Properties

Label 300.5.b.a
Level 300
Weight 5
Character orbit 300.b
Analytic conductor 31.011
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 300.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.0109889252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 9 i q^{3} + 94 i q^{7} -81 q^{9} +O(q^{10})\) \( q + 9 i q^{3} + 94 i q^{7} -81 q^{9} + 146 i q^{13} + 46 q^{19} -846 q^{21} -729 i q^{27} + 194 q^{31} + 2062 i q^{37} -1314 q^{39} -3214 i q^{43} -6435 q^{49} + 414 i q^{57} -1966 q^{61} -7614 i q^{63} -5906 i q^{67} -8542 i q^{73} -7682 q^{79} + 6561 q^{81} -13724 q^{91} + 1746 i q^{93} + 18814 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 162q^{9} + O(q^{10}) \) \( 2q - 162q^{9} + 92q^{19} - 1692q^{21} + 388q^{31} - 2628q^{39} - 12870q^{49} - 3932q^{61} - 15364q^{79} + 13122q^{81} - 27448q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.00000i
1.00000i
0 9.00000i 0 0 0 94.0000i 0 −81.0000 0
149.2 0 9.00000i 0 0 0 94.0000i 0 −81.0000 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 CM by \(\Q(\sqrt{-3}) \)
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 300.5.b.a 2
3.b odd 2 1 CM 300.5.b.a 2
5.b even 2 1 inner 300.5.b.a 2
5.c odd 4 1 12.5.c.a 1
5.c odd 4 1 300.5.g.b 1
15.d odd 2 1 inner 300.5.b.a 2
15.e even 4 1 12.5.c.a 1
15.e even 4 1 300.5.g.b 1
20.e even 4 1 48.5.e.a 1
35.f even 4 1 588.5.c.a 1
40.i odd 4 1 192.5.e.a 1
40.k even 4 1 192.5.e.b 1
45.k odd 12 2 324.5.g.b 2
45.l even 12 2 324.5.g.b 2
60.l odd 4 1 48.5.e.a 1
105.k odd 4 1 588.5.c.a 1
120.q odd 4 1 192.5.e.b 1
120.w even 4 1 192.5.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.5.c.a 1 5.c odd 4 1
12.5.c.a 1 15.e even 4 1
48.5.e.a 1 20.e even 4 1
48.5.e.a 1 60.l odd 4 1
192.5.e.a 1 40.i odd 4 1
192.5.e.a 1 120.w even 4 1
192.5.e.b 1 40.k even 4 1
192.5.e.b 1 120.q odd 4 1
300.5.b.a 2 1.a even 1 1 trivial
300.5.b.a 2 3.b odd 2 1 CM
300.5.b.a 2 5.b even 2 1 inner
300.5.b.a 2 15.d odd 2 1 inner
300.5.g.b 1 5.c odd 4 1
300.5.g.b 1 15.e even 4 1
324.5.g.b 2 45.k odd 12 2
324.5.g.b 2 45.l even 12 2
588.5.c.a 1 35.f even 4 1
588.5.c.a 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 8836 \) acting on \(S_{5}^{\mathrm{new}}(300, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 81 T^{2} \)
$5$ 1
$7$ \( 1 + 4034 T^{2} + 5764801 T^{4} \)
$11$ \( ( 1 - 121 T )^{2}( 1 + 121 T )^{2} \)
$13$ \( 1 - 35806 T^{2} + 815730721 T^{4} \)
$17$ \( ( 1 + 83521 T^{2} )^{2} \)
$19$ \( ( 1 - 46 T + 130321 T^{2} )^{2} \)
$23$ \( ( 1 + 279841 T^{2} )^{2} \)
$29$ \( ( 1 - 841 T )^{2}( 1 + 841 T )^{2} \)
$31$ \( ( 1 - 194 T + 923521 T^{2} )^{2} \)
$37$ \( 1 + 503522 T^{2} + 3512479453921 T^{4} \)
$41$ \( ( 1 - 1681 T )^{2}( 1 + 1681 T )^{2} \)
$43$ \( 1 + 3492194 T^{2} + 11688200277601 T^{4} \)
$47$ \( ( 1 + 4879681 T^{2} )^{2} \)
$53$ \( ( 1 + 7890481 T^{2} )^{2} \)
$59$ \( ( 1 - 3481 T )^{2}( 1 + 3481 T )^{2} \)
$61$ \( ( 1 + 1966 T + 13845841 T^{2} )^{2} \)
$67$ \( 1 - 5421406 T^{2} + 406067677556641 T^{4} \)
$71$ \( ( 1 - 5041 T )^{2}( 1 + 5041 T )^{2} \)
$73$ \( 1 + 16169282 T^{2} + 806460091894081 T^{4} \)
$79$ \( ( 1 + 7682 T + 38950081 T^{2} )^{2} \)
$83$ \( ( 1 + 47458321 T^{2} )^{2} \)
$89$ \( ( 1 - 7921 T )^{2}( 1 + 7921 T )^{2} \)
$97$ \( 1 + 176908034 T^{2} + 7837433594376961 T^{4} \)
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