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Properties

Label 912.3.m.a
Level $912$
Weight $3$
Character orbit 912.m
Analytic conductor $24.850$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,3,Mod(799,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.799");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.m (of order \(2\), degree \(1\), 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: \(24.8502001097\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + \cdots + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{11} - \beta_{6}) q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + ( - \beta_{3} - 1) q^{5} + ( - \beta_{11} - \beta_{6}) q^{7} - 3 q^{9} + ( - \beta_{10} - \beta_{9} - \beta_{7}) q^{11} + ( - \beta_1 + 3) q^{13} + (\beta_{9} + \beta_{6}) q^{15} + (\beta_{5} + 2 \beta_{4} + \beta_1 - 1) q^{17} - \beta_{8} q^{19} + ( - \beta_{5} + \beta_{2} - 1) q^{21} + ( - \beta_{10} - \beta_{9} + \cdots - 4 \beta_{6}) q^{23}+ \cdots + (3 \beta_{10} + 3 \beta_{9} + 3 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{5} - 36 q^{9} + 36 q^{13} - 10 q^{17} - 18 q^{21} + 58 q^{25} + 12 q^{29} + 6 q^{33} + 32 q^{37} + 136 q^{41} + 30 q^{45} - 22 q^{49} - 236 q^{53} - 210 q^{61} - 52 q^{65} - 144 q^{69} - 158 q^{73} - 70 q^{77} + 108 q^{81} + 242 q^{85} + 444 q^{89} - 24 q^{93} + 320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + \cdots + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1132830139545 \nu^{11} - 83463703932306 \nu^{10} - 82049035182874 \nu^{9} + \cdots + 63\!\cdots\!20 ) / 52\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3448796555 \nu^{11} - 24132677750 \nu^{10} - 157866815614 \nu^{9} + \cdots - 16\!\cdots\!52 ) / 13\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38271754888553 \nu^{11} - 9342376003726 \nu^{10} + \cdots + 27\!\cdots\!72 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 88301875 \nu^{11} + 184888284 \nu^{10} + 4047789409 \nu^{9} + 4970637175 \nu^{8} + \cdots + 16201567588036 ) / 2134898416716 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 104157093719227 \nu^{11} - 31799241318102 \nu^{10} + \cdots - 19\!\cdots\!44 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2248559103131 \nu^{11} - 4524708578702 \nu^{10} + 112234893734550 \nu^{9} + \cdots + 98\!\cdots\!72 ) / 10\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 451147977297025 \nu^{11} + 972391254937206 \nu^{10} + \cdots - 15\!\cdots\!20 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5084441 \nu^{11} - 10363066 \nu^{10} + 255384002 \nu^{9} - 701038728 \nu^{8} + \cdots + 25239211968 ) / 10877179584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 563617568102453 \nu^{11} + \cdots + 24\!\cdots\!20 ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8386307706881 \nu^{11} - 16626138955879 \nu^{10} + 416130916219964 \nu^{9} + \cdots + 33\!\cdots\!44 ) / 11\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13\!\cdots\!91 \nu^{11} + \cdots - 62\!\cdots\!48 ) / 10\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} + 2 \beta_{10} - 3 \beta_{9} + 8 \beta_{8} - 4 \beta_{7} - 32 \beta_{6} + \beta_{5} + \cdots - 32 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{5} + 14\beta_{3} - 35\beta_{2} + 2\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 45 \beta_{11} - 48 \beta_{10} + 169 \beta_{9} - 336 \beta_{8} + 204 \beta_{7} + 1194 \beta_{6} + \cdots - 1180 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1925 \beta_{11} - 1277 \beta_{10} - 1881 \beta_{9} - 929 \beta_{8} - 1169 \beta_{7} - 3060 \beta_{6} + \cdots - 1351 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2129\beta_{5} - 7604\beta_{4} + 2369\beta_{3} - 3737\beta_{2} - 3450\beta _1 + 48832 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 81757 \beta_{11} + 51697 \beta_{10} + 82931 \beta_{9} + 30307 \beta_{8} + 53779 \beta_{7} + 160202 \beta_{6} + \cdots - 88465 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 171757 \beta_{11} + 8888 \beta_{10} - 393121 \beta_{9} + 540320 \beta_{8} - 420172 \beta_{7} + \cdots - 2069532 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 335060\beta_{5} - 523552\beta_{4} + 1309980\beta_{3} - 2701833\beta_{2} - 119236\beta _1 + 4978623 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9300241 \beta_{11} + 986974 \beta_{10} + 18578035 \beta_{9} - 21771464 \beta_{8} + 18950116 \beta_{7} + \cdots - 88670176 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 150234189 \beta_{11} - 88588545 \beta_{10} - 163747835 \beta_{9} - 24113643 \beta_{8} + \cdots - 261975257 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
−3.37682 + 5.84883i
1.37340 − 2.37879i
−0.0525878 + 0.0910847i
0.816029 − 1.41340i
3.06079 − 5.30145i
−0.820808 + 1.42168i
−3.37682 − 5.84883i
1.37340 + 2.37879i
−0.0525878 − 0.0910847i
0.816029 + 1.41340i
3.06079 + 5.30145i
−0.820808 − 1.42168i
0 − 1.73205i 0 −8.26675 0 − 9.51333i 0 −3.00000 0
799.2 0 − 1.73205i 0 −4.02901 0 − 7.12527i 0 −3.00000 0
799.3 0 − 1.73205i 0 −3.59607 0 9.49604i 0 −3.00000 0
799.4 0 − 1.73205i 0 0.606930 0 − 2.85501i 0 −3.00000 0
799.5 0 − 1.73205i 0 1.38490 0 7.59081i 0 −3.00000 0
799.6 0 − 1.73205i 0 8.90000 0 − 2.78940i 0 −3.00000 0
799.7 0 1.73205i 0 −8.26675 0 9.51333i 0 −3.00000 0
799.8 0 1.73205i 0 −4.02901 0 7.12527i 0 −3.00000 0
799.9 0 1.73205i 0 −3.59607 0 − 9.49604i 0 −3.00000 0
799.10 0 1.73205i 0 0.606930 0 2.85501i 0 −3.00000 0
799.11 0 1.73205i 0 1.38490 0 − 7.59081i 0 −3.00000 0
799.12 0 1.73205i 0 8.90000 0 2.78940i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.m.a ✓ 12
3.b odd 2 1 2736.3.m.f 12
4.b odd 2 1 inner 912.3.m.a ✓ 12
12.b even 2 1 2736.3.m.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.m.a ✓ 12 1.a even 1 1 trivial
912.3.m.a ✓ 12 4.b odd 2 1 inner
2736.3.m.f 12 3.b odd 2 1
2736.3.m.f 12 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 5T_{5}^{5} - 77T_{5}^{4} - 437T_{5}^{3} + 16T_{5}^{2} + 1644T_{5} - 896 \) acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 5 T^{5} + \cdots - 896)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 1514143744 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 187580416 \) Copy content Toggle raw display
$13$ \( (T^{6} - 18 T^{5} + \cdots + 2630656)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 5 T^{5} + \cdots + 8199352)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 99230924406784 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + \cdots + 76842496)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1396965376 \) Copy content Toggle raw display
$37$ \( (T^{6} - 16 T^{5} + \cdots + 6873088)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 68 T^{5} + \cdots + 1137344512)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{6} + 118 T^{5} + \cdots + 2520564736)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} + 105 T^{5} + \cdots + 826249672)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 80\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{6} + 79 T^{5} + \cdots + 1441081768)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} - 222 T^{5} + \cdots + 307161915264)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 160 T^{5} + \cdots + 100074188608)^{2} \) Copy content Toggle raw display
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