Properties

Label 225.6.b.i
Level $225$
Weight $6$
Character orbit 225.b
Analytic conductor $36.086$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.0863594579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Defining polynomial: \(x^{4} + 121 x^{2} + 3600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 25)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( -35 + \beta_{3} ) q^{4} + ( -4 \beta_{1} - 18 \beta_{2} ) q^{7} + ( 15 \beta_{1} - 69 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{2} ) q^{2} + ( -35 + \beta_{3} ) q^{4} + ( -4 \beta_{1} - 18 \beta_{2} ) q^{7} + ( 15 \beta_{1} - 69 \beta_{2} ) q^{8} + ( 103 - 10 \beta_{3} ) q^{11} + ( 32 \beta_{1} - 52 \beta_{2} ) q^{13} + ( 18 - 18 \beta_{3} ) q^{14} + ( 587 - 37 \beta_{3} ) q^{16} + ( -136 \beta_{1} + 217 \beta_{2} ) q^{17} + ( 1583 + 14 \beta_{3} ) q^{19} + ( -223 \beta_{1} + 763 \beta_{2} ) q^{22} + ( -12 \beta_{1} - 150 \beta_{2} ) q^{23} + ( 2404 - 52 \beta_{3} ) q^{26} + ( -362 \beta_{1} + 630 \beta_{2} ) q^{28} + ( -1952 - 16 \beta_{3} ) q^{29} + ( -438 - 220 \beta_{3} ) q^{31} + ( -551 \beta_{1} + 821 \beta_{2} ) q^{32} + ( -10165 + 217 \beta_{3} ) q^{34} + ( -384 \beta_{1} - 10 \beta_{2} ) q^{37} + ( -1415 \beta_{1} + 659 \beta_{2} ) q^{38} + ( -13837 - 80 \beta_{3} ) q^{41} + ( -2128 \beta_{1} + 764 \beta_{2} ) q^{43} + ( -18665 + 443 \beta_{3} ) q^{44} + ( 1302 - 150 \beta_{3} ) q^{46} + ( 1544 \beta_{1} + 1804 \beta_{2} ) q^{47} + ( 5923 - 160 \beta_{3} ) q^{49} + ( -2004 \beta_{1} + 4172 \beta_{2} ) q^{52} + ( -752 \beta_{1} + 3074 \beta_{2} ) q^{53} + ( -27162 + 54 \beta_{3} ) q^{56} + ( 1760 \beta_{1} - 896 \beta_{2} ) q^{58} + ( 6176 - 392 \beta_{3} ) q^{59} + ( -12398 + 400 \beta_{3} ) q^{61} + ( -2202 \beta_{1} + 14082 \beta_{2} ) q^{62} + ( -21643 - 363 \beta_{3} ) q^{64} + ( 1586 \beta_{1} + 3213 \beta_{2} ) q^{67} + ( 8417 \beta_{1} - 17543 \beta_{2} ) q^{68} + ( 43748 - 200 \beta_{3} ) q^{71} + ( 1112 \beta_{1} - 7585 \beta_{2} ) q^{73} + ( -20606 - 10 \beta_{3} ) q^{74} + ( -34321 + 1107 \beta_{3} ) q^{76} + ( 4608 \beta_{1} - 1854 \beta_{2} ) q^{77} + ( -32238 - 1004 \beta_{3} ) q^{79} + ( 12877 \beta_{1} - 8557 \beta_{2} ) q^{82} + ( 858 \beta_{1} - 9687 \beta_{2} ) q^{83} + ( -124844 + 764 \beta_{3} ) q^{86} + ( 16845 \beta_{1} - 23487 \beta_{2} ) q^{88} + ( -35361 - 2088 \beta_{3} ) q^{89} + ( -10536 + 496 \beta_{3} ) q^{91} + ( -3486 \beta_{1} + 6402 \beta_{2} ) q^{92} + ( 59924 + 1804 \beta_{3} ) q^{94} + ( -10944 \beta_{1} + 18086 \beta_{2} ) q^{97} + ( -7843 \beta_{1} + 16483 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 138q^{4} + O(q^{10}) \) \( 4q - 138q^{4} + 392q^{11} + 36q^{14} + 2274q^{16} + 6360q^{19} + 9512q^{26} - 7840q^{29} - 2192q^{31} - 40226q^{34} - 55508q^{41} - 73774q^{44} + 4908q^{46} + 23372q^{49} - 108540q^{56} + 23920q^{59} - 48792q^{61} - 87298q^{64} + 174592q^{71} - 82444q^{74} - 135070q^{76} - 130960q^{79} - 497848q^{86} - 145620q^{89} - 41152q^{91} + 243304q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 121 x^{2} + 3600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 81 \nu \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 61 \nu \)\()/12\)
\(\beta_{3}\)\(=\)\( 5 \nu^{2} + 303 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{2} - 5 \beta_{1}\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 303\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(-243 \beta_{2} + 305 \beta_{1}\)\()/5\)

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
8.26209i
7.26209i
7.26209i
8.26209i
10.2621i 0 −73.3104 0 0 68.9517i 423.931i 0 0
199.2 5.26209i 0 4.31044 0 0 131.048i 191.069i 0 0
199.3 5.26209i 0 4.31044 0 0 131.048i 191.069i 0 0
199.4 10.2621i 0 −73.3104 0 0 68.9517i 423.931i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.6.b.i 4
3.b odd 2 1 25.6.b.b 4
5.b even 2 1 inner 225.6.b.i 4
5.c odd 4 1 225.6.a.l 2
5.c odd 4 1 225.6.a.s 2
12.b even 2 1 400.6.c.n 4
15.d odd 2 1 25.6.b.b 4
15.e even 4 1 25.6.a.b 2
15.e even 4 1 25.6.a.d yes 2
60.h even 2 1 400.6.c.n 4
60.l odd 4 1 400.6.a.o 2
60.l odd 4 1 400.6.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 15.e even 4 1
25.6.a.d yes 2 15.e even 4 1
25.6.b.b 4 3.b odd 2 1
25.6.b.b 4 15.d odd 2 1
225.6.a.l 2 5.c odd 4 1
225.6.a.s 2 5.c odd 4 1
225.6.b.i 4 1.a even 1 1 trivial
225.6.b.i 4 5.b even 2 1 inner
400.6.a.o 2 60.l odd 4 1
400.6.a.w 2 60.l odd 4 1
400.6.c.n 4 12.b even 2 1
400.6.c.n 4 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{4} + 133 T_{2}^{2} + 2916 \)
\( T_{7}^{4} + 21928 T_{7}^{2} + 81649296 \)
\( T_{11}^{2} - 196 T_{11} - 141021 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2916 + 133 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 81649296 + 21928 T^{2} + T^{4} \)
$11$ \( ( -141021 - 196 T + T^{2} )^{2} \)
$13$ \( 858255616 + 188192 T^{2} + T^{4} \)
$17$ \( 312882490881 + 3338818 T^{2} + T^{4} \)
$19$ \( ( 2232875 - 3180 T + T^{2} )^{2} \)
$23$ \( 359668876176 + 1234152 T^{2} + T^{4} \)
$29$ \( ( 3456000 + 3920 T + T^{2} )^{2} \)
$31$ \( ( -72602196 + 1096 T + T^{2} )^{2} \)
$37$ \( 61844446287376 + 19808648 T^{2} + T^{4} \)
$41$ \( ( 182931129 + 27754 T + T^{2} )^{2} \)
$43$ \( 73216315824138496 + 550170272 T^{2} + T^{4} \)
$47$ \( 495608042209536 + 619053088 T^{2} + T^{4} \)
$53$ \( 21876919639178256 + 432103432 T^{2} + T^{4} \)
$59$ \( ( -195696000 - 11960 T + T^{2} )^{2} \)
$61$ \( ( -92208796 + 24396 T + T^{2} )^{2} \)
$67$ \( 62324269199220681 + 1105507018 T^{2} + T^{4} \)
$71$ \( ( 1844897904 - 87296 T + T^{2} )^{2} \)
$73$ \( 1347153105574224001 + 2619345602 T^{2} + T^{4} \)
$79$ \( ( -446416500 + 65480 T + T^{2} )^{2} \)
$83$ \( 4403325447203627961 + 4374235962 T^{2} + T^{4} \)
$89$ \( ( -5241540375 + 72810 T + T^{2} )^{2} \)
$97$ \( 10487144169973969936 + 22388071688 T^{2} + T^{4} \)
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