Properties

Label 675.4.b.c
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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.c 2
3.b odd 2 1 675.4.b.d 2
5.b even 2 1 inner 675.4.b.c 2
5.c odd 4 1 135.4.a.d yes 1
5.c odd 4 1 675.4.a.b 1
15.d odd 2 1 675.4.b.d 2
15.e even 4 1 135.4.a.a 1
15.e even 4 1 675.4.a.i 1
20.e even 4 1 2160.4.a.d 1
45.k odd 12 2 405.4.e.e 2
45.l even 12 2 405.4.e.j 2
60.l odd 4 1 2160.4.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.a 1 15.e even 4 1
135.4.a.d yes 1 5.c odd 4 1
405.4.e.e 2 45.k odd 12 2
405.4.e.j 2 45.l even 12 2
675.4.a.b 1 5.c odd 4 1
675.4.a.i 1 15.e even 4 1
675.4.b.c 2 1.a even 1 1 trivial
675.4.b.c 2 5.b even 2 1 inner
675.4.b.d 2 3.b odd 2 1
675.4.b.d 2 15.d odd 2 1
2160.4.a.d 1 20.e even 4 1
2160.4.a.n 1 60.l odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

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