Properties

Label 5.10.b.a
Level 5
Weight 10
Character orbit 5.b
Analytic conductor 2.575
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 5.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.57517918082\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{3} \) \( + ( -342 - \beta_{3} ) q^{4} \) \( + ( 285 + 19 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{5} \) \( + ( 702 + \beta_{3} ) q^{6} \) \( + ( 133 \beta_{1} - 49 \beta_{2} ) q^{7} \) \( + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} \) \( + ( 2907 + 18 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{3} \) \( + ( -342 - \beta_{3} ) q^{4} \) \( + ( 285 + 19 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{5} \) \( + ( 702 + \beta_{3} ) q^{6} \) \( + ( 133 \beta_{1} - 49 \beta_{2} ) q^{7} \) \( + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} \) \( + ( 2907 + 18 \beta_{3} ) q^{9} \) \( + ( -17290 + 1139 \beta_{1} - 8 \beta_{2} - 19 \beta_{3} ) q^{10} \) \( + ( 27492 - 20 \beta_{3} ) q^{11} \) \( + ( 1044 \beta_{1} + 504 \beta_{2} ) q^{12} \) \( + ( -1918 \beta_{1} - 110 \beta_{2} ) q^{13} \) \( + ( -106134 - 133 \beta_{3} ) q^{14} \) \( + ( -99180 - 987 \beta_{1} - 561 \beta_{2} + 152 \beta_{3} ) q^{15} \) \( + ( 407816 + 172 \beta_{3} ) q^{16} \) \( + ( -7448 \beta_{1} - 1632 \beta_{2} ) q^{17} \) \( + ( 18279 \beta_{1} - 144 \beta_{2} ) q^{18} \) \( + ( -159220 + 476 \beta_{3} ) q^{19} \) \( + ( -825570 - 23788 \beta_{1} + 3736 \beta_{2} - 627 \beta_{3} ) q^{20} \) \( + ( 880992 - 798 \beta_{3} ) q^{21} \) \( + ( 10412 \beta_{1} + 160 \beta_{2} ) q^{22} \) \( + ( 17005 \beta_{1} + 5199 \beta_{2} ) q^{23} \) \( + ( -608760 - 532 \beta_{3} ) q^{24} \) \( + ( -334475 + 45410 \beta_{1} - 8270 \beta_{2} + 1140 \beta_{3} ) q^{25} \) \( + ( 1654692 + 1918 \beta_{3} ) q^{26} \) \( + ( -35226 \beta_{1} + 7362 \beta_{2} ) q^{27} \) \( + ( -151620 \beta_{1} - 24024 \beta_{2} ) q^{28} \) \( + ( -882930 - 1976 \beta_{3} ) q^{29} \) \( + ( 928170 + 30628 \beta_{1} - 1216 \beta_{2} + 987 \beta_{3} ) q^{30} \) \( + ( -2646928 - 760 \beta_{3} ) q^{31} \) \( + ( 204496 \beta_{1} + 2720 \beta_{2} ) q^{32} \) \( + ( -13452 \beta_{1} + 44412 \beta_{2} ) q^{33} \) \( + ( 6608656 + 7448 \beta_{3} ) q^{34} \) \( + ( 3407460 + 144039 \beta_{1} + 26817 \beta_{2} - 9044 \beta_{3} ) q^{35} \) \( + ( -14099994 - 9063 \beta_{3} ) q^{36} \) \( + ( -335370 \beta_{1} - 51378 \beta_{2} ) q^{37} \) \( + ( 247284 \beta_{1} - 3808 \beta_{2} ) q^{38} \) \( + ( 421704 - 4008 \beta_{3} ) q^{39} \) \( + ( 10894600 - 777860 \beta_{1} + 920 \beta_{2} + 14060 \beta_{3} ) q^{40} \) \( + ( -4197138 + 24890 \beta_{3} ) q^{41} \) \( + ( 199500 \beta_{1} + 6384 \beta_{2} ) q^{42} \) \( + ( 719131 \beta_{1} - 132643 \beta_{2} ) q^{43} \) \( + ( 5159736 - 20652 \beta_{3} ) q^{44} \) \( + ( 13934295 + 366453 \beta_{1} - 89991 \beta_{2} + 8037 \beta_{3} ) q^{45} \) \( + ( -15312518 - 17005 \beta_{3} ) q^{46} \) \( + ( -1012111 \beta_{1} + 214259 \beta_{2} ) q^{47} \) \( + ( -528560 \beta_{1} + 262304 \beta_{2} ) q^{48} \) \( + ( -11730257 + 27930 \beta_{3} ) q^{49} \) \( + ( -37523100 + 639085 \beta_{1} - 9120 \beta_{2} - 45410 \beta_{3} ) q^{50} \) \( + ( 21004272 - 38456 \beta_{3} ) q^{51} \) \( + ( 2310648 \beta_{1} - 71664 \beta_{2} ) q^{52} \) \( + ( -1349666 \beta_{1} - 259794 \beta_{2} ) q^{53} \) \( + ( 28963980 + 35226 \beta_{3} ) q^{54} \) \( + ( -6726780 + 176548 \beta_{1} + 315044 \beta_{2} + 21792 \beta_{3} ) q^{55} \) \( + ( 78794520 + 83524 \beta_{3} ) q^{56} \) \( + ( -174932 \beta_{1} - 561916 \beta_{2} ) q^{57} \) \( + ( -2570434 \beta_{1} + 15808 \beta_{2} ) q^{58} \) \( + ( -115207260 - 52972 \beta_{3} ) q^{59} \) \( + ( -76751640 + 1265724 \beta_{1} - 295128 \beta_{2} + 47196 \beta_{3} ) q^{60} \) \( + ( 90122642 - 43150 \beta_{3} ) q^{61} \) \( + ( -3295968 \beta_{1} + 6080 \beta_{2} ) q^{62} \) \( + ( 2297043 \beta_{1} + 591633 \beta_{2} ) q^{63} \) \( + ( 33748768 - 116432 \beta_{3} ) q^{64} \) \( + ( 45973920 - 2201322 \beta_{1} + 77934 \beta_{2} + 21812 \beta_{3} ) q^{65} \) \( + ( 4737384 + 13452 \beta_{3} ) q^{66} \) \( + ( 6669647 \beta_{1} + 1448953 \beta_{2} ) q^{67} \) \( + ( 9155872 \beta_{1} - 895168 \beta_{2} ) q^{68} \) \( + ( -71631216 + 115786 \beta_{3} ) q^{69} \) \( + ( -127085490 - 4316116 \beta_{1} + 72352 \beta_{2} - 144039 \beta_{3} ) q^{70} \) \( + ( -11902968 - 91200 \beta_{3} ) q^{71} \) \( + ( -12480948 \beta_{1} - 1224 \beta_{2} ) q^{72} \) \( + ( -1847940 \beta_{1} + 90564 \beta_{2} ) q^{73} \) \( + ( 294215436 + 335370 \beta_{3} ) q^{74} \) \( + ( 164809800 - 465805 \beta_{1} - 1298915 \beta_{2} - 111720 \beta_{3} ) q^{75} \) \( + ( -292122360 - 3572 \beta_{3} ) q^{76} \) \( + ( 1533756 \beta_{1} - 2162748 \beta_{2} ) q^{77} \) \( + ( -3001128 \beta_{1} + 32064 \beta_{2} ) q^{78} \) \( + ( -182010880 - 267976 \beta_{3} ) q^{79} \) \( + ( 241460760 + 10722384 \beta_{1} + 1800352 \beta_{2} + 456836 \beta_{3} ) q^{80} \) \( + ( -85846959 + 458946 \beta_{3} ) q^{81} \) \( + ( 17058922 \beta_{1} - 199120 \beta_{2} ) q^{82} \) \( + ( -12737657 \beta_{1} + 1180353 \beta_{2} ) q^{83} \) \( + ( 279724536 - 608076 \beta_{3} ) q^{84} \) \( + ( 318854960 - 8731336 \beta_{1} + 988192 \beta_{2} - 75544 \beta_{3} ) q^{85} \) \( + ( -593976138 - 719131 \beta_{3} ) q^{86} \) \( + ( 2270082 \beta_{1} + 788766 \beta_{2} ) q^{87} \) \( + ( -7146128 \beta_{1} + 247136 \beta_{2} ) q^{88} \) \( + ( -395675190 + 185592 \beta_{3} ) q^{89} \) \( + ( -299272230 + 20797893 \beta_{1} - 64296 \beta_{2} - 366453 \beta_{3} ) q^{90} \) \( + ( 118330632 + 357504 \beta_{3} ) q^{91} \) \( + ( -21128228 \beta_{1} + 2797928 \beta_{2} ) q^{92} \) \( + ( 3180448 \beta_{1} - 2003968 \beta_{2} ) q^{93} \) \( + ( 831775426 + 1012111 \beta_{3} ) q^{94} \) \( + ( 301197900 + 5204860 \beta_{1} - 4032420 \beta_{2} - 23560 \beta_{3} ) q^{95} \) \( + ( 99834912 + 256176 \beta_{3} ) q^{96} \) \( + ( 14786464 \beta_{1} - 1954216 \beta_{2} ) q^{97} \) \( + ( 12121963 \beta_{1} - 223440 \beta_{2} ) q^{98} \) \( + ( -182196756 + 436716 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 1368q^{4} \) \(\mathstrut +\mathstrut 1140q^{5} \) \(\mathstrut +\mathstrut 2808q^{6} \) \(\mathstrut +\mathstrut 11628q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 1368q^{4} \) \(\mathstrut +\mathstrut 1140q^{5} \) \(\mathstrut +\mathstrut 2808q^{6} \) \(\mathstrut +\mathstrut 11628q^{9} \) \(\mathstrut -\mathstrut 69160q^{10} \) \(\mathstrut +\mathstrut 109968q^{11} \) \(\mathstrut -\mathstrut 424536q^{14} \) \(\mathstrut -\mathstrut 396720q^{15} \) \(\mathstrut +\mathstrut 1631264q^{16} \) \(\mathstrut -\mathstrut 636880q^{19} \) \(\mathstrut -\mathstrut 3302280q^{20} \) \(\mathstrut +\mathstrut 3523968q^{21} \) \(\mathstrut -\mathstrut 2435040q^{24} \) \(\mathstrut -\mathstrut 1337900q^{25} \) \(\mathstrut +\mathstrut 6618768q^{26} \) \(\mathstrut -\mathstrut 3531720q^{29} \) \(\mathstrut +\mathstrut 3712680q^{30} \) \(\mathstrut -\mathstrut 10587712q^{31} \) \(\mathstrut +\mathstrut 26434624q^{34} \) \(\mathstrut +\mathstrut 13629840q^{35} \) \(\mathstrut -\mathstrut 56399976q^{36} \) \(\mathstrut +\mathstrut 1686816q^{39} \) \(\mathstrut +\mathstrut 43578400q^{40} \) \(\mathstrut -\mathstrut 16788552q^{41} \) \(\mathstrut +\mathstrut 20638944q^{44} \) \(\mathstrut +\mathstrut 55737180q^{45} \) \(\mathstrut -\mathstrut 61250072q^{46} \) \(\mathstrut -\mathstrut 46921028q^{49} \) \(\mathstrut -\mathstrut 150092400q^{50} \) \(\mathstrut +\mathstrut 84017088q^{51} \) \(\mathstrut +\mathstrut 115855920q^{54} \) \(\mathstrut -\mathstrut 26907120q^{55} \) \(\mathstrut +\mathstrut 315178080q^{56} \) \(\mathstrut -\mathstrut 460829040q^{59} \) \(\mathstrut -\mathstrut 307006560q^{60} \) \(\mathstrut +\mathstrut 360490568q^{61} \) \(\mathstrut +\mathstrut 134995072q^{64} \) \(\mathstrut +\mathstrut 183895680q^{65} \) \(\mathstrut +\mathstrut 18949536q^{66} \) \(\mathstrut -\mathstrut 286524864q^{69} \) \(\mathstrut -\mathstrut 508341960q^{70} \) \(\mathstrut -\mathstrut 47611872q^{71} \) \(\mathstrut +\mathstrut 1176861744q^{74} \) \(\mathstrut +\mathstrut 659239200q^{75} \) \(\mathstrut -\mathstrut 1168489440q^{76} \) \(\mathstrut -\mathstrut 728043520q^{79} \) \(\mathstrut +\mathstrut 965843040q^{80} \) \(\mathstrut -\mathstrut 343387836q^{81} \) \(\mathstrut +\mathstrut 1118898144q^{84} \) \(\mathstrut +\mathstrut 1275419840q^{85} \) \(\mathstrut -\mathstrut 2375904552q^{86} \) \(\mathstrut -\mathstrut 1582700760q^{89} \) \(\mathstrut -\mathstrut 1197088920q^{90} \) \(\mathstrut +\mathstrut 473322528q^{91} \) \(\mathstrut +\mathstrut 3327101704q^{94} \) \(\mathstrut +\mathstrut 1204791600q^{95} \) \(\mathstrut +\mathstrut 399339648q^{96} \) \(\mathstrut -\mathstrut 728787024q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(45\) \(x^{2}\mathstrut +\mathstrut \) \(304\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 37 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( -\nu^{3} - 7 \nu \)
\(\beta_{3}\)\(=\)\( 60 \nu^{2} + 1350 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\)\()/30\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(1350\)\()/60\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(37\) \(\beta_{2}\mathstrut -\mathstrut \) \(14\) \(\beta_{1}\)\()/30\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
2.87724i
6.05982i
6.05982i
2.87724i
41.3193i 37.6407i −1195.29 1138.29 810.818i 1555.29 5315.22i 28233.0i 18266.2 −33502.5 47033.3i
4.2 0.843944i 179.263i 511.288 −568.288 1276.78i −151.288 8712.99i 863.597i −12452.2 −1077.53 + 479.603i
4.3 0.843944i 179.263i 511.288 −568.288 + 1276.78i −151.288 8712.99i 863.597i −12452.2 −1077.53 479.603i
4.4 41.3193i 37.6407i −1195.29 1138.29 + 810.818i 1555.29 5315.22i 28233.0i 18266.2 −33502.5 + 47033.3i
\(n\): 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.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{10}^{\mathrm{new}}(5, [\chi])\).