# Properties

 Label 3450.2.d.v Level $3450$ Weight $2$ Character orbit 3450.d Analytic conductor $27.548$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ Defining polynomial: $$x^{4} + 9 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 690) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} - q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} -\beta_{1} q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} - q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} -\beta_{1} q^{8} - q^{9} + ( 1 - \beta_{3} ) q^{11} -\beta_{1} q^{12} -2 \beta_{1} q^{13} + ( 1 - \beta_{3} ) q^{14} + q^{16} + ( 3 \beta_{1} - \beta_{2} ) q^{17} -\beta_{1} q^{18} -4 q^{19} + ( 1 - \beta_{3} ) q^{21} + ( \beta_{1} - \beta_{2} ) q^{22} -\beta_{1} q^{23} + q^{24} + 2 q^{26} -\beta_{1} q^{27} + ( \beta_{1} - \beta_{2} ) q^{28} -2 q^{29} + \beta_{1} q^{32} + ( \beta_{1} - \beta_{2} ) q^{33} + ( -3 + \beta_{3} ) q^{34} + q^{36} + ( -3 \beta_{1} + \beta_{2} ) q^{37} -4 \beta_{1} q^{38} + 2 q^{39} + 2 q^{41} + ( \beta_{1} - \beta_{2} ) q^{42} + ( -1 + \beta_{3} ) q^{44} + q^{46} -8 \beta_{1} q^{47} + \beta_{1} q^{48} + ( -11 + 2 \beta_{3} ) q^{49} + ( -3 + \beta_{3} ) q^{51} + 2 \beta_{1} q^{52} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{53} + q^{54} + ( -1 + \beta_{3} ) q^{56} -4 \beta_{1} q^{57} -2 \beta_{1} q^{58} + ( 6 - 2 \beta_{3} ) q^{59} + ( 5 + \beta_{3} ) q^{61} + ( \beta_{1} - \beta_{2} ) q^{63} - q^{64} + ( -1 + \beta_{3} ) q^{66} -8 \beta_{1} q^{67} + ( -3 \beta_{1} + \beta_{2} ) q^{68} + q^{69} + ( -2 - 2 \beta_{3} ) q^{71} + \beta_{1} q^{72} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 3 - \beta_{3} ) q^{74} + 4 q^{76} + ( -18 \beta_{1} + 2 \beta_{2} ) q^{77} + 2 \beta_{1} q^{78} + ( 1 - \beta_{3} ) q^{79} + q^{81} + 2 \beta_{1} q^{82} + ( \beta_{1} + 3 \beta_{2} ) q^{83} + ( -1 + \beta_{3} ) q^{84} -2 \beta_{1} q^{87} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( 1 + \beta_{3} ) q^{89} + ( -2 + 2 \beta_{3} ) q^{91} + \beta_{1} q^{92} + 8 q^{94} - q^{96} + ( -8 \beta_{1} - 2 \beta_{2} ) q^{97} + ( -11 \beta_{1} + 2 \beta_{2} ) q^{98} + ( -1 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{6} - 4q^{9} + 4q^{11} + 4q^{14} + 4q^{16} - 16q^{19} + 4q^{21} + 4q^{24} + 8q^{26} - 8q^{29} - 12q^{34} + 4q^{36} + 8q^{39} + 8q^{41} - 4q^{44} + 4q^{46} - 44q^{49} - 12q^{51} + 4q^{54} - 4q^{56} + 24q^{59} + 20q^{61} - 4q^{64} - 4q^{66} + 4q^{69} - 8q^{71} + 12q^{74} + 16q^{76} + 4q^{79} + 4q^{81} - 4q^{84} + 4q^{89} - 8q^{91} + 32q^{94} - 4q^{96} - 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 13 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 13 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2899.1
 − 1.56155i 2.56155i − 2.56155i 1.56155i
1.00000i 1.00000i −1.00000 0 −1.00000 3.12311i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 5.12311i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 −1.00000 5.12311i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 −1.00000 3.12311i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3450.2.d.v 4
5.b even 2 1 inner 3450.2.d.v 4
5.c odd 4 1 690.2.a.l 2
5.c odd 4 1 3450.2.a.bi 2
15.e even 4 1 2070.2.a.t 2
20.e even 4 1 5520.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.l 2 5.c odd 4 1
2070.2.a.t 2 15.e even 4 1
3450.2.a.bi 2 5.c odd 4 1
3450.2.d.v 4 1.a even 1 1 trivial
3450.2.d.v 4 5.b even 2 1 inner
5520.2.a.bs 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3450, [\chi])$$:

 $$T_{7}^{4} + 36 T_{7}^{2} + 256$$ $$T_{11}^{2} - 2 T_{11} - 16$$ $$T_{13}^{2} + 4$$ $$T_{17}^{4} + 52 T_{17}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$256 + 36 T^{2} + T^{4}$$
$11$ $$( -16 - 2 T + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$64 + 52 T^{2} + T^{4}$$
$19$ $$( 4 + T )^{4}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( 2 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$64 + 52 T^{2} + T^{4}$$
$41$ $$( -2 + T )^{4}$$
$43$ $$T^{4}$$
$47$ $$( 64 + T^{2} )^{2}$$
$53$ $$2704 + 168 T^{2} + T^{4}$$
$59$ $$( -32 - 12 T + T^{2} )^{2}$$
$61$ $$( 8 - 10 T + T^{2} )^{2}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( -64 + 4 T + T^{2} )^{2}$$
$73$ $$2704 + 168 T^{2} + T^{4}$$
$79$ $$( -16 - 2 T + T^{2} )^{2}$$
$83$ $$23104 + 308 T^{2} + T^{4}$$
$89$ $$( -16 - 2 T + T^{2} )^{2}$$
$97$ $$16 + 264 T^{2} + T^{4}$$