Properties

Label 825.4.c.m
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.245110336.2
Defining polynomial: \(x^{6} + 19 x^{4} + 101 x^{2} + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + 3 \beta_{1} q^{3} + ( -6 - 2 \beta_{3} + \beta_{5} ) q^{4} -3 \beta_{5} q^{6} + ( \beta_{1} - 3 \beta_{2} - \beta_{4} ) q^{7} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} ) q^{8} -9 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + 3 \beta_{1} q^{3} + ( -6 - 2 \beta_{3} + \beta_{5} ) q^{4} -3 \beta_{5} q^{6} + ( \beta_{1} - 3 \beta_{2} - \beta_{4} ) q^{7} + ( 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} ) q^{8} -9 q^{9} + 11 q^{11} + ( -18 \beta_{1} - 3 \beta_{2} + 6 \beta_{4} ) q^{12} + ( 11 \beta_{1} + 13 \beta_{2} - 11 \beta_{4} ) q^{13} + ( -48 - 8 \beta_{3} ) q^{14} + ( 6 - 6 \beta_{3} + 7 \beta_{5} ) q^{16} + ( 9 \beta_{1} - 5 \beta_{2} - 7 \beta_{4} ) q^{17} + 9 \beta_{2} q^{18} + ( -36 + 2 \beta_{3} - 8 \beta_{5} ) q^{19} + ( -3 - 3 \beta_{3} - 9 \beta_{5} ) q^{21} -11 \beta_{2} q^{22} + ( -80 \beta_{1} - 16 \beta_{4} ) q^{23} + ( -6 + 6 \beta_{3} + 9 \beta_{5} ) q^{24} + ( 116 + 4 \beta_{3} - 46 \beta_{5} ) q^{26} -27 \beta_{1} q^{27} + ( -40 \beta_{1} + 40 \beta_{2} + 8 \beta_{4} ) q^{28} + ( -88 + 26 \beta_{5} ) q^{29} + ( 42 - 22 \beta_{3} - 34 \beta_{5} ) q^{31} + ( 78 \beta_{1} + 37 \beta_{2} + 14 \beta_{4} ) q^{32} + 33 \beta_{1} q^{33} + ( -112 - 24 \beta_{3} - 18 \beta_{5} ) q^{34} + ( 54 + 18 \beta_{3} - 9 \beta_{5} ) q^{36} + ( -8 \beta_{1} + 66 \beta_{2} - 42 \beta_{4} ) q^{37} + ( -100 \beta_{1} + 24 \beta_{2} + 12 \beta_{4} ) q^{38} + ( -33 - 33 \beta_{3} + 39 \beta_{5} ) q^{39} + ( -8 - 22 \beta_{5} ) q^{41} + ( -144 \beta_{1} + 24 \beta_{4} ) q^{42} + ( 51 \beta_{1} + 19 \beta_{2} - 27 \beta_{4} ) q^{43} + ( -66 - 22 \beta_{3} + 11 \beta_{5} ) q^{44} + ( -96 - 32 \beta_{3} + 48 \beta_{5} ) q^{46} + ( -54 \beta_{1} + 58 \beta_{2} + 34 \beta_{4} ) q^{47} + ( 18 \beta_{1} - 21 \beta_{2} + 18 \beta_{4} ) q^{48} + ( 157 - 30 \beta_{3} - 14 \beta_{5} ) q^{49} + ( -27 - 21 \beta_{3} - 15 \beta_{5} ) q^{51} + ( -532 \beta_{1} - 66 \beta_{2} - 4 \beta_{4} ) q^{52} + ( 270 \beta_{1} + 12 \beta_{2} - 56 \beta_{4} ) q^{53} + 27 \beta_{5} q^{54} + ( 224 + 32 \beta_{3} + 16 \beta_{5} ) q^{56} + ( -108 \beta_{1} + 24 \beta_{2} - 6 \beta_{4} ) q^{57} + ( 364 \beta_{1} + 114 \beta_{2} - 52 \beta_{4} ) q^{58} + ( -432 + 8 \beta_{3} + 60 \beta_{5} ) q^{59} + ( 140 + 22 \beta_{3} - 78 \beta_{5} ) q^{61} + ( -608 \beta_{1} - 32 \beta_{2} + 112 \beta_{4} ) q^{62} + ( -9 \beta_{1} + 27 \beta_{2} + 9 \beta_{4} ) q^{63} + ( 650 + 54 \beta_{3} - 31 \beta_{5} ) q^{64} -33 \beta_{5} q^{66} + ( 284 \beta_{1} + 88 \beta_{2} - 104 \beta_{4} ) q^{67} + ( -324 \beta_{1} + 102 \beta_{2} + 28 \beta_{4} ) q^{68} + ( 240 - 48 \beta_{3} ) q^{69} + ( 600 + 28 \beta_{3} + 16 \beta_{5} ) q^{71} + ( -18 \beta_{1} - 27 \beta_{2} - 18 \beta_{4} ) q^{72} + ( -19 \beta_{1} + 43 \beta_{2} + 23 \beta_{4} ) q^{73} + ( 672 + 48 \beta_{3} - 142 \beta_{5} ) q^{74} + ( 120 + 88 \beta_{3} + 36 \beta_{5} ) q^{76} + ( 11 \beta_{1} - 33 \beta_{2} - 11 \beta_{4} ) q^{77} + ( 348 \beta_{1} + 138 \beta_{2} - 12 \beta_{4} ) q^{78} + ( 40 + 154 \beta_{3} - 24 \beta_{5} ) q^{79} + 81 q^{81} + ( -308 \beta_{1} - 14 \beta_{2} + 44 \beta_{4} ) q^{82} + ( -289 \beta_{1} - 9 \beta_{2} + 195 \beta_{4} ) q^{83} + ( 120 + 24 \beta_{3} + 120 \beta_{5} ) q^{84} + ( 104 - 16 \beta_{3} - 124 \beta_{5} ) q^{86} + ( -264 \beta_{1} - 78 \beta_{2} ) q^{87} + ( 22 \beta_{1} + 33 \beta_{2} + 22 \beta_{4} ) q^{88} + ( -182 - 292 \beta_{5} ) q^{89} + ( 162 + 38 \beta_{3} - 154 \beta_{5} ) q^{91} + ( -160 \beta_{1} + 208 \beta_{2} - 160 \beta_{4} ) q^{92} + ( 126 \beta_{1} + 102 \beta_{2} + 66 \beta_{4} ) q^{93} + ( 1016 + 184 \beta_{3} + 64 \beta_{5} ) q^{94} + ( -234 + 42 \beta_{3} + 111 \beta_{5} ) q^{96} + ( -374 \beta_{1} + 200 \beta_{2} - 148 \beta_{4} ) q^{97} + ( -376 \beta_{1} - 111 \beta_{2} + 88 \beta_{4} ) q^{98} -99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 34q^{4} - 6q^{6} - 54q^{9} + O(q^{10}) \) \( 6q - 34q^{4} - 6q^{6} - 54q^{9} + 66q^{11} - 288q^{14} + 50q^{16} - 232q^{19} - 36q^{21} - 18q^{24} + 604q^{26} - 476q^{29} + 184q^{31} - 708q^{34} + 306q^{36} - 120q^{39} - 92q^{41} - 374q^{44} - 480q^{46} + 914q^{49} - 192q^{51} + 54q^{54} + 1376q^{56} - 2472q^{59} + 684q^{61} + 3838q^{64} - 66q^{66} + 1440q^{69} + 3632q^{71} + 3748q^{74} + 792q^{76} + 192q^{79} + 486q^{81} + 960q^{84} + 376q^{86} - 1676q^{89} + 664q^{91} + 6224q^{94} - 1182q^{96} - 594q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 19 x^{4} + 101 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 9 \nu^{3} + \nu \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{5} + 37 \nu^{3} + 83 \nu \)\()/10\)
\(\beta_{3}\)\(=\)\( -\nu^{4} - 9 \nu^{2} - 4 \)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{5} + 23 \nu^{3} + 52 \nu \)\()/5\)
\(\beta_{5}\)\(=\)\( \nu^{4} + 11 \nu^{2} + 17 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{3} - 13\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{4} + 5 \beta_{2} + \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(-9 \beta_{5} - 11 \beta_{3} + 109\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(71 \beta_{4} - 89 \beta_{2} + 3 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.12946i
2.91150i
3.04096i
3.04096i
2.91150i
1.12946i
4.59486i 3.00000i −13.1127 0 −13.7846 20.6383i 23.4921i −9.00000 0
199.2 4.38835i 3.00000i −11.2577 0 13.1651 11.7304i 14.2958i −9.00000 0
199.3 0.793499i 3.00000i 7.37036 0 −2.38050 2.90793i 12.1964i −9.00000 0
199.4 0.793499i 3.00000i 7.37036 0 −2.38050 2.90793i 12.1964i −9.00000 0
199.5 4.38835i 3.00000i −11.2577 0 13.1651 11.7304i 14.2958i −9.00000 0
199.6 4.59486i 3.00000i −13.1127 0 −13.7846 20.6383i 23.4921i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.6
Significant digits:
Format:

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 825.4.c.m 6
5.b even 2 1 inner 825.4.c.m 6
5.c odd 4 1 165.4.a.g 3
5.c odd 4 1 825.4.a.p 3
15.e even 4 1 495.4.a.i 3
15.e even 4 1 2475.4.a.z 3
55.e even 4 1 1815.4.a.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.g 3 5.c odd 4 1
495.4.a.i 3 15.e even 4 1
825.4.a.p 3 5.c odd 4 1
825.4.c.m 6 1.a even 1 1 trivial
825.4.c.m 6 5.b even 2 1 inner
1815.4.a.q 3 55.e even 4 1
2475.4.a.z 3 15.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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{6} + 41 T_{2}^{4} + 432 T_{2}^{2} + 256 \)
\( T_{7}^{6} + 572 T_{7}^{4} + 63376 T_{7}^{2} + 495616 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 432 T^{2} + 41 T^{4} + T^{6} \)
$3$ \( ( 9 + T^{2} )^{3} \)
$5$ \( T^{6} \)
$7$ \( 495616 + 63376 T^{2} + 572 T^{4} + T^{6} \)
$11$ \( ( -11 + T )^{6} \)
$13$ \( 6100234816 + 27330560 T^{2} + 10240 T^{4} + T^{6} \)
$17$ \( 502835776 + 5747264 T^{2} + 6384 T^{4} + T^{6} \)
$19$ \( ( 80 + 3356 T + 116 T^{2} + T^{3} )^{2} \)
$23$ \( 32480690176 + 181141504 T^{2} + 38144 T^{4} + T^{6} \)
$29$ \( ( -428416 + 5136 T + 238 T^{2} + T^{3} )^{2} \)
$31$ \( ( 6769664 - 53552 T - 92 T^{2} + T^{3} )^{2} \)
$37$ \( 40502634332224 + 10304040240 T^{2} + 199500 T^{4} + T^{6} \)
$41$ \( ( -245888 - 9136 T + 46 T^{2} + T^{3} )^{2} \)
$43$ \( 5670875449600 + 964657104 T^{2} + 54092 T^{4} + T^{6} \)
$47$ \( 120565319090176 + 13253571840 T^{2} + 317360 T^{4} + T^{6} \)
$53$ \( 298010064169024 + 38946832432 T^{2} + 423308 T^{4} + T^{6} \)
$59$ \( ( 32923904 + 424064 T + 1236 T^{2} + T^{3} )^{2} \)
$61$ \( ( -2655176 - 68308 T - 342 T^{2} + T^{3} )^{2} \)
$67$ \( 23619135781113856 + 267248243456 T^{2} + 943792 T^{4} + T^{6} \)
$71$ \( ( -198158720 + 1056112 T - 1816 T^{2} + T^{3} )^{2} \)
$73$ \( 26347524744256 + 4392718656 T^{2} + 157232 T^{4} + T^{6} \)
$79$ \( ( 167159872 - 812212 T - 96 T^{2} + T^{3} )^{2} \)
$83$ \( 293911633327902976 + 2202879189072 T^{2} + 2992332 T^{4} + T^{6} \)
$89$ \( ( -693013592 - 1499620 T + 838 T^{2} + T^{3} )^{2} \)
$97$ \( 125742050756777536 + 1139364329264 T^{2} + 2646124 T^{4} + T^{6} \)
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