Newform 65.2.o.a

• Label: 65.2.o.a
• Level: 65
• Weight: 2
• Character orbit: 65.o
• Analytic conductor: 0.519
• Analytic rank: 0
• Dimension: 20
• CM: No
• Inner twists: 2

Newspace parameters

Level:	\( N \)	=	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	65.o (of order \(12\) and degree \(4\))

Newform invariants

Self dual:	No
Analytic conductor:	\(0.519027613138\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(5\) over \(\Q(\zeta_{12})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( -\beta_{3} + \beta_{4} ) q^{2} + ( \beta_{2} + \beta_{4} + \beta_{8} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{3} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{18} ) q^{4} + ( -1 - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{16} - \beta_{17} + \beta_{18} ) q^{5} + ( -1 + \beta_{2} + 2 \beta_{5} - \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{16} - \beta_{17} ) q^{6} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{7} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{8} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20q - 4q^{2} - 2q^{3} - 6q^{4} - 6q^{5} - 8q^{6} - 6q^{7} + 12q^{8} - 12q^{9} + O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + 1263 x^{4} + 78 x^{2} + 1\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\((\)\( -20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + 68102 \nu^{6} + 46015 \nu^{4} + 16571 \nu^{2} + 2996 \nu - 409 \)\()/5992\)
\(\beta_{2}\)	\(=\)	\((\)\( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} - 68102 \nu^{6} - 46015 \nu^{4} - 16571 \nu^{2} + 2996 \nu + 409 \)\()/5992\)
\(\beta_{3}\)	\(=\)	\((\)\( -69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} - 257858 \nu^{6} - 105030 \nu^{4} - 19961 \nu^{2} - 539 \)\()/1712\)
\(\beta_{4}\)	\(=\)	\((\)\(321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + 869910 \nu^{13} - 548254 \nu^{12} + 3425391 \nu^{11} - 1503061 \nu^{10} + 7807469 \nu^{9} - 2268567 \nu^{8} + 9753906 \nu^{7} - 1805006 \nu^{6} + 5800042 \nu^{5} - 735210 \nu^{4} + 1183313 \nu^{3} - 139727 \nu^{2} + 39483 \nu - 3773\)\()/23968\)
\(\beta_{5}\)	\(=\)	\((\)\(700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + 912240 \nu^{13} + 1108682 \nu^{12} + 2625448 \nu^{11} + 3846937 \nu^{10} + 3934784 \nu^{9} + 7752327 \nu^{8} + 2192820 \nu^{7} + 8524592 \nu^{6} - 974624 \nu^{5} + 4331072 \nu^{4} - 1268316 \nu^{3} + 669321 \nu^{2} - 170688 \nu + 26961\)\()/11984\)
\(\beta_{6}\)	\(=\)	\((\)\( -409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} - 4719229 \nu^{7} - 2594086 \nu^{5} - 562582 \nu^{3} - 48473 \nu - 2996 \)\()/5992\)
\(\beta_{7}\)	\(=\)	\((\)\(1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + 2423698 \nu^{13} - 319614 \nu^{12} + 7689981 \nu^{11} - 870831 \nu^{10} + 13910023 \nu^{9} - 1440917 \nu^{8} + 13309334 \nu^{7} - 1613590 \nu^{6} + 5445918 \nu^{5} - 1272982 \nu^{4} + 481307 \nu^{3} - 501629 \nu^{2} + 71073 \nu - 30699\)\()/23968\)
\(\beta_{8}\)	\(=\)	\((\)\(700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + 912240 \nu^{13} - 1108682 \nu^{12} + 2625448 \nu^{11} - 3846937 \nu^{10} + 3934784 \nu^{9} - 7752327 \nu^{8} + 2192820 \nu^{7} - 8524592 \nu^{6} - 974624 \nu^{5} - 4331072 \nu^{4} - 1268316 \nu^{3} - 669321 \nu^{2} - 170688 \nu - 26961\)\()/11984\)
\(\beta_{9}\)	\(=\)	\((\)\(-419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} - 1064734 \nu^{13} + 7212374 \nu^{12} - 4108073 \nu^{11} + 23752795 \nu^{10} - 9110155 \nu^{9} + 45500681 \nu^{8} - 10820986 \nu^{7} + 47979462 \nu^{6} - 5625126 \nu^{5} + 23917926 \nu^{4} - 518719 \nu^{3} + 3775361 \nu^{2} + 159919 \nu + 83703\)\()/23968\)
\(\beta_{10}\)	\(=\)	\((\)\(-3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} - 5941306 \nu^{13} - 198098 \nu^{12} - 19648615 \nu^{11} + 414413 \nu^{10} - 37985157 \nu^{9} + 3110895 \nu^{8} - 40898326 \nu^{7} + 5871486 \nu^{6} - 21497042 \nu^{5} + 4145562 \nu^{4} - 4030089 \nu^{3} + 861543 \nu^{2} - 166551 \nu + 81509\)\()/23968\)
\(\beta_{11}\)	\(=\)	\((\)\(3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + 5941306 \nu^{13} + 869910 \nu^{12} + 19648615 \nu^{11} + 3425391 \nu^{10} + 37985157 \nu^{9} + 7807469 \nu^{8} + 40898326 \nu^{7} + 9753906 \nu^{6} + 21497042 \nu^{5} + 5800042 \nu^{4} + 4030089 \nu^{3} + 1183313 \nu^{2} + 154567 \nu + 39483\)\()/23968\)
\(\beta_{12}\)	\(=\)	\((\)\(3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + 5941306 \nu^{13} - 869910 \nu^{12} + 19648615 \nu^{11} - 3425391 \nu^{10} + 37985157 \nu^{9} - 7807469 \nu^{8} + 40898326 \nu^{7} - 9753906 \nu^{6} + 21497042 \nu^{5} - 5800042 \nu^{4} + 4030089 \nu^{3} - 1183313 \nu^{2} + 154567 \nu - 39483\)\()/23968\)
\(\beta_{13}\)	\(=\)	\((\)\(-2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} - 4279914 \nu^{13} - 1404094 \nu^{12} - 14010639 \nu^{11} - 4705845 \nu^{10} - 26598933 \nu^{9} - 9214959 \nu^{8} - 27637370 \nu^{7} - 9971998 \nu^{6} - 13384022 \nu^{5} - 5100714 \nu^{4} - 1923401 \nu^{3} - 798859 \nu^{2} - 7127 \nu - 15221\)\()/11984\)
\(\beta_{14}\)	\(=\)	\((\)\(2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + 4279914 \nu^{13} - 1404094 \nu^{12} + 14010639 \nu^{11} - 4705845 \nu^{10} + 26598933 \nu^{9} - 9214959 \nu^{8} + 27637370 \nu^{7} - 9971998 \nu^{6} + 13384022 \nu^{5} - 5100714 \nu^{4} + 1923401 \nu^{3} - 798859 \nu^{2} + 7127 \nu - 15221\)\()/11984\)
\(\beta_{15}\)	\(=\)	\((\)\(-6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} - 11139534 \nu^{13} + 3217222 \nu^{12} - 37111487 \nu^{11} + 10951069 \nu^{10} - 72399061 \nu^{9} + 21535831 \nu^{8} - 78924566 \nu^{7} + 22849098 \nu^{6} - 42433790 \nu^{5} + 10804742 \nu^{4} - 8585045 \nu^{3} + 1254563 \nu^{2} - 504763 \nu + 1285\)\()/23968\)
\(\beta_{16}\)	\(=\)	\((\)\(8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + 13551722 \nu^{13} + 273770 \nu^{12} + 45488043 \nu^{11} + 1753171 \nu^{10} + 89206401 \nu^{9} + 5197801 \nu^{8} + 97040830 \nu^{7} + 7462350 \nu^{6} + 50921018 \nu^{5} + 4439554 \nu^{4} + 9271077 \nu^{3} + 591213 \nu^{2} + 419203 \nu + 23947\)\()/23968\)
\(\beta_{17}\)	\(=\)	\((\)\(-4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} - 53105203 \nu^{9} - 58320582 \nu^{7} - 31671424 \nu^{5} - 6571905 \nu^{3} + 5992 \nu^{2} - 381235 \nu + 11984\)\()/11984\)
\(\beta_{18}\)	\(=\)	\((\)\(10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + 16210930 \nu^{13} - 548254 \nu^{12} + 53334711 \nu^{11} - 1503061 \nu^{10} + 102576749 \nu^{9} - 2268567 \nu^{8} + 110068986 \nu^{7} - 1805006 \nu^{6} + 58130790 \nu^{5} - 735210 \nu^{4} + 11443805 \nu^{3} - 127743 \nu^{2} + 679767 \nu + 20195\)\()/23968\)
\(\beta_{19}\)	\(=\)	\((\)\(-16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} - 110586 \nu^{14} - 26151990 \nu^{13} - 548254 \nu^{12} - 87742737 \nu^{11} - 1503061 \nu^{10} - 172338195 \nu^{9} - 2268567 \nu^{8} - 188614114 \nu^{7} - 1805006 \nu^{6} - 100694134 \nu^{5} - 735210 \nu^{4} - 19274543 \nu^{3} - 139727 \nu^{2} - 820129 \nu - 15757\)\()/23968\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(-\beta_{11} - \beta_{12}\)	\(-\beta_{11}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2.1

		2.64975i
		1.83163i
		0.493902i
		0.274809i
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		2.08794i
		1.58474i
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0.658935i

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5.39885i

2.2

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1.58624i

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0.512942i

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1.17336i

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1.69810i

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3.50634i

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1.76945i

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0.463873i

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3.86282i

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1.52079i

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0.448637i

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1.83775i

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1.05783i

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0.878433i

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0.611610i

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1.66665i

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1.45395i

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0.193052i

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1.29207i

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0.759023

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1.31467i

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2.15400i

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1.29744i

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1.27123i

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2.42404i

32.1

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1.80821i

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2.66159i

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2.04338i

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2.22760i

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1.48906i

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1.45563i

0.750585

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2.29634i

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1.97411i

32.2

−0.792369

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1.37242i

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0.190588i

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0.442881i

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2.23506i

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0.0809291i

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0.274164i

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2.56436

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1.48053i

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1.67868i

32.3

0.0656513

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0.113711i

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0.332179i

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1.71712i

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2.08297i

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0.0116869i

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1.39069i

0.522947

2.49566

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1.44087i

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0.229216i

32.4

0.511309

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0.885613i

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2.69193i

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0.826407i

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1.45744i

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0.737614i

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0.481787i

3.02107

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2.38337i

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2.24706i

32.5

1.12540

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1.94926i

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1.91913i

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2.65535i

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0.247944i

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0.473740i

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4.05275i

33.1

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0.335680i

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4.34849i

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1.81654i

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3.31955i

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0.0561740i

8.00544

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0.658935i

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5.39885i

33.2

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1.58624i

1.91432

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0.512942i

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1.17336i

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1.69810i

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3.50634i

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1.76945i

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0.463873i

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3.86282i

33.3

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0.427732i

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0.243392i

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1.52079i

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0.284413i

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0.448637i

3.18307

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1.83775i

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1.05783i

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0.878433i

33.4

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0.237991i

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0.611610i

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1.66665i

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1.45395i

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0.627267i

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0.193052i

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2.23793

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1.29207i

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0.204528i

33.5

0.759023

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1.31467i

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0.175069i

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0.263675i

0.600231

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2.15400i

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0.991842i

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1.29744i

2.57390

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1.27123i

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2.42404i

63.1

−1.04397

−

1.80821i

0.713171

−

2.66159i

−1.17974

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2.04338i

−0.194361

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2.22760i

−5.55724

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1.48906i

2.52122

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1.45563i

0.750585

−3.97738

−

2.29634i

4.23088

−

1.97411i

63.2

−0.792369

−

1.37242i

−0.0510678

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0.190588i

−0.255697

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0.442881i

0.0672627

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2.23506i

0.302032

−

0.0809291i

−0.474866

−

0.274164i

−2.35905

2.56436

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1.48053i

−3.12074

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1.67868i

63.3

0.0656513

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0.113711i

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0.332179i

0.991380

−

1.71712i

0.813169

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2.08297i

−0.0436159

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0.0116869i

−2.40874

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1.39069i

0.522947

2.49566

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1.44087i

−0.183472

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0.229216i

63.4

0.511309

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0.885613i

−0.721300

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2.69193i

0.477126

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0.826407i

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1.45744i

−2.75281

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0.737614i

0.834479

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0.481787i

3.02107

−4.12812

−

2.38337i

0.423625

−

2.24706i

63.5

1.12540

+

1.94926i

0.514229

−

1.91913i

−1.53307

+

2.65535i

−2.22228

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0.247944i

4.31958

−

1.15743i

−1.10607

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0.638592i

−2.39966

−0.820542

−

0.473740i

−2.98427

−

4.05275i

Inner twists

Char. orbit	Parity	Mult.	Self Twist	Proved
1.a	Even	1	trivial	yes
65.o	Even	1		yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(65, [\chi])\).