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: \(1\)
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 1.2
Root \(-0.273891\) 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)
 
\(f(q)\) \(=\) \(q+1.27389 q^{3} +0.726109 q^{7} -1.37720 q^{9} +0.273891 q^{11} +5.95328 q^{13} -5.27389 q^{17} -1.00000 q^{19} +0.924984 q^{21} -3.67939 q^{23} -5.57608 q^{27} -2.27389 q^{29} -3.19887 q^{31} +0.348907 q^{33} -8.12386 q^{37} +7.58383 q^{39} -9.43380 q^{41} +9.81100 q^{43} +12.1599 q^{47} -6.47277 q^{49} -6.71836 q^{51} -5.69781 q^{53} -1.27389 q^{57} +4.20662 q^{59} -0.103312 q^{61} -1.00000 q^{63} -11.7827 q^{67} -4.68714 q^{69} -5.75441 q^{71} -6.67939 q^{73} +0.198875 q^{77} -3.87826 q^{79} -2.97170 q^{81} -0.488265 q^{83} -2.89669 q^{87} -16.4338 q^{89} +4.32273 q^{91} -4.07502 q^{93} +4.44447 q^{97} -0.377203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 4 q^{7} + q^{9} - q^{11} - 3 q^{13} - 14 q^{17} - 3 q^{19} - 6 q^{21} + 8 q^{23} - q^{27} - 5 q^{29} + q^{31} + 8 q^{33} - 5 q^{37} + 11 q^{39} + q^{41} - 5 q^{43} + 9 q^{47} - 7 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) 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
\(3\) 1.27389 0.735481 0.367741 − 0.929928i \(-0.380131\pi\)
0.367741 + 0.929928i \(0.380131\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.726109 0.274444 0.137222 − 0.990540i \(-0.456183\pi\)
0.137222 + 0.990540i \(0.456183\pi\)
\(8\) 0 0
\(9\) −1.37720 −0.459068
\(10\) 0 0
\(11\) 0.273891 0.0825811 0.0412906 − 0.999147i \(-0.486853\pi\)
0.0412906 + 0.999147i \(0.486853\pi\)
\(12\) 0 0
\(13\) 5.95328 1.65114 0.825571 − 0.564298i \(-0.190853\pi\)
0.825571 + 0.564298i \(0.190853\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.27389 −1.27911 −0.639553 − 0.768747i \(-0.720881\pi\)
−0.639553 + 0.768747i \(0.720881\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.924984 0.201848
\(22\) 0 0
\(23\) −3.67939 −0.767206 −0.383603 − 0.923498i \(-0.625317\pi\)
−0.383603 + 0.923498i \(0.625317\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.57608 −1.07312
\(28\) 0 0
\(29\) −2.27389 −0.422251 −0.211125 − 0.977459i \(-0.567713\pi\)
−0.211125 + 0.977459i \(0.567713\pi\)
\(30\) 0 0
\(31\) −3.19887 −0.574535 −0.287267 − 0.957850i \(-0.592747\pi\)
−0.287267 + 0.957850i \(0.592747\pi\)
\(32\) 0 0
\(33\) 0.348907 0.0607368
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.12386 −1.33555 −0.667777 − 0.744361i \(-0.732754\pi\)
−0.667777 + 0.744361i \(0.732754\pi\)
\(38\) 0 0
\(39\) 7.58383 1.21438
\(40\) 0 0
\(41\) −9.43380 −1.47331 −0.736656 − 0.676268i \(-0.763596\pi\)
−0.736656 + 0.676268i \(0.763596\pi\)
\(42\) 0 0
\(43\) 9.81100 1.49616 0.748082 − 0.663607i \(-0.230975\pi\)
0.748082 + 0.663607i \(0.230975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.1599 1.77370 0.886852 − 0.462053i \(-0.152887\pi\)
0.886852 + 0.462053i \(0.152887\pi\)
\(48\) 0 0
\(49\) −6.47277 −0.924681
\(50\) 0 0
\(51\) −6.71836 −0.940758
\(52\) 0 0
\(53\) −5.69781 −0.782655 −0.391327 − 0.920252i \(-0.627984\pi\)
−0.391327 + 0.920252i \(0.627984\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.27389 −0.168731
\(58\) 0 0
\(59\) 4.20662 0.547656 0.273828 − 0.961779i \(-0.411710\pi\)
0.273828 + 0.961779i \(0.411710\pi\)
\(60\) 0 0
\(61\) −0.103312 −0.0132278 −0.00661389 − 0.999978i \(-0.502105\pi\)
−0.00661389 + 0.999978i \(0.502105\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.7827 −1.43949 −0.719743 − 0.694241i \(-0.755740\pi\)
−0.719743 + 0.694241i \(0.755740\pi\)
\(68\) 0 0
\(69\) −4.68714 −0.564265
\(70\) 0 0
\(71\) −5.75441 −0.682922 −0.341461 − 0.939896i \(-0.610922\pi\)
−0.341461 + 0.939896i \(0.610922\pi\)
\(72\) 0 0
\(73\) −6.67939 −0.781763 −0.390882 − 0.920441i \(-0.627830\pi\)
−0.390882 + 0.920441i \(0.627830\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.198875 0.0226639
\(78\) 0 0
\(79\) −3.87826 −0.436339 −0.218169 − 0.975911i \(-0.570009\pi\)
−0.218169 + 0.975911i \(0.570009\pi\)
\(80\) 0 0
\(81\) −2.97170 −0.330189
\(82\) 0 0
\(83\) −0.488265 −0.0535941 −0.0267970 − 0.999641i \(-0.508531\pi\)
−0.0267970 + 0.999641i \(0.508531\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.89669 −0.310558
\(88\) 0 0
\(89\) −16.4338 −1.74198 −0.870989 − 0.491302i \(-0.836521\pi\)
−0.870989 + 0.491302i \(0.836521\pi\)
\(90\) 0 0
\(91\) 4.32273 0.453146
\(92\) 0 0
\(93\) −4.07502 −0.422559
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.44447 0.451267 0.225634 − 0.974212i \(-0.427555\pi\)
0.225634 + 0.974212i \(0.427555\pi\)
\(98\) 0 0
\(99\) −0.377203 −0.0379103
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.cc.1.2 3
4.3 odd 2 475.2.a.e.1.1 ✓ 3
5.4 even 2 7600.2.a.bh.1.2 3
12.11 even 2 4275.2.a.bm.1.3 3
20.3 even 4 475.2.b.b.324.6 6
20.7 even 4 475.2.b.b.324.1 6
20.19 odd 2 475.2.a.g.1.3 yes 3
60.59 even 2 4275.2.a.ba.1.1 3
76.75 even 2 9025.2.a.bc.1.3 3
380.379 even 2 9025.2.a.y.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.e.1.1 ✓ 3 4.3 odd 2
475.2.a.g.1.3 yes 3 20.19 odd 2
475.2.b.b.324.1 6 20.7 even 4
475.2.b.b.324.6 6 20.3 even 4
4275.2.a.ba.1.1 3 60.59 even 2
4275.2.a.bm.1.3 3 12.11 even 2
7600.2.a.bh.1.2 3 5.4 even 2
7600.2.a.cc.1.2 3 1.1 even 1 trivial
9025.2.a.y.1.1 3 380.379 even 2
9025.2.a.bc.1.3 3 76.75 even 2