Properties

Label 1200.3.bg.q
Level $1200$
Weight $3$
Character orbit 1200.bg
Analytic conductor $32.698$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.bg (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} + 8 x^{6} + 269 x^{4} - 1116 x^{3} + 2312 x^{2} + 680 x + 100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( \beta_{3} - \beta_{6} ) q^{7} -3 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( \beta_{3} - \beta_{6} ) q^{7} -3 \beta_{2} q^{9} + ( -4 - 3 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{11} + ( -3 \beta_{1} - \beta_{2} + \beta_{7} ) q^{13} + ( -7 - 7 \beta_{2} - 7 \beta_{3} - \beta_{6} ) q^{17} + ( 1 - \beta_{1} + 8 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{19} + ( -3 - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{21} + ( -6 + 4 \beta_{1} + 4 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{23} + 3 \beta_{3} q^{27} + ( 1 - 10 \beta_{1} - 10 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} + ( -12 - 10 \beta_{1} + 12 \beta_{3} - 2 \beta_{4} ) q^{31} + ( -7 - 3 \beta_{1} + 8 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{33} + ( 7 + 7 \beta_{2} - 27 \beta_{3} - \beta_{6} ) q^{37} + ( -1 + \beta_{1} + 8 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{39} + ( -19 + 3 \beta_{1} - 3 \beta_{3} + 3 \beta_{6} + 3 \beta_{7} ) q^{41} + ( -10 + 6 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{43} + ( 11 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} ) q^{47} + ( 1 - 19 \beta_{1} + 39 \beta_{2} - 19 \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{49} + ( 21 - 7 \beta_{1} + 6 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{51} + ( -14 - 21 \beta_{1} + 13 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{53} + ( 4 + 4 \beta_{2} - 7 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{57} + ( -7 \beta_{1} - 12 \beta_{2} - 7 \beta_{3} - \beta_{5} ) q^{59} + ( 33 + 7 \beta_{1} - 3 \beta_{3} - 4 \beta_{4} - \beta_{6} - \beta_{7} ) q^{61} + ( 3 - 3 \beta_{1} - 3 \beta_{7} ) q^{63} + ( -28 - 28 \beta_{2} + 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{67} + ( -3 - 5 \beta_{1} - 12 \beta_{2} - 5 \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{69} + ( 30 - 18 \beta_{1} + 18 \beta_{3} + 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( 29 - 34 \beta_{1} - 23 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 6 \beta_{7} ) q^{73} + ( -16 - 16 \beta_{2} - 47 \beta_{3} - \beta_{4} - \beta_{5} + 10 \beta_{6} ) q^{77} + ( -2 - 16 \beta_{1} - 36 \beta_{2} - 16 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{79} -9 q^{81} + ( 44 - 6 \beta_{1} - 40 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{7} ) q^{83} + ( 32 + 32 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{87} + ( -5 - 17 \beta_{1} - 48 \beta_{2} - 17 \beta_{3} - 2 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{89} + ( -74 - 20 \beta_{1} + 22 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{91} + ( -34 - 14 \beta_{1} + 32 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{93} + ( 23 + 23 \beta_{2} - 12 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} ) q^{97} + ( -9 \beta_{1} + 12 \beta_{2} - 9 \beta_{3} - 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{7} + O(q^{10}) \) \( 8q + 4q^{7} - 32q^{11} + 4q^{13} - 52q^{17} - 24q^{21} - 40q^{23} - 96q^{31} - 60q^{33} + 60q^{37} - 152q^{41} - 88q^{43} - 16q^{47} + 168q^{51} - 108q^{53} + 24q^{57} + 264q^{61} + 12q^{63} - 216q^{67} + 240q^{71} + 208q^{73} - 168q^{77} - 72q^{81} + 336q^{83} + 252q^{87} - 592q^{91} - 264q^{93} + 208q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 8 x^{6} + 269 x^{4} - 1116 x^{3} + 2312 x^{2} + 680 x + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3096 \nu^{7} - 17905 \nu^{6} + 34598 \nu^{5} + 15150 \nu^{4} + 876904 \nu^{3} - 4737945 \nu^{2} + 10010942 \nu + 2940830 \)\()/1277880\)
\(\beta_{2}\)\(=\)\((\)\( -2465 \nu^{7} + 10455 \nu^{6} - 21142 \nu^{5} + 2900 \nu^{4} - 663785 \nu^{3} + 2966245 \nu^{2} - 6107718 \nu - 838660 \)\()/958410\)
\(\beta_{3}\)\(=\)\((\)\( 4775 \nu^{7} - 20987 \nu^{6} + 31148 \nu^{5} - 24410 \nu^{4} + 1286695 \nu^{3} - 5631403 \nu^{2} + 9748972 \nu - 227650 \)\()/1277880\)
\(\beta_{4}\)\(=\)\((\)\( 40034 \nu^{7} - 107007 \nu^{6} + 1294 \nu^{5} - 216230 \nu^{4} + 9466346 \nu^{3} - 29697223 \nu^{2} + 25503006 \nu + 17361370 \)\()/3833640\)
\(\beta_{5}\)\(=\)\((\)\( -17425 \nu^{7} + 50772 \nu^{6} - 167078 \nu^{5} + 20500 \nu^{4} - 4251625 \nu^{3} + 15152828 \nu^{2} - 40601862 \nu - 5575940 \)\()/1277880\)
\(\beta_{6}\)\(=\)\((\)\( 71513 \nu^{7} - 305517 \nu^{6} + 737284 \nu^{5} - 140510 \nu^{4} + 19259897 \nu^{3} - 85711093 \nu^{2} + 213729396 \nu - 3461150 \)\()/3833640\)
\(\beta_{7}\)\(=\)\((\)\( -33016 \nu^{7} + 119837 \nu^{6} - 241278 \nu^{5} + 20050 \nu^{4} - 8904504 \nu^{3} + 33775373 \nu^{2} - 69713302 \nu - 19230750 \)\()/1277880\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{6} + \beta_{5} + \beta_{4} - 5 \beta_{3} + 6 \beta_{2} + 6\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + 19 \beta_{3} + 108 \beta_{2} + 19 \beta_{1} + 3\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-24 \beta_{7} + 17 \beta_{5} - 17 \beta_{4} + 17 \beta_{3} + 52 \beta_{2} - 58 \beta_{1} - 28\)\()/10\)
\(\nu^{4}\)\(=\)\((\)\(-31 \beta_{7} - 31 \beta_{6} + 34 \beta_{4} - 377 \beta_{3} + 343 \beta_{1} - 1505\)\()/10\)
\(\nu^{5}\)\(=\)\((\)\(-308 \beta_{6} - 279 \beta_{5} - 279 \beta_{4} + 1145 \beta_{3} - 224 \beta_{2} - 224\)\()/10\)
\(\nu^{6}\)\(=\)\((\)\(297 \beta_{7} - 297 \beta_{6} + 598 \beta_{5} - 5881 \beta_{3} - 22392 \beta_{2} - 5881 \beta_{1} - 297\)\()/10\)
\(\nu^{7}\)\(=\)\((\)\(4076 \beta_{7} - 4533 \beta_{5} + 4533 \beta_{4} - 4533 \beta_{3} + 4152 \beta_{2} + 13942 \beta_{1} - 8228\)\()/10\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
2.66015 2.66015i
−2.88489 + 2.88489i
2.36263 2.36263i
−0.137883 + 0.137883i
2.66015 + 2.66015i
−2.88489 2.88489i
2.36263 + 2.36263i
−0.137883 0.137883i
0 −1.22474 1.22474i 0 0 0 −2.57407 + 2.57407i 0 3.00000i 0
193.2 0 −1.22474 1.22474i 0 0 0 6.02356 6.02356i 0 3.00000i 0
193.3 0 1.22474 + 1.22474i 0 0 0 −8.78825 + 8.78825i 0 3.00000i 0
193.4 0 1.22474 + 1.22474i 0 0 0 7.33876 7.33876i 0 3.00000i 0
1057.1 0 −1.22474 + 1.22474i 0 0 0 −2.57407 2.57407i 0 3.00000i 0
1057.2 0 −1.22474 + 1.22474i 0 0 0 6.02356 + 6.02356i 0 3.00000i 0
1057.3 0 1.22474 1.22474i 0 0 0 −8.78825 8.78825i 0 3.00000i 0
1057.4 0 1.22474 1.22474i 0 0 0 7.33876 + 7.33876i 0 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1057.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.bg.q 8
4.b odd 2 1 600.3.u.h 8
5.b even 2 1 240.3.bg.e 8
5.c odd 4 1 240.3.bg.e 8
5.c odd 4 1 inner 1200.3.bg.q 8
12.b even 2 1 1800.3.v.t 8
15.d odd 2 1 720.3.bh.o 8
15.e even 4 1 720.3.bh.o 8
20.d odd 2 1 120.3.u.b 8
20.e even 4 1 120.3.u.b 8
20.e even 4 1 600.3.u.h 8
40.e odd 2 1 960.3.bg.l 8
40.f even 2 1 960.3.bg.k 8
40.i odd 4 1 960.3.bg.k 8
40.k even 4 1 960.3.bg.l 8
60.h even 2 1 360.3.v.f 8
60.l odd 4 1 360.3.v.f 8
60.l odd 4 1 1800.3.v.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.b 8 20.d odd 2 1
120.3.u.b 8 20.e even 4 1
240.3.bg.e 8 5.b even 2 1
240.3.bg.e 8 5.c odd 4 1
360.3.v.f 8 60.h even 2 1
360.3.v.f 8 60.l odd 4 1
600.3.u.h 8 4.b odd 2 1
600.3.u.h 8 20.e even 4 1
720.3.bh.o 8 15.d odd 2 1
720.3.bh.o 8 15.e even 4 1
960.3.bg.k 8 40.f even 2 1
960.3.bg.k 8 40.i odd 4 1
960.3.bg.l 8 40.e odd 2 1
960.3.bg.l 8 40.k even 4 1
1200.3.bg.q 8 1.a even 1 1 trivial
1200.3.bg.q 8 5.c odd 4 1 inner
1800.3.v.t 8 12.b even 2 1
1800.3.v.t 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{8} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 9 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 16000000 + 3200000 T + 320000 T^{2} - 120000 T^{3} + 20900 T^{4} - 120 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$11$ \( ( -32288 - 7696 T - 358 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$13$ \( 53824 + 419456 T + 1634432 T^{2} - 393216 T^{3} + 47060 T^{4} - 936 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$17$ \( 2570084416 + 546705664 T + 58147328 T^{2} + 2140128 T^{3} + 103508 T^{4} + 13176 T^{5} + 1352 T^{6} + 52 T^{7} + T^{8} \)
$19$ \( 82591744 + 7498752 T^{2} + 164240 T^{4} + 888 T^{6} + T^{8} \)
$23$ \( 3064729600 - 1512878080 T + 373409792 T^{2} + 18445568 T^{3} + 460816 T^{4} - 2912 T^{5} + 800 T^{6} + 40 T^{7} + T^{8} \)
$29$ \( 619810816 + 617798400 T^{2} + 4760708 T^{4} + 4500 T^{6} + T^{8} \)
$31$ \( ( -1815680 - 140352 T - 2104 T^{2} + 48 T^{3} + T^{4} )^{2} \)
$37$ \( 16133718222400 + 470626362240 T + 6864170112 T^{2} - 211970688 T^{3} + 6912596 T^{4} + 114792 T^{5} + 1800 T^{6} - 60 T^{7} + T^{8} \)
$41$ \( ( -63872 - 115520 T - 948 T^{2} + 76 T^{3} + T^{4} )^{2} \)
$43$ \( 949580087296 - 92550692864 T + 4510220288 T^{2} + 83684352 T^{3} + 1233728 T^{4} - 62016 T^{5} + 3872 T^{6} + 88 T^{7} + T^{8} \)
$47$ \( 287296000000 - 60032000000 T + 6272000000 T^{2} - 356544000 T^{3} + 11699600 T^{4} - 164160 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$53$ \( 5939943840000 + 622246406400 T + 32592108672 T^{2} + 607888800 T^{3} + 6696900 T^{4} + 109512 T^{5} + 5832 T^{6} + 108 T^{7} + T^{8} \)
$59$ \( 861184 + 9775232 T^{2} + 301860 T^{4} + 2348 T^{6} + T^{8} \)
$61$ \( ( -13929728 + 531648 T - 1012 T^{2} - 132 T^{3} + T^{4} )^{2} \)
$67$ \( 3555366682624 + 114401181696 T + 1840545792 T^{2} + 72858624 T^{3} + 34153280 T^{4} + 1251264 T^{5} + 23328 T^{6} + 216 T^{7} + T^{8} \)
$71$ \( ( -4371968 + 271104 T + 176 T^{2} - 120 T^{3} + T^{4} )^{2} \)
$73$ \( 2288349866890000 - 93537969712000 T + 1911716364800 T^{2} - 18811725120 T^{3} + 116212424 T^{4} - 1012704 T^{5} + 21632 T^{6} - 208 T^{7} + T^{8} \)
$79$ \( 1421016580096 + 10878756864 T^{2} + 24554816 T^{4} + 17424 T^{6} + T^{8} \)
$83$ \( 2166784 + 976748544 T + 220150628352 T^{2} - 12233739264 T^{3} + 339889040 T^{4} - 5530944 T^{5} + 56448 T^{6} - 336 T^{7} + T^{8} \)
$89$ \( 693289369600 + 770843096064 T^{2} + 317362448 T^{4} + 37512 T^{6} + T^{8} \)
$97$ \( 10324973344239376 + 287625014890624 T + 4006216114688 T^{2} - 31004823872 T^{3} + 136072648 T^{4} + 1000736 T^{5} + 21632 T^{6} - 208 T^{7} + T^{8} \)
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