LMFDB - Embedded newform 8004.2.a.j.1.12

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			1.12
Root			$1.72426$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.72426 q^{5} +4.78449 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.72426 q^{5} +4.78449 q^{7} +1.00000 q^{9} -0.855035 q^{11} +0.789026 q^{13} +1.72426 q^{15} -0.187005 q^{17} +2.96694 q^{19} +4.78449 q^{21} +1.00000 q^{23} -2.02694 q^{25} +1.00000 q^{27} -1.00000 q^{29} +6.56119 q^{31} -0.855035 q^{33} +8.24969 q^{35} +6.29864 q^{37} +0.789026 q^{39} -5.32568 q^{41} +4.21930 q^{43} +1.72426 q^{45} -5.43940 q^{47} +15.8914 q^{49} -0.187005 q^{51} +0.912333 q^{53} -1.47430 q^{55} +2.96694 q^{57} -4.84785 q^{59} -7.62097 q^{61} +4.78449 q^{63} +1.36048 q^{65} -12.2144 q^{67} +1.00000 q^{69} -5.33636 q^{71} -1.62179 q^{73} -2.02694 q^{75} -4.09091 q^{77} +12.3349 q^{79} +1.00000 q^{81} +13.5924 q^{83} -0.322445 q^{85} -1.00000 q^{87} +8.62527 q^{89} +3.77509 q^{91} +6.56119 q^{93} +5.11577 q^{95} -6.23401 q^{97} -0.855035 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.72426	0.771111	0.385555	−	0.922685i	$-0.374010\pi$
$5$	1.72426	0.771111	0.385555	+	0.922685i	$0.374010\pi$
$6$	0	0
$7$	4.78449	1.80837	0.904184	−	0.427143i	$-0.140480\pi$
$7$	4.78449	1.80837	0.904184	+	0.427143i	$0.140480\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.855035	−0.257803	−0.128901	−	0.991657i	$-0.541145\pi$
$11$	−0.855035	−0.257803	−0.128901	+	0.991657i	$0.541145\pi$
$12$	0	0
$13$	0.789026	0.218836	0.109418	−	0.993996i	$-0.465101\pi$
$13$	0.789026	0.218836	0.109418	+	0.993996i	$0.465101\pi$
$14$	0	0
$15$	1.72426	0.445201
$16$	0	0
$17$	−0.187005	−0.0453555	−0.0226777	−	0.999743i	$-0.507219\pi$
$17$	−0.187005	−0.0453555	−0.0226777	+	0.999743i	$0.507219\pi$
$18$	0	0
$19$	2.96694	0.680663	0.340332	−	0.940305i	$-0.389461\pi$
$19$	2.96694	0.680663	0.340332	+	0.940305i	$0.389461\pi$
$20$	0	0
$21$	4.78449	1.04406
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.02694	−0.405388
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.56119	1.17842	0.589212	−	0.807978i	$-0.299438\pi$
$31$	6.56119	1.17842	0.589212	+	0.807978i	$0.299438\pi$
$32$	0	0
$33$	−0.855035	−0.148843
$34$	0	0
$35$	8.24969	1.39445
$36$	0	0
$37$	6.29864	1.03549	0.517745	−	0.855535i	$-0.326772\pi$
$37$	6.29864	1.03549	0.517745	+	0.855535i	$0.326772\pi$
$38$	0	0
$39$	0.789026	0.126345
$40$	0	0
$41$	−5.32568	−0.831732	−0.415866	−	0.909426i	$-0.636521\pi$
$41$	−5.32568	−0.831732	−0.415866	+	0.909426i	$0.636521\pi$
$42$	0	0
$43$	4.21930	0.643437	0.321718	−	0.946835i	$-0.395740\pi$
$43$	4.21930	0.643437	0.321718	+	0.946835i	$0.395740\pi$
$44$	0	0
$45$	1.72426	0.257037
$46$	0	0
$47$	−5.43940	−0.793418	−0.396709	−	0.917944i	$-0.629848\pi$
$47$	−5.43940	−0.793418	−0.396709	+	0.917944i	$0.629848\pi$
$48$	0	0
$49$	15.8914	2.27020
$50$	0	0
$51$	−0.187005	−0.0261860
$52$	0	0
$53$	0.912333	0.125319	0.0626593	−	0.998035i	$-0.480042\pi$
$53$	0.912333	0.125319	0.0626593	+	0.998035i	$0.480042\pi$
$54$	0	0
$55$	−1.47430	−0.198795
$56$	0	0
$57$	2.96694	0.392981
$58$	0	0
$59$	−4.84785	−0.631137	−0.315568	−	0.948903i	$-0.602195\pi$
$59$	−4.84785	−0.631137	−0.315568	+	0.948903i	$0.602195\pi$
$60$	0	0
$61$	−7.62097	−0.975765	−0.487883	−	0.872909i	$-0.662231\pi$
$61$	−7.62097	−0.975765	−0.487883	+	0.872909i	$0.662231\pi$
$62$	0	0
$63$	4.78449	0.602789
$64$	0	0
$65$	1.36048	0.168747
$66$	0	0
$67$	−12.2144	−1.49223	−0.746114	−	0.665818i	$-0.768083\pi$
$67$	−12.2144	−1.49223	−0.746114	+	0.665818i	$0.768083\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−5.33636	−0.633310	−0.316655	−	0.948541i	$-0.602560\pi$
$71$	−5.33636	−0.633310	−0.316655	+	0.948541i	$0.602560\pi$
$72$	0	0
$73$	−1.62179	−0.189816	−0.0949080	−	0.995486i	$-0.530256\pi$
$73$	−1.62179	−0.189816	−0.0949080	+	0.995486i	$0.530256\pi$
$74$	0	0
$75$	−2.02694	−0.234051
$76$	0	0
$77$	−4.09091	−0.466202
$78$	0	0
$79$	12.3349	1.38779	0.693894	−	0.720077i	$-0.255894\pi$
$79$	12.3349	1.38779	0.693894	+	0.720077i	$0.255894\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.5924	1.49196	0.745981	−	0.665967i	$-0.231981\pi$
$83$	13.5924	1.49196	0.745981	+	0.665967i	$0.231981\pi$
$84$	0	0
$85$	−0.322445	−0.0349741
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	8.62527	0.914277	0.457138	−	0.889396i	$-0.348874\pi$
$89$	8.62527	0.914277	0.457138	+	0.889396i	$0.348874\pi$
$90$	0	0
$91$	3.77509	0.395737
$92$	0	0
$93$	6.56119	0.680364
$94$	0	0
$95$	5.11577	0.524867
$96$	0	0
$97$	−6.23401	−0.632968	−0.316484	−	0.948598i	$-0.602502\pi$
$97$	−6.23401	−0.632968	−0.316484	+	0.948598i	$0.602502\pi$
$98$	0	0
$99$	−0.855035	−0.0859343
$100$	0	0
$101$	3.94404	0.392447	0.196224	−	0.980559i	$-0.437132\pi$
$101$	3.94404	0.392447	0.196224	+	0.980559i	$0.437132\pi$
$102$	0	0
$103$	12.7156	1.25290	0.626452	−	0.779460i	$-0.284507\pi$
$103$	12.7156	1.25290	0.626452	+	0.779460i	$0.284507\pi$
$104$	0	0
$105$	8.24969	0.805087
$106$	0	0
$107$	5.38985	0.521056	0.260528	−	0.965466i	$-0.416103\pi$
$107$	5.38985	0.521056	0.260528	+	0.965466i	$0.416103\pi$
$108$	0	0
$109$	12.3483	1.18275	0.591375	−	0.806397i	$-0.298585\pi$
$109$	12.3483	1.18275	0.591375	+	0.806397i	$0.298585\pi$
$110$	0	0
$111$	6.29864	0.597840
$112$	0	0
$113$	−9.88638	−0.930032	−0.465016	−	0.885302i	$-0.653951\pi$
$113$	−9.88638	−0.930032	−0.465016	+	0.885302i	$0.653951\pi$
$114$	0	0
$115$	1.72426	0.160788
$116$	0	0
$117$	0.789026	0.0729455
$118$	0	0
$119$	−0.894726	−0.0820194
$120$	0	0
$121$	−10.2689	−0.933538
$122$	0	0
$123$	−5.32568	−0.480200
$124$	0	0
$125$	−12.1162	−1.08371
$126$	0	0
$127$	−0.375952	−0.0333604	−0.0166802	−	0.999861i	$-0.505310\pi$
$127$	−0.375952	−0.0333604	−0.0166802	+	0.999861i	$0.505310\pi$
$128$	0	0
$129$	4.21930	0.371488
$130$	0	0
$131$	15.9504	1.39359	0.696795	−	0.717271i	$-0.254609\pi$
$131$	15.9504	1.39359	0.696795	+	0.717271i	$0.254609\pi$
$132$	0	0
$133$	14.1953	1.23089
$134$	0	0
$135$	1.72426	0.148400
$136$	0	0
$137$	−1.97401	−0.168651	−0.0843256	−	0.996438i	$-0.526874\pi$
$137$	−1.97401	−0.168651	−0.0843256	+	0.996438i	$0.526874\pi$
$138$	0	0
$139$	15.1152	1.28205	0.641027	−	0.767518i	$-0.278509\pi$
$139$	15.1152	1.28205	0.641027	+	0.767518i	$0.278509\pi$
$140$	0	0
$141$	−5.43940	−0.458080
$142$	0	0
$143$	−0.674645	−0.0564166
$144$	0	0
$145$	−1.72426	−0.143192
$146$	0	0
$147$	15.8914	1.31070
$148$	0	0
$149$	−9.27676	−0.759982	−0.379991	−	0.924990i	$-0.624073\pi$
$149$	−9.27676	−0.759982	−0.379991	+	0.924990i	$0.624073\pi$
$150$	0	0
$151$	3.16741	0.257760	0.128880	−	0.991660i	$-0.458862\pi$
$151$	3.16741	0.257760	0.128880	+	0.991660i	$0.458862\pi$
$152$	0	0
$153$	−0.187005	−0.0151185
$154$	0	0
$155$	11.3132	0.908696
$156$	0	0
$157$	3.77975	0.301657	0.150828	−	0.988560i	$-0.451806\pi$
$157$	3.77975	0.301657	0.150828	+	0.988560i	$0.451806\pi$
$158$	0	0
$159$	0.912333	0.0723527
$160$	0	0
$161$	4.78449	0.377071
$162$	0	0
$163$	−16.4883	−1.29146	−0.645732	−	0.763565i	$-0.723447\pi$
$163$	−16.4883	−1.29146	−0.645732	+	0.763565i	$0.723447\pi$
$164$	0	0
$165$	−1.47430	−0.114774
$166$	0	0
$167$	−4.69469	−0.363286	−0.181643	−	0.983365i	$-0.558141\pi$
$167$	−4.69469	−0.363286	−0.181643	+	0.983365i	$0.558141\pi$
$168$	0	0
$169$	−12.3774	−0.952111
$170$	0	0
$171$	2.96694	0.226888
$172$	0	0
$173$	9.84890	0.748798	0.374399	−	0.927268i	$-0.377849\pi$
$173$	9.84890	0.748798	0.374399	+	0.927268i	$0.377849\pi$
$174$	0	0
$175$	−9.69788	−0.733091
$176$	0	0
$177$	−4.84785	−0.364387
$178$	0	0
$179$	−20.7742	−1.55274	−0.776370	−	0.630278i	$-0.782941\pi$
$179$	−20.7742	−1.55274	−0.776370	+	0.630278i	$0.782941\pi$
$180$	0	0
$181$	−12.8267	−0.953403	−0.476701	−	0.879065i	$-0.658168\pi$
$181$	−12.8267	−0.953403	−0.476701	+	0.879065i	$0.658168\pi$
$182$	0	0
$183$	−7.62097	−0.563358
$184$	0	0
$185$	10.8605	0.798477
$186$	0	0
$187$	0.159896	0.0116928
$188$	0	0
$189$	4.78449	0.348021
$190$	0	0
$191$	4.62227	0.334456	0.167228	−	0.985918i	$-0.446518\pi$
$191$	4.62227	0.334456	0.167228	+	0.985918i	$0.446518\pi$
$192$	0	0
$193$	9.93990	0.715489	0.357745	−	0.933819i	$-0.383546\pi$
$193$	9.93990	0.715489	0.357745	+	0.933819i	$0.383546\pi$
$194$	0	0
$195$	1.36048	0.0974262
$196$	0	0
$197$	−0.958271	−0.0682740	−0.0341370	−	0.999417i	$-0.510868\pi$
$197$	−0.958271	−0.0682740	−0.0341370	+	0.999417i	$0.510868\pi$
$198$	0	0
$199$	6.03481	0.427797	0.213898	−	0.976856i	$-0.431384\pi$
$199$	6.03481	0.427797	0.213898	+	0.976856i	$0.431384\pi$
$200$	0	0
$201$	−12.2144	−0.861539
$202$	0	0
$203$	−4.78449	−0.335806
$204$	0	0
$205$	−9.18284	−0.641357
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−2.53684	−0.175477
$210$	0	0
$211$	−20.4935	−1.41083	−0.705414	−	0.708796i	$-0.749239\pi$
$211$	−20.4935	−1.41083	−0.705414	+	0.708796i	$0.749239\pi$
$212$	0	0
$213$	−5.33636	−0.365642
$214$	0	0
$215$	7.27515	0.496161
$216$	0	0
$217$	31.3920	2.13103
$218$	0	0
$219$	−1.62179	−0.109590
$220$	0	0
$221$	−0.147552	−0.00992542
$222$	0	0
$223$	−2.12672	−0.142415	−0.0712077	−	0.997462i	$-0.522685\pi$
$223$	−2.12672	−0.142415	−0.0712077	+	0.997462i	$0.522685\pi$
$224$	0	0
$225$	−2.02694	−0.135129
$226$	0	0
$227$	−27.8931	−1.85133	−0.925666	−	0.378341i	$-0.876495\pi$
$227$	−27.8931	−1.85133	−0.925666	+	0.378341i	$0.876495\pi$
$228$	0	0
$229$	19.5863	1.29430	0.647150	−	0.762363i	$-0.275961\pi$
$229$	19.5863	1.29430	0.647150	+	0.762363i	$0.275961\pi$
$230$	0	0
$231$	−4.09091	−0.269162
$232$	0	0
$233$	−18.6241	−1.22011	−0.610053	−	0.792360i	$-0.708852\pi$
$233$	−18.6241	−1.22011	−0.610053	+	0.792360i	$0.708852\pi$
$234$	0	0
$235$	−9.37892	−0.611814
$236$	0	0
$237$	12.3349	0.801240
$238$	0	0
$239$	−0.990350	−0.0640604	−0.0320302	−	0.999487i	$-0.510197\pi$
$239$	−0.990350	−0.0640604	−0.0320302	+	0.999487i	$0.510197\pi$
$240$	0	0
$241$	6.19931	0.399333	0.199666	−	0.979864i	$-0.436014\pi$
$241$	6.19931	0.399333	0.199666	+	0.979864i	$0.436014\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	27.4008	1.75057
$246$	0	0
$247$	2.34099	0.148954
$248$	0	0
$249$	13.5924	0.861385
$250$	0	0
$251$	−2.55696	−0.161394	−0.0806968	−	0.996739i	$-0.525715\pi$
$251$	−2.55696	−0.161394	−0.0806968	+	0.996739i	$0.525715\pi$
$252$	0	0
$253$	−0.855035	−0.0537556
$254$	0	0
$255$	−0.322445	−0.0201923
$256$	0	0
$257$	−3.28746	−0.205066	−0.102533	−	0.994730i	$-0.532695\pi$
$257$	−3.28746	−0.205066	−0.102533	+	0.994730i	$0.532695\pi$
$258$	0	0
$259$	30.1358	1.87255
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	6.29116	0.387930	0.193965	−	0.981008i	$-0.437865\pi$
$263$	6.29116	0.387930	0.193965	+	0.981008i	$0.437865\pi$
$264$	0	0
$265$	1.57310	0.0966345
$266$	0	0
$267$	8.62527	0.527858
$268$	0	0
$269$	−13.4159	−0.817983	−0.408991	−	0.912538i	$-0.634119\pi$
$269$	−13.4159	−0.817983	−0.408991	+	0.912538i	$0.634119\pi$
$270$	0	0
$271$	3.11664	0.189322	0.0946611	−	0.995510i	$-0.469823\pi$
$271$	3.11664	0.189322	0.0946611	+	0.995510i	$0.469823\pi$
$272$	0	0
$273$	3.77509	0.228479
$274$	0	0
$275$	1.73311	0.104510
$276$	0	0
$277$	28.7265	1.72601	0.863004	−	0.505197i	$-0.168580\pi$
$277$	28.7265	1.72601	0.863004	+	0.505197i	$0.168580\pi$
$278$	0	0
$279$	6.56119	0.392808
$280$	0	0
$281$	−0.0574064	−0.00342458	−0.00171229	−	0.999999i	$-0.500545\pi$
$281$	−0.0574064	−0.00342458	−0.00171229	+	0.999999i	$0.500545\pi$
$282$	0	0
$283$	−5.36549	−0.318945	−0.159473	−	0.987202i	$-0.550979\pi$
$283$	−5.36549	−0.318945	−0.159473	+	0.987202i	$0.550979\pi$
$284$	0	0
$285$	5.11577	0.303032
$286$	0	0
$287$	−25.4807	−1.50408
$288$	0	0
$289$	−16.9650	−0.997943
$290$	0	0
$291$	−6.23401	−0.365444
$292$	0	0
$293$	−8.53152	−0.498417	−0.249208	−	0.968450i	$-0.580170\pi$
$293$	−8.53152	−0.498417	−0.249208	+	0.968450i	$0.580170\pi$
$294$	0	0
$295$	−8.35894	−0.486676
$296$	0	0
$297$	−0.855035	−0.0496142
$298$	0	0
$299$	0.789026	0.0456305
$300$	0	0
$301$	20.1872	1.16357
$302$	0	0
$303$	3.94404	0.226579
$304$	0	0
$305$	−13.1405	−0.752423
$306$	0	0
$307$	−24.4870	−1.39755	−0.698774	−	0.715343i	$-0.746270\pi$
$307$	−24.4870	−1.39755	−0.698774	+	0.715343i	$0.746270\pi$
$308$	0	0
$309$	12.7156	0.723364
$310$	0	0
$311$	15.8095	0.896477	0.448238	−	0.893914i	$-0.352052\pi$
$311$	15.8095	0.896477	0.448238	+	0.893914i	$0.352052\pi$
$312$	0	0
$313$	−1.65880	−0.0937610	−0.0468805	−	0.998901i	$-0.514928\pi$
$313$	−1.65880	−0.0937610	−0.0468805	+	0.998901i	$0.514928\pi$
$314$	0	0
$315$	8.24969	0.464817
$316$	0	0
$317$	−19.0431	−1.06957	−0.534785	−	0.844988i	$-0.679607\pi$
$317$	−19.0431	−1.06957	−0.534785	+	0.844988i	$0.679607\pi$
$318$	0	0
$319$	0.855035	0.0478728
$320$	0	0
$321$	5.38985	0.300832
$322$	0	0
$323$	−0.554834	−0.0308718
$324$	0	0
$325$	−1.59931	−0.0887137
$326$	0	0
$327$	12.3483	0.682861
$328$	0	0
$329$	−26.0248	−1.43479
$330$	0	0
$331$	−10.8932	−0.598744	−0.299372	−	0.954136i	$-0.596777\pi$
$331$	−10.8932	−0.598744	−0.299372	+	0.954136i	$0.596777\pi$
$332$	0	0
$333$	6.29864	0.345163
$334$	0	0
$335$	−21.0608	−1.15067
$336$	0	0
$337$	24.5134	1.33533	0.667664	−	0.744463i	$-0.267294\pi$
$337$	24.5134	1.33533	0.667664	+	0.744463i	$0.267294\pi$
$338$	0	0
$339$	−9.88638	−0.536954
$340$	0	0
$341$	−5.61005	−0.303801
$342$	0	0
$343$	42.5407	2.29698
$344$	0	0
$345$	1.72426	0.0928308
$346$	0	0
$347$	5.01255	0.269088	0.134544	−	0.990908i	$-0.457043\pi$
$347$	5.01255	0.269088	0.134544	+	0.990908i	$0.457043\pi$
$348$	0	0
$349$	−26.8924	−1.43952	−0.719760	−	0.694223i	$-0.755748\pi$
$349$	−26.8924	−1.43952	−0.719760	+	0.694223i	$0.755748\pi$
$350$	0	0
$351$	0.789026	0.0421151
$352$	0	0
$353$	−6.79532	−0.361679	−0.180839	−	0.983513i	$-0.557881\pi$
$353$	−6.79532	−0.361679	−0.180839	+	0.983513i	$0.557881\pi$
$354$	0	0
$355$	−9.20126	−0.488352
$356$	0	0
$357$	−0.894726	−0.0473539
$358$	0	0
$359$	−3.11426	−0.164364	−0.0821822	−	0.996617i	$-0.526189\pi$
$359$	−3.11426	−0.164364	−0.0821822	+	0.996617i	$0.526189\pi$
$360$	0	0
$361$	−10.1973	−0.536698
$362$	0	0
$363$	−10.2689	−0.538978
$364$	0	0
$365$	−2.79638	−0.146369
$366$	0	0
$367$	−19.5113	−1.01848	−0.509242	−	0.860624i	$-0.670074\pi$
$367$	−19.5113	−1.01848	−0.509242	+	0.860624i	$0.670074\pi$
$368$	0	0
$369$	−5.32568	−0.277244
$370$	0	0
$371$	4.36505	0.226622
$372$	0	0
$373$	−15.1686	−0.785400	−0.392700	−	0.919667i	$-0.628459\pi$
$373$	−15.1686	−0.785400	−0.392700	+	0.919667i	$0.628459\pi$
$374$	0	0
$375$	−12.1162	−0.625680
$376$	0	0
$377$	−0.789026	−0.0406369
$378$	0	0
$379$	−15.3925	−0.790657	−0.395329	−	0.918540i	$-0.629369\pi$
$379$	−15.3925	−0.790657	−0.395329	+	0.918540i	$0.629369\pi$
$380$	0	0
$381$	−0.375952	−0.0192606
$382$	0	0
$383$	−13.6175	−0.695820	−0.347910	−	0.937528i	$-0.613109\pi$
$383$	−13.6175	−0.695820	−0.347910	+	0.937528i	$0.613109\pi$
$384$	0	0
$385$	−7.05378	−0.359494
$386$	0	0
$387$	4.21930	0.214479
$388$	0	0
$389$	32.8638	1.66626	0.833130	−	0.553077i	$-0.186547\pi$
$389$	32.8638	1.66626	0.833130	+	0.553077i	$0.186547\pi$
$390$	0	0
$391$	−0.187005	−0.00945727
$392$	0	0
$393$	15.9504	0.804589
$394$	0	0
$395$	21.2686	1.07014
$396$	0	0
$397$	−3.54562	−0.177949	−0.0889747	−	0.996034i	$-0.528359\pi$
$397$	−3.54562	−0.177949	−0.0889747	+	0.996034i	$0.528359\pi$
$398$	0	0
$399$	14.1953	0.710654
$400$	0	0
$401$	24.1091	1.20395	0.601975	−	0.798515i	$-0.294381\pi$
$401$	24.1091	1.20395	0.601975	+	0.798515i	$0.294381\pi$
$402$	0	0
$403$	5.17695	0.257882
$404$	0	0
$405$	1.72426	0.0856790
$406$	0	0
$407$	−5.38556	−0.266952
$408$	0	0
$409$	−9.70146	−0.479706	−0.239853	−	0.970809i	$-0.577099\pi$
$409$	−9.70146	−0.479706	−0.239853	+	0.970809i	$0.577099\pi$
$410$	0	0
$411$	−1.97401	−0.0973708
$412$	0	0
$413$	−23.1945	−1.14133
$414$	0	0
$415$	23.4368	1.15047
$416$	0	0
$417$	15.1152	0.740194
$418$	0	0
$419$	2.47064	0.120699	0.0603495	−	0.998177i	$-0.480778\pi$
$419$	2.47064	0.120699	0.0603495	+	0.998177i	$0.480778\pi$
$420$	0	0
$421$	33.7375	1.64426	0.822132	−	0.569297i	$-0.192785\pi$
$421$	33.7375	1.64426	0.822132	+	0.569297i	$0.192785\pi$
$422$	0	0
$423$	−5.43940	−0.264473
$424$	0	0
$425$	0.379049	0.0183866
$426$	0	0
$427$	−36.4625	−1.76454
$428$	0	0
$429$	−0.674645	−0.0325722
$430$	0	0
$431$	27.6550	1.33209	0.666047	−	0.745910i	$-0.267985\pi$
$431$	27.6550	1.33209	0.666047	+	0.745910i	$0.267985\pi$
$432$	0	0
$433$	11.3758	0.546688	0.273344	−	0.961916i	$-0.411870\pi$
$433$	11.3758	0.546688	0.273344	+	0.961916i	$0.411870\pi$
$434$	0	0
$435$	−1.72426	−0.0826718
$436$	0	0
$437$	2.96694	0.141928
$438$	0	0
$439$	−8.93034	−0.426222	−0.213111	−	0.977028i	$-0.568360\pi$
$439$	−8.93034	−0.426222	−0.213111	+	0.977028i	$0.568360\pi$
$440$	0	0
$441$	15.8914	0.756732
$442$	0	0
$443$	40.3086	1.91512	0.957560	−	0.288233i	$-0.0930678\pi$
$443$	40.3086	1.91512	0.957560	+	0.288233i	$0.0930678\pi$
$444$	0	0
$445$	14.8722	0.705009
$446$	0	0
$447$	−9.27676	−0.438776
$448$	0	0
$449$	−0.878885	−0.0414772	−0.0207386	−	0.999785i	$-0.506602\pi$
$449$	−0.878885	−0.0414772	−0.0207386	+	0.999785i	$0.506602\pi$
$450$	0	0
$451$	4.55364	0.214423
$452$	0	0
$453$	3.16741	0.148818
$454$	0	0
$455$	6.50922	0.305157
$456$	0	0
$457$	−6.28449	−0.293976	−0.146988	−	0.989138i	$-0.546958\pi$
$457$	−6.28449	−0.293976	−0.146988	+	0.989138i	$0.546958\pi$
$458$	0	0
$459$	−0.187005	−0.00872866
$460$	0	0
$461$	1.29120	0.0601373	0.0300686	−	0.999548i	$-0.490427\pi$
$461$	1.29120	0.0601373	0.0300686	+	0.999548i	$0.490427\pi$
$462$	0	0
$463$	−16.6541	−0.773981	−0.386990	−	0.922084i	$-0.626485\pi$
$463$	−16.6541	−0.773981	−0.386990	+	0.922084i	$0.626485\pi$
$464$	0	0
$465$	11.3132	0.524636
$466$	0	0
$467$	22.6947	1.05018	0.525092	−	0.851045i	$-0.324031\pi$
$467$	22.6947	1.05018	0.525092	+	0.851045i	$0.324031\pi$
$468$	0	0
$469$	−58.4398	−2.69850
$470$	0	0
$471$	3.77975	0.174162
$472$	0	0
$473$	−3.60765	−0.165880
$474$	0	0
$475$	−6.01381	−0.275933
$476$	0	0
$477$	0.912333	0.0417729
$478$	0	0
$479$	−37.6079	−1.71835	−0.859174	−	0.511684i	$-0.829022\pi$
$479$	−37.6079	−1.71835	−0.859174	+	0.511684i	$0.829022\pi$
$480$	0	0
$481$	4.96979	0.226603
$482$	0	0
$483$	4.78449	0.217702
$484$	0	0
$485$	−10.7490	−0.488089
$486$	0	0
$487$	38.8011	1.75824	0.879122	−	0.476597i	$-0.158130\pi$
$487$	38.8011	1.75824	0.879122	+	0.476597i	$0.158130\pi$
$488$	0	0
$489$	−16.4883	−0.745627
$490$	0	0
$491$	−13.5077	−0.609596	−0.304798	−	0.952417i	$-0.598589\pi$
$491$	−13.5077	−0.609596	−0.304798	+	0.952417i	$0.598589\pi$
$492$	0	0
$493$	0.187005	0.00842230
$494$	0	0
$495$	−1.47430	−0.0662649
$496$	0	0
$497$	−25.5318	−1.14526
$498$	0	0
$499$	−13.2207	−0.591838	−0.295919	−	0.955213i	$-0.595626\pi$
$499$	−13.2207	−0.591838	−0.295919	+	0.955213i	$0.595626\pi$
$500$	0	0
$501$	−4.69469	−0.209743
$502$	0	0
$503$	30.6579	1.36697	0.683485	−	0.729965i	$-0.260464\pi$
$503$	30.6579	1.36697	0.683485	+	0.729965i	$0.260464\pi$
$504$	0	0
$505$	6.80054	0.302620
$506$	0	0
$507$	−12.3774	−0.549701
$508$	0	0
$509$	21.3888	0.948044	0.474022	−	0.880513i	$-0.342802\pi$
$509$	21.3888	0.948044	0.474022	+	0.880513i	$0.342802\pi$
$510$	0	0
$511$	−7.75943	−0.343257
$512$	0	0
$513$	2.96694	0.130994
$514$	0	0
$515$	21.9249	0.966127
$516$	0	0
$517$	4.65088	0.204546
$518$	0	0
$519$	9.84890	0.432319
$520$	0	0
$521$	−10.4546	−0.458024	−0.229012	−	0.973424i	$-0.573549\pi$
$521$	−10.4546	−0.458024	−0.229012	+	0.973424i	$0.573549\pi$
$522$	0	0
$523$	22.7567	0.995082	0.497541	−	0.867440i	$-0.334236\pi$
$523$	22.7567	0.995082	0.497541	+	0.867440i	$0.334236\pi$
$524$	0	0
$525$	−9.69788	−0.423250
$526$	0	0
$527$	−1.22698	−0.0534480
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.84785	−0.210379
$532$	0	0
$533$	−4.20210	−0.182013
$534$	0	0
$535$	9.29348	0.401792
$536$	0	0
$537$	−20.7742	−0.896475
$538$	0	0
$539$	−13.5877	−0.585263
$540$	0	0
$541$	10.2618	0.441188	0.220594	−	0.975366i	$-0.429200\pi$
$541$	10.2618	0.441188	0.220594	+	0.975366i	$0.429200\pi$
$542$	0	0
$543$	−12.8267	−0.550447
$544$	0	0
$545$	21.2916	0.912031
$546$	0	0
$547$	30.4075	1.30013	0.650064	−	0.759879i	$-0.274742\pi$
$547$	30.4075	1.30013	0.650064	+	0.759879i	$0.274742\pi$
$548$	0	0
$549$	−7.62097	−0.325255
$550$	0	0
$551$	−2.96694	−0.126396
$552$	0	0
$553$	59.0164	2.50963
$554$	0	0
$555$	10.8605	0.461001
$556$	0	0
$557$	18.4155	0.780288	0.390144	−	0.920754i	$-0.372425\pi$
$557$	18.4155	0.780288	0.390144	+	0.920754i	$0.372425\pi$
$558$	0	0
$559$	3.32913	0.140807
$560$	0	0
$561$	0.159896	0.00675082
$562$	0	0
$563$	35.3898	1.49150	0.745752	−	0.666224i	$-0.232090\pi$
$563$	35.3898	1.49150	0.745752	+	0.666224i	$0.232090\pi$
$564$	0	0
$565$	−17.0467	−0.717158
$566$	0	0
$567$	4.78449	0.200930
$568$	0	0
$569$	−8.54102	−0.358058	−0.179029	−	0.983844i	$-0.557296\pi$
$569$	−8.54102	−0.358058	−0.179029	+	0.983844i	$0.557296\pi$
$570$	0	0
$571$	−27.8102	−1.16382	−0.581910	−	0.813253i	$-0.697694\pi$
$571$	−27.8102	−1.16382	−0.581910	+	0.813253i	$0.697694\pi$
$572$	0	0
$573$	4.62227	0.193098
$574$	0	0
$575$	−2.02694	−0.0845292
$576$	0	0
$577$	−11.2146	−0.466869	−0.233435	−	0.972372i	$-0.574996\pi$
$577$	−11.2146	−0.466869	−0.233435	+	0.972372i	$0.574996\pi$
$578$	0	0
$579$	9.93990	0.413088
$580$	0	0
$581$	65.0328	2.69802
$582$	0	0
$583$	−0.780077	−0.0323075
$584$	0	0
$585$	1.36048	0.0562490
$586$	0	0
$587$	−34.7689	−1.43506	−0.717532	−	0.696525i	$-0.754728\pi$
$587$	−34.7689	−1.43506	−0.717532	+	0.696525i	$0.754728\pi$
$588$	0	0
$589$	19.4667	0.802110
$590$	0	0
$591$	−0.958271	−0.0394180
$592$	0	0
$593$	−46.3761	−1.90444	−0.952220	−	0.305414i	$-0.901205\pi$
$593$	−46.3761	−1.90444	−0.952220	+	0.305414i	$0.901205\pi$
$594$	0	0
$595$	−1.54274	−0.0632460
$596$	0	0
$597$	6.03481	0.246988
$598$	0	0
$599$	24.1121	0.985192	0.492596	−	0.870258i	$-0.336048\pi$
$599$	24.1121	0.985192	0.492596	+	0.870258i	$0.336048\pi$
$600$	0	0
$601$	11.4601	0.467467	0.233733	−	0.972301i	$-0.424906\pi$
$601$	11.4601	0.467467	0.233733	+	0.972301i	$0.424906\pi$
$602$	0	0
$603$	−12.2144	−0.497410
$604$	0	0
$605$	−17.7062	−0.719861
$606$	0	0
$607$	20.7087	0.840540	0.420270	−	0.907399i	$-0.361935\pi$
$607$	20.7087	0.840540	0.420270	+	0.907399i	$0.361935\pi$
$608$	0	0
$609$	−4.78449	−0.193877
$610$	0	0
$611$	−4.29183	−0.173629
$612$	0	0
$613$	9.31303	0.376150	0.188075	−	0.982155i	$-0.439775\pi$
$613$	9.31303	0.376150	0.188075	+	0.982155i	$0.439775\pi$
$614$	0	0
$615$	−9.18284	−0.370288
$616$	0	0
$617$	25.2876	1.01804	0.509021	−	0.860754i	$-0.330008\pi$
$617$	25.2876	1.01804	0.509021	+	0.860754i	$0.330008\pi$
$618$	0	0
$619$	−18.2419	−0.733202	−0.366601	−	0.930378i	$-0.619479\pi$
$619$	−18.2419	−0.733202	−0.366601	+	0.930378i	$0.619479\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	41.2675	1.65335
$624$	0	0
$625$	−10.7568	−0.430273
$626$	0	0
$627$	−2.53684	−0.101312
$628$	0	0
$629$	−1.17788	−0.0469651
$630$	0	0
$631$	−4.39633	−0.175015	−0.0875075	−	0.996164i	$-0.527890\pi$
$631$	−4.39633	−0.175015	−0.0875075	+	0.996164i	$0.527890\pi$
$632$	0	0
$633$	−20.4935	−0.814542
$634$	0	0
$635$	−0.648238	−0.0257245
$636$	0	0
$637$	12.5387	0.496801
$638$	0	0
$639$	−5.33636	−0.211103
$640$	0	0
$641$	−47.4049	−1.87238	−0.936190	−	0.351495i	$-0.885674\pi$
$641$	−47.4049	−1.87238	−0.936190	+	0.351495i	$0.885674\pi$
$642$	0	0
$643$	−24.9620	−0.984407	−0.492203	−	0.870480i	$-0.663808\pi$
$643$	−24.9620	−0.984407	−0.492203	+	0.870480i	$0.663808\pi$
$644$	0	0
$645$	7.27515	0.286459
$646$	0	0
$647$	39.9094	1.56900	0.784501	−	0.620128i	$-0.212919\pi$
$647$	39.9094	1.56900	0.784501	+	0.620128i	$0.212919\pi$
$648$	0	0
$649$	4.14509	0.162709
$650$	0	0
$651$	31.3920	1.23035
$652$	0	0
$653$	−37.5504	−1.46946	−0.734731	−	0.678359i	$-0.762692\pi$
$653$	−37.5504	−1.46946	−0.734731	+	0.678359i	$0.762692\pi$
$654$	0	0
$655$	27.5025	1.07461
$656$	0	0
$657$	−1.62179	−0.0632720
$658$	0	0
$659$	8.63524	0.336381	0.168191	−	0.985755i	$-0.446208\pi$
$659$	8.63524	0.336381	0.168191	+	0.985755i	$0.446208\pi$
$660$	0	0
$661$	37.8152	1.47084	0.735421	−	0.677610i	$-0.236984\pi$
$661$	37.8152	1.47084	0.735421	+	0.677610i	$0.236984\pi$
$662$	0	0
$663$	−0.147552	−0.00573045
$664$	0	0
$665$	24.4764	0.949152
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−2.12672	−0.0822236
$670$	0	0
$671$	6.51620	0.251555
$672$	0	0
$673$	0.773706	0.0298242	0.0149121	−	0.999889i	$-0.495253\pi$
$673$	0.773706	0.0298242	0.0149121	+	0.999889i	$0.495253\pi$
$674$	0	0
$675$	−2.02694	−0.0780170
$676$	0	0
$677$	26.1809	1.00621	0.503106	−	0.864225i	$-0.332191\pi$
$677$	26.1809	1.00621	0.503106	+	0.864225i	$0.332191\pi$
$678$	0	0
$679$	−29.8266	−1.14464
$680$	0	0
$681$	−27.8931	−1.06887
$682$	0	0
$683$	−30.0959	−1.15159	−0.575793	−	0.817595i	$-0.695307\pi$
$683$	−30.0959	−1.15159	−0.575793	+	0.817595i	$0.695307\pi$
$684$	0	0
$685$	−3.40370	−0.130049
$686$	0	0
$687$	19.5863	0.747264
$688$	0	0
$689$	0.719855	0.0274243
$690$	0	0
$691$	27.1835	1.03411	0.517054	−	0.855953i	$-0.327029\pi$
$691$	27.1835	1.03411	0.517054	+	0.855953i	$0.327029\pi$
$692$	0	0
$693$	−4.09091	−0.155401
$694$	0	0
$695$	26.0625	0.988606
$696$	0	0
$697$	0.995931	0.0377236
$698$	0	0
$699$	−18.6241	−0.704429
$700$	0	0
$701$	11.4391	0.432049	0.216025	−	0.976388i	$-0.430691\pi$
$701$	11.4391	0.432049	0.216025	+	0.976388i	$0.430691\pi$
$702$	0	0
$703$	18.6877	0.704819
$704$	0	0
$705$	−9.37892	−0.353231
$706$	0	0
$707$	18.8703	0.709689
$708$	0	0
$709$	23.3696	0.877665	0.438833	−	0.898569i	$-0.355392\pi$
$709$	23.3696	0.877665	0.438833	+	0.898569i	$0.355392\pi$
$710$	0	0
$711$	12.3349	0.462596
$712$	0	0
$713$	6.56119	0.245719
$714$	0	0
$715$	−1.16326	−0.0435035
$716$	0	0
$717$	−0.990350	−0.0369853
$718$	0	0
$719$	−29.9344	−1.11637	−0.558183	−	0.829718i	$-0.688501\pi$
$719$	−29.9344	−1.11637	−0.558183	+	0.829718i	$0.688501\pi$
$720$	0	0
$721$	60.8376	2.26571
$722$	0	0
$723$	6.19931	0.230555
$724$	0	0
$725$	2.02694	0.0752787
$726$	0	0
$727$	−18.1607	−0.673542	−0.336771	−	0.941587i	$-0.609335\pi$
$727$	−18.1607	−0.673542	−0.336771	+	0.941587i	$0.609335\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.789031	−0.0291834
$732$	0	0
$733$	−20.1354	−0.743718	−0.371859	−	0.928289i	$-0.621280\pi$
$733$	−20.1354	−0.743718	−0.371859	+	0.928289i	$0.621280\pi$
$734$	0	0
$735$	27.4008	1.01069
$736$	0	0
$737$	10.4438	0.384701
$738$	0	0
$739$	−41.6587	−1.53244	−0.766220	−	0.642578i	$-0.777865\pi$
$739$	−41.6587	−1.53244	−0.766220	+	0.642578i	$0.777865\pi$
$740$	0	0
$741$	2.34099	0.0859986
$742$	0	0
$743$	−31.6571	−1.16138	−0.580692	−	0.814123i	$-0.697218\pi$
$743$	−31.6571	−1.16138	−0.580692	+	0.814123i	$0.697218\pi$
$744$	0	0
$745$	−15.9955	−0.586030
$746$	0	0
$747$	13.5924	0.497321
$748$	0	0
$749$	25.7877	0.942262
$750$	0	0
$751$	−1.73648	−0.0633652	−0.0316826	−	0.999498i	$-0.510087\pi$
$751$	−1.73648	−0.0633652	−0.0316826	+	0.999498i	$0.510087\pi$
$752$	0	0
$753$	−2.55696	−0.0931807
$754$	0	0
$755$	5.46142	0.198761
$756$	0	0
$757$	46.3917	1.68613	0.843067	−	0.537809i	$-0.180748\pi$
$757$	46.3917	1.68613	0.843067	+	0.537809i	$0.180748\pi$
$758$	0	0
$759$	−0.855035	−0.0310358
$760$	0	0
$761$	−0.568486	−0.0206076	−0.0103038	−	0.999947i	$-0.503280\pi$
$761$	−0.568486	−0.0206076	−0.0103038	+	0.999947i	$0.503280\pi$
$762$	0	0
$763$	59.0802	2.13885
$764$	0	0
$765$	−0.322445	−0.0116580
$766$	0	0
$767$	−3.82508	−0.138116
$768$	0	0
$769$	2.39867	0.0864984	0.0432492	−	0.999064i	$-0.486229\pi$
$769$	2.39867	0.0864984	0.0432492	+	0.999064i	$0.486229\pi$
$770$	0	0
$771$	−3.28746	−0.118395
$772$	0	0
$773$	12.1579	0.437289	0.218645	−	0.975805i	$-0.429836\pi$
$773$	12.1579	0.437289	0.218645	+	0.975805i	$0.429836\pi$
$774$	0	0
$775$	−13.2991	−0.477719
$776$	0	0
$777$	30.1358	1.08111
$778$	0	0
$779$	−15.8010	−0.566129
$780$	0	0
$781$	4.56278	0.163269
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	6.51725	0.232611
$786$	0	0
$787$	20.6856	0.737362	0.368681	−	0.929556i	$-0.379809\pi$
$787$	20.6856	0.737362	0.368681	+	0.929556i	$0.379809\pi$
$788$	0	0
$789$	6.29116	0.223971
$790$	0	0
$791$	−47.3013	−1.68184
$792$	0	0
$793$	−6.01314	−0.213533
$794$	0	0
$795$	1.57310	0.0557920
$796$	0	0
$797$	−42.1025	−1.49135	−0.745673	−	0.666312i	$-0.767872\pi$
$797$	−42.1025	−1.49135	−0.745673	+	0.666312i	$0.767872\pi$
$798$	0	0
$799$	1.01720	0.0359859
$800$	0	0
$801$	8.62527	0.304759
$802$	0	0
$803$	1.38669	0.0489351
$804$	0	0
$805$	8.24969	0.290763
$806$	0	0
$807$	−13.4159	−0.472262
$808$	0	0
$809$	−1.00091	−0.0351902	−0.0175951	−	0.999845i	$-0.505601\pi$
$809$	−1.00091	−0.0351902	−0.0175951	+	0.999845i	$0.505601\pi$
$810$	0	0
$811$	16.0430	0.563345	0.281673	−	0.959511i	$-0.409111\pi$
$811$	16.0430	0.563345	0.281673	+	0.959511i	$0.409111\pi$
$812$	0	0
$813$	3.11664	0.109305
$814$	0	0
$815$	−28.4301	−0.995861
$816$	0	0
$817$	12.5184	0.437964
$818$	0	0
$819$	3.77509	0.131912
$820$	0	0
$821$	28.6972	1.00154	0.500769	−	0.865581i	$-0.333051\pi$
$821$	28.6972	1.00154	0.500769	+	0.865581i	$0.333051\pi$
$822$	0	0
$823$	−44.7762	−1.56080	−0.780400	−	0.625280i	$-0.784985\pi$
$823$	−44.7762	−1.56080	−0.780400	+	0.625280i	$0.784985\pi$
$824$	0	0
$825$	1.73311	0.0603390
$826$	0	0
$827$	2.69922	0.0938611	0.0469305	−	0.998898i	$-0.485056\pi$
$827$	2.69922	0.0938611	0.0469305	+	0.998898i	$0.485056\pi$
$828$	0	0
$829$	49.0230	1.70264	0.851319	−	0.524648i	$-0.175803\pi$
$829$	49.0230	1.70264	0.851319	+	0.524648i	$0.175803\pi$
$830$	0	0
$831$	28.7265	0.996511
$832$	0	0
$833$	−2.97177	−0.102966
$834$	0	0
$835$	−8.09484	−0.280134
$836$	0	0
$837$	6.56119	0.226788
$838$	0	0
$839$	−29.9215	−1.03300	−0.516502	−	0.856286i	$-0.672766\pi$
$839$	−29.9215	−1.03300	−0.516502	+	0.856286i	$0.672766\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−0.0574064	−0.00197718
$844$	0	0
$845$	−21.3419	−0.734183
$846$	0	0
$847$	−49.1315	−1.68818
$848$	0	0
$849$	−5.36549	−0.184143
$850$	0	0
$851$	6.29864	0.215914
$852$	0	0
$853$	9.79614	0.335414	0.167707	−	0.985837i	$-0.446364\pi$
$853$	9.79614	0.335414	0.167707	+	0.985837i	$0.446364\pi$
$854$	0	0
$855$	5.11577	0.174956
$856$	0	0
$857$	−38.1628	−1.30362	−0.651809	−	0.758383i	$-0.725990\pi$
$857$	−38.1628	−1.30362	−0.651809	+	0.758383i	$0.725990\pi$
$858$	0	0
$859$	−10.8970	−0.371801	−0.185900	−	0.982569i	$-0.559520\pi$
$859$	−10.8970	−0.371801	−0.185900	+	0.982569i	$0.559520\pi$
$860$	0	0
$861$	−25.4807	−0.868379
$862$	0	0
$863$	−33.0375	−1.12461	−0.562304	−	0.826930i	$-0.690085\pi$
$863$	−33.0375	−1.12461	−0.562304	+	0.826930i	$0.690085\pi$
$864$	0	0
$865$	16.9820	0.577406
$866$	0	0
$867$	−16.9650	−0.576163
$868$	0	0
$869$	−10.5468	−0.357776
$870$	0	0
$871$	−9.63749	−0.326554
$872$	0	0
$873$	−6.23401	−0.210989
$874$	0	0
$875$	−57.9701	−1.95975
$876$	0	0
$877$	−14.4941	−0.489431	−0.244716	−	0.969595i	$-0.578695\pi$
$877$	−14.4941	−0.489431	−0.244716	+	0.969595i	$0.578695\pi$
$878$	0	0
$879$	−8.53152	−0.287761
$880$	0	0
$881$	33.5872	1.13158	0.565791	−	0.824549i	$-0.308571\pi$
$881$	33.5872	1.13158	0.565791	+	0.824549i	$0.308571\pi$
$882$	0	0
$883$	−6.58717	−0.221676	−0.110838	−	0.993839i	$-0.535353\pi$
$883$	−6.58717	−0.221676	−0.110838	+	0.993839i	$0.535353\pi$
$884$	0	0
$885$	−8.35894	−0.280983
$886$	0	0
$887$	43.3703	1.45623	0.728116	−	0.685454i	$-0.240396\pi$
$887$	43.3703	1.45623	0.728116	+	0.685454i	$0.240396\pi$
$888$	0	0
$889$	−1.79874	−0.0603278
$890$	0	0
$891$	−0.855035	−0.0286448
$892$	0	0
$893$	−16.1384	−0.540051
$894$	0	0
$895$	−35.8201	−1.19733
$896$	0	0
$897$	0.789026	0.0263448
$898$	0	0
$899$	−6.56119	−0.218828
$900$	0	0
$901$	−0.170611	−0.00568388
$902$	0	0
$903$	20.1872	0.671788
$904$	0	0
$905$	−22.1166	−0.735179
$906$	0	0
$907$	12.5959	0.418240	0.209120	−	0.977890i	$-0.432940\pi$
$907$	12.5959	0.418240	0.209120	+	0.977890i	$0.432940\pi$
$908$	0	0
$909$	3.94404	0.130816
$910$	0	0
$911$	−39.4043	−1.30552	−0.652762	−	0.757563i	$-0.726390\pi$
$911$	−39.4043	−1.30552	−0.652762	+	0.757563i	$0.726390\pi$
$912$	0	0
$913$	−11.6220	−0.384632
$914$	0	0
$915$	−13.1405	−0.434412
$916$	0	0
$917$	76.3144	2.52012
$918$	0	0
$919$	−44.8385	−1.47909	−0.739544	−	0.673109i	$-0.764959\pi$
$919$	−44.8385	−1.47909	−0.739544	+	0.673109i	$0.764959\pi$
$920$	0	0
$921$	−24.4870	−0.806874
$922$	0	0
$923$	−4.21053	−0.138591
$924$	0	0
$925$	−12.7670	−0.419775
$926$	0	0
$927$	12.7156	0.417634
$928$	0	0
$929$	−54.9624	−1.80326	−0.901630	−	0.432509i	$-0.857628\pi$
$929$	−54.9624	−1.80326	−0.901630	+	0.432509i	$0.857628\pi$
$930$	0	0
$931$	47.1488	1.54524
$932$	0	0
$933$	15.8095	0.517581
$934$	0	0
$935$	0.275702	0.00901642
$936$	0	0
$937$	−26.8384	−0.876771	−0.438385	−	0.898787i	$-0.644449\pi$
$937$	−26.8384	−0.876771	−0.438385	+	0.898787i	$0.644449\pi$
$938$	0	0
$939$	−1.65880	−0.0541330
$940$	0	0
$941$	9.20204	0.299978	0.149989	−	0.988688i	$-0.452076\pi$
$941$	9.20204	0.299978	0.149989	+	0.988688i	$0.452076\pi$
$942$	0	0
$943$	−5.32568	−0.173428
$944$	0	0
$945$	8.24969	0.268362
$946$	0	0
$947$	−27.3168	−0.887677	−0.443838	−	0.896107i	$-0.646384\pi$
$947$	−27.3168	−0.887677	−0.443838	+	0.896107i	$0.646384\pi$
$948$	0	0
$949$	−1.27963	−0.0415386
$950$	0	0
$951$	−19.0431	−0.617516
$952$	0	0
$953$	−15.6897	−0.508240	−0.254120	−	0.967173i	$-0.581786\pi$
$953$	−15.6897	−0.508240	−0.254120	+	0.967173i	$0.581786\pi$
$954$	0	0
$955$	7.96999	0.257903
$956$	0	0
$957$	0.855035	0.0276394
$958$	0	0
$959$	−9.44464	−0.304983
$960$	0	0
$961$	12.0492	0.388684
$962$	0	0
$963$	5.38985	0.173685
$964$	0	0
$965$	17.1389	0.551722
$966$	0	0
$967$	−42.0218	−1.35133	−0.675665	−	0.737209i	$-0.736143\pi$
$967$	−42.0218	−1.35133	−0.675665	+	0.737209i	$0.736143\pi$
$968$	0	0
$969$	−0.554834	−0.0178238
$970$	0	0
$971$	−37.8204	−1.21371	−0.606857	−	0.794811i	$-0.707570\pi$
$971$	−37.8204	−1.21371	−0.606857	+	0.794811i	$0.707570\pi$
$972$	0	0
$973$	72.3185	2.31843
$974$	0	0
$975$	−1.59931	−0.0512189
$976$	0	0
$977$	−52.9911	−1.69534	−0.847668	−	0.530526i	$-0.821994\pi$
$977$	−52.9911	−1.69534	−0.847668	+	0.530526i	$0.821994\pi$
$978$	0	0
$979$	−7.37491	−0.235703
$980$	0	0
$981$	12.3483	0.394250
$982$	0	0
$983$	−15.8026	−0.504026	−0.252013	−	0.967724i	$-0.581093\pi$
$983$	−15.8026	−0.504026	−0.252013	+	0.967724i	$0.581093\pi$
$984$	0	0
$985$	−1.65230	−0.0526468
$986$	0	0
$987$	−26.0248	−0.828378
$988$	0	0
$989$	4.21930	0.134166
$990$	0	0
$991$	−27.4667	−0.872508	−0.436254	−	0.899824i	$-0.643695\pi$
$991$	−27.4667	−0.872508	−0.436254	+	0.899824i	$0.643695\pi$
$992$	0	0
$993$	−10.8932	−0.345685
$994$	0	0
$995$	10.4056	0.329879
$996$	0	0
$997$	1.35585	0.0429403	0.0214702	−	0.999769i	$-0.493165\pi$
$997$	1.35585	0.0429403	0.0214702	+	0.999769i	$0.493165\pi$
$998$	0	0
$999$	6.29864	0.199280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.12	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.12	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.72426 q^{5} +4.78449 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.72426 q^{5} +4.78449 q^{7} +1.00000 q^{9} -0.855035 q^{11} +0.789026 q^{13} +1.72426 q^{15} -0.187005 q^{17} +2.96694 q^{19} +4.78449 q^{21} +1.00000 q^{23} -2.02694 q^{25} +1.00000 q^{27} -1.00000 q^{29} +6.56119 q^{31} -0.855035 q^{33} +8.24969 q^{35} +6.29864 q^{37} +0.789026 q^{39} -5.32568 q^{41} +4.21930 q^{43} +1.72426 q^{45} -5.43940 q^{47} +15.8914 q^{49} -0.187005 q^{51} +0.912333 q^{53} -1.47430 q^{55} +2.96694 q^{57} -4.84785 q^{59} -7.62097 q^{61} +4.78449 q^{63} +1.36048 q^{65} -12.2144 q^{67} +1.00000 q^{69} -5.33636 q^{71} -1.62179 q^{73} -2.02694 q^{75} -4.09091 q^{77} +12.3349 q^{79} +1.00000 q^{81} +13.5924 q^{83} -0.322445 q^{85} -1.00000 q^{87} +8.62527 q^{89} +3.77509 q^{91} +6.56119 q^{93} +5.11577 q^{95} -6.23401 q^{97} -0.855035 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 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

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Database

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