563$	−9.75157e6	−1.29659	−0.648296	−	0.761388i	$-0.724518\pi$
$563$	−9.75157e6	−1.29659	−0.648296	+	0.761388i	$0.724518\pi$
$564$	0	0
$565$	2.41482e6	0.318246
$566$	0	0
$567$	778261.	0.101664
$568$	0	0
$569$	1.11118e7	1.43881	0.719407	−	0.694589i	$-0.244414\pi$
$569$	1.11118e7	1.43881	0.719407	+	0.694589i	$0.244414\pi$
$570$	0	0
$571$	9.98460e6	1.28156	0.640782	−	0.767723i	$-0.278610\pi$
$571$	9.98460e6	1.28156	0.640782	+	0.767723i	$0.278610\pi$
$572$	0	0
$573$	923359.	0.117485
$574$	0	0
$575$	533702.	0.0673177
$576$	0	0
$577$	1.35919e6	0.169958	0.0849789	−	0.996383i	$-0.472918\pi$
$577$	1.35919e6	0.169958	0.0849789	+	0.996383i	$0.472918\pi$
$578$	0	0
$579$	7.33179e6	0.908894
$580$	0	0
$581$	7.85222e6	0.965055
$582$	0	0
$583$	985465.	0.120080
$584$	0	0
$585$	−779488.	−0.0941716
$586$	0	0
$587$	1.35858e7	1.62739	0.813693	−	0.581294i	$-0.197453\pi$
$587$	1.35858e7	1.62739	0.813693	+	0.581294i	$0.197453\pi$
$588$	0	0
$589$	7.88875e6	0.936958
$590$	0	0
$591$	7.01505e6	0.826156
$592$	0	0
$593$	461258.	0.0538650	0.0269325	−	0.999637i	$-0.491426\pi$
$593$	461258.	0.0538650	0.0269325	+	0.999637i	$0.491426\pi$
$594$	0	0
$595$	−4.69001e6	−0.543102
$596$	0	0
$597$	−3.63215e6	−0.417088
$598$	0	0
$599$	−1.37101e7	−1.56125	−0.780625	−	0.624999i	$-0.785099\pi$
$599$	−1.37101e7	−1.56125	−0.780625	+	0.624999i	$0.785099\pi$
$600$	0	0
$601$	−1.39728e7	−1.57796	−0.788981	−	0.614417i	$-0.789391\pi$
$601$	−1.39728e7	−1.57796	−0.788981	+	0.614417i	$0.789391\pi$
$602$	0	0
$603$	3.88720e6	0.435355
$604$	0	0
$605$	7.13490e6	0.792500
$606$	0	0
$607$	1.06356e7	1.17163	0.585813	−	0.810447i	$-0.300775\pi$
$607$	1.06356e7	1.17163	0.585813	+	0.810447i	$0.300775\pi$
$608$	0	0
$609$	−17418.7	−0.00190315
$610$	0	0
$611$	2.60450e6	0.282242
$612$	0	0
$613$	−2.05922e6	−0.221336	−0.110668	−	0.993857i	$-0.535299\pi$
$613$	−2.05922e6	−0.221336	−0.110668	+	0.993857i	$0.535299\pi$
$614$	0	0
$615$	−1.91978e6	−0.204675
$616$	0	0
$617$	−7.56906e6	−0.800441	−0.400220	−	0.916419i	$-0.631066\pi$
$617$	−7.56906e6	−0.800441	−0.400220	+	0.916419i	$0.631066\pi$
$618$	0	0
$619$	−1.22790e7	−1.28806	−0.644032	−	0.764999i	$-0.722740\pi$
$619$	−1.22790e7	−1.28806	−0.644032	+	0.764999i	$0.722740\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	1.95269e6	0.201564
$624$	0	0
$625$	−5.59499e6	−0.572927
$626$	0	0
$627$	−826037.	−0.0839132
$628$	0	0
$629$	3.46847e6	0.349552
$630$	0	0
$631$	−2.94433e6	−0.294383	−0.147192	−	0.989108i	$-0.547023\pi$
$631$	−2.94433e6	−0.294383	−0.147192	+	0.989108i	$0.547023\pi$
$632$	0	0
$633$	2.07153e6	0.205486
$634$	0	0
$635$	1.18424e7	1.16549
$636$	0	0
$637$	−572460.	−0.0558981
$638$	0	0
$639$	2.34156e6	0.226858
$640$	0	0
$641$	596572.	0.0573480	0.0286740	−	0.999589i	$-0.490872\pi$
$641$	596572.	0.0573480	0.0286740	+	0.999589i	$0.490872\pi$
$642$	0	0
$643$	−7.46569e6	−0.712102	−0.356051	−	0.934466i	$-0.615877\pi$
$643$	−7.46569e6	−0.712102	−0.356051	+	0.934466i	$0.615877\pi$
$644$	0	0
$645$	7.73563e6	0.732144
$646$	0	0
$647$	1.80127e7	1.69168	0.845838	−	0.533440i	$-0.179101\pi$
$647$	1.80127e7	1.69168	0.845838	+	0.533440i	$0.179101\pi$
$648$	0	0
$649$	−716737.	−0.0667956
$650$	0	0
$651$	7.07712e6	0.654492
$652$	0	0
$653$	−910612.	−0.0835700	−0.0417850	−	0.999127i	$-0.513304\pi$
$653$	−910612.	−0.0835700	−0.0417850	+	0.999127i	$0.513304\pi$
$654$	0	0
$655$	3.32025e6	0.302390
$656$	0	0
$657$	−6.89250e6	−0.622965
$658$	0	0
$659$	1.50558e7	1.35049	0.675244	−	0.737594i	$-0.264038\pi$
$659$	1.50558e7	1.35049	0.675244	+	0.737594i	$0.264038\pi$
$660$	0	0
$661$	−1.21056e7	−1.07766	−0.538831	−	0.842414i	$-0.681134\pi$
$661$	−1.21056e7	−1.07766	−0.538831	+	0.842414i	$0.681134\pi$
$662$	0	0
$663$	−1.61825e6	−0.142976
$664$	0	0
$665$	6.49343e6	0.569403
$666$	0	0
$667$	−8631.26	−0.000751207 0
$668$	0	0
$669$	4.18070e6	0.361147
$670$	0	0
$671$	−1.98306e6	−0.170031
$672$	0	0
$673$	2.08772e6	0.177678	0.0888391	−	0.996046i	$-0.471684\pi$
$673$	2.08772e6	0.177678	0.0888391	+	0.996046i	$0.471684\pi$
$674$	0	0
$675$	735480.	0.0621314
$676$	0	0
$677$	−6.81006e6	−0.571057	−0.285528	−	0.958370i	$-0.592169\pi$
$677$	−6.81006e6	−0.571057	−0.285528	+	0.958370i	$0.592169\pi$
$678$	0	0
$679$	−7.66940e6	−0.638391
$680$	0	0
$681$	−1.10410e7	−0.912310
$682$	0	0
$683$	7.24929e6	0.594626	0.297313	−	0.954780i	$-0.403910\pi$
$683$	7.24929e6	0.594626	0.297313	+	0.954780i	$0.403910\pi$
$684$	0	0
$685$	−6.28278e6	−0.511594
$686$	0	0
$687$	8.03840e6	0.649797
$688$	0	0
$689$	−2.67294e6	−0.214507
$690$	0	0
$691$	2.65049e6	0.211169	0.105585	−	0.994410i	$-0.466329\pi$
$691$	2.65049e6	0.211169	0.105585	+	0.994410i	$0.466329\pi$
$692$	0	0
$693$	−741050.	−0.0586158
$694$	0	0
$695$	−1.56548e7	−1.22938
$696$	0	0
$697$	−3.98556e6	−0.310747
$698$	0	0
$699$	2.69923e6	0.208952
$700$	0	0
$701$	−6.15175e6	−0.472828	−0.236414	−	0.971652i	$-0.575972\pi$
$701$	−6.15175e6	−0.472828	−0.236414	+	0.971652i	$0.575972\pi$
$702$	0	0
$703$	−4.80218e6	−0.366480
$704$	0	0
$705$	5.15443e6	0.390578
$706$	0	0
$707$	6.52857e6	0.491212
$708$	0	0
$709$	193504.	0.0144568	0.00722842	−	0.999974i	$-0.497699\pi$
$709$	193504.	0.0144568	0.00722842	+	0.999974i	$0.497699\pi$
$710$	0	0
$711$	−1.09432e6	−0.0811842
$712$	0	0
$713$	3.50683e6	0.258339
$714$	0	0
$715$	742219.	0.0542959
$716$	0	0
$717$	2.65643e6	0.192975
$718$	0	0
$719$	1.83791e7	1.32587	0.662937	−	0.748675i	$-0.269310\pi$
$719$	1.83791e7	1.32587	0.662937	+	0.748675i	$0.269310\pi$
$720$	0	0
$721$	−1.43988e6	−0.103154
$722$	0	0
$723$	4.24453e6	0.301984
$724$	0	0
$725$	−16461.2	−0.00116310
$726$	0	0
$727$	−2.35434e7	−1.65208	−0.826042	−	0.563608i	$-0.809413\pi$
$727$	−2.35434e7	−1.65208	−0.826042	+	0.563608i	$0.809413\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	1.60595e7	1.11158
$732$	0	0
$733$	−4.37769e6	−0.300944	−0.150472	−	0.988614i	$-0.548079\pi$
$733$	−4.37769e6	−0.300944	−0.150472	+	0.988614i	$0.548079\pi$
$734$	0	0
$735$	−1.13293e6	−0.0773541
$736$	0	0
$737$	−3.70134e6	−0.251010
$738$	0	0
$739$	−1.31869e6	−0.0888242	−0.0444121	−	0.999013i	$-0.514141\pi$
$739$	−1.31869e6	−0.0888242	−0.0444121	+	0.999013i	$0.514141\pi$
$740$	0	0
$741$	2.24051e6	0.149900
$742$	0	0
$743$	−8.55598e6	−0.568588	−0.284294	−	0.958737i	$-0.591759\pi$
$743$	−8.55598e6	−0.568588	−0.284294	+	0.958737i	$0.591759\pi$
$744$	0	0
$745$	−9.25333e6	−0.610812
$746$	0	0
$747$	5.36194e6	0.351577
$748$	0	0
$749$	2.53045e6	0.164814
$750$	0	0
$751$	1.82514e7	1.18085	0.590427	−	0.807091i	$-0.298959\pi$
$751$	1.82514e7	1.18085	0.590427	+	0.807091i	$0.298959\pi$
$752$	0	0
$753$	−5.37013e6	−0.345141
$754$	0	0
$755$	−1.39368e7	−0.889806
$756$	0	0
$757$	−6.87889e6	−0.436293	−0.218147	−	0.975916i	$-0.570001\pi$
$757$	−6.87889e6	−0.436293	−0.218147	+	0.975916i	$0.570001\pi$
$758$	0	0
$759$	−367202.	−0.0231367
$760$	0	0
$761$	2.65597e7	1.66250	0.831248	−	0.555901i	$-0.187627\pi$
$761$	2.65597e7	1.66250	0.831248	+	0.555901i	$0.187627\pi$
$762$	0	0
$763$	9.78042e6	0.608200
$764$	0	0
$765$	−3.20260e6	−0.197856
$766$	0	0
$767$	1.94405e6	0.119322
$768$	0	0
$769$	7.64515e6	0.466198	0.233099	−	0.972453i	$-0.425113\pi$
$769$	7.64515e6	0.466198	0.233099	+	0.972453i	$0.425113\pi$
$770$	0	0
$771$	3.07099e6	0.186056
$772$	0	0
$773$	−6.83125e6	−0.411198	−0.205599	−	0.978636i	$-0.565914\pi$
$773$	−6.83125e6	−0.411198	−0.205599	+	0.978636i	$0.565914\pi$
$774$	0	0
$775$	6.68809e6	0.399989
$776$	0	0
$777$	−4.30811e6	−0.255997
$778$	0	0
$779$	5.51810e6	0.325796
$780$	0	0
$781$	−2.22960e6	−0.130798
$782$	0	0
$783$	−11894.5	−0.000693333 0
$784$	0	0
$785$	1.74877e7	1.01288
$786$	0	0
$787$	−1.08152e7	−0.622440	−0.311220	−	0.950338i	$-0.600738\pi$
$787$	−1.08152e7	−0.622440	−0.311220	+	0.950338i	$0.600738\pi$
$788$	0	0
$789$	8.20242e6	0.469083
$790$	0	0
$791$	−6.22688e6	−0.353858
$792$	0	0
$793$	5.37878e6	0.303739
$794$	0	0
$795$	−5.28988e6	−0.296844
$796$	0	0
$797$	1.46760e7	0.818393	0.409196	−	0.912446i	$-0.365809\pi$
$797$	1.46760e7	0.818393	0.409196	+	0.912446i	$0.365809\pi$
$798$	0	0
$799$	1.07008e7	0.592995
$800$	0	0
$801$	1.33340e6	0.0734312
$802$	0	0
$803$	6.56295e6	0.359178
$804$	0	0
$805$	2.88656e6	0.156997
$806$	0	0
$807$	−2.76075e6	−0.149226
$808$	0	0
$809$	−2.82201e7	−1.51596	−0.757980	−	0.652278i	$-0.773814\pi$
$809$	−2.82201e7	−1.51596	−0.757980	+	0.652278i	$0.773814\pi$
$810$	0	0
$811$	−1.85479e7	−0.990246	−0.495123	−	0.868823i	$-0.664877\pi$
$811$	−1.85479e7	−0.990246	−0.495123	+	0.868823i	$0.664877\pi$
$812$	0	0
$813$	−5.19769e6	−0.275793
$814$	0	0
$815$	2.00430e7	1.05698
$816$	0	0
$817$	−2.22348e7	−1.16541
$818$	0	0
$819$	2.01000e6	0.104709
$820$	0	0
$821$	1.03410e7	0.535435	0.267717	−	0.963497i	$-0.413731\pi$
$821$	1.03410e7	0.535435	0.267717	+	0.963497i	$0.413731\pi$
$822$	0	0
$823$	9.18663e6	0.472778	0.236389	−	0.971659i	$-0.424036\pi$
$823$	9.18663e6	0.472778	0.236389	+	0.971659i	$0.424036\pi$
$824$	0	0
$825$	−700314.	−0.0358227
$826$	0	0
$827$	8.41887e6	0.428045	0.214023	−	0.976829i	$-0.431343\pi$
$827$	8.41887e6	0.428045	0.214023	+	0.976829i	$0.431343\pi$
$828$	0	0
$829$	−2.68167e6	−0.135525	−0.0677625	−	0.997701i	$-0.521586\pi$
$829$	−2.68167e6	−0.135525	−0.0677625	+	0.997701i	$0.521586\pi$
$830$	0	0
$831$	−5.00410e6	−0.251376
$832$	0	0
$833$	−2.35201e6	−0.117443
$834$	0	0
$835$	−815767.	−0.0404902
$836$	0	0
$837$	4.83266e6	0.238437
$838$	0	0
$839$	−8.14119e6	−0.399285	−0.199642	−	0.979869i	$-0.563978\pi$
$839$	−8.14119e6	−0.399285	−0.199642	+	0.979869i	$0.563978\pi$
$840$	0	0
$841$	−2.05109e7	−0.999987
$842$	0	0
$843$	−5.24267e6	−0.254088
$844$	0	0
$845$	1.50668e7	0.725902
$846$	0	0
$847$	−1.83981e7	−0.881181
$848$	0	0
$849$	2.17384e6	0.103504
$850$	0	0
$851$	−2.13474e6	−0.101046
$852$	0	0
$853$	1.73300e7	0.815503	0.407752	−	0.913093i	$-0.366313\pi$
$853$	1.73300e7	0.815503	0.407752	+	0.913093i	$0.366313\pi$
$854$	0	0
$855$	4.43408e6	0.207438
$856$	0	0
$857$	−8.96103e6	−0.416779	−0.208389	−	0.978046i	$-0.566822\pi$
$857$	−8.96103e6	−0.416779	−0.208389	+	0.978046i	$0.566822\pi$
$858$	0	0
$859$	−4.88210e6	−0.225748	−0.112874	−	0.993609i	$-0.536006\pi$
$859$	−4.88210e6	−0.225748	−0.112874	+	0.993609i	$0.536006\pi$
$860$	0	0
$861$	4.95037e6	0.227578
$862$	0	0
$863$	−2.73867e7	−1.25173	−0.625867	−	0.779929i	$-0.715255\pi$
$863$	−2.73867e7	−1.25173	−0.625867	+	0.779929i	$0.715255\pi$
$864$	0	0
$865$	1.45980e7	0.663365
$866$	0	0
$867$	6.12996e6	0.276955
$868$	0	0
$869$	1.04200e6	0.0468078
$870$	0	0
$871$	1.00394e7	0.448397
$872$	0	0
$873$	−5.23710e6	−0.232571
$874$	0	0
$875$	2.25571e7	0.996009
$876$	0	0
$877$	−9.12361e6	−0.400560	−0.200280	−	0.979739i	$-0.564185\pi$
$877$	−9.12361e6	−0.400560	−0.200280	+	0.979739i	$0.564185\pi$
$878$	0	0
$879$	5.79919e6	0.253160
$880$	0	0
$881$	4.20304e7	1.82442	0.912208	−	0.409728i	$-0.134376\pi$
$881$	4.20304e7	1.82442	0.912208	+	0.409728i	$0.134376\pi$
$882$	0	0
$883$	7.87038e6	0.339699	0.169849	−	0.985470i	$-0.445672\pi$
$883$	7.87038e6	0.339699	0.169849	+	0.985470i	$0.445672\pi$
$884$	0	0
$885$	3.84737e6	0.165122
$886$	0	0
$887$	7.39687e6	0.315674	0.157837	−	0.987465i	$-0.449548\pi$
$887$	7.39687e6	0.315674	0.157837	+	0.987465i	$0.449548\pi$
$888$	0	0
$889$	−3.05371e7	−1.29590
$890$	0	0
$891$	−506031.	−0.0213542
$892$	0	0
$893$	−1.48156e7	−0.621713
$894$	0	0
$895$	1.51229e7	0.631069
$896$	0	0
$897$	995987.	0.0413307
$898$	0	0
$899$	−108163.	−0.00446353
$900$	0	0
$901$	−1.09820e7	−0.450683
$902$	0	0
$903$	−1.99472e7	−0.814071
$904$	0	0
$905$	−548715.	−0.0222703
$906$	0	0
$907$	4.58503e7	1.85065	0.925325	−	0.379175i	$-0.123792\pi$
$907$	4.58503e7	1.85065	0.925325	+	0.379175i	$0.123792\pi$
$908$	0	0
$909$	4.45808e6	0.178953
$910$	0	0
$911$	1.18663e7	0.473716	0.236858	−	0.971544i	$-0.423882\pi$
$911$	1.18663e7	0.473716	0.236858	+	0.971544i	$0.423882\pi$
$912$	0	0
$913$	−5.10557e6	−0.202706
$914$	0	0
$915$	1.06449e7	0.420327
$916$	0	0
$917$	−8.56164e6	−0.336228
$918$	0	0
$919$	9.61859e6	0.375684	0.187842	−	0.982199i	$-0.439851\pi$
$919$	9.61859e6	0.375684	0.187842	+	0.982199i	$0.439851\pi$
$920$	0	0
$921$	2.07048e6	0.0804308
$922$	0	0
$923$	6.04750e6	0.233653
$924$	0	0
$925$	−4.07129e6	−0.156451
$926$	0	0
$927$	−983231.	−0.0375799
$928$	0	0
$929$	3.76255e6	0.143035	0.0715176	−	0.997439i	$-0.477216\pi$
$929$	3.76255e6	0.143035	0.0715176	+	0.997439i	$0.477216\pi$
$930$	0	0
$931$	3.25641e6	0.123130
$932$	0	0
$933$	−7.61653e6	−0.286453
$934$	0	0
$935$	3.04948e6	0.114077
$936$	0	0
$937$	−2.59388e7	−0.965165	−0.482583	−	0.875850i	$-0.660301\pi$
$937$	−2.59388e7	−0.965165	−0.482583	+	0.875850i	$0.660301\pi$
$938$	0	0
$939$	1.81196e7	0.670633
$940$	0	0
$941$	1.18056e7	0.434623	0.217311	−	0.976102i	$-0.430271\pi$
$941$	1.18056e7	0.434623	0.217311	+	0.976102i	$0.430271\pi$
$942$	0	0
$943$	2.45299e6	0.0898290
$944$	0	0
$945$	3.97788e6	0.144901
$946$	0	0
$947$	2.73110e7	0.989606	0.494803	−	0.869005i	$-0.335240\pi$
$947$	2.73110e7	0.989606	0.494803	+	0.869005i	$0.335240\pi$
$948$	0	0
$949$	−1.78011e7	−0.641625
$950$	0	0
$951$	1.50636e7	0.540105
$952$	0	0
$953$	9.53286e6	0.340009	0.170005	−	0.985443i	$-0.445622\pi$
$953$	9.53286e6	0.340009	0.170005	+	0.985443i	$0.445622\pi$
$954$	0	0
$955$	4.71952e6	0.167451
$956$	0	0
$957$	11325.8	0.000399750 0
$958$	0	0
$959$	1.62009e7	0.568842
$960$	0	0
$961$	1.53167e7	0.535003
$962$	0	0
$963$	1.72794e6	0.0600430
$964$	0	0
$965$	3.74746e7	1.29544
$966$	0	0
$967$	−1.14113e7	−0.392437	−0.196219	−	0.980560i	$-0.562866\pi$
$967$	−1.14113e7	−0.392437	−0.196219	+	0.980560i	$0.562866\pi$
$968$	0	0
$969$	9.20537e6	0.314943
$970$	0	0
$971$	−2.93621e6	−0.0999401	−0.0499700	−	0.998751i	$-0.515913\pi$
$971$	−2.93621e6	−0.0999401	−0.0499700	+	0.998751i	$0.515913\pi$
$972$	0	0
$973$	4.03676e7	1.36694
$974$	0	0
$975$	1.89951e6	0.0639926
$976$	0	0
$977$	4.19325e7	1.40545	0.702724	−	0.711463i	$-0.251967\pi$
$977$	4.19325e7	1.40545	0.702724	+	0.711463i	$0.251967\pi$
$978$	0	0
$979$	−1.26965e6	−0.0423377
$980$	0	0
$981$	6.67863e6	0.221572
$982$	0	0
$983$	−8.78680e6	−0.290033	−0.145016	−	0.989429i	$-0.546323\pi$
$983$	−8.78680e6	−0.290033	−0.145016	+	0.989429i	$0.546323\pi$
$984$	0	0
$985$	3.58557e7	1.17752
$986$	0	0
$987$	−1.32913e7	−0.434284
$988$	0	0
$989$	−9.88415e6	−0.321328
$990$	0	0
$991$	−3.87330e7	−1.25284	−0.626422	−	0.779484i	$-0.715481\pi$
$991$	−3.87330e7	−1.25284	−0.626422	+	0.779484i	$0.715481\pi$
$992$	0	0
$993$	5.17859e6	0.166663
$994$	0	0
$995$	−1.85648e7	−0.594474
$996$	0	0
$997$	9.54565e6	0.304136	0.152068	−	0.988370i	$-0.451407\pi$
$997$	9.54565e6	0.304136	0.152068	+	0.988370i	$0.451407\pi$
$998$	0	0
$999$	−2.94182e6	−0.0932616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.q.1.2		5
4.3	odd	2		552.6.a.b.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.b.1.2	✓	5	4.3	odd	2
1104.6.a.q.1.2		5	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -46.0012 q^{5} +118.619 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -46.0012 q^{5} +118.619 q^{7} +81.0000 q^{9} -77.1271 q^{11} +209.197 q^{13} +414.011 q^{15} +859.506 q^{17} -1190.01 q^{19} -1067.57 q^{21} -529.000 q^{23} -1008.89 q^{25} -729.000 q^{27} +16.3162 q^{29} -6629.17 q^{31} +694.144 q^{33} -5456.63 q^{35} +4035.42 q^{37} -1882.77 q^{39} -4637.03 q^{41} +18684.6 q^{43} -3726.10 q^{45} +12450.0 q^{47} -2736.47 q^{49} -7735.56 q^{51} -12777.1 q^{53} +3547.94 q^{55} +10710.1 q^{57} +9292.92 q^{59} +25711.5 q^{61} +9608.16 q^{63} -9623.31 q^{65} +47990.2 q^{67} +4761.00 q^{69} +28908.2 q^{71} -85092.6 q^{73} +9080.00 q^{75} -9148.77 q^{77} -13510.2 q^{79} +6561.00 q^{81} +66196.8 q^{83} -39538.3 q^{85} -146.846 q^{87} +16461.8 q^{89} +24814.8 q^{91} +59662.5 q^{93} +54741.8 q^{95} -64655.6 q^{97} -6247.30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} + 16 q^{5} + 134 q^{7} + 405 q^{9}+O(q^{10}) \) 5 * q - 45 * q^3 + 16 * q^5 + 134 * q^7 + 405 * q^9	\( 5 q - 45 q^{3} + 16 q^{5} + 134 q^{7} + 405 q^{9} + 632 q^{11} + 326 q^{13} - 144 q^{15} - 1044 q^{17} + 722 q^{19} - 1206 q^{21} - 2645 q^{23} - 4525 q^{25} - 3645 q^{27} - 7822 q^{29} + 2228 q^{31} - 5688 q^{33} + 3020 q^{35} - 18818 q^{37} - 2934 q^{39} - 4550 q^{41} + 2226 q^{43} + 1296 q^{45} + 16164 q^{47} - 24563 q^{49} + 9396 q^{51} - 8972 q^{53} + 37496 q^{55} - 6498 q^{57} + 56168 q^{59} - 61474 q^{61} + 10854 q^{63} - 32312 q^{65} + 58270 q^{67} + 23805 q^{69} + 75920 q^{71} + 7970 q^{73} + 40725 q^{75} - 86424 q^{77} + 64818 q^{79} + 32805 q^{81} + 92680 q^{83} - 18556 q^{85} + 70398 q^{87} - 52256 q^{89} + 80636 q^{91} - 20052 q^{93} + 132324 q^{95} + 42230 q^{97} + 51192 q^{99}+O(q^{100}) \) 5 * q - 45 * q^3 + 16 * q^5 + 134 * q^7 + 405 * q^9 + 632 * q^11 + 326 * q^13 - 144 * q^15 - 1044 * q^17 + 722 * q^19 - 1206 * q^21 - 2645 * q^23 - 4525 * q^25 - 3645 * q^27 - 7822 * q^29 + 2228 * q^31 - 5688 * q^33 + 3020 * q^35 - 18818 * q^37 - 2934 * q^39 - 4550 * q^41 + 2226 * q^43 + 1296 * q^45 + 16164 * q^47 - 24563 * q^49 + 9396 * q^51 - 8972 * q^53 + 37496 * q^55 - 6498 * q^57 + 56168 * q^59 - 61474 * q^61 + 10854 * q^63 - 32312 * q^65 + 58270 * q^67 + 23805 * q^69 + 75920 * q^71 + 7970 * q^73 + 40725 * q^75 - 86424 * q^77 + 64818 * q^79 + 32805 * q^81 + 92680 * q^83 - 18556 * q^85 + 70398 * q^87 - 52256 * q^89 + 80636 * q^91 - 20052 * q^93 + 132324 * q^95 + 42230 * q^97 + 51192 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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