LMFDB - Embedded newform 450.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, 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("450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.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:	$26.5508595026$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−14.0000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−14.0000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−6.00000	−0.164461	−0.0822304	−	0.996613i	$-0.526204\pi$
$11$	−6.00000	−0.164461	−0.0822304	+	0.996613i	$0.526204\pi$
$12$	0	0
$13$	−68.0000	−1.45075	−0.725377	−	0.688352i	$-0.758335\pi$
$13$	−68.0000	−1.45075	−0.725377	+	0.688352i	$0.758335\pi$
$14$	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	$0.414417\pi$
$20$	0	0
$21$	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
$25$	0	0
$26$	136.000	1.02584
$27$	0	0
$28$	−56.0000	−0.377964
$29$	−126.000	−0.806814	−0.403407	−	0.915021i	$-0.632174\pi$
$29$	−126.000	−0.806814	−0.403407	+	0.915021i	$0.632174\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	304.000	1.35074	0.675369	−	0.737480i	$-0.263984\pi$
$37$	304.000	1.35074	0.675369	+	0.737480i	$0.263984\pi$
$38$	−88.0000	−0.375671
$39$	0	0
$40$	0	0
$41$	480.000	1.82838	0.914188	−	0.405291i	$-0.132830\pi$
$41$	480.000	1.82838	0.914188	+	0.405291i	$0.132830\pi$
$42$	0	0
$43$	−104.000	−0.368834	−0.184417	−	0.982848i	$-0.559040\pi$
$43$	−104.000	−0.368834	−0.184417	+	0.982848i	$0.559040\pi$
$44$	−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
$50$	0	0
$51$	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
$55$	0	0
$56$	112.000	0.267261
$57$	0	0
$58$	252.000	0.570504
$59$	−534.000	−1.17832	−0.589160	−	0.808016i	$-0.700541\pi$
$59$	−534.000	−1.17832	−0.589160	+	0.808016i	$0.700541\pi$
$60$	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
$65$	0	0
$66$	0	0
$67$	268.000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	268.000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	312.000	0.556405
$69$	0	0
$70$	0	0
$71$	972.000	1.62472	0.812360	−	0.583156i	$-0.198182\pi$
$71$	972.000	1.62472	0.812360	+	0.583156i	$0.198182\pi$
$72$	0	0
$73$	−470.000	−0.753553	−0.376776	−	0.926304i	$-0.622967\pi$
$73$	−470.000	−0.753553	−0.376776	+	0.926304i	$0.622967\pi$
$74$	−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	$0.153591\pi$
$80$	0	0
$81$	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
$85$	0	0
$86$	208.000	0.260805
$87$	0	0
$88$	48.0000	0.0581456
$89$	972.000	1.15766	0.578830	−	0.815448i	$-0.303509\pi$
$89$	972.000	1.15766	0.578830	+	0.815448i	$0.303509\pi$
$90$	0	0
$91$	952.000	1.09667
$92$	480.000	0.543951
$93$	0	0
$94$	−1200.00	−1.31671
$95$	0	0
$96$	0	0
$97$	46.0000	0.0481504	0.0240752	−	0.999710i	$-0.492336\pi$
$97$	46.0000	0.0481504	0.0240752	+	0.999710i	$0.492336\pi$
$98$	294.000	0.303046
$99$	0	0
$100$	0	0
$101$	1506.00	1.48369	0.741845	−	0.670572i	$-0.233951\pi$
$101$	1506.00	1.48369	0.741845	+	0.670572i	$0.233951\pi$
$102$	0	0
$103$	1474.00	1.41007	0.705037	−	0.709171i	$-0.250931\pi$
$103$	1474.00	1.41007	0.705037	+	0.709171i	$0.250931\pi$
$104$	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	$0.400782\pi$
$110$	0	0
$111$	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
$115$	0	0
$116$	−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
$125$	0	0
$126$	0	0
$127$	1906.00	1.33173	0.665867	−	0.746071i	$-0.268062\pi$
$127$	1906.00	1.33173	0.665867	+	0.746071i	$0.268062\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−2874.00	−1.91681	−0.958407	−	0.285406i	$-0.907872\pi$
$131$	−2874.00	−1.91681	−0.958407	+	0.285406i	$0.907872\pi$
$132$	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	$0.568510\pi$
$140$	0	0
$141$	0	0
$142$	−1944.00	−1.14885
$143$	408.000	0.238592
$144$	0	0
$145$	0	0
$146$	940.000	0.532842
$147$	0	0
$148$	1216.00	0.675369
$149$	−114.000	−0.0626795	−0.0313397	−	0.999509i	$-0.509977\pi$
$149$	−114.000	−0.0626795	−0.0313397	+	0.999509i	$0.509977\pi$
$150$	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
$155$	0	0
$156$	0	0
$157$	1948.00	0.990238	0.495119	−	0.868825i	$-0.335125\pi$
$157$	1948.00	0.990238	0.495119	+	0.868825i	$0.335125\pi$
$158$	−2488.00	−1.25275
$159$	0	0
$160$	0	0
$161$	−1680.00	−0.822376
$162$	0	0
$163$	−2060.00	−0.989887	−0.494944	−	0.868925i	$-0.664811\pi$
$163$	−2060.00	−0.989887	−0.494944	+	0.868925i	$0.664811\pi$
$164$	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
$170$	0	0
$171$	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
$175$	0	0
$176$	−96.0000	−0.0411152
$177$	0	0
$178$	−1944.00	−0.818590
$179$	1146.00	0.478525	0.239263	−	0.970955i	$-0.423094\pi$
$179$	1146.00	0.478525	0.239263	+	0.970955i	$0.423094\pi$
$180$	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
$185$	0	0
$186$	0	0
$187$	−468.000	−0.183014
$188$	2400.00	0.931053
$189$	0	0
$190$	0	0
$191$	−1692.00	−0.640989	−0.320494	−	0.947250i	$-0.603849\pi$
$191$	−1692.00	−0.640989	−0.320494	+	0.947250i	$0.603849\pi$
$192$	0	0
$193$	−3350.00	−1.24942	−0.624711	−	0.780856i	$-0.714783\pi$
$193$	−3350.00	−1.24942	−0.624711	+	0.780856i	$0.714783\pi$
$194$	−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	$0.340569\pi$
$200$	0	0
$201$	0	0
$202$	−3012.00	−1.04913
$203$	1764.00	0.609894
$204$	0	0
$205$	0	0
$206$	−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
$215$	0	0
$216$	0	0
$217$	3416.00	1.06863
$218$	−1396.00	−0.433711
$219$	0	0
$220$	0	0
$221$	−5304.00	−1.61441
$222$	0	0
$223$	1162.00	0.348938	0.174469	−	0.984663i	$-0.444179\pi$
$223$	1162.00	0.348938	0.174469	+	0.984663i	$0.444179\pi$
$224$	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	$0.608355\pi$
$230$	0	0
$231$	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
$235$	0	0
$236$	−2136.00	−0.589160
$237$	0	0
$238$	2184.00	0.594822
$239$	5868.00	1.58816	0.794078	−	0.607816i	$-0.207954\pi$
$239$	5868.00	1.58816	0.794078	+	0.607816i	$0.207954\pi$
$240$	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
$245$	0	0
$246$	0	0
$247$	−2992.00	−0.770755
$248$	1952.00	0.499807
$249$	0	0
$250$	0	0
$251$	−498.000	−0.125233	−0.0626165	−	0.998038i	$-0.519944\pi$
$251$	−498.000	−0.125233	−0.0626165	+	0.998038i	$0.519944\pi$
$252$	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
$260$	0	0
$261$	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
$265$	0	0
$266$	1232.00	0.283980
$267$	0	0
$268$	1072.00	0.244339
$269$	4218.00	0.956045	0.478022	−	0.878348i	$-0.341354\pi$
$269$	4218.00	0.956045	0.478022	+	0.878348i	$0.341354\pi$
$270$	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
$275$	0	0
$276$	0	0
$277$	1504.00	0.326233	0.163117	−	0.986607i	$-0.447845\pi$
$277$	1504.00	0.326233	0.163117	+	0.986607i	$0.447845\pi$
$278$	1400.00	0.302037
$279$	0	0
$280$	0	0
$281$	1308.00	0.277682	0.138841	−	0.990315i	$-0.455662\pi$
$281$	1308.00	0.277682	0.138841	+	0.990315i	$0.455662\pi$
$282$	0	0
$283$	5932.00	1.24601	0.623005	−	0.782218i	$-0.285912\pi$
$283$	5932.00	1.24601	0.623005	+	0.782218i	$0.285912\pi$
$284$	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
$290$	0	0
$291$	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
$295$	0	0
$296$	−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
$305$	0	0
$306$	0	0
$307$	−4448.00	−0.826908	−0.413454	−	0.910525i	$-0.635678\pi$
$307$	−4448.00	−0.826908	−0.413454	+	0.910525i	$0.635678\pi$
$308$	336.000	0.0621603
$309$	0	0
$310$	0	0
$311$	9132.00	1.66504	0.832521	−	0.553993i	$-0.186897\pi$
$311$	9132.00	1.66504	0.832521	+	0.553993i	$0.186897\pi$
$312$	0	0
$313$	2170.00	0.391871	0.195936	−	0.980617i	$-0.437226\pi$
$313$	2170.00	0.391871	0.195936	+	0.980617i	$0.437226\pi$
$314$	−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
$320$	0	0
$321$	0	0
$322$	3360.00	0.581508
$323$	3432.00	0.591212
$324$	0	0
$325$	0	0
$326$	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
$335$	0	0
$336$	0	0
$337$	−12158.0	−1.96525	−0.982624	−	0.185608i	$-0.940574\pi$
$337$	−12158.0	−1.96525	−0.982624	+	0.185608i	$0.940574\pi$
$338$	−4854.00	−0.781133
$339$	0	0
$340$	0	0
$341$	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	$0.552922\pi$
$350$	0	0
$351$	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
$355$	0	0
$356$	3888.00	0.578830
$357$	0	0
$358$	−2292.00	−0.338369
$359$	8664.00	1.27373	0.636864	−	0.770976i	$-0.280231\pi$
$359$	8664.00	1.27373	0.636864	+	0.770976i	$0.280231\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	236.000	0.0342649
$363$	0	0
$364$	3808.00	0.548334
$365$	0	0
$366$	0	0
$367$	−3782.00	−0.537926	−0.268963	−	0.963151i	$-0.586681\pi$
$367$	−3782.00	−0.537926	−0.268963	+	0.963151i	$0.586681\pi$
$368$	1920.00	0.271975
$369$	0	0
$370$	0	0
$371$	3612.00	0.505460
$372$	0	0
$373$	−11276.0	−1.56528	−0.782640	−	0.622475i	$-0.786127\pi$
$373$	−11276.0	−1.56528	−0.782640	+	0.622475i	$0.786127\pi$
$374$	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	$0.478845\pi$
$380$	0	0
$381$	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
$385$	0	0
$386$	6700.00	0.883474
$387$	0	0
$388$	184.000	0.0240752
$389$	−13338.0	−1.73847	−0.869233	−	0.494402i	$-0.835387\pi$
$389$	−13338.0	−1.73847	−0.869233	+	0.494402i	$0.835387\pi$
$390$	0	0
$391$	9360.00	1.21063
$392$	1176.00	0.151523
$393$	0	0
$394$	7212.00	0.922171
$395$	0	0
$396$	0	0
$397$	7192.00	0.909209	0.454605	−	0.890693i	$-0.349781\pi$
$397$	7192.00	0.909209	0.454605	+	0.890693i	$0.349781\pi$
$398$	−5392.00	−0.679087
$399$	0	0
$400$	0	0
$401$	2316.00	0.288418	0.144209	−	0.989547i	$-0.453936\pi$
$401$	2316.00	0.288418	0.144209	+	0.989547i	$0.453936\pi$
$402$	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	$0.768517\pi$
$410$	0	0
$411$	0	0
$412$	5896.00	0.705037
$413$	7476.00	0.890726
$414$	0	0
$415$	0	0
$416$	2176.00	0.256460
$417$	0	0
$418$	528.000	0.0617831
$419$	−3306.00	−0.385462	−0.192731	−	0.981252i	$-0.561735\pi$
$419$	−3306.00	−0.385462	−0.192731	+	0.981252i	$0.561735\pi$
$420$	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
$425$	0	0
$426$	0	0
$427$	−5068.00	−0.574374
$428$	3696.00	0.417413
$429$	0	0
$430$	0	0
$431$	6480.00	0.724201	0.362100	−	0.932139i	$-0.382060\pi$
$431$	6480.00	0.724201	0.362100	+	0.932139i	$0.382060\pi$
$432$	0	0
$433$	−11894.0	−1.32007	−0.660034	−	0.751236i	$-0.729458\pi$
$433$	−11894.0	−1.32007	−0.660034	+	0.751236i	$0.729458\pi$
$434$	−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	$0.742262\pi$
$440$	0	0
$441$	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
$445$	0	0
$446$	−2324.00	−0.246737
$447$	0	0
$448$	−896.000	−0.0944911
$449$	11508.0	1.20957	0.604784	−	0.796389i	$-0.293259\pi$
$449$	11508.0	1.20957	0.604784	+	0.796389i	$0.293259\pi$
$450$	0	0
$451$	−2880.00	−0.300696
$452$	−888.000	−0.0924071
$453$	0	0
$454$	4800.00	0.496201
$455$	0	0
$456$	0	0
$457$	−1082.00	−0.110752	−0.0553762	−	0.998466i	$-0.517636\pi$
$457$	−1082.00	−0.110752	−0.0553762	+	0.998466i	$0.517636\pi$
$458$	4628.00	0.472166
$459$	0	0
$460$	0	0
$461$	11238.0	1.13537	0.567685	−	0.823246i	$-0.307839\pi$
$461$	11238.0	1.13537	0.567685	+	0.823246i	$0.307839\pi$
$462$	0	0
$463$	2302.00	0.231065	0.115532	−	0.993304i	$-0.463143\pi$
$463$	2302.00	0.231065	0.115532	+	0.993304i	$0.463143\pi$
$464$	−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
$470$	0	0
$471$	0	0
$472$	4272.00	0.416599
$473$	624.000	0.0606587
$474$	0	0
$475$	0	0
$476$	−4368.00	−0.420603
$477$	0	0
$478$	−11736.0	−1.12300
$479$	−4644.00	−0.442985	−0.221492	−	0.975162i	$-0.571093\pi$
$479$	−4644.00	−0.442985	−0.221492	+	0.975162i	$0.571093\pi$
$480$	0	0
$481$	−20672.0	−1.95959
$482$	8660.00	0.818366
$483$	0	0
$484$	−5180.00	−0.486476
$485$	0	0
$486$	0	0
$487$	−2426.00	−0.225734	−0.112867	−	0.993610i	$-0.536003\pi$
$487$	−2426.00	−0.225734	−0.112867	+	0.993610i	$0.536003\pi$
$488$	−2896.00	−0.268639
$489$	0	0
$490$	0	0
$491$	−234.000	−0.0215077	−0.0107538	−	0.999942i	$-0.503423\pi$
$491$	−234.000	−0.0215077	−0.0107538	+	0.999942i	$0.503423\pi$
$492$	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	$0.280120\pi$
$500$	0	0
$501$	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
$505$	0	0
$506$	1440.00	0.126513
$507$	0	0
$508$	7624.00	0.665867
$509$	4458.00	0.388207	0.194104	−	0.980981i	$-0.437820\pi$
$509$	4458.00	0.388207	0.194104	+	0.980981i	$0.437820\pi$
$510$	0	0
$511$	6580.00	0.569632
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−1284.00	−0.110184
$515$	0	0
$516$	0	0
$517$	−3600.00	−0.306243
$518$	8512.00	0.722000
$519$	0	0
$520$	0	0
$521$	−4212.00	−0.354186	−0.177093	−	0.984194i	$-0.556669\pi$
$521$	−4212.00	−0.354186	−0.177093	+	0.984194i	$0.556669\pi$
$522$	0	0
$523$	11212.0	0.937412	0.468706	−	0.883354i	$-0.344720\pi$
$523$	11212.0	0.937412	0.468706	+	0.883354i	$0.344720\pi$
$524$	−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
$530$	0	0
$531$	0	0
$532$	−2464.00	−0.200804
$533$	−32640.0	−2.65252
$534$	0	0
$535$	0	0
$536$	−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
$545$	0	0
$546$	0	0
$547$	−18200.0	−1.42262	−0.711312	−	0.702876i	$-0.751899\pi$
$547$	−18200.0	−1.42262	−0.711312	+	0.702876i	$0.751899\pi$
$548$	−3192.00	−0.248824
$549$	0	0
$550$	0	0
$551$	−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
$560$	0	0
$561$	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
$565$	0	0
$566$	−11864.0	−0.881062
$567$	0	0
$568$	−7776.00	−0.574426
$569$	−3084.00	−0.227220	−0.113610	−	0.993525i	$-0.536241\pi$
$569$	−3084.00	−0.227220	−0.113610	+	0.993525i	$0.536241\pi$
$570$	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
$575$	0	0
$576$	0	0
$577$	11014.0	0.794660	0.397330	−	0.917676i	$-0.369937\pi$
$577$	11014.0	0.794660	0.397330	+	0.917676i	$0.369937\pi$
$578$	−2342.00	−0.168537
$579$	0	0
$580$	0	0
$581$	−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
$590$	0	0
$591$	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
$595$	0	0
$596$	−456.000	−0.0313397
$597$	0	0
$598$	16320.0	1.11601
$599$	8616.00	0.587713	0.293857	−	0.955850i	$-0.405061\pi$
$599$	8616.00	0.587713	0.293857	+	0.955850i	$0.405061\pi$
$600$	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
$605$	0	0
$606$	0	0
$607$	13894.0	0.929061	0.464531	−	0.885557i	$-0.346223\pi$
$607$	13894.0	0.929061	0.464531	+	0.885557i	$0.346223\pi$
$608$	−1408.00	−0.0939177
$609$	0	0
$610$	0	0
$611$	−40800.0	−2.70146
$612$	0	0
$613$	6496.00	0.428011	0.214006	−	0.976832i	$-0.431349\pi$
$613$	6496.00	0.428011	0.214006	+	0.976832i	$0.431349\pi$
$614$	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	$0.522133\pi$
$620$	0	0
$621$	0	0
$622$	−18264.0	−1.17736
$623$	−13608.0	−0.875109
$624$	0	0
$625$	0	0
$626$	−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
$635$	0	0
$636$	0	0
$637$	9996.00	0.621752
$638$	−1512.00	−0.0938255
$639$	0	0
$640$	0	0
$641$	456.000	0.0280982	0.0140491	−	0.999901i	$-0.495528\pi$
$641$	456.000	0.0280982	0.0140491	+	0.999901i	$0.495528\pi$
$642$	0	0
$643$	23452.0	1.43835	0.719173	−	0.694831i	$-0.244521\pi$
$643$	23452.0	1.43835	0.719173	+	0.694831i	$0.244521\pi$
$644$	−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
$650$	0	0
$651$	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
$655$	0	0
$656$	7680.00	0.457094
$657$	0	0
$658$	16800.0	0.995338
$659$	−27810.0	−1.64389	−0.821945	−	0.569567i	$-0.807111\pi$
$659$	−27810.0	−1.64389	−0.821945	+	0.569567i	$0.807111\pi$
$660$	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
$665$	0	0
$666$	0	0
$667$	−15120.0	−0.877734
$668$	−4992.00	−0.289141
$669$	0	0
$670$	0	0
$671$	−2172.00	−0.124961
$672$	0	0
$673$	3778.00	0.216391	0.108196	−	0.994130i	$-0.465493\pi$
$673$	3778.00	0.216391	0.108196	+	0.994130i	$0.465493\pi$
$674$	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
$680$	0	0
$681$	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
$685$	0	0
$686$	−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
$695$	0	0
$696$	0	0
$697$	37440.0	2.03464
$698$	4316.00	0.234044
$699$	0	0
$700$	0	0
$701$	−22782.0	−1.22748	−0.613741	−	0.789508i	$-0.710336\pi$
$701$	−22782.0	−1.22748	−0.613741	+	0.789508i	$0.710336\pi$
$702$	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	$0.257592\pi$
$710$	0	0
$711$	0	0
$712$	−7776.00	−0.409295
$713$	−29280.0	−1.53793
$714$	0	0
$715$	0	0
$716$	4584.00	0.239263
$717$	0	0
$718$	−17328.0	−0.900662
$719$	5976.00	0.309968	0.154984	−	0.987917i	$-0.450467\pi$
$719$	5976.00	0.309968	0.154984	+	0.987917i	$0.450467\pi$
$720$	0	0
$721$	−20636.0	−1.06592
$722$	9846.00	0.507521
$723$	0	0
$724$	−472.000	−0.0242289
$725$	0	0
$726$	0	0
$727$	5110.00	0.260687	0.130343	−	0.991469i	$-0.458392\pi$
$727$	5110.00	0.260687	0.130343	+	0.991469i	$0.458392\pi$
$728$	−7616.00	−0.387730
$729$	0	0
$730$	0	0
$731$	−8112.00	−0.410442
$732$	0	0
$733$	−17336.0	−0.873560	−0.436780	−	0.899568i	$-0.643881\pi$
$733$	−17336.0	−0.873560	−0.436780	+	0.899568i	$0.643881\pi$
$734$	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	$0.610420\pi$
$740$	0	0
$741$	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
$745$	0	0
$746$	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
$755$	0	0
$756$	0	0
$757$	−22160.0	−1.06396	−0.531981	−	0.846756i	$-0.678552\pi$
$757$	−22160.0	−1.06396	−0.531981	+	0.846756i	$0.678552\pi$
$758$	−1960.00	−0.0939187
$759$	0	0
$760$	0	0
$761$	−13116.0	−0.624776	−0.312388	−	0.949955i	$-0.601129\pi$
$761$	−13116.0	−0.624776	−0.312388	+	0.949955i	$0.601129\pi$
$762$	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	$0.220192\pi$
$770$	0	0
$771$	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
$775$	0	0
$776$	−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
$785$	0	0
$786$	0	0
$787$	21076.0	0.954610	0.477305	−	0.878738i	$-0.341614\pi$
$787$	21076.0	0.954610	0.477305	+	0.878738i	$0.341614\pi$
$788$	−14424.0	−0.652073
$789$	0	0
$790$	0	0
$791$	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
$800$	0	0
$801$	0	0
$802$	−4632.00	−0.203942
$803$	2820.00	0.123930
$804$	0	0
$805$	0	0
$806$	−33184.0	−1.45019
$807$	0	0
$808$	−12048.0	−0.524563
$809$	−21384.0	−0.929322	−0.464661	−	0.885489i	$-0.653824\pi$
$809$	−21384.0	−0.929322	−0.464661	+	0.885489i	$0.653824\pi$
$810$	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
$815$	0	0
$816$	0	0
$817$	−4576.00	−0.195953
$818$	24716.0	1.05645
$819$	0	0
$820$	0	0
$821$	38010.0	1.61578	0.807892	−	0.589331i	$-0.200609\pi$
$821$	38010.0	1.61578	0.807892	+	0.589331i	$0.200609\pi$
$822$	0	0
$823$	−38642.0	−1.63667	−0.818333	−	0.574745i	$-0.805101\pi$
$823$	−38642.0	−1.63667	−0.818333	+	0.574745i	$0.805101\pi$
$824$	−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	$0.525940\pi$
$830$	0	0
$831$	0	0
$832$	−4352.00	−0.181344
$833$	−11466.0	−0.476919
$834$	0	0
$835$	0	0
$836$	−1056.00	−0.0436872
$837$	0	0
$838$	6612.00	0.272563
$839$	−27552.0	−1.13373	−0.566866	−	0.823810i	$-0.691844\pi$
$839$	−27552.0	−1.13373	−0.566866	+	0.823810i	$0.691844\pi$
$840$	0	0
$841$	−8513.00	−0.349051
$842$	29012.0	1.18743
$843$	0	0
$844$	−16.0000	−0.000652539 0
$845$	0	0
$846$	0	0
$847$	18130.0	0.735483
$848$	−4128.00	−0.167165
$849$	0	0
$850$	0	0
$851$	36480.0	1.46947
$852$	0	0
$853$	−15104.0	−0.606273	−0.303137	−	0.952947i	$-0.598034\pi$
$853$	−15104.0	−0.606273	−0.303137	+	0.952947i	$0.598034\pi$
$854$	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	$0.892346\pi$
$860$	0	0
$861$	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
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	27796.0	1.07024	0.535122	−	0.844775i	$-0.320266\pi$
$877$	27796.0	1.07024	0.535122	+	0.844775i	$0.320266\pi$
$878$	25376.0	0.975397
$879$	0	0
$880$	0	0
$881$	39996.0	1.52951	0.764756	−	0.644320i	$-0.222860\pi$
$881$	39996.0	1.52951	0.764756	+	0.644320i	$0.222860\pi$
$882$	0	0
$883$	3772.00	0.143758	0.0718788	−	0.997413i	$-0.477101\pi$
$883$	3772.00	0.143758	0.0718788	+	0.997413i	$0.477101\pi$
$884$	−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
$890$	0	0
$891$	0	0
$892$	4648.00	0.174469
$893$	26400.0	0.989297
$894$	0	0
$895$	0	0
$896$	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
$905$	0	0
$906$	0	0
$907$	8440.00	0.308981	0.154490	−	0.987994i	$-0.450626\pi$
$907$	8440.00	0.308981	0.154490	+	0.987994i	$0.450626\pi$
$908$	−9600.00	−0.350867
$909$	0	0
$910$	0	0
$911$	31920.0	1.16087	0.580437	−	0.814305i	$-0.302882\pi$
$911$	31920.0	1.16087	0.580437	+	0.814305i	$0.302882\pi$
$912$	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	$0.286358\pi$
$920$	0	0
$921$	0	0
$922$	−22476.0	−0.802828
$923$	−66096.0	−2.35707
$924$	0	0
$925$	0	0
$926$	−4604.00	−0.163388
$927$	0	0
$928$	4032.00	0.142626
$929$	−1404.00	−0.0495842	−0.0247921	−	0.999693i	$-0.507892\pi$
$929$	−1404.00	−0.0495842	−0.0247921	+	0.999693i	$0.507892\pi$
$930$	0	0
$931$	−6468.00	−0.227691
$932$	−72.0000	−0.00253051
$933$	0	0
$934$	−31752.0	−1.11237
$935$	0	0
$936$	0	0
$937$	7654.00	0.266857	0.133429	−	0.991058i	$-0.457401\pi$
$937$	7654.00	0.266857	0.133429	+	0.991058i	$0.457401\pi$
$938$	7504.00	0.261209
$939$	0	0
$940$	0	0
$941$	−11298.0	−0.391397	−0.195698	−	0.980664i	$-0.562697\pi$
$941$	−11298.0	−0.391397	−0.195698	+	0.980664i	$0.562697\pi$
$942$	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
$950$	0	0
$951$	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
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	41506.0	1.38029	0.690146	−	0.723670i	$-0.257546\pi$
$967$	41506.0	1.38029	0.690146	+	0.723670i	$0.257546\pi$
$968$	10360.0	0.343991
$969$	0	0
$970$	0	0
$971$	18246.0	0.603030	0.301515	−	0.953461i	$-0.402508\pi$
$971$	18246.0	0.603030	0.301515	+	0.953461i	$0.402508\pi$
$972$	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
$980$	0	0
$981$	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
$985$	0	0
$986$	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
$995$	0	0
$996$	0	0
$997$	9052.00	0.287542	0.143771	−	0.989611i	$-0.454077\pi$
$997$	9052.00	0.287542	0.143771	+	0.989611i	$0.454077\pi$
$998$	−28408.0	−0.901042
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.a.c.1.1		1
3.2	odd	2		450.4.a.m.1.1		1
5.2	odd	4		450.4.c.f.199.1		2
5.3	odd	4		450.4.c.f.199.2		2
5.4	even	2		90.4.a.e.1.1	yes	1
15.2	even	4		450.4.c.g.199.2		2
15.8	even	4		450.4.c.g.199.1		2
15.14	odd	2		90.4.a.b.1.1	✓	1
20.19	odd	2		720.4.a.t.1.1		1
45.4	even	6		810.4.e.a.541.1		2
45.14	odd	6		810.4.e.u.541.1		2
45.29	odd	6		810.4.e.u.271.1		2
45.34	even	6		810.4.e.a.271.1		2
60.59	even	2		720.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.4.a.b.1.1	✓	1	15.14	odd	2
90.4.a.e.1.1	yes	1	5.4	even	2
450.4.a.c.1.1		1	1.1	even	1	trivial
450.4.a.m.1.1		1	3.2	odd	2
450.4.c.f.199.1		2	5.2	odd	4
450.4.c.f.199.2		2	5.3	odd	4
450.4.c.g.199.1		2	15.8	even	4
450.4.c.g.199.2		2	15.2	even	4
720.4.a.e.1.1		1	60.59	even	2
720.4.a.t.1.1		1	20.19	odd	2
810.4.e.a.271.1		2	45.34	even	6
810.4.e.a.541.1		2	45.4	even	6
810.4.e.u.271.1		2	45.29	odd	6
810.4.e.u.541.1		2	45.14	odd	6

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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