LMFDB - Embedded newform 4015.2.a.e.1.14

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:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q+0.0645883 q^{2} -2.75103 q^{3} -1.99583 q^{4} -1.00000 q^{5} -0.177684 q^{6} +0.602441 q^{7} -0.258084 q^{8} +4.56815 q^{9} +O(q^{10})$	\(q+0.0645883 q^{2} -2.75103 q^{3} -1.99583 q^{4} -1.00000 q^{5} -0.177684 q^{6} +0.602441 q^{7} -0.258084 q^{8} +4.56815 q^{9} -0.0645883 q^{10} +1.00000 q^{11} +5.49058 q^{12} +2.04657 q^{13} +0.0389107 q^{14} +2.75103 q^{15} +3.97499 q^{16} -3.76651 q^{17} +0.295049 q^{18} +7.17666 q^{19} +1.99583 q^{20} -1.65733 q^{21} +0.0645883 q^{22} +5.12537 q^{23} +0.709996 q^{24} +1.00000 q^{25} +0.132184 q^{26} -4.31401 q^{27} -1.20237 q^{28} -5.52789 q^{29} +0.177684 q^{30} -8.95929 q^{31} +0.772906 q^{32} -2.75103 q^{33} -0.243273 q^{34} -0.602441 q^{35} -9.11723 q^{36} +0.982630 q^{37} +0.463528 q^{38} -5.63016 q^{39} +0.258084 q^{40} +0.595572 q^{41} -0.107044 q^{42} -5.78201 q^{43} -1.99583 q^{44} -4.56815 q^{45} +0.331039 q^{46} +13.0587 q^{47} -10.9353 q^{48} -6.63706 q^{49} +0.0645883 q^{50} +10.3618 q^{51} -4.08459 q^{52} -1.54023 q^{53} -0.278635 q^{54} -1.00000 q^{55} -0.155480 q^{56} -19.7432 q^{57} -0.357037 q^{58} -11.9604 q^{59} -5.49058 q^{60} +8.91313 q^{61} -0.578666 q^{62} +2.75204 q^{63} -7.90005 q^{64} -2.04657 q^{65} -0.177684 q^{66} +8.56167 q^{67} +7.51731 q^{68} -14.1000 q^{69} -0.0389107 q^{70} +1.23278 q^{71} -1.17896 q^{72} -1.00000 q^{73} +0.0634664 q^{74} -2.75103 q^{75} -14.3234 q^{76} +0.602441 q^{77} -0.363642 q^{78} -9.70684 q^{79} -3.97499 q^{80} -1.83648 q^{81} +0.0384670 q^{82} -14.1306 q^{83} +3.30775 q^{84} +3.76651 q^{85} -0.373450 q^{86} +15.2074 q^{87} -0.258084 q^{88} +7.47333 q^{89} -0.295049 q^{90} +1.23293 q^{91} -10.2294 q^{92} +24.6472 q^{93} +0.843443 q^{94} -7.17666 q^{95} -2.12628 q^{96} -7.59454 q^{97} -0.428677 q^{98} +4.56815 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9}+O(q^{10}) $ 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9} - 6 q^{10} + 27 q^{11} + 28 q^{12} + 21 q^{13} + 11 q^{14} - 5 q^{15} + 12 q^{16} + 30 q^{17} + 10 q^{18} + 20 q^{19} - 22 q^{20} - 14 q^{21} + 6 q^{22} + 16 q^{23} + 23 q^{24} + 27 q^{25} + 13 q^{26} + 17 q^{27} + 6 q^{28} - 2 q^{29} - 7 q^{30} - 6 q^{31} + 41 q^{32} + 5 q^{33} + 8 q^{34} - 10 q^{35} + 13 q^{36} + 20 q^{37} + 11 q^{38} - 10 q^{39} - 15 q^{40} + 38 q^{41} - 33 q^{42} + 29 q^{43} + 22 q^{44} - 24 q^{45} + 23 q^{46} + 17 q^{47} + 37 q^{48} + 21 q^{49} + 6 q^{50} + 13 q^{51} + 29 q^{52} + 4 q^{53} + q^{54} - 27 q^{55} + 28 q^{56} + 51 q^{57} - 6 q^{58} + 35 q^{59} - 28 q^{60} - 13 q^{61} + 28 q^{62} + 41 q^{63} - 5 q^{64} - 21 q^{65} + 7 q^{66} + 10 q^{67} + 65 q^{68} - 12 q^{69} - 11 q^{70} + 22 q^{71} - 12 q^{72} - 27 q^{73} - 5 q^{74} + 5 q^{75} + 13 q^{76} + 10 q^{77} - 9 q^{78} - 18 q^{79} - 12 q^{80} + 39 q^{81} + 20 q^{82} + 54 q^{83} - 50 q^{84} - 30 q^{85} + 3 q^{86} + 15 q^{87} + 15 q^{88} + 89 q^{89} - 10 q^{90} - 18 q^{91} + 57 q^{92} + 20 q^{93} - 29 q^{94} - 20 q^{95} + 105 q^{96} + 35 q^{97} + 10 q^{98} + 24 q^{99}+O(q^{100}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9 - 6 * q^10 + 27 * q^11 + 28 * q^12 + 21 * q^13 + 11 * q^14 - 5 * q^15 + 12 * q^16 + 30 * q^17 + 10 * q^18 + 20 * q^19 - 22 * q^20 - 14 * q^21 + 6 * q^22 + 16 * q^23 + 23 * q^24 + 27 * q^25 + 13 * q^26 + 17 * q^27 + 6 * q^28 - 2 * q^29 - 7 * q^30 - 6 * q^31 + 41 * q^32 + 5 * q^33 + 8 * q^34 - 10 * q^35 + 13 * q^36 + 20 * q^37 + 11 * q^38 - 10 * q^39 - 15 * q^40 + 38 * q^41 - 33 * q^42 + 29 * q^43 + 22 * q^44 - 24 * q^45 + 23 * q^46 + 17 * q^47 + 37 * q^48 + 21 * q^49 + 6 * q^50 + 13 * q^51 + 29 * q^52 + 4 * q^53 + q^54 - 27 * q^55 + 28 * q^56 + 51 * q^57 - 6 * q^58 + 35 * q^59 - 28 * q^60 - 13 * q^61 + 28 * q^62 + 41 * q^63 - 5 * q^64 - 21 * q^65 + 7 * q^66 + 10 * q^67 + 65 * q^68 - 12 * q^69 - 11 * q^70 + 22 * q^71 - 12 * q^72 - 27 * q^73 - 5 * q^74 + 5 * q^75 + 13 * q^76 + 10 * q^77 - 9 * q^78 - 18 * q^79 - 12 * q^80 + 39 * q^81 + 20 * q^82 + 54 * q^83 - 50 * q^84 - 30 * q^85 + 3 * q^86 + 15 * q^87 + 15 * q^88 + 89 * q^89 - 10 * q^90 - 18 * q^91 + 57 * q^92 + 20 * q^93 - 29 * q^94 - 20 * q^95 + 105 * q^96 + 35 * q^97 + 10 * q^98 + 24 * 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 $

$2$	0.0645883	0.0456708	0.0228354	−	0.999739i	$-0.492731\pi$
$2$	0.0645883	0.0456708	0.0228354	+	0.999739i	$0.492731\pi$
$3$	−2.75103	−1.58831	−0.794153	−	0.607718i	$-0.792085\pi$
$3$	−2.75103	−1.58831	−0.794153	+	0.607718i	$0.792085\pi$
$4$	−1.99583	−0.997914
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.177684	−0.0725393
$7$	0.602441	0.227701	0.113851	−	0.993498i	$-0.463681\pi$
$7$	0.602441	0.227701	0.113851	+	0.993498i	$0.463681\pi$
$8$	−0.258084	−0.0912464
$9$	4.56815	1.52272
$10$	−0.0645883	−0.0204246
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	5.49058	1.58499
$13$	2.04657	0.567615	0.283808	−	0.958881i	$-0.408402\pi$
$13$	2.04657	0.567615	0.283808	+	0.958881i	$0.408402\pi$
$14$	0.0389107	0.0103993
$15$	2.75103	0.710312
$16$	3.97499	0.993747
$17$	−3.76651	−0.913513	−0.456757	−	0.889592i	$-0.650989\pi$
$17$	−3.76651	−0.913513	−0.456757	+	0.889592i	$0.650989\pi$
$18$	0.295049	0.0695437
$19$	7.17666	1.64644	0.823219	−	0.567724i	$-0.192176\pi$
$19$	7.17666	1.64644	0.823219	+	0.567724i	$0.192176\pi$
$20$	1.99583	0.446281
$21$	−1.65733	−0.361659
$22$	0.0645883	0.0137703
$23$	5.12537	1.06871	0.534357	−	0.845259i	$-0.320554\pi$
$23$	5.12537	1.06871	0.534357	+	0.845259i	$0.320554\pi$
$24$	0.709996	0.144927
$25$	1.00000	0.200000
$26$	0.132184	0.0259235
$27$	−4.31401	−0.830232
$28$	−1.20237	−0.227226
$29$	−5.52789	−1.02650	−0.513252	−	0.858238i	$-0.671559\pi$
$29$	−5.52789	−1.02650	−0.513252	+	0.858238i	$0.671559\pi$
$30$	0.177684	0.0324405
$31$	−8.95929	−1.60914	−0.804568	−	0.593860i	$-0.797603\pi$
$31$	−8.95929	−1.60914	−0.804568	+	0.593860i	$0.797603\pi$
$32$	0.772906	0.136632
$33$	−2.75103	−0.478892
$34$	−0.243273	−0.0417209
$35$	−0.602441	−0.101831
$36$	−9.11723	−1.51954
$37$	0.982630	0.161543	0.0807717	−	0.996733i	$-0.474262\pi$
$37$	0.982630	0.161543	0.0807717	+	0.996733i	$0.474262\pi$
$38$	0.463528	0.0751942
$39$	−5.63016	−0.901546
$40$	0.258084	0.0408066
$41$	0.595572	0.0930128	0.0465064	−	0.998918i	$-0.485191\pi$
$41$	0.595572	0.0930128	0.0465064	+	0.998918i	$0.485191\pi$
$42$	−0.107044	−0.0165173
$43$	−5.78201	−0.881748	−0.440874	−	0.897569i	$-0.645332\pi$
$43$	−5.78201	−0.881748	−0.440874	+	0.897569i	$0.645332\pi$
$44$	−1.99583	−0.300882
$45$	−4.56815	−0.680979
$46$	0.331039	0.0488091
$47$	13.0587	1.90481	0.952407	−	0.304830i	$-0.0985995\pi$
$47$	13.0587	1.90481	0.952407	+	0.304830i	$0.0985995\pi$
$48$	−10.9353	−1.57837
$49$	−6.63706	−0.948152
$50$	0.0645883	0.00913417
$51$	10.3618	1.45094
$52$	−4.08459	−0.566431
$53$	−1.54023	−0.211567	−0.105784	−	0.994389i	$-0.533735\pi$
$53$	−1.54023	−0.211567	−0.105784	+	0.994389i	$0.533735\pi$
$54$	−0.278635	−0.0379174
$55$	−1.00000	−0.134840
$56$	−0.155480	−0.0207769
$57$	−19.7432	−2.61505
$58$	−0.357037	−0.0468813
$59$	−11.9604	−1.55712	−0.778558	−	0.627573i	$-0.784048\pi$
$59$	−11.9604	−1.55712	−0.778558	+	0.627573i	$0.784048\pi$
$60$	−5.49058	−0.708830
$61$	8.91313	1.14121	0.570605	−	0.821225i	$-0.306709\pi$
$61$	8.91313	1.14121	0.570605	+	0.821225i	$0.306709\pi$
$62$	−0.578666	−0.0734906
$63$	2.75204	0.346724
$64$	−7.90005	−0.987507
$65$	−2.04657	−0.253845
$66$	−0.177684	−0.0218714
$67$	8.56167	1.04597	0.522987	−	0.852340i	$-0.324817\pi$
$67$	8.56167	1.04597	0.522987	+	0.852340i	$0.324817\pi$
$68$	7.51731	0.911608
$69$	−14.1000	−1.69744
$70$	−0.0389107	−0.00465071
$71$	1.23278	0.146304	0.0731518	−	0.997321i	$-0.476694\pi$
$71$	1.23278	0.146304	0.0731518	+	0.997321i	$0.476694\pi$
$72$	−1.17896	−0.138942
$73$	−1.00000	−0.117041
$73$	−1.00000	−0.117041
$74$	0.0634664	0.00737782
$75$	−2.75103	−0.317661
$76$	−14.3234	−1.64300
$77$	0.602441	0.0686545
$78$	−0.363642	−0.0411744
$79$	−9.70684	−1.09211	−0.546053	−	0.837751i	$-0.683870\pi$
$79$	−9.70684	−1.09211	−0.546053	+	0.837751i	$0.683870\pi$
$80$	−3.97499	−0.444417
$81$	−1.83648	−0.204054
$82$	0.0384670	0.00424797
$83$	−14.1306	−1.55104	−0.775519	−	0.631324i	$-0.782512\pi$
$83$	−14.1306	−1.55104	−0.775519	+	0.631324i	$0.782512\pi$
$84$	3.30775	0.360905
$85$	3.76651	0.408536
$86$	−0.373450	−0.0402702
$87$	15.2074	1.63040
$88$	−0.258084	−0.0275118
$89$	7.47333	0.792171	0.396085	−	0.918214i	$-0.370368\pi$
$89$	7.47333	0.792171	0.396085	+	0.918214i	$0.370368\pi$
$90$	−0.295049	−0.0311009
$91$	1.23293	0.129247
$92$	−10.2294	−1.06648
$93$	24.6472	2.55580
$94$	0.843443	0.0869945
$95$	−7.17666	−0.736310
$96$	−2.12628	−0.217013
$97$	−7.59454	−0.771108	−0.385554	−	0.922685i	$-0.625990\pi$
$97$	−7.59454	−0.771108	−0.385554	+	0.922685i	$0.625990\pi$
$98$	−0.428677	−0.0433029
$99$	4.56815	0.459116
$100$	−1.99583	−0.199583
$101$	8.21963	0.817884	0.408942	−	0.912560i	$-0.365898\pi$
$101$	8.21963	0.817884	0.408942	+	0.912560i	$0.365898\pi$
$102$	0.669250	0.0662656
$103$	−1.34666	−0.132690	−0.0663451	−	0.997797i	$-0.521134\pi$
$103$	−1.34666	−0.132690	−0.0663451	+	0.997797i	$0.521134\pi$
$104$	−0.528186	−0.0517929
$105$	1.65733	0.161739
$106$	−0.0994810	−0.00966245
$107$	9.73173	0.940802	0.470401	−	0.882453i	$-0.344109\pi$
$107$	9.73173	0.940802	0.470401	+	0.882453i	$0.344109\pi$
$108$	8.61002	0.828500
$109$	−4.02439	−0.385466	−0.192733	−	0.981251i	$-0.561735\pi$
$109$	−4.02439	−0.385466	−0.192733	+	0.981251i	$0.561735\pi$
$110$	−0.0645883	−0.00615826
$111$	−2.70324	−0.256580
$112$	2.39470	0.226277
$113$	11.0246	1.03711	0.518555	−	0.855044i	$-0.326470\pi$
$113$	11.0246	1.03711	0.518555	+	0.855044i	$0.326470\pi$
$114$	−1.27518	−0.119431
$115$	−5.12537	−0.477943
$116$	11.0327	1.02436
$117$	9.34901	0.864316
$118$	−0.772504	−0.0711148
$119$	−2.26910	−0.208008
$120$	−0.709996	−0.0648134
$121$	1.00000	0.0909091
$122$	0.575684	0.0521200
$123$	−1.63844	−0.147733
$124$	17.8812	1.60578
$125$	−1.00000	−0.0894427
$126$	0.177750	0.0158352
$127$	9.45694	0.839168	0.419584	−	0.907717i	$-0.362176\pi$
$127$	9.45694	0.839168	0.419584	+	0.907717i	$0.362176\pi$
$128$	−2.05606	−0.181732
$129$	15.9065	1.40049
$130$	−0.132184	−0.0115933
$131$	6.10339	0.533255	0.266628	−	0.963800i	$-0.414091\pi$
$131$	6.10339	0.533255	0.266628	+	0.963800i	$0.414091\pi$
$132$	5.49058	0.477893
$133$	4.32351	0.374896
$134$	0.552984	0.0477706
$135$	4.31401	0.371291
$136$	0.972076	0.0833548
$137$	1.90727	0.162949	0.0814745	−	0.996675i	$-0.474037\pi$
$137$	1.90727	0.162949	0.0814745	+	0.996675i	$0.474037\pi$
$138$	−0.910697	−0.0775237
$139$	6.73873	0.571572	0.285786	−	0.958293i	$-0.407745\pi$
$139$	6.73873	0.571572	0.285786	+	0.958293i	$0.407745\pi$
$140$	1.20237	0.101619
$141$	−35.9250	−3.02543
$142$	0.0796229	0.00668181
$143$	2.04657	0.171142
$144$	18.1583	1.51319
$145$	5.52789	0.459067
$146$	−0.0645883	−0.00534537
$147$	18.2587	1.50596
$148$	−1.96116	−0.161206
$149$	8.47195	0.694049	0.347025	−	0.937856i	$-0.387192\pi$
$149$	8.47195	0.694049	0.347025	+	0.937856i	$0.387192\pi$
$150$	−0.177684	−0.0145079
$151$	−14.1102	−1.14827	−0.574137	−	0.818759i	$-0.694662\pi$
$151$	−14.1102	−1.14827	−0.574137	+	0.818759i	$0.694662\pi$
$152$	−1.85218	−0.150232
$153$	−17.2060	−1.39102
$154$	0.0389107	0.00313551
$155$	8.95929	0.719628
$156$	11.2368	0.899666
$157$	−1.41393	−0.112844	−0.0564218	−	0.998407i	$-0.517969\pi$
$157$	−1.41393	−0.112844	−0.0564218	+	0.998407i	$0.517969\pi$
$158$	−0.626949	−0.0498774
$159$	4.23722	0.336033
$160$	−0.772906	−0.0611036
$161$	3.08773	0.243347
$162$	−0.118615	−0.00931930
$163$	−15.9319	−1.24788	−0.623942	−	0.781471i	$-0.714470\pi$
$163$	−15.9319	−1.24788	−0.623942	+	0.781471i	$0.714470\pi$
$164$	−1.18866	−0.0928188
$165$	2.75103	0.214167
$166$	−0.912674	−0.0708372
$167$	14.7775	1.14352	0.571761	−	0.820421i	$-0.306261\pi$
$167$	14.7775	1.14352	0.571761	+	0.820421i	$0.306261\pi$
$168$	0.427730	0.0330001
$169$	−8.81157	−0.677813
$170$	0.243273	0.0186582
$171$	32.7840	2.50706
$172$	11.5399	0.879909
$173$	10.9091	0.829407	0.414703	−	0.909957i	$-0.363885\pi$
$173$	10.9091	0.829407	0.414703	+	0.909957i	$0.363885\pi$
$174$	0.982219	0.0744618
$175$	0.602441	0.0455403
$176$	3.97499	0.299626
$177$	32.9035	2.47318
$178$	0.482690	0.0361791
$179$	−1.35072	−0.100957	−0.0504787	−	0.998725i	$-0.516075\pi$
$179$	−1.35072	−0.100957	−0.0504787	+	0.998725i	$0.516075\pi$
$180$	9.11723	0.679559
$181$	−13.2522	−0.985031	−0.492515	−	0.870304i	$-0.663922\pi$
$181$	−13.2522	−0.985031	−0.492515	+	0.870304i	$0.663922\pi$
$182$	0.0796332	0.00590281
$183$	−24.5203	−1.81259
$184$	−1.32278	−0.0975163
$185$	−0.982630	−0.0722444
$186$	1.59192	0.116726
$187$	−3.76651	−0.275435
$188$	−26.0630	−1.90084
$189$	−2.59894	−0.189045
$190$	−0.463528	−0.0336279
$191$	−1.95982	−0.141808	−0.0709040	−	0.997483i	$-0.522588\pi$
$191$	−1.95982	−0.141808	−0.0709040	+	0.997483i	$0.522588\pi$
$192$	21.7333	1.56846
$193$	0.124639	0.00897170	0.00448585	−	0.999990i	$-0.498572\pi$
$193$	0.124639	0.00897170	0.00448585	+	0.999990i	$0.498572\pi$
$194$	−0.490519	−0.0352172
$195$	5.63016	0.403184
$196$	13.2464	0.946174
$197$	20.5233	1.46223	0.731113	−	0.682257i	$-0.239001\pi$
$197$	20.5233	1.46223	0.731113	+	0.682257i	$0.239001\pi$
$198$	0.295049	0.0209682
$199$	13.5839	0.962938	0.481469	−	0.876463i	$-0.340103\pi$
$199$	13.5839	0.962938	0.481469	+	0.876463i	$0.340103\pi$
$200$	−0.258084	−0.0182493
$201$	−23.5534	−1.66133
$202$	0.530892	0.0373535
$203$	−3.33023	−0.233736
$204$	−20.6803	−1.44791
$205$	−0.595572	−0.0415966
$206$	−0.0869784	−0.00606007
$207$	23.4134	1.62735
$208$	8.13507	0.564066
$209$	7.17666	0.496420
$210$	0.107044	0.00738675
$211$	−18.7255	−1.28911	−0.644557	−	0.764556i	$-0.722958\pi$
$211$	−18.7255	−1.28911	−0.644557	+	0.764556i	$0.722958\pi$
$212$	3.07404	0.211126
$213$	−3.39140	−0.232375
$214$	0.628556	0.0429672
$215$	5.78201	0.394330
$216$	1.11338	0.0757557
$217$	−5.39744	−0.366402
$218$	−0.259928	−0.0176046
$219$	2.75103	0.185897
$220$	1.99583	0.134559
$221$	−7.70841	−0.518524
$222$	−0.174598	−0.0117182
$223$	9.10313	0.609590	0.304795	−	0.952418i	$-0.401412\pi$
$223$	9.10313	0.609590	0.304795	+	0.952418i	$0.401412\pi$
$224$	0.465630	0.0311112
$225$	4.56815	0.304543
$226$	0.712063	0.0473657
$227$	−0.00907718	−0.000602473 0	−0.000301237	−	1.00000i	$-0.500096\pi$
$227$	−0.00907718	−0.000602473 0	−0.000301237		1.00000i	$0.500096\pi$
$228$	39.4040	2.60959
$229$	−0.476980	−0.0315197	−0.0157598	−	0.999876i	$-0.505017\pi$
$229$	−0.476980	−0.0315197	−0.0157598	+	0.999876i	$0.505017\pi$
$230$	−0.331039	−0.0218281
$231$	−1.65733	−0.109044
$232$	1.42666	0.0936648
$233$	−6.61265	−0.433209	−0.216604	−	0.976259i	$-0.569498\pi$
$233$	−6.61265	−0.433209	−0.216604	+	0.976259i	$0.569498\pi$
$234$	0.603837	0.0394741
$235$	−13.0587	−0.851859
$236$	23.8710	1.55387
$237$	26.7038	1.73460
$238$	−0.146557	−0.00949991
$239$	17.5819	1.13728	0.568640	−	0.822586i	$-0.307470\pi$
$239$	17.5819	1.13728	0.568640	+	0.822586i	$0.307470\pi$
$240$	10.9353	0.705870
$241$	−4.11938	−0.265353	−0.132676	−	0.991159i	$-0.542357\pi$
$241$	−4.11938	−0.265353	−0.132676	+	0.991159i	$0.542357\pi$
$242$	0.0645883	0.00415190
$243$	17.9942	1.15433
$244$	−17.7891	−1.13883
$245$	6.63706	0.424027
$246$	−0.105824	−0.00674708
$247$	14.6875	0.934543
$248$	2.31225	0.146828
$249$	38.8737	2.46352
$250$	−0.0645883	−0.00408492
$251$	−7.48570	−0.472493	−0.236247	−	0.971693i	$-0.575917\pi$
$251$	−7.48570	−0.472493	−0.236247	+	0.971693i	$0.575917\pi$
$252$	−5.49260	−0.346001
$253$	5.12537	0.322229
$254$	0.610808	0.0383255
$255$	−10.3618	−0.648880
$256$	15.6673	0.979207
$257$	−3.03412	−0.189263	−0.0946315	−	0.995512i	$-0.530167\pi$
$257$	−3.03412	−0.189263	−0.0946315	+	0.995512i	$0.530167\pi$
$258$	1.02737	0.0639614
$259$	0.591977	0.0367836
$260$	4.08459	0.253316
$261$	−25.2522	−1.56307
$262$	0.394208	0.0243542
$263$	−15.9629	−0.984314	−0.492157	−	0.870506i	$-0.663791\pi$
$263$	−15.9629	−0.984314	−0.492157	+	0.870506i	$0.663791\pi$
$264$	0.709996	0.0436972
$265$	1.54023	0.0946157
$266$	0.279248	0.0171218
$267$	−20.5593	−1.25821
$268$	−17.0876	−1.04379
$269$	16.4436	1.00258	0.501292	−	0.865278i	$-0.332858\pi$
$269$	16.4436	1.00258	0.501292	+	0.865278i	$0.332858\pi$
$270$	0.278635	0.0169572
$271$	23.4809	1.42637	0.713183	−	0.700978i	$-0.247253\pi$
$271$	23.4809	1.42637	0.713183	+	0.700978i	$0.247253\pi$
$272$	−14.9718	−0.907801
$273$	−3.39184	−0.205283
$274$	0.123187	0.00744202
$275$	1.00000	0.0603023
$276$	28.1412	1.69390
$277$	18.6117	1.11827	0.559136	−	0.829076i	$-0.311133\pi$
$277$	18.6117	1.11827	0.559136	+	0.829076i	$0.311133\pi$
$278$	0.435244	0.0261042
$279$	−40.9274	−2.45026
$280$	0.155480	0.00929172
$281$	26.3804	1.57373	0.786863	−	0.617128i	$-0.211704\pi$
$281$	26.3804	1.57373	0.786863	+	0.617128i	$0.211704\pi$
$282$	−2.32033	−0.138174
$283$	−15.0492	−0.894584	−0.447292	−	0.894388i	$-0.647612\pi$
$283$	−15.0492	−0.894584	−0.447292	+	0.894388i	$0.647612\pi$
$284$	−2.46041	−0.145998
$285$	19.7432	1.16948
$286$	0.132184	0.00781622
$287$	0.358797	0.0211791
$288$	3.53075	0.208051
$289$	−2.81338	−0.165493
$290$	0.357037	0.0209660
$291$	20.8928	1.22476
$292$	1.99583	0.116797
$293$	3.34774	0.195577	0.0977886	−	0.995207i	$-0.468823\pi$
$293$	3.34774	0.195577	0.0977886	+	0.995207i	$0.468823\pi$
$294$	1.17930	0.0687783
$295$	11.9604	0.696363
$296$	−0.253601	−0.0147403
$297$	−4.31401	−0.250324
$298$	0.547189	0.0316978
$299$	10.4894	0.606618
$300$	5.49058	0.316999
$301$	−3.48332	−0.200775
$302$	−0.911356	−0.0524426
$303$	−22.6124	−1.29905
$304$	28.5271	1.63614
$305$	−8.91313	−0.510365
$306$	−1.11131	−0.0635291
$307$	8.68767	0.495832	0.247916	−	0.968782i	$-0.420254\pi$
$307$	8.68767	0.495832	0.247916	+	0.968782i	$0.420254\pi$
$308$	−1.20237	−0.0685113
$309$	3.70469	0.210753
$310$	0.578666	0.0328660
$311$	−0.0984693	−0.00558368	−0.00279184	−	0.999996i	$-0.500889\pi$
$311$	−0.0984693	−0.00558368	−0.00279184	+	0.999996i	$0.500889\pi$
$312$	1.45305	0.0822629
$313$	25.0152	1.41395	0.706973	−	0.707241i	$-0.250060\pi$
$313$	25.0152	1.41395	0.706973	+	0.707241i	$0.250060\pi$
$314$	−0.0913231	−0.00515366
$315$	−2.75204	−0.155060
$316$	19.3732	1.08983
$317$	20.5462	1.15399	0.576994	−	0.816748i	$-0.304226\pi$
$317$	20.5462	1.15399	0.576994	+	0.816748i	$0.304226\pi$
$318$	0.273675	0.0153469
$319$	−5.52789	−0.309503
$320$	7.90005	0.441626
$321$	−26.7722	−1.49428
$322$	0.199432	0.0111139
$323$	−27.0310	−1.50404
$324$	3.66530	0.203628
$325$	2.04657	0.113523
$326$	−1.02902	−0.0569919
$327$	11.0712	0.612238
$328$	−0.153708	−0.00848709
$329$	7.86712	0.433728
$330$	0.177684	0.00978119
$331$	−29.6718	−1.63091	−0.815455	−	0.578820i	$-0.803513\pi$
$331$	−29.6718	−1.63091	−0.815455	+	0.578820i	$0.803513\pi$
$332$	28.2023	1.54780
$333$	4.48880	0.245985
$334$	0.954457	0.0522256
$335$	−8.56167	−0.467774
$336$	−6.58787	−0.359398
$337$	9.15667	0.498795	0.249398	−	0.968401i	$-0.419767\pi$
$337$	9.15667	0.498795	0.249398	+	0.968401i	$0.419767\pi$
$338$	−0.569125	−0.0309563
$339$	−30.3291	−1.64725
$340$	−7.51731	−0.407684
$341$	−8.95929	−0.485173
$342$	2.11746	0.114499
$343$	−8.21553	−0.443597
$344$	1.49224	0.0804564
$345$	14.1000	0.759120
$346$	0.704604	0.0378797
$347$	−13.1701	−0.707010	−0.353505	−	0.935433i	$-0.615010\pi$
$347$	−13.1701	−0.707010	−0.353505	+	0.935433i	$0.615010\pi$
$348$	−30.3513	−1.62700
$349$	4.77608	0.255658	0.127829	−	0.991796i	$-0.459199\pi$
$349$	4.77608	0.255658	0.127829	+	0.991796i	$0.459199\pi$
$350$	0.0389107	0.00207986
$351$	−8.82890	−0.471252
$352$	0.772906	0.0411960
$353$	13.9216	0.740972	0.370486	−	0.928838i	$-0.379191\pi$
$353$	13.9216	0.740972	0.370486	+	0.928838i	$0.379191\pi$
$354$	2.12518	0.112952
$355$	−1.23278	−0.0654290
$356$	−14.9155	−0.790519
$357$	6.24236	0.330381
$358$	−0.0872406	−0.00461081
$359$	−26.0651	−1.37566	−0.687832	−	0.725870i	$-0.741438\pi$
$359$	−26.0651	−1.37566	−0.687832	+	0.725870i	$0.741438\pi$
$360$	1.17896	0.0621369
$361$	32.5044	1.71076
$362$	−0.855940	−0.0449872
$363$	−2.75103	−0.144391
$364$	−2.46073	−0.128977
$365$	1.00000	0.0523424
$366$	−1.58372	−0.0827825
$367$	−11.1237	−0.580653	−0.290326	−	0.956928i	$-0.593764\pi$
$367$	−11.1237	−0.580653	−0.290326	+	0.956928i	$0.593764\pi$
$368$	20.3733	1.06203
$369$	2.72066	0.141632
$370$	−0.0634664	−0.00329946
$371$	−0.927899	−0.0481741
$372$	−49.1917	−2.55047
$373$	−8.77145	−0.454169	−0.227084	−	0.973875i	$-0.572919\pi$
$373$	−8.77145	−0.454169	−0.227084	+	0.973875i	$0.572919\pi$
$374$	−0.243273	−0.0125793
$375$	2.75103	0.142062
$376$	−3.37025	−0.173807
$377$	−11.3132	−0.582659
$378$	−0.167861	−0.00863384
$379$	20.9005	1.07358	0.536792	−	0.843714i	$-0.319636\pi$
$379$	20.9005	1.07358	0.536792	+	0.843714i	$0.319636\pi$
$380$	14.3234	0.734774
$381$	−26.0163	−1.33285
$382$	−0.126582	−0.00647649
$383$	1.86653	0.0953752	0.0476876	−	0.998862i	$-0.484815\pi$
$383$	1.86653	0.0953752	0.0476876	+	0.998862i	$0.484815\pi$
$384$	5.65628	0.288646
$385$	−0.602441	−0.0307032
$386$	0.00805021	0.000409745 0
$387$	−26.4131	−1.34265
$388$	15.1574	0.769500
$389$	−11.8499	−0.600814	−0.300407	−	0.953811i	$-0.597123\pi$
$389$	−11.8499	−0.600814	−0.300407	+	0.953811i	$0.597123\pi$
$390$	0.363642	0.0184137
$391$	−19.3048	−0.976284
$392$	1.71292	0.0865155
$393$	−16.7906	−0.846973
$394$	1.32557	0.0667811
$395$	9.70684	0.488404
$396$	−9.11723	−0.458158
$397$	11.3557	0.569924	0.284962	−	0.958539i	$-0.408019\pi$
$397$	11.3557	0.569924	0.284962	+	0.958539i	$0.408019\pi$
$398$	0.877362	0.0439782
$399$	−11.8941	−0.595450
$400$	3.97499	0.198749
$401$	−5.44811	−0.272066	−0.136033	−	0.990704i	$-0.543435\pi$
$401$	−5.44811	−0.272066	−0.136033	+	0.990704i	$0.543435\pi$
$402$	−1.52127	−0.0758743
$403$	−18.3358	−0.913370
$404$	−16.4050	−0.816178
$405$	1.83648	0.0912555
$406$	−0.215094	−0.0106749
$407$	0.982630	0.0487072
$408$	−2.67421	−0.132393
$409$	−37.3899	−1.84881	−0.924405	−	0.381412i	$-0.875438\pi$
$409$	−37.3899	−1.84881	−0.924405	+	0.381412i	$0.875438\pi$
$410$	−0.0384670	−0.00189975
$411$	−5.24695	−0.258813
$412$	2.68770	0.132413
$413$	−7.20545	−0.354557
$414$	1.51224	0.0743223
$415$	14.1306	0.693646
$416$	1.58180	0.0775542
$417$	−18.5384	−0.907831
$418$	0.463528	0.0226719
$419$	17.9676	0.877777	0.438888	−	0.898542i	$-0.355372\pi$
$419$	17.9676	0.877777	0.438888	+	0.898542i	$0.355372\pi$
$420$	−3.30775	−0.161402
$421$	18.8716	0.919745	0.459872	−	0.887985i	$-0.347895\pi$
$421$	18.8716	0.919745	0.459872	+	0.887985i	$0.347895\pi$
$422$	−1.20945	−0.0588750
$423$	59.6543	2.90049
$424$	0.397509	0.0193047
$425$	−3.76651	−0.182703
$426$	−0.219045	−0.0106128
$427$	5.36964	0.259855
$428$	−19.4229	−0.938840
$429$	−5.63016	−0.271826
$430$	0.373450	0.0180094
$431$	26.4709	1.27506	0.637529	−	0.770426i	$-0.279957\pi$
$431$	26.4709	1.27506	0.637529	+	0.770426i	$0.279957\pi$
$432$	−17.1481	−0.825040
$433$	36.3073	1.74482	0.872409	−	0.488776i	$-0.162557\pi$
$433$	36.3073	1.74482	0.872409	+	0.488776i	$0.162557\pi$
$434$	−0.348612	−0.0167339
$435$	−15.2074	−0.729138
$436$	8.03198	0.384662
$437$	36.7830	1.75957
$438$	0.177684	0.00849008
$439$	15.0335	0.717511	0.358755	−	0.933432i	$-0.383201\pi$
$439$	15.0335	0.717511	0.358755	+	0.933432i	$0.383201\pi$
$440$	0.258084	0.0123037
$441$	−30.3191	−1.44377
$442$	−0.497874	−0.0236814
$443$	−26.1154	−1.24078	−0.620389	−	0.784294i	$-0.713025\pi$
$443$	−26.1154	−1.24078	−0.620389	+	0.784294i	$0.713025\pi$
$444$	5.39521	0.256045
$445$	−7.47333	−0.354270
$446$	0.587956	0.0278405
$447$	−23.3066	−1.10236
$448$	−4.75932	−0.224857
$449$	28.5214	1.34601	0.673005	−	0.739638i	$-0.265003\pi$
$449$	28.5214	1.34601	0.673005	+	0.739638i	$0.265003\pi$
$450$	0.295049	0.0139087
$451$	0.595572	0.0280444
$452$	−22.0033	−1.03495
$453$	38.8176	1.82381
$454$	−0.000586280 0	−2.75155e−5 0
$455$	−1.23293	−0.0578009
$456$	5.09539	0.238614
$457$	−2.05979	−0.0963529	−0.0481765	−	0.998839i	$-0.515341\pi$
$457$	−2.05979	−0.0963529	−0.0481765	+	0.998839i	$0.515341\pi$
$458$	−0.0308073	−0.00143953
$459$	16.2488	0.758428
$460$	10.2294	0.476946
$461$	−22.3684	−1.04180	−0.520900	−	0.853617i	$-0.674404\pi$
$461$	−22.3684	−1.04180	−0.520900	+	0.853617i	$0.674404\pi$
$462$	−0.107044	−0.00498015
$463$	8.18029	0.380170	0.190085	−	0.981768i	$-0.439124\pi$
$463$	8.18029	0.380170	0.190085	+	0.981768i	$0.439124\pi$
$464$	−21.9733	−1.02009
$465$	−24.6472	−1.14299
$466$	−0.427100	−0.0197850
$467$	−8.12188	−0.375836	−0.187918	−	0.982185i	$-0.560174\pi$
$467$	−8.12188	−0.375836	−0.187918	+	0.982185i	$0.560174\pi$
$468$	−18.6590	−0.862513
$469$	5.15790	0.238170
$470$	−0.843443	−0.0389051
$471$	3.88975	0.179230
$472$	3.08679	0.142081
$473$	−5.78201	−0.265857
$474$	1.72475	0.0792205
$475$	7.17666	0.329288
$476$	4.52874	0.207574
$477$	−7.03600	−0.322156
$478$	1.13559	0.0519406
$479$	−19.4890	−0.890476	−0.445238	−	0.895412i	$-0.646881\pi$
$479$	−19.4890	−0.890476	−0.445238	+	0.895412i	$0.646881\pi$
$480$	2.12628	0.0970511
$481$	2.01102	0.0916945
$482$	−0.266064	−0.0121189
$483$	−8.49443	−0.386510
$484$	−1.99583	−0.0907195
$485$	7.59454	0.344850
$486$	1.16222	0.0527193
$487$	6.80796	0.308498	0.154249	−	0.988032i	$-0.450704\pi$
$487$	6.80796	0.308498	0.154249	+	0.988032i	$0.450704\pi$
$488$	−2.30034	−0.104131
$489$	43.8291	1.98202
$490$	0.428677	0.0193657
$491$	−19.8404	−0.895386	−0.447693	−	0.894187i	$-0.647754\pi$
$491$	−19.8404	−0.895386	−0.447693	+	0.894187i	$0.647754\pi$
$492$	3.27004	0.147425
$493$	20.8209	0.937725
$494$	0.948641	0.0426814
$495$	−4.56815	−0.205323
$496$	−35.6131	−1.59907
$497$	0.742675	0.0333135
$498$	2.51079	0.112511
$499$	24.6660	1.10420	0.552102	−	0.833777i	$-0.313826\pi$
$499$	24.6660	1.10420	0.552102	+	0.833777i	$0.313826\pi$
$500$	1.99583	0.0892562
$501$	−40.6534	−1.81626
$502$	−0.483489	−0.0215792
$503$	−4.17286	−0.186058	−0.0930292	−	0.995663i	$-0.529655\pi$
$503$	−4.17286	−0.186058	−0.0930292	+	0.995663i	$0.529655\pi$
$504$	−0.710257	−0.0316373
$505$	−8.21963	−0.365769
$506$	0.331039	0.0147165
$507$	24.2409	1.07657
$508$	−18.8744	−0.837417
$509$	−6.43669	−0.285301	−0.142651	−	0.989773i	$-0.545563\pi$
$509$	−6.43669	−0.285301	−0.142651	+	0.989773i	$0.545563\pi$
$510$	−0.669250	−0.0296349
$511$	−0.602441	−0.0266504
$512$	5.12405	0.226453
$513$	−30.9602	−1.36692
$514$	−0.195968	−0.00864380
$515$	1.34666	0.0593409
$516$	−31.7466	−1.39756
$517$	13.0587	0.574323
$518$	0.0382348	0.00167994
$519$	−30.0113	−1.31735
$520$	0.528186	0.0231625
$521$	0.295176	0.0129319	0.00646594	−	0.999979i	$-0.497942\pi$
$521$	0.295176	0.0129319	0.00646594	+	0.999979i	$0.497942\pi$
$522$	−1.63100	−0.0713869
$523$	13.7202	0.599941	0.299971	−	0.953948i	$-0.403023\pi$
$523$	13.7202	0.599941	0.299971	+	0.953948i	$0.403023\pi$
$524$	−12.1813	−0.532143
$525$	−1.65733	−0.0723318
$526$	−1.03102	−0.0449545
$527$	33.7453	1.46997
$528$	−10.9353	−0.475898
$529$	3.26943	0.142149
$530$	0.0994810	0.00432118
$531$	−54.6370	−2.37104
$532$	−8.62899	−0.374114
$533$	1.21888	0.0527955
$534$	−1.32789	−0.0574635
$535$	−9.73173	−0.420739
$536$	−2.20963	−0.0954415
$537$	3.71586	0.160351
$538$	1.06206	0.0457888
$539$	−6.63706	−0.285879
$540$	−8.61002	−0.370516
$541$	−23.1498	−0.995286	−0.497643	−	0.867382i	$-0.665801\pi$
$541$	−23.1498	−0.995286	−0.497643	+	0.867382i	$0.665801\pi$
$542$	1.51660	0.0651433
$543$	36.4572	1.56453
$544$	−2.91116	−0.124815
$545$	4.02439	0.172386
$546$	−0.219073	−0.00937546
$547$	38.2188	1.63412	0.817059	−	0.576554i	$-0.195603\pi$
$547$	38.2188	1.63412	0.817059	+	0.576554i	$0.195603\pi$
$548$	−3.80658	−0.162609
$549$	40.7165	1.73774
$550$	0.0645883	0.00275406
$551$	−39.6718	−1.69008
$552$	3.63899	0.154886
$553$	−5.84780	−0.248674
$554$	1.20210	0.0510724
$555$	2.70324	0.114746
$556$	−13.4494	−0.570380
$557$	17.9908	0.762295	0.381147	−	0.924514i	$-0.375529\pi$
$557$	17.9908	0.762295	0.381147	+	0.924514i	$0.375529\pi$
$558$	−2.64343	−0.111905
$559$	−11.8333	−0.500494
$560$	−2.39470	−0.101194
$561$	10.3618	0.437474
$562$	1.70387	0.0718734
$563$	14.0178	0.590778	0.295389	−	0.955377i	$-0.404551\pi$
$563$	14.0178	0.590778	0.295389	+	0.955377i	$0.404551\pi$
$564$	71.7000	3.01912
$565$	−11.0246	−0.463810
$566$	−0.972005	−0.0408564
$567$	−1.10637	−0.0464633
$568$	−0.318160	−0.0133497
$569$	−14.1591	−0.593582	−0.296791	−	0.954943i	$-0.595916\pi$
$569$	−14.1591	−0.593582	−0.296791	+	0.954943i	$0.595916\pi$
$570$	1.27518	0.0534114
$571$	−31.9707	−1.33793	−0.668966	−	0.743293i	$-0.733263\pi$
$571$	−31.9707	−1.33793	−0.668966	+	0.743293i	$0.733263\pi$
$572$	−4.08459	−0.170785
$573$	5.39153	0.225234
$574$	0.0231741	0.000967269 0
$575$	5.12537	0.213743
$576$	−36.0886	−1.50369
$577$	37.1414	1.54621	0.773107	−	0.634275i	$-0.218701\pi$
$577$	37.1414	1.54621	0.773107	+	0.634275i	$0.218701\pi$
$578$	−0.181712	−0.00755821
$579$	−0.342885	−0.0142498
$580$	−11.0327	−0.458109
$581$	−8.51287	−0.353173
$582$	1.34943	0.0559356
$583$	−1.54023	−0.0637899
$584$	0.258084	0.0106796
$585$	−9.34901	−0.386534
$586$	0.216225	0.00893218
$587$	−34.0125	−1.40385	−0.701924	−	0.712252i	$-0.747675\pi$
$587$	−34.0125	−1.40385	−0.701924	+	0.712252i	$0.747675\pi$
$588$	−36.4413	−1.50281
$589$	−64.2978	−2.64934
$590$	0.772504	0.0318035
$591$	−56.4602	−2.32246
$592$	3.90594	0.160533
$593$	23.2781	0.955917	0.477959	−	0.878382i	$-0.341377\pi$
$593$	23.2781	0.955917	0.477959	+	0.878382i	$0.341377\pi$
$594$	−0.278635	−0.0114325
$595$	2.26910	0.0930241
$596$	−16.9086	−0.692602
$597$	−37.3697	−1.52944
$598$	0.677493	0.0277048
$599$	−9.58283	−0.391544	−0.195772	−	0.980649i	$-0.562721\pi$
$599$	−9.58283	−0.391544	−0.195772	+	0.980649i	$0.562721\pi$
$600$	0.709996	0.0289854
$601$	28.8861	1.17829	0.589144	−	0.808028i	$-0.299465\pi$
$601$	28.8861	1.17829	0.589144	+	0.808028i	$0.299465\pi$
$602$	−0.224982	−0.00916957
$603$	39.1110	1.59272
$604$	28.1616	1.14588
$605$	−1.00000	−0.0406558
$606$	−1.46050	−0.0593287
$607$	25.8272	1.04829	0.524147	−	0.851628i	$-0.324384\pi$
$607$	25.8272	1.04829	0.524147	+	0.851628i	$0.324384\pi$
$608$	5.54688	0.224956
$609$	9.16155	0.371245
$610$	−0.575684	−0.0233088
$611$	26.7256	1.08120
$612$	34.3402	1.38812
$613$	−28.1652	−1.13758	−0.568792	−	0.822482i	$-0.692589\pi$
$613$	−28.1652	−1.13758	−0.568792	+	0.822482i	$0.692589\pi$
$614$	0.561122	0.0226451
$615$	1.63844	0.0660681
$616$	−0.155480	−0.00626448
$617$	44.7705	1.80239	0.901195	−	0.433413i	$-0.142691\pi$
$617$	44.7705	1.80239	0.901195	+	0.433413i	$0.142691\pi$
$618$	0.239280	0.00962525
$619$	7.01883	0.282110	0.141055	−	0.990002i	$-0.454950\pi$
$619$	7.01883	0.282110	0.141055	+	0.990002i	$0.454950\pi$
$620$	−17.8812	−0.718127
$621$	−22.1109	−0.887280
$622$	−0.00635996	−0.000255011 0
$623$	4.50224	0.180378
$624$	−22.3798	−0.895909
$625$	1.00000	0.0400000
$626$	1.61569	0.0645761
$627$	−19.7432	−0.788466
$628$	2.82195	0.112608
$629$	−3.70109	−0.147572
$630$	−0.177750	−0.00708171
$631$	29.9044	1.19047	0.595237	−	0.803550i	$-0.297058\pi$
$631$	29.9044	1.19047	0.595237	+	0.803550i	$0.297058\pi$
$632$	2.50518	0.0996507
$633$	51.5143	2.04751
$634$	1.32704	0.0527036
$635$	−9.45694	−0.375287
$636$	−8.45676	−0.335332
$637$	−13.5832	−0.538185
$638$	−0.357037	−0.0141352
$639$	5.63150	0.222779
$640$	2.05606	0.0812730
$641$	22.4454	0.886541	0.443271	−	0.896388i	$-0.353818\pi$
$641$	22.4454	0.886541	0.443271	+	0.896388i	$0.353818\pi$
$642$	−1.72917	−0.0682451
$643$	−25.9948	−1.02513	−0.512567	−	0.858647i	$-0.671305\pi$
$643$	−25.9948	−1.02513	−0.512567	+	0.858647i	$0.671305\pi$
$644$	−6.16259	−0.242840
$645$	−15.9065	−0.626316
$646$	−1.74589	−0.0686909
$647$	−22.0835	−0.868192	−0.434096	−	0.900867i	$-0.642932\pi$
$647$	−22.0835	−0.868192	−0.434096	+	0.900867i	$0.642932\pi$
$648$	0.473966	0.0186192
$649$	−11.9604	−0.469488
$650$	0.132184	0.00518469
$651$	14.8485	0.581959
$652$	31.7973	1.24528
$653$	0.870638	0.0340707	0.0170353	−	0.999855i	$-0.494577\pi$
$653$	0.870638	0.0340707	0.0170353	+	0.999855i	$0.494577\pi$
$654$	0.715070	0.0279614
$655$	−6.10339	−0.238479
$656$	2.36739	0.0924312
$657$	−4.56815	−0.178220
$658$	0.508124	0.0198087
$659$	−22.6473	−0.882213	−0.441106	−	0.897455i	$-0.645414\pi$
$659$	−22.6473	−0.882213	−0.441106	+	0.897455i	$0.645414\pi$
$660$	−5.49058	−0.213720
$661$	19.3755	0.753619	0.376809	−	0.926291i	$-0.377021\pi$
$661$	19.3755	0.753619	0.376809	+	0.926291i	$0.377021\pi$
$662$	−1.91645	−0.0744851
$663$	21.2061	0.823575
$664$	3.64689	0.141527
$665$	−4.32351	−0.167659
$666$	0.289924	0.0112343
$667$	−28.3325	−1.09704
$668$	−29.4935	−1.14114
$669$	−25.0429	−0.968216
$670$	−0.552984	−0.0213636
$671$	8.91313	0.344088
$672$	−1.28096	−0.0494141
$673$	3.52354	0.135822	0.0679112	−	0.997691i	$-0.478367\pi$
$673$	3.52354	0.135822	0.0679112	+	0.997691i	$0.478367\pi$
$674$	0.591414	0.0227804
$675$	−4.31401	−0.166046
$676$	17.5864	0.676399
$677$	9.45562	0.363409	0.181705	−	0.983353i	$-0.441839\pi$
$677$	9.45562	0.363409	0.181705	+	0.983353i	$0.441839\pi$
$678$	−1.95890	−0.0752313
$679$	−4.57526	−0.175582
$680$	−0.972076	−0.0372774
$681$	0.0249716	0.000956912 0
$682$	−0.578666	−0.0221583
$683$	−21.5703	−0.825364	−0.412682	−	0.910875i	$-0.635408\pi$
$683$	−21.5703	−0.825364	−0.412682	+	0.910875i	$0.635408\pi$
$684$	−65.4313	−2.50183
$685$	−1.90727	−0.0728730
$686$	−0.530627	−0.0202594
$687$	1.31218	0.0500629
$688$	−22.9834	−0.876235
$689$	−3.15219	−0.120089
$690$	0.910697	0.0346697
$691$	−46.1515	−1.75569	−0.877843	−	0.478949i	$-0.841018\pi$
$691$	−46.1515	−1.75569	−0.877843	+	0.478949i	$0.841018\pi$
$692$	−21.7728	−0.827677
$693$	2.75204	0.104541
$694$	−0.850637	−0.0322897
$695$	−6.73873	−0.255615
$696$	−3.92478	−0.148768
$697$	−2.24323	−0.0849684
$698$	0.308479	0.0116761
$699$	18.1916	0.688068
$700$	−1.20237	−0.0454453
$701$	0.722161	0.0272756	0.0136378	−	0.999907i	$-0.495659\pi$
$701$	0.722161	0.0272756	0.0136378	+	0.999907i	$0.495659\pi$
$702$	−0.570244	−0.0215225
$703$	7.05200	0.265971
$704$	−7.90005	−0.297744
$705$	35.9250	1.35301
$706$	0.899173	0.0338408
$707$	4.95184	0.186233
$708$	−65.6697	−2.46802
$709$	−17.5176	−0.657888	−0.328944	−	0.944349i	$-0.606693\pi$
$709$	−17.5176	−0.657888	−0.328944	+	0.944349i	$0.606693\pi$
$710$	−0.0796229	−0.00298820
$711$	−44.3423	−1.66297
$712$	−1.92874	−0.0722828
$713$	−45.9197	−1.71971
$714$	0.403183	0.0150888
$715$	−2.04657	−0.0765372
$716$	2.69580	0.100747
$717$	−48.3684	−1.80635
$718$	−1.68350	−0.0628278
$719$	−15.0131	−0.559893	−0.279947	−	0.960016i	$-0.590317\pi$
$719$	−15.0131	−0.559893	−0.279947	+	0.960016i	$0.590317\pi$
$720$	−18.1583	−0.676721
$721$	−0.811282	−0.0302137
$722$	2.09941	0.0781318
$723$	11.3325	0.421461
$724$	26.4492	0.982976
$725$	−5.52789	−0.205301
$726$	−0.177684	−0.00659448
$727$	−8.01826	−0.297381	−0.148690	−	0.988884i	$-0.547506\pi$
$727$	−8.01826	−0.297381	−0.148690	+	0.988884i	$0.547506\pi$
$728$	−0.318201	−0.0117933
$729$	−43.9932	−1.62938
$730$	0.0645883	0.00239052
$731$	21.7780	0.805489
$732$	48.9382	1.80881
$733$	−36.8837	−1.36233	−0.681166	−	0.732129i	$-0.738527\pi$
$733$	−36.8837	−1.36233	−0.681166	+	0.732129i	$0.738527\pi$
$734$	−0.718462	−0.0265189
$735$	−18.2587	−0.673484
$736$	3.96143	0.146020
$737$	8.56167	0.315373
$738$	0.175723	0.00646845
$739$	−28.9826	−1.06614	−0.533071	−	0.846071i	$-0.678962\pi$
$739$	−28.9826	−1.06614	−0.533071	+	0.846071i	$0.678962\pi$
$740$	1.96116	0.0720937
$741$	−40.4057	−1.48434
$742$	−0.0599314	−0.00220015
$743$	46.6353	1.71088	0.855442	−	0.517899i	$-0.173286\pi$
$743$	46.6353	1.71088	0.855442	+	0.517899i	$0.173286\pi$
$744$	−6.36106	−0.233208
$745$	−8.47195	−0.310388
$746$	−0.566533	−0.0207423
$747$	−64.5508	−2.36179
$748$	7.51731	0.274860
$749$	5.86279	0.214222
$750$	0.177684	0.00648811
$751$	41.5636	1.51668	0.758338	−	0.651862i	$-0.226012\pi$
$751$	41.5636	1.51668	0.758338	+	0.651862i	$0.226012\pi$
$752$	51.9083	1.89290
$753$	20.5934	0.750464
$754$	−0.730700	−0.0266105
$755$	14.1102	0.513524
$756$	5.18703	0.188650
$757$	36.2589	1.31785	0.658926	−	0.752208i	$-0.271011\pi$
$757$	36.2589	1.31785	0.658926	+	0.752208i	$0.271011\pi$
$758$	1.34993	0.0490315
$759$	−14.1000	−0.511799
$760$	1.85218	0.0671856
$761$	16.2910	0.590548	0.295274	−	0.955413i	$-0.404589\pi$
$761$	16.2910	0.590548	0.295274	+	0.955413i	$0.404589\pi$
$762$	−1.68035	−0.0608726
$763$	−2.42445	−0.0877712
$764$	3.91147	0.141512
$765$	17.2060	0.622083
$766$	0.120556	0.00435586
$767$	−24.4778	−0.883842
$768$	−43.1012	−1.55528
$769$	−24.4556	−0.881893	−0.440947	−	0.897533i	$-0.645357\pi$
$769$	−24.4556	−0.881893	−0.440947	+	0.897533i	$0.645357\pi$
$770$	−0.0389107	−0.00140224
$771$	8.34693	0.300607
$772$	−0.248758	−0.00895298
$773$	11.3727	0.409048	0.204524	−	0.978862i	$-0.434435\pi$
$773$	11.3727	0.409048	0.204524	+	0.978862i	$0.434435\pi$
$774$	−1.70598	−0.0613200
$775$	−8.95929	−0.321827
$776$	1.96003	0.0703609
$777$	−1.62854	−0.0584237
$778$	−0.765366	−0.0274397
$779$	4.27422	0.153140
$780$	−11.2368	−0.402343
$781$	1.23278	0.0441122
$782$	−1.24686	−0.0445877
$783$	23.8474	0.852236
$784$	−26.3823	−0.942223
$785$	1.41393	0.0504652
$786$	−1.08448	−0.0386820
$787$	40.9485	1.45966	0.729829	−	0.683630i	$-0.239600\pi$
$787$	40.9485	1.45966	0.729829	+	0.683630i	$0.239600\pi$
$788$	−40.9610	−1.45918
$789$	43.9143	1.56339
$790$	0.626949	0.0223058
$791$	6.64169	0.236151
$792$	−1.17896	−0.0418927
$793$	18.2413	0.647768
$794$	0.733443	0.0260289
$795$	−4.23722	−0.150279
$796$	−27.1111	−0.960929
$797$	47.4661	1.68133	0.840667	−	0.541552i	$-0.182163\pi$
$797$	47.4661	1.68133	0.840667	+	0.541552i	$0.182163\pi$
$798$	−0.768220	−0.0271947
$799$	−49.1859	−1.74007
$800$	0.772906	0.0273263
$801$	34.1392	1.20625
$802$	−0.351885	−0.0124255
$803$	−1.00000	−0.0352892
$804$	47.0085	1.65786
$805$	−3.08773	−0.108828
$806$	−1.18428	−0.0417144
$807$	−45.2368	−1.59241
$808$	−2.12136	−0.0746290
$809$	−9.79336	−0.344316	−0.172158	−	0.985069i	$-0.555074\pi$
$809$	−9.79336	−0.344316	−0.172158	+	0.985069i	$0.555074\pi$
$810$	0.118615	0.00416772
$811$	13.3496	0.468769	0.234384	−	0.972144i	$-0.424693\pi$
$811$	13.3496	0.468769	0.234384	+	0.972144i	$0.424693\pi$
$812$	6.64657	0.233249
$813$	−64.5967	−2.26551
$814$	0.0634664	0.00222450
$815$	15.9319	0.558070
$816$	41.1879	1.44187
$817$	−41.4955	−1.45174
$818$	−2.41495	−0.0844367
$819$	5.63223	0.196806
$820$	1.18866	0.0415098
$821$	36.0416	1.25786	0.628931	−	0.777461i	$-0.283493\pi$
$821$	36.0416	1.25786	0.628931	+	0.777461i	$0.283493\pi$
$822$	−0.338891	−0.0118202
$823$	36.6182	1.27643	0.638214	−	0.769859i	$-0.279673\pi$
$823$	36.6182	1.27643	0.638214	+	0.769859i	$0.279673\pi$
$824$	0.347551	0.0121075
$825$	−2.75103	−0.0957784
$826$	−0.465388	−0.0161929
$827$	49.5118	1.72170	0.860848	−	0.508863i	$-0.169934\pi$
$827$	49.5118	1.72170	0.860848	+	0.508863i	$0.169934\pi$
$828$	−46.7292	−1.62395
$829$	42.1244	1.46304	0.731520	−	0.681820i	$-0.238811\pi$
$829$	42.1244	1.46304	0.731520	+	0.681820i	$0.238811\pi$
$830$	0.912674	0.0316794
$831$	−51.2014	−1.77616
$832$	−16.1680	−0.560524
$833$	24.9986	0.866150
$834$	−1.19737	−0.0414614
$835$	−14.7775	−0.511398
$836$	−14.3234	−0.495384
$837$	38.6505	1.33596
$838$	1.16050	0.0400888
$839$	19.0399	0.657330	0.328665	−	0.944447i	$-0.393401\pi$
$839$	19.0399	0.657330	0.328665	+	0.944447i	$0.393401\pi$
$840$	−0.427730	−0.0147581
$841$	1.55760	0.0537104
$842$	1.21888	0.0420055
$843$	−72.5733	−2.49956
$844$	37.3728	1.28643
$845$	8.81157	0.303127
$846$	3.85297	0.132468
$847$	0.602441	0.0207001
$848$	−6.12240	−0.210244
$849$	41.4008	1.42087
$850$	−0.243273	−0.00834419
$851$	5.03634	0.172644
$852$	6.76865	0.231890
$853$	33.1133	1.13378	0.566890	−	0.823794i	$-0.308147\pi$
$853$	33.1133	1.13378	0.566890	+	0.823794i	$0.308147\pi$
$854$	0.346816	0.0118678
$855$	−32.7840	−1.12119
$856$	−2.51160	−0.0858448
$857$	49.1524	1.67901	0.839506	−	0.543350i	$-0.182844\pi$
$857$	49.1524	1.67901	0.839506	+	0.543350i	$0.182844\pi$
$858$	−0.363642	−0.0124145
$859$	19.5024	0.665412	0.332706	−	0.943031i	$-0.392038\pi$
$859$	19.5024	0.665412	0.332706	+	0.943031i	$0.392038\pi$
$860$	−11.5399	−0.393507
$861$	−0.987061	−0.0336389
$862$	1.70971	0.0582330
$863$	−45.8070	−1.55929	−0.779644	−	0.626223i	$-0.784600\pi$
$863$	−45.8070	−1.55929	−0.779644	+	0.626223i	$0.784600\pi$
$864$	−3.33432	−0.113436
$865$	−10.9091	−0.370922
$866$	2.34503	0.0796873
$867$	7.73969	0.262854
$868$	10.7724	0.365638
$869$	−9.70684	−0.329282
$870$	−0.982219	−0.0333004
$871$	17.5220	0.593711
$872$	1.03863	0.0351724
$873$	−34.6930	−1.17418
$874$	2.37575	0.0803611
$875$	−0.602441	−0.0203662
$876$	−5.49058	−0.185509
$877$	−46.7100	−1.57729	−0.788643	−	0.614852i	$-0.789216\pi$
$877$	−46.7100	−1.57729	−0.788643	+	0.614852i	$0.789216\pi$
$878$	0.970990	0.0327693
$879$	−9.20973	−0.310637
$880$	−3.97499	−0.133997
$881$	56.9505	1.91871	0.959355	−	0.282203i	$-0.0910651\pi$
$881$	56.9505	1.91871	0.959355	+	0.282203i	$0.0910651\pi$
$882$	−1.95826	−0.0659380
$883$	8.36139	0.281383	0.140692	−	0.990053i	$-0.455067\pi$
$883$	8.36139	0.281383	0.140692	+	0.990053i	$0.455067\pi$
$884$	15.3847	0.517443
$885$	−32.9035	−1.10604
$886$	−1.68675	−0.0566674
$887$	2.97026	0.0997316	0.0498658	−	0.998756i	$-0.484121\pi$
$887$	2.97026	0.0997316	0.0498658	+	0.998756i	$0.484121\pi$
$888$	0.697663	0.0234120
$889$	5.69725	0.191080
$890$	−0.482690	−0.0161798
$891$	−1.83648	−0.0615245
$892$	−18.1683	−0.608319
$893$	93.7181	3.13616
$894$	−1.50533	−0.0503458
$895$	1.35072	0.0451495
$896$	−1.23866	−0.0413806
$897$	−28.8566	−0.963495
$898$	1.84215	0.0614734
$899$	49.5260	1.65178
$900$	−9.11723	−0.303908
$901$	5.80130	0.193269
$902$	0.0384670	0.00128081
$903$	9.58270	0.318892
$904$	−2.84528	−0.0946327
$905$	13.2522	0.440519
$906$	2.50716	0.0832949
$907$	−40.8111	−1.35511	−0.677556	−	0.735471i	$-0.736961\pi$
$907$	−40.8111	−1.35511	−0.677556	+	0.735471i	$0.736961\pi$
$908$	0.0181165	0.000601217 0
$909$	37.5485	1.24540
$910$	−0.0796332	−0.00263981
$911$	−24.0014	−0.795202	−0.397601	−	0.917559i	$-0.630157\pi$
$911$	−24.0014	−0.795202	−0.397601	+	0.917559i	$0.630157\pi$
$912$	−78.4789	−2.59869
$913$	−14.1306	−0.467656
$914$	−0.133038	−0.00440052
$915$	24.5203	0.810615
$916$	0.951969	0.0314539
$917$	3.67693	0.121423
$918$	1.04948	0.0346380
$919$	2.44187	0.0805500	0.0402750	−	0.999189i	$-0.487177\pi$
$919$	2.44187	0.0805500	0.0402750	+	0.999189i	$0.487177\pi$
$920$	1.32278	0.0436106
$921$	−23.9000	−0.787532
$922$	−1.44474	−0.0475799
$923$	2.52296	0.0830441
$924$	3.30775	0.108817
$925$	0.982630	0.0323087
$926$	0.528351	0.0173627
$927$	−6.15173	−0.202049
$928$	−4.27254	−0.140253
$929$	26.8363	0.880470	0.440235	−	0.897883i	$-0.354895\pi$
$929$	26.8363	0.880470	0.440235	+	0.897883i	$0.354895\pi$
$930$	−1.59192	−0.0522013
$931$	−47.6319	−1.56107
$932$	13.1977	0.432305
$933$	0.270891	0.00886859
$934$	−0.524579	−0.0171647
$935$	3.76651	0.123178
$936$	−2.41283	−0.0788658
$937$	23.6630	0.773037	0.386518	−	0.922282i	$-0.373678\pi$
$937$	23.6630	0.773037	0.386518	+	0.922282i	$0.373678\pi$
$938$	0.333140	0.0108774
$939$	−68.8176	−2.24578
$940$	26.0630	0.850082
$941$	−45.7118	−1.49016	−0.745080	−	0.666975i	$-0.767589\pi$
$941$	−45.7118	−1.49016	−0.745080	+	0.666975i	$0.767589\pi$
$942$	0.251232	0.00818559
$943$	3.05253	0.0994040
$944$	−47.5426	−1.54738
$945$	2.59894	0.0845434
$946$	−0.373450	−0.0121419
$947$	0.501326	0.0162909	0.00814545	−	0.999967i	$-0.497407\pi$
$947$	0.501326	0.0162909	0.00814545	+	0.999967i	$0.497407\pi$
$948$	−53.2962	−1.73098
$949$	−2.04657	−0.0664343
$950$	0.463528	0.0150388
$951$	−56.5231	−1.83289
$952$	0.585618	0.0189800
$953$	6.05798	0.196237	0.0981186	−	0.995175i	$-0.468718\pi$
$953$	6.05798	0.196237	0.0981186	+	0.995175i	$0.468718\pi$
$954$	−0.454444	−0.0147132
$955$	1.95982	0.0634184
$956$	−35.0905	−1.13491
$957$	15.2074	0.491585
$958$	−1.25876	−0.0406688
$959$	1.14902	0.0371037
$960$	−21.7333	−0.701438
$961$	49.2689	1.58932
$962$	0.129888	0.00418776
$963$	44.4560	1.43257
$964$	8.22158	0.264799
$965$	−0.124639	−0.00401227
$966$	−0.548641	−0.0176522
$967$	−15.6458	−0.503134	−0.251567	−	0.967840i	$-0.580946\pi$
$967$	−15.6458	−0.503134	−0.251567	+	0.967840i	$0.580946\pi$
$968$	−0.258084	−0.00829513
$969$	74.3629	2.38888
$970$	0.490519	0.0157496
$971$	18.2037	0.584186	0.292093	−	0.956390i	$-0.405648\pi$
$971$	18.2037	0.584186	0.292093	+	0.956390i	$0.405648\pi$
$972$	−35.9134	−1.15192
$973$	4.05969	0.130148
$974$	0.439715	0.0140894
$975$	−5.63016	−0.180309
$976$	35.4296	1.13407
$977$	18.3890	0.588318	0.294159	−	0.955757i	$-0.404961\pi$
$977$	18.3890	0.588318	0.294159	+	0.955757i	$0.404961\pi$
$978$	2.83085	0.0905205
$979$	7.47333	0.238849
$980$	−13.2464	−0.423142
$981$	−18.3840	−0.586955
$982$	−1.28146	−0.0408930
$983$	0.422567	0.0134778	0.00673890	−	0.999977i	$-0.497855\pi$
$983$	0.422567	0.0134778	0.00673890	+	0.999977i	$0.497855\pi$
$984$	0.422854	0.0134801
$985$	−20.5233	−0.653927
$986$	1.34479	0.0428267
$987$	−21.6427	−0.688893
$988$	−29.3137	−0.932594
$989$	−29.6349	−0.942337
$990$	−0.295049	−0.00937727
$991$	4.54466	0.144366	0.0721829	−	0.997391i	$-0.477003\pi$
$991$	4.54466	0.144366	0.0721829	+	0.997391i	$0.477003\pi$
$992$	−6.92469	−0.219859
$993$	81.6279	2.59038
$994$	0.0479681	0.00152146
$995$	−13.5839	−0.430639
$996$	−77.5853	−2.45838
$997$	−46.4182	−1.47008	−0.735040	−	0.678024i	$-0.762837\pi$
$997$	−46.4182	−1.47008	−0.735040	+	0.678024i	$0.762837\pi$
$998$	1.59314	0.0504299
$999$	−4.23908	−0.134118

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4015.2.a.e.1.14	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4015.2.a.e.1.14	✓	27	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)

\(f(q)\)	\(=\)	\(q+0.0645883 q^{2} -2.75103 q^{3} -1.99583 q^{4} -1.00000 q^{5} -0.177684 q^{6} +0.602441 q^{7} -0.258084 q^{8} +4.56815 q^{9} +O(q^{10})\)	\(q+0.0645883 q^{2} -2.75103 q^{3} -1.99583 q^{4} -1.00000 q^{5} -0.177684 q^{6} +0.602441 q^{7} -0.258084 q^{8} +4.56815 q^{9} -0.0645883 q^{10} +1.00000 q^{11} +5.49058 q^{12} +2.04657 q^{13} +0.0389107 q^{14} +2.75103 q^{15} +3.97499 q^{16} -3.76651 q^{17} +0.295049 q^{18} +7.17666 q^{19} +1.99583 q^{20} -1.65733 q^{21} +0.0645883 q^{22} +5.12537 q^{23} +0.709996 q^{24} +1.00000 q^{25} +0.132184 q^{26} -4.31401 q^{27} -1.20237 q^{28} -5.52789 q^{29} +0.177684 q^{30} -8.95929 q^{31} +0.772906 q^{32} -2.75103 q^{33} -0.243273 q^{34} -0.602441 q^{35} -9.11723 q^{36} +0.982630 q^{37} +0.463528 q^{38} -5.63016 q^{39} +0.258084 q^{40} +0.595572 q^{41} -0.107044 q^{42} -5.78201 q^{43} -1.99583 q^{44} -4.56815 q^{45} +0.331039 q^{46} +13.0587 q^{47} -10.9353 q^{48} -6.63706 q^{49} +0.0645883 q^{50} +10.3618 q^{51} -4.08459 q^{52} -1.54023 q^{53} -0.278635 q^{54} -1.00000 q^{55} -0.155480 q^{56} -19.7432 q^{57} -0.357037 q^{58} -11.9604 q^{59} -5.49058 q^{60} +8.91313 q^{61} -0.578666 q^{62} +2.75204 q^{63} -7.90005 q^{64} -2.04657 q^{65} -0.177684 q^{66} +8.56167 q^{67} +7.51731 q^{68} -14.1000 q^{69} -0.0389107 q^{70} +1.23278 q^{71} -1.17896 q^{72} -1.00000 q^{73} +0.0634664 q^{74} -2.75103 q^{75} -14.3234 q^{76} +0.602441 q^{77} -0.363642 q^{78} -9.70684 q^{79} -3.97499 q^{80} -1.83648 q^{81} +0.0384670 q^{82} -14.1306 q^{83} +3.30775 q^{84} +3.76651 q^{85} -0.373450 q^{86} +15.2074 q^{87} -0.258084 q^{88} +7.47333 q^{89} -0.295049 q^{90} +1.23293 q^{91} -10.2294 q^{92} +24.6472 q^{93} +0.843443 q^{94} -7.17666 q^{95} -2.12628 q^{96} -7.59454 q^{97} -0.428677 q^{98} +4.56815 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9}+O(q^{10}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9	\( 27 q + 6 q^{2} + 5 q^{3} + 22 q^{4} - 27 q^{5} + 7 q^{6} + 10 q^{7} + 15 q^{8} + 24 q^{9} - 6 q^{10} + 27 q^{11} + 28 q^{12} + 21 q^{13} + 11 q^{14} - 5 q^{15} + 12 q^{16} + 30 q^{17} + 10 q^{18} + 20 q^{19} - 22 q^{20} - 14 q^{21} + 6 q^{22} + 16 q^{23} + 23 q^{24} + 27 q^{25} + 13 q^{26} + 17 q^{27} + 6 q^{28} - 2 q^{29} - 7 q^{30} - 6 q^{31} + 41 q^{32} + 5 q^{33} + 8 q^{34} - 10 q^{35} + 13 q^{36} + 20 q^{37} + 11 q^{38} - 10 q^{39} - 15 q^{40} + 38 q^{41} - 33 q^{42} + 29 q^{43} + 22 q^{44} - 24 q^{45} + 23 q^{46} + 17 q^{47} + 37 q^{48} + 21 q^{49} + 6 q^{50} + 13 q^{51} + 29 q^{52} + 4 q^{53} + q^{54} - 27 q^{55} + 28 q^{56} + 51 q^{57} - 6 q^{58} + 35 q^{59} - 28 q^{60} - 13 q^{61} + 28 q^{62} + 41 q^{63} - 5 q^{64} - 21 q^{65} + 7 q^{66} + 10 q^{67} + 65 q^{68} - 12 q^{69} - 11 q^{70} + 22 q^{71} - 12 q^{72} - 27 q^{73} - 5 q^{74} + 5 q^{75} + 13 q^{76} + 10 q^{77} - 9 q^{78} - 18 q^{79} - 12 q^{80} + 39 q^{81} + 20 q^{82} + 54 q^{83} - 50 q^{84} - 30 q^{85} + 3 q^{86} + 15 q^{87} + 15 q^{88} + 89 q^{89} - 10 q^{90} - 18 q^{91} + 57 q^{92} + 20 q^{93} - 29 q^{94} - 20 q^{95} + 105 q^{96} + 35 q^{97} + 10 q^{98} + 24 q^{99}+O(q^{100}) \) 27 * q + 6 * q^2 + 5 * q^3 + 22 * q^4 - 27 * q^5 + 7 * q^6 + 10 * q^7 + 15 * q^8 + 24 * q^9 - 6 * q^10 + 27 * q^11 + 28 * q^12 + 21 * q^13 + 11 * q^14 - 5 * q^15 + 12 * q^16 + 30 * q^17 + 10 * q^18 + 20 * q^19 - 22 * q^20 - 14 * q^21 + 6 * q^22 + 16 * q^23 + 23 * q^24 + 27 * q^25 + 13 * q^26 + 17 * q^27 + 6 * q^28 - 2 * q^29 - 7 * q^30 - 6 * q^31 + 41 * q^32 + 5 * q^33 + 8 * q^34 - 10 * q^35 + 13 * q^36 + 20 * q^37 + 11 * q^38 - 10 * q^39 - 15 * q^40 + 38 * q^41 - 33 * q^42 + 29 * q^43 + 22 * q^44 - 24 * q^45 + 23 * q^46 + 17 * q^47 + 37 * q^48 + 21 * q^49 + 6 * q^50 + 13 * q^51 + 29 * q^52 + 4 * q^53 + q^54 - 27 * q^55 + 28 * q^56 + 51 * q^57 - 6 * q^58 + 35 * q^59 - 28 * q^60 - 13 * q^61 + 28 * q^62 + 41 * q^63 - 5 * q^64 - 21 * q^65 + 7 * q^66 + 10 * q^67 + 65 * q^68 - 12 * q^69 - 11 * q^70 + 22 * q^71 - 12 * q^72 - 27 * q^73 - 5 * q^74 + 5 * q^75 + 13 * q^76 + 10 * q^77 - 9 * q^78 - 18 * q^79 - 12 * q^80 + 39 * q^81 + 20 * q^82 + 54 * q^83 - 50 * q^84 - 30 * q^85 + 3 * q^86 + 15 * q^87 + 15 * q^88 + 89 * q^89 - 10 * q^90 - 18 * q^91 + 57 * q^92 + 20 * q^93 - 29 * q^94 - 20 * q^95 + 105 * q^96 + 35 * q^97 + 10 * q^98 + 24 * q^99

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