LMFDB - Embedded newform 735.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,4,Mod(1,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	735.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:	$43.3664038542$
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	$\chi$	$=$	735.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} -15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -5.00000 q^{5} -3.00000 q^{6} -15.0000 q^{8} +9.00000 q^{9} -5.00000 q^{10} +52.0000 q^{11} +21.0000 q^{12} -22.0000 q^{13} +15.0000 q^{15} +41.0000 q^{16} +14.0000 q^{17} +9.00000 q^{18} +20.0000 q^{19} +35.0000 q^{20} +52.0000 q^{22} -168.000 q^{23} +45.0000 q^{24} +25.0000 q^{25} -22.0000 q^{26} -27.0000 q^{27} +230.000 q^{29} +15.0000 q^{30} +288.000 q^{31} +161.000 q^{32} -156.000 q^{33} +14.0000 q^{34} -63.0000 q^{36} -34.0000 q^{37} +20.0000 q^{38} +66.0000 q^{39} +75.0000 q^{40} -122.000 q^{41} -188.000 q^{43} -364.000 q^{44} -45.0000 q^{45} -168.000 q^{46} -256.000 q^{47} -123.000 q^{48} +25.0000 q^{50} -42.0000 q^{51} +154.000 q^{52} -338.000 q^{53} -27.0000 q^{54} -260.000 q^{55} -60.0000 q^{57} +230.000 q^{58} -100.000 q^{59} -105.000 q^{60} -742.000 q^{61} +288.000 q^{62} -167.000 q^{64} +110.000 q^{65} -156.000 q^{66} -84.0000 q^{67} -98.0000 q^{68} +504.000 q^{69} -328.000 q^{71} -135.000 q^{72} +38.0000 q^{73} -34.0000 q^{74} -75.0000 q^{75} -140.000 q^{76} +66.0000 q^{78} -240.000 q^{79} -205.000 q^{80} +81.0000 q^{81} -122.000 q^{82} -1212.00 q^{83} -70.0000 q^{85} -188.000 q^{86} -690.000 q^{87} -780.000 q^{88} -330.000 q^{89} -45.0000 q^{90} +1176.00 q^{92} -864.000 q^{93} -256.000 q^{94} -100.000 q^{95} -483.000 q^{96} -866.000 q^{97} +468.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	1.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.00000	−0.875000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−3.00000	−0.204124
$7$	0	0
$7$	0	0
$8$	−15.0000	−0.662913
$9$	9.00000	0.333333
$10$	−5.00000	−0.158114
$11$	52.0000	1.42533	0.712663	−	0.701506i	$-0.247489\pi$
$11$	52.0000	1.42533	0.712663	+	0.701506i	$0.247489\pi$
$12$	21.0000	0.505181
$13$	−22.0000	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$13$	−22.0000	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	41.0000	0.640625
$17$	14.0000	0.199735	0.0998676	−	0.995001i	$-0.468158\pi$
$17$	14.0000	0.199735	0.0998676	+	0.995001i	$0.468158\pi$
$18$	9.00000	0.117851
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	35.0000	0.391312
$21$	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	$0.775552\pi$
$24$	45.0000	0.382733
$25$	25.0000	0.200000
$26$	−22.0000	−0.165944
$27$	−27.0000	−0.192450
$28$	0	0
$29$	230.000	1.47276	0.736378	−	0.676570i	$-0.236535\pi$
$29$	230.000	1.47276	0.736378	+	0.676570i	$0.236535\pi$
$30$	15.0000	0.0912871
$31$	288.000	1.66859	0.834296	−	0.551317i	$-0.185875\pi$
$31$	288.000	1.66859	0.834296	+	0.551317i	$0.185875\pi$
$32$	161.000	0.889408
$33$	−156.000	−0.822913
$34$	14.0000	0.0706171
$35$	0	0
$36$	−63.0000	−0.291667
$37$	−34.0000	−0.151069	−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000	−0.151069	−0.0755347	+	0.997143i	$0.524066\pi$
$38$	20.0000	0.0853797
$39$	66.0000	0.270986
$40$	75.0000	0.296464
$41$	−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	$-0.608185\pi$
$43$	−188.000	−0.666738	−0.333369	+	0.942796i	$0.608185\pi$
$44$	−364.000	−1.24716
$45$	−45.0000	−0.149071
$46$	−168.000	−0.538484
$47$	−256.000	−0.794499	−0.397249	−	0.917711i	$-0.630035\pi$
$47$	−256.000	−0.794499	−0.397249	+	0.917711i	$0.630035\pi$
$48$	−123.000	−0.369865
$49$	0	0
$50$	25.0000	0.0707107
$51$	−42.0000	−0.115317
$52$	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	$0.644313\pi$
$54$	−27.0000	−0.0680414
$55$	−260.000	−0.637425
$56$	0	0
$57$	−60.0000	−0.139424
$58$	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	$0.535191\pi$
$60$	−105.000	−0.225924
$61$	−742.000	−1.55743	−0.778716	−	0.627376i	$-0.784129\pi$
$61$	−742.000	−1.55743	−0.778716	+	0.627376i	$0.784129\pi$
$62$	288.000	0.589936
$63$	0	0
$64$	−167.000	−0.326172
$65$	110.000	0.209905
$66$	−156.000	−0.290944
$67$	−84.0000	−0.153168	−0.0765838	−	0.997063i	$-0.524401\pi$
$67$	−84.0000	−0.153168	−0.0765838	+	0.997063i	$0.524401\pi$
$68$	−98.0000	−0.174768
$69$	504.000	0.879340
$70$	0	0
$71$	−328.000	−0.548260	−0.274130	−	0.961693i	$-0.588390\pi$
$71$	−328.000	−0.548260	−0.274130	+	0.961693i	$0.588390\pi$
$72$	−135.000	−0.220971
$73$	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
$75$	−75.0000	−0.115470
$76$	−140.000	−0.211304
$77$	0	0
$78$	66.0000	0.0958081
$79$	−240.000	−0.341799	−0.170899	−	0.985288i	$-0.554667\pi$
$79$	−240.000	−0.341799	−0.170899	+	0.985288i	$0.554667\pi$
$80$	−205.000	−0.286496
$81$	81.0000	0.111111
$82$	−122.000	−0.164301
$83$	−1212.00	−1.60282	−0.801411	−	0.598114i	$-0.795917\pi$
$83$	−1212.00	−1.60282	−0.801411	+	0.598114i	$0.795917\pi$
$84$	0	0
$85$	−70.0000	−0.0893243
$86$	−188.000	−0.235727
$87$	−690.000	−0.850296
$88$	−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	$0.562963\pi$
$90$	−45.0000	−0.0527046
$91$	0	0
$92$	1176.00	1.33268
$93$	−864.000	−0.963362
$94$	−256.000	−0.280898
$95$	−100.000	−0.107998
$96$	−483.000	−0.513500
$97$	−866.000	−0.906484	−0.453242	−	0.891387i	$-0.649733\pi$
$97$	−866.000	−0.906484	−0.453242	+	0.891387i	$0.649733\pi$
$98$	0	0
$99$	468.000	0.475109
$100$	−175.000	−0.175000
$101$	1218.00	1.19996	0.599978	−	0.800017i	$-0.295176\pi$
$101$	1218.00	1.19996	0.599978	+	0.800017i	$0.295176\pi$
$102$	−42.0000	−0.0407708
$103$	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	$0.494823\pi$
$108$	189.000	0.168394
$109$	−970.000	−0.852378	−0.426189	−	0.904634i	$-0.640144\pi$
$109$	−970.000	−0.852378	−0.426189	+	0.904634i	$0.640144\pi$
$110$	−260.000	−0.225364
$111$	102.000	0.0872199
$112$	0	0
$113$	1042.00	0.867461	0.433731	−	0.901043i	$-0.357197\pi$
$113$	1042.00	0.867461	0.433731	+	0.901043i	$0.357197\pi$
$114$	−60.0000	−0.0492940
$115$	840.000	0.681134
$116$	−1610.00	−1.28866
$117$	−198.000	−0.156454
$118$	−100.000	−0.0780148
$119$	0	0
$120$	−225.000	−0.171163
$121$	1373.00	1.03156
$122$	−742.000	−0.550635
$123$	366.000	0.268302
$124$	−2016.00	−1.46002
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1936.00	1.35269	0.676347	−	0.736583i	$-0.263562\pi$
$127$	1936.00	1.35269	0.676347	+	0.736583i	$0.263562\pi$
$128$	−1455.00	−1.00473
$129$	564.000	0.384941
$130$	110.000	0.0742126
$131$	−732.000	−0.488207	−0.244104	−	0.969749i	$-0.578494\pi$
$131$	−732.000	−0.488207	−0.244104	+	0.969749i	$0.578494\pi$
$132$	1092.00	0.720048
$133$	0	0
$134$	−84.0000	−0.0541529
$135$	135.000	0.0860663
$136$	−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	$0.742542\pi$
$138$	504.000	0.310894
$139$	−20.0000	−0.0122042	−0.00610208	−	0.999981i	$-0.501942\pi$
$139$	−20.0000	−0.0122042	−0.00610208	+	0.999981i	$0.501942\pi$
$140$	0	0
$141$	768.000	0.458704
$142$	−328.000	−0.193839
$143$	−1144.00	−0.668994
$144$	369.000	0.213542
$145$	−1150.00	−0.658637
$146$	38.0000	0.0215404
$147$	0	0
$148$	238.000	0.132186
$149$	−1330.00	−0.731261	−0.365630	−	0.930760i	$-0.619147\pi$
$149$	−1330.00	−0.731261	−0.365630	+	0.930760i	$0.619147\pi$
$150$	−75.0000	−0.0408248
$151$	−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
$153$	126.000	0.0665784
$154$	0	0
$155$	−1440.00	−0.746217
$156$	−462.000	−0.237113
$157$	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
$159$	1014.00	0.505757
$160$	−805.000	−0.397755
$161$	0	0
$162$	81.0000	0.0392837
$163$	−2068.00	−0.993732	−0.496866	−	0.867827i	$-0.665516\pi$
$163$	−2068.00	−0.993732	−0.496866	+	0.867827i	$0.665516\pi$
$164$	854.000	0.406623
$165$	780.000	0.368018
$166$	−1212.00	−0.566683
$167$	24.0000	0.0111208	0.00556041	−	0.999985i	$-0.498230\pi$
$167$	24.0000	0.0111208	0.00556041	+	0.999985i	$0.498230\pi$
$168$	0	0
$169$	−1713.00	−0.779700
$170$	−70.0000	−0.0315809
$171$	180.000	0.0804967
$172$	1316.00	0.583396
$173$	618.000	0.271593	0.135797	−	0.990737i	$-0.456641\pi$
$173$	618.000	0.271593	0.135797	+	0.990737i	$0.456641\pi$
$174$	−690.000	−0.300625
$175$	0	0
$176$	2132.00	0.913100
$177$	300.000	0.127398
$178$	−330.000	−0.138958
$179$	3340.00	1.39466	0.697328	−	0.716752i	$-0.254372\pi$
$179$	3340.00	1.39466	0.697328	+	0.716752i	$0.254372\pi$
$180$	315.000	0.130437
$181$	178.000	0.0730974	0.0365487	−	0.999332i	$-0.488364\pi$
$181$	178.000	0.0730974	0.0365487	+	0.999332i	$0.488364\pi$
$182$	0	0
$183$	2226.00	0.899184
$184$	2520.00	1.00966
$185$	170.000	0.0675603
$186$	−864.000	−0.340600
$187$	728.000	0.284688
$188$	1792.00	0.695186
$189$	0	0
$190$	−100.000	−0.0381830
$191$	−1888.00	−0.715240	−0.357620	−	0.933867i	$-0.616412\pi$
$191$	−1888.00	−0.715240	−0.357620	+	0.933867i	$0.616412\pi$
$192$	501.000	0.188315
$193$	1922.00	0.716832	0.358416	−	0.933562i	$-0.383317\pi$
$193$	1922.00	0.716832	0.358416	+	0.933562i	$0.383317\pi$
$194$	−866.000	−0.320491
$195$	−330.000	−0.121189
$196$	0	0
$197$	2526.00	0.913554	0.456777	−	0.889581i	$-0.349004\pi$
$197$	2526.00	0.913554	0.456777	+	0.889581i	$0.349004\pi$
$198$	468.000	0.167976
$199$	1160.00	0.413217	0.206609	−	0.978424i	$-0.433757\pi$
$199$	1160.00	0.413217	0.206609	+	0.978424i	$0.433757\pi$
$200$	−375.000	−0.132583
$201$	252.000	0.0884314
$202$	1218.00	0.424248
$203$	0	0
$204$	294.000	0.100903
$205$	610.000	0.207826
$206$	88.0000	0.0297634
$207$	−1512.00	−0.507687
$208$	−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
$213$	984.000	0.316538
$214$	36.0000	0.0114996
$215$	940.000	0.298174
$216$	405.000	0.127578
$217$	0	0
$218$	−970.000	−0.301361
$219$	−114.000	−0.0351754
$220$	1820.00	0.557747
$221$	−308.000	−0.0937481
$222$	102.000	0.0308369
$223$	−6032.00	−1.81136	−0.905678	−	0.423965i	$-0.860638\pi$
$223$	−6032.00	−1.81136	−0.905678	+	0.423965i	$0.860638\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	1042.00	0.306694
$227$	−2636.00	−0.770738	−0.385369	−	0.922763i	$-0.625926\pi$
$227$	−2636.00	−0.770738	−0.385369	+	0.922763i	$0.625926\pi$
$228$	420.000	0.121996
$229$	−4830.00	−1.39378	−0.696889	−	0.717179i	$-0.745433\pi$
$229$	−4830.00	−1.39378	−0.696889	+	0.717179i	$0.745433\pi$
$230$	840.000	0.240817
$231$	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	$0.376940\pi$
$234$	−198.000	−0.0553148
$235$	1280.00	0.355311
$236$	700.000	0.193077
$237$	720.000	0.197338
$238$	0	0
$239$	2320.00	0.627901	0.313950	−	0.949439i	$-0.398347\pi$
$239$	2320.00	0.627901	0.313950	+	0.949439i	$0.398347\pi$
$240$	615.000	0.165409
$241$	−2002.00	−0.535104	−0.267552	−	0.963543i	$-0.586215\pi$
$241$	−2002.00	−0.535104	−0.267552	+	0.963543i	$0.586215\pi$
$242$	1373.00	0.364710
$243$	−243.000	−0.0641500
$244$	5194.00	1.36275
$245$	0	0
$246$	366.000	0.0948590
$247$	−440.000	−0.113346
$248$	−4320.00	−1.10613
$249$	3636.00	0.925390
$250$	−125.000	−0.0316228
$251$	−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
$255$	210.000	0.0515714
$256$	−119.000	−0.0290527
$257$	7614.00	1.84805	0.924024	−	0.382335i	$-0.124880\pi$
$257$	7614.00	1.84805	0.924024	+	0.382335i	$0.124880\pi$
$258$	564.000	0.136097
$259$	0	0
$260$	−770.000	−0.183667
$261$	2070.00	0.490919
$262$	−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	$0.694227\pi$
$264$	2340.00	0.545519
$265$	1690.00	0.391758
$266$	0	0
$267$	990.000	0.226918
$268$	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	$0.545973\pi$
$270$	135.000	0.0304290
$271$	−1072.00	−0.240293	−0.120146	−	0.992756i	$-0.538336\pi$
$271$	−1072.00	−0.240293	−0.120146	+	0.992756i	$0.538336\pi$
$272$	574.000	0.127955
$273$	0	0
$274$	−2214.00	−0.488148
$275$	1300.00	0.285065
$276$	−3528.00	−0.769423
$277$	−5394.00	−1.17001	−0.585007	−	0.811028i	$-0.698908\pi$
$277$	−5394.00	−1.17001	−0.585007	+	0.811028i	$0.698908\pi$
$278$	−20.0000	−0.00431482
$279$	2592.00	0.556197
$280$	0	0
$281$	2442.00	0.518425	0.259213	−	0.965820i	$-0.416537\pi$
$281$	2442.00	0.518425	0.259213	+	0.965820i	$0.416537\pi$
$282$	768.000	0.162176
$283$	−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
$285$	300.000	0.0623525
$286$	−1144.00	−0.236525
$287$	0	0
$288$	1449.00	0.296469
$289$	−4717.00	−0.960106
$290$	−1150.00	−0.232863
$291$	2598.00	0.523359
$292$	−266.000	−0.0533098
$293$	−4542.00	−0.905619	−0.452810	−	0.891607i	$-0.649578\pi$
$293$	−4542.00	−0.905619	−0.452810	+	0.891607i	$0.649578\pi$
$294$	0	0
$295$	500.000	0.0986818
$296$	510.000	0.100146
$297$	−1404.00	−0.274304
$298$	−1330.00	−0.258540
$299$	3696.00	0.714867
$300$	525.000	0.101036
$301$	0	0
$302$	−1208.00	−0.230174
$303$	−3654.00	−0.692795
$304$	820.000	0.154705
$305$	3710.00	0.696505
$306$	126.000	0.0235390
$307$	−5116.00	−0.951093	−0.475546	−	0.879691i	$-0.657750\pi$
$307$	−5116.00	−0.951093	−0.475546	+	0.879691i	$0.657750\pi$
$308$	0	0
$309$	−264.000	−0.0486034
$310$	−1440.00	−0.263827
$311$	2808.00	0.511984	0.255992	−	0.966679i	$-0.417598\pi$
$311$	2808.00	0.511984	0.255992	+	0.966679i	$0.417598\pi$
$312$	−990.000	−0.179640
$313$	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	$0.436240\pi$
$318$	1014.00	0.178812
$319$	11960.0	2.09916
$320$	835.000	0.145868
$321$	−108.000	−0.0187787
$322$	0	0
$323$	280.000	0.0482341
$324$	−567.000	−0.0972222
$325$	−550.000	−0.0938723
$326$	−2068.00	−0.351337
$327$	2910.00	0.492120
$328$	1830.00	0.308064
$329$	0	0
$330$	780.000	0.130114
$331$	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
$333$	−306.000	−0.0503564
$334$	24.0000	0.00393180
$335$	420.000	0.0684987
$336$	0	0
$337$	−11534.0	−1.86438	−0.932191	−	0.361966i	$-0.882106\pi$
$337$	−11534.0	−1.86438	−0.932191	+	0.361966i	$0.882106\pi$
$338$	−1713.00	−0.275665
$339$	−3126.00	−0.500829
$340$	490.000	0.0781588
$341$	14976.0	2.37829
$342$	180.000	0.0284599
$343$	0	0
$344$	2820.00	0.441989
$345$	−2520.00	−0.393253
$346$	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	$0.124207\pi$
$348$	4830.00	0.744009
$349$	−4870.00	−0.746949	−0.373474	−	0.927640i	$-0.621834\pi$
$349$	−4870.00	−0.746949	−0.373474	+	0.927640i	$0.621834\pi$
$350$	0	0
$351$	594.000	0.0903287
$352$	8372.00	1.26770
$353$	−10722.0	−1.61664	−0.808321	−	0.588742i	$-0.799623\pi$
$353$	−10722.0	−1.61664	−0.808321	+	0.588742i	$0.799623\pi$
$354$	300.000	0.0450419
$355$	1640.00	0.245189
$356$	2310.00	0.343904
$357$	0	0
$358$	3340.00	0.493085
$359$	120.000	0.0176417	0.00882083	−	0.999961i	$-0.497192\pi$
$359$	120.000	0.0176417	0.00882083	+	0.999961i	$0.497192\pi$
$360$	675.000	0.0988212
$361$	−6459.00	−0.941682
$362$	178.000	0.0258438
$363$	−4119.00	−0.595569
$364$	0	0
$365$	−190.000	−0.0272467
$366$	2226.00	0.317910
$367$	−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
$369$	−1098.00	−0.154904
$370$	170.000	0.0238862
$371$	0	0
$372$	6048.00	0.842941
$373$	3022.00	0.419499	0.209750	−	0.977755i	$-0.432735\pi$
$373$	3022.00	0.419499	0.209750	+	0.977755i	$0.432735\pi$
$374$	728.000	0.100652
$375$	375.000	0.0516398
$376$	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	$0.859381\pi$
$380$	700.000	0.0944980
$381$	−5808.00	−0.780979
$382$	−1888.00	−0.252876
$383$	1008.00	0.134481	0.0672407	−	0.997737i	$-0.478580\pi$
$383$	1008.00	0.134481	0.0672407	+	0.997737i	$0.478580\pi$
$384$	4365.00	0.580079
$385$	0	0
$386$	1922.00	0.253438
$387$	−1692.00	−0.222246
$388$	6062.00	0.793174
$389$	9630.00	1.25517	0.627584	−	0.778549i	$-0.284044\pi$
$389$	9630.00	1.25517	0.627584	+	0.778549i	$0.284044\pi$
$390$	−330.000	−0.0428467
$391$	−2352.00	−0.304209
$392$	0	0
$393$	2196.00	0.281867
$394$	2526.00	0.322990
$395$	1200.00	0.152857
$396$	−3276.00	−0.415720
$397$	−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
$400$	1025.00	0.128125
$401$	−8718.00	−1.08568	−0.542838	−	0.839837i	$-0.682650\pi$
$401$	−8718.00	−1.08568	−0.542838	+	0.839837i	$0.682650\pi$
$402$	252.000	0.0312652
$403$	−6336.00	−0.783173
$404$	−8526.00	−1.04996
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−1768.00	−0.215323
$408$	630.000	0.0764452
$409$	10870.0	1.31415	0.657074	−	0.753826i	$-0.271794\pi$
$409$	10870.0	1.31415	0.657074	+	0.753826i	$0.271794\pi$
$410$	610.000	0.0734774
$411$	6642.00	0.797143
$412$	−616.000	−0.0736605
$413$	0	0
$414$	−1512.00	−0.179495
$415$	6060.00	0.716804
$416$	−3542.00	−0.417454
$417$	60.0000	0.00704607
$418$	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
$423$	−2304.00	−0.264833
$424$	5070.00	0.580710
$425$	350.000	0.0399470
$426$	984.000	0.111913
$427$	0	0
$428$	−252.000	−0.0284600
$429$	3432.00	0.386244
$430$	940.000	0.105421
$431$	15792.0	1.76490	0.882452	−	0.470402i	$-0.155891\pi$
$431$	15792.0	1.76490	0.882452	+	0.470402i	$0.155891\pi$
$432$	−1107.00	−0.123288
$433$	−11602.0	−1.28766	−0.643830	−	0.765169i	$-0.722655\pi$
$433$	−11602.0	−1.28766	−0.643830	+	0.765169i	$0.722655\pi$
$434$	0	0
$435$	3450.00	0.380264
$436$	6790.00	0.745830
$437$	−3360.00	−0.367805
$438$	−114.000	−0.0124364
$439$	440.000	0.0478361	0.0239181	−	0.999714i	$-0.492386\pi$
$439$	440.000	0.0478361	0.0239181	+	0.999714i	$0.492386\pi$
$440$	3900.00	0.422557
$441$	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	$0.683975\pi$
$444$	−714.000	−0.0763174
$445$	1650.00	0.175770
$446$	−6032.00	−0.640411
$447$	3990.00	0.422194
$448$	0	0
$449$	−13310.0	−1.39897	−0.699485	−	0.714647i	$-0.746587\pi$
$449$	−13310.0	−1.39897	−0.699485	+	0.714647i	$0.746587\pi$
$450$	225.000	0.0235702
$451$	−6344.00	−0.662367
$452$	−7294.00	−0.759029
$453$	3624.00	0.375873
$454$	−2636.00	−0.272497
$455$	0	0
$456$	900.000	0.0924262
$457$	3226.00	0.330210	0.165105	−	0.986276i	$-0.447204\pi$
$457$	3226.00	0.330210	0.165105	+	0.986276i	$0.447204\pi$
$458$	−4830.00	−0.492775
$459$	−378.000	−0.0384391
$460$	−5880.00	−0.595992
$461$	−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	$-0.226941\pi$
$463$	15072.0	1.51286	0.756431	+	0.654073i	$0.226941\pi$
$464$	9430.00	0.943484
$465$	4320.00	0.430828
$466$	2682.00	0.266612
$467$	−476.000	−0.0471663	−0.0235831	−	0.999722i	$-0.507507\pi$
$467$	−476.000	−0.0471663	−0.0235831	+	0.999722i	$0.507507\pi$
$468$	1386.00	0.136897
$469$	0	0
$470$	1280.00	0.125621
$471$	−10542.0	−1.03132
$472$	1500.00	0.146278
$473$	−9776.00	−0.950319
$474$	720.000	0.0697694
$475$	500.000	0.0482980
$476$	0	0
$477$	−3042.00	−0.291999
$478$	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	$0.112101\pi$
$480$	2415.00	0.229644
$481$	748.000	0.0709062
$482$	−2002.00	−0.189188
$483$	0	0
$484$	−9611.00	−0.902611
$485$	4330.00	0.405392
$486$	−243.000	−0.0226805
$487$	−5944.00	−0.553077	−0.276538	−	0.961003i	$-0.589187\pi$
$487$	−5944.00	−0.553077	−0.276538	+	0.961003i	$0.589187\pi$
$488$	11130.0	1.03244
$489$	6204.00	0.573731
$490$	0	0
$491$	10772.0	0.990089	0.495044	−	0.868868i	$-0.335152\pi$
$491$	10772.0	0.990089	0.495044	+	0.868868i	$0.335152\pi$
$492$	−2562.00	−0.234764
$493$	3220.00	0.294161
$494$	−440.000	−0.0400740
$495$	−2340.00	−0.212475
$496$	11808.0	1.06894
$497$	0	0
$498$	3636.00	0.327175
$499$	8140.00	0.730253	0.365127	−	0.930958i	$-0.381026\pi$
$499$	8140.00	0.730253	0.365127	+	0.930958i	$0.381026\pi$
$500$	875.000	0.0782624
$501$	−72.0000	−0.00642060
$502$	−132.000	−0.0117360
$503$	13768.0	1.22045	0.610223	−	0.792229i	$-0.291080\pi$
$503$	13768.0	1.22045	0.610223	+	0.792229i	$0.291080\pi$
$504$	0	0
$505$	−6090.00	−0.536637
$506$	−8736.00	−0.767515
$507$	5139.00	0.450160
$508$	−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	$0.914837\pi$
$510$	210.000	0.0182332
$511$	0	0
$512$	11521.0	0.994455
$513$	−540.000	−0.0464748
$514$	7614.00	0.653384
$515$	−440.000	−0.0376480
$516$	−3948.00	−0.336824
$517$	−13312.0	−1.13242
$518$	0	0
$519$	−1854.00	−0.156805
$520$	−1650.00	−0.139149
$521$	−1562.00	−0.131348	−0.0656741	−	0.997841i	$-0.520920\pi$
$521$	−1562.00	−0.131348	−0.0656741	+	0.997841i	$0.520920\pi$
$522$	2070.00	0.173566
$523$	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
$528$	−6396.00	−0.527178
$529$	16057.0	1.31972
$530$	1690.00	0.138507
$531$	−900.000	−0.0735531
$532$	0	0
$533$	2684.00	0.218118
$534$	990.000	0.0802275
$535$	−180.000	−0.0145459
$536$	1260.00	0.101537
$537$	−10020.0	−0.805205
$538$	−1270.00	−0.101772
$539$	0	0
$540$	−945.000	−0.0753080
$541$	−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
$543$	−534.000	−0.0422028
$544$	2254.00	0.177646
$545$	4850.00	0.381195
$546$	0	0
$547$	−10484.0	−0.819494	−0.409747	−	0.912199i	$-0.634383\pi$
$547$	−10484.0	−0.819494	−0.409747	+	0.912199i	$0.634383\pi$
$548$	15498.0	1.20811
$549$	−6678.00	−0.519144
$550$	1300.00	0.100786
$551$	4600.00	0.355656
$552$	−7560.00	−0.582926
$553$	0	0
$554$	−5394.00	−0.413663
$555$	−510.000	−0.0390059
$556$	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	$0.456204\pi$
$558$	2592.00	0.196645
$559$	4136.00	0.312941
$560$	0	0
$561$	−2184.00	−0.164365
$562$	2442.00	0.183291
$563$	−12252.0	−0.917159	−0.458579	−	0.888654i	$-0.651641\pi$
$563$	−12252.0	−0.917159	−0.458579	+	0.888654i	$0.651641\pi$
$564$	−5376.00	−0.401366
$565$	−5210.00	−0.387940
$566$	−2772.00	−0.205858
$567$	0	0
$568$	4920.00	0.363448
$569$	−14550.0	−1.07200	−0.536000	−	0.844218i	$-0.680065\pi$
$569$	−14550.0	−1.07200	−0.536000	+	0.844218i	$0.680065\pi$
$570$	300.000	0.0220449
$571$	−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
$573$	5664.00	0.412944
$574$	0	0
$575$	−4200.00	−0.304612
$576$	−1503.00	−0.108724
$577$	−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
$579$	−5766.00	−0.413863
$580$	8050.00	0.576307
$581$	0	0
$582$	2598.00	0.185035
$583$	−17576.0	−1.24858
$584$	−570.000	−0.0403883
$585$	990.000	0.0699683
$586$	−4542.00	−0.320185
$587$	14844.0	1.04374	0.521872	−	0.853024i	$-0.325234\pi$
$587$	14844.0	1.04374	0.521872	+	0.853024i	$0.325234\pi$
$588$	0	0
$589$	5760.00	0.402948
$590$	500.000	0.0348893
$591$	−7578.00	−0.527440
$592$	−1394.00	−0.0967788
$593$	−20402.0	−1.41283	−0.706416	−	0.707797i	$-0.749689\pi$
$593$	−20402.0	−1.41283	−0.706416	+	0.707797i	$0.749689\pi$
$594$	−1404.00	−0.0969812
$595$	0	0
$596$	9310.00	0.639853
$597$	−3480.00	−0.238571
$598$	3696.00	0.252744
$599$	10760.0	0.733959	0.366980	−	0.930229i	$-0.380392\pi$
$599$	10760.0	0.733959	0.366980	+	0.930229i	$0.380392\pi$
$600$	1125.00	0.0765466
$601$	−14282.0	−0.969343	−0.484671	−	0.874696i	$-0.661061\pi$
$601$	−14282.0	−0.969343	−0.484671	+	0.874696i	$0.661061\pi$
$602$	0	0
$603$	−756.000	−0.0510559
$604$	8456.00	0.569652
$605$	−6865.00	−0.461326
$606$	−3654.00	−0.244940
$607$	−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
$610$	3710.00	0.246252
$611$	5632.00	0.372907
$612$	−882.000	−0.0582561
$613$	−16418.0	−1.08176	−0.540878	−	0.841101i	$-0.681908\pi$
$613$	−16418.0	−1.08176	−0.540878	+	0.841101i	$0.681908\pi$
$614$	−5116.00	−0.336262
$615$	−1830.00	−0.119988
$616$	0	0
$617$	−10374.0	−0.676891	−0.338445	−	0.940986i	$-0.609901\pi$
$617$	−10374.0	−0.676891	−0.338445	+	0.940986i	$0.609901\pi$
$618$	−264.000	−0.0171839
$619$	5260.00	0.341546	0.170773	−	0.985310i	$-0.445373\pi$
$619$	5260.00	0.341546	0.170773	+	0.985310i	$0.445373\pi$
$620$	10080.0	0.652940
$621$	4536.00	0.293113
$622$	2808.00	0.181014
$623$	0	0
$624$	2706.00	0.173600
$625$	625.000	0.0400000
$626$	7318.00	0.467230
$627$	−3120.00	−0.198725
$628$	−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
$633$	13404.0	0.841645
$634$	2246.00	0.140694
$635$	−9680.00	−0.604943
$636$	−7098.00	−0.442538
$637$	0	0
$638$	11960.0	0.742164
$639$	−2952.00	−0.182753
$640$	7275.00	0.449328
$641$	−29118.0	−1.79422	−0.897108	−	0.441812i	$-0.854336\pi$
$641$	−29118.0	−1.79422	−0.897108	+	0.441812i	$0.854336\pi$
$642$	−108.000	−0.00663928
$643$	−5772.00	−0.354005	−0.177003	−	0.984210i	$-0.556640\pi$
$643$	−5772.00	−0.354005	−0.177003	+	0.984210i	$0.556640\pi$
$644$	0	0
$645$	−2820.00	−0.172151
$646$	280.000	0.0170533
$647$	14264.0	0.866732	0.433366	−	0.901218i	$-0.357326\pi$
$647$	14264.0	0.866732	0.433366	+	0.901218i	$0.357326\pi$
$648$	−1215.00	−0.0736570
$649$	−5200.00	−0.314511
$650$	−550.000	−0.0331889
$651$	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	$0.433691\pi$
$654$	2910.00	0.173991
$655$	3660.00	0.218333
$656$	−5002.00	−0.297706
$657$	342.000	0.0203085
$658$	0	0
$659$	20140.0	1.19051	0.595253	−	0.803539i	$-0.297052\pi$
$659$	20140.0	1.19051	0.595253	+	0.803539i	$0.297052\pi$
$660$	−5460.00	−0.322015
$661$	3218.00	0.189358	0.0946790	−	0.995508i	$-0.469818\pi$
$661$	3218.00	0.189358	0.0946790	+	0.995508i	$0.469818\pi$
$662$	1332.00	0.0782019
$663$	924.000	0.0541255
$664$	18180.0	1.06253
$665$	0	0
$666$	−306.000	−0.0178037
$667$	−38640.0	−2.24310
$668$	−168.000	−0.00973071
$669$	18096.0	1.04579
$670$	420.000	0.0242179
$671$	−38584.0	−2.21985
$672$	0	0
$673$	−7518.00	−0.430606	−0.215303	−	0.976547i	$-0.569074\pi$
$673$	−7518.00	−0.430606	−0.215303	+	0.976547i	$0.569074\pi$
$674$	−11534.0	−0.659159
$675$	−675.000	−0.0384900
$676$	11991.0	0.682237
$677$	18114.0	1.02833	0.514164	−	0.857692i	$-0.328102\pi$
$677$	18114.0	1.02833	0.514164	+	0.857692i	$0.328102\pi$
$678$	−3126.00	−0.177070
$679$	0	0
$680$	1050.00	0.0592142
$681$	7908.00	0.444986
$682$	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	$0.733099\pi$
$684$	−1260.00	−0.0704347
$685$	11070.0	0.617464
$686$	0	0
$687$	14490.0	0.804699
$688$	−7708.00	−0.427129
$689$	7436.00	0.411160
$690$	−2520.00	−0.139036
$691$	−172.000	−0.00946916	−0.00473458	−	0.999989i	$-0.501507\pi$
$691$	−172.000	−0.00946916	−0.00473458	+	0.999989i	$0.501507\pi$
$692$	−4326.00	−0.237644
$693$	0	0
$694$	11956.0	0.653953
$695$	100.000	0.00545787
$696$	10350.0	0.563672
$697$	−1708.00	−0.0928194
$698$	−4870.00	−0.264086
$699$	−8046.00	−0.435376
$700$	0	0
$701$	−22138.0	−1.19278	−0.596391	−	0.802694i	$-0.703399\pi$
$701$	−22138.0	−1.19278	−0.596391	+	0.802694i	$0.703399\pi$
$702$	594.000	0.0319360
$703$	−680.000	−0.0364818
$704$	−8684.00	−0.464901
$705$	−3840.00	−0.205139
$706$	−10722.0	−0.571569
$707$	0	0
$708$	−2100.00	−0.111473
$709$	3070.00	0.162618	0.0813091	−	0.996689i	$-0.474090\pi$
$709$	3070.00	0.162618	0.0813091	+	0.996689i	$0.474090\pi$
$710$	1640.00	0.0866875
$711$	−2160.00	−0.113933
$712$	4950.00	0.260546
$713$	−48384.0	−2.54137
$714$	0	0
$715$	5720.00	0.299183
$716$	−23380.0	−1.22032
$717$	−6960.00	−0.362519
$718$	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	$0.632581\pi$
$720$	−1845.00	−0.0954987
$721$	0	0
$722$	−6459.00	−0.332935
$723$	6006.00	0.308943
$724$	−1246.00	−0.0639603
$725$	5750.00	0.294551
$726$	−4119.00	−0.210565
$727$	−20696.0	−1.05581	−0.527904	−	0.849304i	$-0.677022\pi$
$727$	−20696.0	−1.05581	−0.527904	+	0.849304i	$0.677022\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−190.000	−0.00963317
$731$	−2632.00	−0.133171
$732$	−15582.0	−0.786786
$733$	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
$738$	−1098.00	−0.0547669
$739$	11740.0	0.584388	0.292194	−	0.956359i	$-0.405615\pi$
$739$	11740.0	0.584388	0.292194	+	0.956359i	$0.405615\pi$
$740$	−1190.00	−0.0591152
$741$	1320.00	0.0654405
$742$	0	0
$743$	2632.00	0.129958	0.0649789	−	0.997887i	$-0.479302\pi$
$743$	2632.00	0.129958	0.0649789	+	0.997887i	$0.479302\pi$
$744$	12960.0	0.638625
$745$	6650.00	0.327030
$746$	3022.00	0.148315
$747$	−10908.0	−0.534274
$748$	−5096.00	−0.249102
$749$	0	0
$750$	375.000	0.0182574
$751$	−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
$753$	396.000	0.0191647
$754$	−5060.00	−0.244396
$755$	6040.00	0.291150
$756$	0	0
$757$	21646.0	1.03928	0.519642	−	0.854384i	$-0.326066\pi$
$757$	21646.0	1.03928	0.519642	+	0.854384i	$0.326066\pi$
$758$	−13340.0	−0.639222
$759$	26208.0	1.25335
$760$	1500.00	0.0715931
$761$	−18282.0	−0.870857	−0.435428	−	0.900223i	$-0.643403\pi$
$761$	−18282.0	−0.870857	−0.435428	+	0.900223i	$0.643403\pi$
$762$	−5808.00	−0.276118
$763$	0	0
$764$	13216.0	0.625835
$765$	−630.000	−0.0297748
$766$	1008.00	0.0475464
$767$	2200.00	0.103569
$768$	357.000	0.0167736
$769$	24190.0	1.13435	0.567174	−	0.823598i	$-0.308037\pi$
$769$	24190.0	1.13435	0.567174	+	0.823598i	$0.308037\pi$
$770$	0	0
$771$	−22842.0	−1.06697
$772$	−13454.0	−0.627228
$773$	25698.0	1.19572	0.597861	−	0.801600i	$-0.296018\pi$
$773$	25698.0	1.19572	0.597861	+	0.801600i	$0.296018\pi$
$774$	−1692.00	−0.0785758
$775$	7200.00	0.333718
$776$	12990.0	0.600920
$777$	0	0
$778$	9630.00	0.443769
$779$	−2440.00	−0.112223
$780$	2310.00	0.106040
$781$	−17056.0	−0.781449
$782$	−2352.00	−0.107554
$783$	−6210.00	−0.283432
$784$	0	0
$785$	−17570.0	−0.798854
$786$	2196.00	0.0996549
$787$	−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
$789$	14664.0	0.661663
$790$	1200.00	0.0540431
$791$	0	0
$792$	−7020.00	−0.314956
$793$	16324.0	0.730999
$794$	−7126.00	−0.318504
$795$	−5070.00	−0.226182
$796$	−8120.00	−0.361565
$797$	37594.0	1.67083	0.835413	−	0.549623i	$-0.185229\pi$
$797$	37594.0	1.67083	0.835413	+	0.549623i	$0.185229\pi$
$798$	0	0
$799$	−3584.00	−0.158689
$800$	4025.00	0.177882
$801$	−2970.00	−0.131011
$802$	−8718.00	−0.383844
$803$	1976.00	0.0868388
$804$	−1764.00	−0.0773775
$805$	0	0
$806$	−6336.00	−0.276893
$807$	3810.00	0.166194
$808$	−18270.0	−0.795466
$809$	4730.00	0.205560	0.102780	−	0.994704i	$-0.467226\pi$
$809$	4730.00	0.205560	0.102780	+	0.994704i	$0.467226\pi$
$810$	−405.000	−0.0175682
$811$	8748.00	0.378772	0.189386	−	0.981903i	$-0.439350\pi$
$811$	8748.00	0.378772	0.189386	+	0.981903i	$0.439350\pi$
$812$	0	0
$813$	3216.00	0.138733
$814$	−1768.00	−0.0761282
$815$	10340.0	0.444410
$816$	−1722.00	−0.0738751
$817$	−3760.00	−0.161011
$818$	10870.0	0.464622
$819$	0	0
$820$	−4270.00	−0.181847
$821$	44142.0	1.87645	0.938226	−	0.346024i	$-0.112468\pi$
$821$	44142.0	1.87645	0.938226	+	0.346024i	$0.112468\pi$
$822$	6642.00	0.281833
$823$	3992.00	0.169079	0.0845397	−	0.996420i	$-0.473058\pi$
$823$	3992.00	0.169079	0.0845397	+	0.996420i	$0.473058\pi$
$824$	−1320.00	−0.0558063
$825$	−3900.00	−0.164583
$826$	0	0
$827$	−14444.0	−0.607336	−0.303668	−	0.952778i	$-0.598211\pi$
$827$	−14444.0	−0.607336	−0.303668	+	0.952778i	$0.598211\pi$
$828$	10584.0	0.444226
$829$	−42150.0	−1.76590	−0.882949	−	0.469468i	$-0.844446\pi$
$829$	−42150.0	−1.76590	−0.882949	+	0.469468i	$0.844446\pi$
$830$	6060.00	0.253429
$831$	16182.0	0.675508
$832$	3674.00	0.153093
$833$	0	0
$834$	60.0000	0.00249116
$835$	−120.000	−0.00497338
$836$	−7280.00	−0.301177
$837$	−7776.00	−0.321121
$838$	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
$843$	−7326.00	−0.299313
$844$	31276.0	1.27555
$845$	8565.00	0.348692
$846$	−2304.00	−0.0936326
$847$	0	0
$848$	−13858.0	−0.561186
$849$	8316.00	0.336165
$850$	350.000	0.0141234
$851$	5712.00	0.230088
$852$	−6888.00	−0.276971
$853$	8658.00	0.347531	0.173766	−	0.984787i	$-0.444406\pi$
$853$	8658.00	0.347531	0.173766	+	0.984787i	$0.444406\pi$
$854$	0	0
$855$	−900.000	−0.0359992
$856$	−540.000	−0.0215617
$857$	−42826.0	−1.70701	−0.853505	−	0.521084i	$-0.825528\pi$
$857$	−42826.0	−1.70701	−0.853505	+	0.521084i	$0.825528\pi$
$858$	3432.00	0.136558
$859$	35900.0	1.42595	0.712976	−	0.701189i	$-0.247347\pi$
$859$	35900.0	1.42595	0.712976	+	0.701189i	$0.247347\pi$
$860$	−6580.00	−0.260902
$861$	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	$0.519398\pi$
$864$	−4347.00	−0.171167
$865$	−3090.00	−0.121460
$866$	−11602.0	−0.455256
$867$	14151.0	0.554317
$868$	0	0
$869$	−12480.0	−0.487175
$870$	3450.00	0.134444
$871$	1848.00	0.0718910
$872$	14550.0	0.565052
$873$	−7794.00	−0.302161
$874$	−3360.00	−0.130039
$875$	0	0
$876$	798.000	0.0307784
$877$	−35274.0	−1.35817	−0.679087	−	0.734058i	$-0.737624\pi$
$877$	−35274.0	−1.35817	−0.679087	+	0.734058i	$0.737624\pi$
$878$	440.000	0.0169126
$879$	13626.0	0.522860
$880$	−10660.0	−0.408351
$881$	−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	$-0.422993\pi$
$883$	12572.0	0.479141	0.239570	+	0.970879i	$0.422993\pi$
$884$	2156.00	0.0820296
$885$	−1500.00	−0.0569740
$886$	−10188.0	−0.386312
$887$	21864.0	0.827645	0.413823	−	0.910358i	$-0.364193\pi$
$887$	21864.0	0.827645	0.413823	+	0.910358i	$0.364193\pi$
$888$	−1530.00	−0.0578192
$889$	0	0
$890$	1650.00	0.0621440
$891$	4212.00	0.158370
$892$	42224.0	1.58494
$893$	−5120.00	−0.191864
$894$	3990.00	0.149268
$895$	−16700.0	−0.623709
$896$	0	0
$897$	−11088.0	−0.412729
$898$	−13310.0	−0.494611
$899$	66240.0	2.45743
$900$	−1575.00	−0.0583333
$901$	−4732.00	−0.174968
$902$	−6344.00	−0.234182
$903$	0	0
$904$	−15630.0	−0.575051
$905$	−890.000	−0.0326902
$906$	3624.00	0.132891
$907$	31236.0	1.14352	0.571761	−	0.820420i	$-0.306260\pi$
$907$	31236.0	1.14352	0.571761	+	0.820420i	$0.306260\pi$
$908$	18452.0	0.674396
$909$	10962.0	0.399985
$910$	0	0
$911$	8272.00	0.300838	0.150419	−	0.988622i	$-0.451938\pi$
$911$	8272.00	0.300838	0.150419	+	0.988622i	$0.451938\pi$
$912$	−2460.00	−0.0893188
$913$	−63024.0	−2.28455
$914$	3226.00	0.116747
$915$	−11130.0	−0.402127
$916$	33810.0	1.21956
$917$	0	0
$918$	−378.000	−0.0135903
$919$	20200.0	0.725067	0.362533	−	0.931971i	$-0.381912\pi$
$919$	20200.0	0.725067	0.362533	+	0.931971i	$0.381912\pi$
$920$	−12600.0	−0.451532
$921$	15348.0	0.549114
$922$	−6582.00	−0.235105
$923$	7216.00	0.257332
$924$	0	0
$925$	−850.000	−0.0302139
$926$	15072.0	0.534878
$927$	792.000	0.0280612
$928$	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	$0.684451\pi$
$930$	4320.00	0.152321
$931$	0	0
$932$	−18774.0	−0.659831
$933$	−8424.00	−0.295594
$934$	−476.000	−0.0166758
$935$	−3640.00	−0.127316
$936$	2970.00	0.103715
$937$	39174.0	1.36580	0.682902	−	0.730510i	$-0.260717\pi$
$937$	39174.0	1.36580	0.682902	+	0.730510i	$0.260717\pi$
$938$	0	0
$939$	−21954.0	−0.762984
$940$	−8960.00	−0.310897
$941$	4138.00	0.143353	0.0716764	−	0.997428i	$-0.477165\pi$
$941$	4138.00	0.143353	0.0716764	+	0.997428i	$0.477165\pi$
$942$	−10542.0	−0.364625
$943$	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	$0.366849\pi$
$948$	−5040.00	−0.172670
$949$	−836.000	−0.0285961
$950$	500.000	0.0170759
$951$	−6738.00	−0.229752
$952$	0	0
$953$	18922.0	0.643173	0.321586	−	0.946880i	$-0.395784\pi$
$953$	18922.0	0.643173	0.321586	+	0.946880i	$0.395784\pi$
$954$	−3042.00	−0.103237
$955$	9440.00	0.319865
$956$	−16240.0	−0.549413
$957$	−35880.0	−1.21195
$958$	19680.0	0.663708
$959$	0	0
$960$	−2505.00	−0.0842172
$961$	53153.0	1.78420
$962$	748.000	0.0250691
$963$	324.000	0.0108419
$964$	14014.0	0.468216
$965$	−9610.00	−0.320577
$966$	0	0
$967$	39656.0	1.31877	0.659385	−	0.751805i	$-0.270817\pi$
$967$	39656.0	1.31877	0.659385	+	0.751805i	$0.270817\pi$
$968$	−20595.0	−0.683831
$969$	−840.000	−0.0278480
$970$	4330.00	0.143328
$971$	33228.0	1.09818	0.549092	−	0.835762i	$-0.314974\pi$
$971$	33228.0	1.09818	0.549092	+	0.835762i	$0.314974\pi$
$972$	1701.00	0.0561313
$973$	0	0
$974$	−5944.00	−0.195542
$975$	1650.00	0.0541972
$976$	−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	$0.505076\pi$
$978$	6204.00	0.202845
$979$	−17160.0	−0.560200
$980$	0	0
$981$	−8730.00	−0.284126
$982$	10772.0	0.350049
$983$	13608.0	0.441534	0.220767	−	0.975327i	$-0.429144\pi$
$983$	13608.0	0.441534	0.220767	+	0.975327i	$0.429144\pi$
$984$	−5490.00	−0.177861
$985$	−12630.0	−0.408554
$986$	3220.00	0.104002
$987$	0	0
$988$	3080.00	0.0991780
$989$	31584.0	1.01548
$990$	−2340.00	−0.0751213
$991$	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
$993$	−3996.00	−0.127703
$994$	0	0
$995$	−5800.00	−0.184796
$996$	−25452.0	−0.809716
$997$	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
$999$	918.000	0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.4.a.e.1.1		1
3.2	odd	2		2205.4.a.l.1.1		1
7.6	odd	2		15.4.a.a.1.1	✓	1
21.20	even	2		45.4.a.c.1.1		1
28.27	even	2		240.4.a.e.1.1		1
35.13	even	4		75.4.b.b.49.1		2
35.27	even	4		75.4.b.b.49.2		2
35.34	odd	2		75.4.a.b.1.1		1
56.13	odd	2		960.4.a.b.1.1		1
56.27	even	2		960.4.a.ba.1.1		1
63.13	odd	6		405.4.e.g.136.1		2
63.20	even	6		405.4.e.i.271.1		2
63.34	odd	6		405.4.e.g.271.1		2
63.41	even	6		405.4.e.i.136.1		2
77.76	even	2		1815.4.a.e.1.1		1
84.83	odd	2		720.4.a.n.1.1		1
105.62	odd	4		225.4.b.e.199.1		2
105.83	odd	4		225.4.b.e.199.2		2
105.104	even	2		225.4.a.f.1.1		1
140.27	odd	4		1200.4.f.b.49.2		2
140.83	odd	4		1200.4.f.b.49.1		2
140.139	even	2		1200.4.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	7.6	odd	2
45.4.a.c.1.1		1	21.20	even	2
75.4.a.b.1.1		1	35.34	odd	2
75.4.b.b.49.1		2	35.13	even	4
75.4.b.b.49.2		2	35.27	even	4
225.4.a.f.1.1		1	105.104	even	2
225.4.b.e.199.1		2	105.62	odd	4
225.4.b.e.199.2		2	105.83	odd	4
240.4.a.e.1.1		1	28.27	even	2
405.4.e.g.136.1		2	63.13	odd	6
405.4.e.g.271.1		2	63.34	odd	6
405.4.e.i.136.1		2	63.41	even	6
405.4.e.i.271.1		2	63.20	even	6
720.4.a.n.1.1		1	84.83	odd	2
735.4.a.e.1.1		1	1.1	even	1	trivial
960.4.a.b.1.1		1	56.13	odd	2
960.4.a.ba.1.1		1	56.27	even	2
1200.4.a.t.1.1		1	140.139	even	2
1200.4.f.b.49.1		2	140.83	odd	4
1200.4.f.b.49.2		2	140.27	odd	4
1815.4.a.e.1.1		1	77.76	even	2
2205.4.a.l.1.1		1	3.2	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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more