Properties

Label 5.8.a.b
Level $5$
Weight $8$
Character orbit 5.a
Self dual yes
Analytic conductor $1.562$
Analytic rank $0$
Dimension $2$
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] = [5,8,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, 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("5.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(1.56192512742\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 10) q^{2} + ( - 8 \beta + 10) q^{3} + (20 \beta + 48) q^{4} - 125 q^{5} + ( - 70 \beta - 508) q^{6} + (56 \beta - 50) q^{7} + (120 \beta + 720) q^{8} + ( - 160 \beta + 2777) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 10) q^{2} + ( - 8 \beta + 10) q^{3} + (20 \beta + 48) q^{4} - 125 q^{5} + ( - 70 \beta - 508) q^{6} + (56 \beta - 50) q^{7} + (120 \beta + 720) q^{8} + ( - 160 \beta + 2777) q^{9} + ( - 125 \beta - 1250) q^{10} + (400 \beta + 2272) q^{11} + ( - 184 \beta - 11680) q^{12} + ( - 608 \beta + 1770) q^{13} + (510 \beta + 3756) q^{14} + (1000 \beta - 1250) q^{15} + ( - 640 \beta + 10176) q^{16} + ( - 1184 \beta - 13670) q^{17} + (1177 \beta + 15610) q^{18} + ( - 320 \beta + 19380) q^{19} + ( - 2500 \beta - 6000) q^{20} + (960 \beta - 34548) q^{21} + (6272 \beta + 53120) q^{22} + ( - 408 \beta - 62070) q^{23} + ( - 4560 \beta - 65760) q^{24} + 15625 q^{25} + ( - 4310 \beta - 28508) q^{26} + ( - 6320 \beta + 103180) q^{27} + (1688 \beta + 82720) q^{28} + (19520 \beta - 36130) q^{29} + (8750 \beta + 63500) q^{30} + ( - 2800 \beta + 153412) q^{31} + ( - 11584 \beta - 39040) q^{32} + ( - 14176 \beta - 220480) q^{33} + ( - 25510 \beta - 226684) q^{34} + ( - 7000 \beta + 6250) q^{35} + (47860 \beta - 109904) q^{36} + (25536 \beta - 61510) q^{37} + (16180 \beta + 169480) q^{38} + ( - 20240 \beta + 387364) q^{39} + ( - 15000 \beta - 90000) q^{40} + ( - 56800 \beta + 132182) q^{41} + ( - 24948 \beta - 272520) q^{42} + (43192 \beta + 211650) q^{43} + (64640 \beta + 717056) q^{44} + (20000 \beta - 347125) q^{45} + ( - 66150 \beta - 651708) q^{46} + (45496 \beta - 52730) q^{47} + ( - 87808 \beta + 490880) q^{48} + ( - 5600 \beta - 582707) q^{49} + (15625 \beta + 156250) q^{50} + (97520 \beta + 583172) q^{51} + (6216 \beta - 839200) q^{52} + ( - 53408 \beta - 1195790) q^{53} + (39980 \beta + 551480) q^{54} + ( - 50000 \beta - 284000) q^{55} + (34320 \beta + 474720) q^{56} + ( - 158240 \beta + 388360) q^{57} + (159070 \beta + 1122220) q^{58} + ( - 227360 \beta - 560060) q^{59} + (23000 \beta + 1460000) q^{60} + (160000 \beta + 1128522) q^{61} + (125412 \beta + 1321320) q^{62} + (163512 \beta - 819810) q^{63} + ( - 72960 \beta - 2573312) q^{64} + (76000 \beta - 221250) q^{65} + ( - 362240 \beta - 3282176) q^{66} + ( - 79384 \beta + 2258230) q^{67} + ( - 330232 \beta - 2455840) q^{68} + (492480 \beta - 372636) q^{69} + ( - 63750 \beta - 469500) q^{70} + ( - 70000 \beta + 310892) q^{71} + (218040 \beta + 540240) q^{72} + ( - 226208 \beta + 2284530) q^{73} + (193850 \beta + 1325636) q^{74} + ( - 125000 \beta + 156250) q^{75} + (372240 \beta + 443840) q^{76} + (107232 \beta + 1588800) q^{77} + (184964 \beta + 2335400) q^{78} + ( - 472480 \beta + 2166520) q^{79} + (80000 \beta - 1272000) q^{80} + ( - 538720 \beta - 1198939) q^{81} + ( - 435818 \beta - 2994980) q^{82} + (490392 \beta - 4896510) q^{83} + ( - 644880 \beta - 199104) q^{84} + (148000 \beta + 1708750) q^{85} + (643570 \beta + 5399092) q^{86} + (484240 \beta - 12229460) q^{87} + (560640 \beta + 5283840) q^{88} + (317760 \beta + 3012810) q^{89} + ( - 147125 \beta - 1951250) q^{90} + (129520 \beta - 2676148) q^{91} + ( - 1260984 \beta - 3599520) q^{92} + ( - 1255296 \beta + 3236520) q^{93} + (402230 \beta + 2930396) q^{94} + (40000 \beta - 2422500) q^{95} + (196480 \beta + 6652672) q^{96} + (561696 \beta + 2304770) q^{97} + ( - 638707 \beta - 6252670) q^{98} + (747280 \beta + 1445344) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{2} + 20 q^{3} + 96 q^{4} - 250 q^{5} - 1016 q^{6} - 100 q^{7} + 1440 q^{8} + 5554 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{2} + 20 q^{3} + 96 q^{4} - 250 q^{5} - 1016 q^{6} - 100 q^{7} + 1440 q^{8} + 5554 q^{9} - 2500 q^{10} + 4544 q^{11} - 23360 q^{12} + 3540 q^{13} + 7512 q^{14} - 2500 q^{15} + 20352 q^{16} - 27340 q^{17} + 31220 q^{18} + 38760 q^{19} - 12000 q^{20} - 69096 q^{21} + 106240 q^{22} - 124140 q^{23} - 131520 q^{24} + 31250 q^{25} - 57016 q^{26} + 206360 q^{27} + 165440 q^{28} - 72260 q^{29} + 127000 q^{30} + 306824 q^{31} - 78080 q^{32} - 440960 q^{33} - 453368 q^{34} + 12500 q^{35} - 219808 q^{36} - 123020 q^{37} + 338960 q^{38} + 774728 q^{39} - 180000 q^{40} + 264364 q^{41} - 545040 q^{42} + 423300 q^{43} + 1434112 q^{44} - 694250 q^{45} - 1303416 q^{46} - 105460 q^{47} + 981760 q^{48} - 1165414 q^{49} + 312500 q^{50} + 1166344 q^{51} - 1678400 q^{52} - 2391580 q^{53} + 1102960 q^{54} - 568000 q^{55} + 949440 q^{56} + 776720 q^{57} + 2244440 q^{58} - 1120120 q^{59} + 2920000 q^{60} + 2257044 q^{61} + 2642640 q^{62} - 1639620 q^{63} - 5146624 q^{64} - 442500 q^{65} - 6564352 q^{66} + 4516460 q^{67} - 4911680 q^{68} - 745272 q^{69} - 939000 q^{70} + 621784 q^{71} + 1080480 q^{72} + 4569060 q^{73} + 2651272 q^{74} + 312500 q^{75} + 887680 q^{76} + 3177600 q^{77} + 4670800 q^{78} + 4333040 q^{79} - 2544000 q^{80} - 2397878 q^{81} - 5989960 q^{82} - 9793020 q^{83} - 398208 q^{84} + 3417500 q^{85} + 10798184 q^{86} - 24458920 q^{87} + 10567680 q^{88} + 6025620 q^{89} - 3902500 q^{90} - 5352296 q^{91} - 7199040 q^{92} + 6473040 q^{93} + 5860792 q^{94} - 4845000 q^{95} + 13305344 q^{96} + 4609540 q^{97} - 12505340 q^{98} + 2890688 q^{99}+O(q^{100}) \) 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
−4.35890
4.35890
1.28220 79.7424 −126.356 −125.000 102.246 −538.197 −326.136 4171.85 −160.275
1.2 18.7178 −59.7424 222.356 −125.000 −1118.25 438.197 1766.14 1382.15 −2339.72
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

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 5.8.a.b 2
3.b odd 2 1 45.8.a.h 2
4.b odd 2 1 80.8.a.g 2
5.b even 2 1 25.8.a.b 2
5.c odd 4 2 25.8.b.c 4
7.b odd 2 1 245.8.a.c 2
8.b even 2 1 320.8.a.l 2
8.d odd 2 1 320.8.a.u 2
11.b odd 2 1 605.8.a.d 2
15.d odd 2 1 225.8.a.w 2
15.e even 4 2 225.8.b.m 4
20.d odd 2 1 400.8.a.bb 2
20.e even 4 2 400.8.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.b 2 1.a even 1 1 trivial
25.8.a.b 2 5.b even 2 1
25.8.b.c 4 5.c odd 4 2
45.8.a.h 2 3.b odd 2 1
80.8.a.g 2 4.b odd 2 1
225.8.a.w 2 15.d odd 2 1
225.8.b.m 4 15.e even 4 2
245.8.a.c 2 7.b odd 2 1
320.8.a.l 2 8.b even 2 1
320.8.a.u 2 8.d odd 2 1
400.8.a.bb 2 20.d odd 2 1
400.8.c.m 4 20.e even 4 2
605.8.a.d 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 20T_{2} + 24 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 20T + 24 \) Copy content Toggle raw display
$3$ \( T^{2} - 20T - 4764 \) Copy content Toggle raw display
$5$ \( (T + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 100T - 235836 \) Copy content Toggle raw display
$11$ \( T^{2} - 4544 T - 6998016 \) Copy content Toggle raw display
$13$ \( T^{2} - 3540 T - 24961564 \) Copy content Toggle raw display
$17$ \( T^{2} + 27340 T + 80327844 \) Copy content Toggle raw display
$19$ \( T^{2} - 38760 T + 367802000 \) Copy content Toggle raw display
$23$ \( T^{2} + 124140 T + 3840033636 \) Copy content Toggle raw display
$29$ \( T^{2} + 72260 T - 27652933500 \) Copy content Toggle raw display
$31$ \( T^{2} - 306824 T + 22939401744 \) Copy content Toggle raw display
$37$ \( T^{2} + 123020 T - 45775154396 \) Copy content Toggle raw display
$41$ \( T^{2} - 264364 T - 227722158876 \) Copy content Toggle raw display
$43$ \( T^{2} - 423300 T - 96985991164 \) Copy content Toggle raw display
$47$ \( T^{2} + 105460 T - 154530884316 \) Copy content Toggle raw display
$53$ \( T^{2} + 2391580 T + 1213130224836 \) Copy content Toggle raw display
$59$ \( T^{2} + 1120120 T - 3614968086000 \) Copy content Toggle raw display
$61$ \( T^{2} - 2257044 T - 672038095516 \) Copy content Toggle raw display
$67$ \( T^{2} - 4516460 T + 4620664454244 \) Copy content Toggle raw display
$71$ \( T^{2} - 621784 T - 275746164336 \) Copy content Toggle raw display
$73$ \( T^{2} - 4569060 T + 1330152816836 \) Copy content Toggle raw display
$79$ \( T^{2} - 4333040 T - 12272229720000 \) Copy content Toggle raw display
$83$ \( T^{2} + 9793020 T + 5699002341636 \) Copy content Toggle raw display
$89$ \( T^{2} - 6025620 T + 1403196358500 \) Copy content Toggle raw display
$97$ \( T^{2} - 4609540 T - 18666217374716 \) Copy content Toggle raw display
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