Properties

Label 10.9.c.a
Level 10
Weight 9
Character orbit 10.c
Analytic conductor 4.074
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(4.07378610061\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{249})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -8 + 8 \beta_{1} ) q^{2} \) \( + ( 13 + 13 \beta_{1} + \beta_{2} ) q^{3} \) \( -128 \beta_{1} q^{4} \) \( + ( 29 + 39 \beta_{1} - 2 \beta_{2} + 11 \beta_{3} ) q^{5} \) \( + ( -208 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6} \) \( + ( -267 + 326 \beta_{1} + 59 \beta_{3} ) q^{7} \) \( + ( 1024 + 1024 \beta_{1} ) q^{8} \) \( + ( -3084 \beta_{1} + 27 \beta_{2} + 27 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -8 + 8 \beta_{1} ) q^{2} \) \( + ( 13 + 13 \beta_{1} + \beta_{2} ) q^{3} \) \( -128 \beta_{1} q^{4} \) \( + ( 29 + 39 \beta_{1} - 2 \beta_{2} + 11 \beta_{3} ) q^{5} \) \( + ( -208 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6} \) \( + ( -267 + 326 \beta_{1} + 59 \beta_{3} ) q^{7} \) \( + ( 1024 + 1024 \beta_{1} ) q^{8} \) \( + ( -3084 \beta_{1} + 27 \beta_{2} + 27 \beta_{3} ) q^{9} \) \( + ( -456 - 96 \beta_{1} - 72 \beta_{2} - 104 \beta_{3} ) q^{10} \) \( + ( -5094 - 131 \beta_{1} + 131 \beta_{2} - 131 \beta_{3} ) q^{11} \) \( + ( 1664 - 1792 \beta_{1} - 128 \beta_{3} ) q^{12} \) \( + ( 18305 + 18305 \beta_{1} + 242 \beta_{2} ) q^{13} \) \( + ( -4744 \beta_{1} - 472 \beta_{2} - 472 \beta_{3} ) q^{14} \) \( + ( -34219 - 5329 \beta_{1} - 153 \beta_{2} + 154 \beta_{3} ) q^{15} \) \( -16384 q^{16} \) \( + ( 49689 - 49783 \beta_{1} - 94 \beta_{3} ) q^{17} \) \( + ( 24888 + 24888 \beta_{1} - 432 \beta_{2} ) q^{18} \) \( + ( 35750 \beta_{1} + 1382 \beta_{2} + 1382 \beta_{3} ) q^{19} \) \( + ( 3584 - 3456 \beta_{1} + 1408 \beta_{2} + 256 \beta_{3} ) q^{20} \) \( + ( -190550 + 1093 \beta_{1} - 1093 \beta_{2} + 1093 \beta_{3} ) q^{21} \) \( + ( 40752 - 38656 \beta_{1} + 2096 \beta_{3} ) q^{22} \) \( + ( 159173 + 159173 \beta_{1} - 2679 \beta_{2} ) q^{23} \) \( + ( 27648 \beta_{1} + 1024 \beta_{2} + 1024 \beta_{3} ) q^{24} \) \( + ( 136985 - 362115 \beta_{1} - 805 \beta_{2} + 365 \beta_{3} ) q^{25} \) \( + ( -292880 + 1936 \beta_{1} - 1936 \beta_{2} + 1936 \beta_{3} ) q^{26} \) \( + ( 41712 - 50628 \beta_{1} - 8916 \beta_{3} ) q^{27} \) \( + ( 34176 + 34176 \beta_{1} + 7552 \beta_{2} ) q^{28} \) \( + ( 623680 \beta_{1} - 7376 \beta_{2} - 7376 \beta_{3} ) q^{29} \) \( + ( 317616 - 232344 \beta_{1} - 8 \beta_{2} - 2456 \beta_{3} ) q^{30} \) \( + ( -592950 - 10477 \beta_{1} + 10477 \beta_{2} - 10477 \beta_{3} ) q^{31} \) \( + ( 131072 - 131072 \beta_{1} ) q^{32} \) \( + ( 341450 + 341450 \beta_{1} - 1426 \beta_{2} ) q^{33} \) \( + ( 795776 \beta_{1} + 752 \beta_{2} + 752 \beta_{3} ) q^{34} \) \( + ( 351997 - 2021948 \beta_{1} - 4586 \beta_{2} - 2527 \beta_{3} ) q^{35} \) \( + ( -398208 - 3456 \beta_{1} + 3456 \beta_{2} - 3456 \beta_{3} ) q^{36} \) \( + ( -221751 + 266427 \beta_{1} + 44676 \beta_{3} ) q^{37} \) \( + ( -274944 - 274944 \beta_{1} - 22112 \beta_{2} ) q^{38} \) \( + ( 1250727 \beta_{1} + 21693 \beta_{2} + 21693 \beta_{3} ) q^{39} \) \( + ( 1024 + 67584 \beta_{1} - 13312 \beta_{2} + 9216 \beta_{3} ) q^{40} \) \( + ( -1031846 + 39317 \beta_{1} - 39317 \beta_{2} + 39317 \beta_{3} ) q^{41} \) \( + ( 1524400 - 1541888 \beta_{1} - 17488 \beta_{3} ) q^{42} \) \( + ( 1758397 + 1758397 \beta_{1} + 56473 \beta_{2} ) q^{43} \) \( + ( 635264 \beta_{1} - 16768 \beta_{2} - 16768 \beta_{3} ) q^{44} \) \( + ( -669108 - 1174878 \beta_{1} + 33654 \beta_{2} + 7653 \beta_{3} ) q^{45} \) \( + ( -2546768 - 21432 \beta_{1} + 21432 \beta_{2} - 21432 \beta_{3} ) q^{46} \) \( + ( -864267 + 785826 \beta_{1} - 78441 \beta_{3} ) q^{47} \) \( + ( -212992 - 212992 \beta_{1} - 16384 \beta_{2} ) q^{48} \) \( + ( -5245636 \beta_{1} - 34987 \beta_{2} - 34987 \beta_{3} ) q^{49} \) \( + ( 1803960 + 3986360 \beta_{1} + 3520 \beta_{2} - 9360 \beta_{3} ) q^{50} \) \( + ( 1584442 - 51005 \beta_{1} + 51005 \beta_{2} - 51005 \beta_{3} ) q^{51} \) \( + ( 2343040 - 2374016 \beta_{1} - 30976 \beta_{3} ) q^{52} \) \( + ( 2576185 + 2576185 \beta_{1} - 157788 \beta_{2} ) q^{53} \) \( + ( 738720 \beta_{1} + 71328 \beta_{2} + 71328 \beta_{3} ) q^{54} \) \( + ( -5447462 + 3473133 \beta_{1} + 17131 \beta_{2} - 53283 \beta_{3} ) q^{55} \) \( + ( -546816 + 60416 \beta_{1} - 60416 \beta_{2} + 60416 \beta_{3} ) q^{56} \) \( + ( -4747568 + 4820632 \beta_{1} + 73064 \beta_{3} ) q^{57} \) \( + ( -5048448 - 5048448 \beta_{1} + 118016 \beta_{2} ) q^{58} \) \( + ( -7994090 \beta_{1} - 40842 \beta_{2} - 40842 \beta_{3} ) q^{59} \) \( + ( -701824 + 4399616 \beta_{1} + 19712 \beta_{2} + 19584 \beta_{3} ) q^{60} \) \( + ( 21358778 + 14501 \beta_{1} - 14501 \beta_{2} + 14501 \beta_{3} ) q^{61} \) \( + ( 4743600 - 4575968 \beta_{1} + 167632 \beta_{3} ) q^{62} \) \( + ( -4126779 - 4126779 \beta_{1} + 165945 \beta_{2} ) q^{63} \) \( + 2097152 \beta_{1} q^{64} \) \( + ( -8265839 - 289124 \beta_{1} - 234093 \beta_{2} + 173699 \beta_{3} ) q^{65} \) \( + ( -5463200 - 11408 \beta_{1} + 11408 \beta_{2} - 11408 \beta_{3} ) q^{66} \) \( + ( -14314243 + 14283490 \beta_{1} - 30753 \beta_{3} ) q^{67} \) \( + ( -6360192 - 6360192 \beta_{1} - 12032 \beta_{2} ) q^{68} \) \( + ( -4076883 \beta_{1} + 121667 \beta_{2} + 121667 \beta_{3} ) q^{69} \) \( + ( 13339392 + 18954872 \beta_{1} + 56904 \beta_{2} - 16472 \beta_{3} ) q^{70} \) \( + ( 28278218 + 27211 \beta_{1} - 27211 \beta_{2} + 27211 \beta_{3} ) q^{71} \) \( + ( 3185664 - 3130368 \beta_{1} + 55296 \beta_{3} ) q^{72} \) \( + ( -15176175 - 15176175 \beta_{1} - 85528 \beta_{2} ) q^{73} \) \( + ( -3905424 \beta_{1} - 357408 \beta_{2} - 357408 \beta_{3} ) q^{74} \) \( + ( 5357165 - 5805235 \beta_{1} + 120605 \beta_{2} - 368640 \beta_{3} ) q^{75} \) \( + ( 4399104 - 176896 \beta_{1} + 176896 \beta_{2} - 176896 \beta_{3} ) q^{76} \) \( + ( -22692550 + 22477416 \beta_{1} - 215134 \beta_{3} ) q^{77} \) \( + ( -9832272 - 9832272 \beta_{1} - 347088 \beta_{2} ) q^{78} \) \( + ( -11147280 \beta_{1} + 180272 \beta_{2} + 180272 \beta_{3} ) q^{79} \) \( + ( -475136 - 638976 \beta_{1} + 32768 \beta_{2} - 180224 \beta_{3} ) q^{80} \) \( + ( 8419833 - 343683 \beta_{1} + 343683 \beta_{2} - 343683 \beta_{3} ) q^{81} \) \( + ( 8254768 - 8883840 \beta_{1} - 629072 \beta_{3} ) q^{82} \) \( + ( 17681821 + 17681821 \beta_{1} + 161965 \beta_{2} ) q^{83} \) \( + ( 24530304 \beta_{1} + 139904 \beta_{2} + 139904 \beta_{3} ) q^{84} \) \( + ( 2247217 + 3812572 \beta_{1} + 450679 \beta_{2} + 644453 \beta_{3} ) q^{85} \) \( + ( -28134352 + 451784 \beta_{1} - 451784 \beta_{2} + 451784 \beta_{3} ) q^{86} \) \( + ( 14750384 - 14325856 \beta_{1} + 424528 \beta_{3} ) q^{87} \) \( + ( -5216256 - 5216256 \beta_{1} + 268288 \beta_{2} ) q^{88} \) \( + ( 2253080 \beta_{1} - 101640 \beta_{2} - 101640 \beta_{3} ) q^{89} \) \( + ( 14813112 + 4315392 \beta_{1} - 330456 \beta_{2} + 208008 \beta_{3} ) q^{90} \) \( + ( -54208006 + 1158887 \beta_{1} - 1158887 \beta_{2} + 1158887 \beta_{3} ) q^{91} \) \( + ( 20374144 - 20031232 \beta_{1} + 342912 \beta_{3} ) q^{92} \) \( + ( 24896074 + 24896074 \beta_{1} - 299594 \beta_{2} ) q^{93} \) \( + ( -13200744 \beta_{1} + 627528 \beta_{2} + 627528 \beta_{3} ) q^{94} \) \( + ( -39669360 - 54909010 \beta_{1} - 407070 \beta_{2} + 4510 \beta_{3} ) q^{95} \) \( + ( 3407872 - 131072 \beta_{1} + 131072 \beta_{2} - 131072 \beta_{3} ) q^{96} \) \( + ( 34986721 - 34829081 \beta_{1} + 157640 \beta_{3} ) q^{97} \) \( + ( 41685192 + 41685192 \beta_{1} + 559792 \beta_{2} ) q^{98} \) \( + ( 37323717 \beta_{1} - 538005 \beta_{2} - 538005 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut +\mathstrut 54q^{3} \) \(\mathstrut +\mathstrut 90q^{5} \) \(\mathstrut -\mathstrut 864q^{6} \) \(\mathstrut -\mathstrut 1186q^{7} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 32q^{2} \) \(\mathstrut +\mathstrut 54q^{3} \) \(\mathstrut +\mathstrut 90q^{5} \) \(\mathstrut -\mathstrut 864q^{6} \) \(\mathstrut -\mathstrut 1186q^{7} \) \(\mathstrut +\mathstrut 4096q^{8} \) \(\mathstrut -\mathstrut 1760q^{10} \) \(\mathstrut -\mathstrut 19852q^{11} \) \(\mathstrut +\mathstrut 6912q^{12} \) \(\mathstrut +\mathstrut 73704q^{13} \) \(\mathstrut -\mathstrut 137490q^{15} \) \(\mathstrut -\mathstrut 65536q^{16} \) \(\mathstrut +\mathstrut 198944q^{17} \) \(\mathstrut +\mathstrut 98688q^{18} \) \(\mathstrut +\mathstrut 16640q^{20} \) \(\mathstrut -\mathstrut 766572q^{21} \) \(\mathstrut +\mathstrut 158816q^{22} \) \(\mathstrut +\mathstrut 631334q^{23} \) \(\mathstrut +\mathstrut 545600q^{25} \) \(\mathstrut -\mathstrut 1179264q^{26} \) \(\mathstrut +\mathstrut 184680q^{27} \) \(\mathstrut +\mathstrut 151808q^{28} \) \(\mathstrut +\mathstrut 1275360q^{30} \) \(\mathstrut -\mathstrut 2329892q^{31} \) \(\mathstrut +\mathstrut 524288q^{32} \) \(\mathstrut +\mathstrut 1362948q^{33} \) \(\mathstrut +\mathstrut 1403870q^{35} \) \(\mathstrut -\mathstrut 1579008q^{36} \) \(\mathstrut -\mathstrut 976356q^{37} \) \(\mathstrut -\mathstrut 1144000q^{38} \) \(\mathstrut -\mathstrut 40960q^{40} \) \(\mathstrut -\mathstrut 4284652q^{41} \) \(\mathstrut +\mathstrut 6132576q^{42} \) \(\mathstrut +\mathstrut 7146534q^{43} \) \(\mathstrut -\mathstrut 2624430q^{45} \) \(\mathstrut -\mathstrut 10101344q^{46} \) \(\mathstrut -\mathstrut 3300186q^{47} \) \(\mathstrut -\mathstrut 884736q^{48} \) \(\mathstrut +\mathstrut 7241600q^{50} \) \(\mathstrut +\mathstrut 6541788q^{51} \) \(\mathstrut +\mathstrut 9434112q^{52} \) \(\mathstrut +\mathstrut 9989164q^{53} \) \(\mathstrut -\mathstrut 21649020q^{55} \) \(\mathstrut -\mathstrut 2428928q^{56} \) \(\mathstrut -\mathstrut 19136400q^{57} \) \(\mathstrut -\mathstrut 19957760q^{58} \) \(\mathstrut -\mathstrut 2807040q^{60} \) \(\mathstrut +\mathstrut 85377108q^{61} \) \(\mathstrut +\mathstrut 18639136q^{62} \) \(\mathstrut -\mathstrut 16175226q^{63} \) \(\mathstrut -\mathstrut 33878940q^{65} \) \(\mathstrut -\mathstrut 21807168q^{66} \) \(\mathstrut -\mathstrut 57195466q^{67} \) \(\mathstrut -\mathstrut 25464832q^{68} \) \(\mathstrut +\mathstrut 53504320q^{70} \) \(\mathstrut +\mathstrut 113004028q^{71} \) \(\mathstrut +\mathstrut 12632064q^{72} \) \(\mathstrut -\mathstrut 60875756q^{73} \) \(\mathstrut +\mathstrut 22407150q^{75} \) \(\mathstrut +\mathstrut 18304000q^{76} \) \(\mathstrut -\mathstrut 90339932q^{77} \) \(\mathstrut -\mathstrut 40023264q^{78} \) \(\mathstrut -\mathstrut 1474560q^{80} \) \(\mathstrut +\mathstrut 35054064q^{81} \) \(\mathstrut +\mathstrut 34277216q^{82} \) \(\mathstrut +\mathstrut 71051214q^{83} \) \(\mathstrut +\mathstrut 8601320q^{85} \) \(\mathstrut -\mathstrut 114344544q^{86} \) \(\mathstrut +\mathstrut 58152480q^{87} \) \(\mathstrut -\mathstrut 20328448q^{88} \) \(\mathstrut +\mathstrut 58175520q^{90} \) \(\mathstrut -\mathstrut 221467572q^{91} \) \(\mathstrut +\mathstrut 80810752q^{92} \) \(\mathstrut +\mathstrut 98985108q^{93} \) \(\mathstrut -\mathstrut 159500600q^{95} \) \(\mathstrut +\mathstrut 14155776q^{96} \) \(\mathstrut +\mathstrut 139631604q^{97} \) \(\mathstrut +\mathstrut 167860352q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(125\) \(x^{2}\mathstrut +\mathstrut \) \(3844\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 63 \nu \)\()/62\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{3} + 310 \nu^{2} + 499 \nu + 19406 \)\()/62\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 155 \nu^{2} + 218 \nu - 9703 \)\()/31\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\)\()/10\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(626\)\()/10\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(63\) \(\beta_{3}\mathstrut -\mathstrut \) \(63\) \(\beta_{2}\mathstrut +\mathstrut \) \(935\) \(\beta_{1}\)\()/10\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
8.38987i
7.38987i
8.38987i
7.38987i
−8.00000 + 8.00000i −25.9493 25.9493i 128.000i 535.341 322.544i 415.189 2031.01 2031.01i 1024.00 + 1024.00i 5214.26i −1702.38 + 6863.08i
3.2 −8.00000 + 8.00000i 52.9493 + 52.9493i 128.000i −490.341 + 387.544i −847.189 −2624.01 + 2624.01i 1024.00 + 1024.00i 953.736i 822.379 7023.08i
7.1 −8.00000 8.00000i −25.9493 + 25.9493i 128.000i 535.341 + 322.544i 415.189 2031.01 + 2031.01i 1024.00 1024.00i 5214.26i −1702.38 6863.08i
7.2 −8.00000 8.00000i 52.9493 52.9493i 128.000i −490.341 387.544i −847.189 −2624.01 2624.01i 1024.00 1024.00i 953.736i 822.379 + 7023.08i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut -\mathstrut 54 T_{3}^{3} \) \(\mathstrut +\mathstrut 1458 T_{3}^{2} \) \(\mathstrut +\mathstrut 148392 T_{3} \) \(\mathstrut +\mathstrut 7551504 \) acting on \(S_{9}^{\mathrm{new}}(10, [\chi])\).