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Properties

Label 7600.2.a.bh.1.3
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
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] = [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 = [3,0,-2,0,0,0,-4,0,1,0,-1,0,3] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.169.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 475)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.65109\) 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)
 
\(f(q)\) \(=\) \(q+1.65109 q^{3} -3.65109 q^{7} -0.273891 q^{9} -2.65109 q^{11} +6.13161 q^{13} +2.34891 q^{17} -1.00000 q^{19} -6.02830 q^{21} -5.48052 q^{23} -5.40550 q^{27} +0.651093 q^{29} +6.67939 q^{31} -4.37720 q^{33} -8.70769 q^{37} +10.1239 q^{39} +1.93273 q^{41} +2.65884 q^{43} -3.71836 q^{47} +6.33048 q^{49} +3.87826 q^{51} +13.7544 q^{53} -1.65109 q^{57} +7.84997 q^{59} -1.92498 q^{61} +1.00000 q^{63} +4.44447 q^{67} -9.04884 q^{69} -3.54778 q^{71} -2.48052 q^{73} +9.67939 q^{77} +15.1599 q^{79} -8.10331 q^{81} +14.7282 q^{83} +1.07502 q^{87} -5.06727 q^{89} -22.3871 q^{91} +11.0283 q^{93} +3.22717 q^{97} +0.726109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 4 q^{7} + q^{9} - q^{11} + 3 q^{13} + 14 q^{17} - 3 q^{19} - 6 q^{21} - 8 q^{23} + q^{27} - 5 q^{29} + q^{31} - 8 q^{33} + 5 q^{37} + 11 q^{39} + q^{41} + 5 q^{43} - 9 q^{47} - 7 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) 1.65109 0.953259 0.476630 − 0.879104i \(-0.341858\pi\)
0.476630 + 0.879104i \(0.341858\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.65109 −1.37998 −0.689992 − 0.723817i \(-0.742386\pi\)
−0.689992 + 0.723817i \(0.742386\pi\)
\(8\) 0 0
\(9\) −0.273891 −0.0912969
\(10\) 0 0
\(11\) −2.65109 −0.799335 −0.399667 − 0.916660i \(-0.630874\pi\)
−0.399667 + 0.916660i \(0.630874\pi\)
\(12\) 0 0
\(13\) 6.13161 1.70060 0.850301 − 0.526297i \(-0.176420\pi\)
0.850301 + 0.526297i \(0.176420\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.34891 0.569694 0.284847 − 0.958573i \(-0.408057\pi\)
0.284847 + 0.958573i \(0.408057\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −6.02830 −1.31548
\(22\) 0 0
\(23\) −5.48052 −1.14277 −0.571383 − 0.820683i \(-0.693593\pi\)
−0.571383 + 0.820683i \(0.693593\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.40550 −1.04029
\(28\) 0 0
\(29\) 0.651093 0.120905 0.0604525 − 0.998171i \(-0.480746\pi\)
0.0604525 + 0.998171i \(0.480746\pi\)
\(30\) 0 0
\(31\) 6.67939 1.19965 0.599827 − 0.800130i \(-0.295236\pi\)
0.599827 + 0.800130i \(0.295236\pi\)
\(32\) 0 0
\(33\) −4.37720 −0.761973
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.70769 −1.43153 −0.715767 − 0.698339i \(-0.753923\pi\)
−0.715767 + 0.698339i \(0.753923\pi\)
\(38\) 0 0
\(39\) 10.1239 1.62111
\(40\) 0 0
\(41\) 1.93273 0.301842 0.150921 − 0.988546i \(-0.451776\pi\)
0.150921 + 0.988546i \(0.451776\pi\)
\(42\) 0 0
\(43\) 2.65884 0.405470 0.202735 − 0.979234i \(-0.435017\pi\)
0.202735 + 0.979234i \(0.435017\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.71836 −0.542378 −0.271189 − 0.962526i \(-0.587417\pi\)
−0.271189 + 0.962526i \(0.587417\pi\)
\(48\) 0 0
\(49\) 6.33048 0.904355
\(50\) 0 0
\(51\) 3.87826 0.543066
\(52\) 0 0
\(53\) 13.7544 1.88931 0.944656 − 0.328061i \(-0.106395\pi\)
0.944656 + 0.328061i \(0.106395\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.65109 −0.218693
\(58\) 0 0
\(59\) 7.84997 1.02198 0.510989 − 0.859587i \(-0.329279\pi\)
0.510989 + 0.859587i \(0.329279\pi\)
\(60\) 0 0
\(61\) −1.92498 −0.246469 −0.123234 − 0.992378i \(-0.539327\pi\)
−0.123234 + 0.992378i \(0.539327\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.44447 0.542978 0.271489 − 0.962442i \(-0.412484\pi\)
0.271489 + 0.962442i \(0.412484\pi\)
\(68\) 0 0
\(69\) −9.04884 −1.08935
\(70\) 0 0
\(71\) −3.54778 −0.421044 −0.210522 − 0.977589i \(-0.567516\pi\)
−0.210522 + 0.977589i \(0.567516\pi\)
\(72\) 0 0
\(73\) −2.48052 −0.290322 −0.145161 − 0.989408i \(-0.546370\pi\)
−0.145161 + 0.989408i \(0.546370\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.67939 1.10307
\(78\) 0 0
\(79\) 15.1599 1.70562 0.852811 − 0.522219i \(-0.174896\pi\)
0.852811 + 0.522219i \(0.174896\pi\)
\(80\) 0 0
\(81\) −8.10331 −0.900368
\(82\) 0 0
\(83\) 14.7282 1.61663 0.808317 − 0.588748i \(-0.200379\pi\)
0.808317 + 0.588748i \(0.200379\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.07502 0.115254
\(88\) 0 0
\(89\) −5.06727 −0.537129 −0.268565 − 0.963262i \(-0.586549\pi\)
−0.268565 + 0.963262i \(0.586549\pi\)
\(90\) 0 0
\(91\) −22.3871 −2.34680
\(92\) 0 0
\(93\) 11.0283 1.14358
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.22717 0.327670 0.163835 − 0.986488i \(-0.447614\pi\)
0.163835 + 0.986488i \(0.447614\pi\)
\(98\) 0 0
\(99\) 0.726109 0.0729767
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.bh.1.3 3
4.3 odd 2 475.2.a.g.1.2 yes 3
5.4 even 2 7600.2.a.cc.1.1 3
12.11 even 2 4275.2.a.ba.1.2 3
20.3 even 4 475.2.b.b.324.3 6
20.7 even 4 475.2.b.b.324.4 6
20.19 odd 2 475.2.a.e.1.2 ✓ 3
60.59 even 2 4275.2.a.bm.1.2 3
76.75 even 2 9025.2.a.y.1.2 3
380.379 even 2 9025.2.a.bc.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.2 ✓ 3 20.19 odd 2
475.2.a.g.1.2 yes 3 4.3 odd 2
475.2.b.b.324.3 6 20.3 even 4
475.2.b.b.324.4 6 20.7 even 4
4275.2.a.ba.1.2 3 12.11 even 2
4275.2.a.bm.1.2 3 60.59 even 2
7600.2.a.bh.1.3 3 1.1 even 1 trivial
7600.2.a.cc.1.1 3 5.4 even 2
9025.2.a.y.1.2 3 76.75 even 2
9025.2.a.bc.1.2 3 380.379 even 2