Properties

Label 23.4.a.b
Level $23$
Weight $4$
Character orbit 23.a
Self dual yes
Analytic conductor $1.357$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.35704393013\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.334189.1
Defining polynomial: \( x^{4} - 2x^{3} - 16x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{4} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{5} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 5) q^{6} + ( - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{7} + (5 \beta_{3} - \beta_{2} + 4 \beta_1 - 15) q^{8} + (2 \beta_{3} + \beta_{2} + 5 \beta_1 - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{4} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{5} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 5) q^{6} + ( - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{7} + (5 \beta_{3} - \beta_{2} + 4 \beta_1 - 15) q^{8} + (2 \beta_{3} + \beta_{2} + 5 \beta_1 - 10) q^{9} + (4 \beta_{3} - 6 \beta_{2} + 4 \beta_1 - 16) q^{10} + (2 \beta_{3} - 4 \beta_{2} - 12 \beta_1 + 10) q^{11} + ( - 6 \beta_{3} + \beta_{2} - 8 \beta_1 - 16) q^{12} + ( - 10 \beta_{3} - 11 \beta_{2} - 11 \beta_1 + 31) q^{13} + (18 \beta_{3} + 24 \beta_{2} + 10 \beta_1 - 38) q^{14} + ( - 6 \beta_{3} + 2 \beta_{2} + 18 \beta_1 - 10) q^{15} + ( - 19 \beta_{3} - 2 \beta_{2} + 8 \beta_1 + 3) q^{16} + (10 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 32) q^{17} + ( - 20 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 6) q^{18} + (24 \beta_{3} + 24 \beta_{2} + 32 \beta_1 + 14) q^{19} + ( - 22 \beta_{3} - 20 \beta_{2} - 34 \beta_1 + 46) q^{20} + (6 \beta_{3} - 8 \beta_{2} - 28 \beta_1 + 64) q^{21} + (12 \beta_{3} - 8 \beta_{2} - 10 \beta_1 + 68) q^{22} - 23 q^{23} + (3 \beta_{3} + 11 \beta_1 - 51) q^{24} + (16 \beta_{3} + 28 \beta_{2} + 8 \beta_1 + 43) q^{25} + (61 \beta_{3} - 13 \beta_{2} + 19 \beta_1 - 33) q^{26} + (4 \beta_{3} - 29 \beta_{2} - 9 \beta_1 - 25) q^{27} + ( - 62 \beta_{3} + 18 \beta_{2} + 2 \beta_1 + 34) q^{28} + ( - 26 \beta_{3} + 23 \beta_{2} + 35 \beta_1 - 31) q^{29} + ( - 8 \beta_{3} + 2 \beta_{2} + 20 \beta_1 - 116) q^{30} + (12 \beta_{3} + 9 \beta_{2} - 19 \beta_1 - 35) q^{31} + (10 \beta_{3} + 30 \beta_{2} + 23 \beta_1 - 122) q^{32} + ( - 6 \beta_{3} + 6 \beta_{2} - 54 \beta_1 - 82) q^{33} + ( - 2 \beta_{3} - 42 \beta_{2} - 36 \beta_1 + 194) q^{34} + ( - 84 \beta_{3} - 48 \beta_{2} - 12 \beta_1 - 212) q^{35} + (50 \beta_{3} + 17 \beta_{2} + 15 \beta_1 - 144) q^{36} + (10 \beta_{3} - 54 \beta_{2} + 30 \beta_1 + 46) q^{37} + ( - 66 \beta_{3} + 16 \beta_{2} - 48 \beta_1 + 174) q^{38} + ( - 32 \beta_{3} + 11 \beta_{2} - 13 \beta_1 - 85) q^{39} + (94 \beta_{3} + 46 \beta_{2} + 14 \beta_1 + 22) q^{40} + ( - 18 \beta_{3} - 7 \beta_{2} + 21 \beta_1 - 49) q^{41} + (66 \beta_{3} - 16 \beta_{2} - 26 \beta_1 + 210) q^{42} + (28 \beta_{3} + 14 \beta_{2} + 70 \beta_1 - 24) q^{43} + (18 \beta_{3} - 14 \beta_{2} + 52 \beta_1 + 194) q^{44} + (52 \beta_{3} + 48 \beta_{2} + 40 \beta_1 + 36) q^{45} + ( - 23 \beta_{3} - 23) q^{46} + (36 \beta_{3} + 13 \beta_{2} - 71 \beta_1 - 119) q^{47} + ( - 23 \beta_{3} - 25 \beta_{2} + 55 \beta_1 + 105) q^{48} + (32 \beta_{3} + 8 \beta_{2} - 92 \beta_1 + 365) q^{49} + (15 \beta_{3} + 72 \beta_{2} - 20 \beta_1 + 103) q^{50} + ( - 2 \beta_{3} + 56 \beta_{2} + 32 \beta_1 - 116) q^{51} + ( - 168 \beta_{3} - 105 \beta_{2} - 108 \beta_1 + 558) q^{52} + ( - 120 \beta_{3} - 26 \beta_{2} + 14 \beta_1 - 118) q^{53} + ( - 57 \beta_{3} - 115 \beta_{2} - 41 \beta_1 + 181) q^{54} + ( - 40 \beta_{3} - 108 \beta_{2} - 120 \beta_1 - 176) q^{55} + (92 \beta_{3} + 2 \beta_{2} + 124 \beta_1 - 560) q^{56} + (72 \beta_{3} + 70 \beta_{2} + 158 \beta_1 + 270) q^{57} + (35 \beta_{3} + 109 \beta_{2} + 101 \beta_1 - 519) q^{58} + (28 \beta_{3} + 60 \beta_{2} + 36 \beta_1 - 304) q^{59} + ( - 62 \beta_{3} - 12 \beta_{2} - 118 \beta_1 - 170) q^{60} + (36 \beta_{3} - 26 \beta_{2} + 14 \beta_1 + 206) q^{61} + ( - 43 \beta_{3} + 31 \beta_{2} - 27 \beta_1 + 95) q^{62} + (44 \beta_{3} - 52 \beta_{2} + 20 \beta_1 - 248) q^{63} + (7 \beta_{3} + 93 \beta_{2} - 64 \beta_1 - 189) q^{64} + ( - 78 \beta_{3} - 66 \beta_{2} - 134 \beta_1 + 374) q^{65} + ( - 4 \beta_{3} + 90 \beta_{2} + 24 \beta_1 - 136) q^{66} + (110 \beta_{3} + 84 \beta_{2} + 92 \beta_1 + 110) q^{67} + (114 \beta_{3} - 80 \beta_{2} - 20 \beta_1 + 158) q^{68} + ( - 23 \beta_{2} - 23 \beta_1 - 23) q^{69} + (4 \beta_{3} - 12 \beta_{2} + 204 \beta_1 - 1052) q^{70} + (68 \beta_{3} - 111 \beta_{2} - 143 \beta_1 + 33) q^{71} + ( - 132 \beta_{3} - 7 \beta_{2} - 93 \beta_1 + 358) q^{72} + ( - 66 \beta_{3} - \beta_{2} + 127 \beta_1 + 281) q^{73} + ( - 68 \beta_{3} - 266 \beta_{2} - 84 \beta_1 + 416) q^{74} + (72 \beta_{3} + 55 \beta_{2} + 35 \beta_1 + 355) q^{75} + (244 \beta_{3} + 52 \beta_{2} - 42 \beta_1 - 828) q^{76} + (36 \beta_{3} + 208 \beta_{2} + 132 \beta_1 + 84) q^{77} + (35 \beta_{3} + 121 \beta_{2} + 107 \beta_1 - 543) q^{78} + ( - 132 \beta_{3} - 50 \beta_{2} - 98 \beta_1 - 214) q^{79} + ( - 52 \beta_{3} + 142 \beta_{2} + 36 \beta_1 + 632) q^{80} + ( - 108 \beta_{3} - 24 \beta_{2} - 156 \beta_1 - 203) q^{81} + ( - 23 \beta_{3} - 13 \beta_{2} + 47 \beta_1 - 269) q^{82} + ( - 22 \beta_{3} + 6 \beta_{2} - 146 \beta_1 + 96) q^{83} + ( - 26 \beta_{3} - 106 \beta_{2} + 10 \beta_1 + 662) q^{84} + (44 \beta_{3} - 96 \beta_{2} + 28 \beta_1 + 60) q^{85} + ( - 164 \beta_{3} - 70 \beta_{2} - 70 \beta_1 + 200) q^{86} + (20 \beta_{3} - 71 \beta_{2} + 133 \beta_1 + 517) q^{87} + ( - 22 \beta_{3} - 80 \beta_{2} + 12 \beta_1 - 98) q^{88} + (160 \beta_{3} + 82 \beta_{2} + 82 \beta_1 + 604) q^{89} + ( - 112 \beta_{3} + 48 \beta_{2} - 108 \beta_1 + 432) q^{90} + ( - 350 \beta_{3} + 56 \beta_{2} - 116) q^{91} + (46 \beta_{3} + 46 \beta_{2} + 69 \beta_1 - 138) q^{92} + (30 \beta_{3} - 39 \beta_{2} - 167 \beta_1 - 19) q^{93} + ( - 143 \beta_{3} + 51 \beta_{2} - 95 \beta_1 + 355) q^{94} + (68 \beta_{3} + 36 \beta_{2} + 540 \beta_1 - 484) q^{95} + (70 \beta_{3} - 109 \beta_{2} - 44 \beta_1 + 284) q^{96} + (282 \beta_{3} + 150 \beta_{2} - 38 \beta_1 + 388) q^{97} + (369 \beta_{3} + 60 \beta_{2} - 88 \beta_1 + 833) q^{98} + ( - 48 \beta_{3} - 46 \beta_{2} - 94 \beta_1 - 340) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 7 q^{3} + 20 q^{4} + 14 q^{5} - 17 q^{6} + 16 q^{7} - 63 q^{8} - 33 q^{9} - 70 q^{10} + 8 q^{11} - 67 q^{12} + 111 q^{13} - 144 q^{14} + 10 q^{15} + 64 q^{16} + 98 q^{17} + 49 q^{18} + 96 q^{19} + 140 q^{20} + 180 q^{21} + 220 q^{22} - 92 q^{23} - 188 q^{24} + 184 q^{25} - 229 q^{26} - 155 q^{27} + 282 q^{28} + 21 q^{29} - 406 q^{30} - 193 q^{31} - 432 q^{32} - 418 q^{33} + 666 q^{34} - 752 q^{35} - 629 q^{36} + 170 q^{37} + 748 q^{38} - 291 q^{39} - 26 q^{40} - 125 q^{41} + 640 q^{42} + 2 q^{43} + 830 q^{44} + 168 q^{45} - 46 q^{46} - 677 q^{47} + 551 q^{48} + 1220 q^{49} + 414 q^{50} - 340 q^{51} + 2247 q^{52} - 230 q^{53} + 641 q^{54} - 972 q^{55} - 2174 q^{56} + 1322 q^{57} - 1835 q^{58} - 1140 q^{59} - 804 q^{60} + 754 q^{61} + 443 q^{62} - 1092 q^{63} - 805 q^{64} + 1318 q^{65} - 398 q^{66} + 488 q^{67} + 284 q^{68} - 161 q^{69} - 3820 q^{70} - 401 q^{71} + 1503 q^{72} + 1509 q^{73} + 1366 q^{74} + 1401 q^{75} - 3832 q^{76} + 736 q^{77} - 1907 q^{78} - 838 q^{79} + 2846 q^{80} - 932 q^{81} - 949 q^{82} + 142 q^{83} + 2614 q^{84} + 112 q^{85} + 918 q^{86} + 2223 q^{87} - 404 q^{88} + 2342 q^{89} + 1784 q^{90} + 292 q^{91} - 460 q^{92} - 509 q^{93} + 1567 q^{94} - 956 q^{95} + 799 q^{96} + 1062 q^{97} + 2478 q^{98} - 1498 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 16x^{2} - 5x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 20\nu - 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 5\nu^{2} + 28\nu - 1 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} + 4\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_{2} + 24\beta _1 + 17 \) Copy content Toggle raw display

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
−0.743529
5.22031
−2.83969
0.362907
−5.07751 1.55870 17.7811 10.0635 −7.91434 24.3381 −49.6639 −24.5704 −51.0976
1.2 −0.0323756 6.42170 −7.99895 14.1026 −0.207906 −14.0109 0.517976 14.2382 −0.456580
1.3 2.86845 3.43737 0.228032 −17.9704 9.85995 32.7301 −22.2935 −15.1845 −51.5473
1.4 4.24143 −4.41777 9.98977 7.80430 −18.7377 −27.0572 8.43948 −7.48328 33.1014
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 23.4.a.b 4
3.b odd 2 1 207.4.a.e 4
4.b odd 2 1 368.4.a.l 4
5.b even 2 1 575.4.a.i 4
5.c odd 4 2 575.4.b.g 8
7.b odd 2 1 1127.4.a.c 4
8.b even 2 1 1472.4.a.y 4
8.d odd 2 1 1472.4.a.bf 4
23.b odd 2 1 529.4.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.4.a.b 4 1.a even 1 1 trivial
207.4.a.e 4 3.b odd 2 1
368.4.a.l 4 4.b odd 2 1
529.4.a.g 4 23.b odd 2 1
575.4.a.i 4 5.b even 2 1
575.4.b.g 8 5.c odd 4 2
1127.4.a.c 4 7.b odd 2 1
1472.4.a.y 4 8.b even 2 1
1472.4.a.bf 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} - 24T_{2}^{2} + 61T_{2} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 24 T^{2} + 61 T + 2 \) Copy content Toggle raw display
$3$ \( T^{4} - 7 T^{3} - 13 T^{2} + 131 T - 152 \) Copy content Toggle raw display
$5$ \( T^{4} - 14 T^{3} - 244 T^{2} + \cdots - 19904 \) Copy content Toggle raw display
$7$ \( T^{4} - 16 T^{3} - 1168 T^{2} + \cdots + 301984 \) Copy content Toggle raw display
$11$ \( T^{4} - 8 T^{3} - 2488 T^{2} + \cdots - 81440 \) Copy content Toggle raw display
$13$ \( T^{4} - 111 T^{3} + 529 T^{2} + \cdots + 1322658 \) Copy content Toggle raw display
$17$ \( T^{4} - 98 T^{3} - 1008 T^{2} + \cdots - 855280 \) Copy content Toggle raw display
$19$ \( T^{4} - 96 T^{3} - 21208 T^{2} + \cdots + 66996944 \) Copy content Toggle raw display
$23$ \( (T + 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 21 T^{3} + \cdots + 325399050 \) Copy content Toggle raw display
$31$ \( T^{4} + 193 T^{3} + \cdots - 58104720 \) Copy content Toggle raw display
$37$ \( T^{4} - 170 T^{3} + \cdots + 2389345472 \) Copy content Toggle raw display
$41$ \( T^{4} + 125 T^{3} + \cdots + 29467114 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} - 80432 T^{2} + \cdots + 78004224 \) Copy content Toggle raw display
$47$ \( T^{4} + 677 T^{3} + \cdots - 3169103456 \) Copy content Toggle raw display
$53$ \( T^{4} + 230 T^{3} + \cdots + 7631805536 \) Copy content Toggle raw display
$59$ \( T^{4} + 1140 T^{3} + \cdots + 1146071296 \) Copy content Toggle raw display
$61$ \( T^{4} - 754 T^{3} + \cdots - 621762112 \) Copy content Toggle raw display
$67$ \( T^{4} - 488 T^{3} + \cdots - 1826338144 \) Copy content Toggle raw display
$71$ \( T^{4} + 401 T^{3} + \cdots - 5581505296 \) Copy content Toggle raw display
$73$ \( T^{4} - 1509 T^{3} + \cdots - 14695752674 \) Copy content Toggle raw display
$79$ \( T^{4} + 838 T^{3} + \cdots - 61908677856 \) Copy content Toggle raw display
$83$ \( T^{4} - 142 T^{3} + \cdots + 7015211408 \) Copy content Toggle raw display
$89$ \( T^{4} - 2342 T^{3} + \cdots - 213195182848 \) Copy content Toggle raw display
$97$ \( T^{4} - 1062 T^{3} + \cdots + 60054540368 \) Copy content Toggle raw display
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