# Properties

 Label 3450.2.d.x 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 138) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{1} q^{3} - q^{4} + q^{6} + 2 \beta_{2} q^{7} -\beta_{1} q^{8} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{1} q^{3} - q^{4} + q^{6} + 2 \beta_{2} q^{7} -\beta_{1} q^{8} - q^{9} + ( -3 + \beta_{3} ) q^{11} + \beta_{1} q^{12} + 2 \beta_{2} q^{13} -2 \beta_{3} q^{14} + q^{16} -4 \beta_{1} q^{17} -\beta_{1} q^{18} + ( 1 + 3 \beta_{3} ) q^{19} + 2 \beta_{3} q^{21} + ( -3 \beta_{1} + \beta_{2} ) q^{22} -\beta_{1} q^{23} - q^{24} -2 \beta_{3} q^{26} + \beta_{1} q^{27} -2 \beta_{2} q^{28} -2 \beta_{3} q^{29} + ( 2 + 2 \beta_{3} ) q^{31} + \beta_{1} q^{32} + ( 3 \beta_{1} - \beta_{2} ) q^{33} + 4 q^{34} + q^{36} + ( 9 \beta_{1} - \beta_{2} ) q^{37} + ( \beta_{1} + 3 \beta_{2} ) q^{38} + 2 \beta_{3} q^{39} -2 q^{41} + 2 \beta_{2} q^{42} + ( 7 \beta_{1} + \beta_{2} ) q^{43} + ( 3 - \beta_{3} ) q^{44} + q^{46} + 4 \beta_{1} q^{47} -\beta_{1} q^{48} -13 q^{49} -4 q^{51} -2 \beta_{2} q^{52} + ( -3 \beta_{1} + \beta_{2} ) q^{53} - q^{54} + 2 \beta_{3} q^{56} + ( -\beta_{1} - 3 \beta_{2} ) q^{57} -2 \beta_{2} q^{58} -4 \beta_{3} q^{59} + ( 3 + \beta_{3} ) q^{61} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{62} -2 \beta_{2} q^{63} - q^{64} + ( -3 + \beta_{3} ) q^{66} + ( 3 \beta_{1} - 3 \beta_{2} ) q^{67} + 4 \beta_{1} q^{68} - q^{69} -4 \beta_{3} q^{71} + \beta_{1} q^{72} -2 \beta_{2} q^{73} + ( -9 + \beta_{3} ) q^{74} + ( -1 - 3 \beta_{3} ) q^{76} + ( 10 \beta_{1} - 6 \beta_{2} ) q^{77} + 2 \beta_{2} q^{78} + 2 \beta_{3} q^{79} + q^{81} -2 \beta_{1} q^{82} + ( -11 \beta_{1} + \beta_{2} ) q^{83} -2 \beta_{3} q^{84} + ( -7 - \beta_{3} ) q^{86} + 2 \beta_{2} q^{87} + ( 3 \beta_{1} - \beta_{2} ) q^{88} + ( 6 - 2 \beta_{3} ) q^{89} -20 q^{91} + \beta_{1} q^{92} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{93} -4 q^{94} + q^{96} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{97} -13 \beta_{1} q^{98} + ( 3 - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 4q^{6} - 4q^{9} - 12q^{11} + 4q^{16} + 4q^{19} - 4q^{24} + 8q^{31} + 16q^{34} + 4q^{36} - 8q^{41} + 12q^{44} + 4q^{46} - 52q^{49} - 16q^{51} - 4q^{54} + 12q^{61} - 4q^{64} - 12q^{66} - 4q^{69} - 36q^{74} - 4q^{76} + 4q^{81} - 28q^{86} + 24q^{89} - 80q^{91} - 16q^{94} + 4q^{96} + 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

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

## 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
 − 0.618034i 1.61803i − 1.61803i 0.618034i
1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 1.00000i −1.00000 0
2899.3 1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 1.00000i −1.00000 0
2899.4 1.00000i 1.00000i −1.00000 0 1.00000 4.47214i 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.x 4
5.b even 2 1 inner 3450.2.d.x 4
5.c odd 4 1 138.2.a.d 2
5.c odd 4 1 3450.2.a.be 2
15.e even 4 1 414.2.a.f 2
20.e even 4 1 1104.2.a.j 2
35.f even 4 1 6762.2.a.cb 2
40.i odd 4 1 4416.2.a.bh 2
40.k even 4 1 4416.2.a.bl 2
60.l odd 4 1 3312.2.a.bc 2
115.e even 4 1 3174.2.a.s 2
345.l odd 4 1 9522.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.d 2 5.c odd 4 1
414.2.a.f 2 15.e even 4 1
1104.2.a.j 2 20.e even 4 1
3174.2.a.s 2 115.e even 4 1
3312.2.a.bc 2 60.l odd 4 1
3450.2.a.be 2 5.c odd 4 1
3450.2.d.x 4 1.a even 1 1 trivial
3450.2.d.x 4 5.b even 2 1 inner
4416.2.a.bh 2 40.i odd 4 1
4416.2.a.bl 2 40.k even 4 1
6762.2.a.cb 2 35.f even 4 1
9522.2.a.q 2 345.l odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 20$$ $$T_{11}^{2} + 6 T_{11} + 4$$ $$T_{13}^{2} + 20$$ $$T_{17}^{2} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 20 + T^{2} )^{2}$$
$11$ $$( 4 + 6 T + T^{2} )^{2}$$
$13$ $$( 20 + T^{2} )^{2}$$
$17$ $$( 16 + T^{2} )^{2}$$
$19$ $$( -44 - 2 T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( -20 + T^{2} )^{2}$$
$31$ $$( -16 - 4 T + T^{2} )^{2}$$
$37$ $$5776 + 172 T^{2} + T^{4}$$
$41$ $$( 2 + T )^{4}$$
$43$ $$1936 + 108 T^{2} + T^{4}$$
$47$ $$( 16 + T^{2} )^{2}$$
$53$ $$16 + 28 T^{2} + T^{4}$$
$59$ $$( -80 + T^{2} )^{2}$$
$61$ $$( 4 - 6 T + T^{2} )^{2}$$
$67$ $$1296 + 108 T^{2} + T^{4}$$
$71$ $$( -80 + T^{2} )^{2}$$
$73$ $$( 20 + T^{2} )^{2}$$
$79$ $$( -20 + T^{2} )^{2}$$
$83$ $$13456 + 252 T^{2} + T^{4}$$
$89$ $$( 16 - 12 T + T^{2} )^{2}$$
$97$ $$16 + 72 T^{2} + T^{4}$$