# Properties

 Label 4400.2.b.bb Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.96668224.1 Defining polynomial: $$x^{6} + 15 x^{4} + 61 x^{2} + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{2} - \beta_{4} ) q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{2} - \beta_{4} ) q^{7} + ( -2 + \beta_{3} ) q^{9} - q^{11} -\beta_{2} q^{13} + ( \beta_{1} - 2 \beta_{2} ) q^{17} + ( 3 - 2 \beta_{5} ) q^{19} + ( -1 - \beta_{3} + 3 \beta_{5} ) q^{21} + ( \beta_{1} + 3 \beta_{2} ) q^{23} + ( -\beta_{1} + \beta_{2} - \beta_{4} ) q^{27} + ( -4 - \beta_{3} + 2 \beta_{5} ) q^{29} + ( 1 - 2 \beta_{3} ) q^{31} -\beta_{1} q^{33} + ( \beta_{1} - 3 \beta_{2} - \beta_{4} ) q^{37} + \beta_{5} q^{39} + ( 1 + \beta_{3} + \beta_{5} ) q^{41} + ( 3 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{43} + ( 3 \beta_{1} - \beta_{2} + \beta_{4} ) q^{47} + ( -3 - \beta_{5} ) q^{49} + ( -5 + \beta_{3} + 2 \beta_{5} ) q^{51} + ( -2 \beta_{1} - \beta_{2} - 3 \beta_{4} ) q^{53} + ( 3 \beta_{1} - 10 \beta_{2} - 2 \beta_{4} ) q^{57} + ( 1 + \beta_{3} - 3 \beta_{5} ) q^{59} + ( -1 - \beta_{3} + 2 \beta_{5} ) q^{61} + ( \beta_{1} + 11 \beta_{2} + \beta_{4} ) q^{63} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{67} + ( -5 + \beta_{3} - 3 \beta_{5} ) q^{69} + ( -6 + \beta_{3} + \beta_{5} ) q^{71} + ( 7 \beta_{2} + 3 \beta_{4} ) q^{73} + ( \beta_{2} + \beta_{4} ) q^{77} + ( 4 + 2 \beta_{3} - \beta_{5} ) q^{79} + ( -2 + \beta_{3} + \beta_{5} ) q^{81} + ( -4 \beta_{1} + 8 \beta_{2} - \beta_{4} ) q^{83} + ( -2 \beta_{1} + 9 \beta_{2} + 3 \beta_{4} ) q^{87} + ( -10 - \beta_{3} ) q^{89} + ( -1 + \beta_{3} ) q^{91} + ( 5 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{93} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{4} ) q^{97} + ( 2 - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{9} + O(q^{10})$$ $$6 q - 12 q^{9} - 6 q^{11} + 14 q^{19} - 20 q^{29} + 6 q^{31} + 2 q^{39} + 8 q^{41} - 20 q^{49} - 26 q^{51} - 2 q^{61} - 36 q^{69} - 34 q^{71} + 22 q^{79} - 10 q^{81} - 60 q^{89} - 6 q^{91} + 12 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 15 x^{4} + 61 x^{2} + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 9 \nu^{3} + 13 \nu$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} + 3 \nu^{3} - 29 \nu$$$$)/6$$ $$\beta_{5}$$ $$=$$ $$\nu^{4} + 8 \nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{2} - 7 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} - 8 \beta_{3} + 34$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{4} - 3 \beta_{2} + 50 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1
 − 2.75153i − 2.59261i − 0.841083i 0.841083i 2.59261i 2.75153i
0 2.75153i 0 0 0 3.57093i 0 −4.57093 0
4049.2 0 2.59261i 0 0 0 2.72165i 0 −3.72165 0
4049.3 0 0.841083i 0 0 0 3.29258i 0 2.29258 0
4049.4 0 0.841083i 0 0 0 3.29258i 0 2.29258 0
4049.5 0 2.59261i 0 0 0 2.72165i 0 −3.72165 0
4049.6 0 2.75153i 0 0 0 3.57093i 0 −4.57093 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4049.6 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 4400.2.b.bb 6
4.b odd 2 1 2200.2.b.m 6
5.b even 2 1 inner 4400.2.b.bb 6
5.c odd 4 1 4400.2.a.by 3
5.c odd 4 1 4400.2.a.bz 3
20.d odd 2 1 2200.2.b.m 6
20.e even 4 1 2200.2.a.u 3
20.e even 4 1 2200.2.a.v yes 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.u 3 20.e even 4 1
2200.2.a.v yes 3 20.e even 4 1
2200.2.b.m 6 4.b odd 2 1
2200.2.b.m 6 20.d odd 2 1
4400.2.a.by 3 5.c odd 4 1
4400.2.a.bz 3 5.c odd 4 1
4400.2.b.bb 6 1.a even 1 1 trivial
4400.2.b.bb 6 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4400, [\chi])$$:

 $$T_{3}^{6} + 15 T_{3}^{4} + 61 T_{3}^{2} + 36$$ $$T_{7}^{6} + 31 T_{7}^{4} + 313 T_{7}^{2} + 1024$$ $$T_{13}^{2} + 1$$ $$T_{17}^{6} + 23 T_{17}^{4} + 41 T_{17}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$36 + 61 T^{2} + 15 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$1024 + 313 T^{2} + 31 T^{4} + T^{6}$$
$11$ $$( 1 + T )^{6}$$
$13$ $$( 1 + T^{2} )^{3}$$
$17$ $$16 + 41 T^{2} + 23 T^{4} + T^{6}$$
$19$ $$( 27 - 13 T - 7 T^{2} + T^{3} )^{2}$$
$23$ $$81 + 496 T^{2} + 48 T^{4} + T^{6}$$
$29$ $$( -201 - 8 T + 10 T^{2} + T^{3} )^{2}$$
$31$ $$( 207 - 53 T - 3 T^{2} + T^{3} )^{2}$$
$37$ $$1936 + 705 T^{2} + 66 T^{4} + T^{6}$$
$41$ $$( 24 - 17 T - 4 T^{2} + T^{3} )^{2}$$
$43$ $$258064 + 12280 T^{2} + 193 T^{4} + T^{6}$$
$47$ $$36864 + 5929 T^{2} + 154 T^{4} + T^{6}$$
$53$ $$20164 + 21301 T^{2} + 307 T^{4} + T^{6}$$
$59$ $$( 192 - 77 T + T^{3} )^{2}$$
$61$ $$( -114 - 41 T + T^{2} + T^{3} )^{2}$$
$67$ $$9216 + 2944 T^{2} + 228 T^{4} + T^{6}$$
$71$ $$( 52 + 74 T + 17 T^{2} + T^{3} )^{2}$$
$73$ $$1106704 + 44625 T^{2} + 399 T^{4} + T^{6}$$
$79$ $$( 144 - 21 T - 11 T^{2} + T^{3} )^{2}$$
$83$ $$687241 + 33196 T^{2} + 388 T^{4} + T^{6}$$
$89$ $$( 879 + 286 T + 30 T^{2} + T^{3} )^{2}$$
$97$ $$529 + 1484 T^{2} + 164 T^{4} + T^{6}$$