# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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