Properties

Label 95.2.e.b
Level $95$
Weight $2$
Character orbit 95.e
Analytic conductor $0.759$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 95.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.758578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.3518667.1
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 3) q^{4} - \beta_{3} q^{5} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{6} + ( - \beta_{4} + 1) q^{7} + ( - \beta_{4} + \beta_{2} - 2) q^{8} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{3} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 3) q^{4} - \beta_{3} q^{5} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{6} + ( - \beta_{4} + 1) q^{7} + ( - \beta_{4} + \beta_{2} - 2) q^{8} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{9} + (\beta_{2} - \beta_1) q^{10} + (\beta_{4} - 2) q^{11} + (\beta_{4} - 2 \beta_{2} - 1) q^{12} + (5 \beta_{3} + 5) q^{13} + (2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{14} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{15} + (\beta_{5} + \beta_{3} + \beta_1) q^{16} + (2 \beta_{5} + \beta_1) q^{17} + ( - \beta_{4} + 5) q^{18} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{4} + \beta_{2} - 3) q^{20} + ( - \beta_{5} - 3 \beta_{3} + 2 \beta_1) q^{21} + ( - 2 \beta_{5} - \beta_{3} + 3 \beta_1) q^{22} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{23} - 7 \beta_{3} q^{24} + ( - \beta_{3} - 1) q^{25} - 5 \beta_{2} q^{26} + (\beta_{2} - 3) q^{27} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1 - 4) q^{28} + (\beta_{5} - \beta_{4} - \beta_1 + 1) q^{29} + ( - 2 \beta_{4} - \beta_{2} + 3) q^{30} - \beta_{2} q^{31} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 3) q^{32} + (2 \beta_{5} + 2 \beta_{3} - 3 \beta_1) q^{33} + ( - \beta_{5} + \beta_{4} + 10 \beta_{3} - 3 \beta_{2} + 4 \beta_1 + 9) q^{34} + ( - \beta_{5} + \beta_1) q^{35} + (2 \beta_{5} - \beta_{3} - 4 \beta_1) q^{36} + (5 \beta_{4} + 2 \beta_{2} - 3) q^{37} + ( - 3 \beta_{5} - 5 \beta_{3} + 2 \beta_{2} + \beta_1 - 7) q^{38} - 5 \beta_{4} q^{39} + ( - \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{40} + (\beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{41} + (3 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 8) q^{42} + (\beta_{5} + 2 \beta_1) q^{43} + (3 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 3 \beta_1 + 7) q^{44} + ( - \beta_{2} - 1) q^{45} + ( - \beta_{4} + 4 \beta_{2} - 9) q^{46} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{47} + ( - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{2} - \beta_1 - 2) q^{48} + ( - 2 \beta_{4} + \beta_{2} - 2) q^{49} + \beta_{2} q^{50} + ( - 4 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + 5 \beta_{2} - \beta_1 - 1) q^{51} + ( - 5 \beta_{5} - 10 \beta_{3}) q^{52} + (4 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{53} + (\beta_{5} + 4 \beta_{3} + 3 \beta_1) q^{54} + (\beta_{5} + \beta_{3} - \beta_1) q^{55} + ( - \beta_{4} + \beta_{2} + 5) q^{56} + (2 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 2) q^{57} + (2 \beta_{4} - 3) q^{58} + ( - \beta_{5} + 2 \beta_{3} - \beta_1) q^{59} + (\beta_{5} - 3 \beta_1) q^{60} + (2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 5) q^{61} + ( - \beta_{5} - 4 \beta_{3}) q^{62} + ( - 3 \beta_{5} + 3 \beta_{4} - \beta_{3} + 3 \beta_1 - 4) q^{63} + (5 \beta_{4} + 2 \beta_{2} - 5) q^{64} + 5 q^{65} + ( - 5 \beta_{5} + 5 \beta_{4} - 6 \beta_{3} - \beta_{2} + 6 \beta_1 - 11) q^{66} + (2 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} + 8) q^{67} + ( - \beta_{4} - 7 \beta_{2} + 14) q^{68} + ( - 3 \beta_{4} - 5 \beta_{2} + 1) q^{69} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_1 + 3) q^{70} + ( - 3 \beta_{5} - 8 \beta_{3} + 2 \beta_1) q^{71} + ( - 4 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{72} + ( - 2 \beta_{5} - 7 \beta_{3} - \beta_1) q^{73} + ( - 8 \beta_{5} + 3 \beta_{3} + 8 \beta_1) q^{74} + \beta_{4} q^{75} + (2 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 1) q^{76} + (3 \beta_{4} - \beta_{2} - 6) q^{77} + (10 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{78} + (2 \beta_{5} - 8 \beta_{3} + 2 \beta_1) q^{79} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{80} + (\beta_{5} - 7 \beta_{3} + \beta_1) q^{81} + ( - 4 \beta_{5} + 4 \beta_{4} - 9 \beta_{3} + 4 \beta_{2} - 13) q^{82} + (3 \beta_{2} - 2) q^{83} + (6 \beta_{4} - \beta_{2} - 11) q^{84} + (2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} + \beta_1 + 2) q^{85} + (\beta_{5} - \beta_{4} + 11 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 12) q^{86} + ( - \beta_{4} + \beta_{2} + 4) q^{87} + (2 \beta_{4} - 2 \beta_{2} - 3) q^{88} + ( - 4 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + 4 \beta_{2} + 2) q^{89} + ( - \beta_{5} - 4 \beta_{3} + \beta_1) q^{90} + (5 \beta_{5} - 5 \beta_{4} - 5 \beta_1 + 5) q^{91} + (2 \beta_{5} + 15 \beta_{3} + 6 \beta_1) q^{92} + (2 \beta_{5} + \beta_{3} - \beta_1) q^{93} + 7 q^{94} + (\beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 1) q^{95} + ( - \beta_{4} + 3 \beta_{2} + 5) q^{96} + ( - 3 \beta_{5} + 3 \beta_{3} - \beta_1) q^{97} + (5 \beta_{5} + 6 \beta_{3}) q^{98} + (3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 4 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} - q^{3} - 7 q^{4} + 3 q^{5} + 6 q^{6} + 4 q^{7} - 12 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} - q^{3} - 7 q^{4} + 3 q^{5} + 6 q^{6} + 4 q^{7} - 12 q^{8} - 4 q^{9} + q^{10} - 10 q^{11} - 8 q^{12} + 15 q^{13} - 7 q^{14} + q^{15} - 3 q^{16} - q^{17} + 28 q^{18} - 14 q^{20} + 12 q^{21} + 8 q^{22} - 4 q^{23} + 21 q^{24} - 3 q^{25} - 10 q^{26} - 16 q^{27} - 11 q^{28} + 2 q^{29} + 12 q^{30} - 2 q^{31} + 6 q^{32} - 11 q^{33} + 25 q^{34} + 2 q^{35} - 3 q^{36} - 4 q^{37} - 19 q^{38} - 10 q^{39} - 6 q^{40} + 2 q^{41} + 23 q^{42} + q^{43} + 18 q^{44} - 8 q^{45} - 48 q^{46} - 6 q^{47} - q^{48} - 14 q^{49} + 2 q^{50} + 6 q^{51} + 35 q^{52} - 11 q^{53} - 10 q^{54} - 5 q^{55} + 30 q^{56} - 19 q^{57} - 14 q^{58} - 6 q^{59} - 4 q^{60} + 9 q^{61} + 13 q^{62} - 9 q^{63} - 16 q^{64} + 30 q^{65} - 29 q^{66} + 20 q^{67} + 68 q^{68} - 10 q^{69} + 7 q^{70} + 29 q^{71} - 11 q^{72} + 22 q^{73} + 7 q^{74} + 2 q^{75} - 19 q^{76} - 32 q^{77} - 30 q^{78} + 24 q^{79} + 3 q^{80} + 21 q^{81} - 31 q^{82} - 6 q^{83} - 56 q^{84} + q^{85} + 32 q^{86} + 24 q^{87} - 18 q^{88} + 14 q^{89} + 14 q^{90} + 10 q^{91} - 41 q^{92} - 6 q^{93} + 42 q^{94} + 34 q^{96} - 7 q^{97} - 23 q^{98} + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 5\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 21\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
1.14257 1.97899i
0.610938 1.05818i
−1.25351 + 2.17114i
1.14257 + 1.97899i
0.610938 + 1.05818i
−1.25351 2.17114i
−1.14257 + 1.97899i 1.25351 2.17114i −1.61094 2.79023i 0.500000 0.866025i 2.86445 + 4.96137i 3.50702 2.79216 −1.64257 2.84502i 1.14257 + 1.97899i
11.2 −0.610938 + 1.05818i −1.14257 + 1.97899i 0.253509 + 0.439091i 0.500000 0.866025i −1.39608 2.41808i −1.28514 −3.06327 −1.11094 1.92420i 0.610938 + 1.05818i
11.3 1.25351 2.17114i −0.610938 + 1.05818i −2.14257 3.71104i 0.500000 0.866025i 1.53163 + 2.65287i −0.221876 −5.72889 0.753509 + 1.30512i −1.25351 2.17114i
26.1 −1.14257 1.97899i 1.25351 + 2.17114i −1.61094 + 2.79023i 0.500000 + 0.866025i 2.86445 4.96137i 3.50702 2.79216 −1.64257 + 2.84502i 1.14257 1.97899i
26.2 −0.610938 1.05818i −1.14257 1.97899i 0.253509 0.439091i 0.500000 + 0.866025i −1.39608 + 2.41808i −1.28514 −3.06327 −1.11094 + 1.92420i 0.610938 1.05818i
26.3 1.25351 + 2.17114i −0.610938 1.05818i −2.14257 + 3.71104i 0.500000 + 0.866025i 1.53163 2.65287i −0.221876 −5.72889 0.753509 1.30512i −1.25351 + 2.17114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.2.e.b 6
3.b odd 2 1 855.2.k.g 6
4.b odd 2 1 1520.2.q.j 6
5.b even 2 1 475.2.e.d 6
5.c odd 4 2 475.2.j.b 12
19.c even 3 1 inner 95.2.e.b 6
19.c even 3 1 1805.2.a.h 3
19.d odd 6 1 1805.2.a.g 3
57.h odd 6 1 855.2.k.g 6
76.g odd 6 1 1520.2.q.j 6
95.h odd 6 1 9025.2.a.ba 3
95.i even 6 1 475.2.e.d 6
95.i even 6 1 9025.2.a.z 3
95.m odd 12 2 475.2.j.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.b 6 1.a even 1 1 trivial
95.2.e.b 6 19.c even 3 1 inner
475.2.e.d 6 5.b even 2 1
475.2.e.d 6 95.i even 6 1
475.2.j.b 12 5.c odd 4 2
475.2.j.b 12 95.m odd 12 2
855.2.k.g 6 3.b odd 2 1
855.2.k.g 6 57.h odd 6 1
1520.2.q.j 6 4.b odd 2 1
1520.2.q.j 6 76.g odd 6 1
1805.2.a.g 3 19.d odd 6 1
1805.2.a.h 3 19.c even 3 1
9025.2.a.z 3 95.i even 6 1
9025.2.a.ba 3 95.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{5} + 7T_{2}^{4} + 8T_{2}^{3} + 43T_{2}^{2} + 42T_{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(95, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + 7 T^{4} + 8 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + 7 T^{4} + 8 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 5 T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 5 T^{2} + 2 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 5 T + 25)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + T^{5} + 45 T^{4} - 30 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$19$ \( T^{6} + 133T^{3} + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 4 T^{5} + 55 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( T^{6} - 2 T^{5} + 9 T^{4} + 12 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{3} + T^{2} - 6 T - 7)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 2 T^{2} - 119 T - 227)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 2 T^{5} + 47 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$43$ \( T^{6} - T^{5} + 45 T^{4} - 198 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( T^{6} + 6 T^{5} + 43 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$53$ \( T^{6} + 11 T^{5} + 163 T^{4} + \cdots + 96721 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + 43 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$61$ \( T^{6} - 9 T^{5} + 130 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$67$ \( T^{6} - 20 T^{5} + 292 T^{4} + \cdots + 7744 \) Copy content Toggle raw display
$71$ \( T^{6} - 29 T^{5} + 605 T^{4} + \cdots + 218089 \) Copy content Toggle raw display
$73$ \( T^{6} - 22 T^{5} + 367 T^{4} + \cdots + 5929 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + 460 T^{4} + \cdots + 61504 \) Copy content Toggle raw display
$83$ \( (T^{3} + 3 T^{2} - 54 T + 77)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 14 T^{5} + 232 T^{4} + \cdots + 3136 \) Copy content Toggle raw display
$97$ \( T^{6} + 7 T^{5} + 115 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
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