LMFDB - Embedded newform 1900.3.e.b.1101.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 29 $ x^2 + 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.2
Root			$5.38516i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.b.1101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.38516i q^{3} +1.00000 q^{7} -20.0000 q^{9} +14.0000 q^{11} +16.1555i q^{13} -23.0000 q^{17} +(10.0000 - 16.1555i) q^{19} +5.38516i q^{21} +1.00000 q^{23} -59.2368i q^{27} +48.4665i q^{29} +32.3110i q^{31} +75.3923i q^{33} +32.3110i q^{37} -87.0000 q^{39} -32.3110i q^{41} -68.0000 q^{43} -26.0000 q^{47} -48.0000 q^{49} -123.859i q^{51} -80.7775i q^{53} +(87.0000 + 53.8516i) q^{57} -16.1555i q^{59} -40.0000 q^{61} -20.0000 q^{63} -16.1555i q^{67} +5.38516i q^{69} +32.3110i q^{71} +7.00000 q^{73} +14.0000 q^{77} -96.9330i q^{79} +139.000 q^{81} -32.0000 q^{83} -261.000 q^{87} +129.244i q^{89} +16.1555i q^{91} -174.000 q^{93} -96.9330i q^{97} -280.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 40 q^{9} + 28 q^{11} - 46 q^{17} + 20 q^{19} + 2 q^{23} - 174 q^{39} - 136 q^{43} - 52 q^{47} - 96 q^{49} + 174 q^{57} - 80 q^{61} - 40 q^{63} + 14 q^{73} + 28 q^{77} + 278 q^{81} - 64 q^{83}+ \cdots - 560 q^{99}+O(q^{100}) $ 2 * q + 2 * q^7 - 40 * q^9 + 28 * q^11 - 46 * q^17 + 20 * q^19 + 2 * q^23 - 174 * q^39 - 136 * q^43 - 52 * q^47 - 96 * q^49 + 174 * q^57 - 80 * q^61 - 40 * q^63 + 14 * q^73 + 28 * q^77 + 278 * q^81 - 64 * q^83 - 522 * q^87 - 348 * q^93 - 560 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			5.38516i			1.79505i	0.440959	+	0.897527i	$0.354639\pi$
$3$			5.38516i			1.79505i	−0.440959	+	0.897527i	$0.645361\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.00000			0.142857			0.0714286	−	0.997446i	$-0.477244\pi$
$7$	1.00000			0.142857			0.0714286	+	0.997446i	$0.477244\pi$
$8$		0			0
$9$	−20.0000			−2.22222
$10$		0			0
$11$	14.0000			1.27273			0.636364	−	0.771389i	$-0.280438\pi$
$11$	14.0000			1.27273			0.636364	+	0.771389i	$0.280438\pi$
$12$		0			0
$13$			16.1555i			1.24273i	0.783521	+	0.621365i	$0.213422\pi$
$13$			16.1555i			1.24273i	−0.783521	+	0.621365i	$0.786578\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−23.0000			−1.35294			−0.676471	−	0.736470i	$-0.736491\pi$
$17$	−23.0000			−1.35294			−0.676471	+	0.736470i	$0.736491\pi$
$18$		0			0
$19$	10.0000	−	16.1555i	0.526316	−	0.850289i
$19$	10.0000	−	16.1555i	0.526316	−	0.850289i
$20$		0			0
$21$			5.38516i			0.256436i
$22$		0			0
$23$	1.00000			0.0434783			0.0217391	−	0.999764i	$-0.493080\pi$
$23$	1.00000			0.0434783			0.0217391	+	0.999764i	$0.493080\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	59.2368i		−	2.19396i
$28$		0			0
$29$			48.4665i			1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$			48.4665i			1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$		0			0
$31$			32.3110i			1.04229i	0.853468	+	0.521145i	$0.174495\pi$
$31$			32.3110i			1.04229i	−0.853468	+	0.521145i	$0.825505\pi$
$32$		0			0
$33$			75.3923i			2.28462i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			32.3110i			0.873270i	0.899639	+	0.436635i	$0.143830\pi$
$37$			32.3110i			0.873270i	−0.899639	+	0.436635i	$0.856170\pi$
$38$		0			0
$39$	−87.0000			−2.23077
$40$		0			0
$41$		−	32.3110i		−	0.788073i	−0.919095	−	0.394036i	$-0.871078\pi$
$41$		−	32.3110i		−	0.788073i	0.919095	−	0.394036i	$-0.128922\pi$
$42$		0			0
$43$	−68.0000			−1.58140			−0.790698	−	0.612207i	$-0.790282\pi$
$43$	−68.0000			−1.58140			−0.790698	+	0.612207i	$0.790282\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−26.0000			−0.553191			−0.276596	−	0.960986i	$-0.589206\pi$
$47$	−26.0000			−0.553191			−0.276596	+	0.960986i	$0.589206\pi$
$48$		0			0
$49$	−48.0000			−0.979592
$50$		0			0
$51$		−	123.859i		−	2.42860i
$52$		0			0
$53$		−	80.7775i		−	1.52410i	−0.647516	−	0.762052i	$-0.724192\pi$
$53$		−	80.7775i		−	1.52410i	0.647516	−	0.762052i	$-0.275808\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	87.0000	+	53.8516i	1.52632	+	0.944766i
$58$		0			0
$59$		−	16.1555i		−	0.273822i	−0.990583	−	0.136911i	$-0.956283\pi$
$59$		−	16.1555i		−	0.273822i	0.990583	−	0.136911i	$-0.0437174\pi$
$60$		0			0
$61$	−40.0000			−0.655738			−0.327869	−	0.944723i	$-0.606330\pi$
$61$	−40.0000			−0.655738			−0.327869	+	0.944723i	$0.606330\pi$
$62$		0			0
$63$	−20.0000			−0.317460
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	16.1555i		−	0.241127i	−0.992706	−	0.120563i	$-0.961530\pi$
$67$		−	16.1555i		−	0.241127i	0.992706	−	0.120563i	$-0.0384701\pi$
$68$		0			0
$69$			5.38516i			0.0780459i
$70$		0			0
$71$			32.3110i			0.455084i	0.973768	+	0.227542i	$0.0730690\pi$
$71$			32.3110i			0.455084i	−0.973768	+	0.227542i	$0.926931\pi$
$72$		0			0
$73$	7.00000			0.0958904			0.0479452	−	0.998850i	$-0.484733\pi$
$73$	7.00000			0.0958904			0.0479452	+	0.998850i	$0.484733\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	14.0000			0.181818
$78$		0			0
$79$		−	96.9330i		−	1.22700i	−0.789695	−	0.613500i	$-0.789761\pi$
$79$		−	96.9330i		−	1.22700i	0.789695	−	0.613500i	$-0.210239\pi$
$80$		0			0
$81$	139.000			1.71605
$82$		0			0
$83$	−32.0000			−0.385542			−0.192771	−	0.981244i	$-0.561747\pi$
$83$	−32.0000			−0.385542			−0.192771	+	0.981244i	$0.561747\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−261.000			−3.00000
$88$		0			0
$89$			129.244i			1.45218i	0.687600	+	0.726090i	$0.258664\pi$
$89$			129.244i			1.45218i	−0.687600	+	0.726090i	$0.741336\pi$
$90$		0			0
$91$			16.1555i			0.177533i
$92$		0			0
$93$	−174.000			−1.87097
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	96.9330i		−	0.999309i	−0.866225	−	0.499654i	$-0.833460\pi$
$97$		−	96.9330i		−	0.999309i	0.866225	−	0.499654i	$-0.166540\pi$
$98$		0			0
$99$	−280.000			−2.82828
$100$		0			0
$101$	14.0000			0.138614			0.0693069	−	0.997595i	$-0.477921\pi$
$101$	14.0000			0.138614			0.0693069	+	0.997595i	$0.477921\pi$
$102$		0			0
$103$			129.244i			1.25480i	0.778699	+	0.627398i	$0.215880\pi$
$103$			129.244i			1.25480i	−0.778699	+	0.627398i	$0.784120\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16.1555i			0.150986i	0.997146	+	0.0754930i	$0.0240530\pi$
$107$			16.1555i			0.150986i	−0.997146	+	0.0754930i	$0.975947\pi$
$108$		0			0
$109$		−	16.1555i		−	0.148216i	−0.997250	−	0.0741078i	$-0.976389\pi$
$109$		−	16.1555i		−	0.148216i	0.997250	−	0.0741078i	$-0.0236109\pi$
$110$		0			0
$111$	−174.000			−1.56757
$112$		0			0
$113$			96.9330i			0.857814i	0.903349	+	0.428907i	$0.141101\pi$
$113$			96.9330i			0.857814i	−0.903349	+	0.428907i	$0.858899\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	323.110i		−	2.76162i
$118$		0			0
$119$	−23.0000			−0.193277
$120$		0			0
$121$	75.0000			0.619835
$122$		0			0
$123$	174.000			1.41463
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	64.6220i		−	0.508834i	−0.967095	−	0.254417i	$-0.918116\pi$
$127$		−	64.6220i		−	0.508834i	0.967095	−	0.254417i	$-0.0818836\pi$
$128$		0			0
$129$		−	366.191i		−	2.83869i
$130$		0			0
$131$	56.0000			0.427481			0.213740	−	0.976890i	$-0.431435\pi$
$131$	56.0000			0.427481			0.213740	+	0.976890i	$0.431435\pi$
$132$		0			0
$133$	10.0000	−	16.1555i	0.0751880	−	0.121470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	19.0000			0.138686			0.0693431	−	0.997593i	$-0.477910\pi$
$137$	19.0000			0.138686			0.0693431	+	0.997593i	$0.477910\pi$
$138$		0			0
$139$	122.000			0.877698			0.438849	−	0.898561i	$-0.355386\pi$
$139$	122.000			0.877698			0.438849	+	0.898561i	$0.355386\pi$
$140$		0			0
$141$		−	140.014i		−	0.993009i
$142$		0			0
$143$			226.177i			1.58166i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	258.488i		−	1.75842i
$148$		0			0
$149$	−82.0000			−0.550336			−0.275168	−	0.961396i	$-0.588733\pi$
$149$	−82.0000			−0.550336			−0.275168	+	0.961396i	$0.588733\pi$
$150$		0			0
$151$			226.177i			1.49786i	0.662649	+	0.748930i	$0.269432\pi$
$151$			226.177i			1.49786i	−0.662649	+	0.748930i	$0.730568\pi$
$152$		0			0
$153$	460.000			3.00654
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−242.000			−1.54140			−0.770701	−	0.637197i	$-0.780094\pi$
$157$	−242.000			−1.54140			−0.770701	+	0.637197i	$0.780094\pi$
$158$		0			0
$159$	435.000			2.73585
$160$		0			0
$161$	1.00000			0.00621118
$162$		0			0
$163$	214.000			1.31288			0.656442	−	0.754377i	$-0.272061\pi$
$163$	214.000			1.31288			0.656442	+	0.754377i	$0.272061\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			32.3110i			0.193479i	0.995310	+	0.0967395i	$0.0308414\pi$
$167$			32.3110i			0.193479i	−0.995310	+	0.0967395i	$0.969159\pi$
$168$		0			0
$169$	−92.0000			−0.544379
$170$		0			0
$171$	−200.000	+	323.110i	−1.16959	+	1.88953i
$172$		0			0
$173$		−	161.555i		−	0.933844i	−0.884299	−	0.466922i	$-0.845363\pi$
$173$		−	161.555i		−	0.933844i	0.884299	−	0.466922i	$-0.154637\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	87.0000			0.491525
$178$		0			0
$179$		−	290.799i		−	1.62457i	−0.583257	−	0.812287i	$-0.698222\pi$
$179$		−	290.799i		−	1.62457i	0.583257	−	0.812287i	$-0.301778\pi$
$180$		0			0
$181$		−	96.9330i		−	0.535541i	−0.963483	−	0.267771i	$-0.913713\pi$
$181$		−	96.9330i		−	0.535541i	0.963483	−	0.267771i	$-0.0862869\pi$
$182$		0			0
$183$		−	215.407i		−	1.17709i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−322.000			−1.72193
$188$		0			0
$189$		−	59.2368i		−	0.313422i
$190$		0			0
$191$	−67.0000			−0.350785			−0.175393	−	0.984499i	$-0.556119\pi$
$191$	−67.0000			−0.350785			−0.175393	+	0.984499i	$0.556119\pi$
$192$		0			0
$193$		−	96.9330i		−	0.502243i	−0.967956	−	0.251122i	$-0.919201\pi$
$193$		−	96.9330i		−	0.502243i	0.967956	−	0.251122i	$-0.0807994\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	142.000			0.720812			0.360406	−	0.932796i	$-0.382638\pi$
$197$	142.000			0.720812			0.360406	+	0.932796i	$0.382638\pi$
$198$		0			0
$199$	263.000			1.32161			0.660804	−	0.750558i	$-0.270215\pi$
$199$	263.000			1.32161			0.660804	+	0.750558i	$0.270215\pi$
$200$		0			0
$201$	87.0000			0.432836
$202$		0			0
$203$			48.4665i			0.238751i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−20.0000			−0.0966184
$208$		0			0
$209$	140.000	−	226.177i	0.669856	−	1.08219i
$210$		0			0
$211$			80.7775i			0.382832i	0.981509	+	0.191416i	$0.0613079\pi$
$211$			80.7775i			0.382832i	−0.981509	+	0.191416i	$0.938692\pi$
$212$		0			0
$213$	−174.000			−0.816901
$214$		0			0
$215$		0			0
$216$		0			0
$217$			32.3110i			0.148899i
$218$		0			0
$219$			37.6962i			0.172129i
$220$		0			0
$221$		−	371.576i		−	1.68134i
$222$		0			0
$223$		−	258.488i		−	1.15914i	−0.814923	−	0.579569i	$-0.803221\pi$
$223$		−	258.488i		−	1.15914i	0.814923	−	0.579569i	$-0.196779\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	80.7775i		−	0.355848i	−0.984044	−	0.177924i	$-0.943062\pi$
$227$		−	80.7775i		−	0.355848i	0.984044	−	0.177924i	$-0.0569381\pi$
$228$		0			0
$229$	−304.000			−1.32751			−0.663755	−	0.747950i	$-0.731038\pi$
$229$	−304.000			−1.32751			−0.663755	+	0.747950i	$0.731038\pi$
$230$		0			0
$231$			75.3923i			0.326374i
$232$		0			0
$233$	−2.00000			−0.00858369			−0.00429185	−	0.999991i	$-0.501366\pi$
$233$	−2.00000			−0.00858369			−0.00429185	+	0.999991i	$0.501366\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	522.000			2.20253
$238$		0			0
$239$	89.0000			0.372385			0.186192	−	0.982513i	$-0.440385\pi$
$239$	89.0000			0.372385			0.186192	+	0.982513i	$0.440385\pi$
$240$		0			0
$241$			161.555i			0.670352i	0.942155	+	0.335176i	$0.108796\pi$
$241$			161.555i			0.670352i	−0.942155	+	0.335176i	$0.891204\pi$
$242$		0			0
$243$			215.407i			0.886447i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	261.000	+	161.555i	1.05668	+	0.654069i
$248$		0			0
$249$		−	172.325i		−	0.692069i
$250$		0			0
$251$	−370.000			−1.47410			−0.737052	−	0.675836i	$-0.763783\pi$
$251$	−370.000			−1.47410			−0.737052	+	0.675836i	$0.763783\pi$
$252$		0			0
$253$	14.0000			0.0553360
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	193.866i		−	0.754342i	−0.926144	−	0.377171i	$-0.876897\pi$
$257$		−	193.866i		−	0.754342i	0.926144	−	0.377171i	$-0.123103\pi$
$258$		0			0
$259$			32.3110i			0.124753i
$260$		0			0
$261$		−	969.330i		−	3.71391i
$262$		0			0
$263$	394.000			1.49810			0.749049	−	0.662514i	$-0.230511\pi$
$263$	394.000			1.49810			0.749049	+	0.662514i	$0.230511\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−696.000			−2.60674
$268$		0			0
$269$		−	96.9330i		−	0.360346i	−0.983635	−	0.180173i	$-0.942334\pi$
$269$		−	96.9330i		−	0.360346i	0.983635	−	0.180173i	$-0.0576657\pi$
$270$		0			0
$271$	−403.000			−1.48708			−0.743542	−	0.668689i	$-0.766856\pi$
$271$	−403.000			−1.48708			−0.743542	+	0.668689i	$0.766856\pi$
$272$		0			0
$273$	−87.0000			−0.318681
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−206.000			−0.743682			−0.371841	−	0.928296i	$-0.621273\pi$
$277$	−206.000			−0.743682			−0.371841	+	0.928296i	$0.621273\pi$
$278$		0			0
$279$		−	646.220i		−	2.31620i
$280$		0			0
$281$			258.488i			0.919886i	0.887948	+	0.459943i	$0.152130\pi$
$281$			258.488i			0.919886i	−0.887948	+	0.459943i	$0.847870\pi$
$282$		0			0
$283$	−56.0000			−0.197880			−0.0989399	−	0.995093i	$-0.531545\pi$
$283$	−56.0000			−0.197880			−0.0989399	+	0.995093i	$0.531545\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	32.3110i		−	0.112582i
$288$		0			0
$289$	240.000			0.830450
$290$		0			0
$291$	522.000			1.79381
$292$		0			0
$293$		−	80.7775i		−	0.275691i	−0.990454	−	0.137846i	$-0.955982\pi$
$293$		−	80.7775i		−	0.275691i	0.990454	−	0.137846i	$-0.0440177\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	829.315i		−	2.79231i
$298$		0			0
$299$			16.1555i			0.0540318i
$300$		0			0
$301$	−68.0000			−0.225914
$302$		0			0
$303$			75.3923i			0.248819i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			420.043i			1.36822i	0.729380	+	0.684109i	$0.239809\pi$
$307$			420.043i			1.36822i	−0.729380	+	0.684109i	$0.760191\pi$
$308$		0			0
$309$	−696.000			−2.25243
$310$		0			0
$311$	−265.000			−0.852090			−0.426045	−	0.904702i	$-0.640093\pi$
$311$	−265.000			−0.852090			−0.426045	+	0.904702i	$0.640093\pi$
$312$		0			0
$313$	−77.0000			−0.246006			−0.123003	−	0.992406i	$-0.539253\pi$
$313$	−77.0000			−0.246006			−0.123003	+	0.992406i	$0.539253\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			403.887i			1.27409i	0.770826	+	0.637046i	$0.219844\pi$
$317$			403.887i			1.27409i	−0.770826	+	0.637046i	$0.780156\pi$
$318$		0			0
$319$			678.531i			2.12706i
$320$		0			0
$321$	−87.0000			−0.271028
$322$		0			0
$323$	−230.000	+	371.576i	−0.712074	+	1.15039i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	87.0000			0.266055
$328$		0			0
$329$	−26.0000			−0.0790274
$330$		0			0
$331$			436.198i			1.31782i	0.752222	+	0.658910i	$0.228982\pi$
$331$			436.198i			1.31782i	−0.752222	+	0.658910i	$0.771018\pi$
$332$		0			0
$333$		−	646.220i		−	1.94060i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	387.732i		−	1.15054i	−0.817964	−	0.575270i	$-0.804897\pi$
$337$		−	387.732i		−	1.15054i	0.817964	−	0.575270i	$-0.195103\pi$
$338$		0			0
$339$	−522.000			−1.53982
$340$		0			0
$341$			452.354i			1.32655i
$342$		0			0
$343$	−97.0000			−0.282799
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−392.000			−1.12968			−0.564841	−	0.825199i	$-0.691063\pi$
$347$	−392.000			−1.12968			−0.564841	+	0.825199i	$0.691063\pi$
$348$		0			0
$349$	410.000			1.17479			0.587393	−	0.809302i	$-0.300154\pi$
$349$	410.000			1.17479			0.587393	+	0.809302i	$0.300154\pi$
$350$		0			0
$351$	957.000			2.72650
$352$		0			0
$353$	481.000			1.36261			0.681303	−	0.732001i	$-0.261414\pi$
$353$	481.000			1.36261			0.681303	+	0.732001i	$0.261414\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	123.859i		−	0.346943i
$358$		0			0
$359$	−109.000			−0.303621			−0.151811	−	0.988410i	$-0.548510\pi$
$359$	−109.000			−0.303621			−0.151811	+	0.988410i	$0.548510\pi$
$360$		0			0
$361$	−161.000	−	323.110i	−0.445983	−	0.895041i
$362$		0			0
$363$			403.887i			1.11264i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−50.0000			−0.136240			−0.0681199	−	0.997677i	$-0.521700\pi$
$367$	−50.0000			−0.136240			−0.0681199	+	0.997677i	$0.521700\pi$
$368$		0			0
$369$			646.220i			1.75127i
$370$		0			0
$371$		−	80.7775i		−	0.217729i
$372$		0			0
$373$			500.820i			1.34268i	0.741149	+	0.671341i	$0.234281\pi$
$373$			500.820i			1.34268i	−0.741149	+	0.671341i	$0.765719\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−783.000			−2.07692
$378$		0			0
$379$		−	210.021i		−	0.554146i	−0.960849	−	0.277073i	$-0.910636\pi$
$379$		−	210.021i		−	0.554146i	0.960849	−	0.277073i	$-0.0893644\pi$
$380$		0			0
$381$	348.000			0.913386
$382$		0			0
$383$			258.488i			0.674903i	0.941343	+	0.337452i	$0.109565\pi$
$383$			258.488i			0.674903i	−0.941343	+	0.337452i	$0.890435\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1360.00			3.51421
$388$		0			0
$389$	578.000			1.48586			0.742931	−	0.669368i	$-0.233435\pi$
$389$	578.000			1.48586			0.742931	+	0.669368i	$0.233435\pi$
$390$		0			0
$391$	−23.0000			−0.0588235
$392$		0			0
$393$			301.569i			0.767352i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	658.000			1.65743			0.828715	−	0.559670i	$-0.189072\pi$
$397$	658.000			1.65743			0.828715	+	0.559670i	$0.189072\pi$
$398$		0			0
$399$	87.0000	+	53.8516i	0.218045	+	0.134967i
$400$		0			0
$401$			775.464i			1.93382i	0.255109	+	0.966912i	$0.417889\pi$
$401$			775.464i			1.93382i	−0.255109	+	0.966912i	$0.582111\pi$
$402$		0			0
$403$	−522.000			−1.29529
$404$		0			0
$405$		0			0
$406$		0			0
$407$			452.354i			1.11143i
$408$		0			0
$409$		−	355.421i		−	0.869000i	−0.900672	−	0.434500i	$-0.856925\pi$
$409$		−	355.421i		−	0.869000i	0.900672	−	0.434500i	$-0.143075\pi$
$410$		0			0
$411$			102.318i			0.248949i
$412$		0			0
$413$		−	16.1555i		−	0.0391174i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			656.990i			1.57552i
$418$		0			0
$419$	8.00000			0.0190931			0.00954654	−	0.999954i	$-0.496961\pi$
$419$	8.00000			0.0190931			0.00954654	+	0.999954i	$0.496961\pi$
$420$		0			0
$421$		−	500.820i		−	1.18960i	−0.803875	−	0.594798i	$-0.797232\pi$
$421$		−	500.820i		−	1.18960i	0.803875	−	0.594798i	$-0.202768\pi$
$422$		0			0
$423$	520.000			1.22931
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−40.0000			−0.0936768
$428$		0			0
$429$	−1218.00			−2.83916
$430$		0			0
$431$			646.220i			1.49935i	0.661806	+	0.749675i	$0.269790\pi$
$431$			646.220i			1.49935i	−0.661806	+	0.749675i	$0.730210\pi$
$432$		0			0
$433$			193.866i			0.447727i	0.974620	+	0.223864i	$0.0718670\pi$
$433$			193.866i			0.447727i	−0.974620	+	0.223864i	$0.928133\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.0000	−	16.1555i	0.0228833	−	0.0369691i
$438$		0			0
$439$			323.110i			0.736013i	0.929823	+	0.368007i	$0.119960\pi$
$439$			323.110i			0.736013i	−0.929823	+	0.368007i	$0.880040\pi$
$440$		0			0
$441$	960.000			2.17687
$442$		0			0
$443$	−182.000			−0.410835			−0.205418	−	0.978674i	$-0.565855\pi$
$443$	−182.000			−0.410835			−0.205418	+	0.978674i	$0.565855\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	441.584i		−	0.987883i
$448$		0			0
$449$			420.043i			0.935507i	0.883859	+	0.467754i	$0.154937\pi$
$449$			420.043i			0.935507i	−0.883859	+	0.467754i	$0.845063\pi$
$450$		0			0
$451$		−	452.354i		−	1.00300i
$452$		0			0
$453$	−1218.00			−2.68874
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−317.000			−0.693654			−0.346827	−	0.937929i	$-0.612741\pi$
$457$	−317.000			−0.693654			−0.346827	+	0.937929i	$0.612741\pi$
$458$		0			0
$459$			1362.45i			2.96829i
$460$		0			0
$461$	−172.000			−0.373102			−0.186551	−	0.982445i	$-0.559731\pi$
$461$	−172.000			−0.373102			−0.186551	+	0.982445i	$0.559731\pi$
$462$		0			0
$463$	−146.000			−0.315335			−0.157667	−	0.987492i	$-0.550397\pi$
$463$	−146.000			−0.315335			−0.157667	+	0.987492i	$0.550397\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−662.000			−1.41756			−0.708779	−	0.705430i	$-0.750754\pi$
$467$	−662.000			−1.41756			−0.708779	+	0.705430i	$0.750754\pi$
$468$		0			0
$469$		−	16.1555i		−	0.0344467i
$470$		0			0
$471$		−	1303.21i		−	2.76690i
$472$		0			0
$473$	−952.000			−2.01268
$474$		0			0
$475$		0			0
$476$		0			0
$477$			1615.55i			3.38690i
$478$		0			0
$479$	−130.000			−0.271399			−0.135699	−	0.990750i	$-0.543328\pi$
$479$	−130.000			−0.271399			−0.135699	+	0.990750i	$0.543328\pi$
$480$		0			0
$481$	−522.000			−1.08524
$482$		0			0
$483$			5.38516i			0.0111494i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			387.732i			0.796164i	0.917350	+	0.398082i	$0.130324\pi$
$487$			387.732i			0.796164i	−0.917350	+	0.398082i	$0.869676\pi$
$488$		0			0
$489$			1152.43i			2.35670i
$490$		0			0
$491$	−280.000			−0.570265			−0.285132	−	0.958488i	$-0.592038\pi$
$491$	−280.000			−0.570265			−0.285132	+	0.958488i	$0.592038\pi$
$492$		0			0
$493$		−	1114.73i		−	2.26111i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			32.3110i			0.0650120i
$498$		0			0
$499$	−460.000			−0.921844			−0.460922	−	0.887441i	$-0.652481\pi$
$499$	−460.000			−0.921844			−0.460922	+	0.887441i	$0.652481\pi$
$500$		0			0
$501$	−174.000			−0.347305
$502$		0			0
$503$	−263.000			−0.522863			−0.261431	−	0.965222i	$-0.584195\pi$
$503$	−263.000			−0.522863			−0.261431	+	0.965222i	$0.584195\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	495.435i		−	0.977190i
$508$		0			0
$509$			290.799i			0.571314i	0.958332	+	0.285657i	$0.0922118\pi$
$509$			290.799i			0.571314i	−0.958332	+	0.285657i	$0.907788\pi$
$510$		0			0
$511$	7.00000			0.0136986
$512$		0			0
$513$	−957.000	−	592.368i	−1.86550	−	1.15471i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−364.000			−0.704062
$518$		0			0
$519$	870.000			1.67630
$520$		0			0
$521$			226.177i			0.434121i	0.976158	+	0.217060i	$0.0696469\pi$
$521$			226.177i			0.434121i	−0.976158	+	0.217060i	$0.930353\pi$
$522$		0			0
$523$			80.7775i			0.154450i	0.997014	+	0.0772251i	$0.0246060\pi$
$523$			80.7775i			0.154450i	−0.997014	+	0.0772251i	$0.975394\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	743.153i		−	1.41016i
$528$		0			0
$529$	−528.000			−0.998110
$530$		0			0
$531$			323.110i			0.608493i
$532$		0			0
$533$	522.000			0.979362
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1566.00			2.91620
$538$		0			0
$539$	−672.000			−1.24675
$540$		0			0
$541$	−640.000			−1.18299			−0.591497	−	0.806307i	$-0.701463\pi$
$541$	−640.000			−1.18299			−0.591497	+	0.806307i	$0.701463\pi$
$542$		0			0
$543$	522.000			0.961326
$544$		0			0
$545$		0			0
$546$		0			0
$547$			32.3110i			0.0590694i	0.999564	+	0.0295347i	$0.00940256\pi$
$547$			32.3110i			0.0590694i	−0.999564	+	0.0295347i	$0.990597\pi$
$548$		0			0
$549$	800.000			1.45719
$550$		0			0
$551$	783.000	+	484.665i	1.42105	+	0.879609i
$552$		0			0
$553$		−	96.9330i		−	0.175286i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	844.000			1.51526			0.757630	−	0.652684i	$-0.226357\pi$
$557$	844.000			1.51526			0.757630	+	0.652684i	$0.226357\pi$
$558$		0			0
$559$		−	1098.57i		−	1.96525i
$560$		0			0
$561$		−	1734.02i		−	3.09095i
$562$		0			0
$563$			420.043i			0.746080i	0.927815	+	0.373040i	$0.121685\pi$
$563$			420.043i			0.746080i	−0.927815	+	0.373040i	$0.878315\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	139.000			0.245150
$568$		0			0
$569$		−	290.799i		−	0.511070i	−0.966800	−	0.255535i	$-0.917748\pi$
$569$		−	290.799i		−	0.511070i	0.966800	−	0.255535i	$-0.0822516\pi$
$570$		0			0
$571$	458.000			0.802102			0.401051	−	0.916056i	$-0.368645\pi$
$571$	458.000			0.802102			0.401051	+	0.916056i	$0.368645\pi$
$572$		0			0
$573$		−	360.806i		−	0.629679i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	85.0000			0.147314			0.0736568	−	0.997284i	$-0.476533\pi$
$577$	85.0000			0.147314			0.0736568	+	0.997284i	$0.476533\pi$
$578$		0			0
$579$	522.000			0.901554
$580$		0			0
$581$	−32.0000			−0.0550775
$582$		0			0
$583$		−	1130.88i		−	1.93977i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−488.000			−0.831346			−0.415673	−	0.909514i	$-0.636454\pi$
$587$	−488.000			−0.831346			−0.415673	+	0.909514i	$0.636454\pi$
$588$		0			0
$589$	522.000	+	323.110i	0.886248	+	0.548574i
$590$		0			0
$591$			764.693i			1.29390i
$592$		0			0
$593$	610.000			1.02867			0.514334	−	0.857590i	$-0.328039\pi$
$593$	610.000			1.02867			0.514334	+	0.857590i	$0.328039\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			1416.30i			2.37236i
$598$		0			0
$599$			323.110i			0.539416i	0.962942	+	0.269708i	$0.0869271\pi$
$599$			323.110i			0.539416i	−0.962942	+	0.269708i	$0.913073\pi$
$600$		0			0
$601$			96.9330i			0.161286i	0.996743	+	0.0806431i	$0.0256974\pi$
$601$			96.9330i			0.161286i	−0.996743	+	0.0806431i	$0.974303\pi$
$602$		0			0
$603$			323.110i			0.535837i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			1001.64i			1.65015i	0.565024	+	0.825075i	$0.308867\pi$
$607$			1001.64i			1.65015i	−0.565024	+	0.825075i	$0.691133\pi$
$608$		0			0
$609$	−261.000			−0.428571
$610$		0			0
$611$		−	420.043i		−	0.687468i
$612$		0			0
$613$	−200.000			−0.326264			−0.163132	−	0.986604i	$-0.552160\pi$
$613$	−200.000			−0.326264			−0.163132	+	0.986604i	$0.552160\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−530.000			−0.858995			−0.429498	−	0.903068i	$-0.641309\pi$
$617$	−530.000			−0.858995			−0.429498	+	0.903068i	$0.641309\pi$
$618$		0			0
$619$	−256.000			−0.413570			−0.206785	−	0.978386i	$-0.566300\pi$
$619$	−256.000			−0.413570			−0.206785	+	0.978386i	$0.566300\pi$
$620$		0			0
$621$		−	59.2368i		−	0.0953894i
$622$		0			0
$623$			129.244i			0.207454i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1218.00	+	753.923i	1.94258	+	1.20243i
$628$		0			0
$629$		−	743.153i		−	1.18148i
$630$		0			0
$631$	−826.000			−1.30903			−0.654517	−	0.756048i	$-0.727128\pi$
$631$	−826.000			−1.30903			−0.654517	+	0.756048i	$0.727128\pi$
$632$		0			0
$633$	−435.000			−0.687204
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	775.464i		−	1.21737i
$638$		0			0
$639$		−	646.220i		−	1.01130i
$640$		0			0
$641$			581.598i			0.907329i	0.891173	+	0.453664i	$0.149884\pi$
$641$			581.598i			0.907329i	−0.891173	+	0.453664i	$0.850116\pi$
$642$		0			0
$643$	−170.000			−0.264386			−0.132193	−	0.991224i	$-0.542202\pi$
$643$	−170.000			−0.264386			−0.132193	+	0.991224i	$0.542202\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	25.0000			0.0386399			0.0193199	−	0.999813i	$-0.493850\pi$
$647$	25.0000			0.0386399			0.0193199	+	0.999813i	$0.493850\pi$
$648$		0			0
$649$		−	226.177i		−	0.348501i
$650$		0			0
$651$	−174.000			−0.267281
$652$		0			0
$653$	1054.00			1.61409			0.807044	−	0.590491i	$-0.201066\pi$
$653$	1054.00			1.61409			0.807044	+	0.590491i	$0.201066\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−140.000			−0.213090
$658$		0			0
$659$			1243.97i			1.88767i	0.330420	+	0.943834i	$0.392810\pi$
$659$			1243.97i			1.88767i	−0.330420	+	0.943834i	$0.607190\pi$
$660$		0			0
$661$			533.131i			0.806553i	0.915078	+	0.403276i	$0.132129\pi$
$661$			533.131i			0.806553i	−0.915078	+	0.403276i	$0.867871\pi$
$662$		0			0
$663$	2001.00			3.01810
$664$		0			0
$665$		0			0
$666$		0			0
$667$			48.4665i			0.0726634i
$668$		0			0
$669$	1392.00			2.08072
$670$		0			0
$671$	−560.000			−0.834575
$672$		0			0
$673$		−	290.799i		−	0.432093i	−0.976383	−	0.216047i	$-0.930684\pi$
$673$		−	290.799i		−	0.432093i	0.976383	−	0.216047i	$-0.0693164\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			953.174i			1.40794i	0.710231	+	0.703969i	$0.248591\pi$
$677$			953.174i			1.40794i	−0.710231	+	0.703969i	$0.751409\pi$
$678$		0			0
$679$		−	96.9330i		−	0.142758i
$680$		0			0
$681$	435.000			0.638767
$682$		0			0
$683$			872.397i			1.27730i	0.769497	+	0.638651i	$0.220507\pi$
$683$			872.397i			1.27730i	−0.769497	+	0.638651i	$0.779493\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	1637.09i		−	2.38296i
$688$		0			0
$689$	1305.00			1.89405
$690$		0			0
$691$	668.000			0.966715			0.483357	−	0.875423i	$-0.339417\pi$
$691$	668.000			0.966715			0.483357	+	0.875423i	$0.339417\pi$
$692$		0			0
$693$	−280.000			−0.404040
$694$		0			0
$695$		0			0
$696$		0			0
$697$			743.153i			1.06622i
$698$		0			0
$699$		−	10.7703i		−	0.0154082i
$700$		0			0
$701$	−700.000			−0.998573			−0.499287	−	0.866437i	$-0.666405\pi$
$701$	−700.000			−0.998573			−0.499287	+	0.866437i	$0.666405\pi$
$702$		0			0
$703$	522.000	+	323.110i	0.742532	+	0.459616i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	14.0000			0.0198020
$708$		0			0
$709$	1292.00			1.82228			0.911142	−	0.412092i	$-0.135202\pi$
$709$	1292.00			1.82228			0.911142	+	0.412092i	$0.135202\pi$
$710$		0			0
$711$			1938.66i			2.72667i
$712$		0			0
$713$			32.3110i			0.0453170i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			479.280i			0.668451i
$718$		0			0
$719$	1061.00			1.47566			0.737830	−	0.674986i	$-0.235850\pi$
$719$	1061.00			1.47566			0.737830	+	0.674986i	$0.235850\pi$
$720$		0			0
$721$			129.244i			0.179257i
$722$		0			0
$723$	−870.000			−1.20332
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−83.0000			−0.114168			−0.0570839	−	0.998369i	$-0.518180\pi$
$727$	−83.0000			−0.114168			−0.0570839	+	0.998369i	$0.518180\pi$
$728$		0			0
$729$	91.0000			0.124829
$730$		0			0
$731$	1564.00			2.13953
$732$		0			0
$733$	−296.000			−0.403820			−0.201910	−	0.979404i	$-0.564715\pi$
$733$	−296.000			−0.403820			−0.201910	+	0.979404i	$0.564715\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	226.177i		−	0.306889i
$738$		0			0
$739$	548.000			0.741543			0.370771	−	0.928724i	$-0.379093\pi$
$739$	548.000			0.741543			0.370771	+	0.928724i	$0.379093\pi$
$740$		0			0
$741$	−870.000	+	1405.53i	−1.17409	+	1.89680i
$742$		0			0
$743$			613.909i			0.826257i	0.910673	+	0.413128i	$0.135564\pi$
$743$			613.909i			0.826257i	−0.910673	+	0.413128i	$0.864436\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	640.000			0.856760
$748$		0			0
$749$			16.1555i			0.0215694i
$750$		0			0
$751$		−	420.043i		−	0.559311i	−0.960100	−	0.279656i	$-0.909780\pi$
$751$		−	420.043i		−	0.559311i	0.960100	−	0.279656i	$-0.0902203\pi$
$752$		0			0
$753$		−	1992.51i		−	2.64610i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−188.000			−0.248349			−0.124174	−	0.992260i	$-0.539628\pi$
$757$	−188.000			−0.248349			−0.124174	+	0.992260i	$0.539628\pi$
$758$		0			0
$759$			75.3923i			0.0993311i
$760$		0			0
$761$	−1183.00			−1.55453			−0.777267	−	0.629171i	$-0.783394\pi$
$761$	−1183.00			−1.55453			−0.777267	+	0.629171i	$0.783394\pi$
$762$		0			0
$763$		−	16.1555i		−	0.0211736i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	261.000			0.340287
$768$		0			0
$769$	−1177.00			−1.53056			−0.765280	−	0.643698i	$-0.777399\pi$
$769$	−1177.00			−1.53056			−0.765280	+	0.643698i	$0.777399\pi$
$770$		0			0
$771$	1044.00			1.35409
$772$		0			0
$773$			1017.80i			1.31668i	0.752719	+	0.658342i	$0.228742\pi$
$773$			1017.80i			1.31668i	−0.752719	+	0.658342i	$0.771258\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−174.000			−0.223938
$778$		0			0
$779$	−522.000	−	323.110i	−0.670090	−	0.414775i
$780$		0			0
$781$			452.354i			0.579198i
$782$		0			0
$783$	2871.00			3.66667
$784$		0			0
$785$		0			0
$786$		0			0
$787$			113.088i			0.143696i	0.997416	+	0.0718478i	$0.0228896\pi$
$787$			113.088i			0.143696i	−0.997416	+	0.0718478i	$0.977110\pi$
$788$		0			0
$789$			2121.75i			2.68917i
$790$		0			0
$791$			96.9330i			0.122545i
$792$		0			0
$793$		−	646.220i		−	0.814905i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	177.710i		−	0.222974i	−0.993766	−	0.111487i	$-0.964439\pi$
$797$		−	177.710i		−	0.222974i	0.993766	−	0.111487i	$-0.0355614\pi$
$798$		0			0
$799$	598.000			0.748436
$800$		0			0
$801$		−	2584.88i		−	3.22707i
$802$		0			0
$803$	98.0000			0.122042
$804$		0			0
$805$		0			0
$806$		0			0
$807$	522.000			0.646840
$808$		0			0
$809$	737.000			0.911001			0.455501	−	0.890235i	$-0.349460\pi$
$809$	737.000			0.911001			0.455501	+	0.890235i	$0.349460\pi$
$810$		0			0
$811$		−	791.619i		−	0.976103i	−0.872815	−	0.488051i	$-0.837708\pi$
$811$		−	791.619i		−	0.976103i	0.872815	−	0.488051i	$-0.162292\pi$
$812$		0			0
$813$		−	2170.22i		−	2.66940i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−680.000	+	1098.57i	−0.832313	+	1.34464i
$818$		0			0
$819$		−	323.110i		−	0.394518i
$820$		0			0
$821$	122.000			0.148599			0.0742996	−	0.997236i	$-0.476328\pi$
$821$	122.000			0.148599			0.0742996	+	0.997236i	$0.476328\pi$
$822$		0			0
$823$	−1499.00			−1.82139			−0.910693	−	0.413085i	$-0.864451\pi$
$823$	−1499.00			−1.82139			−0.910693	+	0.413085i	$0.864451\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	953.174i		−	1.15257i	−0.817249	−	0.576284i	$-0.804502\pi$
$827$		−	953.174i		−	1.15257i	0.817249	−	0.576284i	$-0.195498\pi$
$828$		0			0
$829$		−	80.7775i		−	0.0974397i	−0.998812	−	0.0487198i	$-0.984486\pi$
$829$		−	80.7775i		−	0.0974397i	0.998812	−	0.0487198i	$-0.0155141\pi$
$830$		0			0
$831$		−	1109.34i		−	1.33495i
$832$		0			0
$833$	1104.00			1.32533
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1914.00			2.28674
$838$		0			0
$839$		−	387.732i		−	0.462136i	−0.972938	−	0.231068i	$-0.925778\pi$
$839$		−	387.732i		−	0.462136i	0.972938	−	0.231068i	$-0.0742219\pi$
$840$		0			0
$841$	−1508.00			−1.79310
$842$		0			0
$843$	−1392.00			−1.65125
$844$		0			0
$845$		0			0
$846$		0			0
$847$	75.0000			0.0885478
$848$		0			0
$849$		−	301.569i		−	0.355205i
$850$		0			0
$851$			32.3110i			0.0379683i
$852$		0			0
$853$	−536.000			−0.628370			−0.314185	−	0.949362i	$-0.601731\pi$
$853$	−536.000			−0.628370			−0.314185	+	0.949362i	$0.601731\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1227.82i			1.43269i	0.697745	+	0.716346i	$0.254187\pi$
$857$			1227.82i			1.43269i	−0.697745	+	0.716346i	$0.745813\pi$
$858$		0			0
$859$	−496.000			−0.577416			−0.288708	−	0.957417i	$-0.593226\pi$
$859$	−496.000			−0.577416			−0.288708	+	0.957417i	$0.593226\pi$
$860$		0			0
$861$	174.000			0.202091
$862$		0			0
$863$		−	581.598i		−	0.673926i	−0.941518	−	0.336963i	$-0.890600\pi$
$863$		−	581.598i		−	0.673926i	0.941518	−	0.336963i	$-0.109400\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			1292.44i			1.49070i
$868$		0			0
$869$		−	1357.06i		−	1.56164i
$870$		0			0
$871$	261.000			0.299656
$872$		0			0
$873$			1938.66i			2.22069i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1082.42i		−	1.23423i	−0.786874	−	0.617114i	$-0.788302\pi$
$877$		−	1082.42i		−	1.23423i	0.786874	−	0.617114i	$-0.211698\pi$
$878$		0			0
$879$	435.000			0.494881
$880$		0			0
$881$	302.000			0.342792			0.171396	−	0.985202i	$-0.445172\pi$
$881$	302.000			0.342792			0.171396	+	0.985202i	$0.445172\pi$
$882$		0			0
$883$	1480.00			1.67610			0.838052	−	0.545590i	$-0.183694\pi$
$883$	1480.00			1.67610			0.838052	+	0.545590i	$0.183694\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			969.330i			1.09282i	0.837518	+	0.546409i	$0.184006\pi$
$887$			969.330i			1.09282i	−0.837518	+	0.546409i	$0.815994\pi$
$888$		0			0
$889$		−	64.6220i		−	0.0726906i
$890$		0			0
$891$	1946.00			2.18406
$892$		0			0
$893$	−260.000	+	420.043i	−0.291153	+	0.470373i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−87.0000			−0.0969900
$898$		0			0
$899$	−1566.00			−1.74194
$900$		0			0
$901$			1857.88i			2.06202i
$902$		0			0
$903$		−	366.191i		−	0.405527i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			16.1555i			0.0178120i	0.999960	+	0.00890601i	$0.00283491\pi$
$907$			16.1555i			0.0178120i	−0.999960	+	0.00890601i	$0.997165\pi$
$908$		0			0
$909$	−280.000			−0.308031
$910$		0			0
$911$			452.354i			0.496546i	0.968690	+	0.248273i	$0.0798631\pi$
$911$			452.354i			0.496546i	−0.968690	+	0.248273i	$0.920137\pi$
$912$		0			0
$913$	−448.000			−0.490690
$914$		0			0
$915$		0			0
$916$		0			0
$917$	56.0000			0.0610687
$918$		0			0
$919$	515.000			0.560392			0.280196	−	0.959943i	$-0.409601\pi$
$919$	515.000			0.560392			0.280196	+	0.959943i	$0.409601\pi$
$920$		0			0
$921$	−2262.00			−2.45603
$922$		0			0
$923$	−522.000			−0.565547
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	2584.88i		−	2.78843i
$928$		0			0
$929$	671.000			0.722282			0.361141	−	0.932511i	$-0.382387\pi$
$929$	671.000			0.722282			0.361141	+	0.932511i	$0.382387\pi$
$930$		0			0
$931$	−480.000	+	775.464i	−0.515575	+	0.832936i
$932$		0			0
$933$		−	1427.07i		−	1.52955i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1601.00			−1.70864			−0.854322	−	0.519744i	$-0.826027\pi$
$937$	−1601.00			−1.70864			−0.854322	+	0.519744i	$0.826027\pi$
$938$		0			0
$939$		−	414.658i		−	0.441595i
$940$		0			0
$941$		−	1567.08i		−	1.66534i	−0.553771	−	0.832669i	$-0.686812\pi$
$941$		−	1567.08i		−	1.66534i	0.553771	−	0.832669i	$-0.313188\pi$
$942$		0			0
$943$		−	32.3110i		−	0.0342640i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1192.00			1.25871			0.629356	−	0.777117i	$-0.283319\pi$
$947$	1192.00			1.25871			0.629356	+	0.777117i	$0.283319\pi$
$948$		0			0
$949$			113.088i			0.119166i
$950$		0			0
$951$	−2175.00			−2.28707
$952$		0			0
$953$		−	840.086i		−	0.881517i	−0.897626	−	0.440759i	$-0.854710\pi$
$953$		−	840.086i		−	0.881517i	0.897626	−	0.440759i	$-0.145290\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−3654.00			−3.81818
$958$		0			0
$959$	19.0000			0.0198123
$960$		0			0
$961$	−83.0000			−0.0863684
$962$		0			0
$963$		−	323.110i		−	0.335524i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	514.000			0.531541			0.265770	−	0.964036i	$-0.414374\pi$
$967$	514.000			0.531541			0.265770	+	0.964036i	$0.414374\pi$
$968$		0			0
$969$	−2001.00	−	1238.59i	−2.06502	−	1.27821i
$970$		0			0
$971$			1712.48i			1.76363i	0.471598	+	0.881814i	$0.343677\pi$
$971$			1712.48i			1.76363i	−0.471598	+	0.881814i	$0.656323\pi$
$972$		0			0
$973$	122.000			0.125385
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	161.555i		−	0.165358i	−0.996576	−	0.0826791i	$-0.973652\pi$
$977$		−	161.555i		−	0.165358i	0.996576	−	0.0826791i	$-0.0263476\pi$
$978$		0			0
$979$			1809.42i			1.84823i
$980$		0			0
$981$			323.110i			0.329368i
$982$		0			0
$983$			258.488i			0.262958i	0.991319	+	0.131479i	$0.0419726\pi$
$983$			258.488i			0.262958i	−0.991319	+	0.131479i	$0.958027\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	140.014i		−	0.141858i
$988$		0			0
$989$	−68.0000			−0.0687563
$990$		0			0
$991$		−	904.708i		−	0.912924i	−0.889743	−	0.456462i	$-0.849116\pi$
$991$		−	904.708i		−	0.912924i	0.889743	−	0.456462i	$-0.150884\pi$
$992$		0			0
$993$	−2349.00			−2.36556
$994$		0			0
$995$		0			0
$996$		0			0
$997$	394.000			0.395186			0.197593	−	0.980284i	$-0.436688\pi$
$997$	394.000			0.395186			0.197593	+	0.980284i	$0.436688\pi$
$998$		0			0
$999$	1914.00			1.91592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.b.1101.2		2
5.2	odd	4		1900.3.g.b.949.4		4
5.3	odd	4		1900.3.g.b.949.1		4
5.4	even	2		76.3.c.a.37.1	✓	2
15.14	odd	2		684.3.h.c.37.1		2
19.18	odd	2	inner	1900.3.e.b.1101.1		2
20.19	odd	2		304.3.e.b.113.2		2
40.19	odd	2		1216.3.e.l.1025.1		2
40.29	even	2		1216.3.e.k.1025.2		2
60.59	even	2		2736.3.o.i.721.1		2
95.18	even	4		1900.3.g.b.949.3		4
95.37	even	4		1900.3.g.b.949.2		4
95.94	odd	2		76.3.c.a.37.2	yes	2
285.284	even	2		684.3.h.c.37.2		2
380.379	even	2		304.3.e.b.113.1		2
760.189	odd	2		1216.3.e.k.1025.1		2
760.379	even	2		1216.3.e.l.1025.2		2
1140.1139	odd	2		2736.3.o.i.721.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.c.a.37.1	✓	2	5.4	even	2
76.3.c.a.37.2	yes	2	95.94	odd	2
304.3.e.b.113.1		2	380.379	even	2
304.3.e.b.113.2		2	20.19	odd	2
684.3.h.c.37.1		2	15.14	odd	2
684.3.h.c.37.2		2	285.284	even	2
1216.3.e.k.1025.1		2	760.189	odd	2
1216.3.e.k.1025.2		2	40.29	even	2
1216.3.e.l.1025.1		2	40.19	odd	2
1216.3.e.l.1025.2		2	760.379	even	2
1900.3.e.b.1101.1		2	19.18	odd	2	inner
1900.3.e.b.1101.2		2	1.1	even	1	trivial
1900.3.g.b.949.1		4	5.3	odd	4
1900.3.g.b.949.2		4	95.37	even	4
1900.3.g.b.949.3		4	95.18	even	4
1900.3.g.b.949.4		4	5.2	odd	4
2736.3.o.i.721.1		2	60.59	even	2
2736.3.o.i.721.2		2	1140.1139	odd	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.38516i q^{3} +1.00000 q^{7} -20.0000 q^{9} +14.0000 q^{11} +16.1555i q^{13} -23.0000 q^{17} +(10.0000 - 16.1555i) q^{19} +5.38516i q^{21} +1.00000 q^{23} -59.2368i q^{27} +48.4665i q^{29} +32.3110i q^{31} +75.3923i q^{33} +32.3110i q^{37} -87.0000 q^{39} -32.3110i q^{41} -68.0000 q^{43} -26.0000 q^{47} -48.0000 q^{49} -123.859i q^{51} -80.7775i q^{53} +(87.0000 + 53.8516i) q^{57} -16.1555i q^{59} -40.0000 q^{61} -20.0000 q^{63} -16.1555i q^{67} +5.38516i q^{69} +32.3110i q^{71} +7.00000 q^{73} +14.0000 q^{77} -96.9330i q^{79} +139.000 q^{81} -32.0000 q^{83} -261.000 q^{87} +129.244i q^{89} +16.1555i q^{91} -174.000 q^{93} -96.9330i q^{97} -280.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 40 q^{9} + 28 q^{11} - 46 q^{17} + 20 q^{19} + 2 q^{23} - 174 q^{39} - 136 q^{43} - 52 q^{47} - 96 q^{49} + 174 q^{57} - 80 q^{61} - 40 q^{63} + 14 q^{73} + 28 q^{77} + 278 q^{81} - 64 q^{83}+ \cdots - 560 q^{99}+O(q^{100}) \) 2 * q + 2 * q^7 - 40 * q^9 + 28 * q^11 - 46 * q^17 + 20 * q^19 + 2 * q^23 - 174 * q^39 - 136 * q^43 - 52 * q^47 - 96 * q^49 + 174 * q^57 - 80 * q^61 - 40 * q^63 + 14 * q^73 + 28 * q^77 + 278 * q^81 - 64 * q^83 - 522 * q^87 - 348 * q^93 - 560 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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