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Properties

Label 7600.2.a.ch.1.3
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $1$
Dimension $6$
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] = [7600,2,Mod(1,7600)] 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("7600.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-2,0,0,0,-2,0,6,0,-3,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(60.6863055362\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3800)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.848258\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.848258 q^{3} -1.74484 q^{7} -2.28046 q^{9} -5.92425 q^{11} +6.78582 q^{13} +1.86024 q^{17} +1.00000 q^{19} +1.48007 q^{21} -5.94357 q^{23} +4.47919 q^{27} +3.29977 q^{29} +5.75242 q^{31} +5.02529 q^{33} -4.36379 q^{37} -5.75613 q^{39} +7.12500 q^{41} +6.98455 q^{43} +4.02529 q^{47} -3.95555 q^{49} -1.57796 q^{51} -9.19015 q^{53} -0.848258 q^{57} -2.51553 q^{59} -2.49621 q^{61} +3.97903 q^{63} -6.90485 q^{67} +5.04168 q^{69} -1.27288 q^{71} +12.1217 q^{73} +10.3368 q^{77} +13.8376 q^{79} +3.04187 q^{81} +4.94955 q^{83} -2.79906 q^{87} +15.6067 q^{89} -11.8401 q^{91} -4.87953 q^{93} -15.7764 q^{97} +13.5100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) −0.848258 −0.489742 −0.244871 − 0.969556i \(-0.578746\pi\)
−0.244871 + 0.969556i \(0.578746\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.74484 −0.659486 −0.329743 − 0.944071i \(-0.606962\pi\)
−0.329743 + 0.944071i \(0.606962\pi\)
\(8\) 0 0
\(9\) −2.28046 −0.760153
\(10\) 0 0
\(11\) −5.92425 −1.78623 −0.893115 − 0.449829i \(-0.851485\pi\)
−0.893115 + 0.449829i \(0.851485\pi\)
\(12\) 0 0
\(13\) 6.78582 1.88205 0.941024 − 0.338339i \(-0.109865\pi\)
0.941024 + 0.338339i \(0.109865\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.86024 0.451174 0.225587 − 0.974223i \(-0.427570\pi\)
0.225587 + 0.974223i \(0.427570\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.48007 0.322978
\(22\) 0 0
\(23\) −5.94357 −1.23932 −0.619660 − 0.784871i \(-0.712729\pi\)
−0.619660 + 0.784871i \(0.712729\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.47919 0.862021
\(28\) 0 0
\(29\) 3.29977 0.612752 0.306376 − 0.951911i \(-0.400883\pi\)
0.306376 + 0.951911i \(0.400883\pi\)
\(30\) 0 0
\(31\) 5.75242 1.03316 0.516582 − 0.856238i \(-0.327204\pi\)
0.516582 + 0.856238i \(0.327204\pi\)
\(32\) 0 0
\(33\) 5.02529 0.874791
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.36379 −0.717402 −0.358701 − 0.933453i \(-0.616780\pi\)
−0.358701 + 0.933453i \(0.616780\pi\)
\(38\) 0 0
\(39\) −5.75613 −0.921718
\(40\) 0 0
\(41\) 7.12500 1.11274 0.556369 − 0.830935i \(-0.312194\pi\)
0.556369 + 0.830935i \(0.312194\pi\)
\(42\) 0 0
\(43\) 6.98455 1.06513 0.532567 − 0.846388i \(-0.321227\pi\)
0.532567 + 0.846388i \(0.321227\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.02529 0.587149 0.293575 − 0.955936i \(-0.405155\pi\)
0.293575 + 0.955936i \(0.405155\pi\)
\(48\) 0 0
\(49\) −3.95555 −0.565078
\(50\) 0 0
\(51\) −1.57796 −0.220959
\(52\) 0 0
\(53\) −9.19015 −1.26236 −0.631182 − 0.775635i \(-0.717430\pi\)
−0.631182 + 0.775635i \(0.717430\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.848258 −0.112354
\(58\) 0 0
\(59\) −2.51553 −0.327494 −0.163747 − 0.986502i \(-0.552358\pi\)
−0.163747 + 0.986502i \(0.552358\pi\)
\(60\) 0 0
\(61\) −2.49621 −0.319607 −0.159804 − 0.987149i \(-0.551086\pi\)
−0.159804 + 0.987149i \(0.551086\pi\)
\(62\) 0 0
\(63\) 3.97903 0.501310
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.90485 −0.843561 −0.421781 − 0.906698i \(-0.638595\pi\)
−0.421781 + 0.906698i \(0.638595\pi\)
\(68\) 0 0
\(69\) 5.04168 0.606947
\(70\) 0 0
\(71\) −1.27288 −0.151063 −0.0755314 − 0.997143i \(-0.524065\pi\)
−0.0755314 + 0.997143i \(0.524065\pi\)
\(72\) 0 0
\(73\) 12.1217 1.41874 0.709371 − 0.704836i \(-0.248979\pi\)
0.709371 + 0.704836i \(0.248979\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3368 1.17799
\(78\) 0 0
\(79\) 13.8376 1.55685 0.778424 − 0.627739i \(-0.216020\pi\)
0.778424 + 0.627739i \(0.216020\pi\)
\(80\) 0 0
\(81\) 3.04187 0.337985
\(82\) 0 0
\(83\) 4.94955 0.543283 0.271642 − 0.962398i \(-0.412433\pi\)
0.271642 + 0.962398i \(0.412433\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.79906 −0.300091
\(88\) 0 0
\(89\) 15.6067 1.65430 0.827151 − 0.561980i \(-0.189960\pi\)
0.827151 + 0.561980i \(0.189960\pi\)
\(90\) 0 0
\(91\) −11.8401 −1.24118
\(92\) 0 0
\(93\) −4.87953 −0.505984
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.7764 −1.60185 −0.800924 − 0.598766i \(-0.795658\pi\)
−0.800924 + 0.598766i \(0.795658\pi\)
\(98\) 0 0
\(99\) 13.5100 1.35781
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 7600.2.a.ch.1.3 6
4.3 odd 2 3800.2.a.bc.1.4 yes 6
5.4 even 2 7600.2.a.cl.1.4 6
20.3 even 4 3800.2.d.q.3649.8 12
20.7 even 4 3800.2.d.q.3649.5 12
20.19 odd 2 3800.2.a.ba.1.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.3 ✓ 6 20.19 odd 2
3800.2.a.bc.1.4 yes 6 4.3 odd 2
3800.2.d.q.3649.5 12 20.7 even 4
3800.2.d.q.3649.8 12 20.3 even 4
7600.2.a.ch.1.3 6 1.1 even 1 trivial
7600.2.a.cl.1.4 6 5.4 even 2