Properties

Label 4560.2.d.i
Level $4560$
Weight $2$
Character orbit 4560.d
Analytic conductor $36.412$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} - 784 x^{3} + 800 x^{2} - 320 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + \beta_{5} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} - q^{5} + \beta_{5} q^{7} + q^{9} + ( \beta_{3} - \beta_{5} - \beta_{8} ) q^{11} -\beta_{2} q^{13} + q^{15} + ( -1 + \beta_{1} ) q^{17} + ( -1 - \beta_{3} - \beta_{9} ) q^{19} -\beta_{5} q^{21} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{23} + q^{25} - q^{27} + ( -\beta_{3} + \beta_{5} - \beta_{8} + \beta_{11} ) q^{29} + \beta_{7} q^{31} + ( -\beta_{3} + \beta_{5} + \beta_{8} ) q^{33} -\beta_{5} q^{35} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{9} ) q^{37} + \beta_{2} q^{39} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{41} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{9} + \beta_{11} ) q^{43} - q^{45} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} ) q^{47} + ( -3 + \beta_{1} + 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{49} + ( 1 - \beta_{1} ) q^{51} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{53} + ( -\beta_{3} + \beta_{5} + \beta_{8} ) q^{55} + ( 1 + \beta_{3} + \beta_{9} ) q^{57} + ( -\beta_{1} - \beta_{7} - \beta_{10} ) q^{59} + ( \beta_{1} - \beta_{10} ) q^{61} + \beta_{5} q^{63} + \beta_{2} q^{65} + ( 1 + \beta_{6} + 2 \beta_{7} ) q^{67} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{69} + ( -3 - \beta_{1} - \beta_{7} + 2 \beta_{10} ) q^{71} + ( -1 - \beta_{1} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{73} - q^{75} + ( 5 - 2 \beta_{1} - \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{77} + ( -2 + 2 \beta_{1} + \beta_{4} + \beta_{6} - \beta_{9} - \beta_{10} ) q^{79} + q^{81} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{8} - \beta_{9} - \beta_{11} ) q^{83} + ( 1 - \beta_{1} ) q^{85} + ( \beta_{3} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{87} + ( 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{89} + ( 2 + \beta_{6} - \beta_{7} - \beta_{10} ) q^{91} -\beta_{7} q^{93} + ( 1 + \beta_{3} + \beta_{9} ) q^{95} + ( \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - \beta_{9} ) q^{97} + ( \beta_{3} - \beta_{5} - \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 12q^{5} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 12q^{5} + 12q^{9} + 12q^{15} - 8q^{17} - 8q^{19} + 12q^{25} - 12q^{27} - 12q^{45} - 20q^{49} + 8q^{51} + 8q^{57} - 8q^{59} + 8q^{67} - 32q^{71} - 24q^{73} - 12q^{75} + 56q^{77} - 16q^{79} + 12q^{81} + 8q^{85} + 16q^{91} + 8q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} - 784 x^{3} + 800 x^{2} - 320 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(2101877 \nu^{11} - 9077802 \nu^{10} + 23703524 \nu^{9} - 16487388 \nu^{8} + 110538924 \nu^{7} - 276648264 \nu^{6} + 705909736 \nu^{5} - 913655768 \nu^{4} + 2051812292 \nu^{3} - 1394812968 \nu^{2} + 473528496 \nu - 8758151872\)\()/ 2555640224 \)
\(\beta_{2}\)\(=\)\((\)\(-3940714 \nu^{11} + 13398997 \nu^{10} - 25107184 \nu^{9} - 24324212 \nu^{8} - 172368288 \nu^{7} + 403872146 \nu^{6} - 770790480 \nu^{5} - 647605528 \nu^{4} - 2002990488 \nu^{3} + 1256839928 \nu^{2} - 3217347232 \nu + 792458944\)\()/ 638910056 \)
\(\beta_{3}\)\(=\)\((\)\(13468838 \nu^{11} - 49457227 \nu^{10} + 93612964 \nu^{9} + 77899386 \nu^{8} + 520946664 \nu^{7} - 1529364412 \nu^{6} + 2700529808 \nu^{5} + 2085279108 \nu^{4} + 5438016344 \nu^{3} - 7939856812 \nu^{2} + 9480711840 \nu - 2348307072\)\()/ 638910056 \)
\(\beta_{4}\)\(=\)\((\)\(-27908578 \nu^{11} + 58054709 \nu^{10} - 50596168 \nu^{9} - 384771120 \nu^{8} - 1515562492 \nu^{7} + 1427235004 \nu^{6} - 1224384256 \nu^{5} - 10495968688 \nu^{4} - 22613183792 \nu^{3} - 2051473260 \nu^{2} + 4138399968 \nu - 5758911088\)\()/ 1277820112 \)
\(\beta_{5}\)\(=\)\((\)\(72213471 \nu^{11} - 276433066 \nu^{10} + 528675316 \nu^{9} + 398111644 \nu^{8} + 2588973828 \nu^{7} - 8645816016 \nu^{6} + 15077086984 \nu^{5} + 10709015192 \nu^{4} + 22961507788 \nu^{3} - 49363607352 \nu^{2} + 44433332048 \nu - 11077367072\)\()/ 2555640224 \)
\(\beta_{6}\)\(=\)\((\)\(-19601935 \nu^{11} + 59711704 \nu^{10} - 92253612 \nu^{9} - 182811978 \nu^{8} - 863678932 \nu^{7} + 1785177452 \nu^{6} - 2446008600 \nu^{5} - 4875049852 \nu^{4} - 10479489340 \nu^{3} + 7930481560 \nu^{2} - 2838067024 \nu - 944943624\)\()/ 638910056 \)
\(\beta_{7}\)\(=\)\((\)\(95220313 \nu^{11} - 282494178 \nu^{10} + 403947252 \nu^{9} + 1041253324 \nu^{8} + 4059889564 \nu^{7} - 8435149128 \nu^{6} + 11008990088 \nu^{5} + 28520993784 \nu^{4} + 48690475668 \nu^{3} - 37124235336 \nu^{2} + 13331224688 \nu + 17856922464\)\()/ 2555640224 \)
\(\beta_{8}\)\(=\)\((\)\(101159245 \nu^{11} - 373781562 \nu^{10} + 708970492 \nu^{9} + 580599556 \nu^{8} + 3871197868 \nu^{7} - 11577045064 \nu^{6} + 20456241944 \nu^{5} + 15554634344 \nu^{4} + 39327243748 \nu^{3} - 63031865576 \nu^{2} + 69664703472 \nu - 17273372000\)\()/ 2555640224 \)
\(\beta_{9}\)\(=\)\((\)\(75048949 \nu^{11} - 253862169 \nu^{10} + 423270284 \nu^{9} + 625404004 \nu^{8} + 2978874760 \nu^{7} - 7896232908 \nu^{6} + 11475993800 \nu^{5} + 17019026280 \nu^{4} + 31981959420 \nu^{3} - 43407293348 \nu^{2} + 18945108080 \nu - 68354224\)\()/ 1277820112 \)
\(\beta_{10}\)\(=\)\((\)\(-155955553 \nu^{11} + 464790034 \nu^{10} - 676865564 \nu^{9} - 1628227252 \nu^{8} - 6736017884 \nu^{7} + 13878896504 \nu^{6} - 18054944664 \nu^{5} - 43724741864 \nu^{4} - 80129584308 \nu^{3} + 61113087112 \nu^{2} - 21947888880 \nu - 9750878144\)\()/ 2555640224 \)
\(\beta_{11}\)\(=\)\((\)\(-7567 \nu^{11} + 29316 \nu^{10} - 55332 \nu^{9} - 41748 \nu^{8} - 272116 \nu^{7} + 956992 \nu^{6} - 1582408 \nu^{5} - 1122984 \nu^{4} - 2406636 \nu^{3} + 6211120 \nu^{2} - 4658896 \nu + 1161504\)\()/123664\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_{1} + 3\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{8} + \beta_{5} + 4 \beta_{3} - \beta_{2}\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{10} - 3 \beta_{9} - 6 \beta_{8} + 9 \beta_{7} - 9 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 16 \beta_{3} + 8 \beta_{2} + 3 \beta_{1} - 30\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(10 \beta_{10} - 3 \beta_{9} + 10 \beta_{7} - 16 \beta_{6} + 3 \beta_{4} - 14 \beta_{1} - 90\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(3 \beta_{11} + 22 \beta_{10} - 12 \beta_{9} + 36 \beta_{8} + 37 \beta_{7} - 46 \beta_{6} - 10 \beta_{5} + 24 \beta_{4} - 67 \beta_{3} - 38 \beta_{2} + \beta_{1} - 159\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(36 \beta_{11} + 48 \beta_{9} + 210 \beta_{8} - 94 \beta_{5} + 48 \beta_{4} - 280 \beta_{3} - 110 \beta_{2}\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(54 \beta_{11} - 190 \beta_{10} + 291 \beta_{9} + 390 \beta_{8} - 325 \beta_{7} + 481 \beta_{6} - 194 \beta_{5} - 87 \beta_{4} - 608 \beta_{3} - 388 \beta_{2} + 65 \beta_{1} + 1638\)\()/3\)
\(\nu^{8}\)\(=\)\((\)\(-836 \beta_{10} + 702 \beta_{9} - 1160 \beta_{7} + 2120 \beta_{6} - 702 \beta_{4} + 832 \beta_{1} + 7776\)\()/3\)
\(\nu^{9}\)\(=\)\((\)\(-702 \beta_{11} - 1800 \beta_{10} + 672 \beta_{9} - 4116 \beta_{8} - 3054 \beta_{7} + 5040 \beta_{6} + 2440 \beta_{5} - 3240 \beta_{4} + 5854 \beta_{3} + 4040 \beta_{2} + 1062 \beta_{1} + 16854\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-3912 \beta_{11} - 6480 \beta_{9} - 19788 \beta_{8} + 12052 \beta_{5} - 6480 \beta_{4} + 25984 \beta_{3} + 17084 \beta_{2}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-8088 \beta_{11} + 17876 \beta_{10} - 34854 \beta_{9} - 43068 \beta_{8} + 30074 \beta_{7} - 52730 \beta_{6} + 27476 \beta_{5} + 5718 \beta_{4} + 58448 \beta_{3} + 42232 \beta_{2} - 12994 \beta_{1} - 174132\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
−1.15765 + 1.15765i
2.28268 2.28268i
1.62658 + 1.62658i
0.280289 0.280289i
−1.56297 + 1.56297i
0.531065 0.531065i
0.531065 + 0.531065i
−1.56297 1.56297i
0.280289 + 0.280289i
1.62658 1.62658i
2.28268 + 2.28268i
−1.15765 1.15765i
0 −1.00000 0 −1.00000 0 5.16275i 0 1.00000 0
2431.2 0 −1.00000 0 −1.00000 0 3.67104i 0 1.00000 0
2431.3 0 −1.00000 0 −1.00000 0 2.44391i 0 1.00000 0
2431.4 0 −1.00000 0 −1.00000 0 2.20519i 0 1.00000 0
2431.5 0 −1.00000 0 −1.00000 0 0.855829i 0 1.00000 0
2431.6 0 −1.00000 0 −1.00000 0 0.549104i 0 1.00000 0
2431.7 0 −1.00000 0 −1.00000 0 0.549104i 0 1.00000 0
2431.8 0 −1.00000 0 −1.00000 0 0.855829i 0 1.00000 0
2431.9 0 −1.00000 0 −1.00000 0 2.20519i 0 1.00000 0
2431.10 0 −1.00000 0 −1.00000 0 2.44391i 0 1.00000 0
2431.11 0 −1.00000 0 −1.00000 0 3.67104i 0 1.00000 0
2431.12 0 −1.00000 0 −1.00000 0 5.16275i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2431.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4560.2.d.i 12
4.b odd 2 1 4560.2.d.k yes 12
19.b odd 2 1 4560.2.d.k yes 12
76.d even 2 1 inner 4560.2.d.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.i 12 1.a even 1 1 trivial
4560.2.d.i 12 76.d even 2 1 inner
4560.2.d.k yes 12 4.b odd 2 1
4560.2.d.k yes 12 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4560, [\chi])\):

\( T_{7}^{12} + 52 T_{7}^{10} + 876 T_{7}^{8} + 5920 T_{7}^{6} + 15844 T_{7}^{4} + 11904 T_{7}^{2} + 2304 \)
\( T_{31}^{6} - 48 T_{31}^{4} - 88 T_{31}^{3} + 276 T_{31}^{2} + 576 T_{31} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( 1 + T )^{12} \)
$5$ \( ( 1 + T )^{12} \)
$7$ \( 2304 + 11904 T^{2} + 15844 T^{4} + 5920 T^{6} + 876 T^{8} + 52 T^{10} + T^{12} \)
$11$ \( 3968064 + 2706144 T^{2} + 660100 T^{4} + 71776 T^{6} + 3852 T^{8} + 100 T^{10} + T^{12} \)
$13$ \( 82944 + 146304 T^{2} + 85828 T^{4} + 19456 T^{6} + 1884 T^{8} + 76 T^{10} + T^{12} \)
$17$ \( ( -288 + 240 T + 148 T^{2} - 80 T^{3} - 24 T^{4} + 4 T^{5} + T^{6} )^{2} \)
$19$ \( 47045881 + 19808792 T + 7558618 T^{2} + 1262056 T^{3} + 123823 T^{4} - 53200 T^{5} - 13588 T^{6} - 2800 T^{7} + 343 T^{8} + 184 T^{9} + 58 T^{10} + 8 T^{11} + T^{12} \)
$23$ \( 39538944 + 19243392 T^{2} + 3517456 T^{4} + 296512 T^{6} + 11448 T^{8} + 184 T^{10} + T^{12} \)
$29$ \( 418120704 + 140529024 T^{2} + 16469092 T^{4} + 865840 T^{6} + 21564 T^{8} + 244 T^{10} + T^{12} \)
$31$ \( ( 16 + 576 T + 276 T^{2} - 88 T^{3} - 48 T^{4} + T^{6} )^{2} \)
$37$ \( 2985984 + 2999808 T^{2} + 847300 T^{4} + 100048 T^{6} + 5388 T^{8} + 124 T^{10} + T^{12} \)
$41$ \( 26132544 + 56874528 T^{2} + 10090692 T^{4} + 681984 T^{6} + 20460 T^{8} + 252 T^{10} + T^{12} \)
$43$ \( 1272384 + 1899744 T^{2} + 662308 T^{4} + 89632 T^{6} + 5244 T^{8} + 124 T^{10} + T^{12} \)
$47$ \( 1285652736 + 495220608 T^{2} + 50776720 T^{4} + 2116096 T^{6} + 39384 T^{8} + 328 T^{10} + T^{12} \)
$53$ \( 20846739456 + 4036337664 T^{2} + 249497872 T^{4} + 6804928 T^{6} + 87432 T^{8} + 496 T^{10} + T^{12} \)
$59$ \( ( -1152 + 3072 T - 800 T^{2} - 1040 T^{3} - 156 T^{4} + 4 T^{5} + T^{6} )^{2} \)
$61$ \( ( 1744 - 2544 T + 564 T^{2} + 304 T^{3} - 96 T^{4} + T^{6} )^{2} \)
$67$ \( ( 175488 - 94848 T + 11064 T^{2} + 1512 T^{3} - 272 T^{4} - 4 T^{5} + T^{6} )^{2} \)
$71$ \( ( 497664 + 172800 T - 104 T^{2} - 4424 T^{3} - 240 T^{4} + 16 T^{5} + T^{6} )^{2} \)
$73$ \( ( -32 + 408 T^{2} - 296 T^{3} - 12 T^{4} + 12 T^{5} + T^{6} )^{2} \)
$79$ \( ( -8192 + 14336 T + 6656 T^{2} - 1600 T^{3} - 220 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$83$ \( 103859241984 + 32300176896 T^{2} + 1321989264 T^{4} + 21839616 T^{6} + 174504 T^{8} + 672 T^{10} + T^{12} \)
$89$ \( 627264 + 598934304 T^{2} + 137946276 T^{4} + 7732080 T^{6} + 112908 T^{8} + 588 T^{10} + T^{12} \)
$97$ \( 1214383104 + 3532443264 T^{2} + 327848292 T^{4} + 10174896 T^{6} + 119964 T^{8} + 588 T^{10} + T^{12} \)
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