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Properties

Label 825.6.a.y
Level $825$
Weight $6$
Character orbit 825.a
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 + 16) q^{4} + ( - 9 \beta_1 + 9) q^{6} + (\beta_{7} + \beta_1 + 23) q^{7} + ( - \beta_{3} + \beta_{2} - 16 \beta_1 + 31) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + 9 q^{3} + (\beta_{2} - \beta_1 + 16) q^{4} + ( - 9 \beta_1 + 9) q^{6} + (\beta_{7} + \beta_1 + 23) q^{7} + ( - \beta_{3} + \beta_{2} - 16 \beta_1 + 31) q^{8} + 81 q^{9} + 121 q^{11} + (9 \beta_{2} - 9 \beta_1 + 144) q^{12} + ( - \beta_{11} - \beta_{8} - \beta_{6} + \beta_{2} - 3 \beta_1 + 76) q^{13} + (\beta_{10} + 2 \beta_{7} - \beta_{5} + \beta_{2} - 36 \beta_1 - 48) q^{14} + ( - \beta_{11} - \beta_{7} - 2 \beta_{6} - \beta_{5} + 18 \beta_{2} - 34 \beta_1 + 269) q^{16} + (2 \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{5} + 2 \beta_{3} + 5 \beta_{2} + \cdots + 112) q^{17}+ \cdots + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 47 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 78\nu + 47 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 700402408295 \nu^{12} + 4192326267667 \nu^{11} + 204218759138159 \nu^{10} + \cdots - 10\!\cdots\!00 ) / 89\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 766585621025 \nu^{12} - 1865244494613 \nu^{11} - 224871623640281 \nu^{10} + 405732383014815 \nu^{9} + \cdots + 42\!\cdots\!40 ) / 89\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 891142389387 \nu^{12} - 3560118843527 \nu^{11} - 258925322555763 \nu^{10} + 840913100582581 \nu^{9} + \cdots + 10\!\cdots\!40 ) / 89\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 960809696913 \nu^{12} - 3695209656645 \nu^{11} - 277342519722953 \nu^{10} + 870845035605551 \nu^{9} + \cdots + 10\!\cdots\!40 ) / 89\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38192377867 \nu^{12} + 223973388087 \nu^{11} + 11044992841811 \nu^{10} - 56730273953229 \nu^{9} + \cdots - 58\!\cdots\!00 ) / 27\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 741184491385 \nu^{12} - 2928178310573 \nu^{11} - 213498624550193 \nu^{10} + 689953296380007 \nu^{9} + \cdots + 63\!\cdots\!20 ) / 44\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3500985580757 \nu^{12} - 14835282370393 \nu^{11} + \cdots + 44\!\cdots\!20 ) / 89\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 438710012089 \nu^{12} + 1585086479789 \nu^{11} + 127508098559345 \nu^{10} - 369800452473191 \nu^{9} + \cdots - 62\!\cdots\!00 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6007694208307 \nu^{12} + 26162256334671 \nu^{11} + \cdots - 79\!\cdots\!00 ) / 89\!\cdots\!40 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 80\beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{7} - 2\beta_{6} - \beta_{5} + 4\beta_{3} + 116\beta_{2} + 188\beta _1 + 3758 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{12} + 11 \beta_{10} - 4 \beta_{9} - \beta_{8} + 3 \beta_{7} - 12 \beta_{6} - 4 \beta_{5} + 7 \beta_{4} + 144 \beta_{3} + 401 \beta_{2} + 7639 \beta _1 + 8798 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{12} - 135 \beta_{11} + 64 \beta_{10} + 24 \beta_{9} + 46 \beta_{8} - 213 \beta_{7} - 368 \beta_{6} - 173 \beta_{5} + 878 \beta_{3} + 12992 \beta_{2} + 28982 \beta _1 + 359206 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 521 \beta_{12} - 86 \beta_{11} + 2307 \beta_{10} - 684 \beta_{9} - 79 \beta_{8} - 49 \beta_{7} - 2774 \beta_{6} - 976 \beta_{5} + 1519 \beta_{4} + 18556 \beta_{3} + 64781 \beta_{2} + 799607 \beta _1 + 1355512 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 720 \beta_{12} - 15391 \beta_{11} + 15724 \beta_{10} + 3328 \beta_{9} + 9976 \beta_{8} - 30583 \beta_{7} - 53826 \beta_{6} - 24803 \beta_{5} + 1596 \beta_{4} + 143254 \beta_{3} + 1498742 \beta_{2} + \cdots + 37618998 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 68397 \beta_{12} - 25726 \beta_{11} + 354129 \beta_{10} - 88412 \beta_{9} + 4617 \beta_{8} - 77729 \beta_{7} - 467588 \beta_{6} - 167014 \beta_{5} + 232837 \beta_{4} + 2368506 \beta_{3} + \cdots + 196361842 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 158440 \beta_{12} - 1705535 \beta_{11} + 2743638 \beta_{10} + 284656 \beta_{9} + 1549900 \beta_{8} - 3866735 \beta_{7} - 7335528 \beta_{6} - 3344661 \beta_{5} + 501198 \beta_{4} + \cdots + 4192972336 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8245237 \beta_{12} - 5090222 \beta_{11} + 48984123 \beta_{10} - 10562556 \beta_{9} + 2811285 \beta_{8} - 17696197 \beta_{7} - 69879050 \beta_{6} - 25223916 \beta_{5} + \cdots + 27556846220 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 28040720 \beta_{12} - 190804435 \beta_{11} + 419463104 \beta_{10} + 13305632 \beta_{9} + 213106496 \beta_{8} - 471239355 \beta_{7} - 972828342 \beta_{6} + \cdots + 489536497526 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
11.4306
8.81495
8.39855
4.62206
4.33747
1.88656
−0.317411
−3.97394
−4.09558
−4.72893
−7.57523
−9.04603
−9.75306
−10.4306 9.00000 76.7973 0 −93.8753 21.9028 −467.262 81.0000 0
1.2 −7.81495 9.00000 29.0734 0 −70.3345 11.0326 22.8710 81.0000 0
1.3 −7.39855 9.00000 22.7385 0 −66.5869 150.852 68.5217 81.0000 0
1.4 −3.62206 9.00000 −18.8807 0 −32.5985 −168.040 184.293 81.0000 0
1.5 −3.33747 9.00000 −20.8613 0 −30.0373 103.473 176.423 81.0000 0
1.6 −0.886559 9.00000 −31.2140 0 −7.97903 224.774 56.0430 81.0000 0
1.7 1.31741 9.00000 −30.2644 0 11.8567 −87.4786 −82.0279 81.0000 0
1.8 4.97394 9.00000 −7.25990 0 44.7655 −155.412 −195.276 81.0000 0
1.9 5.09558 9.00000 −6.03505 0 45.8602 170.277 −193.811 81.0000 0
1.10 5.72893 9.00000 0.820586 0 51.5603 −158.171 −178.625 81.0000 0
1.11 8.57523 9.00000 41.5346 0 77.1771 178.462 81.7612 81.0000 0
1.12 10.0460 9.00000 68.9227 0 90.4143 −29.0524 370.927 81.0000 0
1.13 10.7531 9.00000 83.6283 0 96.7775 41.3813 555.162 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.y 13
5.b even 2 1 825.6.a.v 13
5.c odd 4 2 165.6.c.b ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.c.b ✓ 26 5.c odd 4 2
825.6.a.v 13 5.b even 2 1
825.6.a.y 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 13 T_{2}^{12} - 228 T_{2}^{11} + 3286 T_{2}^{10} + 16399 T_{2}^{9} - 292621 T_{2}^{8} - 348208 T_{2}^{7} + 11270528 T_{2}^{6} - 3851488 T_{2}^{5} - 183538464 T_{2}^{4} + \cdots - 1145312512 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(825))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 13 T^{12} + \cdots - 1145312512 \) Copy content Toggle raw display
$3$ \( (T - 9)^{13} \) Copy content Toggle raw display
$5$ \( T^{13} \) Copy content Toggle raw display
$7$ \( T^{13} - 304 T^{12} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( (T - 121)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} - 986 T^{12} + \cdots - 51\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{13} - 1476 T^{12} + \cdots - 48\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{13} - 270 T^{12} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{13} - 9084 T^{12} + \cdots + 44\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{13} - 11952 T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{13} - 19096 T^{12} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{13} - 39964 T^{12} + \cdots - 29\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{13} - 35184 T^{12} + \cdots + 32\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( T^{13} + 96 T^{12} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{13} - 34984 T^{12} + \cdots + 12\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{13} - 22984 T^{12} + \cdots + 45\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{13} + 9192 T^{12} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{13} - 5438 T^{12} + \cdots + 21\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{13} - 71508 T^{12} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{13} - 101700 T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{13} - 77390 T^{12} + \cdots + 28\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{13} - 93954 T^{12} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{13} - 185918 T^{12} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{13} + 18418 T^{12} + \cdots - 97\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{13} - 94312 T^{12} + \cdots - 87\!\cdots\!52 \) Copy content Toggle raw display
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