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Properties

Label 90.4.e.e
Level $90$
Weight $4$
Character orbit 90.e
Analytic conductor $5.310$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [90,4,Mod(31,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 90.e (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(5.31017190052\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} - 30x^{5} + 1440x^{4} - 810x^{3} + 5103x^{2} - 19683x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} - 2) q^{2} - \beta_1 q^{3} + 4 \beta_{3} q^{4} + 5 \beta_{3} q^{5} + 2 \beta_{5} q^{6} + ( - \beta_{6} - \beta_{5} - 6 \beta_{3} + \cdots - 6) q^{7}+ \cdots + (\beta_{7} - 2 \beta_{6} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} - 2) q^{2} - \beta_1 q^{3} + 4 \beta_{3} q^{4} + 5 \beta_{3} q^{5} + 2 \beta_{5} q^{6} + ( - \beta_{6} - \beta_{5} - 6 \beta_{3} + \cdots - 6) q^{7}+ \cdots + (12 \beta_{7} + 9 \beta_{6} + \cdots + 482) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 2 q^{3} - 16 q^{4} - 20 q^{5} - 2 q^{6} - 23 q^{7} + 64 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 2 q^{3} - 16 q^{4} - 20 q^{5} - 2 q^{6} - 23 q^{7} + 64 q^{8} + 8 q^{9} + 80 q^{10} - 39 q^{11} - 4 q^{12} - 56 q^{13} - 46 q^{14} - 5 q^{15} - 64 q^{16} + 162 q^{17} - 26 q^{18} + 166 q^{19} - 80 q^{20} + 16 q^{21} - 78 q^{22} - 75 q^{23} + 16 q^{24} - 100 q^{25} + 224 q^{26} - 70 q^{27} + 184 q^{28} - 183 q^{29} + 20 q^{30} - 218 q^{31} - 128 q^{32} - 27 q^{33} - 162 q^{34} + 230 q^{35} + 20 q^{36} + 292 q^{37} - 166 q^{38} + 202 q^{39} - 160 q^{40} - 204 q^{41} + 434 q^{42} - 167 q^{43} + 312 q^{44} + 25 q^{45} + 300 q^{46} - 459 q^{47} - 16 q^{48} - 51 q^{49} - 200 q^{50} - 339 q^{51} - 224 q^{52} + 1368 q^{53} - 200 q^{54} + 390 q^{55} - 184 q^{56} - 65 q^{57} - 366 q^{58} - 429 q^{59} - 20 q^{60} - 491 q^{61} + 872 q^{62} - 53 q^{63} + 512 q^{64} - 280 q^{65} - 960 q^{66} - 140 q^{67} - 324 q^{68} + 231 q^{69} - 230 q^{70} + 240 q^{71} + 64 q^{72} - 206 q^{73} - 292 q^{74} - 25 q^{75} - 332 q^{76} - 1404 q^{77} + 1628 q^{78} + 412 q^{79} + 640 q^{80} + 2804 q^{81} + 816 q^{82} - 1503 q^{83} - 932 q^{84} - 405 q^{85} - 334 q^{86} + 696 q^{87} - 312 q^{88} + 2346 q^{89} + 80 q^{90} + 1628 q^{91} - 300 q^{92} - 2930 q^{93} - 918 q^{94} - 415 q^{95} - 32 q^{96} + 343 q^{97} + 204 q^{98} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 7x^{6} - 30x^{5} + 1440x^{4} - 810x^{3} + 5103x^{2} - 19683x + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{7} - 190\nu^{6} + 289\nu^{5} + 3137\nu^{4} + 8067\nu^{3} - 119349\nu^{2} + 94608\nu + 2636793 ) / 260982 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 187 \nu^{7} + 268 \nu^{6} + 3227 \nu^{5} + 3747 \nu^{4} - 116919 \nu^{3} + 79299 \nu^{2} + \cdots - 1594323 ) / 7046514 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 187 \nu^{7} - 268 \nu^{6} - 3227 \nu^{5} - 3747 \nu^{4} + 116919 \nu^{3} - 79299 \nu^{2} + \cdots - 5452191 ) / 7046514 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 56\nu^{6} + 23\nu^{5} - 1881\nu^{4} + 891\nu^{3} - 45063\nu^{2} - 21870\nu - 1197909 ) / 28998 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 277 \nu^{7} - 4772 \nu^{6} + 5297 \nu^{5} + 95439 \nu^{4} - 297711 \nu^{3} - 2932443 \nu^{2} + \cdots + 78239925 ) / 7046514 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 649 \nu^{7} + 1783 \nu^{6} - 28033 \nu^{5} + 80382 \nu^{4} - 584883 \nu^{3} - 223722 \nu^{2} + \cdots + 37574847 ) / 3523257 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 61\nu^{7} + 194\nu^{6} + 1819\nu^{5} + 5193\nu^{4} + 42309\nu^{3} + 23031\nu^{2} + 1160082\nu + 4115205 ) / 260982 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} - 3\beta_{6} - 2\beta_{4} + \beta_{3} - 2\beta_{2} - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} + 3\beta_{6} - 81\beta_{5} - 2\beta_{4} - 87\beta_{3} - 3\beta_{2} + 81\beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6\beta_{7} + 15\beta_{6} - 24\beta_{4} - 10\beta_{3} + 5\beta_{2} + 81\beta _1 - 2071 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 120\beta_{7} - 177\beta_{6} + 324\beta_{5} + 11\beta_{4} - 2431\beta_{3} - 745\beta_{2} - 243\beta _1 - 2742 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2031\beta_{7} + 1941\beta_{6} - 2106\beta_{5} + 812\beta_{4} - 5601\beta_{3} + 633\beta_{2} - 567\beta _1 + 5740 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7581 \beta_{7} - 3120 \beta_{6} + 53217 \beta_{5} + 1275 \beta_{4} + 132103 \beta_{3} + 3595 \beta_{2} + \cdots + 24400 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−3.66456 − 3.68388i
3.90023 − 3.43340i
−3.19005 + 4.10166i
3.45439 + 3.88165i
−3.66456 + 3.68388i
3.90023 + 3.43340i
−3.19005 − 4.10166i
3.45439 − 3.88165i
−1.00000 − 1.73205i −5.02262 − 1.33166i −2.00000 + 3.46410i −2.50000 + 4.33013i 2.71610 + 10.0311i −2.69520 − 4.66821i 8.00000 23.4533 + 13.3769i 10.0000
31.2 −1.00000 − 1.73205i −1.02330 + 5.09439i −2.00000 + 3.46410i −2.50000 + 4.33013i 9.84705 − 3.32199i −8.58388 − 14.8677i 8.00000 −24.9057 − 10.4262i 10.0000
31.3 −1.00000 − 1.73205i 1.95711 − 4.81349i −2.00000 + 3.46410i −2.50000 + 4.33013i −10.2943 + 1.42367i −11.8318 − 20.4932i 8.00000 −19.3394 − 18.8411i 10.0000
31.4 −1.00000 − 1.73205i 5.08880 + 1.05076i −2.00000 + 3.46410i −2.50000 + 4.33013i −3.26883 − 9.86482i 11.6109 + 20.1106i 8.00000 24.7918 + 10.6942i 10.0000
61.1 −1.00000 + 1.73205i −5.02262 + 1.33166i −2.00000 − 3.46410i −2.50000 − 4.33013i 2.71610 − 10.0311i −2.69520 + 4.66821i 8.00000 23.4533 − 13.3769i 10.0000
61.2 −1.00000 + 1.73205i −1.02330 − 5.09439i −2.00000 − 3.46410i −2.50000 − 4.33013i 9.84705 + 3.32199i −8.58388 + 14.8677i 8.00000 −24.9057 + 10.4262i 10.0000
61.3 −1.00000 + 1.73205i 1.95711 + 4.81349i −2.00000 − 3.46410i −2.50000 − 4.33013i −10.2943 − 1.42367i −11.8318 + 20.4932i 8.00000 −19.3394 + 18.8411i 10.0000
61.4 −1.00000 + 1.73205i 5.08880 − 1.05076i −2.00000 − 3.46410i −2.50000 − 4.33013i −3.26883 + 9.86482i 11.6109 − 20.1106i 8.00000 24.7918 − 10.6942i 10.0000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.4.e.e ✓ 8
3.b odd 2 1 270.4.e.e 8
9.c even 3 1 inner 90.4.e.e ✓ 8
9.c even 3 1 810.4.a.v 4
9.d odd 6 1 270.4.e.e 8
9.d odd 6 1 810.4.a.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.e.e ✓ 8 1.a even 1 1 trivial
90.4.e.e ✓ 8 9.c even 3 1 inner
270.4.e.e 8 3.b odd 2 1
270.4.e.e 8 9.d odd 6 1
810.4.a.s 4 9.d odd 6 1
810.4.a.v 4 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 23 T_{7}^{7} + 976 T_{7}^{6} + 14429 T_{7}^{5} + 534826 T_{7}^{4} + 7861877 T_{7}^{3} + \cdots + 2585925904 \) acting on \(S_{4}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 2585925904 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 4000815504 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 316275763456 \) Copy content Toggle raw display
$17$ \( (T^{4} - 81 T^{3} + \cdots - 8793540)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 83 T^{3} + \cdots - 49868756)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( (T^{4} - 146 T^{3} + \cdots + 203845072)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 50\!\cdots\!89 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 69\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 54\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{4} - 684 T^{3} + \cdots - 31164536880)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 37\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} - 120 T^{3} + \cdots + 1875123432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 103 T^{3} + \cdots + 12661457536)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 79\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{4} - 1173 T^{3} + \cdots + 35969800734)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
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