Properties

Label 210.12.a.p
Level $210$
Weight $12$
Character orbit 210.a
Self dual yes
Analytic conductor $161.352$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 210.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(161.352067918\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2535805712 x^{2} - 66934369575900 x - 478525314115194389\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5\cdot 7 \)
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.

\(f(q)\) \(=\) \( q + 32 q^{2} + 243 q^{3} + 1024 q^{4} + 3125 q^{5} + 7776 q^{6} + 16807 q^{7} + 32768 q^{8} + 59049 q^{9} +O(q^{10})\) \( q + 32 q^{2} + 243 q^{3} + 1024 q^{4} + 3125 q^{5} + 7776 q^{6} + 16807 q^{7} + 32768 q^{8} + 59049 q^{9} + 100000 q^{10} + ( 114565 - \beta_{1} ) q^{11} + 248832 q^{12} + ( 393579 - \beta_{2} ) q^{13} + 537824 q^{14} + 759375 q^{15} + 1048576 q^{16} + ( 2169518 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + 1889568 q^{18} + ( 3110501 - 3 \beta_{1} - \beta_{3} ) q^{19} + 3200000 q^{20} + 4084101 q^{21} + ( 3666080 - 32 \beta_{1} ) q^{22} + ( 128272 - 3 \beta_{1} - \beta_{2} + 8 \beta_{3} ) q^{23} + 7962624 q^{24} + 9765625 q^{25} + ( 12594528 - 32 \beta_{2} ) q^{26} + 14348907 q^{27} + 17210368 q^{28} + ( 14619174 + 117 \beta_{1} + 57 \beta_{2} + 13 \beta_{3} ) q^{29} + 24300000 q^{30} + ( 36297393 - 124 \beta_{1} + 57 \beta_{2} - 19 \beta_{3} ) q^{31} + 33554432 q^{32} + ( 27839295 - 243 \beta_{1} ) q^{33} + ( 69424576 - 32 \beta_{1} - 32 \beta_{2} - 32 \beta_{3} ) q^{34} + 52521875 q^{35} + 60466176 q^{36} + ( 85228188 + 593 \beta_{1} + 177 \beta_{2} + 10 \beta_{3} ) q^{37} + ( 99536032 - 96 \beta_{1} - 32 \beta_{3} ) q^{38} + ( 95639697 - 243 \beta_{2} ) q^{39} + 102400000 q^{40} + ( 228786842 + 1548 \beta_{1} + 418 \beta_{2} - 65 \beta_{3} ) q^{41} + 130691232 q^{42} + ( 115561006 + 1431 \beta_{1} - 605 \beta_{2} + 144 \beta_{3} ) q^{43} + ( 117314560 - 1024 \beta_{1} ) q^{44} + 184528125 q^{45} + ( 4104704 - 96 \beta_{1} - 32 \beta_{2} + 256 \beta_{3} ) q^{46} + ( 225427510 - 2769 \beta_{1} - 427 \beta_{2} - 149 \beta_{3} ) q^{47} + 254803968 q^{48} + 282475249 q^{49} + 312500000 q^{50} + ( 527192874 - 243 \beta_{1} - 243 \beta_{2} - 243 \beta_{3} ) q^{51} + ( 403024896 - 1024 \beta_{2} ) q^{52} + ( -39486447 - 1933 \beta_{1} + 1672 \beta_{2} + 427 \beta_{3} ) q^{53} + 459165024 q^{54} + ( 358015625 - 3125 \beta_{1} ) q^{55} + 550731776 q^{56} + ( 755851743 - 729 \beta_{1} - 243 \beta_{3} ) q^{57} + ( 467813568 + 3744 \beta_{1} + 1824 \beta_{2} + 416 \beta_{3} ) q^{58} + ( 676747282 + 3340 \beta_{1} - 968 \beta_{2} + 277 \beta_{3} ) q^{59} + 777600000 q^{60} + ( 2185211730 + 3823 \beta_{1} + 229 \beta_{2} - 1510 \beta_{3} ) q^{61} + ( 1161516576 - 3968 \beta_{1} + 1824 \beta_{2} - 608 \beta_{3} ) q^{62} + 992436543 q^{63} + 1073741824 q^{64} + ( 1229934375 - 3125 \beta_{2} ) q^{65} + ( 890857440 - 7776 \beta_{1} ) q^{66} + ( 1470783592 + 1897 \beta_{1} - 969 \beta_{2} - 12 \beta_{3} ) q^{67} + ( 2221586432 - 1024 \beta_{1} - 1024 \beta_{2} - 1024 \beta_{3} ) q^{68} + ( 31170096 - 729 \beta_{1} - 243 \beta_{2} + 1944 \beta_{3} ) q^{69} + 1680700000 q^{70} + ( 86003843 - 14426 \beta_{1} + 11625 \beta_{2} + 1011 \beta_{3} ) q^{71} + 1934917632 q^{72} + ( 2637426561 + 24328 \beta_{1} - 13031 \beta_{2} + 1102 \beta_{3} ) q^{73} + ( 2727302016 + 18976 \beta_{1} + 5664 \beta_{2} + 320 \beta_{3} ) q^{74} + 2373046875 q^{75} + ( 3185153024 - 3072 \beta_{1} - 1024 \beta_{3} ) q^{76} + ( 1925493955 - 16807 \beta_{1} ) q^{77} + ( 3060470304 - 7776 \beta_{2} ) q^{78} + ( -107544494 - 18510 \beta_{1} + 16786 \beta_{2} + 2531 \beta_{3} ) q^{79} + 3276800000 q^{80} + 3486784401 q^{81} + ( 7321178944 + 49536 \beta_{1} + 13376 \beta_{2} - 2080 \beta_{3} ) q^{82} + ( 7126235358 + 21328 \beta_{1} - 18982 \beta_{2} - 2318 \beta_{3} ) q^{83} + 4182119424 q^{84} + ( 6779743750 - 3125 \beta_{1} - 3125 \beta_{2} - 3125 \beta_{3} ) q^{85} + ( 3697952192 + 45792 \beta_{1} - 19360 \beta_{2} + 4608 \beta_{3} ) q^{86} + ( 3552459282 + 28431 \beta_{1} + 13851 \beta_{2} + 3159 \beta_{3} ) q^{87} + ( 3754065920 - 32768 \beta_{1} ) q^{88} + ( 6690946670 - 58004 \beta_{1} - 11958 \beta_{2} - 4611 \beta_{3} ) q^{89} + 5904900000 q^{90} + ( 6614882253 - 16807 \beta_{2} ) q^{91} + ( 131350528 - 3072 \beta_{1} - 1024 \beta_{2} + 8192 \beta_{3} ) q^{92} + ( 8820266499 - 30132 \beta_{1} + 13851 \beta_{2} - 4617 \beta_{3} ) q^{93} + ( 7213680320 - 88608 \beta_{1} - 13664 \beta_{2} - 4768 \beta_{3} ) q^{94} + ( 9720315625 - 9375 \beta_{1} - 3125 \beta_{3} ) q^{95} + 8153726976 q^{96} + ( 15597497619 + 60808 \beta_{1} - 17369 \beta_{2} - 4762 \beta_{3} ) q^{97} + 9039207968 q^{98} + ( 6764948685 - 59049 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 128q^{2} + 972q^{3} + 4096q^{4} + 12500q^{5} + 31104q^{6} + 67228q^{7} + 131072q^{8} + 236196q^{9} + O(q^{10}) \) \( 4q + 128q^{2} + 972q^{3} + 4096q^{4} + 12500q^{5} + 31104q^{6} + 67228q^{7} + 131072q^{8} + 236196q^{9} + 400000q^{10} + 458260q^{11} + 995328q^{12} + 1574316q^{13} + 2151296q^{14} + 3037500q^{15} + 4194304q^{16} + 8678072q^{17} + 7558272q^{18} + 12442004q^{19} + 12800000q^{20} + 16336404q^{21} + 14664320q^{22} + 513088q^{23} + 31850496q^{24} + 39062500q^{25} + 50378112q^{26} + 57395628q^{27} + 68841472q^{28} + 58476696q^{29} + 97200000q^{30} + 145189572q^{31} + 134217728q^{32} + 111357180q^{33} + 277698304q^{34} + 210087500q^{35} + 241864704q^{36} + 340912752q^{37} + 398144128q^{38} + 382558788q^{39} + 409600000q^{40} + 915147368q^{41} + 522764928q^{42} + 462244024q^{43} + 469258240q^{44} + 738112500q^{45} + 16418816q^{46} + 901710040q^{47} + 1019215872q^{48} + 1129900996q^{49} + 1250000000q^{50} + 2108771496q^{51} + 1612099584q^{52} - 157945788q^{53} + 1836660096q^{54} + 1432062500q^{55} + 2202927104q^{56} + 3023406972q^{57} + 1871254272q^{58} + 2706989128q^{59} + 3110400000q^{60} + 8740846920q^{61} + 4646066304q^{62} + 3969746172q^{63} + 4294967296q^{64} + 4919737500q^{65} + 3563429760q^{66} + 5883134368q^{67} + 8886345728q^{68} + 124680384q^{69} + 6722800000q^{70} + 344015372q^{71} + 7739670528q^{72} + 10549706244q^{73} + 10909208064q^{74} + 9492187500q^{75} + 12740612096q^{76} + 7701975820q^{77} + 12241881216q^{78} - 430177976q^{79} + 13107200000q^{80} + 13947137604q^{81} + 29284715776q^{82} + 28504941432q^{83} + 16728477696q^{84} + 27118975000q^{85} + 14791808768q^{86} + 14209837128q^{87} + 15016263680q^{88} + 26763786680q^{89} + 23619600000q^{90} + 26459529012q^{91} + 525402112q^{92} + 35281065996q^{93} + 28854721280q^{94} + 38881262500q^{95} + 32614907904q^{96} + 62389990476q^{97} + 36156831872q^{98} + 27059794740q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2535805712 x^{2} - 66934369575900 x - 478525314115194389\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -17 \nu^{3} + 299755 \nu^{2} + 37791862847 \nu + 473352991492445 \)\()/54032670\)
\(\beta_{2}\)\(=\)\((\)\( 521 \nu^{3} - 11729323 \nu^{2} - 1033756312751 \nu - 11282962781136437 \)\()/ 594359370 \)
\(\beta_{3}\)\(=\)\((\)\( -67 \nu^{3} + 1605173 \nu^{2} + 131984499037 \nu + 1328248640114887 \)\()/9005445\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 11 \beta_{2} + 7 \beta_{1}\)\()/420\)
\(\nu^{2}\)\(=\)\((\)\(3263 \beta_{3} + 29348 \beta_{2} + 4606 \beta_{1} + 35501279968\)\()/28\)
\(\nu^{3}\)\(=\)\((\)\(3086080666 \beta_{3} + 32215802201 \beta_{2} + 15444670087 \beta_{1} + 21084326416408500\)\()/420\)

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
−14006.7
61281.8
−24753.1
−22522.1
32.0000 243.000 1024.00 3125.00 7776.00 16807.0 32768.0 59049.0 100000.
1.2 32.0000 243.000 1024.00 3125.00 7776.00 16807.0 32768.0 59049.0 100000.
1.3 32.0000 243.000 1024.00 3125.00 7776.00 16807.0 32768.0 59049.0 100000.
1.4 32.0000 243.000 1024.00 3125.00 7776.00 16807.0 32768.0 59049.0 100000.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 210.12.a.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.12.a.p 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} - 458260 T_{11}^{3} - 585819000032 T_{11}^{2} + \)\(31\!\cdots\!96\)\( T_{11} - \)\(18\!\cdots\!00\)\( \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(210))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -32 + T )^{4} \)
$3$ \( ( -243 + T )^{4} \)
$5$ \( ( -3125 + T )^{4} \)
$7$ \( ( -16807 + T )^{4} \)
$11$ \( -\)\(18\!\cdots\!00\)\( + 318363769618029696 T - 585819000032 T^{2} - 458260 T^{3} + T^{4} \)
$13$ \( -\)\(84\!\cdots\!16\)\( + 4319474065293120048 T - 2469917037392 T^{2} - 1574316 T^{3} + T^{4} \)
$17$ \( \)\(34\!\cdots\!56\)\( + \)\(13\!\cdots\!08\)\( T - 28257501713768 T^{2} - 8678072 T^{3} + T^{4} \)
$19$ \( -\)\(14\!\cdots\!00\)\( + 83307278547070441600 T - 2026190020800 T^{2} - 12442004 T^{3} + T^{4} \)
$23$ \( \)\(13\!\cdots\!00\)\( - \)\(97\!\cdots\!76\)\( T - 3581027802806240 T^{2} - 513088 T^{3} + T^{4} \)
$29$ \( \)\(16\!\cdots\!00\)\( + \)\(56\!\cdots\!00\)\( T - 23792456041653960 T^{2} - 58476696 T^{3} + T^{4} \)
$31$ \( \)\(24\!\cdots\!48\)\( + \)\(13\!\cdots\!80\)\( T - 36992949765805712 T^{2} - 145189572 T^{3} + T^{4} \)
$37$ \( \)\(20\!\cdots\!04\)\( + \)\(48\!\cdots\!92\)\( T - 261991076271922520 T^{2} - 340912752 T^{3} + T^{4} \)
$41$ \( \)\(26\!\cdots\!32\)\( + \)\(16\!\cdots\!92\)\( T - 2001689030148759272 T^{2} - 915147368 T^{3} + T^{4} \)
$43$ \( \)\(26\!\cdots\!56\)\( + \)\(18\!\cdots\!84\)\( T - 4054674812906585312 T^{2} - 462244024 T^{3} + T^{4} \)
$47$ \( \)\(45\!\cdots\!92\)\( - \)\(36\!\cdots\!72\)\( T - 5942739277581465056 T^{2} - 901710040 T^{3} + T^{4} \)
$53$ \( \)\(45\!\cdots\!00\)\( - \)\(58\!\cdots\!32\)\( T - 21897200153916144864 T^{2} + 157945788 T^{3} + T^{4} \)
$59$ \( \)\(12\!\cdots\!00\)\( + \)\(29\!\cdots\!60\)\( T - 13331378235215084288 T^{2} - 2706989128 T^{3} + T^{4} \)
$61$ \( \)\(13\!\cdots\!52\)\( + \)\(82\!\cdots\!84\)\( T - \)\(11\!\cdots\!44\)\( T^{2} - 8740846920 T^{3} + T^{4} \)
$67$ \( -\)\(10\!\cdots\!80\)\( + \)\(21\!\cdots\!44\)\( T + 6813300833684782528 T^{2} - 5883134368 T^{3} + T^{4} \)
$71$ \( \)\(61\!\cdots\!00\)\( + \)\(18\!\cdots\!68\)\( T - \)\(68\!\cdots\!44\)\( T^{2} - 344015372 T^{3} + T^{4} \)
$73$ \( \)\(26\!\cdots\!84\)\( + \)\(30\!\cdots\!56\)\( T - \)\(11\!\cdots\!40\)\( T^{2} - 10549706244 T^{3} + T^{4} \)
$79$ \( \)\(14\!\cdots\!00\)\( - \)\(42\!\cdots\!80\)\( T - \)\(15\!\cdots\!92\)\( T^{2} + 430177976 T^{3} + T^{4} \)
$83$ \( -\)\(12\!\cdots\!32\)\( + \)\(29\!\cdots\!20\)\( T - \)\(15\!\cdots\!92\)\( T^{2} - 28504941432 T^{3} + T^{4} \)
$89$ \( \)\(79\!\cdots\!00\)\( - \)\(14\!\cdots\!80\)\( T - \)\(31\!\cdots\!96\)\( T^{2} - 26763786680 T^{3} + T^{4} \)
$97$ \( \)\(58\!\cdots\!48\)\( - \)\(32\!\cdots\!20\)\( T - \)\(35\!\cdots\!28\)\( T^{2} - 62389990476 T^{3} + T^{4} \)
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