Properties

Label 251.2.a.a
Level $251$
Weight $2$
Character orbit 251.a
Self dual yes
Analytic conductor $2.004$
Analytic rank $1$
Dimension $4$
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] = [251,2,Mod(1,251)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(251, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("251.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 251.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: \(2.00424509073\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{2} - \beta_{2} q^{3} - \beta_{3} q^{4} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{3} + 1) q^{8} + ( - \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{2} - \beta_{2} q^{3} - \beta_{3} q^{4} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{3} + 1) q^{8} + ( - \beta_{3} + \beta_{2} - 1) q^{9} + ( - \beta_{3} + \beta_1) q^{10} + (\beta_{3} - \beta_1 - 1) q^{11} + \beta_1 q^{12} + (\beta_{3} - \beta_{2} - 3) q^{13} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{14} - q^{15} + (3 \beta_{3} - 3) q^{16} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{17} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{18} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{19} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{20} + (2 \beta_{3} + \beta_1 - 3) q^{21} + ( - \beta_{3} - \beta_{2} + 2) q^{22} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{2} + 2 \beta_1) q^{24} + (4 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 2) q^{25} + ( - 3 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{26} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{27} + (\beta_{3} + \beta_{2}) q^{28} + (\beta_{3} - 4 \beta_{2} - 2 \beta_1 - 1) q^{29} + ( - \beta_{3} + 1) q^{30} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 - 2) q^{31} + (\beta_{3} + 4) q^{32} + (\beta_{3} + \beta_{2} - 1) q^{33} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{34} + ( - \beta_{2} - \beta_1 + 2) q^{35} + (2 \beta_{3} - \beta_1 + 1) q^{36} + (2 \beta_{2} + 3 \beta_1 - 5) q^{37} + ( - 2 \beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{38} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 + 2) q^{39} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{40} + (2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{41} + ( - 3 \beta_{3} + \beta_{2} + 5) q^{42} + ( - \beta_{3} + \beta_{2} + 7 \beta_1 - 3) q^{43} + (\beta_{2} + \beta_1 - 1) q^{44} + (3 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{45} + (2 \beta_1 - 1) q^{46} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{47} + (3 \beta_{2} - 3 \beta_1) q^{48} + ( - 3 \beta_{3} + \beta_1 - 1) q^{49} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 6) q^{50} + (2 \beta_{3} + 2 \beta_1 - 1) q^{51} + (2 \beta_{3} + \beta_1 - 1) q^{52} + ( - 4 \beta_{3} + \beta_1 + 3) q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{54} + ( - 2 \beta_{3} + \beta_1) q^{55} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 - 1) q^{56} + ( - 4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 5) q^{57} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{58} + ( - 5 \beta_{3} - 2 \beta_{2} + 5) q^{59} + \beta_{3} q^{60} + (\beta_{3} + \beta_1 - 6) q^{61} + ( - 2 \beta_{3} - 5 \beta_{2} + 3 \beta_1 + 3) q^{62} + ( - \beta_{3} + 4) q^{63} + ( - 2 \beta_{3} + 3) q^{64} + (\beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{65} + ( - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{66} + ( - 5 \beta_{2} - 5 \beta_1 + 8) q^{67} + ( - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 1) q^{68} + ( - 2 \beta_{2} - \beta_1 - 2) q^{69} + (2 \beta_{3} - \beta_1 - 2) q^{70} + ( - 3 \beta_{3} - \beta_{2} + 6 \beta_1 - 2) q^{71} + (3 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{72} + ( - 4 \beta_{3} - \beta_{2} - 4 \beta_1 + 3) q^{73} + ( - 5 \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{74} + (\beta_{3} + 4 \beta_{2} - \beta_1 + 1) q^{75} + (3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{76} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{77} + (2 \beta_{3} - 5 \beta_{2} + 4 \beta_1 - 3) q^{78} + (8 \beta_{3} - 3 \beta_{2} - 7 \beta_1 - 6) q^{79} + ( - 3 \beta_{3} + 3 \beta_1) q^{80} + (5 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{81} + (3 \beta_{2} - 2 \beta_1 + 2) q^{82} + (4 \beta_{3} - 2 \beta_{2} - 9 \beta_1 + 1) q^{83} + (\beta_{3} - \beta_{2} - \beta_1 - 2) q^{84} + ( - 6 \beta_{3} + 2 \beta_{2} + 3 \beta_1 - 3) q^{85} + ( - 3 \beta_{3} + 6 \beta_{2} + \beta_1 + 2) q^{86} + ( - 2 \beta_{3} + 5 \beta_{2} + \beta_1 + 6) q^{87} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{88} + (6 \beta_{3} + \beta_1 - 2) q^{89} + (3 \beta_{3} - \beta_{2} - 2 \beta_1) q^{90} + (\beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{91} + (\beta_{3} - 2 \beta_{2} - 4 \beta_1 + 1) q^{92} + (5 \beta_{3} - \beta_{2} + \beta_1 - 8) q^{93} + (4 \beta_{3} + 3 \beta_{2} - \beta_1 - 6) q^{94} + (7 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 2) q^{95} + ( - 4 \beta_{2} - \beta_1) q^{96} + ( - 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 5) q^{97} + ( - \beta_{3} + \beta_{2} - 2) q^{98} + ( - 2 \beta_{3} + 2 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{5} + q^{6} - 3 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{5} + q^{6} - 3 q^{7} - 4 q^{9} - q^{10} - 3 q^{11} + q^{12} - 12 q^{13} - q^{14} - 4 q^{15} - 6 q^{16} + q^{17} - 3 q^{18} - 9 q^{19} + 4 q^{20} - 7 q^{21} + 4 q^{22} + 4 q^{23} - 7 q^{25} + 11 q^{26} + q^{27} + 4 q^{28} - 12 q^{29} + 2 q^{30} - 2 q^{31} + 18 q^{32} - 3 q^{34} + 5 q^{35} + 7 q^{36} - 13 q^{37} + 7 q^{38} + 13 q^{39} + 5 q^{40} + q^{41} + 16 q^{42} - 5 q^{43} - q^{44} + 11 q^{45} - 2 q^{46} + 12 q^{47} + 3 q^{48} - 9 q^{49} + 16 q^{50} + 2 q^{51} + q^{52} + 5 q^{53} + 7 q^{54} - 3 q^{55} + 5 q^{56} + 16 q^{57} + 6 q^{58} + 6 q^{59} + 2 q^{60} - 21 q^{61} + q^{62} + 14 q^{63} + 8 q^{64} + q^{65} + 5 q^{66} + 17 q^{67} + 2 q^{68} - 13 q^{69} - 5 q^{70} - 10 q^{71} + 10 q^{72} - 2 q^{73} + 14 q^{74} + 13 q^{75} + 2 q^{76} + 8 q^{77} - 14 q^{78} - 21 q^{79} - 3 q^{80} - 8 q^{81} + 12 q^{82} - q^{83} - 9 q^{84} - 17 q^{85} + 15 q^{86} + 31 q^{87} - 5 q^{88} + 5 q^{89} + 2 q^{90} - 2 q^{91} - 2 q^{92} - 23 q^{93} - 11 q^{94} + 12 q^{95} - 9 q^{96} + 6 q^{97} - 8 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 3x^{2} + x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 2\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 3\beta_1 \) 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
−1.35567
0.737640
2.09529
−0.477260
−1.61803 −2.19353 0.618034 0.455887 3.54920 2.54920 2.23607 1.81156 −0.737640
1.2 −1.61803 1.19353 0.618034 −0.837853 −1.93117 −2.93117 2.23607 −1.57549 1.35567
1.3 0.618034 −1.29496 −1.61803 0.772223 −0.800331 −1.80033 −2.23607 −1.32307 0.477260
1.4 0.618034 0.294963 −1.61803 −3.39026 0.182297 −0.817703 −2.23607 −2.91300 −2.09529
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(251\) \(1\)

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 251.2.a.a 4
3.b odd 2 1 2259.2.a.f 4
4.b odd 2 1 4016.2.a.c 4
5.b even 2 1 6275.2.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
251.2.a.a 4 1.a even 1 1 trivial
2259.2.a.f 4 3.b odd 2 1
4016.2.a.c 4 4.b odd 2 1
6275.2.a.c 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(251))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} - 2 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} - 2 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} - 5 T^{2} - 19 T - 11 \) Copy content Toggle raw display
$11$ \( T^{4} + 3 T^{3} - 4 T - 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 12 T^{3} + 48 T^{2} + 77 T + 41 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} - 24 T^{2} + 34 T + 31 \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} - 3 T^{2} - 89 T + 101 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} - 13 T^{2} + 34 T + 11 \) Copy content Toggle raw display
$29$ \( T^{4} + 12 T^{3} - T^{2} - 222 T + 311 \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} - 50 T^{2} - 51 T - 11 \) Copy content Toggle raw display
$37$ \( T^{4} + 13 T^{3} + 28 T^{2} + \cdots - 659 \) Copy content Toggle raw display
$41$ \( T^{4} - T^{3} - 35 T^{2} + 127 T - 121 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} - 134 T^{2} + \cdots + 1439 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 34 T^{2} - 13 T + 1 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} - 24 T^{2} + 80 T + 139 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 63 T^{2} + 116 T + 551 \) Copy content Toggle raw display
$61$ \( T^{4} + 21 T^{3} + 157 T^{2} + \cdots + 521 \) Copy content Toggle raw display
$67$ \( T^{4} - 17 T^{3} - 26 T^{2} + \cdots - 4159 \) Copy content Toggle raw display
$71$ \( T^{4} + 10 T^{3} - 74 T^{2} + \cdots + 2389 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} - 130 T^{2} + \cdots + 539 \) Copy content Toggle raw display
$79$ \( T^{4} + 21 T^{3} - 20 T^{2} + \cdots - 13051 \) Copy content Toggle raw display
$83$ \( T^{4} + T^{3} - 210 T^{2} + 48 T + 6269 \) Copy content Toggle raw display
$89$ \( T^{4} - 5 T^{3} - 99 T^{2} + \cdots + 2489 \) Copy content Toggle raw display
$97$ \( T^{4} - 6 T^{3} - 80 T^{2} - 58 T + 319 \) Copy content Toggle raw display
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