Properties

Label 225.2.e.b
Level $225$
Weight $2$
Character orbit 225.e
Analytic conductor $1.797$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{2} - \beta_{4} q^{3} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{4} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{7} + (\beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1) q^{8} + ( - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{2} - \beta_{4} q^{3} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}) q^{4} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{7} + (\beta_{5} - 2 \beta_{4} + \beta_{2} - \beta_1) q^{8} + ( - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{9} + ( - \beta_{2} + \beta_1) q^{11} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 4) q^{12} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1) q^{14} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{16} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{17} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{18} + (\beta_{5} - \beta_{4} - \beta_1 + 2) q^{19} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{21} + (3 \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1) q^{22} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + \beta_1) q^{23} + ( - 3 \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_1) q^{24} + (\beta_{5} - \beta_{4} - \beta_1) q^{26} + (\beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 2) q^{27} + (\beta_{5} + 2 \beta_{4} - 3 \beta_{2} - \beta_1 - 2) q^{28} + (2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3) q^{29} + (3 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_1) q^{31} + (\beta_{5} - 6 \beta_{3} + \beta_1) q^{32} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{33} + (2 \beta_{5} + 2 \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{34} + (2 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 1) q^{36} + (\beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1 - 2) q^{37} + (\beta_{2} - \beta_1) q^{38} + ( - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + \beta_1 - 4) q^{39} + ( - 3 \beta_{5} + \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_1) q^{41} + ( - 2 \beta_{4} + 3 \beta_{2} + 2 \beta_1 + 6) q^{42} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 2) q^{43} + (\beta_{5} - 3 \beta_{4} + 2 \beta_{2} - \beta_1 - 6) q^{44} + ( - \beta_{5} + 5 \beta_{4} - 4 \beta_{2} + \beta_1 - 3) q^{46} + ( - 3 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 6) q^{47} + ( - 2 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{48} + ( - 3 \beta_{5} - \beta_{4} - \beta_{2} - 4 \beta_1) q^{49} + (\beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_{2} + 2) q^{51} + ( - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + \beta_{2} - \beta_1 - 4) q^{52} + (2 \beta_{4} - 2 \beta_{2}) q^{53} + (\beta_{5} - 3 \beta_{4} + 7 \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{54} + (3 \beta_{3} + 3) q^{56} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2} - 2) q^{57} + ( - \beta_{5} + 6 \beta_{3} - \beta_1) q^{58} + ( - 2 \beta_{5} - 2 \beta_1) q^{59} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 1) q^{61} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_1 - 12) q^{62} + (\beta_{4} - 3 \beta_{2} - \beta_1 - 6) q^{63} + (\beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1 - 5) q^{64} + ( - 4 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 2) q^{66} + ( - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1) q^{67} + (3 \beta_{5} - \beta_{4} + 6 \beta_{3} - \beta_{2} + 2 \beta_1) q^{68} + (3 \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_1 + 9) q^{69} + ( - 3 \beta_{5} + 5 \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 6) q^{71} + (3 \beta_{5} + \beta_{4} + 6 \beta_{3} - 3 \beta_{2} + 5 \beta_1) q^{72} + ( - 4 \beta_{4} + 4 \beta_{2} + 4) q^{73} + ( - 4 \beta_{5} - 4 \beta_{4} + 6 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 6) q^{74} + ( - \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2} - 2 \beta_1) q^{76} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1) q^{77} + ( - \beta_{5} + \beta_{4} - 4 \beta_{3} - \beta_{2} - 2) q^{78} + (4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 2) q^{79} + (4 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{81} + ( - 3 \beta_{5} + 8 \beta_{4} - 5 \beta_{2} + 3 \beta_1 + 6) q^{82} + (3 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 6) q^{83} + (5 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 8) q^{84} + (7 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} + 4 \beta_1) q^{86} + (4 \beta_{5} - 3 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{87} + ( - 4 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 6) q^{88} - 3 q^{89} + (\beta_{5} - \beta_{4} - \beta_1 + 4) q^{91} + (\beta_{5} + \beta_{4} + 12 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 12) q^{92} + ( - 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 7 \beta_{2} - 5 \beta_1 + 8) q^{93} + (4 \beta_{4} - 9 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{94} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 5 \beta_1 + 1) q^{96} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{97} + ( - 3 \beta_{5} + \beta_{4} + 2 \beta_{2} + 3 \beta_1 + 12) q^{98} + ( - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - q^{3} - 5 q^{4} - 4 q^{6} + 5 q^{7} - 6 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - q^{3} - 5 q^{4} - 4 q^{6} + 5 q^{7} - 6 q^{8} - 7 q^{9} + 2 q^{11} - 17 q^{12} + 4 q^{13} + 9 q^{14} - 5 q^{16} + 4 q^{17} + 23 q^{18} + 8 q^{19} - 4 q^{22} + 3 q^{23} + 15 q^{24} - 4 q^{26} + 2 q^{27} - 10 q^{28} + 7 q^{29} - 8 q^{31} + 17 q^{32} - 20 q^{33} + 4 q^{34} + 10 q^{36} - 12 q^{37} - 2 q^{38} - 14 q^{39} + 13 q^{41} + 33 q^{42} + 10 q^{43} - 44 q^{44} - 6 q^{46} + 13 q^{47} + 10 q^{48} + 2 q^{49} - 4 q^{51} - 12 q^{52} + 4 q^{53} + 5 q^{54} + 9 q^{56} + 2 q^{57} - 17 q^{58} + 2 q^{59} - q^{61} - 84 q^{62} - 33 q^{63} - 30 q^{64} - 2 q^{66} + 11 q^{67} - 22 q^{68} + 39 q^{69} - 20 q^{71} - 15 q^{72} + 16 q^{73} + 16 q^{74} + 12 q^{76} + 4 q^{78} - 2 q^{79} - 19 q^{81} + 58 q^{82} - 15 q^{83} - 27 q^{84} - 28 q^{86} + 26 q^{87} - 24 q^{88} - 18 q^{89} + 20 q^{91} + 39 q^{92} + 42 q^{93} + 31 q^{94} + 13 q^{96} - 18 q^{97} + 80 q^{98} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} - 6\nu - 45 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 2\nu^{3} - 6\nu - 18 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} - 24\nu - 72 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} - 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + \beta_{3} + 4\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + \beta_{4} - 10\beta_{3} - 4\beta_{2} + 6\beta _1 + 4 \) Copy content Toggle raw display

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
1.71903 0.211943i
−1.62241 0.606458i
0.403374 + 1.68443i
1.71903 + 0.211943i
−1.62241 + 0.606458i
0.403374 1.68443i
−1.04307 1.80664i 1.04307 + 1.38276i −1.17597 + 2.03684i 0 1.41016 3.32675i 2.04307 + 3.53869i 0.734191 −0.824030 + 2.88461i 0
76.2 0.285997 + 0.495361i −0.285997 1.70828i 0.836412 1.44871i 0 0.764419 0.630233i 0.714003 + 1.23669i 2.10083 −2.83641 + 0.977122i 0
76.3 1.25707 + 2.17731i −1.25707 + 1.19154i −2.16044 + 3.74200i 0 −4.17458 1.23917i −0.257068 0.445256i −5.83502 0.160442 2.99571i 0
151.1 −1.04307 + 1.80664i 1.04307 1.38276i −1.17597 2.03684i 0 1.41016 + 3.32675i 2.04307 3.53869i 0.734191 −0.824030 2.88461i 0
151.2 0.285997 0.495361i −0.285997 + 1.70828i 0.836412 + 1.44871i 0 0.764419 + 0.630233i 0.714003 1.23669i 2.10083 −2.83641 0.977122i 0
151.3 1.25707 2.17731i −1.25707 1.19154i −2.16044 3.74200i 0 −4.17458 + 1.23917i −0.257068 + 0.445256i −5.83502 0.160442 + 2.99571i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.e.b 6
3.b odd 2 1 675.2.e.b 6
5.b even 2 1 45.2.e.b 6
5.c odd 4 2 225.2.k.b 12
9.c even 3 1 inner 225.2.e.b 6
9.c even 3 1 2025.2.a.n 3
9.d odd 6 1 675.2.e.b 6
9.d odd 6 1 2025.2.a.o 3
15.d odd 2 1 135.2.e.b 6
15.e even 4 2 675.2.k.b 12
20.d odd 2 1 720.2.q.i 6
45.h odd 6 1 135.2.e.b 6
45.h odd 6 1 405.2.a.i 3
45.j even 6 1 45.2.e.b 6
45.j even 6 1 405.2.a.j 3
45.k odd 12 2 225.2.k.b 12
45.k odd 12 2 2025.2.b.l 6
45.l even 12 2 675.2.k.b 12
45.l even 12 2 2025.2.b.m 6
60.h even 2 1 2160.2.q.k 6
180.n even 6 1 2160.2.q.k 6
180.n even 6 1 6480.2.a.bs 3
180.p odd 6 1 720.2.q.i 6
180.p odd 6 1 6480.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 5.b even 2 1
45.2.e.b 6 45.j even 6 1
135.2.e.b 6 15.d odd 2 1
135.2.e.b 6 45.h odd 6 1
225.2.e.b 6 1.a even 1 1 trivial
225.2.e.b 6 9.c even 3 1 inner
225.2.k.b 12 5.c odd 4 2
225.2.k.b 12 45.k odd 12 2
405.2.a.i 3 45.h odd 6 1
405.2.a.j 3 45.j even 6 1
675.2.e.b 6 3.b odd 2 1
675.2.e.b 6 9.d odd 6 1
675.2.k.b 12 15.e even 4 2
675.2.k.b 12 45.l even 12 2
720.2.q.i 6 20.d odd 2 1
720.2.q.i 6 180.p odd 6 1
2025.2.a.n 3 9.c even 3 1
2025.2.a.o 3 9.d odd 6 1
2025.2.b.l 6 45.k odd 12 2
2025.2.b.m 6 45.l even 12 2
2160.2.q.k 6 60.h even 2 1
2160.2.q.k 6 180.n even 6 1
6480.2.a.bs 3 180.n even 6 1
6480.2.a.bv 3 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 6T_{2}^{4} - T_{2}^{3} + 28T_{2}^{2} - 15T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 6 T^{4} - T^{3} + 28 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + 4 T^{4} + 3 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 5 T^{5} + 22 T^{4} - 21 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + 12 T^{4} - 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{6} - 4 T^{5} + 20 T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 4 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + 42 T^{4} + \cdots + 13689 \) Copy content Toggle raw display
$29$ \( T^{6} - 7 T^{5} + 78 T^{4} + \cdots + 2601 \) Copy content Toggle raw display
$31$ \( T^{6} + 8 T^{5} + 124 T^{4} + \cdots + 219024 \) Copy content Toggle raw display
$37$ \( (T^{3} + 6 T^{2} - 12 T - 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 13 T^{5} + 150 T^{4} - 241 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( T^{6} - 10 T^{5} + 104 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( T^{6} - 13 T^{5} + 180 T^{4} + \cdots + 136161 \) Copy content Toggle raw display
$53$ \( (T^{3} - 2 T^{2} - 20 T + 24)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 2 T^{5} + 24 T^{4} - 8 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{6} + T^{5} + 38 T^{4} - 179 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$67$ \( T^{6} - 11 T^{5} + 160 T^{4} + \cdots + 257049 \) Copy content Toggle raw display
$71$ \( (T^{3} + 10 T^{2} - 92 T - 708)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 8 T^{2} - 64 T + 128)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 2 T^{5} + 88 T^{4} - 216 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$83$ \( T^{6} + 15 T^{5} + 198 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$89$ \( (T + 3)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 18 T^{5} + 360 T^{4} + \cdots + 1700416 \) Copy content Toggle raw display
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