LMFDB - Embedded newform 7938.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, 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("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.69963$ of defining polynomial
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.58836 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.58836 q^{5} +1.00000 q^{8} -1.58836 q^{10} +1.58836 q^{11} -4.81089 q^{13} +1.00000 q^{16} -5.39926 q^{17} +7.09888 q^{19} -1.58836 q^{20} +1.58836 q^{22} -0.300372 q^{23} -2.47710 q^{25} -4.81089 q^{26} +8.27561 q^{29} -2.71201 q^{31} +1.00000 q^{32} -5.39926 q^{34} -1.00000 q^{37} +7.09888 q^{38} -1.58836 q^{40} +5.87636 q^{41} +1.66621 q^{43} +1.58836 q^{44} -0.300372 q^{46} -2.66621 q^{47} -2.47710 q^{50} -4.81089 q^{52} +4.88874 q^{53} -2.52290 q^{55} +8.27561 q^{58} -6.47710 q^{59} -4.47710 q^{61} -2.71201 q^{62} +1.00000 q^{64} +7.64145 q^{65} -10.0531 q^{67} -5.39926 q^{68} -12.7207 q^{71} -16.0531 q^{73} -1.00000 q^{74} +7.09888 q^{76} +8.38688 q^{79} -1.58836 q^{80} +5.87636 q^{82} +2.36584 q^{83} +8.57598 q^{85} +1.66621 q^{86} +1.58836 q^{88} +3.21015 q^{89} -0.300372 q^{92} -2.66621 q^{94} -11.2756 q^{95} -1.42402 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} + 3 q^{8} + q^{10} - q^{11} - 8 q^{13} + 3 q^{16} - 4 q^{17} + 3 q^{19} + q^{20} - q^{22} - 7 q^{23} - 2 q^{25} - 8 q^{26} - 5 q^{29} - 20 q^{31} + 3 q^{32} - 4 q^{34} - 3 q^{37} + 3 q^{38} + q^{40} + 6 q^{43} - q^{44} - 7 q^{46} - 9 q^{47} - 2 q^{50} - 8 q^{52} + 15 q^{53} - 13 q^{55} - 5 q^{58} - 14 q^{59} - 8 q^{61} - 20 q^{62} + 3 q^{64} - 12 q^{65} - q^{67} - 4 q^{68} - 7 q^{71} - 19 q^{73} - 3 q^{74} + 3 q^{76} - 5 q^{79} + q^{80} + 2 q^{83} + 2 q^{85} + 6 q^{86} - q^{88} - 9 q^{89} - 7 q^{92} - 9 q^{94} - 4 q^{95} - 28 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 + 3 * q^8 + q^10 - q^11 - 8 * q^13 + 3 * q^16 - 4 * q^17 + 3 * q^19 + q^20 - q^22 - 7 * q^23 - 2 * q^25 - 8 * q^26 - 5 * q^29 - 20 * q^31 + 3 * q^32 - 4 * q^34 - 3 * q^37 + 3 * q^38 + q^40 + 6 * q^43 - q^44 - 7 * q^46 - 9 * q^47 - 2 * q^50 - 8 * q^52 + 15 * q^53 - 13 * q^55 - 5 * q^58 - 14 * q^59 - 8 * q^61 - 20 * q^62 + 3 * q^64 - 12 * q^65 - q^67 - 4 * q^68 - 7 * q^71 - 19 * q^73 - 3 * q^74 + 3 * q^76 - 5 * q^79 + q^80 + 2 * q^83 + 2 * q^85 + 6 * q^86 - q^88 - 9 * q^89 - 7 * q^92 - 9 * q^94 - 4 * q^95 - 28 * q^97

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.58836	−0.710338	−0.355169	−	0.934802i	$-0.615577\pi$
$5$	−1.58836	−0.710338	−0.355169	+	0.934802i	$0.615577\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.58836	−0.502285
$11$	1.58836	0.478910	0.239455	−	0.970907i	$-0.423031\pi$
$11$	1.58836	0.478910	0.239455	+	0.970907i	$0.423031\pi$
$12$	0	0
$13$	−4.81089	−1.33430	−0.667151	−	0.744923i	$-0.732486\pi$
$13$	−4.81089	−1.33430	−0.667151	+	0.744923i	$0.732486\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.39926	−1.30951	−0.654756	−	0.755840i	$-0.727229\pi$
$17$	−5.39926	−1.30951	−0.654756	+	0.755840i	$0.727229\pi$
$18$	0	0
$19$	7.09888	1.62860	0.814298	−	0.580447i	$-0.197122\pi$
$19$	7.09888	1.62860	0.814298	+	0.580447i	$0.197122\pi$
$20$	−1.58836	−0.355169
$21$	0	0
$22$	1.58836	0.338640
$23$	−0.300372	−0.0626319	−0.0313159	−	0.999510i	$-0.509970\pi$
$23$	−0.300372	−0.0626319	−0.0313159	+	0.999510i	$0.509970\pi$
$24$	0	0
$25$	−2.47710	−0.495420
$26$	−4.81089	−0.943494
$27$	0	0
$28$	0	0
$29$	8.27561	1.53674	0.768371	−	0.640004i	$-0.221067\pi$
$29$	8.27561	1.53674	0.768371	+	0.640004i	$0.221067\pi$
$30$	0	0
$31$	−2.71201	−0.487091	−0.243545	−	0.969889i	$-0.578311\pi$
$31$	−2.71201	−0.487091	−0.243545	+	0.969889i	$0.578311\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.39926	−0.925965
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	7.09888	1.15159
$39$	0	0
$40$	−1.58836	−0.251142
$41$	5.87636	0.917733	0.458866	−	0.888505i	$-0.348256\pi$
$41$	5.87636	0.917733	0.458866	+	0.888505i	$0.348256\pi$
$42$	0	0
$43$	1.66621	0.254094	0.127047	−	0.991897i	$-0.459450\pi$
$43$	1.66621	0.254094	0.127047	+	0.991897i	$0.459450\pi$
$44$	1.58836	0.239455
$45$	0	0
$46$	−0.300372	−0.0442874
$47$	−2.66621	−0.388906	−0.194453	−	0.980912i	$-0.562293\pi$
$47$	−2.66621	−0.388906	−0.194453	+	0.980912i	$0.562293\pi$
$48$	0	0
$49$	0	0
$50$	−2.47710	−0.350315
$51$	0	0
$52$	−4.81089	−0.667151
$53$	4.88874	0.671520	0.335760	−	0.941948i	$-0.391007\pi$
$53$	4.88874	0.671520	0.335760	+	0.941948i	$0.391007\pi$
$54$	0	0
$55$	−2.52290	−0.340188
$56$	0	0
$57$	0	0
$58$	8.27561	1.08664
$59$	−6.47710	−0.843247	−0.421623	−	0.906771i	$-0.638540\pi$
$59$	−6.47710	−0.843247	−0.421623	+	0.906771i	$0.638540\pi$
$60$	0	0
$61$	−4.47710	−0.573234	−0.286617	−	0.958045i	$-0.592531\pi$
$61$	−4.47710	−0.573234	−0.286617	+	0.958045i	$0.592531\pi$
$62$	−2.71201	−0.344425
$63$	0	0
$64$	1.00000	0.125000
$65$	7.64145	0.947805
$66$	0	0
$67$	−10.0531	−1.22818	−0.614090	−	0.789236i	$-0.710477\pi$
$67$	−10.0531	−1.22818	−0.614090	+	0.789236i	$0.710477\pi$
$68$	−5.39926	−0.654756
$69$	0	0
$70$	0	0
$71$	−12.7207	−1.50967	−0.754833	−	0.655917i	$-0.772282\pi$
$71$	−12.7207	−1.50967	−0.754833	+	0.655917i	$0.772282\pi$
$72$	0	0
$73$	−16.0531	−1.87887	−0.939436	−	0.342725i	$-0.888650\pi$
$73$	−16.0531	−1.87887	−0.939436	+	0.342725i	$0.888650\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	7.09888	0.814298
$77$	0	0
$78$	0	0
$79$	8.38688	0.943597	0.471799	−	0.881706i	$-0.343605\pi$
$79$	8.38688	0.943597	0.471799	+	0.881706i	$0.343605\pi$
$80$	−1.58836	−0.177584
$81$	0	0
$82$	5.87636	0.648935
$83$	2.36584	0.259684	0.129842	−	0.991535i	$-0.458553\pi$
$83$	2.36584	0.259684	0.129842	+	0.991535i	$0.458553\pi$
$84$	0	0
$85$	8.57598	0.930196
$86$	1.66621	0.179672
$87$	0	0
$88$	1.58836	0.169320
$89$	3.21015	0.340275	0.170138	−	0.985420i	$-0.445579\pi$
$89$	3.21015	0.340275	0.170138	+	0.985420i	$0.445579\pi$
$90$	0	0
$91$	0	0
$92$	−0.300372	−0.0313159
$93$	0	0
$94$	−2.66621	−0.274998
$95$	−11.2756	−1.15685
$96$	0	0
$97$	−1.42402	−0.144587	−0.0722934	−	0.997383i	$-0.523032\pi$
$97$	−1.42402	−0.144587	−0.0722934	+	0.997383i	$0.523032\pi$
$98$	0	0
$99$	0	0
$100$	−2.47710	−0.247710
$101$	−12.0334	−1.19737	−0.598685	−	0.800985i	$-0.704310\pi$
$101$	−12.0334	−1.19737	−0.598685	+	0.800985i	$0.704310\pi$
$102$	0	0
$103$	−6.09888	−0.600941	−0.300470	−	0.953791i	$-0.597144\pi$
$103$	−6.09888	−0.600941	−0.300470	+	0.953791i	$0.597144\pi$
$104$	−4.81089	−0.471747
$105$	0	0
$106$	4.88874	0.474836
$107$	−3.08650	−0.298384	−0.149192	−	0.988808i	$-0.547667\pi$
$107$	−3.08650	−0.298384	−0.149192	+	0.988808i	$0.547667\pi$
$108$	0	0
$109$	−2.28799	−0.219150	−0.109575	−	0.993979i	$-0.534949\pi$
$109$	−2.28799	−0.219150	−0.109575	+	0.993979i	$0.534949\pi$
$110$	−2.52290	−0.240549
$111$	0	0
$112$	0	0
$113$	−19.4647	−1.83109	−0.915543	−	0.402219i	$-0.868239\pi$
$113$	−19.4647	−1.83109	−0.915543	+	0.402219i	$0.868239\pi$
$114$	0	0
$115$	0.477100	0.0444898
$116$	8.27561	0.768371
$117$	0	0
$118$	−6.47710	−0.596265
$119$	0	0
$120$	0	0
$121$	−8.47710	−0.770645
$122$	−4.47710	−0.405338
$123$	0	0
$124$	−2.71201	−0.243545
$125$	11.8764	1.06225
$126$	0	0
$127$	−13.4400	−1.19260	−0.596302	−	0.802760i	$-0.703364\pi$
$127$	−13.4400	−1.19260	−0.596302	+	0.802760i	$0.703364\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	7.64145	0.670199
$131$	3.17673	0.277552	0.138776	−	0.990324i	$-0.455683\pi$
$131$	3.17673	0.277552	0.138776	+	0.990324i	$0.455683\pi$
$132$	0	0
$133$	0	0
$134$	−10.0531	−0.868454
$135$	0	0
$136$	−5.39926	−0.462982
$137$	21.2632	1.81664	0.908320	−	0.418275i	$-0.137365\pi$
$137$	21.2632	1.81664	0.908320	+	0.418275i	$0.137365\pi$
$138$	0	0
$139$	−13.0531	−1.10715	−0.553574	−	0.832800i	$-0.686736\pi$
$139$	−13.0531	−1.10715	−0.553574	+	0.832800i	$0.686736\pi$
$140$	0	0
$141$	0	0
$142$	−12.7207	−1.06749
$143$	−7.64145	−0.639010
$144$	0	0
$145$	−13.1447	−1.09161
$146$	−16.0531	−1.32856
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−5.20877	−0.426719	−0.213360	−	0.976974i	$-0.568441\pi$
$149$	−5.20877	−0.426719	−0.213360	+	0.976974i	$0.568441\pi$
$150$	0	0
$151$	−0.522900	−0.0425530	−0.0212765	−	0.999774i	$-0.506773\pi$
$151$	−0.522900	−0.0425530	−0.0212765	+	0.999774i	$0.506773\pi$
$152$	7.09888	0.575796
$153$	0	0
$154$	0	0
$155$	4.30766	0.345999
$156$	0	0
$157$	8.86398	0.707422	0.353711	−	0.935355i	$-0.384920\pi$
$157$	8.86398	0.707422	0.353711	+	0.935355i	$0.384920\pi$
$158$	8.38688	0.667224
$159$	0	0
$160$	−1.58836	−0.125571
$161$	0	0
$162$	0	0
$163$	−21.9629	−1.72026	−0.860132	−	0.510071i	$-0.829619\pi$
$163$	−21.9629	−1.72026	−0.860132	+	0.510071i	$0.829619\pi$
$164$	5.87636	0.458866
$165$	0	0
$166$	2.36584	0.183624
$167$	3.30037	0.255390	0.127695	−	0.991813i	$-0.459242\pi$
$167$	3.30037	0.255390	0.127695	+	0.991813i	$0.459242\pi$
$168$	0	0
$169$	10.1447	0.780360
$170$	8.57598	0.657748
$171$	0	0
$172$	1.66621	0.127047
$173$	−19.1075	−1.45272	−0.726360	−	0.687315i	$-0.758789\pi$
$173$	−19.1075	−1.45272	−0.726360	+	0.687315i	$0.758789\pi$
$174$	0	0
$175$	0	0
$176$	1.58836	0.119727
$177$	0	0
$178$	3.21015	0.240611
$179$	−16.0741	−1.20144	−0.600718	−	0.799461i	$-0.705119\pi$
$179$	−16.0741	−1.20144	−0.600718	+	0.799461i	$0.705119\pi$
$180$	0	0
$181$	8.05308	0.598581	0.299291	−	0.954162i	$-0.403250\pi$
$181$	8.05308	0.598581	0.299291	+	0.954162i	$0.403250\pi$
$182$	0	0
$183$	0	0
$184$	−0.300372	−0.0221437
$185$	1.58836	0.116779
$186$	0	0
$187$	−8.57598	−0.627138
$188$	−2.66621	−0.194453
$189$	0	0
$190$	−11.2756	−0.818019
$191$	23.9629	1.73389	0.866946	−	0.498402i	$-0.166080\pi$
$191$	23.9629	1.73389	0.866946	+	0.498402i	$0.166080\pi$
$192$	0	0
$193$	9.76509	0.702907	0.351453	−	0.936205i	$-0.385688\pi$
$193$	9.76509	0.702907	0.351453	+	0.936205i	$0.385688\pi$
$194$	−1.42402	−0.102238
$195$	0	0
$196$	0	0
$197$	18.2436	1.29980	0.649900	−	0.760020i	$-0.274811\pi$
$197$	18.2436	1.29980	0.649900	+	0.760020i	$0.274811\pi$
$198$	0	0
$199$	−18.0989	−1.28300	−0.641498	−	0.767125i	$-0.721687\pi$
$199$	−18.0989	−1.28300	−0.641498	+	0.767125i	$0.721687\pi$
$200$	−2.47710	−0.175157
$201$	0	0
$202$	−12.0334	−0.846669
$203$	0	0
$204$	0	0
$205$	−9.33379	−0.651900
$206$	−6.09888	−0.424929
$207$	0	0
$208$	−4.81089	−0.333575
$209$	11.2756	0.779950
$210$	0	0
$211$	−0.332415	−0.0228844	−0.0114422	−	0.999935i	$-0.503642\pi$
$211$	−0.332415	−0.0228844	−0.0114422	+	0.999935i	$0.503642\pi$
$212$	4.88874	0.335760
$213$	0	0
$214$	−3.08650	−0.210989
$215$	−2.64654	−0.180493
$216$	0	0
$217$	0	0
$218$	−2.28799	−0.154962
$219$	0	0
$220$	−2.52290	−0.170094
$221$	25.9752	1.74728
$222$	0	0
$223$	−6.33242	−0.424050	−0.212025	−	0.977264i	$-0.568006\pi$
$223$	−6.33242	−0.424050	−0.212025	+	0.977264i	$0.568006\pi$
$224$	0	0
$225$	0	0
$226$	−19.4647	−1.29477
$227$	23.3090	1.54707	0.773537	−	0.633751i	$-0.218485\pi$
$227$	23.3090	1.54707	0.773537	+	0.633751i	$0.218485\pi$
$228$	0	0
$229$	−4.95420	−0.327383	−0.163691	−	0.986512i	$-0.552340\pi$
$229$	−4.95420	−0.327383	−0.163691	+	0.986512i	$0.552340\pi$
$230$	0.477100	0.0314590
$231$	0	0
$232$	8.27561	0.543321
$233$	−14.2756	−0.935226	−0.467613	−	0.883933i	$-0.654886\pi$
$233$	−14.2756	−0.935226	−0.467613	+	0.883933i	$0.654886\pi$
$234$	0	0
$235$	4.23491	0.276255
$236$	−6.47710	−0.421623
$237$	0	0
$238$	0	0
$239$	4.97524	0.321822	0.160911	−	0.986969i	$-0.448557\pi$
$239$	4.97524	0.321822	0.160911	+	0.986969i	$0.448557\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	−8.47710	−0.544929
$243$	0	0
$244$	−4.47710	−0.286617
$245$	0	0
$246$	0	0
$247$	−34.1520	−2.17304
$248$	−2.71201	−0.172213
$249$	0	0
$250$	11.8764	0.751127
$251$	−2.43268	−0.153549	−0.0767746	−	0.997048i	$-0.524462\pi$
$251$	−2.43268	−0.153549	−0.0767746	+	0.997048i	$0.524462\pi$
$252$	0	0
$253$	−0.477100	−0.0299950
$254$	−13.4400	−0.843298
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.987620	0.0616061	0.0308030	−	0.999525i	$-0.490194\pi$
$257$	0.987620	0.0616061	0.0308030	+	0.999525i	$0.490194\pi$
$258$	0	0
$259$	0	0
$260$	7.64145	0.473902
$261$	0	0
$262$	3.17673	0.196259
$263$	−17.1854	−1.05970	−0.529848	−	0.848092i	$-0.677751\pi$
$263$	−17.1854	−1.05970	−0.529848	+	0.848092i	$0.677751\pi$
$264$	0	0
$265$	−7.76509	−0.477006
$266$	0	0
$267$	0	0
$268$	−10.0531	−0.614090
$269$	22.9047	1.39652	0.698262	−	0.715843i	$-0.253957\pi$
$269$	22.9047	1.39652	0.698262	+	0.715843i	$0.253957\pi$
$270$	0	0
$271$	−14.0073	−0.850882	−0.425441	−	0.904986i	$-0.639881\pi$
$271$	−14.0073	−0.850882	−0.425441	+	0.904986i	$0.639881\pi$
$272$	−5.39926	−0.327378
$273$	0	0
$274$	21.2632	1.28456
$275$	−3.93454	−0.237261
$276$	0	0
$277$	28.2953	1.70010	0.850049	−	0.526703i	$-0.176572\pi$
$277$	28.2953	1.70010	0.850049	+	0.526703i	$0.176572\pi$
$278$	−13.0531	−0.782872
$279$	0	0
$280$	0	0
$281$	−17.5956	−1.04967	−0.524834	−	0.851204i	$-0.675873\pi$
$281$	−17.5956	−1.04967	−0.524834	+	0.851204i	$0.675873\pi$
$282$	0	0
$283$	−18.5229	−1.10107	−0.550536	−	0.834811i	$-0.685577\pi$
$283$	−18.5229	−1.10107	−0.550536	+	0.834811i	$0.685577\pi$
$284$	−12.7207	−0.754833
$285$	0	0
$286$	−7.64145	−0.451848
$287$	0	0
$288$	0	0
$289$	12.1520	0.714822
$290$	−13.1447	−0.771882
$291$	0	0
$292$	−16.0531	−0.939436
$293$	−14.0851	−0.822862	−0.411431	−	0.911441i	$-0.634971\pi$
$293$	−14.0851	−0.822862	−0.411431	+	0.911441i	$0.634971\pi$
$294$	0	0
$295$	10.2880	0.598990
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	−5.20877	−0.301736
$299$	1.44506	0.0835698
$300$	0	0
$301$	0	0
$302$	−0.522900	−0.0300895
$303$	0	0
$304$	7.09888	0.407149
$305$	7.11126	0.407190
$306$	0	0
$307$	−5.85532	−0.334180	−0.167090	−	0.985942i	$-0.553437\pi$
$307$	−5.85532	−0.334180	−0.167090	+	0.985942i	$0.553437\pi$
$308$	0	0
$309$	0	0
$310$	4.30766	0.244658
$311$	−0.810892	−0.0459815	−0.0229907	−	0.999736i	$-0.507319\pi$
$311$	−0.810892	−0.0459815	−0.0229907	+	0.999736i	$0.507319\pi$
$312$	0	0
$313$	10.5760	0.597790	0.298895	−	0.954286i	$-0.403382\pi$
$313$	10.5760	0.597790	0.298895	+	0.954286i	$0.403382\pi$
$314$	8.86398	0.500223
$315$	0	0
$316$	8.38688	0.471799
$317$	−12.1964	−0.685018	−0.342509	−	0.939515i	$-0.611277\pi$
$317$	−12.1964	−0.685018	−0.342509	+	0.939515i	$0.611277\pi$
$318$	0	0
$319$	13.1447	0.735961
$320$	−1.58836	−0.0887922
$321$	0	0
$322$	0	0
$323$	−38.3287	−2.13267
$324$	0	0
$325$	11.9171	0.661040
$326$	−21.9629	−1.21641
$327$	0	0
$328$	5.87636	0.324467
$329$	0	0
$330$	0	0
$331$	−15.6662	−0.861093	−0.430546	−	0.902568i	$-0.641679\pi$
$331$	−15.6662	−0.861093	−0.430546	+	0.902568i	$0.641679\pi$
$332$	2.36584	0.129842
$333$	0	0
$334$	3.30037	0.180588
$335$	15.9680	0.872423
$336$	0	0
$337$	8.42402	0.458885	0.229443	−	0.973322i	$-0.426310\pi$
$337$	8.42402	0.458885	0.229443	+	0.973322i	$0.426310\pi$
$338$	10.1447	0.551798
$339$	0	0
$340$	8.57598	0.465098
$341$	−4.30766	−0.233273
$342$	0	0
$343$	0	0
$344$	1.66621	0.0898359
$345$	0	0
$346$	−19.1075	−1.02723
$347$	−0.567323	−0.0304555	−0.0152277	−	0.999884i	$-0.504847\pi$
$347$	−0.567323	−0.0304555	−0.0152277	+	0.999884i	$0.504847\pi$
$348$	0	0
$349$	0.00728378	0.000389892 0	0.000194946	−	1.00000i	$-0.499938\pi$
$349$	0.00728378	0.000389892 0	0.000194946		1.00000i	$0.499938\pi$
$350$	0	0
$351$	0	0
$352$	1.58836	0.0846601
$353$	−6.65383	−0.354148	−0.177074	−	0.984198i	$-0.556663\pi$
$353$	−6.65383	−0.354148	−0.177074	+	0.984198i	$0.556663\pi$
$354$	0	0
$355$	20.2051	1.07237
$356$	3.21015	0.170138
$357$	0	0
$358$	−16.0741	−0.849544
$359$	−0.797135	−0.0420712	−0.0210356	−	0.999779i	$-0.506696\pi$
$359$	−0.797135	−0.0420712	−0.0210356	+	0.999779i	$0.506696\pi$
$360$	0	0
$361$	31.3942	1.65232
$362$	8.05308	0.423261
$363$	0	0
$364$	0	0
$365$	25.4981	1.33463
$366$	0	0
$367$	−15.4327	−0.805579	−0.402790	−	0.915293i	$-0.631959\pi$
$367$	−15.4327	−0.805579	−0.402790	+	0.915293i	$0.631959\pi$
$368$	−0.300372	−0.0156580
$369$	0	0
$370$	1.58836	0.0825751
$371$	0	0
$372$	0	0
$373$	10.2422	0.530321	0.265160	−	0.964204i	$-0.414575\pi$
$373$	10.2422	0.530321	0.265160	+	0.964204i	$0.414575\pi$
$374$	−8.57598	−0.443454
$375$	0	0
$376$	−2.66621	−0.137499
$377$	−39.8131	−2.05048
$378$	0	0
$379$	25.0087	1.28461	0.642304	−	0.766450i	$-0.277979\pi$
$379$	25.0087	1.28461	0.642304	+	0.766450i	$0.277979\pi$
$380$	−11.2756	−0.578427
$381$	0	0
$382$	23.9629	1.22605
$383$	6.26695	0.320226	0.160113	−	0.987099i	$-0.448814\pi$
$383$	6.26695	0.320226	0.160113	+	0.987099i	$0.448814\pi$
$384$	0	0
$385$	0	0
$386$	9.76509	0.497030
$387$	0	0
$388$	−1.42402	−0.0722934
$389$	21.6342	1.09690	0.548448	−	0.836185i	$-0.315219\pi$
$389$	21.6342	1.09690	0.548448	+	0.836185i	$0.315219\pi$
$390$	0	0
$391$	1.62178	0.0820172
$392$	0	0
$393$	0	0
$394$	18.2436	0.919098
$395$	−13.3214	−0.670273
$396$	0	0
$397$	−4.10617	−0.206083	−0.103041	−	0.994677i	$-0.532857\pi$
$397$	−4.10617	−0.206083	−0.103041	+	0.994677i	$0.532857\pi$
$398$	−18.0989	−0.907215
$399$	0	0
$400$	−2.47710	−0.123855
$401$	−16.7417	−0.836041	−0.418021	−	0.908438i	$-0.637276\pi$
$401$	−16.7417	−0.836041	−0.418021	+	0.908438i	$0.637276\pi$
$402$	0	0
$403$	13.0472	0.649926
$404$	−12.0334	−0.598685
$405$	0	0
$406$	0	0
$407$	−1.58836	−0.0787323
$408$	0	0
$409$	−8.76509	−0.433406	−0.216703	−	0.976238i	$-0.569530\pi$
$409$	−8.76509	−0.433406	−0.216703	+	0.976238i	$0.569530\pi$
$410$	−9.33379	−0.460963
$411$	0	0
$412$	−6.09888	−0.300470
$413$	0	0
$414$	0	0
$415$	−3.75781	−0.184464
$416$	−4.81089	−0.235873
$417$	0	0
$418$	11.2756	0.551508
$419$	−0.420297	−0.0205329	−0.0102664	−	0.999947i	$-0.503268\pi$
$419$	−0.420297	−0.0205329	−0.0102664	+	0.999947i	$0.503268\pi$
$420$	0	0
$421$	−6.57598	−0.320494	−0.160247	−	0.987077i	$-0.551229\pi$
$421$	−6.57598	−0.320494	−0.160247	+	0.987077i	$0.551229\pi$
$422$	−0.332415	−0.0161817
$423$	0	0
$424$	4.88874	0.237418
$425$	13.3745	0.648758
$426$	0	0
$427$	0	0
$428$	−3.08650	−0.149192
$429$	0	0
$430$	−2.64654	−0.127628
$431$	22.0879	1.06394	0.531968	−	0.846765i	$-0.321453\pi$
$431$	22.0879	1.06394	0.531968	+	0.846765i	$0.321453\pi$
$432$	0	0
$433$	−9.43268	−0.453306	−0.226653	−	0.973976i	$-0.572778\pi$
$433$	−9.43268	−0.453306	−0.226653	+	0.973976i	$0.572778\pi$
$434$	0	0
$435$	0	0
$436$	−2.28799	−0.109575
$437$	−2.13231	−0.102002
$438$	0	0
$439$	−31.2064	−1.48940	−0.744701	−	0.667398i	$-0.767408\pi$
$439$	−31.2064	−1.48940	−0.744701	+	0.667398i	$0.767408\pi$
$440$	−2.52290	−0.120275
$441$	0	0
$442$	25.9752	1.23552
$443$	−13.0545	−0.620236	−0.310118	−	0.950698i	$-0.600369\pi$
$443$	−13.0545	−0.620236	−0.310118	+	0.950698i	$0.600369\pi$
$444$	0	0
$445$	−5.09888	−0.241710
$446$	−6.33242	−0.299849
$447$	0	0
$448$	0	0
$449$	9.91706	0.468015	0.234008	−	0.972235i	$-0.424816\pi$
$449$	9.91706	0.468015	0.234008	+	0.972235i	$0.424816\pi$
$450$	0	0
$451$	9.33379	0.439511
$452$	−19.4647	−0.915543
$453$	0	0
$454$	23.3090	1.09395
$455$	0	0
$456$	0	0
$457$	−24.5229	−1.14713	−0.573566	−	0.819159i	$-0.694441\pi$
$457$	−24.5229	−1.14713	−0.573566	+	0.819159i	$0.694441\pi$
$458$	−4.95420	−0.231495
$459$	0	0
$460$	0.477100	0.0222449
$461$	3.51052	0.163501	0.0817506	−	0.996653i	$-0.473949\pi$
$461$	3.51052	0.163501	0.0817506	+	0.996653i	$0.473949\pi$
$462$	0	0
$463$	−17.3883	−0.808101	−0.404050	−	0.914737i	$-0.632398\pi$
$463$	−17.3883	−0.808101	−0.404050	+	0.914737i	$0.632398\pi$
$464$	8.27561	0.384186
$465$	0	0
$466$	−14.2756	−0.661305
$467$	13.3979	0.619980	0.309990	−	0.950740i	$-0.399674\pi$
$467$	13.3979	0.619980	0.309990	+	0.950740i	$0.399674\pi$
$468$	0	0
$469$	0	0
$470$	4.23491	0.195342
$471$	0	0
$472$	−6.47710	−0.298133
$473$	2.64654	0.121688
$474$	0	0
$475$	−17.5846	−0.806839
$476$	0	0
$477$	0	0
$478$	4.97524	0.227562
$479$	−20.8058	−0.950641	−0.475321	−	0.879813i	$-0.657668\pi$
$479$	−20.8058	−0.950641	−0.475321	+	0.879813i	$0.657668\pi$
$480$	0	0
$481$	4.81089	0.219358
$482$	−13.0000	−0.592134
$483$	0	0
$484$	−8.47710	−0.385323
$485$	2.26186	0.102706
$486$	0	0
$487$	−32.4944	−1.47246	−0.736231	−	0.676730i	$-0.763397\pi$
$487$	−32.4944	−1.47246	−0.736231	+	0.676730i	$0.763397\pi$
$488$	−4.47710	−0.202669
$489$	0	0
$490$	0	0
$491$	−19.3214	−0.871963	−0.435982	−	0.899956i	$-0.643599\pi$
$491$	−19.3214	−0.871963	−0.435982	+	0.899956i	$0.643599\pi$
$492$	0	0
$493$	−44.6822	−2.01238
$494$	−34.1520	−1.53657
$495$	0	0
$496$	−2.71201	−0.121773
$497$	0	0
$498$	0	0
$499$	−11.1506	−0.499169	−0.249585	−	0.968353i	$-0.580294\pi$
$499$	−11.1506	−0.499169	−0.249585	+	0.968353i	$0.580294\pi$
$500$	11.8764	0.531127
$501$	0	0
$502$	−2.43268	−0.108576
$503$	40.7651	1.81763	0.908813	−	0.417204i	$-0.136990\pi$
$503$	40.7651	1.81763	0.908813	+	0.417204i	$0.136990\pi$
$504$	0	0
$505$	19.1135	0.850537
$506$	−0.477100	−0.0212097
$507$	0	0
$508$	−13.4400	−0.596302
$509$	−1.44506	−0.0640510	−0.0320255	−	0.999487i	$-0.510196\pi$
$509$	−1.44506	−0.0640510	−0.0320255	+	0.999487i	$0.510196\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	0.987620	0.0435621
$515$	9.68725	0.426871
$516$	0	0
$517$	−4.23491	−0.186251
$518$	0	0
$519$	0	0
$520$	7.64145	0.335100
$521$	19.2843	0.844859	0.422430	−	0.906396i	$-0.361177\pi$
$521$	19.2843	0.844859	0.422430	+	0.906396i	$0.361177\pi$
$522$	0	0
$523$	36.6908	1.60438	0.802189	−	0.597071i	$-0.203669\pi$
$523$	36.6908	1.60438	0.802189	+	0.597071i	$0.203669\pi$
$524$	3.17673	0.138776
$525$	0	0
$526$	−17.1854	−0.749319
$527$	14.6428	0.637851
$528$	0	0
$529$	−22.9098	−0.996077
$530$	−7.76509	−0.337294
$531$	0	0
$532$	0	0
$533$	−28.2705	−1.22453
$534$	0	0
$535$	4.90249	0.211953
$536$	−10.0531	−0.434227
$537$	0	0
$538$	22.9047	0.987491
$539$	0	0
$540$	0	0
$541$	3.25085	0.139765	0.0698825	−	0.997555i	$-0.477738\pi$
$541$	3.25085	0.139765	0.0698825	+	0.997555i	$0.477738\pi$
$542$	−14.0073	−0.601664
$543$	0	0
$544$	−5.39926	−0.231491
$545$	3.63416	0.155670
$546$	0	0
$547$	5.91706	0.252995	0.126498	−	0.991967i	$-0.459626\pi$
$547$	5.91706	0.252995	0.126498	+	0.991967i	$0.459626\pi$
$548$	21.2632	0.908320
$549$	0	0
$550$	−3.93454	−0.167769
$551$	58.7476	2.50273
$552$	0	0
$553$	0	0
$554$	28.2953	1.20215
$555$	0	0
$556$	−13.0531	−0.553574
$557$	25.6080	1.08505	0.542523	−	0.840041i	$-0.317469\pi$
$557$	25.6080	1.08505	0.542523	+	0.840041i	$0.317469\pi$
$558$	0	0
$559$	−8.01594	−0.339038
$560$	0	0
$561$	0	0
$562$	−17.5956	−0.742228
$563$	46.6377	1.96555	0.982773	−	0.184817i	$-0.0591692\pi$
$563$	46.6377	1.96555	0.982773	+	0.184817i	$0.0591692\pi$
$564$	0	0
$565$	30.9171	1.30069
$566$	−18.5229	−0.778576
$567$	0	0
$568$	−12.7207	−0.533747
$569$	−31.1978	−1.30788	−0.653939	−	0.756547i	$-0.726885\pi$
$569$	−31.1978	−1.30788	−0.653939	+	0.756547i	$0.726885\pi$
$570$	0	0
$571$	−15.6762	−0.656030	−0.328015	−	0.944672i	$-0.606380\pi$
$571$	−15.6762	−0.656030	−0.328015	+	0.944672i	$0.606380\pi$
$572$	−7.64145	−0.319505
$573$	0	0
$574$	0	0
$575$	0.744051	0.0310291
$576$	0	0
$577$	−13.9913	−0.582467	−0.291234	−	0.956652i	$-0.594066\pi$
$577$	−13.9913	−0.582467	−0.291234	+	0.956652i	$0.594066\pi$
$578$	12.1520	0.505455
$579$	0	0
$580$	−13.1447	−0.545803
$581$	0	0
$582$	0	0
$583$	7.76509	0.321597
$584$	−16.0531	−0.664281
$585$	0	0
$586$	−14.0851	−0.581851
$587$	2.89602	0.119532	0.0597658	−	0.998212i	$-0.480965\pi$
$587$	2.89602	0.119532	0.0597658	+	0.998212i	$0.480965\pi$
$588$	0	0
$589$	−19.2522	−0.793274
$590$	10.2880	0.423550
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−4.08788	−0.167869	−0.0839346	−	0.996471i	$-0.526749\pi$
$593$	−4.08788	−0.167869	−0.0839346	+	0.996471i	$0.526749\pi$
$594$	0	0
$595$	0	0
$596$	−5.20877	−0.213360
$597$	0	0
$598$	1.44506	0.0590928
$599$	19.7651	0.807580	0.403790	−	0.914852i	$-0.367693\pi$
$599$	19.7651	0.807580	0.403790	+	0.914852i	$0.367693\pi$
$600$	0	0
$601$	26.8640	1.09580	0.547902	−	0.836542i	$-0.315427\pi$
$601$	26.8640	1.09580	0.547902	+	0.836542i	$0.315427\pi$
$602$	0	0
$603$	0	0
$604$	−0.522900	−0.0212765
$605$	13.4647	0.547419
$606$	0	0
$607$	−15.2422	−0.618661	−0.309331	−	0.950955i	$-0.600105\pi$
$607$	−15.2422	−0.618661	−0.309331	+	0.950955i	$0.600105\pi$
$608$	7.09888	0.287898
$609$	0	0
$610$	7.11126	0.287927
$611$	12.8268	0.518918
$612$	0	0
$613$	2.72067	0.109887	0.0549434	−	0.998489i	$-0.482502\pi$
$613$	2.72067	0.109887	0.0549434	+	0.998489i	$0.482502\pi$
$614$	−5.85532	−0.236301
$615$	0	0
$616$	0	0
$617$	−18.4362	−0.742215	−0.371108	−	0.928590i	$-0.621022\pi$
$617$	−18.4362	−0.742215	−0.371108	+	0.928590i	$0.621022\pi$
$618$	0	0
$619$	0.107546	0.00432262	0.00216131	−	0.999998i	$-0.499312\pi$
$619$	0.107546	0.00432262	0.00216131	+	0.999998i	$0.499312\pi$
$620$	4.30766	0.173000
$621$	0	0
$622$	−0.810892	−0.0325138
$623$	0	0
$624$	0	0
$625$	−6.47848	−0.259139
$626$	10.5760	0.422701
$627$	0	0
$628$	8.86398	0.353711
$629$	5.39926	0.215282
$630$	0	0
$631$	35.7266	1.42225	0.711126	−	0.703064i	$-0.248185\pi$
$631$	35.7266	1.42225	0.711126	+	0.703064i	$0.248185\pi$
$632$	8.38688	0.333612
$633$	0	0
$634$	−12.1964	−0.484381
$635$	21.3475	0.847152
$636$	0	0
$637$	0	0
$638$	13.1447	0.520403
$639$	0	0
$640$	−1.58836	−0.0627856
$641$	−17.3128	−0.683813	−0.341906	−	0.939734i	$-0.611073\pi$
$641$	−17.3128	−0.683813	−0.341906	+	0.939734i	$0.611073\pi$
$642$	0	0
$643$	−28.9642	−1.14224	−0.571119	−	0.820867i	$-0.693491\pi$
$643$	−28.9642	−1.14224	−0.571119	+	0.820867i	$0.693491\pi$
$644$	0	0
$645$	0	0
$646$	−38.3287	−1.50802
$647$	2.55632	0.100499	0.0502497	−	0.998737i	$-0.483998\pi$
$647$	2.55632	0.100499	0.0502497	+	0.998737i	$0.483998\pi$
$648$	0	0
$649$	−10.2880	−0.403839
$650$	11.9171	0.467426
$651$	0	0
$652$	−21.9629	−0.860132
$653$	−29.9766	−1.17308	−0.586538	−	0.809922i	$-0.699509\pi$
$653$	−29.9766	−1.17308	−0.586538	+	0.809922i	$0.699509\pi$
$654$	0	0
$655$	−5.04580	−0.197156
$656$	5.87636	0.229433
$657$	0	0
$658$	0	0
$659$	−15.2632	−0.594571	−0.297286	−	0.954789i	$-0.596081\pi$
$659$	−15.2632	−0.594571	−0.297286	+	0.954789i	$0.596081\pi$
$660$	0	0
$661$	−27.2522	−1.05999	−0.529994	−	0.848001i	$-0.677806\pi$
$661$	−27.2522	−1.05999	−0.529994	+	0.848001i	$0.677806\pi$
$662$	−15.6662	−0.608884
$663$	0	0
$664$	2.36584	0.0918122
$665$	0	0
$666$	0	0
$667$	−2.48576	−0.0962491
$668$	3.30037	0.127695
$669$	0	0
$670$	15.9680	0.616896
$671$	−7.11126	−0.274527
$672$	0	0
$673$	−46.4559	−1.79074	−0.895372	−	0.445319i	$-0.853090\pi$
$673$	−46.4559	−1.79074	−0.895372	+	0.445319i	$0.853090\pi$
$674$	8.42402	0.324481
$675$	0	0
$676$	10.1447	0.390180
$677$	−5.09888	−0.195966	−0.0979830	−	0.995188i	$-0.531239\pi$
$677$	−5.09888	−0.195966	−0.0979830	+	0.995188i	$0.531239\pi$
$678$	0	0
$679$	0	0
$680$	8.57598	0.328874
$681$	0	0
$682$	−4.30766	−0.164949
$683$	−15.5439	−0.594772	−0.297386	−	0.954757i	$-0.596115\pi$
$683$	−15.5439	−0.594772	−0.297386	+	0.954757i	$0.596115\pi$
$684$	0	0
$685$	−33.7738	−1.29043
$686$	0	0
$687$	0	0
$688$	1.66621	0.0635236
$689$	−23.5192	−0.896009
$690$	0	0
$691$	23.2967	0.886246	0.443123	−	0.896461i	$-0.353870\pi$
$691$	23.2967	0.886246	0.443123	+	0.896461i	$0.353870\pi$
$692$	−19.1075	−0.726360
$693$	0	0
$694$	−0.567323	−0.0215353
$695$	20.7330	0.786449
$696$	0	0
$697$	−31.7280	−1.20178
$698$	0.00728378	0.000275695 0
$699$	0	0
$700$	0	0
$701$	45.6464	1.72404	0.862020	−	0.506874i	$-0.169199\pi$
$701$	45.6464	1.72404	0.862020	+	0.506874i	$0.169199\pi$
$702$	0	0
$703$	−7.09888	−0.267739
$704$	1.58836	0.0598637
$705$	0	0
$706$	−6.65383	−0.250420
$707$	0	0
$708$	0	0
$709$	18.0014	0.676056	0.338028	−	0.941136i	$-0.390240\pi$
$709$	18.0014	0.676056	0.338028	+	0.941136i	$0.390240\pi$
$710$	20.2051	0.758282
$711$	0	0
$712$	3.21015	0.120305
$713$	0.814611	0.0305074
$714$	0	0
$715$	12.1374	0.453913
$716$	−16.0741	−0.600718
$717$	0	0
$718$	−0.797135	−0.0297488
$719$	36.8777	1.37531	0.687654	−	0.726039i	$-0.258641\pi$
$719$	36.8777	1.37531	0.687654	+	0.726039i	$0.258641\pi$
$720$	0	0
$721$	0	0
$722$	31.3942	1.16837
$723$	0	0
$724$	8.05308	0.299291
$725$	−20.4995	−0.761333
$726$	0	0
$727$	−30.4858	−1.13065	−0.565327	−	0.824867i	$-0.691250\pi$
$727$	−30.4858	−1.13065	−0.565327	+	0.824867i	$0.691250\pi$
$728$	0	0
$729$	0	0
$730$	25.4981	0.943729
$731$	−8.99628	−0.332739
$732$	0	0
$733$	6.15059	0.227177	0.113589	−	0.993528i	$-0.463765\pi$
$733$	6.15059	0.227177	0.113589	+	0.993528i	$0.463765\pi$
$734$	−15.4327	−0.569630
$735$	0	0
$736$	−0.300372	−0.0110719
$737$	−15.9680	−0.588187
$738$	0	0
$739$	40.7824	1.50021	0.750103	−	0.661321i	$-0.230004\pi$
$739$	40.7824	1.50021	0.750103	+	0.661321i	$0.230004\pi$
$740$	1.58836	0.0583894
$741$	0	0
$742$	0	0
$743$	14.5054	0.532152	0.266076	−	0.963952i	$-0.414273\pi$
$743$	14.5054	0.532152	0.266076	+	0.963952i	$0.414273\pi$
$744$	0	0
$745$	8.27342	0.303115
$746$	10.2422	0.374993
$747$	0	0
$748$	−8.57598	−0.313569
$749$	0	0
$750$	0	0
$751$	4.18911	0.152863	0.0764314	−	0.997075i	$-0.475647\pi$
$751$	4.18911	0.152863	0.0764314	+	0.997075i	$0.475647\pi$
$752$	−2.66621	−0.0972266
$753$	0	0
$754$	−39.8131	−1.44991
$755$	0.830556	0.0302270
$756$	0	0
$757$	2.38688	0.0867525	0.0433763	−	0.999059i	$-0.486189\pi$
$757$	2.38688	0.0867525	0.0433763	+	0.999059i	$0.486189\pi$
$758$	25.0087	0.908355
$759$	0	0
$760$	−11.2756	−0.409009
$761$	−3.63416	−0.131738	−0.0658692	−	0.997828i	$-0.520982\pi$
$761$	−3.63416	−0.131738	−0.0658692	+	0.997828i	$0.520982\pi$
$762$	0	0
$763$	0	0
$764$	23.9629	0.866946
$765$	0	0
$766$	6.26695	0.226434
$767$	31.1606	1.12515
$768$	0	0
$769$	39.9344	1.44007	0.720035	−	0.693937i	$-0.244126\pi$
$769$	39.9344	1.44007	0.720035	+	0.693937i	$0.244126\pi$
$770$	0	0
$771$	0	0
$772$	9.76509	0.351453
$773$	36.1396	1.29985	0.649925	−	0.759998i	$-0.274800\pi$
$773$	36.1396	1.29985	0.649925	+	0.759998i	$0.274800\pi$
$774$	0	0
$775$	6.71791	0.241315
$776$	−1.42402	−0.0511192
$777$	0	0
$778$	21.6342	0.775622
$779$	41.7156	1.49462
$780$	0	0
$781$	−20.2051	−0.722994
$782$	1.62178	0.0579949
$783$	0	0
$784$	0	0
$785$	−14.0792	−0.502509
$786$	0	0
$787$	−44.6377	−1.59116	−0.795582	−	0.605846i	$-0.792835\pi$
$787$	−44.6377	−1.59116	−0.795582	+	0.605846i	$0.792835\pi$
$788$	18.2436	0.649900
$789$	0	0
$790$	−13.3214	−0.473955
$791$	0	0
$792$	0	0
$793$	21.5388	0.764867
$794$	−4.10617	−0.145722
$795$	0	0
$796$	−18.0989	−0.641498
$797$	52.5672	1.86202	0.931012	−	0.364988i	$-0.118927\pi$
$797$	52.5672	1.86202	0.931012	+	0.364988i	$0.118927\pi$
$798$	0	0
$799$	14.3955	0.509278
$800$	−2.47710	−0.0875787
$801$	0	0
$802$	−16.7417	−0.591170
$803$	−25.4981	−0.899810
$804$	0	0
$805$	0	0
$806$	13.0472	0.459567
$807$	0	0
$808$	−12.0334	−0.423334
$809$	14.8058	0.520544	0.260272	−	0.965535i	$-0.416188\pi$
$809$	14.8058	0.520544	0.260272	+	0.965535i	$0.416188\pi$
$810$	0	0
$811$	27.0704	0.950571	0.475285	−	0.879832i	$-0.342345\pi$
$811$	27.0704	0.950571	0.475285	+	0.879832i	$0.342345\pi$
$812$	0	0
$813$	0	0
$814$	−1.58836	−0.0556721
$815$	34.8850	1.22197
$816$	0	0
$817$	11.8282	0.413817
$818$	−8.76509	−0.306464
$819$	0	0
$820$	−9.33379	−0.325950
$821$	−43.8182	−1.52926	−0.764632	−	0.644467i	$-0.777079\pi$
$821$	−43.8182	−1.52926	−0.764632	+	0.644467i	$0.777079\pi$
$822$	0	0
$823$	31.3425	1.09253	0.546265	−	0.837613i	$-0.316049\pi$
$823$	31.3425	1.09253	0.546265	+	0.837613i	$0.316049\pi$
$824$	−6.09888	−0.212465
$825$	0	0
$826$	0	0
$827$	14.7665	0.513480	0.256740	−	0.966480i	$-0.417352\pi$
$827$	14.7665	0.513480	0.256740	+	0.966480i	$0.417352\pi$
$828$	0	0
$829$	30.0073	1.04220	0.521098	−	0.853497i	$-0.325523\pi$
$829$	30.0073	1.04220	0.521098	+	0.853497i	$0.325523\pi$
$830$	−3.75781	−0.130435
$831$	0	0
$832$	−4.81089	−0.166788
$833$	0	0
$834$	0	0
$835$	−5.24219	−0.181414
$836$	11.2756	0.389975
$837$	0	0
$838$	−0.420297	−0.0145189
$839$	36.0334	1.24401	0.622006	−	0.783013i	$-0.286318\pi$
$839$	36.0334	1.24401	0.622006	+	0.783013i	$0.286318\pi$
$840$	0	0
$841$	39.4858	1.36158
$842$	−6.57598	−0.226623
$843$	0	0
$844$	−0.332415	−0.0114422
$845$	−16.1135	−0.554320
$846$	0	0
$847$	0	0
$848$	4.88874	0.167880
$849$	0	0
$850$	13.3745	0.458741
$851$	0.300372	0.0102966
$852$	0	0
$853$	24.5316	0.839945	0.419972	−	0.907537i	$-0.362040\pi$
$853$	24.5316	0.839945	0.419972	+	0.907537i	$0.362040\pi$
$854$	0	0
$855$	0	0
$856$	−3.08650	−0.105495
$857$	−29.0480	−0.992260	−0.496130	−	0.868248i	$-0.665246\pi$
$857$	−29.0480	−0.992260	−0.496130	+	0.868248i	$0.665246\pi$
$858$	0	0
$859$	25.2953	0.863064	0.431532	−	0.902098i	$-0.357973\pi$
$859$	25.2953	0.863064	0.431532	+	0.902098i	$0.357973\pi$
$860$	−2.64654	−0.0902464
$861$	0	0
$862$	22.0879	0.752316
$863$	−2.69963	−0.0918964	−0.0459482	−	0.998944i	$-0.514631\pi$
$863$	−2.69963	−0.0918964	−0.0459482	+	0.998944i	$0.514631\pi$
$864$	0	0
$865$	30.3497	1.03192
$866$	−9.43268	−0.320535
$867$	0	0
$868$	0	0
$869$	13.3214	0.451898
$870$	0	0
$871$	48.3643	1.63876
$872$	−2.28799	−0.0774812
$873$	0	0
$874$	−2.13231	−0.0721263
$875$	0	0
$876$	0	0
$877$	−11.0916	−0.374537	−0.187268	−	0.982309i	$-0.559963\pi$
$877$	−11.0916	−0.374537	−0.187268	+	0.982309i	$0.559963\pi$
$878$	−31.2064	−1.05317
$879$	0	0
$880$	−2.52290	−0.0850469
$881$	40.3942	1.36091	0.680457	−	0.732788i	$-0.261781\pi$
$881$	40.3942	1.36091	0.680457	+	0.732788i	$0.261781\pi$
$882$	0	0
$883$	−33.2581	−1.11923	−0.559613	−	0.828754i	$-0.689050\pi$
$883$	−33.2581	−1.11923	−0.559613	+	0.828754i	$0.689050\pi$
$884$	25.9752	0.873642
$885$	0	0
$886$	−13.0545	−0.438573
$887$	−40.5672	−1.36211	−0.681056	−	0.732231i	$-0.738479\pi$
$887$	−40.5672	−1.36211	−0.681056	+	0.732231i	$0.738479\pi$
$888$	0	0
$889$	0	0
$890$	−5.09888	−0.170915
$891$	0	0
$892$	−6.33242	−0.212025
$893$	−18.9271	−0.633371
$894$	0	0
$895$	25.5316	0.853426
$896$	0	0
$897$	0	0
$898$	9.91706	0.330937
$899$	−22.4435	−0.748533
$900$	0	0
$901$	−26.3955	−0.879363
$902$	9.33379	0.310781
$903$	0	0
$904$	−19.4647	−0.647387
$905$	−12.7912	−0.425195
$906$	0	0
$907$	30.1135	0.999901	0.499950	−	0.866054i	$-0.333352\pi$
$907$	30.1135	0.999901	0.499950	+	0.866054i	$0.333352\pi$
$908$	23.3090	0.773537
$909$	0	0
$910$	0	0
$911$	29.2225	0.968186	0.484093	−	0.875017i	$-0.339150\pi$
$911$	29.2225	0.968186	0.484093	+	0.875017i	$0.339150\pi$
$912$	0	0
$913$	3.75781	0.124365
$914$	−24.5229	−0.811145
$915$	0	0
$916$	−4.95420	−0.163691
$917$	0	0
$918$	0	0
$919$	11.0472	0.364413	0.182206	−	0.983260i	$-0.441676\pi$
$919$	11.0472	0.364413	0.182206	+	0.983260i	$0.441676\pi$
$920$	0.477100	0.0157295
$921$	0	0
$922$	3.51052	0.115613
$923$	61.1978	2.01435
$924$	0	0
$925$	2.47710	0.0814465
$926$	−17.3883	−0.571413
$927$	0	0
$928$	8.27561	0.271660
$929$	42.3338	1.38893	0.694463	−	0.719528i	$-0.255642\pi$
$929$	42.3338	1.38893	0.694463	+	0.719528i	$0.255642\pi$
$930$	0	0
$931$	0	0
$932$	−14.2756	−0.467613
$933$	0	0
$934$	13.3979	0.438392
$935$	13.6218	0.445480
$936$	0	0
$937$	−11.7651	−0.384349	−0.192174	−	0.981361i	$-0.561554\pi$
$937$	−11.7651	−0.384349	−0.192174	+	0.981361i	$0.561554\pi$
$938$	0	0
$939$	0	0
$940$	4.23491	0.138127
$941$	−14.5760	−0.475164	−0.237582	−	0.971368i	$-0.576355\pi$
$941$	−14.5760	−0.475164	−0.237582	+	0.971368i	$0.576355\pi$
$942$	0	0
$943$	−1.76509	−0.0574793
$944$	−6.47710	−0.210812
$945$	0	0
$946$	2.64654	0.0860466
$947$	−6.24357	−0.202889	−0.101444	−	0.994841i	$-0.532346\pi$
$947$	−6.24357	−0.202889	−0.101444	+	0.994841i	$0.532346\pi$
$948$	0	0
$949$	77.2297	2.50698
$950$	−17.5846	−0.570521
$951$	0	0
$952$	0	0
$953$	−28.0173	−0.907570	−0.453785	−	0.891111i	$-0.649927\pi$
$953$	−28.0173	−0.907570	−0.453785	+	0.891111i	$0.649927\pi$
$954$	0	0
$955$	−38.0617	−1.23165
$956$	4.97524	0.160911
$957$	0	0
$958$	−20.8058	−0.672205
$959$	0	0
$960$	0	0
$961$	−23.6450	−0.762742
$962$	4.81089	0.155109
$963$	0	0
$964$	−13.0000	−0.418702
$965$	−15.5105	−0.499301
$966$	0	0
$967$	−31.5673	−1.01514	−0.507568	−	0.861612i	$-0.669456\pi$
$967$	−31.5673	−1.01514	−0.507568	+	0.861612i	$0.669456\pi$
$968$	−8.47710	−0.272464
$969$	0	0
$970$	2.26186	0.0726238
$971$	5.64283	0.181087	0.0905434	−	0.995893i	$-0.471140\pi$
$971$	5.64283	0.181087	0.0905434	+	0.995893i	$0.471140\pi$
$972$	0	0
$973$	0	0
$974$	−32.4944	−1.04119
$975$	0	0
$976$	−4.47710	−0.143308
$977$	−6.49304	−0.207731	−0.103865	−	0.994591i	$-0.533121\pi$
$977$	−6.49304	−0.207731	−0.103865	+	0.994591i	$0.533121\pi$
$978$	0	0
$979$	5.09888	0.162961
$980$	0	0
$981$	0	0
$982$	−19.3214	−0.616571
$983$	30.3063	0.966620	0.483310	−	0.875449i	$-0.339434\pi$
$983$	30.3063	0.966620	0.483310	+	0.875449i	$0.339434\pi$
$984$	0	0
$985$	−28.9774	−0.923298
$986$	−44.6822	−1.42297
$987$	0	0
$988$	−34.1520	−1.08652
$989$	−0.500482	−0.0159144
$990$	0	0
$991$	−22.3338	−0.709456	−0.354728	−	0.934969i	$-0.615427\pi$
$991$	−22.3338	−0.709456	−0.354728	+	0.934969i	$0.615427\pi$
$992$	−2.71201	−0.0861063
$993$	0	0
$994$	0	0
$995$	28.7476	0.911361
$996$	0	0
$997$	−8.76509	−0.277593	−0.138797	−	0.990321i	$-0.544323\pi$
$997$	−8.76509	−0.277593	−0.138797	+	0.990321i	$0.544323\pi$
$998$	−11.1506	−0.352966
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.ca.1.1		3
3.2	odd	2		7938.2.a.bv.1.3		3
7.2	even	3		1134.2.g.l.487.3		6
7.4	even	3		1134.2.g.l.163.3		6
7.6	odd	2		7938.2.a.bz.1.3		3
9.2	odd	6		882.2.f.n.589.3		6
9.4	even	3		2646.2.f.l.883.3		6
9.5	odd	6		882.2.f.n.295.3		6
9.7	even	3		2646.2.f.l.1765.3		6
21.2	odd	6		1134.2.g.m.487.1		6
21.11	odd	6		1134.2.g.m.163.1		6
21.20	even	2		7938.2.a.bw.1.1		3
63.2	odd	6		126.2.h.d.67.3	yes	6
63.4	even	3		378.2.h.c.289.1		6
63.5	even	6		882.2.e.o.655.3		6
63.11	odd	6		126.2.e.c.121.1	yes	6
63.13	odd	6		2646.2.f.m.883.1		6
63.16	even	3		378.2.h.c.361.1		6
63.20	even	6		882.2.f.o.589.1		6
63.23	odd	6		126.2.e.c.25.1	✓	6
63.25	even	3		378.2.e.d.37.3		6
63.31	odd	6		2646.2.h.o.667.3		6
63.32	odd	6		126.2.h.d.79.3	yes	6
63.34	odd	6		2646.2.f.m.1765.1		6
63.38	even	6		882.2.e.o.373.3		6
63.40	odd	6		2646.2.e.p.2125.1		6
63.41	even	6		882.2.f.o.295.1		6
63.47	even	6		882.2.h.p.67.1		6
63.52	odd	6		2646.2.e.p.1549.1		6
63.58	even	3		378.2.e.d.235.3		6
63.59	even	6		882.2.h.p.79.1		6
63.61	odd	6		2646.2.h.o.361.3		6
252.11	even	6		1008.2.q.g.625.3		6
252.23	even	6		1008.2.q.g.529.3		6
252.67	odd	6		3024.2.t.h.289.1		6
252.79	odd	6		3024.2.t.h.1873.1		6
252.95	even	6		1008.2.t.h.961.1		6
252.151	odd	6		3024.2.q.g.2305.3		6
252.191	even	6		1008.2.t.h.193.1		6
252.247	odd	6		3024.2.q.g.2881.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.e.c.25.1	✓	6	63.23	odd	6
126.2.e.c.121.1	yes	6	63.11	odd	6
126.2.h.d.67.3	yes	6	63.2	odd	6
126.2.h.d.79.3	yes	6	63.32	odd	6
378.2.e.d.37.3		6	63.25	even	3
378.2.e.d.235.3		6	63.58	even	3
378.2.h.c.289.1		6	63.4	even	3
378.2.h.c.361.1		6	63.16	even	3
882.2.e.o.373.3		6	63.38	even	6
882.2.e.o.655.3		6	63.5	even	6
882.2.f.n.295.3		6	9.5	odd	6
882.2.f.n.589.3		6	9.2	odd	6
882.2.f.o.295.1		6	63.41	even	6
882.2.f.o.589.1		6	63.20	even	6
882.2.h.p.67.1		6	63.47	even	6
882.2.h.p.79.1		6	63.59	even	6
1008.2.q.g.529.3		6	252.23	even	6
1008.2.q.g.625.3		6	252.11	even	6
1008.2.t.h.193.1		6	252.191	even	6
1008.2.t.h.961.1		6	252.95	even	6
1134.2.g.l.163.3		6	7.4	even	3
1134.2.g.l.487.3		6	7.2	even	3
1134.2.g.m.163.1		6	21.11	odd	6
1134.2.g.m.487.1		6	21.2	odd	6
2646.2.e.p.1549.1		6	63.52	odd	6
2646.2.e.p.2125.1		6	63.40	odd	6
2646.2.f.l.883.3		6	9.4	even	3
2646.2.f.l.1765.3		6	9.7	even	3
2646.2.f.m.883.1		6	63.13	odd	6
2646.2.f.m.1765.1		6	63.34	odd	6
2646.2.h.o.361.3		6	63.61	odd	6
2646.2.h.o.667.3		6	63.31	odd	6
3024.2.q.g.2305.3		6	252.151	odd	6
3024.2.q.g.2881.3		6	252.247	odd	6
3024.2.t.h.289.1		6	252.67	odd	6
3024.2.t.h.1873.1		6	252.79	odd	6
7938.2.a.bv.1.3		3	3.2	odd	2
7938.2.a.bw.1.1		3	21.20	even	2
7938.2.a.bz.1.3		3	7.6	odd	2
7938.2.a.ca.1.1		3	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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.58836 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.58836 q^{5} +1.00000 q^{8} -1.58836 q^{10} +1.58836 q^{11} -4.81089 q^{13} +1.00000 q^{16} -5.39926 q^{17} +7.09888 q^{19} -1.58836 q^{20} +1.58836 q^{22} -0.300372 q^{23} -2.47710 q^{25} -4.81089 q^{26} +8.27561 q^{29} -2.71201 q^{31} +1.00000 q^{32} -5.39926 q^{34} -1.00000 q^{37} +7.09888 q^{38} -1.58836 q^{40} +5.87636 q^{41} +1.66621 q^{43} +1.58836 q^{44} -0.300372 q^{46} -2.66621 q^{47} -2.47710 q^{50} -4.81089 q^{52} +4.88874 q^{53} -2.52290 q^{55} +8.27561 q^{58} -6.47710 q^{59} -4.47710 q^{61} -2.71201 q^{62} +1.00000 q^{64} +7.64145 q^{65} -10.0531 q^{67} -5.39926 q^{68} -12.7207 q^{71} -16.0531 q^{73} -1.00000 q^{74} +7.09888 q^{76} +8.38688 q^{79} -1.58836 q^{80} +5.87636 q^{82} +2.36584 q^{83} +8.57598 q^{85} +1.66621 q^{86} +1.58836 q^{88} +3.21015 q^{89} -0.300372 q^{92} -2.66621 q^{94} -11.2756 q^{95} -1.42402 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} + 3 q^{8} + q^{10} - q^{11} - 8 q^{13} + 3 q^{16} - 4 q^{17} + 3 q^{19} + q^{20} - q^{22} - 7 q^{23} - 2 q^{25} - 8 q^{26} - 5 q^{29} - 20 q^{31} + 3 q^{32} - 4 q^{34} - 3 q^{37} + 3 q^{38} + q^{40} + 6 q^{43} - q^{44} - 7 q^{46} - 9 q^{47} - 2 q^{50} - 8 q^{52} + 15 q^{53} - 13 q^{55} - 5 q^{58} - 14 q^{59} - 8 q^{61} - 20 q^{62} + 3 q^{64} - 12 q^{65} - q^{67} - 4 q^{68} - 7 q^{71} - 19 q^{73} - 3 q^{74} + 3 q^{76} - 5 q^{79} + q^{80} + 2 q^{83} + 2 q^{85} + 6 q^{86} - q^{88} - 9 q^{89} - 7 q^{92} - 9 q^{94} - 4 q^{95} - 28 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 + 3 * q^8 + q^10 - q^11 - 8 * q^13 + 3 * q^16 - 4 * q^17 + 3 * q^19 + q^20 - q^22 - 7 * q^23 - 2 * q^25 - 8 * q^26 - 5 * q^29 - 20 * q^31 + 3 * q^32 - 4 * q^34 - 3 * q^37 + 3 * q^38 + q^40 + 6 * q^43 - q^44 - 7 * q^46 - 9 * q^47 - 2 * q^50 - 8 * q^52 + 15 * q^53 - 13 * q^55 - 5 * q^58 - 14 * q^59 - 8 * q^61 - 20 * q^62 + 3 * q^64 - 12 * q^65 - q^67 - 4 * q^68 - 7 * q^71 - 19 * q^73 - 3 * q^74 + 3 * q^76 - 5 * q^79 + q^80 + 2 * q^83 + 2 * q^85 + 6 * q^86 - q^88 - 9 * q^89 - 7 * q^92 - 9 * q^94 - 4 * q^95 - 28 * q^97

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