# Properties

 Label 825.2.n.b Level $825$ Weight $2$ Character orbit 825.n Analytic conductor $6.588$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.n (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} - \zeta_{10}^{2} q^{3} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{6} + (2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10})$$ q + (z^2 - z) * q^2 - z^2 * q^3 + (z^3 + z - 1) * q^4 + (z^2 - z + 1) * q^6 + (2*z^3 - 2*z + 2) * q^7 + (-2*z^3 + z^2 - 2*z) * q^8 + (z^3 - z^2 + z - 1) * q^9 $$q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} - \zeta_{10}^{2} q^{3} + (\zeta_{10}^{3} + \zeta_{10} - 1) q^{4} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{6} + (2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 2 \zeta_{10}) q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{11} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{12} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{13} + ( - 4 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 4 \zeta_{10}) q^{14} + ( - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{16} - 2 \zeta_{10} q^{17} + ( - \zeta_{10} + 1) q^{18} + 5 \zeta_{10}^{2} q^{19} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2) q^{21} + ( - \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{22} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 3) q^{23} + (\zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 1) q^{24} + (2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{26} + \zeta_{10} q^{27} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{28} + (4 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{29} + (7 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 7 \zeta_{10} - 7) q^{31} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 5) q^{32} + ( - 2 \zeta_{10}^{3} + \zeta_{10}^{2} - 4 \zeta_{10} + 2) q^{33} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2}) q^{34} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{36} + (5 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{37} + ( - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 5) q^{38} + ( - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{39} + ( - 4 \zeta_{10}^{3} + 9 \zeta_{10}^{2} - 4 \zeta_{10}) q^{41} + ( - 2 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 2) q^{42} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 1) q^{43} + (5 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3 \zeta_{10} - 3) q^{44} + ( - \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 1) q^{46} + (5 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 5 \zeta_{10}) q^{47} + (3 \zeta_{10} - 3) q^{48} + (12 \zeta_{10}^{2} - 13 \zeta_{10} + 12) q^{49} + 2 \zeta_{10}^{3} q^{51} + 2 \zeta_{10}^{2} q^{52} + ( - 7 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 6 \zeta_{10} + 7) q^{53} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{54} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 2) q^{56} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{57} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10}) q^{58} + (3 \zeta_{10}^{3} + 6 \zeta_{10} - 6) q^{59} - 7 \zeta_{10} q^{61} + ( - 7 \zeta_{10} + 7) q^{62} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{63} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{64} + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + \zeta_{10} - 1) q^{66} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 10) q^{67} + ( - 2 \zeta_{10}^{3} + 2) q^{68} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + \zeta_{10}) q^{69} + 8 \zeta_{10} q^{71} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{72} + (11 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{73} + ( - \zeta_{10}^{3} - 3 \zeta_{10}^{2} - \zeta_{10}) q^{74} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 5) q^{76} + ( - 6 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 6 \zeta_{10} - 2) q^{77} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2) q^{78} + ( - 5 \zeta_{10}^{3} + 5) q^{79} - \zeta_{10}^{3} q^{81} + ( - 9 \zeta_{10}^{2} + 13 \zeta_{10} - 9) q^{82} + (6 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{83} + (2 \zeta_{10} - 2) q^{84} + ( - 5 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11 \zeta_{10} + 5) q^{86} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 4) q^{87} + ( - 4 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{88} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 9) q^{89} + (8 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 8 \zeta_{10}) q^{91} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{92} + 7 \zeta_{10} q^{93} + (3 \zeta_{10}^{2} - 8 \zeta_{10} + 3) q^{94} + ( - \zeta_{10}^{3} - 4 \zeta_{10}^{2} - \zeta_{10}) q^{96} + (9 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 5 \zeta_{10} - 9) q^{97} + ( - 13 \zeta_{10}^{3} + 13 \zeta_{10}^{2} - 12) q^{98} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} - 1) q^{99} +O(q^{100})$$ q + (z^2 - z) * q^2 - z^2 * q^3 + (z^3 + z - 1) * q^4 + (z^2 - z + 1) * q^6 + (2*z^3 - 2*z + 2) * q^7 + (-2*z^3 + z^2 - 2*z) * q^8 + (z^3 - z^2 + z - 1) * q^9 + (-2*z^3 + 4*z^2 - 2*z + 3) * q^11 + (-z^3 + z^2 + 1) * q^12 + (-2*z^2 + 2*z) * q^13 + (-4*z^3 + 6*z^2 - 4*z) * q^14 + (-3*z^2 + 3*z - 3) * q^16 - 2*z * q^17 + (-z + 1) * q^18 + 5*z^2 * q^19 + (2*z^3 - 2*z^2 + 2) * q^21 + (-z^2 + 3*z - 4) * q^22 + (z^3 - z^2 + 3) * q^23 + (z^3 + z^2 - z - 1) * q^24 + (2*z^3 - 2*z + 2) * q^26 + z * q^27 + (-2*z^2 + 2*z - 2) * q^28 + (4*z^3 + 3*z - 3) * q^29 + (7*z^3 - 7*z^2 + 7*z - 7) * q^31 + (-z^3 + z^2 + 5) * q^32 + (-2*z^3 + z^2 - 4*z + 2) * q^33 + (-2*z^3 + 2*z^2) * q^34 + (-z^3 - z) * q^36 + (5*z^3 + 4*z - 4) * q^37 + (-5*z^2 + 5*z - 5) * q^38 + (-2*z^2 + 2*z - 2) * q^39 + (-4*z^3 + 9*z^2 - 4*z) * q^41 + (-2*z^3 + 6*z^2 - 6*z + 2) * q^42 + (5*z^3 - 5*z^2 + 1) * q^43 + (5*z^3 - 2*z^2 + 3*z - 3) * q^44 + (-z^3 + 5*z^2 - 5*z + 1) * q^46 + (5*z^3 - 3*z^2 + 5*z) * q^47 + (3*z - 3) * q^48 + (12*z^2 - 13*z + 12) * q^49 + 2*z^3 * q^51 + 2*z^2 * q^52 + (-7*z^3 + 6*z^2 - 6*z + 7) * q^53 + (z^3 - z^2) * q^54 + (-6*z^3 + 6*z^2 - 2) * q^56 + (-5*z^3 + 5*z^2 - 5*z + 5) * q^57 + (-z^3 - 2*z^2 - z) * q^58 + (3*z^3 + 6*z - 6) * q^59 - 7*z * q^61 + (-7*z + 7) * q^62 + (2*z^3 - 4*z^2 + 2*z) * q^63 + (z^3 - 3*z^2 + 3*z - 1) * q^64 + (-2*z^3 + 3*z^2 + z - 1) * q^66 + (-z^3 + z^2 - 10) * q^67 + (-2*z^3 + 2) * q^68 + (z^3 - 4*z^2 + z) * q^69 + 8*z * q^71 + (2*z^2 - z + 2) * q^72 + (11*z^3 + 4*z - 4) * q^73 + (-z^3 - 3*z^2 - z) * q^74 + (5*z^3 - 5*z^2 - 5) * q^76 + (-6*z^3 + 12*z^2 - 6*z - 2) * q^77 + (2*z^3 - 2*z^2 + 2) * q^78 + (-5*z^3 + 5) * q^79 - z^3 * q^81 + (-9*z^2 + 13*z - 9) * q^82 + (6*z^2 - 4*z + 6) * q^83 + (2*z - 2) * q^84 + (-5*z^3 + 11*z^2 - 11*z + 5) * q^86 + (-3*z^3 + 3*z^2 + 4) * q^87 + (-4*z^3 - 5*z^2 + 2*z - 2) * q^88 + (-3*z^3 + 3*z^2 + 9) * q^89 + (8*z^3 - 12*z^2 + 8*z) * q^91 + (2*z^3 + 3*z - 3) * q^92 + 7*z * q^93 + (3*z^2 - 8*z + 3) * q^94 + (-z^3 - 4*z^2 - z) * q^96 + (9*z^3 - 5*z^2 + 5*z - 9) * q^97 + (-13*z^3 + 13*z^2 - 12) * q^98 + (3*z^3 - z^2 - z - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} + q^{3} - 2 q^{4} + 2 q^{6} + 8 q^{7} - 5 q^{8} - q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 + q^3 - 2 * q^4 + 2 * q^6 + 8 * q^7 - 5 * q^8 - q^9 $$4 q - 2 q^{2} + q^{3} - 2 q^{4} + 2 q^{6} + 8 q^{7} - 5 q^{8} - q^{9} + 4 q^{11} + 2 q^{12} + 4 q^{13} - 14 q^{14} - 6 q^{16} - 2 q^{17} + 3 q^{18} - 5 q^{19} + 12 q^{21} - 12 q^{22} + 14 q^{23} - 5 q^{24} + 8 q^{26} + q^{27} - 4 q^{28} - 5 q^{29} - 7 q^{31} + 18 q^{32} + q^{33} - 4 q^{34} - 2 q^{36} - 7 q^{37} - 10 q^{38} - 4 q^{39} - 17 q^{41} - 6 q^{42} + 14 q^{43} - 2 q^{44} - 7 q^{46} + 13 q^{47} - 9 q^{48} + 23 q^{49} + 2 q^{51} - 2 q^{52} + 9 q^{53} + 2 q^{54} - 20 q^{56} + 5 q^{57} - 15 q^{59} - 7 q^{61} + 21 q^{62} + 8 q^{63} + 3 q^{64} - 8 q^{66} - 42 q^{67} + 6 q^{68} + 6 q^{69} + 8 q^{71} + 5 q^{72} - q^{73} + q^{74} - 10 q^{76} - 32 q^{77} + 12 q^{78} + 15 q^{79} - q^{81} - 14 q^{82} + 14 q^{83} - 6 q^{84} - 7 q^{86} + 10 q^{87} - 5 q^{88} + 30 q^{89} + 28 q^{91} - 7 q^{92} + 7 q^{93} + q^{94} + 2 q^{96} - 17 q^{97} - 74 q^{98} - q^{99}+O(q^{100})$$ 4 * q - 2 * q^2 + q^3 - 2 * q^4 + 2 * q^6 + 8 * q^7 - 5 * q^8 - q^9 + 4 * q^11 + 2 * q^12 + 4 * q^13 - 14 * q^14 - 6 * q^16 - 2 * q^17 + 3 * q^18 - 5 * q^19 + 12 * q^21 - 12 * q^22 + 14 * q^23 - 5 * q^24 + 8 * q^26 + q^27 - 4 * q^28 - 5 * q^29 - 7 * q^31 + 18 * q^32 + q^33 - 4 * q^34 - 2 * q^36 - 7 * q^37 - 10 * q^38 - 4 * q^39 - 17 * q^41 - 6 * q^42 + 14 * q^43 - 2 * q^44 - 7 * q^46 + 13 * q^47 - 9 * q^48 + 23 * q^49 + 2 * q^51 - 2 * q^52 + 9 * q^53 + 2 * q^54 - 20 * q^56 + 5 * q^57 - 15 * q^59 - 7 * q^61 + 21 * q^62 + 8 * q^63 + 3 * q^64 - 8 * q^66 - 42 * q^67 + 6 * q^68 + 6 * q^69 + 8 * q^71 + 5 * q^72 - q^73 + q^74 - 10 * q^76 - 32 * q^77 + 12 * q^78 + 15 * q^79 - q^81 - 14 * q^82 + 14 * q^83 - 6 * q^84 - 7 * q^86 + 10 * q^87 - 5 * q^88 + 30 * q^89 + 28 * q^91 - 7 * q^92 + 7 * q^93 + q^94 + 2 * q^96 - 17 * q^97 - 74 * q^98 - q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
301.1
 0.809017 − 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 + 0.587785i
−0.500000 0.363271i −0.309017 + 0.951057i −0.500000 1.53884i 0 0.500000 0.363271i −0.236068 0.726543i −0.690983 + 2.12663i −0.809017 0.587785i 0
526.1 −0.500000 + 1.53884i 0.809017 0.587785i −0.500000 0.363271i 0 0.500000 + 1.53884i 4.23607 + 3.07768i −1.80902 + 1.31433i 0.309017 0.951057i 0
676.1 −0.500000 1.53884i 0.809017 + 0.587785i −0.500000 + 0.363271i 0 0.500000 1.53884i 4.23607 3.07768i −1.80902 1.31433i 0.309017 + 0.951057i 0
751.1 −0.500000 + 0.363271i −0.309017 0.951057i −0.500000 + 1.53884i 0 0.500000 + 0.363271i −0.236068 + 0.726543i −0.690983 2.12663i −0.809017 + 0.587785i 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.n.b 4
5.b even 2 1 825.2.n.d yes 4
5.c odd 4 2 825.2.bx.c 8
11.c even 5 1 inner 825.2.n.b 4
11.c even 5 1 9075.2.a.z 2
11.d odd 10 1 9075.2.a.bt 2
55.h odd 10 1 9075.2.a.bc 2
55.j even 10 1 825.2.n.d yes 4
55.j even 10 1 9075.2.a.by 2
55.k odd 20 2 825.2.bx.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.b 4 1.a even 1 1 trivial
825.2.n.b 4 11.c even 5 1 inner
825.2.n.d yes 4 5.b even 2 1
825.2.n.d yes 4 55.j even 10 1
825.2.bx.c 8 5.c odd 4 2
825.2.bx.c 8 55.k odd 20 2
9075.2.a.z 2 11.c even 5 1
9075.2.a.bc 2 55.h odd 10 1
9075.2.a.bt 2 11.d odd 10 1
9075.2.a.by 2 55.j even 10 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(825, [\chi])$$:

 $$T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 1$$ T2^4 + 2*T2^3 + 4*T2^2 + 3*T2 + 1 $$T_{13}^{4} - 4T_{13}^{3} + 16T_{13}^{2} - 24T_{13} + 16$$ T13^4 - 4*T13^3 + 16*T13^2 - 24*T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 4 T^{2} + 3 T + 1$$
$3$ $$T^{4} - T^{3} + T^{2} - T + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16$$
$11$ $$T^{4} - 4 T^{3} + 6 T^{2} - 44 T + 121$$
$13$ $$T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16$$
$17$ $$T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16$$
$19$ $$T^{4} + 5 T^{3} + 25 T^{2} + 125 T + 625$$
$23$ $$(T^{2} - 7 T + 11)^{2}$$
$29$ $$T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25$$
$31$ $$T^{4} + 7 T^{3} + 49 T^{2} + \cdots + 2401$$
$37$ $$T^{4} + 7 T^{3} + 69 T^{2} + 143 T + 121$$
$41$ $$T^{4} + 17 T^{3} + 109 T^{2} + \cdots + 841$$
$43$ $$(T^{2} - 7 T - 19)^{2}$$
$47$ $$T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961$$
$53$ $$T^{4} - 9 T^{3} + 46 T^{2} + \cdots + 1681$$
$59$ $$T^{4} + 15 T^{3} + 135 T^{2} + \cdots + 2025$$
$61$ $$T^{4} + 7 T^{3} + 49 T^{2} + \cdots + 2401$$
$67$ $$(T^{2} + 21 T + 109)^{2}$$
$71$ $$T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 4096$$
$73$ $$T^{4} + T^{3} + 141 T^{2} - 1159 T + 3721$$
$79$ $$T^{4} - 15 T^{3} + 100 T^{2} + \cdots + 625$$
$83$ $$T^{4} - 14 T^{3} + 136 T^{2} + \cdots + 1936$$
$89$ $$(T^{2} - 15 T + 45)^{2}$$
$97$ $$T^{4} + 17 T^{3} + 109 T^{2} + \cdots + 841$$
show more
show less