Properties

Label 304.3.e.g
Level $304$
Weight $3$
Character orbit 304.e
Analytic conductor $8.283$
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] = [304,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.113");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(8.28340003655\)
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: \( x^{8} + 34x^{6} + 345x^{4} + 1064x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 152)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{4} - 2) q^{5} + ( - \beta_{3} + 1) q^{7} + (\beta_{6} - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{4} - 2) q^{5} + ( - \beta_{3} + 1) q^{7} + (\beta_{6} - \beta_{4}) q^{9} + ( - 2 \beta_{6} + \beta_{4} + 4) q^{11} + (\beta_{7} - 2 \beta_1) q^{13} + (\beta_{2} - 4 \beta_1) q^{15} + (\beta_{6} - 2 \beta_{4} - 3) q^{17} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 2) q^{19} + (\beta_{7} + 6 \beta_{5} + \beta_{2} + 2 \beta_1) q^{21} + ( - 2 \beta_{6} - \beta_{4} - \beta_{3} - 1) q^{23} + ( - 4 \beta_{6} + \beta_{4} + 2 \beta_{3} + 5) q^{25} + (2 \beta_{5} + \beta_{2} + 5 \beta_1) q^{27} + (\beta_{7} - 4 \beta_{5} + \beta_{2} - 6 \beta_1) q^{29} + (2 \beta_{7} - \beta_{2} + 4 \beta_1) q^{31} + ( - 4 \beta_{5} - \beta_{2} + 10 \beta_1) q^{33} + ( - 2 \beta_{6} + \beta_{4} + 4 \beta_{3} + 6) q^{35} + (6 \beta_{5} + 3 \beta_{2} + 2 \beta_1) q^{37} + ( - 2 \beta_{6} - \beta_{4} + 3 \beta_{3} + 13) q^{39} + ( - 2 \beta_{7} - 4 \beta_{5} + \beta_{2} - 8 \beta_1) q^{41} + ( - 5 \beta_{4} + 4 \beta_{3} - 10) q^{43} + ( - 6 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 22) q^{45} + (8 \beta_{6} - 3 \beta_{4} - 6 \beta_{3} - 4) q^{47} + (\beta_{6} + 6 \beta_{4} - 8 \beta_{3} + 6) q^{49} + (2 \beta_{5} + 2 \beta_{2} - 9 \beta_1) q^{51} + (\beta_{7} - 4 \beta_{5} - \beta_{2} + 8 \beta_1) q^{53} + (8 \beta_{6} - 11 \beta_{4} - 2 \beta_{3} - 26) q^{55} + ( - 3 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{57} + ( - 20 \beta_{5} - 3 \beta_{2} - 13 \beta_1) q^{59} + ( - 6 \beta_{6} - 5 \beta_{4} - 6 \beta_{3} - 18) q^{61} + ( - 6 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 2) q^{63} + ( - 2 \beta_{7} + 4 \beta_{5} - 4 \beta_{2} + 2 \beta_1) q^{65} + (2 \beta_{7} + 16 \beta_{5} - 9 \beta_1) q^{67} + (\beta_{7} + 2 \beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{69} + ( - 2 \beta_{7} + 16 \beta_{5} + 16 \beta_1) q^{71} + (13 \beta_{6} + 8 \beta_{4} - 6 \beta_{3} - 21) q^{73} + ( - 2 \beta_{7} - 20 \beta_{5} - 3 \beta_{2} + 13 \beta_1) q^{75} + (10 \beta_{6} - 5 \beta_{4} - 6 \beta_{3} + 8) q^{77} + (16 \beta_{5} - \beta_{2} - 14 \beta_1) q^{79} + (10 \beta_{6} - 6 \beta_{4} + 4 \beta_{3} - 37) q^{81} + (8 \beta_{6} + 14 \beta_{4} - 10 \beta_{3} + 12) q^{83} + ( - 10 \beta_{6} + 5 \beta_{4} + 4 \beta_{3} + 54) q^{85} + ( - 4 \beta_{6} + 11 \beta_{4} + \beta_{3} + 45) q^{87} + ( - 4 \beta_{7} - 5 \beta_{2} - 12 \beta_1) q^{89} + ( - 2 \beta_{7} + 18 \beta_{5} - \beta_{2} + 19 \beta_1) q^{91} + (6 \beta_{6} - 18 \beta_{4} + 4 \beta_{3} - 50) q^{93} + ( - 4 \beta_{7} + 2 \beta_{6} - 18 \beta_{5} + 7 \beta_{4} - 2 \beta_{3} + \cdots - 26) q^{95}+ \cdots + ( - 2 \beta_{6} - 9 \beta_{4} - 6 \beta_{3} - 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 14 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 14 q^{5} + 6 q^{7} + 4 q^{9} + 26 q^{11} - 18 q^{17} - 16 q^{19} - 12 q^{23} + 34 q^{25} + 50 q^{35} + 108 q^{39} - 62 q^{43} + 162 q^{45} - 22 q^{47} + 22 q^{49} - 174 q^{55} + 4 q^{57} - 158 q^{61} + 2 q^{63} - 170 q^{73} + 82 q^{77} - 256 q^{81} + 64 q^{83} + 410 q^{85} + 332 q^{87} - 344 q^{93} - 222 q^{95} - 526 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 34x^{6} + 345x^{4} + 1064x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 30\nu^{5} - 205\nu^{3} + 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 20\nu^{4} - 35\nu^{2} + 216 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 30\nu^{4} + 205\nu^{2} + 64 ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 30\nu^{5} + 245\nu^{3} + 504\nu ) / 80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 30\nu^{4} + 245\nu^{2} + 424 ) / 40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 40\nu^{5} - 495\nu^{3} - 1784\nu ) / 40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} - 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + \beta_{2} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -17\beta_{6} + 21\beta_{4} + 4\beta_{3} + 125 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 58\beta_{5} - 25\beta_{2} + 197\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 305\beta_{6} - 385\beta_{4} - 120\beta_{3} - 1969 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 120\beta_{7} + 1330\beta_{5} + 505\beta_{2} - 3229\beta_1 \) 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
4.27138i
3.20945i
2.27869i
0.512197i
0.512197i
2.27869i
3.20945i
4.27138i
0 4.27138i 0 −7.92033 0 −5.75693 0 −9.24469 0
113.2 0 3.20945i 0 −3.06310 0 12.3151 0 −1.30054 0
113.3 0 2.27869i 0 6.29008 0 1.03740 0 3.80758 0
113.4 0 0.512197i 0 −2.30665 0 −4.59559 0 8.73765 0
113.5 0 0.512197i 0 −2.30665 0 −4.59559 0 8.73765 0
113.6 0 2.27869i 0 6.29008 0 1.03740 0 3.80758 0
113.7 0 3.20945i 0 −3.06310 0 12.3151 0 −1.30054 0
113.8 0 4.27138i 0 −7.92033 0 −5.75693 0 −9.24469 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 304.3.e.g 8
3.b odd 2 1 2736.3.o.p 8
4.b odd 2 1 152.3.e.b 8
8.b even 2 1 1216.3.e.n 8
8.d odd 2 1 1216.3.e.m 8
12.b even 2 1 1368.3.o.b 8
19.b odd 2 1 inner 304.3.e.g 8
57.d even 2 1 2736.3.o.p 8
76.d even 2 1 152.3.e.b 8
152.b even 2 1 1216.3.e.m 8
152.g odd 2 1 1216.3.e.n 8
228.b odd 2 1 1368.3.o.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.3.e.b 8 4.b odd 2 1
152.3.e.b 8 76.d even 2 1
304.3.e.g 8 1.a even 1 1 trivial
304.3.e.g 8 19.b odd 2 1 inner
1216.3.e.m 8 8.d odd 2 1
1216.3.e.m 8 152.b even 2 1
1216.3.e.n 8 8.b even 2 1
1216.3.e.n 8 152.g odd 2 1
1368.3.o.b 8 12.b even 2 1
1368.3.o.b 8 228.b odd 2 1
2736.3.o.p 8 3.b odd 2 1
2736.3.o.p 8 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(304, [\chi])\):

\( T_{3}^{8} + 34T_{3}^{6} + 345T_{3}^{4} + 1064T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{5}^{4} + 7T_{5}^{3} - 34T_{5}^{2} - 256T_{5} - 352 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 34 T^{6} + 345 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( (T^{4} + 7 T^{3} - 34 T^{2} - 256 T - 352)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 3 T^{3} - 99 T^{2} - 221 T + 338)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 13 T^{3} - 174 T^{2} + 2260 T - 5912)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 778 T^{6} + \cdots + 246866944 \) Copy content Toggle raw display
$17$ \( (T^{4} + 9 T^{3} - 195 T^{2} - 253 T + 4814)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 16 T^{7} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} - 499 T^{2} + 504 T + 27836)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 2498 T^{6} + \cdots + 36014930176 \) Copy content Toggle raw display
$31$ \( T^{8} + 5232 T^{6} + \cdots + 2723080830976 \) Copy content Toggle raw display
$37$ \( T^{8} + 8272 T^{6} + \cdots + 5146582257664 \) Copy content Toggle raw display
$41$ \( T^{8} + 7216 T^{6} + \cdots + 5382400000000 \) Copy content Toggle raw display
$43$ \( (T^{4} + 31 T^{3} - 1982 T^{2} + \cdots + 258656)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 11 T^{3} - 6302 T^{2} + \cdots + 2613512)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 5410 T^{6} + \cdots + 13467138304 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 247119406081024 \) Copy content Toggle raw display
$61$ \( (T^{4} + 79 T^{3} - 7306 T^{2} + \cdots + 17033600)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 15538 T^{6} + \cdots + 273018790144 \) Copy content Toggle raw display
$71$ \( T^{8} + 14696 T^{6} + \cdots + 1596725395456 \) Copy content Toggle raw display
$73$ \( (T^{4} + 85 T^{3} - 12111 T^{2} + \cdots + 28567486)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 19312 T^{6} + \cdots + 787748552704 \) Copy content Toggle raw display
$83$ \( (T^{4} - 32 T^{3} - 19300 T^{2} + \cdots - 454016)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 682745962430464 \) Copy content Toggle raw display
$97$ \( T^{8} + 58184 T^{6} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
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