LMFDB - Embedded newform 7938.2.a.bg.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1134)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.73205 q^{5} -1.00000 q^{8} +3.73205 q^{10} -4.19615 q^{11} -0.464102 q^{13} +1.00000 q^{16} -7.00000 q^{17} +2.73205 q^{19} -3.73205 q^{20} +4.19615 q^{22} +6.19615 q^{23} +8.92820 q^{25} +0.464102 q^{26} +8.46410 q^{29} +2.19615 q^{31} -1.00000 q^{32} +7.00000 q^{34} -6.66025 q^{37} -2.73205 q^{38} +3.73205 q^{40} -9.46410 q^{41} +5.46410 q^{43} -4.19615 q^{44} -6.19615 q^{46} -1.26795 q^{47} -8.92820 q^{50} -0.464102 q^{52} -2.53590 q^{53} +15.6603 q^{55} -8.46410 q^{58} +6.19615 q^{59} +9.92820 q^{61} -2.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} -3.26795 q^{67} -7.00000 q^{68} +13.4641 q^{71} -11.7321 q^{73} +6.66025 q^{74} +2.73205 q^{76} +15.1244 q^{79} -3.73205 q^{80} +9.46410 q^{82} +14.5885 q^{83} +26.1244 q^{85} -5.46410 q^{86} +4.19615 q^{88} +3.92820 q^{89} +6.19615 q^{92} +1.26795 q^{94} -10.1962 q^{95} +2.92820 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{16} - 14 q^{17} + 2 q^{19} - 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} - 6 q^{26} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 4 * q^10 + 2 * q^11 + 6 * q^13 + 2 * q^16 - 14 * q^17 + 2 * q^19 - 4 * q^20 - 2 * q^22 + 2 * q^23 + 4 * q^25 - 6 * q^26 + 10 * q^29 - 6 * q^31 - 2 * q^32 + 14 * q^34 + 4 * q^37 - 2 * q^38 + 4 * q^40 - 12 * q^41 + 4 * q^43 + 2 * q^44 - 2 * q^46 - 6 * q^47 - 4 * q^50 + 6 * q^52 - 12 * q^53 + 14 * q^55 - 10 * q^58 + 2 * q^59 + 6 * q^61 + 6 * q^62 + 2 * q^64 - 10 * q^67 - 14 * q^68 + 20 * q^71 - 20 * q^73 - 4 * q^74 + 2 * q^76 + 6 * q^79 - 4 * q^80 + 12 * q^82 - 2 * q^83 + 28 * q^85 - 4 * q^86 - 2 * q^88 - 6 * q^89 + 2 * q^92 + 6 * q^94 - 10 * q^95 - 8 * q^97

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
$5$	−3.73205	−1.66902	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−3.73205	−1.66902	−0.834512	+	0.550990i	$0.814250\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.73205	1.18018
$11$	−4.19615	−1.26519	−0.632594	−	0.774484i	$-0.718010\pi$
$11$	−4.19615	−1.26519	−0.632594	+	0.774484i	$0.718010\pi$
$12$	0	0
$13$	−0.464102	−0.128719	−0.0643593	−	0.997927i	$-0.520500\pi$
$13$	−0.464102	−0.128719	−0.0643593	+	0.997927i	$0.520500\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.00000	−1.69775	−0.848875	−	0.528594i	$-0.822719\pi$
$17$	−7.00000	−1.69775	−0.848875	+	0.528594i	$0.822719\pi$
$18$	0	0
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	−3.73205	−0.834512
$21$	0	0
$22$	4.19615	0.894623
$23$	6.19615	1.29199	0.645994	−	0.763343i	$-0.276443\pi$
$23$	6.19615	1.29199	0.645994	+	0.763343i	$0.276443\pi$
$24$	0	0
$25$	8.92820	1.78564
$26$	0.464102	0.0910178
$27$	0	0
$28$	0	0
$29$	8.46410	1.57174	0.785872	−	0.618389i	$-0.212214\pi$
$29$	8.46410	1.57174	0.785872	+	0.618389i	$0.212214\pi$
$30$	0	0
$31$	2.19615	0.394441	0.197220	−	0.980359i	$-0.436809\pi$
$31$	2.19615	0.394441	0.197220	+	0.980359i	$0.436809\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	7.00000	1.20049
$35$	0	0
$36$	0	0
$37$	−6.66025	−1.09494	−0.547470	−	0.836826i	$-0.684409\pi$
$37$	−6.66025	−1.09494	−0.547470	+	0.836826i	$0.684409\pi$
$38$	−2.73205	−0.443197
$39$	0	0
$40$	3.73205	0.590089
$41$	−9.46410	−1.47804	−0.739022	−	0.673681i	$-0.764712\pi$
$41$	−9.46410	−1.47804	−0.739022	+	0.673681i	$0.764712\pi$
$42$	0	0
$43$	5.46410	0.833268	0.416634	−	0.909074i	$-0.363210\pi$
$43$	5.46410	0.833268	0.416634	+	0.909074i	$0.363210\pi$
$44$	−4.19615	−0.632594
$45$	0	0
$46$	−6.19615	−0.913573
$47$	−1.26795	−0.184949	−0.0924747	−	0.995715i	$-0.529478\pi$
$47$	−1.26795	−0.184949	−0.0924747	+	0.995715i	$0.529478\pi$
$48$	0	0
$49$	0	0
$50$	−8.92820	−1.26264
$51$	0	0
$52$	−0.464102	−0.0643593
$53$	−2.53590	−0.348332	−0.174166	−	0.984716i	$-0.555723\pi$
$53$	−2.53590	−0.348332	−0.174166	+	0.984716i	$0.555723\pi$
$54$	0	0
$55$	15.6603	2.11163
$56$	0	0
$57$	0	0
$58$	−8.46410	−1.11139
$59$	6.19615	0.806670	0.403335	−	0.915052i	$-0.367851\pi$
$59$	6.19615	0.806670	0.403335	+	0.915052i	$0.367851\pi$
$60$	0	0
$61$	9.92820	1.27118	0.635588	−	0.772028i	$-0.280758\pi$
$61$	9.92820	1.27118	0.635588	+	0.772028i	$0.280758\pi$
$62$	−2.19615	−0.278912
$63$	0	0
$64$	1.00000	0.125000
$65$	1.73205	0.214834
$66$	0	0
$67$	−3.26795	−0.399244	−0.199622	−	0.979873i	$-0.563971\pi$
$67$	−3.26795	−0.399244	−0.199622	+	0.979873i	$0.563971\pi$
$68$	−7.00000	−0.848875
$69$	0	0
$70$	0	0
$71$	13.4641	1.59789	0.798947	−	0.601401i	$-0.205391\pi$
$71$	13.4641	1.59789	0.798947	+	0.601401i	$0.205391\pi$
$72$	0	0
$73$	−11.7321	−1.37313	−0.686566	−	0.727067i	$-0.740883\pi$
$73$	−11.7321	−1.37313	−0.686566	+	0.727067i	$0.740883\pi$
$74$	6.66025	0.774239
$75$	0	0
$76$	2.73205	0.313388
$77$	0	0
$78$	0	0
$79$	15.1244	1.70162	0.850811	−	0.525471i	$-0.176111\pi$
$79$	15.1244	1.70162	0.850811	+	0.525471i	$0.176111\pi$
$80$	−3.73205	−0.417256
$81$	0	0
$82$	9.46410	1.04514
$83$	14.5885	1.60129	0.800646	−	0.599138i	$-0.204490\pi$
$83$	14.5885	1.60129	0.800646	+	0.599138i	$0.204490\pi$
$84$	0	0
$85$	26.1244	2.83358
$86$	−5.46410	−0.589209
$87$	0	0
$88$	4.19615	0.447311
$89$	3.92820	0.416389	0.208194	−	0.978087i	$-0.433241\pi$
$89$	3.92820	0.416389	0.208194	+	0.978087i	$0.433241\pi$
$90$	0	0
$91$	0	0
$92$	6.19615	0.645994
$93$	0	0
$94$	1.26795	0.130779
$95$	−10.1962	−1.04610
$96$	0	0
$97$	2.92820	0.297314	0.148657	−	0.988889i	$-0.452505\pi$
$97$	2.92820	0.297314	0.148657	+	0.988889i	$0.452505\pi$
$98$	0	0
$99$	0	0
$100$	8.92820	0.892820
$101$	4.92820	0.490375	0.245187	−	0.969476i	$-0.421151\pi$
$101$	4.92820	0.490375	0.245187	+	0.969476i	$0.421151\pi$
$102$	0	0
$103$	−12.3923	−1.22105	−0.610525	−	0.791997i	$-0.709042\pi$
$103$	−12.3923	−1.22105	−0.610525	+	0.791997i	$0.709042\pi$
$104$	0.464102	0.0455089
$105$	0	0
$106$	2.53590	0.246308
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−7.19615	−0.689266	−0.344633	−	0.938737i	$-0.611997\pi$
$109$	−7.19615	−0.689266	−0.344633	+	0.938737i	$0.611997\pi$
$110$	−15.6603	−1.49315
$111$	0	0
$112$	0	0
$113$	−2.26795	−0.213351	−0.106675	−	0.994294i	$-0.534021\pi$
$113$	−2.26795	−0.213351	−0.106675	+	0.994294i	$0.534021\pi$
$114$	0	0
$115$	−23.1244	−2.15636
$116$	8.46410	0.785872
$117$	0	0
$118$	−6.19615	−0.570402
$119$	0	0
$120$	0	0
$121$	6.60770	0.600700
$122$	−9.92820	−0.898857
$123$	0	0
$124$	2.19615	0.197220
$125$	−14.6603	−1.31125
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.73205	−0.151911
$131$	−17.4641	−1.52585	−0.762923	−	0.646490i	$-0.776236\pi$
$131$	−17.4641	−1.52585	−0.762923	+	0.646490i	$0.776236\pi$
$132$	0	0
$133$	0	0
$134$	3.26795	0.282308
$135$	0	0
$136$	7.00000	0.600245
$137$	−11.7321	−1.00234	−0.501168	−	0.865350i	$-0.667096\pi$
$137$	−11.7321	−1.00234	−0.501168	+	0.865350i	$0.667096\pi$
$138$	0	0
$139$	6.73205	0.571005	0.285503	−	0.958378i	$-0.407839\pi$
$139$	6.73205	0.571005	0.285503	+	0.958378i	$0.407839\pi$
$140$	0	0
$141$	0	0
$142$	−13.4641	−1.12988
$143$	1.94744	0.162853
$144$	0	0
$145$	−31.5885	−2.62328
$146$	11.7321	0.970951
$147$	0	0
$148$	−6.66025	−0.547470
$149$	9.00000	0.737309	0.368654	−	0.929567i	$-0.379819\pi$
$149$	9.00000	0.737309	0.368654	+	0.929567i	$0.379819\pi$
$150$	0	0
$151$	−16.1962	−1.31802	−0.659012	−	0.752132i	$-0.729025\pi$
$151$	−16.1962	−1.31802	−0.659012	+	0.752132i	$0.729025\pi$
$152$	−2.73205	−0.221599
$153$	0	0
$154$	0	0
$155$	−8.19615	−0.658331
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	−15.1244	−1.20323
$159$	0	0
$160$	3.73205	0.295045
$161$	0	0
$162$	0	0
$163$	−6.53590	−0.511931	−0.255966	−	0.966686i	$-0.582393\pi$
$163$	−6.53590	−0.511931	−0.255966	+	0.966686i	$0.582393\pi$
$164$	−9.46410	−0.739022
$165$	0	0
$166$	−14.5885	−1.13228
$167$	12.1962	0.943767	0.471883	−	0.881661i	$-0.343574\pi$
$167$	12.1962	0.943767	0.471883	+	0.881661i	$0.343574\pi$
$168$	0	0
$169$	−12.7846	−0.983432
$170$	−26.1244	−2.00365
$171$	0	0
$172$	5.46410	0.416634
$173$	−9.73205	−0.739914	−0.369957	−	0.929049i	$-0.620628\pi$
$173$	−9.73205	−0.739914	−0.369957	+	0.929049i	$0.620628\pi$
$174$	0	0
$175$	0	0
$176$	−4.19615	−0.316297
$177$	0	0
$178$	−3.92820	−0.294431
$179$	8.19615	0.612609	0.306305	−	0.951934i	$-0.400907\pi$
$179$	8.19615	0.612609	0.306305	+	0.951934i	$0.400907\pi$
$180$	0	0
$181$	−4.39230	−0.326477	−0.163239	−	0.986587i	$-0.552194\pi$
$181$	−4.39230	−0.326477	−0.163239	+	0.986587i	$0.552194\pi$
$182$	0	0
$183$	0	0
$184$	−6.19615	−0.456786
$185$	24.8564	1.82748
$186$	0	0
$187$	29.3731	2.14797
$188$	−1.26795	−0.0924747
$189$	0	0
$190$	10.1962	0.739707
$191$	11.6603	0.843706	0.421853	−	0.906664i	$-0.361380\pi$
$191$	11.6603	0.843706	0.421853	+	0.906664i	$0.361380\pi$
$192$	0	0
$193$	−8.85641	−0.637498	−0.318749	−	0.947839i	$-0.603263\pi$
$193$	−8.85641	−0.637498	−0.318749	+	0.947839i	$0.603263\pi$
$194$	−2.92820	−0.210233
$195$	0	0
$196$	0	0
$197$	−25.7846	−1.83708	−0.918539	−	0.395331i	$-0.870630\pi$
$197$	−25.7846	−1.83708	−0.918539	+	0.395331i	$0.870630\pi$
$198$	0	0
$199$	−5.12436	−0.363256	−0.181628	−	0.983367i	$-0.558137\pi$
$199$	−5.12436	−0.363256	−0.181628	+	0.983367i	$0.558137\pi$
$200$	−8.92820	−0.631319
$201$	0	0
$202$	−4.92820	−0.346747
$203$	0	0
$204$	0	0
$205$	35.3205	2.46689
$206$	12.3923	0.863413
$207$	0	0
$208$	−0.464102	−0.0321797
$209$	−11.4641	−0.792988
$210$	0	0
$211$	−20.7321	−1.42725	−0.713627	−	0.700526i	$-0.752949\pi$
$211$	−20.7321	−1.42725	−0.713627	+	0.700526i	$0.752949\pi$
$212$	−2.53590	−0.174166
$213$	0	0
$214$	0	0
$215$	−20.3923	−1.39074
$216$	0	0
$217$	0	0
$218$	7.19615	0.487385
$219$	0	0
$220$	15.6603	1.05581
$221$	3.24871	0.218532
$222$	0	0
$223$	18.5359	1.24126	0.620628	−	0.784105i	$-0.286878\pi$
$223$	18.5359	1.24126	0.620628	+	0.784105i	$0.286878\pi$
$224$	0	0
$225$	0	0
$226$	2.26795	0.150862
$227$	5.07180	0.336627	0.168313	−	0.985734i	$-0.446168\pi$
$227$	5.07180	0.336627	0.168313	+	0.985734i	$0.446168\pi$
$228$	0	0
$229$	−4.46410	−0.294996	−0.147498	−	0.989062i	$-0.547122\pi$
$229$	−4.46410	−0.294996	−0.147498	+	0.989062i	$0.547122\pi$
$230$	23.1244	1.52477
$231$	0	0
$232$	−8.46410	−0.555695
$233$	−13.1962	−0.864509	−0.432254	−	0.901752i	$-0.642282\pi$
$233$	−13.1962	−0.864509	−0.432254	+	0.901752i	$0.642282\pi$
$234$	0	0
$235$	4.73205	0.308685
$236$	6.19615	0.403335
$237$	0	0
$238$	0	0
$239$	28.0526	1.81457	0.907285	−	0.420517i	$-0.138151\pi$
$239$	28.0526	1.81457	0.907285	+	0.420517i	$0.138151\pi$
$240$	0	0
$241$	17.7321	1.14222	0.571111	−	0.820873i	$-0.306513\pi$
$241$	17.7321	1.14222	0.571111	+	0.820873i	$0.306513\pi$
$242$	−6.60770	−0.424759
$243$	0	0
$244$	9.92820	0.635588
$245$	0	0
$246$	0	0
$247$	−1.26795	−0.0806777
$248$	−2.19615	−0.139456
$249$	0	0
$250$	14.6603	0.927196
$251$	−16.0526	−1.01323	−0.506614	−	0.862173i	$-0.669103\pi$
$251$	−16.0526	−1.01323	−0.506614	+	0.862173i	$0.669103\pi$
$252$	0	0
$253$	−26.0000	−1.63461
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.464102	0.0289499	0.0144749	−	0.999895i	$-0.495392\pi$
$257$	0.464102	0.0289499	0.0144749	+	0.999895i	$0.495392\pi$
$258$	0	0
$259$	0	0
$260$	1.73205	0.107417
$261$	0	0
$262$	17.4641	1.07894
$263$	−23.6603	−1.45895	−0.729477	−	0.684005i	$-0.760236\pi$
$263$	−23.6603	−1.45895	−0.729477	+	0.684005i	$0.760236\pi$
$264$	0	0
$265$	9.46410	0.581375
$266$	0	0
$267$	0	0
$268$	−3.26795	−0.199622
$269$	25.5885	1.56016	0.780078	−	0.625682i	$-0.215179\pi$
$269$	25.5885	1.56016	0.780078	+	0.625682i	$0.215179\pi$
$270$	0	0
$271$	−25.5167	−1.55003	−0.775013	−	0.631945i	$-0.782257\pi$
$271$	−25.5167	−1.55003	−0.775013	+	0.631945i	$0.782257\pi$
$272$	−7.00000	−0.424437
$273$	0	0
$274$	11.7321	0.708759
$275$	−37.4641	−2.25917
$276$	0	0
$277$	22.7846	1.36899	0.684497	−	0.729015i	$-0.260022\pi$
$277$	22.7846	1.36899	0.684497	+	0.729015i	$0.260022\pi$
$278$	−6.73205	−0.403762
$279$	0	0
$280$	0	0
$281$	−2.80385	−0.167264	−0.0836318	−	0.996497i	$-0.526652\pi$
$281$	−2.80385	−0.167264	−0.0836318	+	0.996497i	$0.526652\pi$
$282$	0	0
$283$	19.3205	1.14848	0.574242	−	0.818685i	$-0.305297\pi$
$283$	19.3205	1.14848	0.574242	+	0.818685i	$0.305297\pi$
$284$	13.4641	0.798947
$285$	0	0
$286$	−1.94744	−0.115155
$287$	0	0
$288$	0	0
$289$	32.0000	1.88235
$290$	31.5885	1.85494
$291$	0	0
$292$	−11.7321	−0.686566
$293$	−20.6603	−1.20698	−0.603492	−	0.797369i	$-0.706225\pi$
$293$	−20.6603	−1.20698	−0.603492	+	0.797369i	$0.706225\pi$
$294$	0	0
$295$	−23.1244	−1.34635
$296$	6.66025	0.387119
$297$	0	0
$298$	−9.00000	−0.521356
$299$	−2.87564	−0.166303
$300$	0	0
$301$	0	0
$302$	16.1962	0.931984
$303$	0	0
$304$	2.73205	0.156694
$305$	−37.0526	−2.12162
$306$	0	0
$307$	−5.85641	−0.334243	−0.167121	−	0.985936i	$-0.553447\pi$
$307$	−5.85641	−0.334243	−0.167121	+	0.985936i	$0.553447\pi$
$308$	0	0
$309$	0	0
$310$	8.19615	0.465510
$311$	0.196152	0.0111228	0.00556139	−	0.999985i	$-0.498230\pi$
$311$	0.196152	0.0111228	0.00556139	+	0.999985i	$0.498230\pi$
$312$	0	0
$313$	5.58846	0.315878	0.157939	−	0.987449i	$-0.449515\pi$
$313$	5.58846	0.315878	0.157939	+	0.987449i	$0.449515\pi$
$314$	−1.00000	−0.0564333
$315$	0	0
$316$	15.1244	0.850811
$317$	−10.6077	−0.595788	−0.297894	−	0.954599i	$-0.596284\pi$
$317$	−10.6077	−0.595788	−0.297894	+	0.954599i	$0.596284\pi$
$318$	0	0
$319$	−35.5167	−1.98855
$320$	−3.73205	−0.208628
$321$	0	0
$322$	0	0
$323$	−19.1244	−1.06411
$324$	0	0
$325$	−4.14359	−0.229845
$326$	6.53590	0.361990
$327$	0	0
$328$	9.46410	0.522568
$329$	0	0
$330$	0	0
$331$	−8.39230	−0.461283	−0.230641	−	0.973039i	$-0.574082\pi$
$331$	−8.39230	−0.461283	−0.230641	+	0.973039i	$0.574082\pi$
$332$	14.5885	0.800646
$333$	0	0
$334$	−12.1962	−0.667344
$335$	12.1962	0.666347
$336$	0	0
$337$	4.39230	0.239264	0.119632	−	0.992818i	$-0.461829\pi$
$337$	4.39230	0.239264	0.119632	+	0.992818i	$0.461829\pi$
$338$	12.7846	0.695391
$339$	0	0
$340$	26.1244	1.41679
$341$	−9.21539	−0.499041
$342$	0	0
$343$	0	0
$344$	−5.46410	−0.294605
$345$	0	0
$346$	9.73205	0.523198
$347$	14.5359	0.780328	0.390164	−	0.920745i	$-0.372418\pi$
$347$	14.5359	0.780328	0.390164	+	0.920745i	$0.372418\pi$
$348$	0	0
$349$	5.46410	0.292487	0.146243	−	0.989249i	$-0.453282\pi$
$349$	5.46410	0.292487	0.146243	+	0.989249i	$0.453282\pi$
$350$	0	0
$351$	0	0
$352$	4.19615	0.223656
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	−50.2487	−2.66692
$356$	3.92820	0.208194
$357$	0	0
$358$	−8.19615	−0.433180
$359$	2.92820	0.154545	0.0772723	−	0.997010i	$-0.475379\pi$
$359$	2.92820	0.154545	0.0772723	+	0.997010i	$0.475379\pi$
$360$	0	0
$361$	−11.5359	−0.607153
$362$	4.39230	0.230854
$363$	0	0
$364$	0	0
$365$	43.7846	2.29179
$366$	0	0
$367$	13.1244	0.685086	0.342543	−	0.939502i	$-0.388712\pi$
$367$	13.1244	0.685086	0.342543	+	0.939502i	$0.388712\pi$
$368$	6.19615	0.322997
$369$	0	0
$370$	−24.8564	−1.29222
$371$	0	0
$372$	0	0
$373$	−33.8564	−1.75302	−0.876509	−	0.481385i	$-0.840134\pi$
$373$	−33.8564	−1.75302	−0.876509	+	0.481385i	$0.840134\pi$
$374$	−29.3731	−1.51885
$375$	0	0
$376$	1.26795	0.0653895
$377$	−3.92820	−0.202313
$378$	0	0
$379$	17.5167	0.899770	0.449885	−	0.893086i	$-0.351465\pi$
$379$	17.5167	0.899770	0.449885	+	0.893086i	$0.351465\pi$
$380$	−10.1962	−0.523052
$381$	0	0
$382$	−11.6603	−0.596590
$383$	35.7128	1.82484	0.912420	−	0.409256i	$-0.134212\pi$
$383$	35.7128	1.82484	0.912420	+	0.409256i	$0.134212\pi$
$384$	0	0
$385$	0	0
$386$	8.85641	0.450779
$387$	0	0
$388$	2.92820	0.148657
$389$	−6.53590	−0.331383	−0.165692	−	0.986178i	$-0.552986\pi$
$389$	−6.53590	−0.331383	−0.165692	+	0.986178i	$0.552986\pi$
$390$	0	0
$391$	−43.3731	−2.19347
$392$	0	0
$393$	0	0
$394$	25.7846	1.29901
$395$	−56.4449	−2.84005
$396$	0	0
$397$	21.0000	1.05396	0.526980	−	0.849878i	$-0.323324\pi$
$397$	21.0000	1.05396	0.526980	+	0.849878i	$0.323324\pi$
$398$	5.12436	0.256861
$399$	0	0
$400$	8.92820	0.446410
$401$	−34.5167	−1.72368	−0.861840	−	0.507180i	$-0.830688\pi$
$401$	−34.5167	−1.72368	−0.861840	+	0.507180i	$0.830688\pi$
$402$	0	0
$403$	−1.01924	−0.0507719
$404$	4.92820	0.245187
$405$	0	0
$406$	0	0
$407$	27.9474	1.38530
$408$	0	0
$409$	−34.6603	−1.71384	−0.856920	−	0.515450i	$-0.827625\pi$
$409$	−34.6603	−1.71384	−0.856920	+	0.515450i	$0.827625\pi$
$410$	−35.3205	−1.74436
$411$	0	0
$412$	−12.3923	−0.610525
$413$	0	0
$414$	0	0
$415$	−54.4449	−2.67259
$416$	0.464102	0.0227545
$417$	0	0
$418$	11.4641	0.560728
$419$	2.53590	0.123887	0.0619434	−	0.998080i	$-0.480270\pi$
$419$	2.53590	0.123887	0.0619434	+	0.998080i	$0.480270\pi$
$420$	0	0
$421$	−24.1244	−1.17575	−0.587875	−	0.808952i	$-0.700035\pi$
$421$	−24.1244	−1.17575	−0.587875	+	0.808952i	$0.700035\pi$
$422$	20.7321	1.00922
$423$	0	0
$424$	2.53590	0.123154
$425$	−62.4974	−3.03157
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	20.3923	0.983404
$431$	−21.4641	−1.03389	−0.516945	−	0.856019i	$-0.672931\pi$
$431$	−21.4641	−1.03389	−0.516945	+	0.856019i	$0.672931\pi$
$432$	0	0
$433$	−12.2679	−0.589560	−0.294780	−	0.955565i	$-0.595246\pi$
$433$	−12.2679	−0.589560	−0.294780	+	0.955565i	$0.595246\pi$
$434$	0	0
$435$	0	0
$436$	−7.19615	−0.344633
$437$	16.9282	0.809786
$438$	0	0
$439$	11.3205	0.540298	0.270149	−	0.962818i	$-0.412927\pi$
$439$	11.3205	0.540298	0.270149	+	0.962818i	$0.412927\pi$
$440$	−15.6603	−0.746573
$441$	0	0
$442$	−3.24871	−0.154525
$443$	−18.7321	−0.889987	−0.444993	−	0.895534i	$-0.646794\pi$
$443$	−18.7321	−0.889987	−0.444993	+	0.895534i	$0.646794\pi$
$444$	0	0
$445$	−14.6603	−0.694963
$446$	−18.5359	−0.877700
$447$	0	0
$448$	0	0
$449$	−11.8564	−0.559538	−0.279769	−	0.960067i	$-0.590258\pi$
$449$	−11.8564	−0.559538	−0.279769	+	0.960067i	$0.590258\pi$
$450$	0	0
$451$	39.7128	1.87000
$452$	−2.26795	−0.106675
$453$	0	0
$454$	−5.07180	−0.238031
$455$	0	0
$456$	0	0
$457$	20.8564	0.975622	0.487811	−	0.872949i	$-0.337796\pi$
$457$	20.8564	0.975622	0.487811	+	0.872949i	$0.337796\pi$
$458$	4.46410	0.208594
$459$	0	0
$460$	−23.1244	−1.07818
$461$	−34.7846	−1.62008	−0.810040	−	0.586374i	$-0.800555\pi$
$461$	−34.7846	−1.62008	−0.810040	+	0.586374i	$0.800555\pi$
$462$	0	0
$463$	−32.5885	−1.51451	−0.757257	−	0.653117i	$-0.773461\pi$
$463$	−32.5885	−1.51451	−0.757257	+	0.653117i	$0.773461\pi$
$464$	8.46410	0.392936
$465$	0	0
$466$	13.1962	0.611300
$467$	−14.5885	−0.675073	−0.337537	−	0.941312i	$-0.609594\pi$
$467$	−14.5885	−0.675073	−0.337537	+	0.941312i	$0.609594\pi$
$468$	0	0
$469$	0	0
$470$	−4.73205	−0.218273
$471$	0	0
$472$	−6.19615	−0.285201
$473$	−22.9282	−1.05424
$474$	0	0
$475$	24.3923	1.11920
$476$	0	0
$477$	0	0
$478$	−28.0526	−1.28309
$479$	−23.5167	−1.07450	−0.537252	−	0.843422i	$-0.680538\pi$
$479$	−23.5167	−1.07450	−0.537252	+	0.843422i	$0.680538\pi$
$480$	0	0
$481$	3.09103	0.140939
$482$	−17.7321	−0.807673
$483$	0	0
$484$	6.60770	0.300350
$485$	−10.9282	−0.496224
$486$	0	0
$487$	−28.5885	−1.29547	−0.647733	−	0.761867i	$-0.724283\pi$
$487$	−28.5885	−1.29547	−0.647733	+	0.761867i	$0.724283\pi$
$488$	−9.92820	−0.449429
$489$	0	0
$490$	0	0
$491$	33.4641	1.51021	0.755107	−	0.655602i	$-0.227585\pi$
$491$	33.4641	1.51021	0.755107	+	0.655602i	$0.227585\pi$
$492$	0	0
$493$	−59.2487	−2.66843
$494$	1.26795	0.0570477
$495$	0	0
$496$	2.19615	0.0986102
$497$	0	0
$498$	0	0
$499$	30.1962	1.35177	0.675883	−	0.737009i	$-0.263763\pi$
$499$	30.1962	1.35177	0.675883	+	0.737009i	$0.263763\pi$
$500$	−14.6603	−0.655626
$501$	0	0
$502$	16.0526	0.716461
$503$	−1.94744	−0.0868321	−0.0434161	−	0.999057i	$-0.513824\pi$
$503$	−1.94744	−0.0868321	−0.0434161	+	0.999057i	$0.513824\pi$
$504$	0	0
$505$	−18.3923	−0.818447
$506$	26.0000	1.15584
$507$	0	0
$508$	12.0000	0.532414
$509$	−4.14359	−0.183662	−0.0918308	−	0.995775i	$-0.529272\pi$
$509$	−4.14359	−0.183662	−0.0918308	+	0.995775i	$0.529272\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−0.464102	−0.0204706
$515$	46.2487	2.03796
$516$	0	0
$517$	5.32051	0.233996
$518$	0	0
$519$	0	0
$520$	−1.73205	−0.0759555
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	29.1769	1.27582	0.637909	−	0.770112i	$-0.279800\pi$
$523$	29.1769	1.27582	0.637909	+	0.770112i	$0.279800\pi$
$524$	−17.4641	−0.762923
$525$	0	0
$526$	23.6603	1.03164
$527$	−15.3731	−0.669661
$528$	0	0
$529$	15.3923	0.669231
$530$	−9.46410	−0.411094
$531$	0	0
$532$	0	0
$533$	4.39230	0.190252
$534$	0	0
$535$	0	0
$536$	3.26795	0.141154
$537$	0	0
$538$	−25.5885	−1.10320
$539$	0	0
$540$	0	0
$541$	20.6603	0.888254	0.444127	−	0.895964i	$-0.353514\pi$
$541$	20.6603	0.888254	0.444127	+	0.895964i	$0.353514\pi$
$542$	25.5167	1.09603
$543$	0	0
$544$	7.00000	0.300123
$545$	26.8564	1.15040
$546$	0	0
$547$	−19.2679	−0.823838	−0.411919	−	0.911220i	$-0.635141\pi$
$547$	−19.2679	−0.823838	−0.411919	+	0.911220i	$0.635141\pi$
$548$	−11.7321	−0.501168
$549$	0	0
$550$	37.4641	1.59747
$551$	23.1244	0.985131
$552$	0	0
$553$	0	0
$554$	−22.7846	−0.968025
$555$	0	0
$556$	6.73205	0.285503
$557$	10.0718	0.426756	0.213378	−	0.976970i	$-0.431553\pi$
$557$	10.0718	0.426756	0.213378	+	0.976970i	$0.431553\pi$
$558$	0	0
$559$	−2.53590	−0.107257
$560$	0	0
$561$	0	0
$562$	2.80385	0.118273
$563$	35.7128	1.50512	0.752558	−	0.658526i	$-0.228820\pi$
$563$	35.7128	1.50512	0.752558	+	0.658526i	$0.228820\pi$
$564$	0	0
$565$	8.46410	0.356087
$566$	−19.3205	−0.812102
$567$	0	0
$568$	−13.4641	−0.564941
$569$	12.8038	0.536765	0.268383	−	0.963312i	$-0.413511\pi$
$569$	12.8038	0.536765	0.268383	+	0.963312i	$0.413511\pi$
$570$	0	0
$571$	−19.2679	−0.806339	−0.403169	−	0.915125i	$-0.632091\pi$
$571$	−19.2679	−0.806339	−0.403169	+	0.915125i	$0.632091\pi$
$572$	1.94744	0.0814266
$573$	0	0
$574$	0	0
$575$	55.3205	2.30702
$576$	0	0
$577$	−7.33975	−0.305558	−0.152779	−	0.988260i	$-0.548822\pi$
$577$	−7.33975	−0.305558	−0.152779	+	0.988260i	$0.548822\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	−31.5885	−1.31164
$581$	0	0
$582$	0	0
$583$	10.6410	0.440706
$584$	11.7321	0.485476
$585$	0	0
$586$	20.6603	0.853467
$587$	−11.2679	−0.465078	−0.232539	−	0.972587i	$-0.574703\pi$
$587$	−11.2679	−0.465078	−0.232539	+	0.972587i	$0.574703\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	23.1244	0.952015
$591$	0	0
$592$	−6.66025	−0.273735
$593$	−40.1769	−1.64987	−0.824934	−	0.565229i	$-0.808788\pi$
$593$	−40.1769	−1.64987	−0.824934	+	0.565229i	$0.808788\pi$
$594$	0	0
$595$	0	0
$596$	9.00000	0.368654
$597$	0	0
$598$	2.87564	0.117594
$599$	9.12436	0.372811	0.186406	−	0.982473i	$-0.440316\pi$
$599$	9.12436	0.372811	0.186406	+	0.982473i	$0.440316\pi$
$600$	0	0
$601$	−8.80385	−0.359116	−0.179558	−	0.983747i	$-0.557467\pi$
$601$	−8.80385	−0.359116	−0.179558	+	0.983747i	$0.557467\pi$
$602$	0	0
$603$	0	0
$604$	−16.1962	−0.659012
$605$	−24.6603	−1.00258
$606$	0	0
$607$	−6.58846	−0.267417	−0.133709	−	0.991021i	$-0.542689\pi$
$607$	−6.58846	−0.267417	−0.133709	+	0.991021i	$0.542689\pi$
$608$	−2.73205	−0.110799
$609$	0	0
$610$	37.0526	1.50021
$611$	0.588457	0.0238064
$612$	0	0
$613$	−14.7846	−0.597145	−0.298572	−	0.954387i	$-0.596510\pi$
$613$	−14.7846	−0.597145	−0.298572	+	0.954387i	$0.596510\pi$
$614$	5.85641	0.236345
$615$	0	0
$616$	0	0
$617$	−39.9808	−1.60956	−0.804782	−	0.593570i	$-0.797718\pi$
$617$	−39.9808	−1.60956	−0.804782	+	0.593570i	$0.797718\pi$
$618$	0	0
$619$	−31.7128	−1.27465	−0.637323	−	0.770597i	$-0.719958\pi$
$619$	−31.7128	−1.27465	−0.637323	+	0.770597i	$0.719958\pi$
$620$	−8.19615	−0.329165
$621$	0	0
$622$	−0.196152	−0.00786500
$623$	0	0
$624$	0	0
$625$	10.0718	0.402872
$626$	−5.58846	−0.223360
$627$	0	0
$628$	1.00000	0.0399043
$629$	46.6218	1.85893
$630$	0	0
$631$	13.6603	0.543806	0.271903	−	0.962325i	$-0.412347\pi$
$631$	13.6603	0.543806	0.271903	+	0.962325i	$0.412347\pi$
$632$	−15.1244	−0.601615
$633$	0	0
$634$	10.6077	0.421285
$635$	−44.7846	−1.77722
$636$	0	0
$637$	0	0
$638$	35.5167	1.40612
$639$	0	0
$640$	3.73205	0.147522
$641$	−19.4449	−0.768026	−0.384013	−	0.923328i	$-0.625458\pi$
$641$	−19.4449	−0.768026	−0.384013	+	0.923328i	$0.625458\pi$
$642$	0	0
$643$	40.5885	1.60065	0.800326	−	0.599565i	$-0.204660\pi$
$643$	40.5885	1.60065	0.800326	+	0.599565i	$0.204660\pi$
$644$	0	0
$645$	0	0
$646$	19.1244	0.752438
$647$	−16.3923	−0.644448	−0.322224	−	0.946663i	$-0.604430\pi$
$647$	−16.3923	−0.644448	−0.322224	+	0.946663i	$0.604430\pi$
$648$	0	0
$649$	−26.0000	−1.02059
$650$	4.14359	0.162525
$651$	0	0
$652$	−6.53590	−0.255966
$653$	18.2487	0.714127	0.357064	−	0.934080i	$-0.383778\pi$
$653$	18.2487	0.714127	0.357064	+	0.934080i	$0.383778\pi$
$654$	0	0
$655$	65.1769	2.54667
$656$	−9.46410	−0.369511
$657$	0	0
$658$	0	0
$659$	−15.6077	−0.607989	−0.303995	−	0.952674i	$-0.598321\pi$
$659$	−15.6077	−0.607989	−0.303995	+	0.952674i	$0.598321\pi$
$660$	0	0
$661$	−14.8564	−0.577847	−0.288924	−	0.957352i	$-0.593297\pi$
$661$	−14.8564	−0.577847	−0.288924	+	0.957352i	$0.593297\pi$
$662$	8.39230	0.326176
$663$	0	0
$664$	−14.5885	−0.566142
$665$	0	0
$666$	0	0
$667$	52.4449	2.03067
$668$	12.1962	0.471883
$669$	0	0
$670$	−12.1962	−0.471178
$671$	−41.6603	−1.60828
$672$	0	0
$673$	16.3205	0.629109	0.314555	−	0.949239i	$-0.398145\pi$
$673$	16.3205	0.629109	0.314555	+	0.949239i	$0.398145\pi$
$674$	−4.39230	−0.169185
$675$	0	0
$676$	−12.7846	−0.491716
$677$	−36.0000	−1.38359	−0.691796	−	0.722093i	$-0.743180\pi$
$677$	−36.0000	−1.38359	−0.691796	+	0.722093i	$0.743180\pi$
$678$	0	0
$679$	0	0
$680$	−26.1244	−1.00182
$681$	0	0
$682$	9.21539	0.352876
$683$	25.8564	0.989368	0.494684	−	0.869073i	$-0.335284\pi$
$683$	25.8564	0.989368	0.494684	+	0.869073i	$0.335284\pi$
$684$	0	0
$685$	43.7846	1.67292
$686$	0	0
$687$	0	0
$688$	5.46410	0.208317
$689$	1.17691	0.0448369
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−9.73205	−0.369957
$693$	0	0
$694$	−14.5359	−0.551775
$695$	−25.1244	−0.953021
$696$	0	0
$697$	66.2487	2.50935
$698$	−5.46410	−0.206819
$699$	0	0
$700$	0	0
$701$	6.60770	0.249569	0.124785	−	0.992184i	$-0.460176\pi$
$701$	6.60770	0.249569	0.124785	+	0.992184i	$0.460176\pi$
$702$	0	0
$703$	−18.1962	−0.686281
$704$	−4.19615	−0.158148
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	0	0
$709$	−28.1244	−1.05623	−0.528116	−	0.849172i	$-0.677101\pi$
$709$	−28.1244	−1.05623	−0.528116	+	0.849172i	$0.677101\pi$
$710$	50.2487	1.88580
$711$	0	0
$712$	−3.92820	−0.147216
$713$	13.6077	0.509612
$714$	0	0
$715$	−7.26795	−0.271806
$716$	8.19615	0.306305
$717$	0	0
$718$	−2.92820	−0.109280
$719$	2.53590	0.0945731	0.0472865	−	0.998881i	$-0.484943\pi$
$719$	2.53590	0.0945731	0.0472865	+	0.998881i	$0.484943\pi$
$720$	0	0
$721$	0	0
$722$	11.5359	0.429322
$723$	0	0
$724$	−4.39230	−0.163239
$725$	75.5692	2.80657
$726$	0	0
$727$	−16.6795	−0.618608	−0.309304	−	0.950963i	$-0.600096\pi$
$727$	−16.6795	−0.618608	−0.309304	+	0.950963i	$0.600096\pi$
$728$	0	0
$729$	0	0
$730$	−43.7846	−1.62054
$731$	−38.2487	−1.41468
$732$	0	0
$733$	−12.6795	−0.468328	−0.234164	−	0.972197i	$-0.575235\pi$
$733$	−12.6795	−0.468328	−0.234164	+	0.972197i	$0.575235\pi$
$734$	−13.1244	−0.484429
$735$	0	0
$736$	−6.19615	−0.228393
$737$	13.7128	0.505118
$738$	0	0
$739$	−16.7321	−0.615498	−0.307749	−	0.951468i	$-0.599576\pi$
$739$	−16.7321	−0.615498	−0.307749	+	0.951468i	$0.599576\pi$
$740$	24.8564	0.913740
$741$	0	0
$742$	0	0
$743$	−19.6077	−0.719337	−0.359668	−	0.933080i	$-0.617110\pi$
$743$	−19.6077	−0.719337	−0.359668	+	0.933080i	$0.617110\pi$
$744$	0	0
$745$	−33.5885	−1.23059
$746$	33.8564	1.23957
$747$	0	0
$748$	29.3731	1.07399
$749$	0	0
$750$	0	0
$751$	−49.8564	−1.81929	−0.909643	−	0.415391i	$-0.863645\pi$
$751$	−49.8564	−1.81929	−0.909643	+	0.415391i	$0.863645\pi$
$752$	−1.26795	−0.0462373
$753$	0	0
$754$	3.92820	0.143057
$755$	60.4449	2.19981
$756$	0	0
$757$	−20.7846	−0.755429	−0.377715	−	0.925922i	$-0.623290\pi$
$757$	−20.7846	−0.755429	−0.377715	+	0.925922i	$0.623290\pi$
$758$	−17.5167	−0.636234
$759$	0	0
$760$	10.1962	0.369853
$761$	−37.0000	−1.34125	−0.670624	−	0.741797i	$-0.733974\pi$
$761$	−37.0000	−1.34125	−0.670624	+	0.741797i	$0.733974\pi$
$762$	0	0
$763$	0	0
$764$	11.6603	0.421853
$765$	0	0
$766$	−35.7128	−1.29036
$767$	−2.87564	−0.103833
$768$	0	0
$769$	35.5885	1.28335	0.641676	−	0.766976i	$-0.278239\pi$
$769$	35.5885	1.28335	0.641676	+	0.766976i	$0.278239\pi$
$770$	0	0
$771$	0	0
$772$	−8.85641	−0.318749
$773$	−20.1244	−0.723823	−0.361911	−	0.932213i	$-0.617876\pi$
$773$	−20.1244	−0.723823	−0.361911	+	0.932213i	$0.617876\pi$
$774$	0	0
$775$	19.6077	0.704329
$776$	−2.92820	−0.105116
$777$	0	0
$778$	6.53590	0.234323
$779$	−25.8564	−0.926402
$780$	0	0
$781$	−56.4974	−2.02164
$782$	43.3731	1.55102
$783$	0	0
$784$	0	0
$785$	−3.73205	−0.133203
$786$	0	0
$787$	7.60770	0.271185	0.135593	−	0.990765i	$-0.456706\pi$
$787$	7.60770	0.271185	0.135593	+	0.990765i	$0.456706\pi$
$788$	−25.7846	−0.918539
$789$	0	0
$790$	56.4449	2.00822
$791$	0	0
$792$	0	0
$793$	−4.60770	−0.163624
$794$	−21.0000	−0.745262
$795$	0	0
$796$	−5.12436	−0.181628
$797$	−29.4449	−1.04299	−0.521495	−	0.853254i	$-0.674626\pi$
$797$	−29.4449	−1.04299	−0.521495	+	0.853254i	$0.674626\pi$
$798$	0	0
$799$	8.87564	0.313998
$800$	−8.92820	−0.315660
$801$	0	0
$802$	34.5167	1.21883
$803$	49.2295	1.73727
$804$	0	0
$805$	0	0
$806$	1.01924	0.0359011
$807$	0	0
$808$	−4.92820	−0.173374
$809$	7.87564	0.276893	0.138446	−	0.990370i	$-0.455789\pi$
$809$	7.87564	0.276893	0.138446	+	0.990370i	$0.455789\pi$
$810$	0	0
$811$	−7.80385	−0.274030	−0.137015	−	0.990569i	$-0.543751\pi$
$811$	−7.80385	−0.274030	−0.137015	+	0.990569i	$0.543751\pi$
$812$	0	0
$813$	0	0
$814$	−27.9474	−0.979557
$815$	24.3923	0.854425
$816$	0	0
$817$	14.9282	0.522272
$818$	34.6603	1.21187
$819$	0	0
$820$	35.3205	1.23345
$821$	12.0718	0.421309	0.210654	−	0.977561i	$-0.432441\pi$
$821$	12.0718	0.421309	0.210654	+	0.977561i	$0.432441\pi$
$822$	0	0
$823$	−40.7846	−1.42166	−0.710831	−	0.703363i	$-0.751681\pi$
$823$	−40.7846	−1.42166	−0.710831	+	0.703363i	$0.751681\pi$
$824$	12.3923	0.431706
$825$	0	0
$826$	0	0
$827$	11.3205	0.393653	0.196826	−	0.980438i	$-0.436936\pi$
$827$	11.3205	0.393653	0.196826	+	0.980438i	$0.436936\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	54.4449	1.88981
$831$	0	0
$832$	−0.464102	−0.0160898
$833$	0	0
$834$	0	0
$835$	−45.5167	−1.57517
$836$	−11.4641	−0.396494
$837$	0	0
$838$	−2.53590	−0.0876012
$839$	5.46410	0.188642	0.0943209	−	0.995542i	$-0.469932\pi$
$839$	5.46410	0.188642	0.0943209	+	0.995542i	$0.469932\pi$
$840$	0	0
$841$	42.6410	1.47038
$842$	24.1244	0.831380
$843$	0	0
$844$	−20.7321	−0.713627
$845$	47.7128	1.64137
$846$	0	0
$847$	0	0
$848$	−2.53590	−0.0870831
$849$	0	0
$850$	62.4974	2.14364
$851$	−41.2679	−1.41465
$852$	0	0
$853$	−5.71281	−0.195603	−0.0978015	−	0.995206i	$-0.531181\pi$
$853$	−5.71281	−0.195603	−0.0978015	+	0.995206i	$0.531181\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.8564	−0.370848	−0.185424	−	0.982659i	$-0.559366\pi$
$857$	−10.8564	−0.370848	−0.185424	+	0.982659i	$0.559366\pi$
$858$	0	0
$859$	−3.60770	−0.123093	−0.0615465	−	0.998104i	$-0.519603\pi$
$859$	−3.60770	−0.123093	−0.0615465	+	0.998104i	$0.519603\pi$
$860$	−20.3923	−0.695372
$861$	0	0
$862$	21.4641	0.731070
$863$	17.1244	0.582920	0.291460	−	0.956583i	$-0.405859\pi$
$863$	17.1244	0.582920	0.291460	+	0.956583i	$0.405859\pi$
$864$	0	0
$865$	36.3205	1.23493
$866$	12.2679	0.416882
$867$	0	0
$868$	0	0
$869$	−63.4641	−2.15287
$870$	0	0
$871$	1.51666	0.0513901
$872$	7.19615	0.243692
$873$	0	0
$874$	−16.9282	−0.572605
$875$	0	0
$876$	0	0
$877$	8.51666	0.287587	0.143794	−	0.989608i	$-0.454070\pi$
$877$	8.51666	0.287587	0.143794	+	0.989608i	$0.454070\pi$
$878$	−11.3205	−0.382049
$879$	0	0
$880$	15.6603	0.527907
$881$	18.2487	0.614815	0.307407	−	0.951578i	$-0.400539\pi$
$881$	18.2487	0.614815	0.307407	+	0.951578i	$0.400539\pi$
$882$	0	0
$883$	7.66025	0.257788	0.128894	−	0.991658i	$-0.458857\pi$
$883$	7.66025	0.257788	0.128894	+	0.991658i	$0.458857\pi$
$884$	3.24871	0.109266
$885$	0	0
$886$	18.7321	0.629316
$887$	−2.44486	−0.0820905	−0.0410452	−	0.999157i	$-0.513069\pi$
$887$	−2.44486	−0.0820905	−0.0410452	+	0.999157i	$0.513069\pi$
$888$	0	0
$889$	0	0
$890$	14.6603	0.491413
$891$	0	0
$892$	18.5359	0.620628
$893$	−3.46410	−0.115922
$894$	0	0
$895$	−30.5885	−1.02246
$896$	0	0
$897$	0	0
$898$	11.8564	0.395653
$899$	18.5885	0.619960
$900$	0	0
$901$	17.7513	0.591381
$902$	−39.7128	−1.32229
$903$	0	0
$904$	2.26795	0.0754309
$905$	16.3923	0.544899
$906$	0	0
$907$	−36.0000	−1.19536	−0.597680	−	0.801735i	$-0.703911\pi$
$907$	−36.0000	−1.19536	−0.597680	+	0.801735i	$0.703911\pi$
$908$	5.07180	0.168313
$909$	0	0
$910$	0	0
$911$	−6.24871	−0.207029	−0.103515	−	0.994628i	$-0.533009\pi$
$911$	−6.24871	−0.207029	−0.103515	+	0.994628i	$0.533009\pi$
$912$	0	0
$913$	−61.2154	−2.02593
$914$	−20.8564	−0.689869
$915$	0	0
$916$	−4.46410	−0.147498
$917$	0	0
$918$	0	0
$919$	−24.9808	−0.824039	−0.412020	−	0.911175i	$-0.635176\pi$
$919$	−24.9808	−0.824039	−0.412020	+	0.911175i	$0.635176\pi$
$920$	23.1244	0.762387
$921$	0	0
$922$	34.7846	1.14557
$923$	−6.24871	−0.205679
$924$	0	0
$925$	−59.4641	−1.95517
$926$	32.5885	1.07092
$927$	0	0
$928$	−8.46410	−0.277848
$929$	−45.4974	−1.49272	−0.746361	−	0.665541i	$-0.768201\pi$
$929$	−45.4974	−1.49272	−0.746361	+	0.665541i	$0.768201\pi$
$930$	0	0
$931$	0	0
$932$	−13.1962	−0.432254
$933$	0	0
$934$	14.5885	0.477349
$935$	−109.622	−3.58502
$936$	0	0
$937$	−53.8372	−1.75878	−0.879392	−	0.476099i	$-0.842050\pi$
$937$	−53.8372	−1.75878	−0.879392	+	0.476099i	$0.842050\pi$
$938$	0	0
$939$	0	0
$940$	4.73205	0.154342
$941$	9.87564	0.321937	0.160968	−	0.986960i	$-0.448538\pi$
$941$	9.87564	0.321937	0.160968	+	0.986960i	$0.448538\pi$
$942$	0	0
$943$	−58.6410	−1.90961
$944$	6.19615	0.201668
$945$	0	0
$946$	22.9282	0.745460
$947$	2.24871	0.0730733	0.0365366	−	0.999332i	$-0.488367\pi$
$947$	2.24871	0.0730733	0.0365366	+	0.999332i	$0.488367\pi$
$948$	0	0
$949$	5.44486	0.176748
$950$	−24.3923	−0.791391
$951$	0	0
$952$	0	0
$953$	−10.4115	−0.337263	−0.168631	−	0.985679i	$-0.553935\pi$
$953$	−10.4115	−0.337263	−0.168631	+	0.985679i	$0.553935\pi$
$954$	0	0
$955$	−43.5167	−1.40817
$956$	28.0526	0.907285
$957$	0	0
$958$	23.5167	0.759789
$959$	0	0
$960$	0	0
$961$	−26.1769	−0.844417
$962$	−3.09103	−0.0996590
$963$	0	0
$964$	17.7321	0.571111
$965$	33.0526	1.06400
$966$	0	0
$967$	13.6603	0.439284	0.219642	−	0.975581i	$-0.429511\pi$
$967$	13.6603	0.439284	0.219642	+	0.975581i	$0.429511\pi$
$968$	−6.60770	−0.212379
$969$	0	0
$970$	10.9282	0.350883
$971$	−33.1244	−1.06301	−0.531506	−	0.847055i	$-0.678374\pi$
$971$	−33.1244	−1.06301	−0.531506	+	0.847055i	$0.678374\pi$
$972$	0	0
$973$	0	0
$974$	28.5885	0.916033
$975$	0	0
$976$	9.92820	0.317794
$977$	2.28719	0.0731736	0.0365868	−	0.999330i	$-0.488351\pi$
$977$	2.28719	0.0731736	0.0365868	+	0.999330i	$0.488351\pi$
$978$	0	0
$979$	−16.4833	−0.526810
$980$	0	0
$981$	0	0
$982$	−33.4641	−1.06788
$983$	10.6410	0.339396	0.169698	−	0.985496i	$-0.445721\pi$
$983$	10.6410	0.339396	0.169698	+	0.985496i	$0.445721\pi$
$984$	0	0
$985$	96.2295	3.06613
$986$	59.2487	1.88686
$987$	0	0
$988$	−1.26795	−0.0403388
$989$	33.8564	1.07657
$990$	0	0
$991$	−10.3397	−0.328453	−0.164226	−	0.986423i	$-0.552513\pi$
$991$	−10.3397	−0.328453	−0.164226	+	0.986423i	$0.552513\pi$
$992$	−2.19615	−0.0697279
$993$	0	0
$994$	0	0
$995$	19.1244	0.606283
$996$	0	0
$997$	7.24871	0.229569	0.114784	−	0.993390i	$-0.463382\pi$
$997$	7.24871	0.229569	0.114784	+	0.993390i	$0.463382\pi$
$998$	−30.1962	−0.955843
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bg.1.1		2
3.2	odd	2		7938.2.a.bt.1.2		2
7.6	odd	2		1134.2.a.l.1.2	✓	2
21.20	even	2		1134.2.a.m.1.1	yes	2
28.27	even	2		9072.2.a.bp.1.2		2
63.13	odd	6		1134.2.f.s.379.1		4
63.20	even	6		1134.2.f.r.757.2		4
63.34	odd	6		1134.2.f.s.757.1		4
63.41	even	6		1134.2.f.r.379.2		4
84.83	odd	2		9072.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.a.l.1.2	✓	2	7.6	odd	2
1134.2.a.m.1.1	yes	2	21.20	even	2
1134.2.f.r.379.2		4	63.41	even	6
1134.2.f.r.757.2		4	63.20	even	6
1134.2.f.s.379.1		4	63.13	odd	6
1134.2.f.s.757.1		4	63.34	odd	6
7938.2.a.bg.1.1		2	1.1	even	1	trivial
7938.2.a.bt.1.2		2	3.2	odd	2
9072.2.a.y.1.1		2	84.83	odd	2
9072.2.a.bp.1.2		2	28.27	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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.73205 q^{5} -1.00000 q^{8} +3.73205 q^{10} -4.19615 q^{11} -0.464102 q^{13} +1.00000 q^{16} -7.00000 q^{17} +2.73205 q^{19} -3.73205 q^{20} +4.19615 q^{22} +6.19615 q^{23} +8.92820 q^{25} +0.464102 q^{26} +8.46410 q^{29} +2.19615 q^{31} -1.00000 q^{32} +7.00000 q^{34} -6.66025 q^{37} -2.73205 q^{38} +3.73205 q^{40} -9.46410 q^{41} +5.46410 q^{43} -4.19615 q^{44} -6.19615 q^{46} -1.26795 q^{47} -8.92820 q^{50} -0.464102 q^{52} -2.53590 q^{53} +15.6603 q^{55} -8.46410 q^{58} +6.19615 q^{59} +9.92820 q^{61} -2.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} -3.26795 q^{67} -7.00000 q^{68} +13.4641 q^{71} -11.7321 q^{73} +6.66025 q^{74} +2.73205 q^{76} +15.1244 q^{79} -3.73205 q^{80} +9.46410 q^{82} +14.5885 q^{83} +26.1244 q^{85} -5.46410 q^{86} +4.19615 q^{88} +3.92820 q^{89} +6.19615 q^{92} +1.26795 q^{94} -10.1962 q^{95} +2.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{16} - 14 q^{17} + 2 q^{19} - 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} - 6 q^{26} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 + 4 * q^10 + 2 * q^11 + 6 * q^13 + 2 * q^16 - 14 * q^17 + 2 * q^19 - 4 * q^20 - 2 * q^22 + 2 * q^23 + 4 * q^25 - 6 * q^26 + 10 * q^29 - 6 * q^31 - 2 * q^32 + 14 * q^34 + 4 * q^37 - 2 * q^38 + 4 * q^40 - 12 * q^41 + 4 * q^43 + 2 * q^44 - 2 * q^46 - 6 * q^47 - 4 * q^50 + 6 * q^52 - 12 * q^53 + 14 * q^55 - 10 * q^58 + 2 * q^59 + 6 * q^61 + 6 * q^62 + 2 * q^64 - 10 * q^67 - 14 * q^68 + 20 * q^71 - 20 * q^73 - 4 * q^74 + 2 * q^76 + 6 * q^79 - 4 * q^80 + 12 * q^82 - 2 * q^83 + 28 * q^85 - 4 * q^86 - 2 * q^88 - 6 * q^89 + 2 * q^92 + 6 * q^94 - 10 * q^95 - 8 * q^97

Introduction

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