LMFDB - Embedded newform 1682.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1682.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1682.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +3.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +3.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} -3.00000 q^{10} +1.00000 q^{11} +3.00000 q^{12} +3.00000 q^{13} -2.00000 q^{14} -9.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{18} +8.00000 q^{19} -3.00000 q^{20} -6.00000 q^{21} +1.00000 q^{22} +3.00000 q^{24} +4.00000 q^{25} +3.00000 q^{26} +9.00000 q^{27} -2.00000 q^{28} -9.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +4.00000 q^{34} +6.00000 q^{35} +6.00000 q^{36} +8.00000 q^{37} +8.00000 q^{38} +9.00000 q^{39} -3.00000 q^{40} +2.00000 q^{41} -6.00000 q^{42} -7.00000 q^{43} +1.00000 q^{44} -18.0000 q^{45} -11.0000 q^{47} +3.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} +12.0000 q^{51} +3.00000 q^{52} +1.00000 q^{53} +9.00000 q^{54} -3.00000 q^{55} -2.00000 q^{56} +24.0000 q^{57} -4.00000 q^{59} -9.00000 q^{60} -4.00000 q^{61} -3.00000 q^{62} -12.0000 q^{63} +1.00000 q^{64} -9.00000 q^{65} +3.00000 q^{66} -4.00000 q^{67} +4.00000 q^{68} +6.00000 q^{70} -2.00000 q^{71} +6.00000 q^{72} +12.0000 q^{73} +8.00000 q^{74} +12.0000 q^{75} +8.00000 q^{76} -2.00000 q^{77} +9.00000 q^{78} +7.00000 q^{79} -3.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} -6.00000 q^{84} -12.0000 q^{85} -7.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -18.0000 q^{90} -6.00000 q^{91} -9.00000 q^{93} -11.0000 q^{94} -24.0000 q^{95} +3.00000 q^{96} +6.00000 q^{97} -3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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
$2$	1.00000	0.707107
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	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	$0.734058\pi$
$6$	3.00000	1.22474
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	−3.00000	−0.948683
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	3.00000	0.866025
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	−2.00000	−0.534522
$15$	−9.00000	−2.32379
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	6.00000	1.41421
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	−3.00000	−0.670820
$21$	−6.00000	−1.30931
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	4.00000	0.800000
$26$	3.00000	0.588348
$27$	9.00000	1.73205
$28$	−2.00000	−0.377964
$29$	0	0
$29$	0	0
$30$	−9.00000	−1.64317
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	1.00000	0.176777
$33$	3.00000	0.522233
$34$	4.00000	0.685994
$35$	6.00000	1.01419
$36$	6.00000	1.00000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	8.00000	1.29777
$39$	9.00000	1.44115
$40$	−3.00000	−0.474342
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−6.00000	−0.925820
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	1.00000	0.150756
$45$	−18.0000	−2.68328
$46$	0	0
$47$	−11.0000	−1.60451	−0.802257	−	0.596978i	$-0.796368\pi$
$47$	−11.0000	−1.60451	−0.802257	+	0.596978i	$0.796368\pi$
$48$	3.00000	0.433013
$49$	−3.00000	−0.428571
$50$	4.00000	0.565685
$51$	12.0000	1.68034
$52$	3.00000	0.416025
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	9.00000	1.22474
$55$	−3.00000	−0.404520
$56$	−2.00000	−0.267261
$57$	24.0000	3.17888
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	−9.00000	−1.16190
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	−3.00000	−0.381000
$63$	−12.0000	−1.51186
$64$	1.00000	0.125000
$65$	−9.00000	−1.11631
$66$	3.00000	0.369274
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	6.00000	0.717137
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	6.00000	0.707107
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	8.00000	0.929981
$75$	12.0000	1.38564
$76$	8.00000	0.917663
$77$	−2.00000	−0.227921
$78$	9.00000	1.01905
$79$	7.00000	0.787562	0.393781	−	0.919204i	$-0.371167\pi$
$79$	7.00000	0.787562	0.393781	+	0.919204i	$0.371167\pi$
$80$	−3.00000	−0.335410
$81$	9.00000	1.00000
$82$	2.00000	0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	−6.00000	−0.654654
$85$	−12.0000	−1.30158
$86$	−7.00000	−0.754829
$87$	0	0
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−18.0000	−1.89737
$91$	−6.00000	−0.628971
$92$	0	0
$93$	−9.00000	−0.933257
$94$	−11.0000	−1.13456
$95$	−24.0000	−2.46235
$96$	3.00000	0.306186
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−3.00000	−0.303046
$99$	6.00000	0.603023
$100$	4.00000	0.400000
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	12.0000	1.18818
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	3.00000	0.294174
$105$	18.0000	1.75662
$106$	1.00000	0.0971286
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	9.00000	0.866025
$109$	1.00000	0.0957826	0.0478913	−	0.998853i	$-0.484750\pi$
$109$	1.00000	0.0957826	0.0478913	+	0.998853i	$0.484750\pi$
$110$	−3.00000	−0.286039
$111$	24.0000	2.27798
$112$	−2.00000	−0.188982
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	24.0000	2.24781
$115$	0	0
$116$	0	0
$117$	18.0000	1.66410
$118$	−4.00000	−0.368230
$119$	−8.00000	−0.733359
$120$	−9.00000	−0.821584
$121$	−10.0000	−0.909091
$122$	−4.00000	−0.362143
$123$	6.00000	0.541002
$124$	−3.00000	−0.269408
$125$	3.00000	0.268328
$126$	−12.0000	−1.06904
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−21.0000	−1.84895
$130$	−9.00000	−0.789352
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	3.00000	0.261116
$133$	−16.0000	−1.38738
$134$	−4.00000	−0.345547
$135$	−27.0000	−2.32379
$136$	4.00000	0.342997
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	6.00000	0.507093
$141$	−33.0000	−2.77910
$142$	−2.00000	−0.167836
$143$	3.00000	0.250873
$144$	6.00000	0.500000
$145$	0	0
$146$	12.0000	0.993127
$147$	−9.00000	−0.742307
$148$	8.00000	0.657596
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	12.0000	0.979796
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	8.00000	0.648886
$153$	24.0000	1.94029
$154$	−2.00000	−0.161165
$155$	9.00000	0.722897
$156$	9.00000	0.720577
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	7.00000	0.556890
$159$	3.00000	0.237915
$160$	−3.00000	−0.237171
$161$	0	0
$162$	9.00000	0.707107
$163$	−19.0000	−1.48819	−0.744097	−	0.668071i	$-0.767120\pi$
$163$	−19.0000	−1.48819	−0.744097	+	0.668071i	$0.767120\pi$
$164$	2.00000	0.156174
$165$	−9.00000	−0.700649
$166$	0	0
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	−6.00000	−0.462910
$169$	−4.00000	−0.307692
$170$	−12.0000	−0.920358
$171$	48.0000	3.67065
$172$	−7.00000	−0.533745
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	1.00000	0.0753778
$177$	−12.0000	−0.901975
$178$	6.00000	0.449719
$179$	−14.0000	−1.04641	−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000	−1.04641	−0.523205	+	0.852207i	$0.675264\pi$
$180$	−18.0000	−1.34164
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	−6.00000	−0.444750
$183$	−12.0000	−0.887066
$184$	0	0
$185$	−24.0000	−1.76452
$186$	−9.00000	−0.659912
$187$	4.00000	0.292509
$188$	−11.0000	−0.802257
$189$	−18.0000	−1.30931
$190$	−24.0000	−1.74114
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	3.00000	0.216506
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	6.00000	0.430775
$195$	−27.0000	−1.93351
$196$	−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
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	4.00000	0.282843
$201$	−12.0000	−0.846415
$202$	−8.00000	−0.562878
$203$	0	0
$204$	12.0000	0.840168
$205$	−6.00000	−0.419058
$206$	−6.00000	−0.418040
$207$	0	0
$208$	3.00000	0.208013
$209$	8.00000	0.553372
$210$	18.0000	1.24212
$211$	25.0000	1.72107	0.860535	−	0.509390i	$-0.170129\pi$
$211$	25.0000	1.72107	0.860535	+	0.509390i	$0.170129\pi$
$212$	1.00000	0.0686803
$213$	−6.00000	−0.411113
$214$	−2.00000	−0.136717
$215$	21.0000	1.43219
$216$	9.00000	0.612372
$217$	6.00000	0.407307
$218$	1.00000	0.0677285
$219$	36.0000	2.43265
$220$	−3.00000	−0.202260
$221$	12.0000	0.807207
$222$	24.0000	1.61077
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	−2.00000	−0.133631
$225$	24.0000	1.60000
$226$	−18.0000	−1.19734
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	24.0000	1.58944
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	−25.0000	−1.63780	−0.818902	−	0.573933i	$-0.805417\pi$
$233$	−25.0000	−1.63780	−0.818902	+	0.573933i	$0.805417\pi$
$234$	18.0000	1.17670
$235$	33.0000	2.15268
$236$	−4.00000	−0.260378
$237$	21.0000	1.36410
$238$	−8.00000	−0.518563
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	−9.00000	−0.580948
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	−10.0000	−0.642824
$243$	0	0
$244$	−4.00000	−0.256074
$245$	9.00000	0.574989
$246$	6.00000	0.382546
$247$	24.0000	1.52708
$248$	−3.00000	−0.190500
$249$	0	0
$250$	3.00000	0.189737
$251$	7.00000	0.441836	0.220918	−	0.975292i	$-0.429095\pi$
$251$	7.00000	0.441836	0.220918	+	0.975292i	$0.429095\pi$
$252$	−12.0000	−0.755929
$253$	0	0
$254$	8.00000	0.501965
$255$	−36.0000	−2.25441
$256$	1.00000	0.0625000
$257$	21.0000	1.30994	0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000	1.30994	0.654972	+	0.755653i	$0.272680\pi$
$258$	−21.0000	−1.30740
$259$	−16.0000	−0.994192
$260$	−9.00000	−0.558156
$261$	0	0
$262$	−12.0000	−0.741362
$263$	17.0000	1.04826	0.524132	−	0.851637i	$-0.324390\pi$
$263$	17.0000	1.04826	0.524132	+	0.851637i	$0.324390\pi$
$264$	3.00000	0.184637
$265$	−3.00000	−0.184289
$266$	−16.0000	−0.981023
$267$	18.0000	1.10158
$268$	−4.00000	−0.244339
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	−27.0000	−1.64317
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	4.00000	0.242536
$273$	−18.0000	−1.08941
$274$	20.0000	1.20824
$275$	4.00000	0.241209
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	6.00000	0.358569
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	−33.0000	−1.96512
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	−2.00000	−0.118678
$285$	−72.0000	−4.26491
$286$	3.00000	0.177394
$287$	−4.00000	−0.236113
$288$	6.00000	0.353553
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	18.0000	1.05518
$292$	12.0000	0.702247
$293$	34.0000	1.98630	0.993151	−	0.116841i	$-0.0372769\pi$
$293$	34.0000	1.98630	0.993151	+	0.116841i	$0.0372769\pi$
$294$	−9.00000	−0.524891
$295$	12.0000	0.698667
$296$	8.00000	0.464991
$297$	9.00000	0.522233
$298$	3.00000	0.173785
$299$	0	0
$300$	12.0000	0.692820
$301$	14.0000	0.806947
$302$	10.0000	0.575435
$303$	−24.0000	−1.37876
$304$	8.00000	0.458831
$305$	12.0000	0.687118
$306$	24.0000	1.37199
$307$	29.0000	1.65512	0.827559	−	0.561379i	$-0.189729\pi$
$307$	29.0000	1.65512	0.827559	+	0.561379i	$0.189729\pi$
$308$	−2.00000	−0.113961
$309$	−18.0000	−1.02398
$310$	9.00000	0.511166
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	9.00000	0.509525
$313$	25.0000	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$313$	25.0000	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$314$	−22.0000	−1.24153
$315$	36.0000	2.02837
$316$	7.00000	0.393781
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	3.00000	0.168232
$319$	0	0
$320$	−3.00000	−0.167705
$321$	−6.00000	−0.334887
$322$	0	0
$323$	32.0000	1.78053
$324$	9.00000	0.500000
$325$	12.0000	0.665640
$326$	−19.0000	−1.05231
$327$	3.00000	0.165900
$328$	2.00000	0.110432
$329$	22.0000	1.21290
$330$	−9.00000	−0.495434
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	0	0
$333$	48.0000	2.63038
$334$	−22.0000	−1.20379
$335$	12.0000	0.655630
$336$	−6.00000	−0.327327
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	−4.00000	−0.217571
$339$	−54.0000	−2.93288
$340$	−12.0000	−0.650791
$341$	−3.00000	−0.162459
$342$	48.0000	2.59554
$343$	20.0000	1.07990
$344$	−7.00000	−0.377415
$345$	0	0
$346$	−14.0000	−0.752645
$347$	18.0000	0.966291	0.483145	−	0.875540i	$-0.339494\pi$
$347$	18.0000	0.966291	0.483145	+	0.875540i	$0.339494\pi$
$348$	0	0
$349$	−19.0000	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$349$	−19.0000	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$350$	−8.00000	−0.427618
$351$	27.0000	1.44115
$352$	1.00000	0.0533002
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	−12.0000	−0.637793
$355$	6.00000	0.318447
$356$	6.00000	0.317999
$357$	−24.0000	−1.27021
$358$	−14.0000	−0.739923
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	−18.0000	−0.948683
$361$	45.0000	2.36842
$362$	−13.0000	−0.683265
$363$	−30.0000	−1.57459
$364$	−6.00000	−0.314485
$365$	−36.0000	−1.88433
$366$	−12.0000	−0.627250
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	−24.0000	−1.24770
$371$	−2.00000	−0.103835
$372$	−9.00000	−0.466628
$373$	−1.00000	−0.0517780	−0.0258890	−	0.999665i	$-0.508242\pi$
$373$	−1.00000	−0.0517780	−0.0258890	+	0.999665i	$0.508242\pi$
$374$	4.00000	0.206835
$375$	9.00000	0.464758
$376$	−11.0000	−0.567282
$377$	0	0
$378$	−18.0000	−0.925820
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	−24.0000	−1.23117
$381$	24.0000	1.22956
$382$	0	0
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	3.00000	0.153093
$385$	6.00000	0.305788
$386$	−10.0000	−0.508987
$387$	−42.0000	−2.13498
$388$	6.00000	0.304604
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	−27.0000	−1.36720
$391$	0	0
$392$	−3.00000	−0.151523
$393$	−36.0000	−1.81596
$394$	2.00000	0.100759
$395$	−21.0000	−1.05662
$396$	6.00000	0.301511
$397$	19.0000	0.953583	0.476791	−	0.879017i	$-0.341800\pi$
$397$	19.0000	0.953583	0.476791	+	0.879017i	$0.341800\pi$
$398$	−2.00000	−0.100251
$399$	−48.0000	−2.40301
$400$	4.00000	0.200000
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	−12.0000	−0.598506
$403$	−9.00000	−0.448322
$404$	−8.00000	−0.398015
$405$	−27.0000	−1.34164
$406$	0	0
$407$	8.00000	0.396545
$408$	12.0000	0.594089
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−6.00000	−0.296319
$411$	60.0000	2.95958
$412$	−6.00000	−0.295599
$413$	8.00000	0.393654
$414$	0	0
$415$	0	0
$416$	3.00000	0.147087
$417$	0	0
$418$	8.00000	0.391293
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	18.0000	0.878310
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	25.0000	1.21698
$423$	−66.0000	−3.20903
$424$	1.00000	0.0485643
$425$	16.0000	0.776114
$426$	−6.00000	−0.290701
$427$	8.00000	0.387147
$428$	−2.00000	−0.0966736
$429$	9.00000	0.434524
$430$	21.0000	1.01271
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	9.00000	0.433013
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	1.00000	0.0478913
$437$	0	0
$438$	36.0000	1.72015
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−3.00000	−0.143019
$441$	−18.0000	−0.857143
$442$	12.0000	0.570782
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	24.0000	1.13899
$445$	−18.0000	−0.853282
$446$	26.0000	1.23114
$447$	9.00000	0.425685
$448$	−2.00000	−0.0944911
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	24.0000	1.13137
$451$	2.00000	0.0941763
$452$	−18.0000	−0.846649
$453$	30.0000	1.40952
$454$	−18.0000	−0.844782
$455$	18.0000	0.843853
$456$	24.0000	1.12390
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−14.0000	−0.654177
$459$	36.0000	1.68034
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	−6.00000	−0.279145
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	27.0000	1.25210
$466$	−25.0000	−1.15810
$467$	−23.0000	−1.06431	−0.532157	−	0.846646i	$-0.678618\pi$
$467$	−23.0000	−1.06431	−0.532157	+	0.846646i	$0.678618\pi$
$468$	18.0000	0.832050
$469$	8.00000	0.369406
$470$	33.0000	1.52218
$471$	−66.0000	−3.04112
$472$	−4.00000	−0.184115
$473$	−7.00000	−0.321860
$474$	21.0000	0.964562
$475$	32.0000	1.46826
$476$	−8.00000	−0.366679
$477$	6.00000	0.274721
$478$	20.0000	0.914779
$479$	11.0000	0.502603	0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000	0.502603	0.251301	+	0.967909i	$0.419141\pi$
$480$	−9.00000	−0.410792
$481$	24.0000	1.09431
$482$	17.0000	0.774329
$483$	0	0
$484$	−10.0000	−0.454545
$485$	−18.0000	−0.817338
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−4.00000	−0.181071
$489$	−57.0000	−2.57763
$490$	9.00000	0.406579
$491$	−5.00000	−0.225647	−0.112823	−	0.993615i	$-0.535989\pi$
$491$	−5.00000	−0.225647	−0.112823	+	0.993615i	$0.535989\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	24.0000	1.07981
$495$	−18.0000	−0.809040
$496$	−3.00000	−0.134704
$497$	4.00000	0.179425
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	3.00000	0.134164
$501$	−66.0000	−2.94866
$502$	7.00000	0.312425
$503$	19.0000	0.847168	0.423584	−	0.905857i	$-0.360772\pi$
$503$	19.0000	0.847168	0.423584	+	0.905857i	$0.360772\pi$
$504$	−12.0000	−0.534522
$505$	24.0000	1.06799
$506$	0	0
$507$	−12.0000	−0.532939
$508$	8.00000	0.354943
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	−36.0000	−1.59411
$511$	−24.0000	−1.06170
$512$	1.00000	0.0441942
$513$	72.0000	3.17888
$514$	21.0000	0.926270
$515$	18.0000	0.793175
$516$	−21.0000	−0.924473
$517$	−11.0000	−0.483779
$518$	−16.0000	−0.703000
$519$	−42.0000	−1.84360
$520$	−9.00000	−0.394676
$521$	27.0000	1.18289	0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000	1.18289	0.591446	+	0.806345i	$0.298557\pi$
$522$	0	0
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−12.0000	−0.524222
$525$	−24.0000	−1.04745
$526$	17.0000	0.741235
$527$	−12.0000	−0.522728
$528$	3.00000	0.130558
$529$	−23.0000	−1.00000
$530$	−3.00000	−0.130312
$531$	−24.0000	−1.04151
$532$	−16.0000	−0.693688
$533$	6.00000	0.259889
$534$	18.0000	0.778936
$535$	6.00000	0.259403
$536$	−4.00000	−0.172774
$537$	−42.0000	−1.81243
$538$	−20.0000	−0.862261
$539$	−3.00000	−0.129219
$540$	−27.0000	−1.16190
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	−13.0000	−0.558398
$543$	−39.0000	−1.67365
$544$	4.00000	0.171499
$545$	−3.00000	−0.128506
$546$	−18.0000	−0.770329
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	20.0000	0.854358
$549$	−24.0000	−1.02430
$550$	4.00000	0.170561
$551$	0	0
$552$	0	0
$553$	−14.0000	−0.595341
$554$	14.0000	0.594803
$555$	−72.0000	−3.05623
$556$	0	0
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−18.0000	−0.762001
$559$	−21.0000	−0.888205
$560$	6.00000	0.253546
$561$	12.0000	0.506640
$562$	−13.0000	−0.548372
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	−33.0000	−1.38955
$565$	54.0000	2.27180
$566$	0	0
$567$	−18.0000	−0.755929
$568$	−2.00000	−0.0839181
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	−72.0000	−3.01575
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	3.00000	0.125436
$573$	0	0
$574$	−4.00000	−0.166957
$575$	0	0
$576$	6.00000	0.250000
$577$	44.0000	1.83174	0.915872	−	0.401470i	$-0.131501\pi$
$577$	44.0000	1.83174	0.915872	+	0.401470i	$0.131501\pi$
$578$	−1.00000	−0.0415945
$579$	−30.0000	−1.24676
$580$	0	0
$581$	0	0
$582$	18.0000	0.746124
$583$	1.00000	0.0414158
$584$	12.0000	0.496564
$585$	−54.0000	−2.23263
$586$	34.0000	1.40453
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	−9.00000	−0.371154
$589$	−24.0000	−0.988903
$590$	12.0000	0.494032
$591$	6.00000	0.246807
$592$	8.00000	0.328798
$593$	−17.0000	−0.698106	−0.349053	−	0.937103i	$-0.613497\pi$
$593$	−17.0000	−0.698106	−0.349053	+	0.937103i	$0.613497\pi$
$594$	9.00000	0.369274
$595$	24.0000	0.983904
$596$	3.00000	0.122885
$597$	−6.00000	−0.245564
$598$	0	0
$599$	−13.0000	−0.531166	−0.265583	−	0.964088i	$-0.585564\pi$
$599$	−13.0000	−0.531166	−0.265583	+	0.964088i	$0.585564\pi$
$600$	12.0000	0.489898
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	14.0000	0.570597
$603$	−24.0000	−0.977356
$604$	10.0000	0.406894
$605$	30.0000	1.21967
$606$	−24.0000	−0.974933
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	12.0000	0.485866
$611$	−33.0000	−1.33504
$612$	24.0000	0.970143
$613$	−27.0000	−1.09052	−0.545260	−	0.838267i	$-0.683569\pi$
$613$	−27.0000	−1.09052	−0.545260	+	0.838267i	$0.683569\pi$
$614$	29.0000	1.17034
$615$	−18.0000	−0.725830
$616$	−2.00000	−0.0805823
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	−18.0000	−0.724066
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	9.00000	0.361449
$621$	0	0
$622$	0	0
$623$	−12.0000	−0.480770
$624$	9.00000	0.360288
$625$	−29.0000	−1.16000
$626$	25.0000	0.999201
$627$	24.0000	0.958468
$628$	−22.0000	−0.877896
$629$	32.0000	1.27592
$630$	36.0000	1.43427
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	7.00000	0.278445
$633$	75.0000	2.98098
$634$	0	0
$635$	−24.0000	−0.952411
$636$	3.00000	0.118958
$637$	−9.00000	−0.356593
$638$	0	0
$639$	−12.0000	−0.474713
$640$	−3.00000	−0.118585
$641$	−36.0000	−1.42191	−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000	−1.42191	−0.710957	+	0.703235i	$0.751738\pi$
$642$	−6.00000	−0.236801
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	0	0
$645$	63.0000	2.48062
$646$	32.0000	1.25902
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	9.00000	0.353553
$649$	−4.00000	−0.157014
$650$	12.0000	0.470679
$651$	18.0000	0.705476
$652$	−19.0000	−0.744097
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	3.00000	0.117309
$655$	36.0000	1.40664
$656$	2.00000	0.0780869
$657$	72.0000	2.80899
$658$	22.0000	0.857649
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	−9.00000	−0.350325
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	−3.00000	−0.116598
$663$	36.0000	1.39812
$664$	0	0
$665$	48.0000	1.86136
$666$	48.0000	1.85996
$667$	0	0
$668$	−22.0000	−0.851206
$669$	78.0000	3.01565
$670$	12.0000	0.463600
$671$	−4.00000	−0.154418
$672$	−6.00000	−0.231455
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−4.00000	−0.154074
$675$	36.0000	1.38564
$676$	−4.00000	−0.153846
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−54.0000	−2.07386
$679$	−12.0000	−0.460518
$680$	−12.0000	−0.460179
$681$	−54.0000	−2.06928
$682$	−3.00000	−0.114876
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	48.0000	1.83533
$685$	−60.0000	−2.29248
$686$	20.0000	0.763604
$687$	−42.0000	−1.60240
$688$	−7.00000	−0.266872
$689$	3.00000	0.114291
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−14.0000	−0.532200
$693$	−12.0000	−0.455842
$694$	18.0000	0.683271
$695$	0	0
$696$	0	0
$697$	8.00000	0.303022
$698$	−19.0000	−0.719161
$699$	−75.0000	−2.83676
$700$	−8.00000	−0.302372
$701$	31.0000	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$701$	31.0000	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$702$	27.0000	1.01905
$703$	64.0000	2.41381
$704$	1.00000	0.0376889
$705$	99.0000	3.72856
$706$	−2.00000	−0.0752710
$707$	16.0000	0.601742
$708$	−12.0000	−0.450988
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	6.00000	0.225176
$711$	42.0000	1.57512
$712$	6.00000	0.224860
$713$	0	0
$714$	−24.0000	−0.898177
$715$	−9.00000	−0.336581
$716$	−14.0000	−0.523205
$717$	60.0000	2.24074
$718$	−9.00000	−0.335877
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−18.0000	−0.670820
$721$	12.0000	0.446903
$722$	45.0000	1.67473
$723$	51.0000	1.89671
$724$	−13.0000	−0.483141
$725$	0	0
$726$	−30.0000	−1.11340
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	−6.00000	−0.222375
$729$	−27.0000	−1.00000
$730$	−36.0000	−1.33242
$731$	−28.0000	−1.03562
$732$	−12.0000	−0.443533
$733$	48.0000	1.77292	0.886460	−	0.462805i	$-0.153157\pi$
$733$	48.0000	1.77292	0.886460	+	0.462805i	$0.153157\pi$
$734$	16.0000	0.590571
$735$	27.0000	0.995910
$736$	0	0
$737$	−4.00000	−0.147342
$738$	12.0000	0.441726
$739$	−1.00000	−0.0367856	−0.0183928	−	0.999831i	$-0.505855\pi$
$739$	−1.00000	−0.0367856	−0.0183928	+	0.999831i	$0.505855\pi$
$740$	−24.0000	−0.882258
$741$	72.0000	2.64499
$742$	−2.00000	−0.0734223
$743$	28.0000	1.02722	0.513610	−	0.858024i	$-0.328308\pi$
$743$	28.0000	1.02722	0.513610	+	0.858024i	$0.328308\pi$
$744$	−9.00000	−0.329956
$745$	−9.00000	−0.329734
$746$	−1.00000	−0.0366126
$747$	0	0
$748$	4.00000	0.146254
$749$	4.00000	0.146157
$750$	9.00000	0.328634
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−11.0000	−0.401129
$753$	21.0000	0.765283
$754$	0	0
$755$	−30.0000	−1.09181
$756$	−18.0000	−0.654654
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	−24.0000	−0.870572
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	24.0000	0.869428
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	−72.0000	−2.60317
$766$	−34.0000	−1.22847
$767$	−12.0000	−0.433295
$768$	3.00000	0.108253
$769$	−48.0000	−1.73092	−0.865462	−	0.500974i	$-0.832975\pi$
$769$	−48.0000	−1.73092	−0.865462	+	0.500974i	$0.832975\pi$
$770$	6.00000	0.216225
$771$	63.0000	2.26889
$772$	−10.0000	−0.359908
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−42.0000	−1.50966
$775$	−12.0000	−0.431053
$776$	6.00000	0.215387
$777$	−48.0000	−1.72199
$778$	−16.0000	−0.573628
$779$	16.0000	0.573259
$780$	−27.0000	−0.966755
$781$	−2.00000	−0.0715656
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	66.0000	2.35564
$786$	−36.0000	−1.28408
$787$	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
$789$	51.0000	1.81565
$790$	−21.0000	−0.747146
$791$	36.0000	1.28001
$792$	6.00000	0.213201
$793$	−12.0000	−0.426132
$794$	19.0000	0.674285
$795$	−9.00000	−0.319197
$796$	−2.00000	−0.0708881
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	−48.0000	−1.69918
$799$	−44.0000	−1.55661
$800$	4.00000	0.141421
$801$	36.0000	1.27200
$802$	−5.00000	−0.176556
$803$	12.0000	0.423471
$804$	−12.0000	−0.423207
$805$	0	0
$806$	−9.00000	−0.317011
$807$	−60.0000	−2.11210
$808$	−8.00000	−0.281439
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	−27.0000	−0.948683
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	0	0
$813$	−39.0000	−1.36779
$814$	8.00000	0.280400
$815$	57.0000	1.99662
$816$	12.0000	0.420084
$817$	−56.0000	−1.95919
$818$	−22.0000	−0.769212
$819$	−36.0000	−1.25794
$820$	−6.00000	−0.209529
$821$	−37.0000	−1.29131	−0.645654	−	0.763630i	$-0.723415\pi$
$821$	−37.0000	−1.29131	−0.645654	+	0.763630i	$0.723415\pi$
$822$	60.0000	2.09274
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	−6.00000	−0.209020
$825$	12.0000	0.417786
$826$	8.00000	0.278356
$827$	−23.0000	−0.799788	−0.399894	−	0.916561i	$-0.630953\pi$
$827$	−23.0000	−0.799788	−0.399894	+	0.916561i	$0.630953\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	42.0000	1.45696
$832$	3.00000	0.104006
$833$	−12.0000	−0.415775
$834$	0	0
$835$	66.0000	2.28402
$836$	8.00000	0.276686
$837$	−27.0000	−0.933257
$838$	−14.0000	−0.483622
$839$	45.0000	1.55357	0.776786	−	0.629764i	$-0.216849\pi$
$839$	45.0000	1.55357	0.776786	+	0.629764i	$0.216849\pi$
$840$	18.0000	0.621059
$841$	0	0
$842$	28.0000	0.964944
$843$	−39.0000	−1.34323
$844$	25.0000	0.860535
$845$	12.0000	0.412813
$846$	−66.0000	−2.26913
$847$	20.0000	0.687208
$848$	1.00000	0.0343401
$849$	0	0
$850$	16.0000	0.548795
$851$	0	0
$852$	−6.00000	−0.205557
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	8.00000	0.273754
$855$	−144.000	−4.92470
$856$	−2.00000	−0.0683586
$857$	−11.0000	−0.375753	−0.187876	−	0.982193i	$-0.560160\pi$
$857$	−11.0000	−0.375753	−0.187876	+	0.982193i	$0.560160\pi$
$858$	9.00000	0.307255
$859$	51.0000	1.74010	0.870049	−	0.492966i	$-0.164087\pi$
$859$	51.0000	1.74010	0.870049	+	0.492966i	$0.164087\pi$
$860$	21.0000	0.716094
$861$	−12.0000	−0.408959
$862$	−20.0000	−0.681203
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	9.00000	0.306186
$865$	42.0000	1.42804
$866$	16.0000	0.543702
$867$	−3.00000	−0.101885
$868$	6.00000	0.203653
$869$	7.00000	0.237459
$870$	0	0
$871$	−12.0000	−0.406604
$872$	1.00000	0.0338643
$873$	36.0000	1.21842
$874$	0	0
$875$	−6.00000	−0.202837
$876$	36.0000	1.21633
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	−8.00000	−0.269987
$879$	102.000	3.44037
$880$	−3.00000	−0.101130
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	−18.0000	−0.606092
$883$	−42.0000	−1.41341	−0.706706	−	0.707507i	$-0.749820\pi$
$883$	−42.0000	−1.41341	−0.706706	+	0.707507i	$0.749820\pi$
$884$	12.0000	0.403604
$885$	36.0000	1.21013
$886$	−12.0000	−0.403148
$887$	1.00000	0.0335767	0.0167884	−	0.999859i	$-0.494656\pi$
$887$	1.00000	0.0335767	0.0167884	+	0.999859i	$0.494656\pi$
$888$	24.0000	0.805387
$889$	−16.0000	−0.536623
$890$	−18.0000	−0.603361
$891$	9.00000	0.301511
$892$	26.0000	0.870544
$893$	−88.0000	−2.94481
$894$	9.00000	0.301005
$895$	42.0000	1.40391
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−22.0000	−0.734150
$899$	0	0
$900$	24.0000	0.800000
$901$	4.00000	0.133259
$902$	2.00000	0.0665927
$903$	42.0000	1.39767
$904$	−18.0000	−0.598671
$905$	39.0000	1.29640
$906$	30.0000	0.996683
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−18.0000	−0.597351
$909$	−48.0000	−1.59206
$910$	18.0000	0.596694
$911$	3.00000	0.0993944	0.0496972	−	0.998764i	$-0.484174\pi$
$911$	3.00000	0.0993944	0.0496972	+	0.998764i	$0.484174\pi$
$912$	24.0000	0.794719
$913$	0	0
$914$	−10.0000	−0.330771
$915$	36.0000	1.19012
$916$	−14.0000	−0.462573
$917$	24.0000	0.792550
$918$	36.0000	1.18818
$919$	−6.00000	−0.197922	−0.0989609	−	0.995091i	$-0.531552\pi$
$919$	−6.00000	−0.197922	−0.0989609	+	0.995091i	$0.531552\pi$
$920$	0	0
$921$	87.0000	2.86675
$922$	30.0000	0.987997
$923$	−6.00000	−0.197492
$924$	−6.00000	−0.197386
$925$	32.0000	1.05215
$926$	−16.0000	−0.525793
$927$	−36.0000	−1.18240
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	27.0000	0.885365
$931$	−24.0000	−0.786568
$932$	−25.0000	−0.818902
$933$	0	0
$934$	−23.0000	−0.752583
$935$	−12.0000	−0.392442
$936$	18.0000	0.588348
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	8.00000	0.261209
$939$	75.0000	2.44753
$940$	33.0000	1.07634
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	−66.0000	−2.15040
$943$	0	0
$944$	−4.00000	−0.130189
$945$	54.0000	1.75662
$946$	−7.00000	−0.227590
$947$	−51.0000	−1.65728	−0.828639	−	0.559784i	$-0.810884\pi$
$947$	−51.0000	−1.65728	−0.828639	+	0.559784i	$0.810884\pi$
$948$	21.0000	0.682048
$949$	36.0000	1.16861
$950$	32.0000	1.03822
$951$	0	0
$952$	−8.00000	−0.259281
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	20.0000	0.646846
$957$	0	0
$958$	11.0000	0.355394
$959$	−40.0000	−1.29167
$960$	−9.00000	−0.290474
$961$	−22.0000	−0.709677
$962$	24.0000	0.773791
$963$	−12.0000	−0.386695
$964$	17.0000	0.547533
$965$	30.0000	0.965734
$966$	0	0
$967$	−59.0000	−1.89731	−0.948656	−	0.316310i	$-0.897556\pi$
$967$	−59.0000	−1.89731	−0.948656	+	0.316310i	$0.897556\pi$
$968$	−10.0000	−0.321412
$969$	96.0000	3.08396
$970$	−18.0000	−0.577945
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	0	0
$973$	0	0
$974$	2.00000	0.0640841
$975$	36.0000	1.15292
$976$	−4.00000	−0.128037
$977$	37.0000	1.18373	0.591867	−	0.806035i	$-0.298391\pi$
$977$	37.0000	1.18373	0.591867	+	0.806035i	$0.298391\pi$
$978$	−57.0000	−1.82266
$979$	6.00000	0.191761
$980$	9.00000	0.287494
$981$	6.00000	0.191565
$982$	−5.00000	−0.159556
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	6.00000	0.191273
$985$	−6.00000	−0.191176
$986$	0	0
$987$	66.0000	2.10080
$988$	24.0000	0.763542
$989$	0	0
$990$	−18.0000	−0.572078
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	−3.00000	−0.0952501
$993$	−9.00000	−0.285606
$994$	4.00000	0.126872
$995$	6.00000	0.190213
$996$	0	0
$997$	−16.0000	−0.506725	−0.253363	−	0.967371i	$-0.581537\pi$
$997$	−16.0000	−0.506725	−0.253363	+	0.967371i	$0.581537\pi$
$998$	−8.00000	−0.253236
$999$	72.0000	2.27798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.a.j.1.1		1
29.12	odd	4		1682.2.b.e.1681.2		2
29.17	odd	4		1682.2.b.e.1681.1		2
29.28	even	2		58.2.a.a.1.1	✓	1
87.86	odd	2		522.2.a.k.1.1		1
116.115	odd	2		464.2.a.f.1.1		1
145.28	odd	4		1450.2.b.f.349.2		2
145.57	odd	4		1450.2.b.f.349.1		2
145.144	even	2		1450.2.a.i.1.1		1
203.202	odd	2		2842.2.a.d.1.1		1
232.115	odd	2		1856.2.a.b.1.1		1
232.173	even	2		1856.2.a.p.1.1		1
319.318	odd	2		7018.2.a.c.1.1		1
348.347	even	2		4176.2.a.bh.1.1		1
377.376	even	2		9802.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.a.a.1.1	✓	1	29.28	even	2
464.2.a.f.1.1		1	116.115	odd	2
522.2.a.k.1.1		1	87.86	odd	2
1450.2.a.i.1.1		1	145.144	even	2
1450.2.b.f.349.1		2	145.57	odd	4
1450.2.b.f.349.2		2	145.28	odd	4
1682.2.a.j.1.1		1	1.1	even	1	trivial
1682.2.b.e.1681.1		2	29.17	odd	4
1682.2.b.e.1681.2		2	29.12	odd	4
1856.2.a.b.1.1		1	232.115	odd	2
1856.2.a.p.1.1		1	232.173	even	2
2842.2.a.d.1.1		1	203.202	odd	2
4176.2.a.bh.1.1		1	348.347	even	2
7018.2.a.c.1.1		1	319.318	odd	2
9802.2.a.d.1.1		1	377.376	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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