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Properties

Label 7600.2.a.ch.1.2
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 1.2
Root \(1.93590\) 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)
 
\(f(q)\) \(=\) \(q-1.93590 q^{3} +1.24708 q^{7} +0.747704 q^{9} +0.513860 q^{11} -6.15670 q^{13} +4.51986 q^{17} +1.00000 q^{19} -2.41421 q^{21} -5.86084 q^{23} +4.36022 q^{27} +6.62700 q^{29} -6.41995 q^{31} -0.994780 q^{33} +1.40671 q^{37} +11.9187 q^{39} +10.6870 q^{41} -3.04878 q^{43} -1.99478 q^{47} -5.44480 q^{49} -8.74998 q^{51} +14.0848 q^{53} -1.93590 q^{57} +4.34261 q^{59} +10.7173 q^{61} +0.932444 q^{63} -9.89978 q^{67} +11.3460 q^{69} -7.42517 q^{71} -12.8079 q^{73} +0.640822 q^{77} -2.56138 q^{79} -10.6841 q^{81} -7.50864 q^{83} -12.8292 q^{87} -7.85353 q^{89} -7.67787 q^{91} +12.4284 q^{93} -6.74797 q^{97} +0.384215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.93590 −1.11769 −0.558846 − 0.829272i \(-0.688756\pi\)
−0.558846 + 0.829272i \(0.688756\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.24708 0.471350 0.235675 − 0.971832i \(-0.424270\pi\)
0.235675 + 0.971832i \(0.424270\pi\)
\(8\) 0 0
\(9\) 0.747704 0.249235
\(10\) 0 0
\(11\) 0.513860 0.154935 0.0774673 − 0.996995i \(-0.475317\pi\)
0.0774673 + 0.996995i \(0.475317\pi\)
\(12\) 0 0
\(13\) −6.15670 −1.70756 −0.853780 − 0.520633i \(-0.825696\pi\)
−0.853780 + 0.520633i \(0.825696\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.51986 1.09623 0.548113 − 0.836404i \(-0.315346\pi\)
0.548113 + 0.836404i \(0.315346\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.41421 −0.526824
\(22\) 0 0
\(23\) −5.86084 −1.22207 −0.611035 − 0.791603i \(-0.709247\pi\)
−0.611035 + 0.791603i \(0.709247\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.36022 0.839124
\(28\) 0 0
\(29\) 6.62700 1.23060 0.615302 − 0.788292i \(-0.289034\pi\)
0.615302 + 0.788292i \(0.289034\pi\)
\(30\) 0 0
\(31\) −6.41995 −1.15306 −0.576528 − 0.817077i \(-0.695593\pi\)
−0.576528 + 0.817077i \(0.695593\pi\)
\(32\) 0 0
\(33\) −0.994780 −0.173169
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.40671 0.231263 0.115631 − 0.993292i \(-0.463111\pi\)
0.115631 + 0.993292i \(0.463111\pi\)
\(38\) 0 0
\(39\) 11.9187 1.90853
\(40\) 0 0
\(41\) 10.6870 1.66903 0.834514 − 0.550987i \(-0.185749\pi\)
0.834514 + 0.550987i \(0.185749\pi\)
\(42\) 0 0
\(43\) −3.04878 −0.464934 −0.232467 − 0.972604i \(-0.574680\pi\)
−0.232467 + 0.972604i \(0.574680\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.99478 −0.290969 −0.145484 − 0.989361i \(-0.546474\pi\)
−0.145484 + 0.989361i \(0.546474\pi\)
\(48\) 0 0
\(49\) −5.44480 −0.777829
\(50\) 0 0
\(51\) −8.74998 −1.22524
\(52\) 0 0
\(53\) 14.0848 1.93469 0.967346 − 0.253459i \(-0.0815685\pi\)
0.967346 + 0.253459i \(0.0815685\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.93590 −0.256416
\(58\) 0 0
\(59\) 4.34261 0.565360 0.282680 − 0.959214i \(-0.408777\pi\)
0.282680 + 0.959214i \(0.408777\pi\)
\(60\) 0 0
\(61\) 10.7173 1.37221 0.686106 − 0.727502i \(-0.259319\pi\)
0.686106 + 0.727502i \(0.259319\pi\)
\(62\) 0 0
\(63\) 0.932444 0.117477
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −9.89978 −1.20945 −0.604725 − 0.796434i \(-0.706717\pi\)
−0.604725 + 0.796434i \(0.706717\pi\)
\(68\) 0 0
\(69\) 11.3460 1.36590
\(70\) 0 0
\(71\) −7.42517 −0.881205 −0.440603 − 0.897702i \(-0.645235\pi\)
−0.440603 + 0.897702i \(0.645235\pi\)
\(72\) 0 0
\(73\) −12.8079 −1.49905 −0.749525 − 0.661976i \(-0.769718\pi\)
−0.749525 + 0.661976i \(0.769718\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.640822 0.0730285
\(78\) 0 0
\(79\) −2.56138 −0.288178 −0.144089 − 0.989565i \(-0.546025\pi\)
−0.144089 + 0.989565i \(0.546025\pi\)
\(80\) 0 0
\(81\) −10.6841 −1.18712
\(82\) 0 0
\(83\) −7.50864 −0.824180 −0.412090 − 0.911143i \(-0.635201\pi\)
−0.412090 + 0.911143i \(0.635201\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −12.8292 −1.37543
\(88\) 0 0
\(89\) −7.85353 −0.832473 −0.416236 − 0.909256i \(-0.636651\pi\)
−0.416236 + 0.909256i \(0.636651\pi\)
\(90\) 0 0
\(91\) −7.67787 −0.804859
\(92\) 0 0
\(93\) 12.4284 1.28876
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.74797 −0.685152 −0.342576 − 0.939490i \(-0.611299\pi\)
−0.342576 + 0.939490i \(0.611299\pi\)
\(98\) 0 0
\(99\) 0.384215 0.0386151
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.1.2 6
4.3 odd 2 3800.2.a.bc.1.5 yes 6
5.4 even 2 7600.2.a.cl.1.5 6
20.3 even 4 3800.2.d.q.3649.10 12
20.7 even 4 3800.2.d.q.3649.3 12
20.19 odd 2 3800.2.a.ba.1.2 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.2 ✓ 6 20.19 odd 2
3800.2.a.bc.1.5 yes 6 4.3 odd 2
3800.2.d.q.3649.3 12 20.7 even 4
3800.2.d.q.3649.10 12 20.3 even 4
7600.2.a.ch.1.2 6 1.1 even 1 trivial
7600.2.a.cl.1.5 6 5.4 even 2