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Properties

Label 112.2.m.d
Level $112$
Weight $2$
Character orbit 112.m
Analytic conductor $0.894$
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] = [112,2,Mod(29,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.m (of order \(4\), 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: \(0.894324502638\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.20138089353117696.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{10} - \beta_{2}) q^{3} + \beta_1 q^{4} + (\beta_{9} - \beta_{6} + \beta_{4} + \cdots + 1) q^{5}+ \cdots + (\beta_{11} + \beta_{8} + \beta_{7} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{10} - \beta_{2}) q^{3} + \beta_1 q^{4} + (\beta_{9} - \beta_{6} + \beta_{4} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{10} - \beta_{9} + 4 \beta_{8} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 4 q^{3} - 6 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 4 q^{3} - 6 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} - 4 q^{10} + 8 q^{12} - 24 q^{15} + 10 q^{16} - 8 q^{17} - 20 q^{20} + 4 q^{21} + 14 q^{22} - 8 q^{24} - 20 q^{26} + 4 q^{27} - 4 q^{29} - 28 q^{30} - 8 q^{31} + 12 q^{32} + 8 q^{34} - 4 q^{35} - 16 q^{36} - 20 q^{37} + 16 q^{38} - 8 q^{40} - 12 q^{42} + 16 q^{43} + 14 q^{44} + 40 q^{45} - 28 q^{46} + 16 q^{47} + 16 q^{48} - 12 q^{49} + 44 q^{50} - 16 q^{51} - 16 q^{52} + 4 q^{53} + 64 q^{54} + 6 q^{56} + 14 q^{58} - 16 q^{59} + 60 q^{60} - 20 q^{61} + 8 q^{62} + 12 q^{63} - 18 q^{64} + 32 q^{65} + 12 q^{66} + 24 q^{67} - 28 q^{68} - 4 q^{69} + 20 q^{70} + 6 q^{72} - 38 q^{74} - 40 q^{75} + 48 q^{76} - 76 q^{78} + 24 q^{79} + 24 q^{80} - 44 q^{81} - 16 q^{82} - 20 q^{83} + 8 q^{84} - 8 q^{85} + 38 q^{86} - 14 q^{88} - 40 q^{90} + 32 q^{92} - 48 q^{93} - 24 q^{94} - 16 q^{96} + 48 q^{97} - 2 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{10} - 2x^{9} + 2x^{8} + 4x^{7} + 2x^{6} + 8x^{5} + 8x^{4} - 16x^{3} - 48x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 3\nu^{8} - 2\nu^{7} + 2\nu^{6} + 4\nu^{5} + 2\nu^{4} + 8\nu^{3} + 8\nu^{2} - 16\nu - 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 10 \nu^{10} + 15 \nu^{9} + 12 \nu^{8} - 10 \nu^{7} - 32 \nu^{6} - 34 \nu^{5} - 52 \nu^{4} + \cdots + 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 3\nu^{9} - 2\nu^{8} + 2\nu^{7} + 4\nu^{6} + 2\nu^{5} + 8\nu^{4} + 8\nu^{3} - 16\nu^{2} - 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{11} + 6 \nu^{10} + 5 \nu^{9} - 4 \nu^{8} - 14 \nu^{7} - 16 \nu^{6} - 22 \nu^{5} - 12 \nu^{4} + \cdots + 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{11} - 10 \nu^{10} - 3 \nu^{9} + 12 \nu^{8} + 18 \nu^{7} + 16 \nu^{6} + 26 \nu^{5} + \cdots + 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3 \nu^{11} - 10 \nu^{10} - 3 \nu^{9} + 12 \nu^{8} + 18 \nu^{7} + 16 \nu^{6} + 26 \nu^{5} + \cdots + 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + \nu^{10} - 3 \nu^{9} - 5 \nu^{8} - 4 \nu^{7} + 2 \nu^{6} + 6 \nu^{5} + 10 \nu^{4} + \cdots - 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13 \nu^{11} + 14 \nu^{10} - 19 \nu^{9} - 44 \nu^{8} - 46 \nu^{7} + 26 \nu^{5} + 196 \nu^{4} + \cdots - 480 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{11} - 14 \nu^{10} + 9 \nu^{9} + 32 \nu^{8} + 34 \nu^{7} + 8 \nu^{6} + 2 \nu^{5} - 52 \nu^{4} + \cdots + 352 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{11} + 26 \nu^{10} - 17 \nu^{9} - 68 \nu^{8} - 58 \nu^{7} - 32 \nu^{6} + 30 \nu^{5} + \cdots - 672 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{11} - 8 \nu^{10} + 9 \nu^{9} + 30 \nu^{8} + 22 \nu^{7} + 4 \nu^{6} - 6 \nu^{5} - 72 \nu^{4} + \cdots + 256 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{9} + 2\beta_{7} - \beta_{6} - \beta_{5} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + 2\beta_{10} - \beta_{6} - \beta_{5} + 2\beta_{4} - 4\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{9} + 2\beta_{8} - \beta_{6} - \beta_{5} - 2\beta_{4} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{11} + 4\beta_{10} + 2\beta_{9} + 2\beta_{7} + \beta_{6} + \beta_{5} + 4\beta_{4} + 2\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} + 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} + 8 \beta_{10} + 10 \beta_{9} - 4 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} - 7 \beta_{5} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} + 8 \beta_{8} + 6 \beta_{7} - 9 \beta_{6} + 3 \beta_{5} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2 \beta_{11} + 4 \beta_{10} + 6 \beta_{9} + 4 \beta_{8} - 10 \beta_{7} - 7 \beta_{6} + 5 \beta_{5} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 6 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 4 \beta_{8} - 6 \beta_{7} - \beta_{6} + 3 \beta_{5} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6 \beta_{11} - 4 \beta_{10} + 6 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 17 \beta_{6} + 5 \beta_{5} + \cdots - 58 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.605558 + 1.27801i
1.35309 + 0.411286i
−1.40471 − 0.163666i
1.37925 − 0.312504i
−1.12465 − 0.857418i
0.402577 − 1.35570i
−0.605558 − 1.27801i
1.35309 − 0.411286i
−1.40471 + 0.163666i
1.37925 + 0.312504i
−1.12465 + 0.857418i
0.402577 + 1.35570i
−1.27801 + 0.605558i 1.39123 + 1.39123i 1.26660 − 1.54781i 2.16478 − 2.16478i −2.62048 − 0.935533i − 1.00000i −0.681431 + 2.74511i 0.871066i −1.45570 + 4.07750i
29.2 −0.411286 − 1.35309i 2.21570 + 2.21570i −1.66169 + 1.11301i −0.393125 + 0.393125i 2.08675 − 3.90932i − 1.00000i 2.18943 + 1.79064i 6.81864i 0.693620 + 0.370246i
29.3 0.163666 + 1.40471i −2.05500 − 2.05500i −1.94643 + 0.459808i 2.72766 − 2.72766i 2.55034 − 3.22301i − 1.00000i −0.964462 − 2.65891i 5.44602i 4.27801 + 3.38515i
29.4 0.312504 − 1.37925i −0.599978 − 0.599978i −1.80468 − 0.862045i 0.974969 − 0.974969i −1.01502 + 0.640026i − 1.00000i −1.75295 + 2.21972i − 2.28005i −1.04005 − 1.64941i
29.5 0.857418 + 1.12465i 0.416854 + 0.416854i −0.529667 + 1.92859i −1.13169 + 1.13169i −0.111396 + 0.826233i − 1.00000i −2.62313 + 1.05792i − 2.65247i −2.24309 − 0.302422i
29.6 1.35570 − 0.402577i 0.631188 + 0.631188i 1.67586 − 1.09155i −2.34259 + 2.34259i 1.10981 + 0.601602i − 1.00000i 1.83254 − 2.15448i − 2.20320i −2.23279 + 4.11894i
85.1 −1.27801 − 0.605558i 1.39123 − 1.39123i 1.26660 + 1.54781i 2.16478 + 2.16478i −2.62048 + 0.935533i 1.00000i −0.681431 − 2.74511i − 0.871066i −1.45570 − 4.07750i
85.2 −0.411286 + 1.35309i 2.21570 − 2.21570i −1.66169 − 1.11301i −0.393125 − 0.393125i 2.08675 + 3.90932i 1.00000i 2.18943 − 1.79064i − 6.81864i 0.693620 − 0.370246i
85.3 0.163666 − 1.40471i −2.05500 + 2.05500i −1.94643 − 0.459808i 2.72766 + 2.72766i 2.55034 + 3.22301i 1.00000i −0.964462 + 2.65891i − 5.44602i 4.27801 − 3.38515i
85.4 0.312504 + 1.37925i −0.599978 + 0.599978i −1.80468 + 0.862045i 0.974969 + 0.974969i −1.01502 − 0.640026i 1.00000i −1.75295 − 2.21972i 2.28005i −1.04005 + 1.64941i
85.5 0.857418 − 1.12465i 0.416854 − 0.416854i −0.529667 − 1.92859i −1.13169 − 1.13169i −0.111396 − 0.826233i 1.00000i −2.62313 − 1.05792i 2.65247i −2.24309 + 0.302422i
85.6 1.35570 + 0.402577i 0.631188 − 0.631188i 1.67586 + 1.09155i −2.34259 − 2.34259i 1.10981 − 0.601602i 1.00000i 1.83254 + 2.15448i 2.20320i −2.23279 − 4.11894i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.2.m.d ✓ 12
4.b odd 2 1 448.2.m.d 12
7.b odd 2 1 784.2.m.h 12
7.c even 3 2 784.2.x.l 24
7.d odd 6 2 784.2.x.m 24
8.b even 2 1 896.2.m.g 12
8.d odd 2 1 896.2.m.h 12
16.e even 4 1 inner 112.2.m.d ✓ 12
16.e even 4 1 896.2.m.g 12
16.f odd 4 1 448.2.m.d 12
16.f odd 4 1 896.2.m.h 12
32.g even 8 2 7168.2.a.bj 12
32.h odd 8 2 7168.2.a.bi 12
112.l odd 4 1 784.2.m.h 12
112.w even 12 2 784.2.x.l 24
112.x odd 12 2 784.2.x.m 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.d ✓ 12 1.a even 1 1 trivial
112.2.m.d ✓ 12 16.e even 4 1 inner
448.2.m.d 12 4.b odd 2 1
448.2.m.d 12 16.f odd 4 1
784.2.m.h 12 7.b odd 2 1
784.2.m.h 12 112.l odd 4 1
784.2.x.l 24 7.c even 3 2
784.2.x.l 24 112.w even 12 2
784.2.x.m 24 7.d odd 6 2
784.2.x.m 24 112.x odd 12 2
896.2.m.g 12 8.b even 2 1
896.2.m.g 12 16.e even 4 1
896.2.m.h 12 8.d odd 2 1
896.2.m.h 12 16.f odd 4 1
7168.2.a.bi 12 32.h odd 8 2
7168.2.a.bj 12 32.g even 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 4 T_{3}^{11} + 8 T_{3}^{10} - 4 T_{3}^{9} + 76 T_{3}^{8} - 288 T_{3}^{7} + 552 T_{3}^{6} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + 32 T^{9} + \cdots + 541696 \) Copy content Toggle raw display
$13$ \( T^{12} + 20 T^{9} + \cdots + 3211264 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} - 44 T^{4} + \cdots - 96)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 76 T^{9} + \cdots + 2849344 \) Copy content Toggle raw display
$23$ \( T^{12} + 136 T^{10} + \cdots + 10137856 \) Copy content Toggle raw display
$29$ \( T^{12} + 4 T^{11} + \cdots + 8620096 \) Copy content Toggle raw display
$31$ \( (T^{6} + 4 T^{5} - 16 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 20 T^{11} + \cdots + 5053504 \) Copy content Toggle raw display
$41$ \( T^{12} + 120 T^{10} + \cdots + 25600 \) Copy content Toggle raw display
$43$ \( T^{12} - 16 T^{11} + \cdots + 23040000 \) Copy content Toggle raw display
$47$ \( (T^{6} - 8 T^{5} + \cdots + 19776)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 4 T^{11} + \cdots + 78400 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 119596096 \) Copy content Toggle raw display
$61$ \( T^{12} + 20 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( T^{12} - 24 T^{11} + \cdots + 3686400 \) Copy content Toggle raw display
$71$ \( T^{12} + 432 T^{10} + \cdots + 6553600 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 3050573824 \) Copy content Toggle raw display
$79$ \( (T^{6} - 12 T^{5} + \cdots - 2240)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 320093429824 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 35476475904 \) Copy content Toggle raw display
$97$ \( (T^{6} - 24 T^{5} + \cdots - 39008)^{2} \) Copy content Toggle raw display
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