LMFDB - Embedded newform 342.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,4,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("342.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	342.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:	$20.1786532220$
Analytic rank:	$1$
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	$\chi$	$=$	342.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +9.00000 q^{5} -31.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +9.00000 q^{5} -31.0000 q^{7} +8.00000 q^{8} +18.0000 q^{10} -57.0000 q^{11} -52.0000 q^{13} -62.0000 q^{14} +16.0000 q^{16} -69.0000 q^{17} +19.0000 q^{19} +36.0000 q^{20} -114.000 q^{22} +72.0000 q^{23} -44.0000 q^{25} -104.000 q^{26} -124.000 q^{28} +150.000 q^{29} +32.0000 q^{31} +32.0000 q^{32} -138.000 q^{34} -279.000 q^{35} -226.000 q^{37} +38.0000 q^{38} +72.0000 q^{40} +258.000 q^{41} -67.0000 q^{43} -228.000 q^{44} +144.000 q^{46} -579.000 q^{47} +618.000 q^{49} -88.0000 q^{50} -208.000 q^{52} +432.000 q^{53} -513.000 q^{55} -248.000 q^{56} +300.000 q^{58} +330.000 q^{59} -13.0000 q^{61} +64.0000 q^{62} +64.0000 q^{64} -468.000 q^{65} -856.000 q^{67} -276.000 q^{68} -558.000 q^{70} -642.000 q^{71} -487.000 q^{73} -452.000 q^{74} +76.0000 q^{76} +1767.00 q^{77} -700.000 q^{79} +144.000 q^{80} +516.000 q^{82} +12.0000 q^{83} -621.000 q^{85} -134.000 q^{86} -456.000 q^{88} +600.000 q^{89} +1612.00 q^{91} +288.000 q^{92} -1158.00 q^{94} +171.000 q^{95} +1424.00 q^{97} +1236.00 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	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	$0.815647\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	18.0000	0.569210
$11$	−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	$-0.687167\pi$
$13$	−52.0000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−62.0000	−1.18359
$15$	0	0
$16$	16.0000	0.250000
$17$	−69.0000	−0.984409	−0.492205	−	0.870480i	$-0.663809\pi$
$17$	−69.0000	−0.984409	−0.492205	+	0.870480i	$0.663809\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	36.0000	0.402492
$21$	0	0
$22$	−114.000	−1.10477
$23$	72.0000	0.652741	0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000	0.652741	0.326370	+	0.945242i	$0.394174\pi$
$24$	0	0
$25$	−44.0000	−0.352000
$26$	−104.000	−0.784465
$27$	0	0
$28$	−124.000	−0.836921
$29$	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	$0.470450\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−138.000	−0.696082
$35$	−279.000	−1.34742
$36$	0	0
$37$	−226.000	−1.00417	−0.502083	−	0.864819i	$-0.667433\pi$
$37$	−226.000	−1.00417	−0.502083	+	0.864819i	$0.667433\pi$
$38$	38.0000	0.162221
$39$	0	0
$40$	72.0000	0.284605
$41$	258.000	0.982752	0.491376	−	0.870948i	$-0.336494\pi$
$41$	258.000	0.982752	0.491376	+	0.870948i	$0.336494\pi$
$42$	0	0
$43$	−67.0000	−0.237614	−0.118807	−	0.992917i	$-0.537907\pi$
$43$	−67.0000	−0.237614	−0.118807	+	0.992917i	$0.537907\pi$
$44$	−228.000	−0.781188
$45$	0	0
$46$	144.000	0.461557
$47$	−579.000	−1.79693	−0.898466	−	0.439043i	$-0.855318\pi$
$47$	−579.000	−1.79693	−0.898466	+	0.439043i	$0.855318\pi$
$48$	0	0
$49$	618.000	1.80175
$50$	−88.0000	−0.248902
$51$	0	0
$52$	−208.000	−0.554700
$53$	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
$56$	−248.000	−0.591793
$57$	0	0
$58$	300.000	0.679171
$59$	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	$-0.504343\pi$
$61$	−13.0000	−0.0272865	−0.0136433	+	0.999907i	$0.504343\pi$
$62$	64.0000	0.131097
$63$	0	0
$64$	64.0000	0.125000
$65$	−468.000	−0.893050
$66$	0	0
$67$	−856.000	−1.56085	−0.780426	−	0.625249i	$-0.784998\pi$
$67$	−856.000	−1.56085	−0.780426	+	0.625249i	$0.784998\pi$
$68$	−276.000	−0.492205
$69$	0	0
$70$	−558.000	−0.952768
$71$	−642.000	−1.07312	−0.536559	−	0.843863i	$-0.680276\pi$
$71$	−642.000	−1.07312	−0.536559	+	0.843863i	$0.680276\pi$
$72$	0	0
$73$	−487.000	−0.780809	−0.390404	−	0.920643i	$-0.627665\pi$
$73$	−487.000	−0.780809	−0.390404	+	0.920643i	$0.627665\pi$
$74$	−452.000	−0.710053
$75$	0	0
$76$	76.0000	0.114708
$77$	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	$0.666100\pi$
$80$	144.000	0.201246
$81$	0	0
$82$	516.000	0.694911
$83$	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
$86$	−134.000	−0.168019
$87$	0	0
$88$	−456.000	−0.552384
$89$	600.000	0.714605	0.357303	−	0.933989i	$-0.383696\pi$
$89$	600.000	0.714605	0.357303	+	0.933989i	$0.383696\pi$
$90$	0	0
$91$	1612.00	1.85696
$92$	288.000	0.326370
$93$	0	0
$94$	−1158.00	−1.27062
$95$	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	$0.232313\pi$
$98$	1236.00	1.27403
$99$	0	0
$100$	−176.000	−0.176000
$101$	−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	$0.309472\pi$
$104$	−416.000	−0.392232
$105$	0	0
$106$	864.000	0.791690
$107$	−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	$-0.278321\pi$
$109$	1460.00	1.28296	0.641480	+	0.767140i	$0.278321\pi$
$110$	−1026.00	−0.889321
$111$	0	0
$112$	−496.000	−0.418461
$113$	822.000	0.684312	0.342156	−	0.939643i	$-0.388843\pi$
$113$	822.000	0.684312	0.342156	+	0.939643i	$0.388843\pi$
$114$	0	0
$115$	648.000	0.525446
$116$	600.000	0.480247
$117$	0	0
$118$	660.000	0.514898
$119$	2139.00	1.64775
$120$	0	0
$121$	1918.00	1.44102
$122$	−26.0000	−0.0192945
$123$	0	0
$124$	128.000	0.0926995
$125$	−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	$0.759898\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−936.000	−0.631482
$131$	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
$134$	−1712.00	−1.10369
$135$	0	0
$136$	−552.000	−0.348041
$137$	−1269.00	−0.791372	−0.395686	−	0.918386i	$-0.629493\pi$
$137$	−1269.00	−0.791372	−0.395686	+	0.918386i	$0.629493\pi$
$138$	0	0
$139$	−1975.00	−1.20516	−0.602580	−	0.798058i	$-0.705861\pi$
$139$	−1975.00	−1.20516	−0.602580	+	0.798058i	$0.705861\pi$
$140$	−1116.00	−0.673709
$141$	0	0
$142$	−1284.00	−0.758809
$143$	2964.00	1.73330
$144$	0	0
$145$	1350.00	0.773182
$146$	−974.000	−0.552115
$147$	0	0
$148$	−904.000	−0.502083
$149$	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	$0.338609\pi$
$152$	152.000	0.0811107
$153$	0	0
$154$	3534.00	1.84921
$155$	288.000	0.149243
$156$	0	0
$157$	−3226.00	−1.63989	−0.819945	−	0.572442i	$-0.805996\pi$
$157$	−3226.00	−1.63989	−0.819945	+	0.572442i	$0.805996\pi$
$158$	−1400.00	−0.704924
$159$	0	0
$160$	288.000	0.142302
$161$	−2232.00	−1.09259
$162$	0	0
$163$	1268.00	0.609309	0.304655	−	0.952463i	$-0.401459\pi$
$163$	1268.00	0.609309	0.304655	+	0.952463i	$0.401459\pi$
$164$	1032.00	0.491376
$165$	0	0
$166$	24.0000	0.0112215
$167$	−654.000	−0.303042	−0.151521	−	0.988454i	$-0.548417\pi$
$167$	−654.000	−0.303042	−0.151521	+	0.988454i	$0.548417\pi$
$168$	0	0
$169$	507.000	0.230769
$170$	−1242.00	−0.560336
$171$	0	0
$172$	−268.000	−0.118807
$173$	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
$176$	−912.000	−0.390594
$177$	0	0
$178$	1200.00	0.505302
$179$	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	$-0.499869\pi$
$181$	2.00000	0.000821319 0	0.000410660		1.00000i	$0.499869\pi$
$182$	3224.00	1.31307
$183$	0	0
$184$	576.000	0.230779
$185$	−2034.00	−0.808339
$186$	0	0
$187$	3933.00	1.53802
$188$	−2316.00	−0.898466
$189$	0	0
$190$	342.000	0.130586
$191$	2643.00	1.00126	0.500630	−	0.865661i	$-0.333102\pi$
$191$	2643.00	1.00126	0.500630	+	0.865661i	$0.333102\pi$
$192$	0	0
$193$	3248.00	1.21138	0.605690	−	0.795701i	$-0.292897\pi$
$193$	3248.00	1.21138	0.605690	+	0.795701i	$0.292897\pi$
$194$	2848.00	1.05399
$195$	0	0
$196$	2472.00	0.900875
$197$	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	$0.679102\pi$
$200$	−352.000	−0.124451
$201$	0	0
$202$	−2124.00	−0.739822
$203$	−4650.00	−1.60771
$204$	0	0
$205$	2322.00	0.791100
$206$	2356.00	0.796846
$207$	0	0
$208$	−832.000	−0.277350
$209$	−1083.00	−0.358434
$210$	0	0
$211$	−4318.00	−1.40883	−0.704416	−	0.709788i	$-0.748791\pi$
$211$	−4318.00	−1.40883	−0.704416	+	0.709788i	$0.748791\pi$
$212$	1728.00	0.559809
$213$	0	0
$214$	−228.000	−0.0728307
$215$	−603.000	−0.191276
$216$	0	0
$217$	−992.000	−0.310329
$218$	2920.00	0.907190
$219$	0	0
$220$	−2052.00	−0.628845
$221$	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	$0.475218\pi$
$224$	−992.000	−0.295896
$225$	0	0
$226$	1644.00	0.483882
$227$	−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	$-0.418986\pi$
$229$	1745.00	0.503550	0.251775	+	0.967786i	$0.418986\pi$
$230$	1296.00	0.371547
$231$	0	0
$232$	1200.00	0.339586
$233$	−5283.00	−1.48541	−0.742706	−	0.669618i	$-0.766458\pi$
$233$	−5283.00	−1.48541	−0.742706	+	0.669618i	$0.766458\pi$
$234$	0	0
$235$	−5211.00	−1.44650
$236$	1320.00	0.364088
$237$	0	0
$238$	4278.00	1.16513
$239$	−465.000	−0.125851	−0.0629254	−	0.998018i	$-0.520043\pi$
$239$	−465.000	−0.125851	−0.0629254	+	0.998018i	$0.520043\pi$
$240$	0	0
$241$	−7078.00	−1.89184	−0.945921	−	0.324396i	$-0.894839\pi$
$241$	−7078.00	−1.89184	−0.945921	+	0.324396i	$0.894839\pi$
$242$	3836.00	1.01896
$243$	0	0
$244$	−52.0000	−0.0136433
$245$	5562.00	1.45038
$246$	0	0
$247$	−988.000	−0.254514
$248$	256.000	0.0655485
$249$	0	0
$250$	−3042.00	−0.769572
$251$	−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
$254$	−4172.00	−1.03061
$255$	0	0
$256$	256.000	0.0625000
$257$	1896.00	0.460192	0.230096	−	0.973168i	$-0.426096\pi$
$257$	1896.00	0.460192	0.230096	+	0.973168i	$0.426096\pi$
$258$	0	0
$259$	7006.00	1.68082
$260$	−1872.00	−0.446525
$261$	0	0
$262$	186.000	0.0438592
$263$	57.0000	0.0133641	0.00668207	−	0.999978i	$-0.497873\pi$
$263$	57.0000	0.0133641	0.00668207	+	0.999978i	$0.497873\pi$
$264$	0	0
$265$	3888.00	0.901275
$266$	−1178.00	−0.271533
$267$	0	0
$268$	−3424.00	−0.780426
$269$	−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	$0.357115\pi$
$272$	−1104.00	−0.246102
$273$	0	0
$274$	−2538.00	−0.559585
$275$	2508.00	0.549957
$276$	0	0
$277$	−7711.00	−1.67260	−0.836298	−	0.548275i	$-0.815285\pi$
$277$	−7711.00	−1.67260	−0.836298	+	0.548275i	$0.815285\pi$
$278$	−3950.00	−0.852177
$279$	0	0
$280$	−2232.00	−0.476384
$281$	6858.00	1.45592	0.727961	−	0.685619i	$-0.240468\pi$
$281$	6858.00	1.45592	0.727961	+	0.685619i	$0.240468\pi$
$282$	0	0
$283$	−1807.00	−0.379558	−0.189779	−	0.981827i	$-0.560777\pi$
$283$	−1807.00	−0.379558	−0.189779	+	0.981827i	$0.560777\pi$
$284$	−2568.00	−0.536559
$285$	0	0
$286$	5928.00	1.22563
$287$	−7998.00	−1.64497
$288$	0	0
$289$	−152.000	−0.0309383
$290$	2700.00	0.546722
$291$	0	0
$292$	−1948.00	−0.390404
$293$	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
$296$	−1808.00	−0.355027
$297$	0	0
$298$	3390.00	0.658985
$299$	−3744.00	−0.724151
$300$	0	0
$301$	2077.00	0.397729
$302$	3604.00	0.686712
$303$	0	0
$304$	304.000	0.0573539
$305$	−117.000	−0.0219652
$306$	0	0
$307$	−1096.00	−0.203753	−0.101876	−	0.994797i	$-0.532485\pi$
$307$	−1096.00	−0.203753	−0.101876	+	0.994797i	$0.532485\pi$
$308$	7068.00	1.30759
$309$	0	0
$310$	576.000	0.105531
$311$	−1947.00	−0.354998	−0.177499	−	0.984121i	$-0.556801\pi$
$311$	−1947.00	−0.354998	−0.177499	+	0.984121i	$0.556801\pi$
$312$	0	0
$313$	7598.00	1.37209	0.686045	−	0.727559i	$-0.259345\pi$
$313$	7598.00	1.37209	0.686045	+	0.727559i	$0.259345\pi$
$314$	−6452.00	−1.15958
$315$	0	0
$316$	−2800.00	−0.498457
$317$	−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
$320$	576.000	0.100623
$321$	0	0
$322$	−4464.00	−0.772575
$323$	−1311.00	−0.225839
$324$	0	0
$325$	2288.00	0.390509
$326$	2536.00	0.430847
$327$	0	0
$328$	2064.00	0.347455
$329$	17949.0	3.00778
$330$	0	0
$331$	−8368.00	−1.38957	−0.694784	−	0.719219i	$-0.744500\pi$
$331$	−8368.00	−1.38957	−0.694784	+	0.719219i	$0.744500\pi$
$332$	48.0000	0.00793477
$333$	0	0
$334$	−1308.00	−0.214283
$335$	−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	$0.814745\pi$
$338$	1014.00	0.163178
$339$	0	0
$340$	−2484.00	−0.396217
$341$	−1824.00	−0.289663
$342$	0	0
$343$	−8525.00	−1.34200
$344$	−536.000	−0.0840093
$345$	0	0
$346$	2724.00	0.423246
$347$	−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	$-0.662039\pi$
$349$	−6355.00	−0.974714	−0.487357	+	0.873203i	$0.662039\pi$
$350$	2728.00	0.416622
$351$	0	0
$352$	−1824.00	−0.276192
$353$	−7218.00	−1.08832	−0.544158	−	0.838983i	$-0.683151\pi$
$353$	−7218.00	−1.08832	−0.544158	+	0.838983i	$0.683151\pi$
$354$	0	0
$355$	−5778.00	−0.863843
$356$	2400.00	0.357303
$357$	0	0
$358$	420.000	0.0620047
$359$	−1665.00	−0.244778	−0.122389	−	0.992482i	$-0.539056\pi$
$359$	−1665.00	−0.244778	−0.122389	+	0.992482i	$0.539056\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	4.00000	0.000580761 0
$363$	0	0
$364$	6448.00	0.928481
$365$	−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	$0.120612\pi$
$368$	1152.00	0.163185
$369$	0	0
$370$	−4068.00	−0.571582
$371$	−13392.0	−1.87406
$372$	0	0
$373$	−10492.0	−1.45645	−0.728224	−	0.685339i	$-0.759654\pi$
$373$	−10492.0	−1.45645	−0.728224	+	0.685339i	$0.759654\pi$
$374$	7866.00	1.08754
$375$	0	0
$376$	−4632.00	−0.635312
$377$	−7800.00	−1.06557
$378$	0	0
$379$	7610.00	1.03140	0.515698	−	0.856770i	$-0.327532\pi$
$379$	7610.00	1.03140	0.515698	+	0.856770i	$0.327532\pi$
$380$	684.000	0.0923381
$381$	0	0
$382$	5286.00	0.707998
$383$	−4008.00	−0.534724	−0.267362	−	0.963596i	$-0.586152\pi$
$383$	−4008.00	−0.534724	−0.267362	+	0.963596i	$0.586152\pi$
$384$	0	0
$385$	15903.0	2.10517
$386$	6496.00	0.856574
$387$	0	0
$388$	5696.00	0.745285
$389$	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
$392$	4944.00	0.637015
$393$	0	0
$394$	6252.00	0.799419
$395$	−6300.00	−0.802500
$396$	0	0
$397$	6629.00	0.838035	0.419018	−	0.907978i	$-0.362375\pi$
$397$	6629.00	0.838035	0.419018	+	0.907978i	$0.362375\pi$
$398$	−5990.00	−0.754401
$399$	0	0
$400$	−704.000	−0.0880000
$401$	10848.0	1.35093	0.675465	−	0.737392i	$-0.263943\pi$
$401$	10848.0	1.35093	0.675465	+	0.737392i	$0.263943\pi$
$402$	0	0
$403$	−1664.00	−0.205682
$404$	−4248.00	−0.523133
$405$	0	0
$406$	−9300.00	−1.13683
$407$	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	$0.558828\pi$
$410$	4644.00	0.559392
$411$	0	0
$412$	4712.00	0.563455
$413$	−10230.0	−1.21885
$414$	0	0
$415$	108.000	0.0127747
$416$	−1664.00	−0.196116
$417$	0	0
$418$	−2166.00	−0.253451
$419$	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	$-0.417800\pi$
$421$	4412.00	0.510755	0.255377	+	0.966841i	$0.417800\pi$
$422$	−8636.00	−0.996194
$423$	0	0
$424$	3456.00	0.395845
$425$	3036.00	0.346512
$426$	0	0
$427$	403.000	0.0456734
$428$	−456.000	−0.0514990
$429$	0	0
$430$	−1206.00	−0.135252
$431$	−432.000	−0.0482801	−0.0241400	−	0.999709i	$-0.507685\pi$
$431$	−432.000	−0.0482801	−0.0241400	+	0.999709i	$0.507685\pi$
$432$	0	0
$433$	−2002.00	−0.222194	−0.111097	−	0.993810i	$-0.535436\pi$
$433$	−2002.00	−0.222194	−0.111097	+	0.993810i	$0.535436\pi$
$434$	−1984.00	−0.219436
$435$	0	0
$436$	5840.00	0.641480
$437$	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	$0.529283\pi$
$440$	−4104.00	−0.444660
$441$	0	0
$442$	7176.00	0.772234
$443$	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
$446$	1036.00	0.109991
$447$	0	0
$448$	−1984.00	−0.209230
$449$	2760.00	0.290095	0.145047	−	0.989425i	$-0.453667\pi$
$449$	2760.00	0.290095	0.145047	+	0.989425i	$0.453667\pi$
$450$	0	0
$451$	−14706.0	−1.53543
$452$	3288.00	0.342156
$453$	0	0
$454$	−5688.00	−0.587998
$455$	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	$0.426044\pi$
$458$	3490.00	0.356063
$459$	0	0
$460$	2592.00	0.262723
$461$	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	$0.524578\pi$
$464$	2400.00	0.240123
$465$	0	0
$466$	−10566.0	−1.05034
$467$	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
$470$	−10422.0	−1.02283
$471$	0	0
$472$	2640.00	0.257449
$473$	3819.00	0.371243
$474$	0	0
$475$	−836.000	−0.0807543
$476$	8556.00	0.823873
$477$	0	0
$478$	−930.000	−0.0889900
$479$	8580.00	0.818435	0.409217	−	0.912437i	$-0.365802\pi$
$479$	8580.00	0.818435	0.409217	+	0.912437i	$0.365802\pi$
$480$	0	0
$481$	11752.0	1.11402
$482$	−14156.0	−1.33773
$483$	0	0
$484$	7672.00	0.720511
$485$	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	$0.309061\pi$
$488$	−104.000	−0.00964725
$489$	0	0
$490$	11124.0	1.02557
$491$	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
$494$	−1976.00	−0.179969
$495$	0	0
$496$	512.000	0.0463498
$497$	19902.0	1.79623
$498$	0	0
$499$	−11905.0	−1.06802	−0.534009	−	0.845479i	$-0.679315\pi$
$499$	−11905.0	−1.06802	−0.534009	+	0.845479i	$0.679315\pi$
$500$	−6084.00	−0.544170
$501$	0	0
$502$	−7134.00	−0.634275
$503$	−9108.00	−0.807367	−0.403684	−	0.914899i	$-0.632270\pi$
$503$	−9108.00	−0.807367	−0.403684	+	0.914899i	$0.632270\pi$
$504$	0	0
$505$	−9558.00	−0.842229
$506$	−8208.00	−0.721127
$507$	0	0
$508$	−8344.00	−0.728750
$509$	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
$512$	512.000	0.0441942
$513$	0	0
$514$	3792.00	0.325405
$515$	10602.0	0.907146
$516$	0	0
$517$	33003.0	2.80749
$518$	14012.0	1.18852
$519$	0	0
$520$	−3744.00	−0.315741
$521$	−21612.0	−1.81735	−0.908675	−	0.417505i	$-0.862905\pi$
$521$	−21612.0	−1.81735	−0.908675	+	0.417505i	$0.862905\pi$
$522$	0	0
$523$	−9022.00	−0.754311	−0.377155	−	0.926150i	$-0.623098\pi$
$523$	−9022.00	−0.754311	−0.377155	+	0.926150i	$0.623098\pi$
$524$	372.000	0.0310132
$525$	0	0
$526$	114.000	0.00944988
$527$	−2208.00	−0.182509
$528$	0	0
$529$	−6983.00	−0.573929
$530$	7776.00	0.637298
$531$	0	0
$532$	−2356.00	−0.192003
$533$	−13416.0	−1.09027
$534$	0	0
$535$	−1026.00	−0.0829119
$536$	−6848.00	−0.551844
$537$	0	0
$538$	−5400.00	−0.432733
$539$	−35226.0	−2.81501
$540$	0	0
$541$	−9253.00	−0.735337	−0.367669	−	0.929957i	$-0.619844\pi$
$541$	−9253.00	−0.735337	−0.367669	+	0.929957i	$0.619844\pi$
$542$	7744.00	0.613715
$543$	0	0
$544$	−2208.00	−0.174021
$545$	13140.0	1.03276
$546$	0	0
$547$	13244.0	1.03523	0.517617	−	0.855613i	$-0.326819\pi$
$547$	13244.0	1.03523	0.517617	+	0.855613i	$0.326819\pi$
$548$	−5076.00	−0.395686
$549$	0	0
$550$	5016.00	0.388878
$551$	2850.00	0.220352
$552$	0	0
$553$	21700.0	1.66868
$554$	−15422.0	−1.18270
$555$	0	0
$556$	−7900.00	−0.602580
$557$	−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
$560$	−4464.00	−0.336854
$561$	0	0
$562$	13716.0	1.02949
$563$	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
$566$	−3614.00	−0.268388
$567$	0	0
$568$	−5136.00	−0.379405
$569$	13800.0	1.01674	0.508371	−	0.861138i	$-0.330248\pi$
$569$	13800.0	1.01674	0.508371	+	0.861138i	$0.330248\pi$
$570$	0	0
$571$	−4348.00	−0.318666	−0.159333	−	0.987225i	$-0.550934\pi$
$571$	−4348.00	−0.318666	−0.159333	+	0.987225i	$0.550934\pi$
$572$	11856.0	0.866651
$573$	0	0
$574$	−15996.0	−1.16317
$575$	−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	$0.459250\pi$
$578$	−304.000	−0.0218767
$579$	0	0
$580$	5400.00	0.386591
$581$	−372.000	−0.0265631
$582$	0	0
$583$	−24624.0	−1.74927
$584$	−3896.00	−0.276058
$585$	0	0
$586$	6024.00	0.424657
$587$	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
$590$	5940.00	0.414485
$591$	0	0
$592$	−3616.00	−0.251042
$593$	−13278.0	−0.919498	−0.459749	−	0.888049i	$-0.652061\pi$
$593$	−13278.0	−0.919498	−0.459749	+	0.888049i	$0.652061\pi$
$594$	0	0
$595$	19251.0	1.32641
$596$	6780.00	0.465973
$597$	0	0
$598$	−7488.00	−0.512052
$599$	−20400.0	−1.39152	−0.695761	−	0.718274i	$-0.744933\pi$
$599$	−20400.0	−1.39152	−0.695761	+	0.718274i	$0.744933\pi$
$600$	0	0
$601$	−22198.0	−1.50661	−0.753307	−	0.657669i	$-0.771543\pi$
$601$	−22198.0	−1.50661	−0.753307	+	0.657669i	$0.771543\pi$
$602$	4154.00	0.281237
$603$	0	0
$604$	7208.00	0.485578
$605$	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	$0.393472\pi$
$608$	608.000	0.0405554
$609$	0	0
$610$	−234.000	−0.0155318
$611$	30108.0	1.99352
$612$	0	0
$613$	−4327.00	−0.285099	−0.142550	−	0.989788i	$-0.545530\pi$
$613$	−4327.00	−0.285099	−0.142550	+	0.989788i	$0.545530\pi$
$614$	−2192.00	−0.144075
$615$	0	0
$616$	14136.0	0.924603
$617$	14151.0	0.923335	0.461668	−	0.887053i	$-0.347251\pi$
$617$	14151.0	0.923335	0.461668	+	0.887053i	$0.347251\pi$
$618$	0	0
$619$	22460.0	1.45839	0.729195	−	0.684306i	$-0.239895\pi$
$619$	22460.0	1.45839	0.729195	+	0.684306i	$0.239895\pi$
$620$	1152.00	0.0746217
$621$	0	0
$622$	−3894.00	−0.251021
$623$	−18600.0	−1.19614
$624$	0	0
$625$	−8189.00	−0.524096
$626$	15196.0	0.970215
$627$	0	0
$628$	−12904.0	−0.819945
$629$	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	$0.672641\pi$
$632$	−5600.00	−0.352462
$633$	0	0
$634$	−16668.0	−1.04412
$635$	−18774.0	−1.17327
$636$	0	0
$637$	−32136.0	−1.99886
$638$	−17100.0	−1.06112
$639$	0	0
$640$	1152.00	0.0711512
$641$	−5592.00	−0.344572	−0.172286	−	0.985047i	$-0.555115\pi$
$641$	−5592.00	−0.344572	−0.172286	+	0.985047i	$0.555115\pi$
$642$	0	0
$643$	16553.0	1.01522	0.507610	−	0.861587i	$-0.330529\pi$
$643$	16553.0	1.01522	0.507610	+	0.861587i	$0.330529\pi$
$644$	−8928.00	−0.546293
$645$	0	0
$646$	−2622.00	−0.159692
$647$	4611.00	0.280181	0.140091	−	0.990139i	$-0.455261\pi$
$647$	4611.00	0.280181	0.140091	+	0.990139i	$0.455261\pi$
$648$	0	0
$649$	−18810.0	−1.13768
$650$	4576.00	0.276132
$651$	0	0
$652$	5072.00	0.304655
$653$	−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
$656$	4128.00	0.245688
$657$	0	0
$658$	35898.0	2.12682
$659$	−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	$-0.209146\pi$
$661$	26912.0	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$662$	−16736.0	−0.982573
$663$	0	0
$664$	96.0000	0.00561073
$665$	−5301.00	−0.309119
$666$	0	0
$667$	10800.0	0.626953
$668$	−2616.00	−0.151521
$669$	0	0
$670$	−15408.0	−0.888452
$671$	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	$0.711854\pi$
$674$	−20672.0	−1.18139
$675$	0	0
$676$	2028.00	0.115385
$677$	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
$680$	−4968.00	−0.280168
$681$	0	0
$682$	−3648.00	−0.204823
$683$	−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
$686$	−17050.0	−0.948939
$687$	0	0
$688$	−1072.00	−0.0594035
$689$	−22464.0	−1.24210
$690$	0	0
$691$	8147.00	0.448519	0.224259	−	0.974529i	$-0.428004\pi$
$691$	8147.00	0.448519	0.224259	+	0.974529i	$0.428004\pi$
$692$	5448.00	0.299280
$693$	0	0
$694$	−13758.0	−0.752517
$695$	−17775.0	−0.970136
$696$	0	0
$697$	−17802.0	−0.967430
$698$	−12710.0	−0.689227
$699$	0	0
$700$	5456.00	0.294596
$701$	−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
$704$	−3648.00	−0.195297
$705$	0	0
$706$	−14436.0	−0.769555
$707$	32922.0	1.75129
$708$	0	0
$709$	21890.0	1.15952	0.579758	−	0.814789i	$-0.303147\pi$
$709$	21890.0	1.15952	0.579758	+	0.814789i	$0.303147\pi$
$710$	−11556.0	−0.610830
$711$	0	0
$712$	4800.00	0.252651
$713$	2304.00	0.121018
$714$	0	0
$715$	26676.0	1.39528
$716$	840.000	0.0438440
$717$	0	0
$718$	−3330.00	−0.173084
$719$	27015.0	1.40124	0.700619	−	0.713536i	$-0.252907\pi$
$719$	27015.0	1.40124	0.700619	+	0.713536i	$0.252907\pi$
$720$	0	0
$721$	−36518.0	−1.88627
$722$	722.000	0.0372161
$723$	0	0
$724$	8.00000	0.000410660 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	$0.607768\pi$
$728$	12896.0	0.656535
$729$	0	0
$730$	−8766.00	−0.444444
$731$	4623.00	0.233909
$732$	0	0
$733$	−6262.00	−0.315542	−0.157771	−	0.987476i	$-0.550431\pi$
$733$	−6262.00	−0.315542	−0.157771	+	0.987476i	$0.550431\pi$
$734$	26128.0	1.31390
$735$	0	0
$736$	2304.00	0.115389
$737$	48792.0	2.43864
$738$	0	0
$739$	−10855.0	−0.540335	−0.270168	−	0.962813i	$-0.587079\pi$
$739$	−10855.0	−0.540335	−0.270168	+	0.962813i	$0.587079\pi$
$740$	−8136.00	−0.404169
$741$	0	0
$742$	−26784.0	−1.32516
$743$	14892.0	0.735309	0.367654	−	0.929962i	$-0.380161\pi$
$743$	14892.0	0.735309	0.367654	+	0.929962i	$0.380161\pi$
$744$	0	0
$745$	15255.0	0.750201
$746$	−20984.0	−1.02986
$747$	0	0
$748$	15732.0	0.769009
$749$	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	$0.251674\pi$
$752$	−9264.00	−0.449233
$753$	0	0
$754$	−15600.0	−0.753473
$755$	16218.0	0.781766
$756$	0	0
$757$	−3541.00	−0.170013	−0.0850065	−	0.996380i	$-0.527091\pi$
$757$	−3541.00	−0.170013	−0.0850065	+	0.996380i	$0.527091\pi$
$758$	15220.0	0.729308
$759$	0	0
$760$	1368.00	0.0652929
$761$	−22617.0	−1.07735	−0.538676	−	0.842513i	$-0.681075\pi$
$761$	−22617.0	−1.07735	−0.538676	+	0.842513i	$0.681075\pi$
$762$	0	0
$763$	−45260.0	−2.14747
$764$	10572.0	0.500630
$765$	0	0
$766$	−8016.00	−0.378107
$767$	−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	$0.413135\pi$
$770$	31806.0	1.48858
$771$	0	0
$772$	12992.0	0.605690
$773$	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
$776$	11392.0	0.526996
$777$	0	0
$778$	7050.00	0.324878
$779$	4902.00	0.225459
$780$	0	0
$781$	36594.0	1.67661
$782$	−9936.00	−0.454361
$783$	0	0
$784$	9888.00	0.450437
$785$	−29034.0	−1.32009
$786$	0	0
$787$	7124.00	0.322672	0.161336	−	0.986900i	$-0.448420\pi$
$787$	7124.00	0.322672	0.161336	+	0.986900i	$0.448420\pi$
$788$	12504.0	0.565275
$789$	0	0
$790$	−12600.0	−0.567453
$791$	−25482.0	−1.14543
$792$	0	0
$793$	676.000	0.0302717
$794$	13258.0	0.592580
$795$	0	0
$796$	−11980.0	−0.533442
$797$	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
$800$	−1408.00	−0.0622254
$801$	0	0
$802$	21696.0	0.955252
$803$	27759.0	1.21992
$804$	0	0
$805$	−20088.0	−0.879514
$806$	−3328.00	−0.145439
$807$	0	0
$808$	−8496.00	−0.369911
$809$	−42855.0	−1.86242	−0.931212	−	0.364477i	$-0.881248\pi$
$809$	−42855.0	−1.86242	−0.931212	+	0.364477i	$0.881248\pi$
$810$	0	0
$811$	−15568.0	−0.674065	−0.337032	−	0.941493i	$-0.609423\pi$
$811$	−15568.0	−0.674065	−0.337032	+	0.941493i	$0.609423\pi$
$812$	−18600.0	−0.803857
$813$	0	0
$814$	25764.0	1.10937
$815$	11412.0	0.490485
$816$	0	0
$817$	−1273.00	−0.0545124
$818$	−6080.00	−0.259880
$819$	0	0
$820$	9288.00	0.395550
$821$	−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	$0.566042\pi$
$824$	9424.00	0.398423
$825$	0	0
$826$	−20460.0	−0.861858
$827$	−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	$-0.479449\pi$
$829$	3080.00	0.129038	0.0645192	+	0.997916i	$0.479449\pi$
$830$	216.000	0.00903310
$831$	0	0
$832$	−3328.00	−0.138675
$833$	−42642.0	−1.77366
$834$	0	0
$835$	−5886.00	−0.243944
$836$	−4332.00	−0.179217
$837$	0	0
$838$	7800.00	0.321535
$839$	−26790.0	−1.10238	−0.551188	−	0.834381i	$-0.685825\pi$
$839$	−26790.0	−1.10238	−0.551188	+	0.834381i	$0.685825\pi$
$840$	0	0
$841$	−1889.00	−0.0774530
$842$	8824.00	0.361158
$843$	0	0
$844$	−17272.0	−0.704416
$845$	4563.00	0.185766
$846$	0	0
$847$	−59458.0	−2.41204
$848$	6912.00	0.279905
$849$	0	0
$850$	6072.00	0.245021
$851$	−16272.0	−0.655461
$852$	0	0
$853$	19178.0	0.769803	0.384902	−	0.922958i	$-0.374235\pi$
$853$	19178.0	0.769803	0.384902	+	0.922958i	$0.374235\pi$
$854$	806.000	0.0322960
$855$	0	0
$856$	−912.000	−0.0364153
$857$	2406.00	0.0959013	0.0479506	−	0.998850i	$-0.484731\pi$
$857$	2406.00	0.0959013	0.0479506	+	0.998850i	$0.484731\pi$
$858$	0	0
$859$	9125.00	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$859$	9125.00	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$860$	−2412.00	−0.0956378
$861$	0	0
$862$	−864.000	−0.0341392
$863$	−8898.00	−0.350975	−0.175488	−	0.984482i	$-0.556150\pi$
$863$	−8898.00	−0.350975	−0.175488	+	0.984482i	$0.556150\pi$
$864$	0	0
$865$	12258.0	0.481832
$866$	−4004.00	−0.157115
$867$	0	0
$868$	−3968.00	−0.155164
$869$	39900.0	1.55755
$870$	0	0
$871$	44512.0	1.73161
$872$	11680.0	0.453595
$873$	0	0
$874$	2736.00	0.105889
$875$	47151.0	1.82171
$876$	0	0
$877$	−15886.0	−0.611667	−0.305834	−	0.952085i	$-0.598935\pi$
$877$	−15886.0	−0.611667	−0.305834	+	0.952085i	$0.598935\pi$
$878$	−3380.00	−0.129920
$879$	0	0
$880$	−8208.00	−0.314422
$881$	25683.0	0.982159	0.491080	−	0.871115i	$-0.336602\pi$
$881$	25683.0	0.982159	0.491080	+	0.871115i	$0.336602\pi$
$882$	0	0
$883$	−28267.0	−1.07730	−0.538652	−	0.842528i	$-0.681066\pi$
$883$	−28267.0	−1.07730	−0.538652	+	0.842528i	$0.681066\pi$
$884$	14352.0	0.546052
$885$	0	0
$886$	3954.00	0.149929
$887$	2466.00	0.0933486	0.0466743	−	0.998910i	$-0.485138\pi$
$887$	2466.00	0.0933486	0.0466743	+	0.998910i	$0.485138\pi$
$888$	0	0
$889$	64666.0	2.43963
$890$	10800.0	0.406760
$891$	0	0
$892$	2072.00	0.0777754
$893$	−11001.0	−0.412245
$894$	0	0
$895$	1890.00	0.0705874
$896$	−3968.00	−0.147948
$897$	0	0
$898$	5520.00	0.205128
$899$	4800.00	0.178074
$900$	0	0
$901$	−29808.0	−1.10216
$902$	−29412.0	−1.08571
$903$	0	0
$904$	6576.00	0.241941
$905$	18.0000	0.000661149 0
$906$	0	0
$907$	29324.0	1.07353	0.536763	−	0.843733i	$-0.319647\pi$
$907$	29324.0	1.07353	0.536763	+	0.843733i	$0.319647\pi$
$908$	−11376.0	−0.415777
$909$	0	0
$910$	29016.0	1.05700
$911$	−47142.0	−1.71447	−0.857236	−	0.514924i	$-0.827820\pi$
$911$	−47142.0	−1.71447	−0.857236	+	0.514924i	$0.827820\pi$
$912$	0	0
$913$	−684.000	−0.0247942
$914$	8998.00	0.325632
$915$	0	0
$916$	6980.00	0.251775
$917$	−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	$0.754399\pi$
$920$	5184.00	0.185773
$921$	0	0
$922$	23286.0	0.831761
$923$	33384.0	1.19052
$924$	0	0
$925$	9944.00	0.353467
$926$	−3074.00	−0.109091
$927$	0	0
$928$	4800.00	0.169793
$929$	4410.00	0.155745	0.0778727	−	0.996963i	$-0.475187\pi$
$929$	4410.00	0.155745	0.0778727	+	0.996963i	$0.475187\pi$
$930$	0	0
$931$	11742.0	0.413350
$932$	−21132.0	−0.742706
$933$	0	0
$934$	15282.0	0.535377
$935$	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	$0.758822\pi$
$938$	53072.0	1.84740
$939$	0	0
$940$	−20844.0	−0.723251
$941$	−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
$944$	5280.00	0.182044
$945$	0	0
$946$	7638.00	0.262508
$947$	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
$950$	−1672.00	−0.0571019
$951$	0	0
$952$	17112.0	0.582566
$953$	−26508.0	−0.901027	−0.450513	−	0.892770i	$-0.648759\pi$
$953$	−26508.0	−0.901027	−0.450513	+	0.892770i	$0.648759\pi$
$954$	0	0
$955$	23787.0	0.805999
$956$	−1860.00	−0.0629254
$957$	0	0
$958$	17160.0	0.578721
$959$	39339.0	1.32463
$960$	0	0
$961$	−28767.0	−0.965627
$962$	23504.0	0.787733
$963$	0	0
$964$	−28312.0	−0.945921
$965$	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	$0.585584\pi$
$968$	15344.0	0.509478
$969$	0	0
$970$	25632.0	0.848447
$971$	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
$974$	24268.0	0.798354
$975$	0	0
$976$	−208.000	−0.00682164
$977$	−21804.0	−0.713994	−0.356997	−	0.934106i	$-0.616199\pi$
$977$	−21804.0	−0.713994	−0.356997	+	0.934106i	$0.616199\pi$
$978$	0	0
$979$	−34200.0	−1.11648
$980$	22248.0	0.725190
$981$	0	0
$982$	11016.0	0.357978
$983$	−11268.0	−0.365609	−0.182804	−	0.983149i	$-0.558517\pi$
$983$	−11268.0	−0.365609	−0.182804	+	0.983149i	$0.558517\pi$
$984$	0	0
$985$	28134.0	0.910075
$986$	−20700.0	−0.668582
$987$	0	0
$988$	−3952.00	−0.127257
$989$	−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	$0.503969\pi$
$992$	1024.00	0.0327742
$993$	0	0
$994$	39804.0	1.27013
$995$	−26955.0	−0.858825
$996$	0	0
$997$	389.000	0.0123568	0.00617841	−	0.999981i	$-0.498033\pi$
$997$	389.000	0.0123568	0.00617841	+	0.999981i	$0.498033\pi$
$998$	−23810.0	−0.755203
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.a.d.1.1		1
3.2	odd	2		38.4.a.a.1.1	✓	1
12.11	even	2		304.4.a.a.1.1		1
15.2	even	4		950.4.b.d.799.1		2
15.8	even	4		950.4.b.d.799.2		2
15.14	odd	2		950.4.a.d.1.1		1
21.20	even	2		1862.4.a.a.1.1		1
24.5	odd	2		1216.4.a.e.1.1		1
24.11	even	2		1216.4.a.b.1.1		1
57.56	even	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	3.2	odd	2
304.4.a.a.1.1		1	12.11	even	2
342.4.a.d.1.1		1	1.1	even	1	trivial
722.4.a.d.1.1		1	57.56	even	2
950.4.a.d.1.1		1	15.14	odd	2
950.4.b.d.799.1		2	15.2	even	4
950.4.b.d.799.2		2	15.8	even	4
1216.4.a.b.1.1		1	24.11	even	2
1216.4.a.e.1.1		1	24.5	odd	2
1862.4.a.a.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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