LMFDB - Embedded newform 8006.2.a.d.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.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:	$63.9282318582$
Analytic rank:	$0$
Dimension:	$98$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.77826 q^{3} +1.00000 q^{4} -3.83157 q^{5} -2.77826 q^{6} +3.52049 q^{7} +1.00000 q^{8} +4.71872 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.77826 q^{3} +1.00000 q^{4} -3.83157 q^{5} -2.77826 q^{6} +3.52049 q^{7} +1.00000 q^{8} +4.71872 q^{9} -3.83157 q^{10} -4.49157 q^{11} -2.77826 q^{12} +6.53575 q^{13} +3.52049 q^{14} +10.6451 q^{15} +1.00000 q^{16} +1.41779 q^{17} +4.71872 q^{18} +4.92967 q^{19} -3.83157 q^{20} -9.78082 q^{21} -4.49157 q^{22} +0.679532 q^{23} -2.77826 q^{24} +9.68091 q^{25} +6.53575 q^{26} -4.77505 q^{27} +3.52049 q^{28} +0.715370 q^{29} +10.6451 q^{30} +1.71760 q^{31} +1.00000 q^{32} +12.4787 q^{33} +1.41779 q^{34} -13.4890 q^{35} +4.71872 q^{36} -1.88775 q^{37} +4.92967 q^{38} -18.1580 q^{39} -3.83157 q^{40} +2.06679 q^{41} -9.78082 q^{42} +5.02792 q^{43} -4.49157 q^{44} -18.0801 q^{45} +0.679532 q^{46} -1.41315 q^{47} -2.77826 q^{48} +5.39383 q^{49} +9.68091 q^{50} -3.93900 q^{51} +6.53575 q^{52} +5.51140 q^{53} -4.77505 q^{54} +17.2098 q^{55} +3.52049 q^{56} -13.6959 q^{57} +0.715370 q^{58} -8.96428 q^{59} +10.6451 q^{60} -12.2543 q^{61} +1.71760 q^{62} +16.6122 q^{63} +1.00000 q^{64} -25.0422 q^{65} +12.4787 q^{66} +0.576951 q^{67} +1.41779 q^{68} -1.88791 q^{69} -13.4890 q^{70} -7.31228 q^{71} +4.71872 q^{72} +3.26706 q^{73} -1.88775 q^{74} -26.8961 q^{75} +4.92967 q^{76} -15.8125 q^{77} -18.1580 q^{78} +10.2176 q^{79} -3.83157 q^{80} -0.889847 q^{81} +2.06679 q^{82} +10.5719 q^{83} -9.78082 q^{84} -5.43238 q^{85} +5.02792 q^{86} -1.98748 q^{87} -4.49157 q^{88} +2.91906 q^{89} -18.0801 q^{90} +23.0090 q^{91} +0.679532 q^{92} -4.77193 q^{93} -1.41315 q^{94} -18.8884 q^{95} -2.77826 q^{96} -5.86059 q^{97} +5.39383 q^{98} -21.1945 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9}+O(q^{10}) $ 98 * q + 98 * q^2 + 16 * q^3 + 98 * q^4 + 4 * q^5 + 16 * q^6 + 29 * q^7 + 98 * q^8 + 130 * q^9	\( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9} + 4 q^{10} + 51 q^{11} + 16 q^{12} + 31 q^{13} + 29 q^{14} + 57 q^{15} + 98 q^{16} + 35 q^{17} + 130 q^{18} + 77 q^{19} + 4 q^{20} + 46 q^{21} + 51 q^{22} + 73 q^{23} + 16 q^{24} + 150 q^{25} + 31 q^{26} + 52 q^{27} + 29 q^{28} + 20 q^{29} + 57 q^{30} + 59 q^{31} + 98 q^{32} + 27 q^{33} + 35 q^{34} + 48 q^{35} + 130 q^{36} + 41 q^{37} + 77 q^{38} + 64 q^{39} + 4 q^{40} + 29 q^{41} + 46 q^{42} + 94 q^{43} + 51 q^{44} - 3 q^{45} + 73 q^{46} + 58 q^{47} + 16 q^{48} + 149 q^{49} + 150 q^{50} + 58 q^{51} + 31 q^{52} - 11 q^{53} + 52 q^{54} + 56 q^{55} + 29 q^{56} + 64 q^{57} + 20 q^{58} + 45 q^{59} + 57 q^{60} + 73 q^{61} + 59 q^{62} + 53 q^{63} + 98 q^{64} + 39 q^{65} + 27 q^{66} + 133 q^{67} + 35 q^{68} + 13 q^{69} + 48 q^{70} + 67 q^{71} + 130 q^{72} + 42 q^{73} + 41 q^{74} + 36 q^{75} + 77 q^{76} - 25 q^{77} + 64 q^{78} + 154 q^{79} + 4 q^{80} + 198 q^{81} + 29 q^{82} + 69 q^{83} + 46 q^{84} + 81 q^{85} + 94 q^{86} + 25 q^{87} + 51 q^{88} + 32 q^{89} - 3 q^{90} + 95 q^{91} + 73 q^{92} - 23 q^{93} + 58 q^{94} + 50 q^{95} + 16 q^{96} + 76 q^{97} + 149 q^{98} + 149 q^{99}+O(q^{100}) \) 98 * q + 98 * q^2 + 16 * q^3 + 98 * q^4 + 4 * q^5 + 16 * q^6 + 29 * q^7 + 98 * q^8 + 130 * q^9 + 4 * q^10 + 51 * q^11 + 16 * q^12 + 31 * q^13 + 29 * q^14 + 57 * q^15 + 98 * q^16 + 35 * q^17 + 130 * q^18 + 77 * q^19 + 4 * q^20 + 46 * q^21 + 51 * q^22 + 73 * q^23 + 16 * q^24 + 150 * q^25 + 31 * q^26 + 52 * q^27 + 29 * q^28 + 20 * q^29 + 57 * q^30 + 59 * q^31 + 98 * q^32 + 27 * q^33 + 35 * q^34 + 48 * q^35 + 130 * q^36 + 41 * q^37 + 77 * q^38 + 64 * q^39 + 4 * q^40 + 29 * q^41 + 46 * q^42 + 94 * q^43 + 51 * q^44 - 3 * q^45 + 73 * q^46 + 58 * q^47 + 16 * q^48 + 149 * q^49 + 150 * q^50 + 58 * q^51 + 31 * q^52 - 11 * q^53 + 52 * q^54 + 56 * q^55 + 29 * q^56 + 64 * q^57 + 20 * q^58 + 45 * q^59 + 57 * q^60 + 73 * q^61 + 59 * q^62 + 53 * q^63 + 98 * q^64 + 39 * q^65 + 27 * q^66 + 133 * q^67 + 35 * q^68 + 13 * q^69 + 48 * q^70 + 67 * q^71 + 130 * q^72 + 42 * q^73 + 41 * q^74 + 36 * q^75 + 77 * q^76 - 25 * q^77 + 64 * q^78 + 154 * q^79 + 4 * q^80 + 198 * q^81 + 29 * q^82 + 69 * q^83 + 46 * q^84 + 81 * q^85 + 94 * q^86 + 25 * q^87 + 51 * q^88 + 32 * q^89 - 3 * q^90 + 95 * q^91 + 73 * q^92 - 23 * q^93 + 58 * q^94 + 50 * q^95 + 16 * q^96 + 76 * q^97 + 149 * q^98 + 149 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.77826	−1.60403	−0.802014	−	0.597305i	$-0.796238\pi$
$3$	−2.77826	−1.60403	−0.802014	+	0.597305i	$0.796238\pi$
$4$	1.00000	0.500000
$5$	−3.83157	−1.71353	−0.856764	−	0.515708i	$-0.827529\pi$
$5$	−3.83157	−1.71353	−0.856764	+	0.515708i	$0.827529\pi$
$6$	−2.77826	−1.13422
$7$	3.52049	1.33062	0.665309	−	0.746568i	$-0.268300\pi$
$7$	3.52049	1.33062	0.665309	+	0.746568i	$0.268300\pi$
$8$	1.00000	0.353553
$9$	4.71872	1.57291
$10$	−3.83157	−1.21165
$11$	−4.49157	−1.35426	−0.677130	−	0.735863i	$-0.736777\pi$
$11$	−4.49157	−1.35426	−0.677130	+	0.735863i	$0.736777\pi$
$12$	−2.77826	−0.802014
$13$	6.53575	1.81269	0.906346	−	0.422537i	$-0.138860\pi$
$13$	6.53575	1.81269	0.906346	+	0.422537i	$0.138860\pi$
$14$	3.52049	0.940890
$15$	10.6451	2.74855
$16$	1.00000	0.250000
$17$	1.41779	0.343866	0.171933	−	0.985109i	$-0.444999\pi$
$17$	1.41779	0.343866	0.171933	+	0.985109i	$0.444999\pi$
$18$	4.71872	1.11221
$19$	4.92967	1.13094	0.565472	−	0.824767i	$-0.308694\pi$
$19$	4.92967	1.13094	0.565472	+	0.824767i	$0.308694\pi$
$20$	−3.83157	−0.856764
$21$	−9.78082	−2.13435
$22$	−4.49157	−0.957607
$23$	0.679532	0.141692	0.0708461	−	0.997487i	$-0.477430\pi$
$23$	0.679532	0.141692	0.0708461	+	0.997487i	$0.477430\pi$
$24$	−2.77826	−0.567110
$25$	9.68091	1.93618
$26$	6.53575	1.28177
$27$	−4.77505	−0.918958
$28$	3.52049	0.665309
$29$	0.715370	0.132841	0.0664204	−	0.997792i	$-0.478842\pi$
$29$	0.715370	0.132841	0.0664204	+	0.997792i	$0.478842\pi$
$30$	10.6451	1.94352
$31$	1.71760	0.308490	0.154245	−	0.988033i	$-0.450706\pi$
$31$	1.71760	0.308490	0.154245	+	0.988033i	$0.450706\pi$
$32$	1.00000	0.176777
$33$	12.4787	2.17227
$34$	1.41779	0.243150
$35$	−13.4890	−2.28005
$36$	4.71872	0.786453
$37$	−1.88775	−0.310345	−0.155172	−	0.987887i	$-0.549593\pi$
$37$	−1.88775	−0.310345	−0.155172	+	0.987887i	$0.549593\pi$
$38$	4.92967	0.799699
$39$	−18.1580	−2.90761
$40$	−3.83157	−0.605824
$41$	2.06679	0.322778	0.161389	−	0.986891i	$-0.448403\pi$
$41$	2.06679	0.322778	0.161389	+	0.986891i	$0.448403\pi$
$42$	−9.78082	−1.50921
$43$	5.02792	0.766751	0.383375	−	0.923593i	$-0.374762\pi$
$43$	5.02792	0.766751	0.383375	+	0.923593i	$0.374762\pi$
$44$	−4.49157	−0.677130
$45$	−18.0801	−2.69522
$46$	0.679532	0.100191
$47$	−1.41315	−0.206129	−0.103064	−	0.994675i	$-0.532865\pi$
$47$	−1.41315	−0.206129	−0.103064	+	0.994675i	$0.532865\pi$
$48$	−2.77826	−0.401007
$49$	5.39383	0.770547
$50$	9.68091	1.36909
$51$	−3.93900	−0.551570
$52$	6.53575	0.906346
$53$	5.51140	0.757049	0.378525	−	0.925591i	$-0.376432\pi$
$53$	5.51140	0.757049	0.378525	+	0.925591i	$0.376432\pi$
$54$	−4.77505	−0.649801
$55$	17.2098	2.32056
$56$	3.52049	0.470445
$57$	−13.6959	−1.81407
$58$	0.715370	0.0939326
$59$	−8.96428	−1.16705	−0.583525	−	0.812095i	$-0.698327\pi$
$59$	−8.96428	−1.16705	−0.583525	+	0.812095i	$0.698327\pi$
$60$	10.6451	1.37427
$61$	−12.2543	−1.56900	−0.784502	−	0.620127i	$-0.787081\pi$
$61$	−12.2543	−1.56900	−0.784502	+	0.620127i	$0.787081\pi$
$62$	1.71760	0.218135
$63$	16.6122	2.09294
$64$	1.00000	0.125000
$65$	−25.0422	−3.10610
$66$	12.4787	1.53603
$67$	0.576951	0.0704858	0.0352429	−	0.999379i	$-0.488780\pi$
$67$	0.576951	0.0704858	0.0352429	+	0.999379i	$0.488780\pi$
$68$	1.41779	0.171933
$69$	−1.88791	−0.227278
$70$	−13.4890	−1.61224
$71$	−7.31228	−0.867808	−0.433904	−	0.900959i	$-0.642864\pi$
$71$	−7.31228	−0.867808	−0.433904	+	0.900959i	$0.642864\pi$
$72$	4.71872	0.556106
$73$	3.26706	0.382380	0.191190	−	0.981553i	$-0.438765\pi$
$73$	3.26706	0.382380	0.191190	+	0.981553i	$0.438765\pi$
$74$	−1.88775	−0.219447
$75$	−26.8961	−3.10569
$76$	4.92967	0.565472
$77$	−15.8125	−1.80200
$78$	−18.1580	−2.05599
$79$	10.2176	1.14958	0.574788	−	0.818303i	$-0.305085\pi$
$79$	10.2176	1.14958	0.574788	+	0.818303i	$0.305085\pi$
$80$	−3.83157	−0.428382
$81$	−0.889847	−0.0988719
$82$	2.06679	0.228239
$83$	10.5719	1.16042	0.580209	−	0.814467i	$-0.302971\pi$
$83$	10.5719	1.16042	0.580209	+	0.814467i	$0.302971\pi$
$84$	−9.78082	−1.06718
$85$	−5.43238	−0.589224
$86$	5.02792	0.542175
$87$	−1.98748	−0.213080
$88$	−4.49157	−0.478803
$89$	2.91906	0.309419	0.154710	−	0.987960i	$-0.450556\pi$
$89$	2.91906	0.309419	0.154710	+	0.987960i	$0.450556\pi$
$90$	−18.0801	−1.90581
$91$	23.0090	2.41200
$92$	0.679532	0.0708461
$93$	−4.77193	−0.494826
$94$	−1.41315	−0.145755
$95$	−18.8884	−1.93791
$96$	−2.77826	−0.283555
$97$	−5.86059	−0.595053	−0.297526	−	0.954714i	$-0.596162\pi$
$97$	−5.86059	−0.595053	−0.297526	+	0.954714i	$0.596162\pi$
$98$	5.39383	0.544859
$99$	−21.1945	−2.13012
$100$	9.68091	0.968091
$101$	−15.7693	−1.56910	−0.784550	−	0.620066i	$-0.787106\pi$
$101$	−15.7693	−1.56910	−0.784550	+	0.620066i	$0.787106\pi$
$102$	−3.93900	−0.390019
$103$	16.9092	1.66611	0.833057	−	0.553188i	$-0.186589\pi$
$103$	16.9092	1.66611	0.833057	+	0.553188i	$0.186589\pi$
$104$	6.53575	0.640883
$105$	37.4759	3.65727
$106$	5.51140	0.535315
$107$	6.85390	0.662592	0.331296	−	0.943527i	$-0.392514\pi$
$107$	6.85390	0.662592	0.331296	+	0.943527i	$0.392514\pi$
$108$	−4.77505	−0.459479
$109$	−18.8369	−1.80425	−0.902126	−	0.431473i	$-0.857994\pi$
$109$	−18.8369	−1.80425	−0.902126	+	0.431473i	$0.857994\pi$
$110$	17.2098	1.64089
$111$	5.24467	0.497802
$112$	3.52049	0.332655
$113$	9.37838	0.882243	0.441122	−	0.897447i	$-0.354581\pi$
$113$	9.37838	0.882243	0.441122	+	0.897447i	$0.354581\pi$
$114$	−13.6959	−1.28274
$115$	−2.60367	−0.242794
$116$	0.715370	0.0664204
$117$	30.8404	2.85119
$118$	−8.96428	−0.825229
$119$	4.99133	0.457554
$120$	10.6451	0.971759
$121$	9.17423	0.834021
$122$	−12.2543	−1.10945
$123$	−5.74207	−0.517745
$124$	1.71760	0.154245
$125$	−17.9352	−1.60417
$126$	16.6122	1.47993
$127$	−14.0405	−1.24589	−0.622947	−	0.782264i	$-0.714065\pi$
$127$	−14.0405	−1.24589	−0.622947	+	0.782264i	$0.714065\pi$
$128$	1.00000	0.0883883
$129$	−13.9689	−1.22989
$130$	−25.0422	−2.19634
$131$	−5.50485	−0.480961	−0.240480	−	0.970654i	$-0.577305\pi$
$131$	−5.50485	−0.480961	−0.240480	+	0.970654i	$0.577305\pi$
$132$	12.4787	1.08614
$133$	17.3548	1.50486
$134$	0.576951	0.0498410
$135$	18.2959	1.57466
$136$	1.41779	0.121575
$137$	8.03982	0.686888	0.343444	−	0.939173i	$-0.388406\pi$
$137$	8.03982	0.686888	0.343444	+	0.939173i	$0.388406\pi$
$138$	−1.88791	−0.160710
$139$	5.14706	0.436568	0.218284	−	0.975885i	$-0.429954\pi$
$139$	5.14706	0.436568	0.218284	+	0.975885i	$0.429954\pi$
$140$	−13.4890	−1.14003
$141$	3.92609	0.330637
$142$	−7.31228	−0.613633
$143$	−29.3558	−2.45486
$144$	4.71872	0.393227
$145$	−2.74099	−0.227627
$146$	3.26706	0.270384
$147$	−14.9854	−1.23598
$148$	−1.88775	−0.155172
$149$	4.15925	0.340739	0.170369	−	0.985380i	$-0.445504\pi$
$149$	4.15925	0.340739	0.170369	+	0.985380i	$0.445504\pi$
$150$	−26.8961	−2.19605
$151$	−11.7420	−0.955554	−0.477777	−	0.878481i	$-0.658557\pi$
$151$	−11.7420	−0.955554	−0.477777	+	0.878481i	$0.658557\pi$
$152$	4.92967	0.399849
$153$	6.69017	0.540869
$154$	−15.8125	−1.27421
$155$	−6.58110	−0.528606
$156$	−18.1580	−1.45380
$157$	−2.26730	−0.180950	−0.0904750	−	0.995899i	$-0.528839\pi$
$157$	−2.26730	−0.180950	−0.0904750	+	0.995899i	$0.528839\pi$
$158$	10.2176	0.812872
$159$	−15.3121	−1.21433
$160$	−3.83157	−0.302912
$161$	2.39228	0.188538
$162$	−0.889847	−0.0699130
$163$	7.17441	0.561943	0.280971	−	0.959716i	$-0.409343\pi$
$163$	7.17441	0.561943	0.280971	+	0.959716i	$0.409343\pi$
$164$	2.06679	0.161389
$165$	−47.8132	−3.72225
$166$	10.5719	0.820540
$167$	−19.3529	−1.49757	−0.748787	−	0.662811i	$-0.769363\pi$
$167$	−19.3529	−1.49757	−0.748787	+	0.662811i	$0.769363\pi$
$168$	−9.78082	−0.754607
$169$	29.7160	2.28585
$170$	−5.43238	−0.416644
$171$	23.2617	1.77887
$172$	5.02792	0.383375
$173$	18.6098	1.41488	0.707439	−	0.706775i	$-0.249851\pi$
$173$	18.6098	1.41488	0.707439	+	0.706775i	$0.249851\pi$
$174$	−1.98748	−0.150671
$175$	34.0815	2.57632
$176$	−4.49157	−0.338565
$177$	24.9051	1.87198
$178$	2.91906	0.218793
$179$	13.6879	1.02308	0.511542	−	0.859259i	$-0.329075\pi$
$179$	13.6879	1.02308	0.511542	+	0.859259i	$0.329075\pi$
$180$	−18.0801	−1.34761
$181$	0.578873	0.0430273	0.0215136	−	0.999769i	$-0.493151\pi$
$181$	0.578873	0.0430273	0.0215136	+	0.999769i	$0.493151\pi$
$182$	23.0090	1.70554
$183$	34.0456	2.51673
$184$	0.679532	0.0500957
$185$	7.23305	0.531785
$186$	−4.77193	−0.349895
$187$	−6.36813	−0.465684
$188$	−1.41315	−0.103064
$189$	−16.8105	−1.22278
$190$	−18.8884	−1.37031
$191$	16.9577	1.22701	0.613507	−	0.789689i	$-0.289758\pi$
$191$	16.9577	1.22701	0.613507	+	0.789689i	$0.289758\pi$
$192$	−2.77826	−0.200504
$193$	7.85041	0.565085	0.282543	−	0.959255i	$-0.408822\pi$
$193$	7.85041	0.565085	0.282543	+	0.959255i	$0.408822\pi$
$194$	−5.86059	−0.420766
$195$	69.5736	4.98227
$196$	5.39383	0.385273
$197$	−19.3794	−1.38072	−0.690362	−	0.723464i	$-0.742549\pi$
$197$	−19.3794	−1.38072	−0.690362	+	0.723464i	$0.742549\pi$
$198$	−21.1945	−1.50623
$199$	3.16798	0.224572	0.112286	−	0.993676i	$-0.464183\pi$
$199$	3.16798	0.224572	0.112286	+	0.993676i	$0.464183\pi$
$200$	9.68091	0.684544
$201$	−1.60292	−0.113061
$202$	−15.7693	−1.10952
$203$	2.51845	0.176760
$204$	−3.93900	−0.275785
$205$	−7.91904	−0.553090
$206$	16.9092	1.17812
$207$	3.20652	0.222869
$208$	6.53575	0.453173
$209$	−22.1420	−1.53159
$210$	37.4759	2.58608
$211$	19.5385	1.34509	0.672543	−	0.740058i	$-0.265202\pi$
$211$	19.5385	1.34509	0.672543	+	0.740058i	$0.265202\pi$
$212$	5.51140	0.378525
$213$	20.3154	1.39199
$214$	6.85390	0.468523
$215$	−19.2648	−1.31385
$216$	−4.77505	−0.324901
$217$	6.04678	0.410482
$218$	−18.8369	−1.27580
$219$	−9.07673	−0.613349
$220$	17.2098	1.16028
$221$	9.26635	0.623322
$222$	5.24467	0.351999
$223$	8.86229	0.593463	0.296731	−	0.954961i	$-0.404103\pi$
$223$	8.86229	0.593463	0.296731	+	0.954961i	$0.404103\pi$
$224$	3.52049	0.235222
$225$	45.6815	3.04543
$226$	9.37838	0.623840
$227$	−8.32613	−0.552625	−0.276312	−	0.961068i	$-0.589112\pi$
$227$	−8.32613	−0.552625	−0.276312	+	0.961068i	$0.589112\pi$
$228$	−13.6959	−0.907033
$229$	−17.1982	−1.13649	−0.568244	−	0.822860i	$-0.692377\pi$
$229$	−17.1982	−1.13649	−0.568244	+	0.822860i	$0.692377\pi$
$230$	−2.60367	−0.171681
$231$	43.9313	2.89047
$232$	0.715370	0.0469663
$233$	28.2405	1.85010	0.925049	−	0.379847i	$-0.124023\pi$
$233$	28.2405	1.85010	0.925049	+	0.379847i	$0.124023\pi$
$234$	30.8404	2.01610
$235$	5.41458	0.353208
$236$	−8.96428	−0.583525
$237$	−28.3873	−1.84395
$238$	4.99133	0.323540
$239$	3.83111	0.247814	0.123907	−	0.992294i	$-0.460458\pi$
$239$	3.83111	0.247814	0.123907	+	0.992294i	$0.460458\pi$
$240$	10.6451	0.687137
$241$	−21.6612	−1.39532	−0.697662	−	0.716427i	$-0.745776\pi$
$241$	−21.6612	−1.39532	−0.697662	+	0.716427i	$0.745776\pi$
$242$	9.17423	0.589742
$243$	16.7974	1.07755
$244$	−12.2543	−0.784502
$245$	−20.6668	−1.32035
$246$	−5.74207	−0.366101
$247$	32.2191	2.05005
$248$	1.71760	0.109068
$249$	−29.3715	−1.86134
$250$	−17.9352	−1.13432
$251$	−24.0307	−1.51681	−0.758404	−	0.651785i	$-0.774020\pi$
$251$	−24.0307	−1.51681	−0.758404	+	0.651785i	$0.774020\pi$
$252$	16.6122	1.04647
$253$	−3.05217	−0.191888
$254$	−14.0405	−0.880981
$255$	15.0925	0.945132
$256$	1.00000	0.0625000
$257$	4.30685	0.268654	0.134327	−	0.990937i	$-0.457113\pi$
$257$	4.30685	0.268654	0.134327	+	0.990937i	$0.457113\pi$
$258$	−13.9689	−0.869663
$259$	−6.64581	−0.412951
$260$	−25.0422	−1.55305
$261$	3.37563	0.208946
$262$	−5.50485	−0.340091
$263$	2.69551	0.166212	0.0831062	−	0.996541i	$-0.473516\pi$
$263$	2.69551	0.166212	0.0831062	+	0.996541i	$0.473516\pi$
$264$	12.4787	0.768014
$265$	−21.1173	−1.29723
$266$	17.3548	1.06409
$267$	−8.10989	−0.496317
$268$	0.576951	0.0352429
$269$	22.3241	1.36113	0.680563	−	0.732689i	$-0.261735\pi$
$269$	22.3241	1.36113	0.680563	+	0.732689i	$0.261735\pi$
$270$	18.2959	1.11345
$271$	−1.22733	−0.0745549	−0.0372774	−	0.999305i	$-0.511869\pi$
$271$	−1.22733	−0.0745549	−0.0372774	+	0.999305i	$0.511869\pi$
$272$	1.41779	0.0859664
$273$	−63.9250	−3.86892
$274$	8.03982	0.485703
$275$	−43.4825	−2.62209
$276$	−1.88791	−0.113639
$277$	4.64649	0.279180	0.139590	−	0.990209i	$-0.455421\pi$
$277$	4.64649	0.279180	0.139590	+	0.990209i	$0.455421\pi$
$278$	5.14706	0.308700
$279$	8.10487	0.485226
$280$	−13.4890	−0.806121
$281$	3.21494	0.191787	0.0958935	−	0.995392i	$-0.469429\pi$
$281$	3.21494	0.191787	0.0958935	+	0.995392i	$0.469429\pi$
$282$	3.92609	0.233795
$283$	−20.0524	−1.19199	−0.595997	−	0.802987i	$-0.703243\pi$
$283$	−20.0524	−1.19199	−0.595997	+	0.802987i	$0.703243\pi$
$284$	−7.31228	−0.433904
$285$	52.4768	3.10846
$286$	−29.3558	−1.73585
$287$	7.27610	0.429495
$288$	4.71872	0.278053
$289$	−14.9899	−0.881756
$290$	−2.74099	−0.160956
$291$	16.2822	0.954481
$292$	3.26706	0.191190
$293$	−7.80987	−0.456258	−0.228129	−	0.973631i	$-0.573261\pi$
$293$	−7.80987	−0.456258	−0.228129	+	0.973631i	$0.573261\pi$
$294$	−14.9854	−0.873969
$295$	34.3473	1.99977
$296$	−1.88775	−0.109723
$297$	21.4475	1.24451
$298$	4.15925	0.240939
$299$	4.44125	0.256844
$300$	−26.8961	−1.55284
$301$	17.7007	1.02025
$302$	−11.7420	−0.675679
$303$	43.8111	2.51688
$304$	4.92967	0.282736
$305$	46.9532	2.68853
$306$	6.69017	0.382452
$307$	17.0379	0.972402	0.486201	−	0.873847i	$-0.338382\pi$
$307$	17.0379	0.972402	0.486201	+	0.873847i	$0.338382\pi$
$308$	−15.8125	−0.901002
$309$	−46.9781	−2.67249
$310$	−6.58110	−0.373781
$311$	19.1412	1.08540	0.542700	−	0.839926i	$-0.317402\pi$
$311$	19.1412	1.08540	0.542700	+	0.839926i	$0.317402\pi$
$312$	−18.1580	−1.02799
$313$	18.7723	1.06108	0.530538	−	0.847661i	$-0.321990\pi$
$313$	18.7723	1.06108	0.530538	+	0.847661i	$0.321990\pi$
$314$	−2.26730	−0.127951
$315$	−63.6507	−3.58631
$316$	10.2176	0.574788
$317$	28.9614	1.62664	0.813318	−	0.581819i	$-0.197659\pi$
$317$	28.9614	1.62664	0.813318	+	0.581819i	$0.197659\pi$
$318$	−15.3121	−0.858660
$319$	−3.21313	−0.179901
$320$	−3.83157	−0.214191
$321$	−19.0419	−1.06282
$322$	2.39228	0.133317
$323$	6.98926	0.388893
$324$	−0.889847	−0.0494360
$325$	63.2720	3.50970
$326$	7.17441	0.397354
$327$	52.3339	2.89407
$328$	2.06679	0.114119
$329$	−4.97497	−0.274279
$330$	−47.8132	−2.63203
$331$	−7.57847	−0.416550	−0.208275	−	0.978070i	$-0.566785\pi$
$331$	−7.57847	−0.416550	−0.208275	+	0.978070i	$0.566785\pi$
$332$	10.5719	0.580209
$333$	−8.90778	−0.488143
$334$	−19.3529	−1.05894
$335$	−2.21063	−0.120779
$336$	−9.78082	−0.533588
$337$	9.05186	0.493086	0.246543	−	0.969132i	$-0.420705\pi$
$337$	9.05186	0.493086	0.246543	+	0.969132i	$0.420705\pi$
$338$	29.7160	1.61634
$339$	−26.0556	−1.41514
$340$	−5.43238	−0.294612
$341$	−7.71472	−0.417776
$342$	23.2617	1.25785
$343$	−5.65451	−0.305315
$344$	5.02792	0.271087
$345$	7.23367	0.389448
$346$	18.6098	1.00047
$347$	6.19268	0.332440	0.166220	−	0.986089i	$-0.446844\pi$
$347$	6.19268	0.332440	0.166220	+	0.986089i	$0.446844\pi$
$348$	−1.98748	−0.106540
$349$	5.97823	0.320007	0.160004	−	0.987116i	$-0.448849\pi$
$349$	5.97823	0.320007	0.160004	+	0.987116i	$0.448849\pi$
$350$	34.0815	1.82173
$351$	−31.2085	−1.66579
$352$	−4.49157	−0.239402
$353$	31.1991	1.66056	0.830281	−	0.557345i	$-0.188180\pi$
$353$	31.1991	1.66056	0.830281	+	0.557345i	$0.188180\pi$
$354$	24.9051	1.32369
$355$	28.0175	1.48701
$356$	2.91906	0.154710
$357$	−13.8672	−0.733930
$358$	13.6879	0.723429
$359$	32.2822	1.70379	0.851894	−	0.523714i	$-0.175454\pi$
$359$	32.2822	1.70379	0.851894	+	0.523714i	$0.175454\pi$
$360$	−18.0801	−0.952904
$361$	5.30168	0.279036
$362$	0.578873	0.0304249
$363$	−25.4884	−1.33779
$364$	23.0090	1.20600
$365$	−12.5180	−0.655220
$366$	34.0456	1.77959
$367$	−7.83315	−0.408887	−0.204444	−	0.978878i	$-0.565538\pi$
$367$	−7.83315	−0.408887	−0.204444	+	0.978878i	$0.565538\pi$
$368$	0.679532	0.0354230
$369$	9.75259	0.507700
$370$	7.23305	0.376029
$371$	19.4028	1.00734
$372$	−4.77193	−0.247413
$373$	−4.28501	−0.221870	−0.110935	−	0.993828i	$-0.535384\pi$
$373$	−4.28501	−0.221870	−0.110935	+	0.993828i	$0.535384\pi$
$374$	−6.36813	−0.329288
$375$	49.8286	2.57314
$376$	−1.41315	−0.0728776
$377$	4.67548	0.240799
$378$	−16.8105	−0.864638
$379$	22.2418	1.14249	0.571243	−	0.820781i	$-0.306461\pi$
$379$	22.2418	1.14249	0.571243	+	0.820781i	$0.306461\pi$
$380$	−18.8884	−0.968953
$381$	39.0082	1.99845
$382$	16.9577	0.867631
$383$	16.9226	0.864703	0.432351	−	0.901705i	$-0.357684\pi$
$383$	16.9226	0.864703	0.432351	+	0.901705i	$0.357684\pi$
$384$	−2.77826	−0.141777
$385$	60.5867	3.08779
$386$	7.85041	0.399576
$387$	23.7253	1.20603
$388$	−5.86059	−0.297526
$389$	−28.2582	−1.43275	−0.716373	−	0.697717i	$-0.754199\pi$
$389$	−28.2582	−1.43275	−0.716373	+	0.697717i	$0.754199\pi$
$390$	69.5736	3.52300
$391$	0.963436	0.0487231
$392$	5.39383	0.272429
$393$	15.2939	0.771475
$394$	−19.3794	−0.976320
$395$	−39.1496	−1.96983
$396$	−21.1945	−1.06506
$397$	−28.1809	−1.41436	−0.707180	−	0.707034i	$-0.750033\pi$
$397$	−28.1809	−1.41436	−0.707180	+	0.707034i	$0.750033\pi$
$398$	3.16798	0.158797
$399$	−48.2163	−2.41383
$400$	9.68091	0.484045
$401$	32.6462	1.63027	0.815137	−	0.579268i	$-0.196662\pi$
$401$	32.6462	1.63027	0.815137	+	0.579268i	$0.196662\pi$
$402$	−1.60292	−0.0799463
$403$	11.2258	0.559197
$404$	−15.7693	−0.784550
$405$	3.40951	0.169420
$406$	2.51845	0.124989
$407$	8.47898	0.420287
$408$	−3.93900	−0.195010
$409$	16.3657	0.809231	0.404616	−	0.914487i	$-0.367405\pi$
$409$	16.3657	0.809231	0.404616	+	0.914487i	$0.367405\pi$
$410$	−7.91904	−0.391093
$411$	−22.3367	−1.10179
$412$	16.9092	0.833057
$413$	−31.5586	−1.55290
$414$	3.20652	0.157592
$415$	−40.5070	−1.98841
$416$	6.53575	0.320442
$417$	−14.2998	−0.700267
$418$	−22.1420	−1.08300
$419$	−15.7417	−0.769031	−0.384516	−	0.923118i	$-0.625632\pi$
$419$	−15.7417	−0.769031	−0.384516	+	0.923118i	$0.625632\pi$
$420$	37.4759	1.82864
$421$	−0.197092	−0.00960567	−0.00480283	−	0.999988i	$-0.501529\pi$
$421$	−0.197092	−0.00960567	−0.00480283	+	0.999988i	$0.501529\pi$
$422$	19.5385	0.951119
$423$	−6.66825	−0.324222
$424$	5.51140	0.267657
$425$	13.7255	0.665786
$426$	20.3154	0.984284
$427$	−43.1411	−2.08775
$428$	6.85390	0.331296
$429$	81.5580	3.93766
$430$	−19.2648	−0.929032
$431$	−14.1304	−0.680636	−0.340318	−	0.940310i	$-0.610535\pi$
$431$	−14.1304	−0.680636	−0.340318	+	0.940310i	$0.610535\pi$
$432$	−4.77505	−0.229739
$433$	17.8792	0.859220	0.429610	−	0.903014i	$-0.358651\pi$
$433$	17.8792	0.859220	0.429610	+	0.903014i	$0.358651\pi$
$434$	6.04678	0.290255
$435$	7.61517	0.365119
$436$	−18.8369	−0.902126
$437$	3.34987	0.160246
$438$	−9.07673	−0.433703
$439$	22.7375	1.08520	0.542601	−	0.839990i	$-0.317439\pi$
$439$	22.7375	1.08520	0.542601	+	0.839990i	$0.317439\pi$
$440$	17.2098	0.820443
$441$	25.4520	1.21200
$442$	9.26635	0.440755
$443$	−8.66400	−0.411639	−0.205820	−	0.978590i	$-0.565986\pi$
$443$	−8.66400	−0.411639	−0.205820	+	0.978590i	$0.565986\pi$
$444$	5.24467	0.248901
$445$	−11.1846	−0.530199
$446$	8.86229	0.419642
$447$	−11.5555	−0.546555
$448$	3.52049	0.166327
$449$	−35.1480	−1.65874	−0.829369	−	0.558702i	$-0.811300\pi$
$449$	−35.1480	−1.65874	−0.829369	+	0.558702i	$0.811300\pi$
$450$	45.6815	2.15345
$451$	−9.28313	−0.437126
$452$	9.37838	0.441122
$453$	32.6224	1.53274
$454$	−8.32613	−0.390765
$455$	−88.1606	−4.13303
$456$	−13.6959	−0.641370
$457$	40.4486	1.89211	0.946053	−	0.324011i	$-0.105032\pi$
$457$	40.4486	1.89211	0.946053	+	0.324011i	$0.105032\pi$
$458$	−17.1982	−0.803618
$459$	−6.77003	−0.315998
$460$	−2.60367	−0.121397
$461$	−34.2818	−1.59666	−0.798331	−	0.602220i	$-0.794283\pi$
$461$	−34.2818	−1.59666	−0.798331	+	0.602220i	$0.794283\pi$
$462$	43.9313	2.04387
$463$	1.63374	0.0759264	0.0379632	−	0.999279i	$-0.487913\pi$
$463$	1.63374	0.0759264	0.0379632	+	0.999279i	$0.487913\pi$
$464$	0.715370	0.0332102
$465$	18.2840	0.847900
$466$	28.2405	1.30822
$467$	3.54032	0.163826	0.0819132	−	0.996639i	$-0.473897\pi$
$467$	3.54032	0.163826	0.0819132	+	0.996639i	$0.473897\pi$
$468$	30.8404	1.42560
$469$	2.03115	0.0937897
$470$	5.41458	0.249756
$471$	6.29913	0.290249
$472$	−8.96428	−0.412615
$473$	−22.5833	−1.03838
$474$	−28.3873	−1.30387
$475$	47.7237	2.18971
$476$	4.99133	0.228777
$477$	26.0068	1.19077
$478$	3.83111	0.175231
$479$	23.4005	1.06920	0.534598	−	0.845106i	$-0.320463\pi$
$479$	23.4005	1.06920	0.534598	+	0.845106i	$0.320463\pi$
$480$	10.6451	0.485879
$481$	−12.3379	−0.562559
$482$	−21.6612	−0.986643
$483$	−6.64638	−0.302421
$484$	9.17423	0.417010
$485$	22.4552	1.01964
$486$	16.7974	0.761944
$487$	37.5374	1.70098	0.850492	−	0.525988i	$-0.176304\pi$
$487$	37.5374	1.70098	0.850492	+	0.525988i	$0.176304\pi$
$488$	−12.2543	−0.554727
$489$	−19.9324	−0.901372
$490$	−20.6668	−0.933631
$491$	37.2102	1.67927	0.839635	−	0.543151i	$-0.182769\pi$
$491$	37.2102	1.67927	0.839635	+	0.543151i	$0.182769\pi$
$492$	−5.74207	−0.258873
$493$	1.01425	0.0456794
$494$	32.2191	1.44961
$495$	81.2080	3.65003
$496$	1.71760	0.0771225
$497$	−25.7428	−1.15472
$498$	−29.3715	−1.31617
$499$	−21.3411	−0.955358	−0.477679	−	0.878534i	$-0.658522\pi$
$499$	−21.3411	−0.955358	−0.477679	+	0.878534i	$0.658522\pi$
$500$	−17.9352	−0.802087
$501$	53.7674	2.40215
$502$	−24.0307	−1.07254
$503$	−20.0458	−0.893796	−0.446898	−	0.894585i	$-0.647471\pi$
$503$	−20.0458	−0.893796	−0.446898	+	0.894585i	$0.647471\pi$
$504$	16.6122	0.739966
$505$	60.4209	2.68870
$506$	−3.05217	−0.135685
$507$	−82.5589	−3.66657
$508$	−14.0405	−0.622947
$509$	−13.3706	−0.592643	−0.296322	−	0.955088i	$-0.595760\pi$
$509$	−13.3706	−0.592643	−0.296322	+	0.955088i	$0.595760\pi$
$510$	15.0925	0.668309
$511$	11.5016	0.508802
$512$	1.00000	0.0441942
$513$	−23.5394	−1.03929
$514$	4.30685	0.189967
$515$	−64.7888	−2.85493
$516$	−13.9689	−0.614945
$517$	6.34726	0.279152
$518$	−6.64581	−0.292000
$519$	−51.7029	−2.26950
$520$	−25.0422	−1.09817
$521$	−26.4674	−1.15956	−0.579778	−	0.814775i	$-0.696861\pi$
$521$	−26.4674	−1.15956	−0.579778	+	0.814775i	$0.696861\pi$
$522$	3.37563	0.147747
$523$	−11.9317	−0.521735	−0.260867	−	0.965375i	$-0.584009\pi$
$523$	−11.9317	−0.521735	−0.260867	+	0.965375i	$0.584009\pi$
$524$	−5.50485	−0.240480
$525$	−94.6872	−4.13249
$526$	2.69551	0.117530
$527$	2.43520	0.106079
$528$	12.4787	0.543068
$529$	−22.5382	−0.979923
$530$	−21.1173	−0.917277
$531$	−42.2999	−1.83566
$532$	17.3548	0.752428
$533$	13.5080	0.585097
$534$	−8.10989	−0.350949
$535$	−26.2612	−1.13537
$536$	0.576951	0.0249205
$537$	−38.0286	−1.64105
$538$	22.3241	0.962462
$539$	−24.2268	−1.04352
$540$	18.2959	0.787331
$541$	−5.82755	−0.250546	−0.125273	−	0.992122i	$-0.539981\pi$
$541$	−5.82755	−0.250546	−0.125273	+	0.992122i	$0.539981\pi$
$542$	−1.22733	−0.0527183
$543$	−1.60826	−0.0690169
$544$	1.41779	0.0607874
$545$	72.1750	3.09164
$546$	−63.9250	−2.73574
$547$	6.98541	0.298674	0.149337	−	0.988786i	$-0.452286\pi$
$547$	6.98541	0.298674	0.149337	+	0.988786i	$0.452286\pi$
$548$	8.03982	0.343444
$549$	−57.8246	−2.46790
$550$	−43.4825	−1.85410
$551$	3.52654	0.150236
$552$	−1.88791	−0.0803550
$553$	35.9711	1.52965
$554$	4.64649	0.197410
$555$	−20.0953	−0.852998
$556$	5.14706	0.218284
$557$	24.1904	1.02498	0.512490	−	0.858693i	$-0.328723\pi$
$557$	24.1904	1.02498	0.512490	+	0.858693i	$0.328723\pi$
$558$	8.10487	0.343106
$559$	32.8612	1.38988
$560$	−13.4890	−0.570014
$561$	17.6923	0.746970
$562$	3.21494	0.135614
$563$	−3.29879	−0.139027	−0.0695137	−	0.997581i	$-0.522145\pi$
$563$	−3.29879	−0.139027	−0.0695137	+	0.997581i	$0.522145\pi$
$564$	3.92609	0.165318
$565$	−35.9339	−1.51175
$566$	−20.0524	−0.842867
$567$	−3.13270	−0.131561
$568$	−7.31228	−0.306816
$569$	−3.30790	−0.138674	−0.0693372	−	0.997593i	$-0.522088\pi$
$569$	−3.30790	−0.138674	−0.0693372	+	0.997593i	$0.522088\pi$
$570$	52.4768	2.19801
$571$	6.54532	0.273913	0.136956	−	0.990577i	$-0.456268\pi$
$571$	6.54532	0.273913	0.136956	+	0.990577i	$0.456268\pi$
$572$	−29.3558	−1.22743
$573$	−47.1128	−1.96817
$574$	7.27610	0.303699
$575$	6.57848	0.274342
$576$	4.71872	0.196613
$577$	38.9987	1.62354	0.811768	−	0.583980i	$-0.198505\pi$
$577$	38.9987	1.62354	0.811768	+	0.583980i	$0.198505\pi$
$578$	−14.9899	−0.623496
$579$	−21.8105	−0.906412
$580$	−2.74099	−0.113813
$581$	37.2183	1.54408
$582$	16.2822	0.674920
$583$	−24.7549	−1.02524
$584$	3.26706	0.135192
$585$	−118.167	−4.88560
$586$	−7.80987	−0.322623
$587$	−24.4537	−1.00931	−0.504655	−	0.863321i	$-0.668380\pi$
$587$	−24.4537	−1.00931	−0.504655	+	0.863321i	$0.668380\pi$
$588$	−14.9854	−0.617989
$589$	8.46720	0.348885
$590$	34.3473	1.41405
$591$	53.8410	2.21472
$592$	−1.88775	−0.0775862
$593$	4.81921	0.197901	0.0989506	−	0.995092i	$-0.468451\pi$
$593$	4.81921	0.197901	0.0989506	+	0.995092i	$0.468451\pi$
$594$	21.4475	0.880000
$595$	−19.1246	−0.784032
$596$	4.15925	0.170369
$597$	−8.80148	−0.360221
$598$	4.44125	0.181616
$599$	−33.7873	−1.38051	−0.690255	−	0.723566i	$-0.742502\pi$
$599$	−33.7873	−1.38051	−0.690255	+	0.723566i	$0.742502\pi$
$600$	−26.8961	−1.09803
$601$	−43.8653	−1.78930	−0.894651	−	0.446766i	$-0.852576\pi$
$601$	−43.8653	−1.78930	−0.894651	+	0.446766i	$0.852576\pi$
$602$	17.7007	0.721428
$603$	2.72247	0.110868
$604$	−11.7420	−0.477777
$605$	−35.1517	−1.42912
$606$	43.8111	1.77970
$607$	−47.3440	−1.92163	−0.960817	−	0.277182i	$-0.910599\pi$
$607$	−47.3440	−1.92163	−0.960817	+	0.277182i	$0.910599\pi$
$608$	4.92967	0.199925
$609$	−6.99690	−0.283529
$610$	46.9532	1.90108
$611$	−9.23599	−0.373648
$612$	6.69017	0.270434
$613$	19.9821	0.807068	0.403534	−	0.914965i	$-0.367782\pi$
$613$	19.9821	0.807068	0.403534	+	0.914965i	$0.367782\pi$
$614$	17.0379	0.687592
$615$	22.0011	0.887171
$616$	−15.8125	−0.637105
$617$	42.1228	1.69580	0.847901	−	0.530155i	$-0.177866\pi$
$617$	42.1228	1.69580	0.847901	+	0.530155i	$0.177866\pi$
$618$	−46.9781	−1.88974
$619$	6.51724	0.261950	0.130975	−	0.991386i	$-0.458189\pi$
$619$	6.51724	0.261950	0.130975	+	0.991386i	$0.458189\pi$
$620$	−6.58110	−0.264303
$621$	−3.24480	−0.130209
$622$	19.1412	0.767494
$623$	10.2765	0.411719
$624$	−18.1580	−0.726902
$625$	20.3154	0.812617
$626$	18.7723	0.750294
$627$	61.5162	2.45672
$628$	−2.26730	−0.0904750
$629$	−2.67645	−0.106717
$630$	−63.6507	−2.53591
$631$	18.8631	0.750929	0.375464	−	0.926837i	$-0.377483\pi$
$631$	18.8631	0.750929	0.375464	+	0.926837i	$0.377483\pi$
$632$	10.2176	0.406436
$633$	−54.2830	−2.15756
$634$	28.9614	1.15021
$635$	53.7972	2.13488
$636$	−15.3121	−0.607164
$637$	35.2527	1.39676
$638$	−3.21313	−0.127209
$639$	−34.5046	−1.36498
$640$	−3.83157	−0.151456
$641$	−28.3164	−1.11843	−0.559216	−	0.829022i	$-0.688898\pi$
$641$	−28.3164	−1.11843	−0.559216	+	0.829022i	$0.688898\pi$
$642$	−19.0419	−0.751524
$643$	15.2123	0.599914	0.299957	−	0.953953i	$-0.403028\pi$
$643$	15.2123	0.599914	0.299957	+	0.953953i	$0.403028\pi$
$644$	2.39228	0.0942691
$645$	53.5226	2.10745
$646$	6.98926	0.274989
$647$	41.5406	1.63313	0.816565	−	0.577253i	$-0.195875\pi$
$647$	41.5406	1.63313	0.816565	+	0.577253i	$0.195875\pi$
$648$	−0.889847	−0.0349565
$649$	40.2637	1.58049
$650$	63.2720	2.48173
$651$	−16.7995	−0.658425
$652$	7.17441	0.280971
$653$	11.6565	0.456154	0.228077	−	0.973643i	$-0.426756\pi$
$653$	11.6565	0.456154	0.228077	+	0.973643i	$0.426756\pi$
$654$	52.3339	2.04642
$655$	21.0922	0.824140
$656$	2.06679	0.0806945
$657$	15.4163	0.601448
$658$	−4.97497	−0.193945
$659$	−24.2901	−0.946206	−0.473103	−	0.881007i	$-0.656866\pi$
$659$	−24.2901	−0.946206	−0.473103	+	0.881007i	$0.656866\pi$
$660$	−47.8132	−1.86113
$661$	1.74915	0.0680341	0.0340171	−	0.999421i	$-0.489170\pi$
$661$	1.74915	0.0680341	0.0340171	+	0.999421i	$0.489170\pi$
$662$	−7.57847	−0.294546
$663$	−25.7443	−0.999827
$664$	10.5719	0.410270
$665$	−66.4963	−2.57861
$666$	−8.90778	−0.345169
$667$	0.486116	0.0188225
$668$	−19.3529	−0.748787
$669$	−24.6217	−0.951931
$670$	−2.21063	−0.0854039
$671$	55.0411	2.12484
$672$	−9.78082	−0.377303
$673$	−23.3391	−0.899656	−0.449828	−	0.893115i	$-0.648515\pi$
$673$	−23.3391	−0.899656	−0.449828	+	0.893115i	$0.648515\pi$
$674$	9.05186	0.348665
$675$	−46.2268	−1.77927
$676$	29.7160	1.14292
$677$	15.6207	0.600351	0.300176	−	0.953884i	$-0.402955\pi$
$677$	15.6207	0.600351	0.300176	+	0.953884i	$0.402955\pi$
$678$	−26.0556	−1.00066
$679$	−20.6321	−0.791788
$680$	−5.43238	−0.208322
$681$	23.1321	0.886425
$682$	−7.71472	−0.295412
$683$	27.1198	1.03771	0.518856	−	0.854862i	$-0.326358\pi$
$683$	27.1198	1.03771	0.518856	+	0.854862i	$0.326358\pi$
$684$	23.2617	0.889435
$685$	−30.8051	−1.17700
$686$	−5.65451	−0.215890
$687$	47.7810	1.82296
$688$	5.02792	0.191688
$689$	36.0211	1.37230
$690$	7.23367	0.275381
$691$	13.5386	0.515033	0.257517	−	0.966274i	$-0.417096\pi$
$691$	13.5386	0.515033	0.257517	+	0.966274i	$0.417096\pi$
$692$	18.6098	0.707439
$693$	−74.6149	−2.83438
$694$	6.19268	0.235071
$695$	−19.7213	−0.748071
$696$	−1.98748	−0.0753353
$697$	2.93028	0.110992
$698$	5.97823	0.226279
$699$	−78.4595	−2.96761
$700$	34.0815	1.28816
$701$	−49.1033	−1.85461	−0.927304	−	0.374309i	$-0.877880\pi$
$701$	−49.1033	−1.85461	−0.927304	+	0.374309i	$0.877880\pi$
$702$	−31.2085	−1.17789
$703$	−9.30601	−0.350983
$704$	−4.49157	−0.169283
$705$	−15.0431	−0.566556
$706$	31.1991	1.17419
$707$	−55.5154	−2.08787
$708$	24.9051	0.935991
$709$	−29.1921	−1.09633	−0.548166	−	0.836370i	$-0.684674\pi$
$709$	−29.1921	−1.09633	−0.548166	+	0.836370i	$0.684674\pi$
$710$	28.0175	1.05148
$711$	48.2142	1.80817
$712$	2.91906	0.109396
$713$	1.16716	0.0437106
$714$	−13.8672	−0.518967
$715$	112.479	4.20647
$716$	13.6879	0.511542
$717$	−10.6438	−0.397501
$718$	32.2822	1.20476
$719$	6.80646	0.253838	0.126919	−	0.991913i	$-0.459491\pi$
$719$	6.80646	0.253838	0.126919	+	0.991913i	$0.459491\pi$
$720$	−18.0801	−0.673805
$721$	59.5286	2.21696
$722$	5.30168	0.197308
$723$	60.1805	2.23814
$724$	0.578873	0.0215136
$725$	6.92543	0.257204
$726$	−25.4884	−0.945962
$727$	9.08172	0.336822	0.168411	−	0.985717i	$-0.446136\pi$
$727$	9.08172	0.336822	0.168411	+	0.985717i	$0.446136\pi$
$728$	23.0090	0.852771
$729$	−43.9979	−1.62955
$730$	−12.5180	−0.463310
$731$	7.12856	0.263659
$732$	34.0456	1.25836
$733$	−28.0741	−1.03694	−0.518470	−	0.855096i	$-0.673498\pi$
$733$	−28.0741	−1.03694	−0.518470	+	0.855096i	$0.673498\pi$
$734$	−7.83315	−0.289127
$735$	57.4177	2.11789
$736$	0.679532	0.0250479
$737$	−2.59142	−0.0954561
$738$	9.75259	0.358998
$739$	50.9215	1.87318	0.936589	−	0.350429i	$-0.113964\pi$
$739$	50.9215	1.87318	0.936589	+	0.350429i	$0.113964\pi$
$740$	7.23305	0.265892
$741$	−89.5130	−3.28834
$742$	19.4028	0.712300
$743$	34.9221	1.28117	0.640584	−	0.767888i	$-0.278692\pi$
$743$	34.9221	1.28117	0.640584	+	0.767888i	$0.278692\pi$
$744$	−4.77193	−0.174948
$745$	−15.9364	−0.583866
$746$	−4.28501	−0.156885
$747$	49.8859	1.82523
$748$	−6.36813	−0.232842
$749$	24.1291	0.881657
$750$	49.8286	1.81948
$751$	19.5514	0.713441	0.356721	−	0.934211i	$-0.383895\pi$
$751$	19.5514	0.713441	0.356721	+	0.934211i	$0.383895\pi$
$752$	−1.41315	−0.0515322
$753$	66.7636	2.43300
$754$	4.67548	0.170271
$755$	44.9904	1.63737
$756$	−16.8105	−0.611391
$757$	28.9647	1.05274	0.526370	−	0.850255i	$-0.323553\pi$
$757$	28.9647	1.05274	0.526370	+	0.850255i	$0.323553\pi$
$758$	22.2418	0.807860
$759$	8.47971	0.307794
$760$	−18.8884	−0.685153
$761$	−6.10069	−0.221150	−0.110575	−	0.993868i	$-0.535269\pi$
$761$	−6.10069	−0.221150	−0.110575	+	0.993868i	$0.535269\pi$
$762$	39.0082	1.41312
$763$	−66.3152	−2.40077
$764$	16.9577	0.613507
$765$	−25.6339	−0.926794
$766$	16.9226	0.611437
$767$	−58.5883	−2.11550
$768$	−2.77826	−0.100252
$769$	29.5981	1.06734	0.533668	−	0.845694i	$-0.320813\pi$
$769$	29.5981	1.06734	0.533668	+	0.845694i	$0.320813\pi$
$770$	60.5867	2.18339
$771$	−11.9655	−0.430928
$772$	7.85041	0.282543
$773$	−18.8349	−0.677445	−0.338723	−	0.940886i	$-0.609995\pi$
$773$	−18.8349	−0.677445	−0.338723	+	0.940886i	$0.609995\pi$
$774$	23.7253	0.852790
$775$	16.6279	0.597292
$776$	−5.86059	−0.210383
$777$	18.4638	0.662384
$778$	−28.2582	−1.01310
$779$	10.1886	0.365044
$780$	69.5736	2.49114
$781$	32.8436	1.17524
$782$	0.963436	0.0344524
$783$	−3.41592	−0.122075
$784$	5.39383	0.192637
$785$	8.68730	0.310063
$786$	15.2939	0.545515
$787$	−48.9756	−1.74579	−0.872895	−	0.487908i	$-0.837760\pi$
$787$	−48.9756	−1.74579	−0.872895	+	0.487908i	$0.837760\pi$
$788$	−19.3794	−0.690362
$789$	−7.48883	−0.266609
$790$	−39.1496	−1.39288
$791$	33.0165	1.17393
$792$	−21.1945	−0.753113
$793$	−80.0911	−2.84412
$794$	−28.1809	−1.00010
$795$	58.6693	2.08079
$796$	3.16798	0.112286
$797$	26.0459	0.922591	0.461296	−	0.887246i	$-0.347385\pi$
$797$	26.0459	0.922591	0.461296	+	0.887246i	$0.347385\pi$
$798$	−48.2163	−1.70684
$799$	−2.00356	−0.0708807
$800$	9.68091	0.342272
$801$	13.7742	0.486688
$802$	32.6462	1.15278
$803$	−14.6742	−0.517842
$804$	−1.60292	−0.0565306
$805$	−9.16619	−0.323066
$806$	11.2258	0.395412
$807$	−62.0222	−2.18329
$808$	−15.7693	−0.554760
$809$	22.4342	0.788743	0.394371	−	0.918951i	$-0.370962\pi$
$809$	22.4342	0.788743	0.394371	+	0.918951i	$0.370962\pi$
$810$	3.40951	0.119798
$811$	−11.0326	−0.387406	−0.193703	−	0.981060i	$-0.562050\pi$
$811$	−11.0326	−0.387406	−0.193703	+	0.981060i	$0.562050\pi$
$812$	2.51845	0.0883802
$813$	3.40983	0.119588
$814$	8.47898	0.297188
$815$	−27.4892	−0.962905
$816$	−3.93900	−0.137893
$817$	24.7860	0.867153
$818$	16.3657	0.572213
$819$	108.573	3.79385
$820$	−7.91904	−0.276545
$821$	52.7610	1.84137	0.920686	−	0.390303i	$-0.127630\pi$
$821$	52.7610	1.84137	0.920686	+	0.390303i	$0.127630\pi$
$822$	−22.3367	−0.779082
$823$	−7.09681	−0.247379	−0.123690	−	0.992321i	$-0.539473\pi$
$823$	−7.09681	−0.247379	−0.123690	+	0.992321i	$0.539473\pi$
$824$	16.9092	0.589060
$825$	120.806	4.20591
$826$	−31.5586	−1.09807
$827$	38.2986	1.33177	0.665886	−	0.746053i	$-0.268054\pi$
$827$	38.2986	1.33177	0.665886	+	0.746053i	$0.268054\pi$
$828$	3.20652	0.111434
$829$	3.27046	0.113588	0.0567939	−	0.998386i	$-0.481912\pi$
$829$	3.27046	0.113588	0.0567939	+	0.998386i	$0.481912\pi$
$830$	−40.5070	−1.40602
$831$	−12.9091	−0.447813
$832$	6.53575	0.226586
$833$	7.64734	0.264965
$834$	−14.2998	−0.495163
$835$	74.1520	2.56614
$836$	−22.1420	−0.765797
$837$	−8.20161	−0.283489
$838$	−15.7417	−0.543787
$839$	44.9118	1.55053	0.775264	−	0.631638i	$-0.217617\pi$
$839$	44.9118	1.55053	0.775264	+	0.631638i	$0.217617\pi$
$840$	37.4759	1.29304
$841$	−28.4882	−0.982353
$842$	−0.197092	−0.00679223
$843$	−8.93192	−0.307632
$844$	19.5385	0.672543
$845$	−113.859	−3.91687
$846$	−6.66825	−0.229259
$847$	32.2977	1.10976
$848$	5.51140	0.189262
$849$	55.7108	1.91199
$850$	13.7255	0.470782
$851$	−1.28279	−0.0439734
$852$	20.3154	0.695994
$853$	−6.16321	−0.211024	−0.105512	−	0.994418i	$-0.533648\pi$
$853$	−6.16321	−0.211024	−0.105512	+	0.994418i	$0.533648\pi$
$854$	−43.1411	−1.47626
$855$	−89.1289	−3.04815
$856$	6.85390	0.234262
$857$	−42.9778	−1.46809	−0.734047	−	0.679099i	$-0.762371\pi$
$857$	−42.9778	−1.46809	−0.734047	+	0.679099i	$0.762371\pi$
$858$	81.5580	2.78434
$859$	0.586848	0.0200230	0.0100115	−	0.999950i	$-0.496813\pi$
$859$	0.586848	0.0200230	0.0100115	+	0.999950i	$0.496813\pi$
$860$	−19.2648	−0.656925
$861$	−20.2149	−0.688921
$862$	−14.1304	−0.481282
$863$	−4.29521	−0.146211	−0.0731053	−	0.997324i	$-0.523291\pi$
$863$	−4.29521	−0.146211	−0.0731053	+	0.997324i	$0.523291\pi$
$864$	−4.77505	−0.162450
$865$	−71.3047	−2.42443
$866$	17.8792	0.607560
$867$	41.6457	1.41436
$868$	6.04678	0.205241
$869$	−45.8933	−1.55682
$870$	7.61517	0.258178
$871$	3.77081	0.127769
$872$	−18.8369	−0.637899
$873$	−27.6545	−0.935962
$874$	3.34987	0.113311
$875$	−63.1407	−2.13454
$876$	−9.07673	−0.306674
$877$	31.3197	1.05759	0.528795	−	0.848750i	$-0.322644\pi$
$877$	31.3197	1.05759	0.528795	+	0.848750i	$0.322644\pi$
$878$	22.7375	0.767354
$879$	21.6978	0.731850
$880$	17.2098	0.580141
$881$	−34.3957	−1.15882	−0.579410	−	0.815036i	$-0.696717\pi$
$881$	−34.3957	−1.15882	−0.579410	+	0.815036i	$0.696717\pi$
$882$	25.4520	0.857012
$883$	42.3759	1.42606	0.713031	−	0.701132i	$-0.247322\pi$
$883$	42.3759	1.42606	0.713031	+	0.701132i	$0.247322\pi$
$884$	9.26635	0.311661
$885$	−95.4255	−3.20769
$886$	−8.66400	−0.291073
$887$	−26.2513	−0.881434	−0.440717	−	0.897646i	$-0.645276\pi$
$887$	−26.2513	−0.881434	−0.440717	+	0.897646i	$0.645276\pi$
$888$	5.24467	0.175999
$889$	−49.4295	−1.65781
$890$	−11.1846	−0.374907
$891$	3.99681	0.133898
$892$	8.86229	0.296731
$893$	−6.96636	−0.233120
$894$	−11.5555	−0.386472
$895$	−52.4462	−1.75308
$896$	3.52049	0.117611
$897$	−12.3389	−0.411985
$898$	−35.1480	−1.17290
$899$	1.22872	0.0409800
$900$	45.6815	1.52272
$901$	7.81403	0.260323
$902$	−9.28313	−0.309094
$903$	−49.1772	−1.63651
$904$	9.37838	0.311920
$905$	−2.21799	−0.0737285
$906$	32.6224	1.08381
$907$	4.92520	0.163539	0.0817693	−	0.996651i	$-0.473943\pi$
$907$	4.92520	0.163539	0.0817693	+	0.996651i	$0.473943\pi$
$908$	−8.32613	−0.276312
$909$	−74.4107	−2.46805
$910$	−88.1606	−2.92250
$911$	−7.60922	−0.252105	−0.126052	−	0.992024i	$-0.540231\pi$
$911$	−7.60922	−0.252105	−0.126052	+	0.992024i	$0.540231\pi$
$912$	−13.6959	−0.453517
$913$	−47.4845	−1.57151
$914$	40.4486	1.33792
$915$	−130.448	−4.31248
$916$	−17.1982	−0.568244
$917$	−19.3797	−0.639975
$918$	−6.77003	−0.223444
$919$	−13.4969	−0.445223	−0.222611	−	0.974907i	$-0.571458\pi$
$919$	−13.4969	−0.445223	−0.222611	+	0.974907i	$0.571458\pi$
$920$	−2.60367	−0.0858405
$921$	−47.3356	−1.55976
$922$	−34.2818	−1.12901
$923$	−47.7912	−1.57307
$924$	43.9313	1.44523
$925$	−18.2752	−0.600884
$926$	1.63374	0.0536880
$927$	79.7898	2.62064
$928$	0.715370	0.0234832
$929$	31.4585	1.03212	0.516059	−	0.856553i	$-0.327398\pi$
$929$	31.4585	1.03212	0.516059	+	0.856553i	$0.327398\pi$
$930$	18.2840	0.599555
$931$	26.5898	0.871446
$932$	28.2405	0.925049
$933$	−53.1793	−1.74101
$934$	3.54032	0.115843
$935$	24.3999	0.797962
$936$	30.8404	1.00805
$937$	−11.4593	−0.374359	−0.187180	−	0.982326i	$-0.559935\pi$
$937$	−11.4593	−0.374359	−0.187180	+	0.982326i	$0.559935\pi$
$938$	2.03115	0.0663193
$939$	−52.1544	−1.70200
$940$	5.41458	0.176604
$941$	−54.5243	−1.77744	−0.888720	−	0.458450i	$-0.848405\pi$
$941$	−54.5243	−1.77744	−0.888720	+	0.458450i	$0.848405\pi$
$942$	6.29913	0.205237
$943$	1.40445	0.0457351
$944$	−8.96428	−0.291763
$945$	64.4105	2.09527
$946$	−22.5833	−0.734246
$947$	48.1703	1.56533	0.782663	−	0.622446i	$-0.213861\pi$
$947$	48.1703	1.56533	0.782663	+	0.622446i	$0.213861\pi$
$948$	−28.3873	−0.921975
$949$	21.3527	0.693137
$950$	47.7237	1.54836
$951$	−80.4623	−2.60917
$952$	4.99133	0.161770
$953$	−15.1716	−0.491456	−0.245728	−	0.969339i	$-0.579027\pi$
$953$	−15.1716	−0.491456	−0.245728	+	0.969339i	$0.579027\pi$
$954$	26.0068	0.842000
$955$	−64.9745	−2.10253
$956$	3.83111	0.123907
$957$	8.92692	0.288566
$958$	23.4005	0.756036
$959$	28.3041	0.913986
$960$	10.6451	0.343569
$961$	−28.0499	−0.904834
$962$	−12.3379	−0.397789
$963$	32.3416	1.04219
$964$	−21.6612	−0.697662
$965$	−30.0794	−0.968290
$966$	−6.64638	−0.213844
$967$	−43.7580	−1.40716	−0.703580	−	0.710616i	$-0.748417\pi$
$967$	−43.7580	−1.40716	−0.703580	+	0.710616i	$0.748417\pi$
$968$	9.17423	0.294871
$969$	−19.4180	−0.623795
$970$	22.4552	0.720994
$971$	25.6454	0.823000	0.411500	−	0.911410i	$-0.365005\pi$
$971$	25.6454	0.823000	0.411500	+	0.911410i	$0.365005\pi$
$972$	16.7974	0.538776
$973$	18.1201	0.580905
$974$	37.5374	1.20278
$975$	−175.786	−5.62966
$976$	−12.2543	−0.392251
$977$	55.8162	1.78572	0.892860	−	0.450335i	$-0.148695\pi$
$977$	55.8162	1.78572	0.892860	+	0.450335i	$0.148695\pi$
$978$	−19.9324	−0.637366
$979$	−13.1112	−0.419034
$980$	−20.6668	−0.660177
$981$	−88.8862	−2.83792
$982$	37.2102	1.18742
$983$	31.2121	0.995510	0.497755	−	0.867318i	$-0.334158\pi$
$983$	31.2121	0.995510	0.497755	+	0.867318i	$0.334158\pi$
$984$	−5.74207	−0.183051
$985$	74.2534	2.36591
$986$	1.01425	0.0323002
$987$	13.8218	0.439951
$988$	32.2191	1.02503
$989$	3.41663	0.108643
$990$	81.2080	2.58096
$991$	7.90918	0.251243	0.125622	−	0.992078i	$-0.459907\pi$
$991$	7.90918	0.251243	0.125622	+	0.992078i	$0.459907\pi$
$992$	1.71760	0.0545338
$993$	21.0550	0.668159
$994$	−25.7428	−0.816512
$995$	−12.1383	−0.384811
$996$	−29.3715	−0.930672
$997$	18.6979	0.592169	0.296084	−	0.955162i	$-0.404319\pi$
$997$	18.6979	0.592169	0.296084	+	0.955162i	$0.404319\pi$
$998$	−21.3411	−0.675540
$999$	9.01411	0.285194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.d.1.9	✓	98

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.d.1.9	✓	98	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:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.77826 q^{3} +1.00000 q^{4} -3.83157 q^{5} -2.77826 q^{6} +3.52049 q^{7} +1.00000 q^{8} +4.71872 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.77826 q^{3} +1.00000 q^{4} -3.83157 q^{5} -2.77826 q^{6} +3.52049 q^{7} +1.00000 q^{8} +4.71872 q^{9} -3.83157 q^{10} -4.49157 q^{11} -2.77826 q^{12} +6.53575 q^{13} +3.52049 q^{14} +10.6451 q^{15} +1.00000 q^{16} +1.41779 q^{17} +4.71872 q^{18} +4.92967 q^{19} -3.83157 q^{20} -9.78082 q^{21} -4.49157 q^{22} +0.679532 q^{23} -2.77826 q^{24} +9.68091 q^{25} +6.53575 q^{26} -4.77505 q^{27} +3.52049 q^{28} +0.715370 q^{29} +10.6451 q^{30} +1.71760 q^{31} +1.00000 q^{32} +12.4787 q^{33} +1.41779 q^{34} -13.4890 q^{35} +4.71872 q^{36} -1.88775 q^{37} +4.92967 q^{38} -18.1580 q^{39} -3.83157 q^{40} +2.06679 q^{41} -9.78082 q^{42} +5.02792 q^{43} -4.49157 q^{44} -18.0801 q^{45} +0.679532 q^{46} -1.41315 q^{47} -2.77826 q^{48} +5.39383 q^{49} +9.68091 q^{50} -3.93900 q^{51} +6.53575 q^{52} +5.51140 q^{53} -4.77505 q^{54} +17.2098 q^{55} +3.52049 q^{56} -13.6959 q^{57} +0.715370 q^{58} -8.96428 q^{59} +10.6451 q^{60} -12.2543 q^{61} +1.71760 q^{62} +16.6122 q^{63} +1.00000 q^{64} -25.0422 q^{65} +12.4787 q^{66} +0.576951 q^{67} +1.41779 q^{68} -1.88791 q^{69} -13.4890 q^{70} -7.31228 q^{71} +4.71872 q^{72} +3.26706 q^{73} -1.88775 q^{74} -26.8961 q^{75} +4.92967 q^{76} -15.8125 q^{77} -18.1580 q^{78} +10.2176 q^{79} -3.83157 q^{80} -0.889847 q^{81} +2.06679 q^{82} +10.5719 q^{83} -9.78082 q^{84} -5.43238 q^{85} +5.02792 q^{86} -1.98748 q^{87} -4.49157 q^{88} +2.91906 q^{89} -18.0801 q^{90} +23.0090 q^{91} +0.679532 q^{92} -4.77193 q^{93} -1.41315 q^{94} -18.8884 q^{95} -2.77826 q^{96} -5.86059 q^{97} +5.39383 q^{98} -21.1945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9}+O(q^{10}) \) 98 * q + 98 * q^2 + 16 * q^3 + 98 * q^4 + 4 * q^5 + 16 * q^6 + 29 * q^7 + 98 * q^8 + 130 * q^9	\( 98 q + 98 q^{2} + 16 q^{3} + 98 q^{4} + 4 q^{5} + 16 q^{6} + 29 q^{7} + 98 q^{8} + 130 q^{9} + 4 q^{10} + 51 q^{11} + 16 q^{12} + 31 q^{13} + 29 q^{14} + 57 q^{15} + 98 q^{16} + 35 q^{17} + 130 q^{18} + 77 q^{19} + 4 q^{20} + 46 q^{21} + 51 q^{22} + 73 q^{23} + 16 q^{24} + 150 q^{25} + 31 q^{26} + 52 q^{27} + 29 q^{28} + 20 q^{29} + 57 q^{30} + 59 q^{31} + 98 q^{32} + 27 q^{33} + 35 q^{34} + 48 q^{35} + 130 q^{36} + 41 q^{37} + 77 q^{38} + 64 q^{39} + 4 q^{40} + 29 q^{41} + 46 q^{42} + 94 q^{43} + 51 q^{44} - 3 q^{45} + 73 q^{46} + 58 q^{47} + 16 q^{48} + 149 q^{49} + 150 q^{50} + 58 q^{51} + 31 q^{52} - 11 q^{53} + 52 q^{54} + 56 q^{55} + 29 q^{56} + 64 q^{57} + 20 q^{58} + 45 q^{59} + 57 q^{60} + 73 q^{61} + 59 q^{62} + 53 q^{63} + 98 q^{64} + 39 q^{65} + 27 q^{66} + 133 q^{67} + 35 q^{68} + 13 q^{69} + 48 q^{70} + 67 q^{71} + 130 q^{72} + 42 q^{73} + 41 q^{74} + 36 q^{75} + 77 q^{76} - 25 q^{77} + 64 q^{78} + 154 q^{79} + 4 q^{80} + 198 q^{81} + 29 q^{82} + 69 q^{83} + 46 q^{84} + 81 q^{85} + 94 q^{86} + 25 q^{87} + 51 q^{88} + 32 q^{89} - 3 q^{90} + 95 q^{91} + 73 q^{92} - 23 q^{93} + 58 q^{94} + 50 q^{95} + 16 q^{96} + 76 q^{97} + 149 q^{98} + 149 q^{99}+O(q^{100}) \) 98 * q + 98 * q^2 + 16 * q^3 + 98 * q^4 + 4 * q^5 + 16 * q^6 + 29 * q^7 + 98 * q^8 + 130 * q^9 + 4 * q^10 + 51 * q^11 + 16 * q^12 + 31 * q^13 + 29 * q^14 + 57 * q^15 + 98 * q^16 + 35 * q^17 + 130 * q^18 + 77 * q^19 + 4 * q^20 + 46 * q^21 + 51 * q^22 + 73 * q^23 + 16 * q^24 + 150 * q^25 + 31 * q^26 + 52 * q^27 + 29 * q^28 + 20 * q^29 + 57 * q^30 + 59 * q^31 + 98 * q^32 + 27 * q^33 + 35 * q^34 + 48 * q^35 + 130 * q^36 + 41 * q^37 + 77 * q^38 + 64 * q^39 + 4 * q^40 + 29 * q^41 + 46 * q^42 + 94 * q^43 + 51 * q^44 - 3 * q^45 + 73 * q^46 + 58 * q^47 + 16 * q^48 + 149 * q^49 + 150 * q^50 + 58 * q^51 + 31 * q^52 - 11 * q^53 + 52 * q^54 + 56 * q^55 + 29 * q^56 + 64 * q^57 + 20 * q^58 + 45 * q^59 + 57 * q^60 + 73 * q^61 + 59 * q^62 + 53 * q^63 + 98 * q^64 + 39 * q^65 + 27 * q^66 + 133 * q^67 + 35 * q^68 + 13 * q^69 + 48 * q^70 + 67 * q^71 + 130 * q^72 + 42 * q^73 + 41 * q^74 + 36 * q^75 + 77 * q^76 - 25 * q^77 + 64 * q^78 + 154 * q^79 + 4 * q^80 + 198 * q^81 + 29 * q^82 + 69 * q^83 + 46 * q^84 + 81 * q^85 + 94 * q^86 + 25 * q^87 + 51 * q^88 + 32 * q^89 - 3 * q^90 + 95 * q^91 + 73 * q^92 - 23 * q^93 + 58 * q^94 + 50 * q^95 + 16 * q^96 + 76 * q^97 + 149 * q^98 + 149 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more