Properties

Label 304.4.a.g
Level $304$
Weight $4$
Character orbit 304.a
Self dual yes
Analytic conductor $17.937$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [304,4,Mod(1,304)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 304.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: \(17.9365806417\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.7057.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 22x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 152)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 2) q^{5} + ( - \beta_{2} - \beta_1 - 2) q^{7} + (3 \beta_{2} - 4 \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} + (\beta_{2} + 2 \beta_1 + 2) q^{5} + ( - \beta_{2} - \beta_1 - 2) q^{7} + (3 \beta_{2} - 4 \beta_1 + 6) q^{9} + (5 \beta_{2} - 2 \beta_1 - 36) q^{11} + (2 \beta_{2} - 9 \beta_1 + 10) q^{13} + (4 \beta_{2} + 2 \beta_1 - 42) q^{15} + (2 \beta_{2} + 8 \beta_1 + 3) q^{17} + 19 q^{19} + ( - 3 \beta_1 + 38) q^{21} + (14 \beta_{2} - 5 \beta_1 - 110) q^{23} + ( - 15 \beta_{2} + 20 \beta_1 + 59) q^{25} + ( - \beta_{2} + 16 \beta_1 - 59) q^{27} + (11 \beta_1 - 46) q^{29} + ( - 6 \beta_{2} + 6 \beta_1 - 138) q^{31} + (18 \beta_{2} + 22 \beta_1 - 116) q^{33} + (9 \beta_{2} - 10 \beta_1 - 114) q^{35} + ( - 18 \beta_{2} - 54 \beta_1 + 40) q^{37} + ( - 50 \beta_{2} + 17 \beta_1 - 38) q^{39} + (14 \beta_{2} - 128) q^{41} + (19 \beta_{2} - 38 \beta_1 - 150) q^{43} + (15 \beta_{2} - 40 \beta_1 - 148) q^{45} + ( - 53 \beta_{2} + 48 \beta_1 + 84) q^{47} + ( - 4 \beta_{2} + 4 \beta_1 - 266) q^{49} + (25 \beta_{2} - 99) q^{51} + ( - 62 \beta_{2} + 67 \beta_1 - 82) q^{53} + ( - 49 \beta_{2} - 112 \beta_1 - 12) q^{55} + ( - 19 \beta_{2} - 19) q^{57} + ( - 13 \beta_{2} + 12 \beta_1 + 67) q^{59} + ( - 45 \beta_{2} - 34 \beta_1 - 188) q^{61} + ( - 23 \beta_{2} + 30 \beta_1 + 28) q^{63} + (54 \beta_{2} - 78 \beta_1 - 530) q^{65} + (57 \beta_{2} + 68 \beta_1 + 63) q^{67} + (62 \beta_{2} + 61 \beta_1 - 318) q^{69} + (18 \beta_{2} - 98 \beta_1 + 312) q^{71} + ( - 104 \beta_{2} - 64 \beta_1 - 175) q^{73} + (51 \beta_{2} - 80 \beta_1 + 341) q^{75} + (31 \beta_{2} + 68 \beta_1 - 34) q^{77} + ( - 8 \beta_{2} + 150 \beta_1 - 148) q^{79} + (44 \beta_{2} + 88 \beta_1 - 135) q^{81} + (168 \beta_{2} + 58 \beta_1 + 204) q^{83} + ( - 55 \beta_{2} + 78 \beta_1 + 646) q^{85} + (90 \beta_{2} - 11 \beta_1 + 2) q^{87} + ( - 30 \beta_{2} - 2 \beta_1 - 442) q^{89} + ( - 53 \beta_{2} + 52 \beta_1 + 241) q^{91} + (174 \beta_{2} - 30 \beta_1 + 306) q^{93} + (19 \beta_{2} + 38 \beta_1 + 38) q^{95} + (226 \beta_{2} - 88 \beta_1 - 258) q^{97} + (33 \beta_{2} + 104 \beta_1 + 424) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} + 7 q^{5} - 7 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} + 7 q^{5} - 7 q^{7} + 21 q^{9} - 103 q^{11} + 32 q^{13} - 122 q^{15} + 11 q^{17} + 57 q^{19} + 114 q^{21} - 316 q^{23} + 162 q^{25} - 178 q^{27} - 138 q^{29} - 420 q^{31} - 330 q^{33} - 333 q^{35} + 102 q^{37} - 164 q^{39} - 370 q^{41} - 431 q^{43} - 429 q^{45} + 199 q^{47} - 802 q^{49} - 272 q^{51} - 308 q^{53} - 85 q^{55} - 76 q^{57} + 188 q^{59} - 609 q^{61} + 61 q^{63} - 1536 q^{65} + 246 q^{67} - 892 q^{69} + 954 q^{71} - 629 q^{73} + 1074 q^{75} - 71 q^{77} - 452 q^{79} - 361 q^{81} + 780 q^{83} + 1883 q^{85} + 96 q^{87} - 1356 q^{89} + 670 q^{91} + 1092 q^{93} + 133 q^{95} - 548 q^{97} + 1305 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 22x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 3\nu - 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - \nu - 14 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{2} + \beta _1 + 29 ) / 2 \) Copy content Toggle raw display

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
−4.86867
4.36181
1.50686
0 −8.28632 0 2.38427 0 −5.83529 0 41.6631 0
1.2 0 −1.33180 0 18.4427 0 −10.3872 0 −25.2263 0
1.3 0 5.61812 0 −13.8269 0 9.22252 0 4.56325 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.4.a.g 3
4.b odd 2 1 152.4.a.c 3
8.b even 2 1 1216.4.a.w 3
8.d odd 2 1 1216.4.a.r 3
12.b even 2 1 1368.4.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.4.a.c 3 4.b odd 2 1
304.4.a.g 3 1.a even 1 1 trivial
1216.4.a.r 3 8.d odd 2 1
1216.4.a.w 3 8.b even 2 1
1368.4.a.d 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 4T_{3}^{2} - 43T_{3} - 62 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 4 T^{2} - 43 T - 62 \) Copy content Toggle raw display
$5$ \( T^{3} - 7 T^{2} - 244 T + 608 \) Copy content Toggle raw display
$7$ \( T^{3} + 7 T^{2} - 89 T - 559 \) Copy content Toggle raw display
$11$ \( T^{3} + 103 T^{2} + 2212 T - 22156 \) Copy content Toggle raw display
$13$ \( T^{3} - 32 T^{2} - 3677 T + 131420 \) Copy content Toggle raw display
$17$ \( T^{3} - 11 T^{2} - 3417 T - 32173 \) Copy content Toggle raw display
$19$ \( (T - 19)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 316 T^{2} + 23147 T - 242336 \) Copy content Toggle raw display
$29$ \( T^{3} + 138 T^{2} + 419 T - 345766 \) Copy content Toggle raw display
$31$ \( T^{3} + 420 T^{2} + 55584 T + 2336256 \) Copy content Toggle raw display
$37$ \( T^{3} - 102 T^{2} + \cdots + 15565400 \) Copy content Toggle raw display
$41$ \( T^{3} + 370 T^{2} + 36160 T + 707456 \) Copy content Toggle raw display
$43$ \( T^{3} + 431 T^{2} - 20508 T - 5420480 \) Copy content Toggle raw display
$47$ \( T^{3} - 199 T^{2} + \cdots + 45306112 \) Copy content Toggle raw display
$53$ \( T^{3} + 308 T^{2} - 340901 T + 6630640 \) Copy content Toggle raw display
$59$ \( T^{3} - 188 T^{2} - 2195 T + 1077242 \) Copy content Toggle raw display
$61$ \( T^{3} + 609 T^{2} + \cdots - 50618548 \) Copy content Toggle raw display
$67$ \( T^{3} - 246 T^{2} + \cdots + 96246632 \) Copy content Toggle raw display
$71$ \( T^{3} - 954 T^{2} + \cdots + 237273448 \) Copy content Toggle raw display
$73$ \( T^{3} + 629 T^{2} + \cdots - 417051529 \) Copy content Toggle raw display
$79$ \( T^{3} + 452 T^{2} + \cdots - 601611824 \) Copy content Toggle raw display
$83$ \( T^{3} - 780 T^{2} + \cdots + 1048786960 \) Copy content Toggle raw display
$89$ \( T^{3} + 1356 T^{2} + \cdots + 71672320 \) Copy content Toggle raw display
$97$ \( T^{3} + 548 T^{2} + \cdots - 2035482752 \) Copy content Toggle raw display
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