# Properties

 Label 3450.2.d.d 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) 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.

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

## 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 2.00000i 1.00000i −1.00000 0
2899.2 1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 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.d 2
5.b even 2 1 inner 3450.2.d.d 2
5.c odd 4 1 690.2.a.g 1
5.c odd 4 1 3450.2.a.l 1
15.e even 4 1 2070.2.a.e 1
20.e even 4 1 5520.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.g 1 5.c odd 4 1
2070.2.a.e 1 15.e even 4 1
3450.2.a.l 1 5.c odd 4 1
3450.2.d.d 2 1.a even 1 1 trivial
3450.2.d.d 2 5.b even 2 1 inner
5520.2.a.z 1 20.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} + 2$$ T11 + 2 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17}^{2} + 16$$ T17^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 16$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 16)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 14)^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 36$$