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Properties

Label 225.7.g.j
Level $225$
Weight $7$
Character orbit 225.g
Analytic conductor $51.762$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.82");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
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: \(51.7621688145\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 34125x^{8} + 197902500x^{4} + 6400000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4}\cdot 5^{10} \)
Twist minimal: no (minimal twist has level 45)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - 21 \beta_{3}) q^{4} + (\beta_{8} - 10 \beta_{3} + 10) q^{7} + (\beta_{9} - 6 \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} - 21 \beta_{3}) q^{4} + (\beta_{8} - 10 \beta_{3} + 10) q^{7} + (\beta_{9} - 6 \beta_{5}) q^{8} + (3 \beta_{11} - \beta_{9} + \cdots + 38 \beta_1) q^{11}+ \cdots + ( - 1800 \beta_{11} + \cdots - 71431 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 120 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 120 q^{7} + 4860 q^{13} + 10188 q^{16} + 39300 q^{22} + 16380 q^{28} + 99696 q^{31} + 141660 q^{37} - 395040 q^{43} + 510360 q^{46} + 1041360 q^{52} - 1409100 q^{58} + 1567824 q^{61} + 2296320 q^{67} - 1468020 q^{73} + 5814744 q^{76} + 7124400 q^{82} - 3728100 q^{88} + 5580000 q^{91} + 8035020 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 34125x^{8} + 197902500x^{4} + 6400000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} - 49575\nu^{4} - 307678750 ) / 3859750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -51\nu^{10} - 1756375\nu^{6} - 10886227500\nu^{2} ) / 61756000000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 49575\nu^{5} - 496806500\nu ) / 3859750 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -51\nu^{11} - 1756375\nu^{7} - 10886227500\nu^{3} ) / 61756000000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -867\nu^{10} - 29858375\nu^{6} - 172714667500\nu^{2} ) / 12351200000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2383 \nu^{10} + 52800 \nu^{8} - 77223875 \nu^{6} + 1382440000 \nu^{4} + \cdots + 2195948000000 ) / 12351200000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2383 \nu^{10} - 52800 \nu^{8} - 77223875 \nu^{6} - 1382440000 \nu^{4} + \cdots - 2195948000000 ) / 12351200000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3417\nu^{11} - 117677125\nu^{7} - 698499242500\nu^{3} ) / 30878000000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 72299 \nu^{11} - 408000 \nu^{9} - 2465667375 \nu^{7} - 14051000000 \nu^{5} + \cdots - 81161244000000 \nu ) / 494048000000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 72299 \nu^{11} - 408000 \nu^{9} + 2465667375 \nu^{7} - 14051000000 \nu^{5} + \cdots - 81161244000000 \nu ) / 494048000000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 85\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 134\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{8} - 5\beta_{7} - 165\beta_{2} - 11375 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 40\beta_{11} + 40\beta_{10} - 255\beta_{4} - 19680\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1275\beta_{8} + 1275\beta_{7} - 26625\beta_{6} + 1667375\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -10200\beta_{11} + 10200\beta_{10} - 49575\beta_{9} + 3028100\beta_{5} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -247875\beta_{8} + 247875\beta_{7} + 4320125\beta_{2} + 256236875 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1983000\beta_{11} - 1983000\beta_{10} + 8781875\beta_{4} + 478829500\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -43909375\beta_{8} - 43909375\beta_{7} + 703475625\beta_{6} - 40489459375\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 351275000\beta_{11} - 351275000\beta_{10} + 1493844375\beta_{9} - 76891777500\beta_{5} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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
−9.04140 − 9.04140i
−6.54993 − 6.54993i
−1.68860 − 1.68860i
1.68860 + 1.68860i
6.54993 + 6.54993i
9.04140 + 9.04140i
−9.04140 + 9.04140i
−6.54993 + 6.54993i
−1.68860 + 1.68860i
1.68860 − 1.68860i
6.54993 − 6.54993i
9.04140 − 9.04140i
−9.04140 − 9.04140i 0 99.4940i 0 0 −230.377 − 230.377i 320.915 − 320.915i 0 0
82.2 −6.54993 − 6.54993i 0 21.8033i 0 0 424.534 + 424.534i −276.386 + 276.386i 0 0
82.3 −1.68860 − 1.68860i 0 − 58.2973i 0 0 −164.157 − 164.157i −206.511 + 206.511i 0 0
82.4 1.68860 + 1.68860i 0 − 58.2973i 0 0 −164.157 − 164.157i 206.511 − 206.511i 0 0
82.5 6.54993 + 6.54993i 0 21.8033i 0 0 424.534 + 424.534i 276.386 − 276.386i 0 0
82.6 9.04140 + 9.04140i 0 99.4940i 0 0 −230.377 − 230.377i −320.915 + 320.915i 0 0
118.1 −9.04140 + 9.04140i 0 − 99.4940i 0 0 −230.377 + 230.377i 320.915 + 320.915i 0 0
118.2 −6.54993 + 6.54993i 0 − 21.8033i 0 0 424.534 − 424.534i −276.386 − 276.386i 0 0
118.3 −1.68860 + 1.68860i 0 58.2973i 0 0 −164.157 + 164.157i −206.511 − 206.511i 0 0
118.4 1.68860 − 1.68860i 0 58.2973i 0 0 −164.157 + 164.157i 206.511 + 206.511i 0 0
118.5 6.54993 − 6.54993i 0 − 21.8033i 0 0 424.534 − 424.534i 276.386 + 276.386i 0 0
118.6 9.04140 − 9.04140i 0 − 99.4940i 0 0 −230.377 + 230.377i −320.915 − 320.915i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.7.g.j 12
3.b odd 2 1 inner 225.7.g.j 12
5.b even 2 1 45.7.g.c ✓ 12
5.c odd 4 1 45.7.g.c ✓ 12
5.c odd 4 1 inner 225.7.g.j 12
15.d odd 2 1 45.7.g.c ✓ 12
15.e even 4 1 45.7.g.c ✓ 12
15.e even 4 1 inner 225.7.g.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.7.g.c ✓ 12 5.b even 2 1
45.7.g.c ✓ 12 5.c odd 4 1
45.7.g.c ✓ 12 15.d odd 2 1
45.7.g.c ✓ 12 15.e even 4 1
225.7.g.j 12 1.a even 1 1 trivial
225.7.g.j 12 3.b odd 2 1 inner
225.7.g.j 12 5.c odd 4 1 inner
225.7.g.j 12 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 34125T_{2}^{8} + 197902500T_{2}^{4} + 6400000000 \) acting on \(S_{7}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 6400000000 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 17\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots + 23406708402688)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 830613704592608)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
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