Properties

Label 1575.2.bk.c
Level $1575$
Weight $2$
Character orbit 1575.bk
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bk (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 + 3 \beta_{2} ) q^{7} -2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 + 3 \beta_{2} ) q^{7} -2 \beta_{3} q^{8} + ( \beta_{1} - \beta_{3} ) q^{11} + ( 3 - 6 \beta_{2} ) q^{13} + ( -\beta_{1} + 3 \beta_{3} ) q^{14} + ( 4 - 4 \beta_{2} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{17} + ( 1 + \beta_{2} ) q^{19} + 2 q^{22} + 4 \beta_{1} q^{23} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{26} -2 \beta_{3} q^{29} + ( 2 - \beta_{2} ) q^{31} + ( 4 - 8 \beta_{2} ) q^{34} + ( -1 + \beta_{2} ) q^{37} + ( \beta_{1} + \beta_{3} ) q^{38} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{41} + q^{43} + 8 \beta_{2} q^{46} + ( 5 \beta_{1} - 10 \beta_{3} ) q^{47} + ( -8 + 3 \beta_{2} ) q^{49} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{53} + ( 6 \beta_{1} - 4 \beta_{3} ) q^{56} + ( 4 - 4 \beta_{2} ) q^{58} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{59} + ( -2 - 2 \beta_{2} ) q^{61} + ( 2 \beta_{1} - \beta_{3} ) q^{62} -8 q^{64} + 11 \beta_{2} q^{67} -5 \beta_{3} q^{71} + ( -2 + \beta_{2} ) q^{73} + ( -\beta_{1} + \beta_{3} ) q^{74} + ( 2 \beta_{1} + \beta_{3} ) q^{77} + ( -5 + 5 \beta_{2} ) q^{79} + ( 6 + 6 \beta_{2} ) q^{82} + ( 6 \beta_{1} - 3 \beta_{3} ) q^{83} + \beta_{1} q^{86} -4 \beta_{2} q^{88} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{89} + ( 15 - 3 \beta_{2} ) q^{91} + ( 20 - 10 \beta_{2} ) q^{94} + ( 6 - 12 \beta_{2} ) q^{97} + ( -8 \beta_{1} + 3 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} + O(q^{10}) \) \( 4q + 2q^{7} + 8q^{16} + 6q^{19} + 8q^{22} + 6q^{31} - 2q^{37} + 4q^{43} + 16q^{46} - 26q^{49} + 8q^{58} - 12q^{61} - 32q^{64} + 22q^{67} - 6q^{73} - 10q^{79} + 36q^{82} - 8q^{88} + 54q^{91} + 60q^{94} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(1\) \(1 - \beta_{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 0 0 0 0 0.500000 + 2.59808i 2.82843i 0 0
26.2 1.22474 + 0.707107i 0 0 0 0 0.500000 + 2.59808i 2.82843i 0 0
1151.1 −1.22474 + 0.707107i 0 0 0 0 0.500000 2.59808i 2.82843i 0 0
1151.2 1.22474 0.707107i 0 0 0 0 0.500000 2.59808i 2.82843i 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
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.bk.c 4
3.b odd 2 1 inner 1575.2.bk.c 4
5.b even 2 1 63.2.p.a 4
5.c odd 4 2 1575.2.bc.a 8
7.d odd 6 1 inner 1575.2.bk.c 4
15.d odd 2 1 63.2.p.a 4
15.e even 4 2 1575.2.bc.a 8
20.d odd 2 1 1008.2.bt.b 4
21.g even 6 1 inner 1575.2.bk.c 4
35.c odd 2 1 441.2.p.a 4
35.i odd 6 1 63.2.p.a 4
35.i odd 6 1 441.2.c.a 4
35.j even 6 1 441.2.c.a 4
35.j even 6 1 441.2.p.a 4
35.k even 12 2 1575.2.bc.a 8
45.h odd 6 1 567.2.i.d 4
45.h odd 6 1 567.2.s.d 4
45.j even 6 1 567.2.i.d 4
45.j even 6 1 567.2.s.d 4
60.h even 2 1 1008.2.bt.b 4
105.g even 2 1 441.2.p.a 4
105.o odd 6 1 441.2.c.a 4
105.o odd 6 1 441.2.p.a 4
105.p even 6 1 63.2.p.a 4
105.p even 6 1 441.2.c.a 4
105.w odd 12 2 1575.2.bc.a 8
140.p odd 6 1 7056.2.k.b 4
140.s even 6 1 1008.2.bt.b 4
140.s even 6 1 7056.2.k.b 4
315.q odd 6 1 567.2.s.d 4
315.u even 6 1 567.2.i.d 4
315.bn odd 6 1 567.2.i.d 4
315.bq even 6 1 567.2.s.d 4
420.ba even 6 1 7056.2.k.b 4
420.be odd 6 1 1008.2.bt.b 4
420.be odd 6 1 7056.2.k.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.p.a 4 5.b even 2 1
63.2.p.a 4 15.d odd 2 1
63.2.p.a 4 35.i odd 6 1
63.2.p.a 4 105.p even 6 1
441.2.c.a 4 35.i odd 6 1
441.2.c.a 4 35.j even 6 1
441.2.c.a 4 105.o odd 6 1
441.2.c.a 4 105.p even 6 1
441.2.p.a 4 35.c odd 2 1
441.2.p.a 4 35.j even 6 1
441.2.p.a 4 105.g even 2 1
441.2.p.a 4 105.o odd 6 1
567.2.i.d 4 45.h odd 6 1
567.2.i.d 4 45.j even 6 1
567.2.i.d 4 315.u even 6 1
567.2.i.d 4 315.bn odd 6 1
567.2.s.d 4 45.h odd 6 1
567.2.s.d 4 45.j even 6 1
567.2.s.d 4 315.q odd 6 1
567.2.s.d 4 315.bq even 6 1
1008.2.bt.b 4 20.d odd 2 1
1008.2.bt.b 4 60.h even 2 1
1008.2.bt.b 4 140.s even 6 1
1008.2.bt.b 4 420.be odd 6 1
1575.2.bc.a 8 5.c odd 4 2
1575.2.bc.a 8 15.e even 4 2
1575.2.bc.a 8 35.k even 12 2
1575.2.bc.a 8 105.w odd 12 2
1575.2.bk.c 4 1.a even 1 1 trivial
1575.2.bk.c 4 3.b odd 2 1 inner
1575.2.bk.c 4 7.d odd 6 1 inner
1575.2.bk.c 4 21.g even 6 1 inner
7056.2.k.b 4 140.p odd 6 1
7056.2.k.b 4 140.s even 6 1
7056.2.k.b 4 420.ba even 6 1
7056.2.k.b 4 420.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\):

\( T_{2}^{4} - 2 T_{2}^{2} + 4 \)
\( T_{11}^{4} - 2 T_{11}^{2} + 4 \)
\( T_{37}^{2} + T_{37} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 7 - T + T^{2} )^{2} \)
$11$ \( 4 - 2 T^{2} + T^{4} \)
$13$ \( ( 27 + T^{2} )^{2} \)
$17$ \( 576 + 24 T^{2} + T^{4} \)
$19$ \( ( 3 - 3 T + T^{2} )^{2} \)
$23$ \( 1024 - 32 T^{2} + T^{4} \)
$29$ \( ( 8 + T^{2} )^{2} \)
$31$ \( ( 3 - 3 T + T^{2} )^{2} \)
$37$ \( ( 1 + T + T^{2} )^{2} \)
$41$ \( ( -54 + T^{2} )^{2} \)
$43$ \( ( -1 + T )^{4} \)
$47$ \( 22500 + 150 T^{2} + T^{4} \)
$53$ \( 64 - 8 T^{2} + T^{4} \)
$59$ \( 576 + 24 T^{2} + T^{4} \)
$61$ \( ( 12 + 6 T + T^{2} )^{2} \)
$67$ \( ( 121 - 11 T + T^{2} )^{2} \)
$71$ \( ( 50 + T^{2} )^{2} \)
$73$ \( ( 3 + 3 T + T^{2} )^{2} \)
$79$ \( ( 25 + 5 T + T^{2} )^{2} \)
$83$ \( ( -54 + T^{2} )^{2} \)
$89$ \( 576 + 24 T^{2} + T^{4} \)
$97$ \( ( 108 + T^{2} )^{2} \)
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