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Properties

Label 1104.3.k.c
Level $1104$
Weight $3$
Character orbit 1104.k
Analytic conductor $30.082$
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] = [1104,3,Mod(415,1104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1104, 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("1104.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1104.k (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: \(30.0818211854\)
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: \( x^{12} - 3 x^{11} + 23 x^{10} - 60 x^{9} + 117 x^{8} - 129 x^{7} - 436 x^{6} + 225 x^{5} + 2611 x^{4} + \cdots + 3627 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + ( - \beta_{3} - 1) q^{5} + \beta_{10} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + ( - \beta_{3} - 1) q^{5} + \beta_{10} q^{7} - 3 q^{9} + (\beta_{10} + \beta_{9} - \beta_{4}) q^{11} + ( - \beta_{5} - \beta_{3} + \beta_{2} + 4) q^{13} + (\beta_{9} + \beta_{4} + \beta_1) q^{15} + ( - \beta_{6} - \beta_{2} + 3) q^{17} + ( - \beta_{9} + 2 \beta_{8} + \cdots + \beta_1) q^{19}+ \cdots + ( - 3 \beta_{10} - 3 \beta_{9} + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{5} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{5} - 36 q^{9} + 48 q^{13} + 44 q^{17} + 12 q^{21} + 44 q^{25} - 104 q^{29} - 36 q^{33} - 44 q^{37} - 8 q^{41} + 24 q^{45} + 44 q^{49} + 48 q^{53} - 24 q^{57} - 148 q^{61} + 176 q^{65} - 368 q^{73} - 392 q^{77} + 108 q^{81} + 240 q^{85} + 92 q^{89} + 192 q^{93} + 136 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 23 x^{10} - 60 x^{9} + 117 x^{8} - 129 x^{7} - 436 x^{6} + 225 x^{5} + 2611 x^{4} + \cdots + 3627 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 434 \nu^{11} - 2382501 \nu^{10} + 4346780 \nu^{9} - 54364258 \nu^{8} + 85948323 \nu^{7} + \cdots + 4380311100 ) / 684935883 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4979735 \nu^{11} + 57297806 \nu^{10} + 88661515 \nu^{9} + 909756391 \nu^{8} + \cdots - 311503833849 ) / 35616665916 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4340354 \nu^{11} - 5699299 \nu^{10} + 84602848 \nu^{9} - 81116705 \nu^{8} + 119476719 \nu^{7} + \cdots + 41809177995 ) / 8904166479 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1821669 \nu^{11} + 5818730 \nu^{10} - 46277093 \nu^{9} + 122719695 \nu^{8} + \cdots - 5379477597 ) / 1874561364 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 76009 \nu^{11} - 199786 \nu^{10} + 1619265 \nu^{9} - 3850371 \nu^{8} + 6397410 \nu^{7} + \cdots + 298095993 ) / 71376084 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 826661 \nu^{11} - 1234754 \nu^{10} + 16459640 \nu^{9} - 21109025 \nu^{8} + 39016032 \nu^{7} + \cdots + 962026611 ) / 684935883 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13206568 \nu^{11} - 23670324 \nu^{10} + 275676568 \nu^{9} - 442914971 \nu^{8} + \cdots + 45286248579 ) / 8904166479 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3254 \nu^{11} + 16070 \nu^{10} - 89784 \nu^{9} + 346124 \nu^{8} - 689874 \nu^{7} + \cdots - 19208121 ) / 1962567 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 83746067 \nu^{11} + 513227466 \nu^{10} - 2590912651 \nu^{9} + 11107328149 \nu^{8} + \cdots - 833411346339 ) / 35616665916 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 107286477 \nu^{11} - 383688462 \nu^{10} + 2688121409 \nu^{9} - 8071332499 \nu^{8} + \cdots + 422248308573 ) / 35616665916 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 95205259 \nu^{11} + 151210612 \nu^{10} - 2080742215 \nu^{9} + 3006592379 \nu^{8} + \cdots + 89341458375 ) / 17808332958 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{8} - \beta_{7} - \beta_{5} + 3\beta_{4} + \beta_{3} - 3\beta _1 + 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} + 4 \beta_{9} - 3 \beta_{8} + \beta_{7} - 9 \beta_{6} + \beta_{5} - 10 \beta_{4} + \cdots - 36 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{11} + 21 \beta_{10} + 12 \beta_{9} + 30 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} - 10 \beta_{5} + \cdots - 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 38 \beta_{11} - 38 \beta_{10} - 56 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + 90 \beta_{6} - 41 \beta_{5} + \cdots + 537 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 114 \beta_{11} - 258 \beta_{10} - 240 \beta_{9} - 339 \beta_{8} - 166 \beta_{7} - 78 \beta_{6} + \cdots - 219 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 577 \beta_{11} + 775 \beta_{10} + 1006 \beta_{9} + 474 \beta_{8} - 353 \beta_{7} - 1053 \beta_{6} + \cdots - 6063 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2022 \beta_{11} + 3678 \beta_{10} + 3948 \beta_{9} + 3945 \beta_{8} + 2027 \beta_{7} + 2574 \beta_{6} + \cdots + 14397 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8732 \beta_{11} - 14780 \beta_{10} - 16496 \beta_{9} - 14259 \beta_{8} + 6316 \beta_{7} + \cdots + 69903 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 28986 \beta_{11} - 48138 \beta_{10} - 54222 \beta_{9} - 45024 \beta_{8} - 24859 \beta_{7} + \cdots - 363000 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 133036 \beta_{11} + 269116 \beta_{10} + 269152 \beta_{9} + 318813 \beta_{8} - 85862 \beta_{7} + \cdots - 783057 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 333723 \beta_{11} + 543099 \beta_{10} + 621984 \beta_{9} + 490089 \beta_{8} + 318965 \beta_{7} + \cdots + 7363086 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(277\) \(415\) \(737\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
415.1
1.27428 + 1.05682i
−0.774281 − 0.190795i
0.914317 − 3.11553i
1.96669 − 0.142469i
−1.46669 + 1.00849i
−0.414317 + 3.98156i
1.27428 − 1.05682i
−0.774281 + 0.190795i
0.914317 + 3.11553i
1.96669 + 0.142469i
−1.46669 − 1.00849i
−0.414317 − 3.98156i
0 − 1.73205i 0 −7.86129 0 5.06458i 0 −3.00000 0
415.2 0 − 1.73205i 0 −5.86891 0 − 6.88910i 0 −3.00000 0
415.3 0 − 1.73205i 0 −1.77084 0 2.72358i 0 −3.00000 0
415.4 0 − 1.73205i 0 −0.507089 0 0.630952i 0 −3.00000 0
415.5 0 − 1.73205i 0 5.63975 0 10.6934i 0 −3.00000 0
415.6 0 − 1.73205i 0 6.36838 0 − 8.75931i 0 −3.00000 0
415.7 0 1.73205i 0 −7.86129 0 − 5.06458i 0 −3.00000 0
415.8 0 1.73205i 0 −5.86891 0 6.88910i 0 −3.00000 0
415.9 0 1.73205i 0 −1.77084 0 − 2.72358i 0 −3.00000 0
415.10 0 1.73205i 0 −0.507089 0 − 0.630952i 0 −3.00000 0
415.11 0 1.73205i 0 5.63975 0 − 10.6934i 0 −3.00000 0
415.12 0 1.73205i 0 6.36838 0 8.75931i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 415.12
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 1104.3.k.c ✓ 12
4.b odd 2 1 inner 1104.3.k.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1104.3.k.c ✓ 12 1.a even 1 1 trivial
1104.3.k.c ✓ 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 4T_{5}^{5} - 78T_{5}^{4} - 248T_{5}^{3} + 1444T_{5}^{2} + 3720T_{5} + 1488 \) acting on \(S_{3}^{\mathrm{new}}(1104, [\chi])\). 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
$5$ \( (T^{6} + 4 T^{5} + \cdots + 1488)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 272 T^{10} + \cdots + 31539456 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 12174032896 \) Copy content Toggle raw display
$13$ \( (T^{6} - 24 T^{5} + \cdots - 149312)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 22 T^{5} + \cdots + 1498176)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 663012833536 \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{6} \) Copy content Toggle raw display
$29$ \( (T^{6} + 52 T^{5} + \cdots + 257663424)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{6} + 22 T^{5} + \cdots - 1734336)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 4 T^{5} + \cdots - 12758023616)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{6} - 24 T^{5} + \cdots + 1494923824)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + 74 T^{5} + \cdots + 4037144256)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{6} + 184 T^{5} + \cdots + 172239417792)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 66\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} - 46 T^{5} + \cdots + 103616342464)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 68 T^{5} + \cdots + 439797184)^{2} \) Copy content Toggle raw display
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