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Properties

Label 225.5.g.m
Level $225$
Weight $5$
Character orbit 225.g
Analytic conductor $23.258$
Analytic rank $0$
Dimension $8$
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] = [225,5,Mod(82,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.82");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), degree \(2\), 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: \(23.2582416939\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{4} + 12 \beta_{2} - \beta_1) q^{4} + (2 \beta_{6} - 2 \beta_{5} - 3 \beta_{3} + \cdots - 5) q^{7}+ \cdots + ( - 2 \beta_{7} - 3 \beta_{6} + \cdots + 20) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{4} + 12 \beta_{2} - \beta_1) q^{4} + (2 \beta_{6} - 2 \beta_{5} - 3 \beta_{3} + \cdots - 5) q^{7}+ \cdots + (364 \beta_{7} + 186 \beta_{6} + \cdots + 6080) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{7} + 180 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{7} + 180 q^{8} + 288 q^{11} + 340 q^{13} + 620 q^{16} + 900 q^{17} + 1100 q^{22} - 1560 q^{23} + 3024 q^{26} - 3580 q^{28} - 512 q^{31} + 4980 q^{32} + 3820 q^{37} - 7680 q^{38} + 2712 q^{41} + 1240 q^{43} + 13528 q^{46} + 4800 q^{47} + 1240 q^{52} + 1020 q^{53} + 30720 q^{56} - 2340 q^{58} - 4760 q^{61} + 28680 q^{62} + 8920 q^{67} - 1920 q^{68} - 7536 q^{71} - 11600 q^{73} + 4344 q^{76} - 360 q^{77} + 27200 q^{82} - 32400 q^{83} - 14592 q^{86} + 14340 q^{88} + 16528 q^{91} - 31800 q^{92} - 58640 q^{97} + 46440 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 60x^{5} + 1973x^{4} - 3300x^{3} + 1800x^{2} + 31560x + 276676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19857 \nu^{7} + 350053 \nu^{6} + 190938 \nu^{5} + 1295568 \nu^{4} - 60124233 \nu^{3} + \cdots - 368125308 ) / 10440376750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 494601 \nu^{7} - 21022379 \nu^{6} - 201291784 \nu^{5} - 708946674 \nu^{4} + \cdots - 271761835006 ) / 20880753500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1331 \nu^{7} + 726 \nu^{6} + 396 \nu^{5} - 79644 \nu^{4} + 3304389 \nu^{3} - 2589906 \nu^{2} + \cdots + 20889564 ) / 39697250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2057 \nu^{7} + 1122 \nu^{6} + 612 \nu^{5} + 598682 \nu^{4} + 5106783 \nu^{3} - 4002582 \nu^{2} + \cdots + 724459558 ) / 39697250 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 906049 \nu^{7} - 9610546 \nu^{6} - 5242116 \nu^{5} - 57222276 \nu^{4} + 2552532831 \nu^{3} + \cdots + 15801463956 ) / 10440376750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9107266 \nu^{7} + 26259761 \nu^{6} + 14323506 \nu^{5} - 1108098234 \nu^{4} + \cdots - 197715968146 ) / 20880753500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + 28\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 3\beta_{6} - 3\beta_{5} + 37\beta_{4} - 25\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 55\beta_{5} - 85\beta_{4} + 85\beta _1 - 959 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 195\beta_{6} - 195\beta_{5} - 110\beta_{3} + 2215\beta_{2} - 1569\beta _1 + 1910 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -60\beta_{7} - 2679\beta_{6} + 5259\beta_{4} + 60\beta_{3} - 42542\beta_{2} + 5259\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4998\beta_{7} + 10797\beta_{6} + 10797\beta_{5} - 71933\beta_{4} + 139095\beta_{2} - 123300 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
−5.02811 − 5.02811i
−2.08045 − 2.08045i
3.30519 + 3.30519i
3.80336 + 3.80336i
−5.02811 + 5.02811i
−2.08045 + 2.08045i
3.30519 − 3.30519i
3.80336 − 3.80336i
−5.02811 − 5.02811i 0 34.5637i 0 0 38.5593 + 38.5593i 93.3405 − 93.3405i 0 0
82.2 −2.08045 − 2.08045i 0 − 7.34348i 0 0 −65.1093 − 65.1093i −48.5649 + 48.5649i 0 0
82.3 3.30519 + 3.30519i 0 5.84858i 0 0 33.1649 + 33.1649i 33.5524 − 33.5524i 0 0
82.4 3.80336 + 3.80336i 0 12.9311i 0 0 −16.6149 − 16.6149i 11.6720 − 11.6720i 0 0
118.1 −5.02811 + 5.02811i 0 − 34.5637i 0 0 38.5593 − 38.5593i 93.3405 + 93.3405i 0 0
118.2 −2.08045 + 2.08045i 0 7.34348i 0 0 −65.1093 + 65.1093i −48.5649 − 48.5649i 0 0
118.3 3.30519 − 3.30519i 0 − 5.84858i 0 0 33.1649 − 33.1649i 33.5524 + 33.5524i 0 0
118.4 3.80336 − 3.80336i 0 − 12.9311i 0 0 −16.6149 + 16.6149i 11.6720 + 11.6720i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.5.g.m 8
3.b odd 2 1 75.5.f.e 8
5.b even 2 1 45.5.g.e 8
5.c odd 4 1 45.5.g.e 8
5.c odd 4 1 inner 225.5.g.m 8
15.d odd 2 1 15.5.f.a ✓ 8
15.e even 4 1 15.5.f.a ✓ 8
15.e even 4 1 75.5.f.e 8
60.h even 2 1 240.5.bg.c 8
60.l odd 4 1 240.5.bg.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.f.a ✓ 8 15.d odd 2 1
15.5.f.a ✓ 8 15.e even 4 1
45.5.g.e 8 5.b even 2 1
45.5.g.e 8 5.c odd 4 1
75.5.f.e 8 3.b odd 2 1
75.5.f.e 8 15.e even 4 1
225.5.g.m 8 1.a even 1 1 trivial
225.5.g.m 8 5.c odd 4 1 inner
240.5.bg.c 8 60.h even 2 1
240.5.bg.c 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 60T_{2}^{5} + 1973T_{2}^{4} - 3300T_{2}^{3} + 1800T_{2}^{2} + 31560T_{2} + 276676 \) acting on \(S_{5}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 60 T^{5} + \cdots + 276676 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 30620728960000 \) Copy content Toggle raw display
$11$ \( (T^{4} - 144 T^{3} + \cdots + 27154144)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 158448579136 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 125636659418176 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + 256 T^{3} + \cdots + 388673200000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 1195607024000)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 51045861284864)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 4392786466304)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
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