Properties

Label 46.2.c.b
Level $46$
Weight $2$
Character orbit 46.c
Analytic conductor $0.367$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 46 = 2 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 46.c (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.367311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/46\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-\zeta_{22}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.142315 + 0.989821i
0.959493 0.281733i
0.654861 + 0.755750i
−0.841254 0.540641i
−0.415415 0.909632i
−0.415415 + 0.909632i
0.142315 0.989821i
−0.841254 + 0.540641i
0.654861 0.755750i
0.959493 + 0.281733i
0.142315 + 0.989821i 0.817178 1.78937i −0.959493 + 0.281733i −0.357685 + 0.412791i 1.88745 + 0.554206i −3.96028 + 2.54512i −0.415415 0.909632i −0.569485 0.657220i −0.459493 0.295298i
9.1 0.959493 0.281733i −1.80075 2.07817i 0.841254 0.540641i 0.459493 + 3.19584i −2.31329 1.48666i −0.497033 + 1.08835i 0.654861 0.755750i −0.649167 + 4.51506i 1.34125 + 2.93694i
13.1 0.654861 + 0.755750i −0.512546 0.329393i −0.142315 + 0.989821i 0.154861 0.339098i −0.0867074 0.603063i −1.97611 0.580239i −0.841254 + 0.540641i −1.09204 2.39124i 0.357685 0.105026i
25.1 −0.841254 0.540641i 0.425839 + 2.96177i 0.415415 + 0.909632i −1.34125 0.393828i 1.24302 2.72183i 2.81051 3.24350i 0.142315 0.989821i −5.71228 + 1.67728i 0.915415 + 1.05645i
27.1 −0.415415 0.909632i 1.07028 0.314261i −0.654861 + 0.755750i −0.915415 0.588302i −0.730471 0.843008i 0.122916 + 0.854902i 0.959493 + 0.281733i −1.47703 + 0.949230i −0.154861 + 1.07708i
29.1 −0.415415 + 0.909632i 1.07028 + 0.314261i −0.654861 0.755750i −0.915415 + 0.588302i −0.730471 + 0.843008i 0.122916 0.854902i 0.959493 0.281733i −1.47703 0.949230i −0.154861 1.07708i
31.1 0.142315 0.989821i 0.817178 + 1.78937i −0.959493 0.281733i −0.357685 0.412791i 1.88745 0.554206i −3.96028 2.54512i −0.415415 + 0.909632i −0.569485 + 0.657220i −0.459493 + 0.295298i
35.1 −0.841254 + 0.540641i 0.425839 2.96177i 0.415415 0.909632i −1.34125 + 0.393828i 1.24302 + 2.72183i 2.81051 + 3.24350i 0.142315 + 0.989821i −5.71228 1.67728i 0.915415 1.05645i
39.1 0.654861 0.755750i −0.512546 + 0.329393i −0.142315 0.989821i 0.154861 + 0.339098i −0.0867074 + 0.603063i −1.97611 + 0.580239i −0.841254 0.540641i −1.09204 + 2.39124i 0.357685 + 0.105026i
41.1 0.959493 + 0.281733i −1.80075 + 2.07817i 0.841254 + 0.540641i 0.459493 3.19584i −2.31329 + 1.48666i −0.497033 1.08835i 0.654861 + 0.755750i −0.649167 4.51506i 1.34125 2.93694i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 46.2.c.b 10
3.b odd 2 1 414.2.i.c 10
4.b odd 2 1 368.2.m.a 10
23.c even 11 1 inner 46.2.c.b 10
23.c even 11 1 1058.2.a.j 5
23.d odd 22 1 1058.2.a.k 5
69.g even 22 1 9522.2.a.bw 5
69.h odd 22 1 414.2.i.c 10
69.h odd 22 1 9522.2.a.bz 5
92.g odd 22 1 368.2.m.a 10
92.g odd 22 1 8464.2.a.bu 5
92.h even 22 1 8464.2.a.bv 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.c.b 10 1.a even 1 1 trivial
46.2.c.b 10 23.c even 11 1 inner
368.2.m.a 10 4.b odd 2 1
368.2.m.a 10 92.g odd 22 1
414.2.i.c 10 3.b odd 2 1
414.2.i.c 10 69.h odd 22 1
1058.2.a.j 5 23.c even 11 1
1058.2.a.k 5 23.d odd 22 1
8464.2.a.bu 5 92.g odd 22 1
8464.2.a.bv 5 92.h even 22 1
9522.2.a.bw 5 69.g even 22 1
9522.2.a.bz 5 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 11T_{3}^{8} + 55T_{3}^{6} - 99T_{3}^{5} + 242T_{3}^{4} - 242T_{3}^{3} - 121T_{3}^{2} + 121T_{3} + 121 \) acting on \(S_{2}^{\mathrm{new}}(46, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} + 11 T^{8} + 55 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + 16 T^{8} + 53 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} + 7 T^{9} + 16 T^{8} + \cdots + 1849 \) Copy content Toggle raw display
$11$ \( T^{10} + 2 T^{9} + 26 T^{8} + 74 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} + 48 T^{8} + \cdots + 436921 \) Copy content Toggle raw display
$17$ \( T^{10} + 9 T^{9} + 81 T^{8} + 454 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + 15 T^{8} + \cdots + 139129 \) Copy content Toggle raw display
$23$ \( T^{10} - 21 T^{9} + 210 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} + 2 T^{9} + 48 T^{8} - 25 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$31$ \( T^{10} - 11 T^{9} + 110 T^{8} + \cdots + 2076481 \) Copy content Toggle raw display
$37$ \( T^{10} + 18 T^{9} + 225 T^{8} + \cdots + 14645929 \) Copy content Toggle raw display
$41$ \( T^{10} - 5 T^{9} + 25 T^{8} + \cdots + 1985281 \) Copy content Toggle raw display
$43$ \( T^{10} + 21 T^{9} + 232 T^{8} + \cdots + 53333809 \) Copy content Toggle raw display
$47$ \( (T^{5} + 11 T^{4} - 66 T^{3} - 726 T^{2} + \cdots + 3883)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 7 T^{9} + 16 T^{8} + \cdots + 1849 \) Copy content Toggle raw display
$59$ \( T^{10} - 43 T^{9} + 859 T^{8} + \cdots + 8300161 \) Copy content Toggle raw display
$61$ \( T^{10} + 3 T^{9} + 53 T^{8} + 104 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$67$ \( T^{10} + T^{9} - 54 T^{8} + \cdots + 157609 \) Copy content Toggle raw display
$71$ \( T^{10} + 11 T^{9} + 110 T^{8} + \cdots + 2076481 \) Copy content Toggle raw display
$73$ \( T^{10} + 28 T^{9} + 366 T^{8} + \cdots + 34774609 \) Copy content Toggle raw display
$79$ \( T^{10} - 34 T^{9} + \cdots + 118613881 \) Copy content Toggle raw display
$83$ \( T^{10} + 3 T^{9} - 277 T^{8} + \cdots + 923126689 \) Copy content Toggle raw display
$89$ \( T^{10} + 49 T^{9} + 1191 T^{8} + \cdots + 380689 \) Copy content Toggle raw display
$97$ \( T^{10} - 16 T^{9} + \cdots + 788093329 \) Copy content Toggle raw display
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