Properties

Label 2475.4.a.z
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1957.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (2 \beta_{2} + \beta_1 + 6) q^{4} + ( - \beta_{2} + 3 \beta_1 - 1) q^{7} + (2 \beta_{2} - 3 \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (2 \beta_{2} + \beta_1 + 6) q^{4} + ( - \beta_{2} + 3 \beta_1 - 1) q^{7} + (2 \beta_{2} - 3 \beta_1 - 2) q^{8} - 11 q^{11} + (11 \beta_{2} + 13 \beta_1 + 11) q^{13} + ( - 8 \beta_{2} - 48) q^{14} + ( - 6 \beta_{2} - 7 \beta_1 + 6) q^{16} + (7 \beta_{2} - 5 \beta_1 + 9) q^{17} + ( - 2 \beta_{2} - 8 \beta_1 + 36) q^{19} + 11 \beta_1 q^{22} + ( - 16 \beta_{2} + 80) q^{23} + ( - 4 \beta_{2} - 46 \beta_1 - 116) q^{26} + ( - 8 \beta_{2} + 40 \beta_1 - 40) q^{28} + ( - 26 \beta_1 - 88) q^{29} + ( - 22 \beta_{2} + 34 \beta_1 + 42) q^{31} + ( - 14 \beta_{2} + 37 \beta_1 + 78) q^{32} + (24 \beta_{2} - 18 \beta_1 + 112) q^{34} + ( - 42 \beta_{2} - 66 \beta_1 + 8) q^{37} + (12 \beta_{2} - 24 \beta_1 + 100) q^{38} + ( - 22 \beta_1 + 8) q^{41} + (27 \beta_{2} + 19 \beta_1 + 51) q^{43} + ( - 22 \beta_{2} - 11 \beta_1 - 66) q^{44} + ( - 32 \beta_{2} - 48 \beta_1 - 96) q^{46} + ( - 34 \beta_{2} + 58 \beta_1 - 54) q^{47} + (30 \beta_{2} - 14 \beta_1 - 157) q^{49} + ( - 4 \beta_{2} + 66 \beta_1 + 532) q^{52} + ( - 56 \beta_{2} - 12 \beta_1 - 270) q^{53} + ( - 32 \beta_{2} + 16 \beta_1 - 224) q^{56} + (52 \beta_{2} + 114 \beta_1 + 364) q^{58} + (8 \beta_{2} - 60 \beta_1 - 432) q^{59} + (22 \beta_{2} + 78 \beta_1 + 140) q^{61} + ( - 112 \beta_{2} - 32 \beta_1 - 608) q^{62} + ( - 54 \beta_{2} - 31 \beta_1 - 650) q^{64} + ( - 104 \beta_{2} - 88 \beta_1 - 284) q^{67} + (28 \beta_{2} - 102 \beta_1 + 324) q^{68} + ( - 28 \beta_{2} + 16 \beta_1 - 600) q^{71} + ( - 23 \beta_{2} + 43 \beta_1 - 19) q^{73} + (48 \beta_{2} + 142 \beta_1 + 672) q^{74} + (88 \beta_{2} - 36 \beta_1 + 120) q^{76} + (11 \beta_{2} - 33 \beta_1 + 11) q^{77} + ( - 154 \beta_{2} - 24 \beta_1 - 40) q^{79} + (44 \beta_{2} + 14 \beta_1 + 308) q^{82} + (195 \beta_{2} + 9 \beta_1 + 289) q^{83} + (16 \beta_{2} - 124 \beta_1 - 104) q^{86} + ( - 22 \beta_{2} + 33 \beta_1 + 22) q^{88} + (292 \beta_1 - 182) q^{89} + (38 \beta_{2} + 154 \beta_1 + 162) q^{91} + (160 \beta_{2} + 208 \beta_1 - 160) q^{92} + ( - 184 \beta_{2} + 64 \beta_1 - 1016) q^{94} + ( - 148 \beta_{2} - 200 \beta_1 + 374) q^{97} + (88 \beta_{2} + 111 \beta_1 + 376) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 17 q^{4} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 17 q^{4} - 6 q^{7} - 3 q^{8} - 33 q^{11} + 20 q^{13} - 144 q^{14} + 25 q^{16} + 32 q^{17} + 116 q^{19} - 11 q^{22} + 240 q^{23} - 302 q^{26} - 160 q^{28} - 238 q^{29} + 92 q^{31} + 197 q^{32} + 354 q^{34} + 90 q^{37} + 324 q^{38} + 46 q^{41} + 134 q^{43} - 187 q^{44} - 240 q^{46} - 220 q^{47} - 457 q^{49} + 1530 q^{52} - 798 q^{53} - 688 q^{56} + 978 q^{58} - 1236 q^{59} + 342 q^{61} - 1792 q^{62} - 1919 q^{64} - 764 q^{67} + 1074 q^{68} - 1816 q^{71} - 100 q^{73} + 1874 q^{74} + 396 q^{76} + 66 q^{77} - 96 q^{79} + 910 q^{82} + 858 q^{83} - 188 q^{86} + 33 q^{88} - 838 q^{89} + 332 q^{91} - 688 q^{92} - 3112 q^{94} + 1322 q^{97} + 1017 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 9x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 7 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + \nu + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + \beta _1 + 13 ) / 2 \) 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
2.91150
−3.04096
1.12946
−4.38835 0 11.2577 0 0 11.7304 −14.2958 0 0
1.2 0.793499 0 −7.37036 0 0 2.90793 −12.1964 0 0
1.3 4.59486 0 13.1127 0 0 −20.6383 23.4921 0 0
\(n\): e.g. 2-40 or 990-1000
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.z 3
3.b odd 2 1 825.4.a.p 3
5.b even 2 1 495.4.a.i 3
15.d odd 2 1 165.4.a.g 3
15.e even 4 2 825.4.c.m 6
165.d even 2 1 1815.4.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.g 3 15.d odd 2 1
495.4.a.i 3 5.b even 2 1
825.4.a.p 3 3.b odd 2 1
825.4.c.m 6 15.e even 4 2
1815.4.a.q 3 165.d even 2 1
2475.4.a.z 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{3} - T_{2}^{2} - 20T_{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{3} + 6T_{7}^{2} - 268T_{7} + 704 \) Copy content Toggle raw display
\( T_{29}^{3} + 238T_{29}^{2} + 5136T_{29} - 428416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 6 T^{2} + \cdots + 704 \) Copy content Toggle raw display
$11$ \( (T + 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 20 T^{2} + \cdots + 78104 \) Copy content Toggle raw display
$17$ \( T^{3} - 32 T^{2} + \cdots - 22424 \) Copy content Toggle raw display
$19$ \( T^{3} - 116 T^{2} + \cdots - 80 \) Copy content Toggle raw display
$23$ \( T^{3} - 240 T^{2} + \cdots + 180224 \) Copy content Toggle raw display
$29$ \( T^{3} + 238 T^{2} + \cdots - 428416 \) Copy content Toggle raw display
$31$ \( T^{3} - 92 T^{2} + \cdots + 6769664 \) Copy content Toggle raw display
$37$ \( T^{3} - 90 T^{2} + \cdots + 6364168 \) Copy content Toggle raw display
$41$ \( T^{3} - 46 T^{2} + \cdots + 245888 \) Copy content Toggle raw display
$43$ \( T^{3} - 134 T^{2} + \cdots + 2381360 \) Copy content Toggle raw display
$47$ \( T^{3} + 220 T^{2} + \cdots + 10980224 \) Copy content Toggle raw display
$53$ \( T^{3} + 798 T^{2} + \cdots - 17262968 \) Copy content Toggle raw display
$59$ \( T^{3} + 1236 T^{2} + \cdots + 32923904 \) Copy content Toggle raw display
$61$ \( T^{3} - 342 T^{2} + \cdots - 2655176 \) Copy content Toggle raw display
$67$ \( T^{3} + 764 T^{2} + \cdots - 153685184 \) Copy content Toggle raw display
$71$ \( T^{3} + 1816 T^{2} + \cdots + 198158720 \) Copy content Toggle raw display
$73$ \( T^{3} + 100 T^{2} + \cdots + 5132984 \) Copy content Toggle raw display
$79$ \( T^{3} + 96 T^{2} + \cdots - 167159872 \) Copy content Toggle raw display
$83$ \( T^{3} - 858 T^{2} + \cdots + 542136176 \) Copy content Toggle raw display
$89$ \( T^{3} + 838 T^{2} + \cdots - 693013592 \) Copy content Toggle raw display
$97$ \( T^{3} - 1322 T^{2} + \cdots + 354601256 \) Copy content Toggle raw display
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