Properties

Label 38.10.a.d
Level $38$
Weight $10$
Character orbit 38.a
Self dual yes
Analytic conductor $19.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [38,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("38.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(19.5713617742\)
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: \( x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 \) 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.

\(f(q)\) \(=\) \( q - 16 q^{2} + (\beta_1 + 21) q^{3} + 256 q^{4} + ( - 4 \beta_{3} - \beta_{2} - 5 \beta_1 - 350) q^{5} + ( - 16 \beta_1 - 336) q^{6} + ( - 9 \beta_{3} + 2 \beta_{2} + 27 \beta_1 + 3075) q^{7} - 4096 q^{8} + (16 \beta_{3} - 18 \beta_{2} + 135 \beta_1 + 4134) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 16 q^{2} + (\beta_1 + 21) q^{3} + 256 q^{4} + ( - 4 \beta_{3} - \beta_{2} - 5 \beta_1 - 350) q^{5} + ( - 16 \beta_1 - 336) q^{6} + ( - 9 \beta_{3} + 2 \beta_{2} + 27 \beta_1 + 3075) q^{7} - 4096 q^{8} + (16 \beta_{3} - 18 \beta_{2} + 135 \beta_1 + 4134) q^{9} + (64 \beta_{3} + 16 \beta_{2} + 80 \beta_1 + 5600) q^{10} + (25 \beta_{2} - 3 \beta_1 - 26056) q^{11} + (256 \beta_1 + 5376) q^{12} + ( - 169 \beta_{3} - 33 \beta_{2} + 766 \beta_1 + 30071) q^{13} + (144 \beta_{3} - 32 \beta_{2} - 432 \beta_1 - 49200) q^{14} + ( - 54 \beta_{3} + 264 \beta_{2} - 1380 \beta_1 - 147720) q^{15} + 65536 q^{16} + (234 \beta_{3} - 587 \beta_{2} - 276 \beta_1 - 103123) q^{17} + ( - 256 \beta_{3} + 288 \beta_{2} - 2160 \beta_1 - 66144) q^{18} + 130321 q^{19} + ( - 1024 \beta_{3} - 256 \beta_{2} - 1280 \beta_1 - 89600) q^{20} + (125 \beta_{3} - 375 \beta_{2} + 2976 \beta_1 + 609189) q^{21} + ( - 400 \beta_{2} + 48 \beta_1 + 416896) q^{22} + (93 \beta_{3} + 1531 \beta_{2} - 678 \beta_1 + 752981) q^{23} + ( - 4096 \beta_1 - 86016) q^{24} + (1350 \beta_{3} + 1845 \beta_{2} + 6795 \beta_1 + 2440945) q^{25} + (2704 \beta_{3} + 528 \beta_{2} - 12256 \beta_1 - 481136) q^{26} + (3948 \beta_{3} - 1674 \beta_{2} + 12769 \beta_1 + 3097563) q^{27} + ( - 2304 \beta_{3} + 512 \beta_{2} + 6912 \beta_1 + 787200) q^{28} + ( - 3481 \beta_{3} - 1027 \beta_{2} - 1436 \beta_1 + 1537183) q^{29} + (864 \beta_{3} - 4224 \beta_{2} + 22080 \beta_1 + 2363520) q^{30} + ( - 2150 \beta_{3} - 2050 \beta_{2} + 9482 \beta_1 + 3192456) q^{31} - 1048576 q^{32} + ( - 2198 \beta_{3} - 1596 \beta_{2} - 38998 \beta_1 - 815454) q^{33} + ( - 3744 \beta_{3} + 9392 \beta_{2} + 4416 \beta_1 + 1649968) q^{34} + ( - 20156 \beta_{3} + 10051 \beta_{2} - 20875 \beta_1 + 2453330) q^{35} + (4096 \beta_{3} - 4608 \beta_{2} + 34560 \beta_1 + 1058304) q^{36} + (14570 \beta_{3} - 6770 \beta_{2} - 40262 \beta_1 + 5128462) q^{37} - 2085136 q^{38} + (12559 \beta_{3} - 7047 \beta_{2} + 93298 \beta_1 + 17471739) q^{39} + (16384 \beta_{3} + 4096 \beta_{2} + 20480 \beta_1 + 1433600) q^{40} + ( - 28480 \beta_{3} - 19020 \beta_{2} - 11750 \beta_1 + 2893200) q^{41} + ( - 2000 \beta_{3} + 6000 \beta_{2} - 47616 \beta_1 - 9747024) q^{42} + (25660 \beta_{3} + 24965 \beta_{2} - 59233 \beta_1 + 1937238) q^{43} + (6400 \beta_{2} - 768 \beta_1 - 6670336) q^{44} + (33138 \beta_{3} + 28557 \beta_{2} - 352695 \beta_1 - 30988530) q^{45} + ( - 1488 \beta_{3} - 24496 \beta_{2} + 10848 \beta_1 - 12047696) q^{46} + (32602 \beta_{3} - 31111 \beta_{2} + 45347 \beta_1 - 7894898) q^{47} + (65536 \beta_1 + 1376256) q^{48} + ( - 100318 \beta_{3} + 12239 \beta_{2} + 110482 \beta_1 + 4720576) q^{49} + ( - 21600 \beta_{3} - 29520 \beta_{2} - 108720 \beta_1 - 39055120) q^{50} + (49576 \beta_{3} + 37392 \beta_{2} + 217655 \beta_1 - 2126961) q^{51} + ( - 43264 \beta_{3} - 8448 \beta_{2} + 196096 \beta_1 + 7698176) q^{52} + ( - 78901 \beta_{3} - 35357 \beta_{2} + 259234 \beta_1 + 18108791) q^{53} + ( - 63168 \beta_{3} + 26784 \beta_{2} - 204304 \beta_1 - 49561008) q^{54} + (51834 \beta_{3} + 18501 \beta_{2} + 321855 \beta_1 + 5350710) q^{55} + (36864 \beta_{3} - 8192 \beta_{2} - 110592 \beta_1 - 12595200) q^{56} + (130321 \beta_1 + 2736741) q^{57} + (55696 \beta_{3} + 16432 \beta_{2} + 22976 \beta_1 - 24594928) q^{58} + ( - 8236 \beta_{3} - 66072 \beta_{2} - 25343 \beta_1 - 37327107) q^{59} + ( - 13824 \beta_{3} + 67584 \beta_{2} - 353280 \beta_1 - 37816320) q^{60} + (44408 \beta_{3} - 115309 \beta_{2} - 7739 \beta_1 + 32233368) q^{61} + (34400 \beta_{3} + 32800 \beta_{2} - 151712 \beta_1 - 51079296) q^{62} + (258888 \beta_{3} - 71559 \beta_{2} + 636137 \beta_1 + 25788720) q^{63} + 16777216 q^{64} + ( - 112430 \beta_{3} + 309130 \beta_{2} + \cdots + 31222100) q^{65}+ \cdots + ( - 519682 \beta_{3} + 374571 \beta_{2} + \cdots - 420494730) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{2} + 84 q^{3} + 1024 q^{4} - 1395 q^{5} - 1344 q^{6} + 12307 q^{7} - 16384 q^{8} + 16538 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 64 q^{2} + 84 q^{3} + 1024 q^{4} - 1395 q^{5} - 1344 q^{6} + 12307 q^{7} - 16384 q^{8} + 16538 q^{9} + 22320 q^{10} - 104249 q^{11} + 21504 q^{12} + 120486 q^{13} - 196912 q^{14} - 591090 q^{15} + 262144 q^{16} - 412139 q^{17} - 264608 q^{18} + 521284 q^{19} - 357120 q^{20} + 2437006 q^{21} + 1667984 q^{22} + 3010300 q^{23} - 344064 q^{24} + 9760585 q^{25} - 1927776 q^{26} + 12387978 q^{27} + 3150592 q^{28} + 6153240 q^{29} + 9457440 q^{30} + 12774024 q^{31} - 4194304 q^{32} - 3258022 q^{33} + 6594224 q^{34} + 9823425 q^{35} + 4233728 q^{36} + 20506048 q^{37} - 8340544 q^{38} + 69881444 q^{39} + 5713920 q^{40} + 11620300 q^{41} - 38992096 q^{42} + 7698327 q^{43} - 26687744 q^{44} - 124015815 q^{45} - 48164800 q^{46} - 31581083 q^{47} + 5505024 q^{48} + 18970383 q^{49} - 156169360 q^{50} - 8594812 q^{51} + 30844416 q^{52} + 72549422 q^{53} - 198207648 q^{54} + 21332505 q^{55} - 50409472 q^{56} + 10946964 q^{57} - 98451840 q^{58} - 149234120 q^{59} - 151319040 q^{60} + 129004373 q^{61} - 204384384 q^{62} + 102967551 q^{63} + 67108864 q^{64} + 124691700 q^{65} + 52128352 q^{66} + 132595266 q^{67} - 105507584 q^{68} - 45529972 q^{69} - 157174800 q^{70} - 47138482 q^{71} - 67739648 q^{72} - 39332795 q^{73} - 328096768 q^{74} + 824627010 q^{75} + 133448704 q^{76} - 165933719 q^{77} - 1118103104 q^{78} - 307010840 q^{79} - 91422720 q^{80} + 1305551744 q^{81} - 185924800 q^{82} - 746568232 q^{83} + 623873536 q^{84} - 105005985 q^{85} - 123173232 q^{86} - 82148208 q^{87} + 427003904 q^{88} + 286943482 q^{89} + 1984253040 q^{90} + 3155781114 q^{91} + 770636800 q^{92} + 1151901596 q^{93} + 505297328 q^{94} - 181797795 q^{95} - 88080384 q^{96} + 793519958 q^{97} - 303526128 q^{98} - 1681833809 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{3} - 736\nu^{2} - 252737\nu - 5880008 ) / 20632 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -17\nu^{3} + 244\nu^{2} + 698613\nu + 30780440 ) / 20632 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{3} - 490\nu^{2} - 413779\nu - 18350856 ) / 10316 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 193\beta_{3} + 85\beta_{2} - 397\beta _1 + 103370 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 43865\beta_{3} + 35033\beta_{2} - 46793\beta _1 + 12429538 ) / 6 \) 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
−124.888
219.264
−26.2676
−67.1081
−16.0000 −140.229 256.000 1263.95 2243.67 −3487.42 −4096.00 −18.7042 −20223.2
1.2 −16.0000 −66.6053 256.000 −2418.56 1065.68 −1533.95 −4096.00 −15246.7 38696.9
1.3 −16.0000 25.2570 256.000 2126.71 −404.112 11469.0 −4096.00 −19045.1 −34027.4
1.4 −16.0000 265.578 256.000 −2367.11 −4249.24 5859.40 −4096.00 50848.5 37873.7
\(n\): 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 38.10.a.d 4
3.b odd 2 1 342.10.a.l 4
4.b odd 2 1 304.10.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.d 4 1.a even 1 1 trivial
304.10.a.e 4 4.b odd 2 1
342.10.a.l 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 84T_{3}^{3} - 44107T_{3}^{2} - 1329018T_{3} + 62650008 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(38))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 84 T^{3} - 44107 T^{2} + \cdots + 62650008 \) Copy content Toggle raw display
$5$ \( T^{4} + 1395 T^{3} + \cdots + 15389064288000 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 359494671206216 \) Copy content Toggle raw display
$11$ \( T^{4} + 104249 T^{3} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{4} - 120486 T^{3} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{4} + 412139 T^{3} + \cdots + 27\!\cdots\!62 \) Copy content Toggle raw display
$19$ \( (T - 130321)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 3010300 T^{3} + \cdots + 22\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{4} - 6153240 T^{3} + \cdots - 34\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{4} - 12774024 T^{3} + \cdots - 14\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{4} - 20506048 T^{3} + \cdots - 58\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( T^{4} - 11620300 T^{3} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} - 7698327 T^{3} + \cdots - 19\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{4} + 31581083 T^{3} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} - 72549422 T^{3} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{4} + 149234120 T^{3} + \cdots - 27\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{4} - 129004373 T^{3} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} - 132595266 T^{3} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{4} + 47138482 T^{3} + \cdots - 32\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{4} + 39332795 T^{3} + \cdots + 35\!\cdots\!34 \) Copy content Toggle raw display
$79$ \( T^{4} + 307010840 T^{3} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + 746568232 T^{3} + \cdots - 94\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{4} - 286943482 T^{3} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{4} - 793519958 T^{3} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
show more
show less