Properties

Label 5.13.c.a
Level $5$
Weight $13$
Character orbit 5.c
Analytic conductor $4.570$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 5.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56996908638\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \( x^{10} + 7950x^{8} + 16939113x^{6} + 4574579500x^{4} + 337520899536x^{2} + 6615595526400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{2}\cdot 5^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (3 \beta_{6} - \beta_{3} - 31 \beta_1 + 31) q^{3} + ( - \beta_{8} + 8 \beta_{6} + 8 \beta_{5} + 1549 \beta_1) q^{4} + (2 \beta_{8} + \beta_{7} - 27 \beta_{6} - 6 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \cdots - 420) q^{5}+ \cdots + ( - 3 \beta_{9} + 55 \beta_{8} + \beta_{7} + 2471 \beta_{6} + 2471 \beta_{5} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (3 \beta_{6} - \beta_{3} - 31 \beta_1 + 31) q^{3} + ( - \beta_{8} + 8 \beta_{6} + 8 \beta_{5} + 1549 \beta_1) q^{4} + (2 \beta_{8} + \beta_{7} - 27 \beta_{6} - 6 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \cdots - 420) q^{5}+ \cdots + (15997938 \beta_{9} + 18816424 \beta_{8} + \cdots + 10665292) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} + 318 q^{3} - 4250 q^{5} - 175080 q^{6} + 279598 q^{7} - 469980 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} + 318 q^{3} - 4250 q^{5} - 175080 q^{6} + 279598 q^{7} - 469980 q^{8} + 1552150 q^{10} + 312620 q^{11} - 2359992 q^{12} + 5290738 q^{13} - 5821650 q^{15} + 30547960 q^{16} - 41269502 q^{17} - 140573742 q^{18} + 334988100 q^{20} + 107493420 q^{21} - 155490544 q^{22} - 510099842 q^{23} + 942201250 q^{25} + 1475846420 q^{26} - 1993958640 q^{27} - 3562106488 q^{28} + 5922516000 q^{30} + 3077089820 q^{31} - 4623883832 q^{32} - 7503698004 q^{33} + 9330787150 q^{35} + 7760793660 q^{36} - 2599618502 q^{37} - 15310240920 q^{38} + 15901243500 q^{40} + 7412079020 q^{41} - 18593270064 q^{42} - 5784410402 q^{43} - 10510145100 q^{45} - 7382547880 q^{46} + 16053249598 q^{47} + 42572492208 q^{48} - 68314688750 q^{50} - 33139878180 q^{51} + 96763417228 q^{52} + 101763514618 q^{53} - 84180068500 q^{55} - 172002747600 q^{56} + 27733489920 q^{57} + 135238672320 q^{58} - 220124568600 q^{60} + 7731718220 q^{61} + 193287375176 q^{62} + 207465112158 q^{63} - 338075024150 q^{65} - 60815472960 q^{66} - 80010636002 q^{67} + 204699541412 q^{68} + 376969924200 q^{70} - 46557252580 q^{71} - 13986370620 q^{72} - 448527032342 q^{73} + 719724648750 q^{75} + 305095930800 q^{76} - 425580405844 q^{77} - 1690993241784 q^{78} + 873032236000 q^{80} + 1107831051810 q^{81} - 671946416464 q^{82} - 91118376722 q^{83} + 543768569650 q^{85} + 414117747320 q^{86} - 2078422804320 q^{87} - 1842434230560 q^{88} + 3098742811350 q^{90} + 2737742572220 q^{91} + 906853941448 q^{92} - 91295366484 q^{93} - 1044695070000 q^{95} - 3473259523680 q^{96} - 1409507601302 q^{97} - 1481746533298 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 7950x^{8} + 16939113x^{6} + 4574579500x^{4} + 337520899536x^{2} + 6615595526400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 26465 \nu^{9} + 176059482 \nu^{7} + 167682655437 \nu^{5} - 471206316514012 \nu^{3} - 74\!\cdots\!92 \nu ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2320992487 \nu^{8} + 18354612359011 \nu^{6} + \cdots + 42\!\cdots\!28 ) / 38\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1460065153745 \nu^{9} + 56782570931856 \nu^{8} + \cdots - 65\!\cdots\!80 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1460065153745 \nu^{9} + 56782570931856 \nu^{8} + \cdots - 65\!\cdots\!80 ) / 14\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5235689538055 \nu^{9} + 55538770561995 \nu^{8} + \cdots + 59\!\cdots\!80 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5235689538055 \nu^{9} - 55538770561995 \nu^{8} + \cdots - 59\!\cdots\!80 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 376177622934695 \nu^{9} + \cdots + 16\!\cdots\!20 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 965372918064815 \nu^{9} + \cdots - 14\!\cdots\!76 \nu ) / 73\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 458260888172395 \nu^{9} + 686396955964080 \nu^{8} + \cdots + 11\!\cdots\!80 ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 3 \beta_{9} - 8 \beta_{8} + \beta_{7} - 115 \beta_{6} - 115 \beta_{5} + 16 \beta_{4} - 18 \beta_{3} - \beta_{2} - 641 \beta _1 - 2 ) / 1500 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{8} + 3 \beta_{7} - 1280 \beta_{6} + 1280 \beta_{5} - 272 \beta_{4} - 274 \beta_{3} - 43 \beta_{2} + 2 \beta _1 - 317606 ) / 200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 23829 \beta_{9} + 54169 \beta_{8} - 7943 \beta_{7} + 1387820 \beta_{6} + 1387820 \beta_{5} - 69338 \beta_{4} + 85224 \beta_{3} + 7943 \beta_{2} + 12075838 \beta _1 + 15886 ) / 3000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 907 \beta_{9} - 907 \beta_{8} - 2721 \beta_{7} + 1004276 \beta_{6} - 1004276 \beta_{5} + 207576 \beta_{4} + 209390 \beta_{3} + 36189 \beta_{2} - 1814 \beta _1 + 234283294 ) / 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 91566861 \beta_{9} - 210463921 \beta_{8} + 30522287 \beta_{7} - 5260650380 \beta_{6} - 5260650380 \beta_{5} + 232020842 \beta_{4} - 293065416 \beta_{3} + \cdots - 61044574 ) / 3000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 21041599 \beta_{9} + 21041599 \beta_{8} + 63124797 \beta_{7} - 18616513460 \beta_{6} + 18616513460 \beta_{5} - 3930827608 \beta_{4} - 3972910806 \beta_{3} + \cdots - 4481712980374 ) / 200 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 351810148749 \beta_{9} + 824142222289 \beta_{8} - 117270049583 \beta_{7} + 19611758189420 \beta_{6} + 19611758189420 \beta_{5} + \cdots + 234540099166 ) / 3000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 94863113647 \beta_{9} - 94863113647 \beta_{8} - 284589340941 \beta_{7} + 68767072032500 \beta_{6} - 68767072032500 \beta_{5} + \cdots + 17\!\cdots\!62 ) / 200 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 13\!\cdots\!41 \beta_{9} + \cdots - 901852006264894 ) / 3000 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
8.55327i
60.9123i
62.7587i
5.61354i
14.0132i
8.55327i
60.9123i
62.7587i
5.61354i
14.0132i
−61.7891 61.7891i 877.187 877.187i 3539.78i −13781.9 7362.02i −108401. 67595.3 + 67595.3i −34368.1 + 34368.1i 1.00747e6i 396679. + 1.30646e6i
2.2 −54.8437 54.8437i −506.429 + 506.429i 1919.66i 15616.9 + 502.591i 55548.9 78559.8 + 78559.8i −119359. + 119359.i 18500.3i −828925. 884053.i
2.3 3.05147 + 3.05147i −299.721 + 299.721i 4077.38i −15614.1 + 583.350i −1829.18 −139850. 139850.i 24940.8 24940.8i 351775.i −49426.1 45865.9i
2.4 34.6956 + 34.6956i 539.020 539.020i 1688.43i 12911.0 + 8800.45i 37403.2 25919.1 + 25919.1i 200694. 200694.i 49644.2i 142616. + 753290.i
2.5 77.8857 + 77.8857i −451.057 + 451.057i 8036.37i −1256.84 15574.4i −70261.7 107575. + 107575.i −306898. + 306898.i 124537.i 1.11513e6 1.31091e6i
3.1 −61.7891 + 61.7891i 877.187 + 877.187i 3539.78i −13781.9 + 7362.02i −108401. 67595.3 67595.3i −34368.1 34368.1i 1.00747e6i 396679. 1.30646e6i
3.2 −54.8437 + 54.8437i −506.429 506.429i 1919.66i 15616.9 502.591i 55548.9 78559.8 78559.8i −119359. 119359.i 18500.3i −828925. + 884053.i
3.3 3.05147 3.05147i −299.721 299.721i 4077.38i −15614.1 583.350i −1829.18 −139850. + 139850.i 24940.8 + 24940.8i 351775.i −49426.1 + 45865.9i
3.4 34.6956 34.6956i 539.020 + 539.020i 1688.43i 12911.0 8800.45i 37403.2 25919.1 25919.1i 200694. + 200694.i 49644.2i 142616. 753290.i
3.5 77.8857 77.8857i −451.057 451.057i 8036.37i −1256.84 + 15574.4i −70261.7 107575. 107575.i −306898. 306898.i 124537.i 1.11513e6 + 1.31091e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.13.c.a 10
3.b odd 2 1 45.13.g.a 10
4.b odd 2 1 80.13.p.c 10
5.b even 2 1 25.13.c.b 10
5.c odd 4 1 inner 5.13.c.a 10
5.c odd 4 1 25.13.c.b 10
15.e even 4 1 45.13.g.a 10
20.e even 4 1 80.13.p.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.13.c.a 10 1.a even 1 1 trivial
5.13.c.a 10 5.c odd 4 1 inner
25.13.c.b 10 5.b even 2 1
25.13.c.b 10 5.c odd 4 1
45.13.g.a 10 3.b odd 2 1
45.13.g.a 10 15.e even 4 1
80.13.p.c 10 4.b odd 2 1
80.13.p.c 10 20.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{10} - 318 T^{9} + \cdots + 33\!\cdots\!32 \) Copy content Toggle raw display
$5$ \( T^{10} + 4250 T^{9} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} - 279598 T^{9} + \cdots + 13\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( (T^{5} - 156310 T^{4} + \cdots - 33\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} - 5290738 T^{9} + \cdots + 12\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{10} + 41269502 T^{9} + \cdots + 18\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + 510099842 T^{9} + \cdots + 29\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} - 1538544910 T^{4} + \cdots - 82\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 2599618502 T^{9} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$41$ \( (T^{5} - 3706039510 T^{4} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 5784410402 T^{9} + \cdots + 22\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{10} - 16053249598 T^{9} + \cdots + 27\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{10} - 101763514618 T^{9} + \cdots + 91\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} - 3865859110 T^{4} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 80010636002 T^{9} + \cdots + 28\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( (T^{5} + 23278626290 T^{4} + \cdots - 61\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 448527032342 T^{9} + \cdots + 16\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + 91118376722 T^{9} + \cdots + 48\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + 1409507601302 T^{9} + \cdots + 32\!\cdots\!32 \) Copy content Toggle raw display
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