# 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 $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.86973631570$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Defining polynomial: $$x^{4} + 3814x^{2} + 2981440$$ x^4 + 3814*x^2 + 2981440 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{6}\cdot 3^{8}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 - 513) q^{3} + (\beta_{3} + 3 \beta_{2} - \beta_1 - 3116) q^{4} + (15 \beta_{3} - 45 \beta_{2} + 95 \beta_1) q^{5} + (81 \beta_{3} - 21 \beta_{2} - 2532 \beta_1 - 100764) q^{6} + (217 \beta_{3} + 651 \beta_{2} - 217 \beta_1 - 785386) q^{7} + (264 \beta_{3} - 792 \beta_{2} + 50528 \beta_1) q^{8} + ( - 729 \beta_{3} + 135 \beta_{2} - 154413 \beta_1 + 4654665) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1 - 513) * q^3 + (b3 + 3*b2 - b1 - 3116) * q^4 + (15*b3 - 45*b2 + 95*b1) * q^5 + (81*b3 - 21*b2 - 2532*b1 - 100764) * q^6 + (217*b3 + 651*b2 - 217*b1 - 785386) * q^7 + (264*b3 - 792*b2 + 50528*b1) * q^8 + (-729*b3 + 135*b2 - 154413*b1 + 4654665) * q^9 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1 - 513) q^{3} + (\beta_{3} + 3 \beta_{2} - \beta_1 - 3116) q^{4} + (15 \beta_{3} - 45 \beta_{2} + 95 \beta_1) q^{5} + (81 \beta_{3} - 21 \beta_{2} - 2532 \beta_1 - 100764) q^{6} + (217 \beta_{3} + 651 \beta_{2} - 217 \beta_1 - 785386) q^{7} + (264 \beta_{3} - 792 \beta_{2} + 50528 \beta_1) q^{8} + ( - 729 \beta_{3} + 135 \beta_{2} - 154413 \beta_1 + 4654665) q^{9} + ( - 3130 \beta_{3} - 9390 \beta_{2} + 3130 \beta_1 - 4661640) q^{10} + ( - 7341 \beta_{3} + 22023 \beta_{2} + 618715 \beta_1) q^{11} + ( - 2187 \beta_{3} + 367 \beta_{2} - 463277 \beta_1 + 143123652) q^{12} + (10972 \beta_{3} + 32916 \beta_{2} - 10972 \beta_1 - 395182606) q^{13} + (57288 \beta_{3} - 171864 \beta_{2} - 3365950 \beta_1) q^{14} + (49005 \beta_{3} - 10275 \beta_{2} + 6442185 \beta_1 + 1707489720) q^{15} + (59304 \beta_{3} + 177912 \beta_{2} - 59304 \beta_1 - 3640317152) q^{16} + ( - 189948 \beta_{3} + 569844 \beta_{2} - 10252764 \beta_1) q^{17} + ( - 161838 \beta_{3} + 56214 \beta_{2} + \cdots + 10576244520) q^{18}+ \cdots + ( - 374876825841 \beta_{3} + \cdots + 47\!\cdots\!40) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1 - 513) * q^3 + (b3 + 3*b2 - b1 - 3116) * q^4 + (15*b3 - 45*b2 + 95*b1) * q^5 + (81*b3 - 21*b2 - 2532*b1 - 100764) * q^6 + (217*b3 + 651*b2 - 217*b1 - 785386) * q^7 + (264*b3 - 792*b2 + 50528*b1) * q^8 + (-729*b3 + 135*b2 - 154413*b1 + 4654665) * q^9 + (-3130*b3 - 9390*b2 + 3130*b1 - 4661640) * q^10 + (-7341*b3 + 22023*b2 + 618715*b1) * q^11 + (-2187*b3 + 367*b2 - 463277*b1 + 143123652) * q^12 + (10972*b3 + 32916*b2 - 10972*b1 - 395182606) * q^13 + (57288*b3 - 171864*b2 - 3365950*b1) * q^14 + (49005*b3 - 10275*b2 + 6442185*b1 + 1707489720) * q^15 + (59304*b3 + 177912*b2 - 59304*b1 - 3640317152) * q^16 + (-189948*b3 + 569844*b2 - 10252764*b1) * q^17 + (-161838*b3 + 56214*b2 + 8793279*b1 + 10576244520) * q^18 + (-642849*b3 - 1928547*b2 + 642849*b1 - 14029279090) * q^19 + (156720*b3 - 470160*b2 + 38786240*b1) * q^20 + (-474579*b3 - 29575*b2 - 100640323*b1 + 31113859266) * q^21 + (2197030*b3 + 6591090*b2 - 2197030*b1 - 43386453000) * q^22 + (269922*b3 - 809766*b2 + 160696274*b1) * q^23 + (4819824*b3 - 1206816*b2 - 10320936*b1 + 25128893088) * q^24 + (-2677100*b3 - 8031300*b2 + 2677100*b1 - 2018512175) * q^25 + (2896608*b3 - 8689824*b2 - 525661630*b1) * q^26 + (-12852999*b3 + 6926958*b2 + 381455082*b1 - 79495916913) * q^27 + (-1461558*b3 - 4384674*b2 + 1461558*b1 + 186713000264) * q^28 + (-19387467*b3 + 58162401*b2 - 67056155*b1) * q^29 + (6845310*b3 - 15563430*b2 + 1435642290*b1 - 440582279400) * q^30 + (-1406435*b3 - 4219305*b2 + 1406435*b1 + 617945289062) * q^31 + (32957760*b3 - 98873280*b2 - 1034157312*b1) * q^32 + (29898801*b3 - 8940783*b2 - 4837111995*b1 - 902674487880) * q^33 + (30586056*b3 + 91758168*b2 - 30586056*b1 + 680315403168) * q^34 + (32370030*b3 - 97110090*b2 + 8406238750*b1) * q^35 + (-38531295*b3 + 149915907*b2 + 1279373103*b1 - 304913531628) * q^36 + (-10171068*b3 - 30513204*b2 + 10171068*b1 + 92640803474) * q^37 + (-169712136*b3 + 509136408*b2 - 6384518782*b1) * q^38 + (-23995764*b3 - 356967130*b2 - 5444069098*b1 + 1755542556846) * q^39 + (-200036240*b3 - 600108720*b2 + 200036240*b1 - 2948821773120) * q^40 + (117316386*b3 - 351949158*b2 - 5574907390*b1) * q^41 + (-114870798*b3 + 39061974*b2 + 34045248720*b1 + 6896970921960) * q^42 + (337207927*b3 + 1011623781*b2 - 337207927*b1 - 7016255515666) * q^43 + (98916144*b3 - 296748432*b2 - 28965427520*b1) * q^44 + (546017355*b3 + 1636632135*b2 - 13469720445*b1 + 4698831610800) * q^45 + (102663044*b3 + 307989132*b2 - 102663044*b1 - 10998644876208) * q^46 + (279906036*b3 - 839718108*b2 + 47334571156*b1) * q^47 + (-129697848*b3 - 3433761320*b2 - 30929705096*b1 + 10260489838752) * q^48 + (-340857524*b3 - 1022572572*b2 + 340857524*b1 + 7369565634291) * q^49 + (-706754400*b3 + 2120263200*b2 + 29817561025*b1) * q^50 + (-1353590676*b3 + 320159340*b2 - 58664684772*b1 - 20710393805664) * q^51 + (-429371358*b3 - 1288114074*b2 + 429371358*b1 + 10548272280104) * q^52 + (-1217891637*b3 + 3653674911*b2 + 58830403419*b1) * q^53 + (614286747*b3 + 10193718537*b2 - 18807312078*b1 - 26765915189076) * q^54 + (-771928300*b3 - 2315784900*b2 + 771928300*b1 + 72563413309200) * q^55 + (3368575056*b3 - 10105725168*b2 - 16497051200*b1) * q^56 + (1405910763*b3 - 16268322157*b2 + 281784759599*b1 - 83782277122086) * q^57 + (4101249250*b3 + 12303747750*b2 - 4101249250*b1 + 2199105495720) * q^58 + (-723473769*b3 + 2170421307*b2 - 709479233185*b1) * q^59 + (3573292320*b3 - 901020480*b2 - 28385648880*b1 + 14031610223040) * q^60 + (-688871180*b3 - 2066613540*b2 + 688871180*b1 + 90567448270802) * q^61 + (-371298840*b3 + 1113896520*b2 + 634670614082*b1) * q^62 + (-8281674009*b3 + 32517007929*b2 + 294488024733*b1 - 66674590946586) * q^63 + (-4233528768*b3 - 12700586304*b2 + 4233528768*b1 - 163487435694848) * q^64 + (-3695375970*b3 + 11086127910*b2 + 391268209150*b1) * q^65 + (-4804904610*b3 - 35734197510*b2 - 1054373974830*b1 + 333192237511320) * q^66 + (-5788376453*b3 - 17365129359*b2 + 5788376453*b1 - 6693601340866) * q^67 + (-4373713344*b3 + 13121140032*b2 - 355339116288*b1) * q^68 + (13759763382*b3 - 3523618698*b2 - 286628665938*b1 + 14705793322704) * q^69 + (1446682300*b3 + 4340046900*b2 - 1446682300*b1 - 573090571544400) * q^70 + (14167417398*b3 - 42502252194*b2 + 1183517442630*b1) * q^71 + (1703148120*b3 + 31551116760*b2 + 192537358992*b1 + 599412438619200) * q^72 + (33109695792*b3 + 99329087376*b2 - 33109695792*b1 + 79407221884994) * q^73 + (-2685161952*b3 + 8055485856*b2 + 213595144130*b1) * q^74 + (5854817700*b3 - 11342851475*b2 + 1229878500925*b1 - 377841466256625) * q^75 + (-12026161606*b3 - 36078484818*b2 + 12026161606*b1 - 502160550127096) * q^76 + (22266543222*b3 - 66799629666*b2 - 6353070111850*b1) * q^77 + (-34601333598*b3 + 9439966614*b2 + 2621457560280*b1 + 384544861463880) * q^78 + (-39564406019*b3 - 118693218057*b2 + 39564406019*b1 + 1198504291296230) * q^79 + (-42538765440*b3 + 127616296320*b2 + 1971904217600*b1) * q^80 + (21122653986*b3 - 101573452278*b2 - 1983465554670*b1 - 1611048213013599) * q^81 + (-30797930380*b3 - 92393791140*b2 + 30797930380*b1 + 397278120330000) * q^82 + (-65301927471*b3 + 195905782413*b2 + 8311670714089*b1) * q^83 + (3196427346*b3 + 181622393750*b2 + 859264991102*b1 - 302630975549784) * q^84 + (62226401520*b3 + 186679204560*b2 - 62226401520*b1 + 1999998523143360) * q^85 + (89022892728*b3 - 267068678184*b2 - 11026332183550*b1) * q^86 + (-58824632673*b3 + 12110061039*b2 - 8467621176045*b1 - 2212542398810520) * q^87 + (93752159600*b3 + 281256478800*b2 - 93752159600*b1 - 842572473526080) * q^88 + (100743455694*b3 - 302230367082*b2 + 17987355800910*b1) * q^89 + (131111284230*b3 - 471182776110*b2 - 1778617653030*b1 + 887283661326840) * q^90 + (-94371880694*b3 - 283115642082*b2 + 94371880694*b1 + 2332134557914252) * q^91 + (44792651808*b3 - 134377955424*b2 - 1688122782592*b1) * q^92 + (3075873345*b3 + 613046675957*b2 + 1265131825097*b1 - 516051849190446) * q^93 + (-12845226584*b3 - 38535679752*b2 + 12845226584*b1 - 3214899032416992) * q^94 + (-341233243890*b3 + 1023699731670*b2 - 26456773272290*b1) * q^95 + (6998928768*b3 + 3524620032*b2 + 17301662760384*b1 + 3876907639949568) * q^96 + (-66747558308*b3 - 200242674924*b2 + 66747558308*b1 - 3458448750650446) * q^97 + (-89986386336*b3 + 269959159008*b2 + 11423043309699*b1) * q^98 + (-374876825841*b3 - 763573764357*b2 + 12441440722815*b1 + 4735794274334640) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2052 q^{3} - 12464 q^{4} - 403056 q^{6} - 3141544 q^{7} + 18618660 q^{9}+O(q^{10})$$ 4 * q - 2052 * q^3 - 12464 * q^4 - 403056 * q^6 - 3141544 * q^7 + 18618660 * q^9 $$4 q - 2052 q^{3} - 12464 q^{4} - 403056 q^{6} - 3141544 q^{7} + 18618660 q^{9} - 18646560 q^{10} + 572494608 q^{12} - 1580730424 q^{13} + 6829958880 q^{15} - 14561268608 q^{16} + 42304978080 q^{18} - 56117116360 q^{19} + 124455437064 q^{21} - 173545812000 q^{22} + 100515572352 q^{24} - 8074048700 q^{25} - 317983667652 q^{27} + 746852001056 q^{28} - 1762329117600 q^{30} + 2471781156248 q^{31} - 3610697951520 q^{33} + 2721261612672 q^{34} - 1219654126512 q^{36} + 370563213896 q^{37} + 7022170227384 q^{39} - 11795287092480 q^{40} + 27587883687840 q^{42} - 28065022062664 q^{43} + 18795326443200 q^{45} - 43994579504832 q^{46} + 41041959355008 q^{48} + 29478262537164 q^{49} - 82841575222656 q^{51} + 42193089120416 q^{52} - 107063660756304 q^{54} + 290253653236800 q^{55} - 335129108488344 q^{57} + 8796421982880 q^{58} + 56126440892160 q^{60} + 362269793083208 q^{61} - 266698363786344 q^{63} - 653949742779392 q^{64} + 13\!\cdots\!80 q^{66}+ \cdots + 18\!\cdots\!60 q^{99}+O(q^{100})$$ 4 * q - 2052 * q^3 - 12464 * q^4 - 403056 * q^6 - 3141544 * q^7 + 18618660 * q^9 - 18646560 * q^10 + 572494608 * q^12 - 1580730424 * q^13 + 6829958880 * q^15 - 14561268608 * q^16 + 42304978080 * q^18 - 56117116360 * q^19 + 124455437064 * q^21 - 173545812000 * q^22 + 100515572352 * q^24 - 8074048700 * q^25 - 317983667652 * q^27 + 746852001056 * q^28 - 1762329117600 * q^30 + 2471781156248 * q^31 - 3610697951520 * q^33 + 2721261612672 * q^34 - 1219654126512 * q^36 + 370563213896 * q^37 + 7022170227384 * q^39 - 11795287092480 * q^40 + 27587883687840 * q^42 - 28065022062664 * q^43 + 18795326443200 * q^45 - 43994579504832 * q^46 + 41041959355008 * q^48 + 29478262537164 * q^49 - 82841575222656 * q^51 + 42193089120416 * q^52 - 107063660756304 * q^54 + 290253653236800 * q^55 - 335129108488344 * q^57 + 8796421982880 * q^58 + 56126440892160 * q^60 + 362269793083208 * q^61 - 266698363786344 * q^63 - 653949742779392 * q^64 + 1332768950045280 * q^66 - 26774405363464 * q^67 + 58823173290816 * q^69 - 2292362286177600 * q^70 + 2397649754476800 * q^72 + 317628887539976 * q^73 - 1511365865026500 * q^75 - 2008642200508384 * q^76 + 1538179445855520 * q^78 + 4794017165184920 * q^79 - 6444192852054396 * q^81 + 1589112481320000 * q^82 - 1210523902199136 * q^84 + 7999994092573440 * q^85 - 8850169595242080 * q^87 - 3370289894104320 * q^88 + 3549134645307360 * q^90 + 9328538231657008 * q^91 - 2064207396761784 * q^93 - 12859596129667968 * q^94 + 15507630559798272 * q^96 - 13833795002601784 * q^97 + 18943177097338560 * q^99

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

 $$\beta_{1}$$ $$=$$ $$6\nu$$ 6*v $$\beta_{2}$$ $$=$$ $$( -3\nu^{3} + 132\nu^{2} - 6690\nu + 251724 ) / 22$$ (-3*v^3 + 132*v^2 - 6690*v + 251724) / 22 $$\beta_{3}$$ $$=$$ $$( 9\nu^{3} + 396\nu^{2} + 20202\nu + 755172 ) / 22$$ (9*v^3 + 396*v^2 + 20202*v + 755172) / 22
 $$\nu$$ $$=$$ $$( \beta_1 ) / 6$$ (b1) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 3\beta_{2} - \beta _1 - 68652 ) / 36$$ (b3 + 3*b2 - b1 - 68652) / 36 $$\nu^{3}$$ $$=$$ $$( 11\beta_{3} - 33\beta_{2} - 3356\beta_1 ) / 9$$ (11*b3 - 33*b2 - 3356*b1) / 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.17.b.a 4
3.b odd 2 1 inner 3.17.b.a 4
4.b odd 2 1 48.17.e.b 4
5.b even 2 1 75.17.c.d 4
5.c odd 4 2 75.17.d.b 8
12.b even 2 1 48.17.e.b 4
15.d odd 2 1 75.17.c.d 4
15.e even 4 2 75.17.d.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.17.b.a 4 1.a even 1 1 trivial
3.17.b.a 4 3.b odd 2 1 inner
48.17.e.b 4 4.b odd 2 1
48.17.e.b 4 12.b even 2 1
75.17.c.d 4 5.b even 2 1
75.17.c.d 4 15.d odd 2 1
75.17.d.b 8 5.c odd 4 2
75.17.d.b 8 15.e even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{17}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 137304 T^{2} + \cdots + 3863946240$$
$3$ $$T^{4} + 2052 T^{3} + \cdots + 18\!\cdots\!41$$
$5$ $$T^{4} + 309212805600 T^{2} + \cdots + 17\!\cdots\!00$$
$7$ $$(T^{2} + 1570772 T - 39368833865900)^{2}$$
$11$ $$T^{4} + \cdots + 12\!\cdots\!00$$
$13$ $$(T^{2} + 790365212 T + 53\!\cdots\!60)^{2}$$
$17$ $$T^{4} + \cdots + 28\!\cdots\!60$$
$19$ $$(T^{2} + 28058558180 T - 15\!\cdots\!64)^{2}$$
$23$ $$T^{4} + \cdots + 32\!\cdots\!40$$
$29$ $$T^{4} + \cdots + 52\!\cdots\!00$$
$31$ $$(T^{2} - 1235890578124 T + 38\!\cdots\!44)^{2}$$
$37$ $$(T^{2} - 185281606948 T - 79\!\cdots\!60)^{2}$$
$41$ $$T^{4} + \cdots + 10\!\cdots\!00$$
$43$ $$(T^{2} + 14032511031332 T - 47\!\cdots\!00)^{2}$$
$47$ $$T^{4} + \cdots + 26\!\cdots\!60$$
$53$ $$T^{4} + \cdots + 12\!\cdots\!60$$
$59$ $$T^{4} + \cdots + 11\!\cdots\!00$$
$61$ $$(T^{2} - 181134896541604 T + 77\!\cdots\!04)^{2}$$
$67$ $$(T^{2} + 13387202681732 T - 28\!\cdots\!20)^{2}$$
$71$ $$T^{4} + \cdots + 29\!\cdots\!00$$
$73$ $$(T^{2} - 158814443769988 T - 92\!\cdots\!60)^{2}$$
$79$ $$(T^{2} + \cdots + 10\!\cdots\!96)^{2}$$
$83$ $$T^{4} + \cdots + 20\!\cdots\!60$$
$89$ $$T^{4} + \cdots + 56\!\cdots\!00$$
$97$ $$(T^{2} + \cdots + 81\!\cdots\!20)^{2}$$