Properties

Label 825.6.a.u
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + 7443872 x^{2} - 12856224 x - 7036608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 22) q^{4} - 9 \beta_1 q^{6} + ( - \beta_{4} - \beta_{2} + \beta_1 - 19) q^{7} + (\beta_{3} + \beta_{2} + 26 \beta_1 - 21) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 22) q^{4} - 9 \beta_1 q^{6} + ( - \beta_{4} - \beta_{2} + \beta_1 - 19) q^{7} + (\beta_{3} + \beta_{2} + 26 \beta_1 - 21) q^{8} + 81 q^{9} + 121 q^{11} + ( - 9 \beta_{2} - 198) q^{12} + ( - \beta_{8} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 22 \beta_1 + 107) q^{13} + (\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} - \beta_{3} - 38 \beta_1 + 72) q^{14} + ( - 2 \beta_{6} + 2 \beta_{5} + 23 \beta_{2} + 14 \beta_1 + 694) q^{16} + (\beta_{9} + \beta_{7} - \beta_{6} - 4 \beta_{4} + \beta_{3} - 6 \beta_{2} + 11 \beta_1 - 112) q^{17} + 81 \beta_1 q^{18} + ( - \beta_{9} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 82 \beta_1 + 268) q^{19} + (9 \beta_{4} + 9 \beta_{2} - 9 \beta_1 + 171) q^{21} + 121 \beta_1 q^{22} + ( - \beta_{9} - \beta_{8} + 4 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots - 223) q^{23}+ \cdots + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9} + 1210 q^{11} - 2007 q^{12} + 1102 q^{13} + 684 q^{14} + 7019 q^{16} - 1106 q^{17} + 81 q^{18} + 2586 q^{19} + 1692 q^{21} + 121 q^{22} - 2206 q^{23} + 1593 q^{24} + 12001 q^{26} - 7290 q^{27} - 14452 q^{28} + 5824 q^{29} - 4586 q^{31} + 10627 q^{32} - 10890 q^{33} + 7426 q^{34} + 18063 q^{36} + 18362 q^{37} - 44001 q^{38} - 9918 q^{39} - 5474 q^{41} - 6156 q^{42} - 20496 q^{43} + 26983 q^{44} + 19981 q^{46} + 14970 q^{47} - 63171 q^{48} + 68582 q^{49} + 9954 q^{51} + 58603 q^{52} - 61980 q^{53} - 729 q^{54} + 7132 q^{56} - 23274 q^{57} - 7161 q^{58} + 61190 q^{59} + 8230 q^{61} + 4509 q^{62} - 15228 q^{63} + 152223 q^{64} - 1089 q^{66} + 11930 q^{67} - 105598 q^{68} + 19854 q^{69} + 59822 q^{71} - 14337 q^{72} + 20680 q^{73} + 132564 q^{74} + 68165 q^{76} - 22748 q^{77} - 108009 q^{78} + 234494 q^{79} + 65610 q^{81} - 151948 q^{82} - 185478 q^{83} + 130068 q^{84} - 17825 q^{86} - 52416 q^{87} - 21417 q^{88} + 181834 q^{89} + 206274 q^{91} - 98373 q^{92} + 41274 q^{93} - 64998 q^{94} - 95643 q^{96} - 304358 q^{97} - 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + 7443872 x^{2} - 12856224 x - 7036608 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 90\nu + 75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 793 \nu^{9} + 4245 \nu^{8} - 218253 \nu^{7} - 1041837 \nu^{6} + 19464294 \nu^{5} + 79330102 \nu^{4} - 594332024 \nu^{3} - 1955331720 \nu^{2} + \cdots + 9482457408 ) / 32138304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1039 \nu^{9} - 2837 \nu^{8} + 241401 \nu^{7} + 667197 \nu^{6} - 16419288 \nu^{5} - 41925694 \nu^{4} + 284331468 \nu^{3} + 402736312 \nu^{2} + \cdots + 3384141600 ) / 16069152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1039 \nu^{9} - 2837 \nu^{8} + 241401 \nu^{7} + 667197 \nu^{6} - 16419288 \nu^{5} - 49960270 \nu^{4} + 284331468 \nu^{3} + 1358850856 \nu^{2} + \cdots - 9246211872 ) / 16069152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12773 \nu^{9} - 45041 \nu^{8} + 3331917 \nu^{7} + 11342781 \nu^{6} - 285383202 \nu^{5} - 900724178 \nu^{4} + 8919161552 \nu^{3} + \cdots - 109330622496 ) / 64276608 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 39585 \nu^{9} + 72205 \nu^{8} - 10562481 \nu^{7} - 16937505 \nu^{6} + 926492130 \nu^{5} + 1150777794 \nu^{4} - 29318748608 \nu^{3} + \cdots + 31465564896 ) / 64276608 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 49661 \nu^{9} - 65505 \nu^{8} + 13136229 \nu^{7} + 16796133 \nu^{6} - 1138140594 \nu^{5} - 1298267594 \nu^{4} + 35398857040 \nu^{3} + \cdots - 92954209248 ) / 64276608 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 90\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 2\beta_{5} + 119\beta_{2} + 14\beta _1 + 4854 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} + 2\beta_{8} - 8\beta_{7} + 6\beta_{6} - 36\beta_{4} + 149\beta_{3} + 131\beta_{2} + 9136\beta _1 - 1721 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30 \beta_{9} + 44 \beta_{8} + 18 \beta_{7} - 410 \beta_{6} + 384 \beta_{5} - 82 \beta_{4} - 6 \beta_{3} + 13559 \beta_{2} + 3328 \beta _1 + 492508 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 420 \beta_{9} + 382 \beta_{8} - 1730 \beta_{7} + 1062 \beta_{6} - 182 \beta_{5} - 8010 \beta_{4} + 19297 \beta_{3} + 15851 \beta_{2} + 982274 \beta _1 - 100793 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7884 \beta_{9} + 10848 \beta_{8} + 2988 \beta_{7} - 62118 \beta_{6} + 55746 \beta_{5} - 16524 \beta_{4} - 1240 \beta_{3} + 1560523 \beta_{2} + 560670 \beta _1 + 52927326 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 63714 \beta_{9} + 55782 \beta_{8} - 272124 \beta_{7} + 138962 \beta_{6} - 44084 \beta_{5} - 1299672 \beta_{4} + 2402009 \beta_{3} + 1917951 \beta_{2} + 109461952 \beta _1 - 1901133 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10.9094
−9.20891
−7.65849
−3.54519
−0.444648
2.70074
3.16483
6.73090
9.15502
11.0151
−10.9094 −9.00000 87.0148 0 98.1845 −180.635 −600.178 81.0000 0
1.2 −9.20891 −9.00000 52.8039 0 82.8802 195.275 −191.582 81.0000 0
1.3 −7.65849 −9.00000 26.6525 0 68.9264 −112.197 40.9539 81.0000 0
1.4 −3.54519 −9.00000 −19.4316 0 31.9067 −15.4153 182.335 81.0000 0
1.5 −0.444648 −9.00000 −31.8023 0 4.00183 −205.804 28.3695 81.0000 0
1.6 2.70074 −9.00000 −24.7060 0 −24.3067 73.4900 −153.148 81.0000 0
1.7 3.16483 −9.00000 −21.9838 0 −28.4835 223.585 −170.850 81.0000 0
1.8 6.73090 −9.00000 13.3050 0 −60.5781 −2.41142 −125.834 81.0000 0
1.9 9.15502 −9.00000 51.8143 0 −82.3952 −226.656 181.401 81.0000 0
1.10 11.0151 −9.00000 89.3332 0 −99.1362 62.7690 631.533 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.6.a.u yes 10
5.b even 2 1 825.6.a.t 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.6.a.t 10 5.b even 2 1
825.6.a.u yes 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - T_{2}^{9} - 271 T_{2}^{8} + 309 T_{2}^{7} + 24456 T_{2}^{6} - 33410 T_{2}^{5} - 822204 T_{2}^{4} + 1367872 T_{2}^{3} + 7443872 T_{2}^{2} - 12856224 T_{2} - 7036608 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} - 271 T^{8} + \cdots - 7036608 \) Copy content Toggle raw display
$3$ \( (T + 9)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + 188 T^{9} + \cdots + 70\!\cdots\!08 \) Copy content Toggle raw display
$11$ \( (T - 121)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 1102 T^{9} + \cdots + 41\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{10} + 1106 T^{9} + \cdots - 64\!\cdots\!52 \) Copy content Toggle raw display
$19$ \( T^{10} - 2586 T^{9} + \cdots - 23\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{10} + 2206 T^{9} + \cdots + 52\!\cdots\!48 \) Copy content Toggle raw display
$29$ \( T^{10} - 5824 T^{9} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + 4586 T^{9} + \cdots - 60\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{10} - 18362 T^{9} + \cdots - 28\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{10} + 5474 T^{9} + \cdots - 20\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{10} + 20496 T^{9} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{10} - 14970 T^{9} + \cdots - 31\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{10} + 61980 T^{9} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{10} - 61190 T^{9} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} - 8230 T^{9} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{10} - 11930 T^{9} + \cdots + 38\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{10} - 59822 T^{9} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} - 20680 T^{9} + \cdots - 62\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{10} - 234494 T^{9} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + 185478 T^{9} + \cdots - 24\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{10} - 181834 T^{9} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + 304358 T^{9} + \cdots - 37\!\cdots\!79 \) Copy content Toggle raw display
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