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Properties

Label 92.11.d.a
Level $92$
Weight $11$
Character orbit 92.d
Analytic conductor $58.453$
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] = [92,11,Mod(45,92)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(92, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("92.45");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 92 = 2^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 92.d (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: \(58.4528672460\)
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: \( x^{20} + 104512336 x^{18} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: multiple of \( 2^{59}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 3) q^{3} + \beta_1 q^{5} + \beta_{10} q^{7} + (\beta_{3} + 33 \beta_{2} + 13723) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 3) q^{3} + \beta_1 q^{5} + \beta_{10} q^{7} + (\beta_{3} + 33 \beta_{2} + 13723) q^{9} + (\beta_{11} - \beta_{10} - 4 \beta_1) q^{11} + ( - \beta_{5} - 95 \beta_{2} - 30785) q^{13} + ( - \beta_{12} - 3 \beta_{10} - 25 \beta_1) q^{15} + ( - \beta_{15} - \beta_{11} + \cdots + 46 \beta_1) q^{17}+ \cdots + (52 \beta_{19} - 52 \beta_{18} + \cdots + 25565 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 62 q^{3} + 274530 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 62 q^{3} + 274530 q^{9} - 615890 q^{13} + 2846044 q^{23} - 13712172 q^{25} + 45676934 q^{27} + 3073902 q^{29} - 24362546 q^{31} + 20182256 q^{35} - 140028138 q^{39} + 148578062 q^{41} + 339564206 q^{47} - 501711836 q^{49} + 762521888 q^{55} - 800417272 q^{59} + 738964846 q^{69} - 5115299938 q^{71} + 4040491870 q^{73} + 5774635646 q^{75} + 6070592528 q^{77} - 679842064 q^{81} - 9567848112 q^{85} - 1773952682 q^{87} + 13690545286 q^{93} + 7534339984 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 104512336 x^{18} + \cdots + 38\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36\!\cdots\!31 \nu^{18} + \cdots + 16\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18\!\cdots\!03 \nu^{18} + \cdots + 11\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!67 \nu^{18} + \cdots - 35\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!31 \nu^{18} + \cdots - 98\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\!\cdots\!99 \nu^{18} + \cdots + 47\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!57 \nu^{18} + \cdots + 30\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!07 \nu^{18} + \cdots + 96\!\cdots\!00 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!29 \nu^{18} + \cdots - 86\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!01 \nu^{19} + \cdots - 11\!\cdots\!00 \nu ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39\!\cdots\!63 \nu^{19} + \cdots + 51\!\cdots\!00 \nu ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 33\!\cdots\!23 \nu^{19} + \cdots - 14\!\cdots\!00 \nu ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!89 \nu^{19} + \cdots + 22\!\cdots\!00 \nu ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 33\!\cdots\!19 \nu^{19} + \cdots - 32\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23\!\cdots\!99 \nu^{19} + \cdots - 11\!\cdots\!00 \nu ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 97\!\cdots\!43 \nu^{19} + \cdots + 38\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 97\!\cdots\!43 \nu^{19} + \cdots - 38\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 89\!\cdots\!33 \nu^{19} + \cdots - 42\!\cdots\!00 \nu ) / 64\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10\!\cdots\!83 \nu^{19} + \cdots - 32\!\cdots\!00 \nu ) / 51\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{5} + \beta_{4} + 22\beta_{3} + 3990\beta_{2} - 10451637 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 39 \beta_{19} + 246 \beta_{18} + 270 \beta_{17} + 270 \beta_{16} + 9803 \beta_{15} + \cdots - 21854540 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6765312 \beta_{17} + 6765312 \beta_{16} + 1373492 \beta_{9} - 10638 \beta_{8} + \cdots + 227887046322306 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2664967446 \beta_{19} - 12055104258 \beta_{18} - 26646697048 \beta_{17} - 26646697048 \beta_{16} + \cdots + 596540845045394 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 368791570817592 \beta_{17} - 368791570817592 \beta_{16} - 69765161418532 \beta_{9} + \cdots - 62\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11\!\cdots\!64 \beta_{19} + \cdots - 18\!\cdots\!64 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15\!\cdots\!16 \beta_{17} + \cdots + 19\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 41\!\cdots\!56 \beta_{19} + \cdots + 61\!\cdots\!52 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 60\!\cdots\!00 \beta_{17} + \cdots - 64\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 15\!\cdots\!84 \beta_{19} + \cdots - 21\!\cdots\!80 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 22\!\cdots\!32 \beta_{17} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 54\!\cdots\!96 \beta_{19} + \cdots + 75\!\cdots\!84 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 81\!\cdots\!32 \beta_{17} + \cdots - 78\!\cdots\!80 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 19\!\cdots\!64 \beta_{19} + \cdots - 26\!\cdots\!44 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 29\!\cdots\!96 \beta_{17} + \cdots + 28\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 70\!\cdots\!56 \beta_{19} + \cdots + 96\!\cdots\!12 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 10\!\cdots\!60 \beta_{17} + \cdots - 10\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 25\!\cdots\!84 \beta_{19} + \cdots - 34\!\cdots\!00 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/92\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(47\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
− 1147.10i
1147.10i
− 5994.76i
5994.76i
− 1778.72i
1778.72i
− 4206.49i
4206.49i
− 810.302i
810.302i
− 1109.85i
1109.85i
− 3396.78i
3396.78i
− 3882.52i
3882.52i
− 4160.32i
4160.32i
− 769.590i
769.590i
0 −417.078 0 − 1147.10i 0 − 7672.70i 0 114905. 0
45.2 0 −417.078 0 1147.10i 0 7672.70i 0 114905. 0
45.3 0 −270.593 0 − 5994.76i 0 − 2498.00i 0 14171.8 0
45.4 0 −270.593 0 5994.76i 0 2498.00i 0 14171.8 0
45.5 0 −251.886 0 − 1778.72i 0 30622.0i 0 4397.71 0
45.6 0 −251.886 0 1778.72i 0 − 30622.0i 0 4397.71 0
45.7 0 −164.589 0 − 4206.49i 0 − 7773.17i 0 −31959.6 0
45.8 0 −164.589 0 4206.49i 0 7773.17i 0 −31959.6 0
45.9 0 −48.8341 0 − 810.302i 0 − 29331.7i 0 −56664.2 0
45.10 0 −48.8341 0 810.302i 0 29331.7i 0 −56664.2 0
45.11 0 10.0567 0 − 1109.85i 0 6579.76i 0 −58947.9 0
45.12 0 10.0567 0 1109.85i 0 − 6579.76i 0 −58947.9 0
45.13 0 150.477 0 − 3396.78i 0 9704.98i 0 −36405.6 0
45.14 0 150.477 0 3396.78i 0 − 9704.98i 0 −36405.6 0
45.15 0 280.465 0 − 3882.52i 0 − 19776.2i 0 19611.4 0
45.16 0 280.465 0 3882.52i 0 19776.2i 0 19611.4 0
45.17 0 299.925 0 − 4160.32i 0 13513.2i 0 30905.8 0
45.18 0 299.925 0 4160.32i 0 − 13513.2i 0 30905.8 0
45.19 0 443.057 0 − 769.590i 0 20996.1i 0 137251. 0
45.20 0 443.057 0 769.590i 0 − 20996.1i 0 137251. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.11.d.a ✓ 20
23.b odd 2 1 inner 92.11.d.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.11.d.a ✓ 20 1.a even 1 1 trivial
92.11.d.a ✓ 20 23.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(92, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 62\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 63\!\cdots\!20)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 15\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 65\!\cdots\!40)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 31\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 66\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 44\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 16\!\cdots\!88 \) Copy content Toggle raw display
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