Properties

Label 675.2.q.a
Level $675$
Weight $2$
Character orbit 675.q
Analytic conductor $5.390$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.q (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + (\beta_{15} + \beta_{8} + \beta_{5}) q^{4} + (\beta_{14} + \beta_{8} + \beta_{5}) q^{7} + ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + 2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{2} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + (\beta_{15} + \beta_{8} + \beta_{5}) q^{4} + (\beta_{14} + \beta_{8} + \beta_{5}) q^{7} + ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + 2 \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{2} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + 2 \beta_{11} + 3 \beta_{9} - \beta_{8} + \beta_{2} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} + 2 q^{7} + 2 q^{13} - 8 q^{16} + 10 q^{22} + 18 q^{23} + 16 q^{28} - 4 q^{31} + 30 q^{32} - 4 q^{37} - 30 q^{38} + 24 q^{41} + 2 q^{43} + 32 q^{46} - 12 q^{47} + 14 q^{52} - 36 q^{56} + 6 q^{58} + 8 q^{61} - 4 q^{67} + 42 q^{68} + 8 q^{73} + 24 q^{76} - 6 q^{77} - 32 q^{82} - 66 q^{83} + 48 q^{86} - 18 q^{88} - 40 q^{91} - 60 q^{92} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} + 34 x^{12} + 18 x^{11} - 72 x^{10} + 132 x^{9} - 93 x^{8} - 102 x^{7} + 144 x^{6} - 432 x^{5} + 502 x^{4} + 288 x^{3} + 72 x^{2} + 12 x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 80029143512 \nu^{15} + 385788744870 \nu^{14} - 820783926284 \nu^{13} + 848040618120 \nu^{12} + 1720995499366 \nu^{11} + \cdots + 33432180594 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 94386116202 \nu^{15} - 619740656932 \nu^{14} + 2033008548312 \nu^{13} - 4441986378774 \nu^{12} + 5386888154268 \nu^{11} + \cdots - 80029143512 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 140020985001 \nu^{15} - 843322184609 \nu^{14} + 2535236829090 \nu^{13} - 5074237635644 \nu^{12} + 4819083120574 \nu^{11} + \cdots - 5622211270526 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 140094553942 \nu^{15} + 840493754711 \nu^{14} - 2524530400854 \nu^{13} + 5054110370148 \nu^{12} - 4783342099524 \nu^{11} + \cdots - 1652709999986 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 190424165289 \nu^{15} - 1059314239720 \nu^{14} + 2941649532255 \nu^{13} - 5449473018186 \nu^{12} + 3809266070290 \nu^{11} + \cdots + 7166890770427 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 250381984552 \nu^{15} + 1889212496921 \nu^{14} - 6927613311171 \nu^{13} + 16582911879478 \nu^{12} - 24294968776849 \nu^{11} + \cdots + 5660844178894 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 254444403226 \nu^{15} + 1391318146672 \nu^{14} - 3674576400880 \nu^{13} + 6138067615014 \nu^{12} - 1944897155536 \nu^{11} + \cdots + 146893504700 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 196858530 \nu^{15} - 1212076232 \nu^{14} + 3734556994 \nu^{13} - 7676123862 \nu^{12} + 7906540260 \nu^{11} + 2288455604 \nu^{10} + \cdots + 785074438 ) / 1973128213 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 677106342206 \nu^{15} + 3890533925017 \nu^{14} - 11114551602021 \nu^{13} + 21019011866538 \nu^{12} + \cdots - 11893459432082 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 785074438 \nu^{15} + 4907305158 \nu^{14} - 15343416116 \nu^{13} + 31997236762 \nu^{12} - 34368654754 \nu^{11} - 6224799624 \nu^{10} + \cdots - 3343669588 ) / 1973128213 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1616321538 \nu^{15} + 9938916556 \nu^{14} - 30594542475 \nu^{13} + 62871582510 \nu^{12} - 64714538406 \nu^{11} - 18641341380 \nu^{10} + \cdots - 10233974547 ) / 3810388399 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2029780580195 \nu^{15} + 12360646755715 \nu^{14} - 37626995630777 \nu^{13} + 76333636000087 \nu^{12} + \cdots - 14351633608601 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2802573231023 \nu^{15} - 17485025240258 \nu^{14} + 54572756591878 \nu^{13} - 113605741903926 \nu^{12} + \cdots + 12411058785072 ) / 3707507912227 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2852772870096 \nu^{15} + 17521813816267 \nu^{14} - 53872302373806 \nu^{13} + 110572117714578 \nu^{12} + \cdots - 17972736434118 ) / 3707507912227 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{3} - 2\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 4\beta_{3} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + 5 \beta_{14} + 2 \beta_{13} + \beta_{12} + 8 \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + 5 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 16 \beta_{15} + 8 \beta_{13} + 14 \beta_{12} + 7 \beta_{11} + 3 \beta_{9} - 8 \beta_{8} - 8 \beta_{6} - 8 \beta_{5} + 8 \beta_{3} + 7 \beta_{2} + 11 \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 35 \beta_{15} - 16 \beta_{14} + 8 \beta_{13} + 30 \beta_{12} - 8 \beta_{11} + 16 \beta_{10} - \beta_{9} - 16 \beta_{8} + 8 \beta_{7} - 16 \beta_{6} - 57 \beta_{5} - 8 \beta_{4} + 34 \beta_{3} + 16 \beta_{2} + 25 \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{15} - \beta_{14} - 10 \beta_{12} - 10 \beta_{11} + 51 \beta_{10} - \beta_{8} + 51 \beta_{7} - 98 \beta_{5} + \beta_{4} + 97 \beta_{3} + 10 \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50 \beta_{15} + 148 \beta_{14} + 50 \beta_{13} - 50 \beta_{12} + 145 \beta_{11} + 50 \beta_{10} - 10 \beta_{9} + 100 \beta_{7} + 50 \beta_{6} + 10 \beta_{5} + 50 \beta_{4} + 128 \beta_{3} - 138 \beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 288 \beta_{15} + 278 \beta_{14} + 298 \beta_{13} + 228 \beta_{12} + 456 \beta_{11} - 55 \beta_{9} - 278 \beta_{8} - 10 \beta_{4} - 10 \beta_{3} + 158 \beta_{2} - 278 \beta _1 + 228 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1385 \beta_{15} - 288 \beta_{14} + 576 \beta_{13} + 1032 \beta_{12} + 288 \beta_{11} + 288 \beta_{10} - 140 \beta_{9} - 1097 \beta_{8} - 288 \beta_{7} - 288 \beta_{6} - 1315 \beta_{5} - 576 \beta_{4} - 218 \beta_{3} + \cdots + 288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1533 \beta_{15} - 1603 \beta_{14} + 817 \beta_{12} - 1245 \beta_{11} + 1673 \beta_{10} - 1603 \beta_{8} - 4331 \beta_{5} - 1533 \beta_{4} + 1245 \beta_{2} - 817 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3206 \beta_{15} - 1603 \beta_{13} - 3206 \beta_{12} - 1603 \beta_{11} + 3206 \beta_{10} + 428 \beta_{9} + 1603 \beta_{7} + 3206 \beta_{6} - 3545 \beta_{5} - 1195 \beta_{4} - 428 \beta_{3} - 3545 \beta _1 - 3923 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8782 \beta_{15} + 8782 \beta_{14} - 6751 \beta_{12} + 6751 \beta_{11} + 428 \beta_{8} + 9210 \beta_{6} + 9210 \beta_{5} + 428 \beta_{4} - 8782 \beta_{3} - 2459 \beta_{2} - 14219 \beta _1 - 6751 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 8782 \beta_{15} + 8782 \beta_{14} + 8782 \beta_{13} + 8782 \beta_{12} + 17564 \beta_{11} - 8782 \beta_{10} - 2459 \beta_{9} - 15075 \beta_{8} - 17564 \beta_{7} + 8782 \beta_{6} + 15105 \beta_{5} + \cdots - 8782 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 47744 \beta_{15} - 45285 \beta_{14} + 36503 \beta_{12} - 22803 \beta_{11} - 45285 \beta_{8} - 50203 \beta_{7} - 45285 \beta_{5} - 47744 \beta_{4} - 78841 \beta_{3} + 22803 \beta_{2} + \cdots - 36503 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1 + \beta_{11}\) \(-\beta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
2.24352 + 0.601150i
1.60599 + 0.430324i
−0.186243 0.0499037i
−1.29724 0.347596i
0.601150 2.24352i
0.430324 1.60599i
−0.0499037 + 0.186243i
−0.347596 + 1.29724i
0.601150 + 2.24352i
0.430324 + 1.60599i
−0.0499037 0.186243i
−0.347596 1.29724i
2.24352 0.601150i
1.60599 0.430324i
−0.186243 + 0.0499037i
−1.29724 + 0.347596i
−2.24352 + 0.601150i 0 2.93996 1.69739i 0 0 0.201351 + 0.751454i −2.29074 + 2.29074i 0 0
143.2 −1.60599 + 0.430324i 0 0.661975 0.382191i 0 0 −0.465559 1.73749i 1.45267 1.45267i 0 0
143.3 0.186243 0.0499037i 0 −1.69985 + 0.981412i 0 0 −0.632007 2.35868i −0.540289 + 0.540289i 0 0
143.4 1.29724 0.347596i 0 −0.170031 + 0.0981673i 0 0 0.530190 + 1.97869i −2.08575 + 2.08575i 0 0
332.1 −0.601150 2.24352i 0 −2.93996 + 1.69739i 0 0 −0.751454 + 0.201351i 2.29074 + 2.29074i 0 0
332.2 −0.430324 1.60599i 0 −0.661975 + 0.382191i 0 0 1.73749 0.465559i −1.45267 1.45267i 0 0
332.3 0.0499037 + 0.186243i 0 1.69985 0.981412i 0 0 2.35868 0.632007i 0.540289 + 0.540289i 0 0
332.4 0.347596 + 1.29724i 0 0.170031 0.0981673i 0 0 −1.97869 + 0.530190i 2.08575 + 2.08575i 0 0
368.1 −0.601150 + 2.24352i 0 −2.93996 1.69739i 0 0 −0.751454 0.201351i 2.29074 2.29074i 0 0
368.2 −0.430324 + 1.60599i 0 −0.661975 0.382191i 0 0 1.73749 + 0.465559i −1.45267 + 1.45267i 0 0
368.3 0.0499037 0.186243i 0 1.69985 + 0.981412i 0 0 2.35868 + 0.632007i 0.540289 0.540289i 0 0
368.4 0.347596 1.29724i 0 0.170031 + 0.0981673i 0 0 −1.97869 0.530190i 2.08575 2.08575i 0 0
557.1 −2.24352 0.601150i 0 2.93996 + 1.69739i 0 0 0.201351 0.751454i −2.29074 2.29074i 0 0
557.2 −1.60599 0.430324i 0 0.661975 + 0.382191i 0 0 −0.465559 + 1.73749i 1.45267 + 1.45267i 0 0
557.3 0.186243 + 0.0499037i 0 −1.69985 0.981412i 0 0 −0.632007 + 2.35868i −0.540289 0.540289i 0 0
557.4 1.29724 + 0.347596i 0 −0.170031 0.0981673i 0 0 0.530190 1.97869i −2.08575 2.08575i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.q.a 16
3.b odd 2 1 225.2.p.b 16
5.b even 2 1 135.2.m.a 16
5.c odd 4 1 135.2.m.a 16
5.c odd 4 1 inner 675.2.q.a 16
9.c even 3 1 225.2.p.b 16
9.d odd 6 1 inner 675.2.q.a 16
15.d odd 2 1 45.2.l.a 16
15.e even 4 1 45.2.l.a 16
15.e even 4 1 225.2.p.b 16
45.h odd 6 1 135.2.m.a 16
45.h odd 6 1 405.2.f.a 16
45.j even 6 1 45.2.l.a 16
45.j even 6 1 405.2.f.a 16
45.k odd 12 1 45.2.l.a 16
45.k odd 12 1 225.2.p.b 16
45.k odd 12 1 405.2.f.a 16
45.l even 12 1 135.2.m.a 16
45.l even 12 1 405.2.f.a 16
45.l even 12 1 inner 675.2.q.a 16
60.h even 2 1 720.2.cu.c 16
60.l odd 4 1 720.2.cu.c 16
180.p odd 6 1 720.2.cu.c 16
180.x even 12 1 720.2.cu.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.l.a 16 15.d odd 2 1
45.2.l.a 16 15.e even 4 1
45.2.l.a 16 45.j even 6 1
45.2.l.a 16 45.k odd 12 1
135.2.m.a 16 5.b even 2 1
135.2.m.a 16 5.c odd 4 1
135.2.m.a 16 45.h odd 6 1
135.2.m.a 16 45.l even 12 1
225.2.p.b 16 3.b odd 2 1
225.2.p.b 16 9.c even 3 1
225.2.p.b 16 15.e even 4 1
225.2.p.b 16 45.k odd 12 1
405.2.f.a 16 45.h odd 6 1
405.2.f.a 16 45.j even 6 1
405.2.f.a 16 45.k odd 12 1
405.2.f.a 16 45.l even 12 1
675.2.q.a 16 1.a even 1 1 trivial
675.2.q.a 16 5.c odd 4 1 inner
675.2.q.a 16 9.d odd 6 1 inner
675.2.q.a 16 45.l even 12 1 inner
720.2.cu.c 16 60.h even 2 1
720.2.cu.c 16 60.l odd 4 1
720.2.cu.c 16 180.p odd 6 1
720.2.cu.c 16 180.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 6 T_{2}^{15} + 18 T_{2}^{14} + 36 T_{2}^{13} + 34 T_{2}^{12} - 18 T_{2}^{11} - 72 T_{2}^{10} - 132 T_{2}^{9} - 93 T_{2}^{8} + 102 T_{2}^{7} + 144 T_{2}^{6} + 432 T_{2}^{5} + 502 T_{2}^{4} - 288 T_{2}^{3} + 72 T_{2}^{2} - 12 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 6 T^{15} + 18 T^{14} + 36 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + 2 T^{14} - 12 T^{13} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{8} - 10 T^{6} + 102 T^{4} - 180 T^{3} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{15} + 2 T^{14} + \cdots + 4477456 \) Copy content Toggle raw display
$17$ \( T^{16} + 964 T^{12} + 6504 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{8} + 60 T^{6} + 864 T^{4} + 1836 T^{2} + \cdots + 324)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 18 T^{15} + 162 T^{14} + \cdots + 62742241 \) Copy content Toggle raw display
$29$ \( T^{16} + 84 T^{14} + \cdots + 981506241 \) Copy content Toggle raw display
$31$ \( (T^{8} + 2 T^{7} + 46 T^{6} + 200 T^{5} + \cdots + 676)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 2 T^{7} + 2 T^{6} - 124 T^{5} + \cdots + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 12 T^{7} + 26 T^{6} + 264 T^{5} + \cdots + 32761)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 2 T^{15} + 2 T^{14} + \cdots + 11316496 \) Copy content Toggle raw display
$47$ \( T^{16} + 12 T^{15} + \cdots + 33243864241 \) Copy content Toggle raw display
$53$ \( T^{16} + 20032 T^{12} + \cdots + 409600000000 \) Copy content Toggle raw display
$59$ \( T^{16} + 144 T^{14} + \cdots + 592240896 \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{7} + 118 T^{6} + 956 T^{5} + \cdots + 11449)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 4 T^{15} + \cdots + 539415333601 \) Copy content Toggle raw display
$71$ \( (T^{8} + 116 T^{6} + 3972 T^{4} + \cdots + 128164)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{7} + 8 T^{6} + 584 T^{5} + \cdots + 270400)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} - 372 T^{14} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{16} + 66 T^{15} + \cdots + 13841287201 \) Copy content Toggle raw display
$89$ \( (T^{8} - 300 T^{6} + 8262 T^{4} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 28 T^{15} + \cdots + 6146560000 \) Copy content Toggle raw display
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