LMFDB - Embedded newform 1104.6.a.q.1.1

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			1.1
Root			$4.49091$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} -53.5417 q^{5} +8.43952 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -53.5417 q^{5} +8.43952 q^{7} +81.0000 q^{9} -245.637 q^{11} +412.287 q^{13} +481.876 q^{15} -1291.09 q^{17} +379.296 q^{19} -75.9556 q^{21} -529.000 q^{23} -258.281 q^{25} -729.000 q^{27} +4135.83 q^{29} +4922.48 q^{31} +2210.74 q^{33} -451.866 q^{35} +1280.77 q^{37} -3710.58 q^{39} +5051.79 q^{41} -18587.4 q^{43} -4336.88 q^{45} +5902.61 q^{47} -16735.8 q^{49} +11619.8 q^{51} +35761.8 q^{53} +13151.9 q^{55} -3413.66 q^{57} +32145.2 q^{59} -48206.5 q^{61} +683.601 q^{63} -22074.6 q^{65} +15720.5 q^{67} +4761.00 q^{69} +81342.2 q^{71} +38636.4 q^{73} +2324.53 q^{75} -2073.06 q^{77} +72705.1 q^{79} +6561.00 q^{81} +15076.3 q^{83} +69127.1 q^{85} -37222.4 q^{87} -36505.4 q^{89} +3479.50 q^{91} -44302.3 q^{93} -20308.2 q^{95} -124575. q^{97} -19896.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−53.5417	−0.957784	−0.478892	−	0.877874i	$-0.658961\pi$
$5$	−53.5417	−0.957784	−0.478892	+	0.877874i	$0.658961\pi$
$6$	0	0
$7$	8.43952	0.0650987	0.0325494	−	0.999470i	$-0.489637\pi$
$7$	8.43952	0.0650987	0.0325494	+	0.999470i	$0.489637\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−245.637	−0.612086	−0.306043	−	0.952018i	$-0.599005\pi$
$11$	−245.637	−0.612086	−0.306043	+	0.952018i	$0.599005\pi$
$12$	0	0
$13$	412.287	0.676614	0.338307	−	0.941036i	$-0.390146\pi$
$13$	412.287	0.676614	0.338307	+	0.941036i	$0.390146\pi$
$14$	0	0
$15$	481.876	0.552977
$16$	0	0
$17$	−1291.09	−1.08351	−0.541756	−	0.840536i	$-0.682240\pi$
$17$	−1291.09	−1.08351	−0.541756	+	0.840536i	$0.682240\pi$
$18$	0	0
$19$	379.296	0.241043	0.120521	−	0.992711i	$-0.461543\pi$
$19$	379.296	0.241043	0.120521	+	0.992711i	$0.461543\pi$
$20$	0	0
$21$	−75.9556	−0.0375848
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−258.281	−0.0826500
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	4135.83	0.913203	0.456601	−	0.889671i	$-0.349067\pi$
$29$	4135.83	0.913203	0.456601	+	0.889671i	$0.349067\pi$
$30$	0	0
$31$	4922.48	0.919982	0.459991	−	0.887924i	$-0.347853\pi$
$31$	4922.48	0.919982	0.459991	+	0.887924i	$0.347853\pi$
$32$	0	0
$33$	2210.74	0.353388
$34$	0	0
$35$	−451.866	−0.0623505
$36$	0	0
$37$	1280.77	0.153804	0.0769018	−	0.997039i	$-0.475497\pi$
$37$	1280.77	0.153804	0.0769018	+	0.997039i	$0.475497\pi$
$38$	0	0
$39$	−3710.58	−0.390643
$40$	0	0
$41$	5051.79	0.469338	0.234669	−	0.972075i	$-0.424599\pi$
$41$	5051.79	0.469338	0.234669	+	0.972075i	$0.424599\pi$
$42$	0	0
$43$	−18587.4	−1.53302	−0.766508	−	0.642235i	$-0.778007\pi$
$43$	−18587.4	−1.53302	−0.766508	+	0.642235i	$0.778007\pi$
$44$	0	0
$45$	−4336.88	−0.319261
$46$	0	0
$47$	5902.61	0.389762	0.194881	−	0.980827i	$-0.437568\pi$
$47$	5902.61	0.389762	0.194881	+	0.980827i	$0.437568\pi$
$48$	0	0
$49$	−16735.8	−0.995762
$50$	0	0
$51$	11619.8	0.625566
$52$	0	0
$53$	35761.8	1.74876	0.874380	−	0.485243i	$-0.161269\pi$
$53$	35761.8	1.74876	0.874380	+	0.485243i	$0.161269\pi$
$54$	0	0
$55$	13151.9	0.586246
$56$	0	0
$57$	−3413.66	−0.139166
$58$	0	0
$59$	32145.2	1.20223	0.601114	−	0.799164i	$-0.294724\pi$
$59$	32145.2	1.20223	0.601114	+	0.799164i	$0.294724\pi$
$60$	0	0
$61$	−48206.5	−1.65875	−0.829375	−	0.558692i	$-0.811303\pi$
$61$	−48206.5	−1.65875	−0.829375	+	0.558692i	$0.811303\pi$
$62$	0	0
$63$	683.601	0.0216996
$64$	0	0
$65$	−22074.6	−0.648050
$66$	0	0
$67$	15720.5	0.427839	0.213919	−	0.976851i	$-0.431377\pi$
$67$	15720.5	0.427839	0.213919	+	0.976851i	$0.431377\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	81342.2	1.91501	0.957503	−	0.288424i	$-0.0931312\pi$
$71$	81342.2	1.91501	0.957503	+	0.288424i	$0.0931312\pi$
$72$	0	0
$73$	38636.4	0.848574	0.424287	−	0.905528i	$-0.360525\pi$
$73$	38636.4	0.848574	0.424287	+	0.905528i	$0.360525\pi$
$74$	0	0
$75$	2324.53	0.0477180
$76$	0	0
$77$	−2073.06	−0.0398460
$78$	0	0
$79$	72705.1	1.31068	0.655341	−	0.755333i	$-0.272525\pi$
$79$	72705.1	1.31068	0.655341	+	0.755333i	$0.272525\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	15076.3	0.240214	0.120107	−	0.992761i	$-0.461676\pi$
$83$	15076.3	0.240214	0.120107	+	0.992761i	$0.461676\pi$
$84$	0	0
$85$	69127.1	1.03777
$86$	0	0
$87$	−37222.4	−0.527238
$88$	0	0
$89$	−36505.4	−0.488520	−0.244260	−	0.969710i	$-0.578545\pi$
$89$	−36505.4	−0.488520	−0.244260	+	0.969710i	$0.578545\pi$
$90$	0	0
$91$	3479.50	0.0440467
$92$	0	0
$93$	−44302.3	−0.531152
$94$	0	0
$95$	−20308.2	−0.230867
$96$	0	0
$97$	−124575.	−1.34432	−0.672160	−	0.740406i	$-0.734633\pi$
$97$	−124575.	−1.34432	−0.672160	+	0.740406i	$0.734633\pi$
$98$	0	0
$99$	−19896.6	−0.204029
$100$	0	0
$101$	39079.3	0.381192	0.190596	−	0.981669i	$-0.438958\pi$
$101$	39079.3	0.381192	0.190596	+	0.981669i	$0.438958\pi$
$102$	0	0
$103$	11295.4	0.104908	0.0524540	−	0.998623i	$-0.483296\pi$
$103$	11295.4	0.104908	0.0524540	+	0.998623i	$0.483296\pi$
$104$	0	0
$105$	4066.80	0.0359981
$106$	0	0
$107$	41460.3	0.350085	0.175042	−	0.984561i	$-0.443994\pi$
$107$	41460.3	0.350085	0.175042	+	0.984561i	$0.443994\pi$
$108$	0	0
$109$	−214233.	−1.72711	−0.863554	−	0.504256i	$-0.831767\pi$
$109$	−214233.	−1.72711	−0.863554	+	0.504256i	$0.831767\pi$
$110$	0	0
$111$	−11526.9	−0.0887986
$112$	0	0
$113$	155728.	1.14728	0.573642	−	0.819106i	$-0.305530\pi$
$113$	155728.	1.14728	0.573642	+	0.819106i	$0.305530\pi$
$114$	0	0
$115$	28323.6	0.199712
$116$	0	0
$117$	33395.2	0.225538
$118$	0	0
$119$	−10896.2	−0.0705352
$120$	0	0
$121$	−100713.	−0.625350
$122$	0	0
$123$	−45466.1	−0.270972
$124$	0	0
$125$	181147.	1.03694
$126$	0	0
$127$	270383.	1.48754	0.743771	−	0.668434i	$-0.233035\pi$
$127$	270383.	1.48754	0.743771	+	0.668434i	$0.233035\pi$
$128$	0	0
$129$	167286.	0.885087
$130$	0	0
$131$	149933.	0.763344	0.381672	−	0.924298i	$-0.375348\pi$
$131$	149933.	0.763344	0.381672	+	0.924298i	$0.375348\pi$
$132$	0	0
$133$	3201.07	0.0156916
$134$	0	0
$135$	39031.9	0.184326
$136$	0	0
$137$	−349534.	−1.59106	−0.795532	−	0.605911i	$-0.792809\pi$
$137$	−349534.	−1.59106	−0.795532	+	0.605911i	$0.792809\pi$
$138$	0	0
$139$	−112848.	−0.495399	−0.247700	−	0.968837i	$-0.579675\pi$
$139$	−112848.	−0.495399	−0.247700	+	0.968837i	$0.579675\pi$
$140$	0	0
$141$	−53123.5	−0.225029
$142$	0	0
$143$	−101273.	−0.414146
$144$	0	0
$145$	−221439.	−0.874651
$146$	0	0
$147$	150622.	0.574904
$148$	0	0
$149$	−58432.2	−0.215619	−0.107809	−	0.994172i	$-0.534384\pi$
$149$	−58432.2	−0.215619	−0.107809	+	0.994172i	$0.534384\pi$
$150$	0	0
$151$	−308185.	−1.09994	−0.549970	−	0.835184i	$-0.685361\pi$
$151$	−308185.	−1.09994	−0.549970	+	0.835184i	$0.685361\pi$
$152$	0	0
$153$	−104578.	−0.361170
$154$	0	0
$155$	−263558.	−0.881144
$156$	0	0
$157$	313928.	1.01644	0.508219	−	0.861228i	$-0.330304\pi$
$157$	313928.	1.01644	0.508219	+	0.861228i	$0.330304\pi$
$158$	0	0
$159$	−321856.	−1.00965
$160$	0	0
$161$	−4464.50	−0.0135740
$162$	0	0
$163$	278379.	0.820667	0.410334	−	0.911935i	$-0.365412\pi$
$163$	278379.	0.820667	0.410334	+	0.911935i	$0.365412\pi$
$164$	0	0
$165$	−118367.	−0.338469
$166$	0	0
$167$	−160251.	−0.444642	−0.222321	−	0.974973i	$-0.571363\pi$
$167$	−160251.	−0.444642	−0.222321	+	0.974973i	$0.571363\pi$
$168$	0	0
$169$	−201312.	−0.542193
$170$	0	0
$171$	30723.0	0.0803476
$172$	0	0
$173$	122894.	0.312187	0.156094	−	0.987742i	$-0.450110\pi$
$173$	122894.	0.312187	0.156094	+	0.987742i	$0.450110\pi$
$174$	0	0
$175$	−2179.77	−0.00538041
$176$	0	0
$177$	−289307.	−0.694106
$178$	0	0
$179$	609108.	1.42089	0.710447	−	0.703750i	$-0.248493\pi$
$179$	609108.	1.42089	0.710447	+	0.703750i	$0.248493\pi$
$180$	0	0
$181$	−267153.	−0.606128	−0.303064	−	0.952970i	$-0.598010\pi$
$181$	−267153.	−0.606128	−0.303064	+	0.952970i	$0.598010\pi$
$182$	0	0
$183$	433859.	0.957680
$184$	0	0
$185$	−68574.6	−0.147311
$186$	0	0
$187$	317139.	0.663202
$188$	0	0
$189$	−6152.41	−0.0125283
$190$	0	0
$191$	−642147.	−1.27365	−0.636826	−	0.771007i	$-0.719753\pi$
$191$	−642147.	−1.27365	−0.636826	+	0.771007i	$0.719753\pi$
$192$	0	0
$193$	−508406.	−0.982466	−0.491233	−	0.871028i	$-0.663454\pi$
$193$	−508406.	−0.982466	−0.491233	+	0.871028i	$0.663454\pi$
$194$	0	0
$195$	198671.	0.374152
$196$	0	0
$197$	−719841.	−1.32151	−0.660756	−	0.750601i	$-0.729764\pi$
$197$	−719841.	−1.32151	−0.660756	+	0.750601i	$0.729764\pi$
$198$	0	0
$199$	−815816.	−1.46036	−0.730180	−	0.683255i	$-0.760564\pi$
$199$	−815816.	−1.46036	−0.730180	+	0.683255i	$0.760564\pi$
$200$	0	0
$201$	−141485.	−0.247013
$202$	0	0
$203$	34904.4	0.0594483
$204$	0	0
$205$	−270482.	−0.449524
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	−93169.2	−0.147539
$210$	0	0
$211$	544502.	0.841964	0.420982	−	0.907069i	$-0.361686\pi$
$211$	544502.	0.841964	0.420982	+	0.907069i	$0.361686\pi$
$212$	0	0
$213$	−732080.	−1.10563
$214$	0	0
$215$	995199.	1.46830
$216$	0	0
$217$	41543.3	0.0598897
$218$	0	0
$219$	−347728.	−0.489924
$220$	0	0
$221$	−532299.	−0.733119
$222$	0	0
$223$	−297692.	−0.400871	−0.200435	−	0.979707i	$-0.564236\pi$
$223$	−297692.	−0.400871	−0.200435	+	0.979707i	$0.564236\pi$
$224$	0	0
$225$	−20920.8	−0.0275500
$226$	0	0
$227$	−458173.	−0.590153	−0.295076	−	0.955474i	$-0.595345\pi$
$227$	−458173.	−0.590153	−0.295076	+	0.955474i	$0.595345\pi$
$228$	0	0
$229$	−749199.	−0.944080	−0.472040	−	0.881577i	$-0.656482\pi$
$229$	−749199.	−0.944080	−0.472040	+	0.881577i	$0.656482\pi$
$230$	0	0
$231$	18657.5	0.0230051
$232$	0	0
$233$	−1.05350e6	−1.27128	−0.635642	−	0.771984i	$-0.719265\pi$
$233$	−1.05350e6	−1.27128	−0.635642	+	0.771984i	$0.719265\pi$
$234$	0	0
$235$	−316036.	−0.373308
$236$	0	0
$237$	−654346.	−0.756723
$238$	0	0
$239$	663817.	0.751716	0.375858	−	0.926677i	$-0.377348\pi$
$239$	663817.	0.751716	0.375858	+	0.926677i	$0.377348\pi$
$240$	0	0
$241$	458071.	0.508031	0.254016	−	0.967200i	$-0.418249\pi$
$241$	458071.	0.508031	0.254016	+	0.967200i	$0.418249\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	896063.	0.953725
$246$	0	0
$247$	156379.	0.163093
$248$	0	0
$249$	−135686.	−0.138688
$250$	0	0
$251$	547538.	0.548568	0.274284	−	0.961649i	$-0.411559\pi$
$251$	547538.	0.548568	0.274284	+	0.961649i	$0.411559\pi$
$252$	0	0
$253$	129942.	0.127629
$254$	0	0
$255$	−622144.	−0.599157
$256$	0	0
$257$	778038.	0.734798	0.367399	−	0.930063i	$-0.380248\pi$
$257$	778038.	0.734798	0.367399	+	0.930063i	$0.380248\pi$
$258$	0	0
$259$	10809.1	0.0100124
$260$	0	0
$261$	335002.	0.304401
$262$	0	0
$263$	−933988.	−0.832629	−0.416315	−	0.909221i	$-0.636679\pi$
$263$	−933988.	−0.832629	−0.416315	+	0.909221i	$0.636679\pi$
$264$	0	0
$265$	−1.91475e6	−1.67493
$266$	0	0
$267$	328549.	0.282047
$268$	0	0
$269$	180458.	0.152053	0.0760265	−	0.997106i	$-0.475777\pi$
$269$	180458.	0.152053	0.0760265	+	0.997106i	$0.475777\pi$
$270$	0	0
$271$	−1.32642e6	−1.09713	−0.548567	−	0.836107i	$-0.684826\pi$
$271$	−1.32642e6	−1.09713	−0.548567	+	0.836107i	$0.684826\pi$
$272$	0	0
$273$	−31315.5	−0.0254304
$274$	0	0
$275$	63443.5	0.0505889
$276$	0	0
$277$	−1.76446e6	−1.38170	−0.690848	−	0.723000i	$-0.742763\pi$
$277$	−1.76446e6	−1.38170	−0.690848	+	0.723000i	$0.742763\pi$
$278$	0	0
$279$	398721.	0.306661
$280$	0	0
$281$	−1.98078e6	−1.49648	−0.748240	−	0.663429i	$-0.769101\pi$
$281$	−1.98078e6	−1.49648	−0.748240	+	0.663429i	$0.769101\pi$
$282$	0	0
$283$	216967.	0.161038	0.0805189	−	0.996753i	$-0.474342\pi$
$283$	216967.	0.161038	0.0805189	+	0.996753i	$0.474342\pi$
$284$	0	0
$285$	182773.	0.133291
$286$	0	0
$287$	42634.6	0.0305533
$288$	0	0
$289$	247050.	0.173997
$290$	0	0
$291$	1.12118e6	0.776143
$292$	0	0
$293$	1.66056e6	1.13002	0.565008	−	0.825085i	$-0.308873\pi$
$293$	1.66056e6	1.13002	0.565008	+	0.825085i	$0.308873\pi$
$294$	0	0
$295$	−1.72111e6	−1.15147
$296$	0	0
$297$	179070.	0.117796
$298$	0	0
$299$	−218100.	−0.141084
$300$	0	0
$301$	−156868.	−0.0997973
$302$	0	0
$303$	−351714.	−0.220081
$304$	0	0
$305$	2.58106e6	1.58872
$306$	0	0
$307$	981503.	0.594355	0.297177	−	0.954822i	$-0.403955\pi$
$307$	981503.	0.594355	0.297177	+	0.954822i	$0.403955\pi$
$308$	0	0
$309$	−101659.	−0.0605687
$310$	0	0
$311$	−2.36204e6	−1.38480	−0.692399	−	0.721515i	$-0.743446\pi$
$311$	−2.36204e6	−1.38480	−0.692399	+	0.721515i	$0.743446\pi$
$312$	0	0
$313$	2.50854e6	1.44731	0.723653	−	0.690164i	$-0.242461\pi$
$313$	2.50854e6	1.44731	0.723653	+	0.690164i	$0.242461\pi$
$314$	0	0
$315$	−36601.2	−0.0207835
$316$	0	0
$317$	1.19964e6	0.670506	0.335253	−	0.942128i	$-0.391178\pi$
$317$	1.19964e6	0.670506	0.335253	+	0.942128i	$0.391178\pi$
$318$	0	0
$319$	−1.01591e6	−0.558959
$320$	0	0
$321$	−373143.	−0.202121
$322$	0	0
$323$	−489704.	−0.261172
$324$	0	0
$325$	−106486.	−0.0559221
$326$	0	0
$327$	1.92809e6	0.997146
$328$	0	0
$329$	49815.2	0.0253730
$330$	0	0
$331$	1.33784e6	0.671175	0.335587	−	0.942009i	$-0.391065\pi$
$331$	1.33784e6	0.671175	0.335587	+	0.942009i	$0.391065\pi$
$332$	0	0
$333$	103742.	0.0512679
$334$	0	0
$335$	−841705.	−0.409777
$336$	0	0
$337$	−2.74697e6	−1.31759	−0.658793	−	0.752324i	$-0.728933\pi$
$337$	−2.74697e6	−1.31759	−0.658793	+	0.752324i	$0.728933\pi$
$338$	0	0
$339$	−1.40155e6	−0.662385
$340$	0	0
$341$	−1.20914e6	−0.563108
$342$	0	0
$343$	−283085.	−0.129922
$344$	0	0
$345$	−254912.	−0.115304
$346$	0	0
$347$	−3.32583e6	−1.48278	−0.741390	−	0.671074i	$-0.765833\pi$
$347$	−3.32583e6	−1.48278	−0.741390	+	0.671074i	$0.765833\pi$
$348$	0	0
$349$	1.25099e6	0.549781	0.274891	−	0.961476i	$-0.411358\pi$
$349$	1.25099e6	0.549781	0.274891	+	0.961476i	$0.411358\pi$
$350$	0	0
$351$	−300557.	−0.130214
$352$	0	0
$353$	1.04651e6	0.446999	0.223500	−	0.974704i	$-0.428252\pi$
$353$	1.04651e6	0.446999	0.223500	+	0.974704i	$0.428252\pi$
$354$	0	0
$355$	−4.35520e6	−1.83416
$356$	0	0
$357$	98065.4	0.0407235
$358$	0	0
$359$	−3.50966e6	−1.43724	−0.718620	−	0.695403i	$-0.755226\pi$
$359$	−3.50966e6	−1.43724	−0.718620	+	0.695403i	$0.755226\pi$
$360$	0	0
$361$	−2.33223e6	−0.941898
$362$	0	0
$363$	906420.	0.361046
$364$	0	0
$365$	−2.06866e6	−0.812751
$366$	0	0
$367$	79651.7	0.0308695	0.0154348	−	0.999881i	$-0.495087\pi$
$367$	79651.7	0.0308695	0.0154348	+	0.999881i	$0.495087\pi$
$368$	0	0
$369$	409195.	0.156446
$370$	0	0
$371$	301812.	0.113842
$372$	0	0
$373$	387822.	0.144331	0.0721656	−	0.997393i	$-0.477009\pi$
$373$	387822.	0.144331	0.0721656	+	0.997393i	$0.477009\pi$
$374$	0	0
$375$	−1.63032e6	−0.598680
$376$	0	0
$377$	1.70515e6	0.617886
$378$	0	0
$379$	−3.10663e6	−1.11094	−0.555472	−	0.831535i	$-0.687462\pi$
$379$	−3.10663e6	−1.11094	−0.555472	+	0.831535i	$0.687462\pi$
$380$	0	0
$381$	−2.43344e6	−0.858833
$382$	0	0
$383$	−1.76143e6	−0.613577	−0.306789	−	0.951778i	$-0.599254\pi$
$383$	−1.76143e6	−0.613577	−0.306789	+	0.951778i	$0.599254\pi$
$384$	0	0
$385$	110995.	0.0381639
$386$	0	0
$387$	−1.50558e6	−0.511005
$388$	0	0
$389$	1.20898e6	0.405084	0.202542	−	0.979274i	$-0.435080\pi$
$389$	1.20898e6	0.405084	0.202542	+	0.979274i	$0.435080\pi$
$390$	0	0
$391$	682985.	0.225928
$392$	0	0
$393$	−1.34940e6	−0.440717
$394$	0	0
$395$	−3.89276e6	−1.25535
$396$	0	0
$397$	−347174.	−0.110553	−0.0552765	−	0.998471i	$-0.517604\pi$
$397$	−347174.	−0.110553	−0.0552765	+	0.998471i	$0.517604\pi$
$398$	0	0
$399$	−28809.6	−0.00905953
$400$	0	0
$401$	778863.	0.241880	0.120940	−	0.992660i	$-0.461409\pi$
$401$	778863.	0.241880	0.120940	+	0.992660i	$0.461409\pi$
$402$	0	0
$403$	2.02947e6	0.622473
$404$	0	0
$405$	−351287.	−0.106420
$406$	0	0
$407$	−314605.	−0.0941411
$408$	0	0
$409$	−5.29479e6	−1.56509	−0.782547	−	0.622591i	$-0.786080\pi$
$409$	−5.29479e6	−1.56509	−0.782547	+	0.622591i	$0.786080\pi$
$410$	0	0
$411$	3.14580e6	0.918602
$412$	0	0
$413$	271290.	0.0782635
$414$	0	0
$415$	−807209.	−0.230073
$416$	0	0
$417$	1.01563e6	0.286019
$418$	0	0
$419$	−7.02933e6	−1.95604	−0.978022	−	0.208501i	$-0.933142\pi$
$419$	−7.02933e6	−1.95604	−0.978022	+	0.208501i	$0.933142\pi$
$420$	0	0
$421$	5.55771e6	1.52824	0.764119	−	0.645076i	$-0.223174\pi$
$421$	5.55771e6	1.52824	0.764119	+	0.645076i	$0.223174\pi$
$422$	0	0
$423$	478112.	0.129921
$424$	0	0
$425$	333464.	0.0895522
$426$	0	0
$427$	−406840.	−0.107983
$428$	0	0
$429$	911457.	0.239107
$430$	0	0
$431$	2.93384e6	0.760751	0.380376	−	0.924832i	$-0.375795\pi$
$431$	2.93384e6	0.760751	0.380376	+	0.924832i	$0.375795\pi$
$432$	0	0
$433$	−7.68731e6	−1.97040	−0.985201	−	0.171405i	$-0.945169\pi$
$433$	−7.68731e6	−1.97040	−0.985201	+	0.171405i	$0.945169\pi$
$434$	0	0
$435$	1.99295e6	0.504980
$436$	0	0
$437$	−200647.	−0.0502609
$438$	0	0
$439$	−2.23310e6	−0.553028	−0.276514	−	0.961010i	$-0.589179\pi$
$439$	−2.23310e6	−0.553028	−0.276514	+	0.961010i	$0.589179\pi$
$440$	0	0
$441$	−1.35560e6	−0.331921
$442$	0	0
$443$	−1.05618e6	−0.255699	−0.127850	−	0.991794i	$-0.540807\pi$
$443$	−1.05618e6	−0.255699	−0.127850	+	0.991794i	$0.540807\pi$
$444$	0	0
$445$	1.95456e6	0.467897
$446$	0	0
$447$	525889.	0.124487
$448$	0	0
$449$	−503953.	−0.117971	−0.0589854	−	0.998259i	$-0.518787\pi$
$449$	−503953.	−0.117971	−0.0589854	+	0.998259i	$0.518787\pi$
$450$	0	0
$451$	−1.24091e6	−0.287275
$452$	0	0
$453$	2.77366e6	0.635051
$454$	0	0
$455$	−186299.	−0.0421872
$456$	0	0
$457$	−1.22252e6	−0.273820	−0.136910	−	0.990583i	$-0.543717\pi$
$457$	−1.22252e6	−0.273820	−0.136910	+	0.990583i	$0.543717\pi$
$458$	0	0
$459$	941203.	0.208522
$460$	0	0
$461$	2.79800e6	0.613191	0.306596	−	0.951840i	$-0.400810\pi$
$461$	2.79800e6	0.613191	0.306596	+	0.951840i	$0.400810\pi$
$462$	0	0
$463$	1.00045e6	0.216892	0.108446	−	0.994102i	$-0.465413\pi$
$463$	1.00045e6	0.216892	0.108446	+	0.994102i	$0.465413\pi$
$464$	0	0
$465$	2.37202e6	0.508729
$466$	0	0
$467$	2.55439e6	0.541995	0.270998	−	0.962580i	$-0.412646\pi$
$467$	2.55439e6	0.541995	0.270998	+	0.962580i	$0.412646\pi$
$468$	0	0
$469$	132674.	0.0278518
$470$	0	0
$471$	−2.82535e6	−0.586840
$472$	0	0
$473$	4.56575e6	0.938337
$474$	0	0
$475$	−97964.9	−0.0199222
$476$	0	0
$477$	2.89671e6	0.582920
$478$	0	0
$479$	−2.73071e6	−0.543798	−0.271899	−	0.962326i	$-0.587652\pi$
$479$	−2.73071e6	−0.543798	−0.271899	+	0.962326i	$0.587652\pi$
$480$	0	0
$481$	528045.	0.104066
$482$	0	0
$483$	40180.5	0.00783696
$484$	0	0
$485$	6.66998e6	1.28757
$486$	0	0
$487$	5.98501e6	1.14352	0.571758	−	0.820422i	$-0.306262\pi$
$487$	5.98501e6	1.14352	0.571758	+	0.820422i	$0.306262\pi$
$488$	0	0
$489$	−2.50541e6	−0.473812
$490$	0	0
$491$	−4.17317e6	−0.781201	−0.390600	−	0.920560i	$-0.627733\pi$
$491$	−4.17317e6	−0.781201	−0.390600	+	0.920560i	$0.627733\pi$
$492$	0	0
$493$	−5.33971e6	−0.989465
$494$	0	0
$495$	1.06530e6	0.195415
$496$	0	0
$497$	686489.	0.124664
$498$	0	0
$499$	2.14757e6	0.386097	0.193048	−	0.981189i	$-0.438163\pi$
$499$	2.14757e6	0.386097	0.193048	+	0.981189i	$0.438163\pi$
$500$	0	0
$501$	1.44226e6	0.256714
$502$	0	0
$503$	−8.19538e6	−1.44427	−0.722136	−	0.691751i	$-0.756839\pi$
$503$	−8.19538e6	−1.44427	−0.722136	+	0.691751i	$0.756839\pi$
$504$	0	0
$505$	−2.09237e6	−0.365099
$506$	0	0
$507$	1.81181e6	0.313035
$508$	0	0
$509$	−6.34369e6	−1.08529	−0.542647	−	0.839961i	$-0.682578\pi$
$509$	−6.34369e6	−1.08529	−0.542647	+	0.839961i	$0.682578\pi$
$510$	0	0
$511$	326073.	0.0552411
$512$	0	0
$513$	−276507.	−0.0463887
$514$	0	0
$515$	−604776.	−0.100479
$516$	0	0
$517$	−1.44990e6	−0.238568
$518$	0	0
$519$	−1.10604e6	−0.180241
$520$	0	0
$521$	5.64374e6	0.910905	0.455452	−	0.890260i	$-0.349478\pi$
$521$	5.64374e6	0.910905	0.455452	+	0.890260i	$0.349478\pi$
$522$	0	0
$523$	1.12790e6	0.180309	0.0901545	−	0.995928i	$-0.471264\pi$
$523$	1.12790e6	0.180309	0.0901545	+	0.995928i	$0.471264\pi$
$524$	0	0
$525$	19617.9	0.00310638
$526$	0	0
$527$	−6.35535e6	−0.996811
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	2.60376e6	0.400742
$532$	0	0
$533$	2.08279e6	0.317561
$534$	0	0
$535$	−2.21986e6	−0.335305
$536$	0	0
$537$	−5.48197e6	−0.820354
$538$	0	0
$539$	4.11093e6	0.609492
$540$	0	0
$541$	4.95838e6	0.728361	0.364181	−	0.931328i	$-0.381349\pi$
$541$	4.95838e6	0.728361	0.364181	+	0.931328i	$0.381349\pi$
$542$	0	0
$543$	2.40438e6	0.349948
$544$	0	0
$545$	1.14704e7	1.65420
$546$	0	0
$547$	6.92427e6	0.989476	0.494738	−	0.869042i	$-0.335264\pi$
$547$	6.92427e6	0.989476	0.494738	+	0.869042i	$0.335264\pi$
$548$	0	0
$549$	−3.90473e6	−0.552917
$550$	0	0
$551$	1.56870e6	0.220121
$552$	0	0
$553$	613596.	0.0853237
$554$	0	0
$555$	617172.	0.0850499
$556$	0	0
$557$	−3.65953e6	−0.499789	−0.249895	−	0.968273i	$-0.580396\pi$
$557$	−3.65953e6	−0.499789	−0.249895	+	0.968273i	$0.580396\pi$
$558$	0	0
$559$	−7.66332e6	−1.03726
$560$	0	0
$561$	−2.85425e6	−0.382900
$562$	0	0
$563$	3.14658e6	0.418376	0.209188	−	0.977875i	$-0.432918\pi$
$563$	3.14658e6	0.418376	0.209188	+	0.977875i	$0.432918\pi$
$564$	0	0
$565$	−8.33796e6	−1.09885
$566$	0	0
$567$	55371.7	0.00723319
$568$	0	0
$569$	−8.10327e6	−1.04925	−0.524626	−	0.851333i	$-0.675795\pi$
$569$	−8.10327e6	−1.04925	−0.524626	+	0.851333i	$0.675795\pi$
$570$	0	0
$571$	−5.04337e6	−0.647337	−0.323669	−	0.946170i	$-0.604916\pi$
$571$	−5.04337e6	−0.647337	−0.323669	+	0.946170i	$0.604916\pi$
$572$	0	0
$573$	5.77932e6	0.735344
$574$	0	0
$575$	136631.	0.0172337
$576$	0	0
$577$	1.37890e7	1.72422	0.862112	−	0.506717i	$-0.169141\pi$
$577$	1.37890e7	1.72422	0.862112	+	0.506717i	$0.169141\pi$
$578$	0	0
$579$	4.57565e6	0.567227
$580$	0	0
$581$	127236.	0.0156376
$582$	0	0
$583$	−8.78444e6	−1.07039
$584$	0	0
$585$	−1.78804e6	−0.216017
$586$	0	0
$587$	−9.96919e6	−1.19417	−0.597083	−	0.802179i	$-0.703674\pi$
$587$	−9.96919e6	−1.19417	−0.597083	+	0.802179i	$0.703674\pi$
$588$	0	0
$589$	1.86707e6	0.221755
$590$	0	0
$591$	6.47857e6	0.762975
$592$	0	0
$593$	−1.11857e7	−1.30626	−0.653128	−	0.757248i	$-0.726544\pi$
$593$	−1.11857e7	−1.30626	−0.653128	+	0.757248i	$0.726544\pi$
$594$	0	0
$595$	583399.	0.0675575
$596$	0	0
$597$	7.34235e6	0.843139
$598$	0	0
$599$	4.11762e6	0.468898	0.234449	−	0.972128i	$-0.424671\pi$
$599$	4.11762e6	0.468898	0.234449	+	0.972128i	$0.424671\pi$
$600$	0	0
$601$	−595995.	−0.0673064	−0.0336532	−	0.999434i	$-0.510714\pi$
$601$	−595995.	−0.0673064	−0.0336532	+	0.999434i	$0.510714\pi$
$602$	0	0
$603$	1.27336e6	0.142613
$604$	0	0
$605$	5.39237e6	0.598951
$606$	0	0
$607$	8.92759e6	0.983473	0.491737	−	0.870744i	$-0.336362\pi$
$607$	8.92759e6	0.983473	0.491737	+	0.870744i	$0.336362\pi$
$608$	0	0
$609$	−314139.	−0.0343225
$610$	0	0
$611$	2.43357e6	0.263719
$612$	0	0
$613$	−1.04068e7	−1.11858	−0.559289	−	0.828973i	$-0.688926\pi$
$613$	−1.04068e7	−1.11858	−0.559289	+	0.828973i	$0.688926\pi$
$614$	0	0
$615$	2.43433e6	0.259533
$616$	0	0
$617$	4.59738e6	0.486181	0.243090	−	0.970004i	$-0.421839\pi$
$617$	4.59738e6	0.486181	0.243090	+	0.970004i	$0.421839\pi$
$618$	0	0
$619$	1.12847e7	1.18375	0.591877	−	0.806028i	$-0.298387\pi$
$619$	1.12847e7	1.18375	0.591877	+	0.806028i	$0.298387\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	−308088.	−0.0318020
$624$	0	0
$625$	−8.89179e6	−0.910519
$626$	0	0
$627$	838523.	0.0851816
$628$	0	0
$629$	−1.65359e6	−0.166648
$630$	0	0
$631$	1.28868e7	1.28846	0.644229	−	0.764833i	$-0.277178\pi$
$631$	1.28868e7	1.28846	0.644229	+	0.764833i	$0.277178\pi$
$632$	0	0
$633$	−4.90052e6	−0.486108
$634$	0	0
$635$	−1.44768e7	−1.42474
$636$	0	0
$637$	−6.89994e6	−0.673747
$638$	0	0
$639$	6.58872e6	0.638335
$640$	0	0
$641$	8.28486e6	0.796417	0.398208	−	0.917295i	$-0.369632\pi$
$641$	8.28486e6	0.796417	0.398208	+	0.917295i	$0.369632\pi$
$642$	0	0
$643$	−186995.	−0.0178362	−0.00891809	−	0.999960i	$-0.502839\pi$
$643$	−186995.	−0.0178362	−0.00891809	+	0.999960i	$0.502839\pi$
$644$	0	0
$645$	−8.95679e6	−0.847722
$646$	0	0
$647$	−1.82808e6	−0.171686	−0.0858430	−	0.996309i	$-0.527358\pi$
$647$	−1.82808e6	−0.171686	−0.0858430	+	0.996309i	$0.527358\pi$
$648$	0	0
$649$	−7.89607e6	−0.735867
$650$	0	0
$651$	−373890.	−0.0345773
$652$	0	0
$653$	1.41528e7	1.29885	0.649427	−	0.760424i	$-0.275009\pi$
$653$	1.41528e7	1.29885	0.649427	+	0.760424i	$0.275009\pi$
$654$	0	0
$655$	−8.02770e6	−0.731118
$656$	0	0
$657$	3.12955e6	0.282858
$658$	0	0
$659$	1.70155e7	1.52627	0.763133	−	0.646241i	$-0.223660\pi$
$659$	1.70155e7	1.52627	0.763133	+	0.646241i	$0.223660\pi$
$660$	0	0
$661$	8.10052e6	0.721123	0.360561	−	0.932736i	$-0.382585\pi$
$661$	8.10052e6	0.721123	0.360561	+	0.932736i	$0.382585\pi$
$662$	0	0
$663$	4.79069e6	0.423267
$664$	0	0
$665$	−171391.	−0.0150291
$666$	0	0
$667$	−2.18785e6	−0.190416
$668$	0	0
$669$	2.67922e6	0.231443
$670$	0	0
$671$	1.18413e7	1.01530
$672$	0	0
$673$	−8.41346e6	−0.716039	−0.358020	−	0.933714i	$-0.616548\pi$
$673$	−8.41346e6	−0.716039	−0.358020	+	0.933714i	$0.616548\pi$
$674$	0	0
$675$	188287.	0.0159060
$676$	0	0
$677$	−1.93255e7	−1.62054	−0.810270	−	0.586057i	$-0.800679\pi$
$677$	−1.93255e7	−1.62054	−0.810270	+	0.586057i	$0.800679\pi$
$678$	0	0
$679$	−1.05135e6	−0.0875134
$680$	0	0
$681$	4.12355e6	0.340725
$682$	0	0
$683$	1.77329e7	1.45455	0.727275	−	0.686346i	$-0.240786\pi$
$683$	1.77329e7	1.45455	0.727275	+	0.686346i	$0.240786\pi$
$684$	0	0
$685$	1.87147e7	1.52390
$686$	0	0
$687$	6.74279e6	0.545065
$688$	0	0
$689$	1.47441e7	1.18324
$690$	0	0
$691$	−1.17239e7	−0.934066	−0.467033	−	0.884240i	$-0.654677\pi$
$691$	−1.17239e7	−0.934066	−0.467033	+	0.884240i	$0.654677\pi$
$692$	0	0
$693$	−167918.	−0.0132820
$694$	0	0
$695$	6.04206e6	0.474486
$696$	0	0
$697$	−6.52230e6	−0.508533
$698$	0	0
$699$	9.48146e6	0.733976
$700$	0	0
$701$	1.45534e7	1.11859	0.559293	−	0.828970i	$-0.311073\pi$
$701$	1.45534e7	1.11859	0.559293	+	0.828970i	$0.311073\pi$
$702$	0	0
$703$	485790.	0.0370733
$704$	0	0
$705$	2.84433e6	0.215529
$706$	0	0
$707$	329810.	0.0248151
$708$	0	0
$709$	−1.28662e7	−0.961245	−0.480622	−	0.876928i	$-0.659589\pi$
$709$	−1.28662e7	−0.961245	−0.480622	+	0.876928i	$0.659589\pi$
$710$	0	0
$711$	5.88912e6	0.436894
$712$	0	0
$713$	−2.60399e6	−0.191830
$714$	0	0
$715$	5.42234e6	0.396663
$716$	0	0
$717$	−5.97435e6	−0.434003
$718$	0	0
$719$	−2.09145e7	−1.50878	−0.754389	−	0.656427i	$-0.772067\pi$
$719$	−2.09145e7	−1.50878	−0.754389	+	0.656427i	$0.772067\pi$
$720$	0	0
$721$	95327.7	0.00682938
$722$	0	0
$723$	−4.12264e6	−0.293312
$724$	0	0
$725$	−1.06821e6	−0.0754762
$726$	0	0
$727$	2.83278e6	0.198782	0.0993909	−	0.995048i	$-0.468311\pi$
$727$	2.83278e6	0.198782	0.0993909	+	0.995048i	$0.468311\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.39979e7	1.66104
$732$	0	0
$733$	−6.70043e6	−0.460620	−0.230310	−	0.973117i	$-0.573974\pi$
$733$	−6.70043e6	−0.460620	−0.230310	+	0.973117i	$0.573974\pi$
$734$	0	0
$735$	−8.06456e6	−0.550633
$736$	0	0
$737$	−3.86155e6	−0.261874
$738$	0	0
$739$	−2.17913e6	−0.146782	−0.0733910	−	0.997303i	$-0.523382\pi$
$739$	−2.17913e6	−0.146782	−0.0733910	+	0.997303i	$0.523382\pi$
$740$	0	0
$741$	−1.40741e6	−0.0941617
$742$	0	0
$743$	−4.25702e6	−0.282900	−0.141450	−	0.989945i	$-0.545177\pi$
$743$	−4.25702e6	−0.282900	−0.141450	+	0.989945i	$0.545177\pi$
$744$	0	0
$745$	3.12856e6	0.206516
$746$	0	0
$747$	1.22118e6	0.0800713
$748$	0	0
$749$	349905.	0.0227901
$750$	0	0
$751$	1.14378e7	0.740017	0.370008	−	0.929028i	$-0.379355\pi$
$751$	1.14378e7	0.740017	0.370008	+	0.929028i	$0.379355\pi$
$752$	0	0
$753$	−4.92785e6	−0.316716
$754$	0	0
$755$	1.65008e7	1.05351
$756$	0	0
$757$	1.01311e7	0.642566	0.321283	−	0.946983i	$-0.395886\pi$
$757$	1.01311e7	0.642566	0.321283	+	0.946983i	$0.395886\pi$
$758$	0	0
$759$	−1.16948e6	−0.0736865
$760$	0	0
$761$	6.21714e6	0.389161	0.194580	−	0.980887i	$-0.437665\pi$
$761$	6.21714e6	0.389161	0.194580	+	0.980887i	$0.437665\pi$
$762$	0	0
$763$	−1.80802e6	−0.112433
$764$	0	0
$765$	5.59929e6	0.345923
$766$	0	0
$767$	1.32531e7	0.813444
$768$	0	0
$769$	−244173.	−0.0148895	−0.00744477	−	0.999972i	$-0.502370\pi$
$769$	−244173.	−0.0148895	−0.00744477	+	0.999972i	$0.502370\pi$
$770$	0	0
$771$	−7.00234e6	−0.424236
$772$	0	0
$773$	6.00999e6	0.361763	0.180882	−	0.983505i	$-0.442105\pi$
$773$	6.00999e6	0.361763	0.180882	+	0.983505i	$0.442105\pi$
$774$	0	0
$775$	−1.27138e6	−0.0760365
$776$	0	0
$777$	−97281.7	−0.00578067
$778$	0	0
$779$	1.91612e6	0.113130
$780$	0	0
$781$	−1.99807e7	−1.17215
$782$	0	0
$783$	−3.01502e6	−0.175746
$784$	0	0
$785$	−1.68082e7	−0.973527
$786$	0	0
$787$	−2.14817e7	−1.23632	−0.618160	−	0.786052i	$-0.712122\pi$
$787$	−2.14817e7	−1.23632	−0.618160	+	0.786052i	$0.712122\pi$
$788$	0	0
$789$	8.40589e6	0.480719
$790$	0	0
$791$	1.31427e6	0.0746868
$792$	0	0
$793$	−1.98749e7	−1.12233
$794$	0	0
$795$	1.72328e7	0.967023
$796$	0	0
$797$	3.35186e6	0.186913	0.0934567	−	0.995623i	$-0.470208\pi$
$797$	3.35186e6	0.186913	0.0934567	+	0.995623i	$0.470208\pi$
$798$	0	0
$799$	−7.62079e6	−0.422312
$800$	0	0
$801$	−2.95694e6	−0.162840
$802$	0	0
$803$	−9.49055e6	−0.519400
$804$	0	0
$805$	239037.	0.0130010
$806$	0	0
$807$	−1.62412e6	−0.0877879
$808$	0	0
$809$	1.65962e7	0.891530	0.445765	−	0.895150i	$-0.352932\pi$
$809$	1.65962e7	0.891530	0.445765	+	0.895150i	$0.352932\pi$
$810$	0	0
$811$	−9.94295e6	−0.530839	−0.265420	−	0.964133i	$-0.585510\pi$
$811$	−9.94295e6	−0.530839	−0.265420	+	0.964133i	$0.585510\pi$
$812$	0	0
$813$	1.19378e7	0.633430
$814$	0	0
$815$	−1.49049e7	−0.786022
$816$	0	0
$817$	−7.05010e6	−0.369522
$818$	0	0
$819$	281840.	0.0146822
$820$	0	0
$821$	−3.12374e7	−1.61740	−0.808700	−	0.588222i	$-0.799828\pi$
$821$	−3.12374e7	−1.61740	−0.808700	+	0.588222i	$0.799828\pi$
$822$	0	0
$823$	−2.10337e7	−1.08247	−0.541236	−	0.840871i	$-0.682043\pi$
$823$	−2.10337e7	−1.08247	−0.541236	+	0.840871i	$0.682043\pi$
$824$	0	0
$825$	−570991.	−0.0292075
$826$	0	0
$827$	−2.06958e7	−1.05225	−0.526124	−	0.850408i	$-0.676355\pi$
$827$	−2.06958e7	−1.05225	−0.526124	+	0.850408i	$0.676355\pi$
$828$	0	0
$829$	−6.00787e6	−0.303623	−0.151811	−	0.988409i	$-0.548511\pi$
$829$	−6.00787e6	−0.303623	−0.151811	+	0.988409i	$0.548511\pi$
$830$	0	0
$831$	1.58801e7	0.797723
$832$	0	0
$833$	2.16074e7	1.07892
$834$	0	0
$835$	8.58014e6	0.425871
$836$	0	0
$837$	−3.58849e6	−0.177051
$838$	0	0
$839$	2.38590e7	1.17017	0.585083	−	0.810973i	$-0.301062\pi$
$839$	2.38590e7	1.17017	0.585083	+	0.810973i	$0.301062\pi$
$840$	0	0
$841$	−3.40610e6	−0.166061
$842$	0	0
$843$	1.78270e7	0.863993
$844$	0	0
$845$	1.07786e7	0.519304
$846$	0	0
$847$	−849972.	−0.0407095
$848$	0	0
$849$	−1.95270e6	−0.0929753
$850$	0	0
$851$	−677527.	−0.0320703
$852$	0	0
$853$	1.64429e7	0.773760	0.386880	−	0.922130i	$-0.373553\pi$
$853$	1.64429e7	0.773760	0.386880	+	0.922130i	$0.373553\pi$
$854$	0	0
$855$	−1.64496e6	−0.0769556
$856$	0	0
$857$	2.65743e7	1.23598	0.617988	−	0.786188i	$-0.287948\pi$
$857$	2.65743e7	1.23598	0.617988	+	0.786188i	$0.287948\pi$
$858$	0	0
$859$	3.10783e7	1.43706	0.718529	−	0.695497i	$-0.244816\pi$
$859$	3.10783e7	1.43706	0.718529	+	0.695497i	$0.244816\pi$
$860$	0	0
$861$	−383712.	−0.0176399
$862$	0	0
$863$	2.59995e7	1.18833	0.594165	−	0.804343i	$-0.297482\pi$
$863$	2.59995e7	1.18833	0.594165	+	0.804343i	$0.297482\pi$
$864$	0	0
$865$	−6.57995e6	−0.299008
$866$	0	0
$867$	−2.22345e6	−0.100457
$868$	0	0
$869$	−1.78591e7	−0.802250
$870$	0	0
$871$	6.48137e6	0.289482
$872$	0	0
$873$	−1.00906e7	−0.448106
$874$	0	0
$875$	1.52879e6	0.0675038
$876$	0	0
$877$	−2.32478e7	−1.02066	−0.510332	−	0.859978i	$-0.670477\pi$
$877$	−2.32478e7	−1.02066	−0.510332	+	0.859978i	$0.670477\pi$
$878$	0	0
$879$	−1.49450e7	−0.652415
$880$	0	0
$881$	2.38838e7	1.03673	0.518363	−	0.855161i	$-0.326542\pi$
$881$	2.38838e7	1.03673	0.518363	+	0.855161i	$0.326542\pi$
$882$	0	0
$883$	1.10275e7	0.475965	0.237982	−	0.971269i	$-0.423514\pi$
$883$	1.10275e7	0.475965	0.237982	+	0.971269i	$0.423514\pi$
$884$	0	0
$885$	1.54900e7	0.664804
$886$	0	0
$887$	1.94887e7	0.831715	0.415858	−	0.909430i	$-0.363481\pi$
$887$	1.94887e7	0.831715	0.415858	+	0.909430i	$0.363481\pi$
$888$	0	0
$889$	2.28190e6	0.0968371
$890$	0	0
$891$	−1.61163e6	−0.0680096
$892$	0	0
$893$	2.23884e6	0.0939493
$894$	0	0
$895$	−3.26127e7	−1.36091
$896$	0	0
$897$	1.96290e6	0.0814548
$898$	0	0
$899$	2.03585e7	0.840130
$900$	0	0
$901$	−4.61716e7	−1.89480
$902$	0	0
$903$	1.41181e6	0.0576180
$904$	0	0
$905$	1.43039e7	0.580539
$906$	0	0
$907$	1.35982e7	0.548861	0.274430	−	0.961607i	$-0.411511\pi$
$907$	1.35982e7	0.548861	0.274430	+	0.961607i	$0.411511\pi$
$908$	0	0
$909$	3.16542e6	0.127064
$910$	0	0
$911$	1.14025e7	0.455202	0.227601	−	0.973754i	$-0.426912\pi$
$911$	1.14025e7	0.455202	0.227601	+	0.973754i	$0.426912\pi$
$912$	0	0
$913$	−3.70329e6	−0.147032
$914$	0	0
$915$	−2.32295e7	−0.917251
$916$	0	0
$917$	1.26537e6	0.0496927
$918$	0	0
$919$	206772.	0.00807613	0.00403807	−	0.999992i	$-0.498715\pi$
$919$	206772.	0.00807613	0.00403807	+	0.999992i	$0.498715\pi$
$920$	0	0
$921$	−8.83353e6	−0.343151
$922$	0	0
$923$	3.35363e7	1.29572
$924$	0	0
$925$	−330799.	−0.0127119
$926$	0	0
$927$	914928.	0.0349693
$928$	0	0
$929$	1.83152e7	0.696261	0.348131	−	0.937446i	$-0.386817\pi$
$929$	1.83152e7	0.696261	0.348131	+	0.937446i	$0.386817\pi$
$930$	0	0
$931$	−6.34781e6	−0.240021
$932$	0	0
$933$	2.12584e7	0.799513
$934$	0	0
$935$	−1.69802e7	−0.635204
$936$	0	0
$937$	−8.14218e6	−0.302964	−0.151482	−	0.988460i	$-0.548405\pi$
$937$	−8.14218e6	−0.302964	−0.151482	+	0.988460i	$0.548405\pi$
$938$	0	0
$939$	−2.25769e7	−0.835603
$940$	0	0
$941$	−826698.	−0.0304350	−0.0152175	−	0.999884i	$-0.504844\pi$
$941$	−826698.	−0.0304350	−0.0152175	+	0.999884i	$0.504844\pi$
$942$	0	0
$943$	−2.67240e6	−0.0978637
$944$	0	0
$945$	329411.	0.0119994
$946$	0	0
$947$	1.08701e7	0.393875	0.196938	−	0.980416i	$-0.436900\pi$
$947$	1.08701e7	0.393875	0.196938	+	0.980416i	$0.436900\pi$
$948$	0	0
$949$	1.59293e7	0.574157
$950$	0	0
$951$	−1.07968e7	−0.387117
$952$	0	0
$953$	4.10365e6	0.146365	0.0731827	−	0.997319i	$-0.476684\pi$
$953$	4.10365e6	0.146365	0.0731827	+	0.997319i	$0.476684\pi$
$954$	0	0
$955$	3.43817e7	1.21988
$956$	0	0
$957$	9.14322e6	0.322715
$958$	0	0
$959$	−2.94990e6	−0.103576
$960$	0	0
$961$	−4.39837e6	−0.153633
$962$	0	0
$963$	3.35828e6	0.116695
$964$	0	0
$965$	2.72209e7	0.940990
$966$	0	0
$967$	4.55792e7	1.56748	0.783738	−	0.621092i	$-0.213311\pi$
$967$	4.55792e7	1.56748	0.783738	+	0.621092i	$0.213311\pi$
$968$	0	0
$969$	4.40734e6	0.150788
$970$	0	0
$971$	4.44072e7	1.51149	0.755745	−	0.654867i	$-0.227275\pi$
$971$	4.44072e7	1.51149	0.755745	+	0.654867i	$0.227275\pi$
$972$	0	0
$973$	−952379.	−0.0322499
$974$	0	0
$975$	958373.	0.0322867
$976$	0	0
$977$	−4.71041e6	−0.157878	−0.0789390	−	0.996879i	$-0.525153\pi$
$977$	−4.71041e6	−0.157878	−0.0789390	+	0.996879i	$0.525153\pi$
$978$	0	0
$979$	8.96709e6	0.299016
$980$	0	0
$981$	−1.73528e7	−0.575703
$982$	0	0
$983$	−1.94997e7	−0.643641	−0.321821	−	0.946801i	$-0.604295\pi$
$983$	−1.94997e7	−0.643641	−0.321821	+	0.946801i	$0.604295\pi$
$984$	0	0
$985$	3.85415e7	1.26572
$986$	0	0
$987$	−448337.	−0.0146491
$988$	0	0
$989$	9.83271e6	0.319656
$990$	0	0
$991$	−5.51747e7	−1.78466	−0.892331	−	0.451382i	$-0.850931\pi$
$991$	−5.51747e7	−1.78466	−0.892331	+	0.451382i	$0.850931\pi$
$992$	0	0
$993$	−1.20406e7	−0.387503
$994$	0	0
$995$	4.36802e7	1.39871
$996$	0	0
$997$	−1.85357e7	−0.590569	−0.295284	−	0.955409i	$-0.595414\pi$
$997$	−1.85357e7	−0.590569	−0.295284	+	0.955409i	$0.595414\pi$
$998$	0	0
$999$	−933681.	−0.0295995

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.b.1.1	✓	5	4.3	odd	2
1104.6.a.q.1.1		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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -53.5417 q^{5} +8.43952 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -53.5417 q^{5} +8.43952 q^{7} +81.0000 q^{9} -245.637 q^{11} +412.287 q^{13} +481.876 q^{15} -1291.09 q^{17} +379.296 q^{19} -75.9556 q^{21} -529.000 q^{23} -258.281 q^{25} -729.000 q^{27} +4135.83 q^{29} +4922.48 q^{31} +2210.74 q^{33} -451.866 q^{35} +1280.77 q^{37} -3710.58 q^{39} +5051.79 q^{41} -18587.4 q^{43} -4336.88 q^{45} +5902.61 q^{47} -16735.8 q^{49} +11619.8 q^{51} +35761.8 q^{53} +13151.9 q^{55} -3413.66 q^{57} +32145.2 q^{59} -48206.5 q^{61} +683.601 q^{63} -22074.6 q^{65} +15720.5 q^{67} +4761.00 q^{69} +81342.2 q^{71} +38636.4 q^{73} +2324.53 q^{75} -2073.06 q^{77} +72705.1 q^{79} +6561.00 q^{81} +15076.3 q^{83} +69127.1 q^{85} -37222.4 q^{87} -36505.4 q^{89} +3479.50 q^{91} -44302.3 q^{93} -20308.2 q^{95} -124575. q^{97} -19896.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} + 16 q^{5} + 134 q^{7} + 405 q^{9}+O(q^{10}) \) 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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