Properties

 Label 4900.2.e.q Level $4900$ Weight $2$ Character orbit 4900.e Analytic conductor $39.127$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4900.e (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$39.1266969904$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 980) 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{9} + ( -1 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{13} + ( \zeta_{8} - 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{17} + ( -2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{19} + ( -\zeta_{8} - 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{23} + ( \zeta_{8} - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{27} + ( -1 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( -6 - \zeta_{8} + \zeta_{8}^{3} ) q^{31} + ( \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{33} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{37} + ( -7 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{39} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{41} + ( 4 \zeta_{8} - 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{43} + ( -7 \zeta_{8} - \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{47} + ( 3 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{51} + ( 3 \zeta_{8} + 8 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{53} + ( -6 \zeta_{8} - 10 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{57} + ( 2 - \zeta_{8} + \zeta_{8}^{3} ) q^{59} + ( -8 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{61} + ( 5 \zeta_{8} - 4 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{67} + ( 4 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{69} + ( -2 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + ( 2 \zeta_{8} + 8 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{73} + ( 1 + 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{79} + ( -1 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{81} -8 \zeta_{8}^{2} q^{83} + ( 3 \zeta_{8} + 7 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{87} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{89} + ( -7 \zeta_{8} - 8 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{93} + ( 9 \zeta_{8} - 3 \zeta_{8}^{2} + 9 \zeta_{8}^{3} ) q^{97} + ( -8 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q - 4 q^{11} - 8 q^{19} - 4 q^{29} - 24 q^{31} - 28 q^{39} - 8 q^{41} + 12 q^{51} + 8 q^{59} - 32 q^{61} + 16 q^{69} - 8 q^{71} + 4 q^{79} - 4 q^{81} - 32 q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$1177$$ $$2451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2549.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
0 2.41421i 0 0 0 0 0 −2.82843 0
2549.2 0 0.414214i 0 0 0 0 0 2.82843 0
2549.3 0 0.414214i 0 0 0 0 0 2.82843 0
2549.4 0 2.41421i 0 0 0 0 0 −2.82843 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 4900.2.e.q 4
5.b even 2 1 inner 4900.2.e.q 4
5.c odd 4 1 980.2.a.j 2
5.c odd 4 1 4900.2.a.z 2
7.b odd 2 1 4900.2.e.r 4
15.e even 4 1 8820.2.a.bg 2
20.e even 4 1 3920.2.a.bx 2
35.c odd 2 1 4900.2.e.r 4
35.f even 4 1 980.2.a.k yes 2
35.f even 4 1 4900.2.a.x 2
35.k even 12 2 980.2.i.k 4
35.l odd 12 2 980.2.i.l 4
105.k odd 4 1 8820.2.a.bl 2
140.j odd 4 1 3920.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 5.c odd 4 1
980.2.a.k yes 2 35.f even 4 1
980.2.i.k 4 35.k even 12 2
980.2.i.l 4 35.l odd 12 2
3920.2.a.bo 2 140.j odd 4 1
3920.2.a.bx 2 20.e even 4 1
4900.2.a.x 2 35.f even 4 1
4900.2.a.z 2 5.c odd 4 1
4900.2.e.q 4 1.a even 1 1 trivial
4900.2.e.q 4 5.b even 2 1 inner
4900.2.e.r 4 7.b odd 2 1
4900.2.e.r 4 35.c odd 2 1
8820.2.a.bg 2 15.e even 4 1
8820.2.a.bl 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 $$S_{2}^{\mathrm{new}}(4900, [\chi])$$:

 $$T_{3}^{4} + 6 T_{3}^{2} + 1$$ $$T_{11}^{2} + 2 T_{11} - 7$$ $$T_{19}^{2} + 4 T_{19} - 28$$ $$T_{31}^{2} + 12 T_{31} + 34$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + 6 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -7 + 2 T + T^{2} )^{2}$$
$13$ $$529 + 54 T^{2} + T^{4}$$
$17$ $$529 + 54 T^{2} + T^{4}$$
$19$ $$( -28 + 4 T + T^{2} )^{2}$$
$23$ $$4 + 12 T^{2} + T^{4}$$
$29$ $$( -31 + 2 T + T^{2} )^{2}$$
$31$ $$( 34 + 12 T + T^{2} )^{2}$$
$37$ $$4 + 12 T^{2} + T^{4}$$
$41$ $$( 2 + 4 T + T^{2} )^{2}$$
$43$ $$16 + 136 T^{2} + T^{4}$$
$47$ $$9409 + 198 T^{2} + T^{4}$$
$53$ $$2116 + 164 T^{2} + T^{4}$$
$59$ $$( 2 - 4 T + T^{2} )^{2}$$
$61$ $$( 56 + 16 T + T^{2} )^{2}$$
$67$ $$1156 + 132 T^{2} + T^{4}$$
$71$ $$( -68 + 4 T + T^{2} )^{2}$$
$73$ $$3136 + 144 T^{2} + T^{4}$$
$79$ $$( -199 - 2 T + T^{2} )^{2}$$
$83$ $$( 64 + T^{2} )^{2}$$
$89$ $$( -288 + T^{2} )^{2}$$
$97$ $$23409 + 342 T^{2} + T^{4}$$