LMFDB - Embedded newform 675.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-3,0,1,0,0,25] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} +1.00000 q^{4} +25.0000 q^{7} +21.0000 q^{8} -15.0000 q^{11} -20.0000 q^{13} -75.0000 q^{14} -71.0000 q^{16} -72.0000 q^{17} +2.00000 q^{19} +45.0000 q^{22} -114.000 q^{23} +60.0000 q^{26} +25.0000 q^{28} +30.0000 q^{29} +101.000 q^{31} +45.0000 q^{32} +216.000 q^{34} +430.000 q^{37} -6.00000 q^{38} -30.0000 q^{41} -110.000 q^{43} -15.0000 q^{44} +342.000 q^{46} +330.000 q^{47} +282.000 q^{49} -20.0000 q^{52} -621.000 q^{53} +525.000 q^{56} -90.0000 q^{58} -660.000 q^{59} -376.000 q^{61} -303.000 q^{62} +433.000 q^{64} +250.000 q^{67} -72.0000 q^{68} -360.000 q^{71} -785.000 q^{73} -1290.00 q^{74} +2.00000 q^{76} -375.000 q^{77} +488.000 q^{79} +90.0000 q^{82} -489.000 q^{83} +330.000 q^{86} -315.000 q^{88} -450.000 q^{89} -500.000 q^{91} -114.000 q^{92} -990.000 q^{94} +1105.00 q^{97} -846.000 q^{98} +O(q^{100})$

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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	25.0000	1.34987	0.674937	−	0.737876i	$-0.264171\pi$
$7$	25.0000	1.34987	0.674937	+	0.737876i	$0.264171\pi$
$8$	21.0000	0.928078
$9$	0	0
$10$	0	0
$11$	−15.0000	−0.411152	−0.205576	−	0.978641i	$-0.565907\pi$
$11$	−15.0000	−0.411152	−0.205576	+	0.978641i	$0.565907\pi$
$12$	0	0
$13$	−20.0000	−0.426692	−0.213346	−	0.976977i	$-0.568436\pi$
$13$	−20.0000	−0.426692	−0.213346	+	0.976977i	$0.568436\pi$
$14$	−75.0000	−1.43176
$15$	0	0
$16$	−71.0000	−1.10938
$17$	−72.0000	−1.02721	−0.513605	−	0.858027i	$-0.671690\pi$
$17$	−72.0000	−1.02721	−0.513605	+	0.858027i	$0.671690\pi$
$18$	0	0
$19$	2.00000	0.0241490	0.0120745	−	0.999927i	$-0.496156\pi$
$19$	2.00000	0.0241490	0.0120745	+	0.999927i	$0.496156\pi$
$20$	0	0
$21$	0	0
$22$	45.0000	0.436092
$23$	−114.000	−1.03351	−0.516753	−	0.856134i	$-0.672859\pi$
$23$	−114.000	−1.03351	−0.516753	+	0.856134i	$0.672859\pi$
$24$	0	0
$25$	0	0
$26$	60.0000	0.452576
$27$	0	0
$28$	25.0000	0.168734
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	0	0
$31$	101.000	0.585166	0.292583	−	0.956240i	$-0.405485\pi$
$31$	101.000	0.585166	0.292583	+	0.956240i	$0.405485\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	216.000	1.08952
$35$	0	0
$36$	0	0
$37$	430.000	1.91058	0.955291	−	0.295666i	$-0.0955415\pi$
$37$	430.000	1.91058	0.955291	+	0.295666i	$0.0955415\pi$
$38$	−6.00000	−0.0256139
$39$	0	0
$40$	0	0
$41$	−30.0000	−0.114273	−0.0571367	−	0.998366i	$-0.518197\pi$
$41$	−30.0000	−0.114273	−0.0571367	+	0.998366i	$0.518197\pi$
$42$	0	0
$43$	−110.000	−0.390113	−0.195056	−	0.980792i	$-0.562489\pi$
$43$	−110.000	−0.390113	−0.195056	+	0.980792i	$0.562489\pi$
$44$	−15.0000	−0.0513940
$45$	0	0
$46$	342.000	1.09620
$47$	330.000	1.02416	0.512079	−	0.858938i	$-0.328875\pi$
$47$	330.000	1.02416	0.512079	+	0.858938i	$0.328875\pi$
$48$	0	0
$49$	282.000	0.822157
$50$	0	0
$51$	0	0
$52$	−20.0000	−0.0533366
$53$	−621.000	−1.60945	−0.804726	−	0.593647i	$-0.797688\pi$
$53$	−621.000	−1.60945	−0.804726	+	0.593647i	$0.797688\pi$
$54$	0	0
$55$	0	0
$56$	525.000	1.25279
$57$	0	0
$58$	−90.0000	−0.203751
$59$	−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
$61$	−376.000	−0.789211	−0.394605	−	0.918851i	$-0.629119\pi$
$61$	−376.000	−0.789211	−0.394605	+	0.918851i	$0.629119\pi$
$62$	−303.000	−0.620662
$63$	0	0
$64$	433.000	0.845703
$65$	0	0
$66$	0	0
$67$	250.000	0.455856	0.227928	−	0.973678i	$-0.426805\pi$
$67$	250.000	0.455856	0.227928	+	0.973678i	$0.426805\pi$
$68$	−72.0000	−0.128401
$69$	0	0
$70$	0	0
$71$	−360.000	−0.601748	−0.300874	−	0.953664i	$-0.597278\pi$
$71$	−360.000	−0.601748	−0.300874	+	0.953664i	$0.597278\pi$
$72$	0	0
$73$	−785.000	−1.25859	−0.629297	−	0.777165i	$-0.716657\pi$
$73$	−785.000	−1.25859	−0.629297	+	0.777165i	$0.716657\pi$
$74$	−1290.00	−2.02648
$75$	0	0
$76$	2.00000	0.00301863
$77$	−375.000	−0.555003
$78$	0	0
$79$	488.000	0.694991	0.347496	−	0.937682i	$-0.387032\pi$
$79$	488.000	0.694991	0.347496	+	0.937682i	$0.387032\pi$
$80$	0	0
$81$	0	0
$82$	90.0000	0.121205
$83$	−489.000	−0.646683	−0.323342	−	0.946282i	$-0.604806\pi$
$83$	−489.000	−0.646683	−0.323342	+	0.946282i	$0.604806\pi$
$84$	0	0
$85$	0	0
$86$	330.000	0.413777
$87$	0	0
$88$	−315.000	−0.381581
$89$	−450.000	−0.535954	−0.267977	−	0.963425i	$-0.586355\pi$
$89$	−450.000	−0.535954	−0.267977	+	0.963425i	$0.586355\pi$
$90$	0	0
$91$	−500.000	−0.575981
$92$	−114.000	−0.129188
$93$	0	0
$94$	−990.000	−1.08628
$95$	0	0
$96$	0	0
$97$	1105.00	1.15666	0.578329	−	0.815804i	$-0.303705\pi$
$97$	1105.00	1.15666	0.578329	+	0.815804i	$0.303705\pi$
$98$	−846.000	−0.872030
$99$	0	0
$100$	0	0
$101$	1425.00	1.40389	0.701945	−	0.712232i	$-0.252315\pi$
$101$	1425.00	1.40389	0.701945	+	0.712232i	$0.252315\pi$
$102$	0	0
$103$	1060.00	1.01403	0.507014	−	0.861938i	$-0.330749\pi$
$103$	1060.00	1.01403	0.507014	+	0.861938i	$0.330749\pi$
$104$	−420.000	−0.396004
$105$	0	0
$106$	1863.00	1.70708
$107$	−1485.00	−1.34169	−0.670843	−	0.741600i	$-0.734067\pi$
$107$	−1485.00	−1.34169	−0.670843	+	0.741600i	$0.734067\pi$
$108$	0	0
$109$	−862.000	−0.757474	−0.378737	−	0.925504i	$-0.623641\pi$
$109$	−862.000	−0.757474	−0.378737	+	0.925504i	$0.623641\pi$
$110$	0	0
$111$	0	0
$112$	−1775.00	−1.49752
$113$	−690.000	−0.574422	−0.287211	−	0.957867i	$-0.592728\pi$
$113$	−690.000	−0.574422	−0.287211	+	0.957867i	$0.592728\pi$
$114$	0	0
$115$	0	0
$116$	30.0000	0.0240123
$117$	0	0
$118$	1980.00	1.54469
$119$	−1800.00	−1.38660
$120$	0	0
$121$	−1106.00	−0.830954
$122$	1128.00	0.837085
$123$	0	0
$124$	101.000	0.0731457
$125$	0	0
$126$	0	0
$127$	−1865.00	−1.30309	−0.651543	−	0.758611i	$-0.725878\pi$
$127$	−1865.00	−1.30309	−0.651543	+	0.758611i	$0.725878\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	0	0
$131$	−1155.00	−0.770327	−0.385163	−	0.922848i	$-0.625855\pi$
$131$	−1155.00	−0.770327	−0.385163	+	0.922848i	$0.625855\pi$
$132$	0	0
$133$	50.0000	0.0325981
$134$	−750.000	−0.483508
$135$	0	0
$136$	−1512.00	−0.953330
$137$	2778.00	1.73241	0.866206	−	0.499686i	$-0.166551\pi$
$137$	2778.00	1.73241	0.866206	+	0.499686i	$0.166551\pi$
$138$	0	0
$139$	−1924.00	−1.17404	−0.587020	−	0.809572i	$-0.699699\pi$
$139$	−1924.00	−1.17404	−0.587020	+	0.809572i	$0.699699\pi$
$140$	0	0
$141$	0	0
$142$	1080.00	0.638251
$143$	300.000	0.175435
$144$	0	0
$145$	0	0
$146$	2355.00	1.33494
$147$	0	0
$148$	430.000	0.238823
$149$	1455.00	0.799988	0.399994	−	0.916518i	$-0.369012\pi$
$149$	1455.00	0.799988	0.399994	+	0.916518i	$0.369012\pi$
$150$	0	0
$151$	−727.000	−0.391804	−0.195902	−	0.980623i	$-0.562763\pi$
$151$	−727.000	−0.391804	−0.195902	+	0.980623i	$0.562763\pi$
$152$	42.0000	0.0224122
$153$	0	0
$154$	1125.00	0.588669
$155$	0	0
$156$	0	0
$157$	−3260.00	−1.65717	−0.828587	−	0.559860i	$-0.810855\pi$
$157$	−3260.00	−1.65717	−0.828587	+	0.559860i	$0.810855\pi$
$158$	−1464.00	−0.737149
$159$	0	0
$160$	0	0
$161$	−2850.00	−1.39510
$162$	0	0
$163$	−2540.00	−1.22054	−0.610270	−	0.792193i	$-0.708939\pi$
$163$	−2540.00	−1.22054	−0.610270	+	0.792193i	$0.708939\pi$
$164$	−30.0000	−0.0142842
$165$	0	0
$166$	1467.00	0.685911
$167$	−3498.00	−1.62086	−0.810429	−	0.585837i	$-0.800766\pi$
$167$	−3498.00	−1.62086	−0.810429	+	0.585837i	$0.800766\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	0	0
$171$	0	0
$172$	−110.000	−0.0487641
$173$	1149.00	0.504953	0.252476	−	0.967603i	$-0.418755\pi$
$173$	1149.00	0.504953	0.252476	+	0.967603i	$0.418755\pi$
$174$	0	0
$175$	0	0
$176$	1065.00	0.456122
$177$	0	0
$178$	1350.00	0.568465
$179$	315.000	0.131532	0.0657659	−	0.997835i	$-0.479051\pi$
$179$	315.000	0.131532	0.0657659	+	0.997835i	$0.479051\pi$
$180$	0	0
$181$	1136.00	0.466509	0.233255	−	0.972416i	$-0.425062\pi$
$181$	1136.00	0.466509	0.233255	+	0.972416i	$0.425062\pi$
$182$	1500.00	0.610920
$183$	0	0
$184$	−2394.00	−0.959174
$185$	0	0
$186$	0	0
$187$	1080.00	0.422339
$188$	330.000	0.128020
$189$	0	0
$190$	0	0
$191$	2460.00	0.931934	0.465967	−	0.884802i	$-0.345707\pi$
$191$	2460.00	0.931934	0.465967	+	0.884802i	$0.345707\pi$
$192$	0	0
$193$	−965.000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−965.000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	−3315.00	−1.22682
$195$	0	0
$196$	282.000	0.102770
$197$	−2493.00	−0.901619	−0.450809	−	0.892620i	$-0.648865\pi$
$197$	−2493.00	−0.901619	−0.450809	+	0.892620i	$0.648865\pi$
$198$	0	0
$199$	−511.000	−0.182029	−0.0910146	−	0.995850i	$-0.529011\pi$
$199$	−511.000	−0.182029	−0.0910146	+	0.995850i	$0.529011\pi$
$200$	0	0
$201$	0	0
$202$	−4275.00	−1.48905
$203$	750.000	0.259309
$204$	0	0
$205$	0	0
$206$	−3180.00	−1.07554
$207$	0	0
$208$	1420.00	0.473362
$209$	−30.0000	−0.00992892
$210$	0	0
$211$	−2086.00	−0.680598	−0.340299	−	0.940317i	$-0.610528\pi$
$211$	−2086.00	−0.680598	−0.340299	+	0.940317i	$0.610528\pi$
$212$	−621.000	−0.201181
$213$	0	0
$214$	4455.00	1.42307
$215$	0	0
$216$	0	0
$217$	2525.00	0.789899
$218$	2586.00	0.803422
$219$	0	0
$220$	0	0
$221$	1440.00	0.438303
$222$	0	0
$223$	−5240.00	−1.57353	−0.786763	−	0.617255i	$-0.788245\pi$
$223$	−5240.00	−1.57353	−0.786763	+	0.617255i	$0.788245\pi$
$224$	1125.00	0.335568
$225$	0	0
$226$	2070.00	0.609267
$227$	−2388.00	−0.698225	−0.349113	−	0.937081i	$-0.613517\pi$
$227$	−2388.00	−0.698225	−0.349113	+	0.937081i	$0.613517\pi$
$228$	0	0
$229$	182.000	0.0525192	0.0262596	−	0.999655i	$-0.491640\pi$
$229$	182.000	0.0525192	0.0262596	+	0.999655i	$0.491640\pi$
$230$	0	0
$231$	0	0
$232$	630.000	0.178282
$233$	−450.000	−0.126526	−0.0632628	−	0.997997i	$-0.520151\pi$
$233$	−450.000	−0.126526	−0.0632628	+	0.997997i	$0.520151\pi$
$234$	0	0
$235$	0	0
$236$	−660.000	−0.182044
$237$	0	0
$238$	5400.00	1.47071
$239$	5190.00	1.40466	0.702329	−	0.711853i	$-0.252144\pi$
$239$	5190.00	1.40466	0.702329	+	0.711853i	$0.252144\pi$
$240$	0	0
$241$	−2266.00	−0.605668	−0.302834	−	0.953043i	$-0.597933\pi$
$241$	−2266.00	−0.605668	−0.302834	+	0.953043i	$0.597933\pi$
$242$	3318.00	0.881360
$243$	0	0
$244$	−376.000	−0.0986514
$245$	0	0
$246$	0	0
$247$	−40.0000	−0.0103042
$248$	2121.00	0.543079
$249$	0	0
$250$	0	0
$251$	−2880.00	−0.724239	−0.362119	−	0.932132i	$-0.617947\pi$
$251$	−2880.00	−0.724239	−0.362119	+	0.932132i	$0.617947\pi$
$252$	0	0
$253$	1710.00	0.424928
$254$	5595.00	1.38213
$255$	0	0
$256$	1513.00	0.369385
$257$	4188.00	1.01650	0.508250	−	0.861210i	$-0.330293\pi$
$257$	4188.00	1.01650	0.508250	+	0.861210i	$0.330293\pi$
$258$	0	0
$259$	10750.0	2.57904
$260$	0	0
$261$	0	0
$262$	3465.00	0.817055
$263$	3030.00	0.710410	0.355205	−	0.934788i	$-0.384411\pi$
$263$	3030.00	0.710410	0.355205	+	0.934788i	$0.384411\pi$
$264$	0	0
$265$	0	0
$266$	−150.000	−0.0345755
$267$	0	0
$268$	250.000	0.0569820
$269$	3510.00	0.795571	0.397785	−	0.917479i	$-0.369779\pi$
$269$	3510.00	0.795571	0.397785	+	0.917479i	$0.369779\pi$
$270$	0	0
$271$	2999.00	0.672237	0.336119	−	0.941820i	$-0.390886\pi$
$271$	2999.00	0.672237	0.336119	+	0.941820i	$0.390886\pi$
$272$	5112.00	1.13956
$273$	0	0
$274$	−8334.00	−1.83750
$275$	0	0
$276$	0	0
$277$	7720.00	1.67455	0.837274	−	0.546783i	$-0.184148\pi$
$277$	7720.00	1.67455	0.837274	+	0.546783i	$0.184148\pi$
$278$	5772.00	1.24526
$279$	0	0
$280$	0	0
$281$	−7440.00	−1.57948	−0.789739	−	0.613443i	$-0.789784\pi$
$281$	−7440.00	−1.57948	−0.789739	+	0.613443i	$0.789784\pi$
$282$	0	0
$283$	−830.000	−0.174341	−0.0871703	−	0.996193i	$-0.527782\pi$
$283$	−830.000	−0.174341	−0.0871703	+	0.996193i	$0.527782\pi$
$284$	−360.000	−0.0752186
$285$	0	0
$286$	−900.000	−0.186077
$287$	−750.000	−0.154255
$288$	0	0
$289$	271.000	0.0551598
$290$	0	0
$291$	0	0
$292$	−785.000	−0.157324
$293$	−546.000	−0.108866	−0.0544329	−	0.998517i	$-0.517335\pi$
$293$	−546.000	−0.108866	−0.0544329	+	0.998517i	$0.517335\pi$
$294$	0	0
$295$	0	0
$296$	9030.00	1.77317
$297$	0	0
$298$	−4365.00	−0.848516
$299$	2280.00	0.440989
$300$	0	0
$301$	−2750.00	−0.526603
$302$	2181.00	0.415571
$303$	0	0
$304$	−142.000	−0.0267903
$305$	0	0
$306$	0	0
$307$	5560.00	1.03364	0.516818	−	0.856096i	$-0.327117\pi$
$307$	5560.00	1.03364	0.516818	+	0.856096i	$0.327117\pi$
$308$	−375.000	−0.0693754
$309$	0	0
$310$	0	0
$311$	−8670.00	−1.58081	−0.790403	−	0.612587i	$-0.790129\pi$
$311$	−8670.00	−1.58081	−0.790403	+	0.612587i	$0.790129\pi$
$312$	0	0
$313$	−4565.00	−0.824374	−0.412187	−	0.911099i	$-0.635235\pi$
$313$	−4565.00	−0.824374	−0.412187	+	0.911099i	$0.635235\pi$
$314$	9780.00	1.75770
$315$	0	0
$316$	488.000	0.0868739
$317$	−4233.00	−0.749997	−0.374998	−	0.927025i	$-0.622357\pi$
$317$	−4233.00	−0.749997	−0.374998	+	0.927025i	$0.622357\pi$
$318$	0	0
$319$	−450.000	−0.0789817
$320$	0	0
$321$	0	0
$322$	8550.00	1.47973
$323$	−144.000	−0.0248061
$324$	0	0
$325$	0	0
$326$	7620.00	1.29458
$327$	0	0
$328$	−630.000	−0.106055
$329$	8250.00	1.38248
$330$	0	0
$331$	542.000	0.0900031	0.0450015	−	0.998987i	$-0.485671\pi$
$331$	542.000	0.0900031	0.0450015	+	0.998987i	$0.485671\pi$
$332$	−489.000	−0.0808354
$333$	0	0
$334$	10494.0	1.71918
$335$	0	0
$336$	0	0
$337$	−5690.00	−0.919745	−0.459872	−	0.887985i	$-0.652105\pi$
$337$	−5690.00	−0.919745	−0.459872	+	0.887985i	$0.652105\pi$
$338$	5391.00	0.867550
$339$	0	0
$340$	0	0
$341$	−1515.00	−0.240592
$342$	0	0
$343$	−1525.00	−0.240065
$344$	−2310.00	−0.362055
$345$	0	0
$346$	−3447.00	−0.535583
$347$	−5055.00	−0.782036	−0.391018	−	0.920383i	$-0.627877\pi$
$347$	−5055.00	−0.782036	−0.391018	+	0.920383i	$0.627877\pi$
$348$	0	0
$349$	1622.00	0.248778	0.124389	−	0.992234i	$-0.460303\pi$
$349$	1622.00	0.248778	0.124389	+	0.992234i	$0.460303\pi$
$350$	0	0
$351$	0	0
$352$	−675.000	−0.102209
$353$	−30.0000	−0.00452334	−0.00226167	−	0.999997i	$-0.500720\pi$
$353$	−30.0000	−0.00452334	−0.00226167	+	0.999997i	$0.500720\pi$
$354$	0	0
$355$	0	0
$356$	−450.000	−0.0669942
$357$	0	0
$358$	−945.000	−0.139511
$359$	−7470.00	−1.09819	−0.549097	−	0.835759i	$-0.685028\pi$
$359$	−7470.00	−1.09819	−0.549097	+	0.835759i	$0.685028\pi$
$360$	0	0
$361$	−6855.00	−0.999417
$362$	−3408.00	−0.494808
$363$	0	0
$364$	−500.000	−0.0719976
$365$	0	0
$366$	0	0
$367$	1375.00	0.195571	0.0977853	−	0.995208i	$-0.468824\pi$
$367$	1375.00	0.195571	0.0977853	+	0.995208i	$0.468824\pi$
$368$	8094.00	1.14655
$369$	0	0
$370$	0	0
$371$	−15525.0	−2.17255
$372$	0	0
$373$	4840.00	0.671865	0.335933	−	0.941886i	$-0.390949\pi$
$373$	4840.00	0.671865	0.335933	+	0.941886i	$0.390949\pi$
$374$	−3240.00	−0.447958
$375$	0	0
$376$	6930.00	0.950499
$377$	−600.000	−0.0819670
$378$	0	0
$379$	1892.00	0.256426	0.128213	−	0.991747i	$-0.459076\pi$
$379$	1892.00	0.256426	0.128213	+	0.991747i	$0.459076\pi$
$380$	0	0
$381$	0	0
$382$	−7380.00	−0.988465
$383$	10704.0	1.42806	0.714032	−	0.700113i	$-0.246867\pi$
$383$	10704.0	1.42806	0.714032	+	0.700113i	$0.246867\pi$
$384$	0	0
$385$	0	0
$386$	2895.00	0.381740
$387$	0	0
$388$	1105.00	0.144582
$389$	7815.00	1.01860	0.509301	−	0.860588i	$-0.329904\pi$
$389$	7815.00	1.01860	0.509301	+	0.860588i	$0.329904\pi$
$390$	0	0
$391$	8208.00	1.06163
$392$	5922.00	0.763026
$393$	0	0
$394$	7479.00	0.956311
$395$	0	0
$396$	0	0
$397$	−4700.00	−0.594172	−0.297086	−	0.954851i	$-0.596015\pi$
$397$	−4700.00	−0.594172	−0.297086	+	0.954851i	$0.596015\pi$
$398$	1533.00	0.193071
$399$	0	0
$400$	0	0
$401$	−2100.00	−0.261519	−0.130759	−	0.991414i	$-0.541742\pi$
$401$	−2100.00	−0.261519	−0.130759	+	0.991414i	$0.541742\pi$
$402$	0	0
$403$	−2020.00	−0.249686
$404$	1425.00	0.175486
$405$	0	0
$406$	−2250.00	−0.275038
$407$	−6450.00	−0.785540
$408$	0	0
$409$	−10753.0	−1.30000	−0.650002	−	0.759933i	$-0.725232\pi$
$409$	−10753.0	−1.30000	−0.650002	+	0.759933i	$0.725232\pi$
$410$	0	0
$411$	0	0
$412$	1060.00	0.126754
$413$	−16500.0	−1.96589
$414$	0	0
$415$	0	0
$416$	−900.000	−0.106072
$417$	0	0
$418$	90.0000	0.0105312
$419$	2940.00	0.342789	0.171394	−	0.985203i	$-0.445173\pi$
$419$	2940.00	0.342789	0.171394	+	0.985203i	$0.445173\pi$
$420$	0	0
$421$	8696.00	1.00669	0.503346	−	0.864085i	$-0.332102\pi$
$421$	8696.00	1.00669	0.503346	+	0.864085i	$0.332102\pi$
$422$	6258.00	0.721883
$423$	0	0
$424$	−13041.0	−1.49370
$425$	0	0
$426$	0	0
$427$	−9400.00	−1.06533
$428$	−1485.00	−0.167711
$429$	0	0
$430$	0	0
$431$	8370.00	0.935426	0.467713	−	0.883880i	$-0.345078\pi$
$431$	8370.00	0.935426	0.467713	+	0.883880i	$0.345078\pi$
$432$	0	0
$433$	5155.00	0.572133	0.286066	−	0.958210i	$-0.407652\pi$
$433$	5155.00	0.572133	0.286066	+	0.958210i	$0.407652\pi$
$434$	−7575.00	−0.837815
$435$	0	0
$436$	−862.000	−0.0946842
$437$	−228.000	−0.0249582
$438$	0	0
$439$	−10987.0	−1.19449	−0.597245	−	0.802059i	$-0.703738\pi$
$439$	−10987.0	−1.19449	−0.597245	+	0.802059i	$0.703738\pi$
$440$	0	0
$441$	0	0
$442$	−4320.00	−0.464890
$443$	−1956.00	−0.209780	−0.104890	−	0.994484i	$-0.533449\pi$
$443$	−1956.00	−0.209780	−0.104890	+	0.994484i	$0.533449\pi$
$444$	0	0
$445$	0	0
$446$	15720.0	1.66898
$447$	0	0
$448$	10825.0	1.14159
$449$	−8730.00	−0.917582	−0.458791	−	0.888544i	$-0.651717\pi$
$449$	−8730.00	−0.917582	−0.458791	+	0.888544i	$0.651717\pi$
$450$	0	0
$451$	450.000	0.0469838
$452$	−690.000	−0.0718028
$453$	0	0
$454$	7164.00	0.740580
$455$	0	0
$456$	0	0
$457$	8665.00	0.886940	0.443470	−	0.896289i	$-0.353747\pi$
$457$	8665.00	0.886940	0.443470	+	0.896289i	$0.353747\pi$
$458$	−546.000	−0.0557050
$459$	0	0
$460$	0	0
$461$	−9825.00	−0.992616	−0.496308	−	0.868147i	$-0.665311\pi$
$461$	−9825.00	−0.992616	−0.496308	+	0.868147i	$0.665311\pi$
$462$	0	0
$463$	5245.00	0.526470	0.263235	−	0.964732i	$-0.415210\pi$
$463$	5245.00	0.526470	0.263235	+	0.964732i	$0.415210\pi$
$464$	−2130.00	−0.213109
$465$	0	0
$466$	1350.00	0.134201
$467$	11007.0	1.09067	0.545335	−	0.838218i	$-0.316402\pi$
$467$	11007.0	1.09067	0.545335	+	0.838218i	$0.316402\pi$
$468$	0	0
$469$	6250.00	0.615348
$470$	0	0
$471$	0	0
$472$	−13860.0	−1.35161
$473$	1650.00	0.160396
$474$	0	0
$475$	0	0
$476$	−1800.00	−0.173325
$477$	0	0
$478$	−15570.0	−1.48986
$479$	16950.0	1.61684	0.808419	−	0.588608i	$-0.200324\pi$
$479$	16950.0	1.61684	0.808419	+	0.588608i	$0.200324\pi$
$480$	0	0
$481$	−8600.00	−0.815231
$482$	6798.00	0.642408
$483$	0	0
$484$	−1106.00	−0.103869
$485$	0	0
$486$	0	0
$487$	−10640.0	−0.990030	−0.495015	−	0.868885i	$-0.664837\pi$
$487$	−10640.0	−0.990030	−0.495015	+	0.868885i	$0.664837\pi$
$488$	−7896.00	−0.732449
$489$	0	0
$490$	0	0
$491$	1635.00	0.150278	0.0751390	−	0.997173i	$-0.476060\pi$
$491$	1635.00	0.150278	0.0751390	+	0.997173i	$0.476060\pi$
$492$	0	0
$493$	−2160.00	−0.197326
$494$	120.000	0.0109293
$495$	0	0
$496$	−7171.00	−0.649168
$497$	−9000.00	−0.812284
$498$	0	0
$499$	−15802.0	−1.41762	−0.708812	−	0.705397i	$-0.750769\pi$
$499$	−15802.0	−1.41762	−0.708812	+	0.705397i	$0.750769\pi$
$500$	0	0
$501$	0	0
$502$	8640.00	0.768171
$503$	7866.00	0.697272	0.348636	−	0.937258i	$-0.386645\pi$
$503$	7866.00	0.697272	0.348636	+	0.937258i	$0.386645\pi$
$504$	0	0
$505$	0	0
$506$	−5130.00	−0.450704
$507$	0	0
$508$	−1865.00	−0.162886
$509$	−11955.0	−1.04105	−0.520527	−	0.853845i	$-0.674264\pi$
$509$	−11955.0	−1.04105	−0.520527	+	0.853845i	$0.674264\pi$
$510$	0	0
$511$	−19625.0	−1.69894
$512$	8733.00	0.753804
$513$	0	0
$514$	−12564.0	−1.07816
$515$	0	0
$516$	0	0
$517$	−4950.00	−0.421085
$518$	−32250.0	−2.73549
$519$	0	0
$520$	0	0
$521$	19260.0	1.61957	0.809785	−	0.586727i	$-0.199584\pi$
$521$	19260.0	1.61957	0.809785	+	0.586727i	$0.199584\pi$
$522$	0	0
$523$	18520.0	1.54842	0.774209	−	0.632930i	$-0.218148\pi$
$523$	18520.0	1.54842	0.774209	+	0.632930i	$0.218148\pi$
$524$	−1155.00	−0.0962909
$525$	0	0
$526$	−9090.00	−0.753503
$527$	−7272.00	−0.601088
$528$	0	0
$529$	829.000	0.0681351
$530$	0	0
$531$	0	0
$532$	50.0000	0.00407476
$533$	600.000	0.0487596
$534$	0	0
$535$	0	0
$536$	5250.00	0.423070
$537$	0	0
$538$	−10530.0	−0.843830
$539$	−4230.00	−0.338032
$540$	0	0
$541$	8372.00	0.665324	0.332662	−	0.943046i	$-0.392053\pi$
$541$	8372.00	0.665324	0.332662	+	0.943046i	$0.392053\pi$
$542$	−8997.00	−0.713015
$543$	0	0
$544$	−3240.00	−0.255356
$545$	0	0
$546$	0	0
$547$	−17120.0	−1.33821	−0.669103	−	0.743170i	$-0.733321\pi$
$547$	−17120.0	−1.33821	−0.669103	+	0.743170i	$0.733321\pi$
$548$	2778.00	0.216552
$549$	0	0
$550$	0	0
$551$	60.0000	0.00463899
$552$	0	0
$553$	12200.0	0.938150
$554$	−23160.0	−1.77613
$555$	0	0
$556$	−1924.00	−0.146755
$557$	−10575.0	−0.804447	−0.402224	−	0.915541i	$-0.631763\pi$
$557$	−10575.0	−0.804447	−0.402224	+	0.915541i	$0.631763\pi$
$558$	0	0
$559$	2200.00	0.166458
$560$	0	0
$561$	0	0
$562$	22320.0	1.67529
$563$	−10455.0	−0.782639	−0.391319	−	0.920255i	$-0.627981\pi$
$563$	−10455.0	−0.782639	−0.391319	+	0.920255i	$0.627981\pi$
$564$	0	0
$565$	0	0
$566$	2490.00	0.184916
$567$	0	0
$568$	−7560.00	−0.558469
$569$	−24540.0	−1.80803	−0.904016	−	0.427498i	$-0.859395\pi$
$569$	−24540.0	−1.80803	−0.904016	+	0.427498i	$0.859395\pi$
$570$	0	0
$571$	24644.0	1.80616	0.903082	−	0.429469i	$-0.141299\pi$
$571$	24644.0	1.80616	0.903082	+	0.429469i	$0.141299\pi$
$572$	300.000	0.0219294
$573$	0	0
$574$	2250.00	0.163612
$575$	0	0
$576$	0	0
$577$	9610.00	0.693361	0.346681	−	0.937983i	$-0.387309\pi$
$577$	9610.00	0.693361	0.346681	+	0.937983i	$0.387309\pi$
$578$	−813.000	−0.0585058
$579$	0	0
$580$	0	0
$581$	−12225.0	−0.872941
$582$	0	0
$583$	9315.00	0.661729
$584$	−16485.0	−1.16807
$585$	0	0
$586$	1638.00	0.115470
$587$	−4017.00	−0.282452	−0.141226	−	0.989977i	$-0.545104\pi$
$587$	−4017.00	−0.282452	−0.141226	+	0.989977i	$0.545104\pi$
$588$	0	0
$589$	202.000	0.0141312
$590$	0	0
$591$	0	0
$592$	−30530.0	−2.11955
$593$	−594.000	−0.0411343	−0.0205672	−	0.999788i	$-0.506547\pi$
$593$	−594.000	−0.0411343	−0.0205672	+	0.999788i	$0.506547\pi$
$594$	0	0
$595$	0	0
$596$	1455.00	0.0999985
$597$	0	0
$598$	−6840.00	−0.467740
$599$	8790.00	0.599582	0.299791	−	0.954005i	$-0.403083\pi$
$599$	8790.00	0.599582	0.299791	+	0.954005i	$0.403083\pi$
$600$	0	0
$601$	9371.00	0.636025	0.318013	−	0.948087i	$-0.396985\pi$
$601$	9371.00	0.636025	0.318013	+	0.948087i	$0.396985\pi$
$602$	8250.00	0.558546
$603$	0	0
$604$	−727.000	−0.0489755
$605$	0	0
$606$	0	0
$607$	14560.0	0.973595	0.486798	−	0.873515i	$-0.338165\pi$
$607$	14560.0	0.973595	0.486798	+	0.873515i	$0.338165\pi$
$608$	90.0000	0.00600326
$609$	0	0
$610$	0	0
$611$	−6600.00	−0.437001
$612$	0	0
$613$	18250.0	1.20246	0.601232	−	0.799074i	$-0.294677\pi$
$613$	18250.0	1.20246	0.601232	+	0.799074i	$0.294677\pi$
$614$	−16680.0	−1.09634
$615$	0	0
$616$	−7875.00	−0.515086
$617$	−19662.0	−1.28292	−0.641461	−	0.767156i	$-0.721671\pi$
$617$	−19662.0	−1.28292	−0.641461	+	0.767156i	$0.721671\pi$
$618$	0	0
$619$	12044.0	0.782050	0.391025	−	0.920380i	$-0.372121\pi$
$619$	12044.0	0.782050	0.391025	+	0.920380i	$0.372121\pi$
$620$	0	0
$621$	0	0
$622$	26010.0	1.67670
$623$	−11250.0	−0.723470
$624$	0	0
$625$	0	0
$626$	13695.0	0.874381
$627$	0	0
$628$	−3260.00	−0.207147
$629$	−30960.0	−1.96257
$630$	0	0
$631$	14879.0	0.938706	0.469353	−	0.883011i	$-0.344487\pi$
$631$	14879.0	0.938706	0.469353	+	0.883011i	$0.344487\pi$
$632$	10248.0	0.645006
$633$	0	0
$634$	12699.0	0.795492
$635$	0	0
$636$	0	0
$637$	−5640.00	−0.350808
$638$	1350.00	0.0837727
$639$	0	0
$640$	0	0
$641$	8850.00	0.545326	0.272663	−	0.962110i	$-0.412096\pi$
$641$	8850.00	0.545326	0.272663	+	0.962110i	$0.412096\pi$
$642$	0	0
$643$	−18380.0	−1.12727	−0.563636	−	0.826023i	$-0.690598\pi$
$643$	−18380.0	−1.12727	−0.563636	+	0.826023i	$0.690598\pi$
$644$	−2850.00	−0.174388
$645$	0	0
$646$	432.000	0.0263109
$647$	−3888.00	−0.236249	−0.118124	−	0.992999i	$-0.537688\pi$
$647$	−3888.00	−0.236249	−0.118124	+	0.992999i	$0.537688\pi$
$648$	0	0
$649$	9900.00	0.598781
$650$	0	0
$651$	0	0
$652$	−2540.00	−0.152568
$653$	6789.00	0.406852	0.203426	−	0.979090i	$-0.434792\pi$
$653$	6789.00	0.406852	0.203426	+	0.979090i	$0.434792\pi$
$654$	0	0
$655$	0	0
$656$	2130.00	0.126772
$657$	0	0
$658$	−24750.0	−1.46635
$659$	28335.0	1.67492	0.837462	−	0.546496i	$-0.184038\pi$
$659$	28335.0	1.67492	0.837462	+	0.546496i	$0.184038\pi$
$660$	0	0
$661$	−6082.00	−0.357886	−0.178943	−	0.983859i	$-0.557268\pi$
$661$	−6082.00	−0.357886	−0.178943	+	0.983859i	$0.557268\pi$
$662$	−1626.00	−0.0954627
$663$	0	0
$664$	−10269.0	−0.600172
$665$	0	0
$666$	0	0
$667$	−3420.00	−0.198535
$668$	−3498.00	−0.202607
$669$	0	0
$670$	0	0
$671$	5640.00	0.324486
$672$	0	0
$673$	−9965.00	−0.570762	−0.285381	−	0.958414i	$-0.592120\pi$
$673$	−9965.00	−0.570762	−0.285381	+	0.958414i	$0.592120\pi$
$674$	17070.0	0.975537
$675$	0	0
$676$	−1797.00	−0.102242
$677$	−8130.00	−0.461538	−0.230769	−	0.973009i	$-0.574124\pi$
$677$	−8130.00	−0.461538	−0.230769	+	0.973009i	$0.574124\pi$
$678$	0	0
$679$	27625.0	1.56134
$680$	0	0
$681$	0	0
$682$	4545.00	0.255186
$683$	−33516.0	−1.87768	−0.938839	−	0.344356i	$-0.888097\pi$
$683$	−33516.0	−1.87768	−0.938839	+	0.344356i	$0.888097\pi$
$684$	0	0
$685$	0	0
$686$	4575.00	0.254627
$687$	0	0
$688$	7810.00	0.432781
$689$	12420.0	0.686741
$690$	0	0
$691$	−22084.0	−1.21580	−0.607898	−	0.794015i	$-0.707987\pi$
$691$	−22084.0	−1.21580	−0.607898	+	0.794015i	$0.707987\pi$
$692$	1149.00	0.0631191
$693$	0	0
$694$	15165.0	0.829475
$695$	0	0
$696$	0	0
$697$	2160.00	0.117383
$698$	−4866.00	−0.263869
$699$	0	0
$700$	0	0
$701$	−10395.0	−0.560077	−0.280038	−	0.959989i	$-0.590347\pi$
$701$	−10395.0	−0.560077	−0.280038	+	0.959989i	$0.590347\pi$
$702$	0	0
$703$	860.000	0.0461387
$704$	−6495.00	−0.347712
$705$	0	0
$706$	90.0000	0.00479773
$707$	35625.0	1.89507
$708$	0	0
$709$	−4804.00	−0.254468	−0.127234	−	0.991873i	$-0.540610\pi$
$709$	−4804.00	−0.254468	−0.127234	+	0.991873i	$0.540610\pi$
$710$	0	0
$711$	0	0
$712$	−9450.00	−0.497407
$713$	−11514.0	−0.604772
$714$	0	0
$715$	0	0
$716$	315.000	0.0164415
$717$	0	0
$718$	22410.0	1.16481
$719$	10980.0	0.569520	0.284760	−	0.958599i	$-0.408086\pi$
$719$	10980.0	0.569520	0.284760	+	0.958599i	$0.408086\pi$
$720$	0	0
$721$	26500.0	1.36881
$722$	20565.0	1.06004
$723$	0	0
$724$	1136.00	0.0583137
$725$	0	0
$726$	0	0
$727$	25945.0	1.32359	0.661793	−	0.749687i	$-0.269796\pi$
$727$	25945.0	1.32359	0.661793	+	0.749687i	$0.269796\pi$
$728$	−10500.0	−0.534555
$729$	0	0
$730$	0	0
$731$	7920.00	0.400727
$732$	0	0
$733$	−18650.0	−0.939773	−0.469886	−	0.882727i	$-0.655705\pi$
$733$	−18650.0	−0.939773	−0.469886	+	0.882727i	$0.655705\pi$
$734$	−4125.00	−0.207434
$735$	0	0
$736$	−5130.00	−0.256922
$737$	−3750.00	−0.187426
$738$	0	0
$739$	−5128.00	−0.255259	−0.127630	−	0.991822i	$-0.540737\pi$
$739$	−5128.00	−0.255259	−0.127630	+	0.991822i	$0.540737\pi$
$740$	0	0
$741$	0	0
$742$	46575.0	2.30434
$743$	32700.0	1.61460	0.807299	−	0.590142i	$-0.200928\pi$
$743$	32700.0	1.61460	0.807299	+	0.590142i	$0.200928\pi$
$744$	0	0
$745$	0	0
$746$	−14520.0	−0.712621
$747$	0	0
$748$	1080.00	0.0527924
$749$	−37125.0	−1.81111
$750$	0	0
$751$	21161.0	1.02820	0.514098	−	0.857731i	$-0.328127\pi$
$751$	21161.0	1.02820	0.514098	+	0.857731i	$0.328127\pi$
$752$	−23430.0	−1.13618
$753$	0	0
$754$	1800.00	0.0869392
$755$	0	0
$756$	0	0
$757$	−7130.00	−0.342331	−0.171165	−	0.985242i	$-0.554753\pi$
$757$	−7130.00	−0.342331	−0.171165	+	0.985242i	$0.554753\pi$
$758$	−5676.00	−0.271981
$759$	0	0
$760$	0	0
$761$	−3360.00	−0.160052	−0.0800262	−	0.996793i	$-0.525500\pi$
$761$	−3360.00	−0.160052	−0.0800262	+	0.996793i	$0.525500\pi$
$762$	0	0
$763$	−21550.0	−1.02249
$764$	2460.00	0.116492
$765$	0	0
$766$	−32112.0	−1.51469
$767$	13200.0	0.621414
$768$	0	0
$769$	33473.0	1.56966	0.784829	−	0.619712i	$-0.212751\pi$
$769$	33473.0	1.56966	0.784829	+	0.619712i	$0.212751\pi$
$770$	0	0
$771$	0	0
$772$	−965.000	−0.0449885
$773$	3546.00	0.164995	0.0824973	−	0.996591i	$-0.473710\pi$
$773$	3546.00	0.164995	0.0824973	+	0.996591i	$0.473710\pi$
$774$	0	0
$775$	0	0
$776$	23205.0	1.07347
$777$	0	0
$778$	−23445.0	−1.08039
$779$	−60.0000	−0.00275959
$780$	0	0
$781$	5400.00	0.247410
$782$	−24624.0	−1.12603
$783$	0	0
$784$	−20022.0	−0.912081
$785$	0	0
$786$	0	0
$787$	31840.0	1.44215	0.721076	−	0.692856i	$-0.243648\pi$
$787$	31840.0	1.44215	0.721076	+	0.692856i	$0.243648\pi$
$788$	−2493.00	−0.112702
$789$	0	0
$790$	0	0
$791$	−17250.0	−0.775397
$792$	0	0
$793$	7520.00	0.336750
$794$	14100.0	0.630214
$795$	0	0
$796$	−511.000	−0.0227537
$797$	15717.0	0.698525	0.349263	−	0.937025i	$-0.386432\pi$
$797$	15717.0	0.698525	0.349263	+	0.937025i	$0.386432\pi$
$798$	0	0
$799$	−23760.0	−1.05203
$800$	0	0
$801$	0	0
$802$	6300.00	0.277382
$803$	11775.0	0.517473
$804$	0	0
$805$	0	0
$806$	6060.00	0.264832
$807$	0	0
$808$	29925.0	1.30292
$809$	10530.0	0.457621	0.228810	−	0.973471i	$-0.426516\pi$
$809$	10530.0	0.457621	0.228810	+	0.973471i	$0.426516\pi$
$810$	0	0
$811$	−26782.0	−1.15961	−0.579805	−	0.814755i	$-0.696871\pi$
$811$	−26782.0	−1.15961	−0.579805	+	0.814755i	$0.696871\pi$
$812$	750.000	0.0324136
$813$	0	0
$814$	19350.0	0.833191
$815$	0	0
$816$	0	0
$817$	−220.000	−0.00942084
$818$	32259.0	1.37886
$819$	0	0
$820$	0	0
$821$	10110.0	0.429770	0.214885	−	0.976639i	$-0.431062\pi$
$821$	10110.0	0.429770	0.214885	+	0.976639i	$0.431062\pi$
$822$	0	0
$823$	12535.0	0.530914	0.265457	−	0.964123i	$-0.414477\pi$
$823$	12535.0	0.530914	0.265457	+	0.964123i	$0.414477\pi$
$824$	22260.0	0.941097
$825$	0	0
$826$	49500.0	2.08514
$827$	−9792.00	−0.411731	−0.205865	−	0.978580i	$-0.566001\pi$
$827$	−9792.00	−0.411731	−0.205865	+	0.978580i	$0.566001\pi$
$828$	0	0
$829$	−4534.00	−0.189955	−0.0949773	−	0.995479i	$-0.530278\pi$
$829$	−4534.00	−0.189955	−0.0949773	+	0.995479i	$0.530278\pi$
$830$	0	0
$831$	0	0
$832$	−8660.00	−0.360855
$833$	−20304.0	−0.844528
$834$	0	0
$835$	0	0
$836$	−30.0000	−0.00124111
$837$	0	0
$838$	−8820.00	−0.363582
$839$	−8880.00	−0.365401	−0.182701	−	0.983169i	$-0.558484\pi$
$839$	−8880.00	−0.365401	−0.182701	+	0.983169i	$0.558484\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	−26088.0	−1.06776
$843$	0	0
$844$	−2086.00	−0.0850747
$845$	0	0
$846$	0	0
$847$	−27650.0	−1.12168
$848$	44091.0	1.78548
$849$	0	0
$850$	0	0
$851$	−49020.0	−1.97460
$852$	0	0
$853$	−2270.00	−0.0911176	−0.0455588	−	0.998962i	$-0.514507\pi$
$853$	−2270.00	−0.0911176	−0.0455588	+	0.998962i	$0.514507\pi$
$854$	28200.0	1.12996
$855$	0	0
$856$	−31185.0	−1.24519
$857$	19608.0	0.781560	0.390780	−	0.920484i	$-0.372205\pi$
$857$	19608.0	0.781560	0.390780	+	0.920484i	$0.372205\pi$
$858$	0	0
$859$	−952.000	−0.0378135	−0.0189068	−	0.999821i	$-0.506019\pi$
$859$	−952.000	−0.0378135	−0.0189068	+	0.999821i	$0.506019\pi$
$860$	0	0
$861$	0	0
$862$	−25110.0	−0.992169
$863$	17604.0	0.694377	0.347188	−	0.937795i	$-0.387136\pi$
$863$	17604.0	0.694377	0.347188	+	0.937795i	$0.387136\pi$
$864$	0	0
$865$	0	0
$866$	−15465.0	−0.606838
$867$	0	0
$868$	2525.00	0.0987374
$869$	−7320.00	−0.285747
$870$	0	0
$871$	−5000.00	−0.194510
$872$	−18102.0	−0.702994
$873$	0	0
$874$	684.000	0.0264721
$875$	0	0
$876$	0	0
$877$	−21890.0	−0.842842	−0.421421	−	0.906865i	$-0.638469\pi$
$877$	−21890.0	−0.842842	−0.421421	+	0.906865i	$0.638469\pi$
$878$	32961.0	1.26695
$879$	0	0
$880$	0	0
$881$	23940.0	0.915504	0.457752	−	0.889080i	$-0.348655\pi$
$881$	23940.0	0.915504	0.457752	+	0.889080i	$0.348655\pi$
$882$	0	0
$883$	34990.0	1.33353	0.666765	−	0.745268i	$-0.267678\pi$
$883$	34990.0	1.33353	0.666765	+	0.745268i	$0.267678\pi$
$884$	1440.00	0.0547878
$885$	0	0
$886$	5868.00	0.222505
$887$	22188.0	0.839910	0.419955	−	0.907545i	$-0.362046\pi$
$887$	22188.0	0.839910	0.419955	+	0.907545i	$0.362046\pi$
$888$	0	0
$889$	−46625.0	−1.75900
$890$	0	0
$891$	0	0
$892$	−5240.00	−0.196691
$893$	660.000	0.0247324
$894$	0	0
$895$	0	0
$896$	−41475.0	−1.54641
$897$	0	0
$898$	26190.0	0.973242
$899$	3030.00	0.112410
$900$	0	0
$901$	44712.0	1.65324
$902$	−1350.00	−0.0498338
$903$	0	0
$904$	−14490.0	−0.533109
$905$	0	0
$906$	0	0
$907$	−37370.0	−1.36808	−0.684041	−	0.729444i	$-0.739779\pi$
$907$	−37370.0	−1.36808	−0.684041	+	0.729444i	$0.739779\pi$
$908$	−2388.00	−0.0872782
$909$	0	0
$910$	0	0
$911$	40710.0	1.48055	0.740276	−	0.672303i	$-0.234695\pi$
$911$	40710.0	1.48055	0.740276	+	0.672303i	$0.234695\pi$
$912$	0	0
$913$	7335.00	0.265885
$914$	−25995.0	−0.940742
$915$	0	0
$916$	182.000	0.00656490
$917$	−28875.0	−1.03984
$918$	0	0
$919$	20981.0	0.753100	0.376550	−	0.926396i	$-0.377110\pi$
$919$	20981.0	0.753100	0.376550	+	0.926396i	$0.377110\pi$
$920$	0	0
$921$	0	0
$922$	29475.0	1.05283
$923$	7200.00	0.256762
$924$	0	0
$925$	0	0
$926$	−15735.0	−0.558406
$927$	0	0
$928$	1350.00	0.0477542
$929$	20100.0	0.709860	0.354930	−	0.934893i	$-0.384505\pi$
$929$	20100.0	0.709860	0.354930	+	0.934893i	$0.384505\pi$
$930$	0	0
$931$	564.000	0.0198543
$932$	−450.000	−0.0158157
$933$	0	0
$934$	−33021.0	−1.15683
$935$	0	0
$936$	0	0
$937$	−15635.0	−0.545115	−0.272558	−	0.962139i	$-0.587870\pi$
$937$	−15635.0	−0.545115	−0.272558	+	0.962139i	$0.587870\pi$
$938$	−18750.0	−0.652675
$939$	0	0
$940$	0	0
$941$	23955.0	0.829873	0.414937	−	0.909850i	$-0.363804\pi$
$941$	23955.0	0.829873	0.414937	+	0.909850i	$0.363804\pi$
$942$	0	0
$943$	3420.00	0.118102
$944$	46860.0	1.61564
$945$	0	0
$946$	−4950.00	−0.170125
$947$	36393.0	1.24880	0.624400	−	0.781105i	$-0.285344\pi$
$947$	36393.0	1.24880	0.624400	+	0.781105i	$0.285344\pi$
$948$	0	0
$949$	15700.0	0.537032
$950$	0	0
$951$	0	0
$952$	−37800.0	−1.28688
$953$	−43020.0	−1.46228	−0.731141	−	0.682227i	$-0.761012\pi$
$953$	−43020.0	−1.46228	−0.731141	+	0.682227i	$0.761012\pi$
$954$	0	0
$955$	0	0
$956$	5190.00	0.175582
$957$	0	0
$958$	−50850.0	−1.71492
$959$	69450.0	2.33854
$960$	0	0
$961$	−19590.0	−0.657581
$962$	25800.0	0.864683
$963$	0	0
$964$	−2266.00	−0.0757084
$965$	0	0
$966$	0	0
$967$	43585.0	1.44943	0.724715	−	0.689049i	$-0.241971\pi$
$967$	43585.0	1.44943	0.724715	+	0.689049i	$0.241971\pi$
$968$	−23226.0	−0.771190
$969$	0	0
$970$	0	0
$971$	−43335.0	−1.43222	−0.716110	−	0.697987i	$-0.754079\pi$
$971$	−43335.0	−1.43222	−0.716110	+	0.697987i	$0.754079\pi$
$972$	0	0
$973$	−48100.0	−1.58480
$974$	31920.0	1.05008
$975$	0	0
$976$	26696.0	0.875531
$977$	−30390.0	−0.995151	−0.497575	−	0.867421i	$-0.665776\pi$
$977$	−30390.0	−0.995151	−0.497575	+	0.867421i	$0.665776\pi$
$978$	0	0
$979$	6750.00	0.220358
$980$	0	0
$981$	0	0
$982$	−4905.00	−0.159394
$983$	59226.0	1.92168	0.960842	−	0.277096i	$-0.0893719\pi$
$983$	59226.0	1.92168	0.960842	+	0.277096i	$0.0893719\pi$
$984$	0	0
$985$	0	0
$986$	6480.00	0.209295
$987$	0	0
$988$	−40.0000	−0.00128803
$989$	12540.0	0.403184
$990$	0	0
$991$	8399.00	0.269226	0.134613	−	0.990898i	$-0.457021\pi$
$991$	8399.00	0.269226	0.134613	+	0.990898i	$0.457021\pi$
$992$	4545.00	0.145468
$993$	0	0
$994$	27000.0	0.861557
$995$	0	0
$996$	0	0
$997$	−13340.0	−0.423753	−0.211877	−	0.977296i	$-0.567958\pi$
$997$	−13340.0	−0.423753	−0.211877	+	0.977296i	$0.567958\pi$
$998$	47406.0	1.50362
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.a.1.1		1
3.2	odd	2		675.4.a.j.1.1		1
5.2	odd	4		675.4.b.a.649.1		2
5.3	odd	4		675.4.b.a.649.2		2
5.4	even	2		27.4.a.b.1.1	yes	1
15.2	even	4		675.4.b.b.649.2		2
15.8	even	4		675.4.b.b.649.1		2
15.14	odd	2		27.4.a.a.1.1	✓	1
20.19	odd	2		432.4.a.n.1.1		1
35.34	odd	2		1323.4.a.k.1.1		1
40.19	odd	2		1728.4.a.d.1.1		1
40.29	even	2		1728.4.a.c.1.1		1
45.4	even	6		81.4.c.a.55.1		2
45.14	odd	6		81.4.c.c.55.1		2
45.29	odd	6		81.4.c.c.28.1		2
45.34	even	6		81.4.c.a.28.1		2
60.59	even	2		432.4.a.a.1.1		1
105.104	even	2		1323.4.a.d.1.1		1
120.29	odd	2		1728.4.a.bc.1.1		1
120.59	even	2		1728.4.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.a.1.1	✓	1	15.14	odd	2
27.4.a.b.1.1	yes	1	5.4	even	2
81.4.c.a.28.1		2	45.34	even	6
81.4.c.a.55.1		2	45.4	even	6
81.4.c.c.28.1		2	45.29	odd	6
81.4.c.c.55.1		2	45.14	odd	6
432.4.a.a.1.1		1	60.59	even	2
432.4.a.n.1.1		1	20.19	odd	2
675.4.a.a.1.1		1	1.1	even	1	trivial
675.4.a.j.1.1		1	3.2	odd	2
675.4.b.a.649.1		2	5.2	odd	4
675.4.b.a.649.2		2	5.3	odd	4
675.4.b.b.649.1		2	15.8	even	4
675.4.b.b.649.2		2	15.2	even	4
1323.4.a.d.1.1		1	105.104	even	2
1323.4.a.k.1.1		1	35.34	odd	2
1728.4.a.c.1.1		1	40.29	even	2
1728.4.a.d.1.1		1	40.19	odd	2
1728.4.a.bc.1.1		1	120.29	odd	2
1728.4.a.bd.1.1		1	120.59	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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