Properties

Label 304.5.e.e
Level $304$
Weight $5$
Character orbit 304.e
Analytic conductor $31.424$
Analytic rank $0$
Dimension $8$
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] = [304,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(31.4244687775\)
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: \( x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{5} + 2) q^{5} + ( - \beta_{4} + 20) q^{7} + (\beta_{5} + \beta_{4} - \beta_{3} - 33) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{5} + 2) q^{5} + ( - \beta_{4} + 20) q^{7} + (\beta_{5} + \beta_{4} - \beta_{3} - 33) q^{9} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2) q^{11} + (\beta_{6} + 7 \beta_1) q^{13} + (2 \beta_{6} - 3 \beta_{2} - 12 \beta_1) q^{15} + (\beta_{4} + 7 \beta_{3} + 64) q^{17} + (\beta_{7} - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} - 4 \beta_{2} - 14 \beta_1 + 1) q^{19} + ( - 2 \beta_{7} + \beta_{6} + \beta_{2} - 47 \beta_1) q^{21} + ( - 7 \beta_{5} + \beta_{4} - 14 \beta_{3} + 48) q^{23} + ( - 3 \beta_{5} - 2 \beta_{4} + 12 \beta_{3} + 431) q^{25} + (2 \beta_{7} - 4 \beta_{6} + 11 \beta_{2} - 3 \beta_1) q^{27} + (4 \beta_{7} - \beta_{6} - \beta_{2} + 15 \beta_1) q^{29} + ( - 21 \beta_{2} + 30 \beta_1) q^{31} + (4 \beta_{7} - 6 \beta_{6} - 11 \beta_{2} + 66 \beta_1) q^{33} + ( - 9 \beta_{5} - 2 \beta_{4} + 58 \beta_{3} - 128) q^{35} + ( - 2 \beta_{7} + 4 \beta_{6} - 13 \beta_{2} - 76 \beta_1) q^{37} + (45 \beta_{5} - 3 \beta_{4} - 6 \beta_{3} + 834) q^{39} + (4 \beta_{7} - 4 \beta_{6} - 25 \beta_{2} + 42 \beta_1) q^{41} + ( - 15 \beta_{5} - 48 \beta_{3} + 1078) q^{43} + (35 \beta_{5} + 20 \beta_{4} - 74 \beta_{3} - 1278) q^{45} + (31 \beta_{5} - 18 \beta_{4} + 24 \beta_{3} - 398) q^{47} + ( - 24 \beta_{5} - 39 \beta_{4} - 57 \beta_{3} + 1137) q^{49} + (2 \beta_{7} + 6 \beta_{6} - 64 \beta_{2} - 93 \beta_1) q^{51} + (4 \beta_{7} + 11 \beta_{6} + 5 \beta_{2} - 183 \beta_1) q^{53} + ( - 45 \beta_{5} - 22 \beta_{4} - 144 \beta_{3} - 2160) q^{55} + (2 \beta_{7} + 8 \beta_{6} + 27 \beta_{5} - 21 \beta_{4} - 69 \beta_{3} + \cdots - 1758) q^{57}+ \cdots + ( - 83 \beta_{5} - 68 \beta_{4} + 146 \beta_{3} + 7062) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 18 q^{5} + 162 q^{7} - 268 q^{9} + 6 q^{11} + 510 q^{17} + 12 q^{19} + 396 q^{23} + 3458 q^{25} - 1002 q^{35} + 6588 q^{39} + 8654 q^{43} - 10334 q^{45} - 3210 q^{47} + 9222 q^{49} - 17146 q^{55} - 14076 q^{57} + 1314 q^{61} - 29938 q^{63} + 23398 q^{73} - 44622 q^{77} - 20368 q^{81} + 10440 q^{83} + 21274 q^{85} + 14316 q^{87} + 19416 q^{93} + 34686 q^{95} + 56798 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 450x^{6} + 68229x^{4} + 4001228x^{2} + 77475204 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} - 450\nu^{5} - 59427\nu^{3} - 3270662\nu ) / 1249884 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} - 900\nu^{5} - 118854\nu^{3} - 4041556\nu ) / 104157 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 225\nu^{2} + 8802 ) / 71 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 352\nu^{4} - 33685\nu^{2} - 733602 ) / 3408 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 352\nu^{4} + 37093\nu^{2} + 1115298 ) / 3408 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -56\nu^{7} - 20799\nu^{5} - 2337687\nu^{3} - 75675850\nu ) / 312471 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -475\nu^{7} - 169740\nu^{5} - 17075691\nu^{3} - 401259422\nu ) / 2499768 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 24\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 112 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24\beta_{7} - 30\beta_{6} - 31\beta_{2} + 1764\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -225\beta_{5} - 225\beta_{4} + 71\beta_{3} + 16398 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2700\beta_{7} + 3801\beta_{6} - 2618\beta_{2} - 147342\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 45515\beta_{5} + 42107\beta_{4} - 24992\beta_{3} - 2732978 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 501876\beta_{7} - 819045\beta_{6} + 1281553\beta_{2} + 26013954\beta_1 ) / 6 \) Copy content Toggle raw display

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
13.9305i
12.2418i
6.38941i
8.07810i
8.07810i
6.38941i
12.2418i
13.9305i
0 15.3447i 0 41.6240 0 62.4342 0 −154.460 0
113.2 0 10.8276i 0 −26.2296 0 86.0910 0 −36.2364 0
113.3 0 7.80363i 0 −33.0971 0 −16.0783 0 20.1034 0
113.4 0 6.66389i 0 26.7027 0 −51.4469 0 36.5926 0
113.5 0 6.66389i 0 26.7027 0 −51.4469 0 36.5926 0
113.6 0 7.80363i 0 −33.0971 0 −16.0783 0 20.1034 0
113.7 0 10.8276i 0 −26.2296 0 86.0910 0 −36.2364 0
113.8 0 15.3447i 0 41.6240 0 62.4342 0 −154.460 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.8
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 304.5.e.e 8
4.b odd 2 1 38.5.b.a 8
12.b even 2 1 342.5.d.a 8
19.b odd 2 1 inner 304.5.e.e 8
76.d even 2 1 38.5.b.a 8
228.b odd 2 1 342.5.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.5.b.a 8 4.b odd 2 1
38.5.b.a 8 76.d even 2 1
304.5.e.e 8 1.a even 1 1 trivial
304.5.e.e 8 19.b odd 2 1 inner
342.5.d.a 8 12.b even 2 1
342.5.d.a 8 228.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 \(S_{5}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{8} + 458T_{3}^{6} + 67449T_{3}^{4} + 3860640T_{3}^{2} + 74649600 \) Copy content Toggle raw display
\( T_{5}^{4} - 9T_{5}^{3} - 2074T_{5}^{2} + 6624T_{5} + 964896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 458 T^{6} + \cdots + 74649600 \) Copy content Toggle raw display
$5$ \( (T^{4} - 9 T^{3} - 2074 T^{2} + \cdots + 964896)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 81 T^{3} - 3827 T^{2} + \cdots + 4446118)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 3 T^{3} - 37690 T^{2} + \cdots + 76301016)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 140586 T^{6} + \cdots + 57\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( (T^{4} - 255 T^{3} - 67555 T^{2} + \cdots + 403423998)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$23$ \( (T^{4} - 198 T^{3} + \cdots + 18350234964)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 3774042 T^{6} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + 2202408 T^{6} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{8} + 6003528 T^{6} + \cdots + 61\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{8} + 7187688 T^{6} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{4} - 4327 T^{3} + \cdots + 407751532960)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1605 T^{3} + \cdots - 98774187816)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 36847098 T^{6} + \cdots + 82\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{8} + 25952346 T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} - 657 T^{3} + \cdots + 195962902247296)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 149621370 T^{6} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{8} + 175804200 T^{6} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{4} - 11699 T^{3} + \cdots - 15\!\cdots\!50)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 113259624 T^{6} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( (T^{4} - 5220 T^{3} + \cdots + 57645106800768)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 282427200 T^{6} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{8} + 436011432 T^{6} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
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