# Properties

 Label 4950.2.c.m Level $4950$ Weight $2$ Character orbit 4950.c Analytic conductor $39.526$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4950.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.5259490005$$ 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 110) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} + 3 i q^{7} + i q^{8} +O(q^{10})$$ q - i * q^2 - q^4 + 3*i * q^7 + i * q^8 $$q - i q^{2} - q^{4} + 3 i q^{7} + i q^{8} - q^{11} + 6 i q^{13} + 3 q^{14} + q^{16} + 7 i q^{17} - 5 q^{19} + i q^{22} - 6 i q^{23} + 6 q^{26} - 3 i q^{28} + 5 q^{29} - 3 q^{31} - i q^{32} + 7 q^{34} + 3 i q^{37} + 5 i q^{38} - 2 q^{41} - 4 i q^{43} + q^{44} - 6 q^{46} + 2 i q^{47} - 2 q^{49} - 6 i q^{52} - i q^{53} - 3 q^{56} - 5 i q^{58} - 10 q^{59} + 7 q^{61} + 3 i q^{62} - q^{64} + 8 i q^{67} - 7 i q^{68} - 7 q^{71} - 14 i q^{73} + 3 q^{74} + 5 q^{76} - 3 i q^{77} - 10 q^{79} + 2 i q^{82} - 6 i q^{83} - 4 q^{86} - i q^{88} - 15 q^{89} - 18 q^{91} + 6 i q^{92} + 2 q^{94} - 12 i q^{97} + 2 i q^{98} +O(q^{100})$$ q - i * q^2 - q^4 + 3*i * q^7 + i * q^8 - q^11 + 6*i * q^13 + 3 * q^14 + q^16 + 7*i * q^17 - 5 * q^19 + i * q^22 - 6*i * q^23 + 6 * q^26 - 3*i * q^28 + 5 * q^29 - 3 * q^31 - i * q^32 + 7 * q^34 + 3*i * q^37 + 5*i * q^38 - 2 * q^41 - 4*i * q^43 + q^44 - 6 * q^46 + 2*i * q^47 - 2 * q^49 - 6*i * q^52 - i * q^53 - 3 * q^56 - 5*i * q^58 - 10 * q^59 + 7 * q^61 + 3*i * q^62 - q^64 + 8*i * q^67 - 7*i * q^68 - 7 * q^71 - 14*i * q^73 + 3 * q^74 + 5 * q^76 - 3*i * q^77 - 10 * q^79 + 2*i * q^82 - 6*i * q^83 - 4 * q^86 - i * q^88 - 15 * q^89 - 18 * q^91 + 6*i * q^92 + 2 * q^94 - 12*i * q^97 + 2*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 2 q^{11} + 6 q^{14} + 2 q^{16} - 10 q^{19} + 12 q^{26} + 10 q^{29} - 6 q^{31} + 14 q^{34} - 4 q^{41} + 2 q^{44} - 12 q^{46} - 4 q^{49} - 6 q^{56} - 20 q^{59} + 14 q^{61} - 2 q^{64} - 14 q^{71} + 6 q^{74} + 10 q^{76} - 20 q^{79} - 8 q^{86} - 30 q^{89} - 36 q^{91} + 4 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^11 + 6 * q^14 + 2 * q^16 - 10 * q^19 + 12 * q^26 + 10 * q^29 - 6 * q^31 + 14 * q^34 - 4 * q^41 + 2 * q^44 - 12 * q^46 - 4 * q^49 - 6 * q^56 - 20 * q^59 + 14 * q^61 - 2 * q^64 - 14 * q^71 + 6 * q^74 + 10 * q^76 - 20 * q^79 - 8 * q^86 - 30 * q^89 - 36 * q^91 + 4 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2377$$ $$4501$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 0 0 3.00000i 1.00000i 0 0
199.2 1.00000i 0 −1.00000 0 0 3.00000i 1.00000i 0 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 4950.2.c.m 2
3.b odd 2 1 550.2.b.a 2
5.b even 2 1 inner 4950.2.c.m 2
5.c odd 4 1 990.2.a.d 1
5.c odd 4 1 4950.2.a.bc 1
12.b even 2 1 4400.2.b.i 2
15.d odd 2 1 550.2.b.a 2
15.e even 4 1 110.2.a.b 1
15.e even 4 1 550.2.a.f 1
20.e even 4 1 7920.2.a.d 1
60.h even 2 1 4400.2.b.i 2
60.l odd 4 1 880.2.a.i 1
60.l odd 4 1 4400.2.a.l 1
105.k odd 4 1 5390.2.a.bf 1
120.q odd 4 1 3520.2.a.h 1
120.w even 4 1 3520.2.a.y 1
165.l odd 4 1 1210.2.a.b 1
165.l odd 4 1 6050.2.a.bj 1
660.q even 4 1 9680.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.b 1 15.e even 4 1
550.2.a.f 1 15.e even 4 1
550.2.b.a 2 3.b odd 2 1
550.2.b.a 2 15.d odd 2 1
880.2.a.i 1 60.l odd 4 1
990.2.a.d 1 5.c odd 4 1
1210.2.a.b 1 165.l odd 4 1
3520.2.a.h 1 120.q odd 4 1
3520.2.a.y 1 120.w even 4 1
4400.2.a.l 1 60.l odd 4 1
4400.2.b.i 2 12.b even 2 1
4400.2.b.i 2 60.h even 2 1
4950.2.a.bc 1 5.c odd 4 1
4950.2.c.m 2 1.a even 1 1 trivial
4950.2.c.m 2 5.b even 2 1 inner
5390.2.a.bf 1 105.k odd 4 1
6050.2.a.bj 1 165.l odd 4 1
7920.2.a.d 1 20.e even 4 1
9680.2.a.x 1 660.q even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4950, [\chi])$$:

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17}^{2} + 49$$ T17^2 + 49 $$T_{19} + 5$$ T19 + 5 $$T_{29} - 5$$ T29 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 49$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 5)^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 1$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 7)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 15)^{2}$$
$97$ $$T^{2} + 144$$