LMFDB - Embedded newform 2205.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 245)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -7.00000 q^{4} +5.00000 q^{5} +15.0000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -7.00000 q^{4} +5.00000 q^{5} +15.0000 q^{8} -5.00000 q^{10} +44.0000 q^{11} +6.00000 q^{13} +41.0000 q^{16} +24.0000 q^{17} -114.000 q^{19} -35.0000 q^{20} -44.0000 q^{22} +52.0000 q^{23} +25.0000 q^{25} -6.00000 q^{26} -146.000 q^{29} -276.000 q^{31} -161.000 q^{32} -24.0000 q^{34} -210.000 q^{37} +114.000 q^{38} +75.0000 q^{40} -444.000 q^{41} +492.000 q^{43} -308.000 q^{44} -52.0000 q^{46} +612.000 q^{47} -25.0000 q^{50} -42.0000 q^{52} -50.0000 q^{53} +220.000 q^{55} +146.000 q^{58} -294.000 q^{59} +450.000 q^{61} +276.000 q^{62} -167.000 q^{64} +30.0000 q^{65} -668.000 q^{67} -168.000 q^{68} +308.000 q^{71} +12.0000 q^{73} +210.000 q^{74} +798.000 q^{76} +596.000 q^{79} +205.000 q^{80} +444.000 q^{82} +966.000 q^{83} +120.000 q^{85} -492.000 q^{86} +660.000 q^{88} +408.000 q^{89} -364.000 q^{92} -612.000 q^{94} -570.000 q^{95} -1200.00 q^{97} +O(q^{100})$

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 $

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	15.0000	0.662913
$9$	0	0
$10$	−5.00000	−0.158114
$11$	44.0000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	44.0000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	6.00000	0.128008	0.0640039	−	0.997950i	$-0.479613\pi$
$13$	6.00000	0.128008	0.0640039	+	0.997950i	$0.479613\pi$
$14$	0	0
$15$	0	0
$16$	41.0000	0.640625
$17$	24.0000	0.342403	0.171202	−	0.985236i	$-0.445235\pi$
$17$	24.0000	0.342403	0.171202	+	0.985236i	$0.445235\pi$
$18$	0	0
$19$	−114.000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−114.000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−35.0000	−0.391312
$21$	0	0
$22$	−44.0000	−0.426401
$23$	52.0000	0.471424	0.235712	−	0.971823i	$-0.424258\pi$
$23$	52.0000	0.471424	0.235712	+	0.971823i	$0.424258\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−6.00000	−0.0452576
$27$	0	0
$28$	0	0
$29$	−146.000	−0.934880	−0.467440	−	0.884025i	$-0.654824\pi$
$29$	−146.000	−0.934880	−0.467440	+	0.884025i	$0.654824\pi$
$30$	0	0
$31$	−276.000	−1.59907	−0.799533	−	0.600622i	$-0.794920\pi$
$31$	−276.000	−1.59907	−0.799533	+	0.600622i	$0.794920\pi$
$32$	−161.000	−0.889408
$33$	0	0
$34$	−24.0000	−0.121058
$35$	0	0
$36$	0	0
$37$	−210.000	−0.933075	−0.466538	−	0.884501i	$-0.654499\pi$
$37$	−210.000	−0.933075	−0.466538	+	0.884501i	$0.654499\pi$
$38$	114.000	0.486664
$39$	0	0
$40$	75.0000	0.296464
$41$	−444.000	−1.69125	−0.845624	−	0.533779i	$-0.820771\pi$
$41$	−444.000	−1.69125	−0.845624	+	0.533779i	$0.820771\pi$
$42$	0	0
$43$	492.000	1.74487	0.872434	−	0.488733i	$-0.162541\pi$
$43$	492.000	1.74487	0.872434	+	0.488733i	$0.162541\pi$
$44$	−308.000	−1.05529
$45$	0	0
$46$	−52.0000	−0.166674
$47$	612.000	1.89935	0.949674	−	0.313239i	$-0.101414\pi$
$47$	612.000	1.89935	0.949674	+	0.313239i	$0.101414\pi$
$48$	0	0
$49$	0	0
$50$	−25.0000	−0.0707107
$51$	0	0
$52$	−42.0000	−0.112007
$53$	−50.0000	−0.129585	−0.0647927	−	0.997899i	$-0.520639\pi$
$53$	−50.0000	−0.129585	−0.0647927	+	0.997899i	$0.520639\pi$
$54$	0	0
$55$	220.000	0.539360
$56$	0	0
$57$	0	0
$58$	146.000	0.330530
$59$	−294.000	−0.648738	−0.324369	−	0.945931i	$-0.605152\pi$
$59$	−294.000	−0.648738	−0.324369	+	0.945931i	$0.605152\pi$
$60$	0	0
$61$	450.000	0.944534	0.472267	−	0.881455i	$-0.343436\pi$
$61$	450.000	0.944534	0.472267	+	0.881455i	$0.343436\pi$
$62$	276.000	0.565355
$63$	0	0
$64$	−167.000	−0.326172
$65$	30.0000	0.0572468
$66$	0	0
$67$	−668.000	−1.21805	−0.609024	−	0.793152i	$-0.708439\pi$
$67$	−668.000	−1.21805	−0.609024	+	0.793152i	$0.708439\pi$
$68$	−168.000	−0.299603
$69$	0	0
$70$	0	0
$71$	308.000	0.514829	0.257415	−	0.966301i	$-0.417129\pi$
$71$	308.000	0.514829	0.257415	+	0.966301i	$0.417129\pi$
$72$	0	0
$73$	12.0000	0.0192396	0.00961982	−	0.999954i	$-0.496938\pi$
$73$	12.0000	0.0192396	0.00961982	+	0.999954i	$0.496938\pi$
$74$	210.000	0.329892
$75$	0	0
$76$	798.000	1.20443
$77$	0	0
$78$	0	0
$79$	596.000	0.848800	0.424400	−	0.905475i	$-0.360485\pi$
$79$	596.000	0.848800	0.424400	+	0.905475i	$0.360485\pi$
$80$	205.000	0.286496
$81$	0	0
$82$	444.000	0.597946
$83$	966.000	1.27750	0.638749	−	0.769415i	$-0.279452\pi$
$83$	966.000	1.27750	0.638749	+	0.769415i	$0.279452\pi$
$84$	0	0
$85$	120.000	0.153127
$86$	−492.000	−0.616904
$87$	0	0
$88$	660.000	0.799503
$89$	408.000	0.485932	0.242966	−	0.970035i	$-0.421880\pi$
$89$	408.000	0.485932	0.242966	+	0.970035i	$0.421880\pi$
$90$	0	0
$91$	0	0
$92$	−364.000	−0.412496
$93$	0	0
$94$	−612.000	−0.671521
$95$	−570.000	−0.615587
$96$	0	0
$97$	−1200.00	−1.25610	−0.628049	−	0.778174i	$-0.716146\pi$
$97$	−1200.00	−1.25610	−0.628049	+	0.778174i	$0.716146\pi$
$98$	0	0
$99$	0	0
$100$	−175.000	−0.175000
$101$	−1098.00	−1.08173	−0.540867	−	0.841108i	$-0.681904\pi$
$101$	−1098.00	−1.08173	−0.540867	+	0.841108i	$0.681904\pi$
$102$	0	0
$103$	972.000	0.929845	0.464922	−	0.885351i	$-0.346082\pi$
$103$	972.000	0.929845	0.464922	+	0.885351i	$0.346082\pi$
$104$	90.0000	0.0848579
$105$	0	0
$106$	50.0000	0.0458154
$107$	−1516.00	−1.36969	−0.684847	−	0.728687i	$-0.740131\pi$
$107$	−1516.00	−1.36969	−0.684847	+	0.728687i	$0.740131\pi$
$108$	0	0
$109$	930.000	0.817228	0.408614	−	0.912707i	$-0.366012\pi$
$109$	930.000	0.817228	0.408614	+	0.912707i	$0.366012\pi$
$110$	−220.000	−0.190693
$111$	0	0
$112$	0	0
$113$	−1694.00	−1.41025	−0.705124	−	0.709084i	$-0.749109\pi$
$113$	−1694.00	−1.41025	−0.705124	+	0.709084i	$0.749109\pi$
$114$	0	0
$115$	260.000	0.210827
$116$	1022.00	0.818020
$117$	0	0
$118$	294.000	0.229364
$119$	0	0
$120$	0	0
$121$	605.000	0.454545
$122$	−450.000	−0.333943
$123$	0	0
$124$	1932.00	1.39918
$125$	125.000	0.0894427
$126$	0	0
$127$	916.000	0.640015	0.320007	−	0.947415i	$-0.396315\pi$
$127$	916.000	0.640015	0.320007	+	0.947415i	$0.396315\pi$
$128$	1455.00	1.00473
$129$	0	0
$130$	−30.0000	−0.0202398
$131$	1002.00	0.668284	0.334142	−	0.942523i	$-0.391554\pi$
$131$	1002.00	0.668284	0.334142	+	0.942523i	$0.391554\pi$
$132$	0	0
$133$	0	0
$134$	668.000	0.430645
$135$	0	0
$136$	360.000	0.226983
$137$	274.000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	274.000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−270.000	−0.164756	−0.0823781	−	0.996601i	$-0.526251\pi$
$139$	−270.000	−0.164756	−0.0823781	+	0.996601i	$0.526251\pi$
$140$	0	0
$141$	0	0
$142$	−308.000	−0.182020
$143$	264.000	0.154383
$144$	0	0
$145$	−730.000	−0.418091
$146$	−12.0000	−0.00680224
$147$	0	0
$148$	1470.00	0.816441
$149$	530.000	0.291405	0.145702	−	0.989328i	$-0.453456\pi$
$149$	530.000	0.291405	0.145702	+	0.989328i	$0.453456\pi$
$150$	0	0
$151$	−3120.00	−1.68147	−0.840735	−	0.541447i	$-0.817877\pi$
$151$	−3120.00	−1.68147	−0.840735	+	0.541447i	$0.817877\pi$
$152$	−1710.00	−0.912495
$153$	0	0
$154$	0	0
$155$	−1380.00	−0.715124
$156$	0	0
$157$	−2106.00	−1.07055	−0.535277	−	0.844676i	$-0.679793\pi$
$157$	−2106.00	−1.07055	−0.535277	+	0.844676i	$0.679793\pi$
$158$	−596.000	−0.300096
$159$	0	0
$160$	−805.000	−0.397755
$161$	0	0
$162$	0	0
$163$	628.000	0.301772	0.150886	−	0.988551i	$-0.451787\pi$
$163$	628.000	0.301772	0.150886	+	0.988551i	$0.451787\pi$
$164$	3108.00	1.47984
$165$	0	0
$166$	−966.000	−0.451663
$167$	−1284.00	−0.594963	−0.297482	−	0.954728i	$-0.596147\pi$
$167$	−1284.00	−0.594963	−0.297482	+	0.954728i	$0.596147\pi$
$168$	0	0
$169$	−2161.00	−0.983614
$170$	−120.000	−0.0541387
$171$	0	0
$172$	−3444.00	−1.52676
$173$	906.000	0.398161	0.199081	−	0.979983i	$-0.436204\pi$
$173$	906.000	0.398161	0.199081	+	0.979983i	$0.436204\pi$
$174$	0	0
$175$	0	0
$176$	1804.00	0.772623
$177$	0	0
$178$	−408.000	−0.171803
$179$	2084.00	0.870198	0.435099	−	0.900383i	$-0.356713\pi$
$179$	2084.00	0.870198	0.435099	+	0.900383i	$0.356713\pi$
$180$	0	0
$181$	−4674.00	−1.91942	−0.959712	−	0.280986i	$-0.909339\pi$
$181$	−4674.00	−1.91942	−0.959712	+	0.280986i	$0.909339\pi$
$182$	0	0
$183$	0	0
$184$	780.000	0.312513
$185$	−1050.00	−0.417284
$186$	0	0
$187$	1056.00	0.412954
$188$	−4284.00	−1.66193
$189$	0	0
$190$	570.000	0.217643
$191$	−2012.00	−0.762216	−0.381108	−	0.924531i	$-0.624457\pi$
$191$	−2012.00	−0.762216	−0.381108	+	0.924531i	$0.624457\pi$
$192$	0	0
$193$	4206.00	1.56868	0.784338	−	0.620334i	$-0.213003\pi$
$193$	4206.00	1.56868	0.784338	+	0.620334i	$0.213003\pi$
$194$	1200.00	0.444098
$195$	0	0
$196$	0	0
$197$	1574.00	0.569253	0.284627	−	0.958638i	$-0.408130\pi$
$197$	1574.00	0.569253	0.284627	+	0.958638i	$0.408130\pi$
$198$	0	0
$199$	−2724.00	−0.970348	−0.485174	−	0.874418i	$-0.661244\pi$
$199$	−2724.00	−0.970348	−0.485174	+	0.874418i	$0.661244\pi$
$200$	375.000	0.132583
$201$	0	0
$202$	1098.00	0.382451
$203$	0	0
$204$	0	0
$205$	−2220.00	−0.756349
$206$	−972.000	−0.328750
$207$	0	0
$208$	246.000	0.0820050
$209$	−5016.00	−1.66011
$210$	0	0
$211$	−180.000	−0.0587285	−0.0293642	−	0.999569i	$-0.509348\pi$
$211$	−180.000	−0.0587285	−0.0293642	+	0.999569i	$0.509348\pi$
$212$	350.000	0.113387
$213$	0	0
$214$	1516.00	0.484260
$215$	2460.00	0.780328
$216$	0	0
$217$	0	0
$218$	−930.000	−0.288934
$219$	0	0
$220$	−1540.00	−0.471940
$221$	144.000	0.0438303
$222$	0	0
$223$	−4584.00	−1.37654	−0.688268	−	0.725457i	$-0.741628\pi$
$223$	−4584.00	−1.37654	−0.688268	+	0.725457i	$0.741628\pi$
$224$	0	0
$225$	0	0
$226$	1694.00	0.498598
$227$	1686.00	0.492968	0.246484	−	0.969147i	$-0.420725\pi$
$227$	1686.00	0.492968	0.246484	+	0.969147i	$0.420725\pi$
$228$	0	0
$229$	−4026.00	−1.16177	−0.580886	−	0.813985i	$-0.697294\pi$
$229$	−4026.00	−1.16177	−0.580886	+	0.813985i	$0.697294\pi$
$230$	−260.000	−0.0745387
$231$	0	0
$232$	−2190.00	−0.619744
$233$	6074.00	1.70782	0.853908	−	0.520425i	$-0.174226\pi$
$233$	6074.00	1.70782	0.853908	+	0.520425i	$0.174226\pi$
$234$	0	0
$235$	3060.00	0.849414
$236$	2058.00	0.567646
$237$	0	0
$238$	0	0
$239$	−3928.00	−1.06310	−0.531551	−	0.847027i	$-0.678390\pi$
$239$	−3928.00	−1.06310	−0.531551	+	0.847027i	$0.678390\pi$
$240$	0	0
$241$	−1236.00	−0.330364	−0.165182	−	0.986263i	$-0.552821\pi$
$241$	−1236.00	−0.330364	−0.165182	+	0.986263i	$0.552821\pi$
$242$	−605.000	−0.160706
$243$	0	0
$244$	−3150.00	−0.826468
$245$	0	0
$246$	0	0
$247$	−684.000	−0.176202
$248$	−4140.00	−1.06004
$249$	0	0
$250$	−125.000	−0.0316228
$251$	78.0000	0.0196148	0.00980740	−	0.999952i	$-0.496878\pi$
$251$	78.0000	0.0196148	0.00980740	+	0.999952i	$0.496878\pi$
$252$	0	0
$253$	2288.00	0.568559
$254$	−916.000	−0.226279
$255$	0	0
$256$	−119.000	−0.0290527
$257$	3276.00	0.795141	0.397571	−	0.917572i	$-0.369853\pi$
$257$	3276.00	0.795141	0.397571	+	0.917572i	$0.369853\pi$
$258$	0	0
$259$	0	0
$260$	−210.000	−0.0500910
$261$	0	0
$262$	−1002.00	−0.236274
$263$	−2240.00	−0.525188	−0.262594	−	0.964906i	$-0.584578\pi$
$263$	−2240.00	−0.525188	−0.262594	+	0.964906i	$0.584578\pi$
$264$	0	0
$265$	−250.000	−0.0579524
$266$	0	0
$267$	0	0
$268$	4676.00	1.06579
$269$	−4494.00	−1.01860	−0.509301	−	0.860588i	$-0.670096\pi$
$269$	−4494.00	−1.01860	−0.509301	+	0.860588i	$0.670096\pi$
$270$	0	0
$271$	3216.00	0.720879	0.360439	−	0.932783i	$-0.382627\pi$
$271$	3216.00	0.720879	0.360439	+	0.932783i	$0.382627\pi$
$272$	984.000	0.219352
$273$	0	0
$274$	−274.000	−0.0604122
$275$	1100.00	0.241209
$276$	0	0
$277$	1514.00	0.328402	0.164201	−	0.986427i	$-0.447495\pi$
$277$	1514.00	0.328402	0.164201	+	0.986427i	$0.447495\pi$
$278$	270.000	0.0582501
$279$	0	0
$280$	0	0
$281$	5690.00	1.20796	0.603980	−	0.796999i	$-0.293581\pi$
$281$	5690.00	1.20796	0.603980	+	0.796999i	$0.293581\pi$
$282$	0	0
$283$	−7518.00	−1.57915	−0.789574	−	0.613656i	$-0.789698\pi$
$283$	−7518.00	−1.57915	−0.789574	+	0.613656i	$0.789698\pi$
$284$	−2156.00	−0.450476
$285$	0	0
$286$	−264.000	−0.0545827
$287$	0	0
$288$	0	0
$289$	−4337.00	−0.882760
$290$	730.000	0.147818
$291$	0	0
$292$	−84.0000	−0.0168347
$293$	702.000	0.139970	0.0699851	−	0.997548i	$-0.477705\pi$
$293$	702.000	0.139970	0.0699851	+	0.997548i	$0.477705\pi$
$294$	0	0
$295$	−1470.00	−0.290124
$296$	−3150.00	−0.618547
$297$	0	0
$298$	−530.000	−0.103027
$299$	312.000	0.0603459
$300$	0	0
$301$	0	0
$302$	3120.00	0.594489
$303$	0	0
$304$	−4674.00	−0.881817
$305$	2250.00	0.422409
$306$	0	0
$307$	−10374.0	−1.92858	−0.964292	−	0.264840i	$-0.914681\pi$
$307$	−10374.0	−1.92858	−0.964292	+	0.264840i	$0.914681\pi$
$308$	0	0
$309$	0	0
$310$	1380.00	0.252835
$311$	2784.00	0.507608	0.253804	−	0.967256i	$-0.418318\pi$
$311$	2784.00	0.507608	0.253804	+	0.967256i	$0.418318\pi$
$312$	0	0
$313$	6216.00	1.12252	0.561261	−	0.827639i	$-0.310317\pi$
$313$	6216.00	1.12252	0.561261	+	0.827639i	$0.310317\pi$
$314$	2106.00	0.378498
$315$	0	0
$316$	−4172.00	−0.742700
$317$	−5066.00	−0.897586	−0.448793	−	0.893636i	$-0.648146\pi$
$317$	−5066.00	−0.897586	−0.448793	+	0.893636i	$0.648146\pi$
$318$	0	0
$319$	−6424.00	−1.12751
$320$	−835.000	−0.145868
$321$	0	0
$322$	0	0
$323$	−2736.00	−0.471316
$324$	0	0
$325$	150.000	0.0256015
$326$	−628.000	−0.106692
$327$	0	0
$328$	−6660.00	−1.12115
$329$	0	0
$330$	0	0
$331$	−6468.00	−1.07406	−0.537029	−	0.843564i	$-0.680454\pi$
$331$	−6468.00	−1.07406	−0.537029	+	0.843564i	$0.680454\pi$
$332$	−6762.00	−1.11781
$333$	0	0
$334$	1284.00	0.210351
$335$	−3340.00	−0.544727
$336$	0	0
$337$	−3438.00	−0.555726	−0.277863	−	0.960621i	$-0.589626\pi$
$337$	−3438.00	−0.555726	−0.277863	+	0.960621i	$0.589626\pi$
$338$	2161.00	0.347760
$339$	0	0
$340$	−840.000	−0.133986
$341$	−12144.0	−1.92855
$342$	0	0
$343$	0	0
$344$	7380.00	1.15669
$345$	0	0
$346$	−906.000	−0.140771
$347$	−2212.00	−0.342209	−0.171104	−	0.985253i	$-0.554734\pi$
$347$	−2212.00	−0.342209	−0.171104	+	0.985253i	$0.554734\pi$
$348$	0	0
$349$	2910.00	0.446329	0.223164	−	0.974781i	$-0.428361\pi$
$349$	2910.00	0.446329	0.223164	+	0.974781i	$0.428361\pi$
$350$	0	0
$351$	0	0
$352$	−7084.00	−1.07267
$353$	−8364.00	−1.26111	−0.630554	−	0.776146i	$-0.717172\pi$
$353$	−8364.00	−1.26111	−0.630554	+	0.776146i	$0.717172\pi$
$354$	0	0
$355$	1540.00	0.230239
$356$	−2856.00	−0.425190
$357$	0	0
$358$	−2084.00	−0.307662
$359$	6712.00	0.986757	0.493379	−	0.869815i	$-0.335762\pi$
$359$	6712.00	0.986757	0.493379	+	0.869815i	$0.335762\pi$
$360$	0	0
$361$	6137.00	0.894737
$362$	4674.00	0.678619
$363$	0	0
$364$	0	0
$365$	60.0000	0.00860423
$366$	0	0
$367$	456.000	0.0648583	0.0324292	−	0.999474i	$-0.489676\pi$
$367$	456.000	0.0648583	0.0324292	+	0.999474i	$0.489676\pi$
$368$	2132.00	0.302006
$369$	0	0
$370$	1050.00	0.147532
$371$	0	0
$372$	0	0
$373$	−2558.00	−0.355089	−0.177545	−	0.984113i	$-0.556815\pi$
$373$	−2558.00	−0.355089	−0.177545	+	0.984113i	$0.556815\pi$
$374$	−1056.00	−0.146001
$375$	0	0
$376$	9180.00	1.25910
$377$	−876.000	−0.119672
$378$	0	0
$379$	2004.00	0.271606	0.135803	−	0.990736i	$-0.456639\pi$
$379$	2004.00	0.271606	0.135803	+	0.990736i	$0.456639\pi$
$380$	3990.00	0.538639
$381$	0	0
$382$	2012.00	0.269484
$383$	−11340.0	−1.51292	−0.756458	−	0.654042i	$-0.773072\pi$
$383$	−11340.0	−1.51292	−0.756458	+	0.654042i	$0.773072\pi$
$384$	0	0
$385$	0	0
$386$	−4206.00	−0.554611
$387$	0	0
$388$	8400.00	1.09909
$389$	−10522.0	−1.37143	−0.685715	−	0.727870i	$-0.740511\pi$
$389$	−10522.0	−1.37143	−0.685715	+	0.727870i	$0.740511\pi$
$390$	0	0
$391$	1248.00	0.161417
$392$	0	0
$393$	0	0
$394$	−1574.00	−0.201261
$395$	2980.00	0.379595
$396$	0	0
$397$	−2898.00	−0.366364	−0.183182	−	0.983079i	$-0.558640\pi$
$397$	−2898.00	−0.366364	−0.183182	+	0.983079i	$0.558640\pi$
$398$	2724.00	0.343070
$399$	0	0
$400$	1025.00	0.128125
$401$	−3026.00	−0.376836	−0.188418	−	0.982089i	$-0.560336\pi$
$401$	−3026.00	−0.376836	−0.188418	+	0.982089i	$0.560336\pi$
$402$	0	0
$403$	−1656.00	−0.204693
$404$	7686.00	0.946517
$405$	0	0
$406$	0	0
$407$	−9240.00	−1.12533
$408$	0	0
$409$	−8940.00	−1.08082	−0.540409	−	0.841402i	$-0.681730\pi$
$409$	−8940.00	−1.08082	−0.540409	+	0.841402i	$0.681730\pi$
$410$	2220.00	0.267410
$411$	0	0
$412$	−6804.00	−0.813614
$413$	0	0
$414$	0	0
$415$	4830.00	0.571314
$416$	−966.000	−0.113851
$417$	0	0
$418$	5016.00	0.586939
$419$	−2994.00	−0.349085	−0.174542	−	0.984650i	$-0.555845\pi$
$419$	−2994.00	−0.349085	−0.174542	+	0.984650i	$0.555845\pi$
$420$	0	0
$421$	15766.0	1.82515	0.912575	−	0.408910i	$-0.134091\pi$
$421$	15766.0	1.82515	0.912575	+	0.408910i	$0.134091\pi$
$422$	180.000	0.0207637
$423$	0	0
$424$	−750.000	−0.0859038
$425$	600.000	0.0684806
$426$	0	0
$427$	0	0
$428$	10612.0	1.19848
$429$	0	0
$430$	−2460.00	−0.275888
$431$	−14720.0	−1.64510	−0.822549	−	0.568694i	$-0.807449\pi$
$431$	−14720.0	−1.64510	−0.822549	+	0.568694i	$0.807449\pi$
$432$	0	0
$433$	4440.00	0.492778	0.246389	−	0.969171i	$-0.420756\pi$
$433$	4440.00	0.492778	0.246389	+	0.969171i	$0.420756\pi$
$434$	0	0
$435$	0	0
$436$	−6510.00	−0.715074
$437$	−5928.00	−0.648912
$438$	0	0
$439$	−1488.00	−0.161773	−0.0808865	−	0.996723i	$-0.525775\pi$
$439$	−1488.00	−0.161773	−0.0808865	+	0.996723i	$0.525775\pi$
$440$	3300.00	0.357548
$441$	0	0
$442$	−144.000	−0.0154963
$443$	5828.00	0.625049	0.312524	−	0.949910i	$-0.398825\pi$
$443$	5828.00	0.625049	0.312524	+	0.949910i	$0.398825\pi$
$444$	0	0
$445$	2040.00	0.217315
$446$	4584.00	0.486679
$447$	0	0
$448$	0	0
$449$	6958.00	0.731333	0.365666	−	0.930746i	$-0.380841\pi$
$449$	6958.00	0.731333	0.365666	+	0.930746i	$0.380841\pi$
$450$	0	0
$451$	−19536.0	−2.03972
$452$	11858.0	1.23397
$453$	0	0
$454$	−1686.00	−0.174291
$455$	0	0
$456$	0	0
$457$	18102.0	1.85290	0.926451	−	0.376416i	$-0.122844\pi$
$457$	18102.0	1.85290	0.926451	+	0.376416i	$0.122844\pi$
$458$	4026.00	0.410748
$459$	0	0
$460$	−1820.00	−0.184474
$461$	−2574.00	−0.260050	−0.130025	−	0.991511i	$-0.541506\pi$
$461$	−2574.00	−0.260050	−0.130025	+	0.991511i	$0.541506\pi$
$462$	0	0
$463$	−12832.0	−1.28802	−0.644010	−	0.765017i	$-0.722731\pi$
$463$	−12832.0	−1.28802	−0.644010	+	0.765017i	$0.722731\pi$
$464$	−5986.00	−0.598907
$465$	0	0
$466$	−6074.00	−0.603804
$467$	−7170.00	−0.710467	−0.355233	−	0.934778i	$-0.615599\pi$
$467$	−7170.00	−0.710467	−0.355233	+	0.934778i	$0.615599\pi$
$468$	0	0
$469$	0	0
$470$	−3060.00	−0.300313
$471$	0	0
$472$	−4410.00	−0.430057
$473$	21648.0	2.10439
$474$	0	0
$475$	−2850.00	−0.275299
$476$	0	0
$477$	0	0
$478$	3928.00	0.375863
$479$	10164.0	0.969530	0.484765	−	0.874644i	$-0.338905\pi$
$479$	10164.0	0.969530	0.484765	+	0.874644i	$0.338905\pi$
$480$	0	0
$481$	−1260.00	−0.119441
$482$	1236.00	0.116801
$483$	0	0
$484$	−4235.00	−0.397727
$485$	−6000.00	−0.561744
$486$	0	0
$487$	10212.0	0.950205	0.475103	−	0.879930i	$-0.342411\pi$
$487$	10212.0	0.950205	0.475103	+	0.879930i	$0.342411\pi$
$488$	6750.00	0.626144
$489$	0	0
$490$	0	0
$491$	7972.00	0.732732	0.366366	−	0.930471i	$-0.380602\pi$
$491$	7972.00	0.732732	0.366366	+	0.930471i	$0.380602\pi$
$492$	0	0
$493$	−3504.00	−0.320106
$494$	684.000	0.0622968
$495$	0	0
$496$	−11316.0	−1.02440
$497$	0	0
$498$	0	0
$499$	−1548.00	−0.138874	−0.0694369	−	0.997586i	$-0.522120\pi$
$499$	−1548.00	−0.138874	−0.0694369	+	0.997586i	$0.522120\pi$
$500$	−875.000	−0.0782624
$501$	0	0
$502$	−78.0000	−0.00693488
$503$	−1368.00	−0.121265	−0.0606323	−	0.998160i	$-0.519312\pi$
$503$	−1368.00	−0.121265	−0.0606323	+	0.998160i	$0.519312\pi$
$504$	0	0
$505$	−5490.00	−0.483766
$506$	−2288.00	−0.201016
$507$	0	0
$508$	−6412.00	−0.560013
$509$	8274.00	0.720508	0.360254	−	0.932854i	$-0.382690\pi$
$509$	8274.00	0.720508	0.360254	+	0.932854i	$0.382690\pi$
$510$	0	0
$511$	0	0
$512$	−11521.0	−0.994455
$513$	0	0
$514$	−3276.00	−0.281125
$515$	4860.00	0.415839
$516$	0	0
$517$	26928.0	2.29070
$518$	0	0
$519$	0	0
$520$	450.000	0.0379496
$521$	−20268.0	−1.70433	−0.852166	−	0.523271i	$-0.824711\pi$
$521$	−20268.0	−1.70433	−0.852166	+	0.523271i	$0.824711\pi$
$522$	0	0
$523$	−1302.00	−0.108858	−0.0544288	−	0.998518i	$-0.517334\pi$
$523$	−1302.00	−0.108858	−0.0544288	+	0.998518i	$0.517334\pi$
$524$	−7014.00	−0.584748
$525$	0	0
$526$	2240.00	0.185682
$527$	−6624.00	−0.547526
$528$	0	0
$529$	−9463.00	−0.777760
$530$	250.000	0.0204893
$531$	0	0
$532$	0	0
$533$	−2664.00	−0.216493
$534$	0	0
$535$	−7580.00	−0.612546
$536$	−10020.0	−0.807459
$537$	0	0
$538$	4494.00	0.360130
$539$	0	0
$540$	0	0
$541$	−8314.00	−0.660715	−0.330357	−	0.943856i	$-0.607169\pi$
$541$	−8314.00	−0.660715	−0.330357	+	0.943856i	$0.607169\pi$
$542$	−3216.00	−0.254869
$543$	0	0
$544$	−3864.00	−0.304536
$545$	4650.00	0.365475
$546$	0	0
$547$	−9484.00	−0.741328	−0.370664	−	0.928767i	$-0.620870\pi$
$547$	−9484.00	−0.741328	−0.370664	+	0.928767i	$0.620870\pi$
$548$	−1918.00	−0.149513
$549$	0	0
$550$	−1100.00	−0.0852803
$551$	16644.0	1.28686
$552$	0	0
$553$	0	0
$554$	−1514.00	−0.116108
$555$	0	0
$556$	1890.00	0.144162
$557$	−23218.0	−1.76621	−0.883104	−	0.469177i	$-0.844551\pi$
$557$	−23218.0	−1.76621	−0.883104	+	0.469177i	$0.844551\pi$
$558$	0	0
$559$	2952.00	0.223357
$560$	0	0
$561$	0	0
$562$	−5690.00	−0.427079
$563$	−5334.00	−0.399292	−0.199646	−	0.979868i	$-0.563979\pi$
$563$	−5334.00	−0.399292	−0.199646	+	0.979868i	$0.563979\pi$
$564$	0	0
$565$	−8470.00	−0.630682
$566$	7518.00	0.558313
$567$	0	0
$568$	4620.00	0.341287
$569$	182.000	0.0134092	0.00670460	−	0.999978i	$-0.497866\pi$
$569$	182.000	0.0134092	0.00670460	+	0.999978i	$0.497866\pi$
$570$	0	0
$571$	14164.0	1.03808	0.519041	−	0.854749i	$-0.326289\pi$
$571$	14164.0	1.03808	0.519041	+	0.854749i	$0.326289\pi$
$572$	−1848.00	−0.135085
$573$	0	0
$574$	0	0
$575$	1300.00	0.0942848
$576$	0	0
$577$	−13740.0	−0.991341	−0.495670	−	0.868511i	$-0.665078\pi$
$577$	−13740.0	−0.991341	−0.495670	+	0.868511i	$0.665078\pi$
$578$	4337.00	0.312103
$579$	0	0
$580$	5110.00	0.365830
$581$	0	0
$582$	0	0
$583$	−2200.00	−0.156286
$584$	180.000	0.0127542
$585$	0	0
$586$	−702.000	−0.0494870
$587$	−9174.00	−0.645062	−0.322531	−	0.946559i	$-0.604534\pi$
$587$	−9174.00	−0.645062	−0.322531	+	0.946559i	$0.604534\pi$
$588$	0	0
$589$	31464.0	2.20111
$590$	1470.00	0.102574
$591$	0	0
$592$	−8610.00	−0.597751
$593$	−14580.0	−1.00966	−0.504830	−	0.863219i	$-0.668445\pi$
$593$	−14580.0	−1.00966	−0.504830	+	0.863219i	$0.668445\pi$
$594$	0	0
$595$	0	0
$596$	−3710.00	−0.254979
$597$	0	0
$598$	−312.000	−0.0213355
$599$	−1988.00	−0.135605	−0.0678026	−	0.997699i	$-0.521599\pi$
$599$	−1988.00	−0.135605	−0.0678026	+	0.997699i	$0.521599\pi$
$600$	0	0
$601$	−7800.00	−0.529399	−0.264699	−	0.964331i	$-0.585273\pi$
$601$	−7800.00	−0.529399	−0.264699	+	0.964331i	$0.585273\pi$
$602$	0	0
$603$	0	0
$604$	21840.0	1.47129
$605$	3025.00	0.203279
$606$	0	0
$607$	24288.0	1.62408	0.812042	−	0.583598i	$-0.198356\pi$
$607$	24288.0	1.62408	0.812042	+	0.583598i	$0.198356\pi$
$608$	18354.0	1.22426
$609$	0	0
$610$	−2250.00	−0.149344
$611$	3672.00	0.243131
$612$	0	0
$613$	−9866.00	−0.650055	−0.325028	−	0.945704i	$-0.605374\pi$
$613$	−9866.00	−0.650055	−0.325028	+	0.945704i	$0.605374\pi$
$614$	10374.0	0.681858
$615$	0	0
$616$	0	0
$617$	−22858.0	−1.49146	−0.745728	−	0.666250i	$-0.767898\pi$
$617$	−22858.0	−1.49146	−0.745728	+	0.666250i	$0.767898\pi$
$618$	0	0
$619$	−19074.0	−1.23853	−0.619264	−	0.785183i	$-0.712569\pi$
$619$	−19074.0	−1.23853	−0.619264	+	0.785183i	$0.712569\pi$
$620$	9660.00	0.625734
$621$	0	0
$622$	−2784.00	−0.179467
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−6216.00	−0.396871
$627$	0	0
$628$	14742.0	0.936735
$629$	−5040.00	−0.319488
$630$	0	0
$631$	22084.0	1.39326	0.696632	−	0.717428i	$-0.254681\pi$
$631$	22084.0	1.39326	0.696632	+	0.717428i	$0.254681\pi$
$632$	8940.00	0.562681
$633$	0	0
$634$	5066.00	0.317345
$635$	4580.00	0.286223
$636$	0	0
$637$	0	0
$638$	6424.00	0.398634
$639$	0	0
$640$	7275.00	0.449328
$641$	16622.0	1.02423	0.512114	−	0.858918i	$-0.328863\pi$
$641$	16622.0	1.02423	0.512114	+	0.858918i	$0.328863\pi$
$642$	0	0
$643$	−12906.0	−0.791544	−0.395772	−	0.918349i	$-0.629523\pi$
$643$	−12906.0	−0.791544	−0.395772	+	0.918349i	$0.629523\pi$
$644$	0	0
$645$	0	0
$646$	2736.00	0.166635
$647$	−3156.00	−0.191770	−0.0958850	−	0.995392i	$-0.530568\pi$
$647$	−3156.00	−0.191770	−0.0958850	+	0.995392i	$0.530568\pi$
$648$	0	0
$649$	−12936.0	−0.782407
$650$	−150.000	−0.00905151
$651$	0	0
$652$	−4396.00	−0.264050
$653$	3658.00	0.219217	0.109608	−	0.993975i	$-0.465040\pi$
$653$	3658.00	0.219217	0.109608	+	0.993975i	$0.465040\pi$
$654$	0	0
$655$	5010.00	0.298865
$656$	−18204.0	−1.08346
$657$	0	0
$658$	0	0
$659$	12316.0	0.728017	0.364009	−	0.931396i	$-0.381408\pi$
$659$	12316.0	0.728017	0.364009	+	0.931396i	$0.381408\pi$
$660$	0	0
$661$	32298.0	1.90052	0.950262	−	0.311451i	$-0.100815\pi$
$661$	32298.0	1.90052	0.950262	+	0.311451i	$0.100815\pi$
$662$	6468.00	0.379737
$663$	0	0
$664$	14490.0	0.846869
$665$	0	0
$666$	0	0
$667$	−7592.00	−0.440725
$668$	8988.00	0.520593
$669$	0	0
$670$	3340.00	0.192590
$671$	19800.0	1.13915
$672$	0	0
$673$	−23274.0	−1.33306	−0.666528	−	0.745480i	$-0.732220\pi$
$673$	−23274.0	−1.33306	−0.666528	+	0.745480i	$0.732220\pi$
$674$	3438.00	0.196479
$675$	0	0
$676$	15127.0	0.860662
$677$	−4518.00	−0.256486	−0.128243	−	0.991743i	$-0.540934\pi$
$677$	−4518.00	−0.256486	−0.128243	+	0.991743i	$0.540934\pi$
$678$	0	0
$679$	0	0
$680$	1800.00	0.101510
$681$	0	0
$682$	12144.0	0.681844
$683$	19636.0	1.10007	0.550037	−	0.835140i	$-0.314614\pi$
$683$	19636.0	1.10007	0.550037	+	0.835140i	$0.314614\pi$
$684$	0	0
$685$	1370.00	0.0764161
$686$	0	0
$687$	0	0
$688$	20172.0	1.11781
$689$	−300.000	−0.0165879
$690$	0	0
$691$	−17226.0	−0.948347	−0.474174	−	0.880431i	$-0.657253\pi$
$691$	−17226.0	−0.948347	−0.474174	+	0.880431i	$0.657253\pi$
$692$	−6342.00	−0.348391
$693$	0	0
$694$	2212.00	0.120989
$695$	−1350.00	−0.0736812
$696$	0	0
$697$	−10656.0	−0.579089
$698$	−2910.00	−0.157801
$699$	0	0
$700$	0	0
$701$	−21362.0	−1.15097	−0.575486	−	0.817812i	$-0.695187\pi$
$701$	−21362.0	−1.15097	−0.575486	+	0.817812i	$0.695187\pi$
$702$	0	0
$703$	23940.0	1.28437
$704$	−7348.00	−0.393378
$705$	0	0
$706$	8364.00	0.445869
$707$	0	0
$708$	0	0
$709$	8658.00	0.458615	0.229307	−	0.973354i	$-0.426354\pi$
$709$	8658.00	0.458615	0.229307	+	0.973354i	$0.426354\pi$
$710$	−1540.00	−0.0814016
$711$	0	0
$712$	6120.00	0.322130
$713$	−14352.0	−0.753838
$714$	0	0
$715$	1320.00	0.0690422
$716$	−14588.0	−0.761423
$717$	0	0
$718$	−6712.00	−0.348871
$719$	−29484.0	−1.52930	−0.764651	−	0.644445i	$-0.777088\pi$
$719$	−29484.0	−1.52930	−0.764651	+	0.644445i	$0.777088\pi$
$720$	0	0
$721$	0	0
$722$	−6137.00	−0.316337
$723$	0	0
$724$	32718.0	1.67950
$725$	−3650.00	−0.186976
$726$	0	0
$727$	−28260.0	−1.44169	−0.720843	−	0.693099i	$-0.756245\pi$
$727$	−28260.0	−1.44169	−0.720843	+	0.693099i	$0.756245\pi$
$728$	0	0
$729$	0	0
$730$	−60.0000	−0.00304205
$731$	11808.0	0.597448
$732$	0	0
$733$	−10950.0	−0.551770	−0.275885	−	0.961191i	$-0.588971\pi$
$733$	−10950.0	−0.551770	−0.275885	+	0.961191i	$0.588971\pi$
$734$	−456.000	−0.0229309
$735$	0	0
$736$	−8372.00	−0.419288
$737$	−29392.0	−1.46902
$738$	0	0
$739$	−9772.00	−0.486426	−0.243213	−	0.969973i	$-0.578201\pi$
$739$	−9772.00	−0.486426	−0.243213	+	0.969973i	$0.578201\pi$
$740$	7350.00	0.365123
$741$	0	0
$742$	0	0
$743$	−7844.00	−0.387306	−0.193653	−	0.981070i	$-0.562034\pi$
$743$	−7844.00	−0.387306	−0.193653	+	0.981070i	$0.562034\pi$
$744$	0	0
$745$	2650.00	0.130320
$746$	2558.00	0.125543
$747$	0	0
$748$	−7392.00	−0.361335
$749$	0	0
$750$	0	0
$751$	1800.00	0.0874606	0.0437303	−	0.999043i	$-0.486076\pi$
$751$	1800.00	0.0874606	0.0437303	+	0.999043i	$0.486076\pi$
$752$	25092.0	1.21677
$753$	0	0
$754$	876.000	0.0423104
$755$	−15600.0	−0.751976
$756$	0	0
$757$	−11274.0	−0.541295	−0.270648	−	0.962678i	$-0.587238\pi$
$757$	−11274.0	−0.541295	−0.270648	+	0.962678i	$0.587238\pi$
$758$	−2004.00	−0.0960271
$759$	0	0
$760$	−8550.00	−0.408080
$761$	−22668.0	−1.07978	−0.539891	−	0.841735i	$-0.681535\pi$
$761$	−22668.0	−1.07978	−0.539891	+	0.841735i	$0.681535\pi$
$762$	0	0
$763$	0	0
$764$	14084.0	0.666939
$765$	0	0
$766$	11340.0	0.534897
$767$	−1764.00	−0.0830435
$768$	0	0
$769$	−15468.0	−0.725345	−0.362673	−	0.931917i	$-0.618136\pi$
$769$	−15468.0	−0.725345	−0.362673	+	0.931917i	$0.618136\pi$
$770$	0	0
$771$	0	0
$772$	−29442.0	−1.37259
$773$	−28986.0	−1.34871	−0.674356	−	0.738407i	$-0.735579\pi$
$773$	−28986.0	−1.34871	−0.674356	+	0.738407i	$0.735579\pi$
$774$	0	0
$775$	−6900.00	−0.319813
$776$	−18000.0	−0.832683
$777$	0	0
$778$	10522.0	0.484874
$779$	50616.0	2.32799
$780$	0	0
$781$	13552.0	0.620907
$782$	−1248.00	−0.0570696
$783$	0	0
$784$	0	0
$785$	−10530.0	−0.478767
$786$	0	0
$787$	20562.0	0.931329	0.465665	−	0.884961i	$-0.345815\pi$
$787$	20562.0	0.931329	0.465665	+	0.884961i	$0.345815\pi$
$788$	−11018.0	−0.498096
$789$	0	0
$790$	−2980.00	−0.134207
$791$	0	0
$792$	0	0
$793$	2700.00	0.120908
$794$	2898.00	0.129529
$795$	0	0
$796$	19068.0	0.849054
$797$	20826.0	0.925589	0.462795	−	0.886465i	$-0.346847\pi$
$797$	20826.0	0.925589	0.462795	+	0.886465i	$0.346847\pi$
$798$	0	0
$799$	14688.0	0.650343
$800$	−4025.00	−0.177882
$801$	0	0
$802$	3026.00	0.133232
$803$	528.000	0.0232039
$804$	0	0
$805$	0	0
$806$	1656.00	0.0723699
$807$	0	0
$808$	−16470.0	−0.717095
$809$	4250.00	0.184700	0.0923498	−	0.995727i	$-0.470562\pi$
$809$	4250.00	0.184700	0.0923498	+	0.995727i	$0.470562\pi$
$810$	0	0
$811$	24342.0	1.05396	0.526981	−	0.849877i	$-0.323324\pi$
$811$	24342.0	1.05396	0.526981	+	0.849877i	$0.323324\pi$
$812$	0	0
$813$	0	0
$814$	9240.00	0.397865
$815$	3140.00	0.134956
$816$	0	0
$817$	−56088.0	−2.40180
$818$	8940.00	0.382127
$819$	0	0
$820$	15540.0	0.661805
$821$	2138.00	0.0908852	0.0454426	−	0.998967i	$-0.485530\pi$
$821$	2138.00	0.0908852	0.0454426	+	0.998967i	$0.485530\pi$
$822$	0	0
$823$	9560.00	0.404910	0.202455	−	0.979292i	$-0.435108\pi$
$823$	9560.00	0.404910	0.202455	+	0.979292i	$0.435108\pi$
$824$	14580.0	0.616406
$825$	0	0
$826$	0	0
$827$	−7748.00	−0.325785	−0.162893	−	0.986644i	$-0.552082\pi$
$827$	−7748.00	−0.325785	−0.162893	+	0.986644i	$0.552082\pi$
$828$	0	0
$829$	−5334.00	−0.223471	−0.111736	−	0.993738i	$-0.535641\pi$
$829$	−5334.00	−0.223471	−0.111736	+	0.993738i	$0.535641\pi$
$830$	−4830.00	−0.201990
$831$	0	0
$832$	−1002.00	−0.0417525
$833$	0	0
$834$	0	0
$835$	−6420.00	−0.266076
$836$	35112.0	1.45260
$837$	0	0
$838$	2994.00	0.123420
$839$	36444.0	1.49963	0.749813	−	0.661650i	$-0.230143\pi$
$839$	36444.0	1.49963	0.749813	+	0.661650i	$0.230143\pi$
$840$	0	0
$841$	−3073.00	−0.125999
$842$	−15766.0	−0.645288
$843$	0	0
$844$	1260.00	0.0513874
$845$	−10805.0	−0.439886
$846$	0	0
$847$	0	0
$848$	−2050.00	−0.0830157
$849$	0	0
$850$	−600.000	−0.0242116
$851$	−10920.0	−0.439874
$852$	0	0
$853$	21030.0	0.844142	0.422071	−	0.906563i	$-0.361303\pi$
$853$	21030.0	0.844142	0.422071	+	0.906563i	$0.361303\pi$
$854$	0	0
$855$	0	0
$856$	−22740.0	−0.907987
$857$	32328.0	1.28857	0.644284	−	0.764786i	$-0.277155\pi$
$857$	32328.0	1.28857	0.644284	+	0.764786i	$0.277155\pi$
$858$	0	0
$859$	−43518.0	−1.72854	−0.864269	−	0.503029i	$-0.832219\pi$
$859$	−43518.0	−1.72854	−0.864269	+	0.503029i	$0.832219\pi$
$860$	−17220.0	−0.682787
$861$	0	0
$862$	14720.0	0.581630
$863$	5032.00	0.198484	0.0992418	−	0.995063i	$-0.468358\pi$
$863$	5032.00	0.198484	0.0992418	+	0.995063i	$0.468358\pi$
$864$	0	0
$865$	4530.00	0.178063
$866$	−4440.00	−0.174223
$867$	0	0
$868$	0	0
$869$	26224.0	1.02369
$870$	0	0
$871$	−4008.00	−0.155920
$872$	13950.0	0.541751
$873$	0	0
$874$	5928.00	0.229425
$875$	0	0
$876$	0	0
$877$	4286.00	0.165026	0.0825131	−	0.996590i	$-0.473705\pi$
$877$	4286.00	0.165026	0.0825131	+	0.996590i	$0.473705\pi$
$878$	1488.00	0.0571954
$879$	0	0
$880$	9020.00	0.345527
$881$	−19080.0	−0.729650	−0.364825	−	0.931076i	$-0.618871\pi$
$881$	−19080.0	−0.729650	−0.364825	+	0.931076i	$0.618871\pi$
$882$	0	0
$883$	26580.0	1.01301	0.506505	−	0.862237i	$-0.330937\pi$
$883$	26580.0	1.01301	0.506505	+	0.862237i	$0.330937\pi$
$884$	−1008.00	−0.0383515
$885$	0	0
$886$	−5828.00	−0.220988
$887$	−12588.0	−0.476509	−0.238255	−	0.971203i	$-0.576575\pi$
$887$	−12588.0	−0.476509	−0.238255	+	0.971203i	$0.576575\pi$
$888$	0	0
$889$	0	0
$890$	−2040.00	−0.0768325
$891$	0	0
$892$	32088.0	1.20447
$893$	−69768.0	−2.61444
$894$	0	0
$895$	10420.0	0.389164
$896$	0	0
$897$	0	0
$898$	−6958.00	−0.258565
$899$	40296.0	1.49494
$900$	0	0
$901$	−1200.00	−0.0443705
$902$	19536.0	0.721150
$903$	0	0
$904$	−25410.0	−0.934872
$905$	−23370.0	−0.858392
$906$	0	0
$907$	−19332.0	−0.707727	−0.353864	−	0.935297i	$-0.615132\pi$
$907$	−19332.0	−0.707727	−0.353864	+	0.935297i	$0.615132\pi$
$908$	−11802.0	−0.431347
$909$	0	0
$910$	0	0
$911$	43640.0	1.58711	0.793555	−	0.608498i	$-0.208228\pi$
$911$	43640.0	1.58711	0.793555	+	0.608498i	$0.208228\pi$
$912$	0	0
$913$	42504.0	1.54072
$914$	−18102.0	−0.655099
$915$	0	0
$916$	28182.0	1.01655
$917$	0	0
$918$	0	0
$919$	9084.00	0.326065	0.163032	−	0.986621i	$-0.447872\pi$
$919$	9084.00	0.326065	0.163032	+	0.986621i	$0.447872\pi$
$920$	3900.00	0.139760
$921$	0	0
$922$	2574.00	0.0919416
$923$	1848.00	0.0659021
$924$	0	0
$925$	−5250.00	−0.186615
$926$	12832.0	0.455384
$927$	0	0
$928$	23506.0	0.831489
$929$	48228.0	1.70324	0.851620	−	0.524160i	$-0.175621\pi$
$929$	48228.0	1.70324	0.851620	+	0.524160i	$0.175621\pi$
$930$	0	0
$931$	0	0
$932$	−42518.0	−1.49434
$933$	0	0
$934$	7170.00	0.251188
$935$	5280.00	0.184679
$936$	0	0
$937$	39204.0	1.36685	0.683425	−	0.730021i	$-0.260490\pi$
$937$	39204.0	1.36685	0.683425	+	0.730021i	$0.260490\pi$
$938$	0	0
$939$	0	0
$940$	−21420.0	−0.743238
$941$	−4770.00	−0.165247	−0.0826236	−	0.996581i	$-0.526330\pi$
$941$	−4770.00	−0.165247	−0.0826236	+	0.996581i	$0.526330\pi$
$942$	0	0
$943$	−23088.0	−0.797295
$944$	−12054.0	−0.415598
$945$	0	0
$946$	−21648.0	−0.744014
$947$	−56236.0	−1.92970	−0.964849	−	0.262804i	$-0.915353\pi$
$947$	−56236.0	−1.92970	−0.964849	+	0.262804i	$0.915353\pi$
$948$	0	0
$949$	72.0000	0.00246282
$950$	2850.00	0.0973329
$951$	0	0
$952$	0	0
$953$	52814.0	1.79519	0.897594	−	0.440824i	$-0.145314\pi$
$953$	52814.0	1.79519	0.897594	+	0.440824i	$0.145314\pi$
$954$	0	0
$955$	−10060.0	−0.340873
$956$	27496.0	0.930214
$957$	0	0
$958$	−10164.0	−0.342781
$959$	0	0
$960$	0	0
$961$	46385.0	1.55701
$962$	1260.00	0.0422287
$963$	0	0
$964$	8652.00	0.289069
$965$	21030.0	0.701533
$966$	0	0
$967$	−39364.0	−1.30906	−0.654530	−	0.756036i	$-0.727133\pi$
$967$	−39364.0	−1.30906	−0.654530	+	0.756036i	$0.727133\pi$
$968$	9075.00	0.301324
$969$	0	0
$970$	6000.00	0.198607
$971$	29322.0	0.969091	0.484546	−	0.874766i	$-0.338985\pi$
$971$	29322.0	0.969091	0.484546	+	0.874766i	$0.338985\pi$
$972$	0	0
$973$	0	0
$974$	−10212.0	−0.335948
$975$	0	0
$976$	18450.0	0.605092
$977$	35882.0	1.17499	0.587496	−	0.809227i	$-0.300114\pi$
$977$	35882.0	1.17499	0.587496	+	0.809227i	$0.300114\pi$
$978$	0	0
$979$	17952.0	0.586056
$980$	0	0
$981$	0	0
$982$	−7972.00	−0.259060
$983$	32580.0	1.05711	0.528556	−	0.848899i	$-0.322734\pi$
$983$	32580.0	1.05711	0.528556	+	0.848899i	$0.322734\pi$
$984$	0	0
$985$	7870.00	0.254578
$986$	3504.00	0.113175
$987$	0	0
$988$	4788.00	0.154177
$989$	25584.0	0.822572
$990$	0	0
$991$	−30036.0	−0.962790	−0.481395	−	0.876504i	$-0.659870\pi$
$991$	−30036.0	−0.962790	−0.481395	+	0.876504i	$0.659870\pi$
$992$	44436.0	1.42222
$993$	0	0
$994$	0	0
$995$	−13620.0	−0.433953
$996$	0	0
$997$	−25134.0	−0.798397	−0.399198	−	0.916865i	$-0.630712\pi$
$997$	−25134.0	−0.798397	−0.399198	+	0.916865i	$0.630712\pi$
$998$	1548.00	0.0490993
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.n.1.1		1
3.2	odd	2		245.4.a.c.1.1	yes	1
7.6	odd	2		2205.4.a.k.1.1		1
15.14	odd	2		1225.4.a.f.1.1		1
21.2	odd	6		245.4.e.c.116.1		2
21.5	even	6		245.4.e.d.116.1		2
21.11	odd	6		245.4.e.c.226.1		2
21.17	even	6		245.4.e.d.226.1		2
21.20	even	2		245.4.a.b.1.1	✓	1
105.104	even	2		1225.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
245.4.a.b.1.1	✓	1	21.20	even	2
245.4.a.c.1.1	yes	1	3.2	odd	2
245.4.e.c.116.1		2	21.2	odd	6
245.4.e.c.226.1		2	21.11	odd	6
245.4.e.d.116.1		2	21.5	even	6
245.4.e.d.226.1		2	21.17	even	6
1225.4.a.f.1.1		1	15.14	odd	2
1225.4.a.g.1.1		1	105.104	even	2
2205.4.a.k.1.1		1	7.6	odd	2
2205.4.a.n.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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