Properties

Label 3332.1.cc.a.1971.1
Level $3332$
Weight $1$
Character 3332.1971
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 1971.1
Root \(0.826239 - 0.563320i\) of defining polynomial
Character \(\chi\) \(=\) 3332.1971
Dual form 3332.1.cc.a.1087.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.733052 + 0.680173i) q^{2} +(-1.95557 + 0.294755i) q^{3} +(0.0747301 - 0.997204i) q^{4} +(1.23305 - 1.54620i) q^{6} +(-0.623490 - 0.781831i) q^{7} +(0.623490 + 0.781831i) q^{8} +(2.78181 - 0.858075i) q^{9} +(1.40097 + 0.432142i) q^{11} +(0.147791 + 1.97213i) q^{12} +(0.326239 + 1.42935i) q^{13} +(0.988831 + 0.149042i) q^{14} +(-0.988831 - 0.149042i) q^{16} +(0.826239 - 0.563320i) q^{17} +(-1.45557 + 2.52113i) q^{18} +(1.44973 + 1.34515i) q^{21} +(-1.32091 + 0.636119i) q^{22} +(0.367711 + 0.250701i) q^{23} +(-1.44973 - 1.34515i) q^{24} +(-0.733052 - 0.680173i) q^{25} +(-1.21135 - 0.825886i) q^{26} +(-3.40530 + 1.63991i) q^{27} +(-0.826239 + 0.563320i) q^{28} +(0.623490 - 1.07992i) q^{31} +(0.826239 - 0.563320i) q^{32} +(-2.86707 - 0.432142i) q^{33} +(-0.222521 + 0.974928i) q^{34} +(-0.647791 - 2.83816i) q^{36} +(-1.05929 - 2.69903i) q^{39} -1.97766 q^{42} +(0.535628 - 1.36476i) q^{44} +(-0.440071 + 0.0663300i) q^{46} +1.97766 q^{48} +(-0.222521 + 0.974928i) q^{49} +1.00000 q^{50} +(-1.44973 + 1.34515i) q^{51} +(1.44973 - 0.218511i) q^{52} +(0.0546039 - 0.728639i) q^{53} +(1.38084 - 3.51833i) q^{54} +(0.222521 - 0.974928i) q^{56} +(0.277479 + 1.21572i) q^{62} +(-2.40530 - 1.63991i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(2.39564 - 1.63332i) q^{66} +(-0.500000 - 0.866025i) q^{68} +(-0.792981 - 0.381879i) q^{69} +(0.658322 - 0.317031i) q^{71} +(2.40530 + 1.63991i) q^{72} +(1.63402 + 1.11406i) q^{75} +(-0.535628 - 1.36476i) q^{77} +(2.61232 + 1.25803i) q^{78} +(0.955573 + 1.65510i) q^{79} +(3.77064 - 2.57078i) q^{81} +(1.44973 - 1.34515i) q^{84} +(0.535628 + 1.36476i) q^{88} +(-1.88980 + 0.582926i) q^{89} +(0.914101 - 1.14625i) q^{91} +(0.277479 - 0.347948i) q^{92} +(-0.900969 + 2.29563i) q^{93} +(-1.44973 + 1.34515i) q^{96} +(-0.500000 - 0.866025i) q^{98} +4.26804 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - 13 q^{3} + q^{4} + 5 q^{6} + 2 q^{7} - 2 q^{8} + 14 q^{9} + 8 q^{11} + q^{12} - 5 q^{13} - q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} - 2 q^{22} - 2 q^{23} + q^{24} + q^{25} - q^{26}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(e\left(\frac{20}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.733052 + 0.680173i −0.733052 + 0.680173i
\(3\) −1.95557 + 0.294755i −1.95557 + 0.294755i −0.955573 + 0.294755i \(0.904762\pi\)
−1.00000 \(1.00000\pi\)
\(4\) 0.0747301 − 0.997204i 0.0747301 − 0.997204i
\(5\) 0 0 0.365341 − 0.930874i \(-0.380952\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(6\) 1.23305 − 1.54620i 1.23305 − 1.54620i
\(7\) −0.623490 − 0.781831i −0.623490 − 0.781831i
\(8\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(9\) 2.78181 − 0.858075i 2.78181 − 0.858075i
\(10\) 0 0
\(11\) 1.40097 + 0.432142i 1.40097 + 0.432142i 0.900969 − 0.433884i \(-0.142857\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0.147791 + 1.97213i 0.147791 + 1.97213i
\(13\) 0.326239 + 1.42935i 0.326239 + 1.42935i 0.826239 + 0.563320i \(0.190476\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0.988831 + 0.149042i 0.988831 + 0.149042i
\(15\) 0 0
\(16\) −0.988831 − 0.149042i −0.988831 − 0.149042i
\(17\) 0.826239 − 0.563320i 0.826239 − 0.563320i
\(18\) −1.45557 + 2.52113i −1.45557 + 2.52113i
\(19\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 1.44973 + 1.34515i 1.44973 + 1.34515i
\(22\) −1.32091 + 0.636119i −1.32091 + 0.636119i
\(23\) 0.367711 + 0.250701i 0.367711 + 0.250701i 0.733052 − 0.680173i \(-0.238095\pi\)
−0.365341 + 0.930874i \(0.619048\pi\)
\(24\) −1.44973 − 1.34515i −1.44973 − 1.34515i
\(25\) −0.733052 − 0.680173i −0.733052 − 0.680173i
\(26\) −1.21135 − 0.825886i −1.21135 − 0.825886i
\(27\) −3.40530 + 1.63991i −3.40530 + 1.63991i
\(28\) −0.826239 + 0.563320i −0.826239 + 0.563320i
\(29\) 0 0 −0.900969 − 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(30\) 0 0
\(31\) 0.623490 − 1.07992i 0.623490 − 1.07992i −0.365341 − 0.930874i \(-0.619048\pi\)
0.988831 − 0.149042i \(-0.0476190\pi\)
\(32\) 0.826239 − 0.563320i 0.826239 − 0.563320i
\(33\) −2.86707 − 0.432142i −2.86707 − 0.432142i
\(34\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(35\) 0 0
\(36\) −0.647791 − 2.83816i −0.647791 − 2.83816i
\(37\) 0 0 −0.0747301 − 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(38\) 0 0
\(39\) −1.05929 − 2.69903i −1.05929 − 2.69903i
\(40\) 0 0
\(41\) 0 0 −0.623490 − 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(42\) −1.97766 −1.97766
\(43\) 0 0 0.623490 − 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(44\) 0.535628 − 1.36476i 0.535628 − 1.36476i
\(45\) 0 0
\(46\) −0.440071 + 0.0663300i −0.440071 + 0.0663300i
\(47\) 0 0 0.733052 − 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(48\) 1.97766 1.97766
\(49\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(50\) 1.00000 1.00000
\(51\) −1.44973 + 1.34515i −1.44973 + 1.34515i
\(52\) 1.44973 − 0.218511i 1.44973 − 0.218511i
\(53\) 0.0546039 − 0.728639i 0.0546039 − 0.728639i −0.900969 − 0.433884i \(-0.857143\pi\)
0.955573 − 0.294755i \(-0.0952381\pi\)
\(54\) 1.38084 − 3.51833i 1.38084 − 3.51833i
\(55\) 0 0
\(56\) 0.222521 − 0.974928i 0.222521 − 0.974928i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.365341 − 0.930874i \(-0.619048\pi\)
0.365341 + 0.930874i \(0.380952\pi\)
\(60\) 0 0
\(61\) 0 0 −0.0747301 − 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(62\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(63\) −2.40530 − 1.63991i −2.40530 − 1.63991i
\(64\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(65\) 0 0
\(66\) 2.39564 − 1.63332i 2.39564 − 1.63332i
\(67\) 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
\(69\) −0.792981 − 0.381879i −0.792981 − 0.381879i
\(70\) 0 0
\(71\) 0.658322 − 0.317031i 0.658322 − 0.317031i −0.0747301 − 0.997204i \(-0.523810\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(72\) 2.40530 + 1.63991i 2.40530 + 1.63991i
\(73\) 0 0 −0.733052 − 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(74\) 0 0
\(75\) 1.63402 + 1.11406i 1.63402 + 1.11406i
\(76\) 0 0
\(77\) −0.535628 − 1.36476i −0.535628 − 1.36476i
\(78\) 2.61232 + 1.25803i 2.61232 + 1.25803i
\(79\) 0.955573 + 1.65510i 0.955573 + 1.65510i 0.733052 + 0.680173i \(0.238095\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(80\) 0 0
\(81\) 3.77064 − 2.57078i 3.77064 − 2.57078i
\(82\) 0 0
\(83\) 0 0 0.222521 − 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(84\) 1.44973 − 1.34515i 1.44973 − 1.34515i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.535628 + 1.36476i 0.535628 + 1.36476i
\(89\) −1.88980 + 0.582926i −1.88980 + 0.582926i −0.900969 + 0.433884i \(0.857143\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(90\) 0 0
\(91\) 0.914101 − 1.14625i 0.914101 − 1.14625i
\(92\) 0.277479 − 0.347948i 0.277479 − 0.347948i
\(93\) −0.900969 + 2.29563i −0.900969 + 2.29563i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.44973 + 1.34515i −1.44973 + 1.34515i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(99\) 4.26804 4.26804
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.1971.1 yes 12
4.3 odd 2 3332.1.cc.b.1971.1 yes 12
17.16 even 2 3332.1.cc.b.1971.1 yes 12
49.9 even 21 inner 3332.1.cc.a.1087.1 ✓ 12
68.67 odd 2 CM 3332.1.cc.a.1971.1 yes 12
196.107 odd 42 3332.1.cc.b.1087.1 yes 12
833.254 even 42 3332.1.cc.b.1087.1 yes 12
3332.1087 odd 42 inner 3332.1.cc.a.1087.1 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.cc.a.1087.1 ✓ 12 49.9 even 21 inner
3332.1.cc.a.1087.1 ✓ 12 3332.1087 odd 42 inner
3332.1.cc.a.1971.1 yes 12 1.1 even 1 trivial
3332.1.cc.a.1971.1 yes 12 68.67 odd 2 CM
3332.1.cc.b.1087.1 yes 12 196.107 odd 42
3332.1.cc.b.1087.1 yes 12 833.254 even 42
3332.1.cc.b.1971.1 yes 12 4.3 odd 2
3332.1.cc.b.1971.1 yes 12 17.16 even 2