Properties

Label 3332.1.cc.a.2923.1
Level $3332$
Weight $1$
Character 3332.2923
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 2923.1
Root \(0.365341 + 0.930874i\) of defining polynomial
Character \(\chi\) \(=\) 3332.2923
Dual form 3332.1.cc.a.611.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0747301 + 0.997204i) q^{2} +(-1.82624 - 0.563320i) q^{3} +(-0.988831 + 0.149042i) q^{4} +(0.425270 - 1.86323i) q^{6} +(0.222521 + 0.974928i) q^{7} +(-0.222521 - 0.974928i) q^{8} +(2.19158 + 1.49419i) q^{9} +(-0.123490 + 0.0841939i) q^{11} +(1.88980 + 0.284841i) q^{12} +(-0.134659 - 0.0648483i) q^{13} +(-0.955573 + 0.294755i) q^{14} +(0.955573 - 0.294755i) q^{16} +(0.365341 + 0.930874i) q^{17} +(-1.32624 + 2.29711i) q^{18} +(0.142820 - 1.90580i) q^{21} +(-0.0931869 - 0.116853i) q^{22} +(0.658322 - 1.67738i) q^{23} +(-0.142820 + 1.90580i) q^{24} +(0.0747301 - 0.997204i) q^{25} +(0.0546039 - 0.139129i) q^{26} +(-1.96906 - 2.46912i) q^{27} +(-0.365341 - 0.930874i) q^{28} +(-0.222521 + 0.385418i) q^{31} +(0.365341 + 0.930874i) q^{32} +(0.272950 - 0.0841939i) q^{33} +(-0.900969 + 0.433884i) q^{34} +(-2.38980 - 1.15087i) q^{36} +(0.209389 + 0.194285i) q^{39} +1.91115 q^{42} +(0.109562 - 0.101659i) q^{44} +(1.72188 + 0.531130i) q^{46} -1.91115 q^{48} +(-0.900969 + 0.433884i) q^{49} +1.00000 q^{50} +(-0.142820 - 1.90580i) q^{51} +(0.142820 + 0.0440542i) q^{52} +(1.44973 - 0.218511i) q^{53} +(2.31507 - 2.14807i) q^{54} +(0.900969 - 0.433884i) q^{56} +(-0.400969 - 0.193096i) q^{62} +(-0.969059 + 2.46912i) q^{63} +(-0.900969 + 0.433884i) q^{64} +(0.104356 + 0.265895i) q^{66} +(-0.500000 - 0.866025i) q^{68} +(-2.14715 + 2.69244i) q^{69} +(0.914101 + 1.14625i) q^{71} +(0.969059 - 2.46912i) q^{72} +(-0.698220 + 1.77904i) q^{75} +(-0.109562 - 0.101659i) q^{77} +(-0.178094 + 0.223322i) q^{78} +(0.826239 + 1.43109i) q^{79} +(1.23601 + 3.14929i) q^{81} +(0.142820 + 1.90580i) q^{84} +(0.109562 + 0.101659i) q^{88} +(1.57906 + 1.07659i) q^{89} +(0.0332580 - 0.145713i) q^{91} +(-0.400969 + 1.75676i) q^{92} +(0.623490 - 0.578514i) q^{93} +(-0.142820 - 1.90580i) q^{96} +(-0.500000 - 0.866025i) q^{98} -0.396440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} - 13 q^{3} + q^{4} + 5 q^{6} + 2 q^{7} - 2 q^{8} + 14 q^{9} + 8 q^{11} + q^{12} - 5 q^{13} - q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} - 2 q^{22} - 2 q^{23} + q^{24} + q^{25} - q^{26}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{21}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0747301 + 0.997204i 0.0747301 + 0.997204i
\(3\) −1.82624 − 0.563320i −1.82624 − 0.563320i −0.826239 − 0.563320i \(-0.809524\pi\)
−1.00000 \(\pi\)
\(4\) −0.988831 + 0.149042i −0.988831 + 0.149042i
\(5\) 0 0 0.733052 − 0.680173i \(-0.238095\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(6\) 0.425270 − 1.86323i 0.425270 − 1.86323i
\(7\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(8\) −0.222521 − 0.974928i −0.222521 − 0.974928i
\(9\) 2.19158 + 1.49419i 2.19158 + 1.49419i
\(10\) 0 0
\(11\) −0.123490 + 0.0841939i −0.123490 + 0.0841939i −0.623490 − 0.781831i \(-0.714286\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 1.88980 + 0.284841i 1.88980 + 0.284841i
\(13\) −0.134659 − 0.0648483i −0.134659 − 0.0648483i 0.365341 − 0.930874i \(-0.380952\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) −0.955573 + 0.294755i −0.955573 + 0.294755i
\(15\) 0 0
\(16\) 0.955573 − 0.294755i 0.955573 − 0.294755i
\(17\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(18\) −1.32624 + 2.29711i −1.32624 + 2.29711i
\(19\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0.142820 − 1.90580i 0.142820 − 1.90580i
\(22\) −0.0931869 − 0.116853i −0.0931869 − 0.116853i
\(23\) 0.658322 − 1.67738i 0.658322 − 1.67738i −0.0747301 − 0.997204i \(-0.523810\pi\)
0.733052 − 0.680173i \(-0.238095\pi\)
\(24\) −0.142820 + 1.90580i −0.142820 + 1.90580i
\(25\) 0.0747301 − 0.997204i 0.0747301 − 0.997204i
\(26\) 0.0546039 − 0.139129i 0.0546039 − 0.139129i
\(27\) −1.96906 − 2.46912i −1.96906 − 2.46912i
\(28\) −0.365341 − 0.930874i −0.365341 − 0.930874i
\(29\) 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 −0.955573 − 0.294755i \(-0.904762\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(32\) 0.365341 + 0.930874i 0.365341 + 0.930874i
\(33\) 0.272950 − 0.0841939i 0.272950 − 0.0841939i
\(34\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(35\) 0 0
\(36\) −2.38980 − 1.15087i −2.38980 − 1.15087i
\(37\) 0 0 −0.988831 − 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(38\) 0 0
\(39\) 0.209389 + 0.194285i 0.209389 + 0.194285i
\(40\) 0 0
\(41\) 0 0 −0.222521 − 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(42\) 1.91115 1.91115
\(43\) 0 0 0.222521 − 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(44\) 0.109562 − 0.101659i 0.109562 − 0.101659i
\(45\) 0 0
\(46\) 1.72188 + 0.531130i 1.72188 + 0.531130i
\(47\) 0 0 −0.0747301 − 0.997204i \(-0.523810\pi\)
0.0747301 + 0.997204i \(0.476190\pi\)
\(48\) −1.91115 −1.91115
\(49\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(50\) 1.00000 1.00000
\(51\) −0.142820 − 1.90580i −0.142820 − 1.90580i
\(52\) 0.142820 + 0.0440542i 0.142820 + 0.0440542i
\(53\) 1.44973 − 0.218511i 1.44973 − 0.218511i 0.623490 − 0.781831i \(-0.285714\pi\)
0.826239 + 0.563320i \(0.190476\pi\)
\(54\) 2.31507 − 2.14807i 2.31507 − 2.14807i
\(55\) 0 0
\(56\) 0.900969 − 0.433884i 0.900969 − 0.433884i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.733052 − 0.680173i \(-0.761905\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(60\) 0 0
\(61\) 0 0 −0.988831 − 0.149042i \(-0.952381\pi\)
0.988831 + 0.149042i \(0.0476190\pi\)
\(62\) −0.400969 − 0.193096i −0.400969 − 0.193096i
\(63\) −0.969059 + 2.46912i −0.969059 + 2.46912i
\(64\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(65\) 0 0
\(66\) 0.104356 + 0.265895i 0.104356 + 0.265895i
\(67\) 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
\(69\) −2.14715 + 2.69244i −2.14715 + 2.69244i
\(70\) 0 0
\(71\) 0.914101 + 1.14625i 0.914101 + 1.14625i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(72\) 0.969059 − 2.46912i 0.969059 − 2.46912i
\(73\) 0 0 0.0747301 − 0.997204i \(-0.476190\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(74\) 0 0
\(75\) −0.698220 + 1.77904i −0.698220 + 1.77904i
\(76\) 0 0
\(77\) −0.109562 − 0.101659i −0.109562 − 0.101659i
\(78\) −0.178094 + 0.223322i −0.178094 + 0.223322i
\(79\) 0.826239 + 1.43109i 0.826239 + 1.43109i 0.900969 + 0.433884i \(0.142857\pi\)
−0.0747301 + 0.997204i \(0.523810\pi\)
\(80\) 0 0
\(81\) 1.23601 + 3.14929i 1.23601 + 3.14929i
\(82\) 0 0
\(83\) 0 0 0.900969 − 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(84\) 0.142820 + 1.90580i 0.142820 + 1.90580i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.109562 + 0.101659i 0.109562 + 0.101659i
\(89\) 1.57906 + 1.07659i 1.57906 + 1.07659i 0.955573 + 0.294755i \(0.0952381\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(90\) 0 0
\(91\) 0.0332580 − 0.145713i 0.0332580 − 0.145713i
\(92\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(93\) 0.623490 − 0.578514i 0.623490 − 0.578514i
\(94\) 0 0
\(95\) 0 0
\(96\) −0.142820 − 1.90580i −0.142820 − 1.90580i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(99\) −0.396440 −0.396440
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.2923.1 yes 12
4.3 odd 2 3332.1.cc.b.2923.1 yes 12
17.16 even 2 3332.1.cc.b.2923.1 yes 12
49.23 even 21 inner 3332.1.cc.a.611.1 ✓ 12
68.67 odd 2 CM 3332.1.cc.a.2923.1 yes 12
196.23 odd 42 3332.1.cc.b.611.1 yes 12
833.611 even 42 3332.1.cc.b.611.1 yes 12
3332.611 odd 42 inner 3332.1.cc.a.611.1 ✓ 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.cc.a.611.1 ✓ 12 49.23 even 21 inner
3332.1.cc.a.611.1 ✓ 12 3332.611 odd 42 inner
3332.1.cc.a.2923.1 yes 12 1.1 even 1 trivial
3332.1.cc.a.2923.1 yes 12 68.67 odd 2 CM
3332.1.cc.b.611.1 yes 12 196.23 odd 42
3332.1.cc.b.611.1 yes 12 833.611 even 42
3332.1.cc.b.2923.1 yes 12 4.3 odd 2
3332.1.cc.b.2923.1 yes 12 17.16 even 2