Properties

Label 99.2.m.a
Level $99$
Weight $2$
Character orbit 99.m
Analytic conductor $0.791$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.m (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + 1) q^{2} + ( - 2 \zeta_{15}^{6} - \zeta_{15}) q^{3} + ( - 3 \zeta_{15}^{7} - 3 \zeta_{15}) q^{4} + (2 \zeta_{15}^{5} - 2 \zeta_{15} + 2) q^{5} + (2 \zeta_{15}^{7} + \zeta_{15}^{6} - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} + 4 \zeta_{15} - 3) q^{6} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{7} + (4 \zeta_{15}^{7} - 5 \zeta_{15}^{6} + 4 \zeta_{15}^{2} - 4) q^{8} - 3 \zeta_{15}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{15}^{7} + \zeta_{15}^{6} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{3} - 2 \zeta_{15}^{2} + 1) q^{2} + ( - 2 \zeta_{15}^{6} - \zeta_{15}) q^{3} + ( - 3 \zeta_{15}^{7} - 3 \zeta_{15}) q^{4} + (2 \zeta_{15}^{5} - 2 \zeta_{15} + 2) q^{5} + (2 \zeta_{15}^{7} + \zeta_{15}^{6} - 4 \zeta_{15}^{5} + 2 \zeta_{15}^{4} - \zeta_{15}^{3} + 4 \zeta_{15} - 3) q^{6} + (\zeta_{15}^{7} - \zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1) q^{7} + (4 \zeta_{15}^{7} - 5 \zeta_{15}^{6} + 4 \zeta_{15}^{2} - 4) q^{8} - 3 \zeta_{15}^{2} q^{9} + ( - 4 \zeta_{15}^{7} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{2} - 2) q^{10} + ( - 2 \zeta_{15}^{7} + 4 \zeta_{15}^{6} - 2 \zeta_{15}^{4} + 2 \zeta_{15}^{3} - \zeta_{15}^{2} + 2 \zeta_{15} + 2) q^{11} + (3 \zeta_{15}^{7} + 3 \zeta_{15}^{5} - 3 \zeta_{15}^{4} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{2} - 3 \zeta_{15} + 3) q^{12} + ( - 4 \zeta_{15}^{7} + 4 \zeta_{15}^{5} - 4 \zeta_{15}^{4} + 4) q^{13} + (\zeta_{15}^{7} - \zeta_{15}^{5} + 2 \zeta_{15} - 1) q^{14} + (4 \zeta_{15}^{7} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{2} + 2 \zeta_{15}) q^{15} + (5 \zeta_{15}^{7} + 3 \zeta_{15}^{6} - 8 \zeta_{15}^{5} + 5 \zeta_{15}^{4} - 5 \zeta_{15}^{3} + 8 \zeta_{15} - 5) q^{16} + (3 \zeta_{15}^{6} + 3 \zeta_{15}^{3} + 3) q^{17} + ( - 3 \zeta_{15}^{7} + 3 \zeta_{15}^{4} - 3 \zeta_{15}) q^{18} + \zeta_{15}^{6} q^{19} + (6 \zeta_{15}^{7} - 6 \zeta_{15}^{6} - 6 \zeta_{15}^{5} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{3} + 12 \zeta_{15}^{2} + \cdots - 6) q^{20} + \cdots + ( - 6 \zeta_{15}^{7} + 6 \zeta_{15}^{5} - 9 \zeta_{15}^{4} - 12 \zeta_{15} + 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 3 q^{3} - 6 q^{4} + 6 q^{5} - q^{7} - 14 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 3 q^{3} - 6 q^{4} + 6 q^{5} - q^{7} - 14 q^{8} - 3 q^{9} - 32 q^{10} + q^{11} + 18 q^{12} + 8 q^{13} - q^{14} + 12 q^{15} + 14 q^{16} + 12 q^{17} - 3 q^{18} - 2 q^{19} + 24 q^{20} + 11 q^{22} - 14 q^{23} - 39 q^{24} - 9 q^{25} - 16 q^{26} + 18 q^{28} - 9 q^{29} + 18 q^{30} - 3 q^{31} + 30 q^{32} - 6 q^{34} + 12 q^{35} - 36 q^{36} + 24 q^{37} - 4 q^{38} - 24 q^{39} + 12 q^{40} + 3 q^{41} - 12 q^{42} + 6 q^{43} - 48 q^{44} - 24 q^{45} + 18 q^{46} + 23 q^{47} - 6 q^{49} - 24 q^{50} + 27 q^{51} + 12 q^{52} + 28 q^{53} + 54 q^{54} - 32 q^{55} - 12 q^{56} + 3 q^{57} + 6 q^{58} + 3 q^{59} + 6 q^{62} + 3 q^{63} - 34 q^{64} - 32 q^{65} + 12 q^{66} - 24 q^{67} - 9 q^{68} - 18 q^{69} - 6 q^{70} + 42 q^{71} + 39 q^{72} - 8 q^{73} + 13 q^{74} - 6 q^{76} - 11 q^{77} + 24 q^{78} - 22 q^{79} + 52 q^{80} + 9 q^{81} - 96 q^{82} - 17 q^{83} + 18 q^{84} - 6 q^{85} + 21 q^{86} + 37 q^{88} - 72 q^{89} + 42 q^{90} - 24 q^{91} - 21 q^{92} + 9 q^{93} + 28 q^{94} - 4 q^{95} + 6 q^{97} + 72 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\) \(-1 - \zeta_{15}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.669131 + 0.743145i
−0.104528 + 0.994522i
0.669131 0.743145i
−0.104528 0.994522i
0.913545 0.406737i
−0.978148 0.207912i
−0.978148 + 0.207912i
0.913545 + 0.406737i
0.273659 2.60369i −1.28716 + 1.15897i −4.74803 1.00922i −0.338261 3.21834i 2.66535 + 3.66854i −0.669131 + 0.743145i −2.30902 + 7.10642i 0.313585 2.98357i −8.47214
16.1 0.373619 + 0.0794152i 1.72256 + 0.181049i −1.69381 0.754131i 1.20906 0.256993i 0.629204 + 0.204441i 0.104528 + 0.994522i −1.19098 0.865300i 2.93444 + 0.623735i 0.472136
25.1 0.273659 + 2.60369i −1.28716 1.15897i −4.74803 + 1.00922i −0.338261 + 3.21834i 2.66535 3.66854i −0.669131 0.743145i −2.30902 7.10642i 0.313585 + 2.98357i −8.47214
31.1 0.373619 0.0794152i 1.72256 0.181049i −1.69381 + 0.754131i 1.20906 + 0.256993i 0.629204 0.204441i 0.104528 0.994522i −1.19098 + 0.865300i 2.93444 0.623735i 0.472136
49.1 −0.255585 + 0.283856i 0.704489 + 1.58231i 0.193806 + 1.84395i −0.827091 0.918578i −0.629204 0.204441i −0.913545 0.406737i −1.19098 0.865300i −2.00739 + 2.22943i 0.472136
58.1 −2.39169 1.06485i 0.360114 1.69420i 3.24803 + 3.60730i 2.95630 1.31623i −2.66535 + 3.66854i 0.978148 0.207912i −2.30902 7.10642i −2.74064 1.22021i −8.47214
70.1 −2.39169 + 1.06485i 0.360114 + 1.69420i 3.24803 3.60730i 2.95630 + 1.31623i −2.66535 3.66854i 0.978148 + 0.207912i −2.30902 + 7.10642i −2.74064 + 1.22021i −8.47214
97.1 −0.255585 0.283856i 0.704489 1.58231i 0.193806 1.84395i −0.827091 + 0.918578i −0.629204 + 0.204441i −0.913545 + 0.406737i −1.19098 + 0.865300i −2.00739 2.22943i 0.472136
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.m.a 8
3.b odd 2 1 297.2.n.a 8
9.c even 3 1 inner 99.2.m.a 8
9.c even 3 1 891.2.f.b 4
9.d odd 6 1 297.2.n.a 8
9.d odd 6 1 891.2.f.a 4
11.c even 5 1 inner 99.2.m.a 8
11.c even 5 1 1089.2.e.g 4
11.d odd 10 1 1089.2.e.d 4
33.h odd 10 1 297.2.n.a 8
99.m even 15 1 inner 99.2.m.a 8
99.m even 15 1 891.2.f.b 4
99.m even 15 1 1089.2.e.g 4
99.m even 15 1 9801.2.a.n 2
99.n odd 30 1 297.2.n.a 8
99.n odd 30 1 891.2.f.a 4
99.n odd 30 1 9801.2.a.bb 2
99.o odd 30 1 1089.2.e.d 4
99.o odd 30 1 9801.2.a.bc 2
99.p even 30 1 9801.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.m.a 8 1.a even 1 1 trivial
99.2.m.a 8 9.c even 3 1 inner
99.2.m.a 8 11.c even 5 1 inner
99.2.m.a 8 99.m even 15 1 inner
297.2.n.a 8 3.b odd 2 1
297.2.n.a 8 9.d odd 6 1
297.2.n.a 8 33.h odd 10 1
297.2.n.a 8 99.n odd 30 1
891.2.f.a 4 9.d odd 6 1
891.2.f.a 4 99.n odd 30 1
891.2.f.b 4 9.c even 3 1
891.2.f.b 4 99.m even 15 1
1089.2.e.d 4 11.d odd 10 1
1089.2.e.d 4 99.o odd 30 1
1089.2.e.g 4 11.c even 5 1
1089.2.e.g 4 99.m even 15 1
9801.2.a.m 2 99.p even 30 1
9801.2.a.n 2 99.m even 15 1
9801.2.a.bb 2 99.n odd 30 1
9801.2.a.bc 2 99.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{7} + 10T_{2}^{6} + 26T_{2}^{5} + 39T_{2}^{4} - 14T_{2}^{3} - 5T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + 10 T^{6} + 26 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + 6 T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + 20 T^{6} - 64 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} - T^{5} - T^{4} - T^{3} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{8} - T^{7} - 20 T^{6} + T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} - 128 T^{5} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( (T^{4} - 6 T^{3} + 36 T^{2} - 81 T + 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 7 T^{3} + 38 T^{2} + 77 T + 121)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 9 T^{7} + 45 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + 5 T^{6} + 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( (T^{4} - 12 T^{3} + 94 T^{2} - 403 T + 961)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 3 T^{7} - 100 T^{6} + \cdots + 707281 \) Copy content Toggle raw display
$43$ \( (T^{4} - 3 T^{3} + 18 T^{2} + 27 T + 81)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 23 T^{7} + 280 T^{6} + \cdots + 25411681 \) Copy content Toggle raw display
$53$ \( (T^{4} - 14 T^{3} + 96 T^{2} - 319 T + 841)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 3 T^{7} + 5 T^{6} - 8 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{8} - 90 T^{6} + 1350 T^{5} + \cdots + 4100625 \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T + 36)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 21 T^{3} + 166 T^{2} + 164 T + 1681)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} + 46 T^{2} - 11 T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 22 T^{7} + \cdots + 104060401 \) Copy content Toggle raw display
$83$ \( T^{8} + 17 T^{7} + \cdots + 373301041 \) Copy content Toggle raw display
$89$ \( (T^{2} + 18 T + 76)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 6 T^{7} + 216 T^{5} + \cdots + 1679616 \) Copy content Toggle raw display
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