Properties

Label 668.2.a.b
Level $668$
Weight $2$
Character orbit 668.a
Self dual yes
Analytic conductor $5.334$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.826865.1
Defining polynomial: \(x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} ) q^{3} + 2 q^{5} + ( 2 - \beta_{3} ) q^{7} + ( 2 - \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{2} ) q^{3} + 2 q^{5} + ( 2 - \beta_{3} ) q^{7} + ( 2 - \beta_{2} + \beta_{4} ) q^{9} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} -2 \beta_{1} q^{13} + ( 2 - 2 \beta_{2} ) q^{15} + ( -2 + 2 \beta_{1} - 2 \beta_{4} ) q^{17} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{19} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{21} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{23} - q^{25} + ( 4 - \beta_{1} + 3 \beta_{4} ) q^{27} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{29} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{31} + ( -3 \beta_{1} - \beta_{2} ) q^{33} + ( 4 - 2 \beta_{3} ) q^{35} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{37} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{39} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{41} + 4 \beta_{1} q^{43} + ( 4 - 2 \beta_{2} + 2 \beta_{4} ) q^{45} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{47} + ( 4 - \beta_{1} - \beta_{2} - 4 \beta_{3} - \beta_{4} ) q^{49} + ( -2 + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{51} + ( 2 - 4 \beta_{1} + 2 \beta_{4} ) q^{53} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{55} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{57} + ( 2 - 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{59} + ( 3 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -1 + \beta_{1} - \beta_{4} ) q^{63} -4 \beta_{1} q^{65} + ( 6 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{67} + ( -2 - 4 \beta_{1} + 2 \beta_{3} ) q^{69} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{71} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{73} + ( -1 + \beta_{2} ) q^{75} + ( -4 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{77} + ( -6 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{79} + ( -2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{81} + ( -2 + 6 \beta_{1} ) q^{83} + ( -4 + 4 \beta_{1} - 4 \beta_{4} ) q^{85} + ( 3 - 4 \beta_{1} - \beta_{3} + \beta_{4} ) q^{87} + ( -5 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{89} + ( -6 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{91} + ( -3 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{93} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{95} + ( -6 - \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} ) q^{97} + ( 1 + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + O(q^{10}) \) \( 5q + 3q^{3} + 10q^{5} + 9q^{7} + 6q^{9} + 5q^{11} - 4q^{13} + 6q^{15} - 2q^{17} + 5q^{19} - 4q^{21} + 6q^{23} - 5q^{25} + 12q^{27} - 5q^{29} + 9q^{31} - 8q^{33} + 18q^{35} + 8q^{37} - 4q^{41} + 8q^{43} + 12q^{45} + 13q^{47} + 14q^{49} - 2q^{53} + 10q^{55} - 12q^{57} + 4q^{59} + 11q^{61} - q^{63} - 8q^{65} + 28q^{67} - 16q^{69} + 2q^{71} + 8q^{73} - 3q^{75} - 12q^{77} - 10q^{79} - 15q^{81} + 2q^{83} - 4q^{85} + 4q^{87} - 17q^{89} - 12q^{91} - 12q^{93} + 10q^{95} - 27q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 5 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 2 \beta_{3} + 8 \beta_{2} + 9 \beta_{1} + 8\)

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
2.75474
−1.69135
−0.873948
1.66287
0.147687
0 −1.83385 0 2.00000 0 1.53681 0 0.363011 0
1.2 0 −1.55202 0 2.00000 0 4.48567 0 −0.591244 0
1.3 0 1.36227 0 2.00000 0 −1.42676 0 −1.14423 0
1.4 0 1.89773 0 2.00000 0 4.92082 0 0.601369 0
1.5 0 3.12588 0 2.00000 0 −0.516539 0 6.77110 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.b 5
3.b odd 2 1 6012.2.a.d 5
4.b odd 2 1 2672.2.a.j 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
668.2.a.b 5 1.a even 1 1 trivial
2672.2.a.j 5 4.b odd 2 1
6012.2.a.d 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} - 3 T_{3}^{4} - 6 T_{3}^{3} + 17 T_{3}^{2} + 9 T_{3} - 23 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(668))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 3 T + 9 T^{2} - 19 T^{3} + 45 T^{4} - 83 T^{5} + 135 T^{6} - 171 T^{7} + 243 T^{8} - 243 T^{9} + 243 T^{10} \)
$5$ \( ( 1 - 2 T + 5 T^{2} )^{5} \)
$7$ \( 1 - 9 T + 51 T^{2} - 223 T^{3} + 787 T^{4} - 2265 T^{5} + 5509 T^{6} - 10927 T^{7} + 17493 T^{8} - 21609 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 5 T + 43 T^{2} - 191 T^{3} + 849 T^{4} - 3017 T^{5} + 9339 T^{6} - 23111 T^{7} + 57233 T^{8} - 73205 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 4 T + 45 T^{2} + 160 T^{3} + 1006 T^{4} + 2840 T^{5} + 13078 T^{6} + 27040 T^{7} + 98865 T^{8} + 114244 T^{9} + 371293 T^{10} \)
$17$ \( 1 + 2 T + 25 T^{2} + 96 T^{3} + 662 T^{4} + 1308 T^{5} + 11254 T^{6} + 27744 T^{7} + 122825 T^{8} + 167042 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 5 T + 53 T^{2} - 279 T^{3} + 1821 T^{4} - 6547 T^{5} + 34599 T^{6} - 100719 T^{7} + 363527 T^{8} - 651605 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 6 T + 63 T^{2} - 192 T^{3} + 1414 T^{4} - 2452 T^{5} + 32522 T^{6} - 101568 T^{7} + 766521 T^{8} - 1679046 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + 5 T + 117 T^{2} + 541 T^{3} + 6005 T^{4} + 22981 T^{5} + 174145 T^{6} + 454981 T^{7} + 2853513 T^{8} + 3536405 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 9 T + 119 T^{2} - 909 T^{3} + 6257 T^{4} - 39425 T^{5} + 193967 T^{6} - 873549 T^{7} + 3545129 T^{8} - 8311689 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 8 T + 141 T^{2} - 1104 T^{3} + 8998 T^{4} - 60080 T^{5} + 332926 T^{6} - 1511376 T^{7} + 7142073 T^{8} - 14993288 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 4 T + 109 T^{2} + 424 T^{3} + 6474 T^{4} + 20392 T^{5} + 265434 T^{6} + 712744 T^{7} + 7512389 T^{8} + 11303044 T^{9} + 115856201 T^{10} \)
$43$ \( 1 - 8 T + 135 T^{2} - 992 T^{3} + 9706 T^{4} - 56752 T^{5} + 417358 T^{6} - 1834208 T^{7} + 10733445 T^{8} - 27350408 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 13 T + 261 T^{2} - 2283 T^{3} + 25333 T^{4} - 157087 T^{5} + 1190651 T^{6} - 5043147 T^{7} + 27097803 T^{8} - 63435853 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 2 T + 141 T^{2} + 216 T^{3} + 11910 T^{4} + 17068 T^{5} + 631230 T^{6} + 606744 T^{7} + 20991657 T^{8} + 15780962 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 4 T + 59 T^{2} - 120 T^{3} + 4414 T^{4} - 36008 T^{5} + 260426 T^{6} - 417720 T^{7} + 12117361 T^{8} - 48469444 T^{9} + 714924299 T^{10} \)
$61$ \( 1 - 11 T + 229 T^{2} - 2491 T^{3} + 24489 T^{4} - 221163 T^{5} + 1493829 T^{6} - 9269011 T^{7} + 51978649 T^{8} - 152304251 T^{9} + 844596301 T^{10} \)
$67$ \( 1 - 28 T + 491 T^{2} - 5616 T^{3} + 53302 T^{4} - 436232 T^{5} + 3571234 T^{6} - 25210224 T^{7} + 147674633 T^{8} - 564231388 T^{9} + 1350125107 T^{10} \)
$71$ \( 1 - 2 T + 175 T^{2} - 320 T^{3} + 17542 T^{4} - 36060 T^{5} + 1245482 T^{6} - 1613120 T^{7} + 62634425 T^{8} - 50823362 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 - 8 T + 241 T^{2} - 616 T^{3} + 19206 T^{4} + 4320 T^{5} + 1402038 T^{6} - 3282664 T^{7} + 93753097 T^{8} - 227185928 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 10 T + 139 T^{2} + 1552 T^{3} + 15962 T^{4} + 195500 T^{5} + 1260998 T^{6} + 9686032 T^{7} + 68532421 T^{8} + 389500810 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 2 T + 179 T^{2} - 656 T^{3} + 20622 T^{4} - 69980 T^{5} + 1711626 T^{6} - 4519184 T^{7} + 102349873 T^{8} - 94916642 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 17 T + 485 T^{2} + 5689 T^{3} + 89097 T^{4} + 743149 T^{5} + 7929633 T^{6} + 45062569 T^{7} + 341909965 T^{8} + 1066618097 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 27 T + 521 T^{2} + 8059 T^{3} + 104425 T^{4} + 1093259 T^{5} + 10129225 T^{6} + 75827131 T^{7} + 475502633 T^{8} + 2390290587 T^{9} + 8587340257 T^{10} \)
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