LMFDB - Embedded newform 7600.2.a.cd.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76156$ 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)

$f(q)$	$=$	$q+2.76156 q^{3} -0.761557 q^{7} +4.62620 q^{9} +O(q^{10})$	$q+2.76156 q^{3} -0.761557 q^{7} +4.62620 q^{9} +0.864641 q^{11} -5.62620 q^{13} +3.62620 q^{17} -1.00000 q^{19} -2.10308 q^{21} -8.01395 q^{23} +4.49084 q^{27} -7.35548 q^{29} -8.11704 q^{31} +2.38776 q^{33} -0.476886 q^{37} -15.5371 q^{39} -2.65847 q^{41} +6.86464 q^{43} +1.25240 q^{47} -6.42003 q^{49} +10.0140 q^{51} -2.37380 q^{53} -2.76156 q^{57} -4.49084 q^{59} -10.8646 q^{61} -3.52311 q^{63} +1.03228 q^{67} -22.1310 q^{69} +10.1816 q^{71} -16.4017 q^{73} -0.658473 q^{77} +12.5693 q^{79} -1.47689 q^{81} -0.270718 q^{83} -20.3126 q^{87} +0.387755 q^{89} +4.28467 q^{91} -22.4157 q^{93} +8.50479 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9	$ 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9} - 8 q^{13} + 2 q^{17} - 3 q^{19} - 10 q^{21} + 2 q^{27} - 8 q^{29} - 4 q^{31} - 8 q^{33} - 14 q^{37} - 10 q^{39} + 2 q^{41} + 18 q^{43} - 14 q^{47} - 3 q^{49} + 6 q^{51} - 16 q^{53} - 2 q^{57} - 2 q^{59} - 30 q^{61} + 2 q^{63} + 2 q^{67} - 22 q^{69} + 8 q^{71} - 10 q^{73} + 8 q^{77} - 17 q^{81} - 6 q^{83} + 6 q^{87} - 14 q^{89} - 6 q^{91} - 4 q^{93} - 10 q^{97} + 12 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9 - 8 * q^13 + 2 * q^17 - 3 * q^19 - 10 * q^21 + 2 * q^27 - 8 * q^29 - 4 * q^31 - 8 * q^33 - 14 * q^37 - 10 * q^39 + 2 * q^41 + 18 * q^43 - 14 * q^47 - 3 * q^49 + 6 * q^51 - 16 * q^53 - 2 * q^57 - 2 * q^59 - 30 * q^61 + 2 * q^63 + 2 * q^67 - 22 * q^69 + 8 * q^71 - 10 * q^73 + 8 * q^77 - 17 * q^81 - 6 * q^83 + 6 * q^87 - 14 * q^89 - 6 * q^91 - 4 * q^93 - 10 * q^97 + 12 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	2.76156	1.59439	0.797193	−	0.603725i	$-0.206317\pi$
$3$	2.76156	1.59439	0.797193	+	0.603725i	$0.206317\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.761557	−0.287842	−0.143921	−	0.989589i	$-0.545971\pi$
$7$	−0.761557	−0.287842	−0.143921	+	0.989589i	$0.545971\pi$
$8$	0	0
$9$	4.62620	1.54207
$10$	0	0
$11$	0.864641	0.260699	0.130350	−	0.991468i	$-0.458390\pi$
$11$	0.864641	0.260699	0.130350	+	0.991468i	$0.458390\pi$
$12$	0	0
$13$	−5.62620	−1.56043	−0.780213	−	0.625514i	$-0.784889\pi$
$13$	−5.62620	−1.56043	−0.780213	+	0.625514i	$0.784889\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.62620	0.879482	0.439741	−	0.898125i	$-0.355070\pi$
$17$	3.62620	0.879482	0.439741	+	0.898125i	$0.355070\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.10308	−0.458930
$22$	0	0
$23$	−8.01395	−1.67102	−0.835512	−	0.549472i	$-0.814829\pi$
$23$	−8.01395	−1.67102	−0.835512	+	0.549472i	$0.814829\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.49084	0.864262
$28$	0	0
$29$	−7.35548	−1.36588	−0.682939	−	0.730475i	$-0.739299\pi$
$29$	−7.35548	−1.36588	−0.682939	+	0.730475i	$0.739299\pi$
$30$	0	0
$31$	−8.11704	−1.45786	−0.728931	−	0.684587i	$-0.759983\pi$
$31$	−8.11704	−1.45786	−0.728931	+	0.684587i	$0.759983\pi$
$32$	0	0
$33$	2.38776	0.415655
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.476886	−0.0783995	−0.0391998	−	0.999231i	$-0.512481\pi$
$37$	−0.476886	−0.0783995	−0.0391998	+	0.999231i	$0.512481\pi$
$38$	0	0
$39$	−15.5371	−2.48792
$40$	0	0
$41$	−2.65847	−0.415184	−0.207592	−	0.978216i	$-0.566563\pi$
$41$	−2.65847	−0.415184	−0.207592	+	0.978216i	$0.566563\pi$
$42$	0	0
$43$	6.86464	1.04685	0.523424	−	0.852072i	$-0.324654\pi$
$43$	6.86464	1.04685	0.523424	+	0.852072i	$0.324654\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.25240	0.182681	0.0913404	−	0.995820i	$-0.470885\pi$
$47$	1.25240	0.182681	0.0913404	+	0.995820i	$0.470885\pi$
$48$	0	0
$49$	−6.42003	−0.917147
$50$	0	0
$51$	10.0140	1.40223
$52$	0	0
$53$	−2.37380	−0.326067	−0.163033	−	0.986621i	$-0.552128\pi$
$53$	−2.37380	−0.326067	−0.163033	+	0.986621i	$0.552128\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.76156	−0.365777
$58$	0	0
$59$	−4.49084	−0.584657	−0.292329	−	0.956318i	$-0.594430\pi$
$59$	−4.49084	−0.584657	−0.292329	+	0.956318i	$0.594430\pi$
$60$	0	0
$61$	−10.8646	−1.39107	−0.695537	−	0.718490i	$-0.744834\pi$
$61$	−10.8646	−1.39107	−0.695537	+	0.718490i	$0.744834\pi$
$62$	0	0
$63$	−3.52311	−0.443871
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.03228	0.126113	0.0630563	−	0.998010i	$-0.479915\pi$
$67$	1.03228	0.126113	0.0630563	+	0.998010i	$0.479915\pi$
$68$	0	0
$69$	−22.1310	−2.66426
$70$	0	0
$71$	10.1816	1.20833	0.604166	−	0.796858i	$-0.293506\pi$
$71$	10.1816	1.20833	0.604166	+	0.796858i	$0.293506\pi$
$72$	0	0
$73$	−16.4017	−1.91967	−0.959837	−	0.280557i	$-0.909481\pi$
$73$	−16.4017	−1.91967	−0.959837	+	0.280557i	$0.909481\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.658473	−0.0750400
$78$	0	0
$79$	12.5693	1.41416	0.707081	−	0.707133i	$-0.250012\pi$
$79$	12.5693	1.41416	0.707081	+	0.707133i	$0.250012\pi$
$80$	0	0
$81$	−1.47689	−0.164098
$82$	0	0
$83$	−0.270718	−0.0297152	−0.0148576	−	0.999890i	$-0.504729\pi$
$83$	−0.270718	−0.0297152	−0.0148576	+	0.999890i	$0.504729\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−20.3126	−2.17774
$88$	0	0
$89$	0.387755	0.0411020	0.0205510	−	0.999789i	$-0.493458\pi$
$89$	0.387755	0.0411020	0.0205510	+	0.999789i	$0.493458\pi$
$90$	0	0
$91$	4.28467	0.449156
$92$	0	0
$93$	−22.4157	−2.32440
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.50479	0.863531	0.431765	−	0.901986i	$-0.357891\pi$
$97$	8.50479	0.863531	0.431765	+	0.901986i	$0.357891\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	16.4157	1.63342	0.816710	−	0.577049i	$-0.195796\pi$
$101$	16.4157	1.63342	0.816710	+	0.577049i	$0.195796\pi$
$102$	0	0
$103$	−9.64015	−0.949872	−0.474936	−	0.880020i	$-0.657529\pi$
$103$	−9.64015	−0.949872	−0.474936	+	0.880020i	$0.657529\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.28467	0.414215	0.207107	−	0.978318i	$-0.433595\pi$
$107$	4.28467	0.414215	0.207107	+	0.978318i	$0.433595\pi$
$108$	0	0
$109$	−13.4200	−1.28541	−0.642703	−	0.766116i	$-0.722187\pi$
$109$	−13.4200	−1.28541	−0.642703	+	0.766116i	$0.722187\pi$
$110$	0	0
$111$	−1.31695	−0.124999
$112$	0	0
$113$	10.3232	0.971125	0.485563	−	0.874202i	$-0.338615\pi$
$113$	10.3232	0.971125	0.485563	+	0.874202i	$0.338615\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−26.0279	−2.40628
$118$	0	0
$119$	−2.76156	−0.253152
$120$	0	0
$121$	−10.2524	−0.932036
$122$	0	0
$123$	−7.34153	−0.661963
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.9817	1.50688	0.753440	−	0.657517i	$-0.228393\pi$
$127$	16.9817	1.50688	0.753440	+	0.657517i	$0.228393\pi$
$128$	0	0
$129$	18.9571	1.66908
$130$	0	0
$131$	−0.541436	−0.0473055	−0.0236528	−	0.999720i	$-0.507530\pi$
$131$	−0.541436	−0.0473055	−0.0236528	+	0.999720i	$0.507530\pi$
$132$	0	0
$133$	0.761557	0.0660354
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.87859	−0.245935	−0.122967	−	0.992411i	$-0.539241\pi$
$137$	−2.87859	−0.245935	−0.122967	+	0.992411i	$0.539241\pi$
$138$	0	0
$139$	−3.58767	−0.304302	−0.152151	−	0.988357i	$-0.548620\pi$
$139$	−3.58767	−0.304302	−0.152151	+	0.988357i	$0.548620\pi$
$140$	0	0
$141$	3.45856	0.291264
$142$	0	0
$143$	−4.86464	−0.406802
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−17.7293	−1.46229
$148$	0	0
$149$	−16.8401	−1.37959	−0.689796	−	0.724004i	$-0.742300\pi$
$149$	−16.8401	−1.37959	−0.689796	+	0.724004i	$0.742300\pi$
$150$	0	0
$151$	16.9817	1.38195	0.690975	−	0.722879i	$-0.257182\pi$
$151$	16.9817	1.38195	0.690975	+	0.722879i	$0.257182\pi$
$152$	0	0
$153$	16.7755	1.35622
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−14.8401	−1.18437	−0.592183	−	0.805804i	$-0.701734\pi$
$157$	−14.8401	−1.18437	−0.592183	+	0.805804i	$0.701734\pi$
$158$	0	0
$159$	−6.55539	−0.519876
$160$	0	0
$161$	6.10308	0.480990
$162$	0	0
$163$	13.3694	1.04717	0.523587	−	0.851972i	$-0.324593\pi$
$163$	13.3694	1.04717	0.523587	+	0.851972i	$0.324593\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.84632	0.761931	0.380966	−	0.924589i	$-0.375592\pi$
$167$	9.84632	0.761931	0.380966	+	0.924589i	$0.375592\pi$
$168$	0	0
$169$	18.6541	1.43493
$170$	0	0
$171$	−4.62620	−0.353774
$172$	0	0
$173$	−2.98168	−0.226693	−0.113346	−	0.993556i	$-0.536157\pi$
$173$	−2.98168	−0.226693	−0.113346	+	0.993556i	$0.536157\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.4017	−0.932169
$178$	0	0
$179$	−11.7938	−0.881512	−0.440756	−	0.897627i	$-0.645290\pi$
$179$	−11.7938	−0.881512	−0.440756	+	0.897627i	$0.645290\pi$
$180$	0	0
$181$	14.5693	1.08293	0.541465	−	0.840723i	$-0.317870\pi$
$181$	14.5693	1.08293	0.541465	+	0.840723i	$0.317870\pi$
$182$	0	0
$183$	−30.0033	−2.21791
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.13536	0.229280
$188$	0	0
$189$	−3.42003	−0.248771
$190$	0	0
$191$	−13.2384	−0.957900	−0.478950	−	0.877842i	$-0.658983\pi$
$191$	−13.2384	−0.957900	−0.478950	+	0.877842i	$0.658983\pi$
$192$	0	0
$193$	−2.54144	−0.182937	−0.0914683	−	0.995808i	$-0.529156\pi$
$193$	−2.54144	−0.182937	−0.0914683	+	0.995808i	$0.529156\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.9109	1.41859	0.709295	−	0.704911i	$-0.249013\pi$
$197$	19.9109	1.41859	0.709295	+	0.704911i	$0.249013\pi$
$198$	0	0
$199$	20.3126	1.43992	0.719960	−	0.694015i	$-0.244160\pi$
$199$	20.3126	1.43992	0.719960	+	0.694015i	$0.244160\pi$
$200$	0	0
$201$	2.85069	0.201072
$202$	0	0
$203$	5.60162	0.393157
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−37.0741	−2.57683
$208$	0	0
$209$	−0.864641	−0.0598085
$210$	0	0
$211$	−18.0419	−1.24205	−0.621026	−	0.783790i	$-0.713284\pi$
$211$	−18.0419	−1.24205	−0.621026	+	0.783790i	$0.713284\pi$
$212$	0	0
$213$	28.1170	1.92655
$214$	0	0
$215$	0	0
$216$	0	0
$217$	6.18159	0.419634
$218$	0	0
$219$	−45.2943	−3.06070
$220$	0	0
$221$	−20.4017	−1.37237
$222$	0	0
$223$	−13.5231	−0.905575	−0.452787	−	0.891619i	$-0.649570\pi$
$223$	−13.5231	−0.905575	−0.452787	+	0.891619i	$0.649570\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.6016	0.902771	0.451386	−	0.892329i	$-0.350930\pi$
$227$	13.6016	0.902771	0.451386	+	0.892329i	$0.350930\pi$
$228$	0	0
$229$	13.5877	0.897898	0.448949	−	0.893557i	$-0.351798\pi$
$229$	13.5877	0.897898	0.448949	+	0.893557i	$0.351798\pi$
$230$	0	0
$231$	−1.81841	−0.119643
$232$	0	0
$233$	−25.5510	−1.67390	−0.836952	−	0.547277i	$-0.815664\pi$
$233$	−25.5510	−1.67390	−0.836952	+	0.547277i	$0.815664\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	34.7110	2.25472
$238$	0	0
$239$	11.3309	0.732935	0.366468	−	0.930431i	$-0.380567\pi$
$239$	11.3309	0.732935	0.366468	+	0.930431i	$0.380567\pi$
$240$	0	0
$241$	−1.25240	−0.0806739	−0.0403370	−	0.999186i	$-0.512843\pi$
$241$	−1.25240	−0.0806739	−0.0403370	+	0.999186i	$0.512843\pi$
$242$	0	0
$243$	−17.5510	−1.12590
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.62620	0.357986
$248$	0	0
$249$	−0.747604	−0.0473775
$250$	0	0
$251$	−10.5939	−0.668682	−0.334341	−	0.942452i	$-0.608514\pi$
$251$	−10.5939	−0.668682	−0.334341	+	0.942452i	$0.608514\pi$
$252$	0	0
$253$	−6.92919	−0.435635
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.153681	0.00958637	0.00479319	−	0.999989i	$-0.498474\pi$
$257$	0.153681	0.00958637	0.00479319	+	0.999989i	$0.498474\pi$
$258$	0	0
$259$	0.363176	0.0225666
$260$	0	0
$261$	−34.0279	−2.10627
$262$	0	0
$263$	0.504792	0.0311268	0.0155634	−	0.999879i	$-0.495046\pi$
$263$	0.504792	0.0311268	0.0155634	+	0.999879i	$0.495046\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.07081	0.0655324
$268$	0	0
$269$	−3.49521	−0.213107	−0.106553	−	0.994307i	$-0.533981\pi$
$269$	−3.49521	−0.213107	−0.106553	+	0.994307i	$0.533981\pi$
$270$	0	0
$271$	−5.47252	−0.332432	−0.166216	−	0.986089i	$-0.553155\pi$
$271$	−5.47252	−0.332432	−0.166216	+	0.986089i	$0.553155\pi$
$272$	0	0
$273$	11.8324	0.716127
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−12.9538	−0.778317	−0.389158	−	0.921171i	$-0.627234\pi$
$277$	−12.9538	−0.778317	−0.389158	+	0.921171i	$0.627234\pi$
$278$	0	0
$279$	−37.5510	−2.24812
$280$	0	0
$281$	0.153681	0.00916785	0.00458393	−	0.999989i	$-0.498541\pi$
$281$	0.153681	0.00916785	0.00458393	+	0.999989i	$0.498541\pi$
$282$	0	0
$283$	−18.2341	−1.08390	−0.541952	−	0.840410i	$-0.682314\pi$
$283$	−18.2341	−1.08390	−0.541952	+	0.840410i	$0.682314\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.02458	0.119507
$288$	0	0
$289$	−3.85069	−0.226511
$290$	0	0
$291$	23.4865	1.37680
$292$	0	0
$293$	−2.03853	−0.119092	−0.0595462	−	0.998226i	$-0.518965\pi$
$293$	−2.03853	−0.119092	−0.0595462	+	0.998226i	$0.518965\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.88296	0.225312
$298$	0	0
$299$	45.0881	2.60751
$300$	0	0
$301$	−5.22782	−0.301326
$302$	0	0
$303$	45.3328	2.60430
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.5414	0.944070	0.472035	−	0.881580i	$-0.343520\pi$
$307$	16.5414	0.944070	0.472035	+	0.881580i	$0.343520\pi$
$308$	0	0
$309$	−26.6218	−1.51446
$310$	0	0
$311$	21.4725	1.21759	0.608797	−	0.793326i	$-0.291652\pi$
$311$	21.4725	1.21759	0.608797	+	0.793326i	$0.291652\pi$
$312$	0	0
$313$	1.12141	0.0633856	0.0316928	−	0.999498i	$-0.489910\pi$
$313$	1.12141	0.0633856	0.0316928	+	0.999498i	$0.489910\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.8882	1.67869	0.839344	−	0.543601i	$-0.182940\pi$
$317$	29.8882	1.67869	0.839344	+	0.543601i	$0.182940\pi$
$318$	0	0
$319$	−6.35985	−0.356083
$320$	0	0
$321$	11.8324	0.660418
$322$	0	0
$323$	−3.62620	−0.201767
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−37.0602	−2.04943
$328$	0	0
$329$	−0.953771	−0.0525831
$330$	0	0
$331$	−32.3126	−1.77606	−0.888030	−	0.459786i	$-0.847926\pi$
$331$	−32.3126	−1.77606	−0.888030	+	0.459786i	$0.847926\pi$
$332$	0	0
$333$	−2.20617	−0.120897
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.3511	−1.43544	−0.717718	−	0.696334i	$-0.754813\pi$
$337$	−26.3511	−1.43544	−0.717718	+	0.696334i	$0.754813\pi$
$338$	0	0
$339$	28.5081	1.54835
$340$	0	0
$341$	−7.01832	−0.380063
$342$	0	0
$343$	10.2201	0.551835
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.77551	−0.148997	−0.0744986	−	0.997221i	$-0.523736\pi$
$347$	−2.77551	−0.148997	−0.0744986	+	0.997221i	$0.523736\pi$
$348$	0	0
$349$	11.5510	0.618312	0.309156	−	0.951011i	$-0.399953\pi$
$349$	11.5510	0.618312	0.309156	+	0.951011i	$0.399953\pi$
$350$	0	0
$351$	−25.2663	−1.34862
$352$	0	0
$353$	8.40171	0.447178	0.223589	−	0.974684i	$-0.428223\pi$
$353$	8.40171	0.447178	0.223589	+	0.974684i	$0.428223\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−7.62620	−0.403621
$358$	0	0
$359$	−22.7895	−1.20278	−0.601391	−	0.798955i	$-0.705387\pi$
$359$	−22.7895	−1.20278	−0.601391	+	0.798955i	$0.705387\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−28.3126	−1.48602
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.06455	0.212168	0.106084	−	0.994357i	$-0.466169\pi$
$367$	4.06455	0.212168	0.106084	+	0.994357i	$0.466169\pi$
$368$	0	0
$369$	−12.2986	−0.640241
$370$	0	0
$371$	1.80779	0.0938556
$372$	0	0
$373$	18.4017	0.952804	0.476402	−	0.879227i	$-0.341941\pi$
$373$	18.4017	0.952804	0.476402	+	0.879227i	$0.341941\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	41.3834	2.13135
$378$	0	0
$379$	−1.23844	−0.0636145	−0.0318073	−	0.999494i	$-0.510126\pi$
$379$	−1.23844	−0.0636145	−0.0318073	+	0.999494i	$0.510126\pi$
$380$	0	0
$381$	46.8959	2.40255
$382$	0	0
$383$	16.8646	0.861743	0.430871	−	0.902413i	$-0.358206\pi$
$383$	16.8646	0.861743	0.430871	+	0.902413i	$0.358206\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	31.7572	1.61431
$388$	0	0
$389$	8.59392	0.435729	0.217865	−	0.975979i	$-0.430091\pi$
$389$	8.59392	0.435729	0.217865	+	0.975979i	$0.430091\pi$
$390$	0	0
$391$	−29.0602	−1.46964
$392$	0	0
$393$	−1.49521	−0.0754233
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.0558	−0.805818	−0.402909	−	0.915240i	$-0.632001\pi$
$397$	−16.0558	−0.805818	−0.402909	+	0.915240i	$0.632001\pi$
$398$	0	0
$399$	2.10308	0.105286
$400$	0	0
$401$	−14.8925	−0.743698	−0.371849	−	0.928293i	$-0.621276\pi$
$401$	−14.8925	−0.743698	−0.371849	+	0.928293i	$0.621276\pi$
$402$	0	0
$403$	45.6681	2.27489
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.412335	−0.0204387
$408$	0	0
$409$	−18.3511	−0.907404	−0.453702	−	0.891153i	$-0.649897\pi$
$409$	−18.3511	−0.907404	−0.453702	+	0.891153i	$0.649897\pi$
$410$	0	0
$411$	−7.94940	−0.392115
$412$	0	0
$413$	3.42003	0.168289
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.90754	−0.485174
$418$	0	0
$419$	−34.7509	−1.69769	−0.848847	−	0.528639i	$-0.822703\pi$
$419$	−34.7509	−1.69769	−0.848847	+	0.528639i	$0.822703\pi$
$420$	0	0
$421$	−40.1589	−1.95722	−0.978612	−	0.205713i	$-0.934049\pi$
$421$	−40.1589	−1.95722	−0.978612	+	0.205713i	$0.934049\pi$
$422$	0	0
$423$	5.79383	0.281706
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.27405	0.400409
$428$	0	0
$429$	−13.4340	−0.648599
$430$	0	0
$431$	−34.9571	−1.68382	−0.841912	−	0.539615i	$-0.818570\pi$
$431$	−34.9571	−1.68382	−0.841912	+	0.539615i	$0.818570\pi$
$432$	0	0
$433$	−1.13536	−0.0545619	−0.0272809	−	0.999628i	$-0.508685\pi$
$433$	−1.13536	−0.0545619	−0.0272809	+	0.999628i	$0.508685\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.01395	0.383359
$438$	0	0
$439$	6.80009	0.324551	0.162275	−	0.986746i	$-0.448117\pi$
$439$	6.80009	0.324551	0.162275	+	0.986746i	$0.448117\pi$
$440$	0	0
$441$	−29.7003	−1.41430
$442$	0	0
$443$	38.0679	1.80866	0.904330	−	0.426835i	$-0.140371\pi$
$443$	38.0679	1.80866	0.904330	+	0.426835i	$0.140371\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−46.5048	−2.19960
$448$	0	0
$449$	−18.5414	−0.875024	−0.437512	−	0.899212i	$-0.644140\pi$
$449$	−18.5414	−0.875024	−0.437512	+	0.899212i	$0.644140\pi$
$450$	0	0
$451$	−2.29862	−0.108238
$452$	0	0
$453$	46.8959	2.20336
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.3738	−0.765934	−0.382967	−	0.923762i	$-0.625098\pi$
$457$	−16.3738	−0.765934	−0.382967	+	0.923762i	$0.625098\pi$
$458$	0	0
$459$	16.2847	0.760103
$460$	0	0
$461$	1.70470	0.0793959	0.0396979	−	0.999212i	$-0.487360\pi$
$461$	1.70470	0.0793959	0.0396979	+	0.999212i	$0.487360\pi$
$462$	0	0
$463$	−10.0279	−0.466036	−0.233018	−	0.972472i	$-0.574860\pi$
$463$	−10.0279	−0.466036	−0.233018	+	0.972472i	$0.574860\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.7509	1.51553	0.757766	−	0.652526i	$-0.226291\pi$
$467$	32.7509	1.51553	0.757766	+	0.652526i	$0.226291\pi$
$468$	0	0
$469$	−0.786137	−0.0363004
$470$	0	0
$471$	−40.9817	−1.88834
$472$	0	0
$473$	5.93545	0.272912
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.9817	−0.502816
$478$	0	0
$479$	−27.2803	−1.24647	−0.623234	−	0.782035i	$-0.714182\pi$
$479$	−27.2803	−1.24647	−0.623234	+	0.782035i	$0.714182\pi$
$480$	0	0
$481$	2.68305	0.122337
$482$	0	0
$483$	16.8540	0.766884
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.0741	−0.501817	−0.250908	−	0.968011i	$-0.580729\pi$
$487$	−11.0741	−0.501817	−0.250908	+	0.968011i	$0.580729\pi$
$488$	0	0
$489$	36.9205	1.66960
$490$	0	0
$491$	−8.11704	−0.366317	−0.183158	−	0.983083i	$-0.558632\pi$
$491$	−8.11704	−0.366317	−0.183158	+	0.983083i	$0.558632\pi$
$492$	0	0
$493$	−26.6724	−1.20127
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.75386	−0.347808
$498$	0	0
$499$	−0.295298	−0.0132193	−0.00660967	−	0.999978i	$-0.502104\pi$
$499$	−0.295298	−0.0132193	−0.00660967	+	0.999978i	$0.502104\pi$
$500$	0	0
$501$	27.1912	1.21481
$502$	0	0
$503$	19.6016	0.873993	0.436996	−	0.899463i	$-0.356042\pi$
$503$	19.6016	0.873993	0.436996	+	0.899463i	$0.356042\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	51.5144	2.28783
$508$	0	0
$509$	−1.79383	−0.0795102	−0.0397551	−	0.999209i	$-0.512658\pi$
$509$	−1.79383	−0.0795102	−0.0397551	+	0.999209i	$0.512658\pi$
$510$	0	0
$511$	12.4908	0.552562
$512$	0	0
$513$	−4.49084	−0.198275
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.08287	0.0476247
$518$	0	0
$519$	−8.23407	−0.361436
$520$	0	0
$521$	−2.61850	−0.114719	−0.0573593	−	0.998354i	$-0.518268\pi$
$521$	−2.61850	−0.114719	−0.0573593	+	0.998354i	$0.518268\pi$
$522$	0	0
$523$	14.9956	0.655713	0.327857	−	0.944728i	$-0.393674\pi$
$523$	14.9956	0.655713	0.327857	+	0.944728i	$0.393674\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.4340	−1.28216
$528$	0	0
$529$	41.2234	1.79232
$530$	0	0
$531$	−20.7755	−0.901580
$532$	0	0
$533$	14.9571	0.647864
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−32.5693	−1.40547
$538$	0	0
$539$	−5.55102	−0.239099
$540$	0	0
$541$	3.40608	0.146439	0.0732194	−	0.997316i	$-0.476673\pi$
$541$	3.40608	0.146439	0.0732194	+	0.997316i	$0.476673\pi$
$542$	0	0
$543$	40.2341	1.72661
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.74760	0.202993	0.101496	−	0.994836i	$-0.467637\pi$
$547$	4.74760	0.202993	0.101496	+	0.994836i	$0.467637\pi$
$548$	0	0
$549$	−50.2620	−2.14513
$550$	0	0
$551$	7.35548	0.313354
$552$	0	0
$553$	−9.57227	−0.407054
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−43.0462	−1.82393	−0.911964	−	0.410271i	$-0.865434\pi$
$557$	−43.0462	−1.82393	−0.911964	+	0.410271i	$0.865434\pi$
$558$	0	0
$559$	−38.6218	−1.63353
$560$	0	0
$561$	8.65847	0.365561
$562$	0	0
$563$	17.0096	0.716869	0.358434	−	0.933555i	$-0.383311\pi$
$563$	17.0096	0.716869	0.358434	+	0.933555i	$0.383311\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.12473	0.0472343
$568$	0	0
$569$	−19.7572	−0.828264	−0.414132	−	0.910217i	$-0.635915\pi$
$569$	−19.7572	−0.828264	−0.414132	+	0.910217i	$0.635915\pi$
$570$	0	0
$571$	11.3973	0.476964	0.238482	−	0.971147i	$-0.423350\pi$
$571$	11.3973	0.476964	0.238482	+	0.971147i	$0.423350\pi$
$572$	0	0
$573$	−36.5587	−1.52726
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.3372	0.763386	0.381693	−	0.924289i	$-0.375341\pi$
$577$	18.3372	0.763386	0.381693	+	0.924289i	$0.375341\pi$
$578$	0	0
$579$	−7.01832	−0.291672
$580$	0	0
$581$	0.206167	0.00855327
$582$	0	0
$583$	−2.05249	−0.0850053
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−11.9475	−0.493127	−0.246563	−	0.969127i	$-0.579301\pi$
$587$	−11.9475	−0.493127	−0.246563	+	0.969127i	$0.579301\pi$
$588$	0	0
$589$	8.11704	0.334457
$590$	0	0
$591$	54.9850	2.26178
$592$	0	0
$593$	24.3911	1.00162	0.500811	−	0.865557i	$-0.333035\pi$
$593$	24.3911	1.00162	0.500811	+	0.865557i	$0.333035\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	56.0943	2.29579
$598$	0	0
$599$	21.0708	0.860930	0.430465	−	0.902607i	$-0.358350\pi$
$599$	21.0708	0.860930	0.430465	+	0.902607i	$0.358350\pi$
$600$	0	0
$601$	32.3878	1.32112	0.660562	−	0.750771i	$-0.270318\pi$
$601$	32.3878	1.32112	0.660562	+	0.750771i	$0.270318\pi$
$602$	0	0
$603$	4.77551	0.194474
$604$	0	0
$605$	0	0
$606$	0	0
$607$	19.0183	0.771930	0.385965	−	0.922513i	$-0.373869\pi$
$607$	19.0183	0.771930	0.385965	+	0.922513i	$0.373869\pi$
$608$	0	0
$609$	15.4692	0.626843
$610$	0	0
$611$	−7.04623	−0.285060
$612$	0	0
$613$	23.5756	0.952210	0.476105	−	0.879389i	$-0.342048\pi$
$613$	23.5756	0.952210	0.476105	+	0.879389i	$0.342048\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.2707	1.05762	0.528810	−	0.848740i	$-0.322639\pi$
$617$	26.2707	1.05762	0.528810	+	0.848740i	$0.322639\pi$
$618$	0	0
$619$	−11.4985	−0.462165	−0.231083	−	0.972934i	$-0.574227\pi$
$619$	−11.4985	−0.462165	−0.231083	+	0.972934i	$0.574227\pi$
$620$	0	0
$621$	−35.9894	−1.44420
$622$	0	0
$623$	−0.295298	−0.0118309
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.38776	−0.0953578
$628$	0	0
$629$	−1.72928	−0.0689510
$630$	0	0
$631$	45.8130	1.82379	0.911893	−	0.410427i	$-0.134620\pi$
$631$	45.8130	1.82379	0.911893	+	0.410427i	$0.134620\pi$
$632$	0	0
$633$	−49.8236	−1.98031
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.1204	1.43114
$638$	0	0
$639$	47.1020	1.86333
$640$	0	0
$641$	1.36943	0.0540894	0.0270447	−	0.999634i	$-0.491390\pi$
$641$	1.36943	0.0540894	0.0270447	+	0.999634i	$0.491390\pi$
$642$	0	0
$643$	−24.7389	−0.975606	−0.487803	−	0.872954i	$-0.662201\pi$
$643$	−24.7389	−0.975606	−0.487803	+	0.872954i	$0.662201\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.82611	−0.268362	−0.134181	−	0.990957i	$-0.542840\pi$
$647$	−6.82611	−0.268362	−0.134181	+	0.990957i	$0.542840\pi$
$648$	0	0
$649$	−3.88296	−0.152420
$650$	0	0
$651$	17.0708	0.669058
$652$	0	0
$653$	−6.91713	−0.270688	−0.135344	−	0.990799i	$-0.543214\pi$
$653$	−6.91713	−0.270688	−0.135344	+	0.990799i	$0.543214\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−75.8776	−2.96027
$658$	0	0
$659$	−9.44461	−0.367910	−0.183955	−	0.982935i	$-0.558890\pi$
$659$	−9.44461	−0.367910	−0.183955	+	0.982935i	$0.558890\pi$
$660$	0	0
$661$	−22.1955	−0.863306	−0.431653	−	0.902040i	$-0.642070\pi$
$661$	−22.1955	−0.863306	−0.431653	+	0.902040i	$0.642070\pi$
$662$	0	0
$663$	−56.3405	−2.18808
$664$	0	0
$665$	0	0
$666$	0	0
$667$	58.9465	2.28242
$668$	0	0
$669$	−37.3449	−1.44384
$670$	0	0
$671$	−9.39401	−0.362652
$672$	0	0
$673$	−12.2986	−0.474077	−0.237039	−	0.971500i	$-0.576177\pi$
$673$	−12.2986	−0.474077	−0.237039	+	0.971500i	$0.576177\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.9527	1.38178	0.690888	−	0.722962i	$-0.257220\pi$
$677$	35.9527	1.38178	0.690888	+	0.722962i	$0.257220\pi$
$678$	0	0
$679$	−6.47689	−0.248560
$680$	0	0
$681$	37.5616	1.43937
$682$	0	0
$683$	−9.00958	−0.344742	−0.172371	−	0.985032i	$-0.555143\pi$
$683$	−9.00958	−0.344742	−0.172371	+	0.985032i	$0.555143\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	37.5231	1.43160
$688$	0	0
$689$	13.3555	0.508803
$690$	0	0
$691$	9.11078	0.346590	0.173295	−	0.984870i	$-0.444559\pi$
$691$	9.11078	0.346590	0.173295	+	0.984870i	$0.444559\pi$
$692$	0	0
$693$	−3.04623	−0.115717
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9.64015	−0.365147
$698$	0	0
$699$	−70.5606	−2.66885
$700$	0	0
$701$	−14.7476	−0.557009	−0.278505	−	0.960435i	$-0.589839\pi$
$701$	−14.7476	−0.557009	−0.278505	+	0.960435i	$0.589839\pi$
$702$	0	0
$703$	0.476886	0.0179861
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.5015	−0.470166
$708$	0	0
$709$	−8.63389	−0.324253	−0.162126	−	0.986770i	$-0.551835\pi$
$709$	−8.63389	−0.324253	−0.162126	+	0.986770i	$0.551835\pi$
$710$	0	0
$711$	58.1483	2.18073
$712$	0	0
$713$	65.0496	2.43613
$714$	0	0
$715$	0	0
$716$	0	0
$717$	31.2909	1.16858
$718$	0	0
$719$	38.2759	1.42745	0.713726	−	0.700425i	$-0.247006\pi$
$719$	38.2759	1.42745	0.713726	+	0.700425i	$0.247006\pi$
$720$	0	0
$721$	7.34153	0.273413
$722$	0	0
$723$	−3.45856	−0.128625
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−31.1893	−1.15675	−0.578373	−	0.815772i	$-0.696312\pi$
$727$	−31.1893	−1.15675	−0.578373	+	0.815772i	$0.696312\pi$
$728$	0	0
$729$	−44.0375	−1.63102
$730$	0	0
$731$	24.8925	0.920684
$732$	0	0
$733$	13.9634	0.515748	0.257874	−	0.966179i	$-0.416978\pi$
$733$	13.9634	0.515748	0.257874	+	0.966179i	$0.416978\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.892548	0.0328774
$738$	0	0
$739$	−9.02165	−0.331867	−0.165933	−	0.986137i	$-0.553064\pi$
$739$	−9.02165	−0.331867	−0.165933	+	0.986137i	$0.553064\pi$
$740$	0	0
$741$	15.5371	0.570768
$742$	0	0
$743$	−15.0342	−0.551550	−0.275775	−	0.961222i	$-0.588934\pi$
$743$	−15.0342	−0.551550	−0.275775	+	0.961222i	$0.588934\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.25240	−0.0458228
$748$	0	0
$749$	−3.26302	−0.119228
$750$	0	0
$751$	−29.6681	−1.08260	−0.541301	−	0.840829i	$-0.682068\pi$
$751$	−29.6681	−1.08260	−0.541301	+	0.840829i	$0.682068\pi$
$752$	0	0
$753$	−29.2557	−1.06614
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−10.5819	−0.384604	−0.192302	−	0.981336i	$-0.561595\pi$
$757$	−10.5819	−0.384604	−0.192302	+	0.981336i	$0.561595\pi$
$758$	0	0
$759$	−19.1354	−0.694570
$760$	0	0
$761$	−0.979789	−0.0355173	−0.0177587	−	0.999842i	$-0.505653\pi$
$761$	−0.979789	−0.0355173	−0.0177587	+	0.999842i	$0.505653\pi$
$762$	0	0
$763$	10.2201	0.369993
$764$	0	0
$765$	0	0
$766$	0	0
$767$	25.2663	0.912315
$768$	0	0
$769$	43.1772	1.55701	0.778505	−	0.627638i	$-0.215978\pi$
$769$	43.1772	1.55701	0.778505	+	0.627638i	$0.215978\pi$
$770$	0	0
$771$	0.424399	0.0152844
$772$	0	0
$773$	−37.5250	−1.34968	−0.674840	−	0.737964i	$-0.735787\pi$
$773$	−37.5250	−1.34968	−0.674840	+	0.737964i	$0.735787\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.00293	0.0359799
$778$	0	0
$779$	2.65847	0.0952497
$780$	0	0
$781$	8.80342	0.315011
$782$	0	0
$783$	−33.0323	−1.18048
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22.5833	0.805008	0.402504	−	0.915418i	$-0.368140\pi$
$787$	22.5833	0.805008	0.402504	+	0.915418i	$0.368140\pi$
$788$	0	0
$789$	1.39401	0.0496282
$790$	0	0
$791$	−7.86171	−0.279530
$792$	0	0
$793$	61.1266	2.17067
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−35.9806	−1.27450	−0.637250	−	0.770657i	$-0.719928\pi$
$797$	−35.9806	−1.27450	−0.637250	+	0.770657i	$0.719928\pi$
$798$	0	0
$799$	4.54144	0.160664
$800$	0	0
$801$	1.79383	0.0633820
$802$	0	0
$803$	−14.1816	−0.500457
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.65222	−0.339774
$808$	0	0
$809$	−0.955660	−0.0335992	−0.0167996	−	0.999859i	$-0.505348\pi$
$809$	−0.955660	−0.0335992	−0.0167996	+	0.999859i	$0.505348\pi$
$810$	0	0
$811$	7.53707	0.264662	0.132331	−	0.991206i	$-0.457754\pi$
$811$	7.53707	0.264662	0.132331	+	0.991206i	$0.457754\pi$
$812$	0	0
$813$	−15.1127	−0.530024
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.86464	−0.240163
$818$	0	0
$819$	19.8217	0.692628
$820$	0	0
$821$	13.3082	0.464460	0.232230	−	0.972661i	$-0.425398\pi$
$821$	13.3082	0.464460	0.232230	+	0.972661i	$0.425398\pi$
$822$	0	0
$823$	24.4050	0.850706	0.425353	−	0.905028i	$-0.360150\pi$
$823$	24.4050	0.850706	0.425353	+	0.905028i	$0.360150\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.6874	−0.406411	−0.203206	−	0.979136i	$-0.565136\pi$
$827$	−11.6874	−0.406411	−0.203206	+	0.979136i	$0.565136\pi$
$828$	0	0
$829$	25.6541	0.891004	0.445502	−	0.895281i	$-0.353025\pi$
$829$	25.6541	0.891004	0.445502	+	0.895281i	$0.353025\pi$
$830$	0	0
$831$	−35.7726	−1.24094
$832$	0	0
$833$	−23.2803	−0.806615
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−36.4523	−1.25998
$838$	0	0
$839$	2.91713	0.100710	0.0503552	−	0.998731i	$-0.483965\pi$
$839$	2.91713	0.100710	0.0503552	+	0.998731i	$0.483965\pi$
$840$	0	0
$841$	25.1031	0.865624
$842$	0	0
$843$	0.424399	0.0146171
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.80779	0.268279
$848$	0	0
$849$	−50.3544	−1.72816
$850$	0	0
$851$	3.82174	0.131008
$852$	0	0
$853$	6.24281	0.213750	0.106875	−	0.994272i	$-0.465916\pi$
$853$	6.24281	0.213750	0.106875	+	0.994272i	$0.465916\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.2158	0.793035	0.396517	−	0.918027i	$-0.370219\pi$
$857$	23.2158	0.793035	0.396517	+	0.918027i	$0.370219\pi$
$858$	0	0
$859$	7.13536	0.243455	0.121728	−	0.992564i	$-0.461157\pi$
$859$	7.13536	0.243455	0.121728	+	0.992564i	$0.461157\pi$
$860$	0	0
$861$	5.59099	0.190541
$862$	0	0
$863$	−7.31362	−0.248959	−0.124479	−	0.992222i	$-0.539726\pi$
$863$	−7.31362	−0.248959	−0.124479	+	0.992222i	$0.539726\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.6339	−0.361146
$868$	0	0
$869$	10.8680	0.368671
$870$	0	0
$871$	−5.80779	−0.196789
$872$	0	0
$873$	39.3449	1.33162
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.0173	0.743471	0.371735	−	0.928339i	$-0.378763\pi$
$877$	22.0173	0.743471	0.371735	+	0.928339i	$0.378763\pi$
$878$	0	0
$879$	−5.62953	−0.189879
$880$	0	0
$881$	11.7572	0.396110	0.198055	−	0.980191i	$-0.436538\pi$
$881$	11.7572	0.396110	0.198055	+	0.980191i	$0.436538\pi$
$882$	0	0
$883$	55.6560	1.87297	0.936487	−	0.350703i	$-0.114057\pi$
$883$	55.6560	1.87297	0.936487	+	0.350703i	$0.114057\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.41566	−0.215417	−0.107708	−	0.994183i	$-0.534351\pi$
$887$	−6.41566	−0.215417	−0.107708	+	0.994183i	$0.534351\pi$
$888$	0	0
$889$	−12.9325	−0.433743
$890$	0	0
$891$	−1.27698	−0.0427803
$892$	0	0
$893$	−1.25240	−0.0419098
$894$	0	0
$895$	0	0
$896$	0	0
$897$	124.513	4.15738
$898$	0	0
$899$	59.7047	1.99126
$900$	0	0
$901$	−8.60788	−0.286770
$902$	0	0
$903$	−14.4369	−0.480430
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−57.1160	−1.89651	−0.948253	−	0.317516i	$-0.897151\pi$
$907$	−57.1160	−1.89651	−0.948253	+	0.317516i	$0.897151\pi$
$908$	0	0
$909$	75.9421	2.51884
$910$	0	0
$911$	26.6339	0.882420	0.441210	−	0.897404i	$-0.354549\pi$
$911$	26.6339	0.882420	0.441210	+	0.897404i	$0.354549\pi$
$912$	0	0
$913$	−0.234074	−0.00774672
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.412335	0.0136165
$918$	0	0
$919$	−6.63246	−0.218785	−0.109392	−	0.993999i	$-0.534890\pi$
$919$	−6.63246	−0.218785	−0.109392	+	0.993999i	$0.534890\pi$
$920$	0	0
$921$	45.6801	1.50521
$922$	0	0
$923$	−57.2836	−1.88551
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−44.5972	−1.46477
$928$	0	0
$929$	−50.0173	−1.64101	−0.820507	−	0.571637i	$-0.806309\pi$
$929$	−50.0173	−1.64101	−0.820507	+	0.571637i	$0.806309\pi$
$930$	0	0
$931$	6.42003	0.210408
$932$	0	0
$933$	59.2976	1.94132
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.8882	1.30309	0.651545	−	0.758610i	$-0.274121\pi$
$937$	39.8882	1.30309	0.651545	+	0.758610i	$0.274121\pi$
$938$	0	0
$939$	3.09683	0.101061
$940$	0	0
$941$	−5.59829	−0.182499	−0.0912495	−	0.995828i	$-0.529086\pi$
$941$	−5.59829	−0.182499	−0.0912495	+	0.995828i	$0.529086\pi$
$942$	0	0
$943$	21.3049	0.693782
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.7110	0.413051	0.206525	−	0.978441i	$-0.433784\pi$
$947$	12.7110	0.413051	0.206525	+	0.978441i	$0.433784\pi$
$948$	0	0
$949$	92.2793	2.99551
$950$	0	0
$951$	82.5379	2.67648
$952$	0	0
$953$	57.0129	1.84683	0.923415	−	0.383804i	$-0.125386\pi$
$953$	57.0129	1.84683	0.923415	+	0.383804i	$0.125386\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−17.5631	−0.567734
$958$	0	0
$959$	2.19221	0.0707903
$960$	0	0
$961$	34.8863	1.12536
$962$	0	0
$963$	19.8217	0.638747
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−33.0183	−1.06180	−0.530899	−	0.847435i	$-0.678146\pi$
$967$	−33.0183	−1.06180	−0.530899	+	0.847435i	$0.678146\pi$
$968$	0	0
$969$	−10.0140	−0.321695
$970$	0	0
$971$	3.04623	0.0977581	0.0488791	−	0.998805i	$-0.484435\pi$
$971$	3.04623	0.0977581	0.0488791	+	0.998805i	$0.484435\pi$
$972$	0	0
$973$	2.73221	0.0875907
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.4465	−0.430192	−0.215096	−	0.976593i	$-0.569006\pi$
$977$	−13.4465	−0.430192	−0.215096	+	0.976593i	$0.569006\pi$
$978$	0	0
$979$	0.335269	0.0107152
$980$	0	0
$981$	−62.0837	−1.98218
$982$	0	0
$983$	−22.0646	−0.703750	−0.351875	−	0.936047i	$-0.614456\pi$
$983$	−22.0646	−0.703750	−0.351875	+	0.936047i	$0.614456\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.63389	−0.0838378
$988$	0	0
$989$	−55.0129	−1.74931
$990$	0	0
$991$	−51.9946	−1.65166	−0.825831	−	0.563917i	$-0.809294\pi$
$991$	−51.9946	−1.65166	−0.825831	+	0.563917i	$0.809294\pi$
$992$	0	0
$993$	−89.2330	−2.83172
$994$	0	0
$995$	0	0
$996$	0	0
$997$	37.7693	1.19616	0.598082	−	0.801435i	$-0.295930\pi$
$997$	37.7693	1.19616	0.598082	+	0.801435i	$0.295930\pi$
$998$	0	0
$999$	−2.14162	−0.0677578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cd.1.3		3
4.3	odd	2		950.2.a.i.1.1		3
5.2	odd	4		1520.2.d.j.609.1		6
5.3	odd	4		1520.2.d.j.609.6		6
5.4	even	2		7600.2.a.bi.1.1		3
12.11	even	2		8550.2.a.cl.1.3		3
20.3	even	4		190.2.b.b.39.4	yes	6
20.7	even	4		190.2.b.b.39.3	✓	6
20.19	odd	2		950.2.a.n.1.3		3
60.23	odd	4		1710.2.d.d.1369.3		6
60.47	odd	4		1710.2.d.d.1369.6		6
60.59	even	2		8550.2.a.ck.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.b.39.3	✓	6	20.7	even	4
190.2.b.b.39.4	yes	6	20.3	even	4
950.2.a.i.1.1		3	4.3	odd	2
950.2.a.n.1.3		3	20.19	odd	2
1520.2.d.j.609.1		6	5.2	odd	4
1520.2.d.j.609.6		6	5.3	odd	4
1710.2.d.d.1369.3		6	60.23	odd	4
1710.2.d.d.1369.6		6	60.47	odd	4
7600.2.a.bi.1.1		3	5.4	even	2
7600.2.a.cd.1.3		3	1.1	even	1	trivial
8550.2.a.ck.1.1		3	60.59	even	2
8550.2.a.cl.1.3		3	12.11	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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76156 q^{3} -0.761557 q^{7} +4.62620 q^{9} +O(q^{10})\)	\(q+2.76156 q^{3} -0.761557 q^{7} +4.62620 q^{9} +0.864641 q^{11} -5.62620 q^{13} +3.62620 q^{17} -1.00000 q^{19} -2.10308 q^{21} -8.01395 q^{23} +4.49084 q^{27} -7.35548 q^{29} -8.11704 q^{31} +2.38776 q^{33} -0.476886 q^{37} -15.5371 q^{39} -2.65847 q^{41} +6.86464 q^{43} +1.25240 q^{47} -6.42003 q^{49} +10.0140 q^{51} -2.37380 q^{53} -2.76156 q^{57} -4.49084 q^{59} -10.8646 q^{61} -3.52311 q^{63} +1.03228 q^{67} -22.1310 q^{69} +10.1816 q^{71} -16.4017 q^{73} -0.658473 q^{77} +12.5693 q^{79} -1.47689 q^{81} -0.270718 q^{83} -20.3126 q^{87} +0.387755 q^{89} +4.28467 q^{91} -22.4157 q^{93} +8.50479 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9	\( 3 q + 2 q^{3} + 4 q^{7} + 5 q^{9} - 8 q^{13} + 2 q^{17} - 3 q^{19} - 10 q^{21} + 2 q^{27} - 8 q^{29} - 4 q^{31} - 8 q^{33} - 14 q^{37} - 10 q^{39} + 2 q^{41} + 18 q^{43} - 14 q^{47} - 3 q^{49} + 6 q^{51} - 16 q^{53} - 2 q^{57} - 2 q^{59} - 30 q^{61} + 2 q^{63} + 2 q^{67} - 22 q^{69} + 8 q^{71} - 10 q^{73} + 8 q^{77} - 17 q^{81} - 6 q^{83} + 6 q^{87} - 14 q^{89} - 6 q^{91} - 4 q^{93} - 10 q^{97} + 12 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 4 * q^7 + 5 * q^9 - 8 * q^13 + 2 * q^17 - 3 * q^19 - 10 * q^21 + 2 * q^27 - 8 * q^29 - 4 * q^31 - 8 * q^33 - 14 * q^37 - 10 * q^39 + 2 * q^41 + 18 * q^43 - 14 * q^47 - 3 * q^49 + 6 * q^51 - 16 * q^53 - 2 * q^57 - 2 * q^59 - 30 * q^61 + 2 * q^63 + 2 * q^67 - 22 * q^69 + 8 * q^71 - 10 * q^73 + 8 * q^77 - 17 * q^81 - 6 * q^83 + 6 * q^87 - 14 * q^89 - 6 * q^91 - 4 * q^93 - 10 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more