LMFDB - Embedded newform 225.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.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:	$13.2754297513$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +8.00000 q^{4} -6.00000 q^{7} -32.0000 q^{11} +38.0000 q^{13} +24.0000 q^{14} -64.0000 q^{16} +26.0000 q^{17} +100.000 q^{19} +128.000 q^{22} -78.0000 q^{23} -152.000 q^{26} -48.0000 q^{28} +50.0000 q^{29} -108.000 q^{31} +256.000 q^{32} -104.000 q^{34} -266.000 q^{37} -400.000 q^{38} -22.0000 q^{41} -442.000 q^{43} -256.000 q^{44} +312.000 q^{46} -514.000 q^{47} -307.000 q^{49} +304.000 q^{52} +2.00000 q^{53} -200.000 q^{58} -500.000 q^{59} -518.000 q^{61} +432.000 q^{62} -512.000 q^{64} -126.000 q^{67} +208.000 q^{68} -412.000 q^{71} +878.000 q^{73} +1064.00 q^{74} +800.000 q^{76} +192.000 q^{77} +600.000 q^{79} +88.0000 q^{82} +282.000 q^{83} +1768.00 q^{86} +150.000 q^{89} -228.000 q^{91} -624.000 q^{92} +2056.00 q^{94} -386.000 q^{97} +1228.00 q^{98} +O(q^{100})$

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 $

$2$	−4.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−4.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	8.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−6.00000	−0.323970	−0.161985	−	0.986793i	$-0.551790\pi$
$7$	−6.00000	−0.323970	−0.161985	+	0.986793i	$0.551790\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−32.0000	−0.877124	−0.438562	−	0.898701i	$-0.644512\pi$
$11$	−32.0000	−0.877124	−0.438562	+	0.898701i	$0.644512\pi$
$12$	0	0
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	24.0000	0.458162
$15$	0	0
$16$	−64.0000	−1.00000
$17$	26.0000	0.370937	0.185468	−	0.982650i	$-0.440620\pi$
$17$	26.0000	0.370937	0.185468	+	0.982650i	$0.440620\pi$
$18$	0	0
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	0	0
$21$	0	0
$22$	128.000	1.24044
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	0	0
$25$	0	0
$26$	−152.000	−1.14653
$27$	0	0
$28$	−48.0000	−0.323970
$29$	50.0000	0.320164	0.160082	−	0.987104i	$-0.448824\pi$
$29$	50.0000	0.320164	0.160082	+	0.987104i	$0.448824\pi$
$30$	0	0
$31$	−108.000	−0.625722	−0.312861	−	0.949799i	$-0.601287\pi$
$31$	−108.000	−0.625722	−0.312861	+	0.949799i	$0.601287\pi$
$32$	256.000	1.41421
$33$	0	0
$34$	−104.000	−0.524584
$35$	0	0
$36$	0	0
$37$	−266.000	−1.18190	−0.590948	−	0.806710i	$-0.701246\pi$
$37$	−266.000	−1.18190	−0.590948	+	0.806710i	$0.701246\pi$
$38$	−400.000	−1.70759
$39$	0	0
$40$	0	0
$41$	−22.0000	−0.0838006	−0.0419003	−	0.999122i	$-0.513341\pi$
$41$	−22.0000	−0.0838006	−0.0419003	+	0.999122i	$0.513341\pi$
$42$	0	0
$43$	−442.000	−1.56754	−0.783772	−	0.621049i	$-0.786707\pi$
$43$	−442.000	−1.56754	−0.783772	+	0.621049i	$0.786707\pi$
$44$	−256.000	−0.877124
$45$	0	0
$46$	312.000	1.00004
$47$	−514.000	−1.59520	−0.797602	−	0.603184i	$-0.793899\pi$
$47$	−514.000	−1.59520	−0.797602	+	0.603184i	$0.793899\pi$
$48$	0	0
$49$	−307.000	−0.895044
$50$	0	0
$51$	0	0
$52$	304.000	0.810716
$53$	2.00000	0.00518342	0.00259171	−	0.999997i	$-0.499175\pi$
$53$	2.00000	0.00518342	0.00259171	+	0.999997i	$0.499175\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−200.000	−0.452781
$59$	−500.000	−1.10330	−0.551648	−	0.834077i	$-0.686001\pi$
$59$	−500.000	−1.10330	−0.551648	+	0.834077i	$0.686001\pi$
$60$	0	0
$61$	−518.000	−1.08726	−0.543632	−	0.839324i	$-0.682951\pi$
$61$	−518.000	−1.08726	−0.543632	+	0.839324i	$0.682951\pi$
$62$	432.000	0.884904
$63$	0	0
$64$	−512.000	−1.00000
$65$	0	0
$66$	0	0
$67$	−126.000	−0.229751	−0.114876	−	0.993380i	$-0.536647\pi$
$67$	−126.000	−0.229751	−0.114876	+	0.993380i	$0.536647\pi$
$68$	208.000	0.370937
$69$	0	0
$70$	0	0
$71$	−412.000	−0.688668	−0.344334	−	0.938847i	$-0.611895\pi$
$71$	−412.000	−0.688668	−0.344334	+	0.938847i	$0.611895\pi$
$72$	0	0
$73$	878.000	1.40770	0.703850	−	0.710348i	$-0.251463\pi$
$73$	878.000	1.40770	0.703850	+	0.710348i	$0.251463\pi$
$74$	1064.00	1.67145
$75$	0	0
$76$	800.000	1.20745
$77$	192.000	0.284161
$78$	0	0
$79$	600.000	0.854497	0.427249	−	0.904134i	$-0.359483\pi$
$79$	600.000	0.854497	0.427249	+	0.904134i	$0.359483\pi$
$80$	0	0
$81$	0	0
$82$	88.0000	0.118512
$83$	282.000	0.372934	0.186467	−	0.982461i	$-0.440296\pi$
$83$	282.000	0.372934	0.186467	+	0.982461i	$0.440296\pi$
$84$	0	0
$85$	0	0
$86$	1768.00	2.21684
$87$	0	0
$88$	0	0
$89$	150.000	0.178651	0.0893257	−	0.996002i	$-0.471529\pi$
$89$	150.000	0.178651	0.0893257	+	0.996002i	$0.471529\pi$
$90$	0	0
$91$	−228.000	−0.262647
$92$	−624.000	−0.707136
$93$	0	0
$94$	2056.00	2.25596
$95$	0	0
$96$	0	0
$97$	−386.000	−0.404045	−0.202022	−	0.979381i	$-0.564751\pi$
$97$	−386.000	−0.404045	−0.202022	+	0.979381i	$0.564751\pi$
$98$	1228.00	1.26578
$99$	0	0
$100$	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
$103$	598.000	0.572065	0.286032	−	0.958220i	$-0.407663\pi$
$103$	598.000	0.572065	0.286032	+	0.958220i	$0.407663\pi$
$104$	0	0
$105$	0	0
$106$	−8.00000	−0.00733046
$107$	−1194.00	−1.07877	−0.539385	−	0.842059i	$-0.681343\pi$
$107$	−1194.00	−1.07877	−0.539385	+	0.842059i	$0.681343\pi$
$108$	0	0
$109$	−550.000	−0.483307	−0.241653	−	0.970363i	$-0.577690\pi$
$109$	−550.000	−0.483307	−0.241653	+	0.970363i	$0.577690\pi$
$110$	0	0
$111$	0	0
$112$	384.000	0.323970
$113$	1562.00	1.30036	0.650180	−	0.759781i	$-0.274694\pi$
$113$	1562.00	1.30036	0.650180	+	0.759781i	$0.274694\pi$
$114$	0	0
$115$	0	0
$116$	400.000	0.320164
$117$	0	0
$118$	2000.00	1.56030
$119$	−156.000	−0.120172
$120$	0	0
$121$	−307.000	−0.230654
$122$	2072.00	1.53762
$123$	0	0
$124$	−864.000	−0.625722
$125$	0	0
$126$	0	0
$127$	−1846.00	−1.28981	−0.644906	−	0.764262i	$-0.723103\pi$
$127$	−1846.00	−1.28981	−0.644906	+	0.764262i	$0.723103\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2208.00	1.47262	0.736312	−	0.676642i	$-0.236565\pi$
$131$	2208.00	1.47262	0.736312	+	0.676642i	$0.236565\pi$
$132$	0	0
$133$	−600.000	−0.391177
$134$	504.000	0.324918
$135$	0	0
$136$	0	0
$137$	−2334.00	−1.45553	−0.727763	−	0.685829i	$-0.759440\pi$
$137$	−2334.00	−1.45553	−0.727763	+	0.685829i	$0.759440\pi$
$138$	0	0
$139$	−700.000	−0.427146	−0.213573	−	0.976927i	$-0.568510\pi$
$139$	−700.000	−0.427146	−0.213573	+	0.976927i	$0.568510\pi$
$140$	0	0
$141$	0	0
$142$	1648.00	0.973923
$143$	−1216.00	−0.711098
$144$	0	0
$145$	0	0
$146$	−3512.00	−1.99079
$147$	0	0
$148$	−2128.00	−1.18190
$149$	−2050.00	−1.12713	−0.563566	−	0.826071i	$-0.690571\pi$
$149$	−2050.00	−1.12713	−0.563566	+	0.826071i	$0.690571\pi$
$150$	0	0
$151$	1852.00	0.998103	0.499052	−	0.866572i	$-0.333682\pi$
$151$	1852.00	0.998103	0.499052	+	0.866572i	$0.333682\pi$
$152$	0	0
$153$	0	0
$154$	−768.000	−0.401865
$155$	0	0
$156$	0	0
$157$	2494.00	1.26779	0.633894	−	0.773420i	$-0.281455\pi$
$157$	2494.00	1.26779	0.633894	+	0.773420i	$0.281455\pi$
$158$	−2400.00	−1.20844
$159$	0	0
$160$	0	0
$161$	468.000	0.229090
$162$	0	0
$163$	−2762.00	−1.32722	−0.663609	−	0.748080i	$-0.730976\pi$
$163$	−2762.00	−1.32722	−0.663609	+	0.748080i	$0.730976\pi$
$164$	−176.000	−0.0838006
$165$	0	0
$166$	−1128.00	−0.527408
$167$	3126.00	1.44849	0.724243	−	0.689545i	$-0.242189\pi$
$167$	3126.00	1.44849	0.724243	+	0.689545i	$0.242189\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	0	0
$171$	0	0
$172$	−3536.00	−1.56754
$173$	−78.0000	−0.0342788	−0.0171394	−	0.999853i	$-0.505456\pi$
$173$	−78.0000	−0.0342788	−0.0171394	+	0.999853i	$0.505456\pi$
$174$	0	0
$175$	0	0
$176$	2048.00	0.877124
$177$	0	0
$178$	−600.000	−0.252651
$179$	1300.00	0.542830	0.271415	−	0.962462i	$-0.412508\pi$
$179$	1300.00	0.542830	0.271415	+	0.962462i	$0.412508\pi$
$180$	0	0
$181$	1742.00	0.715369	0.357685	−	0.933842i	$-0.383566\pi$
$181$	1742.00	0.715369	0.357685	+	0.933842i	$0.383566\pi$
$182$	912.000	0.371439
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−832.000	−0.325358
$188$	−4112.00	−1.59520
$189$	0	0
$190$	0	0
$191$	−3772.00	−1.42897	−0.714483	−	0.699653i	$-0.753338\pi$
$191$	−3772.00	−1.42897	−0.714483	+	0.699653i	$0.753338\pi$
$192$	0	0
$193$	358.000	0.133520	0.0667601	−	0.997769i	$-0.478734\pi$
$193$	358.000	0.133520	0.0667601	+	0.997769i	$0.478734\pi$
$194$	1544.00	0.571406
$195$	0	0
$196$	−2456.00	−0.895044
$197$	−2214.00	−0.800716	−0.400358	−	0.916359i	$-0.631114\pi$
$197$	−2214.00	−0.800716	−0.400358	+	0.916359i	$0.631114\pi$
$198$	0	0
$199$	−2600.00	−0.926176	−0.463088	−	0.886312i	$-0.653259\pi$
$199$	−2600.00	−0.926176	−0.463088	+	0.886312i	$0.653259\pi$
$200$	0	0
$201$	0	0
$202$	2808.00	0.978070
$203$	−300.000	−0.103724
$204$	0	0
$205$	0	0
$206$	−2392.00	−0.809022
$207$	0	0
$208$	−2432.00	−0.810716
$209$	−3200.00	−1.05908
$210$	0	0
$211$	−1168.00	−0.381083	−0.190541	−	0.981679i	$-0.561024\pi$
$211$	−1168.00	−0.381083	−0.190541	+	0.981679i	$0.561024\pi$
$212$	16.0000	0.00518342
$213$	0	0
$214$	4776.00	1.52561
$215$	0	0
$216$	0	0
$217$	648.000	0.202715
$218$	2200.00	0.683499
$219$	0	0
$220$	0	0
$221$	988.000	0.300724
$222$	0	0
$223$	6478.00	1.94529	0.972643	−	0.232303i	$-0.0746262\pi$
$223$	6478.00	1.94529	0.972643	+	0.232303i	$0.0746262\pi$
$224$	−1536.00	−0.458162
$225$	0	0
$226$	−6248.00	−1.83899
$227$	646.000	0.188883	0.0944417	−	0.995530i	$-0.469893\pi$
$227$	646.000	0.188883	0.0944417	+	0.995530i	$0.469893\pi$
$228$	0	0
$229$	3750.00	1.08213	0.541063	−	0.840982i	$-0.318022\pi$
$229$	3750.00	1.08213	0.541063	+	0.840982i	$0.318022\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1482.00	0.416691	0.208346	−	0.978055i	$-0.433192\pi$
$233$	1482.00	0.416691	0.208346	+	0.978055i	$0.433192\pi$
$234$	0	0
$235$	0	0
$236$	−4000.00	−1.10330
$237$	0	0
$238$	624.000	0.169949
$239$	−1400.00	−0.378906	−0.189453	−	0.981890i	$-0.560671\pi$
$239$	−1400.00	−0.378906	−0.189453	+	0.981890i	$0.560671\pi$
$240$	0	0
$241$	3022.00	0.807735	0.403867	−	0.914817i	$-0.367666\pi$
$241$	3022.00	0.807735	0.403867	+	0.914817i	$0.367666\pi$
$242$	1228.00	0.326194
$243$	0	0
$244$	−4144.00	−1.08726
$245$	0	0
$246$	0	0
$247$	3800.00	0.978900
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1248.00	0.313837	0.156918	−	0.987612i	$-0.449844\pi$
$251$	1248.00	0.313837	0.156918	+	0.987612i	$0.449844\pi$
$252$	0	0
$253$	2496.00	0.620246
$254$	7384.00	1.82407
$255$	0	0
$256$	4096.00	1.00000
$257$	2106.00	0.511162	0.255581	−	0.966788i	$-0.417733\pi$
$257$	2106.00	0.511162	0.255581	+	0.966788i	$0.417733\pi$
$258$	0	0
$259$	1596.00	0.382898
$260$	0	0
$261$	0	0
$262$	−8832.00	−2.08261
$263$	−3638.00	−0.852961	−0.426480	−	0.904497i	$-0.640247\pi$
$263$	−3638.00	−0.852961	−0.426480	+	0.904497i	$0.640247\pi$
$264$	0	0
$265$	0	0
$266$	2400.00	0.553208
$267$	0	0
$268$	−1008.00	−0.229751
$269$	6550.00	1.48461	0.742306	−	0.670061i	$-0.233732\pi$
$269$	6550.00	1.48461	0.742306	+	0.670061i	$0.233732\pi$
$270$	0	0
$271$	−4388.00	−0.983587	−0.491793	−	0.870712i	$-0.663658\pi$
$271$	−4388.00	−0.983587	−0.491793	+	0.870712i	$0.663658\pi$
$272$	−1664.00	−0.370937
$273$	0	0
$274$	9336.00	2.05842
$275$	0	0
$276$	0	0
$277$	−546.000	−0.118433	−0.0592165	−	0.998245i	$-0.518860\pi$
$277$	−546.000	−0.118433	−0.0592165	+	0.998245i	$0.518860\pi$
$278$	2800.00	0.604075
$279$	0	0
$280$	0	0
$281$	6858.00	1.45592	0.727961	−	0.685619i	$-0.240468\pi$
$281$	6858.00	1.45592	0.727961	+	0.685619i	$0.240468\pi$
$282$	0	0
$283$	−9282.00	−1.94967	−0.974837	−	0.222920i	$-0.928441\pi$
$283$	−9282.00	−1.94967	−0.974837	+	0.222920i	$0.928441\pi$
$284$	−3296.00	−0.688668
$285$	0	0
$286$	4864.00	1.00564
$287$	132.000	0.0271488
$288$	0	0
$289$	−4237.00	−0.862406
$290$	0	0
$291$	0	0
$292$	7024.00	1.40770
$293$	4842.00	0.965436	0.482718	−	0.875776i	$-0.339650\pi$
$293$	4842.00	0.965436	0.482718	+	0.875776i	$0.339650\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	8200.00	1.59400
$299$	−2964.00	−0.573286
$300$	0	0
$301$	2652.00	0.507836
$302$	−7408.00	−1.41153
$303$	0	0
$304$	−6400.00	−1.20745
$305$	0	0
$306$	0	0
$307$	2594.00	0.482239	0.241120	−	0.970495i	$-0.422485\pi$
$307$	2594.00	0.482239	0.241120	+	0.970495i	$0.422485\pi$
$308$	1536.00	0.284161
$309$	0	0
$310$	0	0
$311$	−7332.00	−1.33685	−0.668424	−	0.743781i	$-0.733031\pi$
$311$	−7332.00	−1.33685	−0.668424	+	0.743781i	$0.733031\pi$
$312$	0	0
$313$	−1562.00	−0.282075	−0.141037	−	0.990004i	$-0.545044\pi$
$313$	−1562.00	−0.282075	−0.141037	+	0.990004i	$0.545044\pi$
$314$	−9976.00	−1.79292
$315$	0	0
$316$	4800.00	0.854497
$317$	1426.00	0.252657	0.126328	−	0.991988i	$-0.459681\pi$
$317$	1426.00	0.252657	0.126328	+	0.991988i	$0.459681\pi$
$318$	0	0
$319$	−1600.00	−0.280824
$320$	0	0
$321$	0	0
$322$	−1872.00	−0.323983
$323$	2600.00	0.447888
$324$	0	0
$325$	0	0
$326$	11048.0	1.87697
$327$	0	0
$328$	0	0
$329$	3084.00	0.516798
$330$	0	0
$331$	−4008.00	−0.665558	−0.332779	−	0.943005i	$-0.607986\pi$
$331$	−4008.00	−0.665558	−0.332779	+	0.943005i	$0.607986\pi$
$332$	2256.00	0.372934
$333$	0	0
$334$	−12504.0	−2.04847
$335$	0	0
$336$	0	0
$337$	−8866.00	−1.43312	−0.716561	−	0.697525i	$-0.754285\pi$
$337$	−8866.00	−1.43312	−0.716561	+	0.697525i	$0.754285\pi$
$338$	3012.00	0.484708
$339$	0	0
$340$	0	0
$341$	3456.00	0.548835
$342$	0	0
$343$	3900.00	0.613936
$344$	0	0
$345$	0	0
$346$	312.000	0.0484775
$347$	−1714.00	−0.265165	−0.132583	−	0.991172i	$-0.542327\pi$
$347$	−1714.00	−0.265165	−0.132583	+	0.991172i	$0.542327\pi$
$348$	0	0
$349$	1150.00	0.176384	0.0881921	−	0.996103i	$-0.471891\pi$
$349$	1150.00	0.176384	0.0881921	+	0.996103i	$0.471891\pi$
$350$	0	0
$351$	0	0
$352$	−8192.00	−1.24044
$353$	−4398.00	−0.663122	−0.331561	−	0.943434i	$-0.607575\pi$
$353$	−4398.00	−0.663122	−0.331561	+	0.943434i	$0.607575\pi$
$354$	0	0
$355$	0	0
$356$	1200.00	0.178651
$357$	0	0
$358$	−5200.00	−0.767677
$359$	−1800.00	−0.264625	−0.132312	−	0.991208i	$-0.542240\pi$
$359$	−1800.00	−0.264625	−0.132312	+	0.991208i	$0.542240\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	−6968.00	−1.01168
$363$	0	0
$364$	−1824.00	−0.262647
$365$	0	0
$366$	0	0
$367$	5874.00	0.835478	0.417739	−	0.908567i	$-0.362823\pi$
$367$	5874.00	0.835478	0.417739	+	0.908567i	$0.362823\pi$
$368$	4992.00	0.707136
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.00167927
$372$	0	0
$373$	2078.00	0.288458	0.144229	−	0.989544i	$-0.453930\pi$
$373$	2078.00	0.288458	0.144229	+	0.989544i	$0.453930\pi$
$374$	3328.00	0.460125
$375$	0	0
$376$	0	0
$377$	1900.00	0.259562
$378$	0	0
$379$	7900.00	1.07070	0.535351	−	0.844630i	$-0.320179\pi$
$379$	7900.00	1.07070	0.535351	+	0.844630i	$0.320179\pi$
$380$	0	0
$381$	0	0
$382$	15088.0	2.02086
$383$	−7518.00	−1.00301	−0.501504	−	0.865155i	$-0.667220\pi$
$383$	−7518.00	−1.00301	−0.501504	+	0.865155i	$0.667220\pi$
$384$	0	0
$385$	0	0
$386$	−1432.00	−0.188826
$387$	0	0
$388$	−3088.00	−0.404045
$389$	1950.00	0.254162	0.127081	−	0.991892i	$-0.459439\pi$
$389$	1950.00	0.254162	0.127081	+	0.991892i	$0.459439\pi$
$390$	0	0
$391$	−2028.00	−0.262303
$392$	0	0
$393$	0	0
$394$	8856.00	1.13238
$395$	0	0
$396$	0	0
$397$	−13786.0	−1.74282	−0.871410	−	0.490555i	$-0.836794\pi$
$397$	−13786.0	−1.74282	−0.871410	+	0.490555i	$0.836794\pi$
$398$	10400.0	1.30981
$399$	0	0
$400$	0	0
$401$	−6402.00	−0.797258	−0.398629	−	0.917112i	$-0.630514\pi$
$401$	−6402.00	−0.797258	−0.398629	+	0.917112i	$0.630514\pi$
$402$	0	0
$403$	−4104.00	−0.507282
$404$	−5616.00	−0.691600
$405$	0	0
$406$	1200.00	0.146687
$407$	8512.00	1.03667
$408$	0	0
$409$	11150.0	1.34800	0.674000	−	0.738731i	$-0.264575\pi$
$409$	11150.0	1.34800	0.674000	+	0.738731i	$0.264575\pi$
$410$	0	0
$411$	0	0
$412$	4784.00	0.572065
$413$	3000.00	0.357434
$414$	0	0
$415$	0	0
$416$	9728.00	1.14653
$417$	0	0
$418$	12800.0	1.49777
$419$	13700.0	1.59735	0.798674	−	0.601764i	$-0.205535\pi$
$419$	13700.0	1.59735	0.798674	+	0.601764i	$0.205535\pi$
$420$	0	0
$421$	−5438.00	−0.629529	−0.314765	−	0.949170i	$-0.601926\pi$
$421$	−5438.00	−0.629529	−0.314765	+	0.949170i	$0.601926\pi$
$422$	4672.00	0.538932
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3108.00	0.352240
$428$	−9552.00	−1.07877
$429$	0	0
$430$	0	0
$431$	−7692.00	−0.859653	−0.429827	−	0.902911i	$-0.641425\pi$
$431$	−7692.00	−0.859653	−0.429827	+	0.902911i	$0.641425\pi$
$432$	0	0
$433$	1118.00	0.124082	0.0620412	−	0.998074i	$-0.480239\pi$
$433$	1118.00	0.124082	0.0620412	+	0.998074i	$0.480239\pi$
$434$	−2592.00	−0.286682
$435$	0	0
$436$	−4400.00	−0.483307
$437$	−7800.00	−0.853832
$438$	0	0
$439$	−2600.00	−0.282668	−0.141334	−	0.989962i	$-0.545139\pi$
$439$	−2600.00	−0.282668	−0.141334	+	0.989962i	$0.545139\pi$
$440$	0	0
$441$	0	0
$442$	−3952.00	−0.425288
$443$	−11958.0	−1.28249	−0.641243	−	0.767337i	$-0.721581\pi$
$443$	−11958.0	−1.28249	−0.641243	+	0.767337i	$0.721581\pi$
$444$	0	0
$445$	0	0
$446$	−25912.0	−2.75105
$447$	0	0
$448$	3072.00	0.323970
$449$	17050.0	1.79207	0.896035	−	0.443984i	$-0.146435\pi$
$449$	17050.0	1.79207	0.896035	+	0.443984i	$0.146435\pi$
$450$	0	0
$451$	704.000	0.0735035
$452$	12496.0	1.30036
$453$	0	0
$454$	−2584.00	−0.267121
$455$	0	0
$456$	0	0
$457$	9494.00	0.971796	0.485898	−	0.874016i	$-0.338493\pi$
$457$	9494.00	0.971796	0.485898	+	0.874016i	$0.338493\pi$
$458$	−15000.0	−1.53036
$459$	0	0
$460$	0	0
$461$	11418.0	1.15356	0.576778	−	0.816901i	$-0.304310\pi$
$461$	11418.0	1.15356	0.576778	+	0.816901i	$0.304310\pi$
$462$	0	0
$463$	−7962.00	−0.799191	−0.399596	−	0.916692i	$-0.630849\pi$
$463$	−7962.00	−0.799191	−0.399596	+	0.916692i	$0.630849\pi$
$464$	−3200.00	−0.320164
$465$	0	0
$466$	−5928.00	−0.589290
$467$	6526.00	0.646654	0.323327	−	0.946287i	$-0.395199\pi$
$467$	6526.00	0.646654	0.323327	+	0.946287i	$0.395199\pi$
$468$	0	0
$469$	756.000	0.0744325
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14144.0	1.37493
$474$	0	0
$475$	0	0
$476$	−1248.00	−0.120172
$477$	0	0
$478$	5600.00	0.535854
$479$	−17400.0	−1.65976	−0.829881	−	0.557940i	$-0.811592\pi$
$479$	−17400.0	−1.65976	−0.829881	+	0.557940i	$0.811592\pi$
$480$	0	0
$481$	−10108.0	−0.958181
$482$	−12088.0	−1.14231
$483$	0	0
$484$	−2456.00	−0.230654
$485$	0	0
$486$	0	0
$487$	−1166.00	−0.108494	−0.0542469	−	0.998528i	$-0.517276\pi$
$487$	−1166.00	−0.108494	−0.0542469	+	0.998528i	$0.517276\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7072.00	−0.650010	−0.325005	−	0.945712i	$-0.605366\pi$
$491$	−7072.00	−0.650010	−0.325005	+	0.945712i	$0.605366\pi$
$492$	0	0
$493$	1300.00	0.118761
$494$	−15200.0	−1.38437
$495$	0	0
$496$	6912.00	0.625722
$497$	2472.00	0.223107
$498$	0	0
$499$	100.000	0.00897117	0.00448559	−	0.999990i	$-0.498572\pi$
$499$	100.000	0.00897117	0.00448559	+	0.999990i	$0.498572\pi$
$500$	0	0
$501$	0	0
$502$	−4992.00	−0.443832
$503$	2602.00	0.230651	0.115325	−	0.993328i	$-0.463209\pi$
$503$	2602.00	0.230651	0.115325	+	0.993328i	$0.463209\pi$
$504$	0	0
$505$	0	0
$506$	−9984.00	−0.877160
$507$	0	0
$508$	−14768.0	−1.28981
$509$	−11150.0	−0.970953	−0.485476	−	0.874250i	$-0.661354\pi$
$509$	−11150.0	−0.970953	−0.485476	+	0.874250i	$0.661354\pi$
$510$	0	0
$511$	−5268.00	−0.456052
$512$	−16384.0	−1.41421
$513$	0	0
$514$	−8424.00	−0.722892
$515$	0	0
$516$	0	0
$517$	16448.0	1.39919
$518$	−6384.00	−0.541500
$519$	0	0
$520$	0	0
$521$	3638.00	0.305919	0.152959	−	0.988232i	$-0.451120\pi$
$521$	3638.00	0.305919	0.152959	+	0.988232i	$0.451120\pi$
$522$	0	0
$523$	2078.00	0.173737	0.0868686	−	0.996220i	$-0.472314\pi$
$523$	2078.00	0.173737	0.0868686	+	0.996220i	$0.472314\pi$
$524$	17664.0	1.47262
$525$	0	0
$526$	14552.0	1.20627
$527$	−2808.00	−0.232103
$528$	0	0
$529$	−6083.00	−0.499959
$530$	0	0
$531$	0	0
$532$	−4800.00	−0.391177
$533$	−836.000	−0.0679384
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	−26200.0	−2.09956
$539$	9824.00	0.785064
$540$	0	0
$541$	5622.00	0.446781	0.223391	−	0.974729i	$-0.428287\pi$
$541$	5622.00	0.446781	0.223391	+	0.974729i	$0.428287\pi$
$542$	17552.0	1.39100
$543$	0	0
$544$	6656.00	0.524584
$545$	0	0
$546$	0	0
$547$	−16486.0	−1.28865	−0.644324	−	0.764753i	$-0.722861\pi$
$547$	−16486.0	−1.28865	−0.644324	+	0.764753i	$0.722861\pi$
$548$	−18672.0	−1.45553
$549$	0	0
$550$	0	0
$551$	5000.00	0.386583
$552$	0	0
$553$	−3600.00	−0.276831
$554$	2184.00	0.167490
$555$	0	0
$556$	−5600.00	−0.427146
$557$	11706.0	0.890483	0.445242	−	0.895410i	$-0.353118\pi$
$557$	11706.0	0.890483	0.445242	+	0.895410i	$0.353118\pi$
$558$	0	0
$559$	−16796.0	−1.27083
$560$	0	0
$561$	0	0
$562$	−27432.0	−2.05898
$563$	−25038.0	−1.87429	−0.937146	−	0.348939i	$-0.886542\pi$
$563$	−25038.0	−1.87429	−0.937146	+	0.348939i	$0.886542\pi$
$564$	0	0
$565$	0	0
$566$	37128.0	2.75725
$567$	0	0
$568$	0	0
$569$	−17550.0	−1.29303	−0.646515	−	0.762901i	$-0.723774\pi$
$569$	−17550.0	−1.29303	−0.646515	+	0.762901i	$0.723774\pi$
$570$	0	0
$571$	10712.0	0.785084	0.392542	−	0.919734i	$-0.371596\pi$
$571$	10712.0	0.785084	0.392542	+	0.919734i	$0.371596\pi$
$572$	−9728.00	−0.711098
$573$	0	0
$574$	−528.000	−0.0383942
$575$	0	0
$576$	0	0
$577$	13654.0	0.985136	0.492568	−	0.870274i	$-0.336058\pi$
$577$	13654.0	0.985136	0.492568	+	0.870274i	$0.336058\pi$
$578$	16948.0	1.21963
$579$	0	0
$580$	0	0
$581$	−1692.00	−0.120819
$582$	0	0
$583$	−64.0000	−0.00454650
$584$	0	0
$585$	0	0
$586$	−19368.0	−1.36533
$587$	14166.0	0.996071	0.498035	−	0.867157i	$-0.334055\pi$
$587$	14166.0	0.996071	0.498035	+	0.867157i	$0.334055\pi$
$588$	0	0
$589$	−10800.0	−0.755528
$590$	0	0
$591$	0	0
$592$	17024.0	1.18190
$593$	17842.0	1.23555	0.617777	−	0.786354i	$-0.288034\pi$
$593$	17842.0	1.23555	0.617777	+	0.786354i	$0.288034\pi$
$594$	0	0
$595$	0	0
$596$	−16400.0	−1.12713
$597$	0	0
$598$	11856.0	0.810749
$599$	17600.0	1.20053	0.600264	−	0.799802i	$-0.295062\pi$
$599$	17600.0	1.20053	0.600264	+	0.799802i	$0.295062\pi$
$600$	0	0
$601$	27302.0	1.85303	0.926516	−	0.376256i	$-0.122789\pi$
$601$	27302.0	1.85303	0.926516	+	0.376256i	$0.122789\pi$
$602$	−10608.0	−0.718189
$603$	0	0
$604$	14816.0	0.998103
$605$	0	0
$606$	0	0
$607$	3794.00	0.253696	0.126848	−	0.991922i	$-0.459514\pi$
$607$	3794.00	0.253696	0.126848	+	0.991922i	$0.459514\pi$
$608$	25600.0	1.70759
$609$	0	0
$610$	0	0
$611$	−19532.0	−1.29326
$612$	0	0
$613$	13238.0	0.872231	0.436116	−	0.899891i	$-0.356354\pi$
$613$	13238.0	0.872231	0.436116	+	0.899891i	$0.356354\pi$
$614$	−10376.0	−0.681989
$615$	0	0
$616$	0	0
$617$	−11574.0	−0.755189	−0.377595	−	0.925971i	$-0.623249\pi$
$617$	−11574.0	−0.755189	−0.377595	+	0.925971i	$0.623249\pi$
$618$	0	0
$619$	8300.00	0.538942	0.269471	−	0.963008i	$-0.413151\pi$
$619$	8300.00	0.538942	0.269471	+	0.963008i	$0.413151\pi$
$620$	0	0
$621$	0	0
$622$	29328.0	1.89059
$623$	−900.000	−0.0578776
$624$	0	0
$625$	0	0
$626$	6248.00	0.398914
$627$	0	0
$628$	19952.0	1.26779
$629$	−6916.00	−0.438409
$630$	0	0
$631$	−7508.00	−0.473675	−0.236837	−	0.971549i	$-0.576111\pi$
$631$	−7508.00	−0.473675	−0.236837	+	0.971549i	$0.576111\pi$
$632$	0	0
$633$	0	0
$634$	−5704.00	−0.357310
$635$	0	0
$636$	0	0
$637$	−11666.0	−0.725626
$638$	6400.00	0.397145
$639$	0	0
$640$	0	0
$641$	27378.0	1.68700	0.843499	−	0.537130i	$-0.180492\pi$
$641$	27378.0	1.68700	0.843499	+	0.537130i	$0.180492\pi$
$642$	0	0
$643$	−1842.00	−0.112973	−0.0564863	−	0.998403i	$-0.517990\pi$
$643$	−1842.00	−0.112973	−0.0564863	+	0.998403i	$0.517990\pi$
$644$	3744.00	0.229090
$645$	0	0
$646$	−10400.0	−0.633409
$647$	−10114.0	−0.614563	−0.307282	−	0.951619i	$-0.599419\pi$
$647$	−10114.0	−0.614563	−0.307282	+	0.951619i	$0.599419\pi$
$648$	0	0
$649$	16000.0	0.967727
$650$	0	0
$651$	0	0
$652$	−22096.0	−1.32722
$653$	10402.0	0.623372	0.311686	−	0.950185i	$-0.399106\pi$
$653$	10402.0	0.623372	0.311686	+	0.950185i	$0.399106\pi$
$654$	0	0
$655$	0	0
$656$	1408.00	0.0838006
$657$	0	0
$658$	−12336.0	−0.730862
$659$	−7100.00	−0.419692	−0.209846	−	0.977734i	$-0.567296\pi$
$659$	−7100.00	−0.419692	−0.209846	+	0.977734i	$0.567296\pi$
$660$	0	0
$661$	−7118.00	−0.418847	−0.209424	−	0.977825i	$-0.567159\pi$
$661$	−7118.00	−0.418847	−0.209424	+	0.977825i	$0.567159\pi$
$662$	16032.0	0.941241
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3900.00	−0.226400
$668$	25008.0	1.44849
$669$	0	0
$670$	0	0
$671$	16576.0	0.953665
$672$	0	0
$673$	31278.0	1.79150	0.895749	−	0.444560i	$-0.146640\pi$
$673$	31278.0	1.79150	0.895749	+	0.444560i	$0.146640\pi$
$674$	35464.0	2.02674
$675$	0	0
$676$	−6024.00	−0.342740
$677$	−30054.0	−1.70616	−0.853079	−	0.521782i	$-0.825268\pi$
$677$	−30054.0	−1.70616	−0.853079	+	0.521782i	$0.825268\pi$
$678$	0	0
$679$	2316.00	0.130898
$680$	0	0
$681$	0	0
$682$	−13824.0	−0.776171
$683$	−4518.00	−0.253113	−0.126557	−	0.991959i	$-0.540393\pi$
$683$	−4518.00	−0.253113	−0.126557	+	0.991959i	$0.540393\pi$
$684$	0	0
$685$	0	0
$686$	−15600.0	−0.868237
$687$	0	0
$688$	28288.0	1.56754
$689$	76.0000	0.00420228
$690$	0	0
$691$	29272.0	1.61152	0.805759	−	0.592243i	$-0.201758\pi$
$691$	29272.0	1.61152	0.805759	+	0.592243i	$0.201758\pi$
$692$	−624.000	−0.0342788
$693$	0	0
$694$	6856.00	0.375000
$695$	0	0
$696$	0	0
$697$	−572.000	−0.0310847
$698$	−4600.00	−0.249445
$699$	0	0
$700$	0	0
$701$	5798.00	0.312393	0.156196	−	0.987726i	$-0.450077\pi$
$701$	5798.00	0.312393	0.156196	+	0.987726i	$0.450077\pi$
$702$	0	0
$703$	−26600.0	−1.42708
$704$	16384.0	0.877124
$705$	0	0
$706$	17592.0	0.937796
$707$	4212.00	0.224057
$708$	0	0
$709$	8950.00	0.474082	0.237041	−	0.971500i	$-0.423822\pi$
$709$	8950.00	0.474082	0.237041	+	0.971500i	$0.423822\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8424.00	0.442470
$714$	0	0
$715$	0	0
$716$	10400.0	0.542830
$717$	0	0
$718$	7200.00	0.374236
$719$	−7800.00	−0.404577	−0.202289	−	0.979326i	$-0.564838\pi$
$719$	−7800.00	−0.404577	−0.202289	+	0.979326i	$0.564838\pi$
$720$	0	0
$721$	−3588.00	−0.185332
$722$	−12564.0	−0.647623
$723$	0	0
$724$	13936.0	0.715369
$725$	0	0
$726$	0	0
$727$	8554.00	0.436383	0.218191	−	0.975906i	$-0.429984\pi$
$727$	8554.00	0.436383	0.218191	+	0.975906i	$0.429984\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11492.0	−0.581460
$732$	0	0
$733$	−2882.00	−0.145224	−0.0726119	−	0.997360i	$-0.523133\pi$
$733$	−2882.00	−0.145224	−0.0726119	+	0.997360i	$0.523133\pi$
$734$	−23496.0	−1.18154
$735$	0	0
$736$	−19968.0	−1.00004
$737$	4032.00	0.201521
$738$	0	0
$739$	18700.0	0.930840	0.465420	−	0.885090i	$-0.345903\pi$
$739$	18700.0	0.930840	0.465420	+	0.885090i	$0.345903\pi$
$740$	0	0
$741$	0	0
$742$	48.0000	0.00237485
$743$	12242.0	0.604462	0.302231	−	0.953235i	$-0.402269\pi$
$743$	12242.0	0.604462	0.302231	+	0.953235i	$0.402269\pi$
$744$	0	0
$745$	0	0
$746$	−8312.00	−0.407941
$747$	0	0
$748$	−6656.00	−0.325358
$749$	7164.00	0.349488
$750$	0	0
$751$	−31148.0	−1.51346	−0.756729	−	0.653729i	$-0.773204\pi$
$751$	−31148.0	−1.51346	−0.756729	+	0.653729i	$0.773204\pi$
$752$	32896.0	1.59520
$753$	0	0
$754$	−7600.00	−0.367076
$755$	0	0
$756$	0	0
$757$	7694.00	0.369410	0.184705	−	0.982794i	$-0.440867\pi$
$757$	7694.00	0.369410	0.184705	+	0.982794i	$0.440867\pi$
$758$	−31600.0	−1.51420
$759$	0	0
$760$	0	0
$761$	4518.00	0.215213	0.107607	−	0.994194i	$-0.465681\pi$
$761$	4518.00	0.215213	0.107607	+	0.994194i	$0.465681\pi$
$762$	0	0
$763$	3300.00	0.156577
$764$	−30176.0	−1.42897
$765$	0	0
$766$	30072.0	1.41847
$767$	−19000.0	−0.894459
$768$	0	0
$769$	−39550.0	−1.85463	−0.927314	−	0.374283i	$-0.877889\pi$
$769$	−39550.0	−1.85463	−0.927314	+	0.374283i	$0.877889\pi$
$770$	0	0
$771$	0	0
$772$	2864.00	0.133520
$773$	22122.0	1.02933	0.514666	−	0.857391i	$-0.327916\pi$
$773$	22122.0	1.02933	0.514666	+	0.857391i	$0.327916\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−7800.00	−0.359439
$779$	−2200.00	−0.101185
$780$	0	0
$781$	13184.0	0.604047
$782$	8112.00	0.370952
$783$	0	0
$784$	19648.0	0.895044
$785$	0	0
$786$	0	0
$787$	16634.0	0.753416	0.376708	−	0.926332i	$-0.377056\pi$
$787$	16634.0	0.753416	0.376708	+	0.926332i	$0.377056\pi$
$788$	−17712.0	−0.800716
$789$	0	0
$790$	0	0
$791$	−9372.00	−0.421277
$792$	0	0
$793$	−19684.0	−0.881462
$794$	55144.0	2.46472
$795$	0	0
$796$	−20800.0	−0.926176
$797$	27586.0	1.22603	0.613015	−	0.790071i	$-0.289956\pi$
$797$	27586.0	1.22603	0.613015	+	0.790071i	$0.289956\pi$
$798$	0	0
$799$	−13364.0	−0.591720
$800$	0	0
$801$	0	0
$802$	25608.0	1.12749
$803$	−28096.0	−1.23473
$804$	0	0
$805$	0	0
$806$	16416.0	0.717406
$807$	0	0
$808$	0	0
$809$	−3850.00	−0.167316	−0.0836581	−	0.996495i	$-0.526660\pi$
$809$	−3850.00	−0.167316	−0.0836581	+	0.996495i	$0.526660\pi$
$810$	0	0
$811$	10032.0	0.434366	0.217183	−	0.976131i	$-0.430313\pi$
$811$	10032.0	0.434366	0.217183	+	0.976131i	$0.430313\pi$
$812$	−2400.00	−0.103724
$813$	0	0
$814$	−34048.0	−1.46607
$815$	0	0
$816$	0	0
$817$	−44200.0	−1.89273
$818$	−44600.0	−1.90636
$819$	0	0
$820$	0	0
$821$	−20562.0	−0.874079	−0.437039	−	0.899442i	$-0.643973\pi$
$821$	−20562.0	−0.874079	−0.437039	+	0.899442i	$0.643973\pi$
$822$	0	0
$823$	−10322.0	−0.437184	−0.218592	−	0.975816i	$-0.570146\pi$
$823$	−10322.0	−0.437184	−0.218592	+	0.975816i	$0.570146\pi$
$824$	0	0
$825$	0	0
$826$	−12000.0	−0.505488
$827$	8846.00	0.371954	0.185977	−	0.982554i	$-0.440455\pi$
$827$	8846.00	0.371954	0.185977	+	0.982554i	$0.440455\pi$
$828$	0	0
$829$	−25350.0	−1.06205	−0.531026	−	0.847355i	$-0.678194\pi$
$829$	−25350.0	−1.06205	−0.531026	+	0.847355i	$0.678194\pi$
$830$	0	0
$831$	0	0
$832$	−19456.0	−0.810716
$833$	−7982.00	−0.332005
$834$	0	0
$835$	0	0
$836$	−25600.0	−1.05908
$837$	0	0
$838$	−54800.0	−2.25899
$839$	−46000.0	−1.89284	−0.946422	−	0.322932i	$-0.895331\pi$
$839$	−46000.0	−1.89284	−0.946422	+	0.322932i	$0.895331\pi$
$840$	0	0
$841$	−21889.0	−0.897495
$842$	21752.0	0.890289
$843$	0	0
$844$	−9344.00	−0.381083
$845$	0	0
$846$	0	0
$847$	1842.00	0.0747248
$848$	−128.000	−0.00518342
$849$	0	0
$850$	0	0
$851$	20748.0	0.835761
$852$	0	0
$853$	16998.0	0.682298	0.341149	−	0.940009i	$-0.389184\pi$
$853$	16998.0	0.682298	0.341149	+	0.940009i	$0.389184\pi$
$854$	−12432.0	−0.498143
$855$	0	0
$856$	0	0
$857$	−26494.0	−1.05603	−0.528015	−	0.849235i	$-0.677064\pi$
$857$	−26494.0	−1.05603	−0.528015	+	0.849235i	$0.677064\pi$
$858$	0	0
$859$	−21500.0	−0.853982	−0.426991	−	0.904256i	$-0.640426\pi$
$859$	−21500.0	−0.853982	−0.426991	+	0.904256i	$0.640426\pi$
$860$	0	0
$861$	0	0
$862$	30768.0	1.21573
$863$	25762.0	1.01616	0.508082	−	0.861309i	$-0.330355\pi$
$863$	25762.0	1.01616	0.508082	+	0.861309i	$0.330355\pi$
$864$	0	0
$865$	0	0
$866$	−4472.00	−0.175479
$867$	0	0
$868$	5184.00	0.202715
$869$	−19200.0	−0.749500
$870$	0	0
$871$	−4788.00	−0.186263
$872$	0	0
$873$	0	0
$874$	31200.0	1.20750
$875$	0	0
$876$	0	0
$877$	−30546.0	−1.17613	−0.588064	−	0.808814i	$-0.700110\pi$
$877$	−30546.0	−1.17613	−0.588064	+	0.808814i	$0.700110\pi$
$878$	10400.0	0.399753
$879$	0	0
$880$	0	0
$881$	−32942.0	−1.25976	−0.629878	−	0.776694i	$-0.716895\pi$
$881$	−32942.0	−1.25976	−0.629878	+	0.776694i	$0.716895\pi$
$882$	0	0
$883$	27118.0	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$883$	27118.0	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$884$	7904.00	0.300724
$885$	0	0
$886$	47832.0	1.81371
$887$	−38634.0	−1.46246	−0.731230	−	0.682131i	$-0.761054\pi$
$887$	−38634.0	−1.46246	−0.731230	+	0.682131i	$0.761054\pi$
$888$	0	0
$889$	11076.0	0.417860
$890$	0	0
$891$	0	0
$892$	51824.0	1.94529
$893$	−51400.0	−1.92613
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−68200.0	−2.53437
$899$	−5400.00	−0.200334
$900$	0	0
$901$	52.0000	0.00192272
$902$	−2816.00	−0.103950
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1794.00	0.0656767	0.0328384	−	0.999461i	$-0.489545\pi$
$907$	1794.00	0.0656767	0.0328384	+	0.999461i	$0.489545\pi$
$908$	5168.00	0.188883
$909$	0	0
$910$	0	0
$911$	−41732.0	−1.51772	−0.758860	−	0.651254i	$-0.774243\pi$
$911$	−41732.0	−1.51772	−0.758860	+	0.651254i	$0.774243\pi$
$912$	0	0
$913$	−9024.00	−0.327109
$914$	−37976.0	−1.37433
$915$	0	0
$916$	30000.0	1.08213
$917$	−13248.0	−0.477086
$918$	0	0
$919$	29200.0	1.04812	0.524058	−	0.851682i	$-0.324417\pi$
$919$	29200.0	1.04812	0.524058	+	0.851682i	$0.324417\pi$
$920$	0	0
$921$	0	0
$922$	−45672.0	−1.63137
$923$	−15656.0	−0.558314
$924$	0	0
$925$	0	0
$926$	31848.0	1.13023
$927$	0	0
$928$	12800.0	0.452781
$929$	48650.0	1.71814	0.859071	−	0.511856i	$-0.171042\pi$
$929$	48650.0	1.71814	0.859071	+	0.511856i	$0.171042\pi$
$930$	0	0
$931$	−30700.0	−1.08072
$932$	11856.0	0.416691
$933$	0	0
$934$	−26104.0	−0.914506
$935$	0	0
$936$	0	0
$937$	11334.0	0.395161	0.197580	−	0.980287i	$-0.436692\pi$
$937$	11334.0	0.395161	0.197580	+	0.980287i	$0.436692\pi$
$938$	−3024.00	−0.105263
$939$	0	0
$940$	0	0
$941$	31178.0	1.08010	0.540050	−	0.841633i	$-0.318405\pi$
$941$	31178.0	1.08010	0.540050	+	0.841633i	$0.318405\pi$
$942$	0	0
$943$	1716.00	0.0592584
$944$	32000.0	1.10330
$945$	0	0
$946$	−56576.0	−1.94444
$947$	4686.00	0.160797	0.0803984	−	0.996763i	$-0.474381\pi$
$947$	4686.00	0.160797	0.0803984	+	0.996763i	$0.474381\pi$
$948$	0	0
$949$	33364.0	1.14124
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−598.000	−0.0203265	−0.0101632	−	0.999948i	$-0.503235\pi$
$953$	−598.000	−0.0203265	−0.0101632	+	0.999948i	$0.503235\pi$
$954$	0	0
$955$	0	0
$956$	−11200.0	−0.378906
$957$	0	0
$958$	69600.0	2.34726
$959$	14004.0	0.471546
$960$	0	0
$961$	−18127.0	−0.608472
$962$	40432.0	1.35507
$963$	0	0
$964$	24176.0	0.807735
$965$	0	0
$966$	0	0
$967$	−41726.0	−1.38761	−0.693804	−	0.720163i	$-0.744067\pi$
$967$	−41726.0	−1.38761	−0.693804	+	0.720163i	$0.744067\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24312.0	−0.803511	−0.401756	−	0.915747i	$-0.631600\pi$
$971$	−24312.0	−0.803511	−0.401756	+	0.915747i	$0.631600\pi$
$972$	0	0
$973$	4200.00	0.138382
$974$	4664.00	0.153433
$975$	0	0
$976$	33152.0	1.08726
$977$	40946.0	1.34082	0.670409	−	0.741992i	$-0.266119\pi$
$977$	40946.0	1.34082	0.670409	+	0.741992i	$0.266119\pi$
$978$	0	0
$979$	−4800.00	−0.156699
$980$	0	0
$981$	0	0
$982$	28288.0	0.919253
$983$	42282.0	1.37191	0.685954	−	0.727645i	$-0.259385\pi$
$983$	42282.0	1.37191	0.685954	+	0.727645i	$0.259385\pi$
$984$	0	0
$985$	0	0
$986$	−5200.00	−0.167953
$987$	0	0
$988$	30400.0	0.978900
$989$	34476.0	1.10847
$990$	0	0
$991$	1172.00	0.0375679	0.0187840	−	0.999824i	$-0.494021\pi$
$991$	1172.00	0.0375679	0.0187840	+	0.999824i	$0.494021\pi$
$992$	−27648.0	−0.884904
$993$	0	0
$994$	−9888.00	−0.315521
$995$	0	0
$996$	0	0
$997$	31614.0	1.00424	0.502119	−	0.864798i	$-0.332554\pi$
$997$	31614.0	1.00424	0.502119	+	0.864798i	$0.332554\pi$
$998$	−400.000	−0.0126872
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.a.b.1.1		1
3.2	odd	2		25.4.a.c.1.1		1
5.2	odd	4		225.4.b.c.199.1		2
5.3	odd	4		225.4.b.c.199.2		2
5.4	even	2		45.4.a.d.1.1		1
12.11	even	2		400.4.a.m.1.1		1
15.2	even	4		25.4.b.a.24.2		2
15.8	even	4		25.4.b.a.24.1		2
15.14	odd	2		5.4.a.a.1.1	✓	1
20.19	odd	2		720.4.a.u.1.1		1
21.20	even	2		1225.4.a.k.1.1		1
24.5	odd	2		1600.4.a.bi.1.1		1
24.11	even	2		1600.4.a.s.1.1		1
35.34	odd	2		2205.4.a.q.1.1		1
45.4	even	6		405.4.e.c.136.1		2
45.14	odd	6		405.4.e.l.136.1		2
45.29	odd	6		405.4.e.l.271.1		2
45.34	even	6		405.4.e.c.271.1		2
60.23	odd	4		400.4.c.k.49.2		2
60.47	odd	4		400.4.c.k.49.1		2
60.59	even	2		80.4.a.d.1.1		1
105.44	odd	6		245.4.e.f.116.1		2
105.59	even	6		245.4.e.g.226.1		2
105.74	odd	6		245.4.e.f.226.1		2
105.89	even	6		245.4.e.g.116.1		2
105.104	even	2		245.4.a.a.1.1		1
120.29	odd	2		320.4.a.g.1.1		1
120.59	even	2		320.4.a.h.1.1		1
165.164	even	2		605.4.a.d.1.1		1
195.194	odd	2		845.4.a.b.1.1		1
240.29	odd	4		1280.4.d.e.641.2		2
240.59	even	4		1280.4.d.l.641.2		2
240.149	odd	4		1280.4.d.e.641.1		2
240.179	even	4		1280.4.d.l.641.1		2
255.254	odd	2		1445.4.a.a.1.1		1
285.284	even	2		1805.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.4.a.a.1.1	✓	1	15.14	odd	2
25.4.a.c.1.1		1	3.2	odd	2
25.4.b.a.24.1		2	15.8	even	4
25.4.b.a.24.2		2	15.2	even	4
45.4.a.d.1.1		1	5.4	even	2
80.4.a.d.1.1		1	60.59	even	2
225.4.a.b.1.1		1	1.1	even	1	trivial
225.4.b.c.199.1		2	5.2	odd	4
225.4.b.c.199.2		2	5.3	odd	4
245.4.a.a.1.1		1	105.104	even	2
245.4.e.f.116.1		2	105.44	odd	6
245.4.e.f.226.1		2	105.74	odd	6
245.4.e.g.116.1		2	105.89	even	6
245.4.e.g.226.1		2	105.59	even	6
320.4.a.g.1.1		1	120.29	odd	2
320.4.a.h.1.1		1	120.59	even	2
400.4.a.m.1.1		1	12.11	even	2
400.4.c.k.49.1		2	60.47	odd	4
400.4.c.k.49.2		2	60.23	odd	4
405.4.e.c.136.1		2	45.4	even	6
405.4.e.c.271.1		2	45.34	even	6
405.4.e.l.136.1		2	45.14	odd	6
405.4.e.l.271.1		2	45.29	odd	6
605.4.a.d.1.1		1	165.164	even	2
720.4.a.u.1.1		1	20.19	odd	2
845.4.a.b.1.1		1	195.194	odd	2
1225.4.a.k.1.1		1	21.20	even	2
1280.4.d.e.641.1		2	240.149	odd	4
1280.4.d.e.641.2		2	240.29	odd	4
1280.4.d.l.641.1		2	240.179	even	4
1280.4.d.l.641.2		2	240.59	even	4
1445.4.a.a.1.1		1	255.254	odd	2
1600.4.a.s.1.1		1	24.11	even	2
1600.4.a.bi.1.1		1	24.5	odd	2
1805.4.a.h.1.1		1	285.284	even	2
2205.4.a.q.1.1		1	35.34	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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