LMFDB - Embedded newform 1225.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,4,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1225.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:	$72.2773397570$
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	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} +8.00000 q^{6} -23.0000 q^{9} +O(q^{10})$

$q+4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} +8.00000 q^{6} -23.0000 q^{9} +32.0000 q^{11} +16.0000 q^{12} -38.0000 q^{13} -64.0000 q^{16} +26.0000 q^{17} -92.0000 q^{18} -100.000 q^{19} +128.000 q^{22} +78.0000 q^{23} -152.000 q^{26} -100.000 q^{27} -50.0000 q^{29} +108.000 q^{31} -256.000 q^{32} +64.0000 q^{33} +104.000 q^{34} -184.000 q^{36} -266.000 q^{37} -400.000 q^{38} -76.0000 q^{39} -22.0000 q^{41} -442.000 q^{43} +256.000 q^{44} +312.000 q^{46} -514.000 q^{47} -128.000 q^{48} +52.0000 q^{51} -304.000 q^{52} -2.00000 q^{53} -400.000 q^{54} -200.000 q^{57} -200.000 q^{58} -500.000 q^{59} +518.000 q^{61} +432.000 q^{62} -512.000 q^{64} +256.000 q^{66} -126.000 q^{67} +208.000 q^{68} +156.000 q^{69} +412.000 q^{71} -878.000 q^{73} -1064.00 q^{74} -800.000 q^{76} -304.000 q^{78} +600.000 q^{79} +421.000 q^{81} -88.0000 q^{82} +282.000 q^{83} -1768.00 q^{86} -100.000 q^{87} +150.000 q^{89} +624.000 q^{92} +216.000 q^{93} -2056.00 q^{94} -512.000 q^{96} +386.000 q^{97} -736.000 q^{99} +O(q^{100})$

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 $

$2$	4.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	4.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	8.00000	1.00000
$5$	0	0
$5$	0	0
$6$	8.00000	0.544331
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	32.0000	0.877124	0.438562	−	0.898701i	$-0.355488\pi$
$11$	32.0000	0.877124	0.438562	+	0.898701i	$0.355488\pi$
$12$	16.0000	0.384900
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	0	0
$15$	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	$0.440620\pi$
$18$	−92.0000	−1.20470
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	0	0
$22$	128.000	1.24044
$23$	78.0000	0.707136	0.353568	−	0.935409i	$-0.384968\pi$
$23$	78.0000	0.707136	0.353568	+	0.935409i	$0.384968\pi$
$24$	0	0
$25$	0	0
$26$	−152.000	−1.14653
$27$	−100.000	−0.712778
$28$	0	0
$29$	−50.0000	−0.320164	−0.160082	−	0.987104i	$-0.551176\pi$
$29$	−50.0000	−0.320164	−0.160082	+	0.987104i	$0.551176\pi$
$30$	0	0
$31$	108.000	0.625722	0.312861	−	0.949799i	$-0.398713\pi$
$31$	108.000	0.625722	0.312861	+	0.949799i	$0.398713\pi$
$32$	−256.000	−1.41421
$33$	64.0000	0.337605
$34$	104.000	0.524584
$35$	0	0
$36$	−184.000	−0.851852
$37$	−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
$39$	−76.0000	−0.312045
$40$	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	$0.793899\pi$
$48$	−128.000	−0.384900
$49$	0	0
$50$	0	0
$51$	52.0000	0.142774
$52$	−304.000	−0.810716
$53$	−2.00000	−0.00518342	−0.00259171	−	0.999997i	$-0.500825\pi$
$53$	−2.00000	−0.00518342	−0.00259171	+	0.999997i	$0.500825\pi$
$54$	−400.000	−1.00802
$55$	0	0
$56$	0	0
$57$	−200.000	−0.464748
$58$	−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	$-0.317049\pi$
$61$	518.000	1.08726	0.543632	+	0.839324i	$0.317049\pi$
$62$	432.000	0.884904
$63$	0	0
$64$	−512.000	−1.00000
$65$	0	0
$66$	256.000	0.477446
$67$	−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
$69$	156.000	0.272177
$70$	0	0
$71$	412.000	0.688668	0.344334	−	0.938847i	$-0.388105\pi$
$71$	412.000	0.688668	0.344334	+	0.938847i	$0.388105\pi$
$72$	0	0
$73$	−878.000	−1.40770	−0.703850	−	0.710348i	$-0.748537\pi$
$73$	−878.000	−1.40770	−0.703850	+	0.710348i	$0.748537\pi$
$74$	−1064.00	−1.67145
$75$	0	0
$76$	−800.000	−1.20745
$77$	0	0
$78$	−304.000	−0.441298
$79$	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
$81$	421.000	0.577503
$82$	−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
$87$	−100.000	−0.123231
$88$	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
$91$	0	0
$92$	624.000	0.707136
$93$	216.000	0.240840
$94$	−2056.00	−2.25596
$95$	0	0
$96$	−512.000	−0.544331
$97$	386.000	0.404045	0.202022	−	0.979381i	$-0.435249\pi$
$97$	386.000	0.404045	0.202022	+	0.979381i	$0.435249\pi$
$98$	0	0
$99$	−736.000	−0.747180
$100$	0	0
$101$	−702.000	−0.691600	−0.345800	−	0.938308i	$-0.612392\pi$
$101$	−702.000	−0.691600	−0.345800	+	0.938308i	$0.612392\pi$
$102$	208.000	0.201912
$103$	−598.000	−0.572065	−0.286032	−	0.958220i	$-0.592337\pi$
$103$	−598.000	−0.572065	−0.286032	+	0.958220i	$0.592337\pi$
$104$	0	0
$105$	0	0
$106$	−8.00000	−0.00733046
$107$	1194.00	1.07877	0.539385	−	0.842059i	$-0.318657\pi$
$107$	1194.00	1.07877	0.539385	+	0.842059i	$0.318657\pi$
$108$	−800.000	−0.712778
$109$	−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
$111$	−532.000	−0.454912
$112$	0	0
$113$	−1562.00	−1.30036	−0.650180	−	0.759781i	$-0.725306\pi$
$113$	−1562.00	−1.30036	−0.650180	+	0.759781i	$0.725306\pi$
$114$	−800.000	−0.657253
$115$	0	0
$116$	−400.000	−0.320164
$117$	874.000	0.690610
$118$	−2000.00	−1.56030
$119$	0	0
$120$	0	0
$121$	−307.000	−0.230654
$122$	2072.00	1.53762
$123$	−44.0000	−0.0322548
$124$	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
$129$	−884.000	−0.603348
$130$	0	0
$131$	2208.00	1.47262	0.736312	−	0.676642i	$-0.236565\pi$
$131$	2208.00	1.47262	0.736312	+	0.676642i	$0.236565\pi$
$132$	512.000	0.337605
$133$	0	0
$134$	−504.000	−0.324918
$135$	0	0
$136$	0	0
$137$	2334.00	1.45553	0.727763	−	0.685829i	$-0.240560\pi$
$137$	2334.00	1.45553	0.727763	+	0.685829i	$0.240560\pi$
$138$	624.000	0.384916
$139$	700.000	0.427146	0.213573	−	0.976927i	$-0.431490\pi$
$139$	700.000	0.427146	0.213573	+	0.976927i	$0.431490\pi$
$140$	0	0
$141$	−1028.00	−0.613994
$142$	1648.00	0.973923
$143$	−1216.00	−0.711098
$144$	1472.00	0.851852
$145$	0	0
$146$	−3512.00	−1.99079
$147$	0	0
$148$	−2128.00	−1.18190
$149$	2050.00	1.12713	0.563566	−	0.826071i	$-0.309429\pi$
$149$	2050.00	1.12713	0.563566	+	0.826071i	$0.309429\pi$
$150$	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
$153$	−598.000	−0.315983
$154$	0	0
$155$	0	0
$156$	−608.000	−0.312045
$157$	−2494.00	−1.26779	−0.633894	−	0.773420i	$-0.718545\pi$
$157$	−2494.00	−1.26779	−0.633894	+	0.773420i	$0.718545\pi$
$158$	2400.00	1.20844
$159$	−4.00000	−0.00199510
$160$	0	0
$161$	0	0
$162$	1684.00	0.816713
$163$	−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
$171$	2300.00	1.02857
$172$	−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	$0.505456\pi$
$174$	−400.000	−0.174275
$175$	0	0
$176$	−2048.00	−0.877124
$177$	−1000.00	−0.424659
$178$	600.000	0.252651
$179$	−1300.00	−0.542830	−0.271415	−	0.962462i	$-0.587492\pi$
$179$	−1300.00	−0.542830	−0.271415	+	0.962462i	$0.587492\pi$
$180$	0	0
$181$	−1742.00	−0.715369	−0.357685	−	0.933842i	$-0.616434\pi$
$181$	−1742.00	−0.715369	−0.357685	+	0.933842i	$0.616434\pi$
$182$	0	0
$183$	1036.00	0.418488
$184$	0	0
$185$	0	0
$186$	864.000	0.340600
$187$	832.000	0.325358
$188$	−4112.00	−1.59520
$189$	0	0
$190$	0	0
$191$	3772.00	1.42897	0.714483	−	0.699653i	$-0.246662\pi$
$191$	3772.00	1.42897	0.714483	+	0.699653i	$0.246662\pi$
$192$	−1024.00	−0.384900
$193$	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
$196$	0	0
$197$	2214.00	0.800716	0.400358	−	0.916359i	$-0.368886\pi$
$197$	2214.00	0.800716	0.400358	+	0.916359i	$0.368886\pi$
$198$	−2944.00	−1.05667
$199$	2600.00	0.926176	0.463088	−	0.886312i	$-0.346741\pi$
$199$	2600.00	0.926176	0.463088	+	0.886312i	$0.346741\pi$
$200$	0	0
$201$	−252.000	−0.0884314
$202$	−2808.00	−0.978070
$203$	0	0
$204$	416.000	0.142774
$205$	0	0
$206$	−2392.00	−0.809022
$207$	−1794.00	−0.602375
$208$	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
$213$	824.000	0.265068
$214$	4776.00	1.52561
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−2200.00	−0.683499
$219$	−1756.00	−0.541824
$220$	0	0
$221$	−988.000	−0.300724
$222$	−2128.00	−0.643342
$223$	−6478.00	−1.94529	−0.972643	−	0.232303i	$-0.925374\pi$
$223$	−6478.00	−1.94529	−0.972643	+	0.232303i	$0.925374\pi$
$224$	0	0
$225$	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	$0.469893\pi$
$228$	−1600.00	−0.464748
$229$	−3750.00	−1.08213	−0.541063	−	0.840982i	$-0.681978\pi$
$229$	−3750.00	−1.08213	−0.541063	+	0.840982i	$0.681978\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1482.00	−0.416691	−0.208346	−	0.978055i	$-0.566808\pi$
$233$	−1482.00	−0.416691	−0.208346	+	0.978055i	$0.566808\pi$
$234$	3496.00	0.976670
$235$	0	0
$236$	−4000.00	−1.10330
$237$	1200.00	0.328896
$238$	0	0
$239$	1400.00	0.378906	0.189453	−	0.981890i	$-0.439329\pi$
$239$	1400.00	0.378906	0.189453	+	0.981890i	$0.439329\pi$
$240$	0	0
$241$	−3022.00	−0.807735	−0.403867	−	0.914817i	$-0.632334\pi$
$241$	−3022.00	−0.807735	−0.403867	+	0.914817i	$0.632334\pi$
$242$	−1228.00	−0.326194
$243$	3542.00	0.935059
$244$	4144.00	1.08726
$245$	0	0
$246$	−176.000	−0.0456152
$247$	3800.00	0.978900
$248$	0	0
$249$	564.000	0.143542
$250$	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	$0.417733\pi$
$258$	−3536.00	−0.853263
$259$	0	0
$260$	0	0
$261$	1150.00	0.272733
$262$	8832.00	2.08261
$263$	3638.00	0.852961	0.426480	−	0.904497i	$-0.359753\pi$
$263$	3638.00	0.852961	0.426480	+	0.904497i	$0.359753\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	300.000	0.0687629
$268$	−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	$-0.336342\pi$
$271$	4388.00	0.983587	0.491793	+	0.870712i	$0.336342\pi$
$272$	−1664.00	−0.370937
$273$	0	0
$274$	9336.00	2.05842
$275$	0	0
$276$	1248.00	0.272177
$277$	−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
$279$	−2484.00	−0.533022
$280$	0	0
$281$	−6858.00	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$281$	−6858.00	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$282$	−4112.00	−0.868319
$283$	9282.00	1.94967	0.974837	−	0.222920i	$-0.0715588\pi$
$283$	9282.00	1.94967	0.974837	+	0.222920i	$0.0715588\pi$
$284$	3296.00	0.688668
$285$	0	0
$286$	−4864.00	−1.00564
$287$	0	0
$288$	5888.00	1.20470
$289$	−4237.00	−0.862406
$290$	0	0
$291$	772.000	0.155517
$292$	−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
$297$	−3200.00	−0.625195
$298$	8200.00	1.59400
$299$	−2964.00	−0.573286
$300$	0	0
$301$	0	0
$302$	7408.00	1.41153
$303$	−1404.00	−0.266197
$304$	6400.00	1.20745
$305$	0	0
$306$	−2392.00	−0.446868
$307$	−2594.00	−0.482239	−0.241120	−	0.970495i	$-0.577515\pi$
$307$	−2594.00	−0.482239	−0.241120	+	0.970495i	$0.577515\pi$
$308$	0	0
$309$	−1196.00	−0.220188
$310$	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	$-0.454956\pi$
$313$	1562.00	0.282075	0.141037	+	0.990004i	$0.454956\pi$
$314$	−9976.00	−1.79292
$315$	0	0
$316$	4800.00	0.854497
$317$	−1426.00	−0.252657	−0.126328	−	0.991988i	$-0.540319\pi$
$317$	−1426.00	−0.252657	−0.126328	+	0.991988i	$0.540319\pi$
$318$	−16.0000	−0.00282150
$319$	−1600.00	−0.280824
$320$	0	0
$321$	2388.00	0.415219
$322$	0	0
$323$	−2600.00	−0.447888
$324$	3368.00	0.577503
$325$	0	0
$326$	−11048.0	−1.87697
$327$	−1100.00	−0.186025
$328$	0	0
$329$	0	0
$330$	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
$333$	6118.00	1.00680
$334$	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
$339$	−3124.00	−0.500509
$340$	0	0
$341$	3456.00	0.548835
$342$	9200.00	1.45462
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−312.000	−0.0484775
$347$	1714.00	0.265165	0.132583	−	0.991172i	$-0.457673\pi$
$347$	1714.00	0.265165	0.132583	+	0.991172i	$0.457673\pi$
$348$	−800.000	−0.123231
$349$	−1150.00	−0.176384	−0.0881921	−	0.996103i	$-0.528109\pi$
$349$	−1150.00	−0.176384	−0.0881921	+	0.996103i	$0.528109\pi$
$350$	0	0
$351$	3800.00	0.577860
$352$	−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	$0.607575\pi$
$354$	−4000.00	−0.600558
$355$	0	0
$356$	1200.00	0.178651
$357$	0	0
$358$	−5200.00	−0.767677
$359$	1800.00	0.264625	0.132312	−	0.991208i	$-0.457760\pi$
$359$	1800.00	0.264625	0.132312	+	0.991208i	$0.457760\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	−6968.00	−1.01168
$363$	−614.000	−0.0887786
$364$	0	0
$365$	0	0
$366$	4144.00	0.591832
$367$	−5874.00	−0.835478	−0.417739	−	0.908567i	$-0.637177\pi$
$367$	−5874.00	−0.835478	−0.417739	+	0.908567i	$0.637177\pi$
$368$	−4992.00	−0.707136
$369$	506.000	0.0713857
$370$	0	0
$371$	0	0
$372$	1728.00	0.240840
$373$	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
$381$	−3692.00	−0.496449
$382$	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
$387$	10166.0	1.33531
$388$	3088.00	0.404045
$389$	−1950.00	−0.254162	−0.127081	−	0.991892i	$-0.540561\pi$
$389$	−1950.00	−0.254162	−0.127081	+	0.991892i	$0.540561\pi$
$390$	0	0
$391$	2028.00	0.262303
$392$	0	0
$393$	4416.00	0.566814
$394$	8856.00	1.13238
$395$	0	0
$396$	−5888.00	−0.747180
$397$	13786.0	1.74282	0.871410	−	0.490555i	$-0.163206\pi$
$397$	13786.0	1.74282	0.871410	+	0.490555i	$0.163206\pi$
$398$	10400.0	1.30981
$399$	0	0
$400$	0	0
$401$	6402.00	0.797258	0.398629	−	0.917112i	$-0.369486\pi$
$401$	6402.00	0.797258	0.398629	+	0.917112i	$0.369486\pi$
$402$	−1008.00	−0.125061
$403$	−4104.00	−0.507282
$404$	−5616.00	−0.691600
$405$	0	0
$406$	0	0
$407$	−8512.00	−1.03667
$408$	0	0
$409$	−11150.0	−1.34800	−0.674000	−	0.738731i	$-0.735425\pi$
$409$	−11150.0	−1.34800	−0.674000	+	0.738731i	$0.735425\pi$
$410$	0	0
$411$	4668.00	0.560232
$412$	−4784.00	−0.572065
$413$	0	0
$414$	−7176.00	−0.851887
$415$	0	0
$416$	9728.00	1.14653
$417$	1400.00	0.164408
$418$	−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
$423$	11822.0	1.35888
$424$	0	0
$425$	0	0
$426$	3296.00	0.374863
$427$	0	0
$428$	9552.00	1.07877
$429$	−2432.00	−0.273702
$430$	0	0
$431$	7692.00	0.859653	0.429827	−	0.902911i	$-0.358575\pi$
$431$	7692.00	0.859653	0.429827	+	0.902911i	$0.358575\pi$
$432$	6400.00	0.712778
$433$	−1118.00	−0.124082	−0.0620412	−	0.998074i	$-0.519761\pi$
$433$	−1118.00	−0.124082	−0.0620412	+	0.998074i	$0.519761\pi$
$434$	0	0
$435$	0	0
$436$	−4400.00	−0.483307
$437$	−7800.00	−0.853832
$438$	−7024.00	−0.766255
$439$	2600.00	0.282668	0.141334	−	0.989962i	$-0.454861\pi$
$439$	2600.00	0.282668	0.141334	+	0.989962i	$0.454861\pi$
$440$	0	0
$441$	0	0
$442$	−3952.00	−0.425288
$443$	11958.0	1.28249	0.641243	−	0.767337i	$-0.278419\pi$
$443$	11958.0	1.28249	0.641243	+	0.767337i	$0.278419\pi$
$444$	−4256.00	−0.454912
$445$	0	0
$446$	−25912.0	−2.75105
$447$	4100.00	0.433833
$448$	0	0
$449$	−17050.0	−1.79207	−0.896035	−	0.443984i	$-0.853565\pi$
$449$	−17050.0	−1.79207	−0.896035	+	0.443984i	$0.853565\pi$
$450$	0	0
$451$	−704.000	−0.0735035
$452$	−12496.0	−1.30036
$453$	3704.00	0.384170
$454$	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
$459$	−2600.00	−0.264396
$460$	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	$0.395199\pi$
$468$	6992.00	0.690610
$469$	0	0
$470$	0	0
$471$	−4988.00	−0.487972
$472$	0	0
$473$	−14144.0	−1.37493
$474$	4800.00	0.465129
$475$	0	0
$476$	0	0
$477$	46.0000	0.00441550
$478$	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
$486$	14168.0	1.32237
$487$	−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
$489$	−5524.00	−0.510846
$490$	0	0
$491$	7072.00	0.650010	0.325005	−	0.945712i	$-0.394634\pi$
$491$	7072.00	0.650010	0.325005	+	0.945712i	$0.394634\pi$
$492$	−352.000	−0.0322548
$493$	−1300.00	−0.118761
$494$	15200.0	1.38437
$495$	0	0
$496$	−6912.00	−0.625722
$497$	0	0
$498$	2256.00	0.203000
$499$	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
$501$	6252.00	0.557522
$502$	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
$507$	−1506.00	−0.131921
$508$	−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
$511$	0	0
$512$	16384.0	1.41421
$513$	10000.0	0.860645
$514$	8424.00	0.722892
$515$	0	0
$516$	−7072.00	−0.603348
$517$	−16448.0	−1.39919
$518$	0	0
$519$	−156.000	−0.0131939
$520$	0	0
$521$	3638.00	0.305919	0.152959	−	0.988232i	$-0.451120\pi$
$521$	3638.00	0.305919	0.152959	+	0.988232i	$0.451120\pi$
$522$	4600.00	0.385702
$523$	−2078.00	−0.173737	−0.0868686	−	0.996220i	$-0.527686\pi$
$523$	−2078.00	−0.173737	−0.0868686	+	0.996220i	$0.527686\pi$
$524$	17664.0	1.47262
$525$	0	0
$526$	14552.0	1.20627
$527$	2808.00	0.232103
$528$	−4096.00	−0.337605
$529$	−6083.00	−0.499959
$530$	0	0
$531$	11500.0	0.939845
$532$	0	0
$533$	836.000	0.0679384
$534$	1200.00	0.0972455
$535$	0	0
$536$	0	0
$537$	−2600.00	−0.208935
$538$	26200.0	2.09956
$539$	0	0
$540$	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
$543$	−3484.00	−0.275346
$544$	−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
$549$	−11914.0	−0.926188
$550$	0	0
$551$	5000.00	0.386583
$552$	0	0
$553$	0	0
$554$	−2184.00	−0.167490
$555$	0	0
$556$	5600.00	0.427146
$557$	−11706.0	−0.890483	−0.445242	−	0.895410i	$-0.646882\pi$
$557$	−11706.0	−0.890483	−0.445242	+	0.895410i	$0.646882\pi$
$558$	−9936.00	−0.753807
$559$	16796.0	1.27083
$560$	0	0
$561$	1664.00	0.125230
$562$	−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	$0.886542\pi$
$564$	−8224.00	−0.613994
$565$	0	0
$566$	37128.0	2.75725
$567$	0	0
$568$	0	0
$569$	17550.0	1.29303	0.646515	−	0.762901i	$-0.276226\pi$
$569$	17550.0	1.29303	0.646515	+	0.762901i	$0.276226\pi$
$570$	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
$573$	7544.00	0.550009
$574$	0	0
$575$	0	0
$576$	11776.0	0.851852
$577$	−13654.0	−0.985136	−0.492568	−	0.870274i	$-0.663942\pi$
$577$	−13654.0	−0.985136	−0.492568	+	0.870274i	$0.663942\pi$
$578$	−16948.0	−1.21963
$579$	716.000	0.0513920
$580$	0	0
$581$	0	0
$582$	3088.00	0.219934
$583$	−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
$591$	4428.00	0.308196
$592$	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	$0.288034\pi$
$594$	−12800.0	−0.884159
$595$	0	0
$596$	16400.0	1.12713
$597$	5200.00	0.356485
$598$	−11856.0	−0.810749
$599$	−17600.0	−1.20053	−0.600264	−	0.799802i	$-0.704938\pi$
$599$	−17600.0	−1.20053	−0.600264	+	0.799802i	$0.704938\pi$
$600$	0	0
$601$	−27302.0	−1.85303	−0.926516	−	0.376256i	$-0.877211\pi$
$601$	−27302.0	−1.85303	−0.926516	+	0.376256i	$0.877211\pi$
$602$	0	0
$603$	2898.00	0.195714
$604$	14816.0	0.998103
$605$	0	0
$606$	−5616.00	−0.376459
$607$	−3794.00	−0.253696	−0.126848	−	0.991922i	$-0.540486\pi$
$607$	−3794.00	−0.253696	−0.126848	+	0.991922i	$0.540486\pi$
$608$	25600.0	1.70759
$609$	0	0
$610$	0	0
$611$	19532.0	1.29326
$612$	−4784.00	−0.315983
$613$	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	$-0.376751\pi$
$617$	11574.0	0.755189	0.377595	+	0.925971i	$0.376751\pi$
$618$	−4784.00	−0.311393
$619$	−8300.00	−0.538942	−0.269471	−	0.963008i	$-0.586849\pi$
$619$	−8300.00	−0.538942	−0.269471	+	0.963008i	$0.586849\pi$
$620$	0	0
$621$	−7800.00	−0.504031
$622$	−29328.0	−1.89059
$623$	0	0
$624$	4864.00	0.312045
$625$	0	0
$626$	6248.00	0.398914
$627$	−6400.00	−0.407642
$628$	−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
$633$	−2336.00	−0.146679
$634$	−5704.00	−0.357310
$635$	0	0
$636$	−32.0000	−0.00199510
$637$	0	0
$638$	−6400.00	−0.397145
$639$	−9476.00	−0.586643
$640$	0	0
$641$	−27378.0	−1.68700	−0.843499	−	0.537130i	$-0.819508\pi$
$641$	−27378.0	−1.68700	−0.843499	+	0.537130i	$0.819508\pi$
$642$	9552.00	0.587208
$643$	1842.00	0.112973	0.0564863	−	0.998403i	$-0.482010\pi$
$643$	1842.00	0.112973	0.0564863	+	0.998403i	$0.482010\pi$
$644$	0	0
$645$	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	$-0.600894\pi$
$653$	−10402.0	−0.623372	−0.311686	+	0.950185i	$0.600894\pi$
$654$	−4400.00	−0.263079
$655$	0	0
$656$	1408.00	0.0838006
$657$	20194.0	1.19915
$658$	0	0
$659$	7100.00	0.419692	0.209846	−	0.977734i	$-0.432704\pi$
$659$	7100.00	0.419692	0.209846	+	0.977734i	$0.432704\pi$
$660$	0	0
$661$	7118.00	0.418847	0.209424	−	0.977825i	$-0.432841\pi$
$661$	7118.00	0.418847	0.209424	+	0.977825i	$0.432841\pi$
$662$	−16032.0	−0.941241
$663$	−1976.00	−0.115749
$664$	0	0
$665$	0	0
$666$	24472.0	1.42383
$667$	−3900.00	−0.226400
$668$	25008.0	1.44849
$669$	−12956.0	−0.748741
$670$	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	$0.825268\pi$
$678$	−12496.0	−0.707826
$679$	0	0
$680$	0	0
$681$	1292.00	0.0727012
$682$	13824.0	0.776171
$683$	4518.00	0.253113	0.126557	−	0.991959i	$-0.459607\pi$
$683$	4518.00	0.253113	0.126557	+	0.991959i	$0.459607\pi$
$684$	18400.0	1.02857
$685$	0	0
$686$	0	0
$687$	−7500.00	−0.416511
$688$	28288.0	1.56754
$689$	76.0000	0.00420228
$690$	0	0
$691$	−29272.0	−1.61152	−0.805759	−	0.592243i	$-0.798242\pi$
$691$	−29272.0	−1.61152	−0.805759	+	0.592243i	$0.798242\pi$
$692$	−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
$699$	−2964.00	−0.160385
$700$	0	0
$701$	−5798.00	−0.312393	−0.156196	−	0.987726i	$-0.549923\pi$
$701$	−5798.00	−0.312393	−0.156196	+	0.987726i	$0.549923\pi$
$702$	15200.0	0.817218
$703$	26600.0	1.42708
$704$	−16384.0	−0.877124
$705$	0	0
$706$	−17592.0	−0.937796
$707$	0	0
$708$	−8000.00	−0.424659
$709$	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
$711$	−13800.0	−0.727905
$712$	0	0
$713$	8424.00	0.442470
$714$	0	0
$715$	0	0
$716$	−10400.0	−0.542830
$717$	2800.00	0.145841
$718$	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
$721$	0	0
$722$	12564.0	0.647623
$723$	−6044.00	−0.310897
$724$	−13936.0	−0.715369
$725$	0	0
$726$	−2456.00	−0.125552
$727$	−8554.00	−0.436383	−0.218191	−	0.975906i	$-0.570016\pi$
$727$	−8554.00	−0.436383	−0.218191	+	0.975906i	$0.570016\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	−11492.0	−0.581460
$732$	8288.00	0.418488
$733$	2882.00	0.145224	0.0726119	−	0.997360i	$-0.476867\pi$
$733$	2882.00	0.145224	0.0726119	+	0.997360i	$0.476867\pi$
$734$	−23496.0	−1.18154
$735$	0	0
$736$	−19968.0	−1.00004
$737$	−4032.00	−0.201521
$738$	2024.00	0.100955
$739$	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
$741$	7600.00	0.376779
$742$	0	0
$743$	−12242.0	−0.604462	−0.302231	−	0.953235i	$-0.597731\pi$
$743$	−12242.0	−0.604462	−0.302231	+	0.953235i	$0.597731\pi$
$744$	0	0
$745$	0	0
$746$	8312.00	0.407941
$747$	−6486.00	−0.317685
$748$	6656.00	0.325358
$749$	0	0
$750$	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
$753$	2496.00	0.120796
$754$	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
$759$	4992.00	0.238733
$760$	0	0
$761$	4518.00	0.215213	0.107607	−	0.994194i	$-0.465681\pi$
$761$	4518.00	0.215213	0.107607	+	0.994194i	$0.465681\pi$
$762$	−14768.0	−0.702084
$763$	0	0
$764$	30176.0	1.42897
$765$	0	0
$766$	−30072.0	−1.41847
$767$	19000.0	0.894459
$768$	8192.00	0.384900
$769$	39550.0	1.85463	0.927314	−	0.374283i	$-0.122111\pi$
$769$	39550.0	1.85463	0.927314	+	0.374283i	$0.122111\pi$
$770$	0	0
$771$	4212.00	0.196746
$772$	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	$0.327916\pi$
$774$	40664.0	1.88842
$775$	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
$783$	5000.00	0.228206
$784$	0	0
$785$	0	0
$786$	17664.0	0.801595
$787$	−16634.0	−0.753416	−0.376708	−	0.926332i	$-0.622944\pi$
$787$	−16634.0	−0.753416	−0.376708	+	0.926332i	$0.622944\pi$
$788$	17712.0	0.800716
$789$	7276.00	0.328305
$790$	0	0
$791$	0	0
$792$	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
$801$	−3450.00	−0.152184
$802$	25608.0	1.12749
$803$	−28096.0	−1.23473
$804$	−2016.00	−0.0884314
$805$	0	0
$806$	−16416.0	−0.717406
$807$	13100.0	0.571427
$808$	0	0
$809$	3850.00	0.167316	0.0836581	−	0.996495i	$-0.473340\pi$
$809$	3850.00	0.167316	0.0836581	+	0.996495i	$0.473340\pi$
$810$	0	0
$811$	−10032.0	−0.434366	−0.217183	−	0.976131i	$-0.569687\pi$
$811$	−10032.0	−0.434366	−0.217183	+	0.976131i	$0.569687\pi$
$812$	0	0
$813$	8776.00	0.378583
$814$	−34048.0	−1.46607
$815$	0	0
$816$	−3328.00	−0.142774
$817$	44200.0	1.89273
$818$	−44600.0	−1.90636
$819$	0	0
$820$	0	0
$821$	20562.0	0.874079	0.437039	−	0.899442i	$-0.356027\pi$
$821$	20562.0	0.874079	0.437039	+	0.899442i	$0.356027\pi$
$822$	18672.0	0.792288
$823$	−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
$826$	0	0
$827$	−8846.00	−0.371954	−0.185977	−	0.982554i	$-0.559545\pi$
$827$	−8846.00	−0.371954	−0.185977	+	0.982554i	$0.559545\pi$
$828$	−14352.0	−0.602375
$829$	25350.0	1.06205	0.531026	−	0.847355i	$-0.321806\pi$
$829$	25350.0	1.06205	0.531026	+	0.847355i	$0.321806\pi$
$830$	0	0
$831$	−1092.00	−0.0455849
$832$	19456.0	0.810716
$833$	0	0
$834$	5600.00	0.232509
$835$	0	0
$836$	−25600.0	−1.05908
$837$	−10800.0	−0.446001
$838$	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
$843$	−13716.0	−0.560385
$844$	−9344.00	−0.381083
$845$	0	0
$846$	47288.0	1.92174
$847$	0	0
$848$	128.000	0.00518342
$849$	18564.0	0.750430
$850$	0	0
$851$	−20748.0	−0.835761
$852$	6592.00	0.265068
$853$	−16998.0	−0.682298	−0.341149	−	0.940009i	$-0.610816\pi$
$853$	−16998.0	−0.682298	−0.341149	+	0.940009i	$0.610816\pi$
$854$	0	0
$855$	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	$0.677064\pi$
$858$	−9728.00	−0.387073
$859$	21500.0	0.853982	0.426991	−	0.904256i	$-0.359574\pi$
$859$	21500.0	0.853982	0.426991	+	0.904256i	$0.359574\pi$
$860$	0	0
$861$	0	0
$862$	30768.0	1.21573
$863$	−25762.0	−1.01616	−0.508082	−	0.861309i	$-0.669645\pi$
$863$	−25762.0	−1.01616	−0.508082	+	0.861309i	$0.669645\pi$
$864$	25600.0	1.00802
$865$	0	0
$866$	−4472.00	−0.175479
$867$	−8474.00	−0.331940
$868$	0	0
$869$	19200.0	0.749500
$870$	0	0
$871$	4788.00	0.186263
$872$	0	0
$873$	−8878.00	−0.344186
$874$	−31200.0	−1.20750
$875$	0	0
$876$	−14048.0	−0.541824
$877$	−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
$879$	9684.00	0.371596
$880$	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
$889$	0	0
$890$	0	0
$891$	13472.0	0.506542
$892$	−51824.0	−1.94529
$893$	51400.0	1.92613
$894$	16400.0	0.613532
$895$	0	0
$896$	0	0
$897$	−5928.00	−0.220658
$898$	−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
$906$	14816.0	0.543299
$907$	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
$909$	16146.0	0.589141
$910$	0	0
$911$	41732.0	1.51772	0.758860	−	0.651254i	$-0.225757\pi$
$911$	41732.0	1.51772	0.758860	+	0.651254i	$0.225757\pi$
$912$	12800.0	0.464748
$913$	9024.00	0.327109
$914$	37976.0	1.37433
$915$	0	0
$916$	−30000.0	−1.08213
$917$	0	0
$918$	−10400.0	−0.373912
$919$	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
$921$	−5188.00	−0.185614
$922$	45672.0	1.63137
$923$	−15656.0	−0.558314
$924$	0	0
$925$	0	0
$926$	−31848.0	−1.13023
$927$	13754.0	0.487315
$928$	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
$931$	0	0
$932$	−11856.0	−0.416691
$933$	−14664.0	−0.514553
$934$	26104.0	0.914506
$935$	0	0
$936$	0	0
$937$	−11334.0	−0.395161	−0.197580	−	0.980287i	$-0.563308\pi$
$937$	−11334.0	−0.395161	−0.197580	+	0.980287i	$0.563308\pi$
$938$	0	0
$939$	3124.00	0.108571
$940$	0	0
$941$	31178.0	1.08010	0.540050	−	0.841633i	$-0.318405\pi$
$941$	31178.0	1.08010	0.540050	+	0.841633i	$0.318405\pi$
$942$	−19952.0	−0.690097
$943$	−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	$-0.525619\pi$
$947$	−4686.00	−0.160797	−0.0803984	+	0.996763i	$0.525619\pi$
$948$	9600.00	0.328896
$949$	33364.0	1.14124
$950$	0	0
$951$	−2852.00	−0.0972476
$952$	0	0
$953$	598.000	0.0203265	0.0101632	−	0.999948i	$-0.496765\pi$
$953$	598.000	0.0203265	0.0101632	+	0.999948i	$0.496765\pi$
$954$	184.000	0.00624447
$955$	0	0
$956$	11200.0	0.378906
$957$	−3200.00	−0.108089
$958$	−69600.0	−2.34726
$959$	0	0
$960$	0	0
$961$	−18127.0	−0.608472
$962$	40432.0	1.35507
$963$	−27462.0	−0.918952
$964$	−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
$969$	−5200.00	−0.172392
$970$	0	0
$971$	−24312.0	−0.803511	−0.401756	−	0.915747i	$-0.631600\pi$
$971$	−24312.0	−0.803511	−0.401756	+	0.915747i	$0.631600\pi$
$972$	28336.0	0.935059
$973$	0	0
$974$	−4664.00	−0.153433
$975$	0	0
$976$	−33152.0	−1.08726
$977$	−40946.0	−1.34082	−0.670409	−	0.741992i	$-0.733881\pi$
$977$	−40946.0	−1.34082	−0.670409	+	0.741992i	$0.733881\pi$
$978$	−22096.0	−0.722446
$979$	4800.00	0.156699
$980$	0	0
$981$	12650.0	0.411706
$982$	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
$993$	−8016.00	−0.256173
$994$	0	0
$995$	0	0
$996$	4512.00	0.143542
$997$	−31614.0	−1.00424	−0.502119	−	0.864798i	$-0.667446\pi$
$997$	−31614.0	−1.00424	−0.502119	+	0.864798i	$0.667446\pi$
$998$	400.000	0.0126872
$999$	26600.0	0.842429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.4.a.k.1.1		1
5.4	even	2		245.4.a.a.1.1		1
7.6	odd	2		25.4.a.c.1.1		1
15.14	odd	2		2205.4.a.q.1.1		1
21.20	even	2		225.4.a.b.1.1		1
28.27	even	2		400.4.a.m.1.1		1
35.4	even	6		245.4.e.g.226.1		2
35.9	even	6		245.4.e.g.116.1		2
35.13	even	4		25.4.b.a.24.1		2
35.19	odd	6		245.4.e.f.116.1		2
35.24	odd	6		245.4.e.f.226.1		2
35.27	even	4		25.4.b.a.24.2		2
35.34	odd	2		5.4.a.a.1.1	✓	1
56.13	odd	2		1600.4.a.bi.1.1		1
56.27	even	2		1600.4.a.s.1.1		1
105.62	odd	4		225.4.b.c.199.1		2
105.83	odd	4		225.4.b.c.199.2		2
105.104	even	2		45.4.a.d.1.1		1
140.27	odd	4		400.4.c.k.49.1		2
140.83	odd	4		400.4.c.k.49.2		2
140.139	even	2		80.4.a.d.1.1		1
280.69	odd	2		320.4.a.g.1.1		1
280.139	even	2		320.4.a.h.1.1		1
315.34	odd	6		405.4.e.l.271.1		2
315.104	even	6		405.4.e.c.136.1		2
315.139	odd	6		405.4.e.l.136.1		2
315.209	even	6		405.4.e.c.271.1		2
385.384	even	2		605.4.a.d.1.1		1
420.419	odd	2		720.4.a.u.1.1		1
455.454	odd	2		845.4.a.b.1.1		1
560.69	odd	4		1280.4.d.e.641.1		2
560.139	even	4		1280.4.d.l.641.2		2
560.349	odd	4		1280.4.d.e.641.2		2
560.419	even	4		1280.4.d.l.641.1		2
595.594	odd	2		1445.4.a.a.1.1		1
665.664	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	35.34	odd	2
25.4.a.c.1.1		1	7.6	odd	2
25.4.b.a.24.1		2	35.13	even	4
25.4.b.a.24.2		2	35.27	even	4
45.4.a.d.1.1		1	105.104	even	2
80.4.a.d.1.1		1	140.139	even	2
225.4.a.b.1.1		1	21.20	even	2
225.4.b.c.199.1		2	105.62	odd	4
225.4.b.c.199.2		2	105.83	odd	4
245.4.a.a.1.1		1	5.4	even	2
245.4.e.f.116.1		2	35.19	odd	6
245.4.e.f.226.1		2	35.24	odd	6
245.4.e.g.116.1		2	35.9	even	6
245.4.e.g.226.1		2	35.4	even	6
320.4.a.g.1.1		1	280.69	odd	2
320.4.a.h.1.1		1	280.139	even	2
400.4.a.m.1.1		1	28.27	even	2
400.4.c.k.49.1		2	140.27	odd	4
400.4.c.k.49.2		2	140.83	odd	4
405.4.e.c.136.1		2	315.104	even	6
405.4.e.c.271.1		2	315.209	even	6
405.4.e.l.136.1		2	315.139	odd	6
405.4.e.l.271.1		2	315.34	odd	6
605.4.a.d.1.1		1	385.384	even	2
720.4.a.u.1.1		1	420.419	odd	2
845.4.a.b.1.1		1	455.454	odd	2
1225.4.a.k.1.1		1	1.1	even	1	trivial
1280.4.d.e.641.1		2	560.69	odd	4
1280.4.d.e.641.2		2	560.349	odd	4
1280.4.d.l.641.1		2	560.419	even	4
1280.4.d.l.641.2		2	560.139	even	4
1445.4.a.a.1.1		1	595.594	odd	2
1600.4.a.s.1.1		1	56.27	even	2
1600.4.a.bi.1.1		1	56.13	odd	2
1805.4.a.h.1.1		1	665.664	even	2
2205.4.a.q.1.1		1	15.14	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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