Properties

Label 23.16.a.a
Level $23$
Weight $16$
Character orbit 23.a
Self dual yes
Analytic conductor $32.820$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 23.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.8195061730\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 4 x^{11} - 68942 x^{10} - 977032 x^{9} + 1644150380 x^{8} + 50352376602 x^{7} - 15614385123802 x^{6} - 702635376772966 x^{5} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{22}\cdot 3^{6}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 21) q^{2} + (\beta_{2} - 145) q^{3} + (\beta_{3} - 2 \beta_{2} + 9 \beta_1 + 13656) q^{4} + (\beta_{6} + 2 \beta_{2} - 257 \beta_1 - 17125) q^{5} + ( - \beta_{6} + \beta_{5} - 3 \beta_{3} + 17 \beta_{2} - 826 \beta_1 - 16888) q^{6} + (\beta_{10} - \beta_{8} - 5 \beta_{6} - 2 \beta_{5} + \beta_{4} - 9 \beta_{3} + \beta_{2} + \cdots - 476268) q^{7}+ \cdots + ( - 6 \beta_{11} - 6 \beta_{10} - \beta_{9} + 5 \beta_{8} + 12 \beta_{7} + \cdots + 2725992) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 21) q^{2} + (\beta_{2} - 145) q^{3} + (\beta_{3} - 2 \beta_{2} + 9 \beta_1 + 13656) q^{4} + (\beta_{6} + 2 \beta_{2} - 257 \beta_1 - 17125) q^{5} + ( - \beta_{6} + \beta_{5} - 3 \beta_{3} + 17 \beta_{2} - 826 \beta_1 - 16888) q^{6} + (\beta_{10} - \beta_{8} - 5 \beta_{6} - 2 \beta_{5} + \beta_{4} - 9 \beta_{3} + \beta_{2} + \cdots - 476268) q^{7}+ \cdots + ( - 249494856 \beta_{11} + \cdots - 270810570431313) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 256 q^{2} - 1745 q^{3} + 163840 q^{4} - 204476 q^{5} - 199430 q^{6} - 5717328 q^{7} + 10323168 q^{8} + 32660341 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 256 q^{2} - 1745 q^{3} + 163840 q^{4} - 204476 q^{5} - 199430 q^{6} - 5717328 q^{7} + 10323168 q^{8} + 32660341 q^{9} - 137846540 q^{10} - 87636002 q^{11} - 398208076 q^{12} - 292496079 q^{13} + 415954912 q^{14} + 548079030 q^{15} + 4273503168 q^{16} - 2462528162 q^{17} + 7261215718 q^{18} + 175321758 q^{19} + 2660811480 q^{20} + 205665472 q^{21} - 21718153768 q^{22} + 40857905364 q^{23} - 63413289624 q^{24} + 20443225284 q^{25} - 137268652810 q^{26} - 151915208903 q^{27} - 325638721712 q^{28} - 164667697193 q^{29} - 356944003956 q^{30} + 20222384151 q^{31} - 369109524032 q^{32} + 132365097022 q^{33} - 582887018988 q^{34} - 1578083373112 q^{35} - 1903913944516 q^{36} - 869669414912 q^{37} - 5525312078376 q^{38} - 5762413466499 q^{39} - 4733269274576 q^{40} - 7510147709883 q^{41} - 7436463221624 q^{42} - 5682603487020 q^{43} - 11849381658176 q^{44} - 10780493432442 q^{45} - 871635314432 q^{46} - 5828073094301 q^{47} - 29418911592496 q^{48} - 6518780198860 q^{49} - 16781003942456 q^{50} - 771327642584 q^{51} - 3841511618340 q^{52} + 1452974784324 q^{53} - 32167598069522 q^{54} - 14882020037092 q^{55} + 416192984288 q^{56} - 12135794354818 q^{57} - 60065613521022 q^{58} - 11503084624084 q^{59} - 6378557828664 q^{60} - 23587566667200 q^{61} + 49359974806402 q^{62} + 87886039196104 q^{63} + 80321007324160 q^{64} + 54548135308138 q^{65} + 316922278045948 q^{66} + 61525019345122 q^{67} + 45114528974104 q^{68} - 5941420405015 q^{69} + 374016699556320 q^{70} + 197895887067063 q^{71} + 439014895837656 q^{72} - 22888563242709 q^{73} + 694696716227036 q^{74} + 612085940395201 q^{75} + 301381886149904 q^{76} + 209007839834200 q^{77} + 350406148895766 q^{78} + 229938065096294 q^{79} + 555529032250016 q^{80} + 37596523177660 q^{81} - 414508112727306 q^{82} + 369402590629184 q^{83} + 559863541234208 q^{84} - 343366303925348 q^{85} + 12\!\cdots\!08 q^{86}+ \cdots - 32\!\cdots\!24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 68942 x^{10} - 977032 x^{9} + 1644150380 x^{8} + 50352376602 x^{7} - 15614385123802 x^{6} - 702635376772966 x^{5} + \cdots + 34\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!41 \nu^{11} + \cdots + 36\!\cdots\!20 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!41 \nu^{11} + \cdots - 47\!\cdots\!40 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!07 \nu^{11} + \cdots - 90\!\cdots\!16 ) / 88\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 54\!\cdots\!55 \nu^{11} + \cdots + 14\!\cdots\!36 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 52\!\cdots\!35 \nu^{11} + \cdots + 48\!\cdots\!56 ) / 88\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 36\!\cdots\!23 \nu^{11} + \cdots - 27\!\cdots\!20 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\!\cdots\!29 \nu^{11} + \cdots - 24\!\cdots\!36 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39\!\cdots\!59 \nu^{11} + \cdots - 13\!\cdots\!12 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 65\!\cdots\!25 \nu^{11} + \cdots - 11\!\cdots\!96 ) / 44\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31\!\cdots\!45 \nu^{11} + \cdots - 52\!\cdots\!56 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2\beta_{2} + 53\beta _1 + 45984 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 3 \beta_{10} + 3 \beta_{8} - 2 \beta_{7} - 7 \beta_{6} - 6 \beta_{5} - 3 \beta_{4} + 36 \beta_{3} - 406 \beta_{2} + 85345 \beta _1 + 2533939 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 83 \beta_{11} - 241 \beta_{10} + 244 \beta_{9} - 25 \beta_{8} + 168 \beta_{7} + 403 \beta_{6} - 464 \beta_{5} - 339 \beta_{4} + 56356 \beta_{3} - 184924 \beta_{2} + 4166111 \beta _1 + 1967634047 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17043 \beta_{11} - 93077 \beta_{10} - 11312 \beta_{9} + 78053 \beta_{8} - 71950 \beta_{7} - 206081 \beta_{6} - 236838 \beta_{5} - 89465 \beta_{4} + 2164592 \beta_{3} - 22115218 \beta_{2} + \cdots + 98912518029 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 761645 \beta_{11} - 8352331 \beta_{10} + 10510980 \beta_{9} + 603293 \beta_{8} + 3078664 \beta_{7} + 15518465 \beta_{6} - 22414892 \beta_{5} - 12190397 \beta_{4} + \cdots + 49619449579813 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 64786283 \beta_{11} - 2332805471 \beta_{10} - 383670104 \beta_{9} + 1685964919 \beta_{8} - 2355025586 \beta_{7} - 4898853667 \beta_{6} + \cdots + 34\!\cdots\!75 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 64803741831 \beta_{11} - 224240419747 \beta_{10} + 354381826612 \beta_{9} + 38523685577 \beta_{8} - 11048048820 \beta_{7} + 527941453577 \beta_{6} + \cdots + 13\!\cdots\!41 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 19433414604075 \beta_{11} - 53468004726027 \beta_{10} - 8195782569304 \beta_{9} + 33490022108623 \beta_{8} - 76637762750094 \beta_{7} + \cdots + 11\!\cdots\!87 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 47\!\cdots\!31 \beta_{11} + \cdots + 37\!\cdots\!53 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10\!\cdots\!55 \beta_{11} + \cdots + 38\!\cdots\!11 ) / 8 \) 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
−155.855
−142.457
−87.2780
−64.3067
−61.7354
−34.7036
3.23741
15.1401
94.1602
98.0924
162.150
177.556
−333.711 −2416.34 78595.0 165276. 806359. −2.92451e6 −1.52930e7 −8.51021e6 −5.51544e7
1.2 −306.914 1666.99 61428.0 64027.3 −511621. 1.64942e6 −8.79615e6 −1.15701e7 −1.96509e7
1.3 −196.556 −4944.68 5866.22 −184879. 971906. −2.97859e6 5.28770e6 1.01009e7 3.63390e7
1.4 −150.613 3985.78 −10083.6 347827. −600312. −3.26400e6 6.45403e6 1.53751e6 −5.23874e7
1.5 −145.471 254.987 −11606.2 −233086. −37093.1 2.20456e6 6.45516e6 −1.42839e7 3.39071e7
1.6 −91.4073 6050.26 −24412.7 −98557.7 −553038. 1.04995e6 5.22673e6 2.22568e7 9.00889e6
1.7 −15.5252 −2381.89 −32527.0 172933. 36979.3 −570604. 1.01372e6 −8.67549e6 −2.68481e6
1.8 8.28020 −6881.62 −32699.4 −91711.3 −56981.2 1.74802e6 −542083. 3.30077e7 −759388.
1.9 166.320 5379.02 −5105.54 −246835. 894641. 345495. −6.29914e6 1.45850e7 −4.10537e7
1.10 174.185 1818.41 −2427.63 106224. 316739. −648096. −6.13055e6 −1.10423e7 1.85026e7
1.11 302.300 1384.15 58617.0 −149481. 418427. −3.04303e6 7.81414e6 −1.24330e7 −4.51879e7
1.12 333.112 −5660.06 78195.9 −56214.4 −1.88544e6 714042. 1.51326e7 1.76874e7 −1.87257e7
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 23.16.a.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.16.a.a 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 256 T_{2}^{11} - 245760 T_{2}^{10} - 66355616 T_{2}^{9} + 18854461712 T_{2}^{8} + 5756110997824 T_{2}^{7} - 415151179919360 T_{2}^{6} + \cdots - 15\!\cdots\!56 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(23))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 256 T^{11} + \cdots - 15\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{12} + 1745 T^{11} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{12} + 204476 T^{11} + \cdots - 54\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + 5717328 T^{11} + \cdots + 52\!\cdots\!72 \) Copy content Toggle raw display
$11$ \( T^{12} + 87636002 T^{11} + \cdots - 58\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( T^{12} + 292496079 T^{11} + \cdots - 45\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{12} + 2462528162 T^{11} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} - 175321758 T^{11} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T - 3404825447)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + 164667697193 T^{11} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} - 20222384151 T^{11} + \cdots + 19\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{12} + 869669414912 T^{11} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{12} + 7510147709883 T^{11} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + 5682603487020 T^{11} + \cdots + 50\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{12} + 5828073094301 T^{11} + \cdots - 29\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{12} - 1452974784324 T^{11} + \cdots - 96\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{12} + 11503084624084 T^{11} + \cdots - 59\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{12} + 23587566667200 T^{11} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{12} - 61525019345122 T^{11} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{12} - 197895887067063 T^{11} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + 22888563242709 T^{11} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{12} - 229938065096294 T^{11} + \cdots - 29\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{12} - 369402590629184 T^{11} + \cdots - 37\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{12} + 85082918654478 T^{11} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
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