LMFDB - Embedded newform 1900.3.g.b.949.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(949,1900)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1900.949");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.g (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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{29})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 15x^{2} + 49 $ x^4 + 15*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			949.3
Root			$2.19258i$ of defining polynomial
Character	$\chi$	$=$	1900.949
Dual form			1900.3.g.b.949.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.38516 q^{3} -1.00000i q^{7} +20.0000 q^{9} +O(q^{10})$	$q+5.38516 q^{3} -1.00000i q^{7} +20.0000 q^{9} +14.0000 q^{11} +16.1555 q^{13} +23.0000i q^{17} +(-10.0000 - 16.1555i) q^{19} -5.38516i q^{21} +1.00000i q^{23} +59.2368 q^{27} +48.4665i q^{29} -32.3110i q^{31} +75.3923 q^{33} -32.3110 q^{37} +87.0000 q^{39} +32.3110i q^{41} -68.0000i q^{43} +26.0000i q^{47} +48.0000 q^{49} +123.859i q^{51} -80.7775 q^{53} +(-53.8516 - 87.0000i) q^{57} -16.1555i q^{59} -40.0000 q^{61} -20.0000i q^{63} +16.1555 q^{67} +5.38516i q^{69} -32.3110i q^{71} +7.00000i q^{73} -14.0000i q^{77} -96.9330i q^{79} +139.000 q^{81} -32.0000i q^{83} +261.000i q^{87} +129.244i q^{89} -16.1555i q^{91} -174.000i q^{93} +96.9330 q^{97} +280.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 80 q^{9}+O(q^{10}) $ 4 * q + 80 * q^9	$ 4 q + 80 q^{9} + 56 q^{11} - 40 q^{19} + 348 q^{39} + 192 q^{49} - 160 q^{61} + 556 q^{81} + 1120 q^{99}+O(q^{100}) $ 4 * q + 80 * q^9 + 56 * q^11 - 40 * q^19 + 348 * q^39 + 192 * q^49 - 160 * q^61 + 556 * q^81 + 1120 * q^99

Display 100 coefficients 10 coefficients

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
$3$	5.38516			1.79505			0.897527	−	0.440959i	$-0.145361\pi$
$3$	5.38516			1.79505			0.897527	+	0.440959i	$0.145361\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	1.00000i		−	0.142857i	−0.997446	−	0.0714286i	$-0.977244\pi$
$7$		−	1.00000i		−	0.142857i	0.997446	−	0.0714286i	$-0.0227558\pi$
$8$		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
$13$	16.1555			1.24273			0.621365	−	0.783521i	$-0.286578\pi$
$13$	16.1555			1.24273			0.621365	+	0.783521i	$0.286578\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			23.0000i			1.35294i	0.736470	+	0.676471i	$0.236491\pi$
$17$			23.0000i			1.35294i	−0.736470	+	0.676471i	$0.763509\pi$
$18$		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
$23$			1.00000i			0.0434783i	0.999764	+	0.0217391i	$0.00692033\pi$
$23$			1.00000i			0.0434783i	−0.999764	+	0.0217391i	$0.993080\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	59.2368			2.19396
$28$		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	$-0.825505\pi$
$31$		−	32.3110i		−	1.04229i	0.853468	−	0.521145i	$-0.174495\pi$
$32$		0			0
$33$	75.3923			2.28462
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−32.3110			−0.873270			−0.436635	−	0.899639i	$-0.643830\pi$
$37$	−32.3110			−0.873270			−0.436635	+	0.899639i	$0.643830\pi$
$38$		0			0
$39$	87.0000			2.23077
$40$		0			0
$41$			32.3110i			0.788073i	0.919095	+	0.394036i	$0.128922\pi$
$41$			32.3110i			0.788073i	−0.919095	+	0.394036i	$0.871078\pi$
$42$		0			0
$43$		−	68.0000i		−	1.58140i	−0.612207	−	0.790698i	$-0.709718\pi$
$43$		−	68.0000i		−	1.58140i	0.612207	−	0.790698i	$-0.290282\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			26.0000i			0.553191i	0.960986	+	0.276596i	$0.0892063\pi$
$47$			26.0000i			0.553191i	−0.960986	+	0.276596i	$0.910794\pi$
$48$		0			0
$49$	48.0000			0.979592
$50$		0			0
$51$			123.859i			2.42860i
$52$		0			0
$53$	−80.7775			−1.52410			−0.762052	−	0.647516i	$-0.775808\pi$
$53$	−80.7775			−1.52410			−0.762052	+	0.647516i	$0.775808\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−53.8516	−	87.0000i	−0.944766	−	1.52632i
$58$		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
$63$		−	20.0000i		−	0.317460i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	16.1555			0.241127			0.120563	−	0.992706i	$-0.461530\pi$
$67$	16.1555			0.241127			0.120563	+	0.992706i	$0.461530\pi$
$68$		0			0
$69$			5.38516i			0.0780459i
$70$		0			0
$71$		−	32.3110i		−	0.455084i	−0.973768	−	0.227542i	$-0.926931\pi$
$71$		−	32.3110i		−	0.455084i	0.973768	−	0.227542i	$-0.0730690\pi$
$72$		0			0
$73$			7.00000i			0.0958904i	0.998850	+	0.0479452i	$0.0152673\pi$
$73$			7.00000i			0.0958904i	−0.998850	+	0.0479452i	$0.984733\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	14.0000i		−	0.181818i
$78$		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
$83$		−	32.0000i		−	0.385542i	−0.981244	−	0.192771i	$-0.938253\pi$
$83$		−	32.0000i		−	0.385542i	0.981244	−	0.192771i	$-0.0617475\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			261.000i			3.00000i
$88$		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
$93$		−	174.000i		−	1.87097i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	96.9330			0.999309			0.499654	−	0.866225i	$-0.333460\pi$
$97$	96.9330			0.999309			0.499654	+	0.866225i	$0.333460\pi$
$98$		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
$103$	129.244			1.25480			0.627398	−	0.778699i	$-0.284120\pi$
$103$	129.244			1.25480			0.627398	+	0.778699i	$0.284120\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−16.1555			−0.150986			−0.0754930	−	0.997146i	$-0.524053\pi$
$107$	−16.1555			−0.150986			−0.0754930	+	0.997146i	$0.524053\pi$
$108$		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
$113$	96.9330			0.857814			0.428907	−	0.903349i	$-0.358899\pi$
$113$	96.9330			0.857814			0.428907	+	0.903349i	$0.358899\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	323.110			2.76162
$118$		0			0
$119$	23.0000			0.193277
$120$		0			0
$121$	75.0000			0.619835
$122$		0			0
$123$			174.000i			1.41463i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	64.6220			0.508834			0.254417	−	0.967095i	$-0.418116\pi$
$127$	64.6220			0.508834			0.254417	+	0.967095i	$0.418116\pi$
$128$		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
$133$	−16.1555	+	10.0000i	−0.121470	+	0.0751880i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	19.0000i		−	0.138686i	−0.997593	−	0.0693431i	$-0.977910\pi$
$137$		−	19.0000i		−	0.138686i	0.997593	−	0.0693431i	$-0.0220903\pi$
$138$		0			0
$139$	−122.000			−0.877698			−0.438849	−	0.898561i	$-0.644614\pi$
$139$	−122.000			−0.877698			−0.438849	+	0.898561i	$0.644614\pi$
$140$		0			0
$141$			140.014i			0.993009i
$142$		0			0
$143$	226.177			1.58166
$144$		0			0
$145$		0			0
$146$		0			0
$147$	258.488			1.75842
$148$		0			0
$149$	82.0000			0.550336			0.275168	−	0.961396i	$-0.411267\pi$
$149$	82.0000			0.550336			0.275168	+	0.961396i	$0.411267\pi$
$150$		0			0
$151$		−	226.177i		−	1.49786i	−0.662649	−	0.748930i	$-0.730568\pi$
$151$		−	226.177i		−	1.49786i	0.662649	−	0.748930i	$-0.269432\pi$
$152$		0			0
$153$			460.000i			3.00654i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			242.000i			1.54140i	0.637197	+	0.770701i	$0.280094\pi$
$157$			242.000i			1.54140i	−0.637197	+	0.770701i	$0.719906\pi$
$158$		0			0
$159$	−435.000			−2.73585
$160$		0			0
$161$	1.00000			0.00621118
$162$		0			0
$163$			214.000i			1.31288i	0.754377	+	0.656442i	$0.227939\pi$
$163$			214.000i			1.31288i	−0.754377	+	0.656442i	$0.772061\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−32.3110			−0.193479			−0.0967395	−	0.995310i	$-0.530841\pi$
$167$	−32.3110			−0.193479			−0.0967395	+	0.995310i	$0.530841\pi$
$168$		0			0
$169$	92.0000			0.544379
$170$		0			0
$171$	−200.000	−	323.110i	−1.16959	−	1.88953i
$172$		0			0
$173$	−161.555			−0.933844			−0.466922	−	0.884299i	$-0.654637\pi$
$173$	−161.555			−0.933844			−0.466922	+	0.884299i	$0.654637\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	87.0000i		−	0.491525i
$178$		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	$0.0862869\pi$
$181$			96.9330i			0.535541i	−0.963483	+	0.267771i	$0.913713\pi$
$182$		0			0
$183$	−215.407			−1.17709
$184$		0			0
$185$		0			0
$186$		0			0
$187$			322.000i			1.72193i
$188$		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
$193$	−96.9330			−0.502243			−0.251122	−	0.967956i	$-0.580799\pi$
$193$	−96.9330			−0.502243			−0.251122	+	0.967956i	$0.580799\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	142.000i		−	0.720812i	−0.932796	−	0.360406i	$-0.882638\pi$
$197$		−	142.000i		−	0.720812i	0.932796	−	0.360406i	$-0.117362\pi$
$198$		0			0
$199$	−263.000			−1.32161			−0.660804	−	0.750558i	$-0.729785\pi$
$199$	−263.000			−1.32161			−0.660804	+	0.750558i	$0.729785\pi$
$200$		0			0
$201$	87.0000			0.432836
$202$		0			0
$203$	48.4665			0.238751
$204$		0			0
$205$		0			0
$206$		0			0
$207$			20.0000i			0.0966184i
$208$		0			0
$209$	−140.000	−	226.177i	−0.669856	−	1.08219i
$210$		0			0
$211$		−	80.7775i		−	0.382832i	−0.981509	−	0.191416i	$-0.938692\pi$
$211$		−	80.7775i		−	0.382832i	0.981509	−	0.191416i	$-0.0613079\pi$
$212$		0			0
$213$		−	174.000i		−	0.816901i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−32.3110			−0.148899
$218$		0			0
$219$			37.6962i			0.172129i
$220$		0			0
$221$			371.576i			1.68134i
$222$		0			0
$223$	−258.488			−1.15914			−0.579569	−	0.814923i	$-0.696779\pi$
$223$	−258.488			−1.15914			−0.579569	+	0.814923i	$0.696779\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	80.7775			0.355848			0.177924	−	0.984044i	$-0.443062\pi$
$227$	80.7775			0.355848			0.177924	+	0.984044i	$0.443062\pi$
$228$		0			0
$229$	304.000			1.32751			0.663755	−	0.747950i	$-0.268962\pi$
$229$	304.000			1.32751			0.663755	+	0.747950i	$0.268962\pi$
$230$		0			0
$231$		−	75.3923i		−	0.326374i
$232$		0			0
$233$		−	2.00000i		−	0.00858369i	−0.999991	−	0.00429185i	$-0.998634\pi$
$233$		−	2.00000i		−	0.00858369i	0.999991	−	0.00429185i	$-0.00136614\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	522.000i		−	2.20253i
$238$		0			0
$239$	−89.0000			−0.372385			−0.186192	−	0.982513i	$-0.559615\pi$
$239$	−89.0000			−0.372385			−0.186192	+	0.982513i	$0.559615\pi$
$240$		0			0
$241$		−	161.555i		−	0.670352i	−0.942155	−	0.335176i	$-0.891204\pi$
$241$		−	161.555i		−	0.670352i	0.942155	−	0.335176i	$-0.108796\pi$
$242$		0			0
$243$	215.407			0.886447
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−161.555	−	261.000i	−0.654069	−	1.05668i
$248$		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
$253$			14.0000i			0.0553360i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	193.866			0.754342			0.377171	−	0.926144i	$-0.376897\pi$
$257$	193.866			0.754342			0.377171	+	0.926144i	$0.376897\pi$
$258$		0			0
$259$			32.3110i			0.124753i
$260$		0			0
$261$			969.330i			3.71391i
$262$		0			0
$263$			394.000i			1.49810i	0.662514	+	0.749049i	$0.269489\pi$
$263$			394.000i			1.49810i	−0.662514	+	0.749049i	$0.730511\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			696.000i			2.60674i
$268$		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
$273$		−	87.0000i		−	0.318681i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			206.000i			0.743682i	0.928296	+	0.371841i	$0.121273\pi$
$277$			206.000i			0.743682i	−0.928296	+	0.371841i	$0.878727\pi$
$278$		0			0
$279$		−	646.220i		−	2.31620i
$280$		0			0
$281$		−	258.488i		−	0.919886i	−0.887948	−	0.459943i	$-0.847870\pi$
$281$		−	258.488i		−	0.919886i	0.887948	−	0.459943i	$-0.152130\pi$
$282$		0			0
$283$		−	56.0000i		−	0.197880i	−0.995093	−	0.0989399i	$-0.968455\pi$
$283$		−	56.0000i		−	0.197880i	0.995093	−	0.0989399i	$-0.0315452\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	32.3110			0.112582
$288$		0			0
$289$	−240.000			−0.830450
$290$		0			0
$291$	522.000			1.79381
$292$		0			0
$293$	−80.7775			−0.275691			−0.137846	−	0.990454i	$-0.544018\pi$
$293$	−80.7775			−0.275691			−0.137846	+	0.990454i	$0.544018\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	829.315			2.79231
$298$		0			0
$299$			16.1555i			0.0540318i
$300$		0			0
$301$	−68.0000			−0.225914
$302$		0			0
$303$	75.3923			0.248819
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−420.043			−1.36822			−0.684109	−	0.729380i	$-0.739809\pi$
$307$	−420.043			−1.36822			−0.684109	+	0.729380i	$0.739809\pi$
$308$		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
$313$		−	77.0000i		−	0.246006i	−0.992406	−	0.123003i	$-0.960747\pi$
$313$		−	77.0000i		−	0.246006i	0.992406	−	0.123003i	$-0.0392525\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−403.887			−1.27409			−0.637046	−	0.770826i	$-0.719844\pi$
$317$	−403.887			−1.27409			−0.637046	+	0.770826i	$0.719844\pi$
$318$		0			0
$319$			678.531i			2.12706i
$320$		0			0
$321$	−87.0000			−0.271028
$322$		0			0
$323$	371.576	−	230.000i	1.15039	−	0.712074i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	87.0000i		−	0.266055i
$328$		0			0
$329$	26.0000			0.0790274
$330$		0			0
$331$		−	436.198i		−	1.31782i	−0.752222	−	0.658910i	$-0.771018\pi$
$331$		−	436.198i		−	1.31782i	0.752222	−	0.658910i	$-0.228982\pi$
$332$		0			0
$333$	−646.220			−1.94060
$334$		0			0
$335$		0			0
$336$		0			0
$337$	387.732			1.15054			0.575270	−	0.817964i	$-0.304897\pi$
$337$	387.732			1.15054			0.575270	+	0.817964i	$0.304897\pi$
$338$		0			0
$339$	522.000			1.53982
$340$		0			0
$341$		−	452.354i		−	1.32655i
$342$		0			0
$343$		−	97.0000i		−	0.282799i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			392.000i			1.12968i	0.825199	+	0.564841i	$0.191063\pi$
$347$			392.000i			1.12968i	−0.825199	+	0.564841i	$0.808937\pi$
$348$		0			0
$349$	−410.000			−1.17479			−0.587393	−	0.809302i	$-0.699846\pi$
$349$	−410.000			−1.17479			−0.587393	+	0.809302i	$0.699846\pi$
$350$		0			0
$351$	957.000			2.72650
$352$		0			0
$353$			481.000i			1.36261i	0.732001	+	0.681303i	$0.238586\pi$
$353$			481.000i			1.36261i	−0.732001	+	0.681303i	$0.761414\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	123.859			0.346943
$358$		0			0
$359$	109.000			0.303621			0.151811	−	0.988410i	$-0.451490\pi$
$359$	109.000			0.303621			0.151811	+	0.988410i	$0.451490\pi$
$360$		0			0
$361$	−161.000	+	323.110i	−0.445983	+	0.895041i
$362$		0			0
$363$	403.887			1.11264
$364$		0			0
$365$		0			0
$366$		0			0
$367$			50.0000i			0.136240i	0.997677	+	0.0681199i	$0.0217000\pi$
$367$			50.0000i			0.136240i	−0.997677	+	0.0681199i	$0.978300\pi$
$368$		0			0
$369$			646.220i			1.75127i
$370$		0			0
$371$			80.7775i			0.217729i
$372$		0			0
$373$	500.820			1.34268			0.671341	−	0.741149i	$-0.265719\pi$
$373$	500.820			1.34268			0.671341	+	0.741149i	$0.265719\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			783.000i			2.07692i
$378$		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
$383$	258.488			0.674903			0.337452	−	0.941343i	$-0.390435\pi$
$383$	258.488			0.674903			0.337452	+	0.941343i	$0.390435\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	1360.00i		−	3.51421i
$388$		0			0
$389$	−578.000			−1.48586			−0.742931	−	0.669368i	$-0.766565\pi$
$389$	−578.000			−1.48586			−0.742931	+	0.669368i	$0.766565\pi$
$390$		0			0
$391$	−23.0000			−0.0588235
$392$		0			0
$393$	301.569			0.767352
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	658.000i		−	1.65743i	−0.559670	−	0.828715i	$-0.689072\pi$
$397$		−	658.000i		−	1.65743i	0.559670	−	0.828715i	$-0.310928\pi$
$398$		0			0
$399$	−87.0000	+	53.8516i	−0.218045	+	0.134967i
$400$		0			0
$401$		−	775.464i		−	1.93382i	−0.255109	−	0.966912i	$-0.582111\pi$
$401$		−	775.464i		−	1.93382i	0.255109	−	0.966912i	$-0.417889\pi$
$402$		0			0
$403$		−	522.000i		−	1.29529i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−452.354			−1.11143
$408$		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
$413$	−16.1555			−0.0391174
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−656.990			−1.57552
$418$		0			0
$419$	−8.00000			−0.0190931			−0.00954654	−	0.999954i	$-0.503039\pi$
$419$	−8.00000			−0.0190931			−0.00954654	+	0.999954i	$0.503039\pi$
$420$		0			0
$421$			500.820i			1.18960i	0.803875	+	0.594798i	$0.202768\pi$
$421$			500.820i			1.18960i	−0.803875	+	0.594798i	$0.797232\pi$
$422$		0			0
$423$			520.000i			1.22931i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			40.0000i			0.0936768i
$428$		0			0
$429$	1218.00			2.83916
$430$		0			0
$431$		−	646.220i		−	1.49935i	−0.661806	−	0.749675i	$-0.730210\pi$
$431$		−	646.220i		−	1.49935i	0.661806	−	0.749675i	$-0.269790\pi$
$432$		0			0
$433$	193.866			0.447727			0.223864	−	0.974620i	$-0.428133\pi$
$433$	193.866			0.447727			0.223864	+	0.974620i	$0.428133\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.1555	−	10.0000i	0.0369691	−	0.0228833i
$438$		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
$443$		−	182.000i		−	0.410835i	−0.978674	−	0.205418i	$-0.934145\pi$
$443$		−	182.000i		−	0.410835i	0.978674	−	0.205418i	$-0.0658553\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	441.584			0.987883
$448$		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
$453$		−	1218.00i		−	2.68874i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			317.000i			0.693654i	0.937929	+	0.346827i	$0.112741\pi$
$457$			317.000i			0.693654i	−0.937929	+	0.346827i	$0.887259\pi$
$458$		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
$463$		−	146.000i		−	0.315335i	−0.987492	−	0.157667i	$-0.949603\pi$
$463$		−	146.000i		−	0.315335i	0.987492	−	0.157667i	$-0.0503974\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			662.000i			1.41756i	0.705430	+	0.708779i	$0.250754\pi$
$467$			662.000i			1.41756i	−0.705430	+	0.708779i	$0.749246\pi$
$468$		0			0
$469$		−	16.1555i		−	0.0344467i
$470$		0			0
$471$			1303.21i			2.76690i
$472$		0			0
$473$		−	952.000i		−	2.01268i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1615.55			−3.38690
$478$		0			0
$479$	130.000			0.271399			0.135699	−	0.990750i	$-0.456672\pi$
$479$	130.000			0.271399			0.135699	+	0.990750i	$0.456672\pi$
$480$		0			0
$481$	−522.000			−1.08524
$482$		0			0
$483$	5.38516			0.0111494
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−387.732			−0.796164			−0.398082	−	0.917350i	$-0.630324\pi$
$487$	−387.732			−0.796164			−0.398082	+	0.917350i	$0.630324\pi$
$488$		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
$493$	−1114.73			−2.26111
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−32.3110			−0.0650120
$498$		0			0
$499$	460.000			0.921844			0.460922	−	0.887441i	$-0.347519\pi$
$499$	460.000			0.921844			0.460922	+	0.887441i	$0.347519\pi$
$500$		0			0
$501$	−174.000			−0.347305
$502$		0			0
$503$		−	263.000i		−	0.522863i	−0.965222	−	0.261431i	$-0.915805\pi$
$503$		−	263.000i		−	0.522863i	0.965222	−	0.261431i	$-0.0841945\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	495.435			0.977190
$508$		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
$513$	−592.368	−	957.000i	−1.15471	−	1.86550i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			364.000i			0.704062i
$518$		0			0
$519$	−870.000			−1.67630
$520$		0			0
$521$		−	226.177i		−	0.434121i	−0.976158	−	0.217060i	$-0.930353\pi$
$521$		−	226.177i		−	0.434121i	0.976158	−	0.217060i	$-0.0696469\pi$
$522$		0			0
$523$	80.7775			0.154450			0.0772251	−	0.997014i	$-0.475394\pi$
$523$	80.7775			0.154450			0.0772251	+	0.997014i	$0.475394\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	743.153			1.41016
$528$		0			0
$529$	528.000			0.998110
$530$		0			0
$531$		−	323.110i		−	0.608493i
$532$		0			0
$533$			522.000i			0.979362i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	1566.00i		−	2.91620i
$538$		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
$543$			522.000i			0.961326i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−32.3110			−0.0590694			−0.0295347	−	0.999564i	$-0.509403\pi$
$547$	−32.3110			−0.0590694			−0.0295347	+	0.999564i	$0.509403\pi$
$548$		0			0
$549$	−800.000			−1.45719
$550$		0			0
$551$	783.000	−	484.665i	1.42105	−	0.879609i
$552$		0			0
$553$	−96.9330			−0.175286
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	844.000i		−	1.51526i	−0.652684	−	0.757630i	$-0.726357\pi$
$557$		−	844.000i		−	1.51526i	0.652684	−	0.757630i	$-0.273643\pi$
$558$		0			0
$559$		−	1098.57i		−	1.96525i
$560$		0			0
$561$			1734.02i			3.09095i
$562$		0			0
$563$	420.043			0.746080			0.373040	−	0.927815i	$-0.378315\pi$
$563$	420.043			0.746080			0.373040	+	0.927815i	$0.378315\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	139.000i		−	0.245150i
$568$		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
$573$	−360.806			−0.629679
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	85.0000i		−	0.147314i	−0.997284	−	0.0736568i	$-0.976533\pi$
$577$		−	85.0000i		−	0.147314i	0.997284	−	0.0736568i	$-0.0234670\pi$
$578$		0			0
$579$	−522.000			−0.901554
$580$		0			0
$581$	−32.0000			−0.0550775
$582$		0			0
$583$	−1130.88			−1.93977
$584$		0			0
$585$		0			0
$586$		0			0
$587$			488.000i			0.831346i	0.909514	+	0.415673i	$0.136454\pi$
$587$			488.000i			0.831346i	−0.909514	+	0.415673i	$0.863546\pi$
$588$		0			0
$589$	−522.000	+	323.110i	−0.886248	+	0.548574i
$590$		0			0
$591$		−	764.693i		−	1.29390i
$592$		0			0
$593$			610.000i			1.02867i	0.857590	+	0.514334i	$0.171961\pi$
$593$			610.000i			1.02867i	−0.857590	+	0.514334i	$0.828039\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1416.30			−2.37236
$598$		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	$-0.974303\pi$
$601$		−	96.9330i		−	0.161286i	0.996743	−	0.0806431i	$-0.0256974\pi$
$602$		0			0
$603$	323.110			0.535837
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−1001.64			−1.65015			−0.825075	−	0.565024i	$-0.808867\pi$
$607$	−1001.64			−1.65015			−0.825075	+	0.565024i	$0.808867\pi$
$608$		0			0
$609$	261.000			0.428571
$610$		0			0
$611$			420.043i			0.687468i
$612$		0			0
$613$		−	200.000i		−	0.326264i	−0.986604	−	0.163132i	$-0.947840\pi$
$613$		−	200.000i		−	0.326264i	0.986604	−	0.163132i	$-0.0521597\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			530.000i			0.858995i	0.903068	+	0.429498i	$0.141309\pi$
$617$			530.000i			0.858995i	−0.903068	+	0.429498i	$0.858691\pi$
$618$		0			0
$619$	256.000			0.413570			0.206785	−	0.978386i	$-0.433700\pi$
$619$	256.000			0.413570			0.206785	+	0.978386i	$0.433700\pi$
$620$		0			0
$621$			59.2368i			0.0953894i
$622$		0			0
$623$	129.244			0.207454
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−753.923	−	1218.00i	−1.20243	−	1.94258i
$628$		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
$633$		−	435.000i		−	0.687204i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	775.464			1.21737
$638$		0			0
$639$		−	646.220i		−	1.01130i
$640$		0			0
$641$		−	581.598i		−	0.907329i	−0.891173	−	0.453664i	$-0.850116\pi$
$641$		−	581.598i		−	0.907329i	0.891173	−	0.453664i	$-0.149884\pi$
$642$		0			0
$643$		−	170.000i		−	0.264386i	−0.991224	−	0.132193i	$-0.957798\pi$
$643$		−	170.000i		−	0.264386i	0.991224	−	0.132193i	$-0.0422018\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	25.0000i		−	0.0386399i	−0.999813	−	0.0193199i	$-0.993850\pi$
$647$		−	25.0000i		−	0.0386399i	0.999813	−	0.0193199i	$-0.00615011\pi$
$648$		0			0
$649$		−	226.177i		−	0.348501i
$650$		0			0
$651$	−174.000			−0.267281
$652$		0			0
$653$			1054.00i			1.61409i	0.590491	+	0.807044i	$0.298934\pi$
$653$			1054.00i			1.61409i	−0.590491	+	0.807044i	$0.701066\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			140.000i			0.213090i
$658$		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	$-0.867871\pi$
$661$		−	533.131i		−	0.806553i	0.915078	−	0.403276i	$-0.132129\pi$
$662$		0			0
$663$			2001.00i			3.01810i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−48.4665			−0.0726634
$668$		0			0
$669$	−1392.00			−2.08072
$670$		0			0
$671$	−560.000			−0.834575
$672$		0			0
$673$	−290.799			−0.432093			−0.216047	−	0.976383i	$-0.569316\pi$
$673$	−290.799			−0.432093			−0.216047	+	0.976383i	$0.569316\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−953.174			−1.40794			−0.703969	−	0.710231i	$-0.748591\pi$
$677$	−953.174			−1.40794			−0.703969	+	0.710231i	$0.748591\pi$
$678$		0			0
$679$		−	96.9330i		−	0.142758i
$680$		0			0
$681$	435.000			0.638767
$682$		0			0
$683$	872.397			1.27730			0.638651	−	0.769497i	$-0.279493\pi$
$683$	872.397			1.27730			0.638651	+	0.769497i	$0.279493\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1637.09			2.38296
$688$		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
$693$		−	280.000i		−	0.404040i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−743.153			−1.06622
$698$		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
$703$	323.110	+	522.000i	0.459616	+	0.742532i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	14.0000i		−	0.0198020i
$708$		0			0
$709$	−1292.00			−1.82228			−0.911142	−	0.412092i	$-0.864798\pi$
$709$	−1292.00			−1.82228			−0.911142	+	0.412092i	$0.864798\pi$
$710$		0			0
$711$		−	1938.66i		−	2.72667i
$712$		0			0
$713$	32.3110			0.0453170
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−479.280			−0.668451
$718$		0			0
$719$	−1061.00			−1.47566			−0.737830	−	0.674986i	$-0.764150\pi$
$719$	−1061.00			−1.47566			−0.737830	+	0.674986i	$0.764150\pi$
$720$		0			0
$721$		−	129.244i		−	0.179257i
$722$		0			0
$723$		−	870.000i		−	1.20332i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			83.0000i			0.114168i	0.998369	+	0.0570839i	$0.0181803\pi$
$727$			83.0000i			0.114168i	−0.998369	+	0.0570839i	$0.981820\pi$
$728$		0			0
$729$	−91.0000			−0.124829
$730$		0			0
$731$	1564.00			2.13953
$732$		0			0
$733$		−	296.000i		−	0.403820i	−0.979404	−	0.201910i	$-0.935285\pi$
$733$		−	296.000i		−	0.403820i	0.979404	−	0.201910i	$-0.0647148\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	226.177			0.306889
$738$		0			0
$739$	−548.000			−0.741543			−0.370771	−	0.928724i	$-0.620907\pi$
$739$	−548.000			−0.741543			−0.370771	+	0.928724i	$0.620907\pi$
$740$		0			0
$741$	−870.000	−	1405.53i	−1.17409	−	1.89680i
$742$		0			0
$743$	613.909			0.826257			0.413128	−	0.910673i	$-0.364436\pi$
$743$	613.909			0.826257			0.413128	+	0.910673i	$0.364436\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	640.000i		−	0.856760i
$748$		0			0
$749$			16.1555i			0.0215694i
$750$		0			0
$751$			420.043i			0.559311i	0.960100	+	0.279656i	$0.0902203\pi$
$751$			420.043i			0.559311i	−0.960100	+	0.279656i	$0.909780\pi$
$752$		0			0
$753$	−1992.51			−2.64610
$754$		0			0
$755$		0			0
$756$		0			0
$757$			188.000i			0.248349i	0.992260	+	0.124174i	$0.0396282\pi$
$757$			188.000i			0.248349i	−0.992260	+	0.124174i	$0.960372\pi$
$758$		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
$763$	−16.1555			−0.0211736
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	261.000i		−	0.340287i
$768$		0			0
$769$	1177.00			1.53056			0.765280	−	0.643698i	$-0.222601\pi$
$769$	1177.00			1.53056			0.765280	+	0.643698i	$0.222601\pi$
$770$		0			0
$771$	1044.00			1.35409
$772$		0			0
$773$	1017.80			1.31668			0.658342	−	0.752719i	$-0.271258\pi$
$773$	1017.80			1.31668			0.658342	+	0.752719i	$0.271258\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			174.000i			0.223938i
$778$		0			0
$779$	522.000	−	323.110i	0.670090	−	0.414775i
$780$		0			0
$781$		−	452.354i		−	0.579198i
$782$		0			0
$783$			2871.00i			3.66667i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−113.088			−0.143696			−0.0718478	−	0.997416i	$-0.522890\pi$
$787$	−113.088			−0.143696			−0.0718478	+	0.997416i	$0.522890\pi$
$788$		0			0
$789$			2121.75i			2.68917i
$790$		0			0
$791$		−	96.9330i		−	0.122545i
$792$		0			0
$793$	−646.220			−0.814905
$794$		0			0
$795$		0			0
$796$		0			0
$797$	177.710			0.222974			0.111487	−	0.993766i	$-0.464439\pi$
$797$	177.710			0.222974			0.111487	+	0.993766i	$0.464439\pi$
$798$		0			0
$799$	−598.000			−0.748436
$800$		0			0
$801$			2584.88i			3.22707i
$802$		0			0
$803$			98.0000i			0.122042i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	522.000i		−	0.646840i
$808$		0			0
$809$	−737.000			−0.911001			−0.455501	−	0.890235i	$-0.650540\pi$
$809$	−737.000			−0.911001			−0.455501	+	0.890235i	$0.650540\pi$
$810$		0			0
$811$			791.619i			0.976103i	0.872815	+	0.488051i	$0.162292\pi$
$811$			791.619i			0.976103i	−0.872815	+	0.488051i	$0.837708\pi$
$812$		0			0
$813$	−2170.22			−2.66940
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1098.57	+	680.000i	−1.34464	+	0.832313i
$818$		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
$823$		−	1499.00i		−	1.82139i	−0.413085	−	0.910693i	$-0.635549\pi$
$823$		−	1499.00i		−	1.82139i	0.413085	−	0.910693i	$-0.364451\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	953.174			1.15257			0.576284	−	0.817249i	$-0.304502\pi$
$827$	953.174			1.15257			0.576284	+	0.817249i	$0.304502\pi$
$828$		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
$833$			1104.00i			1.32533i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	1914.00i		−	2.28674i
$838$		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
$843$		−	1392.00i		−	1.65125i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	75.0000i		−	0.0885478i
$848$		0			0
$849$		−	301.569i		−	0.355205i
$850$		0			0
$851$		−	32.3110i		−	0.0379683i
$852$		0			0
$853$		−	536.000i		−	0.628370i	−0.949362	−	0.314185i	$-0.898269\pi$
$853$		−	536.000i		−	0.628370i	0.949362	−	0.314185i	$-0.101731\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1227.82			−1.43269			−0.716346	−	0.697745i	$-0.754187\pi$
$857$	−1227.82			−1.43269			−0.716346	+	0.697745i	$0.754187\pi$
$858$		0			0
$859$	496.000			0.577416			0.288708	−	0.957417i	$-0.406774\pi$
$859$	496.000			0.577416			0.288708	+	0.957417i	$0.406774\pi$
$860$		0			0
$861$	174.000			0.202091
$862$		0			0
$863$	−581.598			−0.673926			−0.336963	−	0.941518i	$-0.609400\pi$
$863$	−581.598			−0.673926			−0.336963	+	0.941518i	$0.609400\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1292.44			−1.49070
$868$		0			0
$869$		−	1357.06i		−	1.56164i
$870$		0			0
$871$	261.000			0.299656
$872$		0			0
$873$	1938.66			2.22069
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1082.42			1.23423			0.617114	−	0.786874i	$-0.288302\pi$
$877$	1082.42			1.23423			0.617114	+	0.786874i	$0.288302\pi$
$878$		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
$883$			1480.00i			1.67610i	0.545590	+	0.838052i	$0.316306\pi$
$883$			1480.00i			1.67610i	−0.545590	+	0.838052i	$0.683694\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−969.330			−1.09282			−0.546409	−	0.837518i	$-0.684006\pi$
$887$	−969.330			−1.09282			−0.546409	+	0.837518i	$0.684006\pi$
$888$		0			0
$889$		−	64.6220i		−	0.0726906i
$890$		0			0
$891$	1946.00			2.18406
$892$		0			0
$893$	420.043	−	260.000i	0.470373	−	0.291153i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			87.0000i			0.0969900i
$898$		0			0
$899$	1566.00			1.74194
$900$		0			0
$901$		−	1857.88i		−	2.06202i
$902$		0			0
$903$	−366.191			−0.405527
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−16.1555			−0.0178120			−0.00890601	−	0.999960i	$-0.502835\pi$
$907$	−16.1555			−0.0178120			−0.00890601	+	0.999960i	$0.502835\pi$
$908$		0			0
$909$	280.000			0.308031
$910$		0			0
$911$		−	452.354i		−	0.496546i	−0.968690	−	0.248273i	$-0.920137\pi$
$911$		−	452.354i		−	0.496546i	0.968690	−	0.248273i	$-0.0798631\pi$
$912$		0			0
$913$		−	448.000i		−	0.490690i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	56.0000i		−	0.0610687i
$918$		0			0
$919$	−515.000			−0.560392			−0.280196	−	0.959943i	$-0.590399\pi$
$919$	−515.000			−0.560392			−0.280196	+	0.959943i	$0.590399\pi$
$920$		0			0
$921$	−2262.00			−2.45603
$922$		0			0
$923$		−	522.000i		−	0.565547i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	2584.88			2.78843
$928$		0			0
$929$	−671.000			−0.722282			−0.361141	−	0.932511i	$-0.617613\pi$
$929$	−671.000			−0.722282			−0.361141	+	0.932511i	$0.617613\pi$
$930$		0			0
$931$	−480.000	−	775.464i	−0.515575	−	0.832936i
$932$		0			0
$933$	−1427.07			−1.52955
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1601.00i			1.70864i	0.519744	+	0.854322i	$0.326027\pi$
$937$			1601.00i			1.70864i	−0.519744	+	0.854322i	$0.673973\pi$
$938$		0			0
$939$		−	414.658i		−	0.441595i
$940$		0			0
$941$			1567.08i			1.66534i	0.553771	+	0.832669i	$0.313188\pi$
$941$			1567.08i			1.66534i	−0.553771	+	0.832669i	$0.686812\pi$
$942$		0			0
$943$	−32.3110			−0.0342640
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1192.00i		−	1.25871i	−0.777117	−	0.629356i	$-0.783319\pi$
$947$		−	1192.00i		−	1.25871i	0.777117	−	0.629356i	$-0.216681\pi$
$948$		0			0
$949$			113.088i			0.119166i
$950$		0			0
$951$	−2175.00			−2.28707
$952$		0			0
$953$	−840.086			−0.881517			−0.440759	−	0.897626i	$-0.645290\pi$
$953$	−840.086			−0.881517			−0.440759	+	0.897626i	$0.645290\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			3654.00i			3.81818i
$958$		0			0
$959$	−19.0000			−0.0198123
$960$		0			0
$961$	−83.0000			−0.0863684
$962$		0			0
$963$	−323.110			−0.335524
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	514.000i		−	0.531541i	−0.964036	−	0.265770i	$-0.914374\pi$
$967$		−	514.000i		−	0.531541i	0.964036	−	0.265770i	$-0.0856263\pi$
$968$		0			0
$969$	2001.00	−	1238.59i	2.06502	−	1.27821i
$970$		0			0
$971$		−	1712.48i		−	1.76363i	−0.471598	−	0.881814i	$-0.656323\pi$
$971$		−	1712.48i		−	1.76363i	0.471598	−	0.881814i	$-0.343677\pi$
$972$		0			0
$973$			122.000i			0.125385i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	161.555			0.165358			0.0826791	−	0.996576i	$-0.473652\pi$
$977$	161.555			0.165358			0.0826791	+	0.996576i	$0.473652\pi$
$978$		0			0
$979$			1809.42i			1.84823i
$980$		0			0
$981$		−	323.110i		−	0.329368i
$982$		0			0
$983$	258.488			0.262958			0.131479	−	0.991319i	$-0.458027\pi$
$983$	258.488			0.262958			0.131479	+	0.991319i	$0.458027\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	140.014			0.141858
$988$		0			0
$989$	68.0000			0.0687563
$990$		0			0
$991$			904.708i			0.912924i	0.889743	+	0.456462i	$0.150884\pi$
$991$			904.708i			0.912924i	−0.889743	+	0.456462i	$0.849116\pi$
$992$		0			0
$993$		−	2349.00i		−	2.36556i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	394.000i		−	0.395186i	−0.980284	−	0.197593i	$-0.936688\pi$
$997$		−	394.000i		−	0.395186i	0.980284	−	0.197593i	$-0.0633124\pi$
$998$		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.g.b.949.3		4
5.2	odd	4		1900.3.e.b.1101.1		2
5.3	odd	4		76.3.c.a.37.2	yes	2
5.4	even	2	inner	1900.3.g.b.949.2		4
15.8	even	4		684.3.h.c.37.2		2
19.18	odd	2	inner	1900.3.g.b.949.1		4
20.3	even	4		304.3.e.b.113.1		2
40.3	even	4		1216.3.e.l.1025.2		2
40.13	odd	4		1216.3.e.k.1025.1		2
60.23	odd	4		2736.3.o.i.721.2		2
95.18	even	4		76.3.c.a.37.1	✓	2
95.37	even	4		1900.3.e.b.1101.2		2
95.94	odd	2	inner	1900.3.g.b.949.4		4
285.113	odd	4		684.3.h.c.37.1		2
380.303	odd	4		304.3.e.b.113.2		2
760.493	even	4		1216.3.e.k.1025.2		2
760.683	odd	4		1216.3.e.l.1025.1		2
1140.683	even	4		2736.3.o.i.721.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.c.a.37.1	✓	2	95.18	even	4
76.3.c.a.37.2	yes	2	5.3	odd	4
304.3.e.b.113.1		2	20.3	even	4
304.3.e.b.113.2		2	380.303	odd	4
684.3.h.c.37.1		2	285.113	odd	4
684.3.h.c.37.2		2	15.8	even	4
1216.3.e.k.1025.1		2	40.13	odd	4
1216.3.e.k.1025.2		2	760.493	even	4
1216.3.e.l.1025.1		2	760.683	odd	4
1216.3.e.l.1025.2		2	40.3	even	4
1900.3.e.b.1101.1		2	5.2	odd	4
1900.3.e.b.1101.2		2	95.37	even	4
1900.3.g.b.949.1		4	19.18	odd	2	inner
1900.3.g.b.949.2		4	5.4	even	2	inner
1900.3.g.b.949.3		4	1.1	even	1	trivial
1900.3.g.b.949.4		4	95.94	odd	2	inner
2736.3.o.i.721.1		2	1140.683	even	4
2736.3.o.i.721.2		2	60.23	odd	4

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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.38516 q^{3} -1.00000i q^{7} +20.0000 q^{9} +O(q^{10})\)	\(q+5.38516 q^{3} -1.00000i q^{7} +20.0000 q^{9} +14.0000 q^{11} +16.1555 q^{13} +23.0000i q^{17} +(-10.0000 - 16.1555i) q^{19} -5.38516i q^{21} +1.00000i q^{23} +59.2368 q^{27} +48.4665i q^{29} -32.3110i q^{31} +75.3923 q^{33} -32.3110 q^{37} +87.0000 q^{39} +32.3110i q^{41} -68.0000i q^{43} +26.0000i q^{47} +48.0000 q^{49} +123.859i q^{51} -80.7775 q^{53} +(-53.8516 - 87.0000i) q^{57} -16.1555i q^{59} -40.0000 q^{61} -20.0000i q^{63} +16.1555 q^{67} +5.38516i q^{69} -32.3110i q^{71} +7.00000i q^{73} -14.0000i q^{77} -96.9330i q^{79} +139.000 q^{81} -32.0000i q^{83} +261.000i q^{87} +129.244i q^{89} -16.1555i q^{91} -174.000i q^{93} +96.9330 q^{97} +280.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 80 q^{9}+O(q^{10}) \) 4 * q + 80 * q^9	\( 4 q + 80 q^{9} + 56 q^{11} - 40 q^{19} + 348 q^{39} + 192 q^{49} - 160 q^{61} + 556 q^{81} + 1120 q^{99}+O(q^{100}) \) 4 * q + 80 * q^9 + 56 * q^11 - 40 * q^19 + 348 * q^39 + 192 * q^49 - 160 * q^61 + 556 * q^81 + 1120 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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