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Properties

Label 7440.2.a.bx.1.2
Level $7440$
Weight $2$
Character 7440.1
Self dual yes
Analytic conductor $59.409$
Analytic rank $0$
Dimension $4$
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 = [4,0,4,0,-4,0,-1,0,4,0,-1,0,-2,0,-4,0,4] 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: \(4\)
Coefficient field: 4.4.78292.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3720)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.19719\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.19719 q^{7} +1.00000 q^{9} +2.57591 q^{11} +5.11784 q^{13} -1.00000 q^{15} +8.14266 q^{17} +4.57591 q^{19} -2.19719 q^{21} -3.22200 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.02482 q^{29} -1.00000 q^{31} +2.57591 q^{33} +2.19719 q^{35} -7.11784 q^{37} +5.11784 q^{39} +4.00000 q^{41} +4.92065 q^{43} -1.00000 q^{45} -4.09302 q^{47} -2.17237 q^{49} +8.14266 q^{51} +3.17237 q^{53} -2.57591 q^{55} +4.57591 q^{57} -6.52139 q^{59} -5.49657 q^{61} -2.19719 q^{63} -5.11784 q^{65} +1.27653 q^{67} -3.22200 q^{69} -2.10416 q^{71} +1.85245 q^{73} +1.00000 q^{75} -5.65977 q^{77} +13.1129 q^{79} +1.00000 q^{81} -11.9840 q^{83} -8.14266 q^{85} -1.02482 q^{87} +5.55110 q^{89} -11.2449 q^{91} -1.00000 q^{93} -4.57591 q^{95} +14.6301 q^{97} +2.57591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9} - q^{11} - 2 q^{13} - 4 q^{15} + 4 q^{17} + 7 q^{19} - q^{21} + q^{23} + 4 q^{25} + 4 q^{27} + 2 q^{29} - 4 q^{31} - q^{33} + q^{35} - 6 q^{37} - 2 q^{39}+ \cdots - q^{99}+O(q^{100}) \) 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
\(7\) −2.19719 −0.830459 −0.415229 − 0.909717i \(-0.636299\pi\)
−0.415229 + 0.909717i \(0.636299\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.57591 0.776668 0.388334 − 0.921519i \(-0.373051\pi\)
0.388334 + 0.921519i \(0.373051\pi\)
\(12\) 0 0
\(13\) 5.11784 1.41943 0.709717 − 0.704487i \(-0.248823\pi\)
0.709717 + 0.704487i \(0.248823\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 8.14266 1.97488 0.987442 − 0.157980i \(-0.0504981\pi\)
0.987442 + 0.157980i \(0.0504981\pi\)
\(18\) 0 0
\(19\) 4.57591 1.04979 0.524893 − 0.851168i \(-0.324105\pi\)
0.524893 + 0.851168i \(0.324105\pi\)
\(20\) 0 0
\(21\) −2.19719 −0.479466
\(22\) 0 0
\(23\) −3.22200 −0.671834 −0.335917 − 0.941892i \(-0.609046\pi\)
−0.335917 + 0.941892i \(0.609046\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.02482 −0.190304 −0.0951519 − 0.995463i \(-0.530334\pi\)
−0.0951519 + 0.995463i \(0.530334\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) 2.57591 0.448409
\(34\) 0 0
\(35\) 2.19719 0.371392
\(36\) 0 0
\(37\) −7.11784 −1.17017 −0.585083 − 0.810973i \(-0.698938\pi\)
−0.585083 + 0.810973i \(0.698938\pi\)
\(38\) 0 0
\(39\) 5.11784 0.819510
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 4.92065 0.750393 0.375196 − 0.926945i \(-0.377575\pi\)
0.375196 + 0.926945i \(0.377575\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −4.09302 −0.597029 −0.298514 − 0.954405i \(-0.596491\pi\)
−0.298514 + 0.954405i \(0.596491\pi\)
\(48\) 0 0
\(49\) −2.17237 −0.310338
\(50\) 0 0
\(51\) 8.14266 1.14020
\(52\) 0 0
\(53\) 3.17237 0.435758 0.217879 − 0.975976i \(-0.430086\pi\)
0.217879 + 0.975976i \(0.430086\pi\)
\(54\) 0 0
\(55\) −2.57591 −0.347336
\(56\) 0 0
\(57\) 4.57591 0.606095
\(58\) 0 0
\(59\) −6.52139 −0.849012 −0.424506 − 0.905425i \(-0.639552\pi\)
−0.424506 + 0.905425i \(0.639552\pi\)
\(60\) 0 0
\(61\) −5.49657 −0.703763 −0.351882 − 0.936044i \(-0.614458\pi\)
−0.351882 + 0.936044i \(0.614458\pi\)
\(62\) 0 0
\(63\) −2.19719 −0.276820
\(64\) 0 0
\(65\) −5.11784 −0.634790
\(66\) 0 0
\(67\) 1.27653 0.155953 0.0779767 − 0.996955i \(-0.475154\pi\)
0.0779767 + 0.996955i \(0.475154\pi\)
\(68\) 0 0
\(69\) −3.22200 −0.387884
\(70\) 0 0
\(71\) −2.10416 −0.249718 −0.124859 − 0.992174i \(-0.539848\pi\)
−0.124859 + 0.992174i \(0.539848\pi\)
\(72\) 0 0
\(73\) 1.85245 0.216813 0.108406 − 0.994107i \(-0.465425\pi\)
0.108406 + 0.994107i \(0.465425\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −5.65977 −0.644990
\(78\) 0 0
\(79\) 13.1129 1.47532 0.737661 − 0.675171i \(-0.235930\pi\)
0.737661 + 0.675171i \(0.235930\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −11.9840 −1.31541 −0.657706 − 0.753275i \(-0.728473\pi\)
−0.657706 + 0.753275i \(0.728473\pi\)
\(84\) 0 0
\(85\) −8.14266 −0.883195
\(86\) 0 0
\(87\) −1.02482 −0.109872
\(88\) 0 0
\(89\) 5.55110 0.588415 0.294208 − 0.955742i \(-0.404944\pi\)
0.294208 + 0.955742i \(0.404944\pi\)
\(90\) 0 0
\(91\) −11.2449 −1.17878
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −4.57591 −0.469479
\(96\) 0 0
\(97\) 14.6301 1.48546 0.742729 − 0.669593i \(-0.233531\pi\)
0.742729 + 0.669593i \(0.233531\pi\)
\(98\) 0 0
\(99\) 2.57591 0.258889
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.bx.1.2 4
4.3 odd 2 3720.2.a.r.1.3 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.r.1.3 ✓ 4 4.3 odd 2
7440.2.a.bx.1.2 4 1.1 even 1 trivial