LMFDB - Embedded newform 4006.2.a.g.1.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, 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("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.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:	$31.9880710497$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.24
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.388601 q^{3} +1.00000 q^{4} -3.74321 q^{5} -0.388601 q^{6} -4.20575 q^{7} -1.00000 q^{8} -2.84899 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.388601 q^{3} +1.00000 q^{4} -3.74321 q^{5} -0.388601 q^{6} -4.20575 q^{7} -1.00000 q^{8} -2.84899 q^{9} +3.74321 q^{10} +0.0915398 q^{11} +0.388601 q^{12} +3.00728 q^{13} +4.20575 q^{14} -1.45461 q^{15} +1.00000 q^{16} +5.06829 q^{17} +2.84899 q^{18} +4.20126 q^{19} -3.74321 q^{20} -1.63436 q^{21} -0.0915398 q^{22} +3.33740 q^{23} -0.388601 q^{24} +9.01159 q^{25} -3.00728 q^{26} -2.27292 q^{27} -4.20575 q^{28} -3.95314 q^{29} +1.45461 q^{30} -0.812397 q^{31} -1.00000 q^{32} +0.0355724 q^{33} -5.06829 q^{34} +15.7430 q^{35} -2.84899 q^{36} -1.34379 q^{37} -4.20126 q^{38} +1.16863 q^{39} +3.74321 q^{40} -5.81137 q^{41} +1.63436 q^{42} +7.46790 q^{43} +0.0915398 q^{44} +10.6644 q^{45} -3.33740 q^{46} -12.4144 q^{47} +0.388601 q^{48} +10.6883 q^{49} -9.01159 q^{50} +1.96954 q^{51} +3.00728 q^{52} +7.02901 q^{53} +2.27292 q^{54} -0.342652 q^{55} +4.20575 q^{56} +1.63261 q^{57} +3.95314 q^{58} +4.12463 q^{59} -1.45461 q^{60} +7.36788 q^{61} +0.812397 q^{62} +11.9821 q^{63} +1.00000 q^{64} -11.2569 q^{65} -0.0355724 q^{66} +5.07658 q^{67} +5.06829 q^{68} +1.29692 q^{69} -15.7430 q^{70} +3.58433 q^{71} +2.84899 q^{72} -2.39990 q^{73} +1.34379 q^{74} +3.50191 q^{75} +4.20126 q^{76} -0.384993 q^{77} -1.16863 q^{78} +0.844769 q^{79} -3.74321 q^{80} +7.66371 q^{81} +5.81137 q^{82} -14.2517 q^{83} -1.63436 q^{84} -18.9717 q^{85} -7.46790 q^{86} -1.53619 q^{87} -0.0915398 q^{88} +13.6717 q^{89} -10.6644 q^{90} -12.6479 q^{91} +3.33740 q^{92} -0.315698 q^{93} +12.4144 q^{94} -15.7262 q^{95} -0.388601 q^{96} -17.1637 q^{97} -10.6883 q^{98} -0.260796 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) $ 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9 + 23 * q^10 - 28 * q^11 - q^12 - 12 * q^13 - 12 * q^14 - 14 * q^15 + 40 * q^16 - 10 * q^17 - 29 * q^18 + 3 * q^19 - 23 * q^20 - 40 * q^21 + 28 * q^22 - 10 * q^23 + q^24 + 43 * q^25 + 12 * q^26 - 7 * q^27 + 12 * q^28 - 37 * q^29 + 14 * q^30 - 16 * q^31 - 40 * q^32 - 11 * q^33 + 10 * q^34 - 24 * q^35 + 29 * q^36 - 17 * q^37 - 3 * q^38 - 10 * q^39 + 23 * q^40 - 58 * q^41 + 40 * q^42 + 37 * q^43 - 28 * q^44 - 66 * q^45 + 10 * q^46 - 34 * q^47 - q^48 + 28 * q^49 - 43 * q^50 - 7 * q^51 - 12 * q^52 - 43 * q^53 + 7 * q^54 + 28 * q^55 - 12 * q^56 - 25 * q^57 + 37 * q^58 - 92 * q^59 - 14 * q^60 - 37 * q^61 + 16 * q^62 + 14 * q^63 + 40 * q^64 - 54 * q^65 + 11 * q^66 - 10 * q^67 - 10 * q^68 - 49 * q^69 + 24 * q^70 - 87 * q^71 - 29 * q^72 + 12 * q^73 + 17 * q^74 - 31 * q^75 + 3 * q^76 - 53 * q^77 + 10 * q^78 + 14 * q^79 - 23 * q^80 - 4 * q^81 + 58 * q^82 - 42 * q^83 - 40 * q^84 - 24 * q^85 - 37 * q^86 + 24 * q^87 + 28 * q^88 - 118 * q^89 + 66 * q^90 - 35 * q^91 - 10 * q^92 - 22 * q^93 + 34 * q^94 - 18 * q^95 + q^96 + 2 * q^97 - 28 * q^98 - 48 * 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
$3$	0.388601	0.224359	0.112179	−	0.993688i	$-0.464217\pi$
$3$	0.388601	0.224359	0.112179	+	0.993688i	$0.464217\pi$
$4$	1.00000	0.500000
$5$	−3.74321	−1.67401	−0.837006	−	0.547193i	$-0.815696\pi$
$5$	−3.74321	−1.67401	−0.837006	+	0.547193i	$0.815696\pi$
$6$	−0.388601	−0.158646
$7$	−4.20575	−1.58962	−0.794812	−	0.606856i	$-0.792431\pi$
$7$	−4.20575	−1.58962	−0.794812	+	0.606856i	$0.792431\pi$
$8$	−1.00000	−0.353553
$9$	−2.84899	−0.949663
$10$	3.74321	1.18371
$11$	0.0915398	0.0276003	0.0138001	−	0.999905i	$-0.495607\pi$
$11$	0.0915398	0.0276003	0.0138001	+	0.999905i	$0.495607\pi$
$12$	0.388601	0.112179
$13$	3.00728	0.834070	0.417035	−	0.908890i	$-0.363069\pi$
$13$	3.00728	0.834070	0.417035	+	0.908890i	$0.363069\pi$
$14$	4.20575	1.12403
$15$	−1.45461	−0.375579
$16$	1.00000	0.250000
$17$	5.06829	1.22924	0.614620	−	0.788823i	$-0.289309\pi$
$17$	5.06829	1.22924	0.614620	+	0.788823i	$0.289309\pi$
$18$	2.84899	0.671513
$19$	4.20126	0.963834	0.481917	−	0.876217i	$-0.339941\pi$
$19$	4.20126	0.963834	0.481917	+	0.876217i	$0.339941\pi$
$20$	−3.74321	−0.837006
$21$	−1.63436	−0.356646
$22$	−0.0915398	−0.0195163
$23$	3.33740	0.695897	0.347948	−	0.937514i	$-0.386878\pi$
$23$	3.33740	0.695897	0.347948	+	0.937514i	$0.386878\pi$
$24$	−0.388601	−0.0793228
$25$	9.01159	1.80232
$26$	−3.00728	−0.589777
$27$	−2.27292	−0.437424
$28$	−4.20575	−0.794812
$29$	−3.95314	−0.734080	−0.367040	−	0.930205i	$-0.619629\pi$
$29$	−3.95314	−0.734080	−0.367040	+	0.930205i	$0.619629\pi$
$30$	1.45461	0.265575
$31$	−0.812397	−0.145911	−0.0729554	−	0.997335i	$-0.523243\pi$
$31$	−0.812397	−0.145911	−0.0729554	+	0.997335i	$0.523243\pi$
$32$	−1.00000	−0.176777
$33$	0.0355724	0.00619236
$34$	−5.06829	−0.869205
$35$	15.7430	2.66105
$36$	−2.84899	−0.474832
$37$	−1.34379	−0.220918	−0.110459	−	0.993881i	$-0.535232\pi$
$37$	−1.34379	−0.220918	−0.110459	+	0.993881i	$0.535232\pi$
$38$	−4.20126	−0.681534
$39$	1.16863	0.187131
$40$	3.74321	0.591853
$41$	−5.81137	−0.907584	−0.453792	−	0.891108i	$-0.649929\pi$
$41$	−5.81137	−0.907584	−0.453792	+	0.891108i	$0.649929\pi$
$42$	1.63436	0.252187
$43$	7.46790	1.13884	0.569422	−	0.822045i	$-0.307167\pi$
$43$	7.46790	1.13884	0.569422	+	0.822045i	$0.307167\pi$
$44$	0.0915398	0.0138001
$45$	10.6644	1.58975
$46$	−3.33740	−0.492073
$47$	−12.4144	−1.81083	−0.905415	−	0.424528i	$-0.860440\pi$
$47$	−12.4144	−1.81083	−0.905415	+	0.424528i	$0.860440\pi$
$48$	0.388601	0.0560897
$49$	10.6883	1.52690
$50$	−9.01159	−1.27443
$51$	1.96954	0.275791
$52$	3.00728	0.417035
$53$	7.02901	0.965509	0.482754	−	0.875756i	$-0.339636\pi$
$53$	7.02901	0.965509	0.482754	+	0.875756i	$0.339636\pi$
$54$	2.27292	0.309305
$55$	−0.342652	−0.0462032
$56$	4.20575	0.562017
$57$	1.63261	0.216245
$58$	3.95314	0.519073
$59$	4.12463	0.536982	0.268491	−	0.963282i	$-0.413475\pi$
$59$	4.12463	0.536982	0.268491	+	0.963282i	$0.413475\pi$
$60$	−1.45461	−0.187790
$61$	7.36788	0.943361	0.471680	−	0.881770i	$-0.343648\pi$
$61$	7.36788	0.943361	0.471680	+	0.881770i	$0.343648\pi$
$62$	0.812397	0.103174
$63$	11.9821	1.50961
$64$	1.00000	0.125000
$65$	−11.2569	−1.39624
$66$	−0.0355724	−0.00437866
$67$	5.07658	0.620202	0.310101	−	0.950704i	$-0.399637\pi$
$67$	5.07658	0.620202	0.310101	+	0.950704i	$0.399637\pi$
$68$	5.06829	0.614620
$69$	1.29692	0.156130
$70$	−15.7430	−1.88165
$71$	3.58433	0.425382	0.212691	−	0.977120i	$-0.431777\pi$
$71$	3.58433	0.425382	0.212691	+	0.977120i	$0.431777\pi$
$72$	2.84899	0.335757
$73$	−2.39990	−0.280887	−0.140444	−	0.990089i	$-0.544853\pi$
$73$	−2.39990	−0.280887	−0.140444	+	0.990089i	$0.544853\pi$
$74$	1.34379	0.156212
$75$	3.50191	0.404366
$76$	4.20126	0.481917
$77$	−0.384993	−0.0438741
$78$	−1.16863	−0.132322
$79$	0.844769	0.0950440	0.0475220	−	0.998870i	$-0.484868\pi$
$79$	0.844769	0.0950440	0.0475220	+	0.998870i	$0.484868\pi$
$80$	−3.74321	−0.418503
$81$	7.66371	0.851523
$82$	5.81137	0.641759
$83$	−14.2517	−1.56432	−0.782162	−	0.623075i	$-0.785883\pi$
$83$	−14.2517	−1.56432	−0.782162	+	0.623075i	$0.785883\pi$
$84$	−1.63436	−0.178323
$85$	−18.9717	−2.05776
$86$	−7.46790	−0.805285
$87$	−1.53619	−0.164697
$88$	−0.0915398	−0.00975817
$89$	13.6717	1.44920	0.724600	−	0.689169i	$-0.242024\pi$
$89$	13.6717	1.44920	0.724600	+	0.689169i	$0.242024\pi$
$90$	−10.6644	−1.12412
$91$	−12.6479	−1.32586
$92$	3.33740	0.347948
$93$	−0.315698	−0.0327363
$94$	12.4144	1.28045
$95$	−15.7262	−1.61347
$96$	−0.388601	−0.0396614
$97$	−17.1637	−1.74271	−0.871356	−	0.490652i	$-0.836759\pi$
$97$	−17.1637	−1.74271	−0.871356	+	0.490652i	$0.836759\pi$
$98$	−10.6883	−1.07968
$99$	−0.260796	−0.0262110
$100$	9.01159	0.901159
$101$	−10.8056	−1.07519	−0.537597	−	0.843202i	$-0.680668\pi$
$101$	−10.8056	−1.07519	−0.537597	+	0.843202i	$0.680668\pi$
$102$	−1.96954	−0.195014
$103$	8.81100	0.868173	0.434087	−	0.900871i	$-0.357071\pi$
$103$	8.81100	0.868173	0.434087	+	0.900871i	$0.357071\pi$
$104$	−3.00728	−0.294888
$105$	6.11774	0.597030
$106$	−7.02901	−0.682718
$107$	14.7786	1.42871	0.714353	−	0.699786i	$-0.246721\pi$
$107$	14.7786	1.42871	0.714353	+	0.699786i	$0.246721\pi$
$108$	−2.27292	−0.218712
$109$	3.50977	0.336175	0.168088	−	0.985772i	$-0.446241\pi$
$109$	3.50977	0.336175	0.168088	+	0.985772i	$0.446241\pi$
$110$	0.342652	0.0326706
$111$	−0.522197	−0.0495648
$112$	−4.20575	−0.397406
$113$	1.46270	0.137599	0.0687995	−	0.997631i	$-0.478083\pi$
$113$	1.46270	0.137599	0.0687995	+	0.997631i	$0.478083\pi$
$114$	−1.63261	−0.152908
$115$	−12.4926	−1.16494
$116$	−3.95314	−0.367040
$117$	−8.56772	−0.792086
$118$	−4.12463	−0.379703
$119$	−21.3160	−1.95403
$120$	1.45461	0.132787
$121$	−10.9916	−0.999238
$122$	−7.36788	−0.667057
$123$	−2.25830	−0.203624
$124$	−0.812397	−0.0729554
$125$	−15.0162	−1.34309
$126$	−11.9821	−1.06745
$127$	−0.268927	−0.0238634	−0.0119317	−	0.999929i	$-0.503798\pi$
$127$	−0.268927	−0.0238634	−0.0119317	+	0.999929i	$0.503798\pi$
$128$	−1.00000	−0.0883883
$129$	2.90203	0.255510
$130$	11.2569	0.987294
$131$	−11.9461	−1.04373	−0.521866	−	0.853027i	$-0.674764\pi$
$131$	−11.9461	−1.04373	−0.521866	+	0.853027i	$0.674764\pi$
$132$	0.0355724	0.00309618
$133$	−17.6694	−1.53213
$134$	−5.07658	−0.438549
$135$	8.50801	0.732253
$136$	−5.06829	−0.434602
$137$	−0.516454	−0.0441237	−0.0220618	−	0.999757i	$-0.507023\pi$
$137$	−0.516454	−0.0441237	−0.0220618	+	0.999757i	$0.507023\pi$
$138$	−1.29692	−0.110401
$139$	0.869056	0.0737123	0.0368562	−	0.999321i	$-0.488266\pi$
$139$	0.869056	0.0737123	0.0368562	+	0.999321i	$0.488266\pi$
$140$	15.7430	1.33053
$141$	−4.82425	−0.406275
$142$	−3.58433	−0.300790
$143$	0.275286	0.0230206
$144$	−2.84899	−0.237416
$145$	14.7974	1.22886
$146$	2.39990	0.198617
$147$	4.15349	0.342574
$148$	−1.34379	−0.110459
$149$	10.2622	0.840712	0.420356	−	0.907359i	$-0.361905\pi$
$149$	10.2622	0.840712	0.420356	+	0.907359i	$0.361905\pi$
$150$	−3.50191	−0.285930
$151$	0.350136	0.0284937	0.0142468	−	0.999899i	$-0.495465\pi$
$151$	0.350136	0.0284937	0.0142468	+	0.999899i	$0.495465\pi$
$152$	−4.20126	−0.340767
$153$	−14.4395	−1.16736
$154$	0.384993	0.0310236
$155$	3.04097	0.244256
$156$	1.16863	0.0935655
$157$	−19.6845	−1.57099	−0.785497	−	0.618866i	$-0.787592\pi$
$157$	−19.6845	−1.57099	−0.785497	+	0.618866i	$0.787592\pi$
$158$	−0.844769	−0.0672062
$159$	2.73148	0.216620
$160$	3.74321	0.295926
$161$	−14.0363	−1.10621
$162$	−7.66371	−0.602118
$163$	−6.01153	−0.470859	−0.235430	−	0.971891i	$-0.575650\pi$
$163$	−6.01153	−0.470859	−0.235430	+	0.971891i	$0.575650\pi$
$164$	−5.81137	−0.453792
$165$	−0.133155	−0.0103661
$166$	14.2517	1.10614
$167$	0.915006	0.0708053	0.0354026	−	0.999373i	$-0.488729\pi$
$167$	0.915006	0.0708053	0.0354026	+	0.999373i	$0.488729\pi$
$168$	1.63436	0.126093
$169$	−3.95625	−0.304327
$170$	18.9717	1.45506
$171$	−11.9693	−0.915318
$172$	7.46790	0.569422
$173$	−0.829019	−0.0630291	−0.0315146	−	0.999503i	$-0.510033\pi$
$173$	−0.829019	−0.0630291	−0.0315146	+	0.999503i	$0.510033\pi$
$174$	1.53619	0.116458
$175$	−37.9005	−2.86501
$176$	0.0915398	0.00690007
$177$	1.60284	0.120476
$178$	−13.6717	−1.02474
$179$	−14.0986	−1.05378	−0.526889	−	0.849934i	$-0.676642\pi$
$179$	−14.0986	−1.05378	−0.526889	+	0.849934i	$0.676642\pi$
$180$	10.6644	0.794874
$181$	4.22305	0.313897	0.156948	−	0.987607i	$-0.449834\pi$
$181$	4.22305	0.313897	0.156948	+	0.987607i	$0.449834\pi$
$182$	12.6479	0.937523
$183$	2.86316	0.211651
$184$	−3.33740	−0.246037
$185$	5.03008	0.369819
$186$	0.315698	0.0231481
$187$	0.463950	0.0339274
$188$	−12.4144	−0.905415
$189$	9.55934	0.695339
$190$	15.7262	1.14090
$191$	−23.1873	−1.67777	−0.838887	−	0.544306i	$-0.816793\pi$
$191$	−23.1873	−1.67777	−0.838887	+	0.544306i	$0.816793\pi$
$192$	0.388601	0.0280448
$193$	−24.2076	−1.74250	−0.871252	−	0.490837i	$-0.836691\pi$
$193$	−24.2076	−1.74250	−0.871252	+	0.490837i	$0.836691\pi$
$194$	17.1637	1.23228
$195$	−4.37443	−0.313260
$196$	10.6883	0.763452
$197$	−18.6543	−1.32907	−0.664533	−	0.747259i	$-0.731369\pi$
$197$	−18.6543	−1.32907	−0.664533	+	0.747259i	$0.731369\pi$
$198$	0.260796	0.0185340
$199$	6.72753	0.476902	0.238451	−	0.971155i	$-0.423360\pi$
$199$	6.72753	0.476902	0.238451	+	0.971155i	$0.423360\pi$
$200$	−9.01159	−0.637216
$201$	1.97276	0.139148
$202$	10.8056	0.760276
$203$	16.6259	1.16691
$204$	1.96954	0.137895
$205$	21.7532	1.51931
$206$	−8.81100	−0.613891
$207$	−9.50823	−0.660868
$208$	3.00728	0.208518
$209$	0.384582	0.0266021
$210$	−6.11774	−0.422164
$211$	8.83498	0.608225	0.304112	−	0.952636i	$-0.401640\pi$
$211$	8.83498	0.608225	0.304112	+	0.952636i	$0.401640\pi$
$212$	7.02901	0.482754
$213$	1.39287	0.0954381
$214$	−14.7786	−1.01025
$215$	−27.9539	−1.90644
$216$	2.27292	0.154653
$217$	3.41674	0.231943
$218$	−3.50977	−0.237712
$219$	−0.932603	−0.0630194
$220$	−0.342652	−0.0231016
$221$	15.2418	1.02527
$222$	0.522197	0.0350476
$223$	−12.6856	−0.849494	−0.424747	−	0.905312i	$-0.639637\pi$
$223$	−12.6856	−0.849494	−0.424747	+	0.905312i	$0.639637\pi$
$224$	4.20575	0.281008
$225$	−25.6739	−1.71160
$226$	−1.46270	−0.0972972
$227$	−6.91659	−0.459070	−0.229535	−	0.973300i	$-0.573721\pi$
$227$	−6.91659	−0.459070	−0.229535	+	0.973300i	$0.573721\pi$
$228$	1.63261	0.108122
$229$	−6.68836	−0.441979	−0.220990	−	0.975276i	$-0.570929\pi$
$229$	−6.68836	−0.441979	−0.220990	+	0.975276i	$0.570929\pi$
$230$	12.4926	0.823737
$231$	−0.149609	−0.00984353
$232$	3.95314	0.259536
$233$	12.5283	0.820755	0.410378	−	0.911916i	$-0.365397\pi$
$233$	12.5283	0.820755	0.410378	+	0.911916i	$0.365397\pi$
$234$	8.56772	0.560089
$235$	46.4697	3.03135
$236$	4.12463	0.268491
$237$	0.328278	0.0213239
$238$	21.3160	1.38171
$239$	−6.69126	−0.432822	−0.216411	−	0.976302i	$-0.569435\pi$
$239$	−6.69126	−0.432822	−0.216411	+	0.976302i	$0.569435\pi$
$240$	−1.45461	−0.0938948
$241$	23.6655	1.52443	0.762214	−	0.647325i	$-0.224112\pi$
$241$	23.6655	1.52443	0.762214	+	0.647325i	$0.224112\pi$
$242$	10.9916	0.706568
$243$	9.79689	0.628470
$244$	7.36788	0.471680
$245$	−40.0086	−2.55606
$246$	2.25830	0.143984
$247$	12.6344	0.803905
$248$	0.812397	0.0515872
$249$	−5.53821	−0.350970
$250$	15.0162	0.949709
$251$	15.2753	0.964170	0.482085	−	0.876125i	$-0.339880\pi$
$251$	15.2753	0.964170	0.482085	+	0.876125i	$0.339880\pi$
$252$	11.9821	0.754804
$253$	0.305505	0.0192069
$254$	0.268927	0.0168740
$255$	−7.37240	−0.461677
$256$	1.00000	0.0625000
$257$	8.22871	0.513293	0.256646	−	0.966505i	$-0.417382\pi$
$257$	8.22871	0.513293	0.256646	+	0.966505i	$0.417382\pi$
$258$	−2.90203	−0.180673
$259$	5.65164	0.351176
$260$	−11.2569	−0.698122
$261$	11.2625	0.697128
$262$	11.9461	0.738030
$263$	−9.06740	−0.559120	−0.279560	−	0.960128i	$-0.590189\pi$
$263$	−9.06740	−0.559120	−0.279560	+	0.960128i	$0.590189\pi$
$264$	−0.0355724	−0.00218933
$265$	−26.3110	−1.61627
$266$	17.6694	1.08338
$267$	5.31284	0.325141
$268$	5.07658	0.310101
$269$	−27.7047	−1.68919	−0.844593	−	0.535409i	$-0.820158\pi$
$269$	−27.7047	−1.68919	−0.844593	+	0.535409i	$0.820158\pi$
$270$	−8.50801	−0.517781
$271$	−2.04678	−0.124333	−0.0621664	−	0.998066i	$-0.519801\pi$
$271$	−2.04678	−0.124333	−0.0621664	+	0.998066i	$0.519801\pi$
$272$	5.06829	0.307310
$273$	−4.91497	−0.297468
$274$	0.516454	0.0312001
$275$	0.824919	0.0497445
$276$	1.29692	0.0780652
$277$	−7.33292	−0.440592	−0.220296	−	0.975433i	$-0.570702\pi$
$277$	−7.33292	−0.440592	−0.220296	+	0.975433i	$0.570702\pi$
$278$	−0.869056	−0.0521225
$279$	2.31451	0.138566
$280$	−15.7430	−0.940824
$281$	−19.5076	−1.16373	−0.581864	−	0.813286i	$-0.697677\pi$
$281$	−19.5076	−1.16373	−0.581864	+	0.813286i	$0.697677\pi$
$282$	4.82425	0.287280
$283$	13.5260	0.804039	0.402020	−	0.915631i	$-0.368308\pi$
$283$	13.5260	0.804039	0.402020	+	0.915631i	$0.368308\pi$
$284$	3.58433	0.212691
$285$	−6.11120	−0.361996
$286$	−0.275286	−0.0162780
$287$	24.4412	1.44272
$288$	2.84899	0.167878
$289$	8.68756	0.511033
$290$	−14.7974	−0.868934
$291$	−6.66983	−0.390992
$292$	−2.39990	−0.140444
$293$	22.6258	1.32181	0.660906	−	0.750468i	$-0.270172\pi$
$293$	22.6258	1.32181	0.660906	+	0.750468i	$0.270172\pi$
$294$	−4.15349	−0.242237
$295$	−15.4394	−0.898914
$296$	1.34379	0.0781061
$297$	−0.208063	−0.0120730
$298$	−10.2622	−0.594473
$299$	10.0365	0.580427
$300$	3.50191	0.202183
$301$	−31.4081	−1.81033
$302$	−0.350136	−0.0201481
$303$	−4.19905	−0.241229
$304$	4.20126	0.240959
$305$	−27.5795	−1.57920
$306$	14.4395	0.825452
$307$	14.4682	0.825746	0.412873	−	0.910789i	$-0.364525\pi$
$307$	14.4682	0.825746	0.412873	+	0.910789i	$0.364525\pi$
$308$	−0.384993	−0.0219370
$309$	3.42396	0.194782
$310$	−3.04097	−0.172715
$311$	−23.6242	−1.33960	−0.669802	−	0.742540i	$-0.733621\pi$
$311$	−23.6242	−1.33960	−0.669802	+	0.742540i	$0.733621\pi$
$312$	−1.16863	−0.0661608
$313$	16.6261	0.939761	0.469881	−	0.882730i	$-0.344297\pi$
$313$	16.6261	0.939761	0.469881	+	0.882730i	$0.344297\pi$
$314$	19.6845	1.11086
$315$	−44.8516	−2.52710
$316$	0.844769	0.0475220
$317$	3.31091	0.185959	0.0929796	−	0.995668i	$-0.470361\pi$
$317$	3.31091	0.185959	0.0929796	+	0.995668i	$0.470361\pi$
$318$	−2.73148	−0.153174
$319$	−0.361869	−0.0202608
$320$	−3.74321	−0.209252
$321$	5.74299	0.320542
$322$	14.0363	0.782212
$323$	21.2932	1.18478
$324$	7.66371	0.425762
$325$	27.1004	1.50326
$326$	6.01153	0.332948
$327$	1.36390	0.0754238
$328$	5.81137	0.320879
$329$	52.2119	2.87854
$330$	0.133155	0.00732993
$331$	24.8276	1.36465	0.682325	−	0.731049i	$-0.260969\pi$
$331$	24.8276	1.36465	0.682325	+	0.731049i	$0.260969\pi$
$332$	−14.2517	−0.782162
$333$	3.82844	0.209797
$334$	−0.915006	−0.0500669
$335$	−19.0027	−1.03823
$336$	−1.63436	−0.0891615
$337$	6.39084	0.348131	0.174065	−	0.984734i	$-0.444310\pi$
$337$	6.39084	0.348131	0.174065	+	0.984734i	$0.444310\pi$
$338$	3.95625	0.215192
$339$	0.568405	0.0308715
$340$	−18.9717	−1.02888
$341$	−0.0743666	−0.00402718
$342$	11.9693	0.647227
$343$	−15.5122	−0.837579
$344$	−7.46790	−0.402642
$345$	−4.85463	−0.261364
$346$	0.829019	0.0445683
$347$	7.50303	0.402784	0.201392	−	0.979511i	$-0.435454\pi$
$347$	7.50303	0.402784	0.201392	+	0.979511i	$0.435454\pi$
$348$	−1.53619	−0.0823486
$349$	8.23575	0.440850	0.220425	−	0.975404i	$-0.429256\pi$
$349$	8.23575	0.440850	0.220425	+	0.975404i	$0.429256\pi$
$350$	37.9005	2.02587
$351$	−6.83532	−0.364842
$352$	−0.0915398	−0.00487909
$353$	9.31347	0.495706	0.247853	−	0.968798i	$-0.420275\pi$
$353$	9.31347	0.495706	0.247853	+	0.968798i	$0.420275\pi$
$354$	−1.60284	−0.0851897
$355$	−13.4169	−0.712094
$356$	13.6717	0.724600
$357$	−8.28339	−0.438404
$358$	14.0986	0.745133
$359$	−34.4272	−1.81700	−0.908500	−	0.417884i	$-0.862772\pi$
$359$	−34.4272	−1.81700	−0.908500	+	0.417884i	$0.862772\pi$
$360$	−10.6644	−0.562061
$361$	−1.34945	−0.0710239
$362$	−4.22305	−0.221959
$363$	−4.27135	−0.224188
$364$	−12.6479	−0.662929
$365$	8.98332	0.470208
$366$	−2.86316	−0.149660
$367$	−24.0825	−1.25710	−0.628549	−	0.777770i	$-0.716351\pi$
$367$	−24.0825	−1.25710	−0.628549	+	0.777770i	$0.716351\pi$
$368$	3.33740	0.173974
$369$	16.5565	0.861899
$370$	−5.03008	−0.261501
$371$	−29.5623	−1.53480
$372$	−0.315698	−0.0163682
$373$	10.9423	0.566571	0.283285	−	0.959036i	$-0.408576\pi$
$373$	10.9423	0.566571	0.283285	+	0.959036i	$0.408576\pi$
$374$	−0.463950	−0.0239903
$375$	−5.83531	−0.301334
$376$	12.4144	0.640225
$377$	−11.8882	−0.612274
$378$	−9.55934	−0.491679
$379$	15.7641	0.809749	0.404874	−	0.914372i	$-0.367315\pi$
$379$	15.7641	0.809749	0.404874	+	0.914372i	$0.367315\pi$
$380$	−15.7262	−0.806735
$381$	−0.104505	−0.00535396
$382$	23.1873	1.18636
$383$	−8.20678	−0.419347	−0.209673	−	0.977771i	$-0.567240\pi$
$383$	−8.20678	−0.419347	−0.209673	+	0.977771i	$0.567240\pi$
$384$	−0.388601	−0.0198307
$385$	1.44111	0.0734457
$386$	24.2076	1.23214
$387$	−21.2760	−1.08152
$388$	−17.1637	−0.871356
$389$	−8.44794	−0.428328	−0.214164	−	0.976798i	$-0.568703\pi$
$389$	−8.44794	−0.428328	−0.214164	+	0.976798i	$0.568703\pi$
$390$	4.37443	0.221508
$391$	16.9149	0.855425
$392$	−10.6883	−0.539842
$393$	−4.64225	−0.234170
$394$	18.6543	0.939792
$395$	−3.16215	−0.159105
$396$	−0.260796	−0.0131055
$397$	15.4399	0.774908	0.387454	−	0.921889i	$-0.373355\pi$
$397$	15.4399	0.774908	0.387454	+	0.921889i	$0.373355\pi$
$398$	−6.72753	−0.337221
$399$	−6.86635	−0.343747
$400$	9.01159	0.450580
$401$	−23.7927	−1.18815	−0.594074	−	0.804410i	$-0.702481\pi$
$401$	−23.7927	−1.18815	−0.594074	+	0.804410i	$0.702481\pi$
$402$	−1.97276	−0.0983923
$403$	−2.44311	−0.121700
$404$	−10.8056	−0.537597
$405$	−28.6868	−1.42546
$406$	−16.6259	−0.825130
$407$	−0.123010	−0.00609739
$408$	−1.96954	−0.0975068
$409$	15.0838	0.745845	0.372922	−	0.927863i	$-0.378356\pi$
$409$	15.0838	0.745845	0.372922	+	0.927863i	$0.378356\pi$
$410$	−21.7532	−1.07431
$411$	−0.200694	−0.00989952
$412$	8.81100	0.434087
$413$	−17.3472	−0.853599
$414$	9.50823	0.467304
$415$	53.3470	2.61870
$416$	−3.00728	−0.147444
$417$	0.337716	0.0165380
$418$	−0.384582	−0.0188105
$419$	−12.9482	−0.632559	−0.316279	−	0.948666i	$-0.602434\pi$
$419$	−12.9482	−0.632559	−0.316279	+	0.948666i	$0.602434\pi$
$420$	6.11774	0.298515
$421$	−9.33402	−0.454912	−0.227456	−	0.973788i	$-0.573041\pi$
$421$	−9.33402	−0.454912	−0.227456	+	0.973788i	$0.573041\pi$
$422$	−8.83498	−0.430080
$423$	35.3686	1.71968
$424$	−7.02901	−0.341359
$425$	45.6734	2.21548
$426$	−1.39287	−0.0674849
$427$	−30.9875	−1.49959
$428$	14.7786	0.714353
$429$	0.106976	0.00516486
$430$	27.9539	1.34806
$431$	−12.6572	−0.609677	−0.304838	−	0.952404i	$-0.598602\pi$
$431$	−12.6572	−0.609677	−0.304838	+	0.952404i	$0.598602\pi$
$432$	−2.27292	−0.109356
$433$	14.1723	0.681075	0.340538	−	0.940231i	$-0.389391\pi$
$433$	14.1723	0.681075	0.340538	+	0.940231i	$0.389391\pi$
$434$	−3.41674	−0.164009
$435$	5.75028	0.275705
$436$	3.50977	0.168088
$437$	14.0213	0.670729
$438$	0.932603	0.0445615
$439$	25.3836	1.21149	0.605747	−	0.795657i	$-0.292874\pi$
$439$	25.3836	1.21149	0.605747	+	0.795657i	$0.292874\pi$
$440$	0.342652	0.0163353
$441$	−30.4509	−1.45004
$442$	−15.2418	−0.724978
$443$	15.0161	0.713435	0.356717	−	0.934212i	$-0.383896\pi$
$443$	15.0161	0.713435	0.356717	+	0.934212i	$0.383896\pi$
$444$	−0.522197	−0.0247824
$445$	−51.1761	−2.42598
$446$	12.6856	0.600683
$447$	3.98789	0.188621
$448$	−4.20575	−0.198703
$449$	−25.9656	−1.22539	−0.612695	−	0.790319i	$-0.709915\pi$
$449$	−25.9656	−1.22539	−0.612695	+	0.790319i	$0.709915\pi$
$450$	25.6739	1.21028
$451$	−0.531971	−0.0250496
$452$	1.46270	0.0687995
$453$	0.136063	0.00639280
$454$	6.91659	0.324611
$455$	47.3436	2.21950
$456$	−1.63261	−0.0764540
$457$	24.3550	1.13928	0.569639	−	0.821895i	$-0.307083\pi$
$457$	24.3550	1.13928	0.569639	+	0.821895i	$0.307083\pi$
$458$	6.68836	0.312526
$459$	−11.5198	−0.537699
$460$	−12.4926	−0.582470
$461$	8.30201	0.386663	0.193332	−	0.981133i	$-0.438071\pi$
$461$	8.30201	0.386663	0.193332	+	0.981133i	$0.438071\pi$
$462$	0.149609	0.00696042
$463$	11.7172	0.544545	0.272272	−	0.962220i	$-0.412225\pi$
$463$	11.7172	0.544545	0.272272	+	0.962220i	$0.412225\pi$
$464$	−3.95314	−0.183520
$465$	1.18172	0.0548010
$466$	−12.5283	−0.580362
$467$	−20.2147	−0.935424	−0.467712	−	0.883881i	$-0.654922\pi$
$467$	−20.2147	−0.935424	−0.467712	+	0.883881i	$0.654922\pi$
$468$	−8.56772	−0.396043
$469$	−21.3508	−0.985889
$470$	−46.4697	−2.14349
$471$	−7.64940	−0.352466
$472$	−4.12463	−0.189852
$473$	0.683610	0.0314324
$474$	−0.328278	−0.0150783
$475$	37.8600	1.73714
$476$	−21.3160	−0.977015
$477$	−20.0256	−0.916908
$478$	6.69126	0.306051
$479$	−28.5309	−1.30361	−0.651804	−	0.758387i	$-0.725988\pi$
$479$	−28.5309	−1.30361	−0.651804	+	0.758387i	$0.725988\pi$
$480$	1.45461	0.0663937
$481$	−4.04115	−0.184261
$482$	−23.6655	−1.07793
$483$	−5.45451	−0.248189
$484$	−10.9916	−0.499619
$485$	64.2473	2.91732
$486$	−9.79689	−0.444396
$487$	42.9090	1.94439	0.972197	−	0.234166i	$-0.0752359\pi$
$487$	42.9090	1.94439	0.972197	+	0.234166i	$0.0752359\pi$
$488$	−7.36788	−0.333528
$489$	−2.33608	−0.105641
$490$	40.0086	1.80741
$491$	−23.7721	−1.07282	−0.536410	−	0.843957i	$-0.680220\pi$
$491$	−23.7721	−1.07282	−0.536410	+	0.843957i	$0.680220\pi$
$492$	−2.25830	−0.101812
$493$	−20.0357	−0.902361
$494$	−12.6344	−0.568447
$495$	0.976213	0.0438775
$496$	−0.812397	−0.0364777
$497$	−15.0748	−0.676197
$498$	5.53821	0.248173
$499$	24.5021	1.09686	0.548432	−	0.836195i	$-0.315225\pi$
$499$	24.5021	1.09686	0.548432	+	0.836195i	$0.315225\pi$
$500$	−15.0162	−0.671546
$501$	0.355572	0.0158858
$502$	−15.2753	−0.681771
$503$	−30.7080	−1.36920	−0.684601	−	0.728918i	$-0.740024\pi$
$503$	−30.7080	−1.36920	−0.684601	+	0.728918i	$0.740024\pi$
$504$	−11.9821	−0.533727
$505$	40.4474	1.79989
$506$	−0.305505	−0.0135814
$507$	−1.53740	−0.0682783
$508$	−0.268927	−0.0119317
$509$	−5.27520	−0.233819	−0.116910	−	0.993143i	$-0.537299\pi$
$509$	−5.27520	−0.233819	−0.116910	+	0.993143i	$0.537299\pi$
$510$	7.37240	0.326455
$511$	10.0934	0.446505
$512$	−1.00000	−0.0441942
$513$	−9.54912	−0.421604
$514$	−8.22871	−0.362953
$515$	−32.9814	−1.45333
$516$	2.90203	0.127755
$517$	−1.13641	−0.0499794
$518$	−5.65164	−0.248319
$519$	−0.322157	−0.0141411
$520$	11.2569	0.493647
$521$	20.4331	0.895189	0.447594	−	0.894237i	$-0.352281\pi$
$521$	20.4331	0.895189	0.447594	+	0.894237i	$0.352281\pi$
$522$	−11.2625	−0.492944
$523$	−42.3538	−1.85200	−0.926001	−	0.377520i	$-0.876777\pi$
$523$	−42.3538	−1.85200	−0.926001	+	0.377520i	$0.876777\pi$
$524$	−11.9461	−0.521866
$525$	−14.7282	−0.642790
$526$	9.06740	0.395358
$527$	−4.11746	−0.179359
$528$	0.0355724	0.00154809
$529$	−11.8617	−0.515728
$530$	26.3110	1.14288
$531$	−11.7510	−0.509952
$532$	−17.6694	−0.766067
$533$	−17.4764	−0.756989
$534$	−5.31284	−0.229909
$535$	−55.3195	−2.39167
$536$	−5.07658	−0.219275
$537$	−5.47872	−0.236424
$538$	27.7047	1.19443
$539$	0.978407	0.0421430
$540$	8.50801	0.366127
$541$	−38.9436	−1.67432	−0.837158	−	0.546961i	$-0.815785\pi$
$541$	−38.9436	−1.67432	−0.837158	+	0.546961i	$0.815785\pi$
$542$	2.04678	0.0879166
$543$	1.64108	0.0704255
$544$	−5.06829	−0.217301
$545$	−13.1378	−0.562761
$546$	4.91497	0.210341
$547$	−4.02093	−0.171923	−0.0859613	−	0.996298i	$-0.527396\pi$
$547$	−4.02093	−0.171923	−0.0859613	+	0.996298i	$0.527396\pi$
$548$	−0.516454	−0.0220618
$549$	−20.9910	−0.895875
$550$	−0.824919	−0.0351747
$551$	−16.6081	−0.707531
$552$	−1.29692	−0.0552005
$553$	−3.55289	−0.151084
$554$	7.33292	0.311546
$555$	1.95469	0.0829721
$556$	0.869056	0.0368562
$557$	−21.7781	−0.922767	−0.461383	−	0.887201i	$-0.652647\pi$
$557$	−21.7781	−0.922767	−0.461383	+	0.887201i	$0.652647\pi$
$558$	−2.31451	−0.0979810
$559$	22.4581	0.949876
$560$	15.7430	0.665263
$561$	0.180291	0.00761190
$562$	19.5076	0.822880
$563$	14.4286	0.608093	0.304046	−	0.952657i	$-0.401662\pi$
$563$	14.4286	0.608093	0.304046	+	0.952657i	$0.401662\pi$
$564$	−4.82425	−0.203138
$565$	−5.47518	−0.230343
$566$	−13.5260	−0.568542
$567$	−32.2316	−1.35360
$568$	−3.58433	−0.150395
$569$	30.5729	1.28168	0.640842	−	0.767673i	$-0.278586\pi$
$569$	30.5729	1.28168	0.640842	+	0.767673i	$0.278586\pi$
$570$	6.11120	0.255970
$571$	16.3741	0.685234	0.342617	−	0.939475i	$-0.388687\pi$
$571$	16.3741	0.685234	0.342617	+	0.939475i	$0.388687\pi$
$572$	0.275286	0.0115103
$573$	−9.01059	−0.376423
$574$	−24.4412	−1.02015
$575$	30.0753	1.25423
$576$	−2.84899	−0.118708
$577$	8.14678	0.339155	0.169578	−	0.985517i	$-0.445760\pi$
$577$	8.14678	0.339155	0.169578	+	0.985517i	$0.445760\pi$
$578$	−8.68756	−0.361355
$579$	−9.40710	−0.390946
$580$	14.7974	0.614429
$581$	59.9390	2.48669
$582$	6.66983	0.276473
$583$	0.643434	0.0266483
$584$	2.39990	0.0993086
$585$	32.0707	1.32596
$586$	−22.6258	−0.934663
$587$	16.1121	0.665019	0.332509	−	0.943100i	$-0.392105\pi$
$587$	16.1121	0.665019	0.332509	+	0.943100i	$0.392105\pi$
$588$	4.15349	0.171287
$589$	−3.41309	−0.140634
$590$	15.4394	0.635628
$591$	−7.24908	−0.298187
$592$	−1.34379	−0.0552294
$593$	20.3000	0.833620	0.416810	−	0.908994i	$-0.363148\pi$
$593$	20.3000	0.833620	0.416810	+	0.908994i	$0.363148\pi$
$594$	0.208063	0.00853691
$595$	79.7900	3.27107
$596$	10.2622	0.420356
$597$	2.61432	0.106997
$598$	−10.0365	−0.410424
$599$	−6.56979	−0.268434	−0.134217	−	0.990952i	$-0.542852\pi$
$599$	−6.56979	−0.268434	−0.134217	+	0.990952i	$0.542852\pi$
$600$	−3.50191	−0.142965
$601$	−28.3623	−1.15692	−0.578461	−	0.815710i	$-0.696347\pi$
$601$	−28.3623	−1.15692	−0.578461	+	0.815710i	$0.696347\pi$
$602$	31.4081	1.28010
$603$	−14.4631	−0.588983
$604$	0.350136	0.0142468
$605$	41.1439	1.67274
$606$	4.19905	0.170575
$607$	−22.6030	−0.917426	−0.458713	−	0.888584i	$-0.651689\pi$
$607$	−22.6030	−0.917426	−0.458713	+	0.888584i	$0.651689\pi$
$608$	−4.20126	−0.170383
$609$	6.46084	0.261806
$610$	27.5795	1.11666
$611$	−37.3337	−1.51036
$612$	−14.4395	−0.583682
$613$	5.36847	0.216830	0.108415	−	0.994106i	$-0.465422\pi$
$613$	5.36847	0.216830	0.108415	+	0.994106i	$0.465422\pi$
$614$	−14.4682	−0.583890
$615$	8.45329	0.340870
$616$	0.384993	0.0155118
$617$	22.1890	0.893294	0.446647	−	0.894710i	$-0.352618\pi$
$617$	22.1890	0.893294	0.446647	+	0.894710i	$0.352618\pi$
$618$	−3.42396	−0.137732
$619$	−37.1114	−1.49163	−0.745816	−	0.666152i	$-0.767940\pi$
$619$	−37.1114	−1.49163	−0.745816	+	0.666152i	$0.767940\pi$
$620$	3.04097	0.122128
$621$	−7.58566	−0.304402
$622$	23.6242	0.947243
$623$	−57.4999	−2.30368
$624$	1.16863	0.0467827
$625$	11.1509	0.446034
$626$	−16.6261	−0.664512
$627$	0.149449	0.00596841
$628$	−19.6845	−0.785497
$629$	−6.81071	−0.271561
$630$	44.8516	1.78693
$631$	−27.8302	−1.10790	−0.553951	−	0.832549i	$-0.686880\pi$
$631$	−27.8302	−1.10790	−0.553951	+	0.832549i	$0.686880\pi$
$632$	−0.844769	−0.0336031
$633$	3.43328	0.136461
$634$	−3.31091	−0.131493
$635$	1.00665	0.0399476
$636$	2.73148	0.108310
$637$	32.1428	1.27355
$638$	0.361869	0.0143265
$639$	−10.2117	−0.403969
$640$	3.74321	0.147963
$641$	30.0945	1.18866	0.594330	−	0.804221i	$-0.297417\pi$
$641$	30.0945	1.18866	0.594330	+	0.804221i	$0.297417\pi$
$642$	−5.74299	−0.226658
$643$	−9.12069	−0.359685	−0.179842	−	0.983695i	$-0.557559\pi$
$643$	−9.12069	−0.359685	−0.179842	+	0.983695i	$0.557559\pi$
$644$	−14.0363	−0.553107
$645$	−10.8629	−0.427726
$646$	−21.2932	−0.837769
$647$	39.5275	1.55398	0.776992	−	0.629510i	$-0.216744\pi$
$647$	39.5275	1.55398	0.776992	+	0.629510i	$0.216744\pi$
$648$	−7.66371	−0.301059
$649$	0.377568	0.0148208
$650$	−27.1004	−1.06297
$651$	1.32775	0.0520385
$652$	−6.01153	−0.235430
$653$	−31.9578	−1.25061	−0.625304	−	0.780382i	$-0.715025\pi$
$653$	−31.9578	−1.25061	−0.625304	+	0.780382i	$0.715025\pi$
$654$	−1.36390	−0.0533327
$655$	44.7166	1.74722
$656$	−5.81137	−0.226896
$657$	6.83729	0.266748
$658$	−52.2119	−2.03543
$659$	−30.0197	−1.16940	−0.584701	−	0.811249i	$-0.698788\pi$
$659$	−30.0197	−1.16940	−0.584701	+	0.811249i	$0.698788\pi$
$660$	−0.133155	−0.00518305
$661$	−28.5813	−1.11168	−0.555842	−	0.831288i	$-0.687604\pi$
$661$	−28.5813	−1.11168	−0.555842	+	0.831288i	$0.687604\pi$
$662$	−24.8276	−0.964954
$663$	5.92297	0.230029
$664$	14.2517	0.553072
$665$	66.1403	2.56481
$666$	−3.82844	−0.148349
$667$	−13.1932	−0.510844
$668$	0.915006	0.0354026
$669$	−4.92965	−0.190591
$670$	19.0027	0.734137
$671$	0.674454	0.0260370
$672$	1.63436	0.0630467
$673$	19.8095	0.763601	0.381801	−	0.924245i	$-0.375304\pi$
$673$	19.8095	0.763601	0.381801	+	0.924245i	$0.375304\pi$
$674$	−6.39084	−0.246166
$675$	−20.4826	−0.788377
$676$	−3.95625	−0.152163
$677$	−19.8108	−0.761391	−0.380696	−	0.924700i	$-0.624315\pi$
$677$	−19.8108	−0.761391	−0.380696	+	0.924700i	$0.624315\pi$
$678$	−0.568405	−0.0218295
$679$	72.1863	2.77026
$680$	18.9717	0.727530
$681$	−2.68779	−0.102996
$682$	0.0743666	0.00284764
$683$	30.3591	1.16166	0.580829	−	0.814025i	$-0.302728\pi$
$683$	30.3591	1.16166	0.580829	+	0.814025i	$0.302728\pi$
$684$	−11.9693	−0.457659
$685$	1.93319	0.0738636
$686$	15.5122	0.592258
$687$	−2.59910	−0.0991618
$688$	7.46790	0.284711
$689$	21.1382	0.805302
$690$	4.85463	0.184813
$691$	35.4389	1.34816	0.674079	−	0.738659i	$-0.264541\pi$
$691$	35.4389	1.34816	0.674079	+	0.738659i	$0.264541\pi$
$692$	−0.829019	−0.0315146
$693$	1.09684	0.0416656
$694$	−7.50303	−0.284811
$695$	−3.25305	−0.123395
$696$	1.53619	0.0582292
$697$	−29.4537	−1.11564
$698$	−8.23575	−0.311728
$699$	4.86850	0.184144
$700$	−37.9005	−1.43250
$701$	−35.6594	−1.34684	−0.673418	−	0.739262i	$-0.735175\pi$
$701$	−35.6594	−1.34684	−0.673418	+	0.739262i	$0.735175\pi$
$702$	6.83532	0.257982
$703$	−5.64560	−0.212928
$704$	0.0915398	0.00345003
$705$	18.0582	0.680110
$706$	−9.31347	−0.350517
$707$	45.4455	1.70915
$708$	1.60284	0.0602382
$709$	2.54318	0.0955111	0.0477555	−	0.998859i	$-0.484793\pi$
$709$	2.54318	0.0955111	0.0477555	+	0.998859i	$0.484793\pi$
$710$	13.4169	0.503527
$711$	−2.40674	−0.0902598
$712$	−13.6717	−0.512370
$713$	−2.71130	−0.101539
$714$	8.28339	0.309998
$715$	−1.03045	−0.0385367
$716$	−14.0986	−0.526889
$717$	−2.60023	−0.0971073
$718$	34.4272	1.28481
$719$	1.52510	0.0568768	0.0284384	−	0.999596i	$-0.490947\pi$
$719$	1.52510	0.0568768	0.0284384	+	0.999596i	$0.490947\pi$
$720$	10.6644	0.397437
$721$	−37.0568	−1.38007
$722$	1.34945	0.0502215
$723$	9.19642	0.342019
$724$	4.22305	0.156948
$725$	−35.6241	−1.32305
$726$	4.27135	0.158525
$727$	22.0426	0.817514	0.408757	−	0.912643i	$-0.365962\pi$
$727$	22.0426	0.817514	0.408757	+	0.912643i	$0.365962\pi$
$728$	12.6479	0.468762
$729$	−19.1841	−0.710521
$730$	−8.98332	−0.332488
$731$	37.8495	1.39991
$732$	2.86316	0.105826
$733$	32.9452	1.21686	0.608429	−	0.793609i	$-0.291800\pi$
$733$	32.9452	1.21686	0.608429	+	0.793609i	$0.291800\pi$
$734$	24.0825	0.888903
$735$	−15.5474	−0.573474
$736$	−3.33740	−0.123018
$737$	0.464709	0.0171178
$738$	−16.5565	−0.609454
$739$	−19.6114	−0.721416	−0.360708	−	0.932679i	$-0.617465\pi$
$739$	−19.6114	−0.721416	−0.360708	+	0.932679i	$0.617465\pi$
$740$	5.03008	0.184909
$741$	4.90972	0.180363
$742$	29.5623	1.08526
$743$	−46.7342	−1.71451	−0.857255	−	0.514892i	$-0.827832\pi$
$743$	−46.7342	−1.71451	−0.857255	+	0.514892i	$0.827832\pi$
$744$	0.315698	0.0115740
$745$	−38.4135	−1.40736
$746$	−10.9423	−0.400626
$747$	40.6029	1.48558
$748$	0.463950	0.0169637
$749$	−62.1553	−2.27110
$750$	5.83531	0.213076
$751$	23.8700	0.871030	0.435515	−	0.900181i	$-0.356566\pi$
$751$	23.8700	0.871030	0.435515	+	0.900181i	$0.356566\pi$
$752$	−12.4144	−0.452707
$753$	5.93600	0.216320
$754$	11.8882	0.432943
$755$	−1.31063	−0.0476988
$756$	9.55934	0.347670
$757$	51.0578	1.85573	0.927864	−	0.372920i	$-0.121643\pi$
$757$	51.0578	1.85573	0.927864	+	0.372920i	$0.121643\pi$
$758$	−15.7641	−0.572579
$759$	0.118720	0.00430925
$760$	15.7262	0.570448
$761$	−36.7895	−1.33362	−0.666809	−	0.745229i	$-0.732340\pi$
$761$	−36.7895	−1.33362	−0.666809	+	0.745229i	$0.732340\pi$
$762$	0.104505	0.00378582
$763$	−14.7612	−0.534392
$764$	−23.1873	−0.838887
$765$	54.0500	1.95418
$766$	8.20678	0.296523
$767$	12.4039	0.447880
$768$	0.388601	0.0140224
$769$	−39.2111	−1.41399	−0.706994	−	0.707219i	$-0.749949\pi$
$769$	−39.2111	−1.41399	−0.706994	+	0.707219i	$0.749949\pi$
$770$	−1.44111	−0.0519340
$771$	3.19768	0.115162
$772$	−24.2076	−0.871252
$773$	35.4037	1.27338	0.636691	−	0.771119i	$-0.280303\pi$
$773$	35.4037	1.27338	0.636691	+	0.771119i	$0.280303\pi$
$774$	21.2760	0.764749
$775$	−7.32099	−0.262978
$776$	17.1637	0.616142
$777$	2.19623	0.0787893
$778$	8.44794	0.302873
$779$	−24.4151	−0.874760
$780$	−4.37443	−0.156630
$781$	0.328109	0.0117407
$782$	−16.9149	−0.604877
$783$	8.98517	0.321104
$784$	10.6883	0.381726
$785$	73.6831	2.62986
$786$	4.64225	0.165583
$787$	4.06844	0.145024	0.0725122	−	0.997368i	$-0.476898\pi$
$787$	4.06844	0.145024	0.0725122	+	0.997368i	$0.476898\pi$
$788$	−18.6543	−0.664533
$789$	−3.52360	−0.125443
$790$	3.16215	0.112504
$791$	−6.15174	−0.218731
$792$	0.260796	0.00926698
$793$	22.1573	0.786829
$794$	−15.4399	−0.547943
$795$	−10.2245	−0.362625
$796$	6.72753	0.238451
$797$	11.8551	0.419928	0.209964	−	0.977709i	$-0.432665\pi$
$797$	11.8551	0.419928	0.209964	+	0.977709i	$0.432665\pi$
$798$	6.86635	0.243066
$799$	−62.9199	−2.22595
$800$	−9.01159	−0.318608
$801$	−38.9506	−1.37625
$802$	23.7927	0.840148
$803$	−0.219686	−0.00775256
$804$	1.97276	0.0695739
$805$	52.5407	1.85182
$806$	2.44311	0.0860548
$807$	−10.7661	−0.378984
$808$	10.8056	0.380138
$809$	−1.04616	−0.0367811	−0.0183905	−	0.999831i	$-0.505854\pi$
$809$	−1.04616	−0.0367811	−0.0183905	+	0.999831i	$0.505854\pi$
$810$	28.6868	1.00795
$811$	−38.7618	−1.36111	−0.680556	−	0.732696i	$-0.738262\pi$
$811$	−38.7618	−1.36111	−0.680556	+	0.732696i	$0.738262\pi$
$812$	16.6259	0.583455
$813$	−0.795379	−0.0278951
$814$	0.123010	0.00431150
$815$	22.5024	0.788224
$816$	1.96954	0.0689477
$817$	31.3746	1.09766
$818$	−15.0838	−0.527392
$819$	36.0337	1.25912
$820$	21.7532	0.759653
$821$	−39.1091	−1.36492	−0.682458	−	0.730925i	$-0.739089\pi$
$821$	−39.1091	−1.36492	−0.682458	+	0.730925i	$0.739089\pi$
$822$	0.200694	0.00700002
$823$	31.9814	1.11480	0.557401	−	0.830243i	$-0.311799\pi$
$823$	31.9814	1.11480	0.557401	+	0.830243i	$0.311799\pi$
$824$	−8.81100	−0.306946
$825$	0.320564	0.0111606
$826$	17.3472	0.603585
$827$	0.282108	0.00980985	0.00490492	−	0.999988i	$-0.498439\pi$
$827$	0.282108	0.00980985	0.00490492	+	0.999988i	$0.498439\pi$
$828$	−9.50823	−0.330434
$829$	−21.6883	−0.753267	−0.376633	−	0.926362i	$-0.622918\pi$
$829$	−21.6883	−0.753267	−0.376633	+	0.926362i	$0.622918\pi$
$830$	−53.3470	−1.85170
$831$	−2.84958	−0.0988507
$832$	3.00728	0.104259
$833$	54.1715	1.87693
$834$	−0.337716	−0.0116941
$835$	−3.42506	−0.118529
$836$	0.384582	0.0133010
$837$	1.84651	0.0638248
$838$	12.9482	0.447287
$839$	34.0496	1.17552	0.587762	−	0.809034i	$-0.300009\pi$
$839$	34.0496	1.17552	0.587762	+	0.809034i	$0.300009\pi$
$840$	−6.11774	−0.211082
$841$	−13.3727	−0.461127
$842$	9.33402	0.321671
$843$	−7.58068	−0.261093
$844$	8.83498	0.304112
$845$	14.8091	0.509447
$846$	−35.3686	−1.21600
$847$	46.2280	1.58841
$848$	7.02901	0.241377
$849$	5.25622	0.180393
$850$	−45.6734	−1.56658
$851$	−4.48477	−0.153736
$852$	1.39287	0.0477190
$853$	−43.5497	−1.49111	−0.745557	−	0.666441i	$-0.767817\pi$
$853$	−43.5497	−1.49111	−0.745557	+	0.666441i	$0.767817\pi$
$854$	30.9875	1.06037
$855$	44.8037	1.53225
$856$	−14.7786	−0.505124
$857$	−29.0041	−0.990761	−0.495380	−	0.868676i	$-0.664971\pi$
$857$	−29.0041	−0.990761	−0.495380	+	0.868676i	$0.664971\pi$
$858$	−0.106976	−0.00365211
$859$	36.8414	1.25701	0.628507	−	0.777804i	$-0.283666\pi$
$859$	36.8414	1.25701	0.628507	+	0.777804i	$0.283666\pi$
$860$	−27.9539	−0.953220
$861$	9.49785	0.323686
$862$	12.6572	0.431107
$863$	36.6294	1.24688	0.623439	−	0.781872i	$-0.285735\pi$
$863$	36.6294	1.24688	0.623439	+	0.781872i	$0.285735\pi$
$864$	2.27292	0.0773263
$865$	3.10319	0.105512
$866$	−14.1723	−0.481593
$867$	3.37599	0.114655
$868$	3.41674	0.115972
$869$	0.0773300	0.00262324
$870$	−5.75028	−0.194953
$871$	15.2667	0.517292
$872$	−3.50977	−0.118856
$873$	48.8992	1.65499
$874$	−14.0213	−0.474277
$875$	63.1545	2.13501
$876$	−0.932603	−0.0315097
$877$	−42.8569	−1.44717	−0.723587	−	0.690233i	$-0.757508\pi$
$877$	−42.8569	−1.44717	−0.723587	+	0.690233i	$0.757508\pi$
$878$	−25.3836	−0.856655
$879$	8.79239	0.296560
$880$	−0.342652	−0.0115508
$881$	−54.2430	−1.82749	−0.913746	−	0.406285i	$-0.866824\pi$
$881$	−54.2430	−1.82749	−0.913746	+	0.406285i	$0.866824\pi$
$882$	30.4509	1.02534
$883$	−24.0722	−0.810094	−0.405047	−	0.914296i	$-0.632745\pi$
$883$	−24.0722	−0.810094	−0.405047	+	0.914296i	$0.632745\pi$
$884$	15.2418	0.512637
$885$	−5.99974	−0.201679
$886$	−15.0161	−0.504475
$887$	−9.91561	−0.332934	−0.166467	−	0.986047i	$-0.553236\pi$
$887$	−9.91561	−0.332934	−0.166467	+	0.986047i	$0.553236\pi$
$888$	0.522197	0.0175238
$889$	1.13104	0.0379338
$890$	51.1761	1.71543
$891$	0.701534	0.0235023
$892$	−12.6856	−0.424747
$893$	−52.1562	−1.74534
$894$	−3.98789	−0.133375
$895$	52.7739	1.76404
$896$	4.20575	0.140504
$897$	3.90020	0.130224
$898$	25.9656	0.866482
$899$	3.21152	0.107110
$900$	−25.6739	−0.855798
$901$	35.6251	1.18684
$902$	0.531971	0.0177127
$903$	−12.2052	−0.406164
$904$	−1.46270	−0.0486486
$905$	−15.8077	−0.525467
$906$	−0.136063	−0.00452039
$907$	37.2177	1.23579	0.617897	−	0.786259i	$-0.287985\pi$
$907$	37.2177	1.23579	0.617897	+	0.786259i	$0.287985\pi$
$908$	−6.91659	−0.229535
$909$	30.7849	1.02107
$910$	−47.3436	−1.56943
$911$	41.0529	1.36014	0.680072	−	0.733145i	$-0.261949\pi$
$911$	41.0529	1.36014	0.680072	+	0.733145i	$0.261949\pi$
$912$	1.63261	0.0540611
$913$	−1.30459	−0.0431758
$914$	−24.3550	−0.805591
$915$	−10.7174	−0.354307
$916$	−6.68836	−0.220990
$917$	50.2421	1.65914
$918$	11.5198	0.380211
$919$	14.8295	0.489179	0.244589	−	0.969627i	$-0.421347\pi$
$919$	14.8295	0.489179	0.244589	+	0.969627i	$0.421347\pi$
$920$	12.4926	0.411869
$921$	5.62236	0.185263
$922$	−8.30201	−0.273412
$923$	10.7791	0.354798
$924$	−0.149609	−0.00492176
$925$	−12.1097	−0.398164
$926$	−11.7172	−0.385051
$927$	−25.1024	−0.824472
$928$	3.95314	0.129768
$929$	−56.1860	−1.84340	−0.921701	−	0.387900i	$-0.873200\pi$
$929$	−56.1860	−1.84340	−0.921701	+	0.387900i	$0.873200\pi$
$930$	−1.18172	−0.0387502
$931$	44.9044	1.47168
$932$	12.5283	0.410378
$933$	−9.18037	−0.300552
$934$	20.2147	0.661444
$935$	−1.73666	−0.0567949
$936$	8.56772	0.280045
$937$	−36.4830	−1.19185	−0.595924	−	0.803041i	$-0.703214\pi$
$937$	−36.4830	−1.19185	−0.595924	+	0.803041i	$0.703214\pi$
$938$	21.3508	0.697128
$939$	6.46090	0.210844
$940$	46.4697	1.51568
$941$	−22.9368	−0.747717	−0.373859	−	0.927486i	$-0.621965\pi$
$941$	−22.9368	−0.747717	−0.373859	+	0.927486i	$0.621965\pi$
$942$	7.64940	0.249231
$943$	−19.3949	−0.631585
$944$	4.12463	0.134245
$945$	−35.7826	−1.16401
$946$	−0.683610	−0.0222261
$947$	13.8839	0.451167	0.225583	−	0.974224i	$-0.427571\pi$
$947$	13.8839	0.451167	0.225583	+	0.974224i	$0.427571\pi$
$948$	0.328278	0.0106620
$949$	−7.21718	−0.234280
$950$	−37.8600	−1.22834
$951$	1.28662	0.0417215
$952$	21.3160	0.690854
$953$	12.3349	0.399565	0.199783	−	0.979840i	$-0.435976\pi$
$953$	12.3349	0.399565	0.199783	+	0.979840i	$0.435976\pi$
$954$	20.0256	0.648352
$955$	86.7948	2.80861
$956$	−6.69126	−0.216411
$957$	−0.140623	−0.00454569
$958$	28.5309	0.921790
$959$	2.17208	0.0701400
$960$	−1.45461	−0.0469474
$961$	−30.3400	−0.978710
$962$	4.04115	0.130292
$963$	−42.1042	−1.35679
$964$	23.6655	0.762214
$965$	90.6141	2.91697
$966$	5.45451	0.175496
$967$	44.0984	1.41811	0.709055	−	0.705153i	$-0.249122\pi$
$967$	44.0984	1.41811	0.709055	+	0.705153i	$0.249122\pi$
$968$	10.9916	0.353284
$969$	8.27454	0.265817
$970$	−64.2473	−2.06286
$971$	−24.2244	−0.777398	−0.388699	−	0.921365i	$-0.627075\pi$
$971$	−24.2244	−0.777398	−0.388699	+	0.921365i	$0.627075\pi$
$972$	9.79689	0.314235
$973$	−3.65503	−0.117175
$974$	−42.9090	−1.37489
$975$	10.5312	0.337270
$976$	7.36788	0.235840
$977$	−29.6260	−0.947818	−0.473909	−	0.880574i	$-0.657158\pi$
$977$	−29.6260	−0.947818	−0.473909	+	0.880574i	$0.657158\pi$
$978$	2.33608	0.0746997
$979$	1.25151	0.0399983
$980$	−40.0086	−1.27803
$981$	−9.99930	−0.319253
$982$	23.7721	0.758599
$983$	−21.8337	−0.696388	−0.348194	−	0.937422i	$-0.613205\pi$
$983$	−21.8337	−0.696388	−0.348194	+	0.937422i	$0.613205\pi$
$984$	2.25830	0.0719920
$985$	69.8270	2.22487
$986$	20.0357	0.638065
$987$	20.2896	0.645825
$988$	12.6344	0.401953
$989$	24.9234	0.792518
$990$	−0.976213	−0.0310261
$991$	14.8961	0.473190	0.236595	−	0.971608i	$-0.423969\pi$
$991$	14.8961	0.473190	0.236595	+	0.971608i	$0.423969\pi$
$992$	0.812397	0.0257936
$993$	9.64804	0.306171
$994$	15.0748	0.478143
$995$	−25.1825	−0.798340
$996$	−5.53821	−0.175485
$997$	6.48980	0.205534	0.102767	−	0.994705i	$-0.467230\pi$
$997$	6.48980	0.205534	0.102767	+	0.994705i	$0.467230\pi$
$998$	−24.5021	−0.775599
$999$	3.05433	0.0966346

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.g.1.24	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.g.1.24	✓	40	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.388601 q^{3} +1.00000 q^{4} -3.74321 q^{5} -0.388601 q^{6} -4.20575 q^{7} -1.00000 q^{8} -2.84899 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.388601 q^{3} +1.00000 q^{4} -3.74321 q^{5} -0.388601 q^{6} -4.20575 q^{7} -1.00000 q^{8} -2.84899 q^{9} +3.74321 q^{10} +0.0915398 q^{11} +0.388601 q^{12} +3.00728 q^{13} +4.20575 q^{14} -1.45461 q^{15} +1.00000 q^{16} +5.06829 q^{17} +2.84899 q^{18} +4.20126 q^{19} -3.74321 q^{20} -1.63436 q^{21} -0.0915398 q^{22} +3.33740 q^{23} -0.388601 q^{24} +9.01159 q^{25} -3.00728 q^{26} -2.27292 q^{27} -4.20575 q^{28} -3.95314 q^{29} +1.45461 q^{30} -0.812397 q^{31} -1.00000 q^{32} +0.0355724 q^{33} -5.06829 q^{34} +15.7430 q^{35} -2.84899 q^{36} -1.34379 q^{37} -4.20126 q^{38} +1.16863 q^{39} +3.74321 q^{40} -5.81137 q^{41} +1.63436 q^{42} +7.46790 q^{43} +0.0915398 q^{44} +10.6644 q^{45} -3.33740 q^{46} -12.4144 q^{47} +0.388601 q^{48} +10.6883 q^{49} -9.01159 q^{50} +1.96954 q^{51} +3.00728 q^{52} +7.02901 q^{53} +2.27292 q^{54} -0.342652 q^{55} +4.20575 q^{56} +1.63261 q^{57} +3.95314 q^{58} +4.12463 q^{59} -1.45461 q^{60} +7.36788 q^{61} +0.812397 q^{62} +11.9821 q^{63} +1.00000 q^{64} -11.2569 q^{65} -0.0355724 q^{66} +5.07658 q^{67} +5.06829 q^{68} +1.29692 q^{69} -15.7430 q^{70} +3.58433 q^{71} +2.84899 q^{72} -2.39990 q^{73} +1.34379 q^{74} +3.50191 q^{75} +4.20126 q^{76} -0.384993 q^{77} -1.16863 q^{78} +0.844769 q^{79} -3.74321 q^{80} +7.66371 q^{81} +5.81137 q^{82} -14.2517 q^{83} -1.63436 q^{84} -18.9717 q^{85} -7.46790 q^{86} -1.53619 q^{87} -0.0915398 q^{88} +13.6717 q^{89} -10.6644 q^{90} -12.6479 q^{91} +3.33740 q^{92} -0.315698 q^{93} +12.4144 q^{94} -15.7262 q^{95} -0.388601 q^{96} -17.1637 q^{97} -10.6883 q^{98} -0.260796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9}+O(q^{10}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9	\( 40 q - 40 q^{2} - q^{3} + 40 q^{4} - 23 q^{5} + q^{6} + 12 q^{7} - 40 q^{8} + 29 q^{9} + 23 q^{10} - 28 q^{11} - q^{12} - 12 q^{13} - 12 q^{14} - 14 q^{15} + 40 q^{16} - 10 q^{17} - 29 q^{18} + 3 q^{19} - 23 q^{20} - 40 q^{21} + 28 q^{22} - 10 q^{23} + q^{24} + 43 q^{25} + 12 q^{26} - 7 q^{27} + 12 q^{28} - 37 q^{29} + 14 q^{30} - 16 q^{31} - 40 q^{32} - 11 q^{33} + 10 q^{34} - 24 q^{35} + 29 q^{36} - 17 q^{37} - 3 q^{38} - 10 q^{39} + 23 q^{40} - 58 q^{41} + 40 q^{42} + 37 q^{43} - 28 q^{44} - 66 q^{45} + 10 q^{46} - 34 q^{47} - q^{48} + 28 q^{49} - 43 q^{50} - 7 q^{51} - 12 q^{52} - 43 q^{53} + 7 q^{54} + 28 q^{55} - 12 q^{56} - 25 q^{57} + 37 q^{58} - 92 q^{59} - 14 q^{60} - 37 q^{61} + 16 q^{62} + 14 q^{63} + 40 q^{64} - 54 q^{65} + 11 q^{66} - 10 q^{67} - 10 q^{68} - 49 q^{69} + 24 q^{70} - 87 q^{71} - 29 q^{72} + 12 q^{73} + 17 q^{74} - 31 q^{75} + 3 q^{76} - 53 q^{77} + 10 q^{78} + 14 q^{79} - 23 q^{80} - 4 q^{81} + 58 q^{82} - 42 q^{83} - 40 q^{84} - 24 q^{85} - 37 q^{86} + 24 q^{87} + 28 q^{88} - 118 q^{89} + 66 q^{90} - 35 q^{91} - 10 q^{92} - 22 q^{93} + 34 q^{94} - 18 q^{95} + q^{96} + 2 q^{97} - 28 q^{98} - 48 q^{99}+O(q^{100}) \) 40 * q - 40 * q^2 - q^3 + 40 * q^4 - 23 * q^5 + q^6 + 12 * q^7 - 40 * q^8 + 29 * q^9 + 23 * q^10 - 28 * q^11 - q^12 - 12 * q^13 - 12 * q^14 - 14 * q^15 + 40 * q^16 - 10 * q^17 - 29 * q^18 + 3 * q^19 - 23 * q^20 - 40 * q^21 + 28 * q^22 - 10 * q^23 + q^24 + 43 * q^25 + 12 * q^26 - 7 * q^27 + 12 * q^28 - 37 * q^29 + 14 * q^30 - 16 * q^31 - 40 * q^32 - 11 * q^33 + 10 * q^34 - 24 * q^35 + 29 * q^36 - 17 * q^37 - 3 * q^38 - 10 * q^39 + 23 * q^40 - 58 * q^41 + 40 * q^42 + 37 * q^43 - 28 * q^44 - 66 * q^45 + 10 * q^46 - 34 * q^47 - q^48 + 28 * q^49 - 43 * q^50 - 7 * q^51 - 12 * q^52 - 43 * q^53 + 7 * q^54 + 28 * q^55 - 12 * q^56 - 25 * q^57 + 37 * q^58 - 92 * q^59 - 14 * q^60 - 37 * q^61 + 16 * q^62 + 14 * q^63 + 40 * q^64 - 54 * q^65 + 11 * q^66 - 10 * q^67 - 10 * q^68 - 49 * q^69 + 24 * q^70 - 87 * q^71 - 29 * q^72 + 12 * q^73 + 17 * q^74 - 31 * q^75 + 3 * q^76 - 53 * q^77 + 10 * q^78 + 14 * q^79 - 23 * q^80 - 4 * q^81 + 58 * q^82 - 42 * q^83 - 40 * q^84 - 24 * q^85 - 37 * q^86 + 24 * q^87 + 28 * q^88 - 118 * q^89 + 66 * q^90 - 35 * q^91 - 10 * q^92 - 22 * q^93 + 34 * q^94 - 18 * q^95 + q^96 + 2 * q^97 - 28 * q^98 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more