# Properties

 Label 3450.2.d.q 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 $$q + i q^{2} - i q^{3} - q^{4} + q^{6} + 2 i q^{7} - i q^{8} - q^{9} + 2 q^{11} + i q^{12} - 2 i q^{13} - 2 q^{14} + q^{16} - i q^{18} - 8 q^{19} + 2 q^{21} + 2 i q^{22} - i q^{23} - q^{24} + 2 q^{26} + i q^{27} - 2 i q^{28} + 10 q^{29} + 8 q^{31} + i q^{32} - 2 i q^{33} + q^{36} - 8 i q^{37} - 8 i q^{38} - 2 q^{39} - 6 q^{41} + 2 i q^{42} + 12 i q^{43} - 2 q^{44} + q^{46} - 8 i q^{47} - i q^{48} + 3 q^{49} + 2 i q^{52} + 10 i q^{53} - q^{54} + 2 q^{56} + 8 i q^{57} + 10 i q^{58} - 4 q^{59} + 12 q^{61} + 8 i q^{62} - 2 i q^{63} - q^{64} + 2 q^{66} + 4 i q^{67} - q^{69} + 16 q^{71} + i q^{72} - 10 i q^{73} + 8 q^{74} + 8 q^{76} + 4 i q^{77} - 2 i q^{78} - 10 q^{79} + q^{81} - 6 i q^{82} - 10 i q^{83} - 2 q^{84} - 12 q^{86} - 10 i q^{87} - 2 i q^{88} + 4 q^{91} + i q^{92} - 8 i q^{93} + 8 q^{94} + q^{96} - 10 i q^{97} + 3 i q^{98} - 2 q^{99} +O(q^{100})$$ 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 - 2*i * q^13 - 2 * q^14 + q^16 - i * q^18 - 8 * q^19 + 2 * q^21 + 2*i * q^22 - i * q^23 - q^24 + 2 * q^26 + i * q^27 - 2*i * q^28 + 10 * q^29 + 8 * q^31 + i * q^32 - 2*i * q^33 + q^36 - 8*i * q^37 - 8*i * q^38 - 2 * q^39 - 6 * q^41 + 2*i * q^42 + 12*i * q^43 - 2 * q^44 + q^46 - 8*i * q^47 - i * q^48 + 3 * q^49 + 2*i * q^52 + 10*i * q^53 - q^54 + 2 * q^56 + 8*i * q^57 + 10*i * q^58 - 4 * q^59 + 12 * q^61 + 8*i * q^62 - 2*i * q^63 - q^64 + 2 * q^66 + 4*i * q^67 - q^69 + 16 * q^71 + i * q^72 - 10*i * q^73 + 8 * q^74 + 8 * q^76 + 4*i * q^77 - 2*i * q^78 - 10 * q^79 + q^81 - 6*i * q^82 - 10*i * q^83 - 2 * q^84 - 12 * q^86 - 10*i * q^87 - 2*i * q^88 + 4 * q^91 + i * q^92 - 8*i * q^93 + 8 * q^94 + q^96 - 10*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 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + 4 q^{11} - 4 q^{14} + 2 q^{16} - 16 q^{19} + 4 q^{21} - 2 q^{24} + 4 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{36} - 4 q^{39} - 12 q^{41} - 4 q^{44} + 2 q^{46} + 6 q^{49} - 2 q^{54} + 4 q^{56} - 8 q^{59} + 24 q^{61} - 2 q^{64} + 4 q^{66} - 2 q^{69} + 32 q^{71} + 16 q^{74} + 16 q^{76} - 20 q^{79} + 2 q^{81} - 4 q^{84} - 24 q^{86} + 8 q^{91} + 16 q^{94} + 2 q^{96} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 + 4 * q^11 - 4 * q^14 + 2 * q^16 - 16 * q^19 + 4 * q^21 - 2 * q^24 + 4 * q^26 + 20 * q^29 + 16 * q^31 + 2 * q^36 - 4 * q^39 - 12 * q^41 - 4 * q^44 + 2 * q^46 + 6 * q^49 - 2 * q^54 + 4 * q^56 - 8 * q^59 + 24 * q^61 - 2 * q^64 + 4 * q^66 - 2 * q^69 + 32 * q^71 + 16 * q^74 + 16 * q^76 - 20 * q^79 + 2 * q^81 - 4 * q^84 - 24 * q^86 + 8 * q^91 + 16 * q^94 + 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.q 2
5.b even 2 1 inner 3450.2.d.q 2
5.c odd 4 1 690.2.a.c 1
5.c odd 4 1 3450.2.a.z 1
15.e even 4 1 2070.2.a.k 1
20.e even 4 1 5520.2.a.bg 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.a.c 1 5.c odd 4 1
2070.2.a.k 1 15.e even 4 1
3450.2.a.z 1 5.c odd 4 1
3450.2.d.q 2 1.a even 1 1 trivial
3450.2.d.q 2 5.b even 2 1 inner
5520.2.a.bg 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 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2}$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 10)^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 144$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 100$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 16)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 100$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 100$$