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Properties

Label 304.9.e.e
Level $304$
Weight $9$
Character orbit 304.e
Analytic conductor $123.843$
Analytic rank $0$
Dimension $12$
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] = [304,9,Mod(113,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 304.e (of order \(2\), degree \(1\), not 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: \(123.843097459\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{25}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 38)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + ( - \beta_{2} + 47) q^{5} + ( - \beta_{4} - \beta_1 + 452) q^{7} + (\beta_{5} + 2 \beta_{4} - \beta_{2} + \cdots - 1298) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + ( - \beta_{2} + 47) q^{5} + ( - \beta_{4} - \beta_1 + 452) q^{7} + (\beta_{5} + 2 \beta_{4} - \beta_{2} + \cdots - 1298) q^{9}+ \cdots + ( - 3179 \beta_{5} - 2227 \beta_{4} + \cdots + 7164160) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 558 q^{5} + 5422 q^{7} - 15592 q^{9} + 12546 q^{11} + 270810 q^{17} - 41512 q^{19} + 823956 q^{23} + 865538 q^{25} + 1194378 q^{35} - 5786100 q^{39} - 7586646 q^{43} + 2226046 q^{45} + 20260530 q^{47} - 19498842 q^{49} + 14858554 q^{55} + 14430564 q^{57} - 41363266 q^{61} - 84235798 q^{63} + 87906498 q^{73} - 78817962 q^{77} - 100904812 q^{81} + 55944960 q^{83} + 25440254 q^{85} - 119189604 q^{87} + 105500856 q^{93} - 81396774 q^{95} + 85554938 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 627859595 \nu^{10} - 26324430503956 \nu^{8} + \cdots - 13\!\cdots\!44 ) / 30\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!97 \nu^{10} + \cdots - 55\!\cdots\!76 ) / 50\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\!\cdots\!93 \nu^{10} + \cdots + 70\!\cdots\!64 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61\!\cdots\!79 \nu^{10} + \cdots - 14\!\cdots\!92 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 86\!\cdots\!97 \nu^{10} + \cdots + 20\!\cdots\!56 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13946998708921 \nu^{11} + \cdots + 30\!\cdots\!12 \nu ) / 12\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13946998708921 \nu^{11} + \cdots - 29\!\cdots\!28 \nu ) / 59\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 72\!\cdots\!99 \nu^{11} + \cdots - 18\!\cdots\!12 \nu ) / 98\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 34\!\cdots\!87 \nu^{11} + \cdots - 76\!\cdots\!16 \nu ) / 32\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42\!\cdots\!29 \nu^{11} + \cdots + 99\!\cdots\!12 \nu ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 37\!\cdots\!51 \nu^{11} + \cdots - 88\!\cdots\!48 \nu ) / 49\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} + 64\beta_{6} ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 4\beta_{4} - 2\beta_{2} - 13\beta _1 - 15367 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 160\beta_{11} - 192\beta_{10} + 200\beta_{9} - 1584\beta_{8} - 50579\beta_{7} - 451672\beta_{6} ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -19769\beta_{5} - 47248\beta_{4} + 2115\beta_{3} + 37007\beta_{2} + 197287\beta _1 + 108259783 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1606300 \beta_{11} + 3715680 \beta_{10} - 5119076 \beta_{9} + 20304684 \beta_{8} + \cdots + 4024002772 \beta_{6} ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 781327466 \beta_{5} + 1948624906 \beta_{4} - 159513309 \beta_{3} - 1389463040 \beta_{2} + \cdots - 3853054598260 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14676268870 \beta_{11} - 45672679632 \beta_{10} + 81487182554 \beta_{9} + \cdots - 39083007763858 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7899466534745 \beta_{5} - 19941863755858 \beta_{4} + 2150472839598 \beta_{3} + 12265474994369 \beta_{2} + \cdots + 37\!\cdots\!47 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 137489466440515 \beta_{11} + 512387851987704 \beta_{10} + \cdots + 39\!\cdots\!85 \beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 32\!\cdots\!98 \beta_{5} + \cdots - 15\!\cdots\!84 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 26\!\cdots\!15 \beta_{11} + \cdots - 81\!\cdots\!57 \beta_{6} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
− 145.414i
− 111.533i
− 63.2768i
− 61.5968i
− 64.6816i
− 23.4825i
23.4825i
64.6816i
61.5968i
63.2768i
111.533i
145.414i
0 − 132.686i 0 −12.7536 0 1217.09 0 −11044.5 0
113.2 0 − 124.261i 0 −419.091 0 2431.38 0 −8879.72 0
113.3 0 − 76.0047i 0 1155.75 0 2869.50 0 784.279 0
113.4 0 − 74.3247i 0 −154.845 0 −1585.07 0 1036.83 0
113.5 0 − 51.9537i 0 629.221 0 −2565.52 0 3861.81 0
113.6 0 − 10.7546i 0 −919.278 0 343.629 0 6445.34 0
113.7 0 10.7546i 0 −919.278 0 343.629 0 6445.34 0
113.8 0 51.9537i 0 629.221 0 −2565.52 0 3861.81 0
113.9 0 74.3247i 0 −154.845 0 −1585.07 0 1036.83 0
113.10 0 76.0047i 0 1155.75 0 2869.50 0 784.279 0
113.11 0 124.261i 0 −419.091 0 2431.38 0 −8879.72 0
113.12 0 132.686i 0 −12.7536 0 1217.09 0 −11044.5 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.12
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 304.9.e.e 12
4.b odd 2 1 38.9.b.a ✓ 12
12.b even 2 1 342.9.d.a 12
19.b odd 2 1 inner 304.9.e.e 12
76.d even 2 1 38.9.b.a ✓ 12
228.b odd 2 1 342.9.d.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.9.b.a ✓ 12 4.b odd 2 1
38.9.b.a ✓ 12 76.d even 2 1
304.9.e.e 12 1.a even 1 1 trivial
304.9.e.e 12 19.b odd 2 1 inner
342.9.d.a 12 12.b even 2 1
342.9.d.a 12 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{12} + 47162 T_{3}^{10} + 802348665 T_{3}^{8} + 6046694283888 T_{3}^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
\( T_{5}^{6} - 279 T_{5}^{5} - 1349339 T_{5}^{4} + 70794579 T_{5}^{3} + 325516762486 T_{5}^{2} + \cdots + 553286629221600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots + 553286629221600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 11\!\cdots\!70)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 77\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 22\!\cdots\!30)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 23\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 32\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 36\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 67\!\cdots\!50)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 21\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
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