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Properties

Label 975.2.bl.e
Level $975$
Weight $2$
Character orbit 975.bl
Analytic conductor $7.785$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(193,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bl (of order \(12\), degree \(4\), 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.11007531417600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{15} - \beta_{13} + \cdots + \beta_{6}) q^{2}+ \cdots + (\beta_{12} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} - \beta_{13} + \cdots + \beta_{6}) q^{2}+ \cdots + ( - \beta_{12} + \beta_{11} + \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} + 4 q^{6} - 12 q^{11} + 32 q^{19} + 8 q^{21} + 16 q^{24} - 4 q^{26} - 24 q^{29} - 24 q^{31} + 24 q^{34} - 16 q^{39} - 28 q^{41} - 16 q^{44} + 72 q^{46} + 40 q^{49} - 4 q^{54} - 48 q^{56} + 80 q^{59} - 16 q^{61} - 64 q^{64} + 16 q^{66} + 4 q^{69} - 44 q^{71} - 48 q^{74} + 8 q^{81} + 88 q^{84} + 32 q^{86} - 32 q^{89} - 40 q^{91} + 96 q^{94} + 64 q^{96} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{12} + 48x^{8} - 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{13} + 233\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 377\nu^{2} ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{15} - 305\nu^{3} ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} + 13\nu^{12} - 96\nu^{8} + 624\nu^{4} + 233\nu^{2} - 91 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{14} - 13\nu^{12} + 96\nu^{8} - 624\nu^{4} + 610\nu^{2} + 91 ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 21\nu^{13} + 144\nu^{9} - 1008\nu^{5} + 377\nu^{3} + 147\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{12} + 48\nu^{8} - 336\nu^{4} + 1 ) / 48 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} + 7\nu^{13} - 48\nu^{9} + 336\nu^{5} + 329\nu^{3} - 49\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -35\nu^{14} - \nu^{12} + 240\nu^{10} - 1632\nu^{6} + 5\nu^{2} - 233 ) / 144 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35\nu^{13} - 240\nu^{9} + 1632\nu^{5} - 5\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\nu^{14} - 7\nu^{12} - 120\nu^{10} + 48\nu^{8} + 816\nu^{6} - 312\nu^{4} - 119\nu^{2} + 1 ) / 72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7\nu^{14} - 48\nu^{10} + 330\nu^{6} - \nu^{2} ) / 18 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7\nu^{15} - 48\nu^{11} + 330\nu^{7} - \nu^{3} + 18\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -56\nu^{15} + \nu^{13} + 384\nu^{11} - 2640\nu^{7} + 8\nu^{3} + 377\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 89\nu^{15} - 624\nu^{11} + 4272\nu^{7} - 623\nu^{3} ) / 144 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{4} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{11} + 3\beta_{9} - 4\beta_{7} + 3\beta_{5} - 3\beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{14} + 5\beta_{13} - 6\beta_{10} + 5\beta_{8} - 5\beta_{6} + 5\beta_{3} - 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5\beta_{12} - 4\beta_{11} + 4\beta_{9} - 4\beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -16\beta_{15} - 13\beta_{14} + 13\beta_{13} + 16\beta_{3} + 13\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -26\beta_{7} + 21\beta_{5} - 21\beta_{4} - 21\beta_{2} - 26 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -21\beta_{10} + 17\beta_{8} - 17\beta_{6} + 17\beta_{3} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 68\beta_{12} - 55\beta_{11} + 55\beta_{9} + 55\beta_{5} - 123\beta_{2} + 55 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -110\beta_{15} - 89\beta_{14} + 89\beta_{13} - 89\beta_{8} - 89\beta_{6} - 89\beta_{3} + 89\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -72\beta_{11} - 72\beta_{9} - 72\beta_{4} - 161 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -233\beta_{14} - 233\beta_{13} + 521\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 377\beta_{5} + 377\beta_{4} - 843\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -305\beta_{8} - 305\beta_{6} - 682\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-\beta_{2} + \beta_{12}\) \(1\) \(\beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−0.596975 + 0.159959i
−1.56290 + 0.418778i
1.56290 − 0.418778i
0.596975 − 0.159959i
0.159959 + 0.596975i
0.418778 + 1.56290i
−0.418778 − 1.56290i
−0.159959 − 0.596975i
−0.596975 − 0.159959i
−1.56290 − 0.418778i
1.56290 + 0.418778i
0.596975 + 0.159959i
0.159959 − 0.596975i
0.418778 − 1.56290i
−0.418778 + 1.56290i
−0.159959 + 0.596975i
−1.14412 − 1.98168i −0.258819 + 0.965926i −1.61803 + 2.80252i 0 2.21028 − 0.592242i −2.67543 − 1.54466i 2.82843 −0.866025 − 0.500000i 0
193.2 −0.437016 − 0.756934i 0.258819 − 0.965926i 0.618034 − 1.07047i 0 −0.844250 + 0.226216i 0.670631 + 0.387189i −2.82843 −0.866025 − 0.500000i 0
193.3 0.437016 + 0.756934i −0.258819 + 0.965926i 0.618034 − 1.07047i 0 −0.844250 + 0.226216i −0.670631 − 0.387189i 2.82843 −0.866025 − 0.500000i 0
193.4 1.14412 + 1.98168i 0.258819 − 0.965926i −1.61803 + 2.80252i 0 2.21028 − 0.592242i 2.67543 + 1.54466i −2.82843 −0.866025 − 0.500000i 0
457.1 −1.14412 − 1.98168i 0.965926 + 0.258819i −1.61803 + 2.80252i 0 −0.592242 − 2.21028i −4.18930 − 2.41869i 2.82843 0.866025 + 0.500000i 0
457.2 −0.437016 − 0.756934i −0.965926 − 0.258819i 0.618034 − 1.07047i 0 0.226216 + 0.844250i −3.29273 − 1.90106i −2.82843 0.866025 + 0.500000i 0
457.3 0.437016 + 0.756934i 0.965926 + 0.258819i 0.618034 − 1.07047i 0 0.226216 + 0.844250i 3.29273 + 1.90106i 2.82843 0.866025 + 0.500000i 0
457.4 1.14412 + 1.98168i −0.965926 − 0.258819i −1.61803 + 2.80252i 0 −0.592242 − 2.21028i 4.18930 + 2.41869i −2.82843 0.866025 + 0.500000i 0
682.1 −1.14412 + 1.98168i −0.258819 − 0.965926i −1.61803 − 2.80252i 0 2.21028 + 0.592242i −2.67543 + 1.54466i 2.82843 −0.866025 + 0.500000i 0
682.2 −0.437016 + 0.756934i 0.258819 + 0.965926i 0.618034 + 1.07047i 0 −0.844250 − 0.226216i 0.670631 − 0.387189i −2.82843 −0.866025 + 0.500000i 0
682.3 0.437016 − 0.756934i −0.258819 − 0.965926i 0.618034 + 1.07047i 0 −0.844250 − 0.226216i −0.670631 + 0.387189i 2.82843 −0.866025 + 0.500000i 0
682.4 1.14412 − 1.98168i 0.258819 + 0.965926i −1.61803 − 2.80252i 0 2.21028 + 0.592242i 2.67543 − 1.54466i −2.82843 −0.866025 + 0.500000i 0
943.1 −1.14412 + 1.98168i 0.965926 − 0.258819i −1.61803 − 2.80252i 0 −0.592242 + 2.21028i −4.18930 + 2.41869i 2.82843 0.866025 − 0.500000i 0
943.2 −0.437016 + 0.756934i −0.965926 + 0.258819i 0.618034 + 1.07047i 0 0.226216 − 0.844250i −3.29273 + 1.90106i −2.82843 0.866025 − 0.500000i 0
943.3 0.437016 − 0.756934i 0.965926 − 0.258819i 0.618034 + 1.07047i 0 0.226216 − 0.844250i 3.29273 − 1.90106i 2.82843 0.866025 − 0.500000i 0
943.4 1.14412 − 1.98168i −0.965926 + 0.258819i −1.61803 − 2.80252i 0 −0.592242 + 2.21028i 4.18930 − 2.41869i −2.82843 0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.o even 12 1 inner
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bl.e ✓ 16
5.b even 2 1 inner 975.2.bl.e ✓ 16
5.c odd 4 2 975.2.bu.f yes 16
13.f odd 12 1 975.2.bu.f yes 16
65.o even 12 1 inner 975.2.bl.e ✓ 16
65.s odd 12 1 975.2.bu.f yes 16
65.t even 12 1 inner 975.2.bl.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bl.e ✓ 16 1.a even 1 1 trivial
975.2.bl.e ✓ 16 5.b even 2 1 inner
975.2.bl.e ✓ 16 65.o even 12 1 inner
975.2.bl.e ✓ 16 65.t even 12 1 inner
975.2.bu.f yes 16 5.c odd 4 2
975.2.bu.f yes 16 13.f odd 12 1
975.2.bu.f yes 16 65.s odd 12 1

Hecke kernels

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

\( T_{2}^{8} + 6T_{2}^{6} + 32T_{2}^{4} + 24T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{16} - 48 T_{7}^{14} + 1576 T_{7}^{12} - 27648 T_{7}^{10} + 352944 T_{7}^{8} - 2469888 T_{7}^{6} + \cdots + 3748096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 6 T^{6} + 32 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 48 T^{14} + \cdots + 3748096 \) Copy content Toggle raw display
$11$ \( (T^{8} + 6 T^{7} + 18 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 24 T^{6} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 12 T^{14} + \cdots + 3748096 \) Copy content Toggle raw display
$19$ \( (T^{8} - 16 T^{7} + \cdots + 121)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 863591055616 \) Copy content Toggle raw display
$29$ \( (T^{8} + 12 T^{7} + \cdots + 1290496)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 12 T^{7} + \cdots + 185761)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1475789056 \) Copy content Toggle raw display
$41$ \( (T^{8} + 14 T^{7} + \cdots + 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 9250941239296 \) Copy content Toggle raw display
$47$ \( (T^{8} + 156 T^{6} + \cdots + 2062096)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 6505390336 \) Copy content Toggle raw display
$59$ \( (T^{8} - 40 T^{7} + \cdots + 13424896)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 8 T^{7} + \cdots + 57121)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{8} + 22 T^{7} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 216 T^{6} + \cdots + 408321)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 372 T^{6} + \cdots + 22005481)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 432 T^{6} + \cdots + 8202496)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 16 T^{7} + \cdots + 14622976)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 243191580349201 \) Copy content Toggle raw display
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