LMFDB - Embedded newform 225.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.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:	$13.2754297513$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -7.00000 q^{4} +24.0000 q^{7} -15.0000 q^{8} -52.0000 q^{11} -22.0000 q^{13} +24.0000 q^{14} +41.0000 q^{16} -14.0000 q^{17} -20.0000 q^{19} -52.0000 q^{22} -168.000 q^{23} -22.0000 q^{26} -168.000 q^{28} -230.000 q^{29} -288.000 q^{31} +161.000 q^{32} -14.0000 q^{34} +34.0000 q^{37} -20.0000 q^{38} -122.000 q^{41} +188.000 q^{43} +364.000 q^{44} -168.000 q^{46} +256.000 q^{47} +233.000 q^{49} +154.000 q^{52} -338.000 q^{53} -360.000 q^{56} -230.000 q^{58} -100.000 q^{59} +742.000 q^{61} -288.000 q^{62} -167.000 q^{64} +84.0000 q^{67} +98.0000 q^{68} +328.000 q^{71} +38.0000 q^{73} +34.0000 q^{74} +140.000 q^{76} -1248.00 q^{77} -240.000 q^{79} -122.000 q^{82} +1212.00 q^{83} +188.000 q^{86} +780.000 q^{88} -330.000 q^{89} -528.000 q^{91} +1176.00 q^{92} +256.000 q^{94} -866.000 q^{97} +233.000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	1.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	24.0000	1.29588	0.647939	−	0.761692i	$-0.275631\pi$
$7$	24.0000	1.29588	0.647939	+	0.761692i	$0.275631\pi$
$8$	−15.0000	−0.662913
$9$	0	0
$10$	0	0
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	0	0
$13$	−22.0000	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$13$	−22.0000	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$14$	24.0000	0.458162
$15$	0	0
$16$	41.0000	0.640625
$17$	−14.0000	−0.199735	−0.0998676	−	0.995001i	$-0.531842\pi$
$17$	−14.0000	−0.199735	−0.0998676	+	0.995001i	$0.531842\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	0	0
$21$	0	0
$22$	−52.0000	−0.503929
$23$	−168.000	−1.52306	−0.761531	−	0.648129i	$-0.775552\pi$
$23$	−168.000	−1.52306	−0.761531	+	0.648129i	$0.775552\pi$
$24$	0	0
$25$	0	0
$26$	−22.0000	−0.165944
$27$	0	0
$28$	−168.000	−1.13389
$29$	−230.000	−1.47276	−0.736378	−	0.676570i	$-0.763465\pi$
$29$	−230.000	−1.47276	−0.736378	+	0.676570i	$0.763465\pi$
$30$	0	0
$31$	−288.000	−1.66859	−0.834296	−	0.551317i	$-0.814125\pi$
$31$	−288.000	−1.66859	−0.834296	+	0.551317i	$0.814125\pi$
$32$	161.000	0.889408
$33$	0	0
$34$	−14.0000	−0.0706171
$35$	0	0
$36$	0	0
$37$	34.0000	0.151069	0.0755347	−	0.997143i	$-0.475934\pi$
$37$	34.0000	0.151069	0.0755347	+	0.997143i	$0.475934\pi$
$38$	−20.0000	−0.0853797
$39$	0	0
$40$	0	0
$41$	−122.000	−0.464712	−0.232356	−	0.972631i	$-0.574643\pi$
$41$	−122.000	−0.464712	−0.232356	+	0.972631i	$0.574643\pi$
$42$	0	0
$43$	188.000	0.666738	0.333369	−	0.942796i	$-0.391815\pi$
$43$	188.000	0.666738	0.333369	+	0.942796i	$0.391815\pi$
$44$	364.000	1.24716
$45$	0	0
$46$	−168.000	−0.538484
$47$	256.000	0.794499	0.397249	−	0.917711i	$-0.369965\pi$
$47$	256.000	0.794499	0.397249	+	0.917711i	$0.369965\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	0	0
$51$	0	0
$52$	154.000	0.410691
$53$	−338.000	−0.875998	−0.437999	−	0.898976i	$-0.644313\pi$
$53$	−338.000	−0.875998	−0.437999	+	0.898976i	$0.644313\pi$
$54$	0	0
$55$	0	0
$56$	−360.000	−0.859054
$57$	0	0
$58$	−230.000	−0.520698
$59$	−100.000	−0.220659	−0.110330	−	0.993895i	$-0.535191\pi$
$59$	−100.000	−0.220659	−0.110330	+	0.993895i	$0.535191\pi$
$60$	0	0
$61$	742.000	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$61$	742.000	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$62$	−288.000	−0.589936
$63$	0	0
$64$	−167.000	−0.326172
$65$	0	0
$66$	0	0
$67$	84.0000	0.153168	0.0765838	−	0.997063i	$-0.475599\pi$
$67$	84.0000	0.153168	0.0765838	+	0.997063i	$0.475599\pi$
$68$	98.0000	0.174768
$69$	0	0
$70$	0	0
$71$	328.000	0.548260	0.274130	−	0.961693i	$-0.411610\pi$
$71$	328.000	0.548260	0.274130	+	0.961693i	$0.411610\pi$
$72$	0	0
$73$	38.0000	0.0609255	0.0304628	−	0.999536i	$-0.490302\pi$
$73$	38.0000	0.0609255	0.0304628	+	0.999536i	$0.490302\pi$
$74$	34.0000	0.0534111
$75$	0	0
$76$	140.000	0.211304
$77$	−1248.00	−1.84705
$78$	0	0
$79$	−240.000	−0.341799	−0.170899	−	0.985288i	$-0.554667\pi$
$79$	−240.000	−0.341799	−0.170899	+	0.985288i	$0.554667\pi$
$80$	0	0
$81$	0	0
$82$	−122.000	−0.164301
$83$	1212.00	1.60282	0.801411	−	0.598114i	$-0.204083\pi$
$83$	1212.00	1.60282	0.801411	+	0.598114i	$0.204083\pi$
$84$	0	0
$85$	0	0
$86$	188.000	0.235727
$87$	0	0
$88$	780.000	0.944867
$89$	−330.000	−0.393033	−0.196516	−	0.980501i	$-0.562963\pi$
$89$	−330.000	−0.393033	−0.196516	+	0.980501i	$0.562963\pi$
$90$	0	0
$91$	−528.000	−0.608236
$92$	1176.00	1.33268
$93$	0	0
$94$	256.000	0.280898
$95$	0	0
$96$	0	0
$97$	−866.000	−0.906484	−0.453242	−	0.891387i	$-0.649733\pi$
$97$	−866.000	−0.906484	−0.453242	+	0.891387i	$0.649733\pi$
$98$	233.000	0.240169
$99$	0	0
$100$	0	0
$101$	1218.00	1.19996	0.599978	−	0.800017i	$-0.295176\pi$
$101$	1218.00	1.19996	0.599978	+	0.800017i	$0.295176\pi$
$102$	0	0
$103$	88.0000	0.0841835	0.0420917	−	0.999114i	$-0.486598\pi$
$103$	88.0000	0.0841835	0.0420917	+	0.999114i	$0.486598\pi$
$104$	330.000	0.311146
$105$	0	0
$106$	−338.000	−0.309712
$107$	36.0000	0.0325257	0.0162629	−	0.999868i	$-0.494823\pi$
$107$	36.0000	0.0325257	0.0162629	+	0.999868i	$0.494823\pi$
$108$	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
$112$	984.000	0.830172
$113$	1042.00	0.867461	0.433731	−	0.901043i	$-0.357197\pi$
$113$	1042.00	0.867461	0.433731	+	0.901043i	$0.357197\pi$
$114$	0	0
$115$	0	0
$116$	1610.00	1.28866
$117$	0	0
$118$	−100.000	−0.0780148
$119$	−336.000	−0.258833
$120$	0	0
$121$	1373.00	1.03156
$122$	742.000	0.550635
$123$	0	0
$124$	2016.00	1.46002
$125$	0	0
$126$	0	0
$127$	−1936.00	−1.35269	−0.676347	−	0.736583i	$-0.736438\pi$
$127$	−1936.00	−1.35269	−0.676347	+	0.736583i	$0.736438\pi$
$128$	−1455.00	−1.00473
$129$	0	0
$130$	0	0
$131$	−732.000	−0.488207	−0.244104	−	0.969749i	$-0.578494\pi$
$131$	−732.000	−0.488207	−0.244104	+	0.969749i	$0.578494\pi$
$132$	0	0
$133$	−480.000	−0.312942
$134$	84.0000	0.0541529
$135$	0	0
$136$	210.000	0.132407
$137$	−2214.00	−1.38069	−0.690346	−	0.723479i	$-0.742542\pi$
$137$	−2214.00	−1.38069	−0.690346	+	0.723479i	$0.742542\pi$
$138$	0	0
$139$	20.0000	0.0122042	0.00610208	−	0.999981i	$-0.498058\pi$
$139$	20.0000	0.0122042	0.00610208	+	0.999981i	$0.498058\pi$
$140$	0	0
$141$	0	0
$142$	328.000	0.193839
$143$	1144.00	0.668994
$144$	0	0
$145$	0	0
$146$	38.0000	0.0215404
$147$	0	0
$148$	−238.000	−0.132186
$149$	1330.00	0.731261	0.365630	−	0.930760i	$-0.380853\pi$
$149$	1330.00	0.731261	0.365630	+	0.930760i	$0.380853\pi$
$150$	0	0
$151$	−1208.00	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−1208.00	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	300.000	0.160087
$153$	0	0
$154$	−1248.00	−0.653031
$155$	0	0
$156$	0	0
$157$	3514.00	1.78629	0.893146	−	0.449768i	$-0.148493\pi$
$157$	3514.00	1.78629	0.893146	+	0.449768i	$0.148493\pi$
$158$	−240.000	−0.120844
$159$	0	0
$160$	0	0
$161$	−4032.00	−1.97370
$162$	0	0
$163$	2068.00	0.993732	0.496866	−	0.867827i	$-0.334484\pi$
$163$	2068.00	0.993732	0.496866	+	0.867827i	$0.334484\pi$
$164$	854.000	0.406623
$165$	0	0
$166$	1212.00	0.566683
$167$	−24.0000	−0.0111208	−0.00556041	−	0.999985i	$-0.501770\pi$
$167$	−24.0000	−0.0111208	−0.00556041	+	0.999985i	$0.501770\pi$
$168$	0	0
$169$	−1713.00	−0.779700
$170$	0	0
$171$	0	0
$172$	−1316.00	−0.583396
$173$	−618.000	−0.271593	−0.135797	−	0.990737i	$-0.543359\pi$
$173$	−618.000	−0.271593	−0.135797	+	0.990737i	$0.543359\pi$
$174$	0	0
$175$	0	0
$176$	−2132.00	−0.913100
$177$	0	0
$178$	−330.000	−0.138958
$179$	−3340.00	−1.39466	−0.697328	−	0.716752i	$-0.745628\pi$
$179$	−3340.00	−1.39466	−0.697328	+	0.716752i	$0.745628\pi$
$180$	0	0
$181$	−178.000	−0.0730974	−0.0365487	−	0.999332i	$-0.511636\pi$
$181$	−178.000	−0.0730974	−0.0365487	+	0.999332i	$0.511636\pi$
$182$	−528.000	−0.215044
$183$	0	0
$184$	2520.00	1.00966
$185$	0	0
$186$	0	0
$187$	728.000	0.284688
$188$	−1792.00	−0.695186
$189$	0	0
$190$	0	0
$191$	1888.00	0.715240	0.357620	−	0.933867i	$-0.383588\pi$
$191$	1888.00	0.715240	0.357620	+	0.933867i	$0.383588\pi$
$192$	0	0
$193$	−1922.00	−0.716832	−0.358416	−	0.933562i	$-0.616683\pi$
$193$	−1922.00	−0.716832	−0.358416	+	0.933562i	$0.616683\pi$
$194$	−866.000	−0.320491
$195$	0	0
$196$	−1631.00	−0.594388
$197$	2526.00	0.913554	0.456777	−	0.889581i	$-0.349004\pi$
$197$	2526.00	0.913554	0.456777	+	0.889581i	$0.349004\pi$
$198$	0	0
$199$	−1160.00	−0.413217	−0.206609	−	0.978424i	$-0.566243\pi$
$199$	−1160.00	−0.413217	−0.206609	+	0.978424i	$0.566243\pi$
$200$	0	0
$201$	0	0
$202$	1218.00	0.424248
$203$	−5520.00	−1.90851
$204$	0	0
$205$	0	0
$206$	88.0000	0.0297634
$207$	0	0
$208$	−902.000	−0.300685
$209$	1040.00	0.344202
$210$	0	0
$211$	−4468.00	−1.45777	−0.728886	−	0.684635i	$-0.759961\pi$
$211$	−4468.00	−1.45777	−0.728886	+	0.684635i	$0.759961\pi$
$212$	2366.00	0.766498
$213$	0	0
$214$	36.0000	0.0114996
$215$	0	0
$216$	0	0
$217$	−6912.00	−2.16229
$218$	−970.000	−0.301361
$219$	0	0
$220$	0	0
$221$	308.000	0.0937481
$222$	0	0
$223$	−6032.00	−1.81136	−0.905678	−	0.423965i	$-0.860638\pi$
$223$	−6032.00	−1.81136	−0.905678	+	0.423965i	$0.860638\pi$
$224$	3864.00	1.15256
$225$	0	0
$226$	1042.00	0.306694
$227$	2636.00	0.770738	0.385369	−	0.922763i	$-0.374074\pi$
$227$	2636.00	0.770738	0.385369	+	0.922763i	$0.374074\pi$
$228$	0	0
$229$	4830.00	1.39378	0.696889	−	0.717179i	$-0.254567\pi$
$229$	4830.00	1.39378	0.696889	+	0.717179i	$0.254567\pi$
$230$	0	0
$231$	0	0
$232$	3450.00	0.976309
$233$	2682.00	0.754093	0.377046	−	0.926194i	$-0.376940\pi$
$233$	2682.00	0.754093	0.377046	+	0.926194i	$0.376940\pi$
$234$	0	0
$235$	0	0
$236$	700.000	0.193077
$237$	0	0
$238$	−336.000	−0.0915111
$239$	−2320.00	−0.627901	−0.313950	−	0.949439i	$-0.601653\pi$
$239$	−2320.00	−0.627901	−0.313950	+	0.949439i	$0.601653\pi$
$240$	0	0
$241$	2002.00	0.535104	0.267552	−	0.963543i	$-0.413785\pi$
$241$	2002.00	0.535104	0.267552	+	0.963543i	$0.413785\pi$
$242$	1373.00	0.364710
$243$	0	0
$244$	−5194.00	−1.36275
$245$	0	0
$246$	0	0
$247$	440.000	0.113346
$248$	4320.00	1.10613
$249$	0	0
$250$	0	0
$251$	−132.000	−0.0331943	−0.0165971	−	0.999862i	$-0.505283\pi$
$251$	−132.000	−0.0331943	−0.0165971	+	0.999862i	$0.505283\pi$
$252$	0	0
$253$	8736.00	2.17086
$254$	−1936.00	−0.478250
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−7614.00	−1.84805	−0.924024	−	0.382335i	$-0.875120\pi$
$257$	−7614.00	−1.84805	−0.924024	+	0.382335i	$0.875120\pi$
$258$	0	0
$259$	816.000	0.195767
$260$	0	0
$261$	0	0
$262$	−732.000	−0.172607
$263$	−4888.00	−1.14603	−0.573017	−	0.819543i	$-0.694227\pi$
$263$	−4888.00	−1.14603	−0.573017	+	0.819543i	$0.694227\pi$
$264$	0	0
$265$	0	0
$266$	−480.000	−0.110642
$267$	0	0
$268$	−588.000	−0.134022
$269$	−1270.00	−0.287856	−0.143928	−	0.989588i	$-0.545973\pi$
$269$	−1270.00	−0.287856	−0.143928	+	0.989588i	$0.545973\pi$
$270$	0	0
$271$	1072.00	0.240293	0.120146	−	0.992756i	$-0.461664\pi$
$271$	1072.00	0.240293	0.120146	+	0.992756i	$0.461664\pi$
$272$	−574.000	−0.127955
$273$	0	0
$274$	−2214.00	−0.488148
$275$	0	0
$276$	0	0
$277$	5394.00	1.17001	0.585007	−	0.811028i	$-0.301092\pi$
$277$	5394.00	1.17001	0.585007	+	0.811028i	$0.301092\pi$
$278$	20.0000	0.00431482
$279$	0	0
$280$	0	0
$281$	−2442.00	−0.518425	−0.259213	−	0.965820i	$-0.583463\pi$
$281$	−2442.00	−0.518425	−0.259213	+	0.965820i	$0.583463\pi$
$282$	0	0
$283$	−2772.00	−0.582255	−0.291128	−	0.956684i	$-0.594030\pi$
$283$	−2772.00	−0.582255	−0.291128	+	0.956684i	$0.594030\pi$
$284$	−2296.00	−0.479727
$285$	0	0
$286$	1144.00	0.236525
$287$	−2928.00	−0.602210
$288$	0	0
$289$	−4717.00	−0.960106
$290$	0	0
$291$	0	0
$292$	−266.000	−0.0533098
$293$	4542.00	0.905619	0.452810	−	0.891607i	$-0.350422\pi$
$293$	4542.00	0.905619	0.452810	+	0.891607i	$0.350422\pi$
$294$	0	0
$295$	0	0
$296$	−510.000	−0.100146
$297$	0	0
$298$	1330.00	0.258540
$299$	3696.00	0.714867
$300$	0	0
$301$	4512.00	0.864011
$302$	−1208.00	−0.230174
$303$	0	0
$304$	−820.000	−0.154705
$305$	0	0
$306$	0	0
$307$	−5116.00	−0.951093	−0.475546	−	0.879691i	$-0.657750\pi$
$307$	−5116.00	−0.951093	−0.475546	+	0.879691i	$0.657750\pi$
$308$	8736.00	1.61617
$309$	0	0
$310$	0	0
$311$	2808.00	0.511984	0.255992	−	0.966679i	$-0.417598\pi$
$311$	2808.00	0.511984	0.255992	+	0.966679i	$0.417598\pi$
$312$	0	0
$313$	7318.00	1.32153	0.660763	−	0.750594i	$-0.270233\pi$
$313$	7318.00	1.32153	0.660763	+	0.750594i	$0.270233\pi$
$314$	3514.00	0.631549
$315$	0	0
$316$	1680.00	0.299074
$317$	2246.00	0.397943	0.198971	−	0.980005i	$-0.436240\pi$
$317$	2246.00	0.397943	0.198971	+	0.980005i	$0.436240\pi$
$318$	0	0
$319$	11960.0	2.09916
$320$	0	0
$321$	0	0
$322$	−4032.00	−0.697809
$323$	280.000	0.0482341
$324$	0	0
$325$	0	0
$326$	2068.00	0.351337
$327$	0	0
$328$	1830.00	0.308064
$329$	6144.00	1.02957
$330$	0	0
$331$	1332.00	0.221188	0.110594	−	0.993866i	$-0.464725\pi$
$331$	1332.00	0.221188	0.110594	+	0.993866i	$0.464725\pi$
$332$	−8484.00	−1.40247
$333$	0	0
$334$	−24.0000	−0.00393180
$335$	0	0
$336$	0	0
$337$	11534.0	1.86438	0.932191	−	0.361966i	$-0.117894\pi$
$337$	11534.0	1.86438	0.932191	+	0.361966i	$0.117894\pi$
$338$	−1713.00	−0.275665
$339$	0	0
$340$	0	0
$341$	14976.0	2.37829
$342$	0	0
$343$	−2640.00	−0.415588
$344$	−2820.00	−0.441989
$345$	0	0
$346$	−618.000	−0.0960228
$347$	11956.0	1.84966	0.924830	−	0.380382i	$-0.124207\pi$
$347$	11956.0	1.84966	0.924830	+	0.380382i	$0.124207\pi$
$348$	0	0
$349$	4870.00	0.746949	0.373474	−	0.927640i	$-0.378166\pi$
$349$	4870.00	0.746949	0.373474	+	0.927640i	$0.378166\pi$
$350$	0	0
$351$	0	0
$352$	−8372.00	−1.26770
$353$	10722.0	1.61664	0.808321	−	0.588742i	$-0.200377\pi$
$353$	10722.0	1.61664	0.808321	+	0.588742i	$0.200377\pi$
$354$	0	0
$355$	0	0
$356$	2310.00	0.343904
$357$	0	0
$358$	−3340.00	−0.493085
$359$	−120.000	−0.0176417	−0.00882083	−	0.999961i	$-0.502808\pi$
$359$	−120.000	−0.0176417	−0.00882083	+	0.999961i	$0.502808\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	−178.000	−0.0258438
$363$	0	0
$364$	3696.00	0.532206
$365$	0	0
$366$	0	0
$367$	−3936.00	−0.559830	−0.279915	−	0.960025i	$-0.590306\pi$
$367$	−3936.00	−0.559830	−0.279915	+	0.960025i	$0.590306\pi$
$368$	−6888.00	−0.975711
$369$	0	0
$370$	0	0
$371$	−8112.00	−1.13519
$372$	0	0
$373$	−3022.00	−0.419499	−0.209750	−	0.977755i	$-0.567265\pi$
$373$	−3022.00	−0.419499	−0.209750	+	0.977755i	$0.567265\pi$
$374$	728.000	0.100652
$375$	0	0
$376$	−3840.00	−0.526683
$377$	5060.00	0.691255
$378$	0	0
$379$	−13340.0	−1.80799	−0.903997	−	0.427539i	$-0.859381\pi$
$379$	−13340.0	−1.80799	−0.903997	+	0.427539i	$0.859381\pi$
$380$	0	0
$381$	0	0
$382$	1888.00	0.252876
$383$	−1008.00	−0.134481	−0.0672407	−	0.997737i	$-0.521420\pi$
$383$	−1008.00	−0.134481	−0.0672407	+	0.997737i	$0.521420\pi$
$384$	0	0
$385$	0	0
$386$	−1922.00	−0.253438
$387$	0	0
$388$	6062.00	0.793174
$389$	−9630.00	−1.25517	−0.627584	−	0.778549i	$-0.715956\pi$
$389$	−9630.00	−1.25517	−0.627584	+	0.778549i	$0.715956\pi$
$390$	0	0
$391$	2352.00	0.304209
$392$	−3495.00	−0.450317
$393$	0	0
$394$	2526.00	0.322990
$395$	0	0
$396$	0	0
$397$	−7126.00	−0.900866	−0.450433	−	0.892810i	$-0.648730\pi$
$397$	−7126.00	−0.900866	−0.450433	+	0.892810i	$0.648730\pi$
$398$	−1160.00	−0.146094
$399$	0	0
$400$	0	0
$401$	8718.00	1.08568	0.542838	−	0.839837i	$-0.317350\pi$
$401$	8718.00	1.08568	0.542838	+	0.839837i	$0.317350\pi$
$402$	0	0
$403$	6336.00	0.783173
$404$	−8526.00	−1.04996
$405$	0	0
$406$	−5520.00	−0.674761
$407$	−1768.00	−0.215323
$408$	0	0
$409$	−10870.0	−1.31415	−0.657074	−	0.753826i	$-0.728206\pi$
$409$	−10870.0	−1.31415	−0.657074	+	0.753826i	$0.728206\pi$
$410$	0	0
$411$	0	0
$412$	−616.000	−0.0736605
$413$	−2400.00	−0.285947
$414$	0	0
$415$	0	0
$416$	−3542.00	−0.417454
$417$	0	0
$418$	1040.00	0.121694
$419$	9700.00	1.13097	0.565484	−	0.824759i	$-0.308689\pi$
$419$	9700.00	1.13097	0.565484	+	0.824759i	$0.308689\pi$
$420$	0	0
$421$	862.000	0.0997893	0.0498947	−	0.998754i	$-0.484111\pi$
$421$	862.000	0.0997893	0.0498947	+	0.998754i	$0.484111\pi$
$422$	−4468.00	−0.515400
$423$	0	0
$424$	5070.00	0.580710
$425$	0	0
$426$	0	0
$427$	17808.0	2.01824
$428$	−252.000	−0.0284600
$429$	0	0
$430$	0	0
$431$	−15792.0	−1.76490	−0.882452	−	0.470402i	$-0.844109\pi$
$431$	−15792.0	−1.76490	−0.882452	+	0.470402i	$0.844109\pi$
$432$	0	0
$433$	−11602.0	−1.28766	−0.643830	−	0.765169i	$-0.722655\pi$
$433$	−11602.0	−1.28766	−0.643830	+	0.765169i	$0.722655\pi$
$434$	−6912.00	−0.764485
$435$	0	0
$436$	6790.00	0.745830
$437$	3360.00	0.367805
$438$	0	0
$439$	−440.000	−0.0478361	−0.0239181	−	0.999714i	$-0.507614\pi$
$439$	−440.000	−0.0478361	−0.0239181	+	0.999714i	$0.507614\pi$
$440$	0	0
$441$	0	0
$442$	308.000	0.0331449
$443$	−10188.0	−1.09266	−0.546328	−	0.837571i	$-0.683975\pi$
$443$	−10188.0	−1.09266	−0.546328	+	0.837571i	$0.683975\pi$
$444$	0	0
$445$	0	0
$446$	−6032.00	−0.640411
$447$	0	0
$448$	−4008.00	−0.422679
$449$	13310.0	1.39897	0.699485	−	0.714647i	$-0.253413\pi$
$449$	13310.0	1.39897	0.699485	+	0.714647i	$0.253413\pi$
$450$	0	0
$451$	6344.00	0.662367
$452$	−7294.00	−0.759029
$453$	0	0
$454$	2636.00	0.272497
$455$	0	0
$456$	0	0
$457$	−3226.00	−0.330210	−0.165105	−	0.986276i	$-0.552796\pi$
$457$	−3226.00	−0.330210	−0.165105	+	0.986276i	$0.552796\pi$
$458$	4830.00	0.492775
$459$	0	0
$460$	0	0
$461$	−6582.00	−0.664977	−0.332488	−	0.943107i	$-0.607888\pi$
$461$	−6582.00	−0.664977	−0.332488	+	0.943107i	$0.607888\pi$
$462$	0	0
$463$	−15072.0	−1.51286	−0.756431	−	0.654073i	$-0.773059\pi$
$463$	−15072.0	−1.51286	−0.756431	+	0.654073i	$0.773059\pi$
$464$	−9430.00	−0.943484
$465$	0	0
$466$	2682.00	0.266612
$467$	476.000	0.0471663	0.0235831	−	0.999722i	$-0.492493\pi$
$467$	476.000	0.0471663	0.0235831	+	0.999722i	$0.492493\pi$
$468$	0	0
$469$	2016.00	0.198487
$470$	0	0
$471$	0	0
$472$	1500.00	0.146278
$473$	−9776.00	−0.950319
$474$	0	0
$475$	0	0
$476$	2352.00	0.226478
$477$	0	0
$478$	−2320.00	−0.221997
$479$	19680.0	1.87725	0.938624	−	0.344941i	$-0.112101\pi$
$479$	19680.0	1.87725	0.938624	+	0.344941i	$0.112101\pi$
$480$	0	0
$481$	−748.000	−0.0709062
$482$	2002.00	0.189188
$483$	0	0
$484$	−9611.00	−0.902611
$485$	0	0
$486$	0	0
$487$	5944.00	0.553077	0.276538	−	0.961003i	$-0.410813\pi$
$487$	5944.00	0.553077	0.276538	+	0.961003i	$0.410813\pi$
$488$	−11130.0	−1.03244
$489$	0	0
$490$	0	0
$491$	−10772.0	−0.990089	−0.495044	−	0.868868i	$-0.664848\pi$
$491$	−10772.0	−0.990089	−0.495044	+	0.868868i	$0.664848\pi$
$492$	0	0
$493$	3220.00	0.294161
$494$	440.000	0.0400740
$495$	0	0
$496$	−11808.0	−1.06894
$497$	7872.00	0.710478
$498$	0	0
$499$	8140.00	0.730253	0.365127	−	0.930958i	$-0.381026\pi$
$499$	8140.00	0.730253	0.365127	+	0.930958i	$0.381026\pi$
$500$	0	0
$501$	0	0
$502$	−132.000	−0.0117360
$503$	−13768.0	−1.22045	−0.610223	−	0.792229i	$-0.708920\pi$
$503$	−13768.0	−1.22045	−0.610223	+	0.792229i	$0.708920\pi$
$504$	0	0
$505$	0	0
$506$	8736.00	0.767515
$507$	0	0
$508$	13552.0	1.18361
$509$	−22150.0	−1.92884	−0.964422	−	0.264368i	$-0.914837\pi$
$509$	−22150.0	−1.92884	−0.964422	+	0.264368i	$0.914837\pi$
$510$	0	0
$511$	912.000	0.0789521
$512$	11521.0	0.994455
$513$	0	0
$514$	−7614.00	−0.653384
$515$	0	0
$516$	0	0
$517$	−13312.0	−1.13242
$518$	816.000	0.0692143
$519$	0	0
$520$	0	0
$521$	−1562.00	−0.131348	−0.0656741	−	0.997841i	$-0.520920\pi$
$521$	−1562.00	−0.131348	−0.0656741	+	0.997841i	$0.520920\pi$
$522$	0	0
$523$	668.000	0.0558501	0.0279250	−	0.999610i	$-0.491110\pi$
$523$	668.000	0.0558501	0.0279250	+	0.999610i	$0.491110\pi$
$524$	5124.00	0.427181
$525$	0	0
$526$	−4888.00	−0.405184
$527$	4032.00	0.333276
$528$	0	0
$529$	16057.0	1.31972
$530$	0	0
$531$	0	0
$532$	3360.00	0.273824
$533$	2684.00	0.218118
$534$	0	0
$535$	0	0
$536$	−1260.00	−0.101537
$537$	0	0
$538$	−1270.00	−0.101772
$539$	−12116.0	−0.968225
$540$	0	0
$541$	−6138.00	−0.487788	−0.243894	−	0.969802i	$-0.578425\pi$
$541$	−6138.00	−0.487788	−0.243894	+	0.969802i	$0.578425\pi$
$542$	1072.00	0.0849564
$543$	0	0
$544$	−2254.00	−0.177646
$545$	0	0
$546$	0	0
$547$	10484.0	0.819494	0.409747	−	0.912199i	$-0.365617\pi$
$547$	10484.0	0.819494	0.409747	+	0.912199i	$0.365617\pi$
$548$	15498.0	1.20811
$549$	0	0
$550$	0	0
$551$	4600.00	0.355656
$552$	0	0
$553$	−5760.00	−0.442930
$554$	5394.00	0.413663
$555$	0	0
$556$	−140.000	−0.0106786
$557$	3606.00	0.274311	0.137155	−	0.990550i	$-0.456204\pi$
$557$	3606.00	0.274311	0.137155	+	0.990550i	$0.456204\pi$
$558$	0	0
$559$	−4136.00	−0.312941
$560$	0	0
$561$	0	0
$562$	−2442.00	−0.183291
$563$	12252.0	0.917159	0.458579	−	0.888654i	$-0.348359\pi$
$563$	12252.0	0.917159	0.458579	+	0.888654i	$0.348359\pi$
$564$	0	0
$565$	0	0
$566$	−2772.00	−0.205858
$567$	0	0
$568$	−4920.00	−0.363448
$569$	14550.0	1.07200	0.536000	−	0.844218i	$-0.319935\pi$
$569$	14550.0	1.07200	0.536000	+	0.844218i	$0.319935\pi$
$570$	0	0
$571$	−25468.0	−1.86655	−0.933277	−	0.359157i	$-0.883064\pi$
$571$	−25468.0	−1.86655	−0.933277	+	0.359157i	$0.883064\pi$
$572$	−8008.00	−0.585369
$573$	0	0
$574$	−2928.00	−0.212914
$575$	0	0
$576$	0	0
$577$	−12866.0	−0.928282	−0.464141	−	0.885761i	$-0.653637\pi$
$577$	−12866.0	−0.928282	−0.464141	+	0.885761i	$0.653637\pi$
$578$	−4717.00	−0.339449
$579$	0	0
$580$	0	0
$581$	29088.0	2.07706
$582$	0	0
$583$	17576.0	1.24858
$584$	−570.000	−0.0403883
$585$	0	0
$586$	4542.00	0.320185
$587$	−14844.0	−1.04374	−0.521872	−	0.853024i	$-0.674766\pi$
$587$	−14844.0	−1.04374	−0.521872	+	0.853024i	$0.674766\pi$
$588$	0	0
$589$	5760.00	0.402948
$590$	0	0
$591$	0	0
$592$	1394.00	0.0967788
$593$	20402.0	1.41283	0.706416	−	0.707797i	$-0.250311\pi$
$593$	20402.0	1.41283	0.706416	+	0.707797i	$0.250311\pi$
$594$	0	0
$595$	0	0
$596$	−9310.00	−0.639853
$597$	0	0
$598$	3696.00	0.252744
$599$	−10760.0	−0.733959	−0.366980	−	0.930229i	$-0.619608\pi$
$599$	−10760.0	−0.733959	−0.366980	+	0.930229i	$0.619608\pi$
$600$	0	0
$601$	14282.0	0.969343	0.484671	−	0.874696i	$-0.338939\pi$
$601$	14282.0	0.969343	0.484671	+	0.874696i	$0.338939\pi$
$602$	4512.00	0.305474
$603$	0	0
$604$	8456.00	0.569652
$605$	0	0
$606$	0	0
$607$	−11056.0	−0.739290	−0.369645	−	0.929173i	$-0.620521\pi$
$607$	−11056.0	−0.739290	−0.369645	+	0.929173i	$0.620521\pi$
$608$	−3220.00	−0.214783
$609$	0	0
$610$	0	0
$611$	−5632.00	−0.372907
$612$	0	0
$613$	16418.0	1.08176	0.540878	−	0.841101i	$-0.318092\pi$
$613$	16418.0	1.08176	0.540878	+	0.841101i	$0.318092\pi$
$614$	−5116.00	−0.336262
$615$	0	0
$616$	18720.0	1.22443
$617$	−10374.0	−0.676891	−0.338445	−	0.940986i	$-0.609901\pi$
$617$	−10374.0	−0.676891	−0.338445	+	0.940986i	$0.609901\pi$
$618$	0	0
$619$	−5260.00	−0.341546	−0.170773	−	0.985310i	$-0.554627\pi$
$619$	−5260.00	−0.341546	−0.170773	+	0.985310i	$0.554627\pi$
$620$	0	0
$621$	0	0
$622$	2808.00	0.181014
$623$	−7920.00	−0.509323
$624$	0	0
$625$	0	0
$626$	7318.00	0.467230
$627$	0	0
$628$	−24598.0	−1.56300
$629$	−476.000	−0.0301739
$630$	0	0
$631$	21352.0	1.34708	0.673542	−	0.739149i	$-0.264772\pi$
$631$	21352.0	1.34708	0.673542	+	0.739149i	$0.264772\pi$
$632$	3600.00	0.226583
$633$	0	0
$634$	2246.00	0.140694
$635$	0	0
$636$	0	0
$637$	−5126.00	−0.318838
$638$	11960.0	0.742164
$639$	0	0
$640$	0	0
$641$	29118.0	1.79422	0.897108	−	0.441812i	$-0.145664\pi$
$641$	29118.0	1.79422	0.897108	+	0.441812i	$0.145664\pi$
$642$	0	0
$643$	−5772.00	−0.354005	−0.177003	−	0.984210i	$-0.556640\pi$
$643$	−5772.00	−0.354005	−0.177003	+	0.984210i	$0.556640\pi$
$644$	28224.0	1.72699
$645$	0	0
$646$	280.000	0.0170533
$647$	−14264.0	−0.866732	−0.433366	−	0.901218i	$-0.642674\pi$
$647$	−14264.0	−0.866732	−0.433366	+	0.901218i	$0.642674\pi$
$648$	0	0
$649$	5200.00	0.314511
$650$	0	0
$651$	0	0
$652$	−14476.0	−0.869515
$653$	6902.00	0.413623	0.206812	−	0.978381i	$-0.433691\pi$
$653$	6902.00	0.413623	0.206812	+	0.978381i	$0.433691\pi$
$654$	0	0
$655$	0	0
$656$	−5002.00	−0.297706
$657$	0	0
$658$	6144.00	0.364009
$659$	−20140.0	−1.19051	−0.595253	−	0.803539i	$-0.702948\pi$
$659$	−20140.0	−1.19051	−0.595253	+	0.803539i	$0.702948\pi$
$660$	0	0
$661$	−3218.00	−0.189358	−0.0946790	−	0.995508i	$-0.530182\pi$
$661$	−3218.00	−0.189358	−0.0946790	+	0.995508i	$0.530182\pi$
$662$	1332.00	0.0782019
$663$	0	0
$664$	−18180.0	−1.06253
$665$	0	0
$666$	0	0
$667$	38640.0	2.24310
$668$	168.000	0.00973071
$669$	0	0
$670$	0	0
$671$	−38584.0	−2.21985
$672$	0	0
$673$	7518.00	0.430606	0.215303	−	0.976547i	$-0.430926\pi$
$673$	7518.00	0.430606	0.215303	+	0.976547i	$0.430926\pi$
$674$	11534.0	0.659159
$675$	0	0
$676$	11991.0	0.682237
$677$	−18114.0	−1.02833	−0.514164	−	0.857692i	$-0.671898\pi$
$677$	−18114.0	−1.02833	−0.514164	+	0.857692i	$0.671898\pi$
$678$	0	0
$679$	−20784.0	−1.17469
$680$	0	0
$681$	0	0
$682$	14976.0	0.840851
$683$	−23868.0	−1.33716	−0.668582	−	0.743638i	$-0.733099\pi$
$683$	−23868.0	−1.33716	−0.668582	+	0.743638i	$0.733099\pi$
$684$	0	0
$685$	0	0
$686$	−2640.00	−0.146932
$687$	0	0
$688$	7708.00	0.427129
$689$	7436.00	0.411160
$690$	0	0
$691$	172.000	0.00946916	0.00473458	−	0.999989i	$-0.498493\pi$
$691$	172.000	0.00946916	0.00473458	+	0.999989i	$0.498493\pi$
$692$	4326.00	0.237644
$693$	0	0
$694$	11956.0	0.653953
$695$	0	0
$696$	0	0
$697$	1708.00	0.0928194
$698$	4870.00	0.264086
$699$	0	0
$700$	0	0
$701$	22138.0	1.19278	0.596391	−	0.802694i	$-0.296601\pi$
$701$	22138.0	1.19278	0.596391	+	0.802694i	$0.296601\pi$
$702$	0	0
$703$	−680.000	−0.0364818
$704$	8684.00	0.464901
$705$	0	0
$706$	10722.0	0.571569
$707$	29232.0	1.55500
$708$	0	0
$709$	3070.00	0.162618	0.0813091	−	0.996689i	$-0.474090\pi$
$709$	3070.00	0.162618	0.0813091	+	0.996689i	$0.474090\pi$
$710$	0	0
$711$	0	0
$712$	4950.00	0.260546
$713$	48384.0	2.54137
$714$	0	0
$715$	0	0
$716$	23380.0	1.22032
$717$	0	0
$718$	−120.000	−0.00623727
$719$	−15600.0	−0.809154	−0.404577	−	0.914504i	$-0.632581\pi$
$719$	−15600.0	−0.809154	−0.404577	+	0.914504i	$0.632581\pi$
$720$	0	0
$721$	2112.00	0.109092
$722$	−6459.00	−0.332935
$723$	0	0
$724$	1246.00	0.0639603
$725$	0	0
$726$	0	0
$727$	−20696.0	−1.05581	−0.527904	−	0.849304i	$-0.677022\pi$
$727$	−20696.0	−1.05581	−0.527904	+	0.849304i	$0.677022\pi$
$728$	7920.00	0.403207
$729$	0	0
$730$	0	0
$731$	−2632.00	−0.133171
$732$	0	0
$733$	30778.0	1.55090	0.775451	−	0.631408i	$-0.217522\pi$
$733$	30778.0	1.55090	0.775451	+	0.631408i	$0.217522\pi$
$734$	−3936.00	−0.197930
$735$	0	0
$736$	−27048.0	−1.35462
$737$	−4368.00	−0.218314
$738$	0	0
$739$	11740.0	0.584388	0.292194	−	0.956359i	$-0.405615\pi$
$739$	11740.0	0.584388	0.292194	+	0.956359i	$0.405615\pi$
$740$	0	0
$741$	0	0
$742$	−8112.00	−0.401349
$743$	2632.00	0.129958	0.0649789	−	0.997887i	$-0.479302\pi$
$743$	2632.00	0.129958	0.0649789	+	0.997887i	$0.479302\pi$
$744$	0	0
$745$	0	0
$746$	−3022.00	−0.148315
$747$	0	0
$748$	−5096.00	−0.249102
$749$	864.000	0.0421494
$750$	0	0
$751$	−20528.0	−0.997440	−0.498720	−	0.866763i	$-0.666196\pi$
$751$	−20528.0	−0.997440	−0.498720	+	0.866763i	$0.666196\pi$
$752$	10496.0	0.508976
$753$	0	0
$754$	5060.00	0.244396
$755$	0	0
$756$	0	0
$757$	−21646.0	−1.03928	−0.519642	−	0.854384i	$-0.673934\pi$
$757$	−21646.0	−1.03928	−0.519642	+	0.854384i	$0.673934\pi$
$758$	−13340.0	−0.639222
$759$	0	0
$760$	0	0
$761$	−18282.0	−0.870857	−0.435428	−	0.900223i	$-0.643403\pi$
$761$	−18282.0	−0.870857	−0.435428	+	0.900223i	$0.643403\pi$
$762$	0	0
$763$	−23280.0	−1.10458
$764$	−13216.0	−0.625835
$765$	0	0
$766$	−1008.00	−0.0475464
$767$	2200.00	0.103569
$768$	0	0
$769$	−24190.0	−1.13435	−0.567174	−	0.823598i	$-0.691963\pi$
$769$	−24190.0	−1.13435	−0.567174	+	0.823598i	$0.691963\pi$
$770$	0	0
$771$	0	0
$772$	13454.0	0.627228
$773$	−25698.0	−1.19572	−0.597861	−	0.801600i	$-0.703982\pi$
$773$	−25698.0	−1.19572	−0.597861	+	0.801600i	$0.703982\pi$
$774$	0	0
$775$	0	0
$776$	12990.0	0.600920
$777$	0	0
$778$	−9630.00	−0.443769
$779$	2440.00	0.112223
$780$	0	0
$781$	−17056.0	−0.781449
$782$	2352.00	0.107554
$783$	0	0
$784$	9553.00	0.435177
$785$	0	0
$786$	0	0
$787$	−33436.0	−1.51444	−0.757220	−	0.653160i	$-0.773443\pi$
$787$	−33436.0	−1.51444	−0.757220	+	0.653160i	$0.773443\pi$
$788$	−17682.0	−0.799359
$789$	0	0
$790$	0	0
$791$	25008.0	1.12412
$792$	0	0
$793$	−16324.0	−0.730999
$794$	−7126.00	−0.318504
$795$	0	0
$796$	8120.00	0.361565
$797$	−37594.0	−1.67083	−0.835413	−	0.549623i	$-0.814771\pi$
$797$	−37594.0	−1.67083	−0.835413	+	0.549623i	$0.814771\pi$
$798$	0	0
$799$	−3584.00	−0.158689
$800$	0	0
$801$	0	0
$802$	8718.00	0.383844
$803$	−1976.00	−0.0868388
$804$	0	0
$805$	0	0
$806$	6336.00	0.276893
$807$	0	0
$808$	−18270.0	−0.795466
$809$	−4730.00	−0.205560	−0.102780	−	0.994704i	$-0.532774\pi$
$809$	−4730.00	−0.205560	−0.102780	+	0.994704i	$0.532774\pi$
$810$	0	0
$811$	−8748.00	−0.378772	−0.189386	−	0.981903i	$-0.560650\pi$
$811$	−8748.00	−0.378772	−0.189386	+	0.981903i	$0.560650\pi$
$812$	38640.0	1.66995
$813$	0	0
$814$	−1768.00	−0.0761282
$815$	0	0
$816$	0	0
$817$	−3760.00	−0.161011
$818$	−10870.0	−0.464622
$819$	0	0
$820$	0	0
$821$	−44142.0	−1.87645	−0.938226	−	0.346024i	$-0.887532\pi$
$821$	−44142.0	−1.87645	−0.938226	+	0.346024i	$0.887532\pi$
$822$	0	0
$823$	−3992.00	−0.169079	−0.0845397	−	0.996420i	$-0.526942\pi$
$823$	−3992.00	−0.169079	−0.0845397	+	0.996420i	$0.526942\pi$
$824$	−1320.00	−0.0558063
$825$	0	0
$826$	−2400.00	−0.101098
$827$	−14444.0	−0.607336	−0.303668	−	0.952778i	$-0.598211\pi$
$827$	−14444.0	−0.607336	−0.303668	+	0.952778i	$0.598211\pi$
$828$	0	0
$829$	42150.0	1.76590	0.882949	−	0.469468i	$-0.155554\pi$
$829$	42150.0	1.76590	0.882949	+	0.469468i	$0.155554\pi$
$830$	0	0
$831$	0	0
$832$	3674.00	0.153093
$833$	−3262.00	−0.135680
$834$	0	0
$835$	0	0
$836$	−7280.00	−0.301177
$837$	0	0
$838$	9700.00	0.399858
$839$	−13400.0	−0.551394	−0.275697	−	0.961245i	$-0.588909\pi$
$839$	−13400.0	−0.551394	−0.275697	+	0.961245i	$0.588909\pi$
$840$	0	0
$841$	28511.0	1.16901
$842$	862.000	0.0352809
$843$	0	0
$844$	31276.0	1.27555
$845$	0	0
$846$	0	0
$847$	32952.0	1.33677
$848$	−13858.0	−0.561186
$849$	0	0
$850$	0	0
$851$	−5712.00	−0.230088
$852$	0	0
$853$	8658.00	0.347531	0.173766	−	0.984787i	$-0.444406\pi$
$853$	8658.00	0.347531	0.173766	+	0.984787i	$0.444406\pi$
$854$	17808.0	0.713556
$855$	0	0
$856$	−540.000	−0.0215617
$857$	42826.0	1.70701	0.853505	−	0.521084i	$-0.174472\pi$
$857$	42826.0	1.70701	0.853505	+	0.521084i	$0.174472\pi$
$858$	0	0
$859$	−35900.0	−1.42595	−0.712976	−	0.701189i	$-0.752653\pi$
$859$	−35900.0	−1.42595	−0.712976	+	0.701189i	$0.752653\pi$
$860$	0	0
$861$	0	0
$862$	−15792.0	−0.623988
$863$	−3088.00	−0.121804	−0.0609019	−	0.998144i	$-0.519398\pi$
$863$	−3088.00	−0.121804	−0.0609019	+	0.998144i	$0.519398\pi$
$864$	0	0
$865$	0	0
$866$	−11602.0	−0.455256
$867$	0	0
$868$	48384.0	1.89200
$869$	12480.0	0.487175
$870$	0	0
$871$	−1848.00	−0.0718910
$872$	14550.0	0.565052
$873$	0	0
$874$	3360.00	0.130039
$875$	0	0
$876$	0	0
$877$	35274.0	1.35817	0.679087	−	0.734058i	$-0.262376\pi$
$877$	35274.0	1.35817	0.679087	+	0.734058i	$0.262376\pi$
$878$	−440.000	−0.0169126
$879$	0	0
$880$	0	0
$881$	−25042.0	−0.957646	−0.478823	−	0.877911i	$-0.658936\pi$
$881$	−25042.0	−0.957646	−0.478823	+	0.877911i	$0.658936\pi$
$882$	0	0
$883$	−12572.0	−0.479141	−0.239570	−	0.970879i	$-0.577007\pi$
$883$	−12572.0	−0.479141	−0.239570	+	0.970879i	$0.577007\pi$
$884$	−2156.00	−0.0820296
$885$	0	0
$886$	−10188.0	−0.386312
$887$	−21864.0	−0.827645	−0.413823	−	0.910358i	$-0.635807\pi$
$887$	−21864.0	−0.827645	−0.413823	+	0.910358i	$0.635807\pi$
$888$	0	0
$889$	−46464.0	−1.75293
$890$	0	0
$891$	0	0
$892$	42224.0	1.58494
$893$	−5120.00	−0.191864
$894$	0	0
$895$	0	0
$896$	−34920.0	−1.30200
$897$	0	0
$898$	13310.0	0.494611
$899$	66240.0	2.45743
$900$	0	0
$901$	4732.00	0.174968
$902$	6344.00	0.234182
$903$	0	0
$904$	−15630.0	−0.575051
$905$	0	0
$906$	0	0
$907$	−31236.0	−1.14352	−0.571761	−	0.820420i	$-0.693740\pi$
$907$	−31236.0	−1.14352	−0.571761	+	0.820420i	$0.693740\pi$
$908$	−18452.0	−0.674396
$909$	0	0
$910$	0	0
$911$	−8272.00	−0.300838	−0.150419	−	0.988622i	$-0.548062\pi$
$911$	−8272.00	−0.300838	−0.150419	+	0.988622i	$0.548062\pi$
$912$	0	0
$913$	−63024.0	−2.28455
$914$	−3226.00	−0.116747
$915$	0	0
$916$	−33810.0	−1.21956
$917$	−17568.0	−0.632657
$918$	0	0
$919$	20200.0	0.725067	0.362533	−	0.931971i	$-0.381912\pi$
$919$	20200.0	0.725067	0.362533	+	0.931971i	$0.381912\pi$
$920$	0	0
$921$	0	0
$922$	−6582.00	−0.235105
$923$	−7216.00	−0.257332
$924$	0	0
$925$	0	0
$926$	−15072.0	−0.534878
$927$	0	0
$928$	−37030.0	−1.30988
$929$	−31010.0	−1.09516	−0.547581	−	0.836753i	$-0.684451\pi$
$929$	−31010.0	−1.09516	−0.547581	+	0.836753i	$0.684451\pi$
$930$	0	0
$931$	−4660.00	−0.164044
$932$	−18774.0	−0.659831
$933$	0	0
$934$	476.000	0.0166758
$935$	0	0
$936$	0	0
$937$	39174.0	1.36580	0.682902	−	0.730510i	$-0.260717\pi$
$937$	39174.0	1.36580	0.682902	+	0.730510i	$0.260717\pi$
$938$	2016.00	0.0701756
$939$	0	0
$940$	0	0
$941$	4138.00	0.143353	0.0716764	−	0.997428i	$-0.477165\pi$
$941$	4138.00	0.143353	0.0716764	+	0.997428i	$0.477165\pi$
$942$	0	0
$943$	20496.0	0.707785
$944$	−4100.00	−0.141360
$945$	0	0
$946$	−9776.00	−0.335989
$947$	23676.0	0.812425	0.406213	−	0.913779i	$-0.366849\pi$
$947$	23676.0	0.812425	0.406213	+	0.913779i	$0.366849\pi$
$948$	0	0
$949$	−836.000	−0.0285961
$950$	0	0
$951$	0	0
$952$	5040.00	0.171583
$953$	18922.0	0.643173	0.321586	−	0.946880i	$-0.395784\pi$
$953$	18922.0	0.643173	0.321586	+	0.946880i	$0.395784\pi$
$954$	0	0
$955$	0	0
$956$	16240.0	0.549413
$957$	0	0
$958$	19680.0	0.663708
$959$	−53136.0	−1.78921
$960$	0	0
$961$	53153.0	1.78420
$962$	−748.000	−0.0250691
$963$	0	0
$964$	−14014.0	−0.468216
$965$	0	0
$966$	0	0
$967$	−39656.0	−1.31877	−0.659385	−	0.751805i	$-0.729183\pi$
$967$	−39656.0	−1.31877	−0.659385	+	0.751805i	$0.729183\pi$
$968$	−20595.0	−0.683831
$969$	0	0
$970$	0	0
$971$	33228.0	1.09818	0.549092	−	0.835762i	$-0.314974\pi$
$971$	33228.0	1.09818	0.549092	+	0.835762i	$0.314974\pi$
$972$	0	0
$973$	480.000	0.0158151
$974$	5944.00	0.195542
$975$	0	0
$976$	30422.0	0.997730
$977$	−974.000	−0.0318946	−0.0159473	−	0.999873i	$-0.505076\pi$
$977$	−974.000	−0.0318946	−0.0159473	+	0.999873i	$0.505076\pi$
$978$	0	0
$979$	17160.0	0.560200
$980$	0	0
$981$	0	0
$982$	−10772.0	−0.350049
$983$	−13608.0	−0.441534	−0.220767	−	0.975327i	$-0.570856\pi$
$983$	−13608.0	−0.441534	−0.220767	+	0.975327i	$0.570856\pi$
$984$	0	0
$985$	0	0
$986$	3220.00	0.104002
$987$	0	0
$988$	−3080.00	−0.0991780
$989$	−31584.0	−1.01548
$990$	0	0
$991$	13472.0	0.431839	0.215919	−	0.976411i	$-0.430725\pi$
$991$	13472.0	0.431839	0.215919	+	0.976411i	$0.430725\pi$
$992$	−46368.0	−1.48406
$993$	0	0
$994$	7872.00	0.251192
$995$	0	0
$996$	0	0
$997$	3234.00	0.102730	0.0513650	−	0.998680i	$-0.483643\pi$
$997$	3234.00	0.102730	0.0513650	+	0.998680i	$0.483643\pi$
$998$	8140.00	0.258184
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.a.f.1.1		1
3.2	odd	2		75.4.a.b.1.1		1
5.2	odd	4		225.4.b.e.199.2		2
5.3	odd	4		225.4.b.e.199.1		2
5.4	even	2		45.4.a.c.1.1		1
12.11	even	2		1200.4.a.t.1.1		1
15.2	even	4		75.4.b.b.49.1		2
15.8	even	4		75.4.b.b.49.2		2
15.14	odd	2		15.4.a.a.1.1	✓	1
20.19	odd	2		720.4.a.n.1.1		1
35.34	odd	2		2205.4.a.l.1.1		1
45.4	even	6		405.4.e.i.136.1		2
45.14	odd	6		405.4.e.g.136.1		2
45.29	odd	6		405.4.e.g.271.1		2
45.34	even	6		405.4.e.i.271.1		2
60.23	odd	4		1200.4.f.b.49.2		2
60.47	odd	4		1200.4.f.b.49.1		2
60.59	even	2		240.4.a.e.1.1		1
105.104	even	2		735.4.a.e.1.1		1
120.29	odd	2		960.4.a.b.1.1		1
120.59	even	2		960.4.a.ba.1.1		1
165.164	even	2		1815.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	15.14	odd	2
45.4.a.c.1.1		1	5.4	even	2
75.4.a.b.1.1		1	3.2	odd	2
75.4.b.b.49.1		2	15.2	even	4
75.4.b.b.49.2		2	15.8	even	4
225.4.a.f.1.1		1	1.1	even	1	trivial
225.4.b.e.199.1		2	5.3	odd	4
225.4.b.e.199.2		2	5.2	odd	4
240.4.a.e.1.1		1	60.59	even	2
405.4.e.g.136.1		2	45.14	odd	6
405.4.e.g.271.1		2	45.29	odd	6
405.4.e.i.136.1		2	45.4	even	6
405.4.e.i.271.1		2	45.34	even	6
720.4.a.n.1.1		1	20.19	odd	2
735.4.a.e.1.1		1	105.104	even	2
960.4.a.b.1.1		1	120.29	odd	2
960.4.a.ba.1.1		1	120.59	even	2
1200.4.a.t.1.1		1	12.11	even	2
1200.4.f.b.49.1		2	60.47	odd	4
1200.4.f.b.49.2		2	60.23	odd	4
1815.4.a.e.1.1		1	165.164	even	2
2205.4.a.l.1.1		1	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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