# Properties

 Label 3450.2.d.h Level $3450$ Weight $2$ Character orbit 3450.d Analytic conductor $27.548$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## 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: yes 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.

 $$f(q)$$ $$=$$ $$q -i q^{2} -i q^{3} - q^{4} - q^{6} + 5 i q^{7} + i q^{8} - q^{9} +O(q^{10})$$ $$q -i q^{2} -i q^{3} - q^{4} - q^{6} + 5 i q^{7} + i q^{8} - q^{9} + i q^{12} -2 i q^{13} + 5 q^{14} + q^{16} + 3 i q^{17} + i q^{18} -2 q^{19} + 5 q^{21} -i q^{23} + q^{24} -2 q^{26} + i q^{27} -5 i q^{28} -3 q^{29} + 2 q^{31} -i q^{32} + 3 q^{34} + q^{36} -7 i q^{37} + 2 i q^{38} -2 q^{39} -5 i q^{42} -2 i q^{43} - q^{46} -3 i q^{47} -i q^{48} -18 q^{49} + 3 q^{51} + 2 i q^{52} + 12 i q^{53} + q^{54} -5 q^{56} + 2 i q^{57} + 3 i q^{58} -6 q^{59} + 2 q^{61} -2 i q^{62} -5 i q^{63} - q^{64} + 2 i q^{67} -3 i q^{68} - q^{69} -15 q^{71} -i q^{72} -11 i q^{73} -7 q^{74} + 2 q^{76} + 2 i q^{78} -8 q^{79} + q^{81} + 9 i q^{83} -5 q^{84} -2 q^{86} + 3 i q^{87} -3 q^{89} + 10 q^{91} + i q^{92} -2 i q^{93} -3 q^{94} - q^{96} -10 i q^{97} + 18 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + 10q^{14} + 2q^{16} - 4q^{19} + 10q^{21} + 2q^{24} - 4q^{26} - 6q^{29} + 4q^{31} + 6q^{34} + 2q^{36} - 4q^{39} - 2q^{46} - 36q^{49} + 6q^{51} + 2q^{54} - 10q^{56} - 12q^{59} + 4q^{61} - 2q^{64} - 2q^{69} - 30q^{71} - 14q^{74} + 4q^{76} - 16q^{79} + 2q^{81} - 10q^{84} - 4q^{86} - 6q^{89} + 20q^{91} - 6q^{94} - 2q^{96} + O(q^{100})$$

## 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 5.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 5.00000i 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.h 2
5.b even 2 1 inner 3450.2.d.h 2
5.c odd 4 1 3450.2.a.m 1
5.c odd 4 1 3450.2.a.n yes 1

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

 $$T_{7}^{2} + 25$$ $$T_{11}$$ $$T_{13}^{2} + 4$$ $$T_{17}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$25 + T^{2}$$
$11$ $$T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$9 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( 3 + T )^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$49 + T^{2}$$
$41$ $$T^{2}$$
$43$ $$4 + T^{2}$$
$47$ $$9 + T^{2}$$
$53$ $$144 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$4 + T^{2}$$
$71$ $$( 15 + T )^{2}$$
$73$ $$121 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$81 + T^{2}$$
$89$ $$( 3 + T )^{2}$$
$97$ $$100 + T^{2}$$
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