LMFDB - Embedded newform 90.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [90,4,Mod(1,90)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(90, 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("90.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 90 = 2 \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	90.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:	$5.31017190052$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	90.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +14.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000 q^{5} +14.0000 q^{7} -8.00000 q^{8} +10.0000 q^{10} +6.00000 q^{11} +68.0000 q^{13} -28.0000 q^{14} +16.0000 q^{16} +78.0000 q^{17} +44.0000 q^{19} -20.0000 q^{20} -12.0000 q^{22} +120.000 q^{23} +25.0000 q^{25} -136.000 q^{26} +56.0000 q^{28} +126.000 q^{29} -244.000 q^{31} -32.0000 q^{32} -156.000 q^{34} -70.0000 q^{35} -304.000 q^{37} -88.0000 q^{38} +40.0000 q^{40} -480.000 q^{41} +104.000 q^{43} +24.0000 q^{44} -240.000 q^{46} +600.000 q^{47} -147.000 q^{49} -50.0000 q^{50} +272.000 q^{52} -258.000 q^{53} -30.0000 q^{55} -112.000 q^{56} -252.000 q^{58} +534.000 q^{59} +362.000 q^{61} +488.000 q^{62} +64.0000 q^{64} -340.000 q^{65} -268.000 q^{67} +312.000 q^{68} +140.000 q^{70} -972.000 q^{71} +470.000 q^{73} +608.000 q^{74} +176.000 q^{76} +84.0000 q^{77} +1244.00 q^{79} -80.0000 q^{80} +960.000 q^{82} +396.000 q^{83} -390.000 q^{85} -208.000 q^{86} -48.0000 q^{88} -972.000 q^{89} +952.000 q^{91} +480.000 q^{92} -1200.00 q^{94} -220.000 q^{95} -46.0000 q^{97} +294.000 q^{98} +O(q^{100})\)

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	14.0000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	14.0000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	6.00000	0.164461	0.0822304	−	0.996613i	$-0.473796\pi$
$11$	6.00000	0.164461	0.0822304	+	0.996613i	$0.473796\pi$
$12$	0	0
$13$	68.0000	1.45075	0.725377	−	0.688352i	$-0.241665\pi$
$13$	68.0000	1.45075	0.725377	+	0.688352i	$0.241665\pi$
$14$	−28.0000	−0.534522
$15$	0	0
$16$	16.0000	0.250000
$17$	78.0000	1.11281	0.556405	−	0.830911i	$-0.312180\pi$
$17$	78.0000	1.11281	0.556405	+	0.830911i	$0.312180\pi$
$18$	0	0
$19$	44.0000	0.531279	0.265639	−	0.964072i	$-0.414417\pi$
$19$	44.0000	0.531279	0.265639	+	0.964072i	$0.414417\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	−12.0000	−0.116291
$23$	120.000	1.08790	0.543951	−	0.839117i	$-0.316928\pi$
$23$	120.000	1.08790	0.543951	+	0.839117i	$0.316928\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−136.000	−1.02584
$27$	0	0
$28$	56.0000	0.377964
$29$	126.000	0.806814	0.403407	−	0.915021i	$-0.367826\pi$
$29$	126.000	0.806814	0.403407	+	0.915021i	$0.367826\pi$
$30$	0	0
$31$	−244.000	−1.41367	−0.706834	−	0.707380i	$-0.749877\pi$
$31$	−244.000	−1.41367	−0.706834	+	0.707380i	$0.749877\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−156.000	−0.786876
$35$	−70.0000	−0.338062
$36$	0	0
$37$	−304.000	−1.35074	−0.675369	−	0.737480i	$-0.736016\pi$
$37$	−304.000	−1.35074	−0.675369	+	0.737480i	$0.736016\pi$
$38$	−88.0000	−0.375671
$39$	0	0
$40$	40.0000	0.158114
$41$	−480.000	−1.82838	−0.914188	−	0.405291i	$-0.867170\pi$
$41$	−480.000	−1.82838	−0.914188	+	0.405291i	$0.867170\pi$
$42$	0	0
$43$	104.000	0.368834	0.184417	−	0.982848i	$-0.440960\pi$
$43$	104.000	0.368834	0.184417	+	0.982848i	$0.440960\pi$
$44$	24.0000	0.0822304
$45$	0	0
$46$	−240.000	−0.769262
$47$	600.000	1.86211	0.931053	−	0.364884i	$-0.118891\pi$
$47$	600.000	1.86211	0.931053	+	0.364884i	$0.118891\pi$
$48$	0	0
$49$	−147.000	−0.428571
$50$	−50.0000	−0.141421
$51$	0	0
$52$	272.000	0.725377
$53$	−258.000	−0.668661	−0.334330	−	0.942456i	$-0.608510\pi$
$53$	−258.000	−0.668661	−0.334330	+	0.942456i	$0.608510\pi$
$54$	0	0
$55$	−30.0000	−0.0735491
$56$	−112.000	−0.267261
$57$	0	0
$58$	−252.000	−0.570504
$59$	534.000	1.17832	0.589160	−	0.808016i	$-0.299459\pi$
$59$	534.000	1.17832	0.589160	+	0.808016i	$0.299459\pi$
$60$	0	0
$61$	362.000	0.759825	0.379913	−	0.925022i	$-0.375954\pi$
$61$	362.000	0.759825	0.379913	+	0.925022i	$0.375954\pi$
$62$	488.000	0.999614
$63$	0	0
$64$	64.0000	0.125000
$65$	−340.000	−0.648797
$66$	0	0
$67$	−268.000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−268.000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	312.000	0.556405
$69$	0	0
$70$	140.000	0.239046
$71$	−972.000	−1.62472	−0.812360	−	0.583156i	$-0.801818\pi$
$71$	−972.000	−1.62472	−0.812360	+	0.583156i	$0.801818\pi$
$72$	0	0
$73$	470.000	0.753553	0.376776	−	0.926304i	$-0.377033\pi$
$73$	470.000	0.753553	0.376776	+	0.926304i	$0.377033\pi$
$74$	608.000	0.955116
$75$	0	0
$76$	176.000	0.265639
$77$	84.0000	0.124321
$78$	0	0
$79$	1244.00	1.77166	0.885829	−	0.464012i	$-0.153591\pi$
$79$	1244.00	1.77166	0.885829	+	0.464012i	$0.153591\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	960.000	1.29286
$83$	396.000	0.523695	0.261847	−	0.965109i	$-0.415668\pi$
$83$	396.000	0.523695	0.261847	+	0.965109i	$0.415668\pi$
$84$	0	0
$85$	−390.000	−0.497664
$86$	−208.000	−0.260805
$87$	0	0
$88$	−48.0000	−0.0581456
$89$	−972.000	−1.15766	−0.578830	−	0.815448i	$-0.696491\pi$
$89$	−972.000	−1.15766	−0.578830	+	0.815448i	$0.696491\pi$
$90$	0	0
$91$	952.000	1.09667
$92$	480.000	0.543951
$93$	0	0
$94$	−1200.00	−1.31671
$95$	−220.000	−0.237595
$96$	0	0
$97$	−46.0000	−0.0481504	−0.0240752	−	0.999710i	$-0.507664\pi$
$97$	−46.0000	−0.0481504	−0.0240752	+	0.999710i	$0.507664\pi$
$98$	294.000	0.303046
$99$	0	0
$100$	100.000	0.100000
$101$	−1506.00	−1.48369	−0.741845	−	0.670572i	$-0.766049\pi$
$101$	−1506.00	−1.48369	−0.741845	+	0.670572i	$0.766049\pi$
$102$	0	0
$103$	−1474.00	−1.41007	−0.705037	−	0.709171i	$-0.749069\pi$
$103$	−1474.00	−1.41007	−0.705037	+	0.709171i	$0.749069\pi$
$104$	−544.000	−0.512919
$105$	0	0
$106$	516.000	0.472815
$107$	924.000	0.834827	0.417413	−	0.908717i	$-0.362937\pi$
$107$	924.000	0.834827	0.417413	+	0.908717i	$0.362937\pi$
$108$	0	0
$109$	698.000	0.613360	0.306680	−	0.951813i	$-0.400782\pi$
$109$	698.000	0.613360	0.306680	+	0.951813i	$0.400782\pi$
$110$	60.0000	0.0520071
$111$	0	0
$112$	224.000	0.188982
$113$	−222.000	−0.184814	−0.0924071	−	0.995721i	$-0.529456\pi$
$113$	−222.000	−0.184814	−0.0924071	+	0.995721i	$0.529456\pi$
$114$	0	0
$115$	−600.000	−0.486524
$116$	504.000	0.403407
$117$	0	0
$118$	−1068.00	−0.833198
$119$	1092.00	0.841206
$120$	0	0
$121$	−1295.00	−0.972953
$122$	−724.000	−0.537278
$123$	0	0
$124$	−976.000	−0.706834
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1906.00	−1.33173	−0.665867	−	0.746071i	$-0.731938\pi$
$127$	−1906.00	−1.33173	−0.665867	+	0.746071i	$0.731938\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	680.000	0.458769
$131$	2874.00	1.91681	0.958407	−	0.285406i	$-0.0921285\pi$
$131$	2874.00	1.91681	0.958407	+	0.285406i	$0.0921285\pi$
$132$	0	0
$133$	616.000	0.401609
$134$	536.000	0.345547
$135$	0	0
$136$	−624.000	−0.393438
$137$	−798.000	−0.497648	−0.248824	−	0.968549i	$-0.580044\pi$
$137$	−798.000	−0.497648	−0.248824	+	0.968549i	$0.580044\pi$
$138$	0	0
$139$	−700.000	−0.427146	−0.213573	−	0.976927i	$-0.568510\pi$
$139$	−700.000	−0.427146	−0.213573	+	0.976927i	$0.568510\pi$
$140$	−280.000	−0.169031
$141$	0	0
$142$	1944.00	1.14885
$143$	408.000	0.238592
$144$	0	0
$145$	−630.000	−0.360818
$146$	−940.000	−0.532842
$147$	0	0
$148$	−1216.00	−0.675369
$149$	114.000	0.0626795	0.0313397	−	0.999509i	$-0.490023\pi$
$149$	114.000	0.0626795	0.0313397	+	0.999509i	$0.490023\pi$
$150$	0	0
$151$	1064.00	0.573424	0.286712	−	0.958017i	$-0.407438\pi$
$151$	1064.00	0.573424	0.286712	+	0.958017i	$0.407438\pi$
$152$	−352.000	−0.187835
$153$	0	0
$154$	−168.000	−0.0879080
$155$	1220.00	0.632211
$156$	0	0
$157$	−1948.00	−0.990238	−0.495119	−	0.868825i	$-0.664875\pi$
$157$	−1948.00	−0.990238	−0.495119	+	0.868825i	$0.664875\pi$
$158$	−2488.00	−1.25275
$159$	0	0
$160$	160.000	0.0790569
$161$	1680.00	0.822376
$162$	0	0
$163$	2060.00	0.989887	0.494944	−	0.868925i	$-0.335189\pi$
$163$	2060.00	0.989887	0.494944	+	0.868925i	$0.335189\pi$
$164$	−1920.00	−0.914188
$165$	0	0
$166$	−792.000	−0.370308
$167$	−1248.00	−0.578282	−0.289141	−	0.957286i	$-0.593370\pi$
$167$	−1248.00	−0.578282	−0.289141	+	0.957286i	$0.593370\pi$
$168$	0	0
$169$	2427.00	1.10469
$170$	780.000	0.351902
$171$	0	0
$172$	416.000	0.184417
$173$	−1146.00	−0.503634	−0.251817	−	0.967775i	$-0.581028\pi$
$173$	−1146.00	−0.503634	−0.251817	+	0.967775i	$0.581028\pi$
$174$	0	0
$175$	350.000	0.151186
$176$	96.0000	0.0411152
$177$	0	0
$178$	1944.00	0.818590
$179$	−1146.00	−0.478525	−0.239263	−	0.970955i	$-0.576906\pi$
$179$	−1146.00	−0.478525	−0.239263	+	0.970955i	$0.576906\pi$
$180$	0	0
$181$	−118.000	−0.0484579	−0.0242289	−	0.999706i	$-0.507713\pi$
$181$	−118.000	−0.0484579	−0.0242289	+	0.999706i	$0.507713\pi$
$182$	−1904.00	−0.775461
$183$	0	0
$184$	−960.000	−0.384631
$185$	1520.00	0.604068
$186$	0	0
$187$	468.000	0.183014
$188$	2400.00	0.931053
$189$	0	0
$190$	440.000	0.168005
$191$	1692.00	0.640989	0.320494	−	0.947250i	$-0.396151\pi$
$191$	1692.00	0.640989	0.320494	+	0.947250i	$0.396151\pi$
$192$	0	0
$193$	3350.00	1.24942	0.624711	−	0.780856i	$-0.285217\pi$
$193$	3350.00	1.24942	0.624711	+	0.780856i	$0.285217\pi$
$194$	92.0000	0.0340475
$195$	0	0
$196$	−588.000	−0.214286
$197$	−3606.00	−1.30415	−0.652073	−	0.758156i	$-0.726101\pi$
$197$	−3606.00	−1.30415	−0.652073	+	0.758156i	$0.726101\pi$
$198$	0	0
$199$	2696.00	0.960374	0.480187	−	0.877166i	$-0.340569\pi$
$199$	2696.00	0.960374	0.480187	+	0.877166i	$0.340569\pi$
$200$	−200.000	−0.0707107
$201$	0	0
$202$	3012.00	1.04913
$203$	1764.00	0.609894
$204$	0	0
$205$	2400.00	0.817674
$206$	2948.00	0.997072
$207$	0	0
$208$	1088.00	0.362689
$209$	264.000	0.0873745
$210$	0	0
$211$	−4.00000	−0.00130508	−0.000652539	−	1.00000i	$-0.500208\pi$
$211$	−4.00000	−0.00130508	−0.000652539		1.00000i	$0.500208\pi$
$212$	−1032.00	−0.334330
$213$	0	0
$214$	−1848.00	−0.590312
$215$	−520.000	−0.164947
$216$	0	0
$217$	−3416.00	−1.06863
$218$	−1396.00	−0.433711
$219$	0	0
$220$	−120.000	−0.0367745
$221$	5304.00	1.61441
$222$	0	0
$223$	−1162.00	−0.348938	−0.174469	−	0.984663i	$-0.555821\pi$
$223$	−1162.00	−0.348938	−0.174469	+	0.984663i	$0.555821\pi$
$224$	−448.000	−0.133631
$225$	0	0
$226$	444.000	0.130683
$227$	−2400.00	−0.701734	−0.350867	−	0.936425i	$-0.614113\pi$
$227$	−2400.00	−0.701734	−0.350867	+	0.936425i	$0.614113\pi$
$228$	0	0
$229$	−2314.00	−0.667744	−0.333872	−	0.942618i	$-0.608355\pi$
$229$	−2314.00	−0.667744	−0.333872	+	0.942618i	$0.608355\pi$
$230$	1200.00	0.344025
$231$	0	0
$232$	−1008.00	−0.285252
$233$	−18.0000	−0.00506103	−0.00253051	−	0.999997i	$-0.500805\pi$
$233$	−18.0000	−0.00506103	−0.00253051	+	0.999997i	$0.500805\pi$
$234$	0	0
$235$	−3000.00	−0.832759
$236$	2136.00	0.589160
$237$	0	0
$238$	−2184.00	−0.594822
$239$	−5868.00	−1.58816	−0.794078	−	0.607816i	$-0.792046\pi$
$239$	−5868.00	−1.58816	−0.794078	+	0.607816i	$0.792046\pi$
$240$	0	0
$241$	−4330.00	−1.15734	−0.578672	−	0.815560i	$-0.696429\pi$
$241$	−4330.00	−1.15734	−0.578672	+	0.815560i	$0.696429\pi$
$242$	2590.00	0.687981
$243$	0	0
$244$	1448.00	0.379913
$245$	735.000	0.191663
$246$	0	0
$247$	2992.00	0.770755
$248$	1952.00	0.499807
$249$	0	0
$250$	250.000	0.0632456
$251$	498.000	0.125233	0.0626165	−	0.998038i	$-0.480056\pi$
$251$	498.000	0.125233	0.0626165	+	0.998038i	$0.480056\pi$
$252$	0	0
$253$	720.000	0.178917
$254$	3812.00	0.941678
$255$	0	0
$256$	256.000	0.0625000
$257$	642.000	0.155824	0.0779122	−	0.996960i	$-0.475175\pi$
$257$	642.000	0.155824	0.0779122	+	0.996960i	$0.475175\pi$
$258$	0	0
$259$	−4256.00	−1.02106
$260$	−1360.00	−0.324399
$261$	0	0
$262$	−5748.00	−1.35539
$263$	−7968.00	−1.86817	−0.934084	−	0.357055i	$-0.883781\pi$
$263$	−7968.00	−1.86817	−0.934084	+	0.357055i	$0.883781\pi$
$264$	0	0
$265$	1290.00	0.299034
$266$	−1232.00	−0.283980
$267$	0	0
$268$	−1072.00	−0.244339
$269$	−4218.00	−0.956045	−0.478022	−	0.878348i	$-0.658646\pi$
$269$	−4218.00	−0.956045	−0.478022	+	0.878348i	$0.658646\pi$
$270$	0	0
$271$	848.000	0.190082	0.0950412	−	0.995473i	$-0.469702\pi$
$271$	848.000	0.190082	0.0950412	+	0.995473i	$0.469702\pi$
$272$	1248.00	0.278203
$273$	0	0
$274$	1596.00	0.351890
$275$	150.000	0.0328921
$276$	0	0
$277$	−1504.00	−0.326233	−0.163117	−	0.986607i	$-0.552155\pi$
$277$	−1504.00	−0.326233	−0.163117	+	0.986607i	$0.552155\pi$
$278$	1400.00	0.302037
$279$	0	0
$280$	560.000	0.119523
$281$	−1308.00	−0.277682	−0.138841	−	0.990315i	$-0.544338\pi$
$281$	−1308.00	−0.277682	−0.138841	+	0.990315i	$0.544338\pi$
$282$	0	0
$283$	−5932.00	−1.24601	−0.623005	−	0.782218i	$-0.714088\pi$
$283$	−5932.00	−1.24601	−0.623005	+	0.782218i	$0.714088\pi$
$284$	−3888.00	−0.812360
$285$	0	0
$286$	−816.000	−0.168710
$287$	−6720.00	−1.38212
$288$	0	0
$289$	1171.00	0.238347
$290$	1260.00	0.255137
$291$	0	0
$292$	1880.00	0.376776
$293$	5226.00	1.04200	0.521000	−	0.853556i	$-0.325559\pi$
$293$	5226.00	1.04200	0.521000	+	0.853556i	$0.325559\pi$
$294$	0	0
$295$	−2670.00	−0.526961
$296$	2432.00	0.477558
$297$	0	0
$298$	−228.000	−0.0443211
$299$	8160.00	1.57828
$300$	0	0
$301$	1456.00	0.278812
$302$	−2128.00	−0.405472
$303$	0	0
$304$	704.000	0.132820
$305$	−1810.00	−0.339804
$306$	0	0
$307$	4448.00	0.826908	0.413454	−	0.910525i	$-0.364322\pi$
$307$	4448.00	0.826908	0.413454	+	0.910525i	$0.364322\pi$
$308$	336.000	0.0621603
$309$	0	0
$310$	−2440.00	−0.447041
$311$	−9132.00	−1.66504	−0.832521	−	0.553993i	$-0.813103\pi$
$311$	−9132.00	−1.66504	−0.832521	+	0.553993i	$0.813103\pi$
$312$	0	0
$313$	−2170.00	−0.391871	−0.195936	−	0.980617i	$-0.562774\pi$
$313$	−2170.00	−0.391871	−0.195936	+	0.980617i	$0.562774\pi$
$314$	3896.00	0.700204
$315$	0	0
$316$	4976.00	0.885829
$317$	7674.00	1.35967	0.679834	−	0.733366i	$-0.262052\pi$
$317$	7674.00	1.35967	0.679834	+	0.733366i	$0.262052\pi$
$318$	0	0
$319$	756.000	0.132689
$320$	−320.000	−0.0559017
$321$	0	0
$322$	−3360.00	−0.581508
$323$	3432.00	0.591212
$324$	0	0
$325$	1700.00	0.290151
$326$	−4120.00	−0.699956
$327$	0	0
$328$	3840.00	0.646428
$329$	8400.00	1.40762
$330$	0	0
$331$	9596.00	1.59349	0.796743	−	0.604318i	$-0.206554\pi$
$331$	9596.00	1.59349	0.796743	+	0.604318i	$0.206554\pi$
$332$	1584.00	0.261847
$333$	0	0
$334$	2496.00	0.408907
$335$	1340.00	0.218543
$336$	0	0
$337$	12158.0	1.96525	0.982624	−	0.185608i	$-0.0594255\pi$
$337$	12158.0	1.96525	0.982624	+	0.185608i	$0.0594255\pi$
$338$	−4854.00	−0.781133
$339$	0	0
$340$	−1560.00	−0.248832
$341$	−1464.00	−0.232493
$342$	0	0
$343$	−6860.00	−1.07990
$344$	−832.000	−0.130402
$345$	0	0
$346$	2292.00	0.356123
$347$	10320.0	1.59656	0.798280	−	0.602286i	$-0.205743\pi$
$347$	10320.0	1.59656	0.798280	+	0.602286i	$0.205743\pi$
$348$	0	0
$349$	−2158.00	−0.330989	−0.165494	−	0.986211i	$-0.552922\pi$
$349$	−2158.00	−0.330989	−0.165494	+	0.986211i	$0.552922\pi$
$350$	−700.000	−0.106904
$351$	0	0
$352$	−192.000	−0.0290728
$353$	−330.000	−0.0497567	−0.0248784	−	0.999690i	$-0.507920\pi$
$353$	−330.000	−0.0497567	−0.0248784	+	0.999690i	$0.507920\pi$
$354$	0	0
$355$	4860.00	0.726597
$356$	−3888.00	−0.578830
$357$	0	0
$358$	2292.00	0.338369
$359$	−8664.00	−1.27373	−0.636864	−	0.770976i	$-0.719769\pi$
$359$	−8664.00	−1.27373	−0.636864	+	0.770976i	$0.719769\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	236.000	0.0342649
$363$	0	0
$364$	3808.00	0.548334
$365$	−2350.00	−0.336999
$366$	0	0
$367$	3782.00	0.537926	0.268963	−	0.963151i	$-0.413319\pi$
$367$	3782.00	0.537926	0.268963	+	0.963151i	$0.413319\pi$
$368$	1920.00	0.271975
$369$	0	0
$370$	−3040.00	−0.427141
$371$	−3612.00	−0.505460
$372$	0	0
$373$	11276.0	1.56528	0.782640	−	0.622475i	$-0.213873\pi$
$373$	11276.0	1.56528	0.782640	+	0.622475i	$0.213873\pi$
$374$	−936.000	−0.129410
$375$	0	0
$376$	−4800.00	−0.658354
$377$	8568.00	1.17049
$378$	0	0
$379$	980.000	0.132821	0.0664106	−	0.997792i	$-0.478845\pi$
$379$	980.000	0.132821	0.0664106	+	0.997792i	$0.478845\pi$
$380$	−880.000	−0.118797
$381$	0	0
$382$	−3384.00	−0.453247
$383$	−4200.00	−0.560339	−0.280170	−	0.959950i	$-0.590391\pi$
$383$	−4200.00	−0.560339	−0.280170	+	0.959950i	$0.590391\pi$
$384$	0	0
$385$	−420.000	−0.0555979
$386$	−6700.00	−0.883474
$387$	0	0
$388$	−184.000	−0.0240752
$389$	13338.0	1.73847	0.869233	−	0.494402i	$-0.164613\pi$
$389$	13338.0	1.73847	0.869233	+	0.494402i	$0.164613\pi$
$390$	0	0
$391$	9360.00	1.21063
$392$	1176.00	0.151523
$393$	0	0
$394$	7212.00	0.922171
$395$	−6220.00	−0.792309
$396$	0	0
$397$	−7192.00	−0.909209	−0.454605	−	0.890693i	$-0.650219\pi$
$397$	−7192.00	−0.909209	−0.454605	+	0.890693i	$0.650219\pi$
$398$	−5392.00	−0.679087
$399$	0	0
$400$	400.000	0.0500000
$401$	−2316.00	−0.288418	−0.144209	−	0.989547i	$-0.546064\pi$
$401$	−2316.00	−0.288418	−0.144209	+	0.989547i	$0.546064\pi$
$402$	0	0
$403$	−16592.0	−2.05088
$404$	−6024.00	−0.741845
$405$	0	0
$406$	−3528.00	−0.431260
$407$	−1824.00	−0.222143
$408$	0	0
$409$	−12358.0	−1.49404	−0.747022	−	0.664800i	$-0.768517\pi$
$409$	−12358.0	−1.49404	−0.747022	+	0.664800i	$0.768517\pi$
$410$	−4800.00	−0.578183
$411$	0	0
$412$	−5896.00	−0.705037
$413$	7476.00	0.890726
$414$	0	0
$415$	−1980.00	−0.234203
$416$	−2176.00	−0.256460
$417$	0	0
$418$	−528.000	−0.0617831
$419$	3306.00	0.385462	0.192731	−	0.981252i	$-0.438265\pi$
$419$	3306.00	0.385462	0.192731	+	0.981252i	$0.438265\pi$
$420$	0	0
$421$	−14506.0	−1.67929	−0.839643	−	0.543139i	$-0.817236\pi$
$421$	−14506.0	−1.67929	−0.839643	+	0.543139i	$0.817236\pi$
$422$	8.00000	0.000922829 0
$423$	0	0
$424$	2064.00	0.236407
$425$	1950.00	0.222562
$426$	0	0
$427$	5068.00	0.574374
$428$	3696.00	0.417413
$429$	0	0
$430$	1040.00	0.116635
$431$	−6480.00	−0.724201	−0.362100	−	0.932139i	$-0.617940\pi$
$431$	−6480.00	−0.724201	−0.362100	+	0.932139i	$0.617940\pi$
$432$	0	0
$433$	11894.0	1.32007	0.660034	−	0.751236i	$-0.270542\pi$
$433$	11894.0	1.32007	0.660034	+	0.751236i	$0.270542\pi$
$434$	6832.00	0.755637
$435$	0	0
$436$	2792.00	0.306680
$437$	5280.00	0.577979
$438$	0	0
$439$	−12688.0	−1.37942	−0.689710	−	0.724086i	$-0.742262\pi$
$439$	−12688.0	−1.37942	−0.689710	+	0.724086i	$0.742262\pi$
$440$	240.000	0.0260035
$441$	0	0
$442$	−10608.0	−1.14156
$443$	4968.00	0.532814	0.266407	−	0.963861i	$-0.414163\pi$
$443$	4968.00	0.532814	0.266407	+	0.963861i	$0.414163\pi$
$444$	0	0
$445$	4860.00	0.517722
$446$	2324.00	0.246737
$447$	0	0
$448$	896.000	0.0944911
$449$	−11508.0	−1.20957	−0.604784	−	0.796389i	$-0.706741\pi$
$449$	−11508.0	−1.20957	−0.604784	+	0.796389i	$0.706741\pi$
$450$	0	0
$451$	−2880.00	−0.300696
$452$	−888.000	−0.0924071
$453$	0	0
$454$	4800.00	0.496201
$455$	−4760.00	−0.490444
$456$	0	0
$457$	1082.00	0.110752	0.0553762	−	0.998466i	$-0.482364\pi$
$457$	1082.00	0.110752	0.0553762	+	0.998466i	$0.482364\pi$
$458$	4628.00	0.472166
$459$	0	0
$460$	−2400.00	−0.243262
$461$	−11238.0	−1.13537	−0.567685	−	0.823246i	$-0.692161\pi$
$461$	−11238.0	−1.13537	−0.567685	+	0.823246i	$0.692161\pi$
$462$	0	0
$463$	−2302.00	−0.231065	−0.115532	−	0.993304i	$-0.536857\pi$
$463$	−2302.00	−0.231065	−0.115532	+	0.993304i	$0.536857\pi$
$464$	2016.00	0.201704
$465$	0	0
$466$	36.0000	0.00357869
$467$	15876.0	1.57313	0.786567	−	0.617505i	$-0.211856\pi$
$467$	15876.0	1.57313	0.786567	+	0.617505i	$0.211856\pi$
$468$	0	0
$469$	−3752.00	−0.369406
$470$	6000.00	0.588850
$471$	0	0
$472$	−4272.00	−0.416599
$473$	624.000	0.0606587
$474$	0	0
$475$	1100.00	0.106256
$476$	4368.00	0.420603
$477$	0	0
$478$	11736.0	1.12300
$479$	4644.00	0.442985	0.221492	−	0.975162i	$-0.428907\pi$
$479$	4644.00	0.442985	0.221492	+	0.975162i	$0.428907\pi$
$480$	0	0
$481$	−20672.0	−1.95959
$482$	8660.00	0.818366
$483$	0	0
$484$	−5180.00	−0.486476
$485$	230.000	0.0215335
$486$	0	0
$487$	2426.00	0.225734	0.112867	−	0.993610i	$-0.463997\pi$
$487$	2426.00	0.225734	0.112867	+	0.993610i	$0.463997\pi$
$488$	−2896.00	−0.268639
$489$	0	0
$490$	−1470.00	−0.135526
$491$	234.000	0.0215077	0.0107538	−	0.999942i	$-0.496577\pi$
$491$	234.000	0.0215077	0.0107538	+	0.999942i	$0.496577\pi$
$492$	0	0
$493$	9828.00	0.897831
$494$	−5984.00	−0.545006
$495$	0	0
$496$	−3904.00	−0.353417
$497$	−13608.0	−1.22817
$498$	0	0
$499$	14204.0	1.27427	0.637133	−	0.770754i	$-0.280120\pi$
$499$	14204.0	1.27427	0.637133	+	0.770754i	$0.280120\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	−996.000	−0.0885531
$503$	4920.00	0.436127	0.218064	−	0.975935i	$-0.430026\pi$
$503$	4920.00	0.436127	0.218064	+	0.975935i	$0.430026\pi$
$504$	0	0
$505$	7530.00	0.663526
$506$	−1440.00	−0.126513
$507$	0	0
$508$	−7624.00	−0.665867
$509$	−4458.00	−0.388207	−0.194104	−	0.980981i	$-0.562180\pi$
$509$	−4458.00	−0.388207	−0.194104	+	0.980981i	$0.562180\pi$
$510$	0	0
$511$	6580.00	0.569632
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−1284.00	−0.110184
$515$	7370.00	0.630604
$516$	0	0
$517$	3600.00	0.306243
$518$	8512.00	0.722000
$519$	0	0
$520$	2720.00	0.229384
$521$	4212.00	0.354186	0.177093	−	0.984194i	$-0.443331\pi$
$521$	4212.00	0.354186	0.177093	+	0.984194i	$0.443331\pi$
$522$	0	0
$523$	−11212.0	−0.937412	−0.468706	−	0.883354i	$-0.655280\pi$
$523$	−11212.0	−0.937412	−0.468706	+	0.883354i	$0.655280\pi$
$524$	11496.0	0.958407
$525$	0	0
$526$	15936.0	1.32099
$527$	−19032.0	−1.57314
$528$	0	0
$529$	2233.00	0.183529
$530$	−2580.00	−0.211449
$531$	0	0
$532$	2464.00	0.200804
$533$	−32640.0	−2.65252
$534$	0	0
$535$	−4620.00	−0.373346
$536$	2144.00	0.172774
$537$	0	0
$538$	8436.00	0.676026
$539$	−882.000	−0.0704832
$540$	0	0
$541$	14018.0	1.11401	0.557006	−	0.830508i	$-0.311950\pi$
$541$	14018.0	1.11401	0.557006	+	0.830508i	$0.311950\pi$
$542$	−1696.00	−0.134409
$543$	0	0
$544$	−2496.00	−0.196719
$545$	−3490.00	−0.274303
$546$	0	0
$547$	18200.0	1.42262	0.711312	−	0.702876i	$-0.248101\pi$
$547$	18200.0	1.42262	0.711312	+	0.702876i	$0.248101\pi$
$548$	−3192.00	−0.248824
$549$	0	0
$550$	−300.000	−0.0232583
$551$	5544.00	0.428643
$552$	0	0
$553$	17416.0	1.33925
$554$	3008.00	0.230682
$555$	0	0
$556$	−2800.00	−0.213573
$557$	−11826.0	−0.899612	−0.449806	−	0.893126i	$-0.648507\pi$
$557$	−11826.0	−0.899612	−0.449806	+	0.893126i	$0.648507\pi$
$558$	0	0
$559$	7072.00	0.535087
$560$	−1120.00	−0.0845154
$561$	0	0
$562$	2616.00	0.196351
$563$	−2952.00	−0.220980	−0.110490	−	0.993877i	$-0.535242\pi$
$563$	−2952.00	−0.220980	−0.110490	+	0.993877i	$0.535242\pi$
$564$	0	0
$565$	1110.00	0.0826514
$566$	11864.0	0.881062
$567$	0	0
$568$	7776.00	0.574426
$569$	3084.00	0.227220	0.113610	−	0.993525i	$-0.463759\pi$
$569$	3084.00	0.227220	0.113610	+	0.993525i	$0.463759\pi$
$570$	0	0
$571$	−4756.00	−0.348568	−0.174284	−	0.984695i	$-0.555761\pi$
$571$	−4756.00	−0.348568	−0.174284	+	0.984695i	$0.555761\pi$
$572$	1632.00	0.119296
$573$	0	0
$574$	13440.0	0.977308
$575$	3000.00	0.217580
$576$	0	0
$577$	−11014.0	−0.794660	−0.397330	−	0.917676i	$-0.630063\pi$
$577$	−11014.0	−0.794660	−0.397330	+	0.917676i	$0.630063\pi$
$578$	−2342.00	−0.168537
$579$	0	0
$580$	−2520.00	−0.180409
$581$	5544.00	0.395876
$582$	0	0
$583$	−1548.00	−0.109968
$584$	−3760.00	−0.266421
$585$	0	0
$586$	−10452.0	−0.736806
$587$	−852.000	−0.0599077	−0.0299538	−	0.999551i	$-0.509536\pi$
$587$	−852.000	−0.0599077	−0.0299538	+	0.999551i	$0.509536\pi$
$588$	0	0
$589$	−10736.0	−0.751051
$590$	5340.00	0.372617
$591$	0	0
$592$	−4864.00	−0.337684
$593$	15546.0	1.07656	0.538278	−	0.842767i	$-0.319075\pi$
$593$	15546.0	1.07656	0.538278	+	0.842767i	$0.319075\pi$
$594$	0	0
$595$	−5460.00	−0.376199
$596$	456.000	0.0313397
$597$	0	0
$598$	−16320.0	−1.11601
$599$	−8616.00	−0.587713	−0.293857	−	0.955850i	$-0.594939\pi$
$599$	−8616.00	−0.587713	−0.293857	+	0.955850i	$0.594939\pi$
$600$	0	0
$601$	17510.0	1.18843	0.594216	−	0.804305i	$-0.297462\pi$
$601$	17510.0	1.18843	0.594216	+	0.804305i	$0.297462\pi$
$602$	−2912.00	−0.197150
$603$	0	0
$604$	4256.00	0.286712
$605$	6475.00	0.435118
$606$	0	0
$607$	−13894.0	−0.929061	−0.464531	−	0.885557i	$-0.653777\pi$
$607$	−13894.0	−0.929061	−0.464531	+	0.885557i	$0.653777\pi$
$608$	−1408.00	−0.0939177
$609$	0	0
$610$	3620.00	0.240278
$611$	40800.0	2.70146
$612$	0	0
$613$	−6496.00	−0.428011	−0.214006	−	0.976832i	$-0.568651\pi$
$613$	−6496.00	−0.428011	−0.214006	+	0.976832i	$0.568651\pi$
$614$	−8896.00	−0.584712
$615$	0	0
$616$	−672.000	−0.0439540
$617$	−570.000	−0.0371918	−0.0185959	−	0.999827i	$-0.505920\pi$
$617$	−570.000	−0.0371918	−0.0185959	+	0.999827i	$0.505920\pi$
$618$	0	0
$619$	−2140.00	−0.138956	−0.0694781	−	0.997583i	$-0.522133\pi$
$619$	−2140.00	−0.138956	−0.0694781	+	0.997583i	$0.522133\pi$
$620$	4880.00	0.316106
$621$	0	0
$622$	18264.0	1.17736
$623$	−13608.0	−0.875109
$624$	0	0
$625$	625.000	0.0400000
$626$	4340.00	0.277095
$627$	0	0
$628$	−7792.00	−0.495119
$629$	−23712.0	−1.50312
$630$	0	0
$631$	14660.0	0.924890	0.462445	−	0.886648i	$-0.346972\pi$
$631$	14660.0	0.924890	0.462445	+	0.886648i	$0.346972\pi$
$632$	−9952.00	−0.626375
$633$	0	0
$634$	−15348.0	−0.961431
$635$	9530.00	0.595569
$636$	0	0
$637$	−9996.00	−0.621752
$638$	−1512.00	−0.0938255
$639$	0	0
$640$	640.000	0.0395285
$641$	−456.000	−0.0280982	−0.0140491	−	0.999901i	$-0.504472\pi$
$641$	−456.000	−0.0280982	−0.0140491	+	0.999901i	$0.504472\pi$
$642$	0	0
$643$	−23452.0	−1.43835	−0.719173	−	0.694831i	$-0.755479\pi$
$643$	−23452.0	−1.43835	−0.719173	+	0.694831i	$0.755479\pi$
$644$	6720.00	0.411188
$645$	0	0
$646$	−6864.00	−0.418050
$647$	7224.00	0.438956	0.219478	−	0.975617i	$-0.429565\pi$
$647$	7224.00	0.438956	0.219478	+	0.975617i	$0.429565\pi$
$648$	0	0
$649$	3204.00	0.193787
$650$	−3400.00	−0.205168
$651$	0	0
$652$	8240.00	0.494944
$653$	19146.0	1.14738	0.573691	−	0.819072i	$-0.305511\pi$
$653$	19146.0	1.14738	0.573691	+	0.819072i	$0.305511\pi$
$654$	0	0
$655$	−14370.0	−0.857225
$656$	−7680.00	−0.457094
$657$	0	0
$658$	−16800.0	−0.995338
$659$	27810.0	1.64389	0.821945	−	0.569567i	$-0.192889\pi$
$659$	27810.0	1.64389	0.821945	+	0.569567i	$0.192889\pi$
$660$	0	0
$661$	−30598.0	−1.80049	−0.900245	−	0.435383i	$-0.856613\pi$
$661$	−30598.0	−1.80049	−0.900245	+	0.435383i	$0.856613\pi$
$662$	−19192.0	−1.12676
$663$	0	0
$664$	−3168.00	−0.185154
$665$	−3080.00	−0.179605
$666$	0	0
$667$	15120.0	0.877734
$668$	−4992.00	−0.289141
$669$	0	0
$670$	−2680.00	−0.154533
$671$	2172.00	0.124961
$672$	0	0
$673$	−3778.00	−0.216391	−0.108196	−	0.994130i	$-0.534507\pi$
$673$	−3778.00	−0.216391	−0.108196	+	0.994130i	$0.534507\pi$
$674$	−24316.0	−1.38964
$675$	0	0
$676$	9708.00	0.552344
$677$	−27198.0	−1.54402	−0.772012	−	0.635608i	$-0.780749\pi$
$677$	−27198.0	−1.54402	−0.772012	+	0.635608i	$0.780749\pi$
$678$	0	0
$679$	−644.000	−0.0363983
$680$	3120.00	0.175951
$681$	0	0
$682$	2928.00	0.164397
$683$	32316.0	1.81045	0.905225	−	0.424933i	$-0.139702\pi$
$683$	32316.0	1.81045	0.905225	+	0.424933i	$0.139702\pi$
$684$	0	0
$685$	3990.00	0.222555
$686$	13720.0	0.763604
$687$	0	0
$688$	1664.00	0.0922084
$689$	−17544.0	−0.970063
$690$	0	0
$691$	29324.0	1.61438	0.807191	−	0.590291i	$-0.200987\pi$
$691$	29324.0	1.61438	0.807191	+	0.590291i	$0.200987\pi$
$692$	−4584.00	−0.251817
$693$	0	0
$694$	−20640.0	−1.12894
$695$	3500.00	0.191025
$696$	0	0
$697$	−37440.0	−2.03464
$698$	4316.00	0.234044
$699$	0	0
$700$	1400.00	0.0755929
$701$	22782.0	1.22748	0.613741	−	0.789508i	$-0.289664\pi$
$701$	22782.0	1.22748	0.613741	+	0.789508i	$0.289664\pi$
$702$	0	0
$703$	−13376.0	−0.717618
$704$	384.000	0.0205576
$705$	0	0
$706$	660.000	0.0351833
$707$	−21084.0	−1.12156
$708$	0	0
$709$	26054.0	1.38008	0.690041	−	0.723770i	$-0.257592\pi$
$709$	26054.0	1.38008	0.690041	+	0.723770i	$0.257592\pi$
$710$	−9720.00	−0.513782
$711$	0	0
$712$	7776.00	0.409295
$713$	−29280.0	−1.53793
$714$	0	0
$715$	−2040.00	−0.106702
$716$	−4584.00	−0.239263
$717$	0	0
$718$	17328.0	0.900662
$719$	−5976.00	−0.309968	−0.154984	−	0.987917i	$-0.549533\pi$
$719$	−5976.00	−0.309968	−0.154984	+	0.987917i	$0.549533\pi$
$720$	0	0
$721$	−20636.0	−1.06592
$722$	9846.00	0.507521
$723$	0	0
$724$	−472.000	−0.0242289
$725$	3150.00	0.161363
$726$	0	0
$727$	−5110.00	−0.260687	−0.130343	−	0.991469i	$-0.541608\pi$
$727$	−5110.00	−0.260687	−0.130343	+	0.991469i	$0.541608\pi$
$728$	−7616.00	−0.387730
$729$	0	0
$730$	4700.00	0.238294
$731$	8112.00	0.410442
$732$	0	0
$733$	17336.0	0.873560	0.436780	−	0.899568i	$-0.356119\pi$
$733$	17336.0	0.873560	0.436780	+	0.899568i	$0.356119\pi$
$734$	−7564.00	−0.380371
$735$	0	0
$736$	−3840.00	−0.192316
$737$	−1608.00	−0.0803683
$738$	0	0
$739$	−13660.0	−0.679961	−0.339981	−	0.940432i	$-0.610420\pi$
$739$	−13660.0	−0.679961	−0.339981	+	0.940432i	$0.610420\pi$
$740$	6080.00	0.302034
$741$	0	0
$742$	7224.00	0.357414
$743$	1320.00	0.0651765	0.0325882	−	0.999469i	$-0.489625\pi$
$743$	1320.00	0.0651765	0.0325882	+	0.999469i	$0.489625\pi$
$744$	0	0
$745$	−570.000	−0.0280311
$746$	−22552.0	−1.10682
$747$	0	0
$748$	1872.00	0.0915068
$749$	12936.0	0.631070
$750$	0	0
$751$	15860.0	0.770625	0.385313	−	0.922786i	$-0.374094\pi$
$751$	15860.0	0.770625	0.385313	+	0.922786i	$0.374094\pi$
$752$	9600.00	0.465527
$753$	0	0
$754$	−17136.0	−0.827661
$755$	−5320.00	−0.256443
$756$	0	0
$757$	22160.0	1.06396	0.531981	−	0.846756i	$-0.321448\pi$
$757$	22160.0	1.06396	0.531981	+	0.846756i	$0.321448\pi$
$758$	−1960.00	−0.0939187
$759$	0	0
$760$	1760.00	0.0840025
$761$	13116.0	0.624776	0.312388	−	0.949955i	$-0.398871\pi$
$761$	13116.0	0.624776	0.312388	+	0.949955i	$0.398871\pi$
$762$	0	0
$763$	9772.00	0.463657
$764$	6768.00	0.320494
$765$	0	0
$766$	8400.00	0.396220
$767$	36312.0	1.70945
$768$	0	0
$769$	32846.0	1.54026	0.770128	−	0.637889i	$-0.220192\pi$
$769$	32846.0	1.54026	0.770128	+	0.637889i	$0.220192\pi$
$770$	840.000	0.0393136
$771$	0	0
$772$	13400.0	0.624711
$773$	−11982.0	−0.557520	−0.278760	−	0.960361i	$-0.589923\pi$
$773$	−11982.0	−0.557520	−0.278760	+	0.960361i	$0.589923\pi$
$774$	0	0
$775$	−6100.00	−0.282734
$776$	368.000	0.0170238
$777$	0	0
$778$	−26676.0	−1.22928
$779$	−21120.0	−0.971377
$780$	0	0
$781$	−5832.00	−0.267203
$782$	−18720.0	−0.856043
$783$	0	0
$784$	−2352.00	−0.107143
$785$	9740.00	0.442848
$786$	0	0
$787$	−21076.0	−0.954610	−0.477305	−	0.878738i	$-0.658386\pi$
$787$	−21076.0	−0.954610	−0.477305	+	0.878738i	$0.658386\pi$
$788$	−14424.0	−0.652073
$789$	0	0
$790$	12440.0	0.560247
$791$	−3108.00	−0.139706
$792$	0	0
$793$	24616.0	1.10232
$794$	14384.0	0.642908
$795$	0	0
$796$	10784.0	0.480187
$797$	22086.0	0.981589	0.490794	−	0.871275i	$-0.336707\pi$
$797$	22086.0	0.981589	0.490794	+	0.871275i	$0.336707\pi$
$798$	0	0
$799$	46800.0	2.07217
$800$	−800.000	−0.0353553
$801$	0	0
$802$	4632.00	0.203942
$803$	2820.00	0.123930
$804$	0	0
$805$	−8400.00	−0.367778
$806$	33184.0	1.45019
$807$	0	0
$808$	12048.0	0.524563
$809$	21384.0	0.929322	0.464661	−	0.885489i	$-0.346176\pi$
$809$	21384.0	0.929322	0.464661	+	0.885489i	$0.346176\pi$
$810$	0	0
$811$	5228.00	0.226362	0.113181	−	0.993574i	$-0.463896\pi$
$811$	5228.00	0.226362	0.113181	+	0.993574i	$0.463896\pi$
$812$	7056.00	0.304947
$813$	0	0
$814$	3648.00	0.157079
$815$	−10300.0	−0.442691
$816$	0	0
$817$	4576.00	0.195953
$818$	24716.0	1.05645
$819$	0	0
$820$	9600.00	0.408837
$821$	−38010.0	−1.61578	−0.807892	−	0.589331i	$-0.799391\pi$
$821$	−38010.0	−1.61578	−0.807892	+	0.589331i	$0.799391\pi$
$822$	0	0
$823$	38642.0	1.63667	0.818333	−	0.574745i	$-0.194899\pi$
$823$	38642.0	1.63667	0.818333	+	0.574745i	$0.194899\pi$
$824$	11792.0	0.498536
$825$	0	0
$826$	−14952.0	−0.629839
$827$	−15432.0	−0.648879	−0.324440	−	0.945906i	$-0.605176\pi$
$827$	−15432.0	−0.648879	−0.324440	+	0.945906i	$0.605176\pi$
$828$	0	0
$829$	−3886.00	−0.162806	−0.0814031	−	0.996681i	$-0.525940\pi$
$829$	−3886.00	−0.162806	−0.0814031	+	0.996681i	$0.525940\pi$
$830$	3960.00	0.165607
$831$	0	0
$832$	4352.00	0.181344
$833$	−11466.0	−0.476919
$834$	0	0
$835$	6240.00	0.258616
$836$	1056.00	0.0436872
$837$	0	0
$838$	−6612.00	−0.272563
$839$	27552.0	1.13373	0.566866	−	0.823810i	$-0.308156\pi$
$839$	27552.0	1.13373	0.566866	+	0.823810i	$0.308156\pi$
$840$	0	0
$841$	−8513.00	−0.349051
$842$	29012.0	1.18743
$843$	0	0
$844$	−16.0000	−0.000652539 0
$845$	−12135.0	−0.494032
$846$	0	0
$847$	−18130.0	−0.735483
$848$	−4128.00	−0.167165
$849$	0	0
$850$	−3900.00	−0.157375
$851$	−36480.0	−1.46947
$852$	0	0
$853$	15104.0	0.606273	0.303137	−	0.952947i	$-0.401966\pi$
$853$	15104.0	0.606273	0.303137	+	0.952947i	$0.401966\pi$
$854$	−10136.0	−0.406144
$855$	0	0
$856$	−7392.00	−0.295156
$857$	−12306.0	−0.490508	−0.245254	−	0.969459i	$-0.578871\pi$
$857$	−12306.0	−0.490508	−0.245254	+	0.969459i	$0.578871\pi$
$858$	0	0
$859$	−47500.0	−1.88670	−0.943352	−	0.331793i	$-0.892346\pi$
$859$	−47500.0	−1.88670	−0.943352	+	0.331793i	$0.892346\pi$
$860$	−2080.00	−0.0824737
$861$	0	0
$862$	12960.0	0.512087
$863$	4272.00	0.168506	0.0842529	−	0.996444i	$-0.473150\pi$
$863$	4272.00	0.168506	0.0842529	+	0.996444i	$0.473150\pi$
$864$	0	0
$865$	5730.00	0.225232
$866$	−23788.0	−0.933429
$867$	0	0
$868$	−13664.0	−0.534316
$869$	7464.00	0.291368
$870$	0	0
$871$	−18224.0	−0.708951
$872$	−5584.00	−0.216856
$873$	0	0
$874$	−10560.0	−0.408693
$875$	−1750.00	−0.0676123
$876$	0	0
$877$	−27796.0	−1.07024	−0.535122	−	0.844775i	$-0.679734\pi$
$877$	−27796.0	−1.07024	−0.535122	+	0.844775i	$0.679734\pi$
$878$	25376.0	0.975397
$879$	0	0
$880$	−480.000	−0.0183873
$881$	−39996.0	−1.52951	−0.764756	−	0.644320i	$-0.777140\pi$
$881$	−39996.0	−1.52951	−0.764756	+	0.644320i	$0.777140\pi$
$882$	0	0
$883$	−3772.00	−0.143758	−0.0718788	−	0.997413i	$-0.522899\pi$
$883$	−3772.00	−0.143758	−0.0718788	+	0.997413i	$0.522899\pi$
$884$	21216.0	0.807207
$885$	0	0
$886$	−9936.00	−0.376757
$887$	−5784.00	−0.218949	−0.109474	−	0.993990i	$-0.534917\pi$
$887$	−5784.00	−0.218949	−0.109474	+	0.993990i	$0.534917\pi$
$888$	0	0
$889$	−26684.0	−1.00670
$890$	−9720.00	−0.366084
$891$	0	0
$892$	−4648.00	−0.174469
$893$	26400.0	0.989297
$894$	0	0
$895$	5730.00	0.214003
$896$	−1792.00	−0.0668153
$897$	0	0
$898$	23016.0	0.855294
$899$	−30744.0	−1.14057
$900$	0	0
$901$	−20124.0	−0.744093
$902$	5760.00	0.212624
$903$	0	0
$904$	1776.00	0.0653417
$905$	590.000	0.0216710
$906$	0	0
$907$	−8440.00	−0.308981	−0.154490	−	0.987994i	$-0.549374\pi$
$907$	−8440.00	−0.308981	−0.154490	+	0.987994i	$0.549374\pi$
$908$	−9600.00	−0.350867
$909$	0	0
$910$	9520.00	0.346797
$911$	−31920.0	−1.16087	−0.580437	−	0.814305i	$-0.697118\pi$
$911$	−31920.0	−1.16087	−0.580437	+	0.814305i	$0.697118\pi$
$912$	0	0
$913$	2376.00	0.0861272
$914$	−2164.00	−0.0783137
$915$	0	0
$916$	−9256.00	−0.333872
$917$	40236.0	1.44897
$918$	0	0
$919$	34652.0	1.24381	0.621906	−	0.783092i	$-0.286358\pi$
$919$	34652.0	1.24381	0.621906	+	0.783092i	$0.286358\pi$
$920$	4800.00	0.172012
$921$	0	0
$922$	22476.0	0.802828
$923$	−66096.0	−2.35707
$924$	0	0
$925$	−7600.00	−0.270148
$926$	4604.00	0.163388
$927$	0	0
$928$	−4032.00	−0.142626
$929$	1404.00	0.0495842	0.0247921	−	0.999693i	$-0.492108\pi$
$929$	1404.00	0.0495842	0.0247921	+	0.999693i	$0.492108\pi$
$930$	0	0
$931$	−6468.00	−0.227691
$932$	−72.0000	−0.00253051
$933$	0	0
$934$	−31752.0	−1.11237
$935$	−2340.00	−0.0818462
$936$	0	0
$937$	−7654.00	−0.266857	−0.133429	−	0.991058i	$-0.542599\pi$
$937$	−7654.00	−0.266857	−0.133429	+	0.991058i	$0.542599\pi$
$938$	7504.00	0.261209
$939$	0	0
$940$	−12000.0	−0.416380
$941$	11298.0	0.391397	0.195698	−	0.980664i	$-0.437303\pi$
$941$	11298.0	0.391397	0.195698	+	0.980664i	$0.437303\pi$
$942$	0	0
$943$	−57600.0	−1.98909
$944$	8544.00	0.294580
$945$	0	0
$946$	−1248.00	−0.0428922
$947$	28968.0	0.994016	0.497008	−	0.867746i	$-0.334432\pi$
$947$	28968.0	0.994016	0.497008	+	0.867746i	$0.334432\pi$
$948$	0	0
$949$	31960.0	1.09322
$950$	−2200.00	−0.0751341
$951$	0	0
$952$	−8736.00	−0.297411
$953$	46410.0	1.57751	0.788755	−	0.614707i	$-0.210726\pi$
$953$	46410.0	1.57751	0.788755	+	0.614707i	$0.210726\pi$
$954$	0	0
$955$	−8460.00	−0.286659
$956$	−23472.0	−0.794078
$957$	0	0
$958$	−9288.00	−0.313238
$959$	−11172.0	−0.376186
$960$	0	0
$961$	29745.0	0.998456
$962$	41344.0	1.38564
$963$	0	0
$964$	−17320.0	−0.578672
$965$	−16750.0	−0.558758
$966$	0	0
$967$	−41506.0	−1.38029	−0.690146	−	0.723670i	$-0.742454\pi$
$967$	−41506.0	−1.38029	−0.690146	+	0.723670i	$0.742454\pi$
$968$	10360.0	0.343991
$969$	0	0
$970$	−460.000	−0.0152265
$971$	−18246.0	−0.603030	−0.301515	−	0.953461i	$-0.597492\pi$
$971$	−18246.0	−0.603030	−0.301515	+	0.953461i	$0.597492\pi$
$972$	0	0
$973$	−9800.00	−0.322892
$974$	−4852.00	−0.159618
$975$	0	0
$976$	5792.00	0.189956
$977$	25998.0	0.851330	0.425665	−	0.904881i	$-0.360040\pi$
$977$	25998.0	0.851330	0.425665	+	0.904881i	$0.360040\pi$
$978$	0	0
$979$	−5832.00	−0.190390
$980$	2940.00	0.0958315
$981$	0	0
$982$	−468.000	−0.0152082
$983$	14616.0	0.474240	0.237120	−	0.971480i	$-0.423797\pi$
$983$	14616.0	0.474240	0.237120	+	0.971480i	$0.423797\pi$
$984$	0	0
$985$	18030.0	0.583232
$986$	−19656.0	−0.634863
$987$	0	0
$988$	11968.0	0.385377
$989$	12480.0	0.401255
$990$	0	0
$991$	−2968.00	−0.0951379	−0.0475689	−	0.998868i	$-0.515147\pi$
$991$	−2968.00	−0.0951379	−0.0475689	+	0.998868i	$0.515147\pi$
$992$	7808.00	0.249903
$993$	0	0
$994$	27216.0	0.868450
$995$	−13480.0	−0.429492
$996$	0	0
$997$	−9052.00	−0.287542	−0.143771	−	0.989611i	$-0.545923\pi$
$997$	−9052.00	−0.287542	−0.143771	+	0.989611i	$0.545923\pi$
$998$	−28408.0	−0.901042
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	90.4.a.b.1.1	✓	1
3.2	odd	2		90.4.a.e.1.1	yes	1
4.3	odd	2		720.4.a.e.1.1		1
5.2	odd	4		450.4.c.g.199.1		2
5.3	odd	4		450.4.c.g.199.2		2
5.4	even	2		450.4.a.m.1.1		1
9.2	odd	6		810.4.e.a.271.1		2
9.4	even	3		810.4.e.u.541.1		2
9.5	odd	6		810.4.e.a.541.1		2
9.7	even	3		810.4.e.u.271.1		2
12.11	even	2		720.4.a.t.1.1		1
15.2	even	4		450.4.c.f.199.2		2
15.8	even	4		450.4.c.f.199.1		2
15.14	odd	2		450.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.4.a.b.1.1	✓	1	1.1	even	1	trivial
90.4.a.e.1.1	yes	1	3.2	odd	2
450.4.a.c.1.1		1	15.14	odd	2
450.4.a.m.1.1		1	5.4	even	2
450.4.c.f.199.1		2	15.8	even	4
450.4.c.f.199.2		2	15.2	even	4
450.4.c.g.199.1		2	5.2	odd	4
450.4.c.g.199.2		2	5.3	odd	4
720.4.a.e.1.1		1	4.3	odd	2
720.4.a.t.1.1		1	12.11	even	2
810.4.e.a.271.1		2	9.2	odd	6
810.4.e.a.541.1		2	9.5	odd	6
810.4.e.u.271.1		2	9.7	even	3
810.4.e.u.541.1		2	9.4	even	3

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:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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