Properties

 Label 4650.2.d.u Level $4650$ Weight $2$ Character orbit 4650.d Analytic conductor $37.130$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4650.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$37.1304369399$$ Analytic rank: $$1$$ 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 930) 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 $$q + i q^{2} - i q^{3} - q^{4} + q^{6} + 2 i q^{7} - i q^{8} - q^{9} + i q^{12} + 4 i q^{13} - 2 q^{14} + q^{16} + 6 i q^{17} - i q^{18} - 8 q^{19} + 2 q^{21} - q^{24} - 4 q^{26} + i q^{27} - 2 i q^{28} + q^{31} + i q^{32} - 6 q^{34} + q^{36} - 4 i q^{37} - 8 i q^{38} + 4 q^{39} - 6 q^{41} + 2 i q^{42} - 8 i q^{43} - 12 i q^{47} - i q^{48} + 3 q^{49} + 6 q^{51} - 4 i q^{52} + 6 i q^{53} - q^{54} + 2 q^{56} + 8 i q^{57} + 6 q^{59} + 2 q^{61} + i q^{62} - 2 i q^{63} - q^{64} + 2 i q^{67} - 6 i q^{68} - 6 q^{71} + i q^{72} - 8 i q^{73} + 4 q^{74} + 8 q^{76} + 4 i q^{78} - 8 q^{79} + q^{81} - 6 i q^{82} - 12 i q^{83} - 2 q^{84} + 8 q^{86} - 8 q^{91} - i q^{93} + 12 q^{94} + q^{96} - 10 i q^{97} + 3 i q^{98} +O(q^{100})$$ q + i * q^2 - i * q^3 - q^4 + q^6 + 2*i * q^7 - i * q^8 - q^9 + i * q^12 + 4*i * q^13 - 2 * q^14 + q^16 + 6*i * q^17 - i * q^18 - 8 * q^19 + 2 * q^21 - q^24 - 4 * q^26 + i * q^27 - 2*i * q^28 + q^31 + i * q^32 - 6 * q^34 + q^36 - 4*i * q^37 - 8*i * q^38 + 4 * q^39 - 6 * q^41 + 2*i * q^42 - 8*i * q^43 - 12*i * q^47 - i * q^48 + 3 * q^49 + 6 * q^51 - 4*i * q^52 + 6*i * q^53 - q^54 + 2 * q^56 + 8*i * q^57 + 6 * q^59 + 2 * q^61 + i * q^62 - 2*i * q^63 - q^64 + 2*i * q^67 - 6*i * q^68 - 6 * q^71 + i * q^72 - 8*i * q^73 + 4 * q^74 + 8 * q^76 + 4*i * q^78 - 8 * q^79 + q^81 - 6*i * q^82 - 12*i * q^83 - 2 * q^84 + 8 * q^86 - 8 * q^91 - i * q^93 + 12 * q^94 + q^96 - 10*i * q^97 + 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 16 q^{19} + 4 q^{21} - 2 q^{24} - 8 q^{26} + 2 q^{31} - 12 q^{34} + 2 q^{36} + 8 q^{39} - 12 q^{41} + 6 q^{49} + 12 q^{51} - 2 q^{54} + 4 q^{56} + 12 q^{59} + 4 q^{61} - 2 q^{64} - 12 q^{71} + 8 q^{74} + 16 q^{76} - 16 q^{79} + 2 q^{81} - 4 q^{84} + 16 q^{86} - 16 q^{91} + 24 q^{94} + 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 16 * q^19 + 4 * q^21 - 2 * q^24 - 8 * q^26 + 2 * q^31 - 12 * q^34 + 2 * q^36 + 8 * q^39 - 12 * q^41 + 6 * q^49 + 12 * q^51 - 2 * q^54 + 4 * q^56 + 12 * q^59 + 4 * q^61 - 2 * q^64 - 12 * q^71 + 8 * q^74 + 16 * q^76 - 16 * q^79 + 2 * q^81 - 4 * q^84 + 16 * q^86 - 16 * q^91 + 24 * q^94 + 2 * q^96

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times$$.

 $$n$$ $$1801$$ $$2977$$ $$3101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3349.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 0 1.00000 2.00000i 1.00000i −1.00000 0
3349.2 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 4650.2.d.u 2
5.b even 2 1 inner 4650.2.d.u 2
5.c odd 4 1 930.2.a.n 1
5.c odd 4 1 4650.2.a.d 1
15.e even 4 1 2790.2.a.k 1
20.e even 4 1 7440.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.n 1 5.c odd 4 1
2790.2.a.k 1 15.e even 4 1
4650.2.a.d 1 5.c odd 4 1
4650.2.d.u 2 1.a even 1 1 trivial
4650.2.d.u 2 5.b even 2 1 inner
7440.2.a.b 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 $$S_{2}^{\mathrm{new}}(4650, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}$$ T11 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}^{2} + 36$$ T17^2 + 36 $$T_{19} + 8$$ T19 + 8 $$T_{29}$$ T29

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 36$$
$19$ $$(T + 8)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 16$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 64$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 100$$