Properties

Label 425.4.a.j
Level $425$
Weight $4$
Character orbit 425.a
Self dual yes
Analytic conductor $25.076$
Analytic rank $1$
Dimension $6$
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] = [425,4,Mod(1,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, 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("425.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.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: \(25.0758117524\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 30x^{4} + 30x^{3} + 213x^{2} - 109x - 276 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{5} - \beta_{3} - 4) q^{6} + ( - \beta_{5} - 3 \beta_{3} - \beta_{2} + \cdots - 1) q^{7}+ \cdots + (\beta_{5} - \beta_{4} + 3 \beta_{3} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{5} - \beta_{3} - 4) q^{6} + ( - \beta_{5} - 3 \beta_{3} - \beta_{2} + \cdots - 1) q^{7}+ \cdots + ( - 47 \beta_{5} + 17 \beta_{4} + \cdots - 170) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 6 q^{3} + 13 q^{4} - 23 q^{6} - 4 q^{7} + 15 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} + 6 q^{3} + 13 q^{4} - 23 q^{6} - 4 q^{7} + 15 q^{8} - 12 q^{9} - 20 q^{11} - 51 q^{12} - 42 q^{13} - 93 q^{14} - 131 q^{16} - 102 q^{17} - 94 q^{18} - 70 q^{19} - 282 q^{21} + 210 q^{22} + 92 q^{23} + 87 q^{24} - 131 q^{26} + 378 q^{27} - 275 q^{28} - 316 q^{29} - 758 q^{31} + 359 q^{32} - 476 q^{33} + 17 q^{34} - 758 q^{36} - 76 q^{37} - 930 q^{38} - 292 q^{39} - 512 q^{41} + 977 q^{42} - 70 q^{43} - 284 q^{44} - 1162 q^{46} + 448 q^{47} - 273 q^{48} - 84 q^{49} - 102 q^{51} + 1059 q^{52} - 734 q^{53} - 11 q^{54} - 1111 q^{56} + 568 q^{57} - 1178 q^{58} - 238 q^{59} - 1188 q^{61} + 1535 q^{62} - 850 q^{63} - 915 q^{64} - 884 q^{66} + 768 q^{67} - 221 q^{68} - 1736 q^{69} - 1276 q^{71} + 870 q^{72} - 84 q^{73} + 946 q^{74} - 2178 q^{76} - 208 q^{77} - 1417 q^{78} - 3066 q^{79} + 174 q^{81} + 1460 q^{82} - 92 q^{83} + 1255 q^{84} - 740 q^{86} - 38 q^{87} - 850 q^{88} - 1760 q^{89} - 1924 q^{91} + 1096 q^{92} - 774 q^{93} - 1022 q^{94} - 1647 q^{96} + 1164 q^{97} - 794 q^{98} - 950 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 30x^{4} + 30x^{3} + 213x^{2} - 109x - 276 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + \nu^{4} - 23\nu^{3} - 11\nu^{2} + 96\nu + 28 ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{5} + 7\nu^{4} - 31\nu^{3} - 107\nu^{2} - 3\nu + 156 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 6\nu^{4} - 18\nu^{3} - 106\nu^{2} + 51\nu + 208 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 15\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{5} - \beta_{4} - \beta_{3} + 20\beta_{2} - 6\beta _1 + 156 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -26\beta_{5} + 24\beta_{4} - 12\beta_{3} - 32\beta_{2} + 255\beta _1 - 120 \) 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
3.95575
3.74408
1.52037
−1.03585
−2.55730
−4.62705
−3.95575 3.53947 7.64793 0 −14.0012 −13.8166 1.39269 −14.4722 0
1.2 −3.74408 −3.16721 6.01816 0 11.8583 26.6627 7.42016 −16.9688 0
1.3 −1.52037 9.11646 −5.68846 0 −13.8604 −16.3642 20.8116 56.1099 0
1.4 1.03585 −4.77225 −6.92701 0 −4.94335 16.4564 −15.4622 −4.22566 0
1.5 2.55730 3.86147 −1.46020 0 9.87494 5.49760 −24.1926 −12.0891 0
1.6 4.62705 −2.57794 13.4096 0 −11.9282 −22.4360 25.0303 −20.3542 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(17\) \( +1 \)

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 425.4.a.j 6
5.b even 2 1 425.4.a.k yes 6
5.c odd 4 2 425.4.b.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.4.a.j 6 1.a even 1 1 trivial
425.4.a.k yes 6 5.b even 2 1
425.4.b.j 12 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(425))\):

\( T_{2}^{6} + T_{2}^{5} - 30T_{2}^{4} - 30T_{2}^{3} + 213T_{2}^{2} + 109T_{2} - 276 \) Copy content Toggle raw display
\( T_{3}^{6} - 6T_{3}^{5} - 57T_{3}^{4} + 180T_{3}^{3} + 933T_{3}^{2} - 1272T_{3} - 4855 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + \cdots - 276 \) Copy content Toggle raw display
$3$ \( T^{6} - 6 T^{5} + \cdots - 4855 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 4 T^{5} + \cdots - 12236443 \) Copy content Toggle raw display
$11$ \( T^{6} + 20 T^{5} + \cdots - 98280800 \) Copy content Toggle raw display
$13$ \( T^{6} + 42 T^{5} + \cdots + 819022489 \) Copy content Toggle raw display
$17$ \( (T + 17)^{6} \) Copy content Toggle raw display
$19$ \( T^{6} + 70 T^{5} + \cdots - 977808960 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots - 14481037920 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 2867744451360 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots - 4319982391195 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots - 42410952261024 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 15523734022400 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 77226368798496 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 2459602413120 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 16763419208545 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots - 132455594307936 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 5378930561600 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 57\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 61\!\cdots\!07 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
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