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Properties

Label 7600.2.a.ch.1.5
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 1.5
Root \(-1.08999\) 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)
 
\(f(q)\) \(=\) \(q+1.08999 q^{3} -4.19727 q^{7} -1.81192 q^{9} +6.43052 q^{11} +2.24614 q^{13} -7.84744 q^{17} +1.00000 q^{19} -4.57497 q^{21} +0.859601 q^{23} -5.24494 q^{27} +8.38284 q^{29} -1.24541 q^{31} +7.00919 q^{33} -6.79977 q^{37} +2.44827 q^{39} +5.92480 q^{41} -6.81073 q^{43} +6.00919 q^{47} +10.6171 q^{49} -8.55363 q^{51} +13.7594 q^{53} +1.08999 q^{57} -6.88976 q^{59} -1.31884 q^{61} +7.60513 q^{63} -4.73266 q^{67} +0.936955 q^{69} -10.2546 q^{71} -9.86150 q^{73} -26.9906 q^{77} -6.47056 q^{79} -0.281158 q^{81} -5.42133 q^{83} +9.13720 q^{87} +3.10288 q^{89} -9.42766 q^{91} -1.35748 q^{93} +6.76032 q^{97} -11.6516 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.08999 0.629305 0.314653 − 0.949207i \(-0.398112\pi\)
0.314653 + 0.949207i \(0.398112\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.19727 −1.58642 −0.793209 − 0.608949i \(-0.791591\pi\)
−0.793209 + 0.608949i \(0.791591\pi\)
\(8\) 0 0
\(9\) −1.81192 −0.603975
\(10\) 0 0
\(11\) 6.43052 1.93887 0.969437 − 0.245341i \(-0.0788998\pi\)
0.969437 + 0.245341i \(0.0788998\pi\)
\(12\) 0 0
\(13\) 2.24614 0.622967 0.311484 − 0.950251i \(-0.399174\pi\)
0.311484 + 0.950251i \(0.399174\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.84744 −1.90328 −0.951642 − 0.307208i \(-0.900605\pi\)
−0.951642 + 0.307208i \(0.900605\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −4.57497 −0.998341
\(22\) 0 0
\(23\) 0.859601 0.179239 0.0896195 − 0.995976i \(-0.471435\pi\)
0.0896195 + 0.995976i \(0.471435\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.24494 −1.00939
\(28\) 0 0
\(29\) 8.38284 1.55665 0.778327 − 0.627859i \(-0.216068\pi\)
0.778327 + 0.627859i \(0.216068\pi\)
\(30\) 0 0
\(31\) −1.24541 −0.223681 −0.111841 − 0.993726i \(-0.535675\pi\)
−0.111841 + 0.993726i \(0.535675\pi\)
\(32\) 0 0
\(33\) 7.00919 1.22014
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.79977 −1.11787 −0.558937 − 0.829210i \(-0.688791\pi\)
−0.558937 + 0.829210i \(0.688791\pi\)
\(38\) 0 0
\(39\) 2.44827 0.392037
\(40\) 0 0
\(41\) 5.92480 0.925298 0.462649 − 0.886542i \(-0.346899\pi\)
0.462649 + 0.886542i \(0.346899\pi\)
\(42\) 0 0
\(43\) −6.81073 −1.03863 −0.519313 − 0.854584i \(-0.673812\pi\)
−0.519313 + 0.854584i \(0.673812\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00919 0.876531 0.438265 − 0.898846i \(-0.355593\pi\)
0.438265 + 0.898846i \(0.355593\pi\)
\(48\) 0 0
\(49\) 10.6171 1.51672
\(50\) 0 0
\(51\) −8.55363 −1.19775
\(52\) 0 0
\(53\) 13.7594 1.88999 0.944996 − 0.327082i \(-0.106065\pi\)
0.944996 + 0.327082i \(0.106065\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.08999 0.144373
\(58\) 0 0
\(59\) −6.88976 −0.896970 −0.448485 − 0.893790i \(-0.648036\pi\)
−0.448485 + 0.893790i \(0.648036\pi\)
\(60\) 0 0
\(61\) −1.31884 −0.168860 −0.0844301 − 0.996429i \(-0.526907\pi\)
−0.0844301 + 0.996429i \(0.526907\pi\)
\(62\) 0 0
\(63\) 7.60513 0.958156
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.73266 −0.578187 −0.289093 − 0.957301i \(-0.593354\pi\)
−0.289093 + 0.957301i \(0.593354\pi\)
\(68\) 0 0
\(69\) 0.936955 0.112796
\(70\) 0 0
\(71\) −10.2546 −1.21700 −0.608498 − 0.793555i \(-0.708228\pi\)
−0.608498 + 0.793555i \(0.708228\pi\)
\(72\) 0 0
\(73\) −9.86150 −1.15420 −0.577100 − 0.816673i \(-0.695816\pi\)
−0.577100 + 0.816673i \(0.695816\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −26.9906 −3.07586
\(78\) 0 0
\(79\) −6.47056 −0.727995 −0.363997 − 0.931400i \(-0.618588\pi\)
−0.363997 + 0.931400i \(0.618588\pi\)
\(80\) 0 0
\(81\) −0.281158 −0.0312397
\(82\) 0 0
\(83\) −5.42133 −0.595068 −0.297534 − 0.954711i \(-0.596164\pi\)
−0.297534 + 0.954711i \(0.596164\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.13720 0.979611
\(88\) 0 0
\(89\) 3.10288 0.328905 0.164452 − 0.986385i \(-0.447414\pi\)
0.164452 + 0.986385i \(0.447414\pi\)
\(90\) 0 0
\(91\) −9.42766 −0.988287
\(92\) 0 0
\(93\) −1.35748 −0.140764
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.76032 0.686406 0.343203 − 0.939261i \(-0.388488\pi\)
0.343203 + 0.939261i \(0.388488\pi\)
\(98\) 0 0
\(99\) −11.6516 −1.17103
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.1.5 6
4.3 odd 2 3800.2.a.bc.1.2 yes 6
5.4 even 2 7600.2.a.cl.1.2 6
20.3 even 4 3800.2.d.q.3649.4 12
20.7 even 4 3800.2.d.q.3649.9 12
20.19 odd 2 3800.2.a.ba.1.5 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.5 ✓ 6 20.19 odd 2
3800.2.a.bc.1.2 yes 6 4.3 odd 2
3800.2.d.q.3649.4 12 20.3 even 4
3800.2.d.q.3649.9 12 20.7 even 4
7600.2.a.ch.1.5 6 1.1 even 1 trivial
7600.2.a.cl.1.2 6 5.4 even 2