Properties

Label 5.12.b.a
Level 5
Weight 12
Character orbit 5.b
Analytic conductor 3.842
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 5.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(3.8417159028\)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( -\beta_{3} q^{3} \) \( + ( -18 - \beta_{2} ) q^{4} \) \( + ( -75 - 10 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{5} \) \( + ( -438 - 11 \beta_{2} ) q^{6} \) \( + ( -156 \beta_{1} - 119 \beta_{3} ) q^{7} \) \( + ( 1196 \beta_{1} + 112 \beta_{3} ) q^{8} \) \( + ( 6363 + 78 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( -\beta_{3} q^{3} \) \( + ( -18 - \beta_{2} ) q^{4} \) \( + ( -75 - 10 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{5} \) \( + ( -438 - 11 \beta_{2} ) q^{6} \) \( + ( -156 \beta_{1} - 119 \beta_{3} ) q^{7} \) \( + ( 1196 \beta_{1} + 112 \beta_{3} ) q^{8} \) \( + ( 6363 + 78 \beta_{2} ) q^{9} \) \( + ( 22850 - 4245 \beta_{1} + 65 \beta_{2} + 560 \beta_{3} ) q^{10} \) \( + ( -81588 + 220 \beta_{2} ) q^{11} \) \( + ( -9612 \beta_{1} - 816 \beta_{3} ) q^{12} \) \( + ( 39468 \beta_{1} + 110 \beta_{3} ) q^{13} \) \( + ( 270174 - 1153 \beta_{2} ) q^{14} \) \( + ( 858300 - 48060 \beta_{1} - 280 \beta_{2} - 4095 \beta_{3} ) q^{15} \) \( + ( -2458744 - 2012 \beta_{2} ) q^{16} \) \( + ( -29480 \beta_{1} + 12648 \beta_{3} ) q^{17} \) \( + ( 71415 \beta_{1} - 8736 \beta_{3} ) q^{18} \) \( + ( 3865220 + 7436 \beta_{2} ) q^{19} \) \( + ( 8861850 + 56580 \beta_{1} + 165 \beta_{2} + 2960 \beta_{3} ) q^{20} \) \( + ( -20254968 + 10998 \beta_{2} ) q^{21} \) \( + ( 101892 \beta_{1} - 24640 \beta_{3} ) q^{22} \) \( + ( -624404 \beta_{1} + 46641 \beta_{3} ) q^{23} \) \( + ( 18603960 - 21892 \beta_{2} ) q^{24} \) \( + ( 39788125 + 565500 \beta_{1} + 1500 \beta_{2} + 29750 \beta_{3} ) q^{25} \) \( + ( -81492708 - 38258 \beta_{2} ) q^{26} \) \( + ( 749736 \beta_{1} - 118458 \beta_{3} ) q^{27} \) \( + ( -1010916 \beta_{1} - 114576 \beta_{3} ) q^{28} \) \( + ( 54060630 + 4024 \beta_{2} ) q^{29} \) \( + ( 97498350 + 624780 \beta_{1} + 3015 \beta_{2} + 31360 \beta_{3} ) q^{30} \) \( + ( -171010768 + 75560 \beta_{2} ) q^{31} \) \( + ( -1687344 \beta_{1} + 454720 \beta_{3} ) q^{32} \) \( + ( 2114640 \beta_{1} + 265068 \beta_{3} ) q^{33} \) \( + ( 66445504 + 168608 \beta_{2} ) q^{34} \) \( + ( 98573100 - 5056920 \beta_{1} - 43460 \beta_{2} - 574665 \beta_{3} ) q^{35} \) \( + ( -138338334 - 7767 \beta_{2} ) q^{36} \) \( + ( 661212 \beta_{1} - 35238 \beta_{3} ) q^{37} \) \( + ( 10066844 \beta_{1} - 832832 \beta_{3} ) q^{38} \) \( + ( 1499256 - 442728 \beta_{2} ) q^{39} \) \( + ( -68801000 + 305700 \beta_{1} + 109100 \beta_{2} + 1128400 \beta_{3} ) q^{40} \) \( + ( 253218342 - 426490 \beta_{2} ) q^{41} \) \( + ( -11082636 \beta_{1} - 1231776 \beta_{3} ) q^{42} \) \( + ( -4432008 \beta_{1} + 1380883 \beta_{3} ) q^{43} \) \( + ( -388393416 + 77628 \beta_{2} ) q^{44} \) \( + ( -691596225 - 4462830 \beta_{1} - 37665 \beta_{2} - 206085 \beta_{3} ) q^{45} \) \( + ( 1310447422 + 1137455 \beta_{2} ) q^{46} \) \( + ( 32038228 \beta_{1} - 761891 \beta_{3} ) q^{47} \) \( + ( -19339344 \beta_{1} + 780736 \beta_{3} ) q^{48} \) \( + ( -475161593 + 1488630 \beta_{2} ) q^{49} \) \( + ( -1155292500 + 41039125 \beta_{1} - 238250 \beta_{2} - 168000 \beta_{3} ) q^{50} \) \( + ( 2172988272 - 662264 \beta_{2} ) q^{51} \) \( + ( -32569416 \beta_{1} + 4510176 \beta_{3} ) q^{52} \) \( + ( -56138540 \beta_{1} - 4395006 \beta_{3} ) q^{53} \) \( + ( -1600839180 - 2052774 \beta_{2} ) q^{54} \) \( + ( -1943190900 - 11592120 \beta_{1} + 391440 \beta_{2} - 1078940 \beta_{3} ) q^{55} \) \( + ( 2591684520 - 2610764 \beta_{2} ) q^{56} \) \( + ( 71474832 \beta_{1} + 2336404 \beta_{3} ) q^{57} \) \( + ( 57416646 \beta_{1} - 450688 \beta_{3} ) q^{58} \) \( + ( 800242860 - 1049212 \beta_{2} ) q^{59} \) \( + ( 480738600 + 1585980 \beta_{1} - 853260 \beta_{2} - 8724240 \beta_{3} ) q^{60} \) \( + ( -577617838 + 5036150 \beta_{2} ) q^{61} \) \( + ( -107993728 \beta_{1} - 8462720 \beta_{3} ) q^{62} \) \( + ( 78077844 \beta_{1} + 8346807 \beta_{3} ) q^{63} \) \( + ( -1350287648 + 2568688 \beta_{2} ) q^{64} \) \( + ( 807430800 - 162255060 \beta_{1} + 2596220 \beta_{2} + 22552530 \beta_{3} ) q^{65} \) \( + ( -4252746456 + 801108 \beta_{2} ) q^{66} \) \( + ( -36958200 \beta_{1} - 2121217 \beta_{3} ) q^{67} \) \( + ( 146689536 \beta_{1} + 7019008 \beta_{3} ) q^{68} \) \( + ( 8239025496 + 3230446 \beta_{2} ) q^{69} \) \( + ( 10195893450 + 62327460 \beta_{1} - 1264395 \beta_{2} + 4867520 \beta_{3} ) q^{70} \) \( + ( -15083866728 - 7672800 \beta_{2} ) q^{71} \) \( + ( 1441908 \beta_{1} - 17021424 \beta_{3} ) q^{72} \) \( + ( -16271424 \beta_{1} - 35900124 \beta_{3} ) q^{73} \) \( + ( -1381498236 - 1048830 \beta_{2} ) q^{74} \) \( + ( 4833135000 + 14418000 \beta_{1} - 8541000 \beta_{2} - 38537125 \beta_{3} ) q^{75} \) \( + ( -13246909560 - 3999068 \beta_{2} ) q^{76} \) \( + ( 235747008 \beta_{1} + 35386932 \beta_{3} ) q^{77} \) \( + ( -367735896 \beta_{1} + 49585536 \beta_{3} ) q^{78} \) \( + ( 18659442080 - 20151976 \beta_{2} ) q^{79} \) \( + ( 18011731800 + 138064240 \beta_{1} + 12444620 \beta_{2} - 6157120 \beta_{3} ) q^{80} \) \( + ( -19431929079 + 14810094 \beta_{2} ) q^{81} \) \( + ( -102474318 \beta_{1} + 47766880 \beta_{3} ) q^{82} \) \( + ( -478282616 \beta_{1} - 44237433 \beta_{3} ) q^{83} \) \( + ( -19124966376 + 20057004 \beta_{2} ) q^{84} \) \( + ( -11529396400 + 733005480 \beta_{1} + 1625240 \beta_{2} + 35284760 \beta_{3} ) q^{85} \) \( + ( 9761355282 + 19621721 \beta_{2} ) q^{86} \) \( + ( 38678688 \beta_{1} - 50704614 \beta_{3} ) q^{87} \) \( + ( -114976848 \beta_{1} - 59157056 \beta_{3} ) q^{88} \) \( + ( 29568124890 + 24794472 \beta_{2} ) q^{89} \) \( + ( 9129941550 - 723008835 \beta_{1} + 2195895 \beta_{2} + 4218480 \beta_{3} ) q^{90} \) \( + ( 12891273912 - 46716384 \beta_{2} ) q^{91} \) \( + ( 980305500 \beta_{1} - 31874192 \beta_{3} ) q^{92} \) \( + ( 726282720 \beta_{1} + 234027808 \beta_{3} ) q^{93} \) \( + ( -66524687306 - 40419029 \beta_{2} ) q^{94} \) \( + ( -66176569500 - 458042600 \beta_{1} - 19883800 \beta_{2} - 3353700 \beta_{3} ) q^{95} \) \( + ( 78397957152 - 16907376 \beta_{2} ) q^{96} \) \( + ( -2749674072 \beta_{1} - 158351216 \beta_{3} ) q^{97} \) \( + ( 766355827 \beta_{1} - 166726560 \beta_{3} ) q^{98} \) \( + ( 29890091556 - 4964004 \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 72q^{4} \) \(\mathstrut -\mathstrut 300q^{5} \) \(\mathstrut -\mathstrut 1752q^{6} \) \(\mathstrut +\mathstrut 25452q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 72q^{4} \) \(\mathstrut -\mathstrut 300q^{5} \) \(\mathstrut -\mathstrut 1752q^{6} \) \(\mathstrut +\mathstrut 25452q^{9} \) \(\mathstrut +\mathstrut 91400q^{10} \) \(\mathstrut -\mathstrut 326352q^{11} \) \(\mathstrut +\mathstrut 1080696q^{14} \) \(\mathstrut +\mathstrut 3433200q^{15} \) \(\mathstrut -\mathstrut 9834976q^{16} \) \(\mathstrut +\mathstrut 15460880q^{19} \) \(\mathstrut +\mathstrut 35447400q^{20} \) \(\mathstrut -\mathstrut 81019872q^{21} \) \(\mathstrut +\mathstrut 74415840q^{24} \) \(\mathstrut +\mathstrut 159152500q^{25} \) \(\mathstrut -\mathstrut 325970832q^{26} \) \(\mathstrut +\mathstrut 216242520q^{29} \) \(\mathstrut +\mathstrut 389993400q^{30} \) \(\mathstrut -\mathstrut 684043072q^{31} \) \(\mathstrut +\mathstrut 265782016q^{34} \) \(\mathstrut +\mathstrut 394292400q^{35} \) \(\mathstrut -\mathstrut 553353336q^{36} \) \(\mathstrut +\mathstrut 5997024q^{39} \) \(\mathstrut -\mathstrut 275204000q^{40} \) \(\mathstrut +\mathstrut 1012873368q^{41} \) \(\mathstrut -\mathstrut 1553573664q^{44} \) \(\mathstrut -\mathstrut 2766384900q^{45} \) \(\mathstrut +\mathstrut 5241789688q^{46} \) \(\mathstrut -\mathstrut 1900646372q^{49} \) \(\mathstrut -\mathstrut 4621170000q^{50} \) \(\mathstrut +\mathstrut 8691953088q^{51} \) \(\mathstrut -\mathstrut 6403356720q^{54} \) \(\mathstrut -\mathstrut 7772763600q^{55} \) \(\mathstrut +\mathstrut 10366738080q^{56} \) \(\mathstrut +\mathstrut 3200971440q^{59} \) \(\mathstrut +\mathstrut 1922954400q^{60} \) \(\mathstrut -\mathstrut 2310471352q^{61} \) \(\mathstrut -\mathstrut 5401150592q^{64} \) \(\mathstrut +\mathstrut 3229723200q^{65} \) \(\mathstrut -\mathstrut 17010985824q^{66} \) \(\mathstrut +\mathstrut 32956101984q^{69} \) \(\mathstrut +\mathstrut 40783573800q^{70} \) \(\mathstrut -\mathstrut 60335466912q^{71} \) \(\mathstrut -\mathstrut 5525992944q^{74} \) \(\mathstrut +\mathstrut 19332540000q^{75} \) \(\mathstrut -\mathstrut 52987638240q^{76} \) \(\mathstrut +\mathstrut 74637768320q^{79} \) \(\mathstrut +\mathstrut 72046927200q^{80} \) \(\mathstrut -\mathstrut 77727716316q^{81} \) \(\mathstrut -\mathstrut 76499865504q^{84} \) \(\mathstrut -\mathstrut 46117585600q^{85} \) \(\mathstrut +\mathstrut 39045421128q^{86} \) \(\mathstrut +\mathstrut 118272499560q^{89} \) \(\mathstrut +\mathstrut 36519766200q^{90} \) \(\mathstrut +\mathstrut 51565095648q^{91} \) \(\mathstrut -\mathstrut 266098749224q^{94} \) \(\mathstrut -\mathstrut 264706278000q^{95} \) \(\mathstrut +\mathstrut 313591828608q^{96} \) \(\mathstrut +\mathstrut 119560366224q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(142\) \(x^{2}\mathstrut -\mathstrut \) \(2144\) \(x\mathstrut +\mathstrut \) \(28656\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 12 \nu^{2} + 31 \nu + 2562 \)\()/15\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 3 \nu^{2} + 316 \nu + 1422 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{3} + 57 \nu^{2} + 34 \nu - 22932 \)\()/30\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(30\)\()/120\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(38\) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut -\mathstrut \) \(234\) \(\beta_{1}\mathstrut +\mathstrut \) \(8550\)\()/120\)
\(\nu^{3}\)\(=\)\((\)\(518\) \(\beta_{3}\mathstrut -\mathstrut \) \(29\) \(\beta_{2}\mathstrut +\mathstrut \) \(1194\) \(\beta_{1}\mathstrut +\mathstrut \) \(205770\)\()/120\)

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
11.3434 + 1.39818i
−10.8434 10.0894i
−10.8434 + 10.0894i
11.3434 1.39818i
58.2855i 258.747i −1349.20 −6731.01 + 1876.59i −15081.2 21698.4i 40729.8i 110197. 109378. + 392321.i
4.2 27.1071i 524.040i 1313.20 6581.01 2349.13i 14205.2 66589.5i 91112.6i −97470.8 −63678.2 178392.i
4.3 27.1071i 524.040i 1313.20 6581.01 + 2349.13i 14205.2 66589.5i 91112.6i −97470.8 −63678.2 + 178392.i
4.4 58.2855i 258.747i −1349.20 −6731.01 1876.59i −15081.2 21698.4i 40729.8i 110197. 109378. 392321.i
\(n\): 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 \(S_{12}^{\mathrm{new}}(5, [\chi])\).