Properties

Label 225.4.k.a
Level $225$
Weight $4$
Character orbit 225.k
Analytic conductor $13.275$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (2 \beta_{7} + 2 \beta_{5} + \beta_{4} + 3 \beta_1) q^{3} + ( - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{6} + 7 \beta_{3} + \beta_{2} - 9) q^{6} + ( - \beta_{7} - 5 \beta_1) q^{7} + (\beta_{5} - 7 \beta_{4} + \beta_1) q^{8} + (3 \beta_{6} + 3 \beta_{3} - 6 \beta_{2} + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + (2 \beta_{7} + 2 \beta_{5} + \beta_{4} + 3 \beta_1) q^{3} + ( - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{4} + (\beta_{6} + 7 \beta_{3} + \beta_{2} - 9) q^{6} + ( - \beta_{7} - 5 \beta_1) q^{7} + (\beta_{5} - 7 \beta_{4} + \beta_1) q^{8} + (3 \beta_{6} + 3 \beta_{3} - 6 \beta_{2} + 27) q^{9} + (21 \beta_{3} - 5 \beta_{2} + 21) q^{11} + (2 \beta_{7} - 7 \beta_{5} + \beta_{4} - 15 \beta_1) q^{12} + (8 \beta_{7} - 52 \beta_{5} + 8 \beta_{4} + 8 \beta_1) q^{13} + (4 \beta_{6} - 4 \beta_{3} - 4 \beta_{2} + 4) q^{14} + (63 \beta_{3} - 7 \beta_{2} + 63) q^{16} + (51 \beta_{5} + 25 \beta_{4} + 51 \beta_1) q^{17} + ( - 24 \beta_{7} - 51 \beta_{5} - 3 \beta_{4} - 27 \beta_1) q^{18} + ( - 25 \beta_{6} - 30) q^{19} + ( - 9 \beta_{6} + 12 \beta_{3} + 6 \beta_{2} - 9) q^{21} + ( - 21 \beta_{7} - 61 \beta_{5} - 21 \beta_{4} - 21 \beta_1) q^{22} + (23 \beta_{7} - 122 \beta_{5} + 23 \beta_{4} + 23 \beta_1) q^{23} + ( - 6 \beta_{6} - 111 \beta_{3} - 9 \beta_{2} - 63) q^{24} + ( - 52 \beta_{6} - 64) q^{26} + (36 \beta_{7} + 117 \beta_{5} + 18 \beta_{4} + 135 \beta_1) q^{27} + (4 \beta_{5} - 4 \beta_{4} + 4 \beta_1) q^{28} + (182 \beta_{3} - 39 \beta_{2} + 182) q^{29} - 6 \beta_{3} q^{31} + ( - 7 \beta_{7} - 127 \beta_{5} - 7 \beta_{4} - 7 \beta_1) q^{32} + (11 \beta_{7} + 101 \beta_{5} + 37 \beta_{4} + 93 \beta_1) q^{33} + ( - 251 \beta_{3} + 51 \beta_{2} - 251) q^{34} + ( - 24 \beta_{6} + 3 \beta_{3} + 21 \beta_{2} + 27) q^{36} + ( - 324 \beta_{5} - 10 \beta_{4} - 324 \beta_1) q^{37} + (5 \beta_{7} + 200 \beta_1) q^{38} + (68 \beta_{6} - 40 \beta_{3} + 44 \beta_{2} + 144) q^{39} + (60 \beta_{6} + 149 \beta_{3} - 60 \beta_{2} + 60) q^{41} + (36 \beta_{5} - 12 \beta_{4} + 60 \beta_1) q^{42} + ( - 87 \beta_{7} - 92 \beta_1) q^{43} + ( - 21 \beta_{6} + 40) q^{44} + ( - 122 \beta_{6} - 184) q^{46} + ( - 11 \beta_{7} + 445 \beta_1) q^{47} + (49 \beta_{7} + 175 \beta_{5} + 119 \beta_{4} + 231 \beta_1) q^{48} + (9 \beta_{6} + 319 \beta_{3} - 9 \beta_{2} + 9) q^{49} + (76 \beta_{6} + 451 \beta_{3} - 77 \beta_{2} + 225) q^{51} + ( - 52 \beta_{7} - 64 \beta_1) q^{52} + ( - 62 \beta_{5} + 100 \beta_{4} - 62 \beta_1) q^{53} + (18 \beta_{6} + 45 \beta_{3} + 99 \beta_{2} - 243) q^{54} + (60 \beta_{3} + 36 \beta_{2} + 60) q^{56} + ( - 35 \beta_{7} + 190 \beta_{5} + 20 \beta_{4} - 240 \beta_1) q^{57} + ( - 182 \beta_{7} - 494 \beta_{5} - 182 \beta_{4} - 182 \beta_1) q^{58} + (49 \beta_{6} + 67 \beta_{3} - 49 \beta_{2} + 49) q^{59} + (26 \beta_{3} + 195 \beta_{2} + 26) q^{61} + (6 \beta_{5} + 6 \beta_{4} + 6 \beta_1) q^{62} + ( - 9 \beta_{7} - 36 \beta_{5} + 27 \beta_{4} - 117 \beta_1) q^{63} + ( - 71 \beta_{6} - 392) q^{64} + (19 \beta_{6} - 290 \beta_{3} + 82 \beta_{2} - 378) q^{66} + (184 \beta_{7} + 395 \beta_{5} + 184 \beta_{4} + 184 \beta_1) q^{67} + (51 \beta_{7} + 251 \beta_{5} + 51 \beta_{4} + 51 \beta_1) q^{68} + (168 \beta_{6} - 60 \beta_{3} + 99 \beta_{2} + 414) q^{69} + (110 \beta_{6} + 252) q^{71} + ( - 27 \beta_{7} + 189 \beta_{5} - 171 \beta_{4} - 171 \beta_1) q^{72} + (303 \beta_{5} - 205 \beta_{4} + 303 \beta_1) q^{73} + (404 \beta_{3} - 324 \beta_{2} + 404) q^{74} + (5 \beta_{6} - 195 \beta_{3} - 5 \beta_{2} + 5) q^{76} + (4 \beta_{7} + 44 \beta_{5} + 4 \beta_{4} + 4 \beta_1) q^{77} + ( - 76 \beta_{7} + 392 \beta_{5} + 40 \beta_{4} - 504 \beta_1) q^{78} + ( - 458 \beta_{3} + 76 \beta_{2} - 458) q^{79} + (135 \beta_{6} + 135 \beta_{3} - 270 \beta_{2} + 486) q^{81} + ( - 629 \beta_{5} - 149 \beta_{4} - 629 \beta_1) q^{82} + (453 \beta_{7} + 33 \beta_1) q^{83} + ( - 60 \beta_{3} - 12 \beta_{2} - 36) q^{84} + (5 \beta_{6} - 691 \beta_{3} - 5 \beta_{2} + 5) q^{86} + (104 \beta_{7} + 806 \beta_{5} + 325 \beta_{4} + 780 \beta_1) q^{87} + (107 \beta_{7} - 152 \beta_1) q^{88} + ( - 315 \beta_{6} + 375) q^{89} + ( - 92 \beta_{6} - 364) q^{91} + ( - 122 \beta_{7} - 184 \beta_1) q^{92} + (6 \beta_{7} + 6 \beta_{5} - 6 \beta_{4}) q^{93} + ( - 456 \beta_{6} - 544 \beta_{3} + 456 \beta_{2} - 456) q^{94} + (113 \beta_{6} - 310 \beta_{3} + 134 \beta_{2} - 126) q^{96} + ( - 131 \beta_{7} + 450 \beta_1) q^{97} + ( - 391 \beta_{5} - 319 \beta_{4} - 391 \beta_1) q^{98} + (111 \beta_{6} + 624 \beta_{3} - 168 \beta_{2} + 432) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 102 q^{6} + 180 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 102 q^{6} + 180 q^{9} + 74 q^{11} + 24 q^{14} + 238 q^{16} - 140 q^{19} - 72 q^{21} - 54 q^{24} - 304 q^{26} + 650 q^{29} + 24 q^{31} - 902 q^{34} + 342 q^{36} + 1128 q^{39} - 476 q^{41} + 404 q^{44} - 984 q^{46} - 1258 q^{49} - 462 q^{51} - 1998 q^{54} + 312 q^{56} - 170 q^{59} + 494 q^{61} - 2852 q^{64} - 1776 q^{66} + 3078 q^{69} + 1576 q^{71} + 968 q^{74} + 790 q^{76} - 1680 q^{79} + 2268 q^{81} - 72 q^{84} + 2774 q^{86} + 4260 q^{89} - 2544 q^{91} + 1264 q^{94} + 48 q^{96} + 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 16\nu^{4} - 32\nu^{2} - 51 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 5\nu^{4} - 7\nu^{2} - 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} - 3\nu ) / 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + 2\beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 7\beta_{3} - \beta_{2} - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 11\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{6} + 13\beta_{3} - 5\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} - \beta_{5} - 2\beta_{4} + 45\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -16\beta_{6} - 16\beta_{3} + 32\beta_{2} - 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{7} + 118\beta_{5} - 13\beta_{4} ) / 3 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.396143 + 1.68614i
−1.26217 + 1.18614i
1.26217 1.18614i
0.396143 1.68614i
−0.396143 1.68614i
−1.26217 1.18614i
1.26217 + 1.18614i
0.396143 + 1.68614i
−2.92048 + 1.68614i 4.97494 + 1.50000i 1.68614 2.92048i 0 −17.0584 + 4.00772i 1.40965 0.813859i 15.6060i 22.5000 + 14.9248i 0
49.2 −2.05446 + 1.18614i 4.97494 1.50000i −1.18614 + 2.05446i 0 −8.44158 + 8.98266i −6.38458 + 3.68614i 24.6060i 22.5000 14.9248i 0
49.3 2.05446 1.18614i −4.97494 + 1.50000i −1.18614 + 2.05446i 0 −8.44158 + 8.98266i 6.38458 3.68614i 24.6060i 22.5000 14.9248i 0
49.4 2.92048 1.68614i −4.97494 1.50000i 1.68614 2.92048i 0 −17.0584 + 4.00772i −1.40965 + 0.813859i 15.6060i 22.5000 + 14.9248i 0
124.1 −2.92048 1.68614i 4.97494 1.50000i 1.68614 + 2.92048i 0 −17.0584 4.00772i 1.40965 + 0.813859i 15.6060i 22.5000 14.9248i 0
124.2 −2.05446 1.18614i 4.97494 + 1.50000i −1.18614 2.05446i 0 −8.44158 8.98266i −6.38458 3.68614i 24.6060i 22.5000 + 14.9248i 0
124.3 2.05446 + 1.18614i −4.97494 1.50000i −1.18614 2.05446i 0 −8.44158 8.98266i 6.38458 + 3.68614i 24.6060i 22.5000 + 14.9248i 0
124.4 2.92048 + 1.68614i −4.97494 + 1.50000i 1.68614 + 2.92048i 0 −17.0584 4.00772i −1.40965 0.813859i 15.6060i 22.5000 14.9248i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.4
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 225.4.k.a 8
5.b even 2 1 inner 225.4.k.a 8
5.c odd 4 1 45.4.e.a 4
5.c odd 4 1 225.4.e.a 4
9.c even 3 1 inner 225.4.k.a 8
15.e even 4 1 135.4.e.a 4
45.j even 6 1 inner 225.4.k.a 8
45.k odd 12 1 45.4.e.a 4
45.k odd 12 1 225.4.e.a 4
45.k odd 12 1 405.4.a.d 2
45.k odd 12 1 2025.4.a.l 2
45.l even 12 1 135.4.e.a 4
45.l even 12 1 405.4.a.e 2
45.l even 12 1 2025.4.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.a 4 5.c odd 4 1
45.4.e.a 4 45.k odd 12 1
135.4.e.a 4 15.e even 4 1
135.4.e.a 4 45.l even 12 1
225.4.e.a 4 5.c odd 4 1
225.4.e.a 4 45.k odd 12 1
225.4.k.a 8 1.a even 1 1 trivial
225.4.k.a 8 5.b even 2 1 inner
225.4.k.a 8 9.c even 3 1 inner
225.4.k.a 8 45.j even 6 1 inner
405.4.a.d 2 45.k odd 12 1
405.4.a.e 2 45.l even 12 1
2025.4.a.j 2 45.l even 12 1
2025.4.a.l 2 45.k odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 17T_{2}^{6} + 225T_{2}^{4} - 1088T_{2}^{2} + 4096 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 17 T^{6} + 225 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( (T^{4} - 45 T^{2} + 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 57 T^{6} + 3105 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$11$ \( (T^{4} - 37 T^{3} + 1233 T^{2} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 7328 T^{6} + \cdots + 46262633168896 \) Copy content Toggle raw display
$17$ \( (T^{4} + 13277 T^{2} + 13498276)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 35 T - 4850)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 44373 T^{6} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{4} - 325 T^{3} + 91767 T^{2} + \cdots + 192044164)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 36)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 205172 T^{2} + \cdots + 10188076096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 238 T^{3} + 72183 T^{2} + \cdots + 241460521)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 129593 T^{6} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{8} - 407897 T^{6} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + 190088 T^{2} + \cdots + 4893841936)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 85 T^{3} + 25227 T^{2} + \cdots + 324072004)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 247 T^{3} + \cdots + 89074790116)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 742242 T^{6} + \cdots + 12\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{2} - 394 T - 61016)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1022273 T^{2} + \cdots + 33224540176)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 840 T^{3} + \cdots + 16576047504)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 3460833 T^{6} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1065 T - 535050)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 814637 T^{6} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
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