LMFDB - Embedded newform 8004.2.a.j.1.7

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			1.7
Root			$-0.812118$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.812118 q^{5} -4.94593 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.812118 q^{5} -4.94593 q^{7} +1.00000 q^{9} +1.85682 q^{11} -6.88102 q^{13} -0.812118 q^{15} -7.62412 q^{17} +0.765989 q^{19} -4.94593 q^{21} +1.00000 q^{23} -4.34046 q^{25} +1.00000 q^{27} -1.00000 q^{29} -7.16139 q^{31} +1.85682 q^{33} +4.01668 q^{35} -6.94133 q^{37} -6.88102 q^{39} -2.06531 q^{41} +12.9781 q^{43} -0.812118 q^{45} -3.03906 q^{47} +17.4622 q^{49} -7.62412 q^{51} +13.0836 q^{53} -1.50796 q^{55} +0.765989 q^{57} +6.09500 q^{59} -3.49906 q^{61} -4.94593 q^{63} +5.58820 q^{65} -9.90455 q^{67} +1.00000 q^{69} +13.9449 q^{71} -3.62415 q^{73} -4.34046 q^{75} -9.18370 q^{77} +15.4261 q^{79} +1.00000 q^{81} +0.720700 q^{83} +6.19168 q^{85} -1.00000 q^{87} +4.99841 q^{89} +34.0330 q^{91} -7.16139 q^{93} -0.622073 q^{95} -5.22122 q^{97} +1.85682 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.812118	−0.363190	−0.181595	−	0.983373i	$-0.558126\pi$
$5$	−0.812118	−0.363190	−0.181595	+	0.983373i	$0.558126\pi$
$6$	0	0
$7$	−4.94593	−1.86939	−0.934693	−	0.355456i	$-0.884325\pi$
$7$	−4.94593	−1.86939	−0.934693	+	0.355456i	$0.884325\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.85682	0.559852	0.279926	−	0.960022i	$-0.409690\pi$
$11$	1.85682	0.559852	0.279926	+	0.960022i	$0.409690\pi$
$12$	0	0
$13$	−6.88102	−1.90845	−0.954226	−	0.299088i	$-0.903318\pi$
$13$	−6.88102	−1.90845	−0.954226	+	0.299088i	$0.903318\pi$
$14$	0	0
$15$	−0.812118	−0.209688
$16$	0	0
$17$	−7.62412	−1.84912	−0.924560	−	0.381037i	$-0.875567\pi$
$17$	−7.62412	−1.84912	−0.924560	+	0.381037i	$0.875567\pi$
$18$	0	0
$19$	0.765989	0.175730	0.0878650	−	0.996132i	$-0.471996\pi$
$19$	0.765989	0.175730	0.0878650	+	0.996132i	$0.471996\pi$
$20$	0	0
$21$	−4.94593	−1.07929
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.34046	−0.868093
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−7.16139	−1.28622	−0.643112	−	0.765773i	$-0.722357\pi$
$31$	−7.16139	−1.28622	−0.643112	+	0.765773i	$0.722357\pi$
$32$	0	0
$33$	1.85682	0.323231
$34$	0	0
$35$	4.01668	0.678942
$36$	0	0
$37$	−6.94133	−1.14115	−0.570574	−	0.821246i	$-0.693279\pi$
$37$	−6.94133	−1.14115	−0.570574	+	0.821246i	$0.693279\pi$
$38$	0	0
$39$	−6.88102	−1.10184
$40$	0	0
$41$	−2.06531	−0.322548	−0.161274	−	0.986910i	$-0.551560\pi$
$41$	−2.06531	−0.322548	−0.161274	+	0.986910i	$0.551560\pi$
$42$	0	0
$43$	12.9781	1.97914	0.989570	−	0.144051i	$-0.0460130\pi$
$43$	12.9781	1.97914	0.989570	+	0.144051i	$0.0460130\pi$
$44$	0	0
$45$	−0.812118	−0.121063
$46$	0	0
$47$	−3.03906	−0.443292	−0.221646	−	0.975127i	$-0.571143\pi$
$47$	−3.03906	−0.443292	−0.221646	+	0.975127i	$0.571143\pi$
$48$	0	0
$49$	17.4622	2.49460
$50$	0	0
$51$	−7.62412	−1.06759
$52$	0	0
$53$	13.0836	1.79717	0.898586	−	0.438798i	$-0.144596\pi$
$53$	13.0836	1.79717	0.898586	+	0.438798i	$0.144596\pi$
$54$	0	0
$55$	−1.50796	−0.203333
$56$	0	0
$57$	0.765989	0.101458
$58$	0	0
$59$	6.09500	0.793501	0.396750	−	0.917927i	$-0.370138\pi$
$59$	6.09500	0.793501	0.396750	+	0.917927i	$0.370138\pi$
$60$	0	0
$61$	−3.49906	−0.448009	−0.224005	−	0.974588i	$-0.571913\pi$
$61$	−3.49906	−0.448009	−0.224005	+	0.974588i	$0.571913\pi$
$62$	0	0
$63$	−4.94593	−0.623129
$64$	0	0
$65$	5.58820	0.693130
$66$	0	0
$67$	−9.90455	−1.21003	−0.605016	−	0.796213i	$-0.706833\pi$
$67$	−9.90455	−1.21003	−0.605016	+	0.796213i	$0.706833\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	13.9449	1.65495	0.827477	−	0.561500i	$-0.189775\pi$
$71$	13.9449	1.65495	0.827477	+	0.561500i	$0.189775\pi$
$72$	0	0
$73$	−3.62415	−0.424175	−0.212088	−	0.977251i	$-0.568026\pi$
$73$	−3.62415	−0.424175	−0.212088	+	0.977251i	$0.568026\pi$
$74$	0	0
$75$	−4.34046	−0.501194
$76$	0	0
$77$	−9.18370	−1.04658
$78$	0	0
$79$	15.4261	1.73557	0.867783	−	0.496943i	$-0.165544\pi$
$79$	15.4261	1.73557	0.867783	+	0.496943i	$0.165544\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.720700	0.0791071	0.0395535	−	0.999217i	$-0.487406\pi$
$83$	0.720700	0.0791071	0.0395535	+	0.999217i	$0.487406\pi$
$84$	0	0
$85$	6.19168	0.671582
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	4.99841	0.529830	0.264915	−	0.964272i	$-0.414656\pi$
$89$	4.99841	0.529830	0.264915	+	0.964272i	$0.414656\pi$
$90$	0	0
$91$	34.0330	3.56763
$92$	0	0
$93$	−7.16139	−0.742601
$94$	0	0
$95$	−0.622073	−0.0638234
$96$	0	0
$97$	−5.22122	−0.530134	−0.265067	−	0.964230i	$-0.585394\pi$
$97$	−5.22122	−0.530134	−0.265067	+	0.964230i	$0.585394\pi$
$98$	0	0
$99$	1.85682	0.186617
$100$	0	0
$101$	17.5973	1.75100	0.875499	−	0.483220i	$-0.160533\pi$
$101$	17.5973	1.75100	0.875499	+	0.483220i	$0.160533\pi$
$102$	0	0
$103$	−9.42864	−0.929031	−0.464516	−	0.885565i	$-0.653772\pi$
$103$	−9.42864	−0.929031	−0.464516	+	0.885565i	$0.653772\pi$
$104$	0	0
$105$	4.01668	0.391988
$106$	0	0
$107$	6.07940	0.587717	0.293859	−	0.955849i	$-0.405060\pi$
$107$	6.07940	0.587717	0.293859	+	0.955849i	$0.405060\pi$
$108$	0	0
$109$	−17.4110	−1.66767	−0.833834	−	0.552015i	$-0.813859\pi$
$109$	−17.4110	−1.66767	−0.833834	+	0.552015i	$0.813859\pi$
$110$	0	0
$111$	−6.94133	−0.658842
$112$	0	0
$113$	2.07795	0.195477	0.0977386	−	0.995212i	$-0.468839\pi$
$113$	2.07795	0.195477	0.0977386	+	0.995212i	$0.468839\pi$
$114$	0	0
$115$	−0.812118	−0.0757304
$116$	0	0
$117$	−6.88102	−0.636150
$118$	0	0
$119$	37.7083	3.45672
$120$	0	0
$121$	−7.55222	−0.686566
$122$	0	0
$123$	−2.06531	−0.186223
$124$	0	0
$125$	7.58556	0.678473
$126$	0	0
$127$	9.91116	0.879473	0.439737	−	0.898127i	$-0.355072\pi$
$127$	9.91116	0.879473	0.439737	+	0.898127i	$0.355072\pi$
$128$	0	0
$129$	12.9781	1.14266
$130$	0	0
$131$	−9.89560	−0.864583	−0.432291	−	0.901734i	$-0.642295\pi$
$131$	−9.89560	−0.864583	−0.432291	+	0.901734i	$0.642295\pi$
$132$	0	0
$133$	−3.78853	−0.328507
$134$	0	0
$135$	−0.812118	−0.0698960
$136$	0	0
$137$	12.4914	1.06721	0.533607	−	0.845733i	$-0.320836\pi$
$137$	12.4914	1.06721	0.533607	+	0.845733i	$0.320836\pi$
$138$	0	0
$139$	13.0653	1.10819	0.554093	−	0.832455i	$-0.313065\pi$
$139$	13.0653	1.10819	0.554093	+	0.832455i	$0.313065\pi$
$140$	0	0
$141$	−3.03906	−0.255935
$142$	0	0
$143$	−12.7768	−1.06845
$144$	0	0
$145$	0.812118	0.0674427
$146$	0	0
$147$	17.4622	1.44026
$148$	0	0
$149$	−22.9919	−1.88357	−0.941786	−	0.336214i	$-0.890854\pi$
$149$	−22.9919	−1.88357	−0.941786	+	0.336214i	$0.890854\pi$
$150$	0	0
$151$	−7.78139	−0.633240	−0.316620	−	0.948552i	$-0.602548\pi$
$151$	−7.78139	−0.633240	−0.316620	+	0.948552i	$0.602548\pi$
$152$	0	0
$153$	−7.62412	−0.616373
$154$	0	0
$155$	5.81589	0.467143
$156$	0	0
$157$	−4.12748	−0.329409	−0.164704	−	0.986343i	$-0.552667\pi$
$157$	−4.12748	−0.329409	−0.164704	+	0.986343i	$0.552667\pi$
$158$	0	0
$159$	13.0836	1.03760
$160$	0	0
$161$	−4.94593	−0.389794
$162$	0	0
$163$	−15.4265	−1.20829	−0.604147	−	0.796873i	$-0.706486\pi$
$163$	−15.4265	−1.20829	−0.604147	+	0.796873i	$0.706486\pi$
$164$	0	0
$165$	−1.50796	−0.117394
$166$	0	0
$167$	−9.47945	−0.733542	−0.366771	−	0.930311i	$-0.619537\pi$
$167$	−9.47945	−0.733542	−0.366771	+	0.930311i	$0.619537\pi$
$168$	0	0
$169$	34.3484	2.64219
$170$	0	0
$171$	0.765989	0.0585767
$172$	0	0
$173$	−12.7240	−0.967385	−0.483692	−	0.875238i	$-0.660705\pi$
$173$	−12.7240	−0.967385	−0.483692	+	0.875238i	$0.660705\pi$
$174$	0	0
$175$	21.4676	1.62280
$176$	0	0
$177$	6.09500	0.458128
$178$	0	0
$179$	13.5450	1.01240	0.506200	−	0.862416i	$-0.331050\pi$
$179$	13.5450	1.01240	0.506200	+	0.862416i	$0.331050\pi$
$180$	0	0
$181$	−18.9936	−1.41178	−0.705891	−	0.708320i	$-0.749453\pi$
$181$	−18.9936	−1.41178	−0.705891	+	0.708320i	$0.749453\pi$
$182$	0	0
$183$	−3.49906	−0.258658
$184$	0	0
$185$	5.63718	0.414454
$186$	0	0
$187$	−14.1566	−1.03523
$188$	0	0
$189$	−4.94593	−0.359763
$190$	0	0
$191$	−0.0249716	−0.00180688	−0.000903439	−	1.00000i	$-0.500288\pi$
$191$	−0.0249716	−0.00180688	−0.000903439		1.00000i	$0.500288\pi$
$192$	0	0
$193$	9.83124	0.707668	0.353834	−	0.935308i	$-0.384878\pi$
$193$	9.83124	0.707668	0.353834	+	0.935308i	$0.384878\pi$
$194$	0	0
$195$	5.58820	0.400179
$196$	0	0
$197$	4.42935	0.315578	0.157789	−	0.987473i	$-0.449563\pi$
$197$	4.42935	0.315578	0.157789	+	0.987473i	$0.449563\pi$
$198$	0	0
$199$	−1.93109	−0.136891	−0.0684455	−	0.997655i	$-0.521804\pi$
$199$	−1.93109	−0.136891	−0.0684455	+	0.997655i	$0.521804\pi$
$200$	0	0
$201$	−9.90455	−0.698613
$202$	0	0
$203$	4.94593	0.347136
$204$	0	0
$205$	1.67728	0.117146
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	1.42230	0.0983828
$210$	0	0
$211$	9.35328	0.643907	0.321953	−	0.946756i	$-0.395661\pi$
$211$	9.35328	0.643907	0.321953	+	0.946756i	$0.395661\pi$
$212$	0	0
$213$	13.9449	0.955488
$214$	0	0
$215$	−10.5397	−0.718804
$216$	0	0
$217$	35.4197	2.40445
$218$	0	0
$219$	−3.62415	−0.244898
$220$	0	0
$221$	52.4617	3.52895
$222$	0	0
$223$	−1.62611	−0.108892	−0.0544462	−	0.998517i	$-0.517339\pi$
$223$	−1.62611	−0.108892	−0.0544462	+	0.998517i	$0.517339\pi$
$224$	0	0
$225$	−4.34046	−0.289364
$226$	0	0
$227$	−14.0590	−0.933126	−0.466563	−	0.884488i	$-0.654508\pi$
$227$	−14.0590	−0.933126	−0.466563	+	0.884488i	$0.654508\pi$
$228$	0	0
$229$	0.113933	0.00752890	0.00376445	−	0.999993i	$-0.498802\pi$
$229$	0.113933	0.00752890	0.00376445	+	0.999993i	$0.498802\pi$
$230$	0	0
$231$	−9.18370	−0.604243
$232$	0	0
$233$	−5.05724	−0.331311	−0.165655	−	0.986184i	$-0.552974\pi$
$233$	−5.05724	−0.331311	−0.165655	+	0.986184i	$0.552974\pi$
$234$	0	0
$235$	2.46807	0.160999
$236$	0	0
$237$	15.4261	1.00203
$238$	0	0
$239$	−9.38121	−0.606820	−0.303410	−	0.952860i	$-0.598125\pi$
$239$	−9.38121	−0.606820	−0.303410	+	0.952860i	$0.598125\pi$
$240$	0	0
$241$	−19.7142	−1.26990	−0.634952	−	0.772551i	$-0.718980\pi$
$241$	−19.7142	−1.26990	−0.634952	+	0.772551i	$0.718980\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.1814	−0.906015
$246$	0	0
$247$	−5.27079	−0.335372
$248$	0	0
$249$	0.720700	0.0456725
$250$	0	0
$251$	−20.4703	−1.29208	−0.646038	−	0.763305i	$-0.723575\pi$
$251$	−20.4703	−1.29208	−0.646038	+	0.763305i	$0.723575\pi$
$252$	0	0
$253$	1.85682	0.116737
$254$	0	0
$255$	6.19168	0.387738
$256$	0	0
$257$	6.14767	0.383481	0.191740	−	0.981446i	$-0.438587\pi$
$257$	6.14767	0.383481	0.191740	+	0.981446i	$0.438587\pi$
$258$	0	0
$259$	34.3313	2.13325
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−10.0386	−0.619005	−0.309503	−	0.950899i	$-0.600163\pi$
$263$	−10.0386	−0.619005	−0.309503	+	0.950899i	$0.600163\pi$
$264$	0	0
$265$	−10.6254	−0.652715
$266$	0	0
$267$	4.99841	0.305898
$268$	0	0
$269$	28.5233	1.73910	0.869549	−	0.493847i	$-0.164410\pi$
$269$	28.5233	1.73910	0.869549	+	0.493847i	$0.164410\pi$
$270$	0	0
$271$	9.35548	0.568305	0.284152	−	0.958779i	$-0.408288\pi$
$271$	9.35548	0.568305	0.284152	+	0.958779i	$0.408288\pi$
$272$	0	0
$273$	34.0330	2.05977
$274$	0	0
$275$	−8.05946	−0.486004
$276$	0	0
$277$	0.246488	0.0148100	0.00740502	−	0.999973i	$-0.497643\pi$
$277$	0.246488	0.0148100	0.00740502	+	0.999973i	$0.497643\pi$
$278$	0	0
$279$	−7.16139	−0.428741
$280$	0	0
$281$	26.4451	1.57758	0.788791	−	0.614662i	$-0.210708\pi$
$281$	26.4451	1.57758	0.788791	+	0.614662i	$0.210708\pi$
$282$	0	0
$283$	6.59503	0.392034	0.196017	−	0.980601i	$-0.437199\pi$
$283$	6.59503	0.392034	0.196017	+	0.980601i	$0.437199\pi$
$284$	0	0
$285$	−0.622073	−0.0368485
$286$	0	0
$287$	10.2149	0.602966
$288$	0	0
$289$	41.1272	2.41924
$290$	0	0
$291$	−5.22122	−0.306073
$292$	0	0
$293$	16.3626	0.955912	0.477956	−	0.878384i	$-0.341378\pi$
$293$	16.3626	0.955912	0.477956	+	0.878384i	$0.341378\pi$
$294$	0	0
$295$	−4.94985	−0.288192
$296$	0	0
$297$	1.85682	0.107744
$298$	0	0
$299$	−6.88102	−0.397940
$300$	0	0
$301$	−64.1887	−3.69978
$302$	0	0
$303$	17.5973	1.01094
$304$	0	0
$305$	2.84165	0.162712
$306$	0	0
$307$	−17.5073	−0.999197	−0.499599	−	0.866257i	$-0.666519\pi$
$307$	−17.5073	−0.999197	−0.499599	+	0.866257i	$0.666519\pi$
$308$	0	0
$309$	−9.42864	−0.536377
$310$	0	0
$311$	−10.0872	−0.571993	−0.285997	−	0.958231i	$-0.592325\pi$
$311$	−10.0872	−0.571993	−0.285997	+	0.958231i	$0.592325\pi$
$312$	0	0
$313$	21.6111	1.22153	0.610765	−	0.791812i	$-0.290862\pi$
$313$	21.6111	1.22153	0.610765	+	0.791812i	$0.290862\pi$
$314$	0	0
$315$	4.01668	0.226314
$316$	0	0
$317$	−17.6316	−0.990288	−0.495144	−	0.868811i	$-0.664885\pi$
$317$	−17.6316	−0.990288	−0.495144	+	0.868811i	$0.664885\pi$
$318$	0	0
$319$	−1.85682	−0.103962
$320$	0	0
$321$	6.07940	0.339319
$322$	0	0
$323$	−5.83999	−0.324946
$324$	0	0
$325$	29.8668	1.65671
$326$	0	0
$327$	−17.4110	−0.962829
$328$	0	0
$329$	15.0310	0.828684
$330$	0	0
$331$	18.2302	1.00202	0.501012	−	0.865440i	$-0.332961\pi$
$331$	18.2302	1.00202	0.501012	+	0.865440i	$0.332961\pi$
$332$	0	0
$333$	−6.94133	−0.380383
$334$	0	0
$335$	8.04366	0.439472
$336$	0	0
$337$	16.5663	0.902423	0.451211	−	0.892417i	$-0.350992\pi$
$337$	16.5663	0.902423	0.451211	+	0.892417i	$0.350992\pi$
$338$	0	0
$339$	2.07795	0.112859
$340$	0	0
$341$	−13.2974	−0.720094
$342$	0	0
$343$	−51.7454	−2.79399
$344$	0	0
$345$	−0.812118	−0.0437229
$346$	0	0
$347$	−0.993493	−0.0533335	−0.0266668	−	0.999644i	$-0.508489\pi$
$347$	−0.993493	−0.0533335	−0.0266668	+	0.999644i	$0.508489\pi$
$348$	0	0
$349$	−15.2438	−0.815981	−0.407991	−	0.912986i	$-0.633770\pi$
$349$	−15.2438	−0.815981	−0.407991	+	0.912986i	$0.633770\pi$
$350$	0	0
$351$	−6.88102	−0.367282
$352$	0	0
$353$	4.35780	0.231942	0.115971	−	0.993253i	$-0.463002\pi$
$353$	4.35780	0.231942	0.115971	+	0.993253i	$0.463002\pi$
$354$	0	0
$355$	−11.3249	−0.601063
$356$	0	0
$357$	37.7083	1.99574
$358$	0	0
$359$	−15.0026	−0.791808	−0.395904	−	0.918292i	$-0.629569\pi$
$359$	−15.0026	−0.791808	−0.395904	+	0.918292i	$0.629569\pi$
$360$	0	0
$361$	−18.4133	−0.969119
$362$	0	0
$363$	−7.55222	−0.396389
$364$	0	0
$365$	2.94324	0.154056
$366$	0	0
$367$	29.7494	1.55291	0.776454	−	0.630174i	$-0.217016\pi$
$367$	29.7494	1.55291	0.776454	+	0.630174i	$0.217016\pi$
$368$	0	0
$369$	−2.06531	−0.107516
$370$	0	0
$371$	−64.7106	−3.35961
$372$	0	0
$373$	14.3697	0.744036	0.372018	−	0.928226i	$-0.378666\pi$
$373$	14.3697	0.744036	0.372018	+	0.928226i	$0.378666\pi$
$374$	0	0
$375$	7.58556	0.391716
$376$	0	0
$377$	6.88102	0.354390
$378$	0	0
$379$	−22.5341	−1.15750	−0.578749	−	0.815506i	$-0.696459\pi$
$379$	−22.5341	−1.15750	−0.578749	+	0.815506i	$0.696459\pi$
$380$	0	0
$381$	9.91116	0.507764
$382$	0	0
$383$	16.3663	0.836278	0.418139	−	0.908383i	$-0.362682\pi$
$383$	16.3663	0.836278	0.418139	+	0.908383i	$0.362682\pi$
$384$	0	0
$385$	7.45824	0.380107
$386$	0	0
$387$	12.9781	0.659713
$388$	0	0
$389$	17.6023	0.892470	0.446235	−	0.894916i	$-0.352764\pi$
$389$	17.6023	0.892470	0.446235	+	0.894916i	$0.352764\pi$
$390$	0	0
$391$	−7.62412	−0.385568
$392$	0	0
$393$	−9.89560	−0.499167
$394$	0	0
$395$	−12.5278	−0.630340
$396$	0	0
$397$	23.5933	1.18411	0.592056	−	0.805897i	$-0.298316\pi$
$397$	23.5933	1.18411	0.592056	+	0.805897i	$0.298316\pi$
$398$	0	0
$399$	−3.78853	−0.189664
$400$	0	0
$401$	−17.5098	−0.874400	−0.437200	−	0.899364i	$-0.644030\pi$
$401$	−17.5098	−0.874400	−0.437200	+	0.899364i	$0.644030\pi$
$402$	0	0
$403$	49.2776	2.45469
$404$	0	0
$405$	−0.812118	−0.0403544
$406$	0	0
$407$	−12.8888	−0.638874
$408$	0	0
$409$	6.91708	0.342027	0.171014	−	0.985269i	$-0.445296\pi$
$409$	6.91708	0.342027	0.171014	+	0.985269i	$0.445296\pi$
$410$	0	0
$411$	12.4914	0.616156
$412$	0	0
$413$	−30.1454	−1.48336
$414$	0	0
$415$	−0.585293	−0.0287309
$416$	0	0
$417$	13.0653	0.639812
$418$	0	0
$419$	36.1660	1.76683	0.883413	−	0.468595i	$-0.155240\pi$
$419$	36.1660	1.76683	0.883413	+	0.468595i	$0.155240\pi$
$420$	0	0
$421$	−5.14718	−0.250858	−0.125429	−	0.992103i	$-0.540031\pi$
$421$	−5.14718	−0.250858	−0.125429	+	0.992103i	$0.540031\pi$
$422$	0	0
$423$	−3.03906	−0.147764
$424$	0	0
$425$	33.0922	1.60521
$426$	0	0
$427$	17.3061	0.837502
$428$	0	0
$429$	−12.7768	−0.616870
$430$	0	0
$431$	−5.24355	−0.252573	−0.126286	−	0.991994i	$-0.540306\pi$
$431$	−5.24355	−0.252573	−0.126286	+	0.991994i	$0.540306\pi$
$432$	0	0
$433$	25.0784	1.20519	0.602595	−	0.798047i	$-0.294133\pi$
$433$	25.0784	1.20519	0.602595	+	0.798047i	$0.294133\pi$
$434$	0	0
$435$	0.812118	0.0389381
$436$	0	0
$437$	0.765989	0.0366422
$438$	0	0
$439$	11.1658	0.532915	0.266457	−	0.963847i	$-0.414147\pi$
$439$	11.1658	0.532915	0.266457	+	0.963847i	$0.414147\pi$
$440$	0	0
$441$	17.4622	0.831534
$442$	0	0
$443$	−6.99870	−0.332518	−0.166259	−	0.986082i	$-0.553169\pi$
$443$	−6.99870	−0.332518	−0.166259	+	0.986082i	$0.553169\pi$
$444$	0	0
$445$	−4.05930	−0.192429
$446$	0	0
$447$	−22.9919	−1.08748
$448$	0	0
$449$	35.6458	1.68223	0.841114	−	0.540857i	$-0.181900\pi$
$449$	35.6458	1.68223	0.841114	+	0.540857i	$0.181900\pi$
$450$	0	0
$451$	−3.83491	−0.180579
$452$	0	0
$453$	−7.78139	−0.365602
$454$	0	0
$455$	−27.6388	−1.29573
$456$	0	0
$457$	−2.92721	−0.136929	−0.0684647	−	0.997654i	$-0.521810\pi$
$457$	−2.92721	−0.136929	−0.0684647	+	0.997654i	$0.521810\pi$
$458$	0	0
$459$	−7.62412	−0.355863
$460$	0	0
$461$	−8.11556	−0.377979	−0.188990	−	0.981979i	$-0.560521\pi$
$461$	−8.11556	−0.377979	−0.188990	+	0.981979i	$0.560521\pi$
$462$	0	0
$463$	11.8955	0.552830	0.276415	−	0.961038i	$-0.410854\pi$
$463$	11.8955	0.552830	0.276415	+	0.961038i	$0.410854\pi$
$464$	0	0
$465$	5.81589	0.269705
$466$	0	0
$467$	−27.3979	−1.26782	−0.633912	−	0.773405i	$-0.718552\pi$
$467$	−27.3979	−1.26782	−0.633912	+	0.773405i	$0.718552\pi$
$468$	0	0
$469$	48.9872	2.26202
$470$	0	0
$471$	−4.12748	−0.190184
$472$	0	0
$473$	24.0980	1.10803
$474$	0	0
$475$	−3.32475	−0.152550
$476$	0	0
$477$	13.0836	0.599057
$478$	0	0
$479$	18.2273	0.832827	0.416413	−	0.909175i	$-0.363287\pi$
$479$	18.2273	0.832827	0.416413	+	0.909175i	$0.363287\pi$
$480$	0	0
$481$	47.7634	2.17782
$482$	0	0
$483$	−4.94593	−0.225048
$484$	0	0
$485$	4.24024	0.192539
$486$	0	0
$487$	12.2112	0.553344	0.276672	−	0.960964i	$-0.410769\pi$
$487$	12.2112	0.553344	0.276672	+	0.960964i	$0.410769\pi$
$488$	0	0
$489$	−15.4265	−0.697608
$490$	0	0
$491$	24.7324	1.11616	0.558079	−	0.829788i	$-0.311539\pi$
$491$	24.7324	1.11616	0.558079	+	0.829788i	$0.311539\pi$
$492$	0	0
$493$	7.62412	0.343373
$494$	0	0
$495$	−1.50796	−0.0677776
$496$	0	0
$497$	−68.9704	−3.09375
$498$	0	0
$499$	41.7864	1.87062	0.935309	−	0.353832i	$-0.115121\pi$
$499$	41.7864	1.87062	0.935309	+	0.353832i	$0.115121\pi$
$500$	0	0
$501$	−9.47945	−0.423511
$502$	0	0
$503$	−26.4959	−1.18139	−0.590697	−	0.806894i	$-0.701147\pi$
$503$	−26.4959	−1.18139	−0.590697	+	0.806894i	$0.701147\pi$
$504$	0	0
$505$	−14.2911	−0.635945
$506$	0	0
$507$	34.3484	1.52547
$508$	0	0
$509$	−29.6337	−1.31349	−0.656746	−	0.754112i	$-0.728068\pi$
$509$	−29.6337	−1.31349	−0.656746	+	0.754112i	$0.728068\pi$
$510$	0	0
$511$	17.9248	0.792947
$512$	0	0
$513$	0.765989	0.0338193
$514$	0	0
$515$	7.65716	0.337415
$516$	0	0
$517$	−5.64298	−0.248178
$518$	0	0
$519$	−12.7240	−0.558520
$520$	0	0
$521$	5.07091	0.222160	0.111080	−	0.993811i	$-0.464569\pi$
$521$	5.07091	0.222160	0.111080	+	0.993811i	$0.464569\pi$
$522$	0	0
$523$	−3.70485	−0.162002	−0.0810008	−	0.996714i	$-0.525812\pi$
$523$	−3.70485	−0.162002	−0.0810008	+	0.996714i	$0.525812\pi$
$524$	0	0
$525$	21.4676	0.936924
$526$	0	0
$527$	54.5993	2.37838
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.09500	0.264500
$532$	0	0
$533$	14.2114	0.615566
$534$	0	0
$535$	−4.93719	−0.213453
$536$	0	0
$537$	13.5450	0.584509
$538$	0	0
$539$	32.4242	1.39661
$540$	0	0
$541$	4.15109	0.178469	0.0892347	−	0.996011i	$-0.471558\pi$
$541$	4.15109	0.178469	0.0892347	+	0.996011i	$0.471558\pi$
$542$	0	0
$543$	−18.9936	−0.815093
$544$	0	0
$545$	14.1398	0.605681
$546$	0	0
$547$	−13.2304	−0.565691	−0.282846	−	0.959165i	$-0.591278\pi$
$547$	−13.2304	−0.565691	−0.282846	+	0.959165i	$0.591278\pi$
$548$	0	0
$549$	−3.49906	−0.149336
$550$	0	0
$551$	−0.765989	−0.0326322
$552$	0	0
$553$	−76.2962	−3.24444
$554$	0	0
$555$	5.63718	0.239285
$556$	0	0
$557$	−36.6588	−1.55328	−0.776641	−	0.629943i	$-0.783078\pi$
$557$	−36.6588	−1.55328	−0.776641	+	0.629943i	$0.783078\pi$
$558$	0	0
$559$	−89.3025	−3.77709
$560$	0	0
$561$	−14.1566	−0.597692
$562$	0	0
$563$	−19.0766	−0.803982	−0.401991	−	0.915644i	$-0.631682\pi$
$563$	−19.0766	−0.803982	−0.401991	+	0.915644i	$0.631682\pi$
$564$	0	0
$565$	−1.68754	−0.0709953
$566$	0	0
$567$	−4.94593	−0.207710
$568$	0	0
$569$	−42.3700	−1.77624	−0.888121	−	0.459610i	$-0.847989\pi$
$569$	−42.3700	−1.77624	−0.888121	+	0.459610i	$0.847989\pi$
$570$	0	0
$571$	2.63564	0.110298	0.0551490	−	0.998478i	$-0.482437\pi$
$571$	2.63564	0.110298	0.0551490	+	0.998478i	$0.482437\pi$
$572$	0	0
$573$	−0.0249716	−0.00104320
$574$	0	0
$575$	−4.34046	−0.181010
$576$	0	0
$577$	0.740812	0.0308404	0.0154202	−	0.999881i	$-0.495091\pi$
$577$	0.740812	0.0308404	0.0154202	+	0.999881i	$0.495091\pi$
$578$	0	0
$579$	9.83124	0.408573
$580$	0	0
$581$	−3.56453	−0.147882
$582$	0	0
$583$	24.2939	1.00615
$584$	0	0
$585$	5.58820	0.231043
$586$	0	0
$587$	−6.89109	−0.284426	−0.142213	−	0.989836i	$-0.545422\pi$
$587$	−6.89109	−0.284426	−0.142213	+	0.989836i	$0.545422\pi$
$588$	0	0
$589$	−5.48555	−0.226028
$590$	0	0
$591$	4.42935	0.182199
$592$	0	0
$593$	−44.3354	−1.82064	−0.910319	−	0.413907i	$-0.864164\pi$
$593$	−44.3354	−1.82064	−0.910319	+	0.413907i	$0.864164\pi$
$594$	0	0
$595$	−30.6236	−1.25545
$596$	0	0
$597$	−1.93109	−0.0790341
$598$	0	0
$599$	−27.2406	−1.11302	−0.556509	−	0.830841i	$-0.687860\pi$
$599$	−27.2406	−1.11302	−0.556509	+	0.830841i	$0.687860\pi$
$600$	0	0
$601$	−8.15062	−0.332471	−0.166235	−	0.986086i	$-0.553161\pi$
$601$	−8.15062	−0.332471	−0.166235	+	0.986086i	$0.553161\pi$
$602$	0	0
$603$	−9.90455	−0.403344
$604$	0	0
$605$	6.13329	0.249354
$606$	0	0
$607$	26.6056	1.07989	0.539944	−	0.841701i	$-0.318446\pi$
$607$	26.6056	1.07989	0.539944	+	0.841701i	$0.318446\pi$
$608$	0	0
$609$	4.94593	0.200419
$610$	0	0
$611$	20.9118	0.846001
$612$	0	0
$613$	−9.44054	−0.381300	−0.190650	−	0.981658i	$-0.561059\pi$
$613$	−9.44054	−0.381300	−0.190650	+	0.981658i	$0.561059\pi$
$614$	0	0
$615$	1.67728	0.0676343
$616$	0	0
$617$	9.95593	0.400811	0.200405	−	0.979713i	$-0.435774\pi$
$617$	9.95593	0.400811	0.200405	+	0.979713i	$0.435774\pi$
$618$	0	0
$619$	5.11351	0.205529	0.102765	−	0.994706i	$-0.467231\pi$
$619$	5.11351	0.205529	0.102765	+	0.994706i	$0.467231\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−24.7218	−0.990457
$624$	0	0
$625$	15.5420	0.621678
$626$	0	0
$627$	1.42230	0.0568013
$628$	0	0
$629$	52.9215	2.11012
$630$	0	0
$631$	−29.4493	−1.17236	−0.586179	−	0.810182i	$-0.699368\pi$
$631$	−29.4493	−1.17236	−0.586179	+	0.810182i	$0.699368\pi$
$632$	0	0
$633$	9.35328	0.371760
$634$	0	0
$635$	−8.04903	−0.319416
$636$	0	0
$637$	−120.158	−4.76083
$638$	0	0
$639$	13.9449	0.551651
$640$	0	0
$641$	15.3757	0.607303	0.303652	−	0.952783i	$-0.401794\pi$
$641$	15.3757	0.607303	0.303652	+	0.952783i	$0.401794\pi$
$642$	0	0
$643$	35.6822	1.40717	0.703585	−	0.710611i	$-0.251581\pi$
$643$	35.6822	1.40717	0.703585	+	0.710611i	$0.251581\pi$
$644$	0	0
$645$	−10.5397	−0.415002
$646$	0	0
$647$	28.9106	1.13659	0.568297	−	0.822824i	$-0.307603\pi$
$647$	28.9106	1.13659	0.568297	+	0.822824i	$0.307603\pi$
$648$	0	0
$649$	11.3173	0.444243
$650$	0	0
$651$	35.4197	1.38821
$652$	0	0
$653$	6.58135	0.257548	0.128774	−	0.991674i	$-0.458896\pi$
$653$	6.58135	0.257548	0.128774	+	0.991674i	$0.458896\pi$
$654$	0	0
$655$	8.03639	0.314008
$656$	0	0
$657$	−3.62415	−0.141392
$658$	0	0
$659$	−48.8441	−1.90270	−0.951349	−	0.308115i	$-0.900302\pi$
$659$	−48.8441	−1.90270	−0.951349	+	0.308115i	$0.900302\pi$
$660$	0	0
$661$	40.9767	1.59381	0.796904	−	0.604106i	$-0.206470\pi$
$661$	40.9767	1.59381	0.796904	+	0.604106i	$0.206470\pi$
$662$	0	0
$663$	52.4617	2.03744
$664$	0	0
$665$	3.07673	0.119311
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−1.62611	−0.0628691
$670$	0	0
$671$	−6.49713	−0.250819
$672$	0	0
$673$	42.6389	1.64361	0.821804	−	0.569770i	$-0.192968\pi$
$673$	42.6389	1.64361	0.821804	+	0.569770i	$0.192968\pi$
$674$	0	0
$675$	−4.34046	−0.167065
$676$	0	0
$677$	−27.8300	−1.06959	−0.534796	−	0.844981i	$-0.679612\pi$
$677$	−27.8300	−1.06959	−0.534796	+	0.844981i	$0.679612\pi$
$678$	0	0
$679$	25.8238	0.991025
$680$	0	0
$681$	−14.0590	−0.538741
$682$	0	0
$683$	37.2516	1.42539	0.712697	−	0.701472i	$-0.247473\pi$
$683$	37.2516	1.42539	0.712697	+	0.701472i	$0.247473\pi$
$684$	0	0
$685$	−10.1445	−0.387602
$686$	0	0
$687$	0.113933	0.00434681
$688$	0	0
$689$	−90.0285	−3.42981
$690$	0	0
$691$	−9.45465	−0.359672	−0.179836	−	0.983697i	$-0.557557\pi$
$691$	−9.45465	−0.359672	−0.179836	+	0.983697i	$0.557557\pi$
$692$	0	0
$693$	−9.18370	−0.348860
$694$	0	0
$695$	−10.6106	−0.402482
$696$	0	0
$697$	15.7462	0.596429
$698$	0	0
$699$	−5.05724	−0.191282
$700$	0	0
$701$	37.6280	1.42119	0.710596	−	0.703601i	$-0.248426\pi$
$701$	37.6280	1.42119	0.710596	+	0.703601i	$0.248426\pi$
$702$	0	0
$703$	−5.31699	−0.200534
$704$	0	0
$705$	2.46807	0.0929530
$706$	0	0
$707$	−87.0351	−3.27329
$708$	0	0
$709$	25.0874	0.942177	0.471088	−	0.882086i	$-0.343861\pi$
$709$	25.0874	0.942177	0.471088	+	0.882086i	$0.343861\pi$
$710$	0	0
$711$	15.4261	0.578522
$712$	0	0
$713$	−7.16139	−0.268196
$714$	0	0
$715$	10.3763	0.388050
$716$	0	0
$717$	−9.38121	−0.350348
$718$	0	0
$719$	−35.2087	−1.31306	−0.656531	−	0.754299i	$-0.727977\pi$
$719$	−35.2087	−1.31306	−0.656531	+	0.754299i	$0.727977\pi$
$720$	0	0
$721$	46.6334	1.73672
$722$	0	0
$723$	−19.7142	−0.733180
$724$	0	0
$725$	4.34046	0.161201
$726$	0	0
$727$	−18.7656	−0.695976	−0.347988	−	0.937499i	$-0.613135\pi$
$727$	−18.7656	−0.695976	−0.347988	+	0.937499i	$0.613135\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−98.9465	−3.65967
$732$	0	0
$733$	−38.4650	−1.42074	−0.710368	−	0.703830i	$-0.751471\pi$
$733$	−38.4650	−1.42074	−0.710368	+	0.703830i	$0.751471\pi$
$734$	0	0
$735$	−14.1814	−0.523088
$736$	0	0
$737$	−18.3909	−0.677439
$738$	0	0
$739$	−19.5454	−0.718988	−0.359494	−	0.933147i	$-0.617051\pi$
$739$	−19.5454	−0.718988	−0.359494	+	0.933147i	$0.617051\pi$
$740$	0	0
$741$	−5.27079	−0.193627
$742$	0	0
$743$	−22.1029	−0.810878	−0.405439	−	0.914122i	$-0.632881\pi$
$743$	−22.1029	−0.810878	−0.405439	+	0.914122i	$0.632881\pi$
$744$	0	0
$745$	18.6721	0.684094
$746$	0	0
$747$	0.720700	0.0263690
$748$	0	0
$749$	−30.0683	−1.09867
$750$	0	0
$751$	−17.6098	−0.642590	−0.321295	−	0.946979i	$-0.604118\pi$
$751$	−17.6098	−0.642590	−0.321295	+	0.946979i	$0.604118\pi$
$752$	0	0
$753$	−20.4703	−0.745981
$754$	0	0
$755$	6.31940	0.229987
$756$	0	0
$757$	−44.3255	−1.61104	−0.805519	−	0.592570i	$-0.798113\pi$
$757$	−44.3255	−1.61104	−0.805519	+	0.592570i	$0.798113\pi$
$758$	0	0
$759$	1.85682	0.0673983
$760$	0	0
$761$	−7.34078	−0.266103	−0.133052	−	0.991109i	$-0.542478\pi$
$761$	−7.34078	−0.266103	−0.133052	+	0.991109i	$0.542478\pi$
$762$	0	0
$763$	86.1135	3.11752
$764$	0	0
$765$	6.19168	0.223861
$766$	0	0
$767$	−41.9398	−1.51436
$768$	0	0
$769$	17.0004	0.613050	0.306525	−	0.951863i	$-0.400834\pi$
$769$	17.0004	0.613050	0.306525	+	0.951863i	$0.400834\pi$
$770$	0	0
$771$	6.14767	0.221403
$772$	0	0
$773$	−10.6061	−0.381476	−0.190738	−	0.981641i	$-0.561088\pi$
$773$	−10.6061	−0.381476	−0.190738	+	0.981641i	$0.561088\pi$
$774$	0	0
$775$	31.0837	1.11656
$776$	0	0
$777$	34.3313	1.23163
$778$	0	0
$779$	−1.58201	−0.0566813
$780$	0	0
$781$	25.8931	0.926529
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	3.35200	0.119638
$786$	0	0
$787$	−28.6736	−1.02210	−0.511052	−	0.859550i	$-0.670744\pi$
$787$	−28.6736	−1.02210	−0.511052	+	0.859550i	$0.670744\pi$
$788$	0	0
$789$	−10.0386	−0.357383
$790$	0	0
$791$	−10.2774	−0.365422
$792$	0	0
$793$	24.0771	0.855003
$794$	0	0
$795$	−10.6254	−0.376845
$796$	0	0
$797$	5.69586	0.201758	0.100879	−	0.994899i	$-0.467835\pi$
$797$	5.69586	0.201758	0.100879	+	0.994899i	$0.467835\pi$
$798$	0	0
$799$	23.1701	0.819700
$800$	0	0
$801$	4.99841	0.176610
$802$	0	0
$803$	−6.72940	−0.237475
$804$	0	0
$805$	4.01668	0.141569
$806$	0	0
$807$	28.5233	1.00407
$808$	0	0
$809$	30.2084	1.06207	0.531035	−	0.847350i	$-0.321803\pi$
$809$	30.2084	1.06207	0.531035	+	0.847350i	$0.321803\pi$
$810$	0	0
$811$	30.3134	1.06445	0.532224	−	0.846604i	$-0.321357\pi$
$811$	30.3134	1.06445	0.532224	+	0.846604i	$0.321357\pi$
$812$	0	0
$813$	9.35548	0.328111
$814$	0	0
$815$	12.5281	0.438840
$816$	0	0
$817$	9.94108	0.347794
$818$	0	0
$819$	34.0330	1.18921
$820$	0	0
$821$	24.8099	0.865871	0.432935	−	0.901425i	$-0.357478\pi$
$821$	24.8099	0.865871	0.432935	+	0.901425i	$0.357478\pi$
$822$	0	0
$823$	8.77313	0.305812	0.152906	−	0.988241i	$-0.451137\pi$
$823$	8.77313	0.305812	0.152906	+	0.988241i	$0.451137\pi$
$824$	0	0
$825$	−8.05946	−0.280594
$826$	0	0
$827$	−56.7440	−1.97318	−0.986592	−	0.163208i	$-0.947816\pi$
$827$	−56.7440	−1.97318	−0.986592	+	0.163208i	$0.947816\pi$
$828$	0	0
$829$	−6.30091	−0.218840	−0.109420	−	0.993996i	$-0.534899\pi$
$829$	−6.30091	−0.218840	−0.109420	+	0.993996i	$0.534899\pi$
$830$	0	0
$831$	0.246488	0.00855058
$832$	0	0
$833$	−133.134	−4.61282
$834$	0	0
$835$	7.69843	0.266415
$836$	0	0
$837$	−7.16139	−0.247534
$838$	0	0
$839$	−2.38808	−0.0824455	−0.0412228	−	0.999150i	$-0.513125\pi$
$839$	−2.38808	−0.0824455	−0.0412228	+	0.999150i	$0.513125\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	26.4451	0.910817
$844$	0	0
$845$	−27.8950	−0.959615
$846$	0	0
$847$	37.3528	1.28346
$848$	0	0
$849$	6.59503	0.226341
$850$	0	0
$851$	−6.94133	−0.237946
$852$	0	0
$853$	−14.6922	−0.503051	−0.251526	−	0.967851i	$-0.580932\pi$
$853$	−14.6922	−0.503051	−0.251526	+	0.967851i	$0.580932\pi$
$854$	0	0
$855$	−0.622073	−0.0212745
$856$	0	0
$857$	−4.72578	−0.161430	−0.0807148	−	0.996737i	$-0.525720\pi$
$857$	−4.72578	−0.161430	−0.0807148	+	0.996737i	$0.525720\pi$
$858$	0	0
$859$	−37.4858	−1.27900	−0.639500	−	0.768791i	$-0.720858\pi$
$859$	−37.4858	−1.27900	−0.639500	+	0.768791i	$0.720858\pi$
$860$	0	0
$861$	10.2149	0.348122
$862$	0	0
$863$	−32.5637	−1.10848	−0.554240	−	0.832357i	$-0.686991\pi$
$863$	−32.5637	−1.10848	−0.554240	+	0.832357i	$0.686991\pi$
$864$	0	0
$865$	10.3334	0.351344
$866$	0	0
$867$	41.1272	1.39675
$868$	0	0
$869$	28.6434	0.971660
$870$	0	0
$871$	68.1534	2.30929
$872$	0	0
$873$	−5.22122	−0.176711
$874$	0	0
$875$	−37.5176	−1.26833
$876$	0	0
$877$	44.0063	1.48599	0.742994	−	0.669298i	$-0.233405\pi$
$877$	44.0063	1.48599	0.742994	+	0.669298i	$0.233405\pi$
$878$	0	0
$879$	16.3626	0.551896
$880$	0	0
$881$	32.2383	1.08614	0.543069	−	0.839688i	$-0.317262\pi$
$881$	32.2383	1.08614	0.543069	+	0.839688i	$0.317262\pi$
$882$	0	0
$883$	28.1619	0.947724	0.473862	−	0.880599i	$-0.342860\pi$
$883$	28.1619	0.947724	0.473862	+	0.880599i	$0.342860\pi$
$884$	0	0
$885$	−4.94985	−0.166388
$886$	0	0
$887$	22.6636	0.760970	0.380485	−	0.924787i	$-0.375757\pi$
$887$	22.6636	0.760970	0.380485	+	0.924787i	$0.375757\pi$
$888$	0	0
$889$	−49.0199	−1.64408
$890$	0	0
$891$	1.85682	0.0622058
$892$	0	0
$893$	−2.32789	−0.0778997
$894$	0	0
$895$	−11.0001	−0.367694
$896$	0	0
$897$	−6.88102	−0.229751
$898$	0	0
$899$	7.16139	0.238846
$900$	0	0
$901$	−99.7510	−3.32319
$902$	0	0
$903$	−64.1887	−2.13607
$904$	0	0
$905$	15.4250	0.512745
$906$	0	0
$907$	17.9805	0.597033	0.298516	−	0.954404i	$-0.403508\pi$
$907$	17.9805	0.597033	0.298516	+	0.954404i	$0.403508\pi$
$908$	0	0
$909$	17.5973	0.583666
$910$	0	0
$911$	−50.4635	−1.67193	−0.835965	−	0.548783i	$-0.815091\pi$
$911$	−50.4635	−1.67193	−0.835965	+	0.548783i	$0.815091\pi$
$912$	0	0
$913$	1.33821	0.0442882
$914$	0	0
$915$	2.84165	0.0939421
$916$	0	0
$917$	48.9430	1.61624
$918$	0	0
$919$	39.9767	1.31871	0.659355	−	0.751832i	$-0.270830\pi$
$919$	39.9767	1.31871	0.659355	+	0.751832i	$0.270830\pi$
$920$	0	0
$921$	−17.5073	−0.576887
$922$	0	0
$923$	−95.9550	−3.15840
$924$	0	0
$925$	30.1286	0.990622
$926$	0	0
$927$	−9.42864	−0.309677
$928$	0	0
$929$	23.1825	0.760593	0.380297	−	0.924865i	$-0.375822\pi$
$929$	23.1825	0.760593	0.380297	+	0.924865i	$0.375822\pi$
$930$	0	0
$931$	13.3759	0.438377
$932$	0	0
$933$	−10.0872	−0.330240
$934$	0	0
$935$	11.4968	0.375986
$936$	0	0
$937$	−31.7331	−1.03668	−0.518338	−	0.855176i	$-0.673449\pi$
$937$	−31.7331	−1.03668	−0.518338	+	0.855176i	$0.673449\pi$
$938$	0	0
$939$	21.6111	0.705251
$940$	0	0
$941$	59.5878	1.94251	0.971254	−	0.238046i	$-0.0765069\pi$
$941$	59.5878	1.94251	0.971254	+	0.238046i	$0.0765069\pi$
$942$	0	0
$943$	−2.06531	−0.0672558
$944$	0	0
$945$	4.01668	0.130663
$946$	0	0
$947$	33.4587	1.08726	0.543631	−	0.839324i	$-0.317049\pi$
$947$	33.4587	1.08726	0.543631	+	0.839324i	$0.317049\pi$
$948$	0	0
$949$	24.9379	0.809517
$950$	0	0
$951$	−17.6316	−0.571743
$952$	0	0
$953$	15.4240	0.499632	0.249816	−	0.968293i	$-0.419630\pi$
$953$	15.4240	0.499632	0.249816	+	0.968293i	$0.419630\pi$
$954$	0	0
$955$	0.0202798	0.000656240 0
$956$	0	0
$957$	−1.85682	−0.0600224
$958$	0	0
$959$	−61.7817	−1.99503
$960$	0	0
$961$	20.2855	0.654370
$962$	0	0
$963$	6.07940	0.195906
$964$	0	0
$965$	−7.98413	−0.257018
$966$	0	0
$967$	−17.8301	−0.573376	−0.286688	−	0.958024i	$-0.592554\pi$
$967$	−17.8301	−0.573376	−0.286688	+	0.958024i	$0.592554\pi$
$968$	0	0
$969$	−5.83999	−0.187608
$970$	0	0
$971$	50.1093	1.60809	0.804043	−	0.594571i	$-0.202678\pi$
$971$	50.1093	1.60809	0.804043	+	0.594571i	$0.202678\pi$
$972$	0	0
$973$	−64.6202	−2.07163
$974$	0	0
$975$	29.8668	0.956504
$976$	0	0
$977$	−41.2024	−1.31818	−0.659090	−	0.752064i	$-0.729058\pi$
$977$	−41.2024	−1.31818	−0.659090	+	0.752064i	$0.729058\pi$
$978$	0	0
$979$	9.28114	0.296626
$980$	0	0
$981$	−17.4110	−0.555890
$982$	0	0
$983$	35.7778	1.14113	0.570567	−	0.821251i	$-0.306724\pi$
$983$	35.7778	1.14113	0.570567	+	0.821251i	$0.306724\pi$
$984$	0	0
$985$	−3.59715	−0.114615
$986$	0	0
$987$	15.0310	0.478441
$988$	0	0
$989$	12.9781	0.412679
$990$	0	0
$991$	−33.3072	−1.05804	−0.529019	−	0.848610i	$-0.677440\pi$
$991$	−33.3072	−1.05804	−0.529019	+	0.848610i	$0.677440\pi$
$992$	0	0
$993$	18.2302	0.578519
$994$	0	0
$995$	1.56827	0.0497175
$996$	0	0
$997$	12.0657	0.382123	0.191062	−	0.981578i	$-0.438807\pi$
$997$	12.0657	0.382123	0.191062	+	0.981578i	$0.438807\pi$
$998$	0	0
$999$	−6.94133	−0.219614

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.7	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.812118 q^{5} -4.94593 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.812118 q^{5} -4.94593 q^{7} +1.00000 q^{9} +1.85682 q^{11} -6.88102 q^{13} -0.812118 q^{15} -7.62412 q^{17} +0.765989 q^{19} -4.94593 q^{21} +1.00000 q^{23} -4.34046 q^{25} +1.00000 q^{27} -1.00000 q^{29} -7.16139 q^{31} +1.85682 q^{33} +4.01668 q^{35} -6.94133 q^{37} -6.88102 q^{39} -2.06531 q^{41} +12.9781 q^{43} -0.812118 q^{45} -3.03906 q^{47} +17.4622 q^{49} -7.62412 q^{51} +13.0836 q^{53} -1.50796 q^{55} +0.765989 q^{57} +6.09500 q^{59} -3.49906 q^{61} -4.94593 q^{63} +5.58820 q^{65} -9.90455 q^{67} +1.00000 q^{69} +13.9449 q^{71} -3.62415 q^{73} -4.34046 q^{75} -9.18370 q^{77} +15.4261 q^{79} +1.00000 q^{81} +0.720700 q^{83} +6.19168 q^{85} -1.00000 q^{87} +4.99841 q^{89} +34.0330 q^{91} -7.16139 q^{93} -0.622073 q^{95} -5.22122 q^{97} +1.85682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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

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