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Properties

Label 1104.3.g.a
Level $1104$
Weight $3$
Character orbit 1104.g
Analytic conductor $30.082$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1104,3,Mod(737,1104)]
 
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([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1104.737");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1104.g (of order \(2\), degree \(1\), not 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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 4 x^{12} - 10 x^{11} + 82 x^{10} - 124 x^{9} - 303 x^{8} + 948 x^{7} + \cdots + 4782969 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10}\cdot 3 \)
Twist minimal: no (minimal twist has level 276)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{5} q^{5} + \beta_{2} q^{7} + ( - \beta_{4} + \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{5} q^{5} + \beta_{2} q^{7} + ( - \beta_{4} + \beta_{3} + 1) q^{9} + (\beta_{11} + \beta_{10} + \beta_{9} + \cdots - 1) q^{11}+ \cdots + ( - 4 \beta_{13} + 3 \beta_{12} + \cdots - 20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{3} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{3} + 4 q^{7} + 8 q^{9} + 48 q^{13} + 14 q^{15} + 72 q^{19} - 2 q^{21} - 122 q^{25} - 46 q^{27} - 48 q^{31} - 54 q^{33} + 64 q^{37} + 126 q^{39} - 32 q^{43} + 46 q^{45} + 98 q^{49} - 2 q^{51} + 120 q^{55} + 284 q^{57} + 68 q^{61} + 126 q^{63} + 68 q^{67} - 64 q^{73} - 152 q^{75} - 296 q^{79} - 136 q^{81} + 160 q^{85} + 106 q^{87} - 148 q^{91} - 42 q^{93} - 280 q^{97} - 186 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 4 x^{12} - 10 x^{11} + 82 x^{10} - 124 x^{9} - 303 x^{8} + 948 x^{7} + \cdots + 4782969 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4 \nu^{13} - 317 \nu^{12} - 1924 \nu^{11} - 2183 \nu^{10} + 19202 \nu^{9} - 3887 \nu^{8} + \cdots - 162089505 ) / 76527504 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65 \nu^{13} + 1513 \nu^{12} + 2321 \nu^{11} - 9029 \nu^{10} + 51185 \nu^{9} - 20543 \nu^{8} + \cdots - 1035778509 ) / 306110016 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65 \nu^{13} + 1513 \nu^{12} + 2321 \nu^{11} - 9029 \nu^{10} + 51185 \nu^{9} - 20543 \nu^{8} + \cdots - 729668493 ) / 306110016 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 317 \nu^{13} - 710 \nu^{12} + 413 \nu^{11} + 3760 \nu^{10} + 17903 \nu^{9} + 20380 \nu^{8} + \cdots - 421964154 ) / 76527504 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 59 \nu^{13} + 70 \nu^{12} - 502 \nu^{11} + 148 \nu^{10} + 4451 \nu^{9} - 7496 \nu^{8} + \cdots - 57395628 ) / 12754584 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1535 \nu^{13} - 317 \nu^{12} + 263 \nu^{11} - 27455 \nu^{10} + 308615 \nu^{9} + \cdots - 6019632207 ) / 306110016 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{13} - 4 \nu^{12} + 4 \nu^{11} - 10 \nu^{10} + 82 \nu^{9} - 124 \nu^{8} - 303 \nu^{7} + \cdots - 1948617 ) / 177147 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2083 \nu^{13} + 2423 \nu^{12} + 15451 \nu^{11} - 59539 \nu^{10} - 86837 \nu^{9} + \cdots + 806195997 ) / 306110016 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 695 \nu^{13} - 2177 \nu^{12} + 3527 \nu^{11} + 8341 \nu^{10} + 7463 \nu^{9} + 27175 \nu^{8} + \cdots - 766869363 ) / 102036672 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2369 \nu^{13} - 2519 \nu^{12} - 9199 \nu^{11} - 11333 \nu^{10} + 188273 \nu^{9} + \cdots - 4096878669 ) / 306110016 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1025 \nu^{13} - 1715 \nu^{12} - 3415 \nu^{11} - 19745 \nu^{10} + 13625 \nu^{9} + \cdots - 799818705 ) / 102036672 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 524 \nu^{13} + 1241 \nu^{12} + 676 \nu^{11} + 9515 \nu^{10} - 31178 \nu^{9} - 24169 \nu^{8} + \cdots + 889100793 ) / 38263752 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{4} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + 2 \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{5} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{13} + 3\beta_{10} + 3\beta_{9} + 3\beta_{7} - 3\beta_{5} - 7\beta_{4} - 2\beta_{3} + 9\beta_{2} - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{13} + 2 \beta_{12} + 11 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 8 \beta_{7} + \cdots - 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{13} + 12 \beta_{12} + 9 \beta_{11} + 39 \beta_{10} - 9 \beta_{9} - 6 \beta_{8} + 3 \beta_{7} + \cdots + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 32 \beta_{13} + 50 \beta_{12} - 181 \beta_{11} + 14 \beta_{10} - 28 \beta_{9} - 59 \beta_{8} + \cdots - 151 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 201 \beta_{13} - 228 \beta_{12} + 225 \beta_{11} + 183 \beta_{10} + 123 \beta_{9} - 588 \beta_{8} + \cdots + 943 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 85 \beta_{13} - 64 \beta_{12} - 382 \beta_{11} - 157 \beta_{10} - 391 \beta_{9} + 586 \beta_{8} + \cdots + 14702 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1272 \beta_{13} - 1476 \beta_{12} + 981 \beta_{11} + 2046 \beta_{10} - 222 \beta_{9} - 486 \beta_{8} + \cdots - 7577 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1012 \beta_{13} - 4480 \beta_{12} - 1978 \beta_{11} + 986 \beta_{10} + 4682 \beta_{9} + 4795 \beta_{8} + \cdots + 31409 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 29484 \beta_{13} + 34068 \beta_{12} - 15246 \beta_{11} - 5334 \beta_{10} - 4338 \beta_{9} - 726 \beta_{8} + \cdots - 13217 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 57748 \beta_{13} + 6080 \beta_{12} + 42362 \beta_{11} + 77018 \beta_{10} + 104426 \beta_{9} + \cdots - 1006 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1
2.98930 + 0.253200i
2.98930 − 0.253200i
2.47290 + 1.69846i
2.47290 − 1.69846i
2.15079 + 2.09144i
2.15079 − 2.09144i
0.493036 + 2.95921i
0.493036 − 2.95921i
−1.39762 + 2.65456i
−1.39762 − 2.65456i
−1.83360 + 2.37443i
−1.83360 − 2.37443i
−2.87480 + 0.857618i
−2.87480 − 0.857618i
0 −2.98930 − 0.253200i 0 − 8.32463i 0 7.31140 0 8.87178 + 1.51378i 0
737.2 0 −2.98930 + 0.253200i 0 8.32463i 0 7.31140 0 8.87178 − 1.51378i 0
737.3 0 −2.47290 − 1.69846i 0 3.23180i 0 −10.1617 0 3.23045 + 8.40025i 0
737.4 0 −2.47290 + 1.69846i 0 − 3.23180i 0 −10.1617 0 3.23045 − 8.40025i 0
737.5 0 −2.15079 − 2.09144i 0 − 3.80234i 0 1.07296 0 0.251785 + 8.99648i 0
737.6 0 −2.15079 + 2.09144i 0 3.80234i 0 1.07296 0 0.251785 − 8.99648i 0
737.7 0 −0.493036 − 2.95921i 0 2.56183i 0 8.10391 0 −8.51383 + 2.91799i 0
737.8 0 −0.493036 + 2.95921i 0 − 2.56183i 0 8.10391 0 −8.51383 − 2.91799i 0
737.9 0 1.39762 − 2.65456i 0 − 3.78102i 0 −3.09149 0 −5.09334 − 7.42010i 0
737.10 0 1.39762 + 2.65456i 0 3.78102i 0 −3.09149 0 −5.09334 + 7.42010i 0
737.11 0 1.83360 − 2.37443i 0 8.44976i 0 −9.50959 0 −2.27582 − 8.70750i 0
737.12 0 1.83360 + 2.37443i 0 − 8.44976i 0 −9.50959 0 −2.27582 + 8.70750i 0
737.13 0 2.87480 − 0.857618i 0 − 7.03851i 0 8.27455 0 7.52898 − 4.93097i 0
737.14 0 2.87480 + 0.857618i 0 7.03851i 0 8.27455 0 7.52898 + 4.93097i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.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.g.a 14
3.b odd 2 1 inner 1104.3.g.a 14
4.b odd 2 1 276.3.d.a ✓ 14
12.b even 2 1 276.3.d.a ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.3.d.a ✓ 14 4.b odd 2 1
276.3.d.a ✓ 14 12.b even 2 1
1104.3.g.a 14 1.a even 1 1 trivial
1104.3.g.a 14 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 236 T_{5}^{12} + 21388 T_{5}^{10} + 941392 T_{5}^{8} + 21383632 T_{5}^{6} + \cdots + 3472883712 \) acting on \(S_{3}^{\mathrm{new}}(1104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 4 T^{13} + \cdots + 4782969 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 3472883712 \) Copy content Toggle raw display
$7$ \( (T^{7} - 2 T^{6} + \cdots + 157152)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 241444800000000 \) Copy content Toggle raw display
$13$ \( (T^{7} - 24 T^{6} + \cdots + 6166912)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 923332608 \) Copy content Toggle raw display
$19$ \( (T^{7} - 36 T^{6} + \cdots + 94991136)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 23)^{7} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 29\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( (T^{7} + 24 T^{6} + \cdots - 44469223776)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} - 32 T^{6} + \cdots - 99533812704)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( (T^{7} + 16 T^{6} + \cdots + 584564352)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 40\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 81\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( (T^{7} - 34 T^{6} + \cdots + 526961757920)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} - 34 T^{6} + \cdots + 77915707488)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + 32 T^{6} + \cdots + 315380960640)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + 148 T^{6} + \cdots + 57983011488)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 24\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 3172549977216)^{2} \) Copy content Toggle raw display
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