LMFDB - Embedded newform 4015.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4015,2,Mod(1,4015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4015, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4015 = 5 \cdot 11 \cdot 73 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4015.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:	$32.0599364115$
Analytic rank:	$1$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.315426 q^{2} +2.38391 q^{3} -1.90051 q^{4} -1.00000 q^{5} +0.751948 q^{6} +0.00485401 q^{7} -1.23032 q^{8} +2.68302 q^{9} +O(q^{10})$	\(q+0.315426 q^{2} +2.38391 q^{3} -1.90051 q^{4} -1.00000 q^{5} +0.751948 q^{6} +0.00485401 q^{7} -1.23032 q^{8} +2.68302 q^{9} -0.315426 q^{10} +1.00000 q^{11} -4.53064 q^{12} +0.139360 q^{13} +0.00153108 q^{14} -2.38391 q^{15} +3.41294 q^{16} -1.09212 q^{17} +0.846296 q^{18} -4.53407 q^{19} +1.90051 q^{20} +0.0115715 q^{21} +0.315426 q^{22} +1.37855 q^{23} -2.93298 q^{24} +1.00000 q^{25} +0.0439578 q^{26} -0.755641 q^{27} -0.00922507 q^{28} -9.29049 q^{29} -0.751948 q^{30} -2.10380 q^{31} +3.53717 q^{32} +2.38391 q^{33} -0.344484 q^{34} -0.00485401 q^{35} -5.09911 q^{36} +10.9268 q^{37} -1.43016 q^{38} +0.332221 q^{39} +1.23032 q^{40} +5.58866 q^{41} +0.00364996 q^{42} -6.58975 q^{43} -1.90051 q^{44} -2.68302 q^{45} +0.434831 q^{46} -3.69707 q^{47} +8.13613 q^{48} -6.99998 q^{49} +0.315426 q^{50} -2.60352 q^{51} -0.264854 q^{52} -8.56197 q^{53} -0.238349 q^{54} -1.00000 q^{55} -0.00597199 q^{56} -10.8088 q^{57} -2.93046 q^{58} -4.55986 q^{59} +4.53064 q^{60} -10.2471 q^{61} -0.663595 q^{62} +0.0130234 q^{63} -5.71016 q^{64} -0.139360 q^{65} +0.751948 q^{66} +5.74607 q^{67} +2.07558 q^{68} +3.28634 q^{69} -0.00153108 q^{70} +12.5861 q^{71} -3.30098 q^{72} +1.00000 q^{73} +3.44659 q^{74} +2.38391 q^{75} +8.61702 q^{76} +0.00485401 q^{77} +0.104791 q^{78} -2.38764 q^{79} -3.41294 q^{80} -9.85045 q^{81} +1.76281 q^{82} -17.1570 q^{83} -0.0219917 q^{84} +1.09212 q^{85} -2.07858 q^{86} -22.1477 q^{87} -1.23032 q^{88} +0.283948 q^{89} -0.846296 q^{90} +0.000676454 q^{91} -2.61995 q^{92} -5.01528 q^{93} -1.16615 q^{94} +4.53407 q^{95} +8.43230 q^{96} +12.1669 q^{97} -2.20798 q^{98} +2.68302 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9} + 7 q^{10} + 31 q^{11} - 4 q^{12} - 24 q^{13} - 9 q^{14} + 4 q^{15} + 43 q^{16} - 49 q^{17} - 35 q^{18} - 22 q^{19} - 39 q^{20} - 8 q^{21} - 7 q^{22} - q^{23} - 13 q^{24} + 31 q^{25} - 9 q^{26} - 22 q^{27} - 34 q^{28} - 12 q^{29} + 5 q^{30} + 4 q^{31} - 45 q^{32} - 4 q^{33} + 2 q^{34} + 11 q^{35} + 34 q^{36} - 18 q^{37} - 7 q^{38} - q^{39} + 24 q^{40} - 58 q^{41} - 21 q^{42} - 41 q^{43} + 39 q^{44} - 31 q^{45} + 23 q^{46} - 31 q^{47} - 29 q^{48} + 44 q^{49} - 7 q^{50} + 8 q^{51} - 89 q^{52} - 46 q^{53} - 47 q^{54} - 31 q^{55} + 10 q^{56} - 47 q^{57} - 34 q^{58} - 9 q^{59} + 4 q^{60} - 5 q^{61} - 50 q^{62} - 61 q^{63} + 78 q^{64} + 24 q^{65} - 5 q^{66} + q^{67} - 115 q^{68} - 19 q^{69} + 9 q^{70} - 8 q^{71} - 93 q^{72} + 31 q^{73} - 19 q^{74} - 4 q^{75} - 7 q^{76} - 11 q^{77} + 57 q^{78} - 43 q^{80} + 43 q^{81} + 20 q^{82} - 29 q^{83} - 32 q^{84} + 49 q^{85} + 25 q^{86} - 62 q^{87} - 24 q^{88} - 77 q^{89} + 35 q^{90} - 11 q^{91} - 25 q^{92} - 38 q^{94} + 22 q^{95} - 23 q^{96} - 39 q^{97} - 65 q^{98} + 31 q^{99}+O(q^{100}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9 + 7 * q^10 + 31 * q^11 - 4 * q^12 - 24 * q^13 - 9 * q^14 + 4 * q^15 + 43 * q^16 - 49 * q^17 - 35 * q^18 - 22 * q^19 - 39 * q^20 - 8 * q^21 - 7 * q^22 - q^23 - 13 * q^24 + 31 * q^25 - 9 * q^26 - 22 * q^27 - 34 * q^28 - 12 * q^29 + 5 * q^30 + 4 * q^31 - 45 * q^32 - 4 * q^33 + 2 * q^34 + 11 * q^35 + 34 * q^36 - 18 * q^37 - 7 * q^38 - q^39 + 24 * q^40 - 58 * q^41 - 21 * q^42 - 41 * q^43 + 39 * q^44 - 31 * q^45 + 23 * q^46 - 31 * q^47 - 29 * q^48 + 44 * q^49 - 7 * q^50 + 8 * q^51 - 89 * q^52 - 46 * q^53 - 47 * q^54 - 31 * q^55 + 10 * q^56 - 47 * q^57 - 34 * q^58 - 9 * q^59 + 4 * q^60 - 5 * q^61 - 50 * q^62 - 61 * q^63 + 78 * q^64 + 24 * q^65 - 5 * q^66 + q^67 - 115 * q^68 - 19 * q^69 + 9 * q^70 - 8 * q^71 - 93 * q^72 + 31 * q^73 - 19 * q^74 - 4 * q^75 - 7 * q^76 - 11 * q^77 + 57 * q^78 - 43 * q^80 + 43 * q^81 + 20 * q^82 - 29 * q^83 - 32 * q^84 + 49 * q^85 + 25 * q^86 - 62 * q^87 - 24 * q^88 - 77 * q^89 + 35 * q^90 - 11 * q^91 - 25 * q^92 - 38 * q^94 + 22 * q^95 - 23 * q^96 - 39 * q^97 - 65 * q^98 + 31 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.315426	0.223040	0.111520	−	0.993762i	$-0.464428\pi$
$2$	0.315426	0.223040	0.111520	+	0.993762i	$0.464428\pi$
$3$	2.38391	1.37635	0.688175	−	0.725544i	$-0.258412\pi$
$3$	2.38391	1.37635	0.688175	+	0.725544i	$0.258412\pi$
$4$	−1.90051	−0.950253
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.751948	0.306981
$7$	0.00485401	0.00183464	0.000917321	−	1.00000i	$-0.499708\pi$
$7$	0.00485401	0.00183464	0.000917321		1.00000i	$0.499708\pi$
$8$	−1.23032	−0.434985
$9$	2.68302	0.894342
$10$	−0.315426	−0.0997465
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−4.53064	−1.30788
$13$	0.139360	0.0386515	0.0193257	−	0.999813i	$-0.493848\pi$
$13$	0.139360	0.0386515	0.0193257	+	0.999813i	$0.493848\pi$
$14$	0.00153108	0.000409199 0
$15$	−2.38391	−0.615523
$16$	3.41294	0.853234
$17$	−1.09212	−0.264878	−0.132439	−	0.991191i	$-0.542281\pi$
$17$	−1.09212	−0.264878	−0.132439	+	0.991191i	$0.542281\pi$
$18$	0.846296	0.199474
$19$	−4.53407	−1.04019	−0.520093	−	0.854110i	$-0.674103\pi$
$19$	−4.53407	−1.04019	−0.520093	+	0.854110i	$0.674103\pi$
$20$	1.90051	0.424966
$21$	0.0115715	0.00252511
$22$	0.315426	0.0672491
$23$	1.37855	0.287448	0.143724	−	0.989618i	$-0.454092\pi$
$23$	1.37855	0.287448	0.143724	+	0.989618i	$0.454092\pi$
$24$	−2.93298	−0.598691
$25$	1.00000	0.200000
$26$	0.0439578	0.00862083
$27$	−0.755641	−0.145423
$28$	−0.00922507	−0.00174337
$29$	−9.29049	−1.72520	−0.862600	−	0.505886i	$-0.831166\pi$
$29$	−9.29049	−1.72520	−0.862600	+	0.505886i	$0.831166\pi$
$30$	−0.751948	−0.137286
$31$	−2.10380	−0.377854	−0.188927	−	0.981991i	$-0.560501\pi$
$31$	−2.10380	−0.377854	−0.188927	+	0.981991i	$0.560501\pi$
$32$	3.53717	0.625290
$33$	2.38391	0.414985
$34$	−0.344484	−0.0590784
$35$	−0.00485401	−0.000820477 0
$36$	−5.09911	−0.849851
$37$	10.9268	1.79635	0.898175	−	0.439639i	$-0.144894\pi$
$37$	10.9268	1.79635	0.898175	+	0.439639i	$0.144894\pi$
$38$	−1.43016	−0.232003
$39$	0.332221	0.0531980
$40$	1.23032	0.194531
$41$	5.58866	0.872803	0.436401	−	0.899752i	$-0.356253\pi$
$41$	5.58866	0.872803	0.436401	+	0.899752i	$0.356253\pi$
$42$	0.00364996	0.000563201 0
$43$	−6.58975	−1.00493	−0.502464	−	0.864598i	$-0.667573\pi$
$43$	−6.58975	−1.00493	−0.502464	+	0.864598i	$0.667573\pi$
$44$	−1.90051	−0.286512
$45$	−2.68302	−0.399962
$46$	0.434831	0.0641124
$47$	−3.69707	−0.539273	−0.269636	−	0.962962i	$-0.586903\pi$
$47$	−3.69707	−0.539273	−0.269636	+	0.962962i	$0.586903\pi$
$48$	8.13613	1.17435
$49$	−6.99998	−0.999997
$50$	0.315426	0.0446080
$51$	−2.60352	−0.364565
$52$	−0.264854	−0.0367287
$53$	−8.56197	−1.17608	−0.588039	−	0.808833i	$-0.700100\pi$
$53$	−8.56197	−1.17608	−0.588039	+	0.808833i	$0.700100\pi$
$54$	−0.238349	−0.0324352
$55$	−1.00000	−0.134840
$56$	−0.00597199	−0.000798041 0
$57$	−10.8088	−1.43166
$58$	−2.93046	−0.384789
$59$	−4.55986	−0.593643	−0.296821	−	0.954933i	$-0.595927\pi$
$59$	−4.55986	−0.593643	−0.296821	+	0.954933i	$0.595927\pi$
$60$	4.53064	0.584902
$61$	−10.2471	−1.31200	−0.656002	−	0.754759i	$-0.727754\pi$
$61$	−10.2471	−1.31200	−0.656002	+	0.754759i	$0.727754\pi$
$62$	−0.663595	−0.0842766
$63$	0.0130234	0.00164080
$64$	−5.71016	−0.713770
$65$	−0.139360	−0.0172855
$66$	0.751948	0.0925584
$67$	5.74607	0.701994	0.350997	−	0.936377i	$-0.385843\pi$
$67$	5.74607	0.701994	0.350997	+	0.936377i	$0.385843\pi$
$68$	2.07558	0.251701
$69$	3.28634	0.395629
$70$	−0.00153108	−0.000182999 0
$71$	12.5861	1.49369	0.746845	−	0.664998i	$-0.231568\pi$
$71$	12.5861	1.49369	0.746845	+	0.664998i	$0.231568\pi$
$72$	−3.30098	−0.389025
$73$	1.00000	0.117041
$73$	1.00000	0.117041
$74$	3.44659	0.400658
$75$	2.38391	0.275270
$76$	8.61702	0.988440
$77$	0.00485401	0.000553165 0
$78$	0.104791	0.0118653
$79$	−2.38764	−0.268630	−0.134315	−	0.990939i	$-0.542883\pi$
$79$	−2.38764	−0.268630	−0.134315	+	0.990939i	$0.542883\pi$
$80$	−3.41294	−0.381578
$81$	−9.85045	−1.09449
$82$	1.76281	0.194670
$83$	−17.1570	−1.88323	−0.941615	−	0.336693i	$-0.890692\pi$
$83$	−17.1570	−1.88323	−0.941615	+	0.336693i	$0.890692\pi$
$84$	−0.0219917	−0.00239950
$85$	1.09212	0.118457
$86$	−2.07858	−0.224139
$87$	−22.1477	−2.37448
$88$	−1.23032	−0.131153
$89$	0.283948	0.0300984	0.0150492	−	0.999887i	$-0.495210\pi$
$89$	0.283948	0.0300984	0.0150492	+	0.999887i	$0.495210\pi$
$90$	−0.846296	−0.0892075
$91$	0.000676454 0	7.09116e−5 0
$92$	−2.61995	−0.273148
$93$	−5.01528	−0.520060
$94$	−1.16615	−0.120279
$95$	4.53407	0.465185
$96$	8.43230	0.860618
$97$	12.1669	1.23536	0.617680	−	0.786429i	$-0.288073\pi$
$97$	12.1669	1.23536	0.617680	+	0.786429i	$0.288073\pi$
$98$	−2.20798	−0.223039
$99$	2.68302	0.269654
$100$	−1.90051	−0.190051
$101$	−2.70486	−0.269144	−0.134572	−	0.990904i	$-0.542966\pi$
$101$	−2.70486	−0.269144	−0.134572	+	0.990904i	$0.542966\pi$
$102$	−0.821218	−0.0813127
$103$	−11.0497	−1.08875	−0.544377	−	0.838840i	$-0.683234\pi$
$103$	−11.0497	−1.08875	−0.544377	+	0.838840i	$0.683234\pi$
$104$	−0.171458	−0.0168128
$105$	−0.0115715	−0.00112926
$106$	−2.70067	−0.262312
$107$	5.94034	0.574274	0.287137	−	0.957889i	$-0.407296\pi$
$107$	5.94034	0.574274	0.287137	+	0.957889i	$0.407296\pi$
$108$	1.43610	0.138189
$109$	−15.7267	−1.50635	−0.753173	−	0.657822i	$-0.771478\pi$
$109$	−15.7267	−1.50635	−0.753173	+	0.657822i	$0.771478\pi$
$110$	−0.315426	−0.0300747
$111$	26.0484	2.47241
$112$	0.0165664	0.00156538
$113$	−7.35469	−0.691871	−0.345935	−	0.938258i	$-0.612438\pi$
$113$	−7.35469	−0.691871	−0.345935	+	0.938258i	$0.612438\pi$
$114$	−3.40938	−0.319318
$115$	−1.37855	−0.128551
$116$	17.6566	1.63938
$117$	0.373906	0.0345676
$118$	−1.43830	−0.132406
$119$	−0.00530116	−0.000485957 0
$120$	2.93298	0.267743
$121$	1.00000	0.0909091
$122$	−3.23220	−0.292629
$123$	13.3229	1.20128
$124$	3.99829	0.359057
$125$	−1.00000	−0.0894427
$126$	0.00410793	0.000365963 0
$127$	0.465605	0.0413157	0.0206579	−	0.999787i	$-0.493424\pi$
$127$	0.465605	0.0413157	0.0206579	+	0.999787i	$0.493424\pi$
$128$	−8.87548	−0.784489
$129$	−15.7094	−1.38313
$130$	−0.0439578	−0.00385535
$131$	−20.9234	−1.82809	−0.914043	−	0.405617i	$-0.867057\pi$
$131$	−20.9234	−1.82809	−0.914043	+	0.405617i	$0.867057\pi$
$132$	−4.53064	−0.394341
$133$	−0.0220084	−0.00190837
$134$	1.81246	0.156573
$135$	0.755641	0.0650352
$136$	1.34366	0.115218
$137$	−18.2453	−1.55880	−0.779402	−	0.626524i	$-0.784477\pi$
$137$	−18.2453	−1.55880	−0.779402	+	0.626524i	$0.784477\pi$
$138$	1.03660	0.0882411
$139$	10.1370	0.859809	0.429904	−	0.902874i	$-0.358547\pi$
$139$	10.1370	0.859809	0.429904	+	0.902874i	$0.358547\pi$
$140$	0.00922507	0.000779661 0
$141$	−8.81347	−0.742228
$142$	3.96997	0.333153
$143$	0.139360	0.0116539
$144$	9.15699	0.763083
$145$	9.29049	0.771533
$146$	0.315426	0.0261049
$147$	−16.6873	−1.37635
$148$	−20.7664	−1.70699
$149$	18.9733	1.55436	0.777178	−	0.629281i	$-0.216650\pi$
$149$	18.9733	1.55436	0.777178	+	0.629281i	$0.216650\pi$
$150$	0.751948	0.0613963
$151$	−1.70211	−0.138516	−0.0692579	−	0.997599i	$-0.522063\pi$
$151$	−1.70211	−0.138516	−0.0692579	+	0.997599i	$0.522063\pi$
$152$	5.57836	0.452465
$153$	−2.93019	−0.236892
$154$	0.00153108	0.000123378 0
$155$	2.10380	0.168982
$156$	−0.631389	−0.0505516
$157$	8.60002	0.686356	0.343178	−	0.939270i	$-0.388497\pi$
$157$	8.60002	0.686356	0.343178	+	0.939270i	$0.388497\pi$
$158$	−0.753124	−0.0599153
$159$	−20.4110	−1.61869
$160$	−3.53717	−0.279638
$161$	0.00669150	0.000527364 0
$162$	−3.10709	−0.244116
$163$	0.217553	0.0170401	0.00852003	−	0.999964i	$-0.497288\pi$
$163$	0.217553	0.0170401	0.00852003	+	0.999964i	$0.497288\pi$
$164$	−10.6213	−0.829383
$165$	−2.38391	−0.185587
$166$	−5.41178	−0.420036
$167$	10.2115	0.790186	0.395093	−	0.918641i	$-0.370712\pi$
$167$	10.2115	0.790186	0.395093	+	0.918641i	$0.370712\pi$
$168$	−0.0142367	−0.00109838
$169$	−12.9806	−0.998506
$170$	0.344484	0.0264207
$171$	−12.1650	−0.930282
$172$	12.5239	0.954936
$173$	−11.6535	−0.885999	−0.443000	−	0.896522i	$-0.646086\pi$
$173$	−11.6535	−0.885999	−0.443000	+	0.896522i	$0.646086\pi$
$174$	−6.98596	−0.529604
$175$	0.00485401	0.000366928 0
$176$	3.41294	0.257260
$177$	−10.8703	−0.817061
$178$	0.0895645	0.00671314
$179$	18.0084	1.34601	0.673004	−	0.739639i	$-0.265004\pi$
$179$	18.0084	1.34601	0.673004	+	0.739639i	$0.265004\pi$
$180$	5.09911	0.380065
$181$	−2.44675	−0.181865	−0.0909327	−	0.995857i	$-0.528985\pi$
$181$	−2.44675	−0.181865	−0.0909327	+	0.995857i	$0.528985\pi$
$182$	0.000213371 0	1.58161e−5 0
$183$	−24.4281	−1.80578
$184$	−1.69606	−0.125035
$185$	−10.9268	−0.803352
$186$	−1.58195	−0.115994
$187$	−1.09212	−0.0798638
$188$	7.02630	0.512446
$189$	−0.00366789	−0.000266799 0
$190$	1.43016	0.103755
$191$	−7.30975	−0.528915	−0.264458	−	0.964397i	$-0.585193\pi$
$191$	−7.30975	−0.528915	−0.264458	+	0.964397i	$0.585193\pi$
$192$	−13.6125	−0.982397
$193$	−1.53414	−0.110430	−0.0552150	−	0.998474i	$-0.517584\pi$
$193$	−1.53414	−0.110430	−0.0552150	+	0.998474i	$0.517584\pi$
$194$	3.83776	0.275535
$195$	−0.332221	−0.0237909
$196$	13.3035	0.950250
$197$	−2.86076	−0.203820	−0.101910	−	0.994794i	$-0.532495\pi$
$197$	−2.86076	−0.203820	−0.101910	+	0.994794i	$0.532495\pi$
$198$	0.846296	0.0601437
$199$	−21.8748	−1.55066	−0.775332	−	0.631554i	$-0.782418\pi$
$199$	−21.8748	−1.55066	−0.775332	+	0.631554i	$0.782418\pi$
$200$	−1.23032	−0.0869969
$201$	13.6981	0.966190
$202$	−0.853184	−0.0600298
$203$	−0.0450961	−0.00316513
$204$	4.94800	0.346429
$205$	−5.58866	−0.390329
$206$	−3.48535	−0.242836
$207$	3.69869	0.257077
$208$	0.475626	0.0329788
$209$	−4.53407	−0.313628
$210$	−0.00364996	−0.000251871 0
$211$	9.02054	0.621000	0.310500	−	0.950573i	$-0.399504\pi$
$211$	9.02054	0.621000	0.310500	+	0.950573i	$0.399504\pi$
$212$	16.2721	1.11757
$213$	30.0040	2.05584
$214$	1.87374	0.128086
$215$	6.58975	0.449417
$216$	0.929681	0.0632568
$217$	−0.0102119	−0.000693227 0
$218$	−4.96062	−0.335975
$219$	2.38391	0.161090
$220$	1.90051	0.128132
$221$	−0.152198	−0.0102379
$222$	8.21636	0.551446
$223$	15.7203	1.05271	0.526354	−	0.850265i	$-0.323559\pi$
$223$	15.7203	1.05271	0.526354	+	0.850265i	$0.323559\pi$
$224$	0.0171695	0.00114718
$225$	2.68302	0.178868
$226$	−2.31986	−0.154315
$227$	13.6074	0.903155	0.451577	−	0.892232i	$-0.350861\pi$
$227$	13.6074	0.903155	0.451577	+	0.892232i	$0.350861\pi$
$228$	20.5422	1.36044
$229$	18.3939	1.21550	0.607752	−	0.794127i	$-0.292071\pi$
$229$	18.3939	1.21550	0.607752	+	0.794127i	$0.292071\pi$
$230$	−0.434831	−0.0286719
$231$	0.0115715	0.000761350 0
$232$	11.4303	0.750436
$233$	8.50816	0.557388	0.278694	−	0.960380i	$-0.410098\pi$
$233$	8.50816	0.557388	0.278694	+	0.960380i	$0.410098\pi$
$234$	0.117940	0.00770996
$235$	3.69707	0.241170
$236$	8.66604	0.564111
$237$	−5.69192	−0.369730
$238$	−0.00167213	−0.000108388 0
$239$	−11.0629	−0.715599	−0.357799	−	0.933799i	$-0.616473\pi$
$239$	−11.0629	−0.715599	−0.357799	+	0.933799i	$0.616473\pi$
$240$	−8.13613	−0.525185
$241$	−26.8069	−1.72678	−0.863391	−	0.504535i	$-0.831664\pi$
$241$	−26.8069	−1.72678	−0.863391	+	0.504535i	$0.831664\pi$
$242$	0.315426	0.0202764
$243$	−21.2157	−1.36099
$244$	19.4746	1.24674
$245$	6.99998	0.447212
$246$	4.20238	0.267934
$247$	−0.631867	−0.0402047
$248$	2.58836	0.164361
$249$	−40.9008	−2.59198
$250$	−0.315426	−0.0199493
$251$	−14.8704	−0.938610	−0.469305	−	0.883036i	$-0.655496\pi$
$251$	−14.8704	−0.938610	−0.469305	+	0.883036i	$0.655496\pi$
$252$	−0.0247511	−0.00155917
$253$	1.37855	0.0866688
$254$	0.146864	0.00921506
$255$	2.60352	0.163039
$256$	8.62075	0.538797
$257$	−13.0159	−0.811908	−0.405954	−	0.913893i	$-0.633061\pi$
$257$	−13.0159	−0.811908	−0.405954	+	0.913893i	$0.633061\pi$
$258$	−4.95515	−0.308494
$259$	0.0530386	0.00329566
$260$	0.264854	0.0164256
$261$	−24.9266	−1.54292
$262$	−6.59979	−0.407736
$263$	4.55095	0.280624	0.140312	−	0.990107i	$-0.455190\pi$
$263$	4.55095	0.280624	0.140312	+	0.990107i	$0.455190\pi$
$264$	−2.93298	−0.180512
$265$	8.56197	0.525958
$266$	−0.00694202	−0.000425643 0
$267$	0.676905	0.0414259
$268$	−10.9204	−0.667072
$269$	−3.23783	−0.197414	−0.0987069	−	0.995117i	$-0.531471\pi$
$269$	−3.23783	−0.197414	−0.0987069	+	0.995117i	$0.531471\pi$
$270$	0.238349	0.0145055
$271$	−18.5739	−1.12828	−0.564142	−	0.825677i	$-0.690793\pi$
$271$	−18.5739	−1.12828	−0.564142	+	0.825677i	$0.690793\pi$
$272$	−3.72734	−0.226003
$273$	0.00161260	9.75993e−5 0
$274$	−5.75506	−0.347676
$275$	1.00000	0.0603023
$276$	−6.24571	−0.375948
$277$	−8.10302	−0.486863	−0.243431	−	0.969918i	$-0.578273\pi$
$277$	−8.10302	−0.486863	−0.243431	+	0.969918i	$0.578273\pi$
$278$	3.19747	0.191772
$279$	−5.64456	−0.337931
$280$	0.00597199	0.000356895 0
$281$	4.37376	0.260916	0.130458	−	0.991454i	$-0.458355\pi$
$281$	4.37376	0.260916	0.130458	+	0.991454i	$0.458355\pi$
$282$	−2.78000	−0.165547
$283$	15.7094	0.933824	0.466912	−	0.884304i	$-0.345366\pi$
$283$	15.7094	0.933824	0.466912	+	0.884304i	$0.345366\pi$
$284$	−23.9199	−1.41938
$285$	10.8088	0.640258
$286$	0.0439578	0.00259928
$287$	0.0271274	0.00160128
$288$	9.49032	0.559223
$289$	−15.8073	−0.929840
$290$	2.93046	0.172083
$291$	29.0048	1.70029
$292$	−1.90051	−0.111219
$293$	−1.61737	−0.0944880	−0.0472440	−	0.998883i	$-0.515044\pi$
$293$	−1.61737	−0.0944880	−0.0472440	+	0.998883i	$0.515044\pi$
$294$	−5.26362	−0.306980
$295$	4.55986	0.265485
$296$	−13.4434	−0.781384
$297$	−0.755641	−0.0438467
$298$	5.98469	0.346684
$299$	0.192115	0.0111103
$300$	−4.53064	−0.261576
$301$	−0.0319867	−0.00184368
$302$	−0.536890	−0.0308945
$303$	−6.44814	−0.370436
$304$	−15.4745	−0.887522
$305$	10.2471	0.586746
$306$	−0.924258	−0.0528363
$307$	−17.3223	−0.988638	−0.494319	−	0.869281i	$-0.664583\pi$
$307$	−17.3223	−0.988638	−0.494319	+	0.869281i	$0.664583\pi$
$308$	−0.00922507	−0.000525647 0
$309$	−26.3414	−1.49851
$310$	0.663595	0.0376897
$311$	3.45383	0.195849	0.0979244	−	0.995194i	$-0.468780\pi$
$311$	3.45383	0.195849	0.0979244	+	0.995194i	$0.468780\pi$
$312$	−0.408739	−0.0231403
$313$	7.47430	0.422472	0.211236	−	0.977435i	$-0.432251\pi$
$313$	7.47430	0.422472	0.211236	+	0.977435i	$0.432251\pi$
$314$	2.71267	0.153085
$315$	−0.0130234	−0.000733787 0
$316$	4.53772	0.255267
$317$	−1.95137	−0.109600	−0.0547999	−	0.998497i	$-0.517452\pi$
$317$	−1.95137	−0.109600	−0.0547999	+	0.998497i	$0.517452\pi$
$318$	−6.43815	−0.361034
$319$	−9.29049	−0.520168
$320$	5.71016	0.319207
$321$	14.1612	0.790403
$322$	0.00211067	0.000117623 0
$323$	4.95175	0.275523
$324$	18.7208	1.04005
$325$	0.139360	0.00773030
$326$	0.0686219	0.00380062
$327$	−37.4911	−2.07326
$328$	−6.87586	−0.379656
$329$	−0.0179456	−0.000989373 0
$330$	−0.751948	−0.0413934
$331$	11.7265	0.644545	0.322272	−	0.946647i	$-0.395553\pi$
$331$	11.7265	0.644545	0.322272	+	0.946647i	$0.395553\pi$
$332$	32.6071	1.78954
$333$	29.3168	1.60655
$334$	3.22096	0.176243
$335$	−5.74607	−0.313941
$336$	0.0394928	0.00215451
$337$	7.09030	0.386233	0.193117	−	0.981176i	$-0.438140\pi$
$337$	7.09030	0.386233	0.193117	+	0.981176i	$0.438140\pi$
$338$	−4.09442	−0.222707
$339$	−17.5329	−0.952257
$340$	−2.07558	−0.112564
$341$	−2.10380	−0.113927
$342$	−3.83716	−0.207490
$343$	−0.0679560	−0.00366928
$344$	8.10752	0.437128
$345$	−3.28634	−0.176931
$346$	−3.67582	−0.197613
$347$	28.7694	1.54442	0.772212	−	0.635365i	$-0.219150\pi$
$347$	28.7694	1.54442	0.772212	+	0.635365i	$0.219150\pi$
$348$	42.0918	2.25636
$349$	14.5802	0.780462	0.390231	−	0.920717i	$-0.372395\pi$
$349$	14.5802	0.780462	0.390231	+	0.920717i	$0.372395\pi$
$350$	0.00153108	8.18397e−5 0
$351$	−0.105306	−0.00562082
$352$	3.53717	0.188532
$353$	30.8635	1.64270	0.821349	−	0.570426i	$-0.193222\pi$
$353$	30.8635	1.64270	0.821349	+	0.570426i	$0.193222\pi$
$354$	−3.42877	−0.182237
$355$	−12.5861	−0.667998
$356$	−0.539644	−0.0286011
$357$	−0.0126375	−0.000668847 0
$358$	5.68031	0.300214
$359$	−20.5627	−1.08526	−0.542630	−	0.839972i	$-0.682571\pi$
$359$	−20.5627	−1.08526	−0.542630	+	0.839972i	$0.682571\pi$
$360$	3.30098	0.173977
$361$	1.55776	0.0819873
$362$	−0.771768	−0.0405632
$363$	2.38391	0.125123
$364$	−0.00128560	−6.73840e−5 0
$365$	−1.00000	−0.0523424
$366$	−7.70527	−0.402761
$367$	−4.39747	−0.229546	−0.114773	−	0.993392i	$-0.536614\pi$
$367$	−4.39747	−0.229546	−0.114773	+	0.993392i	$0.536614\pi$
$368$	4.70491	0.245260
$369$	14.9945	0.780584
$370$	−3.44659	−0.179180
$371$	−0.0415599	−0.00215768
$372$	9.53157	0.494189
$373$	23.4551	1.21446	0.607230	−	0.794526i	$-0.292281\pi$
$373$	23.4551	1.21446	0.607230	+	0.794526i	$0.292281\pi$
$374$	−0.344484	−0.0178128
$375$	−2.38391	−0.123105
$376$	4.54858	0.234575
$377$	−1.29472	−0.0666816
$378$	−0.00115695	−5.95069e−5 0
$379$	20.2474	1.04004	0.520021	−	0.854154i	$-0.325924\pi$
$379$	20.2474	1.04004	0.520021	+	0.854154i	$0.325924\pi$
$380$	−8.61702	−0.442044
$381$	1.10996	0.0568649
$382$	−2.30569	−0.117969
$383$	25.1613	1.28568	0.642842	−	0.765998i	$-0.277755\pi$
$383$	25.1613	1.28568	0.642842	+	0.765998i	$0.277755\pi$
$384$	−21.1583	−1.07973
$385$	−0.00485401	−0.000247383 0
$386$	−0.483909	−0.0246303
$387$	−17.6805	−0.898749
$388$	−23.1232	−1.17390
$389$	34.6084	1.75471	0.877357	−	0.479839i	$-0.159305\pi$
$389$	34.6084	1.75471	0.877357	+	0.479839i	$0.159305\pi$
$390$	−0.104791	−0.00530632
$391$	−1.50554	−0.0761387
$392$	8.61223	0.434983
$393$	−49.8795	−2.51609
$394$	−0.902357	−0.0454601
$395$	2.38764	0.120135
$396$	−5.09911	−0.256240
$397$	16.8514	0.845745	0.422873	−	0.906189i	$-0.361022\pi$
$397$	16.8514	0.845745	0.422873	+	0.906189i	$0.361022\pi$
$398$	−6.89989	−0.345860
$399$	−0.0524660	−0.00262659
$400$	3.41294	0.170647
$401$	−17.5010	−0.873957	−0.436979	−	0.899472i	$-0.643952\pi$
$401$	−17.5010	−0.873957	−0.436979	+	0.899472i	$0.643952\pi$
$402$	4.32074	0.215499
$403$	−0.293186	−0.0146046
$404$	5.14060	0.255755
$405$	9.85045	0.489473
$406$	−0.0142245	−0.000705950 0
$407$	10.9268	0.541620
$408$	3.20316	0.158580
$409$	18.9926	0.939122	0.469561	−	0.882900i	$-0.344412\pi$
$409$	18.9926	0.939122	0.469561	+	0.882900i	$0.344412\pi$
$410$	−1.76281	−0.0870590
$411$	−43.4952	−2.14546
$412$	20.9999	1.03459
$413$	−0.0221336	−0.00108912
$414$	1.16666	0.0573384
$415$	17.1570	0.842206
$416$	0.492940	0.0241684
$417$	24.1657	1.18340
$418$	−1.43016	−0.0699516
$419$	−34.8838	−1.70418	−0.852092	−	0.523393i	$-0.824666\pi$
$419$	−34.8838	−1.70418	−0.852092	+	0.523393i	$0.824666\pi$
$420$	0.0219917	0.00107309
$421$	22.1520	1.07962	0.539812	−	0.841786i	$-0.318495\pi$
$421$	22.1520	1.07962	0.539812	+	0.841786i	$0.318495\pi$
$422$	2.84531	0.138508
$423$	−9.91932	−0.482294
$424$	10.5340	0.511575
$425$	−1.09212	−0.0529756
$426$	9.46405	0.458535
$427$	−0.0497394	−0.00240706
$428$	−11.2897	−0.545706
$429$	0.332221	0.0160398
$430$	2.07858	0.100238
$431$	21.0720	1.01500	0.507500	−	0.861652i	$-0.330570\pi$
$431$	21.0720	1.01500	0.507500	+	0.861652i	$0.330570\pi$
$432$	−2.57895	−0.124080
$433$	18.0179	0.865884	0.432942	−	0.901422i	$-0.357476\pi$
$433$	18.0179	0.865884	0.432942	+	0.901422i	$0.357476\pi$
$434$	−0.00322109	−0.000154617 0
$435$	22.1477	1.06190
$436$	29.8887	1.43141
$437$	−6.25044	−0.298999
$438$	0.751948	0.0359294
$439$	−22.5790	−1.07764	−0.538819	−	0.842421i	$-0.681130\pi$
$439$	−22.5790	−1.07764	−0.538819	+	0.842421i	$0.681130\pi$
$440$	1.23032	0.0586533
$441$	−18.7811	−0.894339
$442$	−0.0480072	−0.00228347
$443$	−2.30299	−0.109418	−0.0547091	−	0.998502i	$-0.517423\pi$
$443$	−2.30299	−0.109418	−0.0547091	+	0.998502i	$0.517423\pi$
$444$	−49.5052	−2.34941
$445$	−0.283948	−0.0134604
$446$	4.95859	0.234796
$447$	45.2307	2.13934
$448$	−0.0277171	−0.00130951
$449$	−27.4433	−1.29513	−0.647564	−	0.762011i	$-0.724212\pi$
$449$	−27.4433	−1.29513	−0.647564	+	0.762011i	$0.724212\pi$
$450$	0.846296	0.0398948
$451$	5.58866	0.263160
$452$	13.9776	0.657452
$453$	−4.05768	−0.190646
$454$	4.29213	0.201440
$455$	−0.000676454 0	−3.17126e−5 0
$456$	13.2983	0.622750
$457$	11.3488	0.530876	0.265438	−	0.964128i	$-0.414484\pi$
$457$	11.3488	0.530876	0.265438	+	0.964128i	$0.414484\pi$
$458$	5.80192	0.271106
$459$	0.825251	0.0385194
$460$	2.61995	0.122156
$461$	14.8536	0.691799	0.345900	−	0.938272i	$-0.387574\pi$
$461$	14.8536	0.691799	0.345900	+	0.938272i	$0.387574\pi$
$462$	0.00364996	0.000169811 0
$463$	26.6457	1.23833	0.619165	−	0.785261i	$-0.287471\pi$
$463$	26.6457	1.23833	0.619165	+	0.785261i	$0.287471\pi$
$464$	−31.7079	−1.47200
$465$	5.01528	0.232578
$466$	2.68370	0.124320
$467$	−7.88900	−0.365059	−0.182530	−	0.983200i	$-0.558429\pi$
$467$	−7.88900	−0.365059	−0.182530	+	0.983200i	$0.558429\pi$
$468$	−0.710611	−0.0328480
$469$	0.0278915	0.00128791
$470$	1.16615	0.0537906
$471$	20.5017	0.944667
$472$	5.61009	0.258225
$473$	−6.58975	−0.302997
$474$	−1.79538	−0.0824645
$475$	−4.53407	−0.208037
$476$	0.0100749	0.000461782 0
$477$	−22.9720	−1.05181
$478$	−3.48952	−0.159607
$479$	16.5238	0.754991	0.377496	−	0.926011i	$-0.376785\pi$
$479$	16.5238	0.754991	0.377496	+	0.926011i	$0.376785\pi$
$480$	−8.43230	−0.384880
$481$	1.52275	0.0694316
$482$	−8.45559	−0.385142
$483$	0.0159519	0.000725838 0
$484$	−1.90051	−0.0863866
$485$	−12.1669	−0.552470
$486$	−6.69198	−0.303554
$487$	30.4032	1.37770	0.688849	−	0.724905i	$-0.258116\pi$
$487$	30.4032	1.37770	0.688849	+	0.724905i	$0.258116\pi$
$488$	12.6072	0.570702
$489$	0.518627	0.0234531
$490$	2.20798	0.0997462
$491$	37.2373	1.68050	0.840248	−	0.542203i	$-0.182410\pi$
$491$	37.2373	1.68050	0.840248	+	0.542203i	$0.182410\pi$
$492$	−25.3202	−1.14152
$493$	10.1463	0.456968
$494$	−0.199307	−0.00896726
$495$	−2.68302	−0.120593
$496$	−7.18015	−0.322398
$497$	0.0610928	0.00274039
$498$	−12.9012	−0.578116
$499$	−9.64970	−0.431980	−0.215990	−	0.976396i	$-0.569298\pi$
$499$	−9.64970	−0.431980	−0.215990	+	0.976396i	$0.569298\pi$
$500$	1.90051	0.0849932
$501$	24.3432	1.08757
$502$	−4.69051	−0.209348
$503$	28.4325	1.26774	0.633872	−	0.773438i	$-0.281465\pi$
$503$	28.4325	1.26774	0.633872	+	0.773438i	$0.281465\pi$
$504$	−0.0160230	−0.000713721 0
$505$	2.70486	0.120365
$506$	0.434831	0.0193306
$507$	−30.9445	−1.37429
$508$	−0.884885	−0.0392604
$509$	13.6772	0.606232	0.303116	−	0.952954i	$-0.401973\pi$
$509$	13.6772	0.606232	0.303116	+	0.952954i	$0.401973\pi$
$510$	0.821218	0.0363641
$511$	0.00485401	0.000214729 0
$512$	20.4702	0.904662
$513$	3.42613	0.151267
$514$	−4.10555	−0.181088
$515$	11.0497	0.486906
$516$	29.8558	1.31433
$517$	−3.69707	−0.162597
$518$	0.0167298	0.000735064 0
$519$	−27.7809	−1.21945
$520$	0.171458	0.00751891
$521$	20.1311	0.881958	0.440979	−	0.897518i	$-0.354631\pi$
$521$	20.1311	0.881958	0.440979	+	0.897518i	$0.354631\pi$
$522$	−7.86251	−0.344133
$523$	−15.1986	−0.664587	−0.332294	−	0.943176i	$-0.607823\pi$
$523$	−15.1986	−0.664587	−0.332294	+	0.943176i	$0.607823\pi$
$524$	39.7651	1.73715
$525$	0.0115715	0.000505022 0
$526$	1.43549	0.0625903
$527$	2.29761	0.100085
$528$	8.13613	0.354080
$529$	−21.0996	−0.917374
$530$	2.70067	0.117310
$531$	−12.2342	−0.530919
$532$	0.0418271	0.00181343
$533$	0.778836	0.0337351
$534$	0.213514	0.00923964
$535$	−5.94034	−0.256823
$536$	−7.06951	−0.305356
$537$	42.9303	1.85258
$538$	−1.02130	−0.0440312
$539$	−6.99998	−0.301510
$540$	−1.43610	−0.0617999
$541$	12.8571	0.552770	0.276385	−	0.961047i	$-0.410863\pi$
$541$	12.8571	0.552770	0.276385	+	0.961047i	$0.410863\pi$
$542$	−5.85870	−0.251653
$543$	−5.83283	−0.250310
$544$	−3.86302	−0.165626
$545$	15.7267	0.673658
$546$	0.000508658 0	2.17685e−5 0
$547$	0.371232	0.0158727	0.00793637	−	0.999969i	$-0.497474\pi$
$547$	0.371232	0.0158727	0.00793637	+	0.999969i	$0.497474\pi$
$548$	34.6754	1.48126
$549$	−27.4932	−1.17338
$550$	0.315426	0.0134498
$551$	42.1237	1.79453
$552$	−4.04326	−0.172093
$553$	−0.0115896	−0.000492841 0
$554$	−2.55590	−0.108590
$555$	−26.0484	−1.10569
$556$	−19.2654	−0.817036
$557$	−35.9498	−1.52324	−0.761621	−	0.648022i	$-0.775596\pi$
$557$	−35.9498	−1.52324	−0.761621	+	0.648022i	$0.775596\pi$
$558$	−1.78044	−0.0753721
$559$	−0.918347	−0.0388420
$560$	−0.0165664	−0.000700059 0
$561$	−2.60352	−0.109921
$562$	1.37960	0.0581948
$563$	−28.0982	−1.18420	−0.592098	−	0.805866i	$-0.701700\pi$
$563$	−28.0982	−1.18420	−0.592098	+	0.805866i	$0.701700\pi$
$564$	16.7501	0.705305
$565$	7.35469	0.309414
$566$	4.95514	0.208280
$567$	−0.0478142	−0.00200801
$568$	−15.4849	−0.649732
$569$	−30.5378	−1.28021	−0.640105	−	0.768287i	$-0.721109\pi$
$569$	−30.5378	−1.28021	−0.640105	+	0.768287i	$0.721109\pi$
$570$	3.40938	0.142803
$571$	21.6266	0.905047	0.452523	−	0.891752i	$-0.350524\pi$
$571$	21.6266	0.905047	0.452523	+	0.891752i	$0.350524\pi$
$572$	−0.264854	−0.0110741
$573$	−17.4258	−0.727973
$574$	0.00855670	0.000357150 0
$575$	1.37855	0.0574896
$576$	−15.3205	−0.638354
$577$	−6.60725	−0.275064	−0.137532	−	0.990497i	$-0.543917\pi$
$577$	−6.60725	−0.275064	−0.137532	+	0.990497i	$0.543917\pi$
$578$	−4.98603	−0.207391
$579$	−3.65726	−0.151990
$580$	−17.6566	−0.733152
$581$	−0.0832804	−0.00345505
$582$	9.14886	0.379233
$583$	−8.56197	−0.354601
$584$	−1.23032	−0.0509111
$585$	−0.373906	−0.0154591
$586$	−0.510162	−0.0210746
$587$	−12.4065	−0.512070	−0.256035	−	0.966668i	$-0.582416\pi$
$587$	−12.4065	−0.512070	−0.256035	+	0.966668i	$0.582416\pi$
$588$	31.7143	1.30788
$589$	9.53879	0.393039
$590$	1.43830	0.0592138
$591$	−6.81978	−0.280528
$592$	37.2924	1.53271
$593$	−38.3652	−1.57547	−0.787735	−	0.616015i	$-0.788746\pi$
$593$	−38.3652	−1.57547	−0.787735	+	0.616015i	$0.788746\pi$
$594$	−0.238349	−0.00977957
$595$	0.00530116	0.000217326 0
$596$	−36.0589	−1.47703
$597$	−52.1476	−2.13426
$598$	0.0605980	0.00247804
$599$	−38.9043	−1.58959	−0.794793	−	0.606881i	$-0.792420\pi$
$599$	−38.9043	−1.58959	−0.794793	+	0.606881i	$0.792420\pi$
$600$	−2.93298	−0.119738
$601$	8.26541	0.337153	0.168577	−	0.985689i	$-0.446083\pi$
$601$	8.26541	0.337153	0.168577	+	0.985689i	$0.446083\pi$
$602$	−0.0100894	−0.000411215 0
$603$	15.4168	0.627822
$604$	3.23487	0.131625
$605$	−1.00000	−0.0406558
$606$	−2.03391	−0.0826221
$607$	−4.38807	−0.178106	−0.0890531	−	0.996027i	$-0.528384\pi$
$607$	−4.38807	−0.178106	−0.0890531	+	0.996027i	$0.528384\pi$
$608$	−16.0378	−0.650418
$609$	−0.107505	−0.00435632
$610$	3.23220	0.130868
$611$	−0.515223	−0.0208437
$612$	5.56884	0.225107
$613$	23.0162	0.929616	0.464808	−	0.885412i	$-0.346123\pi$
$613$	23.0162	0.929616	0.464808	+	0.885412i	$0.346123\pi$
$614$	−5.46392	−0.220506
$615$	−13.3229	−0.537230
$616$	−0.00597199	−0.000240618 0
$617$	−18.5534	−0.746931	−0.373465	−	0.927644i	$-0.621831\pi$
$617$	−18.5534	−0.746931	−0.373465	+	0.927644i	$0.621831\pi$
$618$	−8.30876	−0.334227
$619$	35.6661	1.43354	0.716770	−	0.697309i	$-0.245620\pi$
$619$	35.6661	1.43354	0.716770	+	0.697309i	$0.245620\pi$
$620$	−3.99829	−0.160575
$621$	−1.04169	−0.0418016
$622$	1.08943	0.0436821
$623$	0.00137828	5.52198e−5 0
$624$	1.13385	0.0453903
$625$	1.00000	0.0400000
$626$	2.35759	0.0942282
$627$	−10.8088	−0.431662
$628$	−16.3444	−0.652212
$629$	−11.9333	−0.475814
$630$	−0.00410793	−0.000163664 0
$631$	−35.7922	−1.42486	−0.712432	−	0.701741i	$-0.752406\pi$
$631$	−35.7922	−1.42486	−0.712432	+	0.701741i	$0.752406\pi$
$632$	2.93757	0.116850
$633$	21.5042	0.854713
$634$	−0.615513	−0.0244452
$635$	−0.465605	−0.0184770
$636$	38.7912	1.53817
$637$	−0.975516	−0.0386513
$638$	−2.93046	−0.116018
$639$	33.7687	1.33587
$640$	8.87548	0.350834
$641$	20.5756	0.812688	0.406344	−	0.913720i	$-0.366804\pi$
$641$	20.5756	0.812688	0.406344	+	0.913720i	$0.366804\pi$
$642$	4.46682	0.176291
$643$	3.28637	0.129602	0.0648009	−	0.997898i	$-0.479359\pi$
$643$	3.28637	0.129602	0.0648009	+	0.997898i	$0.479359\pi$
$644$	−0.0127172	−0.000501129 0
$645$	15.7094	0.618556
$646$	1.56191	0.0614526
$647$	−33.9477	−1.33462	−0.667311	−	0.744779i	$-0.732555\pi$
$647$	−33.9477	−1.33462	−0.667311	+	0.744779i	$0.732555\pi$
$648$	12.1192	0.476088
$649$	−4.55986	−0.178990
$650$	0.0439578	0.00172417
$651$	−0.0243442	−0.000954124 0
$652$	−0.413461	−0.0161924
$653$	33.9537	1.32871	0.664356	−	0.747416i	$-0.268706\pi$
$653$	33.9537	1.32871	0.664356	+	0.747416i	$0.268706\pi$
$654$	−11.8257	−0.462420
$655$	20.9234	0.817545
$656$	19.0738	0.744705
$657$	2.68302	0.104675
$658$	−0.00566051	−0.000220670 0
$659$	−6.02042	−0.234522	−0.117261	−	0.993101i	$-0.537411\pi$
$659$	−6.02042	−0.234522	−0.117261	+	0.993101i	$0.537411\pi$
$660$	4.53064	0.176355
$661$	−8.22504	−0.319917	−0.159958	−	0.987124i	$-0.551136\pi$
$661$	−8.22504	−0.319917	−0.159958	+	0.987124i	$0.551136\pi$
$662$	3.69883	0.143759
$663$	−0.362826	−0.0140910
$664$	21.1087	0.819176
$665$	0.0220084	0.000853449 0
$666$	9.24728	0.358325
$667$	−12.8074	−0.495905
$668$	−19.4069	−0.750877
$669$	37.4758	1.44890
$670$	−1.81246	−0.0700215
$671$	−10.2471	−0.395584
$672$	0.0409305	0.00157893
$673$	−7.11445	−0.274242	−0.137121	−	0.990554i	$-0.543785\pi$
$673$	−7.11445	−0.274242	−0.137121	+	0.990554i	$0.543785\pi$
$674$	2.23647	0.0861455
$675$	−0.755641	−0.0290846
$676$	24.6697	0.948834
$677$	6.18509	0.237712	0.118856	−	0.992911i	$-0.462077\pi$
$677$	6.18509	0.237712	0.118856	+	0.992911i	$0.462077\pi$
$678$	−5.53034	−0.212391
$679$	0.0590582	0.00226644
$680$	−1.34366	−0.0515270
$681$	32.4388	1.24306
$682$	−0.663595	−0.0254104
$683$	5.43431	0.207938	0.103969	−	0.994581i	$-0.466846\pi$
$683$	5.43431	0.207938	0.103969	+	0.994581i	$0.466846\pi$
$684$	23.1197	0.884003
$685$	18.2453	0.697118
$686$	−0.0214351	−0.000818396 0
$687$	43.8494	1.67296
$688$	−22.4904	−0.857439
$689$	−1.19320	−0.0454571
$690$	−1.03660	−0.0394626
$691$	−48.2910	−1.83708	−0.918538	−	0.395333i	$-0.870629\pi$
$691$	−48.2910	−1.83708	−0.918538	+	0.395333i	$0.870629\pi$
$692$	22.1475	0.841923
$693$	0.0130234	0.000494719 0
$694$	9.07463	0.344468
$695$	−10.1370	−0.384518
$696$	27.2488	1.03286
$697$	−6.10350	−0.231186
$698$	4.59899	0.174074
$699$	20.2827	0.767161
$700$	−0.00922507	−0.000348675 0
$701$	25.0342	0.945527	0.472764	−	0.881189i	$-0.343256\pi$
$701$	25.0342	0.945527	0.472764	+	0.881189i	$0.343256\pi$
$702$	−0.0332163	−0.00125367
$703$	−49.5427	−1.86854
$704$	−5.71016	−0.215210
$705$	8.81347	0.331935
$706$	9.73515	0.366387
$707$	−0.0131294	−0.000493782 0
$708$	20.6590	0.776414
$709$	48.7032	1.82909	0.914543	−	0.404489i	$-0.132551\pi$
$709$	48.7032	1.82909	0.914543	+	0.404489i	$0.132551\pi$
$710$	−3.96997	−0.148990
$711$	−6.40610	−0.240247
$712$	−0.349347	−0.0130923
$713$	−2.90020	−0.108613
$714$	−0.00398620	−0.000149180 0
$715$	−0.139360	−0.00521176
$716$	−34.2250	−1.27905
$717$	−26.3729	−0.984915
$718$	−6.48603	−0.242056
$719$	−37.4570	−1.39691	−0.698456	−	0.715653i	$-0.746129\pi$
$719$	−37.4570	−1.39691	−0.698456	+	0.715653i	$0.746129\pi$
$720$	−9.15699	−0.341261
$721$	−0.0536351	−0.00199748
$722$	0.491358	0.0182865
$723$	−63.9052	−2.37666
$724$	4.65006	0.172818
$725$	−9.29049	−0.345040
$726$	0.751948	0.0279074
$727$	−48.7135	−1.80668	−0.903342	−	0.428922i	$-0.858894\pi$
$727$	−48.7135	−1.80668	−0.903342	+	0.428922i	$0.858894\pi$
$728$	−0.000832256 0	−3.08455e−5 0
$729$	−21.0249	−0.778699
$730$	−0.315426	−0.0116744
$731$	7.19681	0.266184
$732$	46.4258	1.71595
$733$	−50.1163	−1.85109	−0.925543	−	0.378642i	$-0.876391\pi$
$733$	−50.1163	−1.85109	−0.925543	+	0.378642i	$0.876391\pi$
$734$	−1.38708	−0.0511979
$735$	16.6873	0.615521
$736$	4.87618	0.179738
$737$	5.74607	0.211659
$738$	4.72967	0.174101
$739$	32.9012	1.21029	0.605145	−	0.796115i	$-0.293115\pi$
$739$	32.9012	1.21029	0.605145	+	0.796115i	$0.293115\pi$
$740$	20.7664	0.763388
$741$	−1.50631	−0.0553358
$742$	−0.0131091	−0.000481249 0
$743$	−18.3917	−0.674727	−0.337363	−	0.941374i	$-0.609535\pi$
$743$	−18.3917	−0.674727	−0.337363	+	0.941374i	$0.609535\pi$
$744$	6.17041	0.226218
$745$	−18.9733	−0.695129
$746$	7.39836	0.270873
$747$	−46.0327	−1.68425
$748$	2.07558	0.0758908
$749$	0.0288345	0.00105359
$750$	−0.751948	−0.0274572
$751$	−0.164906	−0.00601751	−0.00300876	−	0.999995i	$-0.500958\pi$
$751$	−0.164906	−0.00601751	−0.00300876	+	0.999995i	$0.500958\pi$
$752$	−12.6179	−0.460126
$753$	−35.4496	−1.29186
$754$	−0.408389	−0.0148727
$755$	1.70211	0.0619461
$756$	0.00697084	0.000253527 0
$757$	−28.1034	−1.02144	−0.510718	−	0.859748i	$-0.670620\pi$
$757$	−28.1034	−1.02144	−0.510718	+	0.859748i	$0.670620\pi$
$758$	6.38658	0.231971
$759$	3.28634	0.119287
$760$	−5.57836	−0.202348
$761$	8.68736	0.314917	0.157458	−	0.987526i	$-0.449670\pi$
$761$	8.68736	0.314917	0.157458	+	0.987526i	$0.449670\pi$
$762$	0.350110	0.0126832
$763$	−0.0763376	−0.00276361
$764$	13.8922	0.502603
$765$	2.93019	0.105941
$766$	7.93655	0.286759
$767$	−0.635461	−0.0229452
$768$	20.5511	0.741574
$769$	−26.0966	−0.941067	−0.470534	−	0.882382i	$-0.655939\pi$
$769$	−26.0966	−0.941067	−0.470534	+	0.882382i	$0.655939\pi$
$770$	−0.00153108	−5.51763e−5 0
$771$	−31.0287	−1.11747
$772$	2.91565	0.104936
$773$	−36.8793	−1.32645	−0.663227	−	0.748418i	$-0.730814\pi$
$773$	−36.8793	−1.32645	−0.663227	+	0.748418i	$0.730814\pi$
$774$	−5.57688	−0.200457
$775$	−2.10380	−0.0755709
$776$	−14.9692	−0.537363
$777$	0.126439	0.00453598
$778$	10.9164	0.391371
$779$	−25.3394	−0.907877
$780$	0.631389	0.0226073
$781$	12.5861	0.450364
$782$	−0.474888	−0.0169820
$783$	7.02027	0.250884
$784$	−23.8905	−0.853231
$785$	−8.60002	−0.306948
$786$	−15.7333	−0.561188
$787$	8.78418	0.313122	0.156561	−	0.987668i	$-0.449959\pi$
$787$	8.78418	0.313122	0.156561	+	0.987668i	$0.449959\pi$
$788$	5.43688	0.193681
$789$	10.8491	0.386237
$790$	0.753124	0.0267950
$791$	−0.0356997	−0.00126934
$792$	−3.30098	−0.117295
$793$	−1.42803	−0.0507109
$794$	5.31536	0.188635
$795$	20.4110	0.723902
$796$	41.5732	1.47352
$797$	−23.9222	−0.847366	−0.423683	−	0.905810i	$-0.639263\pi$
$797$	−23.9222	−0.847366	−0.423683	+	0.905810i	$0.639263\pi$
$798$	−0.0165492	−0.000585834 0
$799$	4.03764	0.142842
$800$	3.53717	0.125058
$801$	0.761838	0.0269182
$802$	−5.52027	−0.194927
$803$	1.00000	0.0352892
$804$	−26.0333	−0.918125
$805$	−0.00669150	−0.000235844 0
$806$	−0.0924785	−0.00325742
$807$	−7.71868	−0.271711
$808$	3.32785	0.117073
$809$	−42.5025	−1.49431	−0.747154	−	0.664651i	$-0.768580\pi$
$809$	−42.5025	−1.49431	−0.747154	+	0.664651i	$0.768580\pi$
$810$	3.10709	0.109172
$811$	−50.4703	−1.77225	−0.886127	−	0.463443i	$-0.846614\pi$
$811$	−50.4703	−1.77225	−0.886127	+	0.463443i	$0.846614\pi$
$812$	0.0857054	0.00300767
$813$	−44.2785	−1.55292
$814$	3.44659	0.120803
$815$	−0.217553	−0.00762055
$816$	−8.88564	−0.311060
$817$	29.8784	1.04531
$818$	5.99075	0.209462
$819$	0.00181494	6.34192e−5 0
$820$	10.6213	0.370912
$821$	17.3414	0.605219	0.302610	−	0.953115i	$-0.402142\pi$
$821$	17.3414	0.605219	0.302610	+	0.953115i	$0.402142\pi$
$822$	−13.7195	−0.478524
$823$	49.9996	1.74287	0.871437	−	0.490507i	$-0.163188\pi$
$823$	49.9996	1.74287	0.871437	+	0.490507i	$0.163188\pi$
$824$	13.5946	0.473591
$825$	2.38391	0.0829971
$826$	−0.00698151	−0.000242918 0
$827$	−5.86933	−0.204097	−0.102048	−	0.994779i	$-0.532540\pi$
$827$	−5.86933	−0.204097	−0.102048	+	0.994779i	$0.532540\pi$
$828$	−7.02938	−0.244288
$829$	25.0712	0.870759	0.435379	−	0.900247i	$-0.356614\pi$
$829$	25.0712	0.870759	0.435379	+	0.900247i	$0.356614\pi$
$830$	5.41178	0.187846
$831$	−19.3169	−0.670094
$832$	−0.795767	−0.0275882
$833$	7.64482	0.264877
$834$	7.62249	0.263945
$835$	−10.2115	−0.353382
$836$	8.61702	0.298026
$837$	1.58972	0.0549488
$838$	−11.0033	−0.380101
$839$	24.8146	0.856694	0.428347	−	0.903614i	$-0.359096\pi$
$839$	24.8146	0.856694	0.428347	+	0.903614i	$0.359096\pi$
$840$	0.0142367	0.000491212 0
$841$	57.3132	1.97632
$842$	6.98733	0.240799
$843$	10.4266	0.359113
$844$	−17.1436	−0.590107
$845$	12.9806	0.446545
$846$	−3.12881	−0.107571
$847$	0.00485401	0.000166786 0
$848$	−29.2215	−1.00347
$849$	37.4497	1.28527
$850$	−0.344484	−0.0118157
$851$	15.0631	0.516357
$852$	−57.0228	−1.95357
$853$	37.0777	1.26952	0.634758	−	0.772711i	$-0.281100\pi$
$853$	37.0777	1.26952	0.634758	+	0.772711i	$0.281100\pi$
$854$	−0.0156891	−0.000536870 0
$855$	12.1650	0.416035
$856$	−7.30853	−0.249800
$857$	−48.6119	−1.66055	−0.830276	−	0.557353i	$-0.811817\pi$
$857$	−48.6119	−1.66055	−0.830276	+	0.557353i	$0.811817\pi$
$858$	0.104791	0.00357752
$859$	11.0208	0.376026	0.188013	−	0.982166i	$-0.439795\pi$
$859$	11.0208	0.376026	0.188013	+	0.982166i	$0.439795\pi$
$860$	−12.5239	−0.427060
$861$	0.0646693	0.00220392
$862$	6.64665	0.226386
$863$	45.5945	1.55205	0.776027	−	0.630700i	$-0.217232\pi$
$863$	45.5945	1.55205	0.776027	+	0.630700i	$0.217232\pi$
$864$	−2.67283	−0.0909316
$865$	11.6535	0.396231
$866$	5.68331	0.193127
$867$	−37.6831	−1.27979
$868$	0.0194077	0.000658742 0
$869$	−2.38764	−0.0809951
$870$	6.98596	0.236846
$871$	0.800771	0.0271331
$872$	19.3489	0.655237
$873$	32.6441	1.10483
$874$	−1.97155	−0.0666888
$875$	−0.00485401	−0.000164095 0
$876$	−4.53064	−0.153076
$877$	−2.04400	−0.0690208	−0.0345104	−	0.999404i	$-0.510987\pi$
$877$	−2.04400	−0.0690208	−0.0345104	+	0.999404i	$0.510987\pi$
$878$	−7.12202	−0.240357
$879$	−3.85567	−0.130049
$880$	−3.41294	−0.115050
$881$	−9.92953	−0.334534	−0.167267	−	0.985912i	$-0.553494\pi$
$881$	−9.92953	−0.334534	−0.167267	+	0.985912i	$0.553494\pi$
$882$	−5.92405	−0.199473
$883$	36.3679	1.22388	0.611938	−	0.790906i	$-0.290390\pi$
$883$	36.3679	1.22388	0.611938	+	0.790906i	$0.290390\pi$
$884$	0.289253	0.00972863
$885$	10.8703	0.365401
$886$	−0.726422	−0.0244046
$887$	53.0603	1.78159	0.890795	−	0.454405i	$-0.150148\pi$
$887$	53.0603	1.78159	0.890795	+	0.454405i	$0.150148\pi$
$888$	−32.0479	−1.07546
$889$	0.00226005	7.57996e−5 0
$890$	−0.0895645	−0.00300221
$891$	−9.85045	−0.330003
$892$	−29.8765	−1.00034
$893$	16.7627	0.560944
$894$	14.2670	0.477158
$895$	−18.0084	−0.601953
$896$	−0.0430816	−0.00143926
$897$	0.457984	0.0152916
$898$	−8.65633	−0.288866
$899$	19.5454	0.651874
$900$	−5.09911	−0.169970
$901$	9.35071	0.311517
$902$	1.76281	0.0586952
$903$	−0.0762534	−0.00253756
$904$	9.04863	0.300953
$905$	2.44675	0.0813326
$906$	−1.27990	−0.0425217
$907$	34.6596	1.15085	0.575427	−	0.817853i	$-0.304836\pi$
$907$	34.6596	1.15085	0.575427	+	0.817853i	$0.304836\pi$
$908$	−25.8609	−0.858226
$909$	−7.25721	−0.240706
$910$	−0.000213371 0	−7.07319e−6 0
$911$	7.51658	0.249035	0.124518	−	0.992217i	$-0.460262\pi$
$911$	7.51658	0.249035	0.124518	+	0.992217i	$0.460262\pi$
$912$	−36.8898	−1.22154
$913$	−17.1570	−0.567815
$914$	3.57972	0.118407
$915$	24.4281	0.807569
$916$	−34.9578	−1.15504
$917$	−0.101562	−0.00335389
$918$	0.260306	0.00859137
$919$	−48.3136	−1.59372	−0.796859	−	0.604166i	$-0.793506\pi$
$919$	−48.3136	−1.59372	−0.796859	+	0.604166i	$0.793506\pi$
$920$	1.69606	0.0559175
$921$	−41.2949	−1.36071
$922$	4.68520	0.154299
$923$	1.75399	0.0577333
$924$	−0.0219917	−0.000723475 0
$925$	10.9268	0.359270
$926$	8.40474	0.276197
$927$	−29.6465	−0.973719
$928$	−32.8621	−1.07875
$929$	32.6720	1.07193	0.535967	−	0.844239i	$-0.319947\pi$
$929$	32.6720	1.07193	0.535967	+	0.844239i	$0.319947\pi$
$930$	1.58195	0.0518742
$931$	31.7384	1.04018
$932$	−16.1698	−0.529660
$933$	8.23362	0.269557
$934$	−2.48840	−0.0814228
$935$	1.09212	0.0357162
$936$	−0.460025	−0.0150364
$937$	−19.9189	−0.650724	−0.325362	−	0.945590i	$-0.605486\pi$
$937$	−19.9189	−0.650724	−0.325362	+	0.945590i	$0.605486\pi$
$938$	0.00879770	0.000287255 0
$939$	17.8181	0.581470
$940$	−7.02630	−0.229173
$941$	18.9722	0.618476	0.309238	−	0.950985i	$-0.399926\pi$
$941$	18.9722	0.618476	0.309238	+	0.950985i	$0.399926\pi$
$942$	6.46676	0.210698
$943$	7.70426	0.250885
$944$	−15.5625	−0.506516
$945$	0.00366789	0.000119316 0
$946$	−2.07858	−0.0675805
$947$	14.9553	0.485981	0.242991	−	0.970029i	$-0.421872\pi$
$947$	14.9553	0.485981	0.242991	+	0.970029i	$0.421872\pi$
$948$	10.8175	0.351337
$949$	0.139360	0.00452381
$950$	−1.43016	−0.0464006
$951$	−4.65189	−0.150848
$952$	0.00652214	0.000211384 0
$953$	11.7604	0.380956	0.190478	−	0.981691i	$-0.438996\pi$
$953$	11.7604	0.380956	0.190478	+	0.981691i	$0.438996\pi$
$954$	−7.24596	−0.234597
$955$	7.30975	0.236538
$956$	21.0251	0.680000
$957$	−22.1477	−0.715933
$958$	5.21204	0.168393
$959$	−0.0885630	−0.00285985
$960$	13.6125	0.439341
$961$	−26.5740	−0.857226
$962$	0.480316	0.0154860
$963$	15.9381	0.513597
$964$	50.9466	1.64088
$965$	1.53414	0.0493858
$966$	0.00503166	0.000161891 0
$967$	−30.5066	−0.981027	−0.490514	−	0.871434i	$-0.663191\pi$
$967$	−30.5066	−0.981027	−0.490514	+	0.871434i	$0.663191\pi$
$968$	−1.23032	−0.0395440
$969$	11.8045	0.379216
$970$	−3.83776	−0.123223
$971$	47.2156	1.51522	0.757611	−	0.652706i	$-0.226366\pi$
$971$	47.2156	1.51522	0.757611	+	0.652706i	$0.226366\pi$
$972$	40.3205	1.29328
$973$	0.0492050	0.00157744
$974$	9.58995	0.307282
$975$	0.332221	0.0106396
$976$	−34.9726	−1.11945
$977$	−7.27097	−0.232619	−0.116310	−	0.993213i	$-0.537106\pi$
$977$	−7.27097	−0.232619	−0.116310	+	0.993213i	$0.537106\pi$
$978$	0.163588	0.00523098
$979$	0.283948	0.00907500
$980$	−13.3035	−0.424965
$981$	−42.1952	−1.34719
$982$	11.7456	0.374818
$983$	32.1299	1.02478	0.512392	−	0.858752i	$-0.328760\pi$
$983$	32.1299	1.02478	0.512392	+	0.858752i	$0.328760\pi$
$984$	−16.3914	−0.522539
$985$	2.86076	0.0911513
$986$	3.20042	0.101922
$987$	−0.0427807	−0.00136172
$988$	1.20087	0.0382047
$989$	−9.08431	−0.288864
$990$	−0.846296	−0.0268971
$991$	15.8575	0.503732	0.251866	−	0.967762i	$-0.418956\pi$
$991$	15.8575	0.503732	0.251866	+	0.967762i	$0.418956\pi$
$992$	−7.44152	−0.236268
$993$	27.9548	0.887120
$994$	0.0192703	0.000611216 0
$995$	21.8748	0.693478
$996$	77.7323	2.46304
$997$	32.7127	1.03602	0.518011	−	0.855374i	$-0.326673\pi$
$997$	32.7127	1.03602	0.518011	+	0.855374i	$0.326673\pi$
$998$	−3.04377	−0.0963488
$999$	−8.25671	−0.261231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.f.1.18	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.f.1.18	✓	31	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4015 = 5 \cdot 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4015.a (trivial)

\(f(q)\)	\(=\)	\(q+0.315426 q^{2} +2.38391 q^{3} -1.90051 q^{4} -1.00000 q^{5} +0.751948 q^{6} +0.00485401 q^{7} -1.23032 q^{8} +2.68302 q^{9} +O(q^{10})\)	\(q+0.315426 q^{2} +2.38391 q^{3} -1.90051 q^{4} -1.00000 q^{5} +0.751948 q^{6} +0.00485401 q^{7} -1.23032 q^{8} +2.68302 q^{9} -0.315426 q^{10} +1.00000 q^{11} -4.53064 q^{12} +0.139360 q^{13} +0.00153108 q^{14} -2.38391 q^{15} +3.41294 q^{16} -1.09212 q^{17} +0.846296 q^{18} -4.53407 q^{19} +1.90051 q^{20} +0.0115715 q^{21} +0.315426 q^{22} +1.37855 q^{23} -2.93298 q^{24} +1.00000 q^{25} +0.0439578 q^{26} -0.755641 q^{27} -0.00922507 q^{28} -9.29049 q^{29} -0.751948 q^{30} -2.10380 q^{31} +3.53717 q^{32} +2.38391 q^{33} -0.344484 q^{34} -0.00485401 q^{35} -5.09911 q^{36} +10.9268 q^{37} -1.43016 q^{38} +0.332221 q^{39} +1.23032 q^{40} +5.58866 q^{41} +0.00364996 q^{42} -6.58975 q^{43} -1.90051 q^{44} -2.68302 q^{45} +0.434831 q^{46} -3.69707 q^{47} +8.13613 q^{48} -6.99998 q^{49} +0.315426 q^{50} -2.60352 q^{51} -0.264854 q^{52} -8.56197 q^{53} -0.238349 q^{54} -1.00000 q^{55} -0.00597199 q^{56} -10.8088 q^{57} -2.93046 q^{58} -4.55986 q^{59} +4.53064 q^{60} -10.2471 q^{61} -0.663595 q^{62} +0.0130234 q^{63} -5.71016 q^{64} -0.139360 q^{65} +0.751948 q^{66} +5.74607 q^{67} +2.07558 q^{68} +3.28634 q^{69} -0.00153108 q^{70} +12.5861 q^{71} -3.30098 q^{72} +1.00000 q^{73} +3.44659 q^{74} +2.38391 q^{75} +8.61702 q^{76} +0.00485401 q^{77} +0.104791 q^{78} -2.38764 q^{79} -3.41294 q^{80} -9.85045 q^{81} +1.76281 q^{82} -17.1570 q^{83} -0.0219917 q^{84} +1.09212 q^{85} -2.07858 q^{86} -22.1477 q^{87} -1.23032 q^{88} +0.283948 q^{89} -0.846296 q^{90} +0.000676454 q^{91} -2.61995 q^{92} -5.01528 q^{93} -1.16615 q^{94} +4.53407 q^{95} +8.43230 q^{96} +12.1669 q^{97} -2.20798 q^{98} +2.68302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9	\( 31 q - 7 q^{2} - 4 q^{3} + 39 q^{4} - 31 q^{5} - 5 q^{6} - 11 q^{7} - 24 q^{8} + 31 q^{9} + 7 q^{10} + 31 q^{11} - 4 q^{12} - 24 q^{13} - 9 q^{14} + 4 q^{15} + 43 q^{16} - 49 q^{17} - 35 q^{18} - 22 q^{19} - 39 q^{20} - 8 q^{21} - 7 q^{22} - q^{23} - 13 q^{24} + 31 q^{25} - 9 q^{26} - 22 q^{27} - 34 q^{28} - 12 q^{29} + 5 q^{30} + 4 q^{31} - 45 q^{32} - 4 q^{33} + 2 q^{34} + 11 q^{35} + 34 q^{36} - 18 q^{37} - 7 q^{38} - q^{39} + 24 q^{40} - 58 q^{41} - 21 q^{42} - 41 q^{43} + 39 q^{44} - 31 q^{45} + 23 q^{46} - 31 q^{47} - 29 q^{48} + 44 q^{49} - 7 q^{50} + 8 q^{51} - 89 q^{52} - 46 q^{53} - 47 q^{54} - 31 q^{55} + 10 q^{56} - 47 q^{57} - 34 q^{58} - 9 q^{59} + 4 q^{60} - 5 q^{61} - 50 q^{62} - 61 q^{63} + 78 q^{64} + 24 q^{65} - 5 q^{66} + q^{67} - 115 q^{68} - 19 q^{69} + 9 q^{70} - 8 q^{71} - 93 q^{72} + 31 q^{73} - 19 q^{74} - 4 q^{75} - 7 q^{76} - 11 q^{77} + 57 q^{78} - 43 q^{80} + 43 q^{81} + 20 q^{82} - 29 q^{83} - 32 q^{84} + 49 q^{85} + 25 q^{86} - 62 q^{87} - 24 q^{88} - 77 q^{89} + 35 q^{90} - 11 q^{91} - 25 q^{92} - 38 q^{94} + 22 q^{95} - 23 q^{96} - 39 q^{97} - 65 q^{98} + 31 q^{99}+O(q^{100}) \) 31 * q - 7 * q^2 - 4 * q^3 + 39 * q^4 - 31 * q^5 - 5 * q^6 - 11 * q^7 - 24 * q^8 + 31 * q^9 + 7 * q^10 + 31 * q^11 - 4 * q^12 - 24 * q^13 - 9 * q^14 + 4 * q^15 + 43 * q^16 - 49 * q^17 - 35 * q^18 - 22 * q^19 - 39 * q^20 - 8 * q^21 - 7 * q^22 - q^23 - 13 * q^24 + 31 * q^25 - 9 * q^26 - 22 * q^27 - 34 * q^28 - 12 * q^29 + 5 * q^30 + 4 * q^31 - 45 * q^32 - 4 * q^33 + 2 * q^34 + 11 * q^35 + 34 * q^36 - 18 * q^37 - 7 * q^38 - q^39 + 24 * q^40 - 58 * q^41 - 21 * q^42 - 41 * q^43 + 39 * q^44 - 31 * q^45 + 23 * q^46 - 31 * q^47 - 29 * q^48 + 44 * q^49 - 7 * q^50 + 8 * q^51 - 89 * q^52 - 46 * q^53 - 47 * q^54 - 31 * q^55 + 10 * q^56 - 47 * q^57 - 34 * q^58 - 9 * q^59 + 4 * q^60 - 5 * q^61 - 50 * q^62 - 61 * q^63 + 78 * q^64 + 24 * q^65 - 5 * q^66 + q^67 - 115 * q^68 - 19 * q^69 + 9 * q^70 - 8 * q^71 - 93 * q^72 + 31 * q^73 - 19 * q^74 - 4 * q^75 - 7 * q^76 - 11 * q^77 + 57 * q^78 - 43 * q^80 + 43 * q^81 + 20 * q^82 - 29 * q^83 - 32 * q^84 + 49 * q^85 + 25 * q^86 - 62 * q^87 - 24 * q^88 - 77 * q^89 + 35 * q^90 - 11 * q^91 - 25 * q^92 - 38 * q^94 + 22 * q^95 - 23 * q^96 - 39 * q^97 - 65 * q^98 + 31 * q^99

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