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Properties

Label 975.2.bb.j
Level $975$
Weight $2$
Character orbit 975.bb
Analytic conductor $7.785$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(724,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.724");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bb (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_{5}) q^{2} + (\beta_{4} + \beta_{3}) q^{3} + ( - \beta_{11} - \beta_{10} - \beta_{8} + 1) q^{4} - \beta_{11} q^{6} + ( - 2 \beta_{6} + \beta_{5} + \beta_{3}) q^{7} + 2 \beta_{4} q^{8} + ( - \beta_{8} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_{5}) q^{2} + (\beta_{4} + \beta_{3}) q^{3} + ( - \beta_{11} - \beta_{10} - \beta_{8} + 1) q^{4} - \beta_{11} q^{6} + ( - 2 \beta_{6} + \beta_{5} + \beta_{3}) q^{7} + 2 \beta_{4} q^{8} + ( - \beta_{8} + 1) q^{9} + ( - 2 \beta_{11} - 2 \beta_{10} + \cdots + 2 \beta_1) q^{11}+ \cdots + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{4} + 6 q^{9} - 8 q^{11} + 48 q^{14} + 8 q^{16} + 20 q^{21} - 12 q^{24} - 36 q^{26} + 12 q^{29} - 36 q^{31} + 16 q^{34} - 4 q^{36} + 40 q^{41} - 96 q^{44} + 56 q^{46} + 60 q^{49} + 16 q^{51} + 20 q^{56} + 20 q^{59} - 18 q^{61} + 48 q^{64} - 48 q^{66} - 16 q^{69} - 8 q^{71} + 16 q^{74} + 16 q^{76} - 68 q^{79} - 6 q^{81} - 40 q^{86} - 44 q^{89} - 62 q^{91} + 16 q^{94} + 32 q^{96} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + \cdots + 748 ) / 460 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} + \cdots + 612 ) / 460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + \cdots - 12 ) / 460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + \cdots - 142 ) / 230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} + \cdots + 12 ) / 230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + \cdots - 32 ) / 460 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + \cdots - 420 ) / 460 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + \cdots - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} + \cdots + 476 ) / 460 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + \cdots - 1060 ) / 460 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} + \cdots + 612 ) / 230 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{7} + 2\beta_{4} + \beta_{2} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} + \beta_{10} + 7\beta_{8} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} + 3\beta_{10} + 9\beta_{8} - 3\beta_{6} - 8\beta_{5} - 9\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{9} + 6\beta_{7} + 28\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + \cdots - 33 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94\beta_{11} + 28\beta_{10} + 116\beta_{8} - 116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 138\beta_{9} + 44\beta_{7} + 166\beta_{4} - 44\beta_{2} + 138\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 398\beta_{9} + 122\beta_{7} + 122\beta_{6} + 398\beta_{5} + 486\beta_{4} + 486\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 580\beta_{11} + 182\beta_{10} + 702\beta_{8} + 182\beta_{6} + 580\beta_{5} + 702\beta_{3} - 702 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-1 + \beta_{8}\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1
1.98293 − 0.531325i
−0.147520 − 0.550552i
0.312819 + 1.16746i
−1.16746 + 0.312819i
0.550552 − 0.147520i
−0.531325 − 1.98293i
1.98293 + 0.531325i
−0.147520 + 0.550552i
0.312819 − 1.16746i
−1.16746 − 0.312819i
0.550552 + 0.147520i
−0.531325 + 1.98293i
−1.91766 − 1.10716i −0.866025 − 0.500000i 1.45161 + 2.51426i 0 1.10716 + 1.91766i −3.32254 + 1.91827i − 2.00000i 0.500000 + 0.866025i 0
724.2 −1.45071 − 0.837565i 0.866025 + 0.500000i 0.403032 + 0.698071i 0 −0.837565 − 1.45071i −3.15018 + 1.81876i 2.00000i 0.500000 + 0.866025i 0
724.3 −0.466951 − 0.269594i 0.866025 + 0.500000i −0.854638 − 1.48028i 0 −0.269594 − 0.466951i 4.15777 − 2.40049i 2.00000i 0.500000 + 0.866025i 0
724.4 0.466951 + 0.269594i −0.866025 − 0.500000i −0.854638 − 1.48028i 0 −0.269594 − 0.466951i −4.15777 + 2.40049i − 2.00000i 0.500000 + 0.866025i 0
724.5 1.45071 + 0.837565i −0.866025 − 0.500000i 0.403032 + 0.698071i 0 −0.837565 − 1.45071i 3.15018 − 1.81876i − 2.00000i 0.500000 + 0.866025i 0
724.6 1.91766 + 1.10716i 0.866025 + 0.500000i 1.45161 + 2.51426i 0 1.10716 + 1.91766i 3.32254 − 1.91827i 2.00000i 0.500000 + 0.866025i 0
874.1 −1.91766 + 1.10716i −0.866025 + 0.500000i 1.45161 − 2.51426i 0 1.10716 − 1.91766i −3.32254 − 1.91827i 2.00000i 0.500000 − 0.866025i 0
874.2 −1.45071 + 0.837565i 0.866025 − 0.500000i 0.403032 − 0.698071i 0 −0.837565 + 1.45071i −3.15018 − 1.81876i − 2.00000i 0.500000 − 0.866025i 0
874.3 −0.466951 + 0.269594i 0.866025 − 0.500000i −0.854638 + 1.48028i 0 −0.269594 + 0.466951i 4.15777 + 2.40049i − 2.00000i 0.500000 − 0.866025i 0
874.4 0.466951 − 0.269594i −0.866025 + 0.500000i −0.854638 + 1.48028i 0 −0.269594 + 0.466951i −4.15777 − 2.40049i 2.00000i 0.500000 − 0.866025i 0
874.5 1.45071 − 0.837565i −0.866025 + 0.500000i 0.403032 − 0.698071i 0 −0.837565 + 1.45071i 3.15018 + 1.81876i 2.00000i 0.500000 − 0.866025i 0
874.6 1.91766 − 1.10716i 0.866025 − 0.500000i 1.45161 − 2.51426i 0 1.10716 − 1.91766i 3.32254 + 1.91827i − 2.00000i 0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bb.j 12
5.b even 2 1 inner 975.2.bb.j 12
5.c odd 4 1 195.2.i.e ✓ 6
5.c odd 4 1 975.2.i.m 6
13.c even 3 1 inner 975.2.bb.j 12
15.e even 4 1 585.2.j.g 6
65.n even 6 1 inner 975.2.bb.j 12
65.q odd 12 1 195.2.i.e ✓ 6
65.q odd 12 1 975.2.i.m 6
65.q odd 12 1 2535.2.a.y 3
65.r odd 12 1 2535.2.a.z 3
195.bf even 12 1 7605.2.a.bt 3
195.bl even 12 1 585.2.j.g 6
195.bl even 12 1 7605.2.a.bu 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.e ✓ 6 5.c odd 4 1
195.2.i.e ✓ 6 65.q odd 12 1
585.2.j.g 6 15.e even 4 1
585.2.j.g 6 195.bl even 12 1
975.2.i.m 6 5.c odd 4 1
975.2.i.m 6 65.q odd 12 1
975.2.bb.j 12 1.a even 1 1 trivial
975.2.bb.j 12 5.b even 2 1 inner
975.2.bb.j 12 13.c even 3 1 inner
975.2.bb.j 12 65.n even 6 1 inner
2535.2.a.y 3 65.q odd 12 1
2535.2.a.z 3 65.r odd 12 1
7605.2.a.bt 3 195.bf even 12 1
7605.2.a.bu 3 195.bl even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{12} - 8T_{2}^{10} + 48T_{2}^{8} - 120T_{2}^{6} + 224T_{2}^{4} - 64T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{12} - 51T_{7}^{10} + 1762T_{7}^{8} - 33811T_{7}^{6} + 474982T_{7}^{4} - 3766271T_{7}^{2} + 20151121 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 8 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 51 T^{10} + \cdots + 20151121 \) Copy content Toggle raw display
$11$ \( (T^{6} + 4 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 15 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 32 T^{10} + \cdots + 1336336 \) Copy content Toggle raw display
$19$ \( (T^{6} + 40 T^{4} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 8540717056 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 9 T^{2} + \cdots - 109)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 533794816 \) Copy content Toggle raw display
$41$ \( (T^{6} - 20 T^{5} + \cdots + 45796)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 131079601 \) Copy content Toggle raw display
$47$ \( (T^{6} + 8 T^{4} + 16 T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 192 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 10 T^{5} + \cdots + 940900)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 9 T^{5} + \cdots + 11881)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 3722098081 \) Copy content Toggle raw display
$71$ \( (T^{6} + 4 T^{5} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 83 T^{4} + \cdots + 10609)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 17 T^{2} + \cdots + 67)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 28 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 22 T^{5} + \cdots + 17956)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 89526025681 \) Copy content Toggle raw display
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