# Properties

 Label 3450.2.d.w 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: $$\Q(i, \sqrt{73})$$ Defining polynomial: $$x^{4} + 37 x^{2} + 324$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{2} q^{3} - q^{4} - q^{6} -3 \beta_{2} q^{7} + \beta_{2} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{2} q^{3} - q^{4} - q^{6} -3 \beta_{2} q^{7} + \beta_{2} q^{8} - q^{9} + ( 2 - \beta_{3} ) q^{11} + \beta_{2} q^{12} + ( -\beta_{1} - \beta_{2} ) q^{13} -3 q^{14} + q^{16} + ( -\beta_{1} - 4 \beta_{2} ) q^{17} + \beta_{2} q^{18} + ( -4 + \beta_{3} ) q^{19} -3 q^{21} + ( \beta_{1} - \beta_{2} ) q^{22} -\beta_{2} q^{23} + q^{24} -\beta_{3} q^{26} + \beta_{2} q^{27} + 3 \beta_{2} q^{28} -3 q^{29} + ( -2 + 2 \beta_{3} ) q^{31} -\beta_{2} q^{32} + ( \beta_{1} - \beta_{2} ) q^{33} + ( -3 - \beta_{3} ) q^{34} + q^{36} + ( -\beta_{1} + 2 \beta_{2} ) q^{37} + ( -\beta_{1} + 3 \beta_{2} ) q^{38} -\beta_{3} q^{39} + ( -2 + \beta_{3} ) q^{41} + 3 \beta_{2} q^{42} + ( \beta_{1} - 3 \beta_{2} ) q^{43} + ( -2 + \beta_{3} ) q^{44} - q^{46} + ( \beta_{1} - 4 \beta_{2} ) q^{47} -\beta_{2} q^{48} -2 q^{49} + ( -3 - \beta_{3} ) q^{51} + ( \beta_{1} + \beta_{2} ) q^{52} -4 \beta_{2} q^{53} + q^{54} + 3 q^{56} + ( -\beta_{1} + 3 \beta_{2} ) q^{57} + 3 \beta_{2} q^{58} -6 q^{59} + ( -2 + 2 \beta_{3} ) q^{61} -2 \beta_{1} q^{62} + 3 \beta_{2} q^{63} - q^{64} + ( -2 + \beta_{3} ) q^{66} + 2 \beta_{1} q^{67} + ( \beta_{1} + 4 \beta_{2} ) q^{68} - q^{69} + ( -1 + \beta_{3} ) q^{71} -\beta_{2} q^{72} + ( 2 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 3 - \beta_{3} ) q^{74} + ( 4 - \beta_{3} ) q^{76} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{77} + ( \beta_{1} + \beta_{2} ) q^{78} + ( -6 - \beta_{3} ) q^{79} + q^{81} + ( -\beta_{1} + \beta_{2} ) q^{82} + \beta_{2} q^{83} + 3 q^{84} + ( -4 + \beta_{3} ) q^{86} + 3 \beta_{2} q^{87} + ( -\beta_{1} + \beta_{2} ) q^{88} + ( -13 + \beta_{3} ) q^{89} -3 \beta_{3} q^{91} + \beta_{2} q^{92} -2 \beta_{1} q^{93} + ( -5 + \beta_{3} ) q^{94} - q^{96} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{97} + 2 \beta_{2} q^{98} + ( -2 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} - 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} - 4q^{6} - 4q^{9} + 6q^{11} - 12q^{14} + 4q^{16} - 14q^{19} - 12q^{21} + 4q^{24} - 2q^{26} - 12q^{29} - 4q^{31} - 14q^{34} + 4q^{36} - 2q^{39} - 6q^{41} - 6q^{44} - 4q^{46} - 8q^{49} - 14q^{51} + 4q^{54} + 12q^{56} - 24q^{59} - 4q^{61} - 4q^{64} - 6q^{66} - 4q^{69} - 2q^{71} + 10q^{74} + 14q^{76} - 26q^{79} + 4q^{81} + 12q^{84} - 14q^{86} - 50q^{89} - 6q^{91} - 18q^{94} - 4q^{96} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 37 x^{2} + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 19 \nu$$$$)/18$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 19$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 19$$ $$\nu^{3}$$ $$=$$ $$18 \beta_{2} - 19 \beta_{1}$$

## 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
 3.77200i − 4.77200i − 3.77200i 4.77200i
1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 −1.00000 3.00000i 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.w 4
5.b even 2 1 inner 3450.2.d.w 4
5.c odd 4 1 3450.2.a.bh 2
5.c odd 4 1 3450.2.a.bj yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3450.2.a.bh 2 5.c odd 4 1
3450.2.a.bj yes 2 5.c odd 4 1
3450.2.d.w 4 1.a even 1 1 trivial
3450.2.d.w 4 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 $$S_{2}^{\mathrm{new}}(3450, [\chi])$$:

 $$T_{7}^{2} + 9$$ $$T_{11}^{2} - 3 T_{11} - 16$$ $$T_{13}^{4} + 37 T_{13}^{2} + 324$$ $$T_{17}^{4} + 61 T_{17}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 9 + T^{2} )^{2}$$
$11$ $$( -16 - 3 T + T^{2} )^{2}$$
$13$ $$324 + 37 T^{2} + T^{4}$$
$17$ $$36 + 61 T^{2} + T^{4}$$
$19$ $$( -6 + 7 T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( 3 + T )^{4}$$
$31$ $$( -72 + 2 T + T^{2} )^{2}$$
$37$ $$144 + 49 T^{2} + T^{4}$$
$41$ $$( -16 + 3 T + T^{2} )^{2}$$
$43$ $$36 + 61 T^{2} + T^{4}$$
$47$ $$4 + 77 T^{2} + T^{4}$$
$53$ $$( 16 + T^{2} )^{2}$$
$59$ $$( 6 + T )^{4}$$
$61$ $$( -72 + 2 T + T^{2} )^{2}$$
$67$ $$5184 + 148 T^{2} + T^{4}$$
$71$ $$( -18 + T + T^{2} )^{2}$$
$73$ $$4761 + 154 T^{2} + T^{4}$$
$79$ $$( 24 + 13 T + T^{2} )^{2}$$
$83$ $$( 1 + T^{2} )^{2}$$
$89$ $$( 138 + 25 T + T^{2} )^{2}$$
$97$ $$64 + 308 T^{2} + T^{4}$$