# Properties

 Label 2575.1.c.b Level $2575$ Weight $1$ Character orbit 2575.c Analytic conductor $1.285$ Analytic rank $0$ Dimension $4$ Projective image $D_{5}$ CM discriminant -103 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2575 = 5^{2} \cdot 103$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2575.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.28509240753$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 103) Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.10609.1 Artin image: $C_4\times D_5$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{20} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -1 - \beta_{2} ) q^{4} -\beta_{1} q^{7} -\beta_{3} q^{8} - q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{3} ) q^{2} + ( -1 - \beta_{2} ) q^{4} -\beta_{1} q^{7} -\beta_{3} q^{8} - q^{9} + ( -\beta_{1} - \beta_{3} ) q^{13} + q^{14} -\beta_{1} q^{17} + ( -\beta_{1} - \beta_{3} ) q^{18} + ( 1 + \beta_{2} ) q^{19} + ( -\beta_{1} - \beta_{3} ) q^{23} + ( 2 + \beta_{2} ) q^{26} + \beta_{3} q^{28} -\beta_{2} q^{29} -\beta_{3} q^{32} + q^{34} + ( 1 + \beta_{2} ) q^{36} + ( \beta_{1} + 2 \beta_{3} ) q^{38} + ( -1 - \beta_{2} ) q^{41} + ( 2 + \beta_{2} ) q^{46} + \beta_{2} q^{49} + ( \beta_{1} + 2 \beta_{3} ) q^{52} -\beta_{2} q^{56} -\beta_{3} q^{58} -\beta_{2} q^{59} + \beta_{2} q^{61} + \beta_{1} q^{63} + ( 1 + \beta_{2} ) q^{64} + \beta_{3} q^{68} + \beta_{3} q^{72} + ( -2 - \beta_{2} ) q^{76} + ( 1 + \beta_{2} ) q^{79} + q^{81} + ( -\beta_{1} - 2 \beta_{3} ) q^{82} + \beta_{1} q^{83} - q^{91} + ( \beta_{1} + 2 \beta_{3} ) q^{92} -\beta_{1} q^{97} + \beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} - 4 q^{9} + O(q^{10})$$ $$4 q - 2 q^{4} - 4 q^{9} + 4 q^{14} + 2 q^{19} + 6 q^{26} + 2 q^{29} + 4 q^{34} + 2 q^{36} - 2 q^{41} + 6 q^{46} - 2 q^{49} + 2 q^{56} + 2 q^{59} - 2 q^{61} + 2 q^{64} - 6 q^{76} + 2 q^{79} + 4 q^{81} - 4 q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2575\mathbb{Z}\right)^\times$$.

 $$n$$ $$726$$ $$1752$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2574.1
 − 0.618034i − 1.61803i 1.61803i 0.618034i
1.61803i 0 −1.61803 0 0 0.618034i 1.00000i −1.00000 0
2574.2 0.618034i 0 0.618034 0 0 1.61803i 1.00000i −1.00000 0
2574.3 0.618034i 0 0.618034 0 0 1.61803i 1.00000i −1.00000 0
2574.4 1.61803i 0 −1.61803 0 0 0.618034i 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
103.b odd 2 1 CM by $$\Q(\sqrt{-103})$$
5.b even 2 1 inner
515.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2575.1.c.b 4
5.b even 2 1 inner 2575.1.c.b 4
5.c odd 4 1 103.1.b.a 2
5.c odd 4 1 2575.1.d.d 2
15.e even 4 1 927.1.d.b 2
20.e even 4 1 1648.1.c.a 2
103.b odd 2 1 CM 2575.1.c.b 4
515.c odd 2 1 inner 2575.1.c.b 4
515.f even 4 1 103.1.b.a 2
515.f even 4 1 2575.1.d.d 2
1545.m odd 4 1 927.1.d.b 2
2060.m odd 4 1 1648.1.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.1.b.a 2 5.c odd 4 1
103.1.b.a 2 515.f even 4 1
927.1.d.b 2 15.e even 4 1
927.1.d.b 2 1545.m odd 4 1
1648.1.c.a 2 20.e even 4 1
1648.1.c.a 2 2060.m odd 4 1
2575.1.c.b 4 1.a even 1 1 trivial
2575.1.c.b 4 5.b even 2 1 inner
2575.1.c.b 4 103.b odd 2 1 CM
2575.1.c.b 4 515.c odd 2 1 inner
2575.1.d.d 2 5.c odd 4 1
2575.1.d.d 2 515.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2575, [\chi])$$:

 $$T_{2}^{4} + 3 T_{2}^{2} + 1$$ $$T_{7}^{4} + 3 T_{7}^{2} + 1$$

## Hecke characteristic polynomials

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