Properties

Label 3332.1.cc.c.1563.2
Level $3332$
Weight $1$
Character 3332.1563
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 1563.2
Root \(0.997204 + 0.0747301i\) of defining polynomial
Character \(\chi\) \(=\) 3332.1563
Dual form 3332.1.cc.c.1019.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.988831 - 0.149042i) q^{2} +(0.930874 + 0.634659i) q^{3} +(0.955573 - 0.294755i) q^{4} +(1.01507 + 0.488831i) q^{6} +(-0.433884 - 0.900969i) q^{7} +(0.900969 - 0.433884i) q^{8} +(0.0983929 + 0.250701i) q^{9} +(0.108903 - 0.277479i) q^{11} +(1.07659 + 0.332083i) q^{12} +(1.23305 + 1.54620i) q^{13} +(-0.563320 - 0.826239i) q^{14} +(0.826239 - 0.563320i) q^{16} +(-0.733052 + 0.680173i) q^{17} +(0.134659 + 0.233236i) q^{18} +(0.167917 - 1.11406i) q^{21} +(0.0663300 - 0.290611i) q^{22} +(-1.14625 - 1.06356i) q^{23} +(1.11406 + 0.167917i) q^{24} +(-0.988831 - 0.149042i) q^{25} +(1.44973 + 1.34515i) q^{26} +(0.183183 - 0.802576i) q^{27} +(-0.680173 - 0.733052i) q^{28} +(0.433884 + 0.751509i) q^{31} +(0.733052 - 0.680173i) q^{32} +(0.277479 - 0.189182i) q^{33} +(-0.623490 + 0.781831i) q^{34} +(0.167917 + 0.210561i) q^{36} +(0.166507 + 2.22188i) q^{39} -1.12664i q^{42} +(0.0222759 - 0.297251i) q^{44} +(-1.29196 - 0.880843i) q^{46} +1.12664 q^{48} +(-0.623490 + 0.781831i) q^{49} -1.00000 q^{50} +(-1.11406 + 0.167917i) q^{51} +(1.63402 + 1.11406i) q^{52} +(0.142820 - 0.0440542i) q^{53} +(0.0615190 - 0.820914i) q^{54} +(-0.781831 - 0.623490i) q^{56} +(0.541044 + 0.678448i) q^{62} +(0.183183 - 0.197424i) q^{63} +(0.623490 - 0.781831i) q^{64} +(0.246184 - 0.228425i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(-0.392012 - 1.71752i) q^{69} +(-0.443797 + 1.94440i) q^{71} +(0.197424 + 0.183183i) q^{72} +(-0.825886 - 0.766310i) q^{75} +(-0.297251 + 0.0222759i) q^{77} +(0.495802 + 2.17225i) q^{78} +(-0.930874 + 1.61232i) q^{79} +(0.877306 - 0.814021i) q^{81} +(-0.167917 - 1.11406i) q^{84} +(-0.0222759 - 0.297251i) q^{88} +(-0.603718 - 1.53825i) q^{89} +(0.858075 - 1.78181i) q^{91} +(-1.40881 - 0.678448i) q^{92} +(-0.0730607 + 0.974928i) q^{93} +(1.11406 - 0.167917i) q^{96} +(-0.500000 + 0.866025i) q^{98} +0.0802795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(e\left(\frac{4}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.988831 − 0.149042i 0.988831 − 0.149042i
\(3\) 0.930874 + 0.634659i 0.930874 + 0.634659i 0.930874 − 0.365341i \(-0.119048\pi\)
1.00000i \(0.5\pi\)
\(4\) 0.955573 − 0.294755i 0.955573 − 0.294755i
\(5\) 0 0 0.0747301 − 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(6\) 1.01507 + 0.488831i 1.01507 + 0.488831i
\(7\) −0.433884 − 0.900969i −0.433884 − 0.900969i
\(8\) 0.900969 − 0.433884i 0.900969 − 0.433884i
\(9\) 0.0983929 + 0.250701i 0.0983929 + 0.250701i
\(10\) 0 0
\(11\) 0.108903 − 0.277479i 0.108903 − 0.277479i −0.866025 − 0.500000i \(-0.833333\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(12\) 1.07659 + 0.332083i 1.07659 + 0.332083i
\(13\) 1.23305 + 1.54620i 1.23305 + 1.54620i 0.733052 + 0.680173i \(0.238095\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) −0.563320 − 0.826239i −0.563320 − 0.826239i
\(15\) 0 0
\(16\) 0.826239 − 0.563320i 0.826239 − 0.563320i
\(17\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(18\) 0.134659 + 0.233236i 0.134659 + 0.233236i
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0.167917 − 1.11406i 0.167917 − 1.11406i
\(22\) 0.0663300 − 0.290611i 0.0663300 − 0.290611i
\(23\) −1.14625 − 1.06356i −1.14625 − 1.06356i −0.997204 − 0.0747301i \(-0.976190\pi\)
−0.149042 − 0.988831i \(-0.547619\pi\)
\(24\) 1.11406 + 0.167917i 1.11406 + 0.167917i
\(25\) −0.988831 − 0.149042i −0.988831 − 0.149042i
\(26\) 1.44973 + 1.34515i 1.44973 + 1.34515i
\(27\) 0.183183 − 0.802576i 0.183183 − 0.802576i
\(28\) −0.680173 − 0.733052i −0.680173 − 0.733052i
\(29\) 0 0 −0.222521 − 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(30\) 0 0
\(31\) 0.433884 + 0.751509i 0.433884 + 0.751509i 0.997204 − 0.0747301i \(-0.0238095\pi\)
−0.563320 + 0.826239i \(0.690476\pi\)
\(32\) 0.733052 − 0.680173i 0.733052 − 0.680173i
\(33\) 0.277479 − 0.189182i 0.277479 − 0.189182i
\(34\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(35\) 0 0
\(36\) 0.167917 + 0.210561i 0.167917 + 0.210561i
\(37\) 0 0 −0.955573 − 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(38\) 0 0
\(39\) 0.166507 + 2.22188i 0.166507 + 2.22188i
\(40\) 0 0
\(41\) 0 0 0.900969 − 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(42\) − 1.12664i − 1.12664i
\(43\) 0 0 −0.900969 − 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(44\) 0.0222759 − 0.297251i 0.0222759 − 0.297251i
\(45\) 0 0
\(46\) −1.29196 − 0.880843i −1.29196 − 0.880843i
\(47\) 0 0 0.988831 − 0.149042i \(-0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(48\) 1.12664 1.12664
\(49\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(50\) −1.00000 −1.00000
\(51\) −1.11406 + 0.167917i −1.11406 + 0.167917i
\(52\) 1.63402 + 1.11406i 1.63402 + 1.11406i
\(53\) 0.142820 − 0.0440542i 0.142820 − 0.0440542i −0.222521 − 0.974928i \(-0.571429\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(54\) 0.0615190 − 0.820914i 0.0615190 − 0.820914i
\(55\) 0 0
\(56\) −0.781831 − 0.623490i −0.781831 − 0.623490i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.0747301 − 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(60\) 0 0
\(61\) 0 0 −0.955573 − 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(62\) 0.541044 + 0.678448i 0.541044 + 0.678448i
\(63\) 0.183183 − 0.197424i 0.183183 − 0.197424i
\(64\) 0.623490 − 0.781831i 0.623490 − 0.781831i
\(65\) 0 0
\(66\) 0.246184 − 0.228425i 0.246184 − 0.228425i
\(67\) 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
\(69\) −0.392012 − 1.71752i −0.392012 − 1.71752i
\(70\) 0 0
\(71\) −0.443797 + 1.94440i −0.443797 + 1.94440i −0.149042 + 0.988831i \(0.547619\pi\)
−0.294755 + 0.955573i \(0.595238\pi\)
\(72\) 0.197424 + 0.183183i 0.197424 + 0.183183i
\(73\) 0 0 −0.988831 − 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(74\) 0 0
\(75\) −0.825886 − 0.766310i −0.825886 − 0.766310i
\(76\) 0 0
\(77\) −0.297251 + 0.0222759i −0.297251 + 0.0222759i
\(78\) 0.495802 + 2.17225i 0.495802 + 2.17225i
\(79\) −0.930874 + 1.61232i −0.930874 + 1.61232i −0.149042 + 0.988831i \(0.547619\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(80\) 0 0
\(81\) 0.877306 − 0.814021i 0.877306 − 0.814021i
\(82\) 0 0
\(83\) 0 0 0.623490 − 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(84\) −0.167917 − 1.11406i −0.167917 − 1.11406i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.0222759 − 0.297251i −0.0222759 − 0.297251i
\(89\) −0.603718 − 1.53825i −0.603718 − 1.53825i −0.826239 − 0.563320i \(-0.809524\pi\)
0.222521 − 0.974928i \(-0.428571\pi\)
\(90\) 0 0
\(91\) 0.858075 − 1.78181i 0.858075 − 1.78181i
\(92\) −1.40881 − 0.678448i −1.40881 − 0.678448i
\(93\) −0.0730607 + 0.974928i −0.0730607 + 0.974928i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.11406 − 0.167917i 1.11406 − 0.167917i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(99\) 0.0802795 0.0802795
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.1563.2 yes 24
4.3 odd 2 inner 3332.1.cc.c.1563.1 yes 24
17.16 even 2 inner 3332.1.cc.c.1563.1 yes 24
49.39 even 21 inner 3332.1.cc.c.1019.2 yes 24
68.67 odd 2 CM 3332.1.cc.c.1563.2 yes 24
196.39 odd 42 inner 3332.1.cc.c.1019.1 ✓ 24
833.186 even 42 inner 3332.1.cc.c.1019.1 ✓ 24
3332.1019 odd 42 inner 3332.1.cc.c.1019.2 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.cc.c.1019.1 ✓ 24 196.39 odd 42 inner
3332.1.cc.c.1019.1 ✓ 24 833.186 even 42 inner
3332.1.cc.c.1019.2 yes 24 49.39 even 21 inner
3332.1.cc.c.1019.2 yes 24 3332.1019 odd 42 inner
3332.1.cc.c.1563.1 yes 24 4.3 odd 2 inner
3332.1.cc.c.1563.1 yes 24 17.16 even 2 inner
3332.1.cc.c.1563.2 yes 24 1.1 even 1 trivial
3332.1.cc.c.1563.2 yes 24 68.67 odd 2 CM