LMFDB - Embedded newform 222.4.a.b.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:	$0$
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	$\chi$	$=$	222.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -16.0000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +48.0000 q^{11} -12.0000 q^{12} +50.0000 q^{13} -32.0000 q^{14} +16.0000 q^{16} +60.0000 q^{17} +18.0000 q^{18} +20.0000 q^{19} +48.0000 q^{21} +96.0000 q^{22} +162.000 q^{23} -24.0000 q^{24} -125.000 q^{25} +100.000 q^{26} -27.0000 q^{27} -64.0000 q^{28} -264.000 q^{29} +332.000 q^{31} +32.0000 q^{32} -144.000 q^{33} +120.000 q^{34} +36.0000 q^{36} +37.0000 q^{37} +40.0000 q^{38} -150.000 q^{39} +330.000 q^{41} +96.0000 q^{42} +368.000 q^{43} +192.000 q^{44} +324.000 q^{46} -504.000 q^{47} -48.0000 q^{48} -87.0000 q^{49} -250.000 q^{50} -180.000 q^{51} +200.000 q^{52} +354.000 q^{53} -54.0000 q^{54} -128.000 q^{56} -60.0000 q^{57} -528.000 q^{58} +222.000 q^{59} -322.000 q^{61} +664.000 q^{62} -144.000 q^{63} +64.0000 q^{64} -288.000 q^{66} -532.000 q^{67} +240.000 q^{68} -486.000 q^{69} -888.000 q^{71} +72.0000 q^{72} -922.000 q^{73} +74.0000 q^{74} +375.000 q^{75} +80.0000 q^{76} -768.000 q^{77} -300.000 q^{78} +1328.00 q^{79} +81.0000 q^{81} +660.000 q^{82} -696.000 q^{83} +192.000 q^{84} +736.000 q^{86} +792.000 q^{87} +384.000 q^{88} -1488.00 q^{89} -800.000 q^{91} +648.000 q^{92} -996.000 q^{93} -1008.00 q^{94} -96.0000 q^{96} +806.000 q^{97} -174.000 q^{98} +432.000 q^{99} +O(q^{100})\)

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−6.00000	−0.408248
$7$	−16.0000	−0.863919	−0.431959	−	0.901893i	$-0.642178\pi$
$7$	−16.0000	−0.863919	−0.431959	+	0.901893i	$0.642178\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	0	0
$11$	48.0000	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	−12.0000	−0.288675
$13$	50.0000	1.06673	0.533366	−	0.845885i	$-0.320927\pi$
$13$	50.0000	1.06673	0.533366	+	0.845885i	$0.320927\pi$
$14$	−32.0000	−0.610883
$15$	0	0
$16$	16.0000	0.250000
$17$	60.0000	0.856008	0.428004	−	0.903777i	$-0.359217\pi$
$17$	60.0000	0.856008	0.428004	+	0.903777i	$0.359217\pi$
$18$	18.0000	0.235702
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	48.0000	0.498784
$22$	96.0000	0.930330
$23$	162.000	1.46867	0.734333	−	0.678789i	$-0.237495\pi$
$23$	162.000	1.46867	0.734333	+	0.678789i	$0.237495\pi$
$24$	−24.0000	−0.204124
$25$	−125.000	−1.00000
$26$	100.000	0.754293
$27$	−27.0000	−0.192450
$28$	−64.0000	−0.431959
$29$	−264.000	−1.69047	−0.845234	−	0.534396i	$-0.820539\pi$
$29$	−264.000	−1.69047	−0.845234	+	0.534396i	$0.820539\pi$
$30$	0	0
$31$	332.000	1.92351	0.961757	−	0.273903i	$-0.0883146\pi$
$31$	332.000	1.92351	0.961757	+	0.273903i	$0.0883146\pi$
$32$	32.0000	0.176777
$33$	−144.000	−0.759612
$34$	120.000	0.605289
$35$	0	0
$36$	36.0000	0.166667
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	40.0000	0.170759
$39$	−150.000	−0.615878
$40$	0	0
$41$	330.000	1.25701	0.628504	−	0.777806i	$-0.283668\pi$
$41$	330.000	1.25701	0.628504	+	0.777806i	$0.283668\pi$
$42$	96.0000	0.352693
$43$	368.000	1.30510	0.652552	−	0.757744i	$-0.273698\pi$
$43$	368.000	1.30510	0.652552	+	0.757744i	$0.273698\pi$
$44$	192.000	0.657843
$45$	0	0
$46$	324.000	1.03850
$47$	−504.000	−1.56417	−0.782085	−	0.623172i	$-0.785844\pi$
$47$	−504.000	−1.56417	−0.782085	+	0.623172i	$0.785844\pi$
$48$	−48.0000	−0.144338
$49$	−87.0000	−0.253644
$50$	−250.000	−0.707107
$51$	−180.000	−0.494217
$52$	200.000	0.533366
$53$	354.000	0.917465	0.458732	−	0.888574i	$-0.348304\pi$
$53$	354.000	0.917465	0.458732	+	0.888574i	$0.348304\pi$
$54$	−54.0000	−0.136083
$55$	0	0
$56$	−128.000	−0.305441
$57$	−60.0000	−0.139424
$58$	−528.000	−1.19534
$59$	222.000	0.489863	0.244932	−	0.969540i	$-0.421234\pi$
$59$	222.000	0.489863	0.244932	+	0.969540i	$0.421234\pi$
$60$	0	0
$61$	−322.000	−0.675867	−0.337933	−	0.941170i	$-0.609728\pi$
$61$	−322.000	−0.675867	−0.337933	+	0.941170i	$0.609728\pi$
$62$	664.000	1.36013
$63$	−144.000	−0.287973
$64$	64.0000	0.125000
$65$	0	0
$66$	−288.000	−0.537127
$67$	−532.000	−0.970062	−0.485031	−	0.874497i	$-0.661192\pi$
$67$	−532.000	−0.970062	−0.485031	+	0.874497i	$0.661192\pi$
$68$	240.000	0.428004
$69$	−486.000	−0.847935
$70$	0	0
$71$	−888.000	−1.48431	−0.742156	−	0.670227i	$-0.766197\pi$
$71$	−888.000	−1.48431	−0.742156	+	0.670227i	$0.766197\pi$
$72$	72.0000	0.117851
$73$	−922.000	−1.47825	−0.739123	−	0.673571i	$-0.764760\pi$
$73$	−922.000	−1.47825	−0.739123	+	0.673571i	$0.764760\pi$
$74$	74.0000	0.116248
$75$	375.000	0.577350
$76$	80.0000	0.120745
$77$	−768.000	−1.13665
$78$	−300.000	−0.435491
$79$	1328.00	1.89129	0.945644	−	0.325205i	$-0.105433\pi$
$79$	1328.00	1.89129	0.945644	+	0.325205i	$0.105433\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	660.000	0.888839
$83$	−696.000	−0.920433	−0.460216	−	0.887807i	$-0.652228\pi$
$83$	−696.000	−0.920433	−0.460216	+	0.887807i	$0.652228\pi$
$84$	192.000	0.249392
$85$	0	0
$86$	736.000	0.922848
$87$	792.000	0.975992
$88$	384.000	0.465165
$89$	−1488.00	−1.77222	−0.886111	−	0.463474i	$-0.846603\pi$
$89$	−1488.00	−1.77222	−0.886111	+	0.463474i	$0.846603\pi$
$90$	0	0
$91$	−800.000	−0.921569
$92$	648.000	0.734333
$93$	−996.000	−1.11054
$94$	−1008.00	−1.10603
$95$	0	0
$96$	−96.0000	−0.102062
$97$	806.000	0.843679	0.421840	−	0.906670i	$-0.361385\pi$
$97$	806.000	0.843679	0.421840	+	0.906670i	$0.361385\pi$
$98$	−174.000	−0.179354
$99$	432.000	0.438562
$100$	−500.000	−0.500000
$101$	−162.000	−0.159600	−0.0798000	−	0.996811i	$-0.525428\pi$
$101$	−162.000	−0.159600	−0.0798000	+	0.996811i	$0.525428\pi$
$102$	−360.000	−0.349464
$103$	1496.00	1.43112	0.715560	−	0.698552i	$-0.246172\pi$
$103$	1496.00	1.43112	0.715560	+	0.698552i	$0.246172\pi$
$104$	400.000	0.377146
$105$	0	0
$106$	708.000	0.648746
$107$	348.000	0.314415	0.157208	−	0.987566i	$-0.449751\pi$
$107$	348.000	0.314415	0.157208	+	0.987566i	$0.449751\pi$
$108$	−108.000	−0.0962250
$109$	−22.0000	−0.0193323	−0.00966614	−	0.999953i	$-0.503077\pi$
$109$	−22.0000	−0.0193323	−0.00966614	+	0.999953i	$0.503077\pi$
$110$	0	0
$111$	−111.000	−0.0949158
$112$	−256.000	−0.215980
$113$	480.000	0.399598	0.199799	−	0.979837i	$-0.435971\pi$
$113$	480.000	0.399598	0.199799	+	0.979837i	$0.435971\pi$
$114$	−120.000	−0.0985880
$115$	0	0
$116$	−1056.00	−0.845234
$117$	450.000	0.355577
$118$	444.000	0.346386
$119$	−960.000	−0.739521
$120$	0	0
$121$	973.000	0.731029
$122$	−644.000	−0.477910
$123$	−990.000	−0.725734
$124$	1328.00	0.961757
$125$	0	0
$126$	−288.000	−0.203628
$127$	2552.00	1.78310	0.891549	−	0.452925i	$-0.149619\pi$
$127$	2552.00	1.78310	0.891549	+	0.452925i	$0.149619\pi$
$128$	128.000	0.0883883
$129$	−1104.00	−0.753502
$130$	0	0
$131$	−2982.00	−1.98884	−0.994422	−	0.105475i	$-0.966364\pi$
$131$	−2982.00	−1.98884	−0.994422	+	0.105475i	$0.966364\pi$
$132$	−576.000	−0.379806
$133$	−320.000	−0.208628
$134$	−1064.00	−0.685937
$135$	0	0
$136$	480.000	0.302645
$137$	−738.000	−0.460231	−0.230115	−	0.973163i	$-0.573910\pi$
$137$	−738.000	−0.460231	−0.230115	+	0.973163i	$0.573910\pi$
$138$	−972.000	−0.599581
$139$	−2500.00	−1.52552	−0.762760	−	0.646682i	$-0.776156\pi$
$139$	−2500.00	−1.52552	−0.762760	+	0.646682i	$0.776156\pi$
$140$	0	0
$141$	1512.00	0.903074
$142$	−1776.00	−1.04957
$143$	2400.00	1.40348
$144$	144.000	0.0833333
$145$	0	0
$146$	−1844.00	−1.04528
$147$	261.000	0.146442
$148$	148.000	0.0821995
$149$	462.000	0.254017	0.127008	−	0.991902i	$-0.459462\pi$
$149$	462.000	0.254017	0.127008	+	0.991902i	$0.459462\pi$
$150$	750.000	0.408248
$151$	1520.00	0.819178	0.409589	−	0.912270i	$-0.365672\pi$
$151$	1520.00	0.819178	0.409589	+	0.912270i	$0.365672\pi$
$152$	160.000	0.0853797
$153$	540.000	0.285336
$154$	−1536.00	−0.803730
$155$	0	0
$156$	−600.000	−0.307939
$157$	482.000	0.245018	0.122509	−	0.992467i	$-0.460906\pi$
$157$	482.000	0.245018	0.122509	+	0.992467i	$0.460906\pi$
$158$	2656.00	1.33734
$159$	−1062.00	−0.529699
$160$	0	0
$161$	−2592.00	−1.26881
$162$	162.000	0.0785674
$163$	2216.00	1.06485	0.532425	−	0.846477i	$-0.321281\pi$
$163$	2216.00	1.06485	0.532425	+	0.846477i	$0.321281\pi$
$164$	1320.00	0.628504
$165$	0	0
$166$	−1392.00	−0.650844
$167$	−2154.00	−0.998093	−0.499046	−	0.866575i	$-0.666316\pi$
$167$	−2154.00	−0.998093	−0.499046	+	0.866575i	$0.666316\pi$
$168$	384.000	0.176347
$169$	303.000	0.137915
$170$	0	0
$171$	180.000	0.0804967
$172$	1472.00	0.652552
$173$	−1566.00	−0.688213	−0.344106	−	0.938931i	$-0.611818\pi$
$173$	−1566.00	−0.688213	−0.344106	+	0.938931i	$0.611818\pi$
$174$	1584.00	0.690131
$175$	2000.00	0.863919
$176$	768.000	0.328921
$177$	−666.000	−0.282823
$178$	−2976.00	−1.25315
$179$	3030.00	1.26521	0.632606	−	0.774474i	$-0.281985\pi$
$179$	3030.00	1.26521	0.632606	+	0.774474i	$0.281985\pi$
$180$	0	0
$181$	−142.000	−0.0583137	−0.0291568	−	0.999575i	$-0.509282\pi$
$181$	−142.000	−0.0583137	−0.0291568	+	0.999575i	$0.509282\pi$
$182$	−1600.00	−0.651648
$183$	966.000	0.390212
$184$	1296.00	0.519252
$185$	0	0
$186$	−1992.00	−0.785272
$187$	2880.00	1.12624
$188$	−2016.00	−0.782085
$189$	432.000	0.166261
$190$	0	0
$191$	−3090.00	−1.17060	−0.585300	−	0.810817i	$-0.699023\pi$
$191$	−3090.00	−1.17060	−0.585300	+	0.810817i	$0.699023\pi$
$192$	−192.000	−0.0721688
$193$	1706.00	0.636272	0.318136	−	0.948045i	$-0.396943\pi$
$193$	1706.00	0.636272	0.318136	+	0.948045i	$0.396943\pi$
$194$	1612.00	0.596571
$195$	0	0
$196$	−348.000	−0.126822
$197$	−66.0000	−0.0238696	−0.0119348	−	0.999929i	$-0.503799\pi$
$197$	−66.0000	−0.0238696	−0.0119348	+	0.999929i	$0.503799\pi$
$198$	864.000	0.310110
$199$	1700.00	0.605577	0.302788	−	0.953058i	$-0.402082\pi$
$199$	1700.00	0.605577	0.302788	+	0.953058i	$0.402082\pi$
$200$	−1000.00	−0.353553
$201$	1596.00	0.560065
$202$	−324.000	−0.112854
$203$	4224.00	1.46043
$204$	−720.000	−0.247108
$205$	0	0
$206$	2992.00	1.01195
$207$	1458.00	0.489556
$208$	800.000	0.266683
$209$	960.000	0.317725
$210$	0	0
$211$	2012.00	0.656454	0.328227	−	0.944599i	$-0.393549\pi$
$211$	2012.00	0.656454	0.328227	+	0.944599i	$0.393549\pi$
$212$	1416.00	0.458732
$213$	2664.00	0.856968
$214$	696.000	0.222325
$215$	0	0
$216$	−216.000	−0.0680414
$217$	−5312.00	−1.66176
$218$	−44.0000	−0.0136700
$219$	2766.00	0.853466
$220$	0	0
$221$	3000.00	0.913130
$222$	−222.000	−0.0671156
$223$	−4720.00	−1.41737	−0.708687	−	0.705523i	$-0.750712\pi$
$223$	−4720.00	−1.41737	−0.708687	+	0.705523i	$0.750712\pi$
$224$	−512.000	−0.152721
$225$	−1125.00	−0.333333
$226$	960.000	0.282559
$227$	5490.00	1.60522	0.802608	−	0.596507i	$-0.203445\pi$
$227$	5490.00	1.60522	0.802608	+	0.596507i	$0.203445\pi$
$228$	−240.000	−0.0697122
$229$	−370.000	−0.106770	−0.0533849	−	0.998574i	$-0.517001\pi$
$229$	−370.000	−0.106770	−0.0533849	+	0.998574i	$0.517001\pi$
$230$	0	0
$231$	2304.00	0.656243
$232$	−2112.00	−0.597671
$233$	1266.00	0.355959	0.177979	−	0.984034i	$-0.443044\pi$
$233$	1266.00	0.355959	0.177979	+	0.984034i	$0.443044\pi$
$234$	900.000	0.251431
$235$	0	0
$236$	888.000	0.244932
$237$	−3984.00	−1.09194
$238$	−1920.00	−0.522921
$239$	4242.00	1.14808	0.574042	−	0.818826i	$-0.305375\pi$
$239$	4242.00	1.14808	0.574042	+	0.818826i	$0.305375\pi$
$240$	0	0
$241$	−1402.00	−0.374733	−0.187367	−	0.982290i	$-0.559995\pi$
$241$	−1402.00	−0.374733	−0.187367	+	0.982290i	$0.559995\pi$
$242$	1946.00	0.516916
$243$	−243.000	−0.0641500
$244$	−1288.00	−0.337933
$245$	0	0
$246$	−1980.00	−0.513172
$247$	1000.00	0.257605
$248$	2656.00	0.680065
$249$	2088.00	0.531412
$250$	0	0
$251$	2142.00	0.538653	0.269326	−	0.963049i	$-0.413199\pi$
$251$	2142.00	0.538653	0.269326	+	0.963049i	$0.413199\pi$
$252$	−576.000	−0.143986
$253$	7776.00	1.93230
$254$	5104.00	1.26084
$255$	0	0
$256$	256.000	0.0625000
$257$	−2352.00	−0.570871	−0.285435	−	0.958398i	$-0.592138\pi$
$257$	−2352.00	−0.570871	−0.285435	+	0.958398i	$0.592138\pi$
$258$	−2208.00	−0.532806
$259$	−592.000	−0.142027
$260$	0	0
$261$	−2376.00	−0.563489
$262$	−5964.00	−1.40633
$263$	1140.00	0.267283	0.133641	−	0.991030i	$-0.457333\pi$
$263$	1140.00	0.267283	0.133641	+	0.991030i	$0.457333\pi$
$264$	−1152.00	−0.268563
$265$	0	0
$266$	−640.000	−0.147522
$267$	4464.00	1.02319
$268$	−2128.00	−0.485031
$269$	−7026.00	−1.59250	−0.796251	−	0.604967i	$-0.793186\pi$
$269$	−7026.00	−1.59250	−0.796251	+	0.604967i	$0.793186\pi$
$270$	0	0
$271$	−8200.00	−1.83806	−0.919030	−	0.394186i	$-0.871026\pi$
$271$	−8200.00	−1.83806	−0.919030	+	0.394186i	$0.871026\pi$
$272$	960.000	0.214002
$273$	2400.00	0.532068
$274$	−1476.00	−0.325432
$275$	−6000.00	−1.31569
$276$	−1944.00	−0.423968
$277$	−1942.00	−0.421240	−0.210620	−	0.977568i	$-0.567548\pi$
$277$	−1942.00	−0.421240	−0.210620	+	0.977568i	$0.567548\pi$
$278$	−5000.00	−1.07871
$279$	2988.00	0.641172
$280$	0	0
$281$	−7416.00	−1.57438	−0.787191	−	0.616709i	$-0.788466\pi$
$281$	−7416.00	−1.57438	−0.787191	+	0.616709i	$0.788466\pi$
$282$	3024.00	0.638569
$283$	−724.000	−0.152075	−0.0760377	−	0.997105i	$-0.524227\pi$
$283$	−724.000	−0.152075	−0.0760377	+	0.997105i	$0.524227\pi$
$284$	−3552.00	−0.742156
$285$	0	0
$286$	4800.00	0.992412
$287$	−5280.00	−1.08595
$288$	288.000	0.0589256
$289$	−1313.00	−0.267250
$290$	0	0
$291$	−2418.00	−0.487099
$292$	−3688.00	−0.739123
$293$	2682.00	0.534758	0.267379	−	0.963591i	$-0.413842\pi$
$293$	2682.00	0.534758	0.267379	+	0.963591i	$0.413842\pi$
$294$	522.000	0.103550
$295$	0	0
$296$	296.000	0.0581238
$297$	−1296.00	−0.253204
$298$	924.000	0.179617
$299$	8100.00	1.56667
$300$	1500.00	0.288675
$301$	−5888.00	−1.12750
$302$	3040.00	0.579246
$303$	486.000	0.0921451
$304$	320.000	0.0603726
$305$	0	0
$306$	1080.00	0.201763
$307$	668.000	0.124185	0.0620925	−	0.998070i	$-0.480223\pi$
$307$	668.000	0.124185	0.0620925	+	0.998070i	$0.480223\pi$
$308$	−3072.00	−0.568323
$309$	−4488.00	−0.826257
$310$	0	0
$311$	−5430.00	−0.990055	−0.495027	−	0.868877i	$-0.664842\pi$
$311$	−5430.00	−0.990055	−0.495027	+	0.868877i	$0.664842\pi$
$312$	−1200.00	−0.217746
$313$	−2302.00	−0.415708	−0.207854	−	0.978160i	$-0.566648\pi$
$313$	−2302.00	−0.415708	−0.207854	+	0.978160i	$0.566648\pi$
$314$	964.000	0.173254
$315$	0	0
$316$	5312.00	0.945644
$317$	−1398.00	−0.247696	−0.123848	−	0.992301i	$-0.539523\pi$
$317$	−1398.00	−0.247696	−0.123848	+	0.992301i	$0.539523\pi$
$318$	−2124.00	−0.374553
$319$	−12672.0	−2.22412
$320$	0	0
$321$	−1044.00	−0.181528
$322$	−5184.00	−0.897183
$323$	1200.00	0.206718
$324$	324.000	0.0555556
$325$	−6250.00	−1.06673
$326$	4432.00	0.752963
$327$	66.0000	0.0111615
$328$	2640.00	0.444420
$329$	8064.00	1.35132
$330$	0	0
$331$	3644.00	0.605113	0.302556	−	0.953131i	$-0.402160\pi$
$331$	3644.00	0.605113	0.302556	+	0.953131i	$0.402160\pi$
$332$	−2784.00	−0.460216
$333$	333.000	0.0547997
$334$	−4308.00	−0.705758
$335$	0	0
$336$	768.000	0.124696
$337$	−5614.00	−0.907460	−0.453730	−	0.891139i	$-0.649907\pi$
$337$	−5614.00	−0.907460	−0.453730	+	0.891139i	$0.649907\pi$
$338$	606.000	0.0975209
$339$	−1440.00	−0.230708
$340$	0	0
$341$	15936.0	2.53074
$342$	360.000	0.0569198
$343$	6880.00	1.08305
$344$	2944.00	0.461424
$345$	0	0
$346$	−3132.00	−0.486640
$347$	−10866.0	−1.68103	−0.840515	−	0.541788i	$-0.817747\pi$
$347$	−10866.0	−1.68103	−0.840515	+	0.541788i	$0.817747\pi$
$348$	3168.00	0.487996
$349$	4970.00	0.762287	0.381143	−	0.924516i	$-0.375530\pi$
$349$	4970.00	0.762287	0.381143	+	0.924516i	$0.375530\pi$
$350$	4000.00	0.610883
$351$	−1350.00	−0.205293
$352$	1536.00	0.232583
$353$	2112.00	0.318443	0.159222	−	0.987243i	$-0.449102\pi$
$353$	2112.00	0.318443	0.159222	+	0.987243i	$0.449102\pi$
$354$	−1332.00	−0.199986
$355$	0	0
$356$	−5952.00	−0.886111
$357$	2880.00	0.426963
$358$	6060.00	0.894640
$359$	−4548.00	−0.668619	−0.334310	−	0.942463i	$-0.608503\pi$
$359$	−4548.00	−0.668619	−0.334310	+	0.942463i	$0.608503\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	−284.000	−0.0412340
$363$	−2919.00	−0.422060
$364$	−3200.00	−0.460785
$365$	0	0
$366$	1932.00	0.275921
$367$	−5080.00	−0.722545	−0.361272	−	0.932460i	$-0.617658\pi$
$367$	−5080.00	−0.722545	−0.361272	+	0.932460i	$0.617658\pi$
$368$	2592.00	0.367167
$369$	2970.00	0.419003
$370$	0	0
$371$	−5664.00	−0.792615
$372$	−3984.00	−0.555271
$373$	−1198.00	−0.166301	−0.0831503	−	0.996537i	$-0.526498\pi$
$373$	−1198.00	−0.166301	−0.0831503	+	0.996537i	$0.526498\pi$
$374$	5760.00	0.796370
$375$	0	0
$376$	−4032.00	−0.553017
$377$	−13200.0	−1.80327
$378$	864.000	0.117564
$379$	−11284.0	−1.52934	−0.764670	−	0.644422i	$-0.777098\pi$
$379$	−11284.0	−1.52934	−0.764670	+	0.644422i	$0.777098\pi$
$380$	0	0
$381$	−7656.00	−1.02947
$382$	−6180.00	−0.827739
$383$	13074.0	1.74426	0.872128	−	0.489277i	$-0.162739\pi$
$383$	13074.0	1.74426	0.872128	+	0.489277i	$0.162739\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	3412.00	0.449913
$387$	3312.00	0.435035
$388$	3224.00	0.421840
$389$	2124.00	0.276841	0.138420	−	0.990374i	$-0.455797\pi$
$389$	2124.00	0.276841	0.138420	+	0.990374i	$0.455797\pi$
$390$	0	0
$391$	9720.00	1.25719
$392$	−696.000	−0.0896768
$393$	8946.00	1.14826
$394$	−132.000	−0.0168783
$395$	0	0
$396$	1728.00	0.219281
$397$	−3358.00	−0.424517	−0.212258	−	0.977214i	$-0.568082\pi$
$397$	−3358.00	−0.424517	−0.212258	+	0.977214i	$0.568082\pi$
$398$	3400.00	0.428208
$399$	960.000	0.120451
$400$	−2000.00	−0.250000
$401$	2508.00	0.312328	0.156164	−	0.987731i	$-0.450087\pi$
$401$	2508.00	0.312328	0.156164	+	0.987731i	$0.450087\pi$
$402$	3192.00	0.396026
$403$	16600.0	2.05187
$404$	−648.000	−0.0798000
$405$	0	0
$406$	8448.00	1.03268
$407$	1776.00	0.216297
$408$	−1440.00	−0.174732
$409$	−11482.0	−1.38814	−0.694069	−	0.719909i	$-0.744184\pi$
$409$	−11482.0	−1.38814	−0.694069	+	0.719909i	$0.744184\pi$
$410$	0	0
$411$	2214.00	0.265714
$412$	5984.00	0.715560
$413$	−3552.00	−0.423202
$414$	2916.00	0.346168
$415$	0	0
$416$	1600.00	0.188573
$417$	7500.00	0.880759
$418$	1920.00	0.224666
$419$	−5136.00	−0.598831	−0.299415	−	0.954123i	$-0.596792\pi$
$419$	−5136.00	−0.598831	−0.299415	+	0.954123i	$0.596792\pi$
$420$	0	0
$421$	13610.0	1.57556	0.787780	−	0.615957i	$-0.211230\pi$
$421$	13610.0	1.57556	0.787780	+	0.615957i	$0.211230\pi$
$422$	4024.00	0.464183
$423$	−4536.00	−0.521390
$424$	2832.00	0.324373
$425$	−7500.00	−0.856008
$426$	5328.00	0.605968
$427$	5152.00	0.583894
$428$	1392.00	0.157208
$429$	−7200.00	−0.810301
$430$	0	0
$431$	−990.000	−0.110642	−0.0553209	−	0.998469i	$-0.517618\pi$
$431$	−990.000	−0.110642	−0.0553209	+	0.998469i	$0.517618\pi$
$432$	−432.000	−0.0481125
$433$	−13138.0	−1.45813	−0.729067	−	0.684442i	$-0.760046\pi$
$433$	−13138.0	−1.45813	−0.729067	+	0.684442i	$0.760046\pi$
$434$	−10624.0	−1.17504
$435$	0	0
$436$	−88.0000	−0.00966614
$437$	3240.00	0.354669
$438$	5532.00	0.603491
$439$	4196.00	0.456183	0.228091	−	0.973640i	$-0.426752\pi$
$439$	4196.00	0.456183	0.228091	+	0.973640i	$0.426752\pi$
$440$	0	0
$441$	−783.000	−0.0845481
$442$	6000.00	0.645681
$443$	4740.00	0.508362	0.254181	−	0.967157i	$-0.418194\pi$
$443$	4740.00	0.508362	0.254181	+	0.967157i	$0.418194\pi$
$444$	−444.000	−0.0474579
$445$	0	0
$446$	−9440.00	−1.00224
$447$	−1386.00	−0.146657
$448$	−1024.00	−0.107990
$449$	−10848.0	−1.14020	−0.570099	−	0.821576i	$-0.693095\pi$
$449$	−10848.0	−1.14020	−0.570099	+	0.821576i	$0.693095\pi$
$450$	−2250.00	−0.235702
$451$	15840.0	1.65383
$452$	1920.00	0.199799
$453$	−4560.00	−0.472953
$454$	10980.0	1.13506
$455$	0	0
$456$	−480.000	−0.0492940
$457$	2774.00	0.283944	0.141972	−	0.989871i	$-0.454656\pi$
$457$	2774.00	0.283944	0.141972	+	0.989871i	$0.454656\pi$
$458$	−740.000	−0.0754977
$459$	−1620.00	−0.164739
$460$	0	0
$461$	5172.00	0.522525	0.261263	−	0.965268i	$-0.415861\pi$
$461$	5172.00	0.522525	0.261263	+	0.965268i	$0.415861\pi$
$462$	4608.00	0.464034
$463$	−7228.00	−0.725515	−0.362758	−	0.931883i	$-0.618165\pi$
$463$	−7228.00	−0.725515	−0.362758	+	0.931883i	$0.618165\pi$
$464$	−4224.00	−0.422617
$465$	0	0
$466$	2532.00	0.251701
$467$	11406.0	1.13021	0.565104	−	0.825020i	$-0.308836\pi$
$467$	11406.0	1.13021	0.565104	+	0.825020i	$0.308836\pi$
$468$	1800.00	0.177789
$469$	8512.00	0.838055
$470$	0	0
$471$	−1446.00	−0.141461
$472$	1776.00	0.173193
$473$	17664.0	1.71711
$474$	−7968.00	−0.772115
$475$	−2500.00	−0.241490
$476$	−3840.00	−0.369761
$477$	3186.00	0.305822
$478$	8484.00	0.811818
$479$	−18906.0	−1.80342	−0.901709	−	0.432344i	$-0.857687\pi$
$479$	−18906.0	−1.80342	−0.901709	+	0.432344i	$0.857687\pi$
$480$	0	0
$481$	1850.00	0.175370
$482$	−2804.00	−0.264977
$483$	7776.00	0.732547
$484$	3892.00	0.365515
$485$	0	0
$486$	−486.000	−0.0453609
$487$	−12616.0	−1.17389	−0.586946	−	0.809626i	$-0.699670\pi$
$487$	−12616.0	−1.17389	−0.586946	+	0.809626i	$0.699670\pi$
$488$	−2576.00	−0.238955
$489$	−6648.00	−0.614791
$490$	0	0
$491$	−3144.00	−0.288975	−0.144488	−	0.989507i	$-0.546153\pi$
$491$	−3144.00	−0.288975	−0.144488	+	0.989507i	$0.546153\pi$
$492$	−3960.00	−0.362867
$493$	−15840.0	−1.44705
$494$	2000.00	0.182154
$495$	0	0
$496$	5312.00	0.480879
$497$	14208.0	1.28233
$498$	4176.00	0.375765
$499$	11792.0	1.05788	0.528940	−	0.848659i	$-0.322590\pi$
$499$	11792.0	1.05788	0.528940	+	0.848659i	$0.322590\pi$
$500$	0	0
$501$	6462.00	0.576249
$502$	4284.00	0.380885
$503$	−1638.00	−0.145198	−0.0725992	−	0.997361i	$-0.523129\pi$
$503$	−1638.00	−0.145198	−0.0725992	+	0.997361i	$0.523129\pi$
$504$	−1152.00	−0.101814
$505$	0	0
$506$	15552.0	1.36635
$507$	−909.000	−0.0796255
$508$	10208.0	0.891549
$509$	4962.00	0.432096	0.216048	−	0.976383i	$-0.430683\pi$
$509$	4962.00	0.432096	0.216048	+	0.976383i	$0.430683\pi$
$510$	0	0
$511$	14752.0	1.27708
$512$	512.000	0.0441942
$513$	−540.000	−0.0464748
$514$	−4704.00	−0.403666
$515$	0	0
$516$	−4416.00	−0.376751
$517$	−24192.0	−2.05796
$518$	−1184.00	−0.100429
$519$	4698.00	0.397340
$520$	0	0
$521$	−11190.0	−0.940965	−0.470483	−	0.882409i	$-0.655920\pi$
$521$	−11190.0	−0.940965	−0.470483	+	0.882409i	$0.655920\pi$
$522$	−4752.00	−0.398447
$523$	5192.00	0.434092	0.217046	−	0.976161i	$-0.430358\pi$
$523$	5192.00	0.434092	0.217046	+	0.976161i	$0.430358\pi$
$524$	−11928.0	−0.994422
$525$	−6000.00	−0.498784
$526$	2280.00	0.188998
$527$	19920.0	1.64654
$528$	−2304.00	−0.189903
$529$	14077.0	1.15698
$530$	0	0
$531$	1998.00	0.163288
$532$	−1280.00	−0.104314
$533$	16500.0	1.34089
$534$	8928.00	0.723506
$535$	0	0
$536$	−4256.00	−0.342969
$537$	−9090.00	−0.730470
$538$	−14052.0	−1.12607
$539$	−4176.00	−0.333716
$540$	0	0
$541$	18434.0	1.46495	0.732476	−	0.680792i	$-0.238364\pi$
$541$	18434.0	1.46495	0.732476	+	0.680792i	$0.238364\pi$
$542$	−16400.0	−1.29971
$543$	426.000	0.0336674
$544$	1920.00	0.151322
$545$	0	0
$546$	4800.00	0.376229
$547$	12320.0	0.963008	0.481504	−	0.876444i	$-0.340091\pi$
$547$	12320.0	0.963008	0.481504	+	0.876444i	$0.340091\pi$
$548$	−2952.00	−0.230115
$549$	−2898.00	−0.225289
$550$	−12000.0	−0.930330
$551$	−5280.00	−0.408232
$552$	−3888.00	−0.299790
$553$	−21248.0	−1.63392
$554$	−3884.00	−0.297862
$555$	0	0
$556$	−10000.0	−0.762760
$557$	−24804.0	−1.88686	−0.943428	−	0.331576i	$-0.892420\pi$
$557$	−24804.0	−1.88686	−0.943428	+	0.331576i	$0.892420\pi$
$558$	5976.00	0.453377
$559$	18400.0	1.39220
$560$	0	0
$561$	−8640.00	−0.650234
$562$	−14832.0	−1.11326
$563$	1542.00	0.115431	0.0577154	−	0.998333i	$-0.481618\pi$
$563$	1542.00	0.115431	0.0577154	+	0.998333i	$0.481618\pi$
$564$	6048.00	0.451537
$565$	0	0
$566$	−1448.00	−0.107534
$567$	−1296.00	−0.0959910
$568$	−7104.00	−0.524784
$569$	6408.00	0.472122	0.236061	−	0.971738i	$-0.424144\pi$
$569$	6408.00	0.472122	0.236061	+	0.971738i	$0.424144\pi$
$570$	0	0
$571$	−19492.0	−1.42857	−0.714286	−	0.699854i	$-0.753248\pi$
$571$	−19492.0	−1.42857	−0.714286	+	0.699854i	$0.753248\pi$
$572$	9600.00	0.701742
$573$	9270.00	0.675846
$574$	−10560.0	−0.767885
$575$	−20250.0	−1.46867
$576$	576.000	0.0416667
$577$	20162.0	1.45469	0.727344	−	0.686273i	$-0.240754\pi$
$577$	20162.0	1.45469	0.727344	+	0.686273i	$0.240754\pi$
$578$	−2626.00	−0.188974
$579$	−5118.00	−0.367352
$580$	0	0
$581$	11136.0	0.795179
$582$	−4836.00	−0.344431
$583$	16992.0	1.20710
$584$	−7376.00	−0.522639
$585$	0	0
$586$	5364.00	0.378131
$587$	8814.00	0.619749	0.309875	−	0.950777i	$-0.399713\pi$
$587$	8814.00	0.619749	0.309875	+	0.950777i	$0.399713\pi$
$588$	1044.00	0.0732208
$589$	6640.00	0.464510
$590$	0	0
$591$	198.000	0.0137811
$592$	592.000	0.0410997
$593$	−4974.00	−0.344448	−0.172224	−	0.985058i	$-0.555095\pi$
$593$	−4974.00	−0.344448	−0.172224	+	0.985058i	$0.555095\pi$
$594$	−2592.00	−0.179042
$595$	0	0
$596$	1848.00	0.127008
$597$	−5100.00	−0.349630
$598$	16200.0	1.10780
$599$	−15672.0	−1.06902	−0.534508	−	0.845163i	$-0.679503\pi$
$599$	−15672.0	−1.06902	−0.534508	+	0.845163i	$0.679503\pi$
$600$	3000.00	0.204124
$601$	−8482.00	−0.575687	−0.287844	−	0.957677i	$-0.592938\pi$
$601$	−8482.00	−0.575687	−0.287844	+	0.957677i	$0.592938\pi$
$602$	−11776.0	−0.797266
$603$	−4788.00	−0.323354
$604$	6080.00	0.409589
$605$	0	0
$606$	972.000	0.0651564
$607$	−2032.00	−0.135875	−0.0679377	−	0.997690i	$-0.521642\pi$
$607$	−2032.00	−0.135875	−0.0679377	+	0.997690i	$0.521642\pi$
$608$	640.000	0.0426898
$609$	−12672.0	−0.843178
$610$	0	0
$611$	−25200.0	−1.66855
$612$	2160.00	0.142668
$613$	24446.0	1.61071	0.805355	−	0.592793i	$-0.201975\pi$
$613$	24446.0	1.61071	0.805355	+	0.592793i	$0.201975\pi$
$614$	1336.00	0.0878120
$615$	0	0
$616$	−6144.00	−0.401865
$617$	13926.0	0.908654	0.454327	−	0.890835i	$-0.349880\pi$
$617$	13926.0	0.908654	0.454327	+	0.890835i	$0.349880\pi$
$618$	−8976.00	−0.584252
$619$	−1300.00	−0.0844126	−0.0422063	−	0.999109i	$-0.513439\pi$
$619$	−1300.00	−0.0844126	−0.0422063	+	0.999109i	$0.513439\pi$
$620$	0	0
$621$	−4374.00	−0.282645
$622$	−10860.0	−0.700074
$623$	23808.0	1.53106
$624$	−2400.00	−0.153969
$625$	15625.0	1.00000
$626$	−4604.00	−0.293950
$627$	−2880.00	−0.183439
$628$	1928.00	0.122509
$629$	2220.00	0.140727
$630$	0	0
$631$	−15424.0	−0.973090	−0.486545	−	0.873656i	$-0.661743\pi$
$631$	−15424.0	−0.973090	−0.486545	+	0.873656i	$0.661743\pi$
$632$	10624.0	0.668671
$633$	−6036.00	−0.379004
$634$	−2796.00	−0.175147
$635$	0	0
$636$	−4248.00	−0.264849
$637$	−4350.00	−0.270570
$638$	−25344.0	−1.57269
$639$	−7992.00	−0.494771
$640$	0	0
$641$	−18366.0	−1.13169	−0.565845	−	0.824512i	$-0.691450\pi$
$641$	−18366.0	−1.13169	−0.565845	+	0.824512i	$0.691450\pi$
$642$	−2088.00	−0.128359
$643$	−9556.00	−0.586084	−0.293042	−	0.956100i	$-0.594668\pi$
$643$	−9556.00	−0.586084	−0.293042	+	0.956100i	$0.594668\pi$
$644$	−10368.0	−0.634404
$645$	0	0
$646$	2400.00	0.146171
$647$	16986.0	1.03213	0.516065	−	0.856549i	$-0.327396\pi$
$647$	16986.0	1.03213	0.516065	+	0.856549i	$0.327396\pi$
$648$	648.000	0.0392837
$649$	10656.0	0.644506
$650$	−12500.0	−0.754293
$651$	15936.0	0.959418
$652$	8864.00	0.532425
$653$	7356.00	0.440831	0.220415	−	0.975406i	$-0.429259\pi$
$653$	7356.00	0.440831	0.220415	+	0.975406i	$0.429259\pi$
$654$	132.000	0.00789237
$655$	0	0
$656$	5280.00	0.314252
$657$	−8298.00	−0.492749
$658$	16128.0	0.955524
$659$	−20496.0	−1.21155	−0.605775	−	0.795636i	$-0.707137\pi$
$659$	−20496.0	−1.21155	−0.605775	+	0.795636i	$0.707137\pi$
$660$	0	0
$661$	11798.0	0.694235	0.347117	−	0.937822i	$-0.387161\pi$
$661$	11798.0	0.694235	0.347117	+	0.937822i	$0.387161\pi$
$662$	7288.00	0.427879
$663$	−9000.00	−0.527196
$664$	−5568.00	−0.325422
$665$	0	0
$666$	666.000	0.0387492
$667$	−42768.0	−2.48273
$668$	−8616.00	−0.499046
$669$	14160.0	0.818322
$670$	0	0
$671$	−15456.0	−0.889228
$672$	1536.00	0.0881733
$673$	−11050.0	−0.632907	−0.316453	−	0.948608i	$-0.602492\pi$
$673$	−11050.0	−0.632907	−0.316453	+	0.948608i	$0.602492\pi$
$674$	−11228.0	−0.641671
$675$	3375.00	0.192450
$676$	1212.00	0.0689577
$677$	3066.00	0.174056	0.0870280	−	0.996206i	$-0.472263\pi$
$677$	3066.00	0.174056	0.0870280	+	0.996206i	$0.472263\pi$
$678$	−2880.00	−0.163135
$679$	−12896.0	−0.728870
$680$	0	0
$681$	−16470.0	−0.926772
$682$	31872.0	1.78950
$683$	−21642.0	−1.21246	−0.606228	−	0.795291i	$-0.707318\pi$
$683$	−21642.0	−1.21246	−0.606228	+	0.795291i	$0.707318\pi$
$684$	720.000	0.0402484
$685$	0	0
$686$	13760.0	0.765830
$687$	1110.00	0.0616436
$688$	5888.00	0.326276
$689$	17700.0	0.978688
$690$	0	0
$691$	9332.00	0.513757	0.256878	−	0.966444i	$-0.417306\pi$
$691$	9332.00	0.513757	0.256878	+	0.966444i	$0.417306\pi$
$692$	−6264.00	−0.344106
$693$	−6912.00	−0.378882
$694$	−21732.0	−1.18867
$695$	0	0
$696$	6336.00	0.345065
$697$	19800.0	1.07601
$698$	9940.00	0.539018
$699$	−3798.00	−0.205513
$700$	8000.00	0.431959
$701$	34812.0	1.87565	0.937825	−	0.347108i	$-0.112836\pi$
$701$	34812.0	1.87565	0.937825	+	0.347108i	$0.112836\pi$
$702$	−2700.00	−0.145164
$703$	740.000	0.0397008
$704$	3072.00	0.164461
$705$	0	0
$706$	4224.00	0.225173
$707$	2592.00	0.137881
$708$	−2664.00	−0.141411
$709$	28418.0	1.50530	0.752652	−	0.658419i	$-0.228774\pi$
$709$	28418.0	1.50530	0.752652	+	0.658419i	$0.228774\pi$
$710$	0	0
$711$	11952.0	0.630429
$712$	−11904.0	−0.626575
$713$	53784.0	2.82500
$714$	5760.00	0.301908
$715$	0	0
$716$	12120.0	0.632606
$717$	−12726.0	−0.662847
$718$	−9096.00	−0.472785
$719$	−2028.00	−0.105190	−0.0525950	−	0.998616i	$-0.516749\pi$
$719$	−2028.00	−0.105190	−0.0525950	+	0.998616i	$0.516749\pi$
$720$	0	0
$721$	−23936.0	−1.23637
$722$	−12918.0	−0.665870
$723$	4206.00	0.216352
$724$	−568.000	−0.0291568
$725$	33000.0	1.69047
$726$	−5838.00	−0.298441
$727$	−1132.00	−0.0577490	−0.0288745	−	0.999583i	$-0.509192\pi$
$727$	−1132.00	−0.0577490	−0.0288745	+	0.999583i	$0.509192\pi$
$728$	−6400.00	−0.325824
$729$	729.000	0.0370370
$730$	0	0
$731$	22080.0	1.11718
$732$	3864.00	0.195106
$733$	−17146.0	−0.863986	−0.431993	−	0.901877i	$-0.642190\pi$
$733$	−17146.0	−0.863986	−0.431993	+	0.901877i	$0.642190\pi$
$734$	−10160.0	−0.510916
$735$	0	0
$736$	5184.00	0.259626
$737$	−25536.0	−1.27630
$738$	5940.00	0.296280
$739$	1964.00	0.0977631	0.0488815	−	0.998805i	$-0.484434\pi$
$739$	1964.00	0.0977631	0.0488815	+	0.998805i	$0.484434\pi$
$740$	0	0
$741$	−3000.00	−0.148728
$742$	−11328.0	−0.560464
$743$	1332.00	0.0657690	0.0328845	−	0.999459i	$-0.489531\pi$
$743$	1332.00	0.0657690	0.0328845	+	0.999459i	$0.489531\pi$
$744$	−7968.00	−0.392636
$745$	0	0
$746$	−2396.00	−0.117592
$747$	−6264.00	−0.306811
$748$	11520.0	0.563119
$749$	−5568.00	−0.271629
$750$	0	0
$751$	26408.0	1.28314	0.641572	−	0.767063i	$-0.278282\pi$
$751$	26408.0	1.28314	0.641572	+	0.767063i	$0.278282\pi$
$752$	−8064.00	−0.391042
$753$	−6426.00	−0.310991
$754$	−26400.0	−1.27511
$755$	0	0
$756$	1728.00	0.0831306
$757$	12002.0	0.576248	0.288124	−	0.957593i	$-0.406968\pi$
$757$	12002.0	0.576248	0.288124	+	0.957593i	$0.406968\pi$
$758$	−22568.0	−1.08141
$759$	−23328.0	−1.11562
$760$	0	0
$761$	−4674.00	−0.222644	−0.111322	−	0.993784i	$-0.535509\pi$
$761$	−4674.00	−0.222644	−0.111322	+	0.993784i	$0.535509\pi$
$762$	−15312.0	−0.727947
$763$	352.000	0.0167015
$764$	−12360.0	−0.585300
$765$	0	0
$766$	26148.0	1.23338
$767$	11100.0	0.522553
$768$	−768.000	−0.0360844
$769$	−13510.0	−0.633528	−0.316764	−	0.948504i	$-0.602596\pi$
$769$	−13510.0	−0.633528	−0.316764	+	0.948504i	$0.602596\pi$
$770$	0	0
$771$	7056.00	0.329592
$772$	6824.00	0.318136
$773$	23598.0	1.09801	0.549005	−	0.835819i	$-0.315007\pi$
$773$	23598.0	1.09801	0.549005	+	0.835819i	$0.315007\pi$
$774$	6624.00	0.307616
$775$	−41500.0	−1.92351
$776$	6448.00	0.298286
$777$	1776.00	0.0819995
$778$	4248.00	0.195756
$779$	6600.00	0.303555
$780$	0	0
$781$	−42624.0	−1.95289
$782$	19440.0	0.888968
$783$	7128.00	0.325331
$784$	−1392.00	−0.0634111
$785$	0	0
$786$	17892.0	0.811942
$787$	12644.0	0.572694	0.286347	−	0.958126i	$-0.407559\pi$
$787$	12644.0	0.572694	0.286347	+	0.958126i	$0.407559\pi$
$788$	−264.000	−0.0119348
$789$	−3420.00	−0.154316
$790$	0	0
$791$	−7680.00	−0.345220
$792$	3456.00	0.155055
$793$	−16100.0	−0.720968
$794$	−6716.00	−0.300179
$795$	0	0
$796$	6800.00	0.302788
$797$	3576.00	0.158932	0.0794658	−	0.996838i	$-0.474679\pi$
$797$	3576.00	0.158932	0.0794658	+	0.996838i	$0.474679\pi$
$798$	1920.00	0.0851720
$799$	−30240.0	−1.33894
$800$	−4000.00	−0.176777
$801$	−13392.0	−0.590740
$802$	5016.00	0.220849
$803$	−44256.0	−1.94491
$804$	6384.00	0.280033
$805$	0	0
$806$	33200.0	1.45089
$807$	21078.0	0.919431
$808$	−1296.00	−0.0564271
$809$	−24636.0	−1.07065	−0.535325	−	0.844646i	$-0.679811\pi$
$809$	−24636.0	−1.07065	−0.535325	+	0.844646i	$0.679811\pi$
$810$	0	0
$811$	380.000	0.0164533	0.00822664	−	0.999966i	$-0.497381\pi$
$811$	380.000	0.0164533	0.00822664	+	0.999966i	$0.497381\pi$
$812$	16896.0	0.730213
$813$	24600.0	1.06121
$814$	3552.00	0.152945
$815$	0	0
$816$	−2880.00	−0.123554
$817$	7360.00	0.315170
$818$	−22964.0	−0.981562
$819$	−7200.00	−0.307190
$820$	0	0
$821$	33762.0	1.43520	0.717602	−	0.696454i	$-0.245240\pi$
$821$	33762.0	1.43520	0.717602	+	0.696454i	$0.245240\pi$
$822$	4428.00	0.187888
$823$	27752.0	1.17542	0.587712	−	0.809070i	$-0.300029\pi$
$823$	27752.0	1.17542	0.587712	+	0.809070i	$0.300029\pi$
$824$	11968.0	0.505977
$825$	18000.0	0.759612
$826$	−7104.00	−0.299249
$827$	−16698.0	−0.702112	−0.351056	−	0.936355i	$-0.614177\pi$
$827$	−16698.0	−0.702112	−0.351056	+	0.936355i	$0.614177\pi$
$828$	5832.00	0.244778
$829$	−12502.0	−0.523779	−0.261889	−	0.965098i	$-0.584346\pi$
$829$	−12502.0	−0.523779	−0.261889	+	0.965098i	$0.584346\pi$
$830$	0	0
$831$	5826.00	0.243203
$832$	3200.00	0.133341
$833$	−5220.00	−0.217122
$834$	15000.0	0.622791
$835$	0	0
$836$	3840.00	0.158863
$837$	−8964.00	−0.370181
$838$	−10272.0	−0.423437
$839$	17724.0	0.729321	0.364661	−	0.931141i	$-0.381185\pi$
$839$	17724.0	0.729321	0.364661	+	0.931141i	$0.381185\pi$
$840$	0	0
$841$	45307.0	1.85768
$842$	27220.0	1.11409
$843$	22248.0	0.908970
$844$	8048.00	0.328227
$845$	0	0
$846$	−9072.00	−0.368678
$847$	−15568.0	−0.631550
$848$	5664.00	0.229366
$849$	2172.00	0.0878008
$850$	−15000.0	−0.605289
$851$	5994.00	0.241447
$852$	10656.0	0.428484
$853$	19910.0	0.799186	0.399593	−	0.916693i	$-0.369152\pi$
$853$	19910.0	0.799186	0.399593	+	0.916693i	$0.369152\pi$
$854$	10304.0	0.412875
$855$	0	0
$856$	2784.00	0.111163
$857$	4392.00	0.175062	0.0875308	−	0.996162i	$-0.472102\pi$
$857$	4392.00	0.175062	0.0875308	+	0.996162i	$0.472102\pi$
$858$	−14400.0	−0.572970
$859$	43904.0	1.74387	0.871935	−	0.489621i	$-0.162865\pi$
$859$	43904.0	1.74387	0.871935	+	0.489621i	$0.162865\pi$
$860$	0	0
$861$	15840.0	0.626975
$862$	−1980.00	−0.0782356
$863$	−10428.0	−0.411325	−0.205662	−	0.978623i	$-0.565935\pi$
$863$	−10428.0	−0.411325	−0.205662	+	0.978623i	$0.565935\pi$
$864$	−864.000	−0.0340207
$865$	0	0
$866$	−26276.0	−1.03106
$867$	3939.00	0.154297
$868$	−21248.0	−0.830880
$869$	63744.0	2.48834
$870$	0	0
$871$	−26600.0	−1.03480
$872$	−176.000	−0.00683499
$873$	7254.00	0.281226
$874$	6480.00	0.250789
$875$	0	0
$876$	11064.0	0.426733
$877$	−11806.0	−0.454573	−0.227286	−	0.973828i	$-0.572985\pi$
$877$	−11806.0	−0.454573	−0.227286	+	0.973828i	$0.572985\pi$
$878$	8392.00	0.322570
$879$	−8046.00	−0.308743
$880$	0	0
$881$	−10794.0	−0.412780	−0.206390	−	0.978470i	$-0.566172\pi$
$881$	−10794.0	−0.412780	−0.206390	+	0.978470i	$0.566172\pi$
$882$	−1566.00	−0.0597845
$883$	−10132.0	−0.386148	−0.193074	−	0.981184i	$-0.561846\pi$
$883$	−10132.0	−0.386148	−0.193074	+	0.981184i	$0.561846\pi$
$884$	12000.0	0.456565
$885$	0	0
$886$	9480.00	0.359466
$887$	−348.000	−0.0131733	−0.00658664	−	0.999978i	$-0.502097\pi$
$887$	−348.000	−0.0131733	−0.00658664	+	0.999978i	$0.502097\pi$
$888$	−888.000	−0.0335578
$889$	−40832.0	−1.54045
$890$	0	0
$891$	3888.00	0.146187
$892$	−18880.0	−0.708687
$893$	−10080.0	−0.377732
$894$	−2772.00	−0.103702
$895$	0	0
$896$	−2048.00	−0.0763604
$897$	−24300.0	−0.904519
$898$	−21696.0	−0.806242
$899$	−87648.0	−3.25164
$900$	−4500.00	−0.166667
$901$	21240.0	0.785357
$902$	31680.0	1.16943
$903$	17664.0	0.650965
$904$	3840.00	0.141279
$905$	0	0
$906$	−9120.00	−0.334428
$907$	11528.0	0.422030	0.211015	−	0.977483i	$-0.432323\pi$
$907$	11528.0	0.422030	0.211015	+	0.977483i	$0.432323\pi$
$908$	21960.0	0.802608
$909$	−1458.00	−0.0532000
$910$	0	0
$911$	−27270.0	−0.991762	−0.495881	−	0.868390i	$-0.665155\pi$
$911$	−27270.0	−0.991762	−0.495881	+	0.868390i	$0.665155\pi$
$912$	−960.000	−0.0348561
$913$	−33408.0	−1.21100
$914$	5548.00	0.200778
$915$	0	0
$916$	−1480.00	−0.0533849
$917$	47712.0	1.71820
$918$	−3240.00	−0.116488
$919$	6824.00	0.244943	0.122472	−	0.992472i	$-0.460918\pi$
$919$	6824.00	0.244943	0.122472	+	0.992472i	$0.460918\pi$
$920$	0	0
$921$	−2004.00	−0.0716982
$922$	10344.0	0.369481
$923$	−44400.0	−1.58336
$924$	9216.00	0.328121
$925$	−4625.00	−0.164399
$926$	−14456.0	−0.513017
$927$	13464.0	0.477040
$928$	−8448.00	−0.298835
$929$	−17430.0	−0.615565	−0.307782	−	0.951457i	$-0.599587\pi$
$929$	−17430.0	−0.615565	−0.307782	+	0.951457i	$0.599587\pi$
$930$	0	0
$931$	−1740.00	−0.0612526
$932$	5064.00	0.177979
$933$	16290.0	0.571608
$934$	22812.0	0.799177
$935$	0	0
$936$	3600.00	0.125715
$937$	−32326.0	−1.12705	−0.563524	−	0.826100i	$-0.690555\pi$
$937$	−32326.0	−1.12705	−0.563524	+	0.826100i	$0.690555\pi$
$938$	17024.0	0.592594
$939$	6906.00	0.240009
$940$	0	0
$941$	23250.0	0.805450	0.402725	−	0.915321i	$-0.368063\pi$
$941$	23250.0	0.805450	0.402725	+	0.915321i	$0.368063\pi$
$942$	−2892.00	−0.100028
$943$	53460.0	1.84613
$944$	3552.00	0.122466
$945$	0	0
$946$	35328.0	1.21418
$947$	17166.0	0.589039	0.294520	−	0.955645i	$-0.404840\pi$
$947$	17166.0	0.589039	0.294520	+	0.955645i	$0.404840\pi$
$948$	−15936.0	−0.545968
$949$	−46100.0	−1.57689
$950$	−5000.00	−0.170759
$951$	4194.00	0.143007
$952$	−7680.00	−0.261460
$953$	14238.0	0.483960	0.241980	−	0.970281i	$-0.422203\pi$
$953$	14238.0	0.483960	0.241980	+	0.970281i	$0.422203\pi$
$954$	6372.00	0.216249
$955$	0	0
$956$	16968.0	0.574042
$957$	38016.0	1.28410
$958$	−37812.0	−1.27521
$959$	11808.0	0.397602
$960$	0	0
$961$	80433.0	2.69991
$962$	3700.00	0.124005
$963$	3132.00	0.104805
$964$	−5608.00	−0.187367
$965$	0	0
$966$	15552.0	0.517989
$967$	−46156.0	−1.53493	−0.767465	−	0.641091i	$-0.778482\pi$
$967$	−46156.0	−1.53493	−0.767465	+	0.641091i	$0.778482\pi$
$968$	7784.00	0.258458
$969$	−3600.00	−0.119348
$970$	0	0
$971$	−32376.0	−1.07003	−0.535013	−	0.844844i	$-0.679693\pi$
$971$	−32376.0	−1.07003	−0.535013	+	0.844844i	$0.679693\pi$
$972$	−972.000	−0.0320750
$973$	40000.0	1.31793
$974$	−25232.0	−0.830067
$975$	18750.0	0.615878
$976$	−5152.00	−0.168967
$977$	4572.00	0.149715	0.0748573	−	0.997194i	$-0.476150\pi$
$977$	4572.00	0.149715	0.0748573	+	0.997194i	$0.476150\pi$
$978$	−13296.0	−0.434723
$979$	−71424.0	−2.33169
$980$	0	0
$981$	−198.000	−0.00644409
$982$	−6288.00	−0.204336
$983$	−38772.0	−1.25802	−0.629011	−	0.777397i	$-0.716540\pi$
$983$	−38772.0	−1.25802	−0.629011	+	0.777397i	$0.716540\pi$
$984$	−7920.00	−0.256586
$985$	0	0
$986$	−31680.0	−1.02322
$987$	−24192.0	−0.780182
$988$	4000.00	0.128803
$989$	59616.0	1.91676
$990$	0	0
$991$	21332.0	0.683787	0.341894	−	0.939739i	$-0.388932\pi$
$991$	21332.0	0.683787	0.341894	+	0.939739i	$0.388932\pi$
$992$	10624.0	0.340033
$993$	−10932.0	−0.349362
$994$	28416.0	0.906741
$995$	0	0
$996$	8352.00	0.265706
$997$	−36610.0	−1.16294	−0.581470	−	0.813568i	$-0.697522\pi$
$997$	−36610.0	−1.16294	−0.581470	+	0.813568i	$0.697522\pi$
$998$	23584.0	0.748035
$999$	−999.000	−0.0316386

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
222.4.a.b.1.1	✓	1	1.1	even	1	trivial
666.4.a.b.1.1		1	3.2	odd	2
1776.4.a.g.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

Downloads

Learn more