# Properties

 Label 10.9.c.a Level 10 Weight 9 Character orbit 10.c Analytic conductor 4.074 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 10.c (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.07378610061$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{249})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -8 + 8 \beta_{1} ) q^{2} + ( 13 + 13 \beta_{1} + \beta_{2} ) q^{3} -128 \beta_{1} q^{4} + ( 29 + 39 \beta_{1} - 2 \beta_{2} + 11 \beta_{3} ) q^{5} + ( -208 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6} + ( -267 + 326 \beta_{1} + 59 \beta_{3} ) q^{7} + ( 1024 + 1024 \beta_{1} ) q^{8} + ( -3084 \beta_{1} + 27 \beta_{2} + 27 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -8 + 8 \beta_{1} ) q^{2} + ( 13 + 13 \beta_{1} + \beta_{2} ) q^{3} -128 \beta_{1} q^{4} + ( 29 + 39 \beta_{1} - 2 \beta_{2} + 11 \beta_{3} ) q^{5} + ( -208 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6} + ( -267 + 326 \beta_{1} + 59 \beta_{3} ) q^{7} + ( 1024 + 1024 \beta_{1} ) q^{8} + ( -3084 \beta_{1} + 27 \beta_{2} + 27 \beta_{3} ) q^{9} + ( -456 - 96 \beta_{1} - 72 \beta_{2} - 104 \beta_{3} ) q^{10} + ( -5094 - 131 \beta_{1} + 131 \beta_{2} - 131 \beta_{3} ) q^{11} + ( 1664 - 1792 \beta_{1} - 128 \beta_{3} ) q^{12} + ( 18305 + 18305 \beta_{1} + 242 \beta_{2} ) q^{13} + ( -4744 \beta_{1} - 472 \beta_{2} - 472 \beta_{3} ) q^{14} + ( -34219 - 5329 \beta_{1} - 153 \beta_{2} + 154 \beta_{3} ) q^{15} -16384 q^{16} + ( 49689 - 49783 \beta_{1} - 94 \beta_{3} ) q^{17} + ( 24888 + 24888 \beta_{1} - 432 \beta_{2} ) q^{18} + ( 35750 \beta_{1} + 1382 \beta_{2} + 1382 \beta_{3} ) q^{19} + ( 3584 - 3456 \beta_{1} + 1408 \beta_{2} + 256 \beta_{3} ) q^{20} + ( -190550 + 1093 \beta_{1} - 1093 \beta_{2} + 1093 \beta_{3} ) q^{21} + ( 40752 - 38656 \beta_{1} + 2096 \beta_{3} ) q^{22} + ( 159173 + 159173 \beta_{1} - 2679 \beta_{2} ) q^{23} + ( 27648 \beta_{1} + 1024 \beta_{2} + 1024 \beta_{3} ) q^{24} + ( 136985 - 362115 \beta_{1} - 805 \beta_{2} + 365 \beta_{3} ) q^{25} + ( -292880 + 1936 \beta_{1} - 1936 \beta_{2} + 1936 \beta_{3} ) q^{26} + ( 41712 - 50628 \beta_{1} - 8916 \beta_{3} ) q^{27} + ( 34176 + 34176 \beta_{1} + 7552 \beta_{2} ) q^{28} + ( 623680 \beta_{1} - 7376 \beta_{2} - 7376 \beta_{3} ) q^{29} + ( 317616 - 232344 \beta_{1} - 8 \beta_{2} - 2456 \beta_{3} ) q^{30} + ( -592950 - 10477 \beta_{1} + 10477 \beta_{2} - 10477 \beta_{3} ) q^{31} + ( 131072 - 131072 \beta_{1} ) q^{32} + ( 341450 + 341450 \beta_{1} - 1426 \beta_{2} ) q^{33} + ( 795776 \beta_{1} + 752 \beta_{2} + 752 \beta_{3} ) q^{34} + ( 351997 - 2021948 \beta_{1} - 4586 \beta_{2} - 2527 \beta_{3} ) q^{35} + ( -398208 - 3456 \beta_{1} + 3456 \beta_{2} - 3456 \beta_{3} ) q^{36} + ( -221751 + 266427 \beta_{1} + 44676 \beta_{3} ) q^{37} + ( -274944 - 274944 \beta_{1} - 22112 \beta_{2} ) q^{38} + ( 1250727 \beta_{1} + 21693 \beta_{2} + 21693 \beta_{3} ) q^{39} + ( 1024 + 67584 \beta_{1} - 13312 \beta_{2} + 9216 \beta_{3} ) q^{40} + ( -1031846 + 39317 \beta_{1} - 39317 \beta_{2} + 39317 \beta_{3} ) q^{41} + ( 1524400 - 1541888 \beta_{1} - 17488 \beta_{3} ) q^{42} + ( 1758397 + 1758397 \beta_{1} + 56473 \beta_{2} ) q^{43} + ( 635264 \beta_{1} - 16768 \beta_{2} - 16768 \beta_{3} ) q^{44} + ( -669108 - 1174878 \beta_{1} + 33654 \beta_{2} + 7653 \beta_{3} ) q^{45} + ( -2546768 - 21432 \beta_{1} + 21432 \beta_{2} - 21432 \beta_{3} ) q^{46} + ( -864267 + 785826 \beta_{1} - 78441 \beta_{3} ) q^{47} + ( -212992 - 212992 \beta_{1} - 16384 \beta_{2} ) q^{48} + ( -5245636 \beta_{1} - 34987 \beta_{2} - 34987 \beta_{3} ) q^{49} + ( 1803960 + 3986360 \beta_{1} + 3520 \beta_{2} - 9360 \beta_{3} ) q^{50} + ( 1584442 - 51005 \beta_{1} + 51005 \beta_{2} - 51005 \beta_{3} ) q^{51} + ( 2343040 - 2374016 \beta_{1} - 30976 \beta_{3} ) q^{52} + ( 2576185 + 2576185 \beta_{1} - 157788 \beta_{2} ) q^{53} + ( 738720 \beta_{1} + 71328 \beta_{2} + 71328 \beta_{3} ) q^{54} + ( -5447462 + 3473133 \beta_{1} + 17131 \beta_{2} - 53283 \beta_{3} ) q^{55} + ( -546816 + 60416 \beta_{1} - 60416 \beta_{2} + 60416 \beta_{3} ) q^{56} + ( -4747568 + 4820632 \beta_{1} + 73064 \beta_{3} ) q^{57} + ( -5048448 - 5048448 \beta_{1} + 118016 \beta_{2} ) q^{58} + ( -7994090 \beta_{1} - 40842 \beta_{2} - 40842 \beta_{3} ) q^{59} + ( -701824 + 4399616 \beta_{1} + 19712 \beta_{2} + 19584 \beta_{3} ) q^{60} + ( 21358778 + 14501 \beta_{1} - 14501 \beta_{2} + 14501 \beta_{3} ) q^{61} + ( 4743600 - 4575968 \beta_{1} + 167632 \beta_{3} ) q^{62} + ( -4126779 - 4126779 \beta_{1} + 165945 \beta_{2} ) q^{63} + 2097152 \beta_{1} q^{64} + ( -8265839 - 289124 \beta_{1} - 234093 \beta_{2} + 173699 \beta_{3} ) q^{65} + ( -5463200 - 11408 \beta_{1} + 11408 \beta_{2} - 11408 \beta_{3} ) q^{66} + ( -14314243 + 14283490 \beta_{1} - 30753 \beta_{3} ) q^{67} + ( -6360192 - 6360192 \beta_{1} - 12032 \beta_{2} ) q^{68} + ( -4076883 \beta_{1} + 121667 \beta_{2} + 121667 \beta_{3} ) q^{69} + ( 13339392 + 18954872 \beta_{1} + 56904 \beta_{2} - 16472 \beta_{3} ) q^{70} + ( 28278218 + 27211 \beta_{1} - 27211 \beta_{2} + 27211 \beta_{3} ) q^{71} + ( 3185664 - 3130368 \beta_{1} + 55296 \beta_{3} ) q^{72} + ( -15176175 - 15176175 \beta_{1} - 85528 \beta_{2} ) q^{73} + ( -3905424 \beta_{1} - 357408 \beta_{2} - 357408 \beta_{3} ) q^{74} + ( 5357165 - 5805235 \beta_{1} + 120605 \beta_{2} - 368640 \beta_{3} ) q^{75} + ( 4399104 - 176896 \beta_{1} + 176896 \beta_{2} - 176896 \beta_{3} ) q^{76} + ( -22692550 + 22477416 \beta_{1} - 215134 \beta_{3} ) q^{77} + ( -9832272 - 9832272 \beta_{1} - 347088 \beta_{2} ) q^{78} + ( -11147280 \beta_{1} + 180272 \beta_{2} + 180272 \beta_{3} ) q^{79} + ( -475136 - 638976 \beta_{1} + 32768 \beta_{2} - 180224 \beta_{3} ) q^{80} + ( 8419833 - 343683 \beta_{1} + 343683 \beta_{2} - 343683 \beta_{3} ) q^{81} + ( 8254768 - 8883840 \beta_{1} - 629072 \beta_{3} ) q^{82} + ( 17681821 + 17681821 \beta_{1} + 161965 \beta_{2} ) q^{83} + ( 24530304 \beta_{1} + 139904 \beta_{2} + 139904 \beta_{3} ) q^{84} + ( 2247217 + 3812572 \beta_{1} + 450679 \beta_{2} + 644453 \beta_{3} ) q^{85} + ( -28134352 + 451784 \beta_{1} - 451784 \beta_{2} + 451784 \beta_{3} ) q^{86} + ( 14750384 - 14325856 \beta_{1} + 424528 \beta_{3} ) q^{87} + ( -5216256 - 5216256 \beta_{1} + 268288 \beta_{2} ) q^{88} + ( 2253080 \beta_{1} - 101640 \beta_{2} - 101640 \beta_{3} ) q^{89} + ( 14813112 + 4315392 \beta_{1} - 330456 \beta_{2} + 208008 \beta_{3} ) q^{90} + ( -54208006 + 1158887 \beta_{1} - 1158887 \beta_{2} + 1158887 \beta_{3} ) q^{91} + ( 20374144 - 20031232 \beta_{1} + 342912 \beta_{3} ) q^{92} + ( 24896074 + 24896074 \beta_{1} - 299594 \beta_{2} ) q^{93} + ( -13200744 \beta_{1} + 627528 \beta_{2} + 627528 \beta_{3} ) q^{94} + ( -39669360 - 54909010 \beta_{1} - 407070 \beta_{2} + 4510 \beta_{3} ) q^{95} + ( 3407872 - 131072 \beta_{1} + 131072 \beta_{2} - 131072 \beta_{3} ) q^{96} + ( 34986721 - 34829081 \beta_{1} + 157640 \beta_{3} ) q^{97} + ( 41685192 + 41685192 \beta_{1} + 559792 \beta_{2} ) q^{98} + ( 37323717 \beta_{1} - 538005 \beta_{2} - 538005 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 32q^{2} + 54q^{3} + 90q^{5} - 864q^{6} - 1186q^{7} + 4096q^{8} + O(q^{10})$$ $$4q - 32q^{2} + 54q^{3} + 90q^{5} - 864q^{6} - 1186q^{7} + 4096q^{8} - 1760q^{10} - 19852q^{11} + 6912q^{12} + 73704q^{13} - 137490q^{15} - 65536q^{16} + 198944q^{17} + 98688q^{18} + 16640q^{20} - 766572q^{21} + 158816q^{22} + 631334q^{23} + 545600q^{25} - 1179264q^{26} + 184680q^{27} + 151808q^{28} + 1275360q^{30} - 2329892q^{31} + 524288q^{32} + 1362948q^{33} + 1403870q^{35} - 1579008q^{36} - 976356q^{37} - 1144000q^{38} - 40960q^{40} - 4284652q^{41} + 6132576q^{42} + 7146534q^{43} - 2624430q^{45} - 10101344q^{46} - 3300186q^{47} - 884736q^{48} + 7241600q^{50} + 6541788q^{51} + 9434112q^{52} + 9989164q^{53} - 21649020q^{55} - 2428928q^{56} - 19136400q^{57} - 19957760q^{58} - 2807040q^{60} + 85377108q^{61} + 18639136q^{62} - 16175226q^{63} - 33878940q^{65} - 21807168q^{66} - 57195466q^{67} - 25464832q^{68} + 53504320q^{70} + 113004028q^{71} + 12632064q^{72} - 60875756q^{73} + 22407150q^{75} + 18304000q^{76} - 90339932q^{77} - 40023264q^{78} - 1474560q^{80} + 35054064q^{81} + 34277216q^{82} + 71051214q^{83} + 8601320q^{85} - 114344544q^{86} + 58152480q^{87} - 20328448q^{88} + 58175520q^{90} - 221467572q^{91} + 80810752q^{92} + 98985108q^{93} - 159500600q^{95} + 14155776q^{96} + 139631604q^{97} + 167860352q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 125 x^{2} + 3844$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 63 \nu$$$$)/62$$ $$\beta_{2}$$ $$=$$ $$($$$$3 \nu^{3} + 310 \nu^{2} + 499 \nu + 19406$$$$)/62$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 155 \nu^{2} + 218 \nu - 9703$$$$)/31$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1} - 626$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-63 \beta_{3} - 63 \beta_{2} + 935 \beta_{1}$$$$)/10$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 8.38987i 7.38987i 8.38987i − 7.38987i
−8.00000 + 8.00000i −25.9493 25.9493i 128.000i 535.341 322.544i 415.189 2031.01 2031.01i 1024.00 + 1024.00i 5214.26i −1702.38 + 6863.08i
3.2 −8.00000 + 8.00000i 52.9493 + 52.9493i 128.000i −490.341 + 387.544i −847.189 −2624.01 + 2624.01i 1024.00 + 1024.00i 953.736i 822.379 7023.08i
7.1 −8.00000 8.00000i −25.9493 + 25.9493i 128.000i 535.341 + 322.544i 415.189 2031.01 + 2031.01i 1024.00 1024.00i 5214.26i −1702.38 6863.08i
7.2 −8.00000 8.00000i 52.9493 52.9493i 128.000i −490.341 387.544i −847.189 −2624.01 2624.01i 1024.00 1024.00i 953.736i 822.379 + 7023.08i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{4} - 54 T_{3}^{3} + 1458 T_{3}^{2} + 148392 T_{3} + 7551504$$ acting on $$S_{9}^{\mathrm{new}}(10, [\chi])$$.