Properties

Label 46.2.c.a
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.

\(f(q)\) \(=\) \( q - \zeta_{22} q^{2} + ( - \zeta_{22}^{9} - \zeta_{22}^{7} + \zeta_{22}^{4} - \zeta_{22}) q^{3} + \zeta_{22}^{2} q^{4} + ( - \zeta_{22}^{9} - \zeta_{22}^{7} + 2 \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22}^{3}) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22} - 1) q^{6} + (\zeta_{22}^{7} - \zeta_{22}^{6} + \zeta_{22}^{4} - \zeta_{22}^{2} + \zeta_{22}) q^{7} - \zeta_{22}^{3} q^{8} + (2 \zeta_{22}^{9} + \zeta_{22}^{8} + \zeta_{22}^{7} - 2 \zeta_{22}^{6} + \zeta_{22}^{5} - 2 \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}^{2} + 2 \zeta_{22}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22} q^{2} + ( - \zeta_{22}^{9} - \zeta_{22}^{7} + \zeta_{22}^{4} - \zeta_{22}) q^{3} + \zeta_{22}^{2} q^{4} + ( - \zeta_{22}^{9} - \zeta_{22}^{7} + 2 \zeta_{22}^{6} - \zeta_{22}^{5} - \zeta_{22}^{3}) q^{5} + (\zeta_{22}^{9} + \zeta_{22}^{7} - \zeta_{22}^{6} - \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22} - 1) q^{6} + (\zeta_{22}^{7} - \zeta_{22}^{6} + \zeta_{22}^{4} - \zeta_{22}^{2} + \zeta_{22}) q^{7} - \zeta_{22}^{3} q^{8} + (2 \zeta_{22}^{9} + \zeta_{22}^{8} + \zeta_{22}^{7} - 2 \zeta_{22}^{6} + \zeta_{22}^{5} - 2 \zeta_{22}^{4} + \zeta_{22}^{3} + \zeta_{22}^{2} + 2 \zeta_{22}) q^{9} + (\zeta_{22}^{9} - \zeta_{22}^{7} + \zeta_{22}^{5} + \zeta_{22}^{3} - \zeta_{22}^{2} + \zeta_{22} - 1) q^{10} + (2 \zeta_{22}^{9} - \zeta_{22}^{8} + 2 \zeta_{22}^{7} + \zeta_{22}^{6} + \zeta_{22}^{5} - \zeta_{22}^{4} - \zeta_{22}^{3} - 2 \zeta_{22}^{2} + \cdots - 2) q^{11} + \cdots + (5 \zeta_{22}^{8} - \zeta_{22}^{7} + 4 \zeta_{22}^{6} - 9 \zeta_{22}^{5} - 3 \zeta_{22}^{4} - 9 \zeta_{22}^{3} + 4 \zeta_{22}^{2} + \cdots + 5) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} - 4 q^{6} + 3 q^{7} - q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} - 4 q^{6} + 3 q^{7} - q^{8} + 9 q^{9} - 6 q^{10} - 12 q^{11} + 7 q^{12} - 14 q^{13} + 3 q^{14} - 13 q^{15} - q^{16} + 15 q^{17} + 9 q^{18} + 2 q^{19} + 5 q^{20} + q^{21} + 10 q^{22} - q^{23} - 4 q^{24} + 13 q^{25} - 3 q^{26} + 26 q^{27} + 3 q^{28} - 8 q^{29} + 20 q^{30} - 21 q^{31} - q^{32} - 15 q^{33} - 7 q^{34} + 7 q^{35} - 13 q^{36} + 28 q^{37} - 9 q^{38} - 12 q^{39} - 6 q^{40} - 31 q^{41} - 10 q^{42} + 11 q^{43} - 12 q^{44} + 10 q^{45} - 12 q^{46} + 18 q^{47} + 7 q^{48} - 24 q^{49} + 2 q^{50} - 17 q^{51} + 8 q^{52} - 21 q^{53} - 29 q^{54} + 5 q^{55} + 3 q^{56} + 19 q^{57} - 8 q^{58} - 5 q^{59} - 13 q^{60} + 37 q^{61} + q^{62} + 17 q^{63} - q^{64} + 37 q^{65} + 29 q^{66} - 13 q^{67} + 26 q^{68} - 15 q^{69} + 18 q^{70} + 49 q^{71} + 20 q^{72} - 8 q^{73} + 28 q^{74} + 8 q^{75} - 20 q^{76} - 8 q^{77} + 10 q^{78} + 8 q^{79} + 5 q^{80} - 11 q^{81} + 2 q^{82} - 7 q^{83} + q^{84} - 42 q^{85} + 22 q^{86} - 21 q^{87} - q^{88} - 13 q^{89} - 45 q^{90} - 24 q^{91} - 23 q^{92} - 40 q^{93} - 37 q^{94} - 10 q^{95} - 4 q^{96} - 32 q^{97} - 24 q^{98} + 20 q^{99}+O(q^{100}) \) 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 0.580699 1.27155i −0.959493 + 0.281733i −1.66741 + 1.92429i −1.34125 0.393828i 1.75667 1.12894i 0.415415 + 0.909632i 0.684944 + 0.790468i 2.14200 + 1.37658i
9.1 −0.959493 + 0.281733i 0.712591 + 0.822373i 0.841254 0.540641i 0.174863 + 1.21620i −0.915415 0.588302i 0.260554 0.570534i −0.654861 + 0.755750i 0.258432 1.79743i −0.510424 1.11767i
13.1 −0.654861 0.755750i −2.71616 1.74557i −0.142315 + 0.989821i 0.985691 2.15836i 0.459493 + 3.19584i 0.381761 + 0.112095i 0.841254 0.540641i 3.08427 + 6.75361i −2.27667 + 0.668491i
25.1 0.841254 + 0.540641i −0.0530529 0.368991i 0.415415 + 0.909632i −3.26024 0.957293i 0.154861 0.339098i −0.297176 + 0.342959i −0.142315 + 0.989821i 2.74514 0.806046i −2.22514 2.56794i
27.1 0.415415 + 0.909632i −0.524075 + 0.153882i −0.654861 + 0.755750i 0.767092 + 0.492980i −0.357685 0.412791i −0.601808 4.18567i −0.959493 0.281733i −2.27279 + 1.46063i −0.129769 + 0.902563i
29.1 0.415415 0.909632i −0.524075 0.153882i −0.654861 0.755750i 0.767092 0.492980i −0.357685 + 0.412791i −0.601808 + 4.18567i −0.959493 + 0.281733i −2.27279 1.46063i −0.129769 0.902563i
31.1 −0.142315 + 0.989821i 0.580699 + 1.27155i −0.959493 0.281733i −1.66741 1.92429i −1.34125 + 0.393828i 1.75667 + 1.12894i 0.415415 0.909632i 0.684944 0.790468i 2.14200 1.37658i
35.1 0.841254 0.540641i −0.0530529 + 0.368991i 0.415415 0.909632i −3.26024 + 0.957293i 0.154861 + 0.339098i −0.297176 0.342959i −0.142315 0.989821i 2.74514 + 0.806046i −2.22514 + 2.56794i
39.1 −0.654861 + 0.755750i −2.71616 + 1.74557i −0.142315 0.989821i 0.985691 + 2.15836i 0.459493 3.19584i 0.381761 0.112095i 0.841254 + 0.540641i 3.08427 6.75361i −2.27667 0.668491i
41.1 −0.959493 0.281733i 0.712591 0.822373i 0.841254 + 0.540641i 0.174863 1.21620i −0.915415 + 0.588302i 0.260554 + 0.570534i −0.654861 0.755750i 0.258432 + 1.79743i −0.510424 + 1.11767i
\(n\): 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.a 10
3.b odd 2 1 414.2.i.f 10
4.b odd 2 1 368.2.m.b 10
23.c even 11 1 inner 46.2.c.a 10
23.c even 11 1 1058.2.a.m 5
23.d odd 22 1 1058.2.a.l 5
69.g even 22 1 9522.2.a.bu 5
69.h odd 22 1 414.2.i.f 10
69.h odd 22 1 9522.2.a.bp 5
92.g odd 22 1 368.2.m.b 10
92.g odd 22 1 8464.2.a.bx 5
92.h even 22 1 8464.2.a.bw 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.c.a 10 1.a even 1 1 trivial
46.2.c.a 10 23.c even 11 1 inner
368.2.m.b 10 4.b odd 2 1
368.2.m.b 10 92.g odd 22 1
414.2.i.f 10 3.b odd 2 1
414.2.i.f 10 69.h odd 22 1
1058.2.a.l 5 23.d odd 22 1
1058.2.a.m 5 23.c even 11 1
8464.2.a.bw 5 92.h even 22 1
8464.2.a.bx 5 92.g odd 22 1
9522.2.a.bp 5 69.h odd 22 1
9522.2.a.bu 5 69.g even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 4T_{3}^{9} + 5T_{3}^{8} - 2T_{3}^{7} + 25T_{3}^{6} + T_{3}^{5} + 4T_{3}^{4} + 16T_{3}^{3} + 9T_{3}^{2} + 3T_{3} + 1 \) 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
$3$ \( T^{10} + 4 T^{9} + 5 T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + 6 T^{9} + 14 T^{8} + 29 T^{7} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{10} - 3 T^{9} + 20 T^{8} - 71 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} + 12 T^{9} + 56 T^{8} + \cdots + 109561 \) Copy content Toggle raw display
$13$ \( T^{10} + 14 T^{9} + 86 T^{8} + 258 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} - 15 T^{9} + 137 T^{8} + \cdots + 214369 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + 37 T^{8} + \cdots + 4489 \) Copy content Toggle raw display
$23$ \( T^{10} + T^{9} + 78 T^{8} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} - 2 T^{8} + 259 T^{7} + \cdots + 4489 \) Copy content Toggle raw display
$31$ \( T^{10} + 21 T^{9} + 232 T^{8} + \cdots + 20529961 \) Copy content Toggle raw display
$37$ \( T^{10} - 28 T^{9} + 509 T^{8} + \cdots + 49857721 \) Copy content Toggle raw display
$41$ \( T^{10} + 31 T^{9} + \cdots + 172475689 \) Copy content Toggle raw display
$43$ \( T^{10} - 11 T^{9} + 66 T^{8} + \cdots + 7027801 \) Copy content Toggle raw display
$47$ \( (T^{5} - 9 T^{4} - 82 T^{3} + 922 T^{2} + \cdots - 529)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + 21 T^{9} + 144 T^{8} + \cdots + 31236921 \) Copy content Toggle raw display
$59$ \( T^{10} + 5 T^{9} - 41 T^{8} + \cdots + 4489 \) Copy content Toggle raw display
$61$ \( T^{10} - 37 T^{9} + \cdots + 349727401 \) Copy content Toggle raw display
$67$ \( T^{10} + 13 T^{9} + 202 T^{8} + \cdots + 94109401 \) Copy content Toggle raw display
$71$ \( T^{10} - 49 T^{9} + 1180 T^{8} + \cdots + 8300161 \) Copy content Toggle raw display
$73$ \( T^{10} + 8 T^{9} + 42 T^{8} + \cdots + 7921 \) Copy content Toggle raw display
$79$ \( T^{10} - 8 T^{9} + 9 T^{8} + \cdots + 17161 \) Copy content Toggle raw display
$83$ \( T^{10} + 7 T^{9} + 49 T^{8} + \cdots + 667137241 \) Copy content Toggle raw display
$89$ \( T^{10} + 13 T^{9} + 147 T^{8} + \cdots + 26739241 \) Copy content Toggle raw display
$97$ \( T^{10} + 32 T^{9} + 628 T^{8} + \cdots + 5031049 \) Copy content Toggle raw display
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