LMFDB - Embedded newform 6027.2.a.z.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 8x^{6} + 14x^{5} + 18x^{4} - 24x^{3} - 10x^{2} + 10x - 1 $ x^8 - 2x^7 - 8x^6 + 14x^5 + 18x^4 - 24x^3 - 10x^2 + 10*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-2.14356$ 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)

$f(q)$	$=$	$q+2.14356 q^{2} -1.00000 q^{3} +2.59485 q^{4} +1.47286 q^{5} -2.14356 q^{6} +1.27510 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.14356 q^{2} -1.00000 q^{3} +2.59485 q^{4} +1.47286 q^{5} -2.14356 q^{6} +1.27510 q^{8} +1.00000 q^{9} +3.15717 q^{10} -2.07059 q^{11} -2.59485 q^{12} -2.67550 q^{13} -1.47286 q^{15} -2.45645 q^{16} +4.39074 q^{17} +2.14356 q^{18} -0.0316374 q^{19} +3.82186 q^{20} -4.43842 q^{22} -6.77691 q^{23} -1.27510 q^{24} -2.83067 q^{25} -5.73510 q^{26} -1.00000 q^{27} -4.35608 q^{29} -3.15717 q^{30} +0.939046 q^{31} -7.81575 q^{32} +2.07059 q^{33} +9.41181 q^{34} +2.59485 q^{36} -1.46770 q^{37} -0.0678166 q^{38} +2.67550 q^{39} +1.87805 q^{40} +1.00000 q^{41} -3.74000 q^{43} -5.37286 q^{44} +1.47286 q^{45} -14.5267 q^{46} -11.2246 q^{47} +2.45645 q^{48} -6.06772 q^{50} -4.39074 q^{51} -6.94253 q^{52} -3.59696 q^{53} -2.14356 q^{54} -3.04969 q^{55} +0.0316374 q^{57} -9.33752 q^{58} +10.2641 q^{59} -3.82186 q^{60} -2.01549 q^{61} +2.01290 q^{62} -11.8406 q^{64} -3.94065 q^{65} +4.43842 q^{66} -5.92486 q^{67} +11.3933 q^{68} +6.77691 q^{69} +11.9985 q^{71} +1.27510 q^{72} -8.42842 q^{73} -3.14609 q^{74} +2.83067 q^{75} -0.0820943 q^{76} +5.73510 q^{78} +10.7975 q^{79} -3.61802 q^{80} +1.00000 q^{81} +2.14356 q^{82} +12.4855 q^{83} +6.46696 q^{85} -8.01692 q^{86} +4.35608 q^{87} -2.64020 q^{88} -4.60030 q^{89} +3.15717 q^{90} -17.5851 q^{92} -0.939046 q^{93} -24.0607 q^{94} -0.0465976 q^{95} +7.81575 q^{96} -3.89557 q^{97} -2.07059 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 8 * q^9	$ 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 4 q^{13} - 2 q^{15} + 8 q^{17} - 2 q^{18} + 6 q^{19} + 4 q^{20} - 14 q^{22} - 12 q^{23} + 6 q^{24} - 4 q^{25} + 4 q^{26} - 8 q^{27} - 4 q^{29} + 2 q^{30} - 10 q^{31} - 4 q^{32} + 2 q^{33} + 4 q^{34} + 4 q^{36} - 20 q^{37} - 18 q^{38} - 4 q^{39} + 12 q^{40} + 8 q^{41} - 8 q^{43} + 20 q^{44} + 2 q^{45} - 12 q^{46} + 24 q^{47} - 22 q^{50} - 8 q^{51} - 30 q^{52} - 36 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{57} + 14 q^{58} + 10 q^{59} - 4 q^{60} - 22 q^{61} + 30 q^{62} - 24 q^{64} + 8 q^{65} + 14 q^{66} - 14 q^{67} + 38 q^{68} + 12 q^{69} - 10 q^{71} - 6 q^{72} - 12 q^{73} - 2 q^{74} + 4 q^{75} + 32 q^{76} - 4 q^{78} + 16 q^{79} - 14 q^{80} + 8 q^{81} - 2 q^{82} + 24 q^{83} - 44 q^{85} + 36 q^{86} + 4 q^{87} - 34 q^{88} + 2 q^{89} - 2 q^{90} - 48 q^{92} + 10 q^{93} - 34 q^{94} - 24 q^{95} + 4 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) $ 8 * q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 8 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 4 * q^13 - 2 * q^15 + 8 * q^17 - 2 * q^18 + 6 * q^19 + 4 * q^20 - 14 * q^22 - 12 * q^23 + 6 * q^24 - 4 * q^25 + 4 * q^26 - 8 * q^27 - 4 * q^29 + 2 * q^30 - 10 * q^31 - 4 * q^32 + 2 * q^33 + 4 * q^34 + 4 * q^36 - 20 * q^37 - 18 * q^38 - 4 * q^39 + 12 * q^40 + 8 * q^41 - 8 * q^43 + 20 * q^44 + 2 * q^45 - 12 * q^46 + 24 * q^47 - 22 * q^50 - 8 * q^51 - 30 * q^52 - 36 * q^53 + 2 * q^54 + 4 * q^55 - 6 * q^57 + 14 * q^58 + 10 * q^59 - 4 * q^60 - 22 * q^61 + 30 * q^62 - 24 * q^64 + 8 * q^65 + 14 * q^66 - 14 * q^67 + 38 * q^68 + 12 * q^69 - 10 * q^71 - 6 * q^72 - 12 * q^73 - 2 * q^74 + 4 * q^75 + 32 * q^76 - 4 * q^78 + 16 * q^79 - 14 * q^80 + 8 * q^81 - 2 * q^82 + 24 * q^83 - 44 * q^85 + 36 * q^86 + 4 * q^87 - 34 * q^88 + 2 * q^89 - 2 * q^90 - 48 * q^92 + 10 * q^93 - 34 * q^94 - 24 * q^95 + 4 * q^96 + 16 * q^97 - 2 * 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 $

$2$	2.14356	1.51573	0.757863	−	0.652414i	$-0.226243\pi$
$2$	2.14356	1.51573	0.757863	+	0.652414i	$0.226243\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.59485	1.29743
$5$	1.47286	0.658685	0.329342	−	0.944211i	$-0.393173\pi$
$5$	1.47286	0.658685	0.329342	+	0.944211i	$0.393173\pi$
$6$	−2.14356	−0.875105
$7$	0	0
$7$	0	0
$8$	1.27510	0.450816
$9$	1.00000	0.333333
$10$	3.15717	0.998386
$11$	−2.07059	−0.624305	−0.312152	−	0.950032i	$-0.601050\pi$
$11$	−2.07059	−0.624305	−0.312152	+	0.950032i	$0.601050\pi$
$12$	−2.59485	−0.749069
$13$	−2.67550	−0.742050	−0.371025	−	0.928623i	$-0.620994\pi$
$13$	−2.67550	−0.742050	−0.371025	+	0.928623i	$0.620994\pi$
$14$	0	0
$15$	−1.47286	−0.380292
$16$	−2.45645	−0.614112
$17$	4.39074	1.06491	0.532455	−	0.846458i	$-0.321270\pi$
$17$	4.39074	1.06491	0.532455	+	0.846458i	$0.321270\pi$
$18$	2.14356	0.505242
$19$	−0.0316374	−0.00725811	−0.00362906	−	0.999993i	$-0.501155\pi$
$19$	−0.0316374	−0.00725811	−0.00362906	+	0.999993i	$0.501155\pi$
$20$	3.82186	0.854595
$21$	0	0
$22$	−4.43842	−0.946275
$23$	−6.77691	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$23$	−6.77691	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$24$	−1.27510	−0.260279
$25$	−2.83067	−0.566134
$26$	−5.73510	−1.12475
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.35608	−0.808904	−0.404452	−	0.914559i	$-0.632538\pi$
$29$	−4.35608	−0.808904	−0.404452	+	0.914559i	$0.632538\pi$
$30$	−3.15717	−0.576418
$31$	0.939046	0.168658	0.0843288	−	0.996438i	$-0.473125\pi$
$31$	0.939046	0.168658	0.0843288	+	0.996438i	$0.473125\pi$
$32$	−7.81575	−1.38164
$33$	2.07059	0.360443
$34$	9.41181	1.61411
$35$	0	0
$36$	2.59485	0.432475
$37$	−1.46770	−0.241288	−0.120644	−	0.992696i	$-0.538496\pi$
$37$	−1.46770	−0.241288	−0.120644	+	0.992696i	$0.538496\pi$
$38$	−0.0678166	−0.0110013
$39$	2.67550	0.428423
$40$	1.87805	0.296946
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−3.74000	−0.570345	−0.285173	−	0.958476i	$-0.592051\pi$
$43$	−3.74000	−0.570345	−0.285173	+	0.958476i	$0.592051\pi$
$44$	−5.37286	−0.809989
$45$	1.47286	0.219562
$46$	−14.5267	−2.14185
$47$	−11.2246	−1.63728	−0.818640	−	0.574307i	$-0.805271\pi$
$47$	−11.2246	−1.63728	−0.818640	+	0.574307i	$0.805271\pi$
$48$	2.45645	0.354558
$49$	0	0
$50$	−6.06772	−0.858105
$51$	−4.39074	−0.614826
$52$	−6.94253	−0.962755
$53$	−3.59696	−0.494081	−0.247040	−	0.969005i	$-0.579458\pi$
$53$	−3.59696	−0.494081	−0.247040	+	0.969005i	$0.579458\pi$
$54$	−2.14356	−0.291702
$55$	−3.04969	−0.411220
$56$	0	0
$57$	0.0316374	0.00419047
$58$	−9.33752	−1.22608
$59$	10.2641	1.33628	0.668139	−	0.744036i	$-0.267091\pi$
$59$	10.2641	1.33628	0.668139	+	0.744036i	$0.267091\pi$
$60$	−3.82186	−0.493400
$61$	−2.01549	−0.258057	−0.129029	−	0.991641i	$-0.541186\pi$
$61$	−2.01549	−0.258057	−0.129029	+	0.991641i	$0.541186\pi$
$62$	2.01290	0.255639
$63$	0	0
$64$	−11.8406	−1.48008
$65$	−3.94065	−0.488777
$66$	4.43842	0.546332
$67$	−5.92486	−0.723836	−0.361918	−	0.932210i	$-0.617878\pi$
$67$	−5.92486	−0.723836	−0.361918	+	0.932210i	$0.617878\pi$
$68$	11.3933	1.38164
$69$	6.77691	0.815844
$70$	0	0
$71$	11.9985	1.42396	0.711981	−	0.702199i	$-0.247798\pi$
$71$	11.9985	1.42396	0.711981	+	0.702199i	$0.247798\pi$
$72$	1.27510	0.150272
$73$	−8.42842	−0.986472	−0.493236	−	0.869895i	$-0.664186\pi$
$73$	−8.42842	−0.986472	−0.493236	+	0.869895i	$0.664186\pi$
$74$	−3.14609	−0.365726
$75$	2.83067	0.326858
$76$	−0.0820943	−0.00941686
$77$	0	0
$78$	5.73510	0.649372
$79$	10.7975	1.21481	0.607406	−	0.794391i	$-0.292210\pi$
$79$	10.7975	1.21481	0.607406	+	0.794391i	$0.292210\pi$
$80$	−3.61802	−0.404506
$81$	1.00000	0.111111
$82$	2.14356	0.236717
$83$	12.4855	1.37046	0.685230	−	0.728327i	$-0.259702\pi$
$83$	12.4855	1.37046	0.685230	+	0.728327i	$0.259702\pi$
$84$	0	0
$85$	6.46696	0.701440
$86$	−8.01692	−0.864487
$87$	4.35608	0.467021
$88$	−2.64020	−0.281447
$89$	−4.60030	−0.487630	−0.243815	−	0.969822i	$-0.578399\pi$
$89$	−4.60030	−0.487630	−0.243815	+	0.969822i	$0.578399\pi$
$90$	3.15717	0.332795
$91$	0	0
$92$	−17.5851	−1.83337
$93$	−0.939046	−0.0973745
$94$	−24.0607	−2.48167
$95$	−0.0465976	−0.00478081
$96$	7.81575	0.797691
$97$	−3.89557	−0.395535	−0.197767	−	0.980249i	$-0.563369\pi$
$97$	−3.89557	−0.395535	−0.197767	+	0.980249i	$0.563369\pi$
$98$	0	0
$99$	−2.07059	−0.208102
$100$	−7.34517	−0.734517
$101$	−3.27183	−0.325559	−0.162780	−	0.986662i	$-0.552046\pi$
$101$	−3.27183	−0.325559	−0.162780	+	0.986662i	$0.552046\pi$
$102$	−9.41181	−0.931908
$103$	11.8257	1.16522	0.582609	−	0.812753i	$-0.302032\pi$
$103$	11.8257	1.16522	0.582609	+	0.812753i	$0.302032\pi$
$104$	−3.41153	−0.334528
$105$	0	0
$106$	−7.71031	−0.748891
$107$	−18.7370	−1.81138	−0.905689	−	0.423943i	$-0.860646\pi$
$107$	−18.7370	−1.81138	−0.905689	+	0.423943i	$0.860646\pi$
$108$	−2.59485	−0.249690
$109$	−1.21453	−0.116331	−0.0581656	−	0.998307i	$-0.518525\pi$
$109$	−1.21453	−0.116331	−0.0581656	+	0.998307i	$0.518525\pi$
$110$	−6.53720	−0.623297
$111$	1.46770	0.139308
$112$	0	0
$113$	−7.94682	−0.747574	−0.373787	−	0.927515i	$-0.621941\pi$
$113$	−7.94682	−0.747574	−0.373787	+	0.927515i	$0.621941\pi$
$114$	0.0678166	0.00635161
$115$	−9.98147	−0.930777
$116$	−11.3034	−1.04949
$117$	−2.67550	−0.247350
$118$	22.0018	2.02543
$119$	0	0
$120$	−1.87805	−0.171442
$121$	−6.71268	−0.610243
$122$	−4.32033	−0.391144
$123$	−1.00000	−0.0901670
$124$	2.43668	0.218821
$125$	−11.5335	−1.03159
$126$	0	0
$127$	9.86917	0.875747	0.437874	−	0.899037i	$-0.355732\pi$
$127$	9.86917	0.875747	0.437874	+	0.899037i	$0.355732\pi$
$128$	−9.74961	−0.861752
$129$	3.74000	0.329289
$130$	−8.44702	−0.740853
$131$	−13.1425	−1.14827	−0.574134	−	0.818762i	$-0.694661\pi$
$131$	−13.1425	−1.14827	−0.574134	+	0.818762i	$0.694661\pi$
$132$	5.37286	0.467648
$133$	0	0
$134$	−12.7003	−1.09714
$135$	−1.47286	−0.126764
$136$	5.59863	0.480078
$137$	−11.4853	−0.981258	−0.490629	−	0.871369i	$-0.663233\pi$
$137$	−11.4853	−0.981258	−0.490629	+	0.871369i	$0.663233\pi$
$138$	14.5267	1.23660
$139$	−10.1231	−0.858628	−0.429314	−	0.903155i	$-0.641245\pi$
$139$	−10.1231	−0.858628	−0.429314	+	0.903155i	$0.641245\pi$
$140$	0	0
$141$	11.2246	0.945284
$142$	25.7195	2.15834
$143$	5.53985	0.463266
$144$	−2.45645	−0.204704
$145$	−6.41591	−0.532813
$146$	−18.0668	−1.49522
$147$	0	0
$148$	−3.80845	−0.313053
$149$	−7.13896	−0.584846	−0.292423	−	0.956289i	$-0.594462\pi$
$149$	−7.13896	−0.584846	−0.292423	+	0.956289i	$0.594462\pi$
$150$	6.06772	0.495427
$151$	14.1624	1.15252	0.576261	−	0.817266i	$-0.304511\pi$
$151$	14.1624	1.15252	0.576261	+	0.817266i	$0.304511\pi$
$152$	−0.0403408	−0.00327207
$153$	4.39074	0.354970
$154$	0	0
$155$	1.38309	0.111092
$156$	6.94253	0.555847
$157$	−6.86801	−0.548127	−0.274064	−	0.961712i	$-0.588368\pi$
$157$	−6.86801	−0.548127	−0.274064	+	0.961712i	$0.588368\pi$
$158$	23.1451	1.84132
$159$	3.59696	0.285258
$160$	−11.5115	−0.910067
$161$	0	0
$162$	2.14356	0.168414
$163$	−9.17695	−0.718794	−0.359397	−	0.933185i	$-0.617018\pi$
$163$	−9.17695	−0.718794	−0.359397	+	0.933185i	$0.617018\pi$
$164$	2.59485	0.202624
$165$	3.04969	0.237418
$166$	26.7634	2.07724
$167$	14.7293	1.13979	0.569895	−	0.821717i	$-0.306984\pi$
$167$	14.7293	1.13979	0.569895	+	0.821717i	$0.306984\pi$
$168$	0	0
$169$	−5.84170	−0.449361
$170$	13.8623	1.06319
$171$	−0.0316374	−0.00241937
$172$	−9.70475	−0.739980
$173$	−0.400067	−0.0304165	−0.0152082	−	0.999884i	$-0.504841\pi$
$173$	−0.400067	−0.0304165	−0.0152082	+	0.999884i	$0.504841\pi$
$174$	9.33752	0.707876
$175$	0	0
$176$	5.08629	0.383393
$177$	−10.2641	−0.771501
$178$	−9.86101	−0.739114
$179$	0.554138	0.0414182	0.0207091	−	0.999786i	$-0.493408\pi$
$179$	0.554138	0.0414182	0.0207091	+	0.999786i	$0.493408\pi$
$180$	3.82186	0.284865
$181$	19.8440	1.47499	0.737496	−	0.675352i	$-0.236008\pi$
$181$	19.8440	1.47499	0.737496	+	0.675352i	$0.236008\pi$
$182$	0	0
$183$	2.01549	0.148989
$184$	−8.64124	−0.637041
$185$	−2.16172	−0.158933
$186$	−2.01290	−0.147593
$187$	−9.09139	−0.664829
$188$	−29.1262	−2.12425
$189$	0	0
$190$	−0.0998847	−0.00724640
$191$	0.326505	0.0236250	0.0118125	−	0.999930i	$-0.496240\pi$
$191$	0.326505	0.0236250	0.0118125	+	0.999930i	$0.496240\pi$
$192$	11.8406	0.854524
$193$	−7.99202	−0.575278	−0.287639	−	0.957739i	$-0.592870\pi$
$193$	−7.99202	−0.575278	−0.287639	+	0.957739i	$0.592870\pi$
$194$	−8.35038	−0.599522
$195$	3.94065	0.282196
$196$	0	0
$197$	−4.72061	−0.336329	−0.168165	−	0.985759i	$-0.553784\pi$
$197$	−4.72061	−0.336329	−0.168165	+	0.985759i	$0.553784\pi$
$198$	−4.43842	−0.315425
$199$	1.10029	0.0779972	0.0389986	−	0.999239i	$-0.487583\pi$
$199$	1.10029	0.0779972	0.0389986	+	0.999239i	$0.487583\pi$
$200$	−3.60939	−0.255222
$201$	5.92486	0.417907
$202$	−7.01337	−0.493459
$203$	0	0
$204$	−11.3933	−0.797691
$205$	1.47286	0.102869
$206$	25.3490	1.76615
$207$	−6.77691	−0.471028
$208$	6.57223	0.455702
$209$	0.0655079	0.00453128
$210$	0	0
$211$	12.5702	0.865371	0.432685	−	0.901545i	$-0.357566\pi$
$211$	12.5702	0.865371	0.432685	+	0.901545i	$0.357566\pi$
$212$	−9.33358	−0.641033
$213$	−11.9985	−0.822125
$214$	−40.1640	−2.74555
$215$	−5.50852	−0.375678
$216$	−1.27510	−0.0867595
$217$	0	0
$218$	−2.60343	−0.176326
$219$	8.42842	0.569540
$220$	−7.91349	−0.533528
$221$	−11.7474	−0.790217
$222$	3.14609	0.211152
$223$	0.0274795	0.00184016	0.000920080	−	1.00000i	$-0.499707\pi$
$223$	0.0274795	0.00184016	0.000920080		1.00000i	$0.499707\pi$
$224$	0	0
$225$	−2.83067	−0.188711
$226$	−17.0345	−1.13312
$227$	20.5039	1.36089	0.680446	−	0.732798i	$-0.261786\pi$
$227$	20.5039	1.36089	0.680446	+	0.732798i	$0.261786\pi$
$228$	0.0820943	0.00543683
$229$	−8.45982	−0.559040	−0.279520	−	0.960140i	$-0.590175\pi$
$229$	−8.45982	−0.559040	−0.279520	+	0.960140i	$0.590175\pi$
$230$	−21.3959	−1.41080
$231$	0	0
$232$	−5.55444	−0.364667
$233$	20.3483	1.33306	0.666529	−	0.745479i	$-0.267779\pi$
$233$	20.3483	1.33306	0.666529	+	0.745479i	$0.267779\pi$
$234$	−5.73510	−0.374915
$235$	−16.5323	−1.07845
$236$	26.6339	1.73372
$237$	−10.7975	−0.701372
$238$	0	0
$239$	12.2888	0.794899	0.397449	−	0.917624i	$-0.369895\pi$
$239$	12.2888	0.794899	0.397449	+	0.917624i	$0.369895\pi$
$240$	3.61802	0.233542
$241$	−20.9865	−1.35186	−0.675928	−	0.736967i	$-0.736257\pi$
$241$	−20.9865	−1.35186	−0.675928	+	0.736967i	$0.736257\pi$
$242$	−14.3890	−0.924962
$243$	−1.00000	−0.0641500
$244$	−5.22990	−0.334810
$245$	0	0
$246$	−2.14356	−0.136668
$247$	0.0846458	0.00538589
$248$	1.19738	0.0760335
$249$	−12.4855	−0.791235
$250$	−24.7228	−1.56361
$251$	−1.11964	−0.0706711	−0.0353356	−	0.999376i	$-0.511250\pi$
$251$	−1.11964	−0.0706711	−0.0353356	+	0.999376i	$0.511250\pi$
$252$	0	0
$253$	14.0322	0.882195
$254$	21.1552	1.32739
$255$	−6.46696	−0.404977
$256$	2.78239	0.173899
$257$	−2.71524	−0.169372	−0.0846859	−	0.996408i	$-0.526989\pi$
$257$	−2.71524	−0.169372	−0.0846859	+	0.996408i	$0.526989\pi$
$258$	8.01692	0.499112
$259$	0	0
$260$	−10.2254	−0.634152
$261$	−4.35608	−0.269635
$262$	−28.1718	−1.74046
$263$	−25.1848	−1.55296	−0.776482	−	0.630140i	$-0.782997\pi$
$263$	−25.1848	−1.55296	−0.776482	+	0.630140i	$0.782997\pi$
$264$	2.64020	0.162493
$265$	−5.29784	−0.325443
$266$	0	0
$267$	4.60030	0.281534
$268$	−15.3741	−0.939124
$269$	13.1304	0.800575	0.400288	−	0.916390i	$-0.368910\pi$
$269$	13.1304	0.800575	0.400288	+	0.916390i	$0.368910\pi$
$270$	−3.15717	−0.192139
$271$	12.4972	0.759152	0.379576	−	0.925161i	$-0.376070\pi$
$271$	12.4972	0.759152	0.379576	+	0.925161i	$0.376070\pi$
$272$	−10.7856	−0.653974
$273$	0	0
$274$	−24.6195	−1.48732
$275$	5.86115	0.353440
$276$	17.5851	1.05850
$277$	−20.1516	−1.21080	−0.605398	−	0.795923i	$-0.706986\pi$
$277$	−20.1516	−1.21080	−0.605398	+	0.795923i	$0.706986\pi$
$278$	−21.6994	−1.30144
$279$	0.939046	0.0562192
$280$	0	0
$281$	11.9933	0.715460	0.357730	−	0.933825i	$-0.383551\pi$
$281$	11.9933	0.715460	0.357730	+	0.933825i	$0.383551\pi$
$282$	24.0607	1.43279
$283$	16.3923	0.974423	0.487212	−	0.873284i	$-0.338014\pi$
$283$	16.3923	0.974423	0.487212	+	0.873284i	$0.338014\pi$
$284$	31.1344	1.84748
$285$	0.0465976	0.00276020
$286$	11.8750	0.702184
$287$	0	0
$288$	−7.81575	−0.460547
$289$	2.27857	0.134033
$290$	−13.7529	−0.807598
$291$	3.89557	0.228362
$292$	−21.8705	−1.27987
$293$	−5.82846	−0.340502	−0.170251	−	0.985401i	$-0.554458\pi$
$293$	−5.82846	−0.340502	−0.170251	+	0.985401i	$0.554458\pi$
$294$	0	0
$295$	15.1177	0.880186
$296$	−1.87146	−0.108776
$297$	2.07059	0.120148
$298$	−15.3028	−0.886467
$299$	18.1316	1.04858
$300$	7.34517	0.424074
$301$	0	0
$302$	30.3580	1.74691
$303$	3.27183	0.187962
$304$	0.0777156	0.00445730
$305$	−2.96854	−0.169978
$306$	9.41181	0.538037
$307$	−11.0203	−0.628963	−0.314482	−	0.949264i	$-0.601831\pi$
$307$	−11.0203	−0.628963	−0.314482	+	0.949264i	$0.601831\pi$
$308$	0	0
$309$	−11.8257	−0.672738
$310$	2.96473	0.168385
$311$	31.2092	1.76971	0.884857	−	0.465863i	$-0.154256\pi$
$311$	31.2092	1.76971	0.884857	+	0.465863i	$0.154256\pi$
$312$	3.41153	0.193140
$313$	−11.8351	−0.668959	−0.334479	−	0.942403i	$-0.608560\pi$
$313$	−11.8351	−0.668959	−0.334479	+	0.942403i	$0.608560\pi$
$314$	−14.7220	−0.830811
$315$	0	0
$316$	28.0179	1.57613
$317$	13.7250	0.770873	0.385436	−	0.922734i	$-0.374051\pi$
$317$	13.7250	0.770873	0.385436	+	0.922734i	$0.374051\pi$
$318$	7.71031	0.432372
$319$	9.01964	0.505003
$320$	−17.4396	−0.974905
$321$	18.7370	1.04580
$322$	0	0
$323$	−0.138911	−0.00772924
$324$	2.59485	0.144158
$325$	7.57346	0.420100
$326$	−19.6713	−1.08949
$327$	1.21453	0.0671638
$328$	1.27510	0.0704056
$329$	0	0
$330$	6.53720	0.359861
$331$	25.8711	1.42200	0.711002	−	0.703190i	$-0.248242\pi$
$331$	25.8711	1.42200	0.711002	+	0.703190i	$0.248242\pi$
$332$	32.3980	1.77807
$333$	−1.46770	−0.0804292
$334$	31.5732	1.72761
$335$	−8.72651	−0.476780
$336$	0	0
$337$	−6.74672	−0.367517	−0.183759	−	0.982971i	$-0.558826\pi$
$337$	−6.74672	−0.367517	−0.183759	+	0.982971i	$0.558826\pi$
$338$	−12.5220	−0.681109
$339$	7.94682	0.431612
$340$	16.7808	0.910066
$341$	−1.94437	−0.105294
$342$	−0.0678166	−0.00366710
$343$	0	0
$344$	−4.76888	−0.257121
$345$	9.98147	0.537384
$346$	−0.857567	−0.0461031
$347$	−30.0886	−1.61524	−0.807619	−	0.589705i	$-0.799244\pi$
$347$	−30.0886	−1.61524	−0.807619	+	0.589705i	$0.799244\pi$
$348$	11.3034	0.605925
$349$	−6.70515	−0.358918	−0.179459	−	0.983765i	$-0.557435\pi$
$349$	−6.70515	−0.358918	−0.179459	+	0.983765i	$0.557435\pi$
$350$	0	0
$351$	2.67550	0.142808
$352$	16.1832	0.862566
$353$	−10.8937	−0.579814	−0.289907	−	0.957055i	$-0.593624\pi$
$353$	−10.8937	−0.579814	−0.289907	+	0.957055i	$0.593624\pi$
$354$	−22.0018	−1.16938
$355$	17.6722	0.937942
$356$	−11.9371	−0.632664
$357$	0	0
$358$	1.18783	0.0627787
$359$	8.01610	0.423073	0.211537	−	0.977370i	$-0.432153\pi$
$359$	8.01610	0.423073	0.211537	+	0.977370i	$0.432153\pi$
$360$	1.87805	0.0989818
$361$	−18.9990	−0.999947
$362$	42.5368	2.23568
$363$	6.71268	0.352324
$364$	0	0
$365$	−12.4139	−0.649774
$366$	4.32033	0.225827
$367$	−6.45452	−0.336923	−0.168462	−	0.985708i	$-0.553880\pi$
$367$	−6.45452	−0.336923	−0.168462	+	0.985708i	$0.553880\pi$
$368$	16.6471	0.867792
$369$	1.00000	0.0520579
$370$	−4.63377	−0.240898
$371$	0	0
$372$	−2.43668	−0.126336
$373$	23.9346	1.23929	0.619644	−	0.784883i	$-0.287277\pi$
$373$	23.9346	1.23929	0.619644	+	0.784883i	$0.287277\pi$
$374$	−19.4880	−1.00770
$375$	11.5335	0.595588
$376$	−14.3125	−0.738111
$377$	11.6547	0.600247
$378$	0	0
$379$	9.87988	0.507496	0.253748	−	0.967270i	$-0.418337\pi$
$379$	9.87988	0.507496	0.253748	+	0.967270i	$0.418337\pi$
$380$	−0.120914	−0.00620274
$381$	−9.86917	−0.505613
$382$	0.699882	0.0358091
$383$	−28.2948	−1.44580	−0.722899	−	0.690954i	$-0.757191\pi$
$383$	−28.2948	−1.44580	−0.722899	+	0.690954i	$0.757191\pi$
$384$	9.74961	0.497532
$385$	0	0
$386$	−17.1314	−0.871964
$387$	−3.74000	−0.190115
$388$	−10.1084	−0.513177
$389$	−8.68733	−0.440465	−0.220232	−	0.975447i	$-0.570682\pi$
$389$	−8.68733	−0.440465	−0.220232	+	0.975447i	$0.570682\pi$
$390$	8.44702	0.427731
$391$	−29.7556	−1.50481
$392$	0	0
$393$	13.1425	0.662953
$394$	−10.1189	−0.509783
$395$	15.9032	0.800179
$396$	−5.37286	−0.269996
$397$	−14.9679	−0.751217	−0.375608	−	0.926778i	$-0.622566\pi$
$397$	−14.9679	−0.751217	−0.375608	+	0.926778i	$0.622566\pi$
$398$	2.35853	0.118222
$399$	0	0
$400$	6.95340	0.347670
$401$	0.899738	0.0449308	0.0224654	−	0.999748i	$-0.492848\pi$
$401$	0.899738	0.0449308	0.0224654	+	0.999748i	$0.492848\pi$
$402$	12.7003	0.633433
$403$	−2.51242	−0.125152
$404$	−8.48991	−0.422389
$405$	1.47286	0.0731872
$406$	0	0
$407$	3.03899	0.150637
$408$	−5.59863	−0.277173
$409$	−5.23943	−0.259073	−0.129537	−	0.991575i	$-0.541349\pi$
$409$	−5.23943	−0.259073	−0.129537	+	0.991575i	$0.541349\pi$
$410$	3.15717	0.155922
$411$	11.4853	0.566529
$412$	30.6858	1.51178
$413$	0	0
$414$	−14.5267	−0.713949
$415$	18.3894	0.902701
$416$	20.9110	1.02525
$417$	10.1231	0.495729
$418$	0.140420	0.00686817
$419$	−13.1645	−0.643128	−0.321564	−	0.946888i	$-0.604209\pi$
$419$	−13.1645	−0.643128	−0.321564	+	0.946888i	$0.604209\pi$
$420$	0	0
$421$	−12.9446	−0.630881	−0.315440	−	0.948945i	$-0.602152\pi$
$421$	−12.9446	−0.630881	−0.315440	+	0.948945i	$0.602152\pi$
$422$	26.9451	1.31167
$423$	−11.2246	−0.545760
$424$	−4.58649	−0.222739
$425$	−12.4287	−0.602882
$426$	−25.7195	−1.24612
$427$	0	0
$428$	−48.6198	−2.35013
$429$	−5.53985	−0.267467
$430$	−11.8078	−0.569424
$431$	−7.04523	−0.339357	−0.169678	−	0.985499i	$-0.554273\pi$
$431$	−7.04523	−0.339357	−0.169678	+	0.985499i	$0.554273\pi$
$432$	2.45645	0.118186
$433$	4.16142	0.199985	0.0999925	−	0.994988i	$-0.468118\pi$
$433$	4.16142	0.199985	0.0999925	+	0.994988i	$0.468118\pi$
$434$	0	0
$435$	6.41591	0.307620
$436$	−3.15153	−0.150931
$437$	0.214404	0.0102563
$438$	18.0668	0.863267
$439$	−11.9329	−0.569526	−0.284763	−	0.958598i	$-0.591915\pi$
$439$	−11.9329	−0.569526	−0.284763	+	0.958598i	$0.591915\pi$
$440$	−3.88866	−0.185385
$441$	0	0
$442$	−25.1813	−1.19775
$443$	8.67040	0.411943	0.205972	−	0.978558i	$-0.433965\pi$
$443$	8.67040	0.411943	0.205972	+	0.978558i	$0.433965\pi$
$444$	3.80845	0.180741
$445$	−6.77561	−0.321195
$446$	0.0589039	0.00278918
$447$	7.13896	0.337661
$448$	0	0
$449$	34.6316	1.63436	0.817182	−	0.576379i	$-0.195535\pi$
$449$	34.6316	1.63436	0.817182	+	0.576379i	$0.195535\pi$
$450$	−6.06772	−0.286035
$451$	−2.07059	−0.0975001
$452$	−20.6208	−0.969922
$453$	−14.1624	−0.665409
$454$	43.9514	2.06274
$455$	0	0
$456$	0.0403408	0.00188913
$457$	−17.7497	−0.830296	−0.415148	−	0.909754i	$-0.636270\pi$
$457$	−17.7497	−0.830296	−0.415148	+	0.909754i	$0.636270\pi$
$458$	−18.1341	−0.847352
$459$	−4.39074	−0.204942
$460$	−25.9004	−1.20761
$461$	11.1094	0.517418	0.258709	−	0.965955i	$-0.416703\pi$
$461$	11.1094	0.517418	0.258709	+	0.965955i	$0.416703\pi$
$462$	0	0
$463$	29.6797	1.37933	0.689666	−	0.724128i	$-0.257757\pi$
$463$	29.6797	1.37933	0.689666	+	0.724128i	$0.257757\pi$
$464$	10.7005	0.496758
$465$	−1.38309	−0.0641391
$466$	43.6177	2.02055
$467$	−30.5049	−1.41160	−0.705799	−	0.708412i	$-0.749412\pi$
$467$	−30.5049	−1.41160	−0.705799	+	0.708412i	$0.749412\pi$
$468$	−6.94253	−0.320918
$469$	0	0
$470$	−35.4381	−1.63464
$471$	6.86801	0.316461
$472$	13.0878	0.602415
$473$	7.74400	0.356069
$474$	−23.1451	−1.06309
$475$	0.0895550	0.00410907
$476$	0	0
$477$	−3.59696	−0.164694
$478$	26.3419	1.20485
$479$	−26.5624	−1.21367	−0.606835	−	0.794828i	$-0.707561\pi$
$479$	−26.5624	−1.21367	−0.606835	+	0.794828i	$0.707561\pi$
$480$	11.5115	0.525427
$481$	3.92682	0.179048
$482$	−44.9857	−2.04904
$483$	0	0
$484$	−17.4184	−0.791745
$485$	−5.73764	−0.260533
$486$	−2.14356	−0.0972339
$487$	11.4662	0.519585	0.259793	−	0.965664i	$-0.416346\pi$
$487$	11.4662	0.519585	0.259793	+	0.965664i	$0.416346\pi$
$488$	−2.56995	−0.116336
$489$	9.17695	0.414996
$490$	0	0
$491$	−12.7900	−0.577204	−0.288602	−	0.957449i	$-0.593190\pi$
$491$	−12.7900	−0.577204	−0.288602	+	0.957449i	$0.593190\pi$
$492$	−2.59485	−0.116985
$493$	−19.1264	−0.861410
$494$	0.181443	0.00816353
$495$	−3.04969	−0.137073
$496$	−2.30672	−0.103575
$497$	0	0
$498$	−26.7634	−1.19930
$499$	−8.26104	−0.369815	−0.184907	−	0.982756i	$-0.559199\pi$
$499$	−8.26104	−0.369815	−0.184907	+	0.982756i	$0.559199\pi$
$500$	−29.9278	−1.33841
$501$	−14.7293	−0.658058
$502$	−2.40002	−0.107118
$503$	−33.1009	−1.47590	−0.737948	−	0.674857i	$-0.764205\pi$
$503$	−33.1009	−1.47590	−0.737948	+	0.674857i	$0.764205\pi$
$504$	0	0
$505$	−4.81896	−0.214441
$506$	30.0788	1.33717
$507$	5.84170	0.259439
$508$	25.6090	1.13622
$509$	−2.09804	−0.0929940	−0.0464970	−	0.998918i	$-0.514806\pi$
$509$	−2.09804	−0.0929940	−0.0464970	+	0.998918i	$0.514806\pi$
$510$	−13.8623	−0.613834
$511$	0	0
$512$	25.4634	1.12533
$513$	0.0316374	0.00139682
$514$	−5.82028	−0.256721
$515$	17.4176	0.767511
$516$	9.70475	0.427228
$517$	23.2415	1.02216
$518$	0	0
$519$	0.400067	0.0175610
$520$	−5.02472	−0.220349
$521$	27.5876	1.20863	0.604317	−	0.796744i	$-0.293446\pi$
$521$	27.5876	1.20863	0.604317	+	0.796744i	$0.293446\pi$
$522$	−9.33752	−0.408692
$523$	26.2845	1.14934	0.574670	−	0.818385i	$-0.305131\pi$
$523$	26.2845	1.14934	0.574670	+	0.818385i	$0.305131\pi$
$524$	−34.1029	−1.48979
$525$	0	0
$526$	−53.9852	−2.35387
$527$	4.12310	0.179605
$528$	−5.08629	−0.221352
$529$	22.9265	0.996806
$530$	−11.3562	−0.493283
$531$	10.2641	0.445426
$532$	0	0
$533$	−2.67550	−0.115889
$534$	9.86101	0.426728
$535$	−27.5971	−1.19313
$536$	−7.55478	−0.326317
$537$	−0.554138	−0.0239128
$538$	28.1458	1.21345
$539$	0	0
$540$	−3.82186	−0.164467
$541$	−7.82513	−0.336429	−0.168214	−	0.985750i	$-0.553800\pi$
$541$	−7.82513	−0.336429	−0.168214	+	0.985750i	$0.553800\pi$
$542$	26.7885	1.15067
$543$	−19.8440	−0.851587
$544$	−34.3169	−1.47132
$545$	−1.78884	−0.0766256
$546$	0	0
$547$	−16.6793	−0.713157	−0.356579	−	0.934265i	$-0.616057\pi$
$547$	−16.6793	−0.713157	−0.356579	+	0.934265i	$0.616057\pi$
$548$	−29.8027	−1.27311
$549$	−2.01549	−0.0860190
$550$	12.5637	0.535719
$551$	0.137815	0.00587112
$552$	8.64124	0.367796
$553$	0	0
$554$	−43.1963	−1.83523
$555$	2.16172	0.0917598
$556$	−26.2679	−1.11401
$557$	−42.2973	−1.79219	−0.896097	−	0.443859i	$-0.853609\pi$
$557$	−42.2973	−1.79219	−0.896097	+	0.443859i	$0.853609\pi$
$558$	2.01290	0.0852129
$559$	10.0064	0.423225
$560$	0	0
$561$	9.09139	0.383839
$562$	25.7084	1.08444
$563$	35.8542	1.51107	0.755537	−	0.655106i	$-0.227376\pi$
$563$	35.8542	1.51107	0.755537	+	0.655106i	$0.227376\pi$
$564$	29.1262	1.22644
$565$	−11.7046	−0.492416
$566$	35.1380	1.47696
$567$	0	0
$568$	15.2993	0.641945
$569$	−18.1092	−0.759178	−0.379589	−	0.925155i	$-0.623935\pi$
$569$	−18.1092	−0.759178	−0.379589	+	0.925155i	$0.623935\pi$
$570$	0.0998847	0.00418371
$571$	33.5414	1.40366	0.701831	−	0.712344i	$-0.252366\pi$
$571$	33.5414	1.40366	0.701831	+	0.712344i	$0.252366\pi$
$572$	14.3751	0.601053
$573$	−0.326505	−0.0136399
$574$	0	0
$575$	19.1832	0.799995
$576$	−11.8406	−0.493359
$577$	26.1062	1.08682	0.543409	−	0.839468i	$-0.317133\pi$
$577$	26.1062	1.08682	0.543409	+	0.839468i	$0.317133\pi$
$578$	4.88424	0.203158
$579$	7.99202	0.332137
$580$	−16.6483	−0.691285
$581$	0	0
$582$	8.35038	0.346134
$583$	7.44782	0.308457
$584$	−10.7471	−0.444717
$585$	−3.94065	−0.162926
$586$	−12.4936	−0.516108
$587$	−27.1290	−1.11973	−0.559866	−	0.828583i	$-0.689147\pi$
$587$	−27.1290	−1.11973	−0.559866	+	0.828583i	$0.689147\pi$
$588$	0	0
$589$	−0.0297090	−0.00122414
$590$	32.4057	1.33412
$591$	4.72061	0.194180
$592$	3.60532	0.148178
$593$	31.0112	1.27348	0.636738	−	0.771080i	$-0.280283\pi$
$593$	31.0112	1.27348	0.636738	+	0.771080i	$0.280283\pi$
$594$	4.43842	0.182111
$595$	0	0
$596$	−18.5245	−0.758795
$597$	−1.10029	−0.0450317
$598$	38.8662	1.58936
$599$	−35.9927	−1.47062	−0.735311	−	0.677729i	$-0.762964\pi$
$599$	−35.9927	−1.47062	−0.735311	+	0.677729i	$0.762964\pi$
$600$	3.60939	0.147353
$601$	23.5545	0.960809	0.480405	−	0.877047i	$-0.340490\pi$
$601$	23.5545	0.960809	0.480405	+	0.877047i	$0.340490\pi$
$602$	0	0
$603$	−5.92486	−0.241279
$604$	36.7494	1.49531
$605$	−9.88686	−0.401958
$606$	7.01337	0.284899
$607$	−23.9407	−0.971724	−0.485862	−	0.874036i	$-0.661494\pi$
$607$	−23.9407	−0.971724	−0.485862	+	0.874036i	$0.661494\pi$
$608$	0.247270	0.0100281
$609$	0	0
$610$	−6.36325	−0.257641
$611$	30.0315	1.21494
$612$	11.3933	0.460547
$613$	14.8006	0.597791	0.298896	−	0.954286i	$-0.403382\pi$
$613$	14.8006	0.597791	0.298896	+	0.954286i	$0.403382\pi$
$614$	−23.6227	−0.953336
$615$	−1.47286	−0.0593916
$616$	0	0
$617$	25.5480	1.02853	0.514263	−	0.857633i	$-0.328066\pi$
$617$	25.5480	1.02853	0.514263	+	0.857633i	$0.328066\pi$
$618$	−25.3490	−1.01969
$619$	−0.629346	−0.0252956	−0.0126478	−	0.999920i	$-0.504026\pi$
$619$	−0.629346	−0.0252956	−0.0126478	+	0.999920i	$0.504026\pi$
$620$	3.58891	0.144134
$621$	6.77691	0.271948
$622$	66.8989	2.68240
$623$	0	0
$624$	−6.57223	−0.263100
$625$	−2.83394	−0.113358
$626$	−25.3692	−1.01396
$627$	−0.0655079	−0.00261613
$628$	−17.8215	−0.711154
$629$	−6.44427	−0.256950
$630$	0	0
$631$	32.9134	1.31026	0.655132	−	0.755515i	$-0.272613\pi$
$631$	32.9134	1.31026	0.655132	+	0.755515i	$0.272613\pi$
$632$	13.7679	0.547657
$633$	−12.5702	−0.499622
$634$	29.4204	1.16843
$635$	14.5359	0.576841
$636$	9.33358	0.370101
$637$	0	0
$638$	19.3341	0.765446
$639$	11.9985	0.474654
$640$	−14.3598	−0.567623
$641$	28.6986	1.13353	0.566764	−	0.823880i	$-0.308195\pi$
$641$	28.6986	1.13353	0.566764	+	0.823880i	$0.308195\pi$
$642$	40.1640	1.58515
$643$	9.65476	0.380747	0.190373	−	0.981712i	$-0.439030\pi$
$643$	9.65476	0.380747	0.190373	+	0.981712i	$0.439030\pi$
$644$	0	0
$645$	5.50852	0.216898
$646$	−0.297765	−0.0117154
$647$	21.5991	0.849149	0.424574	−	0.905393i	$-0.360424\pi$
$647$	21.5991	0.849149	0.424574	+	0.905393i	$0.360424\pi$
$648$	1.27510	0.0500906
$649$	−21.2528	−0.834245
$650$	16.2342	0.636757
$651$	0	0
$652$	−23.8128	−0.932582
$653$	3.35730	0.131381	0.0656907	−	0.997840i	$-0.479075\pi$
$653$	3.35730	0.131381	0.0656907	+	0.997840i	$0.479075\pi$
$654$	2.60343	0.101802
$655$	−19.3571	−0.756346
$656$	−2.45645	−0.0959082
$657$	−8.42842	−0.328824
$658$	0	0
$659$	5.61536	0.218743	0.109372	−	0.994001i	$-0.465116\pi$
$659$	5.61536	0.218743	0.109372	+	0.994001i	$0.465116\pi$
$660$	7.91349	0.308032
$661$	−5.28225	−0.205456	−0.102728	−	0.994709i	$-0.532757\pi$
$661$	−5.28225	−0.205456	−0.102728	+	0.994709i	$0.532757\pi$
$662$	55.4562	2.15537
$663$	11.7474	0.456232
$664$	15.9202	0.617825
$665$	0	0
$666$	−3.14609	−0.121909
$667$	29.5208	1.14305
$668$	38.2204	1.47879
$669$	−0.0274795	−0.00106242
$670$	−18.7058	−0.722668
$671$	4.17324	0.161106
$672$	0	0
$673$	−10.5427	−0.406392	−0.203196	−	0.979138i	$-0.565133\pi$
$673$	−10.5427	−0.406392	−0.203196	+	0.979138i	$0.565133\pi$
$674$	−14.4620	−0.557055
$675$	2.83067	0.108953
$676$	−15.1583	−0.583013
$677$	42.2570	1.62407	0.812034	−	0.583610i	$-0.198360\pi$
$677$	42.2570	1.62407	0.812034	+	0.583610i	$0.198360\pi$
$678$	17.0345	0.654206
$679$	0	0
$680$	8.24602	0.316220
$681$	−20.5039	−0.785711
$682$	−4.16789	−0.159597
$683$	48.7774	1.86641	0.933207	−	0.359339i	$-0.116998\pi$
$683$	48.7774	1.86641	0.933207	+	0.359339i	$0.116998\pi$
$684$	−0.0820943	−0.00313895
$685$	−16.9163	−0.646340
$686$	0	0
$687$	8.45982	0.322762
$688$	9.18713	0.350256
$689$	9.62367	0.366633
$690$	21.3959	0.814527
$691$	−24.5356	−0.933377	−0.466688	−	0.884422i	$-0.654553\pi$
$691$	−24.5356	−0.933377	−0.466688	+	0.884422i	$0.654553\pi$
$692$	−1.03811	−0.0394631
$693$	0	0
$694$	−64.4966	−2.44826
$695$	−14.9099	−0.565565
$696$	5.55444	0.210540
$697$	4.39074	0.166311
$698$	−14.3729	−0.544022
$699$	−20.3483	−0.769642
$700$	0	0
$701$	30.2324	1.14186	0.570932	−	0.820997i	$-0.306582\pi$
$701$	30.2324	1.14186	0.570932	+	0.820997i	$0.306582\pi$
$702$	5.73510	0.216457
$703$	0.0464340	0.00175129
$704$	24.5170	0.924020
$705$	16.5323	0.622644
$706$	−23.3513	−0.878839
$707$	0	0
$708$	−26.6339	−1.00096
$709$	−18.5506	−0.696682	−0.348341	−	0.937368i	$-0.613255\pi$
$709$	−18.5506	−0.696682	−0.348341	+	0.937368i	$0.613255\pi$
$710$	37.8814	1.42166
$711$	10.7975	0.404937
$712$	−5.86584	−0.219831
$713$	−6.36383	−0.238327
$714$	0	0
$715$	8.15945	0.305146
$716$	1.43791	0.0537371
$717$	−12.2888	−0.458935
$718$	17.1830	0.641264
$719$	−13.8631	−0.517005	−0.258502	−	0.966011i	$-0.583229\pi$
$719$	−13.8631	−0.517005	−0.258502	+	0.966011i	$0.583229\pi$
$720$	−3.61802	−0.134835
$721$	0	0
$722$	−40.7255	−1.51565
$723$	20.9865	0.780495
$724$	51.4922	1.91369
$725$	12.3306	0.457948
$726$	14.3890	0.534027
$727$	46.3572	1.71929	0.859647	−	0.510888i	$-0.170683\pi$
$727$	46.3572	1.71929	0.859647	+	0.510888i	$0.170683\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−26.6100	−0.984880
$731$	−16.4214	−0.607366
$732$	5.22990	0.193303
$733$	51.1163	1.88802	0.944011	−	0.329913i	$-0.107019\pi$
$733$	51.1163	1.88802	0.944011	+	0.329913i	$0.107019\pi$
$734$	−13.8357	−0.510684
$735$	0	0
$736$	52.9666	1.95238
$737$	12.2679	0.451895
$738$	2.14356	0.0789055
$739$	9.53293	0.350674	0.175337	−	0.984508i	$-0.443898\pi$
$739$	9.53293	0.350674	0.175337	+	0.984508i	$0.443898\pi$
$740$	−5.60933	−0.206203
$741$	−0.0846458	−0.00310954
$742$	0	0
$743$	−46.1167	−1.69186	−0.845929	−	0.533295i	$-0.820954\pi$
$743$	−46.1167	−1.69186	−0.845929	+	0.533295i	$0.820954\pi$
$744$	−1.19738	−0.0438980
$745$	−10.5147	−0.385229
$746$	51.3053	1.87842
$747$	12.4855	0.456820
$748$	−23.5908	−0.862566
$749$	0	0
$750$	24.7228	0.902748
$751$	−10.1242	−0.369437	−0.184718	−	0.982791i	$-0.559137\pi$
$751$	−10.1242	−0.369437	−0.184718	+	0.982791i	$0.559137\pi$
$752$	27.5727	1.00547
$753$	1.11964	0.0408020
$754$	24.9825	0.909811
$755$	20.8593	0.759149
$756$	0	0
$757$	10.7627	0.391178	0.195589	−	0.980686i	$-0.437338\pi$
$757$	10.7627	0.391178	0.195589	+	0.980686i	$0.437338\pi$
$758$	21.1781	0.769224
$759$	−14.0322	−0.509336
$760$	−0.0594165	−0.00215526
$761$	−34.5805	−1.25354	−0.626771	−	0.779204i	$-0.715624\pi$
$761$	−34.5805	−1.25354	−0.626771	+	0.779204i	$0.715624\pi$
$762$	−21.1552	−0.766371
$763$	0	0
$764$	0.847231	0.0306517
$765$	6.46696	0.233813
$766$	−60.6517	−2.19143
$767$	−27.4617	−0.991586
$768$	−2.78239	−0.100401
$769$	−16.4163	−0.591986	−0.295993	−	0.955190i	$-0.595650\pi$
$769$	−16.4163	−0.591986	−0.295993	+	0.955190i	$0.595650\pi$
$770$	0	0
$771$	2.71524	0.0977869
$772$	−20.7381	−0.746380
$773$	20.4777	0.736531	0.368265	−	0.929721i	$-0.379952\pi$
$773$	20.4777	0.736531	0.368265	+	0.929721i	$0.379952\pi$
$774$	−8.01692	−0.288162
$775$	−2.65813	−0.0954829
$776$	−4.96723	−0.178313
$777$	0	0
$778$	−18.6218	−0.667624
$779$	−0.0316374	−0.00113353
$780$	10.2254	0.366128
$781$	−24.8440	−0.888987
$782$	−63.7830	−2.28088
$783$	4.35608	0.155674
$784$	0	0
$785$	−10.1156	−0.361043
$786$	28.1718	1.00485
$787$	−25.7917	−0.919374	−0.459687	−	0.888081i	$-0.652038\pi$
$787$	−25.7917	−0.919374	−0.459687	+	0.888081i	$0.652038\pi$
$788$	−12.2493	−0.436362
$789$	25.1848	0.896604
$790$	34.0895	1.21285
$791$	0	0
$792$	−2.64020	−0.0938155
$793$	5.39245	0.191491
$794$	−32.0846	−1.13864
$795$	5.29784	0.187895
$796$	2.85508	0.101196
$797$	−43.0082	−1.52343	−0.761715	−	0.647913i	$-0.775642\pi$
$797$	−43.0082	−1.52343	−0.761715	+	0.647913i	$0.775642\pi$
$798$	0	0
$799$	−49.2844	−1.74356
$800$	22.1238	0.782195
$801$	−4.60030	−0.162543
$802$	1.92864	0.0681027
$803$	17.4518	0.615860
$804$	15.3741	0.542203
$805$	0	0
$806$	−5.38552	−0.189697
$807$	−13.1304	−0.462212
$808$	−4.17191	−0.146767
$809$	30.2475	1.06344	0.531722	−	0.846919i	$-0.321545\pi$
$809$	30.2475	1.06344	0.531722	+	0.846919i	$0.321545\pi$
$810$	3.15717	0.110932
$811$	20.6281	0.724350	0.362175	−	0.932110i	$-0.382034\pi$
$811$	20.6281	0.724350	0.362175	+	0.932110i	$0.382034\pi$
$812$	0	0
$813$	−12.4972	−0.438297
$814$	6.51426	0.228325
$815$	−13.5164	−0.473459
$816$	10.7856	0.377572
$817$	0.118324	0.00413963
$818$	−11.2310	−0.392684
$819$	0	0
$820$	3.82186	0.133465
$821$	−39.3147	−1.37209	−0.686046	−	0.727558i	$-0.740655\pi$
$821$	−39.3147	−1.37209	−0.686046	+	0.727558i	$0.740655\pi$
$822$	24.6195	0.858704
$823$	−37.5494	−1.30889	−0.654446	−	0.756109i	$-0.727098\pi$
$823$	−37.5494	−1.30889	−0.654446	+	0.756109i	$0.727098\pi$
$824$	15.0789	0.525298
$825$	−5.86115	−0.204059
$826$	0	0
$827$	−11.2269	−0.390398	−0.195199	−	0.980764i	$-0.562535\pi$
$827$	−11.2269	−0.390398	−0.195199	+	0.980764i	$0.562535\pi$
$828$	−17.5851	−0.611124
$829$	4.96242	0.172352	0.0861759	−	0.996280i	$-0.472535\pi$
$829$	4.96242	0.172352	0.0861759	+	0.996280i	$0.472535\pi$
$830$	39.4188	1.36825
$831$	20.1516	0.699053
$832$	31.6796	1.09829
$833$	0	0
$834$	21.6994	0.751390
$835$	21.6943	0.750763
$836$	0.169983	0.00587899
$837$	−0.939046	−0.0324582
$838$	−28.2189	−0.974806
$839$	31.8342	1.09904	0.549519	−	0.835481i	$-0.314811\pi$
$839$	31.8342	1.09904	0.549519	+	0.835481i	$0.314811\pi$
$840$	0	0
$841$	−10.0246	−0.345675
$842$	−27.7475	−0.956243
$843$	−11.9933	−0.413071
$844$	32.6179	1.12275
$845$	−8.60402	−0.295987
$846$	−24.0607	−0.827222
$847$	0	0
$848$	8.83576	0.303421
$849$	−16.3923	−0.562583
$850$	−26.6417	−0.913804
$851$	9.94645	0.340960
$852$	−31.1344	−1.06665
$853$	−0.854054	−0.0292423	−0.0146211	−	0.999893i	$-0.504654\pi$
$853$	−0.854054	−0.0292423	−0.0146211	+	0.999893i	$0.504654\pi$
$854$	0	0
$855$	−0.0465976	−0.00159360
$856$	−23.8916	−0.816598
$857$	−28.2228	−0.964072	−0.482036	−	0.876152i	$-0.660102\pi$
$857$	−28.2228	−0.964072	−0.482036	+	0.876152i	$0.660102\pi$
$858$	−11.8750	−0.405406
$859$	27.5370	0.939551	0.469775	−	0.882786i	$-0.344335\pi$
$859$	27.5370	0.939551	0.469775	+	0.882786i	$0.344335\pi$
$860$	−14.2938	−0.487414
$861$	0	0
$862$	−15.1019	−0.514372
$863$	−3.18660	−0.108473	−0.0542366	−	0.998528i	$-0.517273\pi$
$863$	−3.18660	−0.108473	−0.0542366	+	0.998528i	$0.517273\pi$
$864$	7.81575	0.265897
$865$	−0.589244	−0.0200349
$866$	8.92025	0.303123
$867$	−2.27857	−0.0773842
$868$	0	0
$869$	−22.3571	−0.758413
$870$	13.7529	0.466267
$871$	15.8520	0.537123
$872$	−1.54865	−0.0524439
$873$	−3.89557	−0.131845
$874$	0.459587	0.0155458
$875$	0	0
$876$	21.8705	0.738936
$877$	1.10371	0.0372698	0.0186349	−	0.999826i	$-0.494068\pi$
$877$	1.10371	0.0372698	0.0186349	+	0.999826i	$0.494068\pi$
$878$	−25.5789	−0.863245
$879$	5.82846	0.196589
$880$	7.49141	0.252535
$881$	24.4951	0.825259	0.412630	−	0.910899i	$-0.364610\pi$
$881$	24.4951	0.825259	0.412630	+	0.910899i	$0.364610\pi$
$882$	0	0
$883$	22.7712	0.766312	0.383156	−	0.923684i	$-0.374837\pi$
$883$	22.7712	0.766312	0.383156	+	0.923684i	$0.374837\pi$
$884$	−30.4828	−1.02525
$885$	−15.1177	−0.508176
$886$	18.5855	0.624393
$887$	16.8832	0.566881	0.283440	−	0.958990i	$-0.408524\pi$
$887$	16.8832	0.566881	0.283440	+	0.958990i	$0.408524\pi$
$888$	1.87146	0.0628020
$889$	0	0
$890$	−14.5239	−0.486843
$891$	−2.07059	−0.0693672
$892$	0.0713051	0.00238747
$893$	0.355118	0.0118836
$894$	15.3028	0.511802
$895$	0.816170	0.0272816
$896$	0	0
$897$	−18.1316	−0.605398
$898$	74.2349	2.47725
$899$	−4.09056	−0.136428
$900$	−7.34517	−0.244839
$901$	−15.7933	−0.526152
$902$	−4.43842	−0.147783
$903$	0	0
$904$	−10.1330	−0.337018
$905$	29.2275	0.971554
$906$	−30.3580	−1.00858
$907$	−24.6836	−0.819607	−0.409803	−	0.912174i	$-0.634403\pi$
$907$	−24.6836	−0.819607	−0.409803	+	0.912174i	$0.634403\pi$
$908$	53.2046	1.76566
$909$	−3.27183	−0.108520
$910$	0	0
$911$	40.0305	1.32627	0.663135	−	0.748500i	$-0.269226\pi$
$911$	40.0305	1.32627	0.663135	+	0.748500i	$0.269226\pi$
$912$	−0.0777156	−0.00257342
$913$	−25.8522	−0.855584
$914$	−38.0475	−1.25850
$915$	2.96854	0.0981370
$916$	−21.9520	−0.725313
$917$	0	0
$918$	−9.41181	−0.310636
$919$	−34.7312	−1.14568	−0.572838	−	0.819668i	$-0.694158\pi$
$919$	−34.7312	−1.14568	−0.572838	+	0.819668i	$0.694158\pi$
$920$	−12.7274	−0.419609
$921$	11.0203	0.363132
$922$	23.8137	0.784264
$923$	−32.1020	−1.05665
$924$	0	0
$925$	4.15456	0.136601
$926$	63.6202	2.09069
$927$	11.8257	0.388406
$928$	34.0460	1.11762
$929$	53.4159	1.75252	0.876259	−	0.481841i	$-0.160032\pi$
$929$	53.4159	1.75252	0.876259	+	0.481841i	$0.160032\pi$
$930$	−2.96473	−0.0972174
$931$	0	0
$932$	52.8007	1.72954
$933$	−31.2092	−1.02174
$934$	−65.3891	−2.13960
$935$	−13.3904	−0.437913
$936$	−3.41153	−0.111509
$937$	−4.71973	−0.154187	−0.0770934	−	0.997024i	$-0.524564\pi$
$937$	−4.71973	−0.154187	−0.0770934	+	0.997024i	$0.524564\pi$
$938$	0	0
$939$	11.8351	0.386223
$940$	−42.8990	−1.39921
$941$	31.4114	1.02398	0.511991	−	0.858991i	$-0.328908\pi$
$941$	31.4114	1.02398	0.511991	+	0.858991i	$0.328908\pi$
$942$	14.7220	0.479669
$943$	−6.77691	−0.220687
$944$	−25.2134	−0.820625
$945$	0	0
$946$	16.5997	0.539704
$947$	−15.3051	−0.497348	−0.248674	−	0.968587i	$-0.579995\pi$
$947$	−15.3051	−0.497348	−0.248674	+	0.968587i	$0.579995\pi$
$948$	−28.0179	−0.909978
$949$	22.5503	0.732012
$950$	0.191967	0.00622822
$951$	−13.7250	−0.445064
$952$	0	0
$953$	33.8151	1.09538	0.547690	−	0.836681i	$-0.315507\pi$
$953$	33.8151	1.09538	0.547690	+	0.836681i	$0.315507\pi$
$954$	−7.71031	−0.249630
$955$	0.480897	0.0155615
$956$	31.8877	1.03132
$957$	−9.01964	−0.291563
$958$	−56.9382	−1.83959
$959$	0	0
$960$	17.4396	0.562862
$961$	−30.1182	−0.971555
$962$	8.41738	0.271387
$963$	−18.7370	−0.603792
$964$	−54.4567	−1.75393
$965$	−11.7712	−0.378927
$966$	0	0
$967$	25.4310	0.817806	0.408903	−	0.912578i	$-0.365912\pi$
$967$	25.4310	0.817806	0.408903	+	0.912578i	$0.365912\pi$
$968$	−8.55933	−0.275107
$969$	0.138911	0.00446248
$970$	−12.2990	−0.394896
$971$	−2.82296	−0.0905929	−0.0452965	−	0.998974i	$-0.514423\pi$
$971$	−2.82296	−0.0905929	−0.0452965	+	0.998974i	$0.514423\pi$
$972$	−2.59485	−0.0832299
$973$	0	0
$974$	24.5786	0.787549
$975$	−7.57346	−0.242545
$976$	4.95095	0.158476
$977$	32.3383	1.03459	0.517297	−	0.855806i	$-0.326938\pi$
$977$	32.3383	1.03459	0.517297	+	0.855806i	$0.326938\pi$
$978$	19.6713	0.629020
$979$	9.52531	0.304430
$980$	0	0
$981$	−1.21453	−0.0387771
$982$	−27.4161	−0.874882
$983$	44.5912	1.42224	0.711120	−	0.703071i	$-0.248188\pi$
$983$	44.5912	1.42224	0.711120	+	0.703071i	$0.248188\pi$
$984$	−1.27510	−0.0406487
$985$	−6.95281	−0.221535
$986$	−40.9986	−1.30566
$987$	0	0
$988$	0.219643	0.00698779
$989$	25.3457	0.805945
$990$	−6.53720	−0.207766
$991$	−29.5814	−0.939683	−0.469842	−	0.882751i	$-0.655689\pi$
$991$	−29.5814	−0.939683	−0.469842	+	0.882751i	$0.655689\pi$
$992$	−7.33935	−0.233024
$993$	−25.8711	−0.820994
$994$	0	0
$995$	1.62057	0.0513756
$996$	−32.3980	−1.02657
$997$	−18.4768	−0.585167	−0.292583	−	0.956240i	$-0.594515\pi$
$997$	−18.4768	−0.585167	−0.292583	+	0.956240i	$0.594515\pi$
$998$	−17.7080	−0.560538
$999$	1.46770	0.0464358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.z.1.8	✓	8
7.6	odd	2		6027.2.a.ba.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.z.1.8	✓	8	1.1	even	1	trivial
6027.2.a.ba.1.8	yes	8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q+2.14356 q^{2} -1.00000 q^{3} +2.59485 q^{4} +1.47286 q^{5} -2.14356 q^{6} +1.27510 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.14356 q^{2} -1.00000 q^{3} +2.59485 q^{4} +1.47286 q^{5} -2.14356 q^{6} +1.27510 q^{8} +1.00000 q^{9} +3.15717 q^{10} -2.07059 q^{11} -2.59485 q^{12} -2.67550 q^{13} -1.47286 q^{15} -2.45645 q^{16} +4.39074 q^{17} +2.14356 q^{18} -0.0316374 q^{19} +3.82186 q^{20} -4.43842 q^{22} -6.77691 q^{23} -1.27510 q^{24} -2.83067 q^{25} -5.73510 q^{26} -1.00000 q^{27} -4.35608 q^{29} -3.15717 q^{30} +0.939046 q^{31} -7.81575 q^{32} +2.07059 q^{33} +9.41181 q^{34} +2.59485 q^{36} -1.46770 q^{37} -0.0678166 q^{38} +2.67550 q^{39} +1.87805 q^{40} +1.00000 q^{41} -3.74000 q^{43} -5.37286 q^{44} +1.47286 q^{45} -14.5267 q^{46} -11.2246 q^{47} +2.45645 q^{48} -6.06772 q^{50} -4.39074 q^{51} -6.94253 q^{52} -3.59696 q^{53} -2.14356 q^{54} -3.04969 q^{55} +0.0316374 q^{57} -9.33752 q^{58} +10.2641 q^{59} -3.82186 q^{60} -2.01549 q^{61} +2.01290 q^{62} -11.8406 q^{64} -3.94065 q^{65} +4.43842 q^{66} -5.92486 q^{67} +11.3933 q^{68} +6.77691 q^{69} +11.9985 q^{71} +1.27510 q^{72} -8.42842 q^{73} -3.14609 q^{74} +2.83067 q^{75} -0.0820943 q^{76} +5.73510 q^{78} +10.7975 q^{79} -3.61802 q^{80} +1.00000 q^{81} +2.14356 q^{82} +12.4855 q^{83} +6.46696 q^{85} -8.01692 q^{86} +4.35608 q^{87} -2.64020 q^{88} -4.60030 q^{89} +3.15717 q^{90} -17.5851 q^{92} -0.939046 q^{93} -24.0607 q^{94} -0.0465976 q^{95} +7.81575 q^{96} -3.89557 q^{97} -2.07059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 8 * q^9	\( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 4 q^{13} - 2 q^{15} + 8 q^{17} - 2 q^{18} + 6 q^{19} + 4 q^{20} - 14 q^{22} - 12 q^{23} + 6 q^{24} - 4 q^{25} + 4 q^{26} - 8 q^{27} - 4 q^{29} + 2 q^{30} - 10 q^{31} - 4 q^{32} + 2 q^{33} + 4 q^{34} + 4 q^{36} - 20 q^{37} - 18 q^{38} - 4 q^{39} + 12 q^{40} + 8 q^{41} - 8 q^{43} + 20 q^{44} + 2 q^{45} - 12 q^{46} + 24 q^{47} - 22 q^{50} - 8 q^{51} - 30 q^{52} - 36 q^{53} + 2 q^{54} + 4 q^{55} - 6 q^{57} + 14 q^{58} + 10 q^{59} - 4 q^{60} - 22 q^{61} + 30 q^{62} - 24 q^{64} + 8 q^{65} + 14 q^{66} - 14 q^{67} + 38 q^{68} + 12 q^{69} - 10 q^{71} - 6 q^{72} - 12 q^{73} - 2 q^{74} + 4 q^{75} + 32 q^{76} - 4 q^{78} + 16 q^{79} - 14 q^{80} + 8 q^{81} - 2 q^{82} + 24 q^{83} - 44 q^{85} + 36 q^{86} + 4 q^{87} - 34 q^{88} + 2 q^{89} - 2 q^{90} - 48 q^{92} + 10 q^{93} - 34 q^{94} - 24 q^{95} + 4 q^{96} + 16 q^{97} - 2 q^{99}+O(q^{100}) \) 8 * q - 2 * q^2 - 8 * q^3 + 4 * q^4 + 2 * q^5 + 2 * q^6 - 6 * q^8 + 8 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 4 * q^13 - 2 * q^15 + 8 * q^17 - 2 * q^18 + 6 * q^19 + 4 * q^20 - 14 * q^22 - 12 * q^23 + 6 * q^24 - 4 * q^25 + 4 * q^26 - 8 * q^27 - 4 * q^29 + 2 * q^30 - 10 * q^31 - 4 * q^32 + 2 * q^33 + 4 * q^34 + 4 * q^36 - 20 * q^37 - 18 * q^38 - 4 * q^39 + 12 * q^40 + 8 * q^41 - 8 * q^43 + 20 * q^44 + 2 * q^45 - 12 * q^46 + 24 * q^47 - 22 * q^50 - 8 * q^51 - 30 * q^52 - 36 * q^53 + 2 * q^54 + 4 * q^55 - 6 * q^57 + 14 * q^58 + 10 * q^59 - 4 * q^60 - 22 * q^61 + 30 * q^62 - 24 * q^64 + 8 * q^65 + 14 * q^66 - 14 * q^67 + 38 * q^68 + 12 * q^69 - 10 * q^71 - 6 * q^72 - 12 * q^73 - 2 * q^74 + 4 * q^75 + 32 * q^76 - 4 * q^78 + 16 * q^79 - 14 * q^80 + 8 * q^81 - 2 * q^82 + 24 * q^83 - 44 * q^85 + 36 * q^86 + 4 * q^87 - 34 * q^88 + 2 * q^89 - 2 * q^90 - 48 * q^92 + 10 * q^93 - 34 * q^94 - 24 * q^95 + 4 * q^96 + 16 * q^97 - 2 * q^99

Introduction

L-functions

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