# Properties

 Label 3450.2.d.y 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: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} + q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} + \beta_{1} q^{8} - q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{1} q^{3} - q^{4} + q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} + \beta_{1} q^{8} - q^{9} + 2 q^{11} -\beta_{1} q^{12} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{13} + ( 1 + \beta_{3} ) q^{14} + q^{16} + ( 3 \beta_{1} - \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( -2 + 2 \beta_{3} ) q^{19} + ( -1 - \beta_{3} ) q^{21} -2 \beta_{1} q^{22} + \beta_{1} q^{23} - q^{24} + ( 2 + 2 \beta_{3} ) q^{26} -\beta_{1} q^{27} + ( -\beta_{1} - \beta_{2} ) q^{28} -5 q^{29} + 2 q^{31} -\beta_{1} q^{32} + 2 \beta_{1} q^{33} + ( 3 - \beta_{3} ) q^{34} + q^{36} + ( -\beta_{1} + 3 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{38} + ( -2 - 2 \beta_{3} ) q^{39} + 2 \beta_{3} q^{41} + ( \beta_{1} + \beta_{2} ) q^{42} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{43} -2 q^{44} + q^{46} + ( -\beta_{1} - 2 \beta_{2} ) q^{47} + \beta_{1} q^{48} -2 \beta_{3} q^{49} + ( -3 + \beta_{3} ) q^{51} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{52} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{53} - q^{54} + ( -1 - \beta_{3} ) q^{56} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{57} + 5 \beta_{1} q^{58} -10 q^{59} + 2 \beta_{3} q^{61} -2 \beta_{1} q^{62} + ( -\beta_{1} - \beta_{2} ) q^{63} - q^{64} + 2 q^{66} + 2 \beta_{1} q^{67} + ( -3 \beta_{1} + \beta_{2} ) q^{68} - q^{69} + ( 1 - 4 \beta_{3} ) q^{71} -\beta_{1} q^{72} + ( 7 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -1 + 3 \beta_{3} ) q^{74} + ( 2 - 2 \beta_{3} ) q^{76} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{78} -10 q^{79} + q^{81} -2 \beta_{2} q^{82} + ( -5 \beta_{1} - \beta_{2} ) q^{83} + ( 1 + \beta_{3} ) q^{84} + ( 6 - 2 \beta_{3} ) q^{86} -5 \beta_{1} q^{87} + 2 \beta_{1} q^{88} + ( 3 - 3 \beta_{3} ) q^{89} + ( -14 - 4 \beta_{3} ) q^{91} -\beta_{1} q^{92} + 2 \beta_{1} q^{93} + ( -1 - 2 \beta_{3} ) q^{94} + q^{96} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{97} + 2 \beta_{2} q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + 8q^{11} + 4q^{14} + 4q^{16} - 8q^{19} - 4q^{21} - 4q^{24} + 8q^{26} - 20q^{29} + 8q^{31} + 12q^{34} + 4q^{36} - 8q^{39} - 8q^{44} + 4q^{46} - 12q^{51} - 4q^{54} - 4q^{56} - 40q^{59} - 4q^{64} + 8q^{66} - 4q^{69} + 4q^{71} - 4q^{74} + 8q^{76} - 40q^{79} + 4q^{81} + 4q^{84} + 24q^{86} + 12q^{89} - 56q^{91} - 4q^{94} + 4q^{96} - 8q^{99} + O(q^{100})$$

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

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

## 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
 −1.22474 − 1.22474i 1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
1.00000i 1.00000i −1.00000 0 1.00000 1.44949i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 3.44949i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 1.00000 3.44949i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 1.00000 1.44949i 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.y 4
5.b even 2 1 inner 3450.2.d.y 4
5.c odd 4 1 3450.2.a.bf 2
5.c odd 4 1 3450.2.a.bl yes 2

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

 $$T_{7}^{4} + 14 T_{7}^{2} + 25$$ $$T_{11} - 2$$ $$T_{13}^{4} + 56 T_{13}^{2} + 400$$ $$T_{17}^{4} + 30 T_{17}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$25 + 14 T^{2} + T^{4}$$
$11$ $$( -2 + T )^{4}$$
$13$ $$400 + 56 T^{2} + T^{4}$$
$17$ $$9 + 30 T^{2} + T^{4}$$
$19$ $$( -20 + 4 T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( 5 + T )^{4}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$2809 + 110 T^{2} + T^{4}$$
$41$ $$( -24 + T^{2} )^{2}$$
$43$ $$144 + 120 T^{2} + T^{4}$$
$47$ $$529 + 50 T^{2} + T^{4}$$
$53$ $$64 + 80 T^{2} + T^{4}$$
$59$ $$( 10 + T )^{4}$$
$61$ $$( -24 + T^{2} )^{2}$$
$67$ $$( 4 + T^{2} )^{2}$$
$71$ $$( -95 - 2 T + T^{2} )^{2}$$
$73$ $$625 + 146 T^{2} + T^{4}$$
$79$ $$( 10 + T )^{4}$$
$83$ $$361 + 62 T^{2} + T^{4}$$
$89$ $$( -45 - 6 T + T^{2} )^{2}$$
$97$ $$44944 + 440 T^{2} + T^{4}$$