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 $4$

Related objects

Downloads

Learn more

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: \(x^{4} + 1\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( 9 - 9 \zeta_{8}^{2} ) q^{7} -21 \zeta_{8}^{3} q^{8} +O(q^{10})\) \( q + 3 \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} + ( 9 - 9 \zeta_{8}^{2} ) q^{7} -21 \zeta_{8}^{3} q^{8} + ( -27 \zeta_{8} - 27 \zeta_{8}^{3} ) q^{11} + ( -63 - 63 \zeta_{8}^{2} ) q^{13} + ( 27 \zeta_{8} - 27 \zeta_{8}^{3} ) q^{14} + 71 q^{16} + 42 \zeta_{8} q^{17} + 70 \zeta_{8}^{2} q^{19} + ( 81 - 81 \zeta_{8}^{2} ) q^{22} -102 \zeta_{8}^{3} q^{23} + ( -189 \zeta_{8} - 189 \zeta_{8}^{3} ) q^{26} + ( 9 + 9 \zeta_{8}^{2} ) q^{28} + ( 162 \zeta_{8} - 162 \zeta_{8}^{3} ) q^{29} + 196 q^{31} + 45 \zeta_{8} q^{32} + 126 \zeta_{8}^{2} q^{34} + ( -207 + 207 \zeta_{8}^{2} ) q^{37} + 210 \zeta_{8}^{3} q^{38} + ( -189 \zeta_{8} - 189 \zeta_{8}^{3} ) q^{41} + ( 144 + 144 \zeta_{8}^{2} ) q^{43} + ( 27 \zeta_{8} - 27 \zeta_{8}^{3} ) q^{44} + 306 q^{46} -504 \zeta_{8} q^{47} + 181 \zeta_{8}^{2} q^{49} + ( 63 - 63 \zeta_{8}^{2} ) q^{52} + 318 \zeta_{8}^{3} q^{53} + ( -189 \zeta_{8} - 189 \zeta_{8}^{3} ) q^{56} + ( 486 + 486 \zeta_{8}^{2} ) q^{58} + ( 189 \zeta_{8} - 189 \zeta_{8}^{3} ) q^{59} -322 q^{61} + 588 \zeta_{8} q^{62} -433 \zeta_{8}^{2} q^{64} + ( -378 + 378 \zeta_{8}^{2} ) q^{67} + 42 \zeta_{8}^{3} q^{68} + ( 594 \zeta_{8} + 594 \zeta_{8}^{3} ) q^{71} + ( -378 - 378 \zeta_{8}^{2} ) q^{73} + ( -621 \zeta_{8} + 621 \zeta_{8}^{3} ) q^{74} -70 q^{76} -486 \zeta_{8} q^{77} + 488 \zeta_{8}^{2} q^{79} + ( 567 - 567 \zeta_{8}^{2} ) q^{82} -1092 \zeta_{8}^{3} q^{83} + ( 432 \zeta_{8} + 432 \zeta_{8}^{3} ) q^{86} + ( -567 - 567 \zeta_{8}^{2} ) q^{88} + ( 189 \zeta_{8} - 189 \zeta_{8}^{3} ) q^{89} -1134 q^{91} + 102 \zeta_{8} q^{92} -1512 \zeta_{8}^{2} q^{94} + ( 252 - 252 \zeta_{8}^{2} ) q^{97} + 543 \zeta_{8}^{3} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{7} + O(q^{10}) \) \( 4 q + 36 q^{7} - 252 q^{13} + 284 q^{16} + 324 q^{22} + 36 q^{28} + 784 q^{31} - 828 q^{37} + 576 q^{43} + 1224 q^{46} + 252 q^{52} + 1944 q^{58} - 1288 q^{61} - 1512 q^{67} - 1512 q^{73} - 280 q^{76} + 2268 q^{82} - 2268 q^{88} - 4536 q^{91} + 1008 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\)

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 \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{4} + 81 \)
\( T_{7}^{2} - 18 T_{7} + 162 \)

Hecke characteristic polynomials

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