Properties

Label 441.7.d.d
Level $441$
Weight $7$
Character orbit 441.d
Analytic conductor $101.454$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 441.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(101.453850876\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{3} ) q^{2} + ( 43 - \beta_{5} ) q^{4} + ( 3 \beta_{1} + \beta_{4} + \beta_{6} ) q^{5} + ( 61 - 16 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{8} +O(q^{10})\) \( q + ( -1 - \beta_{3} ) q^{2} + ( 43 - \beta_{5} ) q^{4} + ( 3 \beta_{1} + \beta_{4} + \beta_{6} ) q^{5} + ( 61 - 16 \beta_{3} - 3 \beta_{5} + 2 \beta_{7} ) q^{8} + ( -9 \beta_{1} + 34 \beta_{2} - 14 \beta_{4} - 3 \beta_{6} ) q^{10} + ( -265 + \beta_{3} + 9 \beta_{5} + \beta_{7} ) q^{11} + ( -864 \beta_{1} + 14 \beta_{2} + 17 \beta_{4} - 5 \beta_{6} ) q^{13} + ( -933 - 194 \beta_{3} + 5 \beta_{5} - 10 \beta_{7} ) q^{16} + ( 626 \beta_{1} + 70 \beta_{2} - 54 \beta_{4} + 24 \beta_{6} ) q^{17} + ( -1954 \beta_{1} + 360 \beta_{2} - 27 \beta_{4} + \beta_{6} ) q^{19} + ( 3945 \beta_{1} - 24 \beta_{2} + 66 \beta_{4} + 45 \beta_{6} ) q^{20} + ( -169 + 596 \beta_{3} + 11 \beta_{5} - 26 \beta_{7} ) q^{22} + ( -4144 + 1306 \beta_{3} - 54 \beta_{5} + 4 \beta_{7} ) q^{23} + ( -5198 - 1399 \beta_{3} + 81 \beta_{5} + \beta_{7} ) q^{25} + ( 1366 \beta_{1} - 1093 \beta_{2} - 106 \beta_{4} - 103 \beta_{6} ) q^{26} + ( -3949 - 405 \beta_{3} + 183 \beta_{5} - 61 \beta_{7} ) q^{29} + ( -3401 \beta_{1} + 122 \beta_{2} + 44 \beta_{4} + 282 \beta_{6} ) q^{31} + ( 17073 + 1978 \beta_{3} + 183 \beta_{5} - 58 \beta_{7} ) q^{32} + ( 10238 \beta_{1} + 1752 \beta_{2} + 288 \beta_{4} + 466 \beta_{6} ) q^{34} + ( 11970 - 2827 \beta_{3} - 475 \beta_{5} - 71 \beta_{7} ) q^{37} + ( 41396 \beta_{1} - 1451 \beta_{2} + 210 \beta_{4} + 525 \beta_{6} ) q^{38} + ( -7149 \beta_{1} + 3056 \beta_{2} + 98 \beta_{4} - 93 \beta_{6} ) q^{40} + ( 6362 \beta_{1} + 340 \beta_{2} + 256 \beta_{4} + 436 \beta_{6} ) q^{41} + ( -57678 + 6317 \beta_{3} - 1051 \beta_{5} - 257 \beta_{7} ) q^{43} + ( -47319 + 1186 \beta_{3} + 495 \beta_{5} + 122 \beta_{7} ) q^{44} + ( -132004 + 3440 \beta_{3} + 1076 \beta_{5} + 76 \beta_{7} ) q^{46} + ( -6688 \beta_{1} + 1702 \beta_{2} + 770 \beta_{4} + 56 \beta_{6} ) q^{47} + ( 150284 + 6793 \beta_{3} - 1173 \beta_{5} - 170 \beta_{7} ) q^{50} + ( -61172 \beta_{1} - 3804 \beta_{2} + 378 \beta_{4} - 446 \beta_{6} ) q^{52} + ( -3055 + 1947 \beta_{3} + 2463 \beta_{5} + 149 \beta_{7} ) q^{53} + ( -44475 \beta_{1} + 1474 \beta_{2} - 371 \beta_{4} - 345 \beta_{6} ) q^{55} + ( 37607 + 10498 \beta_{3} + 1181 \beta_{5} + 122 \beta_{7} ) q^{58} + ( 151147 \beta_{1} + 1460 \beta_{2} + 703 \beta_{4} - 377 \beta_{6} ) q^{59} + ( -33528 \beta_{1} - 2144 \beta_{2} + 964 \beta_{4} + 52 \beta_{6} ) q^{61} + ( 25589 \beta_{1} + 6415 \beta_{2} - 2044 \beta_{4} + 704 \beta_{6} ) q^{62} + ( -176205 + 4266 \beta_{3} + 3193 \beta_{5} + 738 \beta_{7} ) q^{64} + ( -93648 - 19168 \beta_{3} - 1584 \beta_{5} - 1586 \beta_{7} ) q^{65} + ( -101116 + 18853 \beta_{3} - 1851 \beta_{5} + 167 \beta_{7} ) q^{67} + ( 140802 \beta_{1} + 22668 \beta_{2} - 1644 \beta_{4} - 114 \beta_{6} ) q^{68} + ( -28372 + 5388 \beta_{3} + 1332 \beta_{5} + 1706 \beta_{7} ) q^{71} + ( -77984 \beta_{1} - 10218 \beta_{2} - 195 \beta_{4} + 887 \beta_{6} ) q^{73} + ( 304420 - 32159 \beta_{3} - 3045 \beta_{5} + 1518 \beta_{7} ) q^{74} + ( -58806 \beta_{1} + 34440 \beta_{2} - 3102 \beta_{4} - 1200 \beta_{6} ) q^{76} + ( 127061 + 51202 \beta_{3} + 822 \beta_{5} - 484 \beta_{7} ) q^{79} + ( 70377 \beta_{1} - 6204 \beta_{2} - 4450 \beta_{4} - 691 \beta_{6} ) q^{80} + ( 35342 \beta_{1} + 20938 \beta_{2} - 4664 \beta_{4} + 112 \beta_{6} ) q^{82} + ( -73913 \beta_{1} + 25496 \beta_{2} + 1625 \beta_{4} - 4843 \beta_{6} ) q^{83} + ( 225666 + 75398 \beta_{3} - 1962 \beta_{5} + 1618 \beta_{7} ) q^{85} + ( -578108 + 25879 \beta_{3} + 7533 \beta_{5} + 4158 \beta_{7} ) q^{86} + ( -83477 + 28310 \beta_{3} - 107 \beta_{5} - 302 \beta_{7} ) q^{88} + ( 148486 \beta_{1} + 3980 \beta_{2} - 78 \beta_{4} + 126 \beta_{6} ) q^{89} + ( -8028 + 91444 \beta_{3} + 8832 \beta_{5} - 3016 \beta_{7} ) q^{92} + ( 153920 \beta_{1} - 6106 \beta_{2} - 6496 \beta_{4} - 2750 \beta_{6} ) q^{94} + ( 265014 + 78386 \beta_{3} - 9798 \beta_{5} - 5936 \beta_{7} ) q^{95} + ( -135633 \beta_{1} + 3574 \beta_{2} + 6931 \beta_{4} - 1627 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 10q^{2} + 346q^{4} + 454q^{8} + O(q^{10}) \) \( 8q - 10q^{2} + 346q^{4} + 454q^{8} - 2140q^{11} - 7822q^{16} - 78q^{22} - 30448q^{23} - 44548q^{25} - 32524q^{29} + 140406q^{32} + 91340q^{37} - 445660q^{43} - 377658q^{44} - 1051608q^{46} + 1218884q^{50} - 26068q^{53} + 319002q^{58} - 1410446q^{64} - 778008q^{65} - 768188q^{67} - 225688q^{71} + 2371060q^{74} + 1119184q^{79} + 1953576q^{85} - 4604804q^{86} - 609774q^{88} + 113064q^{92} + 2320224q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1585013359 \nu^{7} - 18232571539 \nu^{6} + 303603349712 \nu^{5} - 4245938445433 \nu^{4} + 65749585575908 \nu^{3} - 780366646056751 \nu^{2} + 2109023702500351 \nu - 10280963027029950\)\()/ 11901315886480818 \)
\(\beta_{2}\)\(=\)\((\)\(8019055 \nu^{7} + 15616321 \nu^{6} + 1444380025 \nu^{5} - 2053828205 \nu^{4} + 305021724904 \nu^{3} + 177516832405 \nu^{2} + 24523492689402 \nu + 29614389599556\)\()/ 11465622241311 \)
\(\beta_{3}\)\(=\)\((\)\(-8019055 \nu^{7} - 15616321 \nu^{6} - 1444380025 \nu^{5} + 2053828205 \nu^{4} - 305021724904 \nu^{3} - 177516832405 \nu^{2} - 1592248206780 \nu - 29614389599556\)\()/ 11465622241311 \)
\(\beta_{4}\)\(=\)\((\)\(27322708193 \nu^{7} + 1000103862583 \nu^{6} + 22563530233273 \nu^{5} + 219959885092027 \nu^{4} + 2723907868150765 \nu^{3} + 19609334703803473 \nu^{2} + 301119122619702170 \nu + 699314583065439780\)\()/ 11901315886480818 \)
\(\beta_{5}\)\(=\)\((\)\(39673486 \nu^{7} - 224426993 \nu^{6} + 7145928130 \nu^{5} - 10161113066 \nu^{4} + 1531993215028 \nu^{3} + 878247070906 \nu^{2} + 7877491417656 \nu + 963541759742958\)\()/ 11465622241311 \)
\(\beta_{6}\)\(=\)\((\)\(-80062726973 \nu^{7} + 817429199804 \nu^{6} - 15380773767166 \nu^{5} + 224024867280275 \nu^{4} - 3322848657824296 \nu^{3} + 29545401640254419 \nu^{2} - 134797977236992055 \nu + 352719998947020942\)\()/ 5950657943240409 \)
\(\beta_{7}\)\(=\)\((\)\(-313586723 \nu^{7} - 1214053715 \nu^{6} - 56482764965 \nu^{5} + 80315355913 \nu^{4} - 8060204009803 \nu^{3} - 6941830646033 \nu^{2} - 62265179296908 \nu - 487133839162452\)\()/ 7643748160874 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 106 \beta_{1} - 106\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} - 6 \beta_{5} - 147 \beta_{3} + 252\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{7} + 205 \beta_{6} + 205 \beta_{5} + 6 \beta_{4} + 774 \beta_{3} + 776 \beta_{2} + 15886 \beta_{1} - 15886\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-424 \beta_{7} - 1560 \beta_{6} + 1560 \beta_{5} + 1272 \beta_{4} + 25013 \beta_{3} - 24589 \beta_{2} - 90180 \beta_{1} - 90180\)\()/2\)
\(\nu^{6}\)\(=\)\(152 \beta_{7} - 38461 \beta_{5} - 199198 \beta_{3} + 2727346\)
\(\nu^{7}\)\(=\)\((\)\(-75858 \beta_{7} + 355626 \beta_{6} + 355626 \beta_{5} - 227574 \beta_{4} + 4549255 \beta_{3} + 4473397 \beta_{2} + 22658292 \beta_{1} - 22658292\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
5.73828 + 9.93899i
5.73828 9.93899i
4.15432 7.19549i
4.15432 + 7.19549i
−2.30325 + 3.98935i
−2.30325 3.98935i
−7.08935 12.2791i
−7.08935 + 12.2791i
−12.4766 0 91.6646 202.496i 0 0 −345.159 0 2526.46i
244.2 −12.4766 0 91.6646 202.496i 0 0 −345.159 0 2526.46i
244.3 −9.30863 0 22.6506 174.756i 0 0 384.906 0 1626.74i
244.4 −9.30863 0 22.6506 174.756i 0 0 384.906 0 1626.74i
244.5 3.60650 0 −50.9932 83.0589i 0 0 −414.723 0 299.552i
244.6 3.60650 0 −50.9932 83.0589i 0 0 −414.723 0 299.552i
244.7 13.1787 0 109.678 79.5668i 0 0 601.976 0 1048.59i
244.8 13.1787 0 109.678 79.5668i 0 0 601.976 0 1048.59i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.7.d.d 8
3.b odd 2 1 147.7.d.a 8
7.b odd 2 1 inner 441.7.d.d 8
7.c even 3 1 63.7.m.c 8
7.d odd 6 1 63.7.m.c 8
21.c even 2 1 147.7.d.a 8
21.g even 6 1 21.7.f.b 8
21.g even 6 1 147.7.f.a 8
21.h odd 6 1 21.7.f.b 8
21.h odd 6 1 147.7.f.a 8
84.j odd 6 1 336.7.bh.b 8
84.n even 6 1 336.7.bh.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b 8 21.g even 6 1
21.7.f.b 8 21.h odd 6 1
63.7.m.c 8 7.c even 3 1
63.7.m.c 8 7.d odd 6 1
147.7.d.a 8 3.b odd 2 1
147.7.d.a 8 21.c even 2 1
147.7.f.a 8 21.g even 6 1
147.7.f.a 8 21.h odd 6 1
336.7.bh.b 8 84.j odd 6 1
336.7.bh.b 8 84.n even 6 1
441.7.d.d 8 1.a even 1 1 trivial
441.7.d.d 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5 T_{2}^{3} - 202 T_{2}^{2} - 914 T_{2} + 5520 \) acting on \(S_{7}^{\mathrm{new}}(441, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 5520 - 914 T - 202 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( 54693259182810000 + 19691805407400 T^{2} + 2242450449 T^{4} + 84774 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 59090060580 - 312885260 T - 264211 T^{2} + 1070 T^{3} + T^{4} )^{2} \)
$13$ \( \)\(34\!\cdots\!00\)\( + 53257929335845404768 T^{2} + 105787465152777 T^{4} + 22315686 T^{6} + T^{8} \)
$17$ \( \)\(22\!\cdots\!00\)\( + \)\(26\!\cdots\!28\)\( T^{2} + 10270113626486736 T^{4} + 169210728 T^{6} + T^{8} \)
$19$ \( \)\(12\!\cdots\!96\)\( + \)\(27\!\cdots\!48\)\( T^{2} + 14949101230916409 T^{4} + 256355982 T^{6} + T^{8} \)
$23$ \( ( 25130113528867200 - 2827786801280 T - 288968680 T^{2} + 15224 T^{3} + T^{4} )^{2} \)
$29$ \( ( -39242852020022400 - 9430137809600 T - 442580539 T^{2} + 16262 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(42\!\cdots\!21\)\( + \)\(24\!\cdots\!12\)\( T^{2} + 5004381354236358894 T^{4} + 4028119908 T^{6} + T^{8} \)
$37$ \( ( 503660415922686500 + 23949968583100 T - 3089311755 T^{2} - 45670 T^{3} + T^{4} )^{2} \)
$41$ \( \)\(53\!\cdots\!16\)\( + \)\(87\!\cdots\!08\)\( T^{2} + 50479416076831845984 T^{4} + 11994986352 T^{6} + T^{8} \)
$43$ \( ( -\)\(14\!\cdots\!84\)\( - 2845100102192540 T - 1577056827 T^{2} + 222830 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(30\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{4} + 27041579112 T^{6} + T^{8} \)
$53$ \( ( \)\(48\!\cdots\!00\)\( - 802901081989208 T - 50576547643 T^{2} + 13034 T^{3} + T^{4} )^{2} \)
$59$ \( \)\(13\!\cdots\!36\)\( + \)\(11\!\cdots\!60\)\( T^{2} + \)\(29\!\cdots\!97\)\( T^{4} + 302515596390 T^{6} + T^{8} \)
$61$ \( \)\(78\!\cdots\!00\)\( + \)\(58\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!96\)\( T^{4} + 58965042912 T^{6} + T^{8} \)
$67$ \( ( \)\(96\!\cdots\!80\)\( - 14714428944980048 T - 44897227815 T^{2} + 384094 T^{3} + T^{4} )^{2} \)
$71$ \( ( -\)\(25\!\cdots\!12\)\( - 44439430271275520 T - 187952640388 T^{2} + 112844 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(12\!\cdots\!24\)\( + \)\(15\!\cdots\!36\)\( T^{2} + \)\(13\!\cdots\!25\)\( T^{4} + 231622808886 T^{6} + T^{8} \)
$79$ \( ( -\)\(76\!\cdots\!35\)\( + 247069108400054368 T - 466345189254 T^{2} - 559592 T^{3} + T^{4} )^{2} \)
$83$ \( \)\(36\!\cdots\!24\)\( + \)\(24\!\cdots\!96\)\( T^{2} + \)\(12\!\cdots\!25\)\( T^{4} + 2038066317246 T^{6} + T^{8} \)
$89$ \( \)\(13\!\cdots\!44\)\( + \)\(10\!\cdots\!04\)\( T^{2} + \)\(26\!\cdots\!60\)\( T^{4} + 282963720024 T^{6} + T^{8} \)
$97$ \( \)\(22\!\cdots\!00\)\( + \)\(33\!\cdots\!12\)\( T^{2} + \)\(14\!\cdots\!01\)\( T^{4} + 2348711138742 T^{6} + T^{8} \)
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