LMFDB - Embedded newform 8033.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8033,2,Mod(1,8033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8033, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8033 = 29 \cdot 277 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8033.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:	$64.1438279437$
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	$\chi$	$=$	8033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} -3.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{14} -3.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} -2.00000 q^{18} -4.00000 q^{19} +3.00000 q^{20} +2.00000 q^{21} -3.00000 q^{22} -4.00000 q^{23} -3.00000 q^{24} +4.00000 q^{25} +1.00000 q^{26} -5.00000 q^{27} -2.00000 q^{28} +1.00000 q^{29} -3.00000 q^{30} -9.00000 q^{31} +5.00000 q^{32} -3.00000 q^{33} -2.00000 q^{34} -6.00000 q^{35} +2.00000 q^{36} +8.00000 q^{37} -4.00000 q^{38} +1.00000 q^{39} +9.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} +13.0000 q^{43} +3.00000 q^{44} +6.00000 q^{45} -4.00000 q^{46} -7.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} -2.00000 q^{51} -1.00000 q^{52} +9.00000 q^{53} -5.00000 q^{54} +9.00000 q^{55} -6.00000 q^{56} -4.00000 q^{57} +1.00000 q^{58} +3.00000 q^{60} +4.00000 q^{61} -9.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} -3.00000 q^{65} -3.00000 q^{66} +2.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} -6.00000 q^{70} -12.0000 q^{71} +6.00000 q^{72} +14.0000 q^{73} +8.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} -6.00000 q^{77} +1.00000 q^{78} +3.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +10.0000 q^{83} -2.00000 q^{84} +6.00000 q^{85} +13.0000 q^{86} +1.00000 q^{87} +9.00000 q^{88} -12.0000 q^{89} +6.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} -9.00000 q^{93} -7.00000 q^{94} +12.0000 q^{95} +5.00000 q^{96} +14.0000 q^{97} -3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	−1.00000	−0.500000
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	1.00000	0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−3.00000	−1.06066
$9$	−2.00000	−0.666667
$10$	−3.00000	−0.948683
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−1.00000	−0.288675
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	2.00000	0.534522
$15$	−3.00000	−0.774597
$16$	−1.00000	−0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−2.00000	−0.471405
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	3.00000	0.670820
$21$	2.00000	0.436436
$22$	−3.00000	−0.639602
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−3.00000	−0.612372
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	−5.00000	−0.962250
$28$	−2.00000	−0.377964
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−3.00000	−0.547723
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	5.00000	0.883883
$33$	−3.00000	−0.522233
$34$	−2.00000	−0.342997
$35$	−6.00000	−1.01419
$36$	2.00000	0.333333
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−4.00000	−0.648886
$39$	1.00000	0.160128
$40$	9.00000	1.42302
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000	0.308607
$43$	13.0000	1.98248	0.991241	−	0.132068i	$-0.0421616\pi$
$43$	13.0000	1.98248	0.991241	+	0.132068i	$0.0421616\pi$
$44$	3.00000	0.452267
$45$	6.00000	0.894427
$46$	−4.00000	−0.589768
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	4.00000	0.565685
$51$	−2.00000	−0.280056
$52$	−1.00000	−0.138675
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−5.00000	−0.680414
$55$	9.00000	1.21356
$56$	−6.00000	−0.801784
$57$	−4.00000	−0.529813
$58$	1.00000	0.131306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	3.00000	0.387298
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	−9.00000	−1.14300
$63$	−4.00000	−0.503953
$64$	7.00000	0.875000
$65$	−3.00000	−0.372104
$66$	−3.00000	−0.369274
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	2.00000	0.242536
$69$	−4.00000	−0.481543
$70$	−6.00000	−0.717137
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	6.00000	0.707107
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	8.00000	0.929981
$75$	4.00000	0.461880
$76$	4.00000	0.458831
$77$	−6.00000	−0.683763
$78$	1.00000	0.113228
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	−2.00000	−0.218218
$85$	6.00000	0.650791
$86$	13.0000	1.40183
$87$	1.00000	0.107211
$88$	9.00000	0.959403
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	6.00000	0.632456
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	−9.00000	−0.933257
$94$	−7.00000	−0.721995
$95$	12.0000	1.23117
$96$	5.00000	0.510310
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	−3.00000	−0.303046
$99$	6.00000	0.603023
$100$	−4.00000	−0.400000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	−2.00000	−0.198030
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	−3.00000	−0.294174
$105$	−6.00000	−0.585540
$106$	9.00000	0.874157
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	5.00000	0.481125
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	9.00000	0.858116
$111$	8.00000	0.759326
$112$	−2.00000	−0.188982
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	−4.00000	−0.374634
$115$	12.0000	1.11901
$116$	−1.00000	−0.0928477
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−4.00000	−0.366679
$120$	9.00000	0.821584
$121$	−2.00000	−0.181818
$122$	4.00000	0.362143
$123$	−6.00000	−0.541002
$124$	9.00000	0.808224
$125$	3.00000	0.268328
$126$	−4.00000	−0.356348
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−3.00000	−0.265165
$129$	13.0000	1.14459
$130$	−3.00000	−0.263117
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000	0.261116
$133$	−8.00000	−0.693688
$134$	2.00000	0.172774
$135$	15.0000	1.29099
$136$	6.00000	0.514496
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	−4.00000	−0.340503
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	6.00000	0.507093
$141$	−7.00000	−0.589506
$142$	−12.0000	−1.00702
$143$	−3.00000	−0.250873
$144$	2.00000	0.166667
$145$	−3.00000	−0.249136
$146$	14.0000	1.15865
$147$	−3.00000	−0.247436
$148$	−8.00000	−0.657596
$149$	−13.0000	−1.06500	−0.532501	−	0.846430i	$-0.678748\pi$
$149$	−13.0000	−1.06500	−0.532501	+	0.846430i	$0.678748\pi$
$150$	4.00000	0.326599
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	12.0000	0.973329
$153$	4.00000	0.323381
$154$	−6.00000	−0.483494
$155$	27.0000	2.16869
$156$	−1.00000	−0.0800641
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	3.00000	0.238667
$159$	9.00000	0.713746
$160$	−15.0000	−1.18585
$161$	−8.00000	−0.630488
$162$	1.00000	0.0785674
$163$	−3.00000	−0.234978	−0.117489	−	0.993074i	$-0.537485\pi$
$163$	−3.00000	−0.234978	−0.117489	+	0.993074i	$0.537485\pi$
$164$	6.00000	0.468521
$165$	9.00000	0.700649
$166$	10.0000	0.776151
$167$	22.0000	1.70241	0.851206	−	0.524832i	$-0.175872\pi$
$167$	22.0000	1.70241	0.851206	+	0.524832i	$0.175872\pi$
$168$	−6.00000	−0.462910
$169$	−12.0000	−0.923077
$170$	6.00000	0.460179
$171$	8.00000	0.611775
$172$	−13.0000	−0.991241
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	1.00000	0.0758098
$175$	8.00000	0.604743
$176$	3.00000	0.226134
$177$	0	0
$178$	−12.0000	−0.899438
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	−6.00000	−0.447214
$181$	−17.0000	−1.26360	−0.631800	−	0.775131i	$-0.717684\pi$
$181$	−17.0000	−1.26360	−0.631800	+	0.775131i	$0.717684\pi$
$182$	2.00000	0.148250
$183$	4.00000	0.295689
$184$	12.0000	0.884652
$185$	−24.0000	−1.76452
$186$	−9.00000	−0.659912
$187$	6.00000	0.438763
$188$	7.00000	0.510527
$189$	−10.0000	−0.727393
$190$	12.0000	0.870572
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	7.00000	0.505181
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	14.0000	1.00514
$195$	−3.00000	−0.214834
$196$	3.00000	0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	6.00000	0.426401
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	−12.0000	−0.848528
$201$	2.00000	0.141069
$202$	2.00000	0.140720
$203$	2.00000	0.140372
$204$	2.00000	0.140028
$205$	18.0000	1.25717
$206$	−14.0000	−0.975426
$207$	8.00000	0.556038
$208$	−1.00000	−0.0693375
$209$	12.0000	0.830057
$210$	−6.00000	−0.414039
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	−9.00000	−0.618123
$213$	−12.0000	−0.822226
$214$	−6.00000	−0.410152
$215$	−39.0000	−2.65978
$216$	15.0000	1.02062
$217$	−18.0000	−1.22192
$218$	1.00000	0.0677285
$219$	14.0000	0.946032
$220$	−9.00000	−0.606780
$221$	−2.00000	−0.134535
$222$	8.00000	0.536925
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	10.0000	0.668153
$225$	−8.00000	−0.533333
$226$	−4.00000	−0.266076
$227$	−10.0000	−0.663723	−0.331862	−	0.943328i	$-0.607677\pi$
$227$	−10.0000	−0.663723	−0.331862	+	0.943328i	$0.607677\pi$
$228$	4.00000	0.264906
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	12.0000	0.791257
$231$	−6.00000	−0.394771
$232$	−3.00000	−0.196960
$233$	5.00000	0.327561	0.163780	−	0.986497i	$-0.447631\pi$
$233$	5.00000	0.327561	0.163780	+	0.986497i	$0.447631\pi$
$234$	−2.00000	−0.130744
$235$	21.0000	1.36989
$236$	0	0
$237$	3.00000	0.194871
$238$	−4.00000	−0.259281
$239$	26.0000	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$239$	26.0000	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$240$	3.00000	0.193649
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	−2.00000	−0.128565
$243$	16.0000	1.02640
$244$	−4.00000	−0.256074
$245$	9.00000	0.574989
$246$	−6.00000	−0.382546
$247$	−4.00000	−0.254514
$248$	27.0000	1.71450
$249$	10.0000	0.633724
$250$	3.00000	0.189737
$251$	19.0000	1.19927	0.599635	−	0.800274i	$-0.295313\pi$
$251$	19.0000	1.19927	0.599635	+	0.800274i	$0.295313\pi$
$252$	4.00000	0.251976
$253$	12.0000	0.754434
$254$	8.00000	0.501965
$255$	6.00000	0.375735
$256$	−17.0000	−1.06250
$257$	19.0000	1.18519	0.592594	−	0.805502i	$-0.298104\pi$
$257$	19.0000	1.18519	0.592594	+	0.805502i	$0.298104\pi$
$258$	13.0000	0.809345
$259$	16.0000	0.994192
$260$	3.00000	0.186052
$261$	−2.00000	−0.123797
$262$	12.0000	0.741362
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	9.00000	0.553912
$265$	−27.0000	−1.65860
$266$	−8.00000	−0.490511
$267$	−12.0000	−0.734388
$268$	−2.00000	−0.122169
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	15.0000	0.912871
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	2.00000	0.121268
$273$	2.00000	0.121046
$274$	−22.0000	−1.32907
$275$	−12.0000	−0.723627
$276$	4.00000	0.240772
$277$	−1.00000	−0.0600842
$277$	−1.00000	−0.0600842
$278$	2.00000	0.119952
$279$	18.0000	1.07763
$280$	18.0000	1.07571
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	−7.00000	−0.416844
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	12.0000	0.712069
$285$	12.0000	0.710819
$286$	−3.00000	−0.177394
$287$	−12.0000	−0.708338
$288$	−10.0000	−0.589256
$289$	−13.0000	−0.764706
$290$	−3.00000	−0.176166
$291$	14.0000	0.820695
$292$	−14.0000	−0.819288
$293$	8.00000	0.467365	0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000	0.467365	0.233682	+	0.972313i	$0.424922\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	−24.0000	−1.39497
$297$	15.0000	0.870388
$298$	−13.0000	−0.753070
$299$	−4.00000	−0.231326
$300$	−4.00000	−0.230940
$301$	26.0000	1.49862
$302$	20.0000	1.15087
$303$	2.00000	0.114897
$304$	4.00000	0.229416
$305$	−12.0000	−0.687118
$306$	4.00000	0.228665
$307$	−29.0000	−1.65512	−0.827559	−	0.561379i	$-0.810271\pi$
$307$	−29.0000	−1.65512	−0.827559	+	0.561379i	$0.810271\pi$
$308$	6.00000	0.341882
$309$	−14.0000	−0.796432
$310$	27.0000	1.53350
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−3.00000	−0.169842
$313$	13.0000	0.734803	0.367402	−	0.930062i	$-0.380247\pi$
$313$	13.0000	0.734803	0.367402	+	0.930062i	$0.380247\pi$
$314$	10.0000	0.564333
$315$	12.0000	0.676123
$316$	−3.00000	−0.168763
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	9.00000	0.504695
$319$	−3.00000	−0.167968
$320$	−21.0000	−1.17394
$321$	−6.00000	−0.334887
$322$	−8.00000	−0.445823
$323$	8.00000	0.445132
$324$	−1.00000	−0.0555556
$325$	4.00000	0.221880
$326$	−3.00000	−0.166155
$327$	1.00000	0.0553001
$328$	18.0000	0.993884
$329$	−14.0000	−0.771845
$330$	9.00000	0.495434
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	−10.0000	−0.548821
$333$	−16.0000	−0.876795
$334$	22.0000	1.20379
$335$	−6.00000	−0.327815
$336$	−2.00000	−0.109109
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−12.0000	−0.652714
$339$	−4.00000	−0.217250
$340$	−6.00000	−0.325396
$341$	27.0000	1.46213
$342$	8.00000	0.432590
$343$	−20.0000	−1.07990
$344$	−39.0000	−2.10274
$345$	12.0000	0.646058
$346$	−6.00000	−0.322562
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	−1.00000	−0.0536056
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	8.00000	0.427618
$351$	−5.00000	−0.266880
$352$	−15.0000	−0.799503
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	12.0000	0.635999
$357$	−4.00000	−0.211702
$358$	6.00000	0.317110
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	−18.0000	−0.948683
$361$	−3.00000	−0.157895
$362$	−17.0000	−0.893500
$363$	−2.00000	−0.104973
$364$	−2.00000	−0.104828
$365$	−42.0000	−2.19838
$366$	4.00000	0.209083
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	4.00000	0.208514
$369$	12.0000	0.624695
$370$	−24.0000	−1.24770
$371$	18.0000	0.934513
$372$	9.00000	0.466628
$373$	−29.0000	−1.50156	−0.750782	−	0.660551i	$-0.770323\pi$
$373$	−29.0000	−1.50156	−0.750782	+	0.660551i	$0.770323\pi$
$374$	6.00000	0.310253
$375$	3.00000	0.154919
$376$	21.0000	1.08299
$377$	1.00000	0.0515026
$378$	−10.0000	−0.514344
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−12.0000	−0.615587
$381$	8.00000	0.409852
$382$	−24.0000	−1.22795
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	−3.00000	−0.153093
$385$	18.0000	0.917365
$386$	−10.0000	−0.508987
$387$	−26.0000	−1.32165
$388$	−14.0000	−0.710742
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	−3.00000	−0.151911
$391$	8.00000	0.404577
$392$	9.00000	0.454569
$393$	12.0000	0.605320
$394$	2.00000	0.100759
$395$	−9.00000	−0.452839
$396$	−6.00000	−0.301511
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	22.0000	1.10276
$399$	−8.00000	−0.400501
$400$	−4.00000	−0.200000
$401$	17.0000	0.848939	0.424470	−	0.905442i	$-0.360461\pi$
$401$	17.0000	0.848939	0.424470	+	0.905442i	$0.360461\pi$
$402$	2.00000	0.0997509
$403$	−9.00000	−0.448322
$404$	−2.00000	−0.0995037
$405$	−3.00000	−0.149071
$406$	2.00000	0.0992583
$407$	−24.0000	−1.18964
$408$	6.00000	0.297044
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	18.0000	0.888957
$411$	−22.0000	−1.08518
$412$	14.0000	0.689730
$413$	0	0
$414$	8.00000	0.393179
$415$	−30.0000	−1.47264
$416$	5.00000	0.245145
$417$	2.00000	0.0979404
$418$	12.0000	0.586939
$419$	−34.0000	−1.66101	−0.830504	−	0.557012i	$-0.811948\pi$
$419$	−34.0000	−1.66101	−0.830504	+	0.557012i	$0.811948\pi$
$420$	6.00000	0.292770
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	15.0000	0.730189
$423$	14.0000	0.680703
$424$	−27.0000	−1.31124
$425$	−8.00000	−0.388057
$426$	−12.0000	−0.581402
$427$	8.00000	0.387147
$428$	6.00000	0.290021
$429$	−3.00000	−0.144841
$430$	−39.0000	−1.88075
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	5.00000	0.240563
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	−18.0000	−0.864028
$435$	−3.00000	−0.143839
$436$	−1.00000	−0.0478913
$437$	16.0000	0.765384
$438$	14.0000	0.668946
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	−27.0000	−1.28717
$441$	6.00000	0.285714
$442$	−2.00000	−0.0951303
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−8.00000	−0.379663
$445$	36.0000	1.70656
$446$	16.0000	0.757622
$447$	−13.0000	−0.614879
$448$	14.0000	0.661438
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−8.00000	−0.377124
$451$	18.0000	0.847587
$452$	4.00000	0.188144
$453$	20.0000	0.939682
$454$	−10.0000	−0.469323
$455$	−6.00000	−0.281284
$456$	12.0000	0.561951
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	10.0000	0.466760
$460$	−12.0000	−0.559503
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	−6.00000	−0.279145
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−1.00000	−0.0464238
$465$	27.0000	1.25210
$466$	5.00000	0.231621
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	2.00000	0.0924500
$469$	4.00000	0.184703
$470$	21.0000	0.968658
$471$	10.0000	0.460776
$472$	0	0
$473$	−39.0000	−1.79322
$474$	3.00000	0.137795
$475$	−16.0000	−0.734130
$476$	4.00000	0.183340
$477$	−18.0000	−0.824163
$478$	26.0000	1.18921
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	−15.0000	−0.684653
$481$	8.00000	0.364769
$482$	1.00000	0.0455488
$483$	−8.00000	−0.364013
$484$	2.00000	0.0909091
$485$	−42.0000	−1.90712
$486$	16.0000	0.725775
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	−12.0000	−0.543214
$489$	−3.00000	−0.135665
$490$	9.00000	0.406579
$491$	1.00000	0.0451294	0.0225647	−	0.999745i	$-0.492817\pi$
$491$	1.00000	0.0451294	0.0225647	+	0.999745i	$0.492817\pi$
$492$	6.00000	0.270501
$493$	−2.00000	−0.0900755
$494$	−4.00000	−0.179969
$495$	−18.0000	−0.809040
$496$	9.00000	0.404112
$497$	−24.0000	−1.07655
$498$	10.0000	0.448111
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	−3.00000	−0.134164
$501$	22.0000	0.982888
$502$	19.0000	0.848012
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	12.0000	0.534522
$505$	−6.00000	−0.266996
$506$	12.0000	0.533465
$507$	−12.0000	−0.532939
$508$	−8.00000	−0.354943
$509$	45.0000	1.99459	0.997295	−	0.0735034i	$-0.0234180\pi$
$509$	45.0000	1.99459	0.997295	+	0.0735034i	$0.0234180\pi$
$510$	6.00000	0.265684
$511$	28.0000	1.23865
$512$	−11.0000	−0.486136
$513$	20.0000	0.883022
$514$	19.0000	0.838054
$515$	42.0000	1.85074
$516$	−13.0000	−0.572293
$517$	21.0000	0.923579
$518$	16.0000	0.703000
$519$	−6.00000	−0.263371
$520$	9.00000	0.394676
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	−2.00000	−0.0875376
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−12.0000	−0.524222
$525$	8.00000	0.349149
$526$	−21.0000	−0.915644
$527$	18.0000	0.784092
$528$	3.00000	0.130558
$529$	−7.00000	−0.304348
$530$	−27.0000	−1.17281
$531$	0	0
$532$	8.00000	0.346844
$533$	−6.00000	−0.259889
$534$	−12.0000	−0.519291
$535$	18.0000	0.778208
$536$	−6.00000	−0.259161
$537$	6.00000	0.258919
$538$	20.0000	0.862261
$539$	9.00000	0.387657
$540$	−15.0000	−0.645497
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−11.0000	−0.472490
$543$	−17.0000	−0.729540
$544$	−10.0000	−0.428746
$545$	−3.00000	−0.128506
$546$	2.00000	0.0855921
$547$	18.0000	0.769624	0.384812	−	0.922995i	$-0.374266\pi$
$547$	18.0000	0.769624	0.384812	+	0.922995i	$0.374266\pi$
$548$	22.0000	0.939793
$549$	−8.00000	−0.341432
$550$	−12.0000	−0.511682
$551$	−4.00000	−0.170406
$552$	12.0000	0.510754
$553$	6.00000	0.255146
$554$	−1.00000	−0.0424859
$555$	−24.0000	−1.01874
$556$	−2.00000	−0.0848189
$557$	−46.0000	−1.94908	−0.974541	−	0.224208i	$-0.928020\pi$
$557$	−46.0000	−1.94908	−0.974541	+	0.224208i	$0.928020\pi$
$558$	18.0000	0.762001
$559$	13.0000	0.549841
$560$	6.00000	0.253546
$561$	6.00000	0.253320
$562$	27.0000	1.13893
$563$	−9.00000	−0.379305	−0.189652	−	0.981851i	$-0.560736\pi$
$563$	−9.00000	−0.379305	−0.189652	+	0.981851i	$0.560736\pi$
$564$	7.00000	0.294753
$565$	12.0000	0.504844
$566$	14.0000	0.588464
$567$	2.00000	0.0839921
$568$	36.0000	1.51053
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	12.0000	0.502625
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	3.00000	0.125436
$573$	−24.0000	−1.00261
$574$	−12.0000	−0.500870
$575$	−16.0000	−0.667246
$576$	−14.0000	−0.583333
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−13.0000	−0.540729
$579$	−10.0000	−0.415586
$580$	3.00000	0.124568
$581$	20.0000	0.829740
$582$	14.0000	0.580319
$583$	−27.0000	−1.11823
$584$	−42.0000	−1.73797
$585$	6.00000	0.248069
$586$	8.00000	0.330477
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	3.00000	0.123718
$589$	36.0000	1.48335
$590$	0	0
$591$	2.00000	0.0822690
$592$	−8.00000	−0.328798
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	15.0000	0.615457
$595$	12.0000	0.491952
$596$	13.0000	0.532501
$597$	22.0000	0.900400
$598$	−4.00000	−0.163572
$599$	−31.0000	−1.26663	−0.633313	−	0.773896i	$-0.718305\pi$
$599$	−31.0000	−1.26663	−0.633313	+	0.773896i	$0.718305\pi$
$600$	−12.0000	−0.489898
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	26.0000	1.05968
$603$	−4.00000	−0.162893
$604$	−20.0000	−0.813788
$605$	6.00000	0.243935
$606$	2.00000	0.0812444
$607$	25.0000	1.01472	0.507359	−	0.861735i	$-0.330622\pi$
$607$	25.0000	1.01472	0.507359	+	0.861735i	$0.330622\pi$
$608$	−20.0000	−0.811107
$609$	2.00000	0.0810441
$610$	−12.0000	−0.485866
$611$	−7.00000	−0.283190
$612$	−4.00000	−0.161690
$613$	−13.0000	−0.525065	−0.262533	−	0.964923i	$-0.584558\pi$
$613$	−13.0000	−0.525065	−0.262533	+	0.964923i	$0.584558\pi$
$614$	−29.0000	−1.17034
$615$	18.0000	0.725830
$616$	18.0000	0.725241
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	−14.0000	−0.563163
$619$	17.0000	0.683288	0.341644	−	0.939829i	$-0.389016\pi$
$619$	17.0000	0.683288	0.341644	+	0.939829i	$0.389016\pi$
$620$	−27.0000	−1.08435
$621$	20.0000	0.802572
$622$	12.0000	0.481156
$623$	−24.0000	−0.961540
$624$	−1.00000	−0.0400320
$625$	−29.0000	−1.16000
$626$	13.0000	0.519584
$627$	12.0000	0.479234
$628$	−10.0000	−0.399043
$629$	−16.0000	−0.637962
$630$	12.0000	0.478091
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−9.00000	−0.358001
$633$	15.0000	0.596196
$634$	18.0000	0.714871
$635$	−24.0000	−0.952411
$636$	−9.00000	−0.356873
$637$	−3.00000	−0.118864
$638$	−3.00000	−0.118771
$639$	24.0000	0.949425
$640$	9.00000	0.355756
$641$	20.0000	0.789953	0.394976	−	0.918691i	$-0.370753\pi$
$641$	20.0000	0.789953	0.394976	+	0.918691i	$0.370753\pi$
$642$	−6.00000	−0.236801
$643$	−2.00000	−0.0788723	−0.0394362	−	0.999222i	$-0.512556\pi$
$643$	−2.00000	−0.0788723	−0.0394362	+	0.999222i	$0.512556\pi$
$644$	8.00000	0.315244
$645$	−39.0000	−1.53562
$646$	8.00000	0.314756
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	−3.00000	−0.117851
$649$	0	0
$650$	4.00000	0.156893
$651$	−18.0000	−0.705476
$652$	3.00000	0.117489
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	1.00000	0.0391031
$655$	−36.0000	−1.40664
$656$	6.00000	0.234261
$657$	−28.0000	−1.09238
$658$	−14.0000	−0.545777
$659$	19.0000	0.740135	0.370067	−	0.929005i	$-0.379335\pi$
$659$	19.0000	0.740135	0.370067	+	0.929005i	$0.379335\pi$
$660$	−9.00000	−0.350325
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	1.00000	0.0388661
$663$	−2.00000	−0.0776736
$664$	−30.0000	−1.16423
$665$	24.0000	0.930680
$666$	−16.0000	−0.619987
$667$	−4.00000	−0.154881
$668$	−22.0000	−0.851206
$669$	16.0000	0.618596
$670$	−6.00000	−0.231800
$671$	−12.0000	−0.463255
$672$	10.0000	0.385758
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	14.0000	0.539260
$675$	−20.0000	−0.769800
$676$	12.0000	0.461538
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	−4.00000	−0.153619
$679$	28.0000	1.07454
$680$	−18.0000	−0.690268
$681$	−10.0000	−0.383201
$682$	27.0000	1.03388
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−8.00000	−0.305888
$685$	66.0000	2.52173
$686$	−20.0000	−0.763604
$687$	0	0
$688$	−13.0000	−0.495620
$689$	9.00000	0.342873
$690$	12.0000	0.456832
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000	0.228086
$693$	12.0000	0.455842
$694$	16.0000	0.607352
$695$	−6.00000	−0.227593
$696$	−3.00000	−0.113715
$697$	12.0000	0.454532
$698$	5.00000	0.189253
$699$	5.00000	0.189117
$700$	−8.00000	−0.302372
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	−5.00000	−0.188713
$703$	−32.0000	−1.20690
$704$	−21.0000	−0.791467
$705$	21.0000	0.790906
$706$	14.0000	0.526897
$707$	4.00000	0.150435
$708$	0	0
$709$	−23.0000	−0.863783	−0.431892	−	0.901926i	$-0.642154\pi$
$709$	−23.0000	−0.863783	−0.431892	+	0.901926i	$0.642154\pi$
$710$	36.0000	1.35106
$711$	−6.00000	−0.225018
$712$	36.0000	1.34916
$713$	36.0000	1.34821
$714$	−4.00000	−0.149696
$715$	9.00000	0.336581
$716$	−6.00000	−0.224231
$717$	26.0000	0.970988
$718$	25.0000	0.932992
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	−6.00000	−0.223607
$721$	−28.0000	−1.04277
$722$	−3.00000	−0.111648
$723$	1.00000	0.0371904
$724$	17.0000	0.631800
$725$	4.00000	0.148556
$726$	−2.00000	−0.0742270
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	−6.00000	−0.222375
$729$	13.0000	0.481481
$730$	−42.0000	−1.55449
$731$	−26.0000	−0.961645
$732$	−4.00000	−0.147844
$733$	10.0000	0.369358	0.184679	−	0.982799i	$-0.440875\pi$
$733$	10.0000	0.369358	0.184679	+	0.982799i	$0.440875\pi$
$734$	−16.0000	−0.590571
$735$	9.00000	0.331970
$736$	−20.0000	−0.737210
$737$	−6.00000	−0.221013
$738$	12.0000	0.441726
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	24.0000	0.882258
$741$	−4.00000	−0.146944
$742$	18.0000	0.660801
$743$	−48.0000	−1.76095	−0.880475	−	0.474093i	$-0.842776\pi$
$743$	−48.0000	−1.76095	−0.880475	+	0.474093i	$0.842776\pi$
$744$	27.0000	0.989868
$745$	39.0000	1.42885
$746$	−29.0000	−1.06177
$747$	−20.0000	−0.731762
$748$	−6.00000	−0.219382
$749$	−12.0000	−0.438470
$750$	3.00000	0.109545
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	7.00000	0.255264
$753$	19.0000	0.692398
$754$	1.00000	0.0364179
$755$	−60.0000	−2.18362
$756$	10.0000	0.363696
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	4.00000	0.145287
$759$	12.0000	0.435572
$760$	−36.0000	−1.30586
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	8.00000	0.289809
$763$	2.00000	0.0724049
$764$	24.0000	0.868290
$765$	−12.0000	−0.433861
$766$	−20.0000	−0.722629
$767$	0	0
$768$	−17.0000	−0.613435
$769$	−8.00000	−0.288487	−0.144244	−	0.989542i	$-0.546075\pi$
$769$	−8.00000	−0.288487	−0.144244	+	0.989542i	$0.546075\pi$
$770$	18.0000	0.648675
$771$	19.0000	0.684268
$772$	10.0000	0.359908
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	−26.0000	−0.934551
$775$	−36.0000	−1.29316
$776$	−42.0000	−1.50771
$777$	16.0000	0.573997
$778$	−18.0000	−0.645331
$779$	24.0000	0.859889
$780$	3.00000	0.107417
$781$	36.0000	1.28818
$782$	8.00000	0.286079
$783$	−5.00000	−0.178685
$784$	3.00000	0.107143
$785$	−30.0000	−1.07075
$786$	12.0000	0.428026
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	−2.00000	−0.0712470
$789$	−21.0000	−0.747620
$790$	−9.00000	−0.320206
$791$	−8.00000	−0.284447
$792$	−18.0000	−0.639602
$793$	4.00000	0.142044
$794$	−7.00000	−0.248421
$795$	−27.0000	−0.957591
$796$	−22.0000	−0.779769
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	−8.00000	−0.283197
$799$	14.0000	0.495284
$800$	20.0000	0.707107
$801$	24.0000	0.847998
$802$	17.0000	0.600291
$803$	−42.0000	−1.48215
$804$	−2.00000	−0.0705346
$805$	24.0000	0.845889
$806$	−9.00000	−0.317011
$807$	20.0000	0.704033
$808$	−6.00000	−0.211079
$809$	−52.0000	−1.82822	−0.914111	−	0.405463i	$-0.867110\pi$
$809$	−52.0000	−1.82822	−0.914111	+	0.405463i	$0.867110\pi$
$810$	−3.00000	−0.105409
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	−2.00000	−0.0701862
$813$	−11.0000	−0.385787
$814$	−24.0000	−0.841200
$815$	9.00000	0.315256
$816$	2.00000	0.0700140
$817$	−52.0000	−1.81925
$818$	−14.0000	−0.489499
$819$	−4.00000	−0.139771
$820$	−18.0000	−0.628587
$821$	21.0000	0.732905	0.366453	−	0.930437i	$-0.380572\pi$
$821$	21.0000	0.732905	0.366453	+	0.930437i	$0.380572\pi$
$822$	−22.0000	−0.767338
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	42.0000	1.46314
$825$	−12.0000	−0.417786
$826$	0	0
$827$	31.0000	1.07798	0.538988	−	0.842314i	$-0.318807\pi$
$827$	31.0000	1.07798	0.538988	+	0.842314i	$0.318807\pi$
$828$	−8.00000	−0.278019
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	−30.0000	−1.04132
$831$	−1.00000	−0.0346896
$832$	7.00000	0.242681
$833$	6.00000	0.207888
$834$	2.00000	0.0692543
$835$	−66.0000	−2.28402
$836$	−12.0000	−0.415029
$837$	45.0000	1.55543
$838$	−34.0000	−1.17451
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	18.0000	0.621059
$841$	1.00000	0.0344828
$842$	34.0000	1.17172
$843$	27.0000	0.929929
$844$	−15.0000	−0.516321
$845$	36.0000	1.23844
$846$	14.0000	0.481330
$847$	−4.00000	−0.137442
$848$	−9.00000	−0.309061
$849$	14.0000	0.480479
$850$	−8.00000	−0.274398
$851$	−32.0000	−1.09695
$852$	12.0000	0.411113
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	8.00000	0.273754
$855$	−24.0000	−0.820783
$856$	18.0000	0.615227
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	−3.00000	−0.102418
$859$	49.0000	1.67186	0.835929	−	0.548837i	$-0.184929\pi$
$859$	49.0000	1.67186	0.835929	+	0.548837i	$0.184929\pi$
$860$	39.0000	1.32989
$861$	−12.0000	−0.408959
$862$	12.0000	0.408722
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	−25.0000	−0.850517
$865$	18.0000	0.612018
$866$	22.0000	0.747590
$867$	−13.0000	−0.441503
$868$	18.0000	0.610960
$869$	−9.00000	−0.305304
$870$	−3.00000	−0.101710
$871$	2.00000	0.0677674
$872$	−3.00000	−0.101593
$873$	−28.0000	−0.947656
$874$	16.0000	0.541208
$875$	6.00000	0.202837
$876$	−14.0000	−0.473016
$877$	−11.0000	−0.371444	−0.185722	−	0.982602i	$-0.559462\pi$
$877$	−11.0000	−0.371444	−0.185722	+	0.982602i	$0.559462\pi$
$878$	4.00000	0.134993
$879$	8.00000	0.269833
$880$	−9.00000	−0.303390
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	6.00000	0.202031
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−36.0000	−1.20944
$887$	55.0000	1.84672	0.923360	−	0.383936i	$-0.125432\pi$
$887$	55.0000	1.84672	0.923360	+	0.383936i	$0.125432\pi$
$888$	−24.0000	−0.805387
$889$	16.0000	0.536623
$890$	36.0000	1.20672
$891$	−3.00000	−0.100504
$892$	−16.0000	−0.535720
$893$	28.0000	0.936984
$894$	−13.0000	−0.434785
$895$	−18.0000	−0.601674
$896$	−6.00000	−0.200446
$897$	−4.00000	−0.133556
$898$	6.00000	0.200223
$899$	−9.00000	−0.300167
$900$	8.00000	0.266667
$901$	−18.0000	−0.599667
$902$	18.0000	0.599334
$903$	26.0000	0.865226
$904$	12.0000	0.399114
$905$	51.0000	1.69530
$906$	20.0000	0.664455
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	10.0000	0.331862
$909$	−4.00000	−0.132672
$910$	−6.00000	−0.198898
$911$	5.00000	0.165657	0.0828287	−	0.996564i	$-0.473605\pi$
$911$	5.00000	0.165657	0.0828287	+	0.996564i	$0.473605\pi$
$912$	4.00000	0.132453
$913$	−30.0000	−0.992855
$914$	18.0000	0.595387
$915$	−12.0000	−0.396708
$916$	0	0
$917$	24.0000	0.792550
$918$	10.0000	0.330049
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	−36.0000	−1.18688
$921$	−29.0000	−0.955582
$922$	−28.0000	−0.922131
$923$	−12.0000	−0.394985
$924$	6.00000	0.197386
$925$	32.0000	1.05215
$926$	−40.0000	−1.31448
$927$	28.0000	0.919641
$928$	5.00000	0.164133
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	27.0000	0.885365
$931$	12.0000	0.393284
$932$	−5.00000	−0.163780
$933$	12.0000	0.392862
$934$	27.0000	0.883467
$935$	−18.0000	−0.588663
$936$	6.00000	0.196116
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	4.00000	0.130605
$939$	13.0000	0.424239
$940$	−21.0000	−0.684944
$941$	−3.00000	−0.0977972	−0.0488986	−	0.998804i	$-0.515571\pi$
$941$	−3.00000	−0.0977972	−0.0488986	+	0.998804i	$0.515571\pi$
$942$	10.0000	0.325818
$943$	24.0000	0.781548
$944$	0	0
$945$	30.0000	0.975900
$946$	−39.0000	−1.26800
$947$	−21.0000	−0.682408	−0.341204	−	0.939989i	$-0.610835\pi$
$947$	−21.0000	−0.682408	−0.341204	+	0.939989i	$0.610835\pi$
$948$	−3.00000	−0.0974355
$949$	14.0000	0.454459
$950$	−16.0000	−0.519109
$951$	18.0000	0.583690
$952$	12.0000	0.388922
$953$	−49.0000	−1.58727	−0.793633	−	0.608397i	$-0.791813\pi$
$953$	−49.0000	−1.58727	−0.793633	+	0.608397i	$0.791813\pi$
$954$	−18.0000	−0.582772
$955$	72.0000	2.32987
$956$	−26.0000	−0.840900
$957$	−3.00000	−0.0969762
$958$	3.00000	0.0969256
$959$	−44.0000	−1.42083
$960$	−21.0000	−0.677772
$961$	50.0000	1.61290
$962$	8.00000	0.257930
$963$	12.0000	0.386695
$964$	−1.00000	−0.0322078
$965$	30.0000	0.965734
$966$	−8.00000	−0.257396
$967$	−39.0000	−1.25416	−0.627078	−	0.778957i	$-0.715749\pi$
$967$	−39.0000	−1.25416	−0.627078	+	0.778957i	$0.715749\pi$
$968$	6.00000	0.192847
$969$	8.00000	0.256997
$970$	−42.0000	−1.34854
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	−16.0000	−0.513200
$973$	4.00000	0.128234
$974$	28.0000	0.897178
$975$	4.00000	0.128103
$976$	−4.00000	−0.128037
$977$	41.0000	1.31171	0.655853	−	0.754889i	$-0.272309\pi$
$977$	41.0000	1.31171	0.655853	+	0.754889i	$0.272309\pi$
$978$	−3.00000	−0.0959294
$979$	36.0000	1.15056
$980$	−9.00000	−0.287494
$981$	−2.00000	−0.0638551
$982$	1.00000	0.0319113
$983$	−49.0000	−1.56286	−0.781429	−	0.623995i	$-0.785509\pi$
$983$	−49.0000	−1.56286	−0.781429	+	0.623995i	$0.785509\pi$
$984$	18.0000	0.573819
$985$	−6.00000	−0.191176
$986$	−2.00000	−0.0636930
$987$	−14.0000	−0.445625
$988$	4.00000	0.127257
$989$	−52.0000	−1.65350
$990$	−18.0000	−0.572078
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	−45.0000	−1.42875
$993$	1.00000	0.0317340
$994$	−24.0000	−0.761234
$995$	−66.0000	−2.09234
$996$	−10.0000	−0.316862
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−34.0000	−1.07625
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8033.2.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8033.2.a.a.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8033 = 29 \cdot 277 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8033.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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