Properties

Label 2450.2.c.w
Level $2450$
Weight $2$
Character orbit 2450.c
Analytic conductor $19.563$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.5633484952\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 490)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{8}^{2} q^{2} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} - q^{4} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{6} + \zeta_{8}^{2} q^{8} + ( -3 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})\) \( q -\zeta_{8}^{2} q^{2} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{3} - q^{4} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{6} + \zeta_{8}^{2} q^{8} + ( -3 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{9} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{12} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{13} + q^{16} + ( -\zeta_{8} - 4 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{17} + ( -4 \zeta_{8} + 3 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{18} + ( 2 + \zeta_{8} - \zeta_{8}^{3} ) q^{19} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{22} + ( 2 \zeta_{8} + 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{23} + ( -2 + \zeta_{8} - \zeta_{8}^{3} ) q^{24} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{26} + ( 8 \zeta_{8} - 8 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{27} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} -\zeta_{8}^{2} q^{32} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{33} + ( -4 - \zeta_{8} + \zeta_{8}^{3} ) q^{34} + ( 3 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{36} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{37} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{38} + ( 8 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{39} + ( -4 - 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{41} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{43} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + ( 4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{46} + ( -2 \zeta_{8} - 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{47} + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{48} + ( 6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{52} + ( 6 \zeta_{8} + 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{53} + ( -8 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{54} + 2 \zeta_{8}^{2} q^{57} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{58} + ( 10 + \zeta_{8} - \zeta_{8}^{3} ) q^{59} + ( 2 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{61} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{62} - q^{64} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{66} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{67} + ( \zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{68} -4 q^{69} + ( 4 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{71} + ( 4 \zeta_{8} - 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{72} + ( \zeta_{8} - 8 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{73} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{74} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{76} + ( 6 \zeta_{8} - 8 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{78} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{79} + ( 23 - 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{81} + ( 5 \zeta_{8} + 4 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{82} + ( 3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{83} + ( 6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{86} + ( -6 \zeta_{8} + 8 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{87} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{88} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{89} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{92} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{93} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{94} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{96} + ( 3 \zeta_{8} + 12 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{97} + ( 10 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 8q^{6} - 12q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 8q^{6} - 12q^{9} + 8q^{11} + 4q^{16} + 8q^{19} - 8q^{24} - 8q^{26} + 8q^{29} - 16q^{34} + 12q^{36} + 32q^{39} - 16q^{41} - 8q^{44} + 16q^{46} + 24q^{51} - 32q^{54} + 40q^{59} + 8q^{61} - 4q^{64} - 16q^{69} + 16q^{71} - 8q^{74} - 8q^{76} + 16q^{79} + 92q^{81} + 24q^{86} - 32q^{94} + 8q^{96} + 40q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
1.00000i 0.585786i −1.00000 0 0.585786 0 1.00000i 2.65685 0
99.2 1.00000i 3.41421i −1.00000 0 3.41421 0 1.00000i −8.65685 0
99.3 1.00000i 3.41421i −1.00000 0 3.41421 0 1.00000i −8.65685 0
99.4 1.00000i 0.585786i −1.00000 0 0.585786 0 1.00000i 2.65685 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 2450.2.c.w 4
5.b even 2 1 inner 2450.2.c.w 4
5.c odd 4 1 490.2.a.l 2
5.c odd 4 1 2450.2.a.bs 2
7.b odd 2 1 2450.2.c.t 4
15.e even 4 1 4410.2.a.by 2
20.e even 4 1 3920.2.a.ca 2
35.c odd 2 1 2450.2.c.t 4
35.f even 4 1 490.2.a.m yes 2
35.f even 4 1 2450.2.a.bn 2
35.k even 12 2 490.2.e.i 4
35.l odd 12 2 490.2.e.j 4
105.k odd 4 1 4410.2.a.bt 2
140.j odd 4 1 3920.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 5.c odd 4 1
490.2.a.m yes 2 35.f even 4 1
490.2.e.i 4 35.k even 12 2
490.2.e.j 4 35.l odd 12 2
2450.2.a.bn 2 35.f even 4 1
2450.2.a.bs 2 5.c odd 4 1
2450.2.c.t 4 7.b odd 2 1
2450.2.c.t 4 35.c odd 2 1
2450.2.c.w 4 1.a even 1 1 trivial
2450.2.c.w 4 5.b even 2 1 inner
3920.2.a.bm 2 140.j odd 4 1
3920.2.a.ca 2 20.e even 4 1
4410.2.a.bt 2 105.k odd 4 1
4410.2.a.by 2 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 \(S_{2}^{\mathrm{new}}(2450, [\chi])\):

\( T_{3}^{4} + 12 T_{3}^{2} + 4 \)
\( T_{11}^{2} - 4 T_{11} - 4 \)
\( T_{13}^{4} + 24 T_{13}^{2} + 16 \)
\( T_{19}^{2} - 4 T_{19} + 2 \)
\( T_{31}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 4 + 12 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -4 - 4 T + T^{2} )^{2} \)
$13$ \( 16 + 24 T^{2} + T^{4} \)
$17$ \( 196 + 36 T^{2} + T^{4} \)
$19$ \( ( 2 - 4 T + T^{2} )^{2} \)
$23$ \( 64 + 48 T^{2} + T^{4} \)
$29$ \( ( -4 - 4 T + T^{2} )^{2} \)
$31$ \( ( -8 + T^{2} )^{2} \)
$37$ \( 784 + 72 T^{2} + T^{4} \)
$41$ \( ( -34 + 8 T + T^{2} )^{2} \)
$43$ \( 784 + 88 T^{2} + T^{4} \)
$47$ \( 3136 + 144 T^{2} + T^{4} \)
$53$ \( 4624 + 152 T^{2} + T^{4} \)
$59$ \( ( 98 - 20 T + T^{2} )^{2} \)
$61$ \( ( -124 - 4 T + T^{2} )^{2} \)
$67$ \( 256 + 96 T^{2} + T^{4} \)
$71$ \( ( -56 - 8 T + T^{2} )^{2} \)
$73$ \( 3844 + 132 T^{2} + T^{4} \)
$79$ \( ( 8 - 8 T + T^{2} )^{2} \)
$83$ \( 196 + 44 T^{2} + T^{4} \)
$89$ \( ( -162 + T^{2} )^{2} \)
$97$ \( 15876 + 324 T^{2} + T^{4} \)
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