Properties

Label 825.2.n.c
Level $825$
Weight $2$
Character orbit 825.n
Analytic conductor $6.588$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{2} q^{3} + ( -3 + 3 \zeta_{10} ) q^{4} + ( 1 - 2 \zeta_{10} + \zeta_{10}^{2} ) q^{6} -\zeta_{10}^{3} q^{7} + ( -4 \zeta_{10} + 5 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{2} + \zeta_{10}^{2} q^{3} + ( -3 + 3 \zeta_{10} ) q^{4} + ( 1 - 2 \zeta_{10} + \zeta_{10}^{2} ) q^{6} -\zeta_{10}^{3} q^{7} + ( -4 \zeta_{10} + 5 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{3} ) q^{11} + ( -3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{12} + ( -3 + \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{13} + ( -\zeta_{10} + 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{14} + ( 3 - 8 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{16} + ( -3 - 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{17} + ( -1 + \zeta_{10} - \zeta_{10}^{3} ) q^{18} + ( -3 \zeta_{10} + 4 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + q^{21} + ( 3 - 6 \zeta_{10} + 7 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{22} + ( 3 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{23} + ( -1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{26} -\zeta_{10} q^{27} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{28} + 6 \zeta_{10}^{3} q^{29} + ( -2 - 3 \zeta_{10} + 3 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{31} + ( 6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{32} + ( 2 - 2 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{33} + 3 q^{34} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{36} + ( -2 + 2 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{37} + ( 7 - 11 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{38} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{2} ) q^{39} + ( -2 \zeta_{10} - \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{41} + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{42} + ( -3 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{43} + ( 12 - 9 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{44} + ( -1 + \zeta_{10}^{3} ) q^{46} + ( -5 \zeta_{10} + 7 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{47} + ( -3 + 3 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{48} + 6 \zeta_{10} q^{49} + ( 3 - 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{51} + ( -3 \zeta_{10} - 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{52} + ( -2 + \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{53} + ( 1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{54} + ( 1 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{56} + ( -1 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{57} + ( 6 \zeta_{10} - 12 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{58} + ( 1 - \zeta_{10} - 9 \zeta_{10}^{3} ) q^{59} + ( -9 + 6 \zeta_{10} - 9 \zeta_{10}^{2} ) q^{61} + ( 8 - 8 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{62} + \zeta_{10}^{2} q^{63} + ( 1 + 5 \zeta_{10} - 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{64} + ( -4 + 7 \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{66} + ( 3 + 3 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{67} + ( 9 - 9 \zeta_{10}^{3} ) q^{68} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{69} + ( 7 - 6 \zeta_{10} + 7 \zeta_{10}^{2} ) q^{71} + ( 4 - 5 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{72} + ( -6 + 6 \zeta_{10} + 4 \zeta_{10}^{3} ) q^{73} + ( -7 \zeta_{10} + 12 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{74} + ( 9 - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{76} + ( -1 - \zeta_{10} - \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{77} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{78} + ( -11 + 11 \zeta_{10} - 11 \zeta_{10}^{2} + 11 \zeta_{10}^{3} ) q^{79} -\zeta_{10}^{3} q^{81} + ( 1 + \zeta_{10}^{2} ) q^{82} + ( -4 - \zeta_{10} - 4 \zeta_{10}^{2} ) q^{83} + ( -3 + 3 \zeta_{10} ) q^{84} + ( -3 + 12 \zeta_{10} - 12 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{86} -6 q^{87} + ( -2 + 12 \zeta_{10} - 18 \zeta_{10}^{2} + 15 \zeta_{10}^{3} ) q^{88} + ( -5 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{89} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{91} + ( -3 + 3 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{92} + ( -5 + 3 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{93} + ( 12 - 19 \zeta_{10} + 12 \zeta_{10}^{2} ) q^{94} + ( -3 \zeta_{10} + 9 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{96} + ( 3 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{97} + ( -6 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{98} + ( 3 - 2 \zeta_{10} + 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{3} - 9 q^{4} + q^{6} - q^{7} - 13 q^{8} - q^{9} + O(q^{10}) \) \( 4 q + q^{2} - q^{3} - 9 q^{4} + q^{6} - q^{7} - 13 q^{8} - q^{9} - 11 q^{11} + 6 q^{12} - 7 q^{13} - 4 q^{14} + q^{16} - 12 q^{17} - 4 q^{18} - 10 q^{19} + 4 q^{21} - 4 q^{22} + 8 q^{23} + 7 q^{24} + 2 q^{26} - q^{27} + 6 q^{28} + 6 q^{29} - 12 q^{31} + 30 q^{32} + 9 q^{33} + 12 q^{34} - 9 q^{36} - 9 q^{37} + 10 q^{38} - 7 q^{39} - 3 q^{41} + q^{42} + 36 q^{44} - 3 q^{46} - 17 q^{47} - 14 q^{48} + 6 q^{49} + 3 q^{51} - 3 q^{52} - 4 q^{53} + 6 q^{54} + 12 q^{56} + 5 q^{57} + 24 q^{58} - 6 q^{59} - 21 q^{61} + 27 q^{62} - q^{63} + 13 q^{64} - 4 q^{66} + 6 q^{67} + 27 q^{68} + 3 q^{69} + 15 q^{71} + 7 q^{72} - 14 q^{73} - 26 q^{74} + 60 q^{76} - q^{77} + 2 q^{78} - 11 q^{79} - q^{81} + 3 q^{82} - 13 q^{83} - 9 q^{84} + 15 q^{86} - 24 q^{87} + 37 q^{88} - 24 q^{89} + 3 q^{91} - 3 q^{92} - 12 q^{93} + 17 q^{94} - 15 q^{96} - 3 q^{97} - 36 q^{98} + 4 q^{99} + O(q^{100}) \)

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
−0.309017 0.224514i 0.309017 0.951057i −0.572949 1.76336i 0 −0.309017 + 0.224514i 0.309017 + 0.951057i −0.454915 + 1.40008i −0.809017 0.587785i 0
526.1 0.809017 2.48990i −0.809017 + 0.587785i −3.92705 2.85317i 0 0.809017 + 2.48990i −0.809017 0.587785i −6.04508 + 4.39201i 0.309017 0.951057i 0
676.1 0.809017 + 2.48990i −0.809017 0.587785i −3.92705 + 2.85317i 0 0.809017 2.48990i −0.809017 + 0.587785i −6.04508 4.39201i 0.309017 + 0.951057i 0
751.1 −0.309017 + 0.224514i 0.309017 + 0.951057i −0.572949 + 1.76336i 0 −0.309017 0.224514i 0.309017 0.951057i −0.454915 1.40008i −0.809017 + 0.587785i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.c 4
5.b even 2 1 33.2.e.b 4
5.c odd 4 2 825.2.bx.d 8
11.c even 5 1 inner 825.2.n.c 4
11.c even 5 1 9075.2.a.cb 2
11.d odd 10 1 9075.2.a.u 2
15.d odd 2 1 99.2.f.a 4
20.d odd 2 1 528.2.y.b 4
45.h odd 6 2 891.2.n.b 8
45.j even 6 2 891.2.n.c 8
55.d odd 2 1 363.2.e.f 4
55.h odd 10 1 363.2.a.i 2
55.h odd 10 2 363.2.e.b 4
55.h odd 10 1 363.2.e.f 4
55.j even 10 1 33.2.e.b 4
55.j even 10 1 363.2.a.d 2
55.j even 10 2 363.2.e.k 4
55.k odd 20 2 825.2.bx.d 8
165.o odd 10 1 99.2.f.a 4
165.o odd 10 1 1089.2.a.t 2
165.r even 10 1 1089.2.a.l 2
220.n odd 10 1 528.2.y.b 4
220.n odd 10 1 5808.2.a.cj 2
220.o even 10 1 5808.2.a.ci 2
495.bl even 30 2 891.2.n.c 8
495.bp odd 30 2 891.2.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.b 4 5.b even 2 1
33.2.e.b 4 55.j even 10 1
99.2.f.a 4 15.d odd 2 1
99.2.f.a 4 165.o odd 10 1
363.2.a.d 2 55.j even 10 1
363.2.a.i 2 55.h odd 10 1
363.2.e.b 4 55.h odd 10 2
363.2.e.f 4 55.d odd 2 1
363.2.e.f 4 55.h odd 10 1
363.2.e.k 4 55.j even 10 2
528.2.y.b 4 20.d odd 2 1
528.2.y.b 4 220.n odd 10 1
825.2.n.c 4 1.a even 1 1 trivial
825.2.n.c 4 11.c even 5 1 inner
825.2.bx.d 8 5.c odd 4 2
825.2.bx.d 8 55.k odd 20 2
891.2.n.b 8 45.h odd 6 2
891.2.n.b 8 495.bp odd 30 2
891.2.n.c 8 45.j even 6 2
891.2.n.c 8 495.bl even 30 2
1089.2.a.l 2 165.r even 10 1
1089.2.a.t 2 165.o odd 10 1
5808.2.a.ci 2 220.o even 10 1
5808.2.a.cj 2 220.n odd 10 1
9075.2.a.u 2 11.d odd 10 1
9075.2.a.cb 2 11.c even 5 1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} + 6 T_{2}^{2} + 4 T_{2} + 1 \)
\( T_{13}^{4} + 7 T_{13}^{3} + 19 T_{13}^{2} + 3 T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 6 T^{2} - T^{3} + T^{4} \)
$3$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$11$ \( 121 + 121 T + 51 T^{2} + 11 T^{3} + T^{4} \)
$13$ \( 1 + 3 T + 19 T^{2} + 7 T^{3} + T^{4} \)
$17$ \( 81 - 27 T + 54 T^{2} + 12 T^{3} + T^{4} \)
$19$ \( 25 + 25 T + 40 T^{2} + 10 T^{3} + T^{4} \)
$23$ \( ( -1 - 4 T + T^{2} )^{2} \)
$29$ \( 1296 - 216 T + 36 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( 961 + 403 T + 94 T^{2} + 12 T^{3} + T^{4} \)
$37$ \( 121 - 11 T + 31 T^{2} + 9 T^{3} + T^{4} \)
$41$ \( 1 + 7 T + 19 T^{2} + 3 T^{3} + T^{4} \)
$43$ \( ( -45 + T^{2} )^{2} \)
$47$ \( 121 + 88 T + 114 T^{2} + 17 T^{3} + T^{4} \)
$53$ \( 1 - T + 6 T^{2} + 4 T^{3} + T^{4} \)
$59$ \( 5041 + 781 T + 76 T^{2} + 6 T^{3} + T^{4} \)
$61$ \( 9801 + 2376 T + 306 T^{2} + 21 T^{3} + T^{4} \)
$67$ \( ( -9 - 3 T + T^{2} )^{2} \)
$71$ \( 3025 - 1100 T + 190 T^{2} - 15 T^{3} + T^{4} \)
$73$ \( 1936 + 704 T + 136 T^{2} + 14 T^{3} + T^{4} \)
$79$ \( 14641 + 1331 T + 121 T^{2} + 11 T^{3} + T^{4} \)
$83$ \( 121 + 77 T + 69 T^{2} + 13 T^{3} + T^{4} \)
$89$ \( ( 31 + 12 T + T^{2} )^{2} \)
$97$ \( 81 - 108 T + 54 T^{2} + 3 T^{3} + T^{4} \)
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