LMFDB - Embedded newform 722.3.b.b.721.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,3,Mod(721,722)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("722.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 722 = 2 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	722.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$19.6730750868$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.1
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	722.721
Dual form			722.3.b.b.721.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -0.317837i q^{3} -2.00000 q^{4} -1.00000 q^{5} -0.449490 q^{6} -2.89898 q^{7} +2.82843i q^{8} +8.89898 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} -0.317837i q^{3} -2.00000 q^{4} -1.00000 q^{5} -0.449490 q^{6} -2.89898 q^{7} +2.82843i q^{8} +8.89898 q^{9} +1.41421i q^{10} -5.10102 q^{11} +0.635674i q^{12} -0.174973i q^{13} +4.09978i q^{14} +0.317837i q^{15} +4.00000 q^{16} -11.8990 q^{17} -12.5851i q^{18} +2.00000 q^{20} +0.921404i q^{21} +7.21393i q^{22} -17.0454 q^{23} +0.898979 q^{24} -24.0000 q^{25} -0.247449 q^{26} -5.68896i q^{27} +5.79796 q^{28} +44.5084i q^{29} +0.449490 q^{30} +31.1769i q^{31} -5.65685i q^{32} +1.62129i q^{33} +16.8277i q^{34} +2.89898 q^{35} -17.7980 q^{36} -21.9917i q^{37} -0.0556128 q^{39} -2.82843i q^{40} +54.2008i q^{41} +1.30306 q^{42} +37.3383 q^{43} +10.2020 q^{44} -8.89898 q^{45} +24.1058i q^{46} +81.5403 q^{47} -1.27135i q^{48} -40.5959 q^{49} +33.9411i q^{50} +3.78194i q^{51} +0.349945i q^{52} +55.7579i q^{53} -8.04541 q^{54} +5.10102 q^{55} -8.19955i q^{56} +62.9444 q^{58} +34.5447i q^{59} -0.635674i q^{60} -76.1918 q^{61} +44.0908 q^{62} -25.7980 q^{63} -8.00000 q^{64} +0.174973i q^{65} +2.29286 q^{66} +118.540i q^{67} +23.7980 q^{68} +5.41767i q^{69} -4.09978i q^{70} -75.6067i q^{71} +25.1701i q^{72} +29.3837 q^{73} -31.1010 q^{74} +7.62809i q^{75} +14.7878 q^{77} +0.0786484i q^{78} +66.0716i q^{79} -4.00000 q^{80} +78.2827 q^{81} +76.6515 q^{82} -30.6969 q^{83} -1.84281i q^{84} +11.8990 q^{85} -52.8043i q^{86} +14.1464 q^{87} -14.4279i q^{88} -10.2173i q^{89} +12.5851i q^{90} +0.507242i q^{91} +34.0908 q^{92} +9.90918 q^{93} -115.315i q^{94} -1.79796 q^{96} +148.781i q^{97} +57.4113i q^{98} -45.3939 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{5} + 8 q^{6} + 8 q^{7} + 16 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 4 * q^5 + 8 * q^6 + 8 * q^7 + 16 * q^9	$ 4 q - 8 q^{4} - 4 q^{5} + 8 q^{6} + 8 q^{7} + 16 q^{9} - 40 q^{11} + 16 q^{16} - 28 q^{17} + 8 q^{20} + 20 q^{23} - 16 q^{24} - 96 q^{25} + 48 q^{26} - 16 q^{28} - 8 q^{30} - 8 q^{35} - 32 q^{36} - 108 q^{39} + 64 q^{42} - 76 q^{43} + 80 q^{44} - 16 q^{45} + 140 q^{47} - 84 q^{49} + 56 q^{54} + 40 q^{55} + 144 q^{58} - 148 q^{61} - 64 q^{63} - 32 q^{64} - 128 q^{66} + 56 q^{68} - 196 q^{73} - 144 q^{74} - 176 q^{77} - 16 q^{80} - 20 q^{81} + 336 q^{82} - 64 q^{83} + 28 q^{85} - 12 q^{87} - 40 q^{92} + 216 q^{93} + 32 q^{96} - 64 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^5 + 8 * q^6 + 8 * q^7 + 16 * q^9 - 40 * q^11 + 16 * q^16 - 28 * q^17 + 8 * q^20 + 20 * q^23 - 16 * q^24 - 96 * q^25 + 48 * q^26 - 16 * q^28 - 8 * q^30 - 8 * q^35 - 32 * q^36 - 108 * q^39 + 64 * q^42 - 76 * q^43 + 80 * q^44 - 16 * q^45 + 140 * q^47 - 84 * q^49 + 56 * q^54 + 40 * q^55 + 144 * q^58 - 148 * q^61 - 64 * q^63 - 32 * q^64 - 128 * q^66 + 56 * q^68 - 196 * q^73 - 144 * q^74 - 176 * q^77 - 16 * q^80 - 20 * q^81 + 336 * q^82 - 64 * q^83 + 28 * q^85 - 12 * q^87 - 40 * q^92 + 216 * q^93 + 32 * q^96 - 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/722\mathbb{Z}\right)^\times$.

$n$	$363$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	− 0.317837i	− 0.105946i	−0.998596	−	0.0529729i	$-0.983130\pi$
$3$	− 0.317837i	− 0.105946i	0.998596	−	0.0529729i	$-0.0168697\pi$
$4$	−2.00000	−0.500000
$5$	−1.00000	−0.200000	−0.100000	−	0.994987i	$-0.531884\pi$
$5$	−1.00000	−0.200000	−0.100000	+	0.994987i	$0.531884\pi$
$6$	−0.449490	−0.0749150
$7$	−2.89898	−0.414140	−0.207070	−	0.978326i	$-0.566393\pi$
$7$	−2.89898	−0.414140	−0.207070	+	0.978326i	$0.566393\pi$
$8$	2.82843i	0.353553i
$9$	8.89898	0.988775
$10$	1.41421i	0.141421i
$11$	−5.10102	−0.463729	−0.231865	−	0.972748i	$-0.574483\pi$
$11$	−5.10102	−0.463729	−0.231865	+	0.972748i	$0.574483\pi$
$12$	0.635674i	0.0529729i
$13$	− 0.174973i	− 0.0134594i	−0.999977	−	0.00672972i	$-0.997858\pi$
$13$	− 0.174973i	− 0.0134594i	0.999977	−	0.00672972i	$-0.00214215\pi$
$14$	4.09978i	0.292841i
$15$	0.317837i	0.0211891i
$16$	4.00000	0.250000
$17$	−11.8990	−0.699940	−0.349970	−	0.936761i	$-0.613808\pi$
$17$	−11.8990	−0.699940	−0.349970	+	0.936761i	$0.613808\pi$
$18$	− 12.5851i	− 0.699170i
$19$	0	0
$19$	0	0
$20$	2.00000	0.100000
$21$	0.921404i	0.0438764i
$22$	7.21393i	0.327906i
$23$	−17.0454	−0.741105	−0.370552	−	0.928812i	$-0.620832\pi$
$23$	−17.0454	−0.741105	−0.370552	+	0.928812i	$0.620832\pi$
$24$	0.898979	0.0374575
$25$	−24.0000	−0.960000
$26$	−0.247449	−0.00951726
$27$	− 5.68896i	− 0.210702i
$28$	5.79796	0.207070
$29$	44.5084i	1.53477i	0.641185	+	0.767386i	$0.278443\pi$
$29$	44.5084i	1.53477i	−0.641185	+	0.767386i	$0.721557\pi$
$30$	0.449490	0.0149830
$31$	31.1769i	1.00571i	0.864372	+	0.502853i	$0.167716\pi$
$31$	31.1769i	1.00571i	−0.864372	+	0.502853i	$0.832284\pi$
$32$	− 5.65685i	− 0.176777i
$33$	1.62129i	0.0491301i
$34$	16.8277i	0.494932i
$35$	2.89898	0.0828280
$36$	−17.7980	−0.494388
$37$	− 21.9917i	− 0.594371i	−0.954820	−	0.297186i	$-0.903952\pi$
$37$	− 21.9917i	− 0.594371i	0.954820	−	0.297186i	$-0.0960480\pi$
$38$	0	0
$39$	−0.0556128	−0.00142597
$40$	− 2.82843i	− 0.0707107i
$41$	54.2008i	1.32197i	0.750399	+	0.660986i	$0.229862\pi$
$41$	54.2008i	1.32197i	−0.750399	+	0.660986i	$0.770138\pi$
$42$	1.30306	0.0310253
$43$	37.3383	0.868332	0.434166	−	0.900833i	$-0.357043\pi$
$43$	37.3383	0.868332	0.434166	+	0.900833i	$0.357043\pi$
$44$	10.2020	0.231865
$45$	−8.89898	−0.197755
$46$	24.1058i	0.524040i
$47$	81.5403	1.73490	0.867450	−	0.497524i	$-0.165757\pi$
$47$	81.5403	1.73490	0.867450	+	0.497524i	$0.165757\pi$
$48$	− 1.27135i	− 0.0264864i
$49$	−40.5959	−0.828488
$50$	33.9411i	0.678823i
$51$	3.78194i	0.0741557i
$52$	0.349945i	0.00672972i
$53$	55.7579i	1.05204i	0.850474	+	0.526018i	$0.176316\pi$
$53$	55.7579i	1.05204i	−0.850474	+	0.526018i	$0.823684\pi$
$54$	−8.04541	−0.148989
$55$	5.10102	0.0927458
$56$	− 8.19955i	− 0.146421i
$57$	0	0
$58$	62.9444	1.08525
$59$	34.5447i	0.585503i	0.956189	+	0.292752i	$0.0945709\pi$
$59$	34.5447i	0.585503i	−0.956189	+	0.292752i	$0.905429\pi$
$60$	− 0.635674i	− 0.0105946i
$61$	−76.1918	−1.24905	−0.624523	−	0.781006i	$-0.714707\pi$
$61$	−76.1918	−1.24905	−0.624523	+	0.781006i	$0.714707\pi$
$62$	44.0908	0.711142
$63$	−25.7980	−0.409491
$64$	−8.00000	−0.125000
$65$	0.174973i	0.00269189i
$66$	2.29286	0.0347402
$67$	118.540i	1.76926i	0.466295	+	0.884629i	$0.345588\pi$
$67$	118.540i	1.76926i	−0.466295	+	0.884629i	$0.654412\pi$
$68$	23.7980	0.349970
$69$	5.41767i	0.0785169i
$70$	− 4.09978i	− 0.0585682i
$71$	− 75.6067i	− 1.06488i	−0.846467	−	0.532441i	$-0.821275\pi$
$71$	− 75.6067i	− 1.06488i	0.846467	−	0.532441i	$-0.178725\pi$
$72$	25.1701i	0.349585i
$73$	29.3837	0.402516	0.201258	−	0.979538i	$-0.435497\pi$
$73$	29.3837	0.402516	0.201258	+	0.979538i	$0.435497\pi$
$74$	−31.1010	−0.420284
$75$	7.62809i	0.101708i
$76$	0	0
$77$	14.7878	0.192049
$78$	0.0786484i	0.00100831i
$79$	66.0716i	0.836349i	0.908367	+	0.418174i	$0.137330\pi$
$79$	66.0716i	0.836349i	−0.908367	+	0.418174i	$0.862670\pi$
$80$	−4.00000	−0.0500000
$81$	78.2827	0.966452
$82$	76.6515	0.934775
$83$	−30.6969	−0.369843	−0.184921	−	0.982753i	$-0.559203\pi$
$83$	−30.6969	−0.369843	−0.184921	+	0.982753i	$0.559203\pi$
$84$	− 1.84281i	− 0.0219382i
$85$	11.8990	0.139988
$86$	− 52.8043i	− 0.614003i
$87$	14.1464	0.162603
$88$	− 14.4279i	− 0.163953i
$89$	− 10.2173i	− 0.114801i	−0.998351	−	0.0574007i	$-0.981719\pi$
$89$	− 10.2173i	− 0.114801i	0.998351	−	0.0574007i	$-0.0182813\pi$
$90$	12.5851i	0.139834i
$91$	0.507242i	0.00557409i
$92$	34.0908	0.370552
$93$	9.90918	0.106550
$94$	− 115.315i	− 1.22676i
$95$	0	0
$96$	−1.79796	−0.0187287
$97$	148.781i	1.53383i	0.641749	+	0.766914i	$0.278209\pi$
$97$	148.781i	1.53383i	−0.641749	+	0.766914i	$0.721791\pi$
$98$	57.4113i	0.585830i
$99$	−45.3939	−0.458524
$100$	48.0000	0.480000
$101$	−140.485	−1.39094	−0.695469	−	0.718556i	$-0.744803\pi$
$101$	−140.485	−1.39094	−0.695469	+	0.718556i	$0.744803\pi$
$102$	5.34847	0.0524360
$103$	− 113.965i	− 1.10646i	−0.833028	−	0.553230i	$-0.813395\pi$
$103$	− 113.965i	− 1.10646i	0.833028	−	0.553230i	$-0.186605\pi$
$104$	0.494897	0.00475863
$105$	− 0.921404i	− 0.00877527i
$106$	78.8536	0.743902
$107$	− 113.965i	− 1.06510i	−0.846399	−	0.532549i	$-0.821234\pi$
$107$	− 113.965i	− 1.06510i	0.846399	−	0.532549i	$-0.178766\pi$
$108$	11.3779i	0.105351i
$109$	− 68.4072i	− 0.627589i	−0.949491	−	0.313794i	$-0.898400\pi$
$109$	− 68.4072i	− 0.627589i	0.949491	−	0.313794i	$-0.101600\pi$
$110$	− 7.21393i	− 0.0655812i
$111$	−6.98979	−0.0629711
$112$	−11.5959	−0.103535
$113$	81.9313i	0.725056i	0.931973	+	0.362528i	$0.118086\pi$
$113$	81.9313i	0.725056i	−0.931973	+	0.362528i	$0.881914\pi$
$114$	0	0
$115$	17.0454	0.148221
$116$	− 89.0168i	− 0.767386i
$117$	− 1.55708i	− 0.0133084i
$118$	48.8536	0.414013
$119$	34.4949	0.289873
$120$	−0.898979	−0.00749150
$121$	−94.9796	−0.784955
$122$	107.752i	0.883209i
$123$	17.2270	0.140057
$124$	− 62.3538i	− 0.502853i
$125$	49.0000	0.392000
$126$	36.4838i	0.289554i
$127$	134.146i	1.05627i	0.849160	+	0.528136i	$0.177109\pi$
$127$	134.146i	1.05627i	−0.849160	+	0.528136i	$0.822891\pi$
$128$	11.3137i	0.0883883i
$129$	− 11.8675i	− 0.0919961i
$130$	0.247449	0.00190345
$131$	82.7526	0.631699	0.315849	−	0.948809i	$-0.397711\pi$
$131$	82.7526	0.631699	0.315849	+	0.948809i	$0.397711\pi$
$132$	− 3.24259i	− 0.0245651i
$133$	0	0
$134$	167.641	1.25105
$135$	5.68896i	0.0421405i
$136$	− 33.6554i	− 0.247466i
$137$	−11.8990	−0.0868539	−0.0434269	−	0.999057i	$-0.513828\pi$
$137$	−11.8990	−0.0868539	−0.0434269	+	0.999057i	$0.513828\pi$
$138$	7.66174	0.0555198
$139$	41.7423	0.300305	0.150152	−	0.988663i	$-0.452024\pi$
$139$	41.7423	0.300305	0.150152	+	0.988663i	$0.452024\pi$
$140$	−5.79796	−0.0414140
$141$	− 25.9165i	− 0.183805i
$142$	−106.924	−0.752986
$143$	0.892539i	0.00624153i
$144$	35.5959	0.247194
$145$	− 44.5084i	− 0.306955i
$146$	− 41.5548i	− 0.284622i
$147$	12.9029i	0.0877748i
$148$	43.9835i	0.297186i
$149$	−35.6765	−0.239440	−0.119720	−	0.992808i	$-0.538200\pi$
$149$	−35.6765	−0.239440	−0.119720	+	0.992808i	$0.538200\pi$
$150$	10.7878	0.0719184
$151$	85.5527i	0.566574i	0.959035	+	0.283287i	$0.0914249\pi$
$151$	85.5527i	0.566574i	−0.959035	+	0.283287i	$0.908575\pi$
$152$	0	0
$153$	−105.889	−0.692083
$154$	− 20.9130i	− 0.135799i
$155$	− 31.1769i	− 0.201141i
$156$	0.111226	0.000712985 0
$157$	−200.151	−1.27485	−0.637424	−	0.770514i	$-0.720000\pi$
$157$	−200.151	−1.27485	−0.637424	+	0.770514i	$0.720000\pi$
$158$	93.4393	0.591388
$159$	17.7219	0.111459
$160$	5.65685i	0.0353553i
$161$	49.4143	0.306921
$162$	− 110.708i	− 0.683385i
$163$	−127.303	−0.781000	−0.390500	−	0.920603i	$-0.627698\pi$
$163$	−127.303	−0.781000	−0.390500	+	0.920603i	$0.627698\pi$
$164$	− 108.402i	− 0.660986i
$165$	− 1.62129i	− 0.00982603i
$166$	43.4120i	0.261518i
$167$	− 183.501i	− 1.09881i	−0.835557	−	0.549404i	$-0.814855\pi$
$167$	− 183.501i	− 1.09881i	0.835557	−	0.549404i	$-0.185145\pi$
$168$	−2.60612	−0.0155126
$169$	168.969	0.999819
$170$	− 16.8277i	− 0.0989865i
$171$	0	0
$172$	−74.6765	−0.434166
$173$	66.5355i	0.384598i	0.981336	+	0.192299i	$0.0615944\pi$
$173$	66.5355i	0.384598i	−0.981336	+	0.192299i	$0.938406\pi$
$174$	− 20.0061i	− 0.114977i
$175$	69.5755	0.397574
$176$	−20.4041	−0.115932
$177$	10.9796	0.0620316
$178$	−14.4495	−0.0811769
$179$	24.2134i	0.135270i	0.997710	+	0.0676351i	$0.0215454\pi$
$179$	24.2134i	0.135270i	−0.997710	+	0.0676351i	$0.978455\pi$
$180$	17.7980	0.0988775
$181$	35.6732i	0.197089i	0.995133	+	0.0985447i	$0.0314187\pi$
$181$	35.6732i	0.197089i	−0.995133	+	0.0985447i	$0.968581\pi$
$182$	0.717349	0.00394148
$183$	24.2166i	0.132331i
$184$	− 48.2117i	− 0.262020i
$185$	21.9917i	0.118874i
$186$	− 14.0137i	− 0.0753425i
$187$	60.6969	0.324583
$188$	−163.081	−0.867450
$189$	16.4922i	0.0872602i
$190$	0	0
$191$	−180.252	−0.943728	−0.471864	−	0.881671i	$-0.656419\pi$
$191$	−180.252	−0.943728	−0.471864	+	0.881671i	$0.656419\pi$
$192$	2.54270i	0.0132432i
$193$	− 218.221i	− 1.13068i	−0.824859	−	0.565339i	$-0.808745\pi$
$193$	− 218.221i	− 1.13068i	0.824859	−	0.565339i	$-0.191255\pi$
$194$	210.409	1.08458
$195$	0.0556128	0.000285194 0
$196$	81.1918	0.414244
$197$	340.091	1.72635	0.863175	−	0.504905i	$-0.168473\pi$
$197$	340.091	1.72635	0.863175	+	0.504905i	$0.168473\pi$
$198$	64.1966i	0.324225i
$199$	−333.388	−1.67532	−0.837659	−	0.546194i	$-0.816076\pi$
$199$	−333.388	−1.67532	−0.837659	+	0.546194i	$0.816076\pi$
$200$	− 67.8823i	− 0.339411i
$201$	37.6765	0.187445
$202$	198.675i	0.983541i
$203$	− 129.029i	− 0.635611i
$204$	− 7.56388i	− 0.0370778i
$205$	− 54.2008i	− 0.264394i
$206$	−161.171	−0.782386
$207$	−151.687	−0.732786
$208$	− 0.699891i	− 0.00336486i
$209$	0	0
$210$	−1.30306	−0.00620505
$211$	− 349.585i	− 1.65680i	−0.560135	−	0.828401i	$-0.689251\pi$
$211$	− 349.585i	− 1.65680i	0.560135	−	0.828401i	$-0.310749\pi$
$212$	− 111.516i	− 0.526018i
$213$	−24.0306	−0.112820
$214$	−161.171	−0.753138
$215$	−37.3383	−0.173666
$216$	16.0908	0.0744945
$217$	− 90.3812i	− 0.416503i
$218$	−96.7423	−0.443772
$219$	− 9.33923i	− 0.0426449i
$220$	−10.2020	−0.0463729
$221$	2.08200i	0.00942080i
$222$	9.88506i	0.0445273i
$223$	31.7805i	0.142513i	0.997458	+	0.0712567i	$0.0227009\pi$
$223$	31.7805i	0.142513i	−0.997458	+	0.0712567i	$0.977299\pi$
$224$	16.3991i	0.0732103i
$225$	−213.576	−0.949224
$226$	115.868	0.512692
$227$	249.730i	1.10013i	0.835121	+	0.550066i	$0.185397\pi$
$227$	249.730i	1.10013i	−0.835121	+	0.550066i	$0.814603\pi$
$228$	0	0
$229$	−259.687	−1.13400	−0.567002	−	0.823717i	$-0.691897\pi$
$229$	−259.687	−1.13400	−0.567002	+	0.823717i	$0.691897\pi$
$230$	− 24.1058i	− 0.104808i
$231$	− 4.70010i	− 0.0203467i
$232$	−125.889	−0.542624
$233$	−390.283	−1.67503	−0.837516	−	0.546412i	$-0.815993\pi$
$233$	−390.283	−1.67503	−0.837516	+	0.546412i	$0.815993\pi$
$234$	−2.20204	−0.00941043
$235$	−81.5403	−0.346980
$236$	− 69.0894i	− 0.292752i
$237$	21.0000	0.0886076
$238$	− 48.7832i	− 0.204971i
$239$	−296.677	−1.24132	−0.620662	−	0.784078i	$-0.713136\pi$
$239$	−296.677	−1.24132	−0.620662	+	0.784078i	$0.713136\pi$
$240$	1.27135i	0.00529729i
$241$	− 115.155i	− 0.477821i	−0.971042	−	0.238911i	$-0.923210\pi$
$241$	− 115.155i	− 0.477821i	0.971042	−	0.238911i	$-0.0767903\pi$
$242$	134.321i	0.555047i
$243$	− 76.0818i	− 0.313094i
$244$	152.384	0.624523
$245$	40.5959	0.165698
$246$	− 24.3627i	− 0.0990354i
$247$	0	0
$248$	−88.1816	−0.355571
$249$	9.75663i	0.0391833i
$250$	− 69.2965i	− 0.277186i
$251$	489.924	1.95189	0.975944	−	0.218021i	$-0.0699600\pi$
$251$	489.924	1.95189	0.975944	+	0.218021i	$0.0699600\pi$
$252$	51.5959	0.204746
$253$	86.9490	0.343672
$254$	189.712	0.746897
$255$	− 3.78194i	− 0.0148311i
$256$	16.0000	0.0625000
$257$	− 227.913i	− 0.886822i	−0.896319	−	0.443411i	$-0.853768\pi$
$257$	− 227.913i	− 0.886822i	0.896319	−	0.443411i	$-0.146232\pi$
$258$	−16.7832	−0.0650510
$259$	63.7536i	0.246153i
$260$	− 0.349945i	− 0.00134594i
$261$	396.079i	1.51755i
$262$	− 117.030i	− 0.446679i
$263$	135.924	0.516821	0.258411	−	0.966035i	$-0.416801\pi$
$263$	135.924	0.516821	0.258411	+	0.966035i	$0.416801\pi$
$264$	−4.58571	−0.0173701
$265$	− 55.7579i	− 0.210407i
$266$	0	0
$267$	−3.24745	−0.0121627
$268$	− 237.081i	− 0.884629i
$269$	− 259.912i	− 0.966215i	−0.875561	−	0.483108i	$-0.839508\pi$
$269$	− 259.912i	− 0.966215i	0.875561	−	0.483108i	$-0.160492\pi$
$270$	8.04541	0.0297978
$271$	−232.984	−0.859720	−0.429860	−	0.902896i	$-0.641437\pi$
$271$	−232.984	−0.859720	−0.429860	+	0.902896i	$0.641437\pi$
$272$	−47.5959	−0.174985
$273$	0.161220	0.000590551 0
$274$	16.8277i	0.0614150i
$275$	122.424	0.445180
$276$	− 10.8353i	− 0.0392584i
$277$	−9.44387	−0.0340934	−0.0170467	−	0.999855i	$-0.505426\pi$
$277$	−9.44387	−0.0340934	−0.0170467	+	0.999855i	$0.505426\pi$
$278$	− 59.0326i	− 0.212347i
$279$	277.443i	0.994418i
$280$	8.19955i	0.0292841i
$281$	156.567i	0.557177i	0.960411	+	0.278589i	$0.0898666\pi$
$281$	156.567i	0.557177i	−0.960411	+	0.278589i	$0.910133\pi$
$282$	−36.6515	−0.129970
$283$	2.93418	0.0103681	0.00518407	−	0.999987i	$-0.498350\pi$
$283$	2.93418	0.0103681	0.00518407	+	0.999987i	$0.498350\pi$
$284$	151.213i	0.532441i
$285$	0	0
$286$	1.26224	0.00441343
$287$	− 157.127i	− 0.547481i
$288$	− 50.3402i	− 0.174792i
$289$	−147.414	−0.510084
$290$	−62.9444	−0.217050
$291$	47.2883	0.162503
$292$	−58.7673	−0.201258
$293$	332.361i	1.13434i	0.823601	+	0.567169i	$0.191961\pi$
$293$	332.361i	1.13434i	−0.823601	+	0.567169i	$0.808039\pi$
$294$	18.2474	0.0620662
$295$	− 34.5447i	− 0.117101i
$296$	62.2020	0.210142
$297$	29.0195i	0.0977088i
$298$	50.4542i	0.169309i
$299$	2.98248i	0.00997485i
$300$	− 15.2562i	− 0.0508540i
$301$	−108.243	−0.359611
$302$	120.990	0.400628
$303$	44.6513i	0.147364i
$304$	0	0
$305$	76.1918	0.249809
$306$	149.749i	0.489377i
$307$	− 257.454i	− 0.838613i	−0.907845	−	0.419307i	$-0.862273\pi$
$307$	− 257.454i	− 0.838613i	0.907845	−	0.419307i	$-0.137727\pi$
$308$	−29.5755	−0.0960244
$309$	−36.2225	−0.117225
$310$	−44.0908	−0.142228
$311$	217.666	0.699892	0.349946	−	0.936770i	$-0.386200\pi$
$311$	217.666	0.699892	0.349946	+	0.936770i	$0.386200\pi$
$312$	− 0.157297i	0 0.000504157i
$313$	−310.242	−0.991188	−0.495594	−	0.868554i	$-0.665050\pi$
$313$	−310.242	−0.991188	−0.495594	+	0.868554i	$0.665050\pi$
$314$	283.056i	0.901453i
$315$	25.7980	0.0818983
$316$	− 132.143i	− 0.418174i
$317$	− 345.185i	− 1.08891i	−0.838789	−	0.544456i	$-0.816736\pi$
$317$	− 345.185i	− 1.08891i	0.838789	−	0.544456i	$-0.183264\pi$
$318$	− 25.0626i	− 0.0788132i
$319$	− 227.038i	− 0.711719i
$320$	8.00000	0.0250000
$321$	−36.2225	−0.112843
$322$	− 69.8824i	− 0.217026i
$323$	0	0
$324$	−156.565	−0.483226
$325$	4.19934i	0.0129211i
$326$	180.034i	0.552251i
$327$	−21.7423	−0.0664904
$328$	−153.303	−0.467387
$329$	−236.384	−0.718491
$330$	−2.29286	−0.00694805
$331$	− 487.318i	− 1.47226i	−0.676841	−	0.736130i	$-0.736651\pi$
$331$	− 487.318i	− 1.47226i	0.676841	−	0.736130i	$-0.263349\pi$
$332$	61.3939	0.184921
$333$	− 195.704i	− 0.587700i
$334$	−259.510	−0.776975
$335$	− 118.540i	− 0.353852i
$336$	3.68561i	0.0109691i
$337$	− 166.731i	− 0.494751i	−0.968920	−	0.247376i	$-0.920432\pi$
$337$	− 166.731i	− 0.494751i	0.968920	−	0.247376i	$-0.0795681\pi$
$338$	− 238.959i	− 0.706979i
$339$	26.0408	0.0768166
$340$	−23.7980	−0.0699940
$341$	− 159.034i	− 0.466376i
$342$	0	0
$343$	259.737	0.757250
$344$	105.609i	0.307002i
$345$	− 5.41767i	− 0.0157034i
$346$	94.0954	0.271952
$347$	377.742	1.08859	0.544297	−	0.838892i	$-0.316796\pi$
$347$	377.742	1.08859	0.544297	+	0.838892i	$0.316796\pi$
$348$	−28.2929	−0.0813013
$349$	−586.434	−1.68033	−0.840163	−	0.542334i	$-0.817541\pi$
$349$	−586.434	−1.68033	−0.840163	+	0.542334i	$0.817541\pi$
$350$	− 98.3946i	− 0.281128i
$351$	−0.995413	−0.00283593
$352$	28.8557i	0.0819765i
$353$	476.817	1.35076	0.675379	−	0.737471i	$-0.263980\pi$
$353$	476.817	1.35076	0.675379	+	0.737471i	$0.263980\pi$
$354$	− 15.5275i	− 0.0438630i
$355$	75.6067i	0.212977i
$356$	20.4347i	0.0574007i
$357$	− 10.9638i	− 0.0307108i
$358$	34.2429	0.0956505
$359$	−247.409	−0.689161	−0.344580	−	0.938757i	$-0.611979\pi$
$359$	−247.409	−0.689161	−0.344580	+	0.938757i	$0.611979\pi$
$360$	− 25.1701i	− 0.0699170i
$361$	0	0
$362$	50.4495	0.139363
$363$	30.1881i	0.0831627i
$364$	− 1.01448i	− 0.00278704i
$365$	−29.3837	−0.0805032
$366$	34.2474	0.0935723
$367$	483.924	1.31859	0.659297	−	0.751883i	$-0.270854\pi$
$367$	483.924	1.31859	0.659297	+	0.751883i	$0.270854\pi$
$368$	−68.1816	−0.185276
$369$	482.332i	1.30713i
$370$	31.1010	0.0840568
$371$	− 161.641i	− 0.435690i
$372$	−19.8184	−0.0532752
$373$	140.908i	0.377768i	0.981999	+	0.188884i	$0.0604871\pi$
$373$	140.908i	0.377768i	−0.981999	+	0.188884i	$0.939513\pi$
$374$	− 85.8384i	− 0.229515i
$375$	− 15.5740i	− 0.0415307i
$376$	230.631i	0.613380i
$377$	7.78775	0.0206572
$378$	23.3235	0.0617023
$379$	− 468.633i	− 1.23650i	−0.785982	−	0.618249i	$-0.787842\pi$
$379$	− 468.633i	− 1.23650i	0.785982	−	0.618249i	$-0.212158\pi$
$380$	0	0
$381$	42.6367	0.111907
$382$	254.915i	0.667316i
$383$	557.203i	1.45484i	0.686193	+	0.727420i	$0.259281\pi$
$383$	557.203i	1.45484i	−0.686193	+	0.727420i	$0.740719\pi$
$384$	3.59592	0.00936437
$385$	−14.7878	−0.0384098
$386$	−308.611	−0.799510
$387$	332.272	0.858585
$388$	− 297.563i	− 0.766914i
$389$	93.1204	0.239384	0.119692	−	0.992811i	$-0.461809\pi$
$389$	93.1204	0.239384	0.119692	+	0.992811i	$0.461809\pi$
$390$	− 0.0786484i	0 0.000201663i
$391$	202.823	0.518729
$392$	− 114.823i	− 0.292915i
$393$	− 26.3018i	− 0.0669258i
$394$	− 480.961i	− 1.22071i
$395$	− 66.0716i	− 0.167270i
$396$	90.7878	0.229262
$397$	259.363	0.653308	0.326654	−	0.945144i	$-0.394079\pi$
$397$	259.363	0.653308	0.326654	+	0.945144i	$0.394079\pi$
$398$	471.482i	1.18463i
$399$	0	0
$400$	−96.0000	−0.240000
$401$	449.738i	1.12154i	0.827972	+	0.560770i	$0.189495\pi$
$401$	449.738i	1.12154i	−0.827972	+	0.560770i	$0.810505\pi$
$402$	− 53.2827i	− 0.132544i
$403$	5.45511	0.0135362
$404$	280.969	0.695469
$405$	−78.2827	−0.193290
$406$	−182.474	−0.449445
$407$	112.180i	0.275627i
$408$	−10.6969	−0.0262180
$409$	104.885i	0.256441i	0.991746	+	0.128221i	$0.0409266\pi$
$409$	104.885i	0.256441i	−0.991746	+	0.128221i	$0.959073\pi$
$410$	−76.6515	−0.186955
$411$	3.78194i	0.00920180i
$412$	227.931i	0.553230i
$413$	− 100.144i	− 0.242480i
$414$	214.517i	0.518158i
$415$	30.6969	0.0739685
$416$	−0.989795	−0.00237931
$417$	− 13.2673i	− 0.0318160i
$418$	0	0
$419$	463.121	1.10530	0.552651	−	0.833413i	$-0.313616\pi$
$419$	463.121	1.10530	0.552651	+	0.833413i	$0.313616\pi$
$420$	1.84281i	0.00438764i
$421$	483.486i	1.14842i	0.818707	+	0.574211i	$0.194691\pi$
$421$	483.486i	1.14842i	−0.818707	+	0.574211i	$0.805309\pi$
$422$	−494.388	−1.17154
$423$	725.626	1.71543
$424$	−157.707	−0.371951
$425$	285.576	0.671942
$426$	33.9844i	0.0797756i
$427$	220.879	0.517280
$428$	227.931i	0.532549i
$429$	0.283682	0.000661264 0
$430$	52.8043i	0.122801i
$431$	453.158i	1.05141i	0.850666	+	0.525706i	$0.176199\pi$
$431$	453.158i	1.05141i	−0.850666	+	0.525706i	$0.823801\pi$
$432$	− 22.7558i	− 0.0526756i
$433$	751.307i	1.73512i	0.497332	+	0.867560i	$0.334313\pi$
$433$	751.307i	1.73512i	−0.497332	+	0.867560i	$0.665687\pi$
$434$	−127.818	−0.294512
$435$	−14.1464	−0.0325205
$436$	136.814i	0.313794i
$437$	0	0
$438$	−13.2077	−0.0301545
$439$	− 627.378i	− 1.42911i	−0.699581	−	0.714553i	$-0.746630\pi$
$439$	− 627.378i	− 1.42911i	0.699581	−	0.714553i	$-0.253370\pi$
$440$	14.4279i	0.0327906i
$441$	−361.262	−0.819189
$442$	2.94439	0.00666151
$443$	441.257	0.996065	0.498032	−	0.867158i	$-0.334056\pi$
$443$	441.257	0.996065	0.498032	+	0.867158i	$0.334056\pi$
$444$	13.9796	0.0314856
$445$	10.2173i	0.0229603i
$446$	44.9444	0.100772
$447$	11.3393i	0.0253676i
$448$	23.1918	0.0517675
$449$	− 347.932i	− 0.774904i	−0.921890	−	0.387452i	$-0.873355\pi$
$449$	− 347.932i	− 0.774904i	0.921890	−	0.387452i	$-0.126645\pi$
$450$	302.041i	0.671203i
$451$	− 276.479i	− 0.613037i
$452$	− 163.863i	− 0.362528i
$453$	27.1918	0.0600261
$454$	353.171	0.777911
$455$	− 0.507242i	− 0.00111482i
$456$	0	0
$457$	443.576	0.970625	0.485312	−	0.874341i	$-0.338706\pi$
$457$	443.576	0.970625	0.485312	+	0.874341i	$0.338706\pi$
$458$	367.253i	0.801861i
$459$	67.6928i	0.147479i
$460$	−34.0908	−0.0741105
$461$	195.615	0.424328	0.212164	−	0.977234i	$-0.431949\pi$
$461$	195.615	0.424328	0.212164	+	0.977234i	$0.431949\pi$
$462$	−6.64694	−0.0143873
$463$	−720.958	−1.55715	−0.778573	−	0.627555i	$-0.784056\pi$
$463$	−720.958	−1.55715	−0.778573	+	0.627555i	$0.784056\pi$
$464$	178.034i	0.383693i
$465$	−9.90918	−0.0213101
$466$	551.943i	1.18443i
$467$	271.889	0.582203	0.291101	−	0.956692i	$-0.405978\pi$
$467$	271.889	0.582203	0.291101	+	0.956692i	$0.405978\pi$
$468$	3.11416i	0.00665418i
$469$	− 343.646i	− 0.732721i
$470$	115.315i	0.245352i
$471$	63.6154i	0.135065i
$472$	−97.7071	−0.207007
$473$	−190.463	−0.402671
$474$	− 29.6985i	− 0.0626550i
$475$	0	0
$476$	−68.9898	−0.144937
$477$	496.188i	1.04023i
$478$	419.564i	0.877749i
$479$	347.197	0.724838	0.362419	−	0.932015i	$-0.381951\pi$
$479$	347.197	0.724838	0.362419	+	0.932015i	$0.381951\pi$
$480$	1.79796	0.00374575
$481$	−3.84795	−0.00799990
$482$	−162.854	−0.337870
$483$	− 15.7057i	− 0.0325170i
$484$	189.959	0.392478
$485$	− 148.781i	− 0.306766i
$486$	−107.596	−0.221391
$487$	495.960i	1.01840i	0.860648	+	0.509200i	$0.170058\pi$
$487$	495.960i	1.01840i	−0.860648	+	0.509200i	$0.829942\pi$
$488$	− 215.503i	− 0.441605i
$489$	40.4617i	0.0827437i
$490$	− 57.4113i	− 0.117166i
$491$	534.348	1.08829	0.544143	−	0.838992i	$-0.316855\pi$
$491$	534.348	1.08829	0.544143	+	0.838992i	$0.316855\pi$
$492$	−34.4541	−0.0700286
$493$	− 529.605i	− 1.07425i
$494$	0	0
$495$	45.3939	0.0917048
$496$	124.708i	0.251427i
$497$	219.182i	0.441010i
$498$	13.7980	0.0277067
$499$	790.914	1.58500	0.792499	−	0.609874i	$-0.208780\pi$
$499$	790.914	1.58500	0.792499	+	0.609874i	$0.208780\pi$
$500$	−98.0000	−0.196000
$501$	−58.3235	−0.116414
$502$	− 692.857i	− 1.38019i
$503$	419.681	0.834356	0.417178	−	0.908825i	$-0.363019\pi$
$503$	419.681	0.834356	0.417178	+	0.908825i	$0.363019\pi$
$504$	− 72.9676i	− 0.144777i
$505$	140.485	0.278188
$506$	− 122.964i	− 0.243013i
$507$	− 53.7048i	− 0.105927i
$508$	− 268.293i	− 0.528136i
$509$	803.111i	1.57782i	0.614508	+	0.788911i	$0.289355\pi$
$509$	803.111i	1.57782i	−0.614508	+	0.788911i	$0.710645\pi$
$510$	−5.34847	−0.0104872
$511$	−85.1827	−0.166698
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−322.318	−0.627078
$515$	113.965i	0.221292i
$516$	23.7350i	0.0459980i
$517$	−415.939	−0.804524
$518$	90.1612	0.174056
$519$	21.1475	0.0407465
$520$	−0.494897	−0.000951726 0
$521$	413.663i	0.793979i	0.917823	+	0.396990i	$0.129945\pi$
$521$	413.663i	0.793979i	−0.917823	+	0.396990i	$0.870055\pi$
$522$	560.141	1.07307
$523$	501.306i	0.958520i	0.877673	+	0.479260i	$0.159095\pi$
$523$	501.306i	0.958520i	−0.877673	+	0.479260i	$0.840905\pi$
$524$	−165.505	−0.315849
$525$	− 22.1137i	− 0.0421213i
$526$	− 192.226i	− 0.365448i
$527$	− 370.973i	− 0.703934i
$528$	6.48518i	0.0122825i
$529$	−238.454	−0.450764
$530$	−78.8536	−0.148780
$531$	307.413i	0.578931i
$532$	0	0
$533$	9.48366	0.0177930
$534$	4.59259i	0.00860035i
$535$	113.965i	0.213019i
$536$	−335.283	−0.625527
$537$	7.69591	0.0143313
$538$	−367.571	−0.683217
$539$	207.081	0.384194
$540$	− 11.3779i	− 0.0210702i
$541$	−393.152	−0.726714	−0.363357	−	0.931650i	$-0.618369\pi$
$541$	−393.152	−0.726714	−0.363357	+	0.931650i	$0.618369\pi$
$542$	329.489i	0.607914i
$543$	11.3383	0.0208808
$544$	67.3108i	0.123733i
$545$	68.4072i	0.125518i
$546$	− 0.228000i	0 0.000417583i
$547$	− 482.849i	− 0.882722i	−0.897330	−	0.441361i	$-0.854496\pi$
$547$	− 482.849i	− 0.882722i	0.897330	−	0.441361i	$-0.145504\pi$
$548$	23.7980	0.0434269
$549$	−678.030	−1.23503
$550$	− 173.134i	− 0.314790i
$551$	0	0
$552$	−15.3235	−0.0277599
$553$	− 191.540i	− 0.346365i
$554$	13.3557i	0.0241077i
$555$	6.98979	0.0125942
$556$	−83.4847	−0.150152
$557$	−151.706	−0.272363	−0.136181	−	0.990684i	$-0.543483\pi$
$557$	−151.706	−0.272363	−0.136181	+	0.990684i	$0.543483\pi$
$558$	392.363	0.703160
$559$	− 6.53318i	− 0.0116873i
$560$	11.5959	0.0207070
$561$	− 19.2917i	− 0.0343881i
$562$	221.419	0.393984
$563$	176.634i	0.313737i	0.987620	+	0.156868i	$0.0501399\pi$
$563$	176.634i	0.313737i	−0.987620	+	0.156868i	$0.949860\pi$
$564$	51.8331i	0.0919026i
$565$	− 81.9313i	− 0.145011i
$566$	− 4.14956i	− 0.00733138i
$567$	−226.940	−0.400247
$568$	213.848	0.376493
$569$	− 957.928i	− 1.68353i	−0.539844	−	0.841765i	$-0.681517\pi$
$569$	− 957.928i	− 1.68353i	0.539844	−	0.841765i	$-0.318483\pi$
$570$	0	0
$571$	833.242	1.45927	0.729634	−	0.683838i	$-0.239691\pi$
$571$	833.242	1.45927	0.729634	+	0.683838i	$0.239691\pi$
$572$	− 1.78508i	− 0.00312077i
$573$	57.2908i	0.0999840i
$574$	−222.211	−0.387128
$575$	409.090	0.711460
$576$	−71.1918	−0.123597
$577$	−484.595	−0.839852	−0.419926	−	0.907558i	$-0.637944\pi$
$577$	−484.595	−0.839852	−0.419926	+	0.907558i	$0.637944\pi$
$578$	208.475i	0.360684i
$579$	−69.3587	−0.119790
$580$	89.0168i	0.153477i
$581$	88.9898	0.153167
$582$	− 66.8757i	− 0.114907i
$583$	− 284.422i	− 0.487860i
$584$	83.1096i	0.142311i
$585$	1.55708i	0.00266167i
$586$	470.030	0.802098
$587$	−618.864	−1.05428	−0.527141	−	0.849778i	$-0.676736\pi$
$587$	−618.864	−1.05428	−0.527141	+	0.849778i	$0.676736\pi$
$588$	− 25.8058i	− 0.0438874i
$589$	0	0
$590$	−48.8536	−0.0828027
$591$	− 108.094i	− 0.182899i
$592$	− 87.9670i	− 0.148593i
$593$	−187.940	−0.316931	−0.158465	−	0.987365i	$-0.550655\pi$
$593$	−187.940	−0.316931	−0.158465	+	0.987365i	$0.550655\pi$
$594$	41.0398	0.0690906
$595$	−34.4949	−0.0579746
$596$	71.3531	0.119720
$597$	105.963i	0.177493i
$598$	4.21786	0.00705328
$599$	− 80.3133i	− 0.134079i	−0.997750	−	0.0670394i	$-0.978645\pi$
$599$	− 80.3133i	− 0.134079i	0.997750	−	0.0670394i	$-0.0213553\pi$
$600$	−21.5755	−0.0359592
$601$	665.929i	1.10804i	0.832505	+	0.554018i	$0.186906\pi$
$601$	665.929i	1.10804i	−0.832505	+	0.554018i	$0.813094\pi$
$602$	153.079i	0.254283i
$603$	1054.89i	1.74940i
$604$	− 171.105i	− 0.283287i
$605$	94.9796	0.156991
$606$	63.1464	0.104202
$607$	− 105.952i	− 0.174550i	−0.996184	−	0.0872751i	$-0.972184\pi$
$607$	− 105.952i	− 0.174550i	0.996184	−	0.0872751i	$-0.0278159\pi$
$608$	0	0
$609$	−41.0102	−0.0673402
$610$	− 107.752i	− 0.176642i
$611$	− 14.2673i	− 0.0233508i
$612$	211.778	0.346042
$613$	−1138.34	−1.85700	−0.928501	−	0.371331i	$-0.878902\pi$
$613$	−1138.34	−1.85700	−0.928501	+	0.371331i	$0.878902\pi$
$614$	−364.095	−0.592989
$615$	−17.2270	−0.0280114
$616$	41.8261i	0.0678995i
$617$	918.262	1.48827	0.744135	−	0.668030i	$-0.232862\pi$
$617$	918.262	1.48827	0.744135	+	0.668030i	$0.232862\pi$
$618$	51.2263i	0.0828904i
$619$	−1123.85	−1.81559	−0.907793	−	0.419418i	$-0.862234\pi$
$619$	−1123.85	−1.81559	−0.907793	+	0.419418i	$0.862234\pi$
$620$	62.3538i	0.100571i
$621$	96.9707i	0.156152i
$622$	− 307.827i	− 0.494898i
$623$	29.6198i	0.0475439i
$624$	−0.222451	−0.000356492 0
$625$	551.000	0.881600
$626$	438.748i	0.700876i
$627$	0	0
$628$	400.302	0.637424
$629$	261.679i	0.416024i
$630$	− 36.4838i	− 0.0579108i
$631$	80.5505	0.127655	0.0638277	−	0.997961i	$-0.479669\pi$
$631$	80.5505	0.127655	0.0638277	+	0.997961i	$0.479669\pi$
$632$	−186.879	−0.295694
$633$	−111.111	−0.175531
$634$	−488.166	−0.769978
$635$	− 134.146i	− 0.211254i
$636$	−35.4439	−0.0557294
$637$	7.10318i	0.0111510i
$638$	−321.081	−0.503261
$639$	− 672.822i	− 1.05293i
$640$	− 11.3137i	− 0.0176777i
$641$	495.243i	0.772610i	0.922371	+	0.386305i	$0.126249\pi$
$641$	495.243i	0.772610i	−0.922371	+	0.386305i	$0.873751\pi$
$642$	51.2263i	0.0797917i
$643$	490.246	0.762436	0.381218	−	0.924485i	$-0.375505\pi$
$643$	490.246	0.762436	0.381218	+	0.924485i	$0.375505\pi$
$644$	−98.8286	−0.153461
$645$	11.8675i	0.0183992i
$646$	0	0
$647$	−494.132	−0.763727	−0.381864	−	0.924219i	$-0.624718\pi$
$647$	−494.132	−0.763727	−0.381864	+	0.924219i	$0.624718\pi$
$648$	221.417i	0.341693i
$649$	− 176.213i	− 0.271515i
$650$	5.93877	0.00913657
$651$	−28.7265	−0.0441268
$652$	254.606	0.390500
$653$	475.383	0.727998	0.363999	−	0.931399i	$-0.381411\pi$
$653$	475.383	0.727998	0.363999	+	0.931399i	$0.381411\pi$
$654$	30.7483i	0.0470158i
$655$	−82.7526	−0.126340
$656$	216.803i	0.330493i
$657$	261.485	0.397998
$658$	334.297i	0.508050i
$659$	570.517i	0.865732i	0.901458	+	0.432866i	$0.142498\pi$
$659$	570.517i	0.865732i	−0.901458	+	0.432866i	$0.857502\pi$
$660$	3.24259i	0.00491301i
$661$	− 21.5888i	− 0.0326608i	−0.999867	−	0.0163304i	$-0.994802\pi$
$661$	− 21.5888i	− 0.0326608i	0.999867	−	0.0163304i	$-0.00519835\pi$
$662$	−689.171	−1.04104
$663$	0.661736	0.000998093 0
$664$	− 86.8241i	− 0.130759i
$665$	0	0
$666$	−276.767	−0.415567
$667$	− 758.664i	− 1.13743i
$668$	367.002i	0.549404i
$669$	10.1010	0.0150987
$670$	−167.641	−0.250211
$671$	388.656	0.579219
$672$	5.21225	0.00775632
$673$	− 256.395i	− 0.380973i	−0.981690	−	0.190486i	$-0.938993\pi$
$673$	− 256.395i	− 0.380973i	0.981690	−	0.190486i	$-0.0610065\pi$
$674$	−235.793	−0.349842
$675$	136.535i	0.202274i
$676$	−337.939	−0.499909
$677$	662.815i	0.979048i	0.871990	+	0.489524i	$0.162829\pi$
$677$	662.815i	0.979048i	−0.871990	+	0.489524i	$0.837171\pi$
$678$	− 36.8273i	− 0.0543175i
$679$	− 431.314i	− 0.635220i
$680$	33.6554i	0.0494932i
$681$	79.3735	0.116554
$682$	−224.908	−0.329777
$683$	310.334i	0.454369i	0.973852	+	0.227184i	$0.0729520\pi$
$683$	310.334i	0.454369i	−0.973852	+	0.227184i	$0.927048\pi$
$684$	0	0
$685$	11.8990	0.0173708
$686$	− 367.323i	− 0.535457i
$687$	82.5381i	0.120143i
$688$	149.353	0.217083
$689$	9.75611	0.0141598
$690$	−7.66174	−0.0111040
$691$	187.789	0.271764	0.135882	−	0.990725i	$-0.456613\pi$
$691$	187.789	0.271764	0.135882	+	0.990725i	$0.456613\pi$
$692$	− 133.071i	− 0.192299i
$693$	131.596	0.189893
$694$	− 534.208i	− 0.769753i
$695$	−41.7423	−0.0600609
$696$	40.0121i	0.0574887i
$697$	− 644.934i	− 0.925300i
$698$	829.342i	1.18817i
$699$	124.046i	0.177463i
$700$	−139.151	−0.198787
$701$	541.636	0.772661	0.386331	−	0.922360i	$-0.373742\pi$
$701$	541.636	0.772661	0.386331	+	0.922360i	$0.373742\pi$
$702$	1.40773i	0.00200531i
$703$	0	0
$704$	40.8082	0.0579661
$705$	25.9165i	0.0367611i
$706$	− 674.322i	− 0.955130i
$707$	407.262	0.576043
$708$	−21.9592	−0.0310158
$709$	−491.243	−0.692867	−0.346434	−	0.938074i	$-0.612607\pi$
$709$	−491.243	−0.692867	−0.346434	+	0.938074i	$0.612607\pi$
$710$	106.924	0.150597
$711$	587.969i	0.826961i
$712$	28.8990	0.0405885
$713$	− 531.423i	− 0.745334i
$714$	−15.5051	−0.0217158
$715$	− 0.892539i	− 0.00124831i
$716$	− 48.4267i	− 0.0676351i
$717$	94.2949i	0.131513i
$718$	349.889i	0.487310i
$719$	−575.468	−0.800373	−0.400186	−	0.916434i	$-0.631055\pi$
$719$	−575.468	−0.800373	−0.400186	+	0.916434i	$0.631055\pi$
$720$	−35.5959	−0.0494388
$721$	330.383i	0.458229i
$722$	0	0
$723$	−36.6005	−0.0506231
$724$	− 71.3464i	− 0.0985447i
$725$	− 1068.20i	− 1.47338i
$726$	42.6924	0.0588049
$727$	−819.105	−1.12669	−0.563346	−	0.826221i	$-0.690486\pi$
$727$	−819.105	−1.12669	−0.563346	+	0.826221i	$0.690486\pi$
$728$	−1.43470	−0.00197074
$729$	680.362	0.933282
$730$	41.5548i	0.0569244i
$731$	−444.287	−0.607780
$732$	− 48.4332i	− 0.0661656i
$733$	804.332	1.09731	0.548657	−	0.836047i	$-0.315139\pi$
$733$	804.332	1.09731	0.548657	+	0.836047i	$0.315139\pi$
$734$	− 684.372i	− 0.932387i
$735$	− 12.9029i	− 0.0175550i
$736$	96.4234i	0.131010i
$737$	− 604.677i	− 0.820457i
$738$	682.120	0.924282
$739$	75.8627	0.102656	0.0513280	−	0.998682i	$-0.483655\pi$
$739$	75.8627	0.102656	0.0513280	+	0.998682i	$0.483655\pi$
$740$	− 43.9835i	− 0.0594371i
$741$	0	0
$742$	−228.595	−0.308079
$743$	− 584.059i	− 0.786082i	−0.919521	−	0.393041i	$-0.871423\pi$
$743$	− 584.059i	− 0.786082i	0.919521	−	0.393041i	$-0.128577\pi$
$744$	28.0274i	0.0376712i
$745$	35.6765	0.0478880
$746$	199.273	0.267123
$747$	−273.171	−0.365691
$748$	−121.394	−0.162291
$749$	330.383i	0.441099i
$750$	−22.0250	−0.0293667
$751$	593.437i	0.790196i	0.918639	+	0.395098i	$0.129289\pi$
$751$	593.437i	0.790196i	−0.918639	+	0.395098i	$0.870711\pi$
$752$	326.161	0.433725
$753$	− 155.716i	− 0.206794i
$754$	− 11.0135i	− 0.0146068i
$755$	− 85.5527i	− 0.113315i
$756$	− 32.9844i	− 0.0436301i
$757$	−556.514	−0.735158	−0.367579	−	0.929992i	$-0.619813\pi$
$757$	−556.514	−0.735158	−0.367579	+	0.929992i	$0.619813\pi$
$758$	−662.747	−0.874336
$759$	− 27.6356i	− 0.0364106i
$760$	0	0
$761$	−1211.85	−1.59244	−0.796219	−	0.605008i	$-0.793170\pi$
$761$	−1211.85	−1.59244	−0.796219	+	0.605008i	$0.793170\pi$
$762$	− 60.2975i	− 0.0791305i
$763$	198.311i	0.259910i
$764$	360.504	0.471864
$765$	105.889	0.138417
$766$	788.005	1.02873
$767$	6.04438	0.00788054
$768$	− 5.08540i	− 0.00662161i
$769$	−958.231	−1.24607	−0.623037	−	0.782193i	$-0.714101\pi$
$769$	−958.231	−1.24607	−0.623037	+	0.782193i	$0.714101\pi$
$770$	20.9130i	0.0271598i
$771$	−72.4393	−0.0939550
$772$	436.441i	0.565339i
$773$	− 466.972i	− 0.604103i	−0.953292	−	0.302051i	$-0.902329\pi$
$773$	− 466.972i	− 0.604103i	0.953292	−	0.302051i	$-0.0976714\pi$
$774$	− 469.904i	− 0.607111i
$775$	− 748.246i	− 0.965479i
$776$	−420.817	−0.542290
$777$	20.2633	0.0260789
$778$	− 131.692i	− 0.169270i
$779$	0	0
$780$	−0.111226	−0.000142597 0
$781$	385.671i	0.493817i
$782$	− 286.835i	− 0.366797i
$783$	253.207	0.323380
$784$	−162.384	−0.207122
$785$	200.151	0.254969
$786$	−37.1964	−0.0473237
$787$	− 1410.48i	− 1.79223i	−0.443824	−	0.896114i	$-0.646379\pi$
$787$	− 1410.48i	− 1.79223i	0.443824	−	0.896114i	$-0.353621\pi$
$788$	−680.182	−0.863175
$789$	− 43.2017i	− 0.0547550i
$790$	−93.4393	−0.118278
$791$	− 237.517i	− 0.300275i
$792$	− 128.393i	− 0.162113i
$793$	13.3315i	0.0168115i
$794$	− 366.795i	− 0.461958i
$795$	−17.7219	−0.0222917
$796$	666.777	0.837659
$797$	774.540i	0.971819i	0.874009	+	0.485909i	$0.161511\pi$
$797$	774.540i	0.971819i	−0.874009	+	0.485909i	$0.838489\pi$
$798$	0	0
$799$	−970.246	−1.21433
$800$	135.765i	0.169706i
$801$	− 90.9238i	− 0.113513i
$802$	636.025	0.793049
$803$	−149.887	−0.186658
$804$	−75.3531	−0.0937227
$805$	−49.4143	−0.0613842
$806$	− 7.71469i	− 0.00957157i
$807$	−82.6097	−0.102366
$808$	− 397.351i	− 0.491771i
$809$	−1114.09	−1.37712	−0.688559	−	0.725180i	$-0.741756\pi$
$809$	−1114.09	−1.37712	−0.688559	+	0.725180i	$0.741756\pi$
$810$	110.708i	0.136677i
$811$	− 15.8951i	− 0.0195994i	−0.999952	−	0.00979970i	$-0.996881\pi$
$811$	− 15.8951i	− 0.0195994i	0.999952	−	0.00979970i	$-0.00311939\pi$
$812$	258.058i	0.317805i
$813$	74.0510i	0.0910837i
$814$	158.647	0.194898
$815$	127.303	0.156200
$816$	15.1278i	0.0185389i
$817$	0	0
$818$	148.329	0.181331
$819$	4.51394i	0.00551152i
$820$	108.402i	0.132197i
$821$	−204.889	−0.249560	−0.124780	−	0.992184i	$-0.539823\pi$
$821$	−204.889	−0.249560	−0.124780	+	0.992184i	$0.539823\pi$
$822$	5.34847	0.00650665
$823$	1242.10	1.50924	0.754619	−	0.656163i	$-0.227822\pi$
$823$	1242.10	1.50924	0.754619	+	0.656163i	$0.227822\pi$
$824$	322.343	0.391193
$825$	− 38.9111i	− 0.0471649i
$826$	−141.626	−0.171459
$827$	− 1143.30i	− 1.38247i	−0.722631	−	0.691234i	$-0.757067\pi$
$827$	− 1143.30i	− 1.38247i	0.722631	−	0.691234i	$-0.242933\pi$
$828$	303.373	0.366393
$829$	− 1080.66i	− 1.30357i	−0.758404	−	0.651784i	$-0.774021\pi$
$829$	− 1080.66i	− 1.30357i	0.758404	−	0.651784i	$-0.225979\pi$
$830$	− 43.4120i	− 0.0523036i
$831$	3.00161i	0.00361205i
$832$	1.39978i	0.00168243i
$833$	483.050	0.579892
$834$	−18.7628	−0.0224973
$835$	183.501i	0.219762i
$836$	0	0
$837$	177.364	0.211905
$838$	− 654.953i	− 0.781566i
$839$	− 672.726i	− 0.801819i	−0.916118	−	0.400909i	$-0.868694\pi$
$839$	− 672.726i	− 0.801819i	0.916118	−	0.400909i	$-0.131306\pi$
$840$	2.60612	0.00310253
$841$	−1140.00	−1.35553
$842$	683.753	0.812058
$843$	49.7628	0.0590306
$844$	699.171i	0.828401i
$845$	−168.969	−0.199964
$846$	− 1026.19i	− 1.21299i
$847$	275.344	0.325081
$848$	223.032i	0.263009i
$849$	− 0.932592i	− 0.00109846i
$850$	− 403.865i	− 0.475135i
$851$	374.858i	0.440491i
$852$	48.0612	0.0564099
$853$	247.241	0.289849	0.144924	−	0.989443i	$-0.453706\pi$
$853$	247.241	0.289849	0.144924	+	0.989443i	$0.453706\pi$
$854$	− 312.369i	− 0.365772i
$855$	0	0
$856$	322.343	0.376569
$857$	710.402i	0.828941i	0.910063	+	0.414470i	$0.136033\pi$
$857$	710.402i	0.828941i	−0.910063	+	0.414470i	$0.863967\pi$
$858$	− 0.401187i	0 0.000467584i
$859$	1010.37	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$859$	1010.37	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$860$	74.6765	0.0868332
$861$	−49.9408	−0.0580033
$862$	640.863	0.743460
$863$	455.826i	0.528188i	0.964497	+	0.264094i	$0.0850729\pi$
$863$	455.826i	0.528188i	−0.964497	+	0.264094i	$0.914927\pi$
$864$	−32.1816	−0.0372473
$865$	− 66.5355i	− 0.0769196i
$866$	1062.51	1.22692
$867$	46.8538i	0.0540412i
$868$	180.762i	0.208252i
$869$	− 337.032i	− 0.387839i
$870$	20.0061i	0.0229955i
$871$	20.7413	0.0238132
$872$	193.485	0.221886
$873$	1324.00i	1.51661i
$874$	0	0
$875$	−142.050	−0.162343
$876$	18.6785i	0.0213224i
$877$	− 651.320i	− 0.742668i	−0.928499	−	0.371334i	$-0.878900\pi$
$877$	− 651.320i	− 0.742668i	0.928499	−	0.371334i	$-0.121100\pi$
$878$	−887.246	−1.01053
$879$	105.637	0.120178
$880$	20.4041	0.0231865
$881$	−518.293	−0.588301	−0.294150	−	0.955759i	$-0.595037\pi$
$881$	−518.293	−0.588301	−0.294150	+	0.955759i	$0.595037\pi$
$882$	510.902i	0.579254i
$883$	−673.064	−0.762247	−0.381123	−	0.924524i	$-0.624463\pi$
$883$	−673.064	−0.762247	−0.381123	+	0.924524i	$0.624463\pi$
$884$	− 4.16399i	− 0.00471040i
$885$	−10.9796	−0.0124063
$886$	− 624.031i	− 0.704324i
$887$	1496.87i	1.68756i	0.536688	+	0.843781i	$0.319675\pi$
$887$	1496.87i	1.68756i	−0.536688	+	0.843781i	$0.680325\pi$
$888$	− 19.7701i	− 0.0222637i
$889$	− 388.888i	− 0.437444i
$890$	14.4495	0.0162354
$891$	−399.321	−0.448172
$892$	− 63.5610i	− 0.0712567i
$893$	0	0
$894$	16.0362	0.0179376
$895$	− 24.2134i	− 0.0270540i
$896$	− 32.7982i	− 0.0366051i
$897$	0.947943	0.00105679
$898$	−492.050	−0.547940
$899$	−1387.63	−1.54353
$900$	427.151	0.474612
$901$	− 663.462i	− 0.736362i
$902$	−391.001	−0.433482
$903$	34.4036i	0.0380992i
$904$	−231.737	−0.256346
$905$	− 35.6732i	− 0.0394179i
$906$	− 38.4551i	− 0.0424449i
$907$	− 280.724i	− 0.309508i	−0.987953	−	0.154754i	$-0.950541\pi$
$907$	− 280.724i	− 0.309508i	0.987953	−	0.154754i	$-0.0494585\pi$
$908$	− 499.460i	− 0.550066i
$909$	−1250.17	−1.37532
$910$	−0.717349	−0.000788295 0
$911$	783.689i	0.860252i	0.902769	+	0.430126i	$0.141531\pi$
$911$	783.689i	0.860252i	−0.902769	+	0.430126i	$0.858469\pi$
$912$	0	0
$913$	156.586	0.171507
$914$	− 627.310i	− 0.686335i
$915$	− 24.2166i	− 0.0264662i
$916$	519.373	0.567002
$917$	−239.898	−0.261612
$918$	95.7321	0.104283
$919$	−220.334	−0.239754	−0.119877	−	0.992789i	$-0.538250\pi$
$919$	−220.334	−0.239754	−0.119877	+	0.992789i	$0.538250\pi$
$920$	48.2117i	0.0524040i
$921$	−81.8286	−0.0888475
$922$	− 276.642i	− 0.300045i
$923$	−13.2291	−0.0143327
$924$	9.40020i	0.0101734i
$925$	527.802i	0.570597i
$926$	1019.59i	1.10107i
$927$	− 1014.18i	− 1.09404i
$928$	251.778	0.271312
$929$	1229.48	1.32345	0.661724	−	0.749747i	$-0.269825\pi$
$929$	1229.48	1.32345	0.661724	+	0.749747i	$0.269825\pi$
$930$	14.0137i	0.0150685i
$931$	0	0
$932$	780.565	0.837516
$933$	− 69.1825i	− 0.0741506i
$934$	− 384.509i	− 0.411680i
$935$	−60.6969	−0.0649165
$936$	4.40408	0.00470522
$937$	−134.212	−0.143236	−0.0716181	−	0.997432i	$-0.522816\pi$
$937$	−134.212	−0.143236	−0.0716181	+	0.997432i	$0.522816\pi$
$938$	−485.989	−0.518112
$939$	98.6064i	0.105012i
$940$	163.081	0.173490
$941$	− 598.537i	− 0.636064i	−0.948080	−	0.318032i	$-0.896978\pi$
$941$	− 598.537i	− 0.636064i	0.948080	−	0.318032i	$-0.103022\pi$
$942$	89.9658	0.0955051
$943$	− 923.875i	− 0.979719i
$944$	138.179i	0.146376i
$945$	− 16.4922i	− 0.0174520i
$946$	269.356i	0.284731i
$947$	820.530	0.866452	0.433226	−	0.901285i	$-0.357375\pi$
$947$	820.530	0.866452	0.433226	+	0.901285i	$0.357375\pi$
$948$	−42.0000	−0.0443038
$949$	− 5.14134i	− 0.00541764i
$950$	0	0
$951$	−109.713	−0.115366
$952$	97.5663i	0.102486i
$953$	78.6775i	0.0825577i	0.999148	+	0.0412789i	$0.0131432\pi$
$953$	78.6775i	0.0825577i	−0.999148	+	0.0412789i	$0.986857\pi$
$954$	701.716	0.735552
$955$	180.252	0.188746
$956$	593.353	0.620662
$957$	−72.1612	−0.0754036
$958$	− 491.011i	− 0.512538i
$959$	34.4949	0.0359697
$960$	− 2.54270i	− 0.00264864i
$961$	−11.0000	−0.0114464
$962$	5.44183i	0.00565679i
$963$	− 1014.18i	− 1.05314i
$964$	230.310i	0.238911i
$965$	218.221i	0.226135i
$966$	−22.2112	−0.0229930
$967$	1329.26	1.37462	0.687310	−	0.726365i	$-0.258792\pi$
$967$	1329.26	1.37462	0.687310	+	0.726365i	$0.258792\pi$
$968$	− 268.643i	− 0.277524i
$969$	0	0
$970$	−210.409	−0.216916
$971$	568.787i	0.585774i	0.956147	+	0.292887i	$0.0946160\pi$
$971$	568.787i	0.585774i	−0.956147	+	0.292887i	$0.905384\pi$
$972$	152.164i	0.156547i
$973$	−121.010	−0.124368
$974$	701.394	0.720117
$975$	1.33471	0.00136893
$976$	−304.767	−0.312262
$977$	− 884.903i	− 0.905735i	−0.891578	−	0.452867i	$-0.850401\pi$
$977$	− 884.903i	− 0.905735i	0.891578	−	0.452867i	$-0.149599\pi$
$978$	57.2214	0.0585086
$979$	52.1188i	0.0532368i
$980$	−81.1918	−0.0828488
$981$	− 608.754i	− 0.620544i
$982$	− 755.683i	− 0.769534i
$983$	− 335.835i	− 0.341643i	−0.985302	−	0.170821i	$-0.945358\pi$
$983$	− 335.835i	− 0.341643i	0.985302	−	0.170821i	$-0.0546421\pi$
$984$	48.7254i	0.0495177i
$985$	−340.091	−0.345270
$986$	−748.974	−0.759608
$987$	75.1315i	0.0761211i
$988$	0	0
$989$	−636.446	−0.643525
$990$	− 64.1966i	− 0.0648451i
$991$	567.458i	0.572611i	0.958138	+	0.286306i	$0.0924273\pi$
$991$	567.458i	0.572611i	−0.958138	+	0.286306i	$0.907573\pi$
$992$	176.363	0.177786
$993$	−154.888	−0.155980
$994$	309.970	0.311841
$995$	333.388	0.335064
$996$	− 19.5133i	− 0.0195916i
$997$	−434.757	−0.436065	−0.218033	−	0.975941i	$-0.569964\pi$
$997$	−434.757	−0.436065	−0.218033	+	0.975941i	$0.569964\pi$
$998$	− 1118.52i	− 1.12076i
$999$	−125.110	−0.125235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.3.b.b.721.1		4
19.7	even	3		38.3.d.a.27.2	✓	4
19.8	odd	6		38.3.d.a.31.2	yes	4
19.18	odd	2	inner	722.3.b.b.721.4		4
57.8	even	6		342.3.m.a.145.1		4
57.26	odd	6		342.3.m.a.217.1		4
76.7	odd	6		304.3.r.a.65.2		4
76.27	even	6		304.3.r.a.145.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.d.a.27.2	✓	4	19.7	even	3
38.3.d.a.31.2	yes	4	19.8	odd	6
304.3.r.a.65.2		4	76.7	odd	6
304.3.r.a.145.2		4	76.27	even	6
342.3.m.a.145.1		4	57.8	even	6
342.3.m.a.217.1		4	57.26	odd	6
722.3.b.b.721.1		4	1.1	even	1	trivial
722.3.b.b.721.4		4	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	722.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -0.317837i q^{3} -2.00000 q^{4} -1.00000 q^{5} -0.449490 q^{6} -2.89898 q^{7} +2.82843i q^{8} +8.89898 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -0.317837i q^{3} -2.00000 q^{4} -1.00000 q^{5} -0.449490 q^{6} -2.89898 q^{7} +2.82843i q^{8} +8.89898 q^{9} +1.41421i q^{10} -5.10102 q^{11} +0.635674i q^{12} -0.174973i q^{13} +4.09978i q^{14} +0.317837i q^{15} +4.00000 q^{16} -11.8990 q^{17} -12.5851i q^{18} +2.00000 q^{20} +0.921404i q^{21} +7.21393i q^{22} -17.0454 q^{23} +0.898979 q^{24} -24.0000 q^{25} -0.247449 q^{26} -5.68896i q^{27} +5.79796 q^{28} +44.5084i q^{29} +0.449490 q^{30} +31.1769i q^{31} -5.65685i q^{32} +1.62129i q^{33} +16.8277i q^{34} +2.89898 q^{35} -17.7980 q^{36} -21.9917i q^{37} -0.0556128 q^{39} -2.82843i q^{40} +54.2008i q^{41} +1.30306 q^{42} +37.3383 q^{43} +10.2020 q^{44} -8.89898 q^{45} +24.1058i q^{46} +81.5403 q^{47} -1.27135i q^{48} -40.5959 q^{49} +33.9411i q^{50} +3.78194i q^{51} +0.349945i q^{52} +55.7579i q^{53} -8.04541 q^{54} +5.10102 q^{55} -8.19955i q^{56} +62.9444 q^{58} +34.5447i q^{59} -0.635674i q^{60} -76.1918 q^{61} +44.0908 q^{62} -25.7980 q^{63} -8.00000 q^{64} +0.174973i q^{65} +2.29286 q^{66} +118.540i q^{67} +23.7980 q^{68} +5.41767i q^{69} -4.09978i q^{70} -75.6067i q^{71} +25.1701i q^{72} +29.3837 q^{73} -31.1010 q^{74} +7.62809i q^{75} +14.7878 q^{77} +0.0786484i q^{78} +66.0716i q^{79} -4.00000 q^{80} +78.2827 q^{81} +76.6515 q^{82} -30.6969 q^{83} -1.84281i q^{84} +11.8990 q^{85} -52.8043i q^{86} +14.1464 q^{87} -14.4279i q^{88} -10.2173i q^{89} +12.5851i q^{90} +0.507242i q^{91} +34.0908 q^{92} +9.90918 q^{93} -115.315i q^{94} -1.79796 q^{96} +148.781i q^{97} +57.4113i q^{98} -45.3939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{5} + 8 q^{6} + 8 q^{7} + 16 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 4 * q^5 + 8 * q^6 + 8 * q^7 + 16 * q^9	\( 4 q - 8 q^{4} - 4 q^{5} + 8 q^{6} + 8 q^{7} + 16 q^{9} - 40 q^{11} + 16 q^{16} - 28 q^{17} + 8 q^{20} + 20 q^{23} - 16 q^{24} - 96 q^{25} + 48 q^{26} - 16 q^{28} - 8 q^{30} - 8 q^{35} - 32 q^{36} - 108 q^{39} + 64 q^{42} - 76 q^{43} + 80 q^{44} - 16 q^{45} + 140 q^{47} - 84 q^{49} + 56 q^{54} + 40 q^{55} + 144 q^{58} - 148 q^{61} - 64 q^{63} - 32 q^{64} - 128 q^{66} + 56 q^{68} - 196 q^{73} - 144 q^{74} - 176 q^{77} - 16 q^{80} - 20 q^{81} + 336 q^{82} - 64 q^{83} + 28 q^{85} - 12 q^{87} - 40 q^{92} + 216 q^{93} + 32 q^{96} - 64 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^5 + 8 * q^6 + 8 * q^7 + 16 * q^9 - 40 * q^11 + 16 * q^16 - 28 * q^17 + 8 * q^20 + 20 * q^23 - 16 * q^24 - 96 * q^25 + 48 * q^26 - 16 * q^28 - 8 * q^30 - 8 * q^35 - 32 * q^36 - 108 * q^39 + 64 * q^42 - 76 * q^43 + 80 * q^44 - 16 * q^45 + 140 * q^47 - 84 * q^49 + 56 * q^54 + 40 * q^55 + 144 * q^58 - 148 * q^61 - 64 * q^63 - 32 * q^64 - 128 * q^66 + 56 * q^68 - 196 * q^73 - 144 * q^74 - 176 * q^77 - 16 * q^80 - 20 * q^81 + 336 * q^82 - 64 * q^83 + 28 * q^85 - 12 * q^87 - 40 * q^92 + 216 * q^93 + 32 * q^96 - 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more