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Properties

Label 3332.1.be.a.1359.1
Level $3332$
Weight $1$
Character 3332.1359
Analytic conductor $1.663$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
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(407,3332)] 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("3332.407"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([7, 10, 7])) 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.be (of order \(14\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-1,-5] 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: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{7} - \cdots)\)

Embedding invariants

Embedding label 1359.1
Root \(0.900969 - 0.433884i\) of defining polynomial
Character \(\chi\) \(=\) 3332.1359
Dual form 3332.1.be.a.407.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.222521 - 0.974928i) q^{2} +(-0.777479 - 0.974928i) q^{3} +(-0.900969 + 0.433884i) q^{4} +(-0.777479 + 0.974928i) q^{6} +(-0.623490 - 0.781831i) q^{7} +(0.623490 + 0.781831i) q^{8} +(-0.123490 + 0.541044i) q^{9} +(-0.0990311 - 0.433884i) q^{11} +(1.12349 + 0.541044i) q^{12} +(0.0990311 + 0.433884i) q^{13} +(-0.623490 + 0.781831i) q^{14} +(0.623490 - 0.781831i) q^{16} +(-0.900969 - 0.433884i) q^{17} +0.554958 q^{18} +(-0.277479 + 1.21572i) q^{21} +(-0.400969 + 0.193096i) q^{22} +(-0.400969 + 0.193096i) q^{23} +(0.277479 - 1.21572i) q^{24} +(-0.222521 + 0.974928i) q^{25} +(0.400969 - 0.193096i) q^{26} +(-0.500000 + 0.240787i) q^{27} +(0.900969 + 0.433884i) q^{28} -1.24698 q^{31} +(-0.900969 - 0.433884i) q^{32} +(-0.346011 + 0.433884i) q^{33} +(-0.222521 + 0.974928i) q^{34} +(-0.123490 - 0.541044i) q^{36} +(0.346011 - 0.433884i) q^{39} +1.24698 q^{42} +(0.277479 + 0.347948i) q^{44} +(0.277479 + 0.347948i) q^{46} -1.24698 q^{48} +(-0.222521 + 0.974928i) q^{49} +1.00000 q^{50} +(0.277479 + 1.21572i) q^{51} +(-0.277479 - 0.347948i) q^{52} +(-1.12349 + 0.541044i) q^{53} +(0.346011 + 0.433884i) q^{54} +(0.222521 - 0.974928i) q^{56} +(0.277479 + 1.21572i) q^{62} +(0.500000 - 0.240787i) q^{63} +(-0.222521 + 0.974928i) q^{64} +(0.500000 + 0.240787i) q^{66} +1.00000 q^{68} +(0.500000 + 0.240787i) q^{69} +(1.12349 - 0.541044i) q^{71} +(-0.500000 + 0.240787i) q^{72} +(1.12349 - 0.541044i) q^{75} +(-0.277479 + 0.347948i) q^{77} +(-0.500000 - 0.240787i) q^{78} +0.445042 q^{79} +(1.12349 + 0.541044i) q^{81} +(-0.277479 - 1.21572i) q^{84} +(0.277479 - 0.347948i) q^{88} +(-0.277479 + 1.21572i) q^{89} +(0.277479 - 0.347948i) q^{91} +(0.277479 - 0.347948i) q^{92} +(0.969501 + 1.21572i) q^{93} +(0.277479 + 1.21572i) q^{96} +1.00000 q^{98} +0.246980 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - 5 q^{3} - q^{4} - 5 q^{6} + q^{7} - q^{8} + 4 q^{9} - 5 q^{11} + 2 q^{12} + 5 q^{13} + q^{14} - q^{16} - q^{17} + 4 q^{18} - 2 q^{21} + 2 q^{22} + 2 q^{23} + 2 q^{24} - q^{25} - 2 q^{26}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{7}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.222521 − 0.974928i −0.222521 − 0.974928i
\(3\) −0.777479 − 0.974928i −0.777479 − 0.974928i 0.222521 − 0.974928i \(-0.428571\pi\)
−1.00000 \(\pi\)
\(4\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(5\) 0 0 −0.623490 − 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(6\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(7\) −0.623490 − 0.781831i −0.623490 − 0.781831i
\(8\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(9\) −0.123490 + 0.541044i −0.123490 + 0.541044i
\(10\) 0 0
\(11\) −0.0990311 − 0.433884i −0.0990311 − 0.433884i 0.900969 − 0.433884i \(-0.142857\pi\)
−1.00000 \(\pi\)
\(12\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(13\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(14\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(15\) 0 0
\(16\) 0.623490 − 0.781831i 0.623490 − 0.781831i
\(17\) −0.900969 − 0.433884i −0.900969 − 0.433884i
\(18\) 0.554958 0.554958
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(22\) −0.400969 + 0.193096i −0.400969 + 0.193096i
\(23\) −0.400969 + 0.193096i −0.400969 + 0.193096i −0.623490 − 0.781831i \(-0.714286\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(24\) 0.277479 − 1.21572i 0.277479 − 1.21572i
\(25\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(26\) 0.400969 − 0.193096i 0.400969 − 0.193096i
\(27\) −0.500000 + 0.240787i −0.500000 + 0.240787i
\(28\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(29\) 0 0 −0.900969 − 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(30\) 0 0
\(31\) −1.24698 −1.24698 −0.623490 − 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(32\) −0.900969 − 0.433884i −0.900969 − 0.433884i
\(33\) −0.346011 + 0.433884i −0.346011 + 0.433884i
\(34\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(35\) 0 0
\(36\) −0.123490 − 0.541044i −0.123490 − 0.541044i
\(37\) 0 0 −0.900969 − 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(38\) 0 0
\(39\) 0.346011 − 0.433884i 0.346011 − 0.433884i
\(40\) 0 0
\(41\) 0 0 −0.623490 − 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(42\) 1.24698 1.24698
\(43\) 0 0 0.623490 − 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(44\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(45\) 0 0
\(46\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(47\) 0 0 −0.222521 − 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(48\) −1.24698 −1.24698
\(49\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(50\) 1.00000 1.00000
\(51\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(52\) −0.277479 − 0.347948i −0.277479 − 0.347948i
\(53\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 − 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(54\) 0.346011 + 0.433884i 0.346011 + 0.433884i
\(55\) 0 0
\(56\) 0.222521 − 0.974928i 0.222521 − 0.974928i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.623490 − 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(60\) 0 0
\(61\) 0 0 −0.900969 − 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(62\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(63\) 0.500000 − 0.240787i 0.500000 − 0.240787i
\(64\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(65\) 0 0
\(66\) 0.500000 + 0.240787i 0.500000 + 0.240787i
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 1.00000 1.00000
\(69\) 0.500000 + 0.240787i 0.500000 + 0.240787i
\(70\) 0 0
\(71\) 1.12349 − 0.541044i 1.12349 − 0.541044i 0.222521 − 0.974928i \(-0.428571\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(72\) −0.500000 + 0.240787i −0.500000 + 0.240787i
\(73\) 0 0 0.222521 − 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(74\) 0 0
\(75\) 1.12349 − 0.541044i 1.12349 − 0.541044i
\(76\) 0 0
\(77\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(78\) −0.500000 − 0.240787i −0.500000 − 0.240787i
\(79\) 0.445042 0.445042 0.222521 − 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(80\) 0 0
\(81\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(82\) 0 0
\(83\) 0 0 0.222521 − 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(84\) −0.277479 − 1.21572i −0.277479 − 1.21572i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.277479 − 0.347948i 0.277479 − 0.347948i
\(89\) −0.277479 + 1.21572i −0.277479 + 1.21572i 0.623490 + 0.781831i \(0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(90\) 0 0
\(91\) 0.277479 − 0.347948i 0.277479 − 0.347948i
\(92\) 0.277479 − 0.347948i 0.277479 − 0.347948i
\(93\) 0.969501 + 1.21572i 0.969501 + 1.21572i
\(94\) 0 0
\(95\) 0 0
\(96\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1.00000 1.00000
\(99\) 0.246980 0.246980
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.be.a.1359.1 yes 6
4.3 odd 2 3332.1.be.b.1359.1 yes 6
17.16 even 2 3332.1.be.b.1359.1 yes 6
49.15 even 7 inner 3332.1.be.a.407.1 ✓ 6
68.67 odd 2 CM 3332.1.be.a.1359.1 yes 6
196.15 odd 14 3332.1.be.b.407.1 yes 6
833.407 even 14 3332.1.be.b.407.1 yes 6
3332.407 odd 14 inner 3332.1.be.a.407.1 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3332.1.be.a.407.1 ✓ 6 49.15 even 7 inner
3332.1.be.a.407.1 ✓ 6 3332.407 odd 14 inner
3332.1.be.a.1359.1 yes 6 1.1 even 1 trivial
3332.1.be.a.1359.1 yes 6 68.67 odd 2 CM
3332.1.be.b.407.1 yes 6 196.15 odd 14
3332.1.be.b.407.1 yes 6 833.407 even 14
3332.1.be.b.1359.1 yes 6 4.3 odd 2
3332.1.be.b.1359.1 yes 6 17.16 even 2