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Properties

Label 7440.2.a.bt.1.1
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7440.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,3,0,-3,0,-4,0,3,0,2,0,4,0,-3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.4086991038\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1860)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 7440.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -4.43807 q^{7} +1.00000 q^{9} -4.17226 q^{11} +4.43807 q^{13} -1.00000 q^{15} -1.35194 q^{17} -2.70388 q^{19} -4.43807 q^{21} +1.35194 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.25839 q^{29} +1.00000 q^{31} -4.17226 q^{33} +4.43807 q^{35} -10.7826 q^{37} +4.43807 q^{39} +2.17226 q^{41} -9.46838 q^{43} -1.00000 q^{45} +4.11644 q^{47} +12.6965 q^{49} -1.35194 q^{51} +12.8203 q^{53} +4.17226 q^{55} -2.70388 q^{57} -5.25839 q^{59} +6.00000 q^{61} -4.43807 q^{63} -4.43807 q^{65} +1.14195 q^{67} +1.35194 q^{69} +10.7268 q^{71} +10.4381 q^{73} +1.00000 q^{75} +18.5168 q^{77} +2.05582 q^{79} +1.00000 q^{81} +6.99258 q^{83} +1.35194 q^{85} -7.25839 q^{87} -2.55451 q^{89} -19.6965 q^{91} +1.00000 q^{93} +2.70388 q^{95} -15.9245 q^{97} -4.17226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 4 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 4 q^{19} - 4 q^{21} + 2 q^{23} + 3 q^{25} + 3 q^{27} + 3 q^{31} + 2 q^{33} + 4 q^{35} + 6 q^{37} + 4 q^{39} - 8 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.43807 −1.67743 −0.838716 − 0.544569i \(-0.816693\pi\)
−0.838716 + 0.544569i \(0.816693\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.17226 −1.25798 −0.628992 − 0.777412i \(-0.716532\pi\)
−0.628992 + 0.777412i \(0.716532\pi\)
\(12\) 0 0
\(13\) 4.43807 1.23090 0.615449 − 0.788176i \(-0.288975\pi\)
0.615449 + 0.788176i \(0.288975\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.35194 −0.327893 −0.163947 − 0.986469i \(-0.552423\pi\)
−0.163947 + 0.986469i \(0.552423\pi\)
\(18\) 0 0
\(19\) −2.70388 −0.620312 −0.310156 − 0.950686i \(-0.600381\pi\)
−0.310156 + 0.950686i \(0.600381\pi\)
\(20\) 0 0
\(21\) −4.43807 −0.968466
\(22\) 0 0
\(23\) 1.35194 0.281899 0.140949 − 0.990017i \(-0.454985\pi\)
0.140949 + 0.990017i \(0.454985\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.25839 −1.34785 −0.673925 − 0.738800i \(-0.735393\pi\)
−0.673925 + 0.738800i \(0.735393\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 0 0
\(33\) −4.17226 −0.726297
\(34\) 0 0
\(35\) 4.43807 0.750171
\(36\) 0 0
\(37\) −10.7826 −1.77265 −0.886323 − 0.463067i \(-0.846749\pi\)
−0.886323 + 0.463067i \(0.846749\pi\)
\(38\) 0 0
\(39\) 4.43807 0.710660
\(40\) 0 0
\(41\) 2.17226 0.339250 0.169625 − 0.985509i \(-0.445744\pi\)
0.169625 + 0.985509i \(0.445744\pi\)
\(42\) 0 0
\(43\) −9.46838 −1.44391 −0.721957 − 0.691938i \(-0.756757\pi\)
−0.721957 + 0.691938i \(0.756757\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 4.11644 0.600445 0.300222 − 0.953869i \(-0.402939\pi\)
0.300222 + 0.953869i \(0.402939\pi\)
\(48\) 0 0
\(49\) 12.6965 1.81378
\(50\) 0 0
\(51\) −1.35194 −0.189309
\(52\) 0 0
\(53\) 12.8203 1.76101 0.880503 − 0.474040i \(-0.157205\pi\)
0.880503 + 0.474040i \(0.157205\pi\)
\(54\) 0 0
\(55\) 4.17226 0.562587
\(56\) 0 0
\(57\) −2.70388 −0.358137
\(58\) 0 0
\(59\) −5.25839 −0.684584 −0.342292 − 0.939594i \(-0.611203\pi\)
−0.342292 + 0.939594i \(0.611203\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −4.43807 −0.559144
\(64\) 0 0
\(65\) −4.43807 −0.550475
\(66\) 0 0
\(67\) 1.14195 0.139511 0.0697556 − 0.997564i \(-0.477778\pi\)
0.0697556 + 0.997564i \(0.477778\pi\)
\(68\) 0 0
\(69\) 1.35194 0.162754
\(70\) 0 0
\(71\) 10.7268 1.27303 0.636517 − 0.771263i \(-0.280375\pi\)
0.636517 + 0.771263i \(0.280375\pi\)
\(72\) 0 0
\(73\) 10.4381 1.22168 0.610842 − 0.791753i \(-0.290831\pi\)
0.610842 + 0.791753i \(0.290831\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 18.5168 2.11018
\(78\) 0 0
\(79\) 2.05582 0.231298 0.115649 − 0.993290i \(-0.463105\pi\)
0.115649 + 0.993290i \(0.463105\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.99258 0.767536 0.383768 − 0.923430i \(-0.374626\pi\)
0.383768 + 0.923430i \(0.374626\pi\)
\(84\) 0 0
\(85\) 1.35194 0.146638
\(86\) 0 0
\(87\) −7.25839 −0.778181
\(88\) 0 0
\(89\) −2.55451 −0.270778 −0.135389 − 0.990793i \(-0.543228\pi\)
−0.135389 + 0.990793i \(0.543228\pi\)
\(90\) 0 0
\(91\) −19.6965 −2.06475
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 2.70388 0.277412
\(96\) 0 0
\(97\) −15.9245 −1.61689 −0.808446 − 0.588570i \(-0.799691\pi\)
−0.808446 + 0.588570i \(0.799691\pi\)
\(98\) 0 0
\(99\) −4.17226 −0.419328
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 7440.2.a.bt.1.1 3
4.3 odd 2 1860.2.a.f.1.3 ✓ 3
12.11 even 2 5580.2.a.l.1.3 3
20.3 even 4 9300.2.g.p.3349.1 6
20.7 even 4 9300.2.g.p.3349.6 6
20.19 odd 2 9300.2.a.v.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.a.f.1.3 ✓ 3 4.3 odd 2
5580.2.a.l.1.3 3 12.11 even 2
7440.2.a.bt.1.1 3 1.1 even 1 trivial
9300.2.a.v.1.1 3 20.19 odd 2
9300.2.g.p.3349.1 6 20.3 even 4
9300.2.g.p.3349.6 6 20.7 even 4