Properties

Label 38.8.a.e
Level $38$
Weight $8$
Character orbit 38.a
Self dual yes
Analytic conductor $11.871$
Analytic rank $0$
Dimension $4$
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] = [38,8,Mod(1,38)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.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: \(11.8706309684\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9097x^{2} - 110520x + 10368000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + ( - \beta_1 + 3) q^{3} + 64 q^{4} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 70) q^{5} + ( - 8 \beta_1 + 24) q^{6} + (2 \beta_{3} + \beta_{2} + \beta_1 + 622) q^{7} + 512 q^{8} + (6 \beta_{3} + 14 \beta_{2} + 17 \beta_1 + 2370) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + ( - \beta_1 + 3) q^{3} + 64 q^{4} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 - 70) q^{5} + ( - 8 \beta_1 + 24) q^{6} + (2 \beta_{3} + \beta_{2} + \beta_1 + 622) q^{7} + 512 q^{8} + (6 \beta_{3} + 14 \beta_{2} + 17 \beta_1 + 2370) q^{9} + ( - 8 \beta_{3} - 8 \beta_{2} - 16 \beta_1 - 560) q^{10} + (7 \beta_{3} - 13 \beta_{2} - 10 \beta_1 + 1324) q^{11} + ( - 64 \beta_1 + 192) q^{12} + ( - 21 \beta_{3} - 20 \beta_{2} + 87 \beta_1 + 1346) q^{13} + (16 \beta_{3} + 8 \beta_{2} + 8 \beta_1 + 4976) q^{14} + ( - 48 \beta_{3} + 42 \beta_{2} + 264 \beta_1 + 6630) q^{15} + 4096 q^{16} + (49 \beta_{3} - 85 \beta_{2} - 43 \beta_1 + 5767) q^{17} + (48 \beta_{3} + 112 \beta_{2} + 136 \beta_1 + 18960) q^{18} + 6859 q^{19} + ( - 64 \beta_{3} - 64 \beta_{2} - 128 \beta_1 - 4480) q^{20} + (117 \beta_{3} - 8 \beta_{2} - 833 \beta_1 + 774) q^{21} + (56 \beta_{3} - 104 \beta_{2} - 80 \beta_1 + 10592) q^{22} + ( - 203 \beta_{3} + 230 \beta_{2} + 137 \beta_1 + 682) q^{23} + ( - 512 \beta_1 + 1536) q^{24} + (9 \beta_{3} + 279 \beta_{2} + 288 \beta_1 + 28075) q^{25} + ( - 168 \beta_{3} - 160 \beta_{2} + 696 \beta_1 + 10768) q^{26} + (234 \beta_{3} - 594 \beta_{2} - 2383 \beta_1 - 54783) q^{27} + (128 \beta_{3} + 64 \beta_{2} + 64 \beta_1 + 39808) q^{28} + ( - 295 \beta_{3} - 238 \beta_{2} + 643 \beta_1 - 30622) q^{29} + ( - 384 \beta_{3} + 336 \beta_{2} + 2112 \beta_1 + 53040) q^{30} + (194 \beta_{3} + 84 \beta_{2} + 2524 \beta_1 + 56446) q^{31} + 32768 q^{32} + (540 \beta_{3} + 722 \beta_{2} + 138 \beta_1 + 44124) q^{33} + (392 \beta_{3} - 680 \beta_{2} - 344 \beta_1 + 46136) q^{34} + ( - 539 \beta_{3} - 899 \beta_{2} - 268 \beta_1 - 196490) q^{35} + (384 \beta_{3} + 896 \beta_{2} + 1088 \beta_1 + 151680) q^{36} + (790 \beta_{3} + 1980 \beta_{2} + 308 \beta_1 + 38424) q^{37} + 54872 q^{38} + ( - 1785 \beta_{3} - 958 \beta_{2} + 31 \beta_1 - 437958) q^{39} + ( - 512 \beta_{3} - 512 \beta_{2} - 1024 \beta_1 - 35840) q^{40} + (468 \beta_{3} - 92 \beta_{2} - 2634 \beta_1 - 263400) q^{41} + (936 \beta_{3} - 64 \beta_{2} - 6664 \beta_1 + 6192) q^{42} + (491 \beta_{3} - 909 \beta_{2} + 6160 \beta_1 - 209726) q^{43} + (448 \beta_{3} - 832 \beta_{2} - 640 \beta_1 + 84736) q^{44} + ( - 2547 \beta_{3} - 3897 \beta_{2} - 10674 \beta_1 - 1040940) q^{45} + ( - 1624 \beta_{3} + 1840 \beta_{2} + 1096 \beta_1 + 5456) q^{46} + (107 \beta_{3} + 1905 \beta_{2} + 4636 \beta_1 - 255892) q^{47} + ( - 4096 \beta_1 + 12288) q^{48} + (2195 \beta_{3} + 1661 \beta_{2} - 1811 \beta_1 - 154678) q^{49} + (72 \beta_{3} + 2232 \beta_{2} + 2304 \beta_1 + 224600) q^{50} + (3600 \beta_{3} + 4472 \beta_{2} + 3225 \beta_1 + 181905) q^{51} + ( - 1344 \beta_{3} - 1280 \beta_{2} + 5568 \beta_1 + 86144) q^{52} + ( - 329 \beta_{3} - 420 \beta_{2} + 6143 \beta_1 - 81770) q^{53} + (1872 \beta_{3} - 4752 \beta_{2} - 19064 \beta_1 - 438264) q^{54} + ( - 57 \beta_{3} - 1587 \beta_{2} + 12036 \beta_1 - 55020) q^{55} + (1024 \beta_{3} + 512 \beta_{2} + 512 \beta_1 + 318464) q^{56} + ( - 6859 \beta_1 + 20577) q^{57} + ( - 2360 \beta_{3} - 1904 \beta_{2} + 5144 \beta_1 - 244976) q^{58} + ( - 4680 \beta_{3} - 1988 \beta_{2} + 15159 \beta_1 + 103503) q^{59} + ( - 3072 \beta_{3} + 2688 \beta_{2} + 16896 \beta_1 + 424320) q^{60} + ( - 4165 \beta_{3} - 4129 \beta_{2} - 5612 \beta_1 + 28156) q^{61} + (1552 \beta_{3} + 672 \beta_{2} + 20192 \beta_1 + 451568) q^{62} + (8019 \beta_{3} + 12087 \beta_{2} + 10306 \beta_1 + 2562444) q^{63} + 262144 q^{64} + (3082 \beta_{3} - 2768 \beta_{2} - 34876 \beta_1 + 1121590) q^{65} + (4320 \beta_{3} + 5776 \beta_{2} + 1104 \beta_1 + 352992) q^{66} + (2254 \beta_{3} - 202 \beta_{2} - 6397 \beta_1 + 1449819) q^{67} + (3136 \beta_{3} - 5440 \beta_{2} - 2752 \beta_1 + 369088) q^{68} + ( - 14301 \beta_{3} - 13798 \beta_{2} - 22821 \beta_1 - 621750) q^{69} + ( - 4312 \beta_{3} - 7192 \beta_{2} - 2144 \beta_1 - 1571920) q^{70} + ( - 420 \beta_{3} - 3046 \beta_{2} + 22494 \beta_1 + 2648208) q^{71} + (3072 \beta_{3} + 7168 \beta_{2} + 8704 \beta_1 + 1213440) q^{72} + ( - 5423 \beta_{3} + 14475 \beta_{2} - 38101 \beta_1 + 986517) q^{73} + (6320 \beta_{3} + 15840 \beta_{2} + 2464 \beta_1 + 307392) q^{74} + ( - 1998 \beta_{3} - 13338 \beta_{2} - 66811 \beta_1 - 920175) q^{75} + 438976 q^{76} + (4505 \beta_{3} - 4857 \beta_{2} - 23774 \beta_1 + 1455110) q^{77} + ( - 14280 \beta_{3} - 7664 \beta_{2} + 248 \beta_1 - 3503664) q^{78} + (14970 \beta_{3} - 820 \beta_{2} - 27438 \beta_1 + 1407140) q^{79} + ( - 4096 \beta_{3} - 4096 \beta_{2} - 8192 \beta_1 - 286720) q^{80} + (17700 \beta_{3} + 27620 \beta_{2} + 126104 \beta_1 + 5143881) q^{81} + (3744 \beta_{3} - 736 \beta_{2} - 21072 \beta_1 - 2107200) q^{82} + ( - 5362 \beta_{3} + 11380 \beta_{2} + 33706 \beta_1 + 1160924) q^{83} + (7488 \beta_{3} - 512 \beta_{2} - 53312 \beta_1 + 49536) q^{84} + (3741 \beta_{3} - 8859 \beta_{2} + 81192 \beta_1 - 499530) q^{85} + (3928 \beta_{3} - 7272 \beta_{2} + 49280 \beta_1 - 1677808) q^{86} + ( - 21729 \beta_{3} - 6810 \beta_{2} + 56523 \beta_1 - 3621558) q^{87} + (3584 \beta_{3} - 6656 \beta_{2} - 5120 \beta_1 + 677888) q^{88} + (6692 \beta_{3} + 31242 \beta_{2} + 29866 \beta_1 - 703018) q^{89} + ( - 20376 \beta_{3} - 31176 \beta_{2} - 85392 \beta_1 - 8327520) q^{90} + ( - 19826 \beta_{3} - 14058 \beta_{2} + 105353 \beta_1 - 2213227) q^{91} + ( - 12992 \beta_{3} + 14720 \beta_{2} + 8768 \beta_1 + 43648) q^{92} + ( - 3174 \beta_{3} - 34312 \beta_{2} - 123932 \beta_1 - 10988310) q^{93} + (856 \beta_{3} + 15240 \beta_{2} + 37088 \beta_1 - 2047136) q^{94} + ( - 6859 \beta_{3} - 6859 \beta_{2} - 13718 \beta_1 - 480130) q^{95} + ( - 32768 \beta_1 + 98304) q^{96} + (19950 \beta_{3} - 20838 \beta_{2} + 27372 \beta_1 - 3596098) q^{97} + (17560 \beta_{3} + 13288 \beta_{2} - 14488 \beta_1 - 1237424) q^{98} + (15717 \beta_{3} + 12751 \beta_{2} - 129468 \beta_1 - 1980408) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9} - 2232 q^{10} + 5269 q^{11} + 768 q^{12} + 5406 q^{13} + 19880 q^{14} + 26658 q^{15} + 16384 q^{16} + 22885 q^{17} + 75856 q^{18} + 27436 q^{19} - 17856 q^{20} + 2854 q^{21} + 42152 q^{22} + 3364 q^{23} + 6144 q^{24} + 112561 q^{25} + 43248 q^{26} - 220194 q^{27} + 159040 q^{28} - 122136 q^{29} + 213264 q^{30} + 225480 q^{31} + 131072 q^{32} + 176138 q^{33} + 183080 q^{34} - 785781 q^{35} + 606848 q^{36} + 154096 q^{37} + 219488 q^{38} - 1749220 q^{39} - 142848 q^{40} - 1054628 q^{41} + 22832 q^{42} - 840795 q^{43} + 337216 q^{44} - 4162563 q^{45} + 26912 q^{46} - 1021877 q^{47} + 49152 q^{48} - 621441 q^{49} + 900488 q^{50} + 724892 q^{51} + 345984 q^{52} - 326842 q^{53} - 1761552 q^{54} - 221553 q^{55} + 1272320 q^{56} + 82308 q^{57} - 977088 q^{58} + 421384 q^{59} + 1706112 q^{60} + 116825 q^{61} + 1803840 q^{62} + 10245825 q^{63} + 1048576 q^{64} + 4477428 q^{65} + 1409104 q^{66} + 5794566 q^{67} + 1464640 q^{68} - 2472196 q^{69} - 6286248 q^{70} + 10590626 q^{71} + 4854784 q^{72} + 3971389 q^{73} + 1232768 q^{74} - 3690042 q^{75} + 1755904 q^{76} + 5806573 q^{77} - 13993760 q^{78} + 5597800 q^{79} - 1142784 q^{80} + 20567744 q^{81} - 8437024 q^{82} + 4665800 q^{83} + 182656 q^{84} - 2014461 q^{85} - 6726360 q^{86} - 14449584 q^{87} + 2697728 q^{88} - 2794214 q^{89} - 33300504 q^{90} - 8827314 q^{91} + 215296 q^{92} - 43981204 q^{93} - 8175016 q^{94} - 1913661 q^{95} + 393216 q^{96} - 14445130 q^{97} - 4971528 q^{98} - 7940315 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 9097x^{2} - 110520x + 10368000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 30\nu^{2} - 7627\nu - 219060 ) / 1140 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{3} + 360\nu^{2} + 40279\nu - 1058940 ) / 3420 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 6\beta_{3} + 14\beta_{2} + 23\beta _1 + 4548 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -180\beta_{3} + 720\beta_{2} + 6937\beta _1 + 82620 \) Copy content Toggle raw display

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
95.4845
29.4051
−48.0684
−76.8211
8.00000 −92.4845 64.0000 −426.367 −739.876 875.686 512.000 6366.38 −3410.93
1.2 8.00000 −26.4051 64.0000 139.358 −211.240 458.899 512.000 −1489.77 1114.87
1.3 8.00000 51.0684 64.0000 338.533 408.547 −143.677 512.000 420.979 2708.27
1.4 8.00000 79.8211 64.0000 −330.525 638.569 1294.09 512.000 4184.42 −2644.20
\(n\): e.g. 2-40 or 990-1000
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 38.8.a.e 4
3.b odd 2 1 342.8.a.o 4
4.b odd 2 1 304.8.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.e 4 1.a even 1 1 trivial
304.8.a.e 4 4.b odd 2 1
342.8.a.o 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 12T_{3}^{3} - 9043T_{3}^{2} + 164994T_{3} + 9954648 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 12 T^{3} - 9043 T^{2} + \cdots + 9954648 \) Copy content Toggle raw display
$5$ \( T^{4} + 279 T^{3} + \cdots + 6648480000 \) Copy content Toggle raw display
$7$ \( T^{4} - 2485 T^{3} + \cdots - 74716579192 \) Copy content Toggle raw display
$11$ \( T^{4} - 5269 T^{3} + \cdots + 88406758195632 \) Copy content Toggle raw display
$13$ \( T^{4} - 5406 T^{3} + \cdots - 31\!\cdots\!80 \) Copy content Toggle raw display
$17$ \( T^{4} - 22885 T^{3} + \cdots + 18\!\cdots\!58 \) Copy content Toggle raw display
$19$ \( (T - 6859)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 3364 T^{3} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{4} + 122136 T^{3} + \cdots + 27\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{4} - 225480 T^{3} + \cdots + 32\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{4} - 154096 T^{3} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{4} + 1054628 T^{3} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} + 840795 T^{3} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{4} + 1021877 T^{3} + \cdots - 57\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{4} + 326842 T^{3} + \cdots + 18\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{4} - 421384 T^{3} + \cdots - 33\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} - 116825 T^{3} + \cdots + 23\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{4} - 5794566 T^{3} + \cdots + 26\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{4} - 10590626 T^{3} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{4} - 3971389 T^{3} + \cdots + 13\!\cdots\!58 \) Copy content Toggle raw display
$79$ \( T^{4} - 5597800 T^{3} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} - 4665800 T^{3} + \cdots + 88\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{4} + 2794214 T^{3} + \cdots + 25\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{4} + 14445130 T^{3} + \cdots - 28\!\cdots\!80 \) Copy content Toggle raw display
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