Properties

Label 10.18.a.b
Level $10$
Weight $18$
Character orbit 10.a
Self dual yes
Analytic conductor $18.322$
Analytic rank $1$
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] = [10,18,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(18.3222087345\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{36061}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 9015 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 5 \)
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 \(\beta = 80\sqrt{36061}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 256 q^{2} + ( - \beta - 3154) q^{3} + 65536 q^{4} - 390625 q^{5} + (256 \beta + 807424) q^{6} + (843 \beta + 3271922) q^{7} - 16777216 q^{8} + (6308 \beta + 111597953) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 256 q^{2} + ( - \beta - 3154) q^{3} + 65536 q^{4} - 390625 q^{5} + (256 \beta + 807424) q^{6} + (843 \beta + 3271922) q^{7} - 16777216 q^{8} + (6308 \beta + 111597953) q^{9} + 100000000 q^{10} + (12738 \beta + 594704352) q^{11} + ( - 65536 \beta - 206700544) q^{12} + ( - 122772 \beta - 1008959614) q^{13} + ( - 215808 \beta - 837612032) q^{14} + (390625 \beta + 1232031250) q^{15} + 4294967296 q^{16} + (1944132 \beta - 9377819718) q^{17} + ( - 1614848 \beta - 28569075968) q^{18} + (2994168 \beta - 68352415300) q^{19} - 25600000000 q^{20} + ( - 5930744 \beta - 204875949188) q^{21} + ( - 3260928 \beta - 152244314112) q^{22} + ( - 22452951 \beta - 324617085354) q^{23} + (16777216 \beta + 52915339264) q^{24} + 152587890625 q^{25} + (31429632 \beta + 258293661184) q^{26} + ( - 2353222 \beta - 1400497712860) q^{27} + (55246848 \beta + 214428680192) q^{28} + ( - 4841544 \beta - 2348271710010) q^{29} + ( - 100000000 \beta - 315400000000) q^{30} + (342178986 \beta + 3560433689372) q^{31} - 1099511627776 q^{32} + ( - 634880004 \beta - 4815505641408) q^{33} + ( - 497697792 \beta + 2400721847808) q^{34} + ( - 329296875 \beta - 1278094531250) q^{35} + (413401088 \beta + 7313683447808) q^{36} + (1485224712 \beta + 4966557318722) q^{37} + ( - 766507008 \beta + 17498218316800) q^{38} + (1396182502 \beta + 31516857611356) q^{39} + 6553600000000 q^{40} + (2262047724 \beta + 4811355940422) q^{41} + (1518270464 \beta + 52448242992128) q^{42} + ( - 8809717953 \beta + 3854218005926) q^{43} + (834797568 \beta + 38974544412672) q^{44} + ( - 2464062500 \beta - 43592950390625) q^{45} + (5747955456 \beta + 83101973850624) q^{46} + ( - 4448723037 \beta - 233036287918998) q^{47} + ( - 4294967296 \beta - 13546326851584) q^{48} + (5516460492 \beta - 57914073443523) q^{49} - 39062500000000 q^{50} + (3246027390 \beta - 419109358542228) q^{51} + ( - 8045985792 \beta - 66123177263104) q^{52} + (27224743188 \beta - 311666560142214) q^{53} + (602424832 \beta + 358527414492160) q^{54} + ( - 4975781250 \beta - 232306387500000) q^{55} + ( - 14143193088 \beta - 54893742129152) q^{56} + (58908809428 \beta - 475441712531000) q^{57} + (1239435264 \beta + 601157557762560) q^{58} + ( - 85208012868 \beta + 327308035997580) q^{59} + (25600000000 \beta + 80742400000000) q^{60} + ( - 118790012448 \beta + 664520192581442) q^{61} + ( - 87597820416 \beta - 911471024479232) q^{62} + (114716358355 \beta + 15\!\cdots\!66) q^{63}+ \cdots + (5172929777730 \beta + 84\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 512 q^{2} - 6308 q^{3} + 131072 q^{4} - 781250 q^{5} + 1614848 q^{6} + 6543844 q^{7} - 33554432 q^{8} + 223195906 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 512 q^{2} - 6308 q^{3} + 131072 q^{4} - 781250 q^{5} + 1614848 q^{6} + 6543844 q^{7} - 33554432 q^{8} + 223195906 q^{9} + 200000000 q^{10} + 1189408704 q^{11} - 413401088 q^{12} - 2017919228 q^{13} - 1675224064 q^{14} + 2464062500 q^{15} + 8589934592 q^{16} - 18755639436 q^{17} - 57138151936 q^{18} - 136704830600 q^{19} - 51200000000 q^{20} - 409751898376 q^{21} - 304488628224 q^{22} - 649234170708 q^{23} + 105830678528 q^{24} + 305175781250 q^{25} + 516587322368 q^{26} - 2800995425720 q^{27} + 428857360384 q^{28} - 4696543420020 q^{29} - 630800000000 q^{30} + 7120867378744 q^{31} - 2199023255552 q^{32} - 9631011282816 q^{33} + 4801443695616 q^{34} - 2556189062500 q^{35} + 14627366895616 q^{36} + 9933114637444 q^{37} + 34996436633600 q^{38} + 63033715222712 q^{39} + 13107200000000 q^{40} + 9622711880844 q^{41} + 104896485984256 q^{42} + 7708436011852 q^{43} + 77949088825344 q^{44} - 87185900781250 q^{45} + 166203947701248 q^{46} - 466072575837996 q^{47} - 27092653703168 q^{48} - 115828146887046 q^{49} - 78125000000000 q^{50} - 838218717084456 q^{51} - 132246354526208 q^{52} - 623333120284428 q^{53} + 717054828984320 q^{54} - 464612775000000 q^{55} - 109787484258304 q^{56} - 950883425062000 q^{57} + 12\!\cdots\!20 q^{58}+ \cdots + 16\!\cdots\!12 q^{99}+O(q^{100}) \) 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
95.4487
−94.4487
−256.000 −18345.8 65536.0 −390625. 4.69652e6 1.60786e7 −1.67772e7 2.07428e8 1.00000e8
1.2 −256.000 12037.8 65536.0 −390625. −3.08167e6 −9.53475e6 −1.67772e7 1.57682e7 1.00000e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)

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 10.18.a.b 2
3.b odd 2 1 90.18.a.n 2
4.b odd 2 1 80.18.a.e 2
5.b even 2 1 50.18.a.g 2
5.c odd 4 2 50.18.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.b 2 1.a even 1 1 trivial
50.18.a.g 2 5.b even 2 1
50.18.b.e 4 5.c odd 4 2
80.18.a.e 2 4.b odd 2 1
90.18.a.n 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 6308T_{3} - 220842684 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6308 T - 220842684 \) Copy content Toggle raw display
$5$ \( (T + 390625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 153305493395516 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 24\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 78\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 10\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 14\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 48\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 73\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 28\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 69\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 95\!\cdots\!84 \) Copy content Toggle raw display
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