LMFDB - Embedded newform 6042.2.a.bf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, 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("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.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:	$48.2456129013$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 27 x^{10} + 134 x^{9} + 294 x^{8} - 1313 x^{7} - 1685 x^{6} + 5910 x^{5} + \cdots + 4048 $ x^12 - 5x^11 - 27x^10 + 134x^9 + 294x^8 - 1313x^7 - 1685x^6 + 5910x^5 + 5237x^4 - 12206x^3 - 7876x^2 + 9264*x + 4048
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.19598$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.19598 q^{5} -1.00000 q^{6} +2.75709 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.19598 q^{5} -1.00000 q^{6} +2.75709 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.19598 q^{10} -5.58341 q^{11} +1.00000 q^{12} -0.357980 q^{13} -2.75709 q^{14} -3.19598 q^{15} +1.00000 q^{16} +6.64295 q^{17} -1.00000 q^{18} +1.00000 q^{19} -3.19598 q^{20} +2.75709 q^{21} +5.58341 q^{22} -8.90349 q^{23} -1.00000 q^{24} +5.21431 q^{25} +0.357980 q^{26} +1.00000 q^{27} +2.75709 q^{28} -8.60587 q^{29} +3.19598 q^{30} -9.83935 q^{31} -1.00000 q^{32} -5.58341 q^{33} -6.64295 q^{34} -8.81161 q^{35} +1.00000 q^{36} +9.10498 q^{37} -1.00000 q^{38} -0.357980 q^{39} +3.19598 q^{40} +6.09268 q^{41} -2.75709 q^{42} +10.9035 q^{43} -5.58341 q^{44} -3.19598 q^{45} +8.90349 q^{46} -2.58860 q^{47} +1.00000 q^{48} +0.601540 q^{49} -5.21431 q^{50} +6.64295 q^{51} -0.357980 q^{52} +1.00000 q^{53} -1.00000 q^{54} +17.8445 q^{55} -2.75709 q^{56} +1.00000 q^{57} +8.60587 q^{58} +9.77834 q^{59} -3.19598 q^{60} +6.88945 q^{61} +9.83935 q^{62} +2.75709 q^{63} +1.00000 q^{64} +1.14410 q^{65} +5.58341 q^{66} -2.61740 q^{67} +6.64295 q^{68} -8.90349 q^{69} +8.81161 q^{70} +12.3429 q^{71} -1.00000 q^{72} -7.87313 q^{73} -9.10498 q^{74} +5.21431 q^{75} +1.00000 q^{76} -15.3940 q^{77} +0.357980 q^{78} -2.48178 q^{79} -3.19598 q^{80} +1.00000 q^{81} -6.09268 q^{82} +1.18756 q^{83} +2.75709 q^{84} -21.2307 q^{85} -10.9035 q^{86} -8.60587 q^{87} +5.58341 q^{88} +3.43485 q^{89} +3.19598 q^{90} -0.986982 q^{91} -8.90349 q^{92} -9.83935 q^{93} +2.58860 q^{94} -3.19598 q^{95} -1.00000 q^{96} -13.8356 q^{97} -0.601540 q^{98} -5.58341 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 7 * q^5 - 12 * q^6 + 5 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} + 10 q^{13} - 5 q^{14} - 7 q^{15} + 12 q^{16} + 6 q^{17} - 12 q^{18} + 12 q^{19} - 7 q^{20} + 5 q^{21} + 5 q^{22} - 12 q^{24} + 21 q^{25} - 10 q^{26} + 12 q^{27} + 5 q^{28} + 7 q^{30} + 2 q^{31} - 12 q^{32} - 5 q^{33} - 6 q^{34} - 17 q^{35} + 12 q^{36} + 16 q^{37} - 12 q^{38} + 10 q^{39} + 7 q^{40} + 7 q^{41} - 5 q^{42} + 24 q^{43} - 5 q^{44} - 7 q^{45} + 4 q^{47} + 12 q^{48} + 29 q^{49} - 21 q^{50} + 6 q^{51} + 10 q^{52} + 12 q^{53} - 12 q^{54} - 2 q^{55} - 5 q^{56} + 12 q^{57} + 2 q^{59} - 7 q^{60} + 29 q^{61} - 2 q^{62} + 5 q^{63} + 12 q^{64} - 17 q^{65} + 5 q^{66} + 36 q^{67} + 6 q^{68} + 17 q^{70} - 3 q^{71} - 12 q^{72} + 7 q^{73} - 16 q^{74} + 21 q^{75} + 12 q^{76} - 35 q^{77} - 10 q^{78} + 40 q^{79} - 7 q^{80} + 12 q^{81} - 7 q^{82} + 24 q^{83} + 5 q^{84} - 22 q^{85} - 24 q^{86} + 5 q^{88} - 45 q^{89} + 7 q^{90} + 71 q^{91} + 2 q^{93} - 4 q^{94} - 7 q^{95} - 12 q^{96} + q^{97} - 29 q^{98} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 7 * q^5 - 12 * q^6 + 5 * q^7 - 12 * q^8 + 12 * q^9 + 7 * q^10 - 5 * q^11 + 12 * q^12 + 10 * q^13 - 5 * q^14 - 7 * q^15 + 12 * q^16 + 6 * q^17 - 12 * q^18 + 12 * q^19 - 7 * q^20 + 5 * q^21 + 5 * q^22 - 12 * q^24 + 21 * q^25 - 10 * q^26 + 12 * q^27 + 5 * q^28 + 7 * q^30 + 2 * q^31 - 12 * q^32 - 5 * q^33 - 6 * q^34 - 17 * q^35 + 12 * q^36 + 16 * q^37 - 12 * q^38 + 10 * q^39 + 7 * q^40 + 7 * q^41 - 5 * q^42 + 24 * q^43 - 5 * q^44 - 7 * q^45 + 4 * q^47 + 12 * q^48 + 29 * q^49 - 21 * q^50 + 6 * q^51 + 10 * q^52 + 12 * q^53 - 12 * q^54 - 2 * q^55 - 5 * q^56 + 12 * q^57 + 2 * q^59 - 7 * q^60 + 29 * q^61 - 2 * q^62 + 5 * q^63 + 12 * q^64 - 17 * q^65 + 5 * q^66 + 36 * q^67 + 6 * q^68 + 17 * q^70 - 3 * q^71 - 12 * q^72 + 7 * q^73 - 16 * q^74 + 21 * q^75 + 12 * q^76 - 35 * q^77 - 10 * q^78 + 40 * q^79 - 7 * q^80 + 12 * q^81 - 7 * q^82 + 24 * q^83 + 5 * q^84 - 22 * q^85 - 24 * q^86 + 5 * q^88 - 45 * q^89 + 7 * q^90 + 71 * q^91 + 2 * q^93 - 4 * q^94 - 7 * q^95 - 12 * q^96 + q^97 - 29 * q^98 - 5 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.19598	−1.42929	−0.714644	−	0.699489i	$-0.753411\pi$
$5$	−3.19598	−1.42929	−0.714644	+	0.699489i	$0.753411\pi$
$6$	−1.00000	−0.408248
$7$	2.75709	1.04208	0.521041	−	0.853532i	$-0.325544\pi$
$7$	2.75709	1.04208	0.521041	+	0.853532i	$0.325544\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.19598	1.01066
$11$	−5.58341	−1.68346	−0.841731	−	0.539897i	$-0.818463\pi$
$11$	−5.58341	−1.68346	−0.841731	+	0.539897i	$0.818463\pi$
$12$	1.00000	0.288675
$13$	−0.357980	−0.0992857	−0.0496428	−	0.998767i	$-0.515808\pi$
$13$	−0.357980	−0.0992857	−0.0496428	+	0.998767i	$0.515808\pi$
$14$	−2.75709	−0.736863
$15$	−3.19598	−0.825199
$16$	1.00000	0.250000
$17$	6.64295	1.61115	0.805575	−	0.592493i	$-0.201856\pi$
$17$	6.64295	1.61115	0.805575	+	0.592493i	$0.201856\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.19598	−0.714644
$21$	2.75709	0.601646
$22$	5.58341	1.19039
$23$	−8.90349	−1.85651	−0.928253	−	0.371949i	$-0.878690\pi$
$23$	−8.90349	−1.85651	−0.928253	+	0.371949i	$0.878690\pi$
$24$	−1.00000	−0.204124
$25$	5.21431	1.04286
$26$	0.357980	0.0702056
$27$	1.00000	0.192450
$28$	2.75709	0.521041
$29$	−8.60587	−1.59807	−0.799035	−	0.601285i	$-0.794656\pi$
$29$	−8.60587	−1.59807	−0.799035	+	0.601285i	$0.794656\pi$
$30$	3.19598	0.583504
$31$	−9.83935	−1.76720	−0.883600	−	0.468243i	$-0.844887\pi$
$31$	−9.83935	−1.76720	−0.883600	+	0.468243i	$0.844887\pi$
$32$	−1.00000	−0.176777
$33$	−5.58341	−0.971947
$34$	−6.64295	−1.13926
$35$	−8.81161	−1.48943
$36$	1.00000	0.166667
$37$	9.10498	1.49685	0.748425	−	0.663219i	$-0.230810\pi$
$37$	9.10498	1.49685	0.748425	+	0.663219i	$0.230810\pi$
$38$	−1.00000	−0.162221
$39$	−0.357980	−0.0573226
$40$	3.19598	0.505329
$41$	6.09268	0.951516	0.475758	−	0.879576i	$-0.342174\pi$
$41$	6.09268	0.951516	0.475758	+	0.879576i	$0.342174\pi$
$42$	−2.75709	−0.425428
$43$	10.9035	1.66277	0.831383	−	0.555699i	$-0.187549\pi$
$43$	10.9035	1.66277	0.831383	+	0.555699i	$0.187549\pi$
$44$	−5.58341	−0.841731
$45$	−3.19598	−0.476429
$46$	8.90349	1.31275
$47$	−2.58860	−0.377586	−0.188793	−	0.982017i	$-0.560457\pi$
$47$	−2.58860	−0.377586	−0.188793	+	0.982017i	$0.560457\pi$
$48$	1.00000	0.144338
$49$	0.601540	0.0859343
$50$	−5.21431	−0.737415
$51$	6.64295	0.930198
$52$	−0.357980	−0.0496428
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	−1.00000	−0.136083
$55$	17.8445	2.40615
$56$	−2.75709	−0.368432
$57$	1.00000	0.132453
$58$	8.60587	1.13001
$59$	9.77834	1.27303	0.636515	−	0.771264i	$-0.280375\pi$
$59$	9.77834	1.27303	0.636515	+	0.771264i	$0.280375\pi$
$60$	−3.19598	−0.412600
$61$	6.88945	0.882104	0.441052	−	0.897482i	$-0.354606\pi$
$61$	6.88945	0.882104	0.441052	+	0.897482i	$0.354606\pi$
$62$	9.83935	1.24960
$63$	2.75709	0.347361
$64$	1.00000	0.125000
$65$	1.14410	0.141908
$66$	5.58341	0.687271
$67$	−2.61740	−0.319766	−0.159883	−	0.987136i	$-0.551112\pi$
$67$	−2.61740	−0.319766	−0.159883	+	0.987136i	$0.551112\pi$
$68$	6.64295	0.805575
$69$	−8.90349	−1.07185
$70$	8.81161	1.05319
$71$	12.3429	1.46484	0.732418	−	0.680856i	$-0.238392\pi$
$71$	12.3429	1.46484	0.732418	+	0.680856i	$0.238392\pi$
$72$	−1.00000	−0.117851
$73$	−7.87313	−0.921481	−0.460740	−	0.887535i	$-0.652416\pi$
$73$	−7.87313	−0.921481	−0.460740	+	0.887535i	$0.652416\pi$
$74$	−9.10498	−1.05843
$75$	5.21431	0.602097
$76$	1.00000	0.114708
$77$	−15.3940	−1.75431
$78$	0.357980	0.0405332
$79$	−2.48178	−0.279223	−0.139611	−	0.990206i	$-0.544585\pi$
$79$	−2.48178	−0.279223	−0.139611	+	0.990206i	$0.544585\pi$
$80$	−3.19598	−0.357322
$81$	1.00000	0.111111
$82$	−6.09268	−0.672824
$83$	1.18756	0.130351	0.0651757	−	0.997874i	$-0.479239\pi$
$83$	1.18756	0.130351	0.0651757	+	0.997874i	$0.479239\pi$
$84$	2.75709	0.300823
$85$	−21.2307	−2.30280
$86$	−10.9035	−1.17575
$87$	−8.60587	−0.922646
$88$	5.58341	0.595194
$89$	3.43485	0.364094	0.182047	−	0.983290i	$-0.441728\pi$
$89$	3.43485	0.364094	0.182047	+	0.983290i	$0.441728\pi$
$90$	3.19598	0.336886
$91$	−0.986982	−0.103464
$92$	−8.90349	−0.928253
$93$	−9.83935	−1.02029
$94$	2.58860	0.266993
$95$	−3.19598	−0.327901
$96$	−1.00000	−0.102062
$97$	−13.8356	−1.40479	−0.702395	−	0.711788i	$-0.747886\pi$
$97$	−13.8356	−1.40479	−0.702395	+	0.711788i	$0.747886\pi$
$98$	−0.601540	−0.0607647
$99$	−5.58341	−0.561154
$100$	5.21431	0.521431
$101$	−5.06221	−0.503709	−0.251855	−	0.967765i	$-0.581040\pi$
$101$	−5.06221	−0.503709	−0.251855	+	0.967765i	$0.581040\pi$
$102$	−6.64295	−0.657750
$103$	3.86336	0.380668	0.190334	−	0.981719i	$-0.439043\pi$
$103$	3.86336	0.380668	0.190334	+	0.981719i	$0.439043\pi$
$104$	0.357980	0.0351028
$105$	−8.81161	−0.859925
$106$	−1.00000	−0.0971286
$107$	14.6809	1.41925	0.709626	−	0.704578i	$-0.248864\pi$
$107$	14.6809	1.41925	0.709626	+	0.704578i	$0.248864\pi$
$108$	1.00000	0.0962250
$109$	−1.70725	−0.163525	−0.0817625	−	0.996652i	$-0.526055\pi$
$109$	−1.70725	−0.163525	−0.0817625	+	0.996652i	$0.526055\pi$
$110$	−17.8445	−1.70141
$111$	9.10498	0.864207
$112$	2.75709	0.260520
$113$	−15.9461	−1.50008	−0.750040	−	0.661392i	$-0.769966\pi$
$113$	−15.9461	−1.50008	−0.750040	+	0.661392i	$0.769966\pi$
$114$	−1.00000	−0.0936586
$115$	28.4554	2.65348
$116$	−8.60587	−0.799035
$117$	−0.357980	−0.0330952
$118$	−9.77834	−0.900169
$119$	18.3152	1.67895
$120$	3.19598	0.291752
$121$	20.1745	1.83405
$122$	−6.88945	−0.623741
$123$	6.09268	0.549358
$124$	−9.83935	−0.883600
$125$	−0.684934	−0.0612624
$126$	−2.75709	−0.245621
$127$	−9.11335	−0.808679	−0.404339	−	0.914609i	$-0.632499\pi$
$127$	−9.11335	−0.808679	−0.404339	+	0.914609i	$0.632499\pi$
$128$	−1.00000	−0.0883883
$129$	10.9035	0.959999
$130$	−1.14410	−0.100344
$131$	12.8570	1.12332	0.561662	−	0.827366i	$-0.310162\pi$
$131$	12.8570	1.12332	0.561662	+	0.827366i	$0.310162\pi$
$132$	−5.58341	−0.485974
$133$	2.75709	0.239070
$134$	2.61740	0.226108
$135$	−3.19598	−0.275066
$136$	−6.64295	−0.569628
$137$	16.1837	1.38267	0.691333	−	0.722537i	$-0.257024\pi$
$137$	16.1837	1.38267	0.691333	+	0.722537i	$0.257024\pi$
$138$	8.90349	0.757916
$139$	13.2750	1.12597	0.562987	−	0.826465i	$-0.309652\pi$
$139$	13.2750	1.12597	0.562987	+	0.826465i	$0.309652\pi$
$140$	−8.81161	−0.744717
$141$	−2.58860	−0.217999
$142$	−12.3429	−1.03580
$143$	1.99875	0.167144
$144$	1.00000	0.0833333
$145$	27.5042	2.28410
$146$	7.87313	0.651585
$147$	0.601540	0.0496142
$148$	9.10498	0.748425
$149$	−12.6140	−1.03338	−0.516690	−	0.856173i	$-0.672836\pi$
$149$	−12.6140	−1.03338	−0.516690	+	0.856173i	$0.672836\pi$
$150$	−5.21431	−0.425747
$151$	−4.96964	−0.404424	−0.202212	−	0.979342i	$-0.564813\pi$
$151$	−4.96964	−0.404424	−0.202212	+	0.979342i	$0.564813\pi$
$152$	−1.00000	−0.0811107
$153$	6.64295	0.537050
$154$	15.3940	1.24048
$155$	31.4464	2.52584
$156$	−0.357980	−0.0286613
$157$	−6.86928	−0.548228	−0.274114	−	0.961697i	$-0.588385\pi$
$157$	−6.86928	−0.548228	−0.274114	+	0.961697i	$0.588385\pi$
$158$	2.48178	0.197440
$159$	1.00000	0.0793052
$160$	3.19598	0.252665
$161$	−24.5477	−1.93463
$162$	−1.00000	−0.0785674
$163$	13.7890	1.08003	0.540017	−	0.841654i	$-0.318418\pi$
$163$	13.7890	1.08003	0.540017	+	0.841654i	$0.318418\pi$
$164$	6.09268	0.475758
$165$	17.8445	1.38919
$166$	−1.18756	−0.0921724
$167$	−8.03372	−0.621668	−0.310834	−	0.950464i	$-0.600608\pi$
$167$	−8.03372	−0.621668	−0.310834	+	0.950464i	$0.600608\pi$
$168$	−2.75709	−0.212714
$169$	−12.8719	−0.990142
$170$	21.2307	1.62832
$171$	1.00000	0.0764719
$172$	10.9035	0.831383
$173$	0.738936	0.0561803	0.0280901	−	0.999605i	$-0.491057\pi$
$173$	0.738936	0.0561803	0.0280901	+	0.999605i	$0.491057\pi$
$174$	8.60587	0.652409
$175$	14.3763	1.08675
$176$	−5.58341	−0.420866
$177$	9.77834	0.734985
$178$	−3.43485	−0.257453
$179$	12.5297	0.936514	0.468257	−	0.883592i	$-0.344882\pi$
$179$	12.5297	0.936514	0.468257	+	0.883592i	$0.344882\pi$
$180$	−3.19598	−0.238215
$181$	24.6253	1.83038	0.915192	−	0.403019i	$-0.132039\pi$
$181$	24.6253	1.83038	0.915192	+	0.403019i	$0.132039\pi$
$182$	0.986982	0.0731600
$183$	6.88945	0.509283
$184$	8.90349	0.656374
$185$	−29.0994	−2.13943
$186$	9.83935	0.721456
$187$	−37.0903	−2.71231
$188$	−2.58860	−0.188793
$189$	2.75709	0.200549
$190$	3.19598	0.231861
$191$	−12.2282	−0.884803	−0.442401	−	0.896817i	$-0.645873\pi$
$191$	−12.2282	−0.884803	−0.442401	+	0.896817i	$0.645873\pi$
$192$	1.00000	0.0721688
$193$	14.0920	1.01437	0.507183	−	0.861838i	$-0.330687\pi$
$193$	14.0920	1.01437	0.507183	+	0.861838i	$0.330687\pi$
$194$	13.8356	0.993336
$195$	1.14410	0.0819305
$196$	0.601540	0.0429672
$197$	2.07606	0.147913	0.0739566	−	0.997261i	$-0.476437\pi$
$197$	2.07606	0.147913	0.0739566	+	0.997261i	$0.476437\pi$
$198$	5.58341	0.396796
$199$	−10.2695	−0.727986	−0.363993	−	0.931402i	$-0.618587\pi$
$199$	−10.2695	−0.727986	−0.363993	+	0.931402i	$0.618587\pi$
$200$	−5.21431	−0.368707
$201$	−2.61740	−0.184617
$202$	5.06221	0.356176
$203$	−23.7271	−1.66532
$204$	6.64295	0.465099
$205$	−19.4721	−1.35999
$206$	−3.86336	−0.269173
$207$	−8.90349	−0.618835
$208$	−0.357980	−0.0248214
$209$	−5.58341	−0.386213
$210$	8.81161	0.608059
$211$	20.4818	1.41002	0.705011	−	0.709196i	$-0.250942\pi$
$211$	20.4818	1.41002	0.705011	+	0.709196i	$0.250942\pi$
$212$	1.00000	0.0686803
$213$	12.3429	0.845723
$214$	−14.6809	−1.00356
$215$	−34.8474	−2.37657
$216$	−1.00000	−0.0680414
$217$	−27.1280	−1.84157
$218$	1.70725	0.115630
$219$	−7.87313	−0.532017
$220$	17.8445	1.20308
$221$	−2.37804	−0.159964
$222$	−9.10498	−0.611086
$223$	3.48319	0.233252	0.116626	−	0.993176i	$-0.462792\pi$
$223$	3.48319	0.233252	0.116626	+	0.993176i	$0.462792\pi$
$224$	−2.75709	−0.184216
$225$	5.21431	0.347621
$226$	15.9461	1.06072
$227$	9.68301	0.642684	0.321342	−	0.946963i	$-0.395866\pi$
$227$	9.68301	0.642684	0.321342	+	0.946963i	$0.395866\pi$
$228$	1.00000	0.0662266
$229$	−8.30561	−0.548850	−0.274425	−	0.961609i	$-0.588488\pi$
$229$	−8.30561	−0.548850	−0.274425	+	0.961609i	$0.588488\pi$
$230$	−28.4554	−1.87629
$231$	−15.3940	−1.01285
$232$	8.60587	0.565003
$233$	18.4922	1.21146	0.605731	−	0.795669i	$-0.292881\pi$
$233$	18.4922	1.21146	0.605731	+	0.795669i	$0.292881\pi$
$234$	0.357980	0.0234019
$235$	8.27311	0.539678
$236$	9.77834	0.636515
$237$	−2.48178	−0.161209
$238$	−18.3152	−1.18720
$239$	−12.1846	−0.788154	−0.394077	−	0.919077i	$-0.628936\pi$
$239$	−12.1846	−0.788154	−0.394077	+	0.919077i	$0.628936\pi$
$240$	−3.19598	−0.206300
$241$	24.2270	1.56060	0.780300	−	0.625405i	$-0.215066\pi$
$241$	24.2270	1.56060	0.780300	+	0.625405i	$0.215066\pi$
$242$	−20.1745	−1.29687
$243$	1.00000	0.0641500
$244$	6.88945	0.441052
$245$	−1.92251	−0.122825
$246$	−6.09268	−0.388455
$247$	−0.357980	−0.0227777
$248$	9.83935	0.624799
$249$	1.18756	0.0752584
$250$	0.684934	0.0433191
$251$	12.6617	0.799200	0.399600	−	0.916690i	$-0.369149\pi$
$251$	12.6617	0.799200	0.399600	+	0.916690i	$0.369149\pi$
$252$	2.75709	0.173680
$253$	49.7119	3.12536
$254$	9.11335	0.571822
$255$	−21.2307	−1.32952
$256$	1.00000	0.0625000
$257$	−25.8070	−1.60980	−0.804899	−	0.593411i	$-0.797781\pi$
$257$	−25.8070	−1.60980	−0.804899	+	0.593411i	$0.797781\pi$
$258$	−10.9035	−0.678822
$259$	25.1032	1.55984
$260$	1.14410	0.0709539
$261$	−8.60587	−0.532690
$262$	−12.8570	−0.794311
$263$	25.4126	1.56701	0.783503	−	0.621388i	$-0.213431\pi$
$263$	25.4126	1.56701	0.783503	+	0.621388i	$0.213431\pi$
$264$	5.58341	0.343635
$265$	−3.19598	−0.196328
$266$	−2.75709	−0.169048
$267$	3.43485	0.210210
$268$	−2.61740	−0.159883
$269$	30.6074	1.86617	0.933083	−	0.359661i	$-0.117108\pi$
$269$	30.6074	1.86617	0.933083	+	0.359661i	$0.117108\pi$
$270$	3.19598	0.194501
$271$	10.1469	0.616380	0.308190	−	0.951325i	$-0.400277\pi$
$271$	10.1469	0.616380	0.308190	+	0.951325i	$0.400277\pi$
$272$	6.64295	0.402788
$273$	−0.986982	−0.0597349
$274$	−16.1837	−0.977692
$275$	−29.1137	−1.75562
$276$	−8.90349	−0.535927
$277$	−10.3668	−0.622882	−0.311441	−	0.950266i	$-0.600812\pi$
$277$	−10.3668	−0.622882	−0.311441	+	0.950266i	$0.600812\pi$
$278$	−13.2750	−0.796184
$279$	−9.83935	−0.589067
$280$	8.81161	0.526595
$281$	24.5562	1.46490	0.732450	−	0.680820i	$-0.238377\pi$
$281$	24.5562	1.46490	0.732450	+	0.680820i	$0.238377\pi$
$282$	2.58860	0.154149
$283$	−3.88067	−0.230682	−0.115341	−	0.993326i	$-0.536796\pi$
$283$	−3.88067	−0.230682	−0.115341	+	0.993326i	$0.536796\pi$
$284$	12.3429	0.732418
$285$	−3.19598	−0.189314
$286$	−1.99875	−0.118188
$287$	16.7981	0.991558
$288$	−1.00000	−0.0589256
$289$	27.1287	1.59581
$290$	−27.5042	−1.61510
$291$	−13.8356	−0.811055
$292$	−7.87313	−0.460740
$293$	0.651074	0.0380362	0.0190181	−	0.999819i	$-0.493946\pi$
$293$	0.651074	0.0380362	0.0190181	+	0.999819i	$0.493946\pi$
$294$	−0.601540	−0.0350825
$295$	−31.2514	−1.81953
$296$	−9.10498	−0.529216
$297$	−5.58341	−0.323982
$298$	12.6140	0.730710
$299$	3.18727	0.184324
$300$	5.21431	0.301048
$301$	30.0619	1.73274
$302$	4.96964	0.285971
$303$	−5.06221	−0.290817
$304$	1.00000	0.0573539
$305$	−22.0186	−1.26078
$306$	−6.64295	−0.379752
$307$	−29.3170	−1.67321	−0.836606	−	0.547805i	$-0.815464\pi$
$307$	−29.3170	−1.67321	−0.836606	+	0.547805i	$0.815464\pi$
$308$	−15.3940	−0.877153
$309$	3.86336	0.219779
$310$	−31.4464	−1.78604
$311$	31.5013	1.78627	0.893136	−	0.449786i	$-0.148500\pi$
$311$	31.5013	1.78627	0.893136	+	0.449786i	$0.148500\pi$
$312$	0.357980	0.0202666
$313$	15.6047	0.882028	0.441014	−	0.897500i	$-0.354619\pi$
$313$	15.6047	0.882028	0.441014	+	0.897500i	$0.354619\pi$
$314$	6.86928	0.387656
$315$	−8.81161	−0.496478
$316$	−2.48178	−0.139611
$317$	−0.382368	−0.0214759	−0.0107380	−	0.999942i	$-0.503418\pi$
$317$	−0.382368	−0.0214759	−0.0107380	+	0.999942i	$0.503418\pi$
$318$	−1.00000	−0.0560772
$319$	48.0501	2.69029
$320$	−3.19598	−0.178661
$321$	14.6809	0.819406
$322$	24.5477	1.36799
$323$	6.64295	0.369623
$324$	1.00000	0.0555556
$325$	−1.86662	−0.103541
$326$	−13.7890	−0.763700
$327$	−1.70725	−0.0944112
$328$	−6.09268	−0.336412
$329$	−7.13699	−0.393475
$330$	−17.8445	−0.982307
$331$	6.40900	0.352271	0.176135	−	0.984366i	$-0.443640\pi$
$331$	6.40900	0.352271	0.176135	+	0.984366i	$0.443640\pi$
$332$	1.18756	0.0651757
$333$	9.10498	0.498950
$334$	8.03372	0.439586
$335$	8.36515	0.457037
$336$	2.75709	0.150412
$337$	−12.8720	−0.701182	−0.350591	−	0.936529i	$-0.614019\pi$
$337$	−12.8720	−0.701182	−0.350591	+	0.936529i	$0.614019\pi$
$338$	12.8719	0.700136
$339$	−15.9461	−0.866072
$340$	−21.2307	−1.15140
$341$	54.9372	2.97501
$342$	−1.00000	−0.0540738
$343$	−17.6411	−0.952531
$344$	−10.9035	−0.587877
$345$	28.4554	1.53199
$346$	−0.738936	−0.0397254
$347$	9.81481	0.526887	0.263443	−	0.964675i	$-0.415142\pi$
$347$	9.81481	0.526887	0.263443	+	0.964675i	$0.415142\pi$
$348$	−8.60587	−0.461323
$349$	−12.6940	−0.679495	−0.339748	−	0.940517i	$-0.610342\pi$
$349$	−12.6940	−0.679495	−0.339748	+	0.940517i	$0.610342\pi$
$350$	−14.3763	−0.768447
$351$	−0.357980	−0.0191075
$352$	5.58341	0.297597
$353$	−20.6310	−1.09808	−0.549038	−	0.835797i	$-0.685006\pi$
$353$	−20.6310	−1.09808	−0.549038	+	0.835797i	$0.685006\pi$
$354$	−9.77834	−0.519713
$355$	−39.4478	−2.09367
$356$	3.43485	0.182047
$357$	18.3152	0.969343
$358$	−12.5297	−0.662216
$359$	1.70948	0.0902228	0.0451114	−	0.998982i	$-0.485636\pi$
$359$	1.70948	0.0902228	0.0451114	+	0.998982i	$0.485636\pi$
$360$	3.19598	0.168443
$361$	1.00000	0.0526316
$362$	−24.6253	−1.29428
$363$	20.1745	1.05889
$364$	−0.986982	−0.0517319
$365$	25.1624	1.31706
$366$	−6.88945	−0.360117
$367$	20.8011	1.08581	0.542904	−	0.839794i	$-0.317325\pi$
$367$	20.8011	1.08581	0.542904	+	0.839794i	$0.317325\pi$
$368$	−8.90349	−0.464127
$369$	6.09268	0.317172
$370$	29.0994	1.51280
$371$	2.75709	0.143141
$372$	−9.83935	−0.510147
$373$	4.77836	0.247414	0.123707	−	0.992319i	$-0.460522\pi$
$373$	4.77836	0.247414	0.123707	+	0.992319i	$0.460522\pi$
$374$	37.0903	1.91789
$375$	−0.684934	−0.0353699
$376$	2.58860	0.133497
$377$	3.08072	0.158665
$378$	−2.75709	−0.141809
$379$	−8.26753	−0.424675	−0.212337	−	0.977196i	$-0.568108\pi$
$379$	−8.26753	−0.424675	−0.212337	+	0.977196i	$0.568108\pi$
$380$	−3.19598	−0.163950
$381$	−9.11335	−0.466891
$382$	12.2282	0.625650
$383$	13.3559	0.682456	0.341228	−	0.939981i	$-0.389157\pi$
$383$	13.3559	0.682456	0.341228	+	0.939981i	$0.389157\pi$
$384$	−1.00000	−0.0510310
$385$	49.1989	2.50741
$386$	−14.0920	−0.717265
$387$	10.9035	0.554256
$388$	−13.8356	−0.702395
$389$	7.43961	0.377203	0.188602	−	0.982054i	$-0.439605\pi$
$389$	7.43961	0.377203	0.188602	+	0.982054i	$0.439605\pi$
$390$	−1.14410	−0.0579336
$391$	−59.1454	−2.99111
$392$	−0.601540	−0.0303824
$393$	12.8570	0.648552
$394$	−2.07606	−0.104590
$395$	7.93174	0.399089
$396$	−5.58341	−0.280577
$397$	−0.341516	−0.0171402	−0.00857011	−	0.999963i	$-0.502728\pi$
$397$	−0.341516	−0.0171402	−0.00857011	+	0.999963i	$0.502728\pi$
$398$	10.2695	0.514764
$399$	2.75709	0.138027
$400$	5.21431	0.260716
$401$	−16.5150	−0.824721	−0.412360	−	0.911021i	$-0.635296\pi$
$401$	−16.5150	−0.824721	−0.412360	+	0.911021i	$0.635296\pi$
$402$	2.61740	0.130544
$403$	3.52229	0.175458
$404$	−5.06221	−0.251855
$405$	−3.19598	−0.158810
$406$	23.7271	1.17756
$407$	−50.8369	−2.51989
$408$	−6.64295	−0.328875
$409$	−4.64338	−0.229600	−0.114800	−	0.993389i	$-0.536623\pi$
$409$	−4.64338	−0.229600	−0.114800	+	0.993389i	$0.536623\pi$
$410$	19.4721	0.961658
$411$	16.1837	0.798282
$412$	3.86336	0.190334
$413$	26.9597	1.32660
$414$	8.90349	0.437583
$415$	−3.79542	−0.186310
$416$	0.357980	0.0175514
$417$	13.2750	0.650082
$418$	5.58341	0.273094
$419$	−4.43298	−0.216565	−0.108283	−	0.994120i	$-0.534535\pi$
$419$	−4.43298	−0.216565	−0.108283	+	0.994120i	$0.534535\pi$
$420$	−8.81161	−0.429963
$421$	−25.1895	−1.22766	−0.613832	−	0.789437i	$-0.710373\pi$
$421$	−25.1895	−1.22766	−0.613832	+	0.789437i	$0.710373\pi$
$422$	−20.4818	−0.997036
$423$	−2.58860	−0.125862
$424$	−1.00000	−0.0485643
$425$	34.6384	1.68021
$426$	−12.3429	−0.598017
$427$	18.9948	0.919224
$428$	14.6809	0.709626
$429$	1.99875	0.0965005
$430$	34.8474	1.68049
$431$	23.9368	1.15300	0.576499	−	0.817098i	$-0.304419\pi$
$431$	23.9368	1.15300	0.576499	+	0.817098i	$0.304419\pi$
$432$	1.00000	0.0481125
$433$	−6.80969	−0.327253	−0.163626	−	0.986522i	$-0.552319\pi$
$433$	−6.80969	−0.327253	−0.163626	+	0.986522i	$0.552319\pi$
$434$	27.1280	1.30218
$435$	27.5042	1.31873
$436$	−1.70725	−0.0817625
$437$	−8.90349	−0.425912
$438$	7.87313	0.376193
$439$	23.9015	1.14076	0.570379	−	0.821382i	$-0.306796\pi$
$439$	23.9015	1.14076	0.570379	+	0.821382i	$0.306796\pi$
$440$	−17.8445	−0.850703
$441$	0.601540	0.0286448
$442$	2.37804	0.113112
$443$	18.8973	0.897838	0.448919	−	0.893572i	$-0.351809\pi$
$443$	18.8973	0.897838	0.448919	+	0.893572i	$0.351809\pi$
$444$	9.10498	0.432103
$445$	−10.9777	−0.520394
$446$	−3.48319	−0.164934
$447$	−12.6140	−0.596622
$448$	2.75709	0.130260
$449$	25.5614	1.20632	0.603158	−	0.797622i	$-0.293909\pi$
$449$	25.5614	1.20632	0.603158	+	0.797622i	$0.293909\pi$
$450$	−5.21431	−0.245805
$451$	−34.0179	−1.60184
$452$	−15.9461	−0.750040
$453$	−4.96964	−0.233494
$454$	−9.68301	−0.454446
$455$	3.15438	0.147879
$456$	−1.00000	−0.0468293
$457$	31.9762	1.49578	0.747892	−	0.663821i	$-0.231066\pi$
$457$	31.9762	1.49578	0.747892	+	0.663821i	$0.231066\pi$
$458$	8.30561	0.388096
$459$	6.64295	0.310066
$460$	28.4554	1.32674
$461$	−4.96285	−0.231143	−0.115571	−	0.993299i	$-0.536870\pi$
$461$	−4.96285	−0.231143	−0.115571	+	0.993299i	$0.536870\pi$
$462$	15.3940	0.716192
$463$	35.1756	1.63475	0.817375	−	0.576106i	$-0.195428\pi$
$463$	35.1756	1.63475	0.817375	+	0.576106i	$0.195428\pi$
$464$	−8.60587	−0.399517
$465$	31.4464	1.45829
$466$	−18.4922	−0.856633
$467$	−33.6594	−1.55757	−0.778786	−	0.627290i	$-0.784164\pi$
$467$	−33.6594	−1.55757	−0.778786	+	0.627290i	$0.784164\pi$
$468$	−0.357980	−0.0165476
$469$	−7.21639	−0.333222
$470$	−8.27311	−0.381610
$471$	−6.86928	−0.316520
$472$	−9.77834	−0.450084
$473$	−60.8787	−2.79921
$474$	2.48178	0.113992
$475$	5.21431	0.239249
$476$	18.3152	0.839475
$477$	1.00000	0.0457869
$478$	12.1846	0.557309
$479$	−12.8983	−0.589338	−0.294669	−	0.955599i	$-0.595209\pi$
$479$	−12.8983	−0.589338	−0.294669	+	0.955599i	$0.595209\pi$
$480$	3.19598	0.145876
$481$	−3.25940	−0.148616
$482$	−24.2270	−1.10351
$483$	−24.5477	−1.11696
$484$	20.1745	0.917023
$485$	44.2183	2.00785
$486$	−1.00000	−0.0453609
$487$	10.9949	0.498229	0.249114	−	0.968474i	$-0.419861\pi$
$487$	10.9949	0.498229	0.249114	+	0.968474i	$0.419861\pi$
$488$	−6.88945	−0.311871
$489$	13.7890	0.623558
$490$	1.92251	0.0868503
$491$	24.1993	1.09210	0.546050	−	0.837753i	$-0.316131\pi$
$491$	24.1993	1.09210	0.546050	+	0.837753i	$0.316131\pi$
$492$	6.09268	0.274679
$493$	−57.1683	−2.57473
$494$	0.357980	0.0161063
$495$	17.8445	0.802050
$496$	−9.83935	−0.441800
$497$	34.0305	1.52648
$498$	−1.18756	−0.0532157
$499$	4.93749	0.221032	0.110516	−	0.993874i	$-0.464750\pi$
$499$	4.93749	0.221032	0.110516	+	0.993874i	$0.464750\pi$
$500$	−0.684934	−0.0306312
$501$	−8.03372	−0.358920
$502$	−12.6617	−0.565120
$503$	23.1525	1.03232	0.516159	−	0.856493i	$-0.327362\pi$
$503$	23.1525	1.03232	0.516159	+	0.856493i	$0.327362\pi$
$504$	−2.75709	−0.122811
$505$	16.1788	0.719945
$506$	−49.7119	−2.20996
$507$	−12.8719	−0.571659
$508$	−9.11335	−0.404339
$509$	22.7562	1.00865	0.504326	−	0.863513i	$-0.331741\pi$
$509$	22.7562	1.00865	0.504326	+	0.863513i	$0.331741\pi$
$510$	21.2307	0.940113
$511$	−21.7069	−0.960258
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	25.8070	1.13830
$515$	−12.3472	−0.544084
$516$	10.9035	0.479999
$517$	14.4532	0.635651
$518$	−25.1032	−1.10297
$519$	0.738936	0.0324357
$520$	−1.14410	−0.0501720
$521$	−5.08899	−0.222953	−0.111476	−	0.993767i	$-0.535558\pi$
$521$	−5.08899	−0.222953	−0.111476	+	0.993767i	$0.535558\pi$
$522$	8.60587	0.376669
$523$	10.4812	0.458310	0.229155	−	0.973390i	$-0.426404\pi$
$523$	10.4812	0.458310	0.229155	+	0.973390i	$0.426404\pi$
$524$	12.8570	0.561662
$525$	14.3763	0.627434
$526$	−25.4126	−1.10804
$527$	−65.3623	−2.84723
$528$	−5.58341	−0.242987
$529$	56.2722	2.44662
$530$	3.19598	0.138825
$531$	9.77834	0.424344
$532$	2.75709	0.119535
$533$	−2.18105	−0.0944720
$534$	−3.43485	−0.148641
$535$	−46.9198	−2.02852
$536$	2.61740	0.113054
$537$	12.5297	0.540697
$538$	−30.6074	−1.31958
$539$	−3.35865	−0.144667
$540$	−3.19598	−0.137533
$541$	−21.1175	−0.907910	−0.453955	−	0.891025i	$-0.649987\pi$
$541$	−21.1175	−0.907910	−0.453955	+	0.891025i	$0.649987\pi$
$542$	−10.1469	−0.435846
$543$	24.6253	1.05677
$544$	−6.64295	−0.284814
$545$	5.45635	0.233724
$546$	0.986982	0.0422389
$547$	−3.99702	−0.170900	−0.0854501	−	0.996342i	$-0.527233\pi$
$547$	−3.99702	−0.170900	−0.0854501	+	0.996342i	$0.527233\pi$
$548$	16.1837	0.691333
$549$	6.88945	0.294035
$550$	29.1137	1.24141
$551$	−8.60587	−0.366622
$552$	8.90349	0.378958
$553$	−6.84250	−0.290973
$554$	10.3668	0.440444
$555$	−29.0994	−1.23520
$556$	13.2750	0.562987
$557$	−21.1608	−0.896611	−0.448306	−	0.893880i	$-0.647972\pi$
$557$	−21.1608	−0.896611	−0.448306	+	0.893880i	$0.647972\pi$
$558$	9.83935	0.416533
$559$	−3.90323	−0.165089
$560$	−8.81161	−0.372359
$561$	−37.0903	−1.56595
$562$	−24.5562	−1.03584
$563$	−14.6078	−0.615644	−0.307822	−	0.951444i	$-0.599600\pi$
$563$	−14.6078	−0.615644	−0.307822	+	0.951444i	$0.599600\pi$
$564$	−2.58860	−0.109000
$565$	50.9634	2.14405
$566$	3.88067	0.163117
$567$	2.75709	0.115787
$568$	−12.3429	−0.517898
$569$	−31.8948	−1.33710	−0.668550	−	0.743667i	$-0.733085\pi$
$569$	−31.8948	−1.33710	−0.668550	+	0.743667i	$0.733085\pi$
$570$	3.19598	0.133865
$571$	−20.3280	−0.850700	−0.425350	−	0.905029i	$-0.639849\pi$
$571$	−20.3280	−0.850700	−0.425350	+	0.905029i	$0.639849\pi$
$572$	1.99875	0.0835719
$573$	−12.2282	−0.510841
$574$	−16.7981	−0.701137
$575$	−46.4256	−1.93608
$576$	1.00000	0.0416667
$577$	−36.4775	−1.51858	−0.759290	−	0.650753i	$-0.774453\pi$
$577$	−36.4775	−1.51858	−0.759290	+	0.650753i	$0.774453\pi$
$578$	−27.1287	−1.12841
$579$	14.0920	0.585645
$580$	27.5042	1.14205
$581$	3.27420	0.135837
$582$	13.8356	0.573503
$583$	−5.58341	−0.231241
$584$	7.87313	0.325793
$585$	1.14410	0.0473026
$586$	−0.651074	−0.0268956
$587$	−14.2352	−0.587551	−0.293775	−	0.955874i	$-0.594912\pi$
$587$	−14.2352	−0.587551	−0.293775	+	0.955874i	$0.594912\pi$
$588$	0.601540	0.0248071
$589$	−9.83935	−0.405423
$590$	31.2514	1.28660
$591$	2.07606	0.0853977
$592$	9.10498	0.374212
$593$	−3.48580	−0.143144	−0.0715722	−	0.997435i	$-0.522802\pi$
$593$	−3.48580	−0.143144	−0.0715722	+	0.997435i	$0.522802\pi$
$594$	5.58341	0.229090
$595$	−58.5351	−2.39970
$596$	−12.6140	−0.516690
$597$	−10.2695	−0.420303
$598$	−3.18727	−0.130337
$599$	45.3303	1.85215	0.926073	−	0.377343i	$-0.123162\pi$
$599$	45.3303	1.85215	0.926073	+	0.377343i	$0.123162\pi$
$600$	−5.21431	−0.212873
$601$	15.2670	0.622753	0.311376	−	0.950287i	$-0.399210\pi$
$601$	15.2670	0.622753	0.311376	+	0.950287i	$0.399210\pi$
$602$	−30.0619	−1.22523
$603$	−2.61740	−0.106589
$604$	−4.96964	−0.202212
$605$	−64.4774	−2.62138
$606$	5.06221	0.205638
$607$	11.9203	0.483832	0.241916	−	0.970297i	$-0.422224\pi$
$607$	11.9203	0.483832	0.241916	+	0.970297i	$0.422224\pi$
$608$	−1.00000	−0.0405554
$609$	−23.7271	−0.961472
$610$	22.0186	0.891506
$611$	0.926665	0.0374889
$612$	6.64295	0.268525
$613$	14.7958	0.597598	0.298799	−	0.954316i	$-0.403414\pi$
$613$	14.7958	0.597598	0.298799	+	0.954316i	$0.403414\pi$
$614$	29.3170	1.18314
$615$	−19.4721	−0.785191
$616$	15.3940	0.620241
$617$	13.7611	0.554000	0.277000	−	0.960870i	$-0.410660\pi$
$617$	13.7611	0.554000	0.277000	+	0.960870i	$0.410660\pi$
$618$	−3.86336	−0.155407
$619$	15.1920	0.610618	0.305309	−	0.952253i	$-0.401240\pi$
$619$	15.1920	0.610618	0.305309	+	0.952253i	$0.401240\pi$
$620$	31.4464	1.26292
$621$	−8.90349	−0.357285
$622$	−31.5013	−1.26309
$623$	9.47019	0.379415
$624$	−0.357980	−0.0143307
$625$	−23.8825	−0.955301
$626$	−15.6047	−0.623688
$627$	−5.58341	−0.222980
$628$	−6.86928	−0.274114
$629$	60.4839	2.41165
$630$	8.81161	0.351063
$631$	−11.2276	−0.446965	−0.223482	−	0.974708i	$-0.571743\pi$
$631$	−11.2276	−0.446965	−0.223482	+	0.974708i	$0.571743\pi$
$632$	2.48178	0.0987201
$633$	20.4818	0.814077
$634$	0.382368	0.0151858
$635$	29.1261	1.15583
$636$	1.00000	0.0396526
$637$	−0.215339	−0.00853205
$638$	−48.0501	−1.90232
$639$	12.3429	0.488279
$640$	3.19598	0.126332
$641$	−13.5370	−0.534680	−0.267340	−	0.963602i	$-0.586145\pi$
$641$	−13.5370	−0.534680	−0.267340	+	0.963602i	$0.586145\pi$
$642$	−14.6809	−0.579407
$643$	32.4768	1.28076	0.640380	−	0.768058i	$-0.278777\pi$
$643$	32.4768	1.28076	0.640380	+	0.768058i	$0.278777\pi$
$644$	−24.5477	−0.967316
$645$	−34.8474	−1.37211
$646$	−6.64295	−0.261363
$647$	7.94870	0.312496	0.156248	−	0.987718i	$-0.450060\pi$
$647$	7.94870	0.312496	0.156248	+	0.987718i	$0.450060\pi$
$648$	−1.00000	−0.0392837
$649$	−54.5965	−2.14310
$650$	1.86662	0.0732147
$651$	−27.1280	−1.06323
$652$	13.7890	0.540017
$653$	39.7282	1.55468	0.777342	−	0.629079i	$-0.216568\pi$
$653$	39.7282	1.55468	0.777342	+	0.629079i	$0.216568\pi$
$654$	1.70725	0.0667588
$655$	−41.0909	−1.60555
$656$	6.09268	0.237879
$657$	−7.87313	−0.307160
$658$	7.13699	0.278229
$659$	−22.1802	−0.864018	−0.432009	−	0.901869i	$-0.642195\pi$
$659$	−22.1802	−0.864018	−0.432009	+	0.901869i	$0.642195\pi$
$660$	17.8445	0.694596
$661$	29.0166	1.12862	0.564308	−	0.825565i	$-0.309143\pi$
$661$	29.0166	1.12862	0.564308	+	0.825565i	$0.309143\pi$
$662$	−6.40900	−0.249093
$663$	−2.37804	−0.0923554
$664$	−1.18756	−0.0460862
$665$	−8.81161	−0.341700
$666$	−9.10498	−0.352811
$667$	76.6223	2.96683
$668$	−8.03372	−0.310834
$669$	3.48319	0.134668
$670$	−8.36515	−0.323174
$671$	−38.4666	−1.48499
$672$	−2.75709	−0.106357
$673$	−23.1625	−0.892847	−0.446424	−	0.894822i	$-0.647303\pi$
$673$	−23.1625	−0.892847	−0.446424	+	0.894822i	$0.647303\pi$
$674$	12.8720	0.495811
$675$	5.21431	0.200699
$676$	−12.8719	−0.495071
$677$	25.3183	0.973062	0.486531	−	0.873663i	$-0.338262\pi$
$677$	25.3183	0.973062	0.486531	+	0.873663i	$0.338262\pi$
$678$	15.9461	0.612405
$679$	−38.1459	−1.46391
$680$	21.2307	0.814162
$681$	9.68301	0.371054
$682$	−54.9372	−2.10365
$683$	42.2066	1.61499	0.807496	−	0.589874i	$-0.200822\pi$
$683$	42.2066	1.61499	0.807496	+	0.589874i	$0.200822\pi$
$684$	1.00000	0.0382360
$685$	−51.7228	−1.97623
$686$	17.6411	0.673541
$687$	−8.30561	−0.316879
$688$	10.9035	0.415692
$689$	−0.357980	−0.0136379
$690$	−28.4554	−1.08328
$691$	14.2157	0.540792	0.270396	−	0.962749i	$-0.412845\pi$
$691$	14.2157	0.540792	0.270396	+	0.962749i	$0.412845\pi$
$692$	0.738936	0.0280901
$693$	−15.3940	−0.584768
$694$	−9.81481	−0.372565
$695$	−42.4268	−1.60934
$696$	8.60587	0.326205
$697$	40.4733	1.53304
$698$	12.6940	0.480476
$699$	18.4922	0.699438
$700$	14.3763	0.543374
$701$	−20.2810	−0.766002	−0.383001	−	0.923748i	$-0.625109\pi$
$701$	−20.2810	−0.766002	−0.383001	+	0.923748i	$0.625109\pi$
$702$	0.357980	0.0135111
$703$	9.10498	0.343401
$704$	−5.58341	−0.210433
$705$	8.27311	0.311583
$706$	20.6310	0.776457
$707$	−13.9570	−0.524906
$708$	9.77834	0.367492
$709$	−16.4197	−0.616654	−0.308327	−	0.951280i	$-0.599769\pi$
$709$	−16.4197	−0.616654	−0.308327	+	0.951280i	$0.599769\pi$
$710$	39.4478	1.48045
$711$	−2.48178	−0.0930742
$712$	−3.43485	−0.128727
$713$	87.6046	3.28082
$714$	−18.3152	−0.685429
$715$	−6.38797	−0.238896
$716$	12.5297	0.468257
$717$	−12.1846	−0.455041
$718$	−1.70948	−0.0637972
$719$	−8.51579	−0.317585	−0.158793	−	0.987312i	$-0.550760\pi$
$719$	−8.51579	−0.317585	−0.158793	+	0.987312i	$0.550760\pi$
$720$	−3.19598	−0.119107
$721$	10.6516	0.396687
$722$	−1.00000	−0.0372161
$723$	24.2270	0.901013
$724$	24.6253	0.915192
$725$	−44.8737	−1.66657
$726$	−20.1745	−0.748746
$727$	26.7034	0.990374	0.495187	−	0.868786i	$-0.335099\pi$
$727$	26.7034	0.990374	0.495187	+	0.868786i	$0.335099\pi$
$728$	0.986982	0.0365800
$729$	1.00000	0.0370370
$730$	−25.1624	−0.931302
$731$	72.4313	2.67897
$732$	6.88945	0.254641
$733$	10.9155	0.403173	0.201586	−	0.979471i	$-0.435390\pi$
$733$	10.9155	0.403173	0.201586	+	0.979471i	$0.435390\pi$
$734$	−20.8011	−0.767783
$735$	−1.92251	−0.0709129
$736$	8.90349	0.328187
$737$	14.6140	0.538314
$738$	−6.09268	−0.224275
$739$	18.6359	0.685532	0.342766	−	0.939421i	$-0.388636\pi$
$739$	18.6359	0.685532	0.342766	+	0.939421i	$0.388636\pi$
$740$	−29.0994	−1.06971
$741$	−0.357980	−0.0131507
$742$	−2.75709	−0.101216
$743$	−26.0026	−0.953942	−0.476971	−	0.878919i	$-0.658265\pi$
$743$	−26.0026	−0.953942	−0.476971	+	0.878919i	$0.658265\pi$
$744$	9.83935	0.360728
$745$	40.3141	1.47700
$746$	−4.77836	−0.174948
$747$	1.18756	0.0434505
$748$	−37.0903	−1.35616
$749$	40.4764	1.47898
$750$	0.684934	0.0250103
$751$	26.2038	0.956192	0.478096	−	0.878308i	$-0.341327\pi$
$751$	26.2038	0.956192	0.478096	+	0.878308i	$0.341327\pi$
$752$	−2.58860	−0.0943964
$753$	12.6617	0.461418
$754$	−3.08072	−0.112193
$755$	15.8829	0.578038
$756$	2.75709	0.100274
$757$	49.4907	1.79877	0.899385	−	0.437157i	$-0.144015\pi$
$757$	49.4907	1.79877	0.899385	+	0.437157i	$0.144015\pi$
$758$	8.26753	0.300290
$759$	49.7119	1.80443
$760$	3.19598	0.115931
$761$	16.0794	0.582877	0.291438	−	0.956590i	$-0.405866\pi$
$761$	16.0794	0.582877	0.291438	+	0.956590i	$0.405866\pi$
$762$	9.11335	0.330142
$763$	−4.70704	−0.170406
$764$	−12.2282	−0.442401
$765$	−21.2307	−0.767599
$766$	−13.3559	−0.482569
$767$	−3.50045	−0.126394
$768$	1.00000	0.0360844
$769$	−42.1681	−1.52062	−0.760311	−	0.649559i	$-0.774954\pi$
$769$	−42.1681	−1.52062	−0.760311	+	0.649559i	$0.774954\pi$
$770$	−49.1989	−1.77300
$771$	−25.8070	−0.929418
$772$	14.0920	0.507183
$773$	−40.7415	−1.46537	−0.732685	−	0.680568i	$-0.761733\pi$
$773$	−40.7415	−1.46537	−0.732685	+	0.680568i	$0.761733\pi$
$774$	−10.9035	−0.391918
$775$	−51.3054	−1.84295
$776$	13.8356	0.496668
$777$	25.1032	0.900574
$778$	−7.43961	−0.266723
$779$	6.09268	0.218293
$780$	1.14410	0.0409652
$781$	−68.9156	−2.46600
$782$	59.1454	2.11504
$783$	−8.60587	−0.307549
$784$	0.601540	0.0214836
$785$	21.9541	0.783575
$786$	−12.8570	−0.458595
$787$	−41.2425	−1.47014	−0.735069	−	0.677993i	$-0.762850\pi$
$787$	−41.2425	−1.47014	−0.735069	+	0.677993i	$0.762850\pi$
$788$	2.07606	0.0739566
$789$	25.4126	0.904712
$790$	−7.93174	−0.282199
$791$	−43.9648	−1.56321
$792$	5.58341	0.198398
$793$	−2.46628	−0.0875803
$794$	0.341516	0.0121200
$795$	−3.19598	−0.113350
$796$	−10.2695	−0.363993
$797$	−3.42639	−0.121369	−0.0606845	−	0.998157i	$-0.519328\pi$
$797$	−3.42639	−0.121369	−0.0606845	+	0.998157i	$0.519328\pi$
$798$	−2.75709	−0.0975999
$799$	−17.1959	−0.608348
$800$	−5.21431	−0.184354
$801$	3.43485	0.121365
$802$	16.5150	0.583166
$803$	43.9590	1.55128
$804$	−2.61740	−0.0923084
$805$	78.4541	2.76514
$806$	−3.52229	−0.124067
$807$	30.6074	1.07743
$808$	5.06221	0.178088
$809$	−30.7429	−1.08086	−0.540431	−	0.841389i	$-0.681739\pi$
$809$	−30.7429	−1.08086	−0.540431	+	0.841389i	$0.681739\pi$
$810$	3.19598	0.112295
$811$	−15.0023	−0.526803	−0.263401	−	0.964686i	$-0.584844\pi$
$811$	−15.0023	−0.526803	−0.263401	+	0.964686i	$0.584844\pi$
$812$	−23.7271	−0.832659
$813$	10.1469	0.355867
$814$	50.8369	1.78183
$815$	−44.0693	−1.54368
$816$	6.64295	0.232550
$817$	10.9035	0.381465
$818$	4.64338	0.162352
$819$	−0.986982	−0.0344879
$820$	−19.4721	−0.679995
$821$	−2.40832	−0.0840510	−0.0420255	−	0.999117i	$-0.513381\pi$
$821$	−2.40832	−0.0840510	−0.0420255	+	0.999117i	$0.513381\pi$
$822$	−16.1837	−0.564471
$823$	3.92610	0.136855	0.0684276	−	0.997656i	$-0.478202\pi$
$823$	3.92610	0.136855	0.0684276	+	0.997656i	$0.478202\pi$
$824$	−3.86336	−0.134586
$825$	−29.1137	−1.01361
$826$	−26.9597	−0.938049
$827$	4.51611	0.157041	0.0785203	−	0.996913i	$-0.474980\pi$
$827$	4.51611	0.157041	0.0785203	+	0.996913i	$0.474980\pi$
$828$	−8.90349	−0.309418
$829$	−27.1203	−0.941928	−0.470964	−	0.882152i	$-0.656094\pi$
$829$	−27.1203	−0.941928	−0.470964	+	0.882152i	$0.656094\pi$
$830$	3.79542	0.131741
$831$	−10.3668	−0.359621
$832$	−0.357980	−0.0124107
$833$	3.99600	0.138453
$834$	−13.2750	−0.459677
$835$	25.6756	0.888542
$836$	−5.58341	−0.193106
$837$	−9.83935	−0.340098
$838$	4.43298	0.153135
$839$	−31.8198	−1.09854	−0.549271	−	0.835644i	$-0.685094\pi$
$839$	−31.8198	−1.09854	−0.549271	+	0.835644i	$0.685094\pi$
$840$	8.81161	0.304029
$841$	45.0609	1.55383
$842$	25.1895	0.868089
$843$	24.5562	0.845761
$844$	20.4818	0.705011
$845$	41.1382	1.41520
$846$	2.58860	0.0889978
$847$	55.6229	1.91123
$848$	1.00000	0.0343401
$849$	−3.88067	−0.133184
$850$	−34.6384	−1.18809
$851$	−81.0661	−2.77891
$852$	12.3429	0.422862
$853$	−10.0348	−0.343585	−0.171793	−	0.985133i	$-0.554956\pi$
$853$	−10.0348	−0.343585	−0.171793	+	0.985133i	$0.554956\pi$
$854$	−18.9948	−0.649990
$855$	−3.19598	−0.109300
$856$	−14.6809	−0.501781
$857$	−44.3334	−1.51440	−0.757200	−	0.653183i	$-0.773433\pi$
$857$	−44.3334	−1.51440	−0.757200	+	0.653183i	$0.773433\pi$
$858$	−1.99875	−0.0682361
$859$	51.0509	1.74183	0.870916	−	0.491431i	$-0.163526\pi$
$859$	51.0509	1.74183	0.870916	+	0.491431i	$0.163526\pi$
$860$	−34.8474	−1.18829
$861$	16.7981	0.572476
$862$	−23.9368	−0.815292
$863$	24.4866	0.833533	0.416767	−	0.909014i	$-0.363163\pi$
$863$	24.4866	0.833533	0.416767	+	0.909014i	$0.363163\pi$
$864$	−1.00000	−0.0340207
$865$	−2.36163	−0.0802977
$866$	6.80969	0.231403
$867$	27.1287	0.921340
$868$	−27.1280	−0.920783
$869$	13.8568	0.470061
$870$	−27.5042	−0.932480
$871$	0.936974	0.0317482
$872$	1.70725	0.0578148
$873$	−13.8356	−0.468263
$874$	8.90349	0.301165
$875$	−1.88843	−0.0638404
$876$	−7.87313	−0.266009
$877$	2.33869	0.0789720	0.0394860	−	0.999220i	$-0.487428\pi$
$877$	2.33869	0.0789720	0.0394860	+	0.999220i	$0.487428\pi$
$878$	−23.9015	−0.806638
$879$	0.651074	0.0219602
$880$	17.8445	0.601538
$881$	−8.34677	−0.281210	−0.140605	−	0.990066i	$-0.544905\pi$
$881$	−8.34677	−0.281210	−0.140605	+	0.990066i	$0.544905\pi$
$882$	−0.601540	−0.0202549
$883$	−27.0459	−0.910168	−0.455084	−	0.890449i	$-0.650391\pi$
$883$	−27.0459	−0.910168	−0.455084	+	0.890449i	$0.650391\pi$
$884$	−2.37804	−0.0799821
$885$	−31.2514	−1.05050
$886$	−18.8973	−0.634867
$887$	52.4484	1.76104	0.880522	−	0.474006i	$-0.157192\pi$
$887$	52.4484	1.76104	0.880522	+	0.474006i	$0.157192\pi$
$888$	−9.10498	−0.305543
$889$	−25.1263	−0.842710
$890$	10.9777	0.367974
$891$	−5.58341	−0.187051
$892$	3.48319	0.116626
$893$	−2.58860	−0.0866241
$894$	12.6140	0.421875
$895$	−40.0447	−1.33855
$896$	−2.75709	−0.0921079
$897$	3.18727	0.106420
$898$	−25.5614	−0.852994
$899$	84.6761	2.82411
$900$	5.21431	0.173810
$901$	6.64295	0.221309
$902$	34.0179	1.13267
$903$	30.0619	1.00040
$904$	15.9461	0.530359
$905$	−78.7020	−2.61614
$906$	4.96964	0.165105
$907$	−46.5775	−1.54658	−0.773290	−	0.634053i	$-0.781390\pi$
$907$	−46.5775	−1.54658	−0.773290	+	0.634053i	$0.781390\pi$
$908$	9.68301	0.321342
$909$	−5.06221	−0.167903
$910$	−3.15438	−0.104567
$911$	−9.44780	−0.313019	−0.156510	−	0.987676i	$-0.550024\pi$
$911$	−9.44780	−0.313019	−0.156510	+	0.987676i	$0.550024\pi$
$912$	1.00000	0.0331133
$913$	−6.63063	−0.219442
$914$	−31.9762	−1.05768
$915$	−22.0186	−0.727911
$916$	−8.30561	−0.274425
$917$	35.4480	1.17060
$918$	−6.64295	−0.219250
$919$	−52.8887	−1.74464	−0.872318	−	0.488939i	$-0.837384\pi$
$919$	−52.8887	−1.74464	−0.872318	+	0.488939i	$0.837384\pi$
$920$	−28.4554	−0.938147
$921$	−29.3170	−0.966029
$922$	4.96285	0.163443
$923$	−4.41852	−0.145437
$924$	−15.3940	−0.506424
$925$	47.4762	1.56101
$926$	−35.1756	−1.15594
$927$	3.86336	0.126889
$928$	8.60587	0.282501
$929$	−18.4509	−0.605354	−0.302677	−	0.953093i	$-0.597880\pi$
$929$	−18.4509	−0.605354	−0.302677	+	0.953093i	$0.597880\pi$
$930$	−31.4464	−1.03117
$931$	0.601540	0.0197147
$932$	18.4922	0.605731
$933$	31.5013	1.03131
$934$	33.6594	1.10137
$935$	118.540	3.87667
$936$	0.357980	0.0117009
$937$	12.8605	0.420133	0.210067	−	0.977687i	$-0.432632\pi$
$937$	12.8605	0.420133	0.210067	+	0.977687i	$0.432632\pi$
$938$	7.21639	0.235624
$939$	15.6047	0.509239
$940$	8.27311	0.269839
$941$	21.4951	0.700719	0.350359	−	0.936615i	$-0.386059\pi$
$941$	21.4951	0.700719	0.350359	+	0.936615i	$0.386059\pi$
$942$	6.86928	0.223813
$943$	−54.2461	−1.76650
$944$	9.77834	0.318258
$945$	−8.81161	−0.286642
$946$	60.8787	1.97934
$947$	−37.6752	−1.22428	−0.612139	−	0.790750i	$-0.709691\pi$
$947$	−37.6752	−1.22428	−0.612139	+	0.790750i	$0.709691\pi$
$948$	−2.48178	−0.0806046
$949$	2.81842	0.0914898
$950$	−5.21431	−0.169175
$951$	−0.382368	−0.0123991
$952$	−18.3152	−0.593599
$953$	−9.17488	−0.297203	−0.148602	−	0.988897i	$-0.547477\pi$
$953$	−9.17488	−0.297203	−0.148602	+	0.988897i	$0.547477\pi$
$954$	−1.00000	−0.0323762
$955$	39.0812	1.26464
$956$	−12.1846	−0.394077
$957$	48.0501	1.55324
$958$	12.8983	0.416725
$959$	44.6198	1.44085
$960$	−3.19598	−0.103150
$961$	65.8128	2.12299
$962$	3.25940	0.105087
$963$	14.6809	0.473084
$964$	24.2270	0.780300
$965$	−45.0379	−1.44982
$966$	24.5477	0.789810
$967$	−1.87569	−0.0603182	−0.0301591	−	0.999545i	$-0.509601\pi$
$967$	−1.87569	−0.0603182	−0.0301591	+	0.999545i	$0.509601\pi$
$968$	−20.1745	−0.648433
$969$	6.64295	0.213402
$970$	−44.2183	−1.41976
$971$	26.9574	0.865105	0.432552	−	0.901609i	$-0.357613\pi$
$971$	26.9574	0.865105	0.432552	+	0.901609i	$0.357613\pi$
$972$	1.00000	0.0320750
$973$	36.6005	1.17336
$974$	−10.9949	−0.352301
$975$	−1.86662	−0.0597796
$976$	6.88945	0.220526
$977$	−29.2765	−0.936637	−0.468318	−	0.883560i	$-0.655140\pi$
$977$	−29.2765	−0.936637	−0.468318	+	0.883560i	$0.655140\pi$
$978$	−13.7890	−0.440922
$979$	−19.1782	−0.612938
$980$	−1.92251	−0.0614124
$981$	−1.70725	−0.0545083
$982$	−24.1993	−0.772231
$983$	−58.7435	−1.87363	−0.936814	−	0.349829i	$-0.886240\pi$
$983$	−58.7435	−1.87363	−0.936814	+	0.349829i	$0.886240\pi$
$984$	−6.09268	−0.194227
$985$	−6.63505	−0.211410
$986$	57.1683	1.82061
$987$	−7.13699	−0.227173
$988$	−0.357980	−0.0113888
$989$	−97.0791	−3.08694
$990$	−17.8445	−0.567135
$991$	−19.7187	−0.626384	−0.313192	−	0.949690i	$-0.601398\pi$
$991$	−19.7187	−0.626384	−0.313192	+	0.949690i	$0.601398\pi$
$992$	9.83935	0.312400
$993$	6.40900	0.203383
$994$	−34.0305	−1.07938
$995$	32.8212	1.04050
$996$	1.18756	0.0376292
$997$	−41.6077	−1.31773	−0.658864	−	0.752262i	$-0.728963\pi$
$997$	−41.6077	−1.31773	−0.658864	+	0.752262i	$0.728963\pi$
$998$	−4.93749	−0.156294
$999$	9.10498	0.288069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.bf.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.bf.1.3	✓	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.19598 q^{5} -1.00000 q^{6} +2.75709 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.19598 q^{5} -1.00000 q^{6} +2.75709 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.19598 q^{10} -5.58341 q^{11} +1.00000 q^{12} -0.357980 q^{13} -2.75709 q^{14} -3.19598 q^{15} +1.00000 q^{16} +6.64295 q^{17} -1.00000 q^{18} +1.00000 q^{19} -3.19598 q^{20} +2.75709 q^{21} +5.58341 q^{22} -8.90349 q^{23} -1.00000 q^{24} +5.21431 q^{25} +0.357980 q^{26} +1.00000 q^{27} +2.75709 q^{28} -8.60587 q^{29} +3.19598 q^{30} -9.83935 q^{31} -1.00000 q^{32} -5.58341 q^{33} -6.64295 q^{34} -8.81161 q^{35} +1.00000 q^{36} +9.10498 q^{37} -1.00000 q^{38} -0.357980 q^{39} +3.19598 q^{40} +6.09268 q^{41} -2.75709 q^{42} +10.9035 q^{43} -5.58341 q^{44} -3.19598 q^{45} +8.90349 q^{46} -2.58860 q^{47} +1.00000 q^{48} +0.601540 q^{49} -5.21431 q^{50} +6.64295 q^{51} -0.357980 q^{52} +1.00000 q^{53} -1.00000 q^{54} +17.8445 q^{55} -2.75709 q^{56} +1.00000 q^{57} +8.60587 q^{58} +9.77834 q^{59} -3.19598 q^{60} +6.88945 q^{61} +9.83935 q^{62} +2.75709 q^{63} +1.00000 q^{64} +1.14410 q^{65} +5.58341 q^{66} -2.61740 q^{67} +6.64295 q^{68} -8.90349 q^{69} +8.81161 q^{70} +12.3429 q^{71} -1.00000 q^{72} -7.87313 q^{73} -9.10498 q^{74} +5.21431 q^{75} +1.00000 q^{76} -15.3940 q^{77} +0.357980 q^{78} -2.48178 q^{79} -3.19598 q^{80} +1.00000 q^{81} -6.09268 q^{82} +1.18756 q^{83} +2.75709 q^{84} -21.2307 q^{85} -10.9035 q^{86} -8.60587 q^{87} +5.58341 q^{88} +3.43485 q^{89} +3.19598 q^{90} -0.986982 q^{91} -8.90349 q^{92} -9.83935 q^{93} +2.58860 q^{94} -3.19598 q^{95} -1.00000 q^{96} -13.8356 q^{97} -0.601540 q^{98} -5.58341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 7 * q^5 - 12 * q^6 + 5 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} + 10 q^{13} - 5 q^{14} - 7 q^{15} + 12 q^{16} + 6 q^{17} - 12 q^{18} + 12 q^{19} - 7 q^{20} + 5 q^{21} + 5 q^{22} - 12 q^{24} + 21 q^{25} - 10 q^{26} + 12 q^{27} + 5 q^{28} + 7 q^{30} + 2 q^{31} - 12 q^{32} - 5 q^{33} - 6 q^{34} - 17 q^{35} + 12 q^{36} + 16 q^{37} - 12 q^{38} + 10 q^{39} + 7 q^{40} + 7 q^{41} - 5 q^{42} + 24 q^{43} - 5 q^{44} - 7 q^{45} + 4 q^{47} + 12 q^{48} + 29 q^{49} - 21 q^{50} + 6 q^{51} + 10 q^{52} + 12 q^{53} - 12 q^{54} - 2 q^{55} - 5 q^{56} + 12 q^{57} + 2 q^{59} - 7 q^{60} + 29 q^{61} - 2 q^{62} + 5 q^{63} + 12 q^{64} - 17 q^{65} + 5 q^{66} + 36 q^{67} + 6 q^{68} + 17 q^{70} - 3 q^{71} - 12 q^{72} + 7 q^{73} - 16 q^{74} + 21 q^{75} + 12 q^{76} - 35 q^{77} - 10 q^{78} + 40 q^{79} - 7 q^{80} + 12 q^{81} - 7 q^{82} + 24 q^{83} + 5 q^{84} - 22 q^{85} - 24 q^{86} + 5 q^{88} - 45 q^{89} + 7 q^{90} + 71 q^{91} + 2 q^{93} - 4 q^{94} - 7 q^{95} - 12 q^{96} + q^{97} - 29 q^{98} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^3 + 12 * q^4 - 7 * q^5 - 12 * q^6 + 5 * q^7 - 12 * q^8 + 12 * q^9 + 7 * q^10 - 5 * q^11 + 12 * q^12 + 10 * q^13 - 5 * q^14 - 7 * q^15 + 12 * q^16 + 6 * q^17 - 12 * q^18 + 12 * q^19 - 7 * q^20 + 5 * q^21 + 5 * q^22 - 12 * q^24 + 21 * q^25 - 10 * q^26 + 12 * q^27 + 5 * q^28 + 7 * q^30 + 2 * q^31 - 12 * q^32 - 5 * q^33 - 6 * q^34 - 17 * q^35 + 12 * q^36 + 16 * q^37 - 12 * q^38 + 10 * q^39 + 7 * q^40 + 7 * q^41 - 5 * q^42 + 24 * q^43 - 5 * q^44 - 7 * q^45 + 4 * q^47 + 12 * q^48 + 29 * q^49 - 21 * q^50 + 6 * q^51 + 10 * q^52 + 12 * q^53 - 12 * q^54 - 2 * q^55 - 5 * q^56 + 12 * q^57 + 2 * q^59 - 7 * q^60 + 29 * q^61 - 2 * q^62 + 5 * q^63 + 12 * q^64 - 17 * q^65 + 5 * q^66 + 36 * q^67 + 6 * q^68 + 17 * q^70 - 3 * q^71 - 12 * q^72 + 7 * q^73 - 16 * q^74 + 21 * q^75 + 12 * q^76 - 35 * q^77 - 10 * q^78 + 40 * q^79 - 7 * q^80 + 12 * q^81 - 7 * q^82 + 24 * q^83 + 5 * q^84 - 22 * q^85 - 24 * q^86 + 5 * q^88 - 45 * q^89 + 7 * q^90 + 71 * q^91 + 2 * q^93 - 4 * q^94 - 7 * q^95 - 12 * q^96 + q^97 - 29 * q^98 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more