Properties

Label 92.10.a.b
Level $92$
Weight $10$
Character orbit 92.a
Self dual yes
Analytic conductor $47.383$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [92,10,Mod(1,92)]
 
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, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("92.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 92 = 2^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 92.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: \(47.3832969271\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 115691 x^{6} - 1021622 x^{5} + 3687087652 x^{4} + 64518632368 x^{3} - 23061241322016 x^{2} + 272828694011200 x + 25\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{15}\cdot 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} + 176) q^{5} + (\beta_{4} + 2255) q^{7} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_{2} + 16 \beta_1 + 9244) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} + 176) q^{5} + (\beta_{4} + 2255) q^{7} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 7 \beta_{3} + \beta_{2} + 16 \beta_1 + 9244) q^{9} + (3 \beta_{6} + 4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 144) q^{11} + (\beta_{7} - \beta_{6} - 29 \beta_{3} - 6 \beta_{2} + 104 \beta_1 + 9801) q^{13} + ( - 3 \beta_{7} - 15 \beta_{6} + 10 \beta_{5} - 17 \beta_{4} - 83 \beta_{3} + \cdots + 444) q^{15}+ \cdots + (16737 \beta_{7} + 23526 \beta_{6} - 48324 \beta_{5} + \cdots + 250086246) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 1404 q^{5} + 18042 q^{7} + 73926 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 1404 q^{5} + 18042 q^{7} + 73926 q^{9} + 1148 q^{11} + 78302 q^{13} + 3186 q^{15} - 801082 q^{17} + 730958 q^{19} + 38998 q^{21} - 2238728 q^{23} + 5319768 q^{25} + 3444092 q^{27} - 4923594 q^{29} + 15832004 q^{31} + 21114 q^{33} + 9507032 q^{35} + 3867160 q^{37} + 24454740 q^{39} - 1581522 q^{41} + 35686230 q^{43} + 165668970 q^{45} + 74101900 q^{47} + 125546836 q^{49} + 37577854 q^{51} + 186799998 q^{53} + 90851100 q^{55} + 198553864 q^{57} + 149329196 q^{59} + 383250150 q^{61} + 825904752 q^{63} + 603870638 q^{65} + 667836556 q^{67} - 2238728 q^{69} + 402817056 q^{71} + 693473370 q^{73} + 1393408692 q^{75} + 726060200 q^{77} + 1307850720 q^{79} + 1470127920 q^{81} + 529007816 q^{83} + 627434204 q^{85} + 1068391628 q^{87} + 828643896 q^{89} + 1545925794 q^{91} + 1685501414 q^{93} + 1524181588 q^{95} + 1232759070 q^{97} + 2002116066 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 115691 x^{6} - 1021622 x^{5} + 3687087652 x^{4} + 64518632368 x^{3} - 23061241322016 x^{2} + 272828694011200 x + 25\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 381158857937222 \nu^{7} + \cdots - 23\!\cdots\!70 ) / 34\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 877519312298923 \nu^{7} + \cdots + 75\!\cdots\!80 ) / 11\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 720214156629755 \nu^{7} + \cdots - 73\!\cdots\!74 ) / 68\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67\!\cdots\!69 \nu^{7} + \cdots - 67\!\cdots\!60 ) / 34\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 77\!\cdots\!17 \nu^{7} + \cdots + 29\!\cdots\!80 ) / 34\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 51\!\cdots\!54 \nu^{7} + \cdots + 54\!\cdots\!40 ) / 34\!\cdots\!90 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{5} + \beta_{4} - 7\beta_{3} + \beta_{2} + 14\beta _1 + 28926 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 37\beta_{7} - 255\beta_{6} + 21\beta_{5} - 107\beta_{4} + 173\beta_{3} + 52031\beta _1 + 383063 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5602 \beta_{7} - 43894 \beta_{6} - 71203 \beta_{5} + 49593 \beta_{4} - 494250 \beta_{3} + 45721 \beta_{2} + 1203307 \beta _1 + 1502785000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2262416 \beta_{7} - 20127078 \beta_{6} + 3749061 \beta_{5} - 10520821 \beta_{4} + 8835028 \beta_{3} + 1161345 \beta_{2} + 2940887541 \beta _1 + 33546154918 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 590178536 \beta_{7} - 2035541368 \beta_{6} - 4460149717 \beta_{5} + 2911755047 \beta_{4} - 33131319152 \beta_{3} + 1953147295 \beta_{2} + \cdots + 84894114544112 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 124769793168 \beta_{7} - 1368698749098 \beta_{6} + 344695609281 \beta_{5} - 849440649279 \beta_{4} + 311283668208 \beta_{3} + 113055420399 \beta_{2} + \cdots + 18\!\cdots\!34 \) Copy content Toggle raw display

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
−252.470
−188.722
−116.040
−6.14769
19.7494
67.2771
229.524
246.829
0 −251.470 0 1803.22 0 8767.74 0 43554.0 0
1.2 0 −187.722 0 36.3653 0 −4026.70 0 15556.4 0
1.3 0 −115.040 0 −889.807 0 11621.4 0 −6448.71 0
1.4 0 −5.14769 0 357.420 0 −9708.70 0 −19656.5 0
1.5 0 20.7494 0 −2517.67 0 −4003.17 0 −19252.5 0
1.6 0 68.2771 0 1428.14 0 4424.00 0 −15021.2 0
1.7 0 230.524 0 −1367.75 0 9371.73 0 33458.4 0
1.8 0 247.829 0 2554.08 0 1595.72 0 41736.1 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 92.10.a.b 8
4.b odd 2 1 368.10.a.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
92.10.a.b 8 1.a even 1 1 trivial
368.10.a.g 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 8 T_{3}^{7} - 115663 T_{3}^{6} - 327532 T_{3}^{5} + 3690460467 T_{3}^{4} + 49762379304 T_{3}^{3} - 23232666212325 T_{3}^{2} + 319129979787756 T_{3} + 22\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(92))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{8} - 1404 T^{7} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} - 18042 T^{7} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{8} - 1148 T^{7} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} - 78302 T^{7} + \cdots - 52\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + 801082 T^{7} + \cdots - 46\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{8} - 730958 T^{7} + \cdots + 13\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( (T + 279841)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 4923594 T^{7} + \cdots + 31\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{8} - 15832004 T^{7} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} - 3867160 T^{7} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{8} + 1581522 T^{7} + \cdots - 31\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( T^{8} - 35686230 T^{7} + \cdots + 27\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{8} - 74101900 T^{7} + \cdots - 19\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{8} - 186799998 T^{7} + \cdots - 83\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{8} - 149329196 T^{7} + \cdots + 67\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{8} - 383250150 T^{7} + \cdots - 32\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{8} - 667836556 T^{7} + \cdots + 72\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{8} - 402817056 T^{7} + \cdots - 92\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{8} - 693473370 T^{7} + \cdots + 23\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{8} - 1307850720 T^{7} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} - 529007816 T^{7} + \cdots - 13\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{8} - 828643896 T^{7} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} - 1232759070 T^{7} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
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