Properties

Label 225.4.f.b
Level $225$
Weight $4$
Character orbit 225.f
Analytic conductor $13.275$
Analytic rank $0$
Dimension $4$
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.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - 9 \beta_{2} + 9) q^{7} - 7 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - 9 \beta_{2} + 9) q^{7} - 7 \beta_{3} q^{8} + ( - 9 \beta_{3} - 9 \beta_1) q^{11} + ( - 63 \beta_{2} - 63) q^{13} + ( - 9 \beta_{3} + 9 \beta_1) q^{14} + 71 q^{16} + 14 \beta_1 q^{17} + 70 \beta_{2} q^{19} + ( - 81 \beta_{2} + 81) q^{22} - 34 \beta_{3} q^{23} + ( - 63 \beta_{3} - 63 \beta_1) q^{26} + (9 \beta_{2} + 9) q^{28} + ( - 54 \beta_{3} + 54 \beta_1) q^{29} + 196 q^{31} + 15 \beta_1 q^{32} + 126 \beta_{2} q^{34} + (207 \beta_{2} - 207) q^{37} + 70 \beta_{3} q^{38} + ( - 63 \beta_{3} - 63 \beta_1) q^{41} + (144 \beta_{2} + 144) q^{43} + ( - 9 \beta_{3} + 9 \beta_1) q^{44} + 306 q^{46} - 168 \beta_1 q^{47} + 181 \beta_{2} q^{49} + ( - 63 \beta_{2} + 63) q^{52} + 106 \beta_{3} q^{53} + ( - 63 \beta_{3} - 63 \beta_1) q^{56} + (486 \beta_{2} + 486) q^{58} + ( - 63 \beta_{3} + 63 \beta_1) q^{59} - 322 q^{61} + 196 \beta_1 q^{62} - 433 \beta_{2} q^{64} + (378 \beta_{2} - 378) q^{67} + 14 \beta_{3} q^{68} + (198 \beta_{3} + 198 \beta_1) q^{71} + ( - 378 \beta_{2} - 378) q^{73} + (207 \beta_{3} - 207 \beta_1) q^{74} - 70 q^{76} - 162 \beta_1 q^{77} + 488 \beta_{2} q^{79} + ( - 567 \beta_{2} + 567) q^{82} - 364 \beta_{3} q^{83} + (144 \beta_{3} + 144 \beta_1) q^{86} + ( - 567 \beta_{2} - 567) q^{88} + ( - 63 \beta_{3} + 63 \beta_1) q^{89} - 1134 q^{91} + 34 \beta_1 q^{92} - 1512 \beta_{2} q^{94} + ( - 252 \beta_{2} + 252) q^{97} + 181 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{7} - 252 q^{13} + 284 q^{16} + 324 q^{22} + 36 q^{28} + 784 q^{31} - 828 q^{37} + 576 q^{43} + 1224 q^{46} + 252 q^{52} + 1944 q^{58} - 1288 q^{61} - 1512 q^{67} - 1512 q^{73} - 280 q^{76} + 2268 q^{82} - 2268 q^{88} - 4536 q^{91} + 1008 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\zeta_{8}^{3} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( \beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−2.12132 + 2.12132i 0 1.00000i 0 0 9.00000 + 9.00000i −14.8492 14.8492i 0 0
107.2 2.12132 2.12132i 0 1.00000i 0 0 9.00000 + 9.00000i 14.8492 + 14.8492i 0 0
143.1 −2.12132 2.12132i 0 1.00000i 0 0 9.00000 9.00000i −14.8492 + 14.8492i 0 0
143.2 2.12132 + 2.12132i 0 1.00000i 0 0 9.00000 9.00000i 14.8492 14.8492i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.f.b yes 4
3.b odd 2 1 inner 225.4.f.b yes 4
5.b even 2 1 225.4.f.a 4
5.c odd 4 1 225.4.f.a 4
5.c odd 4 1 inner 225.4.f.b yes 4
15.d odd 2 1 225.4.f.a 4
15.e even 4 1 225.4.f.a 4
15.e even 4 1 inner 225.4.f.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.f.a 4 5.b even 2 1
225.4.f.a 4 5.c odd 4 1
225.4.f.a 4 15.d odd 2 1
225.4.f.a 4 15.e even 4 1
225.4.f.b yes 4 1.a even 1 1 trivial
225.4.f.b yes 4 3.b odd 2 1 inner
225.4.f.b yes 4 5.c odd 4 1 inner
225.4.f.b yes 4 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{4} + 81 \) Copy content Toggle raw display
\( T_{7}^{2} - 18T_{7} + 162 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 81 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 18 T + 162)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1458)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 126 T + 7938)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 3111696 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4900)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 108243216 \) Copy content Toggle raw display
$29$ \( (T^{2} - 52488)^{2} \) Copy content Toggle raw display
$31$ \( (T - 196)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 414 T + 85698)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 71442)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 288 T + 41472)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 64524128256 \) Copy content Toggle raw display
$53$ \( T^{4} + 10226063376 \) Copy content Toggle raw display
$59$ \( (T^{2} - 71442)^{2} \) Copy content Toggle raw display
$61$ \( (T + 322)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 756 T + 285768)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 705672)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 756 T + 285768)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 238144)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1421970391296 \) Copy content Toggle raw display
$89$ \( (T^{2} - 71442)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 504 T + 127008)^{2} \) Copy content Toggle raw display
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