# Properties

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

# 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 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 3.41421i −1.00000 0 −3.41421 0 1.00000i −8.65685 0
99.2 1.00000i 0.585786i −1.00000 0 −0.585786 0 1.00000i 2.65685 0
99.3 1.00000i 0.585786i −1.00000 0 −0.585786 0 1.00000i 2.65685 0
99.4 1.00000i 3.41421i −1.00000 0 −3.41421 0 1.00000i −8.65685 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2450.2.c.t 4
5.b even 2 1 inner 2450.2.c.t 4
5.c odd 4 1 490.2.a.m yes 2
5.c odd 4 1 2450.2.a.bn 2
7.b odd 2 1 2450.2.c.w 4
15.e even 4 1 4410.2.a.bt 2
20.e even 4 1 3920.2.a.bm 2
35.c odd 2 1 2450.2.c.w 4
35.f even 4 1 490.2.a.l 2
35.f even 4 1 2450.2.a.bs 2
35.k even 12 2 490.2.e.j 4
35.l odd 12 2 490.2.e.i 4
105.k odd 4 1 4410.2.a.by 2
140.j odd 4 1 3920.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 35.f even 4 1
490.2.a.m yes 2 5.c odd 4 1
490.2.e.i 4 35.l odd 12 2
490.2.e.j 4 35.k even 12 2
2450.2.a.bn 2 5.c odd 4 1
2450.2.a.bs 2 35.f even 4 1
2450.2.c.t 4 1.a even 1 1 trivial
2450.2.c.t 4 5.b even 2 1 inner
2450.2.c.w 4 7.b odd 2 1
2450.2.c.w 4 35.c odd 2 1
3920.2.a.bm 2 20.e even 4 1
3920.2.a.ca 2 140.j odd 4 1
4410.2.a.bt 2 15.e even 4 1
4410.2.a.by 2 105.k odd 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

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