LMFDB - Embedded newform 45.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, 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("45.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.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:	$2.65508595026$
Analytic rank:	$0$
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	$\chi$	$=$	45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +8.00000 q^{4} +5.00000 q^{5} +6.00000 q^{7} +O(q^{10})$

$q+4.00000 q^{2} +8.00000 q^{4} +5.00000 q^{5} +6.00000 q^{7} +20.0000 q^{10} -32.0000 q^{11} -38.0000 q^{13} +24.0000 q^{14} -64.0000 q^{16} -26.0000 q^{17} +100.000 q^{19} +40.0000 q^{20} -128.000 q^{22} +78.0000 q^{23} +25.0000 q^{25} -152.000 q^{26} +48.0000 q^{28} +50.0000 q^{29} -108.000 q^{31} -256.000 q^{32} -104.000 q^{34} +30.0000 q^{35} +266.000 q^{37} +400.000 q^{38} -22.0000 q^{41} +442.000 q^{43} -256.000 q^{44} +312.000 q^{46} +514.000 q^{47} -307.000 q^{49} +100.000 q^{50} -304.000 q^{52} -2.00000 q^{53} -160.000 q^{55} +200.000 q^{58} -500.000 q^{59} -518.000 q^{61} -432.000 q^{62} -512.000 q^{64} -190.000 q^{65} +126.000 q^{67} -208.000 q^{68} +120.000 q^{70} -412.000 q^{71} -878.000 q^{73} +1064.00 q^{74} +800.000 q^{76} -192.000 q^{77} +600.000 q^{79} -320.000 q^{80} -88.0000 q^{82} -282.000 q^{83} -130.000 q^{85} +1768.00 q^{86} +150.000 q^{89} -228.000 q^{91} +624.000 q^{92} +2056.00 q^{94} +500.000 q^{95} +386.000 q^{97} -1228.00 q^{98} +O(q^{100})$

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	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	8.00000	1.00000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	6.00000	0.323970	0.161985	−	0.986793i	$-0.448210\pi$
$7$	6.00000	0.323970	0.161985	+	0.986793i	$0.448210\pi$
$8$	0	0
$9$	0	0
$10$	20.0000	0.632456
$11$	−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	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	24.0000	0.458162
$15$	0	0
$16$	−64.0000	−1.00000
$17$	−26.0000	−0.370937	−0.185468	−	0.982650i	$-0.559380\pi$
$17$	−26.0000	−0.370937	−0.185468	+	0.982650i	$0.559380\pi$
$18$	0	0
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	40.0000	0.447214
$21$	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
$25$	25.0000	0.200000
$26$	−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
$35$	30.0000	0.144884
$36$	0	0
$37$	266.000	1.18190	0.590948	−	0.806710i	$-0.298754\pi$
$37$	266.000	1.18190	0.590948	+	0.806710i	$0.298754\pi$
$38$	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	$-0.213293\pi$
$43$	442.000	1.56754	0.783772	+	0.621049i	$0.213293\pi$
$44$	−256.000	−0.877124
$45$	0	0
$46$	312.000	1.00004
$47$	514.000	1.59520	0.797602	−	0.603184i	$-0.206101\pi$
$47$	514.000	1.59520	0.797602	+	0.603184i	$0.206101\pi$
$48$	0	0
$49$	−307.000	−0.895044
$50$	100.000	0.282843
$51$	0	0
$52$	−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	$0.500825\pi$
$54$	0	0
$55$	−160.000	−0.392262
$56$	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
$65$	−190.000	−0.362563
$66$	0	0
$67$	126.000	0.229751	0.114876	−	0.993380i	$-0.463353\pi$
$67$	126.000	0.229751	0.114876	+	0.993380i	$0.463353\pi$
$68$	−208.000	−0.370937
$69$	0	0
$70$	120.000	0.204896
$71$	−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	$-0.748537\pi$
$73$	−878.000	−1.40770	−0.703850	+	0.710348i	$0.748537\pi$
$74$	1064.00	1.67145
$75$	0	0
$76$	800.000	1.20745
$77$	−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	$0.359483\pi$
$80$	−320.000	−0.447214
$81$	0	0
$82$	−88.0000	−0.118512
$83$	−282.000	−0.372934	−0.186467	−	0.982461i	$-0.559704\pi$
$83$	−282.000	−0.372934	−0.186467	+	0.982461i	$0.559704\pi$
$84$	0	0
$85$	−130.000	−0.165888
$86$	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
$95$	500.000	0.539989
$96$	0	0
$97$	386.000	0.404045	0.202022	−	0.979381i	$-0.435249\pi$
$97$	386.000	0.404045	0.202022	+	0.979381i	$0.435249\pi$
$98$	−1228.00	−1.26578
$99$	0	0
$100$	200.000	0.200000
$101$	−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	$-0.592337\pi$
$103$	−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	$0.318657\pi$
$108$	0	0
$109$	−550.000	−0.483307	−0.241653	−	0.970363i	$-0.577690\pi$
$109$	−550.000	−0.483307	−0.241653	+	0.970363i	$0.577690\pi$
$110$	−640.000	−0.554742
$111$	0	0
$112$	−384.000	−0.323970
$113$	−1562.00	−1.30036	−0.650180	−	0.759781i	$-0.725306\pi$
$113$	−1562.00	−1.30036	−0.650180	+	0.759781i	$0.725306\pi$
$114$	0	0
$115$	390.000	0.316241
$116$	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
$125$	125.000	0.0894427
$126$	0	0
$127$	1846.00	1.28981	0.644906	−	0.764262i	$-0.276897\pi$
$127$	1846.00	1.28981	0.644906	+	0.764262i	$0.276897\pi$
$128$	0	0
$129$	0	0
$130$	−760.000	−0.512742
$131$	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	$-0.240560\pi$
$137$	2334.00	1.45553	0.727763	+	0.685829i	$0.240560\pi$
$138$	0	0
$139$	−700.000	−0.427146	−0.213573	−	0.976927i	$-0.568510\pi$
$139$	−700.000	−0.427146	−0.213573	+	0.976927i	$0.568510\pi$
$140$	240.000	0.144884
$141$	0	0
$142$	−1648.00	−0.973923
$143$	1216.00	0.711098
$144$	0	0
$145$	250.000	0.143182
$146$	−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
$155$	−540.000	−0.279831
$156$	0	0
$157$	−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
$159$	0	0
$160$	−1280.00	−0.632456
$161$	468.000	0.229090
$162$	0	0
$163$	2762.00	1.32722	0.663609	−	0.748080i	$-0.269024\pi$
$163$	2762.00	1.32722	0.663609	+	0.748080i	$0.269024\pi$
$164$	−176.000	−0.0838006
$165$	0	0
$166$	−1128.00	−0.527408
$167$	−3126.00	−1.44849	−0.724243	−	0.689545i	$-0.757811\pi$
$167$	−3126.00	−1.44849	−0.724243	+	0.689545i	$0.757811\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	−520.000	−0.234601
$171$	0	0
$172$	3536.00	1.56754
$173$	78.0000	0.0342788	0.0171394	−	0.999853i	$-0.494544\pi$
$173$	78.0000	0.0342788	0.0171394	+	0.999853i	$0.494544\pi$
$174$	0	0
$175$	150.000	0.0647939
$176$	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
$185$	1330.00	0.528560
$186$	0	0
$187$	832.000	0.325358
$188$	4112.00	1.59520
$189$	0	0
$190$	2000.00	0.763659
$191$	−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	$-0.521266\pi$
$193$	−358.000	−0.133520	−0.0667601	+	0.997769i	$0.521266\pi$
$194$	1544.00	0.571406
$195$	0	0
$196$	−2456.00	−0.895044
$197$	2214.00	0.800716	0.400358	−	0.916359i	$-0.368886\pi$
$197$	2214.00	0.800716	0.400358	+	0.916359i	$0.368886\pi$
$198$	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
$205$	−110.000	−0.0374767
$206$	−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
$215$	2210.00	0.701027
$216$	0	0
$217$	−648.000	−0.202715
$218$	−2200.00	−0.683499
$219$	0	0
$220$	−1280.00	−0.392262
$221$	988.000	0.300724
$222$	0	0
$223$	−6478.00	−1.94529	−0.972643	−	0.232303i	$-0.925374\pi$
$223$	−6478.00	−1.94529	−0.972643	+	0.232303i	$0.925374\pi$
$224$	−1536.00	−0.458162
$225$	0	0
$226$	−6248.00	−1.83899
$227$	−646.000	−0.188883	−0.0944417	−	0.995530i	$-0.530107\pi$
$227$	−646.000	−0.188883	−0.0944417	+	0.995530i	$0.530107\pi$
$228$	0	0
$229$	3750.00	1.08213	0.541063	−	0.840982i	$-0.318022\pi$
$229$	3750.00	1.08213	0.541063	+	0.840982i	$0.318022\pi$
$230$	1560.00	0.447232
$231$	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	$0.566808\pi$
$234$	0	0
$235$	2570.00	0.713397
$236$	−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
$245$	−1535.00	−0.400276
$246$	0	0
$247$	−3800.00	−0.978900
$248$	0	0
$249$	0	0
$250$	500.000	0.126491
$251$	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	$-0.582267\pi$
$257$	−2106.00	−0.511162	−0.255581	+	0.966788i	$0.582267\pi$
$258$	0	0
$259$	1596.00	0.382898
$260$	−1520.00	−0.362563
$261$	0	0
$262$	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
$265$	−10.0000	−0.00231809
$266$	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
$275$	−800.000	−0.175425
$276$	0	0
$277$	546.000	0.118433	0.0592165	−	0.998245i	$-0.481140\pi$
$277$	546.000	0.118433	0.0592165	+	0.998245i	$0.481140\pi$
$278$	−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	$-0.0715588\pi$
$283$	9282.00	1.94967	0.974837	+	0.222920i	$0.0715588\pi$
$284$	−3296.00	−0.688668
$285$	0	0
$286$	4864.00	1.00564
$287$	−132.000	−0.0271488
$288$	0	0
$289$	−4237.00	−0.862406
$290$	1000.00	0.202490
$291$	0	0
$292$	−7024.00	−1.40770
$293$	−4842.00	−0.965436	−0.482718	−	0.875776i	$-0.660350\pi$
$293$	−4842.00	−0.965436	−0.482718	+	0.875776i	$0.660350\pi$
$294$	0	0
$295$	−2500.00	−0.493409
$296$	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
$305$	−2590.00	−0.486239
$306$	0	0
$307$	−2594.00	−0.482239	−0.241120	−	0.970495i	$-0.577515\pi$
$307$	−2594.00	−0.482239	−0.241120	+	0.970495i	$0.577515\pi$
$308$	−1536.00	−0.284161
$309$	0	0
$310$	−2160.00	−0.395741
$311$	−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	$0.540319\pi$
$318$	0	0
$319$	−1600.00	−0.280824
$320$	−2560.00	−0.447214
$321$	0	0
$322$	1872.00	0.323983
$323$	−2600.00	−0.447888
$324$	0	0
$325$	−950.000	−0.162143
$326$	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
$335$	630.000	0.102748
$336$	0	0
$337$	8866.00	1.43312	0.716561	−	0.697525i	$-0.245715\pi$
$337$	8866.00	1.43312	0.716561	+	0.697525i	$0.245715\pi$
$338$	−3012.00	−0.484708
$339$	0	0
$340$	−1040.00	−0.165888
$341$	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	$-0.457673\pi$
$347$	1714.00	0.265165	0.132583	+	0.991172i	$0.457673\pi$
$348$	0	0
$349$	1150.00	0.176384	0.0881921	−	0.996103i	$-0.471891\pi$
$349$	1150.00	0.176384	0.0881921	+	0.996103i	$0.471891\pi$
$350$	600.000	0.0916324
$351$	0	0
$352$	8192.00	1.24044
$353$	4398.00	0.663122	0.331561	−	0.943434i	$-0.392425\pi$
$353$	4398.00	0.663122	0.331561	+	0.943434i	$0.392425\pi$
$354$	0	0
$355$	−2060.00	−0.307982
$356$	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
$365$	−4390.00	−0.629543
$366$	0	0
$367$	−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
$369$	0	0
$370$	5320.00	0.747496
$371$	−12.0000	−0.00167927
$372$	0	0
$373$	−2078.00	−0.288458	−0.144229	−	0.989544i	$-0.546070\pi$
$373$	−2078.00	−0.288458	−0.144229	+	0.989544i	$0.546070\pi$
$374$	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	$0.320179\pi$
$380$	4000.00	0.539989
$381$	0	0
$382$	−15088.0	−2.02086
$383$	7518.00	1.00301	0.501504	−	0.865155i	$-0.332780\pi$
$383$	7518.00	1.00301	0.501504	+	0.865155i	$0.332780\pi$
$384$	0	0
$385$	−960.000	−0.127081
$386$	−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
$395$	3000.00	0.382143
$396$	0	0
$397$	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
$400$	−1600.00	−0.200000
$401$	−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	$0.264575\pi$
$410$	−440.000	−0.0530001
$411$	0	0
$412$	−4784.00	−0.572065
$413$	−3000.00	−0.357434
$414$	0	0
$415$	−1410.00	−0.166781
$416$	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
$425$	−650.000	−0.0741874
$426$	0	0
$427$	−3108.00	−0.352240
$428$	9552.00	1.07877
$429$	0	0
$430$	8840.00	0.991402
$431$	−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	$-0.519761\pi$
$433$	−1118.00	−0.124082	−0.0620412	+	0.998074i	$0.519761\pi$
$434$	−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	$-0.278419\pi$
$443$	11958.0	1.28249	0.641243	+	0.767337i	$0.278419\pi$
$444$	0	0
$445$	750.000	0.0798953
$446$	−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
$455$	−1140.00	−0.117459
$456$	0	0
$457$	−9494.00	−0.971796	−0.485898	−	0.874016i	$-0.661507\pi$
$457$	−9494.00	−0.971796	−0.485898	+	0.874016i	$0.661507\pi$
$458$	15000.0	1.53036
$459$	0	0
$460$	3120.00	0.316241
$461$	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	$-0.369151\pi$
$463$	7962.00	0.799191	0.399596	+	0.916692i	$0.369151\pi$
$464$	−3200.00	−0.320164
$465$	0	0
$466$	−5928.00	−0.589290
$467$	−6526.00	−0.646654	−0.323327	−	0.946287i	$-0.604801\pi$
$467$	−6526.00	−0.646654	−0.323327	+	0.946287i	$0.604801\pi$
$468$	0	0
$469$	756.000	0.0744325
$470$	10280.0	1.00890
$471$	0	0
$472$	0	0
$473$	−14144.0	−1.37493
$474$	0	0
$475$	2500.00	0.241490
$476$	−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
$485$	1930.00	0.180694
$486$	0	0
$487$	1166.00	0.108494	0.0542469	−	0.998528i	$-0.482724\pi$
$487$	1166.00	0.108494	0.0542469	+	0.998528i	$0.482724\pi$
$488$	0	0
$489$	0	0
$490$	−6140.00	−0.566075
$491$	−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	$0.498572\pi$
$500$	1000.00	0.0894427
$501$	0	0
$502$	4992.00	0.443832
$503$	−2602.00	−0.230651	−0.115325	−	0.993328i	$-0.536791\pi$
$503$	−2602.00	−0.230651	−0.115325	+	0.993328i	$0.536791\pi$
$504$	0	0
$505$	−3510.00	−0.309293
$506$	−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
$515$	−2990.00	−0.255835
$516$	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	$-0.527686\pi$
$523$	−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
$528$	0	0
$529$	−6083.00	−0.499959
$530$	−40.0000	−0.00327828
$531$	0	0
$532$	4800.00	0.391177
$533$	836.000	0.0679384
$534$	0	0
$535$	5970.00	0.482440
$536$	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
$545$	−2750.00	−0.216141
$546$	0	0
$547$	16486.0	1.28865	0.644324	−	0.764753i	$-0.277139\pi$
$547$	16486.0	1.28865	0.644324	+	0.764753i	$0.277139\pi$
$548$	18672.0	1.45553
$549$	0	0
$550$	−3200.00	−0.248088
$551$	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	$-0.646882\pi$
$557$	−11706.0	−0.890483	−0.445242	+	0.895410i	$0.646882\pi$
$558$	0	0
$559$	−16796.0	−1.27083
$560$	−1920.00	−0.144884
$561$	0	0
$562$	27432.0	2.05898
$563$	25038.0	1.87429	0.937146	−	0.348939i	$-0.113458\pi$
$563$	25038.0	1.87429	0.937146	+	0.348939i	$0.113458\pi$
$564$	0	0
$565$	−7810.00	−0.581538
$566$	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
$575$	1950.00	0.141427
$576$	0	0
$577$	−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
$579$	0	0
$580$	2000.00	0.143182
$581$	−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	$-0.665945\pi$
$587$	−14166.0	−0.996071	−0.498035	+	0.867157i	$0.665945\pi$
$588$	0	0
$589$	−10800.0	−0.755528
$590$	−10000.0	−0.697786
$591$	0	0
$592$	−17024.0	−1.18190
$593$	−17842.0	−1.23555	−0.617777	−	0.786354i	$-0.711966\pi$
$593$	−17842.0	−1.23555	−0.617777	+	0.786354i	$0.711966\pi$
$594$	0	0
$595$	−780.000	−0.0537427
$596$	−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
$605$	−1535.00	−0.103151
$606$	0	0
$607$	−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
$610$	−10360.0	−0.687646
$611$	−19532.0	−1.29326
$612$	0	0
$613$	−13238.0	−0.872231	−0.436116	−	0.899891i	$-0.643646\pi$
$613$	−13238.0	−0.872231	−0.436116	+	0.899891i	$0.643646\pi$
$614$	−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	$0.376751\pi$
$618$	0	0
$619$	8300.00	0.538942	0.269471	−	0.963008i	$-0.413151\pi$
$619$	8300.00	0.538942	0.269471	+	0.963008i	$0.413151\pi$
$620$	−4320.00	−0.279831
$621$	0	0
$622$	−29328.0	−1.89059
$623$	900.000	0.0578776
$624$	0	0
$625$	625.000	0.0400000
$626$	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
$635$	9230.00	0.576821
$636$	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	$-0.482010\pi$
$643$	1842.00	0.112973	0.0564863	+	0.998403i	$0.482010\pi$
$644$	3744.00	0.229090
$645$	0	0
$646$	−10400.0	−0.633409
$647$	10114.0	0.614563	0.307282	−	0.951619i	$-0.400581\pi$
$647$	10114.0	0.614563	0.307282	+	0.951619i	$0.400581\pi$
$648$	0	0
$649$	16000.0	0.967727
$650$	−3800.00	−0.229305
$651$	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	$0.600894\pi$
$654$	0	0
$655$	11040.0	0.658578
$656$	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
$665$	3000.00	0.174940
$666$	0	0
$667$	3900.00	0.226400
$668$	−25008.0	−1.44849
$669$	0	0
$670$	2520.00	0.145308
$671$	16576.0	0.953665
$672$	0	0
$673$	−31278.0	−1.79150	−0.895749	−	0.444560i	$-0.853360\pi$
$673$	−31278.0	−1.79150	−0.895749	+	0.444560i	$0.853360\pi$
$674$	35464.0	2.02674
$675$	0	0
$676$	−6024.00	−0.342740
$677$	30054.0	1.70616	0.853079	−	0.521782i	$-0.174732\pi$
$677$	30054.0	1.70616	0.853079	+	0.521782i	$0.174732\pi$
$678$	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	$-0.459607\pi$
$683$	4518.00	0.253113	0.126557	+	0.991959i	$0.459607\pi$
$684$	0	0
$685$	11670.0	0.650931
$686$	−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
$695$	−3500.00	−0.191025
$696$	0	0
$697$	572.000	0.0310847
$698$	4600.00	0.249445
$699$	0	0
$700$	1200.00	0.0647939
$701$	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	$0.423822\pi$
$710$	−8240.00	−0.435552
$711$	0	0
$712$	0	0
$713$	−8424.00	−0.442470
$714$	0	0
$715$	6080.00	0.318013
$716$	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
$725$	1250.00	0.0640329
$726$	0	0
$727$	−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
$729$	0	0
$730$	−17560.0	−0.890308
$731$	−11492.0	−0.581460
$732$	0	0
$733$	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
$738$	0	0
$739$	18700.0	0.930840	0.465420	−	0.885090i	$-0.345903\pi$
$739$	18700.0	0.930840	0.465420	+	0.885090i	$0.345903\pi$
$740$	10640.0	0.528560
$741$	0	0
$742$	−48.0000	−0.00237485
$743$	−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
$745$	−10250.0	−0.504068
$746$	−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
$755$	9260.00	0.446365
$756$	0	0
$757$	−7694.00	−0.369410	−0.184705	−	0.982794i	$-0.559133\pi$
$757$	−7694.00	−0.369410	−0.184705	+	0.982794i	$0.559133\pi$
$758$	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	$0.877889\pi$
$770$	−3840.00	−0.179719
$771$	0	0
$772$	−2864.00	−0.133520
$773$	−22122.0	−1.02933	−0.514666	−	0.857391i	$-0.672084\pi$
$773$	−22122.0	−1.02933	−0.514666	+	0.857391i	$0.672084\pi$
$774$	0	0
$775$	−2700.00	−0.125144
$776$	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
$785$	−12470.0	−0.566972
$786$	0	0
$787$	−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
$789$	0	0
$790$	12000.0	0.540431
$791$	−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	$-0.710044\pi$
$797$	−27586.0	−1.22603	−0.613015	+	0.790071i	$0.710044\pi$
$798$	0	0
$799$	−13364.0	−0.591720
$800$	−6400.00	−0.282843
$801$	0	0
$802$	−25608.0	−1.12749
$803$	28096.0	1.23473
$804$	0	0
$805$	2340.00	0.102452
$806$	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
$815$	13810.0	0.593550
$816$	0	0
$817$	44200.0	1.89273
$818$	44600.0	1.90636
$819$	0	0
$820$	−880.000	−0.0374767
$821$	−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	$-0.429854\pi$
$823$	10322.0	0.437184	0.218592	+	0.975816i	$0.429854\pi$
$824$	0	0
$825$	0	0
$826$	−12000.0	−0.505488
$827$	−8846.00	−0.371954	−0.185977	−	0.982554i	$-0.559545\pi$
$827$	−8846.00	−0.371954	−0.185977	+	0.982554i	$0.559545\pi$
$828$	0	0
$829$	−25350.0	−1.06205	−0.531026	−	0.847355i	$-0.678194\pi$
$829$	−25350.0	−1.06205	−0.531026	+	0.847355i	$0.678194\pi$
$830$	−5640.00	−0.235864
$831$	0	0
$832$	19456.0	0.810716
$833$	7982.00	0.332005
$834$	0	0
$835$	−15630.0	−0.647783
$836$	−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
$845$	−3765.00	−0.153278
$846$	0	0
$847$	−1842.00	−0.0747248
$848$	128.000	0.00518342
$849$	0	0
$850$	−2600.00	−0.104917
$851$	20748.0	0.835761
$852$	0	0
$853$	−16998.0	−0.682298	−0.341149	−	0.940009i	$-0.610816\pi$
$853$	−16998.0	−0.682298	−0.341149	+	0.940009i	$0.610816\pi$
$854$	−12432.0	−0.498143
$855$	0	0
$856$	0	0
$857$	26494.0	1.05603	0.528015	−	0.849235i	$-0.322936\pi$
$857$	26494.0	1.05603	0.528015	+	0.849235i	$0.322936\pi$
$858$	0	0
$859$	−21500.0	−0.853982	−0.426991	−	0.904256i	$-0.640426\pi$
$859$	−21500.0	−0.853982	−0.426991	+	0.904256i	$0.640426\pi$
$860$	17680.0	0.701027
$861$	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	$0.669645\pi$
$864$	0	0
$865$	390.000	0.0153299
$866$	−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
$875$	750.000	0.0289767
$876$	0	0
$877$	30546.0	1.17613	0.588064	−	0.808814i	$-0.299890\pi$
$877$	30546.0	1.17613	0.588064	+	0.808814i	$0.299890\pi$
$878$	−10400.0	−0.399753
$879$	0	0
$880$	10240.0	0.392262
$881$	−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	$-0.672861\pi$
$883$	−27118.0	−1.03351	−0.516757	+	0.856132i	$0.672861\pi$
$884$	7904.00	0.300724
$885$	0	0
$886$	47832.0	1.81371
$887$	38634.0	1.46246	0.731230	−	0.682131i	$-0.238946\pi$
$887$	38634.0	1.46246	0.731230	+	0.682131i	$0.238946\pi$
$888$	0	0
$889$	11076.0	0.417860
$890$	3000.00	0.112989
$891$	0	0
$892$	−51824.0	−1.94529
$893$	51400.0	1.92613
$894$	0	0
$895$	6500.00	0.242761
$896$	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
$905$	8710.00	0.319923
$906$	0	0
$907$	−1794.00	−0.0656767	−0.0328384	−	0.999461i	$-0.510455\pi$
$907$	−1794.00	−0.0656767	−0.0328384	+	0.999461i	$0.510455\pi$
$908$	−5168.00	−0.188883
$909$	0	0
$910$	−4560.00	−0.166113
$911$	−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
$925$	6650.00	0.236379
$926$	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
$935$	4160.00	0.145504
$936$	0	0
$937$	−11334.0	−0.395161	−0.197580	−	0.980287i	$-0.563308\pi$
$937$	−11334.0	−0.395161	−0.197580	+	0.980287i	$0.563308\pi$
$938$	3024.00	0.105263
$939$	0	0
$940$	20560.0	0.713397
$941$	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	$-0.525619\pi$
$947$	−4686.00	−0.160797	−0.0803984	+	0.996763i	$0.525619\pi$
$948$	0	0
$949$	33364.0	1.14124
$950$	10000.0	0.341519
$951$	0	0
$952$	0	0
$953$	598.000	0.0203265	0.0101632	−	0.999948i	$-0.496765\pi$
$953$	598.000	0.0203265	0.0101632	+	0.999948i	$0.496765\pi$
$954$	0	0
$955$	−18860.0	−0.639053
$956$	−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
$965$	−1790.00	−0.0597121
$966$	0	0
$967$	41726.0	1.38761	0.693804	−	0.720163i	$-0.255933\pi$
$967$	41726.0	1.38761	0.693804	+	0.720163i	$0.255933\pi$
$968$	0	0
$969$	0	0
$970$	7720.00	0.255540
$971$	−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	$-0.733881\pi$
$977$	−40946.0	−1.34082	−0.670409	+	0.741992i	$0.733881\pi$
$978$	0	0
$979$	−4800.00	−0.156699
$980$	−12280.0	−0.400276
$981$	0	0
$982$	−28288.0	−0.919253
$983$	−42282.0	−1.37191	−0.685954	−	0.727645i	$-0.740615\pi$
$983$	−42282.0	−1.37191	−0.685954	+	0.727645i	$0.740615\pi$
$984$	0	0
$985$	11070.0	0.358091
$986$	−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
$995$	−13000.0	−0.414199
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.a.d.1.1		1
3.2	odd	2		5.4.a.a.1.1	✓	1
4.3	odd	2		720.4.a.u.1.1		1
5.2	odd	4		225.4.b.c.199.2		2
5.3	odd	4		225.4.b.c.199.1		2
5.4	even	2		225.4.a.b.1.1		1
7.6	odd	2		2205.4.a.q.1.1		1
9.2	odd	6		405.4.e.l.271.1		2
9.4	even	3		405.4.e.c.136.1		2
9.5	odd	6		405.4.e.l.136.1		2
9.7	even	3		405.4.e.c.271.1		2
12.11	even	2		80.4.a.d.1.1		1
15.2	even	4		25.4.b.a.24.1		2
15.8	even	4		25.4.b.a.24.2		2
15.14	odd	2		25.4.a.c.1.1		1
21.2	odd	6		245.4.e.f.116.1		2
21.5	even	6		245.4.e.g.116.1		2
21.11	odd	6		245.4.e.f.226.1		2
21.17	even	6		245.4.e.g.226.1		2
21.20	even	2		245.4.a.a.1.1		1
24.5	odd	2		320.4.a.g.1.1		1
24.11	even	2		320.4.a.h.1.1		1
33.32	even	2		605.4.a.d.1.1		1
39.38	odd	2		845.4.a.b.1.1		1
48.5	odd	4		1280.4.d.e.641.1		2
48.11	even	4		1280.4.d.l.641.2		2
48.29	odd	4		1280.4.d.e.641.2		2
48.35	even	4		1280.4.d.l.641.1		2
51.50	odd	2		1445.4.a.a.1.1		1
57.56	even	2		1805.4.a.h.1.1		1
60.23	odd	4		400.4.c.k.49.1		2
60.47	odd	4		400.4.c.k.49.2		2
60.59	even	2		400.4.a.m.1.1		1
105.104	even	2		1225.4.a.k.1.1		1
120.29	odd	2		1600.4.a.bi.1.1		1
120.59	even	2		1600.4.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.4.a.a.1.1	✓	1	3.2	odd	2
25.4.a.c.1.1		1	15.14	odd	2
25.4.b.a.24.1		2	15.2	even	4
25.4.b.a.24.2		2	15.8	even	4
45.4.a.d.1.1		1	1.1	even	1	trivial
80.4.a.d.1.1		1	12.11	even	2
225.4.a.b.1.1		1	5.4	even	2
225.4.b.c.199.1		2	5.3	odd	4
225.4.b.c.199.2		2	5.2	odd	4
245.4.a.a.1.1		1	21.20	even	2
245.4.e.f.116.1		2	21.2	odd	6
245.4.e.f.226.1		2	21.11	odd	6
245.4.e.g.116.1		2	21.5	even	6
245.4.e.g.226.1		2	21.17	even	6
320.4.a.g.1.1		1	24.5	odd	2
320.4.a.h.1.1		1	24.11	even	2
400.4.a.m.1.1		1	60.59	even	2
400.4.c.k.49.1		2	60.23	odd	4
400.4.c.k.49.2		2	60.47	odd	4
405.4.e.c.136.1		2	9.4	even	3
405.4.e.c.271.1		2	9.7	even	3
405.4.e.l.136.1		2	9.5	odd	6
405.4.e.l.271.1		2	9.2	odd	6
605.4.a.d.1.1		1	33.32	even	2
720.4.a.u.1.1		1	4.3	odd	2
845.4.a.b.1.1		1	39.38	odd	2
1225.4.a.k.1.1		1	105.104	even	2
1280.4.d.e.641.1		2	48.5	odd	4
1280.4.d.e.641.2		2	48.29	odd	4
1280.4.d.l.641.1		2	48.35	even	4
1280.4.d.l.641.2		2	48.11	even	4
1445.4.a.a.1.1		1	51.50	odd	2
1600.4.a.s.1.1		1	120.59	even	2
1600.4.a.bi.1.1		1	120.29	odd	2
1805.4.a.h.1.1		1	57.56	even	2
2205.4.a.q.1.1		1	7.6	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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