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Properties

Label 14.13.b.a
Level $14$
Weight $13$
Character orbit 14.b
Analytic conductor $12.796$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [14,13,Mod(13,14)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(14, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("14.13");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 14.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: \(12.7959134419\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 154710x^{6} + 8245426887x^{4} + 174724076278260x^{2} + 1264170035276291934 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{3} q^{3} + 2048 q^{4} + ( - \beta_{4} - 11 \beta_{3}) q^{5} + ( - \beta_{7} - 2 \beta_{4} - 5 \beta_{3}) q^{6} + ( - \beta_{7} - 3 \beta_{6} + \cdots + 24395) q^{7}+ \cdots + ( - 5 \beta_{6} - 10 \beta_{5} + \cdots - 184863) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{3} q^{3} + 2048 q^{4} + ( - \beta_{4} - 11 \beta_{3}) q^{5} + ( - \beta_{7} - 2 \beta_{4} - 5 \beta_{3}) q^{6} + ( - \beta_{7} - 3 \beta_{6} + \cdots + 24395) q^{7}+ \cdots + ( - 2129490 \beta_{6} + \cdots - 225119282058) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16384 q^{4} + 195160 q^{7} - 1478904 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16384 q^{4} + 195160 q^{7} - 1478904 q^{9} - 213840 q^{11} - 8418816 q^{14} + 65882304 q^{15} + 33554432 q^{16} + 32547840 q^{18} - 4449984 q^{21} - 221337600 q^{22} + 156731760 q^{23} + 191237000 q^{25} + 399687680 q^{28} + 308853648 q^{29} - 2203567104 q^{30} - 3764734848 q^{35} - 3028795392 q^{36} - 3243600880 q^{37} + 13521315264 q^{39} - 12108579840 q^{42} + 21006302000 q^{43} - 437944320 q^{44} + 9664610304 q^{46} - 19258758904 q^{49} + 26259489792 q^{50} - 80965832832 q^{51} + 180445637520 q^{53} - 17241735168 q^{56} - 63145962240 q^{57} - 94193264640 q^{58} + 134926958592 q^{60} - 402706514280 q^{63} + 68719476736 q^{64} - 424890168192 q^{65} + 369211259440 q^{67} - 137936354304 q^{70} + 574058144304 q^{71} + 66657976320 q^{72} + 450517137408 q^{74} - 73915435440 q^{77} - 251000847360 q^{78} - 607826610128 q^{79} + 919051941384 q^{81} - 9113567232 q^{84} - 247202260608 q^{85} - 413092638720 q^{86} - 453299404800 q^{88} + 144527421696 q^{91} + 320986644480 q^{92} + 2292312458880 q^{93} - 1053641981376 q^{95} - 290797516800 q^{98} - 1800954256464 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 154710x^{6} + 8245426887x^{4} + 174724076278260x^{2} + 1264170035276291934 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -125\nu^{6} - 8963839\nu^{4} - 86616988470\nu^{2} + 824985433394766 ) / 7033679917677 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38659002 \nu^{6} + 4172748382238 \nu^{4} + \cdots + 13\!\cdots\!88 ) / 218044077447987 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1774768547 \nu^{7} + 176258268664353 \nu^{5} + \cdots + 42\!\cdots\!34 \nu ) / 51\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3573248699 \nu^{7} + 538494861420537 \nu^{5} + \cdots + 29\!\cdots\!26 \nu ) / 25\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1886431118 \nu^{7} + 1294978085403 \nu^{6} + 240235006799442 \nu^{5} + \cdots + 32\!\cdots\!70 ) / 12\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1324158591 \nu^{7} - 140815055259077 \nu^{5} + \cdots - 72\!\cdots\!38 \nu ) / 86\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 31567664251 \nu^{7} + \cdots - 19\!\cdots\!18 \nu ) / 51\!\cdots\!52 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -3\beta_{7} - 4\beta_{6} - 10\beta_{4} - 31\beta_{3} ) / 896 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -8\beta_{6} - 16\beta_{5} + 8\beta_{4} + \beta_{2} + 982\beta _1 - 2165940 ) / 56 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 183441\beta_{7} + 86196\beta_{6} + 587382\beta_{4} + 1283493\beta_{3} ) / 896 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 616590\beta_{6} + 1233180\beta_{5} - 616590\beta_{4} - 68355\beta_{2} + 11296332\beta _1 + 104220624564 ) / 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -10991030859\beta_{7} - 2214913356\beta_{6} - 33479955810\beta_{4} - 70597450215\beta_{3} ) / 896 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 38672620650 \beta_{6} - 77345241300 \beta_{5} + 38672620650 \beta_{4} + 4208849811 \beta_{2} + \cdots - 56\!\cdots\!80 ) / 56 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 646664008432761 \beta_{7} + 72705176403156 \beta_{6} + \cdots + 43\!\cdots\!45 \beta_{3} ) / 896 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
242.361i
130.480i
− 130.480i
− 242.361i
− 237.947i
149.422i
− 149.422i
237.947i
−45.2548 − 1265.70i 2048.00 23188.3i 57279.1i 109682. + 42556.3i −92681.9 −1.07056e6 − 1.04938e6i
13.2 −45.2548 − 102.042i 2048.00 − 6916.58i 4617.87i −14384.7 + 116766.i −92681.9 521029. 313009.i
13.3 −45.2548 102.042i 2048.00 6916.58i − 4617.87i −14384.7 − 116766.i −92681.9 521029. − 313009.i
13.4 −45.2548 1265.70i 2048.00 − 23188.3i − 57279.1i 109682. − 42556.3i −92681.9 −1.07056e6 1.04938e6i
13.5 45.2548 − 1072.00i 2048.00 − 1140.63i − 48513.4i 74038.7 − 91430.6i 92681.9 −617752. − 51619.1i
13.6 45.2548 − 321.888i 2048.00 17149.5i − 14567.0i −71756.5 + 93232.5i 92681.9 427829. 776099.i
13.7 45.2548 321.888i 2048.00 − 17149.5i 14567.0i −71756.5 − 93232.5i 92681.9 427829. − 776099.i
13.8 45.2548 1072.00i 2048.00 1140.63i 48513.4i 74038.7 + 91430.6i 92681.9 −617752. 51619.1i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.13.b.a ✓ 8
3.b odd 2 1 126.13.c.a 8
4.b odd 2 1 112.13.c.c 8
7.b odd 2 1 inner 14.13.b.a ✓ 8
7.c even 3 2 98.13.d.b 16
7.d odd 6 2 98.13.d.b 16
21.c even 2 1 126.13.c.a 8
28.d even 2 1 112.13.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.13.b.a ✓ 8 1.a even 1 1 trivial
14.13.b.a ✓ 8 7.b odd 2 1 inner
98.13.d.b 16 7.c even 3 2
98.13.d.b 16 7.d odd 6 2
112.13.c.c 8 4.b odd 2 1
112.13.c.c 8 28.d even 2 1
126.13.c.a 8 3.b odd 2 1
126.13.c.a 8 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2048)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 26\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 16\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 24\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 70\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 40\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 49\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 12\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 53\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
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