Properties

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

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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 98)
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.

\(f(q)\) \(=\) \( q -\zeta_{8}^{2} q^{2} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{3} - q^{4} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{6} + \zeta_{8}^{2} q^{8} + q^{9} +O(q^{10})\) \( q -\zeta_{8}^{2} q^{2} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{3} - q^{4} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{6} + \zeta_{8}^{2} q^{8} + q^{9} -2 q^{11} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{12} + q^{16} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} -\zeta_{8}^{2} q^{18} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{19} + 2 \zeta_{8}^{2} q^{22} -4 \zeta_{8}^{2} q^{23} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{24} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{27} -2 q^{29} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{31} -\zeta_{8}^{2} q^{32} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{33} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{34} - q^{36} -10 \zeta_{8}^{2} q^{37} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{38} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{41} + 2 \zeta_{8}^{2} q^{43} + 2 q^{44} -4 q^{46} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{47} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{48} + 2 q^{51} -2 \zeta_{8}^{2} q^{53} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{54} -10 \zeta_{8}^{2} q^{57} + 2 \zeta_{8}^{2} q^{58} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{61} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{62} - q^{64} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{66} -12 \zeta_{8}^{2} q^{67} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{68} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{69} -12 q^{71} + \zeta_{8}^{2} q^{72} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{73} -10 q^{74} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{76} + 4 q^{79} -5 q^{81} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{82} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{83} + 2 q^{86} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{87} -2 \zeta_{8}^{2} q^{88} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{89} + 4 \zeta_{8}^{2} q^{92} -12 \zeta_{8}^{2} q^{93} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{94} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{96} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} + 4 q^{9} - 8 q^{11} + 4 q^{16} - 8 q^{29} - 4 q^{36} + 8 q^{44} - 16 q^{46} + 8 q^{51} - 4 q^{64} - 48 q^{71} - 40 q^{74} + 16 q^{79} - 20 q^{81} + 8 q^{86} - 8 q^{99} + O(q^{100}) \)

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 1.41421i −1.00000 0 −1.41421 0 1.00000i 1.00000 0
99.2 1.00000i 1.41421i −1.00000 0 1.41421 0 1.00000i 1.00000 0
99.3 1.00000i 1.41421i −1.00000 0 1.41421 0 1.00000i 1.00000 0
99.4 1.00000i 1.41421i −1.00000 0 −1.41421 0 1.00000i 1.00000 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
7.b odd 2 1 inner
35.c odd 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.v 4
5.b even 2 1 inner 2450.2.c.v 4
5.c odd 4 1 98.2.a.b 2
5.c odd 4 1 2450.2.a.bj 2
7.b odd 2 1 inner 2450.2.c.v 4
15.e even 4 1 882.2.a.n 2
20.e even 4 1 784.2.a.l 2
35.c odd 2 1 inner 2450.2.c.v 4
35.f even 4 1 98.2.a.b 2
35.f even 4 1 2450.2.a.bj 2
35.k even 12 2 98.2.c.c 4
35.l odd 12 2 98.2.c.c 4
40.i odd 4 1 3136.2.a.bn 2
40.k even 4 1 3136.2.a.bm 2
60.l odd 4 1 7056.2.a.cl 2
105.k odd 4 1 882.2.a.n 2
105.w odd 12 2 882.2.g.l 4
105.x even 12 2 882.2.g.l 4
140.j odd 4 1 784.2.a.l 2
140.w even 12 2 784.2.i.m 4
140.x odd 12 2 784.2.i.m 4
280.s even 4 1 3136.2.a.bn 2
280.y odd 4 1 3136.2.a.bm 2
420.w even 4 1 7056.2.a.cl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.2.a.b 2 5.c odd 4 1
98.2.a.b 2 35.f even 4 1
98.2.c.c 4 35.k even 12 2
98.2.c.c 4 35.l odd 12 2
784.2.a.l 2 20.e even 4 1
784.2.a.l 2 140.j odd 4 1
784.2.i.m 4 140.w even 12 2
784.2.i.m 4 140.x odd 12 2
882.2.a.n 2 15.e even 4 1
882.2.a.n 2 105.k odd 4 1
882.2.g.l 4 105.w odd 12 2
882.2.g.l 4 105.x even 12 2
2450.2.a.bj 2 5.c odd 4 1
2450.2.a.bj 2 35.f even 4 1
2450.2.c.v 4 1.a even 1 1 trivial
2450.2.c.v 4 5.b even 2 1 inner
2450.2.c.v 4 7.b odd 2 1 inner
2450.2.c.v 4 35.c odd 2 1 inner
3136.2.a.bm 2 40.k even 4 1
3136.2.a.bm 2 280.y odd 4 1
3136.2.a.bn 2 40.i odd 4 1
3136.2.a.bn 2 280.s even 4 1
7056.2.a.cl 2 60.l odd 4 1
7056.2.a.cl 2 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2450, [\chi])\):

\( T_{3}^{2} + 2 \)
\( T_{11} + 2 \)
\( T_{13} \)
\( T_{19}^{2} - 50 \)
\( T_{31}^{2} - 72 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 2 + T^{2} )^{2} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 2 + T )^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 2 + T^{2} )^{2} \)
$19$ \( ( -50 + T^{2} )^{2} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( ( 2 + T )^{4} \)
$31$ \( ( -72 + T^{2} )^{2} \)
$37$ \( ( 100 + T^{2} )^{2} \)
$41$ \( ( -98 + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( ( 8 + T^{2} )^{2} \)
$53$ \( ( 4 + T^{2} )^{2} \)
$59$ \( ( -2 + T^{2} )^{2} \)
$61$ \( ( -8 + T^{2} )^{2} \)
$67$ \( ( 144 + T^{2} )^{2} \)
$71$ \( ( 12 + T )^{4} \)
$73$ \( ( 2 + T^{2} )^{2} \)
$79$ \( ( -4 + T )^{4} \)
$83$ \( ( 98 + T^{2} )^{2} \)
$89$ \( ( -50 + T^{2} )^{2} \)
$97$ \( ( 98 + T^{2} )^{2} \)
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