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Properties

Label 912.3.m.c
Level $912$
Weight $3$
Character orbit 912.m
Analytic conductor $24.850$
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] = [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: \( x^{12} + 61x^{10} + 1243x^{8} + 9566x^{6} + 25219x^{4} + 13245x^{2} + 841 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16} \)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} - \beta_{4} q^{5} + \beta_{7} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} - \beta_{4} q^{5} + \beta_{7} q^{7} - 3 q^{9} + (2 \beta_{9} - \beta_{7} + 2 \beta_{6}) q^{11} + (\beta_{4} - \beta_{3} - \beta_{2} - 1) q^{13} + (\beta_{11} + \beta_{10}) q^{15} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{17} + \beta_{9} q^{19} - \beta_1 q^{21} + ( - \beta_{11} + \beta_{10} + \cdots - 4 \beta_{6}) q^{23}+ \cdots + ( - 6 \beta_{9} + 3 \beta_{7} - 6 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{5} - 36 q^{9} + 20 q^{17} + 88 q^{25} - 24 q^{29} - 72 q^{33} - 40 q^{37} - 32 q^{41} + 12 q^{45} + 128 q^{49} + 184 q^{53} - 276 q^{61} - 232 q^{65} + 120 q^{69} - 92 q^{73} + 308 q^{77} + 108 q^{81} - 244 q^{85} + 72 q^{89} + 144 q^{93} - 280 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 61x^{10} + 1243x^{8} + 9566x^{6} + 25219x^{4} + 13245x^{2} + 841 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -807\nu^{10} - 43510\nu^{8} - 672699\nu^{6} - 2008769\nu^{4} + 6846830\nu^{2} + 8707883 ) / 1555456 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -1501\nu^{10} - 89842\nu^{8} - 1715961\nu^{6} - 10642331\nu^{4} - 9433446\nu^{2} + 19669961 ) / 1555456 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1783\nu^{10} + 111190\nu^{8} + 2297611\nu^{6} + 17407089\nu^{4} + 40942514\nu^{2} + 18125445 ) / 1555456 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 971\nu^{10} + 60062\nu^{8} + 1248143\nu^{6} + 9903229\nu^{4} + 26741194\nu^{2} + 8468161 ) / 777728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -71\nu^{10} - 4214\nu^{8} - 82203\nu^{6} - 581281\nu^{4} - 1279762\nu^{2} - 330613 ) / 31744 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 463\nu^{11} + 28678\nu^{9} + 601667\nu^{7} + 4922841\nu^{5} + 14657858\nu^{3} + 10347469\nu ) / 1455104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11097\nu^{11} + 677178\nu^{9} + 13768341\nu^{7} + 104817727\nu^{5} + 265601598\nu^{3} + 145475419\nu ) / 11277056 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 92311 \nu^{11} - 5553686 \nu^{9} - 109761707 \nu^{7} - 771157457 \nu^{5} + \cdots + 1346698523 \nu ) / 45108224 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -6597\nu^{11} - 401286\nu^{9} - 8133197\nu^{7} - 61790099\nu^{5} - 156244306\nu^{3} - 56482115\nu ) / 2819264 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -985\nu^{11} - 59215\nu^{9} - 1176302\nu^{7} - 8575623\nu^{5} - 19844073\nu^{3} - 5976154\nu ) / 352408 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 208681 \nu^{11} - 12727018 \nu^{9} - 258929557 \nu^{7} - 1979791471 \nu^{5} + \cdots - 2207855835 \nu ) / 45108224 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{9} + 2\beta_{7} + \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - 3\beta_{4} + 3\beta_{3} + 2\beta_{2} - 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{11} - \beta_{10} + 9\beta_{9} - 9\beta_{7} - 2\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 33\beta_{5} + 89\beta_{4} - 71\beta_{3} - 36\beta_{2} - 18\beta _1 + 758 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 563\beta_{11} + 120\beta_{10} - 935\beta_{9} + 18\beta_{8} + 804\beta_{7} - 7\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -232\beta_{5} - 572\beta_{4} + 408\beta_{3} + 192\beta_{2} + 168\beta _1 - 4311 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13263\beta_{11} - 3280\beta_{10} + 23183\beta_{9} - 672\beta_{8} - 19166\beta_{7} + 3793\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 25303\beta_{5} + 57797\beta_{4} - 37541\beta_{3} - 17822\beta_{2} - 19792\beta _1 + 407850 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 78434\beta_{11} + 22171\beta_{10} - 142343\beta_{9} + 4948\beta_{8} + 116715\beta_{7} - 40798\beta_{6} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -681295\beta_{5} - 1455927\beta_{4} + 865833\beta_{3} + 427276\beta_{2} + 544030\beta _1 - 9781322 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7446069 \beta_{11} - 2379960 \beta_{10} + 13964417 \beta_{9} - 544030 \beta_{8} + \cdots + 5496961 \beta_{6} ) / 4 \) 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
− 0.271103i
5.01087i
− 2.92820i
− 4.76380i
− 0.769253i
1.98943i
0.271103i
− 5.01087i
2.92820i
4.76380i
0.769253i
− 1.98943i
0 − 1.73205i 0 −8.42938 0 − 3.04146i 0 −3.00000 0
799.2 0 − 1.73205i 0 −6.20134 0 7.77305i 0 −3.00000 0
799.3 0 − 1.73205i 0 −0.121187 0 − 10.6149i 0 −3.00000 0
799.4 0 − 1.73205i 0 −0.0633322 0 0.702827i 0 −3.00000 0
799.5 0 − 1.73205i 0 5.32253 0 − 1.51700i 0 −3.00000 0
799.6 0 − 1.73205i 0 7.49272 0 6.69753i 0 −3.00000 0
799.7 0 1.73205i 0 −8.42938 0 3.04146i 0 −3.00000 0
799.8 0 1.73205i 0 −6.20134 0 − 7.77305i 0 −3.00000 0
799.9 0 1.73205i 0 −0.121187 0 10.6149i 0 −3.00000 0
799.10 0 1.73205i 0 −0.0633322 0 − 0.702827i 0 −3.00000 0
799.11 0 1.73205i 0 5.32253 0 1.51700i 0 −3.00000 0
799.12 0 1.73205i 0 7.49272 0 − 6.69753i 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.c ✓ 12
3.b odd 2 1 2736.3.m.d 12
4.b odd 2 1 inner 912.3.m.c ✓ 12
12.b even 2 1 2736.3.m.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.m.c ✓ 12 1.a even 1 1 trivial
912.3.m.c ✓ 12 4.b odd 2 1 inner
2736.3.m.d 12 3.b odd 2 1
2736.3.m.d 12 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 2T_{5}^{5} - 95T_{5}^{4} - 104T_{5}^{3} + 2068T_{5}^{2} + 384T_{5} + 16 \) 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
$5$ \( (T^{6} + 2 T^{5} - 95 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 230 T^{10} + \cdots + 3211264 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 330366976 \) Copy content Toggle raw display
$13$ \( (T^{6} - 680 T^{4} + \cdots + 102592)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 10 T^{5} + \cdots + 19600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 276072314699776 \) Copy content Toggle raw display
$29$ \( (T^{6} + 12 T^{5} + \cdots - 89489600)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 59\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{6} + 20 T^{5} + \cdots - 2250752)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 16 T^{5} + \cdots - 2028879872)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 39\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{6} - 92 T^{5} + \cdots + 114216256)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{6} + 138 T^{5} + \cdots + 3171014224)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{6} + 46 T^{5} + \cdots - 180480268400)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 92\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{6} - 36 T^{5} + \cdots - 7968369600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 140 T^{5} + \cdots - 8659749056)^{2} \) Copy content Toggle raw display
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