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Properties

Label 507.2.k.h.188.1
Level $507$
Weight $2$
Character 507.188
Analytic conductor $4.048$
Analytic rank $0$
Dimension $8$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [507,2,Mod(80,507)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("507.80"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.k (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,12,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.56070144.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 188.1
Root \(0.500000 + 2.19293i\) of defining polynomial
Character \(\chi\) \(=\) 507.188
Dual form 507.2.k.h.89.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.31259 + 0.619657i) q^{2} +(0.866025 - 1.50000i) q^{3} +(3.23205 - 1.86603i) q^{4} +(1.23931 + 1.23931i) q^{5} +(-1.07328 + 4.00552i) q^{6} +(-2.93225 + 2.93225i) q^{8} +(-1.50000 - 2.59808i) q^{9} +(-3.63397 - 2.09808i) q^{10} +(-1.69293 - 6.31812i) q^{11} -6.46410i q^{12} +(2.93225 - 0.785693i) q^{15} +(1.23205 - 2.13397i) q^{16} +(5.07880 + 5.07880i) q^{18} +(6.31812 + 1.69293i) q^{20} +(7.83013 + 13.5622i) q^{22} +(1.85897 + 6.93777i) q^{24} -1.92820i q^{25} -5.19615 q^{27} +(-6.29423 + 3.63397i) q^{30} +(0.619657 - 2.31259i) q^{32} +(-10.9433 - 2.93225i) q^{33} +(-9.69615 - 5.59808i) q^{36} -7.26795 q^{40} +(-7.55743 + 2.02501i) q^{41} +(3.46410 - 2.00000i) q^{43} +(-17.2614 - 17.2614i) q^{44} +(1.36086 - 5.07880i) q^{45} +(7.10381 - 7.10381i) q^{47} +(-2.13397 - 3.69615i) q^{48} +(6.06218 + 3.50000i) q^{49} +(1.19482 + 4.45915i) q^{50} +(12.0166 - 3.21983i) q^{54} +(5.73205 - 9.92820i) q^{55} +(-0.453620 - 0.121547i) q^{59} +(8.01105 - 8.01105i) q^{60} +(-6.92820 - 12.0000i) q^{61} +10.6603i q^{64} +27.1244 q^{66} +(4.17156 - 15.5685i) q^{71} +(12.0166 + 3.21983i) q^{72} +(-2.89230 - 1.66987i) q^{75} -10.3923 q^{79} +(4.17156 - 1.11777i) q^{80} +(-4.50000 + 7.79423i) q^{81} +(16.2224 - 9.36603i) q^{82} +(8.91829 + 8.91829i) q^{83} +(-6.77174 + 6.77174i) q^{86} +(23.4904 + 13.5622i) q^{88} +(1.11777 + 4.17156i) q^{89} +12.5885i q^{90} +(-12.0263 + 20.8301i) q^{94} +(-2.93225 - 2.93225i) q^{96} +(-16.1881 - 4.33760i) q^{98} +(-13.8755 + 13.8755i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4} - 12 q^{9} - 36 q^{10} - 4 q^{16} + 28 q^{22} + 12 q^{30} - 36 q^{36} - 72 q^{40} - 24 q^{48} + 32 q^{55} + 120 q^{66} + 60 q^{75} - 36 q^{81} + 12 q^{82} + 84 q^{88} - 20 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{12}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.31259 + 0.619657i −1.63525 + 0.438164i −0.955430 − 0.295217i \(-0.904608\pi\)
−0.679818 + 0.733380i \(0.737941\pi\)
\(3\) 0.866025 − 1.50000i 0.500000 − 0.866025i
\(4\) 3.23205 − 1.86603i 1.61603 − 0.933013i
\(5\) 1.23931 + 1.23931i 0.554238 + 0.554238i 0.927661 − 0.373423i \(-0.121816\pi\)
−0.373423 + 0.927661i \(0.621816\pi\)
\(6\) −1.07328 + 4.00552i −0.438164 + 1.63525i
\(7\) 0 0 −0.965926 − 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(8\) −2.93225 + 2.93225i −1.03671 + 1.03671i
\(9\) −1.50000 − 2.59808i −0.500000 − 0.866025i
\(10\) −3.63397 − 2.09808i −1.14916 − 0.663470i
\(11\) −1.69293 − 6.31812i −0.510439 − 1.90498i −0.415756 − 0.909476i \(-0.636483\pi\)
−0.0946823 − 0.995508i \(-0.530184\pi\)
\(12\) − 6.46410i − 1.86603i
\(13\) 0 0
\(14\) 0 0
\(15\) 2.93225 − 0.785693i 0.757103 − 0.202865i
\(16\) 1.23205 − 2.13397i 0.308013 − 0.533494i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 5.07880 + 5.07880i 1.19709 + 1.19709i
\(19\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(20\) 6.31812 + 1.69293i 1.41277 + 0.378552i
\(21\) 0 0
\(22\) 7.83013 + 13.5622i 1.66939 + 2.89147i
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 1.85897 + 6.93777i 0.379461 + 1.41617i
\(25\) − 1.92820i − 0.385641i
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) −6.29423 + 3.63397i −1.14916 + 0.663470i
\(31\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(32\) 0.619657 − 2.31259i 0.109541 − 0.408812i
\(33\) −10.9433 − 2.93225i −1.90498 − 0.510439i
\(34\) 0 0
\(35\) 0 0
\(36\) −9.69615 − 5.59808i −1.61603 − 0.933013i
\(37\) 0 0 −0.258819 − 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −7.26795 −1.14916
\(41\) −7.55743 + 2.02501i −1.18027 + 0.316253i −0.795034 − 0.606564i \(-0.792547\pi\)
−0.385238 + 0.922817i \(0.625881\pi\)
\(42\) 0 0
\(43\) 3.46410 − 2.00000i 0.528271 − 0.304997i −0.212041 − 0.977261i \(-0.568011\pi\)
0.740312 + 0.672264i \(0.234678\pi\)
\(44\) −17.2614 − 17.2614i −2.60226 − 2.60226i
\(45\) 1.36086 − 5.07880i 0.202865 − 0.757103i
\(46\) 0 0
\(47\) 7.10381 − 7.10381i 1.03620 − 1.03620i 0.0368772 − 0.999320i \(-0.488259\pi\)
0.999320 − 0.0368772i \(-0.0117410\pi\)
\(48\) −2.13397 − 3.69615i −0.308013 − 0.533494i
\(49\) 6.06218 + 3.50000i 0.866025 + 0.500000i
\(50\) 1.19482 + 4.45915i 0.168974 + 0.630618i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 12.0166 − 3.21983i 1.63525 − 0.438164i
\(55\) 5.73205 − 9.92820i 0.772910 − 1.33872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.453620 − 0.121547i −0.0590563 − 0.0158241i 0.229170 − 0.973386i \(-0.426399\pi\)
−0.288226 + 0.957562i \(0.593066\pi\)
\(60\) 8.01105 − 8.01105i 1.03422 − 1.03422i
\(61\) −6.92820 − 12.0000i −0.887066 − 1.53644i −0.843328 − 0.537400i \(-0.819407\pi\)
−0.0437377 − 0.999043i \(-0.513927\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 10.6603i 1.33253i
\(65\) 0 0
\(66\) 27.1244 3.33878
\(67\) 0 0 0.965926 − 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.17156 − 15.5685i 0.495073 − 1.84764i −0.0345462 − 0.999403i \(-0.510999\pi\)
0.529619 − 0.848235i \(-0.322335\pi\)
\(72\) 12.0166 + 3.21983i 1.41617 + 0.379461i
\(73\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) −2.89230 − 1.66987i −0.333975 − 0.192820i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 − 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) 4.17156 − 1.11777i 0.466395 − 0.124970i
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 16.2224 − 9.36603i 1.79147 − 1.03430i
\(83\) 8.91829 + 8.91829i 0.978910 + 0.978910i 0.999782 − 0.0208726i \(-0.00664445\pi\)
−0.0208726 + 0.999782i \(0.506644\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.77174 + 6.77174i −0.730215 + 0.730215i
\(87\) 0 0
\(88\) 23.4904 + 13.5622i 2.50408 + 1.44573i
\(89\) 1.11777 + 4.17156i 0.118483 + 0.442185i 0.999524 − 0.0308556i \(-0.00982320\pi\)
−0.881041 + 0.473040i \(0.843157\pi\)
\(90\) 12.5885i 1.32694i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −12.0263 + 20.8301i −1.24042 + 2.14846i
\(95\) 0 0
\(96\) −2.93225 − 2.93225i −0.299271 − 0.299271i
\(97\) 0 0 0.258819 − 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(98\) −16.1881 − 4.33760i −1.63525 − 0.438164i
\(99\) −13.8755 + 13.8755i −1.39454 + 1.39454i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 507.2.k.h.188.1 8
3.2 odd 2 inner 507.2.k.h.188.2 8
13.2 odd 12 inner 507.2.k.h.89.1 8
13.3 even 3 507.2.k.g.80.2 8
13.4 even 6 507.2.f.d.239.1 ✓ 8
13.5 odd 4 507.2.k.g.488.2 8
13.6 odd 12 507.2.f.d.437.4 yes 8
13.7 odd 12 507.2.f.d.437.1 yes 8
13.8 odd 4 507.2.k.g.488.1 8
13.9 even 3 507.2.f.d.239.4 yes 8
13.10 even 6 507.2.k.g.80.1 8
13.11 odd 12 inner 507.2.k.h.89.2 8
13.12 even 2 inner 507.2.k.h.188.2 8
39.2 even 12 inner 507.2.k.h.89.2 8
39.5 even 4 507.2.k.g.488.1 8
39.8 even 4 507.2.k.g.488.2 8
39.11 even 12 inner 507.2.k.h.89.1 8
39.17 odd 6 507.2.f.d.239.4 yes 8
39.20 even 12 507.2.f.d.437.4 yes 8
39.23 odd 6 507.2.k.g.80.2 8
39.29 odd 6 507.2.k.g.80.1 8
39.32 even 12 507.2.f.d.437.1 yes 8
39.35 odd 6 507.2.f.d.239.1 ✓ 8
39.38 odd 2 CM 507.2.k.h.188.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.f.d.239.1 ✓ 8 13.4 even 6
507.2.f.d.239.1 ✓ 8 39.35 odd 6
507.2.f.d.239.4 yes 8 13.9 even 3
507.2.f.d.239.4 yes 8 39.17 odd 6
507.2.f.d.437.1 yes 8 13.7 odd 12
507.2.f.d.437.1 yes 8 39.32 even 12
507.2.f.d.437.4 yes 8 13.6 odd 12
507.2.f.d.437.4 yes 8 39.20 even 12
507.2.k.g.80.1 8 13.10 even 6
507.2.k.g.80.1 8 39.29 odd 6
507.2.k.g.80.2 8 13.3 even 3
507.2.k.g.80.2 8 39.23 odd 6
507.2.k.g.488.1 8 13.8 odd 4
507.2.k.g.488.1 8 39.5 even 4
507.2.k.g.488.2 8 13.5 odd 4
507.2.k.g.488.2 8 39.8 even 4
507.2.k.h.89.1 8 13.2 odd 12 inner
507.2.k.h.89.1 8 39.11 even 12 inner
507.2.k.h.89.2 8 13.11 odd 12 inner
507.2.k.h.89.2 8 39.2 even 12 inner
507.2.k.h.188.1 8 1.1 even 1 trivial
507.2.k.h.188.1 8 39.38 odd 2 CM
507.2.k.h.188.2 8 3.2 odd 2 inner
507.2.k.h.188.2 8 13.12 even 2 inner