Properties

Label 3600.3.c.k
Level $3600$
Weight $3$
Character orbit 3600.c
Analytic conductor $98.093$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3600.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(98.0928951697\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 2 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 2 \beta_1) q^{7} + ( - \beta_{5} - \beta_{2}) q^{11} + ( - \beta_{3} - \beta_1) q^{13} + 2 \beta_{4} q^{17} + (\beta_{6} + 8) q^{19} + (\beta_{7} - \beta_{4}) q^{23} + ( - \beta_{5} - 8 \beta_{2}) q^{29} + ( - \beta_{6} - 2) q^{31} + ( - 3 \beta_{3} - 17 \beta_1) q^{37} + ( - 4 \beta_{5} - 3 \beta_{2}) q^{41} + ( - 2 \beta_{3} + 10 \beta_1) q^{43} + ( - 2 \beta_{7} - 7 \beta_{4}) q^{47} + (4 \beta_{6} - 57) q^{49} + (4 \beta_{7} + 5 \beta_{4}) q^{53} + ( - 3 \beta_{5} + 17 \beta_{2}) q^{59} + ( - 3 \beta_{6} - 10) q^{61} + 38 \beta_1 q^{67} + ( - 2 \beta_{5} - 12 \beta_{2}) q^{71} + (6 \beta_{3} - 19 \beta_1) q^{73} + ( - 7 \beta_{7} + 17 \beta_{4}) q^{77} + (3 \beta_{6} + 50) q^{79} + (4 \beta_{7} - \beta_{4}) q^{83} + (2 \beta_{5} - 21 \beta_{2}) q^{89} + (\beta_{6} - 82) q^{91} + (2 \beta_{3} - 53 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 64 q^{19} - 16 q^{31} - 456 q^{49} - 80 q^{61} + 400 q^{79} - 656 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} + 16\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{7} + \nu^{5} + 13\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{7} + \nu^{5} + 29\nu^{3} + 11\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -4\nu^{7} + 2\nu^{5} - 26\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -6\nu^{6} - 36\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -8\nu^{7} + 2\nu^{5} - 58\nu^{3} + 22\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 4\nu^{4} + 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + 2\beta_{3} - 2\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 9\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} + 2\beta_{4} + 2\beta_{3} - 4\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{6} + 11\beta_{4} - 10\beta_{3} + 22\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{5} - 27\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{6} - 29\beta_{4} - 26\beta_{3} + 58\beta_{2} ) / 24 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.437016 + 0.437016i
−1.14412 + 1.14412i
1.14412 + 1.14412i
0.437016 + 0.437016i
0.437016 0.437016i
1.14412 1.14412i
−1.14412 1.14412i
−0.437016 0.437016i
0 0 0 0 0 13.4868i 0 0 0
449.2 0 0 0 0 0 13.4868i 0 0 0
449.3 0 0 0 0 0 5.48683i 0 0 0
449.4 0 0 0 0 0 5.48683i 0 0 0
449.5 0 0 0 0 0 5.48683i 0 0 0
449.6 0 0 0 0 0 5.48683i 0 0 0
449.7 0 0 0 0 0 13.4868i 0 0 0
449.8 0 0 0 0 0 13.4868i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.3.c.k 8
3.b odd 2 1 inner 3600.3.c.k 8
4.b odd 2 1 900.3.b.b 8
5.b even 2 1 inner 3600.3.c.k 8
5.c odd 4 1 720.3.l.c 4
5.c odd 4 1 3600.3.l.n 4
12.b even 2 1 900.3.b.b 8
15.d odd 2 1 inner 3600.3.c.k 8
15.e even 4 1 720.3.l.c 4
15.e even 4 1 3600.3.l.n 4
20.d odd 2 1 900.3.b.b 8
20.e even 4 1 180.3.g.a 4
20.e even 4 1 900.3.g.d 4
40.i odd 4 1 2880.3.l.f 4
40.k even 4 1 2880.3.l.b 4
60.h even 2 1 900.3.b.b 8
60.l odd 4 1 180.3.g.a 4
60.l odd 4 1 900.3.g.d 4
120.q odd 4 1 2880.3.l.b 4
120.w even 4 1 2880.3.l.f 4
180.v odd 12 2 1620.3.o.f 8
180.x even 12 2 1620.3.o.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.g.a 4 20.e even 4 1
180.3.g.a 4 60.l odd 4 1
720.3.l.c 4 5.c odd 4 1
720.3.l.c 4 15.e even 4 1
900.3.b.b 8 4.b odd 2 1
900.3.b.b 8 12.b even 2 1
900.3.b.b 8 20.d odd 2 1
900.3.b.b 8 60.h even 2 1
900.3.g.d 4 20.e even 4 1
900.3.g.d 4 60.l odd 4 1
1620.3.o.f 8 180.v odd 12 2
1620.3.o.f 8 180.x even 12 2
2880.3.l.b 4 40.k even 4 1
2880.3.l.b 4 120.q odd 4 1
2880.3.l.f 4 40.i odd 4 1
2880.3.l.f 4 120.w even 4 1
3600.3.c.k 8 1.a even 1 1 trivial
3600.3.c.k 8 3.b odd 2 1 inner
3600.3.c.k 8 5.b even 2 1 inner
3600.3.c.k 8 15.d odd 2 1 inner
3600.3.l.n 4 5.c odd 4 1
3600.3.l.n 4 15.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 \(S_{3}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{4} + 212T_{7}^{2} + 5476 \) Copy content Toggle raw display
\( T_{13}^{4} + 188T_{13}^{2} + 7396 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 212 T^{2} + 5476)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 396 T^{2} + 26244)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 188 T^{2} + 7396)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 288)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 16 T - 296)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 504 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2664 T^{2} + 944784)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 356)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3932 T^{2} + 119716)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 6084 T^{2} + 7387524)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1520 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 8496 T^{2} + 7884864)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 9360 T^{2} + 1166400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 13644 T^{2} + 12830724)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 20 T - 3140)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 5776)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6624 T^{2} + 3504384)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 9368 T^{2} + 3225616)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 100 T - 740)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 5904 T^{2} + 7884864)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 17316 T^{2} + 52099524)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 23192 T^{2} + \cdots + 118287376)^{2} \) Copy content Toggle raw display
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