LMFDB - Embedded newform 2240.4.a.l.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 280)
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)

$f(q)$

$=$

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

$q-4.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} -11.0000 q^{9} +20.0000 q^{11} +10.0000 q^{13} +20.0000 q^{15} -14.0000 q^{17} +12.0000 q^{19} +28.0000 q^{21} -104.000 q^{23} +25.0000 q^{25} +152.000 q^{27} +122.000 q^{29} -224.000 q^{31} -80.0000 q^{33} +35.0000 q^{35} -158.000 q^{37} -40.0000 q^{39} +378.000 q^{41} +404.000 q^{43} +55.0000 q^{45} -112.000 q^{47} +49.0000 q^{49} +56.0000 q^{51} -270.000 q^{53} -100.000 q^{55} -48.0000 q^{57} +324.000 q^{59} +186.000 q^{61} +77.0000 q^{63} -50.0000 q^{65} +156.000 q^{67} +416.000 q^{69} +360.000 q^{71} -102.000 q^{73} -100.000 q^{75} -140.000 q^{77} +912.000 q^{79} -311.000 q^{81} +1068.00 q^{83} +70.0000 q^{85} -488.000 q^{87} -1590.00 q^{89} -70.0000 q^{91} +896.000 q^{93} -60.0000 q^{95} +866.000 q^{97} -220.000 q^{99} +O(q^{100})$

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	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	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
$11$	20.0000	0.548202	0.274101	−	0.961701i	$-0.411620\pi$
$11$	20.0000	0.548202	0.274101	+	0.961701i	$0.411620\pi$
$12$	0	0
$13$	10.0000	0.213346	0.106673	−	0.994294i	$-0.465980\pi$
$13$	10.0000	0.213346	0.106673	+	0.994294i	$0.465980\pi$
$14$	0	0
$15$	20.0000	0.344265
$16$	0	0
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	12.0000	0.144894	0.0724471	−	0.997372i	$-0.476919\pi$
$19$	12.0000	0.144894	0.0724471	+	0.997372i	$0.476919\pi$
$20$	0	0
$21$	28.0000	0.290957
$22$	0	0
$23$	−104.000	−0.942848	−0.471424	−	0.881907i	$-0.656260\pi$
$23$	−104.000	−0.942848	−0.471424	+	0.881907i	$0.656260\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	122.000	0.781201	0.390601	−	0.920560i	$-0.372267\pi$
$29$	122.000	0.781201	0.390601	+	0.920560i	$0.372267\pi$
$30$	0	0
$31$	−224.000	−1.29779	−0.648897	−	0.760877i	$-0.724769\pi$
$31$	−224.000	−1.29779	−0.648897	+	0.760877i	$0.724769\pi$
$32$	0	0
$33$	−80.0000	−0.422006
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−158.000	−0.702028	−0.351014	−	0.936370i	$-0.614163\pi$
$37$	−158.000	−0.702028	−0.351014	+	0.936370i	$0.614163\pi$
$38$	0	0
$39$	−40.0000	−0.164234
$40$	0	0
$41$	378.000	1.43985	0.719923	−	0.694054i	$-0.244177\pi$
$41$	378.000	1.43985	0.719923	+	0.694054i	$0.244177\pi$
$42$	0	0
$43$	404.000	1.43278	0.716389	−	0.697701i	$-0.245794\pi$
$43$	404.000	1.43278	0.716389	+	0.697701i	$0.245794\pi$
$44$	0	0
$45$	55.0000	0.182198
$46$	0	0
$47$	−112.000	−0.347593	−0.173797	−	0.984782i	$-0.555604\pi$
$47$	−112.000	−0.347593	−0.173797	+	0.984782i	$0.555604\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	56.0000	0.153756
$52$	0	0
$53$	−270.000	−0.699761	−0.349881	−	0.936794i	$-0.613778\pi$
$53$	−270.000	−0.699761	−0.349881	+	0.936794i	$0.613778\pi$
$54$	0	0
$55$	−100.000	−0.245164
$56$	0	0
$57$	−48.0000	−0.111540
$58$	0	0
$59$	324.000	0.714936	0.357468	−	0.933925i	$-0.383640\pi$
$59$	324.000	0.714936	0.357468	+	0.933925i	$0.383640\pi$
$60$	0	0
$61$	186.000	0.390408	0.195204	−	0.980763i	$-0.437463\pi$
$61$	186.000	0.390408	0.195204	+	0.980763i	$0.437463\pi$
$62$	0	0
$63$	77.0000	0.153986
$64$	0	0
$65$	−50.0000	−0.0954113
$66$	0	0
$67$	156.000	0.284454	0.142227	−	0.989834i	$-0.454574\pi$
$67$	156.000	0.284454	0.142227	+	0.989834i	$0.454574\pi$
$68$	0	0
$69$	416.000	0.725805
$70$	0	0
$71$	360.000	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$72$	0	0
$73$	−102.000	−0.163537	−0.0817685	−	0.996651i	$-0.526057\pi$
$73$	−102.000	−0.163537	−0.0817685	+	0.996651i	$0.526057\pi$
$74$	0	0
$75$	−100.000	−0.153960
$76$	0	0
$77$	−140.000	−0.207201
$78$	0	0
$79$	912.000	1.29884	0.649418	−	0.760432i	$-0.275013\pi$
$79$	912.000	1.29884	0.649418	+	0.760432i	$0.275013\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	1068.00	1.41239	0.706194	−	0.708018i	$-0.250411\pi$
$83$	1068.00	1.41239	0.706194	+	0.708018i	$0.250411\pi$
$84$	0	0
$85$	70.0000	0.0893243
$86$	0	0
$87$	−488.000	−0.601369
$88$	0	0
$89$	−1590.00	−1.89370	−0.946852	−	0.321669i	$-0.895756\pi$
$89$	−1590.00	−1.89370	−0.946852	+	0.321669i	$0.895756\pi$
$90$	0	0
$91$	−70.0000	−0.0806373
$92$	0	0
$93$	896.000	0.999042
$94$	0	0
$95$	−60.0000	−0.0647986
$96$	0	0
$97$	866.000	0.906484	0.453242	−	0.891387i	$-0.350267\pi$
$97$	866.000	0.906484	0.453242	+	0.891387i	$0.350267\pi$
$98$	0	0
$99$	−220.000	−0.223342
$100$	0	0
$101$	−702.000	−0.691600	−0.345800	−	0.938308i	$-0.612392\pi$
$101$	−702.000	−0.691600	−0.345800	+	0.938308i	$0.612392\pi$
$102$	0	0
$103$	296.000	0.283163	0.141581	−	0.989927i	$-0.454781\pi$
$103$	296.000	0.283163	0.141581	+	0.989927i	$0.454781\pi$
$104$	0	0
$105$	−140.000	−0.130120
$106$	0	0
$107$	−44.0000	−0.0397537	−0.0198768	−	0.999802i	$-0.506327\pi$
$107$	−44.0000	−0.0397537	−0.0198768	+	0.999802i	$0.506327\pi$
$108$	0	0
$109$	650.000	0.571181	0.285590	−	0.958352i	$-0.407810\pi$
$109$	650.000	0.571181	0.285590	+	0.958352i	$0.407810\pi$
$110$	0	0
$111$	632.000	0.540421
$112$	0	0
$113$	−942.000	−0.784212	−0.392106	−	0.919920i	$-0.628253\pi$
$113$	−942.000	−0.784212	−0.392106	+	0.919920i	$0.628253\pi$
$114$	0	0
$115$	520.000	0.421654
$116$	0	0
$117$	−110.000	−0.0869188
$118$	0	0
$119$	98.0000	0.0754928
$120$	0	0
$121$	−931.000	−0.699474
$122$	0	0
$123$	−1512.00	−1.10839
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−736.000	−0.514248	−0.257124	−	0.966378i	$-0.582775\pi$
$127$	−736.000	−0.514248	−0.257124	+	0.966378i	$0.582775\pi$
$128$	0	0
$129$	−1616.00	−1.10295
$130$	0	0
$131$	924.000	0.616261	0.308131	−	0.951344i	$-0.400297\pi$
$131$	924.000	0.616261	0.308131	+	0.951344i	$0.400297\pi$
$132$	0	0
$133$	−84.0000	−0.0547648
$134$	0	0
$135$	−760.000	−0.484521
$136$	0	0
$137$	474.000	0.295595	0.147798	−	0.989018i	$-0.452782\pi$
$137$	474.000	0.295595	0.147798	+	0.989018i	$0.452782\pi$
$138$	0	0
$139$	1012.00	0.617530	0.308765	−	0.951138i	$-0.400084\pi$
$139$	1012.00	0.617530	0.308765	+	0.951138i	$0.400084\pi$
$140$	0	0
$141$	448.000	0.267577
$142$	0	0
$143$	200.000	0.116957
$144$	0	0
$145$	−610.000	−0.349364
$146$	0	0
$147$	−196.000	−0.109971
$148$	0	0
$149$	−302.000	−0.166046	−0.0830228	−	0.996548i	$-0.526457\pi$
$149$	−302.000	−0.166046	−0.0830228	+	0.996548i	$0.526457\pi$
$150$	0	0
$151$	2104.00	1.13391	0.566957	−	0.823747i	$-0.308120\pi$
$151$	2104.00	1.13391	0.566957	+	0.823747i	$0.308120\pi$
$152$	0	0
$153$	154.000	0.0813736
$154$	0	0
$155$	1120.00	0.580391
$156$	0	0
$157$	−1190.00	−0.604919	−0.302460	−	0.953162i	$-0.597808\pi$
$157$	−1190.00	−0.604919	−0.302460	+	0.953162i	$0.597808\pi$
$158$	0	0
$159$	1080.00	0.538677
$160$	0	0
$161$	728.000	0.356363
$162$	0	0
$163$	−1732.00	−0.832274	−0.416137	−	0.909302i	$-0.636616\pi$
$163$	−1732.00	−0.832274	−0.416137	+	0.909302i	$0.636616\pi$
$164$	0	0
$165$	400.000	0.188727
$166$	0	0
$167$	−2264.00	−1.04906	−0.524532	−	0.851391i	$-0.675760\pi$
$167$	−2264.00	−1.04906	−0.524532	+	0.851391i	$0.675760\pi$
$168$	0	0
$169$	−2097.00	−0.954483
$170$	0	0
$171$	−132.000	−0.0590309
$172$	0	0
$173$	1066.00	0.468477	0.234238	−	0.972179i	$-0.424740\pi$
$173$	1066.00	0.468477	0.234238	+	0.972179i	$0.424740\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−1296.00	−0.550358
$178$	0	0
$179$	−2356.00	−0.983775	−0.491887	−	0.870659i	$-0.663693\pi$
$179$	−2356.00	−0.983775	−0.491887	+	0.870659i	$0.663693\pi$
$180$	0	0
$181$	−2158.00	−0.886204	−0.443102	−	0.896471i	$-0.646122\pi$
$181$	−2158.00	−0.886204	−0.443102	+	0.896471i	$0.646122\pi$
$182$	0	0
$183$	−744.000	−0.300536
$184$	0	0
$185$	790.000	0.313957
$186$	0	0
$187$	−280.000	−0.109495
$188$	0	0
$189$	−1064.00	−0.409495
$190$	0	0
$191$	2688.00	1.01831	0.509154	−	0.860675i	$-0.329958\pi$
$191$	2688.00	1.01831	0.509154	+	0.860675i	$0.329958\pi$
$192$	0	0
$193$	−4414.00	−1.64625	−0.823126	−	0.567859i	$-0.807772\pi$
$193$	−4414.00	−1.64625	−0.823126	+	0.567859i	$0.807772\pi$
$194$	0	0
$195$	200.000	0.0734477
$196$	0	0
$197$	−3774.00	−1.36491	−0.682453	−	0.730930i	$-0.739087\pi$
$197$	−3774.00	−1.36491	−0.682453	+	0.730930i	$0.739087\pi$
$198$	0	0
$199$	−664.000	−0.236531	−0.118266	−	0.992982i	$-0.537733\pi$
$199$	−664.000	−0.236531	−0.118266	+	0.992982i	$0.537733\pi$
$200$	0	0
$201$	−624.000	−0.218973
$202$	0	0
$203$	−854.000	−0.295266
$204$	0	0
$205$	−1890.00	−0.643919
$206$	0	0
$207$	1144.00	0.384123
$208$	0	0
$209$	240.000	0.0794313
$210$	0	0
$211$	1260.00	0.411099	0.205550	−	0.978647i	$-0.434102\pi$
$211$	1260.00	0.411099	0.205550	+	0.978647i	$0.434102\pi$
$212$	0	0
$213$	−1440.00	−0.463226
$214$	0	0
$215$	−2020.00	−0.640757
$216$	0	0
$217$	1568.00	0.490520
$218$	0	0
$219$	408.000	0.125891
$220$	0	0
$221$	−140.000	−0.0426128
$222$	0	0
$223$	1152.00	0.345936	0.172968	−	0.984927i	$-0.444664\pi$
$223$	1152.00	0.345936	0.172968	+	0.984927i	$0.444664\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	2588.00	0.756703	0.378352	−	0.925662i	$-0.376491\pi$
$227$	2588.00	0.756703	0.378352	+	0.925662i	$0.376491\pi$
$228$	0	0
$229$	−766.000	−0.221042	−0.110521	−	0.993874i	$-0.535252\pi$
$229$	−766.000	−0.221042	−0.110521	+	0.993874i	$0.535252\pi$
$230$	0	0
$231$	560.000	0.159503
$232$	0	0
$233$	6138.00	1.72581	0.862905	−	0.505366i	$-0.168643\pi$
$233$	6138.00	1.72581	0.862905	+	0.505366i	$0.168643\pi$
$234$	0	0
$235$	560.000	0.155448
$236$	0	0
$237$	−3648.00	−0.999844
$238$	0	0
$239$	−3856.00	−1.04361	−0.521807	−	0.853063i	$-0.674742\pi$
$239$	−3856.00	−1.04361	−0.521807	+	0.853063i	$0.674742\pi$
$240$	0	0
$241$	2578.00	0.689060	0.344530	−	0.938775i	$-0.388038\pi$
$241$	2578.00	0.689060	0.344530	+	0.938775i	$0.388038\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	120.000	0.0309126
$248$	0	0
$249$	−4272.00	−1.08726
$250$	0	0
$251$	−1404.00	−0.353067	−0.176533	−	0.984295i	$-0.556488\pi$
$251$	−1404.00	−0.353067	−0.176533	+	0.984295i	$0.556488\pi$
$252$	0	0
$253$	−2080.00	−0.516871
$254$	0	0
$255$	−280.000	−0.0687619
$256$	0	0
$257$	2946.00	0.715044	0.357522	−	0.933905i	$-0.383622\pi$
$257$	2946.00	0.715044	0.357522	+	0.933905i	$0.383622\pi$
$258$	0	0
$259$	1106.00	0.265342
$260$	0	0
$261$	−1342.00	−0.318267
$262$	0	0
$263$	3080.00	0.722133	0.361066	−	0.932540i	$-0.382413\pi$
$263$	3080.00	0.722133	0.361066	+	0.932540i	$0.382413\pi$
$264$	0	0
$265$	1350.00	0.312943
$266$	0	0
$267$	6360.00	1.45777
$268$	0	0
$269$	−2294.00	−0.519954	−0.259977	−	0.965615i	$-0.583715\pi$
$269$	−2294.00	−0.519954	−0.259977	+	0.965615i	$0.583715\pi$
$270$	0	0
$271$	3728.00	0.835645	0.417823	−	0.908529i	$-0.362793\pi$
$271$	3728.00	0.835645	0.417823	+	0.908529i	$0.362793\pi$
$272$	0	0
$273$	280.000	0.0620746
$274$	0	0
$275$	500.000	0.109640
$276$	0	0
$277$	−2734.00	−0.593033	−0.296516	−	0.955028i	$-0.595825\pi$
$277$	−2734.00	−0.593033	−0.296516	+	0.955028i	$0.595825\pi$
$278$	0	0
$279$	2464.00	0.528731
$280$	0	0
$281$	1034.00	0.219513	0.109757	−	0.993958i	$-0.464993\pi$
$281$	1034.00	0.219513	0.109757	+	0.993958i	$0.464993\pi$
$282$	0	0
$283$	−5180.00	−1.08805	−0.544027	−	0.839068i	$-0.683101\pi$
$283$	−5180.00	−1.08805	−0.544027	+	0.839068i	$0.683101\pi$
$284$	0	0
$285$	240.000	0.0498820
$286$	0	0
$287$	−2646.00	−0.544211
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	−3464.00	−0.697812
$292$	0	0
$293$	−7934.00	−1.58194	−0.790971	−	0.611853i	$-0.790424\pi$
$293$	−7934.00	−1.58194	−0.790971	+	0.611853i	$0.790424\pi$
$294$	0	0
$295$	−1620.00	−0.319729
$296$	0	0
$297$	3040.00	0.593935
$298$	0	0
$299$	−1040.00	−0.201153
$300$	0	0
$301$	−2828.00	−0.541539
$302$	0	0
$303$	2808.00	0.532394
$304$	0	0
$305$	−930.000	−0.174596
$306$	0	0
$307$	−3444.00	−0.640259	−0.320129	−	0.947374i	$-0.603726\pi$
$307$	−3444.00	−0.640259	−0.320129	+	0.947374i	$0.603726\pi$
$308$	0	0
$309$	−1184.00	−0.217979
$310$	0	0
$311$	−7656.00	−1.39592	−0.697961	−	0.716135i	$-0.745909\pi$
$311$	−7656.00	−1.39592	−0.697961	+	0.716135i	$0.745909\pi$
$312$	0	0
$313$	−3798.00	−0.685865	−0.342932	−	0.939360i	$-0.611420\pi$
$313$	−3798.00	−0.685865	−0.342932	+	0.939360i	$0.611420\pi$
$314$	0	0
$315$	−385.000	−0.0688644
$316$	0	0
$317$	−6822.00	−1.20871	−0.604356	−	0.796714i	$-0.706570\pi$
$317$	−6822.00	−1.20871	−0.604356	+	0.796714i	$0.706570\pi$
$318$	0	0
$319$	2440.00	0.428256
$320$	0	0
$321$	176.000	0.0306024
$322$	0	0
$323$	−168.000	−0.0289405
$324$	0	0
$325$	250.000	0.0426692
$326$	0	0
$327$	−2600.00	−0.439695
$328$	0	0
$329$	784.000	0.131378
$330$	0	0
$331$	−6572.00	−1.09133	−0.545664	−	0.838004i	$-0.683723\pi$
$331$	−6572.00	−1.09133	−0.545664	+	0.838004i	$0.683723\pi$
$332$	0	0
$333$	1738.00	0.286011
$334$	0	0
$335$	−780.000	−0.127212
$336$	0	0
$337$	−11022.0	−1.78162	−0.890811	−	0.454374i	$-0.849863\pi$
$337$	−11022.0	−1.78162	−0.890811	+	0.454374i	$0.849863\pi$
$338$	0	0
$339$	3768.00	0.603686
$340$	0	0
$341$	−4480.00	−0.711453
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−2080.00	−0.324590
$346$	0	0
$347$	−10908.0	−1.68753	−0.843764	−	0.536715i	$-0.819665\pi$
$347$	−10908.0	−1.68753	−0.843764	+	0.536715i	$0.819665\pi$
$348$	0	0
$349$	12058.0	1.84943	0.924713	−	0.380664i	$-0.124305\pi$
$349$	12058.0	1.84943	0.924713	+	0.380664i	$0.124305\pi$
$350$	0	0
$351$	1520.00	0.231144
$352$	0	0
$353$	1570.00	0.236721	0.118361	−	0.992971i	$-0.462236\pi$
$353$	1570.00	0.236721	0.118361	+	0.992971i	$0.462236\pi$
$354$	0	0
$355$	−1800.00	−0.269110
$356$	0	0
$357$	−392.000	−0.0581144
$358$	0	0
$359$	−120.000	−0.0176417	−0.00882083	−	0.999961i	$-0.502808\pi$
$359$	−120.000	−0.0176417	−0.00882083	+	0.999961i	$0.502808\pi$
$360$	0	0
$361$	−6715.00	−0.979006
$362$	0	0
$363$	3724.00	0.538455
$364$	0	0
$365$	510.000	0.0731359
$366$	0	0
$367$	−4336.00	−0.616723	−0.308362	−	0.951269i	$-0.599781\pi$
$367$	−4336.00	−0.616723	−0.308362	+	0.951269i	$0.599781\pi$
$368$	0	0
$369$	−4158.00	−0.586604
$370$	0	0
$371$	1890.00	0.264485
$372$	0	0
$373$	−7758.00	−1.07693	−0.538464	−	0.842649i	$-0.680995\pi$
$373$	−7758.00	−1.07693	−0.538464	+	0.842649i	$0.680995\pi$
$374$	0	0
$375$	500.000	0.0688530
$376$	0	0
$377$	1220.00	0.166666
$378$	0	0
$379$	10660.0	1.44477	0.722384	−	0.691492i	$-0.243046\pi$
$379$	10660.0	1.44477	0.722384	+	0.691492i	$0.243046\pi$
$380$	0	0
$381$	2944.00	0.395868
$382$	0	0
$383$	14304.0	1.90836	0.954178	−	0.299240i	$-0.0967331\pi$
$383$	14304.0	1.90836	0.954178	+	0.299240i	$0.0967331\pi$
$384$	0	0
$385$	700.000	0.0926631
$386$	0	0
$387$	−4444.00	−0.583724
$388$	0	0
$389$	11970.0	1.56016	0.780081	−	0.625678i	$-0.215178\pi$
$389$	11970.0	1.56016	0.780081	+	0.625678i	$0.215178\pi$
$390$	0	0
$391$	1456.00	0.188320
$392$	0	0
$393$	−3696.00	−0.474398
$394$	0	0
$395$	−4560.00	−0.580857
$396$	0	0
$397$	4106.00	0.519079	0.259539	−	0.965733i	$-0.416429\pi$
$397$	4106.00	0.519079	0.259539	+	0.965733i	$0.416429\pi$
$398$	0	0
$399$	336.000	0.0421580
$400$	0	0
$401$	−3790.00	−0.471979	−0.235989	−	0.971756i	$-0.575833\pi$
$401$	−3790.00	−0.471979	−0.235989	+	0.971756i	$0.575833\pi$
$402$	0	0
$403$	−2240.00	−0.276879
$404$	0	0
$405$	1555.00	0.190787
$406$	0	0
$407$	−3160.00	−0.384854
$408$	0	0
$409$	−13366.0	−1.61591	−0.807954	−	0.589246i	$-0.799425\pi$
$409$	−13366.0	−1.61591	−0.807954	+	0.589246i	$0.799425\pi$
$410$	0	0
$411$	−1896.00	−0.227549
$412$	0	0
$413$	−2268.00	−0.270220
$414$	0	0
$415$	−5340.00	−0.631639
$416$	0	0
$417$	−4048.00	−0.475375
$418$	0	0
$419$	6524.00	0.760664	0.380332	−	0.924850i	$-0.375810\pi$
$419$	6524.00	0.760664	0.380332	+	0.924850i	$0.375810\pi$
$420$	0	0
$421$	2978.00	0.344748	0.172374	−	0.985032i	$-0.444856\pi$
$421$	2978.00	0.344748	0.172374	+	0.985032i	$0.444856\pi$
$422$	0	0
$423$	1232.00	0.141612
$424$	0	0
$425$	−350.000	−0.0399470
$426$	0	0
$427$	−1302.00	−0.147560
$428$	0	0
$429$	−800.000	−0.0900335
$430$	0	0
$431$	−15760.0	−1.76133	−0.880664	−	0.473741i	$-0.842903\pi$
$431$	−15760.0	−1.76133	−0.880664	+	0.473741i	$0.842903\pi$
$432$	0	0
$433$	10322.0	1.14560	0.572799	−	0.819696i	$-0.305858\pi$
$433$	10322.0	1.14560	0.572799	+	0.819696i	$0.305858\pi$
$434$	0	0
$435$	2440.00	0.268940
$436$	0	0
$437$	−1248.00	−0.136613
$438$	0	0
$439$	344.000	0.0373991	0.0186996	−	0.999825i	$-0.494047\pi$
$439$	344.000	0.0373991	0.0186996	+	0.999825i	$0.494047\pi$
$440$	0	0
$441$	−539.000	−0.0582011
$442$	0	0
$443$	−252.000	−0.0270268	−0.0135134	−	0.999909i	$-0.504302\pi$
$443$	−252.000	−0.0270268	−0.0135134	+	0.999909i	$0.504302\pi$
$444$	0	0
$445$	7950.00	0.846890
$446$	0	0
$447$	1208.00	0.127822
$448$	0	0
$449$	−11966.0	−1.25771	−0.628854	−	0.777524i	$-0.716475\pi$
$449$	−11966.0	−1.25771	−0.628854	+	0.777524i	$0.716475\pi$
$450$	0	0
$451$	7560.00	0.789327
$452$	0	0
$453$	−8416.00	−0.872888
$454$	0	0
$455$	350.000	0.0360621
$456$	0	0
$457$	−15846.0	−1.62198	−0.810990	−	0.585060i	$-0.801071\pi$
$457$	−15846.0	−1.62198	−0.810990	+	0.585060i	$0.801071\pi$
$458$	0	0
$459$	−2128.00	−0.216398
$460$	0	0
$461$	−5430.00	−0.548591	−0.274295	−	0.961645i	$-0.588445\pi$
$461$	−5430.00	−0.548591	−0.274295	+	0.961645i	$0.588445\pi$
$462$	0	0
$463$	−8912.00	−0.894548	−0.447274	−	0.894397i	$-0.647605\pi$
$463$	−8912.00	−0.894548	−0.447274	+	0.894397i	$0.647605\pi$
$464$	0	0
$465$	−4480.00	−0.446785
$466$	0	0
$467$	−11028.0	−1.09275	−0.546376	−	0.837540i	$-0.683993\pi$
$467$	−11028.0	−1.09275	−0.546376	+	0.837540i	$0.683993\pi$
$468$	0	0
$469$	−1092.00	−0.107514
$470$	0	0
$471$	4760.00	0.465667
$472$	0	0
$473$	8080.00	0.785452
$474$	0	0
$475$	300.000	0.0289788
$476$	0	0
$477$	2970.00	0.285088
$478$	0	0
$479$	−5856.00	−0.558596	−0.279298	−	0.960204i	$-0.590102\pi$
$479$	−5856.00	−0.558596	−0.279298	+	0.960204i	$0.590102\pi$
$480$	0	0
$481$	−1580.00	−0.149775
$482$	0	0
$483$	−2912.00	−0.274328
$484$	0	0
$485$	−4330.00	−0.405392
$486$	0	0
$487$	−10264.0	−0.955044	−0.477522	−	0.878620i	$-0.658465\pi$
$487$	−10264.0	−0.955044	−0.477522	+	0.878620i	$0.658465\pi$
$488$	0	0
$489$	6928.00	0.640685
$490$	0	0
$491$	2804.00	0.257725	0.128862	−	0.991663i	$-0.458867\pi$
$491$	2804.00	0.257725	0.128862	+	0.991663i	$0.458867\pi$
$492$	0	0
$493$	−1708.00	−0.156033
$494$	0	0
$495$	1100.00	0.0998815
$496$	0	0
$497$	−2520.00	−0.227440
$498$	0	0
$499$	8716.00	0.781927	0.390964	−	0.920406i	$-0.372142\pi$
$499$	8716.00	0.781927	0.390964	+	0.920406i	$0.372142\pi$
$500$	0	0
$501$	9056.00	0.807569
$502$	0	0
$503$	−2504.00	−0.221964	−0.110982	−	0.993822i	$-0.535400\pi$
$503$	−2504.00	−0.221964	−0.110982	+	0.993822i	$0.535400\pi$
$504$	0	0
$505$	3510.00	0.309293
$506$	0	0
$507$	8388.00	0.734762
$508$	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
$511$	714.000	0.0618112
$512$	0	0
$513$	1824.00	0.156982
$514$	0	0
$515$	−1480.00	−0.126634
$516$	0	0
$517$	−2240.00	−0.190551
$518$	0	0
$519$	−4264.00	−0.360634
$520$	0	0
$521$	9882.00	0.830976	0.415488	−	0.909599i	$-0.363611\pi$
$521$	9882.00	0.830976	0.415488	+	0.909599i	$0.363611\pi$
$522$	0	0
$523$	−7532.00	−0.629735	−0.314867	−	0.949136i	$-0.601960\pi$
$523$	−7532.00	−0.629735	−0.314867	+	0.949136i	$0.601960\pi$
$524$	0	0
$525$	700.000	0.0581914
$526$	0	0
$527$	3136.00	0.259215
$528$	0	0
$529$	−1351.00	−0.111038
$530$	0	0
$531$	−3564.00	−0.291270
$532$	0	0
$533$	3780.00	0.307186
$534$	0	0
$535$	220.000	0.0177784
$536$	0	0
$537$	9424.00	0.757310
$538$	0	0
$539$	980.000	0.0783146
$540$	0	0
$541$	17338.0	1.37785	0.688927	−	0.724831i	$-0.258082\pi$
$541$	17338.0	1.37785	0.688927	+	0.724831i	$0.258082\pi$
$542$	0	0
$543$	8632.00	0.682200
$544$	0	0
$545$	−3250.00	−0.255440
$546$	0	0
$547$	764.000	0.0597190	0.0298595	−	0.999554i	$-0.490494\pi$
$547$	764.000	0.0597190	0.0298595	+	0.999554i	$0.490494\pi$
$548$	0	0
$549$	−2046.00	−0.159055
$550$	0	0
$551$	1464.00	0.113191
$552$	0	0
$553$	−6384.00	−0.490914
$554$	0	0
$555$	−3160.00	−0.241684
$556$	0	0
$557$	2442.00	0.185765	0.0928823	−	0.995677i	$-0.470392\pi$
$557$	2442.00	0.185765	0.0928823	+	0.995677i	$0.470392\pi$
$558$	0	0
$559$	4040.00	0.305678
$560$	0	0
$561$	1120.00	0.0842895
$562$	0	0
$563$	8972.00	0.671625	0.335812	−	0.941929i	$-0.390989\pi$
$563$	8972.00	0.671625	0.335812	+	0.941929i	$0.390989\pi$
$564$	0	0
$565$	4710.00	0.350710
$566$	0	0
$567$	2177.00	0.161244
$568$	0	0
$569$	8682.00	0.639663	0.319832	−	0.947474i	$-0.396374\pi$
$569$	8682.00	0.639663	0.319832	+	0.947474i	$0.396374\pi$
$570$	0	0
$571$	15524.0	1.13776	0.568878	−	0.822422i	$-0.307377\pi$
$571$	15524.0	1.13776	0.568878	+	0.822422i	$0.307377\pi$
$572$	0	0
$573$	−10752.0	−0.783894
$574$	0	0
$575$	−2600.00	−0.188570
$576$	0	0
$577$	−19774.0	−1.42669	−0.713347	−	0.700811i	$-0.752822\pi$
$577$	−19774.0	−1.42669	−0.713347	+	0.700811i	$0.752822\pi$
$578$	0	0
$579$	17656.0	1.26729
$580$	0	0
$581$	−7476.00	−0.533833
$582$	0	0
$583$	−5400.00	−0.383611
$584$	0	0
$585$	550.000	0.0388713
$586$	0	0
$587$	−6700.00	−0.471105	−0.235552	−	0.971862i	$-0.575690\pi$
$587$	−6700.00	−0.471105	−0.235552	+	0.971862i	$0.575690\pi$
$588$	0	0
$589$	−2688.00	−0.188043
$590$	0	0
$591$	15096.0	1.05070
$592$	0	0
$593$	−1294.00	−0.0896091	−0.0448046	−	0.998996i	$-0.514267\pi$
$593$	−1294.00	−0.0896091	−0.0448046	+	0.998996i	$0.514267\pi$
$594$	0	0
$595$	−490.000	−0.0337614
$596$	0	0
$597$	2656.00	0.182082
$598$	0	0
$599$	−19720.0	−1.34514	−0.672569	−	0.740035i	$-0.734809\pi$
$599$	−19720.0	−1.34514	−0.672569	+	0.740035i	$0.734809\pi$
$600$	0	0
$601$	21450.0	1.45585	0.727923	−	0.685659i	$-0.240486\pi$
$601$	21450.0	1.45585	0.727923	+	0.685659i	$0.240486\pi$
$602$	0	0
$603$	−1716.00	−0.115889
$604$	0	0
$605$	4655.00	0.312814
$606$	0	0
$607$	19264.0	1.28814	0.644071	−	0.764966i	$-0.277244\pi$
$607$	19264.0	1.28814	0.644071	+	0.764966i	$0.277244\pi$
$608$	0	0
$609$	3416.00	0.227296
$610$	0	0
$611$	−1120.00	−0.0741577
$612$	0	0
$613$	−13150.0	−0.866433	−0.433217	−	0.901290i	$-0.642621\pi$
$613$	−13150.0	−0.866433	−0.433217	+	0.901290i	$0.642621\pi$
$614$	0	0
$615$	7560.00	0.495689
$616$	0	0
$617$	−15238.0	−0.994261	−0.497130	−	0.867676i	$-0.665613\pi$
$617$	−15238.0	−0.994261	−0.497130	+	0.867676i	$0.665613\pi$
$618$	0	0
$619$	23764.0	1.54306	0.771531	−	0.636191i	$-0.219491\pi$
$619$	23764.0	1.54306	0.771531	+	0.636191i	$0.219491\pi$
$620$	0	0
$621$	−15808.0	−1.02150
$622$	0	0
$623$	11130.0	0.715753
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−960.000	−0.0611463
$628$	0	0
$629$	2212.00	0.140220
$630$	0	0
$631$	2008.00	0.126683	0.0633417	−	0.997992i	$-0.479824\pi$
$631$	2008.00	0.126683	0.0633417	+	0.997992i	$0.479824\pi$
$632$	0	0
$633$	−5040.00	−0.316464
$634$	0	0
$635$	3680.00	0.229978
$636$	0	0
$637$	490.000	0.0304780
$638$	0	0
$639$	−3960.00	−0.245157
$640$	0	0
$641$	13058.0	0.804618	0.402309	−	0.915504i	$-0.368208\pi$
$641$	13058.0	0.804618	0.402309	+	0.915504i	$0.368208\pi$
$642$	0	0
$643$	24188.0	1.48349	0.741743	−	0.670684i	$-0.233999\pi$
$643$	24188.0	1.48349	0.741743	+	0.670684i	$0.233999\pi$
$644$	0	0
$645$	8080.00	0.493255
$646$	0	0
$647$	−6648.00	−0.403956	−0.201978	−	0.979390i	$-0.564737\pi$
$647$	−6648.00	−0.403956	−0.201978	+	0.979390i	$0.564737\pi$
$648$	0	0
$649$	6480.00	0.391930
$650$	0	0
$651$	−6272.00	−0.377602
$652$	0	0
$653$	−9814.00	−0.588134	−0.294067	−	0.955785i	$-0.595009\pi$
$653$	−9814.00	−0.588134	−0.294067	+	0.955785i	$0.595009\pi$
$654$	0	0
$655$	−4620.00	−0.275601
$656$	0	0
$657$	1122.00	0.0666262
$658$	0	0
$659$	26540.0	1.56882	0.784409	−	0.620243i	$-0.212966\pi$
$659$	26540.0	1.56882	0.784409	+	0.620243i	$0.212966\pi$
$660$	0	0
$661$	1330.00	0.0782617	0.0391309	−	0.999234i	$-0.487541\pi$
$661$	1330.00	0.0782617	0.0391309	+	0.999234i	$0.487541\pi$
$662$	0	0
$663$	560.000	0.0328033
$664$	0	0
$665$	420.000	0.0244916
$666$	0	0
$667$	−12688.0	−0.736554
$668$	0	0
$669$	−4608.00	−0.266301
$670$	0	0
$671$	3720.00	0.214022
$672$	0	0
$673$	−17950.0	−1.02812	−0.514058	−	0.857756i	$-0.671858\pi$
$673$	−17950.0	−1.02812	−0.514058	+	0.857756i	$0.671858\pi$
$674$	0	0
$675$	3800.00	0.216685
$676$	0	0
$677$	−23742.0	−1.34783	−0.673914	−	0.738810i	$-0.735388\pi$
$677$	−23742.0	−1.34783	−0.673914	+	0.738810i	$0.735388\pi$
$678$	0	0
$679$	−6062.00	−0.342619
$680$	0	0
$681$	−10352.0	−0.582510
$682$	0	0
$683$	6356.00	0.356084	0.178042	−	0.984023i	$-0.443024\pi$
$683$	6356.00	0.356084	0.178042	+	0.984023i	$0.443024\pi$
$684$	0	0
$685$	−2370.00	−0.132194
$686$	0	0
$687$	3064.00	0.170159
$688$	0	0
$689$	−2700.00	−0.149291
$690$	0	0
$691$	−3156.00	−0.173748	−0.0868740	−	0.996219i	$-0.527688\pi$
$691$	−3156.00	−0.173748	−0.0868740	+	0.996219i	$0.527688\pi$
$692$	0	0
$693$	1540.00	0.0844152
$694$	0	0
$695$	−5060.00	−0.276168
$696$	0	0
$697$	−5292.00	−0.287588
$698$	0	0
$699$	−24552.0	−1.32853
$700$	0	0
$701$	18330.0	0.987610	0.493805	−	0.869573i	$-0.335606\pi$
$701$	18330.0	0.987610	0.493805	+	0.869573i	$0.335606\pi$
$702$	0	0
$703$	−1896.00	−0.101720
$704$	0	0
$705$	−2240.00	−0.119664
$706$	0	0
$707$	4914.00	0.261400
$708$	0	0
$709$	−19070.0	−1.01014	−0.505070	−	0.863079i	$-0.668533\pi$
$709$	−19070.0	−1.01014	−0.505070	+	0.863079i	$0.668533\pi$
$710$	0	0
$711$	−10032.0	−0.529155
$712$	0	0
$713$	23296.0	1.22362
$714$	0	0
$715$	−1000.00	−0.0523047
$716$	0	0
$717$	15424.0	0.803375
$718$	0	0
$719$	4176.00	0.216604	0.108302	−	0.994118i	$-0.465459\pi$
$719$	4176.00	0.216604	0.108302	+	0.994118i	$0.465459\pi$
$720$	0	0
$721$	−2072.00	−0.107025
$722$	0	0
$723$	−10312.0	−0.530439
$724$	0	0
$725$	3050.00	0.156240
$726$	0	0
$727$	10520.0	0.536678	0.268339	−	0.963324i	$-0.413525\pi$
$727$	10520.0	0.536678	0.268339	+	0.963324i	$0.413525\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	−5656.00	−0.286176
$732$	0	0
$733$	6874.00	0.346381	0.173190	−	0.984888i	$-0.444592\pi$
$733$	6874.00	0.346381	0.173190	+	0.984888i	$0.444592\pi$
$734$	0	0
$735$	980.000	0.0491807
$736$	0	0
$737$	3120.00	0.155939
$738$	0	0
$739$	11804.0	0.587574	0.293787	−	0.955871i	$-0.405084\pi$
$739$	11804.0	0.587574	0.293787	+	0.955871i	$0.405084\pi$
$740$	0	0
$741$	−480.000	−0.0237965
$742$	0	0
$743$	20648.0	1.01952	0.509759	−	0.860317i	$-0.329735\pi$
$743$	20648.0	1.01952	0.509759	+	0.860317i	$0.329735\pi$
$744$	0	0
$745$	1510.00	0.0742579
$746$	0	0
$747$	−11748.0	−0.575417
$748$	0	0
$749$	308.000	0.0150255
$750$	0	0
$751$	−1232.00	−0.0598619	−0.0299310	−	0.999552i	$-0.509529\pi$
$751$	−1232.00	−0.0598619	−0.0299310	+	0.999552i	$0.509529\pi$
$752$	0	0
$753$	5616.00	0.271791
$754$	0	0
$755$	−10520.0	−0.507102
$756$	0	0
$757$	11250.0	0.540143	0.270071	−	0.962840i	$-0.412953\pi$
$757$	11250.0	0.540143	0.270071	+	0.962840i	$0.412953\pi$
$758$	0	0
$759$	8320.00	0.397888
$760$	0	0
$761$	14698.0	0.700134	0.350067	−	0.936725i	$-0.386159\pi$
$761$	14698.0	0.700134	0.350067	+	0.936725i	$0.386159\pi$
$762$	0	0
$763$	−4550.00	−0.215886
$764$	0	0
$765$	−770.000	−0.0363914
$766$	0	0
$767$	3240.00	0.152529
$768$	0	0
$769$	−30014.0	−1.40745	−0.703727	−	0.710470i	$-0.748482\pi$
$769$	−30014.0	−1.40745	−0.703727	+	0.710470i	$0.748482\pi$
$770$	0	0
$771$	−11784.0	−0.550441
$772$	0	0
$773$	34018.0	1.58285	0.791425	−	0.611267i	$-0.209340\pi$
$773$	34018.0	1.58285	0.791425	+	0.611267i	$0.209340\pi$
$774$	0	0
$775$	−5600.00	−0.259559
$776$	0	0
$777$	−4424.00	−0.204260
$778$	0	0
$779$	4536.00	0.208625
$780$	0	0
$781$	7200.00	0.329880
$782$	0	0
$783$	18544.0	0.846371
$784$	0	0
$785$	5950.00	0.270528
$786$	0	0
$787$	−18516.0	−0.838658	−0.419329	−	0.907834i	$-0.637735\pi$
$787$	−18516.0	−0.838658	−0.419329	+	0.907834i	$0.637735\pi$
$788$	0	0
$789$	−12320.0	−0.555898
$790$	0	0
$791$	6594.00	0.296404
$792$	0	0
$793$	1860.00	0.0832920
$794$	0	0
$795$	−5400.00	−0.240903
$796$	0	0
$797$	38618.0	1.71634	0.858168	−	0.513369i	$-0.171603\pi$
$797$	38618.0	1.71634	0.858168	+	0.513369i	$0.171603\pi$
$798$	0	0
$799$	1568.00	0.0694266
$800$	0	0
$801$	17490.0	0.771509
$802$	0	0
$803$	−2040.00	−0.0896514
$804$	0	0
$805$	−3640.00	−0.159370
$806$	0	0
$807$	9176.00	0.400261
$808$	0	0
$809$	19514.0	0.848054	0.424027	−	0.905650i	$-0.360616\pi$
$809$	19514.0	0.848054	0.424027	+	0.905650i	$0.360616\pi$
$810$	0	0
$811$	−31020.0	−1.34311	−0.671553	−	0.740956i	$-0.734373\pi$
$811$	−31020.0	−1.34311	−0.671553	+	0.740956i	$0.734373\pi$
$812$	0	0
$813$	−14912.0	−0.643280
$814$	0	0
$815$	8660.00	0.372204
$816$	0	0
$817$	4848.00	0.207601
$818$	0	0
$819$	770.000	0.0328522
$820$	0	0
$821$	−3726.00	−0.158390	−0.0791951	−	0.996859i	$-0.525235\pi$
$821$	−3726.00	−0.158390	−0.0791951	+	0.996859i	$0.525235\pi$
$822$	0	0
$823$	−32904.0	−1.39363	−0.696817	−	0.717249i	$-0.745401\pi$
$823$	−32904.0	−1.39363	−0.696817	+	0.717249i	$0.745401\pi$
$824$	0	0
$825$	−2000.00	−0.0844013
$826$	0	0
$827$	−39868.0	−1.67636	−0.838178	−	0.545397i	$-0.816379\pi$
$827$	−39868.0	−1.67636	−0.838178	+	0.545397i	$0.816379\pi$
$828$	0	0
$829$	26810.0	1.12322	0.561610	−	0.827402i	$-0.310182\pi$
$829$	26810.0	1.12322	0.561610	+	0.827402i	$0.310182\pi$
$830$	0	0
$831$	10936.0	0.456517
$832$	0	0
$833$	−686.000	−0.0285336
$834$	0	0
$835$	11320.0	0.469155
$836$	0	0
$837$	−34048.0	−1.40606
$838$	0	0
$839$	37096.0	1.52646	0.763228	−	0.646130i	$-0.223613\pi$
$839$	37096.0	1.52646	0.763228	+	0.646130i	$0.223613\pi$
$840$	0	0
$841$	−9505.00	−0.389725
$842$	0	0
$843$	−4136.00	−0.168982
$844$	0	0
$845$	10485.0	0.426858
$846$	0	0
$847$	6517.00	0.264376
$848$	0	0
$849$	20720.0	0.837584
$850$	0	0
$851$	16432.0	0.661906
$852$	0	0
$853$	14386.0	0.577453	0.288726	−	0.957412i	$-0.406768\pi$
$853$	14386.0	0.577453	0.288726	+	0.957412i	$0.406768\pi$
$854$	0	0
$855$	660.000	0.0263994
$856$	0	0
$857$	−118.000	−0.00470339	−0.00235169	−	0.999997i	$-0.500749\pi$
$857$	−118.000	−0.00470339	−0.00235169	+	0.999997i	$0.500749\pi$
$858$	0	0
$859$	−25116.0	−0.997610	−0.498805	−	0.866714i	$-0.666228\pi$
$859$	−25116.0	−0.997610	−0.498805	+	0.866714i	$0.666228\pi$
$860$	0	0
$861$	10584.0	0.418934
$862$	0	0
$863$	11776.0	0.464496	0.232248	−	0.972657i	$-0.425392\pi$
$863$	11776.0	0.464496	0.232248	+	0.972657i	$0.425392\pi$
$864$	0	0
$865$	−5330.00	−0.209509
$866$	0	0
$867$	18868.0	0.739090
$868$	0	0
$869$	18240.0	0.712025
$870$	0	0
$871$	1560.00	0.0606872
$872$	0	0
$873$	−9526.00	−0.369308
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−24694.0	−0.950806	−0.475403	−	0.879768i	$-0.657698\pi$
$877$	−24694.0	−0.950806	−0.475403	+	0.879768i	$0.657698\pi$
$878$	0	0
$879$	31736.0	1.21778
$880$	0	0
$881$	32850.0	1.25624	0.628118	−	0.778118i	$-0.283825\pi$
$881$	32850.0	1.25624	0.628118	+	0.778118i	$0.283825\pi$
$882$	0	0
$883$	47340.0	1.80421	0.902105	−	0.431516i	$-0.142021\pi$
$883$	47340.0	1.80421	0.902105	+	0.431516i	$0.142021\pi$
$884$	0	0
$885$	6480.00	0.246127
$886$	0	0
$887$	−1992.00	−0.0754057	−0.0377028	−	0.999289i	$-0.512004\pi$
$887$	−1992.00	−0.0754057	−0.0377028	+	0.999289i	$0.512004\pi$
$888$	0	0
$889$	5152.00	0.194367
$890$	0	0
$891$	−6220.00	−0.233870
$892$	0	0
$893$	−1344.00	−0.0503642
$894$	0	0
$895$	11780.0	0.439958
$896$	0	0
$897$	4160.00	0.154848
$898$	0	0
$899$	−27328.0	−1.01384
$900$	0	0
$901$	3780.00	0.139767
$902$	0	0
$903$	11312.0	0.416877
$904$	0	0
$905$	10790.0	0.396322
$906$	0	0
$907$	−13516.0	−0.494809	−0.247404	−	0.968912i	$-0.579578\pi$
$907$	−13516.0	−0.494809	−0.247404	+	0.968912i	$0.579578\pi$
$908$	0	0
$909$	7722.00	0.281763
$910$	0	0
$911$	3408.00	0.123943	0.0619715	−	0.998078i	$-0.480261\pi$
$911$	3408.00	0.123943	0.0619715	+	0.998078i	$0.480261\pi$
$912$	0	0
$913$	21360.0	0.774275
$914$	0	0
$915$	3720.00	0.134404
$916$	0	0
$917$	−6468.00	−0.232925
$918$	0	0
$919$	−39240.0	−1.40850	−0.704248	−	0.709954i	$-0.748716\pi$
$919$	−39240.0	−1.40850	−0.704248	+	0.709954i	$0.748716\pi$
$920$	0	0
$921$	13776.0	0.492871
$922$	0	0
$923$	3600.00	0.128381
$924$	0	0
$925$	−3950.00	−0.140406
$926$	0	0
$927$	−3256.00	−0.115363
$928$	0	0
$929$	21922.0	0.774206	0.387103	−	0.922036i	$-0.373476\pi$
$929$	21922.0	0.774206	0.387103	+	0.922036i	$0.373476\pi$
$930$	0	0
$931$	588.000	0.0206992
$932$	0	0
$933$	30624.0	1.07458
$934$	0	0
$935$	1400.00	0.0489678
$936$	0	0
$937$	186.000	0.00648490	0.00324245	−	0.999995i	$-0.498968\pi$
$937$	186.000	0.00648490	0.00324245	+	0.999995i	$0.498968\pi$
$938$	0	0
$939$	15192.0	0.527979
$940$	0	0
$941$	18282.0	0.633343	0.316672	−	0.948535i	$-0.397435\pi$
$941$	18282.0	0.633343	0.316672	+	0.948535i	$0.397435\pi$
$942$	0	0
$943$	−39312.0	−1.35756
$944$	0	0
$945$	5320.00	0.183132
$946$	0	0
$947$	−16980.0	−0.582657	−0.291328	−	0.956623i	$-0.594097\pi$
$947$	−16980.0	−0.582657	−0.291328	+	0.956623i	$0.594097\pi$
$948$	0	0
$949$	−1020.00	−0.0348900
$950$	0	0
$951$	27288.0	0.930467
$952$	0	0
$953$	−44310.0	−1.50613	−0.753065	−	0.657946i	$-0.771425\pi$
$953$	−44310.0	−1.50613	−0.753065	+	0.657946i	$0.771425\pi$
$954$	0	0
$955$	−13440.0	−0.455401
$956$	0	0
$957$	−9760.00	−0.329672
$958$	0	0
$959$	−3318.00	−0.111725
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	484.000	0.0161959
$964$	0	0
$965$	22070.0	0.736226
$966$	0	0
$967$	−51512.0	−1.71304	−0.856522	−	0.516110i	$-0.827380\pi$
$967$	−51512.0	−1.71304	−0.856522	+	0.516110i	$0.827380\pi$
$968$	0	0
$969$	672.000	0.0222784
$970$	0	0
$971$	1844.00	0.0609442	0.0304721	−	0.999536i	$-0.490299\pi$
$971$	1844.00	0.0609442	0.0304721	+	0.999536i	$0.490299\pi$
$972$	0	0
$973$	−7084.00	−0.233405
$974$	0	0
$975$	−1000.00	−0.0328468
$976$	0	0
$977$	34162.0	1.11867	0.559334	−	0.828942i	$-0.311057\pi$
$977$	34162.0	1.11867	0.559334	+	0.828942i	$0.311057\pi$
$978$	0	0
$979$	−31800.0	−1.03813
$980$	0	0
$981$	−7150.00	−0.232703
$982$	0	0
$983$	216.000	0.00700847	0.00350424	−	0.999994i	$-0.498885\pi$
$983$	216.000	0.00700847	0.00350424	+	0.999994i	$0.498885\pi$
$984$	0	0
$985$	18870.0	0.610404
$986$	0	0
$987$	−3136.00	−0.101135
$988$	0	0
$989$	−42016.0	−1.35089
$990$	0	0
$991$	−52832.0	−1.69351	−0.846753	−	0.531987i	$-0.821446\pi$
$991$	−52832.0	−1.69351	−0.846753	+	0.531987i	$0.821446\pi$
$992$	0	0
$993$	26288.0	0.840105
$994$	0	0
$995$	3320.00	0.105780
$996$	0	0
$997$	−5054.00	−0.160543	−0.0802717	−	0.996773i	$-0.525579\pi$
$997$	−5054.00	−0.160543	−0.0802717	+	0.996773i	$0.525579\pi$
$998$	0	0
$999$	−24016.0	−0.760593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.l.1.1		1
4.3	odd	2		2240.4.a.z.1.1		1
8.3	odd	2		280.4.a.a.1.1	✓	1
8.5	even	2		560.4.a.l.1.1		1
40.3	even	4		1400.4.g.e.449.1		2
40.19	odd	2		1400.4.a.g.1.1		1
40.27	even	4		1400.4.g.e.449.2		2
56.27	even	2		1960.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.a.1.1	✓	1	8.3	odd	2
560.4.a.l.1.1		1	8.5	even	2
1400.4.a.g.1.1		1	40.19	odd	2
1400.4.g.e.449.1		2	40.3	even	4
1400.4.g.e.449.2		2	40.27	even	4
1960.4.a.g.1.1		1	56.27	even	2
2240.4.a.l.1.1		1	1.1	even	1	trivial
2240.4.a.z.1.1		1	4.3	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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