Properties

Label 23.12.a.b
Level $23$
Weight $12$
Character orbit 23.a
Self dual yes
Analytic conductor $17.672$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6718931529\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \( x^{11} - x^{10} - 16849 x^{9} - 2148 x^{8} + 97176782 x^{7} + 169360278 x^{6} - 226650696110 x^{5} - 940430954112 x^{4} + 180101325169073 x^{3} + \cdots - 51\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 3) q^{2} + ( - \beta_{2} + \beta_1 - 2) q^{3} + (\beta_{3} + \beta_{2} - 5 \beta_1 + 1025) q^{4} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + 19 \beta_1 + 93) q^{5} + ( - \beta_{9} - \beta_{7} - \beta_{5} + 2 \beta_{3} - 12 \beta_{2} + 53 \beta_1 - 2039) q^{6} + (2 \beta_{8} + 2 \beta_{7} + \beta_{5} - \beta_{4} + 4 \beta_{3} - 23 \beta_{2} + \cdots + 14516) q^{7}+ \cdots + ( - \beta_{10} + 5 \beta_{9} + 8 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + \cdots + 55661) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 3) q^{2} + ( - \beta_{2} + \beta_1 - 2) q^{3} + (\beta_{3} + \beta_{2} - 5 \beta_1 + 1025) q^{4} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + 19 \beta_1 + 93) q^{5} + ( - \beta_{9} - \beta_{7} - \beta_{5} + 2 \beta_{3} - 12 \beta_{2} + 53 \beta_1 - 2039) q^{6} + (2 \beta_{8} + 2 \beta_{7} + \beta_{5} - \beta_{4} + 4 \beta_{3} - 23 \beta_{2} + \cdots + 14516) q^{7}+ \cdots + ( - 1371161 \beta_{10} - 288486 \beta_{9} + \cdots + 8825524717) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 32 q^{2} - 20 q^{3} + 11264 q^{4} + 1034 q^{5} - 22385 q^{6} + 159584 q^{7} + 115497 q^{8} + 611943 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 32 q^{2} - 20 q^{3} + 11264 q^{4} + 1034 q^{5} - 22385 q^{6} + 159584 q^{7} + 115497 q^{8} + 611943 q^{9} - 627650 q^{10} - 771396 q^{11} - 1720771 q^{12} + 3433434 q^{13} + 4585896 q^{14} + 5551840 q^{15} + 18802384 q^{16} + 29035398 q^{17} + 26169127 q^{18} + 21398428 q^{19} + 72466260 q^{20} + 61896432 q^{21} + 100463524 q^{22} - 70799773 q^{23} + 161844076 q^{24} + 233562509 q^{25} + 328796191 q^{26} + 356379712 q^{27} + 499445210 q^{28} + 226699042 q^{29} + 510413234 q^{30} + 251932328 q^{31} + 806116648 q^{32} + 221442992 q^{33} + 325378622 q^{34} - 355232072 q^{35} - 1034240009 q^{36} + 573876170 q^{37} - 770782036 q^{38} - 1199522184 q^{39} - 1009699226 q^{40} - 1733596378 q^{41} - 6499506824 q^{42} + 647370308 q^{43} - 4662321170 q^{44} - 5023123422 q^{45} - 205962976 q^{46} - 5436527248 q^{47} - 4050950401 q^{48} + 4356386219 q^{49} - 10296502416 q^{50} - 1050918064 q^{51} - 5616607137 q^{52} - 3387203910 q^{53} - 21748294807 q^{54} - 10571441512 q^{55} + 635842210 q^{56} - 4678697728 q^{57} + 1991171353 q^{58} + 15113662084 q^{59} + 19934836476 q^{60} + 23895772578 q^{61} + 7557529251 q^{62} + 56666471160 q^{63} + 32993181147 q^{64} + 3660035708 q^{65} + 33342886858 q^{66} + 46806014468 q^{67} + 40754169364 q^{68} + 128726860 q^{69} - 12274756860 q^{70} + 45541532768 q^{71} + 12763980783 q^{72} + 63786612542 q^{73} - 41720765910 q^{74} + 88573702476 q^{75} + 5581739704 q^{76} + 19147126968 q^{77} - 111443125499 q^{78} + 21847812496 q^{79} - 67715736674 q^{80} + 122793712411 q^{81} - 30216129401 q^{82} + 40153340788 q^{83} - 221098994762 q^{84} + 116854272412 q^{85} - 47307463306 q^{86} - 5595049008 q^{87} - 263050721364 q^{88} + 37300228382 q^{89} - 419483354956 q^{90} + 109416811256 q^{91} - 72498967552 q^{92} - 224700035960 q^{93} - 378035850441 q^{94} - 255722421456 q^{95} - 96864379937 q^{96} - 243602730 q^{97} - 514347061348 q^{98} + 97029276404 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - x^{10} - 16849 x^{9} - 2148 x^{8} + 97176782 x^{7} + 169360278 x^{6} - 226650696110 x^{5} - 940430954112 x^{4} + 180101325169073 x^{3} + \cdots - 51\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!07 \nu^{10} + \cdots + 98\!\cdots\!72 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!07 \nu^{10} + \cdots - 44\!\cdots\!92 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!73 \nu^{10} + \cdots + 45\!\cdots\!68 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\!\cdots\!91 \nu^{10} + \cdots - 51\!\cdots\!24 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 71\!\cdots\!05 \nu^{10} + \cdots - 39\!\cdots\!16 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\!\cdots\!61 \nu^{10} + \cdots - 24\!\cdots\!64 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26\!\cdots\!27 \nu^{10} + \cdots - 23\!\cdots\!88 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30\!\cdots\!93 \nu^{10} + \cdots + 27\!\cdots\!08 ) / 71\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 78\!\cdots\!17 \nu^{10} + \cdots + 54\!\cdots\!08 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3064 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{9} + 8\beta_{8} - \beta_{7} - 8\beta_{5} - \beta_{4} - 8\beta_{3} + 73\beta_{2} + 5289\beta _1 + 4705 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 123 \beta_{10} - 162 \beta_{9} + 264 \beta_{8} - 189 \beta_{7} + 234 \beta_{6} - 124 \beta_{5} - 321 \beta_{4} + 7141 \beta_{3} + 8737 \beta_{2} + 2388 \beta _1 + 16289825 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3733 \beta_{10} - 59250 \beta_{9} + 83429 \beta_{8} - 5821 \beta_{7} - 4688 \beta_{6} - 59739 \beta_{5} - 2981 \beta_{4} - 63782 \beta_{3} + 612759 \beta_{2} + 32523523 \beta _1 + 21419939 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1512389 \beta_{10} - 1822149 \beta_{9} + 2714334 \beta_{8} - 1441240 \beta_{7} + 2377899 \beta_{6} - 987435 \beta_{5} - 4037051 \beta_{4} + 50752667 \beta_{3} + \cdots + 100373583678 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 52746322 \beta_{10} - 493295655 \beta_{9} + 698593673 \beta_{8} - 7308298 \beta_{7} - 56097359 \beta_{6} - 395621922 \beta_{5} - 142499 \beta_{4} - 504926440 \beta_{3} + \cdots + 122660780986 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14256502838 \beta_{10} - 15916027128 \beta_{9} + 22639183841 \beta_{8} - 8914331330 \beta_{7} + 19770844626 \beta_{6} - 5854537559 \beta_{5} + \cdots + 671079396927896 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 532754398126 \beta_{10} - 3887491812723 \beta_{9} + 5495463885961 \beta_{8} + 192652823785 \beta_{7} - 508821621377 \beta_{6} - 2638046791224 \beta_{5} + \cdots + 677468729008055 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 120945483015933 \beta_{10} - 128300335955160 \beta_{9} + 177936231166533 \beta_{8} - 52787109586795 \beta_{7} + 155178826135498 \beta_{6} + \cdots + 47\!\cdots\!97 \) 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
86.4321
69.7231
52.3794
46.0566
6.40598
−3.80280
−13.8911
−30.7175
−62.3523
−62.6325
−86.6009
−83.4321 −194.005 4912.92 1374.36 16186.3 62218.9 −239026. −139509. −114666.
1.2 −66.7231 −326.423 2403.97 10955.3 21780.0 −74379.3 −23751.4 −70595.0 −730973.
1.3 −49.3794 698.433 390.321 10492.6 −34488.2 69616.8 81855.1 310661. −518116.
1.4 −43.0566 78.8729 −194.133 −11277.9 −3395.99 −9281.65 96538.5 −170926. 485588.
1.5 −3.40598 746.589 −2036.40 −11427.6 −2542.87 38639.7 13911.4 380248. 38922.2
1.6 6.80280 −489.531 −2001.72 −3091.03 −3330.18 −55047.7 −27549.4 62493.6 −21027.7
1.7 16.8911 −378.630 −1762.69 −6541.27 −6395.47 −3800.70 −64366.7 −33786.5 −110489.
1.8 33.7175 −5.72695 −911.129 12636.0 −193.099 22981.2 −99774.5 −177114. 426055.
1.9 65.3523 −763.826 2222.93 −7377.16 −49917.8 78344.9 11431.9 406283. −482115.
1.10 65.6325 631.031 2259.62 3124.13 41416.2 7787.45 13889.4 221053. 205044.
1.11 89.6009 −16.7832 5980.32 2166.57 −1503.79 22504.4 352339. −176865. 194127.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 23.12.a.b 11
3.b odd 2 1 207.12.a.d 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.12.a.b 11 1.a even 1 1 trivial
207.12.a.d 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} - 32 T_{2}^{10} - 16384 T_{2}^{9} + 453021 T_{2}^{8} + 91689644 T_{2}^{7} - 2171413128 T_{2}^{6} - 205410732416 T_{2}^{5} + 4226023192464 T_{2}^{4} + \cdots + 60\!\cdots\!68 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} - 32 T^{10} + \cdots + 60\!\cdots\!68 \) Copy content Toggle raw display
$3$ \( T^{11} + 20 T^{10} + \cdots + 22\!\cdots\!80 \) Copy content Toggle raw display
$5$ \( T^{11} - 1034 T^{10} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{11} - 159584 T^{10} + \cdots - 76\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{11} + 771396 T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{11} - 3433434 T^{10} + \cdots - 52\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{11} - 29035398 T^{10} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{11} - 21398428 T^{10} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( (T + 6436343)^{11} \) Copy content Toggle raw display
$29$ \( T^{11} - 226699042 T^{10} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{11} - 251932328 T^{10} + \cdots - 71\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{11} - 573876170 T^{10} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{11} + 1733596378 T^{10} + \cdots - 88\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{11} - 647370308 T^{10} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{11} + 5436527248 T^{10} + \cdots + 17\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{11} + 3387203910 T^{10} + \cdots - 49\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{11} - 15113662084 T^{10} + \cdots + 46\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{11} - 23895772578 T^{10} + \cdots + 14\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{11} - 46806014468 T^{10} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{11} - 45541532768 T^{10} + \cdots - 46\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{11} - 63786612542 T^{10} + \cdots - 66\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{11} - 21847812496 T^{10} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{11} - 40153340788 T^{10} + \cdots - 18\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{11} - 37300228382 T^{10} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{11} + 243602730 T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
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