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Properties

Label 2475.4.a.bs
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
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] = [2475,4,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(146.029727264\)
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: \( x^{7} - 2x^{6} - 31x^{5} + 50x^{4} + 272x^{3} - 322x^{2} - 704x + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 2) q^{4} + ( - \beta_{4} + \beta_{2} - 5) q^{7} + (\beta_{5} - \beta_{4} + 2 \beta_{2} + \cdots + 10) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 2) q^{4} + ( - \beta_{4} + \beta_{2} - 5) q^{7} + (\beta_{5} - \beta_{4} + 2 \beta_{2} + \cdots + 10) q^{8}+ \cdots + (18 \beta_{6} + 19 \beta_{5} + \cdots + 393) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 5 q^{2} + 13 q^{4} - 34 q^{7} + 75 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 5 q^{2} + 13 q^{4} - 34 q^{7} + 75 q^{8} - 77 q^{11} - 80 q^{13} - 42 q^{14} - 43 q^{16} + 162 q^{17} + 58 q^{19} - 55 q^{22} + 324 q^{23} + 200 q^{26} + 168 q^{28} - 64 q^{29} - 348 q^{31} - 75 q^{32} + 206 q^{34} - 664 q^{37} + 334 q^{38} + 332 q^{41} - 774 q^{43} - 143 q^{44} - 328 q^{46} + 872 q^{47} - 417 q^{49} - 134 q^{52} + 1628 q^{53} + 1618 q^{56} - 1568 q^{58} + 332 q^{59} + 22 q^{61} - 260 q^{62} + 561 q^{64} - 1524 q^{67} + 2324 q^{68} + 516 q^{71} - 1700 q^{73} - 1628 q^{74} + 2794 q^{76} + 374 q^{77} + 1746 q^{79} - 364 q^{82} + 2344 q^{83} - 1270 q^{86} - 825 q^{88} + 2226 q^{89} + 1072 q^{91} + 4184 q^{92} + 4736 q^{94} - 1048 q^{97} + 3057 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 31x^{5} + 50x^{4} + 272x^{3} - 322x^{2} - 704x + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + 15\nu^{4} - 34\nu^{3} + 80\nu^{2} + 98\nu - 576 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 6\nu^{5} + 77\nu^{4} - 102\nu^{3} - 432\nu^{2} + 166\nu + 288 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} + 6\nu^{5} + 77\nu^{4} - 134\nu^{3} - 400\nu^{2} + 614\nu + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 6\nu^{5} - 39\nu^{4} - 158\nu^{3} + 384\nu^{2} + 814\nu - 768 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{2} + 14\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{3} + 21\beta_{2} + 4\beta _1 + 126 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} - 24\beta_{5} + 27\beta_{4} - 5\beta_{3} + 29\beta_{2} + 234\beta _1 + 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{6} - 14\beta_{5} + 35\beta_{4} - 87\beta_{3} + 419\beta_{2} + 150\beta _1 + 2110 \) 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
4.57884
3.20690
2.41169
0.647712
−1.98683
−2.70507
−4.15324
−3.57884 0 4.80807 0 0 −7.85216 11.4234 0 0
1.2 −2.20690 0 −3.12958 0 0 −1.50972 24.5619 0 0
1.3 −1.41169 0 −6.00714 0 0 15.2844 19.7737 0 0
1.4 0.352288 0 −7.87589 0 0 −19.8486 −5.59288 0 0
1.5 2.98683 0 0.921158 0 0 −6.37827 −21.1433 0 0
1.6 3.70507 0 5.72754 0 0 −30.9104 −8.41961 0 0
1.7 5.15324 0 18.5558 0 0 17.2148 54.3968 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 2475.4.a.bs 7
3.b odd 2 1 825.4.a.ba 7
5.b even 2 1 2475.4.a.bo 7
5.c odd 4 2 495.4.c.d 14
15.d odd 2 1 825.4.a.bd 7
15.e even 4 2 165.4.c.b ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.c.b ✓ 14 15.e even 4 2
495.4.c.d 14 5.c odd 4 2
825.4.a.ba 7 3.b odd 2 1
825.4.a.bd 7 15.d odd 2 1
2475.4.a.bo 7 5.b even 2 1
2475.4.a.bs 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{7} - 5T_{2}^{6} - 22T_{2}^{5} + 100T_{2}^{4} + 157T_{2}^{3} - 475T_{2}^{2} - 492T_{2} + 224 \) Copy content Toggle raw display
\( T_{7}^{7} + 34T_{7}^{6} - 414T_{7}^{5} - 17368T_{7}^{4} + 3864T_{7}^{3} + 2011320T_{7}^{2} + 11055312T_{7} + 12206016 \) Copy content Toggle raw display
\( T_{29}^{7} + 64 T_{29}^{6} - 70264 T_{29}^{5} - 1539392 T_{29}^{4} + 1292528592 T_{29}^{3} + \cdots + 38672535237120 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 5 T^{6} + \cdots + 224 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + 34 T^{6} + \cdots + 12206016 \) Copy content Toggle raw display
$11$ \( (T + 11)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 49306392576 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots + 21646098142208 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 46691778560000 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 636625874944 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 38672535237120 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 197687760429056 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 26\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 30\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots + 39\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 765697934131200 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 76\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 70\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
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