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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(751,975)]
 
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, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bc (of order \(6\), degree \(2\), 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: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 150x^{8} + 520x^{6} + 825x^{4} + 512x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{2} + \beta_{6} q^{3} + (\beta_{10} + \beta_{6} - \beta_{2} + 1) q^{4} + \beta_{3} q^{6} + (\beta_{9} + \beta_{8} + \cdots + \beta_{4}) q^{7}+ \cdots + ( - \beta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{2} + \beta_{6} q^{3} + (\beta_{10} + \beta_{6} - \beta_{2} + 1) q^{4} + \beta_{3} q^{6} + (\beta_{9} + \beta_{8} + \cdots + \beta_{4}) q^{7}+ \cdots + ( - \beta_{11} + \beta_{9} + \cdots + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 8 q^{4} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} + 8 q^{4} + 3 q^{7} - 6 q^{9} + 9 q^{11} - 16 q^{12} + 3 q^{13} + 10 q^{14} - 4 q^{16} - 9 q^{19} - 15 q^{22} - q^{23} - 5 q^{26} + 12 q^{27} + 39 q^{28} - 16 q^{29} - 30 q^{32} - 9 q^{33} + 8 q^{36} + 15 q^{37} - 30 q^{38} - 3 q^{39} + 18 q^{41} - 5 q^{42} - q^{43} - 45 q^{46} - 4 q^{48} + 31 q^{49} + 40 q^{52} + 16 q^{53} + 25 q^{56} + 30 q^{58} + 54 q^{59} - 11 q^{61} + 20 q^{62} - 3 q^{63} + 16 q^{64} + 30 q^{66} - 45 q^{67} + 30 q^{68} - q^{69} - 33 q^{71} + 5 q^{74} - 27 q^{76} + 28 q^{77} - 20 q^{78} + 18 q^{79} - 6 q^{81} + 20 q^{82} - 39 q^{84} - 16 q^{87} + 5 q^{88} - 48 q^{89} - 29 q^{91} - 46 q^{92} + 12 q^{93} - 30 q^{94} + 9 q^{97} - 105 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 150x^{8} + 520x^{6} + 825x^{4} + 512x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 25\nu^{2} + 2\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{8} + 15\nu^{6} + 75\nu^{4} + 2\nu^{3} + 135\nu^{2} + 10\nu + 50 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} - 15\nu^{6} - 75\nu^{4} + 2\nu^{3} - 135\nu^{2} + 10\nu - 50 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} - 20\nu^{9} - 150\nu^{7} - 510\nu^{5} - 725\nu^{3} - 262\nu - 20 ) / 40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 5 \nu^{10} + 15 \nu^{9} + 85 \nu^{8} + 75 \nu^{7} + 505 \nu^{6} + 145 \nu^{5} + 1215 \nu^{4} + \cdots + 200 ) / 40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} - 20 \nu^{9} - 5 \nu^{8} - 145 \nu^{7} - 70 \nu^{6} - 450 \nu^{5} - 315 \nu^{4} + \cdots - 160 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 20 \nu^{9} - 5 \nu^{8} + 145 \nu^{7} - 75 \nu^{6} + 450 \nu^{5} - 365 \nu^{4} + 520 \nu^{3} + \cdots - 210 ) / 20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{11} + 60\nu^{9} + 440\nu^{7} + 1430\nu^{5} + 1925\nu^{3} + 20\nu^{2} + 686\nu + 60 ) / 40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{11} + 5 \nu^{10} - 55 \nu^{9} + 85 \nu^{8} - 365 \nu^{7} + 515 \nu^{6} - 1045 \nu^{5} + \cdots + 380 ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} - 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{6} - 8\beta_{5} - 8\beta_{4} - \beta_{3} - \beta_{2} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{9} - 10\beta_{8} + 10\beta_{5} - 10\beta_{4} + 14\beta_{3} + 35\beta_{2} - 12\beta _1 - 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 24 \beta_{10} + 10 \beta_{9} - 10 \beta_{8} - 32 \beta_{6} + 55 \beta_{5} + 55 \beta_{4} + 10 \beta_{3} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 75\beta_{9} + 75\beta_{8} - 77\beta_{5} + 77\beta_{4} - 135\beta_{3} - 210\beta_{2} + 105\beta _1 + 430 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4 \beta_{11} + 214 \beta_{10} - 77 \beta_{9} + 73 \beta_{8} - 4 \beta_{7} + 326 \beta_{6} - 358 \beta_{5} + \cdots + 163 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4 \beta_{11} - 508 \beta_{9} - 512 \beta_{8} + 4 \beta_{7} + 546 \beta_{5} - 542 \beta_{4} + 1124 \beta_{3} + \cdots - 2571 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 80 \beta_{11} - 1700 \beta_{10} + 550 \beta_{9} - 470 \beta_{8} + 80 \beta_{7} - 2780 \beta_{6} + \cdots - 1390 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1 + \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
2.36051i
2.03964i
0.606747i
− 0.834094i
− 1.58745i
− 2.58535i
− 2.36051i
− 2.03964i
− 0.606747i
0.834094i
1.58745i
2.58535i
−2.04426 + 1.18025i −0.500000 − 0.866025i 1.78599 − 3.09343i 0 2.04426 + 1.18025i 3.93440 + 2.27152i 3.71068i −0.500000 + 0.866025i 0
751.2 −1.76638 + 1.01982i −0.500000 − 0.866025i 1.08006 − 1.87073i 0 1.76638 + 1.01982i −4.48917 − 2.59182i 0.326607i −0.500000 + 0.866025i 0
751.3 −0.525459 + 0.303374i −0.500000 − 0.866025i −0.815929 + 1.41323i 0 0.525459 + 0.303374i 1.68150 + 0.970816i − 2.20362i −0.500000 + 0.866025i 0
751.4 0.722346 − 0.417047i −0.500000 − 0.866025i −0.652144 + 1.12955i 0 −0.722346 − 0.417047i −2.05120 − 1.18426i 2.75609i −0.500000 + 0.866025i 0
751.5 1.37477 − 0.793724i −0.500000 − 0.866025i 0.259994 − 0.450324i 0 −1.37477 − 0.793724i −0.925994 − 0.534623i 2.34944i −0.500000 + 0.866025i 0
751.6 2.23898 − 1.29268i −0.500000 − 0.866025i 2.34202 − 4.05650i 0 −2.23898 − 1.29268i 3.35046 + 1.93439i − 6.93919i −0.500000 + 0.866025i 0
901.1 −2.04426 − 1.18025i −0.500000 + 0.866025i 1.78599 + 3.09343i 0 2.04426 − 1.18025i 3.93440 − 2.27152i − 3.71068i −0.500000 − 0.866025i 0
901.2 −1.76638 − 1.01982i −0.500000 + 0.866025i 1.08006 + 1.87073i 0 1.76638 − 1.01982i −4.48917 + 2.59182i − 0.326607i −0.500000 − 0.866025i 0
901.3 −0.525459 − 0.303374i −0.500000 + 0.866025i −0.815929 − 1.41323i 0 0.525459 − 0.303374i 1.68150 − 0.970816i 2.20362i −0.500000 − 0.866025i 0
901.4 0.722346 + 0.417047i −0.500000 + 0.866025i −0.652144 − 1.12955i 0 −0.722346 + 0.417047i −2.05120 + 1.18426i − 2.75609i −0.500000 − 0.866025i 0
901.5 1.37477 + 0.793724i −0.500000 + 0.866025i 0.259994 + 0.450324i 0 −1.37477 + 0.793724i −0.925994 + 0.534623i − 2.34944i −0.500000 − 0.866025i 0
901.6 2.23898 + 1.29268i −0.500000 + 0.866025i 2.34202 + 4.05650i 0 −2.23898 + 1.29268i 3.35046 − 1.93439i 6.93919i −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.bc.k ✓ 12
5.b even 2 1 975.2.bc.l yes 12
5.c odd 4 2 975.2.w.k 24
13.e even 6 1 inner 975.2.bc.k ✓ 12
65.l even 6 1 975.2.bc.l yes 12
65.r odd 12 2 975.2.w.k 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.w.k 24 5.c odd 4 2
975.2.w.k 24 65.r odd 12 2
975.2.bc.k ✓ 12 1.a even 1 1 trivial
975.2.bc.k ✓ 12 13.e even 6 1 inner
975.2.bc.l yes 12 5.b even 2 1
975.2.bc.l yes 12 65.l even 6 1

Hecke kernels

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

\( T_{2}^{12} - 10T_{2}^{10} + 75T_{2}^{8} + 6T_{2}^{7} - 230T_{2}^{6} + 525T_{2}^{4} - 150T_{2}^{3} - 238T_{2}^{2} + 60T_{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{12} - 3 T_{7}^{11} - 32 T_{7}^{10} + 105 T_{7}^{9} + 860 T_{7}^{8} - 3243 T_{7}^{7} + \cdots + 200704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 10 T^{10} + \cdots + 100 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + \cdots + 200704 \) Copy content Toggle raw display
$11$ \( T^{12} - 9 T^{11} + \cdots + 2304 \) Copy content Toggle raw display
$13$ \( T^{12} - 3 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 70 T^{10} + \cdots + 2560000 \) Copy content Toggle raw display
$19$ \( T^{12} + 9 T^{11} + \cdots + 29637136 \) Copy content Toggle raw display
$23$ \( T^{12} + T^{11} + \cdots + 440896 \) Copy content Toggle raw display
$29$ \( T^{12} + 16 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$31$ \( T^{12} + 218 T^{10} + \cdots + 8294400 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 9651097600 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 33703085056 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 5266114624 \) Copy content Toggle raw display
$47$ \( T^{12} + 252 T^{10} + \cdots + 67108864 \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} + \cdots - 10272)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 54 T^{11} + \cdots + 9216 \) Copy content Toggle raw display
$61$ \( T^{12} + 11 T^{11} + \cdots + 5569600 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 64353542400 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2933305600 \) Copy content Toggle raw display
$73$ \( T^{12} + 98 T^{10} + \cdots + 1225 \) Copy content Toggle raw display
$79$ \( (T^{6} - 9 T^{5} + \cdots - 126524)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 307 T^{10} + \cdots + 589824 \) Copy content Toggle raw display
$89$ \( T^{12} + 48 T^{11} + \cdots + 47775744 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 19541803264 \) Copy content Toggle raw display
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