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Properties

Label 3332.1.cc.a.135.1
Level $3332$
Weight $1$
Character 3332.135
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 135.1
Root \(-0.988831 - 0.149042i\) of defining polynomial
Character \(\chi\) \(=\) 3332.135
Dual form 3332.1.cc.a.543.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.826239 - 0.563320i) q^{2} +(-1.07473 + 0.997204i) q^{3} +(0.365341 - 0.930874i) q^{4} +(-0.326239 + 1.42935i) q^{6} +(0.222521 + 0.974928i) q^{7} +(-0.222521 - 0.974928i) q^{8} +(0.0858993 - 1.14625i) q^{9} +(-0.123490 - 1.64786i) q^{11} +(0.535628 + 1.36476i) q^{12} +(-1.48883 - 0.716983i) q^{13} +(0.733052 + 0.680173i) q^{14} +(-0.733052 - 0.680173i) q^{16} +(-0.988831 - 0.149042i) q^{17} +(-0.574730 - 0.995462i) q^{18} +(-1.21135 - 0.825886i) q^{21} +(-1.03030 - 1.29196i) q^{22} +(-1.78181 + 0.268565i) q^{23} +(1.21135 + 0.825886i) q^{24} +(0.826239 + 0.563320i) q^{25} +(-1.63402 + 0.246289i) q^{26} +(0.136622 + 0.171318i) q^{27} +(0.988831 + 0.149042i) q^{28} +(-0.222521 - 0.385418i) q^{31} +(-0.988831 - 0.149042i) q^{32} +(1.77597 + 1.64786i) q^{33} +(-0.900969 + 0.433884i) q^{34} +(-1.03563 - 0.498732i) q^{36} +(2.31507 - 0.714104i) q^{39} -1.46610 q^{42} +(-1.57906 - 0.487076i) q^{44} +(-1.32091 + 1.22563i) q^{46} +1.46610 q^{48} +(-0.900969 + 0.433884i) q^{49} +1.00000 q^{50} +(1.21135 - 0.825886i) q^{51} +(-1.21135 + 1.12397i) q^{52} +(0.698220 - 1.77904i) q^{53} +(0.209389 + 0.0645880i) q^{54} +(0.900969 - 0.433884i) q^{56} +(-0.400969 - 0.193096i) q^{62} +(1.13662 - 0.171318i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(2.39564 + 0.361085i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(1.64715 - 2.06546i) q^{69} +(-1.19158 - 1.49419i) q^{71} +(-1.13662 + 0.171318i) q^{72} +(-1.44973 + 0.218511i) q^{75} +(1.57906 - 0.487076i) q^{77} +(1.51053 - 1.89415i) q^{78} +(0.0747301 - 0.129436i) q^{79} +(0.818951 + 0.123437i) q^{81} +(-1.21135 + 0.825886i) q^{84} +(-1.57906 + 0.487076i) q^{88} +(-0.109562 + 1.46200i) q^{89} +(0.367711 - 1.61105i) q^{91} +(-0.400969 + 1.75676i) q^{92} +(0.623490 + 0.192321i) q^{93} +(1.21135 - 0.825886i) q^{96} +(-0.500000 + 0.866025i) q^{98} -1.89946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - 13 q^{3} + q^{4} + 5 q^{6} + 2 q^{7} - 2 q^{8} + 14 q^{9} + 8 q^{11} + q^{12} - 5 q^{13} - q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} - 2 q^{22} - 2 q^{23} + q^{24} + q^{25} - q^{26}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(e\left(\frac{16}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.826239 − 0.563320i 0.826239 − 0.563320i
\(3\) −1.07473 + 0.997204i −1.07473 + 0.997204i −0.0747301 + 0.997204i \(0.523810\pi\)
−1.00000 \(1.00000\pi\)
\(4\) 0.365341 − 0.930874i 0.365341 − 0.930874i
\(5\) 0 0 −0.955573 − 0.294755i \(-0.904762\pi\)
0.955573 + 0.294755i \(0.0952381\pi\)
\(6\) −0.326239 + 1.42935i −0.326239 + 1.42935i
\(7\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(8\) −0.222521 − 0.974928i −0.222521 − 0.974928i
\(9\) 0.0858993 − 1.14625i 0.0858993 − 1.14625i
\(10\) 0 0
\(11\) −0.123490 − 1.64786i −0.123490 − 1.64786i −0.623490 − 0.781831i \(-0.714286\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(12\) 0.535628 + 1.36476i 0.535628 + 1.36476i
\(13\) −1.48883 − 0.716983i −1.48883 − 0.716983i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(14\) 0.733052 + 0.680173i 0.733052 + 0.680173i
\(15\) 0 0
\(16\) −0.733052 − 0.680173i −0.733052 − 0.680173i
\(17\) −0.988831 − 0.149042i −0.988831 − 0.149042i
\(18\) −0.574730 − 0.995462i −0.574730 − 0.995462i
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) −1.21135 − 0.825886i −1.21135 − 0.825886i
\(22\) −1.03030 − 1.29196i −1.03030 − 1.29196i
\(23\) −1.78181 + 0.268565i −1.78181 + 0.268565i −0.955573 − 0.294755i \(-0.904762\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(24\) 1.21135 + 0.825886i 1.21135 + 0.825886i
\(25\) 0.826239 + 0.563320i 0.826239 + 0.563320i
\(26\) −1.63402 + 0.246289i −1.63402 + 0.246289i
\(27\) 0.136622 + 0.171318i 0.136622 + 0.171318i
\(28\) 0.988831 + 0.149042i 0.988831 + 0.149042i
\(29\) 0 0 0.623490 − 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(30\) 0 0
\(31\) −0.222521 − 0.385418i −0.222521 − 0.385418i 0.733052 − 0.680173i \(-0.238095\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(32\) −0.988831 − 0.149042i −0.988831 − 0.149042i
\(33\) 1.77597 + 1.64786i 1.77597 + 1.64786i
\(34\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(35\) 0 0
\(36\) −1.03563 − 0.498732i −1.03563 − 0.498732i
\(37\) 0 0 −0.365341 − 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(38\) 0 0
\(39\) 2.31507 − 0.714104i 2.31507 − 0.714104i
\(40\) 0 0
\(41\) 0 0 −0.222521 − 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(42\) −1.46610 −1.46610
\(43\) 0 0 0.222521 − 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(44\) −1.57906 − 0.487076i −1.57906 − 0.487076i
\(45\) 0 0
\(46\) −1.32091 + 1.22563i −1.32091 + 1.22563i
\(47\) 0 0 0.826239 − 0.563320i \(-0.190476\pi\)
−0.826239 + 0.563320i \(0.809524\pi\)
\(48\) 1.46610 1.46610
\(49\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(50\) 1.00000 1.00000
\(51\) 1.21135 − 0.825886i 1.21135 − 0.825886i
\(52\) −1.21135 + 1.12397i −1.21135 + 1.12397i
\(53\) 0.698220 − 1.77904i 0.698220 − 1.77904i 0.0747301 − 0.997204i \(-0.476190\pi\)
0.623490 − 0.781831i \(-0.285714\pi\)
\(54\) 0.209389 + 0.0645880i 0.209389 + 0.0645880i
\(55\) 0 0
\(56\) 0.900969 − 0.433884i 0.900969 − 0.433884i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.955573 − 0.294755i \(-0.0952381\pi\)
−0.955573 + 0.294755i \(0.904762\pi\)
\(60\) 0 0
\(61\) 0 0 −0.365341 − 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(62\) −0.400969 − 0.193096i −0.400969 − 0.193096i
\(63\) 1.13662 − 0.171318i 1.13662 − 0.171318i
\(64\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(65\) 0 0
\(66\) 2.39564 + 0.361085i 2.39564 + 0.361085i
\(67\) 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
\(69\) 1.64715 − 2.06546i 1.64715 − 2.06546i
\(70\) 0 0
\(71\) −1.19158 − 1.49419i −1.19158 − 1.49419i −0.826239 − 0.563320i \(-0.809524\pi\)
−0.365341 − 0.930874i \(-0.619048\pi\)
\(72\) −1.13662 + 0.171318i −1.13662 + 0.171318i
\(73\) 0 0 −0.826239 − 0.563320i \(-0.809524\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(74\) 0 0
\(75\) −1.44973 + 0.218511i −1.44973 + 0.218511i
\(76\) 0 0
\(77\) 1.57906 − 0.487076i 1.57906 − 0.487076i
\(78\) 1.51053 − 1.89415i 1.51053 − 1.89415i
\(79\) 0.0747301 − 0.129436i 0.0747301 − 0.129436i −0.826239 − 0.563320i \(-0.809524\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(80\) 0 0
\(81\) 0.818951 + 0.123437i 0.818951 + 0.123437i
\(82\) 0 0
\(83\) 0 0 0.900969 − 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(84\) −1.21135 + 0.825886i −1.21135 + 0.825886i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −1.57906 + 0.487076i −1.57906 + 0.487076i
\(89\) −0.109562 + 1.46200i −0.109562 + 1.46200i 0.623490 + 0.781831i \(0.285714\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(90\) 0 0
\(91\) 0.367711 − 1.61105i 0.367711 − 1.61105i
\(92\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(93\) 0.623490 + 0.192321i 0.623490 + 0.192321i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.21135 − 0.825886i 1.21135 − 0.825886i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(99\) −1.89946 −1.89946
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.135.1 ✓ 12
4.3 odd 2 3332.1.cc.b.135.1 yes 12
17.16 even 2 3332.1.cc.b.135.1 yes 12
49.4 even 21 inner 3332.1.cc.a.543.1 yes 12
68.67 odd 2 CM 3332.1.cc.a.135.1 ✓ 12
196.151 odd 42 3332.1.cc.b.543.1 yes 12
833.543 even 42 3332.1.cc.b.543.1 yes 12
3332.543 odd 42 inner 3332.1.cc.a.543.1 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.cc.a.135.1 ✓ 12 1.1 even 1 trivial
3332.1.cc.a.135.1 ✓ 12 68.67 odd 2 CM
3332.1.cc.a.543.1 yes 12 49.4 even 21 inner
3332.1.cc.a.543.1 yes 12 3332.543 odd 42 inner
3332.1.cc.b.135.1 yes 12 4.3 odd 2
3332.1.cc.b.135.1 yes 12 17.16 even 2
3332.1.cc.b.543.1 yes 12 196.151 odd 42
3332.1.cc.b.543.1 yes 12 833.543 even 42