LMFDB - Embedded newform 6760.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6760,2,Mod(1,6760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6760, 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("6760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$6760 = 2^{3} \cdot 5 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6760.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:	$53.9788717664$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 520)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6760.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.585786 q^{3} -1.00000 q^{5} +2.00000 q^{7} -2.65685 q^{9} +4.24264 q^{11} +0.585786 q^{15} +0.828427 q^{17} -0.242641 q^{19} -1.17157 q^{21} -9.07107 q^{23} +1.00000 q^{25} +3.31371 q^{27} +1.65685 q^{29} -1.41421 q^{31} -2.48528 q^{33} -2.00000 q^{35} +6.82843 q^{37} -4.82843 q^{41} -10.2426 q^{43} +2.65685 q^{45} -2.00000 q^{47} -3.00000 q^{49} -0.485281 q^{51} -8.82843 q^{53} -4.24264 q^{55} +0.142136 q^{57} +2.58579 q^{59} +15.3137 q^{61} -5.31371 q^{63} +4.82843 q^{67} +5.31371 q^{69} -9.89949 q^{71} -1.17157 q^{73} -0.585786 q^{75} +8.48528 q^{77} +1.17157 q^{79} +6.02944 q^{81} +2.00000 q^{83} -0.828427 q^{85} -0.970563 q^{87} +10.0000 q^{89} +0.828427 q^{93} +0.242641 q^{95} -11.6569 q^{97} -11.2721 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{15} - 4 q^{17} + 8 q^{19} - 8 q^{21} - 4 q^{23} + 2 q^{25} - 16 q^{27} - 8 q^{29} + 12 q^{33} - 4 q^{35} + 8 q^{37} - 4 q^{41} - 12 q^{43} - 6 q^{45} - 4 q^{47}+ \cdots - 48 q^{99}+O(q^{100})$ 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^15 - 4 * q^17 + 8 * q^19 - 8 * q^21 - 4 * q^23 + 2 * q^25 - 16 * q^27 - 8 * q^29 + 12 * q^33 - 4 * q^35 + 8 * q^37 - 4 * q^41 - 12 * q^43 - 6 * q^45 - 4 * q^47 - 6 * q^49 + 16 * q^51 - 12 * q^53 - 28 * q^57 + 8 * q^59 + 8 * q^61 + 12 * q^63 + 4 * q^67 - 12 * q^69 - 8 * q^73 - 4 * q^75 + 8 * q^79 + 46 * q^81 + 4 * q^83 + 4 * q^85 + 32 * q^87 + 20 * q^89 - 4 * q^93 - 8 * q^95 - 12 * q^97 - 48 * 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
$3$	−0.585786	−0.338204	−0.169102	−	0.985599i	$-0.554087\pi$
$3$	−0.585786	−0.338204	−0.169102	+	0.985599i	$0.554087\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	−2.65685	−0.885618
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.585786	0.151249
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	−0.242641	−0.0556656	−0.0278328	−	0.999613i	$-0.508861\pi$
$19$	−0.242641	−0.0556656	−0.0278328	+	0.999613i	$0.508861\pi$
$20$	0	0
$21$	−1.17157	−0.255658
$22$	0	0
$23$	−9.07107	−1.89145	−0.945724	−	0.324970i	$-0.894646\pi$
$23$	−9.07107	−1.89145	−0.945724	+	0.324970i	$0.894646\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.31371	0.637723
$28$	0	0
$29$	1.65685	0.307670	0.153835	−	0.988097i	$-0.450838\pi$
$29$	1.65685	0.307670	0.153835	+	0.988097i	$0.450838\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	0	0
$33$	−2.48528	−0.432632
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	6.82843	1.12259	0.561293	−	0.827617i	$-0.310304\pi$
$37$	6.82843	1.12259	0.561293	+	0.827617i	$0.310304\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.82843	−0.754074	−0.377037	−	0.926198i	$-0.623057\pi$
$41$	−4.82843	−0.754074	−0.377037	+	0.926198i	$0.623057\pi$
$42$	0	0
$43$	−10.2426	−1.56199	−0.780994	−	0.624538i	$-0.785287\pi$
$43$	−10.2426	−1.56199	−0.780994	+	0.624538i	$0.785287\pi$
$44$	0	0
$45$	2.65685	0.396060
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−0.485281	−0.0679530
$52$	0	0
$53$	−8.82843	−1.21268	−0.606339	−	0.795206i	$-0.707362\pi$
$53$	−8.82843	−1.21268	−0.606339	+	0.795206i	$0.707362\pi$
$54$	0	0
$55$	−4.24264	−0.572078
$56$	0	0
$57$	0.142136	0.0188263
$58$	0	0
$59$	2.58579	0.336641	0.168320	−	0.985732i	$-0.446166\pi$
$59$	2.58579	0.336641	0.168320	+	0.985732i	$0.446166\pi$
$60$	0	0
$61$	15.3137	1.96072	0.980360	−	0.197218i	$-0.0631906\pi$
$61$	15.3137	1.96072	0.980360	+	0.197218i	$0.0631906\pi$
$62$	0	0
$63$	−5.31371	−0.669464
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.82843	0.589886	0.294943	−	0.955515i	$-0.404699\pi$
$67$	4.82843	0.589886	0.294943	+	0.955515i	$0.404699\pi$
$68$	0	0
$69$	5.31371	0.639695
$70$	0	0
$71$	−9.89949	−1.17485	−0.587427	−	0.809277i	$-0.699859\pi$
$71$	−9.89949	−1.17485	−0.587427	+	0.809277i	$0.699859\pi$
$72$	0	0
$73$	−1.17157	−0.137122	−0.0685611	−	0.997647i	$-0.521841\pi$
$73$	−1.17157	−0.137122	−0.0685611	+	0.997647i	$0.521841\pi$
$74$	0	0
$75$	−0.585786	−0.0676408
$76$	0	0
$77$	8.48528	0.966988
$78$	0	0
$79$	1.17157	0.131812	0.0659061	−	0.997826i	$-0.479006\pi$
$79$	1.17157	0.131812	0.0659061	+	0.997826i	$0.479006\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	0	0
$87$	−0.970563	−0.104055
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.828427	0.0859039
$94$	0	0
$95$	0.242641	0.0248944
$96$	0	0
$97$	−11.6569	−1.18357	−0.591787	−	0.806094i	$-0.701577\pi$
$97$	−11.6569	−1.18357	−0.591787	+	0.806094i	$0.701577\pi$
$98$	0	0
$99$	−11.2721	−1.13289
$100$	0	0
$101$	−3.65685	−0.363871	−0.181935	−	0.983311i	$-0.558236\pi$
$101$	−3.65685	−0.363871	−0.181935	+	0.983311i	$0.558236\pi$
$102$	0	0
$103$	11.4142	1.12468	0.562338	−	0.826908i	$-0.309902\pi$
$103$	11.4142	1.12468	0.562338	+	0.826908i	$0.309902\pi$
$104$	0	0
$105$	1.17157	0.114334
$106$	0	0
$107$	5.07107	0.490239	0.245119	−	0.969493i	$-0.421173\pi$
$107$	5.07107	0.490239	0.245119	+	0.969493i	$0.421173\pi$
$108$	0	0
$109$	−17.3137	−1.65835	−0.829176	−	0.558987i	$-0.811190\pi$
$109$	−17.3137	−1.65835	−0.829176	+	0.558987i	$0.811190\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	−8.82843	−0.830509	−0.415254	−	0.909705i	$-0.636307\pi$
$113$	−8.82843	−0.830509	−0.415254	+	0.909705i	$0.636307\pi$
$114$	0	0
$115$	9.07107	0.845881
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.65685	0.151884
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	2.82843	0.255031
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.8995	1.41085	0.705426	−	0.708784i	$-0.250756\pi$
$127$	15.8995	1.41085	0.705426	+	0.708784i	$0.250756\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	−19.3137	−1.68745	−0.843723	−	0.536778i	$-0.819641\pi$
$131$	−19.3137	−1.68745	−0.843723	+	0.536778i	$0.819641\pi$
$132$	0	0
$133$	−0.485281	−0.0420792
$134$	0	0
$135$	−3.31371	−0.285199
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−8.48528	−0.719712	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	−8.48528	−0.719712	−0.359856	+	0.933008i	$0.617174\pi$
$140$	0	0
$141$	1.17157	0.0986642
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.65685	−0.137594
$146$	0	0
$147$	1.75736	0.144945
$148$	0	0
$149$	22.9706	1.88182	0.940911	−	0.338654i	$-0.109972\pi$
$149$	22.9706	1.88182	0.940911	+	0.338654i	$0.109972\pi$
$150$	0	0
$151$	−15.0711	−1.22647	−0.613233	−	0.789902i	$-0.710131\pi$
$151$	−15.0711	−1.22647	−0.613233	+	0.789902i	$0.710131\pi$
$152$	0	0
$153$	−2.20101	−0.177941
$154$	0	0
$155$	1.41421	0.113592
$156$	0	0
$157$	−17.3137	−1.38178	−0.690892	−	0.722958i	$-0.742782\pi$
$157$	−17.3137	−1.38178	−0.690892	+	0.722958i	$0.742782\pi$
$158$	0	0
$159$	5.17157	0.410132
$160$	0	0
$161$	−18.1421	−1.42980
$162$	0	0
$163$	5.51472	0.431946	0.215973	−	0.976399i	$-0.430708\pi$
$163$	5.51472	0.431946	0.215973	+	0.976399i	$0.430708\pi$
$164$	0	0
$165$	2.48528	0.193479
$166$	0	0
$167$	−9.31371	−0.720716	−0.360358	−	0.932814i	$-0.617346\pi$
$167$	−9.31371	−0.720716	−0.360358	+	0.932814i	$0.617346\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.644661	0.0492985
$172$	0	0
$173$	−15.1716	−1.15347	−0.576737	−	0.816930i	$-0.695674\pi$
$173$	−15.1716	−1.15347	−0.576737	+	0.816930i	$0.695674\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−1.51472	−0.113853
$178$	0	0
$179$	11.3137	0.845626	0.422813	−	0.906217i	$-0.361043\pi$
$179$	11.3137	0.845626	0.422813	+	0.906217i	$0.361043\pi$
$180$	0	0
$181$	0.686292	0.0510116	0.0255058	−	0.999675i	$-0.491880\pi$
$181$	0.686292	0.0510116	0.0255058	+	0.999675i	$0.491880\pi$
$182$	0	0
$183$	−8.97056	−0.663123
$184$	0	0
$185$	−6.82843	−0.502036
$186$	0	0
$187$	3.51472	0.257022
$188$	0	0
$189$	6.62742	0.482074
$190$	0	0
$191$	−10.3431	−0.748404	−0.374202	−	0.927347i	$-0.622083\pi$
$191$	−10.3431	−0.748404	−0.374202	+	0.927347i	$0.622083\pi$
$192$	0	0
$193$	19.6569	1.41493	0.707466	−	0.706748i	$-0.249838\pi$
$193$	19.6569	1.41493	0.707466	+	0.706748i	$0.249838\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.6569	0.830516	0.415258	−	0.909704i	$-0.363691\pi$
$197$	11.6569	0.830516	0.415258	+	0.909704i	$0.363691\pi$
$198$	0	0
$199$	17.6569	1.25166	0.625831	−	0.779959i	$-0.284760\pi$
$199$	17.6569	1.25166	0.625831	+	0.779959i	$0.284760\pi$
$200$	0	0
$201$	−2.82843	−0.199502
$202$	0	0
$203$	3.31371	0.232577
$204$	0	0
$205$	4.82843	0.337232
$206$	0	0
$207$	24.1005	1.67510
$208$	0	0
$209$	−1.02944	−0.0712077
$210$	0	0
$211$	−5.65685	−0.389434	−0.194717	−	0.980859i	$-0.562379\pi$
$211$	−5.65685	−0.389434	−0.194717	+	0.980859i	$0.562379\pi$
$212$	0	0
$213$	5.79899	0.397340
$214$	0	0
$215$	10.2426	0.698542
$216$	0	0
$217$	−2.82843	−0.192006
$218$	0	0
$219$	0.686292	0.0463753
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.68629	−0.179887	−0.0899437	−	0.995947i	$-0.528669\pi$
$223$	−2.68629	−0.179887	−0.0899437	+	0.995947i	$0.528669\pi$
$224$	0	0
$225$	−2.65685	−0.177124
$226$	0	0
$227$	−15.1716	−1.00697	−0.503486	−	0.864003i	$-0.667950\pi$
$227$	−15.1716	−1.00697	−0.503486	+	0.864003i	$0.667950\pi$
$228$	0	0
$229$	−25.7990	−1.70485	−0.852423	−	0.522853i	$-0.824868\pi$
$229$	−25.7990	−1.70485	−0.852423	+	0.522853i	$0.824868\pi$
$230$	0	0
$231$	−4.97056	−0.327039
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	−0.686292	−0.0445794
$238$	0	0
$239$	16.7279	1.08204	0.541020	−	0.841010i	$-0.318038\pi$
$239$	16.7279	1.08204	0.541020	+	0.841010i	$0.318038\pi$
$240$	0	0
$241$	−12.8284	−0.826352	−0.413176	−	0.910651i	$-0.635581\pi$
$241$	−12.8284	−0.826352	−0.413176	+	0.910651i	$0.635581\pi$
$242$	0	0
$243$	−13.4731	−0.864299
$244$	0	0
$245$	3.00000	0.191663
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.17157	−0.0742454
$250$	0	0
$251$	−18.1421	−1.14512	−0.572561	−	0.819862i	$-0.694050\pi$
$251$	−18.1421	−1.14512	−0.572561	+	0.819862i	$0.694050\pi$
$252$	0	0
$253$	−38.4853	−2.41955
$254$	0	0
$255$	0.485281	0.0303895
$256$	0	0
$257$	−18.9706	−1.18335	−0.591676	−	0.806176i	$-0.701533\pi$
$257$	−18.9706	−1.18335	−0.591676	+	0.806176i	$0.701533\pi$
$258$	0	0
$259$	13.6569	0.848596
$260$	0	0
$261$	−4.40202	−0.272478
$262$	0	0
$263$	22.2426	1.37154	0.685770	−	0.727818i	$-0.259466\pi$
$263$	22.2426	1.37154	0.685770	+	0.727818i	$0.259466\pi$
$264$	0	0
$265$	8.82843	0.542326
$266$	0	0
$267$	−5.85786	−0.358495
$268$	0	0
$269$	−28.6274	−1.74544	−0.872722	−	0.488217i	$-0.837647\pi$
$269$	−28.6274	−1.74544	−0.872722	+	0.488217i	$0.837647\pi$
$270$	0	0
$271$	9.89949	0.601351	0.300676	−	0.953726i	$-0.402788\pi$
$271$	9.89949	0.601351	0.300676	+	0.953726i	$0.402788\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.24264	0.255841
$276$	0	0
$277$	−24.1421	−1.45056	−0.725280	−	0.688454i	$-0.758290\pi$
$277$	−24.1421	−1.45056	−0.725280	+	0.688454i	$0.758290\pi$
$278$	0	0
$279$	3.75736	0.224947
$280$	0	0
$281$	5.51472	0.328981	0.164490	−	0.986379i	$-0.447402\pi$
$281$	5.51472	0.328981	0.164490	+	0.986379i	$0.447402\pi$
$282$	0	0
$283$	−15.8995	−0.945127	−0.472563	−	0.881297i	$-0.656671\pi$
$283$	−15.8995	−0.945127	−0.472563	+	0.881297i	$0.656671\pi$
$284$	0	0
$285$	−0.142136	−0.00841939
$286$	0	0
$287$	−9.65685	−0.570026
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	6.82843	0.400289
$292$	0	0
$293$	2.82843	0.165238	0.0826192	−	0.996581i	$-0.473671\pi$
$293$	2.82843	0.165238	0.0826192	+	0.996581i	$0.473671\pi$
$294$	0	0
$295$	−2.58579	−0.150550
$296$	0	0
$297$	14.0589	0.815779
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−20.4853	−1.18075
$302$	0	0
$303$	2.14214	0.123062
$304$	0	0
$305$	−15.3137	−0.876860
$306$	0	0
$307$	−4.34315	−0.247876	−0.123938	−	0.992290i	$-0.539552\pi$
$307$	−4.34315	−0.247876	−0.123938	+	0.992290i	$0.539552\pi$
$308$	0	0
$309$	−6.68629	−0.380370
$310$	0	0
$311$	18.1421	1.02875	0.514373	−	0.857567i	$-0.328025\pi$
$311$	18.1421	1.02875	0.514373	+	0.857567i	$0.328025\pi$
$312$	0	0
$313$	−12.8284	−0.725106	−0.362553	−	0.931963i	$-0.618095\pi$
$313$	−12.8284	−0.725106	−0.362553	+	0.931963i	$0.618095\pi$
$314$	0	0
$315$	5.31371	0.299394
$316$	0	0
$317$	−7.51472	−0.422069	−0.211034	−	0.977479i	$-0.567683\pi$
$317$	−7.51472	−0.422069	−0.211034	+	0.977479i	$0.567683\pi$
$318$	0	0
$319$	7.02944	0.393573
$320$	0	0
$321$	−2.97056	−0.165801
$322$	0	0
$323$	−0.201010	−0.0111845
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.1421	0.560861
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	15.0711	0.828381	0.414190	−	0.910190i	$-0.364065\pi$
$331$	15.0711	0.828381	0.414190	+	0.910190i	$0.364065\pi$
$332$	0	0
$333$	−18.1421	−0.994183
$334$	0	0
$335$	−4.82843	−0.263805
$336$	0	0
$337$	−7.85786	−0.428045	−0.214023	−	0.976829i	$-0.568657\pi$
$337$	−7.85786	−0.428045	−0.214023	+	0.976829i	$0.568657\pi$
$338$	0	0
$339$	5.17157	0.280881
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−5.31371	−0.286080
$346$	0	0
$347$	−32.3848	−1.73851	−0.869253	−	0.494368i	$-0.835400\pi$
$347$	−32.3848	−1.73851	−0.869253	+	0.494368i	$0.835400\pi$
$348$	0	0
$349$	15.4558	0.827332	0.413666	−	0.910429i	$-0.364248\pi$
$349$	15.4558	0.827332	0.413666	+	0.910429i	$0.364248\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.8284	−0.576339	−0.288170	−	0.957579i	$-0.593047\pi$
$353$	−10.8284	−0.576339	−0.288170	+	0.957579i	$0.593047\pi$
$354$	0	0
$355$	9.89949	0.525411
$356$	0	0
$357$	−0.970563	−0.0513676
$358$	0	0
$359$	−3.55635	−0.187697	−0.0938485	−	0.995586i	$-0.529917\pi$
$359$	−3.55635	−0.187697	−0.0938485	+	0.995586i	$0.529917\pi$
$360$	0	0
$361$	−18.9411	−0.996901
$362$	0	0
$363$	−4.10051	−0.215221
$364$	0	0
$365$	1.17157	0.0613229
$366$	0	0
$367$	2.72792	0.142396	0.0711982	−	0.997462i	$-0.477318\pi$
$367$	2.72792	0.142396	0.0711982	+	0.997462i	$0.477318\pi$
$368$	0	0
$369$	12.8284	0.667821
$370$	0	0
$371$	−17.6569	−0.916698
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0.585786	0.0302499
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7.75736	−0.398469	−0.199234	−	0.979952i	$-0.563846\pi$
$379$	−7.75736	−0.398469	−0.199234	+	0.979952i	$0.563846\pi$
$380$	0	0
$381$	−9.31371	−0.477156
$382$	0	0
$383$	−22.0000	−1.12415	−0.562074	−	0.827087i	$-0.689996\pi$
$383$	−22.0000	−1.12415	−0.562074	+	0.827087i	$0.689996\pi$
$384$	0	0
$385$	−8.48528	−0.432450
$386$	0	0
$387$	27.2132	1.38332
$388$	0	0
$389$	9.31371	0.472224	0.236112	−	0.971726i	$-0.424127\pi$
$389$	9.31371	0.472224	0.236112	+	0.971726i	$0.424127\pi$
$390$	0	0
$391$	−7.51472	−0.380036
$392$	0	0
$393$	11.3137	0.570701
$394$	0	0
$395$	−1.17157	−0.0589482
$396$	0	0
$397$	−6.82843	−0.342709	−0.171354	−	0.985209i	$-0.554814\pi$
$397$	−6.82843	−0.342709	−0.171354	+	0.985209i	$0.554814\pi$
$398$	0	0
$399$	0.284271	0.0142314
$400$	0	0
$401$	16.6274	0.830334	0.415167	−	0.909745i	$-0.363723\pi$
$401$	16.6274	0.830334	0.415167	+	0.909745i	$0.363723\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−6.02944	−0.299605
$406$	0	0
$407$	28.9706	1.43602
$408$	0	0
$409$	0.828427	0.0409631	0.0204815	−	0.999790i	$-0.493480\pi$
$409$	0.828427	0.0409631	0.0204815	+	0.999790i	$0.493480\pi$
$410$	0	0
$411$	1.17157	0.0577894
$412$	0	0
$413$	5.17157	0.254476
$414$	0	0
$415$	−2.00000	−0.0981761
$416$	0	0
$417$	4.97056	0.243410
$418$	0	0
$419$	−5.85786	−0.286175	−0.143088	−	0.989710i	$-0.545703\pi$
$419$	−5.85786	−0.286175	−0.143088	+	0.989710i	$0.545703\pi$
$420$	0	0
$421$	−24.3431	−1.18641	−0.593206	−	0.805051i	$-0.702138\pi$
$421$	−24.3431	−1.18641	−0.593206	+	0.805051i	$0.702138\pi$
$422$	0	0
$423$	5.31371	0.258361
$424$	0	0
$425$	0.828427	0.0401846
$426$	0	0
$427$	30.6274	1.48216
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.2132	1.40715	0.703575	−	0.710621i	$-0.251586\pi$
$431$	29.2132	1.40715	0.703575	+	0.710621i	$0.251586\pi$
$432$	0	0
$433$	26.9706	1.29612	0.648061	−	0.761588i	$-0.275580\pi$
$433$	26.9706	1.29612	0.648061	+	0.761588i	$0.275580\pi$
$434$	0	0
$435$	0.970563	0.0465349
$436$	0	0
$437$	2.20101	0.105289
$438$	0	0
$439$	−28.2843	−1.34993	−0.674967	−	0.737848i	$-0.735842\pi$
$439$	−28.2843	−1.34993	−0.674967	+	0.737848i	$0.735842\pi$
$440$	0	0
$441$	7.97056	0.379551
$442$	0	0
$443$	−0.585786	−0.0278316	−0.0139158	−	0.999903i	$-0.504430\pi$
$443$	−0.585786	−0.0278316	−0.0139158	+	0.999903i	$0.504430\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	−13.4558	−0.636440
$448$	0	0
$449$	−36.1421	−1.70565	−0.852826	−	0.522195i	$-0.825114\pi$
$449$	−36.1421	−1.70565	−0.852826	+	0.522195i	$0.825114\pi$
$450$	0	0
$451$	−20.4853	−0.964614
$452$	0	0
$453$	8.82843	0.414796
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.6274	−0.590686	−0.295343	−	0.955391i	$-0.595434\pi$
$457$	−12.6274	−0.590686	−0.295343	+	0.955391i	$0.595434\pi$
$458$	0	0
$459$	2.74517	0.128133
$460$	0	0
$461$	−8.14214	−0.379217	−0.189609	−	0.981860i	$-0.560722\pi$
$461$	−8.14214	−0.379217	−0.189609	+	0.981860i	$0.560722\pi$
$462$	0	0
$463$	−5.51472	−0.256291	−0.128145	−	0.991755i	$-0.540902\pi$
$463$	−5.51472	−0.256291	−0.128145	+	0.991755i	$0.540902\pi$
$464$	0	0
$465$	−0.828427	−0.0384174
$466$	0	0
$467$	5.75736	0.266419	0.133209	−	0.991088i	$-0.457472\pi$
$467$	5.75736	0.266419	0.133209	+	0.991088i	$0.457472\pi$
$468$	0	0
$469$	9.65685	0.445912
$470$	0	0
$471$	10.1421	0.467325
$472$	0	0
$473$	−43.4558	−1.99810
$474$	0	0
$475$	−0.242641	−0.0111331
$476$	0	0
$477$	23.4558	1.07397
$478$	0	0
$479$	23.7574	1.08550	0.542751	−	0.839894i	$-0.317383\pi$
$479$	23.7574	1.08550	0.542751	+	0.839894i	$0.317383\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	10.6274	0.483564
$484$	0	0
$485$	11.6569	0.529310
$486$	0	0
$487$	10.4853	0.475133	0.237567	−	0.971371i	$-0.423650\pi$
$487$	10.4853	0.475133	0.237567	+	0.971371i	$0.423650\pi$
$488$	0	0
$489$	−3.23045	−0.146086
$490$	0	0
$491$	43.1127	1.94565	0.972824	−	0.231544i	$-0.0743777\pi$
$491$	43.1127	1.94565	0.972824	+	0.231544i	$0.0743777\pi$
$492$	0	0
$493$	1.37258	0.0618180
$494$	0	0
$495$	11.2721	0.506642
$496$	0	0
$497$	−19.7990	−0.888106
$498$	0	0
$499$	29.4142	1.31676	0.658381	−	0.752685i	$-0.271242\pi$
$499$	29.4142	1.31676	0.658381	+	0.752685i	$0.271242\pi$
$500$	0	0
$501$	5.45584	0.243749
$502$	0	0
$503$	−24.5858	−1.09623	−0.548113	−	0.836404i	$-0.684654\pi$
$503$	−24.5858	−1.09623	−0.548113	+	0.836404i	$0.684654\pi$
$504$	0	0
$505$	3.65685	0.162728
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.4853	−1.52853	−0.764267	−	0.644900i	$-0.776899\pi$
$509$	−34.4853	−1.52853	−0.764267	+	0.644900i	$0.776899\pi$
$510$	0	0
$511$	−2.34315	−0.103655
$512$	0	0
$513$	−0.804041	−0.0354993
$514$	0	0
$515$	−11.4142	−0.502970
$516$	0	0
$517$	−8.48528	−0.373182
$518$	0	0
$519$	8.88730	0.390109
$520$	0	0
$521$	−35.3137	−1.54712	−0.773561	−	0.633722i	$-0.781526\pi$
$521$	−35.3137	−1.54712	−0.773561	+	0.633722i	$0.781526\pi$
$522$	0	0
$523$	35.6985	1.56099	0.780493	−	0.625165i	$-0.214968\pi$
$523$	35.6985	1.56099	0.780493	+	0.625165i	$0.214968\pi$
$524$	0	0
$525$	−1.17157	−0.0511316
$526$	0	0
$527$	−1.17157	−0.0510345
$528$	0	0
$529$	59.2843	2.57758
$530$	0	0
$531$	−6.87006	−0.298135
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−5.07107	−0.219241
$536$	0	0
$537$	−6.62742	−0.285994
$538$	0	0
$539$	−12.7279	−0.548230
$540$	0	0
$541$	2.48528	0.106851	0.0534253	−	0.998572i	$-0.482986\pi$
$541$	2.48528	0.106851	0.0534253	+	0.998572i	$0.482986\pi$
$542$	0	0
$543$	−0.402020	−0.0172523
$544$	0	0
$545$	17.3137	0.741638
$546$	0	0
$547$	−18.9289	−0.809343	−0.404671	−	0.914462i	$-0.632614\pi$
$547$	−18.9289	−0.809343	−0.404671	+	0.914462i	$0.632614\pi$
$548$	0	0
$549$	−40.6863	−1.73645
$550$	0	0
$551$	−0.402020	−0.0171266
$552$	0	0
$553$	2.34315	0.0996407
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−13.8579	−0.587177	−0.293588	−	0.955932i	$-0.594849\pi$
$557$	−13.8579	−0.587177	−0.293588	+	0.955932i	$0.594849\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.05887	−0.0869257
$562$	0	0
$563$	−13.5563	−0.571332	−0.285666	−	0.958329i	$-0.592215\pi$
$563$	−13.5563	−0.571332	−0.285666	+	0.958329i	$0.592215\pi$
$564$	0	0
$565$	8.82843	0.371415
$566$	0	0
$567$	12.0589	0.506425
$568$	0	0
$569$	−8.68629	−0.364148	−0.182074	−	0.983285i	$-0.558281\pi$
$569$	−8.68629	−0.364148	−0.182074	+	0.983285i	$0.558281\pi$
$570$	0	0
$571$	−29.1716	−1.22079	−0.610396	−	0.792096i	$-0.708990\pi$
$571$	−29.1716	−1.22079	−0.610396	+	0.792096i	$0.708990\pi$
$572$	0	0
$573$	6.05887	0.253113
$574$	0	0
$575$	−9.07107	−0.378290
$576$	0	0
$577$	−5.85786	−0.243866	−0.121933	−	0.992538i	$-0.538909\pi$
$577$	−5.85786	−0.243866	−0.121933	+	0.992538i	$0.538909\pi$
$578$	0	0
$579$	−11.5147	−0.478535
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	−37.4558	−1.55126
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.48528	−0.102579	−0.0512893	−	0.998684i	$-0.516333\pi$
$587$	−2.48528	−0.102579	−0.0512893	+	0.998684i	$0.516333\pi$
$588$	0	0
$589$	0.343146	0.0141391
$590$	0	0
$591$	−6.82843	−0.280884
$592$	0	0
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	−1.65685	−0.0679244
$596$	0	0
$597$	−10.3431	−0.423317
$598$	0	0
$599$	−11.5147	−0.470479	−0.235239	−	0.971937i	$-0.575587\pi$
$599$	−11.5147	−0.470479	−0.235239	+	0.971937i	$0.575587\pi$
$600$	0	0
$601$	35.9411	1.46607	0.733035	−	0.680191i	$-0.238103\pi$
$601$	35.9411	1.46607	0.733035	+	0.680191i	$0.238103\pi$
$602$	0	0
$603$	−12.8284	−0.522414
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−6.44365	−0.261540	−0.130770	−	0.991413i	$-0.541745\pi$
$607$	−6.44365	−0.261540	−0.130770	+	0.991413i	$0.541745\pi$
$608$	0	0
$609$	−1.94113	−0.0786584
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	0	0
$615$	−2.82843	−0.114053
$616$	0	0
$617$	3.65685	0.147219	0.0736097	−	0.997287i	$-0.476548\pi$
$617$	3.65685	0.147219	0.0736097	+	0.997287i	$0.476548\pi$
$618$	0	0
$619$	−36.0416	−1.44864	−0.724318	−	0.689466i	$-0.757845\pi$
$619$	−36.0416	−1.44864	−0.724318	+	0.689466i	$0.757845\pi$
$620$	0	0
$621$	−30.0589	−1.20622
$622$	0	0
$623$	20.0000	0.801283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.603030	0.0240827
$628$	0	0
$629$	5.65685	0.225554
$630$	0	0
$631$	12.7279	0.506691	0.253345	−	0.967376i	$-0.418469\pi$
$631$	12.7279	0.506691	0.253345	+	0.967376i	$0.418469\pi$
$632$	0	0
$633$	3.31371	0.131708
$634$	0	0
$635$	−15.8995	−0.630952
$636$	0	0
$637$	0	0
$638$	0	0
$639$	26.3015	1.04047
$640$	0	0
$641$	28.3431	1.11949	0.559743	−	0.828666i	$-0.310900\pi$
$641$	28.3431	1.11949	0.559743	+	0.828666i	$0.310900\pi$
$642$	0	0
$643$	13.3137	0.525041	0.262521	−	0.964926i	$-0.415446\pi$
$643$	13.3137	0.525041	0.262521	+	0.964926i	$0.415446\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	−37.3553	−1.46859	−0.734295	−	0.678831i	$-0.762487\pi$
$647$	−37.3553	−1.46859	−0.734295	+	0.678831i	$0.762487\pi$
$648$	0	0
$649$	10.9706	0.430632
$650$	0	0
$651$	1.65685	0.0649372
$652$	0	0
$653$	−9.02944	−0.353349	−0.176675	−	0.984269i	$-0.556534\pi$
$653$	−9.02944	−0.353349	−0.176675	+	0.984269i	$0.556534\pi$
$654$	0	0
$655$	19.3137	0.754649
$656$	0	0
$657$	3.11270	0.121438
$658$	0	0
$659$	4.48528	0.174722	0.0873609	−	0.996177i	$-0.472157\pi$
$659$	4.48528	0.174722	0.0873609	+	0.996177i	$0.472157\pi$
$660$	0	0
$661$	9.02944	0.351204	0.175602	−	0.984461i	$-0.443813\pi$
$661$	9.02944	0.351204	0.175602	+	0.984461i	$0.443813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.485281	0.0188184
$666$	0	0
$667$	−15.0294	−0.581942
$668$	0	0
$669$	1.57359	0.0608386
$670$	0	0
$671$	64.9706	2.50816
$672$	0	0
$673$	33.7990	1.30286	0.651428	−	0.758711i	$-0.274170\pi$
$673$	33.7990	1.30286	0.651428	+	0.758711i	$0.274170\pi$
$674$	0	0
$675$	3.31371	0.127545
$676$	0	0
$677$	−20.1421	−0.774125	−0.387063	−	0.922053i	$-0.626510\pi$
$677$	−20.1421	−0.774125	−0.387063	+	0.922053i	$0.626510\pi$
$678$	0	0
$679$	−23.3137	−0.894698
$680$	0	0
$681$	8.88730	0.340562
$682$	0	0
$683$	−37.1127	−1.42008	−0.710039	−	0.704162i	$-0.751323\pi$
$683$	−37.1127	−1.42008	−0.710039	+	0.704162i	$0.751323\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	15.1127	0.576585
$688$	0	0
$689$	0	0
$690$	0	0
$691$	49.4142	1.87981	0.939903	−	0.341443i	$-0.110915\pi$
$691$	49.4142	1.87981	0.939903	+	0.341443i	$0.110915\pi$
$692$	0	0
$693$	−22.5442	−0.856382
$694$	0	0
$695$	8.48528	0.321865
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	12.8873	0.487443
$700$	0	0
$701$	−20.6274	−0.779087	−0.389543	−	0.921008i	$-0.627367\pi$
$701$	−20.6274	−0.779087	−0.389543	+	0.921008i	$0.627367\pi$
$702$	0	0
$703$	−1.65685	−0.0624894
$704$	0	0
$705$	−1.17157	−0.0441240
$706$	0	0
$707$	−7.31371	−0.275060
$708$	0	0
$709$	19.4558	0.730680	0.365340	−	0.930874i	$-0.380953\pi$
$709$	19.4558	0.730680	0.365340	+	0.930874i	$0.380953\pi$
$710$	0	0
$711$	−3.11270	−0.116735
$712$	0	0
$713$	12.8284	0.480428
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.79899	−0.365950
$718$	0	0
$719$	−18.6274	−0.694685	−0.347343	−	0.937738i	$-0.612916\pi$
$719$	−18.6274	−0.694685	−0.347343	+	0.937738i	$0.612916\pi$
$720$	0	0
$721$	22.8284	0.850175
$722$	0	0
$723$	7.51472	0.279475
$724$	0	0
$725$	1.65685	0.0615340
$726$	0	0
$727$	−2.24264	−0.0831749	−0.0415875	−	0.999135i	$-0.513242\pi$
$727$	−2.24264	−0.0831749	−0.0415875	+	0.999135i	$0.513242\pi$
$728$	0	0
$729$	−10.1960	−0.377628
$730$	0	0
$731$	−8.48528	−0.313839
$732$	0	0
$733$	−32.6274	−1.20512	−0.602561	−	0.798073i	$-0.705853\pi$
$733$	−32.6274	−1.20512	−0.602561	+	0.798073i	$0.705853\pi$
$734$	0	0
$735$	−1.75736	−0.0648212
$736$	0	0
$737$	20.4853	0.754585
$738$	0	0
$739$	−18.5858	−0.683689	−0.341845	−	0.939756i	$-0.611052\pi$
$739$	−18.5858	−0.683689	−0.341845	+	0.939756i	$0.611052\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30.9706	−1.13620	−0.568100	−	0.822960i	$-0.692321\pi$
$743$	−30.9706	−1.13620	−0.568100	+	0.822960i	$0.692321\pi$
$744$	0	0
$745$	−22.9706	−0.841576
$746$	0	0
$747$	−5.31371	−0.194418
$748$	0	0
$749$	10.1421	0.370586
$750$	0	0
$751$	8.48528	0.309632	0.154816	−	0.987943i	$-0.450521\pi$
$751$	8.48528	0.309632	0.154816	+	0.987943i	$0.450521\pi$
$752$	0	0
$753$	10.6274	0.387285
$754$	0	0
$755$	15.0711	0.548492
$756$	0	0
$757$	−8.14214	−0.295931	−0.147965	−	0.988993i	$-0.547272\pi$
$757$	−8.14214	−0.295931	−0.147965	+	0.988993i	$0.547272\pi$
$758$	0	0
$759$	22.5442	0.818301
$760$	0	0
$761$	41.3137	1.49762	0.748810	−	0.662784i	$-0.230625\pi$
$761$	41.3137	1.49762	0.748810	+	0.662784i	$0.230625\pi$
$762$	0	0
$763$	−34.6274	−1.25360
$764$	0	0
$765$	2.20101	0.0795777
$766$	0	0
$767$	0	0
$768$	0	0
$769$	19.6569	0.708844	0.354422	−	0.935086i	$-0.384678\pi$
$769$	19.6569	0.708844	0.354422	+	0.935086i	$0.384678\pi$
$770$	0	0
$771$	11.1127	0.400214
$772$	0	0
$773$	−18.1421	−0.652527	−0.326264	−	0.945279i	$-0.605790\pi$
$773$	−18.1421	−0.652527	−0.326264	+	0.945279i	$0.605790\pi$
$774$	0	0
$775$	−1.41421	−0.0508001
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	1.17157	0.0419760
$780$	0	0
$781$	−42.0000	−1.50288
$782$	0	0
$783$	5.49033	0.196208
$784$	0	0
$785$	17.3137	0.617953
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	0	0
$789$	−13.0294	−0.463860
$790$	0	0
$791$	−17.6569	−0.627805
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−5.17157	−0.183417
$796$	0	0
$797$	49.5980	1.75685	0.878425	−	0.477880i	$-0.158595\pi$
$797$	49.5980	1.75685	0.878425	+	0.477880i	$0.158595\pi$
$798$	0	0
$799$	−1.65685	−0.0586153
$800$	0	0
$801$	−26.5685	−0.938753
$802$	0	0
$803$	−4.97056	−0.175407
$804$	0	0
$805$	18.1421	0.639426
$806$	0	0
$807$	16.7696	0.590316
$808$	0	0
$809$	−41.6569	−1.46458	−0.732289	−	0.680995i	$-0.761548\pi$
$809$	−41.6569	−1.46458	−0.732289	+	0.680995i	$0.761548\pi$
$810$	0	0
$811$	−9.89949	−0.347618	−0.173809	−	0.984779i	$-0.555608\pi$
$811$	−9.89949	−0.347618	−0.173809	+	0.984779i	$0.555608\pi$
$812$	0	0
$813$	−5.79899	−0.203379
$814$	0	0
$815$	−5.51472	−0.193172
$816$	0	0
$817$	2.48528	0.0869490
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.68629	−0.233353	−0.116677	−	0.993170i	$-0.537224\pi$
$821$	−6.68629	−0.233353	−0.116677	+	0.993170i	$0.537224\pi$
$822$	0	0
$823$	−16.8701	−0.588053	−0.294027	−	0.955797i	$-0.594995\pi$
$823$	−16.8701	−0.588053	−0.294027	+	0.955797i	$0.594995\pi$
$824$	0	0
$825$	−2.48528	−0.0865264
$826$	0	0
$827$	−20.6274	−0.717286	−0.358643	−	0.933475i	$-0.616760\pi$
$827$	−20.6274	−0.717286	−0.358643	+	0.933475i	$0.616760\pi$
$828$	0	0
$829$	14.3431	0.498158	0.249079	−	0.968483i	$-0.419872\pi$
$829$	14.3431	0.498158	0.249079	+	0.968483i	$0.419872\pi$
$830$	0	0
$831$	14.1421	0.490585
$832$	0	0
$833$	−2.48528	−0.0861099
$834$	0	0
$835$	9.31371	0.322314
$836$	0	0
$837$	−4.68629	−0.161982
$838$	0	0
$839$	37.8995	1.30844	0.654218	−	0.756306i	$-0.272998\pi$
$839$	37.8995	1.30844	0.654218	+	0.756306i	$0.272998\pi$
$840$	0	0
$841$	−26.2548	−0.905339
$842$	0	0
$843$	−3.23045	−0.111263
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	9.31371	0.319646
$850$	0	0
$851$	−61.9411	−2.12331
$852$	0	0
$853$	47.1127	1.61311	0.806554	−	0.591160i	$-0.201330\pi$
$853$	47.1127	1.61311	0.806554	+	0.591160i	$0.201330\pi$
$854$	0	0
$855$	−0.644661	−0.0220469
$856$	0	0
$857$	−52.9117	−1.80743	−0.903714	−	0.428136i	$-0.859170\pi$
$857$	−52.9117	−1.80743	−0.903714	+	0.428136i	$0.859170\pi$
$858$	0	0
$859$	54.8284	1.87072	0.935361	−	0.353695i	$-0.115075\pi$
$859$	54.8284	1.87072	0.935361	+	0.353695i	$0.115075\pi$
$860$	0	0
$861$	5.65685	0.192785
$862$	0	0
$863$	−53.5980	−1.82450	−0.912248	−	0.409638i	$-0.865655\pi$
$863$	−53.5980	−1.82450	−0.912248	+	0.409638i	$0.865655\pi$
$864$	0	0
$865$	15.1716	0.515849
$866$	0	0
$867$	9.55635	0.324551
$868$	0	0
$869$	4.97056	0.168615
$870$	0	0
$871$	0	0
$872$	0	0
$873$	30.9706	1.04819
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	29.3137	0.989854	0.494927	−	0.868935i	$-0.335195\pi$
$877$	29.3137	0.989854	0.494927	+	0.868935i	$0.335195\pi$
$878$	0	0
$879$	−1.65685	−0.0558843
$880$	0	0
$881$	25.9411	0.873979	0.436989	−	0.899467i	$-0.356045\pi$
$881$	25.9411	0.873979	0.436989	+	0.899467i	$0.356045\pi$
$882$	0	0
$883$	−18.2426	−0.613914	−0.306957	−	0.951723i	$-0.599311\pi$
$883$	−18.2426	−0.613914	−0.306957	+	0.951723i	$0.599311\pi$
$884$	0	0
$885$	1.51472	0.0509167
$886$	0	0
$887$	30.0416	1.00870	0.504350	−	0.863500i	$-0.331732\pi$
$887$	30.0416	1.00870	0.504350	+	0.863500i	$0.331732\pi$
$888$	0	0
$889$	31.7990	1.06650
$890$	0	0
$891$	25.5807	0.856987
$892$	0	0
$893$	0.485281	0.0162393
$894$	0	0
$895$	−11.3137	−0.378176
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.34315	−0.0781483
$900$	0	0
$901$	−7.31371	−0.243655
$902$	0	0
$903$	12.0000	0.399335
$904$	0	0
$905$	−0.686292	−0.0228131
$906$	0	0
$907$	1.07107	0.0355642	0.0177821	−	0.999842i	$-0.494339\pi$
$907$	1.07107	0.0355642	0.0177821	+	0.999842i	$0.494339\pi$
$908$	0	0
$909$	9.71573	0.322250
$910$	0	0
$911$	51.5980	1.70952	0.854759	−	0.519026i	$-0.173705\pi$
$911$	51.5980	1.70952	0.854759	+	0.519026i	$0.173705\pi$
$912$	0	0
$913$	8.48528	0.280822
$914$	0	0
$915$	8.97056	0.296558
$916$	0	0
$917$	−38.6274	−1.27559
$918$	0	0
$919$	−3.11270	−0.102678	−0.0513392	−	0.998681i	$-0.516349\pi$
$919$	−3.11270	−0.102678	−0.0513392	+	0.998681i	$0.516349\pi$
$920$	0	0
$921$	2.54416	0.0838328
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.82843	0.224517
$926$	0	0
$927$	−30.3259	−0.996033
$928$	0	0
$929$	41.1127	1.34886	0.674432	−	0.738337i	$-0.264389\pi$
$929$	41.1127	1.34886	0.674432	+	0.738337i	$0.264389\pi$
$930$	0	0
$931$	0.727922	0.0238567
$932$	0	0
$933$	−10.6274	−0.347926
$934$	0	0
$935$	−3.51472	−0.114944
$936$	0	0
$937$	46.2843	1.51204	0.756021	−	0.654548i	$-0.227141\pi$
$937$	46.2843	1.51204	0.756021	+	0.654548i	$0.227141\pi$
$938$	0	0
$939$	7.51472	0.245234
$940$	0	0
$941$	18.4853	0.602603	0.301301	−	0.953529i	$-0.402579\pi$
$941$	18.4853	0.602603	0.301301	+	0.953529i	$0.402579\pi$
$942$	0	0
$943$	43.7990	1.42629
$944$	0	0
$945$	−6.62742	−0.215590
$946$	0	0
$947$	−46.9706	−1.52634	−0.763169	−	0.646199i	$-0.776358\pi$
$947$	−46.9706	−1.52634	−0.763169	+	0.646199i	$0.776358\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	4.40202	0.142745
$952$	0	0
$953$	−19.9411	−0.645956	−0.322978	−	0.946406i	$-0.604684\pi$
$953$	−19.9411	−0.645956	−0.322978	+	0.946406i	$0.604684\pi$
$954$	0	0
$955$	10.3431	0.334696
$956$	0	0
$957$	−4.11775	−0.133108
$958$	0	0
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	−13.4731	−0.434164
$964$	0	0
$965$	−19.6569	−0.632777
$966$	0	0
$967$	44.8284	1.44159	0.720793	−	0.693151i	$-0.243778\pi$
$967$	44.8284	1.44159	0.720793	+	0.693151i	$0.243778\pi$
$968$	0	0
$969$	0.117749	0.00378264
$970$	0	0
$971$	−6.62742	−0.212684	−0.106342	−	0.994330i	$-0.533914\pi$
$971$	−6.62742	−0.212684	−0.106342	+	0.994330i	$0.533914\pi$
$972$	0	0
$973$	−16.9706	−0.544051
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−62.4264	−1.99720	−0.998599	−	0.0529182i	$-0.983148\pi$
$977$	−62.4264	−1.99720	−0.998599	+	0.0529182i	$0.983148\pi$
$978$	0	0
$979$	42.4264	1.35595
$980$	0	0
$981$	46.0000	1.46867
$982$	0	0
$983$	34.4853	1.09991	0.549955	−	0.835194i	$-0.314645\pi$
$983$	34.4853	1.09991	0.549955	+	0.835194i	$0.314645\pi$
$984$	0	0
$985$	−11.6569	−0.371418
$986$	0	0
$987$	2.34315	0.0745832
$988$	0	0
$989$	92.9117	2.95442
$990$	0	0
$991$	55.5980	1.76613	0.883064	−	0.469253i	$-0.155477\pi$
$991$	55.5980	1.76613	0.883064	+	0.469253i	$0.155477\pi$
$992$	0	0
$993$	−8.82843	−0.280162
$994$	0	0
$995$	−17.6569	−0.559760
$996$	0	0
$997$	−3.17157	−0.100445	−0.0502224	−	0.998738i	$-0.515993\pi$
$997$	−3.17157	−0.100445	−0.0502224	+	0.998738i	$0.515993\pi$
$998$	0	0
$999$	22.6274	0.715900

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6760.2.a.n.1.2		2
13.12	even	2		520.2.a.c.1.2	✓	2
39.38	odd	2		4680.2.a.w.1.2		2
52.51	odd	2		1040.2.a.n.1.1		2
65.12	odd	4		2600.2.d.i.1249.3		4
65.38	odd	4		2600.2.d.i.1249.2		4
65.64	even	2		2600.2.a.w.1.1		2
104.51	odd	2		4160.2.a.u.1.2		2
104.77	even	2		4160.2.a.bn.1.1		2
156.155	even	2		9360.2.a.ck.1.1		2
260.259	odd	2		5200.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.a.c.1.2	✓	2	13.12	even	2
1040.2.a.n.1.1		2	52.51	odd	2
2600.2.a.w.1.1		2	65.64	even	2
2600.2.d.i.1249.2		4	65.38	odd	4
2600.2.d.i.1249.3		4	65.12	odd	4
4160.2.a.u.1.2		2	104.51	odd	2
4160.2.a.bn.1.1		2	104.77	even	2
4680.2.a.w.1.2		2	39.38	odd	2
5200.2.a.bl.1.2		2	260.259	odd	2
6760.2.a.n.1.2		2	1.1	even	1	trivial
9360.2.a.ck.1.1		2	156.155	even	2

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Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	6760.2.a.n.1.2

Level	$6760$
Weight	$2$
Character	6760.1
Self dual	yes
Analytic conductor	$53.979$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$