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Properties

Label 722.4.a.t
Level $722$
Weight $4$
Character orbit 722.a
Self dual yes
Analytic conductor $42.599$
Analytic rank $0$
Dimension $9$
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] = [722,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(42.5993790241\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 165x^{7} - 28x^{6} + 8790x^{5} + 7128x^{4} - 179236x^{3} - 272553x^{2} + 1047198x + 1973017 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3\cdot 19^{2} \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + (\beta_{3} - 1) q^{3} + 4 q^{4} - \beta_{6} q^{5} + ( - 2 \beta_{3} + 2) q^{6} + (\beta_{7} - \beta_{4} - \beta_{3} + \cdots + 4) q^{7}+ \cdots + (\beta_{6} - \beta_{4} - 2 \beta_{3} + \cdots + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + (\beta_{3} - 1) q^{3} + 4 q^{4} - \beta_{6} q^{5} + ( - 2 \beta_{3} + 2) q^{6} + (\beta_{7} - \beta_{4} - \beta_{3} + \cdots + 4) q^{7}+ \cdots + (\beta_{8} + 76 \beta_{7} + 19 \beta_{6} + \cdots + 281) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 18 q^{2} - 9 q^{3} + 36 q^{4} + 3 q^{5} + 18 q^{6} + 33 q^{7} - 72 q^{8} + 102 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 18 q^{2} - 9 q^{3} + 36 q^{4} + 3 q^{5} + 18 q^{6} + 33 q^{7} - 72 q^{8} + 102 q^{9} - 6 q^{10} + 75 q^{11} - 36 q^{12} + 90 q^{13} - 66 q^{14} + 144 q^{16} + 237 q^{17} - 204 q^{18} + 12 q^{20} - 273 q^{21} - 150 q^{22} + 336 q^{23} + 72 q^{24} + 534 q^{25} - 180 q^{26} - 678 q^{27} + 132 q^{28} - 159 q^{29} + 42 q^{31} - 288 q^{32} + 78 q^{33} - 474 q^{34} + 555 q^{35} + 408 q^{36} - 528 q^{37} + 906 q^{39} - 24 q^{40} + 180 q^{41} + 546 q^{42} - 165 q^{43} + 300 q^{44} - 1494 q^{45} - 672 q^{46} + 816 q^{47} - 144 q^{48} + 858 q^{49} - 1068 q^{50} - 684 q^{51} + 360 q^{52} - 1074 q^{53} + 1356 q^{54} + 321 q^{55} - 264 q^{56} + 318 q^{58} - 879 q^{59} + 1071 q^{61} - 84 q^{62} + 834 q^{63} + 576 q^{64} - 1755 q^{65} - 156 q^{66} + 2058 q^{67} + 948 q^{68} - 3693 q^{69} - 1110 q^{70} + 2088 q^{71} - 816 q^{72} + 4476 q^{73} + 1056 q^{74} - 2223 q^{75} + 2910 q^{77} - 1812 q^{78} + 2229 q^{79} + 48 q^{80} + 465 q^{81} - 360 q^{82} - 381 q^{83} - 1092 q^{84} + 1146 q^{85} + 330 q^{86} + 4506 q^{87} - 600 q^{88} - 3054 q^{89} + 2988 q^{90} - 1737 q^{91} + 1344 q^{92} - 216 q^{93} - 1632 q^{94} + 288 q^{96} + 2331 q^{97} - 1716 q^{98} + 2358 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 165x^{7} - 28x^{6} + 8790x^{5} + 7128x^{4} - 179236x^{3} - 272553x^{2} + 1047198x + 1973017 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6335134305 \nu^{8} + 934854547295 \nu^{7} - 2746648151780 \nu^{6} - 134642750962488 \nu^{5} + \cdots + 71\!\cdots\!33 ) / 24\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2387438459 \nu^{8} - 237564972437 \nu^{7} - 304617431300 \nu^{6} + \cdots + 11\!\cdots\!85 ) / 613749763851216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 79554421623 \nu^{8} - 234969732911 \nu^{7} - 12008855913676 \nu^{6} + 34275625099368 \nu^{5} + \cdots + 47\!\cdots\!95 ) / 24\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 80263867713 \nu^{8} + 175840747921 \nu^{7} + 11725305911108 \nu^{6} + \cdots - 40\!\cdots\!17 ) / 24\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 208509030407 \nu^{8} + 438012031569 \nu^{7} - 37004990909228 \nu^{6} + \cdots + 22\!\cdots\!31 ) / 24\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53188376307 \nu^{8} - 150897447067 \nu^{7} - 8019998851508 \nu^{6} + 21588493456776 \nu^{5} + \cdots + 19\!\cdots\!39 ) / 306874881925608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 150111808379 \nu^{8} + 456971492163 \nu^{7} + 22746405615292 \nu^{6} + \cdots - 90\!\cdots\!19 ) / 818333018468288 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 338845236613 \nu^{8} + 1244227048585 \nu^{7} + 54754433948364 \nu^{6} + \cdots - 28\!\cdots\!29 ) / 613749763851216 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{7} - 2\beta_{4} - 19\beta_{3} ) / 19 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{8} + 4\beta_{7} + 13\beta_{6} + 12\beta_{5} - 18\beta_{4} + 8\beta_{3} + 15\beta_{2} - 6\beta _1 + 702 ) / 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7 \beta_{8} - 313 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} + 485 \beta_{4} - 1118 \beta_{3} - 49 \beta_{2} + \cdots + 154 ) / 19 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 557 \beta_{8} + 228 \beta_{7} + 640 \beta_{6} + 1076 \beta_{5} - 2782 \beta_{4} + 1946 \beta_{3} + \cdots + 41115 ) / 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 195 \beta_{8} - 26447 \beta_{7} - 2285 \beta_{6} - 370 \beta_{5} + 54011 \beta_{4} - 79369 \beta_{3} + \cdots - 30480 ) / 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 48406 \beta_{8} + 24198 \beta_{7} + 31489 \beta_{6} + 87445 \beta_{5} - 318659 \beta_{4} + \cdots + 2899572 ) / 19 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 25832 \beta_{8} - 2177226 \beta_{7} - 285578 \beta_{6} - 58238 \beta_{5} + 4926932 \beta_{4} + \cdots - 6159757 ) / 19 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4119280 \beta_{8} + 3278671 \beta_{7} + 1921694 \beta_{6} + 7123336 \beta_{5} - 32725040 \beta_{4} + \cdots + 223286606 ) / 19 \) 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
7.42444
8.42478
6.06983
3.09100
−3.16864
−2.23876
−5.78789
−4.30664
−9.50813
−2.00000 −9.95653 4.00000 −5.69887 19.9131 7.83973 −8.00000 72.1324 11.3977
1.2 −2.00000 −7.54540 4.00000 4.50990 15.0908 −8.42169 −8.00000 29.9331 −9.01980
1.3 −2.00000 −7.41713 4.00000 −19.5988 14.8343 31.2078 −8.00000 28.0138 39.1976
1.4 −2.00000 −4.43829 4.00000 21.5062 8.87658 −5.97386 −8.00000 −7.30156 −43.0123
1.5 −2.00000 0.636547 4.00000 17.7533 −1.27309 23.2072 −8.00000 −26.5948 −35.5065
1.6 −2.00000 3.11814 4.00000 10.7989 −6.23628 27.5434 −8.00000 −17.2772 −21.5977
1.7 −2.00000 3.25580 4.00000 −11.0544 −6.51160 −19.6182 −8.00000 −16.3998 22.1088
1.8 −2.00000 5.18603 4.00000 −14.3088 −10.3721 −32.1449 −8.00000 −0.105104 28.6175
1.9 −2.00000 8.16083 4.00000 −0.907363 −16.3217 9.36062 −8.00000 39.5991 1.81473
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 722.4.a.t 9
19.b odd 2 1 722.4.a.u 9
19.e even 9 2 38.4.e.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.e.b ✓ 18 19.e even 9 2
722.4.a.t 9 1.a even 1 1 trivial
722.4.a.u 9 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\):

\( T_{3}^{9} + 9 T_{3}^{8} - 132 T_{3}^{7} - 1043 T_{3}^{6} + 6195 T_{3}^{5} + 34233 T_{3}^{4} + \cdots - 676387 \) Copy content Toggle raw display
\( T_{5}^{9} - 3 T_{5}^{8} - 825 T_{5}^{7} + 851 T_{5}^{6} + 212025 T_{5}^{5} + 148323 T_{5}^{4} + \cdots + 298071768 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 9 T^{8} + \cdots - 676387 \) Copy content Toggle raw display
$5$ \( T^{9} - 3 T^{8} + \cdots + 298071768 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 46444486248 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 14065727044011 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 65320441232824 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 18784252476819 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 20\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 64\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 31\!\cdots\!57 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 18\!\cdots\!27 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 39\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 31\!\cdots\!71 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 15\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 36\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 24\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 20\!\cdots\!07 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 45\!\cdots\!31 \) Copy content Toggle raw display
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