Properties

Label 23.10.a.a
Level $23$
Weight $10$
Character orbit 23.a
Self dual yes
Analytic conductor $11.846$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8458242318\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} - 2 \beta_1 - 13) q^{3} + (\beta_{5} + 2 \beta_{4} + 6 \beta_1 + 220) q^{4} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 13 \beta_1 - 341) q^{5} + (\beta_{6} - 7 \beta_{5} + 12 \beta_{4} + 4 \beta_{3} - \beta_{2} - 81 \beta_1 - 1783) q^{6} + ( - 6 \beta_{6} - 8 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + \cdots - 1410) q^{7}+ \cdots + (25 \beta_{6} + 40 \beta_{5} + 13 \beta_{4} - 17 \beta_{3} - 9 \beta_{2} + \cdots + 4728) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} - 2 \beta_1 - 13) q^{3} + (\beta_{5} + 2 \beta_{4} + 6 \beta_1 + 220) q^{4} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 13 \beta_1 - 341) q^{5} + (\beta_{6} - 7 \beta_{5} + 12 \beta_{4} + 4 \beta_{3} - \beta_{2} - 81 \beta_1 - 1783) q^{6} + ( - 6 \beta_{6} - 8 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + \cdots - 1410) q^{7}+ \cdots + ( - 818753 \beta_{6} - 1142486 \beta_{5} + \cdots - 268003869) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 89 q^{3} + 1536 q^{4} - 2388 q^{5} - 12518 q^{6} - 9896 q^{7} + 34920 q^{8} + 33064 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 89 q^{3} + 1536 q^{4} - 2388 q^{5} - 12518 q^{6} - 9896 q^{7} + 34920 q^{8} + 33064 q^{9} - 60820 q^{10} - 78484 q^{11} - 343492 q^{12} - 296769 q^{13} - 711120 q^{14} - 237870 q^{15} - 253440 q^{16} - 1128820 q^{17} - 499874 q^{18} - 1301252 q^{19} - 3482704 q^{20} - 108908 q^{21} - 1562088 q^{22} - 1958887 q^{23} - 4606464 q^{24} - 1320899 q^{25} + 692230 q^{26} + 2977921 q^{27} - 8371144 q^{28} + 2813849 q^{29} + 25535196 q^{30} + 7334751 q^{31} + 26028800 q^{32} + 646330 q^{33} + 14981564 q^{34} + 23410104 q^{35} + 40211900 q^{36} - 13324320 q^{37} + 37578632 q^{38} + 6304533 q^{39} - 45307920 q^{40} - 15691573 q^{41} + 124523248 q^{42} - 46474818 q^{43} + 43428040 q^{44} - 72736710 q^{45} + 8232227 q^{47} - 163054384 q^{48} + 29219031 q^{49} + 50366304 q^{50} - 136344764 q^{51} - 100922292 q^{52} - 53545400 q^{53} - 26171642 q^{54} - 181608484 q^{55} - 420111696 q^{56} - 218913370 q^{57} - 39304854 q^{58} - 341275144 q^{59} + 420822384 q^{60} - 277157656 q^{61} + 464777594 q^{62} - 574619276 q^{63} + 340566208 q^{64} + 106659278 q^{65} + 258025876 q^{66} + 89654580 q^{67} + 62700400 q^{68} + 24905849 q^{69} + 1187910040 q^{70} - 286098961 q^{71} + 1446323640 q^{72} - 637495039 q^{73} + 189880036 q^{74} - 160733159 q^{75} + 228563936 q^{76} + 511682536 q^{77} + 1199383686 q^{78} + 274469546 q^{79} - 345318560 q^{80} - 237775217 q^{81} - 570256066 q^{82} + 1164579762 q^{83} + 3447171416 q^{84} - 18639492 q^{85} + 415245796 q^{86} - 595368433 q^{87} + 103329440 q^{88} - 504153000 q^{89} - 1414126968 q^{90} - 1692320156 q^{91} - 429835776 q^{92} - 2753858687 q^{93} - 2214048622 q^{94} + 162962164 q^{95} - 3332565856 q^{96} - 3519929016 q^{97} + 2474592568 q^{98} - 1883749262 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3199 \nu^{6} - 79358 \nu^{5} - 1488724 \nu^{4} + 30366663 \nu^{3} + 273707722 \nu^{2} - 2328138428 \nu - 14876298328 ) / 11419280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1391 \nu^{6} - 6842 \nu^{5} - 682136 \nu^{4} - 269473 \nu^{3} + 87308078 \nu^{2} + 269497648 \nu - 2521122692 ) / 2854820 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7713 \nu^{6} + 105666 \nu^{5} + 3328828 \nu^{4} - 32964761 \nu^{3} - 360613014 \nu^{2} + 2265197396 \nu + 6770868456 ) / 11419280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7713 \nu^{6} - 105666 \nu^{5} - 3328828 \nu^{4} + 32964761 \nu^{3} + 383451574 \nu^{2} - 2333713076 \nu - 10950324936 ) / 5709640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29851 \nu^{6} + 497782 \nu^{5} + 11250276 \nu^{4} - 154091107 \nu^{3} - 1009430098 \nu^{2} + 10082064652 \nu + 17652094632 ) / 11419280 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} + 6\beta _1 + 732 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{6} + 15\beta_{5} + 14\beta_{4} - 8\beta_{3} - 6\beta_{2} + 560\beta _1 + 2494 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{6} + 243\beta_{5} + 456\beta_{4} - 32\beta_{3} - 54\beta_{2} + 2205\beta _1 + 103240 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 964\beta_{6} + 9063\beta_{5} + 10258\beta_{4} - 3564\beta_{3} - 3776\beta_{2} + 192494\beta _1 + 1742612 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1206 \beta_{6} + 223049 \beta_{5} + 374874 \beta_{4} - 42256 \beta_{3} - 72698 \beta_{2} + 2453864 \beta _1 + 71615698 ) / 4 \) 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
−15.4224
−15.0730
−10.3773
−2.27545
8.70295
13.1537
21.2915
−30.8448 107.347 439.399 −1512.00 −3311.09 8168.19 2239.35 −8159.64 46637.2
1.2 −30.1460 −193.374 396.783 568.966 5829.45 3614.33 3473.33 17710.4 −17152.1
1.3 −20.7546 245.319 −81.2472 −680.714 −5091.50 −8213.17 12312.6 40498.5 14127.9
1.4 −4.55091 −23.1179 −491.289 2146.10 105.207 615.663 4565.88 −19148.6 −9766.68
1.5 17.4059 86.8015 −209.035 −1221.93 1510.86 −4757.43 −12550.3 −12148.5 −21268.8
1.6 26.3074 −105.922 180.082 92.1465 −2786.54 2337.45 −8731.92 −8463.52 2424.14
1.7 42.5829 −206.054 1301.31 −1780.57 −8774.39 −11661.0 33611.0 22775.3 −75821.7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 23.10.a.a 7
3.b odd 2 1 207.10.a.b 7
4.b odd 2 1 368.10.a.f 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.10.a.a 7 1.a even 1 1 trivial
207.10.a.b 7 3.b odd 2 1
368.10.a.f 7 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 2560T_{2}^{5} - 11640T_{2}^{4} + 1832832T_{2}^{3} + 10289408T_{2}^{2} - 367442944T_{2} - 1712515072 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 2560 T^{5} + \cdots - 1712515072 \) Copy content Toggle raw display
$3$ \( T^{7} + \cdots - 223029139672800 \) Copy content Toggle raw display
$5$ \( T^{7} + 2388 T^{6} + \cdots - 25\!\cdots\!80 \) Copy content Toggle raw display
$7$ \( T^{7} + 9896 T^{6} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( T^{7} + 78484 T^{6} + \cdots - 10\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{7} + 296769 T^{6} + \cdots - 72\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{7} + 1128820 T^{6} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{7} + 1301252 T^{6} + \cdots - 58\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( (T + 279841)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} - 2813849 T^{6} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} - 7334751 T^{6} + \cdots - 89\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{7} + 13324320 T^{6} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{7} + 15691573 T^{6} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{7} + 46474818 T^{6} + \cdots - 77\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{7} - 8232227 T^{6} + \cdots - 30\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{7} + 53545400 T^{6} + \cdots - 18\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{7} + 341275144 T^{6} + \cdots - 23\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{7} + 277157656 T^{6} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{7} - 89654580 T^{6} + \cdots - 12\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{7} + 286098961 T^{6} + \cdots + 17\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{7} + 637495039 T^{6} + \cdots + 95\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{7} - 274469546 T^{6} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{7} - 1164579762 T^{6} + \cdots + 71\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{7} + 504153000 T^{6} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + 3519929016 T^{6} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
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