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Properties

Label 2940.1.dd.a.1739.2
Level $2940$
Weight $1$
Character 2940.1739
Analytic conductor $1.467$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
CM discriminant -20
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] = [2940,1,Mod(59,2940)] 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("2940.59"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 21, 21, 13])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2940.dd (of order \(42\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,-2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.46725113714\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{42})\)
Coefficient field: \(\Q(\zeta_{84})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{42}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

Embedding invariants

Embedding label 1739.2
Root \(-0.930874 - 0.365341i\) of defining polynomial
Character \(\chi\) \(=\) 2940.1739
Dual form 2940.1.dd.a.2399.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.294755 + 0.955573i) q^{2} +(0.974928 - 0.222521i) q^{3} +(-0.826239 + 0.563320i) q^{4} +(0.988831 + 0.149042i) q^{5} +(0.500000 + 0.866025i) q^{6} +(0.563320 + 0.826239i) q^{7} +(-0.781831 - 0.623490i) q^{8} +(0.900969 - 0.433884i) q^{9} +(0.149042 + 0.988831i) q^{10} +(-0.680173 + 0.733052i) q^{12} +(-0.623490 + 0.781831i) q^{14} +(0.997204 - 0.0747301i) q^{15} +(0.365341 - 0.930874i) q^{16} +(0.680173 + 0.733052i) q^{18} +(-0.900969 + 0.433884i) q^{20} +(0.733052 + 0.680173i) q^{21} +(-0.728639 + 0.0546039i) q^{23} +(-0.900969 - 0.433884i) q^{24} +(0.955573 + 0.294755i) q^{25} +(0.781831 - 0.623490i) q^{27} +(-0.930874 - 0.365341i) q^{28} +(-0.865341 - 1.79690i) q^{29} +(0.365341 + 0.930874i) q^{30} +(0.997204 + 0.0747301i) q^{32} +(0.433884 + 0.900969i) q^{35} +(-0.500000 + 0.866025i) q^{36} +(-0.680173 - 0.733052i) q^{40} +(-0.914101 + 1.14625i) q^{41} +(-0.433884 + 0.900969i) q^{42} +(-0.848162 - 1.06356i) q^{43} +(0.955573 - 0.294755i) q^{45} +(-0.266948 - 0.680173i) q^{46} +(-1.49419 + 0.460898i) q^{47} +(0.149042 - 0.988831i) q^{48} +(-0.365341 + 0.930874i) q^{49} +1.00000i q^{50} +(0.826239 + 0.563320i) q^{54} +(0.0747301 - 0.997204i) q^{56} +(1.46200 - 1.35654i) q^{58} +(-0.781831 + 0.623490i) q^{60} +(1.04876 - 1.53825i) q^{61} +(0.866025 + 0.500000i) q^{63} +(0.222521 + 0.974928i) q^{64} +(-0.866025 + 1.50000i) q^{67} +(-0.698220 + 0.215372i) q^{69} +(-0.733052 + 0.680173i) q^{70} +(-0.974928 - 0.222521i) q^{72} +(0.997204 + 0.0747301i) q^{75} +(0.500000 - 0.866025i) q^{80} +(0.623490 - 0.781831i) q^{81} +(-1.36476 - 0.535628i) q^{82} +(0.250701 + 1.09839i) q^{83} +(-0.988831 - 0.149042i) q^{84} +(0.766310 - 1.12397i) q^{86} +(-1.24349 - 1.55929i) q^{87} +(-0.109562 + 0.101659i) q^{89} +(0.563320 + 0.826239i) q^{90} +(0.571270 - 0.455573i) q^{92} +(-0.880843 - 1.29196i) q^{94} +(0.988831 - 0.149042i) q^{96} +(-0.997204 - 0.0747301i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{4} - 2 q^{5} + 12 q^{6} + 4 q^{9} + 4 q^{14} + 2 q^{16} - 4 q^{20} - 2 q^{21} - 4 q^{24} + 2 q^{25} - 14 q^{29} + 2 q^{30} - 12 q^{36} + 4 q^{41} + 2 q^{45} - 26 q^{46} - 2 q^{49} + 2 q^{54}+ \cdots - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(e\left(\frac{37}{42}\right)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.294755 + 0.955573i 0.294755 + 0.955573i
\(3\) 0.974928 − 0.222521i 0.974928 − 0.222521i
\(4\) −0.826239 + 0.563320i −0.826239 + 0.563320i
\(5\) 0.988831 + 0.149042i 0.988831 + 0.149042i
\(6\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(7\) 0.563320 + 0.826239i 0.563320 + 0.826239i
\(8\) −0.781831 − 0.623490i −0.781831 − 0.623490i
\(9\) 0.900969 − 0.433884i 0.900969 − 0.433884i
\(10\) 0.149042 + 0.988831i 0.149042 + 0.988831i
\(11\) 0 0 0.680173 − 0.733052i \(-0.261905\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(12\) −0.680173 + 0.733052i −0.680173 + 0.733052i
\(13\) 0 0 −0.974928 − 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(14\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(15\) 0.997204 − 0.0747301i 0.997204 − 0.0747301i
\(16\) 0.365341 − 0.930874i 0.365341 − 0.930874i
\(17\) 0 0 0.0747301 − 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(18\) 0.680173 + 0.733052i 0.680173 + 0.733052i
\(19\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(20\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(21\) 0.733052 + 0.680173i 0.733052 + 0.680173i
\(22\) 0 0
\(23\) −0.728639 + 0.0546039i −0.728639 + 0.0546039i −0.433884 − 0.900969i \(-0.642857\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(24\) −0.900969 − 0.433884i −0.900969 − 0.433884i
\(25\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(26\) 0 0
\(27\) 0.781831 − 0.623490i 0.781831 − 0.623490i
\(28\) −0.930874 − 0.365341i −0.930874 − 0.365341i
\(29\) −0.865341 − 1.79690i −0.865341 − 1.79690i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.365341 − 0.930874i \(-0.619048\pi\)
\(30\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(31\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) 0.997204 + 0.0747301i 0.997204 + 0.0747301i
\(33\) 0 0
\(34\) 0 0
\(35\) 0.433884 + 0.900969i 0.433884 + 0.900969i
\(36\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(37\) 0 0 −0.826239 − 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.680173 − 0.733052i −0.680173 − 0.733052i
\(41\) −0.914101 + 1.14625i −0.914101 + 1.14625i 0.0747301 + 0.997204i \(0.476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(42\) −0.433884 + 0.900969i −0.433884 + 0.900969i
\(43\) −0.848162 − 1.06356i −0.848162 − 1.06356i −0.997204 − 0.0747301i \(-0.976190\pi\)
0.149042 − 0.988831i \(-0.452381\pi\)
\(44\) 0 0
\(45\) 0.955573 − 0.294755i 0.955573 − 0.294755i
\(46\) −0.266948 − 0.680173i −0.266948 − 0.680173i
\(47\) −1.49419 + 0.460898i −1.49419 + 0.460898i −0.930874 − 0.365341i \(-0.880952\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(48\) 0.149042 − 0.988831i 0.149042 − 0.988831i
\(49\) −0.365341 + 0.930874i −0.365341 + 0.930874i
\(50\) 1.00000i 1.00000i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.563320 − 0.826239i \(-0.690476\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(54\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(55\) 0 0
\(56\) 0.0747301 − 0.997204i 0.0747301 − 0.997204i
\(57\) 0 0
\(58\) 1.46200 − 1.35654i 1.46200 − 1.35654i
\(59\) 0 0 0.988831 − 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(60\) −0.781831 + 0.623490i −0.781831 + 0.623490i
\(61\) 1.04876 − 1.53825i 1.04876 − 1.53825i 0.222521 − 0.974928i \(-0.428571\pi\)
0.826239 − 0.563320i \(-0.190476\pi\)
\(62\) 0 0
\(63\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(64\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(68\) 0 0
\(69\) −0.698220 + 0.215372i −0.698220 + 0.215372i
\(70\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(71\) 0 0 0.433884 − 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(72\) −0.974928 − 0.222521i −0.974928 − 0.222521i
\(73\) 0 0 0.294755 − 0.955573i \(-0.404762\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(74\) 0 0
\(75\) 0.997204 + 0.0747301i 0.997204 + 0.0747301i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(81\) 0.623490 − 0.781831i 0.623490 − 0.781831i
\(82\) −1.36476 − 0.535628i −1.36476 − 0.535628i
\(83\) 0.250701 + 1.09839i 0.250701 + 1.09839i 0.930874 + 0.365341i \(0.119048\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(84\) −0.988831 − 0.149042i −0.988831 − 0.149042i
\(85\) 0 0
\(86\) 0.766310 − 1.12397i 0.766310 − 1.12397i
\(87\) −1.24349 − 1.55929i −1.24349 − 1.55929i
\(88\) 0 0
\(89\) −0.109562 + 0.101659i −0.109562 + 0.101659i −0.733052 − 0.680173i \(-0.761905\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(90\) 0.563320 + 0.826239i 0.563320 + 0.826239i
\(91\) 0 0
\(92\) 0.571270 − 0.455573i 0.571270 − 0.455573i
\(93\) 0 0
\(94\) −0.880843 − 1.29196i −0.880843 − 1.29196i
\(95\) 0 0
\(96\) 0.988831 − 0.149042i 0.988831 − 0.149042i
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −0.997204 − 0.0747301i −0.997204 − 0.0747301i
\(99\) 0 0
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 2940.1.dd.a.1739.2 yes 24
3.2 odd 2 2940.1.dd.b.1739.1 yes 24
4.3 odd 2 inner 2940.1.dd.a.1739.1 ✓ 24
5.4 even 2 inner 2940.1.dd.a.1739.1 ✓ 24
12.11 even 2 2940.1.dd.b.1739.2 yes 24
15.14 odd 2 2940.1.dd.b.1739.2 yes 24
20.19 odd 2 CM 2940.1.dd.a.1739.2 yes 24
49.47 odd 42 2940.1.dd.b.2399.1 yes 24
60.59 even 2 2940.1.dd.b.1739.1 yes 24
147.47 even 42 inner 2940.1.dd.a.2399.2 yes 24
196.47 even 42 2940.1.dd.b.2399.2 yes 24
245.194 odd 42 2940.1.dd.b.2399.2 yes 24
588.47 odd 42 inner 2940.1.dd.a.2399.1 yes 24
735.194 even 42 inner 2940.1.dd.a.2399.1 yes 24
980.439 even 42 2940.1.dd.b.2399.1 yes 24
2940.2399 odd 42 inner 2940.1.dd.a.2399.2 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2940.1.dd.a.1739.1 ✓ 24 4.3 odd 2 inner
2940.1.dd.a.1739.1 ✓ 24 5.4 even 2 inner
2940.1.dd.a.1739.2 yes 24 1.1 even 1 trivial
2940.1.dd.a.1739.2 yes 24 20.19 odd 2 CM
2940.1.dd.a.2399.1 yes 24 588.47 odd 42 inner
2940.1.dd.a.2399.1 yes 24 735.194 even 42 inner
2940.1.dd.a.2399.2 yes 24 147.47 even 42 inner
2940.1.dd.a.2399.2 yes 24 2940.2399 odd 42 inner
2940.1.dd.b.1739.1 yes 24 3.2 odd 2
2940.1.dd.b.1739.1 yes 24 60.59 even 2
2940.1.dd.b.1739.2 yes 24 12.11 even 2
2940.1.dd.b.1739.2 yes 24 15.14 odd 2
2940.1.dd.b.2399.1 yes 24 49.47 odd 42
2940.1.dd.b.2399.1 yes 24 980.439 even 42
2940.1.dd.b.2399.2 yes 24 196.47 even 42
2940.1.dd.b.2399.2 yes 24 245.194 odd 42