LMFDB - Embedded newform 8624.2.a.cg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, 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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 308)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.713538$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49086 q^{3} +0.286462 q^{5} -0.777326 q^{9} -1.00000 q^{11} +3.49086 q^{13} +0.427076 q^{15} -1.77733 q^{17} -6.63148 q^{19} +8.61320 q^{23} -4.91794 q^{25} -5.63148 q^{27} -3.69527 q^{29} -1.85939 q^{31} -1.49086 q^{33} +7.42708 q^{37} +5.20440 q^{39} +2.63671 q^{41} -10.8997 q^{43} -0.222674 q^{45} +7.61320 q^{47} -2.64975 q^{51} -7.49086 q^{53} -0.286462 q^{55} -9.88663 q^{57} -2.36329 q^{59} -1.55465 q^{61} +1.00000 q^{65} +6.12758 q^{67} +12.8411 q^{69} -12.3137 q^{71} +8.45432 q^{73} -7.33198 q^{75} -10.3958 q^{79} -6.06379 q^{81} -16.1861 q^{83} -0.509136 q^{85} -5.50914 q^{87} +10.1406 q^{89} -2.77209 q^{93} -1.89967 q^{95} -3.04551 q^{97} +0.777326 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{5} + 4 q^{9} - 3 q^{11} + 3 q^{13} - q^{15} + q^{17} - 9 q^{19} - 5 q^{25} - 6 q^{27} + 5 q^{29} - 9 q^{31} + 3 q^{33} + 20 q^{37} + 7 q^{39} + 5 q^{41} - 8 q^{43} - 7 q^{45} - 3 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^5 + 4 * q^9 - 3 * q^11 + 3 * q^13 - q^15 + q^17 - 9 * q^19 - 5 * q^25 - 6 * q^27 + 5 * q^29 - 9 * q^31 + 3 * q^33 + 20 * q^37 + 7 * q^39 + 5 * q^41 - 8 * q^43 - 7 * q^45 - 3 * q^47 - 12 * q^51 - 15 * q^53 - 2 * q^55 + 2 * q^57 - 10 * q^59 + 8 * q^61 + 3 * q^65 + 8 * q^67 + 27 * q^69 - 3 * q^71 - 12 * q^73 - 3 * q^75 - 7 * q^79 - 13 * q^81 - 25 * q^83 - 9 * q^85 - 24 * q^87 + 27 * q^89 + 6 * q^93 + 19 * q^95 + 11 * q^97 - 4 * q^99

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
$3$	1.49086	0.860751	0.430375	−	0.902650i	$-0.358381\pi$
$3$	1.49086	0.860751	0.430375	+	0.902650i	$0.358381\pi$
$4$	0	0
$5$	0.286462	0.128110	0.0640549	−	0.997946i	$-0.479597\pi$
$5$	0.286462	0.128110	0.0640549	+	0.997946i	$0.479597\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.777326	−0.259109
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.49086	0.968191	0.484096	−	0.875015i	$-0.339149\pi$
$13$	3.49086	0.968191	0.484096	+	0.875015i	$0.339149\pi$
$14$	0	0
$15$	0.427076	0.110271
$16$	0	0
$17$	−1.77733	−0.431065	−0.215532	−	0.976497i	$-0.569149\pi$
$17$	−1.77733	−0.431065	−0.215532	+	0.976497i	$0.569149\pi$
$18$	0	0
$19$	−6.63148	−1.52137	−0.760683	−	0.649124i	$-0.775136\pi$
$19$	−6.63148	−1.52137	−0.760683	+	0.649124i	$0.775136\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.61320	1.79598	0.897989	−	0.440019i	$-0.145028\pi$
$23$	8.61320	1.79598	0.897989	+	0.440019i	$0.145028\pi$
$24$	0	0
$25$	−4.91794	−0.983588
$26$	0	0
$27$	−5.63148	−1.08378
$28$	0	0
$29$	−3.69527	−0.686194	−0.343097	−	0.939300i	$-0.611476\pi$
$29$	−3.69527	−0.686194	−0.343097	+	0.939300i	$0.611476\pi$
$30$	0	0
$31$	−1.85939	−0.333956	−0.166978	−	0.985961i	$-0.553401\pi$
$31$	−1.85939	−0.333956	−0.166978	+	0.985961i	$0.553401\pi$
$32$	0	0
$33$	−1.49086	−0.259526
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.42708	1.22100	0.610502	−	0.792015i	$-0.290968\pi$
$37$	7.42708	1.22100	0.610502	+	0.792015i	$0.290968\pi$
$38$	0	0
$39$	5.20440	0.833371
$40$	0	0
$41$	2.63671	0.411785	0.205893	−	0.978575i	$-0.433990\pi$
$41$	2.63671	0.411785	0.205893	+	0.978575i	$0.433990\pi$
$42$	0	0
$43$	−10.8997	−1.66218	−0.831092	−	0.556135i	$-0.812284\pi$
$43$	−10.8997	−1.66218	−0.831092	+	0.556135i	$0.812284\pi$
$44$	0	0
$45$	−0.222674	−0.0331943
$46$	0	0
$47$	7.61320	1.11050	0.555250	−	0.831683i	$-0.312623\pi$
$47$	7.61320	1.11050	0.555250	+	0.831683i	$0.312623\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−2.64975	−0.371039
$52$	0	0
$53$	−7.49086	−1.02895	−0.514475	−	0.857506i	$-0.672013\pi$
$53$	−7.49086	−1.02895	−0.514475	+	0.857506i	$0.672013\pi$
$54$	0	0
$55$	−0.286462	−0.0386265
$56$	0	0
$57$	−9.88663	−1.30952
$58$	0	0
$59$	−2.36329	−0.307674	−0.153837	−	0.988096i	$-0.549163\pi$
$59$	−2.36329	−0.307674	−0.153837	+	0.988096i	$0.549163\pi$
$60$	0	0
$61$	−1.55465	−0.199053	−0.0995264	−	0.995035i	$-0.531733\pi$
$61$	−1.55465	−0.199053	−0.0995264	+	0.995035i	$0.531733\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	6.12758	0.748602	0.374301	−	0.927307i	$-0.377883\pi$
$67$	6.12758	0.748602	0.374301	+	0.927307i	$0.377883\pi$
$68$	0	0
$69$	12.8411	1.54589
$70$	0	0
$71$	−12.3137	−1.46137	−0.730684	−	0.682716i	$-0.760799\pi$
$71$	−12.3137	−1.46137	−0.730684	+	0.682716i	$0.760799\pi$
$72$	0	0
$73$	8.45432	0.989503	0.494752	−	0.869034i	$-0.335259\pi$
$73$	8.45432	0.989503	0.494752	+	0.869034i	$0.335259\pi$
$74$	0	0
$75$	−7.33198	−0.846624
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.3958	−1.16961	−0.584807	−	0.811172i	$-0.698830\pi$
$79$	−10.3958	−1.16961	−0.584807	+	0.811172i	$0.698830\pi$
$80$	0	0
$81$	−6.06379	−0.673754
$82$	0	0
$83$	−16.1861	−1.77666	−0.888329	−	0.459207i	$-0.848134\pi$
$83$	−16.1861	−1.77666	−0.888329	+	0.459207i	$0.848134\pi$
$84$	0	0
$85$	−0.509136	−0.0552236
$86$	0	0
$87$	−5.50914	−0.590641
$88$	0	0
$89$	10.1406	1.07490	0.537451	−	0.843295i	$-0.319387\pi$
$89$	10.1406	1.07490	0.537451	+	0.843295i	$0.319387\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.77209	−0.287452
$94$	0	0
$95$	−1.89967	−0.194902
$96$	0	0
$97$	−3.04551	−0.309225	−0.154613	−	0.987975i	$-0.549413\pi$
$97$	−3.04551	−0.309225	−0.154613	+	0.987975i	$0.549413\pi$
$98$	0	0
$99$	0.777326	0.0781242
$100$	0	0
$101$	−4.36852	−0.434684	−0.217342	−	0.976095i	$-0.569739\pi$
$101$	−4.36852	−0.434684	−0.217342	+	0.976095i	$0.569739\pi$
$102$	0	0
$103$	−5.22267	−0.514605	−0.257303	−	0.966331i	$-0.582834\pi$
$103$	−5.22267	−0.514605	−0.257303	+	0.966331i	$0.582834\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.5311	−1.01808	−0.509042	−	0.860742i	$-0.670000\pi$
$107$	−10.5311	−1.01808	−0.509042	+	0.860742i	$0.670000\pi$
$108$	0	0
$109$	3.54942	0.339972	0.169986	−	0.985446i	$-0.445628\pi$
$109$	3.54942	0.339972	0.169986	+	0.985446i	$0.445628\pi$
$110$	0	0
$111$	11.0728	1.05098
$112$	0	0
$113$	−6.31370	−0.593943	−0.296972	−	0.954886i	$-0.595977\pi$
$113$	−6.31370	−0.593943	−0.296972	+	0.954886i	$0.595977\pi$
$114$	0	0
$115$	2.46736	0.230082
$116$	0	0
$117$	−2.71354	−0.250867
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.93098	0.354444
$124$	0	0
$125$	−2.84111	−0.254117
$126$	0	0
$127$	−3.88139	−0.344418	−0.172209	−	0.985060i	$-0.555090\pi$
$127$	−3.88139	−0.344418	−0.172209	+	0.985060i	$0.555090\pi$
$128$	0	0
$129$	−16.2499	−1.43073
$130$	0	0
$131$	−18.2316	−1.59291	−0.796453	−	0.604700i	$-0.793293\pi$
$131$	−18.2316	−1.59291	−0.796453	+	0.604700i	$0.793293\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.61320	−0.138843
$136$	0	0
$137$	−1.38156	−0.118035	−0.0590174	−	0.998257i	$-0.518797\pi$
$137$	−1.38156	−0.118035	−0.0590174	+	0.998257i	$0.518797\pi$
$138$	0	0
$139$	−0.591197	−0.0501447	−0.0250723	−	0.999686i	$-0.507982\pi$
$139$	−0.591197	−0.0501447	−0.0250723	+	0.999686i	$0.507982\pi$
$140$	0	0
$141$	11.3502	0.955863
$142$	0	0
$143$	−3.49086	−0.291921
$144$	0	0
$145$	−1.05855	−0.0879081
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.7993	1.04856	0.524281	−	0.851545i	$-0.324334\pi$
$149$	12.7993	1.04856	0.524281	+	0.851545i	$0.324334\pi$
$150$	0	0
$151$	5.65498	0.460196	0.230098	−	0.973167i	$-0.426095\pi$
$151$	5.65498	0.460196	0.230098	+	0.973167i	$0.426095\pi$
$152$	0	0
$153$	1.38156	0.111693
$154$	0	0
$155$	−0.532644	−0.0427830
$156$	0	0
$157$	19.6535	1.56852	0.784259	−	0.620433i	$-0.213043\pi$
$157$	19.6535	1.56852	0.784259	+	0.620433i	$0.213043\pi$
$158$	0	0
$159$	−11.1679	−0.885669
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	0	0
$165$	−0.427076	−0.0332478
$166$	0	0
$167$	−14.8046	−1.14561	−0.572806	−	0.819691i	$-0.694145\pi$
$167$	−14.8046	−1.14561	−0.572806	+	0.819691i	$0.694145\pi$
$168$	0	0
$169$	−0.813871	−0.0626055
$170$	0	0
$171$	5.15482	0.394199
$172$	0	0
$173$	−15.1992	−1.15557	−0.577786	−	0.816189i	$-0.696083\pi$
$173$	−15.1992	−1.15557	−0.577786	+	0.816189i	$0.696083\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.52334	−0.264830
$178$	0	0
$179$	−18.4543	−1.37934	−0.689670	−	0.724124i	$-0.742245\pi$
$179$	−18.4543	−1.37934	−0.689670	+	0.724124i	$0.742245\pi$
$180$	0	0
$181$	−19.0272	−1.41428	−0.707142	−	0.707072i	$-0.750016\pi$
$181$	−19.0272	−1.41428	−0.707142	+	0.707072i	$0.750016\pi$
$182$	0	0
$183$	−2.31777	−0.171335
$184$	0	0
$185$	2.12758	0.156422
$186$	0	0
$187$	1.77733	0.129971
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.2630	0.742601	0.371301	−	0.928513i	$-0.378912\pi$
$191$	10.2630	0.742601	0.371301	+	0.928513i	$0.378912\pi$
$192$	0	0
$193$	13.7355	0.988706	0.494353	−	0.869261i	$-0.335405\pi$
$193$	13.7355	0.988706	0.494353	+	0.869261i	$0.335405\pi$
$194$	0	0
$195$	1.49086	0.106763
$196$	0	0
$197$	4.05075	0.288604	0.144302	−	0.989534i	$-0.453906\pi$
$197$	4.05075	0.288604	0.144302	+	0.989534i	$0.453906\pi$
$198$	0	0
$199$	−25.7863	−1.82794	−0.913971	−	0.405780i	$-0.867000\pi$
$199$	−25.7863	−1.82794	−0.913971	+	0.405780i	$0.867000\pi$
$200$	0	0
$201$	9.13538	0.644360
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.755318	0.0527537
$206$	0	0
$207$	−6.69527	−0.465353
$208$	0	0
$209$	6.63148	0.458709
$210$	0	0
$211$	11.8359	0.814816	0.407408	−	0.913246i	$-0.366433\pi$
$211$	11.8359	0.814816	0.407408	+	0.913246i	$0.366433\pi$
$212$	0	0
$213$	−18.3581	−1.25787
$214$	0	0
$215$	−3.12234	−0.212942
$216$	0	0
$217$	0	0
$218$	0	0
$219$	12.6042	0.851715
$220$	0	0
$221$	−6.20440	−0.417353
$222$	0	0
$223$	−17.7005	−1.18531	−0.592657	−	0.805455i	$-0.701921\pi$
$223$	−17.7005	−1.18531	−0.592657	+	0.805455i	$0.701921\pi$
$224$	0	0
$225$	3.82284	0.254856
$226$	0	0
$227$	23.7135	1.57392	0.786961	−	0.617002i	$-0.211653\pi$
$227$	23.7135	1.57392	0.786961	+	0.617002i	$0.211653\pi$
$228$	0	0
$229$	18.7993	1.24229	0.621147	−	0.783694i	$-0.286667\pi$
$229$	18.7993	1.24229	0.621147	+	0.783694i	$0.286667\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.9817	1.30905	0.654523	−	0.756042i	$-0.272869\pi$
$233$	19.9817	1.30905	0.654523	+	0.756042i	$0.272869\pi$
$234$	0	0
$235$	2.18089	0.142266
$236$	0	0
$237$	−15.4987	−1.00675
$238$	0	0
$239$	−15.4946	−1.00226	−0.501131	−	0.865371i	$-0.667083\pi$
$239$	−15.4946	−1.00226	−0.501131	+	0.865371i	$0.667083\pi$
$240$	0	0
$241$	4.24618	0.273521	0.136760	−	0.990604i	$-0.456331\pi$
$241$	4.24618	0.273521	0.136760	+	0.990604i	$0.456331\pi$
$242$	0	0
$243$	7.85415	0.503844
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−23.1496	−1.47297
$248$	0	0
$249$	−24.1313	−1.52926
$250$	0	0
$251$	−3.77733	−0.238423	−0.119211	−	0.992869i	$-0.538037\pi$
$251$	−3.77733	−0.238423	−0.119211	+	0.992869i	$0.538037\pi$
$252$	0	0
$253$	−8.61320	−0.541508
$254$	0	0
$255$	−0.759053	−0.0475337
$256$	0	0
$257$	−18.3592	−1.14522	−0.572608	−	0.819829i	$-0.694068\pi$
$257$	−18.3592	−1.14522	−0.572608	+	0.819829i	$0.694068\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.87242	0.177799
$262$	0	0
$263$	−0.496098	−0.0305907	−0.0152954	−	0.999883i	$-0.504869\pi$
$263$	−0.496098	−0.0305907	−0.0152954	+	0.999883i	$0.504869\pi$
$264$	0	0
$265$	−2.14585	−0.131818
$266$	0	0
$267$	15.1183	0.925223
$268$	0	0
$269$	−9.82284	−0.598909	−0.299455	−	0.954111i	$-0.596805\pi$
$269$	−9.82284	−0.598909	−0.299455	+	0.954111i	$0.596805\pi$
$270$	0	0
$271$	11.4998	0.698565	0.349283	−	0.937017i	$-0.386425\pi$
$271$	11.4998	0.698565	0.349283	+	0.937017i	$0.386425\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.91794	0.296563
$276$	0	0
$277$	−2.19917	−0.132135	−0.0660676	−	0.997815i	$-0.521045\pi$
$277$	−2.19917	−0.132135	−0.0660676	+	0.997815i	$0.521045\pi$
$278$	0	0
$279$	1.44535	0.0865308
$280$	0	0
$281$	16.6222	0.991596	0.495798	−	0.868438i	$-0.334876\pi$
$281$	16.6222	0.991596	0.495798	+	0.868438i	$0.334876\pi$
$282$	0	0
$283$	−15.0586	−0.895138	−0.447569	−	0.894249i	$-0.647710\pi$
$283$	−15.0586	−0.895138	−0.447569	+	0.894249i	$0.647710\pi$
$284$	0	0
$285$	−2.83214	−0.167762
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.8411	−0.814183
$290$	0	0
$291$	−4.54045	−0.266166
$292$	0	0
$293$	−26.3905	−1.54175	−0.770876	−	0.636986i	$-0.780181\pi$
$293$	−26.3905	−1.54175	−0.770876	+	0.636986i	$0.780181\pi$
$294$	0	0
$295$	−0.676992	−0.0394160
$296$	0	0
$297$	5.63148	0.326771
$298$	0	0
$299$	30.0675	1.73885
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−6.51287	−0.374155
$304$	0	0
$305$	−0.445349	−0.0255006
$306$	0	0
$307$	−19.2316	−1.09761	−0.548804	−	0.835951i	$-0.684917\pi$
$307$	−19.2316	−1.09761	−0.548804	+	0.835951i	$0.684917\pi$
$308$	0	0
$309$	−7.78630	−0.442947
$310$	0	0
$311$	−25.6770	−1.45601	−0.728004	−	0.685573i	$-0.759552\pi$
$311$	−25.6770	−1.45601	−0.728004	+	0.685573i	$0.759552\pi$
$312$	0	0
$313$	1.36852	0.0773535	0.0386767	−	0.999252i	$-0.487686\pi$
$313$	1.36852	0.0773535	0.0386767	+	0.999252i	$0.487686\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.48306	0.195628	0.0978141	−	0.995205i	$-0.468815\pi$
$317$	3.48306	0.195628	0.0978141	+	0.995205i	$0.468815\pi$
$318$	0	0
$319$	3.69527	0.206895
$320$	0	0
$321$	−15.7005	−0.876316
$322$	0	0
$323$	11.7863	0.655807
$324$	0	0
$325$	−17.1679	−0.952301
$326$	0	0
$327$	5.29170	0.292631
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.9086	0.984348	0.492174	−	0.870497i	$-0.336202\pi$
$331$	17.9086	0.984348	0.492174	+	0.870497i	$0.336202\pi$
$332$	0	0
$333$	−5.77326	−0.316373
$334$	0	0
$335$	1.75532	0.0959033
$336$	0	0
$337$	−34.1496	−1.86025	−0.930123	−	0.367248i	$-0.880300\pi$
$337$	−34.1496	−1.86025	−0.930123	+	0.367248i	$0.880300\pi$
$338$	0	0
$339$	−9.41287	−0.511237
$340$	0	0
$341$	1.85939	0.100691
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.67849	0.198043
$346$	0	0
$347$	−6.30473	−0.338456	−0.169228	−	0.985577i	$-0.554127\pi$
$347$	−6.30473	−0.338456	−0.169228	+	0.985577i	$0.554127\pi$
$348$	0	0
$349$	10.7628	0.576119	0.288059	−	0.957613i	$-0.406990\pi$
$349$	10.7628	0.576119	0.288059	+	0.957613i	$0.406990\pi$
$350$	0	0
$351$	−19.6587	−1.04930
$352$	0	0
$353$	7.86312	0.418512	0.209256	−	0.977861i	$-0.432896\pi$
$353$	7.86312	0.418512	0.209256	+	0.977861i	$0.432896\pi$
$354$	0	0
$355$	−3.52741	−0.187215
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.9959	1.00257	0.501283	−	0.865283i	$-0.332861\pi$
$359$	18.9959	1.00257	0.501283	+	0.865283i	$0.332861\pi$
$360$	0	0
$361$	24.9765	1.31455
$362$	0	0
$363$	1.49086	0.0782500
$364$	0	0
$365$	2.42184	0.126765
$366$	0	0
$367$	18.3958	0.960251	0.480126	−	0.877200i	$-0.340591\pi$
$367$	18.3958	0.960251	0.480126	+	0.877200i	$0.340591\pi$
$368$	0	0
$369$	−2.04958	−0.106697
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−23.2552	−1.20411	−0.602053	−	0.798456i	$-0.705650\pi$
$373$	−23.2552	−1.20411	−0.602053	+	0.798456i	$0.705650\pi$
$374$	0	0
$375$	−4.23571	−0.218731
$376$	0	0
$377$	−12.8997	−0.664367
$378$	0	0
$379$	3.53264	0.181460	0.0907299	−	0.995876i	$-0.471080\pi$
$379$	3.53264	0.181460	0.0907299	+	0.995876i	$0.471080\pi$
$380$	0	0
$381$	−5.78663	−0.296458
$382$	0	0
$383$	24.5636	1.25514	0.627571	−	0.778559i	$-0.284049\pi$
$383$	24.5636	1.25514	0.627571	+	0.778559i	$0.284049\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.47259	0.430686
$388$	0	0
$389$	−1.32301	−0.0670791	−0.0335396	−	0.999437i	$-0.510678\pi$
$389$	−1.32301	−0.0670791	−0.0335396	+	0.999437i	$0.510678\pi$
$390$	0	0
$391$	−15.3085	−0.774183
$392$	0	0
$393$	−27.1809	−1.37109
$394$	0	0
$395$	−2.97799	−0.149839
$396$	0	0
$397$	−15.3540	−0.770594	−0.385297	−	0.922793i	$-0.625901\pi$
$397$	−15.3540	−0.770594	−0.385297	+	0.922793i	$0.625901\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.4543	−0.721814	−0.360907	−	0.932602i	$-0.617533\pi$
$401$	−14.4543	−0.721814	−0.360907	+	0.932602i	$0.617533\pi$
$402$	0	0
$403$	−6.49086	−0.323333
$404$	0	0
$405$	−1.73705	−0.0863145
$406$	0	0
$407$	−7.42708	−0.368146
$408$	0	0
$409$	−32.3122	−1.59774	−0.798868	−	0.601507i	$-0.794567\pi$
$409$	−32.3122	−1.59774	−0.798868	+	0.601507i	$0.794567\pi$
$410$	0	0
$411$	−2.05972	−0.101598
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.63671	−0.227607
$416$	0	0
$417$	−0.881394	−0.0431620
$418$	0	0
$419$	15.5259	0.758490	0.379245	−	0.925296i	$-0.376184\pi$
$419$	15.5259	0.758490	0.379245	+	0.925296i	$0.376184\pi$
$420$	0	0
$421$	10.9049	0.531472	0.265736	−	0.964046i	$-0.414385\pi$
$421$	10.9049	0.531472	0.265736	+	0.964046i	$0.414385\pi$
$422$	0	0
$423$	−5.91794	−0.287740
$424$	0	0
$425$	8.74078	0.423990
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.20440	−0.251271
$430$	0	0
$431$	−16.8254	−0.810451	−0.405226	−	0.914217i	$-0.632807\pi$
$431$	−16.8254	−0.810451	−0.405226	+	0.914217i	$0.632807\pi$
$432$	0	0
$433$	−12.1641	−0.584570	−0.292285	−	0.956331i	$-0.594416\pi$
$433$	−12.1641	−0.584570	−0.292285	+	0.956331i	$0.594416\pi$
$434$	0	0
$435$	−1.57816	−0.0756669
$436$	0	0
$437$	−57.1183	−2.73234
$438$	0	0
$439$	6.98546	0.333398	0.166699	−	0.986008i	$-0.446689\pi$
$439$	6.98546	0.333398	0.166699	+	0.986008i	$0.446689\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	39.3592	1.87001	0.935006	−	0.354631i	$-0.115393\pi$
$443$	39.3592	1.87001	0.935006	+	0.354631i	$0.115393\pi$
$444$	0	0
$445$	2.90490	0.137706
$446$	0	0
$447$	19.0821	0.902550
$448$	0	0
$449$	13.3723	0.631076	0.315538	−	0.948913i	$-0.397815\pi$
$449$	13.3723	0.631076	0.315538	+	0.948913i	$0.397815\pi$
$450$	0	0
$451$	−2.63671	−0.124158
$452$	0	0
$453$	8.43081	0.396114
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.88663	0.322143	0.161071	−	0.986943i	$-0.448505\pi$
$457$	6.88663	0.322143	0.161071	+	0.986943i	$0.448505\pi$
$458$	0	0
$459$	10.0090	0.467179
$460$	0	0
$461$	5.16786	0.240691	0.120346	−	0.992732i	$-0.461600\pi$
$461$	5.16786	0.240691	0.120346	+	0.992732i	$0.461600\pi$
$462$	0	0
$463$	18.6039	0.864597	0.432298	−	0.901731i	$-0.357703\pi$
$463$	18.6039	0.864597	0.432298	+	0.901731i	$0.357703\pi$
$464$	0	0
$465$	−0.794099	−0.0368255
$466$	0	0
$467$	−15.2772	−0.706943	−0.353471	−	0.935445i	$-0.614999\pi$
$467$	−15.2772	−0.706943	−0.353471	+	0.935445i	$0.614999\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	29.3007	1.35010
$472$	0	0
$473$	10.8997	0.501167
$474$	0	0
$475$	32.6132	1.49640
$476$	0	0
$477$	5.82284	0.266610
$478$	0	0
$479$	12.9217	0.590406	0.295203	−	0.955434i	$-0.404613\pi$
$479$	12.9217	0.590406	0.295203	+	0.955434i	$0.404613\pi$
$480$	0	0
$481$	25.9269	1.18217
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.872425	−0.0396148
$486$	0	0
$487$	−14.4648	−0.655462	−0.327731	−	0.944771i	$-0.606284\pi$
$487$	−14.4648	−0.655462	−0.327731	+	0.944771i	$0.606284\pi$
$488$	0	0
$489$	13.4178	0.606773
$490$	0	0
$491$	40.8799	1.84488	0.922442	−	0.386136i	$-0.126190\pi$
$491$	40.8799	1.84488	0.922442	+	0.386136i	$0.126190\pi$
$492$	0	0
$493$	6.56769	0.295794
$494$	0	0
$495$	0.222674	0.0100085
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−17.6848	−0.791680	−0.395840	−	0.918319i	$-0.629547\pi$
$499$	−17.6848	−0.791680	−0.395840	+	0.918319i	$0.629547\pi$
$500$	0	0
$501$	−22.0716	−0.986086
$502$	0	0
$503$	43.8266	1.95413	0.977065	−	0.212940i	$-0.0683040\pi$
$503$	43.8266	1.95413	0.977065	+	0.212940i	$0.0683040\pi$
$504$	0	0
$505$	−1.25142	−0.0556873
$506$	0	0
$507$	−1.21337	−0.0538877
$508$	0	0
$509$	−32.7303	−1.45075	−0.725373	−	0.688356i	$-0.758333\pi$
$509$	−32.7303	−1.45075	−0.725373	+	0.688356i	$0.758333\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	37.3450	1.64882
$514$	0	0
$515$	−1.49610	−0.0659260
$516$	0	0
$517$	−7.61320	−0.334828
$518$	0	0
$519$	−22.6599	−0.994659
$520$	0	0
$521$	40.4670	1.77289	0.886446	−	0.462832i	$-0.153167\pi$
$521$	40.4670	1.77289	0.886446	+	0.462832i	$0.153167\pi$
$522$	0	0
$523$	−34.6860	−1.51671	−0.758356	−	0.651841i	$-0.773997\pi$
$523$	−34.6860	−1.51671	−0.758356	+	0.651841i	$0.773997\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.30473	0.143957
$528$	0	0
$529$	51.1873	2.22553
$530$	0	0
$531$	1.83704	0.0797209
$532$	0	0
$533$	9.20440	0.398687
$534$	0	0
$535$	−3.01677	−0.130426
$536$	0	0
$537$	−27.5129	−1.18727
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.95042	−0.255828	−0.127914	−	0.991785i	$-0.540828\pi$
$541$	−5.95042	−0.255828	−0.127914	+	0.991785i	$0.540828\pi$
$542$	0	0
$543$	−28.3670	−1.21735
$544$	0	0
$545$	1.01677	0.0435538
$546$	0	0
$547$	1.32824	0.0567915	0.0283958	−	0.999597i	$-0.490960\pi$
$547$	1.32824	0.0567915	0.0283958	+	0.999597i	$0.490960\pi$
$548$	0	0
$549$	1.20847	0.0515763
$550$	0	0
$551$	24.5051	1.04395
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.17192	0.134641
$556$	0	0
$557$	−11.1145	−0.470938	−0.235469	−	0.971882i	$-0.575663\pi$
$557$	−11.1145	−0.470938	−0.235469	+	0.971882i	$0.575663\pi$
$558$	0	0
$559$	−38.0492	−1.60931
$560$	0	0
$561$	2.64975	0.111873
$562$	0	0
$563$	−19.4763	−0.820829	−0.410415	−	0.911899i	$-0.634616\pi$
$563$	−19.4763	−0.820829	−0.410415	+	0.911899i	$0.634616\pi$
$564$	0	0
$565$	−1.80864	−0.0760899
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.2223	1.18314	0.591571	−	0.806253i	$-0.298508\pi$
$569$	28.2223	1.18314	0.591571	+	0.806253i	$0.298508\pi$
$570$	0	0
$571$	−23.9310	−1.00148	−0.500740	−	0.865598i	$-0.666939\pi$
$571$	−23.9310	−1.00148	−0.500740	+	0.865598i	$0.666939\pi$
$572$	0	0
$573$	15.3007	0.639194
$574$	0	0
$575$	−42.3592	−1.76650
$576$	0	0
$577$	−27.2954	−1.13632	−0.568162	−	0.822917i	$-0.692345\pi$
$577$	−27.2954	−1.13632	−0.568162	+	0.822917i	$0.692345\pi$
$578$	0	0
$579$	20.4778	0.851029
$580$	0	0
$581$	0	0
$582$	0	0
$583$	7.49086	0.310240
$584$	0	0
$585$	−0.777326	−0.0321385
$586$	0	0
$587$	15.9179	0.657004	0.328502	−	0.944503i	$-0.393456\pi$
$587$	15.9179	0.657004	0.328502	+	0.944503i	$0.393456\pi$
$588$	0	0
$589$	12.3305	0.508068
$590$	0	0
$591$	6.03911	0.248416
$592$	0	0
$593$	6.42184	0.263713	0.131857	−	0.991269i	$-0.457906\pi$
$593$	6.42184	0.263713	0.131857	+	0.991269i	$0.457906\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−38.4438	−1.57340
$598$	0	0
$599$	4.23571	0.173067	0.0865333	−	0.996249i	$-0.472421\pi$
$599$	4.23571	0.173067	0.0865333	+	0.996249i	$0.472421\pi$
$600$	0	0
$601$	−26.4946	−1.08074	−0.540369	−	0.841428i	$-0.681715\pi$
$601$	−26.4946	−1.08074	−0.540369	+	0.841428i	$0.681715\pi$
$602$	0	0
$603$	−4.76312	−0.193969
$604$	0	0
$605$	0.286462	0.0116463
$606$	0	0
$607$	−30.6262	−1.24308	−0.621540	−	0.783382i	$-0.713493\pi$
$607$	−30.6262	−1.24308	−0.621540	+	0.783382i	$0.713493\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26.5767	1.07518
$612$	0	0
$613$	28.3357	1.14447	0.572234	−	0.820090i	$-0.306077\pi$
$613$	28.3357	1.14447	0.572234	+	0.820090i	$0.306077\pi$
$614$	0	0
$615$	1.12608	0.0454078
$616$	0	0
$617$	−49.3279	−1.98587	−0.992933	−	0.118673i	$-0.962136\pi$
$617$	−49.3279	−1.98587	−0.992933	+	0.118673i	$0.962136\pi$
$618$	0	0
$619$	−40.1276	−1.61286	−0.806432	−	0.591327i	$-0.798604\pi$
$619$	−40.1276	−1.61286	−0.806432	+	0.591327i	$0.798604\pi$
$620$	0	0
$621$	−48.5051	−1.94644
$622$	0	0
$623$	0	0
$624$	0	0
$625$	23.7758	0.951033
$626$	0	0
$627$	9.88663	0.394834
$628$	0	0
$629$	−13.2003	−0.526332
$630$	0	0
$631$	39.2354	1.56194	0.780968	−	0.624571i	$-0.214726\pi$
$631$	39.2354	1.56194	0.780968	+	0.624571i	$0.214726\pi$
$632$	0	0
$633$	17.6457	0.701353
$634$	0	0
$635$	−1.11187	−0.0441233
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.57176	0.378653
$640$	0	0
$641$	−4.96869	−0.196251	−0.0981257	−	0.995174i	$-0.531285\pi$
$641$	−4.96869	−0.196251	−0.0981257	+	0.995174i	$0.531285\pi$
$642$	0	0
$643$	−1.88289	−0.0742541	−0.0371270	−	0.999311i	$-0.511821\pi$
$643$	−1.88289	−0.0742541	−0.0371270	+	0.999311i	$0.511821\pi$
$644$	0	0
$645$	−4.65498	−0.183290
$646$	0	0
$647$	−14.9191	−0.586531	−0.293265	−	0.956031i	$-0.594742\pi$
$647$	−14.9191	−0.586531	−0.293265	+	0.956031i	$0.594742\pi$
$648$	0	0
$649$	2.36329	0.0927672
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.5349	1.62539	0.812693	−	0.582693i	$-0.198001\pi$
$653$	41.5349	1.62539	0.812693	+	0.582693i	$0.198001\pi$
$654$	0	0
$655$	−5.22267	−0.204067
$656$	0	0
$657$	−6.57176	−0.256389
$658$	0	0
$659$	−12.4596	−0.485355	−0.242678	−	0.970107i	$-0.578026\pi$
$659$	−12.4596	−0.485355	−0.242678	+	0.970107i	$0.578026\pi$
$660$	0	0
$661$	−5.26819	−0.204909	−0.102454	−	0.994738i	$-0.532670\pi$
$661$	−5.26819	−0.204909	−0.102454	+	0.994738i	$0.532670\pi$
$662$	0	0
$663$	−9.24992	−0.359237
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−31.8281	−1.23239
$668$	0	0
$669$	−26.3890	−1.02026
$670$	0	0
$671$	1.55465	0.0600167
$672$	0	0
$673$	−27.8254	−1.07259	−0.536295	−	0.844030i	$-0.680177\pi$
$673$	−27.8254	−1.07259	−0.536295	+	0.844030i	$0.680177\pi$
$674$	0	0
$675$	27.6953	1.06599
$676$	0	0
$677$	−21.0873	−0.810451	−0.405225	−	0.914217i	$-0.632807\pi$
$677$	−21.0873	−0.810451	−0.405225	+	0.914217i	$0.632807\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	35.3537	1.35475
$682$	0	0
$683$	47.9071	1.83312	0.916558	−	0.399902i	$-0.130956\pi$
$683$	47.9071	1.83312	0.916558	+	0.399902i	$0.130956\pi$
$684$	0	0
$685$	−0.395765	−0.0151214
$686$	0	0
$687$	28.0272	1.06931
$688$	0	0
$689$	−26.1496	−0.996220
$690$	0	0
$691$	−5.47409	−0.208244	−0.104122	−	0.994565i	$-0.533203\pi$
$691$	−5.47409	−0.208244	−0.104122	+	0.994565i	$0.533203\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.169355	−0.00642402
$696$	0	0
$697$	−4.68630	−0.177506
$698$	0	0
$699$	29.7900	1.12676
$700$	0	0
$701$	31.1679	1.17719	0.588597	−	0.808427i	$-0.299681\pi$
$701$	31.1679	1.17719	0.588597	+	0.808427i	$0.299681\pi$
$702$	0	0
$703$	−49.2525	−1.85759
$704$	0	0
$705$	3.25142	0.122455
$706$	0	0
$707$	0	0
$708$	0	0
$709$	48.3667	1.81645	0.908225	−	0.418483i	$-0.137438\pi$
$709$	48.3667	1.81645	0.908225	+	0.418483i	$0.137438\pi$
$710$	0	0
$711$	8.08089	0.303057
$712$	0	0
$713$	−16.0153	−0.599777
$714$	0	0
$715$	−1.00000	−0.0373979
$716$	0	0
$717$	−23.1003	−0.862698
$718$	0	0
$719$	1.78663	0.0666300	0.0333150	−	0.999445i	$-0.489394\pi$
$719$	1.78663	0.0666300	0.0333150	+	0.999445i	$0.489394\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	6.33048	0.235433
$724$	0	0
$725$	18.1731	0.674932
$726$	0	0
$727$	−22.5494	−0.836312	−0.418156	−	0.908375i	$-0.637323\pi$
$727$	−22.5494	−0.836312	−0.418156	+	0.908375i	$0.637323\pi$
$728$	0	0
$729$	29.9008	1.10744
$730$	0	0
$731$	19.3723	0.716509
$732$	0	0
$733$	−5.45955	−0.201653	−0.100827	−	0.994904i	$-0.532149\pi$
$733$	−5.45955	−0.201653	−0.100827	+	0.994904i	$0.532149\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.12758	−0.225712
$738$	0	0
$739$	14.9101	0.548478	0.274239	−	0.961662i	$-0.411574\pi$
$739$	14.9101	0.548478	0.274239	+	0.961662i	$0.411574\pi$
$740$	0	0
$741$	−34.5129	−1.26786
$742$	0	0
$743$	12.7680	0.468413	0.234207	−	0.972187i	$-0.424751\pi$
$743$	12.7680	0.468413	0.234207	+	0.972187i	$0.424751\pi$
$744$	0	0
$745$	3.66652	0.134331
$746$	0	0
$747$	12.5819	0.460347
$748$	0	0
$749$	0	0
$750$	0	0
$751$	31.5129	1.14992	0.574961	−	0.818181i	$-0.305017\pi$
$751$	31.5129	1.14992	0.574961	+	0.818181i	$0.305017\pi$
$752$	0	0
$753$	−5.63148	−0.205222
$754$	0	0
$755$	1.61994	0.0589556
$756$	0	0
$757$	−19.0936	−0.693969	−0.346984	−	0.937871i	$-0.612794\pi$
$757$	−19.0936	−0.693969	−0.346984	+	0.937871i	$0.612794\pi$
$758$	0	0
$759$	−12.8411	−0.466103
$760$	0	0
$761$	13.0220	0.472047	0.236024	−	0.971747i	$-0.424156\pi$
$761$	13.0220	0.472047	0.236024	+	0.971747i	$0.424156\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.395765	0.0143089
$766$	0	0
$767$	−8.24992	−0.297887
$768$	0	0
$769$	25.5103	0.919925	0.459963	−	0.887938i	$-0.347863\pi$
$769$	25.5103	0.919925	0.459963	+	0.887938i	$0.347863\pi$
$770$	0	0
$771$	−27.3711	−0.985746
$772$	0	0
$773$	47.8684	1.72171	0.860853	−	0.508854i	$-0.169931\pi$
$773$	47.8684	1.72171	0.860853	+	0.508854i	$0.169931\pi$
$774$	0	0
$775$	9.14435	0.328475
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.4853	−0.626476
$780$	0	0
$781$	12.3137	0.440619
$782$	0	0
$783$	20.8098	0.743682
$784$	0	0
$785$	5.62998	0.200943
$786$	0	0
$787$	−27.5819	−0.983188	−0.491594	−	0.870824i	$-0.663586\pi$
$787$	−27.5819	−0.983188	−0.491594	+	0.870824i	$0.663586\pi$
$788$	0	0
$789$	−0.739615	−0.0263310
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.42708	−0.192721
$794$	0	0
$795$	−3.19917	−0.113463
$796$	0	0
$797$	26.7773	0.948502	0.474251	−	0.880390i	$-0.342719\pi$
$797$	26.7773	0.948502	0.474251	+	0.880390i	$0.342719\pi$
$798$	0	0
$799$	−13.5311	−0.478697
$800$	0	0
$801$	−7.88256	−0.278517
$802$	0	0
$803$	−8.45432	−0.298346
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.6445	−0.515511
$808$	0	0
$809$	37.1611	1.30652	0.653258	−	0.757136i	$-0.273402\pi$
$809$	37.1611	1.30652	0.653258	+	0.757136i	$0.273402\pi$
$810$	0	0
$811$	15.0414	0.528177	0.264088	−	0.964499i	$-0.414929\pi$
$811$	15.0414	0.528177	0.264088	+	0.964499i	$0.414929\pi$
$812$	0	0
$813$	17.1447	0.601290
$814$	0	0
$815$	2.57816	0.0903090
$816$	0	0
$817$	72.2809	2.52879
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.213371	−0.00744670	−0.00372335	−	0.999993i	$-0.501185\pi$
$821$	−0.213371	−0.00744670	−0.00372335	+	0.999993i	$0.501185\pi$
$822$	0	0
$823$	−40.4398	−1.40964	−0.704821	−	0.709385i	$-0.748973\pi$
$823$	−40.4398	−1.40964	−0.704821	+	0.709385i	$0.748973\pi$
$824$	0	0
$825$	7.33198	0.255267
$826$	0	0
$827$	−50.0713	−1.74115	−0.870574	−	0.492037i	$-0.836252\pi$
$827$	−50.0713	−1.74115	−0.870574	+	0.492037i	$0.836252\pi$
$828$	0	0
$829$	−32.8266	−1.14011	−0.570057	−	0.821605i	$-0.693079\pi$
$829$	−32.8266	−1.14011	−0.570057	+	0.821605i	$0.693079\pi$
$830$	0	0
$831$	−3.27866	−0.113735
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.24095	−0.146764
$836$	0	0
$837$	10.4711	0.361934
$838$	0	0
$839$	−36.7758	−1.26964	−0.634821	−	0.772659i	$-0.718926\pi$
$839$	−36.7758	−1.26964	−0.634821	+	0.772659i	$0.718926\pi$
$840$	0	0
$841$	−15.3450	−0.529138
$842$	0	0
$843$	24.7814	0.853517
$844$	0	0
$845$	−0.233143	−0.00802037
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−22.4502	−0.770491
$850$	0	0
$851$	63.9709	2.19290
$852$	0	0
$853$	21.6613	0.741668	0.370834	−	0.928699i	$-0.379072\pi$
$853$	21.6613	0.741668	0.370834	+	0.928699i	$0.379072\pi$
$854$	0	0
$855$	1.47666	0.0505007
$856$	0	0
$857$	4.36852	0.149226	0.0746129	−	0.997213i	$-0.476228\pi$
$857$	4.36852	0.149226	0.0746129	+	0.997213i	$0.476228\pi$
$858$	0	0
$859$	−13.1626	−0.449103	−0.224551	−	0.974462i	$-0.572092\pi$
$859$	−13.1626	−0.449103	−0.224551	+	0.974462i	$0.572092\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	29.7785	1.01367	0.506836	−	0.862043i	$-0.330815\pi$
$863$	29.7785	1.01367	0.506836	+	0.862043i	$0.330815\pi$
$864$	0	0
$865$	−4.35398	−0.148040
$866$	0	0
$867$	−20.6352	−0.700809
$868$	0	0
$869$	10.3958	0.352652
$870$	0	0
$871$	21.3905	0.724790
$872$	0	0
$873$	2.36736	0.0801229
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.2540	0.447555	0.223778	−	0.974640i	$-0.428161\pi$
$877$	13.2540	0.447555	0.223778	+	0.974640i	$0.428161\pi$
$878$	0	0
$879$	−39.3447	−1.32706
$880$	0	0
$881$	46.3890	1.56289	0.781443	−	0.623977i	$-0.214484\pi$
$881$	46.3890	1.56289	0.781443	+	0.623977i	$0.214484\pi$
$882$	0	0
$883$	−11.6550	−0.392221	−0.196111	−	0.980582i	$-0.562831\pi$
$883$	−11.6550	−0.392221	−0.196111	+	0.980582i	$0.562831\pi$
$884$	0	0
$885$	−1.00930	−0.0339274
$886$	0	0
$887$	27.7198	0.930741	0.465371	−	0.885116i	$-0.345921\pi$
$887$	27.7198	0.930741	0.465371	+	0.885116i	$0.345921\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	6.06379	0.203145
$892$	0	0
$893$	−50.4868	−1.68948
$894$	0	0
$895$	−5.28646	−0.176707
$896$	0	0
$897$	44.8266	1.49672
$898$	0	0
$899$	6.87093	0.229158
$900$	0	0
$901$	13.3137	0.443544
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.45058	−0.181184
$906$	0	0
$907$	−3.72658	−0.123739	−0.0618695	−	0.998084i	$-0.519706\pi$
$907$	−3.72658	−0.123739	−0.0618695	+	0.998084i	$0.519706\pi$
$908$	0	0
$909$	3.39576	0.112630
$910$	0	0
$911$	42.4685	1.40704	0.703522	−	0.710673i	$-0.251609\pi$
$911$	42.4685	1.40704	0.703522	+	0.710673i	$0.251609\pi$
$912$	0	0
$913$	16.1861	0.535683
$914$	0	0
$915$	−0.663954	−0.0219496
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.06229	0.0680286	0.0340143	−	0.999421i	$-0.489171\pi$
$919$	2.06229	0.0680286	0.0340143	+	0.999421i	$0.489171\pi$
$920$	0	0
$921$	−28.6718	−0.944767
$922$	0	0
$923$	−42.9855	−1.41488
$924$	0	0
$925$	−36.5259	−1.20096
$926$	0	0
$927$	4.05972	0.133339
$928$	0	0
$929$	50.6222	1.66086	0.830430	−	0.557123i	$-0.188095\pi$
$929$	50.6222	1.66086	0.830430	+	0.557123i	$0.188095\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−38.2809	−1.25326
$934$	0	0
$935$	0.509136	0.0166505
$936$	0	0
$937$	−32.5129	−1.06215	−0.531075	−	0.847325i	$-0.678212\pi$
$937$	−32.5129	−1.06215	−0.531075	+	0.847325i	$0.678212\pi$
$938$	0	0
$939$	2.04028	0.0665820
$940$	0	0
$941$	−26.1992	−0.854068	−0.427034	−	0.904235i	$-0.640442\pi$
$941$	−26.1992	−0.854068	−0.427034	+	0.904235i	$0.640442\pi$
$942$	0	0
$943$	22.7105	0.739557
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−56.2029	−1.82635	−0.913174	−	0.407569i	$-0.866377\pi$
$947$	−56.2029	−1.82635	−0.913174	+	0.407569i	$0.866377\pi$
$948$	0	0
$949$	29.5129	0.958028
$950$	0	0
$951$	5.19277	0.168387
$952$	0	0
$953$	14.9855	0.485427	0.242713	−	0.970098i	$-0.421963\pi$
$953$	14.9855	0.485427	0.242713	+	0.970098i	$0.421963\pi$
$954$	0	0
$955$	2.93995	0.0951345
$956$	0	0
$957$	5.50914	0.178085
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.5427	−0.888474
$962$	0	0
$963$	8.18613	0.263794
$964$	0	0
$965$	3.93471	0.126663
$966$	0	0
$967$	38.8161	1.24824	0.624121	−	0.781328i	$-0.285457\pi$
$967$	38.8161	1.24824	0.624121	+	0.781328i	$0.285457\pi$
$968$	0	0
$969$	17.5718	0.564486
$970$	0	0
$971$	−27.9672	−0.897510	−0.448755	−	0.893655i	$-0.648132\pi$
$971$	−27.9672	−0.897510	−0.448755	+	0.893655i	$0.648132\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−25.5949	−0.819694
$976$	0	0
$977$	−44.3499	−1.41888	−0.709440	−	0.704766i	$-0.751052\pi$
$977$	−44.3499	−1.41888	−0.709440	+	0.704766i	$0.751052\pi$
$978$	0	0
$979$	−10.1406	−0.324095
$980$	0	0
$981$	−2.75905	−0.0880898
$982$	0	0
$983$	52.6990	1.68084	0.840419	−	0.541938i	$-0.182309\pi$
$983$	52.6990	1.68084	0.840419	+	0.541938i	$0.182309\pi$
$984$	0	0
$985$	1.16039	0.0369730
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−93.8811	−2.98524
$990$	0	0
$991$	32.5479	1.03392	0.516959	−	0.856010i	$-0.327064\pi$
$991$	32.5479	1.03392	0.516959	+	0.856010i	$0.327064\pi$
$992$	0	0
$993$	26.6993	0.847278
$994$	0	0
$995$	−7.38680	−0.234177
$996$	0	0
$997$	−36.4256	−1.15361	−0.576805	−	0.816882i	$-0.695701\pi$
$997$	−36.4256	−1.15361	−0.576805	+	0.816882i	$0.695701\pi$
$998$	0	0
$999$	−41.8254	−1.32330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.cg.1.3		3
4.3	odd	2		2156.2.a.k.1.1		3
7.3	odd	6		1232.2.q.j.177.3		6
7.5	odd	6		1232.2.q.j.529.3		6
7.6	odd	2		8624.2.a.cp.1.1		3
28.3	even	6		308.2.i.b.177.1	✓	6
28.11	odd	6		2156.2.i.j.177.3		6
28.19	even	6		308.2.i.b.221.1	yes	6
28.23	odd	6		2156.2.i.j.1145.3		6
28.27	even	2		2156.2.a.g.1.3		3
84.47	odd	6		2772.2.s.e.2377.2		6
84.59	odd	6		2772.2.s.e.793.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.2.i.b.177.1	✓	6	28.3	even	6
308.2.i.b.221.1	yes	6	28.19	even	6
1232.2.q.j.177.3		6	7.3	odd	6
1232.2.q.j.529.3		6	7.5	odd	6
2156.2.a.g.1.3		3	28.27	even	2
2156.2.a.k.1.1		3	4.3	odd	2
2156.2.i.j.177.3		6	28.11	odd	6
2156.2.i.j.1145.3		6	28.23	odd	6
2772.2.s.e.793.2		6	84.59	odd	6
2772.2.s.e.2377.2		6	84.47	odd	6
8624.2.a.cg.1.3		3	1.1	even	1	trivial
8624.2.a.cp.1.1		3	7.6	odd	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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q+1.49086 q^{3} +0.286462 q^{5} -0.777326 q^{9} -1.00000 q^{11} +3.49086 q^{13} +0.427076 q^{15} -1.77733 q^{17} -6.63148 q^{19} +8.61320 q^{23} -4.91794 q^{25} -5.63148 q^{27} -3.69527 q^{29} -1.85939 q^{31} -1.49086 q^{33} +7.42708 q^{37} +5.20440 q^{39} +2.63671 q^{41} -10.8997 q^{43} -0.222674 q^{45} +7.61320 q^{47} -2.64975 q^{51} -7.49086 q^{53} -0.286462 q^{55} -9.88663 q^{57} -2.36329 q^{59} -1.55465 q^{61} +1.00000 q^{65} +6.12758 q^{67} +12.8411 q^{69} -12.3137 q^{71} +8.45432 q^{73} -7.33198 q^{75} -10.3958 q^{79} -6.06379 q^{81} -16.1861 q^{83} -0.509136 q^{85} -5.50914 q^{87} +10.1406 q^{89} -2.77209 q^{93} -1.89967 q^{95} -3.04551 q^{97} +0.777326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{5} + 4 q^{9} - 3 q^{11} + 3 q^{13} - q^{15} + q^{17} - 9 q^{19} - 5 q^{25} - 6 q^{27} + 5 q^{29} - 9 q^{31} + 3 q^{33} + 20 q^{37} + 7 q^{39} + 5 q^{41} - 8 q^{43} - 7 q^{45} - 3 q^{47}+ \cdots - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^5 + 4 * q^9 - 3 * q^11 + 3 * q^13 - q^15 + q^17 - 9 * q^19 - 5 * q^25 - 6 * q^27 + 5 * q^29 - 9 * q^31 + 3 * q^33 + 20 * q^37 + 7 * q^39 + 5 * q^41 - 8 * q^43 - 7 * q^45 - 3 * q^47 - 12 * q^51 - 15 * q^53 - 2 * q^55 + 2 * q^57 - 10 * q^59 + 8 * q^61 + 3 * q^65 + 8 * q^67 + 27 * q^69 - 3 * q^71 - 12 * q^73 - 3 * q^75 - 7 * q^79 - 13 * q^81 - 25 * q^83 - 9 * q^85 - 24 * q^87 + 27 * q^89 + 6 * q^93 + 19 * q^95 + 11 * q^97 - 4 * q^99

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