LMFDB - Embedded newform 176.5.h.e.65.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,5,Mod(65,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("176.65");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 176 = 2^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	176.h (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$18.1931135028$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{553})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 271x^{2} + 272x + 19602 $ x^4 - 2x^3 - 271x^2 + 272*x + 19602
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.4
Root			$-11.2580 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	176.65
Dual form			176.5.h.e.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.2580 q^{3} -19.2580 q^{5} +33.9411i q^{7} +45.7420 q^{9} +O(q^{10})$	$q+11.2580 q^{3} -19.2580 q^{5} +33.9411i q^{7} +45.7420 q^{9} +(-2.03190 + 120.983i) q^{11} -72.2602i q^{13} -216.806 q^{15} +517.873i q^{17} +411.672i q^{19} +382.108i q^{21} +967.641 q^{23} -254.130 q^{25} -396.933 q^{27} -640.503i q^{29} -716.487 q^{31} +(-22.8751 + 1362.02i) q^{33} -653.637i q^{35} +1209.64 q^{37} -813.504i q^{39} -966.781i q^{41} +2720.75i q^{43} -880.899 q^{45} +177.085 q^{47} +1249.00 q^{49} +5830.20i q^{51} -4023.28 q^{53} +(39.1304 - 2329.89i) q^{55} +4634.59i q^{57} +1988.02 q^{59} +2396.69i q^{61} +1552.54i q^{63} +1391.59i q^{65} +2007.13 q^{67} +10893.7 q^{69} -4255.04 q^{71} -1288.68i q^{73} -2860.99 q^{75} +(-4106.30 - 68.9651i) q^{77} +1480.23i q^{79} -8173.77 q^{81} -10124.2i q^{83} -9973.18i q^{85} -7210.77i q^{87} +5133.72 q^{89} +2452.59 q^{91} -8066.19 q^{93} -7927.96i q^{95} +2239.34 q^{97} +(-92.9434 + 5534.00i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 30 q^{5} + 230 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 30 * q^5 + 230 * q^9	$ 4 q - 2 q^{3} - 30 q^{5} + 230 q^{9} + 180 q^{11} - 538 q^{15} + 1566 q^{23} - 1722 q^{25} - 506 q^{27} - 4418 q^{31} - 2302 q^{33} - 382 q^{37} - 1172 q^{45} - 5688 q^{47} + 4996 q^{49} - 8568 q^{53} + 862 q^{55} + 3390 q^{59} + 8734 q^{67} + 26314 q^{69} - 3522 q^{71} + 9156 q^{75} - 2880 q^{77} - 31096 q^{81} - 8766 q^{89} - 17280 q^{91} + 20458 q^{93} + 17282 q^{97} + 12562 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 30 * q^5 + 230 * q^9 + 180 * q^11 - 538 * q^15 + 1566 * q^23 - 1722 * q^25 - 506 * q^27 - 4418 * q^31 - 2302 * q^33 - 382 * q^37 - 1172 * q^45 - 5688 * q^47 + 4996 * q^49 - 8568 * q^53 + 862 * q^55 + 3390 * q^59 + 8734 * q^67 + 26314 * q^69 - 3522 * q^71 + 9156 * q^75 - 2880 * q^77 - 31096 * q^81 - 8766 * q^89 - 17280 * q^91 + 20458 * q^93 + 17282 * q^97 + 12562 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$.

$n$	$111$	$133$	$145$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	11.2580			1.25089			0.625443	−	0.780270i	$-0.284918\pi$
$3$	11.2580			1.25089			0.625443	+	0.780270i	$0.284918\pi$
$4$		0			0
$5$	−19.2580			−0.770319			−0.385160	−	0.922850i	$-0.625854\pi$
$5$	−19.2580			−0.770319			−0.385160	+	0.922850i	$0.625854\pi$
$6$		0			0
$7$			33.9411i			0.692676i	0.938110	+	0.346338i	$0.112575\pi$
$7$			33.9411i			0.692676i	−0.938110	+	0.346338i	$0.887425\pi$
$8$		0			0
$9$	45.7420			0.564716
$10$		0			0
$11$	−2.03190	+	120.983i	−0.0167926	+	0.999859i
$11$	−2.03190	+	120.983i	−0.0167926	+	0.999859i
$12$		0			0
$13$		−	72.2602i		−	0.427575i	−0.976880	−	0.213788i	$-0.931420\pi$
$13$		−	72.2602i		−	0.427575i	0.976880	−	0.213788i	$-0.0685801\pi$
$14$		0			0
$15$	−216.806			−0.963581
$16$		0			0
$17$			517.873i			1.79195i	0.444107	+	0.895974i	$0.353521\pi$
$17$			517.873i			1.79195i	−0.444107	+	0.895974i	$0.646479\pi$
$18$		0			0
$19$			411.672i			1.14036i	0.821518	+	0.570182i	$0.193127\pi$
$19$			411.672i			1.14036i	−0.821518	+	0.570182i	$0.806873\pi$
$20$		0			0
$21$			382.108i			0.866459i
$22$		0			0
$23$	967.641			1.82919			0.914594	−	0.404373i	$-0.132510\pi$
$23$	967.641			1.82919			0.914594	+	0.404373i	$0.132510\pi$
$24$		0			0
$25$	−254.130			−0.406609
$26$		0			0
$27$	−396.933			−0.544490
$28$		0			0
$29$		−	640.503i		−	0.761597i	−0.924658	−	0.380799i	$-0.875649\pi$
$29$		−	640.503i		−	0.761597i	0.924658	−	0.380799i	$-0.124351\pi$
$30$		0			0
$31$	−716.487			−0.745564			−0.372782	−	0.927919i	$-0.621596\pi$
$31$	−716.487			−0.745564			−0.372782	+	0.927919i	$0.621596\pi$
$32$		0			0
$33$	−22.8751	+	1362.02i	−0.0210056	+	1.25071i
$34$		0			0
$35$		−	653.637i		−	0.533582i
$36$		0			0
$37$	1209.64			0.883590			0.441795	−	0.897116i	$-0.354342\pi$
$37$	1209.64			0.883590			0.441795	+	0.897116i	$0.354342\pi$
$38$		0			0
$39$		−	813.504i		−	0.534848i
$40$		0			0
$41$		−	966.781i		−	0.575122i	−0.957762	−	0.287561i	$-0.907156\pi$
$41$		−	966.781i		−	0.575122i	0.957762	−	0.287561i	$-0.0928445\pi$
$42$		0			0
$43$			2720.75i			1.47147i	0.677269	+	0.735736i	$0.263163\pi$
$43$			2720.75i			1.47147i	−0.677269	+	0.735736i	$0.736837\pi$
$44$		0			0
$45$	−880.899			−0.435012
$46$		0			0
$47$	177.085			0.0801651			0.0400826	−	0.999196i	$-0.487238\pi$
$47$	177.085			0.0801651			0.0400826	+	0.999196i	$0.487238\pi$
$48$		0			0
$49$	1249.00			0.520200
$50$		0			0
$51$			5830.20i			2.24152i
$52$		0			0
$53$	−4023.28			−1.43228			−0.716140	−	0.697956i	$-0.754093\pi$
$53$	−4023.28			−1.43228			−0.716140	+	0.697956i	$0.754093\pi$
$54$		0			0
$55$	39.1304	−	2329.89i	0.0129357	−	0.770210i
$56$		0			0
$57$			4634.59i			1.42647i
$58$		0			0
$59$	1988.02			0.571107			0.285554	−	0.958363i	$-0.407823\pi$
$59$	1988.02			0.571107			0.285554	+	0.958363i	$0.407823\pi$
$60$		0			0
$61$			2396.69i			0.644097i	0.946723	+	0.322049i	$0.104371\pi$
$61$			2396.69i			0.644097i	−0.946723	+	0.322049i	$0.895629\pi$
$62$		0			0
$63$			1552.54i			0.391165i
$64$		0			0
$65$			1391.59i			0.329370i
$66$		0			0
$67$	2007.13			0.447122			0.223561	−	0.974690i	$-0.428232\pi$
$67$	2007.13			0.447122			0.223561	+	0.974690i	$0.428232\pi$
$68$		0			0
$69$	10893.7			2.28811
$70$		0			0
$71$	−4255.04			−0.844086			−0.422043	−	0.906576i	$-0.638687\pi$
$71$	−4255.04			−0.844086			−0.422043	+	0.906576i	$0.638687\pi$
$72$		0			0
$73$		−	1288.68i		−	0.241824i	−0.992663	−	0.120912i	$-0.961418\pi$
$73$		−	1288.68i		−	0.241824i	0.992663	−	0.120912i	$-0.0385819\pi$
$74$		0			0
$75$	−2860.99			−0.508621
$76$		0			0
$77$	−4106.30	−	68.9651i	−0.692578	−	0.0116318i
$78$		0			0
$79$			1480.23i			0.237178i	0.992943	+	0.118589i	$0.0378371\pi$
$79$			1480.23i			0.237178i	−0.992943	+	0.118589i	$0.962163\pi$
$80$		0			0
$81$	−8173.77			−1.24581
$82$		0			0
$83$		−	10124.2i		−	1.46963i	−0.678270	−	0.734813i	$-0.737270\pi$
$83$		−	10124.2i		−	1.46963i	0.678270	−	0.734813i	$-0.262730\pi$
$84$		0			0
$85$		−	9973.18i		−	1.38037i
$86$		0			0
$87$		−	7210.77i		−	0.952672i
$88$		0			0
$89$	5133.72			0.648115			0.324058	−	0.946037i	$-0.394953\pi$
$89$	5133.72			0.648115			0.324058	+	0.946037i	$0.394953\pi$
$90$		0			0
$91$	2452.59			0.296171
$92$		0			0
$93$	−8066.19			−0.932615
$94$		0			0
$95$		−	7927.96i		−	0.878444i
$96$		0			0
$97$	2239.34			0.238000			0.119000	−	0.992894i	$-0.462031\pi$
$97$	2239.34			0.238000			0.119000	+	0.992894i	$0.462031\pi$
$98$		0			0
$99$	−92.9434	+	5534.00i	−0.00948305	+	0.564637i
$100$		0			0
$101$		−	15580.0i		−	1.52730i	−0.645629	−	0.763651i	$-0.723405\pi$
$101$		−	15580.0i		−	1.52730i	0.645629	−	0.763651i	$-0.276595\pi$
$102$		0			0
$103$	9119.47			0.859597			0.429799	−	0.902925i	$-0.358585\pi$
$103$	9119.47			0.859597			0.429799	+	0.902925i	$0.358585\pi$
$104$		0			0
$105$		−	7358.63i		−	0.667450i
$106$		0			0
$107$		−	11204.9i		−	0.978684i	−0.872092	−	0.489342i	$-0.837237\pi$
$107$		−	11204.9i		−	0.978684i	0.872092	−	0.489342i	$-0.162763\pi$
$108$		0			0
$109$		−	10286.3i		−	0.865776i	−0.901448	−	0.432888i	$-0.857495\pi$
$109$		−	10286.3i		−	0.865776i	0.901448	−	0.432888i	$-0.142505\pi$
$110$		0			0
$111$	13618.0			1.10527
$112$		0			0
$113$	17026.9			1.33345			0.666727	−	0.745302i	$-0.267695\pi$
$113$	17026.9			1.33345			0.666727	+	0.745302i	$0.267695\pi$
$114$		0			0
$115$	−18634.8			−1.40906
$116$		0			0
$117$		−	3305.33i		−	0.241459i
$118$		0			0
$119$	−17577.2			−1.24124
$120$		0			0
$121$	−14632.7	−	491.651i	−0.999436	−	0.0335805i
$122$		0			0
$123$		−	10884.0i		−	0.719413i
$124$		0			0
$125$	16930.3			1.08354
$126$		0			0
$127$			27813.0i			1.72441i	0.506558	+	0.862206i	$0.330918\pi$
$127$			27813.0i			1.72441i	−0.506558	+	0.862206i	$0.669082\pi$
$128$		0			0
$129$			30630.1i			1.84064i
$130$		0			0
$131$			6310.74i			0.367738i	0.982951	+	0.183869i	$0.0588621\pi$
$131$			6310.74i			0.367738i	−0.982951	+	0.183869i	$0.941138\pi$
$132$		0			0
$133$	−13972.6			−0.789903
$134$		0			0
$135$	7644.13			0.419431
$136$		0			0
$137$	−14207.1			−0.756944			−0.378472	−	0.925613i	$-0.623550\pi$
$137$	−14207.1			−0.756944			−0.378472	+	0.925613i	$0.623550\pi$
$138$		0			0
$139$		−	27772.6i		−	1.43743i	−0.695306	−	0.718714i	$-0.744731\pi$
$139$		−	27772.6i		−	1.43743i	0.695306	−	0.718714i	$-0.255269\pi$
$140$		0			0
$141$	1993.62			0.100277
$142$		0			0
$143$	8742.26	+	146.826i	0.427515	+	0.00718010i
$144$		0			0
$145$			12334.8i			0.586673i
$146$		0			0
$147$	14061.2			0.650711
$148$		0			0
$149$		−	23436.8i		−	1.05567i	−0.849348	−	0.527833i	$-0.823005\pi$
$149$		−	23436.8i		−	1.05567i	0.849348	−	0.527833i	$-0.176995\pi$
$150$		0			0
$151$			6556.14i			0.287538i	0.989611	+	0.143769i	$0.0459222\pi$
$151$			6556.14i			0.287538i	−0.989611	+	0.143769i	$0.954078\pi$
$152$		0			0
$153$			23688.6i			1.01194i
$154$		0			0
$155$	13798.1			0.574322
$156$		0			0
$157$	2315.17			0.0939254			0.0469627	−	0.998897i	$-0.485046\pi$
$157$	2315.17			0.0939254			0.0469627	+	0.998897i	$0.485046\pi$
$158$		0			0
$159$	−45293.9			−1.79162
$160$		0			0
$161$			32842.8i			1.26704i
$162$		0			0
$163$	21728.0			0.817796			0.408898	−	0.912580i	$-0.365913\pi$
$163$	21728.0			0.817796			0.408898	+	0.912580i	$0.365913\pi$
$164$		0			0
$165$	440.529	−	26229.8i	0.0161810	−	0.963446i
$166$		0			0
$167$			26470.6i			0.949143i	0.880217	+	0.474571i	$0.157397\pi$
$167$			26470.6i			0.949143i	−0.880217	+	0.474571i	$0.842603\pi$
$168$		0			0
$169$	23339.5			0.817179
$170$		0			0
$171$			18830.7i			0.643982i
$172$		0			0
$173$			6136.89i			0.205048i	0.994731	+	0.102524i	$0.0326919\pi$
$173$			6136.89i			0.205048i	−0.994731	+	0.102524i	$0.967308\pi$
$174$		0			0
$175$		−	8625.47i		−	0.281648i
$176$		0			0
$177$	22381.1			0.714390
$178$		0			0
$179$	8164.31			0.254808			0.127404	−	0.991851i	$-0.459335\pi$
$179$	8164.31			0.254808			0.127404	+	0.991851i	$0.459335\pi$
$180$		0			0
$181$	−10130.6			−0.309228			−0.154614	−	0.987975i	$-0.549413\pi$
$181$	−10130.6			−0.309228			−0.154614	+	0.987975i	$0.549413\pi$
$182$		0			0
$183$			26981.8i			0.805692i
$184$		0			0
$185$	−23295.1			−0.680647
$186$		0			0
$187$	−62653.8	−	1052.27i	−1.79170	−	0.0300915i
$188$		0			0
$189$		−	13472.4i		−	0.377155i
$190$		0			0
$191$	−13853.6			−0.379749			−0.189875	−	0.981808i	$-0.560808\pi$
$191$	−13853.6			−0.379749			−0.189875	+	0.981808i	$0.560808\pi$
$192$		0			0
$193$			55752.8i			1.49676i	0.663270	+	0.748380i	$0.269168\pi$
$193$			55752.8i			1.49676i	−0.663270	+	0.748380i	$0.730832\pi$
$194$		0			0
$195$			15666.4i			0.412004i
$196$		0			0
$197$		−	38577.9i		−	0.994046i	−0.867737	−	0.497023i	$-0.834426\pi$
$197$		−	38577.9i		−	0.994046i	0.867737	−	0.497023i	$-0.165574\pi$
$198$		0			0
$199$	43741.2			1.10455			0.552275	−	0.833662i	$-0.313760\pi$
$199$	43741.2			1.10455			0.552275	+	0.833662i	$0.313760\pi$
$200$		0			0
$201$	22596.2			0.559299
$202$		0			0
$203$	21739.4			0.527540
$204$		0			0
$205$			18618.2i			0.443028i
$206$		0			0
$207$	44261.8			1.03297
$208$		0			0
$209$	−49805.2	−	836.477i	−1.14020	−	0.0191497i
$210$		0			0
$211$			40460.8i			0.908803i	0.890797	+	0.454402i	$0.150147\pi$
$211$			40460.8i			0.908803i	−0.890797	+	0.454402i	$0.849853\pi$
$212$		0			0
$213$	−47903.1			−1.05586
$214$		0			0
$215$		−	52396.2i		−	1.13350i
$216$		0			0
$217$		−	24318.4i		−	0.516434i
$218$		0			0
$219$		−	14507.9i		−	0.302494i
$220$		0			0
$221$	37421.6			0.766193
$222$		0			0
$223$	−12067.6			−0.242667			−0.121334	−	0.992612i	$-0.538717\pi$
$223$	−12067.6			−0.242667			−0.121334	+	0.992612i	$0.538717\pi$
$224$		0			0
$225$	−11624.4			−0.229619
$226$		0			0
$227$			38616.4i			0.749411i	0.927144	+	0.374705i	$0.122256\pi$
$227$			38616.4i			0.749411i	−0.927144	+	0.374705i	$0.877744\pi$
$228$		0			0
$229$	74080.9			1.41265			0.706326	−	0.707887i	$-0.250351\pi$
$229$	74080.9			1.41265			0.706326	+	0.707887i	$0.250351\pi$
$230$		0			0
$231$	−46228.6	−	776.408i	−0.866337	−	0.0145501i
$232$		0			0
$233$			51719.5i			0.952670i	0.879264	+	0.476335i	$0.158035\pi$
$233$			51719.5i			0.952670i	−0.879264	+	0.476335i	$0.841965\pi$
$234$		0			0
$235$	−3410.29			−0.0617527
$236$		0			0
$237$			16664.4i			0.296683i
$238$		0			0
$239$		−	16911.3i		−	0.296062i	−0.988983	−	0.148031i	$-0.952706\pi$
$239$		−	16911.3i		−	0.296062i	0.988983	−	0.148031i	$-0.0472935\pi$
$240$		0			0
$241$			59892.0i			1.03118i	0.856835	+	0.515590i	$0.172427\pi$
$241$			59892.0i			1.03118i	−0.856835	+	0.515590i	$0.827573\pi$
$242$		0			0
$243$	−59868.5			−1.01388
$244$		0			0
$245$	−24053.2			−0.400720
$246$		0			0
$247$	29747.5			0.487592
$248$		0			0
$249$		−	113979.i		−	1.83833i
$250$		0			0
$251$	19186.2			0.304538			0.152269	−	0.988339i	$-0.451342\pi$
$251$	19186.2			0.304538			0.152269	+	0.988339i	$0.451342\pi$
$252$		0			0
$253$	−1966.15	+	117068.i	−0.0307168	+	1.82893i
$254$		0			0
$255$		−	112278.i		−	1.72669i
$256$		0			0
$257$	30625.1			0.463672			0.231836	−	0.972755i	$-0.425527\pi$
$257$	30625.1			0.463672			0.231836	+	0.972755i	$0.425527\pi$
$258$		0			0
$259$			41056.4i			0.612042i
$260$		0			0
$261$		−	29297.9i		−	0.430086i
$262$		0			0
$263$			73815.1i			1.06717i	0.845746	+	0.533585i	$0.179156\pi$
$263$			73815.1i			1.06717i	−0.845746	+	0.533585i	$0.820844\pi$
$264$		0			0
$265$	77480.2			1.10331
$266$		0			0
$267$	57795.3			0.810718
$268$		0			0
$269$	−87292.0			−1.20634			−0.603170	−	0.797612i	$-0.706096\pi$
$269$	−87292.0			−1.20634			−0.603170	+	0.797612i	$0.706096\pi$
$270$		0			0
$271$			4518.78i			0.0615294i	0.999527	+	0.0307647i	$0.00979425\pi$
$271$			4518.78i			0.0615294i	−0.999527	+	0.0307647i	$0.990206\pi$
$272$		0			0
$273$	27611.2			0.370477
$274$		0			0
$275$	516.369	−	30745.4i	0.00682801	−	0.406551i
$276$		0			0
$277$			4641.04i			0.0604861i	0.999543	+	0.0302431i	$0.00962813\pi$
$277$			4641.04i			0.0604861i	−0.999543	+	0.0302431i	$0.990372\pi$
$278$		0			0
$279$	−32773.6			−0.421032
$280$		0			0
$281$		−	58621.1i		−	0.742406i	−0.928552	−	0.371203i	$-0.878945\pi$
$281$		−	58621.1i		−	0.742406i	0.928552	−	0.371203i	$-0.121055\pi$
$282$		0			0
$283$			90326.0i			1.12782i	0.825836	+	0.563910i	$0.190704\pi$
$283$			90326.0i			1.12782i	−0.825836	+	0.563910i	$0.809296\pi$
$284$		0			0
$285$		−	89252.8i		−	1.09883i
$286$		0			0
$287$	32813.6			0.398373
$288$		0			0
$289$	−184671.			−2.21108
$290$		0			0
$291$	25210.4			0.297710
$292$		0			0
$293$		−	77687.4i		−	0.904931i	−0.891782	−	0.452465i	$-0.850545\pi$
$293$		−	77687.4i		−	0.904931i	0.891782	−	0.452465i	$-0.149455\pi$
$294$		0			0
$295$	−38285.3			−0.439935
$296$		0			0
$297$	806.531	−	48022.2i	0.00914341	−	0.544414i
$298$		0			0
$299$		−	69922.0i		−	0.782116i
$300$		0			0
$301$	−92345.3			−1.01925
$302$		0			0
$303$		−	175399.i		−	1.91048i
$304$		0			0
$305$		−	46155.3i		−	0.496160i
$306$		0			0
$307$		−	60487.2i		−	0.641781i	−0.947116	−	0.320890i	$-0.896018\pi$
$307$		−	60487.2i		−	0.641781i	0.947116	−	0.320890i	$-0.103982\pi$
$308$		0			0
$309$	102667.			1.07526
$310$		0			0
$311$	−32106.2			−0.331947			−0.165973	−	0.986130i	$-0.553077\pi$
$311$	−32106.2			−0.331947			−0.165973	+	0.986130i	$0.553077\pi$
$312$		0			0
$313$	111475.			1.13786			0.568932	−	0.822384i	$-0.307357\pi$
$313$	111475.			1.13786			0.568932	+	0.822384i	$0.307357\pi$
$314$		0			0
$315$		−	29898.7i		−	0.301322i
$316$		0			0
$317$	139267.			1.38589			0.692945	−	0.720990i	$-0.256313\pi$
$317$	139267.			1.38589			0.692945	+	0.720990i	$0.256313\pi$
$318$		0			0
$319$	77490.0	+	1301.44i	0.761490	+	0.0127892i
$320$		0			0
$321$		−	126145.i		−	1.22422i
$322$		0			0
$323$	−213194.			−2.04347
$324$		0			0
$325$			18363.5i			0.173856i
$326$		0			0
$327$		−	115803.i		−	1.08299i
$328$		0			0
$329$			6010.46i			0.0555285i
$330$		0			0
$331$	60223.2			0.549677			0.274839	−	0.961490i	$-0.411376\pi$
$331$	60223.2			0.549677			0.274839	+	0.961490i	$0.411376\pi$
$332$		0			0
$333$	55331.2			0.498978
$334$		0			0
$335$	−38653.3			−0.344427
$336$		0			0
$337$			85639.9i			0.754078i	0.926197	+	0.377039i	$0.123058\pi$
$337$			85639.9i			0.754078i	−0.926197	+	0.377039i	$0.876942\pi$
$338$		0			0
$339$	191688.			1.66800
$340$		0			0
$341$	1455.83	−	86682.7i	0.0125200	−	0.745459i
$342$		0			0
$343$			123885.i			1.05301i
$344$		0			0
$345$	−209790.			−1.76257
$346$		0			0
$347$		−	99499.7i		−	0.826348i	−0.910652	−	0.413174i	$-0.864420\pi$
$347$		−	99499.7i		−	0.826348i	0.910652	−	0.413174i	$-0.135580\pi$
$348$		0			0
$349$		−	47515.3i		−	0.390106i	−0.980793	−	0.195053i	$-0.937512\pi$
$349$		−	47515.3i		−	0.390106i	0.980793	−	0.195053i	$-0.0624878\pi$
$350$		0			0
$351$			28682.5i			0.232811i
$352$		0			0
$353$	167482.			1.34406			0.672030	−	0.740524i	$-0.265423\pi$
$353$	167482.			1.34406			0.672030	+	0.740524i	$0.265423\pi$
$354$		0			0
$355$	81943.4			0.650216
$356$		0			0
$357$	−197884.			−1.55265
$358$		0			0
$359$			5063.91i			0.0392914i	0.999807	+	0.0196457i	$0.00625382\pi$
$359$			5063.91i			0.0392914i	−0.999807	+	0.0196457i	$0.993746\pi$
$360$		0			0
$361$	−39152.4			−0.300431
$362$		0			0
$363$	−164735.	−	5535.00i	−1.25018	−	0.0420053i
$364$		0			0
$365$			24817.4i			0.186282i
$366$		0			0
$367$	−264021.			−1.96023			−0.980115	−	0.198429i	$-0.936416\pi$
$367$	−264021.			−1.96023			−0.980115	+	0.198429i	$0.936416\pi$
$368$		0			0
$369$		−	44222.5i		−	0.324781i
$370$		0			0
$371$		−	136555.i		−	0.992106i
$372$		0			0
$373$			270510.i			1.94431i	0.234339	+	0.972155i	$0.424707\pi$
$373$			270510.i			1.94431i	−0.234339	+	0.972155i	$0.575293\pi$
$374$		0			0
$375$	190601.			1.35538
$376$		0			0
$377$	−46282.9			−0.325640
$378$		0			0
$379$	38850.0			0.270466			0.135233	−	0.990814i	$-0.456822\pi$
$379$	38850.0			0.270466			0.135233	+	0.990814i	$0.456822\pi$
$380$		0			0
$381$			313118.i			2.15704i
$382$		0			0
$383$	243070.			1.65704			0.828522	−	0.559956i	$-0.189182\pi$
$383$	243070.			1.65704			0.828522	+	0.559956i	$0.189182\pi$
$384$		0			0
$385$	79079.0	+	1328.13i	0.533506	+	0.00896022i
$386$		0			0
$387$			124453.i			0.830964i
$388$		0			0
$389$	−142544.			−0.941995			−0.470998	−	0.882135i	$-0.656106\pi$
$389$	−142544.			−0.941995			−0.470998	+	0.882135i	$0.656106\pi$
$390$		0			0
$391$			501115.i			3.27781i
$392$		0			0
$393$			71046.2i			0.459998i
$394$		0			0
$395$		−	28506.2i		−	0.182703i
$396$		0			0
$397$	−73860.0			−0.468628			−0.234314	−	0.972161i	$-0.575284\pi$
$397$	−73860.0			−0.468628			−0.234314	+	0.972161i	$0.575284\pi$
$398$		0			0
$399$	−157303.			−0.988079
$400$		0			0
$401$	225776.			1.40407			0.702034	−	0.712143i	$-0.252275\pi$
$401$	225776.			1.40407			0.702034	+	0.712143i	$0.252275\pi$
$402$		0			0
$403$			51773.5i			0.318785i
$404$		0			0
$405$	157410.			0.959673
$406$		0			0
$407$	−2457.86	+	146345.i	−0.0148378	+	0.883466i
$408$		0			0
$409$			97972.9i			0.585679i	0.956162	+	0.292839i	$0.0946001\pi$
$409$			97972.9i			0.585679i	−0.956162	+	0.292839i	$0.905400\pi$
$410$		0			0
$411$	−159943.			−0.946851
$412$		0			0
$413$			67475.8i			0.395592i
$414$		0			0
$415$			194973.i			1.13208i
$416$		0			0
$417$		−	312663.i		−	1.79806i
$418$		0			0
$419$	64572.5			0.367807			0.183903	−	0.982944i	$-0.441127\pi$
$419$	64572.5			0.367807			0.183903	+	0.982944i	$0.441127\pi$
$420$		0			0
$421$	203272.			1.14687			0.573434	−	0.819252i	$-0.305611\pi$
$421$	203272.			1.14687			0.573434	+	0.819252i	$0.305611\pi$
$422$		0			0
$423$	8100.21			0.0452706
$424$		0			0
$425$		−	131607.i		−	0.728621i
$426$		0			0
$427$	−81346.2			−0.446151
$428$		0			0
$429$	98420.1	+	1652.96i	0.534773	+	0.00898149i
$430$		0			0
$431$		−	11113.4i		−	0.0598266i	−0.999552	−	0.0299133i	$-0.990477\pi$
$431$		−	11113.4i		−	0.0598266i	0.999552	−	0.0299133i	$-0.00952311\pi$
$432$		0			0
$433$	−288512.			−1.53882			−0.769409	−	0.638756i	$-0.779449\pi$
$433$	−288512.			−1.53882			−0.769409	+	0.638756i	$0.779449\pi$
$434$		0			0
$435$			138865.i			0.733861i
$436$		0			0
$437$			398350.i			2.08594i
$438$		0			0
$439$		−	107467.i		−	0.557630i	−0.960345	−	0.278815i	$-0.910058\pi$
$439$		−	107467.i		−	0.557630i	0.960345	−	0.278815i	$-0.0899417\pi$
$440$		0			0
$441$	57131.8			0.293765
$442$		0			0
$443$	130812.			0.666563			0.333281	−	0.942827i	$-0.391844\pi$
$443$	130812.			0.666563			0.333281	+	0.942827i	$0.391844\pi$
$444$		0			0
$445$	−98865.0			−0.499255
$446$		0			0
$447$		−	263851.i		−	1.32052i
$448$		0			0
$449$	−158898.			−0.788181			−0.394090	−	0.919072i	$-0.628940\pi$
$449$	−158898.			−0.788181			−0.394090	+	0.919072i	$0.628940\pi$
$450$		0			0
$451$	116964.	+	1964.41i	0.575041	+	0.00965780i
$452$		0			0
$453$			73808.9i			0.359677i
$454$		0			0
$455$	−47232.0			−0.228146
$456$		0			0
$457$		−	291561.i		−	1.39604i	−0.716079	−	0.698019i	$-0.754065\pi$
$457$		−	291561.i		−	1.39604i	0.716079	−	0.698019i	$-0.245935\pi$
$458$		0			0
$459$		−	205561.i		−	0.975698i
$460$		0			0
$461$		−	62731.9i		−	0.295180i	−0.989049	−	0.147590i	$-0.952848\pi$
$461$		−	62731.9i		−	0.295180i	0.989049	−	0.147590i	$-0.0471515\pi$
$462$		0			0
$463$	170883.			0.797146			0.398573	−	0.917137i	$-0.369506\pi$
$463$	170883.			0.797146			0.398573	+	0.917137i	$0.369506\pi$
$464$		0			0
$465$	155339.			0.718411
$466$		0			0
$467$	−94559.0			−0.433580			−0.216790	−	0.976218i	$-0.569559\pi$
$467$	−94559.0			−0.433580			−0.216790	+	0.976218i	$0.569559\pi$
$468$		0			0
$469$			68124.3i			0.309711i
$470$		0			0
$471$	26064.1			0.117490
$472$		0			0
$473$	−329164.	−	5528.30i	−1.47126	−	0.0247098i
$474$		0			0
$475$		−	104618.i		−	0.463682i
$476$		0			0
$477$	−184033.			−0.808832
$478$		0			0
$479$		−	154781.i		−	0.674600i	−0.941397	−	0.337300i	$-0.890486\pi$
$479$		−	154781.i		−	0.674600i	0.941397	−	0.337300i	$-0.109514\pi$
$480$		0			0
$481$		−	87408.6i		−	0.377802i
$482$		0			0
$483$			369744.i			1.58492i
$484$		0			0
$485$	−43125.1			−0.183336
$486$		0			0
$487$	−205330.			−0.865754			−0.432877	−	0.901453i	$-0.642502\pi$
$487$	−205330.			−0.865754			−0.432877	+	0.901453i	$0.642502\pi$
$488$		0			0
$489$	244614.			1.02297
$490$		0			0
$491$		−	180799.i		−	0.749950i	−0.927035	−	0.374975i	$-0.877651\pi$
$491$		−	180799.i		−	0.749950i	0.927035	−	0.374975i	$-0.122349\pi$
$492$		0			0
$493$	331699.			1.36474
$494$		0			0
$495$	1789.90	−	106574.i	0.00730498	−	0.434950i
$496$		0			0
$497$		−	144421.i		−	0.584678i
$498$		0			0
$499$	−198403.			−0.796797			−0.398399	−	0.917212i	$-0.630434\pi$
$499$	−198403.			−0.796797			−0.398399	+	0.917212i	$0.630434\pi$
$500$		0			0
$501$			298006.i			1.18727i
$502$		0			0
$503$			278604.i			1.10116i	0.834781	+	0.550582i	$0.185594\pi$
$503$			278604.i			1.10116i	−0.834781	+	0.550582i	$0.814406\pi$
$504$		0			0
$505$			300040.i			1.17651i
$506$		0			0
$507$	262755.			1.02220
$508$		0			0
$509$	−195175.			−0.753336			−0.376668	−	0.926348i	$-0.622930\pi$
$509$	−195175.			−0.753336			−0.376668	+	0.926348i	$0.622930\pi$
$510$		0			0
$511$	43739.2			0.167506
$512$		0			0
$513$		−	163406.i		−	0.620917i
$514$		0			0
$515$	−175622.			−0.662164
$516$		0			0
$517$	−359.819	+	21424.2i	−0.00134618	+	0.0801538i
$518$		0			0
$519$			69089.0i			0.256492i
$520$		0			0
$521$	−2617.52			−0.00964304			−0.00482152	−	0.999988i	$-0.501535\pi$
$521$	−2617.52			−0.00964304			−0.00482152	+	0.999988i	$0.501535\pi$
$522$		0			0
$523$		−	309803.i		−	1.13261i	−0.824194	−	0.566307i	$-0.808372\pi$
$523$		−	309803.i		−	1.13261i	0.824194	−	0.566307i	$-0.191628\pi$
$524$		0			0
$525$		−	97105.3i		−	0.352310i
$526$		0			0
$527$		−	371049.i		−	1.33601i
$528$		0			0
$529$	656488.			2.34593
$530$		0			0
$531$	90936.2			0.322513
$532$		0			0
$533$	−69859.8			−0.245908
$534$		0			0
$535$			215785.i			0.753899i
$536$		0			0
$537$	91913.7			0.318736
$538$		0			0
$539$	−2537.85	+	151108.i	−0.00873551	+	0.520127i
$540$		0			0
$541$		−	197545.i		−	0.674948i	−0.941335	−	0.337474i	$-0.890427\pi$
$541$		−	197545.i		−	0.674948i	0.941335	−	0.337474i	$-0.109573\pi$
$542$		0			0
$543$	−114050.			−0.386809
$544$		0			0
$545$			198093.i			0.666923i
$546$		0			0
$547$			440541.i			1.47235i	0.676790	+	0.736176i	$0.263370\pi$
$547$			440541.i			1.47235i	−0.676790	+	0.736176i	$0.736630\pi$
$548$		0			0
$549$			109629.i			0.363732i
$550$		0			0
$551$	263677.			0.868498
$552$		0			0
$553$	−50240.6			−0.164288
$554$		0			0
$555$	−262256.			−0.851411
$556$		0			0
$557$		−	422793.i		−	1.36275i	−0.731933	−	0.681376i	$-0.761382\pi$
$557$		−	422793.i		−	1.36275i	0.731933	−	0.681376i	$-0.238618\pi$
$558$		0			0
$559$	196602.			0.629165
$560$		0			0
$561$	−705355.	−	11846.4i	−2.24121	−	0.0376410i
$562$		0			0
$563$			52520.6i			0.165696i	0.996562	+	0.0828482i	$0.0264017\pi$
$563$			52520.6i			0.165696i	−0.996562	+	0.0828482i	$0.973598\pi$
$564$		0			0
$565$	−327903.			−1.02718
$566$		0			0
$567$		−	277427.i		−	0.862944i
$568$		0			0
$569$		−	140197.i		−	0.433025i	−0.976280	−	0.216513i	$-0.930532\pi$
$569$		−	140197.i		−	0.433025i	0.976280	−	0.216513i	$-0.0694683\pi$
$570$		0			0
$571$		−	207548.i		−	0.636569i	−0.947995	−	0.318284i	$-0.896893\pi$
$571$		−	207548.i		−	0.636569i	0.947995	−	0.318284i	$-0.103107\pi$
$572$		0			0
$573$	−155964.			−0.475023
$574$		0			0
$575$	−245907.			−0.743764
$576$		0			0
$577$	−273572.			−0.821712			−0.410856	−	0.911700i	$-0.634770\pi$
$577$	−273572.			−0.821712			−0.410856	+	0.911700i	$0.634770\pi$
$578$		0			0
$579$			627664.i			1.87228i
$580$		0			0
$581$	343628.			1.01797
$582$		0			0
$583$	8174.91	−	486748.i	0.0240517	−	1.43208i
$584$		0			0
$585$			63654.0i			0.186000i
$586$		0			0
$587$	576701.			1.67369			0.836843	−	0.547442i	$-0.184398\pi$
$587$	576701.			1.67369			0.836843	+	0.547442i	$0.184398\pi$
$588$		0			0
$589$		−	294957.i		−	0.850214i
$590$		0			0
$591$		−	434309.i		−	1.24344i
$592$		0			0
$593$		−	671956.i		−	1.91087i	−0.295202	−	0.955435i	$-0.595387\pi$
$593$		−	671956.i		−	1.91087i	0.295202	−	0.955435i	$-0.404613\pi$
$594$		0			0
$595$	338501.			0.956150
$596$		0			0
$597$	492438.			1.38167
$598$		0			0
$599$	−582984.			−1.62481			−0.812406	−	0.583092i	$-0.801843\pi$
$599$	−582984.			−1.62481			−0.812406	+	0.583092i	$0.801843\pi$
$600$		0			0
$601$			377092.i			1.04399i	0.852947	+	0.521997i	$0.174813\pi$
$601$			377092.i			1.04399i	−0.852947	+	0.521997i	$0.825187\pi$
$602$		0			0
$603$	91810.2			0.252497
$604$		0			0
$605$	281797.	+	9468.21i	0.769885	+	0.0258677i
$606$		0			0
$607$		−	465535.i		−	1.26350i	−0.775173	−	0.631750i	$-0.782337\pi$
$607$		−	465535.i		−	1.26350i	0.775173	−	0.631750i	$-0.217663\pi$
$608$		0			0
$609$	244742.			0.659893
$610$		0			0
$611$		−	12796.2i		−	0.0342766i
$612$		0			0
$613$		−	623300.i		−	1.65873i	−0.558705	−	0.829366i	$-0.688702\pi$
$613$		−	623300.i		−	1.65873i	0.558705	−	0.829366i	$-0.311298\pi$
$614$		0			0
$615$			209604.i			0.554177i
$616$		0			0
$617$	197189.			0.517980			0.258990	−	0.965880i	$-0.416610\pi$
$617$	197189.			0.517980			0.258990	+	0.965880i	$0.416610\pi$
$618$		0			0
$619$	−214900.			−0.560861			−0.280431	−	0.959874i	$-0.590477\pi$
$619$	−214900.			−0.560861			−0.280431	+	0.959874i	$0.590477\pi$
$620$		0			0
$621$	−384089.			−0.995976
$622$		0			0
$623$			174244.i			0.448934i
$624$		0			0
$625$	−167211.			−0.428061
$626$		0			0
$627$	−560706.	−	9417.04i	−1.42626	−	0.0239541i
$628$		0			0
$629$			626437.i			1.58335i
$630$		0			0
$631$	40334.6			0.101302			0.0506511	−	0.998716i	$-0.483870\pi$
$631$	40334.6			0.101302			0.0506511	+	0.998716i	$0.483870\pi$
$632$		0			0
$633$			455507.i			1.13681i
$634$		0			0
$635$		−	535623.i		−	1.32835i
$636$		0			0
$637$		−	90253.1i		−	0.222425i
$638$		0			0
$639$	−194634.			−0.476669
$640$		0			0
$641$	−19965.1			−0.0485909			−0.0242954	−	0.999705i	$-0.507734\pi$
$641$	−19965.1			−0.0485909			−0.0242954	+	0.999705i	$0.507734\pi$
$642$		0			0
$643$	−21885.5			−0.0529340			−0.0264670	−	0.999650i	$-0.508426\pi$
$643$	−21885.5			−0.0529340			−0.0264670	+	0.999650i	$0.508426\pi$
$644$		0			0
$645$		−	589875.i		−	1.41788i
$646$		0			0
$647$	345969.			0.826472			0.413236	−	0.910624i	$-0.364398\pi$
$647$	345969.			0.826472			0.413236	+	0.910624i	$0.364398\pi$
$648$		0			0
$649$	−4039.47	+	240517.i	−0.00959037	+	0.571027i
$650$		0			0
$651$		−	273776.i		−	0.646000i
$652$		0			0
$653$	380939.			0.893366			0.446683	−	0.894692i	$-0.352605\pi$
$653$	380939.			0.893366			0.446683	+	0.894692i	$0.352605\pi$
$654$		0			0
$655$		−	121532.i		−	0.283275i
$656$		0			0
$657$		−	58946.8i		−	0.136562i
$658$		0			0
$659$			149577.i			0.344424i	0.985060	+	0.172212i	$0.0550914\pi$
$659$			149577.i			0.344424i	−0.985060	+	0.172212i	$0.944909\pi$
$660$		0			0
$661$	−637662.			−1.45945			−0.729723	−	0.683743i	$-0.760351\pi$
$661$	−637662.			−1.45945			−0.729723	+	0.683743i	$0.760351\pi$
$662$		0			0
$663$	421292.			0.958420
$664$		0			0
$665$	269084.			0.608477
$666$		0			0
$667$		−	619777.i		−	1.39311i
$668$		0			0
$669$	−135857.			−0.303549
$670$		0			0
$671$	−289958.	−	4869.84i	−0.644006	−	0.0108161i
$672$		0			0
$673$		−	165463.i		−	0.365318i	−0.983176	−	0.182659i	$-0.941530\pi$
$673$		−	165463.i		−	0.365318i	0.983176	−	0.182659i	$-0.0584704\pi$
$674$		0			0
$675$	100873.			0.221394
$676$		0			0
$677$		−	420292.i		−	0.917009i	−0.888692	−	0.458504i	$-0.848385\pi$
$677$		−	420292.i		−	0.917009i	0.888692	−	0.458504i	$-0.151615\pi$
$678$		0			0
$679$			76005.7i			0.164857i
$680$		0			0
$681$			434742.i			0.937428i
$682$		0			0
$683$	−265281.			−0.568677			−0.284338	−	0.958724i	$-0.591774\pi$
$683$	−265281.			−0.568677			−0.284338	+	0.958724i	$0.591774\pi$
$684$		0			0
$685$	273600.			0.583089
$686$		0			0
$687$	834001.			1.76707
$688$		0			0
$689$			290723.i			0.612408i
$690$		0			0
$691$	72314.2			0.151449			0.0757247	−	0.997129i	$-0.475873\pi$
$691$	72314.2			0.151449			0.0757247	+	0.997129i	$0.475873\pi$
$692$		0			0
$693$	−187830.	−	3154.60i	−0.391110	−	0.00656868i
$694$		0			0
$695$			534843.i			1.10728i
$696$		0			0
$697$	500669.			1.03059
$698$		0			0
$699$			582257.i			1.19168i
$700$		0			0
$701$			142549.i			0.290088i	0.989425	+	0.145044i	$0.0463324\pi$
$701$			142549.i			0.290088i	−0.989425	+	0.145044i	$0.953668\pi$
$702$		0			0
$703$			497972.i			1.00761i
$704$		0			0
$705$	−38393.0			−0.0772456
$706$		0			0
$707$	528803.			1.05793
$708$		0			0
$709$	−588664.			−1.17105			−0.585525	−	0.810654i	$-0.699112\pi$
$709$	−588664.			−1.17105			−0.585525	+	0.810654i	$0.699112\pi$
$710$		0			0
$711$			67708.7i			0.133938i
$712$		0			0
$713$	−693302.			−1.36378
$714$		0			0
$715$	−168358.	−	2827.57i	−0.329323	−	0.00553097i
$716$		0			0
$717$		−	190387.i		−	0.370340i
$718$		0			0
$719$	133520.			0.258280			0.129140	−	0.991626i	$-0.458778\pi$
$719$	133520.			0.258280			0.129140	+	0.991626i	$0.458778\pi$
$720$		0			0
$721$			309525.i			0.595422i
$722$		0			0
$723$			674263.i			1.28989i
$724$		0			0
$725$			162771.i			0.309672i
$726$		0			0
$727$	−224313.			−0.424410			−0.212205	−	0.977225i	$-0.568064\pi$
$727$	−224313.			−0.424410			−0.212205	+	0.977225i	$0.568064\pi$
$728$		0			0
$729$	−11922.8			−0.0224348
$730$		0			0
$731$	−1.40900e6			−2.63680
$732$		0			0
$733$			297252.i			0.553243i	0.960979	+	0.276622i	$0.0892149\pi$
$733$			297252.i			0.553243i	−0.960979	+	0.276622i	$0.910785\pi$
$734$		0			0
$735$	−270790.			−0.501255
$736$		0			0
$737$	−4078.30	+	242829.i	−0.00750834	+	0.447059i
$738$		0			0
$739$			268988.i			0.492543i	0.969201	+	0.246272i	$0.0792056\pi$
$739$			268988.i			0.492543i	−0.969201	+	0.246272i	$0.920794\pi$
$740$		0			0
$741$	334896.			0.609922
$742$		0			0
$743$			133653.i			0.242104i	0.992646	+	0.121052i	$0.0386267\pi$
$743$			133653.i			0.242104i	−0.992646	+	0.121052i	$0.961373\pi$
$744$		0			0
$745$			451346.i			0.813200i
$746$		0			0
$747$		−	463104.i		−	0.829921i
$748$		0			0
$749$	380309.			0.677911
$750$		0			0
$751$	−201267.			−0.356856			−0.178428	−	0.983953i	$-0.557101\pi$
$751$	−201267.			−0.356856			−0.178428	+	0.983953i	$0.557101\pi$
$752$		0			0
$753$	215998.			0.380942
$754$		0			0
$755$		−	126258.i		−	0.221496i
$756$		0			0
$757$	835429.			1.45787			0.728934	−	0.684584i	$-0.240016\pi$
$757$	835429.			1.45787			0.728934	+	0.684584i	$0.240016\pi$
$758$		0			0
$759$	−22134.9	+	1.31795e6i	−0.0384233	+	2.28778i
$760$		0			0
$761$		−	345963.i		−	0.597393i	−0.954348	−	0.298696i	$-0.903448\pi$
$761$		−	345963.i		−	0.597393i	0.954348	−	0.298696i	$-0.0965518\pi$
$762$		0			0
$763$	349128.			0.599702
$764$		0			0
$765$		−	456194.i		−	0.779518i
$766$		0			0
$767$		−	143655.i		−	0.244191i
$768$		0			0
$769$			1.05669e6i			1.78688i	0.449183	+	0.893440i	$0.351715\pi$
$769$			1.05669e6i			1.78688i	−0.449183	+	0.893440i	$0.648285\pi$
$770$		0			0
$771$	344776.			0.580001
$772$		0			0
$773$	−224559.			−0.375813			−0.187907	−	0.982187i	$-0.560170\pi$
$773$	−224559.			−0.375813			−0.187907	+	0.982187i	$0.560170\pi$
$774$		0			0
$775$	182081.			0.303153
$776$		0			0
$777$			462212.i			0.765595i
$778$		0			0
$779$	397996.			0.655849
$780$		0			0
$781$	8645.83	−	514787.i	0.0141744	−	0.843967i
$782$		0			0
$783$			254237.i			0.414682i
$784$		0			0
$785$	−44585.4			−0.0723525
$786$		0			0
$787$		−	227376.i		−	0.367109i	−0.983010	−	0.183554i	$-0.941240\pi$
$787$		−	227376.i		−	0.367109i	0.983010	−	0.183554i	$-0.0587603\pi$
$788$		0			0
$789$			831008.i			1.33491i
$790$		0			0
$791$			577911.i			0.923651i
$792$		0			0
$793$	173185.			0.275400
$794$		0			0
$795$	872270.			1.38012
$796$		0			0
$797$	−972019.			−1.53023			−0.765117	−	0.643891i	$-0.777319\pi$
$797$	−972019.			−1.53023			−0.765117	+	0.643891i	$0.777319\pi$
$798$		0			0
$799$			91707.4i			0.143652i
$800$		0			0
$801$	234827.			0.366001
$802$		0			0
$803$	155908.	+	2618.47i	0.241790	+	0.00406085i
$804$		0			0
$805$		−	632486.i		−	0.976021i
$806$		0			0
$807$	−982732.			−1.50900
$808$		0			0
$809$			56475.6i			0.0862906i	0.999069	+	0.0431453i	$0.0137378\pi$
$809$			56475.6i			0.0862906i	−0.999069	+	0.0431453i	$0.986262\pi$
$810$		0			0
$811$		−	224590.i		−	0.341467i	−0.985317	−	0.170733i	$-0.945386\pi$
$811$		−	224590.i		−	0.341467i	0.985317	−	0.170733i	$-0.0546137\pi$
$812$		0			0
$813$			50872.3i			0.0769663i
$814$		0			0
$815$	−418438.			−0.629964
$816$		0			0
$817$	−1.12006e6			−1.67801
$818$		0			0
$819$	112187.			0.167253
$820$		0			0
$821$			719097.i			1.06684i	0.845849	+	0.533422i	$0.179094\pi$
$821$			719097.i			1.06684i	−0.845849	+	0.533422i	$0.820906\pi$
$822$		0			0
$823$	−52447.6			−0.0774330			−0.0387165	−	0.999250i	$-0.512327\pi$
$823$	−52447.6			−0.0774330			−0.0387165	+	0.999250i	$0.512327\pi$
$824$		0			0
$825$	5813.26	−	346131.i	0.00854107	−	0.508549i
$826$		0			0
$827$			121409.i			0.177517i	0.996053	+	0.0887586i	$0.0282900\pi$
$827$			121409.i			0.177517i	−0.996053	+	0.0887586i	$0.971710\pi$
$828$		0			0
$829$	555136.			0.807774			0.403887	−	0.914809i	$-0.367659\pi$
$829$	555136.			0.807774			0.403887	+	0.914809i	$0.367659\pi$
$830$		0			0
$831$			52248.7i			0.0756612i
$832$		0			0
$833$			646823.i			0.932171i
$834$		0			0
$835$		−	509771.i		−	0.731143i
$836$		0			0
$837$	284398.			0.405952
$838$		0			0
$839$	728123.			1.03438			0.517191	−	0.855870i	$-0.326978\pi$
$839$	728123.			1.03438			0.517191	+	0.855870i	$0.326978\pi$
$840$		0			0
$841$	297036.			0.419969
$842$		0			0
$843$		−	659955.i		−	0.928665i
$844$		0			0
$845$	−449471.			−0.629489
$846$		0			0
$847$	16687.2	−	496652.i	0.0232604	−	0.692285i
$848$		0			0
$849$			1.01689e6i			1.41078i
$850$		0			0
$851$	1.17049e6			1.61625
$852$		0			0
$853$		−	560810.i		−	0.770757i	−0.922759	−	0.385379i	$-0.874071\pi$
$853$		−	560810.i		−	0.770757i	0.922759	−	0.385379i	$-0.125929\pi$
$854$		0			0
$855$		−	362641.i		−	0.496072i
$856$		0			0
$857$		−	1.02077e6i		−	1.38984i	−0.719086	−	0.694921i	$-0.755439\pi$
$857$		−	1.02077e6i		−	1.38984i	0.719086	−	0.694921i	$-0.244561\pi$
$858$		0			0
$859$	−350246.			−0.474665			−0.237332	−	0.971428i	$-0.576273\pi$
$859$	−350246.			−0.474665			−0.237332	+	0.971428i	$0.576273\pi$
$860$		0			0
$861$	369415.			0.498320
$862$		0			0
$863$	516997.			0.694170			0.347085	−	0.937834i	$-0.387171\pi$
$863$	516997.			0.694170			0.347085	+	0.937834i	$0.387171\pi$
$864$		0			0
$865$		−	118184.i		−	0.157953i
$866$		0			0
$867$	−2.07903e6			−2.76581
$868$		0			0
$869$	−179082.	−	3007.68i	−0.237145	−	0.00398284i
$870$		0			0
$871$		−	145036.i		−	0.191178i
$872$		0			0
$873$	102432.			0.134402
$874$		0			0
$875$			574632.i			0.750540i
$876$		0			0
$877$		−	657406.i		−	0.854741i	−0.904077	−	0.427371i	$-0.859440\pi$
$877$		−	657406.i		−	0.854741i	0.904077	−	0.427371i	$-0.140560\pi$
$878$		0			0
$879$		−	874603.i		−	1.13197i
$880$		0			0
$881$	869395.			1.12012			0.560061	−	0.828452i	$-0.310778\pi$
$881$	869395.			1.12012			0.560061	+	0.828452i	$0.310778\pi$
$882$		0			0
$883$	−955758.			−1.22582			−0.612910	−	0.790153i	$-0.710001\pi$
$883$	−955758.			−1.22582			−0.612910	+	0.790153i	$0.710001\pi$
$884$		0			0
$885$	−431015.			−0.550308
$886$		0			0
$887$		−	436265.i		−	0.554503i	−0.960797	−	0.277251i	$-0.910577\pi$
$887$		−	436265.i		−	0.554503i	0.960797	−	0.277251i	$-0.0894235\pi$
$888$		0			0
$889$	−944006.			−1.19446
$890$		0			0
$891$	16608.3	−	988887.i	0.0209204	−	1.24564i
$892$		0			0
$893$			72900.7i			0.0914174i
$894$		0			0
$895$	−157228.			−0.196284
$896$		0			0
$897$		−	787180.i		−	0.978338i
$898$		0			0
$899$			458912.i			0.567819i
$900$		0			0
$901$		−	2.08355e6i		−	2.56657i
$902$		0			0
$903$	−1.03962e6			−1.27497
$904$		0			0
$905$	195095.			0.238204
$906$		0			0
$907$	707203.			0.859666			0.429833	−	0.902909i	$-0.358573\pi$
$907$	707203.			0.859666			0.429833	+	0.902909i	$0.358573\pi$
$908$		0			0
$909$		−	712661.i		−	0.862493i
$910$		0			0
$911$	−132785.			−0.159998			−0.0799989	−	0.996795i	$-0.525492\pi$
$911$	−132785.			−0.159998			−0.0799989	+	0.996795i	$0.525492\pi$
$912$		0			0
$913$	1.22486e6	+	20571.5i	1.46942	+	0.0246788i
$914$		0			0
$915$		−	519615.i		−	0.620640i
$916$		0			0
$917$	−214194.			−0.254723
$918$		0			0
$919$			964589.i			1.14212i	0.820909	+	0.571059i	$0.193467\pi$
$919$			964589.i			1.14212i	−0.820909	+	0.571059i	$0.806533\pi$
$920$		0			0
$921$		−	680963.i		−	0.802795i
$922$		0			0
$923$			307470.i			0.360911i
$924$		0			0
$925$	−307405.			−0.359275
$926$		0			0
$927$	417143.			0.485429
$928$		0			0
$929$	634849.			0.735596			0.367798	−	0.929906i	$-0.380112\pi$
$929$	634849.			0.735596			0.367798	+	0.929906i	$0.380112\pi$
$930$		0			0
$931$			514178.i			0.593217i
$932$		0			0
$933$	−361451.			−0.415228
$934$		0			0
$935$	1.20659e6	+	20264.6i	1.38018	+	0.0231800i
$936$		0			0
$937$			3331.90i			0.00379501i	0.999998	+	0.00189750i	$0.000603995\pi$
$937$			3331.90i			0.00379501i	−0.999998	+	0.00189750i	$0.999396\pi$
$938$		0			0
$939$	1.25499e6			1.42334
$940$		0			0
$941$			1.13619e6i			1.28313i	0.767069	+	0.641564i	$0.221714\pi$
$941$			1.13619e6i			1.28313i	−0.767069	+	0.641564i	$0.778286\pi$
$942$		0			0
$943$		−	935496.i		−	1.05201i
$944$		0			0
$945$			259451.i			0.290530i
$946$		0			0
$947$	−156679.			−0.174707			−0.0873536	−	0.996177i	$-0.527841\pi$
$947$	−156679.			−0.174707			−0.0873536	+	0.996177i	$0.527841\pi$
$948$		0			0
$949$	−93120.3			−0.103398
$950$		0			0
$951$	1.56786e6			1.73359
$952$		0			0
$953$			343607.i			0.378334i	0.981945	+	0.189167i	$0.0605788\pi$
$953$			343607.i			0.378334i	−0.981945	+	0.189167i	$0.939421\pi$
$954$		0			0
$955$	266793.			0.292528
$956$		0			0
$957$	872380.	+	14651.6i	0.952537	+	0.0159978i
$958$		0			0
$959$		−	482205.i		−	0.524317i
$960$		0			0
$961$	−410168.			−0.444135
$962$		0			0
$963$		−	512537.i		−	0.552679i
$964$		0			0
$965$		−	1.07369e6i		−	1.15298i
$966$		0			0
$967$		−	395609.i		−	0.423071i	−0.977370	−	0.211536i	$-0.932154\pi$
$967$		−	395609.i		−	0.423071i	0.977370	−	0.211536i	$-0.0678464\pi$
$968$		0			0
$969$	−2.40013e6			−2.55615
$970$		0			0
$971$	−1.28188e6			−1.35960			−0.679799	−	0.733398i	$-0.737933\pi$
$971$	−1.28188e6			−1.35960			−0.679799	+	0.733398i	$0.737933\pi$
$972$		0			0
$973$	942632.			0.995672
$974$		0			0
$975$			206736.i			0.217474i
$976$		0			0
$977$	−1.02959e6			−1.07863			−0.539316	−	0.842103i	$-0.681317\pi$
$977$	−1.02959e6			−1.07863			−0.539316	+	0.842103i	$0.681317\pi$
$978$		0			0
$979$	−10431.2	+	621092.i	−0.0108835	+	0.648024i
$980$		0			0
$981$		−	470515.i		−	0.488918i
$982$		0			0
$983$	−1.35529e6			−1.40257			−0.701284	−	0.712882i	$-0.747390\pi$
$983$	−1.35529e6			−1.40257			−0.701284	+	0.712882i	$0.747390\pi$
$984$		0			0
$985$			742933.i			0.765732i
$986$		0			0
$987$			67665.6i			0.0694598i
$988$		0			0
$989$			2.63271e6i			2.69160i
$990$		0			0
$991$	47967.0			0.0488422			0.0244211	−	0.999702i	$-0.492226\pi$
$991$	47967.0			0.0488422			0.0244211	+	0.999702i	$0.492226\pi$
$992$		0			0
$993$	677991.			0.687584
$994$		0			0
$995$	−842368.			−0.850855
$996$		0			0
$997$		−	1.71845e6i		−	1.72881i	−0.502798	−	0.864404i	$-0.667696\pi$
$997$		−	1.71845e6i		−	1.72881i	0.502798	−	0.864404i	$-0.332304\pi$
$998$		0			0
$999$	−480145.			−0.481106

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	176.5.h.e.65.4		4
4.3	odd	2		22.5.b.a.21.1	✓	4
8.3	odd	2		704.5.h.i.65.3		4
8.5	even	2		704.5.h.j.65.2		4
11.10	odd	2	inner	176.5.h.e.65.3		4
12.11	even	2		198.5.d.a.109.4		4
20.3	even	4		550.5.c.a.549.2		8
20.7	even	4		550.5.c.a.549.7		8
20.19	odd	2		550.5.d.a.351.4		4
44.43	even	2		22.5.b.a.21.3	yes	4
88.21	odd	2		704.5.h.j.65.1		4
88.43	even	2		704.5.h.i.65.4		4
132.131	odd	2		198.5.d.a.109.2		4
220.43	odd	4		550.5.c.a.549.6		8
220.87	odd	4		550.5.c.a.549.3		8
220.219	even	2		550.5.d.a.351.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.5.b.a.21.1	✓	4	4.3	odd	2
22.5.b.a.21.3	yes	4	44.43	even	2
176.5.h.e.65.3		4	11.10	odd	2	inner
176.5.h.e.65.4		4	1.1	even	1	trivial
198.5.d.a.109.2		4	132.131	odd	2
198.5.d.a.109.4		4	12.11	even	2
550.5.c.a.549.2		8	20.3	even	4
550.5.c.a.549.3		8	220.87	odd	4
550.5.c.a.549.6		8	220.43	odd	4
550.5.c.a.549.7		8	20.7	even	4
550.5.d.a.351.2		4	220.219	even	2
550.5.d.a.351.4		4	20.19	odd	2
704.5.h.i.65.3		4	8.3	odd	2
704.5.h.i.65.4		4	88.43	even	2
704.5.h.j.65.1		4	88.21	odd	2
704.5.h.j.65.2		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	176.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+11.2580 q^{3} -19.2580 q^{5} +33.9411i q^{7} +45.7420 q^{9} +O(q^{10})\)	\(q+11.2580 q^{3} -19.2580 q^{5} +33.9411i q^{7} +45.7420 q^{9} +(-2.03190 + 120.983i) q^{11} -72.2602i q^{13} -216.806 q^{15} +517.873i q^{17} +411.672i q^{19} +382.108i q^{21} +967.641 q^{23} -254.130 q^{25} -396.933 q^{27} -640.503i q^{29} -716.487 q^{31} +(-22.8751 + 1362.02i) q^{33} -653.637i q^{35} +1209.64 q^{37} -813.504i q^{39} -966.781i q^{41} +2720.75i q^{43} -880.899 q^{45} +177.085 q^{47} +1249.00 q^{49} +5830.20i q^{51} -4023.28 q^{53} +(39.1304 - 2329.89i) q^{55} +4634.59i q^{57} +1988.02 q^{59} +2396.69i q^{61} +1552.54i q^{63} +1391.59i q^{65} +2007.13 q^{67} +10893.7 q^{69} -4255.04 q^{71} -1288.68i q^{73} -2860.99 q^{75} +(-4106.30 - 68.9651i) q^{77} +1480.23i q^{79} -8173.77 q^{81} -10124.2i q^{83} -9973.18i q^{85} -7210.77i q^{87} +5133.72 q^{89} +2452.59 q^{91} -8066.19 q^{93} -7927.96i q^{95} +2239.34 q^{97} +(-92.9434 + 5534.00i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 30 q^{5} + 230 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 30 * q^5 + 230 * q^9	\( 4 q - 2 q^{3} - 30 q^{5} + 230 q^{9} + 180 q^{11} - 538 q^{15} + 1566 q^{23} - 1722 q^{25} - 506 q^{27} - 4418 q^{31} - 2302 q^{33} - 382 q^{37} - 1172 q^{45} - 5688 q^{47} + 4996 q^{49} - 8568 q^{53} + 862 q^{55} + 3390 q^{59} + 8734 q^{67} + 26314 q^{69} - 3522 q^{71} + 9156 q^{75} - 2880 q^{77} - 31096 q^{81} - 8766 q^{89} - 17280 q^{91} + 20458 q^{93} + 17282 q^{97} + 12562 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 30 * q^5 + 230 * q^9 + 180 * q^11 - 538 * q^15 + 1566 * q^23 - 1722 * q^25 - 506 * q^27 - 4418 * q^31 - 2302 * q^33 - 382 * q^37 - 1172 * q^45 - 5688 * q^47 + 4996 * q^49 - 8568 * q^53 + 862 * q^55 + 3390 * q^59 + 8734 * q^67 + 26314 * q^69 - 3522 * q^71 + 9156 * q^75 - 2880 * q^77 - 31096 * q^81 - 8766 * q^89 - 17280 * q^91 + 20458 * q^93 + 17282 * q^97 + 12562 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more