Properties

Label 3332.1.cc.b.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.b.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 \(-0.952381\pi\)
1.00000 \(0\)
\(4\) −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 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(12\) −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 0.365341 − 0.930874i \(-0.380952\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(24\) 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 \(-0.476190\pi\)
0.826239 − 0.563320i \(-0.190476\pi\)
\(32\) 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 \(0.380952\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(72\) −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 \(0.285714\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(80\) 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.b.2447.1 yes 12
4.3 odd 2 3332.1.cc.a.2447.1 ✓ 12
17.16 even 2 3332.1.cc.a.2447.1 ✓ 12
49.16 even 21 inner 3332.1.cc.b.2515.1 yes 12
68.67 odd 2 CM 3332.1.cc.b.2447.1 yes 12
196.163 odd 42 3332.1.cc.a.2515.1 yes 12
833.16 even 42 3332.1.cc.a.2515.1 yes 12
3332.2515 odd 42 inner 3332.1.cc.b.2515.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.cc.a.2447.1 ✓ 12 4.3 odd 2
3332.1.cc.a.2447.1 ✓ 12 17.16 even 2
3332.1.cc.a.2515.1 yes 12 196.163 odd 42
3332.1.cc.a.2515.1 yes 12 833.16 even 42
3332.1.cc.b.2447.1 yes 12 1.1 even 1 trivial
3332.1.cc.b.2447.1 yes 12 68.67 odd 2 CM
3332.1.cc.b.2515.1 yes 12 49.16 even 21 inner
3332.1.cc.b.2515.1 yes 12 3332.2515 odd 42 inner