LMFDB - Embedded newform 222.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [222,4,Mod(1,222)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(222, 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("222.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 222 = 2 \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	222.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:	$13.0984240213$
Analytic rank:	$1$
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	$\chi$	$=$	222.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -16.0000 q^{5} +6.00000 q^{6} -24.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -16.0000 q^{5} +6.00000 q^{6} -24.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -32.0000 q^{10} +8.00000 q^{11} +12.0000 q^{12} -78.0000 q^{13} -48.0000 q^{14} -48.0000 q^{15} +16.0000 q^{16} +12.0000 q^{17} +18.0000 q^{18} -16.0000 q^{19} -64.0000 q^{20} -72.0000 q^{21} +16.0000 q^{22} -198.000 q^{23} +24.0000 q^{24} +131.000 q^{25} -156.000 q^{26} +27.0000 q^{27} -96.0000 q^{28} -72.0000 q^{29} -96.0000 q^{30} +280.000 q^{31} +32.0000 q^{32} +24.0000 q^{33} +24.0000 q^{34} +384.000 q^{35} +36.0000 q^{36} +37.0000 q^{37} -32.0000 q^{38} -234.000 q^{39} -128.000 q^{40} -30.0000 q^{41} -144.000 q^{42} +244.000 q^{43} +32.0000 q^{44} -144.000 q^{45} -396.000 q^{46} +56.0000 q^{47} +48.0000 q^{48} +233.000 q^{49} +262.000 q^{50} +36.0000 q^{51} -312.000 q^{52} -654.000 q^{53} +54.0000 q^{54} -128.000 q^{55} -192.000 q^{56} -48.0000 q^{57} -144.000 q^{58} +38.0000 q^{59} -192.000 q^{60} +526.000 q^{61} +560.000 q^{62} -216.000 q^{63} +64.0000 q^{64} +1248.00 q^{65} +48.0000 q^{66} -516.000 q^{67} +48.0000 q^{68} -594.000 q^{69} +768.000 q^{70} -552.000 q^{71} +72.0000 q^{72} -842.000 q^{73} +74.0000 q^{74} +393.000 q^{75} -64.0000 q^{76} -192.000 q^{77} -468.000 q^{78} +588.000 q^{79} -256.000 q^{80} +81.0000 q^{81} -60.0000 q^{82} +368.000 q^{83} -288.000 q^{84} -192.000 q^{85} +488.000 q^{86} -216.000 q^{87} +64.0000 q^{88} +1136.00 q^{89} -288.000 q^{90} +1872.00 q^{91} -792.000 q^{92} +840.000 q^{93} +112.000 q^{94} +256.000 q^{95} +96.0000 q^{96} +726.000 q^{97} +466.000 q^{98} +72.0000 q^{99} +O(q^{100})\)

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
$5$	−16.0000	−1.43108	−0.715542	−	0.698570i	$-0.753820\pi$
$5$	−16.0000	−1.43108	−0.715542	+	0.698570i	$0.753820\pi$
$6$	6.00000	0.408248
$7$	−24.0000	−1.29588	−0.647939	−	0.761692i	$-0.724369\pi$
$7$	−24.0000	−1.29588	−0.647939	+	0.761692i	$0.724369\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−32.0000	−1.01193
$11$	8.00000	0.219281	0.109640	−	0.993971i	$-0.465030\pi$
$11$	8.00000	0.219281	0.109640	+	0.993971i	$0.465030\pi$
$12$	12.0000	0.288675
$13$	−78.0000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−78.0000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−48.0000	−0.916324
$15$	−48.0000	−0.826236
$16$	16.0000	0.250000
$17$	12.0000	0.171202	0.0856008	−	0.996330i	$-0.472719\pi$
$17$	12.0000	0.171202	0.0856008	+	0.996330i	$0.472719\pi$
$18$	18.0000	0.235702
$19$	−16.0000	−0.193192	−0.0965961	−	0.995324i	$-0.530796\pi$
$19$	−16.0000	−0.193192	−0.0965961	+	0.995324i	$0.530796\pi$
$20$	−64.0000	−0.715542
$21$	−72.0000	−0.748176
$22$	16.0000	0.155055
$23$	−198.000	−1.79504	−0.897519	−	0.440977i	$-0.854632\pi$
$23$	−198.000	−1.79504	−0.897519	+	0.440977i	$0.854632\pi$
$24$	24.0000	0.204124
$25$	131.000	1.04800
$26$	−156.000	−1.17670
$27$	27.0000	0.192450
$28$	−96.0000	−0.647939
$29$	−72.0000	−0.461037	−0.230518	−	0.973068i	$-0.574042\pi$
$29$	−72.0000	−0.461037	−0.230518	+	0.973068i	$0.574042\pi$
$30$	−96.0000	−0.584237
$31$	280.000	1.62224	0.811121	−	0.584879i	$-0.198858\pi$
$31$	280.000	1.62224	0.811121	+	0.584879i	$0.198858\pi$
$32$	32.0000	0.176777
$33$	24.0000	0.126602
$34$	24.0000	0.121058
$35$	384.000	1.85451
$36$	36.0000	0.166667
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	−32.0000	−0.136608
$39$	−234.000	−0.960769
$40$	−128.000	−0.505964
$41$	−30.0000	−0.114273	−0.0571367	−	0.998366i	$-0.518197\pi$
$41$	−30.0000	−0.114273	−0.0571367	+	0.998366i	$0.518197\pi$
$42$	−144.000	−0.529040
$43$	244.000	0.865341	0.432670	−	0.901552i	$-0.357571\pi$
$43$	244.000	0.865341	0.432670	+	0.901552i	$0.357571\pi$
$44$	32.0000	0.109640
$45$	−144.000	−0.477028
$46$	−396.000	−1.26928
$47$	56.0000	0.173797	0.0868983	−	0.996217i	$-0.472304\pi$
$47$	56.0000	0.173797	0.0868983	+	0.996217i	$0.472304\pi$
$48$	48.0000	0.144338
$49$	233.000	0.679300
$50$	262.000	0.741048
$51$	36.0000	0.0988433
$52$	−312.000	−0.832050
$53$	−654.000	−1.69498	−0.847489	−	0.530813i	$-0.821887\pi$
$53$	−654.000	−1.69498	−0.847489	+	0.530813i	$0.821887\pi$
$54$	54.0000	0.136083
$55$	−128.000	−0.313809
$56$	−192.000	−0.458162
$57$	−48.0000	−0.111540
$58$	−144.000	−0.326002
$59$	38.0000	0.0838505	0.0419252	−	0.999121i	$-0.486651\pi$
$59$	38.0000	0.0838505	0.0419252	+	0.999121i	$0.486651\pi$
$60$	−192.000	−0.413118
$61$	526.000	1.10406	0.552028	−	0.833826i	$-0.313854\pi$
$61$	526.000	1.10406	0.552028	+	0.833826i	$0.313854\pi$
$62$	560.000	1.14710
$63$	−216.000	−0.431959
$64$	64.0000	0.125000
$65$	1248.00	2.38147
$66$	48.0000	0.0895211
$67$	−516.000	−0.940887	−0.470444	−	0.882430i	$-0.655906\pi$
$67$	−516.000	−0.940887	−0.470444	+	0.882430i	$0.655906\pi$
$68$	48.0000	0.0856008
$69$	−594.000	−1.03637
$70$	768.000	1.31134
$71$	−552.000	−0.922681	−0.461340	−	0.887223i	$-0.652631\pi$
$71$	−552.000	−0.922681	−0.461340	+	0.887223i	$0.652631\pi$
$72$	72.0000	0.117851
$73$	−842.000	−1.34998	−0.674991	−	0.737826i	$-0.735852\pi$
$73$	−842.000	−1.34998	−0.674991	+	0.737826i	$0.735852\pi$
$74$	74.0000	0.116248
$75$	393.000	0.605063
$76$	−64.0000	−0.0965961
$77$	−192.000	−0.284161
$78$	−468.000	−0.679366
$79$	588.000	0.837407	0.418704	−	0.908123i	$-0.362485\pi$
$79$	588.000	0.837407	0.418704	+	0.908123i	$0.362485\pi$
$80$	−256.000	−0.357771
$81$	81.0000	0.111111
$82$	−60.0000	−0.0808036
$83$	368.000	0.486666	0.243333	−	0.969943i	$-0.421759\pi$
$83$	368.000	0.486666	0.243333	+	0.969943i	$0.421759\pi$
$84$	−288.000	−0.374088
$85$	−192.000	−0.245004
$86$	488.000	0.611888
$87$	−216.000	−0.266180
$88$	64.0000	0.0775275
$89$	1136.00	1.35299	0.676493	−	0.736449i	$-0.263499\pi$
$89$	1136.00	1.35299	0.676493	+	0.736449i	$0.263499\pi$
$90$	−288.000	−0.337310
$91$	1872.00	2.15647
$92$	−792.000	−0.897519
$93$	840.000	0.936602
$94$	112.000	0.122893
$95$	256.000	0.276474
$96$	96.0000	0.102062
$97$	726.000	0.759940	0.379970	−	0.924999i	$-0.375934\pi$
$97$	726.000	0.759940	0.379970	+	0.924999i	$0.375934\pi$
$98$	466.000	0.480338
$99$	72.0000	0.0730937
$100$	524.000	0.524000
$101$	−1442.00	−1.42064	−0.710319	−	0.703880i	$-0.751449\pi$
$101$	−1442.00	−1.42064	−0.710319	+	0.703880i	$0.751449\pi$
$102$	72.0000	0.0698928
$103$	1468.00	1.40433	0.702167	−	0.712013i	$-0.252216\pi$
$103$	1468.00	1.40433	0.702167	+	0.712013i	$0.252216\pi$
$104$	−624.000	−0.588348
$105$	1152.00	1.07070
$106$	−1308.00	−1.19853
$107$	−1900.00	−1.71663	−0.858317	−	0.513119i	$-0.828490\pi$
$107$	−1900.00	−1.71663	−0.858317	+	0.513119i	$0.828490\pi$
$108$	108.000	0.0962250
$109$	−70.0000	−0.0615118	−0.0307559	−	0.999527i	$-0.509791\pi$
$109$	−70.0000	−0.0615118	−0.0307559	+	0.999527i	$0.509791\pi$
$110$	−256.000	−0.221897
$111$	111.000	0.0949158
$112$	−384.000	−0.323970
$113$	32.0000	0.0266399	0.0133199	−	0.999911i	$-0.495760\pi$
$113$	32.0000	0.0266399	0.0133199	+	0.999911i	$0.495760\pi$
$114$	−96.0000	−0.0788704
$115$	3168.00	2.56885
$116$	−288.000	−0.230518
$117$	−702.000	−0.554700
$118$	76.0000	0.0592912
$119$	−288.000	−0.221856
$120$	−384.000	−0.292119
$121$	−1267.00	−0.951916
$122$	1052.00	0.780685
$123$	−90.0000	−0.0659758
$124$	1120.00	0.811121
$125$	−96.0000	−0.0686920
$126$	−432.000	−0.305441
$127$	−1592.00	−1.11234	−0.556170	−	0.831069i	$-0.687730\pi$
$127$	−1592.00	−1.11234	−0.556170	+	0.831069i	$0.687730\pi$
$128$	128.000	0.0883883
$129$	732.000	0.499605
$130$	2496.00	1.68395
$131$	−574.000	−0.382829	−0.191415	−	0.981509i	$-0.561307\pi$
$131$	−574.000	−0.382829	−0.191415	+	0.981509i	$0.561307\pi$
$132$	96.0000	0.0633010
$133$	384.000	0.250354
$134$	−1032.00	−0.665308
$135$	−432.000	−0.275412
$136$	96.0000	0.0605289
$137$	−1866.00	−1.16367	−0.581836	−	0.813306i	$-0.697666\pi$
$137$	−1866.00	−1.16367	−0.581836	+	0.813306i	$0.697666\pi$
$138$	−1188.00	−0.732821
$139$	2924.00	1.78425	0.892124	−	0.451791i	$-0.149215\pi$
$139$	2924.00	1.78425	0.892124	+	0.451791i	$0.149215\pi$
$140$	1536.00	0.927255
$141$	168.000	0.100342
$142$	−1104.00	−0.652434
$143$	−624.000	−0.364906
$144$	144.000	0.0833333
$145$	1152.00	0.659782
$146$	−1684.00	−0.954581
$147$	699.000	0.392194
$148$	148.000	0.0821995
$149$	534.000	0.293604	0.146802	−	0.989166i	$-0.453102\pi$
$149$	534.000	0.293604	0.146802	+	0.989166i	$0.453102\pi$
$150$	786.000	0.427844
$151$	264.000	0.142278	0.0711391	−	0.997466i	$-0.477337\pi$
$151$	264.000	0.142278	0.0711391	+	0.997466i	$0.477337\pi$
$152$	−128.000	−0.0683038
$153$	108.000	0.0570672
$154$	−384.000	−0.200932
$155$	−4480.00	−2.32156
$156$	−936.000	−0.480384
$157$	−398.000	−0.202318	−0.101159	−	0.994870i	$-0.532255\pi$
$157$	−398.000	−0.202318	−0.101159	+	0.994870i	$0.532255\pi$
$158$	1176.00	0.592136
$159$	−1962.00	−0.978596
$160$	−512.000	−0.252982
$161$	4752.00	2.32615
$162$	162.000	0.0785674
$163$	−2876.00	−1.38200	−0.690999	−	0.722856i	$-0.742829\pi$
$163$	−2876.00	−1.38200	−0.690999	+	0.722856i	$0.742829\pi$
$164$	−120.000	−0.0571367
$165$	−384.000	−0.181178
$166$	736.000	0.344125
$167$	−2090.00	−0.968437	−0.484219	−	0.874947i	$-0.660896\pi$
$167$	−2090.00	−0.968437	−0.484219	+	0.874947i	$0.660896\pi$
$168$	−576.000	−0.264520
$169$	3887.00	1.76923
$170$	−384.000	−0.173244
$171$	−144.000	−0.0643974
$172$	976.000	0.432670
$173$	1066.00	0.468477	0.234238	−	0.972179i	$-0.424740\pi$
$173$	1066.00	0.468477	0.234238	+	0.972179i	$0.424740\pi$
$174$	−432.000	−0.188217
$175$	−3144.00	−1.35808
$176$	128.000	0.0548202
$177$	114.000	0.0484111
$178$	2272.00	0.956706
$179$	−2778.00	−1.15999	−0.579993	−	0.814622i	$-0.696945\pi$
$179$	−2778.00	−1.15999	−0.579993	+	0.814622i	$0.696945\pi$
$180$	−576.000	−0.238514
$181$	−142.000	−0.0583137	−0.0291568	−	0.999575i	$-0.509282\pi$
$181$	−142.000	−0.0583137	−0.0291568	+	0.999575i	$0.509282\pi$
$182$	3744.00	1.52486
$183$	1578.00	0.637427
$184$	−1584.00	−0.634641
$185$	−592.000	−0.235269
$186$	1680.00	0.662277
$187$	96.0000	0.0375413
$188$	224.000	0.0868983
$189$	−648.000	−0.249392
$190$	512.000	0.195497
$191$	−4418.00	−1.67369	−0.836846	−	0.547438i	$-0.815603\pi$
$191$	−4418.00	−1.67369	−0.836846	+	0.547438i	$0.815603\pi$
$192$	192.000	0.0721688
$193$	−2662.00	−0.992824	−0.496412	−	0.868087i	$-0.665349\pi$
$193$	−2662.00	−0.992824	−0.496412	+	0.868087i	$0.665349\pi$
$194$	1452.00	0.537358
$195$	3744.00	1.37494
$196$	932.000	0.339650
$197$	4278.00	1.54718	0.773591	−	0.633685i	$-0.218459\pi$
$197$	4278.00	1.54718	0.773591	+	0.633685i	$0.218459\pi$
$198$	144.000	0.0516850
$199$	−32.0000	−0.0113991	−0.00569955	−	0.999984i	$-0.501814\pi$
$199$	−32.0000	−0.0113991	−0.00569955	+	0.999984i	$0.501814\pi$
$200$	1048.00	0.370524
$201$	−1548.00	−0.543221
$202$	−2884.00	−1.00454
$203$	1728.00	0.597447
$204$	144.000	0.0494217
$205$	480.000	0.163535
$206$	2936.00	0.993014
$207$	−1782.00	−0.598346
$208$	−1248.00	−0.416025
$209$	−128.000	−0.0423634
$210$	2304.00	0.757100
$211$	−3732.00	−1.21764	−0.608819	−	0.793309i	$-0.708356\pi$
$211$	−3732.00	−1.21764	−0.608819	+	0.793309i	$0.708356\pi$
$212$	−2616.00	−0.847489
$213$	−1656.00	−0.532710
$214$	−3800.00	−1.21384
$215$	−3904.00	−1.23837
$216$	216.000	0.0680414
$217$	−6720.00	−2.10223
$218$	−140.000	−0.0434954
$219$	−2526.00	−0.779412
$220$	−512.000	−0.156905
$221$	−936.000	−0.284897
$222$	222.000	0.0671156
$223$	−5584.00	−1.67683	−0.838413	−	0.545035i	$-0.816516\pi$
$223$	−5584.00	−1.67683	−0.838413	+	0.545035i	$0.816516\pi$
$224$	−768.000	−0.229081
$225$	1179.00	0.349333
$226$	64.0000	0.0188372
$227$	−4494.00	−1.31400	−0.656998	−	0.753892i	$-0.728174\pi$
$227$	−4494.00	−1.31400	−0.656998	+	0.753892i	$0.728174\pi$
$228$	−192.000	−0.0557698
$229$	−3074.00	−0.887055	−0.443528	−	0.896261i	$-0.646273\pi$
$229$	−3074.00	−0.887055	−0.443528	+	0.896261i	$0.646273\pi$
$230$	6336.00	1.81645
$231$	−576.000	−0.164061
$232$	−576.000	−0.163001
$233$	1754.00	0.493169	0.246584	−	0.969121i	$-0.420692\pi$
$233$	1754.00	0.493169	0.246584	+	0.969121i	$0.420692\pi$
$234$	−1404.00	−0.392232
$235$	−896.000	−0.248717
$236$	152.000	0.0419252
$237$	1764.00	0.483477
$238$	−576.000	−0.156876
$239$	−230.000	−0.0622488	−0.0311244	−	0.999516i	$-0.509909\pi$
$239$	−230.000	−0.0622488	−0.0311244	+	0.999516i	$0.509909\pi$
$240$	−768.000	−0.206559
$241$	5654.00	1.51123	0.755614	−	0.655017i	$-0.227338\pi$
$241$	5654.00	1.51123	0.755614	+	0.655017i	$0.227338\pi$
$242$	−2534.00	−0.673106
$243$	243.000	0.0641500
$244$	2104.00	0.552028
$245$	−3728.00	−0.972135
$246$	−180.000	−0.0466520
$247$	1248.00	0.321491
$248$	2240.00	0.573549
$249$	1104.00	0.280977
$250$	−192.000	−0.0485726
$251$	−6418.00	−1.61395	−0.806973	−	0.590588i	$-0.798896\pi$
$251$	−6418.00	−1.61395	−0.806973	+	0.590588i	$0.798896\pi$
$252$	−864.000	−0.215980
$253$	−1584.00	−0.393617
$254$	−3184.00	−0.786543
$255$	−576.000	−0.141453
$256$	256.000	0.0625000
$257$	6592.00	1.59999	0.799995	−	0.600006i	$-0.204835\pi$
$257$	6592.00	1.59999	0.799995	+	0.600006i	$0.204835\pi$
$258$	1464.00	0.353274
$259$	−888.000	−0.213041
$260$	4992.00	1.19073
$261$	−648.000	−0.153679
$262$	−1148.00	−0.270701
$263$	3444.00	0.807476	0.403738	−	0.914875i	$-0.367711\pi$
$263$	3444.00	0.807476	0.403738	+	0.914875i	$0.367711\pi$
$264$	192.000	0.0447605
$265$	10464.0	2.42565
$266$	768.000	0.177027
$267$	3408.00	0.781147
$268$	−2064.00	−0.470444
$269$	−586.000	−0.132822	−0.0664109	−	0.997792i	$-0.521155\pi$
$269$	−586.000	−0.132822	−0.0664109	+	0.997792i	$0.521155\pi$
$270$	−864.000	−0.194746
$271$	5616.00	1.25885	0.629424	−	0.777062i	$-0.283291\pi$
$271$	5616.00	1.25885	0.629424	+	0.777062i	$0.283291\pi$
$272$	192.000	0.0428004
$273$	5616.00	1.24504
$274$	−3732.00	−0.822841
$275$	1048.00	0.229806
$276$	−2376.00	−0.518183
$277$	8058.00	1.74786	0.873932	−	0.486048i	$-0.161562\pi$
$277$	8058.00	1.74786	0.873932	+	0.486048i	$0.161562\pi$
$278$	5848.00	1.26165
$279$	2520.00	0.540747
$280$	3072.00	0.655668
$281$	600.000	0.127377	0.0636886	−	0.997970i	$-0.479714\pi$
$281$	600.000	0.127377	0.0636886	+	0.997970i	$0.479714\pi$
$282$	336.000	0.0709522
$283$	−1904.00	−0.399933	−0.199967	−	0.979803i	$-0.564083\pi$
$283$	−1904.00	−0.399933	−0.199967	+	0.979803i	$0.564083\pi$
$284$	−2208.00	−0.461340
$285$	768.000	0.159622
$286$	−1248.00	−0.258027
$287$	720.000	0.148085
$288$	288.000	0.0589256
$289$	−4769.00	−0.970690
$290$	2304.00	0.466536
$291$	2178.00	0.438751
$292$	−3368.00	−0.674991
$293$	−4262.00	−0.849791	−0.424895	−	0.905242i	$-0.639689\pi$
$293$	−4262.00	−0.849791	−0.424895	+	0.905242i	$0.639689\pi$
$294$	1398.00	0.277323
$295$	−608.000	−0.119997
$296$	296.000	0.0581238
$297$	216.000	0.0422006
$298$	1068.00	0.207609
$299$	15444.0	2.98712
$300$	1572.00	0.302532
$301$	−5856.00	−1.12138
$302$	528.000	0.100606
$303$	−4326.00	−0.820205
$304$	−256.000	−0.0482980
$305$	−8416.00	−1.58000
$306$	216.000	0.0403526
$307$	−652.000	−0.121210	−0.0606052	−	0.998162i	$-0.519303\pi$
$307$	−652.000	−0.121210	−0.0606052	+	0.998162i	$0.519303\pi$
$308$	−768.000	−0.142081
$309$	4404.00	0.810792
$310$	−8960.00	−1.64159
$311$	−7494.00	−1.36639	−0.683193	−	0.730238i	$-0.739409\pi$
$311$	−7494.00	−1.36639	−0.683193	+	0.730238i	$0.739409\pi$
$312$	−1872.00	−0.339683
$313$	4642.00	0.838279	0.419140	−	0.907922i	$-0.362332\pi$
$313$	4642.00	0.838279	0.419140	+	0.907922i	$0.362332\pi$
$314$	−796.000	−0.143060
$315$	3456.00	0.618170
$316$	2352.00	0.418704
$317$	10786.0	1.91105	0.955524	−	0.294914i	$-0.0952910\pi$
$317$	10786.0	1.91105	0.955524	+	0.294914i	$0.0952910\pi$
$318$	−3924.00	−0.691972
$319$	−576.000	−0.101097
$320$	−1024.00	−0.178885
$321$	−5700.00	−0.991100
$322$	9504.00	1.64484
$323$	−192.000	−0.0330748
$324$	324.000	0.0555556
$325$	−10218.0	−1.74398
$326$	−5752.00	−0.977220
$327$	−210.000	−0.0355138
$328$	−240.000	−0.0404018
$329$	−1344.00	−0.225219
$330$	−768.000	−0.128112
$331$	−1832.00	−0.304217	−0.152108	−	0.988364i	$-0.548606\pi$
$331$	−1832.00	−0.304217	−0.152108	+	0.988364i	$0.548606\pi$
$332$	1472.00	0.243333
$333$	333.000	0.0547997
$334$	−4180.00	−0.684789
$335$	8256.00	1.34649
$336$	−1152.00	−0.187044
$337$	8658.00	1.39950	0.699750	−	0.714388i	$-0.253295\pi$
$337$	8658.00	1.39950	0.699750	+	0.714388i	$0.253295\pi$
$338$	7774.00	1.25104
$339$	96.0000	0.0153805
$340$	−768.000	−0.122502
$341$	2240.00	0.355727
$342$	−288.000	−0.0455358
$343$	2640.00	0.415588
$344$	1952.00	0.305944
$345$	9504.00	1.48313
$346$	2132.00	0.331263
$347$	4302.00	0.665543	0.332772	−	0.943007i	$-0.392016\pi$
$347$	4302.00	0.665543	0.332772	+	0.943007i	$0.392016\pi$
$348$	−864.000	−0.133090
$349$	−3542.00	−0.543263	−0.271632	−	0.962401i	$-0.587563\pi$
$349$	−3542.00	−0.543263	−0.271632	+	0.962401i	$0.587563\pi$
$350$	−6288.00	−0.960308
$351$	−2106.00	−0.320256
$352$	256.000	0.0387638
$353$	−8208.00	−1.23759	−0.618793	−	0.785554i	$-0.712378\pi$
$353$	−8208.00	−1.23759	−0.618793	+	0.785554i	$0.712378\pi$
$354$	228.000	0.0342318
$355$	8832.00	1.32043
$356$	4544.00	0.676493
$357$	−864.000	−0.128089
$358$	−5556.00	−0.820234
$359$	4780.00	0.702726	0.351363	−	0.936239i	$-0.385718\pi$
$359$	4780.00	0.702726	0.351363	+	0.936239i	$0.385718\pi$
$360$	−1152.00	−0.168655
$361$	−6603.00	−0.962677
$362$	−284.000	−0.0412340
$363$	−3801.00	−0.549589
$364$	7488.00	1.07824
$365$	13472.0	1.93194
$366$	3156.00	0.450729
$367$	296.000	0.0421010	0.0210505	−	0.999778i	$-0.493299\pi$
$367$	296.000	0.0421010	0.0210505	+	0.999778i	$0.493299\pi$
$368$	−3168.00	−0.448759
$369$	−270.000	−0.0380912
$370$	−1184.00	−0.166360
$371$	15696.0	2.19648
$372$	3360.00	0.468301
$373$	−14206.0	−1.97201	−0.986004	−	0.166723i	$-0.946681\pi$
$373$	−14206.0	−1.97201	−0.986004	+	0.166723i	$0.946681\pi$
$374$	192.000	0.0265457
$375$	−288.000	−0.0396593
$376$	448.000	0.0614464
$377$	5616.00	0.767211
$378$	−1296.00	−0.176347
$379$	4516.00	0.612062	0.306031	−	0.952022i	$-0.400999\pi$
$379$	4516.00	0.612062	0.306031	+	0.952022i	$0.400999\pi$
$380$	1024.00	0.138237
$381$	−4776.00	−0.642210
$382$	−8836.00	−1.18348
$383$	2138.00	0.285239	0.142620	−	0.989778i	$-0.454447\pi$
$383$	2138.00	0.285239	0.142620	+	0.989778i	$0.454447\pi$
$384$	384.000	0.0510310
$385$	3072.00	0.406659
$386$	−5324.00	−0.702032
$387$	2196.00	0.288447
$388$	2904.00	0.379970
$389$	4652.00	0.606339	0.303169	−	0.952937i	$-0.401955\pi$
$389$	4652.00	0.606339	0.303169	+	0.952937i	$0.401955\pi$
$390$	7488.00	0.972230
$391$	−2376.00	−0.307313
$392$	1864.00	0.240169
$393$	−1722.00	−0.221026
$394$	8556.00	1.09402
$395$	−9408.00	−1.19840
$396$	288.000	0.0365468
$397$	7394.00	0.934746	0.467373	−	0.884060i	$-0.345200\pi$
$397$	7394.00	0.934746	0.467373	+	0.884060i	$0.345200\pi$
$398$	−64.0000	−0.00806038
$399$	1152.00	0.144542
$400$	2096.00	0.262000
$401$	−1620.00	−0.201743	−0.100871	−	0.994899i	$-0.532163\pi$
$401$	−1620.00	−0.201743	−0.100871	+	0.994899i	$0.532163\pi$
$402$	−3096.00	−0.384116
$403$	−21840.0	−2.69957
$404$	−5768.00	−0.710319
$405$	−1296.00	−0.159009
$406$	3456.00	0.422459
$407$	296.000	0.0360496
$408$	288.000	0.0349464
$409$	7302.00	0.882789	0.441394	−	0.897313i	$-0.354484\pi$
$409$	7302.00	0.882789	0.441394	+	0.897313i	$0.354484\pi$
$410$	960.000	0.115637
$411$	−5598.00	−0.671847
$412$	5872.00	0.702167
$413$	−912.000	−0.108660
$414$	−3564.00	−0.423094
$415$	−5888.00	−0.696459
$416$	−2496.00	−0.294174
$417$	8772.00	1.03014
$418$	−256.000	−0.0299554
$419$	−6040.00	−0.704232	−0.352116	−	0.935956i	$-0.614538\pi$
$419$	−6040.00	−0.704232	−0.352116	+	0.935956i	$0.614538\pi$
$420$	4608.00	0.535351
$421$	11450.0	1.32551	0.662754	−	0.748837i	$-0.269388\pi$
$421$	11450.0	1.32551	0.662754	+	0.748837i	$0.269388\pi$
$422$	−7464.00	−0.861000
$423$	504.000	0.0579322
$424$	−5232.00	−0.599265
$425$	1572.00	0.179419
$426$	−3312.00	−0.376683
$427$	−12624.0	−1.43072
$428$	−7600.00	−0.858317
$429$	−1872.00	−0.210678
$430$	−7808.00	−0.875663
$431$	12850.0	1.43611	0.718054	−	0.695987i	$-0.245033\pi$
$431$	12850.0	1.43611	0.718054	+	0.695987i	$0.245033\pi$
$432$	432.000	0.0481125
$433$	8238.00	0.914303	0.457151	−	0.889389i	$-0.348870\pi$
$433$	8238.00	0.914303	0.457151	+	0.889389i	$0.348870\pi$
$434$	−13440.0	−1.48650
$435$	3456.00	0.380925
$436$	−280.000	−0.0307559
$437$	3168.00	0.346787
$438$	−5052.00	−0.551128
$439$	−14192.0	−1.54293	−0.771466	−	0.636270i	$-0.780476\pi$
$439$	−14192.0	−1.54293	−0.771466	+	0.636270i	$0.780476\pi$
$440$	−1024.00	−0.110948
$441$	2097.00	0.226433
$442$	−1872.00	−0.201452
$443$	−6900.00	−0.740020	−0.370010	−	0.929028i	$-0.620646\pi$
$443$	−6900.00	−0.740020	−0.370010	+	0.929028i	$0.620646\pi$
$444$	444.000	0.0474579
$445$	−18176.0	−1.93624
$446$	−11168.0	−1.18570
$447$	1602.00	0.169512
$448$	−1536.00	−0.161985
$449$	−10336.0	−1.08638	−0.543192	−	0.839609i	$-0.682784\pi$
$449$	−10336.0	−1.08638	−0.543192	+	0.839609i	$0.682784\pi$
$450$	2358.00	0.247016
$451$	−240.000	−0.0250580
$452$	128.000	0.0133199
$453$	792.000	0.0821444
$454$	−8988.00	−0.929136
$455$	−29952.0	−3.08609
$456$	−384.000	−0.0394352
$457$	−618.000	−0.0632578	−0.0316289	−	0.999500i	$-0.510069\pi$
$457$	−618.000	−0.0632578	−0.0316289	+	0.999500i	$0.510069\pi$
$458$	−6148.00	−0.627243
$459$	324.000	0.0329478
$460$	12672.0	1.28442
$461$	3332.00	0.336631	0.168315	−	0.985733i	$-0.446167\pi$
$461$	3332.00	0.336631	0.168315	+	0.985733i	$0.446167\pi$
$462$	−1152.00	−0.116008
$463$	9928.00	0.996530	0.498265	−	0.867025i	$-0.333971\pi$
$463$	9928.00	0.996530	0.498265	+	0.867025i	$0.333971\pi$
$464$	−1152.00	−0.115259
$465$	−13440.0	−1.34036
$466$	3508.00	0.348723
$467$	2790.00	0.276458	0.138229	−	0.990400i	$-0.455859\pi$
$467$	2790.00	0.276458	0.138229	+	0.990400i	$0.455859\pi$
$468$	−2808.00	−0.277350
$469$	12384.0	1.21928
$470$	−1792.00	−0.175870
$471$	−1194.00	−0.116808
$472$	304.000	0.0296456
$473$	1952.00	0.189753
$474$	3528.00	0.341870
$475$	−2096.00	−0.202465
$476$	−1152.00	−0.110928
$477$	−5886.00	−0.564993
$478$	−460.000	−0.0440165
$479$	−2506.00	−0.239044	−0.119522	−	0.992832i	$-0.538136\pi$
$479$	−2506.00	−0.239044	−0.119522	+	0.992832i	$0.538136\pi$
$480$	−1536.00	−0.146059
$481$	−2886.00	−0.273576
$482$	11308.0	1.06860
$483$	14256.0	1.34300
$484$	−5068.00	−0.475958
$485$	−11616.0	−1.08754
$486$	486.000	0.0453609
$487$	−2924.00	−0.272072	−0.136036	−	0.990704i	$-0.543436\pi$
$487$	−2924.00	−0.272072	−0.136036	+	0.990704i	$0.543436\pi$
$488$	4208.00	0.390343
$489$	−8628.00	−0.797897
$490$	−7456.00	−0.687404
$491$	−12288.0	−1.12943	−0.564715	−	0.825286i	$-0.691014\pi$
$491$	−12288.0	−1.12943	−0.564715	+	0.825286i	$0.691014\pi$
$492$	−360.000	−0.0329879
$493$	−864.000	−0.0789302
$494$	2496.00	0.227329
$495$	−1152.00	−0.104603
$496$	4480.00	0.405560
$497$	13248.0	1.19568
$498$	2208.00	0.198680
$499$	−8868.00	−0.795564	−0.397782	−	0.917480i	$-0.630220\pi$
$499$	−8868.00	−0.795564	−0.397782	+	0.917480i	$0.630220\pi$
$500$	−384.000	−0.0343460
$501$	−6270.00	−0.559128
$502$	−12836.0	−1.14123
$503$	4754.00	0.421412	0.210706	−	0.977549i	$-0.432424\pi$
$503$	4754.00	0.421412	0.210706	+	0.977549i	$0.432424\pi$
$504$	−1728.00	−0.152721
$505$	23072.0	2.03305
$506$	−3168.00	−0.278330
$507$	11661.0	1.02147
$508$	−6368.00	−0.556170
$509$	−2262.00	−0.196977	−0.0984886	−	0.995138i	$-0.531401\pi$
$509$	−2262.00	−0.196977	−0.0984886	+	0.995138i	$0.531401\pi$
$510$	−1152.00	−0.100022
$511$	20208.0	1.74941
$512$	512.000	0.0441942
$513$	−432.000	−0.0371799
$514$	13184.0	1.13136
$515$	−23488.0	−2.00972
$516$	2928.00	0.249802
$517$	448.000	0.0381103
$518$	−1776.00	−0.150643
$519$	3198.00	0.270475
$520$	9984.00	0.841976
$521$	−21438.0	−1.80272	−0.901359	−	0.433073i	$-0.857429\pi$
$521$	−21438.0	−1.80272	−0.901359	+	0.433073i	$0.857429\pi$
$522$	−1296.00	−0.108667
$523$	19164.0	1.60226	0.801131	−	0.598489i	$-0.204232\pi$
$523$	19164.0	1.60226	0.801131	+	0.598489i	$0.204232\pi$
$524$	−2296.00	−0.191415
$525$	−9432.00	−0.784088
$526$	6888.00	0.570972
$527$	3360.00	0.277730
$528$	384.000	0.0316505
$529$	27037.0	2.22216
$530$	20928.0	1.71520
$531$	342.000	0.0279502
$532$	1536.00	0.125177
$533$	2340.00	0.190163
$534$	6816.00	0.552354
$535$	30400.0	2.45665
$536$	−4128.00	−0.332654
$537$	−8334.00	−0.669718
$538$	−1172.00	−0.0939192
$539$	1864.00	0.148958
$540$	−1728.00	−0.137706
$541$	1218.00	0.0967947	0.0483973	−	0.998828i	$-0.484589\pi$
$541$	1218.00	0.0967947	0.0483973	+	0.998828i	$0.484589\pi$
$542$	11232.0	0.890140
$543$	−426.000	−0.0336674
$544$	384.000	0.0302645
$545$	1120.00	0.0880285
$546$	11232.0	0.880376
$547$	1740.00	0.136009	0.0680046	−	0.997685i	$-0.478337\pi$
$547$	1740.00	0.136009	0.0680046	+	0.997685i	$0.478337\pi$
$548$	−7464.00	−0.581836
$549$	4734.00	0.368019
$550$	2096.00	0.162498
$551$	1152.00	0.0890687
$552$	−4752.00	−0.366410
$553$	−14112.0	−1.08518
$554$	16116.0	1.23593
$555$	−1776.00	−0.135832
$556$	11696.0	0.892124
$557$	−2932.00	−0.223039	−0.111520	−	0.993762i	$-0.535572\pi$
$557$	−2932.00	−0.223039	−0.111520	+	0.993762i	$0.535572\pi$
$558$	5040.00	0.382366
$559$	−19032.0	−1.44001
$560$	6144.00	0.463627
$561$	288.000	0.0216745
$562$	1200.00	0.0900693
$563$	−7106.00	−0.531940	−0.265970	−	0.963981i	$-0.585692\pi$
$563$	−7106.00	−0.531940	−0.265970	+	0.963981i	$0.585692\pi$
$564$	672.000	0.0501708
$565$	−512.000	−0.0381239
$566$	−3808.00	−0.282795
$567$	−1944.00	−0.143986
$568$	−4416.00	−0.326217
$569$	21320.0	1.57079	0.785396	−	0.618993i	$-0.212459\pi$
$569$	21320.0	1.57079	0.785396	+	0.618993i	$0.212459\pi$
$570$	1536.00	0.112870
$571$	−14316.0	−1.04922	−0.524611	−	0.851342i	$-0.675789\pi$
$571$	−14316.0	−1.04922	−0.524611	+	0.851342i	$0.675789\pi$
$572$	−2496.00	−0.182453
$573$	−13254.0	−0.966307
$574$	1440.00	0.104712
$575$	−25938.0	−1.88120
$576$	576.000	0.0416667
$577$	−24830.0	−1.79148	−0.895742	−	0.444574i	$-0.853355\pi$
$577$	−24830.0	−1.79148	−0.895742	+	0.444574i	$0.853355\pi$
$578$	−9538.00	−0.686381
$579$	−7986.00	−0.573207
$580$	4608.00	0.329891
$581$	−8832.00	−0.630659
$582$	4356.00	0.310244
$583$	−5232.00	−0.371676
$584$	−6736.00	−0.477291
$585$	11232.0	0.793822
$586$	−8524.00	−0.600893
$587$	17502.0	1.23064	0.615319	−	0.788278i	$-0.289027\pi$
$587$	17502.0	1.23064	0.615319	+	0.788278i	$0.289027\pi$
$588$	2796.00	0.196097
$589$	−4480.00	−0.313404
$590$	−1216.00	−0.0848507
$591$	12834.0	0.893266
$592$	592.000	0.0410997
$593$	20098.0	1.39178	0.695890	−	0.718148i	$-0.255010\pi$
$593$	20098.0	1.39178	0.695890	+	0.718148i	$0.255010\pi$
$594$	432.000	0.0298404
$595$	4608.00	0.317495
$596$	2136.00	0.146802
$597$	−96.0000	−0.00658127
$598$	30888.0	2.11221
$599$	22696.0	1.54814	0.774068	−	0.633103i	$-0.218219\pi$
$599$	22696.0	1.54814	0.774068	+	0.633103i	$0.218219\pi$
$600$	3144.00	0.213922
$601$	−18146.0	−1.23160	−0.615799	−	0.787903i	$-0.711167\pi$
$601$	−18146.0	−1.23160	−0.615799	+	0.787903i	$0.711167\pi$
$602$	−11712.0	−0.792933
$603$	−4644.00	−0.313629
$604$	1056.00	0.0711391
$605$	20272.0	1.36227
$606$	−8652.00	−0.579973
$607$	25108.0	1.67892	0.839458	−	0.543424i	$-0.182872\pi$
$607$	25108.0	1.67892	0.839458	+	0.543424i	$0.182872\pi$
$608$	−512.000	−0.0341519
$609$	5184.00	0.344936
$610$	−16832.0	−1.11723
$611$	−4368.00	−0.289215
$612$	432.000	0.0285336
$613$	3038.00	0.200169	0.100085	−	0.994979i	$-0.468089\pi$
$613$	3038.00	0.200169	0.100085	+	0.994979i	$0.468089\pi$
$614$	−1304.00	−0.0857087
$615$	1440.00	0.0944169
$616$	−1536.00	−0.100466
$617$	12566.0	0.819916	0.409958	−	0.912104i	$-0.365543\pi$
$617$	12566.0	0.819916	0.409958	+	0.912104i	$0.365543\pi$
$618$	8808.00	0.573317
$619$	12724.0	0.826205	0.413102	−	0.910685i	$-0.364445\pi$
$619$	12724.0	0.826205	0.413102	+	0.910685i	$0.364445\pi$
$620$	−17920.0	−1.16078
$621$	−5346.00	−0.345455
$622$	−14988.0	−0.966180
$623$	−27264.0	−1.75331
$624$	−3744.00	−0.240192
$625$	−14839.0	−0.949696
$626$	9284.00	0.592753
$627$	−384.000	−0.0244585
$628$	−1592.00	−0.101159
$629$	444.000	0.0281454
$630$	6912.00	0.437112
$631$	23908.0	1.50834	0.754170	−	0.656679i	$-0.228039\pi$
$631$	23908.0	1.50834	0.754170	+	0.656679i	$0.228039\pi$
$632$	4704.00	0.296068
$633$	−11196.0	−0.703003
$634$	21572.0	1.35131
$635$	25472.0	1.59185
$636$	−7848.00	−0.489298
$637$	−18174.0	−1.13042
$638$	−1152.00	−0.0714861
$639$	−4968.00	−0.307560
$640$	−2048.00	−0.126491
$641$	12370.0	0.762224	0.381112	−	0.924529i	$-0.375541\pi$
$641$	12370.0	0.762224	0.381112	+	0.924529i	$0.375541\pi$
$642$	−11400.0	−0.700813
$643$	7568.00	0.464157	0.232078	−	0.972697i	$-0.425447\pi$
$643$	7568.00	0.464157	0.232078	+	0.972697i	$0.425447\pi$
$644$	19008.0	1.16307
$645$	−11712.0	−0.714976
$646$	−384.000	−0.0233874
$647$	−14150.0	−0.859805	−0.429903	−	0.902875i	$-0.641452\pi$
$647$	−14150.0	−0.859805	−0.429903	+	0.902875i	$0.641452\pi$
$648$	648.000	0.0392837
$649$	304.000	0.0183868
$650$	−20436.0	−1.23318
$651$	−20160.0	−1.21372
$652$	−11504.0	−0.690999
$653$	10188.0	0.610547	0.305274	−	0.952265i	$-0.401252\pi$
$653$	10188.0	0.610547	0.305274	+	0.952265i	$0.401252\pi$
$654$	−420.000	−0.0251121
$655$	9184.00	0.547860
$656$	−480.000	−0.0285684
$657$	−7578.00	−0.449994
$658$	−2688.00	−0.159254
$659$	1720.00	0.101672	0.0508359	−	0.998707i	$-0.483811\pi$
$659$	1720.00	0.101672	0.0508359	+	0.998707i	$0.483811\pi$
$660$	−1536.00	−0.0905890
$661$	9798.00	0.576548	0.288274	−	0.957548i	$-0.406919\pi$
$661$	9798.00	0.576548	0.288274	+	0.957548i	$0.406919\pi$
$662$	−3664.00	−0.215114
$663$	−2808.00	−0.164485
$664$	2944.00	0.172062
$665$	−6144.00	−0.358277
$666$	666.000	0.0387492
$667$	14256.0	0.827578
$668$	−8360.00	−0.484219
$669$	−16752.0	−0.968116
$670$	16512.0	0.952111
$671$	4208.00	0.242098
$672$	−2304.00	−0.132260
$673$	11158.0	0.639093	0.319546	−	0.947571i	$-0.396470\pi$
$673$	11158.0	0.639093	0.319546	+	0.947571i	$0.396470\pi$
$674$	17316.0	0.989596
$675$	3537.00	0.201688
$676$	15548.0	0.884615
$677$	−12334.0	−0.700198	−0.350099	−	0.936713i	$-0.613852\pi$
$677$	−12334.0	−0.700198	−0.350099	+	0.936713i	$0.613852\pi$
$678$	192.000	0.0108757
$679$	−17424.0	−0.984789
$680$	−1536.00	−0.0866219
$681$	−13482.0	−0.758636
$682$	4480.00	0.251537
$683$	16302.0	0.913292	0.456646	−	0.889648i	$-0.349051\pi$
$683$	16302.0	0.913292	0.456646	+	0.889648i	$0.349051\pi$
$684$	−576.000	−0.0321987
$685$	29856.0	1.66531
$686$	5280.00	0.293865
$687$	−9222.00	−0.512142
$688$	3904.00	0.216335
$689$	51012.0	2.82061
$690$	19008.0	1.04873
$691$	−17980.0	−0.989857	−0.494929	−	0.868934i	$-0.664806\pi$
$691$	−17980.0	−0.989857	−0.494929	+	0.868934i	$0.664806\pi$
$692$	4264.00	0.234238
$693$	−1728.00	−0.0947205
$694$	8604.00	0.470610
$695$	−46784.0	−2.55341
$696$	−1728.00	−0.0941087
$697$	−360.000	−0.0195638
$698$	−7084.00	−0.384145
$699$	5262.00	0.284731
$700$	−12576.0	−0.679040
$701$	−7220.00	−0.389009	−0.194505	−	0.980902i	$-0.562310\pi$
$701$	−7220.00	−0.389009	−0.194505	+	0.980902i	$0.562310\pi$
$702$	−4212.00	−0.226455
$703$	−592.000	−0.0317606
$704$	512.000	0.0274101
$705$	−2688.00	−0.143597
$706$	−16416.0	−0.875105
$707$	34608.0	1.84097
$708$	456.000	0.0242056
$709$	786.000	0.0416345	0.0208172	−	0.999783i	$-0.493373\pi$
$709$	786.000	0.0416345	0.0208172	+	0.999783i	$0.493373\pi$
$710$	17664.0	0.933687
$711$	5292.00	0.279136
$712$	9088.00	0.478353
$713$	−55440.0	−2.91198
$714$	−1728.00	−0.0905725
$715$	9984.00	0.522210
$716$	−11112.0	−0.579993
$717$	−690.000	−0.0359394
$718$	9560.00	0.496903
$719$	11844.0	0.614335	0.307167	−	0.951656i	$-0.400619\pi$
$719$	11844.0	0.614335	0.307167	+	0.951656i	$0.400619\pi$
$720$	−2304.00	−0.119257
$721$	−35232.0	−1.81985
$722$	−13206.0	−0.680715
$723$	16962.0	0.872508
$724$	−568.000	−0.0291568
$725$	−9432.00	−0.483166
$726$	−7602.00	−0.388618
$727$	−8272.00	−0.421997	−0.210998	−	0.977486i	$-0.567671\pi$
$727$	−8272.00	−0.421997	−0.210998	+	0.977486i	$0.567671\pi$
$728$	14976.0	0.762428
$729$	729.000	0.0370370
$730$	26944.0	1.36609
$731$	2928.00	0.148148
$732$	6312.00	0.318713
$733$	−19738.0	−0.994597	−0.497299	−	0.867579i	$-0.665675\pi$
$733$	−19738.0	−0.994597	−0.497299	+	0.867579i	$0.665675\pi$
$734$	592.000	0.0297699
$735$	−11184.0	−0.561263
$736$	−6336.00	−0.317321
$737$	−4128.00	−0.206319
$738$	−540.000	−0.0269345
$739$	−16652.0	−0.828895	−0.414448	−	0.910073i	$-0.636025\pi$
$739$	−16652.0	−0.828895	−0.414448	+	0.910073i	$0.636025\pi$
$740$	−2368.00	−0.117634
$741$	3744.00	0.185613
$742$	31392.0	1.55315
$743$	17508.0	0.864477	0.432238	−	0.901759i	$-0.357724\pi$
$743$	17508.0	0.864477	0.432238	+	0.901759i	$0.357724\pi$
$744$	6720.00	0.331139
$745$	−8544.00	−0.420172
$746$	−28412.0	−1.39442
$747$	3312.00	0.162222
$748$	384.000	0.0187706
$749$	45600.0	2.22455
$750$	−576.000	−0.0280434
$751$	30872.0	1.50005	0.750023	−	0.661411i	$-0.230042\pi$
$751$	30872.0	1.50005	0.750023	+	0.661411i	$0.230042\pi$
$752$	896.000	0.0434491
$753$	−19254.0	−0.931812
$754$	11232.0	0.542500
$755$	−4224.00	−0.203612
$756$	−2592.00	−0.124696
$757$	−28670.0	−1.37652	−0.688262	−	0.725462i	$-0.741626\pi$
$757$	−28670.0	−1.37652	−0.688262	+	0.725462i	$0.741626\pi$
$758$	9032.00	0.432793
$759$	−4752.00	−0.227255
$760$	2048.00	0.0977484
$761$	−12850.0	−0.612105	−0.306053	−	0.952015i	$-0.599008\pi$
$761$	−12850.0	−0.612105	−0.306053	+	0.952015i	$0.599008\pi$
$762$	−9552.00	−0.454111
$763$	1680.00	0.0797118
$764$	−17672.0	−0.836846
$765$	−1728.00	−0.0816679
$766$	4276.00	0.201695
$767$	−2964.00	−0.139536
$768$	768.000	0.0360844
$769$	17930.0	0.840796	0.420398	−	0.907340i	$-0.361890\pi$
$769$	17930.0	0.840796	0.420398	+	0.907340i	$0.361890\pi$
$770$	6144.00	0.287551
$771$	19776.0	0.923755
$772$	−10648.0	−0.496412
$773$	−26026.0	−1.21098	−0.605492	−	0.795852i	$-0.707024\pi$
$773$	−26026.0	−1.21098	−0.605492	+	0.795852i	$0.707024\pi$
$774$	4392.00	0.203963
$775$	36680.0	1.70011
$776$	5808.00	0.268679
$777$	−2664.00	−0.122999
$778$	9304.00	0.428746
$779$	480.000	0.0220767
$780$	14976.0	0.687470
$781$	−4416.00	−0.202326
$782$	−4752.00	−0.217303
$783$	−1944.00	−0.0887266
$784$	3728.00	0.169825
$785$	6368.00	0.289533
$786$	−3444.00	−0.156289
$787$	−11612.0	−0.525951	−0.262975	−	0.964803i	$-0.584704\pi$
$787$	−11612.0	−0.525951	−0.262975	+	0.964803i	$0.584704\pi$
$788$	17112.0	0.773591
$789$	10332.0	0.466196
$790$	−18816.0	−0.847397
$791$	−768.000	−0.0345220
$792$	576.000	0.0258425
$793$	−41028.0	−1.83726
$794$	14788.0	0.660965
$795$	31392.0	1.40045
$796$	−128.000	−0.00569955
$797$	23960.0	1.06488	0.532438	−	0.846469i	$-0.321276\pi$
$797$	23960.0	1.06488	0.532438	+	0.846469i	$0.321276\pi$
$798$	2304.00	0.102206
$799$	672.000	0.0297543
$800$	4192.00	0.185262
$801$	10224.0	0.450995
$802$	−3240.00	−0.142654
$803$	−6736.00	−0.296025
$804$	−6192.00	−0.271611
$805$	−76032.0	−3.32891
$806$	−43680.0	−1.90889
$807$	−1758.00	−0.0766847
$808$	−11536.0	−0.502271
$809$	−2604.00	−0.113167	−0.0565833	−	0.998398i	$-0.518021\pi$
$809$	−2604.00	−0.113167	−0.0565833	+	0.998398i	$0.518021\pi$
$810$	−2592.00	−0.112437
$811$	−13788.0	−0.596994	−0.298497	−	0.954411i	$-0.596485\pi$
$811$	−13788.0	−0.596994	−0.298497	+	0.954411i	$0.596485\pi$
$812$	6912.00	0.298724
$813$	16848.0	0.726796
$814$	592.000	0.0254909
$815$	46016.0	1.97775
$816$	576.000	0.0247108
$817$	−3904.00	−0.167177
$818$	14604.0	0.624226
$819$	16848.0	0.718824
$820$	1920.00	0.0817674
$821$	25778.0	1.09581	0.547904	−	0.836541i	$-0.315426\pi$
$821$	25778.0	1.09581	0.547904	+	0.836541i	$0.315426\pi$
$822$	−11196.0	−0.475067
$823$	−39816.0	−1.68639	−0.843195	−	0.537608i	$-0.819328\pi$
$823$	−39816.0	−1.68639	−0.843195	+	0.537608i	$0.819328\pi$
$824$	11744.0	0.496507
$825$	3144.00	0.132679
$826$	−1824.00	−0.0768342
$827$	−31570.0	−1.32744	−0.663722	−	0.747979i	$-0.731024\pi$
$827$	−31570.0	−1.32744	−0.663722	+	0.747979i	$0.731024\pi$
$828$	−7128.00	−0.299173
$829$	1514.00	0.0634299	0.0317150	−	0.999497i	$-0.489903\pi$
$829$	1514.00	0.0634299	0.0317150	+	0.999497i	$0.489903\pi$
$830$	−11776.0	−0.492471
$831$	24174.0	1.00913
$832$	−4992.00	−0.208013
$833$	2796.00	0.116297
$834$	17544.0	0.728416
$835$	33440.0	1.38591
$836$	−512.000	−0.0211817
$837$	7560.00	0.312201
$838$	−12080.0	−0.497967
$839$	16444.0	0.676651	0.338325	−	0.941029i	$-0.390140\pi$
$839$	16444.0	0.676651	0.338325	+	0.941029i	$0.390140\pi$
$840$	9216.00	0.378550
$841$	−19205.0	−0.787445
$842$	22900.0	0.937276
$843$	1800.00	0.0735413
$844$	−14928.0	−0.608819
$845$	−62192.0	−2.53192
$846$	1008.00	0.0409642
$847$	30408.0	1.23357
$848$	−10464.0	−0.423744
$849$	−5712.00	−0.230901
$850$	3144.00	0.126869
$851$	−7326.00	−0.295102
$852$	−6624.00	−0.266355
$853$	−23914.0	−0.959906	−0.479953	−	0.877294i	$-0.659346\pi$
$853$	−23914.0	−0.959906	−0.479953	+	0.877294i	$0.659346\pi$
$854$	−25248.0	−1.01167
$855$	2304.00	0.0921581
$856$	−15200.0	−0.606922
$857$	4216.00	0.168046	0.0840232	−	0.996464i	$-0.473223\pi$
$857$	4216.00	0.168046	0.0840232	+	0.996464i	$0.473223\pi$
$858$	−3744.00	−0.148972
$859$	−44204.0	−1.75579	−0.877893	−	0.478856i	$-0.841052\pi$
$859$	−44204.0	−1.75579	−0.877893	+	0.478856i	$0.841052\pi$
$860$	−15616.0	−0.619187
$861$	2160.00	0.0854966
$862$	25700.0	1.01548
$863$	−28860.0	−1.13836	−0.569181	−	0.822212i	$-0.692740\pi$
$863$	−28860.0	−1.13836	−0.569181	+	0.822212i	$0.692740\pi$
$864$	864.000	0.0340207
$865$	−17056.0	−0.670429
$866$	16476.0	0.646510
$867$	−14307.0	−0.560428
$868$	−26880.0	−1.05111
$869$	4704.00	0.183627
$870$	6912.00	0.269355
$871$	40248.0	1.56573
$872$	−560.000	−0.0217477
$873$	6534.00	0.253313
$874$	6336.00	0.245216
$875$	2304.00	0.0890165
$876$	−10104.0	−0.389706
$877$	19186.0	0.738729	0.369364	−	0.929285i	$-0.379575\pi$
$877$	19186.0	0.738729	0.369364	+	0.929285i	$0.379575\pi$
$878$	−28384.0	−1.09102
$879$	−12786.0	−0.490627
$880$	−2048.00	−0.0784523
$881$	27270.0	1.04285	0.521424	−	0.853298i	$-0.325401\pi$
$881$	27270.0	1.04285	0.521424	+	0.853298i	$0.325401\pi$
$882$	4194.00	0.160113
$883$	−12272.0	−0.467707	−0.233854	−	0.972272i	$-0.575134\pi$
$883$	−12272.0	−0.467707	−0.233854	+	0.972272i	$0.575134\pi$
$884$	−3744.00	−0.142448
$885$	−1824.00	−0.0692803
$886$	−13800.0	−0.523273
$887$	−36668.0	−1.38804	−0.694020	−	0.719956i	$-0.744162\pi$
$887$	−36668.0	−1.38804	−0.694020	+	0.719956i	$0.744162\pi$
$888$	888.000	0.0335578
$889$	38208.0	1.44146
$890$	−36352.0	−1.36913
$891$	648.000	0.0243646
$892$	−22336.0	−0.838413
$893$	−896.000	−0.0335761
$894$	3204.00	0.119863
$895$	44448.0	1.66004
$896$	−3072.00	−0.114541
$897$	46332.0	1.72462
$898$	−20672.0	−0.768189
$899$	−20160.0	−0.747913
$900$	4716.00	0.174667
$901$	−7848.00	−0.290183
$902$	−480.000	−0.0177187
$903$	−17568.0	−0.647427
$904$	256.000	0.00941862
$905$	2272.00	0.0834518
$906$	1584.00	0.0580849
$907$	−52.0000	−0.00190367	−0.000951837	−	1.00000i	$-0.500303\pi$
$907$	−52.0000	−0.00190367	−0.000951837		1.00000i	$0.500303\pi$
$908$	−17976.0	−0.656998
$909$	−12978.0	−0.473546
$910$	−59904.0	−2.18220
$911$	−6670.00	−0.242576	−0.121288	−	0.992617i	$-0.538702\pi$
$911$	−6670.00	−0.242576	−0.121288	+	0.992617i	$0.538702\pi$
$912$	−768.000	−0.0278849
$913$	2944.00	0.106717
$914$	−1236.00	−0.0447300
$915$	−25248.0	−0.912211
$916$	−12296.0	−0.443528
$917$	13776.0	0.496100
$918$	648.000	0.0232976
$919$	−33188.0	−1.19126	−0.595632	−	0.803258i	$-0.703098\pi$
$919$	−33188.0	−1.19126	−0.595632	+	0.803258i	$0.703098\pi$
$920$	25344.0	0.908225
$921$	−1956.00	−0.0699809
$922$	6664.00	0.238034
$923$	43056.0	1.53543
$924$	−2304.00	−0.0820303
$925$	4847.00	0.172290
$926$	19856.0	0.704653
$927$	13212.0	0.468111
$928$	−2304.00	−0.0815005
$929$	22034.0	0.778162	0.389081	−	0.921204i	$-0.372793\pi$
$929$	22034.0	0.778162	0.389081	+	0.921204i	$0.372793\pi$
$930$	−26880.0	−0.947774
$931$	−3728.00	−0.131236
$932$	7016.00	0.246584
$933$	−22482.0	−0.788883
$934$	5580.00	0.195485
$935$	−1536.00	−0.0537247
$936$	−5616.00	−0.196116
$937$	38570.0	1.34475	0.672373	−	0.740213i	$-0.265275\pi$
$937$	38570.0	1.34475	0.672373	+	0.740213i	$0.265275\pi$
$938$	24768.0	0.862158
$939$	13926.0	0.483981
$940$	−3584.00	−0.124359
$941$	21282.0	0.737272	0.368636	−	0.929574i	$-0.379825\pi$
$941$	21282.0	0.737272	0.368636	+	0.929574i	$0.379825\pi$
$942$	−2388.00	−0.0825958
$943$	5940.00	0.205125
$944$	608.000	0.0209626
$945$	10368.0	0.356901
$946$	3904.00	0.134175
$947$	−47994.0	−1.64688	−0.823440	−	0.567403i	$-0.807948\pi$
$947$	−47994.0	−1.64688	−0.823440	+	0.567403i	$0.807948\pi$
$948$	7056.00	0.241739
$949$	65676.0	2.24650
$950$	−4192.00	−0.143165
$951$	32358.0	1.10334
$952$	−2304.00	−0.0784381
$953$	−52946.0	−1.79967	−0.899837	−	0.436226i	$-0.856315\pi$
$953$	−52946.0	−1.79967	−0.899837	+	0.436226i	$0.856315\pi$
$954$	−11772.0	−0.399510
$955$	70688.0	2.39519
$956$	−920.000	−0.0311244
$957$	−1728.00	−0.0583681
$958$	−5012.00	−0.169030
$959$	44784.0	1.50798
$960$	−3072.00	−0.103280
$961$	48609.0	1.63167
$962$	−5772.00	−0.193448
$963$	−17100.0	−0.572212
$964$	22616.0	0.755614
$965$	42592.0	1.42081
$966$	28512.0	0.949647
$967$	−38392.0	−1.27674	−0.638368	−	0.769731i	$-0.720390\pi$
$967$	−38392.0	−1.27674	−0.638368	+	0.769731i	$0.720390\pi$
$968$	−10136.0	−0.336553
$969$	−576.000	−0.0190958
$970$	−23232.0	−0.769005
$971$	−4224.00	−0.139603	−0.0698016	−	0.997561i	$-0.522237\pi$
$971$	−4224.00	−0.139603	−0.0698016	+	0.997561i	$0.522237\pi$
$972$	972.000	0.0320750
$973$	−70176.0	−2.31217
$974$	−5848.00	−0.192384
$975$	−30654.0	−1.00689
$976$	8416.00	0.276014
$977$	36396.0	1.19182	0.595912	−	0.803050i	$-0.296791\pi$
$977$	36396.0	1.19182	0.595912	+	0.803050i	$0.296791\pi$
$978$	−17256.0	−0.564198
$979$	9088.00	0.296684
$980$	−14912.0	−0.486068
$981$	−630.000	−0.0205039
$982$	−24576.0	−0.798627
$983$	5244.00	0.170150	0.0850751	−	0.996375i	$-0.472887\pi$
$983$	5244.00	0.170150	0.0850751	+	0.996375i	$0.472887\pi$
$984$	−720.000	−0.0233260
$985$	−68448.0	−2.21415
$986$	−1728.00	−0.0558121
$987$	−4032.00	−0.130030
$988$	4992.00	0.160746
$989$	−48312.0	−1.55332
$990$	−2304.00	−0.0739656
$991$	−51296.0	−1.64427	−0.822135	−	0.569293i	$-0.807217\pi$
$991$	−51296.0	−1.64427	−0.822135	+	0.569293i	$0.807217\pi$
$992$	8960.00	0.286774
$993$	−5496.00	−0.175640
$994$	26496.0	0.845475
$995$	512.000	0.0163131
$996$	4416.00	0.140488
$997$	−20946.0	−0.665363	−0.332681	−	0.943039i	$-0.607953\pi$
$997$	−20946.0	−0.665363	−0.332681	+	0.943039i	$0.607953\pi$
$998$	−17736.0	−0.562548
$999$	999.000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	222.4.a.c.1.1	✓	1
3.2	odd	2		666.4.a.d.1.1		1
4.3	odd	2		1776.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.4.a.c.1.1	✓	1	1.1	even	1	trivial
666.4.a.d.1.1		1	3.2	odd	2
1776.4.a.a.1.1		1	4.3	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:	\( N \)	\(=\)	\( 222 = 2 \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	222.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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