LMFDB - Embedded newform 7616.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, 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("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.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:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 238)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{3} +1.23607 q^{5} +1.00000 q^{7} +7.47214 q^{9} +O(q^{10})$	$q+3.23607 q^{3} +1.23607 q^{5} +1.00000 q^{7} +7.47214 q^{9} +5.23607 q^{11} +2.47214 q^{13} +4.00000 q^{15} +1.00000 q^{17} -8.47214 q^{19} +3.23607 q^{21} -8.00000 q^{23} -3.47214 q^{25} +14.4721 q^{27} +5.70820 q^{29} +1.52786 q^{31} +16.9443 q^{33} +1.23607 q^{35} +3.23607 q^{37} +8.00000 q^{39} -3.52786 q^{41} +2.47214 q^{43} +9.23607 q^{45} +2.47214 q^{47} +1.00000 q^{49} +3.23607 q^{51} +4.47214 q^{53} +6.47214 q^{55} -27.4164 q^{57} -6.00000 q^{59} -1.23607 q^{61} +7.47214 q^{63} +3.05573 q^{65} -1.52786 q^{67} -25.8885 q^{69} +2.47214 q^{71} +13.4164 q^{73} -11.2361 q^{75} +5.23607 q^{77} +10.4721 q^{79} +24.4164 q^{81} +2.00000 q^{83} +1.23607 q^{85} +18.4721 q^{87} +2.00000 q^{89} +2.47214 q^{91} +4.94427 q^{93} -10.4721 q^{95} -8.47214 q^{97} +39.1246 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} + 6 q^{11} - 4 q^{13} + 8 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 16 q^{23} + 2 q^{25} + 20 q^{27} - 2 q^{29} + 12 q^{31} + 16 q^{33} - 2 q^{35} + 2 q^{37} + 16 q^{39} - 16 q^{41} - 4 q^{43} + 14 q^{45} - 4 q^{47} + 2 q^{49} + 2 q^{51} + 4 q^{55} - 28 q^{57} - 12 q^{59} + 2 q^{61} + 6 q^{63} + 24 q^{65} - 12 q^{67} - 16 q^{69} - 4 q^{71} - 18 q^{75} + 6 q^{77} + 12 q^{79} + 22 q^{81} + 4 q^{83} - 2 q^{85} + 28 q^{87} + 4 q^{89} - 4 q^{91} - 8 q^{93} - 12 q^{95} - 8 q^{97} + 38 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9 + 6 * q^11 - 4 * q^13 + 8 * q^15 + 2 * q^17 - 8 * q^19 + 2 * q^21 - 16 * q^23 + 2 * q^25 + 20 * q^27 - 2 * q^29 + 12 * q^31 + 16 * q^33 - 2 * q^35 + 2 * q^37 + 16 * q^39 - 16 * q^41 - 4 * q^43 + 14 * q^45 - 4 * q^47 + 2 * q^49 + 2 * q^51 + 4 * q^55 - 28 * q^57 - 12 * q^59 + 2 * q^61 + 6 * q^63 + 24 * q^65 - 12 * q^67 - 16 * q^69 - 4 * q^71 - 18 * q^75 + 6 * q^77 + 12 * q^79 + 22 * q^81 + 4 * q^83 - 2 * q^85 + 28 * q^87 + 4 * q^89 - 4 * q^91 - 8 * q^93 - 12 * q^95 - 8 * q^97 + 38 * 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
$3$	3.23607	1.86834	0.934172	−	0.356822i	$-0.116140\pi$
$3$	3.23607	1.86834	0.934172	+	0.356822i	$0.116140\pi$
$4$	0	0
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	5.23607	1.57873	0.789367	−	0.613922i	$-0.210409\pi$
$11$	5.23607	1.57873	0.789367	+	0.613922i	$0.210409\pi$
$12$	0	0
$13$	2.47214	0.685647	0.342824	−	0.939400i	$-0.388617\pi$
$13$	2.47214	0.685647	0.342824	+	0.939400i	$0.388617\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−8.47214	−1.94364	−0.971821	−	0.235722i	$-0.924255\pi$
$19$	−8.47214	−1.94364	−0.971821	+	0.235722i	$0.924255\pi$
$20$	0	0
$21$	3.23607	0.706168
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	14.4721	2.78516
$28$	0	0
$29$	5.70820	1.05999	0.529993	−	0.848002i	$-0.322194\pi$
$29$	5.70820	1.05999	0.529993	+	0.848002i	$0.322194\pi$
$30$	0	0
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	0	0
$33$	16.9443	2.94962
$34$	0	0
$35$	1.23607	0.208934
$36$	0	0
$37$	3.23607	0.532006	0.266003	−	0.963972i	$-0.414297\pi$
$37$	3.23607	0.532006	0.266003	+	0.963972i	$0.414297\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	−3.52786	−0.550960	−0.275480	−	0.961307i	$-0.588837\pi$
$41$	−3.52786	−0.550960	−0.275480	+	0.961307i	$0.588837\pi$
$42$	0	0
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	0	0
$45$	9.23607	1.37683
$46$	0	0
$47$	2.47214	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.23607	0.453140
$52$	0	0
$53$	4.47214	0.614295	0.307148	−	0.951662i	$-0.400625\pi$
$53$	4.47214	0.614295	0.307148	+	0.951662i	$0.400625\pi$
$54$	0	0
$55$	6.47214	0.872703
$56$	0	0
$57$	−27.4164	−3.63139
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−1.23607	−0.158262	−0.0791311	−	0.996864i	$-0.525215\pi$
$61$	−1.23607	−0.158262	−0.0791311	+	0.996864i	$0.525215\pi$
$62$	0	0
$63$	7.47214	0.941401
$64$	0	0
$65$	3.05573	0.379016
$66$	0	0
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	0	0
$69$	−25.8885	−3.11661
$70$	0	0
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	13.4164	1.57027	0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164	1.57027	0.785136	+	0.619324i	$0.212593\pi$
$74$	0	0
$75$	−11.2361	−1.29743
$76$	0	0
$77$	5.23607	0.596705
$78$	0	0
$79$	10.4721	1.17821	0.589104	−	0.808057i	$-0.299481\pi$
$79$	10.4721	1.17821	0.589104	+	0.808057i	$0.299481\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	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
$85$	1.23607	0.134070
$86$	0	0
$87$	18.4721	1.98042
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	2.47214	0.259150
$92$	0	0
$93$	4.94427	0.512697
$94$	0	0
$95$	−10.4721	−1.07442
$96$	0	0
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	0	0
$99$	39.1246	3.93217
$100$	0	0
$101$	6.47214	0.644002	0.322001	−	0.946739i	$-0.395645\pi$
$101$	6.47214	0.644002	0.322001	+	0.946739i	$0.395645\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−1.23607	−0.119495	−0.0597476	−	0.998214i	$-0.519030\pi$
$107$	−1.23607	−0.119495	−0.0597476	+	0.998214i	$0.519030\pi$
$108$	0	0
$109$	−12.1803	−1.16666	−0.583332	−	0.812233i	$-0.698252\pi$
$109$	−12.1803	−1.16666	−0.583332	+	0.812233i	$0.698252\pi$
$110$	0	0
$111$	10.4721	0.993971
$112$	0	0
$113$	13.4164	1.26211	0.631055	−	0.775738i	$-0.282622\pi$
$113$	13.4164	1.26211	0.631055	+	0.775738i	$0.282622\pi$
$114$	0	0
$115$	−9.88854	−0.922111
$116$	0	0
$117$	18.4721	1.70775
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	16.4164	1.49240
$122$	0	0
$123$	−11.4164	−1.02938
$124$	0	0
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	1.70820	0.149246	0.0746232	−	0.997212i	$-0.476225\pi$
$131$	1.70820	0.149246	0.0746232	+	0.997212i	$0.476225\pi$
$132$	0	0
$133$	−8.47214	−0.734627
$134$	0	0
$135$	17.8885	1.53960
$136$	0	0
$137$	3.52786	0.301406	0.150703	−	0.988579i	$-0.451846\pi$
$137$	3.52786	0.301406	0.150703	+	0.988579i	$0.451846\pi$
$138$	0	0
$139$	−12.1803	−1.03312	−0.516561	−	0.856250i	$-0.672788\pi$
$139$	−12.1803	−1.03312	−0.516561	+	0.856250i	$0.672788\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	12.9443	1.08245
$144$	0	0
$145$	7.05573	0.585946
$146$	0	0
$147$	3.23607	0.266906
$148$	0	0
$149$	−18.9443	−1.55198	−0.775988	−	0.630748i	$-0.782748\pi$
$149$	−18.9443	−1.55198	−0.775988	+	0.630748i	$0.782748\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	7.47214	0.604086
$154$	0	0
$155$	1.88854	0.151691
$156$	0	0
$157$	−20.9443	−1.67153	−0.835767	−	0.549084i	$-0.814977\pi$
$157$	−20.9443	−1.67153	−0.835767	+	0.549084i	$0.814977\pi$
$158$	0	0
$159$	14.4721	1.14772
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−0.291796	−0.0228552	−0.0114276	−	0.999935i	$-0.503638\pi$
$163$	−0.291796	−0.0228552	−0.0114276	+	0.999935i	$0.503638\pi$
$164$	0	0
$165$	20.9443	1.63051
$166$	0	0
$167$	14.4721	1.11989	0.559944	−	0.828531i	$-0.310823\pi$
$167$	14.4721	1.11989	0.559944	+	0.828531i	$0.310823\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	0	0
$171$	−63.3050	−4.84105
$172$	0	0
$173$	−20.6525	−1.57018	−0.785089	−	0.619383i	$-0.787383\pi$
$173$	−20.6525	−1.57018	−0.785089	+	0.619383i	$0.787383\pi$
$174$	0	0
$175$	−3.47214	−0.262469
$176$	0	0
$177$	−19.4164	−1.45943
$178$	0	0
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	0	0
$181$	2.76393	0.205441	0.102721	−	0.994710i	$-0.467245\pi$
$181$	2.76393	0.205441	0.102721	+	0.994710i	$0.467245\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	5.23607	0.382899
$188$	0	0
$189$	14.4721	1.05269
$190$	0	0
$191$	−12.9443	−0.936615	−0.468307	−	0.883566i	$-0.655136\pi$
$191$	−12.9443	−0.936615	−0.468307	+	0.883566i	$0.655136\pi$
$192$	0	0
$193$	−11.8885	−0.855756	−0.427878	−	0.903836i	$-0.640739\pi$
$193$	−11.8885	−0.855756	−0.427878	+	0.903836i	$0.640739\pi$
$194$	0	0
$195$	9.88854	0.708133
$196$	0	0
$197$	−21.7082	−1.54665	−0.773323	−	0.634013i	$-0.781407\pi$
$197$	−21.7082	−1.54665	−0.773323	+	0.634013i	$0.781407\pi$
$198$	0	0
$199$	−14.4721	−1.02590	−0.512951	−	0.858418i	$-0.671448\pi$
$199$	−14.4721	−1.02590	−0.512951	+	0.858418i	$0.671448\pi$
$200$	0	0
$201$	−4.94427	−0.348742
$202$	0	0
$203$	5.70820	0.400637
$204$	0	0
$205$	−4.36068	−0.304563
$206$	0	0
$207$	−59.7771	−4.15479
$208$	0	0
$209$	−44.3607	−3.06849
$210$	0	0
$211$	−27.7082	−1.90751	−0.953756	−	0.300583i	$-0.902819\pi$
$211$	−27.7082	−1.90751	−0.953756	+	0.300583i	$0.902819\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	3.05573	0.208399
$216$	0	0
$217$	1.52786	0.103718
$218$	0	0
$219$	43.4164	2.93381
$220$	0	0
$221$	2.47214	0.166294
$222$	0	0
$223$	−1.52786	−0.102313	−0.0511567	−	0.998691i	$-0.516291\pi$
$223$	−1.52786	−0.102313	−0.0511567	+	0.998691i	$0.516291\pi$
$224$	0	0
$225$	−25.9443	−1.72962
$226$	0	0
$227$	27.5967	1.83166	0.915830	−	0.401566i	$-0.131534\pi$
$227$	27.5967	1.83166	0.915830	+	0.401566i	$0.131534\pi$
$228$	0	0
$229$	4.94427	0.326727	0.163363	−	0.986566i	$-0.447766\pi$
$229$	4.94427	0.326727	0.163363	+	0.986566i	$0.447766\pi$
$230$	0	0
$231$	16.9443	1.11485
$232$	0	0
$233$	19.8885	1.30294	0.651471	−	0.758674i	$-0.274152\pi$
$233$	19.8885	1.30294	0.651471	+	0.758674i	$0.274152\pi$
$234$	0	0
$235$	3.05573	0.199334
$236$	0	0
$237$	33.8885	2.20130
$238$	0	0
$239$	4.94427	0.319818	0.159909	−	0.987132i	$-0.448880\pi$
$239$	4.94427	0.319818	0.159909	+	0.987132i	$0.448880\pi$
$240$	0	0
$241$	11.8885	0.765808	0.382904	−	0.923788i	$-0.374924\pi$
$241$	11.8885	0.765808	0.382904	+	0.923788i	$0.374924\pi$
$242$	0	0
$243$	35.5967	2.28353
$244$	0	0
$245$	1.23607	0.0789695
$246$	0	0
$247$	−20.9443	−1.33265
$248$	0	0
$249$	6.47214	0.410155
$250$	0	0
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	0	0
$253$	−41.8885	−2.63351
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	3.23607	0.201079
$260$	0	0
$261$	42.6525	2.64012
$262$	0	0
$263$	−7.05573	−0.435075	−0.217537	−	0.976052i	$-0.569802\pi$
$263$	−7.05573	−0.435075	−0.217537	+	0.976052i	$0.569802\pi$
$264$	0	0
$265$	5.52786	0.339574
$266$	0	0
$267$	6.47214	0.396088
$268$	0	0
$269$	0.291796	0.0177911	0.00889556	−	0.999960i	$-0.497168\pi$
$269$	0.291796	0.0177911	0.00889556	+	0.999960i	$0.497168\pi$
$270$	0	0
$271$	−19.4164	−1.17946	−0.589731	−	0.807599i	$-0.700766\pi$
$271$	−19.4164	−1.17946	−0.589731	+	0.807599i	$0.700766\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	−18.1803	−1.09632
$276$	0	0
$277$	22.6525	1.36106	0.680528	−	0.732722i	$-0.261751\pi$
$277$	22.6525	1.36106	0.680528	+	0.732722i	$0.261751\pi$
$278$	0	0
$279$	11.4164	0.683482
$280$	0	0
$281$	3.52786	0.210455	0.105227	−	0.994448i	$-0.466443\pi$
$281$	3.52786	0.210455	0.105227	+	0.994448i	$0.466443\pi$
$282$	0	0
$283$	10.2918	0.611784	0.305892	−	0.952066i	$-0.401045\pi$
$283$	10.2918	0.611784	0.305892	+	0.952066i	$0.401045\pi$
$284$	0	0
$285$	−33.8885	−2.00738
$286$	0	0
$287$	−3.52786	−0.208243
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−27.4164	−1.60718
$292$	0	0
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	0	0
$295$	−7.41641	−0.431800
$296$	0	0
$297$	75.7771	4.39703
$298$	0	0
$299$	−19.7771	−1.14374
$300$	0	0
$301$	2.47214	0.142492
$302$	0	0
$303$	20.9443	1.20322
$304$	0	0
$305$	−1.52786	−0.0874852
$306$	0	0
$307$	−3.52786	−0.201346	−0.100673	−	0.994920i	$-0.532100\pi$
$307$	−3.52786	−0.201346	−0.100673	+	0.994920i	$0.532100\pi$
$308$	0	0
$309$	12.9443	0.736374
$310$	0	0
$311$	20.9443	1.18764	0.593820	−	0.804598i	$-0.297619\pi$
$311$	20.9443	1.18764	0.593820	+	0.804598i	$0.297619\pi$
$312$	0	0
$313$	−5.41641	−0.306153	−0.153077	−	0.988214i	$-0.548918\pi$
$313$	−5.41641	−0.306153	−0.153077	+	0.988214i	$0.548918\pi$
$314$	0	0
$315$	9.23607	0.520393
$316$	0	0
$317$	28.1803	1.58277	0.791383	−	0.611321i	$-0.209362\pi$
$317$	28.1803	1.58277	0.791383	+	0.611321i	$0.209362\pi$
$318$	0	0
$319$	29.8885	1.67344
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−8.47214	−0.471402
$324$	0	0
$325$	−8.58359	−0.476132
$326$	0	0
$327$	−39.4164	−2.17973
$328$	0	0
$329$	2.47214	0.136293
$330$	0	0
$331$	−20.3607	−1.11912	−0.559562	−	0.828788i	$-0.689031\pi$
$331$	−20.3607	−1.11912	−0.559562	+	0.828788i	$0.689031\pi$
$332$	0	0
$333$	24.1803	1.32507
$334$	0	0
$335$	−1.88854	−0.103182
$336$	0	0
$337$	−20.4721	−1.11519	−0.557594	−	0.830114i	$-0.688275\pi$
$337$	−20.4721	−1.11519	−0.557594	+	0.830114i	$0.688275\pi$
$338$	0	0
$339$	43.4164	2.35806
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−32.0000	−1.72282
$346$	0	0
$347$	4.65248	0.249758	0.124879	−	0.992172i	$-0.460146\pi$
$347$	4.65248	0.249758	0.124879	+	0.992172i	$0.460146\pi$
$348$	0	0
$349$	21.5279	1.15236	0.576180	−	0.817323i	$-0.304543\pi$
$349$	21.5279	1.15236	0.576180	+	0.817323i	$0.304543\pi$
$350$	0	0
$351$	35.7771	1.90964
$352$	0	0
$353$	−11.8885	−0.632763	−0.316382	−	0.948632i	$-0.602468\pi$
$353$	−11.8885	−0.632763	−0.316382	+	0.948632i	$0.602468\pi$
$354$	0	0
$355$	3.05573	0.162181
$356$	0	0
$357$	3.23607	0.171271
$358$	0	0
$359$	−22.8328	−1.20507	−0.602535	−	0.798092i	$-0.705843\pi$
$359$	−22.8328	−1.20507	−0.602535	+	0.798092i	$0.705843\pi$
$360$	0	0
$361$	52.7771	2.77774
$362$	0	0
$363$	53.1246	2.78832
$364$	0	0
$365$	16.5836	0.868025
$366$	0	0
$367$	28.9443	1.51088	0.755439	−	0.655219i	$-0.227423\pi$
$367$	28.9443	1.51088	0.755439	+	0.655219i	$0.227423\pi$
$368$	0	0
$369$	−26.3607	−1.37228
$370$	0	0
$371$	4.47214	0.232182
$372$	0	0
$373$	−13.4164	−0.694675	−0.347338	−	0.937740i	$-0.612914\pi$
$373$	−13.4164	−0.694675	−0.347338	+	0.937740i	$0.612914\pi$
$374$	0	0
$375$	−33.8885	−1.75000
$376$	0	0
$377$	14.1115	0.726777
$378$	0	0
$379$	−17.5967	−0.903884	−0.451942	−	0.892047i	$-0.649269\pi$
$379$	−17.5967	−0.903884	−0.451942	+	0.892047i	$0.649269\pi$
$380$	0	0
$381$	−38.8328	−1.98947
$382$	0	0
$383$	8.94427	0.457031	0.228515	−	0.973540i	$-0.426613\pi$
$383$	8.94427	0.457031	0.228515	+	0.973540i	$0.426613\pi$
$384$	0	0
$385$	6.47214	0.329851
$386$	0	0
$387$	18.4721	0.938991
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	0	0
$393$	5.52786	0.278844
$394$	0	0
$395$	12.9443	0.651297
$396$	0	0
$397$	−18.1803	−0.912445	−0.456223	−	0.889866i	$-0.650798\pi$
$397$	−18.1803	−0.912445	−0.456223	+	0.889866i	$0.650798\pi$
$398$	0	0
$399$	−27.4164	−1.37254
$400$	0	0
$401$	2.94427	0.147030	0.0735150	−	0.997294i	$-0.476578\pi$
$401$	2.94427	0.147030	0.0735150	+	0.997294i	$0.476578\pi$
$402$	0	0
$403$	3.77709	0.188150
$404$	0	0
$405$	30.1803	1.49967
$406$	0	0
$407$	16.9443	0.839896
$408$	0	0
$409$	14.9443	0.738947	0.369473	−	0.929241i	$-0.379538\pi$
$409$	14.9443	0.738947	0.369473	+	0.929241i	$0.379538\pi$
$410$	0	0
$411$	11.4164	0.563130
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	2.47214	0.121352
$416$	0	0
$417$	−39.4164	−1.93023
$418$	0	0
$419$	−30.0689	−1.46896	−0.734481	−	0.678630i	$-0.762574\pi$
$419$	−30.0689	−1.46896	−0.734481	+	0.678630i	$0.762574\pi$
$420$	0	0
$421$	15.5279	0.756782	0.378391	−	0.925646i	$-0.376478\pi$
$421$	15.5279	0.756782	0.378391	+	0.925646i	$0.376478\pi$
$422$	0	0
$423$	18.4721	0.898146
$424$	0	0
$425$	−3.47214	−0.168423
$426$	0	0
$427$	−1.23607	−0.0598175
$428$	0	0
$429$	41.8885	2.02240
$430$	0	0
$431$	12.9443	0.623504	0.311752	−	0.950164i	$-0.399084\pi$
$431$	12.9443	0.623504	0.311752	+	0.950164i	$0.399084\pi$
$432$	0	0
$433$	10.9443	0.525948	0.262974	−	0.964803i	$-0.415297\pi$
$433$	10.9443	0.525948	0.262974	+	0.964803i	$0.415297\pi$
$434$	0	0
$435$	22.8328	1.09475
$436$	0	0
$437$	67.7771	3.24222
$438$	0	0
$439$	3.05573	0.145842	0.0729210	−	0.997338i	$-0.476768\pi$
$439$	3.05573	0.145842	0.0729210	+	0.997338i	$0.476768\pi$
$440$	0	0
$441$	7.47214	0.355816
$442$	0	0
$443$	−8.36068	−0.397228	−0.198614	−	0.980078i	$-0.563644\pi$
$443$	−8.36068	−0.397228	−0.198614	+	0.980078i	$0.563644\pi$
$444$	0	0
$445$	2.47214	0.117190
$446$	0	0
$447$	−61.3050	−2.89962
$448$	0	0
$449$	9.41641	0.444388	0.222194	−	0.975003i	$-0.428678\pi$
$449$	9.41641	0.444388	0.222194	+	0.975003i	$0.428678\pi$
$450$	0	0
$451$	−18.4721	−0.869819
$452$	0	0
$453$	−12.9443	−0.608175
$454$	0	0
$455$	3.05573	0.143255
$456$	0	0
$457$	15.8885	0.743235	0.371617	−	0.928386i	$-0.378803\pi$
$457$	15.8885	0.743235	0.371617	+	0.928386i	$0.378803\pi$
$458$	0	0
$459$	14.4721	0.675501
$460$	0	0
$461$	5.52786	0.257458	0.128729	−	0.991680i	$-0.458910\pi$
$461$	5.52786	0.257458	0.128729	+	0.991680i	$0.458910\pi$
$462$	0	0
$463$	−32.9443	−1.53105	−0.765525	−	0.643406i	$-0.777521\pi$
$463$	−32.9443	−1.53105	−0.765525	+	0.643406i	$0.777521\pi$
$464$	0	0
$465$	6.11146	0.283412
$466$	0	0
$467$	−10.3607	−0.479435	−0.239718	−	0.970843i	$-0.577055\pi$
$467$	−10.3607	−0.479435	−0.239718	+	0.970843i	$0.577055\pi$
$468$	0	0
$469$	−1.52786	−0.0705502
$470$	0	0
$471$	−67.7771	−3.12300
$472$	0	0
$473$	12.9443	0.595178
$474$	0	0
$475$	29.4164	1.34972
$476$	0	0
$477$	33.4164	1.53003
$478$	0	0
$479$	19.4164	0.887158	0.443579	−	0.896235i	$-0.353709\pi$
$479$	19.4164	0.887158	0.443579	+	0.896235i	$0.353709\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	−25.8885	−1.17797
$484$	0	0
$485$	−10.4721	−0.475515
$486$	0	0
$487$	−3.05573	−0.138468	−0.0692341	−	0.997600i	$-0.522056\pi$
$487$	−3.05573	−0.138468	−0.0692341	+	0.997600i	$0.522056\pi$
$488$	0	0
$489$	−0.944272	−0.0427015
$490$	0	0
$491$	−32.9443	−1.48675	−0.743377	−	0.668873i	$-0.766777\pi$
$491$	−32.9443	−1.48675	−0.743377	+	0.668873i	$0.766777\pi$
$492$	0	0
$493$	5.70820	0.257085
$494$	0	0
$495$	48.3607	2.17365
$496$	0	0
$497$	2.47214	0.110890
$498$	0	0
$499$	−40.6525	−1.81985	−0.909927	−	0.414768i	$-0.863863\pi$
$499$	−40.6525	−1.81985	−0.909927	+	0.414768i	$0.863863\pi$
$500$	0	0
$501$	46.8328	2.09234
$502$	0	0
$503$	−12.9443	−0.577157	−0.288578	−	0.957456i	$-0.593183\pi$
$503$	−12.9443	−0.577157	−0.288578	+	0.957456i	$0.593183\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	0	0
$507$	−22.2918	−0.990013
$508$	0	0
$509$	3.05573	0.135443	0.0677214	−	0.997704i	$-0.478427\pi$
$509$	3.05573	0.135443	0.0677214	+	0.997704i	$0.478427\pi$
$510$	0	0
$511$	13.4164	0.593507
$512$	0	0
$513$	−122.610	−5.41336
$514$	0	0
$515$	4.94427	0.217871
$516$	0	0
$517$	12.9443	0.569288
$518$	0	0
$519$	−66.8328	−2.93364
$520$	0	0
$521$	−1.05573	−0.0462523	−0.0231261	−	0.999733i	$-0.507362\pi$
$521$	−1.05573	−0.0462523	−0.0231261	+	0.999733i	$0.507362\pi$
$522$	0	0
$523$	−42.9443	−1.87782	−0.938911	−	0.344160i	$-0.888164\pi$
$523$	−42.9443	−1.87782	−0.938911	+	0.344160i	$0.888164\pi$
$524$	0	0
$525$	−11.2361	−0.490382
$526$	0	0
$527$	1.52786	0.0665548
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−44.8328	−1.94558
$532$	0	0
$533$	−8.72136	−0.377764
$534$	0	0
$535$	−1.52786	−0.0660553
$536$	0	0
$537$	−28.9443	−1.24904
$538$	0	0
$539$	5.23607	0.225533
$540$	0	0
$541$	23.2361	0.998997	0.499498	−	0.866315i	$-0.333518\pi$
$541$	23.2361	0.998997	0.499498	+	0.866315i	$0.333518\pi$
$542$	0	0
$543$	8.94427	0.383835
$544$	0	0
$545$	−15.0557	−0.644917
$546$	0	0
$547$	41.5967	1.77855	0.889274	−	0.457374i	$-0.151210\pi$
$547$	41.5967	1.77855	0.889274	+	0.457374i	$0.151210\pi$
$548$	0	0
$549$	−9.23607	−0.394186
$550$	0	0
$551$	−48.3607	−2.06023
$552$	0	0
$553$	10.4721	0.445321
$554$	0	0
$555$	12.9443	0.549454
$556$	0	0
$557$	−34.3607	−1.45591	−0.727954	−	0.685626i	$-0.759529\pi$
$557$	−34.3607	−1.45591	−0.727954	+	0.685626i	$0.759529\pi$
$558$	0	0
$559$	6.11146	0.258487
$560$	0	0
$561$	16.9443	0.715388
$562$	0	0
$563$	−32.8328	−1.38374	−0.691869	−	0.722023i	$-0.743212\pi$
$563$	−32.8328	−1.38374	−0.691869	+	0.722023i	$0.743212\pi$
$564$	0	0
$565$	16.5836	0.697677
$566$	0	0
$567$	24.4164	1.02539
$568$	0	0
$569$	15.8885	0.666082	0.333041	−	0.942912i	$-0.391925\pi$
$569$	15.8885	0.666082	0.333041	+	0.942912i	$0.391925\pi$
$570$	0	0
$571$	−13.2361	−0.553912	−0.276956	−	0.960883i	$-0.589326\pi$
$571$	−13.2361	−0.553912	−0.276956	+	0.960883i	$0.589326\pi$
$572$	0	0
$573$	−41.8885	−1.74992
$574$	0	0
$575$	27.7771	1.15838
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	−38.4721	−1.59885
$580$	0	0
$581$	2.00000	0.0829740
$582$	0	0
$583$	23.4164	0.969809
$584$	0	0
$585$	22.8328	0.944021
$586$	0	0
$587$	21.4164	0.883950	0.441975	−	0.897027i	$-0.354278\pi$
$587$	21.4164	0.883950	0.441975	+	0.897027i	$0.354278\pi$
$588$	0	0
$589$	−12.9443	−0.533359
$590$	0	0
$591$	−70.2492	−2.88967
$592$	0	0
$593$	−2.94427	−0.120907	−0.0604534	−	0.998171i	$-0.519255\pi$
$593$	−2.94427	−0.120907	−0.0604534	+	0.998171i	$0.519255\pi$
$594$	0	0
$595$	1.23607	0.0506738
$596$	0	0
$597$	−46.8328	−1.91674
$598$	0	0
$599$	−31.7771	−1.29838	−0.649188	−	0.760628i	$-0.724891\pi$
$599$	−31.7771	−1.29838	−0.649188	+	0.760628i	$0.724891\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	−11.4164	−0.464912
$604$	0	0
$605$	20.2918	0.824979
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	18.4721	0.748529
$610$	0	0
$611$	6.11146	0.247243
$612$	0	0
$613$	44.8328	1.81078	0.905390	−	0.424581i	$-0.139578\pi$
$613$	44.8328	1.81078	0.905390	+	0.424581i	$0.139578\pi$
$614$	0	0
$615$	−14.1115	−0.569029
$616$	0	0
$617$	10.3607	0.417105	0.208553	−	0.978011i	$-0.433125\pi$
$617$	10.3607	0.417105	0.208553	+	0.978011i	$0.433125\pi$
$618$	0	0
$619$	25.7082	1.03330	0.516650	−	0.856197i	$-0.327179\pi$
$619$	25.7082	1.03330	0.516650	+	0.856197i	$0.327179\pi$
$620$	0	0
$621$	−115.777	−4.64597
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	−143.554	−5.73300
$628$	0	0
$629$	3.23607	0.129030
$630$	0	0
$631$	−3.05573	−0.121647	−0.0608233	−	0.998149i	$-0.519373\pi$
$631$	−3.05573	−0.121647	−0.0608233	+	0.998149i	$0.519373\pi$
$632$	0	0
$633$	−89.6656	−3.56389
$634$	0	0
$635$	−14.8328	−0.588622
$636$	0	0
$637$	2.47214	0.0979496
$638$	0	0
$639$	18.4721	0.730746
$640$	0	0
$641$	9.05573	0.357680	0.178840	−	0.983878i	$-0.442766\pi$
$641$	9.05573	0.357680	0.178840	+	0.983878i	$0.442766\pi$
$642$	0	0
$643$	13.1246	0.517584	0.258792	−	0.965933i	$-0.416676\pi$
$643$	13.1246	0.517584	0.258792	+	0.965933i	$0.416676\pi$
$644$	0	0
$645$	9.88854	0.389361
$646$	0	0
$647$	−20.3607	−0.800461	−0.400230	−	0.916415i	$-0.631070\pi$
$647$	−20.3607	−0.800461	−0.400230	+	0.916415i	$0.631070\pi$
$648$	0	0
$649$	−31.4164	−1.23320
$650$	0	0
$651$	4.94427	0.193781
$652$	0	0
$653$	7.59675	0.297284	0.148642	−	0.988891i	$-0.452510\pi$
$653$	7.59675	0.297284	0.148642	+	0.988891i	$0.452510\pi$
$654$	0	0
$655$	2.11146	0.0825014
$656$	0	0
$657$	100.249	3.91109
$658$	0	0
$659$	−20.9443	−0.815873	−0.407936	−	0.913010i	$-0.633752\pi$
$659$	−20.9443	−0.815873	−0.407936	+	0.913010i	$0.633752\pi$
$660$	0	0
$661$	3.41641	0.132883	0.0664414	−	0.997790i	$-0.478835\pi$
$661$	3.41641	0.132883	0.0664414	+	0.997790i	$0.478835\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	−10.4721	−0.406092
$666$	0	0
$667$	−45.6656	−1.76818
$668$	0	0
$669$	−4.94427	−0.191157
$670$	0	0
$671$	−6.47214	−0.249854
$672$	0	0
$673$	−23.3050	−0.898340	−0.449170	−	0.893446i	$-0.648280\pi$
$673$	−23.3050	−0.898340	−0.449170	+	0.893446i	$0.648280\pi$
$674$	0	0
$675$	−50.2492	−1.93409
$676$	0	0
$677$	−34.1803	−1.31366	−0.656829	−	0.754040i	$-0.728102\pi$
$677$	−34.1803	−1.31366	−0.656829	+	0.754040i	$0.728102\pi$
$678$	0	0
$679$	−8.47214	−0.325131
$680$	0	0
$681$	89.3050	3.42217
$682$	0	0
$683$	46.1803	1.76704	0.883521	−	0.468392i	$-0.155166\pi$
$683$	46.1803	1.76704	0.883521	+	0.468392i	$0.155166\pi$
$684$	0	0
$685$	4.36068	0.166613
$686$	0	0
$687$	16.0000	0.610438
$688$	0	0
$689$	11.0557	0.421190
$690$	0	0
$691$	−34.6525	−1.31824	−0.659121	−	0.752037i	$-0.729072\pi$
$691$	−34.6525	−1.31824	−0.659121	+	0.752037i	$0.729072\pi$
$692$	0	0
$693$	39.1246	1.48622
$694$	0	0
$695$	−15.0557	−0.571096
$696$	0	0
$697$	−3.52786	−0.133627
$698$	0	0
$699$	64.3607	2.43434
$700$	0	0
$701$	36.2492	1.36911	0.684557	−	0.728959i	$-0.259996\pi$
$701$	36.2492	1.36911	0.684557	+	0.728959i	$0.259996\pi$
$702$	0	0
$703$	−27.4164	−1.03403
$704$	0	0
$705$	9.88854	0.372424
$706$	0	0
$707$	6.47214	0.243410
$708$	0	0
$709$	23.5967	0.886194	0.443097	−	0.896474i	$-0.353880\pi$
$709$	23.5967	0.886194	0.443097	+	0.896474i	$0.353880\pi$
$710$	0	0
$711$	78.2492	2.93458
$712$	0	0
$713$	−12.2229	−0.457752
$714$	0	0
$715$	16.0000	0.598366
$716$	0	0
$717$	16.0000	0.597531
$718$	0	0
$719$	27.4164	1.02246	0.511230	−	0.859444i	$-0.329190\pi$
$719$	27.4164	1.02246	0.511230	+	0.859444i	$0.329190\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	38.4721	1.43079
$724$	0	0
$725$	−19.8197	−0.736084
$726$	0	0
$727$	37.5279	1.39183	0.695916	−	0.718123i	$-0.254999\pi$
$727$	37.5279	1.39183	0.695916	+	0.718123i	$0.254999\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	2.47214	0.0914353
$732$	0	0
$733$	4.58359	0.169299	0.0846494	−	0.996411i	$-0.473023\pi$
$733$	4.58359	0.169299	0.0846494	+	0.996411i	$0.473023\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	46.8328	1.72277	0.861386	−	0.507950i	$-0.169597\pi$
$739$	46.8328	1.72277	0.861386	+	0.507950i	$0.169597\pi$
$740$	0	0
$741$	−67.7771	−2.48985
$742$	0	0
$743$	−29.5279	−1.08327	−0.541636	−	0.840613i	$-0.682195\pi$
$743$	−29.5279	−1.08327	−0.541636	+	0.840613i	$0.682195\pi$
$744$	0	0
$745$	−23.4164	−0.857911
$746$	0	0
$747$	14.9443	0.546782
$748$	0	0
$749$	−1.23607	−0.0451649
$750$	0	0
$751$	−28.9443	−1.05619	−0.528096	−	0.849185i	$-0.677094\pi$
$751$	−28.9443	−1.05619	−0.528096	+	0.849185i	$0.677094\pi$
$752$	0	0
$753$	45.3050	1.65100
$754$	0	0
$755$	−4.94427	−0.179940
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	0	0
$759$	−135.554	−4.92030
$760$	0	0
$761$	3.88854	0.140960	0.0704798	−	0.997513i	$-0.477547\pi$
$761$	3.88854	0.140960	0.0704798	+	0.997513i	$0.477547\pi$
$762$	0	0
$763$	−12.1803	−0.440958
$764$	0	0
$765$	9.23607	0.333931
$766$	0	0
$767$	−14.8328	−0.535582
$768$	0	0
$769$	−4.83282	−0.174276	−0.0871379	−	0.996196i	$-0.527772\pi$
$769$	−4.83282	−0.174276	−0.0871379	+	0.996196i	$0.527772\pi$
$770$	0	0
$771$	−6.47214	−0.233088
$772$	0	0
$773$	−4.00000	−0.143870	−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000	−0.143870	−0.0719350	+	0.997409i	$0.522917\pi$
$774$	0	0
$775$	−5.30495	−0.190559
$776$	0	0
$777$	10.4721	0.375686
$778$	0	0
$779$	29.8885	1.07087
$780$	0	0
$781$	12.9443	0.463182
$782$	0	0
$783$	82.6099	2.95224
$784$	0	0
$785$	−25.8885	−0.924002
$786$	0	0
$787$	9.70820	0.346060	0.173030	−	0.984917i	$-0.444644\pi$
$787$	9.70820	0.346060	0.173030	+	0.984917i	$0.444644\pi$
$788$	0	0
$789$	−22.8328	−0.812870
$790$	0	0
$791$	13.4164	0.477033
$792$	0	0
$793$	−3.05573	−0.108512
$794$	0	0
$795$	17.8885	0.634441
$796$	0	0
$797$	−39.4164	−1.39620	−0.698100	−	0.716000i	$-0.745971\pi$
$797$	−39.4164	−1.39620	−0.698100	+	0.716000i	$0.745971\pi$
$798$	0	0
$799$	2.47214	0.0874579
$800$	0	0
$801$	14.9443	0.528030
$802$	0	0
$803$	70.2492	2.47904
$804$	0	0
$805$	−9.88854	−0.348525
$806$	0	0
$807$	0.944272	0.0332399
$808$	0	0
$809$	1.05573	0.0371174	0.0185587	−	0.999828i	$-0.494092\pi$
$809$	1.05573	0.0371174	0.0185587	+	0.999828i	$0.494092\pi$
$810$	0	0
$811$	−19.8197	−0.695962	−0.347981	−	0.937502i	$-0.613133\pi$
$811$	−19.8197	−0.695962	−0.347981	+	0.937502i	$0.613133\pi$
$812$	0	0
$813$	−62.8328	−2.20364
$814$	0	0
$815$	−0.360680	−0.0126341
$816$	0	0
$817$	−20.9443	−0.732747
$818$	0	0
$819$	18.4721	0.645469
$820$	0	0
$821$	43.0132	1.50117	0.750585	−	0.660774i	$-0.229772\pi$
$821$	43.0132	1.50117	0.750585	+	0.660774i	$0.229772\pi$
$822$	0	0
$823$	2.47214	0.0861732	0.0430866	−	0.999071i	$-0.486281\pi$
$823$	2.47214	0.0861732	0.0430866	+	0.999071i	$0.486281\pi$
$824$	0	0
$825$	−58.8328	−2.04830
$826$	0	0
$827$	−37.5967	−1.30737	−0.653684	−	0.756768i	$-0.726777\pi$
$827$	−37.5967	−1.30737	−0.653684	+	0.756768i	$0.726777\pi$
$828$	0	0
$829$	−40.9443	−1.42205	−0.711027	−	0.703165i	$-0.751769\pi$
$829$	−40.9443	−1.42205	−0.711027	+	0.703165i	$0.751769\pi$
$830$	0	0
$831$	73.3050	2.54292
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	17.8885	0.619059
$836$	0	0
$837$	22.1115	0.764284
$838$	0	0
$839$	−7.63932	−0.263739	−0.131869	−	0.991267i	$-0.542098\pi$
$839$	−7.63932	−0.263739	−0.131869	+	0.991267i	$0.542098\pi$
$840$	0	0
$841$	3.58359	0.123572
$842$	0	0
$843$	11.4164	0.393202
$844$	0	0
$845$	−8.51471	−0.292915
$846$	0	0
$847$	16.4164	0.564074
$848$	0	0
$849$	33.3050	1.14302
$850$	0	0
$851$	−25.8885	−0.887448
$852$	0	0
$853$	21.2361	0.727109	0.363555	−	0.931573i	$-0.381563\pi$
$853$	21.2361	0.727109	0.363555	+	0.931573i	$0.381563\pi$
$854$	0	0
$855$	−78.2492	−2.67607
$856$	0	0
$857$	−3.52786	−0.120510	−0.0602548	−	0.998183i	$-0.519191\pi$
$857$	−3.52786	−0.120510	−0.0602548	+	0.998183i	$0.519191\pi$
$858$	0	0
$859$	−35.8885	−1.22450	−0.612251	−	0.790664i	$-0.709736\pi$
$859$	−35.8885	−1.22450	−0.612251	+	0.790664i	$0.709736\pi$
$860$	0	0
$861$	−11.4164	−0.389070
$862$	0	0
$863$	−16.9443	−0.576790	−0.288395	−	0.957512i	$-0.593122\pi$
$863$	−16.9443	−0.576790	−0.288395	+	0.957512i	$0.593122\pi$
$864$	0	0
$865$	−25.5279	−0.867973
$866$	0	0
$867$	3.23607	0.109903
$868$	0	0
$869$	54.8328	1.86008
$870$	0	0
$871$	−3.77709	−0.127982
$872$	0	0
$873$	−63.3050	−2.14255
$874$	0	0
$875$	−10.4721	−0.354023
$876$	0	0
$877$	−9.12461	−0.308116	−0.154058	−	0.988062i	$-0.549234\pi$
$877$	−9.12461	−0.308116	−0.154058	+	0.988062i	$0.549234\pi$
$878$	0	0
$879$	90.6099	3.05620
$880$	0	0
$881$	−46.9443	−1.58159	−0.790796	−	0.612079i	$-0.790333\pi$
$881$	−46.9443	−1.58159	−0.790796	+	0.612079i	$0.790333\pi$
$882$	0	0
$883$	3.41641	0.114971	0.0574856	−	0.998346i	$-0.481692\pi$
$883$	3.41641	0.114971	0.0574856	+	0.998346i	$0.481692\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	24.3607	0.817952	0.408976	−	0.912545i	$-0.365886\pi$
$887$	24.3607	0.817952	0.408976	+	0.912545i	$0.365886\pi$
$888$	0	0
$889$	−12.0000	−0.402467
$890$	0	0
$891$	127.846	4.28300
$892$	0	0
$893$	−20.9443	−0.700873
$894$	0	0
$895$	−11.0557	−0.369552
$896$	0	0
$897$	−64.0000	−2.13690
$898$	0	0
$899$	8.72136	0.290874
$900$	0	0
$901$	4.47214	0.148988
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	3.41641	0.113565
$906$	0	0
$907$	−6.76393	−0.224593	−0.112296	−	0.993675i	$-0.535821\pi$
$907$	−6.76393	−0.224593	−0.112296	+	0.993675i	$0.535821\pi$
$908$	0	0
$909$	48.3607	1.60402
$910$	0	0
$911$	−12.3607	−0.409528	−0.204764	−	0.978811i	$-0.565643\pi$
$911$	−12.3607	−0.409528	−0.204764	+	0.978811i	$0.565643\pi$
$912$	0	0
$913$	10.4721	0.346577
$914$	0	0
$915$	−4.94427	−0.163453
$916$	0	0
$917$	1.70820	0.0564099
$918$	0	0
$919$	27.0557	0.892486	0.446243	−	0.894912i	$-0.352762\pi$
$919$	27.0557	0.892486	0.446243	+	0.894912i	$0.352762\pi$
$920$	0	0
$921$	−11.4164	−0.376183
$922$	0	0
$923$	6.11146	0.201161
$924$	0	0
$925$	−11.2361	−0.369440
$926$	0	0
$927$	29.8885	0.981669
$928$	0	0
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−8.47214	−0.277663
$932$	0	0
$933$	67.7771	2.21892
$934$	0	0
$935$	6.47214	0.211661
$936$	0	0
$937$	20.1115	0.657013	0.328506	−	0.944502i	$-0.393455\pi$
$937$	20.1115	0.657013	0.328506	+	0.944502i	$0.393455\pi$
$938$	0	0
$939$	−17.5279	−0.572000
$940$	0	0
$941$	−59.1246	−1.92741	−0.963704	−	0.266974i	$-0.913976\pi$
$941$	−59.1246	−1.92741	−0.963704	+	0.266974i	$0.913976\pi$
$942$	0	0
$943$	28.2229	0.919064
$944$	0	0
$945$	17.8885	0.581914
$946$	0	0
$947$	−47.1246	−1.53134	−0.765672	−	0.643231i	$-0.777593\pi$
$947$	−47.1246	−1.53134	−0.765672	+	0.643231i	$0.777593\pi$
$948$	0	0
$949$	33.1672	1.07665
$950$	0	0
$951$	91.1935	2.95715
$952$	0	0
$953$	5.05573	0.163771	0.0818855	−	0.996642i	$-0.473906\pi$
$953$	5.05573	0.163771	0.0818855	+	0.996642i	$0.473906\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	0	0
$957$	96.7214	3.12656
$958$	0	0
$959$	3.52786	0.113921
$960$	0	0
$961$	−28.6656	−0.924698
$962$	0	0
$963$	−9.23607	−0.297628
$964$	0	0
$965$	−14.6950	−0.473050
$966$	0	0
$967$	−44.9443	−1.44531	−0.722655	−	0.691209i	$-0.757079\pi$
$967$	−44.9443	−1.44531	−0.722655	+	0.691209i	$0.757079\pi$
$968$	0	0
$969$	−27.4164	−0.880742
$970$	0	0
$971$	24.8328	0.796923	0.398461	−	0.917185i	$-0.369544\pi$
$971$	24.8328	0.796923	0.398461	+	0.917185i	$0.369544\pi$
$972$	0	0
$973$	−12.1803	−0.390484
$974$	0	0
$975$	−27.7771	−0.889579
$976$	0	0
$977$	−58.9443	−1.88579	−0.942897	−	0.333084i	$-0.891911\pi$
$977$	−58.9443	−1.88579	−0.942897	+	0.333084i	$0.891911\pi$
$978$	0	0
$979$	10.4721	0.334691
$980$	0	0
$981$	−91.0132	−2.90583
$982$	0	0
$983$	−54.4721	−1.73739	−0.868696	−	0.495346i	$-0.835041\pi$
$983$	−54.4721	−1.73739	−0.868696	+	0.495346i	$0.835041\pi$
$984$	0	0
$985$	−26.8328	−0.854965
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	−19.7771	−0.628875
$990$	0	0
$991$	22.2492	0.706770	0.353385	−	0.935478i	$-0.385031\pi$
$991$	22.2492	0.706770	0.353385	+	0.935478i	$0.385031\pi$
$992$	0	0
$993$	−65.8885	−2.09091
$994$	0	0
$995$	−17.8885	−0.567105
$996$	0	0
$997$	47.4853	1.50387	0.751937	−	0.659235i	$-0.229120\pi$
$997$	47.4853	1.50387	0.751937	+	0.659235i	$0.229120\pi$
$998$	0	0
$999$	46.8328	1.48172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.y.1.2		2
4.3	odd	2		7616.2.a.n.1.1		2
8.3	odd	2		238.2.a.f.1.2	✓	2
8.5	even	2		1904.2.a.f.1.1		2
24.11	even	2		2142.2.a.x.1.2		2
40.19	odd	2		5950.2.a.x.1.1		2
56.27	even	2		1666.2.a.o.1.1		2
136.67	odd	2		4046.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.a.f.1.2	✓	2	8.3	odd	2
1666.2.a.o.1.1		2	56.27	even	2
1904.2.a.f.1.1		2	8.5	even	2
2142.2.a.x.1.2		2	24.11	even	2
4046.2.a.v.1.1		2	136.67	odd	2
5950.2.a.x.1.1		2	40.19	odd	2
7616.2.a.n.1.1		2	4.3	odd	2
7616.2.a.y.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23607 q^{3} +1.23607 q^{5} +1.00000 q^{7} +7.47214 q^{9} +O(q^{10})\)	\(q+3.23607 q^{3} +1.23607 q^{5} +1.00000 q^{7} +7.47214 q^{9} +5.23607 q^{11} +2.47214 q^{13} +4.00000 q^{15} +1.00000 q^{17} -8.47214 q^{19} +3.23607 q^{21} -8.00000 q^{23} -3.47214 q^{25} +14.4721 q^{27} +5.70820 q^{29} +1.52786 q^{31} +16.9443 q^{33} +1.23607 q^{35} +3.23607 q^{37} +8.00000 q^{39} -3.52786 q^{41} +2.47214 q^{43} +9.23607 q^{45} +2.47214 q^{47} +1.00000 q^{49} +3.23607 q^{51} +4.47214 q^{53} +6.47214 q^{55} -27.4164 q^{57} -6.00000 q^{59} -1.23607 q^{61} +7.47214 q^{63} +3.05573 q^{65} -1.52786 q^{67} -25.8885 q^{69} +2.47214 q^{71} +13.4164 q^{73} -11.2361 q^{75} +5.23607 q^{77} +10.4721 q^{79} +24.4164 q^{81} +2.00000 q^{83} +1.23607 q^{85} +18.4721 q^{87} +2.00000 q^{89} +2.47214 q^{91} +4.94427 q^{93} -10.4721 q^{95} -8.47214 q^{97} +39.1246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 6 q^{9} + 6 q^{11} - 4 q^{13} + 8 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} - 16 q^{23} + 2 q^{25} + 20 q^{27} - 2 q^{29} + 12 q^{31} + 16 q^{33} - 2 q^{35} + 2 q^{37} + 16 q^{39} - 16 q^{41} - 4 q^{43} + 14 q^{45} - 4 q^{47} + 2 q^{49} + 2 q^{51} + 4 q^{55} - 28 q^{57} - 12 q^{59} + 2 q^{61} + 6 q^{63} + 24 q^{65} - 12 q^{67} - 16 q^{69} - 4 q^{71} - 18 q^{75} + 6 q^{77} + 12 q^{79} + 22 q^{81} + 4 q^{83} - 2 q^{85} + 28 q^{87} + 4 q^{89} - 4 q^{91} - 8 q^{93} - 12 q^{95} - 8 q^{97} + 38 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 6 * q^9 + 6 * q^11 - 4 * q^13 + 8 * q^15 + 2 * q^17 - 8 * q^19 + 2 * q^21 - 16 * q^23 + 2 * q^25 + 20 * q^27 - 2 * q^29 + 12 * q^31 + 16 * q^33 - 2 * q^35 + 2 * q^37 + 16 * q^39 - 16 * q^41 - 4 * q^43 + 14 * q^45 - 4 * q^47 + 2 * q^49 + 2 * q^51 + 4 * q^55 - 28 * q^57 - 12 * q^59 + 2 * q^61 + 6 * q^63 + 24 * q^65 - 12 * q^67 - 16 * q^69 - 4 * q^71 - 18 * q^75 + 6 * q^77 + 12 * q^79 + 22 * q^81 + 4 * q^83 - 2 * q^85 + 28 * q^87 + 4 * q^89 - 4 * q^91 - 8 * q^93 - 12 * q^95 - 8 * q^97 + 38 * q^99

Introduction

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