Properties

Label 304.4.a.f
Level $304$
Weight $4$
Character orbit 304.a
Self dual yes
Analytic conductor $17.937$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 76)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{3} - 5 \beta q^{5} + ( - 4 \beta + 17) q^{7} + ( - 5 \beta - 10) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 3) q^{3} - 5 \beta q^{5} + ( - 4 \beta + 17) q^{7} + ( - 5 \beta - 10) q^{9} + ( - 5 \beta + 38) q^{11} + (11 \beta - 23) q^{13} + ( - 10 \beta + 40) q^{15} + ( - 6 \beta + 3) q^{17} + 19 q^{19} + ( - 25 \beta + 83) q^{21} + (59 \beta - 27) q^{23} + (25 \beta + 75) q^{25} + (27 \beta - 71) q^{27} + (65 \beta + 45) q^{29} + ( - 80 \beta + 84) q^{31} + ( - 48 \beta + 154) q^{33} + ( - 65 \beta + 160) q^{35} + (8 \beta + 186) q^{37} + (45 \beta - 157) q^{39} + ( - 30 \beta - 56) q^{41} + (117 \beta - 136) q^{43} + (75 \beta + 200) q^{45} + (23 \beta + 216) q^{47} + ( - 120 \beta + 74) q^{49} + ( - 15 \beta + 57) q^{51} + (123 \beta - 199) q^{53} + ( - 165 \beta + 200) q^{55} + ( - 19 \beta + 57) q^{57} + (35 \beta + 419) q^{59} + ( - 175 \beta + 310) q^{61} + ( - 25 \beta - 10) q^{63} + (60 \beta - 440) q^{65} + (61 \beta - 353) q^{67} + (145 \beta - 553) q^{69} + (20 \beta + 846) q^{71} + ( - 64 \beta - 463) q^{73} + ( - 25 \beta + 25) q^{75} + ( - 217 \beta + 806) q^{77} + ( - 10 \beta + 642) q^{79} + (260 \beta - 159) q^{81} + ( - 114 \beta + 102) q^{83} + (15 \beta + 240) q^{85} + (85 \beta - 385) q^{87} + (80 \beta - 484) q^{89} + (235 \beta - 743) q^{91} + ( - 244 \beta + 892) q^{93} - 95 \beta q^{95} + (458 \beta + 126) q^{97} + ( - 115 \beta - 180) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} - 5 q^{5} + 30 q^{7} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} - 5 q^{5} + 30 q^{7} - 25 q^{9} + 71 q^{11} - 35 q^{13} + 70 q^{15} + 38 q^{19} + 141 q^{21} + 5 q^{23} + 175 q^{25} - 115 q^{27} + 155 q^{29} + 88 q^{31} + 260 q^{33} + 255 q^{35} + 380 q^{37} - 269 q^{39} - 142 q^{41} - 155 q^{43} + 475 q^{45} + 455 q^{47} + 28 q^{49} + 99 q^{51} - 275 q^{53} + 235 q^{55} + 95 q^{57} + 873 q^{59} + 445 q^{61} - 45 q^{63} - 820 q^{65} - 645 q^{67} - 961 q^{69} + 1712 q^{71} - 990 q^{73} + 25 q^{75} + 1395 q^{77} + 1274 q^{79} - 58 q^{81} + 90 q^{83} + 495 q^{85} - 685 q^{87} - 888 q^{89} - 1251 q^{91} + 1540 q^{93} - 95 q^{95} + 710 q^{97} - 475 q^{99}+O(q^{100}) \) 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
3.37228
−2.37228
0 −0.372281 0 −16.8614 0 3.51087 0 −26.8614 0
1.2 0 5.37228 0 11.8614 0 26.4891 0 1.86141 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.f 2
4.b odd 2 1 76.4.a.a 2
8.b even 2 1 1216.4.a.h 2
8.d odd 2 1 1216.4.a.o 2
12.b even 2 1 684.4.a.g 2
20.d odd 2 1 1900.4.a.b 2
20.e even 4 2 1900.4.c.b 4
76.d even 2 1 1444.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.a 2 4.b odd 2 1
304.4.a.f 2 1.a even 1 1 trivial
684.4.a.g 2 12.b even 2 1
1216.4.a.h 2 8.b even 2 1
1216.4.a.o 2 8.d odd 2 1
1444.4.a.d 2 76.d even 2 1
1900.4.a.b 2 20.d odd 2 1
1900.4.c.b 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 5T_{3} - 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(304))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T - 200 \) Copy content Toggle raw display
$7$ \( T^{2} - 30T + 93 \) Copy content Toggle raw display
$11$ \( T^{2} - 71T + 1054 \) Copy content Toggle raw display
$13$ \( T^{2} + 35T - 692 \) Copy content Toggle raw display
$17$ \( T^{2} - 297 \) Copy content Toggle raw display
$19$ \( (T - 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 5T - 28712 \) Copy content Toggle raw display
$29$ \( T^{2} - 155T - 28850 \) Copy content Toggle raw display
$31$ \( T^{2} - 88T - 50864 \) Copy content Toggle raw display
$37$ \( T^{2} - 380T + 35572 \) Copy content Toggle raw display
$41$ \( T^{2} + 142T - 2384 \) Copy content Toggle raw display
$43$ \( T^{2} + 155T - 106928 \) Copy content Toggle raw display
$47$ \( T^{2} - 455T + 47392 \) Copy content Toggle raw display
$53$ \( T^{2} + 275T - 105908 \) Copy content Toggle raw display
$59$ \( T^{2} - 873T + 180426 \) Copy content Toggle raw display
$61$ \( T^{2} - 445T - 203150 \) Copy content Toggle raw display
$67$ \( T^{2} + 645T + 73308 \) Copy content Toggle raw display
$71$ \( T^{2} - 1712 T + 729436 \) Copy content Toggle raw display
$73$ \( T^{2} + 990T + 211233 \) Copy content Toggle raw display
$79$ \( T^{2} - 1274 T + 404944 \) Copy content Toggle raw display
$83$ \( T^{2} - 90T - 105192 \) Copy content Toggle raw display
$89$ \( T^{2} + 888T + 144336 \) Copy content Toggle raw display
$97$ \( T^{2} - 710 T - 1604528 \) Copy content Toggle raw display
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