↑

Properties

Label 7581.2.a.bi
Level $7581$
Weight $2$
Character orbit 7581.a
Self dual yes
Analytic conductor $60.535$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7581,2,Mod(1,7581)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7581, 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("7581.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7581 = 3 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7581.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: \(60.5345897723\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 6 x^{10} + 36 x^{9} + 2 x^{8} - 110 x^{7} + 45 x^{6} + 128 x^{5} - 87 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 6 x^{10} + 36 x^{9} + 2 x^{8} - 110 x^{7} + 45 x^{6} + 128 x^{5} - 87 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( - \nu^{11} + 4 \nu^{10} + 6 \nu^{9} - 36 \nu^{8} - 2 \nu^{7} + 110 \nu^{6} - 44 \nu^{5} - 130 \nu^{4} + \cdots + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 36 \nu^{8} + 2 \nu^{7} - 110 \nu^{6} + 45 \nu^{5} + 128 \nu^{4} + \cdots - 11 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 36 \nu^{8} + 2 \nu^{7} - 110 \nu^{6} + 45 \nu^{5} + 128 \nu^{4} + \cdots - 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 6 \nu^{11} - 23 \nu^{10} - 40 \nu^{9} + 210 \nu^{8} + 48 \nu^{7} - 658 \nu^{6} + 160 \nu^{5} + 812 \nu^{4} + \cdots - 31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - 6 \nu^{11} + 24 \nu^{10} + 37 \nu^{9} - 219 \nu^{8} - 21 \nu^{7} + 687 \nu^{6} - 241 \nu^{5} + \cdots + 37 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 8 \nu^{11} - 31 \nu^{10} - 52 \nu^{9} + 282 \nu^{8} + 52 \nu^{7} - 878 \nu^{6} + 250 \nu^{5} + 1069 \nu^{4} + \cdots - 49 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( - 12 \nu^{11} + 45 \nu^{10} + 83 \nu^{9} - 410 \nu^{8} - 126 \nu^{7} + 1279 \nu^{6} - 215 \nu^{5} + \cdots + 50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( - 14 \nu^{11} + 53 \nu^{10} + 96 \nu^{9} - 486 \nu^{8} - 135 \nu^{7} + 1530 \nu^{6} - 304 \nu^{5} + \cdots + 67 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( - 24 \nu^{11} + 90 \nu^{10} + 166 \nu^{9} - 821 \nu^{8} - 249 \nu^{7} + 2565 \nu^{6} - 452 \nu^{5} + \cdots + 105 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{6} - 7\beta_{5} + 5\beta_{4} + \beta_{2} + 8\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} - 2\beta_{6} - 11\beta_{5} + 8\beta_{4} + \beta_{3} + 7\beta_{2} + 28\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{11} + \beta_{9} + 9 \beta_{8} - \beta_{7} - 12 \beta_{6} - 45 \beta_{5} + 29 \beta_{4} + \cdots + 52 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{11} + \beta_{10} + 4 \beta_{9} + 19 \beta_{8} - 4 \beta_{7} - 28 \beta_{6} - 91 \beta_{5} + \cdots + 99 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 20 \beta_{11} + 3 \beta_{10} + 21 \beta_{9} + 61 \beta_{8} - 19 \beta_{7} - 109 \beta_{6} + \cdots + 329 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 69 \beta_{11} + 18 \beta_{10} + 72 \beta_{9} + 134 \beta_{8} - 65 \beta_{7} - 278 \beta_{6} + \cdots + 735 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 250 \beta_{11} + 54 \beta_{10} + 268 \beta_{9} + 376 \beta_{8} - 228 \beta_{7} - 909 \beta_{6} + \cdots + 2169 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 796 \beta_{11} + 214 \beta_{10} + 850 \beta_{9} + 846 \beta_{8} - 720 \beta_{7} - 2426 \beta_{6} + \cdots + 5246 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
2.68629
2.50665
1.72220
1.23246
0.792003
0.482134
0.229751
0.197602
−1.02447
−1.20434
−1.73640
−1.88389
−2.68629 1.00000 5.21614 −0.366506 −2.68629 1.00000 −8.63947 1.00000 0.984540
1.2 −2.50665 1.00000 4.28328 1.07393 −2.50665 1.00000 −5.72337 1.00000 −2.69196
1.3 −1.72220 1.00000 0.965979 2.24042 −1.72220 1.00000 1.78079 1.00000 −3.85845
1.4 −1.23246 1.00000 −0.481030 −2.65861 −1.23246 1.00000 3.05778 1.00000 3.27665
1.5 −0.792003 1.00000 −1.37273 2.97136 −0.792003 1.00000 2.67121 1.00000 −2.35333
1.6 −0.482134 1.00000 −1.76755 −3.12443 −0.482134 1.00000 1.81646 1.00000 1.50639
1.7 −0.229751 1.00000 −1.94721 −2.10066 −0.229751 1.00000 0.906877 1.00000 0.482630
1.8 −0.197602 1.00000 −1.96095 0.770234 −0.197602 1.00000 0.782694 1.00000 −0.152200
1.9 1.02447 1.00000 −0.950465 0.305282 1.02447 1.00000 −3.02266 1.00000 0.312752
1.10 1.20434 1.00000 −0.549574 −0.734023 1.20434 1.00000 −3.07054 1.00000 −0.884011
1.11 1.73640 1.00000 1.01508 −0.591301 1.73640 1.00000 −1.71022 1.00000 −1.02673
1.12 1.88389 1.00000 1.54904 0.214306 1.88389 1.00000 −0.849559 1.00000 0.403728
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(19\) \(1\)

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 7581.2.a.bi ✓ 12
19.b odd 2 1 7581.2.a.bp yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7581.2.a.bi ✓ 12 1.a even 1 1 trivial
7581.2.a.bp yes 12 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7581))\):

\( T_{2}^{12} + 4 T_{2}^{11} - 6 T_{2}^{10} - 36 T_{2}^{9} + 2 T_{2}^{8} + 110 T_{2}^{7} + 45 T_{2}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{12} + 2 T_{5}^{11} - 17 T_{5}^{10} - 30 T_{5}^{9} + 89 T_{5}^{8} + 124 T_{5}^{7} - 161 T_{5}^{6} + \cdots + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 44 T^{10} + \cdots - 64 \) Copy content Toggle raw display
$13$ \( T^{12} + 24 T^{11} + \cdots - 304 \) Copy content Toggle raw display
$17$ \( T^{12} - 2 T^{11} + \cdots + 656 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 4 T^{11} + \cdots - 122819 \) Copy content Toggle raw display
$29$ \( T^{12} + 14 T^{11} + \cdots + 416656 \) Copy content Toggle raw display
$31$ \( T^{12} + 28 T^{11} + \cdots - 1265219 \) Copy content Toggle raw display
$37$ \( T^{12} + 16 T^{11} + \cdots + 32281 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 858365779 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 112339105 \) Copy content Toggle raw display
$47$ \( T^{12} - 2 T^{11} + \cdots + 4894976 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 557410561 \) Copy content Toggle raw display
$59$ \( T^{12} + 52 T^{11} + \cdots + 20945605 \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + \cdots - 46756795 \) Copy content Toggle raw display
$67$ \( T^{12} + 28 T^{11} + \cdots + 10087081 \) Copy content Toggle raw display
$71$ \( T^{12} + 50 T^{11} + \cdots + 2529721 \) Copy content Toggle raw display
$73$ \( T^{12} + 24 T^{11} + \cdots - 15456439 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 378598047920 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 4703170601 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 4176813259 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1180012781 \) Copy content Toggle raw display
show more
show less