LMFDB - Embedded newform 5239.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5239,2,Mod(1,5239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5239, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5239.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.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:	$41.8336256189$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 17 x^{14} + 80 x^{13} + 98 x^{12} - 628 x^{11} - 158 x^{10} + 2458 x^{9} + \cdots - 147 $ x^16 - 4x^15 - 17x^14 + 80x^13 + 98x^12 - 628x^11 - 158x^10 + 2458x^9 - 510x^8 - 5030x^7 + 2318x^6 + 5112x^5 - 3154x^4 - 2086x^3 + 1542x^2 + 124*x - 147
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.44935$ of defining polynomial
Character	$\chi$	$=$	5239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.44935 q^{2} +0.0382575 q^{3} +0.100611 q^{4} +3.85977 q^{5} -0.0554484 q^{6} -1.66255 q^{7} +2.75288 q^{8} -2.99854 q^{9} +O(q^{10})$	\(q-1.44935 q^{2} +0.0382575 q^{3} +0.100611 q^{4} +3.85977 q^{5} -0.0554484 q^{6} -1.66255 q^{7} +2.75288 q^{8} -2.99854 q^{9} -5.59415 q^{10} -1.88038 q^{11} +0.00384914 q^{12} +2.40961 q^{14} +0.147665 q^{15} -4.19110 q^{16} -3.75015 q^{17} +4.34592 q^{18} +3.26949 q^{19} +0.388337 q^{20} -0.0636049 q^{21} +2.72532 q^{22} -5.31401 q^{23} +0.105318 q^{24} +9.89783 q^{25} -0.229489 q^{27} -0.167271 q^{28} +3.11168 q^{29} -0.214018 q^{30} +1.00000 q^{31} +0.568613 q^{32} -0.0719384 q^{33} +5.43528 q^{34} -6.41705 q^{35} -0.301687 q^{36} +11.5806 q^{37} -4.73863 q^{38} +10.6255 q^{40} -7.94532 q^{41} +0.0921856 q^{42} +9.24319 q^{43} -0.189187 q^{44} -11.5737 q^{45} +7.70185 q^{46} -6.97995 q^{47} -0.160341 q^{48} -4.23593 q^{49} -14.3454 q^{50} -0.143471 q^{51} +6.11470 q^{53} +0.332609 q^{54} -7.25782 q^{55} -4.57679 q^{56} +0.125082 q^{57} -4.50991 q^{58} +12.1104 q^{59} +0.0148568 q^{60} +1.75416 q^{61} -1.44935 q^{62} +4.98521 q^{63} +7.55808 q^{64} +0.104264 q^{66} +7.94479 q^{67} -0.377308 q^{68} -0.203300 q^{69} +9.30055 q^{70} +0.735383 q^{71} -8.25460 q^{72} -13.5423 q^{73} -16.7843 q^{74} +0.378666 q^{75} +0.328948 q^{76} +3.12622 q^{77} -5.66333 q^{79} -16.1767 q^{80} +8.98683 q^{81} +11.5155 q^{82} -13.7729 q^{83} -0.00639938 q^{84} -14.4747 q^{85} -13.3966 q^{86} +0.119045 q^{87} -5.17644 q^{88} -10.2781 q^{89} +16.7743 q^{90} -0.534650 q^{92} +0.0382575 q^{93} +10.1164 q^{94} +12.6195 q^{95} +0.0217537 q^{96} -8.34364 q^{97} +6.13935 q^{98} +5.63838 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} - 2 q^{3} + 18 q^{4} - 4 q^{5} - 6 q^{6} - 2 q^{7} - 12 q^{8} + 10 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 - 2 * q^3 + 18 * q^4 - 4 * q^5 - 6 * q^6 - 2 * q^7 - 12 * q^8 + 10 * q^9	$ 16 q - 4 q^{2} - 2 q^{3} + 18 q^{4} - 4 q^{5} - 6 q^{6} - 2 q^{7} - 12 q^{8} + 10 q^{9} - 2 q^{10} - 14 q^{11} + 8 q^{12} - 8 q^{14} + 14 q^{16} + 4 q^{17} + 28 q^{18} - 22 q^{19} - 28 q^{20} - 12 q^{21} - 8 q^{22} + 4 q^{23} + 8 q^{24} - 2 q^{25} + 10 q^{27} - 16 q^{28} - 8 q^{29} - 20 q^{30} + 16 q^{31} - 48 q^{32} - 10 q^{33} - 8 q^{34} - 2 q^{35} + 22 q^{36} - 16 q^{37} - 6 q^{38} + 14 q^{40} - 44 q^{41} + 14 q^{42} + 16 q^{43} - 4 q^{44} - 56 q^{45} - 10 q^{47} + 32 q^{49} - 2 q^{50} - 6 q^{53} - 24 q^{54} + 22 q^{55} - 4 q^{56} + 8 q^{57} - 74 q^{58} - 2 q^{59} - 40 q^{60} + 8 q^{61} - 4 q^{62} - 56 q^{63} + 38 q^{64} - 34 q^{66} + 8 q^{67} + 32 q^{68} - 10 q^{69} + 108 q^{70} - 50 q^{71} + 44 q^{72} - 14 q^{73} + 8 q^{74} - 44 q^{76} + 16 q^{77} + 32 q^{79} - 68 q^{80} - 8 q^{81} - 6 q^{82} + 20 q^{83} - 136 q^{84} + 32 q^{85} - 8 q^{86} - 36 q^{87} - 40 q^{88} - 52 q^{89} - 34 q^{90} + 14 q^{92} - 2 q^{93} + 44 q^{94} - 2 q^{95} + 80 q^{96} - 18 q^{97} - 12 q^{98} - 38 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 - 2 * q^3 + 18 * q^4 - 4 * q^5 - 6 * q^6 - 2 * q^7 - 12 * q^8 + 10 * q^9 - 2 * q^10 - 14 * q^11 + 8 * q^12 - 8 * q^14 + 14 * q^16 + 4 * q^17 + 28 * q^18 - 22 * q^19 - 28 * q^20 - 12 * q^21 - 8 * q^22 + 4 * q^23 + 8 * q^24 - 2 * q^25 + 10 * q^27 - 16 * q^28 - 8 * q^29 - 20 * q^30 + 16 * q^31 - 48 * q^32 - 10 * q^33 - 8 * q^34 - 2 * q^35 + 22 * q^36 - 16 * q^37 - 6 * q^38 + 14 * q^40 - 44 * q^41 + 14 * q^42 + 16 * q^43 - 4 * q^44 - 56 * q^45 - 10 * q^47 + 32 * q^49 - 2 * q^50 - 6 * q^53 - 24 * q^54 + 22 * q^55 - 4 * q^56 + 8 * q^57 - 74 * q^58 - 2 * q^59 - 40 * q^60 + 8 * q^61 - 4 * q^62 - 56 * q^63 + 38 * q^64 - 34 * q^66 + 8 * q^67 + 32 * q^68 - 10 * q^69 + 108 * q^70 - 50 * q^71 + 44 * q^72 - 14 * q^73 + 8 * q^74 - 44 * q^76 + 16 * q^77 + 32 * q^79 - 68 * q^80 - 8 * q^81 - 6 * q^82 + 20 * q^83 - 136 * q^84 + 32 * q^85 - 8 * q^86 - 36 * q^87 - 40 * q^88 - 52 * q^89 - 34 * q^90 + 14 * q^92 - 2 * q^93 + 44 * q^94 - 2 * q^95 + 80 * q^96 - 18 * q^97 - 12 * q^98 - 38 * 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 $

$2$	−1.44935	−1.02484	−0.512422	−	0.858734i	$-0.671252\pi$
$2$	−1.44935	−1.02484	−0.512422	+	0.858734i	$0.671252\pi$
$3$	0.0382575	0.0220880	0.0110440	−	0.999939i	$-0.496485\pi$
$3$	0.0382575	0.0220880	0.0110440	+	0.999939i	$0.496485\pi$
$4$	0.100611	0.0503057
$5$	3.85977	1.72614	0.863071	−	0.505083i	$-0.168538\pi$
$5$	3.85977	1.72614	0.863071	+	0.505083i	$0.168538\pi$
$6$	−0.0554484	−0.0226367
$7$	−1.66255	−0.628384	−0.314192	−	0.949359i	$-0.601734\pi$
$7$	−1.66255	−0.628384	−0.314192	+	0.949359i	$0.601734\pi$
$8$	2.75288	0.973289
$9$	−2.99854	−0.999512
$10$	−5.59415	−1.76903
$11$	−1.88038	−0.566955	−0.283477	−	0.958979i	$-0.591488\pi$
$11$	−1.88038	−0.566955	−0.283477	+	0.958979i	$0.591488\pi$
$12$	0.00384914	0.00111115
$13$	0	0
$13$	0	0
$14$	2.40961	0.643996
$15$	0.147665	0.0381270
$16$	−4.19110	−1.04778
$17$	−3.75015	−0.909546	−0.454773	−	0.890607i	$-0.650280\pi$
$17$	−3.75015	−0.909546	−0.454773	+	0.890607i	$0.650280\pi$
$18$	4.34592	1.02434
$19$	3.26949	0.750073	0.375036	−	0.927010i	$-0.377630\pi$
$19$	3.26949	0.750073	0.375036	+	0.927010i	$0.377630\pi$
$20$	0.388337	0.0868349
$21$	−0.0636049	−0.0138797
$22$	2.72532	0.581040
$23$	−5.31401	−1.10805	−0.554023	−	0.832501i	$-0.686908\pi$
$23$	−5.31401	−1.10805	−0.554023	+	0.832501i	$0.686908\pi$
$24$	0.105318	0.0214980
$25$	9.89783	1.97957
$26$	0	0
$27$	−0.229489	−0.0441651
$28$	−0.167271	−0.0316113
$29$	3.11168	0.577824	0.288912	−	0.957356i	$-0.406706\pi$
$29$	3.11168	0.577824	0.288912	+	0.957356i	$0.406706\pi$
$30$	−0.214018	−0.0390742
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0.568613	0.100518
$33$	−0.0719384	−0.0125229
$34$	5.43528	0.932143
$35$	−6.41705	−1.08468
$36$	−0.301687	−0.0502812
$37$	11.5806	1.90383	0.951915	−	0.306361i	$-0.0991115\pi$
$37$	11.5806	1.90383	0.951915	+	0.306361i	$0.0991115\pi$
$38$	−4.73863	−0.768708
$39$	0	0
$40$	10.6255	1.68003
$41$	−7.94532	−1.24085	−0.620425	−	0.784266i	$-0.713040\pi$
$41$	−7.94532	−1.24085	−0.620425	+	0.784266i	$0.713040\pi$
$42$	0.0921856	0.0142246
$43$	9.24319	1.40957	0.704786	−	0.709420i	$-0.251043\pi$
$43$	9.24319	1.40957	0.704786	+	0.709420i	$0.251043\pi$
$44$	−0.189187	−0.0285211
$45$	−11.5737	−1.72530
$46$	7.70185	1.13558
$47$	−6.97995	−1.01813	−0.509065	−	0.860728i	$-0.670009\pi$
$47$	−6.97995	−1.01813	−0.509065	+	0.860728i	$0.670009\pi$
$48$	−0.160341	−0.0231432
$49$	−4.23593	−0.605133
$50$	−14.3454	−2.02875
$51$	−0.143471	−0.0200900
$52$	0	0
$53$	6.11470	0.839918	0.419959	−	0.907543i	$-0.362044\pi$
$53$	6.11470	0.839918	0.419959	+	0.907543i	$0.362044\pi$
$54$	0.332609	0.0452624
$55$	−7.25782	−0.978644
$56$	−4.57679	−0.611599
$57$	0.125082	0.0165676
$58$	−4.50991	−0.592180
$59$	12.1104	1.57664	0.788318	−	0.615268i	$-0.210952\pi$
$59$	12.1104	1.57664	0.788318	+	0.615268i	$0.210952\pi$
$60$	0.0148568	0.00191801
$61$	1.75416	0.224598	0.112299	−	0.993674i	$-0.464179\pi$
$61$	1.75416	0.224598	0.112299	+	0.993674i	$0.464179\pi$
$62$	−1.44935	−0.184067
$63$	4.98521	0.628078
$64$	7.55808	0.944760
$65$	0	0
$66$	0.104264	0.0128340
$67$	7.94479	0.970611	0.485306	−	0.874345i	$-0.338708\pi$
$67$	7.94479	0.970611	0.485306	+	0.874345i	$0.338708\pi$
$68$	−0.377308	−0.0457554
$69$	−0.203300	−0.0244745
$70$	9.30055	1.11163
$71$	0.735383	0.0872739	0.0436369	−	0.999047i	$-0.486106\pi$
$71$	0.735383	0.0872739	0.0436369	+	0.999047i	$0.486106\pi$
$72$	−8.25460	−0.972814
$73$	−13.5423	−1.58500	−0.792501	−	0.609870i	$-0.791222\pi$
$73$	−13.5423	−1.58500	−0.792501	+	0.609870i	$0.791222\pi$
$74$	−16.7843	−1.95113
$75$	0.378666	0.0437246
$76$	0.328948	0.0377330
$77$	3.12622	0.356265
$78$	0	0
$79$	−5.66333	−0.637174	−0.318587	−	0.947894i	$-0.603208\pi$
$79$	−5.66333	−0.637174	−0.318587	+	0.947894i	$0.603208\pi$
$80$	−16.1767	−1.80861
$81$	8.98683	0.998537
$82$	11.5155	1.27168
$83$	−13.7729	−1.51177	−0.755886	−	0.654704i	$-0.772793\pi$
$83$	−13.7729	−1.51177	−0.755886	+	0.654704i	$0.772793\pi$
$84$	−0.00639938	−0.000698230 0
$85$	−14.4747	−1.57001
$86$	−13.3966	−1.44459
$87$	0.119045	0.0127630
$88$	−5.17644	−0.551811
$89$	−10.2781	−1.08948	−0.544738	−	0.838606i	$-0.683371\pi$
$89$	−10.2781	−1.08948	−0.544738	+	0.838606i	$0.683371\pi$
$90$	16.7743	1.76816
$91$	0	0
$92$	−0.534650	−0.0557411
$93$	0.0382575	0.00396711
$94$	10.1164	1.04342
$95$	12.6195	1.29473
$96$	0.0217537	0.00222023
$97$	−8.34364	−0.847169	−0.423584	−	0.905857i	$-0.639228\pi$
$97$	−8.34364	−0.847169	−0.423584	+	0.905857i	$0.639228\pi$
$98$	6.13935	0.620168
$99$	5.63838	0.566678
$100$	0.995836	0.0995836
$101$	−8.36630	−0.832478	−0.416239	−	0.909255i	$-0.636652\pi$
$101$	−8.36630	−0.832478	−0.416239	+	0.909255i	$0.636652\pi$
$102$	0.207940	0.0205891
$103$	−14.0180	−1.38123	−0.690616	−	0.723222i	$-0.742660\pi$
$103$	−14.0180	−1.38123	−0.690616	+	0.723222i	$0.742660\pi$
$104$	0	0
$105$	−0.245500	−0.0239584
$106$	−8.86233	−0.860785
$107$	−0.569178	−0.0550245	−0.0275122	−	0.999621i	$-0.508759\pi$
$107$	−0.569178	−0.0550245	−0.0275122	+	0.999621i	$0.508759\pi$
$108$	−0.0230892	−0.00222176
$109$	−10.7415	−1.02885	−0.514423	−	0.857537i	$-0.671994\pi$
$109$	−10.7415	−1.02885	−0.514423	+	0.857537i	$0.671994\pi$
$110$	10.5191	1.00296
$111$	0.443043	0.0420517
$112$	6.96791	0.658405
$113$	−7.41542	−0.697584	−0.348792	−	0.937200i	$-0.613408\pi$
$113$	−7.41542	−0.697584	−0.348792	+	0.937200i	$0.613408\pi$
$114$	−0.181288	−0.0169792
$115$	−20.5108	−1.91265
$116$	0.313071	0.0290679
$117$	0	0
$118$	−17.5522	−1.61581
$119$	6.23481	0.571544
$120$	0.406504	0.0371085
$121$	−7.46419	−0.678562
$122$	−2.54240	−0.230178
$123$	−0.303968	−0.0274079
$124$	0.100611	0.00903518
$125$	18.9045	1.69087
$126$	−7.22531	−0.643682
$127$	18.2144	1.61627	0.808133	−	0.589000i	$-0.200478\pi$
$127$	18.2144	1.61627	0.808133	+	0.589000i	$0.200478\pi$
$128$	−12.0915	−1.06875
$129$	0.353621	0.0311346
$130$	0	0
$131$	18.5465	1.62042	0.810209	−	0.586141i	$-0.199354\pi$
$131$	18.5465	1.62042	0.810209	+	0.586141i	$0.199354\pi$
$132$	−0.00723783	−0.000629972 0
$133$	−5.43568	−0.471334
$134$	−11.5148	−0.994725
$135$	−0.885774	−0.0762353
$136$	−10.3237	−0.885251
$137$	−3.12119	−0.266661	−0.133331	−	0.991072i	$-0.542567\pi$
$137$	−3.12119	−0.266661	−0.133331	+	0.991072i	$0.542567\pi$
$138$	0.294653	0.0250825
$139$	−16.0910	−1.36482	−0.682410	−	0.730970i	$-0.739068\pi$
$139$	−16.0910	−1.36482	−0.682410	+	0.730970i	$0.739068\pi$
$140$	−0.645629	−0.0545657
$141$	−0.267035	−0.0224884
$142$	−1.06583	−0.0894421
$143$	0	0
$144$	12.5672	1.04726
$145$	12.0104	0.997407
$146$	19.6275	1.62438
$147$	−0.162056	−0.0133662
$148$	1.16514	0.0957736
$149$	−18.8430	−1.54368	−0.771838	−	0.635820i	$-0.780662\pi$
$149$	−18.8430	−1.54368	−0.771838	+	0.635820i	$0.780662\pi$
$150$	−0.548819	−0.0448109
$151$	0.726888	0.0591533	0.0295767	−	0.999563i	$-0.490584\pi$
$151$	0.726888	0.0591533	0.0295767	+	0.999563i	$0.490584\pi$
$152$	9.00050	0.730037
$153$	11.2450	0.909102
$154$	−4.53098	−0.365116
$155$	3.85977	0.310024
$156$	0	0
$157$	−5.75901	−0.459619	−0.229809	−	0.973236i	$-0.573810\pi$
$157$	−5.75901	−0.459619	−0.229809	+	0.973236i	$0.573810\pi$
$158$	8.20814	0.653004
$159$	0.233933	0.0185521
$160$	2.19472	0.173508
$161$	8.83479	0.696279
$162$	−13.0250	−1.02334
$163$	3.17279	0.248512	0.124256	−	0.992250i	$-0.460346\pi$
$163$	3.17279	0.248512	0.124256	+	0.992250i	$0.460346\pi$
$164$	−0.799390	−0.0624219
$165$	−0.277666	−0.0216163
$166$	19.9617	1.54933
$167$	−4.97051	−0.384630	−0.192315	−	0.981333i	$-0.561600\pi$
$167$	−4.97051	−0.384630	−0.192315	+	0.981333i	$0.561600\pi$
$168$	−0.175096	−0.0135090
$169$	0	0
$170$	20.9789	1.60901
$171$	−9.80369	−0.749707
$172$	0.929971	0.0709096
$173$	1.97560	0.150202	0.0751009	−	0.997176i	$-0.476072\pi$
$173$	1.97560	0.150202	0.0751009	+	0.997176i	$0.476072\pi$
$174$	−0.172538	−0.0130800
$175$	−16.4556	−1.24393
$176$	7.88084	0.594041
$177$	0.463312	0.0348247
$178$	14.8965	1.11654
$179$	−6.07427	−0.454012	−0.227006	−	0.973893i	$-0.572894\pi$
$179$	−6.07427	−0.454012	−0.227006	+	0.973893i	$0.572894\pi$
$180$	−1.16444	−0.0867925
$181$	−20.6084	−1.53181	−0.765906	−	0.642953i	$-0.777709\pi$
$181$	−20.6084	−1.53181	−0.765906	+	0.642953i	$0.777709\pi$
$182$	0	0
$183$	0.0671099	0.00496091
$184$	−14.6288	−1.07845
$185$	44.6983	3.28628
$186$	−0.0554484	−0.00406567
$187$	7.05170	0.515671
$188$	−0.702263	−0.0512178
$189$	0.381536	0.0277527
$190$	−18.2900	−1.32690
$191$	6.78433	0.490897	0.245448	−	0.969410i	$-0.421065\pi$
$191$	6.78433	0.490897	0.245448	+	0.969410i	$0.421065\pi$
$192$	0.289153	0.0208678
$193$	−22.3446	−1.60840	−0.804198	−	0.594361i	$-0.797405\pi$
$193$	−22.3446	−1.60840	−0.804198	+	0.594361i	$0.797405\pi$
$194$	12.0928	0.868216
$195$	0	0
$196$	−0.426184	−0.0304417
$197$	−6.85560	−0.488441	−0.244221	−	0.969720i	$-0.578532\pi$
$197$	−6.85560	−0.488441	−0.244221	+	0.969720i	$0.578532\pi$
$198$	−8.17197	−0.580757
$199$	6.55363	0.464575	0.232287	−	0.972647i	$-0.425379\pi$
$199$	6.55363	0.464575	0.232287	+	0.972647i	$0.425379\pi$
$200$	27.2475	1.92669
$201$	0.303948	0.0214388
$202$	12.1257	0.853160
$203$	−5.17332	−0.363096
$204$	−0.0144349	−0.00101064
$205$	−30.6671	−2.14188
$206$	20.3169	1.41555
$207$	15.9342	1.10751
$208$	0	0
$209$	−6.14787	−0.425257
$210$	0.355815	0.0245536
$211$	21.8282	1.50271	0.751356	−	0.659897i	$-0.229400\pi$
$211$	21.8282	1.50271	0.751356	+	0.659897i	$0.229400\pi$
$212$	0.615209	0.0422527
$213$	0.0281339	0.00192770
$214$	0.824937	0.0563915
$215$	35.6766	2.43312
$216$	−0.631754	−0.0429854
$217$	−1.66255	−0.112861
$218$	15.5681	1.05441
$219$	−0.518093	−0.0350095
$220$	−0.730220	−0.0492314
$221$	0	0
$222$	−0.642123	−0.0430965
$223$	−1.22551	−0.0820659	−0.0410330	−	0.999158i	$-0.513065\pi$
$223$	−1.22551	−0.0820659	−0.0410330	+	0.999158i	$0.513065\pi$
$224$	−0.945347	−0.0631636
$225$	−29.6790	−1.97860
$226$	10.7475	0.714915
$227$	−9.82473	−0.652090	−0.326045	−	0.945354i	$-0.605716\pi$
$227$	−9.82473	−0.652090	−0.326045	+	0.945354i	$0.605716\pi$
$228$	0.0125847	0.000833444 0
$229$	1.46932	0.0970954	0.0485477	−	0.998821i	$-0.484541\pi$
$229$	1.46932	0.0970954	0.0485477	+	0.998821i	$0.484541\pi$
$230$	29.7274	1.96016
$231$	0.119601	0.00786917
$232$	8.56607	0.562390
$233$	−14.1766	−0.928737	−0.464369	−	0.885642i	$-0.653719\pi$
$233$	−14.1766	−0.928737	−0.464369	+	0.885642i	$0.653719\pi$
$234$	0	0
$235$	−26.9410	−1.75744
$236$	1.21844	0.0793139
$237$	−0.216665	−0.0140739
$238$	−9.03641	−0.585744
$239$	−12.4101	−0.802745	−0.401372	−	0.915915i	$-0.631467\pi$
$239$	−12.4101	−0.802745	−0.401372	+	0.915915i	$0.631467\pi$
$240$	−0.618879	−0.0399485
$241$	9.96203	0.641711	0.320855	−	0.947128i	$-0.396030\pi$
$241$	9.96203	0.641711	0.320855	+	0.947128i	$0.396030\pi$
$242$	10.8182	0.695421
$243$	1.03228	0.0662208
$244$	0.176489	0.0112986
$245$	−16.3497	−1.04455
$246$	0.440555	0.0280888
$247$	0	0
$248$	2.75288	0.174808
$249$	−0.526916	−0.0333919
$250$	−27.3992	−1.73288
$251$	2.57983	0.162837	0.0814187	−	0.996680i	$-0.474055\pi$
$251$	2.57983	0.162837	0.0814187	+	0.996680i	$0.474055\pi$
$252$	0.501570	0.0315959
$253$	9.99233	0.628212
$254$	−26.3990	−1.65642
$255$	−0.553767	−0.0346782
$256$	2.40867	0.150542
$257$	−8.69963	−0.542668	−0.271334	−	0.962485i	$-0.587465\pi$
$257$	−8.69963	−0.542668	−0.271334	+	0.962485i	$0.587465\pi$
$258$	−0.512520	−0.0319081
$259$	−19.2532	−1.19634
$260$	0	0
$261$	−9.33048	−0.577542
$262$	−26.8804	−1.66068
$263$	4.64597	0.286483	0.143241	−	0.989688i	$-0.454248\pi$
$263$	4.64597	0.286483	0.143241	+	0.989688i	$0.454248\pi$
$264$	−0.198038	−0.0121884
$265$	23.6013	1.44982
$266$	7.87820	0.483044
$267$	−0.393214	−0.0240643
$268$	0.799338	0.0488273
$269$	−1.50231	−0.0915976	−0.0457988	−	0.998951i	$-0.514583\pi$
$269$	−1.50231	−0.0915976	−0.0457988	+	0.998951i	$0.514583\pi$
$270$	1.28380	0.0781293
$271$	8.71423	0.529352	0.264676	−	0.964337i	$-0.414735\pi$
$271$	8.71423	0.529352	0.264676	+	0.964337i	$0.414735\pi$
$272$	15.7173	0.952999
$273$	0	0
$274$	4.52369	0.273286
$275$	−18.6116	−1.12232
$276$	−0.0204544	−0.00123121
$277$	−4.24495	−0.255054	−0.127527	−	0.991835i	$-0.540704\pi$
$277$	−4.24495	−0.255054	−0.127527	+	0.991835i	$0.540704\pi$
$278$	23.3214	1.39873
$279$	−2.99854	−0.179518
$280$	−17.6654	−1.05571
$281$	5.08775	0.303510	0.151755	−	0.988418i	$-0.451508\pi$
$281$	5.08775	0.303510	0.151755	+	0.988418i	$0.451508\pi$
$282$	0.387027	0.0230471
$283$	32.6740	1.94227	0.971134	−	0.238533i	$-0.0766664\pi$
$283$	32.6740	1.94227	0.971134	+	0.238533i	$0.0766664\pi$
$284$	0.0739880	0.00439038
$285$	0.482789	0.0285980
$286$	0	0
$287$	13.2095	0.779731
$288$	−1.70501	−0.100469
$289$	−2.93635	−0.172727
$290$	−17.4072	−1.02219
$291$	−0.319207	−0.0187122
$292$	−1.36251	−0.0797347
$293$	−10.6513	−0.622254	−0.311127	−	0.950368i	$-0.600707\pi$
$293$	−10.6513	−0.622254	−0.311127	+	0.950368i	$0.600707\pi$
$294$	0.234876	0.0136982
$295$	46.7433	2.72150
$296$	31.8798	1.85298
$297$	0.431525	0.0250396
$298$	27.3100	1.58203
$299$	0	0
$300$	0.0380982	0.00219960
$301$	−15.3672	−0.885753
$302$	−1.05351	−0.0606229
$303$	−0.320073	−0.0183877
$304$	−13.7028	−0.785907
$305$	6.77067	0.387688
$306$	−16.2979	−0.931688
$307$	−11.4536	−0.653691	−0.326845	−	0.945078i	$-0.605986\pi$
$307$	−11.4536	−0.653691	−0.326845	+	0.945078i	$0.605986\pi$
$308$	0.314533	0.0179222
$309$	−0.536292	−0.0305086
$310$	−5.59415	−0.317727
$311$	14.2666	0.808983	0.404492	−	0.914542i	$-0.367448\pi$
$311$	14.2666	0.808983	0.404492	+	0.914542i	$0.367448\pi$
$312$	0	0
$313$	10.3671	0.585983	0.292991	−	0.956115i	$-0.405349\pi$
$313$	10.3671	0.585983	0.292991	+	0.956115i	$0.405349\pi$
$314$	8.34681	0.471038
$315$	19.2418	1.08415
$316$	−0.569796	−0.0320535
$317$	5.67935	0.318984	0.159492	−	0.987199i	$-0.449014\pi$
$317$	5.67935	0.318984	0.159492	+	0.987199i	$0.449014\pi$
$318$	−0.339050	−0.0190130
$319$	−5.85113	−0.327600
$320$	29.1725	1.63079
$321$	−0.0217753	−0.00121538
$322$	−12.8047	−0.713577
$323$	−12.2611	−0.682225
$324$	0.904178	0.0502321
$325$	0	0
$326$	−4.59848	−0.254686
$327$	−0.410941	−0.0227251
$328$	−21.8725	−1.20771
$329$	11.6045	0.639777
$330$	0.402435	0.0221533
$331$	0.676975	0.0372099	0.0186050	−	0.999827i	$-0.494078\pi$
$331$	0.676975	0.0372099	0.0186050	+	0.999827i	$0.494078\pi$
$332$	−1.38571	−0.0760508
$333$	−34.7247	−1.90290
$334$	7.20401	0.394186
$335$	30.6651	1.67541
$336$	0.266574	0.0145428
$337$	−4.73721	−0.258052	−0.129026	−	0.991641i	$-0.541185\pi$
$337$	−4.73721	−0.258052	−0.129026	+	0.991641i	$0.541185\pi$
$338$	0	0
$339$	−0.283695	−0.0154082
$340$	−1.45632	−0.0789803
$341$	−1.88038	−0.101828
$342$	14.2090	0.768332
$343$	18.6803	1.00864
$344$	25.4453	1.37192
$345$	−0.784693	−0.0422464
$346$	−2.86333	−0.153933
$347$	−31.6786	−1.70059	−0.850297	−	0.526303i	$-0.823578\pi$
$347$	−31.6786	−1.70059	−0.850297	+	0.526303i	$0.823578\pi$
$348$	0.0119773	0.000642050 0
$349$	8.64433	0.462720	0.231360	−	0.972868i	$-0.425682\pi$
$349$	8.64433	0.462720	0.231360	+	0.972868i	$0.425682\pi$
$350$	23.8499	1.27483
$351$	0	0
$352$	−1.06921	−0.0569889
$353$	−11.6266	−0.618820	−0.309410	−	0.950929i	$-0.600131\pi$
$353$	−11.6266	−0.618820	−0.309410	+	0.950929i	$0.600131\pi$
$354$	−0.671501	−0.0356899
$355$	2.83841	0.150647
$356$	−1.03409	−0.0548069
$357$	0.238528	0.0126242
$358$	8.80373	0.465292
$359$	5.12526	0.270501	0.135250	−	0.990811i	$-0.456816\pi$
$359$	5.12526	0.270501	0.135250	+	0.990811i	$0.456816\pi$
$360$	−31.8609	−1.67921
$361$	−8.31043	−0.437391
$362$	29.8688	1.56987
$363$	−0.285561	−0.0149881
$364$	0	0
$365$	−52.2701	−2.73594
$366$	−0.0972656	−0.00508416
$367$	−35.1284	−1.83369	−0.916844	−	0.399246i	$-0.869272\pi$
$367$	−35.1284	−1.83369	−0.916844	+	0.399246i	$0.869272\pi$
$368$	22.2715	1.16098
$369$	23.8243	1.24024
$370$	−64.7834	−3.36793
$371$	−10.1660	−0.527791
$372$	0.00384914	0.000199569 0
$373$	3.37056	0.174521	0.0872604	−	0.996186i	$-0.472189\pi$
$373$	3.37056	0.174521	0.0872604	+	0.996186i	$0.472189\pi$
$374$	−10.2204	−0.528483
$375$	0.723239	0.0373479
$376$	−19.2149	−0.990934
$377$	0	0
$378$	−0.552979	−0.0284422
$379$	−15.5733	−0.799945	−0.399972	−	0.916527i	$-0.630980\pi$
$379$	−15.5733	−0.799945	−0.399972	+	0.916527i	$0.630980\pi$
$380$	1.26967	0.0651324
$381$	0.696837	0.0357000
$382$	−9.83285	−0.503092
$383$	14.2746	0.729396	0.364698	−	0.931126i	$-0.381172\pi$
$383$	14.2746	0.729396	0.364698	+	0.931126i	$0.381172\pi$
$384$	−0.462591	−0.0236065
$385$	12.0665	0.614964
$386$	32.3851	1.64836
$387$	−27.7160	−1.40888
$388$	−0.839467	−0.0426175
$389$	−10.3188	−0.523185	−0.261592	−	0.965178i	$-0.584248\pi$
$389$	−10.3188	−0.523185	−0.261592	+	0.965178i	$0.584248\pi$
$390$	0	0
$391$	19.9283	1.00782
$392$	−11.6610	−0.588970
$393$	0.709543	0.0357917
$394$	9.93615	0.500576
$395$	−21.8591	−1.09985
$396$	0.567285	0.0285072
$397$	24.3801	1.22360	0.611801	−	0.791012i	$-0.290446\pi$
$397$	24.3801	1.22360	0.611801	+	0.791012i	$0.290446\pi$
$398$	−9.49850	−0.476117
$399$	−0.207956	−0.0104108
$400$	−41.4828	−2.07414
$401$	−15.7425	−0.786141	−0.393071	−	0.919508i	$-0.628587\pi$
$401$	−15.7425	−0.786141	−0.393071	+	0.919508i	$0.628587\pi$
$402$	−0.440526	−0.0219714
$403$	0	0
$404$	−0.841746	−0.0418784
$405$	34.6871	1.72362
$406$	7.49794	0.372116
$407$	−21.7758	−1.07939
$408$	−0.394959	−0.0195534
$409$	−13.4932	−0.667196	−0.333598	−	0.942716i	$-0.608263\pi$
$409$	−13.4932	−0.667196	−0.333598	+	0.942716i	$0.608263\pi$
$410$	44.4473	2.19510
$411$	−0.119409	−0.00589000
$412$	−1.41037	−0.0694839
$413$	−20.1341	−0.990733
$414$	−23.0943	−1.13502
$415$	−53.1602	−2.60953
$416$	0	0
$417$	−0.615600	−0.0301461
$418$	8.91041	0.435822
$419$	−30.1496	−1.47290	−0.736451	−	0.676491i	$-0.763500\pi$
$419$	−30.1496	−1.47290	−0.736451	+	0.676491i	$0.763500\pi$
$420$	−0.0247001	−0.00120524
$421$	−29.2902	−1.42752	−0.713758	−	0.700393i	$-0.753008\pi$
$421$	−29.2902	−1.42752	−0.713758	+	0.700393i	$0.753008\pi$
$422$	−31.6366	−1.54005
$423$	20.9296	1.01763
$424$	16.8330	0.817483
$425$	−37.1184	−1.80051
$426$	−0.0407758	−0.00197559
$427$	−2.91638	−0.141134
$428$	−0.0572658	−0.00276805
$429$	0	0
$430$	−51.7078	−2.49357
$431$	−35.3794	−1.70416	−0.852082	−	0.523408i	$-0.824660\pi$
$431$	−35.3794	−1.70416	−0.852082	+	0.523408i	$0.824660\pi$
$432$	0.961811	0.0462751
$433$	−12.7072	−0.610671	−0.305335	−	0.952245i	$-0.598769\pi$
$433$	−12.7072	−0.610671	−0.305335	+	0.952245i	$0.598769\pi$
$434$	2.40961	0.115665
$435$	0.459486	0.0220307
$436$	−1.08071	−0.0517568
$437$	−17.3741	−0.831115
$438$	0.750897	0.0358793
$439$	−18.4508	−0.880609	−0.440305	−	0.897849i	$-0.645130\pi$
$439$	−18.4508	−0.880609	−0.440305	+	0.897849i	$0.645130\pi$
$440$	−19.9799	−0.952503
$441$	12.7016	0.604838
$442$	0	0
$443$	10.8749	0.516680	0.258340	−	0.966054i	$-0.416824\pi$
$443$	10.8749	0.516680	0.258340	+	0.966054i	$0.416824\pi$
$444$	0.0445752	0.00211544
$445$	−39.6711	−1.88059
$446$	1.77619	0.0841048
$447$	−0.720884	−0.0340966
$448$	−12.5657	−0.593672
$449$	−11.8359	−0.558569	−0.279284	−	0.960208i	$-0.590097\pi$
$449$	−11.8359	−0.558569	−0.279284	+	0.960208i	$0.590097\pi$
$450$	43.0152	2.02776
$451$	14.9402	0.703506
$452$	−0.746076	−0.0350925
$453$	0.0278089	0.00130658
$454$	14.2395	0.668291
$455$	0	0
$456$	0.344336	0.0161250
$457$	32.9139	1.53965	0.769823	−	0.638258i	$-0.220345\pi$
$457$	32.9139	1.53965	0.769823	+	0.638258i	$0.220345\pi$
$458$	−2.12956	−0.0995077
$459$	0.860618	0.0401702
$460$	−2.06363	−0.0962171
$461$	25.2638	1.17665	0.588325	−	0.808625i	$-0.299788\pi$
$461$	25.2638	1.17665	0.588325	+	0.808625i	$0.299788\pi$
$462$	−0.173344	−0.00806468
$463$	−19.8975	−0.924714	−0.462357	−	0.886694i	$-0.652996\pi$
$463$	−19.8975	−0.924714	−0.462357	+	0.886694i	$0.652996\pi$
$464$	−13.0414	−0.605430
$465$	0.147665	0.00684780
$466$	20.5468	0.951811
$467$	19.7000	0.911608	0.455804	−	0.890080i	$-0.349352\pi$
$467$	19.7000	0.911608	0.455804	+	0.890080i	$0.349352\pi$
$468$	0	0
$469$	−13.2086	−0.609917
$470$	39.0469	1.80110
$471$	−0.220325	−0.0101520
$472$	33.3384	1.53452
$473$	−17.3807	−0.799164
$474$	0.314022	0.0144235
$475$	32.3609	1.48482
$476$	0.627294	0.0287520
$477$	−18.3351	−0.839508
$478$	17.9866	0.822688
$479$	−24.1566	−1.10374	−0.551872	−	0.833929i	$-0.686086\pi$
$479$	−24.1566	−1.10374	−0.551872	+	0.833929i	$0.686086\pi$
$480$	0.0839643	0.00383243
$481$	0	0
$482$	−14.4385	−0.657653
$483$	0.337997	0.0153794
$484$	−0.750983	−0.0341356
$485$	−32.2046	−1.46233
$486$	−1.49613	−0.0678660
$487$	21.5211	0.975213	0.487607	−	0.873063i	$-0.337870\pi$
$487$	21.5211	0.975213	0.487607	+	0.873063i	$0.337870\pi$
$488$	4.82900	0.218598
$489$	0.121383	0.00548913
$490$	23.6965	1.07050
$491$	−1.06521	−0.0480723	−0.0240361	−	0.999711i	$-0.507652\pi$
$491$	−1.06521	−0.0480723	−0.0240361	+	0.999711i	$0.507652\pi$
$492$	−0.0305827	−0.00137877
$493$	−11.6693	−0.525558
$494$	0	0
$495$	21.7628	0.978167
$496$	−4.19110	−0.188186
$497$	−1.22261	−0.0548415
$498$	0.763685	0.0342215
$499$	−21.7718	−0.974641	−0.487321	−	0.873223i	$-0.662026\pi$
$499$	−21.7718	−0.974641	−0.487321	+	0.873223i	$0.662026\pi$
$500$	1.90201	0.0850605
$501$	−0.190159	−0.00849569
$502$	−3.73907	−0.166883
$503$	−19.9950	−0.891531	−0.445766	−	0.895150i	$-0.647068\pi$
$503$	−19.9950	−0.891531	−0.445766	+	0.895150i	$0.647068\pi$
$504$	13.7237	0.611301
$505$	−32.2920	−1.43697
$506$	−14.4824	−0.643820
$507$	0	0
$508$	1.83258	0.0813075
$509$	18.7005	0.828883	0.414442	−	0.910076i	$-0.363977\pi$
$509$	18.7005	0.828883	0.414442	+	0.910076i	$0.363977\pi$
$510$	0.802601	0.0355398
$511$	22.5147	0.995990
$512$	20.6920	0.914468
$513$	−0.750311	−0.0331271
$514$	12.6088	0.556150
$515$	−54.1062	−2.38420
$516$	0.0355783	0.00156625
$517$	13.1249	0.577234
$518$	27.9046	1.22606
$519$	0.0755813	0.00331765
$520$	0	0
$521$	21.5470	0.943992	0.471996	−	0.881601i	$-0.343534\pi$
$521$	21.5470	0.943992	0.471996	+	0.881601i	$0.343534\pi$
$522$	13.5231	0.591891
$523$	−35.8946	−1.56956	−0.784781	−	0.619773i	$-0.787225\pi$
$523$	−35.8946	−1.56956	−0.784781	+	0.619773i	$0.787225\pi$
$524$	1.86599	0.0815163
$525$	−0.629550	−0.0274758
$526$	−6.73363	−0.293600
$527$	−3.75015	−0.163359
$528$	0.301501	0.0131212
$529$	5.23865	0.227768
$530$	−34.2065	−1.48584
$531$	−36.3134	−1.57587
$532$	−0.546892	−0.0237108
$533$	0	0
$534$	0.569904	0.0246622
$535$	−2.19690	−0.0949801
$536$	21.8710	0.944685
$537$	−0.232386	−0.0100282
$538$	2.17737	0.0938733
$539$	7.96515	0.343083
$540$	−0.0891191	−0.00383507
$541$	12.8955	0.554420	0.277210	−	0.960809i	$-0.410590\pi$
$541$	12.8955	0.554420	0.277210	+	0.960809i	$0.410590\pi$
$542$	−12.6300	−0.542503
$543$	−0.788426	−0.0338346
$544$	−2.13239	−0.0914253
$545$	−41.4596	−1.77593
$546$	0	0
$547$	28.8880	1.23516	0.617582	−	0.786507i	$-0.288113\pi$
$547$	28.8880	1.23516	0.617582	+	0.786507i	$0.288113\pi$
$548$	−0.314027	−0.0134146
$549$	−5.25993	−0.224488
$550$	26.9748	1.15021
$551$	10.1736	0.433410
$552$	−0.559661	−0.0238207
$553$	9.41555	0.400390
$554$	6.15241	0.261391
$555$	1.71004	0.0725873
$556$	−1.61894	−0.0686583
$557$	−8.61442	−0.365005	−0.182502	−	0.983205i	$-0.558420\pi$
$557$	−8.61442	−0.365005	−0.182502	+	0.983205i	$0.558420\pi$
$558$	4.34592	0.183978
$559$	0	0
$560$	26.8945	1.13650
$561$	0.269780	0.0113901
$562$	−7.37392	−0.311050
$563$	−5.36118	−0.225947	−0.112973	−	0.993598i	$-0.536037\pi$
$563$	−5.36118	−0.225947	−0.112973	+	0.993598i	$0.536037\pi$
$564$	−0.0268668	−0.00113130
$565$	−28.6218	−1.20413
$566$	−47.3560	−1.99052
$567$	−14.9410	−0.627465
$568$	2.02442	0.0849427
$569$	13.1854	0.552760	0.276380	−	0.961048i	$-0.410865\pi$
$569$	13.1854	0.552760	0.276380	+	0.961048i	$0.410865\pi$
$570$	−0.699730	−0.0293085
$571$	−36.6258	−1.53274	−0.766372	−	0.642397i	$-0.777940\pi$
$571$	−36.6258	−1.53274	−0.766372	+	0.642397i	$0.777940\pi$
$572$	0	0
$573$	0.259551	0.0108429
$574$	−19.1451	−0.799102
$575$	−52.5971	−2.19345
$576$	−22.6632	−0.944299
$577$	26.2411	1.09243	0.546215	−	0.837645i	$-0.316068\pi$
$577$	26.2411	1.09243	0.546215	+	0.837645i	$0.316068\pi$
$578$	4.25580	0.177018
$579$	−0.854846	−0.0355262
$580$	1.20838	0.0501753
$581$	22.8981	0.949973
$582$	0.462642	0.0191771
$583$	−11.4979	−0.476195
$584$	−37.2802	−1.54267
$585$	0	0
$586$	15.4374	0.637714
$587$	−15.9650	−0.658945	−0.329472	−	0.944165i	$-0.606871\pi$
$587$	−15.9650	−0.658945	−0.329472	+	0.944165i	$0.606871\pi$
$588$	−0.0163047	−0.000672395 0
$589$	3.26949	0.134717
$590$	−67.7473	−2.78911
$591$	−0.262278	−0.0107887
$592$	−48.5352	−1.99479
$593$	20.7033	0.850181	0.425091	−	0.905151i	$-0.360242\pi$
$593$	20.7033	0.850181	0.425091	+	0.905151i	$0.360242\pi$
$594$	−0.625430	−0.0256617
$595$	24.0649	0.986566
$596$	−1.89582	−0.0776558
$597$	0.250725	0.0102615
$598$	0	0
$599$	−31.4587	−1.28537	−0.642685	−	0.766131i	$-0.722179\pi$
$599$	−31.4587	−1.28537	−0.642685	+	0.766131i	$0.722179\pi$
$600$	1.04242	0.0425566
$601$	14.5688	0.594274	0.297137	−	0.954835i	$-0.403968\pi$
$601$	14.5688	0.594274	0.297137	+	0.954835i	$0.403968\pi$
$602$	22.2725	0.907759
$603$	−23.8228	−0.970137
$604$	0.0731333	0.00297575
$605$	−28.8101	−1.17130
$606$	0.463898	0.0188446
$607$	−8.19708	−0.332709	−0.166355	−	0.986066i	$-0.553200\pi$
$607$	−8.19708	−0.332709	−0.166355	+	0.986066i	$0.553200\pi$
$608$	1.85908	0.0753955
$609$	−0.197918	−0.00802004
$610$	−9.81307	−0.397319
$611$	0	0
$612$	1.13137	0.0457331
$613$	5.02425	0.202927	0.101464	−	0.994839i	$-0.467647\pi$
$613$	5.02425	0.202927	0.101464	+	0.994839i	$0.467647\pi$
$614$	16.6002	0.669931
$615$	−1.17325	−0.0473098
$616$	8.60608	0.346749
$617$	−20.3301	−0.818460	−0.409230	−	0.912431i	$-0.634203\pi$
$617$	−20.3301	−0.818460	−0.409230	+	0.912431i	$0.634203\pi$
$618$	0.777274	0.0312666
$619$	9.73194	0.391160	0.195580	−	0.980688i	$-0.437341\pi$
$619$	9.73194	0.391160	0.195580	+	0.980688i	$0.437341\pi$
$620$	0.388337	0.0155960
$621$	1.21950	0.0489370
$622$	−20.6772	−0.829082
$623$	17.0878	0.684609
$624$	0	0
$625$	23.4779	0.939117
$626$	−15.0255	−0.600541
$627$	−0.235202	−0.00939306
$628$	−0.579422	−0.0231215
$629$	−43.4288	−1.73162
$630$	−27.8880	−1.11109
$631$	−46.9356	−1.86848	−0.934238	−	0.356651i	$-0.883919\pi$
$631$	−46.9356	−1.86848	−0.934238	+	0.356651i	$0.883919\pi$
$632$	−15.5904	−0.620154
$633$	0.835090	0.0331919
$634$	−8.23136	−0.326909
$635$	70.3034	2.78991
$636$	0.0235363	0.000933276 0
$637$	0	0
$638$	8.48032	0.335739
$639$	−2.20507	−0.0872313
$640$	−46.6705	−1.84481
$641$	27.2007	1.07436	0.537181	−	0.843467i	$-0.319489\pi$
$641$	27.2007	1.07436	0.537181	+	0.843467i	$0.319489\pi$
$642$	0.0315600	0.00124557
$643$	41.5545	1.63875	0.819375	−	0.573257i	$-0.194320\pi$
$643$	41.5545	1.63875	0.819375	+	0.573257i	$0.194320\pi$
$644$	0.888881	0.0350268
$645$	1.36490	0.0537427
$646$	17.7706	0.699175
$647$	12.4684	0.490184	0.245092	−	0.969500i	$-0.421182\pi$
$647$	12.4684	0.490184	0.245092	+	0.969500i	$0.421182\pi$
$648$	24.7396	0.971864
$649$	−22.7720	−0.893881
$650$	0	0
$651$	−0.0636049	−0.00249287
$652$	0.319219	0.0125016
$653$	48.1876	1.88572	0.942862	−	0.333182i	$-0.108122\pi$
$653$	48.1876	1.88572	0.942862	+	0.333182i	$0.108122\pi$
$654$	0.595597	0.0232897
$655$	71.5854	2.79707
$656$	33.2996	1.30013
$657$	40.6070	1.58423
$658$	−16.8190	−0.655672
$659$	−8.64394	−0.336720	−0.168360	−	0.985726i	$-0.553847\pi$
$659$	−8.64394	−0.336720	−0.168360	+	0.985726i	$0.553847\pi$
$660$	−0.0279364	−0.00108742
$661$	−25.8464	−1.00531	−0.502654	−	0.864487i	$-0.667643\pi$
$661$	−25.8464	−1.00531	−0.502654	+	0.864487i	$0.667643\pi$
$662$	−0.981173	−0.0381344
$663$	0	0
$664$	−37.9151	−1.47139
$665$	−20.9805	−0.813589
$666$	50.3282	1.95018
$667$	−16.5355	−0.640256
$668$	−0.500091	−0.0193491
$669$	−0.0468847	−0.00181267
$670$	−44.4444	−1.71704
$671$	−3.29849	−0.127337
$672$	−0.0361666	−0.00139516
$673$	13.1020	0.505044	0.252522	−	0.967591i	$-0.418740\pi$
$673$	13.1020	0.505044	0.252522	+	0.967591i	$0.418740\pi$
$674$	6.86586	0.264463
$675$	−2.27144	−0.0874278
$676$	0	0
$677$	−44.8541	−1.72388	−0.861941	−	0.507008i	$-0.830752\pi$
$677$	−44.8541	−1.72388	−0.861941	+	0.507008i	$0.830752\pi$
$678$	0.411173	0.0157910
$679$	13.8717	0.532347
$680$	−39.8471	−1.52807
$681$	−0.375869	−0.0144033
$682$	2.72532	0.104358
$683$	−0.979482	−0.0374788	−0.0187394	−	0.999824i	$-0.505965\pi$
$683$	−0.979482	−0.0374788	−0.0187394	+	0.999824i	$0.505965\pi$
$684$	−0.986363	−0.0377146
$685$	−12.0471	−0.460295
$686$	−27.0742	−1.03370
$687$	0.0562125	0.00214464
$688$	−38.7391	−1.47692
$689$	0	0
$690$	1.13729	0.0432960
$691$	40.0083	1.52199	0.760994	−	0.648759i	$-0.224711\pi$
$691$	40.0083	1.52199	0.760994	+	0.648759i	$0.224711\pi$
$692$	0.198768	0.00755602
$693$	−9.37407	−0.356091
$694$	45.9133	1.74284
$695$	−62.1075	−2.35587
$696$	0.327716	0.0124220
$697$	29.7962	1.12861
$698$	−12.5287	−0.474216
$699$	−0.542359	−0.0205139
$700$	−1.65562	−0.0625767
$701$	37.1969	1.40491	0.702454	−	0.711729i	$-0.252088\pi$
$701$	37.1969	1.40491	0.702454	+	0.711729i	$0.252088\pi$
$702$	0	0
$703$	37.8625	1.42801
$704$	−14.2120	−0.535636
$705$	−1.03069	−0.0388182
$706$	16.8509	0.634194
$707$	13.9094	0.523116
$708$	0.0466145	0.00175188
$709$	33.3437	1.25225	0.626125	−	0.779722i	$-0.284640\pi$
$709$	33.3437	1.25225	0.626125	+	0.779722i	$0.284640\pi$
$710$	−4.11384	−0.154390
$711$	16.9817	0.636863
$712$	−28.2943	−1.06037
$713$	−5.31401	−0.199011
$714$	−0.345710	−0.0129379
$715$	0	0
$716$	−0.611141	−0.0228394
$717$	−0.474780	−0.0177310
$718$	−7.42828	−0.277221
$719$	23.1357	0.862815	0.431407	−	0.902157i	$-0.358017\pi$
$719$	23.1357	0.862815	0.431407	+	0.902157i	$0.358017\pi$
$720$	48.5064	1.80773
$721$	23.3056	0.867944
$722$	12.0447	0.448258
$723$	0.381122	0.0141741
$724$	−2.07344	−0.0770589
$725$	30.7989	1.14384
$726$	0.413877	0.0153604
$727$	4.47177	0.165849	0.0829243	−	0.996556i	$-0.473574\pi$
$727$	4.47177	0.165849	0.0829243	+	0.996556i	$0.473574\pi$
$728$	0	0
$729$	−26.9210	−0.997074
$730$	75.7575	2.80391
$731$	−34.6634	−1.28207
$732$	0.00675203	0.000249562 0
$733$	−50.4881	−1.86482	−0.932410	−	0.361402i	$-0.882298\pi$
$733$	−50.4881	−1.86482	−0.932410	+	0.361402i	$0.882298\pi$
$734$	50.9133	1.87924
$735$	−0.625499	−0.0230719
$736$	−3.02161	−0.111378
$737$	−14.9392	−0.550292
$738$	−34.5298	−1.27106
$739$	51.2533	1.88538	0.942691	−	0.333666i	$-0.108286\pi$
$739$	51.2533	1.88538	0.942691	+	0.333666i	$0.108286\pi$
$740$	4.49716	0.165319
$741$	0	0
$742$	14.7340	0.540904
$743$	−26.3425	−0.966411	−0.483206	−	0.875507i	$-0.660528\pi$
$743$	−26.3425	−0.966411	−0.483206	+	0.875507i	$0.660528\pi$
$744$	0.105318	0.00386115
$745$	−72.7295	−2.66460
$746$	−4.88511	−0.178857
$747$	41.2985	1.51103
$748$	0.709482	0.0259412
$749$	0.946285	0.0345765
$750$	−1.04823	−0.0382758
$751$	34.0846	1.24377	0.621883	−	0.783110i	$-0.286368\pi$
$751$	34.0846	1.24377	0.621883	+	0.783110i	$0.286368\pi$
$752$	29.2537	1.06677
$753$	0.0986978	0.00359675
$754$	0	0
$755$	2.80562	0.102107
$756$	0.0383869	0.00139612
$757$	−50.7996	−1.84634	−0.923171	−	0.384389i	$-0.874412\pi$
$757$	−50.7996	−1.84634	−0.923171	+	0.384389i	$0.874412\pi$
$758$	22.5711	0.819819
$759$	0.382281	0.0138759
$760$	34.7399	1.26015
$761$	−9.08920	−0.329483	−0.164742	−	0.986337i	$-0.552679\pi$
$761$	−9.08920	−0.329483	−0.164742	+	0.986337i	$0.552679\pi$
$762$	−1.00996	−0.0365870
$763$	17.8582	0.646510
$764$	0.682581	0.0246949
$765$	43.4030	1.56924
$766$	−20.6888	−0.747518
$767$	0	0
$768$	0.0921495	0.00332516
$769$	−11.9781	−0.431942	−0.215971	−	0.976400i	$-0.569292\pi$
$769$	−11.9781	−0.431942	−0.215971	+	0.976400i	$0.569292\pi$
$770$	−17.4885	−0.630243
$771$	−0.332826	−0.0119864
$772$	−2.24812	−0.0809116
$773$	−29.6744	−1.06731	−0.533657	−	0.845701i	$-0.679183\pi$
$773$	−29.6744	−1.06731	−0.533657	+	0.845701i	$0.679183\pi$
$774$	40.1702	1.44389
$775$	9.89783	0.355541
$776$	−22.9690	−0.824540
$777$	−0.736579	−0.0264246
$778$	14.9556	0.536183
$779$	−25.9771	−0.930728
$780$	0	0
$781$	−1.38280	−0.0494803
$782$	−28.8831	−1.03286
$783$	−0.714095	−0.0255197
$784$	17.7532	0.634044
$785$	−22.2284	−0.793367
$786$	−1.02838	−0.0366809
$787$	−31.7791	−1.13280	−0.566402	−	0.824129i	$-0.691665\pi$
$787$	−31.7791	−1.13280	−0.566402	+	0.824129i	$0.691665\pi$
$788$	−0.689752	−0.0245714
$789$	0.177743	0.00632782
$790$	31.6815	1.12718
$791$	12.3285	0.438351
$792$	15.5217	0.551541
$793$	0	0
$794$	−35.3352	−1.25400
$795$	0.902927	0.0320235
$796$	0.659371	0.0233708
$797$	−3.59563	−0.127364	−0.0636819	−	0.997970i	$-0.520284\pi$
$797$	−3.59563	−0.127364	−0.0636819	+	0.997970i	$0.520284\pi$
$798$	0.301400	0.0106694
$799$	26.1759	0.926036
$800$	5.62804	0.198981
$801$	30.8192	1.08894
$802$	22.8163	0.805672
$803$	25.4646	0.898625
$804$	0.0305806	0.00107850
$805$	34.1003	1.20188
$806$	0	0
$807$	−0.0574747	−0.00202320
$808$	−23.0314	−0.810241
$809$	13.4626	0.473321	0.236660	−	0.971592i	$-0.423947\pi$
$809$	13.4626	0.473321	0.236660	+	0.971592i	$0.423947\pi$
$810$	−50.2737	−1.76644
$811$	−24.6877	−0.866902	−0.433451	−	0.901177i	$-0.642704\pi$
$811$	−24.6877	−0.866902	−0.433451	+	0.901177i	$0.642704\pi$
$812$	−0.520495	−0.0182658
$813$	0.333384	0.0116923
$814$	31.5607	1.10620
$815$	12.2462	0.428967
$816$	0.601303	0.0210498
$817$	30.2205	1.05728
$818$	19.5563	0.683771
$819$	0	0
$820$	−3.08546	−0.107749
$821$	0.766000	0.0267336	0.0133668	−	0.999911i	$-0.495745\pi$
$821$	0.766000	0.0267336	0.0133668	+	0.999911i	$0.495745\pi$
$822$	0.173065	0.00603633
$823$	10.3978	0.362445	0.181222	−	0.983442i	$-0.441995\pi$
$823$	10.3978	0.362445	0.181222	+	0.983442i	$0.441995\pi$
$824$	−38.5897	−1.34434
$825$	−0.712034	−0.0247899
$826$	29.1813	1.01535
$827$	−28.3555	−0.986017	−0.493009	−	0.870024i	$-0.664103\pi$
$827$	−28.3555	−0.986017	−0.493009	+	0.870024i	$0.664103\pi$
$828$	1.60317	0.0557139
$829$	35.5293	1.23398	0.616991	−	0.786970i	$-0.288351\pi$
$829$	35.5293	1.23398	0.616991	+	0.786970i	$0.288351\pi$
$830$	77.0477	2.67436
$831$	−0.162401	−0.00563363
$832$	0	0
$833$	15.8854	0.550396
$834$	0.892220	0.0308950
$835$	−19.1850	−0.663926
$836$	−0.618546	−0.0213929
$837$	−0.229489	−0.00793229
$838$	43.6972	1.50950
$839$	−19.1269	−0.660333	−0.330167	−	0.943923i	$-0.607105\pi$
$839$	−19.1269	−0.660333	−0.330167	+	0.943923i	$0.607105\pi$
$840$	−0.675832	−0.0233184
$841$	−19.3175	−0.666119
$842$	42.4517	1.46298
$843$	0.194644	0.00670391
$844$	2.19616	0.0755951
$845$	0	0
$846$	−30.3343	−1.04292
$847$	12.4096	0.426398
$848$	−25.6273	−0.880045
$849$	1.25003	0.0429008
$850$	53.7975	1.84524
$851$	−61.5391	−2.10953
$852$	0.00283059	9.69745e−5 0
$853$	45.3232	1.55184	0.775919	−	0.630833i	$-0.217287\pi$
$853$	45.3232	1.55184	0.775919	+	0.630833i	$0.217287\pi$
$854$	4.22686	0.144640
$855$	−37.8400	−1.29410
$856$	−1.56688	−0.0535547
$857$	−35.9135	−1.22678	−0.613391	−	0.789779i	$-0.710195\pi$
$857$	−35.9135	−1.22678	−0.613391	+	0.789779i	$0.710195\pi$
$858$	0	0
$859$	32.4843	1.10835	0.554176	−	0.832400i	$-0.313034\pi$
$859$	32.4843	1.10835	0.554176	+	0.832400i	$0.313034\pi$
$860$	3.58947	0.122400
$861$	0.505361	0.0172227
$862$	51.2771	1.74650
$863$	6.12078	0.208354	0.104177	−	0.994559i	$-0.466779\pi$
$863$	6.12078	0.208354	0.104177	+	0.994559i	$0.466779\pi$
$864$	−0.130490	−0.00443937
$865$	7.62535	0.259270
$866$	18.4172	0.625842
$867$	−0.112337	−0.00381518
$868$	−0.167271	−0.00567756
$869$	10.6492	0.361249
$870$	−0.665956	−0.0225780
$871$	0	0
$872$	−29.5699	−1.00136
$873$	25.0187	0.846755
$874$	25.1811	0.851764
$875$	−31.4297	−1.06252
$876$	−0.0521261	−0.00176118
$877$	13.2533	0.447533	0.223766	−	0.974643i	$-0.428165\pi$
$877$	13.2533	0.447533	0.223766	+	0.974643i	$0.428165\pi$
$878$	26.7417	0.902487
$879$	−0.407491	−0.0137443
$880$	30.4183	1.02540
$881$	−4.71128	−0.158727	−0.0793635	−	0.996846i	$-0.525289\pi$
$881$	−4.71128	−0.158727	−0.0793635	+	0.996846i	$0.525289\pi$
$882$	−18.4090	−0.619865
$883$	37.7426	1.27014	0.635070	−	0.772455i	$-0.280971\pi$
$883$	37.7426	1.27014	0.635070	+	0.772455i	$0.280971\pi$
$884$	0	0
$885$	1.78828	0.0601123
$886$	−15.7615	−0.529517
$887$	57.0438	1.91534	0.957671	−	0.287864i	$-0.0929450\pi$
$887$	57.0438	1.91534	0.957671	+	0.287864i	$0.0929450\pi$
$888$	1.21964	0.0409285
$889$	−30.2823	−1.01564
$890$	57.4973	1.92731
$891$	−16.8986	−0.566125
$892$	−0.123300	−0.00412839
$893$	−22.8209	−0.763671
$894$	1.04481	0.0349437
$895$	−23.4453	−0.783690
$896$	20.1027	0.671585
$897$	0	0
$898$	17.1543	0.572446
$899$	3.11168	0.103780
$900$	−2.98605	−0.0995350
$901$	−22.9310	−0.763944
$902$	−21.6535	−0.720984
$903$	−0.587912	−0.0195645
$904$	−20.4137	−0.678950
$905$	−79.5438	−2.64412
$906$	−0.0403048	−0.00133904
$907$	22.9446	0.761863	0.380931	−	0.924603i	$-0.375603\pi$
$907$	22.9446	0.761863	0.380931	+	0.924603i	$0.375603\pi$
$908$	−0.988481	−0.0328039
$909$	25.0866	0.832072
$910$	0	0
$911$	11.9059	0.394460	0.197230	−	0.980357i	$-0.436805\pi$
$911$	11.9059	0.394460	0.197230	+	0.980357i	$0.436805\pi$
$912$	−0.524233	−0.0173591
$913$	25.8982	0.857106
$914$	−47.7037	−1.57790
$915$	0.259029	0.00856323
$916$	0.147830	0.00488446
$917$	−30.8345	−1.01824
$918$	−1.24734	−0.0411682
$919$	−3.50789	−0.115714	−0.0578572	−	0.998325i	$-0.518427\pi$
$919$	−3.50789	−0.115714	−0.0578572	+	0.998325i	$0.518427\pi$
$920$	−56.4638	−1.86156
$921$	−0.438185	−0.0144387
$922$	−36.6160	−1.20588
$923$	0	0
$924$	0.0120332	0.000395865 0
$925$	114.622	3.76876
$926$	28.8384	0.947688
$927$	42.0334	1.38056
$928$	1.76934	0.0580815
$929$	12.0272	0.394599	0.197300	−	0.980343i	$-0.436783\pi$
$929$	12.0272	0.394599	0.197300	+	0.980343i	$0.436783\pi$
$930$	−0.214018	−0.00701793
$931$	−13.8493	−0.453894
$932$	−1.42632	−0.0467208
$933$	0.545803	0.0178688
$934$	−28.5522	−0.934256
$935$	27.2179	0.890122
$936$	0	0
$937$	15.4940	0.506167	0.253083	−	0.967445i	$-0.418555\pi$
$937$	15.4940	0.506167	0.253083	+	0.967445i	$0.418555\pi$
$938$	19.1439	0.625069
$939$	0.396619	0.0129432
$940$	−2.71057	−0.0884092
$941$	4.92655	0.160601	0.0803005	−	0.996771i	$-0.474412\pi$
$941$	4.92655	0.160601	0.0803005	+	0.996771i	$0.474412\pi$
$942$	0.319328	0.0104043
$943$	42.2215	1.37492
$944$	−50.7558	−1.65196
$945$	1.47264	0.0479051
$946$	25.1906	0.819018
$947$	−21.8513	−0.710072	−0.355036	−	0.934853i	$-0.615531\pi$
$947$	−21.8513	−0.710072	−0.355036	+	0.934853i	$0.615531\pi$
$948$	−0.0217989	−0.000707997 0
$949$	0	0
$950$	−46.9022	−1.52171
$951$	0.217278	0.00704571
$952$	17.1637	0.556277
$953$	26.5138	0.858866	0.429433	−	0.903099i	$-0.358714\pi$
$953$	26.5138	0.858866	0.429433	+	0.903099i	$0.358714\pi$
$954$	26.5740	0.860365
$955$	26.1859	0.847357
$956$	−1.24860	−0.0403827
$957$	−0.223849	−0.00723602
$958$	35.0113	1.13117
$959$	5.18913	0.167566
$960$	1.11606	0.0360208
$961$	1.00000	0.0322581
$962$	0	0
$963$	1.70670	0.0549976
$964$	1.00229	0.0322817
$965$	−86.2449	−2.77632
$966$	−0.489875	−0.0157615
$967$	−16.7775	−0.539528	−0.269764	−	0.962927i	$-0.586946\pi$
$967$	−16.7775	−0.539528	−0.269764	+	0.962927i	$0.586946\pi$
$968$	−20.5480	−0.660437
$969$	−0.469078	−0.0150690
$970$	46.6756	1.49866
$971$	6.43434	0.206488	0.103244	−	0.994656i	$-0.467078\pi$
$971$	6.43434	0.206488	0.103244	+	0.994656i	$0.467078\pi$
$972$	0.103859	0.00333129
$973$	26.7520	0.857631
$974$	−31.1915	−0.999442
$975$	0	0
$976$	−7.35188	−0.235328
$977$	−38.4778	−1.23101	−0.615506	−	0.788132i	$-0.711048\pi$
$977$	−38.4778	−1.23101	−0.615506	+	0.788132i	$0.711048\pi$
$978$	−0.175926	−0.00562550
$979$	19.3267	0.617683
$980$	−1.64497	−0.0525467
$981$	32.2087	1.02834
$982$	1.54386	0.0492666
$983$	28.3956	0.905678	0.452839	−	0.891592i	$-0.350411\pi$
$983$	28.3956	0.905678	0.452839	+	0.891592i	$0.350411\pi$
$984$	−0.836786	−0.0266758
$985$	−26.4610	−0.843119
$986$	16.9128	0.538615
$987$	0.443959	0.0141314
$988$	0	0
$989$	−49.1183	−1.56187
$990$	−31.5419	−1.00247
$991$	−33.6179	−1.06791	−0.533954	−	0.845513i	$-0.679295\pi$
$991$	−33.6179	−1.06791	−0.533954	+	0.845513i	$0.679295\pi$
$992$	0.568613	0.0180535
$993$	0.0258994	0.000821891 0
$994$	1.77199	0.0562040
$995$	25.2955	0.801922
$996$	−0.0530138	−0.00167981
$997$	−57.9223	−1.83442	−0.917208	−	0.398408i	$-0.869563\pi$
$997$	−57.9223	−1.83442	−0.917208	+	0.398408i	$0.869563\pi$
$998$	31.5550	0.998855
$999$	−2.65761	−0.0840830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.k.1.6		16
13.5	odd	4		403.2.c.b.311.24	yes	32
13.8	odd	4		403.2.c.b.311.9	✓	32
13.12	even	2		5239.2.a.l.1.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.c.b.311.9	✓	32	13.8	odd	4
403.2.c.b.311.24	yes	32	13.5	odd	4
5239.2.a.k.1.6		16	1.1	even	1	trivial
5239.2.a.l.1.11		16	13.12	even	2

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:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44935 q^{2} +0.0382575 q^{3} +0.100611 q^{4} +3.85977 q^{5} -0.0554484 q^{6} -1.66255 q^{7} +2.75288 q^{8} -2.99854 q^{9} +O(q^{10})\)	\(q-1.44935 q^{2} +0.0382575 q^{3} +0.100611 q^{4} +3.85977 q^{5} -0.0554484 q^{6} -1.66255 q^{7} +2.75288 q^{8} -2.99854 q^{9} -5.59415 q^{10} -1.88038 q^{11} +0.00384914 q^{12} +2.40961 q^{14} +0.147665 q^{15} -4.19110 q^{16} -3.75015 q^{17} +4.34592 q^{18} +3.26949 q^{19} +0.388337 q^{20} -0.0636049 q^{21} +2.72532 q^{22} -5.31401 q^{23} +0.105318 q^{24} +9.89783 q^{25} -0.229489 q^{27} -0.167271 q^{28} +3.11168 q^{29} -0.214018 q^{30} +1.00000 q^{31} +0.568613 q^{32} -0.0719384 q^{33} +5.43528 q^{34} -6.41705 q^{35} -0.301687 q^{36} +11.5806 q^{37} -4.73863 q^{38} +10.6255 q^{40} -7.94532 q^{41} +0.0921856 q^{42} +9.24319 q^{43} -0.189187 q^{44} -11.5737 q^{45} +7.70185 q^{46} -6.97995 q^{47} -0.160341 q^{48} -4.23593 q^{49} -14.3454 q^{50} -0.143471 q^{51} +6.11470 q^{53} +0.332609 q^{54} -7.25782 q^{55} -4.57679 q^{56} +0.125082 q^{57} -4.50991 q^{58} +12.1104 q^{59} +0.0148568 q^{60} +1.75416 q^{61} -1.44935 q^{62} +4.98521 q^{63} +7.55808 q^{64} +0.104264 q^{66} +7.94479 q^{67} -0.377308 q^{68} -0.203300 q^{69} +9.30055 q^{70} +0.735383 q^{71} -8.25460 q^{72} -13.5423 q^{73} -16.7843 q^{74} +0.378666 q^{75} +0.328948 q^{76} +3.12622 q^{77} -5.66333 q^{79} -16.1767 q^{80} +8.98683 q^{81} +11.5155 q^{82} -13.7729 q^{83} -0.00639938 q^{84} -14.4747 q^{85} -13.3966 q^{86} +0.119045 q^{87} -5.17644 q^{88} -10.2781 q^{89} +16.7743 q^{90} -0.534650 q^{92} +0.0382575 q^{93} +10.1164 q^{94} +12.6195 q^{95} +0.0217537 q^{96} -8.34364 q^{97} +6.13935 q^{98} +5.63838 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} - 2 q^{3} + 18 q^{4} - 4 q^{5} - 6 q^{6} - 2 q^{7} - 12 q^{8} + 10 q^{9}+O(q^{10}) \) 16 * q - 4 * q^2 - 2 * q^3 + 18 * q^4 - 4 * q^5 - 6 * q^6 - 2 * q^7 - 12 * q^8 + 10 * q^9	\( 16 q - 4 q^{2} - 2 q^{3} + 18 q^{4} - 4 q^{5} - 6 q^{6} - 2 q^{7} - 12 q^{8} + 10 q^{9} - 2 q^{10} - 14 q^{11} + 8 q^{12} - 8 q^{14} + 14 q^{16} + 4 q^{17} + 28 q^{18} - 22 q^{19} - 28 q^{20} - 12 q^{21} - 8 q^{22} + 4 q^{23} + 8 q^{24} - 2 q^{25} + 10 q^{27} - 16 q^{28} - 8 q^{29} - 20 q^{30} + 16 q^{31} - 48 q^{32} - 10 q^{33} - 8 q^{34} - 2 q^{35} + 22 q^{36} - 16 q^{37} - 6 q^{38} + 14 q^{40} - 44 q^{41} + 14 q^{42} + 16 q^{43} - 4 q^{44} - 56 q^{45} - 10 q^{47} + 32 q^{49} - 2 q^{50} - 6 q^{53} - 24 q^{54} + 22 q^{55} - 4 q^{56} + 8 q^{57} - 74 q^{58} - 2 q^{59} - 40 q^{60} + 8 q^{61} - 4 q^{62} - 56 q^{63} + 38 q^{64} - 34 q^{66} + 8 q^{67} + 32 q^{68} - 10 q^{69} + 108 q^{70} - 50 q^{71} + 44 q^{72} - 14 q^{73} + 8 q^{74} - 44 q^{76} + 16 q^{77} + 32 q^{79} - 68 q^{80} - 8 q^{81} - 6 q^{82} + 20 q^{83} - 136 q^{84} + 32 q^{85} - 8 q^{86} - 36 q^{87} - 40 q^{88} - 52 q^{89} - 34 q^{90} + 14 q^{92} - 2 q^{93} + 44 q^{94} - 2 q^{95} + 80 q^{96} - 18 q^{97} - 12 q^{98} - 38 q^{99}+O(q^{100}) \) 16 * q - 4 * q^2 - 2 * q^3 + 18 * q^4 - 4 * q^5 - 6 * q^6 - 2 * q^7 - 12 * q^8 + 10 * q^9 - 2 * q^10 - 14 * q^11 + 8 * q^12 - 8 * q^14 + 14 * q^16 + 4 * q^17 + 28 * q^18 - 22 * q^19 - 28 * q^20 - 12 * q^21 - 8 * q^22 + 4 * q^23 + 8 * q^24 - 2 * q^25 + 10 * q^27 - 16 * q^28 - 8 * q^29 - 20 * q^30 + 16 * q^31 - 48 * q^32 - 10 * q^33 - 8 * q^34 - 2 * q^35 + 22 * q^36 - 16 * q^37 - 6 * q^38 + 14 * q^40 - 44 * q^41 + 14 * q^42 + 16 * q^43 - 4 * q^44 - 56 * q^45 - 10 * q^47 + 32 * q^49 - 2 * q^50 - 6 * q^53 - 24 * q^54 + 22 * q^55 - 4 * q^56 + 8 * q^57 - 74 * q^58 - 2 * q^59 - 40 * q^60 + 8 * q^61 - 4 * q^62 - 56 * q^63 + 38 * q^64 - 34 * q^66 + 8 * q^67 + 32 * q^68 - 10 * q^69 + 108 * q^70 - 50 * q^71 + 44 * q^72 - 14 * q^73 + 8 * q^74 - 44 * q^76 + 16 * q^77 + 32 * q^79 - 68 * q^80 - 8 * q^81 - 6 * q^82 + 20 * q^83 - 136 * q^84 + 32 * q^85 - 8 * q^86 - 36 * q^87 - 40 * q^88 - 52 * q^89 - 34 * q^90 + 14 * q^92 - 2 * q^93 + 44 * q^94 - 2 * q^95 + 80 * q^96 - 18 * q^97 - 12 * q^98 - 38 * q^99

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