# Properties

 Label 2925.2.c.r Level $2925$ Weight $2$ Character orbit 2925.c Analytic conductor $23.356$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$23.3562425912$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 1) q^{4} + (2 \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8}+O(q^{10})$$ q + (b2 + b1) * q^2 + (-2*b3 - 1) * q^4 + (2*b2 + 2*b1) * q^7 + (-b2 - 3*b1) * q^8 $$q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 1) q^{4} + (2 \beta_{2} + 2 \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} + ( - \beta_{3} - 2) q^{11} + \beta_1 q^{13} + ( - 4 \beta_{3} - 6) q^{14} + 3 q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{3} - 2) q^{19} + ( - 3 \beta_{2} - 4 \beta_1) q^{22} + \beta_{2} q^{23} + ( - \beta_{3} - 1) q^{26} + ( - 6 \beta_{2} - 10 \beta_1) q^{28} - 4 \beta_{3} q^{29} + ( - 3 \beta_{3} + 6) q^{31} + (\beta_{2} - 3 \beta_1) q^{32} + 2 q^{34} - 6 \beta_{2} q^{37} - \beta_{2} q^{38} + ( - 2 \beta_{3} + 6) q^{41} + (5 \beta_{2} + 4 \beta_1) q^{43} + (5 \beta_{3} + 6) q^{44} + ( - \beta_{3} - 2) q^{46} + (2 \beta_{2} + 2 \beta_1) q^{47} + ( - 8 \beta_{3} - 5) q^{49} + ( - 2 \beta_{2} - \beta_1) q^{52} + (6 \beta_{2} - 6 \beta_1) q^{53} + (8 \beta_{3} + 10) q^{56} + ( - 4 \beta_{2} - 8 \beta_1) q^{58} + ( - 3 \beta_{3} + 6) q^{59} - 8 q^{61} + 3 \beta_{2} q^{62} + (2 \beta_{3} + 7) q^{64} - 2 \beta_1 q^{67} + ( - 2 \beta_{2} + 6 \beta_1) q^{68} + ( - 7 \beta_{3} - 2) q^{71} - 6 \beta_{2} q^{73} + (6 \beta_{3} + 12) q^{74} + (3 \beta_{3} - 2) q^{76} + ( - 6 \beta_{2} - 8 \beta_1) q^{77} + 6 \beta_{3} q^{79} + (4 \beta_{2} + 2 \beta_1) q^{82} + (2 \beta_{2} - 6 \beta_1) q^{83} + ( - 9 \beta_{3} - 14) q^{86} + (5 \beta_{2} + 8 \beta_1) q^{88} + 6 q^{89} + ( - 2 \beta_{3} - 2) q^{91} + ( - \beta_{2} - 4 \beta_1) q^{92} + ( - 4 \beta_{3} - 6) q^{94} + ( - 4 \beta_{2} - 2 \beta_1) q^{97} + ( - 13 \beta_{2} - 21 \beta_1) q^{98}+O(q^{100})$$ q + (b2 + b1) * q^2 + (-2*b3 - 1) * q^4 + (2*b2 + 2*b1) * q^7 + (-b2 - 3*b1) * q^8 + (-b3 - 2) * q^11 + b1 * q^13 + (-4*b3 - 6) * q^14 + 3 * q^16 + (-2*b2 + 2*b1) * q^17 + (b3 - 2) * q^19 + (-3*b2 - 4*b1) * q^22 + b2 * q^23 + (-b3 - 1) * q^26 + (-6*b2 - 10*b1) * q^28 - 4*b3 * q^29 + (-3*b3 + 6) * q^31 + (b2 - 3*b1) * q^32 + 2 * q^34 - 6*b2 * q^37 - b2 * q^38 + (-2*b3 + 6) * q^41 + (5*b2 + 4*b1) * q^43 + (5*b3 + 6) * q^44 + (-b3 - 2) * q^46 + (2*b2 + 2*b1) * q^47 + (-8*b3 - 5) * q^49 + (-2*b2 - b1) * q^52 + (6*b2 - 6*b1) * q^53 + (8*b3 + 10) * q^56 + (-4*b2 - 8*b1) * q^58 + (-3*b3 + 6) * q^59 - 8 * q^61 + 3*b2 * q^62 + (2*b3 + 7) * q^64 - 2*b1 * q^67 + (-2*b2 + 6*b1) * q^68 + (-7*b3 - 2) * q^71 - 6*b2 * q^73 + (6*b3 + 12) * q^74 + (3*b3 - 2) * q^76 + (-6*b2 - 8*b1) * q^77 + 6*b3 * q^79 + (4*b2 + 2*b1) * q^82 + (2*b2 - 6*b1) * q^83 + (-9*b3 - 14) * q^86 + (5*b2 + 8*b1) * q^88 + 6 * q^89 + (-2*b3 - 2) * q^91 + (-b2 - 4*b1) * q^92 + (-4*b3 - 6) * q^94 + (-4*b2 - 2*b1) * q^97 + (-13*b2 - 21*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} - 8 q^{11} - 24 q^{14} + 12 q^{16} - 8 q^{19} - 4 q^{26} + 24 q^{31} + 8 q^{34} + 24 q^{41} + 24 q^{44} - 8 q^{46} - 20 q^{49} + 40 q^{56} + 24 q^{59} - 32 q^{61} + 28 q^{64} - 8 q^{71} + 48 q^{74} - 8 q^{76} - 56 q^{86} + 24 q^{89} - 8 q^{91} - 24 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 - 8 * q^11 - 24 * q^14 + 12 * q^16 - 8 * q^19 - 4 * q^26 + 24 * q^31 + 8 * q^34 + 24 * q^41 + 24 * q^44 - 8 * q^46 - 20 * q^49 + 40 * q^56 + 24 * q^59 - 32 * q^61 + 28 * q^64 - 8 * q^71 + 48 * q^74 - 8 * q^76 - 56 * q^86 + 24 * q^89 - 8 * q^91 - 24 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$2251$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2224.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 0 −3.82843 0 0 4.82843i 4.41421i 0 0
2224.2 0.414214i 0 1.82843 0 0 0.828427i 1.58579i 0 0
2224.3 0.414214i 0 1.82843 0 0 0.828427i 1.58579i 0 0
2224.4 2.41421i 0 −3.82843 0 0 4.82843i 4.41421i 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 2925.2.c.r 4
3.b odd 2 1 325.2.b.f 4
5.b even 2 1 inner 2925.2.c.r 4
5.c odd 4 1 585.2.a.m 2
5.c odd 4 1 2925.2.a.u 2
15.d odd 2 1 325.2.b.f 4
15.e even 4 1 65.2.a.b 2
15.e even 4 1 325.2.a.i 2
20.e even 4 1 9360.2.a.cd 2
60.l odd 4 1 1040.2.a.j 2
60.l odd 4 1 5200.2.a.bu 2
65.h odd 4 1 7605.2.a.x 2
105.k odd 4 1 3185.2.a.j 2
120.q odd 4 1 4160.2.a.z 2
120.w even 4 1 4160.2.a.bf 2
165.l odd 4 1 7865.2.a.j 2
195.j odd 4 1 845.2.c.b 4
195.s even 4 1 845.2.a.g 2
195.s even 4 1 4225.2.a.r 2
195.u odd 4 1 845.2.c.b 4
195.bc odd 12 2 845.2.m.f 8
195.bf even 12 2 845.2.e.c 4
195.bl even 12 2 845.2.e.h 4
195.bn odd 12 2 845.2.m.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 15.e even 4 1
325.2.a.i 2 15.e even 4 1
325.2.b.f 4 3.b odd 2 1
325.2.b.f 4 15.d odd 2 1
585.2.a.m 2 5.c odd 4 1
845.2.a.g 2 195.s even 4 1
845.2.c.b 4 195.j odd 4 1
845.2.c.b 4 195.u odd 4 1
845.2.e.c 4 195.bf even 12 2
845.2.e.h 4 195.bl even 12 2
845.2.m.f 8 195.bc odd 12 2
845.2.m.f 8 195.bn odd 12 2
1040.2.a.j 2 60.l odd 4 1
2925.2.a.u 2 5.c odd 4 1
2925.2.c.r 4 1.a even 1 1 trivial
2925.2.c.r 4 5.b even 2 1 inner
3185.2.a.j 2 105.k odd 4 1
4160.2.a.z 2 120.q odd 4 1
4160.2.a.bf 2 120.w even 4 1
4225.2.a.r 2 195.s even 4 1
5200.2.a.bu 2 60.l odd 4 1
7605.2.a.x 2 65.h odd 4 1
7865.2.a.j 2 165.l odd 4 1
9360.2.a.cd 2 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 $$S_{2}^{\mathrm{new}}(2925, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{7}^{4} + 24T_{7}^{2} + 16$$ T7^4 + 24*T7^2 + 16 $$T_{11}^{2} + 4T_{11} + 2$$ T11^2 + 4*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T^{2} + 4 T + 2)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4} + 24T^{2} + 16$$
$19$ $$(T^{2} + 4 T + 2)^{2}$$
$23$ $$(T^{2} + 2)^{2}$$
$29$ $$(T^{2} - 32)^{2}$$
$31$ $$(T^{2} - 12 T + 18)^{2}$$
$37$ $$(T^{2} + 72)^{2}$$
$41$ $$(T^{2} - 12 T + 28)^{2}$$
$43$ $$T^{4} + 132T^{2} + 1156$$
$47$ $$T^{4} + 24T^{2} + 16$$
$53$ $$T^{4} + 216T^{2} + 1296$$
$59$ $$(T^{2} - 12 T + 18)^{2}$$
$61$ $$(T + 8)^{4}$$
$67$ $$(T^{2} + 4)^{2}$$
$71$ $$(T^{2} + 4 T - 94)^{2}$$
$73$ $$(T^{2} + 72)^{2}$$
$79$ $$(T^{2} - 72)^{2}$$
$83$ $$T^{4} + 88T^{2} + 784$$
$89$ $$(T - 6)^{4}$$
$97$ $$T^{4} + 72T^{2} + 784$$