# Properties

 Label 3450.2.d.z 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: $$\Q(i, \sqrt{57})$$ Defining polynomial: $$x^{4} + 29 x^{2} + 196$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 29 x^{2} + 196$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 15 \nu$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 15$$ $$\nu^{3}$$ $$=$$ $$14 \beta_{2} - 15 \beta_{1}$$

## 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
 3.27492i − 4.27492i − 3.27492i 4.27492i
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.z 4
5.b even 2 1 inner 3450.2.d.z 4
5.c odd 4 1 3450.2.a.bd 2
5.c odd 4 1 3450.2.a.bm yes 2

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

 $$T_{7}^{2} + 9$$ $$T_{11}^{2} - 5 T_{11} - 8$$ $$T_{13}^{4} + 29 T_{13}^{2} + 196$$ $$T_{17}^{4} + 33 T_{17}^{2} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 9 + T^{2} )^{2}$$
$11$ $$( -8 - 5 T + T^{2} )^{2}$$
$13$ $$196 + 29 T^{2} + T^{4}$$
$17$ $$144 + 33 T^{2} + T^{4}$$
$19$ $$( -14 - T + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( -53 - 4 T + T^{2} )^{2}$$
$31$ $$( -6 + T )^{4}$$
$37$ $$16384 + 257 T^{2} + T^{4}$$
$41$ $$( 28 + 13 T + T^{2} )^{2}$$
$43$ $$4 + 53 T^{2} + T^{4}$$
$47$ $$36 + 69 T^{2} + T^{4}$$
$53$ $$2304 + 132 T^{2} + T^{4}$$
$59$ $$( -32 + 10 T + T^{2} )^{2}$$
$61$ $$( -8 + 14 T + T^{2} )^{2}$$
$67$ $$2304 + 132 T^{2} + T^{4}$$
$71$ $$( -122 - 5 T + T^{2} )^{2}$$
$73$ $$49 + 242 T^{2} + T^{4}$$
$79$ $$( -128 - T + T^{2} )^{2}$$
$83$ $$1849 + 314 T^{2} + T^{4}$$
$89$ $$( 6 + 9 T + T^{2} )^{2}$$
$97$ $$( 196 + T^{2} )^{2}$$