LMFDB - Embedded newform 450.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,4,Mod(1,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [1,-2,0,4,0,0,-32,-8,0,0,60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.5508595026$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
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)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} -32.0000 q^{7} -8.00000 q^{8} +60.0000 q^{11} +34.0000 q^{13} +64.0000 q^{14} +16.0000 q^{16} +42.0000 q^{17} -76.0000 q^{19} -120.000 q^{22} -68.0000 q^{26} -128.000 q^{28} -6.00000 q^{29} -232.000 q^{31} -32.0000 q^{32} -84.0000 q^{34} -134.000 q^{37} +152.000 q^{38} -234.000 q^{41} +412.000 q^{43} +240.000 q^{44} -360.000 q^{47} +681.000 q^{49} +136.000 q^{52} +222.000 q^{53} +256.000 q^{56} +12.0000 q^{58} -660.000 q^{59} -490.000 q^{61} +464.000 q^{62} +64.0000 q^{64} -812.000 q^{67} +168.000 q^{68} -120.000 q^{71} -746.000 q^{73} +268.000 q^{74} -304.000 q^{76} -1920.00 q^{77} +152.000 q^{79} +468.000 q^{82} -804.000 q^{83} -824.000 q^{86} -480.000 q^{88} +678.000 q^{89} -1088.00 q^{91} +720.000 q^{94} -194.000 q^{97} -1362.00 q^{98} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−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
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	60.0000	1.64461	0.822304	−	0.569049i	$-0.192689\pi$
$11$	60.0000	1.64461	0.822304	+	0.569049i	$0.192689\pi$
$12$	0	0
$13$	34.0000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	34.0000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	64.0000	1.22177
$15$	0	0
$16$	16.0000	0.250000
$17$	42.0000	0.599206	0.299603	−	0.954064i	$-0.403146\pi$
$17$	42.0000	0.599206	0.299603	+	0.954064i	$0.403146\pi$
$18$	0	0
$19$	−76.0000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−76.0000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	−120.000	−1.16291
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	−68.0000	−0.512919
$27$	0	0
$28$	−128.000	−0.863919
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	−232.000	−1.34414	−0.672071	−	0.740486i	$-0.734595\pi$
$31$	−232.000	−1.34414	−0.672071	+	0.740486i	$0.734595\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−84.0000	−0.423702
$35$	0	0
$36$	0	0
$37$	−134.000	−0.595391	−0.297695	−	0.954661i	$-0.596218\pi$
$37$	−134.000	−0.595391	−0.297695	+	0.954661i	$0.596218\pi$
$38$	152.000	0.648886
$39$	0	0
$40$	0	0
$41$	−234.000	−0.891333	−0.445667	−	0.895199i	$-0.647033\pi$
$41$	−234.000	−0.891333	−0.445667	+	0.895199i	$0.647033\pi$
$42$	0	0
$43$	412.000	1.46115	0.730575	−	0.682833i	$-0.239252\pi$
$43$	412.000	1.46115	0.730575	+	0.682833i	$0.239252\pi$
$44$	240.000	0.822304
$45$	0	0
$46$	0	0
$47$	−360.000	−1.11726	−0.558632	−	0.829416i	$-0.688674\pi$
$47$	−360.000	−1.11726	−0.558632	+	0.829416i	$0.688674\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	0	0
$52$	136.000	0.362689
$53$	222.000	0.575359	0.287680	−	0.957727i	$-0.407116\pi$
$53$	222.000	0.575359	0.287680	+	0.957727i	$0.407116\pi$
$54$	0	0
$55$	0	0
$56$	256.000	0.610883
$57$	0	0
$58$	12.0000	0.0271668
$59$	−660.000	−1.45635	−0.728175	−	0.685391i	$-0.759631\pi$
$59$	−660.000	−1.45635	−0.728175	+	0.685391i	$0.759631\pi$
$60$	0	0
$61$	−490.000	−1.02849	−0.514246	−	0.857642i	$-0.671928\pi$
$61$	−490.000	−1.02849	−0.514246	+	0.857642i	$0.671928\pi$
$62$	464.000	0.950453
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−812.000	−1.48062	−0.740310	−	0.672265i	$-0.765321\pi$
$67$	−812.000	−1.48062	−0.740310	+	0.672265i	$0.765321\pi$
$68$	168.000	0.299603
$69$	0	0
$70$	0	0
$71$	−120.000	−0.200583	−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000	−0.200583	−0.100291	+	0.994958i	$0.531978\pi$
$72$	0	0
$73$	−746.000	−1.19606	−0.598032	−	0.801472i	$-0.704051\pi$
$73$	−746.000	−1.19606	−0.598032	+	0.801472i	$0.704051\pi$
$74$	268.000	0.421005
$75$	0	0
$76$	−304.000	−0.458831
$77$	−1920.00	−2.84161
$78$	0	0
$79$	152.000	0.216473	0.108236	−	0.994125i	$-0.465480\pi$
$79$	152.000	0.216473	0.108236	+	0.994125i	$0.465480\pi$
$80$	0	0
$81$	0	0
$82$	468.000	0.630268
$83$	−804.000	−1.06326	−0.531629	−	0.846977i	$-0.678420\pi$
$83$	−804.000	−1.06326	−0.531629	+	0.846977i	$0.678420\pi$
$84$	0	0
$85$	0	0
$86$	−824.000	−1.03319
$87$	0	0
$88$	−480.000	−0.581456
$89$	678.000	0.807504	0.403752	−	0.914868i	$-0.367706\pi$
$89$	678.000	0.807504	0.403752	+	0.914868i	$0.367706\pi$
$90$	0	0
$91$	−1088.00	−1.25333
$92$	0	0
$93$	0	0
$94$	720.000	0.790025
$95$	0	0
$96$	0	0
$97$	−194.000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−194.000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	−1362.00	−1.40391
$99$	0	0
$100$	0	0
$101$	−798.000	−0.786178	−0.393089	−	0.919500i	$-0.628594\pi$
$101$	−798.000	−0.786178	−0.393089	+	0.919500i	$0.628594\pi$
$102$	0	0
$103$	−1088.00	−1.04081	−0.520407	−	0.853918i	$-0.674220\pi$
$103$	−1088.00	−1.04081	−0.520407	+	0.853918i	$0.674220\pi$
$104$	−272.000	−0.256460
$105$	0	0
$106$	−444.000	−0.406840
$107$	1716.00	1.55039	0.775196	−	0.631721i	$-0.217651\pi$
$107$	1716.00	1.55039	0.775196	+	0.631721i	$0.217651\pi$
$108$	0	0
$109$	−970.000	−0.852378	−0.426189	−	0.904634i	$-0.640144\pi$
$109$	−970.000	−0.852378	−0.426189	+	0.904634i	$0.640144\pi$
$110$	0	0
$111$	0	0
$112$	−512.000	−0.431959
$113$	426.000	0.354643	0.177322	−	0.984153i	$-0.443257\pi$
$113$	426.000	0.354643	0.177322	+	0.984153i	$0.443257\pi$
$114$	0	0
$115$	0	0
$116$	−24.0000	−0.0192099
$117$	0	0
$118$	1320.00	1.02980
$119$	−1344.00	−1.03533
$120$	0	0
$121$	2269.00	1.70473
$122$	980.000	0.727254
$123$	0	0
$124$	−928.000	−0.672071
$125$	0	0
$126$	0	0
$127$	−200.000	−0.139741	−0.0698706	−	0.997556i	$-0.522259\pi$
$127$	−200.000	−0.139741	−0.0698706	+	0.997556i	$0.522259\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−60.0000	−0.0400170	−0.0200085	−	0.999800i	$-0.506369\pi$
$131$	−60.0000	−0.0400170	−0.0200085	+	0.999800i	$0.506369\pi$
$132$	0	0
$133$	2432.00	1.58557
$134$	1624.00	1.04696
$135$	0	0
$136$	−336.000	−0.211851
$137$	642.000	0.400363	0.200182	−	0.979759i	$-0.435847\pi$
$137$	642.000	0.400363	0.200182	+	0.979759i	$0.435847\pi$
$138$	0	0
$139$	−2836.00	−1.73055	−0.865275	−	0.501298i	$-0.832856\pi$
$139$	−2836.00	−1.73055	−0.865275	+	0.501298i	$0.832856\pi$
$140$	0	0
$141$	0	0
$142$	240.000	0.141833
$143$	2040.00	1.19296
$144$	0	0
$145$	0	0
$146$	1492.00	0.845745
$147$	0	0
$148$	−536.000	−0.297695
$149$	1554.00	0.854420	0.427210	−	0.904152i	$-0.359496\pi$
$149$	1554.00	0.854420	0.427210	+	0.904152i	$0.359496\pi$
$150$	0	0
$151$	−2272.00	−1.22446	−0.612228	−	0.790682i	$-0.709726\pi$
$151$	−2272.00	−1.22446	−0.612228	+	0.790682i	$0.709726\pi$
$152$	608.000	0.324443
$153$	0	0
$154$	3840.00	2.00932
$155$	0	0
$156$	0	0
$157$	−1694.00	−0.861120	−0.430560	−	0.902562i	$-0.641684\pi$
$157$	−1694.00	−0.861120	−0.430560	+	0.902562i	$0.641684\pi$
$158$	−304.000	−0.153069
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	52.0000	0.0249874	0.0124937	−	0.999922i	$-0.496023\pi$
$163$	52.0000	0.0249874	0.0124937	+	0.999922i	$0.496023\pi$
$164$	−936.000	−0.445667
$165$	0	0
$166$	1608.00	0.751837
$167$	−1200.00	−0.556041	−0.278020	−	0.960575i	$-0.589678\pi$
$167$	−1200.00	−0.556041	−0.278020	+	0.960575i	$0.589678\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	0	0
$172$	1648.00	0.730575
$173$	54.0000	0.0237315	0.0118657	−	0.999930i	$-0.496223\pi$
$173$	54.0000	0.0237315	0.0118657	+	0.999930i	$0.496223\pi$
$174$	0	0
$175$	0	0
$176$	960.000	0.411152
$177$	0	0
$178$	−1356.00	−0.570992
$179$	−876.000	−0.365784	−0.182892	−	0.983133i	$-0.558546\pi$
$179$	−876.000	−0.365784	−0.182892	+	0.983133i	$0.558546\pi$
$180$	0	0
$181$	3854.00	1.58268	0.791341	−	0.611375i	$-0.209383\pi$
$181$	3854.00	1.58268	0.791341	+	0.611375i	$0.209383\pi$
$182$	2176.00	0.886241
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2520.00	0.985458
$188$	−1440.00	−0.558632
$189$	0	0
$190$	0	0
$191$	2784.00	1.05468	0.527338	−	0.849656i	$-0.323190\pi$
$191$	2784.00	1.05468	0.527338	+	0.849656i	$0.323190\pi$
$192$	0	0
$193$	−914.000	−0.340887	−0.170443	−	0.985367i	$-0.554520\pi$
$193$	−914.000	−0.340887	−0.170443	+	0.985367i	$0.554520\pi$
$194$	388.000	0.143592
$195$	0	0
$196$	2724.00	0.992711
$197$	−5202.00	−1.88136	−0.940678	−	0.339300i	$-0.889810\pi$
$197$	−5202.00	−1.88136	−0.940678	+	0.339300i	$0.889810\pi$
$198$	0	0
$199$	3152.00	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$199$	3152.00	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$200$	0	0
$201$	0	0
$202$	1596.00	0.555912
$203$	192.000	0.0663830
$204$	0	0
$205$	0	0
$206$	2176.00	0.735967
$207$	0	0
$208$	544.000	0.181344
$209$	−4560.00	−1.50920
$210$	0	0
$211$	740.000	0.241439	0.120720	−	0.992687i	$-0.461480\pi$
$211$	740.000	0.241439	0.120720	+	0.992687i	$0.461480\pi$
$212$	888.000	0.287680
$213$	0	0
$214$	−3432.00	−1.09629
$215$	0	0
$216$	0	0
$217$	7424.00	2.32246
$218$	1940.00	0.602722
$219$	0	0
$220$	0	0
$221$	1428.00	0.434650
$222$	0	0
$223$	520.000	0.156151	0.0780757	−	0.996947i	$-0.475122\pi$
$223$	520.000	0.156151	0.0780757	+	0.996947i	$0.475122\pi$
$224$	1024.00	0.305441
$225$	0	0
$226$	−852.000	−0.250771
$227$	396.000	0.115786	0.0578930	−	0.998323i	$-0.481562\pi$
$227$	396.000	0.115786	0.0578930	+	0.998323i	$0.481562\pi$
$228$	0	0
$229$	−1330.00	−0.383794	−0.191897	−	0.981415i	$-0.561464\pi$
$229$	−1330.00	−0.383794	−0.191897	+	0.981415i	$0.561464\pi$
$230$	0	0
$231$	0	0
$232$	48.0000	0.0135834
$233$	4866.00	1.36816	0.684082	−	0.729405i	$-0.260203\pi$
$233$	4866.00	1.36816	0.684082	+	0.729405i	$0.260203\pi$
$234$	0	0
$235$	0	0
$236$	−2640.00	−0.728175
$237$	0	0
$238$	2688.00	0.732089
$239$	1824.00	0.493660	0.246830	−	0.969059i	$-0.420611\pi$
$239$	1824.00	0.493660	0.246830	+	0.969059i	$0.420611\pi$
$240$	0	0
$241$	6482.00	1.73254	0.866270	−	0.499575i	$-0.166511\pi$
$241$	6482.00	1.73254	0.866270	+	0.499575i	$0.166511\pi$
$242$	−4538.00	−1.20543
$243$	0	0
$244$	−1960.00	−0.514246
$245$	0	0
$246$	0	0
$247$	−2584.00	−0.665652
$248$	1856.00	0.475226
$249$	0	0
$250$	0	0
$251$	−1476.00	−0.371172	−0.185586	−	0.982628i	$-0.559418\pi$
$251$	−1476.00	−0.371172	−0.185586	+	0.982628i	$0.559418\pi$
$252$	0	0
$253$	0	0
$254$	400.000	0.0988119
$255$	0	0
$256$	256.000	0.0625000
$257$	4314.00	1.04708	0.523541	−	0.852001i	$-0.324611\pi$
$257$	4314.00	1.04708	0.523541	+	0.852001i	$0.324611\pi$
$258$	0	0
$259$	4288.00	1.02874
$260$	0	0
$261$	0	0
$262$	120.000	0.0282963
$263$	−5280.00	−1.23794	−0.618971	−	0.785414i	$-0.712450\pi$
$263$	−5280.00	−1.23794	−0.618971	+	0.785414i	$0.712450\pi$
$264$	0	0
$265$	0	0
$266$	−4864.00	−1.12117
$267$	0	0
$268$	−3248.00	−0.740310
$269$	−5526.00	−1.25251	−0.626257	−	0.779617i	$-0.715414\pi$
$269$	−5526.00	−1.25251	−0.626257	+	0.779617i	$0.715414\pi$
$270$	0	0
$271$	2024.00	0.453687	0.226844	−	0.973931i	$-0.427159\pi$
$271$	2024.00	0.453687	0.226844	+	0.973931i	$0.427159\pi$
$272$	672.000	0.149801
$273$	0	0
$274$	−1284.00	−0.283100
$275$	0	0
$276$	0	0
$277$	−2054.00	−0.445534	−0.222767	−	0.974872i	$-0.571509\pi$
$277$	−2054.00	−0.445534	−0.222767	+	0.974872i	$0.571509\pi$
$278$	5672.00	1.22368
$279$	0	0
$280$	0	0
$281$	7302.00	1.55018	0.775090	−	0.631850i	$-0.217704\pi$
$281$	7302.00	1.55018	0.775090	+	0.631850i	$0.217704\pi$
$282$	0	0
$283$	3724.00	0.782222	0.391111	−	0.920344i	$-0.372091\pi$
$283$	3724.00	0.782222	0.391111	+	0.920344i	$0.372091\pi$
$284$	−480.000	−0.100291
$285$	0	0
$286$	−4080.00	−0.843551
$287$	7488.00	1.54008
$288$	0	0
$289$	−3149.00	−0.640953
$290$	0	0
$291$	0	0
$292$	−2984.00	−0.598032
$293$	−7218.00	−1.43918	−0.719591	−	0.694399i	$-0.755670\pi$
$293$	−7218.00	−1.43918	−0.719591	+	0.694399i	$0.755670\pi$
$294$	0	0
$295$	0	0
$296$	1072.00	0.210502
$297$	0	0
$298$	−3108.00	−0.604166
$299$	0	0
$300$	0	0
$301$	−13184.0	−2.52463
$302$	4544.00	0.865821
$303$	0	0
$304$	−1216.00	−0.229416
$305$	0	0
$306$	0	0
$307$	−2540.00	−0.472200	−0.236100	−	0.971729i	$-0.575869\pi$
$307$	−2540.00	−0.472200	−0.236100	+	0.971729i	$0.575869\pi$
$308$	−7680.00	−1.42081
$309$	0	0
$310$	0	0
$311$	−1560.00	−0.284436	−0.142218	−	0.989835i	$-0.545423\pi$
$311$	−1560.00	−0.284436	−0.142218	+	0.989835i	$0.545423\pi$
$312$	0	0
$313$	934.000	0.168667	0.0843335	−	0.996438i	$-0.473124\pi$
$313$	934.000	0.168667	0.0843335	+	0.996438i	$0.473124\pi$
$314$	3388.00	0.608904
$315$	0	0
$316$	608.000	0.108236
$317$	−1674.00	−0.296597	−0.148298	−	0.988943i	$-0.547380\pi$
$317$	−1674.00	−0.296597	−0.148298	+	0.988943i	$0.547380\pi$
$318$	0	0
$319$	−360.000	−0.0631854
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3192.00	−0.549869
$324$	0	0
$325$	0	0
$326$	−104.000	−0.0176688
$327$	0	0
$328$	1872.00	0.315134
$329$	11520.0	1.93045
$330$	0	0
$331$	−3988.00	−0.662237	−0.331118	−	0.943589i	$-0.607426\pi$
$331$	−3988.00	−0.662237	−0.331118	+	0.943589i	$0.607426\pi$
$332$	−3216.00	−0.531629
$333$	0	0
$334$	2400.00	0.393180
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.000323285 0	−0.000161642	−	1.00000i	$-0.500051\pi$
$337$	−2.00000	−0.000323285 0	−0.000161642		1.00000i	$0.500051\pi$
$338$	2082.00	0.335047
$339$	0	0
$340$	0	0
$341$	−13920.0	−2.21059
$342$	0	0
$343$	−10816.0	−1.70265
$344$	−3296.00	−0.516594
$345$	0	0
$346$	−108.000	−0.0167807
$347$	1764.00	0.272901	0.136450	−	0.990647i	$-0.456431\pi$
$347$	1764.00	0.272901	0.136450	+	0.990647i	$0.456431\pi$
$348$	0	0
$349$	4310.00	0.661057	0.330529	−	0.943796i	$-0.392773\pi$
$349$	4310.00	0.661057	0.330529	+	0.943796i	$0.392773\pi$
$350$	0	0
$351$	0	0
$352$	−1920.00	−0.290728
$353$	138.000	0.0208074	0.0104037	−	0.999946i	$-0.496688\pi$
$353$	138.000	0.0208074	0.0104037	+	0.999946i	$0.496688\pi$
$354$	0	0
$355$	0	0
$356$	2712.00	0.403752
$357$	0	0
$358$	1752.00	0.258648
$359$	11976.0	1.76064	0.880319	−	0.474382i	$-0.157328\pi$
$359$	11976.0	1.76064	0.880319	+	0.474382i	$0.157328\pi$
$360$	0	0
$361$	−1083.00	−0.157895
$362$	−7708.00	−1.11913
$363$	0	0
$364$	−4352.00	−0.626667
$365$	0	0
$366$	0	0
$367$	−9704.00	−1.38023	−0.690115	−	0.723699i	$-0.742440\pi$
$367$	−9704.00	−1.38023	−0.690115	+	0.723699i	$0.742440\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7104.00	−0.994128
$372$	0	0
$373$	8122.00	1.12746	0.563728	−	0.825960i	$-0.309367\pi$
$373$	8122.00	1.12746	0.563728	+	0.825960i	$0.309367\pi$
$374$	−5040.00	−0.696824
$375$	0	0
$376$	2880.00	0.395012
$377$	−204.000	−0.0278688
$378$	0	0
$379$	3404.00	0.461350	0.230675	−	0.973031i	$-0.425907\pi$
$379$	3404.00	0.461350	0.230675	+	0.973031i	$0.425907\pi$
$380$	0	0
$381$	0	0
$382$	−5568.00	−0.745769
$383$	−2520.00	−0.336204	−0.168102	−	0.985770i	$-0.553764\pi$
$383$	−2520.00	−0.336204	−0.168102	+	0.985770i	$0.553764\pi$
$384$	0	0
$385$	0	0
$386$	1828.00	0.241043
$387$	0	0
$388$	−776.000	−0.101535
$389$	−1566.00	−0.204111	−0.102056	−	0.994779i	$-0.532542\pi$
$389$	−1566.00	−0.204111	−0.102056	+	0.994779i	$0.532542\pi$
$390$	0	0
$391$	0	0
$392$	−5448.00	−0.701953
$393$	0	0
$394$	10404.0	1.33032
$395$	0	0
$396$	0	0
$397$	4354.00	0.550431	0.275215	−	0.961383i	$-0.411251\pi$
$397$	4354.00	0.550431	0.275215	+	0.961383i	$0.411251\pi$
$398$	−6304.00	−0.793947
$399$	0	0
$400$	0	0
$401$	8046.00	1.00199	0.500995	−	0.865450i	$-0.332967\pi$
$401$	8046.00	1.00199	0.500995	+	0.865450i	$0.332967\pi$
$402$	0	0
$403$	−7888.00	−0.975011
$404$	−3192.00	−0.393089
$405$	0	0
$406$	−384.000	−0.0469399
$407$	−8040.00	−0.979184
$408$	0	0
$409$	−2806.00	−0.339237	−0.169618	−	0.985510i	$-0.554253\pi$
$409$	−2806.00	−0.339237	−0.169618	+	0.985510i	$0.554253\pi$
$410$	0	0
$411$	0	0
$412$	−4352.00	−0.520407
$413$	21120.0	2.51634
$414$	0	0
$415$	0	0
$416$	−1088.00	−0.128230
$417$	0	0
$418$	9120.00	1.06716
$419$	−11580.0	−1.35017	−0.675084	−	0.737741i	$-0.735892\pi$
$419$	−11580.0	−1.35017	−0.675084	+	0.737741i	$0.735892\pi$
$420$	0	0
$421$	−370.000	−0.0428330	−0.0214165	−	0.999771i	$-0.506818\pi$
$421$	−370.000	−0.0428330	−0.0214165	+	0.999771i	$0.506818\pi$
$422$	−1480.00	−0.170723
$423$	0	0
$424$	−1776.00	−0.203420
$425$	0	0
$426$	0	0
$427$	15680.0	1.77707
$428$	6864.00	0.775196
$429$	0	0
$430$	0	0
$431$	−5040.00	−0.563267	−0.281634	−	0.959522i	$-0.590876\pi$
$431$	−5040.00	−0.563267	−0.281634	+	0.959522i	$0.590876\pi$
$432$	0	0
$433$	3742.00	0.415310	0.207655	−	0.978202i	$-0.433417\pi$
$433$	3742.00	0.415310	0.207655	+	0.978202i	$0.433417\pi$
$434$	−14848.0	−1.64223
$435$	0	0
$436$	−3880.00	−0.426189
$437$	0	0
$438$	0	0
$439$	−6208.00	−0.674924	−0.337462	−	0.941339i	$-0.609568\pi$
$439$	−6208.00	−0.674924	−0.337462	+	0.941339i	$0.609568\pi$
$440$	0	0
$441$	0	0
$442$	−2856.00	−0.307344
$443$	−15564.0	−1.66923	−0.834614	−	0.550835i	$-0.814309\pi$
$443$	−15564.0	−1.66923	−0.834614	+	0.550835i	$0.814309\pi$
$444$	0	0
$445$	0	0
$446$	−1040.00	−0.110416
$447$	0	0
$448$	−2048.00	−0.215980
$449$	15774.0	1.65795	0.828977	−	0.559283i	$-0.188924\pi$
$449$	15774.0	1.65795	0.828977	+	0.559283i	$0.188924\pi$
$450$	0	0
$451$	−14040.0	−1.46589
$452$	1704.00	0.177322
$453$	0	0
$454$	−792.000	−0.0818731
$455$	0	0
$456$	0	0
$457$	−9722.00	−0.995133	−0.497567	−	0.867426i	$-0.665773\pi$
$457$	−9722.00	−0.995133	−0.497567	+	0.867426i	$0.665773\pi$
$458$	2660.00	0.271383
$459$	0	0
$460$	0	0
$461$	10890.0	1.10021	0.550106	−	0.835095i	$-0.314587\pi$
$461$	10890.0	1.10021	0.550106	+	0.835095i	$0.314587\pi$
$462$	0	0
$463$	−15128.0	−1.51848	−0.759242	−	0.650809i	$-0.774430\pi$
$463$	−15128.0	−1.51848	−0.759242	+	0.650809i	$0.774430\pi$
$464$	−96.0000	−0.00960493
$465$	0	0
$466$	−9732.00	−0.967438
$467$	10668.0	1.05708	0.528540	−	0.848909i	$-0.322740\pi$
$467$	10668.0	1.05708	0.528540	+	0.848909i	$0.322740\pi$
$468$	0	0
$469$	25984.0	2.55827
$470$	0	0
$471$	0	0
$472$	5280.00	0.514898
$473$	24720.0	2.40302
$474$	0	0
$475$	0	0
$476$	−5376.00	−0.517665
$477$	0	0
$478$	−3648.00	−0.349070
$479$	−15264.0	−1.45601	−0.728006	−	0.685571i	$-0.759553\pi$
$479$	−15264.0	−1.45601	−0.728006	+	0.685571i	$0.759553\pi$
$480$	0	0
$481$	−4556.00	−0.431883
$482$	−12964.0	−1.22509
$483$	0	0
$484$	9076.00	0.852367
$485$	0	0
$486$	0	0
$487$	5776.00	0.537445	0.268722	−	0.963218i	$-0.413399\pi$
$487$	5776.00	0.537445	0.268722	+	0.963218i	$0.413399\pi$
$488$	3920.00	0.363627
$489$	0	0
$490$	0	0
$491$	−14244.0	−1.30921	−0.654606	−	0.755971i	$-0.727165\pi$
$491$	−14244.0	−1.30921	−0.654606	+	0.755971i	$0.727165\pi$
$492$	0	0
$493$	−252.000	−0.0230213
$494$	5168.00	0.470687
$495$	0	0
$496$	−3712.00	−0.336036
$497$	3840.00	0.346575
$498$	0	0
$499$	−17116.0	−1.53551	−0.767753	−	0.640746i	$-0.778625\pi$
$499$	−17116.0	−1.53551	−0.767753	+	0.640746i	$0.778625\pi$
$500$	0	0
$501$	0	0
$502$	2952.00	0.262459
$503$	−16848.0	−1.49347	−0.746735	−	0.665122i	$-0.768380\pi$
$503$	−16848.0	−1.49347	−0.746735	+	0.665122i	$0.768380\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−800.000	−0.0698706
$509$	3834.00	0.333868	0.166934	−	0.985968i	$-0.446613\pi$
$509$	3834.00	0.333868	0.166934	+	0.985968i	$0.446613\pi$
$510$	0	0
$511$	23872.0	2.06660
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−8628.00	−0.740398
$515$	0	0
$516$	0	0
$517$	−21600.0	−1.83746
$518$	−8576.00	−0.727428
$519$	0	0
$520$	0	0
$521$	18822.0	1.58274	0.791369	−	0.611338i	$-0.209369\pi$
$521$	18822.0	1.58274	0.791369	+	0.611338i	$0.209369\pi$
$522$	0	0
$523$	15340.0	1.28255	0.641273	−	0.767313i	$-0.278407\pi$
$523$	15340.0	1.28255	0.641273	+	0.767313i	$0.278407\pi$
$524$	−240.000	−0.0200085
$525$	0	0
$526$	10560.0	0.875357
$527$	−9744.00	−0.805418
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	9728.00	0.792786
$533$	−7956.00	−0.646553
$534$	0	0
$535$	0	0
$536$	6496.00	0.523478
$537$	0	0
$538$	11052.0	0.885661
$539$	40860.0	3.26524
$540$	0	0
$541$	18950.0	1.50596	0.752980	−	0.658044i	$-0.228616\pi$
$541$	18950.0	1.50596	0.752980	+	0.658044i	$0.228616\pi$
$542$	−4048.00	−0.320805
$543$	0	0
$544$	−1344.00	−0.105926
$545$	0	0
$546$	0	0
$547$	10036.0	0.784476	0.392238	−	0.919864i	$-0.371701\pi$
$547$	10036.0	0.784476	0.392238	+	0.919864i	$0.371701\pi$
$548$	2568.00	0.200182
$549$	0	0
$550$	0	0
$551$	456.000	0.0352564
$552$	0	0
$553$	−4864.00	−0.374030
$554$	4108.00	0.315040
$555$	0	0
$556$	−11344.0	−0.865275
$557$	10326.0	0.785506	0.392753	−	0.919644i	$-0.371523\pi$
$557$	10326.0	0.785506	0.392753	+	0.919644i	$0.371523\pi$
$558$	0	0
$559$	14008.0	1.05988
$560$	0	0
$561$	0	0
$562$	−14604.0	−1.09614
$563$	4524.00	0.338657	0.169328	−	0.985560i	$-0.445840\pi$
$563$	4524.00	0.338657	0.169328	+	0.985560i	$0.445840\pi$
$564$	0	0
$565$	0	0
$566$	−7448.00	−0.553114
$567$	0	0
$568$	960.000	0.0709167
$569$	−16362.0	−1.20550	−0.602751	−	0.797929i	$-0.705929\pi$
$569$	−16362.0	−1.20550	−0.602751	+	0.797929i	$0.705929\pi$
$570$	0	0
$571$	6620.00	0.485181	0.242591	−	0.970129i	$-0.422003\pi$
$571$	6620.00	0.485181	0.242591	+	0.970129i	$0.422003\pi$
$572$	8160.00	0.596480
$573$	0	0
$574$	−14976.0	−1.08900
$575$	0	0
$576$	0	0
$577$	−8834.00	−0.637373	−0.318687	−	0.947860i	$-0.603242\pi$
$577$	−8834.00	−0.637373	−0.318687	+	0.947860i	$0.603242\pi$
$578$	6298.00	0.453222
$579$	0	0
$580$	0	0
$581$	25728.0	1.83714
$582$	0	0
$583$	13320.0	0.946240
$584$	5968.00	0.422873
$585$	0	0
$586$	14436.0	1.01765
$587$	3636.00	0.255662	0.127831	−	0.991796i	$-0.459198\pi$
$587$	3636.00	0.255662	0.127831	+	0.991796i	$0.459198\pi$
$588$	0	0
$589$	17632.0	1.23347
$590$	0	0
$591$	0	0
$592$	−2144.00	−0.148848
$593$	6570.00	0.454971	0.227485	−	0.973782i	$-0.426950\pi$
$593$	6570.00	0.454971	0.227485	+	0.973782i	$0.426950\pi$
$594$	0	0
$595$	0	0
$596$	6216.00	0.427210
$597$	0	0
$598$	0	0
$599$	−16584.0	−1.13123	−0.565613	−	0.824671i	$-0.691360\pi$
$599$	−16584.0	−1.13123	−0.565613	+	0.824671i	$0.691360\pi$
$600$	0	0
$601$	−502.000	−0.0340716	−0.0170358	−	0.999855i	$-0.505423\pi$
$601$	−502.000	−0.0340716	−0.0170358	+	0.999855i	$0.505423\pi$
$602$	26368.0	1.78518
$603$	0	0
$604$	−9088.00	−0.612228
$605$	0	0
$606$	0	0
$607$	18568.0	1.24160	0.620801	−	0.783969i	$-0.286808\pi$
$607$	18568.0	1.24160	0.620801	+	0.783969i	$0.286808\pi$
$608$	2432.00	0.162221
$609$	0	0
$610$	0	0
$611$	−12240.0	−0.810438
$612$	0	0
$613$	13114.0	0.864061	0.432031	−	0.901859i	$-0.357797\pi$
$613$	13114.0	0.864061	0.432031	+	0.901859i	$0.357797\pi$
$614$	5080.00	0.333896
$615$	0	0
$616$	15360.0	1.00466
$617$	5250.00	0.342556	0.171278	−	0.985223i	$-0.445210\pi$
$617$	5250.00	0.342556	0.171278	+	0.985223i	$0.445210\pi$
$618$	0	0
$619$	−10804.0	−0.701534	−0.350767	−	0.936463i	$-0.614079\pi$
$619$	−10804.0	−0.701534	−0.350767	+	0.936463i	$0.614079\pi$
$620$	0	0
$621$	0	0
$622$	3120.00	0.201126
$623$	−21696.0	−1.39524
$624$	0	0
$625$	0	0
$626$	−1868.00	−0.119266
$627$	0	0
$628$	−6776.00	−0.430560
$629$	−5628.00	−0.356762
$630$	0	0
$631$	−27088.0	−1.70896	−0.854482	−	0.519481i	$-0.826125\pi$
$631$	−27088.0	−1.70896	−0.854482	+	0.519481i	$0.826125\pi$
$632$	−1216.00	−0.0765346
$633$	0	0
$634$	3348.00	0.209726
$635$	0	0
$636$	0	0
$637$	23154.0	1.44018
$638$	720.000	0.0446788
$639$	0	0
$640$	0	0
$641$	−18930.0	−1.16644	−0.583222	−	0.812313i	$-0.698208\pi$
$641$	−18930.0	−1.16644	−0.583222	+	0.812313i	$0.698208\pi$
$642$	0	0
$643$	−20108.0	−1.23325	−0.616627	−	0.787256i	$-0.711501\pi$
$643$	−20108.0	−1.23325	−0.616627	+	0.787256i	$0.711501\pi$
$644$	0	0
$645$	0	0
$646$	6384.00	0.388816
$647$	−7152.00	−0.434581	−0.217291	−	0.976107i	$-0.569722\pi$
$647$	−7152.00	−0.434581	−0.217291	+	0.976107i	$0.569722\pi$
$648$	0	0
$649$	−39600.0	−2.39512
$650$	0	0
$651$	0	0
$652$	208.000	0.0124937
$653$	−31626.0	−1.89528	−0.947642	−	0.319333i	$-0.896541\pi$
$653$	−31626.0	−1.89528	−0.947642	+	0.319333i	$0.896541\pi$
$654$	0	0
$655$	0	0
$656$	−3744.00	−0.222833
$657$	0	0
$658$	−23040.0	−1.36503
$659$	−28092.0	−1.66056	−0.830280	−	0.557347i	$-0.811819\pi$
$659$	−28092.0	−1.66056	−0.830280	+	0.557347i	$0.811819\pi$
$660$	0	0
$661$	−13186.0	−0.775909	−0.387955	−	0.921678i	$-0.626818\pi$
$661$	−13186.0	−0.775909	−0.387955	+	0.921678i	$0.626818\pi$
$662$	7976.00	0.468272
$663$	0	0
$664$	6432.00	0.375919
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−4800.00	−0.278020
$669$	0	0
$670$	0	0
$671$	−29400.0	−1.69147
$672$	0	0
$673$	−5138.00	−0.294287	−0.147144	−	0.989115i	$-0.547008\pi$
$673$	−5138.00	−0.294287	−0.147144	+	0.989115i	$0.547008\pi$
$674$	4.00000	0.000228597 0
$675$	0	0
$676$	−4164.00	−0.236914
$677$	6078.00	0.345047	0.172523	−	0.985005i	$-0.444808\pi$
$677$	6078.00	0.345047	0.172523	+	0.985005i	$0.444808\pi$
$678$	0	0
$679$	6208.00	0.350871
$680$	0	0
$681$	0	0
$682$	27840.0	1.56312
$683$	32244.0	1.80642	0.903208	−	0.429203i	$-0.141205\pi$
$683$	32244.0	1.80642	0.903208	+	0.429203i	$0.141205\pi$
$684$	0	0
$685$	0	0
$686$	21632.0	1.20396
$687$	0	0
$688$	6592.00	0.365287
$689$	7548.00	0.417353
$690$	0	0
$691$	4484.00	0.246859	0.123429	−	0.992353i	$-0.460611\pi$
$691$	4484.00	0.246859	0.123429	+	0.992353i	$0.460611\pi$
$692$	216.000	0.0118657
$693$	0	0
$694$	−3528.00	−0.192970
$695$	0	0
$696$	0	0
$697$	−9828.00	−0.534092
$698$	−8620.00	−0.467438
$699$	0	0
$700$	0	0
$701$	30426.0	1.63934	0.819668	−	0.572839i	$-0.194158\pi$
$701$	30426.0	1.63934	0.819668	+	0.572839i	$0.194158\pi$
$702$	0	0
$703$	10184.0	0.546368
$704$	3840.00	0.205576
$705$	0	0
$706$	−276.000	−0.0147130
$707$	25536.0	1.35839
$708$	0	0
$709$	13262.0	0.702489	0.351245	−	0.936284i	$-0.385759\pi$
$709$	13262.0	0.702489	0.351245	+	0.936284i	$0.385759\pi$
$710$	0	0
$711$	0	0
$712$	−5424.00	−0.285496
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3504.00	−0.182892
$717$	0	0
$718$	−23952.0	−1.24496
$719$	−13920.0	−0.722014	−0.361007	−	0.932563i	$-0.617567\pi$
$719$	−13920.0	−0.722014	−0.361007	+	0.932563i	$0.617567\pi$
$720$	0	0
$721$	34816.0	1.79836
$722$	2166.00	0.111648
$723$	0	0
$724$	15416.0	0.791341
$725$	0	0
$726$	0	0
$727$	9376.00	0.478317	0.239159	−	0.970981i	$-0.423128\pi$
$727$	9376.00	0.478317	0.239159	+	0.970981i	$0.423128\pi$
$728$	8704.00	0.443120
$729$	0	0
$730$	0	0
$731$	17304.0	0.875529
$732$	0	0
$733$	−6014.00	−0.303045	−0.151523	−	0.988454i	$-0.548418\pi$
$733$	−6014.00	−0.303045	−0.151523	+	0.988454i	$0.548418\pi$
$734$	19408.0	0.975971
$735$	0	0
$736$	0	0
$737$	−48720.0	−2.43504
$738$	0	0
$739$	−7468.00	−0.371739	−0.185869	−	0.982574i	$-0.559510\pi$
$739$	−7468.00	−0.371739	−0.185869	+	0.982574i	$0.559510\pi$
$740$	0	0
$741$	0	0
$742$	14208.0	0.702954
$743$	31248.0	1.54290	0.771452	−	0.636287i	$-0.219531\pi$
$743$	31248.0	1.54290	0.771452	+	0.636287i	$0.219531\pi$
$744$	0	0
$745$	0	0
$746$	−16244.0	−0.797232
$747$	0	0
$748$	10080.0	0.492729
$749$	−54912.0	−2.67883
$750$	0	0
$751$	32840.0	1.59567	0.797835	−	0.602875i	$-0.205978\pi$
$751$	32840.0	1.59567	0.797835	+	0.602875i	$0.205978\pi$
$752$	−5760.00	−0.279316
$753$	0	0
$754$	408.000	0.0197062
$755$	0	0
$756$	0	0
$757$	19066.0	0.915410	0.457705	−	0.889104i	$-0.348672\pi$
$757$	19066.0	0.915410	0.457705	+	0.889104i	$0.348672\pi$
$758$	−6808.00	−0.326224
$759$	0	0
$760$	0	0
$761$	−6858.00	−0.326678	−0.163339	−	0.986570i	$-0.552227\pi$
$761$	−6858.00	−0.326678	−0.163339	+	0.986570i	$0.552227\pi$
$762$	0	0
$763$	31040.0	1.47277
$764$	11136.0	0.527338
$765$	0	0
$766$	5040.00	0.237732
$767$	−22440.0	−1.05640
$768$	0	0
$769$	22178.0	1.04000	0.519999	−	0.854167i	$-0.325932\pi$
$769$	22178.0	1.04000	0.519999	+	0.854167i	$0.325932\pi$
$770$	0	0
$771$	0	0
$772$	−3656.00	−0.170443
$773$	14286.0	0.664724	0.332362	−	0.943152i	$-0.392154\pi$
$773$	14286.0	0.664724	0.332362	+	0.943152i	$0.392154\pi$
$774$	0	0
$775$	0	0
$776$	1552.00	0.0717958
$777$	0	0
$778$	3132.00	0.144329
$779$	17784.0	0.817943
$780$	0	0
$781$	−7200.00	−0.329880
$782$	0	0
$783$	0	0
$784$	10896.0	0.496356
$785$	0	0
$786$	0	0
$787$	18868.0	0.854602	0.427301	−	0.904109i	$-0.359465\pi$
$787$	18868.0	0.854602	0.427301	+	0.904109i	$0.359465\pi$
$788$	−20808.0	−0.940678
$789$	0	0
$790$	0	0
$791$	−13632.0	−0.612766
$792$	0	0
$793$	−16660.0	−0.746045
$794$	−8708.00	−0.389213
$795$	0	0
$796$	12608.0	0.561405
$797$	−21690.0	−0.963989	−0.481994	−	0.876174i	$-0.660087\pi$
$797$	−21690.0	−0.963989	−0.481994	+	0.876174i	$0.660087\pi$
$798$	0	0
$799$	−15120.0	−0.669471
$800$	0	0
$801$	0	0
$802$	−16092.0	−0.708514
$803$	−44760.0	−1.96706
$804$	0	0
$805$	0	0
$806$	15776.0	0.689437
$807$	0	0
$808$	6384.00	0.277956
$809$	24726.0	1.07456	0.537281	−	0.843404i	$-0.319452\pi$
$809$	24726.0	1.07456	0.537281	+	0.843404i	$0.319452\pi$
$810$	0	0
$811$	−2644.00	−0.114480	−0.0572401	−	0.998360i	$-0.518230\pi$
$811$	−2644.00	−0.114480	−0.0572401	+	0.998360i	$0.518230\pi$
$812$	768.000	0.0331915
$813$	0	0
$814$	16080.0	0.692388
$815$	0	0
$816$	0	0
$817$	−31312.0	−1.34084
$818$	5612.00	0.239877
$819$	0	0
$820$	0	0
$821$	37842.0	1.60864	0.804321	−	0.594195i	$-0.202529\pi$
$821$	37842.0	1.60864	0.804321	+	0.594195i	$0.202529\pi$
$822$	0	0
$823$	880.000	0.0372720	0.0186360	−	0.999826i	$-0.494068\pi$
$823$	880.000	0.0372720	0.0186360	+	0.999826i	$0.494068\pi$
$824$	8704.00	0.367983
$825$	0	0
$826$	−42240.0	−1.77932
$827$	−12876.0	−0.541406	−0.270703	−	0.962663i	$-0.587256\pi$
$827$	−12876.0	−0.541406	−0.270703	+	0.962663i	$0.587256\pi$
$828$	0	0
$829$	−25498.0	−1.06825	−0.534127	−	0.845404i	$-0.679359\pi$
$829$	−25498.0	−1.06825	−0.534127	+	0.845404i	$0.679359\pi$
$830$	0	0
$831$	0	0
$832$	2176.00	0.0906721
$833$	28602.0	1.18968
$834$	0	0
$835$	0	0
$836$	−18240.0	−0.754598
$837$	0	0
$838$	23160.0	0.954712
$839$	40584.0	1.66998	0.834991	−	0.550263i	$-0.185473\pi$
$839$	40584.0	1.66998	0.834991	+	0.550263i	$0.185473\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	740.000	0.0302875
$843$	0	0
$844$	2960.00	0.120720
$845$	0	0
$846$	0	0
$847$	−72608.0	−2.94550
$848$	3552.00	0.143840
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	25738.0	1.03312	0.516561	−	0.856251i	$-0.327212\pi$
$853$	25738.0	1.03312	0.516561	+	0.856251i	$0.327212\pi$
$854$	−31360.0	−1.25658
$855$	0	0
$856$	−13728.0	−0.548146
$857$	13314.0	0.530686	0.265343	−	0.964154i	$-0.414515\pi$
$857$	13314.0	0.530686	0.265343	+	0.964154i	$0.414515\pi$
$858$	0	0
$859$	24524.0	0.974096	0.487048	−	0.873375i	$-0.338074\pi$
$859$	24524.0	0.974096	0.487048	+	0.873375i	$0.338074\pi$
$860$	0	0
$861$	0	0
$862$	10080.0	0.398290
$863$	5592.00	0.220572	0.110286	−	0.993900i	$-0.464823\pi$
$863$	5592.00	0.220572	0.110286	+	0.993900i	$0.464823\pi$
$864$	0	0
$865$	0	0
$866$	−7484.00	−0.293668
$867$	0	0
$868$	29696.0	1.16123
$869$	9120.00	0.356012
$870$	0	0
$871$	−27608.0	−1.07401
$872$	7760.00	0.301361
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14386.0	0.553912	0.276956	−	0.960883i	$-0.410674\pi$
$877$	14386.0	0.553912	0.276956	+	0.960883i	$0.410674\pi$
$878$	12416.0	0.477243
$879$	0	0
$880$	0	0
$881$	−47106.0	−1.80141	−0.900705	−	0.434432i	$-0.856949\pi$
$881$	−47106.0	−1.80141	−0.900705	+	0.434432i	$0.856949\pi$
$882$	0	0
$883$	−51548.0	−1.96458	−0.982292	−	0.187354i	$-0.940009\pi$
$883$	−51548.0	−1.96458	−0.982292	+	0.187354i	$0.940009\pi$
$884$	5712.00	0.217325
$885$	0	0
$886$	31128.0	1.18032
$887$	34080.0	1.29007	0.645036	−	0.764152i	$-0.276842\pi$
$887$	34080.0	1.29007	0.645036	+	0.764152i	$0.276842\pi$
$888$	0	0
$889$	6400.00	0.241450
$890$	0	0
$891$	0	0
$892$	2080.00	0.0780757
$893$	27360.0	1.02527
$894$	0	0
$895$	0	0
$896$	4096.00	0.152721
$897$	0	0
$898$	−31548.0	−1.17235
$899$	1392.00	0.0516416
$900$	0	0
$901$	9324.00	0.344759
$902$	28080.0	1.03654
$903$	0	0
$904$	−3408.00	−0.125385
$905$	0	0
$906$	0	0
$907$	−25748.0	−0.942611	−0.471306	−	0.881970i	$-0.656217\pi$
$907$	−25748.0	−0.942611	−0.471306	+	0.881970i	$0.656217\pi$
$908$	1584.00	0.0578930
$909$	0	0
$910$	0	0
$911$	24768.0	0.900769	0.450384	−	0.892835i	$-0.351287\pi$
$911$	24768.0	0.900769	0.450384	+	0.892835i	$0.351287\pi$
$912$	0	0
$913$	−48240.0	−1.74864
$914$	19444.0	0.703666
$915$	0	0
$916$	−5320.00	−0.191897
$917$	1920.00	0.0691428
$918$	0	0
$919$	−31264.0	−1.12220	−0.561101	−	0.827747i	$-0.689622\pi$
$919$	−31264.0	−1.12220	−0.561101	+	0.827747i	$0.689622\pi$
$920$	0	0
$921$	0	0
$922$	−21780.0	−0.777968
$923$	−4080.00	−0.145498
$924$	0	0
$925$	0	0
$926$	30256.0	1.07373
$927$	0	0
$928$	192.000	0.00679171
$929$	6174.00	0.218043	0.109022	−	0.994039i	$-0.465228\pi$
$929$	6174.00	0.218043	0.109022	+	0.994039i	$0.465228\pi$
$930$	0	0
$931$	−51756.0	−1.82195
$932$	19464.0	0.684082
$933$	0	0
$934$	−21336.0	−0.747468
$935$	0	0
$936$	0	0
$937$	−28922.0	−1.00837	−0.504184	−	0.863596i	$-0.668207\pi$
$937$	−28922.0	−1.00837	−0.504184	+	0.863596i	$0.668207\pi$
$938$	−51968.0	−1.80897
$939$	0	0
$940$	0	0
$941$	−29238.0	−1.01289	−0.506446	−	0.862272i	$-0.669041\pi$
$941$	−29238.0	−1.01289	−0.506446	+	0.862272i	$0.669041\pi$
$942$	0	0
$943$	0	0
$944$	−10560.0	−0.364088
$945$	0	0
$946$	−49440.0	−1.69919
$947$	−2868.00	−0.0984134	−0.0492067	−	0.998789i	$-0.515669\pi$
$947$	−2868.00	−0.0984134	−0.0492067	+	0.998789i	$0.515669\pi$
$948$	0	0
$949$	−25364.0	−0.867598
$950$	0	0
$951$	0	0
$952$	10752.0	0.366044
$953$	24018.0	0.816390	0.408195	−	0.912895i	$-0.366158\pi$
$953$	24018.0	0.816390	0.408195	+	0.912895i	$0.366158\pi$
$954$	0	0
$955$	0	0
$956$	7296.00	0.246830
$957$	0	0
$958$	30528.0	1.02956
$959$	−20544.0	−0.691763
$960$	0	0
$961$	24033.0	0.806720
$962$	9112.00	0.305387
$963$	0	0
$964$	25928.0	0.866270
$965$	0	0
$966$	0	0
$967$	−25712.0	−0.855059	−0.427530	−	0.904001i	$-0.640616\pi$
$967$	−25712.0	−0.855059	−0.427530	+	0.904001i	$0.640616\pi$
$968$	−18152.0	−0.602714
$969$	0	0
$970$	0	0
$971$	12396.0	0.409688	0.204844	−	0.978795i	$-0.434331\pi$
$971$	12396.0	0.409688	0.204844	+	0.978795i	$0.434331\pi$
$972$	0	0
$973$	90752.0	2.99011
$974$	−11552.0	−0.380031
$975$	0	0
$976$	−7840.00	−0.257123
$977$	−46614.0	−1.52642	−0.763211	−	0.646150i	$-0.776378\pi$
$977$	−46614.0	−1.52642	−0.763211	+	0.646150i	$0.776378\pi$
$978$	0	0
$979$	40680.0	1.32803
$980$	0	0
$981$	0	0
$982$	28488.0	0.925752
$983$	−672.000	−0.0218041	−0.0109021	−	0.999941i	$-0.503470\pi$
$983$	−672.000	−0.0218041	−0.0109021	+	0.999941i	$0.503470\pi$
$984$	0	0
$985$	0	0
$986$	504.000	0.0162785
$987$	0	0
$988$	−10336.0	−0.332826
$989$	0	0
$990$	0	0
$991$	−38776.0	−1.24295	−0.621473	−	0.783435i	$-0.713466\pi$
$991$	−38776.0	−1.24295	−0.621473	+	0.783435i	$0.713466\pi$
$992$	7424.00	0.237613
$993$	0	0
$994$	−7680.00	−0.245065
$995$	0	0
$996$	0	0
$997$	−30422.0	−0.966374	−0.483187	−	0.875517i	$-0.660521\pi$
$997$	−30422.0	−0.966374	−0.483187	+	0.875517i	$0.660521\pi$
$998$	34232.0	1.08577
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.a.b.1.1		1
3.2	odd	2		150.4.a.e.1.1		1
5.2	odd	4		450.4.c.k.199.1		2
5.3	odd	4		450.4.c.k.199.2		2
5.4	even	2		90.4.a.d.1.1		1
12.11	even	2		1200.4.a.bk.1.1		1
15.2	even	4		150.4.c.a.49.2		2
15.8	even	4		150.4.c.a.49.1		2
15.14	odd	2		30.4.a.a.1.1	✓	1
20.19	odd	2		720.4.a.b.1.1		1
45.4	even	6		810.4.e.e.541.1		2
45.14	odd	6		810.4.e.m.541.1		2
45.29	odd	6		810.4.e.m.271.1		2
45.34	even	6		810.4.e.e.271.1		2
60.23	odd	4		1200.4.f.u.49.2		2
60.47	odd	4		1200.4.f.u.49.1		2
60.59	even	2		240.4.a.c.1.1		1
105.104	even	2		1470.4.a.a.1.1		1
120.29	odd	2		960.4.a.j.1.1		1
120.59	even	2		960.4.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.4.a.a.1.1	✓	1	15.14	odd	2
90.4.a.d.1.1		1	5.4	even	2
150.4.a.e.1.1		1	3.2	odd	2
150.4.c.a.49.1		2	15.8	even	4
150.4.c.a.49.2		2	15.2	even	4
240.4.a.c.1.1		1	60.59	even	2
450.4.a.b.1.1		1	1.1	even	1	trivial
450.4.c.k.199.1		2	5.2	odd	4
450.4.c.k.199.2		2	5.3	odd	4
720.4.a.b.1.1		1	20.19	odd	2
810.4.e.e.271.1		2	45.34	even	6
810.4.e.e.541.1		2	45.4	even	6
810.4.e.m.271.1		2	45.29	odd	6
810.4.e.m.541.1		2	45.14	odd	6
960.4.a.j.1.1		1	120.29	odd	2
960.4.a.s.1.1		1	120.59	even	2
1200.4.a.bk.1.1		1	12.11	even	2
1200.4.f.u.49.1		2	60.47	odd	4
1200.4.f.u.49.2		2	60.23	odd	4
1470.4.a.a.1.1		1	105.104	even	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 $\F_{q}$
Belyi maps

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

Introduction

L-functions

Modular forms

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Fields

Representations

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Database

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