LMFDB - Embedded newform 1216.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1216.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:	$71.7463225670$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{3} +9.00000 q^{5} -31.0000 q^{7} -23.0000 q^{9} +O(q^{10})$

$q+2.00000 q^{3} +9.00000 q^{5} -31.0000 q^{7} -23.0000 q^{9} -57.0000 q^{11} +52.0000 q^{13} +18.0000 q^{15} +69.0000 q^{17} -19.0000 q^{19} -62.0000 q^{21} -72.0000 q^{23} -44.0000 q^{25} -100.000 q^{27} +150.000 q^{29} +32.0000 q^{31} -114.000 q^{33} -279.000 q^{35} +226.000 q^{37} +104.000 q^{39} -258.000 q^{41} +67.0000 q^{43} -207.000 q^{45} +579.000 q^{47} +618.000 q^{49} +138.000 q^{51} +432.000 q^{53} -513.000 q^{55} -38.0000 q^{57} +330.000 q^{59} +13.0000 q^{61} +713.000 q^{63} +468.000 q^{65} +856.000 q^{67} -144.000 q^{69} +642.000 q^{71} -487.000 q^{73} -88.0000 q^{75} +1767.00 q^{77} -700.000 q^{79} +421.000 q^{81} +12.0000 q^{83} +621.000 q^{85} +300.000 q^{87} -600.000 q^{89} -1612.00 q^{91} +64.0000 q^{93} -171.000 q^{95} +1424.00 q^{97} +1311.00 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	0	0
$5$	9.00000	0.804984	0.402492	−	0.915423i	$-0.368144\pi$
$5$	9.00000	0.804984	0.402492	+	0.915423i	$0.368144\pi$
$6$	0	0
$7$	−31.0000	−1.67384	−0.836921	−	0.547323i	$-0.815647\pi$
$7$	−31.0000	−1.67384	−0.836921	+	0.547323i	$0.815647\pi$
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	−57.0000	−1.56238	−0.781188	−	0.624295i	$-0.785386\pi$
$11$	−57.0000	−1.56238	−0.781188	+	0.624295i	$0.785386\pi$
$12$	0	0
$13$	52.0000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	52.0000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	18.0000	0.309839
$16$	0	0
$17$	69.0000	0.984409	0.492205	−	0.870480i	$-0.336191\pi$
$17$	69.0000	0.984409	0.492205	+	0.870480i	$0.336191\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−62.0000	−0.644262
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	0	0
$27$	−100.000	−0.712778
$28$	0	0
$29$	150.000	0.960493	0.480247	−	0.877134i	$-0.340547\pi$
$29$	150.000	0.960493	0.480247	+	0.877134i	$0.340547\pi$
$30$	0	0
$31$	32.0000	0.185399	0.0926995	−	0.995694i	$-0.470450\pi$
$31$	32.0000	0.185399	0.0926995	+	0.995694i	$0.470450\pi$
$32$	0	0
$33$	−114.000	−0.601359
$34$	0	0
$35$	−279.000	−1.34742
$36$	0	0
$37$	226.000	1.00417	0.502083	−	0.864819i	$-0.332567\pi$
$37$	226.000	1.00417	0.502083	+	0.864819i	$0.332567\pi$
$38$	0	0
$39$	104.000	0.427008
$40$	0	0
$41$	−258.000	−0.982752	−0.491376	−	0.870948i	$-0.663506\pi$
$41$	−258.000	−0.982752	−0.491376	+	0.870948i	$0.663506\pi$
$42$	0	0
$43$	67.0000	0.237614	0.118807	−	0.992917i	$-0.462093\pi$
$43$	67.0000	0.237614	0.118807	+	0.992917i	$0.462093\pi$
$44$	0	0
$45$	−207.000	−0.685728
$46$	0	0
$47$	579.000	1.79693	0.898466	−	0.439043i	$-0.144682\pi$
$47$	579.000	1.79693	0.898466	+	0.439043i	$0.144682\pi$
$48$	0	0
$49$	618.000	1.80175
$50$	0	0
$51$	138.000	0.378899
$52$	0	0
$53$	432.000	1.11962	0.559809	−	0.828622i	$-0.310874\pi$
$53$	432.000	1.11962	0.559809	+	0.828622i	$0.310874\pi$
$54$	0	0
$55$	−513.000	−1.25769
$56$	0	0
$57$	−38.0000	−0.0883022
$58$	0	0
$59$	330.000	0.728175	0.364088	−	0.931365i	$-0.381381\pi$
$59$	330.000	0.728175	0.364088	+	0.931365i	$0.381381\pi$
$60$	0	0
$61$	13.0000	0.0272865	0.0136433	−	0.999907i	$-0.495657\pi$
$61$	13.0000	0.0272865	0.0136433	+	0.999907i	$0.495657\pi$
$62$	0	0
$63$	713.000	1.42587
$64$	0	0
$65$	468.000	0.893050
$66$	0	0
$67$	856.000	1.56085	0.780426	−	0.625249i	$-0.215002\pi$
$67$	856.000	1.56085	0.780426	+	0.625249i	$0.215002\pi$
$68$	0	0
$69$	−144.000	−0.251240
$70$	0	0
$71$	642.000	1.07312	0.536559	−	0.843863i	$-0.319724\pi$
$71$	642.000	1.07312	0.536559	+	0.843863i	$0.319724\pi$
$72$	0	0
$73$	−487.000	−0.780809	−0.390404	−	0.920643i	$-0.627665\pi$
$73$	−487.000	−0.780809	−0.390404	+	0.920643i	$0.627665\pi$
$74$	0	0
$75$	−88.0000	−0.135485
$76$	0	0
$77$	1767.00	2.61517
$78$	0	0
$79$	−700.000	−0.996913	−0.498457	−	0.866915i	$-0.666100\pi$
$79$	−700.000	−0.996913	−0.498457	+	0.866915i	$0.666100\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	12.0000	0.0158695	0.00793477	−	0.999969i	$-0.497474\pi$
$83$	12.0000	0.0158695	0.00793477	+	0.999969i	$0.497474\pi$
$84$	0	0
$85$	621.000	0.792434
$86$	0	0
$87$	300.000	0.369694
$88$	0	0
$89$	−600.000	−0.714605	−0.357303	−	0.933989i	$-0.616304\pi$
$89$	−600.000	−0.714605	−0.357303	+	0.933989i	$0.616304\pi$
$90$	0	0
$91$	−1612.00	−1.85696
$92$	0	0
$93$	64.0000	0.0713601
$94$	0	0
$95$	−171.000	−0.184676
$96$	0	0
$97$	1424.00	1.49057	0.745285	−	0.666746i	$-0.232313\pi$
$97$	1424.00	1.49057	0.745285	+	0.666746i	$0.232313\pi$
$98$	0	0
$99$	1311.00	1.33091
$100$	0	0
$101$	−1062.00	−1.04627	−0.523133	−	0.852251i	$-0.675237\pi$
$101$	−1062.00	−1.04627	−0.523133	+	0.852251i	$0.675237\pi$
$102$	0	0
$103$	1178.00	1.12691	0.563455	−	0.826147i	$-0.309472\pi$
$103$	1178.00	1.12691	0.563455	+	0.826147i	$0.309472\pi$
$104$	0	0
$105$	−558.000	−0.518621
$106$	0	0
$107$	−114.000	−0.102998	−0.0514990	−	0.998673i	$-0.516400\pi$
$107$	−114.000	−0.102998	−0.0514990	+	0.998673i	$0.516400\pi$
$108$	0	0
$109$	−1460.00	−1.28296	−0.641480	−	0.767140i	$-0.721679\pi$
$109$	−1460.00	−1.28296	−0.641480	+	0.767140i	$0.721679\pi$
$110$	0	0
$111$	452.000	0.386504
$112$	0	0
$113$	−822.000	−0.684312	−0.342156	−	0.939643i	$-0.611157\pi$
$113$	−822.000	−0.684312	−0.342156	+	0.939643i	$0.611157\pi$
$114$	0	0
$115$	−648.000	−0.525446
$116$	0	0
$117$	−1196.00	−0.945045
$118$	0	0
$119$	−2139.00	−1.64775
$120$	0	0
$121$	1918.00	1.44102
$122$	0	0
$123$	−516.000	−0.378261
$124$	0	0
$125$	−1521.00	−1.08834
$126$	0	0
$127$	−2086.00	−1.45750	−0.728750	−	0.684780i	$-0.759898\pi$
$127$	−2086.00	−1.45750	−0.728750	+	0.684780i	$0.759898\pi$
$128$	0	0
$129$	134.000	0.0914577
$130$	0	0
$131$	93.0000	0.0620263	0.0310132	−	0.999519i	$-0.490127\pi$
$131$	93.0000	0.0620263	0.0310132	+	0.999519i	$0.490127\pi$
$132$	0	0
$133$	589.000	0.384006
$134$	0	0
$135$	−900.000	−0.573775
$136$	0	0
$137$	1269.00	0.791372	0.395686	−	0.918386i	$-0.370507\pi$
$137$	1269.00	0.791372	0.395686	+	0.918386i	$0.370507\pi$
$138$	0	0
$139$	1975.00	1.20516	0.602580	−	0.798058i	$-0.294139\pi$
$139$	1975.00	1.20516	0.602580	+	0.798058i	$0.294139\pi$
$140$	0	0
$141$	1158.00	0.691640
$142$	0	0
$143$	−2964.00	−1.73330
$144$	0	0
$145$	1350.00	0.773182
$146$	0	0
$147$	1236.00	0.693494
$148$	0	0
$149$	1695.00	0.931945	0.465973	−	0.884799i	$-0.345705\pi$
$149$	1695.00	0.931945	0.465973	+	0.884799i	$0.345705\pi$
$150$	0	0
$151$	1802.00	0.971157	0.485578	−	0.874193i	$-0.338609\pi$
$151$	1802.00	0.971157	0.485578	+	0.874193i	$0.338609\pi$
$152$	0	0
$153$	−1587.00	−0.838571
$154$	0	0
$155$	288.000	0.149243
$156$	0	0
$157$	3226.00	1.63989	0.819945	−	0.572442i	$-0.194004\pi$
$157$	3226.00	1.63989	0.819945	+	0.572442i	$0.194004\pi$
$158$	0	0
$159$	864.000	0.430941
$160$	0	0
$161$	2232.00	1.09259
$162$	0	0
$163$	−1268.00	−0.609309	−0.304655	−	0.952463i	$-0.598541\pi$
$163$	−1268.00	−0.609309	−0.304655	+	0.952463i	$0.598541\pi$
$164$	0	0
$165$	−1026.00	−0.484085
$166$	0	0
$167$	654.000	0.303042	0.151521	−	0.988454i	$-0.451583\pi$
$167$	654.000	0.303042	0.151521	+	0.988454i	$0.451583\pi$
$168$	0	0
$169$	507.000	0.230769
$170$	0	0
$171$	437.000	0.195428
$172$	0	0
$173$	1362.00	0.598560	0.299280	−	0.954165i	$-0.403253\pi$
$173$	1362.00	0.598560	0.299280	+	0.954165i	$0.403253\pi$
$174$	0	0
$175$	1364.00	0.589193
$176$	0	0
$177$	660.000	0.280275
$178$	0	0
$179$	210.000	0.0876879	0.0438440	−	0.999038i	$-0.486040\pi$
$179$	210.000	0.0876879	0.0438440	+	0.999038i	$0.486040\pi$
$180$	0	0
$181$	−2.00000	−0.000821319 0	−0.000410660	−	1.00000i	$-0.500131\pi$
$181$	−2.00000	−0.000821319 0	−0.000410660		1.00000i	$0.500131\pi$
$182$	0	0
$183$	26.0000	0.0105026
$184$	0	0
$185$	2034.00	0.808339
$186$	0	0
$187$	−3933.00	−1.53802
$188$	0	0
$189$	3100.00	1.19308
$190$	0	0
$191$	−2643.00	−1.00126	−0.500630	−	0.865661i	$-0.666898\pi$
$191$	−2643.00	−1.00126	−0.500630	+	0.865661i	$0.666898\pi$
$192$	0	0
$193$	3248.00	1.21138	0.605690	−	0.795701i	$-0.292897\pi$
$193$	3248.00	1.21138	0.605690	+	0.795701i	$0.292897\pi$
$194$	0	0
$195$	936.000	0.343735
$196$	0	0
$197$	3126.00	1.13055	0.565275	−	0.824903i	$-0.308770\pi$
$197$	3126.00	1.13055	0.565275	+	0.824903i	$0.308770\pi$
$198$	0	0
$199$	−2995.00	−1.06688	−0.533442	−	0.845837i	$-0.679102\pi$
$199$	−2995.00	−1.06688	−0.533442	+	0.845837i	$0.679102\pi$
$200$	0	0
$201$	1712.00	0.600772
$202$	0	0
$203$	−4650.00	−1.60771
$204$	0	0
$205$	−2322.00	−0.791100
$206$	0	0
$207$	1656.00	0.556038
$208$	0	0
$209$	1083.00	0.358434
$210$	0	0
$211$	4318.00	1.40883	0.704416	−	0.709788i	$-0.251209\pi$
$211$	4318.00	1.40883	0.704416	+	0.709788i	$0.251209\pi$
$212$	0	0
$213$	1284.00	0.413043
$214$	0	0
$215$	603.000	0.191276
$216$	0	0
$217$	−992.000	−0.310329
$218$	0	0
$219$	−974.000	−0.300533
$220$	0	0
$221$	3588.00	1.09210
$222$	0	0
$223$	518.000	0.155551	0.0777754	−	0.996971i	$-0.475218\pi$
$223$	518.000	0.155551	0.0777754	+	0.996971i	$0.475218\pi$
$224$	0	0
$225$	1012.00	0.299852
$226$	0	0
$227$	−2844.00	−0.831555	−0.415777	−	0.909466i	$-0.636490\pi$
$227$	−2844.00	−0.831555	−0.415777	+	0.909466i	$0.636490\pi$
$228$	0	0
$229$	−1745.00	−0.503550	−0.251775	−	0.967786i	$-0.581014\pi$
$229$	−1745.00	−0.503550	−0.251775	+	0.967786i	$0.581014\pi$
$230$	0	0
$231$	3534.00	1.00658
$232$	0	0
$233$	5283.00	1.48541	0.742706	−	0.669618i	$-0.233542\pi$
$233$	5283.00	1.48541	0.742706	+	0.669618i	$0.233542\pi$
$234$	0	0
$235$	5211.00	1.44650
$236$	0	0
$237$	−1400.00	−0.383712
$238$	0	0
$239$	465.000	0.125851	0.0629254	−	0.998018i	$-0.479957\pi$
$239$	465.000	0.125851	0.0629254	+	0.998018i	$0.479957\pi$
$240$	0	0
$241$	−7078.00	−1.89184	−0.945921	−	0.324396i	$-0.894839\pi$
$241$	−7078.00	−1.89184	−0.945921	+	0.324396i	$0.894839\pi$
$242$	0	0
$243$	3542.00	0.935059
$244$	0	0
$245$	5562.00	1.45038
$246$	0	0
$247$	−988.000	−0.254514
$248$	0	0
$249$	24.0000	0.00610819
$250$	0	0
$251$	−3567.00	−0.897000	−0.448500	−	0.893783i	$-0.648042\pi$
$251$	−3567.00	−0.897000	−0.448500	+	0.893783i	$0.648042\pi$
$252$	0	0
$253$	4104.00	1.01983
$254$	0	0
$255$	1242.00	0.305008
$256$	0	0
$257$	−1896.00	−0.460192	−0.230096	−	0.973168i	$-0.573904\pi$
$257$	−1896.00	−0.460192	−0.230096	+	0.973168i	$0.573904\pi$
$258$	0	0
$259$	−7006.00	−1.68082
$260$	0	0
$261$	−3450.00	−0.818198
$262$	0	0
$263$	−57.0000	−0.0133641	−0.00668207	−	0.999978i	$-0.502127\pi$
$263$	−57.0000	−0.0133641	−0.00668207	+	0.999978i	$0.502127\pi$
$264$	0	0
$265$	3888.00	0.901275
$266$	0	0
$267$	−1200.00	−0.275052
$268$	0	0
$269$	−2700.00	−0.611977	−0.305989	−	0.952035i	$-0.598987\pi$
$269$	−2700.00	−0.611977	−0.305989	+	0.952035i	$0.598987\pi$
$270$	0	0
$271$	3872.00	0.867923	0.433962	−	0.900931i	$-0.357115\pi$
$271$	3872.00	0.867923	0.433962	+	0.900931i	$0.357115\pi$
$272$	0	0
$273$	−3224.00	−0.714745
$274$	0	0
$275$	2508.00	0.549957
$276$	0	0
$277$	7711.00	1.67260	0.836298	−	0.548275i	$-0.184715\pi$
$277$	7711.00	1.67260	0.836298	+	0.548275i	$0.184715\pi$
$278$	0	0
$279$	−736.000	−0.157932
$280$	0	0
$281$	−6858.00	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$281$	−6858.00	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$282$	0	0
$283$	1807.00	0.379558	0.189779	−	0.981827i	$-0.439223\pi$
$283$	1807.00	0.379558	0.189779	+	0.981827i	$0.439223\pi$
$284$	0	0
$285$	−342.000	−0.0710819
$286$	0	0
$287$	7998.00	1.64497
$288$	0	0
$289$	−152.000	−0.0309383
$290$	0	0
$291$	2848.00	0.573721
$292$	0	0
$293$	3012.00	0.600556	0.300278	−	0.953852i	$-0.402921\pi$
$293$	3012.00	0.600556	0.300278	+	0.953852i	$0.402921\pi$
$294$	0	0
$295$	2970.00	0.586170
$296$	0	0
$297$	5700.00	1.11363
$298$	0	0
$299$	−3744.00	−0.724151
$300$	0	0
$301$	−2077.00	−0.397729
$302$	0	0
$303$	−2124.00	−0.402708
$304$	0	0
$305$	117.000	0.0219652
$306$	0	0
$307$	1096.00	0.203753	0.101876	−	0.994797i	$-0.467515\pi$
$307$	1096.00	0.203753	0.101876	+	0.994797i	$0.467515\pi$
$308$	0	0
$309$	2356.00	0.433748
$310$	0	0
$311$	1947.00	0.354998	0.177499	−	0.984121i	$-0.443199\pi$
$311$	1947.00	0.354998	0.177499	+	0.984121i	$0.443199\pi$
$312$	0	0
$313$	7598.00	1.37209	0.686045	−	0.727559i	$-0.259345\pi$
$313$	7598.00	1.37209	0.686045	+	0.727559i	$0.259345\pi$
$314$	0	0
$315$	6417.00	1.14780
$316$	0	0
$317$	−8334.00	−1.47661	−0.738303	−	0.674469i	$-0.764373\pi$
$317$	−8334.00	−1.47661	−0.738303	+	0.674469i	$0.764373\pi$
$318$	0	0
$319$	−8550.00	−1.50065
$320$	0	0
$321$	−228.000	−0.0396440
$322$	0	0
$323$	−1311.00	−0.225839
$324$	0	0
$325$	−2288.00	−0.390509
$326$	0	0
$327$	−2920.00	−0.493812
$328$	0	0
$329$	−17949.0	−3.00778
$330$	0	0
$331$	8368.00	1.38957	0.694784	−	0.719219i	$-0.255500\pi$
$331$	8368.00	1.38957	0.694784	+	0.719219i	$0.255500\pi$
$332$	0	0
$333$	−5198.00	−0.855401
$334$	0	0
$335$	7704.00	1.25646
$336$	0	0
$337$	−10336.0	−1.67074	−0.835368	−	0.549692i	$-0.814745\pi$
$337$	−10336.0	−1.67074	−0.835368	+	0.549692i	$0.814745\pi$
$338$	0	0
$339$	−1644.00	−0.263392
$340$	0	0
$341$	−1824.00	−0.289663
$342$	0	0
$343$	−8525.00	−1.34200
$344$	0	0
$345$	−1296.00	−0.202244
$346$	0	0
$347$	−6879.00	−1.06422	−0.532110	−	0.846675i	$-0.678601\pi$
$347$	−6879.00	−1.06422	−0.532110	+	0.846675i	$0.678601\pi$
$348$	0	0
$349$	6355.00	0.974714	0.487357	−	0.873203i	$-0.337961\pi$
$349$	6355.00	0.974714	0.487357	+	0.873203i	$0.337961\pi$
$350$	0	0
$351$	−5200.00	−0.790756
$352$	0	0
$353$	7218.00	1.08832	0.544158	−	0.838983i	$-0.316849\pi$
$353$	7218.00	1.08832	0.544158	+	0.838983i	$0.316849\pi$
$354$	0	0
$355$	5778.00	0.863843
$356$	0	0
$357$	−4278.00	−0.634218
$358$	0	0
$359$	1665.00	0.244778	0.122389	−	0.992482i	$-0.460944\pi$
$359$	1665.00	0.244778	0.122389	+	0.992482i	$0.460944\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	3836.00	0.554650
$364$	0	0
$365$	−4383.00	−0.628539
$366$	0	0
$367$	13064.0	1.85813	0.929067	−	0.369911i	$-0.120612\pi$
$367$	13064.0	1.85813	0.929067	+	0.369911i	$0.120612\pi$
$368$	0	0
$369$	5934.00	0.837159
$370$	0	0
$371$	−13392.0	−1.87406
$372$	0	0
$373$	10492.0	1.45645	0.728224	−	0.685339i	$-0.240346\pi$
$373$	10492.0	1.45645	0.728224	+	0.685339i	$0.240346\pi$
$374$	0	0
$375$	−3042.00	−0.418902
$376$	0	0
$377$	7800.00	1.06557
$378$	0	0
$379$	−7610.00	−1.03140	−0.515698	−	0.856770i	$-0.672468\pi$
$379$	−7610.00	−1.03140	−0.515698	+	0.856770i	$0.672468\pi$
$380$	0	0
$381$	−4172.00	−0.560992
$382$	0	0
$383$	4008.00	0.534724	0.267362	−	0.963596i	$-0.413848\pi$
$383$	4008.00	0.534724	0.267362	+	0.963596i	$0.413848\pi$
$384$	0	0
$385$	15903.0	2.10517
$386$	0	0
$387$	−1541.00	−0.202412
$388$	0	0
$389$	3525.00	0.459446	0.229723	−	0.973256i	$-0.426218\pi$
$389$	3525.00	0.459446	0.229723	+	0.973256i	$0.426218\pi$
$390$	0	0
$391$	−4968.00	−0.642564
$392$	0	0
$393$	186.000	0.0238739
$394$	0	0
$395$	−6300.00	−0.802500
$396$	0	0
$397$	−6629.00	−0.838035	−0.419018	−	0.907978i	$-0.637625\pi$
$397$	−6629.00	−0.838035	−0.419018	+	0.907978i	$0.637625\pi$
$398$	0	0
$399$	1178.00	0.147804
$400$	0	0
$401$	−10848.0	−1.35093	−0.675465	−	0.737392i	$-0.736057\pi$
$401$	−10848.0	−1.35093	−0.675465	+	0.737392i	$0.736057\pi$
$402$	0	0
$403$	1664.00	0.205682
$404$	0	0
$405$	3789.00	0.464881
$406$	0	0
$407$	−12882.0	−1.56889
$408$	0	0
$409$	−3040.00	−0.367526	−0.183763	−	0.982971i	$-0.558828\pi$
$409$	−3040.00	−0.367526	−0.183763	+	0.982971i	$0.558828\pi$
$410$	0	0
$411$	2538.00	0.304599
$412$	0	0
$413$	−10230.0	−1.21885
$414$	0	0
$415$	108.000	0.0127747
$416$	0	0
$417$	3950.00	0.463867
$418$	0	0
$419$	3900.00	0.454719	0.227360	−	0.973811i	$-0.426991\pi$
$419$	3900.00	0.454719	0.227360	+	0.973811i	$0.426991\pi$
$420$	0	0
$421$	−4412.00	−0.510755	−0.255377	−	0.966841i	$-0.582200\pi$
$421$	−4412.00	−0.510755	−0.255377	+	0.966841i	$0.582200\pi$
$422$	0	0
$423$	−13317.0	−1.53072
$424$	0	0
$425$	−3036.00	−0.346512
$426$	0	0
$427$	−403.000	−0.0456734
$428$	0	0
$429$	−5928.00	−0.667148
$430$	0	0
$431$	432.000	0.0482801	0.0241400	−	0.999709i	$-0.492315\pi$
$431$	432.000	0.0482801	0.0241400	+	0.999709i	$0.492315\pi$
$432$	0	0
$433$	−2002.00	−0.222194	−0.111097	−	0.993810i	$-0.535436\pi$
$433$	−2002.00	−0.222194	−0.111097	+	0.993810i	$0.535436\pi$
$434$	0	0
$435$	2700.00	0.297598
$436$	0	0
$437$	1368.00	0.149749
$438$	0	0
$439$	−1690.00	−0.183734	−0.0918671	−	0.995771i	$-0.529283\pi$
$439$	−1690.00	−0.183734	−0.0918671	+	0.995771i	$0.529283\pi$
$440$	0	0
$441$	−14214.0	−1.53482
$442$	0	0
$443$	1977.00	0.212032	0.106016	−	0.994364i	$-0.466191\pi$
$443$	1977.00	0.212032	0.106016	+	0.994364i	$0.466191\pi$
$444$	0	0
$445$	−5400.00	−0.575246
$446$	0	0
$447$	3390.00	0.358706
$448$	0	0
$449$	−2760.00	−0.290095	−0.145047	−	0.989425i	$-0.546333\pi$
$449$	−2760.00	−0.290095	−0.145047	+	0.989425i	$0.546333\pi$
$450$	0	0
$451$	14706.0	1.53543
$452$	0	0
$453$	3604.00	0.373798
$454$	0	0
$455$	−14508.0	−1.49483
$456$	0	0
$457$	4499.00	0.460513	0.230256	−	0.973130i	$-0.426044\pi$
$457$	4499.00	0.460513	0.230256	+	0.973130i	$0.426044\pi$
$458$	0	0
$459$	−6900.00	−0.701665
$460$	0	0
$461$	11643.0	1.17629	0.588144	−	0.808756i	$-0.299859\pi$
$461$	11643.0	1.17629	0.588144	+	0.808756i	$0.299859\pi$
$462$	0	0
$463$	−1537.00	−0.154277	−0.0771387	−	0.997020i	$-0.524578\pi$
$463$	−1537.00	−0.154277	−0.0771387	+	0.997020i	$0.524578\pi$
$464$	0	0
$465$	576.000	0.0574438
$466$	0	0
$467$	7641.00	0.757138	0.378569	−	0.925573i	$-0.376416\pi$
$467$	7641.00	0.757138	0.378569	+	0.925573i	$0.376416\pi$
$468$	0	0
$469$	−26536.0	−2.61262
$470$	0	0
$471$	6452.00	0.631194
$472$	0	0
$473$	−3819.00	−0.371243
$474$	0	0
$475$	836.000	0.0807543
$476$	0	0
$477$	−9936.00	−0.953749
$478$	0	0
$479$	−8580.00	−0.818435	−0.409217	−	0.912437i	$-0.634198\pi$
$479$	−8580.00	−0.818435	−0.409217	+	0.912437i	$0.634198\pi$
$480$	0	0
$481$	11752.0	1.11402
$482$	0	0
$483$	4464.00	0.420536
$484$	0	0
$485$	12816.0	1.19989
$486$	0	0
$487$	12134.0	1.12904	0.564522	−	0.825418i	$-0.309061\pi$
$487$	12134.0	1.12904	0.564522	+	0.825418i	$0.309061\pi$
$488$	0	0
$489$	−2536.00	−0.234523
$490$	0	0
$491$	5508.00	0.506258	0.253129	−	0.967433i	$-0.418540\pi$
$491$	5508.00	0.506258	0.253129	+	0.967433i	$0.418540\pi$
$492$	0	0
$493$	10350.0	0.945518
$494$	0	0
$495$	11799.0	1.07136
$496$	0	0
$497$	−19902.0	−1.79623
$498$	0	0
$499$	11905.0	1.06802	0.534009	−	0.845479i	$-0.320685\pi$
$499$	11905.0	1.06802	0.534009	+	0.845479i	$0.320685\pi$
$500$	0	0
$501$	1308.00	0.116641
$502$	0	0
$503$	9108.00	0.807367	0.403684	−	0.914899i	$-0.367730\pi$
$503$	9108.00	0.807367	0.403684	+	0.914899i	$0.367730\pi$
$504$	0	0
$505$	−9558.00	−0.842229
$506$	0	0
$507$	1014.00	0.0888231
$508$	0	0
$509$	2520.00	0.219444	0.109722	−	0.993962i	$-0.465004\pi$
$509$	2520.00	0.219444	0.109722	+	0.993962i	$0.465004\pi$
$510$	0	0
$511$	15097.0	1.30695
$512$	0	0
$513$	1900.00	0.163523
$514$	0	0
$515$	10602.0	0.907146
$516$	0	0
$517$	−33003.0	−2.80749
$518$	0	0
$519$	2724.00	0.230386
$520$	0	0
$521$	21612.0	1.81735	0.908675	−	0.417505i	$-0.137095\pi$
$521$	21612.0	1.81735	0.908675	+	0.417505i	$0.137095\pi$
$522$	0	0
$523$	9022.00	0.754311	0.377155	−	0.926150i	$-0.376902\pi$
$523$	9022.00	0.754311	0.377155	+	0.926150i	$0.376902\pi$
$524$	0	0
$525$	2728.00	0.226780
$526$	0	0
$527$	2208.00	0.182509
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	−7590.00	−0.620297
$532$	0	0
$533$	−13416.0	−1.09027
$534$	0	0
$535$	−1026.00	−0.0829119
$536$	0	0
$537$	420.000	0.0337511
$538$	0	0
$539$	−35226.0	−2.81501
$540$	0	0
$541$	9253.00	0.735337	0.367669	−	0.929957i	$-0.380156\pi$
$541$	9253.00	0.735337	0.367669	+	0.929957i	$0.380156\pi$
$542$	0	0
$543$	−4.00000	−0.000316126 0
$544$	0	0
$545$	−13140.0	−1.03276
$546$	0	0
$547$	−13244.0	−1.03523	−0.517617	−	0.855613i	$-0.673181\pi$
$547$	−13244.0	−1.03523	−0.517617	+	0.855613i	$0.673181\pi$
$548$	0	0
$549$	−299.000	−0.0232441
$550$	0	0
$551$	−2850.00	−0.220352
$552$	0	0
$553$	21700.0	1.66868
$554$	0	0
$555$	4068.00	0.311130
$556$	0	0
$557$	−1569.00	−0.119355	−0.0596774	−	0.998218i	$-0.519007\pi$
$557$	−1569.00	−0.119355	−0.0596774	+	0.998218i	$0.519007\pi$
$558$	0	0
$559$	3484.00	0.263609
$560$	0	0
$561$	−7866.00	−0.591984
$562$	0	0
$563$	15762.0	1.17991	0.589955	−	0.807436i	$-0.299146\pi$
$563$	15762.0	1.17991	0.589955	+	0.807436i	$0.299146\pi$
$564$	0	0
$565$	−7398.00	−0.550861
$566$	0	0
$567$	−13051.0	−0.966650
$568$	0	0
$569$	−13800.0	−1.01674	−0.508371	−	0.861138i	$-0.669752\pi$
$569$	−13800.0	−1.01674	−0.508371	+	0.861138i	$0.669752\pi$
$570$	0	0
$571$	4348.00	0.318666	0.159333	−	0.987225i	$-0.449066\pi$
$571$	4348.00	0.318666	0.159333	+	0.987225i	$0.449066\pi$
$572$	0	0
$573$	−5286.00	−0.385385
$574$	0	0
$575$	3168.00	0.229765
$576$	0	0
$577$	3539.00	0.255339	0.127669	−	0.991817i	$-0.459250\pi$
$577$	3539.00	0.255339	0.127669	+	0.991817i	$0.459250\pi$
$578$	0	0
$579$	6496.00	0.466260
$580$	0	0
$581$	−372.000	−0.0265631
$582$	0	0
$583$	−24624.0	−1.74927
$584$	0	0
$585$	−10764.0	−0.760746
$586$	0	0
$587$	6321.00	0.444456	0.222228	−	0.974995i	$-0.428667\pi$
$587$	6321.00	0.444456	0.222228	+	0.974995i	$0.428667\pi$
$588$	0	0
$589$	−608.000	−0.0425335
$590$	0	0
$591$	6252.00	0.435149
$592$	0	0
$593$	13278.0	0.919498	0.459749	−	0.888049i	$-0.347939\pi$
$593$	13278.0	0.919498	0.459749	+	0.888049i	$0.347939\pi$
$594$	0	0
$595$	−19251.0	−1.32641
$596$	0	0
$597$	−5990.00	−0.410644
$598$	0	0
$599$	20400.0	1.39152	0.695761	−	0.718274i	$-0.255067\pi$
$599$	20400.0	1.39152	0.695761	+	0.718274i	$0.255067\pi$
$600$	0	0
$601$	−22198.0	−1.50661	−0.753307	−	0.657669i	$-0.771543\pi$
$601$	−22198.0	−1.50661	−0.753307	+	0.657669i	$0.771543\pi$
$602$	0	0
$603$	−19688.0	−1.32961
$604$	0	0
$605$	17262.0	1.16000
$606$	0	0
$607$	9824.00	0.656909	0.328455	−	0.944520i	$-0.393472\pi$
$607$	9824.00	0.656909	0.328455	+	0.944520i	$0.393472\pi$
$608$	0	0
$609$	−9300.00	−0.618810
$610$	0	0
$611$	30108.0	1.99352
$612$	0	0
$613$	4327.00	0.285099	0.142550	−	0.989788i	$-0.454470\pi$
$613$	4327.00	0.285099	0.142550	+	0.989788i	$0.454470\pi$
$614$	0	0
$615$	−4644.00	−0.304495
$616$	0	0
$617$	−14151.0	−0.923335	−0.461668	−	0.887053i	$-0.652749\pi$
$617$	−14151.0	−0.923335	−0.461668	+	0.887053i	$0.652749\pi$
$618$	0	0
$619$	−22460.0	−1.45839	−0.729195	−	0.684306i	$-0.760105\pi$
$619$	−22460.0	−1.45839	−0.729195	+	0.684306i	$0.760105\pi$
$620$	0	0
$621$	7200.00	0.465259
$622$	0	0
$623$	18600.0	1.19614
$624$	0	0
$625$	−8189.00	−0.524096
$626$	0	0
$627$	2166.00	0.137961
$628$	0	0
$629$	15594.0	0.988511
$630$	0	0
$631$	−16363.0	−1.03233	−0.516165	−	0.856489i	$-0.672641\pi$
$631$	−16363.0	−1.03233	−0.516165	+	0.856489i	$0.672641\pi$
$632$	0	0
$633$	8636.00	0.542259
$634$	0	0
$635$	−18774.0	−1.17327
$636$	0	0
$637$	32136.0	1.99886
$638$	0	0
$639$	−14766.0	−0.914138
$640$	0	0
$641$	5592.00	0.344572	0.172286	−	0.985047i	$-0.444885\pi$
$641$	5592.00	0.344572	0.172286	+	0.985047i	$0.444885\pi$
$642$	0	0
$643$	−16553.0	−1.01522	−0.507610	−	0.861587i	$-0.669471\pi$
$643$	−16553.0	−1.01522	−0.507610	+	0.861587i	$0.669471\pi$
$644$	0	0
$645$	1206.00	0.0736220
$646$	0	0
$647$	−4611.00	−0.280181	−0.140091	−	0.990139i	$-0.544739\pi$
$647$	−4611.00	−0.280181	−0.140091	+	0.990139i	$0.544739\pi$
$648$	0	0
$649$	−18810.0	−1.13768
$650$	0	0
$651$	−1984.00	−0.119446
$652$	0	0
$653$	−16413.0	−0.983599	−0.491800	−	0.870708i	$-0.663661\pi$
$653$	−16413.0	−0.983599	−0.491800	+	0.870708i	$0.663661\pi$
$654$	0	0
$655$	837.000	0.0499302
$656$	0	0
$657$	11201.0	0.665133
$658$	0	0
$659$	−27390.0	−1.61906	−0.809532	−	0.587076i	$-0.800279\pi$
$659$	−27390.0	−1.61906	−0.809532	+	0.587076i	$0.800279\pi$
$660$	0	0
$661$	−26912.0	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$661$	−26912.0	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$662$	0	0
$663$	7176.00	0.420351
$664$	0	0
$665$	5301.00	0.309119
$666$	0	0
$667$	−10800.0	−0.626953
$668$	0	0
$669$	1036.00	0.0598716
$670$	0	0
$671$	−741.000	−0.0426319
$672$	0	0
$673$	−21562.0	−1.23500	−0.617499	−	0.786571i	$-0.711854\pi$
$673$	−21562.0	−1.23500	−0.617499	+	0.786571i	$0.711854\pi$
$674$	0	0
$675$	4400.00	0.250898
$676$	0	0
$677$	21966.0	1.24700	0.623502	−	0.781822i	$-0.285709\pi$
$677$	21966.0	1.24700	0.623502	+	0.781822i	$0.285709\pi$
$678$	0	0
$679$	−44144.0	−2.49498
$680$	0	0
$681$	−5688.00	−0.320066
$682$	0	0
$683$	−15348.0	−0.859846	−0.429923	−	0.902866i	$-0.641459\pi$
$683$	−15348.0	−0.859846	−0.429923	+	0.902866i	$0.641459\pi$
$684$	0	0
$685$	11421.0	0.637042
$686$	0	0
$687$	−3490.00	−0.193816
$688$	0	0
$689$	22464.0	1.24210
$690$	0	0
$691$	−8147.00	−0.448519	−0.224259	−	0.974529i	$-0.571996\pi$
$691$	−8147.00	−0.448519	−0.224259	+	0.974529i	$0.571996\pi$
$692$	0	0
$693$	−40641.0	−2.22774
$694$	0	0
$695$	17775.0	0.970136
$696$	0	0
$697$	−17802.0	−0.967430
$698$	0	0
$699$	10566.0	0.571735
$700$	0	0
$701$	−14982.0	−0.807222	−0.403611	−	0.914931i	$-0.632245\pi$
$701$	−14982.0	−0.807222	−0.403611	+	0.914931i	$0.632245\pi$
$702$	0	0
$703$	−4294.00	−0.230372
$704$	0	0
$705$	10422.0	0.556759
$706$	0	0
$707$	32922.0	1.75129
$708$	0	0
$709$	−21890.0	−1.15952	−0.579758	−	0.814789i	$-0.696853\pi$
$709$	−21890.0	−1.15952	−0.579758	+	0.814789i	$0.696853\pi$
$710$	0	0
$711$	16100.0	0.849222
$712$	0	0
$713$	−2304.00	−0.121018
$714$	0	0
$715$	−26676.0	−1.39528
$716$	0	0
$717$	930.000	0.0484400
$718$	0	0
$719$	−27015.0	−1.40124	−0.700619	−	0.713536i	$-0.747093\pi$
$719$	−27015.0	−1.40124	−0.700619	+	0.713536i	$0.747093\pi$
$720$	0	0
$721$	−36518.0	−1.88627
$722$	0	0
$723$	−14156.0	−0.728171
$724$	0	0
$725$	−6600.00	−0.338094
$726$	0	0
$727$	−13021.0	−0.664267	−0.332134	−	0.943232i	$-0.607768\pi$
$727$	−13021.0	−0.664267	−0.332134	+	0.943232i	$0.607768\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	4623.00	0.233909
$732$	0	0
$733$	6262.00	0.315542	0.157771	−	0.987476i	$-0.449569\pi$
$733$	6262.00	0.315542	0.157771	+	0.987476i	$0.449569\pi$
$734$	0	0
$735$	11124.0	0.558252
$736$	0	0
$737$	−48792.0	−2.43864
$738$	0	0
$739$	10855.0	0.540335	0.270168	−	0.962813i	$-0.412921\pi$
$739$	10855.0	0.540335	0.270168	+	0.962813i	$0.412921\pi$
$740$	0	0
$741$	−1976.00	−0.0979624
$742$	0	0
$743$	−14892.0	−0.735309	−0.367654	−	0.929962i	$-0.619839\pi$
$743$	−14892.0	−0.735309	−0.367654	+	0.929962i	$0.619839\pi$
$744$	0	0
$745$	15255.0	0.750201
$746$	0	0
$747$	−276.000	−0.0135185
$748$	0	0
$749$	3534.00	0.172403
$750$	0	0
$751$	28952.0	1.40676	0.703378	−	0.710816i	$-0.251674\pi$
$751$	28952.0	1.40676	0.703378	+	0.710816i	$0.251674\pi$
$752$	0	0
$753$	−7134.00	−0.345256
$754$	0	0
$755$	16218.0	0.781766
$756$	0	0
$757$	3541.00	0.170013	0.0850065	−	0.996380i	$-0.472909\pi$
$757$	3541.00	0.170013	0.0850065	+	0.996380i	$0.472909\pi$
$758$	0	0
$759$	8208.00	0.392532
$760$	0	0
$761$	22617.0	1.07735	0.538676	−	0.842513i	$-0.318925\pi$
$761$	22617.0	1.07735	0.538676	+	0.842513i	$0.318925\pi$
$762$	0	0
$763$	45260.0	2.14747
$764$	0	0
$765$	−14283.0	−0.675037
$766$	0	0
$767$	17160.0	0.807838
$768$	0	0
$769$	11495.0	0.539038	0.269519	−	0.962995i	$-0.413135\pi$
$769$	11495.0	0.539038	0.269519	+	0.962995i	$0.413135\pi$
$770$	0	0
$771$	−3792.00	−0.177128
$772$	0	0
$773$	14622.0	0.680358	0.340179	−	0.940361i	$-0.389512\pi$
$773$	14622.0	0.680358	0.340179	+	0.940361i	$0.389512\pi$
$774$	0	0
$775$	−1408.00	−0.0652605
$776$	0	0
$777$	−14012.0	−0.646947
$778$	0	0
$779$	4902.00	0.225459
$780$	0	0
$781$	−36594.0	−1.67661
$782$	0	0
$783$	−15000.0	−0.684618
$784$	0	0
$785$	29034.0	1.32009
$786$	0	0
$787$	−7124.00	−0.322672	−0.161336	−	0.986900i	$-0.551580\pi$
$787$	−7124.00	−0.322672	−0.161336	+	0.986900i	$0.551580\pi$
$788$	0	0
$789$	−114.000	−0.00514386
$790$	0	0
$791$	25482.0	1.14543
$792$	0	0
$793$	676.000	0.0302717
$794$	0	0
$795$	7776.00	0.346901
$796$	0	0
$797$	3576.00	0.158932	0.0794658	−	0.996838i	$-0.474679\pi$
$797$	3576.00	0.158932	0.0794658	+	0.996838i	$0.474679\pi$
$798$	0	0
$799$	39951.0	1.76892
$800$	0	0
$801$	13800.0	0.608738
$802$	0	0
$803$	27759.0	1.21992
$804$	0	0
$805$	20088.0	0.879514
$806$	0	0
$807$	−5400.00	−0.235550
$808$	0	0
$809$	42855.0	1.86242	0.931212	−	0.364477i	$-0.118752\pi$
$809$	42855.0	1.86242	0.931212	+	0.364477i	$0.118752\pi$
$810$	0	0
$811$	15568.0	0.674065	0.337032	−	0.941493i	$-0.390577\pi$
$811$	15568.0	0.674065	0.337032	+	0.941493i	$0.390577\pi$
$812$	0	0
$813$	7744.00	0.334064
$814$	0	0
$815$	−11412.0	−0.490485
$816$	0	0
$817$	−1273.00	−0.0545124
$818$	0	0
$819$	37076.0	1.58186
$820$	0	0
$821$	−2517.00	−0.106996	−0.0534981	−	0.998568i	$-0.517037\pi$
$821$	−2517.00	−0.106996	−0.0534981	+	0.998568i	$0.517037\pi$
$822$	0	0
$823$	−9727.00	−0.411983	−0.205991	−	0.978554i	$-0.566042\pi$
$823$	−9727.00	−0.411983	−0.205991	+	0.978554i	$0.566042\pi$
$824$	0	0
$825$	5016.00	0.211678
$826$	0	0
$827$	−28224.0	−1.18675	−0.593376	−	0.804925i	$-0.702205\pi$
$827$	−28224.0	−1.18675	−0.593376	+	0.804925i	$0.702205\pi$
$828$	0	0
$829$	−3080.00	−0.129038	−0.0645192	−	0.997916i	$-0.520551\pi$
$829$	−3080.00	−0.129038	−0.0645192	+	0.997916i	$0.520551\pi$
$830$	0	0
$831$	15422.0	0.643782
$832$	0	0
$833$	42642.0	1.77366
$834$	0	0
$835$	5886.00	0.243944
$836$	0	0
$837$	−3200.00	−0.132148
$838$	0	0
$839$	26790.0	1.10238	0.551188	−	0.834381i	$-0.314175\pi$
$839$	26790.0	1.10238	0.551188	+	0.834381i	$0.314175\pi$
$840$	0	0
$841$	−1889.00	−0.0774530
$842$	0	0
$843$	−13716.0	−0.560385
$844$	0	0
$845$	4563.00	0.185766
$846$	0	0
$847$	−59458.0	−2.41204
$848$	0	0
$849$	3614.00	0.146092
$850$	0	0
$851$	−16272.0	−0.655461
$852$	0	0
$853$	−19178.0	−0.769803	−0.384902	−	0.922958i	$-0.625765\pi$
$853$	−19178.0	−0.769803	−0.384902	+	0.922958i	$0.625765\pi$
$854$	0	0
$855$	3933.00	0.157317
$856$	0	0
$857$	−2406.00	−0.0959013	−0.0479506	−	0.998850i	$-0.515269\pi$
$857$	−2406.00	−0.0959013	−0.0479506	+	0.998850i	$0.515269\pi$
$858$	0	0
$859$	−9125.00	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$859$	−9125.00	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$860$	0	0
$861$	15996.0	0.633150
$862$	0	0
$863$	8898.00	0.350975	0.175488	−	0.984482i	$-0.443850\pi$
$863$	8898.00	0.350975	0.175488	+	0.984482i	$0.443850\pi$
$864$	0	0
$865$	12258.0	0.481832
$866$	0	0
$867$	−304.000	−0.0119082
$868$	0	0
$869$	39900.0	1.55755
$870$	0	0
$871$	44512.0	1.73161
$872$	0	0
$873$	−32752.0	−1.26974
$874$	0	0
$875$	47151.0	1.82171
$876$	0	0
$877$	15886.0	0.611667	0.305834	−	0.952085i	$-0.401065\pi$
$877$	15886.0	0.611667	0.305834	+	0.952085i	$0.401065\pi$
$878$	0	0
$879$	6024.00	0.231154
$880$	0	0
$881$	−25683.0	−0.982159	−0.491080	−	0.871115i	$-0.663398\pi$
$881$	−25683.0	−0.982159	−0.491080	+	0.871115i	$0.663398\pi$
$882$	0	0
$883$	28267.0	1.07730	0.538652	−	0.842528i	$-0.318934\pi$
$883$	28267.0	1.07730	0.538652	+	0.842528i	$0.318934\pi$
$884$	0	0
$885$	5940.00	0.225617
$886$	0	0
$887$	−2466.00	−0.0933486	−0.0466743	−	0.998910i	$-0.514862\pi$
$887$	−2466.00	−0.0933486	−0.0466743	+	0.998910i	$0.514862\pi$
$888$	0	0
$889$	64666.0	2.43963
$890$	0	0
$891$	−23997.0	−0.902278
$892$	0	0
$893$	−11001.0	−0.412245
$894$	0	0
$895$	1890.00	0.0705874
$896$	0	0
$897$	−7488.00	−0.278726
$898$	0	0
$899$	4800.00	0.178074
$900$	0	0
$901$	29808.0	1.10216
$902$	0	0
$903$	−4154.00	−0.153086
$904$	0	0
$905$	−18.0000	−0.000661149 0
$906$	0	0
$907$	−29324.0	−1.07353	−0.536763	−	0.843733i	$-0.680353\pi$
$907$	−29324.0	−1.07353	−0.536763	+	0.843733i	$0.680353\pi$
$908$	0	0
$909$	24426.0	0.891264
$910$	0	0
$911$	47142.0	1.71447	0.857236	−	0.514924i	$-0.172180\pi$
$911$	47142.0	1.71447	0.857236	+	0.514924i	$0.172180\pi$
$912$	0	0
$913$	−684.000	−0.0247942
$914$	0	0
$915$	234.000	0.00845443
$916$	0	0
$917$	−2883.00	−0.103822
$918$	0	0
$919$	−39940.0	−1.43362	−0.716811	−	0.697267i	$-0.754399\pi$
$919$	−39940.0	−1.43362	−0.716811	+	0.697267i	$0.754399\pi$
$920$	0	0
$921$	2192.00	0.0784244
$922$	0	0
$923$	33384.0	1.19052
$924$	0	0
$925$	−9944.00	−0.353467
$926$	0	0
$927$	−27094.0	−0.959961
$928$	0	0
$929$	−4410.00	−0.155745	−0.0778727	−	0.996963i	$-0.524813\pi$
$929$	−4410.00	−0.155745	−0.0778727	+	0.996963i	$0.524813\pi$
$930$	0	0
$931$	−11742.0	−0.413350
$932$	0	0
$933$	3894.00	0.136639
$934$	0	0
$935$	−35397.0	−1.23808
$936$	0	0
$937$	−41671.0	−1.45286	−0.726431	−	0.687239i	$-0.758822\pi$
$937$	−41671.0	−1.45286	−0.726431	+	0.687239i	$0.758822\pi$
$938$	0	0
$939$	15196.0	0.528118
$940$	0	0
$941$	−4062.00	−0.140720	−0.0703599	−	0.997522i	$-0.522415\pi$
$941$	−4062.00	−0.140720	−0.0703599	+	0.997522i	$0.522415\pi$
$942$	0	0
$943$	18576.0	0.641482
$944$	0	0
$945$	27900.0	0.960410
$946$	0	0
$947$	45036.0	1.54538	0.772689	−	0.634785i	$-0.218911\pi$
$947$	45036.0	1.54538	0.772689	+	0.634785i	$0.218911\pi$
$948$	0	0
$949$	−25324.0	−0.866230
$950$	0	0
$951$	−16668.0	−0.568346
$952$	0	0
$953$	26508.0	0.901027	0.450513	−	0.892770i	$-0.351241\pi$
$953$	26508.0	0.901027	0.450513	+	0.892770i	$0.351241\pi$
$954$	0	0
$955$	−23787.0	−0.805999
$956$	0	0
$957$	−17100.0	−0.577601
$958$	0	0
$959$	−39339.0	−1.32463
$960$	0	0
$961$	−28767.0	−0.965627
$962$	0	0
$963$	2622.00	0.0877391
$964$	0	0
$965$	29232.0	0.975141
$966$	0	0
$967$	−15976.0	−0.531286	−0.265643	−	0.964071i	$-0.585584\pi$
$967$	−15976.0	−0.531286	−0.265643	+	0.964071i	$0.585584\pi$
$968$	0	0
$969$	−2622.00	−0.0869255
$970$	0	0
$971$	39468.0	1.30442	0.652208	−	0.758040i	$-0.273843\pi$
$971$	39468.0	1.30442	0.652208	+	0.758040i	$0.273843\pi$
$972$	0	0
$973$	−61225.0	−2.01725
$974$	0	0
$975$	−4576.00	−0.150307
$976$	0	0
$977$	21804.0	0.713994	0.356997	−	0.934106i	$-0.383801\pi$
$977$	21804.0	0.713994	0.356997	+	0.934106i	$0.383801\pi$
$978$	0	0
$979$	34200.0	1.11648
$980$	0	0
$981$	33580.0	1.09289
$982$	0	0
$983$	11268.0	0.365609	0.182804	−	0.983149i	$-0.441483\pi$
$983$	11268.0	0.365609	0.182804	+	0.983149i	$0.441483\pi$
$984$	0	0
$985$	28134.0	0.910075
$986$	0	0
$987$	−35898.0	−1.15770
$988$	0	0
$989$	−4824.00	−0.155100
$990$	0	0
$991$	−778.000	−0.0249384	−0.0124692	−	0.999922i	$-0.503969\pi$
$991$	−778.000	−0.0249384	−0.0124692	+	0.999922i	$0.503969\pi$
$992$	0	0
$993$	16736.0	0.534845
$994$	0	0
$995$	−26955.0	−0.858825
$996$	0	0
$997$	−389.000	−0.0123568	−0.00617841	−	0.999981i	$-0.501967\pi$
$997$	−389.000	−0.0123568	−0.00617841	+	0.999981i	$0.501967\pi$
$998$	0	0
$999$	−22600.0	−0.715748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.4.a.e.1.1		1
4.3	odd	2		1216.4.a.b.1.1		1
8.3	odd	2		304.4.a.a.1.1		1
8.5	even	2		38.4.a.a.1.1	✓	1
24.5	odd	2		342.4.a.d.1.1		1
40.13	odd	4		950.4.b.d.799.2		2
40.29	even	2		950.4.a.d.1.1		1
40.37	odd	4		950.4.b.d.799.1		2
56.13	odd	2		1862.4.a.a.1.1		1
152.37	odd	2		722.4.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.a.a.1.1	✓	1	8.5	even	2
304.4.a.a.1.1		1	8.3	odd	2
342.4.a.d.1.1		1	24.5	odd	2
722.4.a.d.1.1		1	152.37	odd	2
950.4.a.d.1.1		1	40.29	even	2
950.4.b.d.799.1		2	40.37	odd	4
950.4.b.d.799.2		2	40.13	odd	4
1216.4.a.b.1.1		1	4.3	odd	2
1216.4.a.e.1.1		1	1.1	even	1	trivial
1862.4.a.a.1.1		1	56.13	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more