Properties

Label 225.2.e.c
Level $225$
Weight $2$
Character orbit 225.e
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1223810289.2
Defining polynomial: \( x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{7} - \beta_{4}) q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 - 1) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{2} + 1) q^{6} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + \beta_{2} + 3) q^{8} + (\beta_{7} - 2 \beta_{5} + \beta_{3} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{7} - \beta_{4}) q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 - 1) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{2} + 1) q^{6} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + \beta_{2} + 3) q^{8} + (\beta_{7} - 2 \beta_{5} + \beta_{3} + \beta_1 + 1) q^{9} + (\beta_{7} + \beta_{6} - 2 \beta_{4} - \beta_{2} - \beta_1) q^{11} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{12} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_1 - 2) q^{13} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{2}) q^{14} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{2} - \beta_1) q^{16} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 3) q^{17} + (\beta_{7} - 4 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 + 2) q^{18} + ( - \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 1) q^{19} + (\beta_{6} - \beta_{5} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{21} + (\beta_{7} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{22} + (\beta_{7} + \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{2} - 3) q^{23} + (\beta_{7} - 2 \beta_{4} - 3 \beta_{3} - 3) q^{24} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 3) q^{26} + ( - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{27} + ( - 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{28} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{29} + ( - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{31} + ( - 3 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} - 2 \beta_{3} + 2 \beta_1 - 3) q^{32} + (\beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{3} + 2 \beta_{2} - 5) q^{33} + ( - \beta_{6} + \beta_{4} - 4 \beta_1) q^{34} + ( - \beta_{7} + 5 \beta_{5} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 4) q^{36} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} - 1) q^{37} + (\beta_{7} + \beta_{6} - 6 \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1) q^{38} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5} + 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 1) q^{39} + (\beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{41} + ( - \beta_{7} + 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 3 \beta_1 - 3) q^{42} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + 4 \beta_1) q^{43} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + 6) q^{44} + (2 \beta_{7} - 4 \beta_{6} - 4 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 3) q^{46} + (\beta_{7} - 2 \beta_{6} - 6 \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1) q^{47} + ( - 2 \beta_{7} + \beta_{6} + 5 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 5) q^{48}+ \cdots + ( - 4 \beta_{7} - 6 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} - \beta_{3} - 3 \beta_{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + q^{3} - 4 q^{4} + 8 q^{6} + q^{7} + 18 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} + q^{3} - 4 q^{4} + 8 q^{6} + q^{7} + 18 q^{8} + 5 q^{9} + q^{11} + 11 q^{12} - 2 q^{13} - 3 q^{14} - 4 q^{16} + 22 q^{17} - 5 q^{18} + 4 q^{19} - 15 q^{21} - 3 q^{22} - 15 q^{23} - 33 q^{24} - 20 q^{26} - 2 q^{27} - 8 q^{28} - q^{29} + 4 q^{31} - 10 q^{32} - 28 q^{33} - 9 q^{34} - 14 q^{36} - 2 q^{37} - 23 q^{38} + 25 q^{39} + 5 q^{41} - 21 q^{42} + 10 q^{43} + 44 q^{44} - 20 q^{47} + 53 q^{48} + 3 q^{49} + 11 q^{51} - 17 q^{52} + 40 q^{53} + 26 q^{54} + 30 q^{56} - 8 q^{57} + 18 q^{58} - 17 q^{59} + 13 q^{61} - 12 q^{62} + 9 q^{63} + 38 q^{64} - 8 q^{66} - 17 q^{67} - 34 q^{68} - 27 q^{69} - 16 q^{71} + 18 q^{72} + 4 q^{73} - 40 q^{74} - 11 q^{76} - 12 q^{77} - 61 q^{78} + 7 q^{79} + 17 q^{81} + 24 q^{82} - 30 q^{83} + 27 q^{84} + 34 q^{86} - 23 q^{87} - 9 q^{88} - 18 q^{89} - 34 q^{91} + 12 q^{92} + 15 q^{93} - 3 q^{94} + 34 q^{96} + 19 q^{97} + 26 q^{98} - 17 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - 67\nu^{6} + 79\nu^{5} - 664\nu^{4} + 181\nu^{3} - 2269\nu^{2} + 54\nu - 2367 ) / 1233 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -76\nu^{7} - 32\nu^{6} - 361\nu^{5} - 722\nu^{4} - 3496\nu^{3} - 1691\nu^{2} - 741\nu - 2934 ) / 6165 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 108\nu^{7} - 279\nu^{6} + 1198\nu^{5} - 1029\nu^{4} + 3598\nu^{3} - 1707\nu^{2} + 5848\nu - 1563 ) / 2055 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -326\nu^{7} + 728\nu^{6} - 2576\nu^{5} + 1013\nu^{4} - 6776\nu^{3} + 6104\nu^{2} - 10371\nu + 2016 ) / 6165 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -407\nu^{7} + 1451\nu^{6} - 4502\nu^{5} + 5381\nu^{4} - 10502\nu^{3} + 14063\nu^{2} - 18867\nu + 1647 ) / 6165 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -33\nu^{7} + 51\nu^{6} - 191\nu^{5} - 108\nu^{4} - 422\nu^{3} - 15\nu^{2} - 356\nu - 744 ) / 411 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} + 3\beta_{5} - \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} - 2\beta_{4} - 4\beta_{3} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{7} - 2\beta_{6} - 15\beta_{5} - 6\beta_{4} - 8\beta_{2} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{7} + 11\beta_{6} - 27\beta_{5} + 8\beta_{4} + 22\beta_{3} - 8\beta_{2} - 22\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{7} + 57\beta_{6} + 57\beta_{4} + 49\beta_{3} + 38\beta_{2} + 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -144\beta_{7} + 57\beta_{6} + 204\beta_{5} + 87\beta_{4} + 144\beta_{2} + 139\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\)

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
1.31686 + 2.28087i
0.736627 + 1.27588i
−0.236627 0.409850i
−0.816862 1.41485i
1.31686 2.28087i
0.736627 1.27588i
−0.236627 + 0.409850i
−0.816862 + 1.41485i
−1.31686 2.28087i −1.71558 + 0.238330i −2.46825 + 4.27513i 0 2.80278 + 3.59916i 0.898714 + 1.55662i 7.73393 2.88640 0.817746i 0
76.2 −0.736627 1.27588i 1.69629 + 0.350156i −0.0852394 + 0.147639i 0 −0.802776 2.42219i −1.93291 3.34791i −2.69535 2.75478 + 1.18793i 0
76.3 0.236627 + 0.409850i −0.544899 + 1.64411i 0.888015 1.53809i 0 −0.802776 + 0.165713i 1.28153 + 2.21967i 1.78702 −2.40617 1.79175i 0
76.4 0.816862 + 1.41485i 1.06419 1.36657i −0.334526 + 0.579416i 0 2.80278 + 0.389365i 0.252674 + 0.437645i 2.17440 −0.735010 2.90857i 0
151.1 −1.31686 + 2.28087i −1.71558 0.238330i −2.46825 4.27513i 0 2.80278 3.59916i 0.898714 1.55662i 7.73393 2.88640 + 0.817746i 0
151.2 −0.736627 + 1.27588i 1.69629 0.350156i −0.0852394 0.147639i 0 −0.802776 + 2.42219i −1.93291 + 3.34791i −2.69535 2.75478 1.18793i 0
151.3 0.236627 0.409850i −0.544899 1.64411i 0.888015 + 1.53809i 0 −0.802776 0.165713i 1.28153 2.21967i 1.78702 −2.40617 + 1.79175i 0
151.4 0.816862 1.41485i 1.06419 + 1.36657i −0.334526 0.579416i 0 2.80278 0.389365i 0.252674 0.437645i 2.17440 −0.735010 + 2.90857i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.4
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.c 8
3.b odd 2 1 675.2.e.e 8
5.b even 2 1 225.2.e.e yes 8
5.c odd 4 2 225.2.k.c 16
9.c even 3 1 inner 225.2.e.c 8
9.c even 3 1 2025.2.a.y 4
9.d odd 6 1 675.2.e.e 8
9.d odd 6 1 2025.2.a.p 4
15.d odd 2 1 675.2.e.c 8
15.e even 4 2 675.2.k.c 16
45.h odd 6 1 675.2.e.c 8
45.h odd 6 1 2025.2.a.z 4
45.j even 6 1 225.2.e.e yes 8
45.j even 6 1 2025.2.a.q 4
45.k odd 12 2 225.2.k.c 16
45.k odd 12 2 2025.2.b.n 8
45.l even 12 2 675.2.k.c 16
45.l even 12 2 2025.2.b.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c 8 1.a even 1 1 trivial
225.2.e.c 8 9.c even 3 1 inner
225.2.e.e yes 8 5.b even 2 1
225.2.e.e yes 8 45.j even 6 1
225.2.k.c 16 5.c odd 4 2
225.2.k.c 16 45.k odd 12 2
675.2.e.c 8 15.d odd 2 1
675.2.e.c 8 45.h odd 6 1
675.2.e.e 8 3.b odd 2 1
675.2.e.e 8 9.d odd 6 1
675.2.k.c 16 15.e even 4 2
675.2.k.c 16 45.l even 12 2
2025.2.a.p 4 9.d odd 6 1
2025.2.a.q 4 45.j even 6 1
2025.2.a.y 4 9.c even 3 1
2025.2.a.z 4 45.h odd 6 1
2025.2.b.n 8 45.k odd 12 2
2025.2.b.o 8 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 8T_{2}^{6} + 2T_{2}^{5} + 23T_{2}^{4} + 8T_{2}^{3} + 37T_{2}^{2} - 15T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + 8 T^{6} + 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} - 2 T^{6} + 3 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + 13 T^{6} - 36 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{8} - T^{7} + 26 T^{6} + 107 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + 34 T^{6} + \cdots + 11449 \) Copy content Toggle raw display
$17$ \( (T^{4} - 11 T^{3} + 20 T^{2} + 110 T - 303)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} - 27 T^{2} + 80 T - 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 15 T^{7} + 168 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} + 41 T^{6} + 244 T^{5} + \cdots + 16641 \) Copy content Toggle raw display
$31$ \( T^{8} - 4 T^{7} + 58 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$37$ \( (T^{4} + T^{3} - 99 T^{2} - 503 T - 647)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 5 T^{7} + 50 T^{6} + \cdots + 42849 \) Copy content Toggle raw display
$43$ \( T^{8} - 10 T^{7} + 148 T^{6} + \cdots + 452929 \) Copy content Toggle raw display
$47$ \( T^{8} + 20 T^{7} + 293 T^{6} + \cdots + 145161 \) Copy content Toggle raw display
$53$ \( (T^{4} - 20 T^{3} + 86 T^{2} + 179 T - 471)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 17 T^{7} + 287 T^{6} + \cdots + 5349969 \) Copy content Toggle raw display
$61$ \( T^{8} - 13 T^{7} + 172 T^{6} - 143 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} + 17 T^{7} + 253 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$71$ \( (T^{4} + 8 T^{3} - 40 T^{2} - 263 T + 381)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{3} - 96 T^{2} - 241 T + 113)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 7 T^{7} + 82 T^{6} + \cdots + 42849 \) Copy content Toggle raw display
$83$ \( T^{8} + 30 T^{7} + 612 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$89$ \( (T^{4} + 9 T^{3} - 99 T^{2} - 405 T + 2025)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 19 T^{7} + 280 T^{6} + \cdots + 908209 \) Copy content Toggle raw display
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