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Properties

Label 684.5.h.f
Level $684$
Weight $5$
Character orbit 684.h
Analytic conductor $70.705$
Analytic rank $0$
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] = [684,5,Mod(37,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.37");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 684.h (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: \(70.7050547493\)
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: \( x^{14} - 2 x^{13} + 2574 x^{12} - 7948 x^{11} + 5136095 x^{10} - 4313010 x^{9} + 3526383758 x^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{13} \)
Twist minimal: no (minimal twist has level 228)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{5} + (\beta_{2} - 6) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{5} + (\beta_{2} - 6) q^{7} + ( - \beta_{7} - 18) q^{11} - \beta_{10} q^{13} + ( - \beta_{4} - 2 \beta_{2} - 36) q^{17} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \cdots + 12) q^{19}+ \cdots + ( - 12 \beta_{13} + 9 \beta_{12} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 18 q^{5} - 86 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 18 q^{5} - 86 q^{7} - 258 q^{11} - 498 q^{17} + 170 q^{19} + 588 q^{23} + 1560 q^{25} - 534 q^{35} + 1882 q^{43} + 222 q^{47} + 4104 q^{49} + 2702 q^{55} - 2462 q^{61} - 5774 q^{73} + 4578 q^{77} - 17988 q^{83} + 2342 q^{85} + 18270 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 2574 x^{12} - 7948 x^{11} + 5136095 x^{10} - 4313010 x^{9} + 3526383758 x^{8} + \cdots + 13\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 22\!\cdots\!81 \nu^{13} + \cdots - 42\!\cdots\!40 ) / 56\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!39 \nu^{13} + \cdots - 35\!\cdots\!20 ) / 27\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 15\!\cdots\!32 \nu^{13} + \cdots - 80\!\cdots\!60 ) / 37\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 67\!\cdots\!46 \nu^{13} + \cdots - 65\!\cdots\!40 ) / 75\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 41\!\cdots\!39 \nu^{13} + \cdots - 39\!\cdots\!80 ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 69\!\cdots\!45 \nu^{13} + \cdots - 29\!\cdots\!40 ) / 27\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\!\cdots\!74 \nu^{13} + \cdots + 64\!\cdots\!56 ) / 54\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42\!\cdots\!78 \nu^{13} + \cdots - 15\!\cdots\!56 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18\!\cdots\!51 \nu^{13} + \cdots + 52\!\cdots\!40 ) / 43\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67\!\cdots\!76 \nu^{13} + \cdots + 21\!\cdots\!80 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 93\!\cdots\!64 \nu^{13} + \cdots + 48\!\cdots\!20 ) / 72\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 71\!\cdots\!24 \nu^{13} + \cdots + 11\!\cdots\!20 ) / 33\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 22\!\cdots\!26 \nu^{13} + \cdots - 58\!\cdots\!20 ) / 58\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{12} + \beta_{11} + 4 \beta_{10} - \beta_{9} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \cdots - 2 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 24 \beta_{13} - 7 \beta_{12} - 5 \beta_{11} - 104 \beta_{10} - 31 \beta_{9} - 36 \beta_{8} + \cdots - 26450 ) / 72 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 45\beta_{8} + 23\beta_{7} - 21\beta_{6} + 35\beta_{4} - 173\beta_{2} - 1409\beta _1 + 983 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 46296 \beta_{13} + 9407 \beta_{12} + 28609 \beta_{11} + 228952 \beta_{10} + 41303 \beta_{9} + \cdots - 37683410 ) / 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 863016 \beta_{13} + 991727 \beta_{12} - 2905319 \beta_{11} - 10400708 \beta_{10} + 2078435 \beta_{9} + \cdots - 353051078 ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4544613 \beta_{8} - 2420077 \beta_{7} - 9644119 \beta_{6} + 6283026 \beta_{4} - 10078737 \beta_{2} + \cdots + 1722108184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2120399784 \beta_{13} - 1482920419 \beta_{12} + 5357485147 \beta_{11} + 20386861036 \beta_{10} + \cdots - 1145186615330 ) / 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 153385280088 \beta_{13} + 534425945 \beta_{12} - 164343382361 \beta_{11} - 935806597352 \beta_{10} + \cdots - 107931367345934 ) / 72 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 423364313469 \beta_{8} + 29044509743 \beta_{7} - 508249174131 \beta_{6} + 418221177491 \beta_{4} + \cdots + 77214393561671 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 287955400352472 \beta_{13} - 21709435456765 \beta_{12} + 350797857066109 \beta_{11} + \cdots - 19\!\cdots\!78 ) / 72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10\!\cdots\!88 \beta_{13} + \cdots - 61\!\cdots\!50 ) / 72 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 36\!\cdots\!41 \beta_{8} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 20\!\cdots\!24 \beta_{13} + \cdots - 13\!\cdots\!86 ) / 72 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
17.9577 − 31.1036i
17.9577 + 31.1036i
17.5385 + 30.3776i
17.5385 − 30.3776i
6.18424 + 10.7114i
6.18424 − 10.7114i
−4.56664 + 7.90965i
−4.56664 − 7.90965i
−5.60642 − 9.71061i
−5.60642 + 9.71061i
−8.18111 + 14.1701i
−8.18111 − 14.1701i
−22.3263 + 38.6702i
−22.3263 − 38.6702i
0 0 0 −36.9153 0 −44.6594 0 0 0
37.2 0 0 0 −36.9153 0 −44.6594 0 0 0
37.3 0 0 0 −36.0770 0 53.4819 0 0 0
37.4 0 0 0 −36.0770 0 53.4819 0 0 0
37.5 0 0 0 −13.3685 0 −81.9550 0 0 0
37.6 0 0 0 −13.3685 0 −81.9550 0 0 0
37.7 0 0 0 8.13328 0 −13.7213 0 0 0
37.8 0 0 0 8.13328 0 −13.7213 0 0 0
37.9 0 0 0 10.2128 0 41.1350 0 0 0
37.10 0 0 0 10.2128 0 41.1350 0 0 0
37.11 0 0 0 15.3622 0 53.3379 0 0 0
37.12 0 0 0 15.3622 0 53.3379 0 0 0
37.13 0 0 0 43.6525 0 −50.6192 0 0 0
37.14 0 0 0 43.6525 0 −50.6192 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.5.h.f 14
3.b odd 2 1 228.5.h.a ✓ 14
12.b even 2 1 912.5.o.c 14
19.b odd 2 1 inner 684.5.h.f 14
57.d even 2 1 228.5.h.a ✓ 14
228.b odd 2 1 912.5.o.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.5.h.a ✓ 14 3.b odd 2 1
228.5.h.a ✓ 14 57.d even 2 1
684.5.h.f 14 1.a even 1 1 trivial
684.5.h.f 14 19.b odd 2 1 inner
912.5.o.c 14 12.b even 2 1
912.5.o.c 14 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} + 9T_{5}^{6} - 2537T_{5}^{5} - 19329T_{5}^{4} + 1430092T_{5}^{3} - 2196732T_{5}^{2} - 177778416T_{5} + 991733472 \) acting on \(S_{5}^{\mathrm{new}}(684, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T^{7} + 9 T^{6} + \cdots + 991733472)^{2} \) Copy content Toggle raw display
$7$ \( (T^{7} + 43 T^{6} + \cdots - 298299152800)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots - 71429510530152)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 47\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 139564838688336)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 63\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 18\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 46\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 47\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 14\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 68\!\cdots\!20)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 35\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 68\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots + 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 69\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 16\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 58\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
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