Properties

Label 304.4.a.c
Level $304$
Weight $4$
Character orbit 304.a
Self dual yes
Analytic conductor $17.937$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) 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 38)
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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 4) q^{3} + ( - 3 \beta - 3) q^{5} + (4 \beta + 7) q^{7} + (9 \beta + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 4) q^{3} + ( - 3 \beta - 3) q^{5} + (4 \beta + 7) q^{7} + (9 \beta + 7) q^{9} + ( - \beta + 9) q^{11} + (13 \beta + 2) q^{13} + (18 \beta + 66) q^{15} + ( - 2 \beta - 39) q^{17} - 19 q^{19} + ( - 27 \beta - 100) q^{21} + ( - 13 \beta - 30) q^{23} + (27 \beta + 46) q^{25} + ( - 25 \beta - 82) q^{27} + ( - 21 \beta + 12) q^{29} + (44 \beta - 128) q^{31} + ( - 4 \beta - 18) q^{33} + ( - 45 \beta - 237) q^{35} + ( - 28 \beta + 110) q^{37} + ( - 67 \beta - 242) q^{39} + (10 \beta - 30) q^{41} + ( - 7 \beta - 335) q^{43} + ( - 75 \beta - 507) q^{45} + (71 \beta + 159) q^{47} + (72 \beta - 6) q^{49} + (49 \beta + 192) q^{51} + (17 \beta - 618) q^{53} + ( - 21 \beta + 27) q^{55} + (19 \beta + 76) q^{57} + ( - 25 \beta + 156) q^{59} + (111 \beta + 101) q^{61} + (127 \beta + 697) q^{63} + ( - 84 \beta - 708) q^{65} + (77 \beta - 650) q^{67} + (95 \beta + 354) q^{69} + ( - 116 \beta - 42) q^{71} + ( - 184 \beta + 281) q^{73} + ( - 181 \beta - 670) q^{75} + (25 \beta - 9) q^{77} + (58 \beta - 704) q^{79} + ( - 36 \beta + 589) q^{81} + ( - 194 \beta + 432) q^{83} + (129 \beta + 225) q^{85} + (93 \beta + 330) q^{87} + ( - 188 \beta - 24) q^{89} + (151 \beta + 950) q^{91} + ( - 92 \beta - 280) q^{93} + (57 \beta + 57) q^{95} + (102 \beta + 596) q^{97} + (65 \beta - 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} - 9 q^{5} + 18 q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{3} - 9 q^{5} + 18 q^{7} + 23 q^{9} + 17 q^{11} + 17 q^{13} + 150 q^{15} - 80 q^{17} - 38 q^{19} - 227 q^{21} - 73 q^{23} + 119 q^{25} - 189 q^{27} + 3 q^{29} - 212 q^{31} - 40 q^{33} - 519 q^{35} + 192 q^{37} - 551 q^{39} - 50 q^{41} - 677 q^{43} - 1089 q^{45} + 389 q^{47} + 60 q^{49} + 433 q^{51} - 1219 q^{53} + 33 q^{55} + 171 q^{57} + 287 q^{59} + 313 q^{61} + 1521 q^{63} - 1500 q^{65} - 1223 q^{67} + 803 q^{69} - 200 q^{71} + 378 q^{73} - 1521 q^{75} + 7 q^{77} - 1350 q^{79} + 1142 q^{81} + 670 q^{83} + 579 q^{85} + 753 q^{87} - 236 q^{89} + 2051 q^{91} - 652 q^{93} + 171 q^{95} + 1294 q^{97} - 133 q^{99}+O(q^{100}) \) 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
4.77200
−3.77200
0 −8.77200 0 −17.3160 0 26.0880 0 49.9480 0
1.2 0 −0.227998 0 8.31601 0 −8.08801 0 −26.9480 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.c 2
4.b odd 2 1 38.4.a.c 2
8.b even 2 1 1216.4.a.p 2
8.d odd 2 1 1216.4.a.g 2
12.b even 2 1 342.4.a.h 2
20.d odd 2 1 950.4.a.e 2
20.e even 4 2 950.4.b.i 4
28.d even 2 1 1862.4.a.e 2
76.d even 2 1 722.4.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.c 2 4.b odd 2 1
304.4.a.c 2 1.a even 1 1 trivial
342.4.a.h 2 12.b even 2 1
722.4.a.f 2 76.d even 2 1
950.4.a.e 2 20.d odd 2 1
950.4.b.i 4 20.e even 4 2
1216.4.a.g 2 8.d odd 2 1
1216.4.a.p 2 8.b even 2 1
1862.4.a.e 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 9T_{3} + 2 \) 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
$3$ \( T^{2} + 9T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 9T - 144 \) Copy content Toggle raw display
$7$ \( T^{2} - 18T - 211 \) Copy content Toggle raw display
$11$ \( T^{2} - 17T + 54 \) Copy content Toggle raw display
$13$ \( T^{2} - 17T - 3012 \) Copy content Toggle raw display
$17$ \( T^{2} + 80T + 1527 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 73T - 1752 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 8046 \) Copy content Toggle raw display
$31$ \( T^{2} + 212T - 24096 \) Copy content Toggle raw display
$37$ \( T^{2} - 192T - 5092 \) Copy content Toggle raw display
$41$ \( T^{2} + 50T - 1200 \) Copy content Toggle raw display
$43$ \( T^{2} + 677T + 113688 \) Copy content Toggle raw display
$47$ \( T^{2} - 389T - 54168 \) Copy content Toggle raw display
$53$ \( T^{2} + 1219 T + 366216 \) Copy content Toggle raw display
$59$ \( T^{2} - 287T + 9186 \) Copy content Toggle raw display
$61$ \( T^{2} - 313T - 200366 \) Copy content Toggle raw display
$67$ \( T^{2} + 1223 T + 265728 \) Copy content Toggle raw display
$71$ \( T^{2} + 200T - 235572 \) Copy content Toggle raw display
$73$ \( T^{2} - 378T - 582151 \) Copy content Toggle raw display
$79$ \( T^{2} + 1350 T + 394232 \) Copy content Toggle raw display
$83$ \( T^{2} - 670T - 574632 \) Copy content Toggle raw display
$89$ \( T^{2} + 236T - 631104 \) Copy content Toggle raw display
$97$ \( T^{2} - 1294 T + 228736 \) Copy content Toggle raw display
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