# Properties

 Label 325.2.n.b Level $325$ Weight $2$ Character orbit 325.n Analytic conductor $2.595$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2x + 4$$ x^4 - x^3 - x^2 - 2*x + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 1) q^{2} + \beta_{2} q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_1 - 1) q^{6} + (\beta_{2} + 1) q^{7} + (2 \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 2) q^{9}+O(q^{10})$$ q + (b3 - 1) * q^2 + b2 * q^3 + (-2*b3 + b2 + b1) * q^4 + (b1 - 1) * q^6 + (b2 + 1) * q^7 + (2*b2 - 1) * q^8 + (-2*b2 + 2) * q^9 $$q + (\beta_{3} - 1) q^{2} + \beta_{2} q^{3} + ( - 2 \beta_{3} + \beta_{2} + \beta_1) q^{4} + (\beta_1 - 1) q^{6} + (\beta_{2} + 1) q^{7} + (2 \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 2) q^{9} + ( - 2 \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{3} - \beta_1 + 1) q^{12} + (4 \beta_{2} - 1) q^{13} + (\beta_{3} + \beta_1 - 2) q^{14} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{16} + ( - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{17} + (2 \beta_{3} - 2 \beta_1) q^{18} + (\beta_{2} + 1) q^{19} + (2 \beta_{2} - 1) q^{21} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{22} + ( - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 2) q^{23} + (\beta_{2} - 2) q^{24} + ( - \beta_{3} + 4 \beta_1 - 3) q^{26} + 5 q^{27} + ( - 3 \beta_{3} + \beta_{2} + 1) q^{28} + (2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 2) q^{29} + (6 \beta_{3} - 4 \beta_{2} - 6 \beta_1 + 2) q^{31} + ( - 3 \beta_{2} + \beta_1 - 4) q^{32} + ( - \beta_{2} - 2 \beta_1 + 1) q^{33} + (3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 2) q^{34} + ( - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{36} + (6 \beta_{3} - 3 \beta_{2}) q^{37} + (\beta_{3} + \beta_1 - 2) q^{38} + (3 \beta_{2} - 4) q^{39} + (2 \beta_{3} - \beta_{2}) q^{41} + ( - \beta_{3} + 2 \beta_1 - 1) q^{42} + (4 \beta_{3} - 9 \beta_{2} - 2 \beta_1 + 7) q^{43} + ( - \beta_{3} - 6 \beta_{2} + \beta_1 + 3) q^{44} + (2 \beta_{2} - 3 \beta_1 + 5) q^{46} + (4 \beta_{3} - 4 \beta_1) q^{47} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{48} - 4 \beta_{2} q^{49} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{51} + ( - 2 \beta_{3} - \beta_{2} - 5 \beta_1 + 4) q^{52} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{53} + (5 \beta_{3} - 5) q^{54} + (3 \beta_{2} - 3) q^{56} + (2 \beta_{2} - 1) q^{57} + ( - 2 \beta_{2} + 3 \beta_1 - 5) q^{58} + (7 \beta_{2} + 4 \beta_1 + 3) q^{59} + ( - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 5) q^{61} + ( - 4 \beta_{3} - 6 \beta_{2} + 8 \beta_1 - 4) q^{62} + ( - 2 \beta_{2} + 4) q^{63} + ( - 2 \beta_{3} - 2 \beta_1 + 9) q^{64} + (\beta_{3} + \beta_1 - 4) q^{66} + ( - 6 \beta_{3} - \beta_{2} + 8) q^{67} + ( - \beta_{3} - 9 \beta_{2} + 2 \beta_1 - 1) q^{68} + ( - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{69} + (3 \beta_{2} + 4 \beta_1 - 1) q^{71} + (2 \beta_{2} + 2) q^{72} + ( - 6 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 9) q^{74} + ( - 3 \beta_{3} + \beta_{2} + 1) q^{76} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{77} + ( - 4 \beta_{3} + 3 \beta_1 + 1) q^{78} - 6 q^{79} - \beta_{2} q^{81} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{82} + ( - 2 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 4) q^{83} + ( - \beta_{2} - 3 \beta_1 + 2) q^{84} + (3 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 2) q^{86} + (4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{87} + (2 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 2) q^{88} + ( - 2 \beta_{3} + 5 \beta_{2} - 8) q^{89} + (7 \beta_{2} - 5) q^{91} + (\beta_{3} + \beta_1 - 11) q^{92} + (6 \beta_{3} - 2 \beta_{2} - 2) q^{93} + ( - 4 \beta_{3} - 4 \beta_{2} + 8 \beta_1 - 4) q^{94} + ( - \beta_{3} - 6 \beta_{2} + \beta_1 + 3) q^{96} + (5 \beta_{2} + 6 \beta_1 - 1) q^{97} + ( - 4 \beta_1 + 4) q^{98} + ( - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{99}+O(q^{100})$$ q + (b3 - 1) * q^2 + b2 * q^3 + (-2*b3 + b2 + b1) * q^4 + (b1 - 1) * q^6 + (b2 + 1) * q^7 + (2*b2 - 1) * q^8 + (-2*b2 + 2) * q^9 + (-2*b3 + b2) * q^11 + (-b3 - b1 + 1) * q^12 + (4*b2 - 1) * q^13 + (b3 + b1 - 2) * q^14 + (b3 - 2*b2 - 2*b1 + 1) * q^16 + (-4*b3 + 3*b2 + 2*b1 - 1) * q^17 + (2*b3 - 2*b1) * q^18 + (b2 + 1) * q^19 + (2*b2 - 1) * q^21 + (2*b3 + 2*b2 - b1 - 3) * q^22 + (-2*b3 + 3*b2 + 4*b1 - 2) * q^23 + (b2 - 2) * q^24 + (-b3 + 4*b1 - 3) * q^26 + 5 * q^27 + (-3*b3 + b2 + 1) * q^28 + (2*b3 - 3*b2 - 4*b1 + 2) * q^29 + (6*b3 - 4*b2 - 6*b1 + 2) * q^31 + (-3*b2 + b1 - 4) * q^32 + (-b2 - 2*b1 + 1) * q^33 + (3*b3 + 4*b2 - 3*b1 - 2) * q^34 + (-2*b3 + 2*b2 + 4*b1 - 2) * q^36 + (6*b3 - 3*b2) * q^37 + (b3 + b1 - 2) * q^38 + (3*b2 - 4) * q^39 + (2*b3 - b2) * q^41 + (-b3 + 2*b1 - 1) * q^42 + (4*b3 - 9*b2 - 2*b1 + 7) * q^43 + (-b3 - 6*b2 + b1 + 3) * q^44 + (2*b2 - 3*b1 + 5) * q^46 + (4*b3 - 4*b1) * q^47 + (2*b3 - 2*b2 - b1 + 1) * q^48 - 4*b2 * q^49 + (-2*b3 - 2*b1 + 1) * q^51 + (-2*b3 - b2 - 5*b1 + 4) * q^52 + (-2*b3 - 2*b1 - 2) * q^53 + (5*b3 - 5) * q^54 + (3*b2 - 3) * q^56 + (2*b2 - 1) * q^57 + (-2*b2 + 3*b1 - 5) * q^58 + (7*b2 + 4*b1 + 3) * q^59 + (-4*b3 - 3*b2 + 2*b1 + 5) * q^61 + (-4*b3 - 6*b2 + 8*b1 - 4) * q^62 + (-2*b2 + 4) * q^63 + (-2*b3 - 2*b1 + 9) * q^64 + (b3 + b1 - 4) * q^66 + (-6*b3 - b2 + 8) * q^67 + (-b3 - 9*b2 + 2*b1 - 1) * q^68 + (-4*b3 + 3*b2 + 2*b1 - 1) * q^69 + (3*b2 + 4*b1 - 1) * q^71 + (2*b2 + 2) * q^72 + (-6*b3 - 6*b2 + 3*b1 + 9) * q^74 + (-3*b3 + b2 + 1) * q^76 + (-2*b3 - 2*b1 + 1) * q^77 + (-4*b3 + 3*b1 + 1) * q^78 - 6 * q^79 - b2 * q^81 + (-2*b3 - 2*b2 + b1 + 3) * q^82 + (-2*b3 - 8*b2 + 2*b1 + 4) * q^83 + (-b2 - 3*b1 + 2) * q^84 + (3*b3 - 4*b2 - 3*b1 + 2) * q^86 + (4*b3 - 3*b2 - 2*b1 + 1) * q^87 + (2*b3 - 3*b2 - 4*b1 + 2) * q^88 + (-2*b3 + 5*b2 - 8) * q^89 + (7*b2 - 5) * q^91 + (b3 + b1 - 11) * q^92 + (6*b3 - 2*b2 - 2) * q^93 + (-4*b3 - 4*b2 + 8*b1 - 4) * q^94 + (-b3 - 6*b2 + b1 + 3) * q^96 + (5*b2 + 6*b1 - 1) * q^97 + (-4*b1 + 4) * q^98 + (-4*b3 + 4*b2 + 4*b1 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 3 q^{2} + 2 q^{3} + q^{4} - 3 q^{6} + 6 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 3 * q^2 + 2 * q^3 + q^4 - 3 * q^6 + 6 * q^7 + 4 * q^9 $$4 q - 3 q^{2} + 2 q^{3} + q^{4} - 3 q^{6} + 6 q^{7} + 4 q^{9} + 2 q^{12} + 4 q^{13} - 6 q^{14} - q^{16} + 6 q^{19} - 7 q^{22} - 6 q^{24} - 9 q^{26} + 20 q^{27} + 3 q^{28} - 21 q^{32} - 2 q^{36} - 6 q^{38} - 10 q^{39} - 3 q^{42} + 12 q^{43} + 21 q^{46} + q^{48} - 8 q^{49} + 7 q^{52} - 12 q^{53} - 15 q^{54} - 6 q^{56} - 21 q^{58} + 30 q^{59} + 12 q^{61} - 24 q^{62} + 12 q^{63} + 32 q^{64} - 14 q^{66} + 24 q^{67} - 21 q^{68} + 6 q^{71} + 12 q^{72} + 21 q^{74} + 3 q^{76} + 3 q^{78} - 24 q^{79} - 2 q^{81} + 7 q^{82} + 3 q^{84} - 24 q^{89} - 6 q^{91} - 42 q^{92} - 6 q^{93} - 20 q^{94} + 12 q^{97} + 12 q^{98}+O(q^{100})$$ 4 * q - 3 * q^2 + 2 * q^3 + q^4 - 3 * q^6 + 6 * q^7 + 4 * q^9 + 2 * q^12 + 4 * q^13 - 6 * q^14 - q^16 + 6 * q^19 - 7 * q^22 - 6 * q^24 - 9 * q^26 + 20 * q^27 + 3 * q^28 - 21 * q^32 - 2 * q^36 - 6 * q^38 - 10 * q^39 - 3 * q^42 + 12 * q^43 + 21 * q^46 + q^48 - 8 * q^49 + 7 * q^52 - 12 * q^53 - 15 * q^54 - 6 * q^56 - 21 * q^58 + 30 * q^59 + 12 * q^61 - 24 * q^62 + 12 * q^63 + 32 * q^64 - 14 * q^66 + 24 * q^67 - 21 * q^68 + 6 * q^71 + 12 * q^72 + 21 * q^74 + 3 * q^76 + 3 * q^78 - 24 * q^79 - 2 * q^81 + 7 * q^82 + 3 * q^84 - 24 * q^89 - 6 * q^91 - 42 * q^92 - 6 * q^93 - 20 * q^94 + 12 * q^97 + 12 * q^98

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + \nu^{2} - \nu - 2 ) / 2$$ (v^3 + v^2 - v - 2) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + \nu + 2 ) / 2$$ (-v^3 + v^2 + v + 2) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2}$$ b3 + b2 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta _1 + 2$$ -b3 + b2 + b1 + 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 −0.895644 − 1.09445i 1.39564 + 0.228425i −0.895644 + 1.09445i 1.39564 − 0.228425i
−1.89564 + 1.09445i 0.500000 + 0.866025i 1.39564 2.41733i 0 −1.89564 1.09445i 1.50000 + 0.866025i 1.73205i 1.00000 1.73205i 0
101.2 0.395644 0.228425i 0.500000 + 0.866025i −0.895644 + 1.55130i 0 0.395644 + 0.228425i 1.50000 + 0.866025i 1.73205i 1.00000 1.73205i 0
251.1 −1.89564 1.09445i 0.500000 0.866025i 1.39564 + 2.41733i 0 −1.89564 + 1.09445i 1.50000 0.866025i 1.73205i 1.00000 + 1.73205i 0
251.2 0.395644 + 0.228425i 0.500000 0.866025i −0.895644 1.55130i 0 0.395644 0.228425i 1.50000 0.866025i 1.73205i 1.00000 + 1.73205i 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.n.b 4
5.b even 2 1 325.2.n.c 4
5.c odd 4 2 65.2.l.a 8
13.e even 6 1 inner 325.2.n.b 4
13.f odd 12 2 4225.2.a.bj 4
15.e even 4 2 585.2.bf.a 8
20.e even 4 2 1040.2.df.b 8
65.f even 4 2 845.2.n.d 8
65.h odd 4 2 845.2.l.c 8
65.k even 4 2 845.2.n.c 8
65.l even 6 1 325.2.n.c 4
65.o even 12 2 845.2.b.f 8
65.o even 12 2 845.2.n.d 8
65.q odd 12 2 845.2.d.c 8
65.q odd 12 2 845.2.l.c 8
65.r odd 12 2 65.2.l.a 8
65.r odd 12 2 845.2.d.c 8
65.s odd 12 2 4225.2.a.bk 4
65.t even 12 2 845.2.b.f 8
65.t even 12 2 845.2.n.c 8
195.bf even 12 2 585.2.bf.a 8
260.bg even 12 2 1040.2.df.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a 8 5.c odd 4 2
65.2.l.a 8 65.r odd 12 2
325.2.n.b 4 1.a even 1 1 trivial
325.2.n.b 4 13.e even 6 1 inner
325.2.n.c 4 5.b even 2 1
325.2.n.c 4 65.l even 6 1
585.2.bf.a 8 15.e even 4 2
585.2.bf.a 8 195.bf even 12 2
845.2.b.f 8 65.o even 12 2
845.2.b.f 8 65.t even 12 2
845.2.d.c 8 65.q odd 12 2
845.2.d.c 8 65.r odd 12 2
845.2.l.c 8 65.h odd 4 2
845.2.l.c 8 65.q odd 12 2
845.2.n.c 8 65.k even 4 2
845.2.n.c 8 65.t even 12 2
845.2.n.d 8 65.f even 4 2
845.2.n.d 8 65.o even 12 2
1040.2.df.b 8 20.e even 4 2
1040.2.df.b 8 260.bg even 12 2
4225.2.a.bj 4 13.f odd 12 2
4225.2.a.bk 4 65.s odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{3} + 2T_{2}^{2} - 3T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3 T^{3} + 2 T^{2} - 3 T + 1$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 3 T + 3)^{2}$$
$11$ $$T^{4} - 7T^{2} + 49$$
$13$ $$(T^{2} - 2 T + 13)^{2}$$
$17$ $$T^{4} + 21T^{2} + 441$$
$19$ $$(T^{2} - 3 T + 3)^{2}$$
$23$ $$T^{4} + 21T^{2} + 441$$
$29$ $$T^{4} + 21T^{2} + 441$$
$31$ $$T^{4} + 132T^{2} + 3600$$
$37$ $$T^{4} - 63T^{2} + 3969$$
$41$ $$T^{4} - 7T^{2} + 49$$
$43$ $$T^{4} - 12 T^{3} + 129 T^{2} + \cdots + 225$$
$47$ $$T^{4} + 80T^{2} + 256$$
$53$ $$(T^{2} + 6 T - 12)^{2}$$
$59$ $$T^{4} - 30 T^{3} + 347 T^{2} + \cdots + 2209$$
$61$ $$T^{4} - 12 T^{3} + 129 T^{2} + \cdots + 225$$
$67$ $$T^{4} - 24 T^{3} + 177 T^{2} + \cdots + 225$$
$71$ $$T^{4} - 6 T^{3} - 13 T^{2} + 150 T + 625$$
$73$ $$T^{4}$$
$79$ $$(T + 6)^{4}$$
$83$ $$T^{4} + 164T^{2} + 4624$$
$89$ $$T^{4} + 24 T^{3} + 233 T^{2} + \cdots + 1681$$
$97$ $$T^{4} - 12 T^{3} - 3 T^{2} + \cdots + 2601$$