Properties

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

Related objects

Downloads

Learn more

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: 6.0.2230106176.1
Defining polynomial: \(x^{6} + 41 x^{4} + 452 x^{2} + 676\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} + ( -7 + \beta_{3} + \beta_{5} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{7} + ( -3 \beta_{1} + 15 \beta_{2} + 4 \beta_{4} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} + ( -7 + \beta_{3} + \beta_{5} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{7} + ( -3 \beta_{1} + 15 \beta_{2} + 4 \beta_{4} ) q^{8} -9 q^{9} + 11 q^{11} + ( 3 \beta_{1} - 21 \beta_{2} - 3 \beta_{4} ) q^{12} + ( 12 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{13} + ( 22 + 10 \beta_{3} + 4 \beta_{5} ) q^{14} + ( 9 - 23 \beta_{3} - 7 \beta_{5} ) q^{16} + ( -10 \beta_{1} - 76 \beta_{2} + 6 \beta_{4} ) q^{17} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{18} + ( -48 + 2 \beta_{3} + 4 \beta_{5} ) q^{19} + ( 6 + 6 \beta_{3} - 6 \beta_{5} ) q^{21} + ( 11 \beta_{1} - 11 \beta_{2} ) q^{22} + ( -20 \beta_{1} + 60 \beta_{2} + 8 \beta_{4} ) q^{23} + ( -45 + 9 \beta_{3} + 12 \beta_{5} ) q^{24} + ( -168 + 4 \beta_{3} + 18 \beta_{5} ) q^{26} -27 \beta_{2} q^{27} + ( -10 \beta_{1} + 110 \beta_{2} + 6 \beta_{4} ) q^{28} + ( -34 - 34 \beta_{3} + 20 \beta_{5} ) q^{29} + ( -24 - 4 \beta_{3} + 16 \beta_{5} ) q^{31} + ( 13 \beta_{1} - 225 \beta_{2} - 12 \beta_{4} ) q^{32} + 33 \beta_{2} q^{33} + ( 76 + 52 \beta_{3} - 28 \beta_{5} ) q^{34} + ( 63 - 9 \beta_{3} - 9 \beta_{5} ) q^{36} + ( 24 \beta_{1} - 122 \beta_{2} - 4 \beta_{4} ) q^{37} + ( -64 \beta_{1} + 84 \beta_{2} + 14 \beta_{4} ) q^{38} + ( -12 - 36 \beta_{3} - 6 \beta_{5} ) q^{39} + ( -66 - 2 \beta_{3} + 20 \beta_{5} ) q^{41} + ( 30 \beta_{1} + 66 \beta_{2} - 12 \beta_{4} ) q^{42} + ( -74 \beta_{1} + 150 \beta_{2} - 14 \beta_{4} ) q^{43} + ( -77 + 11 \beta_{3} + 11 \beta_{5} ) q^{44} + ( 356 - 92 \beta_{3} - 44 \beta_{5} ) q^{46} + ( -48 \beta_{1} - 36 \beta_{2} + 28 \beta_{4} ) q^{47} + ( -69 \beta_{1} + 27 \beta_{2} + 21 \beta_{4} ) q^{48} + ( 51 - 4 \beta_{3} - 20 \beta_{5} ) q^{49} + ( 228 + 30 \beta_{3} + 18 \beta_{5} ) q^{51} + ( -144 \beta_{1} + 292 \beta_{2} + 42 \beta_{4} ) q^{52} + ( -32 \beta_{1} + 42 \beta_{2} - 52 \beta_{4} ) q^{53} + ( -27 + 27 \beta_{3} ) q^{54} + ( 438 - 54 \beta_{3} + 4 \beta_{5} ) q^{56} + ( 6 \beta_{1} - 144 \beta_{2} - 12 \beta_{4} ) q^{57} + ( -114 \beta_{1} - 402 \beta_{2} + 26 \beta_{4} ) q^{58} + ( 308 - 120 \beta_{3} - 4 \beta_{5} ) q^{59} + ( 218 + 12 \beta_{3} + 44 \beta_{5} ) q^{61} + ( -88 \beta_{1} + 44 \beta_{4} ) q^{62} + ( 18 \beta_{1} + 18 \beta_{2} + 18 \beta_{4} ) q^{63} + ( -359 + 89 \beta_{3} - 7 \beta_{5} ) q^{64} + ( 33 - 33 \beta_{3} ) q^{66} + ( -148 \beta_{1} - 128 \beta_{2} - 28 \beta_{4} ) q^{67} + ( 108 \beta_{1} - 12 \beta_{2} + 16 \beta_{4} ) q^{68} + ( -180 + 60 \beta_{3} + 24 \beta_{5} ) q^{69} + ( -216 - 104 \beta_{3} + 16 \beta_{5} ) q^{71} + ( 27 \beta_{1} - 135 \beta_{2} - 36 \beta_{4} ) q^{72} + ( -168 \beta_{1} - 356 \beta_{2} - 14 \beta_{4} ) q^{73} + ( -466 + 138 \beta_{3} + 36 \beta_{5} ) q^{74} + ( 624 - 124 \beta_{3} - 74 \beta_{5} ) q^{76} + ( -22 \beta_{1} - 22 \beta_{2} - 22 \beta_{4} ) q^{77} + ( 12 \beta_{1} - 504 \beta_{2} - 54 \beta_{4} ) q^{78} + ( 524 - 14 \beta_{3} - 52 \beta_{5} ) q^{79} + 81 q^{81} + ( -146 \beta_{1} + 78 \beta_{2} + 58 \beta_{4} ) q^{82} + ( 164 \beta_{1} + 582 \beta_{2} - 70 \beta_{4} ) q^{83} + ( -330 + 30 \beta_{3} + 18 \beta_{5} ) q^{84} + ( 1158 - 94 \beta_{3} - 32 \beta_{5} ) q^{86} + ( -102 \beta_{1} - 102 \beta_{2} - 60 \beta_{4} ) q^{87} + ( -33 \beta_{1} + 165 \beta_{2} + 44 \beta_{4} ) q^{88} + ( 730 + 68 \beta_{3} ) q^{89} + ( 56 + 176 \beta_{3} - 4 \beta_{5} ) q^{91} + ( 372 \beta_{1} - 1252 \beta_{2} - 160 \beta_{4} ) q^{92} + ( -12 \beta_{1} - 72 \beta_{2} - 48 \beta_{4} ) q^{93} + ( 692 - 76 \beta_{3} - 132 \beta_{5} ) q^{94} + ( 675 - 39 \beta_{3} - 36 \beta_{5} ) q^{96} + ( 240 \beta_{1} + 286 \beta_{2} + 64 \beta_{4} ) q^{97} + ( 131 \beta_{1} - 147 \beta_{2} - 64 \beta_{4} ) q^{98} -99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + O(q^{10}) \) \( 6 q - 44 q^{4} + 24 q^{6} - 54 q^{9} + 66 q^{11} + 112 q^{14} + 100 q^{16} - 292 q^{19} + 24 q^{21} - 288 q^{24} - 1016 q^{26} - 136 q^{29} - 136 q^{31} + 352 q^{34} + 396 q^{36} - 392 q^{41} - 484 q^{44} + 2320 q^{46} + 314 q^{49} + 1308 q^{51} - 216 q^{54} + 2736 q^{56} + 2088 q^{59} + 1284 q^{61} - 2332 q^{64} + 264 q^{66} - 1200 q^{69} - 1088 q^{71} - 3072 q^{74} + 3992 q^{76} + 3172 q^{79} + 486 q^{81} - 2040 q^{84} + 7136 q^{86} + 4244 q^{89} - 16 q^{91} + 4304 q^{94} + 4128 q^{96} - 594 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 41 x^{4} + 452 x^{2} + 676\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 15 \nu^{3} - 94 \nu \)\()/156\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 21 \nu^{2} + 26 \)\()/6\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 28 \nu^{3} + 153 \nu \)\()/13\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{4} + 27 \nu^{2} + 110 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{3} - 14\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 12 \beta_{2} - 19 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-21 \beta_{5} + 27 \beta_{3} + 268\)
\(\nu^{5}\)\(=\)\(-15 \beta_{4} + 336 \beta_{2} + 379 \beta_{1}\)

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
4.26150i
4.59056i
1.32906i
1.32906i
4.59056i
4.26150i
5.26150i 3.00000i −19.6833 0 15.7845 10.3207i 61.4719i −9.00000 0
199.2 3.59056i 3.00000i −4.89212 0 −10.7717 16.1465i 11.1590i −9.00000 0
199.3 2.32906i 3.00000i 2.57547 0 6.98719 22.4672i 24.6309i −9.00000 0
199.4 2.32906i 3.00000i 2.57547 0 6.98719 22.4672i 24.6309i −9.00000 0
199.5 3.59056i 3.00000i −4.89212 0 −10.7717 16.1465i 11.1590i −9.00000 0
199.6 5.26150i 3.00000i −19.6833 0 15.7845 10.3207i 61.4719i −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.l 6
5.b even 2 1 inner 825.4.c.l 6
5.c odd 4 1 165.4.a.d 3
5.c odd 4 1 825.4.a.s 3
15.e even 4 1 495.4.a.l 3
15.e even 4 1 2475.4.a.s 3
55.e even 4 1 1815.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.d 3 5.c odd 4 1
495.4.a.l 3 15.e even 4 1
825.4.a.s 3 5.c odd 4 1
825.4.c.l 6 1.a even 1 1 trivial
825.4.c.l 6 5.b even 2 1 inner
1815.4.a.s 3 55.e even 4 1
2475.4.a.s 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{6} + 46 T_{2}^{4} + 577 T_{2}^{2} + 1936 \)
\( T_{7}^{6} + 872 T_{7}^{4} + 213136 T_{7}^{2} + 14017536 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1936 + 577 T^{2} + 46 T^{4} + T^{6} \)
$3$ \( ( 9 + T^{2} )^{3} \)
$5$ \( T^{6} \)
$7$ \( 14017536 + 213136 T^{2} + 872 T^{4} + T^{6} \)
$11$ \( ( -11 + T )^{6} \)
$13$ \( 1165812736 + 12673600 T^{2} + 7120 T^{4} + T^{6} \)
$17$ \( 55273890816 + 196207744 T^{2} + 28164 T^{4} + T^{6} \)
$19$ \( ( 30960 + 5376 T + 146 T^{2} + T^{3} )^{2} \)
$23$ \( 2768896 + 6473984 T^{2} + 45344 T^{4} + T^{6} \)
$29$ \( ( -3163056 - 54364 T + 68 T^{2} + T^{3} )^{2} \)
$31$ \( ( -1812096 - 23232 T + 68 T^{2} + T^{3} )^{2} \)
$37$ \( 383101578304 + 846549040 T^{2} + 79180 T^{4} + T^{6} \)
$41$ \( ( -4364208 - 26076 T + 196 T^{2} + T^{3} )^{2} \)
$43$ \( 978058072166400 + 33596922384 T^{2} + 331912 T^{4} + T^{6} \)
$47$ \( 439614082584576 + 20990283520 T^{2} + 275440 T^{4} + T^{6} \)
$53$ \( 1566176825802304 + 55538081072 T^{2} + 546668 T^{4} + T^{6} \)
$59$ \( ( 84227264 + 64144 T - 1044 T^{2} + T^{3} )^{2} \)
$61$ \( ( 22757384 - 60548 T - 642 T^{2} + T^{3} )^{2} \)
$67$ \( 7662842618576896 + 172318457856 T^{2} + 980112 T^{4} + T^{6} \)
$71$ \( ( 6553600 - 129728 T + 544 T^{2} + T^{3} )^{2} \)
$73$ \( 31464740110336 + 79648960576 T^{2} + 1409152 T^{4} + T^{6} \)
$79$ \( ( 14694992 + 562208 T - 1586 T^{2} + T^{3} )^{2} \)
$83$ \( 854969086152507456 + 3020222825712 T^{2} + 3118012 T^{4} + T^{6} \)
$89$ \( ( -293444632 + 1406940 T - 2122 T^{2} + T^{3} )^{2} \)
$97$ \( 49971801904837696 + 1392188832304 T^{2} + 2965324 T^{4} + T^{6} \)
show more
show less