Properties

Label 668.2.a.c
Level $668$
Weight $2$
Character orbit 668.a
Self dual yes
Analytic conductor $5.334$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 668 = 2^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 668.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.33400685502\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 11 x^{5} - 7 x^{4} + 21 x^{3} + 17 x^{2} - 4 x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{3} ) q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{3} ) q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -2 - \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{9} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{11} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{13} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{15} + ( -2 \beta_{1} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{17} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} ) q^{19} + ( -1 + 3 \beta_{1} + \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{21} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{23} + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{25} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{27} + ( -2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{29} + ( -4 + 2 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{31} + ( -1 - \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{33} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{35} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} ) q^{37} + ( -2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{39} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{41} + ( -4 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{43} + ( -2 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{45} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{47} + ( 3 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{49} + ( 5 \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{51} + ( -2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{55} + ( 3 + 3 \beta_{1} + \beta_{2} + 3 \beta_{5} + \beta_{6} ) q^{57} + ( -2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{6} ) q^{59} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{61} + ( -2 - 3 \beta_{1} - \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 5 \beta_{5} - \beta_{6} ) q^{63} + ( -3 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{65} + ( -6 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} ) q^{67} + ( 4 - \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{69} + ( 4 - \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{71} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{73} + ( 5 - 3 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{75} + ( -5 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - \beta_{5} - 5 \beta_{6} ) q^{77} + ( \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} ) q^{79} + ( 6 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{81} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{83} + ( -4 + 2 \beta_{2} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{87} + ( 3 - 3 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{89} + ( \beta_{1} - 7 \beta_{2} + \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{91} + ( -1 - 8 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - 8 \beta_{5} - 5 \beta_{6} ) q^{93} + ( -2 - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{95} + ( -1 - \beta_{1} + 4 \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{97} + ( -1 + 7 \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 4q^{3} - 2q^{5} - 12q^{7} + 7q^{9} + O(q^{10}) \) \( 7q - 4q^{3} - 2q^{5} - 12q^{7} + 7q^{9} - 7q^{11} - 9q^{13} - 17q^{15} - q^{17} - 11q^{19} - 4q^{21} - 19q^{23} + 3q^{25} - 16q^{27} - 5q^{29} - 13q^{31} - 8q^{33} - 7q^{35} - 26q^{37} - 17q^{39} - 2q^{41} - 24q^{43} - 7q^{45} - 11q^{47} + 19q^{49} + 8q^{51} + 4q^{53} - 4q^{55} + 14q^{57} - 4q^{59} - 5q^{61} - 21q^{63} + 13q^{65} - 42q^{67} + 24q^{69} + 9q^{71} + 27q^{73} + 25q^{75} + 12q^{77} - 8q^{79} + 35q^{81} + 16q^{83} - 27q^{85} + 3q^{87} + 9q^{89} - 2q^{91} - 10q^{93} + 10q^{95} - 8q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 11 x^{5} - 7 x^{4} + 21 x^{3} + 17 x^{2} - 4 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 7 \nu^{6} - 6 \nu^{5} - 73 \nu^{4} + 13 \nu^{3} + 149 \nu^{2} + 5 \nu - 50 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( 9 \nu^{6} - 6 \nu^{5} - 95 \nu^{4} - \nu^{3} + 191 \nu^{2} + 35 \nu - 58 \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -13 \nu^{6} + 10 \nu^{5} + 135 \nu^{4} - 11 \nu^{3} - 263 \nu^{2} - 31 \nu + 70 \)\()/4\)
\(\beta_{5}\)\(=\)\( -6 \nu^{6} + 4 \nu^{5} + 63 \nu^{4} - 123 \nu^{2} - 18 \nu + 33 \)
\(\beta_{6}\)\(=\)\((\)\( 27 \nu^{6} - 18 \nu^{5} - 285 \nu^{4} + \nu^{3} + 565 \nu^{2} + 81 \nu - 154 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-11 \beta_{6} - 12 \beta_{5} + 9 \beta_{4} + 7 \beta_{3} + 9 \beta_{2} + 4 \beta_{1} + 29\)
\(\nu^{5}\)\(=\)\(-18 \beta_{6} - 29 \beta_{5} + 27 \beta_{4} - 3 \beta_{3} + 24 \beta_{2} + 48 \beta_{1} + 48\)
\(\nu^{6}\)\(=\)\(-107 \beta_{6} - 125 \beta_{5} + 92 \beta_{4} + 51 \beta_{3} + 90 \beta_{2} + 71 \beta_{1} + 260\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.38961
1.47217
0.495342
3.27771
−1.47685
−0.721798
−0.656969
0 −3.06430 0 −0.836956 0 1.84506 0 6.38996 0
1.2 0 −3.04732 0 1.35854 0 −3.73540 0 6.28618 0
1.3 0 −1.07866 0 4.03761 0 −3.68648 0 −1.83648 0
1.4 0 −0.685166 0 0.610181 0 −0.184306 0 −2.53055 0
1.5 0 0.568565 0 −1.35423 0 −2.86706 0 −2.67673 0
1.6 0 0.678805 0 −2.77086 0 1.71364 0 −2.53922 0
1.7 0 2.62809 0 −3.04428 0 −5.08545 0 3.90685 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(167\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 668.2.a.c 7
3.b odd 2 1 6012.2.a.g 7
4.b odd 2 1 2672.2.a.k 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.a.c 7 1.a even 1 1 trivial
2672.2.a.k 7 4.b odd 2 1
6012.2.a.g 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 4 T_{3}^{6} - 6 T_{3}^{5} - 32 T_{3}^{4} - 6 T_{3}^{3} + 29 T_{3}^{2} + 4 T_{3} - 7 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(668))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4 T + 15 T^{2} + 40 T^{3} + 93 T^{4} + 185 T^{5} + 355 T^{6} + 599 T^{7} + 1065 T^{8} + 1665 T^{9} + 2511 T^{10} + 3240 T^{11} + 3645 T^{12} + 2916 T^{13} + 2187 T^{14} \)
$5$ \( 1 + 2 T + 18 T^{2} + 18 T^{3} + 128 T^{4} - 2 T^{5} + 545 T^{6} - 452 T^{7} + 2725 T^{8} - 50 T^{9} + 16000 T^{10} + 11250 T^{11} + 56250 T^{12} + 31250 T^{13} + 78125 T^{14} \)
$7$ \( 1 + 12 T + 87 T^{2} + 466 T^{3} + 2053 T^{4} + 7675 T^{5} + 24829 T^{6} + 70131 T^{7} + 173803 T^{8} + 376075 T^{9} + 704179 T^{10} + 1118866 T^{11} + 1462209 T^{12} + 1411788 T^{13} + 823543 T^{14} \)
$11$ \( 1 + 7 T + 49 T^{2} + 211 T^{3} + 1044 T^{4} + 3820 T^{5} + 15876 T^{6} + 49868 T^{7} + 174636 T^{8} + 462220 T^{9} + 1389564 T^{10} + 3089251 T^{11} + 7891499 T^{12} + 12400927 T^{13} + 19487171 T^{14} \)
$13$ \( 1 + 9 T + 68 T^{2} + 356 T^{3} + 1658 T^{4} + 6631 T^{5} + 26213 T^{6} + 92808 T^{7} + 340769 T^{8} + 1120639 T^{9} + 3642626 T^{10} + 10167716 T^{11} + 25247924 T^{12} + 43441281 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + T + 66 T^{2} + 26 T^{3} + 2212 T^{4} + 199 T^{5} + 51705 T^{6} + 1532 T^{7} + 878985 T^{8} + 57511 T^{9} + 10867556 T^{10} + 2171546 T^{11} + 93710562 T^{12} + 24137569 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 11 T + 123 T^{2} + 943 T^{3} + 6780 T^{4} + 38558 T^{5} + 207830 T^{6} + 937252 T^{7} + 3948770 T^{8} + 13919438 T^{9} + 46504020 T^{10} + 122892703 T^{11} + 304560177 T^{12} + 517504691 T^{13} + 893871739 T^{14} \)
$23$ \( 1 + 19 T + 232 T^{2} + 2074 T^{3} + 15558 T^{4} + 100741 T^{5} + 582135 T^{6} + 2958428 T^{7} + 13389105 T^{8} + 53291989 T^{9} + 189294186 T^{10} + 580390234 T^{11} + 1493231576 T^{12} + 2812681891 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 + 5 T + 162 T^{2} + 702 T^{3} + 12203 T^{4} + 45007 T^{5} + 550271 T^{6} + 1671295 T^{7} + 15957859 T^{8} + 37850887 T^{9} + 297618967 T^{10} + 496511262 T^{11} + 3322806138 T^{12} + 2974116605 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 + 13 T + 95 T^{2} + 305 T^{3} + 648 T^{4} - 560 T^{5} + 20132 T^{6} + 105568 T^{7} + 624092 T^{8} - 538160 T^{9} + 19304568 T^{10} + 281673905 T^{11} + 2719769345 T^{12} + 11537547853 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 + 26 T + 404 T^{2} + 4432 T^{3} + 39662 T^{4} + 308446 T^{5} + 2171097 T^{6} + 13871632 T^{7} + 80330589 T^{8} + 422262574 T^{9} + 2008999286 T^{10} + 8306281552 T^{11} + 28014958628 T^{12} + 66708886634 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 + 2 T + 146 T^{2} + 266 T^{3} + 11268 T^{4} + 21566 T^{5} + 619985 T^{6} + 1112908 T^{7} + 25419385 T^{8} + 36252446 T^{9} + 776601828 T^{10} + 751652426 T^{11} + 16915005346 T^{12} + 9500208482 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 24 T + 514 T^{2} + 7062 T^{3} + 86260 T^{4} + 816360 T^{5} + 6931999 T^{6} + 47907988 T^{7} + 298075957 T^{8} + 1509449640 T^{9} + 6858273820 T^{10} + 24143572662 T^{11} + 75562339702 T^{12} + 151712713176 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + 11 T + 154 T^{2} + 1124 T^{3} + 10727 T^{4} + 58321 T^{5} + 473997 T^{6} + 2317571 T^{7} + 22277859 T^{8} + 128831089 T^{9} + 1113709321 T^{10} + 5484761444 T^{11} + 35319131078 T^{12} + 118571368619 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 - 4 T + 207 T^{2} - 920 T^{3} + 21673 T^{4} - 86044 T^{5} + 1568415 T^{6} - 5154512 T^{7} + 83125995 T^{8} - 241697596 T^{9} + 3226611221 T^{10} - 7259242520 T^{11} + 86566467051 T^{12} - 88657444516 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 + 4 T + 321 T^{2} + 944 T^{3} + 46921 T^{4} + 103612 T^{5} + 4159569 T^{6} + 7292384 T^{7} + 245414571 T^{8} + 360673372 T^{9} + 9636588059 T^{10} + 11438788784 T^{11} + 229490699979 T^{12} + 168722134564 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 + 5 T + 210 T^{2} + 398 T^{3} + 22363 T^{4} + 26651 T^{5} + 1945003 T^{6} + 2595919 T^{7} + 118645183 T^{8} + 99168371 T^{9} + 5075976103 T^{10} + 5510644718 T^{11} + 177365223210 T^{12} + 257601871805 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 + 42 T + 1060 T^{2} + 19258 T^{3} + 278042 T^{4} + 3324230 T^{5} + 33867347 T^{6} + 297649804 T^{7} + 2269112249 T^{8} + 14922468470 T^{9} + 83624746046 T^{10} + 388070288218 T^{11} + 1431132613420 T^{12} + 3799252051098 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 9 T + 320 T^{2} - 2246 T^{3} + 49410 T^{4} - 293863 T^{5} + 5051659 T^{6} - 25462260 T^{7} + 358667789 T^{8} - 1481363383 T^{9} + 17684382510 T^{10} - 57074635526 T^{11} + 577353392320 T^{12} - 1152902555289 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 - 27 T + 614 T^{2} - 10216 T^{3} + 143660 T^{4} - 1707773 T^{5} + 17858693 T^{6} - 161388576 T^{7} + 1303684589 T^{8} - 9100722317 T^{9} + 55886182220 T^{10} - 290116430056 T^{11} + 1272865958102 T^{12} - 4086024109803 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 + 8 T + 314 T^{2} + 2854 T^{3} + 52244 T^{4} + 477640 T^{5} + 5822187 T^{6} + 47351668 T^{7} + 459952773 T^{8} + 2980951240 T^{9} + 25758329516 T^{10} + 111163531174 T^{11} + 966195709286 T^{12} + 1944699644168 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - 16 T + 238 T^{2} - 1826 T^{3} + 19504 T^{4} - 186096 T^{5} + 2712391 T^{6} - 24177788 T^{7} + 225128453 T^{8} - 1282015344 T^{9} + 11152133648 T^{10} - 86658894146 T^{11} + 937491673034 T^{12} - 5231045973904 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 - 9 T + 261 T^{2} - 1817 T^{3} + 33900 T^{4} - 261500 T^{5} + 4225720 T^{6} - 30821292 T^{7} + 376089080 T^{8} - 2071341500 T^{9} + 23898449100 T^{10} - 114002651897 T^{11} + 1457439516189 T^{12} - 4472831618649 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 + 8 T + 417 T^{2} + 3412 T^{3} + 94457 T^{4} + 671935 T^{5} + 13479577 T^{6} + 82425871 T^{7} + 1307518969 T^{8} + 6322236415 T^{9} + 86208353561 T^{10} + 302061906772 T^{11} + 3580920887169 T^{12} + 6663776039432 T^{13} + 80798284478113 T^{14} \)
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