LMFDB - Embedded newform 2240.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.164278413$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q+4.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} -11.0000 q^{9} +60.0000 q^{11} -38.0000 q^{13} -20.0000 q^{15} +42.0000 q^{17} -52.0000 q^{19} -28.0000 q^{21} -120.000 q^{23} +25.0000 q^{25} -152.000 q^{27} +234.000 q^{29} +304.000 q^{31} +240.000 q^{33} +35.0000 q^{35} +106.000 q^{37} -152.000 q^{39} -54.0000 q^{41} -196.000 q^{43} +55.0000 q^{45} -336.000 q^{47} +49.0000 q^{49} +168.000 q^{51} -438.000 q^{53} -300.000 q^{55} -208.000 q^{57} -444.000 q^{59} -38.0000 q^{61} +77.0000 q^{63} +190.000 q^{65} -988.000 q^{67} -480.000 q^{69} +720.000 q^{71} +146.000 q^{73} +100.000 q^{75} -420.000 q^{77} +808.000 q^{79} -311.000 q^{81} +612.000 q^{83} -210.000 q^{85} +936.000 q^{87} +1146.00 q^{89} +266.000 q^{91} +1216.00 q^{93} +260.000 q^{95} -70.0000 q^{97} -660.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	4.00000	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$3$	4.00000	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	60.0000	1.64461	0.822304	−	0.569049i	$-0.192689\pi$
$11$	60.0000	1.64461	0.822304	+	0.569049i	$0.192689\pi$
$12$	0	0
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	0	0
$15$	−20.0000	−0.344265
$16$	0	0
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	0	0
$19$	−52.0000	−0.627875	−0.313937	−	0.949444i	$-0.601648\pi$
$19$	−52.0000	−0.627875	−0.313937	+	0.949444i	$0.601648\pi$
$20$	0	0
$21$	−28.0000	−0.290957
$22$	0	0
$23$	−120.000	−1.08790	−0.543951	−	0.839117i	$-0.683072\pi$
$23$	−120.000	−1.08790	−0.543951	+	0.839117i	$0.683072\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−152.000	−1.08342
$28$	0	0
$29$	234.000	1.49837	0.749185	−	0.662361i	$-0.230446\pi$
$29$	234.000	1.49837	0.749185	+	0.662361i	$0.230446\pi$
$30$	0	0
$31$	304.000	1.76129	0.880645	−	0.473776i	$-0.157109\pi$
$31$	304.000	1.76129	0.880645	+	0.473776i	$0.157109\pi$
$32$	0	0
$33$	240.000	1.26602
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	106.000	0.470981	0.235490	−	0.971877i	$-0.424330\pi$
$37$	106.000	0.470981	0.235490	+	0.971877i	$0.424330\pi$
$38$	0	0
$39$	−152.000	−0.624089
$40$	0	0
$41$	−54.0000	−0.205692	−0.102846	−	0.994697i	$-0.532795\pi$
$41$	−54.0000	−0.205692	−0.102846	+	0.994697i	$0.532795\pi$
$42$	0	0
$43$	−196.000	−0.695110	−0.347555	−	0.937660i	$-0.612988\pi$
$43$	−196.000	−0.695110	−0.347555	+	0.937660i	$0.612988\pi$
$44$	0	0
$45$	55.0000	0.182198
$46$	0	0
$47$	−336.000	−1.04278	−0.521390	−	0.853319i	$-0.674586\pi$
$47$	−336.000	−1.04278	−0.521390	+	0.853319i	$0.674586\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	168.000	0.461269
$52$	0	0
$53$	−438.000	−1.13517	−0.567584	−	0.823315i	$-0.692122\pi$
$53$	−438.000	−1.13517	−0.567584	+	0.823315i	$0.692122\pi$
$54$	0	0
$55$	−300.000	−0.735491
$56$	0	0
$57$	−208.000	−0.483338
$58$	0	0
$59$	−444.000	−0.979727	−0.489863	−	0.871799i	$-0.662953\pi$
$59$	−444.000	−0.979727	−0.489863	+	0.871799i	$0.662953\pi$
$60$	0	0
$61$	−38.0000	−0.0797607	−0.0398803	−	0.999204i	$-0.512698\pi$
$61$	−38.0000	−0.0797607	−0.0398803	+	0.999204i	$0.512698\pi$
$62$	0	0
$63$	77.0000	0.153986
$64$	0	0
$65$	190.000	0.362563
$66$	0	0
$67$	−988.000	−1.80154	−0.900772	−	0.434293i	$-0.856998\pi$
$67$	−988.000	−1.80154	−0.900772	+	0.434293i	$0.856998\pi$
$68$	0	0
$69$	−480.000	−0.837467
$70$	0	0
$71$	720.000	1.20350	0.601748	−	0.798686i	$-0.294471\pi$
$71$	720.000	1.20350	0.601748	+	0.798686i	$0.294471\pi$
$72$	0	0
$73$	146.000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	146.000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	100.000	0.153960
$76$	0	0
$77$	−420.000	−0.621603
$78$	0	0
$79$	808.000	1.15072	0.575361	−	0.817899i	$-0.304861\pi$
$79$	808.000	1.15072	0.575361	+	0.817899i	$0.304861\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	612.000	0.809346	0.404673	−	0.914461i	$-0.367385\pi$
$83$	612.000	0.809346	0.404673	+	0.914461i	$0.367385\pi$
$84$	0	0
$85$	−210.000	−0.267973
$86$	0	0
$87$	936.000	1.15345
$88$	0	0
$89$	1146.00	1.36490	0.682448	−	0.730934i	$-0.260915\pi$
$89$	1146.00	1.36490	0.682448	+	0.730934i	$0.260915\pi$
$90$	0	0
$91$	266.000	0.306422
$92$	0	0
$93$	1216.00	1.35584
$94$	0	0
$95$	260.000	0.280794
$96$	0	0
$97$	−70.0000	−0.0732724	−0.0366362	−	0.999329i	$-0.511664\pi$
$97$	−70.0000	−0.0732724	−0.0366362	+	0.999329i	$0.511664\pi$
$98$	0	0
$99$	−660.000	−0.670025
$100$	0	0
$101$	−1950.00	−1.92111	−0.960556	−	0.278088i	$-0.910299\pi$
$101$	−1950.00	−1.92111	−0.960556	+	0.278088i	$0.910299\pi$
$102$	0	0
$103$	−488.000	−0.466836	−0.233418	−	0.972377i	$-0.574991\pi$
$103$	−488.000	−0.466836	−0.233418	+	0.972377i	$0.574991\pi$
$104$	0	0
$105$	140.000	0.130120
$106$	0	0
$107$	−996.000	−0.899878	−0.449939	−	0.893059i	$-0.648554\pi$
$107$	−996.000	−0.899878	−0.449939	+	0.893059i	$0.648554\pi$
$108$	0	0
$109$	−758.000	−0.666085	−0.333042	−	0.942912i	$-0.608075\pi$
$109$	−758.000	−0.666085	−0.333042	+	0.942912i	$0.608075\pi$
$110$	0	0
$111$	424.000	0.362561
$112$	0	0
$113$	−1422.00	−1.18381	−0.591905	−	0.806008i	$-0.701624\pi$
$113$	−1422.00	−1.18381	−0.591905	+	0.806008i	$0.701624\pi$
$114$	0	0
$115$	600.000	0.486524
$116$	0	0
$117$	418.000	0.330292
$118$	0	0
$119$	−294.000	−0.226478
$120$	0	0
$121$	2269.00	1.70473
$122$	0	0
$123$	−216.000	−0.158342
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−848.000	−0.592503	−0.296251	−	0.955110i	$-0.595737\pi$
$127$	−848.000	−0.592503	−0.296251	+	0.955110i	$0.595737\pi$
$128$	0	0
$129$	−784.000	−0.535096
$130$	0	0
$131$	588.000	0.392166	0.196083	−	0.980587i	$-0.437178\pi$
$131$	588.000	0.392166	0.196083	+	0.980587i	$0.437178\pi$
$132$	0	0
$133$	364.000	0.237314
$134$	0	0
$135$	760.000	0.484521
$136$	0	0
$137$	−1254.00	−0.782018	−0.391009	−	0.920387i	$-0.627874\pi$
$137$	−1254.00	−0.782018	−0.391009	+	0.920387i	$0.627874\pi$
$138$	0	0
$139$	−1564.00	−0.954365	−0.477183	−	0.878804i	$-0.658342\pi$
$139$	−1564.00	−0.954365	−0.477183	+	0.878804i	$0.658342\pi$
$140$	0	0
$141$	−1344.00	−0.802732
$142$	0	0
$143$	−2280.00	−1.33331
$144$	0	0
$145$	−1170.00	−0.670091
$146$	0	0
$147$	196.000	0.109971
$148$	0	0
$149$	−1038.00	−0.570713	−0.285357	−	0.958421i	$-0.592112\pi$
$149$	−1038.00	−0.570713	−0.285357	+	0.958421i	$0.592112\pi$
$150$	0	0
$151$	−3008.00	−1.62111	−0.810555	−	0.585663i	$-0.800834\pi$
$151$	−3008.00	−1.62111	−0.810555	+	0.585663i	$0.800834\pi$
$152$	0	0
$153$	−462.000	−0.244121
$154$	0	0
$155$	−1520.00	−0.787673
$156$	0	0
$157$	−3062.00	−1.55652	−0.778262	−	0.627940i	$-0.783898\pi$
$157$	−3062.00	−1.55652	−0.778262	+	0.627940i	$0.783898\pi$
$158$	0	0
$159$	−1752.00	−0.873853
$160$	0	0
$161$	840.000	0.411188
$162$	0	0
$163$	−988.000	−0.474762	−0.237381	−	0.971417i	$-0.576289\pi$
$163$	−988.000	−0.474762	−0.237381	+	0.971417i	$0.576289\pi$
$164$	0	0
$165$	−1200.00	−0.566181
$166$	0	0
$167$	−1656.00	−0.767336	−0.383668	−	0.923471i	$-0.625339\pi$
$167$	−1656.00	−0.767336	−0.383668	+	0.923471i	$0.625339\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	0	0
$171$	572.000	0.255801
$172$	0	0
$173$	3066.00	1.34742	0.673710	−	0.738996i	$-0.264700\pi$
$173$	3066.00	1.34742	0.673710	+	0.738996i	$0.264700\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−1776.00	−0.754194
$178$	0	0
$179$	−2028.00	−0.846815	−0.423407	−	0.905939i	$-0.639166\pi$
$179$	−2028.00	−0.846815	−0.423407	+	0.905939i	$0.639166\pi$
$180$	0	0
$181$	4210.00	1.72888	0.864439	−	0.502738i	$-0.167674\pi$
$181$	4210.00	1.72888	0.864439	+	0.502738i	$0.167674\pi$
$182$	0	0
$183$	−152.000	−0.0613998
$184$	0	0
$185$	−530.000	−0.210629
$186$	0	0
$187$	2520.00	0.985458
$188$	0	0
$189$	1064.00	0.409495
$190$	0	0
$191$	2616.00	0.991032	0.495516	−	0.868599i	$-0.334979\pi$
$191$	2616.00	0.991032	0.495516	+	0.868599i	$0.334979\pi$
$192$	0	0
$193$	722.000	0.269278	0.134639	−	0.990895i	$-0.457012\pi$
$193$	722.000	0.269278	0.134639	+	0.990895i	$0.457012\pi$
$194$	0	0
$195$	760.000	0.279101
$196$	0	0
$197$	−2934.00	−1.06111	−0.530555	−	0.847650i	$-0.678017\pi$
$197$	−2934.00	−1.06111	−0.530555	+	0.847650i	$0.678017\pi$
$198$	0	0
$199$	520.000	0.185235	0.0926176	−	0.995702i	$-0.470477\pi$
$199$	520.000	0.185235	0.0926176	+	0.995702i	$0.470477\pi$
$200$	0	0
$201$	−3952.00	−1.38683
$202$	0	0
$203$	−1638.00	−0.566330
$204$	0	0
$205$	270.000	0.0919884
$206$	0	0
$207$	1320.00	0.443219
$208$	0	0
$209$	−3120.00	−1.03261
$210$	0	0
$211$	3476.00	1.13411	0.567056	−	0.823679i	$-0.308082\pi$
$211$	3476.00	1.13411	0.567056	+	0.823679i	$0.308082\pi$
$212$	0	0
$213$	2880.00	0.926452
$214$	0	0
$215$	980.000	0.310863
$216$	0	0
$217$	−2128.00	−0.665705
$218$	0	0
$219$	584.000	0.180197
$220$	0	0
$221$	−1596.00	−0.485785
$222$	0	0
$223$	1456.00	0.437224	0.218612	−	0.975812i	$-0.429847\pi$
$223$	1456.00	0.437224	0.218612	+	0.975812i	$0.429847\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	−3324.00	−0.971901	−0.485951	−	0.873986i	$-0.661527\pi$
$227$	−3324.00	−0.971901	−0.485951	+	0.873986i	$0.661527\pi$
$228$	0	0
$229$	−1694.00	−0.488833	−0.244416	−	0.969670i	$-0.578596\pi$
$229$	−1694.00	−0.488833	−0.244416	+	0.969670i	$0.578596\pi$
$230$	0	0
$231$	−1680.00	−0.478510
$232$	0	0
$233$	−6198.00	−1.74268	−0.871340	−	0.490680i	$-0.836749\pi$
$233$	−6198.00	−1.74268	−0.871340	+	0.490680i	$0.836749\pi$
$234$	0	0
$235$	1680.00	0.466345
$236$	0	0
$237$	3232.00	0.885827
$238$	0	0
$239$	−936.000	−0.253326	−0.126663	−	0.991946i	$-0.540427\pi$
$239$	−936.000	−0.253326	−0.126663	+	0.991946i	$0.540427\pi$
$240$	0	0
$241$	1730.00	0.462403	0.231201	−	0.972906i	$-0.425734\pi$
$241$	1730.00	0.462403	0.231201	+	0.972906i	$0.425734\pi$
$242$	0	0
$243$	2860.00	0.755017
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	1976.00	0.509028
$248$	0	0
$249$	2448.00	0.623035
$250$	0	0
$251$	−3708.00	−0.932458	−0.466229	−	0.884664i	$-0.654388\pi$
$251$	−3708.00	−0.932458	−0.466229	+	0.884664i	$0.654388\pi$
$252$	0	0
$253$	−7200.00	−1.78917
$254$	0	0
$255$	−840.000	−0.206286
$256$	0	0
$257$	4746.00	1.15194	0.575968	−	0.817473i	$-0.304625\pi$
$257$	4746.00	1.15194	0.575968	+	0.817473i	$0.304625\pi$
$258$	0	0
$259$	−742.000	−0.178014
$260$	0	0
$261$	−2574.00	−0.610447
$262$	0	0
$263$	−4200.00	−0.984727	−0.492363	−	0.870390i	$-0.663867\pi$
$263$	−4200.00	−0.984727	−0.492363	+	0.870390i	$0.663867\pi$
$264$	0	0
$265$	2190.00	0.507663
$266$	0	0
$267$	4584.00	1.05070
$268$	0	0
$269$	−8502.00	−1.92705	−0.963524	−	0.267621i	$-0.913763\pi$
$269$	−8502.00	−1.92705	−0.963524	+	0.267621i	$0.913763\pi$
$270$	0	0
$271$	4336.00	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	4336.00	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	1064.00	0.235884
$274$	0	0
$275$	1500.00	0.328921
$276$	0	0
$277$	−3350.00	−0.726650	−0.363325	−	0.931663i	$-0.618359\pi$
$277$	−3350.00	−0.726650	−0.363325	+	0.931663i	$0.618359\pi$
$278$	0	0
$279$	−3344.00	−0.717563
$280$	0	0
$281$	−8406.00	−1.78455	−0.892277	−	0.451488i	$-0.850894\pi$
$281$	−8406.00	−1.78455	−0.892277	+	0.451488i	$0.850894\pi$
$282$	0	0
$283$	6284.00	1.31995	0.659974	−	0.751289i	$-0.270567\pi$
$283$	6284.00	1.31995	0.659974	+	0.751289i	$0.270567\pi$
$284$	0	0
$285$	1040.00	0.216155
$286$	0	0
$287$	378.000	0.0777444
$288$	0	0
$289$	−3149.00	−0.640953
$290$	0	0
$291$	−280.000	−0.0564051
$292$	0	0
$293$	−3582.00	−0.714207	−0.357104	−	0.934065i	$-0.616236\pi$
$293$	−3582.00	−0.714207	−0.357104	+	0.934065i	$0.616236\pi$
$294$	0	0
$295$	2220.00	0.438147
$296$	0	0
$297$	−9120.00	−1.78180
$298$	0	0
$299$	4560.00	0.881979
$300$	0	0
$301$	1372.00	0.262727
$302$	0	0
$303$	−7800.00	−1.47887
$304$	0	0
$305$	190.000	0.0356701
$306$	0	0
$307$	8804.00	1.63671	0.818356	−	0.574711i	$-0.194886\pi$
$307$	8804.00	1.63671	0.818356	+	0.574711i	$0.194886\pi$
$308$	0	0
$309$	−1952.00	−0.359370
$310$	0	0
$311$	2424.00	0.441969	0.220985	−	0.975277i	$-0.429073\pi$
$311$	2424.00	0.441969	0.220985	+	0.975277i	$0.429073\pi$
$312$	0	0
$313$	−9070.00	−1.63791	−0.818956	−	0.573856i	$-0.805447\pi$
$313$	−9070.00	−1.63791	−0.818956	+	0.573856i	$0.805447\pi$
$314$	0	0
$315$	−385.000	−0.0688644
$316$	0	0
$317$	6738.00	1.19383	0.596914	−	0.802305i	$-0.296393\pi$
$317$	6738.00	1.19383	0.596914	+	0.802305i	$0.296393\pi$
$318$	0	0
$319$	14040.0	2.46423
$320$	0	0
$321$	−3984.00	−0.692726
$322$	0	0
$323$	−2184.00	−0.376226
$324$	0	0
$325$	−950.000	−0.162143
$326$	0	0
$327$	−3032.00	−0.512752
$328$	0	0
$329$	2352.00	0.394134
$330$	0	0
$331$	7868.00	1.30654	0.653269	−	0.757125i	$-0.273397\pi$
$331$	7868.00	1.30654	0.653269	+	0.757125i	$0.273397\pi$
$332$	0	0
$333$	−1166.00	−0.191881
$334$	0	0
$335$	4940.00	0.805675
$336$	0	0
$337$	9218.00	1.49002	0.745010	−	0.667054i	$-0.232445\pi$
$337$	9218.00	1.49002	0.745010	+	0.667054i	$0.232445\pi$
$338$	0	0
$339$	−5688.00	−0.911297
$340$	0	0
$341$	18240.0	2.89663
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	2400.00	0.374527
$346$	0	0
$347$	1212.00	0.187503	0.0937515	−	0.995596i	$-0.470114\pi$
$347$	1212.00	0.187503	0.0937515	+	0.995596i	$0.470114\pi$
$348$	0	0
$349$	−10118.0	−1.55187	−0.775937	−	0.630810i	$-0.782723\pi$
$349$	−10118.0	−1.55187	−0.775937	+	0.630810i	$0.782723\pi$
$350$	0	0
$351$	5776.00	0.878348
$352$	0	0
$353$	1194.00	0.180029	0.0900145	−	0.995940i	$-0.471309\pi$
$353$	1194.00	0.180029	0.0900145	+	0.995940i	$0.471309\pi$
$354$	0	0
$355$	−3600.00	−0.538220
$356$	0	0
$357$	−1176.00	−0.174343
$358$	0	0
$359$	−2064.00	−0.303437	−0.151718	−	0.988424i	$-0.548481\pi$
$359$	−2064.00	−0.303437	−0.151718	+	0.988424i	$0.548481\pi$
$360$	0	0
$361$	−4155.00	−0.605773
$362$	0	0
$363$	9076.00	1.31230
$364$	0	0
$365$	−730.000	−0.104685
$366$	0	0
$367$	7504.00	1.06732	0.533659	−	0.845700i	$-0.320817\pi$
$367$	7504.00	1.06732	0.533659	+	0.845700i	$0.320817\pi$
$368$	0	0
$369$	594.000	0.0838006
$370$	0	0
$371$	3066.00	0.429053
$372$	0	0
$373$	−902.000	−0.125211	−0.0626056	−	0.998038i	$-0.519941\pi$
$373$	−902.000	−0.125211	−0.0626056	+	0.998038i	$0.519941\pi$
$374$	0	0
$375$	−500.000	−0.0688530
$376$	0	0
$377$	−8892.00	−1.21475
$378$	0	0
$379$	−12004.0	−1.62692	−0.813462	−	0.581618i	$-0.802420\pi$
$379$	−12004.0	−1.62692	−0.813462	+	0.581618i	$0.802420\pi$
$380$	0	0
$381$	−3392.00	−0.456109
$382$	0	0
$383$	−10848.0	−1.44728	−0.723638	−	0.690179i	$-0.757532\pi$
$383$	−10848.0	−1.44728	−0.723638	+	0.690179i	$0.757532\pi$
$384$	0	0
$385$	2100.00	0.277989
$386$	0	0
$387$	2156.00	0.283193
$388$	0	0
$389$	5538.00	0.721819	0.360910	−	0.932601i	$-0.382466\pi$
$389$	5538.00	0.721819	0.360910	+	0.932601i	$0.382466\pi$
$390$	0	0
$391$	−5040.00	−0.651877
$392$	0	0
$393$	2352.00	0.301890
$394$	0	0
$395$	−4040.00	−0.514619
$396$	0	0
$397$	−11270.0	−1.42475	−0.712374	−	0.701800i	$-0.752380\pi$
$397$	−11270.0	−1.42475	−0.712374	+	0.701800i	$0.752380\pi$
$398$	0	0
$399$	1456.00	0.182685
$400$	0	0
$401$	2802.00	0.348941	0.174470	−	0.984662i	$-0.444179\pi$
$401$	2802.00	0.348941	0.174470	+	0.984662i	$0.444179\pi$
$402$	0	0
$403$	−11552.0	−1.42791
$404$	0	0
$405$	1555.00	0.190787
$406$	0	0
$407$	6360.00	0.774579
$408$	0	0
$409$	4106.00	0.496403	0.248201	−	0.968708i	$-0.420161\pi$
$409$	4106.00	0.496403	0.248201	+	0.968708i	$0.420161\pi$
$410$	0	0
$411$	−5016.00	−0.601998
$412$	0	0
$413$	3108.00	0.370302
$414$	0	0
$415$	−3060.00	−0.361951
$416$	0	0
$417$	−6256.00	−0.734671
$418$	0	0
$419$	4284.00	0.499492	0.249746	−	0.968311i	$-0.419653\pi$
$419$	4284.00	0.499492	0.249746	+	0.968311i	$0.419653\pi$
$420$	0	0
$421$	−1550.00	−0.179436	−0.0897178	−	0.995967i	$-0.528597\pi$
$421$	−1550.00	−0.179436	−0.0897178	+	0.995967i	$0.528597\pi$
$422$	0	0
$423$	3696.00	0.424836
$424$	0	0
$425$	1050.00	0.119841
$426$	0	0
$427$	266.000	0.0301467
$428$	0	0
$429$	−9120.00	−1.02638
$430$	0	0
$431$	−16584.0	−1.85342	−0.926709	−	0.375780i	$-0.877375\pi$
$431$	−16584.0	−1.85342	−0.926709	+	0.375780i	$0.877375\pi$
$432$	0	0
$433$	−8422.00	−0.934724	−0.467362	−	0.884066i	$-0.654796\pi$
$433$	−8422.00	−0.934724	−0.467362	+	0.884066i	$0.654796\pi$
$434$	0	0
$435$	−4680.00	−0.515836
$436$	0	0
$437$	6240.00	0.683066
$438$	0	0
$439$	2968.00	0.322676	0.161338	−	0.986899i	$-0.448419\pi$
$439$	2968.00	0.322676	0.161338	+	0.986899i	$0.448419\pi$
$440$	0	0
$441$	−539.000	−0.0582011
$442$	0	0
$443$	−2436.00	−0.261259	−0.130630	−	0.991431i	$-0.541700\pi$
$443$	−2436.00	−0.261259	−0.130630	+	0.991431i	$0.541700\pi$
$444$	0	0
$445$	−5730.00	−0.610400
$446$	0	0
$447$	−4152.00	−0.439335
$448$	0	0
$449$	7362.00	0.773796	0.386898	−	0.922123i	$-0.373547\pi$
$449$	7362.00	0.773796	0.386898	+	0.922123i	$0.373547\pi$
$450$	0	0
$451$	−3240.00	−0.338283
$452$	0	0
$453$	−12032.0	−1.24793
$454$	0	0
$455$	−1330.00	−0.137036
$456$	0	0
$457$	9290.00	0.950914	0.475457	−	0.879739i	$-0.342283\pi$
$457$	9290.00	0.950914	0.475457	+	0.879739i	$0.342283\pi$
$458$	0	0
$459$	−6384.00	−0.649193
$460$	0	0
$461$	−11862.0	−1.19841	−0.599207	−	0.800594i	$-0.704517\pi$
$461$	−11862.0	−1.19841	−0.599207	+	0.800594i	$0.704517\pi$
$462$	0	0
$463$	13264.0	1.33138	0.665692	−	0.746227i	$-0.268137\pi$
$463$	13264.0	1.33138	0.665692	+	0.746227i	$0.268137\pi$
$464$	0	0
$465$	−6080.00	−0.606351
$466$	0	0
$467$	14628.0	1.44947	0.724736	−	0.689027i	$-0.241962\pi$
$467$	14628.0	1.44947	0.724736	+	0.689027i	$0.241962\pi$
$468$	0	0
$469$	6916.00	0.680919
$470$	0	0
$471$	−12248.0	−1.19821
$472$	0	0
$473$	−11760.0	−1.14318
$474$	0	0
$475$	−1300.00	−0.125575
$476$	0	0
$477$	4818.00	0.462476
$478$	0	0
$479$	−4416.00	−0.421236	−0.210618	−	0.977568i	$-0.567548\pi$
$479$	−4416.00	−0.421236	−0.210618	+	0.977568i	$0.567548\pi$
$480$	0	0
$481$	−4028.00	−0.381832
$482$	0	0
$483$	3360.00	0.316533
$484$	0	0
$485$	350.000	0.0327684
$486$	0	0
$487$	232.000	0.0215871	0.0107936	−	0.999942i	$-0.496564\pi$
$487$	232.000	0.0215871	0.0107936	+	0.999942i	$0.496564\pi$
$488$	0	0
$489$	−3952.00	−0.365472
$490$	0	0
$491$	−17892.0	−1.64451	−0.822255	−	0.569119i	$-0.807284\pi$
$491$	−17892.0	−1.64451	−0.822255	+	0.569119i	$0.807284\pi$
$492$	0	0
$493$	9828.00	0.897831
$494$	0	0
$495$	3300.00	0.299644
$496$	0	0
$497$	−5040.00	−0.454879
$498$	0	0
$499$	−6316.00	−0.566619	−0.283310	−	0.959029i	$-0.591432\pi$
$499$	−6316.00	−0.566619	−0.283310	+	0.959029i	$0.591432\pi$
$500$	0	0
$501$	−6624.00	−0.590696
$502$	0	0
$503$	12600.0	1.11691	0.558455	−	0.829534i	$-0.311394\pi$
$503$	12600.0	1.11691	0.558455	+	0.829534i	$0.311394\pi$
$504$	0	0
$505$	9750.00	0.859147
$506$	0	0
$507$	−3012.00	−0.263841
$508$	0	0
$509$	−17094.0	−1.48856	−0.744281	−	0.667866i	$-0.767208\pi$
$509$	−17094.0	−1.48856	−0.744281	+	0.667866i	$0.767208\pi$
$510$	0	0
$511$	−1022.00	−0.0884748
$512$	0	0
$513$	7904.00	0.680254
$514$	0	0
$515$	2440.00	0.208775
$516$	0	0
$517$	−20160.0	−1.71496
$518$	0	0
$519$	12264.0	1.03724
$520$	0	0
$521$	14586.0	1.22653	0.613267	−	0.789876i	$-0.289855\pi$
$521$	14586.0	1.22653	0.613267	+	0.789876i	$0.289855\pi$
$522$	0	0
$523$	−16324.0	−1.36482	−0.682408	−	0.730972i	$-0.739067\pi$
$523$	−16324.0	−1.36482	−0.682408	+	0.730972i	$0.739067\pi$
$524$	0	0
$525$	−700.000	−0.0581914
$526$	0	0
$527$	12768.0	1.05538
$528$	0	0
$529$	2233.00	0.183529
$530$	0	0
$531$	4884.00	0.399148
$532$	0	0
$533$	2052.00	0.166758
$534$	0	0
$535$	4980.00	0.402438
$536$	0	0
$537$	−8112.00	−0.651878
$538$	0	0
$539$	2940.00	0.234944
$540$	0	0
$541$	7162.00	0.569165	0.284583	−	0.958652i	$-0.408145\pi$
$541$	7162.00	0.569165	0.284583	+	0.958652i	$0.408145\pi$
$542$	0	0
$543$	16840.0	1.33089
$544$	0	0
$545$	3790.00	0.297882
$546$	0	0
$547$	9956.00	0.778223	0.389111	−	0.921191i	$-0.372782\pi$
$547$	9956.00	0.778223	0.389111	+	0.921191i	$0.372782\pi$
$548$	0	0
$549$	418.000	0.0324951
$550$	0	0
$551$	−12168.0	−0.940788
$552$	0	0
$553$	−5656.00	−0.434932
$554$	0	0
$555$	−2120.00	−0.162142
$556$	0	0
$557$	−15342.0	−1.16708	−0.583538	−	0.812086i	$-0.698332\pi$
$557$	−15342.0	−1.16708	−0.583538	+	0.812086i	$0.698332\pi$
$558$	0	0
$559$	7448.00	0.563536
$560$	0	0
$561$	10080.0	0.758606
$562$	0	0
$563$	3300.00	0.247031	0.123515	−	0.992343i	$-0.460583\pi$
$563$	3300.00	0.247031	0.123515	+	0.992343i	$0.460583\pi$
$564$	0	0
$565$	7110.00	0.529416
$566$	0	0
$567$	2177.00	0.161244
$568$	0	0
$569$	2154.00	0.158700	0.0793501	−	0.996847i	$-0.474716\pi$
$569$	2154.00	0.158700	0.0793501	+	0.996847i	$0.474716\pi$
$570$	0	0
$571$	−1348.00	−0.0987952	−0.0493976	−	0.998779i	$-0.515730\pi$
$571$	−1348.00	−0.0987952	−0.0493976	+	0.998779i	$0.515730\pi$
$572$	0	0
$573$	10464.0	0.762897
$574$	0	0
$575$	−3000.00	−0.217580
$576$	0	0
$577$	−16486.0	−1.18946	−0.594732	−	0.803924i	$-0.702742\pi$
$577$	−16486.0	−1.18946	−0.594732	+	0.803924i	$0.702742\pi$
$578$	0	0
$579$	2888.00	0.207290
$580$	0	0
$581$	−4284.00	−0.305904
$582$	0	0
$583$	−26280.0	−1.86691
$584$	0	0
$585$	−2090.00	−0.147711
$586$	0	0
$587$	9756.00	0.685985	0.342993	−	0.939338i	$-0.388559\pi$
$587$	9756.00	0.685985	0.342993	+	0.939338i	$0.388559\pi$
$588$	0	0
$589$	−15808.0	−1.10587
$590$	0	0
$591$	−11736.0	−0.816844
$592$	0	0
$593$	8202.00	0.567986	0.283993	−	0.958826i	$-0.408341\pi$
$593$	8202.00	0.567986	0.283993	+	0.958826i	$0.408341\pi$
$594$	0	0
$595$	1470.00	0.101284
$596$	0	0
$597$	2080.00	0.142594
$598$	0	0
$599$	−20544.0	−1.40134	−0.700672	−	0.713484i	$-0.747116\pi$
$599$	−20544.0	−1.40134	−0.700672	+	0.713484i	$0.747116\pi$
$600$	0	0
$601$	−5254.00	−0.356598	−0.178299	−	0.983976i	$-0.557059\pi$
$601$	−5254.00	−0.356598	−0.178299	+	0.983976i	$0.557059\pi$
$602$	0	0
$603$	10868.0	0.733962
$604$	0	0
$605$	−11345.0	−0.762380
$606$	0	0
$607$	19456.0	1.30098	0.650490	−	0.759515i	$-0.274564\pi$
$607$	19456.0	1.30098	0.650490	+	0.759515i	$0.274564\pi$
$608$	0	0
$609$	−6552.00	−0.435961
$610$	0	0
$611$	12768.0	0.845398
$612$	0	0
$613$	−3926.00	−0.258678	−0.129339	−	0.991600i	$-0.541286\pi$
$613$	−3926.00	−0.258678	−0.129339	+	0.991600i	$0.541286\pi$
$614$	0	0
$615$	1080.00	0.0708127
$616$	0	0
$617$	22122.0	1.44343	0.721717	−	0.692189i	$-0.243353\pi$
$617$	22122.0	1.44343	0.721717	+	0.692189i	$0.243353\pi$
$618$	0	0
$619$	−12796.0	−0.830880	−0.415440	−	0.909621i	$-0.636372\pi$
$619$	−12796.0	−0.830880	−0.415440	+	0.909621i	$0.636372\pi$
$620$	0	0
$621$	18240.0	1.17866
$622$	0	0
$623$	−8022.00	−0.515882
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−12480.0	−0.794901
$628$	0	0
$629$	4452.00	0.282214
$630$	0	0
$631$	16576.0	1.04577	0.522884	−	0.852404i	$-0.324856\pi$
$631$	16576.0	1.04577	0.522884	+	0.852404i	$0.324856\pi$
$632$	0	0
$633$	13904.0	0.873040
$634$	0	0
$635$	4240.00	0.264975
$636$	0	0
$637$	−1862.00	−0.115817
$638$	0	0
$639$	−7920.00	−0.490314
$640$	0	0
$641$	11202.0	0.690253	0.345127	−	0.938556i	$-0.387836\pi$
$641$	11202.0	0.690253	0.345127	+	0.938556i	$0.387836\pi$
$642$	0	0
$643$	4052.00	0.248515	0.124258	−	0.992250i	$-0.460345\pi$
$643$	4052.00	0.248515	0.124258	+	0.992250i	$0.460345\pi$
$644$	0	0
$645$	3920.00	0.239302
$646$	0	0
$647$	17544.0	1.06604	0.533018	−	0.846104i	$-0.321058\pi$
$647$	17544.0	1.06604	0.533018	+	0.846104i	$0.321058\pi$
$648$	0	0
$649$	−26640.0	−1.61127
$650$	0	0
$651$	−8512.00	−0.512460
$652$	0	0
$653$	12066.0	0.723092	0.361546	−	0.932354i	$-0.382249\pi$
$653$	12066.0	0.723092	0.361546	+	0.932354i	$0.382249\pi$
$654$	0	0
$655$	−2940.00	−0.175382
$656$	0	0
$657$	−1606.00	−0.0953669
$658$	0	0
$659$	−7596.00	−0.449011	−0.224505	−	0.974473i	$-0.572077\pi$
$659$	−7596.00	−0.449011	−0.224505	+	0.974473i	$0.572077\pi$
$660$	0	0
$661$	−11630.0	−0.684349	−0.342174	−	0.939636i	$-0.611163\pi$
$661$	−11630.0	−0.684349	−0.342174	+	0.939636i	$0.611163\pi$
$662$	0	0
$663$	−6384.00	−0.373958
$664$	0	0
$665$	−1820.00	−0.106130
$666$	0	0
$667$	−28080.0	−1.63008
$668$	0	0
$669$	5824.00	0.336575
$670$	0	0
$671$	−2280.00	−0.131175
$672$	0	0
$673$	7778.00	0.445498	0.222749	−	0.974876i	$-0.428497\pi$
$673$	7778.00	0.445498	0.222749	+	0.974876i	$0.428497\pi$
$674$	0	0
$675$	−3800.00	−0.216685
$676$	0	0
$677$	28914.0	1.64144	0.820720	−	0.571330i	$-0.193572\pi$
$677$	28914.0	1.64144	0.820720	+	0.571330i	$0.193572\pi$
$678$	0	0
$679$	490.000	0.0276944
$680$	0	0
$681$	−13296.0	−0.748170
$682$	0	0
$683$	25404.0	1.42322	0.711608	−	0.702576i	$-0.247967\pi$
$683$	25404.0	1.42322	0.711608	+	0.702576i	$0.247967\pi$
$684$	0	0
$685$	6270.00	0.349729
$686$	0	0
$687$	−6776.00	−0.376304
$688$	0	0
$689$	16644.0	0.920299
$690$	0	0
$691$	1244.00	0.0684862	0.0342431	−	0.999414i	$-0.489098\pi$
$691$	1244.00	0.0684862	0.0342431	+	0.999414i	$0.489098\pi$
$692$	0	0
$693$	4620.00	0.253246
$694$	0	0
$695$	7820.00	0.426805
$696$	0	0
$697$	−2268.00	−0.123252
$698$	0	0
$699$	−24792.0	−1.34152
$700$	0	0
$701$	−3798.00	−0.204634	−0.102317	−	0.994752i	$-0.532626\pi$
$701$	−3798.00	−0.204634	−0.102317	+	0.994752i	$0.532626\pi$
$702$	0	0
$703$	−5512.00	−0.295717
$704$	0	0
$705$	6720.00	0.358993
$706$	0	0
$707$	13650.0	0.726112
$708$	0	0
$709$	15154.0	0.802709	0.401354	−	0.915923i	$-0.368540\pi$
$709$	15154.0	0.802709	0.401354	+	0.915923i	$0.368540\pi$
$710$	0	0
$711$	−8888.00	−0.468813
$712$	0	0
$713$	−36480.0	−1.91611
$714$	0	0
$715$	11400.0	0.596274
$716$	0	0
$717$	−3744.00	−0.195010
$718$	0	0
$719$	10032.0	0.520348	0.260174	−	0.965562i	$-0.416220\pi$
$719$	10032.0	0.520348	0.260174	+	0.965562i	$0.416220\pi$
$720$	0	0
$721$	3416.00	0.176447
$722$	0	0
$723$	6920.00	0.355958
$724$	0	0
$725$	5850.00	0.299674
$726$	0	0
$727$	−17768.0	−0.906436	−0.453218	−	0.891400i	$-0.649724\pi$
$727$	−17768.0	−0.906436	−0.453218	+	0.891400i	$0.649724\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	−8232.00	−0.416514
$732$	0	0
$733$	−9830.00	−0.495333	−0.247667	−	0.968845i	$-0.579664\pi$
$733$	−9830.00	−0.495333	−0.247667	+	0.968845i	$0.579664\pi$
$734$	0	0
$735$	−980.000	−0.0491807
$736$	0	0
$737$	−59280.0	−2.96283
$738$	0	0
$739$	−33244.0	−1.65480	−0.827402	−	0.561610i	$-0.810182\pi$
$739$	−33244.0	−1.65480	−0.827402	+	0.561610i	$0.810182\pi$
$740$	0	0
$741$	7904.00	0.391850
$742$	0	0
$743$	−3672.00	−0.181309	−0.0906545	−	0.995882i	$-0.528896\pi$
$743$	−3672.00	−0.181309	−0.0906545	+	0.995882i	$0.528896\pi$
$744$	0	0
$745$	5190.00	0.255231
$746$	0	0
$747$	−6732.00	−0.329734
$748$	0	0
$749$	6972.00	0.340122
$750$	0	0
$751$	6280.00	0.305140	0.152570	−	0.988293i	$-0.451245\pi$
$751$	6280.00	0.305140	0.152570	+	0.988293i	$0.451245\pi$
$752$	0	0
$753$	−14832.0	−0.717806
$754$	0	0
$755$	15040.0	0.724982
$756$	0	0
$757$	826.000	0.0396585	0.0198292	−	0.999803i	$-0.493688\pi$
$757$	826.000	0.0396585	0.0198292	+	0.999803i	$0.493688\pi$
$758$	0	0
$759$	−28800.0	−1.37730
$760$	0	0
$761$	16842.0	0.802263	0.401131	−	0.916021i	$-0.368617\pi$
$761$	16842.0	0.802263	0.401131	+	0.916021i	$0.368617\pi$
$762$	0	0
$763$	5306.00	0.251756
$764$	0	0
$765$	2310.00	0.109174
$766$	0	0
$767$	16872.0	0.794280
$768$	0	0
$769$	−21454.0	−1.00605	−0.503024	−	0.864272i	$-0.667779\pi$
$769$	−21454.0	−1.00605	−0.503024	+	0.864272i	$0.667779\pi$
$770$	0	0
$771$	18984.0	0.886760
$772$	0	0
$773$	−13182.0	−0.613355	−0.306678	−	0.951813i	$-0.599217\pi$
$773$	−13182.0	−0.613355	−0.306678	+	0.951813i	$0.599217\pi$
$774$	0	0
$775$	7600.00	0.352258
$776$	0	0
$777$	−2968.00	−0.137035
$778$	0	0
$779$	2808.00	0.129149
$780$	0	0
$781$	43200.0	1.97928
$782$	0	0
$783$	−35568.0	−1.62337
$784$	0	0
$785$	15310.0	0.696099
$786$	0	0
$787$	24356.0	1.10317	0.551587	−	0.834117i	$-0.314023\pi$
$787$	24356.0	1.10317	0.551587	+	0.834117i	$0.314023\pi$
$788$	0	0
$789$	−16800.0	−0.758043
$790$	0	0
$791$	9954.00	0.447438
$792$	0	0
$793$	1444.00	0.0646632
$794$	0	0
$795$	8760.00	0.390799
$796$	0	0
$797$	13194.0	0.586393	0.293197	−	0.956052i	$-0.405281\pi$
$797$	13194.0	0.586393	0.293197	+	0.956052i	$0.405281\pi$
$798$	0	0
$799$	−14112.0	−0.624839
$800$	0	0
$801$	−12606.0	−0.556069
$802$	0	0
$803$	8760.00	0.384973
$804$	0	0
$805$	−4200.00	−0.183889
$806$	0	0
$807$	−34008.0	−1.48344
$808$	0	0
$809$	29274.0	1.27221	0.636106	−	0.771602i	$-0.280544\pi$
$809$	29274.0	1.27221	0.636106	+	0.771602i	$0.280544\pi$
$810$	0	0
$811$	9812.00	0.424841	0.212420	−	0.977178i	$-0.431865\pi$
$811$	9812.00	0.424841	0.212420	+	0.977178i	$0.431865\pi$
$812$	0	0
$813$	17344.0	0.748193
$814$	0	0
$815$	4940.00	0.212320
$816$	0	0
$817$	10192.0	0.436442
$818$	0	0
$819$	−2926.00	−0.124838
$820$	0	0
$821$	41154.0	1.74943	0.874717	−	0.484635i	$-0.161048\pi$
$821$	41154.0	1.74943	0.874717	+	0.484635i	$0.161048\pi$
$822$	0	0
$823$	−37640.0	−1.59423	−0.797113	−	0.603830i	$-0.793640\pi$
$823$	−37640.0	−1.59423	−0.797113	+	0.603830i	$0.793640\pi$
$824$	0	0
$825$	6000.00	0.253204
$826$	0	0
$827$	−13572.0	−0.570671	−0.285335	−	0.958428i	$-0.592105\pi$
$827$	−13572.0	−0.570671	−0.285335	+	0.958428i	$0.592105\pi$
$828$	0	0
$829$	−13574.0	−0.568691	−0.284345	−	0.958722i	$-0.591776\pi$
$829$	−13574.0	−0.568691	−0.284345	+	0.958722i	$0.591776\pi$
$830$	0	0
$831$	−13400.0	−0.559375
$832$	0	0
$833$	2058.00	0.0856008
$834$	0	0
$835$	8280.00	0.343163
$836$	0	0
$837$	−46208.0	−1.90822
$838$	0	0
$839$	5112.00	0.210353	0.105176	−	0.994454i	$-0.466459\pi$
$839$	5112.00	0.210353	0.105176	+	0.994454i	$0.466459\pi$
$840$	0	0
$841$	30367.0	1.24511
$842$	0	0
$843$	−33624.0	−1.37375
$844$	0	0
$845$	3765.00	0.153278
$846$	0	0
$847$	−15883.0	−0.644329
$848$	0	0
$849$	25136.0	1.01610
$850$	0	0
$851$	−12720.0	−0.512381
$852$	0	0
$853$	−20702.0	−0.830977	−0.415488	−	0.909599i	$-0.636389\pi$
$853$	−20702.0	−0.830977	−0.415488	+	0.909599i	$0.636389\pi$
$854$	0	0
$855$	−2860.00	−0.114398
$856$	0	0
$857$	−5646.00	−0.225045	−0.112523	−	0.993649i	$-0.535893\pi$
$857$	−5646.00	−0.225045	−0.112523	+	0.993649i	$0.535893\pi$
$858$	0	0
$859$	20756.0	0.824430	0.412215	−	0.911087i	$-0.364755\pi$
$859$	20756.0	0.824430	0.412215	+	0.911087i	$0.364755\pi$
$860$	0	0
$861$	1512.00	0.0598476
$862$	0	0
$863$	24432.0	0.963702	0.481851	−	0.876253i	$-0.339965\pi$
$863$	24432.0	0.963702	0.481851	+	0.876253i	$0.339965\pi$
$864$	0	0
$865$	−15330.0	−0.602585
$866$	0	0
$867$	−12596.0	−0.493406
$868$	0	0
$869$	48480.0	1.89249
$870$	0	0
$871$	37544.0	1.46054
$872$	0	0
$873$	770.000	0.0298517
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−43166.0	−1.66204	−0.831022	−	0.556240i	$-0.812244\pi$
$877$	−43166.0	−1.66204	−0.831022	+	0.556240i	$0.812244\pi$
$878$	0	0
$879$	−14328.0	−0.549797
$880$	0	0
$881$	−1854.00	−0.0708999	−0.0354500	−	0.999371i	$-0.511286\pi$
$881$	−1854.00	−0.0708999	−0.0354500	+	0.999371i	$0.511286\pi$
$882$	0	0
$883$	−23740.0	−0.904773	−0.452387	−	0.891822i	$-0.649427\pi$
$883$	−23740.0	−0.904773	−0.452387	+	0.891822i	$0.649427\pi$
$884$	0	0
$885$	8880.00	0.337286
$886$	0	0
$887$	−32088.0	−1.21467	−0.607333	−	0.794447i	$-0.707761\pi$
$887$	−32088.0	−1.21467	−0.607333	+	0.794447i	$0.707761\pi$
$888$	0	0
$889$	5936.00	0.223945
$890$	0	0
$891$	−18660.0	−0.701609
$892$	0	0
$893$	17472.0	0.654735
$894$	0	0
$895$	10140.0	0.378707
$896$	0	0
$897$	18240.0	0.678947
$898$	0	0
$899$	71136.0	2.63906
$900$	0	0
$901$	−18396.0	−0.680199
$902$	0	0
$903$	5488.00	0.202247
$904$	0	0
$905$	−21050.0	−0.773178
$906$	0	0
$907$	4268.00	0.156248	0.0781238	−	0.996944i	$-0.475107\pi$
$907$	4268.00	0.156248	0.0781238	+	0.996944i	$0.475107\pi$
$908$	0	0
$909$	21450.0	0.782675
$910$	0	0
$911$	19512.0	0.709617	0.354809	−	0.934939i	$-0.384546\pi$
$911$	19512.0	0.709617	0.354809	+	0.934939i	$0.384546\pi$
$912$	0	0
$913$	36720.0	1.33106
$914$	0	0
$915$	760.000	0.0274588
$916$	0	0
$917$	−4116.00	−0.148225
$918$	0	0
$919$	6496.00	0.233170	0.116585	−	0.993181i	$-0.462805\pi$
$919$	6496.00	0.233170	0.116585	+	0.993181i	$0.462805\pi$
$920$	0	0
$921$	35216.0	1.25994
$922$	0	0
$923$	−27360.0	−0.975694
$924$	0	0
$925$	2650.00	0.0941962
$926$	0	0
$927$	5368.00	0.190192
$928$	0	0
$929$	−16926.0	−0.597765	−0.298883	−	0.954290i	$-0.596614\pi$
$929$	−16926.0	−0.597765	−0.298883	+	0.954290i	$0.596614\pi$
$930$	0	0
$931$	−2548.00	−0.0896964
$932$	0	0
$933$	9696.00	0.340228
$934$	0	0
$935$	−12600.0	−0.440710
$936$	0	0
$937$	−14686.0	−0.512029	−0.256014	−	0.966673i	$-0.582409\pi$
$937$	−14686.0	−0.512029	−0.256014	+	0.966673i	$0.582409\pi$
$938$	0	0
$939$	−36280.0	−1.26087
$940$	0	0
$941$	33834.0	1.17211	0.586056	−	0.810271i	$-0.300680\pi$
$941$	33834.0	1.17211	0.586056	+	0.810271i	$0.300680\pi$
$942$	0	0
$943$	6480.00	0.223773
$944$	0	0
$945$	−5320.00	−0.183132
$946$	0	0
$947$	17508.0	0.600775	0.300387	−	0.953817i	$-0.402884\pi$
$947$	17508.0	0.600775	0.300387	+	0.953817i	$0.402884\pi$
$948$	0	0
$949$	−5548.00	−0.189774
$950$	0	0
$951$	26952.0	0.919010
$952$	0	0
$953$	−12582.0	−0.427672	−0.213836	−	0.976870i	$-0.568596\pi$
$953$	−12582.0	−0.427672	−0.213836	+	0.976870i	$0.568596\pi$
$954$	0	0
$955$	−13080.0	−0.443203
$956$	0	0
$957$	56160.0	1.89696
$958$	0	0
$959$	8778.00	0.295575
$960$	0	0
$961$	62625.0	2.10214
$962$	0	0
$963$	10956.0	0.366617
$964$	0	0
$965$	−3610.00	−0.120425
$966$	0	0
$967$	19096.0	0.635042	0.317521	−	0.948251i	$-0.397150\pi$
$967$	19096.0	0.635042	0.317521	+	0.948251i	$0.397150\pi$
$968$	0	0
$969$	−8736.00	−0.289619
$970$	0	0
$971$	50484.0	1.66850	0.834248	−	0.551390i	$-0.185902\pi$
$971$	50484.0	1.66850	0.834248	+	0.551390i	$0.185902\pi$
$972$	0	0
$973$	10948.0	0.360716
$974$	0	0
$975$	−3800.00	−0.124818
$976$	0	0
$977$	26898.0	0.880802	0.440401	−	0.897801i	$-0.354836\pi$
$977$	26898.0	0.880802	0.440401	+	0.897801i	$0.354836\pi$
$978$	0	0
$979$	68760.0	2.24472
$980$	0	0
$981$	8338.00	0.271368
$982$	0	0
$983$	13368.0	0.433747	0.216873	−	0.976200i	$-0.430414\pi$
$983$	13368.0	0.433747	0.216873	+	0.976200i	$0.430414\pi$
$984$	0	0
$985$	14670.0	0.474543
$986$	0	0
$987$	9408.00	0.303404
$988$	0	0
$989$	23520.0	0.756211
$990$	0	0
$991$	44296.0	1.41989	0.709944	−	0.704258i	$-0.248720\pi$
$991$	44296.0	1.41989	0.709944	+	0.704258i	$0.248720\pi$
$992$	0	0
$993$	31472.0	1.00577
$994$	0	0
$995$	−2600.00	−0.0828397
$996$	0	0
$997$	4354.00	0.138307	0.0691537	−	0.997606i	$-0.477970\pi$
$997$	4354.00	0.138307	0.0691537	+	0.997606i	$0.477970\pi$
$998$	0	0
$999$	−16112.0	−0.510271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.y.1.1		1
4.3	odd	2		2240.4.a.m.1.1		1
8.3	odd	2		70.4.a.d.1.1	✓	1
8.5	even	2		560.4.a.g.1.1		1
24.11	even	2		630.4.a.o.1.1		1
40.3	even	4		350.4.c.d.99.2		2
40.19	odd	2		350.4.a.o.1.1		1
40.27	even	4		350.4.c.d.99.1		2
56.3	even	6		490.4.e.q.471.1		2
56.11	odd	6		490.4.e.k.471.1		2
56.19	even	6		490.4.e.q.361.1		2
56.27	even	2		490.4.a.b.1.1		1
56.51	odd	6		490.4.e.k.361.1		2
280.139	even	2		2450.4.a.bm.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.a.d.1.1	✓	1	8.3	odd	2
350.4.a.o.1.1		1	40.19	odd	2
350.4.c.d.99.1		2	40.27	even	4
350.4.c.d.99.2		2	40.3	even	4
490.4.a.b.1.1		1	56.27	even	2
490.4.e.k.361.1		2	56.51	odd	6
490.4.e.k.471.1		2	56.11	odd	6
490.4.e.q.361.1		2	56.19	even	6
490.4.e.q.471.1		2	56.3	even	6
560.4.a.g.1.1		1	8.5	even	2
630.4.a.o.1.1		1	24.11	even	2
2240.4.a.m.1.1		1	4.3	odd	2
2240.4.a.y.1.1		1	1.1	even	1	trivial
2450.4.a.bm.1.1		1	280.139	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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