# Properties

 Label 425.2.b.b Level $425$ Weight $2$ Character orbit 425.b Analytic conductor $3.394$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$425 = 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 425.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.39364208590$$ 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 17) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} - 4 i q^{7} + 3 i q^{8} + 3 q^{9} +O(q^{10})$$ q + i * q^2 + q^4 - 4*i * q^7 + 3*i * q^8 + 3 * q^9 $$q + i q^{2} + q^{4} - 4 i q^{7} + 3 i q^{8} + 3 q^{9} - 2 i q^{13} + 4 q^{14} - q^{16} - i q^{17} + 3 i q^{18} + 4 q^{19} + 4 i q^{23} + 2 q^{26} - 4 i q^{28} - 6 q^{29} + 4 q^{31} + 5 i q^{32} + q^{34} + 3 q^{36} + 2 i q^{37} + 4 i q^{38} - 6 q^{41} + 4 i q^{43} - 4 q^{46} - 9 q^{49} - 2 i q^{52} + 6 i q^{53} + 12 q^{56} - 6 i q^{58} + 12 q^{59} - 10 q^{61} + 4 i q^{62} - 12 i q^{63} - 7 q^{64} - 4 i q^{67} - i q^{68} - 4 q^{71} + 9 i q^{72} - 6 i q^{73} - 2 q^{74} + 4 q^{76} - 12 q^{79} + 9 q^{81} - 6 i q^{82} - 4 i q^{83} - 4 q^{86} - 10 q^{89} - 8 q^{91} + 4 i q^{92} - 2 i q^{97} - 9 i q^{98} +O(q^{100})$$ q + i * q^2 + q^4 - 4*i * q^7 + 3*i * q^8 + 3 * q^9 - 2*i * q^13 + 4 * q^14 - q^16 - i * q^17 + 3*i * q^18 + 4 * q^19 + 4*i * q^23 + 2 * q^26 - 4*i * q^28 - 6 * q^29 + 4 * q^31 + 5*i * q^32 + q^34 + 3 * q^36 + 2*i * q^37 + 4*i * q^38 - 6 * q^41 + 4*i * q^43 - 4 * q^46 - 9 * q^49 - 2*i * q^52 + 6*i * q^53 + 12 * q^56 - 6*i * q^58 + 12 * q^59 - 10 * q^61 + 4*i * q^62 - 12*i * q^63 - 7 * q^64 - 4*i * q^67 - i * q^68 - 4 * q^71 + 9*i * q^72 - 6*i * q^73 - 2 * q^74 + 4 * q^76 - 12 * q^79 + 9 * q^81 - 6*i * q^82 - 4*i * q^83 - 4 * q^86 - 10 * q^89 - 8 * q^91 + 4*i * q^92 - 2*i * q^97 - 9*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^4 + 6 * q^9 $$2 q + 2 q^{4} + 6 q^{9} + 8 q^{14} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{34} + 6 q^{36} - 12 q^{41} - 8 q^{46} - 18 q^{49} + 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{71} - 4 q^{74} + 8 q^{76} - 24 q^{79} + 18 q^{81} - 8 q^{86} - 20 q^{89} - 16 q^{91}+O(q^{100})$$ 2 * q + 2 * q^4 + 6 * q^9 + 8 * q^14 - 2 * q^16 + 8 * q^19 + 4 * q^26 - 12 * q^29 + 8 * q^31 + 2 * q^34 + 6 * q^36 - 12 * q^41 - 8 * q^46 - 18 * q^49 + 24 * q^56 + 24 * q^59 - 20 * q^61 - 14 * q^64 - 8 * q^71 - 4 * q^74 + 8 * q^76 - 24 * q^79 + 18 * q^81 - 8 * q^86 - 20 * q^89 - 16 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$$.

 $$n$$ $$52$$ $$326$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 0 0 4.00000i 3.00000i 3.00000 0
324.2 1.00000i 0 1.00000 0 0 4.00000i 3.00000i 3.00000 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 425.2.b.b 2
5.b even 2 1 inner 425.2.b.b 2
5.c odd 4 1 17.2.a.a 1
5.c odd 4 1 425.2.a.d 1
15.e even 4 1 153.2.a.c 1
15.e even 4 1 3825.2.a.d 1
20.e even 4 1 272.2.a.b 1
20.e even 4 1 6800.2.a.n 1
35.f even 4 1 833.2.a.a 1
35.k even 12 2 833.2.e.a 2
35.l odd 12 2 833.2.e.b 2
40.i odd 4 1 1088.2.a.i 1
40.k even 4 1 1088.2.a.h 1
55.e even 4 1 2057.2.a.e 1
60.l odd 4 1 2448.2.a.o 1
65.h odd 4 1 2873.2.a.c 1
85.f odd 4 1 289.2.b.a 2
85.g odd 4 1 289.2.a.a 1
85.g odd 4 1 7225.2.a.g 1
85.i odd 4 1 289.2.b.a 2
85.k odd 8 2 289.2.c.a 4
85.n odd 8 2 289.2.c.a 4
85.o even 16 4 289.2.d.d 8
85.r even 16 4 289.2.d.d 8
95.g even 4 1 6137.2.a.b 1
105.k odd 4 1 7497.2.a.l 1
115.e even 4 1 8993.2.a.a 1
120.q odd 4 1 9792.2.a.i 1
120.w even 4 1 9792.2.a.n 1
255.o even 4 1 2601.2.a.g 1
340.r even 4 1 4624.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 5.c odd 4 1
153.2.a.c 1 15.e even 4 1
272.2.a.b 1 20.e even 4 1
289.2.a.a 1 85.g odd 4 1
289.2.b.a 2 85.f odd 4 1
289.2.b.a 2 85.i odd 4 1
289.2.c.a 4 85.k odd 8 2
289.2.c.a 4 85.n odd 8 2
289.2.d.d 8 85.o even 16 4
289.2.d.d 8 85.r even 16 4
425.2.a.d 1 5.c odd 4 1
425.2.b.b 2 1.a even 1 1 trivial
425.2.b.b 2 5.b even 2 1 inner
833.2.a.a 1 35.f even 4 1
833.2.e.a 2 35.k even 12 2
833.2.e.b 2 35.l odd 12 2
1088.2.a.h 1 40.k even 4 1
1088.2.a.i 1 40.i odd 4 1
2057.2.a.e 1 55.e even 4 1
2448.2.a.o 1 60.l odd 4 1
2601.2.a.g 1 255.o even 4 1
2873.2.a.c 1 65.h odd 4 1
3825.2.a.d 1 15.e even 4 1
4624.2.a.d 1 340.r even 4 1
6137.2.a.b 1 95.g even 4 1
6800.2.a.n 1 20.e even 4 1
7225.2.a.g 1 85.g odd 4 1
7497.2.a.l 1 105.k odd 4 1
8993.2.a.a 1 115.e even 4 1
9792.2.a.i 1 120.q odd 4 1
9792.2.a.n 1 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(425, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{3}$$ T3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 1$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T + 12)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 4$$