Properties

Label 90.18.a.n
Level $90$
Weight $18$
Character orbit 90.a
Self dual yes
Analytic conductor $164.900$
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] = [90,18,Mod(1,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 90.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: \(164.899878610\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
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 \(\beta = 240\sqrt{36061}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 256 q^{2} + 65536 q^{4} + 390625 q^{5} + ( - 281 \beta + 3271922) q^{7} + 16777216 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 256 q^{2} + 65536 q^{4} + 390625 q^{5} + ( - 281 \beta + 3271922) q^{7} + 16777216 q^{8} + 100000000 q^{10} + (4246 \beta - 594704352) q^{11} + (40924 \beta - 1008959614) q^{13} + ( - 71936 \beta + 837612032) q^{14} + 4294967296 q^{16} + (648044 \beta + 9377819718) q^{17} + ( - 998056 \beta - 68352415300) q^{19} + 25600000000 q^{20} + (1086976 \beta - 152244314112) q^{22} + ( - 7484317 \beta + 324617085354) q^{23} + 152587890625 q^{25} + (10476544 \beta - 258293661184) q^{26} + ( - 18415616 \beta + 214428680192) q^{28} + ( - 1613848 \beta + 2348271710010) q^{29} + ( - 114059662 \beta + 3560433689372) q^{31} + 1099511627776 q^{32} + (165899264 \beta + 2400721847808) q^{34} + ( - 109765625 \beta + 1278094531250) q^{35} + ( - 495074904 \beta + 4966557318722) q^{37} + ( - 255502336 \beta - 17498218316800) q^{38} + 6553600000000 q^{40} + (754015908 \beta - 4811355940422) q^{41} + (2936572651 \beta + 3854218005926) q^{43} + (278265856 \beta - 38974544412672) q^{44} + ( - 1915985152 \beta + 83101973850624) q^{46} + ( - 1482907679 \beta + 233036287918998) q^{47} + ( - 1838820164 \beta - 57914073443523) q^{49} + 39062500000000 q^{50} + (2681995264 \beta - 66123177263104) q^{52} + (9074914396 \beta + 311666560142214) q^{53} + (1658593750 \beta - 232306387500000) q^{55} + ( - 4714397696 \beta + 54893742129152) q^{56} + ( - 413145088 \beta + 601157557762560) q^{58} + ( - 28402670956 \beta - 327308035997580) q^{59} + (39596670816 \beta + 664520192581442) q^{61} + ( - 29199273472 \beta + 911471024479232) q^{62} + 281474976710656 q^{64} + (15985937500 \beta - 394124849218750) q^{65} + (42155770497 \beta - 21\!\cdots\!98) q^{67}+ \cdots + ( - 470737961984 \beta - 14\!\cdots\!88) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 512 q^{2} + 131072 q^{4} + 781250 q^{5} + 6543844 q^{7} + 33554432 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 512 q^{2} + 131072 q^{4} + 781250 q^{5} + 6543844 q^{7} + 33554432 q^{8} + 200000000 q^{10} - 1189408704 q^{11} - 2017919228 q^{13} + 1675224064 q^{14} + 8589934592 q^{16} + 18755639436 q^{17} - 136704830600 q^{19} + 51200000000 q^{20} - 304488628224 q^{22} + 649234170708 q^{23} + 305175781250 q^{25} - 516587322368 q^{26} + 428857360384 q^{28} + 4696543420020 q^{29} + 7120867378744 q^{31} + 2199023255552 q^{32} + 4801443695616 q^{34} + 2556189062500 q^{35} + 9933114637444 q^{37} - 34996436633600 q^{38} + 13107200000000 q^{40} - 9622711880844 q^{41} + 7708436011852 q^{43} - 77949088825344 q^{44} + 166203947701248 q^{46} + 466072575837996 q^{47} - 115828146887046 q^{49} + 78125000000000 q^{50} - 132246354526208 q^{52} + 623333120284428 q^{53} - 464612775000000 q^{55} + 109787484258304 q^{56} + 12\!\cdots\!20 q^{58}+ \cdots - 29\!\cdots\!76 q^{98}+O(q^{100}) \) 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 0 65536.0 390625. 0 −9.53475e6 1.67772e7 0 1.00000e8
1.2 256.000 0 65536.0 390625. 0 1.60786e7 1.67772e7 0 1.00000e8
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

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 90.18.a.n 2
3.b odd 2 1 10.18.a.b 2
12.b even 2 1 80.18.a.e 2
15.d odd 2 1 50.18.a.g 2
15.e even 4 2 50.18.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.b 2 3.b odd 2 1
50.18.a.g 2 15.d odd 2 1
50.18.b.e 4 15.e even 4 2
80.18.a.e 2 12.b even 2 1
90.18.a.n 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(90))\):

\( T_{7}^{2} - 6543844T_{7} - 153305493395516 \) Copy content Toggle raw display
\( T_{11}^{2} + 1189408704T_{11} + 316225990516322304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 390625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 153305493395516 \) Copy content Toggle raw display
$11$ \( T^{2} + 1189408704 T + 31\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{2} + 2017919228 T - 24\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{2} - 18755639436 T - 78\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + 136704830600 T + 26\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} - 649234170708 T - 10\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{2} - 4696543420020 T + 55\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} - 7120867378744 T - 14\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{2} - 9933114637444 T - 48\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{2} + 9622711880844 T - 11\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{2} - 7708436011852 T - 17\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} - 466072575837996 T + 49\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} - 623333120284428 T - 73\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + 654616071995160 T - 15\!\cdots\!00 \) 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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