Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,10,Mod(12,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
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("13.12");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 13.b (of order \(2\), degree \(1\), 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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 3841x^{8} + 5134480x^{6} + 2823572208x^{4} + 614223235584x^{2} + 43308450164736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4}\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 256) q^{4} + (\beta_{5} - 6 \beta_1) q^{5} + (\beta_{8} - 6 \beta_1) q^{6} + (\beta_{9} - \beta_{5} - 22 \beta_1) q^{7} + ( - \beta_{9} - \beta_{8} + \cdots - 144 \beta_1) q^{8}+ \cdots + ( - \beta_{4} - 33 \beta_{3} + \cdots + 6727) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 256) q^{4} + (\beta_{5} - 6 \beta_1) q^{5} + (\beta_{8} - 6 \beta_1) q^{6} + (\beta_{9} - \beta_{5} - 22 \beta_1) q^{7} + ( - \beta_{9} - \beta_{8} + \cdots - 144 \beta_1) q^{8}+ \cdots + ( - 45438 \beta_{9} + \cdots + 7727926 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{3} - 2562 q^{4} + 67304 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{3} - 2562 q^{4} + 67304 q^{9} + 45562 q^{10} + 46298 q^{12} - 111488 q^{13} + 169350 q^{14} - 209470 q^{16} - 1189686 q^{17} + 3516760 q^{22} + 2210916 q^{23} - 10109316 q^{25} + 7627230 q^{26} - 7880006 q^{27} + 7039224 q^{29} - 7573410 q^{30} + 39493506 q^{35} + 29007196 q^{36} - 50219652 q^{38} + 14502332 q^{39} - 23263606 q^{40} + 400842 q^{42} - 92334370 q^{43} + 102213790 q^{48} - 100714448 q^{49} + 77969322 q^{51} - 113165052 q^{52} + 29507556 q^{53} + 138590608 q^{55} + 437436342 q^{56} - 238375816 q^{61} - 565735320 q^{62} + 272247742 q^{64} - 351372138 q^{65} - 495101088 q^{66} + 550420050 q^{68} + 505082484 q^{69} + 1257672774 q^{74} - 1443387976 q^{75} + 530898576 q^{77} - 1745232606 q^{78} - 441679756 q^{79} + 1486784810 q^{81} + 518795296 q^{82} + 4793242032 q^{87} - 3927690208 q^{88} - 2656725444 q^{90} - 777339186 q^{91} - 4121025444 q^{92} + 6226353478 q^{94} + 1047837276 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 3841x^{8} + 5134480x^{6} + 2823572208x^{4} + 614223235584x^{2} + 43308450164736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 768 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -101\nu^{8} - 335285\nu^{6} - 359803808\nu^{4} - 134078143536\nu^{2} - 13050018918144 ) / 12944968704 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 128\nu^{8} + 466637\nu^{6} + 584699369\nu^{4} + 281289968700\nu^{2} + 36497540643552 ) / 202265136 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 103037 \nu^{9} + 407466221 \nu^{7} + 517098453440 \nu^{5} + 225182669584944 \nu^{3} + 24\!\cdots\!24 \nu ) / 49242660950016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3871\nu^{8} + 20860879\nu^{6} + 32030013856\nu^{4} + 15796747994256\nu^{2} + 1898872235880192 ) / 6472484352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18149 \nu^{9} + 73154197 \nu^{7} + 101232420736 \nu^{5} + 55804164878768 \nu^{3} + 10\!\cdots\!36 \nu ) / 2735703386112 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -101\nu^{9} - 335285\nu^{7} - 359803808\nu^{5} - 134078143536\nu^{3} - 12972349105920\nu ) / 12944968704 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 118481 \nu^{9} + 432033281 \nu^{7} + 531812876480 \nu^{5} + 244210927479792 \nu^{3} + 29\!\cdots\!40 \nu ) / 8207110158336 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} - 1168\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + 6\beta_{4} + 410\beta_{3} - 1657\beta_{2} + 896615 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2045\beta_{9} + 2005\beta_{8} - 1653\beta_{7} - 1392\beta_{5} + 1556172\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3085\beta_{6} - 13662\beta_{4} - 871634\beta_{3} + 2442489\beta_{2} - 1194377627 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3338893\beta_{9} - 3045285\beta_{8} + 2439845\beta_{7} + 3945264\beta_{5} - 2164532636\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -6678717\beta_{6} + 23978622\beta_{4} + 1304765106\beta_{3} - 3532803545\beta_{2} + 1661122838379 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5126337581\beta_{9} + 4165989317\beta_{8} - 3538276037\beta_{7} - 8138029104\beta_{5} + 3063857781212\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
38.5911i
36.6220i
24.9226i
15.7959i
11.8282i
11.8282i
15.7959i
24.9226i
36.6220i
38.5911i
38.5911i −58.0612 −977.274 2510.84i 2240.65i 8007.46i 17955.4i −16311.9 96896.0
12.2 36.6220i 126.985 −829.173 2095.65i 4650.45i 3353.27i 11615.5i −3557.77 −76746.8
12.3 24.9226i −252.690 −109.136 825.595i 6297.70i 9476.84i 10040.4i 44169.4 −20576.0
12.4 15.7959i 217.907 262.489 1801.57i 3442.04i 6896.47i 12233.8i 27800.4 28457.4
12.5 11.8282i −35.1406 372.094 443.831i 415.650i 6276.11i 10457.2i −18448.1 −5249.72
12.6 11.8282i −35.1406 372.094 443.831i 415.650i 6276.11i 10457.2i −18448.1 −5249.72
12.7 15.7959i 217.907 262.489 1801.57i 3442.04i 6896.47i 12233.8i 27800.4 28457.4
12.8 24.9226i −252.690 −109.136 825.595i 6297.70i 9476.84i 10040.4i 44169.4 −20576.0
12.9 36.6220i 126.985 −829.173 2095.65i 4650.45i 3353.27i 11615.5i −3557.77 −76746.8
12.10 38.5911i −58.0612 −977.274 2510.84i 2240.65i 8007.46i 17955.4i −16311.9 96896.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.10.b.a 10
3.b odd 2 1 117.10.b.c 10
4.b odd 2 1 208.10.f.b 10
13.b even 2 1 inner 13.10.b.a 10
13.d odd 4 2 169.10.a.e 10
39.d odd 2 1 117.10.b.c 10
52.b odd 2 1 208.10.f.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.10.b.a 10 1.a even 1 1 trivial
13.10.b.a 10 13.b even 2 1 inner
117.10.b.c 10 3.b odd 2 1
117.10.b.c 10 39.d odd 2 1
169.10.a.e 10 13.d odd 4 2
208.10.f.b 10 4.b odd 2 1
208.10.f.b 10 52.b odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 43308450164736 \) Copy content Toggle raw display
$3$ \( (T^{5} + T^{4} + \cdots + 14266185264)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 13\!\cdots\!93 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots + 94\!\cdots\!60)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 20\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 22\!\cdots\!60)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 56\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 61\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 58\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 51\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 26\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
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