Properties

Label 225.4.b.i
Level $225$
Weight $4$
Character orbit 225.b
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.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{10})\)
Defining polynomial: \(x^{4} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} -2 q^{4} + 3 \beta_{1} q^{7} -6 \beta_{2} q^{8} +O(q^{10})\) \( q -\beta_{2} q^{2} -2 q^{4} + 3 \beta_{1} q^{7} -6 \beta_{2} q^{8} -4 \beta_{3} q^{11} -7 \beta_{1} q^{13} + 3 \beta_{3} q^{14} -76 q^{16} -28 \beta_{2} q^{17} -91 q^{19} + 40 \beta_{1} q^{22} -36 \beta_{2} q^{23} -7 \beta_{3} q^{26} -6 \beta_{1} q^{28} -4 \beta_{3} q^{29} -147 q^{31} + 28 \beta_{2} q^{32} -280 q^{34} + 74 \beta_{1} q^{37} + 91 \beta_{2} q^{38} + 28 \beta_{3} q^{41} -67 \beta_{1} q^{43} + 8 \beta_{3} q^{44} -360 q^{46} + 56 \beta_{2} q^{47} + 118 q^{49} + 14 \beta_{1} q^{52} + 28 \beta_{2} q^{53} + 18 \beta_{3} q^{56} + 40 \beta_{1} q^{58} -56 \beta_{3} q^{59} + 427 q^{61} + 147 \beta_{2} q^{62} -328 q^{64} + 3 \beta_{1} q^{67} + 56 \beta_{2} q^{68} + 4 \beta_{3} q^{71} + 14 \beta_{1} q^{73} + 74 \beta_{3} q^{74} + 182 q^{76} -300 \beta_{2} q^{77} + 876 q^{79} -280 \beta_{1} q^{82} -168 \beta_{2} q^{83} -67 \beta_{3} q^{86} + 240 \beta_{1} q^{88} + 525 q^{91} + 72 \beta_{2} q^{92} + 560 q^{94} -217 \beta_{1} q^{97} -118 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + O(q^{10}) \) \( 4 q - 8 q^{4} - 304 q^{16} - 364 q^{19} - 588 q^{31} - 1120 q^{34} - 1440 q^{46} + 472 q^{49} + 1708 q^{61} - 1312 q^{64} + 728 q^{76} + 3504 q^{79} + 2100 q^{91} + 2240 q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 5 \beta_{2}\)\()/10\)
\(\nu^{2}\)\(=\)\(\beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} + 5 \beta_{2}\)\()/2\)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−1.58114 + 1.58114i
1.58114 + 1.58114i
1.58114 1.58114i
−1.58114 1.58114i
3.16228i 0 −2.00000 0 0 15.0000i 18.9737i 0 0
199.2 3.16228i 0 −2.00000 0 0 15.0000i 18.9737i 0 0
199.3 3.16228i 0 −2.00000 0 0 15.0000i 18.9737i 0 0
199.4 3.16228i 0 −2.00000 0 0 15.0000i 18.9737i 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.b even 2 1 inner
15.d odd 2 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{2} + 10 \)
\( T_{7}^{2} + 225 \)
\( T_{11}^{2} - 4000 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 10 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 225 + T^{2} )^{2} \)
$11$ \( ( -4000 + T^{2} )^{2} \)
$13$ \( ( 1225 + T^{2} )^{2} \)
$17$ \( ( 7840 + T^{2} )^{2} \)
$19$ \( ( 91 + T )^{4} \)
$23$ \( ( 12960 + T^{2} )^{2} \)
$29$ \( ( -4000 + T^{2} )^{2} \)
$31$ \( ( 147 + T )^{4} \)
$37$ \( ( 136900 + T^{2} )^{2} \)
$41$ \( ( -196000 + T^{2} )^{2} \)
$43$ \( ( 112225 + T^{2} )^{2} \)
$47$ \( ( 31360 + T^{2} )^{2} \)
$53$ \( ( 7840 + T^{2} )^{2} \)
$59$ \( ( -784000 + T^{2} )^{2} \)
$61$ \( ( -427 + T )^{4} \)
$67$ \( ( 225 + T^{2} )^{2} \)
$71$ \( ( -4000 + T^{2} )^{2} \)
$73$ \( ( 4900 + T^{2} )^{2} \)
$79$ \( ( -876 + T )^{4} \)
$83$ \( ( 282240 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 1177225 + T^{2} )^{2} \)
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