Properties

Label 675.4.b.d
Level $675$
Weight $4$
Character orbit 675.b
Analytic conductor $39.826$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 4 q^{4} + 24 i q^{8} +O(q^{10})\) \( q + 2 i q^{2} + 4 q^{4} + 24 i q^{8} + 10 q^{11} -80 i q^{13} -16 q^{16} -7 i q^{17} + 113 q^{19} + 20 i q^{22} -81 i q^{23} + 160 q^{26} + 220 q^{29} -189 q^{31} + 160 i q^{32} + 14 q^{34} -170 i q^{37} + 226 i q^{38} -130 q^{41} + 10 i q^{43} + 40 q^{44} + 162 q^{46} -160 i q^{47} + 343 q^{49} -320 i q^{52} + 631 i q^{53} + 440 i q^{58} + 560 q^{59} + 229 q^{61} -378 i q^{62} -448 q^{64} -750 i q^{67} -28 i q^{68} + 890 q^{71} -890 i q^{73} + 340 q^{74} + 452 q^{76} + 27 q^{79} -260 i q^{82} + 429 i q^{83} -20 q^{86} + 240 i q^{88} + 750 q^{89} -324 i q^{92} + 320 q^{94} + 1480 i q^{97} + 686 i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4} + O(q^{10}) \) \( 2 q + 8 q^{4} + 20 q^{11} - 32 q^{16} + 226 q^{19} + 320 q^{26} + 440 q^{29} - 378 q^{31} + 28 q^{34} - 260 q^{41} + 80 q^{44} + 324 q^{46} + 686 q^{49} + 1120 q^{59} + 458 q^{61} - 896 q^{64} + 1780 q^{71} + 680 q^{74} + 904 q^{76} + 54 q^{79} - 40 q^{86} + 1500 q^{89} + 640 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
2.00000i 0 4.00000 0 0 0 24.0000i 0 0
649.2 2.00000i 0 4.00000 0 0 0 24.0000i 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 675.4.b.d 2
3.b odd 2 1 675.4.b.c 2
5.b even 2 1 inner 675.4.b.d 2
5.c odd 4 1 135.4.a.a 1
5.c odd 4 1 675.4.a.i 1
15.d odd 2 1 675.4.b.c 2
15.e even 4 1 135.4.a.d yes 1
15.e even 4 1 675.4.a.b 1
20.e even 4 1 2160.4.a.n 1
45.k odd 12 2 405.4.e.j 2
45.l even 12 2 405.4.e.e 2
60.l odd 4 1 2160.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.a 1 5.c odd 4 1
135.4.a.d yes 1 15.e even 4 1
405.4.e.e 2 45.l even 12 2
405.4.e.j 2 45.k odd 12 2
675.4.a.b 1 15.e even 4 1
675.4.a.i 1 5.c odd 4 1
675.4.b.c 2 3.b odd 2 1
675.4.b.c 2 15.d odd 2 1
675.4.b.d 2 1.a even 1 1 trivial
675.4.b.d 2 5.b even 2 1 inner
2160.4.a.d 1 60.l odd 4 1
2160.4.a.n 1 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(675, [\chi])\):

\( T_{2}^{2} + 4 \)
\( T_{7} \)
\( T_{11} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( ( -10 + T )^{2} \)
$13$ \( 6400 + T^{2} \)
$17$ \( 49 + T^{2} \)
$19$ \( ( -113 + T )^{2} \)
$23$ \( 6561 + T^{2} \)
$29$ \( ( -220 + T )^{2} \)
$31$ \( ( 189 + T )^{2} \)
$37$ \( 28900 + T^{2} \)
$41$ \( ( 130 + T )^{2} \)
$43$ \( 100 + T^{2} \)
$47$ \( 25600 + T^{2} \)
$53$ \( 398161 + T^{2} \)
$59$ \( ( -560 + T )^{2} \)
$61$ \( ( -229 + T )^{2} \)
$67$ \( 562500 + T^{2} \)
$71$ \( ( -890 + T )^{2} \)
$73$ \( 792100 + T^{2} \)
$79$ \( ( -27 + T )^{2} \)
$83$ \( 184041 + T^{2} \)
$89$ \( ( -750 + T )^{2} \)
$97$ \( 2190400 + T^{2} \)
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