Properties

Label 3332.1.cc.a.2447.1
Level $3332$
Weight $1$
Character 3332.2447
Analytic conductor $1.663$
Analytic rank $0$
Dimension $12$
Projective image $D_{21}$
CM discriminant -68
Inner twists $4$

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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] = [3332,1,Mod(135,3332)] 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("3332.135"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([21, 32, 21])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.cc (of order \(42\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,1,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.66288462209\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{21}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{21} + \cdots)\)

Embedding invariants

Embedding label 2447.1
Root \(0.955573 + 0.294755i\) of defining polynomial
Character \(\chi\) \(=\) 3332.2447
Dual form 3332.1.cc.a.2515.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.365341 - 0.930874i) q^{2} +(-0.0111692 + 0.149042i) q^{3} +(-0.733052 - 0.680173i) q^{4} +(0.134659 + 0.0648483i) q^{6} +(0.900969 - 0.433884i) q^{7} +(-0.900969 + 0.433884i) q^{8} +(0.966742 + 0.145713i) q^{9} +(0.722521 - 0.108903i) q^{11} +(0.109562 - 0.101659i) q^{12} +(0.455573 + 0.571270i) q^{13} +(-0.0747301 - 0.997204i) q^{14} +(0.0747301 + 0.997204i) q^{16} +(0.955573 + 0.294755i) q^{17} +(0.488831 - 0.846680i) q^{18} +(0.0546039 + 0.139129i) q^{21} +(0.162592 - 0.712362i) q^{22} +(-1.19158 + 0.367554i) q^{23} +(-0.0546039 - 0.139129i) q^{24} +(0.365341 + 0.930874i) q^{25} +(0.698220 - 0.215372i) q^{26} +(-0.0657731 + 0.288171i) q^{27} +(-0.955573 - 0.294755i) q^{28} +(-0.900969 + 1.56052i) q^{31} +(0.955573 + 0.294755i) q^{32} +(0.00816111 + 0.108903i) q^{33} +(0.623490 - 0.781831i) q^{34} +(-0.609562 - 0.764367i) q^{36} +(-0.0902318 + 0.0615190i) q^{39} +0.149460 q^{42} +(-0.603718 - 0.411608i) q^{44} +(-0.0931869 + 1.24349i) q^{46} -0.149460 q^{48} +(0.623490 - 0.781831i) q^{49} +1.00000 q^{50} +(-0.0546039 + 0.139129i) q^{51} +(0.0546039 - 0.728639i) q^{52} +(-1.21135 - 1.12397i) q^{53} +(0.244221 + 0.166507i) q^{54} +(-0.623490 + 0.781831i) q^{56} +(1.12349 + 1.40881i) q^{62} +(0.934227 - 0.288171i) q^{63} +(0.623490 - 0.781831i) q^{64} +(0.104356 + 0.0321896i) q^{66} +(-0.500000 - 0.866025i) q^{68} +(-0.0414721 - 0.181701i) q^{69} +(0.367711 - 1.61105i) q^{71} +(-0.934227 + 0.288171i) q^{72} +(-0.142820 + 0.0440542i) q^{75} +(0.603718 - 0.411608i) q^{77} +(0.0243010 + 0.106470i) q^{78} +(-0.988831 - 1.71271i) q^{79} +(0.892012 + 0.275149i) q^{81} +(0.0546039 - 0.139129i) q^{84} +(-0.603718 + 0.411608i) q^{88} +(-0.147791 - 0.0222759i) q^{89} +(0.658322 + 0.317031i) q^{91} +(1.12349 + 0.541044i) q^{92} +(-0.222521 - 0.151712i) q^{93} +(-0.0546039 + 0.139129i) q^{96} +(-0.500000 - 0.866025i) q^{98} +0.714360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - 13 q^{3} + q^{4} + 5 q^{6} + 2 q^{7} - 2 q^{8} + 14 q^{9} + 8 q^{11} + q^{12} - 5 q^{13} - q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} - 2 q^{22} - 2 q^{23} + q^{24} + q^{25} - q^{26}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(e\left(\frac{11}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.365341 − 0.930874i 0.365341 − 0.930874i
\(3\) −0.0111692 + 0.149042i −0.0111692 + 0.149042i 0.988831 + 0.149042i \(0.0476190\pi\)
−1.00000 \(\pi\)
\(4\) −0.733052 − 0.680173i −0.733052 − 0.680173i
\(5\) 0 0 −0.826239 − 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(6\) 0.134659 + 0.0648483i 0.134659 + 0.0648483i
\(7\) 0.900969 − 0.433884i 0.900969 − 0.433884i
\(8\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(9\) 0.966742 + 0.145713i 0.966742 + 0.145713i
\(10\) 0 0
\(11\) 0.722521 − 0.108903i 0.722521 − 0.108903i 0.222521 − 0.974928i \(-0.428571\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0.109562 − 0.101659i 0.109562 − 0.101659i
\(13\) 0.455573 + 0.571270i 0.455573 + 0.571270i 0.955573 − 0.294755i \(-0.0952381\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) −0.0747301 − 0.997204i −0.0747301 − 0.997204i
\(15\) 0 0
\(16\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(17\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(18\) 0.488831 − 0.846680i 0.488831 − 0.846680i
\(19\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0.0546039 + 0.139129i 0.0546039 + 0.139129i
\(22\) 0.162592 − 0.712362i 0.162592 − 0.712362i
\(23\) −1.19158 + 0.367554i −1.19158 + 0.367554i −0.826239 − 0.563320i \(-0.809524\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(24\) −0.0546039 − 0.139129i −0.0546039 − 0.139129i
\(25\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(26\) 0.698220 − 0.215372i 0.698220 − 0.215372i
\(27\) −0.0657731 + 0.288171i −0.0657731 + 0.288171i
\(28\) −0.955573 − 0.294755i −0.955573 − 0.294755i
\(29\) 0 0 −0.222521 − 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(30\) 0 0
\(31\) −0.900969 + 1.56052i −0.900969 + 1.56052i −0.0747301 + 0.997204i \(0.523810\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(32\) 0.955573 + 0.294755i 0.955573 + 0.294755i
\(33\) 0.00816111 + 0.108903i 0.00816111 + 0.108903i
\(34\) 0.623490 − 0.781831i 0.623490 − 0.781831i
\(35\) 0 0
\(36\) −0.609562 − 0.764367i −0.609562 − 0.764367i
\(37\) 0 0 0.733052 − 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(38\) 0 0
\(39\) −0.0902318 + 0.0615190i −0.0902318 + 0.0615190i
\(40\) 0 0
\(41\) 0 0 0.900969 − 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(42\) 0.149460 0.149460
\(43\) 0 0 −0.900969 − 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(44\) −0.603718 − 0.411608i −0.603718 − 0.411608i
\(45\) 0 0
\(46\) −0.0931869 + 1.24349i −0.0931869 + 1.24349i
\(47\) 0 0 0.365341 − 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(48\) −0.149460 −0.149460
\(49\) 0.623490 − 0.781831i 0.623490 − 0.781831i
\(50\) 1.00000 1.00000
\(51\) −0.0546039 + 0.139129i −0.0546039 + 0.139129i
\(52\) 0.0546039 − 0.728639i 0.0546039 − 0.728639i
\(53\) −1.21135 − 1.12397i −1.21135 − 1.12397i −0.988831 − 0.149042i \(-0.952381\pi\)
−0.222521 − 0.974928i \(-0.571429\pi\)
\(54\) 0.244221 + 0.166507i 0.244221 + 0.166507i
\(55\) 0 0
\(56\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.826239 − 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(60\) 0 0
\(61\) 0 0 0.733052 − 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(62\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(63\) 0.934227 − 0.288171i 0.934227 − 0.288171i
\(64\) 0.623490 − 0.781831i 0.623490 − 0.781831i
\(65\) 0 0
\(66\) 0.104356 + 0.0321896i 0.104356 + 0.0321896i
\(67\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(69\) −0.0414721 − 0.181701i −0.0414721 − 0.181701i
\(70\) 0 0
\(71\) 0.367711 − 1.61105i 0.367711 − 1.61105i −0.365341 − 0.930874i \(-0.619048\pi\)
0.733052 − 0.680173i \(-0.238095\pi\)
\(72\) −0.934227 + 0.288171i −0.934227 + 0.288171i
\(73\) 0 0 −0.365341 − 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(74\) 0 0
\(75\) −0.142820 + 0.0440542i −0.142820 + 0.0440542i
\(76\) 0 0
\(77\) 0.603718 − 0.411608i 0.603718 − 0.411608i
\(78\) 0.0243010 + 0.106470i 0.0243010 + 0.106470i
\(79\) −0.988831 − 1.71271i −0.988831 − 1.71271i −0.623490 − 0.781831i \(-0.714286\pi\)
−0.365341 − 0.930874i \(-0.619048\pi\)
\(80\) 0 0
\(81\) 0.892012 + 0.275149i 0.892012 + 0.275149i
\(82\) 0 0
\(83\) 0 0 0.623490 − 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(84\) 0.0546039 − 0.139129i 0.0546039 − 0.139129i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.603718 + 0.411608i −0.603718 + 0.411608i
\(89\) −0.147791 − 0.0222759i −0.147791 − 0.0222759i 0.0747301 − 0.997204i \(-0.476190\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(90\) 0 0
\(91\) 0.658322 + 0.317031i 0.658322 + 0.317031i
\(92\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(93\) −0.222521 − 0.151712i −0.222521 − 0.151712i
\(94\) 0 0
\(95\) 0 0
\(96\) −0.0546039 + 0.139129i −0.0546039 + 0.139129i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(99\) 0.714360 0.714360
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 3332.1.cc.a.2447.1 ✓ 12
4.3 odd 2 3332.1.cc.b.2447.1 yes 12
17.16 even 2 3332.1.cc.b.2447.1 yes 12
49.16 even 21 inner 3332.1.cc.a.2515.1 yes 12
68.67 odd 2 CM 3332.1.cc.a.2447.1 ✓ 12
196.163 odd 42 3332.1.cc.b.2515.1 yes 12
833.16 even 42 3332.1.cc.b.2515.1 yes 12
3332.2515 odd 42 inner 3332.1.cc.a.2515.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.cc.a.2447.1 ✓ 12 1.1 even 1 trivial
3332.1.cc.a.2447.1 ✓ 12 68.67 odd 2 CM
3332.1.cc.a.2515.1 yes 12 49.16 even 21 inner
3332.1.cc.a.2515.1 yes 12 3332.2515 odd 42 inner
3332.1.cc.b.2447.1 yes 12 4.3 odd 2
3332.1.cc.b.2447.1 yes 12 17.16 even 2
3332.1.cc.b.2515.1 yes 12 196.163 odd 42
3332.1.cc.b.2515.1 yes 12 833.16 even 42