Properties

Label 624.4.q.i
Level $624$
Weight $4$
Character orbit 624.q
Analytic conductor $36.817$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - 3 \beta_{2} q^{3} + (\beta_{4} - 2) q^{5} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{7} + (9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_{2} q^{3} + (\beta_{4} - 2) q^{5} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{7} + (9 \beta_{2} - 9) q^{9} + (\beta_{7} + 10 \beta_{2} - \beta_1) q^{11} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} - 23 \beta_{2} - 2 \beta_1 + 4) q^{13} + ( - 3 \beta_{6} - 3 \beta_{4} + 6 \beta_{2}) q^{15} + ( - 5 \beta_{6} - 5 \beta_{5} - \beta_{3} + 29 \beta_{2} - 29) q^{17} + ( - 4 \beta_{6} + 3 \beta_{5} - \beta_{3} - 31 \beta_{2} + 31) q^{19} + ( - 3 \beta_{7} + 3 \beta_{3} + 3 \beta_{2} + 9) q^{21} + (3 \beta_{7} + \beta_{6} + \beta_{4} + 23 \beta_{2} + 2 \beta_1) q^{23} + (2 \beta_{7} + 3 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 8) q^{25} + 27 q^{27} + (6 \beta_{7} + 5 \beta_{6} + 5 \beta_{4} - 56 \beta_{2} + 4 \beta_1) q^{29} + ( - 15 \beta_{5} + 8 \beta_{4} + 15 \beta_1 - 18) q^{31} + (3 \beta_{5} - 3 \beta_{3} - 33 \beta_{2} + 33) q^{33} + ( - 13 \beta_{6} - 12 \beta_{5} - 5 \beta_{3} - 12 \beta_{2} + 12) q^{35} + (\beta_{7} + 4 \beta_{6} + 4 \beta_{4} - 31 \beta_{2} + 6 \beta_1) q^{37} + ( - 6 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} + 60 \beta_{2} + \cdots - 72) q^{39}+ \cdots + ( - 9 \beta_{7} - 9 \beta_{5} + 9 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{3} - 12 q^{5} - 14 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 12 q^{3} - 12 q^{5} - 14 q^{7} - 36 q^{9} + 40 q^{11} - 60 q^{13} + 18 q^{15} - 98 q^{17} + 124 q^{19} + 84 q^{21} + 104 q^{23} - 116 q^{25} + 216 q^{27} - 194 q^{29} - 52 q^{31} + 120 q^{33} + 88 q^{35} - 102 q^{37} - 342 q^{39} + 1054 q^{41} + 450 q^{43} + 54 q^{45} + 192 q^{47} - 1070 q^{49} + 588 q^{51} + 524 q^{53} + 204 q^{55} - 744 q^{57} + 308 q^{59} + 928 q^{61} - 126 q^{63} + 2346 q^{65} - 1134 q^{67} + 312 q^{69} + 1064 q^{71} + 1904 q^{73} + 174 q^{75} + 5016 q^{77} + 1492 q^{79} - 324 q^{81} + 808 q^{83} + 1394 q^{85} - 582 q^{87} - 1620 q^{89} - 3278 q^{91} + 78 q^{93} + 2204 q^{95} - 2166 q^{97} - 720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5924 \nu^{7} + 261773 \nu^{6} - 568842 \nu^{5} + 5976355 \nu^{4} + 5492514 \nu^{3} + 104173009 \nu^{2} + 54255204 \nu + 87160788 ) / 37839126 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5984 \nu^{7} - 26943 \nu^{6} + 182186 \nu^{5} - 346833 \nu^{4} + 3020182 \nu^{3} - 11402021 \nu^{2} + 10968108 \nu + 5315076 ) / 37839126 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4151 \nu^{7} - 166803 \nu^{6} + 1127906 \nu^{5} - 5302947 \nu^{4} + 18697822 \nu^{3} - 70589441 \nu^{2} + 343464684 \nu - 201355050 ) / 18919563 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3550 \nu^{7} - 12579 \nu^{6} + 85058 \nu^{5} + 128084 \nu^{4} + 1410046 \nu^{3} + 983208 \nu^{2} + 1395576 \nu + 73106644 ) / 6306521 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44524 \nu^{7} + 220899 \nu^{6} - 1493698 \nu^{5} + 4583667 \nu^{4} - 24761726 \nu^{3} + 93482353 \nu^{2} + 39080772 \nu + 266656650 ) / 37839126 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31238 \nu^{7} - 150864 \nu^{6} + 1020128 \nu^{5} - 2812083 \nu^{4} + 16911136 \nu^{3} - 63844208 \nu^{2} + 72590028 \nu - 182114400 ) / 18919563 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 108812 \nu^{7} - 222399 \nu^{6} + 3305704 \nu^{5} + 103215 \nu^{4} + 64570858 \nu^{3} - 22042613 \nu^{2} + 273536628 \nu + 201818916 ) / 12613042 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - 3\beta_{2} + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{4} - 14\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 9\beta_{5} + 13\beta_{4} + 2\beta_{3} + 2\beta_{2} + 9\beta _1 - 55 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -32\beta_{6} - 3\beta_{5} + 2\beta_{3} + 309\beta_{2} - 309 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 29\beta_{7} - 183\beta_{6} - 183\beta_{4} + 882\beta_{2} - 99\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 84\beta_{7} + 165\beta_{5} - 932\beta_{4} - 84\beta_{3} - 84\beta_{2} - 165\beta _1 + 7795 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5164\beta_{6} + 2382\beta_{5} - 767\beta_{3} - 28804\beta_{2} + 28804 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
2.66520 4.61626i
−2.11303 + 3.65987i
1.18088 2.04535i
−0.733051 + 1.26968i
2.66520 + 4.61626i
−2.11303 3.65987i
1.18088 + 2.04535i
−0.733051 1.26968i
0 −1.50000 2.59808i 0 −16.4131 0 4.83984 8.38285i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 2.59808i 0 −5.85953 0 −12.0627 + 20.8932i 0 −4.50000 + 7.79423i 0
289.3 0 −1.50000 2.59808i 0 6.42208 0 −14.7469 + 25.5424i 0 −4.50000 + 7.79423i 0
289.4 0 −1.50000 2.59808i 0 9.85055 0 14.9698 25.9285i 0 −4.50000 + 7.79423i 0
529.1 0 −1.50000 + 2.59808i 0 −16.4131 0 4.83984 + 8.38285i 0 −4.50000 7.79423i 0
529.2 0 −1.50000 + 2.59808i 0 −5.85953 0 −12.0627 20.8932i 0 −4.50000 7.79423i 0
529.3 0 −1.50000 + 2.59808i 0 6.42208 0 −14.7469 25.5424i 0 −4.50000 7.79423i 0
529.4 0 −1.50000 + 2.59808i 0 9.85055 0 14.9698 + 25.9285i 0 −4.50000 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.q.i 8
4.b odd 2 1 39.4.e.c 8
12.b even 2 1 117.4.g.e 8
13.c even 3 1 inner 624.4.q.i 8
52.i odd 6 1 507.4.a.i 4
52.j odd 6 1 39.4.e.c 8
52.j odd 6 1 507.4.a.m 4
52.l even 12 2 507.4.b.h 8
156.p even 6 1 117.4.g.e 8
156.p even 6 1 1521.4.a.v 4
156.r even 6 1 1521.4.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 4.b odd 2 1
39.4.e.c 8 52.j odd 6 1
117.4.g.e 8 12.b even 2 1
117.4.g.e 8 156.p even 6 1
507.4.a.i 4 52.i odd 6 1
507.4.a.m 4 52.j odd 6 1
507.4.b.h 8 52.l even 12 2
624.4.q.i 8 1.a even 1 1 trivial
624.4.q.i 8 13.c even 3 1 inner
1521.4.a.v 4 156.p even 6 1
1521.4.a.bb 4 156.r even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{4} + 6T_{5}^{3} - 203T_{5}^{2} - 156T_{5} + 6084 \) Copy content Toggle raw display
\( T_{7}^{8} + 14 T_{7}^{7} + 1319 T_{7}^{6} + 9582 T_{7}^{5} + 1232045 T_{7}^{4} + 8434260 T_{7}^{3} + 391649180 T_{7}^{2} - 2608994224 T_{7} + 42523388944 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 6 T^{3} - 203 T^{2} - 156 T + 6084)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 14 T^{7} + \cdots + 42523388944 \) Copy content Toggle raw display
$11$ \( T^{8} - 40 T^{7} + \cdots + 751198464 \) Copy content Toggle raw display
$13$ \( T^{8} + 60 T^{7} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 509493017090304 \) Copy content Toggle raw display
$19$ \( T^{8} - 124 T^{7} + \cdots + 2959721107456 \) Copy content Toggle raw display
$23$ \( T^{8} - 104 T^{7} + \cdots + 6612632822016 \) Copy content Toggle raw display
$29$ \( T^{8} + 194 T^{7} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{4} + 26 T^{3} - 80975 T^{2} + \cdots + 328187792)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 738573457717264 \) Copy content Toggle raw display
$41$ \( T^{8} - 1054 T^{7} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{8} - 450 T^{7} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{4} - 96 T^{3} - 434600 T^{2} + \cdots + 42871452048)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 262 T^{3} - 111719 T^{2} + \cdots + 744728256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 308 T^{7} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} - 928 T^{7} + \cdots + 27\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{8} + 1134 T^{7} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{8} - 1064 T^{7} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} - 952 T^{3} + \cdots - 120133390247)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 746 T^{3} + 184337 T^{2} + \cdots + 680937616)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 404 T^{3} + \cdots + 58964273856)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 1620 T^{7} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{8} + 2166 T^{7} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
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