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Properties

Label 2100.2.d.j.1301.5
Level $2100$
Weight $2$
Character 2100.1301
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
CM discriminant -35
Inner twists $8$

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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] = [2100,2,Mod(1301,2100)] 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("2100.1301"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-14,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,-38,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(43)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5}, \sqrt{-7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1301.5
Root \(-0.862555 - 0.141174i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1301
Dual form 2100.2.d.j.1301.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.586627 - 1.62968i) q^{3} -2.64575i q^{7} +(-2.31174 - 1.91203i) q^{9} -0.359964i q^{11} -4.48660i q^{13} -7.99190 q^{17} +(-4.31174 - 1.55207i) q^{21} +(-4.47214 + 2.64575i) q^{27} +10.7523i q^{29} +(-0.586627 - 0.211164i) q^{33} +(-7.31174 - 2.63196i) q^{39} +12.4640 q^{47} -7.00000 q^{49} +(-4.68826 + 13.0243i) q^{51} +(-5.05876 + 6.11628i) q^{63} -11.8322i q^{71} -10.5830i q^{73} -0.952374 q^{77} -15.8704 q^{79} +(1.68826 + 8.84024i) q^{81} -8.94427 q^{83} +(17.5228 + 6.30757i) q^{87} -11.8704 q^{91} -15.0696i q^{97} +(-0.688262 + 0.832142i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{9} - 14 q^{21} - 38 q^{39} - 56 q^{49} - 58 q^{51} - 4 q^{79} + 34 q^{81} + 28 q^{91} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) 0.586627 − 1.62968i 0.338689 − 0.940898i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.64575i − 1.00000i
\(8\) 0 0
\(9\) −2.31174 − 1.91203i −0.770579 − 0.637344i
\(10\) 0 0
\(11\) − 0.359964i − 0.108533i −0.998526 − 0.0542666i \(-0.982718\pi\)
0.998526 − 0.0542666i \(-0.0172821\pi\)
\(12\) 0 0
\(13\) − 4.48660i − 1.24436i −0.782875 − 0.622179i \(-0.786247\pi\)
0.782875 − 0.622179i \(-0.213753\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.99190 −1.93832 −0.969160 − 0.246433i \(-0.920742\pi\)
−0.969160 + 0.246433i \(0.920742\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −4.31174 − 1.55207i −0.940898 − 0.338689i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.47214 + 2.64575i −0.860663 + 0.509175i
\(28\) 0 0
\(29\) 10.7523i 1.99665i 0.0578882 + 0.998323i \(0.481563\pi\)
−0.0578882 + 0.998323i \(0.518437\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) −0.586627 − 0.211164i −0.102119 − 0.0367590i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −7.31174 − 2.63196i −1.17082 − 0.421451i
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.4640 1.81807 0.909033 − 0.416724i \(-0.136822\pi\)
0.909033 + 0.416724i \(0.136822\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −4.68826 + 13.0243i −0.656488 + 1.82376i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −5.05876 + 6.11628i −0.637344 + 0.770579i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 11.8322i − 1.40422i −0.712069 − 0.702109i \(-0.752242\pi\)
0.712069 − 0.702109i \(-0.247758\pi\)
\(72\) 0 0
\(73\) − 10.5830i − 1.23865i −0.785136 − 0.619324i \(-0.787407\pi\)
0.785136 − 0.619324i \(-0.212593\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.952374 −0.108533
\(78\) 0 0
\(79\) −15.8704 −1.78556 −0.892781 − 0.450490i \(-0.851249\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) 0 0
\(81\) 1.68826 + 8.84024i 0.187585 + 0.982248i
\(82\) 0 0
\(83\) −8.94427 −0.981761 −0.490881 − 0.871227i \(-0.663325\pi\)
−0.490881 + 0.871227i \(0.663325\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 17.5228 + 6.30757i 1.87864 + 0.676243i
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −11.8704 −1.24436
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 15.0696i − 1.53009i −0.643979 − 0.765043i \(-0.722718\pi\)
0.643979 − 0.765043i \(-0.277282\pi\)
\(98\) 0 0
\(99\) −0.688262 + 0.832142i −0.0691730 + 0.0836334i
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 2100.2.d.j.1301.5 8
3.2 odd 2 inner 2100.2.d.j.1301.3 8
5.2 odd 4 420.2.f.a.209.1 ✓ 8
5.3 odd 4 420.2.f.a.209.8 yes 8
5.4 even 2 inner 2100.2.d.j.1301.4 8
7.6 odd 2 inner 2100.2.d.j.1301.4 8
15.2 even 4 420.2.f.a.209.2 yes 8
15.8 even 4 420.2.f.a.209.7 yes 8
15.14 odd 2 inner 2100.2.d.j.1301.6 8
20.3 even 4 1680.2.k.d.209.1 8
20.7 even 4 1680.2.k.d.209.8 8
21.20 even 2 inner 2100.2.d.j.1301.6 8
35.13 even 4 420.2.f.a.209.1 ✓ 8
35.27 even 4 420.2.f.a.209.8 yes 8
35.34 odd 2 CM 2100.2.d.j.1301.5 8
60.23 odd 4 1680.2.k.d.209.2 8
60.47 odd 4 1680.2.k.d.209.7 8
105.62 odd 4 420.2.f.a.209.7 yes 8
105.83 odd 4 420.2.f.a.209.2 yes 8
105.104 even 2 inner 2100.2.d.j.1301.3 8
140.27 odd 4 1680.2.k.d.209.1 8
140.83 odd 4 1680.2.k.d.209.8 8
420.83 even 4 1680.2.k.d.209.7 8
420.167 even 4 1680.2.k.d.209.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.f.a.209.1 ✓ 8 5.2 odd 4
420.2.f.a.209.1 ✓ 8 35.13 even 4
420.2.f.a.209.2 yes 8 15.2 even 4
420.2.f.a.209.2 yes 8 105.83 odd 4
420.2.f.a.209.7 yes 8 15.8 even 4
420.2.f.a.209.7 yes 8 105.62 odd 4
420.2.f.a.209.8 yes 8 5.3 odd 4
420.2.f.a.209.8 yes 8 35.27 even 4
1680.2.k.d.209.1 8 20.3 even 4
1680.2.k.d.209.1 8 140.27 odd 4
1680.2.k.d.209.2 8 60.23 odd 4
1680.2.k.d.209.2 8 420.167 even 4
1680.2.k.d.209.7 8 60.47 odd 4
1680.2.k.d.209.7 8 420.83 even 4
1680.2.k.d.209.8 8 20.7 even 4
1680.2.k.d.209.8 8 140.83 odd 4
2100.2.d.j.1301.3 8 3.2 odd 2 inner
2100.2.d.j.1301.3 8 105.104 even 2 inner
2100.2.d.j.1301.4 8 5.4 even 2 inner
2100.2.d.j.1301.4 8 7.6 odd 2 inner
2100.2.d.j.1301.5 8 1.1 even 1 trivial
2100.2.d.j.1301.5 8 35.34 odd 2 CM
2100.2.d.j.1301.6 8 15.14 odd 2 inner
2100.2.d.j.1301.6 8 21.20 even 2 inner