LMFDB - Embedded newform 9280.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9280,2,Mod(1,9280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, 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("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.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:	$74.1011730757$
Analytic rank:	$1$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 4640)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{3} +1.00000 q^{5} +2.85410 q^{7} -2.61803 q^{9} -3.23607 q^{11} +5.09017 q^{13} +0.618034 q^{15} -3.61803 q^{17} +2.76393 q^{19} +1.76393 q^{21} -5.85410 q^{23} +1.00000 q^{25} -3.47214 q^{27} -1.00000 q^{29} +1.61803 q^{31} -2.00000 q^{33} +2.85410 q^{35} -9.23607 q^{37} +3.14590 q^{39} -3.70820 q^{41} -7.61803 q^{43} -2.61803 q^{45} +8.00000 q^{47} +1.14590 q^{49} -2.23607 q^{51} -9.32624 q^{53} -3.23607 q^{55} +1.70820 q^{57} -9.38197 q^{59} -2.14590 q^{61} -7.47214 q^{63} +5.09017 q^{65} -2.47214 q^{67} -3.61803 q^{69} -12.9443 q^{71} +1.90983 q^{73} +0.618034 q^{75} -9.23607 q^{77} +2.90983 q^{79} +5.70820 q^{81} +15.7082 q^{83} -3.61803 q^{85} -0.618034 q^{87} -1.70820 q^{89} +14.5279 q^{91} +1.00000 q^{93} +2.76393 q^{95} +8.79837 q^{97} +8.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9} - 2 q^{11} - q^{13} - q^{15} - 5 q^{17} + 10 q^{19} + 8 q^{21} - 5 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} + q^{31} - 4 q^{33} - q^{35} - 14 q^{37} + 13 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 - q^7 - 3 * q^9 - 2 * q^11 - q^13 - q^15 - 5 * q^17 + 10 * q^19 + 8 * q^21 - 5 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 + q^31 - 4 * q^33 - q^35 - 14 * q^37 + 13 * q^39 + 6 * q^41 - 13 * q^43 - 3 * q^45 + 16 * q^47 + 9 * q^49 - 3 * q^53 - 2 * q^55 - 10 * q^57 - 21 * q^59 - 11 * q^61 - 6 * q^63 - q^65 + 4 * q^67 - 5 * q^69 - 8 * q^71 + 15 * q^73 - q^75 - 14 * q^77 + 17 * q^79 - 2 * q^81 + 18 * q^83 - 5 * q^85 + q^87 + 10 * q^89 + 38 * q^91 + 2 * q^93 + 10 * q^95 - 7 * q^97 + 8 * 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−3.23607	−0.975711	−0.487856	−	0.872924i	$-0.662221\pi$
$11$	−3.23607	−0.975711	−0.487856	+	0.872924i	$0.662221\pi$
$12$	0	0
$13$	5.09017	1.41176	0.705880	−	0.708332i	$-0.250552\pi$
$13$	5.09017	1.41176	0.705880	+	0.708332i	$0.250552\pi$
$14$	0	0
$15$	0.618034	0.159576
$16$	0	0
$17$	−3.61803	−0.877502	−0.438751	−	0.898609i	$-0.644579\pi$
$17$	−3.61803	−0.877502	−0.438751	+	0.898609i	$0.644579\pi$
$18$	0	0
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	0	0
$21$	1.76393	0.384922
$22$	0	0
$23$	−5.85410	−1.22066	−0.610332	−	0.792145i	$-0.708964\pi$
$23$	−5.85410	−1.22066	−0.610332	+	0.792145i	$0.708964\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.47214	−0.668213
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.61803	0.290607	0.145304	−	0.989387i	$-0.453584\pi$
$31$	1.61803	0.290607	0.145304	+	0.989387i	$0.453584\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	2.85410	0.482431
$36$	0	0
$37$	−9.23607	−1.51840	−0.759200	−	0.650857i	$-0.774410\pi$
$37$	−9.23607	−1.51840	−0.759200	+	0.650857i	$0.774410\pi$
$38$	0	0
$39$	3.14590	0.503747
$40$	0	0
$41$	−3.70820	−0.579124	−0.289562	−	0.957159i	$-0.593510\pi$
$41$	−3.70820	−0.579124	−0.289562	+	0.957159i	$0.593510\pi$
$42$	0	0
$43$	−7.61803	−1.16174	−0.580870	−	0.813997i	$-0.697287\pi$
$43$	−7.61803	−1.16174	−0.580870	+	0.813997i	$0.697287\pi$
$44$	0	0
$45$	−2.61803	−0.390273
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	1.14590	0.163700
$50$	0	0
$51$	−2.23607	−0.313112
$52$	0	0
$53$	−9.32624	−1.28106	−0.640529	−	0.767934i	$-0.721285\pi$
$53$	−9.32624	−1.28106	−0.640529	+	0.767934i	$0.721285\pi$
$54$	0	0
$55$	−3.23607	−0.436351
$56$	0	0
$57$	1.70820	0.226257
$58$	0	0
$59$	−9.38197	−1.22143	−0.610714	−	0.791851i	$-0.709117\pi$
$59$	−9.38197	−1.22143	−0.610714	+	0.791851i	$0.709117\pi$
$60$	0	0
$61$	−2.14590	−0.274754	−0.137377	−	0.990519i	$-0.543867\pi$
$61$	−2.14590	−0.274754	−0.137377	+	0.990519i	$0.543867\pi$
$62$	0	0
$63$	−7.47214	−0.941401
$64$	0	0
$65$	5.09017	0.631358
$66$	0	0
$67$	−2.47214	−0.302019	−0.151010	−	0.988532i	$-0.548252\pi$
$67$	−2.47214	−0.302019	−0.151010	+	0.988532i	$0.548252\pi$
$68$	0	0
$69$	−3.61803	−0.435560
$70$	0	0
$71$	−12.9443	−1.53620	−0.768101	−	0.640328i	$-0.778798\pi$
$71$	−12.9443	−1.53620	−0.768101	+	0.640328i	$0.778798\pi$
$72$	0	0
$73$	1.90983	0.223529	0.111764	−	0.993735i	$-0.464350\pi$
$73$	1.90983	0.223529	0.111764	+	0.993735i	$0.464350\pi$
$74$	0	0
$75$	0.618034	0.0713644
$76$	0	0
$77$	−9.23607	−1.05255
$78$	0	0
$79$	2.90983	0.327381	0.163691	−	0.986512i	$-0.447660\pi$
$79$	2.90983	0.327381	0.163691	+	0.986512i	$0.447660\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	15.7082	1.72420	0.862100	−	0.506739i	$-0.169149\pi$
$83$	15.7082	1.72420	0.862100	+	0.506739i	$0.169149\pi$
$84$	0	0
$85$	−3.61803	−0.392431
$86$	0	0
$87$	−0.618034	−0.0662602
$88$	0	0
$89$	−1.70820	−0.181069	−0.0905346	−	0.995893i	$-0.528858\pi$
$89$	−1.70820	−0.181069	−0.0905346	+	0.995893i	$0.528858\pi$
$90$	0	0
$91$	14.5279	1.52293
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	2.76393	0.283573
$96$	0	0
$97$	8.79837	0.893340	0.446670	−	0.894699i	$-0.352610\pi$
$97$	8.79837	0.893340	0.446670	+	0.894699i	$0.352610\pi$
$98$	0	0
$99$	8.47214	0.851482
$100$	0	0
$101$	7.09017	0.705498	0.352749	−	0.935718i	$-0.385247\pi$
$101$	7.09017	0.705498	0.352749	+	0.935718i	$0.385247\pi$
$102$	0	0
$103$	−8.94427	−0.881305	−0.440653	−	0.897678i	$-0.645253\pi$
$103$	−8.94427	−0.881305	−0.440653	+	0.897678i	$0.645253\pi$
$104$	0	0
$105$	1.76393	0.172142
$106$	0	0
$107$	0.291796	0.0282090	0.0141045	−	0.999901i	$-0.495510\pi$
$107$	0.291796	0.0282090	0.0141045	+	0.999901i	$0.495510\pi$
$108$	0	0
$109$	−11.7082	−1.12144	−0.560721	−	0.828005i	$-0.689476\pi$
$109$	−11.7082	−1.12144	−0.560721	+	0.828005i	$0.689476\pi$
$110$	0	0
$111$	−5.70820	−0.541799
$112$	0	0
$113$	8.79837	0.827681	0.413841	−	0.910349i	$-0.364187\pi$
$113$	8.79837	0.827681	0.413841	+	0.910349i	$0.364187\pi$
$114$	0	0
$115$	−5.85410	−0.545898
$116$	0	0
$117$	−13.3262	−1.23201
$118$	0	0
$119$	−10.3262	−0.946605
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−2.29180	−0.206644
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.944272	0.0837906	0.0418953	−	0.999122i	$-0.486660\pi$
$127$	0.944272	0.0837906	0.0418953	+	0.999122i	$0.486660\pi$
$128$	0	0
$129$	−4.70820	−0.414534
$130$	0	0
$131$	−18.9443	−1.65517	−0.827584	−	0.561341i	$-0.810285\pi$
$131$	−18.9443	−1.65517	−0.827584	+	0.561341i	$0.810285\pi$
$132$	0	0
$133$	7.88854	0.684023
$134$	0	0
$135$	−3.47214	−0.298834
$136$	0	0
$137$	−0.145898	−0.0124649	−0.00623246	−	0.999981i	$-0.501984\pi$
$137$	−0.145898	−0.0124649	−0.00623246	+	0.999981i	$0.501984\pi$
$138$	0	0
$139$	5.61803	0.476515	0.238258	−	0.971202i	$-0.423424\pi$
$139$	5.61803	0.476515	0.238258	+	0.971202i	$0.423424\pi$
$140$	0	0
$141$	4.94427	0.416383
$142$	0	0
$143$	−16.4721	−1.37747
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0.708204	0.0584117
$148$	0	0
$149$	−16.1803	−1.32555	−0.662773	−	0.748821i	$-0.730620\pi$
$149$	−16.1803	−1.32555	−0.662773	+	0.748821i	$0.730620\pi$
$150$	0	0
$151$	−1.23607	−0.100590	−0.0502949	−	0.998734i	$-0.516016\pi$
$151$	−1.23607	−0.100590	−0.0502949	+	0.998734i	$0.516016\pi$
$152$	0	0
$153$	9.47214	0.765777
$154$	0	0
$155$	1.61803	0.129964
$156$	0	0
$157$	−5.05573	−0.403491	−0.201746	−	0.979438i	$-0.564661\pi$
$157$	−5.05573	−0.403491	−0.201746	+	0.979438i	$0.564661\pi$
$158$	0	0
$159$	−5.76393	−0.457110
$160$	0	0
$161$	−16.7082	−1.31679
$162$	0	0
$163$	−13.5279	−1.05958	−0.529792	−	0.848128i	$-0.677730\pi$
$163$	−13.5279	−1.05958	−0.529792	+	0.848128i	$0.677730\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	4.38197	0.339087	0.169543	−	0.985523i	$-0.445771\pi$
$167$	4.38197	0.339087	0.169543	+	0.985523i	$0.445771\pi$
$168$	0	0
$169$	12.9098	0.993064
$170$	0	0
$171$	−7.23607	−0.553356
$172$	0	0
$173$	4.79837	0.364814	0.182407	−	0.983223i	$-0.441611\pi$
$173$	4.79837	0.364814	0.182407	+	0.983223i	$0.441611\pi$
$174$	0	0
$175$	2.85410	0.215750
$176$	0	0
$177$	−5.79837	−0.435832
$178$	0	0
$179$	−1.09017	−0.0814831	−0.0407416	−	0.999170i	$-0.512972\pi$
$179$	−1.09017	−0.0814831	−0.0407416	+	0.999170i	$0.512972\pi$
$180$	0	0
$181$	−0.291796	−0.0216890	−0.0108445	−	0.999941i	$-0.503452\pi$
$181$	−0.291796	−0.0216890	−0.0108445	+	0.999941i	$0.503452\pi$
$182$	0	0
$183$	−1.32624	−0.0980383
$184$	0	0
$185$	−9.23607	−0.679049
$186$	0	0
$187$	11.7082	0.856189
$188$	0	0
$189$	−9.90983	−0.720834
$190$	0	0
$191$	20.8541	1.50895	0.754475	−	0.656329i	$-0.227892\pi$
$191$	20.8541	1.50895	0.754475	+	0.656329i	$0.227892\pi$
$192$	0	0
$193$	0.145898	0.0105020	0.00525099	−	0.999986i	$-0.498329\pi$
$193$	0.145898	0.0105020	0.00525099	+	0.999986i	$0.498329\pi$
$194$	0	0
$195$	3.14590	0.225282
$196$	0	0
$197$	−18.8541	−1.34330	−0.671650	−	0.740869i	$-0.734414\pi$
$197$	−18.8541	−1.34330	−0.671650	+	0.740869i	$0.734414\pi$
$198$	0	0
$199$	8.76393	0.621259	0.310629	−	0.950531i	$-0.399460\pi$
$199$	8.76393	0.621259	0.310629	+	0.950531i	$0.399460\pi$
$200$	0	0
$201$	−1.52786	−0.107767
$202$	0	0
$203$	−2.85410	−0.200319
$204$	0	0
$205$	−3.70820	−0.258992
$206$	0	0
$207$	15.3262	1.06525
$208$	0	0
$209$	−8.94427	−0.618688
$210$	0	0
$211$	15.2361	1.04889	0.524447	−	0.851443i	$-0.324272\pi$
$211$	15.2361	1.04889	0.524447	+	0.851443i	$0.324272\pi$
$212$	0	0
$213$	−8.00000	−0.548151
$214$	0	0
$215$	−7.61803	−0.519546
$216$	0	0
$217$	4.61803	0.313493
$218$	0	0
$219$	1.18034	0.0797600
$220$	0	0
$221$	−18.4164	−1.23882
$222$	0	0
$223$	24.9787	1.67270	0.836349	−	0.548197i	$-0.184686\pi$
$223$	24.9787	1.67270	0.836349	+	0.548197i	$0.184686\pi$
$224$	0	0
$225$	−2.61803	−0.174536
$226$	0	0
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	−4.09017	−0.270286	−0.135143	−	0.990826i	$-0.543149\pi$
$229$	−4.09017	−0.270286	−0.135143	+	0.990826i	$0.543149\pi$
$230$	0	0
$231$	−5.70820	−0.375572
$232$	0	0
$233$	16.4721	1.07913	0.539563	−	0.841945i	$-0.318590\pi$
$233$	16.4721	1.07913	0.539563	+	0.841945i	$0.318590\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	1.79837	0.116817
$238$	0	0
$239$	−11.5279	−0.745676	−0.372838	−	0.927897i	$-0.621615\pi$
$239$	−11.5279	−0.745676	−0.372838	+	0.927897i	$0.621615\pi$
$240$	0	0
$241$	−1.20163	−0.0774035	−0.0387018	−	0.999251i	$-0.512322\pi$
$241$	−1.20163	−0.0774035	−0.0387018	+	0.999251i	$0.512322\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	0	0
$245$	1.14590	0.0732087
$246$	0	0
$247$	14.0689	0.895182
$248$	0	0
$249$	9.70820	0.615232
$250$	0	0
$251$	−18.1803	−1.14753	−0.573766	−	0.819019i	$-0.694518\pi$
$251$	−18.1803	−1.14753	−0.573766	+	0.819019i	$0.694518\pi$
$252$	0	0
$253$	18.9443	1.19102
$254$	0	0
$255$	−2.23607	−0.140028
$256$	0	0
$257$	−28.0689	−1.75089	−0.875444	−	0.483319i	$-0.839431\pi$
$257$	−28.0689	−1.75089	−0.875444	+	0.483319i	$0.839431\pi$
$258$	0	0
$259$	−26.3607	−1.63797
$260$	0	0
$261$	2.61803	0.162052
$262$	0	0
$263$	17.5279	1.08081	0.540407	−	0.841404i	$-0.318270\pi$
$263$	17.5279	1.08081	0.540407	+	0.841404i	$0.318270\pi$
$264$	0	0
$265$	−9.32624	−0.572906
$266$	0	0
$267$	−1.05573	−0.0646095
$268$	0	0
$269$	−26.5623	−1.61953	−0.809766	−	0.586753i	$-0.800406\pi$
$269$	−26.5623	−1.61953	−0.809766	+	0.586753i	$0.800406\pi$
$270$	0	0
$271$	15.4164	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$271$	15.4164	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$272$	0	0
$273$	8.97871	0.543416
$274$	0	0
$275$	−3.23607	−0.195142
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	0	0
$279$	−4.23607	−0.253607
$280$	0	0
$281$	−4.85410	−0.289571	−0.144786	−	0.989463i	$-0.546249\pi$
$281$	−4.85410	−0.289571	−0.144786	+	0.989463i	$0.546249\pi$
$282$	0	0
$283$	22.1803	1.31848	0.659242	−	0.751931i	$-0.270877\pi$
$283$	22.1803	1.31848	0.659242	+	0.751931i	$0.270877\pi$
$284$	0	0
$285$	1.70820	0.101185
$286$	0	0
$287$	−10.5836	−0.624730
$288$	0	0
$289$	−3.90983	−0.229990
$290$	0	0
$291$	5.43769	0.318763
$292$	0	0
$293$	−14.1803	−0.828424	−0.414212	−	0.910180i	$-0.635943\pi$
$293$	−14.1803	−0.828424	−0.414212	+	0.910180i	$0.635943\pi$
$294$	0	0
$295$	−9.38197	−0.546239
$296$	0	0
$297$	11.2361	0.651983
$298$	0	0
$299$	−29.7984	−1.72328
$300$	0	0
$301$	−21.7426	−1.25323
$302$	0	0
$303$	4.38197	0.251737
$304$	0	0
$305$	−2.14590	−0.122874
$306$	0	0
$307$	−9.52786	−0.543784	−0.271892	−	0.962328i	$-0.587649\pi$
$307$	−9.52786	−0.543784	−0.271892	+	0.962328i	$0.587649\pi$
$308$	0	0
$309$	−5.52786	−0.314469
$310$	0	0
$311$	−6.32624	−0.358728	−0.179364	−	0.983783i	$-0.557404\pi$
$311$	−6.32624	−0.358728	−0.179364	+	0.983783i	$0.557404\pi$
$312$	0	0
$313$	−3.88854	−0.219793	−0.109897	−	0.993943i	$-0.535052\pi$
$313$	−3.88854	−0.219793	−0.109897	+	0.993943i	$0.535052\pi$
$314$	0	0
$315$	−7.47214	−0.421007
$316$	0	0
$317$	4.76393	0.267569	0.133785	−	0.991010i	$-0.457287\pi$
$317$	4.76393	0.267569	0.133785	+	0.991010i	$0.457287\pi$
$318$	0	0
$319$	3.23607	0.181185
$320$	0	0
$321$	0.180340	0.0100656
$322$	0	0
$323$	−10.0000	−0.556415
$324$	0	0
$325$	5.09017	0.282352
$326$	0	0
$327$	−7.23607	−0.400155
$328$	0	0
$329$	22.8328	1.25881
$330$	0	0
$331$	−10.2918	−0.565688	−0.282844	−	0.959166i	$-0.591278\pi$
$331$	−10.2918	−0.565688	−0.282844	+	0.959166i	$0.591278\pi$
$332$	0	0
$333$	24.1803	1.32507
$334$	0	0
$335$	−2.47214	−0.135067
$336$	0	0
$337$	17.1459	0.933997	0.466998	−	0.884258i	$-0.345335\pi$
$337$	17.1459	0.933997	0.466998	+	0.884258i	$0.345335\pi$
$338$	0	0
$339$	5.43769	0.295335
$340$	0	0
$341$	−5.23607	−0.283549
$342$	0	0
$343$	−16.7082	−0.902158
$344$	0	0
$345$	−3.61803	−0.194788
$346$	0	0
$347$	30.7639	1.65149	0.825747	−	0.564040i	$-0.190754\pi$
$347$	30.7639	1.65149	0.825747	+	0.564040i	$0.190754\pi$
$348$	0	0
$349$	12.9443	0.692891	0.346445	−	0.938070i	$-0.387389\pi$
$349$	12.9443	0.692891	0.346445	+	0.938070i	$0.387389\pi$
$350$	0	0
$351$	−17.6738	−0.943356
$352$	0	0
$353$	−18.4721	−0.983173	−0.491586	−	0.870829i	$-0.663583\pi$
$353$	−18.4721	−0.983173	−0.491586	+	0.870829i	$0.663583\pi$
$354$	0	0
$355$	−12.9443	−0.687011
$356$	0	0
$357$	−6.38197	−0.337769
$358$	0	0
$359$	−14.9098	−0.786911	−0.393455	−	0.919344i	$-0.628720\pi$
$359$	−14.9098	−0.786911	−0.393455	+	0.919344i	$0.628720\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	0	0
$363$	−0.326238	−0.0171231
$364$	0	0
$365$	1.90983	0.0999651
$366$	0	0
$367$	18.1803	0.949006	0.474503	−	0.880254i	$-0.342628\pi$
$367$	18.1803	0.949006	0.474503	+	0.880254i	$0.342628\pi$
$368$	0	0
$369$	9.70820	0.505389
$370$	0	0
$371$	−26.6180	−1.38194
$372$	0	0
$373$	−26.2705	−1.36024	−0.680118	−	0.733103i	$-0.738071\pi$
$373$	−26.2705	−1.36024	−0.680118	+	0.733103i	$0.738071\pi$
$374$	0	0
$375$	0.618034	0.0319151
$376$	0	0
$377$	−5.09017	−0.262157
$378$	0	0
$379$	38.0689	1.95547	0.977734	−	0.209850i	$-0.0672975\pi$
$379$	38.0689	1.95547	0.977734	+	0.209850i	$0.0672975\pi$
$380$	0	0
$381$	0.583592	0.0298983
$382$	0	0
$383$	26.2705	1.34236	0.671180	−	0.741294i	$-0.265788\pi$
$383$	26.2705	1.34236	0.671180	+	0.741294i	$0.265788\pi$
$384$	0	0
$385$	−9.23607	−0.470714
$386$	0	0
$387$	19.9443	1.01382
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	21.1803	1.07114
$392$	0	0
$393$	−11.7082	−0.590601
$394$	0	0
$395$	2.90983	0.146409
$396$	0	0
$397$	−5.14590	−0.258265	−0.129133	−	0.991627i	$-0.541219\pi$
$397$	−5.14590	−0.258265	−0.129133	+	0.991627i	$0.541219\pi$
$398$	0	0
$399$	4.87539	0.244075
$400$	0	0
$401$	8.32624	0.415792	0.207896	−	0.978151i	$-0.433338\pi$
$401$	8.32624	0.415792	0.207896	+	0.978151i	$0.433338\pi$
$402$	0	0
$403$	8.23607	0.410268
$404$	0	0
$405$	5.70820	0.283643
$406$	0	0
$407$	29.8885	1.48152
$408$	0	0
$409$	7.41641	0.366718	0.183359	−	0.983046i	$-0.441303\pi$
$409$	7.41641	0.366718	0.183359	+	0.983046i	$0.441303\pi$
$410$	0	0
$411$	−0.0901699	−0.00444776
$412$	0	0
$413$	−26.7771	−1.31761
$414$	0	0
$415$	15.7082	0.771085
$416$	0	0
$417$	3.47214	0.170031
$418$	0	0
$419$	3.90983	0.191008	0.0955038	−	0.995429i	$-0.469554\pi$
$419$	3.90983	0.191008	0.0955038	+	0.995429i	$0.469554\pi$
$420$	0	0
$421$	−7.52786	−0.366886	−0.183443	−	0.983030i	$-0.558724\pi$
$421$	−7.52786	−0.366886	−0.183443	+	0.983030i	$0.558724\pi$
$422$	0	0
$423$	−20.9443	−1.01835
$424$	0	0
$425$	−3.61803	−0.175500
$426$	0	0
$427$	−6.12461	−0.296391
$428$	0	0
$429$	−10.1803	−0.491511
$430$	0	0
$431$	−23.7082	−1.14198	−0.570992	−	0.820956i	$-0.693441\pi$
$431$	−23.7082	−1.14198	−0.570992	+	0.820956i	$0.693441\pi$
$432$	0	0
$433$	4.47214	0.214917	0.107459	−	0.994210i	$-0.465729\pi$
$433$	4.47214	0.214917	0.107459	+	0.994210i	$0.465729\pi$
$434$	0	0
$435$	−0.618034	−0.0296325
$436$	0	0
$437$	−16.1803	−0.774011
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	25.2705	1.20064	0.600319	−	0.799761i	$-0.295040\pi$
$443$	25.2705	1.20064	0.600319	+	0.799761i	$0.295040\pi$
$444$	0	0
$445$	−1.70820	−0.0809766
$446$	0	0
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−24.0689	−1.13588	−0.567940	−	0.823070i	$-0.692260\pi$
$449$	−24.0689	−1.13588	−0.567940	+	0.823070i	$0.692260\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	0	0
$453$	−0.763932	−0.0358927
$454$	0	0
$455$	14.5279	0.681077
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	12.5623	0.586358
$460$	0	0
$461$	19.3262	0.900113	0.450056	−	0.893000i	$-0.351404\pi$
$461$	19.3262	0.900113	0.450056	+	0.893000i	$0.351404\pi$
$462$	0	0
$463$	38.8328	1.80471	0.902357	−	0.430989i	$-0.141835\pi$
$463$	38.8328	1.80471	0.902357	+	0.430989i	$0.141835\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−22.6738	−1.04922	−0.524608	−	0.851344i	$-0.675788\pi$
$467$	−22.6738	−1.04922	−0.524608	+	0.851344i	$0.675788\pi$
$468$	0	0
$469$	−7.05573	−0.325803
$470$	0	0
$471$	−3.12461	−0.143975
$472$	0	0
$473$	24.6525	1.13352
$474$	0	0
$475$	2.76393	0.126818
$476$	0	0
$477$	24.4164	1.11795
$478$	0	0
$479$	−0.673762	−0.0307850	−0.0153925	−	0.999882i	$-0.504900\pi$
$479$	−0.673762	−0.0307850	−0.0153925	+	0.999882i	$0.504900\pi$
$480$	0	0
$481$	−47.0132	−2.14362
$482$	0	0
$483$	−10.3262	−0.469860
$484$	0	0
$485$	8.79837	0.399514
$486$	0	0
$487$	21.2705	0.963859	0.481929	−	0.876210i	$-0.339936\pi$
$487$	21.2705	0.963859	0.481929	+	0.876210i	$0.339936\pi$
$488$	0	0
$489$	−8.36068	−0.378083
$490$	0	0
$491$	−41.3050	−1.86407	−0.932033	−	0.362373i	$-0.881967\pi$
$491$	−41.3050	−1.86407	−0.932033	+	0.362373i	$0.881967\pi$
$492$	0	0
$493$	3.61803	0.162948
$494$	0	0
$495$	8.47214	0.380794
$496$	0	0
$497$	−36.9443	−1.65718
$498$	0	0
$499$	−1.67376	−0.0749279	−0.0374639	−	0.999298i	$-0.511928\pi$
$499$	−1.67376	−0.0749279	−0.0374639	+	0.999298i	$0.511928\pi$
$500$	0	0
$501$	2.70820	0.120994
$502$	0	0
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	7.09017	0.315508
$506$	0	0
$507$	7.97871	0.354347
$508$	0	0
$509$	3.70820	0.164363	0.0821816	−	0.996617i	$-0.473811\pi$
$509$	3.70820	0.164363	0.0821816	+	0.996617i	$0.473811\pi$
$510$	0	0
$511$	5.45085	0.241131
$512$	0	0
$513$	−9.59675	−0.423707
$514$	0	0
$515$	−8.94427	−0.394132
$516$	0	0
$517$	−25.8885	−1.13858
$518$	0	0
$519$	2.96556	0.130174
$520$	0	0
$521$	−35.3262	−1.54767	−0.773835	−	0.633387i	$-0.781664\pi$
$521$	−35.3262	−1.54767	−0.773835	+	0.633387i	$0.781664\pi$
$522$	0	0
$523$	−10.1803	−0.445155	−0.222578	−	0.974915i	$-0.571447\pi$
$523$	−10.1803	−0.445155	−0.222578	+	0.974915i	$0.571447\pi$
$524$	0	0
$525$	1.76393	0.0769843
$526$	0	0
$527$	−5.85410	−0.255009
$528$	0	0
$529$	11.2705	0.490022
$530$	0	0
$531$	24.5623	1.06591
$532$	0	0
$533$	−18.8754	−0.817584
$534$	0	0
$535$	0.291796	0.0126154
$536$	0	0
$537$	−0.673762	−0.0290750
$538$	0	0
$539$	−3.70820	−0.159724
$540$	0	0
$541$	−5.03444	−0.216448	−0.108224	−	0.994127i	$-0.534516\pi$
$541$	−5.03444	−0.216448	−0.108224	+	0.994127i	$0.534516\pi$
$542$	0	0
$543$	−0.180340	−0.00773913
$544$	0	0
$545$	−11.7082	−0.501524
$546$	0	0
$547$	19.1246	0.817709	0.408855	−	0.912600i	$-0.365928\pi$
$547$	19.1246	0.817709	0.408855	+	0.912600i	$0.365928\pi$
$548$	0	0
$549$	5.61803	0.239772
$550$	0	0
$551$	−2.76393	−0.117747
$552$	0	0
$553$	8.30495	0.353162
$554$	0	0
$555$	−5.70820	−0.242300
$556$	0	0
$557$	−24.1459	−1.02309	−0.511547	−	0.859255i	$-0.670927\pi$
$557$	−24.1459	−1.02309	−0.511547	+	0.859255i	$0.670927\pi$
$558$	0	0
$559$	−38.7771	−1.64010
$560$	0	0
$561$	7.23607	0.305507
$562$	0	0
$563$	28.3951	1.19671	0.598356	−	0.801230i	$-0.295821\pi$
$563$	28.3951	1.19671	0.598356	+	0.801230i	$0.295821\pi$
$564$	0	0
$565$	8.79837	0.370150
$566$	0	0
$567$	16.2918	0.684191
$568$	0	0
$569$	32.3607	1.35663	0.678315	−	0.734771i	$-0.262710\pi$
$569$	32.3607	1.35663	0.678315	+	0.734771i	$0.262710\pi$
$570$	0	0
$571$	−20.1459	−0.843080	−0.421540	−	0.906810i	$-0.638510\pi$
$571$	−20.1459	−0.843080	−0.421540	+	0.906810i	$0.638510\pi$
$572$	0	0
$573$	12.8885	0.538427
$574$	0	0
$575$	−5.85410	−0.244133
$576$	0	0
$577$	12.7426	0.530483	0.265242	−	0.964182i	$-0.414548\pi$
$577$	12.7426	0.530483	0.265242	+	0.964182i	$0.414548\pi$
$578$	0	0
$579$	0.0901699	0.00374733
$580$	0	0
$581$	44.8328	1.85998
$582$	0	0
$583$	30.1803	1.24994
$584$	0	0
$585$	−13.3262	−0.550972
$586$	0	0
$587$	−8.83282	−0.364569	−0.182285	−	0.983246i	$-0.558349\pi$
$587$	−8.83282	−0.364569	−0.182285	+	0.983246i	$0.558349\pi$
$588$	0	0
$589$	4.47214	0.184271
$590$	0	0
$591$	−11.6525	−0.479319
$592$	0	0
$593$	−6.65248	−0.273184	−0.136592	−	0.990627i	$-0.543615\pi$
$593$	−6.65248	−0.273184	−0.136592	+	0.990627i	$0.543615\pi$
$594$	0	0
$595$	−10.3262	−0.423334
$596$	0	0
$597$	5.41641	0.221679
$598$	0	0
$599$	0.673762	0.0275292	0.0137646	−	0.999905i	$-0.495618\pi$
$599$	0.673762	0.0275292	0.0137646	+	0.999905i	$0.495618\pi$
$600$	0	0
$601$	20.5410	0.837886	0.418943	−	0.908013i	$-0.362401\pi$
$601$	20.5410	0.837886	0.418943	+	0.908013i	$0.362401\pi$
$602$	0	0
$603$	6.47214	0.263566
$604$	0	0
$605$	−0.527864	−0.0214607
$606$	0	0
$607$	2.47214	0.100341	0.0501705	−	0.998741i	$-0.484024\pi$
$607$	2.47214	0.100341	0.0501705	+	0.998741i	$0.484024\pi$
$608$	0	0
$609$	−1.76393	−0.0714781
$610$	0	0
$611$	40.7214	1.64741
$612$	0	0
$613$	−32.7426	−1.32246	−0.661232	−	0.750182i	$-0.729966\pi$
$613$	−32.7426	−1.32246	−0.661232	+	0.750182i	$0.729966\pi$
$614$	0	0
$615$	−2.29180	−0.0924141
$616$	0	0
$617$	−31.7984	−1.28015	−0.640077	−	0.768311i	$-0.721098\pi$
$617$	−31.7984	−1.28015	−0.640077	+	0.768311i	$0.721098\pi$
$618$	0	0
$619$	41.7771	1.67916	0.839581	−	0.543234i	$-0.182800\pi$
$619$	41.7771	1.67916	0.839581	+	0.543234i	$0.182800\pi$
$620$	0	0
$621$	20.3262	0.815664
$622$	0	0
$623$	−4.87539	−0.195328
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.52786	−0.220762
$628$	0	0
$629$	33.4164	1.33240
$630$	0	0
$631$	−38.3607	−1.52711	−0.763557	−	0.645740i	$-0.776549\pi$
$631$	−38.3607	−1.52711	−0.763557	+	0.645740i	$0.776549\pi$
$632$	0	0
$633$	9.41641	0.374269
$634$	0	0
$635$	0.944272	0.0374723
$636$	0	0
$637$	5.83282	0.231105
$638$	0	0
$639$	33.8885	1.34061
$640$	0	0
$641$	−38.7639	−1.53108	−0.765542	−	0.643386i	$-0.777529\pi$
$641$	−38.7639	−1.53108	−0.765542	+	0.643386i	$0.777529\pi$
$642$	0	0
$643$	−25.7082	−1.01383	−0.506916	−	0.861995i	$-0.669215\pi$
$643$	−25.7082	−1.01383	−0.506916	+	0.861995i	$0.669215\pi$
$644$	0	0
$645$	−4.70820	−0.185385
$646$	0	0
$647$	−37.3050	−1.46661	−0.733304	−	0.679900i	$-0.762023\pi$
$647$	−37.3050	−1.46661	−0.733304	+	0.679900i	$0.762023\pi$
$648$	0	0
$649$	30.3607	1.19176
$650$	0	0
$651$	2.85410	0.111861
$652$	0	0
$653$	−39.5967	−1.54954	−0.774770	−	0.632243i	$-0.782134\pi$
$653$	−39.5967	−1.54954	−0.774770	+	0.632243i	$0.782134\pi$
$654$	0	0
$655$	−18.9443	−0.740214
$656$	0	0
$657$	−5.00000	−0.195069
$658$	0	0
$659$	−15.8197	−0.616246	−0.308123	−	0.951346i	$-0.599701\pi$
$659$	−15.8197	−0.616246	−0.308123	+	0.951346i	$0.599701\pi$
$660$	0	0
$661$	−27.5967	−1.07339	−0.536695	−	0.843777i	$-0.680327\pi$
$661$	−27.5967	−1.07339	−0.536695	+	0.843777i	$0.680327\pi$
$662$	0	0
$663$	−11.3820	−0.442039
$664$	0	0
$665$	7.88854	0.305905
$666$	0	0
$667$	5.85410	0.226672
$668$	0	0
$669$	15.4377	0.596856
$670$	0	0
$671$	6.94427	0.268081
$672$	0	0
$673$	18.7639	0.723296	0.361648	−	0.932315i	$-0.382214\pi$
$673$	18.7639	0.723296	0.361648	+	0.932315i	$0.382214\pi$
$674$	0	0
$675$	−3.47214	−0.133643
$676$	0	0
$677$	13.8885	0.533780	0.266890	−	0.963727i	$-0.414004\pi$
$677$	13.8885	0.533780	0.266890	+	0.963727i	$0.414004\pi$
$678$	0	0
$679$	25.1115	0.963689
$680$	0	0
$681$	1.23607	0.0473662
$682$	0	0
$683$	15.3475	0.587257	0.293628	−	0.955920i	$-0.405137\pi$
$683$	15.3475	0.587257	0.293628	+	0.955920i	$0.405137\pi$
$684$	0	0
$685$	−0.145898	−0.00557448
$686$	0	0
$687$	−2.52786	−0.0964440
$688$	0	0
$689$	−47.4721	−1.80854
$690$	0	0
$691$	13.0344	0.495854	0.247927	−	0.968779i	$-0.420251\pi$
$691$	13.0344	0.495854	0.247927	+	0.968779i	$0.420251\pi$
$692$	0	0
$693$	24.1803	0.918535
$694$	0	0
$695$	5.61803	0.213104
$696$	0	0
$697$	13.4164	0.508183
$698$	0	0
$699$	10.1803	0.385056
$700$	0	0
$701$	13.1246	0.495710	0.247855	−	0.968797i	$-0.420274\pi$
$701$	13.1246	0.495710	0.247855	+	0.968797i	$0.420274\pi$
$702$	0	0
$703$	−25.5279	−0.962802
$704$	0	0
$705$	4.94427	0.186212
$706$	0	0
$707$	20.2361	0.761056
$708$	0	0
$709$	18.2918	0.686963	0.343481	−	0.939159i	$-0.388394\pi$
$709$	18.2918	0.686963	0.343481	+	0.939159i	$0.388394\pi$
$710$	0	0
$711$	−7.61803	−0.285699
$712$	0	0
$713$	−9.47214	−0.354734
$714$	0	0
$715$	−16.4721	−0.616023
$716$	0	0
$717$	−7.12461	−0.266074
$718$	0	0
$719$	−20.6525	−0.770207	−0.385104	−	0.922873i	$-0.625834\pi$
$719$	−20.6525	−0.770207	−0.385104	+	0.922873i	$0.625834\pi$
$720$	0	0
$721$	−25.5279	−0.950707
$722$	0	0
$723$	−0.742646	−0.0276193
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−44.7639	−1.66020	−0.830101	−	0.557613i	$-0.811717\pi$
$727$	−44.7639	−1.66020	−0.830101	+	0.557613i	$0.811717\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	27.5623	1.01943
$732$	0	0
$733$	−35.7771	−1.32146	−0.660728	−	0.750625i	$-0.729753\pi$
$733$	−35.7771	−1.32146	−0.660728	+	0.750625i	$0.729753\pi$
$734$	0	0
$735$	0.708204	0.0261225
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	−3.41641	−0.125675	−0.0628373	−	0.998024i	$-0.520015\pi$
$739$	−3.41641	−0.125675	−0.0628373	+	0.998024i	$0.520015\pi$
$740$	0	0
$741$	8.69505	0.319421
$742$	0	0
$743$	7.41641	0.272082	0.136041	−	0.990703i	$-0.456562\pi$
$743$	7.41641	0.272082	0.136041	+	0.990703i	$0.456562\pi$
$744$	0	0
$745$	−16.1803	−0.592802
$746$	0	0
$747$	−41.1246	−1.50467
$748$	0	0
$749$	0.832816	0.0304304
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	−11.2361	−0.409465
$754$	0	0
$755$	−1.23607	−0.0449851
$756$	0	0
$757$	−22.1803	−0.806158	−0.403079	−	0.915165i	$-0.632060\pi$
$757$	−22.1803	−0.806158	−0.403079	+	0.915165i	$0.632060\pi$
$758$	0	0
$759$	11.7082	0.424981
$760$	0	0
$761$	−25.0344	−0.907498	−0.453749	−	0.891130i	$-0.649914\pi$
$761$	−25.0344	−0.907498	−0.453749	+	0.891130i	$0.649914\pi$
$762$	0	0
$763$	−33.4164	−1.20976
$764$	0	0
$765$	9.47214	0.342466
$766$	0	0
$767$	−47.7558	−1.72436
$768$	0	0
$769$	50.6525	1.82657	0.913287	−	0.407316i	$-0.133535\pi$
$769$	50.6525	1.82657	0.913287	+	0.407316i	$0.133535\pi$
$770$	0	0
$771$	−17.3475	−0.624756
$772$	0	0
$773$	52.8328	1.90026	0.950132	−	0.311848i	$-0.100948\pi$
$773$	52.8328	1.90026	0.950132	+	0.311848i	$0.100948\pi$
$774$	0	0
$775$	1.61803	0.0581215
$776$	0	0
$777$	−16.2918	−0.584465
$778$	0	0
$779$	−10.2492	−0.367217
$780$	0	0
$781$	41.8885	1.49889
$782$	0	0
$783$	3.47214	0.124084
$784$	0	0
$785$	−5.05573	−0.180447
$786$	0	0
$787$	−44.6525	−1.59169	−0.795844	−	0.605501i	$-0.792973\pi$
$787$	−44.6525	−1.59169	−0.795844	+	0.605501i	$0.792973\pi$
$788$	0	0
$789$	10.8328	0.385658
$790$	0	0
$791$	25.1115	0.892861
$792$	0	0
$793$	−10.9230	−0.387887
$794$	0	0
$795$	−5.76393	−0.204426
$796$	0	0
$797$	−30.8328	−1.09215	−0.546077	−	0.837735i	$-0.683879\pi$
$797$	−30.8328	−1.09215	−0.546077	+	0.837735i	$0.683879\pi$
$798$	0	0
$799$	−28.9443	−1.02397
$800$	0	0
$801$	4.47214	0.158015
$802$	0	0
$803$	−6.18034	−0.218099
$804$	0	0
$805$	−16.7082	−0.588887
$806$	0	0
$807$	−16.4164	−0.577885
$808$	0	0
$809$	11.4164	0.401380	0.200690	−	0.979655i	$-0.435682\pi$
$809$	11.4164	0.401380	0.200690	+	0.979655i	$0.435682\pi$
$810$	0	0
$811$	−0.673762	−0.0236590	−0.0118295	−	0.999930i	$-0.503766\pi$
$811$	−0.673762	−0.0236590	−0.0118295	+	0.999930i	$0.503766\pi$
$812$	0	0
$813$	9.52786	0.334157
$814$	0	0
$815$	−13.5279	−0.473860
$816$	0	0
$817$	−21.0557	−0.736647
$818$	0	0
$819$	−38.0344	−1.32903
$820$	0	0
$821$	34.0689	1.18901	0.594506	−	0.804091i	$-0.297348\pi$
$821$	34.0689	1.18901	0.594506	+	0.804091i	$0.297348\pi$
$822$	0	0
$823$	45.7771	1.59569	0.797844	−	0.602863i	$-0.205974\pi$
$823$	45.7771	1.59569	0.797844	+	0.602863i	$0.205974\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	−1.90983	−0.0664113	−0.0332056	−	0.999449i	$-0.510572\pi$
$827$	−1.90983	−0.0664113	−0.0332056	+	0.999449i	$0.510572\pi$
$828$	0	0
$829$	45.7984	1.59064	0.795322	−	0.606188i	$-0.207302\pi$
$829$	45.7984	1.59064	0.795322	+	0.606188i	$0.207302\pi$
$830$	0	0
$831$	−18.5410	−0.643181
$832$	0	0
$833$	−4.14590	−0.143647
$834$	0	0
$835$	4.38197	0.151644
$836$	0	0
$837$	−5.61803	−0.194188
$838$	0	0
$839$	28.3607	0.979119	0.489560	−	0.871970i	$-0.337158\pi$
$839$	28.3607	0.979119	0.489560	+	0.871970i	$0.337158\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.00000	−0.103325
$844$	0	0
$845$	12.9098	0.444112
$846$	0	0
$847$	−1.50658	−0.0517666
$848$	0	0
$849$	13.7082	0.470464
$850$	0	0
$851$	54.0689	1.85346
$852$	0	0
$853$	−1.81966	−0.0623040	−0.0311520	−	0.999515i	$-0.509918\pi$
$853$	−1.81966	−0.0623040	−0.0311520	+	0.999515i	$0.509918\pi$
$854$	0	0
$855$	−7.23607	−0.247468
$856$	0	0
$857$	33.7082	1.15145	0.575725	−	0.817643i	$-0.304720\pi$
$857$	33.7082	1.15145	0.575725	+	0.817643i	$0.304720\pi$
$858$	0	0
$859$	−16.6525	−0.568175	−0.284088	−	0.958798i	$-0.591691\pi$
$859$	−16.6525	−0.568175	−0.284088	+	0.958798i	$0.591691\pi$
$860$	0	0
$861$	−6.54102	−0.222917
$862$	0	0
$863$	−4.03444	−0.137334	−0.0686670	−	0.997640i	$-0.521875\pi$
$863$	−4.03444	−0.137334	−0.0686670	+	0.997640i	$0.521875\pi$
$864$	0	0
$865$	4.79837	0.163150
$866$	0	0
$867$	−2.41641	−0.0820655
$868$	0	0
$869$	−9.41641	−0.319430
$870$	0	0
$871$	−12.5836	−0.426379
$872$	0	0
$873$	−23.0344	−0.779598
$874$	0	0
$875$	2.85410	0.0964863
$876$	0	0
$877$	−50.6180	−1.70925	−0.854625	−	0.519246i	$-0.826213\pi$
$877$	−50.6180	−1.70925	−0.854625	+	0.519246i	$0.826213\pi$
$878$	0	0
$879$	−8.76393	−0.295600
$880$	0	0
$881$	10.1803	0.342984	0.171492	−	0.985185i	$-0.445141\pi$
$881$	10.1803	0.342984	0.171492	+	0.985185i	$0.445141\pi$
$882$	0	0
$883$	21.5967	0.726788	0.363394	−	0.931635i	$-0.381618\pi$
$883$	21.5967	0.726788	0.363394	+	0.931635i	$0.381618\pi$
$884$	0	0
$885$	−5.79837	−0.194910
$886$	0	0
$887$	−31.4164	−1.05486	−0.527430	−	0.849599i	$-0.676844\pi$
$887$	−31.4164	−1.05486	−0.527430	+	0.849599i	$0.676844\pi$
$888$	0	0
$889$	2.69505	0.0903890
$890$	0	0
$891$	−18.4721	−0.618840
$892$	0	0
$893$	22.1115	0.739932
$894$	0	0
$895$	−1.09017	−0.0364404
$896$	0	0
$897$	−18.4164	−0.614906
$898$	0	0
$899$	−1.61803	−0.0539645
$900$	0	0
$901$	33.7426	1.12413
$902$	0	0
$903$	−13.4377	−0.447178
$904$	0	0
$905$	−0.291796	−0.00969963
$906$	0	0
$907$	−25.9098	−0.860322	−0.430161	−	0.902752i	$-0.641543\pi$
$907$	−25.9098	−0.860322	−0.430161	+	0.902752i	$0.641543\pi$
$908$	0	0
$909$	−18.5623	−0.615673
$910$	0	0
$911$	−39.2705	−1.30109	−0.650545	−	0.759468i	$-0.725459\pi$
$911$	−39.2705	−1.30109	−0.650545	+	0.759468i	$0.725459\pi$
$912$	0	0
$913$	−50.8328	−1.68232
$914$	0	0
$915$	−1.32624	−0.0438441
$916$	0	0
$917$	−54.0689	−1.78551
$918$	0	0
$919$	−26.4721	−0.873235	−0.436618	−	0.899647i	$-0.643824\pi$
$919$	−26.4721	−0.873235	−0.436618	+	0.899647i	$0.643824\pi$
$920$	0	0
$921$	−5.88854	−0.194034
$922$	0	0
$923$	−65.8885	−2.16875
$924$	0	0
$925$	−9.23607	−0.303680
$926$	0	0
$927$	23.4164	0.769096
$928$	0	0
$929$	−44.4508	−1.45839	−0.729193	−	0.684309i	$-0.760104\pi$
$929$	−44.4508	−1.45839	−0.729193	+	0.684309i	$0.760104\pi$
$930$	0	0
$931$	3.16718	0.103800
$932$	0	0
$933$	−3.90983	−0.128002
$934$	0	0
$935$	11.7082	0.382899
$936$	0	0
$937$	43.1246	1.40882	0.704410	−	0.709793i	$-0.251212\pi$
$937$	43.1246	1.40882	0.704410	+	0.709793i	$0.251212\pi$
$938$	0	0
$939$	−2.40325	−0.0784272
$940$	0	0
$941$	−32.3607	−1.05493	−0.527464	−	0.849577i	$-0.676857\pi$
$941$	−32.3607	−1.05493	−0.527464	+	0.849577i	$0.676857\pi$
$942$	0	0
$943$	21.7082	0.706916
$944$	0	0
$945$	−9.90983	−0.322367
$946$	0	0
$947$	55.0902	1.79019	0.895095	−	0.445875i	$-0.147108\pi$
$947$	55.0902	1.79019	0.895095	+	0.445875i	$0.147108\pi$
$948$	0	0
$949$	9.72136	0.315569
$950$	0	0
$951$	2.94427	0.0954746
$952$	0	0
$953$	12.8328	0.415696	0.207848	−	0.978161i	$-0.433354\pi$
$953$	12.8328	0.415696	0.207848	+	0.978161i	$0.433354\pi$
$954$	0	0
$955$	20.8541	0.674823
$956$	0	0
$957$	2.00000	0.0646508
$958$	0	0
$959$	−0.416408	−0.0134465
$960$	0	0
$961$	−28.3820	−0.915547
$962$	0	0
$963$	−0.763932	−0.0246174
$964$	0	0
$965$	0.145898	0.00469662
$966$	0	0
$967$	23.5967	0.758820	0.379410	−	0.925229i	$-0.376127\pi$
$967$	23.5967	0.758820	0.379410	+	0.925229i	$0.376127\pi$
$968$	0	0
$969$	−6.18034	−0.198541
$970$	0	0
$971$	52.0689	1.67097	0.835485	−	0.549513i	$-0.185187\pi$
$971$	52.0689	1.67097	0.835485	+	0.549513i	$0.185187\pi$
$972$	0	0
$973$	16.0344	0.514041
$974$	0	0
$975$	3.14590	0.100749
$976$	0	0
$977$	10.6525	0.340803	0.170401	−	0.985375i	$-0.445494\pi$
$977$	10.6525	0.340803	0.170401	+	0.985375i	$0.445494\pi$
$978$	0	0
$979$	5.52786	0.176671
$980$	0	0
$981$	30.6525	0.978658
$982$	0	0
$983$	13.0557	0.416413	0.208207	−	0.978085i	$-0.433237\pi$
$983$	13.0557	0.416413	0.208207	+	0.978085i	$0.433237\pi$
$984$	0	0
$985$	−18.8541	−0.600742
$986$	0	0
$987$	14.1115	0.449173
$988$	0	0
$989$	44.5967	1.41809
$990$	0	0
$991$	10.0689	0.319849	0.159924	−	0.987129i	$-0.448875\pi$
$991$	10.0689	0.319849	0.159924	+	0.987129i	$0.448875\pi$
$992$	0	0
$993$	−6.36068	−0.201850
$994$	0	0
$995$	8.76393	0.277835
$996$	0	0
$997$	−44.1803	−1.39921	−0.699603	−	0.714532i	$-0.746640\pi$
$997$	−44.1803	−1.39921	−0.699603	+	0.714532i	$0.746640\pi$
$998$	0	0
$999$	32.0689	1.01461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.y.1.2		2
4.3	odd	2		9280.2.a.bd.1.1		2
8.3	odd	2		4640.2.a.g.1.2	✓	2
8.5	even	2		4640.2.a.i.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.g.1.2	✓	2	8.3	odd	2
4640.2.a.i.1.1	yes	2	8.5	even	2
9280.2.a.y.1.2		2	1.1	even	1	trivial
9280.2.a.bd.1.1		2	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{3} +1.00000 q^{5} +2.85410 q^{7} -2.61803 q^{9} -3.23607 q^{11} +5.09017 q^{13} +0.618034 q^{15} -3.61803 q^{17} +2.76393 q^{19} +1.76393 q^{21} -5.85410 q^{23} +1.00000 q^{25} -3.47214 q^{27} -1.00000 q^{29} +1.61803 q^{31} -2.00000 q^{33} +2.85410 q^{35} -9.23607 q^{37} +3.14590 q^{39} -3.70820 q^{41} -7.61803 q^{43} -2.61803 q^{45} +8.00000 q^{47} +1.14590 q^{49} -2.23607 q^{51} -9.32624 q^{53} -3.23607 q^{55} +1.70820 q^{57} -9.38197 q^{59} -2.14590 q^{61} -7.47214 q^{63} +5.09017 q^{65} -2.47214 q^{67} -3.61803 q^{69} -12.9443 q^{71} +1.90983 q^{73} +0.618034 q^{75} -9.23607 q^{77} +2.90983 q^{79} +5.70820 q^{81} +15.7082 q^{83} -3.61803 q^{85} -0.618034 q^{87} -1.70820 q^{89} +14.5279 q^{91} +1.00000 q^{93} +2.76393 q^{95} +8.79837 q^{97} +8.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9} - 2 q^{11} - q^{13} - q^{15} - 5 q^{17} + 10 q^{19} + 8 q^{21} - 5 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{29} + q^{31} - 4 q^{33} - q^{35} - 14 q^{37} + 13 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 - q^7 - 3 * q^9 - 2 * q^11 - q^13 - q^15 - 5 * q^17 + 10 * q^19 + 8 * q^21 - 5 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^29 + q^31 - 4 * q^33 - q^35 - 14 * q^37 + 13 * q^39 + 6 * q^41 - 13 * q^43 - 3 * q^45 + 16 * q^47 + 9 * q^49 - 3 * q^53 - 2 * q^55 - 10 * q^57 - 21 * q^59 - 11 * q^61 - 6 * q^63 - q^65 + 4 * q^67 - 5 * q^69 - 8 * q^71 + 15 * q^73 - q^75 - 14 * q^77 + 17 * q^79 - 2 * q^81 + 18 * q^83 - 5 * q^85 + q^87 + 10 * q^89 + 38 * q^91 + 2 * q^93 + 10 * q^95 - 7 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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