LMFDB - Embedded newform 33.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,8,Mod(1,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("33.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	33.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:	$10.3087058410$
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	$\chi$	$=$	33.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+10.0000 q^{2} +27.0000 q^{3} -28.0000 q^{4} -410.000 q^{5} +270.000 q^{6} -1028.00 q^{7} -1560.00 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+10.0000 q^{2} +27.0000 q^{3} -28.0000 q^{4} -410.000 q^{5} +270.000 q^{6} -1028.00 q^{7} -1560.00 q^{8} +729.000 q^{9} -4100.00 q^{10} -1331.00 q^{11} -756.000 q^{12} +12958.0 q^{13} -10280.0 q^{14} -11070.0 q^{15} -12016.0 q^{16} +17062.0 q^{17} +7290.00 q^{18} -54168.0 q^{19} +11480.0 q^{20} -27756.0 q^{21} -13310.0 q^{22} -11488.0 q^{23} -42120.0 q^{24} +89975.0 q^{25} +129580. q^{26} +19683.0 q^{27} +28784.0 q^{28} -186654. q^{29} -110700. q^{30} -188672. q^{31} +79520.0 q^{32} -35937.0 q^{33} +170620. q^{34} +421480. q^{35} -20412.0 q^{36} +395886. q^{37} -541680. q^{38} +349866. q^{39} +639600. q^{40} -47546.0 q^{41} -277560. q^{42} +602088. q^{43} +37268.0 q^{44} -298890. q^{45} -114880. q^{46} -647200. q^{47} -324432. q^{48} +233241. q^{49} +899750. q^{50} +460674. q^{51} -362824. q^{52} -1.31272e6 q^{53} +196830. q^{54} +545710. q^{55} +1.60368e6 q^{56} -1.46254e6 q^{57} -1.86654e6 q^{58} -2.68114e6 q^{59} +309960. q^{60} +551190. q^{61} -1.88672e6 q^{62} -749412. q^{63} +2.33325e6 q^{64} -5.31278e6 q^{65} -359370. q^{66} +459260. q^{67} -477736. q^{68} -310176. q^{69} +4.21480e6 q^{70} -18072.0 q^{71} -1.13724e6 q^{72} -426062. q^{73} +3.95886e6 q^{74} +2.42932e6 q^{75} +1.51670e6 q^{76} +1.36827e6 q^{77} +3.49866e6 q^{78} +297764. q^{79} +4.92656e6 q^{80} +531441. q^{81} -475460. q^{82} +5.68403e6 q^{83} +777168. q^{84} -6.99542e6 q^{85} +6.02088e6 q^{86} -5.03966e6 q^{87} +2.07636e6 q^{88} -6.34297e6 q^{89} -2.98890e6 q^{90} -1.33208e7 q^{91} +321664. q^{92} -5.09414e6 q^{93} -6.47200e6 q^{94} +2.22089e7 q^{95} +2.14704e6 q^{96} +1.66516e7 q^{97} +2.33241e6 q^{98} -970299. q^{99} +O(q^{100})\)

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 $

$2$	10.0000	0.883883	0.441942	−	0.897044i	$-0.354290\pi$
$2$	10.0000	0.883883	0.441942	+	0.897044i	$0.354290\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−28.0000	−0.218750
$5$	−410.000	−1.46686	−0.733430	−	0.679765i	$-0.762082\pi$
$5$	−410.000	−1.46686	−0.733430	+	0.679765i	$0.762082\pi$
$6$	270.000	0.510310
$7$	−1028.00	−1.13279	−0.566396	−	0.824133i	$-0.691663\pi$
$7$	−1028.00	−1.13279	−0.566396	+	0.824133i	$0.691663\pi$
$8$	−1560.00	−1.07723
$9$	729.000	0.333333
$10$	−4100.00	−1.29653
$11$	−1331.00	−0.301511
$11$	−1331.00	−0.301511
$12$	−756.000	−0.126295
$13$	12958.0	1.63582	0.817911	−	0.575344i	$-0.195132\pi$
$13$	12958.0	1.63582	0.817911	+	0.575344i	$0.195132\pi$
$14$	−10280.0	−1.00126
$15$	−11070.0	−0.846892
$16$	−12016.0	−0.733398
$17$	17062.0	0.842284	0.421142	−	0.906995i	$-0.361629\pi$
$17$	17062.0	0.842284	0.421142	+	0.906995i	$0.361629\pi$
$18$	7290.00	0.294628
$19$	−54168.0	−1.81178	−0.905889	−	0.423514i	$-0.860796\pi$
$19$	−54168.0	−1.81178	−0.905889	+	0.423514i	$0.860796\pi$
$20$	11480.0	0.320876
$21$	−27756.0	−0.654017
$22$	−13310.0	−0.266501
$23$	−11488.0	−0.196878	−0.0984390	−	0.995143i	$-0.531385\pi$
$23$	−11488.0	−0.196878	−0.0984390	+	0.995143i	$0.531385\pi$
$24$	−42120.0	−0.621941
$25$	89975.0	1.15168
$26$	129580.	1.44588
$27$	19683.0	0.192450
$28$	28784.0	0.247798
$29$	−186654.	−1.42116	−0.710582	−	0.703614i	$-0.751568\pi$
$29$	−186654.	−1.42116	−0.710582	+	0.703614i	$0.751568\pi$
$30$	−110700.	−0.748554
$31$	−188672.	−1.13747	−0.568737	−	0.822519i	$-0.692568\pi$
$31$	−188672.	−1.13747	−0.568737	+	0.822519i	$0.692568\pi$
$32$	79520.0	0.428994
$33$	−35937.0	−0.174078
$34$	170620.	0.744481
$35$	421480.	1.66165
$36$	−20412.0	−0.0729167
$37$	395886.	1.28488	0.642442	−	0.766334i	$-0.277921\pi$
$37$	395886.	1.28488	0.642442	+	0.766334i	$0.277921\pi$
$38$	−541680.	−1.60140
$39$	349866.	0.944443
$40$	639600.	1.58015
$41$	−47546.0	−0.107738	−0.0538692	−	0.998548i	$-0.517155\pi$
$41$	−47546.0	−0.107738	−0.0538692	+	0.998548i	$0.517155\pi$
$42$	−277560.	−0.578075
$43$	602088.	1.15484	0.577418	−	0.816449i	$-0.304060\pi$
$43$	602088.	1.15484	0.577418	+	0.816449i	$0.304060\pi$
$44$	37268.0	0.0659556
$45$	−298890.	−0.488954
$46$	−114880.	−0.174017
$47$	−647200.	−0.909277	−0.454638	−	0.890676i	$-0.650231\pi$
$47$	−647200.	−0.909277	−0.454638	+	0.890676i	$0.650231\pi$
$48$	−324432.	−0.423428
$49$	233241.	0.283217
$50$	899750.	1.01795
$51$	460674.	0.486293
$52$	−362824.	−0.357836
$53$	−1.31272e6	−1.21118	−0.605588	−	0.795778i	$-0.707062\pi$
$53$	−1.31272e6	−1.21118	−0.605588	+	0.795778i	$0.707062\pi$
$54$	196830.	0.170103
$55$	545710.	0.442275
$56$	1.60368e6	1.22028
$57$	−1.46254e6	−1.04603
$58$	−1.86654e6	−1.25614
$59$	−2.68114e6	−1.69956	−0.849782	−	0.527135i	$-0.823266\pi$
$59$	−2.68114e6	−1.69956	−0.849782	+	0.527135i	$0.823266\pi$
$60$	309960.	0.185258
$61$	551190.	0.310919	0.155459	−	0.987842i	$-0.450314\pi$
$61$	551190.	0.310919	0.155459	+	0.987842i	$0.450314\pi$
$62$	−1.88672e6	−1.00539
$63$	−749412.	−0.377597
$64$	2.33325e6	1.11258
$65$	−5.31278e6	−2.39952
$66$	−359370.	−0.153864
$67$	459260.	0.186551	0.0932753	−	0.995640i	$-0.470266\pi$
$67$	459260.	0.186551	0.0932753	+	0.995640i	$0.470266\pi$
$68$	−477736.	−0.184250
$69$	−310176.	−0.113668
$70$	4.21480e6	1.46870
$71$	−18072.0	−0.00599242	−0.00299621	−	0.999996i	$-0.500954\pi$
$71$	−18072.0	−0.00599242	−0.00299621	+	0.999996i	$0.500954\pi$
$72$	−1.13724e6	−0.359078
$73$	−426062.	−0.128187	−0.0640933	−	0.997944i	$-0.520416\pi$
$73$	−426062.	−0.128187	−0.0640933	+	0.997944i	$0.520416\pi$
$74$	3.95886e6	1.13569
$75$	2.42932e6	0.664923
$76$	1.51670e6	0.396327
$77$	1.36827e6	0.341549
$78$	3.49866e6	0.834777
$79$	297764.	0.0679481	0.0339741	−	0.999423i	$-0.489184\pi$
$79$	297764.	0.0679481	0.0339741	+	0.999423i	$0.489184\pi$
$80$	4.92656e6	1.07579
$81$	531441.	0.111111
$82$	−475460.	−0.0952282
$83$	5.68403e6	1.09115	0.545573	−	0.838063i	$-0.316312\pi$
$83$	5.68403e6	1.09115	0.545573	+	0.838063i	$0.316312\pi$
$84$	777168.	0.143066
$85$	−6.99542e6	−1.23551
$86$	6.02088e6	1.02074
$87$	−5.03966e6	−0.820510
$88$	2.07636e6	0.324798
$89$	−6.34297e6	−0.953734	−0.476867	−	0.878975i	$-0.658228\pi$
$89$	−6.34297e6	−0.953734	−0.476867	+	0.878975i	$0.658228\pi$
$90$	−2.98890e6	−0.432178
$91$	−1.33208e7	−1.85305
$92$	321664.	0.0430670
$93$	−5.09414e6	−0.656721
$94$	−6.47200e6	−0.803695
$95$	2.22089e7	2.65763
$96$	2.14704e6	0.247680
$97$	1.66516e7	1.85248	0.926242	−	0.376929i	$-0.123020\pi$
$97$	1.66516e7	1.85248	0.926242	+	0.376929i	$0.123020\pi$
$98$	2.33241e6	0.250330
$99$	−970299.	−0.100504
$100$	−2.51930e6	−0.251930
$101$	−2.08327e6	−0.201197	−0.100598	−	0.994927i	$-0.532076\pi$
$101$	−2.08327e6	−0.201197	−0.100598	+	0.994927i	$0.532076\pi$
$102$	4.60674e6	0.429826
$103$	−2.39046e6	−0.215552	−0.107776	−	0.994175i	$-0.534373\pi$
$103$	−2.39046e6	−0.215552	−0.107776	+	0.994175i	$0.534373\pi$
$104$	−2.02145e7	−1.76216
$105$	1.13800e7	0.959352
$106$	−1.31272e7	−1.07054
$107$	−1.40615e7	−1.10965	−0.554827	−	0.831966i	$-0.687215\pi$
$107$	−1.40615e7	−1.10965	−0.554827	+	0.831966i	$0.687215\pi$
$108$	−551124.	−0.0420985
$109$	−1.11321e7	−0.823347	−0.411674	−	0.911331i	$-0.635056\pi$
$109$	−1.11321e7	−0.823347	−0.411674	+	0.911331i	$0.635056\pi$
$110$	5.45710e6	0.390920
$111$	1.06889e7	0.741828
$112$	1.23524e7	0.830788
$113$	5.66903e6	0.369602	0.184801	−	0.982776i	$-0.440836\pi$
$113$	5.66903e6	0.369602	0.184801	+	0.982776i	$0.440836\pi$
$114$	−1.46254e7	−0.924570
$115$	4.71008e6	0.288792
$116$	5.22631e6	0.310880
$117$	9.44638e6	0.545274
$118$	−2.68114e7	−1.50222
$119$	−1.75397e7	−0.954132
$120$	1.72692e7	0.912300
$121$	1.77156e6	0.0909091
$122$	5.51190e6	0.274816
$123$	−1.28374e6	−0.0622028
$124$	5.28282e6	0.248822
$125$	−4.85850e6	−0.222493
$126$	−7.49412e6	−0.333752
$127$	−2.09170e7	−0.906123	−0.453061	−	0.891479i	$-0.649668\pi$
$127$	−2.09170e7	−0.906123	−0.453061	+	0.891479i	$0.649668\pi$
$128$	1.31539e7	0.554396
$129$	1.62564e7	0.666745
$130$	−5.31278e7	−2.12090
$131$	−1.12649e7	−0.437802	−0.218901	−	0.975747i	$-0.570247\pi$
$131$	−1.12649e7	−0.437802	−0.218901	+	0.975747i	$0.570247\pi$
$132$	1.00624e6	0.0380795
$133$	5.56847e7	2.05237
$134$	4.59260e6	0.164889
$135$	−8.07003e6	−0.282297
$136$	−2.66167e7	−0.907336
$137$	444290.	0.0147620	0.00738099	−	0.999973i	$-0.497651\pi$
$137$	444290.	0.0147620	0.00738099	+	0.999973i	$0.497651\pi$
$138$	−3.10176e6	−0.100469
$139$	3.42613e7	1.08206	0.541030	−	0.841003i	$-0.318034\pi$
$139$	3.42613e7	1.08206	0.541030	+	0.841003i	$0.318034\pi$
$140$	−1.18014e7	−0.363485
$141$	−1.74744e7	−0.524971
$142$	−180720.	−0.00529660
$143$	−1.72471e7	−0.493219
$144$	−8.75966e6	−0.244466
$145$	7.65281e7	2.08465
$146$	−4.26062e6	−0.113302
$147$	6.29751e6	0.163515
$148$	−1.10848e7	−0.281068
$149$	−4.82211e7	−1.19422	−0.597112	−	0.802158i	$-0.703685\pi$
$149$	−4.82211e7	−1.19422	−0.597112	+	0.802158i	$0.703685\pi$
$150$	2.42932e7	0.587714
$151$	−4.48693e7	−1.06055	−0.530273	−	0.847827i	$-0.677911\pi$
$151$	−4.48693e7	−1.06055	−0.530273	+	0.847827i	$0.677911\pi$
$152$	8.45021e7	1.95171
$153$	1.24382e7	0.280761
$154$	1.36827e7	0.301890
$155$	7.73555e7	1.66852
$156$	−9.79625e6	−0.206597
$157$	−5.38907e6	−0.111139	−0.0555693	−	0.998455i	$-0.517697\pi$
$157$	−5.38907e6	−0.111139	−0.0555693	+	0.998455i	$0.517697\pi$
$158$	2.97764e6	0.0600582
$159$	−3.54435e7	−0.699273
$160$	−3.26032e7	−0.629275
$161$	1.18097e7	0.223022
$162$	5.31441e6	0.0982093
$163$	9.81674e7	1.77546	0.887730	−	0.460365i	$-0.152281\pi$
$163$	9.81674e7	1.77546	0.887730	+	0.460365i	$0.152281\pi$
$164$	1.33129e6	0.0235678
$165$	1.47342e7	0.255348
$166$	5.68403e7	0.964446
$167$	−4.40611e7	−0.732062	−0.366031	−	0.930603i	$-0.619284\pi$
$167$	−4.40611e7	−0.732062	−0.366031	+	0.930603i	$0.619284\pi$
$168$	4.32994e7	0.704529
$169$	1.05161e8	1.67592
$170$	−6.99542e7	−1.09205
$171$	−3.94885e7	−0.603926
$172$	−1.68585e7	−0.252620
$173$	6.71087e7	0.985411	0.492706	−	0.870196i	$-0.336008\pi$
$173$	6.71087e7	0.985411	0.492706	+	0.870196i	$0.336008\pi$
$174$	−5.03966e7	−0.725235
$175$	−9.24943e7	−1.30461
$176$	1.59933e7	0.221128
$177$	−7.23908e7	−0.981244
$178$	−6.34297e7	−0.842990
$179$	4.34929e6	0.0566804	0.0283402	−	0.999598i	$-0.490978\pi$
$179$	4.34929e6	0.0566804	0.0283402	+	0.999598i	$0.490978\pi$
$180$	8.36892e6	0.106959
$181$	−1.20238e7	−0.150719	−0.0753593	−	0.997156i	$-0.524010\pi$
$181$	−1.20238e7	−0.150719	−0.0753593	+	0.997156i	$0.524010\pi$
$182$	−1.33208e8	−1.63788
$183$	1.48821e7	0.179509
$184$	1.79213e7	0.212083
$185$	−1.62313e8	−1.88475
$186$	−5.09414e7	−0.580465
$187$	−2.27095e7	−0.253958
$188$	1.81216e7	0.198904
$189$	−2.02341e7	−0.218006
$190$	2.22089e8	2.34903
$191$	5.96399e7	0.619327	0.309664	−	0.950846i	$-0.399784\pi$
$191$	5.96399e7	0.619327	0.309664	+	0.950846i	$0.399784\pi$
$192$	6.29977e7	0.642348
$193$	−9.81036e7	−0.982278	−0.491139	−	0.871081i	$-0.663419\pi$
$193$	−9.81036e7	−0.982278	−0.491139	+	0.871081i	$0.663419\pi$
$194$	1.66516e8	1.63738
$195$	−1.43445e8	−1.38537
$196$	−6.53075e6	−0.0619536
$197$	−1.09317e8	−1.01872	−0.509361	−	0.860553i	$-0.670118\pi$
$197$	−1.09317e8	−1.01872	−0.509361	+	0.860553i	$0.670118\pi$
$198$	−9.70299e6	−0.0888336
$199$	−3.64317e7	−0.327713	−0.163857	−	0.986484i	$-0.552393\pi$
$199$	−3.64317e7	−0.327713	−0.163857	+	0.986484i	$0.552393\pi$
$200$	−1.40361e8	−1.24063
$201$	1.24000e7	0.107705
$202$	−2.08327e7	−0.177834
$203$	1.91880e8	1.60988
$204$	−1.28989e7	−0.106377
$205$	1.94939e7	0.158037
$206$	−2.39046e7	−0.190523
$207$	−8.37475e6	−0.0656260
$208$	−1.55703e8	−1.19971
$209$	7.20976e7	0.546272
$210$	1.13800e8	0.847956
$211$	1.38637e7	0.101599	0.0507997	−	0.998709i	$-0.483823\pi$
$211$	1.38637e7	0.101599	0.0507997	+	0.998709i	$0.483823\pi$
$212$	3.67562e7	0.264945
$213$	−487944.	−0.00345972
$214$	−1.40615e8	−0.980805
$215$	−2.46856e8	−1.69398
$216$	−3.07055e7	−0.207314
$217$	1.93955e8	1.28852
$218$	−1.11321e8	−0.727743
$219$	−1.15037e7	−0.0740086
$220$	−1.52799e7	−0.0967477
$221$	2.21089e8	1.37783
$222$	1.06889e8	0.655690
$223$	1.35935e8	0.820850	0.410425	−	0.911894i	$-0.365380\pi$
$223$	1.35935e8	0.820850	0.410425	+	0.911894i	$0.365380\pi$
$224$	−8.17466e7	−0.485961
$225$	6.55918e7	0.383893
$226$	5.66903e7	0.326685
$227$	2.82203e7	0.160129	0.0800646	−	0.996790i	$-0.474487\pi$
$227$	2.82203e7	0.160129	0.0800646	+	0.996790i	$0.474487\pi$
$228$	4.09510e7	0.228819
$229$	−5.31215e7	−0.292312	−0.146156	−	0.989262i	$-0.546690\pi$
$229$	−5.31215e7	−0.292312	−0.146156	+	0.989262i	$0.546690\pi$
$230$	4.71008e7	0.255259
$231$	3.69432e7	0.197194
$232$	2.91180e8	1.53093
$233$	1.54589e8	0.800631	0.400316	−	0.916377i	$-0.368901\pi$
$233$	1.54589e8	0.800631	0.400316	+	0.916377i	$0.368901\pi$
$234$	9.44638e7	0.481959
$235$	2.65352e8	1.33378
$236$	7.50719e7	0.371780
$237$	8.03963e6	0.0392299
$238$	−1.75397e8	−0.843342
$239$	−1.86143e8	−0.881972	−0.440986	−	0.897514i	$-0.645371\pi$
$239$	−1.86143e8	−0.881972	−0.440986	+	0.897514i	$0.645371\pi$
$240$	1.33017e8	0.621110
$241$	2.62107e8	1.20620	0.603100	−	0.797666i	$-0.293932\pi$
$241$	2.62107e8	1.20620	0.603100	+	0.797666i	$0.293932\pi$
$242$	1.77156e7	0.0803530
$243$	1.43489e7	0.0641500
$244$	−1.54333e7	−0.0680135
$245$	−9.56288e7	−0.415439
$246$	−1.28374e7	−0.0549800
$247$	−7.01909e8	−2.96375
$248$	2.94328e8	1.22532
$249$	1.53469e8	0.629973
$250$	−4.85850e7	−0.196658
$251$	−2.75827e8	−1.10098	−0.550489	−	0.834842i	$-0.685559\pi$
$251$	−2.75827e8	−1.10098	−0.550489	+	0.834842i	$0.685559\pi$
$252$	2.09835e7	0.0825994
$253$	1.52905e7	0.0593609
$254$	−2.09170e8	−0.800907
$255$	−1.88876e8	−0.713324
$256$	−1.67117e8	−0.622558
$257$	1.06856e6	0.00392675	0.00196338	−	0.999998i	$-0.499375\pi$
$257$	1.06856e6	0.00392675	0.00196338	+	0.999998i	$0.499375\pi$
$258$	1.62564e8	0.589325
$259$	−4.06971e8	−1.45551
$260$	1.48758e8	0.524896
$261$	−1.36071e8	−0.473721
$262$	−1.12649e8	−0.386966
$263$	−7.92924e7	−0.268774	−0.134387	−	0.990929i	$-0.542906\pi$
$263$	−7.92924e7	−0.268774	−0.134387	+	0.990929i	$0.542906\pi$
$264$	5.60617e7	0.187522
$265$	5.38216e8	1.77663
$266$	5.56847e8	1.81405
$267$	−1.71260e8	−0.550639
$268$	−1.28593e7	−0.0408080
$269$	2.10170e8	0.658321	0.329160	−	0.944274i	$-0.393234\pi$
$269$	2.10170e8	0.658321	0.329160	+	0.944274i	$0.393234\pi$
$270$	−8.07003e7	−0.249518
$271$	−2.65510e8	−0.810378	−0.405189	−	0.914233i	$-0.632794\pi$
$271$	−2.65510e8	−0.810378	−0.405189	+	0.914233i	$0.632794\pi$
$272$	−2.05017e8	−0.617730
$273$	−3.59662e8	−1.06986
$274$	4.44290e6	0.0130479
$275$	−1.19757e8	−0.347245
$276$	8.68493e6	0.0248648
$277$	−6.23529e8	−1.76270	−0.881349	−	0.472466i	$-0.843364\pi$
$277$	−6.23529e8	−1.76270	−0.881349	+	0.472466i	$0.843364\pi$
$278$	3.42613e8	0.956415
$279$	−1.37542e8	−0.379158
$280$	−6.57509e8	−1.78998
$281$	1.30611e8	0.351162	0.175581	−	0.984465i	$-0.443820\pi$
$281$	1.30611e8	0.351162	0.175581	+	0.984465i	$0.443820\pi$
$282$	−1.74744e8	−0.464013
$283$	−2.20874e7	−0.0579283	−0.0289642	−	0.999580i	$-0.509221\pi$
$283$	−2.20874e7	−0.0579283	−0.0289642	+	0.999580i	$0.509221\pi$
$284$	506016.	0.00131084
$285$	5.99640e8	1.53438
$286$	−1.72471e8	−0.435948
$287$	4.88773e7	0.122045
$288$	5.79701e7	0.142998
$289$	−1.19227e8	−0.290557
$290$	7.65281e8	1.84259
$291$	4.49593e8	1.06953
$292$	1.19297e7	0.0280408
$293$	2.00188e8	0.464944	0.232472	−	0.972603i	$-0.425319\pi$
$293$	2.00188e8	0.464944	0.232472	+	0.972603i	$0.425319\pi$
$294$	6.29751e7	0.144528
$295$	1.09927e9	2.49302
$296$	−6.17582e8	−1.38412
$297$	−2.61981e7	−0.0580259
$298$	−4.82211e8	−1.05555
$299$	−1.48862e8	−0.322057
$300$	−6.80211e7	−0.145452
$301$	−6.18946e8	−1.30819
$302$	−4.48693e8	−0.937399
$303$	−5.62483e7	−0.116161
$304$	6.50883e8	1.32876
$305$	−2.25988e8	−0.456074
$306$	1.24382e8	0.248160
$307$	−4.79736e8	−0.946276	−0.473138	−	0.880988i	$-0.656879\pi$
$307$	−4.79736e8	−0.946276	−0.473138	+	0.880988i	$0.656879\pi$
$308$	−3.83115e7	−0.0747139
$309$	−6.45425e7	−0.124449
$310$	7.73555e8	1.47477
$311$	−5.19734e8	−0.979761	−0.489880	−	0.871790i	$-0.662960\pi$
$311$	−5.19734e8	−0.979761	−0.489880	+	0.871790i	$0.662960\pi$
$312$	−5.45791e8	−1.01738
$313$	9.69759e8	1.78755	0.893776	−	0.448514i	$-0.148047\pi$
$313$	9.69759e8	1.78755	0.893776	+	0.448514i	$0.148047\pi$
$314$	−5.38907e7	−0.0982335
$315$	3.07259e8	0.553882
$316$	−8.33739e6	−0.0148636
$317$	7.56875e8	1.33450	0.667248	−	0.744836i	$-0.267472\pi$
$317$	7.56875e8	1.33450	0.667248	+	0.744836i	$0.267472\pi$
$318$	−3.54435e8	−0.618076
$319$	2.48436e8	0.428497
$320$	−9.56632e8	−1.63200
$321$	−3.79660e8	−0.640659
$322$	1.18097e8	0.197125
$323$	−9.24214e8	−1.52603
$324$	−1.48803e7	−0.0243056
$325$	1.16590e9	1.88394
$326$	9.81674e8	1.56930
$327$	−3.00566e8	−0.475360
$328$	7.41718e7	0.116059
$329$	6.65322e8	1.03002
$330$	1.47342e8	0.225698
$331$	−1.79867e8	−0.272618	−0.136309	−	0.990666i	$-0.543524\pi$
$331$	−1.79867e8	−0.272618	−0.136309	+	0.990666i	$0.543524\pi$
$332$	−1.59153e8	−0.238688
$333$	2.88601e8	0.428295
$334$	−4.40611e8	−0.647058
$335$	−1.88297e8	−0.273644
$336$	3.33516e8	0.479655
$337$	−1.38092e9	−1.96546	−0.982728	−	0.185054i	$-0.940754\pi$
$337$	−1.38092e9	−1.96546	−0.982728	+	0.185054i	$0.940754\pi$
$338$	1.05161e9	1.48131
$339$	1.53064e8	0.213390
$340$	1.95872e8	0.270269
$341$	2.51122e8	0.342961
$342$	−3.94885e8	−0.533800
$343$	6.06830e8	0.811966
$344$	−9.39257e8	−1.24403
$345$	1.27172e8	0.166734
$346$	6.71087e8	0.870989
$347$	7.66253e8	0.984507	0.492254	−	0.870452i	$-0.336173\pi$
$347$	7.66253e8	0.984507	0.492254	+	0.870452i	$0.336173\pi$
$348$	1.41110e8	0.179486
$349$	−2.68852e8	−0.338552	−0.169276	−	0.985569i	$-0.554143\pi$
$349$	−2.68852e8	−0.338552	−0.169276	+	0.985569i	$0.554143\pi$
$350$	−9.24943e8	−1.15313
$351$	2.55052e8	0.314814
$352$	−1.05841e8	−0.129347
$353$	−3.95002e8	−0.477956	−0.238978	−	0.971025i	$-0.576812\pi$
$353$	−3.95002e8	−0.477956	−0.238978	+	0.971025i	$0.576812\pi$
$354$	−7.23908e8	−0.867305
$355$	7.40952e6	0.00879004
$356$	1.77603e8	0.208629
$357$	−4.73573e8	−0.550869
$358$	4.34929e7	0.0500989
$359$	−4.25768e7	−0.0485671	−0.0242836	−	0.999705i	$-0.507730\pi$
$359$	−4.25768e7	−0.0485671	−0.0242836	+	0.999705i	$0.507730\pi$
$360$	4.66268e8	0.526717
$361$	2.04030e9	2.28254
$362$	−1.20238e8	−0.133218
$363$	4.78321e7	0.0524864
$364$	3.72983e8	0.405354
$365$	1.74685e8	0.188032
$366$	1.48821e8	0.158665
$367$	1.85295e9	1.95673	0.978366	−	0.206882i	$-0.0663315\pi$
$367$	1.85295e9	1.95673	0.978366	+	0.206882i	$0.0663315\pi$
$368$	1.38040e8	0.144390
$369$	−3.46610e7	−0.0359128
$370$	−1.62313e9	−1.66590
$371$	1.34948e9	1.37201
$372$	1.42636e8	0.143658
$373$	−4.83602e7	−0.0482511	−0.0241256	−	0.999709i	$-0.507680\pi$
$373$	−4.83602e7	−0.0482511	−0.0241256	+	0.999709i	$0.507680\pi$
$374$	−2.27095e8	−0.224470
$375$	−1.31180e8	−0.128457
$376$	1.00963e9	0.979503
$377$	−2.41866e9	−2.32477
$378$	−2.02341e8	−0.192692
$379$	−2.26078e8	−0.213315	−0.106658	−	0.994296i	$-0.534015\pi$
$379$	−2.26078e8	−0.213315	−0.106658	+	0.994296i	$0.534015\pi$
$380$	−6.21849e8	−0.581356
$381$	−5.64760e8	−0.523150
$382$	5.96399e8	0.547413
$383$	−1.35198e9	−1.22963	−0.614815	−	0.788671i	$-0.710769\pi$
$383$	−1.35198e9	−1.22963	−0.614815	+	0.788671i	$0.710769\pi$
$384$	3.55156e8	0.320081
$385$	−5.60990e8	−0.501005
$386$	−9.81036e8	−0.868219
$387$	4.38922e8	0.384945
$388$	−4.66244e8	−0.405231
$389$	−1.09107e9	−0.939789	−0.469894	−	0.882723i	$-0.655708\pi$
$389$	−1.09107e9	−0.939789	−0.469894	+	0.882723i	$0.655708\pi$
$390$	−1.43445e9	−1.22450
$391$	−1.96008e8	−0.165827
$392$	−3.63856e8	−0.305090
$393$	−3.04152e8	−0.252765
$394$	−1.09317e9	−0.900431
$395$	−1.22083e8	−0.0996704
$396$	2.71684e7	0.0219852
$397$	−6.97868e8	−0.559766	−0.279883	−	0.960034i	$-0.590296\pi$
$397$	−6.97868e8	−0.559766	−0.279883	+	0.960034i	$0.590296\pi$
$398$	−3.64317e8	−0.289660
$399$	1.50349e9	1.18494
$400$	−1.08114e9	−0.844640
$401$	1.74689e9	1.35288	0.676441	−	0.736497i	$-0.263521\pi$
$401$	1.74689e9	1.35288	0.676441	+	0.736497i	$0.263521\pi$
$402$	1.24000e8	0.0951987
$403$	−2.44481e9	−1.86071
$404$	5.83316e7	0.0440118
$405$	−2.17891e8	−0.162985
$406$	1.91880e9	1.42295
$407$	−5.26924e8	−0.387407
$408$	−7.18651e8	−0.523851
$409$	−1.30304e9	−0.941729	−0.470865	−	0.882205i	$-0.656058\pi$
$409$	−1.30304e9	−0.941729	−0.470865	+	0.882205i	$0.656058\pi$
$410$	1.94939e8	0.139686
$411$	1.19958e7	0.00852283
$412$	6.69330e7	0.0471520
$413$	2.75621e9	1.92525
$414$	−8.37475e7	−0.0580057
$415$	−2.33045e9	−1.60056
$416$	1.03042e9	0.701759
$417$	9.25054e8	0.624728
$418$	7.20976e8	0.482841
$419$	2.87139e9	1.90697	0.953484	−	0.301443i	$-0.0974684\pi$
$419$	2.87139e9	1.90697	0.953484	+	0.301443i	$0.0974684\pi$
$420$	−3.18639e8	−0.209858
$421$	1.15946e9	0.757299	0.378650	−	0.925540i	$-0.376389\pi$
$421$	1.15946e9	0.757299	0.378650	+	0.925540i	$0.376389\pi$
$422$	1.38637e8	0.0898020
$423$	−4.71809e8	−0.303092
$424$	2.04785e9	1.30472
$425$	1.53515e9	0.970042
$426$	−4.87944e6	−0.00305799
$427$	−5.66623e8	−0.352206
$428$	3.93721e8	0.242737
$429$	−4.65672e8	−0.284760
$430$	−2.46856e9	−1.49728
$431$	−1.66703e9	−1.00294	−0.501468	−	0.865176i	$-0.667207\pi$
$431$	−1.66703e9	−1.00294	−0.501468	+	0.865176i	$0.667207\pi$
$432$	−2.36511e8	−0.141143
$433$	6.34094e8	0.375358	0.187679	−	0.982230i	$-0.439903\pi$
$433$	6.34094e8	0.375358	0.187679	+	0.982230i	$0.439903\pi$
$434$	1.93955e9	1.13890
$435$	2.06626e9	1.20357
$436$	3.11698e8	0.180107
$437$	6.22282e8	0.356699
$438$	−1.15037e8	−0.0654150
$439$	−1.22368e9	−0.690307	−0.345154	−	0.938546i	$-0.612173\pi$
$439$	−1.22368e9	−0.690307	−0.345154	+	0.938546i	$0.612173\pi$
$440$	−8.51308e8	−0.476433
$441$	1.70033e8	0.0944055
$442$	2.21089e9	1.21784
$443$	−1.23213e9	−0.673355	−0.336677	−	0.941620i	$-0.609303\pi$
$443$	−1.23213e9	−0.673355	−0.336677	+	0.941620i	$0.609303\pi$
$444$	−2.99290e8	−0.162275
$445$	2.60062e9	1.39900
$446$	1.35935e9	0.725536
$447$	−1.30197e9	−0.689485
$448$	−2.39858e9	−1.26032
$449$	−3.07511e9	−1.60324	−0.801621	−	0.597833i	$-0.796029\pi$
$449$	−3.07511e9	−1.60324	−0.801621	+	0.597833i	$0.796029\pi$
$450$	6.55918e8	0.339317
$451$	6.32837e7	0.0324843
$452$	−1.58733e8	−0.0808505
$453$	−1.21147e9	−0.612307
$454$	2.82203e8	0.141536
$455$	5.46154e9	2.71816
$456$	2.28156e9	1.12682
$457$	−2.44730e9	−1.19945	−0.599723	−	0.800207i	$-0.704723\pi$
$457$	−2.44730e9	−1.19945	−0.599723	+	0.800207i	$0.704723\pi$
$458$	−5.31215e8	−0.258370
$459$	3.35831e8	0.162098
$460$	−1.31882e8	−0.0631733
$461$	9.52419e8	0.452767	0.226383	−	0.974038i	$-0.427310\pi$
$461$	9.52419e8	0.452767	0.226383	+	0.974038i	$0.427310\pi$
$462$	3.69432e8	0.174296
$463$	−6.05200e8	−0.283378	−0.141689	−	0.989911i	$-0.545253\pi$
$463$	−6.05200e8	−0.283378	−0.141689	+	0.989911i	$0.545253\pi$
$464$	2.24283e9	1.04228
$465$	2.08860e9	0.963318
$466$	1.54589e9	0.707665
$467$	−1.37708e9	−0.625676	−0.312838	−	0.949806i	$-0.601280\pi$
$467$	−1.37708e9	−0.625676	−0.312838	+	0.949806i	$0.601280\pi$
$468$	−2.64499e8	−0.119279
$469$	−4.72119e8	−0.211323
$470$	2.65352e9	1.17891
$471$	−1.45505e8	−0.0641659
$472$	4.18258e9	1.83083
$473$	−8.01379e8	−0.348196
$474$	8.03963e7	0.0346746
$475$	−4.87377e9	−2.08659
$476$	4.91113e8	0.208716
$477$	−9.56974e8	−0.403725
$478$	−1.86143e9	−0.779561
$479$	−4.00222e9	−1.66390	−0.831949	−	0.554851i	$-0.812775\pi$
$479$	−4.00222e9	−1.66390	−0.831949	+	0.554851i	$0.812775\pi$
$480$	−8.80286e8	−0.363312
$481$	5.12989e9	2.10184
$482$	2.62107e9	1.06614
$483$	3.18861e8	0.128762
$484$	−4.96037e7	−0.0198864
$485$	−6.82715e9	−2.71734
$486$	1.43489e8	0.0567012
$487$	−2.88677e9	−1.13256	−0.566279	−	0.824214i	$-0.691617\pi$
$487$	−2.88677e9	−1.13256	−0.566279	+	0.824214i	$0.691617\pi$
$488$	−8.59856e8	−0.334932
$489$	2.65052e9	1.02506
$490$	−9.56288e8	−0.367200
$491$	1.19743e8	0.0456525	0.0228262	−	0.999739i	$-0.492734\pi$
$491$	1.19743e8	0.0456525	0.0228262	+	0.999739i	$0.492734\pi$
$492$	3.59448e7	0.0136069
$493$	−3.18469e9	−1.19702
$494$	−7.01909e9	−2.61961
$495$	3.97823e8	0.147425
$496$	2.26708e9	0.834222
$497$	1.85780e7	0.00678816
$498$	1.53469e9	0.556823
$499$	−4.78950e9	−1.72559	−0.862796	−	0.505552i	$-0.831289\pi$
$499$	−4.78950e9	−1.72559	−0.862796	+	0.505552i	$0.831289\pi$
$500$	1.36038e8	0.0486704
$501$	−1.18965e9	−0.422656
$502$	−2.75827e9	−0.973137
$503$	3.83047e9	1.34203	0.671017	−	0.741442i	$-0.265858\pi$
$503$	3.83047e9	1.34203	0.671017	+	0.741442i	$0.265858\pi$
$504$	1.16908e9	0.406760
$505$	8.54141e8	0.295127
$506$	1.52905e8	0.0524681
$507$	2.83935e9	0.967591
$508$	5.85677e8	0.198214
$509$	−2.34385e9	−0.787803	−0.393902	−	0.919153i	$-0.628875\pi$
$509$	−2.34385e9	−0.787803	−0.393902	+	0.919153i	$0.628875\pi$
$510$	−1.88876e9	−0.630495
$511$	4.37992e8	0.145209
$512$	−3.35487e9	−1.10466
$513$	−1.06619e9	−0.348677
$514$	1.06856e7	0.00347079
$515$	9.80090e8	0.316185
$516$	−4.55179e8	−0.145850
$517$	8.61423e8	0.274157
$518$	−4.06971e9	−1.28650
$519$	1.81194e9	0.568928
$520$	8.28794e9	2.58485
$521$	5.77085e9	1.78775	0.893877	−	0.448313i	$-0.147975\pi$
$521$	5.77085e9	1.78775	0.893877	+	0.448313i	$0.147975\pi$
$522$	−1.36071e9	−0.418715
$523$	−3.49411e8	−0.106802	−0.0534012	−	0.998573i	$-0.517006\pi$
$523$	−3.49411e8	−0.106802	−0.0534012	+	0.998573i	$0.517006\pi$
$524$	3.15417e8	0.0957691
$525$	−2.49735e9	−0.753219
$526$	−7.92924e8	−0.237565
$527$	−3.21912e9	−0.958077
$528$	4.31819e8	0.127668
$529$	−3.27285e9	−0.961239
$530$	5.38216e9	1.57033
$531$	−1.95455e9	−0.566521
$532$	−1.55917e9	−0.448955
$533$	−6.16101e8	−0.176241
$534$	−1.71260e9	−0.486700
$535$	5.76520e9	1.62771
$536$	−7.16446e8	−0.200959
$537$	1.17431e8	0.0327244
$538$	2.10170e9	0.581879
$539$	−3.10444e8	−0.0853930
$540$	2.25961e8	0.0617526
$541$	−5.10025e9	−1.38484	−0.692422	−	0.721493i	$-0.743456\pi$
$541$	−5.10025e9	−1.38484	−0.692422	+	0.721493i	$0.743456\pi$
$542$	−2.65510e9	−0.716280
$543$	−3.24643e8	−0.0870175
$544$	1.35677e9	0.361335
$545$	4.56415e9	1.20774
$546$	−3.59662e9	−0.945629
$547$	4.96217e9	1.29633	0.648166	−	0.761499i	$-0.275536\pi$
$547$	4.96217e9	1.29633	0.648166	+	0.761499i	$0.275536\pi$
$548$	−1.24401e7	−0.00322918
$549$	4.01818e8	0.103640
$550$	−1.19757e9	−0.306924
$551$	1.01107e10	2.57484
$552$	4.83875e8	0.122446
$553$	−3.06101e8	−0.0769710
$554$	−6.23529e9	−1.55802
$555$	−4.38246e9	−1.08816
$556$	−9.59315e8	−0.236701
$557$	1.42590e9	0.349620	0.174810	−	0.984602i	$-0.444069\pi$
$557$	1.42590e9	0.349620	0.174810	+	0.984602i	$0.444069\pi$
$558$	−1.37542e9	−0.335132
$559$	7.80186e9	1.88911
$560$	−5.06450e9	−1.21865
$561$	−6.13157e8	−0.146623
$562$	1.30611e9	0.310386
$563$	5.96929e9	1.40975	0.704876	−	0.709330i	$-0.251002\pi$
$563$	5.96929e9	1.40975	0.704876	+	0.709330i	$0.251002\pi$
$564$	4.89283e8	0.114837
$565$	−2.32430e9	−0.542155
$566$	−2.20874e8	−0.0512019
$567$	−5.46321e8	−0.125866
$568$	2.81923e7	0.00645523
$569$	−3.51616e9	−0.800158	−0.400079	−	0.916481i	$-0.631017\pi$
$569$	−3.51616e9	−0.800158	−0.400079	+	0.916481i	$0.631017\pi$
$570$	5.99640e9	1.35621
$571$	6.44706e8	0.144922	0.0724611	−	0.997371i	$-0.476915\pi$
$571$	6.44706e8	0.144922	0.0724611	+	0.997371i	$0.476915\pi$
$572$	4.82919e8	0.107892
$573$	1.61028e9	0.357569
$574$	4.88773e8	0.107874
$575$	−1.03363e9	−0.226740
$576$	1.70094e9	0.370860
$577$	−2.63322e9	−0.570652	−0.285326	−	0.958430i	$-0.592102\pi$
$577$	−2.63322e9	−0.570652	−0.285326	+	0.958430i	$0.592102\pi$
$578$	−1.19227e9	−0.256819
$579$	−2.64880e9	−0.567118
$580$	−2.14279e9	−0.456017
$581$	−5.84318e9	−1.23604
$582$	4.49593e9	0.945342
$583$	1.74723e9	0.365183
$584$	6.64657e8	0.138087
$585$	−3.87302e9	−0.799841
$586$	2.00188e9	0.410956
$587$	6.76347e9	1.38018	0.690090	−	0.723723i	$-0.257571\pi$
$587$	6.76347e9	1.38018	0.690090	+	0.723723i	$0.257571\pi$
$588$	−1.76330e8	−0.0357689
$589$	1.02200e10	2.06085
$590$	1.09927e10	2.20354
$591$	−2.95156e9	−0.588159
$592$	−4.75697e9	−0.942332
$593$	−4.22718e9	−0.832452	−0.416226	−	0.909261i	$-0.636648\pi$
$593$	−4.22718e9	−0.832452	−0.416226	+	0.909261i	$0.636648\pi$
$594$	−2.61981e8	−0.0512881
$595$	7.19129e9	1.39958
$596$	1.35019e9	0.261236
$597$	−9.83657e8	−0.189205
$598$	−1.48862e9	−0.284661
$599$	4.00299e9	0.761010	0.380505	−	0.924779i	$-0.375750\pi$
$599$	4.00299e9	0.761010	0.380505	+	0.924779i	$0.375750\pi$
$600$	−3.78975e9	−0.716277
$601$	6.67554e9	1.25437	0.627185	−	0.778870i	$-0.284207\pi$
$601$	6.67554e9	1.25437	0.627185	+	0.778870i	$0.284207\pi$
$602$	−6.18946e9	−1.15629
$603$	3.34801e8	0.0621836
$604$	1.25634e9	0.231994
$605$	−7.26340e8	−0.133351
$606$	−5.62483e8	−0.102673
$607$	−5.30634e9	−0.963018	−0.481509	−	0.876441i	$-0.659911\pi$
$607$	−5.30634e9	−0.963018	−0.481509	+	0.876441i	$0.659911\pi$
$608$	−4.30744e9	−0.777243
$609$	5.18077e9	0.929466
$610$	−2.25988e9	−0.403117
$611$	−8.38642e9	−1.48742
$612$	−3.48270e8	−0.0614166
$613$	−8.65802e9	−1.51812	−0.759061	−	0.651019i	$-0.774342\pi$
$613$	−8.65802e9	−1.51812	−0.759061	+	0.651019i	$0.774342\pi$
$614$	−4.79736e9	−0.836398
$615$	5.26334e8	0.0912428
$616$	−2.13450e9	−0.367928
$617$	−7.38891e9	−1.26643	−0.633217	−	0.773974i	$-0.718266\pi$
$617$	−7.38891e9	−1.26643	−0.633217	+	0.773974i	$0.718266\pi$
$618$	−6.45425e8	−0.109998
$619$	9.99141e9	1.69321	0.846603	−	0.532225i	$-0.178644\pi$
$619$	9.99141e9	1.69321	0.846603	+	0.532225i	$0.178644\pi$
$620$	−2.16595e9	−0.364988
$621$	−2.26118e8	−0.0378892
$622$	−5.19734e9	−0.865994
$623$	6.52057e9	1.08038
$624$	−4.20399e9	−0.692653
$625$	−5.03731e9	−0.825313
$626$	9.69759e9	1.57999
$627$	1.94664e9	0.315390
$628$	1.50894e8	0.0243116
$629$	6.75461e9	1.08224
$630$	3.07259e9	0.489567
$631$	−3.29834e9	−0.522628	−0.261314	−	0.965254i	$-0.584156\pi$
$631$	−3.29834e9	−0.522628	−0.261314	+	0.965254i	$0.584156\pi$
$632$	−4.64512e8	−0.0731959
$633$	3.74320e8	0.0586584
$634$	7.56875e9	1.17954
$635$	8.57598e9	1.32916
$636$	9.92418e8	0.152966
$637$	3.02234e9	0.463292
$638$	2.48436e9	0.378742
$639$	−1.31745e7	−0.00199747
$640$	−5.39311e9	−0.813222
$641$	−9.76971e9	−1.46514	−0.732569	−	0.680692i	$-0.761679\pi$
$641$	−9.76971e9	−1.46514	−0.732569	+	0.680692i	$0.761679\pi$
$642$	−3.79660e9	−0.566268
$643$	−4.18444e9	−0.620724	−0.310362	−	0.950618i	$-0.600450\pi$
$643$	−4.18444e9	−0.620724	−0.310362	+	0.950618i	$0.600450\pi$
$644$	−3.30671e8	−0.0487860
$645$	−6.66511e9	−0.978022
$646$	−9.24214e9	−1.34884
$647$	−6.96085e8	−0.101041	−0.0505204	−	0.998723i	$-0.516088\pi$
$647$	−6.96085e8	−0.101041	−0.0505204	+	0.998723i	$0.516088\pi$
$648$	−8.29048e8	−0.119693
$649$	3.56860e9	0.512438
$650$	1.16590e10	1.66519
$651$	5.23678e9	0.743928
$652$	−2.74869e9	−0.388382
$653$	6.20046e9	0.871420	0.435710	−	0.900087i	$-0.356497\pi$
$653$	6.20046e9	0.871420	0.435710	+	0.900087i	$0.356497\pi$
$654$	−3.00566e9	−0.420163
$655$	4.61861e9	0.642194
$656$	5.71313e8	0.0790152
$657$	−3.10599e8	−0.0427289
$658$	6.65322e9	0.910418
$659$	−1.11404e10	−1.51636	−0.758178	−	0.652047i	$-0.773910\pi$
$659$	−1.11404e10	−1.51636	−0.758178	+	0.652047i	$0.773910\pi$
$660$	−4.12557e8	−0.0558573
$661$	4.56096e9	0.614258	0.307129	−	0.951668i	$-0.400632\pi$
$661$	4.56096e9	0.614258	0.307129	+	0.951668i	$0.400632\pi$
$662$	−1.79867e9	−0.240963
$663$	5.96941e9	0.795489
$664$	−8.86708e9	−1.17542
$665$	−2.28307e10	−3.01054
$666$	2.88601e9	0.378563
$667$	2.14428e9	0.279796
$668$	1.23371e9	0.160139
$669$	3.67024e9	0.473918
$670$	−1.88297e9	−0.241869
$671$	−7.33634e8	−0.0937455
$672$	−2.20716e9	−0.280570
$673$	5.82879e9	0.737099	0.368550	−	0.929608i	$-0.379854\pi$
$673$	5.82879e9	0.737099	0.368550	+	0.929608i	$0.379854\pi$
$674$	−1.38092e10	−1.73723
$675$	1.77098e9	0.221641
$676$	−2.94451e9	−0.366607
$677$	4.99624e9	0.618846	0.309423	−	0.950924i	$-0.399864\pi$
$677$	4.99624e9	0.618846	0.309423	+	0.950924i	$0.399864\pi$
$678$	1.53064e9	0.188612
$679$	−1.71178e10	−2.09848
$680$	1.09129e10	1.33094
$681$	7.61947e8	0.0924507
$682$	2.51122e9	0.303138
$683$	1.21371e10	1.45762	0.728808	−	0.684718i	$-0.240075\pi$
$683$	1.21371e10	1.45762	0.728808	+	0.684718i	$0.240075\pi$
$684$	1.10568e9	0.132109
$685$	−1.82159e8	−0.0216538
$686$	6.06830e9	0.717684
$687$	−1.43428e9	−0.168766
$688$	−7.23469e9	−0.846955
$689$	−1.70103e10	−1.98127
$690$	1.27172e9	0.147374
$691$	−9.23403e9	−1.06468	−0.532339	−	0.846531i	$-0.678687\pi$
$691$	−9.23403e9	−1.06468	−0.532339	+	0.846531i	$0.678687\pi$
$692$	−1.87904e9	−0.215559
$693$	9.97467e8	0.113850
$694$	7.66253e9	0.870190
$695$	−1.40471e10	−1.58723
$696$	7.86187e9	0.883880
$697$	−8.11230e8	−0.0907464
$698$	−2.68852e9	−0.299240
$699$	4.17390e9	0.462245
$700$	2.58984e9	0.285384
$701$	4.74530e9	0.520296	0.260148	−	0.965569i	$-0.416229\pi$
$701$	4.74530e9	0.520296	0.260148	+	0.965569i	$0.416229\pi$
$702$	2.55052e9	0.278259
$703$	−2.14444e10	−2.32793
$704$	−3.10555e9	−0.335455
$705$	7.16450e9	0.770059
$706$	−3.95002e9	−0.422458
$707$	2.14160e9	0.227914
$708$	2.02694e9	0.214647
$709$	1.34547e10	1.41779	0.708894	−	0.705315i	$-0.249195\pi$
$709$	1.34547e10	1.41779	0.708894	+	0.705315i	$0.249195\pi$
$710$	7.40952e7	0.00776937
$711$	2.17070e8	0.0226494
$712$	9.89503e9	1.02739
$713$	2.16746e9	0.223944
$714$	−4.73573e9	−0.486904
$715$	7.07131e9	0.723484
$716$	−1.21780e8	−0.0123988
$717$	−5.02587e9	−0.509207
$718$	−4.25768e8	−0.0429277
$719$	2.63976e9	0.264858	0.132429	−	0.991192i	$-0.457722\pi$
$719$	2.63976e9	0.264858	0.132429	+	0.991192i	$0.457722\pi$
$720$	3.59146e9	0.358598
$721$	2.45740e9	0.244175
$722$	2.04030e10	2.01750
$723$	7.07689e9	0.696400
$724$	3.36667e8	0.0329697
$725$	−1.67942e10	−1.63673
$726$	4.78321e8	0.0463919
$727$	−3.52707e9	−0.340442	−0.170221	−	0.985406i	$-0.554448\pi$
$727$	−3.52707e9	−0.340442	−0.170221	+	0.985406i	$0.554448\pi$
$728$	2.07805e10	1.99616
$729$	3.87420e8	0.0370370
$730$	1.74685e9	0.166198
$731$	1.02728e10	0.972700
$732$	−4.16700e8	−0.0392676
$733$	−1.03828e10	−0.973760	−0.486880	−	0.873469i	$-0.661865\pi$
$733$	−1.03828e10	−0.973760	−0.486880	+	0.873469i	$0.661865\pi$
$734$	1.85295e10	1.72952
$735$	−2.58198e9	−0.239854
$736$	−9.13526e8	−0.0844595
$737$	−6.11275e8	−0.0562471
$738$	−3.46610e8	−0.0317427
$739$	2.05418e9	0.187233	0.0936164	−	0.995608i	$-0.470157\pi$
$739$	2.05418e9	0.187233	0.0936164	+	0.995608i	$0.470157\pi$
$740$	4.54477e9	0.412288
$741$	−1.89515e10	−1.71112
$742$	1.34948e10	1.21270
$743$	4.87476e9	0.436006	0.218003	−	0.975948i	$-0.430046\pi$
$743$	4.87476e9	0.436006	0.218003	+	0.975948i	$0.430046\pi$
$744$	7.94686e9	0.707442
$745$	1.97707e10	1.75176
$746$	−4.83602e8	−0.0426484
$747$	4.14366e9	0.363715
$748$	6.35867e8	0.0555534
$749$	1.44552e10	1.25701
$750$	−1.31180e9	−0.113541
$751$	1.15809e10	0.997705	0.498853	−	0.866687i	$-0.333755\pi$
$751$	1.15809e10	0.997705	0.498853	+	0.866687i	$0.333755\pi$
$752$	7.77676e9	0.666862
$753$	−7.44733e9	−0.635650
$754$	−2.41866e10	−2.05483
$755$	1.83964e10	1.55567
$756$	5.66555e8	0.0476888
$757$	3.46735e9	0.290511	0.145255	−	0.989394i	$-0.453600\pi$
$757$	3.46735e9	0.290511	0.145255	+	0.989394i	$0.453600\pi$
$758$	−2.26078e9	−0.188546
$759$	4.12844e8	0.0342720
$760$	−3.46459e10	−2.86288
$761$	−1.14023e10	−0.937877	−0.468938	−	0.883231i	$-0.655363\pi$
$761$	−1.14023e10	−0.937877	−0.468938	+	0.883231i	$0.655363\pi$
$762$	−5.64760e9	−0.462404
$763$	1.14438e10	0.932681
$764$	−1.66992e9	−0.135478
$765$	−5.09966e9	−0.411838
$766$	−1.35198e10	−1.08685
$767$	−3.47422e10	−2.78018
$768$	−4.51215e9	−0.359434
$769$	2.30715e10	1.82951	0.914754	−	0.404012i	$-0.132384\pi$
$769$	2.30715e10	1.82951	0.914754	+	0.404012i	$0.132384\pi$
$770$	−5.60990e9	−0.442830
$771$	2.88512e7	0.00226711
$772$	2.74690e9	0.214873
$773$	2.15091e10	1.67492	0.837461	−	0.546497i	$-0.184039\pi$
$773$	2.15091e10	1.67492	0.837461	+	0.546497i	$0.184039\pi$
$774$	4.38922e9	0.340247
$775$	−1.69758e10	−1.31001
$776$	−2.59765e10	−1.99556
$777$	−1.09882e10	−0.840337
$778$	−1.09107e10	−0.830664
$779$	2.57547e9	0.195198
$780$	4.01646e9	0.303049
$781$	2.40538e7	0.00180678
$782$	−1.96008e9	−0.146572
$783$	−3.67391e9	−0.273503
$784$	−2.80262e9	−0.207711
$785$	2.20952e9	0.163025
$786$	−3.04152e9	−0.223415
$787$	−4.46678e9	−0.326651	−0.163325	−	0.986572i	$-0.552222\pi$
$787$	−4.46678e9	−0.326651	−0.163325	+	0.986572i	$0.552222\pi$
$788$	3.06087e9	0.222845
$789$	−2.14090e9	−0.155176
$790$	−1.22083e9	−0.0880970
$791$	−5.82777e9	−0.418682
$792$	1.51367e9	0.108266
$793$	7.14232e9	0.508608
$794$	−6.97868e9	−0.494768
$795$	1.45318e10	1.02574
$796$	1.02009e9	0.0716873
$797$	2.43899e10	1.70650	0.853248	−	0.521505i	$-0.174629\pi$
$797$	2.43899e10	1.70650	0.853248	+	0.521505i	$0.174629\pi$
$798$	1.50349e10	1.04734
$799$	−1.10425e10	−0.765869
$800$	7.15481e9	0.494064
$801$	−4.62402e9	−0.317911
$802$	1.74689e10	1.19579
$803$	5.67089e8	0.0386497
$804$	−3.47201e8	−0.0235605
$805$	−4.84196e9	−0.327142
$806$	−2.44481e10	−1.64465
$807$	5.67459e9	0.380082
$808$	3.24990e9	0.216736
$809$	9.88857e9	0.656620	0.328310	−	0.944570i	$-0.393521\pi$
$809$	9.88857e9	0.656620	0.328310	+	0.944570i	$0.393521\pi$
$810$	−2.17891e9	−0.144059
$811$	−1.15204e10	−0.758395	−0.379198	−	0.925316i	$-0.623800\pi$
$811$	−1.15204e10	−0.758395	−0.379198	+	0.925316i	$0.623800\pi$
$812$	−5.37265e9	−0.352162
$813$	−7.16876e9	−0.467872
$814$	−5.26924e9	−0.342423
$815$	−4.02487e10	−2.60435
$816$	−5.53546e9	−0.356647
$817$	−3.26139e10	−2.09231
$818$	−1.30304e10	−0.832379
$819$	−9.71088e9	−0.617682
$820$	−5.45828e8	−0.0345706
$821$	2.63516e9	0.166191	0.0830953	−	0.996542i	$-0.473519\pi$
$821$	2.63516e9	0.166191	0.0830953	+	0.996542i	$0.473519\pi$
$822$	1.19958e8	0.00753319
$823$	−1.27039e10	−0.794400	−0.397200	−	0.917732i	$-0.630018\pi$
$823$	−1.27039e10	−0.794400	−0.397200	+	0.917732i	$0.630018\pi$
$824$	3.72912e9	0.232200
$825$	−3.23343e9	−0.200482
$826$	2.75621e10	1.70170
$827$	−1.11339e10	−0.684504	−0.342252	−	0.939608i	$-0.611190\pi$
$827$	−1.11339e10	−0.684504	−0.342252	+	0.939608i	$0.611190\pi$
$828$	2.34493e8	0.0143557
$829$	2.79852e10	1.70604	0.853018	−	0.521881i	$-0.174770\pi$
$829$	2.79852e10	1.70604	0.853018	+	0.521881i	$0.174770\pi$
$830$	−2.33045e10	−1.41471
$831$	−1.68353e10	−1.01769
$832$	3.02342e10	1.81998
$833$	3.97956e9	0.238549
$834$	9.25054e9	0.552187
$835$	1.80651e10	1.07383
$836$	−2.01873e9	−0.119497
$837$	−3.71363e9	−0.218907
$838$	2.87139e10	1.68554
$839$	2.71170e9	0.158517	0.0792583	−	0.996854i	$-0.474745\pi$
$839$	2.71170e9	0.158517	0.0792583	+	0.996854i	$0.474745\pi$
$840$	−1.77527e10	−1.03345
$841$	1.75898e10	1.01971
$842$	1.15946e10	0.669364
$843$	3.52650e9	0.202744
$844$	−3.88184e8	−0.0222249
$845$	−4.31161e10	−2.45834
$846$	−4.71809e9	−0.267898
$847$	−1.82116e9	−0.102981
$848$	1.57737e10	0.888275
$849$	−5.96359e8	−0.0334449
$850$	1.53515e10	0.857404
$851$	−4.54794e9	−0.252965
$852$	1.36624e7	0.000756814 0
$853$	1.97175e10	1.08775	0.543877	−	0.839165i	$-0.316956\pi$
$853$	1.97175e10	1.08775	0.543877	+	0.839165i	$0.316956\pi$
$854$	−5.66623e9	−0.311309
$855$	1.61903e10	0.885876
$856$	2.19359e10	1.19536
$857$	1.89411e10	1.02795	0.513976	−	0.857804i	$-0.328172\pi$
$857$	1.89411e10	1.02795	0.513976	+	0.857804i	$0.328172\pi$
$858$	−4.65672e9	−0.251695
$859$	−6.77637e9	−0.364772	−0.182386	−	0.983227i	$-0.558382\pi$
$859$	−6.77637e9	−0.364772	−0.182386	+	0.983227i	$0.558382\pi$
$860$	6.91197e9	0.370559
$861$	1.31969e9	0.0704628
$862$	−1.66703e10	−0.886479
$863$	2.80635e10	1.48629	0.743146	−	0.669129i	$-0.233333\pi$
$863$	2.80635e10	1.48629	0.743146	+	0.669129i	$0.233333\pi$
$864$	1.56519e9	0.0825600
$865$	−2.75146e10	−1.44546
$866$	6.34094e9	0.331773
$867$	−3.21912e9	−0.167753
$868$	−5.43073e9	−0.281864
$869$	−3.96324e8	−0.0204871
$870$	2.06626e10	1.06382
$871$	5.95109e9	0.305164
$872$	1.73660e10	0.886937
$873$	1.21390e10	0.617495
$874$	6.22282e9	0.315281
$875$	4.99454e9	0.252039
$876$	3.22103e8	0.0161894
$877$	−1.01559e10	−0.508418	−0.254209	−	0.967149i	$-0.581815\pi$
$877$	−1.01559e10	−0.508418	−0.254209	+	0.967149i	$0.581815\pi$
$878$	−1.22368e10	−0.610151
$879$	5.40507e9	0.268436
$880$	−6.55725e9	−0.324364
$881$	−2.78023e10	−1.36982	−0.684912	−	0.728626i	$-0.740159\pi$
$881$	−2.78023e10	−1.36982	−0.684912	+	0.728626i	$0.740159\pi$
$882$	1.70033e9	0.0834435
$883$	2.15199e10	1.05191	0.525954	−	0.850513i	$-0.323709\pi$
$883$	2.15199e10	1.05191	0.525954	+	0.850513i	$0.323709\pi$
$884$	−6.19050e9	−0.301400
$885$	2.96802e10	1.43935
$886$	−1.23213e10	−0.595167
$887$	−3.13376e10	−1.50776	−0.753881	−	0.657011i	$-0.771820\pi$
$887$	−3.13376e10	−1.50776	−0.753881	+	0.657011i	$0.771820\pi$
$888$	−1.66747e10	−0.799122
$889$	2.15027e10	1.02645
$890$	2.60062e10	1.23655
$891$	−7.07348e8	−0.0335013
$892$	−3.80618e9	−0.179561
$893$	3.50575e10	1.64741
$894$	−1.30197e10	−0.609424
$895$	−1.78321e9	−0.0831423
$896$	−1.35222e10	−0.628015
$897$	−4.01926e9	−0.185940
$898$	−3.07511e10	−1.41708
$899$	3.52164e10	1.61654
$900$	−1.83657e9	−0.0839767
$901$	−2.23977e10	−1.02015
$902$	6.32837e8	0.0287124
$903$	−1.67116e10	−0.755283
$904$	−8.84369e9	−0.398148
$905$	4.92976e9	0.221083
$906$	−1.21147e10	−0.541208
$907$	−1.62459e10	−0.722966	−0.361483	−	0.932379i	$-0.617730\pi$
$907$	−1.62459e10	−0.722966	−0.361483	+	0.932379i	$0.617730\pi$
$908$	−7.90168e8	−0.0350283
$909$	−1.51870e9	−0.0670656
$910$	5.46154e10	2.40254
$911$	−3.15726e9	−0.138356	−0.0691778	−	0.997604i	$-0.522038\pi$
$911$	−3.15726e9	−0.138356	−0.0691778	+	0.997604i	$0.522038\pi$
$912$	1.75738e10	0.767158
$913$	−7.56544e9	−0.328993
$914$	−2.44730e10	−1.06017
$915$	−6.10167e9	−0.263315
$916$	1.48740e9	0.0639432
$917$	1.15803e10	0.495938
$918$	3.35831e9	0.143275
$919$	2.53655e9	0.107805	0.0539026	−	0.998546i	$-0.482834\pi$
$919$	2.53655e9	0.107805	0.0539026	+	0.998546i	$0.482834\pi$
$920$	−7.34772e9	−0.311097
$921$	−1.29529e10	−0.546333
$922$	9.52419e9	0.400193
$923$	−2.34177e8	−0.00980253
$924$	−1.03441e9	−0.0431361
$925$	3.56198e10	1.47978
$926$	−6.05200e9	−0.250473
$927$	−1.74265e9	−0.0718506
$928$	−1.48427e10	−0.609671
$929$	−2.10282e10	−0.860494	−0.430247	−	0.902711i	$-0.641574\pi$
$929$	−2.10282e10	−0.860494	−0.430247	+	0.902711i	$0.641574\pi$
$930$	2.08860e10	0.851461
$931$	−1.26342e10	−0.513126
$932$	−4.32849e9	−0.175138
$933$	−1.40328e10	−0.565665
$934$	−1.37708e10	−0.553025
$935$	9.31090e9	0.372521
$936$	−1.47364e10	−0.587387
$937$	3.12893e10	1.24253	0.621265	−	0.783601i	$-0.286619\pi$
$937$	3.12893e10	1.24253	0.621265	+	0.783601i	$0.286619\pi$
$938$	−4.72119e9	−0.186785
$939$	2.61835e10	1.03204
$940$	−7.42986e9	−0.291765
$941$	7.91706e9	0.309742	0.154871	−	0.987935i	$-0.450504\pi$
$941$	7.91706e9	0.309742	0.154871	+	0.987935i	$0.450504\pi$
$942$	−1.45505e9	−0.0567152
$943$	5.46208e8	0.0212113
$944$	3.22166e10	1.24646
$945$	8.29599e9	0.319784
$946$	−8.01379e9	−0.307765
$947$	8.55849e9	0.327471	0.163735	−	0.986504i	$-0.447646\pi$
$947$	8.55849e9	0.327471	0.163735	+	0.986504i	$0.447646\pi$
$948$	−2.25110e8	−0.00858153
$949$	−5.52091e9	−0.209691
$950$	−4.87377e10	−1.84430
$951$	2.04356e10	0.770471
$952$	2.73620e10	1.02782
$953$	−8.49661e9	−0.317995	−0.158998	−	0.987279i	$-0.550826\pi$
$953$	−8.49661e9	−0.317995	−0.158998	+	0.987279i	$0.550826\pi$
$954$	−9.56974e9	−0.356846
$955$	−2.44524e10	−0.908466
$956$	5.21202e9	0.192931
$957$	6.70778e9	0.247393
$958$	−4.00222e10	−1.47069
$959$	−4.56730e8	−0.0167222
$960$	−2.58291e10	−0.942235
$961$	8.08451e9	0.293847
$962$	5.12989e10	1.85778
$963$	−1.02508e10	−0.369885
$964$	−7.33900e9	−0.263856
$965$	4.02225e10	1.44086
$966$	3.18861e9	0.113810
$967$	−1.47988e10	−0.526300	−0.263150	−	0.964755i	$-0.584761\pi$
$967$	−1.47988e10	−0.526300	−0.263150	+	0.964755i	$0.584761\pi$
$968$	−2.76364e9	−0.0979303
$969$	−2.49538e10	−0.881056
$970$	−6.82715e10	−2.40181
$971$	−2.86157e10	−1.00308	−0.501542	−	0.865133i	$-0.667234\pi$
$971$	−2.86157e10	−1.00308	−0.501542	+	0.865133i	$0.667234\pi$
$972$	−4.01769e8	−0.0140328
$973$	−3.52206e10	−1.22575
$974$	−2.88677e10	−1.00105
$975$	3.14792e10	1.08770
$976$	−6.62310e9	−0.228027
$977$	3.37991e10	1.15951	0.579755	−	0.814791i	$-0.303148\pi$
$977$	3.37991e10	1.15951	0.579755	+	0.814791i	$0.303148\pi$
$978$	2.65052e10	0.906036
$979$	8.44249e9	0.287562
$980$	2.67761e9	0.0908773
$981$	−8.11528e9	−0.274449
$982$	1.19743e9	0.0403515
$983$	1.03134e9	0.0346308	0.0173154	−	0.999850i	$-0.494488\pi$
$983$	1.03134e9	0.0346308	0.0173154	+	0.999850i	$0.494488\pi$
$984$	2.00264e9	0.0670069
$985$	4.48199e10	1.49432
$986$	−3.18469e10	−1.05803
$987$	1.79637e10	0.594683
$988$	1.96535e10	0.648320
$989$	−6.91679e9	−0.227362
$990$	3.97823e9	0.130307
$991$	−5.63139e10	−1.83805	−0.919027	−	0.394195i	$-0.871023\pi$
$991$	−5.63139e10	−1.83805	−0.919027	+	0.394195i	$0.871023\pi$
$992$	−1.50032e10	−0.487970
$993$	−4.85642e9	−0.157396
$994$	1.85780e8	0.00599994
$995$	1.49370e10	0.480710
$996$	−4.29713e9	−0.137807
$997$	2.55531e10	0.816603	0.408301	−	0.912847i	$-0.366121\pi$
$997$	2.55531e10	0.816603	0.408301	+	0.912847i	$0.366121\pi$
$998$	−4.78950e10	−1.52522
$999$	7.79222e9	0.247276

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.a.a.1.1	✓	1
3.2	odd	2		99.8.a.a.1.1		1
4.3	odd	2		528.8.a.a.1.1		1
11.10	odd	2		363.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.a.a.1.1	✓	1	1.1	even	1	trivial
99.8.a.a.1.1		1	3.2	odd	2
363.8.a.a.1.1		1	11.10	odd	2
528.8.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:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more