# Properties

 Label 6.15.b.a Level $6$ Weight $15$ Character orbit 6.b Analytic conductor $7.460$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6 = 2 \cdot 3$$ Weight: $$k$$ $$=$$ $$15$$ Character orbit: $$[\chi]$$ $$=$$ 6.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.45973808911$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-35})$$ Defining polynomial: $$x^{4} - 2 x^{3} + 23 x^{2} - 22 x + 51$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{15}\cdot 3^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -819 - 7 \beta_{1} - \beta_{2} ) q^{3} -8192 q^{4} + ( 351 \beta_{1} + 38 \beta_{2} - 19 \beta_{3} ) q^{5} + ( 59904 - 811 \beta_{1} + 32 \beta_{2} - 72 \beta_{3} ) q^{6} + ( -413518 - 111 \beta_{1} + 148 \beta_{2} + 259 \beta_{3} ) q^{7} -8192 \beta_{1} q^{8} + ( -538407 - 28341 \beta_{1} + 1170 \beta_{2} + 1053 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -819 - 7 \beta_{1} - \beta_{2} ) q^{3} -8192 q^{4} + ( 351 \beta_{1} + 38 \beta_{2} - 19 \beta_{3} ) q^{5} + ( 59904 - 811 \beta_{1} + 32 \beta_{2} - 72 \beta_{3} ) q^{6} + ( -413518 - 111 \beta_{1} + 148 \beta_{2} + 259 \beta_{3} ) q^{7} -8192 \beta_{1} q^{8} + ( -538407 - 28341 \beta_{1} + 1170 \beta_{2} + 1053 \beta_{3} ) q^{9} + ( -2933760 - 912 \beta_{1} + 1216 \beta_{2} + 2128 \beta_{3} ) q^{10} + ( -85071 \beta_{1} + 12830 \beta_{2} - 6415 \beta_{3} ) q^{11} + ( 6709248 + 57344 \beta_{1} + 8192 \beta_{2} ) q^{12} + ( -62053030 - 780 \beta_{1} + 1040 \beta_{2} + 1820 \beta_{3} ) q^{13} + ( -406414 \beta_{1} - 37888 \beta_{2} + 18944 \beta_{3} ) q^{14} + ( 130044960 + 691635 \beta_{1} - 28554 \beta_{2} - 25785 \beta_{3} ) q^{15} + 67108864 q^{16} + ( 2236668 \beta_{1} - 17912 \beta_{2} + 8956 \beta_{3} ) q^{17} + ( 227017728 - 514071 \beta_{1} - 172224 \beta_{2} + 117936 \beta_{3} ) q^{18} + ( -313016902 + 155127 \beta_{1} - 206836 \beta_{2} - 361963 \beta_{3} ) q^{19} + ( -2875392 \beta_{1} - 311296 \beta_{2} + 155648 \beta_{3} ) q^{20} + ( -628041078 + 16150357 \beta_{1} + 569362 \beta_{2} - 350649 \beta_{3} ) q^{21} + ( 677194752 - 307920 \beta_{1} + 410560 \beta_{2} + 718480 \beta_{3} ) q^{22} + ( -43220418 \beta_{1} + 1492132 \beta_{2} - 746066 \beta_{3} ) q^{23} + ( -490733568 + 6643712 \beta_{1} - 262144 \beta_{2} + 589824 \beta_{3} ) q^{24} + ( -252624215 - 653220 \beta_{1} + 870960 \beta_{2} + 1524180 \beta_{3} ) q^{25} + ( -62003110 \beta_{1} - 266240 \beta_{2} + 133120 \beta_{3} ) q^{26} + ( -4711196763 + 102870324 \beta_{1} - 47223 \beta_{2} + 270459 \beta_{3} ) q^{27} + ( 3387539456 + 909312 \beta_{1} - 1212416 \beta_{2} - 2121728 \beta_{3} ) q^{28} + ( -120388323 \beta_{1} - 1523374 \beta_{2} + 761687 \beta_{3} ) q^{29} + ( -5539968000 + 129448272 \beta_{1} + 4214208 \beta_{2} - 2881008 \beta_{3} ) q^{30} + ( 12136121378 + 2683845 \beta_{1} - 3578460 \beta_{2} - 6262305 \beta_{3} ) q^{31} + 67108864 \beta_{1} q^{32} + ( 31712047776 + 398620941 \beta_{1} - 16155282 \beta_{2} + 5951907 \beta_{3} ) q^{33} + ( -18295271424 + 429888 \beta_{1} - 573184 \beta_{2} - 1003072 \beta_{3} ) q^{34} + ( -788334318 \beta_{1} - 29282324 \beta_{2} + 14641162 \beta_{3} ) q^{35} + ( 4410630144 + 232169472 \beta_{1} - 9584640 \beta_{2} - 8626176 \beta_{3} ) q^{36} + ( -57340274902 - 11636436 \beta_{1} + 15515248 \beta_{2} + 27151684 \beta_{3} ) q^{37} + ( -322945030 \beta_{1} + 52950016 \beta_{2} - 26475008 \beta_{3} ) q^{38} + ( 44028317970 + 527519590 \beta_{1} + 63148150 \beta_{2} - 2464020 \beta_{3} ) q^{39} + ( 24033361920 + 7471104 \beta_{1} - 9961472 \beta_{2} - 17432576 \beta_{3} ) q^{40} + ( -405730710 \beta_{1} + 56996612 \beta_{2} - 28498306 \beta_{3} ) q^{41} + ( -133043162112 - 643816742 \beta_{1} + 26663488 \beta_{2} + 29773296 \beta_{3} ) q^{42} + ( -118780954006 + 410247 \beta_{1} - 546996 \beta_{2} - 957243 \beta_{3} ) q^{43} + ( 696901632 \beta_{1} - 105103360 \beta_{2} + 52551680 \beta_{3} ) q^{44} + ( 19589532480 - 3331303389 \beta_{1} - 115835130 \beta_{2} - 6505839 \beta_{3} ) q^{45} + ( 351769749504 - 35811168 \beta_{1} + 47748224 \beta_{2} + 83559392 \beta_{3} ) q^{46} + ( 6692871900 \beta_{1} + 29988840 \beta_{2} - 14994420 \beta_{3} ) q^{47} + ( -54962159616 - 469762048 \beta_{1} - 67108864 \beta_{2} ) q^{48} + ( 136604468595 + 91800996 \beta_{1} - 122401328 \beta_{2} - 214202324 \beta_{3} ) q^{49} + ( -210818135 \beta_{1} - 222965760 \beta_{2} + 111482880 \beta_{3} ) q^{50} + ( 82597408128 - 2274132852 \beta_{1} + 90327240 \beta_{2} - 160798284 \beta_{3} ) q^{51} + ( 508338421760 + 6389760 \beta_{1} - 8519680 \beta_{2} - 14909440 \beta_{3} ) q^{52} + ( 10733013003 \beta_{1} + 38071838 \beta_{2} - 19035919 \beta_{3} ) q^{53} + ( -843146703360 - 4702164291 \beta_{1} - 33107616 \beta_{2} + 5254632 \beta_{3} ) q^{54} + ( -1548779595840 - 34883028 \beta_{1} + 46510704 \beta_{2} + 81393732 \beta_{3} ) q^{55} + ( 3329343488 \beta_{1} + 310378496 \beta_{2} - 155189248 \beta_{3} ) q^{56} + ( 1607380500978 - 16334303153 \beta_{1} + 95218594 \beta_{2} + 490046193 \beta_{3} ) q^{57} + ( 988561044480 + 36560976 \beta_{1} - 48747968 \beta_{2} - 85308944 \beta_{3} ) q^{58} + ( -8598314571 \beta_{1} + 273042214 \beta_{2} - 136521107 \beta_{3} ) q^{59} + ( -1065328312320 - 5665873920 \beta_{1} + 233914368 \beta_{2} + 211230720 \beta_{3} ) q^{60} + ( -2002126733446 - 517172004 \beta_{1} + 689562672 \beta_{2} + 1206734676 \beta_{3} ) q^{61} + ( 11964355298 \beta_{1} + 916085760 \beta_{2} - 458042880 \beta_{3} ) q^{62} + ( 3389590546146 + 4071508299 \beta_{1} + 486456696 \beta_{2} - 1099860363 \beta_{3} ) q^{63} -549755813888 q^{64} + ( -26300323530 \beta_{1} - 2453362340 \beta_{2} + 1226681170 \beta_{3} ) q^{65} + ( -3236334732288 + 32031751056 \beta_{1} - 244875072 \beta_{2} - 972719280 \beta_{3} ) q^{66} + ( -1426624547302 + 840366747 \beta_{1} - 1120488996 \beta_{2} - 1960855743 \beta_{3} ) q^{67} + ( -18322784256 \beta_{1} + 146735104 \beta_{2} - 73367552 \beta_{3} ) q^{68} + ( 1691719480512 + 73387614342 \beta_{1} - 2945315580 \beta_{2} + 3091726314 \beta_{3} ) q^{69} + ( 6503012382720 + 702775776 \beta_{1} - 937034368 \beta_{2} - 1639810144 \beta_{3} ) q^{70} + ( -5260134918 \beta_{1} + 2352370636 \beta_{2} - 1176185318 \beta_{3} ) q^{71} + ( -1859729227776 + 4211269632 \beta_{1} + 1410859008 \beta_{2} - 966131712 \beta_{3} ) q^{72} + ( 7033949980850 - 378277776 \beta_{1} + 504370368 \beta_{2} + 882648144 \beta_{3} ) q^{73} + ( -56595542998 \beta_{1} - 3971903488 \beta_{2} + 1985951744 \beta_{3} ) q^{74} + ( -5482072134315 + 79776555125 \beta_{1} + 1169745095 \beta_{2} - 2063521980 \beta_{3} ) q^{75} + ( 2564234461184 - 1270800384 \beta_{1} + 1694400512 \beta_{2} + 2965200896 \beta_{3} ) q^{76} + ( -183428926746 \beta_{1} - 2173410212 \beta_{2} + 1086705106 \beta_{3} ) q^{77} + ( -4478053432320 + 43444284130 \beta_{1} - 1705346240 \beta_{2} + 4467818160 \beta_{3} ) q^{78} + ( -20073013180798 + 303345813 \beta_{1} - 404461084 \beta_{2} - 707806897 \beta_{3} ) q^{79} + ( 23555211264 \beta_{1} + 2550136832 \beta_{2} - 1275068416 \beta_{3} ) q^{80} + ( 8858907955377 - 44721850734 \beta_{1} + 8285513652 \beta_{2} - 7670399490 \beta_{3} ) q^{81} + ( 3236199180288 - 1367918688 \beta_{1} + 1823891584 \beta_{2} + 3191810272 \beta_{3} ) q^{82} + ( 144414004035 \beta_{1} + 20943453402 \beta_{2} - 10471726701 \beta_{3} ) q^{83} + ( 5144912510976 - 132303724544 \beta_{1} - 4664213504 \beta_{2} + 2872516608 \beta_{3} ) q^{84} + ( -4051155029760 - 1882824624 \beta_{1} + 2510432832 \beta_{2} + 4393257456 \beta_{3} ) q^{85} + ( -118807209814 \beta_{1} + 140030976 \beta_{2} - 70015488 \beta_{3} ) q^{86} + ( -11582168050080 + 58496405145 \beta_{1} - 2257453758 \beta_{2} + 8688524805 \beta_{3} ) q^{87} + ( -5547579408384 + 2522480640 \beta_{1} - 3363307520 \beta_{2} - 5885788160 \beta_{3} ) q^{88} + ( 60399863214 \beta_{1} - 34631630756 \beta_{2} + 17315815378 \beta_{3} ) q^{89} + ( 27599899253760 + 20308026672 \beta_{1} + 4539471552 \beta_{2} - 8548316208 \beta_{3} ) q^{90} + ( 30184258517140 + 7210430370 \beta_{1} - 9613907160 \beta_{2} - 16824337530 \beta_{3} ) q^{91} + ( 354061664256 \beta_{1} - 12223545344 \beta_{2} + 6111772672 \beta_{3} ) q^{92} + ( 13434444757818 - 405460303391 \beta_{1} - 15904239758 \beta_{2} + 8478266355 \beta_{3} ) q^{93} + ( -54874069463040 - 719732160 \beta_{1} + 959642880 \beta_{2} + 1679375040 \beta_{3} ) q^{94} + ( 789014468898 \beta_{1} + 7068082204 \beta_{2} - 3534041102 \beta_{3} ) q^{95} + ( 4020089389056 - 54425288704 \beta_{1} + 2147483648 \beta_{2} - 4831838208 \beta_{3} ) q^{96} + ( 44561041643714 - 5945558604 \beta_{1} + 7927411472 \beta_{2} + 13872970076 \beta_{3} ) q^{97} + ( 130729204851 \beta_{1} + 31334739968 \beta_{2} - 15667369984 \beta_{3} ) q^{98} + ( -39602152594368 - 1020098811339 \beta_{1} - 4048290306 \beta_{2} - 26205950331 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3276q^{3} - 32768q^{4} + 239616q^{6} - 1654072q^{7} - 2153628q^{9} + O(q^{10})$$ $$4q - 3276q^{3} - 32768q^{4} + 239616q^{6} - 1654072q^{7} - 2153628q^{9} - 11735040q^{10} + 26836992q^{12} - 248212120q^{13} + 520179840q^{15} + 268435456q^{16} + 908070912q^{18} - 1252067608q^{19} - 2512164312q^{21} + 2708779008q^{22} - 1962934272q^{24} - 1010496860q^{25} - 18844787052q^{27} + 13550157824q^{28} - 22159872000q^{30} + 48544485512q^{31} + 126848191104q^{33} - 73181085696q^{34} + 17642520576q^{36} - 229361099608q^{37} + 176113271880q^{39} + 96133447680q^{40} - 532172648448q^{42} - 475123816024q^{43} + 78358129920q^{45} + 1407078998016q^{46} - 219848638464q^{48} + 546417874380q^{49} + 330389632512q^{51} + 2033353687040q^{52} - 3372586813440q^{54} - 6195118383360q^{55} + 6429522003912q^{57} + 3954244177920q^{58} - 4261313249280q^{60} - 8008506933784q^{61} + 13558362184584q^{63} - 2199023255552q^{64} - 12945338929152q^{66} - 5706498189208q^{67} + 6766877922048q^{69} + 26012049530880q^{70} - 7438916911104q^{72} + 28135799923400q^{73} - 21928288537260q^{75} + 10256937844736q^{76} - 17912213729280q^{78} - 80292052723192q^{79} + 35435631821508q^{81} + 12944796721152q^{82} + 20579650043904q^{84} - 16204620119040q^{85} - 46328672200320q^{87} - 22190317633536q^{88} + 110399597015040q^{90} + 120737034068560q^{91} + 53737779031272q^{93} - 219496277852160q^{94} + 16080357556224q^{96} + 178244166574856q^{97} - 158408610377472q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} + 23 x^{2} - 22 x + 51$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$128 \nu^{3} - 192 \nu^{2} + 1984 \nu - 960$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$-968 \nu^{3} + 5340 \nu^{2} - 32500 \nu + 56832$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$608 \nu^{3} + 6864 \nu^{2} + 9424 \nu + 77088$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$8 \beta_{3} - 16 \beta_{2} - 159 \beta_{1} + 5184$$$$)/10368$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{3} - 19 \beta_{1} - 12096$$$$)/1152$$ $$\nu^{3}$$ $$=$$ $$($$$$-70 \beta_{3} + 248 \beta_{2} + 4395 \beta_{1} - 165888$$$$)/10368$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 − 1.54383i 0.5 + 4.37225i 0.5 + 1.54383i 0.5 − 4.37225i
90.5097i −2023.79 829.000i −8192.00 40425.0i −75032.5 + 183173.i 388872. 741455.i 3.40849e6 + 3.35545e6i 3.65885e6
5.2 90.5097i 385.790 + 2152.70i −8192.00 105253.i 194841. 34917.8i −1.21591e6 741455.i −4.48530e6 + 1.66099e6i −9.52637e6
5.3 90.5097i −2023.79 + 829.000i −8192.00 40425.0i −75032.5 183173.i 388872. 741455.i 3.40849e6 3.35545e6i 3.65885e6
5.4 90.5097i 385.790 2152.70i −8192.00 105253.i 194841. + 34917.8i −1.21591e6 741455.i −4.48530e6 1.66099e6i −9.52637e6
 $$n$$: 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 6.15.b.a 4
3.b odd 2 1 inner 6.15.b.a 4
4.b odd 2 1 48.15.e.d 4
5.b even 2 1 150.15.d.a 4
5.c odd 4 2 150.15.b.a 8
12.b even 2 1 48.15.e.d 4
15.d odd 2 1 150.15.d.a 4
15.e even 4 2 150.15.b.a 8

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{15}^{\mathrm{new}}(6, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8192 + T^{2} )^{2}$$
$3$ $$22876792454961 + 15669006444 T + 6442902 T^{2} + 3276 T^{3} + T^{4}$$
$5$ $$18103614409876377600 + 12712279680 T^{2} + T^{4}$$
$7$ $$( -472833268796 + 827036 T + T^{2} )^{2}$$
$11$ $$30\!\cdots\!44$$$$+ 1321559988292224 T^{2} + T^{4}$$
$13$ $$( 3818786760532900 + 124106060 T + T^{2} )^{2}$$
$17$ $$15\!\cdots\!44$$$$+ 84075640060225536 T^{2} + T^{4}$$
$19$ $$( -1159498178205101276 + 626033804 T + T^{2} )^{2}$$
$23$ $$47\!\cdots\!64$$$$+ 46571163794771544576 T^{2} + T^{4}$$
$29$ $$12\!\cdots\!00$$$$+$$$$25\!\cdots\!20$$$$T^{2} + T^{4}$$
$31$ $$( -$$$$22\!\cdots\!16$$$$- 24272242756 T + T^{2} )^{2}$$
$37$ $$( -$$$$37\!\cdots\!16$$$$+ 114680549804 T + T^{2} )^{2}$$
$41$ $$11\!\cdots\!64$$$$+$$$$26\!\cdots\!44$$$$T^{2} + T^{4}$$
$43$ $$($$$$14\!\cdots\!56$$$$+ 237561908012 T + T^{2} )^{2}$$
$47$ $$13\!\cdots\!00$$$$+$$$$74\!\cdots\!00$$$$T^{2} + T^{4}$$
$53$ $$88\!\cdots\!24$$$$+$$$$19\!\cdots\!96$$$$T^{2} + T^{4}$$
$59$ $$10\!\cdots\!64$$$$+$$$$17\!\cdots\!24$$$$T^{2} + T^{4}$$
$61$ $$( -$$$$99\!\cdots\!04$$$$+ 4004253466892 T + T^{2} )^{2}$$
$67$ $$( -$$$$34\!\cdots\!76$$$$+ 2853249094604 T + T^{2} )^{2}$$
$71$ $$40\!\cdots\!00$$$$+$$$$41\!\cdots\!20$$$$T^{2} + T^{4}$$
$73$ $$($$$$41\!\cdots\!80$$$$- 14067899961700 T + T^{2} )^{2}$$
$79$ $$($$$$39\!\cdots\!24$$$$+ 40146026361596 T + T^{2} )^{2}$$
$83$ $$20\!\cdots\!04$$$$+$$$$35\!\cdots\!64$$$$T^{2} + T^{4}$$
$89$ $$19\!\cdots\!64$$$$+$$$$88\!\cdots\!24$$$$T^{2} + T^{4}$$
$97$ $$($$$$13\!\cdots\!76$$$$- 89122083287428 T + T^{2} )^{2}$$