LMFDB - Embedded newform 124.5.c.a.61.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,5,Mod(61,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("124.61");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	124.c (of order $2$, degree $1$, 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:	$12.8178754224$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 478x^{8} + 69668x^{6} + 4198200x^{4} + 101304000x^{2} + 622080000 $ x^10 + 478x^8 + 69668x^6 + 4198200x^4 + 101304000x^2 + 622080000
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			61.10
Root			$16.6119i$ of defining polynomial
Character	$\chi$	$=$	124.61
Dual form			124.5.c.a.61.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.6119i q^{3} -1.78751 q^{5} -31.2283 q^{7} -194.955 q^{9} +O(q^{10})$	$q+16.6119i q^{3} -1.78751 q^{5} -31.2283 q^{7} -194.955 q^{9} +10.8065i q^{11} +112.377i q^{13} -29.6939i q^{15} -202.456i q^{17} +53.3820 q^{19} -518.761i q^{21} -617.863i q^{23} -621.805 q^{25} -1893.00i q^{27} +1183.36i q^{29} +(657.758 - 700.625i) q^{31} -179.516 q^{33} +55.8208 q^{35} +836.523i q^{37} -1866.80 q^{39} -1892.58 q^{41} +781.250i q^{43} +348.483 q^{45} -3018.62 q^{47} -1425.79 q^{49} +3363.18 q^{51} +1982.72i q^{53} -19.3167i q^{55} +886.775i q^{57} +4395.49 q^{59} +2315.03i q^{61} +6088.10 q^{63} -200.875i q^{65} -5510.08 q^{67} +10263.9 q^{69} +860.100 q^{71} -1905.51i q^{73} -10329.4i q^{75} -337.468i q^{77} +5474.24i q^{79} +15655.0 q^{81} +11264.2i q^{83} +361.892i q^{85} -19657.8 q^{87} +8327.32i q^{89} -3509.35i q^{91} +(11638.7 + 10926.6i) q^{93} -95.4207 q^{95} +6014.40 q^{97} -2106.78i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9}+O(q^{10}) $ 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9	$ 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9} - 310 q^{19} - 756 q^{25} + 734 q^{31} - 372 q^{33} - 666 q^{35} - 1924 q^{39} - 210 q^{41} + 650 q^{45} - 1992 q^{47} + 188 q^{49} + 1352 q^{51} + 5610 q^{59} + 3478 q^{63} - 5420 q^{67} + 10160 q^{69} - 1734 q^{71} + 11598 q^{81} - 13244 q^{87} + 3088 q^{93} + 4302 q^{95} + 2942 q^{97}+O(q^{100}) $ 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9 - 310 * q^19 - 756 * q^25 + 734 * q^31 - 372 * q^33 - 666 * q^35 - 1924 * q^39 - 210 * q^41 + 650 * q^45 - 1992 * q^47 + 188 * q^49 + 1352 * q^51 + 5610 * q^59 + 3478 * q^63 - 5420 * q^67 + 10160 * q^69 - 1734 * q^71 + 11598 * q^81 - 13244 * q^87 + 3088 * q^93 + 4302 * q^95 + 2942 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/124\mathbb{Z}\right)^\times$.

$n$	$63$	$65$
$\chi(n)$	$1$	$-1$

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
$3$			16.6119i			1.84577i	0.385081	+	0.922883i	$0.374173\pi$
$3$			16.6119i			1.84577i	−0.385081	+	0.922883i	$0.625827\pi$
$4$		0			0
$5$	−1.78751			−0.0715003			−0.0357501	−	0.999361i	$-0.511382\pi$
$5$	−1.78751			−0.0715003			−0.0357501	+	0.999361i	$0.511382\pi$
$6$		0			0
$7$	−31.2283			−0.637312			−0.318656	−	0.947870i	$-0.603231\pi$
$7$	−31.2283			−0.637312			−0.318656	+	0.947870i	$0.603231\pi$
$8$		0			0
$9$	−194.955			−2.40685
$10$		0			0
$11$			10.8065i			0.0893098i	0.999002	+	0.0446549i	$0.0142188\pi$
$11$			10.8065i			0.0893098i	−0.999002	+	0.0446549i	$0.985781\pi$
$12$		0			0
$13$			112.377i			0.664955i	0.943111	+	0.332478i	$0.107885\pi$
$13$			112.377i			0.664955i	−0.943111	+	0.332478i	$0.892115\pi$
$14$		0			0
$15$		−	29.6939i		−	0.131973i
$16$		0			0
$17$		−	202.456i		−	0.700540i	−0.936649	−	0.350270i	$-0.886090\pi$
$17$		−	202.456i		−	0.700540i	0.936649	−	0.350270i	$-0.113910\pi$
$18$		0			0
$19$	53.3820			0.147873			0.0739363	−	0.997263i	$-0.476444\pi$
$19$	53.3820			0.147873			0.0739363	+	0.997263i	$0.476444\pi$
$20$		0			0
$21$		−	518.761i		−	1.17633i
$22$		0			0
$23$		−	617.863i		−	1.16798i	−0.811760	−	0.583992i	$-0.801490\pi$
$23$		−	617.863i		−	1.16798i	0.811760	−	0.583992i	$-0.198510\pi$
$24$		0			0
$25$	−621.805			−0.994888
$26$		0			0
$27$		−	1893.00i		−	2.59671i
$28$		0			0
$29$			1183.36i			1.40708i	0.710655	+	0.703541i	$0.248399\pi$
$29$			1183.36i			1.40708i	−0.710655	+	0.703541i	$0.751601\pi$
$30$		0			0
$31$	657.758	−	700.625i	0.684452	−	0.729058i
$31$	657.758	−	700.625i	0.684452	−	0.729058i
$32$		0			0
$33$	−179.516			−0.164845
$34$		0			0
$35$	55.8208			0.0455680
$36$		0			0
$37$			836.523i			0.611047i	0.952185	+	0.305523i	$0.0988314\pi$
$37$			836.523i			0.611047i	−0.952185	+	0.305523i	$0.901169\pi$
$38$		0			0
$39$	−1866.80			−1.22735
$40$		0			0
$41$	−1892.58			−1.12587			−0.562933	−	0.826502i	$-0.690327\pi$
$41$	−1892.58			−1.12587			−0.562933	+	0.826502i	$0.690327\pi$
$42$		0			0
$43$			781.250i			0.422526i	0.977429	+	0.211263i	$0.0677576\pi$
$43$			781.250i			0.422526i	−0.977429	+	0.211263i	$0.932242\pi$
$44$		0			0
$45$	348.483			0.172090
$46$		0			0
$47$	−3018.62			−1.36651			−0.683255	−	0.730179i	$-0.739436\pi$
$47$	−3018.62			−1.36651			−0.683255	+	0.730179i	$0.739436\pi$
$48$		0			0
$49$	−1425.79			−0.593834
$50$		0			0
$51$	3363.18			1.29303
$52$		0			0
$53$			1982.72i			0.705846i	0.935652	+	0.352923i	$0.114812\pi$
$53$			1982.72i			0.705846i	−0.935652	+	0.352923i	$0.885188\pi$
$54$		0			0
$55$		−	19.3167i		−	0.00638567i
$56$		0			0
$57$			886.775i			0.272938i
$58$		0			0
$59$	4395.49			1.26271			0.631355	−	0.775494i	$-0.282499\pi$
$59$	4395.49			1.26271			0.631355	+	0.775494i	$0.282499\pi$
$60$		0			0
$61$			2315.03i			0.622151i	0.950385	+	0.311076i	$0.100689\pi$
$61$			2315.03i			0.622151i	−0.950385	+	0.311076i	$0.899311\pi$
$62$		0			0
$63$	6088.10			1.53391
$64$		0			0
$65$		−	200.875i		−	0.0475445i
$66$		0			0
$67$	−5510.08			−1.22746			−0.613731	−	0.789515i	$-0.710332\pi$
$67$	−5510.08			−1.22746			−0.613731	+	0.789515i	$0.710332\pi$
$68$		0			0
$69$	10263.9			2.15582
$70$		0			0
$71$	860.100			0.170621			0.0853104	−	0.996354i	$-0.472812\pi$
$71$	860.100			0.170621			0.0853104	+	0.996354i	$0.472812\pi$
$72$		0			0
$73$		−	1905.51i		−	0.357573i	−0.983888	−	0.178787i	$-0.942783\pi$
$73$		−	1905.51i		−	0.357573i	0.983888	−	0.178787i	$-0.0572172\pi$
$74$		0			0
$75$		−	10329.4i		−	1.83633i
$76$		0			0
$77$		−	337.468i		−	0.0569182i
$78$		0			0
$79$			5474.24i			0.877141i	0.898697	+	0.438571i	$0.144515\pi$
$79$			5474.24i			0.877141i	−0.898697	+	0.438571i	$0.855485\pi$
$80$		0			0
$81$	15655.0			2.38608
$82$		0			0
$83$			11264.2i			1.63510i	0.575859	+	0.817549i	$0.304668\pi$
$83$			11264.2i			1.63510i	−0.575859	+	0.817549i	$0.695332\pi$
$84$		0			0
$85$			361.892i			0.0500888i
$86$		0			0
$87$	−19657.8			−2.59714
$88$		0			0
$89$			8327.32i			1.05130i	0.850702	+	0.525648i	$0.176177\pi$
$89$			8327.32i			1.05130i	−0.850702	+	0.525648i	$0.823823\pi$
$90$		0			0
$91$		−	3509.35i		−	0.423784i
$92$		0			0
$93$	11638.7	+	10926.6i	1.34567	+	1.26334i
$94$		0			0
$95$	−95.4207			−0.0105729
$96$		0			0
$97$	6014.40			0.639217			0.319609	−	0.947550i	$-0.396449\pi$
$97$	6014.40			0.639217			0.319609	+	0.947550i	$0.396449\pi$
$98$		0			0
$99$		−	2106.78i		−	0.214955i
$100$		0			0
$101$	−11483.5			−1.12572			−0.562861	−	0.826551i	$-0.690300\pi$
$101$	−11483.5			−1.12572			−0.562861	+	0.826551i	$0.690300\pi$
$102$		0			0
$103$	−17468.9			−1.64661			−0.823304	−	0.567601i	$-0.807872\pi$
$103$	−17468.9			−1.64661			−0.823304	+	0.567601i	$0.807872\pi$
$104$		0			0
$105$			927.288i			0.0841078i
$106$		0			0
$107$	−1415.86			−0.123666			−0.0618331	−	0.998087i	$-0.519695\pi$
$107$	−1415.86			−0.123666			−0.0618331	+	0.998087i	$0.519695\pi$
$108$		0			0
$109$	−14679.2			−1.23552			−0.617760	−	0.786367i	$-0.711960\pi$
$109$	−14679.2			−1.23552			−0.617760	+	0.786367i	$0.711960\pi$
$110$		0			0
$111$	−13896.2			−1.12785
$112$		0			0
$113$	11386.9			0.891764			0.445882	−	0.895092i	$-0.352890\pi$
$113$	11386.9			0.891764			0.445882	+	0.895092i	$0.352890\pi$
$114$		0			0
$115$			1104.44i			0.0835112i
$116$		0			0
$117$		−	21908.5i		−	1.60045i
$118$		0			0
$119$			6322.35i			0.446462i
$120$		0			0
$121$	14524.2			0.992024
$122$		0			0
$123$		−	31439.3i		−	2.07808i
$124$		0			0
$125$	2228.67			0.142635
$126$		0			0
$127$			17806.6i			1.10401i	0.833839	+	0.552007i	$0.186138\pi$
$127$			17806.6i			1.10401i	−0.833839	+	0.552007i	$0.813862\pi$
$128$		0			0
$129$	−12978.0			−0.779884
$130$		0			0
$131$	22205.3			1.29394			0.646971	−	0.762515i	$-0.276036\pi$
$131$	22205.3			1.29394			0.646971	+	0.762515i	$0.276036\pi$
$132$		0			0
$133$	−1667.03			−0.0942409
$134$		0			0
$135$			3383.76i			0.185666i
$136$		0			0
$137$			3098.75i			0.165100i	0.996587	+	0.0825498i	$0.0263063\pi$
$137$			3098.75i			0.165100i	−0.996587	+	0.0825498i	$0.973694\pi$
$138$		0			0
$139$		−	24453.1i		−	1.26562i	−0.774306	−	0.632812i	$-0.781901\pi$
$139$		−	24453.1i		−	1.26562i	0.774306	−	0.632812i	$-0.218099\pi$
$140$		0			0
$141$		−	50145.0i		−	2.52226i
$142$		0			0
$143$	−1214.40			−0.0593870
$144$		0			0
$145$		−	2115.26i		−	0.100607i
$146$		0			0
$147$		−	23685.1i		−	1.09608i
$148$		0			0
$149$	−12493.9			−0.562764			−0.281382	−	0.959596i	$-0.590793\pi$
$149$	−12493.9			−0.562764			−0.281382	+	0.959596i	$0.590793\pi$
$150$		0			0
$151$		−	32070.2i		−	1.40653i	−0.710929	−	0.703264i	$-0.751725\pi$
$151$		−	32070.2i		−	1.40653i	0.710929	−	0.703264i	$-0.248275\pi$
$152$		0			0
$153$			39469.8i			1.68609i
$154$		0			0
$155$	−1175.75	+	1252.37i	−0.0489385	+	0.0521279i
$156$		0			0
$157$	−20885.7			−0.847323			−0.423661	−	0.905821i	$-0.639255\pi$
$157$	−20885.7			−0.847323			−0.423661	+	0.905821i	$0.639255\pi$
$158$		0			0
$159$	−32936.7			−1.30283
$160$		0			0
$161$			19294.8i			0.744370i
$162$		0			0
$163$	1716.19			0.0645937			0.0322969	−	0.999478i	$-0.489718\pi$
$163$	1716.19			0.0645937			0.0322969	+	0.999478i	$0.489718\pi$
$164$		0			0
$165$	320.886			0.0117865
$166$		0			0
$167$			31561.4i			1.13168i	0.824516	+	0.565839i	$0.191448\pi$
$167$			31561.4i			1.13168i	−0.824516	+	0.565839i	$0.808552\pi$
$168$		0			0
$169$	15932.3			0.557835
$170$		0			0
$171$	−10407.1			−0.355907
$172$		0			0
$173$	21918.1			0.732337			0.366168	−	0.930549i	$-0.380669\pi$
$173$	21918.1			0.732337			0.366168	+	0.930549i	$0.380669\pi$
$174$		0			0
$175$	19417.9			0.634054
$176$		0			0
$177$			73017.4i			2.33067i
$178$		0			0
$179$		−	20453.9i		−	0.638367i	−0.947693	−	0.319184i	$-0.896591\pi$
$179$		−	20453.9i		−	0.638367i	0.947693	−	0.319184i	$-0.103409\pi$
$180$		0			0
$181$			40228.6i			1.22794i	0.789329	+	0.613970i	$0.210429\pi$
$181$			40228.6i			1.22794i	−0.789329	+	0.613970i	$0.789571\pi$
$182$		0			0
$183$	−38456.9			−1.14835
$184$		0			0
$185$		−	1495.29i		−	0.0436900i
$186$		0			0
$187$	2187.84			0.0625651
$188$		0			0
$189$			59115.3i			1.65492i
$190$		0			0
$191$	58795.3			1.61167			0.805834	−	0.592141i	$-0.201717\pi$
$191$	58795.3			1.61167			0.805834	+	0.592141i	$0.201717\pi$
$192$		0			0
$193$	−57041.6			−1.53136			−0.765680	−	0.643222i	$-0.777597\pi$
$193$	−57041.6			−1.53136			−0.765680	+	0.643222i	$0.777597\pi$
$194$		0			0
$195$	3336.92			0.0877559
$196$		0			0
$197$			3782.01i			0.0974518i	0.998812	+	0.0487259i	$0.0155161\pi$
$197$			3782.01i			0.0974518i	−0.998812	+	0.0487259i	$0.984484\pi$
$198$		0			0
$199$		−	61781.8i		−	1.56011i	−0.625712	−	0.780054i	$-0.715192\pi$
$199$		−	61781.8i		−	1.56011i	0.625712	−	0.780054i	$-0.284808\pi$
$200$		0			0
$201$		−	91532.8i		−	2.26561i
$202$		0			0
$203$		−	36954.2i		−	0.896750i
$204$		0			0
$205$	3383.00			0.0804998
$206$		0			0
$207$			120455.i			2.81116i
$208$		0			0
$209$			576.871i			0.0132065i
$210$		0			0
$211$	70489.2			1.58328			0.791640	−	0.610988i	$-0.209228\pi$
$211$	70489.2			1.58328			0.791640	+	0.610988i	$0.209228\pi$
$212$		0			0
$213$			14287.9i			0.314926i
$214$		0			0
$215$		−	1396.49i		−	0.0302107i
$216$		0			0
$217$	−20540.7	+	21879.3i	−0.436209	+	0.464637i
$218$		0			0
$219$	31654.1			0.659996
$220$		0			0
$221$	22751.5			0.465828
$222$		0			0
$223$		−	16523.3i		−	0.332266i	−0.986103	−	0.166133i	$-0.946872\pi$
$223$		−	16523.3i		−	0.332266i	0.986103	−	0.166133i	$-0.0531281\pi$
$224$		0			0
$225$	121224.			2.39455
$226$		0			0
$227$	90269.0			1.75181			0.875905	−	0.482484i	$-0.160265\pi$
$227$	90269.0			1.75181			0.875905	+	0.482484i	$0.160265\pi$
$228$		0			0
$229$		−	39070.6i		−	0.745039i	−0.928024	−	0.372520i	$-0.878494\pi$
$229$		−	39070.6i		−	0.745039i	0.928024	−	0.372520i	$-0.121506\pi$
$230$		0			0
$231$	5605.98			0.105058
$232$		0			0
$233$	92977.5			1.71264			0.856320	−	0.516446i	$-0.172745\pi$
$233$	92977.5			1.71264			0.856320	+	0.516446i	$0.172745\pi$
$234$		0			0
$235$	5395.81			0.0977059
$236$		0			0
$237$	−90937.4			−1.61900
$238$		0			0
$239$			102028.i			1.78618i	0.449882	+	0.893088i	$0.351466\pi$
$239$			102028.i			1.78618i	−0.449882	+	0.893088i	$0.648534\pi$
$240$		0			0
$241$		−	113079.i		−	1.94692i	−0.228864	−	0.973458i	$-0.573501\pi$
$241$		−	113079.i		−	1.94692i	0.228864	−	0.973458i	$-0.426499\pi$
$242$		0			0
$243$			106726.i			1.80742i
$244$		0			0
$245$	2548.62			0.0424593
$246$		0			0
$247$			5998.93i			0.0983286i
$248$		0			0
$249$	−187119.			−3.01801
$250$		0			0
$251$			45849.8i			0.727763i	0.931445	+	0.363882i	$0.118549\pi$
$251$			45849.8i			0.727763i	−0.931445	+	0.363882i	$0.881451\pi$
$252$		0			0
$253$	6676.93			0.104312
$254$		0			0
$255$	−6011.70			−0.0924522
$256$		0			0
$257$	−21707.3			−0.328654			−0.164327	−	0.986406i	$-0.552545\pi$
$257$	−21707.3			−0.328654			−0.164327	+	0.986406i	$0.552545\pi$
$258$		0			0
$259$		−	26123.2i		−	0.389427i
$260$		0			0
$261$		−	230701.i		−	3.38663i
$262$		0			0
$263$		−	32027.5i		−	0.463033i	−0.972831	−	0.231516i	$-0.925631\pi$
$263$		−	32027.5i		−	0.463033i	0.972831	−	0.231516i	$-0.0743687\pi$
$264$		0			0
$265$		−	3544.13i		−	0.0504682i
$266$		0			0
$267$	−138333.			−1.94045
$268$		0			0
$269$			115732.i			1.59937i	0.600423	+	0.799683i	$0.294999\pi$
$269$			115732.i			1.59937i	−0.600423	+	0.799683i	$0.705001\pi$
$270$		0			0
$271$		−	53818.5i		−	0.732812i	−0.930455	−	0.366406i	$-0.880588\pi$
$271$		−	53818.5i		−	0.732812i	0.930455	−	0.366406i	$-0.119412\pi$
$272$		0			0
$273$	58297.0			0.782205
$274$		0			0
$275$		−	6719.52i		−	0.0888532i
$276$		0			0
$277$			64639.9i			0.842444i	0.906958	+	0.421222i	$0.138399\pi$
$277$			64639.9i			0.842444i	−0.906958	+	0.421222i	$0.861601\pi$
$278$		0			0
$279$	−128233.	+	136590.i	−1.64737	+	1.75473i
$280$		0			0
$281$	−49752.2			−0.630086			−0.315043	−	0.949077i	$-0.602019\pi$
$281$	−49752.2			−0.630086			−0.315043	+	0.949077i	$0.602019\pi$
$282$		0			0
$283$	−109383.			−1.36577			−0.682884	−	0.730527i	$-0.739274\pi$
$283$	−109383.			−1.36577			−0.682884	+	0.730527i	$0.739274\pi$
$284$		0			0
$285$		−	1585.12i		−	0.0195151i
$286$		0			0
$287$	59102.1			0.717528
$288$		0			0
$289$	42532.5			0.509244
$290$		0			0
$291$			99910.5i			1.17985i
$292$		0			0
$293$	16019.7			0.186604			0.0933018	−	0.995638i	$-0.470258\pi$
$293$	16019.7			0.186604			0.0933018	+	0.995638i	$0.470258\pi$
$294$		0			0
$295$	−7856.97			−0.0902841
$296$		0			0
$297$	20456.7			0.231912
$298$		0			0
$299$	69433.9			0.776657
$300$		0			0
$301$		−	24397.1i		−	0.269281i
$302$		0			0
$303$		−	190763.i		−	2.07782i
$304$		0			0
$305$		−	4138.12i		−	0.0444840i
$306$		0			0
$307$	−120195.			−1.27529			−0.637644	−	0.770331i	$-0.720091\pi$
$307$	−120195.			−1.27529			−0.637644	+	0.770331i	$0.720091\pi$
$308$		0			0
$309$		−	290191.i		−	3.03925i
$310$		0			0
$311$	79909.5			0.826185			0.413093	−	0.910689i	$-0.364449\pi$
$311$	79909.5			0.826185			0.413093	+	0.910689i	$0.364449\pi$
$312$		0			0
$313$			176806.i			1.80471i	0.430994	+	0.902355i	$0.358163\pi$
$313$			176806.i			1.80471i	−0.430994	+	0.902355i	$0.641837\pi$
$314$		0			0
$315$	−10882.5			−0.109675
$316$		0			0
$317$	−134865.			−1.34209			−0.671045	−	0.741417i	$-0.734154\pi$
$317$	−134865.			−1.34209			−0.671045	+	0.741417i	$0.734154\pi$
$318$		0			0
$319$	−12787.9			−0.125666
$320$		0			0
$321$		−	23520.0i		−	0.228259i
$322$		0			0
$323$		−	10807.5i		−	0.103591i
$324$		0			0
$325$		−	69876.8i		−	0.661556i
$326$		0			0
$327$		−	243850.i		−	2.28048i
$328$		0			0
$329$	94266.4			0.870894
$330$		0			0
$331$		−	48491.9i		−	0.442602i	−0.975206	−	0.221301i	$-0.928970\pi$
$331$		−	48491.9i		−	0.442602i	0.975206	−	0.221301i	$-0.0710303\pi$
$332$		0			0
$333$		−	163084.i		−	1.47070i
$334$		0			0
$335$	9849.30			0.0877638
$336$		0			0
$337$		−	159306.i		−	1.40272i	−0.712805	−	0.701362i	$-0.752576\pi$
$337$		−	159306.i		−	1.40272i	0.712805	−	0.701362i	$-0.247424\pi$
$338$		0			0
$339$			189159.i			1.64599i
$340$		0			0
$341$	7571.29	+	7108.05i	0.0651120	+	0.0611282i
$342$		0			0
$343$	119504.			1.01577
$344$		0			0
$345$	−18346.8			−0.154142
$346$		0			0
$347$			70839.2i			0.588322i	0.955756	+	0.294161i	$0.0950401\pi$
$347$			70839.2i			0.588322i	−0.955756	+	0.294161i	$0.904960\pi$
$348$		0			0
$349$	−19624.1			−0.161116			−0.0805581	−	0.996750i	$-0.525670\pi$
$349$	−19624.1			−0.161116			−0.0805581	+	0.996750i	$0.525670\pi$
$350$		0			0
$351$	212731.			1.72670
$352$		0			0
$353$			182114.i			1.46149i	0.682653	+	0.730743i	$0.260826\pi$
$353$			182114.i			1.46149i	−0.682653	+	0.730743i	$0.739174\pi$
$354$		0			0
$355$	−1537.43			−0.0121994
$356$		0			0
$357$	−105026.			−0.824065
$358$		0			0
$359$	12324.4			0.0956264			0.0478132	−	0.998856i	$-0.484775\pi$
$359$	12324.4			0.0956264			0.0478132	+	0.998856i	$0.484775\pi$
$360$		0			0
$361$	−127471.			−0.978134
$362$		0			0
$363$			241275.i			1.83104i
$364$		0			0
$365$			3406.11i			0.0255666i
$366$		0			0
$367$			146976.i			1.09123i	0.838037	+	0.545613i	$0.183703\pi$
$367$			146976.i			1.09123i	−0.838037	+	0.545613i	$0.816297\pi$
$368$		0			0
$369$	368968.			2.70979
$370$		0			0
$371$		−	61917.0i		−	0.449844i
$372$		0			0
$373$	−73150.0			−0.525771			−0.262885	−	0.964827i	$-0.584674\pi$
$373$	−73150.0			−0.525771			−0.262885	+	0.964827i	$0.584674\pi$
$374$		0			0
$375$			37022.5i			0.263271i
$376$		0			0
$377$	−132982.			−0.935646
$378$		0			0
$379$	242822.			1.69048			0.845240	−	0.534386i	$-0.179457\pi$
$379$	242822.			1.69048			0.845240	+	0.534386i	$0.179457\pi$
$380$		0			0
$381$	−295802.			−2.03775
$382$		0			0
$383$			31084.7i			0.211909i	0.994371	+	0.105954i	$0.0337898\pi$
$383$			31084.7i			0.211909i	−0.994371	+	0.105954i	$0.966210\pi$
$384$		0			0
$385$			603.226i			0.00406966i
$386$		0			0
$387$		−	152309.i		−	1.01696i
$388$		0			0
$389$			24085.4i			0.159168i	0.996828	+	0.0795839i	$0.0253592\pi$
$389$			24085.4i			0.159168i	−0.996828	+	0.0795839i	$0.974641\pi$
$390$		0			0
$391$	−125090.			−0.818219
$392$		0			0
$393$			368872.i			2.38831i
$394$		0			0
$395$		−	9785.24i		−	0.0627158i
$396$		0			0
$397$	−191617.			−1.21577			−0.607886	−	0.794024i	$-0.707982\pi$
$397$	−191617.			−1.21577			−0.607886	+	0.794024i	$0.707982\pi$
$398$		0			0
$399$		−	27692.5i		−	0.173947i
$400$		0			0
$401$			9565.74i			0.0594880i	0.999558	+	0.0297440i	$0.00946921\pi$
$401$			9565.74i			0.0594880i	−0.999558	+	0.0297440i	$0.990531\pi$
$402$		0			0
$403$	78734.4	+	73917.2i	0.484791	+	0.455130i
$404$		0			0
$405$	−27983.5			−0.170605
$406$		0			0
$407$	−9039.87			−0.0545724
$408$		0			0
$409$			252461.i			1.50921i	0.656182	+	0.754603i	$0.272170\pi$
$409$			252461.i			1.50921i	−0.656182	+	0.754603i	$0.727830\pi$
$410$		0			0
$411$	−51476.2			−0.304735
$412$		0			0
$413$	−137264.			−0.804740
$414$		0			0
$415$		−	20134.8i		−	0.116910i
$416$		0			0
$417$	406212.			2.33604
$418$		0			0
$419$	−240528.			−1.37006			−0.685028	−	0.728517i	$-0.740210\pi$
$419$	−240528.			−1.37006			−0.685028	+	0.728517i	$0.740210\pi$
$420$		0			0
$421$	−68479.6			−0.386364			−0.193182	−	0.981163i	$-0.561881\pi$
$421$	−68479.6			−0.386364			−0.193182	+	0.981163i	$0.561881\pi$
$422$		0			0
$423$	588495.			3.28899
$424$		0			0
$425$			125888.i			0.696959i
$426$		0			0
$427$		−	72294.3i		−	0.396504i
$428$		0			0
$429$		−	20173.5i		−	0.109614i
$430$		0			0
$431$	161315.			0.868400			0.434200	−	0.900816i	$-0.357031\pi$
$431$	161315.			0.868400			0.434200	+	0.900816i	$0.357031\pi$
$432$		0			0
$433$			297458.i			1.58654i	0.608872	+	0.793269i	$0.291622\pi$
$433$			297458.i			1.58654i	−0.608872	+	0.793269i	$0.708378\pi$
$434$		0			0
$435$	35138.4			0.185696
$436$		0			0
$437$		−	32982.8i		−	0.172713i
$438$		0			0
$439$	−128308.			−0.665772			−0.332886	−	0.942967i	$-0.608022\pi$
$439$	−128308.			−0.665772			−0.332886	+	0.942967i	$0.608022\pi$
$440$		0			0
$441$	277966.			1.42927
$442$		0			0
$443$	−24343.1			−0.124042			−0.0620208	−	0.998075i	$-0.519755\pi$
$443$	−24343.1			−0.124042			−0.0620208	+	0.998075i	$0.519755\pi$
$444$		0			0
$445$		−	14885.1i		−	0.0751680i
$446$		0			0
$447$		−	207548.i		−	1.03873i
$448$		0			0
$449$		−	273383.i		−	1.35606i	−0.735034	−	0.678030i	$-0.762834\pi$
$449$		−	273383.i		−	1.35606i	0.735034	−	0.678030i	$-0.237166\pi$
$450$		0			0
$451$		−	20452.1i		−	0.100551i
$452$		0			0
$453$	532747.			2.59612
$454$		0			0
$455$			6272.99i			0.0303007i
$456$		0			0
$457$			60844.6i			0.291333i	0.989334	+	0.145666i	$0.0465326\pi$
$457$			60844.6i			0.291333i	−0.989334	+	0.145666i	$0.953467\pi$
$458$		0			0
$459$	−383250.			−1.81910
$460$		0			0
$461$			257674.i			1.21246i	0.795289	+	0.606231i	$0.207319\pi$
$461$			257674.i			1.21246i	−0.795289	+	0.606231i	$0.792681\pi$
$462$		0			0
$463$		−	337335.i		−	1.57362i	−0.617197	−	0.786809i	$-0.711732\pi$
$463$		−	337335.i		−	1.57362i	0.617197	−	0.786809i	$-0.288268\pi$
$464$		0			0
$465$	−20804.3	−	19531.4i	−0.0962158	−	0.0903290i
$466$		0			0
$467$	118376.			0.542788			0.271394	−	0.962468i	$-0.412515\pi$
$467$	118376.			0.542788			0.271394	+	0.962468i	$0.412515\pi$
$468$		0			0
$469$	172070.			0.782276
$470$		0			0
$471$		−	346950.i		−	1.56396i
$472$		0			0
$473$	−8442.57			−0.0377357
$474$		0			0
$475$	−33193.2			−0.147117
$476$		0			0
$477$		−	386541.i		−	1.69887i
$478$		0			0
$479$	−237053.			−1.03318			−0.516588	−	0.856234i	$-0.672798\pi$
$479$	−237053.			−1.03318			−0.516588	+	0.856234i	$0.672798\pi$
$480$		0			0
$481$	−94006.3			−0.406319
$482$		0			0
$483$	−320523.			−1.37393
$484$		0			0
$485$	−10750.8			−0.0457042
$486$		0			0
$487$			36121.6i			0.152303i	0.997096	+	0.0761516i	$0.0242633\pi$
$487$			36121.6i			0.152303i	−0.997096	+	0.0761516i	$0.975737\pi$
$488$		0			0
$489$			28509.2i			0.119225i
$490$		0			0
$491$			46104.7i			0.191241i	0.995418	+	0.0956207i	$0.0304836\pi$
$491$			46104.7i			0.191241i	−0.995418	+	0.0956207i	$0.969516\pi$
$492$		0			0
$493$	239578.			0.985717
$494$		0			0
$495$			3765.88i			0.0153694i
$496$		0			0
$497$	−26859.4			−0.108739
$498$		0			0
$499$		−	18661.7i		−	0.0749462i	−0.999298	−	0.0374731i	$-0.988069\pi$
$499$		−	18661.7i		−	0.0749462i	0.999298	−	0.0374731i	$-0.0119308\pi$
$500$		0			0
$501$	−524294.			−2.08881
$502$		0			0
$503$	305825.			1.20875			0.604376	−	0.796699i	$-0.293423\pi$
$503$	305825.			1.20875			0.604376	+	0.796699i	$0.293423\pi$
$504$		0			0
$505$	20526.8			0.0804895
$506$		0			0
$507$			264666.i			1.02963i
$508$		0			0
$509$		−	58492.2i		−	0.225768i	−0.993608	−	0.112884i	$-0.963991\pi$
$509$		−	58492.2i		−	0.225768i	0.993608	−	0.112884i	$-0.0360089\pi$
$510$		0			0
$511$			59505.7i			0.227886i
$512$		0			0
$513$		−	101052.i		−	0.383983i
$514$		0			0
$515$	31225.7			0.117733
$516$		0			0
$517$		−	32620.7i		−	0.122043i
$518$		0			0
$519$			364101.i			1.35172i
$520$		0			0
$521$	−385543.			−1.42036			−0.710179	−	0.704022i	$-0.751386\pi$
$521$	−385543.			−1.42036			−0.710179	+	0.704022i	$0.751386\pi$
$522$		0			0
$523$		−	81833.2i		−	0.299176i	−0.988748	−	0.149588i	$-0.952205\pi$
$523$		−	81833.2i		−	0.299176i	0.988748	−	0.149588i	$-0.0477947\pi$
$524$		0			0
$525$			322568.i			1.17031i
$526$		0			0
$527$	−141846.	−	133167.i	−0.510734	−	0.479486i
$528$		0			0
$529$	−101914.			−0.364186
$530$		0			0
$531$	−856922.			−3.03915
$532$		0			0
$533$		−	212683.i		−	0.748650i
$534$		0			0
$535$	2530.85			0.00884217
$536$		0			0
$537$	339778.			1.17828
$538$		0			0
$539$		−	15407.8i		−	0.0530351i
$540$		0			0
$541$	87657.9			0.299500			0.149750	−	0.988724i	$-0.452153\pi$
$541$	87657.9			0.299500			0.149750	+	0.988724i	$0.452153\pi$
$542$		0			0
$543$	−668272.			−2.26649
$544$		0			0
$545$	26239.2			0.0883400
$546$		0			0
$547$	117754.			0.393552			0.196776	−	0.980448i	$-0.436953\pi$
$547$	117754.			0.393552			0.196776	+	0.980448i	$0.436953\pi$
$548$		0			0
$549$		−	451325.i		−	1.49742i
$550$		0			0
$551$			63169.9i			0.208069i
$552$		0			0
$553$		−	170951.i		−	0.559012i
$554$		0			0
$555$	24839.6			0.0806415
$556$		0			0
$557$		−	61220.3i		−	0.197326i	−0.995121	−	0.0986632i	$-0.968543\pi$
$557$		−	61220.3i		−	0.197326i	0.995121	−	0.0986632i	$-0.0314566\pi$
$558$		0			0
$559$	−87794.9			−0.280961
$560$		0			0
$561$			36344.1i			0.115480i
$562$		0			0
$563$	116386.			0.367185			0.183592	−	0.983002i	$-0.441227\pi$
$563$	116386.			0.367185			0.183592	+	0.983002i	$0.441227\pi$
$564$		0			0
$565$	−20354.2			−0.0637614
$566$		0			0
$567$	−488880.			−1.52067
$568$		0			0
$569$		−	600299.i		−	1.85414i	−0.374885	−	0.927071i	$-0.622318\pi$
$569$		−	600299.i		−	1.85414i	0.374885	−	0.927071i	$-0.377682\pi$
$570$		0			0
$571$			293030.i			0.898753i	0.893342	+	0.449376i	$0.148354\pi$
$571$			293030.i			0.898753i	−0.893342	+	0.449376i	$0.851646\pi$
$572$		0			0
$573$			976701.i			2.97476i
$574$		0			0
$575$			384190.i			1.16201i
$576$		0			0
$577$	257181.			0.772479			0.386239	−	0.922399i	$-0.373774\pi$
$577$	257181.			0.772479			0.386239	+	0.922399i	$0.373774\pi$
$578$		0			0
$579$		−	947569.i		−	2.82653i
$580$		0			0
$581$		−	351761.i		−	1.04207i
$582$		0			0
$583$	−21426.2			−0.0630389
$584$		0			0
$585$			39161.6i			0.114432i
$586$		0			0
$587$		−	142372.i		−	0.413189i	−0.978427	−	0.206595i	$-0.933762\pi$
$587$		−	142372.i		−	0.413189i	0.978427	−	0.206595i	$-0.0662381\pi$
$588$		0			0
$589$	35112.4	−	37400.7i	0.101212	−	0.107808i
$590$		0			0
$591$	−62826.3			−0.179873
$592$		0			0
$593$	24509.1			0.0696977			0.0348489	−	0.999393i	$-0.488905\pi$
$593$	24509.1			0.0696977			0.0348489	+	0.999393i	$0.488905\pi$
$594$		0			0
$595$		−	11301.3i		−	0.0319222i
$596$		0			0
$597$	1.02631e6			2.87959
$598$		0			0
$599$	−90130.8			−0.251200			−0.125600	−	0.992081i	$-0.540086\pi$
$599$	−90130.8			−0.251200			−0.125600	+	0.992081i	$0.540086\pi$
$600$		0			0
$601$			193343.i			0.535277i	0.963519	+	0.267638i	$0.0862433\pi$
$601$			193343.i			0.535277i	−0.963519	+	0.267638i	$0.913757\pi$
$602$		0			0
$603$	1.07422e6			2.95432
$604$		0			0
$605$	−25962.1			−0.0709300
$606$		0			0
$607$	74303.7			0.201666			0.100833	−	0.994903i	$-0.467849\pi$
$607$	74303.7			0.201666			0.100833	+	0.994903i	$0.467849\pi$
$608$		0			0
$609$	613879.			1.65519
$610$		0			0
$611$		−	339225.i		−	0.908668i
$612$		0			0
$613$			193854.i			0.515887i	0.966160	+	0.257944i	$0.0830449\pi$
$613$			193854.i			0.515887i	−0.966160	+	0.257944i	$0.916955\pi$
$614$		0			0
$615$			56198.1i			0.148584i
$616$		0			0
$617$	−21846.3			−0.0573862			−0.0286931	−	0.999588i	$-0.509135\pi$
$617$	−21846.3			−0.0573862			−0.0286931	+	0.999588i	$0.509135\pi$
$618$		0			0
$619$		−	619082.i		−	1.61572i	−0.589372	−	0.807862i	$-0.700625\pi$
$619$		−	619082.i		−	1.61572i	0.589372	−	0.807862i	$-0.299375\pi$
$620$		0			0
$621$	−1.16962e6			−3.03292
$622$		0			0
$623$		−	260048.i		−	0.670004i
$624$		0			0
$625$	384644.			0.984689
$626$		0			0
$627$	−9582.92			−0.0243760
$628$		0			0
$629$	169359.			0.428063
$630$		0			0
$631$			374450.i			0.940450i	0.882547	+	0.470225i	$0.155827\pi$
$631$			374450.i			0.940450i	−0.882547	+	0.470225i	$0.844173\pi$
$632$		0			0
$633$			1.17096e6i			2.92236i
$634$		0			0
$635$		−	31829.5i		−	0.0789373i
$636$		0			0
$637$		−	160227.i		−	0.394873i
$638$		0			0
$639$	−167681.			−0.410659
$640$		0			0
$641$		−	270904.i		−	0.659324i	−0.944099	−	0.329662i	$-0.893065\pi$
$641$		−	270904.i		−	0.659324i	0.944099	−	0.329662i	$-0.106935\pi$
$642$		0			0
$643$			235278.i			0.569062i	0.958667	+	0.284531i	$0.0918379\pi$
$643$			235278.i			0.569062i	−0.958667	+	0.284531i	$0.908162\pi$
$644$		0			0
$645$	23198.3			0.0557619
$646$		0			0
$647$		−	129051.i		−	0.308286i	−0.988049	−	0.154143i	$-0.950738\pi$
$647$		−	129051.i		−	0.308286i	0.988049	−	0.154143i	$-0.0492617\pi$
$648$		0			0
$649$			47499.8i			0.112772i
$650$		0			0
$651$	−363457.	−	341219.i	−0.857612	−	0.805140i
$652$		0			0
$653$	−267010.			−0.626183			−0.313092	−	0.949723i	$-0.601365\pi$
$653$	−267010.			−0.626183			−0.313092	+	0.949723i	$0.601365\pi$
$654$		0			0
$655$	−39692.2			−0.0925172
$656$		0			0
$657$			371488.i			0.860625i
$658$		0			0
$659$	325900.			0.750436			0.375218	−	0.926937i	$-0.377568\pi$
$659$	325900.			0.750436			0.375218	+	0.926937i	$0.377568\pi$
$660$		0			0
$661$	487942.			1.11677			0.558387	−	0.829581i	$-0.311421\pi$
$661$	487942.			1.11677			0.558387	+	0.829581i	$0.311421\pi$
$662$		0			0
$663$			377945.i			0.859808i
$664$		0			0
$665$	2979.82			0.00673825
$666$		0			0
$667$	731152.			1.64345
$668$		0			0
$669$	274482.			0.613285
$670$		0			0
$671$	−25017.3			−0.0555642
$672$		0			0
$673$		−	555478.i		−	1.22641i	−0.789922	−	0.613207i	$-0.789879\pi$
$673$		−	555478.i		−	1.22641i	0.789922	−	0.613207i	$-0.210121\pi$
$674$		0			0
$675$			1.17708e6i			2.58344i
$676$		0			0
$677$		−	437954.i		−	0.955545i	−0.878484	−	0.477772i	$-0.841444\pi$
$677$		−	437954.i		−	0.955545i	0.878484	−	0.477772i	$-0.158556\pi$
$678$		0			0
$679$	−187819.			−0.407381
$680$		0			0
$681$			1.49954e6i			3.23343i
$682$		0			0
$683$	−604717.			−1.29632			−0.648158	−	0.761506i	$-0.724460\pi$
$683$	−604717.			−1.29632			−0.648158	+	0.761506i	$0.724460\pi$
$684$		0			0
$685$		−	5539.04i		−	0.0118047i
$686$		0			0
$687$	649036.			1.37517
$688$		0			0
$689$	−222813.			−0.469356
$690$		0			0
$691$	−249705.			−0.522964			−0.261482	−	0.965208i	$-0.584211\pi$
$691$	−249705.			−0.522964			−0.261482	+	0.965208i	$0.584211\pi$
$692$		0			0
$693$			65791.0i			0.136993i
$694$		0			0
$695$			43710.1i			0.0904924i
$696$		0			0
$697$			383165.i			0.788714i
$698$		0			0
$699$			1.54453e6i			3.16113i
$700$		0			0
$701$	−720468.			−1.46615			−0.733076	−	0.680147i	$-0.761916\pi$
$701$	−720468.			−1.46615			−0.733076	+	0.680147i	$0.761916\pi$
$702$		0			0
$703$			44655.3i			0.0903570i
$704$		0			0
$705$			89634.6i			0.180342i
$706$		0			0
$707$	358610.			0.717436
$708$		0			0
$709$			626159.i			1.24564i	0.782365	+	0.622820i	$0.214013\pi$
$709$			626159.i			1.24564i	−0.782365	+	0.622820i	$0.785987\pi$
$710$		0			0
$711$		−	1.06723e6i		−	2.11115i
$712$		0			0
$713$	−432890.	−	406405.i	−0.851528	−	0.799429i
$714$		0			0
$715$	2170.76			0.00424619
$716$		0			0
$717$	−1.69488e6			−3.29686
$718$		0			0
$719$			95222.3i			0.184196i	0.995750	+	0.0920981i	$0.0293573\pi$
$719$			95222.3i			0.184196i	−0.995750	+	0.0920981i	$0.970643\pi$
$720$		0			0
$721$	545523.			1.04940
$722$		0			0
$723$	1.87845e6			3.59355
$724$		0			0
$725$		−	735816.i		−	1.39989i
$726$		0			0
$727$	251949.			0.476699			0.238350	−	0.971179i	$-0.423394\pi$
$727$	251949.			0.476699			0.238350	+	0.971179i	$0.423394\pi$
$728$		0			0
$729$	−504869.			−0.950000
$730$		0			0
$731$	158169.			0.295996
$732$		0			0
$733$	180221.			0.335427			0.167714	−	0.985836i	$-0.446362\pi$
$733$	180221.			0.335427			0.167714	+	0.985836i	$0.446362\pi$
$734$		0			0
$735$			42337.4i			0.0783699i
$736$		0			0
$737$		−	59544.5i		−	0.109624i
$738$		0			0
$739$		−	782110.i		−	1.43212i	−0.698040	−	0.716059i	$-0.745944\pi$
$739$		−	782110.i		−	1.43212i	0.698040	−	0.716059i	$-0.254056\pi$
$740$		0			0
$741$	−99653.5			−0.181491
$742$		0			0
$743$			612084.i			1.10875i	0.832267	+	0.554375i	$0.187043\pi$
$743$			612084.i			1.10875i	−0.832267	+	0.554375i	$0.812957\pi$
$744$		0			0
$745$	22333.0			0.0402378
$746$		0			0
$747$		−	2.19601e6i		−	3.93543i
$748$		0			0
$749$	44214.7			0.0788140
$750$		0			0
$751$	−739583.			−1.31131			−0.655657	−	0.755059i	$-0.727608\pi$
$751$	−739583.			−1.31131			−0.655657	+	0.755059i	$0.727608\pi$
$752$		0			0
$753$	−761652.			−1.34328
$754$		0			0
$755$			57325.8i			0.100567i
$756$		0			0
$757$			4565.83i			0.00796761i	0.999992	+	0.00398380i	$0.00126809\pi$
$757$			4565.83i			0.00796761i	−0.999992	+	0.00398380i	$0.998732\pi$
$758$		0			0
$759$			110916.i			0.192536i
$760$		0			0
$761$			139759.i			0.241329i	0.992693	+	0.120665i	$0.0385025\pi$
$761$			139759.i			0.241329i	−0.992693	+	0.120665i	$0.961497\pi$
$762$		0			0
$763$	458407.			0.787412
$764$		0			0
$765$		−	70552.5i		−	0.120556i
$766$		0			0
$767$			493954.i			0.839645i
$768$		0			0
$769$	646499.			1.09324			0.546620	−	0.837381i	$-0.315914\pi$
$769$	646499.			1.09324			0.546620	+	0.837381i	$0.315914\pi$
$770$		0			0
$771$		−	360599.i		−	0.606618i
$772$		0			0
$773$		−	608409.i		−	1.01821i	−0.860705	−	0.509104i	$-0.829977\pi$
$773$		−	608409.i		−	1.01821i	0.860705	−	0.509104i	$-0.170023\pi$
$774$		0			0
$775$	−408997.	+	435652.i	−0.680953	+	0.725331i
$776$		0			0
$777$	433955.			0.718792
$778$		0			0
$779$	−101030.			−0.166485
$780$		0			0
$781$			9294.65i			0.0152381i
$782$		0			0
$783$	2.24010e6			3.65379
$784$		0			0
$785$	37333.3			0.0605838
$786$		0			0
$787$		−	28049.6i		−	0.0452874i	−0.999744	−	0.0226437i	$-0.992792\pi$
$787$		−	28049.6i		−	0.0452874i	0.999744	−	0.0226437i	$-0.00720833\pi$
$788$		0			0
$789$	532037.			0.854650
$790$		0			0
$791$	−355594.			−0.568332
$792$		0			0
$793$	−260157.			−0.413703
$794$		0			0
$795$	58874.7			0.0931524
$796$		0			0
$797$		−	1.02153e6i		−	1.60818i	−0.594508	−	0.804090i	$-0.702653\pi$
$797$		−	1.02153e6i		−	1.60818i	0.594508	−	0.804090i	$-0.297347\pi$
$798$		0			0
$799$			611138.i			0.957296i
$800$		0			0
$801$		−	1.62345e6i		−	2.53031i
$802$		0			0
$803$	20591.8			0.0319348
$804$		0			0
$805$		−	34489.6i		−	0.0532227i
$806$		0			0
$807$	−1.92252e6			−2.95205
$808$		0			0
$809$		−	10460.0i		−	0.0159821i	−0.999968	−	0.00799104i	$-0.997456\pi$
$809$		−	10460.0i		−	0.0159821i	0.999968	−	0.00799104i	$-0.00254365\pi$
$810$		0			0
$811$	−615528.			−0.935850			−0.467925	−	0.883768i	$-0.654998\pi$
$811$	−615528.			−0.935850			−0.467925	+	0.883768i	$0.654998\pi$
$812$		0			0
$813$	894026.			1.35260
$814$		0			0
$815$	−3067.70			−0.00461847
$816$		0			0
$817$			41704.7i			0.0624800i
$818$		0			0
$819$			684165.i			1.01998i
$820$		0			0
$821$			692194.i			1.02693i	0.858110	+	0.513466i	$0.171639\pi$
$821$			692194.i			1.02693i	−0.858110	+	0.513466i	$0.828361\pi$
$822$		0			0
$823$		−	161312.i		−	0.238159i	−0.992885	−	0.119080i	$-0.962006\pi$
$823$		−	161312.i		−	0.238159i	0.992885	−	0.119080i	$-0.0379944\pi$
$824$		0			0
$825$	111624.			0.164002
$826$		0			0
$827$		−	990799.i		−	1.44869i	−0.689439	−	0.724343i	$-0.742143\pi$
$827$		−	990799.i		−	1.44869i	0.689439	−	0.724343i	$-0.257857\pi$
$828$		0			0
$829$			579231.i			0.842836i	0.906867	+	0.421418i	$0.138467\pi$
$829$			579231.i			0.842836i	−0.906867	+	0.421418i	$0.861533\pi$
$830$		0			0
$831$	−1.07379e6			−1.55495
$832$		0			0
$833$			288661.i			0.416004i
$834$		0			0
$835$		−	56416.1i		−	0.0809153i
$836$		0			0
$837$	−1.32629e6	−	1.24514e6i	−1.89316	−	1.77733i
$838$		0			0
$839$	−454727.			−0.645991			−0.322996	−	0.946400i	$-0.604690\pi$
$839$	−454727.			−0.645991			−0.322996	+	0.946400i	$0.604690\pi$
$840$		0			0
$841$	−693050.			−0.979880
$842$		0			0
$843$		−	826478.i		−	1.16299i
$844$		0			0
$845$	−28479.1			−0.0398853
$846$		0			0
$847$	−453566.			−0.632228
$848$		0			0
$849$		−	1.81706e6i		−	2.52089i
$850$		0			0
$851$	516857.			0.713693
$852$		0			0
$853$	−352196.			−0.484046			−0.242023	−	0.970271i	$-0.577811\pi$
$853$	−352196.			−0.484046			−0.242023	+	0.970271i	$0.577811\pi$
$854$		0			0
$855$	18602.7			0.0254474
$856$		0			0
$857$	−232881.			−0.317083			−0.158541	−	0.987352i	$-0.550679\pi$
$857$	−232881.			−0.317083			−0.158541	+	0.987352i	$0.550679\pi$
$858$		0			0
$859$			365673.i			0.495571i	0.968815	+	0.247786i	$0.0797029\pi$
$859$			365673.i			0.495571i	−0.968815	+	0.247786i	$0.920297\pi$
$860$		0			0
$861$			981797.i			1.32439i
$862$		0			0
$863$			1.16428e6i			1.56327i	0.623735	+	0.781636i	$0.285614\pi$
$863$			1.16428e6i			1.56327i	−0.623735	+	0.781636i	$0.714386\pi$
$864$		0			0
$865$	−39178.8			−0.0523623
$866$		0			0
$867$			706546.i			0.939944i
$868$		0			0
$869$	−59157.2			−0.0783373
$870$		0			0
$871$		−	619208.i		−	0.816207i
$872$		0			0
$873$	−1.17254e6			−1.53850
$874$		0			0
$875$	−69597.6			−0.0909030
$876$		0			0
$877$	−150993.			−0.196317			−0.0981587	−	0.995171i	$-0.531295\pi$
$877$	−150993.			−0.196317			−0.0981587	+	0.995171i	$0.531295\pi$
$878$		0			0
$879$			266118.i			0.344426i
$880$		0			0
$881$			486000.i			0.626159i	0.949727	+	0.313079i	$0.101361\pi$
$881$			486000.i			0.626159i	−0.949727	+	0.313079i	$0.898639\pi$
$882$		0			0
$883$			1.18536e6i			1.52029i	0.649752	+	0.760146i	$0.274873\pi$
$883$			1.18536e6i			1.52029i	−0.649752	+	0.760146i	$0.725127\pi$
$884$		0			0
$885$		−	130519.i		−	0.166643i
$886$		0			0
$887$	−451532.			−0.573907			−0.286953	−	0.957945i	$-0.592642\pi$
$887$	−451532.			−0.573907			−0.286953	+	0.957945i	$0.592642\pi$
$888$		0			0
$889$		−	556071.i		−	0.703601i
$890$		0			0
$891$			169176.i			0.213100i
$892$		0			0
$893$	−161140.			−0.202069
$894$		0			0
$895$			36561.5i			0.0456435i
$896$		0			0
$897$			1.15343e6i			1.43353i
$898$		0			0
$899$	829089.	+	778362.i	1.02584	+	0.963080i
$900$		0			0
$901$	401414.			0.494473
$902$		0			0
$903$	405282.			0.497029
$904$		0			0
$905$		−	71908.8i		−	0.0877981i
$906$		0			0
$907$	−849740.			−1.03293			−0.516466	−	0.856308i	$-0.672753\pi$
$907$	−849740.			−1.03293			−0.516466	+	0.856308i	$0.672753\pi$
$908$		0			0
$909$	2.23876e6			2.70945
$910$		0			0
$911$			1.25785e6i			1.51562i	0.652473	+	0.757812i	$0.273731\pi$
$911$			1.25785e6i			1.51562i	−0.652473	+	0.757812i	$0.726269\pi$
$912$		0			0
$913$	−121726.			−0.146030
$914$		0			0
$915$	68742.1			0.0821070
$916$		0			0
$917$	−693434.			−0.824644
$918$		0			0
$919$	441681.			0.522971			0.261485	−	0.965207i	$-0.415788\pi$
$919$	441681.			0.522971			0.261485	+	0.965207i	$0.415788\pi$
$920$		0			0
$921$		−	1.99666e6i		−	2.35388i
$922$		0			0
$923$			96655.8i			0.113455i
$924$		0			0
$925$		−	520154.i		−	0.607923i
$926$		0			0
$927$	3.40564e6			3.96314
$928$		0			0
$929$		−	1.25042e6i		−	1.44885i	−0.689355	−	0.724424i	$-0.742106\pi$
$929$		−	1.25042e6i		−	1.44885i	0.689355	−	0.724424i	$-0.257894\pi$
$930$		0			0
$931$	−76111.7			−0.0878117
$932$		0			0
$933$			1.32745e6i			1.52494i
$934$		0			0
$935$	−3910.77			−0.00447342
$936$		0			0
$937$	894389.			1.01870			0.509351	−	0.860559i	$-0.329885\pi$
$937$	894389.			1.01870			0.509351	+	0.860559i	$0.329885\pi$
$938$		0			0
$939$	−2.93708e6			−3.33107
$940$		0			0
$941$		−	80180.5i		−	0.0905502i	−0.998975	−	0.0452751i	$-0.985584\pi$
$941$		−	80180.5i		−	0.0905502i	0.998975	−	0.0452751i	$-0.0144164\pi$
$942$		0			0
$943$			1.16936e6i			1.31499i
$944$		0			0
$945$		−	105669.i		−	0.118327i
$946$		0			0
$947$			636275.i			0.709487i	0.934964	+	0.354744i	$0.115432\pi$
$947$			636275.i			0.709487i	−0.934964	+	0.354744i	$0.884568\pi$
$948$		0			0
$949$	214136.			0.237770
$950$		0			0
$951$		−	2.24037e6i		−	2.47718i
$952$		0			0
$953$		−	589013.i		−	0.648543i	−0.945964	−	0.324272i	$-0.894881\pi$
$953$		−	589013.i		−	0.648543i	0.945964	−	0.324272i	$-0.105119\pi$
$954$		0			0
$955$	−105097.			−0.115235
$956$		0			0
$957$		−	212431.i		−	0.231950i
$958$		0			0
$959$		−	96768.7i		−	0.105220i
$960$		0			0
$961$	−58229.5	−	921683.i	−0.0630516	−	0.998010i
$962$		0			0
$963$	276028.			0.297646
$964$		0			0
$965$	101962.			0.109493
$966$		0			0
$967$		−	698622.i		−	0.747118i	−0.927606	−	0.373559i	$-0.878137\pi$
$967$		−	698622.i		−	0.747118i	0.927606	−	0.373559i	$-0.121863\pi$
$968$		0			0
$969$	179533.			0.191204
$970$		0			0
$971$	−129190.			−0.137022			−0.0685111	−	0.997650i	$-0.521825\pi$
$971$	−129190.			−0.137022			−0.0685111	+	0.997650i	$0.521825\pi$
$972$		0			0
$973$			763629.i			0.806597i
$974$		0			0
$975$	1.16079e6			1.22108
$976$		0			0
$977$	−1.42574e6			−1.49366			−0.746828	−	0.665017i	$-0.768424\pi$
$977$	−1.42574e6			−1.49366			−0.746828	+	0.665017i	$0.768424\pi$
$978$		0			0
$979$	−89989.0			−0.0938910
$980$		0			0
$981$	2.86178e6			2.97371
$982$		0			0
$983$			344602.i			0.356625i	0.983974	+	0.178312i	$0.0570637\pi$
$983$			344602.i			0.356625i	−0.983974	+	0.178312i	$0.942936\pi$
$984$		0			0
$985$		−	6760.36i		−	0.00696783i
$986$		0			0
$987$			1.56594e6i			1.60747i
$988$		0			0
$989$	482706.			0.493503
$990$		0			0
$991$		−	669895.i		−	0.682118i	−0.940042	−	0.341059i	$-0.889214\pi$
$991$		−	669895.i		−	0.682118i	0.940042	−	0.341059i	$-0.110786\pi$
$992$		0			0
$993$	805542.			0.816939
$994$		0			0
$995$			110435.i			0.111548i
$996$		0			0
$997$	−103519.			−0.104143			−0.0520716	−	0.998643i	$-0.516582\pi$
$997$	−103519.			−0.104143			−0.0520716	+	0.998643i	$0.516582\pi$
$998$		0			0
$999$	1.58354e6			1.58671

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.5.c.a.61.10	yes	10
3.2	odd	2		1116.5.h.b.433.5		10
4.3	odd	2		496.5.e.c.433.1		10
31.30	odd	2	inner	124.5.c.a.61.1	✓	10
93.92	even	2		1116.5.h.b.433.6		10
124.123	even	2		496.5.e.c.433.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.5.c.a.61.1	✓	10	31.30	odd	2	inner
124.5.c.a.61.10	yes	10	1.1	even	1	trivial
496.5.e.c.433.1		10	4.3	odd	2
496.5.e.c.433.10		10	124.123	even	2
1116.5.h.b.433.5		10	3.2	odd	2
1116.5.h.b.433.6		10	93.92	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:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	124.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+16.6119i q^{3} -1.78751 q^{5} -31.2283 q^{7} -194.955 q^{9} +O(q^{10})\)	\(q+16.6119i q^{3} -1.78751 q^{5} -31.2283 q^{7} -194.955 q^{9} +10.8065i q^{11} +112.377i q^{13} -29.6939i q^{15} -202.456i q^{17} +53.3820 q^{19} -518.761i q^{21} -617.863i q^{23} -621.805 q^{25} -1893.00i q^{27} +1183.36i q^{29} +(657.758 - 700.625i) q^{31} -179.516 q^{33} +55.8208 q^{35} +836.523i q^{37} -1866.80 q^{39} -1892.58 q^{41} +781.250i q^{43} +348.483 q^{45} -3018.62 q^{47} -1425.79 q^{49} +3363.18 q^{51} +1982.72i q^{53} -19.3167i q^{55} +886.775i q^{57} +4395.49 q^{59} +2315.03i q^{61} +6088.10 q^{63} -200.875i q^{65} -5510.08 q^{67} +10263.9 q^{69} +860.100 q^{71} -1905.51i q^{73} -10329.4i q^{75} -337.468i q^{77} +5474.24i q^{79} +15655.0 q^{81} +11264.2i q^{83} +361.892i q^{85} -19657.8 q^{87} +8327.32i q^{89} -3509.35i q^{91} +(11638.7 + 10926.6i) q^{93} -95.4207 q^{95} +6014.40 q^{97} -2106.78i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9}+O(q^{10}) \) 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9	\( 10 q - 6 q^{5} - 2 q^{7} - 146 q^{9} - 310 q^{19} - 756 q^{25} + 734 q^{31} - 372 q^{33} - 666 q^{35} - 1924 q^{39} - 210 q^{41} + 650 q^{45} - 1992 q^{47} + 188 q^{49} + 1352 q^{51} + 5610 q^{59} + 3478 q^{63} - 5420 q^{67} + 10160 q^{69} - 1734 q^{71} + 11598 q^{81} - 13244 q^{87} + 3088 q^{93} + 4302 q^{95} + 2942 q^{97}+O(q^{100}) \) 10 * q - 6 * q^5 - 2 * q^7 - 146 * q^9 - 310 * q^19 - 756 * q^25 + 734 * q^31 - 372 * q^33 - 666 * q^35 - 1924 * q^39 - 210 * q^41 + 650 * q^45 - 1992 * q^47 + 188 * q^49 + 1352 * q^51 + 5610 * q^59 + 3478 * q^63 - 5420 * q^67 + 10160 * q^69 - 1734 * q^71 + 11598 * q^81 - 13244 * q^87 + 3088 * q^93 + 4302 * q^95 + 2942 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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