Properties

Label 90.6.c.c
Level $90$
Weight $6$
Character orbit 90.c
Analytic conductor $14.435$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 90.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4345437832\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{1249})\)
Defining polynomial: \( x^{4} + 625x^{2} + 97344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 5 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 16 q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{3} + 3 \beta_{2} - 15 \beta_1 - 1) q^{7} - 16 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 16 q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{3} + 3 \beta_{2} - 15 \beta_1 - 1) q^{7} - 16 \beta_1 q^{8} + ( - 4 \beta_{3} - 3 \beta_1 + 4) q^{10} + (3 \beta_{3} + \beta_{2} + \beta_1 - 89) q^{11} + ( - 3 \beta_{3} + 9 \beta_{2} + 78 \beta_1 - 3) q^{13} + (12 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 220) q^{14} + 256 q^{16} + ( - 3 \beta_{3} + 9 \beta_{2} + 187 \beta_1 - 3) q^{17} + (36 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 36) q^{19} + (16 \beta_{2} + 16) q^{20} + (4 \beta_{3} - 12 \beta_{2} - 84 \beta_1 + 4) q^{22} + ( - 12 \beta_{3} + 36 \beta_{2} - 553 \beta_1 - 12) q^{23} + (\beta_{3} + 3 \beta_{2} - 468 \beta_1 - 2498) q^{25} + (36 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 1308) q^{26} + (16 \beta_{3} - 48 \beta_{2} + 240 \beta_1 + 16) q^{28} + (81 \beta_{3} + 27 \beta_{2} + 27 \beta_1 - 1269) q^{29} + (18 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 5830) q^{31} + 256 \beta_1 q^{32} + (36 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 3052) q^{34} + (57 \beta_{3} - 5 \beta_{2} + 824 \beta_1 + 9313) q^{35} + ( - 19 \beta_{3} + 57 \beta_{2} + 2184 \beta_1 - 19) q^{37} + (48 \beta_{3} - 144 \beta_{2} + 96 \beta_1 + 48) q^{38} + (64 \beta_{3} + 48 \beta_1 - 64) q^{40} + (102 \beta_{3} + 34 \beta_{2} + 34 \beta_1 - 12560) q^{41} + ( - 56 \beta_{3} + 168 \beta_{2} - 3063 \beta_1 - 56) q^{43} + ( - 48 \beta_{3} - 16 \beta_{2} - 16 \beta_1 + 1424) q^{44} + (144 \beta_{3} + 48 \beta_{2} + 48 \beta_1 + 8608) q^{46} + ( - 6 \beta_{3} + 18 \beta_{2} - 1669 \beta_1 - 6) q^{47} + ( - 342 \beta_{3} - 114 \beta_{2} - 114 \beta_1 - 17439) q^{49} + (12 \beta_{3} - 4 \beta_{2} - 2491 \beta_1 + 7484) q^{50} + (48 \beta_{3} - 144 \beta_{2} - 1248 \beta_1 + 48) q^{52} + ( - 111 \beta_{3} + 333 \beta_{2} + 1629 \beta_1 - 111) q^{53} + ( - 5 \beta_{3} + 87 \beta_{2} + 2340 \beta_1 - 3033) q^{55} + ( - 192 \beta_{3} - 64 \beta_{2} - 64 \beta_1 - 3520) q^{56} + (108 \beta_{3} - 324 \beta_{2} - 1134 \beta_1 + 108) q^{58} + ( - 555 \beta_{3} - 185 \beta_{2} - 185 \beta_1 - 11495) q^{59} + ( - 288 \beta_{3} - 96 \beta_{2} - 96 \beta_1 + 28754) q^{61} + (24 \beta_{3} - 72 \beta_{2} + 5860 \beta_1 + 24) q^{62} - 4096 q^{64} + ( - 321 \beta_{3} - 15 \beta_{2} + 2103 \beta_1 + 28431) q^{65} + ( - 86 \beta_{3} + 258 \beta_{2} + 4725 \beta_1 - 86) q^{67} + (48 \beta_{3} - 144 \beta_{2} - 2992 \beta_1 + 48) q^{68} + ( - 20 \beta_{3} - 228 \beta_{2} + 9360 \beta_1 - 12708) q^{70} + ( - 882 \beta_{3} - 294 \beta_{2} - 294 \beta_1 + 6126) q^{71} + ( - 98 \beta_{3} + 294 \beta_{2} - 11142 \beta_1 - 98) q^{73} + (228 \beta_{3} + 76 \beta_{2} + 76 \beta_1 - 35324) q^{74} + ( - 576 \beta_{3} - 192 \beta_{2} - 192 \beta_1 - 576) q^{76} + (30 \beta_{3} - 90 \beta_{2} - 6544 \beta_1 + 30) q^{77} + (594 \beta_{3} + 198 \beta_{2} + 198 \beta_1 - 7506) q^{79} + ( - 256 \beta_{2} - 256) q^{80} + (136 \beta_{3} - 408 \beta_{2} - 12390 \beta_1 + 136) q^{82} + (432 \beta_{3} - 1296 \beta_{2} - 6025 \beta_1 + 432) q^{83} + ( - 757 \beta_{3} - 15 \beta_{2} + 1776 \beta_1 + 28867) q^{85} + (672 \beta_{3} + 224 \beta_{2} + 224 \beta_1 + 47888) q^{86} + ( - 64 \beta_{3} + 192 \beta_{2} + 1344 \beta_1 - 64) q^{88} + (600 \beta_{3} + 200 \beta_{2} + 200 \beta_1 - 30610) q^{89} + (450 \beta_{3} + 150 \beta_{2} + 150 \beta_1 - 75678) q^{91} + (192 \beta_{3} - 576 \beta_{2} + 8848 \beta_1 + 192) q^{92} + (72 \beta_{3} + 24 \beta_{2} + 24 \beta_1 + 26584) q^{94} + ( - 60 \beta_{3} - 60 \beta_{2} + 28080 \beta_1 - 37500) q^{95} + (16 \beta_{3} - 48 \beta_{2} + 21414 \beta_1 + 16) q^{97} + ( - 456 \beta_{3} + 1368 \beta_{2} - 18009 \beta_1 - 456) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 64 q^{4} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 64 q^{4} - 6 q^{5} + 8 q^{10} - 348 q^{11} + 912 q^{14} + 1024 q^{16} + 240 q^{19} + 96 q^{20} - 9984 q^{25} - 5136 q^{26} - 4860 q^{29} + 23368 q^{31} - 12112 q^{34} + 37356 q^{35} - 128 q^{40} - 49968 q^{41} + 5568 q^{44} + 34816 q^{46} - 70668 q^{49} + 29952 q^{50} - 11968 q^{55} - 14592 q^{56} - 47460 q^{59} + 114248 q^{61} - 16384 q^{64} + 113052 q^{65} - 51328 q^{70} + 22152 q^{71} - 140688 q^{74} - 3840 q^{76} - 28440 q^{79} - 1536 q^{80} + 113924 q^{85} + 193344 q^{86} - 120840 q^{89} - 301512 q^{91} + 106528 q^{94} - 150240 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 625x^{2} + 97344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} - 313\nu ) / 78 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 312\nu^{2} + 1249\nu + 97656 ) / 312 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 936\nu^{2} + \nu + 292656 ) / 312 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{3} + 6\beta_{2} + \beta _1 - 2 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} + \beta_{2} + \beta _1 - 3127 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 626\beta_{3} - 1878\beta_{2} - 1873\beta _1 + 626 ) / 20 \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
17.1706i
18.1706i
17.1706i
18.1706i
4.00000i 0 −16.0000 −19.1706 52.5118i 0 233.706i 64.0000i 0 −210.047 + 76.6824i
19.2 4.00000i 0 −16.0000 16.1706 + 53.5118i 0 119.706i 64.0000i 0 214.047 64.6824i
19.3 4.00000i 0 −16.0000 −19.1706 + 52.5118i 0 233.706i 64.0000i 0 −210.047 76.6824i
19.4 4.00000i 0 −16.0000 16.1706 53.5118i 0 119.706i 64.0000i 0 214.047 + 64.6824i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.6.c.c 4
3.b odd 2 1 30.6.c.b 4
4.b odd 2 1 720.6.f.i 4
5.b even 2 1 inner 90.6.c.c 4
5.c odd 4 1 450.6.a.bb 2
5.c odd 4 1 450.6.a.bc 2
12.b even 2 1 240.6.f.b 4
15.d odd 2 1 30.6.c.b 4
15.e even 4 1 150.6.a.n 2
15.e even 4 1 150.6.a.o 2
20.d odd 2 1 720.6.f.i 4
60.h even 2 1 240.6.f.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.b 4 3.b odd 2 1
30.6.c.b 4 15.d odd 2 1
90.6.c.c 4 1.a even 1 1 trivial
90.6.c.c 4 5.b even 2 1 inner
150.6.a.n 2 15.e even 4 1
150.6.a.o 2 15.e even 4 1
240.6.f.b 4 12.b even 2 1
240.6.f.b 4 60.h even 2 1
450.6.a.bb 2 5.c odd 4 1
450.6.a.bc 2 5.c odd 4 1
720.6.f.i 4 4.b odd 2 1
720.6.f.i 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 68948T_{7}^{2} + 782656576 \) acting on \(S_{6}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + 5010 T^{2} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{4} + 68948 T^{2} + \cdots + 782656576 \) Copy content Toggle raw display
$11$ \( (T^{2} + 174 T - 23656)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 768132 T^{2} + \cdots + 31678304256 \) Copy content Toggle raw display
$17$ \( T^{4} + 1708148 T^{2} + \cdots + 85278016576 \) Copy content Toggle raw display
$19$ \( (T^{2} - 120 T - 4492800)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18462752 T^{2} + \cdots + 56918507776 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2430 T - 21286800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 11684 T + 33004864)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 177178148 T^{2} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{2} + 24984 T + 119953964)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 487889312 T^{2} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{4} + 90906128 T^{2} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{4} + 863264052 T^{2} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{2} + 23730 T - 927897400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 57124 T + 528018244)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 1195938128 T^{2} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{2} - 11076 T - 2668294656)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 4520143952 T^{2} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{2} + 14220 T - 1173592800)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 12944582432 T^{2} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{2} + 60420 T - 336355900)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 14673446528 T^{2} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
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