LMFDB - Embedded newform 8004.2.a.j.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$4.08902$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.08902 q^{5} +2.57555 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +4.08902 q^{5} +2.57555 q^{7} +1.00000 q^{9} +4.61172 q^{11} -4.79639 q^{13} +4.08902 q^{15} -1.83258 q^{17} +4.89968 q^{19} +2.57555 q^{21} +1.00000 q^{23} +11.7201 q^{25} +1.00000 q^{27} -1.00000 q^{29} +8.03980 q^{31} +4.61172 q^{33} +10.5315 q^{35} -8.43013 q^{37} -4.79639 q^{39} +8.64925 q^{41} -7.46071 q^{43} +4.08902 q^{45} -5.52609 q^{47} -0.366553 q^{49} -1.83258 q^{51} -7.81069 q^{53} +18.8574 q^{55} +4.89968 q^{57} +7.57493 q^{59} -10.4358 q^{61} +2.57555 q^{63} -19.6125 q^{65} +8.30407 q^{67} +1.00000 q^{69} -0.0870476 q^{71} +9.17475 q^{73} +11.7201 q^{75} +11.8777 q^{77} -8.38478 q^{79} +1.00000 q^{81} -9.19932 q^{83} -7.49346 q^{85} -1.00000 q^{87} -11.9049 q^{89} -12.3533 q^{91} +8.03980 q^{93} +20.0349 q^{95} +6.00162 q^{97} +4.61172 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	4.08902	1.82866	0.914332	−	0.404965i	$-0.132716\pi$
$5$	4.08902	1.82866	0.914332	+	0.404965i	$0.132716\pi$
$6$	0	0
$7$	2.57555	0.973466	0.486733	−	0.873551i	$-0.338189\pi$
$7$	2.57555	0.973466	0.486733	+	0.873551i	$0.338189\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.61172	1.39049	0.695243	−	0.718775i	$-0.255297\pi$
$11$	4.61172	1.39049	0.695243	+	0.718775i	$0.255297\pi$
$12$	0	0
$13$	−4.79639	−1.33028	−0.665140	−	0.746719i	$-0.731628\pi$
$13$	−4.79639	−1.33028	−0.665140	+	0.746719i	$0.731628\pi$
$14$	0	0
$15$	4.08902	1.05578
$16$	0	0
$17$	−1.83258	−0.444466	−0.222233	−	0.974994i	$-0.571335\pi$
$17$	−1.83258	−0.444466	−0.222233	+	0.974994i	$0.571335\pi$
$18$	0	0
$19$	4.89968	1.12406	0.562032	−	0.827116i	$-0.310020\pi$
$19$	4.89968	1.12406	0.562032	+	0.827116i	$0.310020\pi$
$20$	0	0
$21$	2.57555	0.562031
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	11.7201	2.34401
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	8.03980	1.44399	0.721996	−	0.691898i	$-0.243225\pi$
$31$	8.03980	1.44399	0.721996	+	0.691898i	$0.243225\pi$
$32$	0	0
$33$	4.61172	0.802797
$34$	0	0
$35$	10.5315	1.78014
$36$	0	0
$37$	−8.43013	−1.38591	−0.692953	−	0.720983i	$-0.743691\pi$
$37$	−8.43013	−1.38591	−0.692953	+	0.720983i	$0.743691\pi$
$38$	0	0
$39$	−4.79639	−0.768037
$40$	0	0
$41$	8.64925	1.35079	0.675393	−	0.737458i	$-0.263974\pi$
$41$	8.64925	1.35079	0.675393	+	0.737458i	$0.263974\pi$
$42$	0	0
$43$	−7.46071	−1.13775	−0.568874	−	0.822425i	$-0.692621\pi$
$43$	−7.46071	−1.13775	−0.568874	+	0.822425i	$0.692621\pi$
$44$	0	0
$45$	4.08902	0.609555
$46$	0	0
$47$	−5.52609	−0.806063	−0.403032	−	0.915186i	$-0.632044\pi$
$47$	−5.52609	−0.806063	−0.403032	+	0.915186i	$0.632044\pi$
$48$	0	0
$49$	−0.366553	−0.0523647
$50$	0	0
$51$	−1.83258	−0.256613
$52$	0	0
$53$	−7.81069	−1.07288	−0.536441	−	0.843938i	$-0.680231\pi$
$53$	−7.81069	−1.07288	−0.536441	+	0.843938i	$0.680231\pi$
$54$	0	0
$55$	18.8574	2.54273
$56$	0	0
$57$	4.89968	0.648978
$58$	0	0
$59$	7.57493	0.986172	0.493086	−	0.869981i	$-0.335869\pi$
$59$	7.57493	0.986172	0.493086	+	0.869981i	$0.335869\pi$
$60$	0	0
$61$	−10.4358	−1.33616	−0.668082	−	0.744088i	$-0.732885\pi$
$61$	−10.4358	−1.33616	−0.668082	+	0.744088i	$0.732885\pi$
$62$	0	0
$63$	2.57555	0.324489
$64$	0	0
$65$	−19.6125	−2.43263
$66$	0	0
$67$	8.30407	1.01450	0.507252	−	0.861798i	$-0.330661\pi$
$67$	8.30407	1.01450	0.507252	+	0.861798i	$0.330661\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−0.0870476	−0.0103306	−0.00516532	−	0.999987i	$-0.501644\pi$
$71$	−0.0870476	−0.0103306	−0.00516532	+	0.999987i	$0.501644\pi$
$72$	0	0
$73$	9.17475	1.07382	0.536912	−	0.843639i	$-0.319591\pi$
$73$	9.17475	1.07382	0.536912	+	0.843639i	$0.319591\pi$
$74$	0	0
$75$	11.7201	1.35332
$76$	0	0
$77$	11.8777	1.35359
$78$	0	0
$79$	−8.38478	−0.943361	−0.471680	−	0.881770i	$-0.656352\pi$
$79$	−8.38478	−0.943361	−0.471680	+	0.881770i	$0.656352\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.19932	−1.00976	−0.504878	−	0.863191i	$-0.668463\pi$
$83$	−9.19932	−1.00976	−0.504878	+	0.863191i	$0.668463\pi$
$84$	0	0
$85$	−7.49346	−0.812780
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−11.9049	−1.26191	−0.630957	−	0.775818i	$-0.717337\pi$
$89$	−11.9049	−1.26191	−0.630957	+	0.775818i	$0.717337\pi$
$90$	0	0
$91$	−12.3533	−1.29498
$92$	0	0
$93$	8.03980	0.833689
$94$	0	0
$95$	20.0349	2.05553
$96$	0	0
$97$	6.00162	0.609372	0.304686	−	0.952453i	$-0.401448\pi$
$97$	6.00162	0.609372	0.304686	+	0.952453i	$0.401448\pi$
$98$	0	0
$99$	4.61172	0.463495
$100$	0	0
$101$	14.2557	1.41849	0.709246	−	0.704961i	$-0.249036\pi$
$101$	14.2557	1.41849	0.709246	+	0.704961i	$0.249036\pi$
$102$	0	0
$103$	−15.9300	−1.56963	−0.784817	−	0.619727i	$-0.787243\pi$
$103$	−15.9300	−1.56963	−0.784817	+	0.619727i	$0.787243\pi$
$104$	0	0
$105$	10.5315	1.02777
$106$	0	0
$107$	13.6477	1.31937	0.659686	−	0.751542i	$-0.270689\pi$
$107$	13.6477	1.31937	0.659686	+	0.751542i	$0.270689\pi$
$108$	0	0
$109$	−10.7879	−1.03329	−0.516646	−	0.856199i	$-0.672820\pi$
$109$	−10.7879	−1.03329	−0.516646	+	0.856199i	$0.672820\pi$
$110$	0	0
$111$	−8.43013	−0.800153
$112$	0	0
$113$	14.8279	1.39490	0.697448	−	0.716635i	$-0.254319\pi$
$113$	14.8279	1.39490	0.697448	+	0.716635i	$0.254319\pi$
$114$	0	0
$115$	4.08902	0.381303
$116$	0	0
$117$	−4.79639	−0.443426
$118$	0	0
$119$	−4.71990	−0.432673
$120$	0	0
$121$	10.2679	0.933450
$122$	0	0
$123$	8.64925	0.779876
$124$	0	0
$125$	27.4785	2.45775
$126$	0	0
$127$	−6.83178	−0.606223	−0.303111	−	0.952955i	$-0.598025\pi$
$127$	−6.83178	−0.606223	−0.303111	+	0.952955i	$0.598025\pi$
$128$	0	0
$129$	−7.46071	−0.656879
$130$	0	0
$131$	13.1242	1.14667	0.573335	−	0.819321i	$-0.305649\pi$
$131$	13.1242	1.14667	0.573335	+	0.819321i	$0.305649\pi$
$132$	0	0
$133$	12.6194	1.09424
$134$	0	0
$135$	4.08902	0.351927
$136$	0	0
$137$	−16.4552	−1.40587	−0.702933	−	0.711256i	$-0.748127\pi$
$137$	−16.4552	−1.40587	−0.702933	+	0.711256i	$0.748127\pi$
$138$	0	0
$139$	17.7348	1.50425	0.752123	−	0.659023i	$-0.229030\pi$
$139$	17.7348	1.50425	0.752123	+	0.659023i	$0.229030\pi$
$140$	0	0
$141$	−5.52609	−0.465381
$142$	0	0
$143$	−22.1196	−1.84973
$144$	0	0
$145$	−4.08902	−0.339574
$146$	0	0
$147$	−0.366553	−0.0302328
$148$	0	0
$149$	−13.4929	−1.10538	−0.552689	−	0.833388i	$-0.686398\pi$
$149$	−13.4929	−1.10538	−0.552689	+	0.833388i	$0.686398\pi$
$150$	0	0
$151$	−6.07542	−0.494410	−0.247205	−	0.968963i	$-0.579512\pi$
$151$	−6.07542	−0.494410	−0.247205	+	0.968963i	$0.579512\pi$
$152$	0	0
$153$	−1.83258	−0.148155
$154$	0	0
$155$	32.8749	2.64057
$156$	0	0
$157$	−7.80818	−0.623161	−0.311580	−	0.950220i	$-0.600858\pi$
$157$	−7.80818	−0.623161	−0.311580	+	0.950220i	$0.600858\pi$
$158$	0	0
$159$	−7.81069	−0.619428
$160$	0	0
$161$	2.57555	0.202982
$162$	0	0
$163$	−9.01307	−0.705958	−0.352979	−	0.935631i	$-0.614831\pi$
$163$	−9.01307	−0.705958	−0.352979	+	0.935631i	$0.614831\pi$
$164$	0	0
$165$	18.8574	1.46805
$166$	0	0
$167$	16.9939	1.31503	0.657513	−	0.753443i	$-0.271609\pi$
$167$	16.9939	1.31503	0.657513	+	0.753443i	$0.271609\pi$
$168$	0	0
$169$	10.0054	0.769643
$170$	0	0
$171$	4.89968	0.374688
$172$	0	0
$173$	20.5627	1.56335	0.781676	−	0.623685i	$-0.214365\pi$
$173$	20.5627	1.56335	0.781676	+	0.623685i	$0.214365\pi$
$174$	0	0
$175$	30.1856	2.28182
$176$	0	0
$177$	7.57493	0.569367
$178$	0	0
$179$	5.34572	0.399558	0.199779	−	0.979841i	$-0.435978\pi$
$179$	5.34572	0.399558	0.199779	+	0.979841i	$0.435978\pi$
$180$	0	0
$181$	−5.46362	−0.406108	−0.203054	−	0.979168i	$-0.565087\pi$
$181$	−5.46362	−0.406108	−0.203054	+	0.979168i	$0.565087\pi$
$182$	0	0
$183$	−10.4358	−0.771435
$184$	0	0
$185$	−34.4710	−2.53436
$186$	0	0
$187$	−8.45135	−0.618024
$188$	0	0
$189$	2.57555	0.187344
$190$	0	0
$191$	−8.74290	−0.632614	−0.316307	−	0.948657i	$-0.602443\pi$
$191$	−8.74290	−0.632614	−0.316307	+	0.948657i	$0.602443\pi$
$192$	0	0
$193$	−21.2254	−1.52784	−0.763920	−	0.645312i	$-0.776728\pi$
$193$	−21.2254	−1.52784	−0.763920	+	0.645312i	$0.776728\pi$
$194$	0	0
$195$	−19.6125	−1.40448
$196$	0	0
$197$	−19.3805	−1.38080	−0.690400	−	0.723428i	$-0.742565\pi$
$197$	−19.3805	−1.38080	−0.690400	+	0.723428i	$0.742565\pi$
$198$	0	0
$199$	21.6802	1.53687	0.768435	−	0.639928i	$-0.221036\pi$
$199$	21.6802	1.53687	0.768435	+	0.639928i	$0.221036\pi$
$200$	0	0
$201$	8.30407	0.585724
$202$	0	0
$203$	−2.57555	−0.180768
$204$	0	0
$205$	35.3669	2.47013
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	22.5959	1.56299
$210$	0	0
$211$	25.2974	1.74155	0.870774	−	0.491684i	$-0.163619\pi$
$211$	25.2974	1.74155	0.870774	+	0.491684i	$0.163619\pi$
$212$	0	0
$213$	−0.0870476	−0.00596440
$214$	0	0
$215$	−30.5070	−2.08056
$216$	0	0
$217$	20.7069	1.40568
$218$	0	0
$219$	9.17475	0.619972
$220$	0	0
$221$	8.78978	0.591264
$222$	0	0
$223$	−22.4964	−1.50647	−0.753234	−	0.657753i	$-0.771507\pi$
$223$	−22.4964	−1.50647	−0.753234	+	0.657753i	$0.771507\pi$
$224$	0	0
$225$	11.7201	0.781338
$226$	0	0
$227$	0.0204943	0.00136026	0.000680128	−	1.00000i	$-0.499784\pi$
$227$	0.0204943	0.00136026	0.000680128		1.00000i	$0.499784\pi$
$228$	0	0
$229$	−14.9723	−0.989399	−0.494700	−	0.869064i	$-0.664722\pi$
$229$	−14.9723	−0.989399	−0.494700	+	0.869064i	$0.664722\pi$
$230$	0	0
$231$	11.8777	0.781495
$232$	0	0
$233$	−7.81673	−0.512091	−0.256045	−	0.966665i	$-0.582420\pi$
$233$	−7.81673	−0.512091	−0.256045	+	0.966665i	$0.582420\pi$
$234$	0	0
$235$	−22.5963	−1.47402
$236$	0	0
$237$	−8.38478	−0.544650
$238$	0	0
$239$	−8.24516	−0.533335	−0.266668	−	0.963789i	$-0.585923\pi$
$239$	−8.24516	−0.533335	−0.266668	+	0.963789i	$0.585923\pi$
$240$	0	0
$241$	2.39246	0.154112	0.0770560	−	0.997027i	$-0.475448\pi$
$241$	2.39246	0.154112	0.0770560	+	0.997027i	$0.475448\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.49884	−0.0957575
$246$	0	0
$247$	−23.5008	−1.49532
$248$	0	0
$249$	−9.19932	−0.582983
$250$	0	0
$251$	5.14853	0.324972	0.162486	−	0.986711i	$-0.448049\pi$
$251$	5.14853	0.324972	0.162486	+	0.986711i	$0.448049\pi$
$252$	0	0
$253$	4.61172	0.289936
$254$	0	0
$255$	−7.49346	−0.469258
$256$	0	0
$257$	19.8047	1.23538	0.617690	−	0.786421i	$-0.288068\pi$
$257$	19.8047	1.23538	0.617690	+	0.786421i	$0.288068\pi$
$258$	0	0
$259$	−21.7122	−1.34913
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−29.0507	−1.79135	−0.895673	−	0.444714i	$-0.853305\pi$
$263$	−29.0507	−1.79135	−0.895673	+	0.444714i	$0.853305\pi$
$264$	0	0
$265$	−31.9381	−1.96194
$266$	0	0
$267$	−11.9049	−0.728566
$268$	0	0
$269$	−8.83872	−0.538906	−0.269453	−	0.963013i	$-0.586843\pi$
$269$	−8.83872	−0.538906	−0.269453	+	0.963013i	$0.586843\pi$
$270$	0	0
$271$	−10.4526	−0.634951	−0.317476	−	0.948266i	$-0.602835\pi$
$271$	−10.4526	−0.634951	−0.317476	+	0.948266i	$0.602835\pi$
$272$	0	0
$273$	−12.3533	−0.747658
$274$	0	0
$275$	54.0496	3.25932
$276$	0	0
$277$	−11.7721	−0.707317	−0.353658	−	0.935375i	$-0.615062\pi$
$277$	−11.7721	−0.707317	−0.353658	+	0.935375i	$0.615062\pi$
$278$	0	0
$279$	8.03980	0.481330
$280$	0	0
$281$	−30.2251	−1.80308	−0.901539	−	0.432698i	$-0.857562\pi$
$281$	−30.2251	−1.80308	−0.901539	+	0.432698i	$0.857562\pi$
$282$	0	0
$283$	−18.5425	−1.10224	−0.551118	−	0.834427i	$-0.685799\pi$
$283$	−18.5425	−1.10224	−0.551118	+	0.834427i	$0.685799\pi$
$284$	0	0
$285$	20.0349	1.18676
$286$	0	0
$287$	22.2765	1.31494
$288$	0	0
$289$	−13.6416	−0.802450
$290$	0	0
$291$	6.00162	0.351821
$292$	0	0
$293$	12.1800	0.711563	0.355781	−	0.934569i	$-0.384215\pi$
$293$	12.1800	0.711563	0.355781	+	0.934569i	$0.384215\pi$
$294$	0	0
$295$	30.9740	1.80338
$296$	0	0
$297$	4.61172	0.267599
$298$	0	0
$299$	−4.79639	−0.277382
$300$	0	0
$301$	−19.2154	−1.10756
$302$	0	0
$303$	14.2557	0.818966
$304$	0	0
$305$	−42.6721	−2.44340
$306$	0	0
$307$	23.3621	1.33335	0.666673	−	0.745351i	$-0.267718\pi$
$307$	23.3621	1.33335	0.666673	+	0.745351i	$0.267718\pi$
$308$	0	0
$309$	−15.9300	−0.906229
$310$	0	0
$311$	−4.26341	−0.241756	−0.120878	−	0.992667i	$-0.538571\pi$
$311$	−4.26341	−0.241756	−0.120878	+	0.992667i	$0.538571\pi$
$312$	0	0
$313$	8.27089	0.467498	0.233749	−	0.972297i	$-0.424901\pi$
$313$	8.27089	0.467498	0.233749	+	0.972297i	$0.424901\pi$
$314$	0	0
$315$	10.5315	0.593381
$316$	0	0
$317$	−33.9633	−1.90757	−0.953783	−	0.300495i	$-0.902848\pi$
$317$	−33.9633	−1.90757	−0.953783	+	0.300495i	$0.902848\pi$
$318$	0	0
$319$	−4.61172	−0.258207
$320$	0	0
$321$	13.6477	0.761739
$322$	0	0
$323$	−8.97906	−0.499608
$324$	0	0
$325$	−56.2140	−3.11819
$326$	0	0
$327$	−10.7879	−0.596572
$328$	0	0
$329$	−14.2327	−0.784675
$330$	0	0
$331$	−21.1396	−1.16194	−0.580969	−	0.813926i	$-0.697326\pi$
$331$	−21.1396	−1.16194	−0.580969	+	0.813926i	$0.697326\pi$
$332$	0	0
$333$	−8.43013	−0.461968
$334$	0	0
$335$	33.9555	1.85519
$336$	0	0
$337$	15.1873	0.827303	0.413651	−	0.910435i	$-0.364253\pi$
$337$	15.1873	0.827303	0.413651	+	0.910435i	$0.364253\pi$
$338$	0	0
$339$	14.8279	0.805344
$340$	0	0
$341$	37.0773	2.00785
$342$	0	0
$343$	−18.9729	−1.02444
$344$	0	0
$345$	4.08902	0.220145
$346$	0	0
$347$	−19.3836	−1.04056	−0.520282	−	0.853994i	$-0.674173\pi$
$347$	−19.3836	−1.04056	−0.520282	+	0.853994i	$0.674173\pi$
$348$	0	0
$349$	34.6528	1.85492	0.927461	−	0.373920i	$-0.121987\pi$
$349$	34.6528	1.85492	0.927461	+	0.373920i	$0.121987\pi$
$350$	0	0
$351$	−4.79639	−0.256012
$352$	0	0
$353$	7.36647	0.392078	0.196039	−	0.980596i	$-0.437192\pi$
$353$	7.36647	0.392078	0.196039	+	0.980596i	$0.437192\pi$
$354$	0	0
$355$	−0.355939	−0.0188913
$356$	0	0
$357$	−4.71990	−0.249804
$358$	0	0
$359$	12.8569	0.678559	0.339280	−	0.940686i	$-0.389817\pi$
$359$	12.8569	0.678559	0.339280	+	0.940686i	$0.389817\pi$
$360$	0	0
$361$	5.00684	0.263518
$362$	0	0
$363$	10.2679	0.538927
$364$	0	0
$365$	37.5157	1.96366
$366$	0	0
$367$	−8.53494	−0.445520	−0.222760	−	0.974873i	$-0.571507\pi$
$367$	−8.53494	−0.445520	−0.222760	+	0.974873i	$0.571507\pi$
$368$	0	0
$369$	8.64925	0.450262
$370$	0	0
$371$	−20.1168	−1.04441
$372$	0	0
$373$	−3.58762	−0.185760	−0.0928799	−	0.995677i	$-0.529607\pi$
$373$	−3.58762	−0.185760	−0.0928799	+	0.995677i	$0.529607\pi$
$374$	0	0
$375$	27.4785	1.41898
$376$	0	0
$377$	4.79639	0.247027
$378$	0	0
$379$	1.08870	0.0559229	0.0279614	−	0.999609i	$-0.491098\pi$
$379$	1.08870	0.0559229	0.0279614	+	0.999609i	$0.491098\pi$
$380$	0	0
$381$	−6.83178	−0.350003
$382$	0	0
$383$	−2.77293	−0.141690	−0.0708450	−	0.997487i	$-0.522570\pi$
$383$	−2.77293	−0.141690	−0.0708450	+	0.997487i	$0.522570\pi$
$384$	0	0
$385$	48.5681	2.47526
$386$	0	0
$387$	−7.46071	−0.379249
$388$	0	0
$389$	13.3371	0.676216	0.338108	−	0.941107i	$-0.390213\pi$
$389$	13.3371	0.676216	0.338108	+	0.941107i	$0.390213\pi$
$390$	0	0
$391$	−1.83258	−0.0926776
$392$	0	0
$393$	13.1242	0.662030
$394$	0	0
$395$	−34.2855	−1.72509
$396$	0	0
$397$	31.3261	1.57221	0.786107	−	0.618090i	$-0.212093\pi$
$397$	31.3261	1.57221	0.786107	+	0.618090i	$0.212093\pi$
$398$	0	0
$399$	12.6194	0.631758
$400$	0	0
$401$	35.8572	1.79063	0.895313	−	0.445438i	$-0.146952\pi$
$401$	35.8572	1.79063	0.895313	+	0.445438i	$0.146952\pi$
$402$	0	0
$403$	−38.5620	−1.92091
$404$	0	0
$405$	4.08902	0.203185
$406$	0	0
$407$	−38.8774	−1.92708
$408$	0	0
$409$	33.1034	1.63686	0.818430	−	0.574607i	$-0.194845\pi$
$409$	33.1034	1.63686	0.818430	+	0.574607i	$0.194845\pi$
$410$	0	0
$411$	−16.4552	−0.811677
$412$	0	0
$413$	19.5096	0.960004
$414$	0	0
$415$	−37.6162	−1.84651
$416$	0	0
$417$	17.7348	0.868477
$418$	0	0
$419$	−27.2056	−1.32908	−0.664540	−	0.747253i	$-0.731373\pi$
$419$	−27.2056	−1.32908	−0.664540	+	0.747253i	$0.731373\pi$
$420$	0	0
$421$	6.97246	0.339817	0.169908	−	0.985460i	$-0.445653\pi$
$421$	6.97246	0.339817	0.169908	+	0.985460i	$0.445653\pi$
$422$	0	0
$423$	−5.52609	−0.268688
$424$	0	0
$425$	−21.4780	−1.04183
$426$	0	0
$427$	−26.8778	−1.30071
$428$	0	0
$429$	−22.1196	−1.06794
$430$	0	0
$431$	−14.5307	−0.699921	−0.349960	−	0.936764i	$-0.613805\pi$
$431$	−14.5307	−0.699921	−0.349960	+	0.936764i	$0.613805\pi$
$432$	0	0
$433$	36.5362	1.75582	0.877909	−	0.478828i	$-0.158938\pi$
$433$	36.5362	1.75582	0.877909	+	0.478828i	$0.158938\pi$
$434$	0	0
$435$	−4.08902	−0.196053
$436$	0	0
$437$	4.89968	0.234383
$438$	0	0
$439$	−12.4792	−0.595600	−0.297800	−	0.954628i	$-0.596253\pi$
$439$	−12.4792	−0.595600	−0.297800	+	0.954628i	$0.596253\pi$
$440$	0	0
$441$	−0.366553	−0.0174549
$442$	0	0
$443$	28.1712	1.33845	0.669226	−	0.743059i	$-0.266626\pi$
$443$	28.1712	1.33845	0.669226	+	0.743059i	$0.266626\pi$
$444$	0	0
$445$	−48.6792	−2.30762
$446$	0	0
$447$	−13.4929	−0.638190
$448$	0	0
$449$	20.9088	0.986749	0.493374	−	0.869817i	$-0.335763\pi$
$449$	20.9088	0.986749	0.493374	+	0.869817i	$0.335763\pi$
$450$	0	0
$451$	39.8879	1.87825
$452$	0	0
$453$	−6.07542	−0.285448
$454$	0	0
$455$	−50.5130	−2.36809
$456$	0	0
$457$	−22.1453	−1.03591	−0.517956	−	0.855407i	$-0.673307\pi$
$457$	−22.1453	−1.03591	−0.517956	+	0.855407i	$0.673307\pi$
$458$	0	0
$459$	−1.83258	−0.0855376
$460$	0	0
$461$	24.7242	1.15152	0.575760	−	0.817619i	$-0.304706\pi$
$461$	24.7242	1.15152	0.575760	+	0.817619i	$0.304706\pi$
$462$	0	0
$463$	1.91742	0.0891101	0.0445550	−	0.999007i	$-0.485813\pi$
$463$	1.91742	0.0891101	0.0445550	+	0.999007i	$0.485813\pi$
$464$	0	0
$465$	32.8749	1.52454
$466$	0	0
$467$	−5.84323	−0.270393	−0.135196	−	0.990819i	$-0.543167\pi$
$467$	−5.84323	−0.270393	−0.135196	+	0.990819i	$0.543167\pi$
$468$	0	0
$469$	21.3875	0.987584
$470$	0	0
$471$	−7.80818	−0.359782
$472$	0	0
$473$	−34.4067	−1.58202
$474$	0	0
$475$	57.4245	2.63482
$476$	0	0
$477$	−7.81069	−0.357627
$478$	0	0
$479$	18.8889	0.863057	0.431529	−	0.902099i	$-0.357974\pi$
$479$	18.8889	0.863057	0.431529	+	0.902099i	$0.357974\pi$
$480$	0	0
$481$	40.4342	1.84364
$482$	0	0
$483$	2.57555	0.117191
$484$	0	0
$485$	24.5407	1.11434
$486$	0	0
$487$	17.8317	0.808030	0.404015	−	0.914752i	$-0.367614\pi$
$487$	17.8317	0.808030	0.404015	+	0.914752i	$0.367614\pi$
$488$	0	0
$489$	−9.01307	−0.407585
$490$	0	0
$491$	20.8005	0.938712	0.469356	−	0.883009i	$-0.344486\pi$
$491$	20.8005	0.938712	0.469356	+	0.883009i	$0.344486\pi$
$492$	0	0
$493$	1.83258	0.0825353
$494$	0	0
$495$	18.8574	0.847577
$496$	0	0
$497$	−0.224195	−0.0100565
$498$	0	0
$499$	−14.3965	−0.644478	−0.322239	−	0.946658i	$-0.604435\pi$
$499$	−14.3965	−0.644478	−0.322239	+	0.946658i	$0.604435\pi$
$500$	0	0
$501$	16.9939	0.759231
$502$	0	0
$503$	−32.9726	−1.47018	−0.735089	−	0.677971i	$-0.762859\pi$
$503$	−32.9726	−1.47018	−0.735089	+	0.677971i	$0.762859\pi$
$504$	0	0
$505$	58.2917	2.59394
$506$	0	0
$507$	10.0054	0.444354
$508$	0	0
$509$	−14.8224	−0.656991	−0.328496	−	0.944506i	$-0.606542\pi$
$509$	−14.8224	−0.656991	−0.328496	+	0.944506i	$0.606542\pi$
$510$	0	0
$511$	23.6300	1.04533
$512$	0	0
$513$	4.89968	0.216326
$514$	0	0
$515$	−65.1382	−2.87033
$516$	0	0
$517$	−25.4848	−1.12082
$518$	0	0
$519$	20.5627	0.902601
$520$	0	0
$521$	36.0062	1.57746	0.788730	−	0.614740i	$-0.210739\pi$
$521$	36.0062	1.57746	0.788730	+	0.614740i	$0.210739\pi$
$522$	0	0
$523$	−2.29678	−0.100431	−0.0502157	−	0.998738i	$-0.515991\pi$
$523$	−2.29678	−0.100431	−0.0502157	+	0.998738i	$0.515991\pi$
$524$	0	0
$525$	30.1856	1.31741
$526$	0	0
$527$	−14.7336	−0.641805
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	7.57493	0.328724
$532$	0	0
$533$	−41.4852	−1.79692
$534$	0	0
$535$	55.8056	2.41269
$536$	0	0
$537$	5.34572	0.230685
$538$	0	0
$539$	−1.69044	−0.0728124
$540$	0	0
$541$	37.7612	1.62348	0.811741	−	0.584018i	$-0.198520\pi$
$541$	37.7612	1.62348	0.811741	+	0.584018i	$0.198520\pi$
$542$	0	0
$543$	−5.46362	−0.234466
$544$	0	0
$545$	−44.1119	−1.88955
$546$	0	0
$547$	−0.134339	−0.00574390	−0.00287195	−	0.999996i	$-0.500914\pi$
$547$	−0.134339	−0.00574390	−0.00287195	+	0.999996i	$0.500914\pi$
$548$	0	0
$549$	−10.4358	−0.445388
$550$	0	0
$551$	−4.89968	−0.208733
$552$	0	0
$553$	−21.5954	−0.918329
$554$	0	0
$555$	−34.4710	−1.46321
$556$	0	0
$557$	−18.8767	−0.799833	−0.399917	−	0.916552i	$-0.630961\pi$
$557$	−18.8767	−0.799833	−0.399917	+	0.916552i	$0.630961\pi$
$558$	0	0
$559$	35.7845	1.51352
$560$	0	0
$561$	−8.45135	−0.356816
$562$	0	0
$563$	−43.1815	−1.81988	−0.909941	−	0.414737i	$-0.863874\pi$
$563$	−43.1815	−1.81988	−0.909941	+	0.414737i	$0.863874\pi$
$564$	0	0
$565$	60.6317	2.55080
$566$	0	0
$567$	2.57555	0.108163
$568$	0	0
$569$	−34.4190	−1.44292	−0.721459	−	0.692457i	$-0.756528\pi$
$569$	−34.4190	−1.44292	−0.721459	+	0.692457i	$0.756528\pi$
$570$	0	0
$571$	−7.69661	−0.322093	−0.161047	−	0.986947i	$-0.551487\pi$
$571$	−7.69661	−0.322093	−0.161047	+	0.986947i	$0.551487\pi$
$572$	0	0
$573$	−8.74290	−0.365240
$574$	0	0
$575$	11.7201	0.488760
$576$	0	0
$577$	18.7060	0.778740	0.389370	−	0.921082i	$-0.372693\pi$
$577$	18.7060	0.778740	0.389370	+	0.921082i	$0.372693\pi$
$578$	0	0
$579$	−21.2254	−0.882098
$580$	0	0
$581$	−23.6933	−0.982963
$582$	0	0
$583$	−36.0207	−1.49183
$584$	0	0
$585$	−19.6125	−0.810878
$586$	0	0
$587$	10.1984	0.420934	0.210467	−	0.977601i	$-0.432502\pi$
$587$	10.1984	0.420934	0.210467	+	0.977601i	$0.432502\pi$
$588$	0	0
$589$	39.3924	1.62314
$590$	0	0
$591$	−19.3805	−0.797205
$592$	0	0
$593$	−27.5308	−1.13055	−0.565277	−	0.824901i	$-0.691231\pi$
$593$	−27.5308	−1.13055	−0.565277	+	0.824901i	$0.691231\pi$
$594$	0	0
$595$	−19.2998	−0.791213
$596$	0	0
$597$	21.6802	0.887312
$598$	0	0
$599$	−6.81479	−0.278445	−0.139222	−	0.990261i	$-0.544460\pi$
$599$	−6.81479	−0.278445	−0.139222	+	0.990261i	$0.544460\pi$
$600$	0	0
$601$	8.18356	0.333814	0.166907	−	0.985973i	$-0.446622\pi$
$601$	8.18356	0.333814	0.166907	+	0.985973i	$0.446622\pi$
$602$	0	0
$603$	8.30407	0.338168
$604$	0	0
$605$	41.9858	1.70697
$606$	0	0
$607$	−13.2929	−0.539543	−0.269771	−	0.962924i	$-0.586948\pi$
$607$	−13.2929	−0.539543	−0.269771	+	0.962924i	$0.586948\pi$
$608$	0	0
$609$	−2.57555	−0.104366
$610$	0	0
$611$	26.5053	1.07229
$612$	0	0
$613$	−2.91465	−0.117722	−0.0588608	−	0.998266i	$-0.518747\pi$
$613$	−2.91465	−0.117722	−0.0588608	+	0.998266i	$0.518747\pi$
$614$	0	0
$615$	35.3669	1.42613
$616$	0	0
$617$	−48.6323	−1.95786	−0.978931	−	0.204190i	$-0.934544\pi$
$617$	−48.6323	−1.95786	−0.978931	+	0.204190i	$0.934544\pi$
$618$	0	0
$619$	−6.13839	−0.246723	−0.123361	−	0.992362i	$-0.539367\pi$
$619$	−6.13839	−0.246723	−0.123361	+	0.992362i	$0.539367\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−30.6616	−1.22843
$624$	0	0
$625$	53.7596	2.15038
$626$	0	0
$627$	22.5959	0.902395
$628$	0	0
$629$	15.4489	0.615988
$630$	0	0
$631$	−3.23694	−0.128860	−0.0644302	−	0.997922i	$-0.520523\pi$
$631$	−3.23694	−0.128860	−0.0644302	+	0.997922i	$0.520523\pi$
$632$	0	0
$633$	25.2974	1.00548
$634$	0	0
$635$	−27.9353	−1.10858
$636$	0	0
$637$	1.75813	0.0696597
$638$	0	0
$639$	−0.0870476	−0.00344355
$640$	0	0
$641$	32.5722	1.28652	0.643262	−	0.765646i	$-0.277581\pi$
$641$	32.5722	1.28652	0.643262	+	0.765646i	$0.277581\pi$
$642$	0	0
$643$	−3.35815	−0.132433	−0.0662163	−	0.997805i	$-0.521093\pi$
$643$	−3.35815	−0.132433	−0.0662163	+	0.997805i	$0.521093\pi$
$644$	0	0
$645$	−30.5070	−1.20121
$646$	0	0
$647$	25.7591	1.01270	0.506348	−	0.862329i	$-0.330995\pi$
$647$	25.7591	1.01270	0.506348	+	0.862329i	$0.330995\pi$
$648$	0	0
$649$	34.9334	1.37126
$650$	0	0
$651$	20.7069	0.811567
$652$	0	0
$653$	−13.9565	−0.546162	−0.273081	−	0.961991i	$-0.588043\pi$
$653$	−13.9565	−0.546162	−0.273081	+	0.961991i	$0.588043\pi$
$654$	0	0
$655$	53.6652	2.09687
$656$	0	0
$657$	9.17475	0.357941
$658$	0	0
$659$	−40.1400	−1.56363	−0.781815	−	0.623510i	$-0.785706\pi$
$659$	−40.1400	−1.56363	−0.781815	+	0.623510i	$0.785706\pi$
$660$	0	0
$661$	19.1524	0.744944	0.372472	−	0.928043i	$-0.378510\pi$
$661$	19.1524	0.744944	0.372472	+	0.928043i	$0.378510\pi$
$662$	0	0
$663$	8.78978	0.341367
$664$	0	0
$665$	51.6008	2.00099
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−22.4964	−0.869760
$670$	0	0
$671$	−48.1269	−1.85792
$672$	0	0
$673$	21.5637	0.831221	0.415610	−	0.909543i	$-0.363568\pi$
$673$	21.5637	0.831221	0.415610	+	0.909543i	$0.363568\pi$
$674$	0	0
$675$	11.7201	0.451105
$676$	0	0
$677$	−4.25248	−0.163436	−0.0817180	−	0.996655i	$-0.526041\pi$
$677$	−4.25248	−0.163436	−0.0817180	+	0.996655i	$0.526041\pi$
$678$	0	0
$679$	15.4575	0.593203
$680$	0	0
$681$	0.0204943	0.000785344 0
$682$	0	0
$683$	21.3257	0.816006	0.408003	−	0.912981i	$-0.366225\pi$
$683$	21.3257	0.816006	0.408003	+	0.912981i	$0.366225\pi$
$684$	0	0
$685$	−67.2858	−2.57086
$686$	0	0
$687$	−14.9723	−0.571230
$688$	0	0
$689$	37.4631	1.42723
$690$	0	0
$691$	−5.87470	−0.223484	−0.111742	−	0.993737i	$-0.535643\pi$
$691$	−5.87470	−0.223484	−0.111742	+	0.993737i	$0.535643\pi$
$692$	0	0
$693$	11.8777	0.451197
$694$	0	0
$695$	72.5179	2.75076
$696$	0	0
$697$	−15.8504	−0.600378
$698$	0	0
$699$	−7.81673	−0.295656
$700$	0	0
$701$	−18.4384	−0.696410	−0.348205	−	0.937419i	$-0.613209\pi$
$701$	−18.4384	−0.696410	−0.348205	+	0.937419i	$0.613209\pi$
$702$	0	0
$703$	−41.3049	−1.55785
$704$	0	0
$705$	−22.5963	−0.851025
$706$	0	0
$707$	36.7161	1.38085
$708$	0	0
$709$	44.5791	1.67421	0.837103	−	0.547046i	$-0.184248\pi$
$709$	44.5791	1.67421	0.837103	+	0.547046i	$0.184248\pi$
$710$	0	0
$711$	−8.38478	−0.314454
$712$	0	0
$713$	8.03980	0.301093
$714$	0	0
$715$	−90.4474	−3.38254
$716$	0	0
$717$	−8.24516	−0.307921
$718$	0	0
$719$	20.6171	0.768888	0.384444	−	0.923148i	$-0.374393\pi$
$719$	20.6171	0.768888	0.384444	+	0.923148i	$0.374393\pi$
$720$	0	0
$721$	−41.0286	−1.52798
$722$	0	0
$723$	2.39246	0.0889766
$724$	0	0
$725$	−11.7201	−0.435272
$726$	0	0
$727$	17.2029	0.638021	0.319010	−	0.947751i	$-0.396650\pi$
$727$	17.2029	0.638021	0.319010	+	0.947751i	$0.396650\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.6724	0.505691
$732$	0	0
$733$	−27.3497	−1.01018	−0.505092	−	0.863065i	$-0.668541\pi$
$733$	−27.3497	−1.01018	−0.505092	+	0.863065i	$0.668541\pi$
$734$	0	0
$735$	−1.49884	−0.0552856
$736$	0	0
$737$	38.2960	1.41065
$738$	0	0
$739$	5.64790	0.207761	0.103881	−	0.994590i	$-0.466874\pi$
$739$	5.64790	0.207761	0.103881	+	0.994590i	$0.466874\pi$
$740$	0	0
$741$	−23.5008	−0.863322
$742$	0	0
$743$	33.4297	1.22642	0.613208	−	0.789922i	$-0.289879\pi$
$743$	33.4297	1.22642	0.613208	+	0.789922i	$0.289879\pi$
$744$	0	0
$745$	−55.1725	−2.02136
$746$	0	0
$747$	−9.19932	−0.336586
$748$	0	0
$749$	35.1503	1.28436
$750$	0	0
$751$	−5.76461	−0.210354	−0.105177	−	0.994454i	$-0.533541\pi$
$751$	−5.76461	−0.210354	−0.105177	+	0.994454i	$0.533541\pi$
$752$	0	0
$753$	5.14853	0.187623
$754$	0	0
$755$	−24.8425	−0.904111
$756$	0	0
$757$	35.7253	1.29846	0.649229	−	0.760593i	$-0.275092\pi$
$757$	35.7253	1.29846	0.649229	+	0.760593i	$0.275092\pi$
$758$	0	0
$759$	4.61172	0.167395
$760$	0	0
$761$	−43.5930	−1.58025	−0.790123	−	0.612948i	$-0.789983\pi$
$761$	−43.5930	−1.58025	−0.790123	+	0.612948i	$0.789983\pi$
$762$	0	0
$763$	−27.7847	−1.00588
$764$	0	0
$765$	−7.49346	−0.270927
$766$	0	0
$767$	−36.3323	−1.31188
$768$	0	0
$769$	2.90207	0.104651	0.0523257	−	0.998630i	$-0.483337\pi$
$769$	2.90207	0.104651	0.0523257	+	0.998630i	$0.483337\pi$
$770$	0	0
$771$	19.8047	0.713248
$772$	0	0
$773$	−18.6284	−0.670016	−0.335008	−	0.942215i	$-0.608739\pi$
$773$	−18.6284	−0.670016	−0.335008	+	0.942215i	$0.608739\pi$
$774$	0	0
$775$	94.2270	3.38473
$776$	0	0
$777$	−21.7122	−0.778921
$778$	0	0
$779$	42.3785	1.51837
$780$	0	0
$781$	−0.401439	−0.0143646
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−31.9278	−1.13955
$786$	0	0
$787$	−17.0913	−0.609239	−0.304619	−	0.952474i	$-0.598529\pi$
$787$	−17.0913	−0.609239	−0.304619	+	0.952474i	$0.598529\pi$
$788$	0	0
$789$	−29.0507	−1.03423
$790$	0	0
$791$	38.1901	1.35788
$792$	0	0
$793$	50.0541	1.77747
$794$	0	0
$795$	−31.9381	−1.13273
$796$	0	0
$797$	14.4626	0.512291	0.256146	−	0.966638i	$-0.417547\pi$
$797$	14.4626	0.512291	0.256146	+	0.966638i	$0.417547\pi$
$798$	0	0
$799$	10.1270	0.358268
$800$	0	0
$801$	−11.9049	−0.420638
$802$	0	0
$803$	42.3114	1.49314
$804$	0	0
$805$	10.5315	0.371185
$806$	0	0
$807$	−8.83872	−0.311138
$808$	0	0
$809$	11.8020	0.414937	0.207468	−	0.978242i	$-0.433478\pi$
$809$	11.8020	0.414937	0.207468	+	0.978242i	$0.433478\pi$
$810$	0	0
$811$	−14.9607	−0.525342	−0.262671	−	0.964885i	$-0.584603\pi$
$811$	−14.9607	−0.525342	−0.262671	+	0.964885i	$0.584603\pi$
$812$	0	0
$813$	−10.4526	−0.366589
$814$	0	0
$815$	−36.8546	−1.29096
$816$	0	0
$817$	−36.5551	−1.27890
$818$	0	0
$819$	−12.3533	−0.431660
$820$	0	0
$821$	−1.87938	−0.0655908	−0.0327954	−	0.999462i	$-0.510441\pi$
$821$	−1.87938	−0.0655908	−0.0327954	+	0.999462i	$0.510441\pi$
$822$	0	0
$823$	37.3394	1.30157	0.650785	−	0.759262i	$-0.274440\pi$
$823$	37.3394	1.30157	0.650785	+	0.759262i	$0.274440\pi$
$824$	0	0
$825$	54.0496	1.88177
$826$	0	0
$827$	−5.86439	−0.203925	−0.101962	−	0.994788i	$-0.532512\pi$
$827$	−5.86439	−0.203925	−0.101962	+	0.994788i	$0.532512\pi$
$828$	0	0
$829$	51.8217	1.79984	0.899920	−	0.436054i	$-0.143624\pi$
$829$	51.8217	1.79984	0.899920	+	0.436054i	$0.143624\pi$
$830$	0	0
$831$	−11.7721	−0.408370
$832$	0	0
$833$	0.671738	0.0232744
$834$	0	0
$835$	69.4883	2.40474
$836$	0	0
$837$	8.03980	0.277896
$838$	0	0
$839$	−43.0640	−1.48674	−0.743368	−	0.668883i	$-0.766773\pi$
$839$	−43.0640	−1.48674	−0.743368	+	0.668883i	$0.766773\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−30.2251	−1.04101
$844$	0	0
$845$	40.9121	1.40742
$846$	0	0
$847$	26.4456	0.908681
$848$	0	0
$849$	−18.5425	−0.636376
$850$	0	0
$851$	−8.43013	−0.288981
$852$	0	0
$853$	−5.44120	−0.186303	−0.0931515	−	0.995652i	$-0.529694\pi$
$853$	−5.44120	−0.186303	−0.0931515	+	0.995652i	$0.529694\pi$
$854$	0	0
$855$	20.0349	0.685178
$856$	0	0
$857$	2.25114	0.0768974	0.0384487	−	0.999261i	$-0.487758\pi$
$857$	2.25114	0.0768974	0.0384487	+	0.999261i	$0.487758\pi$
$858$	0	0
$859$	−22.2512	−0.759203	−0.379601	−	0.925150i	$-0.623939\pi$
$859$	−22.2512	−0.759203	−0.379601	+	0.925150i	$0.623939\pi$
$860$	0	0
$861$	22.2765	0.759183
$862$	0	0
$863$	14.2820	0.486164	0.243082	−	0.970006i	$-0.421842\pi$
$863$	14.2820	0.486164	0.243082	+	0.970006i	$0.421842\pi$
$864$	0	0
$865$	84.0812	2.85884
$866$	0	0
$867$	−13.6416	−0.463295
$868$	0	0
$869$	−38.6682	−1.31173
$870$	0	0
$871$	−39.8295	−1.34957
$872$	0	0
$873$	6.00162	0.203124
$874$	0	0
$875$	70.7721	2.39253
$876$	0	0
$877$	−1.50627	−0.0508631	−0.0254315	−	0.999677i	$-0.508096\pi$
$877$	−1.50627	−0.0508631	−0.0254315	+	0.999677i	$0.508096\pi$
$878$	0	0
$879$	12.1800	0.410821
$880$	0	0
$881$	−1.41483	−0.0476669	−0.0238335	−	0.999716i	$-0.507587\pi$
$881$	−1.41483	−0.0476669	−0.0238335	+	0.999716i	$0.507587\pi$
$882$	0	0
$883$	31.0284	1.04419	0.522095	−	0.852888i	$-0.325151\pi$
$883$	31.0284	1.04419	0.522095	+	0.852888i	$0.325151\pi$
$884$	0	0
$885$	30.9740	1.04118
$886$	0	0
$887$	53.7505	1.80477	0.902383	−	0.430935i	$-0.141816\pi$
$887$	53.7505	1.80477	0.902383	+	0.430935i	$0.141816\pi$
$888$	0	0
$889$	−17.5956	−0.590137
$890$	0	0
$891$	4.61172	0.154498
$892$	0	0
$893$	−27.0761	−0.906066
$894$	0	0
$895$	21.8587	0.730657
$896$	0	0
$897$	−4.79639	−0.160147
$898$	0	0
$899$	−8.03980	−0.268142
$900$	0	0
$901$	14.3137	0.476859
$902$	0	0
$903$	−19.2154	−0.639449
$904$	0	0
$905$	−22.3408	−0.742635
$906$	0	0
$907$	−43.5740	−1.44685	−0.723426	−	0.690402i	$-0.757434\pi$
$907$	−43.5740	−1.44685	−0.723426	+	0.690402i	$0.757434\pi$
$908$	0	0
$909$	14.2557	0.472830
$910$	0	0
$911$	−26.4435	−0.876114	−0.438057	−	0.898947i	$-0.644333\pi$
$911$	−26.4435	−0.876114	−0.438057	+	0.898947i	$0.644333\pi$
$912$	0	0
$913$	−42.4247	−1.40405
$914$	0	0
$915$	−42.6721	−1.41070
$916$	0	0
$917$	33.8021	1.11624
$918$	0	0
$919$	−54.2766	−1.79042	−0.895209	−	0.445646i	$-0.852974\pi$
$919$	−54.2766	−1.79042	−0.895209	+	0.445646i	$0.852974\pi$
$920$	0	0
$921$	23.3621	0.769807
$922$	0	0
$923$	0.417514	0.0137426
$924$	0	0
$925$	−98.8017	−3.24858
$926$	0	0
$927$	−15.9300	−0.523211
$928$	0	0
$929$	−32.9093	−1.07972	−0.539859	−	0.841755i	$-0.681522\pi$
$929$	−32.9093	−1.07972	−0.539859	+	0.841755i	$0.681522\pi$
$930$	0	0
$931$	−1.79599	−0.0588613
$932$	0	0
$933$	−4.26341	−0.139578
$934$	0	0
$935$	−34.5577	−1.13016
$936$	0	0
$937$	−52.6248	−1.71918	−0.859589	−	0.510986i	$-0.829281\pi$
$937$	−52.6248	−1.71918	−0.859589	+	0.510986i	$0.829281\pi$
$938$	0	0
$939$	8.27089	0.269910
$940$	0	0
$941$	31.9101	1.04024	0.520121	−	0.854093i	$-0.325887\pi$
$941$	31.9101	1.04024	0.520121	+	0.854093i	$0.325887\pi$
$942$	0	0
$943$	8.64925	0.281658
$944$	0	0
$945$	10.5315	0.342588
$946$	0	0
$947$	37.9610	1.23357	0.616783	−	0.787134i	$-0.288436\pi$
$947$	37.9610	1.23357	0.616783	+	0.787134i	$0.288436\pi$
$948$	0	0
$949$	−44.0057	−1.42848
$950$	0	0
$951$	−33.9633	−1.10133
$952$	0	0
$953$	2.24358	0.0726765	0.0363383	−	0.999340i	$-0.488431\pi$
$953$	2.24358	0.0726765	0.0363383	+	0.999340i	$0.488431\pi$
$954$	0	0
$955$	−35.7499	−1.15684
$956$	0	0
$957$	−4.61172	−0.149076
$958$	0	0
$959$	−42.3813	−1.36856
$960$	0	0
$961$	33.6384	1.08511
$962$	0	0
$963$	13.6477	0.439790
$964$	0	0
$965$	−86.7911	−2.79390
$966$	0	0
$967$	−1.55361	−0.0499608	−0.0249804	−	0.999688i	$-0.507952\pi$
$967$	−1.55361	−0.0499608	−0.0249804	+	0.999688i	$0.507952\pi$
$968$	0	0
$969$	−8.97906	−0.288449
$970$	0	0
$971$	−42.2327	−1.35531	−0.677655	−	0.735380i	$-0.737004\pi$
$971$	−42.2327	−1.35531	−0.677655	+	0.735380i	$0.737004\pi$
$972$	0	0
$973$	45.6768	1.46433
$974$	0	0
$975$	−56.2140	−1.80029
$976$	0	0
$977$	−13.1262	−0.419946	−0.209973	−	0.977707i	$-0.567338\pi$
$977$	−13.1262	−0.419946	−0.209973	+	0.977707i	$0.567338\pi$
$978$	0	0
$979$	−54.9019	−1.75467
$980$	0	0
$981$	−10.7879	−0.344431
$982$	0	0
$983$	44.3672	1.41510	0.707548	−	0.706666i	$-0.249801\pi$
$983$	44.3672	1.41510	0.707548	+	0.706666i	$0.249801\pi$
$984$	0	0
$985$	−79.2470	−2.52502
$986$	0	0
$987$	−14.2327	−0.453032
$988$	0	0
$989$	−7.46071	−0.237237
$990$	0	0
$991$	−12.8996	−0.409770	−0.204885	−	0.978786i	$-0.565682\pi$
$991$	−12.8996	−0.409770	−0.204885	+	0.978786i	$0.565682\pi$
$992$	0	0
$993$	−21.1396	−0.670845
$994$	0	0
$995$	88.6508	2.81042
$996$	0	0
$997$	35.7713	1.13289	0.566445	−	0.824100i	$-0.308319\pi$
$997$	35.7713	1.13289	0.566445	+	0.824100i	$0.308319\pi$
$998$	0	0
$999$	−8.43013	−0.266718

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.15	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.15	✓	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.08902 q^{5} +2.57555 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +4.08902 q^{5} +2.57555 q^{7} +1.00000 q^{9} +4.61172 q^{11} -4.79639 q^{13} +4.08902 q^{15} -1.83258 q^{17} +4.89968 q^{19} +2.57555 q^{21} +1.00000 q^{23} +11.7201 q^{25} +1.00000 q^{27} -1.00000 q^{29} +8.03980 q^{31} +4.61172 q^{33} +10.5315 q^{35} -8.43013 q^{37} -4.79639 q^{39} +8.64925 q^{41} -7.46071 q^{43} +4.08902 q^{45} -5.52609 q^{47} -0.366553 q^{49} -1.83258 q^{51} -7.81069 q^{53} +18.8574 q^{55} +4.89968 q^{57} +7.57493 q^{59} -10.4358 q^{61} +2.57555 q^{63} -19.6125 q^{65} +8.30407 q^{67} +1.00000 q^{69} -0.0870476 q^{71} +9.17475 q^{73} +11.7201 q^{75} +11.8777 q^{77} -8.38478 q^{79} +1.00000 q^{81} -9.19932 q^{83} -7.49346 q^{85} -1.00000 q^{87} -11.9049 q^{89} -12.3533 q^{91} +8.03980 q^{93} +20.0349 q^{95} +6.00162 q^{97} +4.61172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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