Properties

Label 10.8.b.a
Level 10
Weight 8
Character orbit 10.b
Analytic conductor 3.124
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 10.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.12385025484\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{31})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 3 \beta_{1} - \beta_{3} ) q^{3} -64 q^{4} + ( 15 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{5} + ( 224 + 2 \beta_{2} ) q^{6} + ( -29 \beta_{1} - 7 \beta_{3} ) q^{7} + 64 \beta_{1} q^{8} + ( -1697 - 14 \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 3 \beta_{1} - \beta_{3} ) q^{3} -64 q^{4} + ( 15 - 4 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{5} + ( 224 + 2 \beta_{2} ) q^{6} + ( -29 \beta_{1} - 7 \beta_{3} ) q^{7} + 64 \beta_{1} q^{8} + ( -1697 - 14 \beta_{2} ) q^{9} + ( -160 + \beta_{1} + 6 \beta_{2} + 32 \beta_{3} ) q^{10} + ( 4452 - 4 \beta_{2} ) q^{11} + ( -192 \beta_{1} + 64 \beta_{3} ) q^{12} + ( -180 \beta_{1} + 78 \beta_{3} ) q^{13} + ( -1632 + 14 \beta_{2} ) q^{14} + ( -8740 + 1539 \beta_{1} - 16 \beta_{2} - 127 \beta_{3} ) q^{15} + 4096 q^{16} + ( -2812 \beta_{1} - 168 \beta_{3} ) q^{17} + ( 1473 \beta_{1} - 448 \beta_{3} ) q^{18} + ( 15900 + 156 \beta_{2} ) q^{19} + ( -960 + 256 \beta_{1} - 64 \beta_{2} + 192 \beta_{3} ) q^{20} + ( -15988 + 2 \beta_{2} ) q^{21} + ( -4516 \beta_{1} - 128 \beta_{3} ) q^{22} + ( 4903 \beta_{1} + 849 \beta_{3} ) q^{23} + ( -14336 - 128 \beta_{2} ) q^{24} + ( 21525 + 9260 \beta_{1} + 60 \beta_{2} + 70 \beta_{3} ) q^{25} + ( -14016 - 156 \beta_{2} ) q^{26} + ( -19446 \beta_{1} + 1078 \beta_{3} ) q^{27} + ( 1856 \beta_{1} + 448 \beta_{3} ) q^{28} + ( 42390 - 648 \beta_{2} ) q^{29} + ( 102560 + 8484 \beta_{1} + 254 \beta_{2} - 512 \beta_{3} ) q^{30} + ( -98528 + 312 \beta_{2} ) q^{31} -4096 \beta_{1} q^{32} + ( 7380 \beta_{1} - 4004 \beta_{3} ) q^{33} + ( -174592 + 336 \beta_{2} ) q^{34} + ( -69180 + 10823 \beta_{1} + 188 \beta_{2} + 711 \beta_{3} ) q^{35} + ( 108608 + 896 \beta_{2} ) q^{36} + ( -14432 \beta_{1} - 4438 \beta_{3} ) q^{37} + ( -13404 \beta_{1} + 4992 \beta_{3} ) q^{38} + ( 290856 + 984 \beta_{2} ) q^{39} + ( 10240 - 64 \beta_{1} - 384 \beta_{2} - 2048 \beta_{3} ) q^{40} + ( 58122 - 1942 \beta_{2} ) q^{41} + ( 16020 \beta_{1} + 64 \beta_{3} ) q^{42} + ( 16907 \beta_{1} - 653 \beta_{3} ) q^{43} + ( -284928 + 256 \beta_{2} ) q^{44} + ( -719855 - 58872 \beta_{1} - 1907 \beta_{2} + 3971 \beta_{3} ) q^{45} + ( 286624 - 1698 \beta_{2} ) q^{46} + ( 68911 \beta_{1} + 16413 \beta_{3} ) q^{47} + ( 12288 \beta_{1} - 4096 \beta_{3} ) q^{48} + ( 630027 + 714 \beta_{2} ) q^{49} + ( 590400 - 20565 \beta_{1} - 140 \beta_{2} + 1920 \beta_{3} ) q^{50} + ( 90272 + 4280 \beta_{2} ) q^{51} + ( 11520 \beta_{1} - 4992 \beta_{3} ) q^{52} + ( -119100 \beta_{1} + 12738 \beta_{3} ) q^{53} + ( -1279040 - 2156 \beta_{2} ) q^{54} + ( -131620 - 36568 \beta_{1} + 4392 \beta_{2} - 13676 \beta_{3} ) q^{55} + ( 104448 - 896 \beta_{2} ) q^{56} + ( 280764 \beta_{1} - 33372 \beta_{3} ) q^{57} + ( -52758 \beta_{1} - 20736 \beta_{3} ) q^{58} + ( 523380 - 2956 \beta_{2} ) q^{59} + ( 559360 - 98496 \beta_{1} + 1024 \beta_{2} + 8128 \beta_{3} ) q^{60} + ( -1312858 - 366 \beta_{2} ) q^{61} + ( 103520 \beta_{1} + 9984 \beta_{3} ) q^{62} + ( -108399 \beta_{1} + 455 \beta_{3} ) q^{63} -262144 q^{64} + ( 690360 - 120096 \beta_{1} + 924 \beta_{2} + 8178 \beta_{3} ) q^{65} + ( 600448 + 8008 \beta_{2} ) q^{66} + ( -89041 \beta_{1} + 31759 \beta_{3} ) q^{67} + ( 179968 \beta_{1} + 10752 \beta_{3} ) q^{68} + ( 1628716 - 3014 \beta_{2} ) q^{69} + ( 669920 + 72188 \beta_{1} - 1422 \beta_{2} + 6016 \beta_{3} ) q^{70} + ( -1958088 - 13296 \beta_{2} ) q^{71} + ( -94272 \beta_{1} + 28672 \beta_{3} ) q^{72} + ( 173540 \beta_{1} + 21268 \beta_{3} ) q^{73} + ( -781632 + 8876 \beta_{2} ) q^{74} + ( -1849400 + 154215 \beta_{1} - 17960 \beta_{2} - 28245 \beta_{3} ) q^{75} + ( -1017600 - 9984 \beta_{2} ) q^{76} + ( -174140 \beta_{1} - 34428 \beta_{3} ) q^{77} + ( -275112 \beta_{1} + 31488 \beta_{3} ) q^{78} + ( 1931760 + 12456 \beta_{2} ) q^{79} + ( 61440 - 16384 \beta_{1} + 4096 \beta_{2} - 12288 \beta_{3} ) q^{80} + ( 4107101 + 16898 \beta_{2} ) q^{81} + ( -89194 \beta_{1} - 62144 \beta_{3} ) q^{82} + ( 264631 \beta_{1} - 118665 \beta_{3} ) q^{83} + ( 1023232 - 128 \beta_{2} ) q^{84} + ( -1998880 + 261868 \beta_{1} + 17208 \beta_{2} + 84776 \beta_{3} ) q^{85} + ( 1102944 + 1306 \beta_{2} ) q^{86} + ( -840942 \beta_{1} + 30186 \beta_{3} ) q^{87} + ( 289024 \beta_{1} + 8192 \beta_{3} ) q^{88} + ( -8367510 + 8680 \beta_{2} ) q^{89} + ( -3894880 + 689343 \beta_{1} - 7942 \beta_{2} - 61024 \beta_{3} ) q^{90} + ( 1335192 - 912 \beta_{2} ) q^{91} + ( -313792 \beta_{1} - 54336 \beta_{3} ) q^{92} + ( 170544 \beta_{1} + 63584 \beta_{3} ) q^{93} + ( 3885088 - 32826 \beta_{2} ) q^{94} + ( 7976100 + 668040 \beta_{1} + 18240 \beta_{2} - 35220 \beta_{3} ) q^{95} + ( 917504 + 8192 \beta_{2} ) q^{96} + ( -730348 \beta_{1} + 114880 \beta_{3} ) q^{97} + ( -618603 \beta_{1} + 22848 \beta_{3} ) q^{98} + ( -4777444 - 55540 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 256q^{4} + 60q^{5} + 896q^{6} - 6788q^{9} + O(q^{10}) \) \( 4q - 256q^{4} + 60q^{5} + 896q^{6} - 6788q^{9} - 640q^{10} + 17808q^{11} - 6528q^{14} - 34960q^{15} + 16384q^{16} + 63600q^{19} - 3840q^{20} - 63952q^{21} - 57344q^{24} + 86100q^{25} - 56064q^{26} + 169560q^{29} + 410240q^{30} - 394112q^{31} - 698368q^{34} - 276720q^{35} + 434432q^{36} + 1163424q^{39} + 40960q^{40} + 232488q^{41} - 1139712q^{44} - 2879420q^{45} + 1146496q^{46} + 2520108q^{49} + 2361600q^{50} + 361088q^{51} - 5116160q^{54} - 526480q^{55} + 417792q^{56} + 2093520q^{59} + 2237440q^{60} - 5251432q^{61} - 1048576q^{64} + 2761440q^{65} + 2401792q^{66} + 6514864q^{69} + 2679680q^{70} - 7832352q^{71} - 3126528q^{74} - 7397600q^{75} - 4070400q^{76} + 7727040q^{79} + 245760q^{80} + 16428404q^{81} + 4092928q^{84} - 7995520q^{85} + 4411776q^{86} - 33470040q^{89} - 15579520q^{90} + 5340768q^{91} + 15540352q^{94} + 31904400q^{95} + 3670016q^{96} - 19109776q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 15 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{3} + 7 \nu \)
\(\beta_{2}\)\(=\)\( -5 \nu^{3} + 115 \nu \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 40 \nu^{2} - 7 \nu - 300 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 5 \beta_{1}\)\()/80\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{1} + 300\)\()/40\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{2} - 115 \beta_{1}\)\()/80\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−2.78388 0.500000i
2.78388 0.500000i
2.78388 + 0.500000i
−2.78388 + 0.500000i
8.00000i 27.6776i −64.0000 −207.711 187.033i −221.421 593.744i 512.000i 1420.95 −1496.26 + 1661.68i
9.2 8.00000i 83.6776i −64.0000 237.711 + 147.033i 669.421 185.744i 512.000i −4814.95 1176.26 1901.68i
9.3 8.00000i 83.6776i −64.0000 237.711 147.033i 669.421 185.744i 512.000i −4814.95 1176.26 + 1901.68i
9.4 8.00000i 27.6776i −64.0000 −207.711 + 187.033i −221.421 593.744i 512.000i 1420.95 −1496.26 1661.68i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.8.b.a 4
3.b odd 2 1 90.8.c.c 4
4.b odd 2 1 80.8.c.d 4
5.b even 2 1 inner 10.8.b.a 4
5.c odd 4 1 50.8.a.i 2
5.c odd 4 1 50.8.a.j 2
8.b even 2 1 320.8.c.f 4
8.d odd 2 1 320.8.c.g 4
15.d odd 2 1 90.8.c.c 4
15.e even 4 1 450.8.a.bd 2
15.e even 4 1 450.8.a.bi 2
20.d odd 2 1 80.8.c.d 4
20.e even 4 1 400.8.a.t 2
20.e even 4 1 400.8.a.bf 2
40.e odd 2 1 320.8.c.g 4
40.f even 2 1 320.8.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.8.b.a 4 1.a even 1 1 trivial
10.8.b.a 4 5.b even 2 1 inner
50.8.a.i 2 5.c odd 4 1
50.8.a.j 2 5.c odd 4 1
80.8.c.d 4 4.b odd 2 1
80.8.c.d 4 20.d odd 2 1
90.8.c.c 4 3.b odd 2 1
90.8.c.c 4 15.d odd 2 1
320.8.c.f 4 8.b even 2 1
320.8.c.f 4 40.f even 2 1
320.8.c.g 4 8.d odd 2 1
320.8.c.g 4 40.e odd 2 1
400.8.a.t 2 20.e even 4 1
400.8.a.bf 2 20.e even 4 1
450.8.a.bd 2 15.e even 4 1
450.8.a.bi 2 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 64 T^{2} )^{2} \)
$3$ \( 1 - 980 T^{2} + 84438 T^{4} - 4687309620 T^{6} + 22876792454961 T^{8} \)
$5$ \( 1 - 60 T - 41250 T^{2} - 4687500 T^{3} + 6103515625 T^{4} \)
$7$ \( 1 - 2907140 T^{2} + 3444026008998 T^{4} - 1971689424002241860 T^{6} + \)\(45\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 - 8904 T + 58001046 T^{2} - 173513770584 T^{3} + 379749833583241 T^{4} )^{2} \)
$13$ \( 1 - 207134260 T^{2} + 18369334894869078 T^{4} - \)\(81\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \)
$17$ \( 1 - 513791940 T^{2} + 236061419749305158 T^{4} - \)\(86\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \)
$19$ \( ( 1 - 31800 T + 833487878 T^{2} - 28425121300200 T^{3} + 799006685782884121 T^{4} )^{2} \)
$23$ \( 1 - 6583044420 T^{2} + 22546653652650262118 T^{4} - \)\(76\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!81\)\( T^{8} \)
$29$ \( ( 1 - 84780 T + 15469426318 T^{2} - 1462444513477020 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} )^{2} \)
$31$ \( ( 1 + 197056 T + 59904732606 T^{2} + 5421525686257216 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$37$ \( 1 - 238521132500 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{6} + \)\(81\!\cdots\!21\)\( T^{8} \)
$41$ \( ( 1 - 116244 T + 205827060246 T^{2} - 22639015813022964 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} )^{2} \)
$43$ \( 1 - 1046615537780 T^{2} + \)\(42\!\cdots\!98\)\( T^{4} - \)\(77\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 115388894940 T^{2} - \)\(27\!\cdots\!62\)\( T^{4} + \)\(29\!\cdots\!60\)\( T^{6} + \)\(65\!\cdots\!61\)\( T^{8} \)
$53$ \( 1 - 1677817211540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(23\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!61\)\( T^{8} \)
$59$ \( ( 1 - 1046760 T + 4817827968438 T^{2} - 2605020828249136440 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} )^{2} \)
$61$ \( ( 1 + 2625716 T + 8002437582606 T^{2} + 8251950148425716036 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} )^{2} \)
$67$ \( 1 - 16580251270100 T^{2} + \)\(13\!\cdots\!58\)\( T^{4} - \)\(60\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!41\)\( T^{8} \)
$71$ \( ( 1 + 3916176 T + 13255881578926 T^{2} + 35618091281407032816 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} )^{2} \)
$73$ \( 1 - 37988250868580 T^{2} + \)\(59\!\cdots\!18\)\( T^{4} - \)\(46\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!81\)\( T^{8} \)
$79$ \( ( 1 - 3863520 T + 34443978644318 T^{2} - 74194686446205019680 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} )^{2} \)
$83$ \( 1 - 7805733448980 T^{2} + \)\(31\!\cdots\!58\)\( T^{4} - \)\(57\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!41\)\( T^{8} \)
$89$ \( ( 1 + 16735020 T + 154740910351158 T^{2} + \)\(74\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} )^{2} \)
$97$ \( 1 - 161931097215620 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(42\!\cdots\!61\)\( T^{8} \)
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