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Properties

Label 91.2.e.c
Level $91$
Weight $2$
Character orbit 91.e
Analytic conductor $0.727$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_1) q^{2} + (\beta_{9} - \beta_{4}) q^{3} + (\beta_{9} + 2 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} - \beta_{8} q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{9} + \beta_{6} + \beta_{4}) q^{7} + (\beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 2) q^{8} + (\beta_{8} - \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_1) q^{2} + (\beta_{9} - \beta_{4}) q^{3} + (\beta_{9} + 2 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} - \beta_{8} q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{9} + \beta_{6} + \beta_{4}) q^{7} + (\beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 2) q^{8} + (\beta_{8} - \beta_{7}) q^{9} + (\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{10} + (2 \beta_{7} + \beta_{6} - \beta_{3} - 2) q^{11} + ( - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_1) q^{12} + q^{13} + ( - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - 2 \beta_{2} - \beta_1 + 1) q^{14} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{2}) q^{15} + ( - 2 \beta_{8} - 2 \beta_{7} + \beta_{6} + 3 \beta_1) q^{16} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 1) q^{17} + ( - \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - 2) q^{18} + (\beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} - 2 \beta_1) q^{19} + ( - 3 \beta_{3} + \beta_{2} + 1) q^{20} + ( - \beta_{8} + 4 \beta_{7} - \beta_{4} - 2 \beta_{2} - 2) q^{21} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{22} + ( - \beta_{9} - 2 \beta_{7}) q^{23} + ( - \beta_{9} - 2 \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - 2 \beta_{3} + \cdots + 2 \beta_1) q^{24}+ \cdots + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{2} - 8 q^{4} - 2 q^{5} - 10 q^{6} + q^{7} + 18 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{2} - 8 q^{4} - 2 q^{5} - 10 q^{6} + q^{7} + 18 q^{8} - 3 q^{9} + 5 q^{10} - 11 q^{11} - 5 q^{12} + 10 q^{13} + 10 q^{14} - 10 q^{16} + 5 q^{17} - 9 q^{18} - 9 q^{19} + 2 q^{20} + 2 q^{21} + 16 q^{22} - 10 q^{23} - 9 q^{25} - 4 q^{26} + 37 q^{28} - 6 q^{29} + 13 q^{30} + 6 q^{31} - 22 q^{32} - 8 q^{33} - 44 q^{34} - 4 q^{35} + 14 q^{36} - 4 q^{37} + 10 q^{38} - 28 q^{40} + 28 q^{41} + 52 q^{42} + 4 q^{43} + 32 q^{45} - 3 q^{46} - q^{47} - 46 q^{48} - 11 q^{49} + 18 q^{50} + 8 q^{51} - 8 q^{52} - 17 q^{53} - 23 q^{54} - 21 q^{56} - 32 q^{57} + 27 q^{58} - 11 q^{59} + 29 q^{60} + 11 q^{61} - 46 q^{62} + 5 q^{63} + 18 q^{64} - 2 q^{65} - 21 q^{66} - 13 q^{67} + 32 q^{68} + 36 q^{69} + 49 q^{70} + 30 q^{71} + 19 q^{72} + 33 q^{74} + 20 q^{75} + 16 q^{76} - 46 q^{77} - 10 q^{78} - 2 q^{79} - 55 q^{80} + 19 q^{81} - 34 q^{82} + 12 q^{83} - 23 q^{84} - 44 q^{85} - 28 q^{86} + 8 q^{87} + 3 q^{88} + 4 q^{89} - 68 q^{90} + q^{91} + 42 q^{92} - 18 q^{93} - 20 q^{94} + 12 q^{95} + 37 q^{96} - 24 q^{97} - 7 q^{98} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 364 \nu^{9} + 176 \nu^{8} - 220 \nu^{7} - 5913 \nu^{6} + 880 \nu^{5} + 6908 \nu^{4} + 84549 \nu^{3} + 9416 \nu^{2} + 2376 \nu + 30518 ) / 118350 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 983 \nu^{9} - 7328 \nu^{8} + 9160 \nu^{7} - 87336 \nu^{6} - 36640 \nu^{5} - 287624 \nu^{4} - 39747 \nu^{3} - 392048 \nu^{2} - 98928 \nu - 22604 ) / 118350 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + 25194 \nu^{3} - 526654 \nu^{2} - 132894 \nu - 348592 ) / 118350 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} - 129531 \nu^{3} - 162854 \nu^{2} - 41094 \nu - 180842 ) / 59175 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2249 \nu^{9} - 9541 \nu^{8} + 26720 \nu^{7} - 34617 \nu^{6} + 31195 \nu^{5} - 152578 \nu^{4} + 39066 \nu^{3} - 46906 \nu^{2} - 20316 \nu - 3988 ) / 78900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} - 871206 \nu^{3} - 258154 \nu^{2} - 957744 \nu - 2692 ) / 236700 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18139 \nu^{9} - 21776 \nu^{8} + 145570 \nu^{7} + 96813 \nu^{6} + 719570 \nu^{5} + 92092 \nu^{4} + 981276 \nu^{3} + 255184 \nu^{2} + 1362924 \nu + 532 ) / 118350 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1058 \nu^{9} - 1552 \nu^{8} + 9041 \nu^{7} + 3726 \nu^{6} + 39580 \nu^{5} + 2993 \nu^{4} + 53832 \nu^{3} + 11648 \nu^{2} + 50058 \nu - 136 ) / 4734 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} + 2\beta_{3} + 6\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{9} + 2\beta_{8} - 19\beta_{7} + 9\beta_{6} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 22 \beta_{9} + 9 \beta_{8} - 45 \beta_{7} + 23 \beta_{6} - 9 \beta_{5} + 22 \beta_{4} - 23 \beta_{3} - 47 \beta_{2} - 47 \beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -23\beta_{5} + 70\beta_{4} - 78\beta_{3} - 128\beta_{2} + 154 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 206\beta_{9} - 78\beta_{8} + 431\beta_{7} - 221\beta_{6} + 407\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 628 \beta_{9} - 221 \beta_{8} + 1349 \beta_{7} - 691 \beta_{6} + 221 \beta_{5} - 628 \beta_{4} + 691 \beta_{3} + 1187 \beta_{2} + 1187 \beta _1 - 1349 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 691\beta_{5} - 1878\beta_{4} + 2036\beta_{3} + 3634\beta_{2} - 3968 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.862625 − 1.49411i
−0.606661 − 1.05077i
−0.132804 − 0.230024i
0.597828 + 1.03547i
1.50426 + 2.60546i
−0.862625 + 1.49411i
−0.606661 + 1.05077i
−0.132804 + 0.230024i
0.597828 − 1.03547i
1.50426 − 2.60546i
−1.36263 − 2.36014i 0.673208 − 1.16603i −2.71349 + 4.69991i −1.09358 − 1.89414i −3.66932 −2.19729 − 1.47375i 9.33940 0.593582 + 1.02811i −2.98028 + 5.16200i
53.2 −1.10666 − 1.91679i −1.23721 + 2.14292i −1.44940 + 2.51043i 1.06140 + 1.83839i 5.47671 2.63169 + 0.272389i 1.98932 −1.56140 − 2.70442i 2.34921 − 4.06896i
53.3 −0.632804 − 1.09605i 1.31364 − 2.27529i 0.199118 − 0.344882i 1.45130 + 2.51373i −3.32511 −1.29536 + 2.30696i −3.03523 −1.95130 − 3.37975i 1.83678 − 3.18139i
53.4 0.0978281 + 0.169443i 0.129894 − 0.224983i 0.980859 − 1.69890i −1.96625 − 3.40565i 0.0508292 1.12324 + 2.39548i 0.775135 1.46625 + 2.53963i 0.384710 − 0.666337i
53.5 1.00426 + 1.73943i −0.879528 + 1.52339i −1.01709 + 1.76164i −0.452861 − 0.784378i −3.53311 0.237709 − 2.63505i −0.0686323 −0.0471392 − 0.0816475i 0.909582 − 1.57544i
79.1 −1.36263 + 2.36014i 0.673208 + 1.16603i −2.71349 − 4.69991i −1.09358 + 1.89414i −3.66932 −2.19729 + 1.47375i 9.33940 0.593582 − 1.02811i −2.98028 − 5.16200i
79.2 −1.10666 + 1.91679i −1.23721 − 2.14292i −1.44940 − 2.51043i 1.06140 − 1.83839i 5.47671 2.63169 − 0.272389i 1.98932 −1.56140 + 2.70442i 2.34921 + 4.06896i
79.3 −0.632804 + 1.09605i 1.31364 + 2.27529i 0.199118 + 0.344882i 1.45130 − 2.51373i −3.32511 −1.29536 − 2.30696i −3.03523 −1.95130 + 3.37975i 1.83678 + 3.18139i
79.4 0.0978281 − 0.169443i 0.129894 + 0.224983i 0.980859 + 1.69890i −1.96625 + 3.40565i 0.0508292 1.12324 − 2.39548i 0.775135 1.46625 − 2.53963i 0.384710 + 0.666337i
79.5 1.00426 − 1.73943i −0.879528 − 1.52339i −1.01709 − 1.76164i −0.452861 + 0.784378i −3.53311 0.237709 + 2.63505i −0.0686323 −0.0471392 + 0.0816475i 0.909582 + 1.57544i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.e.c ✓ 10
3.b odd 2 1 819.2.j.h 10
4.b odd 2 1 1456.2.r.p 10
7.b odd 2 1 637.2.e.m 10
7.c even 3 1 inner 91.2.e.c ✓ 10
7.c even 3 1 637.2.a.l 5
7.d odd 6 1 637.2.a.k 5
7.d odd 6 1 637.2.e.m 10
13.b even 2 1 1183.2.e.f 10
21.g even 6 1 5733.2.a.bm 5
21.h odd 6 1 819.2.j.h 10
21.h odd 6 1 5733.2.a.bl 5
28.g odd 6 1 1456.2.r.p 10
91.r even 6 1 1183.2.e.f 10
91.r even 6 1 8281.2.a.bw 5
91.s odd 6 1 8281.2.a.bx 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c ✓ 10 1.a even 1 1 trivial
91.2.e.c ✓ 10 7.c even 3 1 inner
637.2.a.k 5 7.d odd 6 1
637.2.a.l 5 7.c even 3 1
637.2.e.m 10 7.b odd 2 1
637.2.e.m 10 7.d odd 6 1
819.2.j.h 10 3.b odd 2 1
819.2.j.h 10 21.h odd 6 1
1183.2.e.f 10 13.b even 2 1
1183.2.e.f 10 91.r even 6 1
1456.2.r.p 10 4.b odd 2 1
1456.2.r.p 10 28.g odd 6 1
5733.2.a.bl 5 21.h odd 6 1
5733.2.a.bm 5 21.g even 6 1
8281.2.a.bw 5 91.r even 6 1
8281.2.a.bx 5 91.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 4T_{2}^{9} + 17T_{2}^{8} + 30T_{2}^{7} + 81T_{2}^{6} + 116T_{2}^{5} + 265T_{2}^{4} + 210T_{2}^{3} + 195T_{2}^{2} - 36T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 4 T^{9} + 17 T^{8} + 30 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{10} + 9 T^{8} + 65 T^{6} - 4 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + 19 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{10} - T^{9} + 6 T^{8} - 17 T^{7} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 11 T^{9} + 85 T^{8} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( (T - 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 5 T^{9} + 47 T^{8} + \cdots + 184041 \) Copy content Toggle raw display
$19$ \( T^{10} + 9 T^{9} + 95 T^{8} + \cdots + 49729 \) Copy content Toggle raw display
$23$ \( T^{10} + 10 T^{9} + 69 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( (T^{5} + 3 T^{4} - 25 T^{3} - 19 T^{2} + \cdots - 108)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} - 6 T^{9} + 97 T^{8} + \cdots + 126736 \) Copy content Toggle raw display
$37$ \( T^{10} + 4 T^{9} + 127 T^{8} + \cdots + 49505296 \) Copy content Toggle raw display
$41$ \( (T^{5} - 14 T^{4} - 28 T^{3} + 940 T^{2} + \cdots - 1584)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 2 T^{4} - 72 T^{3} + 308 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + T^{9} + 125 T^{8} + \cdots + 26718561 \) Copy content Toggle raw display
$53$ \( T^{10} + 17 T^{9} + \cdots + 398361681 \) Copy content Toggle raw display
$59$ \( T^{10} + 11 T^{9} + 85 T^{8} + \cdots + 1089 \) Copy content Toggle raw display
$61$ \( T^{10} - 11 T^{9} + 243 T^{8} + \cdots + 71588521 \) Copy content Toggle raw display
$67$ \( T^{10} + 13 T^{9} + \cdots + 515244601 \) Copy content Toggle raw display
$71$ \( (T^{5} - 15 T^{4} - 25 T^{3} + 853 T^{2} + \cdots - 6336)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 75 T^{8} + 84 T^{7} + \cdots + 506944 \) Copy content Toggle raw display
$79$ \( T^{10} + 2 T^{9} + 141 T^{8} + \cdots + 1000000 \) Copy content Toggle raw display
$83$ \( (T^{5} - 6 T^{4} - 124 T^{3} + 308 T^{2} + \cdots - 7488)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} - 4 T^{9} + 171 T^{8} + \cdots + 59166864 \) Copy content Toggle raw display
$97$ \( (T^{5} + 12 T^{4} - 16 T^{3} - 612 T^{2} + \cdots - 2384)^{2} \) Copy content Toggle raw display
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