LMFDB - Embedded newform 7600.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.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:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{3} +4.47214 q^{7} +7.47214 q^{9} +1.23607 q^{13} +2.47214 q^{17} -1.00000 q^{19} -14.4721 q^{21} -2.00000 q^{23} -14.4721 q^{27} +2.00000 q^{29} +10.4721 q^{31} +9.23607 q^{37} -4.00000 q^{39} +12.4721 q^{41} -8.47214 q^{43} -3.52786 q^{47} +13.0000 q^{49} -8.00000 q^{51} +6.76393 q^{53} +3.23607 q^{57} +1.52786 q^{59} +4.47214 q^{61} +33.4164 q^{63} +3.23607 q^{67} +6.47214 q^{69} +10.4721 q^{71} -2.47214 q^{73} -2.47214 q^{79} +24.4164 q^{81} -2.94427 q^{83} -6.47214 q^{87} +10.9443 q^{89} +5.52786 q^{91} -33.8885 q^{93} -18.1803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 20 q^{21} - 4 q^{23} - 20 q^{27} + 4 q^{29} + 12 q^{31} + 14 q^{37} - 8 q^{39} + 16 q^{41} - 8 q^{43} - 16 q^{47} + 26 q^{49} - 16 q^{51} + 18 q^{53}+ \cdots - 14 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^9 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 20 * q^21 - 4 * q^23 - 20 * q^27 + 4 * q^29 + 12 * q^31 + 14 * q^37 - 8 * q^39 + 16 * q^41 - 8 * q^43 - 16 * q^47 + 26 * q^49 - 16 * q^51 + 18 * q^53 + 2 * q^57 + 12 * q^59 + 40 * q^63 + 2 * q^67 + 4 * q^69 + 12 * q^71 + 4 * q^73 + 4 * q^79 + 22 * q^81 + 12 * q^83 - 4 * q^87 + 4 * q^89 + 20 * q^91 - 32 * q^93 - 14 * 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 $

$2$	0	0
$2$	0	0
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−14.4721	−3.15808
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−14.4721	−2.78516
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.23607	1.51840	0.759200	−	0.650857i	$-0.225590\pi$
$37$	9.23607	1.51840	0.759200	+	0.650857i	$0.225590\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	12.4721	1.94782	0.973910	−	0.226934i	$-0.0728701\pi$
$41$	12.4721	1.94782	0.973910	+	0.226934i	$0.0728701\pi$
$42$	0	0
$43$	−8.47214	−1.29199	−0.645994	−	0.763342i	$-0.723557\pi$
$43$	−8.47214	−1.29199	−0.645994	+	0.763342i	$0.723557\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.52786	−0.514592	−0.257296	−	0.966333i	$-0.582832\pi$
$47$	−3.52786	−0.514592	−0.257296	+	0.966333i	$0.582832\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	6.76393	0.929098	0.464549	−	0.885548i	$-0.346217\pi$
$53$	6.76393	0.929098	0.464549	+	0.885548i	$0.346217\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.23607	0.428628
$58$	0	0
$59$	1.52786	0.198911	0.0994555	−	0.995042i	$-0.468290\pi$
$59$	1.52786	0.198911	0.0994555	+	0.995042i	$0.468290\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	0	0
$63$	33.4164	4.21007
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.23607	0.395349	0.197674	−	0.980268i	$-0.436661\pi$
$67$	3.23607	0.395349	0.197674	+	0.980268i	$0.436661\pi$
$68$	0	0
$69$	6.47214	0.779154
$70$	0	0
$71$	10.4721	1.24281	0.621407	−	0.783488i	$-0.286561\pi$
$71$	10.4721	1.24281	0.621407	+	0.783488i	$0.286561\pi$
$72$	0	0
$73$	−2.47214	−0.289342	−0.144671	−	0.989480i	$-0.546212\pi$
$73$	−2.47214	−0.289342	−0.144671	+	0.989480i	$0.546212\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.47214	−0.278137	−0.139069	−	0.990283i	$-0.544411\pi$
$79$	−2.47214	−0.278137	−0.139069	+	0.990283i	$0.544411\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	−2.94427	−0.323176	−0.161588	−	0.986858i	$-0.551662\pi$
$83$	−2.94427	−0.323176	−0.161588	+	0.986858i	$0.551662\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.47214	−0.693886
$88$	0	0
$89$	10.9443	1.16009	0.580045	−	0.814584i	$-0.303035\pi$
$89$	10.9443	1.16009	0.580045	+	0.814584i	$0.303035\pi$
$90$	0	0
$91$	5.52786	0.579478
$92$	0	0
$93$	−33.8885	−3.51408
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.1803	−1.84593	−0.922967	−	0.384879i	$-0.874243\pi$
$97$	−18.1803	−1.84593	−0.922967	+	0.384879i	$0.874243\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.472136	−0.0469793	−0.0234896	−	0.999724i	$-0.507478\pi$
$101$	−0.472136	−0.0469793	−0.0234896	+	0.999724i	$0.507478\pi$
$102$	0	0
$103$	−12.7639	−1.25767	−0.628834	−	0.777540i	$-0.716467\pi$
$103$	−12.7639	−1.25767	−0.628834	+	0.777540i	$0.716467\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.70820	0.551833	0.275916	−	0.961182i	$-0.411019\pi$
$107$	5.70820	0.551833	0.275916	+	0.961182i	$0.411019\pi$
$108$	0	0
$109$	−13.4164	−1.28506	−0.642529	−	0.766261i	$-0.722115\pi$
$109$	−13.4164	−1.28506	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	−29.8885	−2.83690
$112$	0	0
$113$	10.7639	1.01259	0.506293	−	0.862362i	$-0.331016\pi$
$113$	10.7639	1.01259	0.506293	+	0.862362i	$0.331016\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	9.23607	0.853875
$118$	0	0
$119$	11.0557	1.01348
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−40.3607	−3.63920
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.2361	−0.997040	−0.498520	−	0.866878i	$-0.666123\pi$
$127$	−11.2361	−0.997040	−0.498520	+	0.866878i	$0.666123\pi$
$128$	0	0
$129$	27.4164	2.41388
$130$	0	0
$131$	−8.94427	−0.781465	−0.390732	−	0.920504i	$-0.627778\pi$
$131$	−8.94427	−0.781465	−0.390732	+	0.920504i	$0.627778\pi$
$132$	0	0
$133$	−4.47214	−0.387783
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.4164	−1.31711	−0.658556	−	0.752531i	$-0.728833\pi$
$137$	−15.4164	−1.31711	−0.658556	+	0.752531i	$0.728833\pi$
$138$	0	0
$139$	12.9443	1.09792	0.548959	−	0.835849i	$-0.315024\pi$
$139$	12.9443	1.09792	0.548959	+	0.835849i	$0.315024\pi$
$140$	0	0
$141$	11.4164	0.961435
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−42.0689	−3.46978
$148$	0	0
$149$	5.41641	0.443729	0.221865	−	0.975077i	$-0.428786\pi$
$149$	5.41641	0.443729	0.221865	+	0.975077i	$0.428786\pi$
$150$	0	0
$151$	17.8885	1.45575	0.727875	−	0.685710i	$-0.240508\pi$
$151$	17.8885	1.45575	0.727875	+	0.685710i	$0.240508\pi$
$152$	0	0
$153$	18.4721	1.49338
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.4164	−0.911129	−0.455564	−	0.890203i	$-0.650562\pi$
$157$	−11.4164	−0.911129	−0.455564	+	0.890203i	$0.650562\pi$
$158$	0	0
$159$	−21.8885	−1.73587
$160$	0	0
$161$	−8.94427	−0.704907
$162$	0	0
$163$	−0.472136	−0.0369805	−0.0184903	−	0.999829i	$-0.505886\pi$
$163$	−0.472136	−0.0369805	−0.0184903	+	0.999829i	$0.505886\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.1803	0.942543	0.471271	−	0.881988i	$-0.343795\pi$
$167$	12.1803	0.942543	0.471271	+	0.881988i	$0.343795\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	−7.47214	−0.571409
$172$	0	0
$173$	16.6525	1.26606	0.633032	−	0.774126i	$-0.281810\pi$
$173$	16.6525	1.26606	0.633032	+	0.774126i	$0.281810\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.94427	−0.371634
$178$	0	0
$179$	14.4721	1.08170	0.540849	−	0.841120i	$-0.318103\pi$
$179$	14.4721	1.08170	0.540849	+	0.841120i	$0.318103\pi$
$180$	0	0
$181$	−11.8885	−0.883669	−0.441834	−	0.897097i	$-0.645672\pi$
$181$	−11.8885	−0.883669	−0.441834	+	0.897097i	$0.645672\pi$
$182$	0	0
$183$	−14.4721	−1.06981
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−64.7214	−4.70779
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	7.70820	0.554849	0.277424	−	0.960747i	$-0.410519\pi$
$193$	7.70820	0.554849	0.277424	+	0.960747i	$0.410519\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.9443	1.20723	0.603615	−	0.797276i	$-0.293727\pi$
$197$	16.9443	1.20723	0.603615	+	0.797276i	$0.293727\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	−10.4721	−0.738648
$202$	0	0
$203$	8.94427	0.627765
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−14.9443	−1.03870
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−24.9443	−1.71723	−0.858617	−	0.512617i	$-0.828676\pi$
$211$	−24.9443	−1.71723	−0.858617	+	0.512617i	$0.828676\pi$
$212$	0	0
$213$	−33.8885	−2.32200
$214$	0	0
$215$	0	0
$216$	0	0
$217$	46.8328	3.17922
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	3.05573	0.205551
$222$	0	0
$223$	−20.7639	−1.39046	−0.695228	−	0.718789i	$-0.744697\pi$
$223$	−20.7639	−1.39046	−0.695228	+	0.718789i	$0.744697\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.1803	−0.808438	−0.404219	−	0.914662i	$-0.632457\pi$
$227$	−12.1803	−0.808438	−0.404219	+	0.914662i	$0.632457\pi$
$228$	0	0
$229$	−9.41641	−0.622254	−0.311127	−	0.950368i	$-0.600706\pi$
$229$	−9.41641	−0.622254	−0.311127	+	0.950368i	$0.600706\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.94427	−0.323910	−0.161955	−	0.986798i	$-0.551780\pi$
$233$	−4.94427	−0.323910	−0.161955	+	0.986798i	$0.551780\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−11.0557	−0.715136	−0.357568	−	0.933887i	$-0.616394\pi$
$239$	−11.0557	−0.715136	−0.357568	+	0.933887i	$0.616394\pi$
$240$	0	0
$241$	−5.41641	−0.348902	−0.174451	−	0.984666i	$-0.555815\pi$
$241$	−5.41641	−0.348902	−0.174451	+	0.984666i	$0.555815\pi$
$242$	0	0
$243$	−35.5967	−2.28353
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.23607	−0.0786491
$248$	0	0
$249$	9.52786	0.603804
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.7082	−1.47888	−0.739439	−	0.673224i	$-0.764909\pi$
$257$	−23.7082	−1.47888	−0.739439	+	0.673224i	$0.764909\pi$
$258$	0	0
$259$	41.3050	2.56656
$260$	0	0
$261$	14.9443	0.925027
$262$	0	0
$263$	23.8885	1.47303	0.736515	−	0.676421i	$-0.236470\pi$
$263$	23.8885	1.47303	0.736515	+	0.676421i	$0.236470\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−35.4164	−2.16745
$268$	0	0
$269$	1.41641	0.0863599	0.0431800	−	0.999067i	$-0.486251\pi$
$269$	1.41641	0.0863599	0.0431800	+	0.999067i	$0.486251\pi$
$270$	0	0
$271$	16.9443	1.02929	0.514646	−	0.857403i	$-0.327924\pi$
$271$	16.9443	1.02929	0.514646	+	0.857403i	$0.327924\pi$
$272$	0	0
$273$	−17.8885	−1.08266
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.5279	1.05315	0.526574	−	0.850130i	$-0.323476\pi$
$277$	17.5279	1.05315	0.526574	+	0.850130i	$0.323476\pi$
$278$	0	0
$279$	78.2492	4.68466
$280$	0	0
$281$	−5.41641	−0.323116	−0.161558	−	0.986863i	$-0.551652\pi$
$281$	−5.41641	−0.323116	−0.161558	+	0.986863i	$0.551652\pi$
$282$	0	0
$283$	−15.8885	−0.944476	−0.472238	−	0.881471i	$-0.656554\pi$
$283$	−15.8885	−0.944476	−0.472238	+	0.881471i	$0.656554\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	55.7771	3.29242
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	58.8328	3.44884
$292$	0	0
$293$	−4.29180	−0.250729	−0.125365	−	0.992111i	$-0.540010\pi$
$293$	−4.29180	−0.250729	−0.125365	+	0.992111i	$0.540010\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.47214	−0.142967
$300$	0	0
$301$	−37.8885	−2.18386
$302$	0	0
$303$	1.52786	0.0877735
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.76393	0.500184	0.250092	−	0.968222i	$-0.419539\pi$
$307$	8.76393	0.500184	0.250092	+	0.968222i	$0.419539\pi$
$308$	0	0
$309$	41.3050	2.34976
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	0	0
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.1246	1.07415	0.537073	−	0.843536i	$-0.319530\pi$
$317$	19.1246	1.07415	0.537073	+	0.843536i	$0.319530\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−18.4721	−1.03101
$322$	0	0
$323$	−2.47214	−0.137553
$324$	0	0
$325$	0	0
$326$	0	0
$327$	43.4164	2.40093
$328$	0	0
$329$	−15.7771	−0.869819
$330$	0	0
$331$	15.0557	0.827538	0.413769	−	0.910382i	$-0.364212\pi$
$331$	15.0557	0.827538	0.413769	+	0.910382i	$0.364212\pi$
$332$	0	0
$333$	69.0132	3.78190
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.7639	1.45792	0.728962	−	0.684554i	$-0.240003\pi$
$337$	26.7639	1.45792	0.728962	+	0.684554i	$0.240003\pi$
$338$	0	0
$339$	−34.8328	−1.89186
$340$	0	0
$341$	0	0
$342$	0	0
$343$	26.8328	1.44884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.94427	−0.372788	−0.186394	−	0.982475i	$-0.559680\pi$
$347$	−6.94427	−0.372788	−0.186394	+	0.982475i	$0.559680\pi$
$348$	0	0
$349$	27.8885	1.49284	0.746420	−	0.665475i	$-0.231771\pi$
$349$	27.8885	1.49284	0.746420	+	0.665475i	$0.231771\pi$
$350$	0	0
$351$	−17.8885	−0.954820
$352$	0	0
$353$	−13.5279	−0.720016	−0.360008	−	0.932949i	$-0.617226\pi$
$353$	−13.5279	−0.720016	−0.360008	+	0.932949i	$0.617226\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−35.7771	−1.89352
$358$	0	0
$359$	29.8885	1.57746	0.788729	−	0.614742i	$-0.210740\pi$
$359$	29.8885	1.57746	0.788729	+	0.614742i	$0.210740\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	35.5967	1.86834
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.3607	0.749621	0.374811	−	0.927101i	$-0.377708\pi$
$367$	14.3607	0.749621	0.374811	+	0.927101i	$0.377708\pi$
$368$	0	0
$369$	93.1935	4.85146
$370$	0	0
$371$	30.2492	1.57046
$372$	0	0
$373$	−19.1246	−0.990235	−0.495117	−	0.868826i	$-0.664875\pi$
$373$	−19.1246	−0.990235	−0.495117	+	0.868826i	$0.664875\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.47214	0.127321
$378$	0	0
$379$	13.8885	0.713407	0.356703	−	0.934218i	$-0.383901\pi$
$379$	13.8885	0.713407	0.356703	+	0.934218i	$0.383901\pi$
$380$	0	0
$381$	36.3607	1.86281
$382$	0	0
$383$	10.6525	0.544316	0.272158	−	0.962253i	$-0.412263\pi$
$383$	10.6525	0.544316	0.272158	+	0.962253i	$0.412263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−63.3050	−3.21797
$388$	0	0
$389$	−33.7771	−1.71257	−0.856283	−	0.516507i	$-0.827232\pi$
$389$	−33.7771	−1.71257	−0.856283	+	0.516507i	$0.827232\pi$
$390$	0	0
$391$	−4.94427	−0.250043
$392$	0	0
$393$	28.9443	1.46005
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.41641	−0.171465	−0.0857323	−	0.996318i	$-0.527323\pi$
$397$	−3.41641	−0.171465	−0.0857323	+	0.996318i	$0.527323\pi$
$398$	0	0
$399$	14.4721	0.724513
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	12.9443	0.644800
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−15.5279	−0.767803	−0.383902	−	0.923374i	$-0.625420\pi$
$409$	−15.5279	−0.767803	−0.383902	+	0.923374i	$0.625420\pi$
$410$	0	0
$411$	49.8885	2.46082
$412$	0	0
$413$	6.83282	0.336221
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−41.8885	−2.05129
$418$	0	0
$419$	−5.88854	−0.287674	−0.143837	−	0.989601i	$-0.545944\pi$
$419$	−5.88854	−0.287674	−0.143837	+	0.989601i	$0.545944\pi$
$420$	0	0
$421$	11.5279	0.561834	0.280917	−	0.959732i	$-0.409361\pi$
$421$	11.5279	0.561834	0.280917	+	0.959732i	$0.409361\pi$
$422$	0	0
$423$	−26.3607	−1.28170
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.0000	0.967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.8885	−0.861661	−0.430830	−	0.902433i	$-0.641779\pi$
$431$	−17.8885	−0.861661	−0.430830	+	0.902433i	$0.641779\pi$
$432$	0	0
$433$	−10.1803	−0.489236	−0.244618	−	0.969620i	$-0.578663\pi$
$433$	−10.1803	−0.489236	−0.244618	+	0.969620i	$0.578663\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−13.5279	−0.645650	−0.322825	−	0.946459i	$-0.604632\pi$
$439$	−13.5279	−0.645650	−0.322825	+	0.946459i	$0.604632\pi$
$440$	0	0
$441$	97.1378	4.62561
$442$	0	0
$443$	11.8885	0.564842	0.282421	−	0.959291i	$-0.408863\pi$
$443$	11.8885	0.564842	0.282421	+	0.959291i	$0.408863\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−17.5279	−0.829040
$448$	0	0
$449$	−17.4164	−0.821931	−0.410966	−	0.911651i	$-0.634808\pi$
$449$	−17.4164	−0.821931	−0.410966	+	0.911651i	$0.634808\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−57.8885	−2.71984
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.88854	0.0883424	0.0441712	−	0.999024i	$-0.485935\pi$
$457$	1.88854	0.0883424	0.0441712	+	0.999024i	$0.485935\pi$
$458$	0	0
$459$	−35.7771	−1.66993
$460$	0	0
$461$	23.8885	1.11260	0.556300	−	0.830981i	$-0.312220\pi$
$461$	23.8885	1.11260	0.556300	+	0.830981i	$0.312220\pi$
$462$	0	0
$463$	−3.88854	−0.180716	−0.0903580	−	0.995909i	$-0.528801\pi$
$463$	−3.88854	−0.180716	−0.0903580	+	0.995909i	$0.528801\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	14.4721	0.668261
$470$	0	0
$471$	36.9443	1.70230
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	50.5410	2.31411
$478$	0	0
$479$	7.05573	0.322384	0.161192	−	0.986923i	$-0.448466\pi$
$479$	7.05573	0.322384	0.161192	+	0.986923i	$0.448466\pi$
$480$	0	0
$481$	11.4164	0.520543
$482$	0	0
$483$	28.9443	1.31701
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18.6525	−0.845224	−0.422612	−	0.906311i	$-0.638887\pi$
$487$	−18.6525	−0.845224	−0.422612	+	0.906311i	$0.638887\pi$
$488$	0	0
$489$	1.52786	0.0690924
$490$	0	0
$491$	−24.9443	−1.12572	−0.562860	−	0.826553i	$-0.690299\pi$
$491$	−24.9443	−1.12572	−0.562860	+	0.826553i	$0.690299\pi$
$492$	0	0
$493$	4.94427	0.222679
$494$	0	0
$495$	0	0
$496$	0	0
$497$	46.8328	2.10074
$498$	0	0
$499$	−38.8328	−1.73840	−0.869198	−	0.494465i	$-0.835364\pi$
$499$	−38.8328	−1.73840	−0.869198	+	0.494465i	$0.835364\pi$
$500$	0	0
$501$	−39.4164	−1.76100
$502$	0	0
$503$	33.7771	1.50605	0.753023	−	0.657994i	$-0.228595\pi$
$503$	33.7771	1.50605	0.753023	+	0.657994i	$0.228595\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	37.1246	1.64876
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−11.0557	−0.489077
$512$	0	0
$513$	14.4721	0.638960
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−53.8885	−2.36544
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	−16.7639	−0.733036	−0.366518	−	0.930411i	$-0.619450\pi$
$523$	−16.7639	−0.733036	−0.366518	+	0.930411i	$0.619450\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	25.8885	1.12772
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	11.4164	0.495430
$532$	0	0
$533$	15.4164	0.667759
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−46.8328	−2.02099
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.3607	0.617414	0.308707	−	0.951157i	$-0.400104\pi$
$541$	14.3607	0.617414	0.308707	+	0.951157i	$0.400104\pi$
$542$	0	0
$543$	38.4721	1.65100
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.7082	0.586120	0.293060	−	0.956094i	$-0.405326\pi$
$547$	13.7082	0.586120	0.293060	+	0.956094i	$0.405326\pi$
$548$	0	0
$549$	33.4164	1.42618
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	−11.0557	−0.470137
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12.5836	−0.533184	−0.266592	−	0.963809i	$-0.585898\pi$
$557$	−12.5836	−0.533184	−0.266592	+	0.963809i	$0.585898\pi$
$558$	0	0
$559$	−10.4721	−0.442924
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.7639	−1.04368	−0.521838	−	0.853045i	$-0.674753\pi$
$563$	−24.7639	−1.04368	−0.521838	+	0.853045i	$0.674753\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	109.193	4.58570
$568$	0	0
$569$	3.52786	0.147896	0.0739479	−	0.997262i	$-0.476440\pi$
$569$	3.52786	0.147896	0.0739479	+	0.997262i	$0.476440\pi$
$570$	0	0
$571$	20.9443	0.876491	0.438245	−	0.898855i	$-0.355600\pi$
$571$	20.9443	0.876491	0.438245	+	0.898855i	$0.355600\pi$
$572$	0	0
$573$	25.8885	1.08151
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.05573	−0.127212	−0.0636058	−	0.997975i	$-0.520260\pi$
$577$	−3.05573	−0.127212	−0.0636058	+	0.997975i	$0.520260\pi$
$578$	0	0
$579$	−24.9443	−1.03665
$580$	0	0
$581$	−13.1672	−0.546267
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.7771	−1.88942	−0.944711	−	0.327903i	$-0.893658\pi$
$587$	−45.7771	−1.88942	−0.944711	+	0.327903i	$0.893658\pi$
$588$	0	0
$589$	−10.4721	−0.431497
$590$	0	0
$591$	−54.8328	−2.25552
$592$	0	0
$593$	36.9443	1.51712	0.758560	−	0.651604i	$-0.225903\pi$
$593$	36.9443	1.51712	0.758560	+	0.651604i	$0.225903\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	29.5279	1.20648	0.603238	−	0.797561i	$-0.293877\pi$
$599$	29.5279	1.20648	0.603238	+	0.797561i	$0.293877\pi$
$600$	0	0
$601$	−2.36068	−0.0962941	−0.0481471	−	0.998840i	$-0.515332\pi$
$601$	−2.36068	−0.0962941	−0.0481471	+	0.998840i	$0.515332\pi$
$602$	0	0
$603$	24.1803	0.984700
$604$	0	0
$605$	0	0
$606$	0	0
$607$	25.1246	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$607$	25.1246	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$608$	0	0
$609$	−28.9443	−1.17288
$610$	0	0
$611$	−4.36068	−0.176414
$612$	0	0
$613$	12.5836	0.508246	0.254123	−	0.967172i	$-0.418213\pi$
$613$	12.5836	0.508246	0.254123	+	0.967172i	$0.418213\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.2492	1.53986	0.769928	−	0.638131i	$-0.220292\pi$
$617$	38.2492	1.53986	0.769928	+	0.638131i	$0.220292\pi$
$618$	0	0
$619$	28.9443	1.16337	0.581684	−	0.813415i	$-0.302394\pi$
$619$	28.9443	1.16337	0.581684	+	0.813415i	$0.302394\pi$
$620$	0	0
$621$	28.9443	1.16149
$622$	0	0
$623$	48.9443	1.96091
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22.8328	0.910404
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	80.7214	3.20839
$634$	0	0
$635$	0	0
$636$	0	0
$637$	16.0689	0.636672
$638$	0	0
$639$	78.2492	3.09549
$640$	0	0
$641$	−10.3607	−0.409222	−0.204611	−	0.978843i	$-0.565593\pi$
$641$	−10.3607	−0.409222	−0.204611	+	0.978843i	$0.565593\pi$
$642$	0	0
$643$	−6.00000	−0.236617	−0.118308	−	0.992977i	$-0.537747\pi$
$643$	−6.00000	−0.236617	−0.118308	+	0.992977i	$0.537747\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.9443	−1.05929	−0.529644	−	0.848220i	$-0.677675\pi$
$647$	−26.9443	−1.05929	−0.529644	+	0.848220i	$0.677675\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−151.554	−5.93988
$652$	0	0
$653$	30.4721	1.19247	0.596233	−	0.802811i	$-0.296663\pi$
$653$	30.4721	1.19247	0.596233	+	0.802811i	$0.296663\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−18.4721	−0.720667
$658$	0	0
$659$	16.3607	0.637322	0.318661	−	0.947869i	$-0.396767\pi$
$659$	16.3607	0.637322	0.318661	+	0.947869i	$0.396767\pi$
$660$	0	0
$661$	−36.4721	−1.41860	−0.709301	−	0.704906i	$-0.750989\pi$
$661$	−36.4721	−1.41860	−0.709301	+	0.704906i	$0.750989\pi$
$662$	0	0
$663$	−9.88854	−0.384039
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.00000	−0.154881
$668$	0	0
$669$	67.1935	2.59785
$670$	0	0
$671$	0	0
$672$	0	0
$673$	42.1803	1.62593	0.812966	−	0.582311i	$-0.197851\pi$
$673$	42.1803	1.62593	0.812966	+	0.582311i	$0.197851\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.1246	−1.04248	−0.521242	−	0.853409i	$-0.674531\pi$
$677$	−27.1246	−1.04248	−0.521242	+	0.853409i	$0.674531\pi$
$678$	0	0
$679$	−81.3050	−3.12020
$680$	0	0
$681$	39.4164	1.51044
$682$	0	0
$683$	36.1803	1.38440	0.692201	−	0.721705i	$-0.256641\pi$
$683$	36.1803	1.38440	0.692201	+	0.721705i	$0.256641\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	30.4721	1.16258
$688$	0	0
$689$	8.36068	0.318517
$690$	0	0
$691$	−17.8885	−0.680512	−0.340256	−	0.940333i	$-0.610514\pi$
$691$	−17.8885	−0.680512	−0.340256	+	0.940333i	$0.610514\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	30.8328	1.16788
$698$	0	0
$699$	16.0000	0.605176
$700$	0	0
$701$	−39.3050	−1.48453	−0.742264	−	0.670108i	$-0.766248\pi$
$701$	−39.3050	−1.48453	−0.742264	+	0.670108i	$0.766248\pi$
$702$	0	0
$703$	−9.23607	−0.348345
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.11146	−0.0794095
$708$	0	0
$709$	35.8885	1.34782	0.673911	−	0.738812i	$-0.264613\pi$
$709$	35.8885	1.34782	0.673911	+	0.738812i	$0.264613\pi$
$710$	0	0
$711$	−18.4721	−0.692759
$712$	0	0
$713$	−20.9443	−0.784369
$714$	0	0
$715$	0	0
$716$	0	0
$717$	35.7771	1.33612
$718$	0	0
$719$	10.1115	0.377094	0.188547	−	0.982064i	$-0.439622\pi$
$719$	10.1115	0.377094	0.188547	+	0.982064i	$0.439622\pi$
$720$	0	0
$721$	−57.0820	−2.12585
$722$	0	0
$723$	17.5279	0.651868
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.5836	0.689227	0.344614	−	0.938745i	$-0.388010\pi$
$727$	18.5836	0.689227	0.344614	+	0.938745i	$0.388010\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	−20.9443	−0.774652
$732$	0	0
$733$	0.944272	0.0348775	0.0174387	−	0.999848i	$-0.494449\pi$
$733$	0.944272	0.0348775	0.0174387	+	0.999848i	$0.494449\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	−1.12461	−0.0412580	−0.0206290	−	0.999787i	$-0.506567\pi$
$743$	−1.12461	−0.0412580	−0.0206290	+	0.999787i	$0.506567\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−22.0000	−0.804938
$748$	0	0
$749$	25.5279	0.932768
$750$	0	0
$751$	−3.05573	−0.111505	−0.0557526	−	0.998445i	$-0.517756\pi$
$751$	−3.05573	−0.111505	−0.0557526	+	0.998445i	$0.517756\pi$
$752$	0	0
$753$	12.9443	0.471715
$754$	0	0
$755$	0	0
$756$	0	0
$757$	12.0000	0.436147	0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000	0.436147	0.218074	+	0.975932i	$0.430023\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.47214	0.307115	0.153557	−	0.988140i	$-0.450927\pi$
$761$	8.47214	0.307115	0.153557	+	0.988140i	$0.450927\pi$
$762$	0	0
$763$	−60.0000	−2.17215
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.88854	0.0681914
$768$	0	0
$769$	−29.4164	−1.06078	−0.530391	−	0.847753i	$-0.677955\pi$
$769$	−29.4164	−1.06078	−0.530391	+	0.847753i	$0.677955\pi$
$770$	0	0
$771$	76.7214	2.76305
$772$	0	0
$773$	−27.7082	−0.996595	−0.498297	−	0.867006i	$-0.666041\pi$
$773$	−27.7082	−0.996595	−0.498297	+	0.867006i	$0.666041\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−133.666	−4.79523
$778$	0	0
$779$	−12.4721	−0.446861
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−28.9443	−1.03438
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.5410	0.447039	0.223520	−	0.974699i	$-0.428245\pi$
$787$	12.5410	0.447039	0.223520	+	0.974699i	$0.428245\pi$
$788$	0	0
$789$	−77.3050	−2.75213
$790$	0	0
$791$	48.1378	1.71158
$792$	0	0
$793$	5.52786	0.196300
$794$	0	0
$795$	0	0
$796$	0	0
$797$	16.6525	0.589861	0.294930	−	0.955519i	$-0.404704\pi$
$797$	16.6525	0.589861	0.294930	+	0.955519i	$0.404704\pi$
$798$	0	0
$799$	−8.72136	−0.308540
$800$	0	0
$801$	81.7771	2.88945
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.58359	−0.161350
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	21.3050	0.748118	0.374059	−	0.927405i	$-0.377966\pi$
$811$	21.3050	0.748118	0.374059	+	0.927405i	$0.377966\pi$
$812$	0	0
$813$	−54.8328	−1.92307
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.47214	0.296403
$818$	0	0
$819$	41.3050	1.44331
$820$	0	0
$821$	25.0557	0.874451	0.437225	−	0.899352i	$-0.355961\pi$
$821$	25.0557	0.874451	0.437225	+	0.899352i	$0.355961\pi$
$822$	0	0
$823$	−49.4164	−1.72255	−0.861274	−	0.508141i	$-0.830333\pi$
$823$	−49.4164	−1.72255	−0.861274	+	0.508141i	$0.830333\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.2361	1.50347	0.751733	−	0.659468i	$-0.229218\pi$
$827$	43.2361	1.50347	0.751733	+	0.659468i	$0.229218\pi$
$828$	0	0
$829$	17.4164	0.604897	0.302448	−	0.953166i	$-0.402196\pi$
$829$	17.4164	0.604897	0.302448	+	0.953166i	$0.402196\pi$
$830$	0	0
$831$	−56.7214	−1.96764
$832$	0	0
$833$	32.1378	1.11351
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−151.554	−5.23848
$838$	0	0
$839$	26.4721	0.913920	0.456960	−	0.889487i	$-0.348938\pi$
$839$	26.4721	0.913920	0.456960	+	0.889487i	$0.348938\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	17.5279	0.603692
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−49.1935	−1.69031
$848$	0	0
$849$	51.4164	1.76461
$850$	0	0
$851$	−18.4721	−0.633217
$852$	0	0
$853$	23.0557	0.789413	0.394707	−	0.918807i	$-0.370846\pi$
$853$	23.0557	0.789413	0.394707	+	0.918807i	$0.370846\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−52.6525	−1.79857	−0.899287	−	0.437359i	$-0.855914\pi$
$857$	−52.6525	−1.79857	−0.899287	+	0.437359i	$0.855914\pi$
$858$	0	0
$859$	2.11146	0.0720420	0.0360210	−	0.999351i	$-0.488532\pi$
$859$	2.11146	0.0720420	0.0360210	+	0.999351i	$0.488532\pi$
$860$	0	0
$861$	−180.498	−6.15137
$862$	0	0
$863$	16.7639	0.570651	0.285325	−	0.958431i	$-0.407898\pi$
$863$	16.7639	0.570651	0.285325	+	0.958431i	$0.407898\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	35.2361	1.19668
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	−135.846	−4.59769
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.1803	−0.478836	−0.239418	−	0.970917i	$-0.576957\pi$
$877$	−14.1803	−0.478836	−0.239418	+	0.970917i	$0.576957\pi$
$878$	0	0
$879$	13.8885	0.468449
$880$	0	0
$881$	−40.2492	−1.35603	−0.678015	−	0.735048i	$-0.737160\pi$
$881$	−40.2492	−1.35603	−0.678015	+	0.735048i	$0.737160\pi$
$882$	0	0
$883$	44.4721	1.49661	0.748303	−	0.663357i	$-0.230869\pi$
$883$	44.4721	1.49661	0.748303	+	0.663357i	$0.230869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.7082	1.66904	0.834519	−	0.550979i	$-0.185746\pi$
$887$	49.7082	1.66904	0.834519	+	0.550979i	$0.185746\pi$
$888$	0	0
$889$	−50.2492	−1.68530
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.52786	0.118055
$894$	0	0
$895$	0	0
$896$	0	0
$897$	8.00000	0.267112
$898$	0	0
$899$	20.9443	0.698531
$900$	0	0
$901$	16.7214	0.557069
$902$	0	0
$903$	122.610	4.08020
$904$	0	0
$905$	0	0
$906$	0	0
$907$	11.8197	0.392465	0.196233	−	0.980557i	$-0.437129\pi$
$907$	11.8197	0.392465	0.196233	+	0.980557i	$0.437129\pi$
$908$	0	0
$909$	−3.52786	−0.117012
$910$	0	0
$911$	54.8328	1.81669	0.908346	−	0.418219i	$-0.137346\pi$
$911$	54.8328	1.81669	0.908346	+	0.418219i	$0.137346\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−40.0000	−1.32092
$918$	0	0
$919$	1.88854	0.0622973	0.0311487	−	0.999515i	$-0.490083\pi$
$919$	1.88854	0.0622973	0.0311487	+	0.999515i	$0.490083\pi$
$920$	0	0
$921$	−28.3607	−0.934516
$922$	0	0
$923$	12.9443	0.426066
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−95.3738	−3.13249
$928$	0	0
$929$	−29.7771	−0.976955	−0.488477	−	0.872577i	$-0.662447\pi$
$929$	−29.7771	−0.976955	−0.488477	+	0.872577i	$0.662447\pi$
$930$	0	0
$931$	−13.0000	−0.426058
$932$	0	0
$933$	−64.7214	−2.11888
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8.58359	0.280414	0.140207	−	0.990122i	$-0.455223\pi$
$937$	8.58359	0.280414	0.140207	+	0.990122i	$0.455223\pi$
$938$	0	0
$939$	51.7771	1.68968
$940$	0	0
$941$	36.8328	1.20072	0.600358	−	0.799732i	$-0.295025\pi$
$941$	36.8328	1.20072	0.600358	+	0.799732i	$0.295025\pi$
$942$	0	0
$943$	−24.9443	−0.812297
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.9443	0.875571	0.437786	−	0.899079i	$-0.355763\pi$
$947$	26.9443	0.875571	0.437786	+	0.899079i	$0.355763\pi$
$948$	0	0
$949$	−3.05573	−0.0991931
$950$	0	0
$951$	−61.8885	−2.00687
$952$	0	0
$953$	−15.7082	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$953$	−15.7082	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−68.9443	−2.22633
$960$	0	0
$961$	78.6656	2.53760
$962$	0	0
$963$	42.6525	1.37446
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−20.8328	−0.669938	−0.334969	−	0.942229i	$-0.608726\pi$
$967$	−20.8328	−0.669938	−0.334969	+	0.942229i	$0.608726\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	−0.360680	−0.0115748	−0.00578738	−	0.999983i	$-0.501842\pi$
$971$	−0.360680	−0.0115748	−0.00578738	+	0.999983i	$0.501842\pi$
$972$	0	0
$973$	57.8885	1.85582
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.1803	0.837583	0.418792	−	0.908082i	$-0.362454\pi$
$977$	26.1803	0.837583	0.418792	+	0.908082i	$0.362454\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−100.249	−3.20071
$982$	0	0
$983$	−39.2361	−1.25144	−0.625718	−	0.780049i	$-0.715194\pi$
$983$	−39.2361	−1.25144	−0.625718	+	0.780049i	$0.715194\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	51.0557	1.62512
$988$	0	0
$989$	16.9443	0.538797
$990$	0	0
$991$	25.8885	0.822377	0.411188	−	0.911550i	$-0.365114\pi$
$991$	25.8885	0.822377	0.411188	+	0.911550i	$0.365114\pi$
$992$	0	0
$993$	−48.7214	−1.54613
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−29.3050	−0.928097	−0.464049	−	0.885810i	$-0.653604\pi$
$997$	−29.3050	−0.928097	−0.464049	+	0.885810i	$0.653604\pi$
$998$	0	0
$999$	−133.666	−4.22899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.w.1.1		2
4.3	odd	2		3800.2.a.q.1.2		2
5.2	odd	4		1520.2.d.c.609.4		4
5.3	odd	4		1520.2.d.c.609.1		4
5.4	even	2		7600.2.a.be.1.2		2
20.3	even	4		760.2.d.b.609.4	yes	4
20.7	even	4		760.2.d.b.609.1	✓	4
20.19	odd	2		3800.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.b.609.1	✓	4	20.7	even	4
760.2.d.b.609.4	yes	4	20.3	even	4
1520.2.d.c.609.1		4	5.3	odd	4
1520.2.d.c.609.4		4	5.2	odd	4
3800.2.a.k.1.1		2	20.19	odd	2
3800.2.a.q.1.2		2	4.3	odd	2
7600.2.a.w.1.1		2	1.1	even	1	trivial
7600.2.a.be.1.2		2	5.4	even	2

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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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23607 q^{3} +4.47214 q^{7} +7.47214 q^{9} +1.23607 q^{13} +2.47214 q^{17} -1.00000 q^{19} -14.4721 q^{21} -2.00000 q^{23} -14.4721 q^{27} +2.00000 q^{29} +10.4721 q^{31} +9.23607 q^{37} -4.00000 q^{39} +12.4721 q^{41} -8.47214 q^{43} -3.52786 q^{47} +13.0000 q^{49} -8.00000 q^{51} +6.76393 q^{53} +3.23607 q^{57} +1.52786 q^{59} +4.47214 q^{61} +33.4164 q^{63} +3.23607 q^{67} +6.47214 q^{69} +10.4721 q^{71} -2.47214 q^{73} -2.47214 q^{79} +24.4164 q^{81} -2.94427 q^{83} -6.47214 q^{87} +10.9443 q^{89} +5.52786 q^{91} -33.8885 q^{93} -18.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{9} - 2 q^{13} - 4 q^{17} - 2 q^{19} - 20 q^{21} - 4 q^{23} - 20 q^{27} + 4 q^{29} + 12 q^{31} + 14 q^{37} - 8 q^{39} + 16 q^{41} - 8 q^{43} - 16 q^{47} + 26 q^{49} - 16 q^{51} + 18 q^{53}+ \cdots - 14 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^9 - 2 * q^13 - 4 * q^17 - 2 * q^19 - 20 * q^21 - 4 * q^23 - 20 * q^27 + 4 * q^29 + 12 * q^31 + 14 * q^37 - 8 * q^39 + 16 * q^41 - 8 * q^43 - 16 * q^47 + 26 * q^49 - 16 * q^51 + 18 * q^53 + 2 * q^57 + 12 * q^59 + 40 * q^63 + 2 * q^67 + 4 * q^69 + 12 * q^71 + 4 * q^73 + 4 * q^79 + 22 * q^81 + 12 * q^83 - 4 * q^87 + 4 * q^89 + 20 * q^91 - 32 * q^93 - 14 * q^97

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