# Properties

 Label 275.2.b.a Level $275$ Weight $2$ Character orbit 275.b Analytic conductor $2.196$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 275.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.19588605559$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 11) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - i q^{3} - 2 q^{4} + 2 q^{6} + 2 i q^{7} + 2 q^{9} +O(q^{10})$$ q + 2*i * q^2 - i * q^3 - 2 * q^4 + 2 * q^6 + 2*i * q^7 + 2 * q^9 $$q + 2 i q^{2} - i q^{3} - 2 q^{4} + 2 q^{6} + 2 i q^{7} + 2 q^{9} + q^{11} + 2 i q^{12} + 4 i q^{13} - 4 q^{14} - 4 q^{16} + 2 i q^{17} + 4 i q^{18} + 2 q^{21} + 2 i q^{22} - i q^{23} - 8 q^{26} - 5 i q^{27} - 4 i q^{28} + 7 q^{31} - 8 i q^{32} - i q^{33} - 4 q^{34} - 4 q^{36} - 3 i q^{37} + 4 q^{39} - 8 q^{41} + 4 i q^{42} - 6 i q^{43} - 2 q^{44} + 2 q^{46} - 8 i q^{47} + 4 i q^{48} + 3 q^{49} + 2 q^{51} - 8 i q^{52} - 6 i q^{53} + 10 q^{54} - 5 q^{59} + 12 q^{61} + 14 i q^{62} + 4 i q^{63} + 8 q^{64} + 2 q^{66} + 7 i q^{67} - 4 i q^{68} - q^{69} - 3 q^{71} + 4 i q^{73} + 6 q^{74} + 2 i q^{77} + 8 i q^{78} + 10 q^{79} + q^{81} - 16 i q^{82} - 6 i q^{83} - 4 q^{84} + 12 q^{86} - 15 q^{89} - 8 q^{91} + 2 i q^{92} - 7 i q^{93} + 16 q^{94} - 8 q^{96} + 7 i q^{97} + 6 i q^{98} + 2 q^{99} +O(q^{100})$$ q + 2*i * q^2 - i * q^3 - 2 * q^4 + 2 * q^6 + 2*i * q^7 + 2 * q^9 + q^11 + 2*i * q^12 + 4*i * q^13 - 4 * q^14 - 4 * q^16 + 2*i * q^17 + 4*i * q^18 + 2 * q^21 + 2*i * q^22 - i * q^23 - 8 * q^26 - 5*i * q^27 - 4*i * q^28 + 7 * q^31 - 8*i * q^32 - i * q^33 - 4 * q^34 - 4 * q^36 - 3*i * q^37 + 4 * q^39 - 8 * q^41 + 4*i * q^42 - 6*i * q^43 - 2 * q^44 + 2 * q^46 - 8*i * q^47 + 4*i * q^48 + 3 * q^49 + 2 * q^51 - 8*i * q^52 - 6*i * q^53 + 10 * q^54 - 5 * q^59 + 12 * q^61 + 14*i * q^62 + 4*i * q^63 + 8 * q^64 + 2 * q^66 + 7*i * q^67 - 4*i * q^68 - q^69 - 3 * q^71 + 4*i * q^73 + 6 * q^74 + 2*i * q^77 + 8*i * q^78 + 10 * q^79 + q^81 - 16*i * q^82 - 6*i * q^83 - 4 * q^84 + 12 * q^86 - 15 * q^89 - 8 * q^91 + 2*i * q^92 - 7*i * q^93 + 16 * q^94 - 8 * q^96 + 7*i * q^97 + 6*i * q^98 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 4 q^{6} + 4 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 + 4 * q^6 + 4 * q^9 $$2 q - 4 q^{4} + 4 q^{6} + 4 q^{9} + 2 q^{11} - 8 q^{14} - 8 q^{16} + 4 q^{21} - 16 q^{26} + 14 q^{31} - 8 q^{34} - 8 q^{36} + 8 q^{39} - 16 q^{41} - 4 q^{44} + 4 q^{46} + 6 q^{49} + 4 q^{51} + 20 q^{54} - 10 q^{59} + 24 q^{61} + 16 q^{64} + 4 q^{66} - 2 q^{69} - 6 q^{71} + 12 q^{74} + 20 q^{79} + 2 q^{81} - 8 q^{84} + 24 q^{86} - 30 q^{89} - 16 q^{91} + 32 q^{94} - 16 q^{96} + 4 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^6 + 4 * q^9 + 2 * q^11 - 8 * q^14 - 8 * q^16 + 4 * q^21 - 16 * q^26 + 14 * q^31 - 8 * q^34 - 8 * q^36 + 8 * q^39 - 16 * q^41 - 4 * q^44 + 4 * q^46 + 6 * q^49 + 4 * q^51 + 20 * q^54 - 10 * q^59 + 24 * q^61 + 16 * q^64 + 4 * q^66 - 2 * q^69 - 6 * q^71 + 12 * q^74 + 20 * q^79 + 2 * q^81 - 8 * q^84 + 24 * q^86 - 30 * q^89 - 16 * q^91 + 32 * q^94 - 16 * q^96 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
2.00000i 1.00000i −2.00000 0 2.00000 2.00000i 0 2.00000 0
199.2 2.00000i 1.00000i −2.00000 0 2.00000 2.00000i 0 2.00000 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 275.2.b.a 2
3.b odd 2 1 2475.2.c.a 2
4.b odd 2 1 4400.2.b.h 2
5.b even 2 1 inner 275.2.b.a 2
5.c odd 4 1 11.2.a.a 1
5.c odd 4 1 275.2.a.b 1
15.d odd 2 1 2475.2.c.a 2
15.e even 4 1 99.2.a.d 1
15.e even 4 1 2475.2.a.a 1
20.d odd 2 1 4400.2.b.h 2
20.e even 4 1 176.2.a.b 1
20.e even 4 1 4400.2.a.i 1
35.f even 4 1 539.2.a.a 1
35.k even 12 2 539.2.e.g 2
35.l odd 12 2 539.2.e.h 2
40.i odd 4 1 704.2.a.h 1
40.k even 4 1 704.2.a.c 1
45.k odd 12 2 891.2.e.k 2
45.l even 12 2 891.2.e.b 2
55.e even 4 1 121.2.a.d 1
55.e even 4 1 3025.2.a.a 1
55.k odd 20 4 121.2.c.e 4
55.l even 20 4 121.2.c.a 4
60.l odd 4 1 1584.2.a.g 1
65.h odd 4 1 1859.2.a.b 1
80.i odd 4 1 2816.2.c.j 2
80.j even 4 1 2816.2.c.f 2
80.s even 4 1 2816.2.c.f 2
80.t odd 4 1 2816.2.c.j 2
85.g odd 4 1 3179.2.a.a 1
95.g even 4 1 3971.2.a.b 1
105.k odd 4 1 4851.2.a.t 1
115.e even 4 1 5819.2.a.a 1
120.q odd 4 1 6336.2.a.bu 1
120.w even 4 1 6336.2.a.br 1
140.j odd 4 1 8624.2.a.j 1
145.h odd 4 1 9251.2.a.d 1
165.l odd 4 1 1089.2.a.b 1
220.i odd 4 1 1936.2.a.i 1
385.l odd 4 1 5929.2.a.h 1
440.t even 4 1 7744.2.a.x 1
440.w odd 4 1 7744.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.2.a.a 1 5.c odd 4 1
99.2.a.d 1 15.e even 4 1
121.2.a.d 1 55.e even 4 1
121.2.c.a 4 55.l even 20 4
121.2.c.e 4 55.k odd 20 4
176.2.a.b 1 20.e even 4 1
275.2.a.b 1 5.c odd 4 1
275.2.b.a 2 1.a even 1 1 trivial
275.2.b.a 2 5.b even 2 1 inner
539.2.a.a 1 35.f even 4 1
539.2.e.g 2 35.k even 12 2
539.2.e.h 2 35.l odd 12 2
704.2.a.c 1 40.k even 4 1
704.2.a.h 1 40.i odd 4 1
891.2.e.b 2 45.l even 12 2
891.2.e.k 2 45.k odd 12 2
1089.2.a.b 1 165.l odd 4 1
1584.2.a.g 1 60.l odd 4 1
1859.2.a.b 1 65.h odd 4 1
1936.2.a.i 1 220.i odd 4 1
2475.2.a.a 1 15.e even 4 1
2475.2.c.a 2 3.b odd 2 1
2475.2.c.a 2 15.d odd 2 1
2816.2.c.f 2 80.j even 4 1
2816.2.c.f 2 80.s even 4 1
2816.2.c.j 2 80.i odd 4 1
2816.2.c.j 2 80.t odd 4 1
3025.2.a.a 1 55.e even 4 1
3179.2.a.a 1 85.g odd 4 1
3971.2.a.b 1 95.g even 4 1
4400.2.a.i 1 20.e even 4 1
4400.2.b.h 2 4.b odd 2 1
4400.2.b.h 2 20.d odd 2 1
4851.2.a.t 1 105.k odd 4 1
5819.2.a.a 1 115.e even 4 1
5929.2.a.h 1 385.l odd 4 1
6336.2.a.br 1 120.w even 4 1
6336.2.a.bu 1 120.q odd 4 1
7744.2.a.k 1 440.w odd 4 1
7744.2.a.x 1 440.t even 4 1
8624.2.a.j 1 140.j odd 4 1
9251.2.a.d 1 145.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$T^{2}$$
$31$ $$(T - 7)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 5)^{2}$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T + 15)^{2}$$
$97$ $$T^{2} + 49$$