Properties

Label 6.13.b.a
Level $6$
Weight $13$
Character orbit 6.b
Analytic conductor $5.484$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 6.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.48396290366\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{1009})\)
Defining polynomial: \(x^{4} - 2 x^{3} - 499 x^{2} + 500 x + 64518\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 195 - \beta_{1} - \beta_{3} ) q^{3} -2048 q^{4} + ( 203 \beta_{1} + \beta_{2} - 10 \beta_{3} ) q^{5} + ( -2496 - 201 \beta_{1} - 16 \beta_{2} ) q^{6} + ( 38270 - 17 \beta_{1} - 85 \beta_{2} - 68 \beta_{3} ) q^{7} + 2048 \beta_{1} q^{8} + ( -382743 + 5271 \beta_{1} - 39 \beta_{2} - 390 \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 195 - \beta_{1} - \beta_{3} ) q^{3} -2048 q^{4} + ( 203 \beta_{1} + \beta_{2} - 10 \beta_{3} ) q^{5} + ( -2496 - 201 \beta_{1} - 16 \beta_{2} ) q^{6} + ( 38270 - 17 \beta_{1} - 85 \beta_{2} - 68 \beta_{3} ) q^{7} + 2048 \beta_{1} q^{8} + ( -382743 + 5271 \beta_{1} - 39 \beta_{2} - 390 \beta_{3} ) q^{9} + ( 410496 - 32 \beta_{1} - 160 \beta_{2} - 128 \beta_{3} ) q^{10} + ( -19129 \beta_{1} + 121 \beta_{2} - 1210 \beta_{3} ) q^{11} + ( -399360 + 2048 \beta_{1} + 2048 \beta_{3} ) q^{12} + ( 1813250 + 500 \beta_{1} + 2500 \beta_{2} + 2000 \beta_{3} ) q^{13} + ( -41058 \beta_{1} - 1088 \beta_{2} + 10880 \beta_{3} ) q^{14} + ( -4403448 + 71397 \beta_{1} + 3207 \beta_{2} - 2106 \beta_{3} ) q^{15} + 4194304 q^{16} + ( -370412 \beta_{1} + 3020 \beta_{2} - 30200 \beta_{3} ) q^{17} + ( 10650240 + 379311 \beta_{1} - 6240 \beta_{2} + 4992 \beta_{3} ) q^{18} + ( -30067018 - 2935 \beta_{1} - 14675 \beta_{2} - 11740 \beta_{3} ) q^{19} + ( -415744 \beta_{1} - 2048 \beta_{2} + 20480 \beta_{3} ) q^{20} + ( 40808082 + 2560129 \beta_{1} - 17901 \beta_{2} - 38270 \beta_{3} ) q^{21} + ( -39811200 - 3872 \beta_{1} - 19360 \beta_{2} - 15488 \beta_{3} ) q^{22} + ( -3667862 \beta_{1} - 6730 \beta_{2} + 67300 \beta_{3} ) q^{23} + ( 5111808 + 411648 \beta_{1} + 32768 \beta_{2} ) q^{24} + ( 108901441 + 12828 \beta_{1} + 64140 \beta_{2} + 51312 \beta_{3} ) q^{25} + ( -1731250 \beta_{1} + 32000 \beta_{2} - 320000 \beta_{3} ) q^{26} + ( -196235325 + 4669308 \beta_{1} + 67995 \beta_{2} + 312777 \beta_{3} ) q^{27} + ( -78376960 + 34816 \beta_{1} + 174080 \beta_{2} + 139264 \beta_{3} ) q^{28} + ( -7796455 \beta_{1} - 72245 \beta_{2} + 722450 \beta_{3} ) q^{29} + ( 142814592 + 4480608 \beta_{1} - 33696 \beta_{2} - 410496 \beta_{3} ) q^{30} + ( 682931918 - 43125 \beta_{1} - 215625 \beta_{2} - 172500 \beta_{3} ) q^{31} -4194304 \beta_{1} q^{32} + ( -641872440 - 143055 \beta_{1} - 311025 \beta_{2} - 254826 \beta_{3} ) q^{33} + ( -774452736 - 96640 \beta_{1} - 483200 \beta_{2} - 386560 \beta_{3} ) q^{34} + ( 36462838 \beta_{1} + 256346 \beta_{2} - 2563460 \beta_{3} ) q^{35} + ( 783857664 - 10795008 \beta_{1} + 79872 \beta_{2} + 798720 \beta_{3} ) q^{36} + ( -3820030 - 10388 \beta_{1} - 51940 \beta_{2} - 41552 \beta_{3} ) q^{37} + ( 29585678 \beta_{1} - 187840 \beta_{2} + 1878400 \beta_{3} ) q^{38} + ( -627164250 - 78236750 \beta_{1} + 526500 \beta_{2} - 1813250 \beta_{3} ) q^{39} + ( -840695808 + 65536 \beta_{1} + 327680 \beta_{2} + 262144 \beta_{3} ) q^{40} + ( 27961274 \beta_{1} - 264146 \beta_{2} + 2641460 \beta_{3} ) q^{41} + ( 5239747200 - 41538930 \beta_{1} - 612320 \beta_{2} + 2291328 \beta_{3} ) q^{42} + ( 407279990 + 552129 \beta_{1} + 2760645 \beta_{2} + 2208516 \beta_{3} ) q^{43} + ( 39176192 \beta_{1} - 247808 \beta_{2} + 2478080 \beta_{3} ) q^{44} + ( -3894169392 - 52936785 \beta_{1} + 1700037 \beta_{2} + 3492486 \beta_{3} ) q^{45} + ( -7476462336 + 215360 \beta_{1} + 1076800 \beta_{2} + 861440 \beta_{3} ) q^{46} + ( 97272900 \beta_{1} + 1009500 \beta_{2} - 10095000 \beta_{3} ) q^{47} + ( 817889280 - 4194304 \beta_{1} - 4194304 \beta_{3} ) q^{48} + ( 18234412275 - 1301180 \beta_{1} - 6505900 \beta_{2} - 5204720 \beta_{3} ) q^{49} + ( -106797649 \beta_{1} + 820992 \beta_{2} - 8209920 \beta_{3} ) q^{50} + ( -15753159072 + 17941068 \beta_{1} - 6050412 \beta_{2} - 6360120 \beta_{3} ) q^{51} + ( -3713536000 - 1024000 \beta_{1} - 5120000 \beta_{2} - 4096000 \beta_{3} ) q^{52} + ( 40828247 \beta_{1} - 3483995 \beta_{2} + 34839950 \beta_{3} ) q^{53} + ( 9650646720 + 200015847 \beta_{1} + 5004432 \beta_{2} - 8703360 \beta_{3} ) q^{54} + ( 1571449968 + 154044 \beta_{1} + 770220 \beta_{2} + 616176 \beta_{3} ) q^{55} + ( 84086784 \beta_{1} + 2228224 \beta_{2} - 22282240 \beta_{3} ) q^{56} + ( -106077750 + 478672963 \beta_{1} - 3090555 \beta_{2} + 30067018 \beta_{3} ) q^{57} + ( -15587998080 + 2311840 \beta_{1} + 11559200 \beta_{2} + 9247360 \beta_{3} ) q^{58} + ( -1214785349 \beta_{1} + 2880341 \beta_{2} - 28803410 \beta_{3} ) q^{59} + ( 9018261504 - 146221056 \beta_{1} - 6567936 \beta_{2} + 4313088 \beta_{3} ) q^{60} + ( 11369266274 + 2488380 \beta_{1} + 12441900 \beta_{2} + 9953520 \beta_{3} ) q^{61} + ( -690004418 \beta_{1} - 2760000 \beta_{2} + 27600000 \beta_{3} ) q^{62} + ( 11361862350 + 1157440413 \beta_{1} + 36698565 \beta_{2} - 45478176 \beta_{3} ) q^{63} -8589934592 q^{64} + ( -475852250 \beta_{1} - 4600750 \beta_{2} + 46007500 \beta_{3} ) q^{65} + ( -168271488 + 631634784 \beta_{1} - 4077216 \beta_{2} + 39811200 \beta_{3} ) q^{66} + ( -53358402490 - 537123 \beta_{1} - 2685615 \beta_{2} - 2148492 \beta_{3} ) q^{67} + ( 758603776 \beta_{1} - 6184960 \beta_{2} + 61849600 \beta_{3} ) q^{68} + ( 23890231728 - 943137882 \beta_{1} - 58409862 \beta_{2} + 14173380 \beta_{3} ) q^{69} + ( 73330588416 - 8203072 \beta_{1} - 41015360 \beta_{2} - 32812288 \beta_{3} ) q^{70} + ( 1643647630 \beta_{1} + 24011570 \beta_{2} - 240115700 \beta_{3} ) q^{71} + ( -21811691520 - 776828928 \beta_{1} + 12779520 \beta_{2} - 10223616 \beta_{3} ) q^{72} + ( -63595906270 - 6709680 \beta_{1} - 33548400 \beta_{2} - 26838720 \beta_{3} ) q^{73} + ( 2116398 \beta_{1} - 664832 \beta_{2} + 6648320 \beta_{3} ) q^{74} + ( -3926289693 - 2069622757 \beta_{1} + 13507884 \beta_{2} - 108901441 \beta_{3} ) q^{75} + ( 61577252864 + 6010880 \beta_{1} + 30054400 \beta_{2} + 24043520 \beta_{3} ) q^{76} + ( 2618097262 \beta_{1} - 16519030 \beta_{2} + 165190300 \beta_{3} ) q^{77} + ( -161445552000 + 631026750 \beta_{1} - 29012000 \beta_{2} - 67392000 \beta_{3} ) q^{78} + ( 77145039758 + 7813835 \beta_{1} + 39069175 \beta_{2} + 31255340 \beta_{3} ) q^{79} + ( 851443712 \beta_{1} + 4194304 \beta_{2} - 41943040 \beta_{3} ) q^{80} + ( 54753914817 - 2410778646 \beta_{1} + 94753854 \beta_{2} + 246619620 \beta_{3} ) q^{81} + ( 58650927360 + 8452672 \beta_{1} + 42263360 \beta_{2} + 33810688 \beta_{3} ) q^{82} + ( 1543479261 \beta_{1} - 31430205 \beta_{2} + 314302050 \beta_{3} ) q^{83} + ( -83574951936 - 5243144192 \beta_{1} + 36661248 \beta_{2} + 78376960 \beta_{3} ) q^{84} + ( -4711013568 + 7269456 \beta_{1} + 36347280 \beta_{2} + 29077824 \beta_{3} ) q^{85} + ( -316730834 \beta_{1} + 35336256 \beta_{2} - 353362560 \beta_{3} ) q^{86} + ( 335272823640 - 3777350985 \beta_{1} - 121781235 \beta_{2} + 152147970 \beta_{3} ) q^{87} + ( 81533337600 + 7929856 \beta_{1} + 39649280 \beta_{2} + 31719424 \beta_{3} ) q^{88} + ( 854639446 \beta_{1} - 2079214 \beta_{2} + 20792140 \beta_{3} ) q^{89} + ( -108155530368 + 3962725344 \beta_{1} + 55879776 \beta_{2} - 217604736 \beta_{3} ) q^{90} + ( -830933586500 - 11690250 \beta_{1} - 58451250 \beta_{2} - 46761000 \beta_{3} ) q^{91} + ( 7511781376 \beta_{1} + 13783040 \beta_{2} - 137830400 \beta_{3} ) q^{92} + ( 217761239010 + 5908594957 \beta_{1} - 45410625 \beta_{2} - 682931918 \beta_{3} ) q^{93} + ( 193917043200 - 32304000 \beta_{1} - 161520000 \beta_{2} - 129216000 \beta_{3} ) q^{94} + ( -1149665114 \beta_{1} + 7583162 \beta_{2} - 75831620 \beta_{3} ) q^{95} + ( -10468982784 - 843055104 \beta_{1} - 67108864 \beta_{2} ) q^{96} + ( 319057118930 - 43978444 \beta_{1} - 219892220 \beta_{2} - 175913776 \beta_{3} ) q^{97} + ( -18447805795 \beta_{1} - 83275520 \beta_{2} + 832755200 \beta_{3} ) q^{98} + ( -5864210352 + 10167505227 \beta_{1} - 66933603 \beta_{2} + 640701270 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 780q^{3} - 8192q^{4} - 9984q^{6} + 153080q^{7} - 1530972q^{9} + O(q^{10}) \) \( 4q + 780q^{3} - 8192q^{4} - 9984q^{6} + 153080q^{7} - 1530972q^{9} + 1641984q^{10} - 1597440q^{12} + 7253000q^{13} - 17613792q^{15} + 16777216q^{16} + 42600960q^{18} - 120268072q^{19} + 163232328q^{21} - 159244800q^{22} + 20447232q^{24} + 435605764q^{25} - 784941300q^{27} - 313507840q^{28} + 571258368q^{30} + 2731727672q^{31} - 2567489760q^{33} - 3097810944q^{34} + 3135430656q^{36} - 15280120q^{37} - 2508657000q^{39} - 3362783232q^{40} + 20958988800q^{42} + 1629119960q^{43} - 15576677568q^{45} - 29905849344q^{46} + 3271557120q^{48} + 72937649100q^{49} - 63012636288q^{51} - 14854144000q^{52} + 38602586880q^{54} + 6285799872q^{55} - 424311000q^{57} - 62351992320q^{58} + 36073046016q^{60} + 45477065096q^{61} + 45447449400q^{63} - 34359738368q^{64} - 673085952q^{66} - 213433609960q^{67} + 95560926912q^{69} + 293322353664q^{70} - 87246766080q^{72} - 254383625080q^{73} - 15705158772q^{75} + 246309011456q^{76} - 645782208000q^{78} + 308580159032q^{79} + 219015659268q^{81} + 234603709440q^{82} - 334299807744q^{84} - 18844054272q^{85} + 1341091294560q^{87} + 326133350400q^{88} - 432622121472q^{90} - 3323734346000q^{91} + 871044956040q^{93} + 775668172800q^{94} - 41875931136q^{96} + 1276228475720q^{97} - 23456841408q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 499 x^{2} + 500 x + 64518\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 64 \nu^{3} - 96 \nu^{2} - 15712 \nu + 7872 \)\()/1017\)
\(\beta_{2}\)\(=\)\((\)\( -88 \nu^{3} - 3936 \nu^{2} + 66352 \nu + 985836 \)\()/339\)
\(\beta_{3}\)\(=\)\((\)\( -10 \nu^{3} + 15270 \nu^{2} - 596 \nu - 3821082 \)\()/1017\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(16 \beta_{3} + 20 \beta_{2} + 85 \beta_{1} + 1296\)\()/2592\)
\(\nu^{2}\)\(=\)\((\)\(44 \beta_{3} + \beta_{2} + 11 \beta_{1} + 162324\)\()/648\)
\(\nu^{3}\)\(=\)\((\)\(2096 \beta_{3} + 2458 \beta_{2} + 31061 \beta_{1} + 486648\)\()/1296\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
16.3824 + 1.41421i
−15.3824 + 1.41421i
16.3824 1.41421i
−15.3824 1.41421i
45.2548i 4.41144 728.987i −2048.00 1793.38i −32990.2 199.639i −136690. 92681.9i −531402. 6431.76i 81159.0
5.2 45.2548i 385.589 + 618.678i −2048.00 16348.2i 27998.2 17449.7i 213230. 92681.9i −234084. + 477110.i 739833.
5.3 45.2548i 4.41144 + 728.987i −2048.00 1793.38i −32990.2 + 199.639i −136690. 92681.9i −531402. + 6431.76i 81159.0
5.4 45.2548i 385.589 618.678i −2048.00 16348.2i 27998.2 + 17449.7i 213230. 92681.9i −234084. 477110.i 739833.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.13.b.a 4
3.b odd 2 1 inner 6.13.b.a 4
4.b odd 2 1 48.13.e.c 4
5.b even 2 1 150.13.d.a 4
5.c odd 4 2 150.13.b.a 8
8.b even 2 1 192.13.e.e 4
8.d odd 2 1 192.13.e.h 4
9.c even 3 2 162.13.d.d 8
9.d odd 6 2 162.13.d.d 8
12.b even 2 1 48.13.e.c 4
15.d odd 2 1 150.13.d.a 4
15.e even 4 2 150.13.b.a 8
24.f even 2 1 192.13.e.h 4
24.h odd 2 1 192.13.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.13.b.a 4 1.a even 1 1 trivial
6.13.b.a 4 3.b odd 2 1 inner
48.13.e.c 4 4.b odd 2 1
48.13.e.c 4 12.b even 2 1
150.13.b.a 8 5.c odd 4 2
150.13.b.a 8 15.e even 4 2
150.13.d.a 4 5.b even 2 1
150.13.d.a 4 15.d odd 2 1
162.13.d.d 8 9.c even 3 2
162.13.d.d 8 9.d odd 6 2
192.13.e.e 4 8.b even 2 1
192.13.e.e 4 24.h odd 2 1
192.13.e.h 4 8.d odd 2 1
192.13.e.h 4 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{13}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 2048 + T^{2} )^{2} \)
$3$ \( 282429536481 - 414523980 T + 1069686 T^{2} - 780 T^{3} + T^{4} \)
$5$ \( 859568578560000 + 270478368 T^{2} + T^{4} \)
$7$ \( ( -29146513676 - 76540 T + T^{2} )^{2} \)
$11$ \( 2252083298013609984 + 3098571008544 T^{2} + T^{4} \)
$13$ \( ( -23192320437500 - 3626500 T + T^{2} )^{2} \)
$17$ \( \)\(36\!\cdots\!04\)\( + 1551759684309504 T^{2} + T^{4} \)
$19$ \( ( -8399894140076 + 60134036 T + T^{2} )^{2} \)
$23$ \( \)\(61\!\cdots\!04\)\( + 59384851139378304 T^{2} + T^{4} \)
$29$ \( \)\(24\!\cdots\!00\)\( + 790127294580583200 T^{2} + T^{4} \)
$31$ \( ( 269408171566908724 - 1365863836 T + T^{2} )^{2} \)
$37$ \( ( -11415376793145596 + 7640060 T + T^{2} )^{2} \)
$41$ \( \)\(40\!\cdots\!84\)\( + 10749730331677014144 T^{2} + T^{4} \)
$43$ \( ( -32123696154683510444 - 814559980 T + T^{2} )^{2} \)
$47$ \( \)\(12\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( T^{2} + T^{4} \)
$53$ \( \)\(40\!\cdots\!84\)\( + \)\(12\!\cdots\!44\)\( T^{2} + T^{4} \)
$59$ \( \)\(68\!\cdots\!04\)\( + \)\(69\!\cdots\!04\)\( T^{2} + T^{4} \)
$61$ \( ( -\)\(52\!\cdots\!24\)\( - 22738532548 T + T^{2} )^{2} \)
$67$ \( ( \)\(28\!\cdots\!64\)\( + 106716804980 T + T^{2} )^{2} \)
$71$ \( \)\(64\!\cdots\!00\)\( + \)\(71\!\cdots\!00\)\( T^{2} + T^{4} \)
$73$ \( ( -\)\(72\!\cdots\!00\)\( + 127191812540 T + T^{2} )^{2} \)
$79$ \( ( -\)\(51\!\cdots\!36\)\( - 154290079516 T + T^{2} )^{2} \)
$83$ \( \)\(22\!\cdots\!44\)\( + \)\(11\!\cdots\!76\)\( T^{2} + T^{4} \)
$89$ \( \)\(16\!\cdots\!24\)\( + \)\(34\!\cdots\!64\)\( T^{2} + T^{4} \)
$97$ \( ( -\)\(10\!\cdots\!24\)\( - 638114237860 T + T^{2} )^{2} \)
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