Properties

Label 4.25.b.b
Level $4$
Weight $25$
Character orbit 4.b
Analytic conductor $14.599$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4,25,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(14.5986860903\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} + 1157886 x^{8} - 1182314620 x^{7} + 1715110302918 x^{6} + \cdots + 10\!\cdots\!86 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{90}\cdot 3^{8}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 621) q^{2} + (\beta_{2} - 15 \beta_1 - 3) q^{3} + ( - \beta_{3} - 4 \beta_{2} - 761 \beta_1 - 3319485) q^{4} + (\beta_{5} + \beta_{3} - \beta_{2} - 1492 \beta_1 + 5675512) q^{5} + ( - \beta_{8} - \beta_{7} - \beta_{4} + 10 \beta_{3} + 359 \beta_{2} + \cdots + 242821904) q^{6}+ \cdots + (91 \beta_{8} - 92 \beta_{6} - 160 \beta_{5} + 586 \beta_{4} + \cdots - 93909906177) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 621) q^{2} + (\beta_{2} - 15 \beta_1 - 3) q^{3} + ( - \beta_{3} - 4 \beta_{2} - 761 \beta_1 - 3319485) q^{4} + (\beta_{5} + \beta_{3} - \beta_{2} - 1492 \beta_1 + 5675512) q^{5} + ( - \beta_{8} - \beta_{7} - \beta_{4} + 10 \beta_{3} + 359 \beta_{2} + \cdots + 242821904) q^{6}+ \cdots + ( - 20783640488088 \beta_{9} + \cdots + 29\!\cdots\!67) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 6212 q^{2} - 33193328 q^{4} + 56758100 q^{5} + 2428200768 q^{6} - 159817477952 q^{8} - 939054565686 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 6212 q^{2} - 33193328 q^{4} + 56758100 q^{5} + 2428200768 q^{6} - 159817477952 q^{8} - 939054565686 q^{9} - 279847745800 q^{10} + 12433669328640 q^{12} - 24798065342764 q^{13} - 3829480368768 q^{14} - 588864378801920 q^{16} - 16\!\cdots\!64 q^{17}+ \cdots + 25\!\cdots\!08 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} + 1157886 x^{8} - 1182314620 x^{7} + 1715110302918 x^{6} + \cdots + 10\!\cdots\!86 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} - 158 \nu^{8} + 1182376 \nu^{7} - 1365582900 \nu^{6} + 1926775652418 \nu^{5} + \cdots - 13\!\cdots\!74 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 116131909 \nu^{9} - 164190034582 \nu^{8} + 258894897225544 \nu^{7} + \cdots - 38\!\cdots\!78 ) / 85\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1071558713 \nu^{9} + 1596009168082 \nu^{8} + 410701483466600 \nu^{7} + \cdots - 64\!\cdots\!70 ) / 85\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 732859613 \nu^{9} + 436776275866 \nu^{8} + 845431847719688 \nu^{7} + \cdots + 61\!\cdots\!18 ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1915129217 \nu^{9} - 485066975842 \nu^{8} + 5621882820952 \nu^{7} + \cdots - 46\!\cdots\!30 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9465306485 \nu^{9} - 3666444407242 \nu^{8} + \cdots - 63\!\cdots\!34 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 153798607993 \nu^{9} + 377481835692718 \nu^{8} + \cdots - 18\!\cdots\!86 ) / 85\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 399504925291 \nu^{9} + 14488026038774 \nu^{8} + \cdots + 43\!\cdots\!42 ) / 85\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5798090099 \nu^{9} - 3295412110086 \nu^{8} + \cdots - 12\!\cdots\!22 ) / 59\!\cdots\!24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 5\beta_{3} - 34\beta_{2} + 14820\beta _1 + 81607 ) / 262144 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 128 \beta_{6} - 128 \beta_{5} + 95 \beta_{4} + 6821 \beta_{3} + 54370 \beta_{2} - 22637284 \beta _1 - 60710865063 ) / 262144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8192 \beta_{9} + 23808 \beta_{8} + 24576 \beta_{7} + 49152 \beta_{6} + 132608 \beta_{5} - 622897 \beta_{4} - 7779595 \beta_{3} - 24084606 \beta_{2} + \cdots + 92713157194985 ) / 262144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2531328 \beta_{9} + 29634816 \beta_{8} + 31449088 \beta_{7} - 72194176 \beta_{6} + 505853568 \beta_{5} + 509668355 \beta_{4} - 5960046479 \beta_{3} + \cdots - 10\!\cdots\!87 ) / 262144 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10051059712 \beta_{9} - 14366767104 \beta_{8} + 39610155008 \beta_{7} + 42742323456 \beta_{6} + 20704057088 \beta_{5} - 263573653265 \beta_{4} + \cdots + 11\!\cdots\!85 ) / 262144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1016716279808 \beta_{9} - 11861240065536 \beta_{8} + 38129545363456 \beta_{7} - 21442424033152 \beta_{6} - 462667894707328 \beta_{5} + \cdots + 10\!\cdots\!97 ) / 262144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 70\!\cdots\!32 \beta_{9} + \cdots - 24\!\cdots\!39 ) / 262144 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 34\!\cdots\!92 \beta_{9} + \cdots + 20\!\cdots\!45 ) / 262144 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 37\!\cdots\!16 \beta_{9} + \cdots + 88\!\cdots\!85 ) / 262144 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
848.477 203.983i
848.477 + 203.983i
472.934 808.873i
472.934 + 808.873i
112.122 988.545i
112.122 + 988.545i
−697.391 868.555i
−697.391 + 868.555i
−734.642 844.151i
−734.642 + 844.151i
−4013.91 815.932i 666259.i 1.54457e7 + 6.55016e6i 9.84947e7 −5.43622e8 + 2.67430e9i 1.39617e10i −5.66533e10 3.88944e10i −1.61471e11 −3.95349e11 8.03650e10i
3.2 −4013.91 + 815.932i 666259.i 1.54457e7 6.55016e6i 9.84947e7 −5.43622e8 2.67430e9i 1.39617e10i −5.66533e10 + 3.88944e10i −1.61471e11 −3.95349e11 + 8.03650e10i
3.3 −2511.74 3235.49i 386500.i −4.15959e6 + 1.62534e7i −3.41994e8 1.25052e9 9.70785e8i 1.51829e10i 6.30355e10 2.73659e10i 1.33047e11 8.58998e11 + 1.10652e12i
3.4 −2511.74 + 3235.49i 386500.i −4.15959e6 1.62534e7i −3.41994e8 1.25052e9 + 9.70785e8i 1.51829e10i 6.30355e10 + 2.73659e10i 1.33047e11 8.58998e11 1.10652e12i
3.5 −1068.49 3954.18i 193889.i −1.44939e7 + 8.44998e6i 3.74285e8 −7.66672e8 + 2.07168e8i 2.47654e10i 4.88993e10 + 4.82828e10i 2.44837e11 −3.99918e11 1.47999e12i
3.6 −1068.49 + 3954.18i 193889.i −1.44939e7 8.44998e6i 3.74285e8 −7.66672e8 2.07168e8i 2.47654e10i 4.88993e10 4.82828e10i 2.44837e11 −3.99918e11 + 1.47999e12i
3.7 2169.56 3474.22i 569070.i −7.36320e6 1.50751e7i −2.26723e8 −1.97707e9 1.23463e9i 6.45252e9i −6.83491e10 7.12502e9i −4.14111e10 −4.91891e11 + 7.87687e11i
3.8 2169.56 + 3474.22i 569070.i −7.36320e6 + 1.50751e7i −2.26723e8 −1.97707e9 + 1.23463e9i 6.45252e9i −6.83491e10 + 7.12502e9i −4.14111e10 −4.91891e11 7.87687e11i
3.9 2318.57 3376.61i 962787.i −6.02571e6 1.56578e7i 1.24317e8 3.25095e9 + 2.23229e9i 3.87327e9i −6.68411e10 1.59571e10i −6.44529e11 2.88236e11 4.19768e11i
3.10 2318.57 + 3376.61i 962787.i −6.02571e6 + 1.56578e7i 1.24317e8 3.25095e9 2.23229e9i 3.87327e9i −6.68411e10 + 1.59571e10i −6.44529e11 2.88236e11 + 4.19768e11i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.25.b.b 10
3.b odd 2 1 36.25.d.b 10
4.b odd 2 1 inner 4.25.b.b 10
8.b even 2 1 64.25.c.d 10
8.d odd 2 1 64.25.c.d 10
12.b even 2 1 36.25.d.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.25.b.b 10 1.a even 1 1 trivial
4.25.b.b 10 4.b odd 2 1 inner
36.25.d.b 10 3.b odd 2 1
36.25.d.b 10 12.b even 2 1
64.25.c.d 10 8.b even 2 1
64.25.c.d 10 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 1881674965248 T_{3}^{8} + \cdots + 74\!\cdots\!00 \) acting on \(S_{25}^{\mathrm{new}}(4, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 6212 T^{9} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{10} + 1881674965248 T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{5} - 28379050 T^{4} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + 12399032671382 T^{4} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + 812519065310582 T^{4} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 91\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 17\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 19\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 26\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
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