Properties

Label 13.10.e.a
Level $13$
Weight $10$
Character orbit 13.e
Analytic conductor $6.695$
Analytic rank $0$
Dimension $20$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,10,Mod(4,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.69546587013\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 7679 x^{18} + 24599364 x^{16} + 42662336000 x^{14} + 43527566862400 x^{12} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{10}\cdot 13^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{6} + 16 \beta_{5}) q^{3} + ( - \beta_{7} - \beta_{6} + 256 \beta_{5} + \cdots + 256) q^{4}+ \cdots + (\beta_{19} + \beta_{15} + \beta_{12} + \cdots - 6458) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{6} + 16 \beta_{5}) q^{3} + ( - \beta_{7} - \beta_{6} + 256 \beta_{5} + \cdots + 256) q^{4}+ \cdots + ( - 8884 \beta_{19} + 15570 \beta_{18} + \cdots + 24334027) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 3 q^{2} - 163 q^{3} + 2559 q^{4} - 3288 q^{6} + 1023 q^{7} - 64763 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 3 q^{2} - 163 q^{3} + 2559 q^{4} - 3288 q^{6} + 1023 q^{7} - 64763 q^{9} + 44195 q^{10} - 94599 q^{11} - 427652 q^{12} + 187013 q^{13} + 473556 q^{14} + 754950 q^{15} - 594809 q^{16} - 49656 q^{17} - 3879 q^{19} - 154815 q^{20} + 2536766 q^{22} + 3126189 q^{23} - 6472626 q^{24} - 1529274 q^{25} - 13931889 q^{26} + 18052718 q^{27} + 4918980 q^{28} - 2712414 q^{29} + 15022758 q^{30} - 14390595 q^{32} - 34050309 q^{33} + 7549080 q^{35} + 27039443 q^{36} - 36102006 q^{37} - 39021096 q^{38} - 32365163 q^{39} + 134360674 q^{40} + 30265110 q^{41} - 60800034 q^{42} + 26621029 q^{43} + 31870857 q^{45} + 121485864 q^{46} - 41385538 q^{48} + 7027649 q^{49} - 58518456 q^{50} - 10208934 q^{51} - 109817238 q^{52} - 36429786 q^{53} - 393493974 q^{54} + 65727080 q^{55} + 63636336 q^{56} + 470295633 q^{58} + 3987867 q^{59} - 268212896 q^{61} - 445379898 q^{62} + 506780166 q^{63} + 343337066 q^{64} + 292006563 q^{65} - 445758060 q^{66} - 944780397 q^{67} - 168297045 q^{68} + 541942791 q^{69} + 1318764849 q^{71} + 1409393649 q^{72} - 541934631 q^{74} - 609413441 q^{75} - 594798654 q^{76} - 1470187374 q^{77} - 2076950874 q^{78} + 1556703616 q^{79} - 968911845 q^{80} - 1069563758 q^{81} + 1859661377 q^{82} + 7275171300 q^{84} - 318942363 q^{85} - 644123073 q^{87} - 2054580464 q^{88} - 2287100109 q^{89} - 6489878934 q^{90} - 4972612749 q^{91} + 10148843820 q^{92} - 1470324870 q^{93} - 2447356414 q^{94} + 4625601270 q^{95} + 7273214547 q^{97} - 1557570597 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 7679 x^{18} + 24599364 x^{16} + 42662336000 x^{14} + 43527566862400 x^{12} + \cdots + 25\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!39 \nu^{18} + \cdots - 75\!\cdots\!92 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!19 \nu^{18} + \cdots + 74\!\cdots\!32 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!79 \nu^{18} + \cdots + 35\!\cdots\!72 ) / 33\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!09 \nu^{19} + \cdots - 36\!\cdots\!00 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18\!\cdots\!91 \nu^{19} + \cdots + 74\!\cdots\!24 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45\!\cdots\!19 \nu^{19} + \cdots + 27\!\cdots\!24 ) / 52\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!99 \nu^{19} + \cdots + 49\!\cdots\!52 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!99 \nu^{19} + \cdots - 49\!\cdots\!52 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!77 \nu^{19} + \cdots - 32\!\cdots\!72 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!65 \nu^{19} + \cdots + 20\!\cdots\!12 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 24\!\cdots\!65 \nu^{19} + \cdots + 20\!\cdots\!12 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 24\!\cdots\!65 \nu^{19} + \cdots - 10\!\cdots\!56 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!95 \nu^{19} + \cdots + 14\!\cdots\!96 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!95 \nu^{19} + \cdots - 14\!\cdots\!96 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 21\!\cdots\!33 \nu^{19} + \cdots + 27\!\cdots\!16 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 21\!\cdots\!33 \nu^{19} + \cdots + 27\!\cdots\!16 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 17\!\cdots\!27 \nu^{19} + \cdots - 50\!\cdots\!88 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 54\!\cdots\!75 \nu^{19} + \cdots + 19\!\cdots\!84 ) / 69\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} - 768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - 3 \beta_{9} - 3 \beta_{8} + 2 \beta_{7} + 16 \beta_{6} - 848 \beta_{5} + \cdots - 424 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14 \beta_{17} + 14 \beta_{16} - \beta_{15} + \beta_{14} + 2 \beta_{13} - 24 \beta_{12} - 16 \beta_{11} + \cdots + 977278 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 52 \beta_{19} - 36 \beta_{18} + 278 \beta_{17} - 314 \beta_{16} - 2005 \beta_{15} - 1989 \beta_{14} + \cdots + 1344534 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 30318 \beta_{17} - 30318 \beta_{16} + 4059 \beta_{15} - 4059 \beta_{14} - 6530 \beta_{13} + \cdots - 1411087822 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 213068 \beta_{19} + 54428 \beta_{18} - 1009194 \beta_{17} + 1063622 \beta_{16} + 3530711 \beta_{15} + \cdots - 3180751082 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 53380522 \beta_{17} + 53380522 \beta_{16} - 9996305 \beta_{15} + 9996305 \beta_{14} + \cdots + 2149209310794 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 562436644 \beta_{19} - 29436692 \beta_{18} + 2428197214 \beta_{17} - 2457633906 \beta_{16} + \cdots + 6394139827358 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 88545991966 \beta_{17} - 88545991966 \beta_{16} + 20251605251 \beta_{15} - 20251605251 \beta_{14} + \cdots - 33\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1244556001196 \beta_{19} - 78429393540 \beta_{18} - 4998619321978 \beta_{17} + 4920189928438 \beta_{16} + \cdots - 11\!\cdots\!42 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 143624723962938 \beta_{17} + 143624723962938 \beta_{16} - 37408814632761 \beta_{15} + \cdots + 53\!\cdots\!34 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 25\!\cdots\!68 \beta_{19} + 342327078908876 \beta_{18} + \cdots + 21\!\cdots\!66 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 23\!\cdots\!06 \beta_{17} + \cdots - 86\!\cdots\!70 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 48\!\cdots\!96 \beta_{19} + \cdots - 38\!\cdots\!38 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 36\!\cdots\!42 \beta_{17} + \cdots + 14\!\cdots\!94 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 88\!\cdots\!08 \beta_{19} + \cdots + 67\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 59\!\cdots\!26 \beta_{17} + \cdots - 23\!\cdots\!54 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 16\!\cdots\!40 \beta_{19} + \cdots - 11\!\cdots\!46 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 + \beta_{5}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
40.5664i
34.2355i
27.6771i
13.9570i
5.95885i
2.58166i
19.7704i
20.2564i
36.6943i
41.3599i
40.5664i
34.2355i
27.6771i
13.9570i
5.95885i
2.58166i
19.7704i
20.2564i
36.6943i
41.3599i
−35.1315 20.2832i 50.0772 86.7363i 566.815 + 981.752i 1429.76i −3518.58 + 2031.45i 5656.27 3265.65i 25217.2i 4826.04 + 8358.95i 29000.2 50229.8i
4.2 −29.6488 17.1177i −132.003 + 228.636i 330.033 + 571.634i 402.267i 7827.45 4519.18i −8113.97 + 4684.60i 5069.12i −25008.0 43315.2i 6885.89 11926.7i
4.3 −23.9691 13.8386i 22.1194 38.3120i 127.012 + 219.991i 2467.22i −1060.37 + 612.203i −3100.70 + 1790.19i 7140.04i 8862.96 + 15351.1i −34142.7 + 59136.9i
4.4 −12.0871 6.97851i −12.8075 + 22.1833i −158.601 274.705i 792.583i 309.613 178.755i 780.110 450.397i 11573.2i 9513.43 + 16477.8i 5531.05 9580.06i
4.5 −5.16052 2.97943i 125.329 217.077i −238.246 412.654i 841.807i −1293.53 + 746.819i −1098.20 + 634.044i 5890.28i −21573.3 37366.1i 2508.10 4344.16i
4.6 2.23578 + 1.29083i −92.1011 + 159.524i −252.668 437.633i 581.690i −411.836 + 237.773i 9459.43 5461.41i 2626.41i −7123.72 12338.6i 750.863 1300.53i
4.7 17.1217 + 9.88522i −49.7554 + 86.1790i −60.5648 104.901i 1521.53i −1703.80 + 983.687i −8005.36 + 4621.90i 12517.3i 4890.29 + 8470.23i −15040.7 + 26051.2i
4.8 17.5426 + 10.1282i 49.6540 86.0032i −50.8382 88.0544i 1799.37i 1742.12 1005.81i −1675.98 + 967.628i 12430.9i 4910.47 + 8505.18i 18224.4 31565.5i
4.9 31.7782 + 18.3471i 65.0132 112.606i 417.236 + 722.673i 1417.57i 4132.00 2385.61i 6502.05 3753.96i 11832.9i 1388.07 + 2404.21i −26008.3 + 45047.7i
4.10 35.8188 + 20.6800i −107.026 + 185.375i 599.322 + 1038.06i 1662.90i −7667.08 + 4426.59i 107.839 62.2611i 28399.6i −13067.7 22633.9i 34388.7 59563.0i
10.1 −35.1315 + 20.2832i 50.0772 + 86.7363i 566.815 981.752i 1429.76i −3518.58 2031.45i 5656.27 + 3265.65i 25217.2i 4826.04 8358.95i 29000.2 + 50229.8i
10.2 −29.6488 + 17.1177i −132.003 228.636i 330.033 571.634i 402.267i 7827.45 + 4519.18i −8113.97 4684.60i 5069.12i −25008.0 + 43315.2i 6885.89 + 11926.7i
10.3 −23.9691 + 13.8386i 22.1194 + 38.3120i 127.012 219.991i 2467.22i −1060.37 612.203i −3100.70 1790.19i 7140.04i 8862.96 15351.1i −34142.7 59136.9i
10.4 −12.0871 + 6.97851i −12.8075 22.1833i −158.601 + 274.705i 792.583i 309.613 + 178.755i 780.110 + 450.397i 11573.2i 9513.43 16477.8i 5531.05 + 9580.06i
10.5 −5.16052 + 2.97943i 125.329 + 217.077i −238.246 + 412.654i 841.807i −1293.53 746.819i −1098.20 634.044i 5890.28i −21573.3 + 37366.1i 2508.10 + 4344.16i
10.6 2.23578 1.29083i −92.1011 159.524i −252.668 + 437.633i 581.690i −411.836 237.773i 9459.43 + 5461.41i 2626.41i −7123.72 + 12338.6i 750.863 + 1300.53i
10.7 17.1217 9.88522i −49.7554 86.1790i −60.5648 + 104.901i 1521.53i −1703.80 983.687i −8005.36 4621.90i 12517.3i 4890.29 8470.23i −15040.7 26051.2i
10.8 17.5426 10.1282i 49.6540 + 86.0032i −50.8382 + 88.0544i 1799.37i 1742.12 + 1005.81i −1675.98 967.628i 12430.9i 4910.47 8505.18i 18224.4 + 31565.5i
10.9 31.7782 18.3471i 65.0132 + 112.606i 417.236 722.673i 1417.57i 4132.00 + 2385.61i 6502.05 + 3753.96i 11832.9i 1388.07 2404.21i −26008.3 45047.7i
10.10 35.8188 20.6800i −107.026 185.375i 599.322 1038.06i 1662.90i −7667.08 4426.59i 107.839 + 62.2611i 28399.6i −13067.7 + 22633.9i 34388.7 + 59563.0i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.10.e.a 20
13.e even 6 1 inner 13.10.e.a 20
13.f odd 12 2 169.10.a.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.e.a 20 1.a even 1 1 trivial
13.10.e.a 20 13.e even 6 1 inner
169.10.a.f 20 13.f odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 17\!\cdots\!49 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 85\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 30\!\cdots\!49 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 64\!\cdots\!49 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 30\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 52\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
show more
show less