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Properties

Label 900.2.i.f
Level $900$
Weight $2$
Character orbit 900.i
Analytic conductor $7.187$
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] = [900,2,Mod(301,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.i (of order \(3\), 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 4x^{10} + 10x^{8} + 6x^{6} + 90x^{4} + 324x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{11} + \beta_{5}) q^{7} + (\beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{11} + \beta_{5}) q^{7} + (\beta_{3} + \beta_{2} - 1) q^{9} + ( - \beta_{10} - \beta_{9}) q^{11} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{13}+ \cdots + ( - 3 \beta_{10} + 2 \beta_{9} + \cdots - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{9} + 2 q^{11} + 10 q^{21} + 18 q^{29} + 6 q^{31} + 42 q^{39} + 14 q^{41} + 16 q^{51} + 34 q^{59} + 6 q^{61} - 14 q^{69} + 6 q^{79} - 8 q^{81} - 112 q^{89} + 12 q^{91} - 82 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 4x^{10} + 10x^{8} + 6x^{6} + 90x^{4} + 324x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} - 31\nu^{8} + 17\nu^{6} + 21\nu^{4} + 45\nu^{2} - 1701 ) / 1296 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 31\nu^{8} - 17\nu^{6} - 21\nu^{4} + 1251\nu^{2} + 2997 ) / 1296 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 31\nu^{9} + 17\nu^{7} + 21\nu^{5} + 45\nu^{3} - 2997\nu ) / 1296 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + 5\nu^{9} - \nu^{7} + 57\nu^{5} - 225\nu^{3} + 405\nu ) / 486 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{10} + 11\nu^{8} - 13\nu^{6} + 75\nu^{4} + 531\nu^{2} + 1053 ) / 324 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 4\nu^{9} + 10\nu^{7} + 6\nu^{5} + 90\nu^{3} + 324\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{11} - 25\nu^{9} + 23\nu^{7} - 93\nu^{5} - 693\nu^{3} - 2403\nu ) / 1296 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{10} + \nu^{8} - 11\nu^{6} - 39\nu^{4} + 657\nu^{2} + 243 ) / 324 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -37\nu^{10} - 139\nu^{8} - 91\nu^{6} - 375\nu^{4} - 3519\nu^{2} - 11097 ) / 1296 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{11} - 9\nu^{9} - 9\nu^{7} - 29\nu^{5} - 405\nu^{3} - 819\nu ) / 432 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - 2\beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{9} + 3\beta_{6} - \beta_{3} + 3\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 5\beta_{8} + \beta_{7} + 5\beta_{5} + 6\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{10} - \beta_{9} - 9\beta_{6} + 14\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} + 10\beta_{8} + 19\beta_{7} + 2\beta_{5} + 5\beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -3\beta_{10} - 3\beta_{9} - 3\beta_{6} + 4\beta_{3} - 29\beta_{2} - 61 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7\beta_{11} - 5\beta_{8} + 16\beta_{7} + 2\beta_{5} - 32\beta_{4} - 93\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -9\beta_{10} + 34\beta_{9} + 3\beta_{6} - 100\beta_{3} - 51\beta_{2} + 52 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -134\beta_{11} + 130\beta_{8} - 107\beta_{7} + 32\beta_{5} - 48\beta_{4} - 8\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
301.1
−1.58848 + 0.690466i
−0.783067 − 1.54493i
−0.602950 + 1.62372i
0.602950 − 1.62372i
0.783067 + 1.54493i
1.58848 − 0.690466i
−1.58848 − 0.690466i
−0.783067 + 1.54493i
−0.602950 − 1.62372i
0.602950 + 1.62372i
0.783067 − 1.54493i
1.58848 + 0.690466i
0 −1.58848 + 0.690466i 0 0 0 0.644175 + 1.11574i 0 2.04651 − 2.19358i 0
301.2 0 −0.783067 − 1.54493i 0 0 0 −1.13248 − 1.96151i 0 −1.77361 + 2.41957i 0
301.3 0 −0.602950 + 1.62372i 0 0 0 −1.88482 − 3.26460i 0 −2.27290 − 1.95804i 0
301.4 0 0.602950 − 1.62372i 0 0 0 1.88482 + 3.26460i 0 −2.27290 − 1.95804i 0
301.5 0 0.783067 + 1.54493i 0 0 0 1.13248 + 1.96151i 0 −1.77361 + 2.41957i 0
301.6 0 1.58848 − 0.690466i 0 0 0 −0.644175 − 1.11574i 0 2.04651 − 2.19358i 0
601.1 0 −1.58848 − 0.690466i 0 0 0 0.644175 − 1.11574i 0 2.04651 + 2.19358i 0
601.2 0 −0.783067 + 1.54493i 0 0 0 −1.13248 + 1.96151i 0 −1.77361 − 2.41957i 0
601.3 0 −0.602950 − 1.62372i 0 0 0 −1.88482 + 3.26460i 0 −2.27290 + 1.95804i 0
601.4 0 0.602950 + 1.62372i 0 0 0 1.88482 − 3.26460i 0 −2.27290 + 1.95804i 0
601.5 0 0.783067 − 1.54493i 0 0 0 1.13248 − 1.96151i 0 −1.77361 − 2.41957i 0
601.6 0 1.58848 + 0.690466i 0 0 0 −0.644175 + 1.11574i 0 2.04651 + 2.19358i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 301.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.i.f 12
3.b odd 2 1 2700.2.i.f 12
5.b even 2 1 inner 900.2.i.f 12
5.c odd 4 2 180.2.r.a ✓ 12
9.c even 3 1 inner 900.2.i.f 12
9.c even 3 1 8100.2.a.bc 6
9.d odd 6 1 2700.2.i.f 12
9.d odd 6 1 8100.2.a.bd 6
15.d odd 2 1 2700.2.i.f 12
15.e even 4 2 540.2.r.a 12
20.e even 4 2 720.2.by.e 12
45.h odd 6 1 2700.2.i.f 12
45.h odd 6 1 8100.2.a.bd 6
45.j even 6 1 inner 900.2.i.f 12
45.j even 6 1 8100.2.a.bc 6
45.k odd 12 2 180.2.r.a ✓ 12
45.k odd 12 2 1620.2.d.c 6
45.l even 12 2 540.2.r.a 12
45.l even 12 2 1620.2.d.d 6
60.l odd 4 2 2160.2.by.e 12
180.v odd 12 2 2160.2.by.e 12
180.x even 12 2 720.2.by.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.r.a ✓ 12 5.c odd 4 2
180.2.r.a ✓ 12 45.k odd 12 2
540.2.r.a 12 15.e even 4 2
540.2.r.a 12 45.l even 12 2
720.2.by.e 12 20.e even 4 2
720.2.by.e 12 180.x even 12 2
900.2.i.f 12 1.a even 1 1 trivial
900.2.i.f 12 5.b even 2 1 inner
900.2.i.f 12 9.c even 3 1 inner
900.2.i.f 12 45.j even 6 1 inner
1620.2.d.c 6 45.k odd 12 2
1620.2.d.d 6 45.l even 12 2
2160.2.by.e 12 60.l odd 4 2
2160.2.by.e 12 180.v odd 12 2
2700.2.i.f 12 3.b odd 2 1
2700.2.i.f 12 9.d odd 6 1
2700.2.i.f 12 15.d odd 2 1
2700.2.i.f 12 45.h odd 6 1
8100.2.a.bc 6 9.c even 3 1
8100.2.a.bc 6 45.j even 6 1
8100.2.a.bd 6 9.d odd 6 1
8100.2.a.bd 6 45.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{12} + 21T_{7}^{10} + 336T_{7}^{8} + 1963T_{7}^{6} + 8484T_{7}^{4} + 12705T_{7}^{2} + 14641 \) Copy content Toggle raw display
\( T_{11}^{6} - T_{11}^{5} + 23T_{11}^{4} - 70T_{11}^{3} + 530T_{11}^{2} - 1012T_{11} + 2116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 21 T^{10} + \cdots + 14641 \) Copy content Toggle raw display
$11$ \( (T^{6} - T^{5} + 23 T^{4} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 39 T^{10} + \cdots + 104976 \) Copy content Toggle raw display
$17$ \( (T^{6} - 36 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 42 T - 72)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 815730721 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{6} \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} + \cdots + 128164)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 60 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 7 T^{5} + \cdots + 9409)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 8318169616 \) Copy content Toggle raw display
$47$ \( T^{12} + 105 T^{10} + \cdots + 923521 \) Copy content Toggle raw display
$53$ \( (T^{6} - 264 T^{4} + \cdots - 331776)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 17 T^{5} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + \cdots + 961)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 214358881 \) Copy content Toggle raw display
$71$ \( (T^{3} - 42 T - 72)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 384 T^{4} + \cdots - 369664)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 3 T^{5} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 3637263079921 \) Copy content Toggle raw display
$89$ \( (T^{3} + 28 T^{2} + \cdots + 662)^{4} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 3102044416 \) Copy content Toggle raw display
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