LMFDB - Embedded newform 2205.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,4,Mod(1,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, 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("2205.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2205.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:	$130.099211563$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 735)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2205.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +8.00000 q^{4} +5.00000 q^{5} +O(q^{10})$

$q+4.00000 q^{2} +8.00000 q^{4} +5.00000 q^{5} +20.0000 q^{10} +10.0000 q^{11} -24.0000 q^{13} -64.0000 q^{16} -54.0000 q^{17} -12.0000 q^{19} +40.0000 q^{20} +40.0000 q^{22} +134.000 q^{23} +25.0000 q^{25} -96.0000 q^{26} -118.000 q^{29} +144.000 q^{31} -256.000 q^{32} -216.000 q^{34} -378.000 q^{37} -48.0000 q^{38} -330.000 q^{41} +204.000 q^{43} +80.0000 q^{44} +536.000 q^{46} -312.000 q^{47} +100.000 q^{50} -192.000 q^{52} -142.000 q^{53} +50.0000 q^{55} -472.000 q^{58} -108.000 q^{59} -168.000 q^{61} +576.000 q^{62} -512.000 q^{64} -120.000 q^{65} +448.000 q^{67} -432.000 q^{68} -146.000 q^{71} -360.000 q^{73} -1512.00 q^{74} -96.0000 q^{76} +236.000 q^{79} -320.000 q^{80} -1320.00 q^{82} -324.000 q^{83} -270.000 q^{85} +816.000 q^{86} -1194.00 q^{89} +1072.00 q^{92} -1248.00 q^{94} -60.0000 q^{95} -1728.00 q^{97} +O(q^{100})$

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 $

$2$	4.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	4.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	8.00000	1.00000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	20.0000	0.632456
$11$	10.0000	0.274101	0.137051	−	0.990564i	$-0.456238\pi$
$11$	10.0000	0.274101	0.137051	+	0.990564i	$0.456238\pi$
$12$	0	0
$13$	−24.0000	−0.512031	−0.256015	−	0.966673i	$-0.582410\pi$
$13$	−24.0000	−0.512031	−0.256015	+	0.966673i	$0.582410\pi$
$14$	0	0
$15$	0	0
$16$	−64.0000	−1.00000
$17$	−54.0000	−0.770407	−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000	−0.770407	−0.385204	+	0.922832i	$0.625869\pi$
$18$	0	0
$19$	−12.0000	−0.144894	−0.0724471	−	0.997372i	$-0.523081\pi$
$19$	−12.0000	−0.144894	−0.0724471	+	0.997372i	$0.523081\pi$
$20$	40.0000	0.447214
$21$	0	0
$22$	40.0000	0.387638
$23$	134.000	1.21482	0.607412	−	0.794387i	$-0.292208\pi$
$23$	134.000	1.21482	0.607412	+	0.794387i	$0.292208\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−96.0000	−0.724121
$27$	0	0
$28$	0	0
$29$	−118.000	−0.755588	−0.377794	−	0.925890i	$-0.623317\pi$
$29$	−118.000	−0.755588	−0.377794	+	0.925890i	$0.623317\pi$
$30$	0	0
$31$	144.000	0.834296	0.417148	−	0.908839i	$-0.363030\pi$
$31$	144.000	0.834296	0.417148	+	0.908839i	$0.363030\pi$
$32$	−256.000	−1.41421
$33$	0	0
$34$	−216.000	−1.08952
$35$	0	0
$36$	0	0
$37$	−378.000	−1.67954	−0.839768	−	0.542946i	$-0.817309\pi$
$37$	−378.000	−1.67954	−0.839768	+	0.542946i	$0.817309\pi$
$38$	−48.0000	−0.204911
$39$	0	0
$40$	0	0
$41$	−330.000	−1.25701	−0.628504	−	0.777806i	$-0.716332\pi$
$41$	−330.000	−1.25701	−0.628504	+	0.777806i	$0.716332\pi$
$42$	0	0
$43$	204.000	0.723482	0.361741	−	0.932279i	$-0.382183\pi$
$43$	204.000	0.723482	0.361741	+	0.932279i	$0.382183\pi$
$44$	80.0000	0.274101
$45$	0	0
$46$	536.000	1.71802
$47$	−312.000	−0.968295	−0.484148	−	0.874986i	$-0.660870\pi$
$47$	−312.000	−0.968295	−0.484148	+	0.874986i	$0.660870\pi$
$48$	0	0
$49$	0	0
$50$	100.000	0.282843
$51$	0	0
$52$	−192.000	−0.512031
$53$	−142.000	−0.368023	−0.184011	−	0.982924i	$-0.558908\pi$
$53$	−142.000	−0.368023	−0.184011	+	0.982924i	$0.558908\pi$
$54$	0	0
$55$	50.0000	0.122582
$56$	0	0
$57$	0	0
$58$	−472.000	−1.06856
$59$	−108.000	−0.238312	−0.119156	−	0.992876i	$-0.538019\pi$
$59$	−108.000	−0.238312	−0.119156	+	0.992876i	$0.538019\pi$
$60$	0	0
$61$	−168.000	−0.352626	−0.176313	−	0.984334i	$-0.556417\pi$
$61$	−168.000	−0.352626	−0.176313	+	0.984334i	$0.556417\pi$
$62$	576.000	1.17987
$63$	0	0
$64$	−512.000	−1.00000
$65$	−120.000	−0.228987
$66$	0	0
$67$	448.000	0.816894	0.408447	−	0.912782i	$-0.366070\pi$
$67$	448.000	0.816894	0.408447	+	0.912782i	$0.366070\pi$
$68$	−432.000	−0.770407
$69$	0	0
$70$	0	0
$71$	−146.000	−0.244042	−0.122021	−	0.992527i	$-0.538938\pi$
$71$	−146.000	−0.244042	−0.122021	+	0.992527i	$0.538938\pi$
$72$	0	0
$73$	−360.000	−0.577189	−0.288595	−	0.957451i	$-0.593188\pi$
$73$	−360.000	−0.577189	−0.288595	+	0.957451i	$0.593188\pi$
$74$	−1512.00	−2.37522
$75$	0	0
$76$	−96.0000	−0.144894
$77$	0	0
$78$	0	0
$79$	236.000	0.336102	0.168051	−	0.985778i	$-0.446253\pi$
$79$	236.000	0.336102	0.168051	+	0.985778i	$0.446253\pi$
$80$	−320.000	−0.447214
$81$	0	0
$82$	−1320.00	−1.77768
$83$	−324.000	−0.428477	−0.214239	−	0.976781i	$-0.568727\pi$
$83$	−324.000	−0.428477	−0.214239	+	0.976781i	$0.568727\pi$
$84$	0	0
$85$	−270.000	−0.344537
$86$	816.000	1.02316
$87$	0	0
$88$	0	0
$89$	−1194.00	−1.42206	−0.711032	−	0.703159i	$-0.751772\pi$
$89$	−1194.00	−1.42206	−0.711032	+	0.703159i	$0.751772\pi$
$90$	0	0
$91$	0	0
$92$	1072.00	1.21482
$93$	0	0
$94$	−1248.00	−1.36938
$95$	−60.0000	−0.0647986
$96$	0	0
$97$	−1728.00	−1.80878	−0.904391	−	0.426705i	$-0.859674\pi$
$97$	−1728.00	−1.80878	−0.904391	+	0.426705i	$0.859674\pi$
$98$	0	0
$99$	0	0
$100$	200.000	0.200000
$101$	−1374.00	−1.35364	−0.676822	−	0.736146i	$-0.736643\pi$
$101$	−1374.00	−1.35364	−0.676822	+	0.736146i	$0.736643\pi$
$102$	0	0
$103$	−948.000	−0.906886	−0.453443	−	0.891285i	$-0.649804\pi$
$103$	−948.000	−0.906886	−0.453443	+	0.891285i	$0.649804\pi$
$104$	0	0
$105$	0	0
$106$	−568.000	−0.520463
$107$	550.000	0.496921	0.248460	−	0.968642i	$-0.420075\pi$
$107$	550.000	0.496921	0.248460	+	0.968642i	$0.420075\pi$
$108$	0	0
$109$	402.000	0.353253	0.176627	−	0.984278i	$-0.443481\pi$
$109$	402.000	0.353253	0.176627	+	0.984278i	$0.443481\pi$
$110$	200.000	0.173357
$111$	0	0
$112$	0	0
$113$	1490.00	1.24042	0.620210	−	0.784436i	$-0.287047\pi$
$113$	1490.00	1.24042	0.620210	+	0.784436i	$0.287047\pi$
$114$	0	0
$115$	670.000	0.543285
$116$	−944.000	−0.755588
$117$	0	0
$118$	−432.000	−0.337024
$119$	0	0
$120$	0	0
$121$	−1231.00	−0.924869
$122$	−672.000	−0.498689
$123$	0	0
$124$	1152.00	0.834296
$125$	125.000	0.0894427
$126$	0	0
$127$	124.000	0.0866395	0.0433198	−	0.999061i	$-0.486207\pi$
$127$	124.000	0.0866395	0.0433198	+	0.999061i	$0.486207\pi$
$128$	0	0
$129$	0	0
$130$	−480.000	−0.323837
$131$	−228.000	−0.152065	−0.0760323	−	0.997105i	$-0.524225\pi$
$131$	−228.000	−0.152065	−0.0760323	+	0.997105i	$0.524225\pi$
$132$	0	0
$133$	0	0
$134$	1792.00	1.15526
$135$	0	0
$136$	0	0
$137$	2726.00	1.69998	0.849992	−	0.526795i	$-0.176606\pi$
$137$	2726.00	1.69998	0.849992	+	0.526795i	$0.176606\pi$
$138$	0	0
$139$	1848.00	1.12766	0.563832	−	0.825889i	$-0.309327\pi$
$139$	1848.00	1.12766	0.563832	+	0.825889i	$0.309327\pi$
$140$	0	0
$141$	0	0
$142$	−584.000	−0.345128
$143$	−240.000	−0.140348
$144$	0	0
$145$	−590.000	−0.337909
$146$	−1440.00	−0.816269
$147$	0	0
$148$	−3024.00	−1.67954
$149$	1030.00	0.566315	0.283157	−	0.959073i	$-0.408618\pi$
$149$	1030.00	0.566315	0.283157	+	0.959073i	$0.408618\pi$
$150$	0	0
$151$	−2040.00	−1.09942	−0.549711	−	0.835355i	$-0.685262\pi$
$151$	−2040.00	−1.09942	−0.549711	+	0.835355i	$0.685262\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	720.000	0.373108
$156$	0	0
$157$	−2844.00	−1.44571	−0.722853	−	0.691002i	$-0.757170\pi$
$157$	−2844.00	−1.44571	−0.722853	+	0.691002i	$0.757170\pi$
$158$	944.000	0.475320
$159$	0	0
$160$	−1280.00	−0.632456
$161$	0	0
$162$	0	0
$163$	−416.000	−0.199900	−0.0999498	−	0.994992i	$-0.531868\pi$
$163$	−416.000	−0.199900	−0.0999498	+	0.994992i	$0.531868\pi$
$164$	−2640.00	−1.25701
$165$	0	0
$166$	−1296.00	−0.605958
$167$	4056.00	1.87942	0.939709	−	0.341976i	$-0.111096\pi$
$167$	4056.00	1.87942	0.939709	+	0.341976i	$0.111096\pi$
$168$	0	0
$169$	−1621.00	−0.737824
$170$	−1080.00	−0.487248
$171$	0	0
$172$	1632.00	0.723482
$173$	−3534.00	−1.55309	−0.776546	−	0.630060i	$-0.783030\pi$
$173$	−3534.00	−1.55309	−0.776546	+	0.630060i	$0.783030\pi$
$174$	0	0
$175$	0	0
$176$	−640.000	−0.274101
$177$	0	0
$178$	−4776.00	−2.01110
$179$	4198.00	1.75292	0.876462	−	0.481472i	$-0.159898\pi$
$179$	4198.00	1.75292	0.876462	+	0.481472i	$0.159898\pi$
$180$	0	0
$181$	1812.00	0.744115	0.372058	−	0.928210i	$-0.378652\pi$
$181$	1812.00	0.744115	0.372058	+	0.928210i	$0.378652\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1890.00	−0.751111
$186$	0	0
$187$	−540.000	−0.211170
$188$	−2496.00	−0.968295
$189$	0	0
$190$	−240.000	−0.0916391
$191$	3158.00	1.19636	0.598180	−	0.801362i	$-0.295891\pi$
$191$	3158.00	1.19636	0.598180	+	0.801362i	$0.295891\pi$
$192$	0	0
$193$	−4530.00	−1.68952	−0.844758	−	0.535149i	$-0.820256\pi$
$193$	−4530.00	−1.68952	−0.844758	+	0.535149i	$0.820256\pi$
$194$	−6912.00	−2.55800
$195$	0	0
$196$	0	0
$197$	−782.000	−0.282818	−0.141409	−	0.989951i	$-0.545163\pi$
$197$	−782.000	−0.282818	−0.141409	+	0.989951i	$0.545163\pi$
$198$	0	0
$199$	−2796.00	−0.995996	−0.497998	−	0.867178i	$-0.665931\pi$
$199$	−2796.00	−0.995996	−0.497998	+	0.867178i	$0.665931\pi$
$200$	0	0
$201$	0	0
$202$	−5496.00	−1.91434
$203$	0	0
$204$	0	0
$205$	−1650.00	−0.562151
$206$	−3792.00	−1.28253
$207$	0	0
$208$	1536.00	0.512031
$209$	−120.000	−0.0397157
$210$	0	0
$211$	−1644.00	−0.536387	−0.268193	−	0.963365i	$-0.586427\pi$
$211$	−1644.00	−0.536387	−0.268193	+	0.963365i	$0.586427\pi$
$212$	−1136.00	−0.368023
$213$	0	0
$214$	2200.00	0.702752
$215$	1020.00	0.323551
$216$	0	0
$217$	0	0
$218$	1608.00	0.499576
$219$	0	0
$220$	400.000	0.122582
$221$	1296.00	0.394472
$222$	0	0
$223$	3840.00	1.15312	0.576559	−	0.817055i	$-0.304395\pi$
$223$	3840.00	1.15312	0.576559	+	0.817055i	$0.304395\pi$
$224$	0	0
$225$	0	0
$226$	5960.00	1.75422
$227$	−5532.00	−1.61750	−0.808748	−	0.588155i	$-0.799855\pi$
$227$	−5532.00	−1.61750	−0.808748	+	0.588155i	$0.799855\pi$
$228$	0	0
$229$	6900.00	1.99111	0.995556	−	0.0941671i	$-0.0300188\pi$
$229$	6900.00	1.99111	0.995556	+	0.0941671i	$0.0300188\pi$
$230$	2680.00	0.768322
$231$	0	0
$232$	0	0
$233$	−5402.00	−1.51887	−0.759435	−	0.650583i	$-0.774525\pi$
$233$	−5402.00	−1.51887	−0.759435	+	0.650583i	$0.774525\pi$
$234$	0	0
$235$	−1560.00	−0.433035
$236$	−864.000	−0.238312
$237$	0	0
$238$	0	0
$239$	−2702.00	−0.731288	−0.365644	−	0.930755i	$-0.619151\pi$
$239$	−2702.00	−0.731288	−0.365644	+	0.930755i	$0.619151\pi$
$240$	0	0
$241$	−5196.00	−1.38881	−0.694406	−	0.719583i	$-0.744333\pi$
$241$	−5196.00	−1.38881	−0.694406	+	0.719583i	$0.744333\pi$
$242$	−4924.00	−1.30796
$243$	0	0
$244$	−1344.00	−0.352626
$245$	0	0
$246$	0	0
$247$	288.000	0.0741903
$248$	0	0
$249$	0	0
$250$	500.000	0.126491
$251$	1836.00	0.461702	0.230851	−	0.972989i	$-0.425849\pi$
$251$	1836.00	0.461702	0.230851	+	0.972989i	$0.425849\pi$
$252$	0	0
$253$	1340.00	0.332984
$254$	496.000	0.122527
$255$	0	0
$256$	4096.00	1.00000
$257$	6630.00	1.60921	0.804607	−	0.593808i	$-0.202376\pi$
$257$	6630.00	1.60921	0.804607	+	0.593808i	$0.202376\pi$
$258$	0	0
$259$	0	0
$260$	−960.000	−0.228987
$261$	0	0
$262$	−912.000	−0.215052
$263$	−3250.00	−0.761991	−0.380995	−	0.924577i	$-0.624419\pi$
$263$	−3250.00	−0.761991	−0.380995	+	0.924577i	$0.624419\pi$
$264$	0	0
$265$	−710.000	−0.164585
$266$	0	0
$267$	0	0
$268$	3584.00	0.816894
$269$	−1290.00	−0.292389	−0.146195	−	0.989256i	$-0.546703\pi$
$269$	−1290.00	−0.292389	−0.146195	+	0.989256i	$0.546703\pi$
$270$	0	0
$271$	960.000	0.215188	0.107594	−	0.994195i	$-0.465685\pi$
$271$	960.000	0.215188	0.107594	+	0.994195i	$0.465685\pi$
$272$	3456.00	0.770407
$273$	0	0
$274$	10904.0	2.40414
$275$	250.000	0.0548202
$276$	0	0
$277$	434.000	0.0941391	0.0470696	−	0.998892i	$-0.485012\pi$
$277$	434.000	0.0941391	0.0470696	+	0.998892i	$0.485012\pi$
$278$	7392.00	1.59476
$279$	0	0
$280$	0	0
$281$	−3278.00	−0.695904	−0.347952	−	0.937512i	$-0.613123\pi$
$281$	−3278.00	−0.695904	−0.347952	+	0.937512i	$0.613123\pi$
$282$	0	0
$283$	−5040.00	−1.05865	−0.529323	−	0.848420i	$-0.677554\pi$
$283$	−5040.00	−1.05865	−0.529323	+	0.848420i	$0.677554\pi$
$284$	−1168.00	−0.244042
$285$	0	0
$286$	−960.000	−0.198482
$287$	0	0
$288$	0	0
$289$	−1997.00	−0.406473
$290$	−2360.00	−0.477876
$291$	0	0
$292$	−2880.00	−0.577189
$293$	−1398.00	−0.278744	−0.139372	−	0.990240i	$-0.544508\pi$
$293$	−1398.00	−0.278744	−0.139372	+	0.990240i	$0.544508\pi$
$294$	0	0
$295$	−540.000	−0.106576
$296$	0	0
$297$	0	0
$298$	4120.00	0.800890
$299$	−3216.00	−0.622027
$300$	0	0
$301$	0	0
$302$	−8160.00	−1.55482
$303$	0	0
$304$	768.000	0.144894
$305$	−840.000	−0.157699
$306$	0	0
$307$	−5016.00	−0.932502	−0.466251	−	0.884652i	$-0.654396\pi$
$307$	−5016.00	−0.932502	−0.466251	+	0.884652i	$0.654396\pi$
$308$	0	0
$309$	0	0
$310$	2880.00	0.527655
$311$	5352.00	0.975833	0.487917	−	0.872890i	$-0.337757\pi$
$311$	5352.00	0.975833	0.487917	+	0.872890i	$0.337757\pi$
$312$	0	0
$313$	−972.000	−0.175529	−0.0877647	−	0.996141i	$-0.527972\pi$
$313$	−972.000	−0.175529	−0.0877647	+	0.996141i	$0.527972\pi$
$314$	−11376.0	−2.04454
$315$	0	0
$316$	1888.00	0.336102
$317$	2018.00	0.357546	0.178773	−	0.983890i	$-0.442787\pi$
$317$	2018.00	0.357546	0.178773	+	0.983890i	$0.442787\pi$
$318$	0	0
$319$	−1180.00	−0.207108
$320$	−2560.00	−0.447214
$321$	0	0
$322$	0	0
$323$	648.000	0.111628
$324$	0	0
$325$	−600.000	−0.102406
$326$	−1664.00	−0.282701
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4308.00	0.715375	0.357687	−	0.933841i	$-0.383565\pi$
$331$	4308.00	0.715375	0.357687	+	0.933841i	$0.383565\pi$
$332$	−2592.00	−0.428477
$333$	0	0
$334$	16224.0	2.65790
$335$	2240.00	0.365326
$336$	0	0
$337$	7494.00	1.21135	0.605674	−	0.795713i	$-0.292904\pi$
$337$	7494.00	1.21135	0.605674	+	0.795713i	$0.292904\pi$
$338$	−6484.00	−1.04344
$339$	0	0
$340$	−2160.00	−0.344537
$341$	1440.00	0.228681
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−14136.0	−2.19640
$347$	9106.00	1.40875	0.704374	−	0.709829i	$-0.251228\pi$
$347$	9106.00	1.40875	0.704374	+	0.709829i	$0.251228\pi$
$348$	0	0
$349$	5280.00	0.809834	0.404917	−	0.914354i	$-0.367300\pi$
$349$	5280.00	0.809834	0.404917	+	0.914354i	$0.367300\pi$
$350$	0	0
$351$	0	0
$352$	−2560.00	−0.387638
$353$	−558.000	−0.0841341	−0.0420671	−	0.999115i	$-0.513394\pi$
$353$	−558.000	−0.0841341	−0.0420671	+	0.999115i	$0.513394\pi$
$354$	0	0
$355$	−730.000	−0.109139
$356$	−9552.00	−1.42206
$357$	0	0
$358$	16792.0	2.47901
$359$	−2794.00	−0.410757	−0.205378	−	0.978683i	$-0.565843\pi$
$359$	−2794.00	−0.410757	−0.205378	+	0.978683i	$0.565843\pi$
$360$	0	0
$361$	−6715.00	−0.979006
$362$	7248.00	1.05234
$363$	0	0
$364$	0	0
$365$	−1800.00	−0.258127
$366$	0	0
$367$	8952.00	1.27327	0.636636	−	0.771165i	$-0.280326\pi$
$367$	8952.00	1.27327	0.636636	+	0.771165i	$0.280326\pi$
$368$	−8576.00	−1.21482
$369$	0	0
$370$	−7560.00	−1.06223
$371$	0	0
$372$	0	0
$373$	−6194.00	−0.859821	−0.429910	−	0.902872i	$-0.641455\pi$
$373$	−6194.00	−0.859821	−0.429910	+	0.902872i	$0.641455\pi$
$374$	−2160.00	−0.298639
$375$	0	0
$376$	0	0
$377$	2832.00	0.386884
$378$	0	0
$379$	3588.00	0.486288	0.243144	−	0.969990i	$-0.421821\pi$
$379$	3588.00	0.486288	0.243144	+	0.969990i	$0.421821\pi$
$380$	−480.000	−0.0647986
$381$	0	0
$382$	12632.0	1.69191
$383$	11928.0	1.59136	0.795682	−	0.605715i	$-0.207113\pi$
$383$	11928.0	1.59136	0.795682	+	0.605715i	$0.207113\pi$
$384$	0	0
$385$	0	0
$386$	−18120.0	−2.38934
$387$	0	0
$388$	−13824.0	−1.80878
$389$	2986.00	0.389193	0.194597	−	0.980883i	$-0.437660\pi$
$389$	2986.00	0.389193	0.194597	+	0.980883i	$0.437660\pi$
$390$	0	0
$391$	−7236.00	−0.935909
$392$	0	0
$393$	0	0
$394$	−3128.00	−0.399965
$395$	1180.00	0.150309
$396$	0	0
$397$	−9636.00	−1.21818	−0.609089	−	0.793102i	$-0.708465\pi$
$397$	−9636.00	−1.21818	−0.609089	+	0.793102i	$0.708465\pi$
$398$	−11184.0	−1.40855
$399$	0	0
$400$	−1600.00	−0.200000
$401$	8690.00	1.08219	0.541095	−	0.840962i	$-0.318010\pi$
$401$	8690.00	1.08219	0.541095	+	0.840962i	$0.318010\pi$
$402$	0	0
$403$	−3456.00	−0.427185
$404$	−10992.0	−1.35364
$405$	0	0
$406$	0	0
$407$	−3780.00	−0.460363
$408$	0	0
$409$	−1044.00	−0.126216	−0.0631082	−	0.998007i	$-0.520101\pi$
$409$	−1044.00	−0.126216	−0.0631082	+	0.998007i	$0.520101\pi$
$410$	−6600.00	−0.795002
$411$	0	0
$412$	−7584.00	−0.906886
$413$	0	0
$414$	0	0
$415$	−1620.00	−0.191621
$416$	6144.00	0.724121
$417$	0	0
$418$	−480.000	−0.0561664
$419$	−468.000	−0.0545663	−0.0272832	−	0.999628i	$-0.508686\pi$
$419$	−468.000	−0.0545663	−0.0272832	+	0.999628i	$0.508686\pi$
$420$	0	0
$421$	−15518.0	−1.79644	−0.898220	−	0.439547i	$-0.855139\pi$
$421$	−15518.0	−1.79644	−0.898220	+	0.439547i	$0.855139\pi$
$422$	−6576.00	−0.758566
$423$	0	0
$424$	0	0
$425$	−1350.00	−0.154081
$426$	0	0
$427$	0	0
$428$	4400.00	0.496921
$429$	0	0
$430$	4080.00	0.457570
$431$	3830.00	0.428039	0.214019	−	0.976829i	$-0.431344\pi$
$431$	3830.00	0.428039	0.214019	+	0.976829i	$0.431344\pi$
$432$	0	0
$433$	−852.000	−0.0945601	−0.0472800	−	0.998882i	$-0.515055\pi$
$433$	−852.000	−0.0945601	−0.0472800	+	0.998882i	$0.515055\pi$
$434$	0	0
$435$	0	0
$436$	3216.00	0.353253
$437$	−1608.00	−0.176021
$438$	0	0
$439$	2916.00	0.317023	0.158511	−	0.987357i	$-0.449331\pi$
$439$	2916.00	0.317023	0.158511	+	0.987357i	$0.449331\pi$
$440$	0	0
$441$	0	0
$442$	5184.00	0.557868
$443$	−12542.0	−1.34512	−0.672560	−	0.740042i	$-0.734805\pi$
$443$	−12542.0	−1.34512	−0.672560	+	0.740042i	$0.734805\pi$
$444$	0	0
$445$	−5970.00	−0.635967
$446$	15360.0	1.63076
$447$	0	0
$448$	0	0
$449$	14054.0	1.47717	0.738585	−	0.674160i	$-0.235494\pi$
$449$	14054.0	1.47717	0.738585	+	0.674160i	$0.235494\pi$
$450$	0	0
$451$	−3300.00	−0.344548
$452$	11920.0	1.24042
$453$	0	0
$454$	−22128.0	−2.28749
$455$	0	0
$456$	0	0
$457$	4674.00	0.478426	0.239213	−	0.970967i	$-0.423111\pi$
$457$	4674.00	0.478426	0.239213	+	0.970967i	$0.423111\pi$
$458$	27600.0	2.81586
$459$	0	0
$460$	5360.00	0.543285
$461$	12594.0	1.27237	0.636183	−	0.771538i	$-0.280512\pi$
$461$	12594.0	1.27237	0.636183	+	0.771538i	$0.280512\pi$
$462$	0	0
$463$	−6388.00	−0.641200	−0.320600	−	0.947215i	$-0.603885\pi$
$463$	−6388.00	−0.641200	−0.320600	+	0.947215i	$0.603885\pi$
$464$	7552.00	0.755588
$465$	0	0
$466$	−21608.0	−2.14801
$467$	15132.0	1.49941	0.749706	−	0.661771i	$-0.230195\pi$
$467$	15132.0	1.49941	0.749706	+	0.661771i	$0.230195\pi$
$468$	0	0
$469$	0	0
$470$	−6240.00	−0.612404
$471$	0	0
$472$	0	0
$473$	2040.00	0.198307
$474$	0	0
$475$	−300.000	−0.0289788
$476$	0	0
$477$	0	0
$478$	−10808.0	−1.03420
$479$	−16056.0	−1.53156	−0.765780	−	0.643102i	$-0.777647\pi$
$479$	−16056.0	−1.53156	−0.765780	+	0.643102i	$0.777647\pi$
$480$	0	0
$481$	9072.00	0.859974
$482$	−20784.0	−1.96408
$483$	0	0
$484$	−9848.00	−0.924869
$485$	−8640.00	−0.808912
$486$	0	0
$487$	5184.00	0.482360	0.241180	−	0.970480i	$-0.422466\pi$
$487$	5184.00	0.482360	0.241180	+	0.970480i	$0.422466\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12374.0	1.13733	0.568667	−	0.822568i	$-0.307459\pi$
$491$	12374.0	1.13733	0.568667	+	0.822568i	$0.307459\pi$
$492$	0	0
$493$	6372.00	0.582110
$494$	1152.00	0.104921
$495$	0	0
$496$	−9216.00	−0.834296
$497$	0	0
$498$	0	0
$499$	21492.0	1.92808	0.964042	−	0.265749i	$-0.0856194\pi$
$499$	21492.0	1.92808	0.964042	+	0.265749i	$0.0856194\pi$
$500$	1000.00	0.0894427
$501$	0	0
$502$	7344.00	0.652946
$503$	−11688.0	−1.03607	−0.518034	−	0.855360i	$-0.673336\pi$
$503$	−11688.0	−1.03607	−0.518034	+	0.855360i	$0.673336\pi$
$504$	0	0
$505$	−6870.00	−0.605368
$506$	5360.00	0.470911
$507$	0	0
$508$	992.000	0.0866395
$509$	3774.00	0.328644	0.164322	−	0.986407i	$-0.447456\pi$
$509$	3774.00	0.328644	0.164322	+	0.986407i	$0.447456\pi$
$510$	0	0
$511$	0	0
$512$	16384.0	1.41421
$513$	0	0
$514$	26520.0	2.27577
$515$	−4740.00	−0.405572
$516$	0	0
$517$	−3120.00	−0.265411
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18282.0	1.53733	0.768665	−	0.639652i	$-0.220921\pi$
$521$	18282.0	1.53733	0.768665	+	0.639652i	$0.220921\pi$
$522$	0	0
$523$	−9372.00	−0.783574	−0.391787	−	0.920056i	$-0.628143\pi$
$523$	−9372.00	−0.783574	−0.391787	+	0.920056i	$0.628143\pi$
$524$	−1824.00	−0.152065
$525$	0	0
$526$	−13000.0	−1.07762
$527$	−7776.00	−0.642747
$528$	0	0
$529$	5789.00	0.475795
$530$	−2840.00	−0.232758
$531$	0	0
$532$	0	0
$533$	7920.00	0.643627
$534$	0	0
$535$	2750.00	0.222230
$536$	0	0
$537$	0	0
$538$	−5160.00	−0.413501
$539$	0	0
$540$	0	0
$541$	14330.0	1.13881	0.569404	−	0.822058i	$-0.307174\pi$
$541$	14330.0	1.13881	0.569404	+	0.822058i	$0.307174\pi$
$542$	3840.00	0.304321
$543$	0	0
$544$	13824.0	1.08952
$545$	2010.00	0.157980
$546$	0	0
$547$	−6712.00	−0.524652	−0.262326	−	0.964979i	$-0.584490\pi$
$547$	−6712.00	−0.524652	−0.262326	+	0.964979i	$0.584490\pi$
$548$	21808.0	1.69998
$549$	0	0
$550$	1000.00	0.0775275
$551$	1416.00	0.109480
$552$	0	0
$553$	0	0
$554$	1736.00	0.133133
$555$	0	0
$556$	14784.0	1.12766
$557$	19066.0	1.45036	0.725182	−	0.688558i	$-0.241756\pi$
$557$	19066.0	1.45036	0.725182	+	0.688558i	$0.241756\pi$
$558$	0	0
$559$	−4896.00	−0.370445
$560$	0	0
$561$	0	0
$562$	−13112.0	−0.984157
$563$	−15492.0	−1.15970	−0.579849	−	0.814724i	$-0.696888\pi$
$563$	−15492.0	−1.15970	−0.579849	+	0.814724i	$0.696888\pi$
$564$	0	0
$565$	7450.00	0.554732
$566$	−20160.0	−1.49715
$567$	0	0
$568$	0	0
$569$	−8870.00	−0.653514	−0.326757	−	0.945108i	$-0.605956\pi$
$569$	−8870.00	−0.653514	−0.326757	+	0.945108i	$0.605956\pi$
$570$	0	0
$571$	−19136.0	−1.40248	−0.701241	−	0.712925i	$-0.747370\pi$
$571$	−19136.0	−1.40248	−0.701241	+	0.712925i	$0.747370\pi$
$572$	−1920.00	−0.140348
$573$	0	0
$574$	0	0
$575$	3350.00	0.242965
$576$	0	0
$577$	−17532.0	−1.26493	−0.632467	−	0.774587i	$-0.717958\pi$
$577$	−17532.0	−1.26493	−0.632467	+	0.774587i	$0.717958\pi$
$578$	−7988.00	−0.574839
$579$	0	0
$580$	−4720.00	−0.337909
$581$	0	0
$582$	0	0
$583$	−1420.00	−0.100875
$584$	0	0
$585$	0	0
$586$	−5592.00	−0.394204
$587$	5700.00	0.400791	0.200395	−	0.979715i	$-0.435777\pi$
$587$	5700.00	0.400791	0.200395	+	0.979715i	$0.435777\pi$
$588$	0	0
$589$	−1728.00	−0.120885
$590$	−2160.00	−0.150722
$591$	0	0
$592$	24192.0	1.67954
$593$	27042.0	1.87265	0.936325	−	0.351134i	$-0.114204\pi$
$593$	27042.0	1.87265	0.936325	+	0.351134i	$0.114204\pi$
$594$	0	0
$595$	0	0
$596$	8240.00	0.566315
$597$	0	0
$598$	−12864.0	−0.879679
$599$	−12514.0	−0.853603	−0.426801	−	0.904345i	$-0.640360\pi$
$599$	−12514.0	−0.853603	−0.426801	+	0.904345i	$0.640360\pi$
$600$	0	0
$601$	4608.00	0.312752	0.156376	−	0.987698i	$-0.450019\pi$
$601$	4608.00	0.312752	0.156376	+	0.987698i	$0.450019\pi$
$602$	0	0
$603$	0	0
$604$	−16320.0	−1.09942
$605$	−6155.00	−0.413614
$606$	0	0
$607$	15456.0	1.03351	0.516754	−	0.856134i	$-0.327140\pi$
$607$	15456.0	1.03351	0.516754	+	0.856134i	$0.327140\pi$
$608$	3072.00	0.204911
$609$	0	0
$610$	−3360.00	−0.223020
$611$	7488.00	0.495797
$612$	0	0
$613$	2050.00	0.135071	0.0675357	−	0.997717i	$-0.478486\pi$
$613$	2050.00	0.135071	0.0675357	+	0.997717i	$0.478486\pi$
$614$	−20064.0	−1.31876
$615$	0	0
$616$	0	0
$617$	7318.00	0.477490	0.238745	−	0.971082i	$-0.423264\pi$
$617$	7318.00	0.477490	0.238745	+	0.971082i	$0.423264\pi$
$618$	0	0
$619$	5472.00	0.355312	0.177656	−	0.984093i	$-0.443149\pi$
$619$	5472.00	0.355312	0.177656	+	0.984093i	$0.443149\pi$
$620$	5760.00	0.373108
$621$	0	0
$622$	21408.0	1.38004
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	−3888.00	−0.248236
$627$	0	0
$628$	−22752.0	−1.44571
$629$	20412.0	1.29393
$630$	0	0
$631$	−22124.0	−1.39579	−0.697894	−	0.716201i	$-0.745879\pi$
$631$	−22124.0	−1.39579	−0.697894	+	0.716201i	$0.745879\pi$
$632$	0	0
$633$	0	0
$634$	8072.00	0.505647
$635$	620.000	0.0387464
$636$	0	0
$637$	0	0
$638$	−4720.00	−0.292894
$639$	0	0
$640$	0	0
$641$	1282.00	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	1282.00	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	9192.00	0.563759	0.281880	−	0.959450i	$-0.409042\pi$
$643$	9192.00	0.563759	0.281880	+	0.959450i	$0.409042\pi$
$644$	0	0
$645$	0	0
$646$	2592.00	0.157865
$647$	17352.0	1.05437	0.527185	−	0.849750i	$-0.323247\pi$
$647$	17352.0	1.05437	0.527185	+	0.849750i	$0.323247\pi$
$648$	0	0
$649$	−1080.00	−0.0653216
$650$	−2400.00	−0.144824
$651$	0	0
$652$	−3328.00	−0.199900
$653$	−658.000	−0.0394327	−0.0197163	−	0.999806i	$-0.506276\pi$
$653$	−658.000	−0.0394327	−0.0197163	+	0.999806i	$0.506276\pi$
$654$	0	0
$655$	−1140.00	−0.0680053
$656$	21120.0	1.25701
$657$	0	0
$658$	0	0
$659$	17666.0	1.04426	0.522132	−	0.852865i	$-0.325137\pi$
$659$	17666.0	1.04426	0.522132	+	0.852865i	$0.325137\pi$
$660$	0	0
$661$	−22392.0	−1.31762	−0.658811	−	0.752309i	$-0.728940\pi$
$661$	−22392.0	−1.31762	−0.658811	+	0.752309i	$0.728940\pi$
$662$	17232.0	1.01169
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−15812.0	−0.917906
$668$	32448.0	1.87942
$669$	0	0
$670$	8960.00	0.516649
$671$	−1680.00	−0.0966553
$672$	0	0
$673$	5010.00	0.286956	0.143478	−	0.989654i	$-0.454171\pi$
$673$	5010.00	0.286956	0.143478	+	0.989654i	$0.454171\pi$
$674$	29976.0	1.71310
$675$	0	0
$676$	−12968.0	−0.737824
$677$	−23286.0	−1.32194	−0.660970	−	0.750412i	$-0.729855\pi$
$677$	−23286.0	−1.32194	−0.660970	+	0.750412i	$0.729855\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	5760.00	0.323404
$683$	−23398.0	−1.31083	−0.655417	−	0.755267i	$-0.727507\pi$
$683$	−23398.0	−1.31083	−0.655417	+	0.755267i	$0.727507\pi$
$684$	0	0
$685$	13630.0	0.760256
$686$	0	0
$687$	0	0
$688$	−13056.0	−0.723482
$689$	3408.00	0.188439
$690$	0	0
$691$	−15696.0	−0.864116	−0.432058	−	0.901846i	$-0.642212\pi$
$691$	−15696.0	−0.864116	−0.432058	+	0.901846i	$0.642212\pi$
$692$	−28272.0	−1.55309
$693$	0	0
$694$	36424.0	1.99227
$695$	9240.00	0.504307
$696$	0	0
$697$	17820.0	0.968408
$698$	21120.0	1.14528
$699$	0	0
$700$	0	0
$701$	−21502.0	−1.15852	−0.579258	−	0.815144i	$-0.696658\pi$
$701$	−21502.0	−1.15852	−0.579258	+	0.815144i	$0.696658\pi$
$702$	0	0
$703$	4536.00	0.243355
$704$	−5120.00	−0.274101
$705$	0	0
$706$	−2232.00	−0.118984
$707$	0	0
$708$	0	0
$709$	−8886.00	−0.470692	−0.235346	−	0.971912i	$-0.575622\pi$
$709$	−8886.00	−0.470692	−0.235346	+	0.971912i	$0.575622\pi$
$710$	−2920.00	−0.154346
$711$	0	0
$712$	0	0
$713$	19296.0	1.01352
$714$	0	0
$715$	−1200.00	−0.0627657
$716$	33584.0	1.75292
$717$	0	0
$718$	−11176.0	−0.580898
$719$	21768.0	1.12908	0.564541	−	0.825405i	$-0.309053\pi$
$719$	21768.0	1.12908	0.564541	+	0.825405i	$0.309053\pi$
$720$	0	0
$721$	0	0
$722$	−26860.0	−1.38452
$723$	0	0
$724$	14496.0	0.744115
$725$	−2950.00	−0.151118
$726$	0	0
$727$	22932.0	1.16988	0.584939	−	0.811078i	$-0.301119\pi$
$727$	22932.0	1.16988	0.584939	+	0.811078i	$0.301119\pi$
$728$	0	0
$729$	0	0
$730$	−7200.00	−0.365047
$731$	−11016.0	−0.557375
$732$	0	0
$733$	−6624.00	−0.333783	−0.166892	−	0.985975i	$-0.553373\pi$
$733$	−6624.00	−0.333783	−0.166892	+	0.985975i	$0.553373\pi$
$734$	35808.0	1.80068
$735$	0	0
$736$	−34304.0	−1.71802
$737$	4480.00	0.223912
$738$	0	0
$739$	29144.0	1.45072	0.725358	−	0.688372i	$-0.241674\pi$
$739$	29144.0	1.45072	0.725358	+	0.688372i	$0.241674\pi$
$740$	−15120.0	−0.751111
$741$	0	0
$742$	0	0
$743$	8954.00	0.442114	0.221057	−	0.975261i	$-0.429049\pi$
$743$	8954.00	0.442114	0.221057	+	0.975261i	$0.429049\pi$
$744$	0	0
$745$	5150.00	0.253264
$746$	−24776.0	−1.21597
$747$	0	0
$748$	−4320.00	−0.211170
$749$	0	0
$750$	0	0
$751$	24096.0	1.17081	0.585403	−	0.810742i	$-0.300936\pi$
$751$	24096.0	1.17081	0.585403	+	0.810742i	$0.300936\pi$
$752$	19968.0	0.968295
$753$	0	0
$754$	11328.0	0.547137
$755$	−10200.0	−0.491677
$756$	0	0
$757$	23442.0	1.12551	0.562757	−	0.826622i	$-0.309741\pi$
$757$	23442.0	1.12551	0.562757	+	0.826622i	$0.309741\pi$
$758$	14352.0	0.687715
$759$	0	0
$760$	0	0
$761$	−7746.00	−0.368978	−0.184489	−	0.982835i	$-0.559063\pi$
$761$	−7746.00	−0.368978	−0.184489	+	0.982835i	$0.559063\pi$
$762$	0	0
$763$	0	0
$764$	25264.0	1.19636
$765$	0	0
$766$	47712.0	2.25053
$767$	2592.00	0.122023
$768$	0	0
$769$	−19320.0	−0.905978	−0.452989	−	0.891516i	$-0.649642\pi$
$769$	−19320.0	−0.905978	−0.452989	+	0.891516i	$0.649642\pi$
$770$	0	0
$771$	0	0
$772$	−36240.0	−1.68952
$773$	−13638.0	−0.634573	−0.317286	−	0.948330i	$-0.602772\pi$
$773$	−13638.0	−0.634573	−0.317286	+	0.948330i	$0.602772\pi$
$774$	0	0
$775$	3600.00	0.166859
$776$	0	0
$777$	0	0
$778$	11944.0	0.550403
$779$	3960.00	0.182133
$780$	0	0
$781$	−1460.00	−0.0668923
$782$	−28944.0	−1.32357
$783$	0	0
$784$	0	0
$785$	−14220.0	−0.646540
$786$	0	0
$787$	−39636.0	−1.79526	−0.897631	−	0.440748i	$-0.854713\pi$
$787$	−39636.0	−1.79526	−0.897631	+	0.440748i	$0.854713\pi$
$788$	−6256.00	−0.282818
$789$	0	0
$790$	4720.00	0.212570
$791$	0	0
$792$	0	0
$793$	4032.00	0.180556
$794$	−38544.0	−1.72276
$795$	0	0
$796$	−22368.0	−0.995996
$797$	20490.0	0.910656	0.455328	−	0.890324i	$-0.349522\pi$
$797$	20490.0	0.910656	0.455328	+	0.890324i	$0.349522\pi$
$798$	0	0
$799$	16848.0	0.745982
$800$	−6400.00	−0.282843
$801$	0	0
$802$	34760.0	1.53045
$803$	−3600.00	−0.158208
$804$	0	0
$805$	0	0
$806$	−13824.0	−0.604131
$807$	0	0
$808$	0	0
$809$	8638.00	0.375397	0.187698	−	0.982227i	$-0.439897\pi$
$809$	8638.00	0.375397	0.187698	+	0.982227i	$0.439897\pi$
$810$	0	0
$811$	44472.0	1.92555	0.962776	−	0.270300i	$-0.0871227\pi$
$811$	44472.0	1.92555	0.962776	+	0.270300i	$0.0871227\pi$
$812$	0	0
$813$	0	0
$814$	−15120.0	−0.651051
$815$	−2080.00	−0.0893978
$816$	0	0
$817$	−2448.00	−0.104828
$818$	−4176.00	−0.178497
$819$	0	0
$820$	−13200.0	−0.562151
$821$	−290.000	−0.0123277	−0.00616387	−	0.999981i	$-0.501962\pi$
$821$	−290.000	−0.0123277	−0.00616387	+	0.999981i	$0.501962\pi$
$822$	0	0
$823$	30692.0	1.29995	0.649973	−	0.759957i	$-0.274780\pi$
$823$	30692.0	1.29995	0.649973	+	0.759957i	$0.274780\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3446.00	0.144896	0.0724481	−	0.997372i	$-0.476919\pi$
$827$	3446.00	0.144896	0.0724481	+	0.997372i	$0.476919\pi$
$828$	0	0
$829$	−7332.00	−0.307178	−0.153589	−	0.988135i	$-0.549083\pi$
$829$	−7332.00	−0.307178	−0.153589	+	0.988135i	$0.549083\pi$
$830$	−6480.00	−0.270993
$831$	0	0
$832$	12288.0	0.512031
$833$	0	0
$834$	0	0
$835$	20280.0	0.840501
$836$	−960.000	−0.0397157
$837$	0	0
$838$	−1872.00	−0.0771685
$839$	48024.0	1.97613	0.988065	−	0.154039i	$-0.0492282\pi$
$839$	48024.0	1.97613	0.988065	+	0.154039i	$0.0492282\pi$
$840$	0	0
$841$	−10465.0	−0.429087
$842$	−62072.0	−2.54055
$843$	0	0
$844$	−13152.0	−0.536387
$845$	−8105.00	−0.329965
$846$	0	0
$847$	0	0
$848$	9088.00	0.368023
$849$	0	0
$850$	−5400.00	−0.217904
$851$	−50652.0	−2.04034
$852$	0	0
$853$	46380.0	1.86169	0.930845	−	0.365415i	$-0.119073\pi$
$853$	46380.0	1.86169	0.930845	+	0.365415i	$0.119073\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6210.00	−0.247526	−0.123763	−	0.992312i	$-0.539496\pi$
$857$	−6210.00	−0.247526	−0.123763	+	0.992312i	$0.539496\pi$
$858$	0	0
$859$	−6492.00	−0.257863	−0.128931	−	0.991654i	$-0.541155\pi$
$859$	−6492.00	−0.257863	−0.128931	+	0.991654i	$0.541155\pi$
$860$	8160.00	0.323551
$861$	0	0
$862$	15320.0	0.605338
$863$	−16354.0	−0.645071	−0.322536	−	0.946557i	$-0.604535\pi$
$863$	−16354.0	−0.645071	−0.322536	+	0.946557i	$0.604535\pi$
$864$	0	0
$865$	−17670.0	−0.694564
$866$	−3408.00	−0.133728
$867$	0	0
$868$	0	0
$869$	2360.00	0.0921260
$870$	0	0
$871$	−10752.0	−0.418275
$872$	0	0
$873$	0	0
$874$	−6432.00	−0.248931
$875$	0	0
$876$	0	0
$877$	−35926.0	−1.38328	−0.691639	−	0.722243i	$-0.743111\pi$
$877$	−35926.0	−1.38328	−0.691639	+	0.722243i	$0.743111\pi$
$878$	11664.0	0.448338
$879$	0	0
$880$	−3200.00	−0.122582
$881$	7098.00	0.271439	0.135719	−	0.990747i	$-0.456665\pi$
$881$	7098.00	0.271439	0.135719	+	0.990747i	$0.456665\pi$
$882$	0	0
$883$	−11340.0	−0.432187	−0.216094	−	0.976373i	$-0.569332\pi$
$883$	−11340.0	−0.432187	−0.216094	+	0.976373i	$0.569332\pi$
$884$	10368.0	0.394472
$885$	0	0
$886$	−50168.0	−1.90229
$887$	−22560.0	−0.853992	−0.426996	−	0.904254i	$-0.640428\pi$
$887$	−22560.0	−0.853992	−0.426996	+	0.904254i	$0.640428\pi$
$888$	0	0
$889$	0	0
$890$	−23880.0	−0.899393
$891$	0	0
$892$	30720.0	1.15312
$893$	3744.00	0.140300
$894$	0	0
$895$	20990.0	0.783931
$896$	0	0
$897$	0	0
$898$	56216.0	2.08903
$899$	−16992.0	−0.630384
$900$	0	0
$901$	7668.00	0.283527
$902$	−13200.0	−0.487264
$903$	0	0
$904$	0	0
$905$	9060.00	0.332779
$906$	0	0
$907$	20676.0	0.756930	0.378465	−	0.925616i	$-0.376452\pi$
$907$	20676.0	0.756930	0.378465	+	0.925616i	$0.376452\pi$
$908$	−44256.0	−1.61750
$909$	0	0
$910$	0	0
$911$	−12962.0	−0.471405	−0.235703	−	0.971825i	$-0.575739\pi$
$911$	−12962.0	−0.471405	−0.235703	+	0.971825i	$0.575739\pi$
$912$	0	0
$913$	−3240.00	−0.117446
$914$	18696.0	0.676596
$915$	0	0
$916$	55200.0	1.99111
$917$	0	0
$918$	0	0
$919$	17496.0	0.628008	0.314004	−	0.949422i	$-0.398329\pi$
$919$	17496.0	0.628008	0.314004	+	0.949422i	$0.398329\pi$
$920$	0	0
$921$	0	0
$922$	50376.0	1.79940
$923$	3504.00	0.124957
$924$	0	0
$925$	−9450.00	−0.335907
$926$	−25552.0	−0.906794
$927$	0	0
$928$	30208.0	1.06856
$929$	−19698.0	−0.695662	−0.347831	−	0.937557i	$-0.613082\pi$
$929$	−19698.0	−0.695662	−0.347831	+	0.937557i	$0.613082\pi$
$930$	0	0
$931$	0	0
$932$	−43216.0	−1.51887
$933$	0	0
$934$	60528.0	2.12049
$935$	−2700.00	−0.0944379
$936$	0	0
$937$	26676.0	0.930061	0.465030	−	0.885295i	$-0.346043\pi$
$937$	26676.0	0.930061	0.465030	+	0.885295i	$0.346043\pi$
$938$	0	0
$939$	0	0
$940$	−12480.0	−0.433035
$941$	−20034.0	−0.694038	−0.347019	−	0.937858i	$-0.612806\pi$
$941$	−20034.0	−0.694038	−0.347019	+	0.937858i	$0.612806\pi$
$942$	0	0
$943$	−44220.0	−1.52704
$944$	6912.00	0.238312
$945$	0	0
$946$	8160.00	0.280449
$947$	−42962.0	−1.47421	−0.737105	−	0.675778i	$-0.763808\pi$
$947$	−42962.0	−1.47421	−0.737105	+	0.675778i	$0.763808\pi$
$948$	0	0
$949$	8640.00	0.295539
$950$	−1200.00	−0.0409823
$951$	0	0
$952$	0	0
$953$	24202.0	0.822644	0.411322	−	0.911490i	$-0.365067\pi$
$953$	24202.0	0.822644	0.411322	+	0.911490i	$0.365067\pi$
$954$	0	0
$955$	15790.0	0.535029
$956$	−21616.0	−0.731288
$957$	0	0
$958$	−64224.0	−2.16595
$959$	0	0
$960$	0	0
$961$	−9055.00	−0.303951
$962$	36288.0	1.21619
$963$	0	0
$964$	−41568.0	−1.38881
$965$	−22650.0	−0.755574
$966$	0	0
$967$	27236.0	0.905740	0.452870	−	0.891577i	$-0.350400\pi$
$967$	27236.0	0.905740	0.452870	+	0.891577i	$0.350400\pi$
$968$	0	0
$969$	0	0
$970$	−34560.0	−1.14397
$971$	−35820.0	−1.18385	−0.591925	−	0.805993i	$-0.701632\pi$
$971$	−35820.0	−1.18385	−0.591925	+	0.805993i	$0.701632\pi$
$972$	0	0
$973$	0	0
$974$	20736.0	0.682160
$975$	0	0
$976$	10752.0	0.352626
$977$	10798.0	0.353591	0.176796	−	0.984248i	$-0.443427\pi$
$977$	10798.0	0.353591	0.176796	+	0.984248i	$0.443427\pi$
$978$	0	0
$979$	−11940.0	−0.389790
$980$	0	0
$981$	0	0
$982$	49496.0	1.60843
$983$	−42576.0	−1.38145	−0.690724	−	0.723118i	$-0.742708\pi$
$983$	−42576.0	−1.38145	−0.690724	+	0.723118i	$0.742708\pi$
$984$	0	0
$985$	−3910.00	−0.126480
$986$	25488.0	0.823228
$987$	0	0
$988$	2304.00	0.0741903
$989$	27336.0	0.878902
$990$	0	0
$991$	−45504.0	−1.45861	−0.729305	−	0.684189i	$-0.760156\pi$
$991$	−45504.0	−1.45861	−0.729305	+	0.684189i	$0.760156\pi$
$992$	−36864.0	−1.17987
$993$	0	0
$994$	0	0
$995$	−13980.0	−0.445423
$996$	0	0
$997$	4548.00	0.144470	0.0722350	−	0.997388i	$-0.476987\pi$
$997$	4548.00	0.144470	0.0722350	+	0.997388i	$0.476987\pi$
$998$	85968.0	2.72672
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.4.a.s.1.1		1
3.2	odd	2		735.4.a.a.1.1	✓	1
7.6	odd	2		2205.4.a.r.1.1		1
21.20	even	2		735.4.a.b.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.a.1.1	✓	1	3.2	odd	2
735.4.a.b.1.1	yes	1	21.20	even	2
2205.4.a.r.1.1		1	7.6	odd	2
2205.4.a.s.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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