Properties

Label 320.6.d.a
Level 320
Weight 6
Character orbit 320.d
Analytic conductor 51.323
Analytic rank 0
Dimension 8
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 384 x^{6} + 506 x^{5} + 49869 x^{4} + 29654 x^{3} - 2235516 x^{2} - 1528906 x + 34180205\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{3} + 25 \beta_{1} q^{5} + ( -40 - 3 \beta_{2} - \beta_{6} ) q^{7} + ( -149 - 5 \beta_{2} + \beta_{3} - \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{4} q^{3} + 25 \beta_{1} q^{5} + ( -40 - 3 \beta_{2} - \beta_{6} ) q^{7} + ( -149 - 5 \beta_{2} + \beta_{3} - \beta_{6} ) q^{9} + ( -92 \beta_{1} + 2 \beta_{5} - 3 \beta_{7} ) q^{11} + ( 220 \beta_{1} + 3 \beta_{4} - 5 \beta_{5} + \beta_{7} ) q^{13} -25 \beta_{2} q^{15} + ( -300 - 3 \beta_{2} - 5 \beta_{3} - 5 \beta_{6} ) q^{17} + ( 204 \beta_{1} - 30 \beta_{4} - 14 \beta_{5} + \beta_{7} ) q^{19} + ( -1090 \beta_{1} + 120 \beta_{4} + 20 \beta_{5} - 10 \beta_{7} ) q^{21} + ( 220 - 117 \beta_{2} + 10 \beta_{3} + 11 \beta_{6} ) q^{23} -625 q^{25} + ( -1980 \beta_{1} + 58 \beta_{4} + 30 \beta_{5} ) q^{27} + ( -1480 \beta_{1} + 210 \beta_{4} - 10 \beta_{5} + 20 \beta_{7} ) q^{29} + ( 3880 - 150 \beta_{2} - 40 \beta_{3} ) q^{31} + ( 470 + 21 \beta_{2} + 35 \beta_{3} + 55 \beta_{6} ) q^{33} + ( -1000 \beta_{1} + 75 \beta_{4} + 25 \beta_{7} ) q^{35} + ( -1230 \beta_{1} + 312 \beta_{4} - 40 \beta_{5} + 4 \beta_{7} ) q^{37} + ( 560 - 450 \beta_{2} + 20 \beta_{3} - 50 \beta_{6} ) q^{39} + ( 2698 + 165 \beta_{2} - 33 \beta_{3} - 47 \beta_{6} ) q^{41} + ( 7020 \beta_{1} - 135 \beta_{4} - 70 \beta_{5} + 50 \beta_{7} ) q^{43} + ( -3725 \beta_{1} + 125 \beta_{4} + 25 \beta_{5} + 25 \beta_{7} ) q^{45} + ( 5960 - 321 \beta_{2} + 80 \beta_{3} + 43 \beta_{6} ) q^{47} + ( 10353 + 195 \beta_{2} - 95 \beta_{3} + 35 \beta_{6} ) q^{49} + ( -216 \beta_{1} + 330 \beta_{4} + 48 \beta_{5} - 102 \beta_{7} ) q^{51} + ( -10020 \beta_{1} + 141 \beta_{4} + 45 \beta_{5} - 153 \beta_{7} ) q^{53} + ( 2300 - 50 \beta_{3} - 75 \beta_{6} ) q^{55} + ( -13330 - 1157 \beta_{2} + 125 \beta_{3} - 155 \beta_{6} ) q^{57} + ( 13524 \beta_{1} - 570 \beta_{4} + 30 \beta_{5} - 135 \beta_{7} ) q^{59} + ( -17350 \beta_{1} - 1290 \beta_{4} - 50 \beta_{5} - 30 \beta_{7} ) q^{61} + ( 40300 + 1551 \beta_{2} - 110 \beta_{3} + 167 \beta_{6} ) q^{63} + ( -5500 + 75 \beta_{2} + 125 \beta_{3} + 25 \beta_{6} ) q^{65} + ( -18300 \beta_{1} - 723 \beta_{4} - 10 \beta_{5} + 110 \beta_{7} ) q^{67} + ( -47870 \beta_{1} + 270 \beta_{4} + 10 \beta_{5} + 340 \beta_{7} ) q^{69} + ( 30840 + 1290 \beta_{2} + 180 \beta_{3} + 180 \beta_{6} ) q^{71} + ( 5800 + 141 \beta_{2} - 205 \beta_{3} - 5 \beta_{6} ) q^{73} + 625 \beta_{4} q^{75} + ( -49990 \beta_{1} - 2709 \beta_{4} + 155 \beta_{5} - 103 \beta_{7} ) q^{77} + ( 40720 + 1020 \beta_{2} - 60 \beta_{3} - 410 \beta_{6} ) q^{79} + ( -10291 + 2915 \beta_{2} - 55 \beta_{3} + 55 \beta_{6} ) q^{81} + ( -14300 \beta_{1} - 69 \beta_{4} + 110 \beta_{5} - 60 \beta_{7} ) q^{83} + ( -7500 \beta_{1} + 75 \beta_{4} - 125 \beta_{5} + 125 \beta_{7} ) q^{85} + ( 79540 + 3210 \beta_{2} - 470 \beta_{3} - 130 \beta_{6} ) q^{87} + ( -9774 - 1470 \beta_{2} + 254 \beta_{3} + 206 \beta_{6} ) q^{89} + ( -16240 \beta_{1} + 4530 \beta_{4} - 220 \beta_{5} + 440 \beta_{7} ) q^{91} + ( -54560 \beta_{1} - 5610 \beta_{4} - 170 \beta_{5} - 170 \beta_{7} ) q^{93} + ( -5100 - 750 \beta_{2} + 350 \beta_{3} + 25 \beta_{6} ) q^{95} + ( 3120 - 3015 \beta_{2} - 145 \beta_{3} + 475 \beta_{6} ) q^{97} + ( -22564 \beta_{1} - 1980 \beta_{4} - 190 \beta_{5} + 245 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 320q^{7} - 1192q^{9} + O(q^{10}) \) \( 8q - 320q^{7} - 1192q^{9} - 2400q^{17} + 1760q^{23} - 5000q^{25} + 31040q^{31} + 3760q^{33} + 4480q^{39} + 21584q^{41} + 47680q^{47} + 82824q^{49} + 18400q^{55} - 106640q^{57} + 322400q^{63} - 44000q^{65} + 246720q^{71} + 46400q^{73} + 325760q^{79} - 82328q^{81} + 636320q^{87} - 78192q^{89} - 40800q^{95} + 24960q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 384 x^{6} + 506 x^{5} + 49869 x^{4} + 29654 x^{3} - 2235516 x^{2} - 1528906 x + 34180205\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1351312 \nu^{7} - 13653287 \nu^{6} - 394224929 \nu^{5} + 2862879190 \nu^{4} + 38223976863 \nu^{3} - 163085600918 \nu^{2} - 987814371481 \nu + 3699160158415\)\()/ 281446960630 \)
\(\beta_{2}\)\(=\)\((\)\(-1351312 \nu^{7} + 13653287 \nu^{6} + 394224929 \nu^{5} - 2862879190 \nu^{4} - 38223976863 \nu^{3} + 163085600918 \nu^{2} + 1550708292741 \nu - 3980607119045\)\()/ 281446960630 \)
\(\beta_{3}\)\(=\)\((\)\(46680604 \nu^{7} - 595364489 \nu^{6} - 11237259467 \nu^{5} + 129626329330 \nu^{4} + 680074862701 \nu^{3} - 8470823112146 \nu^{2} + 6684424237957 \nu + 233594355624595\)\()/ 844340881890 \)
\(\beta_{4}\)\(=\)\((\)\(324498 \nu^{7} - 4782019 \nu^{6} - 86408283 \nu^{5} + 1112583304 \nu^{4} + 8082523469 \nu^{3} - 76894837308 \nu^{2} - 227603504321 \nu + 1741708282831\)\()/ 5117217466 \)
\(\beta_{5}\)\(=\)\((\)\(-8770602 \nu^{7} + 204376019 \nu^{6} + 1922738643 \nu^{5} - 61548799060 \nu^{4} - 143197714893 \nu^{3} + 5939121203576 \nu^{2} + 7063580525697 \nu - 148559382301075\)\()/ 76758261990 \)
\(\beta_{6}\)\(=\)\((\)\(87017312 \nu^{7} - 1189115032 \nu^{6} - 22832683396 \nu^{5} + 269861003750 \nu^{4} + 1960333630208 \nu^{3} - 16457519954998 \nu^{2} - 47531360053324 \nu + 269387125903700\)\()/ 422170440945 \)
\(\beta_{7}\)\(=\)\((\)\(109345770 \nu^{7} - 502754801 \nu^{6} - 43714650957 \nu^{5} + 138978838510 \nu^{4} + 5044385010495 \nu^{3} - 8601384396614 \nu^{2} - 140104059489243 \nu + 305925739736305\)\()/ 422170440945 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} - 2 \beta_{4} - \beta_{3} + 7 \beta_{2} + 2 \beta_{1} + 392\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 21 \beta_{4} - 33 \beta_{3} + 559 \beta_{2} + 1178 \beta_{1} + 3154\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{7} + 196 \beta_{6} - 33 \beta_{5} - 561 \beta_{4} - 226 \beta_{3} + 2182 \beta_{2} + 3156 \beta_{1} + 56086\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-985 \beta_{7} + 1635 \beta_{6} - 1135 \beta_{5} - 10945 \beta_{4} - 7515 \beta_{3} + 95369 \beta_{2} + 282394 \beta_{1} + 915374\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-2460 \beta_{7} + 33721 \beta_{6} - 11355 \beta_{5} - 144457 \beta_{4} - 47416 \beta_{3} + 512212 \beta_{2} + 1380952 \beta_{1} + 9668402\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-119749 \beta_{7} + 242214 \beta_{6} - 167944 \beta_{5} - 1811908 \beta_{4} - 750429 \beta_{3} + 8864437 \beta_{2} + 34334284 \beta_{1} + 105600322\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
14.2856 0.500000i
−10.3099 + 0.500000i
−7.33903 + 0.500000i
5.36332 0.500000i
5.36332 + 0.500000i
−7.33903 0.500000i
−10.3099 0.500000i
14.2856 + 0.500000i
0 27.5713i 0 25.0000i 0 −220.359 0 −517.174 0
161.2 0 21.6198i 0 25.0000i 0 −169.466 0 −224.417 0
161.3 0 15.6781i 0 25.0000i 0 164.681 0 −2.80186 0
161.4 0 9.72664i 0 25.0000i 0 65.1432 0 148.393 0
161.5 0 9.72664i 0 25.0000i 0 65.1432 0 148.393 0
161.6 0 15.6781i 0 25.0000i 0 164.681 0 −2.80186 0
161.7 0 21.6198i 0 25.0000i 0 −169.466 0 −224.417 0
161.8 0 27.5713i 0 25.0000i 0 −220.359 0 −517.174 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.d.a 8
4.b odd 2 1 320.6.d.b yes 8
8.b even 2 1 inner 320.6.d.a 8
8.d odd 2 1 320.6.d.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.d.a 8 1.a even 1 1 trivial
320.6.d.a 8 8.b even 2 1 inner
320.6.d.b yes 8 4.b odd 2 1
320.6.d.b yes 8 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{8} + 1568 T_{3}^{6} + 796456 T_{3}^{4} + 149500800 T_{3}^{2} + 8262810000 \)
\( T_{7}^{4} + 160 T_{7}^{3} - 41520 T_{7}^{2} - 4400400 T_{7} + 400612500 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 376 T^{2} + 163684 T^{4} - 39360744 T^{6} + 11878187814 T^{8} - 2324212572456 T^{10} + 570730817893284 T^{12} - 77415065667588024 T^{14} + 12157665459056928801 T^{16} \)
$5$ \( ( 1 + 625 T^{2} )^{4} \)
$7$ \( ( 1 + 160 T + 25708 T^{2} + 3666960 T^{3} + 699810714 T^{4} + 61630596720 T^{5} + 7261873701292 T^{6} + 759609841590880 T^{7} + 79792266297612001 T^{8} )^{2} \)
$11$ \( 1 - 461928 T^{2} + 80897449276 T^{4} - 9751521734727448 T^{6} + \)\(14\!\cdots\!94\)\( T^{8} - \)\(25\!\cdots\!48\)\( T^{10} + \)\(54\!\cdots\!76\)\( T^{12} - \)\(80\!\cdots\!28\)\( T^{14} + \)\(45\!\cdots\!01\)\( T^{16} \)
$13$ \( 1 - 1173304 T^{2} + 912998161852 T^{4} - 514796045425358792 T^{6} + \)\(21\!\cdots\!70\)\( T^{8} - \)\(70\!\cdots\!08\)\( T^{10} + \)\(17\!\cdots\!52\)\( T^{12} - \)\(30\!\cdots\!96\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} \)
$17$ \( ( 1 + 1200 T + 4842092 T^{2} + 4006103760 T^{3} + 9533675124390 T^{4} + 5688094466362320 T^{5} + 9761627937412899308 T^{6} + \)\(34\!\cdots\!00\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} )^{2} \)
$19$ \( 1 - 4628552 T^{2} + 15129915669436 T^{4} - 43933554813794877112 T^{6} + \)\(13\!\cdots\!54\)\( T^{8} - \)\(26\!\cdots\!12\)\( T^{10} + \)\(56\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!01\)\( T^{16} \)
$23$ \( ( 1 - 880 T + 8141852 T^{2} - 31216146720 T^{3} + 19333174339674 T^{4} - 200917827428244960 T^{5} + \)\(33\!\cdots\!48\)\( T^{6} - \)\(23\!\cdots\!60\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \)
$29$ \( 1 - 57762792 T^{2} + 1773990452604028 T^{4} - \)\(43\!\cdots\!44\)\( T^{6} + \)\(95\!\cdots\!70\)\( T^{8} - \)\(18\!\cdots\!44\)\( T^{10} + \)\(31\!\cdots\!28\)\( T^{12} - \)\(43\!\cdots\!92\)\( T^{14} + \)\(31\!\cdots\!01\)\( T^{16} \)
$31$ \( ( 1 - 15520 T + 133260604 T^{2} - 763025830560 T^{3} + 3983065379572806 T^{4} - 21844781720002654560 T^{5} + \)\(10\!\cdots\!04\)\( T^{6} - \)\(36\!\cdots\!20\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \)
$37$ \( 1 - 301131416 T^{2} + 47822074973756092 T^{4} - \)\(50\!\cdots\!08\)\( T^{6} + \)\(40\!\cdots\!70\)\( T^{8} - \)\(24\!\cdots\!92\)\( T^{10} + \)\(11\!\cdots\!92\)\( T^{12} - \)\(33\!\cdots\!84\)\( T^{14} + \)\(53\!\cdots\!01\)\( T^{16} \)
$41$ \( ( 1 - 10792 T + 381689636 T^{2} - 2595640517112 T^{3} + 58822716652823334 T^{4} - \)\(30\!\cdots\!12\)\( T^{5} + \)\(51\!\cdots\!36\)\( T^{6} - \)\(16\!\cdots\!92\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \)
$43$ \( 1 - 542148344 T^{2} + 153567024284830372 T^{4} - \)\(32\!\cdots\!92\)\( T^{6} + \)\(53\!\cdots\!70\)\( T^{8} - \)\(69\!\cdots\!08\)\( T^{10} + \)\(71\!\cdots\!72\)\( T^{12} - \)\(54\!\cdots\!56\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \)
$47$ \( ( 1 - 23840 T + 811201148 T^{2} - 13600000745040 T^{3} + 275458842496788474 T^{4} - \)\(31\!\cdots\!80\)\( T^{5} + \)\(42\!\cdots\!52\)\( T^{6} - \)\(28\!\cdots\!20\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} )^{2} \)
$53$ \( 1 - 984292984 T^{2} + 650040420165524092 T^{4} - \)\(31\!\cdots\!92\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(55\!\cdots\!08\)\( T^{10} + \)\(19\!\cdots\!92\)\( T^{12} - \)\(52\!\cdots\!16\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} \)
$59$ \( 1 - 3042039688 T^{2} + 4949915655184301308 T^{4} - \)\(54\!\cdots\!56\)\( T^{6} + \)\(44\!\cdots\!70\)\( T^{8} - \)\(27\!\cdots\!56\)\( T^{10} + \)\(12\!\cdots\!08\)\( T^{12} - \)\(40\!\cdots\!88\)\( T^{14} + \)\(68\!\cdots\!01\)\( T^{16} \)
$61$ \( 1 - 2582224408 T^{2} + 4867570283316136828 T^{4} - \)\(60\!\cdots\!56\)\( T^{6} + \)\(59\!\cdots\!70\)\( T^{8} - \)\(42\!\cdots\!56\)\( T^{10} + \)\(24\!\cdots\!28\)\( T^{12} - \)\(93\!\cdots\!08\)\( T^{14} + \)\(25\!\cdots\!01\)\( T^{16} \)
$67$ \( 1 - 7537851704 T^{2} + 25204334234273950564 T^{4} - \)\(51\!\cdots\!16\)\( T^{6} + \)\(78\!\cdots\!94\)\( T^{8} - \)\(94\!\cdots\!84\)\( T^{10} + \)\(83\!\cdots\!64\)\( T^{12} - \)\(45\!\cdots\!96\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \)
$71$ \( ( 1 - 123360 T + 10006413404 T^{2} - 529309926298080 T^{3} + 24936343678058479206 T^{4} - \)\(95\!\cdots\!80\)\( T^{5} + \)\(32\!\cdots\!04\)\( T^{6} - \)\(72\!\cdots\!60\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \)
$73$ \( ( 1 - 23200 T + 7007019628 T^{2} - 115342151623520 T^{3} + 20469715216228479110 T^{4} - \)\(23\!\cdots\!60\)\( T^{5} + \)\(30\!\cdots\!72\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \)
$79$ \( ( 1 - 162880 T + 14181869596 T^{2} - 858599705367360 T^{3} + 48361267115636995206 T^{4} - \)\(26\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!96\)\( T^{6} - \)\(47\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \)
$83$ \( 1 - 29848826136 T^{2} + \)\(39\!\cdots\!24\)\( T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!54\)\( T^{8} - \)\(47\!\cdots\!36\)\( T^{10} + \)\(95\!\cdots\!24\)\( T^{12} - \)\(11\!\cdots\!64\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} \)
$89$ \( ( 1 + 39096 T + 18076226204 T^{2} + 489497603288904 T^{3} + \)\(13\!\cdots\!94\)\( T^{4} + \)\(27\!\cdots\!96\)\( T^{5} + \)\(56\!\cdots\!04\)\( T^{6} + \)\(68\!\cdots\!04\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \)
$97$ \( ( 1 - 12480 T + 14325999628 T^{2} - 824172284150080 T^{3} + 95392462085617046694 T^{4} - \)\(70\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(79\!\cdots\!40\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} )^{2} \)
show more
show less