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Properties

Label 7104.2.a.bc.1.1
Level $7104$
Weight $2$
Character 7104.1
Self dual yes
Analytic conductor $56.726$
Analytic rank $0$
Dimension $2$
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] = [7104,2,Mod(1,7104)] 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("7104.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7104 = 2^{6} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7104.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,-2,0,-8,0,2,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.7257255959\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 888)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7104.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.23607 q^{5} -4.00000 q^{7} +1.00000 q^{9} +6.47214 q^{11} +4.47214 q^{13} +3.23607 q^{15} -0.763932 q^{17} +4.00000 q^{21} +1.23607 q^{23} +5.47214 q^{25} -1.00000 q^{27} +9.70820 q^{29} -1.52786 q^{31} -6.47214 q^{33} +12.9443 q^{35} -1.00000 q^{37} -4.47214 q^{39} -10.9443 q^{41} -2.47214 q^{43} -3.23607 q^{45} -8.00000 q^{47} +9.00000 q^{49} +0.763932 q^{51} -8.47214 q^{53} -20.9443 q^{55} -10.1803 q^{59} -2.94427 q^{61} -4.00000 q^{63} -14.4721 q^{65} +12.9443 q^{67} -1.23607 q^{69} +4.94427 q^{71} +8.47214 q^{73} -5.47214 q^{75} -25.8885 q^{77} +8.94427 q^{79} +1.00000 q^{81} -14.4721 q^{83} +2.47214 q^{85} -9.70820 q^{87} +17.7082 q^{89} -17.8885 q^{91} +1.52786 q^{93} -13.4164 q^{97} +6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 8 q^{7} + 2 q^{9} + 4 q^{11} + 2 q^{15} - 6 q^{17} + 8 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 12 q^{31} - 4 q^{33} + 8 q^{35} - 2 q^{37} - 4 q^{41} + 4 q^{43} - 2 q^{45}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(5\) −3.23607 −1.44721 −0.723607 − 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.47214 1.95142 0.975711 − 0.219061i \(-0.0702993\pi\)
0.975711 + 0.219061i \(0.0702993\pi\)
\(12\) 0 0
\(13\) 4.47214 1.24035 0.620174 − 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 0 0
\(15\) 3.23607 0.835549
\(16\) 0 0
\(17\) −0.763932 −0.185281 −0.0926404 − 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 1.23607 0.257738 0.128869 − 0.991662i \(-0.458865\pi\)
0.128869 + 0.991662i \(0.458865\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.70820 1.80277 0.901384 − 0.433020i \(-0.142552\pi\)
0.901384 + 0.433020i \(0.142552\pi\)
\(30\) 0 0
\(31\) −1.52786 −0.274412 −0.137206 − 0.990543i \(-0.543812\pi\)
−0.137206 + 0.990543i \(0.543812\pi\)
\(32\) 0 0
\(33\) −6.47214 −1.12665
\(34\) 0 0
\(35\) 12.9443 2.18798
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) −4.47214 −0.716115
\(40\) 0 0
\(41\) −10.9443 −1.70921 −0.854604 − 0.519280i \(-0.826200\pi\)
−0.854604 + 0.519280i \(0.826200\pi\)
\(42\) 0 0
\(43\) −2.47214 −0.376997 −0.188499 − 0.982073i \(-0.560362\pi\)
−0.188499 + 0.982073i \(0.560362\pi\)
\(44\) 0 0
\(45\) −3.23607 −0.482405
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0.763932 0.106972
\(52\) 0 0
\(53\) −8.47214 −1.16374 −0.581869 − 0.813283i \(-0.697678\pi\)
−0.581869 + 0.813283i \(0.697678\pi\)
\(54\) 0 0
\(55\) −20.9443 −2.82413
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.1803 −1.32537 −0.662684 − 0.748899i \(-0.730583\pi\)
−0.662684 + 0.748899i \(0.730583\pi\)
\(60\) 0 0
\(61\) −2.94427 −0.376975 −0.188488 − 0.982076i \(-0.560359\pi\)
−0.188488 + 0.982076i \(0.560359\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) −14.4721 −1.79505
\(66\) 0 0
\(67\) 12.9443 1.58139 0.790697 − 0.612207i \(-0.209718\pi\)
0.790697 + 0.612207i \(0.209718\pi\)
\(68\) 0 0
\(69\) −1.23607 −0.148805
\(70\) 0 0
\(71\) 4.94427 0.586777 0.293389 − 0.955993i \(-0.405217\pi\)
0.293389 + 0.955993i \(0.405217\pi\)
\(72\) 0 0
\(73\) 8.47214 0.991589 0.495794 − 0.868440i \(-0.334877\pi\)
0.495794 + 0.868440i \(0.334877\pi\)
\(74\) 0 0
\(75\) −5.47214 −0.631868
\(76\) 0 0
\(77\) −25.8885 −2.95027
\(78\) 0 0
\(79\) 8.94427 1.00631 0.503155 − 0.864196i \(-0.332173\pi\)
0.503155 + 0.864196i \(0.332173\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.4721 −1.58852 −0.794262 − 0.607576i \(-0.792142\pi\)
−0.794262 + 0.607576i \(0.792142\pi\)
\(84\) 0 0
\(85\) 2.47214 0.268141
\(86\) 0 0
\(87\) −9.70820 −1.04083
\(88\) 0 0
\(89\) 17.7082 1.87707 0.938533 − 0.345190i \(-0.112185\pi\)
0.938533 + 0.345190i \(0.112185\pi\)
\(90\) 0 0
\(91\) −17.8885 −1.87523
\(92\) 0 0
\(93\) 1.52786 0.158432
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −13.4164 −1.36223 −0.681115 − 0.732177i \(-0.738505\pi\)
−0.681115 + 0.732177i \(0.738505\pi\)
\(98\) 0 0
\(99\) 6.47214 0.650474
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 7104.2.a.bc.1.1 2
4.3 odd 2 7104.2.a.bj.1.1 2
8.3 odd 2 888.2.a.f.1.2 ✓ 2
8.5 even 2 1776.2.a.p.1.2 2
24.5 odd 2 5328.2.a.ba.1.1 2
24.11 even 2 2664.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.a.f.1.2 ✓ 2 8.3 odd 2
1776.2.a.p.1.2 2 8.5 even 2
2664.2.a.j.1.1 2 24.11 even 2
5328.2.a.ba.1.1 2 24.5 odd 2
7104.2.a.bc.1.1 2 1.1 even 1 trivial
7104.2.a.bj.1.1 2 4.3 odd 2