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Properties

Label 225.3.d.b
Level $225$
Weight $3$
Character orbit 225.d
Analytic conductor $6.131$
Analytic rank $0$
Dimension $8$
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] = [225,3,Mod(224,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.224");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.d (of order \(2\), degree \(1\), not 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: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 45)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{6} + 3) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{7} + (\beta_{7} + 5 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{6} + 3) q^{4} + ( - \beta_{3} + 2 \beta_1) q^{7} + (\beta_{7} + 5 \beta_{4}) q^{8} + ( - 2 \beta_{5} + \beta_{2}) q^{11} + (5 \beta_{3} + 3 \beta_1) q^{13} + (4 \beta_{5} + 3 \beta_{2}) q^{14} + ( - 2 \beta_{6} + 21) q^{16} + (2 \beta_{7} - 4 \beta_{4}) q^{17} + \beta_{6} q^{19} + (5 \beta_{3} - 2 \beta_1) q^{22} + (4 \beta_{7} + 2 \beta_{4}) q^{23} + ( - 7 \beta_{5} - 2 \beta_{2}) q^{26} + ( - 11 \beta_{3} + 16 \beta_1) q^{28} + ( - 7 \beta_{5} - 5 \beta_{2}) q^{29} + (7 \beta_{6} - 14) q^{31} + ( - 2 \beta_{7} + 13 \beta_{4}) q^{32} + (2 \beta_{6} - 32) q^{34} + ( - 9 \beta_{3} + 19 \beta_1) q^{37} + ( - \beta_{7} - 6 \beta_{4}) q^{38} + ( - \beta_{5} - 4 \beta_{2}) q^{41} + ( - 10 \beta_{3} + 6 \beta_1) q^{43} + ( - 4 \beta_{5} - 11 \beta_{2}) q^{44} + ( - 6 \beta_{6} + 6) q^{46} + (3 \beta_{7} + 10 \beta_{4}) q^{47} + (4 \beta_{6} + 23) q^{49} + (3 \beta_{3} - 41 \beta_1) q^{52} + ( - 5 \beta_{7} - 2 \beta_{4}) q^{53} + (22 \beta_{5} + 15 \beta_{2}) q^{56} + (26 \beta_{3} - 41 \beta_1) q^{58} + ( - 2 \beta_{5} + 11 \beta_{2}) q^{59} + ( - \beta_{6} - 42) q^{61} + ( - 7 \beta_{7} - 56 \beta_{4}) q^{62} + ( - 3 \beta_{6} + 11) q^{64} + (8 \beta_{3} + 26 \beta_1) q^{67} + ( - 10 \beta_{7} - 28 \beta_{4}) q^{68} + ( - 6 \beta_{5} - 22 \beta_{2}) q^{71} + (2 \beta_{3} - 27 \beta_1) q^{73} + (37 \beta_{5} + 28 \beta_{2}) q^{74} + (3 \beta_{6} - 40) q^{76} + (2 \beta_{7} + 10 \beta_{4}) q^{77} + ( - 13 \beta_{6} + 14) q^{79} + (7 \beta_{3} - 19 \beta_1) q^{82} + ( - \beta_{7} + 22 \beta_{4}) q^{83} + (26 \beta_{5} + 16 \beta_{2}) q^{86} + (3 \beta_{3} - 48 \beta_1) q^{88} + ( - 21 \beta_{5} - 6 \beta_{2}) q^{89} + ( - 7 \beta_{6} + 26) q^{91} + ( - 10 \beta_{7} + 34 \beta_{4}) q^{92} + ( - 13 \beta_{6} + 64) q^{94} + ( - 18 \beta_{3} - 29 \beta_1) q^{97} + ( - 4 \beta_{7} - \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{4} + 168 q^{16} - 112 q^{31} - 256 q^{34} + 48 q^{46} + 184 q^{49} - 336 q^{61} + 88 q^{64} - 320 q^{76} + 112 q^{79} + 208 q^{91} + 512 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} + 16\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} - 6\nu^{6} - \nu^{5} - 13\nu^{3} - 36\nu^{2} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} + \nu^{5} + 29\nu^{3} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} + \nu^{5} - 2\nu^{4} - 13\nu^{3} + 5\nu - 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{7} + \nu^{5} + 13\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 2\nu^{5} - 58\nu^{3} + 22\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{7} + 4\nu^{5} + 4\nu^{4} - 52\nu^{3} + 20\nu + 14 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + 3\beta_{6} - 2\beta_{5} - 2\beta_{4} + 6\beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 3\beta_{2} + 9\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} - 3\beta_{6} - 4\beta_{5} + 4\beta_{4} + 6\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 4\beta_{4} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} - 15\beta_{6} + 22\beta_{5} + 22\beta_{4} - 30\beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{5} - 12\beta_{2} - 27\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -29\beta_{7} + 39\beta_{6} + 58\beta_{5} - 58\beta_{4} - 78\beta_{3} ) / 24 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
224.1
−0.437016 + 0.437016i
−0.437016 − 0.437016i
0.437016 − 0.437016i
0.437016 + 0.437016i
1.14412 − 1.14412i
1.14412 + 1.14412i
−1.14412 + 1.14412i
−1.14412 − 1.14412i
−3.65028 0 9.32456 0 0 − 7.16228i −19.4361 0 0
224.2 −3.65028 0 9.32456 0 0 7.16228i −19.4361 0 0
224.3 −0.821854 0 −3.32456 0 0 − 0.837722i 6.01972 0 0
224.4 −0.821854 0 −3.32456 0 0 0.837722i 6.01972 0 0
224.5 0.821854 0 −3.32456 0 0 − 0.837722i −6.01972 0 0
224.6 0.821854 0 −3.32456 0 0 0.837722i −6.01972 0 0
224.7 3.65028 0 9.32456 0 0 − 7.16228i 19.4361 0 0
224.8 3.65028 0 9.32456 0 0 7.16228i 19.4361 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 224.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.3.d.b 8
3.b odd 2 1 inner 225.3.d.b 8
4.b odd 2 1 3600.3.c.i 8
5.b even 2 1 inner 225.3.d.b 8
5.c odd 4 1 45.3.c.a ✓ 4
5.c odd 4 1 225.3.c.c 4
12.b even 2 1 3600.3.c.i 8
15.d odd 2 1 inner 225.3.d.b 8
15.e even 4 1 45.3.c.a ✓ 4
15.e even 4 1 225.3.c.c 4
20.d odd 2 1 3600.3.c.i 8
20.e even 4 1 720.3.l.a 4
20.e even 4 1 3600.3.l.v 4
40.i odd 4 1 2880.3.l.g 4
40.k even 4 1 2880.3.l.c 4
45.k odd 12 2 405.3.i.d 8
45.l even 12 2 405.3.i.d 8
60.h even 2 1 3600.3.c.i 8
60.l odd 4 1 720.3.l.a 4
60.l odd 4 1 3600.3.l.v 4
120.q odd 4 1 2880.3.l.c 4
120.w even 4 1 2880.3.l.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.c.a ✓ 4 5.c odd 4 1
45.3.c.a ✓ 4 15.e even 4 1
225.3.c.c 4 5.c odd 4 1
225.3.c.c 4 15.e even 4 1
225.3.d.b 8 1.a even 1 1 trivial
225.3.d.b 8 3.b odd 2 1 inner
225.3.d.b 8 5.b even 2 1 inner
225.3.d.b 8 15.d odd 2 1 inner
405.3.i.d 8 45.k odd 12 2
405.3.i.d 8 45.l even 12 2
720.3.l.a 4 20.e even 4 1
720.3.l.a 4 60.l odd 4 1
2880.3.l.c 4 40.k even 4 1
2880.3.l.c 4 120.q odd 4 1
2880.3.l.g 4 40.i odd 4 1
2880.3.l.g 4 120.w even 4 1
3600.3.c.i 8 4.b odd 2 1
3600.3.c.i 8 12.b even 2 1
3600.3.c.i 8 20.d odd 2 1
3600.3.c.i 8 60.h even 2 1
3600.3.l.v 4 20.e even 4 1
3600.3.l.v 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 14T_{2}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 14 T^{2} + 9)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 52 T^{2} + 36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 236 T^{2} + 6084)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 572 T^{2} + 45796)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 704 T^{2} + 82944)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1656 T^{2} + 219024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2024 T^{2} + 144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 28 T - 1764)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4508 T^{2} + 401956)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 644 T^{2} + 101124)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2288 T^{2} + 732736)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2096 T^{2} + 788544)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2576 T^{2} + 419904)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 5996 T^{2} + 3392964)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 84 T + 1724)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 6688 T^{2} + 4260096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 19424 T^{2} + 93083904)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 5912 T^{2} + 8271376)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 28 T - 6564)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 7056 T^{2} + 4981824)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 14436 T^{2} + 33385284)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 13208 T^{2} + 15376)^{2} \) Copy content Toggle raw display
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