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Properties

Label 1156.4.a.l
Level $1156$
Weight $4$
Character orbit 1156.a
Self dual yes
Analytic conductor $68.206$
Analytic rank $0$
Dimension $20$
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] = [1156,4,Mod(1,1156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1156, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1156.a (trivial)

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: yes
Analytic conductor: \(68.2062079666\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 396 x^{18} + 63110 x^{16} - 5172304 x^{14} + 232007180 x^{12} - 5694034864 x^{10} + \cdots + 151170810368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 17^{6} \)
Twist minimal: no (minimal twist has level 68)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{13} q^{5} - \beta_{15} q^{7} + (\beta_{2} + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{13} q^{5} - \beta_{15} q^{7} + (\beta_{2} + 13) q^{9} + (\beta_{13} + \beta_{12} + \beta_1) q^{11} + ( - \beta_{10} + \beta_{2} + 10) q^{13} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots + 2) q^{15}+ \cdots + ( - 7 \beta_{19} - 7 \beta_{18} + \cdots + 85 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 252 q^{9} + 200 q^{13} + 48 q^{15} + 104 q^{19} + 256 q^{21} + 508 q^{25} + 640 q^{33} + 64 q^{35} - 392 q^{43} + 368 q^{47} + 2084 q^{49} + 2952 q^{53} - 1536 q^{55} + 2152 q^{59} + 376 q^{67} - 80 q^{69} + 3152 q^{77} + 11620 q^{81} + 6632 q^{83} + 6752 q^{87} + 4064 q^{89} + 7008 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 396 x^{18} + 63110 x^{16} - 5172304 x^{14} + 232007180 x^{12} - 5694034864 x^{10} + \cdots + 151170810368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 53954009105 \nu^{18} + 20790412140019 \nu^{16} + \cdots - 91\!\cdots\!88 ) / 59\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 170163594324665 \nu^{18} + \cdots + 74\!\cdots\!56 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 377239552597045 \nu^{18} + \cdots - 53\!\cdots\!12 ) / 54\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 393130122732187 \nu^{18} + \cdots - 10\!\cdots\!00 ) / 54\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!57 \nu^{18} + \cdots - 20\!\cdots\!40 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!65 \nu^{18} + \cdots + 46\!\cdots\!04 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 24\!\cdots\!49 \nu^{18} + \cdots - 49\!\cdots\!36 ) / 16\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!05 \nu^{18} + \cdots - 24\!\cdots\!92 ) / 81\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!45 \nu^{19} + \cdots + 30\!\cdots\!08 \nu ) / 55\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45\!\cdots\!97 \nu^{19} + \cdots + 21\!\cdots\!12 \nu ) / 18\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 93\!\cdots\!89 \nu^{19} + \cdots - 56\!\cdots\!44 \nu ) / 18\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38\!\cdots\!63 \nu^{19} + \cdots - 18\!\cdots\!36 \nu ) / 26\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14\!\cdots\!50 \nu^{19} + \cdots + 46\!\cdots\!40 \nu ) / 69\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 47\!\cdots\!85 \nu^{19} + \cdots - 48\!\cdots\!20 \nu ) / 20\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 79\!\cdots\!19 \nu^{19} + \cdots - 30\!\cdots\!68 \nu ) / 27\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 42\!\cdots\!31 \nu^{19} + \cdots - 32\!\cdots\!28 \nu ) / 92\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 34\!\cdots\!69 \nu^{19} + \cdots + 96\!\cdots\!64 \nu ) / 55\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} - 4\beta_{17} - \beta_{16} - 2\beta_{15} + 2\beta_{14} - 2\beta_{13} - 2\beta_{11} + 78\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{10} - 3\beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - 8\beta_{5} + 12\beta_{4} + 90\beta_{2} + 3100 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 85 \beta_{19} + 163 \beta_{18} - 500 \beta_{17} - 146 \beta_{16} - 155 \beta_{15} + 124 \beta_{14} + \cdots + 6743 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 888 \beta_{10} - 684 \beta_{9} - 236 \beta_{8} + 28 \beta_{7} + 340 \beta_{6} - 1552 \beta_{5} + \cdots + 268144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6122 \beta_{19} + 20510 \beta_{18} - 54904 \beta_{17} - 17626 \beta_{16} - 8944 \beta_{15} + \cdots + 600168 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 105972 \beta_{10} - 105042 \beta_{9} - 40870 \beta_{8} + 21914 \beta_{7} + 72890 \beta_{6} + \cdots + 23923432 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 392502 \beta_{19} + 2348874 \beta_{18} - 5791384 \beta_{17} - 2005012 \beta_{16} - 284130 \beta_{15} + \cdots + 54327930 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11552256 \beta_{10} - 13582608 \beta_{9} - 5943376 \beta_{8} + 4118528 \beta_{7} + 10229696 \beta_{6} + \cdots + 2170937392 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 20064212 \beta_{19} + 256397948 \beta_{18} - 598454256 \beta_{17} - 221433852 \beta_{16} + \cdots + 4985945800 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1196647800 \beta_{10} - 1606327836 \beta_{9} - 779002740 \beta_{8} + 591477324 \beta_{7} + \cdots + 199659462928 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 352424804 \beta_{19} + 27210069884 \beta_{18} - 61074720272 \beta_{17} - 23993058824 \beta_{16} + \cdots + 463191106316 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 120052574496 \beta_{10} - 180349501584 \beta_{9} - 95584600208 \beta_{8} + 75563850640 \beta_{7} + \cdots + 18579906296320 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 106605133112 \beta_{19} + 2836864988440 \beta_{18} - 6183387371616 \beta_{17} + \cdots + 43501666151488 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 11791700106960 \beta_{10} - 19594854841224 \beta_{9} - 11215433582424 \beta_{8} + 9049076206248 \beta_{7} + \cdots + 17\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 22792673853864 \beta_{19} + 292356220359720 \beta_{18} - 622815566933088 \beta_{17} + \cdots + 41\!\cdots\!28 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 11\!\cdots\!20 \beta_{10} + \cdots + 16\!\cdots\!92 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 33\!\cdots\!72 \beta_{19} + \cdots + 39\!\cdots\!76 \beta_1 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10.0526
−9.36784
−9.04017
−8.47923
−4.88608
−3.17869
−3.00577
−2.97468
−1.26969
−0.305476
0.305476
1.26969
2.97468
3.00577
3.17869
4.88608
8.47923
9.04017
9.36784
10.0526
0 −10.0526 0 −12.2096 0 12.6197 0 74.0548 0
1.2 0 −9.36784 0 17.0087 0 −11.7386 0 60.7564 0
1.3 0 −9.04017 0 −2.88132 0 −32.9337 0 54.7247 0
1.4 0 −8.47923 0 −8.79335 0 −4.45860 0 44.8974 0
1.5 0 −4.88608 0 −14.4412 0 16.2581 0 −3.12619 0
1.6 0 −3.17869 0 8.50017 0 30.0382 0 −16.8959 0
1.7 0 −3.00577 0 21.1277 0 23.6765 0 −17.9653 0
1.8 0 −2.97468 0 11.3425 0 −23.1233 0 −18.1513 0
1.9 0 −1.26969 0 −11.0765 0 3.71948 0 −25.3879 0
1.10 0 −0.305476 0 1.21818 0 28.1908 0 −26.9067 0
1.11 0 0.305476 0 −1.21818 0 −28.1908 0 −26.9067 0
1.12 0 1.26969 0 11.0765 0 −3.71948 0 −25.3879 0
1.13 0 2.97468 0 −11.3425 0 23.1233 0 −18.1513 0
1.14 0 3.00577 0 −21.1277 0 −23.6765 0 −17.9653 0
1.15 0 3.17869 0 −8.50017 0 −30.0382 0 −16.8959 0
1.16 0 4.88608 0 14.4412 0 −16.2581 0 −3.12619 0
1.17 0 8.47923 0 8.79335 0 4.45860 0 44.8974 0
1.18 0 9.04017 0 2.88132 0 32.9337 0 54.7247 0
1.19 0 9.36784 0 −17.0087 0 11.7386 0 60.7564 0
1.20 0 10.0526 0 12.2096 0 −12.6197 0 74.0548 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.4.a.l 20
17.b even 2 1 inner 1156.4.a.l 20
17.c even 4 2 1156.4.b.g 20
17.e odd 16 2 68.4.h.a ✓ 20
51.i even 16 2 612.4.w.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.4.h.a ✓ 20 17.e odd 16 2
612.4.w.a 20 51.i even 16 2
1156.4.a.l 20 1.a even 1 1 trivial
1156.4.a.l 20 17.b even 2 1 inner
1156.4.b.g 20 17.c even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 396 T_{3}^{18} + 63110 T_{3}^{16} - 5172304 T_{3}^{14} + 232007180 T_{3}^{12} + \cdots + 151170810368 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1156))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 151170810368 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 43\!\cdots\!48 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 37\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 366648118513408)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 34\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 40\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 14\!\cdots\!48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 82\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 26\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 50\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 49\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 92\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 66\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 10\!\cdots\!28 \) Copy content Toggle raw display
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