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Properties

Label 7600.2.a.ch.1.6
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 1.6
Root \(-2.77008\) 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)
 
\(f(q)\) \(=\) \(q+2.77008 q^{3} +2.31077 q^{7} +4.67334 q^{9} -2.16026 q^{11} -6.25643 q^{13} -7.10756 q^{17} +1.00000 q^{19} +6.40100 q^{21} -8.99784 q^{23} +4.63527 q^{27} +3.16424 q^{29} +9.95490 q^{31} -5.98410 q^{33} -9.43207 q^{37} -17.3308 q^{39} -10.8193 q^{41} +1.05217 q^{43} -6.98410 q^{47} -1.66036 q^{49} -19.6885 q^{51} +2.69517 q^{53} +2.77008 q^{57} -11.2021 q^{59} -4.36457 q^{61} +10.7990 q^{63} +6.25408 q^{67} -24.9247 q^{69} +13.9390 q^{71} +8.64016 q^{73} -4.99186 q^{77} +9.93758 q^{79} -1.17994 q^{81} -9.82384 q^{83} +8.76520 q^{87} -1.63686 q^{89} -14.4572 q^{91} +27.5759 q^{93} -9.27994 q^{97} -10.0956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} - 3 q^{11} - q^{13} - 14 q^{17} + 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} - 5 q^{31} + 2 q^{33} - 8 q^{37} - 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(3\) 2.77008 1.59931 0.799653 − 0.600463i \(-0.205017\pi\)
0.799653 + 0.600463i \(0.205017\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.31077 0.873387 0.436694 − 0.899610i \(-0.356149\pi\)
0.436694 + 0.899610i \(0.356149\pi\)
\(8\) 0 0
\(9\) 4.67334 1.55778
\(10\) 0 0
\(11\) −2.16026 −0.651344 −0.325672 − 0.945483i \(-0.605591\pi\)
−0.325672 + 0.945483i \(0.605591\pi\)
\(12\) 0 0
\(13\) −6.25643 −1.73522 −0.867611 − 0.497243i \(-0.834346\pi\)
−0.867611 + 0.497243i \(0.834346\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.10756 −1.72384 −0.861918 − 0.507047i \(-0.830737\pi\)
−0.861918 + 0.507047i \(0.830737\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 6.40100 1.39681
\(22\) 0 0
\(23\) −8.99784 −1.87618 −0.938090 − 0.346392i \(-0.887407\pi\)
−0.938090 + 0.346392i \(0.887407\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.63527 0.892059
\(28\) 0 0
\(29\) 3.16424 0.587585 0.293793 − 0.955869i \(-0.405083\pi\)
0.293793 + 0.955869i \(0.405083\pi\)
\(30\) 0 0
\(31\) 9.95490 1.78795 0.893977 − 0.448114i \(-0.147904\pi\)
0.893977 + 0.448114i \(0.147904\pi\)
\(32\) 0 0
\(33\) −5.98410 −1.04170
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.43207 −1.55062 −0.775311 − 0.631579i \(-0.782407\pi\)
−0.775311 + 0.631579i \(0.782407\pi\)
\(38\) 0 0
\(39\) −17.3308 −2.77515
\(40\) 0 0
\(41\) −10.8193 −1.68970 −0.844848 − 0.535007i \(-0.820309\pi\)
−0.844848 + 0.535007i \(0.820309\pi\)
\(42\) 0 0
\(43\) 1.05217 0.160455 0.0802275 − 0.996777i \(-0.474435\pi\)
0.0802275 + 0.996777i \(0.474435\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.98410 −1.01874 −0.509368 − 0.860549i \(-0.670121\pi\)
−0.509368 + 0.860549i \(0.670121\pi\)
\(48\) 0 0
\(49\) −1.66036 −0.237194
\(50\) 0 0
\(51\) −19.6885 −2.75694
\(52\) 0 0
\(53\) 2.69517 0.370210 0.185105 − 0.982719i \(-0.440738\pi\)
0.185105 + 0.982719i \(0.440738\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.77008 0.366906
\(58\) 0 0
\(59\) −11.2021 −1.45840 −0.729198 − 0.684303i \(-0.760106\pi\)
−0.729198 + 0.684303i \(0.760106\pi\)
\(60\) 0 0
\(61\) −4.36457 −0.558826 −0.279413 − 0.960171i \(-0.590140\pi\)
−0.279413 + 0.960171i \(0.590140\pi\)
\(62\) 0 0
\(63\) 10.7990 1.36054
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.25408 0.764058 0.382029 − 0.924150i \(-0.375225\pi\)
0.382029 + 0.924150i \(0.375225\pi\)
\(68\) 0 0
\(69\) −24.9247 −3.00059
\(70\) 0 0
\(71\) 13.9390 1.65426 0.827128 − 0.562014i \(-0.189973\pi\)
0.827128 + 0.562014i \(0.189973\pi\)
\(72\) 0 0
\(73\) 8.64016 1.01125 0.505627 − 0.862752i \(-0.331261\pi\)
0.505627 + 0.862752i \(0.331261\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.99186 −0.568876
\(78\) 0 0
\(79\) 9.93758 1.11806 0.559032 − 0.829146i \(-0.311173\pi\)
0.559032 + 0.829146i \(0.311173\pi\)
\(80\) 0 0
\(81\) −1.17994 −0.131104
\(82\) 0 0
\(83\) −9.82384 −1.07831 −0.539153 − 0.842208i \(-0.681256\pi\)
−0.539153 + 0.842208i \(0.681256\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.76520 0.939728
\(88\) 0 0
\(89\) −1.63686 −0.173507 −0.0867533 − 0.996230i \(-0.527649\pi\)
−0.0867533 + 0.996230i \(0.527649\pi\)
\(90\) 0 0
\(91\) −14.4572 −1.51552
\(92\) 0 0
\(93\) 27.5759 2.85948
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.27994 −0.942235 −0.471117 − 0.882070i \(-0.656149\pi\)
−0.471117 + 0.882070i \(0.656149\pi\)
\(98\) 0 0
\(99\) −10.0956 −1.01465
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.1.6 6
4.3 odd 2 3800.2.a.bc.1.1 yes 6
5.4 even 2 7600.2.a.cl.1.1 6
20.3 even 4 3800.2.d.q.3649.2 12
20.7 even 4 3800.2.d.q.3649.11 12
20.19 odd 2 3800.2.a.ba.1.6 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.6 ✓ 6 20.19 odd 2
3800.2.a.bc.1.1 yes 6 4.3 odd 2
3800.2.d.q.3649.2 12 20.3 even 4
3800.2.d.q.3649.11 12 20.7 even 4
7600.2.a.ch.1.6 6 1.1 even 1 trivial
7600.2.a.cl.1.1 6 5.4 even 2