Properties

Label 3.17.b.a
Level 3
Weight 17
Character orbit 3.b
Analytic conductor 4.870
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.8697363157\)
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: \( 2^{6}\cdot 3^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -513 + \beta_{1} + \beta_{2} ) q^{3} + ( -3116 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{4} + ( 95 \beta_{1} - 45 \beta_{2} + 15 \beta_{3} ) q^{5} + ( -100764 - 2532 \beta_{1} - 21 \beta_{2} + 81 \beta_{3} ) q^{6} + ( -785386 - 217 \beta_{1} + 651 \beta_{2} + 217 \beta_{3} ) q^{7} + ( 50528 \beta_{1} - 792 \beta_{2} + 264 \beta_{3} ) q^{8} + ( 4654665 - 154413 \beta_{1} + 135 \beta_{2} - 729 \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -513 + \beta_{1} + \beta_{2} ) q^{3} + ( -3116 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{4} + ( 95 \beta_{1} - 45 \beta_{2} + 15 \beta_{3} ) q^{5} + ( -100764 - 2532 \beta_{1} - 21 \beta_{2} + 81 \beta_{3} ) q^{6} + ( -785386 - 217 \beta_{1} + 651 \beta_{2} + 217 \beta_{3} ) q^{7} + ( 50528 \beta_{1} - 792 \beta_{2} + 264 \beta_{3} ) q^{8} + ( 4654665 - 154413 \beta_{1} + 135 \beta_{2} - 729 \beta_{3} ) q^{9} + ( -4661640 + 3130 \beta_{1} - 9390 \beta_{2} - 3130 \beta_{3} ) q^{10} + ( 618715 \beta_{1} + 22023 \beta_{2} - 7341 \beta_{3} ) q^{11} + ( 143123652 - 463277 \beta_{1} + 367 \beta_{2} - 2187 \beta_{3} ) q^{12} + ( -395182606 - 10972 \beta_{1} + 32916 \beta_{2} + 10972 \beta_{3} ) q^{13} + ( -3365950 \beta_{1} - 171864 \beta_{2} + 57288 \beta_{3} ) q^{14} + ( 1707489720 + 6442185 \beta_{1} - 10275 \beta_{2} + 49005 \beta_{3} ) q^{15} + ( -3640317152 - 59304 \beta_{1} + 177912 \beta_{2} + 59304 \beta_{3} ) q^{16} + ( -10252764 \beta_{1} + 569844 \beta_{2} - 189948 \beta_{3} ) q^{17} + ( 10576244520 + 8793279 \beta_{1} + 56214 \beta_{2} - 161838 \beta_{3} ) q^{18} + ( -14029279090 + 642849 \beta_{1} - 1928547 \beta_{2} - 642849 \beta_{3} ) q^{19} + ( 38786240 \beta_{1} - 470160 \beta_{2} + 156720 \beta_{3} ) q^{20} + ( 31113859266 - 100640323 \beta_{1} - 29575 \beta_{2} - 474579 \beta_{3} ) q^{21} + ( -43386453000 - 2197030 \beta_{1} + 6591090 \beta_{2} + 2197030 \beta_{3} ) q^{22} + ( 160696274 \beta_{1} - 809766 \beta_{2} + 269922 \beta_{3} ) q^{23} + ( 25128893088 - 10320936 \beta_{1} - 1206816 \beta_{2} + 4819824 \beta_{3} ) q^{24} + ( -2018512175 + 2677100 \beta_{1} - 8031300 \beta_{2} - 2677100 \beta_{3} ) q^{25} + ( -525661630 \beta_{1} - 8689824 \beta_{2} + 2896608 \beta_{3} ) q^{26} + ( -79495916913 + 381455082 \beta_{1} + 6926958 \beta_{2} - 12852999 \beta_{3} ) q^{27} + ( 186713000264 + 1461558 \beta_{1} - 4384674 \beta_{2} - 1461558 \beta_{3} ) q^{28} + ( -67056155 \beta_{1} + 58162401 \beta_{2} - 19387467 \beta_{3} ) q^{29} + ( -440582279400 + 1435642290 \beta_{1} - 15563430 \beta_{2} + 6845310 \beta_{3} ) q^{30} + ( 617945289062 + 1406435 \beta_{1} - 4219305 \beta_{2} - 1406435 \beta_{3} ) q^{31} + ( -1034157312 \beta_{1} - 98873280 \beta_{2} + 32957760 \beta_{3} ) q^{32} + ( -902674487880 - 4837111995 \beta_{1} - 8940783 \beta_{2} + 29898801 \beta_{3} ) q^{33} + ( 680315403168 - 30586056 \beta_{1} + 91758168 \beta_{2} + 30586056 \beta_{3} ) q^{34} + ( 8406238750 \beta_{1} - 97110090 \beta_{2} + 32370030 \beta_{3} ) q^{35} + ( -304913531628 + 1279373103 \beta_{1} + 149915907 \beta_{2} - 38531295 \beta_{3} ) q^{36} + ( 92640803474 + 10171068 \beta_{1} - 30513204 \beta_{2} - 10171068 \beta_{3} ) q^{37} + ( -6384518782 \beta_{1} + 509136408 \beta_{2} - 169712136 \beta_{3} ) q^{38} + ( 1755542556846 - 5444069098 \beta_{1} - 356967130 \beta_{2} - 23995764 \beta_{3} ) q^{39} + ( -2948821773120 + 200036240 \beta_{1} - 600108720 \beta_{2} - 200036240 \beta_{3} ) q^{40} + ( -5574907390 \beta_{1} - 351949158 \beta_{2} + 117316386 \beta_{3} ) q^{41} + ( 6896970921960 + 34045248720 \beta_{1} + 39061974 \beta_{2} - 114870798 \beta_{3} ) q^{42} + ( -7016255515666 - 337207927 \beta_{1} + 1011623781 \beta_{2} + 337207927 \beta_{3} ) q^{43} + ( -28965427520 \beta_{1} - 296748432 \beta_{2} + 98916144 \beta_{3} ) q^{44} + ( 4698831610800 - 13469720445 \beta_{1} + 1636632135 \beta_{2} + 546017355 \beta_{3} ) q^{45} + ( -10998644876208 - 102663044 \beta_{1} + 307989132 \beta_{2} + 102663044 \beta_{3} ) q^{46} + ( 47334571156 \beta_{1} - 839718108 \beta_{2} + 279906036 \beta_{3} ) q^{47} + ( 10260489838752 - 30929705096 \beta_{1} - 3433761320 \beta_{2} - 129697848 \beta_{3} ) q^{48} + ( 7369565634291 + 340857524 \beta_{1} - 1022572572 \beta_{2} - 340857524 \beta_{3} ) q^{49} + ( 29817561025 \beta_{1} + 2120263200 \beta_{2} - 706754400 \beta_{3} ) q^{50} + ( -20710393805664 - 58664684772 \beta_{1} + 320159340 \beta_{2} - 1353590676 \beta_{3} ) q^{51} + ( 10548272280104 + 429371358 \beta_{1} - 1288114074 \beta_{2} - 429371358 \beta_{3} ) q^{52} + ( 58830403419 \beta_{1} + 3653674911 \beta_{2} - 1217891637 \beta_{3} ) q^{53} + ( -26765915189076 - 18807312078 \beta_{1} + 10193718537 \beta_{2} + 614286747 \beta_{3} ) q^{54} + ( 72563413309200 + 771928300 \beta_{1} - 2315784900 \beta_{2} - 771928300 \beta_{3} ) q^{55} + ( -16497051200 \beta_{1} - 10105725168 \beta_{2} + 3368575056 \beta_{3} ) q^{56} + ( -83782277122086 + 281784759599 \beta_{1} - 16268322157 \beta_{2} + 1405910763 \beta_{3} ) q^{57} + ( 2199105495720 - 4101249250 \beta_{1} + 12303747750 \beta_{2} + 4101249250 \beta_{3} ) q^{58} + ( -709479233185 \beta_{1} + 2170421307 \beta_{2} - 723473769 \beta_{3} ) q^{59} + ( 14031610223040 - 28385648880 \beta_{1} - 901020480 \beta_{2} + 3573292320 \beta_{3} ) q^{60} + ( 90567448270802 + 688871180 \beta_{1} - 2066613540 \beta_{2} - 688871180 \beta_{3} ) q^{61} + ( 634670614082 \beta_{1} + 1113896520 \beta_{2} - 371298840 \beta_{3} ) q^{62} + ( -66674590946586 + 294488024733 \beta_{1} + 32517007929 \beta_{2} - 8281674009 \beta_{3} ) q^{63} + ( -163487435694848 + 4233528768 \beta_{1} - 12700586304 \beta_{2} - 4233528768 \beta_{3} ) q^{64} + ( 391268209150 \beta_{1} + 11086127910 \beta_{2} - 3695375970 \beta_{3} ) q^{65} + ( 333192237511320 - 1054373974830 \beta_{1} - 35734197510 \beta_{2} - 4804904610 \beta_{3} ) q^{66} + ( -6693601340866 + 5788376453 \beta_{1} - 17365129359 \beta_{2} - 5788376453 \beta_{3} ) q^{67} + ( -355339116288 \beta_{1} + 13121140032 \beta_{2} - 4373713344 \beta_{3} ) q^{68} + ( 14705793322704 - 286628665938 \beta_{1} - 3523618698 \beta_{2} + 13759763382 \beta_{3} ) q^{69} + ( -573090571544400 - 1446682300 \beta_{1} + 4340046900 \beta_{2} + 1446682300 \beta_{3} ) q^{70} + ( 1183517442630 \beta_{1} - 42502252194 \beta_{2} + 14167417398 \beta_{3} ) q^{71} + ( 599412438619200 + 192537358992 \beta_{1} + 31551116760 \beta_{2} + 1703148120 \beta_{3} ) q^{72} + ( 79407221884994 - 33109695792 \beta_{1} + 99329087376 \beta_{2} + 33109695792 \beta_{3} ) q^{73} + ( 213595144130 \beta_{1} + 8055485856 \beta_{2} - 2685161952 \beta_{3} ) q^{74} + ( -377841466256625 + 1229878500925 \beta_{1} - 11342851475 \beta_{2} + 5854817700 \beta_{3} ) q^{75} + ( -502160550127096 + 12026161606 \beta_{1} - 36078484818 \beta_{2} - 12026161606 \beta_{3} ) q^{76} + ( -6353070111850 \beta_{1} - 66799629666 \beta_{2} + 22266543222 \beta_{3} ) q^{77} + ( 384544861463880 + 2621457560280 \beta_{1} + 9439966614 \beta_{2} - 34601333598 \beta_{3} ) q^{78} + ( 1198504291296230 + 39564406019 \beta_{1} - 118693218057 \beta_{2} - 39564406019 \beta_{3} ) q^{79} + ( 1971904217600 \beta_{1} + 127616296320 \beta_{2} - 42538765440 \beta_{3} ) q^{80} + ( -1611048213013599 - 1983465554670 \beta_{1} - 101573452278 \beta_{2} + 21122653986 \beta_{3} ) q^{81} + ( 397278120330000 + 30797930380 \beta_{1} - 92393791140 \beta_{2} - 30797930380 \beta_{3} ) q^{82} + ( 8311670714089 \beta_{1} + 195905782413 \beta_{2} - 65301927471 \beta_{3} ) q^{83} + ( -302630975549784 + 859264991102 \beta_{1} + 181622393750 \beta_{2} + 3196427346 \beta_{3} ) q^{84} + ( 1999998523143360 - 62226401520 \beta_{1} + 186679204560 \beta_{2} + 62226401520 \beta_{3} ) q^{85} + ( -11026332183550 \beta_{1} - 267068678184 \beta_{2} + 89022892728 \beta_{3} ) q^{86} + ( -2212542398810520 - 8467621176045 \beta_{1} + 12110061039 \beta_{2} - 58824632673 \beta_{3} ) q^{87} + ( -842572473526080 - 93752159600 \beta_{1} + 281256478800 \beta_{2} + 93752159600 \beta_{3} ) q^{88} + ( 17987355800910 \beta_{1} - 302230367082 \beta_{2} + 100743455694 \beta_{3} ) q^{89} + ( 887283661326840 - 1778617653030 \beta_{1} - 471182776110 \beta_{2} + 131111284230 \beta_{3} ) q^{90} + ( 2332134557914252 + 94371880694 \beta_{1} - 283115642082 \beta_{2} - 94371880694 \beta_{3} ) q^{91} + ( -1688122782592 \beta_{1} - 134377955424 \beta_{2} + 44792651808 \beta_{3} ) q^{92} + ( -516051849190446 + 1265131825097 \beta_{1} + 613046675957 \beta_{2} + 3075873345 \beta_{3} ) q^{93} + ( -3214899032416992 + 12845226584 \beta_{1} - 38535679752 \beta_{2} - 12845226584 \beta_{3} ) q^{94} + ( -26456773272290 \beta_{1} + 1023699731670 \beta_{2} - 341233243890 \beta_{3} ) q^{95} + ( 3876907639949568 + 17301662760384 \beta_{1} + 3524620032 \beta_{2} + 6998928768 \beta_{3} ) q^{96} + ( -3458448750650446 + 66747558308 \beta_{1} - 200242674924 \beta_{2} - 66747558308 \beta_{3} ) q^{97} + ( 11423043309699 \beta_{1} + 269959159008 \beta_{2} - 89986386336 \beta_{3} ) q^{98} + ( 4735794274334640 + 12441440722815 \beta_{1} - 763573764357 \beta_{2} - 374876825841 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2052q^{3} - 12464q^{4} - 403056q^{6} - 3141544q^{7} + 18618660q^{9} + O(q^{10}) \) \( 4q - 2052q^{3} - 12464q^{4} - 403056q^{6} - 3141544q^{7} + 18618660q^{9} - 18646560q^{10} + 572494608q^{12} - 1580730424q^{13} + 6829958880q^{15} - 14561268608q^{16} + 42304978080q^{18} - 56117116360q^{19} + 124455437064q^{21} - 173545812000q^{22} + 100515572352q^{24} - 8074048700q^{25} - 317983667652q^{27} + 746852001056q^{28} - 1762329117600q^{30} + 2471781156248q^{31} - 3610697951520q^{33} + 2721261612672q^{34} - 1219654126512q^{36} + 370563213896q^{37} + 7022170227384q^{39} - 11795287092480q^{40} + 27587883687840q^{42} - 28065022062664q^{43} + 18795326443200q^{45} - 43994579504832q^{46} + 41041959355008q^{48} + 29478262537164q^{49} - 82841575222656q^{51} + 42193089120416q^{52} - 107063660756304q^{54} + 290253653236800q^{55} - 335129108488344q^{57} + 8796421982880q^{58} + 56126440892160q^{60} + 362269793083208q^{61} - 266698363786344q^{63} - 653949742779392q^{64} + 1332768950045280q^{66} - 26774405363464q^{67} + 58823173290816q^{69} - 2292362286177600q^{70} + 2397649754476800q^{72} + 317628887539976q^{73} - 1511365865026500q^{75} - 2008642200508384q^{76} + 1538179445855520q^{78} + 4794017165184920q^{79} - 6444192852054396q^{81} + 1589112481320000q^{82} - 1210523902199136q^{84} + 7999994092573440q^{85} - 8850169595242080q^{87} - 3370289894104320q^{88} + 3549134645307360q^{90} + 9328538231657008q^{91} - 2064207396761784q^{93} - 12859596129667968q^{94} + 15507630559798272q^{96} - 13833795002601784q^{97} + 18943177097338560q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3814 x^{2} + 2981440\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 6 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{3} + 132 \nu^{2} - 6690 \nu + 251724 \)\()/22\)
\(\beta_{3}\)\(=\)\((\)\( 9 \nu^{3} + 396 \nu^{2} + 20202 \nu + 755172 \)\()/22\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} - \beta_{1} - 68652\)\()/36\)
\(\nu^{3}\)\(=\)\((\)\(11 \beta_{3} - 33 \beta_{2} - 3356 \beta_{1}\)\()/9\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
52.1196i
33.1293i
33.1293i
52.1196i
312.717i −5369.70 3770.02i −32256.2 276758.i −1.17895e6 + 1.67920e6i −7.10881e6 1.04072e7i 1.46206e7 + 4.04878e7i 8.65472e7
2.2 198.776i 4343.70 + 4917.21i 26024.2 482304.i 977423. 863422.i 5.53804e6 1.81999e7i −5.31128e6 + 4.27178e7i −9.58704e7
2.3 198.776i 4343.70 4917.21i 26024.2 482304.i 977423. + 863422.i 5.53804e6 1.81999e7i −5.31128e6 4.27178e7i −9.58704e7
2.4 312.717i −5369.70 + 3770.02i −32256.2 276758.i −1.17895e6 1.67920e6i −7.10881e6 1.04072e7i 1.46206e7 4.04878e7i 8.65472e7
\(n\): 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
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{17}^{\mathrm{new}}(3, [\chi])\).