Properties

Label 225.2.m.a
Level $225$
Weight $2$
Character orbit 225.m
Analytic conductor $1.797$
Analytic rank $0$
Dimension $8$
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.m (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.58140625.2
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{4} + (\beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{4} + 2 \beta_{2} - 1) q^{7} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{4} + (\beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{4} + 2 \beta_{2} - 1) q^{7} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{8} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{10} + 2 \beta_{4} q^{11} + (\beta_{5} + \beta_{4} + \beta_1 - 1) q^{13} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 2) q^{14} + (\beta_{7} + \beta_{6} + 2 \beta_{2} + 2 \beta_1 - 1) q^{16} + (\beta_{7} + \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \beta_1 - 1) q^{17} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 2) q^{19} + (\beta_{7} - 2 \beta_{6} - 3 \beta_{3} - \beta_{2} + 2) q^{20} + (2 \beta_{5} + 2 \beta_{4}) q^{22} + (\beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{23} + (2 \beta_{7} + \beta_{5} - 3 \beta_{2}) q^{25} + (\beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{26} + ( - 3 \beta_{7} - \beta_{5} - \beta_{4} - 3 \beta_{3} - 3) q^{28} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 5 \beta_{2} - 4 \beta_1 + 3) q^{29} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} + \beta_1) q^{31} + (\beta_{7} + \beta_{6} - 6 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{32} + ( - \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{34} + (\beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{35} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 4 \beta_{3} + \beta_1 + 2) q^{37} + (\beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{38} + (3 \beta_{7} - 3 \beta_{6} + \beta_{5} + 3 \beta_{3} + \beta_1 + 3) q^{40} + (\beta_{7} + \beta_{6} - \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{41} + (\beta_{7} - \beta_{6} - 6 \beta_{4} + 5 \beta_{3} - \beta_{2} + \beta_1 + 3) q^{43} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5}) q^{44} + ( - \beta_{7} + 4 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} + 2 \beta_1 - 3) q^{46} + ( - 2 \beta_{7} + 3 \beta_{5} + 3 \beta_{4} - \beta_{2}) q^{47} + ( - \beta_{7} - \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 4) q^{49} + (4 \beta_{7} + \beta_{6} + 3 \beta_{5} - 5 \beta_{4} + 4 \beta_{3} + 3 \beta_1 + 4) q^{50} + (2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{52} + ( - 2 \beta_{7} - 3 \beta_{5} - 6 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{53} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{2} - 4 \beta_1) q^{55} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_1 - 2) q^{56} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 3) q^{58} + ( - \beta_{7} - \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_1 - 3) q^{59} + ( - 6 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - \beta_{2} - 3 \beta_1 - 2) q^{61} + (\beta_{7} - \beta_{6} + \beta_{5} + 5 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + \beta_1 + 3) q^{62} + ( - 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} - 9 \beta_{4} - 4 \beta_{2} - 2 \beta_1 + 2) q^{64} + ( - 2 \beta_{7} + 2 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{65} + (4 \beta_{7} + 4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{67} + ( - 4 \beta_{7} - \beta_{6} - 3 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 4) q^{68} + (3 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 4) q^{70} + ( - 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} + 6 \beta_{3} - \beta_{2} + \cdots + 4) q^{71}+ \cdots + ( - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{2} - q^{4} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{2} - q^{4} - 10 q^{8} - 5 q^{10} + 4 q^{11} - 5 q^{13} - 13 q^{14} + 3 q^{16} + 10 q^{17} - 5 q^{19} + 15 q^{20} - 5 q^{23} - 10 q^{25} - 6 q^{26} - 15 q^{28} + 5 q^{29} - 9 q^{31} + 13 q^{34} - 15 q^{35} + 30 q^{37} - 15 q^{38} + 10 q^{40} + 4 q^{41} + 2 q^{44} - 19 q^{46} + 14 q^{49} + 15 q^{50} - 10 q^{52} + 10 q^{53} - 10 q^{55} - 10 q^{56} + 20 q^{58} - 9 q^{61} + 30 q^{62} + 4 q^{64} - 5 q^{65} + 20 q^{67} + 30 q^{70} - 6 q^{71} + 15 q^{73} + 12 q^{74} - 20 q^{76} - 10 q^{77} + 15 q^{79} - 20 q^{80} + 45 q^{83} - 15 q^{85} + 9 q^{86} - 20 q^{88} + 25 q^{89} + 6 q^{91} - 30 q^{92} - 27 q^{94} - 15 q^{95} - 60 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 420\nu^{7} - 776\nu^{6} + 698\nu^{5} - 1924\nu^{4} + 2297\nu^{3} + 5129\nu^{2} + 1055\nu - 10265 ) / 1355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 728\nu^{7} - 1327\nu^{6} + 1246\nu^{5} - 3353\nu^{4} + 3584\nu^{3} + 8547\nu^{2} + 2190\nu - 15715 ) / 1355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -857\nu^{7} + 1666\nu^{6} - 1743\nu^{5} + 4424\nu^{4} - 4907\nu^{3} - 9470\nu^{2} - 2485\nu + 18200 ) / 1355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 891\nu^{7} - 1623\nu^{6} + 1624\nu^{5} - 4492\nu^{4} + 4991\nu^{3} + 9520\nu^{2} + 3235\nu - 18960 ) / 1355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 955\nu^{7} - 1829\nu^{6} + 1942\nu^{5} - 4891\nu^{4} + 5723\nu^{3} + 9646\nu^{2} + 2415\nu - 20550 ) / 1355 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{5} - 3\beta_{4} + 4\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{7} - 5\beta_{6} + 4\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{6} - 7\beta_{3} + 11\beta_{2} + 4\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 21\beta_{2} - 2\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{7} + 38\beta_{5} + 23\beta_{4} - 23\beta_{3} + 12\beta_{2} \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.17421 0.0566033i
−0.983224 + 0.644389i
1.66637 + 0.917186i
−0.357358 1.86824i
1.66637 0.917186i
−0.357358 + 1.86824i
1.17421 + 0.0566033i
−0.983224 0.644389i
−0.174207 + 0.0566033i 0 −1.59089 + 1.15585i −0.107666 2.23347i 0 3.26086i 0.427051 0.587785i 0 0.145178 + 0.382993i
19.2 1.98322 0.644389i 0 1.89991 1.38036i 1.22570 + 1.87020i 0 0.992398i 0.427051 0.587785i 0 3.63597 + 2.91920i
64.1 −0.666375 0.917186i 0 0.220859 0.679734i 1.07822 1.95894i 0 0.407162i −2.92705 + 0.951057i 0 −2.51521 + 0.316463i
64.2 1.35736 + 1.86824i 0 −1.02988 + 3.16963i −2.19625 + 0.420099i 0 3.03582i −2.92705 + 0.951057i 0 −3.76594 3.53290i
109.1 −0.666375 + 0.917186i 0 0.220859 + 0.679734i 1.07822 + 1.95894i 0 0.407162i −2.92705 0.951057i 0 −2.51521 0.316463i
109.2 1.35736 1.86824i 0 −1.02988 3.16963i −2.19625 0.420099i 0 3.03582i −2.92705 0.951057i 0 −3.76594 + 3.53290i
154.1 −0.174207 0.0566033i 0 −1.59089 1.15585i −0.107666 + 2.23347i 0 3.26086i 0.427051 + 0.587785i 0 0.145178 0.382993i
154.2 1.98322 + 0.644389i 0 1.89991 + 1.38036i 1.22570 1.87020i 0 0.992398i 0.427051 + 0.587785i 0 3.63597 2.91920i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 154.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.2.m.a 8
3.b odd 2 1 25.2.e.a 8
12.b even 2 1 400.2.y.c 8
15.d odd 2 1 125.2.e.b 8
15.e even 4 2 125.2.d.b 16
25.e even 10 1 inner 225.2.m.a 8
25.f odd 20 2 5625.2.a.x 8
75.h odd 10 1 25.2.e.a 8
75.h odd 10 1 625.2.b.c 8
75.h odd 10 1 625.2.e.a 8
75.h odd 10 1 625.2.e.i 8
75.j odd 10 1 125.2.e.b 8
75.j odd 10 1 625.2.b.c 8
75.j odd 10 1 625.2.e.a 8
75.j odd 10 1 625.2.e.i 8
75.l even 20 2 125.2.d.b 16
75.l even 20 2 625.2.a.f 8
75.l even 20 4 625.2.d.o 16
300.r even 10 1 400.2.y.c 8
300.u odd 20 2 10000.2.a.bj 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 3.b odd 2 1
25.2.e.a 8 75.h odd 10 1
125.2.d.b 16 15.e even 4 2
125.2.d.b 16 75.l even 20 2
125.2.e.b 8 15.d odd 2 1
125.2.e.b 8 75.j odd 10 1
225.2.m.a 8 1.a even 1 1 trivial
225.2.m.a 8 25.e even 10 1 inner
400.2.y.c 8 12.b even 2 1
400.2.y.c 8 300.r even 10 1
625.2.a.f 8 75.l even 20 2
625.2.b.c 8 75.h odd 10 1
625.2.b.c 8 75.j odd 10 1
625.2.d.o 16 75.l even 20 4
625.2.e.a 8 75.h odd 10 1
625.2.e.a 8 75.j odd 10 1
625.2.e.i 8 75.h odd 10 1
625.2.e.i 8 75.j odd 10 1
5625.2.a.x 8 25.f odd 20 2
10000.2.a.bj 8 300.u odd 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 5T_{2}^{7} + 11T_{2}^{6} - 10T_{2}^{5} + T_{2}^{4} - 10T_{2}^{3} + 26T_{2}^{2} + 10T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 5 T^{7} + 11 T^{6} - 10 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{6} + 20 T^{5} + 5 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 21 T^{6} + 121 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{7} + 4 T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} - 10 T^{7} + 56 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} + 30 T^{6} + 40 T^{5} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{8} + 5 T^{7} - T^{6} + 15 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{8} - 5 T^{7} + 30 T^{6} + \cdots + 483025 \) Copy content Toggle raw display
$31$ \( T^{8} + 9 T^{7} + 117 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$37$ \( T^{8} - 30 T^{7} + 406 T^{6} + \cdots + 116281 \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} + 52 T^{6} + \cdots + 13456 \) Copy content Toggle raw display
$43$ \( T^{8} + 129 T^{6} + 4421 T^{4} + \cdots + 246016 \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{6} - 615 T^{5} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} - 10 T^{7} - 6 T^{6} + \cdots + 8755681 \) Copy content Toggle raw display
$59$ \( T^{8} - 15 T^{5} + 5635 T^{4} + \cdots + 4080400 \) Copy content Toggle raw display
$61$ \( T^{8} + 9 T^{7} - 43 T^{6} + \cdots + 116281 \) Copy content Toggle raw display
$67$ \( T^{8} - 20 T^{7} + 116 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} + 142 T^{6} + \cdots + 24245776 \) Copy content Toggle raw display
$73$ \( T^{8} - 15 T^{7} + 49 T^{6} + 120 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} - 15 T^{7} + 100 T^{6} + \cdots + 33408400 \) Copy content Toggle raw display
$83$ \( T^{8} - 45 T^{7} + 949 T^{6} + \cdots + 99856 \) Copy content Toggle raw display
$89$ \( T^{8} - 25 T^{7} + 520 T^{6} + \cdots + 1392400 \) Copy content Toggle raw display
$97$ \( T^{8} + 60 T^{7} + \cdots + 301334881 \) Copy content Toggle raw display
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