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Properties

Label 2025.4.a.be
Level $2025$
Weight $4$
Character orbit 2025.a
Self dual yes
Analytic conductor $119.479$
Analytic rank $1$
Dimension $12$
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] = [2025,4,Mod(1,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2025.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2025.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: \(119.478867762\)
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: \( x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + \cdots + 73008 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 225)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 4) q^{4} + (\beta_{6} - 1) q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 4) q^{4} + (\beta_{6} - 1) q^{7} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 2) q^{8} + ( - \beta_{6} + \beta_{5} - \beta_1 - 2) q^{11} + ( - \beta_{11} - \beta_{5} - \beta_1 - 1) q^{13} + (\beta_{8} + \beta_{7} - 2 \beta_{6} + \cdots + 7) q^{14}+ \cdots + ( - 11 \beta_{11} - 7 \beta_{10} + \cdots - 590) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8} - 29 q^{11} - 24 q^{13} + 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} + 18 q^{22} - 318 q^{23} + 154 q^{26} - 96 q^{28} - 106 q^{29} + 60 q^{31} - 914 q^{32} - 108 q^{34} + 84 q^{37} - 640 q^{38} + 353 q^{41} + 426 q^{43} - 571 q^{44} + 270 q^{46} - 1210 q^{47} + 666 q^{49} + 75 q^{52} - 448 q^{53} + 570 q^{56} - 594 q^{58} - 482 q^{59} + 402 q^{61} - 2544 q^{62} + 975 q^{64} + 201 q^{67} - 3437 q^{68} + 944 q^{71} + 453 q^{73} - 10 q^{74} - 462 q^{76} - 2652 q^{77} + 258 q^{79} - 873 q^{82} - 3012 q^{83} - 1952 q^{86} + 216 q^{88} + 738 q^{89} - 618 q^{91} - 5232 q^{92} + 63 q^{94} + 318 q^{97} - 7511 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + \cdots + 73008 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 20\nu + 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37591 \nu^{11} + 423747 \nu^{10} - 4004629 \nu^{9} - 25985670 \nu^{8} + 140077854 \nu^{7} + \cdots + 346418736 ) / 158287872 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59961 \nu^{11} + 592499 \nu^{10} + 2246427 \nu^{9} - 35963750 \nu^{8} + 8257662 \nu^{7} + \cdots - 16398173904 ) / 158287872 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 65605 \nu^{11} + 48265 \nu^{10} - 4932735 \nu^{9} - 4652306 \nu^{8} + 134284506 \nu^{7} + \cdots - 8996491632 ) / 158287872 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6097 \nu^{11} + 10547 \nu^{10} + 390691 \nu^{9} - 623630 \nu^{8} - 8238762 \nu^{7} + \cdots - 264990672 ) / 9892992 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 59757 \nu^{11} + 119023 \nu^{10} + 4175607 \nu^{9} - 6183454 \nu^{8} - 108583914 \nu^{7} + \cdots - 4259211024 ) / 79143936 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 196583 \nu^{11} + 1709165 \nu^{10} + 9232965 \nu^{9} - 103544538 \nu^{8} - 97918782 \nu^{7} + \cdots - 38296603440 ) / 158287872 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 232351 \nu^{11} - 1184469 \nu^{10} - 13940461 \nu^{9} + 70745130 \nu^{8} + 281453742 \nu^{7} + \cdots + 5029485744 ) / 158287872 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 86739 \nu^{11} + 184561 \nu^{10} + 5649033 \nu^{9} - 9691522 \nu^{8} - 129243510 \nu^{7} + \cdots + 2385982224 ) / 39571968 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 20\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{5} + \beta_{4} + 2\beta_{3} + 28\beta_{2} + 8\beta _1 + 237 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 3 \beta_{5} + \cdots + 141 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 5 \beta_{11} + 4 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} + 50 \beta_{7} - 12 \beta_{6} - 43 \beta_{5} + \cdots + 5427 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 69 \beta_{11} + 49 \beta_{10} + 90 \beta_{9} - 29 \beta_{8} + 133 \beta_{7} - 291 \beta_{6} + \cdots + 6568 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 318 \beta_{11} + 245 \beta_{10} + 177 \beta_{9} + 345 \beta_{8} + 1839 \beta_{7} - 909 \beta_{6} + \cdots + 134213 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3213 \beta_{11} + 1736 \beta_{10} + 3075 \beta_{9} - 354 \beta_{8} + 5988 \beta_{7} - 12174 \beta_{6} + \cdots + 252055 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 14481 \beta_{11} + 10121 \beta_{10} + 7626 \beta_{9} + 14103 \beta_{8} + 61153 \beta_{7} + \cdots + 3497582 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 126046 \beta_{11} + 56109 \beta_{10} + 96569 \beta_{9} + 9401 \beta_{8} + 230378 \beta_{7} + \cdots + 8807654 \) 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
5.54419
4.78880
3.63978
2.97659
2.14580
1.38827
−0.476035
−1.33965
−1.66434
−3.81662
−4.33545
−4.85133
−5.54419 0 22.7380 0 0 −25.7127 −81.7101 0 0
1.2 −4.78880 0 14.9326 0 0 30.6698 −33.1990 0 0
1.3 −3.63978 0 5.24803 0 0 3.72054 10.0166 0 0
1.4 −2.97659 0 0.860098 0 0 −24.9651 21.2526 0 0
1.5 −2.14580 0 −3.39555 0 0 23.2564 24.4526 0 0
1.6 −1.38827 0 −6.07270 0 0 −26.4332 19.5367 0 0
1.7 0.476035 0 −7.77339 0 0 −12.6809 −7.50869 0 0
1.8 1.33965 0 −6.20534 0 0 22.3290 −19.0302 0 0
1.9 1.66434 0 −5.22997 0 0 10.5672 −22.0192 0 0
1.10 3.81662 0 6.56661 0 0 −9.98804 −5.47071 0 0
1.11 4.33545 0 10.7961 0 0 −13.0015 12.1226 0 0
1.12 4.85133 0 15.5354 0 0 16.2385 36.5569 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

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

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 2025.4.a.be 12
3.b odd 2 1 2025.4.a.bi 12
5.b even 2 1 2025.4.a.bj 12
9.d odd 6 2 225.4.e.e ✓ 24
15.d odd 2 1 2025.4.a.bf 12
45.h odd 6 2 225.4.e.f yes 24
45.l even 12 4 225.4.k.e 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.e.e ✓ 24 9.d odd 6 2
225.4.e.f yes 24 45.h odd 6 2
225.4.k.e 48 45.l even 12 4
2025.4.a.be 12 1.a even 1 1 trivial
2025.4.a.bf 12 15.d odd 2 1
2025.4.a.bi 12 3.b odd 2 1
2025.4.a.bj 12 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2025))\):

\( T_{2}^{12} + 4 T_{2}^{11} - 64 T_{2}^{10} - 241 T_{2}^{9} + 1452 T_{2}^{8} + 5130 T_{2}^{7} + \cdots + 73008 \) Copy content Toggle raw display
\( T_{7}^{12} + 6 T_{7}^{11} - 2373 T_{7}^{10} - 14914 T_{7}^{9} + 2090268 T_{7}^{8} + \cdots + 284111126595504 \) Copy content Toggle raw display
\( T_{11}^{12} + 29 T_{11}^{11} - 8485 T_{11}^{10} - 189035 T_{11}^{9} + 25135263 T_{11}^{8} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 4 T^{11} + \cdots + 73008 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 284111126595504 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 38\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 71\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 18\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 43\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 95\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 69\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 92\!\cdots\!30 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 47\!\cdots\!87 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 32\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 47\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 33\!\cdots\!79 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 15\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 53\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 10\!\cdots\!02 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 68\!\cdots\!39 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 27\!\cdots\!97 \) Copy content Toggle raw display
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