LMFDB - Embedded newform 42.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [42,4,Mod(1,42)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, 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("42.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	42.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:	$2.47808022024$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	42.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +2.00000 q^{5} +6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +4.00000 q^{10} -8.00000 q^{11} +12.0000 q^{12} -42.0000 q^{13} -14.0000 q^{14} +6.00000 q^{15} +16.0000 q^{16} -2.00000 q^{17} +18.0000 q^{18} -124.000 q^{19} +8.00000 q^{20} -21.0000 q^{21} -16.0000 q^{22} +76.0000 q^{23} +24.0000 q^{24} -121.000 q^{25} -84.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} +254.000 q^{29} +12.0000 q^{30} -72.0000 q^{31} +32.0000 q^{32} -24.0000 q^{33} -4.00000 q^{34} -14.0000 q^{35} +36.0000 q^{36} +398.000 q^{37} -248.000 q^{38} -126.000 q^{39} +16.0000 q^{40} +462.000 q^{41} -42.0000 q^{42} +212.000 q^{43} -32.0000 q^{44} +18.0000 q^{45} +152.000 q^{46} -264.000 q^{47} +48.0000 q^{48} +49.0000 q^{49} -242.000 q^{50} -6.00000 q^{51} -168.000 q^{52} -162.000 q^{53} +54.0000 q^{54} -16.0000 q^{55} -56.0000 q^{56} -372.000 q^{57} +508.000 q^{58} -772.000 q^{59} +24.0000 q^{60} +30.0000 q^{61} -144.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -84.0000 q^{65} -48.0000 q^{66} -764.000 q^{67} -8.00000 q^{68} +228.000 q^{69} -28.0000 q^{70} -236.000 q^{71} +72.0000 q^{72} +418.000 q^{73} +796.000 q^{74} -363.000 q^{75} -496.000 q^{76} +56.0000 q^{77} -252.000 q^{78} +552.000 q^{79} +32.0000 q^{80} +81.0000 q^{81} +924.000 q^{82} +1036.00 q^{83} -84.0000 q^{84} -4.00000 q^{85} +424.000 q^{86} +762.000 q^{87} -64.0000 q^{88} +30.0000 q^{89} +36.0000 q^{90} +294.000 q^{91} +304.000 q^{92} -216.000 q^{93} -528.000 q^{94} -248.000 q^{95} +96.0000 q^{96} -1190.00 q^{97} +98.0000 q^{98} -72.0000 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	2.00000	0.178885	0.0894427	−	0.995992i	$-0.471491\pi$
$5$	2.00000	0.178885	0.0894427	+	0.995992i	$0.471491\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	4.00000	0.126491
$11$	−8.00000	−0.219281	−0.109640	−	0.993971i	$-0.534970\pi$
$11$	−8.00000	−0.219281	−0.109640	+	0.993971i	$0.534970\pi$
$12$	12.0000	0.288675
$13$	−42.0000	−0.896054	−0.448027	−	0.894020i	$-0.647873\pi$
$13$	−42.0000	−0.896054	−0.448027	+	0.894020i	$0.647873\pi$
$14$	−14.0000	−0.267261
$15$	6.00000	0.103280
$16$	16.0000	0.250000
$17$	−2.00000	−0.0285336	−0.0142668	−	0.999898i	$-0.504541\pi$
$17$	−2.00000	−0.0285336	−0.0142668	+	0.999898i	$0.504541\pi$
$18$	18.0000	0.235702
$19$	−124.000	−1.49724	−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000	−1.49724	−0.748620	+	0.663000i	$0.769283\pi$
$20$	8.00000	0.0894427
$21$	−21.0000	−0.218218
$22$	−16.0000	−0.155055
$23$	76.0000	0.689004	0.344502	−	0.938786i	$-0.388048\pi$
$23$	76.0000	0.689004	0.344502	+	0.938786i	$0.388048\pi$
$24$	24.0000	0.204124
$25$	−121.000	−0.968000
$26$	−84.0000	−0.633606
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	254.000	1.62644	0.813218	−	0.581960i	$-0.197714\pi$
$29$	254.000	1.62644	0.813218	+	0.581960i	$0.197714\pi$
$30$	12.0000	0.0730297
$31$	−72.0000	−0.417148	−0.208574	−	0.978007i	$-0.566882\pi$
$31$	−72.0000	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$32$	32.0000	0.176777
$33$	−24.0000	−0.126602
$34$	−4.00000	−0.0201763
$35$	−14.0000	−0.0676123
$36$	36.0000	0.166667
$37$	398.000	1.76840	0.884200	−	0.467109i	$-0.154704\pi$
$37$	398.000	1.76840	0.884200	+	0.467109i	$0.154704\pi$
$38$	−248.000	−1.05871
$39$	−126.000	−0.517337
$40$	16.0000	0.0632456
$41$	462.000	1.75981	0.879906	−	0.475148i	$-0.157606\pi$
$41$	462.000	1.75981	0.879906	+	0.475148i	$0.157606\pi$
$42$	−42.0000	−0.154303
$43$	212.000	0.751853	0.375927	−	0.926649i	$-0.377324\pi$
$43$	212.000	0.751853	0.375927	+	0.926649i	$0.377324\pi$
$44$	−32.0000	−0.109640
$45$	18.0000	0.0596285
$46$	152.000	0.487200
$47$	−264.000	−0.819327	−0.409663	−	0.912237i	$-0.634354\pi$
$47$	−264.000	−0.819327	−0.409663	+	0.912237i	$0.634354\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	−242.000	−0.684479
$51$	−6.00000	−0.0164739
$52$	−168.000	−0.448027
$53$	−162.000	−0.419857	−0.209928	−	0.977717i	$-0.567323\pi$
$53$	−162.000	−0.419857	−0.209928	+	0.977717i	$0.567323\pi$
$54$	54.0000	0.136083
$55$	−16.0000	−0.0392262
$56$	−56.0000	−0.133631
$57$	−372.000	−0.864432
$58$	508.000	1.15006
$59$	−772.000	−1.70349	−0.851744	−	0.523958i	$-0.824455\pi$
$59$	−772.000	−1.70349	−0.851744	+	0.523958i	$0.824455\pi$
$60$	24.0000	0.0516398
$61$	30.0000	0.0629690	0.0314845	−	0.999504i	$-0.489977\pi$
$61$	30.0000	0.0629690	0.0314845	+	0.999504i	$0.489977\pi$
$62$	−144.000	−0.294968
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−84.0000	−0.160291
$66$	−48.0000	−0.0895211
$67$	−764.000	−1.39310	−0.696548	−	0.717510i	$-0.745282\pi$
$67$	−764.000	−1.39310	−0.696548	+	0.717510i	$0.745282\pi$
$68$	−8.00000	−0.0142668
$69$	228.000	0.397797
$70$	−28.0000	−0.0478091
$71$	−236.000	−0.394480	−0.197240	−	0.980355i	$-0.563198\pi$
$71$	−236.000	−0.394480	−0.197240	+	0.980355i	$0.563198\pi$
$72$	72.0000	0.117851
$73$	418.000	0.670181	0.335090	−	0.942186i	$-0.391233\pi$
$73$	418.000	0.670181	0.335090	+	0.942186i	$0.391233\pi$
$74$	796.000	1.25045
$75$	−363.000	−0.558875
$76$	−496.000	−0.748620
$77$	56.0000	0.0828804
$78$	−252.000	−0.365813
$79$	552.000	0.786137	0.393069	−	0.919509i	$-0.371413\pi$
$79$	552.000	0.786137	0.393069	+	0.919509i	$0.371413\pi$
$80$	32.0000	0.0447214
$81$	81.0000	0.111111
$82$	924.000	1.24437
$83$	1036.00	1.37007	0.685035	−	0.728510i	$-0.259787\pi$
$83$	1036.00	1.37007	0.685035	+	0.728510i	$0.259787\pi$
$84$	−84.0000	−0.109109
$85$	−4.00000	−0.00510425
$86$	424.000	0.531641
$87$	762.000	0.939023
$88$	−64.0000	−0.0775275
$89$	30.0000	0.0357303	0.0178651	−	0.999840i	$-0.494313\pi$
$89$	30.0000	0.0357303	0.0178651	+	0.999840i	$0.494313\pi$
$90$	36.0000	0.0421637
$91$	294.000	0.338677
$92$	304.000	0.344502
$93$	−216.000	−0.240840
$94$	−528.000	−0.579352
$95$	−248.000	−0.267834
$96$	96.0000	0.102062
$97$	−1190.00	−1.24563	−0.622815	−	0.782369i	$-0.714011\pi$
$97$	−1190.00	−1.24563	−0.622815	+	0.782369i	$0.714011\pi$
$98$	98.0000	0.101015
$99$	−72.0000	−0.0730937
$100$	−484.000	−0.484000
$101$	1370.00	1.34970	0.674852	−	0.737953i	$-0.264207\pi$
$101$	1370.00	1.34970	0.674852	+	0.737953i	$0.264207\pi$
$102$	−12.0000	−0.0116488
$103$	464.000	0.443876	0.221938	−	0.975061i	$-0.428762\pi$
$103$	464.000	0.443876	0.221938	+	0.975061i	$0.428762\pi$
$104$	−336.000	−0.316803
$105$	−42.0000	−0.0390360
$106$	−324.000	−0.296884
$107$	−2136.00	−1.92986	−0.964930	−	0.262509i	$-0.915450\pi$
$107$	−2136.00	−1.92986	−0.964930	+	0.262509i	$0.915450\pi$
$108$	108.000	0.0962250
$109$	−1226.00	−1.07733	−0.538667	−	0.842518i	$-0.681072\pi$
$109$	−1226.00	−1.07733	−0.538667	+	0.842518i	$0.681072\pi$
$110$	−32.0000	−0.0277371
$111$	1194.00	1.02099
$112$	−112.000	−0.0944911
$113$	338.000	0.281384	0.140692	−	0.990053i	$-0.455067\pi$
$113$	338.000	0.281384	0.140692	+	0.990053i	$0.455067\pi$
$114$	−744.000	−0.611245
$115$	152.000	0.123253
$116$	1016.00	0.813218
$117$	−378.000	−0.298685
$118$	−1544.00	−1.20455
$119$	14.0000	0.0107847
$120$	48.0000	0.0365148
$121$	−1267.00	−0.951916
$122$	60.0000	0.0445258
$123$	1386.00	1.01603
$124$	−288.000	−0.208574
$125$	−492.000	−0.352047
$126$	−126.000	−0.0890871
$127$	2088.00	1.45890	0.729449	−	0.684035i	$-0.239777\pi$
$127$	2088.00	1.45890	0.729449	+	0.684035i	$0.239777\pi$
$128$	128.000	0.0883883
$129$	636.000	0.434083
$130$	−168.000	−0.113343
$131$	−292.000	−0.194749	−0.0973747	−	0.995248i	$-0.531045\pi$
$131$	−292.000	−0.194749	−0.0973747	+	0.995248i	$0.531045\pi$
$132$	−96.0000	−0.0633010
$133$	868.000	0.565903
$134$	−1528.00	−0.985068
$135$	54.0000	0.0344265
$136$	−16.0000	−0.0100882
$137$	818.000	0.510120	0.255060	−	0.966925i	$-0.417905\pi$
$137$	818.000	0.510120	0.255060	+	0.966925i	$0.417905\pi$
$138$	456.000	0.281285
$139$	−2156.00	−1.31561	−0.657804	−	0.753189i	$-0.728515\pi$
$139$	−2156.00	−1.31561	−0.657804	+	0.753189i	$0.728515\pi$
$140$	−56.0000	−0.0338062
$141$	−792.000	−0.473039
$142$	−472.000	−0.278939
$143$	336.000	0.196488
$144$	144.000	0.0833333
$145$	508.000	0.290946
$146$	836.000	0.473889
$147$	147.000	0.0824786
$148$	1592.00	0.884200
$149$	−2850.00	−1.56699	−0.783494	−	0.621400i	$-0.786564\pi$
$149$	−2850.00	−1.56699	−0.783494	+	0.621400i	$0.786564\pi$
$150$	−726.000	−0.395184
$151$	1672.00	0.901096	0.450548	−	0.892752i	$-0.351229\pi$
$151$	1672.00	0.901096	0.450548	+	0.892752i	$0.351229\pi$
$152$	−992.000	−0.529354
$153$	−18.0000	−0.00951120
$154$	112.000	0.0586053
$155$	−144.000	−0.0746217
$156$	−504.000	−0.258669
$157$	446.000	0.226718	0.113359	−	0.993554i	$-0.463839\pi$
$157$	446.000	0.226718	0.113359	+	0.993554i	$0.463839\pi$
$158$	1104.00	0.555883
$159$	−486.000	−0.242404
$160$	64.0000	0.0316228
$161$	−532.000	−0.260419
$162$	162.000	0.0785674
$163$	2708.00	1.30127	0.650635	−	0.759391i	$-0.274503\pi$
$163$	2708.00	1.30127	0.650635	+	0.759391i	$0.274503\pi$
$164$	1848.00	0.879906
$165$	−48.0000	−0.0226472
$166$	2072.00	0.968785
$167$	896.000	0.415177	0.207589	−	0.978216i	$-0.433439\pi$
$167$	896.000	0.415177	0.207589	+	0.978216i	$0.433439\pi$
$168$	−168.000	−0.0771517
$169$	−433.000	−0.197087
$170$	−8.00000	−0.00360925
$171$	−1116.00	−0.499080
$172$	848.000	0.375927
$173$	4034.00	1.77283	0.886414	−	0.462893i	$-0.153189\pi$
$173$	4034.00	1.77283	0.886414	+	0.462893i	$0.153189\pi$
$174$	1524.00	0.663989
$175$	847.000	0.365870
$176$	−128.000	−0.0548202
$177$	−2316.00	−0.983510
$178$	60.0000	0.0252651
$179$	−3480.00	−1.45311	−0.726557	−	0.687106i	$-0.758881\pi$
$179$	−3480.00	−1.45311	−0.726557	+	0.687106i	$0.758881\pi$
$180$	72.0000	0.0298142
$181$	−2898.00	−1.19009	−0.595046	−	0.803692i	$-0.702866\pi$
$181$	−2898.00	−1.19009	−0.595046	+	0.803692i	$0.702866\pi$
$182$	588.000	0.239481
$183$	90.0000	0.0363551
$184$	608.000	0.243600
$185$	796.000	0.316341
$186$	−432.000	−0.170300
$187$	16.0000	0.00625688
$188$	−1056.00	−0.409663
$189$	−189.000	−0.0727393
$190$	−496.000	−0.189387
$191$	2652.00	1.00467	0.502335	−	0.864673i	$-0.332474\pi$
$191$	2652.00	1.00467	0.502335	+	0.864673i	$0.332474\pi$
$192$	192.000	0.0721688
$193$	146.000	0.0544524	0.0272262	−	0.999629i	$-0.491333\pi$
$193$	146.000	0.0544524	0.0272262	+	0.999629i	$0.491333\pi$
$194$	−2380.00	−0.880794
$195$	−252.000	−0.0925441
$196$	196.000	0.0714286
$197$	−2546.00	−0.920787	−0.460393	−	0.887715i	$-0.652292\pi$
$197$	−2546.00	−0.920787	−0.460393	+	0.887715i	$0.652292\pi$
$198$	−144.000	−0.0516850
$199$	−2536.00	−0.903378	−0.451689	−	0.892175i	$-0.649178\pi$
$199$	−2536.00	−0.903378	−0.451689	+	0.892175i	$0.649178\pi$
$200$	−968.000	−0.342240
$201$	−2292.00	−0.804305
$202$	2740.00	0.954385
$203$	−1778.00	−0.614735
$204$	−24.0000	−0.00823694
$205$	924.000	0.314805
$206$	928.000	0.313868
$207$	684.000	0.229668
$208$	−672.000	−0.224014
$209$	992.000	0.328316
$210$	−84.0000	−0.0276026
$211$	−1300.00	−0.424150	−0.212075	−	0.977253i	$-0.568022\pi$
$211$	−1300.00	−0.424150	−0.212075	+	0.977253i	$0.568022\pi$
$212$	−648.000	−0.209928
$213$	−708.000	−0.227753
$214$	−4272.00	−1.36462
$215$	424.000	0.134496
$216$	216.000	0.0680414
$217$	504.000	0.157667
$218$	−2452.00	−0.761791
$219$	1254.00	0.386929
$220$	−64.0000	−0.0196131
$221$	84.0000	0.0255677
$222$	2388.00	0.721946
$223$	2576.00	0.773550	0.386775	−	0.922174i	$-0.373589\pi$
$223$	2576.00	0.773550	0.386775	+	0.922174i	$0.373589\pi$
$224$	−224.000	−0.0668153
$225$	−1089.00	−0.322667
$226$	676.000	0.198968
$227$	−1836.00	−0.536826	−0.268413	−	0.963304i	$-0.586499\pi$
$227$	−1836.00	−0.536826	−0.268413	+	0.963304i	$0.586499\pi$
$228$	−1488.00	−0.432216
$229$	−1874.00	−0.540775	−0.270387	−	0.962752i	$-0.587152\pi$
$229$	−1874.00	−0.540775	−0.270387	+	0.962752i	$0.587152\pi$
$230$	304.000	0.0871529
$231$	168.000	0.0478510
$232$	2032.00	0.575032
$233$	3730.00	1.04876	0.524379	−	0.851485i	$-0.324298\pi$
$233$	3730.00	1.04876	0.524379	+	0.851485i	$0.324298\pi$
$234$	−756.000	−0.211202
$235$	−528.000	−0.146566
$236$	−3088.00	−0.851744
$237$	1656.00	0.453877
$238$	28.0000	0.00762593
$239$	2004.00	0.542377	0.271188	−	0.962526i	$-0.412583\pi$
$239$	2004.00	0.542377	0.271188	+	0.962526i	$0.412583\pi$
$240$	96.0000	0.0258199
$241$	−646.000	−0.172666	−0.0863330	−	0.996266i	$-0.527515\pi$
$241$	−646.000	−0.172666	−0.0863330	+	0.996266i	$0.527515\pi$
$242$	−2534.00	−0.673106
$243$	243.000	0.0641500
$244$	120.000	0.0314845
$245$	98.0000	0.0255551
$246$	2772.00	0.718440
$247$	5208.00	1.34161
$248$	−576.000	−0.147484
$249$	3108.00	0.791010
$250$	−984.000	−0.248934
$251$	1260.00	0.316855	0.158427	−	0.987371i	$-0.449358\pi$
$251$	1260.00	0.316855	0.158427	+	0.987371i	$0.449358\pi$
$252$	−252.000	−0.0629941
$253$	−608.000	−0.151086
$254$	4176.00	1.03160
$255$	−12.0000	−0.00294694
$256$	256.000	0.0625000
$257$	5910.00	1.43446	0.717229	−	0.696838i	$-0.245410\pi$
$257$	5910.00	1.43446	0.717229	+	0.696838i	$0.245410\pi$
$258$	1272.00	0.306943
$259$	−2786.00	−0.668392
$260$	−336.000	−0.0801455
$261$	2286.00	0.542145
$262$	−584.000	−0.137709
$263$	2988.00	0.700563	0.350281	−	0.936645i	$-0.386086\pi$
$263$	2988.00	0.700563	0.350281	+	0.936645i	$0.386086\pi$
$264$	−192.000	−0.0447605
$265$	−324.000	−0.0751063
$266$	1736.00	0.400154
$267$	90.0000	0.0206289
$268$	−3056.00	−0.696548
$269$	−1318.00	−0.298736	−0.149368	−	0.988782i	$-0.547724\pi$
$269$	−1318.00	−0.298736	−0.149368	+	0.988782i	$0.547724\pi$
$270$	108.000	0.0243432
$271$	−5640.00	−1.26423	−0.632114	−	0.774876i	$-0.717812\pi$
$271$	−5640.00	−1.26423	−0.632114	+	0.774876i	$0.717812\pi$
$272$	−32.0000	−0.00713340
$273$	882.000	0.195535
$274$	1636.00	0.360709
$275$	968.000	0.212264
$276$	912.000	0.198898
$277$	6446.00	1.39820	0.699102	−	0.715022i	$-0.253583\pi$
$277$	6446.00	1.39820	0.699102	+	0.715022i	$0.253583\pi$
$278$	−4312.00	−0.930275
$279$	−648.000	−0.139049
$280$	−112.000	−0.0239046
$281$	4930.00	1.04662	0.523308	−	0.852144i	$-0.324698\pi$
$281$	4930.00	1.04662	0.523308	+	0.852144i	$0.324698\pi$
$282$	−1584.00	−0.334489
$283$	−6260.00	−1.31491	−0.657453	−	0.753496i	$-0.728366\pi$
$283$	−6260.00	−1.31491	−0.657453	+	0.753496i	$0.728366\pi$
$284$	−944.000	−0.197240
$285$	−744.000	−0.154634
$286$	672.000	0.138938
$287$	−3234.00	−0.665146
$288$	288.000	0.0589256
$289$	−4909.00	−0.999186
$290$	1016.00	0.205730
$291$	−3570.00	−0.719165
$292$	1672.00	0.335090
$293$	−2310.00	−0.460586	−0.230293	−	0.973121i	$-0.573968\pi$
$293$	−2310.00	−0.460586	−0.230293	+	0.973121i	$0.573968\pi$
$294$	294.000	0.0583212
$295$	−1544.00	−0.304729
$296$	3184.00	0.625224
$297$	−216.000	−0.0422006
$298$	−5700.00	−1.10803
$299$	−3192.00	−0.617385
$300$	−1452.00	−0.279438
$301$	−1484.00	−0.284174
$302$	3344.00	0.637171
$303$	4110.00	0.779252
$304$	−1984.00	−0.374310
$305$	60.0000	0.0112642
$306$	−36.0000	−0.00672543
$307$	196.000	0.0364375	0.0182187	−	0.999834i	$-0.494200\pi$
$307$	196.000	0.0364375	0.0182187	+	0.999834i	$0.494200\pi$
$308$	224.000	0.0414402
$309$	1392.00	0.256272
$310$	−288.000	−0.0527655
$311$	−6736.00	−1.22818	−0.614089	−	0.789237i	$-0.710477\pi$
$311$	−6736.00	−1.22818	−0.614089	+	0.789237i	$0.710477\pi$
$312$	−1008.00	−0.182906
$313$	394.000	0.0711508	0.0355754	−	0.999367i	$-0.488674\pi$
$313$	394.000	0.0711508	0.0355754	+	0.999367i	$0.488674\pi$
$314$	892.000	0.160314
$315$	−126.000	−0.0225374
$316$	2208.00	0.393069
$317$	−6714.00	−1.18958	−0.594788	−	0.803882i	$-0.702764\pi$
$317$	−6714.00	−1.18958	−0.594788	+	0.803882i	$0.702764\pi$
$318$	−972.000	−0.171406
$319$	−2032.00	−0.356646
$320$	128.000	0.0223607
$321$	−6408.00	−1.11420
$322$	−1064.00	−0.184144
$323$	248.000	0.0427216
$324$	324.000	0.0555556
$325$	5082.00	0.867380
$326$	5416.00	0.920136
$327$	−3678.00	−0.622000
$328$	3696.00	0.622187
$329$	1848.00	0.309676
$330$	−96.0000	−0.0160140
$331$	692.000	0.114912	0.0574558	−	0.998348i	$-0.481701\pi$
$331$	692.000	0.114912	0.0574558	+	0.998348i	$0.481701\pi$
$332$	4144.00	0.685035
$333$	3582.00	0.589467
$334$	1792.00	0.293574
$335$	−1528.00	−0.249205
$336$	−336.000	−0.0545545
$337$	−1566.00	−0.253132	−0.126566	−	0.991958i	$-0.540396\pi$
$337$	−1566.00	−0.253132	−0.126566	+	0.991958i	$0.540396\pi$
$338$	−866.000	−0.139362
$339$	1014.00	0.162457
$340$	−16.0000	−0.00255212
$341$	576.000	0.0914726
$342$	−2232.00	−0.352903
$343$	−343.000	−0.0539949
$344$	1696.00	0.265820
$345$	456.000	0.0711600
$346$	8068.00	1.25358
$347$	−5328.00	−0.824271	−0.412135	−	0.911123i	$-0.635217\pi$
$347$	−5328.00	−0.824271	−0.412135	+	0.911123i	$0.635217\pi$
$348$	3048.00	0.469511
$349$	11326.0	1.73715	0.868577	−	0.495554i	$-0.165035\pi$
$349$	11326.0	1.73715	0.868577	+	0.495554i	$0.165035\pi$
$350$	1694.00	0.258709
$351$	−1134.00	−0.172446
$352$	−256.000	−0.0387638
$353$	−2130.00	−0.321157	−0.160579	−	0.987023i	$-0.551336\pi$
$353$	−2130.00	−0.321157	−0.160579	+	0.987023i	$0.551336\pi$
$354$	−4632.00	−0.695446
$355$	−472.000	−0.0705666
$356$	120.000	0.0178651
$357$	42.0000	0.00622654
$358$	−6960.00	−1.02751
$359$	3044.00	0.447510	0.223755	−	0.974645i	$-0.428168\pi$
$359$	3044.00	0.447510	0.223755	+	0.974645i	$0.428168\pi$
$360$	144.000	0.0210819
$361$	8517.00	1.24173
$362$	−5796.00	−0.841522
$363$	−3801.00	−0.549589
$364$	1176.00	0.169338
$365$	836.000	0.119886
$366$	180.000	0.0257070
$367$	12416.0	1.76597	0.882984	−	0.469404i	$-0.155531\pi$
$367$	12416.0	1.76597	0.882984	+	0.469404i	$0.155531\pi$
$368$	1216.00	0.172251
$369$	4158.00	0.586604
$370$	1592.00	0.223687
$371$	1134.00	0.158691
$372$	−864.000	−0.120420
$373$	−7442.00	−1.03306	−0.516531	−	0.856268i	$-0.672777\pi$
$373$	−7442.00	−1.03306	−0.516531	+	0.856268i	$0.672777\pi$
$374$	32.0000	0.00442428
$375$	−1476.00	−0.203254
$376$	−2112.00	−0.289676
$377$	−10668.0	−1.45737
$378$	−378.000	−0.0514344
$379$	100.000	0.0135532	0.00677659	−	0.999977i	$-0.497843\pi$
$379$	100.000	0.0135532	0.00677659	+	0.999977i	$0.497843\pi$
$380$	−992.000	−0.133917
$381$	6264.00	0.842295
$382$	5304.00	0.710409
$383$	8080.00	1.07799	0.538993	−	0.842310i	$-0.318805\pi$
$383$	8080.00	1.07799	0.538993	+	0.842310i	$0.318805\pi$
$384$	384.000	0.0510310
$385$	112.000	0.0148261
$386$	292.000	0.0385036
$387$	1908.00	0.250618
$388$	−4760.00	−0.622815
$389$	−5482.00	−0.714520	−0.357260	−	0.934005i	$-0.616289\pi$
$389$	−5482.00	−0.714520	−0.357260	+	0.934005i	$0.616289\pi$
$390$	−504.000	−0.0654385
$391$	−152.000	−0.0196598
$392$	392.000	0.0505076
$393$	−876.000	−0.112439
$394$	−5092.00	−0.651095
$395$	1104.00	0.140629
$396$	−288.000	−0.0365468
$397$	10446.0	1.32058	0.660289	−	0.751011i	$-0.270434\pi$
$397$	10446.0	1.32058	0.660289	+	0.751011i	$0.270434\pi$
$398$	−5072.00	−0.638785
$399$	2604.00	0.326724
$400$	−1936.00	−0.242000
$401$	−11334.0	−1.41145	−0.705727	−	0.708484i	$-0.749379\pi$
$401$	−11334.0	−1.41145	−0.705727	+	0.708484i	$0.749379\pi$
$402$	−4584.00	−0.568729
$403$	3024.00	0.373787
$404$	5480.00	0.674852
$405$	162.000	0.0198762
$406$	−3556.00	−0.434683
$407$	−3184.00	−0.387776
$408$	−48.0000	−0.00582440
$409$	8594.00	1.03899	0.519494	−	0.854474i	$-0.326121\pi$
$409$	8594.00	1.03899	0.519494	+	0.854474i	$0.326121\pi$
$410$	1848.00	0.222601
$411$	2454.00	0.294518
$412$	1856.00	0.221938
$413$	5404.00	0.643858
$414$	1368.00	0.162400
$415$	2072.00	0.245085
$416$	−1344.00	−0.158401
$417$	−6468.00	−0.759567
$418$	1984.00	0.232155
$419$	10500.0	1.22424	0.612122	−	0.790763i	$-0.290316\pi$
$419$	10500.0	1.22424	0.612122	+	0.790763i	$0.290316\pi$
$420$	−168.000	−0.0195180
$421$	−12066.0	−1.39682	−0.698410	−	0.715698i	$-0.746109\pi$
$421$	−12066.0	−1.39682	−0.698410	+	0.715698i	$0.746109\pi$
$422$	−2600.00	−0.299919
$423$	−2376.00	−0.273109
$424$	−1296.00	−0.148442
$425$	242.000	0.0276205
$426$	−1416.00	−0.161046
$427$	−210.000	−0.0238000
$428$	−8544.00	−0.964930
$429$	1008.00	0.113442
$430$	848.000	0.0951028
$431$	4332.00	0.484142	0.242071	−	0.970259i	$-0.422173\pi$
$431$	4332.00	0.484142	0.242071	+	0.970259i	$0.422173\pi$
$432$	432.000	0.0481125
$433$	−1918.00	−0.212871	−0.106436	−	0.994320i	$-0.533944\pi$
$433$	−1918.00	−0.212871	−0.106436	+	0.994320i	$0.533944\pi$
$434$	1008.00	0.111487
$435$	1524.00	0.167977
$436$	−4904.00	−0.538667
$437$	−9424.00	−1.03160
$438$	2508.00	0.273600
$439$	−7992.00	−0.868878	−0.434439	−	0.900701i	$-0.643053\pi$
$439$	−7992.00	−0.868878	−0.434439	+	0.900701i	$0.643053\pi$
$440$	−128.000	−0.0138685
$441$	441.000	0.0476190
$442$	168.000	0.0180791
$443$	3184.00	0.341482	0.170741	−	0.985316i	$-0.445384\pi$
$443$	3184.00	0.341482	0.170741	+	0.985316i	$0.445384\pi$
$444$	4776.00	0.510493
$445$	60.0000	0.00639162
$446$	5152.00	0.546983
$447$	−8550.00	−0.904700
$448$	−448.000	−0.0472456
$449$	11426.0	1.20095	0.600475	−	0.799644i	$-0.294978\pi$
$449$	11426.0	1.20095	0.600475	+	0.799644i	$0.294978\pi$
$450$	−2178.00	−0.228160
$451$	−3696.00	−0.385893
$452$	1352.00	0.140692
$453$	5016.00	0.520248
$454$	−3672.00	−0.379594
$455$	588.000	0.0605843
$456$	−2976.00	−0.305623
$457$	−16934.0	−1.73335	−0.866673	−	0.498877i	$-0.833746\pi$
$457$	−16934.0	−1.73335	−0.866673	+	0.498877i	$0.833746\pi$
$458$	−3748.00	−0.382385
$459$	−54.0000	−0.00549129
$460$	608.000	0.0616264
$461$	−17038.0	−1.72134	−0.860671	−	0.509161i	$-0.829956\pi$
$461$	−17038.0	−1.72134	−0.860671	+	0.509161i	$0.829956\pi$
$462$	336.000	0.0338358
$463$	−13592.0	−1.36431	−0.682153	−	0.731209i	$-0.738956\pi$
$463$	−13592.0	−1.36431	−0.682153	+	0.731209i	$0.738956\pi$
$464$	4064.00	0.406609
$465$	−432.000	−0.0430828
$466$	7460.00	0.741583
$467$	8612.00	0.853353	0.426676	−	0.904404i	$-0.359684\pi$
$467$	8612.00	0.853353	0.426676	+	0.904404i	$0.359684\pi$
$468$	−1512.00	−0.149342
$469$	5348.00	0.526541
$470$	−1056.00	−0.103638
$471$	1338.00	0.130896
$472$	−6176.00	−0.602274
$473$	−1696.00	−0.164867
$474$	3312.00	0.320939
$475$	15004.0	1.44933
$476$	56.0000	0.00539234
$477$	−1458.00	−0.139952
$478$	4008.00	0.383518
$479$	7432.00	0.708928	0.354464	−	0.935070i	$-0.384663\pi$
$479$	7432.00	0.708928	0.354464	+	0.935070i	$0.384663\pi$
$480$	192.000	0.0182574
$481$	−16716.0	−1.58458
$482$	−1292.00	−0.122093
$483$	−1596.00	−0.150353
$484$	−5068.00	−0.475958
$485$	−2380.00	−0.222825
$486$	486.000	0.0453609
$487$	−6616.00	−0.615605	−0.307802	−	0.951450i	$-0.599594\pi$
$487$	−6616.00	−0.615605	−0.307802	+	0.951450i	$0.599594\pi$
$488$	240.000	0.0222629
$489$	8124.00	0.751288
$490$	196.000	0.0180702
$491$	17040.0	1.56620	0.783100	−	0.621896i	$-0.213637\pi$
$491$	17040.0	1.56620	0.783100	+	0.621896i	$0.213637\pi$
$492$	5544.00	0.508014
$493$	−508.000	−0.0464081
$494$	10416.0	0.948660
$495$	−144.000	−0.0130754
$496$	−1152.00	−0.104287
$497$	1652.00	0.149099
$498$	6216.00	0.559329
$499$	−2948.00	−0.264470	−0.132235	−	0.991218i	$-0.542215\pi$
$499$	−2948.00	−0.264470	−0.132235	+	0.991218i	$0.542215\pi$
$500$	−1968.00	−0.176023
$501$	2688.00	0.239703
$502$	2520.00	0.224050
$503$	17304.0	1.53389	0.766946	−	0.641712i	$-0.221776\pi$
$503$	17304.0	1.53389	0.766946	+	0.641712i	$0.221776\pi$
$504$	−504.000	−0.0445435
$505$	2740.00	0.241442
$506$	−1216.00	−0.106834
$507$	−1299.00	−0.113788
$508$	8352.00	0.729449
$509$	4650.00	0.404927	0.202463	−	0.979290i	$-0.435105\pi$
$509$	4650.00	0.404927	0.202463	+	0.979290i	$0.435105\pi$
$510$	−24.0000	−0.00208380
$511$	−2926.00	−0.253305
$512$	512.000	0.0441942
$513$	−3348.00	−0.288144
$514$	11820.0	1.01431
$515$	928.000	0.0794030
$516$	2544.00	0.217041
$517$	2112.00	0.179663
$518$	−5572.00	−0.472625
$519$	12102.0	1.02354
$520$	−672.000	−0.0566714
$521$	16854.0	1.41725	0.708625	−	0.705585i	$-0.249316\pi$
$521$	16854.0	1.41725	0.708625	+	0.705585i	$0.249316\pi$
$522$	4572.00	0.383354
$523$	−124.000	−0.0103674	−0.00518369	−	0.999987i	$-0.501650\pi$
$523$	−124.000	−0.0103674	−0.00518369	+	0.999987i	$0.501650\pi$
$524$	−1168.00	−0.0973747
$525$	2541.00	0.211235
$526$	5976.00	0.495373
$527$	144.000	0.0119027
$528$	−384.000	−0.0316505
$529$	−6391.00	−0.525273
$530$	−648.000	−0.0531082
$531$	−6948.00	−0.567830
$532$	3472.00	0.282952
$533$	−19404.0	−1.57689
$534$	180.000	0.0145868
$535$	−4272.00	−0.345224
$536$	−6112.00	−0.492534
$537$	−10440.0	−0.838956
$538$	−2636.00	−0.211238
$539$	−392.000	−0.0313259
$540$	216.000	0.0172133
$541$	5382.00	0.427708	0.213854	−	0.976866i	$-0.431398\pi$
$541$	5382.00	0.427708	0.213854	+	0.976866i	$0.431398\pi$
$542$	−11280.0	−0.893944
$543$	−8694.00	−0.687100
$544$	−64.0000	−0.00504408
$545$	−2452.00	−0.192720
$546$	1764.00	0.138264
$547$	17460.0	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$547$	17460.0	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$548$	3272.00	0.255060
$549$	270.000	0.0209897
$550$	1936.00	0.150093
$551$	−31496.0	−2.43516
$552$	1824.00	0.140642
$553$	−3864.00	−0.297132
$554$	12892.0	0.988680
$555$	2388.00	0.182640
$556$	−8624.00	−0.657804
$557$	−9514.00	−0.723736	−0.361868	−	0.932229i	$-0.617861\pi$
$557$	−9514.00	−0.723736	−0.361868	+	0.932229i	$0.617861\pi$
$558$	−1296.00	−0.0983227
$559$	−8904.00	−0.673701
$560$	−224.000	−0.0169031
$561$	48.0000	0.00361241
$562$	9860.00	0.740069
$563$	3988.00	0.298533	0.149267	−	0.988797i	$-0.452309\pi$
$563$	3988.00	0.298533	0.149267	+	0.988797i	$0.452309\pi$
$564$	−3168.00	−0.236519
$565$	676.000	0.0503355
$566$	−12520.0	−0.929779
$567$	−567.000	−0.0419961
$568$	−1888.00	−0.139470
$569$	11346.0	0.835939	0.417969	−	0.908461i	$-0.362742\pi$
$569$	11346.0	0.835939	0.417969	+	0.908461i	$0.362742\pi$
$570$	−1488.00	−0.109343
$571$	−8436.00	−0.618276	−0.309138	−	0.951017i	$-0.600041\pi$
$571$	−8436.00	−0.618276	−0.309138	+	0.951017i	$0.600041\pi$
$572$	1344.00	0.0982438
$573$	7956.00	0.580047
$574$	−6468.00	−0.470329
$575$	−9196.00	−0.666956
$576$	576.000	0.0416667
$577$	2098.00	0.151371	0.0756853	−	0.997132i	$-0.475886\pi$
$577$	2098.00	0.151371	0.0756853	+	0.997132i	$0.475886\pi$
$578$	−9818.00	−0.706531
$579$	438.000	0.0314381
$580$	2032.00	0.145473
$581$	−7252.00	−0.517838
$582$	−7140.00	−0.508527
$583$	1296.00	0.0920666
$584$	3344.00	0.236945
$585$	−756.000	−0.0534303
$586$	−4620.00	−0.325683
$587$	9436.00	0.663484	0.331742	−	0.943370i	$-0.392364\pi$
$587$	9436.00	0.663484	0.331742	+	0.943370i	$0.392364\pi$
$588$	588.000	0.0412393
$589$	8928.00	0.624570
$590$	−3088.00	−0.215476
$591$	−7638.00	−0.531616
$592$	6368.00	0.442100
$593$	−1314.00	−0.0909941	−0.0454971	−	0.998964i	$-0.514487\pi$
$593$	−1314.00	−0.0909941	−0.0454971	+	0.998964i	$0.514487\pi$
$594$	−432.000	−0.0298404
$595$	28.0000	0.00192922
$596$	−11400.0	−0.783494
$597$	−7608.00	−0.521566
$598$	−6384.00	−0.436557
$599$	−8940.00	−0.609814	−0.304907	−	0.952382i	$-0.598625\pi$
$599$	−8940.00	−0.609814	−0.304907	+	0.952382i	$0.598625\pi$
$600$	−2904.00	−0.197592
$601$	16058.0	1.08988	0.544941	−	0.838474i	$-0.316552\pi$
$601$	16058.0	1.08988	0.544941	+	0.838474i	$0.316552\pi$
$602$	−2968.00	−0.200941
$603$	−6876.00	−0.464365
$604$	6688.00	0.450548
$605$	−2534.00	−0.170284
$606$	8220.00	0.551014
$607$	3936.00	0.263192	0.131596	−	0.991303i	$-0.457990\pi$
$607$	3936.00	0.263192	0.131596	+	0.991303i	$0.457990\pi$
$608$	−3968.00	−0.264677
$609$	−5334.00	−0.354917
$610$	120.000	0.00796501
$611$	11088.0	0.734161
$612$	−72.0000	−0.00475560
$613$	174.000	0.0114646	0.00573230	−	0.999984i	$-0.498175\pi$
$613$	174.000	0.0114646	0.00573230	+	0.999984i	$0.498175\pi$
$614$	392.000	0.0257652
$615$	2772.00	0.181753
$616$	448.000	0.0293027
$617$	16018.0	1.04515	0.522577	−	0.852592i	$-0.324971\pi$
$617$	16018.0	1.04515	0.522577	+	0.852592i	$0.324971\pi$
$618$	2784.00	0.181212
$619$	−3068.00	−0.199214	−0.0996069	−	0.995027i	$-0.531759\pi$
$619$	−3068.00	−0.199214	−0.0996069	+	0.995027i	$0.531759\pi$
$620$	−576.000	−0.0373108
$621$	2052.00	0.132599
$622$	−13472.0	−0.868453
$623$	−210.000	−0.0135048
$624$	−2016.00	−0.129334
$625$	14141.0	0.905024
$626$	788.000	0.0503112
$627$	2976.00	0.189553
$628$	1784.00	0.113359
$629$	−796.000	−0.0504588
$630$	−252.000	−0.0159364
$631$	24656.0	1.55553	0.777765	−	0.628555i	$-0.216353\pi$
$631$	24656.0	1.55553	0.777765	+	0.628555i	$0.216353\pi$
$632$	4416.00	0.277942
$633$	−3900.00	−0.244883
$634$	−13428.0	−0.841158
$635$	4176.00	0.260976
$636$	−1944.00	−0.121202
$637$	−2058.00	−0.128008
$638$	−4064.00	−0.252187
$639$	−2124.00	−0.131493
$640$	256.000	0.0158114
$641$	7594.00	0.467933	0.233966	−	0.972245i	$-0.424829\pi$
$641$	7594.00	0.467933	0.233966	+	0.972245i	$0.424829\pi$
$642$	−12816.0	−0.787862
$643$	−3724.00	−0.228398	−0.114199	−	0.993458i	$-0.536430\pi$
$643$	−3724.00	−0.228398	−0.114199	+	0.993458i	$0.536430\pi$
$644$	−2128.00	−0.130210
$645$	1272.00	0.0776511
$646$	496.000	0.0302088
$647$	3792.00	0.230416	0.115208	−	0.993341i	$-0.463247\pi$
$647$	3792.00	0.230416	0.115208	+	0.993341i	$0.463247\pi$
$648$	648.000	0.0392837
$649$	6176.00	0.373543
$650$	10164.0	0.613331
$651$	1512.00	0.0910291
$652$	10832.0	0.650635
$653$	24702.0	1.48034	0.740171	−	0.672418i	$-0.234744\pi$
$653$	24702.0	1.48034	0.740171	+	0.672418i	$0.234744\pi$
$654$	−7356.00	−0.439820
$655$	−584.000	−0.0348378
$656$	7392.00	0.439953
$657$	3762.00	0.223394
$658$	3696.00	0.218974
$659$	−20144.0	−1.19074	−0.595371	−	0.803451i	$-0.702995\pi$
$659$	−20144.0	−1.19074	−0.595371	+	0.803451i	$0.702995\pi$
$660$	−192.000	−0.0113236
$661$	−2522.00	−0.148403	−0.0742015	−	0.997243i	$-0.523641\pi$
$661$	−2522.00	−0.148403	−0.0742015	+	0.997243i	$0.523641\pi$
$662$	1384.00	0.0812548
$663$	252.000	0.0147615
$664$	8288.00	0.484393
$665$	1736.00	0.101232
$666$	7164.00	0.416816
$667$	19304.0	1.12062
$668$	3584.00	0.207589
$669$	7728.00	0.446609
$670$	−3056.00	−0.176214
$671$	−240.000	−0.0138079
$672$	−672.000	−0.0385758
$673$	−10414.0	−0.596479	−0.298239	−	0.954491i	$-0.596399\pi$
$673$	−10414.0	−0.596479	−0.298239	+	0.954491i	$0.596399\pi$
$674$	−3132.00	−0.178991
$675$	−3267.00	−0.186292
$676$	−1732.00	−0.0985435
$677$	−22230.0	−1.26199	−0.630996	−	0.775786i	$-0.717353\pi$
$677$	−22230.0	−1.26199	−0.630996	+	0.775786i	$0.717353\pi$
$678$	2028.00	0.114874
$679$	8330.00	0.470804
$680$	−32.0000	−0.00180462
$681$	−5508.00	−0.309937
$682$	1152.00	0.0646809
$683$	18192.0	1.01918	0.509588	−	0.860418i	$-0.329798\pi$
$683$	18192.0	1.01918	0.509588	+	0.860418i	$0.329798\pi$
$684$	−4464.00	−0.249540
$685$	1636.00	0.0912531
$686$	−686.000	−0.0381802
$687$	−5622.00	−0.312216
$688$	3392.00	0.187963
$689$	6804.00	0.376214
$690$	912.000	0.0503177
$691$	8108.00	0.446372	0.223186	−	0.974776i	$-0.428354\pi$
$691$	8108.00	0.446372	0.223186	+	0.974776i	$0.428354\pi$
$692$	16136.0	0.886414
$693$	504.000	0.0276268
$694$	−10656.0	−0.582848
$695$	−4312.00	−0.235343
$696$	6096.00	0.331995
$697$	−924.000	−0.0502138
$698$	22652.0	1.22835
$699$	11190.0	0.605500
$700$	3388.00	0.182935
$701$	−5794.00	−0.312177	−0.156089	−	0.987743i	$-0.549889\pi$
$701$	−5794.00	−0.312177	−0.156089	+	0.987743i	$0.549889\pi$
$702$	−2268.00	−0.121938
$703$	−49352.0	−2.64772
$704$	−512.000	−0.0274101
$705$	−1584.00	−0.0846197
$706$	−4260.00	−0.227092
$707$	−9590.00	−0.510140
$708$	−9264.00	−0.491755
$709$	−1954.00	−0.103504	−0.0517518	−	0.998660i	$-0.516480\pi$
$709$	−1954.00	−0.103504	−0.0517518	+	0.998660i	$0.516480\pi$
$710$	−944.000	−0.0498982
$711$	4968.00	0.262046
$712$	240.000	0.0126326
$713$	−5472.00	−0.287417
$714$	84.0000	0.00440283
$715$	672.000	0.0351488
$716$	−13920.0	−0.726557
$717$	6012.00	0.313141
$718$	6088.00	0.316438
$719$	−32016.0	−1.66063	−0.830317	−	0.557292i	$-0.811840\pi$
$719$	−32016.0	−1.66063	−0.830317	+	0.557292i	$0.811840\pi$
$720$	288.000	0.0149071
$721$	−3248.00	−0.167770
$722$	17034.0	0.878033
$723$	−1938.00	−0.0996888
$724$	−11592.0	−0.595046
$725$	−30734.0	−1.57439
$726$	−7602.00	−0.388618
$727$	−23072.0	−1.17702	−0.588510	−	0.808490i	$-0.700285\pi$
$727$	−23072.0	−1.17702	−0.588510	+	0.808490i	$0.700285\pi$
$728$	2352.00	0.119740
$729$	729.000	0.0370370
$730$	1672.00	0.0847719
$731$	−424.000	−0.0214531
$732$	360.000	0.0181776
$733$	31782.0	1.60149	0.800747	−	0.599003i	$-0.204436\pi$
$733$	31782.0	1.60149	0.800747	+	0.599003i	$0.204436\pi$
$734$	24832.0	1.24873
$735$	294.000	0.0147542
$736$	2432.00	0.121800
$737$	6112.00	0.305480
$738$	8316.00	0.414792
$739$	−24396.0	−1.21437	−0.607186	−	0.794559i	$-0.707702\pi$
$739$	−24396.0	−1.21437	−0.607186	+	0.794559i	$0.707702\pi$
$740$	3184.00	0.158170
$741$	15624.0	0.774578
$742$	2268.00	0.112211
$743$	−32604.0	−1.60986	−0.804929	−	0.593371i	$-0.797797\pi$
$743$	−32604.0	−1.60986	−0.804929	+	0.593371i	$0.797797\pi$
$744$	−1728.00	−0.0851499
$745$	−5700.00	−0.280311
$746$	−14884.0	−0.730485
$747$	9324.00	0.456690
$748$	64.0000	0.00312844
$749$	14952.0	0.729418
$750$	−2952.00	−0.143722
$751$	−7680.00	−0.373165	−0.186583	−	0.982439i	$-0.559741\pi$
$751$	−7680.00	−0.373165	−0.186583	+	0.982439i	$0.559741\pi$
$752$	−4224.00	−0.204832
$753$	3780.00	0.182936
$754$	−21336.0	−1.03052
$755$	3344.00	0.161193
$756$	−756.000	−0.0363696
$757$	366.000	0.0175727	0.00878633	−	0.999961i	$-0.497203\pi$
$757$	366.000	0.0175727	0.00878633	+	0.999961i	$0.497203\pi$
$758$	200.000	0.00958355
$759$	−1824.00	−0.0872293
$760$	−1984.00	−0.0946937
$761$	29374.0	1.39922	0.699610	−	0.714525i	$-0.253357\pi$
$761$	29374.0	1.39922	0.699610	+	0.714525i	$0.253357\pi$
$762$	12528.0	0.595593
$763$	8582.00	0.407194
$764$	10608.0	0.502335
$765$	−36.0000	−0.00170142
$766$	16160.0	0.762251
$767$	32424.0	1.52642
$768$	768.000	0.0360844
$769$	−38990.0	−1.82837	−0.914184	−	0.405299i	$-0.867167\pi$
$769$	−38990.0	−1.82837	−0.914184	+	0.405299i	$0.867167\pi$
$770$	224.000	0.0104836
$771$	17730.0	0.828185
$772$	584.000	0.0272262
$773$	−20470.0	−0.952464	−0.476232	−	0.879320i	$-0.657998\pi$
$773$	−20470.0	−0.952464	−0.476232	+	0.879320i	$0.657998\pi$
$774$	3816.00	0.177214
$775$	8712.00	0.403799
$776$	−9520.00	−0.440397
$777$	−8358.00	−0.385896
$778$	−10964.0	−0.505242
$779$	−57288.0	−2.63486
$780$	−1008.00	−0.0462720
$781$	1888.00	0.0865019
$782$	−304.000	−0.0139016
$783$	6858.00	0.313008
$784$	784.000	0.0357143
$785$	892.000	0.0405565
$786$	−1752.00	−0.0795061
$787$	29916.0	1.35501	0.677503	−	0.735520i	$-0.263062\pi$
$787$	29916.0	1.35501	0.677503	+	0.735520i	$0.263062\pi$
$788$	−10184.0	−0.460393
$789$	8964.00	0.404470
$790$	2208.00	0.0994394
$791$	−2366.00	−0.106353
$792$	−576.000	−0.0258425
$793$	−1260.00	−0.0564236
$794$	20892.0	0.933790
$795$	−972.000	−0.0433626
$796$	−10144.0	−0.451689
$797$	4914.00	0.218398	0.109199	−	0.994020i	$-0.465172\pi$
$797$	4914.00	0.218398	0.109199	+	0.994020i	$0.465172\pi$
$798$	5208.00	0.231029
$799$	528.000	0.0233783
$800$	−3872.00	−0.171120
$801$	270.000	0.0119101
$802$	−22668.0	−0.998049
$803$	−3344.00	−0.146958
$804$	−9168.00	−0.402152
$805$	−1064.00	−0.0465852
$806$	6048.00	0.264307
$807$	−3954.00	−0.172475
$808$	10960.0	0.477192
$809$	34250.0	1.48846	0.744231	−	0.667922i	$-0.232816\pi$
$809$	34250.0	1.48846	0.744231	+	0.667922i	$0.232816\pi$
$810$	324.000	0.0140546
$811$	−41804.0	−1.81003	−0.905017	−	0.425376i	$-0.860142\pi$
$811$	−41804.0	−1.81003	−0.905017	+	0.425376i	$0.860142\pi$
$812$	−7112.00	−0.307367
$813$	−16920.0	−0.729902
$814$	−6368.00	−0.274199
$815$	5416.00	0.232778
$816$	−96.0000	−0.00411847
$817$	−26288.0	−1.12570
$818$	17188.0	0.734675
$819$	2646.00	0.112892
$820$	3696.00	0.157402
$821$	30862.0	1.31193	0.655963	−	0.754793i	$-0.272263\pi$
$821$	30862.0	1.31193	0.655963	+	0.754793i	$0.272263\pi$
$822$	4908.00	0.208256
$823$	10576.0	0.447942	0.223971	−	0.974596i	$-0.428098\pi$
$823$	10576.0	0.447942	0.223971	+	0.974596i	$0.428098\pi$
$824$	3712.00	0.156934
$825$	2904.00	0.122551
$826$	10808.0	0.455277
$827$	−10680.0	−0.449069	−0.224534	−	0.974466i	$-0.572086\pi$
$827$	−10680.0	−0.449069	−0.224534	+	0.974466i	$0.572086\pi$
$828$	2736.00	0.114834
$829$	−1178.00	−0.0493530	−0.0246765	−	0.999695i	$-0.507856\pi$
$829$	−1178.00	−0.0493530	−0.0246765	+	0.999695i	$0.507856\pi$
$830$	4144.00	0.173302
$831$	19338.0	0.807254
$832$	−2688.00	−0.112007
$833$	−98.0000	−0.00407623
$834$	−12936.0	−0.537095
$835$	1792.00	0.0742691
$836$	3968.00	0.164158
$837$	−1944.00	−0.0802801
$838$	21000.0	0.865672
$839$	−5600.00	−0.230433	−0.115217	−	0.993340i	$-0.536756\pi$
$839$	−5600.00	−0.230433	−0.115217	+	0.993340i	$0.536756\pi$
$840$	−336.000	−0.0138013
$841$	40127.0	1.64529
$842$	−24132.0	−0.987700
$843$	14790.0	0.604264
$844$	−5200.00	−0.212075
$845$	−866.000	−0.0352560
$846$	−4752.00	−0.193117
$847$	8869.00	0.359790
$848$	−2592.00	−0.104964
$849$	−18780.0	−0.759161
$850$	484.000	0.0195307
$851$	30248.0	1.21843
$852$	−2832.00	−0.113876
$853$	−826.000	−0.0331556	−0.0165778	−	0.999863i	$-0.505277\pi$
$853$	−826.000	−0.0331556	−0.0165778	+	0.999863i	$0.505277\pi$
$854$	−420.000	−0.0168292
$855$	−2232.00	−0.0892781
$856$	−17088.0	−0.682308
$857$	45918.0	1.83026	0.915128	−	0.403164i	$-0.132090\pi$
$857$	45918.0	1.83026	0.915128	+	0.403164i	$0.132090\pi$
$858$	2016.00	0.0802157
$859$	42380.0	1.68334	0.841669	−	0.539994i	$-0.181574\pi$
$859$	42380.0	1.68334	0.841669	+	0.539994i	$0.181574\pi$
$860$	1696.00	0.0672478
$861$	−9702.00	−0.384022
$862$	8664.00	0.342340
$863$	−26524.0	−1.04622	−0.523110	−	0.852265i	$-0.675228\pi$
$863$	−26524.0	−1.04622	−0.523110	+	0.852265i	$0.675228\pi$
$864$	864.000	0.0340207
$865$	8068.00	0.317133
$866$	−3836.00	−0.150523
$867$	−14727.0	−0.576880
$868$	2016.00	0.0788335
$869$	−4416.00	−0.172385
$870$	3048.00	0.118778
$871$	32088.0	1.24829
$872$	−9808.00	−0.380895
$873$	−10710.0	−0.415210
$874$	−18848.0	−0.729454
$875$	3444.00	0.133061
$876$	5016.00	0.193465
$877$	20614.0	0.793712	0.396856	−	0.917881i	$-0.370101\pi$
$877$	20614.0	0.793712	0.396856	+	0.917881i	$0.370101\pi$
$878$	−15984.0	−0.614389
$879$	−6930.00	−0.265919
$880$	−256.000	−0.00980654
$881$	−23730.0	−0.907473	−0.453737	−	0.891136i	$-0.649909\pi$
$881$	−23730.0	−0.907473	−0.453737	+	0.891136i	$0.649909\pi$
$882$	882.000	0.0336718
$883$	−9028.00	−0.344073	−0.172036	−	0.985091i	$-0.555035\pi$
$883$	−9028.00	−0.344073	−0.172036	+	0.985091i	$0.555035\pi$
$884$	336.000	0.0127838
$885$	−4632.00	−0.175936
$886$	6368.00	0.241464
$887$	37200.0	1.40818	0.704089	−	0.710112i	$-0.251356\pi$
$887$	37200.0	1.40818	0.704089	+	0.710112i	$0.251356\pi$
$888$	9552.00	0.360973
$889$	−14616.0	−0.551412
$890$	120.000	0.00451956
$891$	−648.000	−0.0243646
$892$	10304.0	0.386775
$893$	32736.0	1.22673
$894$	−17100.0	−0.639720
$895$	−6960.00	−0.259941
$896$	−896.000	−0.0334077
$897$	−9576.00	−0.356447
$898$	22852.0	0.849199
$899$	−18288.0	−0.678464
$900$	−4356.00	−0.161333
$901$	324.000	0.0119800
$902$	−7392.00	−0.272868
$903$	−4452.00	−0.164068
$904$	2704.00	0.0994842
$905$	−5796.00	−0.212890
$906$	10032.0	0.367871
$907$	23988.0	0.878179	0.439090	−	0.898443i	$-0.355301\pi$
$907$	23988.0	0.878179	0.439090	+	0.898443i	$0.355301\pi$
$908$	−7344.00	−0.268413
$909$	12330.0	0.449901
$910$	1176.00	0.0428396
$911$	15276.0	0.555561	0.277781	−	0.960645i	$-0.410401\pi$
$911$	15276.0	0.555561	0.277781	+	0.960645i	$0.410401\pi$
$912$	−5952.00	−0.216108
$913$	−8288.00	−0.300430
$914$	−33868.0	−1.22566
$915$	180.000	0.00650341
$916$	−7496.00	−0.270387
$917$	2044.00	0.0736083
$918$	−108.000	−0.00388293
$919$	−10760.0	−0.386224	−0.193112	−	0.981177i	$-0.561858\pi$
$919$	−10760.0	−0.386224	−0.193112	+	0.981177i	$0.561858\pi$
$920$	1216.00	0.0435764
$921$	588.000	0.0210372
$922$	−34076.0	−1.21717
$923$	9912.00	0.353475
$924$	672.000	0.0239255
$925$	−48158.0	−1.71181
$926$	−27184.0	−0.964710
$927$	4176.00	0.147959
$928$	8128.00	0.287516
$929$	−52890.0	−1.86788	−0.933942	−	0.357424i	$-0.883655\pi$
$929$	−52890.0	−1.86788	−0.933942	+	0.357424i	$0.883655\pi$
$930$	−864.000	−0.0304642
$931$	−6076.00	−0.213891
$932$	14920.0	0.524379
$933$	−20208.0	−0.709089
$934$	17224.0	0.603412
$935$	32.0000	0.00111926
$936$	−3024.00	−0.105601
$937$	−6118.00	−0.213305	−0.106652	−	0.994296i	$-0.534013\pi$
$937$	−6118.00	−0.213305	−0.106652	+	0.994296i	$0.534013\pi$
$938$	10696.0	0.372321
$939$	1182.00	0.0410789
$940$	−2112.00	−0.0732828
$941$	−32230.0	−1.11654	−0.558272	−	0.829658i	$-0.688535\pi$
$941$	−32230.0	−1.11654	−0.558272	+	0.829658i	$0.688535\pi$
$942$	2676.00	0.0925571
$943$	35112.0	1.21252
$944$	−12352.0	−0.425872
$945$	−378.000	−0.0130120
$946$	−3392.00	−0.116579
$947$	−18544.0	−0.636324	−0.318162	−	0.948036i	$-0.603066\pi$
$947$	−18544.0	−0.636324	−0.318162	+	0.948036i	$0.603066\pi$
$948$	6624.00	0.226938
$949$	−17556.0	−0.600518
$950$	30008.0	1.02483
$951$	−20142.0	−0.686802
$952$	112.000	0.00381296
$953$	25930.0	0.881380	0.440690	−	0.897659i	$-0.354734\pi$
$953$	25930.0	0.881380	0.440690	+	0.897659i	$0.354734\pi$
$954$	−2916.00	−0.0989612
$955$	5304.00	0.179721
$956$	8016.00	0.271188
$957$	−6096.00	−0.205910
$958$	14864.0	0.501288
$959$	−5726.00	−0.192807
$960$	384.000	0.0129099
$961$	−24607.0	−0.825988
$962$	−33432.0	−1.12047
$963$	−19224.0	−0.643286
$964$	−2584.00	−0.0863330
$965$	292.000	0.00974074
$966$	−3192.00	−0.106316
$967$	8192.00	0.272427	0.136214	−	0.990680i	$-0.456507\pi$
$967$	8192.00	0.272427	0.136214	+	0.990680i	$0.456507\pi$
$968$	−10136.0	−0.336553
$969$	744.000	0.0246653
$970$	−4760.00	−0.157561
$971$	−54444.0	−1.79937	−0.899686	−	0.436537i	$-0.856205\pi$
$971$	−54444.0	−1.79937	−0.899686	+	0.436537i	$0.856205\pi$
$972$	972.000	0.0320750
$973$	15092.0	0.497253
$974$	−13232.0	−0.435298
$975$	15246.0	0.500782
$976$	480.000	0.0157422
$977$	−25446.0	−0.833255	−0.416627	−	0.909077i	$-0.636788\pi$
$977$	−25446.0	−0.833255	−0.416627	+	0.909077i	$0.636788\pi$
$978$	16248.0	0.531241
$979$	−240.000	−0.00783497
$980$	392.000	0.0127775
$981$	−11034.0	−0.359112
$982$	34080.0	1.10747
$983$	33192.0	1.07697	0.538484	−	0.842635i	$-0.318997\pi$
$983$	33192.0	1.07697	0.538484	+	0.842635i	$0.318997\pi$
$984$	11088.0	0.359220
$985$	−5092.00	−0.164715
$986$	−1016.00	−0.0328154
$987$	5544.00	0.178792
$988$	20832.0	0.670804
$989$	16112.0	0.518030
$990$	−288.000	−0.00924570
$991$	11024.0	0.353369	0.176685	−	0.984268i	$-0.443463\pi$
$991$	11024.0	0.353369	0.176685	+	0.984268i	$0.443463\pi$
$992$	−2304.00	−0.0737420
$993$	2076.00	0.0663443
$994$	3304.00	0.105429
$995$	−5072.00	−0.161601
$996$	12432.0	0.395505
$997$	−40714.0	−1.29331	−0.646653	−	0.762785i	$-0.723832\pi$
$997$	−40714.0	−1.29331	−0.646653	+	0.762785i	$0.723832\pi$
$998$	−5896.00	−0.187009
$999$	10746.0	0.340329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	42.4.a.b.1.1	✓	1
3.2	odd	2		126.4.a.c.1.1		1
4.3	odd	2		336.4.a.d.1.1		1
5.2	odd	4		1050.4.g.n.799.2		2
5.3	odd	4		1050.4.g.n.799.1		2
5.4	even	2		1050.4.a.d.1.1		1
7.2	even	3		294.4.e.a.67.1		2
7.3	odd	6		294.4.e.d.79.1		2
7.4	even	3		294.4.e.a.79.1		2
7.5	odd	6		294.4.e.d.67.1		2
7.6	odd	2		294.4.a.h.1.1		1
8.3	odd	2		1344.4.a.t.1.1		1
8.5	even	2		1344.4.a.f.1.1		1
12.11	even	2		1008.4.a.j.1.1		1
21.2	odd	6		882.4.g.s.361.1		2
21.5	even	6		882.4.g.r.361.1		2
21.11	odd	6		882.4.g.s.667.1		2
21.17	even	6		882.4.g.r.667.1		2
21.20	even	2		882.4.a.d.1.1		1
28.27	even	2		2352.4.a.ba.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.a.b.1.1	✓	1	1.1	even	1	trivial
126.4.a.c.1.1		1	3.2	odd	2
294.4.a.h.1.1		1	7.6	odd	2
294.4.e.a.67.1		2	7.2	even	3
294.4.e.a.79.1		2	7.4	even	3
294.4.e.d.67.1		2	7.5	odd	6
294.4.e.d.79.1		2	7.3	odd	6
336.4.a.d.1.1		1	4.3	odd	2
882.4.a.d.1.1		1	21.20	even	2
882.4.g.r.361.1		2	21.5	even	6
882.4.g.r.667.1		2	21.17	even	6
882.4.g.s.361.1		2	21.2	odd	6
882.4.g.s.667.1		2	21.11	odd	6
1008.4.a.j.1.1		1	12.11	even	2
1050.4.a.d.1.1		1	5.4	even	2
1050.4.g.n.799.1		2	5.3	odd	4
1050.4.g.n.799.2		2	5.2	odd	4
1344.4.a.f.1.1		1	8.5	even	2
1344.4.a.t.1.1		1	8.3	odd	2
2352.4.a.ba.1.1		1	28.27	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:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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