Properties

Label 5915.2.a.bb
Level $5915$
Weight $2$
Character orbit 5915.a
Self dual yes
Analytic conductor $47.232$
Analytic rank $0$
Dimension $7$
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] = [5915,2,Mod(1,5915)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5915, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5915.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5915 = 5 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5915.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: \(47.2315127956\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 12x^{5} + 11x^{4} + 37x^{3} - 25x^{2} - 27x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 455)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{6} - q^{7} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{6} - q^{7} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} + 3) q^{9} + \beta_1 q^{10} + ( - \beta_{5} + \beta_{2} + 1) q^{11} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{12} - \beta_1 q^{14} - \beta_{3} q^{15} + (\beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 + 4) q^{16} + (\beta_{5} - \beta_{4} + 1) q^{17} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 1) q^{18} + ( - \beta_{6} + \beta_{5} + \beta_{3} - 1) q^{19} + (\beta_{2} + 2) q^{20} + \beta_{3} q^{21} + ( - \beta_{6} + \beta_{4} + 3 \beta_1 - 1) q^{22} + (\beta_{6} + \beta_{5} + \beta_{3} + 1) q^{23} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 7) q^{24} + q^{25} + ( - \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{27} + ( - \beta_{2} - 2) q^{28} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{29} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{30} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{31} + (\beta_{6} + 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 3) q^{32} + ( - 2 \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{33} + (\beta_{6} + 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{34} - q^{35} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - \beta_1 + 7) q^{36} + (\beta_{6} + \beta_{5} - 3 \beta_1 + 1) q^{37} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 - 1) q^{38} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{40} + (\beta_{6} + \beta_{2} + 1) q^{41} + (\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{42} + (\beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{43} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 8) q^{44} + ( - \beta_{5} - \beta_{3} + 3) q^{45} + (\beta_{6} + 3 \beta_{5} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 3) q^{46} + ( - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{47} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 8 \beta_1 + 10) q^{48} + q^{49} + \beta_1 q^{50} + (2 \beta_{6} - \beta_{3} - \beta_{2} - 5 \beta_1 + 4) q^{51} + (\beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 1) q^{53} + ( - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 6 \beta_1 + 6) q^{54} + ( - \beta_{5} + \beta_{2} + 1) q^{55} + ( - \beta_{4} - \beta_{3} - 2 \beta_1) q^{56} + (2 \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 2) q^{57} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 + 8) q^{58} + (2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_1 + 2) q^{59} + ( - \beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{60} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{61} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 5) q^{62} + (\beta_{5} + \beta_{3} - 3) q^{63} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 7 \beta_1 + 10) q^{64} + ( - 5 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 5 \beta_1) q^{66} + (\beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{67} + (2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 4) q^{68} + (\beta_{5} - 2 \beta_{4} - 3 \beta_{2} - \beta_1 - 4) q^{69} - \beta_1 q^{70} + (\beta_{6} - \beta_{4} + 2 \beta_{3} + \beta_1 + 6) q^{71} + ( - 4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 7 \beta_1 - 6) q^{72} + ( - \beta_{6} + 3 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{73} + (\beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 9) q^{74} - \beta_{3} q^{75} + (\beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 - 5) q^{76} + (\beta_{5} - \beta_{2} - 1) q^{77} + ( - \beta_{5} - \beta_{3} + 3 \beta_{2} + \beta_1 + 4) q^{79} + (\beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 + 4) q^{80} + ( - \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{81} + (2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{82} + ( - \beta_{5} - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 7) q^{83} + (\beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1 - 3) q^{84} + (\beta_{5} - \beta_{4} + 1) q^{85} + (\beta_{6} + \beta_{5} + 3 \beta_{3} + 3 \beta_{2} - \beta_1 + 9) q^{86} + (2 \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 4) q^{87} + (2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - 3 \beta_{2} + 6 \beta_1 - 2) q^{88} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1 + 3) q^{89} + ( - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 1) q^{90} + (\beta_{6} + 3 \beta_{5} - \beta_{4} + 8 \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 3) q^{92} + ( - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_1 + 5) q^{93} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 3 \beta_1 - 5) q^{94} + ( - \beta_{6} + \beta_{5} + \beta_{3} - 1) q^{95} + (\beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} - 8 \beta_{2} + 7 \beta_1 - 19) q^{96} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2) q^{97} + \beta_1 q^{98} + ( - \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 11 q^{4} + 7 q^{5} + 2 q^{6} - 7 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 11 q^{4} + 7 q^{5} + 2 q^{6} - 7 q^{7} + 21 q^{9} + q^{10} + 4 q^{11} + 20 q^{12} - q^{14} + 23 q^{16} + 9 q^{17} + q^{18} - 4 q^{19} + 11 q^{20} - 3 q^{22} + 4 q^{23} - 37 q^{24} + 7 q^{25} + 21 q^{27} - 11 q^{28} + 2 q^{30} - 14 q^{32} - q^{33} + 8 q^{34} - 7 q^{35} + 43 q^{36} + q^{37} - 10 q^{38} + q^{41} - 2 q^{42} - 15 q^{43} + 52 q^{44} + 21 q^{45} + 14 q^{46} + 3 q^{47} + 58 q^{48} + 7 q^{49} + q^{50} + 20 q^{51} + 9 q^{53} + 41 q^{54} + 4 q^{55} - 26 q^{57} + 54 q^{58} + 7 q^{59} + 20 q^{60} + 10 q^{61} - 28 q^{62} - 21 q^{63} + 58 q^{64} + 2 q^{66} - 6 q^{67} + 15 q^{68} - 16 q^{69} - q^{70} + 42 q^{71} - 34 q^{72} - 5 q^{73} - 59 q^{74} - 35 q^{76} - 4 q^{77} + 20 q^{79} + 23 q^{80} - q^{81} + 12 q^{82} - 38 q^{83} - 20 q^{84} + 9 q^{85} + 50 q^{86} + 25 q^{87} - 7 q^{88} + 24 q^{89} + q^{90} + 15 q^{92} + 42 q^{93} - 28 q^{94} - 4 q^{95} - 111 q^{96} - 7 q^{97} + q^{98} + 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 12x^{5} + 11x^{4} + 37x^{3} - 25x^{2} - 27x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} - 12\nu^{4} + 37\nu^{2} + 5\nu - 22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 12\nu^{4} + \nu^{3} - 37\nu^{2} - 11\nu + 22 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{6} + 13\nu^{4} + \nu^{3} - 45\nu^{2} - 10\nu + 30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\nu^{6} + \nu^{5} + 24\nu^{4} - 8\nu^{3} - 72\nu^{2} - 2\nu + 39 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{4} + 8\beta_{2} - \beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + 8\beta_{4} + 10\beta_{3} - 2\beta_{2} + 40\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{5} - 12\beta_{4} + \beta_{3} + 59\beta_{2} - 17\beta _1 + 162 \) 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
−2.79387
−1.60096
−1.09774
0.721017
0.837791
2.33769
2.59607
−2.79387 2.71207 5.80572 1.00000 −7.57717 −1.00000 −10.6327 4.35531 −2.79387
1.2 −1.60096 −2.83441 0.563086 1.00000 4.53778 −1.00000 2.30045 5.03387 −1.60096
1.3 −1.09774 −1.42206 −0.794971 1.00000 1.56105 −1.00000 3.06815 −0.977748 −1.09774
1.4 0.721017 2.26250 −1.48013 1.00000 1.63130 −1.00000 −2.50924 2.11892 0.721017
1.5 0.837791 −2.59295 −1.29811 1.00000 −2.17235 −1.00000 −2.76312 3.72341 0.837791
1.6 2.33769 3.28143 3.46480 1.00000 7.67097 −1.00000 3.42426 7.76778 2.33769
1.7 2.59607 −1.40658 4.73960 1.00000 −3.65158 −1.00000 7.11220 −1.02153 2.59607
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

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 5915.2.a.bb 7
13.b even 2 1 5915.2.a.x 7
13.e even 6 2 455.2.i.h 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
455.2.i.h 14 13.e even 6 2
5915.2.a.x 7 13.b even 2 1
5915.2.a.bb 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):

\( T_{2}^{7} - T_{2}^{6} - 12T_{2}^{5} + 11T_{2}^{4} + 37T_{2}^{3} - 25T_{2}^{2} - 27T_{2} + 18 \) Copy content Toggle raw display
\( T_{3}^{7} - 21T_{3}^{5} - 7T_{3}^{4} + 142T_{3}^{3} + 92T_{3}^{2} - 307T_{3} - 296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} - 12 T^{5} + 11 T^{4} + \cdots + 18 \) Copy content Toggle raw display
$3$ \( T^{7} - 21 T^{5} - 7 T^{4} + 142 T^{3} + \cdots - 296 \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( (T + 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 4 T^{6} - 42 T^{5} + \cdots - 1692 \) Copy content Toggle raw display
$13$ \( T^{7} \) Copy content Toggle raw display
$17$ \( T^{7} - 9 T^{6} - 28 T^{5} + \cdots + 1881 \) Copy content Toggle raw display
$19$ \( T^{7} + 4 T^{6} - 79 T^{5} + \cdots + 1676 \) Copy content Toggle raw display
$23$ \( T^{7} - 4 T^{6} - 89 T^{5} + \cdots - 2736 \) Copy content Toggle raw display
$29$ \( T^{7} - 137 T^{5} - 75 T^{4} + \cdots - 26847 \) Copy content Toggle raw display
$31$ \( T^{7} - 111 T^{5} + 317 T^{4} + \cdots - 8408 \) Copy content Toggle raw display
$37$ \( T^{7} - T^{6} - 166 T^{5} + 541 T^{4} + \cdots - 7696 \) Copy content Toggle raw display
$41$ \( T^{7} - T^{6} - 106 T^{5} + 238 T^{4} + \cdots - 6336 \) Copy content Toggle raw display
$43$ \( T^{7} + 15 T^{6} - 7 T^{5} + \cdots - 11504 \) Copy content Toggle raw display
$47$ \( T^{7} - 3 T^{6} - 61 T^{5} + 134 T^{4} + \cdots + 333 \) Copy content Toggle raw display
$53$ \( T^{7} - 9 T^{6} - 103 T^{5} + \cdots - 23328 \) Copy content Toggle raw display
$59$ \( T^{7} - 7 T^{6} - 255 T^{5} + \cdots + 2195568 \) Copy content Toggle raw display
$61$ \( T^{7} - 10 T^{6} - 45 T^{5} + \cdots - 43264 \) Copy content Toggle raw display
$67$ \( T^{7} + 6 T^{6} - 285 T^{5} + \cdots - 1708936 \) Copy content Toggle raw display
$71$ \( T^{7} - 42 T^{6} + 589 T^{5} + \cdots + 61407 \) Copy content Toggle raw display
$73$ \( T^{7} + 5 T^{6} - 386 T^{5} + \cdots - 6724649 \) Copy content Toggle raw display
$79$ \( T^{7} - 20 T^{6} - 152 T^{5} + \cdots + 77296 \) Copy content Toggle raw display
$83$ \( T^{7} + 38 T^{6} + 260 T^{5} + \cdots - 8710839 \) Copy content Toggle raw display
$89$ \( T^{7} - 24 T^{6} - 4 T^{5} + \cdots - 27072 \) Copy content Toggle raw display
$97$ \( T^{7} + 7 T^{6} - 300 T^{5} + \cdots + 146546 \) Copy content Toggle raw display
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