# Properties

 Label 3450.2.d.f Level $3450$ Weight $2$ Character orbit 3450.d Analytic conductor $27.548$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 0 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 0 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.f 2
5.b even 2 1 inner 3450.2.d.f 2
5.c odd 4 1 138.2.a.c 1
5.c odd 4 1 3450.2.a.k 1
15.e even 4 1 414.2.a.a 1
20.e even 4 1 1104.2.a.g 1
35.f even 4 1 6762.2.a.bg 1
40.i odd 4 1 4416.2.a.s 1
40.k even 4 1 4416.2.a.c 1
60.l odd 4 1 3312.2.a.d 1
115.e even 4 1 3174.2.a.e 1
345.l odd 4 1 9522.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.c 1 5.c odd 4 1
414.2.a.a 1 15.e even 4 1
1104.2.a.g 1 20.e even 4 1
3174.2.a.e 1 115.e even 4 1
3312.2.a.d 1 60.l odd 4 1
3450.2.a.k 1 5.c odd 4 1
3450.2.d.f 2 1.a even 1 1 trivial
3450.2.d.f 2 5.b even 2 1 inner
4416.2.a.c 1 40.k even 4 1
4416.2.a.s 1 40.i odd 4 1
6762.2.a.bg 1 35.f even 4 1
9522.2.a.d 1 345.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 $$S_{2}^{\mathrm{new}}(3450, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -8 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$( -4 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$64 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$256 + T^{2}$$
$89$ $$( 18 + T )^{2}$$
$97$ $$100 + T^{2}$$