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Properties

Label 225.2.k.b
Level $225$
Weight $2$
Character orbit 225.k
Analytic conductor $1.797$
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] = [225,2,Mod(49,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.k (of order \(6\), degree \(2\), 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: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{6} + \beta_1) q^{2} + \beta_{6} q^{3} + ( - \beta_{11} + 2 \beta_{8} + 2) q^{4} + (2 \beta_{8} + \beta_{5} - \beta_{2}) q^{6} + ( - \beta_{10} + \beta_{7} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{11} + \beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{6} + \beta_1) q^{2} + \beta_{6} q^{3} + ( - \beta_{11} + 2 \beta_{8} + 2) q^{4} + (2 \beta_{8} + \beta_{5} - \beta_{2}) q^{6} + ( - \beta_{10} + \beta_{7} + \cdots - \beta_1) q^{7}+ \cdots + (2 \beta_{11} + 2 \beta_{5} - \beta_{4} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{4} - 8 q^{6} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{4} - 8 q^{6} + 14 q^{9} + 4 q^{11} - 18 q^{14} - 10 q^{16} - 16 q^{19} - 30 q^{24} - 8 q^{26} - 14 q^{29} - 16 q^{31} - 8 q^{34} + 20 q^{36} + 28 q^{39} + 26 q^{41} + 88 q^{44} - 12 q^{46} - 4 q^{49} - 8 q^{51} - 10 q^{54} + 18 q^{56} - 4 q^{59} - 2 q^{61} + 60 q^{64} - 4 q^{66} - 78 q^{69} - 40 q^{71} - 32 q^{74} + 24 q^{76} + 4 q^{79} - 38 q^{81} + 54 q^{84} - 56 q^{86} + 36 q^{89} + 40 q^{91} - 62 q^{94} + 26 q^{96} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1330 \nu^{11} + 3816 \nu^{10} - 5925 \nu^{9} + 4140 \nu^{8} - 21730 \nu^{7} - 93918 \nu^{6} + \cdots + 195120 ) / 103944 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 748 \nu^{11} + 816 \nu^{10} - 1768 \nu^{9} + 4162 \nu^{8} + 6311 \nu^{7} + 8976 \nu^{6} + \cdots + 98004 ) / 51972 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 962 \nu^{11} + 2392 \nu^{10} - 2887 \nu^{9} + 2220 \nu^{8} + 13990 \nu^{7} - 14442 \nu^{6} + \cdots - 15984 ) / 51972 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1833 \nu^{11} + 1380 \nu^{10} - 718 \nu^{9} + 112 \nu^{8} + 30103 \nu^{7} + 21996 \nu^{6} + \cdots - 25344 ) / 51972 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1054 \nu^{11} + 840 \nu^{10} - 684 \nu^{9} + 630 \nu^{8} + 15925 \nu^{7} + 12648 \nu^{6} + \cdots - 60768 ) / 25986 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5984 \nu^{11} - 7522 \nu^{10} + 8393 \nu^{9} - 6600 \nu^{8} - 90816 \nu^{7} - 28498 \nu^{6} + \cdots + 272448 ) / 103944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6823 \nu^{11} - 186 \nu^{10} + 2391 \nu^{9} - 2376 \nu^{8} + 113464 \nu^{7} + 159834 \nu^{6} + \cdots - 548280 ) / 103944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 118 \nu^{11} - 90 \nu^{10} + 53 \nu^{9} - 32 \nu^{8} - 1866 \nu^{7} - 1416 \nu^{6} + 1364 \nu^{5} + \cdots + 3468 ) / 1704 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9709 \nu^{11} + 6990 \nu^{10} - 6270 \nu^{9} + 4284 \nu^{8} + 155434 \nu^{7} + 116508 \nu^{6} + \cdots - 596232 ) / 103944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5790 \nu^{11} + 1824 \nu^{10} - 473 \nu^{9} + 516 \nu^{8} + 91522 \nu^{7} + 112790 \nu^{6} + \cdots - 404496 ) / 51972 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17172 \nu^{11} + 12498 \nu^{10} - 4217 \nu^{9} - 3016 \nu^{8} + 280340 \nu^{7} + 206064 \nu^{6} + \cdots - 737076 ) / 103944 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{7} + \beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} + 2\beta_{7} + 3\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{10} - 4\beta_{9} - 4\beta_{8} + 4\beta_{7} - 4\beta_{6} - 4\beta_{5} - 4\beta_{4} + \beta_{3} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{11} - 10\beta_{8} - 3\beta_{5} - 6\beta_{4} - 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{11} + 14 \beta_{10} - 2 \beta_{9} - 13 \beta_{8} - 2 \beta_{7} + 20 \beta_{6} - 19 \beta_{4} + \cdots + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{10} - 24\beta_{9} + 8\beta_{7} + 24\beta_{6} + 27\beta_{3} + 32\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 4 \beta_{11} + 20 \beta_{10} - 51 \beta_{9} - 36 \beta_{8} + 43 \beta_{7} + 8 \beta_{6} - 47 \beta_{5} + \cdots - 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -35\beta_{11} - 223\beta_{8} - 166\beta_{5} - 83\beta_{4} - 223 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 48 \beta_{11} + 107 \beta_{10} + 212 \beta_{9} - 380 \beta_{8} - 260 \beta_{7} + 212 \beta_{6} + \cdots - 190 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 636\beta_{10} + 262\beta_{9} - 590\beta_{7} + 852\beta_{6} + 590\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 131 \beta_{11} + 1114 \beta_{10} - 262 \beta_{9} + 983 \beta_{8} - 262 \beta_{7} + 1324 \beta_{6} + \cdots - 983 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 - \beta_{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.180407 + 0.673288i
2.17840 + 0.583700i
−0.403293 + 1.50511i
−1.50511 − 0.403293i
0.583700 − 2.17840i
−0.673288 − 0.180407i
−0.180407 − 0.673288i
2.17840 − 0.583700i
−0.403293 − 1.50511i
−1.50511 + 0.403293i
0.583700 + 2.17840i
−0.673288 + 0.180407i
−2.17731 + 1.25707i 1.19154 + 1.25707i 2.16044 − 3.74200i 0 −4.17458 − 1.23917i 0.445256 − 0.257068i 5.83502i −0.160442 + 2.99571i 0
49.2 −1.80664 + 1.04307i −1.38276 + 1.04307i 1.17597 − 2.03684i 0 1.41016 − 3.32675i 3.53869 − 2.04307i 0.734191i 0.824030 − 2.88461i 0
49.3 −0.495361 + 0.285997i −1.70828 + 0.285997i −0.836412 + 1.44871i 0 0.764419 − 0.630233i −1.23669 + 0.714003i − 2.10083i 2.83641 − 0.977122i 0
49.4 0.495361 − 0.285997i 1.70828 − 0.285997i −0.836412 + 1.44871i 0 0.764419 − 0.630233i 1.23669 − 0.714003i 2.10083i 2.83641 − 0.977122i 0
49.5 1.80664 − 1.04307i 1.38276 − 1.04307i 1.17597 − 2.03684i 0 1.41016 − 3.32675i −3.53869 + 2.04307i − 0.734191i 0.824030 − 2.88461i 0
49.6 2.17731 − 1.25707i −1.19154 − 1.25707i 2.16044 − 3.74200i 0 −4.17458 − 1.23917i −0.445256 + 0.257068i − 5.83502i −0.160442 + 2.99571i 0
124.1 −2.17731 − 1.25707i 1.19154 − 1.25707i 2.16044 + 3.74200i 0 −4.17458 + 1.23917i 0.445256 + 0.257068i − 5.83502i −0.160442 − 2.99571i 0
124.2 −1.80664 − 1.04307i −1.38276 − 1.04307i 1.17597 + 2.03684i 0 1.41016 + 3.32675i 3.53869 + 2.04307i − 0.734191i 0.824030 + 2.88461i 0
124.3 −0.495361 − 0.285997i −1.70828 − 0.285997i −0.836412 − 1.44871i 0 0.764419 + 0.630233i −1.23669 − 0.714003i 2.10083i 2.83641 + 0.977122i 0
124.4 0.495361 + 0.285997i 1.70828 + 0.285997i −0.836412 − 1.44871i 0 0.764419 + 0.630233i 1.23669 + 0.714003i − 2.10083i 2.83641 + 0.977122i 0
124.5 1.80664 + 1.04307i 1.38276 + 1.04307i 1.17597 + 2.03684i 0 1.41016 + 3.32675i −3.53869 − 2.04307i 0.734191i 0.824030 + 2.88461i 0
124.6 2.17731 + 1.25707i −1.19154 + 1.25707i 2.16044 + 3.74200i 0 −4.17458 + 1.23917i −0.445256 − 0.257068i 5.83502i −0.160442 − 2.99571i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
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 225.2.k.b 12
3.b odd 2 1 675.2.k.b 12
5.b even 2 1 inner 225.2.k.b 12
5.c odd 4 1 45.2.e.b ✓ 6
5.c odd 4 1 225.2.e.b 6
9.c even 3 1 inner 225.2.k.b 12
9.c even 3 1 2025.2.b.l 6
9.d odd 6 1 675.2.k.b 12
9.d odd 6 1 2025.2.b.m 6
15.d odd 2 1 675.2.k.b 12
15.e even 4 1 135.2.e.b 6
15.e even 4 1 675.2.e.b 6
20.e even 4 1 720.2.q.i 6
45.h odd 6 1 675.2.k.b 12
45.h odd 6 1 2025.2.b.m 6
45.j even 6 1 inner 225.2.k.b 12
45.j even 6 1 2025.2.b.l 6
45.k odd 12 1 45.2.e.b ✓ 6
45.k odd 12 1 225.2.e.b 6
45.k odd 12 1 405.2.a.j 3
45.k odd 12 1 2025.2.a.n 3
45.l even 12 1 135.2.e.b 6
45.l even 12 1 405.2.a.i 3
45.l even 12 1 675.2.e.b 6
45.l even 12 1 2025.2.a.o 3
60.l odd 4 1 2160.2.q.k 6
180.v odd 12 1 2160.2.q.k 6
180.v odd 12 1 6480.2.a.bs 3
180.x even 12 1 720.2.q.i 6
180.x even 12 1 6480.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b ✓ 6 5.c odd 4 1
45.2.e.b ✓ 6 45.k odd 12 1
135.2.e.b 6 15.e even 4 1
135.2.e.b 6 45.l even 12 1
225.2.e.b 6 5.c odd 4 1
225.2.e.b 6 45.k odd 12 1
225.2.k.b 12 1.a even 1 1 trivial
225.2.k.b 12 5.b even 2 1 inner
225.2.k.b 12 9.c even 3 1 inner
225.2.k.b 12 45.j even 6 1 inner
405.2.a.i 3 45.l even 12 1
405.2.a.j 3 45.k odd 12 1
675.2.e.b 6 15.e even 4 1
675.2.e.b 6 45.l even 12 1
675.2.k.b 12 3.b odd 2 1
675.2.k.b 12 9.d odd 6 1
675.2.k.b 12 15.d odd 2 1
675.2.k.b 12 45.h odd 6 1
720.2.q.i 6 20.e even 4 1
720.2.q.i 6 180.x even 12 1
2025.2.a.n 3 45.k odd 12 1
2025.2.a.o 3 45.l even 12 1
2025.2.b.l 6 9.c even 3 1
2025.2.b.l 6 45.j even 6 1
2025.2.b.m 6 9.d odd 6 1
2025.2.b.m 6 45.h odd 6 1
2160.2.q.k 6 60.l odd 4 1
2160.2.q.k 6 180.v odd 12 1
6480.2.a.bs 3 180.v odd 12 1
6480.2.a.bv 3 180.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 11T_{2}^{10} + 90T_{2}^{8} - 323T_{2}^{6} + 862T_{2}^{4} - 279T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 11 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{12} - 7 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 19 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T^{6} - 2 T^{5} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 24 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{6} + 20 T^{4} + \cdots + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 4 T^{2} - 4 T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 187388721 \) Copy content Toggle raw display
$29$ \( (T^{6} + 7 T^{5} + \cdots + 2601)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots + 219024)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 60 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 13 T^{5} + 150 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 108 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 18539817921 \) Copy content Toggle raw display
$53$ \( (T^{6} + 44 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 2 T^{5} + \cdots + 576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + T^{5} + 38 T^{4} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 66074188401 \) Copy content Toggle raw display
$71$ \( (T^{3} + 10 T^{2} + \cdots - 708)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 192 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 2 T^{5} + \cdots + 576)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 171 T^{10} + \cdots + 43046721 \) Copy content Toggle raw display
$89$ \( (T - 3)^{12} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2891414573056 \) Copy content Toggle raw display
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