LMFDB - Embedded newform 9600.2.a.dp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9600,2,Mod(1,9600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9600, 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("9600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9600 = 2^{7} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9600.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:	$76.6563859404$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 3x + 1$ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 1920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	9600.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.34017 q^{7} +1.00000 q^{9} +3.07838 q^{11} -0.921622 q^{13} +7.75872 q^{17} +4.00000 q^{19} -2.34017 q^{21} +2.15676 q^{23} -1.00000 q^{27} +6.49693 q^{29} -2.00000 q^{31} -3.07838 q^{33} -3.07838 q^{37} +0.921622 q^{39} +10.6803 q^{41} +2.15676 q^{43} +8.68035 q^{47} -1.52359 q^{49} -7.75872 q^{51} -2.92162 q^{53} -4.00000 q^{57} +11.7587 q^{59} -6.00000 q^{61} +2.34017 q^{63} -6.83710 q^{67} -2.15676 q^{69} -11.5174 q^{71} +6.52359 q^{73} +7.20394 q^{77} +15.3607 q^{79} +1.00000 q^{81} -1.84324 q^{83} -6.49693 q^{87} +6.00000 q^{89} -2.15676 q^{91} +2.00000 q^{93} +3.07838 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 3 q^{3} - 4 q^{7} + 3 q^{9} + 6 q^{11} - 6 q^{13} - 2 q^{17} + 12 q^{19} + 4 q^{21} - 3 q^{27} + 2 q^{29} - 6 q^{31} - 6 q^{33} - 6 q^{37} + 6 q^{39} + 10 q^{41} + 4 q^{47} + 11 q^{49} + 2 q^{51}+ \cdots + 6 q^{99}+O(q^{100})$ 3 * q - 3 * q^3 - 4 * q^7 + 3 * q^9 + 6 * q^11 - 6 * q^13 - 2 * q^17 + 12 * q^19 + 4 * q^21 - 3 * q^27 + 2 * q^29 - 6 * q^31 - 6 * q^33 - 6 * q^37 + 6 * q^39 + 10 * q^41 + 4 * q^47 + 11 * q^49 + 2 * q^51 - 12 * q^53 - 12 * q^57 + 10 * q^59 - 18 * q^61 - 4 * q^63 + 8 * q^67 + 16 * q^71 + 4 * q^73 - 16 * q^77 + 2 * q^79 + 3 * q^81 - 12 * q^83 - 2 * q^87 + 18 * q^89 + 6 * q^93 + 6 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.34017	0.884502	0.442251	−	0.896891i	$-0.354180\pi$
$7$	2.34017	0.884502	0.442251	+	0.896891i	$0.354180\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.07838	0.928166	0.464083	−	0.885792i	$-0.346384\pi$
$11$	3.07838	0.928166	0.464083	+	0.885792i	$0.346384\pi$
$12$	0	0
$13$	−0.921622	−0.255612	−0.127806	−	0.991799i	$-0.540793\pi$
$13$	−0.921622	−0.255612	−0.127806	+	0.991799i	$0.540793\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.75872	1.88177	0.940883	−	0.338730i	$-0.109997\pi$
$17$	7.75872	1.88177	0.940883	+	0.338730i	$0.109997\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−2.34017	−0.510668
$22$	0	0
$23$	2.15676	0.449715	0.224857	−	0.974392i	$-0.427808\pi$
$23$	2.15676	0.449715	0.224857	+	0.974392i	$0.427808\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.49693	1.20645	0.603225	−	0.797571i	$-0.293882\pi$
$29$	6.49693	1.20645	0.603225	+	0.797571i	$0.293882\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	−3.07838	−0.535877
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.07838	−0.506082	−0.253041	−	0.967456i	$-0.581431\pi$
$37$	−3.07838	−0.506082	−0.253041	+	0.967456i	$0.581431\pi$
$38$	0	0
$39$	0.921622	0.147578
$40$	0	0
$41$	10.6803	1.66799	0.833995	−	0.551772i	$-0.186048\pi$
$41$	10.6803	1.66799	0.833995	+	0.551772i	$0.186048\pi$
$42$	0	0
$43$	2.15676	0.328902	0.164451	−	0.986385i	$-0.447415\pi$
$43$	2.15676	0.328902	0.164451	+	0.986385i	$0.447415\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.68035	1.26616	0.633079	−	0.774087i	$-0.281791\pi$
$47$	8.68035	1.26616	0.633079	+	0.774087i	$0.281791\pi$
$48$	0	0
$49$	−1.52359	−0.217656
$50$	0	0
$51$	−7.75872	−1.08644
$52$	0	0
$53$	−2.92162	−0.401316	−0.200658	−	0.979661i	$-0.564308\pi$
$53$	−2.92162	−0.401316	−0.200658	+	0.979661i	$0.564308\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	11.7587	1.53086	0.765428	−	0.643522i	$-0.222527\pi$
$59$	11.7587	1.53086	0.765428	+	0.643522i	$0.222527\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	2.34017	0.294834
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.83710	−0.835285	−0.417642	−	0.908611i	$-0.637144\pi$
$67$	−6.83710	−0.835285	−0.417642	+	0.908611i	$0.637144\pi$
$68$	0	0
$69$	−2.15676	−0.259643
$70$	0	0
$71$	−11.5174	−1.36687	−0.683435	−	0.730012i	$-0.739515\pi$
$71$	−11.5174	−1.36687	−0.683435	+	0.730012i	$0.739515\pi$
$72$	0	0
$73$	6.52359	0.763529	0.381764	−	0.924260i	$-0.375317\pi$
$73$	6.52359	0.763529	0.381764	+	0.924260i	$0.375317\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.20394	0.820965
$78$	0	0
$79$	15.3607	1.72821	0.864106	−	0.503309i	$-0.167884\pi$
$79$	15.3607	1.72821	0.864106	+	0.503309i	$0.167884\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.84324	−0.202322	−0.101161	−	0.994870i	$-0.532256\pi$
$83$	−1.84324	−0.202322	−0.101161	+	0.994870i	$0.532256\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.49693	−0.696544
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−2.15676	−0.226089
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	3.07838	0.309389
$100$	0	0
$101$	−8.34017	−0.829878	−0.414939	−	0.909849i	$-0.636197\pi$
$101$	−8.34017	−0.829878	−0.414939	+	0.909849i	$0.636197\pi$
$102$	0	0
$103$	10.3402	1.01885	0.509424	−	0.860516i	$-0.329859\pi$
$103$	10.3402	1.01885	0.509424	+	0.860516i	$0.329859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.3607	−1.29163	−0.645813	−	0.763495i	$-0.723482\pi$
$107$	−13.3607	−1.29163	−0.645813	+	0.763495i	$0.723482\pi$
$108$	0	0
$109$	−5.31965	−0.509530	−0.254765	−	0.967003i	$-0.581998\pi$
$109$	−5.31965	−0.509530	−0.254765	+	0.967003i	$0.581998\pi$
$110$	0	0
$111$	3.07838	0.292187
$112$	0	0
$113$	−19.7587	−1.85874	−0.929372	−	0.369144i	$-0.879651\pi$
$113$	−19.7587	−1.85874	−0.929372	+	0.369144i	$0.879651\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.921622	−0.0852040
$118$	0	0
$119$	18.1568	1.66443
$120$	0	0
$121$	−1.52359	−0.138508
$122$	0	0
$123$	−10.6803	−0.963014
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−19.3340	−1.71562	−0.857809	−	0.513969i	$-0.828175\pi$
$127$	−19.3340	−1.71562	−0.857809	+	0.513969i	$0.828175\pi$
$128$	0	0
$129$	−2.15676	−0.189892
$130$	0	0
$131$	−7.44521	−0.650491	−0.325246	−	0.945630i	$-0.605447\pi$
$131$	−7.44521	−0.650491	−0.325246	+	0.945630i	$0.605447\pi$
$132$	0	0
$133$	9.36069	0.811675
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.44521	−0.294344	−0.147172	−	0.989111i	$-0.547017\pi$
$137$	−3.44521	−0.294344	−0.147172	+	0.989111i	$0.547017\pi$
$138$	0	0
$139$	19.2039	1.62886	0.814428	−	0.580264i	$-0.197051\pi$
$139$	19.2039	1.62886	0.814428	+	0.580264i	$0.197051\pi$
$140$	0	0
$141$	−8.68035	−0.731017
$142$	0	0
$143$	−2.83710	−0.237250
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.52359	0.125664
$148$	0	0
$149$	−9.81658	−0.804206	−0.402103	−	0.915594i	$-0.631721\pi$
$149$	−9.81658	−0.804206	−0.402103	+	0.915594i	$0.631721\pi$
$150$	0	0
$151$	23.6742	1.92658	0.963290	−	0.268464i	$-0.0865161\pi$
$151$	23.6742	1.92658	0.963290	+	0.268464i	$0.0865161\pi$
$152$	0	0
$153$	7.75872	0.627256
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.0784	0.884151	0.442075	−	0.896978i	$-0.354242\pi$
$157$	11.0784	0.884151	0.442075	+	0.896978i	$0.354242\pi$
$158$	0	0
$159$	2.92162	0.231700
$160$	0	0
$161$	5.04718	0.397774
$162$	0	0
$163$	−12.9939	−1.01776	−0.508879	−	0.860838i	$-0.669940\pi$
$163$	−12.9939	−1.01776	−0.508879	+	0.860838i	$0.669940\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−12.1506	−0.934662
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	−18.4391	−1.40190	−0.700948	−	0.713212i	$-0.747240\pi$
$173$	−18.4391	−1.40190	−0.700948	+	0.713212i	$0.747240\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.7587	−0.883840
$178$	0	0
$179$	−1.91548	−0.143170	−0.0715848	−	0.997435i	$-0.522806\pi$
$179$	−1.91548	−0.143170	−0.0715848	+	0.997435i	$0.522806\pi$
$180$	0	0
$181$	6.99386	0.519849	0.259925	−	0.965629i	$-0.416302\pi$
$181$	6.99386	0.519849	0.259925	+	0.965629i	$0.416302\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	0	0
$186$	0	0
$187$	23.8843	1.74659
$188$	0	0
$189$	−2.34017	−0.170223
$190$	0	0
$191$	−6.15676	−0.445487	−0.222744	−	0.974877i	$-0.571501\pi$
$191$	−6.15676	−0.445487	−0.222744	+	0.974877i	$0.571501\pi$
$192$	0	0
$193$	−14.5236	−1.04543	−0.522715	−	0.852507i	$-0.675081\pi$
$193$	−14.5236	−1.04543	−0.522715	+	0.852507i	$0.675081\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.5958	−1.75238	−0.876190	−	0.481966i	$-0.839923\pi$
$197$	−24.5958	−1.75238	−0.876190	+	0.481966i	$0.839923\pi$
$198$	0	0
$199$	−3.36069	−0.238233	−0.119117	−	0.992880i	$-0.538006\pi$
$199$	−3.36069	−0.238233	−0.119117	+	0.992880i	$0.538006\pi$
$200$	0	0
$201$	6.83710	0.482252
$202$	0	0
$203$	15.2039	1.06711
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.15676	0.149905
$208$	0	0
$209$	12.3135	0.851743
$210$	0	0
$211$	11.2039	0.771311	0.385655	−	0.922643i	$-0.373975\pi$
$211$	11.2039	0.771311	0.385655	+	0.922643i	$0.373975\pi$
$212$	0	0
$213$	11.5174	0.789162
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.68035	−0.317723
$218$	0	0
$219$	−6.52359	−0.440823
$220$	0	0
$221$	−7.15061	−0.481002
$222$	0	0
$223$	6.65368	0.445564	0.222782	−	0.974868i	$-0.428486\pi$
$223$	6.65368	0.445564	0.222782	+	0.974868i	$0.428486\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.84324	−0.653319	−0.326660	−	0.945142i	$-0.605923\pi$
$227$	−9.84324	−0.653319	−0.326660	+	0.945142i	$0.605923\pi$
$228$	0	0
$229$	−1.31965	−0.0872052	−0.0436026	−	0.999049i	$-0.513884\pi$
$229$	−1.31965	−0.0872052	−0.0436026	+	0.999049i	$0.513884\pi$
$230$	0	0
$231$	−7.20394	−0.473984
$232$	0	0
$233$	2.39803	0.157100	0.0785501	−	0.996910i	$-0.474971\pi$
$233$	2.39803	0.157100	0.0785501	+	0.996910i	$0.474971\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−15.3607	−0.997784
$238$	0	0
$239$	6.15676	0.398247	0.199124	−	0.979974i	$-0.436190\pi$
$239$	6.15676	0.398247	0.199124	+	0.979974i	$0.436190\pi$
$240$	0	0
$241$	0.639308	0.0411815	0.0205907	−	0.999788i	$-0.493445\pi$
$241$	0.639308	0.0411815	0.0205907	+	0.999788i	$0.493445\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.68649	−0.234566
$248$	0	0
$249$	1.84324	0.116811
$250$	0	0
$251$	12.9216	0.815606	0.407803	−	0.913070i	$-0.366295\pi$
$251$	12.9216	0.815606	0.407803	+	0.913070i	$0.366295\pi$
$252$	0	0
$253$	6.63931	0.417410
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.9627	0.933345	0.466673	−	0.884430i	$-0.345453\pi$
$257$	14.9627	0.933345	0.466673	+	0.884430i	$0.345453\pi$
$258$	0	0
$259$	−7.20394	−0.447631
$260$	0	0
$261$	6.49693	0.402150
$262$	0	0
$263$	−13.3607	−0.823856	−0.411928	−	0.911216i	$-0.635144\pi$
$263$	−13.3607	−0.823856	−0.411928	+	0.911216i	$0.635144\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	0	0
$269$	−27.5441	−1.67939	−0.839697	−	0.543055i	$-0.817267\pi$
$269$	−27.5441	−1.67939	−0.839697	+	0.543055i	$0.817267\pi$
$270$	0	0
$271$	−10.0000	−0.607457	−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000	−0.607457	−0.303728	+	0.952759i	$0.598232\pi$
$272$	0	0
$273$	2.15676	0.130533
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.12556	−0.247881	−0.123940	−	0.992290i	$-0.539553\pi$
$277$	−4.12556	−0.247881	−0.123940	+	0.992290i	$0.539553\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−11.3607	−0.677722	−0.338861	−	0.940836i	$-0.610042\pi$
$281$	−11.3607	−0.677722	−0.338861	+	0.940836i	$0.610042\pi$
$282$	0	0
$283$	−10.5236	−0.625563	−0.312781	−	0.949825i	$-0.601261\pi$
$283$	−10.5236	−0.625563	−0.312781	+	0.949825i	$0.601261\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	24.9939	1.47534
$288$	0	0
$289$	43.1978	2.54105
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.5958	−1.90427	−0.952134	−	0.305680i	$-0.901116\pi$
$293$	−32.5958	−1.90427	−0.952134	+	0.305680i	$0.901116\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.07838	−0.178626
$298$	0	0
$299$	−1.98771	−0.114952
$300$	0	0
$301$	5.04718	0.290915
$302$	0	0
$303$	8.34017	0.479130
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.9877	1.71149	0.855745	−	0.517398i	$-0.173099\pi$
$307$	29.9877	1.71149	0.855745	+	0.517398i	$0.173099\pi$
$308$	0	0
$309$	−10.3402	−0.588232
$310$	0	0
$311$	27.2039	1.54259	0.771297	−	0.636476i	$-0.219608\pi$
$311$	27.2039	1.54259	0.771297	+	0.636476i	$0.219608\pi$
$312$	0	0
$313$	26.8371	1.51692	0.758461	−	0.651718i	$-0.225951\pi$
$313$	26.8371	1.51692	0.758461	+	0.651718i	$0.225951\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.5958	1.38144	0.690720	−	0.723123i	$-0.257294\pi$
$317$	24.5958	1.38144	0.690720	+	0.723123i	$0.257294\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	0	0
$321$	13.3607	0.745721
$322$	0	0
$323$	31.0349	1.72683
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.31965	0.294178
$328$	0	0
$329$	20.3135	1.11992
$330$	0	0
$331$	12.3135	0.676812	0.338406	−	0.941000i	$-0.390112\pi$
$331$	12.3135	0.676812	0.338406	+	0.941000i	$0.390112\pi$
$332$	0	0
$333$	−3.07838	−0.168694
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.47641	0.516213	0.258106	−	0.966116i	$-0.416901\pi$
$337$	9.47641	0.516213	0.258106	+	0.966116i	$0.416901\pi$
$338$	0	0
$339$	19.7587	1.07315
$340$	0	0
$341$	−6.15676	−0.333407
$342$	0	0
$343$	−19.9467	−1.07702
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.51745	0.403558	0.201779	−	0.979431i	$-0.435328\pi$
$347$	7.51745	0.403558	0.201779	+	0.979431i	$0.435328\pi$
$348$	0	0
$349$	−27.6742	−1.48137	−0.740683	−	0.671855i	$-0.765498\pi$
$349$	−27.6742	−1.48137	−0.740683	+	0.671855i	$0.765498\pi$
$350$	0	0
$351$	0.921622	0.0491926
$352$	0	0
$353$	−21.6020	−1.14976	−0.574878	−	0.818239i	$-0.694951\pi$
$353$	−21.6020	−1.14976	−0.574878	+	0.818239i	$0.694951\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−18.1568	−0.960957
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	1.52359	0.0799678
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−23.0205	−1.20166	−0.600831	−	0.799376i	$-0.705163\pi$
$367$	−23.0205	−1.20166	−0.600831	+	0.799376i	$0.705163\pi$
$368$	0	0
$369$	10.6803	0.555997
$370$	0	0
$371$	−6.83710	−0.354965
$372$	0	0
$373$	−26.5958	−1.37708	−0.688540	−	0.725199i	$-0.741748\pi$
$373$	−26.5958	−1.37708	−0.688540	+	0.725199i	$0.741748\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.98771	−0.308383
$378$	0	0
$379$	36.1445	1.85662	0.928308	−	0.371811i	$-0.121263\pi$
$379$	36.1445	1.85662	0.928308	+	0.371811i	$0.121263\pi$
$380$	0	0
$381$	19.3340	0.990512
$382$	0	0
$383$	−6.83710	−0.349360	−0.174680	−	0.984625i	$-0.555889\pi$
$383$	−6.83710	−0.349360	−0.174680	+	0.984625i	$0.555889\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.15676	0.109634
$388$	0	0
$389$	−16.7070	−0.847079	−0.423539	−	0.905878i	$-0.639213\pi$
$389$	−16.7070	−0.847079	−0.423539	+	0.905878i	$0.639213\pi$
$390$	0	0
$391$	16.7337	0.846258
$392$	0	0
$393$	7.44521	0.375561
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.56093	0.178718	0.0893590	−	0.995999i	$-0.471518\pi$
$397$	3.56093	0.178718	0.0893590	+	0.995999i	$0.471518\pi$
$398$	0	0
$399$	−9.36069	−0.468621
$400$	0	0
$401$	−25.7152	−1.28416	−0.642079	−	0.766639i	$-0.721928\pi$
$401$	−25.7152	−1.28416	−0.642079	+	0.766639i	$0.721928\pi$
$402$	0	0
$403$	1.84324	0.0918185
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.47641	−0.469728
$408$	0	0
$409$	28.8371	1.42590	0.712951	−	0.701213i	$-0.247358\pi$
$409$	28.8371	1.42590	0.712951	+	0.701213i	$0.247358\pi$
$410$	0	0
$411$	3.44521	0.169940
$412$	0	0
$413$	27.5174	1.35405
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−19.2039	−0.940421
$418$	0	0
$419$	6.28231	0.306911	0.153456	−	0.988156i	$-0.450960\pi$
$419$	6.28231	0.306911	0.153456	+	0.988156i	$0.450960\pi$
$420$	0	0
$421$	10.9939	0.535808	0.267904	−	0.963446i	$-0.413669\pi$
$421$	10.9939	0.535808	0.267904	+	0.963446i	$0.413669\pi$
$422$	0	0
$423$	8.68035	0.422053
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−14.0410	−0.679493
$428$	0	0
$429$	2.83710	0.136977
$430$	0	0
$431$	21.3607	1.02891	0.514454	−	0.857518i	$-0.327995\pi$
$431$	21.3607	1.02891	0.514454	+	0.857518i	$0.327995\pi$
$432$	0	0
$433$	−18.4703	−0.887624	−0.443812	−	0.896120i	$-0.646374\pi$
$433$	−18.4703	−0.887624	−0.443812	+	0.896120i	$0.646374\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.62702	0.412686
$438$	0	0
$439$	−6.31351	−0.301327	−0.150664	−	0.988585i	$-0.548141\pi$
$439$	−6.31351	−0.301327	−0.150664	+	0.988585i	$0.548141\pi$
$440$	0	0
$441$	−1.52359	−0.0725519
$442$	0	0
$443$	−27.8310	−1.32229	−0.661144	−	0.750259i	$-0.729929\pi$
$443$	−27.8310	−1.32229	−0.661144	+	0.750259i	$0.729929\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	9.81658	0.464308
$448$	0	0
$449$	14.6803	0.692808	0.346404	−	0.938085i	$-0.387403\pi$
$449$	14.6803	0.692808	0.346404	+	0.938085i	$0.387403\pi$
$450$	0	0
$451$	32.8781	1.54817
$452$	0	0
$453$	−23.6742	−1.11231
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.6865	0.920895	0.460448	−	0.887687i	$-0.347689\pi$
$457$	19.6865	0.920895	0.460448	+	0.887687i	$0.347689\pi$
$458$	0	0
$459$	−7.75872	−0.362146
$460$	0	0
$461$	−13.5031	−0.628901	−0.314450	−	0.949274i	$-0.601820\pi$
$461$	−13.5031	−0.628901	−0.314450	+	0.949274i	$0.601820\pi$
$462$	0	0
$463$	−6.02666	−0.280083	−0.140041	−	0.990146i	$-0.544724\pi$
$463$	−6.02666	−0.280083	−0.140041	+	0.990146i	$0.544724\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.8310	−0.917667	−0.458834	−	0.888522i	$-0.651733\pi$
$467$	−19.8310	−0.917667	−0.458834	+	0.888522i	$0.651733\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−11.0784	−0.510465
$472$	0	0
$473$	6.63931	0.305276
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.92162	−0.133772
$478$	0	0
$479$	−23.2039	−1.06021	−0.530107	−	0.847930i	$-0.677848\pi$
$479$	−23.2039	−1.06021	−0.530107	+	0.847930i	$0.677848\pi$
$480$	0	0
$481$	2.83710	0.129361
$482$	0	0
$483$	−5.04718	−0.229655
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−23.3874	−1.05978	−0.529891	−	0.848066i	$-0.677767\pi$
$487$	−23.3874	−1.05978	−0.529891	+	0.848066i	$0.677767\pi$
$488$	0	0
$489$	12.9939	0.587603
$490$	0	0
$491$	30.2290	1.36422	0.682108	−	0.731252i	$-0.261064\pi$
$491$	30.2290	1.36422	0.682108	+	0.731252i	$0.261064\pi$
$492$	0	0
$493$	50.4079	2.27026
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−26.9528	−1.20900
$498$	0	0
$499$	−1.36069	−0.0609129	−0.0304565	−	0.999536i	$-0.509696\pi$
$499$	−1.36069	−0.0609129	−0.0304565	+	0.999536i	$0.509696\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−16.6803	−0.743740	−0.371870	−	0.928285i	$-0.621283\pi$
$503$	−16.6803	−0.743740	−0.371870	+	0.928285i	$0.621283\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.1506	0.539628
$508$	0	0
$509$	14.0144	0.621176	0.310588	−	0.950545i	$-0.399474\pi$
$509$	14.0144	0.621176	0.310588	+	0.950545i	$0.399474\pi$
$510$	0	0
$511$	15.2663	0.675343
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	26.7214	1.17521
$518$	0	0
$519$	18.4391	0.809385
$520$	0	0
$521$	6.68035	0.292671	0.146336	−	0.989235i	$-0.453252\pi$
$521$	6.68035	0.292671	0.146336	+	0.989235i	$0.453252\pi$
$522$	0	0
$523$	26.0410	1.13870	0.569348	−	0.822097i	$-0.307196\pi$
$523$	26.0410	1.13870	0.569348	+	0.822097i	$0.307196\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.5174	−0.675951
$528$	0	0
$529$	−18.3484	−0.797757
$530$	0	0
$531$	11.7587	0.510285
$532$	0	0
$533$	−9.84324	−0.426358
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.91548	0.0826590
$538$	0	0
$539$	−4.69019	−0.202021
$540$	0	0
$541$	14.9939	0.644636	0.322318	−	0.946631i	$-0.395538\pi$
$541$	14.9939	0.644636	0.322318	+	0.946631i	$0.395538\pi$
$542$	0	0
$543$	−6.99386	−0.300135
$544$	0	0
$545$	0	0
$546$	0	0
$547$	26.1568	1.11838	0.559191	−	0.829039i	$-0.311112\pi$
$547$	26.1568	1.11838	0.559191	+	0.829039i	$0.311112\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	25.9877	1.10711
$552$	0	0
$553$	35.9467	1.52861
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.7526	0.625087	0.312543	−	0.949903i	$-0.398819\pi$
$557$	14.7526	0.625087	0.312543	+	0.949903i	$0.398819\pi$
$558$	0	0
$559$	−1.98771	−0.0840713
$560$	0	0
$561$	−23.8843	−1.00840
$562$	0	0
$563$	−38.5523	−1.62479	−0.812394	−	0.583109i	$-0.801836\pi$
$563$	−38.5523	−1.62479	−0.812394	+	0.583109i	$0.801836\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.34017	0.0982780
$568$	0	0
$569$	14.6803	0.615432	0.307716	−	0.951478i	$-0.400435\pi$
$569$	14.6803	0.615432	0.307716	+	0.951478i	$0.400435\pi$
$570$	0	0
$571$	−13.6742	−0.572248	−0.286124	−	0.958193i	$-0.592367\pi$
$571$	−13.6742	−0.572248	−0.286124	+	0.958193i	$0.592367\pi$
$572$	0	0
$573$	6.15676	0.257202
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−22.6681	−0.943684	−0.471842	−	0.881683i	$-0.656411\pi$
$577$	−22.6681	−0.943684	−0.471842	+	0.881683i	$0.656411\pi$
$578$	0	0
$579$	14.5236	0.603580
$580$	0	0
$581$	−4.31351	−0.178955
$582$	0	0
$583$	−8.99386	−0.372487
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.3607	−1.54204	−0.771020	−	0.636810i	$-0.780253\pi$
$587$	−37.3607	−1.54204	−0.771020	+	0.636810i	$0.780253\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	24.5958	1.01174
$592$	0	0
$593$	15.5897	0.640192	0.320096	−	0.947385i	$-0.396285\pi$
$593$	15.5897	0.640192	0.320096	+	0.947385i	$0.396285\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.36069	0.137544
$598$	0	0
$599$	45.3607	1.85339	0.926694	−	0.375817i	$-0.122638\pi$
$599$	45.3607	1.85339	0.926694	+	0.375817i	$0.122638\pi$
$600$	0	0
$601$	16.5236	0.674011	0.337006	−	0.941503i	$-0.390586\pi$
$601$	16.5236	0.674011	0.337006	+	0.941503i	$0.390586\pi$
$602$	0	0
$603$	−6.83710	−0.278428
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.0554	1.21991	0.609956	−	0.792435i	$-0.291187\pi$
$607$	30.0554	1.21991	0.609956	+	0.792435i	$0.291187\pi$
$608$	0	0
$609$	−15.2039	−0.616095
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	24.2700	0.980257	0.490129	−	0.871650i	$-0.336950\pi$
$613$	24.2700	0.980257	0.490129	+	0.871650i	$0.336950\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0.0722347	0.00290806	0.00145403	−	0.999999i	$-0.499537\pi$
$617$	0.0722347	0.00290806	0.00145403	+	0.999999i	$0.499537\pi$
$618$	0	0
$619$	−29.1917	−1.17331	−0.586656	−	0.809836i	$-0.699556\pi$
$619$	−29.1917	−1.17331	−0.586656	+	0.809836i	$0.699556\pi$
$620$	0	0
$621$	−2.15676	−0.0865476
$622$	0	0
$623$	14.0410	0.562542
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−12.3135	−0.491754
$628$	0	0
$629$	−23.8843	−0.952329
$630$	0	0
$631$	−3.36069	−0.133787	−0.0668935	−	0.997760i	$-0.521309\pi$
$631$	−3.36069	−0.133787	−0.0668935	+	0.997760i	$0.521309\pi$
$632$	0	0
$633$	−11.2039	−0.445316
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.40417	0.0556354
$638$	0	0
$639$	−11.5174	−0.455623
$640$	0	0
$641$	36.6681	1.44830	0.724151	−	0.689642i	$-0.242232\pi$
$641$	36.6681	1.44830	0.724151	+	0.689642i	$0.242232\pi$
$642$	0	0
$643$	40.6803	1.60428	0.802138	−	0.597139i	$-0.203696\pi$
$643$	40.6803	1.60428	0.802138	+	0.597139i	$0.203696\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.7275	1.48322	0.741611	−	0.670830i	$-0.234062\pi$
$647$	37.7275	1.48322	0.741611	+	0.670830i	$0.234062\pi$
$648$	0	0
$649$	36.1978	1.42089
$650$	0	0
$651$	4.68035	0.183437
$652$	0	0
$653$	38.9216	1.52312	0.761560	−	0.648094i	$-0.224434\pi$
$653$	38.9216	1.52312	0.761560	+	0.648094i	$0.224434\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.52359	0.254510
$658$	0	0
$659$	32.9504	1.28356	0.641782	−	0.766887i	$-0.278195\pi$
$659$	32.9504	1.28356	0.641782	+	0.766887i	$0.278195\pi$
$660$	0	0
$661$	−36.3012	−1.41195	−0.705977	−	0.708235i	$-0.749492\pi$
$661$	−36.3012	−1.41195	−0.705977	+	0.708235i	$0.749492\pi$
$662$	0	0
$663$	7.15061	0.277707
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.0123	0.542558
$668$	0	0
$669$	−6.65368	−0.257246
$670$	0	0
$671$	−18.4703	−0.713037
$672$	0	0
$673$	37.1917	1.43363	0.716816	−	0.697262i	$-0.245599\pi$
$673$	37.1917	1.43363	0.716816	+	0.697262i	$0.245599\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.07838	−0.348910	−0.174455	−	0.984665i	$-0.555816\pi$
$677$	−9.07838	−0.348910	−0.174455	+	0.984665i	$0.555816\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.84324	0.377194
$682$	0	0
$683$	9.84324	0.376641	0.188321	−	0.982108i	$-0.439696\pi$
$683$	9.84324	0.376641	0.188321	+	0.982108i	$0.439696\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.31965	0.0503479
$688$	0	0
$689$	2.69263	0.102581
$690$	0	0
$691$	−1.67420	−0.0636897	−0.0318448	−	0.999493i	$-0.510138\pi$
$691$	−1.67420	−0.0636897	−0.0318448	+	0.999493i	$0.510138\pi$
$692$	0	0
$693$	7.20394	0.273655
$694$	0	0
$695$	0	0
$696$	0	0
$697$	82.8659	3.13877
$698$	0	0
$699$	−2.39803	−0.0907019
$700$	0	0
$701$	8.96719	0.338686	0.169343	−	0.985557i	$-0.445835\pi$
$701$	8.96719	0.338686	0.169343	+	0.985557i	$0.445835\pi$
$702$	0	0
$703$	−12.3135	−0.464413
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.5174	−0.734029
$708$	0	0
$709$	−25.7152	−0.965756	−0.482878	−	0.875688i	$-0.660409\pi$
$709$	−25.7152	−0.965756	−0.482878	+	0.875688i	$0.660409\pi$
$710$	0	0
$711$	15.3607	0.576071
$712$	0	0
$713$	−4.31351	−0.161542
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.15676	−0.229928
$718$	0	0
$719$	−18.4079	−0.686498	−0.343249	−	0.939244i	$-0.611527\pi$
$719$	−18.4079	−0.686498	−0.343249	+	0.939244i	$0.611527\pi$
$720$	0	0
$721$	24.1978	0.901173
$722$	0	0
$723$	−0.639308	−0.0237761
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−42.3402	−1.57031	−0.785155	−	0.619299i	$-0.787417\pi$
$727$	−42.3402	−1.57031	−0.785155	+	0.619299i	$0.787417\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.7337	0.618917
$732$	0	0
$733$	−4.75258	−0.175541	−0.0877703	−	0.996141i	$-0.527974\pi$
$733$	−4.75258	−0.175541	−0.0877703	+	0.996141i	$0.527974\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.0472	−0.775283
$738$	0	0
$739$	21.0472	0.774233	0.387117	−	0.922031i	$-0.373471\pi$
$739$	21.0472	0.774233	0.387117	+	0.922031i	$0.373471\pi$
$740$	0	0
$741$	3.68649	0.135427
$742$	0	0
$743$	49.5585	1.81812	0.909062	−	0.416660i	$-0.136800\pi$
$743$	49.5585	1.81812	0.909062	+	0.416660i	$0.136800\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.84324	−0.0674408
$748$	0	0
$749$	−31.2663	−1.14245
$750$	0	0
$751$	−41.6619	−1.52026	−0.760132	−	0.649768i	$-0.774866\pi$
$751$	−41.6619	−1.52026	−0.760132	+	0.649768i	$0.774866\pi$
$752$	0	0
$753$	−12.9216	−0.470890
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.9688	0.798470	0.399235	−	0.916849i	$-0.369276\pi$
$757$	21.9688	0.798470	0.399235	+	0.916849i	$0.369276\pi$
$758$	0	0
$759$	−6.63931	−0.240992
$760$	0	0
$761$	−11.3074	−0.409892	−0.204946	−	0.978773i	$-0.565702\pi$
$761$	−11.3074	−0.409892	−0.204946	+	0.978773i	$0.565702\pi$
$762$	0	0
$763$	−12.4489	−0.450681
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.8371	−0.391305
$768$	0	0
$769$	6.19779	0.223498	0.111749	−	0.993736i	$-0.464355\pi$
$769$	6.19779	0.223498	0.111749	+	0.993736i	$0.464355\pi$
$770$	0	0
$771$	−14.9627	−0.538867
$772$	0	0
$773$	−30.4391	−1.09482	−0.547409	−	0.836865i	$-0.684386\pi$
$773$	−30.4391	−1.09482	−0.547409	+	0.836865i	$0.684386\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.20394	0.258440
$778$	0	0
$779$	42.7214	1.53065
$780$	0	0
$781$	−35.4551	−1.26868
$782$	0	0
$783$	−6.49693	−0.232181
$784$	0	0
$785$	0	0
$786$	0	0
$787$	36.9939	1.31869	0.659344	−	0.751841i	$-0.270834\pi$
$787$	36.9939	1.31869	0.659344	+	0.751841i	$0.270834\pi$
$788$	0	0
$789$	13.3607	0.475653
$790$	0	0
$791$	−46.2388	−1.64406
$792$	0	0
$793$	5.52973	0.196367
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.7526	−1.08931	−0.544656	−	0.838659i	$-0.683340\pi$
$797$	−30.7526	−1.08931	−0.544656	+	0.838659i	$0.683340\pi$
$798$	0	0
$799$	67.3484	2.38262
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	20.0821	0.708681
$804$	0	0
$805$	0	0
$806$	0	0
$807$	27.5441	0.969599
$808$	0	0
$809$	−5.31965	−0.187029	−0.0935145	−	0.995618i	$-0.529810\pi$
$809$	−5.31965	−0.187029	−0.0935145	+	0.995618i	$0.529810\pi$
$810$	0	0
$811$	−49.9253	−1.75312	−0.876558	−	0.481297i	$-0.840166\pi$
$811$	−49.9253	−1.75312	−0.876558	+	0.481297i	$0.840166\pi$
$812$	0	0
$813$	10.0000	0.350715
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.62702	0.301821
$818$	0	0
$819$	−2.15676	−0.0753631
$820$	0	0
$821$	44.1711	1.54158	0.770792	−	0.637087i	$-0.219861\pi$
$821$	44.1711	1.54158	0.770792	+	0.637087i	$0.219861\pi$
$822$	0	0
$823$	−19.0738	−0.664872	−0.332436	−	0.943126i	$-0.607871\pi$
$823$	−19.0738	−0.664872	−0.332436	+	0.943126i	$0.607871\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.0349	1.49647	0.748235	−	0.663434i	$-0.230902\pi$
$827$	43.0349	1.49647	0.748235	+	0.663434i	$0.230902\pi$
$828$	0	0
$829$	36.0410	1.25176	0.625878	−	0.779921i	$-0.284741\pi$
$829$	36.0410	1.25176	0.625878	+	0.779921i	$0.284741\pi$
$830$	0	0
$831$	4.12556	0.143114
$832$	0	0
$833$	−11.8211	−0.409577
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.00000	0.0691301
$838$	0	0
$839$	43.0349	1.48573	0.742865	−	0.669441i	$-0.233467\pi$
$839$	43.0349	1.48573	0.742865	+	0.669441i	$0.233467\pi$
$840$	0	0
$841$	13.2101	0.455520
$842$	0	0
$843$	11.3607	0.391283
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.56547	−0.122511
$848$	0	0
$849$	10.5236	0.361169
$850$	0	0
$851$	−6.63931	−0.227593
$852$	0	0
$853$	−47.1605	−1.61474	−0.807372	−	0.590043i	$-0.799111\pi$
$853$	−47.1605	−1.61474	−0.807372	+	0.590043i	$0.799111\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.8059	0.437441	0.218721	−	0.975788i	$-0.429812\pi$
$857$	12.8059	0.437441	0.218721	+	0.975788i	$0.429812\pi$
$858$	0	0
$859$	−14.4703	−0.493719	−0.246860	−	0.969051i	$-0.579399\pi$
$859$	−14.4703	−0.493719	−0.246860	+	0.969051i	$0.579399\pi$
$860$	0	0
$861$	−24.9939	−0.851788
$862$	0	0
$863$	−24.3135	−0.827642	−0.413821	−	0.910358i	$-0.635806\pi$
$863$	−24.3135	−0.827642	−0.413821	+	0.910358i	$0.635806\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−43.1978	−1.46707
$868$	0	0
$869$	47.2860	1.60407
$870$	0	0
$871$	6.30122	0.213509
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	44.4391	1.50060	0.750300	−	0.661097i	$-0.229909\pi$
$877$	44.4391	1.50060	0.750300	+	0.661097i	$0.229909\pi$
$878$	0	0
$879$	32.5958	1.09943
$880$	0	0
$881$	6.62702	0.223270	0.111635	−	0.993749i	$-0.464391\pi$
$881$	6.62702	0.223270	0.111635	+	0.993749i	$0.464391\pi$
$882$	0	0
$883$	−7.31965	−0.246326	−0.123163	−	0.992386i	$-0.539304\pi$
$883$	−7.31965	−0.246326	−0.123163	+	0.992386i	$0.539304\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	−45.2450	−1.51747
$890$	0	0
$891$	3.07838	0.103130
$892$	0	0
$893$	34.7214	1.16191
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.98771	0.0663678
$898$	0	0
$899$	−12.9939	−0.433369
$900$	0	0
$901$	−22.6681	−0.755183
$902$	0	0
$903$	−5.04718	−0.167960
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.48255	0.148841	0.0744204	−	0.997227i	$-0.476289\pi$
$907$	4.48255	0.148841	0.0744204	+	0.997227i	$0.476289\pi$
$908$	0	0
$909$	−8.34017	−0.276626
$910$	0	0
$911$	−8.73367	−0.289359	−0.144680	−	0.989479i	$-0.546215\pi$
$911$	−8.73367	−0.289359	−0.144680	+	0.989479i	$0.546215\pi$
$912$	0	0
$913$	−5.67420	−0.187789
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.4231	−0.575361
$918$	0	0
$919$	−18.7337	−0.617967	−0.308983	−	0.951067i	$-0.599989\pi$
$919$	−18.7337	−0.617967	−0.308983	+	0.951067i	$0.599989\pi$
$920$	0	0
$921$	−29.9877	−0.988129
$922$	0	0
$923$	10.6147	0.349388
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.3402	0.339616
$928$	0	0
$929$	−20.0410	−0.657525	−0.328763	−	0.944413i	$-0.606632\pi$
$929$	−20.0410	−0.657525	−0.328763	+	0.944413i	$0.606632\pi$
$930$	0	0
$931$	−6.09436	−0.199735
$932$	0	0
$933$	−27.2039	−0.890617
$934$	0	0
$935$	0	0
$936$	0	0
$937$	45.0472	1.47163	0.735814	−	0.677184i	$-0.236800\pi$
$937$	45.0472	1.47163	0.735814	+	0.677184i	$0.236800\pi$
$938$	0	0
$939$	−26.8371	−0.875796
$940$	0	0
$941$	19.3751	0.631609	0.315805	−	0.948824i	$-0.397726\pi$
$941$	19.3751	0.631609	0.315805	+	0.948824i	$0.397726\pi$
$942$	0	0
$943$	23.0349	0.750119
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.95282	−0.225936	−0.112968	−	0.993599i	$-0.536036\pi$
$947$	−6.95282	−0.225936	−0.112968	+	0.993599i	$0.536036\pi$
$948$	0	0
$949$	−6.01229	−0.195167
$950$	0	0
$951$	−24.5958	−0.797574
$952$	0	0
$953$	29.2885	0.948746	0.474373	−	0.880324i	$-0.342675\pi$
$953$	29.2885	0.948746	0.474373	+	0.880324i	$0.342675\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−20.0000	−0.646508
$958$	0	0
$959$	−8.06239	−0.260348
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−13.3607	−0.430542
$964$	0	0
$965$	0	0
$966$	0	0
$967$	47.3874	1.52387	0.761937	−	0.647651i	$-0.224248\pi$
$967$	47.3874	1.52387	0.761937	+	0.647651i	$0.224248\pi$
$968$	0	0
$969$	−31.0349	−0.996984
$970$	0	0
$971$	2.59583	0.0833040	0.0416520	−	0.999132i	$-0.486738\pi$
$971$	2.59583	0.0833040	0.0416520	+	0.999132i	$0.486738\pi$
$972$	0	0
$973$	44.9405	1.44073
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.3234	−0.906144	−0.453072	−	0.891474i	$-0.649672\pi$
$977$	−28.3234	−0.906144	−0.453072	+	0.891474i	$0.649672\pi$
$978$	0	0
$979$	18.4703	0.590312
$980$	0	0
$981$	−5.31965	−0.169843
$982$	0	0
$983$	1.16290	0.0370907	0.0185454	−	0.999828i	$-0.494096\pi$
$983$	1.16290	0.0370907	0.0185454	+	0.999828i	$0.494096\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−20.3135	−0.646586
$988$	0	0
$989$	4.65159	0.147912
$990$	0	0
$991$	12.9528	0.411460	0.205730	−	0.978609i	$-0.434043\pi$
$991$	12.9528	0.411460	0.205730	+	0.978609i	$0.434043\pi$
$992$	0	0
$993$	−12.3135	−0.390757
$994$	0	0
$995$	0	0
$996$	0	0
$997$	19.0784	0.604218	0.302109	−	0.953273i	$-0.402309\pi$
$997$	19.0784	0.604218	0.302109	+	0.953273i	$0.402309\pi$
$998$	0	0
$999$	3.07838	0.0973956

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a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9600.2.a.dp.1.3		3
4.3	odd	2		9600.2.a.du.1.1		3
5.2	odd	4		1920.2.f.n.769.4	yes	6
5.3	odd	4		1920.2.f.n.769.1	yes	6
5.4	even	2		9600.2.a.dv.1.1		3
8.3	odd	2		9600.2.a.dr.1.1		3
8.5	even	2		9600.2.a.ds.1.3		3
20.3	even	4		1920.2.f.m.769.4	yes	6
20.7	even	4		1920.2.f.m.769.1	✓	6
20.19	odd	2		9600.2.a.do.1.3		3
40.3	even	4		1920.2.f.p.769.3	yes	6
40.13	odd	4		1920.2.f.o.769.6	yes	6
40.19	odd	2		9600.2.a.dt.1.3		3
40.27	even	4		1920.2.f.p.769.6	yes	6
40.29	even	2		9600.2.a.dq.1.1		3
40.37	odd	4		1920.2.f.o.769.3	yes	6
80.3	even	4		3840.2.d.bj.2689.4		6
80.13	odd	4		3840.2.d.bl.2689.4		6
80.27	even	4		3840.2.d.bj.2689.3		6
80.37	odd	4		3840.2.d.bl.2689.3		6
80.43	even	4		3840.2.d.bk.2689.3		6
80.53	odd	4		3840.2.d.bi.2689.3		6
80.67	even	4		3840.2.d.bk.2689.4		6
80.77	odd	4		3840.2.d.bi.2689.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.f.m.769.1	✓	6	20.7	even	4
1920.2.f.m.769.4	yes	6	20.3	even	4
1920.2.f.n.769.1	yes	6	5.3	odd	4
1920.2.f.n.769.4	yes	6	5.2	odd	4
1920.2.f.o.769.3	yes	6	40.37	odd	4
1920.2.f.o.769.6	yes	6	40.13	odd	4
1920.2.f.p.769.3	yes	6	40.3	even	4
1920.2.f.p.769.6	yes	6	40.27	even	4
3840.2.d.bi.2689.3		6	80.53	odd	4
3840.2.d.bi.2689.4		6	80.77	odd	4
3840.2.d.bj.2689.3		6	80.27	even	4
3840.2.d.bj.2689.4		6	80.3	even	4
3840.2.d.bk.2689.3		6	80.43	even	4
3840.2.d.bk.2689.4		6	80.67	even	4
3840.2.d.bl.2689.3		6	80.37	odd	4
3840.2.d.bl.2689.4		6	80.13	odd	4
9600.2.a.do.1.3		3	20.19	odd	2
9600.2.a.dp.1.3		3	1.1	even	1	trivial
9600.2.a.dq.1.1		3	40.29	even	2
9600.2.a.dr.1.1		3	8.3	odd	2
9600.2.a.ds.1.3		3	8.5	even	2
9600.2.a.dt.1.3		3	40.19	odd	2
9600.2.a.du.1.1		3	4.3	odd	2
9600.2.a.dv.1.1		3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

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	9600.2.a.dp.1.3

Level	$9600$
Weight	$2$
Character	9600.1
Self dual	yes
Analytic conductor	$76.656$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$