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Properties

Label 2312.4.a.m
Level $2312$
Weight $4$
Character orbit 2312.a
Self dual yes
Analytic conductor $136.412$
Analytic rank $1$
Dimension $14$
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] = [2312,4,Mod(1,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2312.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2312.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: \(136.412415933\)
Analytic rank: \(1\)
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: \( x^{14} - 4 x^{13} - 148 x^{12} + 474 x^{11} + 8325 x^{10} - 20424 x^{9} - 224201 x^{8} + 401234 x^{7} + \cdots - 5899068 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: no (minimal twist has level 136)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{9} - \beta_1) q^{5} + (\beta_{5} + \beta_1) q^{7} + (\beta_{8} + \beta_{4} + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{9} - \beta_1) q^{5} + (\beta_{5} + \beta_1) q^{7} + (\beta_{8} + \beta_{4} + 14) q^{9} + (\beta_{11} - \beta_{9} - 2 \beta_{3} + \beta_1) q^{11} + ( - \beta_{13} - \beta_{6} + \beta_{4} + \cdots - 4) q^{13}+ \cdots + (19 \beta_{11} - 3 \beta_{10} + \cdots + 190 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 190 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 190 q^{9} - 72 q^{13} - 60 q^{15} - 468 q^{19} + 60 q^{21} + 306 q^{25} - 904 q^{33} - 804 q^{35} - 896 q^{43} - 448 q^{47} + 866 q^{49} - 108 q^{53} - 1612 q^{55} - 1760 q^{59} - 284 q^{67} - 2532 q^{69} + 180 q^{77} + 950 q^{81} - 4256 q^{83} - 868 q^{87} + 4516 q^{89} - 7156 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 148 x^{12} + 474 x^{11} + 8325 x^{10} - 20424 x^{9} - 224201 x^{8} + 401234 x^{7} + \cdots - 5899068 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 87\!\cdots\!77 \nu^{13} + \cdots - 15\!\cdots\!32 ) / 94\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 85\!\cdots\!64 \nu^{13} + \cdots - 40\!\cdots\!00 ) / 70\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 47\!\cdots\!51 \nu^{13} + \cdots + 45\!\cdots\!72 ) / 31\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 87\!\cdots\!77 \nu^{13} + \cdots + 10\!\cdots\!44 ) / 47\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!83 \nu^{13} + \cdots - 24\!\cdots\!68 ) / 35\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 61\!\cdots\!83 \nu^{13} + \cdots + 87\!\cdots\!60 ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 27\!\cdots\!55 \nu^{13} + \cdots - 23\!\cdots\!84 ) / 35\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!83 \nu^{13} + \cdots - 19\!\cdots\!72 ) / 47\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30\!\cdots\!02 \nu^{13} + \cdots - 32\!\cdots\!86 ) / 23\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 75\!\cdots\!13 \nu^{13} + \cdots - 24\!\cdots\!04 ) / 47\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!17 \nu^{13} + \cdots + 12\!\cdots\!10 ) / 78\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35\!\cdots\!85 \nu^{13} + \cdots - 55\!\cdots\!40 ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 40\!\cdots\!74 \nu^{13} + \cdots - 47\!\cdots\!08 ) / 35\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{4} - 4\beta_{3} + 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{12} + 4\beta_{8} - 6\beta_{7} + 6\beta_{5} + 77\beta_{4} - 12\beta_{3} - 3\beta_{2} + 248\beta _1 + 152 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{13} + 6 \beta_{12} + 24 \beta_{11} - 24 \beta_{9} + 99 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} + \cdots + 3680 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15 \beta_{13} + 228 \beta_{12} + 45 \beta_{11} - 15 \beta_{10} - 135 \beta_{9} + 293 \beta_{8} + \cdots + 11080 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 375 \beta_{13} + 777 \beta_{12} + 2166 \beta_{11} - 90 \beta_{10} - 3570 \beta_{9} + 5171 \beta_{8} + \cdots + 186702 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2010 \beta_{13} + 15726 \beta_{12} + 6993 \beta_{11} - 2205 \beta_{10} - 19215 \beta_{9} + 20058 \beta_{8} + \cdots + 789466 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 17007 \beta_{13} + 37011 \beta_{12} + 79128 \beta_{11} - 6360 \beta_{10} - 163944 \beta_{9} + \cdots + 5300411 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 196752 \beta_{13} + 1076529 \beta_{12} + 702702 \beta_{11} - 204318 \beta_{10} - 1891458 \beta_{9} + \cdots + 55646948 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2697789 \beta_{13} + 6189324 \beta_{12} + 10976628 \beta_{11} - 1244640 \beta_{10} - 25680180 \beta_{9} + \cdots + 643635192 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 16858515 \beta_{13} + 73962888 \beta_{12} + 59627601 \beta_{11} - 16160199 \beta_{10} + \cdots + 3903848312 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 200530071 \beta_{13} + 482152563 \beta_{12} + 751206396 \beta_{11} - 105192420 \beta_{10} + \cdots + 40811052002 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1341721704 \beta_{13} + 5099230548 \beta_{12} + 4661281755 \beta_{11} - 1194392979 \beta_{10} + \cdots + 273162195442 ) / 4 \) 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
8.33215
−6.97807
6.76451
−5.95163
−4.48768
3.25589
2.47905
−0.349380
0.427464
−1.65925
−3.12320
3.93608
−4.14964
5.50372
0 −9.78344 0 −11.5964 0 25.6146 0 68.7157 0
1.2 0 −7.86848 0 −4.75624 0 −27.5696 0 34.9130 0
1.3 0 −7.56646 0 15.6924 0 −27.2848 0 30.2512 0
1.4 0 −6.41688 0 20.6261 0 13.3791 0 14.1763 0
1.5 0 −4.34654 0 −12.2701 0 12.7607 0 −8.10760 0
1.6 0 −2.60453 0 −6.93188 0 −8.68894 0 −20.2164 0
1.7 0 −1.50590 0 0.778787 0 16.0004 0 −24.7323 0
1.8 0 1.50590 0 −0.778787 0 −16.0004 0 −24.7323 0
1.9 0 2.60453 0 6.93188 0 8.68894 0 −20.2164 0
1.10 0 4.34654 0 12.2701 0 −12.7607 0 −8.10760 0
1.11 0 6.41688 0 −20.6261 0 −13.3791 0 14.1763 0
1.12 0 7.56646 0 −15.6924 0 27.2848 0 30.2512 0
1.13 0 7.86848 0 4.75624 0 27.5696 0 34.9130 0
1.14 0 9.78344 0 11.5964 0 −25.6146 0 68.7157 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 2312.4.a.m 14
17.b even 2 1 inner 2312.4.a.m 14
17.d even 8 2 136.4.k.a ✓ 14
68.g odd 8 2 272.4.o.g 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.k.a ✓ 14 17.d even 8 2
272.4.o.g 14 68.g odd 8 2
2312.4.a.m 14 1.a even 1 1 trivial
2312.4.a.m 14 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 284 T_{3}^{12} + 31140 T_{3}^{10} - 1667912 T_{3}^{8} + 45162976 T_{3}^{6} + \cdots - 4060086272 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2312))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots - 4060086272 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 1398380605952 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 20\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 17\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( (T^{7} + 36 T^{6} + \cdots + 2332419584)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 3000100123648)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 54\!\cdots\!68 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 44\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots - 65\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 45\!\cdots\!48 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 12\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 65\!\cdots\!32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots + 143786480194048)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 39\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots - 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 43\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 31\!\cdots\!72 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 68\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 98\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 53\!\cdots\!92 \) Copy content Toggle raw display
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