Properties

Label 225.4.b.h
Level $225$
Weight $4$
Character orbit 225.b
Analytic conductor $13.275$
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: \( k \) \(=\) \( 4 \)
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: \(\Q(i, \sqrt{19})\)
Defining polynomial: \( x^{4} - 9x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 75)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_1) q^{2} + (2 \beta_{2} - 12) q^{4} + ( - 4 \beta_{3} - 13 \beta_1) q^{7} + ( - 6 \beta_{3} + 42 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_1) q^{2} + (2 \beta_{2} - 12) q^{4} + ( - 4 \beta_{3} - 13 \beta_1) q^{7} + ( - 6 \beta_{3} + 42 \beta_1) q^{8} + ( - 4 \beta_{2} - 14) q^{11} + ( - 16 \beta_{3} + 9 \beta_1) q^{13} + (9 \beta_{2} + 63) q^{14} + ( - 32 \beta_{2} + 60) q^{16} + ( - 20 \beta_{3} - 34 \beta_1) q^{17} + (4 \beta_{2} - 3) q^{19} + ( - 10 \beta_{3} - 62 \beta_1) q^{22} + (12 \beta_{3} - 66 \beta_1) q^{23} + ( - 25 \beta_{2} + 313) q^{26} + (22 \beta_{3} + 4 \beta_1) q^{28} + ( - 28 \beta_{2} + 46) q^{29} + (28 \beta_{2} + 61) q^{31} + (44 \beta_{3} - 332 \beta_1) q^{32} + (14 \beta_{2} + 346) q^{34} + ( - 24 \beta_{3} + 142 \beta_1) q^{37} + ( - 7 \beta_{3} + 79 \beta_1) q^{38} + (52 \beta_{2} - 196) q^{41} + ( - 4 \beta_{3} + 345 \beta_1) q^{43} + (20 \beta_{2} + 16) q^{44} + (78 \beta_{2} - 294) q^{46} + ( - 32 \beta_{3} - 310 \beta_1) q^{47} + ( - 104 \beta_{2} - 130) q^{49} + (210 \beta_{3} - 716 \beta_1) q^{52} + ( - 28 \beta_{3} + 424 \beta_1) q^{53} + (90 \beta_{2} + 90) q^{56} + (74 \beta_{3} - 578 \beta_1) q^{58} + (64 \beta_{2} + 62) q^{59} + (56 \beta_{2} + 375) q^{61} + (33 \beta_{3} + 471 \beta_1) q^{62} + (120 \beta_{2} - 688) q^{64} + ( - 100 \beta_{3} + 179 \beta_1) q^{67} + (172 \beta_{3} - 352 \beta_1) q^{68} + ( - 20 \beta_{2} - 412) q^{71} + (8 \beta_{3} - 54 \beta_1) q^{73} + ( - 166 \beta_{2} + 598) q^{74} + ( - 54 \beta_{2} + 188) q^{76} + (108 \beta_{3} + 486 \beta_1) q^{77} + ( - 160 \beta_{2} + 440) q^{79} + ( - 248 \beta_{3} + 1184 \beta_1) q^{82} + ( - 192 \beta_{3} - 78 \beta_1) q^{83} + ( - 349 \beta_{2} + 421) q^{86} + ( - 84 \beta_{3} - 132 \beta_1) q^{88} + ( - 144 \beta_{2} - 432) q^{89} + ( - 172 \beta_{2} - 1099) q^{91} + ( - 276 \beta_{3} + 1248 \beta_1) q^{92} + (278 \beta_{2} + 298) q^{94} - 521 \beta_1 q^{97} + ( - 26 \beta_{3} - 1846 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 48 q^{4} - 56 q^{11} + 252 q^{14} + 240 q^{16} - 12 q^{19} + 1252 q^{26} + 184 q^{29} + 244 q^{31} + 1384 q^{34} - 784 q^{41} + 64 q^{44} - 1176 q^{46} - 520 q^{49} + 360 q^{56} + 248 q^{59} + 1500 q^{61} - 2752 q^{64} - 1648 q^{71} + 2392 q^{74} + 752 q^{76} + 1760 q^{79} + 1684 q^{86} - 1728 q^{89} - 4396 q^{91} + 1192 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 9x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 4\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 14\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 7\beta_1 \) Copy content Toggle raw display

Character values

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

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

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
−2.17945 + 0.500000i
2.17945 0.500000i
2.17945 + 0.500000i
−2.17945 0.500000i
5.35890i 0 −20.7178 0 0 4.43560i 68.1534i 0 0
199.2 3.35890i 0 −3.28220 0 0 30.4356i 15.8466i 0 0
199.3 3.35890i 0 −3.28220 0 0 30.4356i 15.8466i 0 0
199.4 5.35890i 0 −20.7178 0 0 4.43560i 68.1534i 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 225.4.b.h 4
3.b odd 2 1 75.4.b.c 4
5.b even 2 1 inner 225.4.b.h 4
5.c odd 4 1 225.4.a.j 2
5.c odd 4 1 225.4.a.n 2
12.b even 2 1 1200.4.f.v 4
15.d odd 2 1 75.4.b.c 4
15.e even 4 1 75.4.a.d 2
15.e even 4 1 75.4.a.e yes 2
60.h even 2 1 1200.4.f.v 4
60.l odd 4 1 1200.4.a.bl 2
60.l odd 4 1 1200.4.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 15.e even 4 1
75.4.a.e yes 2 15.e even 4 1
75.4.b.c 4 3.b odd 2 1
75.4.b.c 4 15.d odd 2 1
225.4.a.j 2 5.c odd 4 1
225.4.a.n 2 5.c odd 4 1
225.4.b.h 4 1.a even 1 1 trivial
225.4.b.h 4 5.b even 2 1 inner
1200.4.a.bl 2 60.l odd 4 1
1200.4.a.bu 2 60.l odd 4 1
1200.4.f.v 4 12.b even 2 1
1200.4.f.v 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 \(S_{4}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{4} + 40T_{2}^{2} + 324 \) Copy content Toggle raw display
\( T_{7}^{4} + 946T_{7}^{2} + 18225 \) Copy content Toggle raw display
\( T_{11}^{2} + 28T_{11} - 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 40T^{2} + 324 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 946 T^{2} + 18225 \) Copy content Toggle raw display
$11$ \( (T^{2} + 28 T - 108)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 9890 T^{2} + \cdots + 22877089 \) Copy content Toggle raw display
$17$ \( T^{4} + 17512 T^{2} + \cdots + 41525136 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T - 295)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 14184 T^{2} + \cdots + 2624400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 92 T - 12780)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 122 T - 11175)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 62216 T^{2} + \cdots + 85008400 \) Copy content Toggle raw display
$41$ \( (T^{2} + 392 T - 12960)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 238658 T^{2} + \cdots + 14094675841 \) Copy content Toggle raw display
$47$ \( T^{4} + 231112 T^{2} + \cdots + 5874302736 \) Copy content Toggle raw display
$53$ \( T^{4} + 389344 T^{2} + \cdots + 27185414400 \) Copy content Toggle raw display
$59$ \( (T^{2} - 124 T - 73980)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 750 T + 81041)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 444082 T^{2} + \cdots + 24951045681 \) Copy content Toggle raw display
$71$ \( (T^{2} + 824 T + 162144)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8264 T^{2} + \cdots + 2890000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 880 T - 292800)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1413000 T^{2} + \cdots + 482096926224 \) Copy content Toggle raw display
$89$ \( (T^{2} + 864 T - 207360)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 271441)^{2} \) Copy content Toggle raw display
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