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Properties

Label 3332.1.cc.c.2991.1
Level $3332$
Weight $1$
Character 3332.2991
Analytic conductor $1.663$
Analytic rank $0$
Dimension $24$
Projective image $D_{42}$
CM discriminant -68
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] = [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 = [24,-2,0] 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: \(24\)
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: \(\Z[a_1, a_2, a_3]\)
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 2991.1
Root \(-0.149042 - 0.988831i\) of defining polynomial
Character \(\chi\) \(=\) 3332.2991
Dual form 3332.1.cc.c.1495.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.955573 - 0.294755i) q^{2} +(-0.680173 + 1.73305i) q^{3} +(0.826239 + 0.563320i) q^{4} +(1.16078 - 1.45557i) q^{6} +(-0.781831 + 0.623490i) q^{7} +(-0.623490 - 0.781831i) q^{8} +(-1.80778 - 1.67738i) q^{9} +(-0.432142 + 0.400969i) q^{11} +(-1.53825 + 1.04876i) q^{12} +(0.425270 + 1.86323i) q^{13} +(0.930874 - 0.365341i) q^{14} +(0.365341 + 0.930874i) q^{16} +(0.0747301 + 0.997204i) q^{17} +(1.23305 + 2.13571i) q^{18} +(-0.548760 - 1.77904i) q^{21} +(0.531130 - 0.255779i) q^{22} +(-0.145713 + 1.94440i) q^{23} +(1.77904 - 0.548760i) q^{24} +(0.955573 - 0.294755i) q^{25} +(0.142820 - 1.90580i) q^{26} +(2.45921 - 1.18429i) q^{27} +(-0.997204 + 0.0747301i) q^{28} +(0.781831 + 1.35417i) q^{31} +(-0.0747301 - 0.997204i) q^{32} +(-0.400969 - 1.02165i) q^{33} +(0.222521 - 0.974928i) q^{34} +(-0.548760 - 2.40427i) q^{36} +(-3.51833 - 0.530303i) q^{39} +1.86175i q^{42} +(-0.582926 + 0.0878620i) q^{44} +(0.712362 - 1.81507i) q^{46} -1.86175 q^{48} +(0.222521 - 0.974928i) q^{49} -1.00000 q^{50} +(-1.77904 - 0.548760i) q^{51} +(-0.698220 + 1.77904i) q^{52} +(-1.63402 - 1.11406i) q^{53} +(-2.69903 + 0.406813i) q^{54} +(0.974928 + 0.222521i) q^{56} +(-0.347948 - 1.52446i) q^{62} +(2.45921 + 0.184292i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(0.0820177 + 1.09445i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(-3.27064 - 1.57506i) q^{69} +(0.268565 - 0.129334i) q^{71} +(-0.184292 + 2.45921i) q^{72} +(-0.139129 + 1.85654i) q^{75} +(0.0878620 - 0.582926i) q^{77} +(3.20571 + 1.54379i) q^{78} +(0.680173 - 1.17809i) q^{79} +(0.195461 + 2.60825i) q^{81} +(0.548760 - 1.77904i) q^{84} +(0.582926 + 0.0878620i) q^{88} +(0.535628 + 0.496990i) q^{89} +(-1.49419 - 1.19158i) q^{91} +(-1.21572 + 1.52446i) q^{92} +(-2.87863 + 0.433884i) q^{93} +(1.77904 + 0.548760i) q^{96} +(-0.500000 + 0.866025i) q^{98} +1.45379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.955573 − 0.294755i −0.955573 − 0.294755i
\(3\) −0.680173 + 1.73305i −0.680173 + 1.73305i 1.00000i \(0.5\pi\)
−0.680173 + 0.733052i \(0.738095\pi\)
\(4\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(5\) 0 0 0.988831 − 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(6\) 1.16078 − 1.45557i 1.16078 − 1.45557i
\(7\) −0.781831 + 0.623490i −0.781831 + 0.623490i
\(8\) −0.623490 − 0.781831i −0.623490 − 0.781831i
\(9\) −1.80778 − 1.67738i −1.80778 − 1.67738i
\(10\) 0 0
\(11\) −0.432142 + 0.400969i −0.432142 + 0.400969i −0.866025 − 0.500000i \(-0.833333\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(12\) −1.53825 + 1.04876i −1.53825 + 1.04876i
\(13\) 0.425270 + 1.86323i 0.425270 + 1.86323i 0.500000 + 0.866025i \(0.333333\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(14\) 0.930874 − 0.365341i 0.930874 − 0.365341i
\(15\) 0 0
\(16\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(17\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(18\) 1.23305 + 2.13571i 1.23305 + 2.13571i
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) −0.548760 − 1.77904i −0.548760 − 1.77904i
\(22\) 0.531130 − 0.255779i 0.531130 − 0.255779i
\(23\) −0.145713 + 1.94440i −0.145713 + 1.94440i 0.149042 + 0.988831i \(0.452381\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(24\) 1.77904 − 0.548760i 1.77904 − 0.548760i
\(25\) 0.955573 − 0.294755i 0.955573 − 0.294755i
\(26\) 0.142820 − 1.90580i 0.142820 − 1.90580i
\(27\) 2.45921 − 1.18429i 2.45921 − 1.18429i
\(28\) −0.997204 + 0.0747301i −0.997204 + 0.0747301i
\(29\) 0 0 −0.900969 − 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(30\) 0 0
\(31\) 0.781831 + 1.35417i 0.781831 + 1.35417i 0.930874 + 0.365341i \(0.119048\pi\)
−0.149042 + 0.988831i \(0.547619\pi\)
\(32\) −0.0747301 − 0.997204i −0.0747301 − 0.997204i
\(33\) −0.400969 − 1.02165i −0.400969 − 1.02165i
\(34\) 0.222521 − 0.974928i 0.222521 − 0.974928i
\(35\) 0 0
\(36\) −0.548760 − 2.40427i −0.548760 − 2.40427i
\(37\) 0 0 0.826239 − 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(38\) 0 0
\(39\) −3.51833 − 0.530303i −3.51833 − 0.530303i
\(40\) 0 0
\(41\) 0 0 −0.623490 − 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(42\) 1.86175i 1.86175i
\(43\) 0 0 0.623490 − 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(44\) −0.582926 + 0.0878620i −0.582926 + 0.0878620i
\(45\) 0 0
\(46\) 0.712362 − 1.81507i 0.712362 − 1.81507i
\(47\) 0 0 −0.955573 − 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(48\) −1.86175 −1.86175
\(49\) 0.222521 − 0.974928i 0.222521 − 0.974928i
\(50\) −1.00000 −1.00000
\(51\) −1.77904 − 0.548760i −1.77904 − 0.548760i
\(52\) −0.698220 + 1.77904i −0.698220 + 1.77904i
\(53\) −1.63402 − 1.11406i −1.63402 − 1.11406i −0.900969 − 0.433884i \(-0.857143\pi\)
−0.733052 − 0.680173i \(-0.761905\pi\)
\(54\) −2.69903 + 0.406813i −2.69903 + 0.406813i
\(55\) 0 0
\(56\) 0.974928 + 0.222521i 0.974928 + 0.222521i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.988831 − 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(60\) 0 0
\(61\) 0 0 0.826239 − 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(62\) −0.347948 − 1.52446i −0.347948 − 1.52446i
\(63\) 2.45921 + 0.184292i 2.45921 + 0.184292i
\(64\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(65\) 0 0
\(66\) 0.0820177 + 1.09445i 0.0820177 + 1.09445i
\(67\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(69\) −3.27064 − 1.57506i −3.27064 − 1.57506i
\(70\) 0 0
\(71\) 0.268565 − 0.129334i 0.268565 − 0.129334i −0.294755 − 0.955573i \(-0.595238\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(72\) −0.184292 + 2.45921i −0.184292 + 2.45921i
\(73\) 0 0 0.955573 − 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(74\) 0 0
\(75\) −0.139129 + 1.85654i −0.139129 + 1.85654i
\(76\) 0 0
\(77\) 0.0878620 − 0.582926i 0.0878620 − 0.582926i
\(78\) 3.20571 + 1.54379i 3.20571 + 1.54379i
\(79\) 0.680173 − 1.17809i 0.680173 − 1.17809i −0.294755 − 0.955573i \(-0.595238\pi\)
0.974928 − 0.222521i \(-0.0714286\pi\)
\(80\) 0 0
\(81\) 0.195461 + 2.60825i 0.195461 + 2.60825i
\(82\) 0 0
\(83\) 0 0 0.222521 − 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(84\) 0.548760 − 1.77904i 0.548760 − 1.77904i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.582926 + 0.0878620i 0.582926 + 0.0878620i
\(89\) 0.535628 + 0.496990i 0.535628 + 0.496990i 0.900969 − 0.433884i \(-0.142857\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(90\) 0 0
\(91\) −1.49419 − 1.19158i −1.49419 − 1.19158i
\(92\) −1.21572 + 1.52446i −1.21572 + 1.52446i
\(93\) −2.87863 + 0.433884i −2.87863 + 0.433884i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.77904 + 0.548760i 1.77904 + 0.548760i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(99\) 1.45379 1.45379
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.c.2991.1 yes 24
4.3 odd 2 inner 3332.1.cc.c.2991.2 yes 24
17.16 even 2 inner 3332.1.cc.c.2991.2 yes 24
49.25 even 21 inner 3332.1.cc.c.1495.1 ✓ 24
68.67 odd 2 CM 3332.1.cc.c.2991.1 yes 24
196.123 odd 42 inner 3332.1.cc.c.1495.2 yes 24
833.662 even 42 inner 3332.1.cc.c.1495.2 yes 24
3332.1495 odd 42 inner 3332.1.cc.c.1495.1 ✓ 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.cc.c.1495.1 ✓ 24 49.25 even 21 inner
3332.1.cc.c.1495.1 ✓ 24 3332.1495 odd 42 inner
3332.1.cc.c.1495.2 yes 24 196.123 odd 42 inner
3332.1.cc.c.1495.2 yes 24 833.662 even 42 inner
3332.1.cc.c.2991.1 yes 24 1.1 even 1 trivial
3332.1.cc.c.2991.1 yes 24 68.67 odd 2 CM
3332.1.cc.c.2991.2 yes 24 4.3 odd 2 inner
3332.1.cc.c.2991.2 yes 24 17.16 even 2 inner