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Properties

Label 722.6.a.p
Level $722$
Weight $6$
Character orbit 722.a
Self dual yes
Analytic conductor $115.797$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [722,6,Mod(1,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

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: yes
Analytic conductor: \(115.797117905\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 1707 x^{10} - 2102 x^{9} + 996375 x^{8} + 3286746 x^{7} - 234371509 x^{6} - 1273502622 x^{5} + \cdots + 125341931625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 19^{3} \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} - \beta_1 q^{3} + 16 q^{4} + (\beta_{7} - 3) q^{5} - 4 \beta_1 q^{6} + (\beta_{6} + \beta_{5} + 3 \beta_1 - 37) q^{7} + 64 q^{8} + (\beta_{10} + \beta_{7} - 2 \beta_{6} + \cdots + 48) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} - \beta_1 q^{3} + 16 q^{4} + (\beta_{7} - 3) q^{5} - 4 \beta_1 q^{6} + (\beta_{6} + \beta_{5} + 3 \beta_1 - 37) q^{7} + 64 q^{8} + (\beta_{10} + \beta_{7} - 2 \beta_{6} + \cdots + 48) q^{9}+ \cdots + ( - 269 \beta_{11} - 273 \beta_{10} + \cdots - 34732) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 48 q^{2} + 192 q^{4} - 42 q^{5} - 438 q^{7} + 768 q^{8} + 546 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 48 q^{2} + 192 q^{4} - 42 q^{5} - 438 q^{7} + 768 q^{8} + 546 q^{9} - 168 q^{10} - 600 q^{11} + 444 q^{13} - 1752 q^{14} + 3072 q^{16} - 2220 q^{17} + 2184 q^{18} - 672 q^{20} - 11046 q^{21} - 2400 q^{22} - 7464 q^{23} + 4146 q^{25} + 1776 q^{26} - 5130 q^{27} - 7008 q^{28} - 6606 q^{29} + 294 q^{31} + 12288 q^{32} - 14886 q^{33} - 8880 q^{34} - 2064 q^{35} + 8736 q^{36} - 14304 q^{37} - 19434 q^{39} - 2688 q^{40} - 19512 q^{41} - 44184 q^{42} - 8634 q^{43} - 9600 q^{44} + 34344 q^{45} - 29856 q^{46} - 17964 q^{47} - 1422 q^{49} + 16584 q^{50} + 27456 q^{51} + 7104 q^{52} - 44112 q^{53} - 20520 q^{54} - 121596 q^{55} - 28032 q^{56} - 26424 q^{58} - 34002 q^{59} - 27582 q^{61} + 1176 q^{62} - 204222 q^{63} + 49152 q^{64} - 202674 q^{65} - 59544 q^{66} - 33702 q^{67} - 35520 q^{68} - 19938 q^{69} - 8256 q^{70} - 2904 q^{71} + 34944 q^{72} - 336072 q^{73} - 57216 q^{74} - 87480 q^{75} - 57528 q^{77} - 77736 q^{78} - 40098 q^{79} - 10752 q^{80} + 97476 q^{81} - 78048 q^{82} + 148440 q^{83} - 176736 q^{84} - 252912 q^{85} - 34536 q^{86} - 182874 q^{87} - 38400 q^{88} - 123774 q^{89} + 137376 q^{90} - 178866 q^{91} - 119424 q^{92} - 118584 q^{93} - 71856 q^{94} - 317346 q^{97} - 5688 q^{98} - 409860 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 1707 x^{10} - 2102 x^{9} + 996375 x^{8} + 3286746 x^{7} - 234371509 x^{6} - 1273502622 x^{5} + \cdots + 125341931625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 21\!\cdots\!68 \nu^{11} + \cdots + 20\!\cdots\!75 ) / 45\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21\!\cdots\!68 \nu^{11} + \cdots + 20\!\cdots\!75 ) / 24\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14\!\cdots\!89 \nu^{11} + \cdots - 13\!\cdots\!50 ) / 45\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 39\!\cdots\!06 \nu^{11} + \cdots - 81\!\cdots\!75 ) / 13\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 74\!\cdots\!78 \nu^{11} + \cdots - 72\!\cdots\!25 ) / 24\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!64 \nu^{11} + \cdots + 16\!\cdots\!75 ) / 13\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54\!\cdots\!70 \nu^{11} + \cdots - 77\!\cdots\!13 ) / 54\!\cdots\!83 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34\!\cdots\!49 \nu^{11} + \cdots - 68\!\cdots\!55 ) / 27\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!66 \nu^{11} + \cdots - 79\!\cdots\!75 ) / 13\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27\!\cdots\!07 \nu^{11} + \cdots + 33\!\cdots\!75 ) / 13\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\!\cdots\!16 \nu^{11} + \cdots - 31\!\cdots\!75 ) / 13\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + 19\beta_1 ) / 19 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 13 \beta_{10} - 6 \beta_{8} + 19 \beta_{7} - 38 \beta_{6} - 45 \beta_{5} + 13 \beta_{4} - 61 \beta_{3} + \cdots + 5455 ) / 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 342 \beta_{11} - 165 \beta_{10} + 171 \beta_{9} - 51 \beta_{8} - 57 \beta_{7} + 2436 \beta_{5} + \cdots + 9707 ) / 19 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2964 \beta_{11} + 9424 \beta_{10} - 2280 \beta_{9} - 2508 \beta_{8} + 13813 \beta_{7} - 21470 \beta_{6} + \cdots + 2957236 ) / 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 327636 \beta_{11} - 118575 \beta_{10} + 181526 \beta_{9} - 15500 \beta_{8} - 117743 \beta_{7} + \cdots + 2012238 ) / 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3596928 \beta_{11} + 6376129 \beta_{10} - 2833926 \beta_{9} - 831540 \beta_{8} + 8967373 \beta_{7} + \cdots + 1871590201 ) / 19 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 268585539 \beta_{11} - 95049031 \beta_{10} + 154217338 \beta_{9} + 3193243 \beta_{8} + \cdots - 3777516815 ) / 19 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3541410190 \beta_{11} + 4519754739 \beta_{10} - 2752987748 \beta_{9} - 259712646 \beta_{8} + \cdots + 1280366628970 ) / 19 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 211448646945 \beta_{11} - 81295622520 \beta_{10} + 123517948236 \beta_{9} + 7623679800 \beta_{8} + \cdots - 6820263419047 ) / 19 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3241090592790 \beta_{11} + 3338450890354 \beta_{10} - 2464232773554 \beta_{9} - 107436663264 \beta_{8} + \cdots + 916006393506175 ) / 19 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 164840599797582 \beta_{11} - 70121769641712 \beta_{10} + 97414080212510 \beta_{9} + \cdots - 79\!\cdots\!88 ) / 19 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
24.7757
25.4218
20.2034
12.2127
0.446425
0.918020
−4.08620
−4.63163
−12.2467
−17.3772
−17.1572
−28.4791
4.00000 −26.6551 16.0000 19.2457 −106.620 −9.05816 64.0000 467.495 76.9829
1.2 4.00000 −23.8897 16.0000 −25.0477 −95.5587 58.4363 64.0000 327.717 −100.191
1.3 4.00000 −19.8561 16.0000 82.4631 −79.4244 −62.1960 64.0000 151.265 329.852
1.4 4.00000 −11.8654 16.0000 −92.2261 −47.4616 −154.597 64.0000 −102.212 −368.904
1.5 4.00000 −2.32581 16.0000 −94.8034 −9.30324 111.531 64.0000 −237.591 −379.214
1.6 4.00000 0.614069 16.0000 12.2873 2.45627 153.180 64.0000 −242.623 49.1492
1.7 4.00000 2.20681 16.0000 69.9924 8.82725 19.0915 64.0000 −238.130 279.970
1.8 4.00000 4.97893 16.0000 0.812786 19.9157 105.563 64.0000 −218.210 3.25115
1.9 4.00000 13.7788 16.0000 −75.4755 55.1153 −79.0554 64.0000 −53.1437 −301.902
1.10 4.00000 15.4978 16.0000 −8.43466 61.9912 −220.990 64.0000 −2.81800 −33.7386
1.11 4.00000 18.6893 16.0000 74.2358 74.7572 −150.516 64.0000 106.290 296.943
1.12 4.00000 28.8264 16.0000 −5.04982 115.305 −209.389 64.0000 587.960 −20.1993
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.6.a.p 12
19.b odd 2 1 722.6.a.o 12
19.f odd 18 2 38.6.e.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.e.a ✓ 24 19.f odd 18 2
722.6.a.o 12 19.b odd 2 1
722.6.a.p 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 1731 T_{3}^{10} + 1710 T_{3}^{9} + 1020009 T_{3}^{8} - 2763954 T_{3}^{7} + \cdots - 270848511261 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots - 270848511261 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 57\!\cdots\!53 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots - 96\!\cdots\!89 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 15\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 25\!\cdots\!67 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots - 60\!\cdots\!07 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 22\!\cdots\!27 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 53\!\cdots\!03 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 21\!\cdots\!03 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 12\!\cdots\!91 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 12\!\cdots\!13 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 44\!\cdots\!77 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 18\!\cdots\!39 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 28\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 21\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 65\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 30\!\cdots\!37 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 44\!\cdots\!17 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 60\!\cdots\!37 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 12\!\cdots\!51 \) Copy content Toggle raw display
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