LMFDB - Embedded newform 6027.2.a.z.1.4

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			1.4
Root			$0.487949$ 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)

$f(q)$	$=$	$q-0.487949 q^{2} -1.00000 q^{3} -1.76191 q^{4} -1.53225 q^{5} +0.487949 q^{6} +1.83562 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.487949 q^{2} -1.00000 q^{3} -1.76191 q^{4} -1.53225 q^{5} +0.487949 q^{6} +1.83562 q^{8} +1.00000 q^{9} +0.747657 q^{10} -3.51053 q^{11} +1.76191 q^{12} -2.32735 q^{13} +1.53225 q^{15} +2.62812 q^{16} +4.08881 q^{17} -0.487949 q^{18} -1.32044 q^{19} +2.69967 q^{20} +1.71296 q^{22} +1.38676 q^{23} -1.83562 q^{24} -2.65223 q^{25} +1.13563 q^{26} -1.00000 q^{27} -5.15628 q^{29} -0.747657 q^{30} -0.813042 q^{31} -4.95363 q^{32} +3.51053 q^{33} -1.99513 q^{34} -1.76191 q^{36} +10.0334 q^{37} +0.644308 q^{38} +2.32735 q^{39} -2.81262 q^{40} +1.00000 q^{41} -2.79727 q^{43} +6.18521 q^{44} -1.53225 q^{45} -0.676670 q^{46} +10.6010 q^{47} -2.62812 q^{48} +1.29415 q^{50} -4.08881 q^{51} +4.10057 q^{52} -8.05270 q^{53} +0.487949 q^{54} +5.37898 q^{55} +1.32044 q^{57} +2.51600 q^{58} -0.693332 q^{59} -2.69967 q^{60} +1.29889 q^{61} +0.396723 q^{62} -2.83913 q^{64} +3.56607 q^{65} -1.71296 q^{66} +8.82262 q^{67} -7.20409 q^{68} -1.38676 q^{69} +7.16765 q^{71} +1.83562 q^{72} +1.53683 q^{73} -4.89578 q^{74} +2.65223 q^{75} +2.32649 q^{76} -1.13563 q^{78} -5.48031 q^{79} -4.02693 q^{80} +1.00000 q^{81} -0.487949 q^{82} +5.90345 q^{83} -6.26506 q^{85} +1.36492 q^{86} +5.15628 q^{87} -6.44398 q^{88} +12.6977 q^{89} +0.747657 q^{90} -2.44335 q^{92} +0.813042 q^{93} -5.17277 q^{94} +2.02324 q^{95} +4.95363 q^{96} +14.2954 q^{97} -3.51053 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.487949	−0.345032	−0.172516	−	0.985007i	$-0.555190\pi$
$2$	−0.487949	−0.345032	−0.172516	+	0.985007i	$0.555190\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.76191	−0.880953
$5$	−1.53225	−0.685241	−0.342620	−	0.939474i	$-0.611314\pi$
$5$	−1.53225	−0.685241	−0.342620	+	0.939474i	$0.611314\pi$
$6$	0.487949	0.199204
$7$	0	0
$7$	0	0
$8$	1.83562	0.648989
$9$	1.00000	0.333333
$10$	0.747657	0.236430
$11$	−3.51053	−1.05846	−0.529232	−	0.848477i	$-0.677520\pi$
$11$	−3.51053	−1.05846	−0.529232	+	0.848477i	$0.677520\pi$
$12$	1.76191	0.508618
$13$	−2.32735	−0.645490	−0.322745	−	0.946486i	$-0.604606\pi$
$13$	−2.32735	−0.645490	−0.322745	+	0.946486i	$0.604606\pi$
$14$	0	0
$15$	1.53225	0.395624
$16$	2.62812	0.657031
$17$	4.08881	0.991682	0.495841	−	0.868413i	$-0.334860\pi$
$17$	4.08881	0.991682	0.495841	+	0.868413i	$0.334860\pi$
$18$	−0.487949	−0.115011
$19$	−1.32044	−0.302930	−0.151465	−	0.988463i	$-0.548399\pi$
$19$	−1.32044	−0.302930	−0.151465	+	0.988463i	$0.548399\pi$
$20$	2.69967	0.603665
$21$	0	0
$22$	1.71296	0.365204
$23$	1.38676	0.289160	0.144580	−	0.989493i	$-0.453817\pi$
$23$	1.38676	0.289160	0.144580	+	0.989493i	$0.453817\pi$
$24$	−1.83562	−0.374694
$25$	−2.65223	−0.530445
$26$	1.13563	0.222715
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.15628	−0.957496	−0.478748	−	0.877952i	$-0.658909\pi$
$29$	−5.15628	−0.957496	−0.478748	+	0.877952i	$0.658909\pi$
$30$	−0.747657	−0.136503
$31$	−0.813042	−0.146027	−0.0730133	−	0.997331i	$-0.523262\pi$
$31$	−0.813042	−0.146027	−0.0730133	+	0.997331i	$0.523262\pi$
$32$	−4.95363	−0.875686
$33$	3.51053	0.611104
$34$	−1.99513	−0.342162
$35$	0	0
$36$	−1.76191	−0.293651
$37$	10.0334	1.64948	0.824739	−	0.565514i	$-0.191322\pi$
$37$	10.0334	1.64948	0.824739	+	0.565514i	$0.191322\pi$
$38$	0.644308	0.104521
$39$	2.32735	0.372674
$40$	−2.81262	−0.444714
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−2.79727	−0.426579	−0.213290	−	0.976989i	$-0.568418\pi$
$43$	−2.79727	−0.426579	−0.213290	+	0.976989i	$0.568418\pi$
$44$	6.18521	0.932456
$45$	−1.53225	−0.228414
$46$	−0.676670	−0.0997696
$47$	10.6010	1.54632	0.773160	−	0.634211i	$-0.218675\pi$
$47$	10.6010	1.54632	0.773160	+	0.634211i	$0.218675\pi$
$48$	−2.62812	−0.379337
$49$	0	0
$50$	1.29415	0.183021
$51$	−4.08881	−0.572548
$52$	4.10057	0.568647
$53$	−8.05270	−1.10612	−0.553062	−	0.833140i	$-0.686541\pi$
$53$	−8.05270	−1.10612	−0.553062	+	0.833140i	$0.686541\pi$
$54$	0.487949	0.0664015
$55$	5.37898	0.725302
$56$	0	0
$57$	1.32044	0.174897
$58$	2.51600	0.330367
$59$	−0.693332	−0.0902641	−0.0451321	−	0.998981i	$-0.514371\pi$
$59$	−0.693332	−0.0902641	−0.0451321	+	0.998981i	$0.514371\pi$
$60$	−2.69967	−0.348526
$61$	1.29889	0.166306	0.0831532	−	0.996537i	$-0.473501\pi$
$61$	1.29889	0.166306	0.0831532	+	0.996537i	$0.473501\pi$
$62$	0.396723	0.0503839
$63$	0	0
$64$	−2.83913	−0.354891
$65$	3.56607	0.442316
$66$	−1.71296	−0.210850
$67$	8.82262	1.07785	0.538927	−	0.842352i	$-0.318830\pi$
$67$	8.82262	1.07785	0.538927	+	0.842352i	$0.318830\pi$
$68$	−7.20409	−0.873625
$69$	−1.38676	−0.166947
$70$	0	0
$71$	7.16765	0.850644	0.425322	−	0.905042i	$-0.360161\pi$
$71$	7.16765	0.850644	0.425322	+	0.905042i	$0.360161\pi$
$72$	1.83562	0.216330
$73$	1.53683	0.179872	0.0899360	−	0.995948i	$-0.471334\pi$
$73$	1.53683	0.179872	0.0899360	+	0.995948i	$0.471334\pi$
$74$	−4.89578	−0.569123
$75$	2.65223	0.306253
$76$	2.32649	0.266867
$77$	0	0
$78$	−1.13563	−0.128584
$79$	−5.48031	−0.616583	−0.308292	−	0.951292i	$-0.599757\pi$
$79$	−5.48031	−0.616583	−0.308292	+	0.951292i	$0.599757\pi$
$80$	−4.02693	−0.450224
$81$	1.00000	0.111111
$82$	−0.487949	−0.0538850
$83$	5.90345	0.647987	0.323994	−	0.946059i	$-0.394974\pi$
$83$	5.90345	0.647987	0.323994	+	0.946059i	$0.394974\pi$
$84$	0	0
$85$	−6.26506	−0.679541
$86$	1.36492	0.147184
$87$	5.15628	0.552811
$88$	−6.44398	−0.686931
$89$	12.6977	1.34595	0.672977	−	0.739664i	$-0.265015\pi$
$89$	12.6977	1.34595	0.672977	+	0.739664i	$0.265015\pi$
$90$	0.747657	0.0788100
$91$	0	0
$92$	−2.44335	−0.254737
$93$	0.813042	0.0843085
$94$	−5.17277	−0.533530
$95$	2.02324	0.207580
$96$	4.95363	0.505577
$97$	14.2954	1.45148	0.725740	−	0.687969i	$-0.241498\pi$
$97$	14.2954	1.45148	0.725740	+	0.687969i	$0.241498\pi$
$98$	0	0
$99$	−3.51053	−0.352821
$100$	4.67297	0.467297
$101$	−9.76287	−0.971442	−0.485721	−	0.874114i	$-0.661443\pi$
$101$	−9.76287	−0.971442	−0.485721	+	0.874114i	$0.661443\pi$
$102$	1.99513	0.197547
$103$	6.73133	0.663257	0.331629	−	0.943410i	$-0.392402\pi$
$103$	6.73133	0.663257	0.331629	+	0.943410i	$0.392402\pi$
$104$	−4.27212	−0.418916
$105$	0	0
$106$	3.92931	0.381648
$107$	3.40497	0.329171	0.164585	−	0.986363i	$-0.447371\pi$
$107$	3.40497	0.329171	0.164585	+	0.986363i	$0.447371\pi$
$108$	1.76191	0.169539
$109$	2.90920	0.278651	0.139326	−	0.990247i	$-0.455507\pi$
$109$	2.90920	0.278651	0.139326	+	0.990247i	$0.455507\pi$
$110$	−2.62467	−0.250253
$111$	−10.0334	−0.952326
$112$	0	0
$113$	3.89937	0.366822	0.183411	−	0.983036i	$-0.441286\pi$
$113$	3.89937	0.366822	0.183411	+	0.983036i	$0.441286\pi$
$114$	−0.644308	−0.0603449
$115$	−2.12486	−0.198144
$116$	9.08487	0.843509
$117$	−2.32735	−0.215163
$118$	0.338311	0.0311440
$119$	0	0
$120$	2.81262	0.256756
$121$	1.32379	0.120344
$122$	−0.633794	−0.0573810
$123$	−1.00000	−0.0901670
$124$	1.43250	0.128643
$125$	11.7251	1.04872
$126$	0	0
$127$	−5.42195	−0.481120	−0.240560	−	0.970634i	$-0.577331\pi$
$127$	−5.42195	−0.481120	−0.240560	+	0.970634i	$0.577331\pi$
$128$	11.2926	0.998135
$129$	2.79727	0.246286
$130$	−1.74006	−0.152613
$131$	3.23920	0.283010	0.141505	−	0.989938i	$-0.454806\pi$
$131$	3.23920	0.283010	0.141505	+	0.989938i	$0.454806\pi$
$132$	−6.18521	−0.538354
$133$	0	0
$134$	−4.30499	−0.371894
$135$	1.53225	0.131875
$136$	7.50549	0.643591
$137$	−7.01858	−0.599638	−0.299819	−	0.953996i	$-0.596926\pi$
$137$	−7.01858	−0.599638	−0.299819	+	0.953996i	$0.596926\pi$
$138$	0.676670	0.0576020
$139$	−14.7100	−1.24769	−0.623843	−	0.781550i	$-0.714430\pi$
$139$	−14.7100	−1.24769	−0.623843	+	0.781550i	$0.714430\pi$
$140$	0	0
$141$	−10.6010	−0.892768
$142$	−3.49745	−0.293499
$143$	8.17022	0.683228
$144$	2.62812	0.219010
$145$	7.90068	0.656116
$146$	−0.749893	−0.0620616
$147$	0	0
$148$	−17.6779	−1.45311
$149$	9.98658	0.818132	0.409066	−	0.912505i	$-0.365855\pi$
$149$	9.98658	0.818132	0.409066	+	0.912505i	$0.365855\pi$
$150$	−1.29415	−0.105667
$151$	−18.2139	−1.48223	−0.741114	−	0.671380i	$-0.765702\pi$
$151$	−18.2139	−1.48223	−0.741114	+	0.671380i	$0.765702\pi$
$152$	−2.42382	−0.196598
$153$	4.08881	0.330561
$154$	0	0
$155$	1.24578	0.100063
$156$	−4.10057	−0.328308
$157$	−4.40579	−0.351620	−0.175810	−	0.984424i	$-0.556254\pi$
$157$	−4.40579	−0.351620	−0.175810	+	0.984424i	$0.556254\pi$
$158$	2.67411	0.212741
$159$	8.05270	0.638621
$160$	7.59017	0.600056
$161$	0	0
$162$	−0.487949	−0.0383369
$163$	4.66216	0.365169	0.182584	−	0.983190i	$-0.441554\pi$
$163$	4.66216	0.365169	0.182584	+	0.983190i	$0.441554\pi$
$164$	−1.76191	−0.137582
$165$	−5.37898	−0.418753
$166$	−2.88058	−0.223576
$167$	7.14687	0.553041	0.276521	−	0.961008i	$-0.410819\pi$
$167$	7.14687	0.553041	0.276521	+	0.961008i	$0.410819\pi$
$168$	0	0
$169$	−7.58345	−0.583342
$170$	3.05703	0.234463
$171$	−1.32044	−0.100977
$172$	4.92852	0.375796
$173$	−10.9222	−0.830397	−0.415198	−	0.909731i	$-0.636288\pi$
$173$	−10.9222	−0.830397	−0.415198	+	0.909731i	$0.636288\pi$
$174$	−2.51600	−0.190737
$175$	0	0
$176$	−9.22609	−0.695443
$177$	0.693332	0.0521140
$178$	−6.19583	−0.464397
$179$	−3.51144	−0.262458	−0.131229	−	0.991352i	$-0.541892\pi$
$179$	−3.51144	−0.262458	−0.131229	+	0.991352i	$0.541892\pi$
$180$	2.69967	0.201222
$181$	3.38587	0.251669	0.125835	−	0.992051i	$-0.459839\pi$
$181$	3.38587	0.251669	0.125835	+	0.992051i	$0.459839\pi$
$182$	0	0
$183$	−1.29889	−0.0960170
$184$	2.54557	0.187662
$185$	−15.3736	−1.13029
$186$	−0.396723	−0.0290891
$187$	−14.3539	−1.04966
$188$	−18.6780	−1.36224
$189$	0	0
$190$	−0.987237	−0.0716217
$191$	−9.56641	−0.692201	−0.346100	−	0.938197i	$-0.612494\pi$
$191$	−9.56641	−0.692201	−0.346100	+	0.938197i	$0.612494\pi$
$192$	2.83913	0.204897
$193$	−3.73602	−0.268925	−0.134462	−	0.990919i	$-0.542931\pi$
$193$	−3.73602	−0.268925	−0.134462	+	0.990919i	$0.542931\pi$
$194$	−6.97543	−0.500807
$195$	−3.56607	−0.255371
$196$	0	0
$197$	3.86710	0.275520	0.137760	−	0.990466i	$-0.456010\pi$
$197$	3.86710	0.275520	0.137760	+	0.990466i	$0.456010\pi$
$198$	1.71296	0.121735
$199$	9.78954	0.693962	0.346981	−	0.937872i	$-0.387207\pi$
$199$	9.78954	0.693962	0.346981	+	0.937872i	$0.387207\pi$
$200$	−4.86847	−0.344253
$201$	−8.82262	−0.622300
$202$	4.76378	0.335179
$203$	0	0
$204$	7.20409	0.504388
$205$	−1.53225	−0.107017
$206$	−3.28454	−0.228845
$207$	1.38676	0.0963868
$208$	−6.11656	−0.424107
$209$	4.63544	0.320640
$210$	0	0
$211$	0.378702	0.0260710	0.0130355	−	0.999915i	$-0.495851\pi$
$211$	0.378702	0.0260710	0.0130355	+	0.999915i	$0.495851\pi$
$212$	14.1881	0.974443
$213$	−7.16765	−0.491119
$214$	−1.66145	−0.113574
$215$	4.28610	0.292310
$216$	−1.83562	−0.124898
$217$	0	0
$218$	−1.41954	−0.0961436
$219$	−1.53683	−0.103849
$220$	−9.47726	−0.638957
$221$	−9.51608	−0.640121
$222$	4.89578	0.328583
$223$	7.17776	0.480659	0.240329	−	0.970691i	$-0.422745\pi$
$223$	7.17776	0.480659	0.240329	+	0.970691i	$0.422745\pi$
$224$	0	0
$225$	−2.65223	−0.176815
$226$	−1.90269	−0.126565
$227$	16.0821	1.06741	0.533704	−	0.845671i	$-0.320800\pi$
$227$	16.0821	1.06741	0.533704	+	0.845671i	$0.320800\pi$
$228$	−2.32649	−0.154076
$229$	−27.5061	−1.81766	−0.908828	−	0.417172i	$-0.863021\pi$
$229$	−27.5061	−1.81766	−0.908828	+	0.417172i	$0.863021\pi$
$230$	1.03682	0.0683662
$231$	0	0
$232$	−9.46495	−0.621405
$233$	−24.4620	−1.60256	−0.801280	−	0.598289i	$-0.795847\pi$
$233$	−24.4620	−1.60256	−0.801280	+	0.598289i	$0.795847\pi$
$234$	1.13563	0.0742383
$235$	−16.2434	−1.05960
$236$	1.22159	0.0795185
$237$	5.48031	0.355985
$238$	0	0
$239$	−18.4572	−1.19390	−0.596950	−	0.802279i	$-0.703621\pi$
$239$	−18.4572	−1.19390	−0.596950	+	0.802279i	$0.703621\pi$
$240$	4.02693	0.259937
$241$	26.5183	1.70819	0.854096	−	0.520115i	$-0.174111\pi$
$241$	26.5183	1.70819	0.854096	+	0.520115i	$0.174111\pi$
$242$	−0.645941	−0.0415227
$243$	−1.00000	−0.0641500
$244$	−2.28853	−0.146508
$245$	0	0
$246$	0.487949	0.0311105
$247$	3.07313	0.195538
$248$	−1.49243	−0.0947697
$249$	−5.90345	−0.374116
$250$	−5.72124	−0.361843
$251$	−6.46587	−0.408122	−0.204061	−	0.978958i	$-0.565414\pi$
$251$	−6.46587	−0.408122	−0.204061	+	0.978958i	$0.565414\pi$
$252$	0	0
$253$	−4.86827	−0.306065
$254$	2.64563	0.166002
$255$	6.26506	0.392333
$256$	0.168043	0.0105027
$257$	20.8940	1.30333	0.651666	−	0.758506i	$-0.274071\pi$
$257$	20.8940	1.30333	0.651666	+	0.758506i	$0.274071\pi$
$258$	−1.36492	−0.0849765
$259$	0	0
$260$	−6.28308	−0.389660
$261$	−5.15628	−0.319165
$262$	−1.58056	−0.0976476
$263$	22.8930	1.41164	0.705822	−	0.708389i	$-0.250578\pi$
$263$	22.8930	1.41164	0.705822	+	0.708389i	$0.250578\pi$
$264$	6.44398	0.396600
$265$	12.3387	0.757961
$266$	0	0
$267$	−12.6977	−0.777086
$268$	−15.5446	−0.949539
$269$	1.07667	0.0656459	0.0328229	−	0.999461i	$-0.489550\pi$
$269$	1.07667	0.0656459	0.0328229	+	0.999461i	$0.489550\pi$
$270$	−0.747657	−0.0455010
$271$	−20.4648	−1.24315	−0.621574	−	0.783355i	$-0.713507\pi$
$271$	−20.4648	−1.24315	−0.621574	+	0.783355i	$0.713507\pi$
$272$	10.7459	0.651565
$273$	0	0
$274$	3.42471	0.206894
$275$	9.31070	0.561457
$276$	2.44335	0.147072
$277$	−19.0318	−1.14351	−0.571755	−	0.820425i	$-0.693737\pi$
$277$	−19.0318	−1.14351	−0.571755	+	0.820425i	$0.693737\pi$
$278$	7.17773	0.430492
$279$	−0.813042	−0.0486756
$280$	0	0
$281$	5.92479	0.353443	0.176722	−	0.984261i	$-0.443451\pi$
$281$	5.92479	0.353443	0.176722	+	0.984261i	$0.443451\pi$
$282$	5.17277	0.308034
$283$	−19.1146	−1.13625	−0.568123	−	0.822944i	$-0.692330\pi$
$283$	−19.1146	−1.13625	−0.568123	+	0.822944i	$0.692330\pi$
$284$	−12.6287	−0.749377
$285$	−2.02324	−0.119846
$286$	−3.98665	−0.235735
$287$	0	0
$288$	−4.95363	−0.291895
$289$	−0.281647	−0.0165675
$290$	−3.85513	−0.226381
$291$	−14.2954	−0.838012
$292$	−2.70774	−0.158459
$293$	−4.67356	−0.273032	−0.136516	−	0.990638i	$-0.543591\pi$
$293$	−4.67356	−0.273032	−0.136516	+	0.990638i	$0.543591\pi$
$294$	0	0
$295$	1.06235	0.0618527
$296$	18.4175	1.07049
$297$	3.51053	0.203701
$298$	−4.87294	−0.282282
$299$	−3.22748	−0.186650
$300$	−4.67297	−0.269794
$301$	0	0
$302$	8.88746	0.511416
$303$	9.76287	0.560862
$304$	−3.47028	−0.199034
$305$	−1.99022	−0.113960
$306$	−1.99513	−0.114054
$307$	32.9595	1.88110	0.940548	−	0.339660i	$-0.110312\pi$
$307$	32.9595	1.88110	0.940548	+	0.339660i	$0.110312\pi$
$308$	0	0
$309$	−6.73133	−0.382932
$310$	−0.607877	−0.0345251
$311$	−1.39024	−0.0788331	−0.0394165	−	0.999223i	$-0.512550\pi$
$311$	−1.39024	−0.0788331	−0.0394165	+	0.999223i	$0.512550\pi$
$312$	4.27212	0.241861
$313$	−30.8397	−1.74316	−0.871580	−	0.490252i	$-0.836905\pi$
$313$	−30.8397	−1.74316	−0.871580	+	0.490252i	$0.836905\pi$
$314$	2.14980	0.121320
$315$	0	0
$316$	9.65580	0.543181
$317$	−4.30009	−0.241517	−0.120758	−	0.992682i	$-0.538533\pi$
$317$	−4.30009	−0.241517	−0.120758	+	0.992682i	$0.538533\pi$
$318$	−3.92931	−0.220345
$319$	18.1012	1.01347
$320$	4.35024	0.243186
$321$	−3.40497	−0.190047
$322$	0	0
$323$	−5.39903	−0.300410
$324$	−1.76191	−0.0978837
$325$	6.17265	0.342397
$326$	−2.27490	−0.125995
$327$	−2.90920	−0.160879
$328$	1.83562	0.101355
$329$	0	0
$330$	2.62467	0.144483
$331$	12.5388	0.689196	0.344598	−	0.938750i	$-0.388015\pi$
$331$	12.5388	0.689196	0.344598	+	0.938750i	$0.388015\pi$
$332$	−10.4013	−0.570846
$333$	10.0334	0.549826
$334$	−3.48731	−0.190817
$335$	−13.5184	−0.738590
$336$	0	0
$337$	−9.90601	−0.539615	−0.269807	−	0.962914i	$-0.586960\pi$
$337$	−9.90601	−0.539615	−0.269807	+	0.962914i	$0.586960\pi$
$338$	3.70034	0.201272
$339$	−3.89937	−0.211785
$340$	11.0384	0.598643
$341$	2.85420	0.154564
$342$	0.644308	0.0348402
$343$	0	0
$344$	−5.13472	−0.276845
$345$	2.12486	0.114399
$346$	5.32946	0.286514
$347$	−14.4682	−0.776692	−0.388346	−	0.921514i	$-0.626953\pi$
$347$	−14.4682	−0.776692	−0.388346	+	0.921514i	$0.626953\pi$
$348$	−9.08487	−0.487000
$349$	−1.75459	−0.0939213	−0.0469606	−	0.998897i	$-0.514954\pi$
$349$	−1.75459	−0.0939213	−0.0469606	+	0.998897i	$0.514954\pi$
$350$	0	0
$351$	2.32735	0.124225
$352$	17.3898	0.926881
$353$	−12.1456	−0.646446	−0.323223	−	0.946323i	$-0.604766\pi$
$353$	−12.1456	−0.646446	−0.323223	+	0.946323i	$0.604766\pi$
$354$	−0.338311	−0.0179810
$355$	−10.9826	−0.582896
$356$	−22.3721	−1.18572
$357$	0	0
$358$	1.71341	0.0905563
$359$	16.5245	0.872130	0.436065	−	0.899915i	$-0.356372\pi$
$359$	16.5245	0.872130	0.436065	+	0.899915i	$0.356372\pi$
$360$	−2.81262	−0.148238
$361$	−17.2564	−0.908234
$362$	−1.65213	−0.0868340
$363$	−1.32379	−0.0694808
$364$	0	0
$365$	−2.35480	−0.123256
$366$	0.633794	0.0331289
$367$	−8.81154	−0.459959	−0.229979	−	0.973196i	$-0.573866\pi$
$367$	−8.81154	−0.459959	−0.229979	+	0.973196i	$0.573866\pi$
$368$	3.64459	0.189987
$369$	1.00000	0.0520579
$370$	7.50153	0.389986
$371$	0	0
$372$	−1.43250	−0.0742718
$373$	−1.82834	−0.0946679	−0.0473340	−	0.998879i	$-0.515072\pi$
$373$	−1.82834	−0.0946679	−0.0473340	+	0.998879i	$0.515072\pi$
$374$	7.00395	0.362166
$375$	−11.7251	−0.605481
$376$	19.4595	1.00355
$377$	12.0004	0.618055
$378$	0	0
$379$	−27.6184	−1.41866	−0.709332	−	0.704875i	$-0.751003\pi$
$379$	−27.6184	−1.41866	−0.709332	+	0.704875i	$0.751003\pi$
$380$	−3.56476	−0.182868
$381$	5.42195	0.277775
$382$	4.66792	0.238832
$383$	25.2343	1.28941	0.644705	−	0.764431i	$-0.276980\pi$
$383$	25.2343	1.28941	0.644705	+	0.764431i	$0.276980\pi$
$384$	−11.2926	−0.576273
$385$	0	0
$386$	1.82299	0.0927877
$387$	−2.79727	−0.142193
$388$	−25.1872	−1.27868
$389$	−35.0906	−1.77917	−0.889583	−	0.456774i	$-0.849005\pi$
$389$	−35.0906	−1.77917	−0.889583	+	0.456774i	$0.849005\pi$
$390$	1.74006	0.0881113
$391$	5.67021	0.286755
$392$	0	0
$393$	−3.23920	−0.163396
$394$	−1.88695	−0.0950631
$395$	8.39718	0.422508
$396$	6.18521	0.310819
$397$	−25.5787	−1.28376	−0.641878	−	0.766807i	$-0.721844\pi$
$397$	−25.5787	−1.28376	−0.641878	+	0.766807i	$0.721844\pi$
$398$	−4.77680	−0.239439
$399$	0	0
$400$	−6.97037	−0.348519
$401$	−15.7251	−0.785275	−0.392638	−	0.919693i	$-0.628437\pi$
$401$	−15.7251	−0.785275	−0.392638	+	0.919693i	$0.628437\pi$
$402$	4.30499	0.214713
$403$	1.89223	0.0942588
$404$	17.2013	0.855795
$405$	−1.53225	−0.0761379
$406$	0	0
$407$	−35.2224	−1.74591
$408$	−7.50549	−0.371577
$409$	−0.206464	−0.0102090	−0.00510448	−	0.999987i	$-0.501625\pi$
$409$	−0.206464	−0.0102090	−0.00510448	+	0.999987i	$0.501625\pi$
$410$	0.747657	0.0369242
$411$	7.01858	0.346201
$412$	−11.8600	−0.584298
$413$	0	0
$414$	−0.676670	−0.0332565
$415$	−9.04553	−0.444027
$416$	11.5288	0.565247
$417$	14.7100	0.720352
$418$	−2.26186	−0.110631
$419$	−15.2919	−0.747056	−0.373528	−	0.927619i	$-0.621852\pi$
$419$	−15.2919	−0.747056	−0.373528	+	0.927619i	$0.621852\pi$
$420$	0	0
$421$	−34.0348	−1.65875	−0.829377	−	0.558690i	$-0.811304\pi$
$421$	−34.0348	−1.65875	−0.829377	+	0.558690i	$0.811304\pi$
$422$	−0.184787	−0.00899531
$423$	10.6010	0.515440
$424$	−14.7817	−0.717862
$425$	−10.8444	−0.526033
$426$	3.49745	0.169452
$427$	0	0
$428$	−5.99923	−0.289984
$429$	−8.17022	−0.394462
$430$	−2.09140	−0.100856
$431$	−16.6176	−0.800442	−0.400221	−	0.916419i	$-0.631067\pi$
$431$	−16.6176	−0.800442	−0.400221	+	0.916419i	$0.631067\pi$
$432$	−2.62812	−0.126446
$433$	1.16266	0.0558738	0.0279369	−	0.999610i	$-0.491106\pi$
$433$	1.16266	0.0558738	0.0279369	+	0.999610i	$0.491106\pi$
$434$	0	0
$435$	−7.90068	−0.378808
$436$	−5.12574	−0.245479
$437$	−1.83114	−0.0875953
$438$	0.749893	0.0358313
$439$	10.6534	0.508457	0.254228	−	0.967144i	$-0.418178\pi$
$439$	10.6534	0.508457	0.254228	+	0.967144i	$0.418178\pi$
$440$	9.87376	0.470713
$441$	0	0
$442$	4.64336	0.220862
$443$	−1.10628	−0.0525610	−0.0262805	−	0.999655i	$-0.508366\pi$
$443$	−1.10628	−0.0525610	−0.0262805	+	0.999655i	$0.508366\pi$
$444$	17.6779	0.838955
$445$	−19.4560	−0.922302
$446$	−3.50238	−0.165843
$447$	−9.98658	−0.472349
$448$	0	0
$449$	13.3468	0.629872	0.314936	−	0.949113i	$-0.398017\pi$
$449$	13.3468	0.629872	0.314936	+	0.949113i	$0.398017\pi$
$450$	1.29415	0.0610069
$451$	−3.51053	−0.165304
$452$	−6.87032	−0.323153
$453$	18.2139	0.855764
$454$	−7.84726	−0.368290
$455$	0	0
$456$	2.42382	0.113506
$457$	−41.0559	−1.92051	−0.960256	−	0.279120i	$-0.909957\pi$
$457$	−41.0559	−1.92051	−0.960256	+	0.279120i	$0.909957\pi$
$458$	13.4216	0.627149
$459$	−4.08881	−0.190849
$460$	3.74381	0.174556
$461$	−30.8859	−1.43850	−0.719251	−	0.694751i	$-0.755515\pi$
$461$	−30.8859	−1.43850	−0.719251	+	0.694751i	$0.755515\pi$
$462$	0	0
$463$	14.4244	0.670357	0.335178	−	0.942155i	$-0.391203\pi$
$463$	14.4244	0.670357	0.335178	+	0.942155i	$0.391203\pi$
$464$	−13.5513	−0.629105
$465$	−1.24578	−0.0577716
$466$	11.9362	0.552935
$467$	26.9128	1.24538	0.622689	−	0.782470i	$-0.286040\pi$
$467$	26.9128	1.24538	0.622689	+	0.782470i	$0.286040\pi$
$468$	4.10057	0.189549
$469$	0	0
$470$	7.92595	0.365597
$471$	4.40579	0.203008
$472$	−1.27269	−0.0585804
$473$	9.81988	0.451519
$474$	−2.67411	−0.122826
$475$	3.50211	0.160688
$476$	0	0
$477$	−8.05270	−0.368708
$478$	9.00619	0.411933
$479$	15.0420	0.687288	0.343644	−	0.939100i	$-0.388339\pi$
$479$	15.0420	0.687288	0.343644	+	0.939100i	$0.388339\pi$
$480$	−7.59017	−0.346442
$481$	−23.3512	−1.06472
$482$	−12.9396	−0.589381
$483$	0	0
$484$	−2.33239	−0.106018
$485$	−21.9041	−0.994613
$486$	0.487949	0.0221338
$487$	−10.0306	−0.454529	−0.227265	−	0.973833i	$-0.572978\pi$
$487$	−10.0306	−0.454529	−0.227265	+	0.973833i	$0.572978\pi$
$488$	2.38427	0.107931
$489$	−4.66216	−0.210830
$490$	0	0
$491$	−26.0046	−1.17357	−0.586786	−	0.809742i	$-0.699607\pi$
$491$	−26.0046	−1.17357	−0.586786	+	0.809742i	$0.699607\pi$
$492$	1.76191	0.0794328
$493$	−21.0830	−0.949531
$494$	−1.49953	−0.0674670
$495$	5.37898	0.241767
$496$	−2.13677	−0.0959440
$497$	0	0
$498$	2.88058	0.129082
$499$	−16.9912	−0.760631	−0.380315	−	0.924857i	$-0.624185\pi$
$499$	−16.9912	−0.760631	−0.380315	+	0.924857i	$0.624185\pi$
$500$	−20.6585	−0.923876
$501$	−7.14687	−0.319298
$502$	3.15502	0.140815
$503$	−13.3920	−0.597119	−0.298559	−	0.954391i	$-0.596506\pi$
$503$	−13.3920	−0.597119	−0.298559	+	0.954391i	$0.596506\pi$
$504$	0	0
$505$	14.9591	0.665672
$506$	2.37547	0.105602
$507$	7.58345	0.336793
$508$	9.55296	0.423844
$509$	37.2055	1.64910	0.824552	−	0.565786i	$-0.191427\pi$
$509$	37.2055	1.64910	0.824552	+	0.565786i	$0.191427\pi$
$510$	−3.05703	−0.135367
$511$	0	0
$512$	−22.6672	−1.00176
$513$	1.32044	0.0582989
$514$	−10.1952	−0.449691
$515$	−10.3140	−0.454491
$516$	−4.92852	−0.216966
$517$	−37.2152	−1.63672
$518$	0	0
$519$	10.9222	0.479430
$520$	6.54594	0.287058
$521$	−14.3750	−0.629778	−0.314889	−	0.949128i	$-0.601967\pi$
$521$	−14.3750	−0.629778	−0.314889	+	0.949128i	$0.601967\pi$
$522$	2.51600	0.110122
$523$	−23.0126	−1.00627	−0.503135	−	0.864208i	$-0.667820\pi$
$523$	−23.0126	−1.00627	−0.503135	+	0.864208i	$0.667820\pi$
$524$	−5.70717	−0.249319
$525$	0	0
$526$	−11.1706	−0.487063
$527$	−3.32437	−0.144812
$528$	9.22609	0.401514
$529$	−21.0769	−0.916386
$530$	−6.02066	−0.261521
$531$	−0.693332	−0.0300880
$532$	0	0
$533$	−2.32735	−0.100809
$534$	6.19583	0.268120
$535$	−5.21724	−0.225561
$536$	16.1950	0.699516
$537$	3.51144	0.151530
$538$	−0.525361	−0.0226499
$539$	0	0
$540$	−2.69967	−0.116175
$541$	7.00491	0.301165	0.150582	−	0.988597i	$-0.451885\pi$
$541$	7.00491	0.301165	0.150582	+	0.988597i	$0.451885\pi$
$542$	9.98578	0.428926
$543$	−3.38587	−0.145301
$544$	−20.2544	−0.868402
$545$	−4.45761	−0.190943
$546$	0	0
$547$	18.4358	0.788256	0.394128	−	0.919056i	$-0.371047\pi$
$547$	18.4358	0.788256	0.394128	+	0.919056i	$0.371047\pi$
$548$	12.3661	0.528252
$549$	1.29889	0.0554354
$550$	−4.54315	−0.193721
$551$	6.80856	0.290054
$552$	−2.54557	−0.108347
$553$	0	0
$554$	9.28654	0.394547
$555$	15.3736	0.652573
$556$	25.9176	1.09915
$557$	7.62468	0.323068	0.161534	−	0.986867i	$-0.448356\pi$
$557$	7.62468	0.323068	0.161534	+	0.986867i	$0.448356\pi$
$558$	0.396723	0.0167946
$559$	6.51022	0.275353
$560$	0	0
$561$	14.3539	0.606021
$562$	−2.89100	−0.121949
$563$	31.2475	1.31693	0.658464	−	0.752613i	$-0.271207\pi$
$563$	31.2475	1.31693	0.658464	+	0.752613i	$0.271207\pi$
$564$	18.6780	0.786487
$565$	−5.97479	−0.251361
$566$	9.32696	0.392041
$567$	0	0
$568$	13.1571	0.552059
$569$	15.3290	0.642626	0.321313	−	0.946973i	$-0.395876\pi$
$569$	15.3290	0.642626	0.321313	+	0.946973i	$0.395876\pi$
$570$	0.987237	0.0413508
$571$	−20.7415	−0.868006	−0.434003	−	0.900911i	$-0.642899\pi$
$571$	−20.7415	−0.868006	−0.434003	+	0.900911i	$0.642899\pi$
$572$	−14.3952	−0.601891
$573$	9.56641	0.399642
$574$	0	0
$575$	−3.67801	−0.153384
$576$	−2.83913	−0.118297
$577$	29.1178	1.21219	0.606094	−	0.795393i	$-0.292735\pi$
$577$	29.1178	1.21219	0.606094	+	0.795393i	$0.292735\pi$
$578$	0.137430	0.00571632
$579$	3.73602	0.155264
$580$	−13.9202	−0.578007
$581$	0	0
$582$	6.97543	0.289141
$583$	28.2692	1.17079
$584$	2.82103	0.116735
$585$	3.56607	0.147439
$586$	2.28046	0.0942048
$587$	−8.40584	−0.346946	−0.173473	−	0.984839i	$-0.555499\pi$
$587$	−8.40584	−0.346946	−0.173473	+	0.984839i	$0.555499\pi$
$588$	0	0
$589$	1.07357	0.0442358
$590$	−0.518375	−0.0213412
$591$	−3.86710	−0.159071
$592$	26.3690	1.08376
$593$	8.52390	0.350035	0.175017	−	0.984565i	$-0.444002\pi$
$593$	8.52390	0.350035	0.175017	+	0.984565i	$0.444002\pi$
$594$	−1.71296	−0.0702835
$595$	0	0
$596$	−17.5954	−0.720736
$597$	−9.78954	−0.400659
$598$	1.57485	0.0644003
$599$	1.76430	0.0720872	0.0360436	−	0.999350i	$-0.488524\pi$
$599$	1.76430	0.0720872	0.0360436	+	0.999350i	$0.488524\pi$
$600$	4.86847	0.198755
$601$	43.6074	1.77878	0.889392	−	0.457145i	$-0.151128\pi$
$601$	43.6074	1.77878	0.889392	+	0.457145i	$0.151128\pi$
$602$	0	0
$603$	8.82262	0.359285
$604$	32.0912	1.30577
$605$	−2.02837	−0.0824649
$606$	−4.76378	−0.193515
$607$	−24.2662	−0.984936	−0.492468	−	0.870330i	$-0.663905\pi$
$607$	−24.2662	−0.984936	−0.492468	+	0.870330i	$0.663905\pi$
$608$	6.54097	0.265271
$609$	0	0
$610$	0.971128	0.0393198
$611$	−24.6723	−0.998135
$612$	−7.20409	−0.291208
$613$	28.8716	1.16611	0.583056	−	0.812432i	$-0.301857\pi$
$613$	28.8716	1.16611	0.583056	+	0.812432i	$0.301857\pi$
$614$	−16.0825	−0.649039
$615$	1.53225	0.0617861
$616$	0	0
$617$	−12.0451	−0.484918	−0.242459	−	0.970162i	$-0.577954\pi$
$617$	−12.0451	−0.484918	−0.242459	+	0.970162i	$0.577954\pi$
$618$	3.28454	0.132124
$619$	−29.3883	−1.18121	−0.590607	−	0.806960i	$-0.701112\pi$
$619$	−29.3883	−1.18121	−0.590607	+	0.806960i	$0.701112\pi$
$620$	−2.19495	−0.0881512
$621$	−1.38676	−0.0556489
$622$	0.678364	0.0271999
$623$	0	0
$624$	6.11656	0.244858
$625$	−4.70458	−0.188183
$626$	15.0482	0.601446
$627$	−4.63544	−0.185122
$628$	7.76259	0.309761
$629$	41.0246	1.63576
$630$	0	0
$631$	40.0954	1.59617	0.798087	−	0.602543i	$-0.205846\pi$
$631$	40.0954	1.59617	0.798087	+	0.602543i	$0.205846\pi$
$632$	−10.0598	−0.400156
$633$	−0.378702	−0.0150521
$634$	2.09822	0.0833311
$635$	8.30775	0.329683
$636$	−14.1881	−0.562595
$637$	0	0
$638$	−8.83248	−0.349681
$639$	7.16765	0.283548
$640$	−17.3030	−0.683963
$641$	−19.7554	−0.780293	−0.390146	−	0.920753i	$-0.627576\pi$
$641$	−19.7554	−0.780293	−0.390146	+	0.920753i	$0.627576\pi$
$642$	1.66145	0.0655722
$643$	−30.7410	−1.21231	−0.606154	−	0.795347i	$-0.707289\pi$
$643$	−30.7410	−1.21231	−0.606154	+	0.795347i	$0.707289\pi$
$644$	0	0
$645$	−4.28610	−0.168765
$646$	2.63445	0.103651
$647$	9.16424	0.360283	0.180142	−	0.983641i	$-0.442344\pi$
$647$	9.16424	0.360283	0.180142	+	0.983641i	$0.442344\pi$
$648$	1.83562	0.0721099
$649$	2.43396	0.0955413
$650$	−3.01194	−0.118138
$651$	0	0
$652$	−8.21429	−0.321696
$653$	−46.8033	−1.83155	−0.915777	−	0.401687i	$-0.868424\pi$
$653$	−46.8033	−1.83155	−0.915777	+	0.401687i	$0.868424\pi$
$654$	1.41954	0.0555085
$655$	−4.96325	−0.193930
$656$	2.62812	0.102611
$657$	1.53683	0.0599573
$658$	0	0
$659$	11.7930	0.459389	0.229695	−	0.973263i	$-0.426227\pi$
$659$	11.7930	0.459389	0.229695	+	0.973263i	$0.426227\pi$
$660$	9.47726	0.368902
$661$	−0.712807	−0.0277250	−0.0138625	−	0.999904i	$-0.504413\pi$
$661$	−0.712807	−0.0277250	−0.0138625	+	0.999904i	$0.504413\pi$
$662$	−6.11830	−0.237795
$663$	9.51608	0.369574
$664$	10.8365	0.420537
$665$	0	0
$666$	−4.89578	−0.189708
$667$	−7.15054	−0.276870
$668$	−12.5921	−0.487203
$669$	−7.17776	−0.277508
$670$	6.59630	0.254837
$671$	−4.55980	−0.176029
$672$	0	0
$673$	15.2796	0.588984	0.294492	−	0.955654i	$-0.404850\pi$
$673$	15.2796	0.588984	0.294492	+	0.955654i	$0.404850\pi$
$674$	4.83363	0.186184
$675$	2.65223	0.102084
$676$	13.3613	0.513897
$677$	−14.7878	−0.568343	−0.284172	−	0.958773i	$-0.591719\pi$
$677$	−14.7878	−0.568343	−0.284172	+	0.958773i	$0.591719\pi$
$678$	1.90269	0.0730725
$679$	0	0
$680$	−11.5003	−0.441015
$681$	−16.0821	−0.616269
$682$	−1.39271	−0.0533295
$683$	7.33388	0.280623	0.140312	−	0.990107i	$-0.455190\pi$
$683$	7.33388	0.280623	0.140312	+	0.990107i	$0.455190\pi$
$684$	2.32649	0.0889556
$685$	10.7542	0.410896
$686$	0	0
$687$	27.5061	1.04942
$688$	−7.35157	−0.280276
$689$	18.7414	0.713992
$690$	−1.03682	−0.0394712
$691$	−9.80122	−0.372856	−0.186428	−	0.982469i	$-0.559691\pi$
$691$	−9.80122	−0.372856	−0.186428	+	0.982469i	$0.559691\pi$
$692$	19.2438	0.731541
$693$	0	0
$694$	7.05973	0.267984
$695$	22.5393	0.854965
$696$	9.46495	0.358768
$697$	4.08881	0.154875
$698$	0.856153	0.0324059
$699$	24.4620	0.925239
$700$	0	0
$701$	38.1869	1.44230	0.721151	−	0.692778i	$-0.243614\pi$
$701$	38.1869	1.44230	0.721151	+	0.692778i	$0.243614\pi$
$702$	−1.13563	−0.0428615
$703$	−13.2485	−0.499676
$704$	9.96683	0.375639
$705$	16.2434	0.611761
$706$	5.92645	0.223045
$707$	0	0
$708$	−1.22159	−0.0459100
$709$	49.2240	1.84864	0.924322	−	0.381613i	$-0.124631\pi$
$709$	49.2240	1.84864	0.924322	+	0.381613i	$0.124631\pi$
$710$	5.35895	0.201118
$711$	−5.48031	−0.205528
$712$	23.3081	0.873509
$713$	−1.12750	−0.0422251
$714$	0	0
$715$	−12.5188	−0.468176
$716$	6.18683	0.231213
$717$	18.4572	0.689298
$718$	−8.06311	−0.300913
$719$	−2.98460	−0.111307	−0.0556533	−	0.998450i	$-0.517724\pi$
$719$	−2.98460	−0.111307	−0.0556533	+	0.998450i	$0.517724\pi$
$720$	−4.02693	−0.150075
$721$	0	0
$722$	8.42026	0.313370
$723$	−26.5183	−0.986225
$724$	−5.96558	−0.221709
$725$	13.6756	0.507899
$726$	0.645941	0.0239731
$727$	−32.4202	−1.20240	−0.601199	−	0.799099i	$-0.705310\pi$
$727$	−32.4202	−1.20240	−0.601199	+	0.799099i	$0.705310\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	1.14902	0.0425272
$731$	−11.4375	−0.423031
$732$	2.28853	0.0845865
$733$	−19.5731	−0.722948	−0.361474	−	0.932382i	$-0.617726\pi$
$733$	−19.5731	−0.722948	−0.361474	+	0.932382i	$0.617726\pi$
$734$	4.29958	0.158701
$735$	0	0
$736$	−6.86951	−0.253214
$737$	−30.9720	−1.14087
$738$	−0.487949	−0.0179617
$739$	20.0765	0.738526	0.369263	−	0.929325i	$-0.379610\pi$
$739$	20.0765	0.738526	0.369263	+	0.929325i	$0.379610\pi$
$740$	27.0868	0.995732
$741$	−3.07313	−0.112894
$742$	0	0
$743$	39.5791	1.45202	0.726008	−	0.687686i	$-0.241374\pi$
$743$	39.5791	1.45202	0.726008	+	0.687686i	$0.241374\pi$
$744$	1.49243	0.0547153
$745$	−15.3019	−0.560618
$746$	0.892137	0.0326635
$747$	5.90345	0.215996
$748$	25.2902	0.924700
$749$	0	0
$750$	5.72124	0.208910
$751$	−23.0778	−0.842121	−0.421061	−	0.907033i	$-0.638342\pi$
$751$	−23.0778	−0.842121	−0.421061	+	0.907033i	$0.638342\pi$
$752$	27.8608	1.01598
$753$	6.46587	0.235629
$754$	−5.85561	−0.213249
$755$	27.9082	1.01568
$756$	0	0
$757$	−3.38807	−0.123141	−0.0615707	−	0.998103i	$-0.519611\pi$
$757$	−3.38807	−0.123141	−0.0615707	+	0.998103i	$0.519611\pi$
$758$	13.4764	0.489485
$759$	4.86827	0.176707
$760$	3.71389	0.134717
$761$	36.0382	1.30638	0.653191	−	0.757193i	$-0.273430\pi$
$761$	36.0382	1.30638	0.653191	+	0.757193i	$0.273430\pi$
$762$	−2.64563	−0.0958412
$763$	0	0
$764$	16.8551	0.609796
$765$	−6.26506	−0.226514
$766$	−12.3130	−0.444888
$767$	1.61363	0.0582646
$768$	−0.168043	−0.00606374
$769$	−29.7848	−1.07407	−0.537034	−	0.843561i	$-0.680455\pi$
$769$	−29.7848	−1.07407	−0.537034	+	0.843561i	$0.680455\pi$
$770$	0	0
$771$	−20.8940	−0.752479
$772$	6.58252	0.236910
$773$	27.4004	0.985525	0.492763	−	0.870164i	$-0.335987\pi$
$773$	27.4004	0.985525	0.492763	+	0.870164i	$0.335987\pi$
$774$	1.36492	0.0490612
$775$	2.15637	0.0774591
$776$	26.2409	0.941994
$777$	0	0
$778$	17.1224	0.613869
$779$	−1.32044	−0.0473097
$780$	6.28308	0.224970
$781$	−25.1622	−0.900375
$782$	−2.76677	−0.0989396
$783$	5.15628	0.184270
$784$	0	0
$785$	6.75075	0.240945
$786$	1.58056	0.0563769
$787$	−16.1782	−0.576690	−0.288345	−	0.957527i	$-0.593105\pi$
$787$	−16.1782	−0.576690	−0.288345	+	0.957527i	$0.593105\pi$
$788$	−6.81347	−0.242720
$789$	−22.8930	−0.815013
$790$	−4.09740	−0.145779
$791$	0	0
$792$	−6.44398	−0.228977
$793$	−3.02298	−0.107349
$794$	12.4811	0.442937
$795$	−12.3387	−0.437609
$796$	−17.2482	−0.611348
$797$	38.1592	1.35167	0.675834	−	0.737054i	$-0.263784\pi$
$797$	38.1592	1.35167	0.675834	+	0.737054i	$0.263784\pi$
$798$	0	0
$799$	43.3456	1.53346
$800$	13.1381	0.464503
$801$	12.6977	0.448651
$802$	7.67306	0.270945
$803$	−5.39507	−0.190388
$804$	15.5446	0.548217
$805$	0	0
$806$	−0.923313	−0.0325223
$807$	−1.07667	−0.0379007
$808$	−17.9209	−0.630455
$809$	−27.0936	−0.952561	−0.476281	−	0.879293i	$-0.658015\pi$
$809$	−27.0936	−0.952561	−0.476281	+	0.879293i	$0.658015\pi$
$810$	0.747657	0.0262700
$811$	−50.5845	−1.77626	−0.888131	−	0.459591i	$-0.847996\pi$
$811$	−50.5845	−1.77626	−0.888131	+	0.459591i	$0.847996\pi$
$812$	0	0
$813$	20.4648	0.717732
$814$	17.1868	0.602396
$815$	−7.14357	−0.250228
$816$	−10.7459	−0.376181
$817$	3.69363	0.129224
$818$	0.100744	0.00352242
$819$	0	0
$820$	2.69967	0.0942766
$821$	24.3297	0.849114	0.424557	−	0.905401i	$-0.360430\pi$
$821$	24.3297	0.849114	0.424557	+	0.905401i	$0.360430\pi$
$822$	−3.42471	−0.119450
$823$	−9.84814	−0.343285	−0.171642	−	0.985159i	$-0.554907\pi$
$823$	−9.84814	−0.343285	−0.171642	+	0.985159i	$0.554907\pi$
$824$	12.3561	0.430447
$825$	−9.31070	−0.324157
$826$	0	0
$827$	41.1328	1.43033	0.715164	−	0.698956i	$-0.246352\pi$
$827$	41.1328	1.43033	0.715164	+	0.698956i	$0.246352\pi$
$828$	−2.44335	−0.0849122
$829$	−43.7460	−1.51936	−0.759681	−	0.650296i	$-0.774645\pi$
$829$	−43.7460	−1.51936	−0.759681	+	0.650296i	$0.774645\pi$
$830$	4.41376	0.153204
$831$	19.0318	0.660205
$832$	6.60764	0.229079
$833$	0	0
$834$	−7.17773	−0.248545
$835$	−10.9508	−0.378966
$836$	−8.16721	−0.282469
$837$	0.813042	0.0281028
$838$	7.46165	0.257758
$839$	−31.7703	−1.09683	−0.548417	−	0.836205i	$-0.684769\pi$
$839$	−31.7703	−1.09683	−0.548417	+	0.836205i	$0.684769\pi$
$840$	0	0
$841$	−2.41283	−0.0832009
$842$	16.6072	0.572323
$843$	−5.92479	−0.204061
$844$	−0.667238	−0.0229673
$845$	11.6197	0.399730
$846$	−5.17277	−0.177843
$847$	0	0
$848$	−21.1635	−0.726757
$849$	19.1146	0.656012
$850$	5.29153	0.181498
$851$	13.9139	0.476963
$852$	12.6287	0.432653
$853$	1.20173	0.0411463	0.0205732	−	0.999788i	$-0.493451\pi$
$853$	1.20173	0.0411463	0.0205732	+	0.999788i	$0.493451\pi$
$854$	0	0
$855$	2.02324	0.0691933
$856$	6.25022	0.213628
$857$	11.0723	0.378222	0.189111	−	0.981956i	$-0.439439\pi$
$857$	11.0723	0.378222	0.189111	+	0.981956i	$0.439439\pi$
$858$	3.98665	0.136102
$859$	−8.79105	−0.299947	−0.149973	−	0.988690i	$-0.547919\pi$
$859$	−8.79105	−0.299947	−0.149973	+	0.988690i	$0.547919\pi$
$860$	−7.55170	−0.257511
$861$	0	0
$862$	8.10855	0.276178
$863$	32.4019	1.10297	0.551487	−	0.834183i	$-0.314060\pi$
$863$	32.4019	1.10297	0.551487	+	0.834183i	$0.314060\pi$
$864$	4.95363	0.168526
$865$	16.7354	0.569022
$866$	−0.567318	−0.0192782
$867$	0.281647	0.00956525
$868$	0	0
$869$	19.2388	0.652631
$870$	3.85513	0.130701
$871$	−20.5333	−0.695745
$872$	5.34019	0.180842
$873$	14.2954	0.483826
$874$	0.893503	0.0302232
$875$	0	0
$876$	2.70774	0.0914862
$877$	−38.4772	−1.29928	−0.649641	−	0.760241i	$-0.725081\pi$
$877$	−38.4772	−1.29928	−0.649641	+	0.760241i	$0.725081\pi$
$878$	−5.19829	−0.175434
$879$	4.67356	0.157635
$880$	14.1366	0.476546
$881$	38.0482	1.28188	0.640939	−	0.767592i	$-0.278545\pi$
$881$	38.0482	1.28188	0.640939	+	0.767592i	$0.278545\pi$
$882$	0	0
$883$	−1.84034	−0.0619323	−0.0309661	−	0.999520i	$-0.509858\pi$
$883$	−1.84034	−0.0619323	−0.0309661	+	0.999520i	$0.509858\pi$
$884$	16.7664	0.563916
$885$	−1.06235	−0.0357107
$886$	0.539809	0.0181352
$887$	−22.1892	−0.745040	−0.372520	−	0.928024i	$-0.621506\pi$
$887$	−22.1892	−0.745040	−0.372520	+	0.928024i	$0.621506\pi$
$888$	−18.4175	−0.618049
$889$	0	0
$890$	9.49353	0.318224
$891$	−3.51053	−0.117607
$892$	−12.6465	−0.423438
$893$	−13.9980	−0.468427
$894$	4.87294	0.162976
$895$	5.38039	0.179847
$896$	0	0
$897$	3.22748	0.107763
$898$	−6.51254	−0.217326
$899$	4.19227	0.139820
$900$	4.67297	0.155766
$901$	−32.9259	−1.09692
$902$	1.71296	0.0570352
$903$	0	0
$904$	7.15775	0.238063
$905$	−5.18798	−0.172454
$906$	−8.88746	−0.295266
$907$	−12.1000	−0.401773	−0.200887	−	0.979615i	$-0.564382\pi$
$907$	−12.1000	−0.401773	−0.200887	+	0.979615i	$0.564382\pi$
$908$	−28.3352	−0.940336
$909$	−9.76287	−0.323814
$910$	0	0
$911$	−4.33503	−0.143626	−0.0718130	−	0.997418i	$-0.522878\pi$
$911$	−4.33503	−0.143626	−0.0718130	+	0.997418i	$0.522878\pi$
$912$	3.47028	0.114912
$913$	−20.7242	−0.685871
$914$	20.0332	0.662638
$915$	1.99022	0.0657948
$916$	48.4632	1.60127
$917$	0	0
$918$	1.99513	0.0658491
$919$	7.90732	0.260838	0.130419	−	0.991459i	$-0.458368\pi$
$919$	7.90732	0.260838	0.130419	+	0.991459i	$0.458368\pi$
$920$	−3.90044	−0.128594
$921$	−32.9595	−1.08605
$922$	15.0708	0.496329
$923$	−16.6816	−0.549082
$924$	0	0
$925$	−26.6108	−0.874957
$926$	−7.03835	−0.231295
$927$	6.73133	0.221086
$928$	25.5423	0.838466
$929$	7.05313	0.231406	0.115703	−	0.993284i	$-0.463088\pi$
$929$	7.05313	0.231406	0.115703	+	0.993284i	$0.463088\pi$
$930$	0.607877	0.0199331
$931$	0	0
$932$	43.0998	1.41178
$933$	1.39024	0.0455143
$934$	−13.1321	−0.429695
$935$	21.9936	0.719269
$936$	−4.27212	−0.139639
$937$	27.5399	0.899689	0.449845	−	0.893107i	$-0.351479\pi$
$937$	27.5399	0.899689	0.449845	+	0.893107i	$0.351479\pi$
$938$	0	0
$939$	30.8397	1.00641
$940$	28.6193	0.933459
$941$	24.9906	0.814671	0.407335	−	0.913279i	$-0.366458\pi$
$941$	24.9906	0.814671	0.407335	+	0.913279i	$0.366458\pi$
$942$	−2.14980	−0.0700443
$943$	1.38676	0.0451592
$944$	−1.82216	−0.0593063
$945$	0	0
$946$	−4.79160	−0.155788
$947$	−54.5580	−1.77290	−0.886448	−	0.462827i	$-0.846835\pi$
$947$	−54.5580	−1.77290	−0.886448	+	0.462827i	$0.846835\pi$
$948$	−9.65580	−0.313606
$949$	−3.57673	−0.116106
$950$	−1.70885	−0.0554424
$951$	4.30009	0.139440
$952$	0	0
$953$	−23.2096	−0.751832	−0.375916	−	0.926654i	$-0.622672\pi$
$953$	−23.2096	−0.751832	−0.375916	+	0.926654i	$0.622672\pi$
$954$	3.92931	0.127216
$955$	14.6581	0.474324
$956$	32.5199	1.05177
$957$	−18.1012	−0.585130
$958$	−7.33975	−0.237136
$959$	0	0
$960$	−4.35024	−0.140403
$961$	−30.3390	−0.978676
$962$	11.3942	0.367363
$963$	3.40497	0.109724
$964$	−46.7227	−1.50484
$965$	5.72450	0.184278
$966$	0	0
$967$	57.5186	1.84967	0.924837	−	0.380364i	$-0.124201\pi$
$967$	57.5186	1.84967	0.924837	+	0.380364i	$0.124201\pi$
$968$	2.42997	0.0781022
$969$	5.39903	0.173442
$970$	10.6881	0.343173
$971$	−1.14481	−0.0367386	−0.0183693	−	0.999831i	$-0.505847\pi$
$971$	−1.14481	−0.0367386	−0.0183693	+	0.999831i	$0.505847\pi$
$972$	1.76191	0.0565132
$973$	0	0
$974$	4.89441	0.156827
$975$	−6.17265	−0.197683
$976$	3.41365	0.109268
$977$	55.3288	1.77013	0.885063	−	0.465472i	$-0.154115\pi$
$977$	55.3288	1.77013	0.885063	+	0.465472i	$0.154115\pi$
$978$	2.27490	0.0727432
$979$	−44.5756	−1.42464
$980$	0	0
$981$	2.90920	0.0928837
$982$	12.6889	0.404920
$983$	42.3937	1.35215	0.676075	−	0.736833i	$-0.263680\pi$
$983$	42.3937	1.35215	0.676075	+	0.736833i	$0.263680\pi$
$984$	−1.83562	−0.0585174
$985$	−5.92535	−0.188797
$986$	10.2874	0.327619
$987$	0	0
$988$	−5.41456	−0.172260
$989$	−3.87915	−0.123350
$990$	−2.62467	−0.0834175
$991$	−41.8777	−1.33029	−0.665144	−	0.746715i	$-0.731630\pi$
$991$	−41.8777	−1.33029	−0.665144	+	0.746715i	$0.731630\pi$
$992$	4.02751	0.127873
$993$	−12.5388	−0.397907
$994$	0	0
$995$	−15.0000	−0.475531
$996$	10.4013	0.329578
$997$	−23.3614	−0.739863	−0.369932	−	0.929059i	$-0.620619\pi$
$997$	−23.3614	−0.739863	−0.369932	+	0.929059i	$0.620619\pi$
$998$	8.29084	0.262442
$999$	−10.0334	−0.317442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.z.1.4	✓	8
7.6	odd	2		6027.2.a.ba.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6027.2.a.z.1.4	✓	8	1.1	even	1	trivial
6027.2.a.ba.1.4	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)

\(f(q)\)	\(=\)	\(q-0.487949 q^{2} -1.00000 q^{3} -1.76191 q^{4} -1.53225 q^{5} +0.487949 q^{6} +1.83562 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.487949 q^{2} -1.00000 q^{3} -1.76191 q^{4} -1.53225 q^{5} +0.487949 q^{6} +1.83562 q^{8} +1.00000 q^{9} +0.747657 q^{10} -3.51053 q^{11} +1.76191 q^{12} -2.32735 q^{13} +1.53225 q^{15} +2.62812 q^{16} +4.08881 q^{17} -0.487949 q^{18} -1.32044 q^{19} +2.69967 q^{20} +1.71296 q^{22} +1.38676 q^{23} -1.83562 q^{24} -2.65223 q^{25} +1.13563 q^{26} -1.00000 q^{27} -5.15628 q^{29} -0.747657 q^{30} -0.813042 q^{31} -4.95363 q^{32} +3.51053 q^{33} -1.99513 q^{34} -1.76191 q^{36} +10.0334 q^{37} +0.644308 q^{38} +2.32735 q^{39} -2.81262 q^{40} +1.00000 q^{41} -2.79727 q^{43} +6.18521 q^{44} -1.53225 q^{45} -0.676670 q^{46} +10.6010 q^{47} -2.62812 q^{48} +1.29415 q^{50} -4.08881 q^{51} +4.10057 q^{52} -8.05270 q^{53} +0.487949 q^{54} +5.37898 q^{55} +1.32044 q^{57} +2.51600 q^{58} -0.693332 q^{59} -2.69967 q^{60} +1.29889 q^{61} +0.396723 q^{62} -2.83913 q^{64} +3.56607 q^{65} -1.71296 q^{66} +8.82262 q^{67} -7.20409 q^{68} -1.38676 q^{69} +7.16765 q^{71} +1.83562 q^{72} +1.53683 q^{73} -4.89578 q^{74} +2.65223 q^{75} +2.32649 q^{76} -1.13563 q^{78} -5.48031 q^{79} -4.02693 q^{80} +1.00000 q^{81} -0.487949 q^{82} +5.90345 q^{83} -6.26506 q^{85} +1.36492 q^{86} +5.15628 q^{87} -6.44398 q^{88} +12.6977 q^{89} +0.747657 q^{90} -2.44335 q^{92} +0.813042 q^{93} -5.17277 q^{94} +2.02324 q^{95} +4.95363 q^{96} +14.2954 q^{97} -3.51053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} - 8 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 6 q^{8} + 8 q^{9}+O(q^{10}) \) 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

Modular forms

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Database

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