LMFDB - Embedded newform 9280.2.a.cb.1.1

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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 4x^{2} + 1$ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 4640)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.93185$ 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)

$f(q)$	$=$	$q-1.41421 q^{3} -1.00000 q^{5} -1.41421 q^{7} -1.00000 q^{9} -1.41421 q^{11} -1.46410 q^{13} +1.41421 q^{15} -6.19615 q^{17} +2.44949 q^{19} +2.00000 q^{21} -0.378937 q^{23} +1.00000 q^{25} +5.65685 q^{27} -1.00000 q^{29} -3.20736 q^{31} +2.00000 q^{33} +1.41421 q^{35} -0.732051 q^{37} +2.07055 q^{39} -4.00000 q^{41} -4.24264 q^{43} +1.00000 q^{45} -11.2122 q^{47} -5.00000 q^{49} +8.76268 q^{51} +2.53590 q^{53} +1.41421 q^{55} -3.46410 q^{57} -8.76268 q^{59} +12.3923 q^{61} +1.41421 q^{63} +1.46410 q^{65} -15.0759 q^{67} +0.535898 q^{69} -7.45001 q^{71} -8.19615 q^{73} -1.41421 q^{75} +2.00000 q^{77} -8.38375 q^{79} -5.00000 q^{81} -9.89949 q^{83} +6.19615 q^{85} +1.41421 q^{87} +11.8564 q^{89} +2.07055 q^{91} +4.53590 q^{93} -2.44949 q^{95} -12.1962 q^{97} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{5} - 4 q^{9} + 8 q^{13} - 4 q^{17} + 8 q^{21} + 4 q^{25} - 4 q^{29} + 8 q^{33} + 4 q^{37} - 16 q^{41} + 4 q^{45} - 20 q^{49} + 24 q^{53} + 8 q^{61} - 8 q^{65} + 16 q^{69} - 12 q^{73} + 8 q^{77}+ \cdots - 28 q^{97}+O(q^{100})$ 4 * q - 4 * q^5 - 4 * q^9 + 8 * q^13 - 4 * q^17 + 8 * q^21 + 4 * q^25 - 4 * q^29 + 8 * q^33 + 4 * q^37 - 16 * q^41 + 4 * q^45 - 20 * q^49 + 24 * q^53 + 8 * q^61 - 8 * q^65 + 16 * q^69 - 12 * q^73 + 8 * q^77 - 20 * q^81 + 4 * q^85 - 8 * q^89 + 32 * q^93 - 28 * q^97

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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	0	0
$15$	1.41421	0.365148
$16$	0	0
$17$	−6.19615	−1.50279	−0.751394	−	0.659854i	$-0.770618\pi$
$17$	−6.19615	−1.50279	−0.751394	+	0.659854i	$0.770618\pi$
$18$	0	0
$19$	2.44949	0.561951	0.280976	−	0.959715i	$-0.409342\pi$
$19$	2.44949	0.561951	0.280976	+	0.959715i	$0.409342\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−0.378937	−0.0790139	−0.0395070	−	0.999219i	$-0.512579\pi$
$23$	−0.378937	−0.0790139	−0.0395070	+	0.999219i	$0.512579\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.20736	−0.576060	−0.288030	−	0.957621i	$-0.593000\pi$
$31$	−3.20736	−0.576060	−0.288030	+	0.957621i	$0.593000\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	1.41421	0.239046
$36$	0	0
$37$	−0.732051	−0.120348	−0.0601742	−	0.998188i	$-0.519166\pi$
$37$	−0.732051	−0.120348	−0.0601742	+	0.998188i	$0.519166\pi$
$38$	0	0
$39$	2.07055	0.331554
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−4.24264	−0.646997	−0.323498	−	0.946229i	$-0.604859\pi$
$43$	−4.24264	−0.646997	−0.323498	+	0.946229i	$0.604859\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−11.2122	−1.63546	−0.817732	−	0.575600i	$-0.804769\pi$
$47$	−11.2122	−1.63546	−0.817732	+	0.575600i	$0.804769\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	8.76268	1.22702
$52$	0	0
$53$	2.53590	0.348332	0.174166	−	0.984716i	$-0.444277\pi$
$53$	2.53590	0.348332	0.174166	+	0.984716i	$0.444277\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	−8.76268	−1.14080	−0.570402	−	0.821366i	$-0.693213\pi$
$59$	−8.76268	−1.14080	−0.570402	+	0.821366i	$0.693213\pi$
$60$	0	0
$61$	12.3923	1.58667	0.793336	−	0.608784i	$-0.208342\pi$
$61$	12.3923	1.58667	0.793336	+	0.608784i	$0.208342\pi$
$62$	0	0
$63$	1.41421	0.178174
$64$	0	0
$65$	1.46410	0.181599
$66$	0	0
$67$	−15.0759	−1.84181	−0.920906	−	0.389785i	$-0.872549\pi$
$67$	−15.0759	−1.84181	−0.920906	+	0.389785i	$0.872549\pi$
$68$	0	0
$69$	0.535898	0.0645146
$70$	0	0
$71$	−7.45001	−0.884153	−0.442076	−	0.896977i	$-0.645758\pi$
$71$	−7.45001	−0.884153	−0.442076	+	0.896977i	$0.645758\pi$
$72$	0	0
$73$	−8.19615	−0.959287	−0.479644	−	0.877463i	$-0.659234\pi$
$73$	−8.19615	−0.959287	−0.479644	+	0.877463i	$0.659234\pi$
$74$	0	0
$75$	−1.41421	−0.163299
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−8.38375	−0.943245	−0.471623	−	0.881801i	$-0.656331\pi$
$79$	−8.38375	−0.943245	−0.471623	+	0.881801i	$0.656331\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−9.89949	−1.08661	−0.543305	−	0.839535i	$-0.682827\pi$
$83$	−9.89949	−1.08661	−0.543305	+	0.839535i	$0.682827\pi$
$84$	0	0
$85$	6.19615	0.672067
$86$	0	0
$87$	1.41421	0.151620
$88$	0	0
$89$	11.8564	1.25678	0.628388	−	0.777900i	$-0.283715\pi$
$89$	11.8564	1.25678	0.628388	+	0.777900i	$0.283715\pi$
$90$	0	0
$91$	2.07055	0.217053
$92$	0	0
$93$	4.53590	0.470351
$94$	0	0
$95$	−2.44949	−0.251312
$96$	0	0
$97$	−12.1962	−1.23833	−0.619166	−	0.785260i	$-0.712529\pi$
$97$	−12.1962	−1.23833	−0.619166	+	0.785260i	$0.712529\pi$
$98$	0	0
$99$	1.41421	0.142134
$100$	0	0
$101$	−2.39230	−0.238043	−0.119022	−	0.992892i	$-0.537976\pi$
$101$	−2.39230	−0.238043	−0.119022	+	0.992892i	$0.537976\pi$
$102$	0	0
$103$	−17.9043	−1.76416	−0.882082	−	0.471096i	$-0.843858\pi$
$103$	−17.9043	−1.76416	−0.882082	+	0.471096i	$0.843858\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	13.2827	1.28409	0.642045	−	0.766667i	$-0.278086\pi$
$107$	13.2827	1.28409	0.642045	+	0.766667i	$0.278086\pi$
$108$	0	0
$109$	6.92820	0.663602	0.331801	−	0.943349i	$-0.392344\pi$
$109$	6.92820	0.663602	0.331801	+	0.943349i	$0.392344\pi$
$110$	0	0
$111$	1.03528	0.0982641
$112$	0	0
$113$	20.5885	1.93680	0.968400	−	0.249404i	$-0.0802347\pi$
$113$	20.5885	1.93680	0.968400	+	0.249404i	$0.0802347\pi$
$114$	0	0
$115$	0.378937	0.0353361
$116$	0	0
$117$	1.46410	0.135356
$118$	0	0
$119$	8.76268	0.803274
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	5.65685	0.510061
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.1817	1.61337	0.806683	−	0.590985i	$-0.201261\pi$
$127$	18.1817	1.61337	0.806683	+	0.590985i	$0.201261\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	0	0
$131$	−1.69161	−0.147797	−0.0738985	−	0.997266i	$-0.523544\pi$
$131$	−1.69161	−0.147797	−0.0738985	+	0.997266i	$0.523544\pi$
$132$	0	0
$133$	−3.46410	−0.300376
$134$	0	0
$135$	−5.65685	−0.486864
$136$	0	0
$137$	4.73205	0.404286	0.202143	−	0.979356i	$-0.435209\pi$
$137$	4.73205	0.404286	0.202143	+	0.979356i	$0.435209\pi$
$138$	0	0
$139$	−15.4548	−1.31086	−0.655430	−	0.755256i	$-0.727513\pi$
$139$	−15.4548	−1.31086	−0.655430	+	0.755256i	$0.727513\pi$
$140$	0	0
$141$	15.8564	1.33535
$142$	0	0
$143$	2.07055	0.173148
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	7.07107	0.583212
$148$	0	0
$149$	12.5359	1.02698	0.513490	−	0.858095i	$-0.328352\pi$
$149$	12.5359	1.02698	0.513490	+	0.858095i	$0.328352\pi$
$150$	0	0
$151$	−6.69213	−0.544598	−0.272299	−	0.962213i	$-0.587784\pi$
$151$	−6.69213	−0.544598	−0.272299	+	0.962213i	$0.587784\pi$
$152$	0	0
$153$	6.19615	0.500929
$154$	0	0
$155$	3.20736	0.257622
$156$	0	0
$157$	−20.1962	−1.61183	−0.805914	−	0.592032i	$-0.798326\pi$
$157$	−20.1962	−1.61183	−0.805914	+	0.592032i	$0.798326\pi$
$158$	0	0
$159$	−3.58630	−0.284412
$160$	0	0
$161$	0.535898	0.0422347
$162$	0	0
$163$	−7.07107	−0.553849	−0.276924	−	0.960892i	$-0.589315\pi$
$163$	−7.07107	−0.553849	−0.276924	+	0.960892i	$0.589315\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−17.3495	−1.34254	−0.671272	−	0.741211i	$-0.734252\pi$
$167$	−17.3495	−1.34254	−0.671272	+	0.741211i	$0.734252\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−2.44949	−0.187317
$172$	0	0
$173$	21.8564	1.66171	0.830856	−	0.556488i	$-0.187851\pi$
$173$	21.8564	1.66171	0.830856	+	0.556488i	$0.187851\pi$
$174$	0	0
$175$	−1.41421	−0.106904
$176$	0	0
$177$	12.3923	0.931463
$178$	0	0
$179$	9.04008	0.675688	0.337844	−	0.941202i	$-0.390302\pi$
$179$	9.04008	0.675688	0.337844	+	0.941202i	$0.390302\pi$
$180$	0	0
$181$	18.5359	1.37776	0.688881	−	0.724874i	$-0.258102\pi$
$181$	18.5359	1.37776	0.688881	+	0.724874i	$0.258102\pi$
$182$	0	0
$183$	−17.5254	−1.29551
$184$	0	0
$185$	0.732051	0.0538214
$186$	0	0
$187$	8.76268	0.640791
$188$	0	0
$189$	−8.00000	−0.581914
$190$	0	0
$191$	−0.656339	−0.0474910	−0.0237455	−	0.999718i	$-0.507559\pi$
$191$	−0.656339	−0.0474910	−0.0237455	+	0.999718i	$0.507559\pi$
$192$	0	0
$193$	−21.1244	−1.52056	−0.760282	−	0.649593i	$-0.774939\pi$
$193$	−21.1244	−1.52056	−0.760282	+	0.649593i	$0.774939\pi$
$194$	0	0
$195$	−2.07055	−0.148275
$196$	0	0
$197$	12.9282	0.921096	0.460548	−	0.887635i	$-0.347653\pi$
$197$	12.9282	0.921096	0.460548	+	0.887635i	$0.347653\pi$
$198$	0	0
$199$	11.0363	0.782343	0.391172	−	0.920318i	$-0.372070\pi$
$199$	11.0363	0.782343	0.391172	+	0.920318i	$0.372070\pi$
$200$	0	0
$201$	21.3205	1.50383
$202$	0	0
$203$	1.41421	0.0992583
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	0.378937	0.0263380
$208$	0	0
$209$	−3.46410	−0.239617
$210$	0	0
$211$	−14.0406	−0.966595	−0.483297	−	0.875456i	$-0.660561\pi$
$211$	−14.0406	−0.966595	−0.483297	+	0.875456i	$0.660561\pi$
$212$	0	0
$213$	10.5359	0.721908
$214$	0	0
$215$	4.24264	0.289346
$216$	0	0
$217$	4.53590	0.307917
$218$	0	0
$219$	11.5911	0.783255
$220$	0	0
$221$	9.07180	0.610235
$222$	0	0
$223$	29.2180	1.95658	0.978291	−	0.207234i	$-0.0664462\pi$
$223$	29.2180	1.95658	0.978291	+	0.207234i	$0.0664462\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	19.9749	1.32578	0.662889	−	0.748718i	$-0.269330\pi$
$227$	19.9749	1.32578	0.662889	+	0.748718i	$0.269330\pi$
$228$	0	0
$229$	4.92820	0.325665	0.162832	−	0.986654i	$-0.447937\pi$
$229$	4.92820	0.325665	0.162832	+	0.986654i	$0.447937\pi$
$230$	0	0
$231$	−2.82843	−0.186097
$232$	0	0
$233$	−20.2487	−1.32654	−0.663269	−	0.748381i	$-0.730831\pi$
$233$	−20.2487	−1.32654	−0.663269	+	0.748381i	$0.730831\pi$
$234$	0	0
$235$	11.2122	0.731401
$236$	0	0
$237$	11.8564	0.770156
$238$	0	0
$239$	8.20788	0.530924	0.265462	−	0.964121i	$-0.414476\pi$
$239$	8.20788	0.530924	0.265462	+	0.964121i	$0.414476\pi$
$240$	0	0
$241$	3.85641	0.248413	0.124206	−	0.992256i	$-0.460361\pi$
$241$	3.85641	0.248413	0.124206	+	0.992256i	$0.460361\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	5.00000	0.319438
$246$	0	0
$247$	−3.58630	−0.228191
$248$	0	0
$249$	14.0000	0.887214
$250$	0	0
$251$	−16.1112	−1.01693	−0.508463	−	0.861084i	$-0.669786\pi$
$251$	−16.1112	−1.01693	−0.508463	+	0.861084i	$0.669786\pi$
$252$	0	0
$253$	0.535898	0.0336916
$254$	0	0
$255$	−8.76268	−0.548740
$256$	0	0
$257$	−2.92820	−0.182656	−0.0913281	−	0.995821i	$-0.529111\pi$
$257$	−2.92820	−0.182656	−0.0913281	+	0.995821i	$0.529111\pi$
$258$	0	0
$259$	1.03528	0.0643289
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	4.79744	0.295823	0.147912	−	0.989001i	$-0.452745\pi$
$263$	4.79744	0.295823	0.147912	+	0.989001i	$0.452745\pi$
$264$	0	0
$265$	−2.53590	−0.155779
$266$	0	0
$267$	−16.7675	−1.02615
$268$	0	0
$269$	−1.85641	−0.113187	−0.0565935	−	0.998397i	$-0.518024\pi$
$269$	−1.85641	−0.113187	−0.0565935	+	0.998397i	$0.518024\pi$
$270$	0	0
$271$	16.3142	0.991019	0.495509	−	0.868603i	$-0.334981\pi$
$271$	16.3142	0.991019	0.495509	+	0.868603i	$0.334981\pi$
$272$	0	0
$273$	−2.92820	−0.177223
$274$	0	0
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	−0.392305	−0.0235713	−0.0117857	−	0.999931i	$-0.503752\pi$
$277$	−0.392305	−0.0235713	−0.0117857	+	0.999931i	$0.503752\pi$
$278$	0	0
$279$	3.20736	0.192020
$280$	0	0
$281$	−13.8564	−0.826604	−0.413302	−	0.910594i	$-0.635625\pi$
$281$	−13.8564	−0.826604	−0.413302	+	0.910594i	$0.635625\pi$
$282$	0	0
$283$	−29.2180	−1.73683	−0.868415	−	0.495838i	$-0.834861\pi$
$283$	−29.2180	−1.73683	−0.868415	+	0.495838i	$0.834861\pi$
$284$	0	0
$285$	3.46410	0.205196
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	21.3923	1.25837
$290$	0	0
$291$	17.2480	1.01109
$292$	0	0
$293$	−1.66025	−0.0969931	−0.0484965	−	0.998823i	$-0.515443\pi$
$293$	−1.66025	−0.0969931	−0.0484965	+	0.998823i	$0.515443\pi$
$294$	0	0
$295$	8.76268	0.510183
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	0.554803	0.0320851
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	3.38323	0.194361
$304$	0	0
$305$	−12.3923	−0.709581
$306$	0	0
$307$	1.41421	0.0807134	0.0403567	−	0.999185i	$-0.487151\pi$
$307$	1.41421	0.0807134	0.0403567	+	0.999185i	$0.487151\pi$
$308$	0	0
$309$	25.3205	1.44043
$310$	0	0
$311$	13.7632	0.780439	0.390220	−	0.920722i	$-0.372399\pi$
$311$	13.7632	0.780439	0.390220	+	0.920722i	$0.372399\pi$
$312$	0	0
$313$	2.53590	0.143337	0.0716687	−	0.997428i	$-0.477168\pi$
$313$	2.53590	0.143337	0.0716687	+	0.997428i	$0.477168\pi$
$314$	0	0
$315$	−1.41421	−0.0796819
$316$	0	0
$317$	−25.5167	−1.43316	−0.716579	−	0.697506i	$-0.754293\pi$
$317$	−25.5167	−1.43316	−0.716579	+	0.697506i	$0.754293\pi$
$318$	0	0
$319$	1.41421	0.0791808
$320$	0	0
$321$	−18.7846	−1.04845
$322$	0	0
$323$	−15.1774	−0.844494
$324$	0	0
$325$	−1.46410	−0.0812137
$326$	0	0
$327$	−9.79796	−0.541828
$328$	0	0
$329$	15.8564	0.874192
$330$	0	0
$331$	22.8033	1.25338	0.626691	−	0.779268i	$-0.284409\pi$
$331$	22.8033	1.25338	0.626691	+	0.779268i	$0.284409\pi$
$332$	0	0
$333$	0.732051	0.0401161
$334$	0	0
$335$	15.0759	0.823683
$336$	0	0
$337$	12.5885	0.685737	0.342868	−	0.939383i	$-0.388601\pi$
$337$	12.5885	0.685737	0.342868	+	0.939383i	$0.388601\pi$
$338$	0	0
$339$	−29.1165	−1.58139
$340$	0	0
$341$	4.53590	0.245633
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	−0.535898	−0.0288518
$346$	0	0
$347$	−14.0406	−0.753739	−0.376869	−	0.926266i	$-0.622999\pi$
$347$	−14.0406	−0.753739	−0.376869	+	0.926266i	$0.622999\pi$
$348$	0	0
$349$	−7.85641	−0.420544	−0.210272	−	0.977643i	$-0.567435\pi$
$349$	−7.85641	−0.420544	−0.210272	+	0.977643i	$0.567435\pi$
$350$	0	0
$351$	−8.28221	−0.442072
$352$	0	0
$353$	26.5359	1.41236	0.706182	−	0.708031i	$-0.250416\pi$
$353$	26.5359	1.41236	0.706182	+	0.708031i	$0.250416\pi$
$354$	0	0
$355$	7.45001	0.395405
$356$	0	0
$357$	−12.3923	−0.655870
$358$	0	0
$359$	−9.41902	−0.497117	−0.248558	−	0.968617i	$-0.579957\pi$
$359$	−9.41902	−0.497117	−0.248558	+	0.968617i	$0.579957\pi$
$360$	0	0
$361$	−13.0000	−0.684211
$362$	0	0
$363$	12.7279	0.668043
$364$	0	0
$365$	8.19615	0.429006
$366$	0	0
$367$	−35.9101	−1.87449	−0.937247	−	0.348666i	$-0.886635\pi$
$367$	−35.9101	−1.87449	−0.937247	+	0.348666i	$0.886635\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	−3.58630	−0.186192
$372$	0	0
$373$	18.3923	0.952317	0.476159	−	0.879359i	$-0.342029\pi$
$373$	18.3923	0.952317	0.476159	+	0.879359i	$0.342029\pi$
$374$	0	0
$375$	1.41421	0.0730297
$376$	0	0
$377$	1.46410	0.0754051
$378$	0	0
$379$	9.69642	0.498072	0.249036	−	0.968494i	$-0.419886\pi$
$379$	9.69642	0.498072	0.249036	+	0.968494i	$0.419886\pi$
$380$	0	0
$381$	−25.7128	−1.31731
$382$	0	0
$383$	−13.2827	−0.678716	−0.339358	−	0.940657i	$-0.610210\pi$
$383$	−13.2827	−0.678716	−0.339358	+	0.940657i	$0.610210\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	0	0
$387$	4.24264	0.215666
$388$	0	0
$389$	32.7846	1.66225	0.831123	−	0.556089i	$-0.187699\pi$
$389$	32.7846	1.66225	0.831123	+	0.556089i	$0.187699\pi$
$390$	0	0
$391$	2.34795	0.118741
$392$	0	0
$393$	2.39230	0.120676
$394$	0	0
$395$	8.38375	0.421832
$396$	0	0
$397$	−15.8564	−0.795810	−0.397905	−	0.917427i	$-0.630263\pi$
$397$	−15.8564	−0.795810	−0.397905	+	0.917427i	$0.630263\pi$
$398$	0	0
$399$	4.89898	0.245256
$400$	0	0
$401$	0.392305	0.0195908	0.00979538	−	0.999952i	$-0.496882\pi$
$401$	0.392305	0.0195908	0.00979538	+	0.999952i	$0.496882\pi$
$402$	0	0
$403$	4.69591	0.233920
$404$	0	0
$405$	5.00000	0.248452
$406$	0	0
$407$	1.03528	0.0513167
$408$	0	0
$409$	−15.3205	−0.757550	−0.378775	−	0.925489i	$-0.623655\pi$
$409$	−15.3205	−0.757550	−0.378775	+	0.925489i	$0.623655\pi$
$410$	0	0
$411$	−6.69213	−0.330098
$412$	0	0
$413$	12.3923	0.609785
$414$	0	0
$415$	9.89949	0.485947
$416$	0	0
$417$	21.8564	1.07031
$418$	0	0
$419$	−12.1459	−0.593367	−0.296683	−	0.954976i	$-0.595881\pi$
$419$	−12.1459	−0.593367	−0.296683	+	0.954976i	$0.595881\pi$
$420$	0	0
$421$	34.9282	1.70230	0.851148	−	0.524925i	$-0.175907\pi$
$421$	34.9282	1.70230	0.851148	+	0.524925i	$0.175907\pi$
$422$	0	0
$423$	11.2122	0.545154
$424$	0	0
$425$	−6.19615	−0.300558
$426$	0	0
$427$	−17.5254	−0.848112
$428$	0	0
$429$	−2.92820	−0.141375
$430$	0	0
$431$	15.4548	0.744432	0.372216	−	0.928146i	$-0.378598\pi$
$431$	15.4548	0.744432	0.372216	+	0.928146i	$0.378598\pi$
$432$	0	0
$433$	6.19615	0.297768	0.148884	−	0.988855i	$-0.452432\pi$
$433$	6.19615	0.297768	0.148884	+	0.988855i	$0.452432\pi$
$434$	0	0
$435$	−1.41421	−0.0678064
$436$	0	0
$437$	−0.928203	−0.0444020
$438$	0	0
$439$	38.9144	1.85728	0.928642	−	0.370976i	$-0.120977\pi$
$439$	38.9144	1.85728	0.928642	+	0.370976i	$0.120977\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	0	0
$443$	10.6574	0.506347	0.253173	−	0.967421i	$-0.418526\pi$
$443$	10.6574	0.506347	0.253173	+	0.967421i	$0.418526\pi$
$444$	0	0
$445$	−11.8564	−0.562048
$446$	0	0
$447$	−17.7284	−0.838526
$448$	0	0
$449$	22.5359	1.06353	0.531767	−	0.846890i	$-0.321528\pi$
$449$	22.5359	1.06353	0.531767	+	0.846890i	$0.321528\pi$
$450$	0	0
$451$	5.65685	0.266371
$452$	0	0
$453$	9.46410	0.444662
$454$	0	0
$455$	−2.07055	−0.0970690
$456$	0	0
$457$	−1.46410	−0.0684878	−0.0342439	−	0.999414i	$-0.510902\pi$
$457$	−1.46410	−0.0684878	−0.0342439	+	0.999414i	$0.510902\pi$
$458$	0	0
$459$	−35.0507	−1.63603
$460$	0	0
$461$	−3.46410	−0.161339	−0.0806696	−	0.996741i	$-0.525706\pi$
$461$	−3.46410	−0.161339	−0.0806696	+	0.996741i	$0.525706\pi$
$462$	0	0
$463$	−0.175865	−0.00817316	−0.00408658	−	0.999992i	$-0.501301\pi$
$463$	−0.175865	−0.00817316	−0.00408658	+	0.999992i	$0.501301\pi$
$464$	0	0
$465$	−4.53590	−0.210347
$466$	0	0
$467$	−30.8081	−1.42563	−0.712814	−	0.701353i	$-0.752580\pi$
$467$	−30.8081	−1.42563	−0.712814	+	0.701353i	$0.752580\pi$
$468$	0	0
$469$	21.3205	0.984490
$470$	0	0
$471$	28.5617	1.31605
$472$	0	0
$473$	6.00000	0.275880
$474$	0	0
$475$	2.44949	0.112390
$476$	0	0
$477$	−2.53590	−0.116111
$478$	0	0
$479$	29.6985	1.35696	0.678479	−	0.734620i	$-0.262639\pi$
$479$	29.6985	1.35696	0.678479	+	0.734620i	$0.262639\pi$
$480$	0	0
$481$	1.07180	0.0488697
$482$	0	0
$483$	−0.757875	−0.0344845
$484$	0	0
$485$	12.1962	0.553799
$486$	0	0
$487$	−39.2934	−1.78055	−0.890276	−	0.455421i	$-0.849489\pi$
$487$	−39.2934	−1.78055	−0.890276	+	0.455421i	$0.849489\pi$
$488$	0	0
$489$	10.0000	0.452216
$490$	0	0
$491$	−18.6622	−0.842212	−0.421106	−	0.907011i	$-0.638358\pi$
$491$	−18.6622	−0.842212	−0.421106	+	0.907011i	$0.638358\pi$
$492$	0	0
$493$	6.19615	0.279061
$494$	0	0
$495$	−1.41421	−0.0635642
$496$	0	0
$497$	10.5359	0.472600
$498$	0	0
$499$	43.7391	1.95803	0.979015	−	0.203787i	$-0.0653251\pi$
$499$	43.7391	1.95803	0.979015	+	0.203787i	$0.0653251\pi$
$500$	0	0
$501$	24.5359	1.09618
$502$	0	0
$503$	−1.96902	−0.0877941	−0.0438971	−	0.999036i	$-0.513977\pi$
$503$	−1.96902	−0.0877941	−0.0438971	+	0.999036i	$0.513977\pi$
$504$	0	0
$505$	2.39230	0.106456
$506$	0	0
$507$	15.3533	0.681863
$508$	0	0
$509$	16.7846	0.743965	0.371982	−	0.928240i	$-0.378678\pi$
$509$	16.7846	0.743965	0.371982	+	0.928240i	$0.378678\pi$
$510$	0	0
$511$	11.5911	0.512761
$512$	0	0
$513$	13.8564	0.611775
$514$	0	0
$515$	17.9043	0.788958
$516$	0	0
$517$	15.8564	0.697364
$518$	0	0
$519$	−30.9096	−1.35678
$520$	0	0
$521$	−1.07180	−0.0469563	−0.0234781	−	0.999724i	$-0.507474\pi$
$521$	−1.07180	−0.0469563	−0.0234781	+	0.999724i	$0.507474\pi$
$522$	0	0
$523$	−1.13681	−0.0497093	−0.0248547	−	0.999691i	$-0.507912\pi$
$523$	−1.13681	−0.0497093	−0.0248547	+	0.999691i	$0.507912\pi$
$524$	0	0
$525$	2.00000	0.0872872
$526$	0	0
$527$	19.8733	0.865695
$528$	0	0
$529$	−22.8564	−0.993757
$530$	0	0
$531$	8.76268	0.380268
$532$	0	0
$533$	5.85641	0.253669
$534$	0	0
$535$	−13.2827	−0.574262
$536$	0	0
$537$	−12.7846	−0.551697
$538$	0	0
$539$	7.07107	0.304572
$540$	0	0
$541$	−0.143594	−0.00617357	−0.00308678	−	0.999995i	$-0.500983\pi$
$541$	−0.143594	−0.00617357	−0.00308678	+	0.999995i	$0.500983\pi$
$542$	0	0
$543$	−26.2137	−1.12494
$544$	0	0
$545$	−6.92820	−0.296772
$546$	0	0
$547$	23.8386	1.01926	0.509632	−	0.860393i	$-0.329782\pi$
$547$	23.8386	1.01926	0.509632	+	0.860393i	$0.329782\pi$
$548$	0	0
$549$	−12.3923	−0.528891
$550$	0	0
$551$	−2.44949	−0.104352
$552$	0	0
$553$	11.8564	0.504186
$554$	0	0
$555$	−1.03528	−0.0439450
$556$	0	0
$557$	28.2487	1.19694	0.598468	−	0.801147i	$-0.295776\pi$
$557$	28.2487	1.19694	0.598468	+	0.801147i	$0.295776\pi$
$558$	0	0
$559$	6.21166	0.262725
$560$	0	0
$561$	−12.3923	−0.523204
$562$	0	0
$563$	35.3553	1.49005	0.745025	−	0.667037i	$-0.232438\pi$
$563$	35.3553	1.49005	0.745025	+	0.667037i	$0.232438\pi$
$564$	0	0
$565$	−20.5885	−0.866163
$566$	0	0
$567$	7.07107	0.296957
$568$	0	0
$569$	17.8564	0.748580	0.374290	−	0.927312i	$-0.377887\pi$
$569$	17.8564	0.748580	0.374290	+	0.927312i	$0.377887\pi$
$570$	0	0
$571$	−22.6274	−0.946928	−0.473464	−	0.880813i	$-0.656997\pi$
$571$	−22.6274	−0.946928	−0.473464	+	0.880813i	$0.656997\pi$
$572$	0	0
$573$	0.928203	0.0387762
$574$	0	0
$575$	−0.378937	−0.0158028
$576$	0	0
$577$	41.5167	1.72836	0.864181	−	0.503182i	$-0.167837\pi$
$577$	41.5167	1.72836	0.864181	+	0.503182i	$0.167837\pi$
$578$	0	0
$579$	29.8744	1.24154
$580$	0	0
$581$	14.0000	0.580818
$582$	0	0
$583$	−3.58630	−0.148529
$584$	0	0
$585$	−1.46410	−0.0605332
$586$	0	0
$587$	7.34847	0.303304	0.151652	−	0.988434i	$-0.451541\pi$
$587$	7.34847	0.303304	0.151652	+	0.988434i	$0.451541\pi$
$588$	0	0
$589$	−7.85641	−0.323718
$590$	0	0
$591$	−18.2832	−0.752072
$592$	0	0
$593$	10.1436	0.416547	0.208274	−	0.978071i	$-0.433216\pi$
$593$	10.1436	0.416547	0.208274	+	0.978071i	$0.433216\pi$
$594$	0	0
$595$	−8.76268	−0.359235
$596$	0	0
$597$	−15.6077	−0.638780
$598$	0	0
$599$	−24.0416	−0.982314	−0.491157	−	0.871071i	$-0.663426\pi$
$599$	−24.0416	−0.982314	−0.491157	+	0.871071i	$0.663426\pi$
$600$	0	0
$601$	−34.9282	−1.42475	−0.712376	−	0.701798i	$-0.752381\pi$
$601$	−34.9282	−1.42475	−0.712376	+	0.701798i	$0.752381\pi$
$602$	0	0
$603$	15.0759	0.613937
$604$	0	0
$605$	9.00000	0.365902
$606$	0	0
$607$	32.5269	1.32023	0.660113	−	0.751166i	$-0.270508\pi$
$607$	32.5269	1.32023	0.660113	+	0.751166i	$0.270508\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	16.4158	0.664111
$612$	0	0
$613$	−11.8564	−0.478876	−0.239438	−	0.970912i	$-0.576963\pi$
$613$	−11.8564	−0.478876	−0.239438	+	0.970912i	$0.576963\pi$
$614$	0	0
$615$	−5.65685	−0.228106
$616$	0	0
$617$	−11.1244	−0.447850	−0.223925	−	0.974606i	$-0.571887\pi$
$617$	−11.1244	−0.447850	−0.223925	+	0.974606i	$0.571887\pi$
$618$	0	0
$619$	−13.2827	−0.533878	−0.266939	−	0.963713i	$-0.586012\pi$
$619$	−13.2827	−0.533878	−0.266939	+	0.963713i	$0.586012\pi$
$620$	0	0
$621$	−2.14359	−0.0860194
$622$	0	0
$623$	−16.7675	−0.671775
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.89898	0.195646
$628$	0	0
$629$	4.53590	0.180858
$630$	0	0
$631$	−26.7685	−1.06564	−0.532819	−	0.846229i	$-0.678867\pi$
$631$	−26.7685	−1.06564	−0.532819	+	0.846229i	$0.678867\pi$
$632$	0	0
$633$	19.8564	0.789221
$634$	0	0
$635$	−18.1817	−0.721519
$636$	0	0
$637$	7.32051	0.290049
$638$	0	0
$639$	7.45001	0.294718
$640$	0	0
$641$	−2.92820	−0.115657	−0.0578285	−	0.998327i	$-0.518418\pi$
$641$	−2.92820	−0.115657	−0.0578285	+	0.998327i	$0.518418\pi$
$642$	0	0
$643$	−34.8749	−1.37533	−0.687665	−	0.726028i	$-0.741364\pi$
$643$	−34.8749	−1.37533	−0.687665	+	0.726028i	$0.741364\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	7.62587	0.299804	0.149902	−	0.988701i	$-0.452104\pi$
$647$	7.62587	0.299804	0.149902	+	0.988701i	$0.452104\pi$
$648$	0	0
$649$	12.3923	0.486441
$650$	0	0
$651$	−6.41473	−0.251413
$652$	0	0
$653$	−8.58846	−0.336092	−0.168046	−	0.985779i	$-0.553746\pi$
$653$	−8.58846	−0.336092	−0.168046	+	0.985779i	$0.553746\pi$
$654$	0	0
$655$	1.69161	0.0660969
$656$	0	0
$657$	8.19615	0.319762
$658$	0	0
$659$	8.66115	0.337390	0.168695	−	0.985668i	$-0.446045\pi$
$659$	8.66115	0.337390	0.168695	+	0.985668i	$0.446045\pi$
$660$	0	0
$661$	21.8564	0.850116	0.425058	−	0.905166i	$-0.360254\pi$
$661$	21.8564	0.850116	0.425058	+	0.905166i	$0.360254\pi$
$662$	0	0
$663$	−12.8295	−0.498255
$664$	0	0
$665$	3.46410	0.134332
$666$	0	0
$667$	0.378937	0.0146725
$668$	0	0
$669$	−41.3205	−1.59754
$670$	0	0
$671$	−17.5254	−0.676559
$672$	0	0
$673$	27.8564	1.07379	0.536893	−	0.843650i	$-0.319598\pi$
$673$	27.8564	1.07379	0.536893	+	0.843650i	$0.319598\pi$
$674$	0	0
$675$	5.65685	0.217732
$676$	0	0
$677$	−9.94744	−0.382311	−0.191156	−	0.981560i	$-0.561224\pi$
$677$	−9.94744	−0.382311	−0.191156	+	0.981560i	$0.561224\pi$
$678$	0	0
$679$	17.2480	0.661916
$680$	0	0
$681$	−28.2487	−1.08249
$682$	0	0
$683$	−34.6718	−1.32668	−0.663340	−	0.748318i	$-0.730862\pi$
$683$	−34.6718	−1.32668	−0.663340	+	0.748318i	$0.730862\pi$
$684$	0	0
$685$	−4.73205	−0.180802
$686$	0	0
$687$	−6.96953	−0.265904
$688$	0	0
$689$	−3.71281	−0.141447
$690$	0	0
$691$	−3.66063	−0.139257	−0.0696285	−	0.997573i	$-0.522181\pi$
$691$	−3.66063	−0.139257	−0.0696285	+	0.997573i	$0.522181\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	15.4548	0.586234
$696$	0	0
$697$	24.7846	0.938784
$698$	0	0
$699$	28.6360	1.08311
$700$	0	0
$701$	2.92820	0.110597	0.0552984	−	0.998470i	$-0.482389\pi$
$701$	2.92820	0.110597	0.0552984	+	0.998470i	$0.482389\pi$
$702$	0	0
$703$	−1.79315	−0.0676300
$704$	0	0
$705$	−15.8564	−0.597187
$706$	0	0
$707$	3.38323	0.127239
$708$	0	0
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	8.38375	0.314415
$712$	0	0
$713$	1.21539	0.0455167
$714$	0	0
$715$	−2.07055	−0.0774343
$716$	0	0
$717$	−11.6077	−0.433497
$718$	0	0
$719$	39.8754	1.48710	0.743550	−	0.668680i	$-0.233140\pi$
$719$	39.8754	1.48710	0.743550	+	0.668680i	$0.233140\pi$
$720$	0	0
$721$	25.3205	0.942985
$722$	0	0
$723$	−5.45378	−0.202828
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−40.8091	−1.51353	−0.756763	−	0.653689i	$-0.773220\pi$
$727$	−40.8091	−1.51353	−0.756763	+	0.653689i	$0.773220\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	26.2880	0.972299
$732$	0	0
$733$	13.4115	0.495367	0.247683	−	0.968841i	$-0.420331\pi$
$733$	13.4115	0.495367	0.247683	+	0.968841i	$0.420331\pi$
$734$	0	0
$735$	−7.07107	−0.260820
$736$	0	0
$737$	21.3205	0.785351
$738$	0	0
$739$	28.6632	1.05439	0.527197	−	0.849743i	$-0.323243\pi$
$739$	28.6632	1.05439	0.527197	+	0.849743i	$0.323243\pi$
$740$	0	0
$741$	5.07180	0.186317
$742$	0	0
$743$	6.11012	0.224159	0.112079	−	0.993699i	$-0.464249\pi$
$743$	6.11012	0.224159	0.112079	+	0.993699i	$0.464249\pi$
$744$	0	0
$745$	−12.5359	−0.459280
$746$	0	0
$747$	9.89949	0.362204
$748$	0	0
$749$	−18.7846	−0.686375
$750$	0	0
$751$	1.48854	0.0543177	0.0271589	−	0.999631i	$-0.491354\pi$
$751$	1.48854	0.0543177	0.0271589	+	0.999631i	$0.491354\pi$
$752$	0	0
$753$	22.7846	0.830317
$754$	0	0
$755$	6.69213	0.243552
$756$	0	0
$757$	−14.3397	−0.521187	−0.260593	−	0.965449i	$-0.583918\pi$
$757$	−14.3397	−0.521187	−0.260593	+	0.965449i	$0.583918\pi$
$758$	0	0
$759$	−0.757875	−0.0275091
$760$	0	0
$761$	5.71281	0.207089	0.103545	−	0.994625i	$-0.466982\pi$
$761$	5.71281	0.207089	0.103545	+	0.994625i	$0.466982\pi$
$762$	0	0
$763$	−9.79796	−0.354710
$764$	0	0
$765$	−6.19615	−0.224022
$766$	0	0
$767$	12.8295	0.463245
$768$	0	0
$769$	−3.46410	−0.124919	−0.0624593	−	0.998048i	$-0.519894\pi$
$769$	−3.46410	−0.124919	−0.0624593	+	0.998048i	$0.519894\pi$
$770$	0	0
$771$	4.14110	0.149138
$772$	0	0
$773$	−41.9090	−1.50736	−0.753680	−	0.657241i	$-0.771723\pi$
$773$	−41.9090	−1.50736	−0.753680	+	0.657241i	$0.771723\pi$
$774$	0	0
$775$	−3.20736	−0.115212
$776$	0	0
$777$	−1.46410	−0.0525244
$778$	0	0
$779$	−9.79796	−0.351048
$780$	0	0
$781$	10.5359	0.377004
$782$	0	0
$783$	−5.65685	−0.202159
$784$	0	0
$785$	20.1962	0.720832
$786$	0	0
$787$	34.5975	1.23327	0.616633	−	0.787251i	$-0.288496\pi$
$787$	34.5975	1.23327	0.616633	+	0.787251i	$0.288496\pi$
$788$	0	0
$789$	−6.78461	−0.241539
$790$	0	0
$791$	−29.1165	−1.03526
$792$	0	0
$793$	−18.1436	−0.644298
$794$	0	0
$795$	3.58630	0.127193
$796$	0	0
$797$	−39.1244	−1.38586	−0.692928	−	0.721007i	$-0.743680\pi$
$797$	−39.1244	−1.38586	−0.692928	+	0.721007i	$0.743680\pi$
$798$	0	0
$799$	69.4723	2.45775
$800$	0	0
$801$	−11.8564	−0.418926
$802$	0	0
$803$	11.5911	0.409041
$804$	0	0
$805$	−0.535898	−0.0188879
$806$	0	0
$807$	2.62536	0.0924169
$808$	0	0
$809$	36.9282	1.29833	0.649163	−	0.760649i	$-0.275119\pi$
$809$	36.9282	1.29833	0.649163	+	0.760649i	$0.275119\pi$
$810$	0	0
$811$	−34.6990	−1.21845	−0.609223	−	0.792999i	$-0.708519\pi$
$811$	−34.6990	−1.21845	−0.609223	+	0.792999i	$0.708519\pi$
$812$	0	0
$813$	−23.0718	−0.809163
$814$	0	0
$815$	7.07107	0.247689
$816$	0	0
$817$	−10.3923	−0.363581
$818$	0	0
$819$	−2.07055	−0.0723510
$820$	0	0
$821$	12.5359	0.437506	0.218753	−	0.975780i	$-0.429801\pi$
$821$	12.5359	0.437506	0.218753	+	0.975780i	$0.429801\pi$
$822$	0	0
$823$	−17.6269	−0.614435	−0.307218	−	0.951639i	$-0.599398\pi$
$823$	−17.6269	−0.614435	−0.307218	+	0.951639i	$0.599398\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	33.0817	1.15036	0.575182	−	0.818025i	$-0.304931\pi$
$827$	33.0817	1.15036	0.575182	+	0.818025i	$0.304931\pi$
$828$	0	0
$829$	−17.8564	−0.620179	−0.310089	−	0.950707i	$-0.600359\pi$
$829$	−17.8564	−0.620179	−0.310089	+	0.950707i	$0.600359\pi$
$830$	0	0
$831$	0.554803	0.0192459
$832$	0	0
$833$	30.9808	1.07342
$834$	0	0
$835$	17.3495	0.600404
$836$	0	0
$837$	−18.1436	−0.627134
$838$	0	0
$839$	−9.69642	−0.334758	−0.167379	−	0.985893i	$-0.553530\pi$
$839$	−9.69642	−0.334758	−0.167379	+	0.985893i	$0.553530\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	19.5959	0.674919
$844$	0	0
$845$	10.8564	0.373472
$846$	0	0
$847$	12.7279	0.437337
$848$	0	0
$849$	41.3205	1.41812
$850$	0	0
$851$	0.277401	0.00950920
$852$	0	0
$853$	0.0525589	0.00179958	0.000899791	−	1.00000i	$-0.499714\pi$
$853$	0.0525589	0.00179958	0.000899791		1.00000i	$0.499714\pi$
$854$	0	0
$855$	2.44949	0.0837708
$856$	0	0
$857$	44.7846	1.52981	0.764907	−	0.644141i	$-0.222785\pi$
$857$	44.7846	1.52981	0.764907	+	0.644141i	$0.222785\pi$
$858$	0	0
$859$	1.61729	0.0551811	0.0275905	−	0.999619i	$-0.491217\pi$
$859$	1.61729	0.0551811	0.0275905	+	0.999619i	$0.491217\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	−18.9396	−0.644711	−0.322355	−	0.946619i	$-0.604475\pi$
$863$	−18.9396	−0.644711	−0.322355	+	0.946619i	$0.604475\pi$
$864$	0	0
$865$	−21.8564	−0.743140
$866$	0	0
$867$	−30.2533	−1.02746
$868$	0	0
$869$	11.8564	0.402201
$870$	0	0
$871$	22.0726	0.747902
$872$	0	0
$873$	12.1962	0.412777
$874$	0	0
$875$	1.41421	0.0478091
$876$	0	0
$877$	−43.0333	−1.45313	−0.726566	−	0.687097i	$-0.758885\pi$
$877$	−43.0333	−1.45313	−0.726566	+	0.687097i	$0.758885\pi$
$878$	0	0
$879$	2.34795	0.0791945
$880$	0	0
$881$	2.53590	0.0854366	0.0427183	−	0.999087i	$-0.486398\pi$
$881$	2.53590	0.0854366	0.0427183	+	0.999087i	$0.486398\pi$
$882$	0	0
$883$	30.2533	1.01810	0.509052	−	0.860736i	$-0.329996\pi$
$883$	30.2533	1.01810	0.509052	+	0.860736i	$0.329996\pi$
$884$	0	0
$885$	−12.3923	−0.416563
$886$	0	0
$887$	31.2142	1.04807	0.524036	−	0.851696i	$-0.324426\pi$
$887$	31.2142	1.04807	0.524036	+	0.851696i	$0.324426\pi$
$888$	0	0
$889$	−25.7128	−0.862380
$890$	0	0
$891$	7.07107	0.236890
$892$	0	0
$893$	−27.4641	−0.919051
$894$	0	0
$895$	−9.04008	−0.302177
$896$	0	0
$897$	−0.784610	−0.0261974
$898$	0	0
$899$	3.20736	0.106972
$900$	0	0
$901$	−15.7128	−0.523470
$902$	0	0
$903$	−8.48528	−0.282372
$904$	0	0
$905$	−18.5359	−0.616154
$906$	0	0
$907$	−5.55532	−0.184461	−0.0922307	−	0.995738i	$-0.529400\pi$
$907$	−5.55532	−0.184461	−0.0922307	+	0.995738i	$0.529400\pi$
$908$	0	0
$909$	2.39230	0.0793477
$910$	0	0
$911$	−20.1779	−0.668525	−0.334262	−	0.942480i	$-0.608487\pi$
$911$	−20.1779	−0.668525	−0.334262	+	0.942480i	$0.608487\pi$
$912$	0	0
$913$	14.0000	0.463332
$914$	0	0
$915$	17.5254	0.579371
$916$	0	0
$917$	2.39230	0.0790009
$918$	0	0
$919$	−45.8840	−1.51357	−0.756786	−	0.653662i	$-0.773232\pi$
$919$	−45.8840	−1.51357	−0.756786	+	0.653662i	$0.773232\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	10.9076	0.359027
$924$	0	0
$925$	−0.732051	−0.0240697
$926$	0	0
$927$	17.9043	0.588054
$928$	0	0
$929$	−16.2487	−0.533103	−0.266551	−	0.963821i	$-0.585884\pi$
$929$	−16.2487	−0.533103	−0.266551	+	0.963821i	$0.585884\pi$
$930$	0	0
$931$	−12.2474	−0.401394
$932$	0	0
$933$	−19.4641	−0.637226
$934$	0	0
$935$	−8.76268	−0.286570
$936$	0	0
$937$	−40.7846	−1.33238	−0.666188	−	0.745784i	$-0.732075\pi$
$937$	−40.7846	−1.33238	−0.666188	+	0.745784i	$0.732075\pi$
$938$	0	0
$939$	−3.58630	−0.117035
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	1.51575	0.0493596
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	−5.55532	−0.180524	−0.0902618	−	0.995918i	$-0.528770\pi$
$947$	−5.55532	−0.180524	−0.0902618	+	0.995918i	$0.528770\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	36.0860	1.17017
$952$	0	0
$953$	−16.6410	−0.539055	−0.269528	−	0.962993i	$-0.586868\pi$
$953$	−16.6410	−0.539055	−0.269528	+	0.962993i	$0.586868\pi$
$954$	0	0
$955$	0.656339	0.0212386
$956$	0	0
$957$	−2.00000	−0.0646508
$958$	0	0
$959$	−6.69213	−0.216100
$960$	0	0
$961$	−20.7128	−0.668155
$962$	0	0
$963$	−13.2827	−0.428030
$964$	0	0
$965$	21.1244	0.680017
$966$	0	0
$967$	10.4543	0.336188	0.168094	−	0.985771i	$-0.446239\pi$
$967$	10.4543	0.336188	0.168094	+	0.985771i	$0.446239\pi$
$968$	0	0
$969$	21.4641	0.689526
$970$	0	0
$971$	−43.3601	−1.39149	−0.695747	−	0.718287i	$-0.744926\pi$
$971$	−43.3601	−1.39149	−0.695747	+	0.718287i	$0.744926\pi$
$972$	0	0
$973$	21.8564	0.700684
$974$	0	0
$975$	2.07055	0.0663107
$976$	0	0
$977$	−15.7128	−0.502697	−0.251349	−	0.967897i	$-0.580874\pi$
$977$	−15.7128	−0.502697	−0.251349	+	0.967897i	$0.580874\pi$
$978$	0	0
$979$	−16.7675	−0.535891
$980$	0	0
$981$	−6.92820	−0.221201
$982$	0	0
$983$	7.62587	0.243227	0.121614	−	0.992578i	$-0.461193\pi$
$983$	7.62587	0.243227	0.121614	+	0.992578i	$0.461193\pi$
$984$	0	0
$985$	−12.9282	−0.411927
$986$	0	0
$987$	−22.4243	−0.713775
$988$	0	0
$989$	1.60770	0.0511217
$990$	0	0
$991$	−32.4997	−1.03239	−0.516194	−	0.856472i	$-0.672651\pi$
$991$	−32.4997	−1.03239	−0.516194	+	0.856472i	$0.672651\pi$
$992$	0	0
$993$	−32.2487	−1.02338
$994$	0	0
$995$	−11.0363	−0.349874
$996$	0	0
$997$	−25.6603	−0.812668	−0.406334	−	0.913725i	$-0.633193\pi$
$997$	−25.6603	−0.812668	−0.406334	+	0.913725i	$0.633193\pi$
$998$	0	0
$999$	−4.14110	−0.131019

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.cb.1.1		4
4.3	odd	2	inner	9280.2.a.cb.1.3		4
8.3	odd	2		4640.2.a.p.1.2	✓	4
8.5	even	2		4640.2.a.p.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.p.1.2	✓	4	8.3	odd	2
4640.2.a.p.1.4	yes	4	8.5	even	2
9280.2.a.cb.1.1		4	1.1	even	1	trivial
9280.2.a.cb.1.3		4	4.3	odd	2	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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	9280.2.a.cb.1.1

Level	$9280$
Weight	$2$
Character	9280.1
Self dual	yes
Analytic conductor	$74.101$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$