LMFDB - Embedded newform 6027.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 14 x^{14} + 68 x^{13} + 64 x^{12} - 456 x^{11} - 54 x^{10} + 1532 x^{9} - 400 x^{8} + \cdots - 49 $ x^16 - 4x^15 - 14x^14 + 68x^13 + 64x^12 - 456x^11 - 54x^10 + 1532x^9 - 400x^8 - 2708x^7 + 1218x^6 + 2424x^5 - 1276x^4 - 960x^3 + 500x^2 + 112*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.72726$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72726 q^{2} +1.00000 q^{3} +5.43794 q^{4} -0.416230 q^{5} -2.72726 q^{6} -9.37616 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.72726 q^{2} +1.00000 q^{3} +5.43794 q^{4} -0.416230 q^{5} -2.72726 q^{6} -9.37616 q^{8} +1.00000 q^{9} +1.13517 q^{10} -0.0467863 q^{11} +5.43794 q^{12} +3.00540 q^{13} -0.416230 q^{15} +14.6953 q^{16} -5.85587 q^{17} -2.72726 q^{18} +6.05832 q^{19} -2.26343 q^{20} +0.127598 q^{22} +0.952936 q^{23} -9.37616 q^{24} -4.82675 q^{25} -8.19650 q^{26} +1.00000 q^{27} +2.00826 q^{29} +1.13517 q^{30} -1.80066 q^{31} -21.3257 q^{32} -0.0467863 q^{33} +15.9705 q^{34} +5.43794 q^{36} +3.23743 q^{37} -16.5226 q^{38} +3.00540 q^{39} +3.90264 q^{40} +1.00000 q^{41} -9.90222 q^{43} -0.254421 q^{44} -0.416230 q^{45} -2.59890 q^{46} -9.28958 q^{47} +14.6953 q^{48} +13.1638 q^{50} -5.85587 q^{51} +16.3432 q^{52} -2.46073 q^{53} -2.72726 q^{54} +0.0194739 q^{55} +6.05832 q^{57} -5.47703 q^{58} -6.18781 q^{59} -2.26343 q^{60} -10.1483 q^{61} +4.91086 q^{62} +28.7700 q^{64} -1.25094 q^{65} +0.127598 q^{66} -9.49723 q^{67} -31.8439 q^{68} +0.952936 q^{69} +4.70904 q^{71} -9.37616 q^{72} -5.34672 q^{73} -8.82930 q^{74} -4.82675 q^{75} +32.9448 q^{76} -8.19650 q^{78} +14.0370 q^{79} -6.11664 q^{80} +1.00000 q^{81} -2.72726 q^{82} -3.23850 q^{83} +2.43739 q^{85} +27.0059 q^{86} +2.00826 q^{87} +0.438676 q^{88} +4.51179 q^{89} +1.13517 q^{90} +5.18201 q^{92} -1.80066 q^{93} +25.3351 q^{94} -2.52165 q^{95} -21.3257 q^{96} +12.3847 q^{97} -0.0467863 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	$ 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) $ 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * 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 $

$2$	−2.72726	−1.92846	−0.964232	−	0.265061i	$-0.914608\pi$
$2$	−2.72726	−1.92846	−0.964232	+	0.265061i	$0.914608\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.43794	2.71897
$5$	−0.416230	−0.186144	−0.0930718	−	0.995659i	$-0.529669\pi$
$5$	−0.416230	−0.186144	−0.0930718	+	0.995659i	$0.529669\pi$
$6$	−2.72726	−1.11340
$7$	0	0
$7$	0	0
$8$	−9.37616	−3.31497
$9$	1.00000	0.333333
$10$	1.13517	0.358971
$11$	−0.0467863	−0.0141066	−0.00705330	−	0.999975i	$-0.502245\pi$
$11$	−0.0467863	−0.0141066	−0.00705330	+	0.999975i	$0.502245\pi$
$12$	5.43794	1.56980
$13$	3.00540	0.833548	0.416774	−	0.909010i	$-0.363161\pi$
$13$	3.00540	0.833548	0.416774	+	0.909010i	$0.363161\pi$
$14$	0	0
$15$	−0.416230	−0.107470
$16$	14.6953	3.67383
$17$	−5.85587	−1.42026	−0.710129	−	0.704071i	$-0.751363\pi$
$17$	−5.85587	−1.42026	−0.710129	+	0.704071i	$0.751363\pi$
$18$	−2.72726	−0.642821
$19$	6.05832	1.38987	0.694937	−	0.719071i	$-0.255432\pi$
$19$	6.05832	1.38987	0.694937	+	0.719071i	$0.255432\pi$
$20$	−2.26343	−0.506119
$21$	0	0
$22$	0.127598	0.0272041
$23$	0.952936	0.198701	0.0993504	−	0.995053i	$-0.468324\pi$
$23$	0.952936	0.198701	0.0993504	+	0.995053i	$0.468324\pi$
$24$	−9.37616	−1.91390
$25$	−4.82675	−0.965351
$26$	−8.19650	−1.60747
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00826	0.372924	0.186462	−	0.982462i	$-0.440298\pi$
$29$	2.00826	0.372924	0.186462	+	0.982462i	$0.440298\pi$
$30$	1.13517	0.207252
$31$	−1.80066	−0.323407	−0.161704	−	0.986839i	$-0.551699\pi$
$31$	−1.80066	−0.323407	−0.161704	+	0.986839i	$0.551699\pi$
$32$	−21.3257	−3.76988
$33$	−0.0467863	−0.00814445
$34$	15.9705	2.73892
$35$	0	0
$36$	5.43794	0.906324
$37$	3.23743	0.532230	0.266115	−	0.963941i	$-0.414260\pi$
$37$	3.23743	0.532230	0.266115	+	0.963941i	$0.414260\pi$
$38$	−16.5226	−2.68032
$39$	3.00540	0.481249
$40$	3.90264	0.617061
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−9.90222	−1.51007	−0.755037	−	0.655682i	$-0.772381\pi$
$43$	−9.90222	−1.51007	−0.755037	+	0.655682i	$0.772381\pi$
$44$	−0.254421	−0.0383555
$45$	−0.416230	−0.0620479
$46$	−2.59890	−0.383187
$47$	−9.28958	−1.35503	−0.677513	−	0.735511i	$-0.736942\pi$
$47$	−9.28958	−1.35503	−0.677513	+	0.735511i	$0.736942\pi$
$48$	14.6953	2.12109
$49$	0	0
$50$	13.1638	1.86164
$51$	−5.85587	−0.819987
$52$	16.3432	2.26639
$53$	−2.46073	−0.338007	−0.169004	−	0.985615i	$-0.554055\pi$
$53$	−2.46073	−0.338007	−0.169004	+	0.985615i	$0.554055\pi$
$54$	−2.72726	−0.371133
$55$	0.0194739	0.00262586
$56$	0	0
$57$	6.05832	0.802444
$58$	−5.47703	−0.719170
$59$	−6.18781	−0.805584	−0.402792	−	0.915292i	$-0.631960\pi$
$59$	−6.18781	−0.805584	−0.402792	+	0.915292i	$0.631960\pi$
$60$	−2.26343	−0.292208
$61$	−10.1483	−1.29935	−0.649676	−	0.760212i	$-0.725095\pi$
$61$	−10.1483	−1.29935	−0.649676	+	0.760212i	$0.725095\pi$
$62$	4.91086	0.623679
$63$	0	0
$64$	28.7700	3.59625
$65$	−1.25094	−0.155160
$66$	0.127598	0.0157063
$67$	−9.49723	−1.16027	−0.580136	−	0.814520i	$-0.697001\pi$
$67$	−9.49723	−1.16027	−0.580136	+	0.814520i	$0.697001\pi$
$68$	−31.8439	−3.86164
$69$	0.952936	0.114720
$70$	0	0
$71$	4.70904	0.558860	0.279430	−	0.960166i	$-0.409854\pi$
$71$	4.70904	0.558860	0.279430	+	0.960166i	$0.409854\pi$
$72$	−9.37616	−1.10499
$73$	−5.34672	−0.625786	−0.312893	−	0.949788i	$-0.601298\pi$
$73$	−5.34672	−0.625786	−0.312893	+	0.949788i	$0.601298\pi$
$74$	−8.82930	−1.02639
$75$	−4.82675	−0.557345
$76$	32.9448	3.77903
$77$	0	0
$78$	−8.19650	−0.928071
$79$	14.0370	1.57928	0.789641	−	0.613570i	$-0.210267\pi$
$79$	14.0370	1.57928	0.789641	+	0.613570i	$0.210267\pi$
$80$	−6.11664	−0.683861
$81$	1.00000	0.111111
$82$	−2.72726	−0.301175
$83$	−3.23850	−0.355472	−0.177736	−	0.984078i	$-0.556877\pi$
$83$	−3.23850	−0.355472	−0.177736	+	0.984078i	$0.556877\pi$
$84$	0	0
$85$	2.43739	0.264372
$86$	27.0059	2.91212
$87$	2.00826	0.215308
$88$	0.438676	0.0467630
$89$	4.51179	0.478249	0.239125	−	0.970989i	$-0.423140\pi$
$89$	4.51179	0.478249	0.239125	+	0.970989i	$0.423140\pi$
$90$	1.13517	0.119657
$91$	0	0
$92$	5.18201	0.540262
$93$	−1.80066	−0.186719
$94$	25.3351	2.61312
$95$	−2.52165	−0.258716
$96$	−21.3257	−2.17654
$97$	12.3847	1.25748	0.628739	−	0.777617i	$-0.283572\pi$
$97$	12.3847	1.25748	0.628739	+	0.777617i	$0.283572\pi$
$98$	0	0
$99$	−0.0467863	−0.00470220
$100$	−26.2476	−2.62476
$101$	5.93765	0.590819	0.295409	−	0.955371i	$-0.404544\pi$
$101$	5.93765	0.590819	0.295409	+	0.955371i	$0.404544\pi$
$102$	15.9705	1.58131
$103$	11.4718	1.13035	0.565177	−	0.824970i	$-0.308808\pi$
$103$	11.4718	1.13035	0.565177	+	0.824970i	$0.308808\pi$
$104$	−28.1791	−2.76319
$105$	0	0
$106$	6.71105	0.651834
$107$	12.0824	1.16805	0.584027	−	0.811734i	$-0.301476\pi$
$107$	12.0824	1.16805	0.584027	+	0.811734i	$0.301476\pi$
$108$	5.43794	0.523266
$109$	−9.98086	−0.955993	−0.477996	−	0.878362i	$-0.658637\pi$
$109$	−9.98086	−0.955993	−0.477996	+	0.878362i	$0.658637\pi$
$110$	−0.0531103	−0.00506387
$111$	3.23743	0.307283
$112$	0	0
$113$	−14.0482	−1.32154	−0.660770	−	0.750589i	$-0.729770\pi$
$113$	−14.0482	−1.32154	−0.660770	+	0.750589i	$0.729770\pi$
$114$	−16.5226	−1.54748
$115$	−0.396640	−0.0369869
$116$	10.9208	1.01397
$117$	3.00540	0.277849
$118$	16.8758	1.55354
$119$	0	0
$120$	3.90264	0.356261
$121$	−10.9978	−0.999801
$122$	27.6769	2.50575
$123$	1.00000	0.0901670
$124$	−9.79187	−0.879336
$125$	4.09019	0.365837
$126$	0	0
$127$	−2.80952	−0.249304	−0.124652	−	0.992201i	$-0.539782\pi$
$127$	−2.80952	−0.249304	−0.124652	+	0.992201i	$0.539782\pi$
$128$	−35.8118	−3.16535
$129$	−9.90222	−0.871842
$130$	3.41163	0.299220
$131$	−8.38214	−0.732351	−0.366176	−	0.930546i	$-0.619333\pi$
$131$	−8.38214	−0.732351	−0.366176	+	0.930546i	$0.619333\pi$
$132$	−0.254421	−0.0221445
$133$	0	0
$134$	25.9014	2.23754
$135$	−0.416230	−0.0358234
$136$	54.9056	4.70812
$137$	−22.1052	−1.88857	−0.944285	−	0.329128i	$-0.893245\pi$
$137$	−22.1052	−1.88857	−0.944285	+	0.329128i	$0.893245\pi$
$138$	−2.59890	−0.221233
$139$	−13.0683	−1.10844	−0.554219	−	0.832371i	$-0.686983\pi$
$139$	−13.0683	−1.10844	−0.554219	+	0.832371i	$0.686983\pi$
$140$	0	0
$141$	−9.28958	−0.782324
$142$	−12.8428	−1.07774
$143$	−0.140612	−0.0117585
$144$	14.6953	1.22461
$145$	−0.835896	−0.0694174
$146$	14.5819	1.20681
$147$	0	0
$148$	17.6049	1.44712
$149$	18.2459	1.49476	0.747382	−	0.664395i	$-0.231311\pi$
$149$	18.2459	1.49476	0.747382	+	0.664395i	$0.231311\pi$
$150$	13.1638	1.07482
$151$	3.74369	0.304657	0.152329	−	0.988330i	$-0.451323\pi$
$151$	3.74369	0.304657	0.152329	+	0.988330i	$0.451323\pi$
$152$	−56.8038	−4.60740
$153$	−5.85587	−0.473419
$154$	0	0
$155$	0.749487	0.0602002
$156$	16.3432	1.30850
$157$	2.17613	0.173674	0.0868370	−	0.996223i	$-0.472324\pi$
$157$	2.17613	0.173674	0.0868370	+	0.996223i	$0.472324\pi$
$158$	−38.2824	−3.04559
$159$	−2.46073	−0.195149
$160$	8.87638	0.701740
$161$	0	0
$162$	−2.72726	−0.214274
$163$	−3.84343	−0.301041	−0.150520	−	0.988607i	$-0.548095\pi$
$163$	−3.84343	−0.301041	−0.150520	+	0.988607i	$0.548095\pi$
$164$	5.43794	0.424632
$165$	0.0194739	0.00151604
$166$	8.83223	0.685514
$167$	−10.5022	−0.812688	−0.406344	−	0.913720i	$-0.633197\pi$
$167$	−10.5022	−0.812688	−0.406344	+	0.913720i	$0.633197\pi$
$168$	0	0
$169$	−3.96758	−0.305199
$170$	−6.64739	−0.509832
$171$	6.05832	0.463291
$172$	−53.8477	−4.10585
$173$	12.1600	0.924504	0.462252	−	0.886749i	$-0.347042\pi$
$173$	12.1600	0.924504	0.462252	+	0.886749i	$0.347042\pi$
$174$	−5.47703	−0.415213
$175$	0	0
$176$	−0.687541	−0.0518253
$177$	−6.18781	−0.465104
$178$	−12.3048	−0.922286
$179$	7.77291	0.580974	0.290487	−	0.956879i	$-0.406183\pi$
$179$	7.77291	0.580974	0.290487	+	0.956879i	$0.406183\pi$
$180$	−2.26343	−0.168706
$181$	11.0255	0.819515	0.409758	−	0.912194i	$-0.365613\pi$
$181$	11.0255	0.819515	0.409758	+	0.912194i	$0.365613\pi$
$182$	0	0
$183$	−10.1483	−0.750181
$184$	−8.93488	−0.658688
$185$	−1.34751	−0.0990712
$186$	4.91086	0.360081
$187$	0.273975	0.0200350
$188$	−50.5162	−3.68427
$189$	0	0
$190$	6.87720	0.498925
$191$	−7.32365	−0.529921	−0.264960	−	0.964259i	$-0.585359\pi$
$191$	−7.32365	−0.529921	−0.264960	+	0.964259i	$0.585359\pi$
$192$	28.7700	2.07629
$193$	1.51930	0.109362	0.0546809	−	0.998504i	$-0.482586\pi$
$193$	1.51930	0.109362	0.0546809	+	0.998504i	$0.482586\pi$
$194$	−33.7763	−2.42500
$195$	−1.25094	−0.0895814
$196$	0	0
$197$	−15.3181	−1.09137	−0.545685	−	0.837990i	$-0.683730\pi$
$197$	−15.3181	−1.09137	−0.545685	+	0.837990i	$0.683730\pi$
$198$	0.127598	0.00906803
$199$	−20.2178	−1.43320	−0.716599	−	0.697485i	$-0.754302\pi$
$199$	−20.2178	−1.43320	−0.716599	+	0.697485i	$0.754302\pi$
$200$	45.2564	3.20011
$201$	−9.49723	−0.669883
$202$	−16.1935	−1.13937
$203$	0	0
$204$	−31.8439	−2.22952
$205$	−0.416230	−0.0290708
$206$	−31.2867	−2.17985
$207$	0.952936	0.0662336
$208$	44.1653	3.06232
$209$	−0.283447	−0.0196064
$210$	0	0
$211$	−19.1529	−1.31854	−0.659271	−	0.751906i	$-0.729135\pi$
$211$	−19.1529	−1.31854	−0.659271	+	0.751906i	$0.729135\pi$
$212$	−13.3813	−0.919032
$213$	4.70904	0.322658
$214$	−32.9520	−2.25255
$215$	4.12160	0.281091
$216$	−9.37616	−0.637967
$217$	0	0
$218$	27.2204	1.84360
$219$	−5.34672	−0.361298
$220$	0.105898	0.00713963
$221$	−17.5992	−1.18385
$222$	−8.82930	−0.592584
$223$	−10.8935	−0.729481	−0.364740	−	0.931109i	$-0.618842\pi$
$223$	−10.8935	−0.729481	−0.364740	+	0.931109i	$0.618842\pi$
$224$	0	0
$225$	−4.82675	−0.321784
$226$	38.3130	2.54854
$227$	1.93678	0.128549	0.0642744	−	0.997932i	$-0.479527\pi$
$227$	1.93678	0.128549	0.0642744	+	0.997932i	$0.479527\pi$
$228$	32.9448	2.18182
$229$	−23.9066	−1.57980	−0.789898	−	0.613238i	$-0.789866\pi$
$229$	−23.9066	−1.57980	−0.789898	+	0.613238i	$0.789866\pi$
$230$	1.08174	0.0713279
$231$	0	0
$232$	−18.8297	−1.23623
$233$	−13.0814	−0.856989	−0.428495	−	0.903544i	$-0.640956\pi$
$233$	−13.0814	−0.856989	−0.428495	+	0.903544i	$0.640956\pi$
$234$	−8.19650	−0.535822
$235$	3.86660	0.252229
$236$	−33.6490	−2.19036
$237$	14.0370	0.911798
$238$	0	0
$239$	−17.5892	−1.13775	−0.568876	−	0.822423i	$-0.692622\pi$
$239$	−17.5892	−1.13775	−0.568876	+	0.822423i	$0.692622\pi$
$240$	−6.11664	−0.394827
$241$	24.9349	1.60620	0.803099	−	0.595846i	$-0.203183\pi$
$241$	24.9349	1.60620	0.803099	+	0.595846i	$0.203183\pi$
$242$	29.9939	1.92808
$243$	1.00000	0.0641500
$244$	−55.1856	−3.53290
$245$	0	0
$246$	−2.72726	−0.173884
$247$	18.2077	1.15853
$248$	16.8832	1.07209
$249$	−3.23850	−0.205232
$250$	−11.1550	−0.705504
$251$	16.0506	1.01310	0.506552	−	0.862209i	$-0.330920\pi$
$251$	16.0506	1.01310	0.506552	+	0.862209i	$0.330920\pi$
$252$	0	0
$253$	−0.0445844	−0.00280300
$254$	7.66228	0.480774
$255$	2.43739	0.152635
$256$	40.1282	2.50801
$257$	3.53643	0.220596	0.110298	−	0.993899i	$-0.464819\pi$
$257$	3.53643	0.220596	0.110298	+	0.993899i	$0.464819\pi$
$258$	27.0059	1.68131
$259$	0	0
$260$	−6.80252	−0.421874
$261$	2.00826	0.124308
$262$	22.8603	1.41231
$263$	21.2018	1.30736	0.653681	−	0.756770i	$-0.273224\pi$
$263$	21.2018	1.30736	0.653681	+	0.756770i	$0.273224\pi$
$264$	0.438676	0.0269987
$265$	1.02423	0.0629179
$266$	0	0
$267$	4.51179	0.276117
$268$	−51.6454	−3.15475
$269$	6.80296	0.414784	0.207392	−	0.978258i	$-0.433503\pi$
$269$	6.80296	0.414784	0.207392	+	0.978258i	$0.433503\pi$
$270$	1.13517	0.0690840
$271$	−0.369791	−0.0224632	−0.0112316	−	0.999937i	$-0.503575\pi$
$271$	−0.369791	−0.0224632	−0.0112316	+	0.999937i	$0.503575\pi$
$272$	−86.0541	−5.21779
$273$	0	0
$274$	60.2865	3.64204
$275$	0.225826	0.0136178
$276$	5.18201	0.311920
$277$	20.8105	1.25038	0.625191	−	0.780472i	$-0.285021\pi$
$277$	20.8105	1.25038	0.625191	+	0.780472i	$0.285021\pi$
$278$	35.6406	2.13758
$279$	−1.80066	−0.107802
$280$	0	0
$281$	28.3373	1.69046	0.845232	−	0.534400i	$-0.179462\pi$
$281$	28.3373	1.69046	0.845232	+	0.534400i	$0.179462\pi$
$282$	25.3351	1.50868
$283$	−20.3351	−1.20879	−0.604397	−	0.796683i	$-0.706586\pi$
$283$	−20.3351	−1.20879	−0.604397	+	0.796683i	$0.706586\pi$
$284$	25.6075	1.51952
$285$	−2.52165	−0.149370
$286$	0.383484	0.0226759
$287$	0	0
$288$	−21.3257	−1.25663
$289$	17.2913	1.01713
$290$	2.27970	0.133869
$291$	12.3847	0.726005
$292$	−29.0751	−1.70149
$293$	6.54345	0.382272	0.191136	−	0.981564i	$-0.438783\pi$
$293$	6.54345	0.382272	0.191136	+	0.981564i	$0.438783\pi$
$294$	0	0
$295$	2.57555	0.149954
$296$	−30.3546	−1.76433
$297$	−0.0467863	−0.00271482
$298$	−49.7613	−2.88260
$299$	2.86395	0.165627
$300$	−26.2476	−1.51541
$301$	0	0
$302$	−10.2100	−0.587520
$303$	5.93765	0.341109
$304$	89.0291	5.10617
$305$	4.22401	0.241866
$306$	15.9705	0.912972
$307$	27.3197	1.55922	0.779608	−	0.626268i	$-0.215419\pi$
$307$	27.3197	1.55922	0.779608	+	0.626268i	$0.215419\pi$
$308$	0	0
$309$	11.4718	0.652610
$310$	−2.04404	−0.116094
$311$	−9.86057	−0.559142	−0.279571	−	0.960125i	$-0.590192\pi$
$311$	−9.86057	−0.559142	−0.279571	+	0.960125i	$0.590192\pi$
$312$	−28.1791	−1.59533
$313$	−10.1230	−0.572187	−0.286094	−	0.958202i	$-0.592357\pi$
$313$	−10.1230	−0.572187	−0.286094	+	0.958202i	$0.592357\pi$
$314$	−5.93487	−0.334924
$315$	0	0
$316$	76.3322	4.29402
$317$	−25.6530	−1.44082	−0.720409	−	0.693549i	$-0.756046\pi$
$317$	−25.6530	−1.44082	−0.720409	+	0.693549i	$0.756046\pi$
$318$	6.71105	0.376337
$319$	−0.0939589	−0.00526069
$320$	−11.9749	−0.669418
$321$	12.0824	0.674376
$322$	0	0
$323$	−35.4768	−1.97398
$324$	5.43794	0.302108
$325$	−14.5063	−0.804666
$326$	10.4820	0.580546
$327$	−9.98086	−0.551943
$328$	−9.37616	−0.517712
$329$	0	0
$330$	−0.0531103	−0.00292362
$331$	−1.38318	−0.0760264	−0.0380132	−	0.999277i	$-0.512103\pi$
$331$	−1.38318	−0.0760264	−0.0380132	+	0.999277i	$0.512103\pi$
$332$	−17.6108	−0.966517
$333$	3.23743	0.177410
$334$	28.6423	1.56724
$335$	3.95303	0.215977
$336$	0	0
$337$	15.4787	0.843177	0.421588	−	0.906787i	$-0.361473\pi$
$337$	15.4787	0.843177	0.421588	+	0.906787i	$0.361473\pi$
$338$	10.8206	0.588564
$339$	−14.0482	−0.762991
$340$	13.2544	0.718820
$341$	0.0842461	0.00456218
$342$	−16.5226	−0.893440
$343$	0	0
$344$	92.8448	5.00586
$345$	−0.396640	−0.0213544
$346$	−33.1633	−1.78287
$347$	−10.2007	−0.547604	−0.273802	−	0.961786i	$-0.588281\pi$
$347$	−10.2007	−0.547604	−0.273802	+	0.961786i	$0.588281\pi$
$348$	10.9208	0.585415
$349$	−19.2750	−1.03177	−0.515883	−	0.856659i	$-0.672536\pi$
$349$	−19.2750	−1.03177	−0.515883	+	0.856659i	$0.672536\pi$
$350$	0	0
$351$	3.00540	0.160416
$352$	0.997750	0.0531803
$353$	−36.6606	−1.95125	−0.975624	−	0.219450i	$-0.929574\pi$
$353$	−36.6606	−1.95125	−0.975624	+	0.219450i	$0.929574\pi$
$354$	16.8758	0.896937
$355$	−1.96004	−0.104028
$356$	24.5349	1.30035
$357$	0	0
$358$	−21.1987	−1.12039
$359$	30.4236	1.60570	0.802849	−	0.596182i	$-0.203316\pi$
$359$	30.4236	1.60570	0.802849	+	0.596182i	$0.203316\pi$
$360$	3.90264	0.205687
$361$	17.7032	0.931750
$362$	−30.0693	−1.58041
$363$	−10.9978	−0.577235
$364$	0	0
$365$	2.22546	0.116486
$366$	27.6769	1.44670
$367$	10.5725	0.551882	0.275941	−	0.961175i	$-0.411011\pi$
$367$	10.5725	0.551882	0.275941	+	0.961175i	$0.411011\pi$
$368$	14.0037	0.729994
$369$	1.00000	0.0520579
$370$	3.67502	0.191055
$371$	0	0
$372$	−9.79187	−0.507685
$373$	12.4721	0.645783	0.322892	−	0.946436i	$-0.395345\pi$
$373$	12.4721	0.645783	0.322892	+	0.946436i	$0.395345\pi$
$374$	−0.747200	−0.0386368
$375$	4.09019	0.211216
$376$	87.1006	4.49187
$377$	6.03561	0.310850
$378$	0	0
$379$	−5.18412	−0.266290	−0.133145	−	0.991097i	$-0.542508\pi$
$379$	−5.18412	−0.266290	−0.133145	+	0.991097i	$0.542508\pi$
$380$	−13.7126	−0.703442
$381$	−2.80952	−0.143936
$382$	19.9735	1.02193
$383$	−25.0431	−1.27964	−0.639821	−	0.768524i	$-0.720992\pi$
$383$	−25.0431	−1.27964	−0.639821	+	0.768524i	$0.720992\pi$
$384$	−35.8118	−1.82751
$385$	0	0
$386$	−4.14353	−0.210900
$387$	−9.90222	−0.503358
$388$	67.3474	3.41904
$389$	26.7641	1.35699	0.678496	−	0.734604i	$-0.262632\pi$
$389$	26.7641	1.35699	0.678496	+	0.734604i	$0.262632\pi$
$390$	3.41163	0.172754
$391$	−5.58027	−0.282207
$392$	0	0
$393$	−8.38214	−0.422823
$394$	41.7764	2.10467
$395$	−5.84260	−0.293973
$396$	−0.254421	−0.0127852
$397$	24.8660	1.24799	0.623993	−	0.781430i	$-0.285509\pi$
$397$	24.8660	1.24799	0.623993	+	0.781430i	$0.285509\pi$
$398$	55.1391	2.76387
$399$	0	0
$400$	−70.9308	−3.54654
$401$	−35.9364	−1.79458	−0.897288	−	0.441446i	$-0.854466\pi$
$401$	−35.9364	−1.79458	−0.897288	+	0.441446i	$0.854466\pi$
$402$	25.9014	1.29185
$403$	−5.41169	−0.269575
$404$	32.2886	1.60642
$405$	−0.416230	−0.0206826
$406$	0	0
$407$	−0.151467	−0.00750796
$408$	54.9056	2.71823
$409$	−17.2575	−0.853328	−0.426664	−	0.904410i	$-0.640311\pi$
$409$	−17.2575	−0.853328	−0.426664	+	0.904410i	$0.640311\pi$
$410$	1.13517	0.0560619
$411$	−22.1052	−1.09037
$412$	62.3832	3.07340
$413$	0	0
$414$	−2.59890	−0.127729
$415$	1.34796	0.0661688
$416$	−64.0921	−3.14238
$417$	−13.0683	−0.639957
$418$	0.773032	0.0378102
$419$	−5.11135	−0.249706	−0.124853	−	0.992175i	$-0.539846\pi$
$419$	−5.11135	−0.249706	−0.124853	+	0.992175i	$0.539846\pi$
$420$	0	0
$421$	−19.6763	−0.958965	−0.479482	−	0.877551i	$-0.659176\pi$
$421$	−19.6763	−0.958965	−0.479482	+	0.877551i	$0.659176\pi$
$422$	52.2350	2.54276
$423$	−9.28958	−0.451675
$424$	23.0722	1.12048
$425$	28.2649	1.37105
$426$	−12.8428	−0.622234
$427$	0	0
$428$	65.7036	3.17591
$429$	−0.140612	−0.00678879
$430$	−11.2407	−0.542073
$431$	−41.3547	−1.99198	−0.995992	−	0.0894383i	$-0.971493\pi$
$431$	−41.3547	−1.99198	−0.995992	+	0.0894383i	$0.971493\pi$
$432$	14.6953	0.707030
$433$	27.9849	1.34487	0.672435	−	0.740156i	$-0.265249\pi$
$433$	27.9849	1.34487	0.672435	+	0.740156i	$0.265249\pi$
$434$	0	0
$435$	−0.835896	−0.0400781
$436$	−54.2753	−2.59932
$437$	5.77319	0.276169
$438$	14.5819	0.696749
$439$	15.6955	0.749104	0.374552	−	0.927206i	$-0.377797\pi$
$439$	15.6955	0.749104	0.374552	+	0.927206i	$0.377797\pi$
$440$	−0.182590	−0.00870464
$441$	0	0
$442$	47.9977	2.28302
$443$	−35.8815	−1.70478	−0.852390	−	0.522906i	$-0.824848\pi$
$443$	−35.8815	−1.70478	−0.852390	+	0.522906i	$0.824848\pi$
$444$	17.6049	0.835494
$445$	−1.87794	−0.0890230
$446$	29.7093	1.40678
$447$	18.2459	0.863002
$448$	0	0
$449$	12.8859	0.608123	0.304061	−	0.952652i	$-0.401657\pi$
$449$	12.8859	0.608123	0.304061	+	0.952652i	$0.401657\pi$
$450$	13.1638	0.620548
$451$	−0.0467863	−0.00220308
$452$	−76.3931	−3.59323
$453$	3.74369	0.175894
$454$	−5.28211	−0.247902
$455$	0	0
$456$	−56.8038	−2.66008
$457$	−33.7488	−1.57870	−0.789351	−	0.613942i	$-0.789583\pi$
$457$	−33.7488	−1.57870	−0.789351	+	0.613942i	$0.789583\pi$
$458$	65.1996	3.04658
$459$	−5.85587	−0.273329
$460$	−2.15691	−0.100566
$461$	14.3109	0.666523	0.333262	−	0.942834i	$-0.391851\pi$
$461$	14.3109	0.666523	0.333262	+	0.942834i	$0.391851\pi$
$462$	0	0
$463$	−26.1933	−1.21731	−0.608654	−	0.793436i	$-0.708290\pi$
$463$	−26.1933	−1.21731	−0.608654	+	0.793436i	$0.708290\pi$
$464$	29.5120	1.37006
$465$	0.749487	0.0347566
$466$	35.6763	1.65267
$467$	22.3888	1.03603	0.518014	−	0.855372i	$-0.326671\pi$
$467$	22.3888	1.03603	0.518014	+	0.855372i	$0.326671\pi$
$468$	16.3432	0.755464
$469$	0	0
$470$	−10.5452	−0.486415
$471$	2.17613	0.100271
$472$	58.0179	2.67049
$473$	0.463288	0.0213020
$474$	−38.2824	−1.75837
$475$	−29.2420	−1.34172
$476$	0	0
$477$	−2.46073	−0.112669
$478$	47.9703	2.19411
$479$	−12.0786	−0.551887	−0.275943	−	0.961174i	$-0.588990\pi$
$479$	−12.0786	−0.551887	−0.275943	+	0.961174i	$0.588990\pi$
$480$	8.87638	0.405150
$481$	9.72976	0.443639
$482$	−68.0039	−3.09749
$483$	0	0
$484$	−59.8055	−2.71843
$485$	−5.15489	−0.234071
$486$	−2.72726	−0.123711
$487$	24.1912	1.09621	0.548104	−	0.836410i	$-0.315350\pi$
$487$	24.1912	1.09621	0.548104	+	0.836410i	$0.315350\pi$
$488$	95.1517	4.30731
$489$	−3.84343	−0.173806
$490$	0	0
$491$	6.30741	0.284650	0.142325	−	0.989820i	$-0.454542\pi$
$491$	6.30741	0.284650	0.142325	+	0.989820i	$0.454542\pi$
$492$	5.43794	0.245161
$493$	−11.7601	−0.529648
$494$	−49.6570	−2.23418
$495$	0.0194739	0.000875285 0
$496$	−26.4613	−1.18815
$497$	0	0
$498$	8.83223	0.395782
$499$	−37.1859	−1.66467	−0.832335	−	0.554273i	$-0.812996\pi$
$499$	−37.1859	−1.66467	−0.832335	+	0.554273i	$0.812996\pi$
$500$	22.2422	0.994702
$501$	−10.5022	−0.469206
$502$	−43.7741	−1.95373
$503$	6.02057	0.268444	0.134222	−	0.990951i	$-0.457147\pi$
$503$	6.02057	0.268444	0.134222	+	0.990951i	$0.457147\pi$
$504$	0	0
$505$	−2.47143	−0.109977
$506$	0.121593	0.00540547
$507$	−3.96758	−0.176206
$508$	−15.2780	−0.677851
$509$	19.2491	0.853200	0.426600	−	0.904440i	$-0.359711\pi$
$509$	19.2491	0.853200	0.426600	+	0.904440i	$0.359711\pi$
$510$	−6.64739	−0.294352
$511$	0	0
$512$	−37.8163	−1.67126
$513$	6.05832	0.267481
$514$	−9.64476	−0.425412
$515$	−4.77492	−0.210408
$516$	−53.8477	−2.37051
$517$	0.434625	0.0191148
$518$	0	0
$519$	12.1600	0.533763
$520$	11.7290	0.514350
$521$	−27.9157	−1.22301	−0.611505	−	0.791241i	$-0.709436\pi$
$521$	−27.9157	−1.22301	−0.611505	+	0.791241i	$0.709436\pi$
$522$	−5.47703	−0.239723
$523$	14.1196	0.617408	0.308704	−	0.951158i	$-0.400105\pi$
$523$	14.1196	0.617408	0.308704	+	0.951158i	$0.400105\pi$
$524$	−45.5816	−1.99124
$525$	0	0
$526$	−57.8229	−2.52120
$527$	10.5444	0.459322
$528$	−0.687541	−0.0299214
$529$	−22.0919	−0.960518
$530$	−2.79334	−0.121335
$531$	−6.18781	−0.268528
$532$	0	0
$533$	3.00540	0.130178
$534$	−12.3048	−0.532482
$535$	−5.02907	−0.217426
$536$	89.0476	3.84627
$537$	7.77291	0.335426
$538$	−18.5534	−0.799895
$539$	0	0
$540$	−2.26343	−0.0974027
$541$	32.0222	1.37674	0.688370	−	0.725360i	$-0.258327\pi$
$541$	32.0222	1.37674	0.688370	+	0.725360i	$0.258327\pi$
$542$	1.00852	0.0433195
$543$	11.0255	0.473147
$544$	124.880	5.35421
$545$	4.15433	0.177952
$546$	0	0
$547$	−16.2757	−0.695897	−0.347949	−	0.937514i	$-0.613122\pi$
$547$	−16.2757	−0.695897	−0.347949	+	0.937514i	$0.613122\pi$
$548$	−120.207	−5.13497
$549$	−10.1483	−0.433117
$550$	−0.615886	−0.0262615
$551$	12.1667	0.518317
$552$	−8.93488	−0.380294
$553$	0	0
$554$	−56.7557	−2.41132
$555$	−1.34751	−0.0571988
$556$	−71.0646	−3.01381
$557$	−13.4419	−0.569552	−0.284776	−	0.958594i	$-0.591919\pi$
$557$	−13.4419	−0.569552	−0.284776	+	0.958594i	$0.591919\pi$
$558$	4.91086	0.207893
$559$	−29.7601	−1.25872
$560$	0	0
$561$	0.273975	0.0115672
$562$	−77.2832	−3.26000
$563$	−16.0633	−0.676988	−0.338494	−	0.940969i	$-0.609917\pi$
$563$	−16.0633	−0.676988	−0.338494	+	0.940969i	$0.609917\pi$
$564$	−50.5162	−2.12712
$565$	5.84726	0.245996
$566$	55.4590	2.33112
$567$	0	0
$568$	−44.1527	−1.85261
$569$	14.8609	0.623001	0.311501	−	0.950246i	$-0.399168\pi$
$569$	14.8609	0.623001	0.311501	+	0.950246i	$0.399168\pi$
$570$	6.87720	0.288054
$571$	10.1204	0.423526	0.211763	−	0.977321i	$-0.432079\pi$
$571$	10.1204	0.423526	0.211763	+	0.977321i	$0.432079\pi$
$572$	−0.764638	−0.0319711
$573$	−7.32365	−0.305950
$574$	0	0
$575$	−4.59959	−0.191816
$576$	28.7700	1.19875
$577$	6.51473	0.271212	0.135606	−	0.990763i	$-0.456702\pi$
$577$	6.51473	0.271212	0.135606	+	0.990763i	$0.456702\pi$
$578$	−47.1578	−1.96151
$579$	1.51930	0.0631401
$580$	−4.54555	−0.188744
$581$	0	0
$582$	−33.7763	−1.40007
$583$	0.115128	0.00476813
$584$	50.1317	2.07446
$585$	−1.25094	−0.0517199
$586$	−17.8457	−0.737198
$587$	43.9501	1.81402	0.907008	−	0.421113i	$-0.138360\pi$
$587$	43.9501	1.81402	0.907008	+	0.421113i	$0.138360\pi$
$588$	0	0
$589$	−10.9090	−0.449496
$590$	−7.02419	−0.289182
$591$	−15.3181	−0.630103
$592$	47.5751	1.95532
$593$	18.3336	0.752873	0.376436	−	0.926443i	$-0.377149\pi$
$593$	18.3336	0.752873	0.376436	+	0.926443i	$0.377149\pi$
$594$	0.127598	0.00523543
$595$	0	0
$596$	99.2202	4.06422
$597$	−20.2178	−0.827458
$598$	−7.81074	−0.319405
$599$	−33.2968	−1.36047	−0.680234	−	0.732995i	$-0.738122\pi$
$599$	−33.2968	−1.36047	−0.680234	+	0.732995i	$0.738122\pi$
$600$	45.2564	1.84759
$601$	−10.6328	−0.433720	−0.216860	−	0.976203i	$-0.569582\pi$
$601$	−10.6328	−0.433720	−0.216860	+	0.976203i	$0.569582\pi$
$602$	0	0
$603$	−9.49723	−0.386757
$604$	20.3580	0.828354
$605$	4.57762	0.186107
$606$	−16.1935	−0.657817
$607$	−40.0459	−1.62541	−0.812706	−	0.582674i	$-0.802007\pi$
$607$	−40.0459	−1.62541	−0.812706	+	0.582674i	$0.802007\pi$
$608$	−129.198	−5.23966
$609$	0	0
$610$	−11.5200	−0.466430
$611$	−27.9189	−1.12948
$612$	−31.8439	−1.28721
$613$	−29.4237	−1.18841	−0.594207	−	0.804312i	$-0.702534\pi$
$613$	−29.4237	−1.18841	−0.594207	+	0.804312i	$0.702534\pi$
$614$	−74.5078	−3.00689
$615$	−0.416230	−0.0167840
$616$	0	0
$617$	17.7795	0.715775	0.357888	−	0.933765i	$-0.383497\pi$
$617$	17.7795	0.715775	0.357888	+	0.933765i	$0.383497\pi$
$618$	−31.2867	−1.25853
$619$	42.7487	1.71822	0.859109	−	0.511793i	$-0.171019\pi$
$619$	42.7487	1.71822	0.859109	+	0.511793i	$0.171019\pi$
$620$	4.07567	0.163683
$621$	0.952936	0.0382400
$622$	26.8923	1.07828
$623$	0	0
$624$	44.1653	1.76803
$625$	22.4313	0.897252
$626$	27.6081	1.10344
$627$	−0.283447	−0.0113198
$628$	11.8337	0.472215
$629$	−18.9580	−0.755904
$630$	0	0
$631$	−12.1892	−0.485245	−0.242622	−	0.970121i	$-0.578008\pi$
$631$	−12.1892	−0.485245	−0.242622	+	0.970121i	$0.578008\pi$
$632$	−131.613	−5.23528
$633$	−19.1529	−0.761260
$634$	69.9625	2.77857
$635$	1.16940	0.0464064
$636$	−13.3813	−0.530603
$637$	0	0
$638$	0.256250	0.0101450
$639$	4.70904	0.186287
$640$	14.9059	0.589209
$641$	−30.9625	−1.22295	−0.611473	−	0.791265i	$-0.709423\pi$
$641$	−30.9625	−1.22295	−0.611473	+	0.791265i	$0.709423\pi$
$642$	−32.9520	−1.30051
$643$	31.7205	1.25094	0.625468	−	0.780250i	$-0.284908\pi$
$643$	31.7205	1.25094	0.625468	+	0.780250i	$0.284908\pi$
$644$	0	0
$645$	4.12160	0.162288
$646$	96.7543	3.80675
$647$	−4.13242	−0.162462	−0.0812311	−	0.996695i	$-0.525885\pi$
$647$	−4.13242	−0.162462	−0.0812311	+	0.996695i	$0.525885\pi$
$648$	−9.37616	−0.368330
$649$	0.289505	0.0113641
$650$	39.5625	1.55177
$651$	0	0
$652$	−20.9004	−0.818521
$653$	−21.6772	−0.848293	−0.424147	−	0.905593i	$-0.639426\pi$
$653$	−21.6772	−0.848293	−0.424147	+	0.905593i	$0.639426\pi$
$654$	27.2204	1.06440
$655$	3.48890	0.136323
$656$	14.6953	0.573757
$657$	−5.34672	−0.208595
$658$	0	0
$659$	38.7978	1.51135	0.755673	−	0.654949i	$-0.227310\pi$
$659$	38.7978	1.51135	0.755673	+	0.654949i	$0.227310\pi$
$660$	0.105898	0.00412206
$661$	−20.2968	−0.789455	−0.394728	−	0.918798i	$-0.629161\pi$
$661$	−20.2968	−0.789455	−0.394728	+	0.918798i	$0.629161\pi$
$662$	3.77229	0.146614
$663$	−17.5992	−0.683498
$664$	30.3647	1.17838
$665$	0	0
$666$	−8.82930	−0.342129
$667$	1.91374	0.0741003
$668$	−57.1106	−2.20968
$669$	−10.8935	−0.421166
$670$	−10.7809	−0.416504
$671$	0.474800	0.0183294
$672$	0	0
$673$	−38.4819	−1.48337	−0.741684	−	0.670750i	$-0.765972\pi$
$673$	−38.4819	−1.48337	−0.741684	+	0.670750i	$0.765972\pi$
$674$	−42.2143	−1.62604
$675$	−4.82675	−0.185782
$676$	−21.5755	−0.829826
$677$	34.3408	1.31982	0.659911	−	0.751343i	$-0.270594\pi$
$677$	34.3408	1.31982	0.659911	+	0.751343i	$0.270594\pi$
$678$	38.3130	1.47140
$679$	0	0
$680$	−22.8534	−0.876386
$681$	1.93678	0.0742176
$682$	−0.229761	−0.00879800
$683$	−46.7531	−1.78896	−0.894479	−	0.447109i	$-0.852454\pi$
$683$	−46.7531	−1.78896	−0.894479	+	0.447109i	$0.852454\pi$
$684$	32.9448	1.25968
$685$	9.20082	0.351545
$686$	0	0
$687$	−23.9066	−0.912096
$688$	−145.516	−5.54776
$689$	−7.39547	−0.281745
$690$	1.08174	0.0411812
$691$	0.631161	0.0240105	0.0120052	−	0.999928i	$-0.496179\pi$
$691$	0.631161	0.0240105	0.0120052	+	0.999928i	$0.496179\pi$
$692$	66.1251	2.51370
$693$	0	0
$694$	27.8200	1.05603
$695$	5.43941	0.206329
$696$	−18.8297	−0.713739
$697$	−5.85587	−0.221807
$698$	52.5678	1.98972
$699$	−13.0814	−0.494783
$700$	0	0
$701$	−18.3514	−0.693122	−0.346561	−	0.938027i	$-0.612651\pi$
$701$	−18.3514	−0.693122	−0.346561	+	0.938027i	$0.612651\pi$
$702$	−8.19650	−0.309357
$703$	19.6134	0.739732
$704$	−1.34604	−0.0507308
$705$	3.86660	0.145625
$706$	99.9830	3.76291
$707$	0	0
$708$	−33.6490	−1.26461
$709$	−50.9531	−1.91358	−0.956791	−	0.290776i	$-0.906086\pi$
$709$	−50.9531	−1.91358	−0.956791	+	0.290776i	$0.906086\pi$
$710$	5.34554	0.200615
$711$	14.0370	0.526427
$712$	−42.3033	−1.58538
$713$	−1.71591	−0.0642613
$714$	0	0
$715$	0.0585267	0.00218878
$716$	42.2686	1.57965
$717$	−17.5892	−0.656881
$718$	−82.9731	−3.09653
$719$	−31.7735	−1.18495	−0.592475	−	0.805588i	$-0.701849\pi$
$719$	−31.7735	−1.18495	−0.592475	+	0.805588i	$0.701849\pi$
$720$	−6.11664	−0.227954
$721$	0	0
$722$	−48.2813	−1.79685
$723$	24.9349	0.927339
$724$	59.9558	2.22824
$725$	−9.69335	−0.360002
$726$	29.9939	1.11318
$727$	−14.2783	−0.529551	−0.264776	−	0.964310i	$-0.585298\pi$
$727$	−14.2783	−0.529551	−0.264776	+	0.964310i	$0.585298\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.06942	−0.224639
$731$	57.9861	2.14469
$732$	−55.1856	−2.03972
$733$	−34.2139	−1.26372	−0.631861	−	0.775082i	$-0.717708\pi$
$733$	−34.2139	−1.26372	−0.631861	+	0.775082i	$0.717708\pi$
$734$	−28.8340	−1.06428
$735$	0	0
$736$	−20.3220	−0.749079
$737$	0.444341	0.0163675
$738$	−2.72726	−0.100392
$739$	−29.2298	−1.07524	−0.537619	−	0.843188i	$-0.680676\pi$
$739$	−29.2298	−1.07524	−0.537619	+	0.843188i	$0.680676\pi$
$740$	−7.32770	−0.269372
$741$	18.2077	0.668875
$742$	0	0
$743$	−28.0271	−1.02821	−0.514107	−	0.857726i	$-0.671876\pi$
$743$	−28.0271	−1.02821	−0.514107	+	0.857726i	$0.671876\pi$
$744$	16.8832	0.618970
$745$	−7.59449	−0.278241
$746$	−34.0148	−1.24537
$747$	−3.23850	−0.118491
$748$	1.48986	0.0544747
$749$	0	0
$750$	−11.1550	−0.407323
$751$	34.5204	1.25967	0.629834	−	0.776730i	$-0.283123\pi$
$751$	34.5204	1.25967	0.629834	+	0.776730i	$0.283123\pi$
$752$	−136.514	−4.97814
$753$	16.0506	0.584916
$754$	−16.4607	−0.599462
$755$	−1.55824	−0.0567100
$756$	0	0
$757$	−43.9296	−1.59665	−0.798325	−	0.602227i	$-0.794280\pi$
$757$	−43.9296	−1.59665	−0.798325	+	0.602227i	$0.794280\pi$
$758$	14.1384	0.513531
$759$	−0.0445844	−0.00161831
$760$	23.6434	0.857637
$761$	−31.2348	−1.13226	−0.566131	−	0.824315i	$-0.691560\pi$
$761$	−31.2348	−1.13226	−0.566131	+	0.824315i	$0.691560\pi$
$762$	7.66228	0.277575
$763$	0	0
$764$	−39.8256	−1.44084
$765$	2.43739	0.0881240
$766$	68.2990	2.46774
$767$	−18.5968	−0.671493
$768$	40.1282	1.44800
$769$	40.3776	1.45606	0.728028	−	0.685548i	$-0.240437\pi$
$769$	40.3776	1.45606	0.728028	+	0.685548i	$0.240437\pi$
$770$	0	0
$771$	3.53643	0.127361
$772$	8.26188	0.297352
$773$	28.0706	1.00963	0.504814	−	0.863228i	$-0.331561\pi$
$773$	28.0706	1.00963	0.504814	+	0.863228i	$0.331561\pi$
$774$	27.0059	0.970707
$775$	8.69132	0.312202
$776$	−116.121	−4.16850
$777$	0	0
$778$	−72.9926	−2.61691
$779$	6.05832	0.217062
$780$	−6.80252	−0.243569
$781$	−0.220319	−0.00788362
$782$	15.2189	0.544225
$783$	2.00826	0.0717692
$784$	0	0
$785$	−0.905770	−0.0323283
$786$	22.8603	0.815399
$787$	−40.3706	−1.43906	−0.719529	−	0.694462i	$-0.755642\pi$
$787$	−40.3706	−1.43906	−0.719529	+	0.694462i	$0.755642\pi$
$788$	−83.2990	−2.96740
$789$	21.2018	0.754806
$790$	15.9343	0.566916
$791$	0	0
$792$	0.438676	0.0155877
$793$	−30.4996	−1.08307
$794$	−67.8159	−2.40670
$795$	1.02423	0.0363257
$796$	−109.943	−3.89683
$797$	−8.74108	−0.309625	−0.154812	−	0.987944i	$-0.549477\pi$
$797$	−8.74108	−0.309625	−0.154812	+	0.987944i	$0.549477\pi$
$798$	0	0
$799$	54.3986	1.92449
$800$	102.934	3.63926
$801$	4.51179	0.159416
$802$	98.0077	3.46077
$803$	0.250153	0.00882772
$804$	−51.6454	−1.82139
$805$	0	0
$806$	14.7591	0.519866
$807$	6.80296	0.239475
$808$	−55.6724	−1.95855
$809$	6.74812	0.237251	0.118626	−	0.992939i	$-0.462151\pi$
$809$	6.74812	0.237251	0.118626	+	0.992939i	$0.462151\pi$
$810$	1.13517	0.0398857
$811$	30.3742	1.06658	0.533291	−	0.845932i	$-0.320955\pi$
$811$	30.3742	1.06658	0.533291	+	0.845932i	$0.320955\pi$
$812$	0	0
$813$	−0.369791	−0.0129691
$814$	0.413091	0.0144788
$815$	1.59975	0.0560368
$816$	−86.0541	−3.01249
$817$	−59.9908	−2.09881
$818$	47.0657	1.64561
$819$	0	0
$820$	−2.26343	−0.0790425
$821$	28.1089	0.981008	0.490504	−	0.871439i	$-0.336813\pi$
$821$	28.1089	0.981008	0.490504	+	0.871439i	$0.336813\pi$
$822$	60.2865	2.10273
$823$	29.3386	1.02268	0.511339	−	0.859379i	$-0.329150\pi$
$823$	29.3386	1.02268	0.511339	+	0.859379i	$0.329150\pi$
$824$	−107.562	−3.74709
$825$	0.225826	0.00786225
$826$	0	0
$827$	−6.75490	−0.234891	−0.117445	−	0.993079i	$-0.537471\pi$
$827$	−6.75490	−0.234891	−0.117445	+	0.993079i	$0.537471\pi$
$828$	5.18201	0.180087
$829$	46.2748	1.60719	0.803595	−	0.595176i	$-0.202918\pi$
$829$	46.2748	1.60719	0.803595	+	0.595176i	$0.202918\pi$
$830$	−3.67624	−0.127604
$831$	20.8105	0.721909
$832$	86.4652	2.99764
$833$	0	0
$834$	35.6406	1.23413
$835$	4.37135	0.151277
$836$	−1.54137	−0.0533093
$837$	−1.80066	−0.0622398
$838$	13.9400	0.481548
$839$	−22.2264	−0.767340	−0.383670	−	0.923470i	$-0.625340\pi$
$839$	−22.2264	−0.767340	−0.383670	+	0.923470i	$0.625340\pi$
$840$	0	0
$841$	−24.9669	−0.860928
$842$	53.6624	1.84933
$843$	28.3373	0.975989
$844$	−104.152	−3.58508
$845$	1.65143	0.0568108
$846$	25.3351	0.871039
$847$	0	0
$848$	−36.1613	−1.24178
$849$	−20.3351	−0.697898
$850$	−77.0856	−2.64401
$851$	3.08506	0.105755
$852$	25.6075	0.877298
$853$	−32.4363	−1.11060	−0.555298	−	0.831651i	$-0.687396\pi$
$853$	−32.4363	−1.11060	−0.555298	+	0.831651i	$0.687396\pi$
$854$	0	0
$855$	−2.52165	−0.0862387
$856$	−113.287	−3.87207
$857$	18.7376	0.640063	0.320031	−	0.947407i	$-0.396307\pi$
$857$	18.7376	0.640063	0.320031	+	0.947407i	$0.396307\pi$
$858$	0.383484	0.0130919
$859$	6.87509	0.234575	0.117288	−	0.993098i	$-0.462580\pi$
$859$	6.87509	0.234575	0.117288	+	0.993098i	$0.462580\pi$
$860$	22.4130	0.764277
$861$	0	0
$862$	112.785	3.84147
$863$	−23.0107	−0.783292	−0.391646	−	0.920116i	$-0.628094\pi$
$863$	−23.0107	−0.783292	−0.391646	+	0.920116i	$0.628094\pi$
$864$	−21.3257	−0.725514
$865$	−5.06133	−0.172091
$866$	−76.3221	−2.59353
$867$	17.2913	0.587242
$868$	0	0
$869$	−0.656738	−0.0222783
$870$	2.27970	0.0772892
$871$	−28.5430	−0.967142
$872$	93.5821	3.16909
$873$	12.3847	0.419159
$874$	−15.7450	−0.532582
$875$	0	0
$876$	−29.0751	−0.982358
$877$	−46.3732	−1.56591	−0.782955	−	0.622078i	$-0.786289\pi$
$877$	−46.3732	−1.56591	−0.782955	+	0.622078i	$0.786289\pi$
$878$	−42.8056	−1.44462
$879$	6.54345	0.220705
$880$	0.286175	0.00964696
$881$	10.6982	0.360431	0.180216	−	0.983627i	$-0.442320\pi$
$881$	10.6982	0.360431	0.180216	+	0.983627i	$0.442320\pi$
$882$	0	0
$883$	21.2931	0.716569	0.358285	−	0.933612i	$-0.383362\pi$
$883$	21.2931	0.716569	0.358285	+	0.933612i	$0.383362\pi$
$884$	−95.7036	−3.21886
$885$	2.57555	0.0865762
$886$	97.8581	3.28761
$887$	−16.7878	−0.563678	−0.281839	−	0.959462i	$-0.590944\pi$
$887$	−16.7878	−0.563678	−0.281839	+	0.959462i	$0.590944\pi$
$888$	−30.3546	−1.01864
$889$	0	0
$890$	5.12164	0.171678
$891$	−0.0467863	−0.00156740
$892$	−59.2381	−1.98344
$893$	−56.2793	−1.88331
$894$	−49.7613	−1.66427
$895$	−3.23532	−0.108145
$896$	0	0
$897$	2.86395	0.0956246
$898$	−35.1432	−1.17274
$899$	−3.61618	−0.120606
$900$	−26.2476	−0.874920
$901$	14.4097	0.480057
$902$	0.127598	0.00424856
$903$	0	0
$904$	131.718	4.38087
$905$	−4.58912	−0.152548
$906$	−10.2100	−0.339205
$907$	32.9902	1.09542	0.547711	−	0.836667i	$-0.315499\pi$
$907$	32.9902	1.09542	0.547711	+	0.836667i	$0.315499\pi$
$908$	10.5321	0.349520
$909$	5.93765	0.196940
$910$	0	0
$911$	39.6694	1.31431	0.657154	−	0.753757i	$-0.271760\pi$
$911$	39.6694	1.31431	0.657154	+	0.753757i	$0.271760\pi$
$912$	89.0291	2.94805
$913$	0.151518	0.00501450
$914$	92.0417	3.04447
$915$	4.22401	0.139641
$916$	−130.003	−4.29542
$917$	0	0
$918$	15.9705	0.527105
$919$	−36.6457	−1.20883	−0.604415	−	0.796669i	$-0.706593\pi$
$919$	−36.6457	−1.20883	−0.604415	+	0.796669i	$0.706593\pi$
$920$	3.71896	0.122611
$921$	27.3197	0.900213
$922$	−39.0294	−1.28537
$923$	14.1525	0.465837
$924$	0	0
$925$	−15.6263	−0.513788
$926$	71.4361	2.34753
$927$	11.4718	0.376785
$928$	−42.8274	−1.40588
$929$	−48.7399	−1.59911	−0.799553	−	0.600596i	$-0.794930\pi$
$929$	−48.7399	−1.59911	−0.799553	+	0.600596i	$0.794930\pi$
$930$	−2.04404	−0.0670269
$931$	0	0
$932$	−71.1358	−2.33013
$933$	−9.86057	−0.322821
$934$	−61.0600	−1.99794
$935$	−0.114037	−0.00372939
$936$	−28.1791	−0.921063
$937$	40.9631	1.33821	0.669103	−	0.743170i	$-0.266679\pi$
$937$	40.9631	1.33821	0.669103	+	0.743170i	$0.266679\pi$
$938$	0	0
$939$	−10.1230	−0.330352
$940$	21.0264	0.685804
$941$	10.7987	0.352026	0.176013	−	0.984388i	$-0.443680\pi$
$941$	10.7987	0.352026	0.176013	+	0.984388i	$0.443680\pi$
$942$	−5.93487	−0.193368
$943$	0.952936	0.0310319
$944$	−90.9320	−2.95958
$945$	0	0
$946$	−1.26351	−0.0410802
$947$	10.4430	0.339351	0.169676	−	0.985500i	$-0.445728\pi$
$947$	10.4430	0.339351	0.169676	+	0.985500i	$0.445728\pi$
$948$	76.3322	2.47915
$949$	−16.0690	−0.521622
$950$	79.7506	2.58745
$951$	−25.6530	−0.831857
$952$	0	0
$953$	−9.99144	−0.323655	−0.161827	−	0.986819i	$-0.551739\pi$
$953$	−9.99144	−0.323655	−0.161827	+	0.986819i	$0.551739\pi$
$954$	6.71105	0.217278
$955$	3.04832	0.0986413
$956$	−95.6491	−3.09351
$957$	−0.0939589	−0.00303726
$958$	32.9415	1.06429
$959$	0	0
$960$	−11.9749	−0.386489
$961$	−27.7576	−0.895408
$962$	−26.5356	−0.855541
$963$	12.0824	0.389351
$964$	135.595	4.36721
$965$	−0.632379	−0.0203570
$966$	0	0
$967$	−47.6624	−1.53272	−0.766360	−	0.642411i	$-0.777934\pi$
$967$	−47.6624	−1.53272	−0.766360	+	0.642411i	$0.777934\pi$
$968$	103.117	3.31431
$969$	−35.4768	−1.13968
$970$	14.0587	0.451398
$971$	55.4444	1.77929	0.889647	−	0.456649i	$-0.150950\pi$
$971$	55.4444	1.77929	0.889647	+	0.456649i	$0.150950\pi$
$972$	5.43794	0.174422
$973$	0	0
$974$	−65.9757	−2.11400
$975$	−14.5063	−0.464574
$976$	−149.132	−4.77360
$977$	2.47040	0.0790352	0.0395176	−	0.999219i	$-0.487418\pi$
$977$	2.47040	0.0790352	0.0395176	+	0.999219i	$0.487418\pi$
$978$	10.4820	0.335178
$979$	−0.211090	−0.00674647
$980$	0	0
$981$	−9.98086	−0.318664
$982$	−17.2019	−0.548936
$983$	−7.12844	−0.227362	−0.113681	−	0.993517i	$-0.536264\pi$
$983$	−7.12844	−0.227362	−0.113681	+	0.993517i	$0.536264\pi$
$984$	−9.37616	−0.298901
$985$	6.37585	0.203152
$986$	32.0728	1.02141
$987$	0	0
$988$	99.0123	3.15000
$989$	−9.43618	−0.300053
$990$	−0.0531103	−0.00168796
$991$	−28.3333	−0.900037	−0.450019	−	0.893019i	$-0.648583\pi$
$991$	−28.3333	−0.900037	−0.450019	+	0.893019i	$0.648583\pi$
$992$	38.4002	1.21921
$993$	−1.38318	−0.0438939
$994$	0	0
$995$	8.41523	0.266781
$996$	−17.6108	−0.558019
$997$	−42.5700	−1.34821	−0.674103	−	0.738637i	$-0.735470\pi$
$997$	−42.5700	−1.34821	−0.674103	+	0.738637i	$0.735470\pi$
$998$	101.416	3.21026
$999$	3.23743	0.102428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bm.1.1	yes	16
7.6	odd	2		6027.2.a.bl.1.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.bl.1.1	✓	16	7.6	odd	2
6027.2.a.bm.1.1	yes	16	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72726 q^{2} +1.00000 q^{3} +5.43794 q^{4} -0.416230 q^{5} -2.72726 q^{6} -9.37616 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.72726 q^{2} +1.00000 q^{3} +5.43794 q^{4} -0.416230 q^{5} -2.72726 q^{6} -9.37616 q^{8} +1.00000 q^{9} +1.13517 q^{10} -0.0467863 q^{11} +5.43794 q^{12} +3.00540 q^{13} -0.416230 q^{15} +14.6953 q^{16} -5.85587 q^{17} -2.72726 q^{18} +6.05832 q^{19} -2.26343 q^{20} +0.127598 q^{22} +0.952936 q^{23} -9.37616 q^{24} -4.82675 q^{25} -8.19650 q^{26} +1.00000 q^{27} +2.00826 q^{29} +1.13517 q^{30} -1.80066 q^{31} -21.3257 q^{32} -0.0467863 q^{33} +15.9705 q^{34} +5.43794 q^{36} +3.23743 q^{37} -16.5226 q^{38} +3.00540 q^{39} +3.90264 q^{40} +1.00000 q^{41} -9.90222 q^{43} -0.254421 q^{44} -0.416230 q^{45} -2.59890 q^{46} -9.28958 q^{47} +14.6953 q^{48} +13.1638 q^{50} -5.85587 q^{51} +16.3432 q^{52} -2.46073 q^{53} -2.72726 q^{54} +0.0194739 q^{55} +6.05832 q^{57} -5.47703 q^{58} -6.18781 q^{59} -2.26343 q^{60} -10.1483 q^{61} +4.91086 q^{62} +28.7700 q^{64} -1.25094 q^{65} +0.127598 q^{66} -9.49723 q^{67} -31.8439 q^{68} +0.952936 q^{69} +4.70904 q^{71} -9.37616 q^{72} -5.34672 q^{73} -8.82930 q^{74} -4.82675 q^{75} +32.9448 q^{76} -8.19650 q^{78} +14.0370 q^{79} -6.11664 q^{80} +1.00000 q^{81} -2.72726 q^{82} -3.23850 q^{83} +2.43739 q^{85} +27.0059 q^{86} +2.00826 q^{87} +0.438676 q^{88} +4.51179 q^{89} +1.13517 q^{90} +5.18201 q^{92} -1.80066 q^{93} +25.3351 q^{94} -2.52165 q^{95} -21.3257 q^{96} +12.3847 q^{97} -0.0467863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9	\( 16 q - 4 q^{2} + 16 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 12 q^{8} + 16 q^{9} - 4 q^{10} - 4 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{17} - 4 q^{18} + 4 q^{19} - 20 q^{20} - 16 q^{22} - 12 q^{23} - 12 q^{24} - 8 q^{25} - 8 q^{26} + 16 q^{27} - 16 q^{29} - 4 q^{30} - 4 q^{31} - 48 q^{32} - 4 q^{33} + 16 q^{34} + 12 q^{36} - 48 q^{37} - 4 q^{38} + 56 q^{40} + 16 q^{41} - 16 q^{43} - 12 q^{45} - 4 q^{46} - 36 q^{47} - 8 q^{50} - 8 q^{51} - 60 q^{53} - 4 q^{54} + 8 q^{55} + 4 q^{57} - 36 q^{58} - 36 q^{59} - 20 q^{60} - 4 q^{61} - 12 q^{62} + 52 q^{64} - 36 q^{65} - 16 q^{66} - 52 q^{67} - 8 q^{68} - 12 q^{69} - 12 q^{71} - 12 q^{72} - 16 q^{73} + 4 q^{74} - 8 q^{75} + 16 q^{76} - 8 q^{78} - 36 q^{79} - 68 q^{80} + 16 q^{81} - 4 q^{82} - 32 q^{83} - 28 q^{85} - 8 q^{86} - 16 q^{87} - 36 q^{88} - 12 q^{89} - 4 q^{90} - 36 q^{92} - 4 q^{93} + 24 q^{94} - 20 q^{95} - 48 q^{96} + 48 q^{97} - 4 q^{99}+O(q^{100}) \) 16 * q - 4 * q^2 + 16 * q^3 + 12 * q^4 - 12 * q^5 - 4 * q^6 - 12 * q^8 + 16 * q^9 - 4 * q^10 - 4 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^17 - 4 * q^18 + 4 * q^19 - 20 * q^20 - 16 * q^22 - 12 * q^23 - 12 * q^24 - 8 * q^25 - 8 * q^26 + 16 * q^27 - 16 * q^29 - 4 * q^30 - 4 * q^31 - 48 * q^32 - 4 * q^33 + 16 * q^34 + 12 * q^36 - 48 * q^37 - 4 * q^38 + 56 * q^40 + 16 * q^41 - 16 * q^43 - 12 * q^45 - 4 * q^46 - 36 * q^47 - 8 * q^50 - 8 * q^51 - 60 * q^53 - 4 * q^54 + 8 * q^55 + 4 * q^57 - 36 * q^58 - 36 * q^59 - 20 * q^60 - 4 * q^61 - 12 * q^62 + 52 * q^64 - 36 * q^65 - 16 * q^66 - 52 * q^67 - 8 * q^68 - 12 * q^69 - 12 * q^71 - 12 * q^72 - 16 * q^73 + 4 * q^74 - 8 * q^75 + 16 * q^76 - 8 * q^78 - 36 * q^79 - 68 * q^80 + 16 * q^81 - 4 * q^82 - 32 * q^83 - 28 * q^85 - 8 * q^86 - 16 * q^87 - 36 * q^88 - 12 * q^89 - 4 * q^90 - 36 * q^92 - 4 * q^93 + 24 * q^94 - 20 * q^95 - 48 * q^96 + 48 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

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