Properties

Label 91.8.a.c
Level $91$
Weight $8$
Character orbit 91.a
Self dual yes
Analytic conductor $28.427$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + 11067840 x^{3} + 1840757376 x^{2} + 541859072 x - 4516262912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + (\beta_{3} - \beta_1 - 8) q^{3} + (\beta_{2} - 3 \beta_1 + 68) q^{4} + ( - \beta_{6} + 4 \beta_1 - 93) q^{5} + ( - \beta_{8} + \beta_{4} - 4 \beta_{3} - 10 \beta_1 - 140) q^{6} + 343 q^{7} + (4 \beta_{8} + 4 \beta_{6} + \beta_{5} - \beta_{4} + 9 \beta_{3} - 5 \beta_{2} + 59 \beta_1 - 503) q^{8} + (2 \beta_{9} + \beta_{8} + 2 \beta_{6} - \beta_{5} - 5 \beta_{4} - 15 \beta_{3} - 5 \beta_{2} + \cdots + 365) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{2} + (\beta_{3} - \beta_1 - 8) q^{3} + (\beta_{2} - 3 \beta_1 + 68) q^{4} + ( - \beta_{6} + 4 \beta_1 - 93) q^{5} + ( - \beta_{8} + \beta_{4} - 4 \beta_{3} - 10 \beta_1 - 140) q^{6} + 343 q^{7} + (4 \beta_{8} + 4 \beta_{6} + \beta_{5} - \beta_{4} + 9 \beta_{3} - 5 \beta_{2} + 59 \beta_1 - 503) q^{8} + (2 \beta_{9} + \beta_{8} + 2 \beta_{6} - \beta_{5} - 5 \beta_{4} - 15 \beta_{3} - 5 \beta_{2} + \cdots + 365) q^{9}+ \cdots + (5845 \beta_{9} - 7463 \beta_{8} - 6618 \beta_{7} + \cdots + 1894757) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9} + 9420 q^{10} + 876 q^{11} - 8765 q^{12} - 21970 q^{13} - 6174 q^{14} - 5320 q^{15} + 41370 q^{16} + 6294 q^{17} - 16027 q^{18} - 97401 q^{19} - 166650 q^{20} - 27440 q^{21} + 74171 q^{22} - 15255 q^{23} + 196187 q^{24} + 162145 q^{25} + 39546 q^{26} - 181820 q^{27} + 229810 q^{28} - 340533 q^{29} - 325020 q^{30} - 148675 q^{31} - 642762 q^{32} - 624400 q^{33} - 1161518 q^{34} - 317961 q^{35} - 773917 q^{36} - 621782 q^{37} - 805092 q^{38} + 175760 q^{39} - 350478 q^{40} - 2043336 q^{41} - 486717 q^{42} - 1801391 q^{43} - 3953667 q^{44} - 1908807 q^{45} - 2707731 q^{46} - 1624701 q^{47} - 6068625 q^{48} + 1176490 q^{49} - 6891516 q^{50} + 1811700 q^{51} - 1471990 q^{52} - 199965 q^{53} - 2895913 q^{54} + 739086 q^{55} - 1673154 q^{56} + 2159088 q^{57} + 2071092 q^{58} - 8098908 q^{59} + 8096436 q^{60} + 2271618 q^{61} - 8910225 q^{62} + 1238916 q^{63} + 8099930 q^{64} + 2036619 q^{65} - 5999191 q^{66} + 1970272 q^{67} - 1766238 q^{68} - 4622962 q^{69} + 3231060 q^{70} - 7145820 q^{71} + 984975 q^{72} + 1409431 q^{73} - 5498643 q^{74} - 8857892 q^{75} - 2749534 q^{76} + 300468 q^{77} + 3117543 q^{78} - 9011055 q^{79} - 23850522 q^{80} + 11613490 q^{81} + 27962597 q^{82} - 15006567 q^{83} - 3006395 q^{84} - 9416628 q^{85} + 38357850 q^{86} - 15828996 q^{87} + 42205269 q^{88} - 11472777 q^{89} + 53425712 q^{90} - 7535710 q^{91} + 16755837 q^{92} + 36339848 q^{93} + 5133371 q^{94} + 29637939 q^{95} + 65329611 q^{96} + 3228571 q^{97} - 2117682 q^{98} + 19367194 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + 11067840 x^{3} + 1840757376 x^{2} + 541859072 x - 4516262912 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1176111762173 \nu^{9} - 12574063170642 \nu^{8} + 876995328306269 \nu^{7} + \cdots - 25\!\cdots\!60 ) / 59\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2790653758009 \nu^{9} - 499024731176506 \nu^{8} + \cdots + 30\!\cdots\!76 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6214679484235 \nu^{9} - 367159236899342 \nu^{8} + \cdots - 13\!\cdots\!72 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 711607147759 \nu^{9} - 687274083168 \nu^{8} + 619904788708757 \nu^{7} + \cdots - 33\!\cdots\!24 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25960034868425 \nu^{9} + 56774085353906 \nu^{8} + \cdots + 80\!\cdots\!60 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31766366680659 \nu^{9} + 48358777692646 \nu^{8} + \cdots + 32\!\cdots\!00 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13936442615075 \nu^{9} - 2356424471438 \nu^{8} + \cdots + 61\!\cdots\!84 ) / 19\!\cdots\!88 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{8} + 4\beta_{6} + \beta_{5} - \beta_{4} + 9\beta_{3} + \beta_{2} + 309\beta _1 + 145 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{9} + 22 \beta_{8} + 18 \beta_{7} + 44 \beta_{6} + 15 \beta_{5} + \beta_{4} - 202 \beta_{3} + 435 \beta_{2} + 762 \beta _1 + 59730 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 109 \beta_{9} + 2446 \beta_{8} + 118 \beta_{7} + 2192 \beta_{6} + 851 \beta_{5} - 751 \beta_{4} + 5230 \beta_{3} + 1353 \beta_{2} + 112976 \beta _1 + 119766 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1541 \beta_{9} + 16218 \beta_{8} + 11722 \beta_{7} + 34984 \beta_{6} + 12813 \beta_{5} - 2329 \beta_{4} - 118488 \beta_{3} + 184491 \beta_{2} + 481814 \beta _1 + 21929496 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 86067 \beta_{9} + 1226770 \beta_{8} + 89618 \beta_{7} + 1040712 \beta_{6} + 516339 \beta_{5} - 420631 \beta_{4} + 2297336 \beta_{3} + 979737 \beta_{2} + 45245650 \beta _1 + 80561872 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 384913 \beta_{9} + 9584946 \beta_{8} + 6145098 \beta_{7} + 20408544 \beta_{6} + 7766169 \beta_{5} - 2514621 \beta_{4} - 56223408 \beta_{3} + 79325107 \beta_{2} + \cdots + 8791834440 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 47808719 \beta_{9} + 580883410 \beta_{8} + 52781794 \beta_{7} + 484274136 \beta_{6} + 273774531 \beta_{5} - 211377687 \beta_{4} + 930228476 \beta_{3} + \cdots + 48382466052 \) 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
−20.0501
−16.8829
−11.8703
−9.51136
−1.80018
1.45069
10.4944
12.2698
16.1785
21.7215
−22.0501 −47.1151 358.208 −467.623 1038.89 343.000 −5076.12 32.8291 10311.1
1.2 −18.8829 14.3398 228.563 382.689 −270.777 343.000 −1898.92 −1981.37 −7226.26
1.3 −13.8703 79.8279 64.3848 −355.678 −1107.24 343.000 882.360 4185.49 4933.36
1.4 −11.5114 23.9058 4.51150 −301.396 −275.188 343.000 1421.52 −1615.51 3469.48
1.5 −3.80018 −76.5407 −113.559 −339.464 290.868 343.000 917.967 3671.47 1290.03
1.6 −0.549314 −7.31646 −127.698 243.344 4.01903 343.000 140.459 −2133.47 −133.672
1.7 8.49443 45.4613 −55.8447 −130.405 386.168 343.000 −1561.66 −120.267 −1107.72
1.8 10.2698 −81.6519 −22.5320 313.060 −838.546 343.000 −1545.93 4480.04 3215.05
1.9 14.1785 6.78261 73.0290 −4.23520 96.1671 343.000 −779.405 −2141.00 −60.0487
1.10 19.7215 −37.6933 260.937 −267.290 −743.369 343.000 2621.72 −766.212 −5271.36
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(13\) \(1\)

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 91.8.a.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.8.a.c 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 18 T_{2}^{9} - 813 T_{2}^{8} - 13416 T_{2}^{7} + 222070 T_{2}^{6} + 3157452 T_{2}^{5} - 24903776 T_{2}^{4} - 272954592 T_{2}^{3} + 992532576 T_{2}^{2} + 6785706432 T_{2} + 3385168384 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 18 T^{9} + \cdots + 3385168384 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots - 685212736352256 \) Copy content Toggle raw display
$5$ \( T^{10} + 927 T^{9} + \cdots - 73\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T - 343)^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 876 T^{9} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( (T + 2197)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 6294 T^{9} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{10} + 97401 T^{9} + \cdots - 12\!\cdots\!68 \) Copy content Toggle raw display
$23$ \( T^{10} + 15255 T^{9} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{10} + 340533 T^{9} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{10} + 148675 T^{9} + \cdots - 13\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{10} + 621782 T^{9} + \cdots + 91\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{10} + 2043336 T^{9} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{10} + 1801391 T^{9} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{10} + 1624701 T^{9} + \cdots + 85\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{10} + 199965 T^{9} + \cdots + 63\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{10} + 8098908 T^{9} + \cdots - 58\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{10} - 2271618 T^{9} + \cdots - 13\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{10} - 1970272 T^{9} + \cdots + 57\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{10} + 7145820 T^{9} + \cdots + 80\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{10} - 1409431 T^{9} + \cdots + 39\!\cdots\!54 \) Copy content Toggle raw display
$79$ \( T^{10} + 9011055 T^{9} + \cdots + 40\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{10} + 15006567 T^{9} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{10} + 11472777 T^{9} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{10} - 3228571 T^{9} + \cdots + 52\!\cdots\!18 \) Copy content Toggle raw display
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