Properties

Label 23.2.a.a
Level 23
Weight 2
Character orbit 23.a
Self dual yes
Analytic conductor 0.184
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.183655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
−1.61803 2.23607 0.618034 −3.23607 −3.61803 −1.23607 2.23607 2.00000 5.23607
1.2 0.618034 −2.23607 −1.61803 1.23607 −1.38197 3.23607 −2.23607 2.00000 0.763932
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 23.2.a.a 2
3.b odd 2 1 207.2.a.d 2
4.b odd 2 1 368.2.a.h 2
5.b even 2 1 575.2.a.f 2
5.c odd 4 2 575.2.b.d 4
7.b odd 2 1 1127.2.a.c 2
8.b even 2 1 1472.2.a.t 2
8.d odd 2 1 1472.2.a.s 2
11.b odd 2 1 2783.2.a.c 2
12.b even 2 1 3312.2.a.ba 2
13.b even 2 1 3887.2.a.i 2
15.d odd 2 1 5175.2.a.be 2
17.b even 2 1 6647.2.a.b 2
19.b odd 2 1 8303.2.a.e 2
20.d odd 2 1 9200.2.a.bt 2
23.b odd 2 1 529.2.a.a 2
23.c even 11 10 529.2.c.o 20
23.d odd 22 10 529.2.c.n 20
69.c even 2 1 4761.2.a.w 2
92.b even 2 1 8464.2.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.2.a.a 2 1.a even 1 1 trivial
207.2.a.d 2 3.b odd 2 1
368.2.a.h 2 4.b odd 2 1
529.2.a.a 2 23.b odd 2 1
529.2.c.n 20 23.d odd 22 10
529.2.c.o 20 23.c even 11 10
575.2.a.f 2 5.b even 2 1
575.2.b.d 4 5.c odd 4 2
1127.2.a.c 2 7.b odd 2 1
1472.2.a.s 2 8.d odd 2 1
1472.2.a.t 2 8.b even 2 1
2783.2.a.c 2 11.b odd 2 1
3312.2.a.ba 2 12.b even 2 1
3887.2.a.i 2 13.b even 2 1
4761.2.a.w 2 69.c even 2 1
5175.2.a.be 2 15.d odd 2 1
6647.2.a.b 2 17.b even 2 1
8303.2.a.e 2 19.b odd 2 1
8464.2.a.bb 2 92.b even 2 1
9200.2.a.bt 2 20.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(23\) \(-1\)

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4} \)
$3$ \( 1 + T^{2} + 9 T^{4} \)
$5$ \( 1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( 1 + 6 T + 26 T^{2} + 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} \)
$19$ \( ( 1 + 2 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - T )^{2} \)
$29$ \( ( 1 + 3 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 17 T^{2} + 961 T^{4} \)
$37$ \( 1 - 2 T + 70 T^{2} - 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 2 T + 63 T^{2} - 82 T^{3} + 1681 T^{4} \)
$43$ \( ( 1 + 43 T^{2} )^{2} \)
$47$ \( 1 + 89 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 8 T + 102 T^{2} + 424 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 4 T + 102 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 4 T + 46 T^{2} - 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 10 T + 154 T^{2} + 670 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 20 T + 237 T^{2} - 1420 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 22 T + 247 T^{2} - 1606 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 4 T + 82 T^{2} + 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 22 T + 282 T^{2} + 1826 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 12 T + 194 T^{2} + 1068 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 22 T + 270 T^{2} - 2134 T^{3} + 9409 T^{4} \)
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