Properties

 Label 10.9.c.b 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: $$\Q(i, \sqrt{601})$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 8 + 8 \beta_{1} ) q^{2}$$ $$+ ( 22 - 21 \beta_{1} + \beta_{3} ) q^{3}$$ $$+ 128 \beta_{1} q^{4}$$ $$+ ( -216 - 71 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5}$$ $$+ ( 352 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6}$$ $$+ ( 1442 + 1442 \beta_{1} - 21 \beta_{2} ) q^{7}$$ $$+ ( -1024 + 1024 \beta_{1} ) q^{8}$$ $$+ ( -1876 \beta_{1} + 43 \beta_{2} + 43 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 8 + 8 \beta_{1} ) q^{2}$$ $$+ ( 22 - 21 \beta_{1} + \beta_{3} ) q^{3}$$ $$+ 128 \beta_{1} q^{4}$$ $$+ ( -216 - 71 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5}$$ $$+ ( 352 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6}$$ $$+ ( 1442 + 1442 \beta_{1} - 21 \beta_{2} ) q^{7}$$ $$+ ( -1024 + 1024 \beta_{1} ) q^{8}$$ $$+ ( -1876 \beta_{1} + 43 \beta_{2} + 43 \beta_{3} ) q^{9}$$ $$+ ( -1112 - 2272 \beta_{1} - 24 \beta_{2} + 72 \beta_{3} ) q^{10}$$ $$+ ( -3824 - 141 \beta_{1} + 141 \beta_{2} - 141 \beta_{3} ) q^{11}$$ $$+ ( 2816 + 2816 \beta_{1} - 128 \beta_{2} ) q^{12}$$ $$+ ( 11235 - 11553 \beta_{1} - 318 \beta_{3} ) q^{13}$$ $$+ ( 22904 \beta_{1} - 168 \beta_{2} - 168 \beta_{3} ) q^{14}$$ $$+ ( -28982 - 42167 \beta_{1} + 266 \beta_{2} - 153 \beta_{3} ) q^{15}$$ $$-16384 q^{16}$$ $$+ ( 871 + 871 \beta_{1} + 66 \beta_{2} ) q^{17}$$ $$+ ( 15352 - 14664 \beta_{1} + 688 \beta_{3} ) q^{18}$$ $$+ ( 122890 \beta_{1} + 438 \beta_{2} + 438 \beta_{3} ) q^{19}$$ $$+ ( 9856 - 27264 \beta_{1} - 768 \beta_{2} + 384 \beta_{3} ) q^{20}$$ $$+ ( 221200 + 1883 \beta_{1} - 1883 \beta_{2} + 1883 \beta_{3} ) q^{21}$$ $$+ ( -30592 - 30592 \beta_{1} + 2256 \beta_{2} ) q^{22}$$ $$+ ( -113258 + 115139 \beta_{1} + 1881 \beta_{3} ) q^{23}$$ $$+ ( 44032 \beta_{1} - 1024 \beta_{2} - 1024 \beta_{3} ) q^{24}$$ $$+ ( -229705 - 172605 \beta_{1} - 435 \beta_{2} - 3045 \beta_{3} ) q^{25}$$ $$+ ( 179760 - 2544 \beta_{1} + 2544 \beta_{2} - 2544 \beta_{3} ) q^{26}$$ $$+ ( -220892 - 220892 \beta_{1} - 2836 \beta_{2} ) q^{27}$$ $$+ ( -184576 + 181888 \beta_{1} - 2688 \beta_{3} ) q^{28}$$ $$+ ( 132400 \beta_{1} + 96 \beta_{2} + 96 \beta_{3} ) q^{29}$$ $$+ ( 104256 - 567064 \beta_{1} + 3352 \beta_{2} + 904 \beta_{3} ) q^{30}$$ $$+ ( 1165300 - 2787 \beta_{1} + 2787 \beta_{2} - 2787 \beta_{3} ) q^{31}$$ $$+ ( -131072 - 131072 \beta_{1} ) q^{32}$$ $$+ ( -1143320 + 1133574 \beta_{1} - 9746 \beta_{3} ) q^{33}$$ $$+ ( 14464 \beta_{1} + 528 \beta_{2} + 528 \beta_{3} ) q^{34}$$ $$+ ( 746074 - 881356 \beta_{1} + 273 \beta_{2} + 14406 \beta_{3} ) q^{35}$$ $$+ ( 245632 + 5504 \beta_{1} - 5504 \beta_{2} + 5504 \beta_{3} ) q^{36}$$ $$+ ( -1386939 - 1386939 \beta_{1} - 3564 \beta_{2} ) q^{37}$$ $$+ ( -979616 + 986624 \beta_{1} + 7008 \beta_{3} ) q^{38}$$ $$+ ( 1899033 \beta_{1} + 4557 \beta_{2} + 4557 \beta_{3} ) q^{39}$$ $$+ ( 300032 - 145408 \beta_{1} - 9216 \beta_{2} - 3072 \beta_{3} ) q^{40}$$ $$+ ( 2677024 - 7653 \beta_{1} + 7653 \beta_{2} - 7653 \beta_{3} ) q^{41}$$ $$+ ( 1769600 + 1769600 \beta_{1} - 30128 \beta_{2} ) q^{42}$$ $$+ ( 1243038 - 1213605 \beta_{1} + 29433 \beta_{3} ) q^{43}$$ $$+ ( -471424 \beta_{1} + 18048 \beta_{2} + 18048 \beta_{3} ) q^{44}$$ $$+ ( -3054907 - 572642 \beta_{1} + 5021 \beta_{2} - 18098 \beta_{3} ) q^{45}$$ $$+ ( -1812128 + 15048 \beta_{1} - 15048 \beta_{2} + 15048 \beta_{3} ) q^{46}$$ $$+ ( -1398618 - 1398618 \beta_{1} + 72999 \beta_{2} ) q^{47}$$ $$+ ( -360448 + 344064 \beta_{1} - 16384 \beta_{3} ) q^{48}$$ $$+ ( 1646596 \beta_{1} - 60123 \beta_{2} - 60123 \beta_{3} ) q^{49}$$ $$+ ( -481160 - 3221960 \beta_{1} + 20880 \beta_{2} - 27840 \beta_{3} ) q^{50}$$ $$+ ( -457468 - 515 \beta_{1} + 515 \beta_{2} - 515 \beta_{3} ) q^{51}$$ $$+ ( 1438080 + 1438080 \beta_{1} + 40704 \beta_{2} ) q^{52}$$ $$+ ( 5047325 - 5097833 \beta_{1} - 50508 \beta_{3} ) q^{53}$$ $$+ ( -3556960 \beta_{1} - 22688 \beta_{2} - 22688 \beta_{3} ) q^{54}$$ $$+ ( -2351592 + 9825523 \beta_{1} - 51939 \beta_{2} - 1653 \beta_{3} ) q^{55}$$ $$+ ( -2953216 - 21504 \beta_{1} + 21504 \beta_{2} - 21504 \beta_{3} ) q^{56}$$ $$+ ( -596312 - 596312 \beta_{1} - 104056 \beta_{2} ) q^{57}$$ $$+ ( -1058432 + 1059968 \beta_{1} + 1536 \beta_{3} ) q^{58}$$ $$+ ( -172550 \beta_{1} + 127782 \beta_{2} + 127782 \beta_{3} ) q^{59}$$ $$+ ( 5377792 - 3675648 \beta_{1} + 19584 \beta_{2} + 34048 \beta_{3} ) q^{60}$$ $$+ ( -10824672 + 68331 \beta_{1} - 68331 \beta_{2} + 68331 \beta_{3} ) q^{61}$$ $$+ ( 9322400 + 9322400 \beta_{1} + 44592 \beta_{2} ) q^{62}$$ $$+ ( 9550534 - 9388029 \beta_{1} + 162505 \beta_{3} ) q^{63}$$ $$-2097152 \beta_{1} q^{64}$$ $$+ ( 3874593 + 15997908 \beta_{1} + 79491 \beta_{2} + 103347 \beta_{3} ) q^{65}$$ $$+ ( -18293120 - 77968 \beta_{1} + 77968 \beta_{2} - 77968 \beta_{3} ) q^{66}$$ $$+ ( 2480678 + 2480678 \beta_{1} - 202113 \beta_{2} ) q^{67}$$ $$+ ( -111488 + 119936 \beta_{1} + 8448 \beta_{3} ) q^{68}$$ $$+ ( -9220477 \beta_{1} - 73757 \beta_{2} - 73757 \beta_{3} ) q^{69}$$ $$+ ( 13134688 - 1080072 \beta_{1} - 113064 \beta_{2} + 117432 \beta_{3} ) q^{70}$$ $$+ ( 4993028 - 108699 \beta_{1} + 108699 \beta_{2} - 108699 \beta_{3} ) q^{71}$$ $$+ ( 1965056 + 1965056 \beta_{1} - 88064 \beta_{2} ) q^{72}$$ $$+ ( -3126215 + 2677967 \beta_{1} - 448248 \beta_{3} ) q^{73}$$ $$+ ( -22219536 \beta_{1} - 28512 \beta_{2} - 28512 \beta_{3} ) q^{74}$$ $$+ ( -5516110 + 23912715 \beta_{1} + 96480 \beta_{2} - 284515 \beta_{3} ) q^{75}$$ $$+ ( -15673856 + 56064 \beta_{1} - 56064 \beta_{2} + 56064 \beta_{3} ) q^{76}$$ $$+ ( -27757240 - 27757240 \beta_{1} + 481026 \beta_{2} ) q^{77}$$ $$+ ( -15155808 + 15228720 \beta_{1} + 72912 \beta_{3} ) q^{78}$$ $$+ ( -34376880 \beta_{1} + 102768 \beta_{2} + 102768 \beta_{3} ) q^{79}$$ $$+ ( 3538944 + 1163264 \beta_{1} - 49152 \beta_{2} - 98304 \beta_{3} ) q^{80}$$ $$+ ( 24175343 + 120787 \beta_{1} - 120787 \beta_{2} + 120787 \beta_{3} ) q^{81}$$ $$+ ( 21416192 + 21416192 \beta_{1} + 122448 \beta_{2} ) q^{82}$$ $$+ ( 16164594 - 15967749 \beta_{1} + 196845 \beta_{3} ) q^{83}$$ $$+ ( 28072576 \beta_{1} - 241024 \beta_{2} - 241024 \beta_{3} ) q^{84}$$ $$+ ( -3095821 + 1235524 \beta_{1} - 17067 \beta_{2} + 3351 \beta_{3} ) q^{85}$$ $$+ ( 19888608 + 235464 \beta_{1} - 235464 \beta_{2} + 235464 \beta_{3} ) q^{86}$$ $$+ ( 2189536 + 2189536 \beta_{1} - 128272 \beta_{2} ) q^{87}$$ $$+ ( 3915776 - 3627008 \beta_{1} + 288768 \beta_{3} ) q^{88}$$ $$+ ( 70322120 \beta_{1} - 72360 \beta_{2} - 72360 \beta_{3} ) q^{89}$$ $$+ ( -20002904 - 28980224 \beta_{1} + 184952 \beta_{2} - 104616 \beta_{3} ) q^{90}$$ $$+ ( -17763396 - 215943 \beta_{1} + 215943 \beta_{2} - 215943 \beta_{3} ) q^{91}$$ $$+ ( -14497024 - 14497024 \beta_{1} - 240768 \beta_{2} ) q^{92}$$ $$+ ( 4700656 - 3652410 \beta_{1} + 1048246 \beta_{3} ) q^{93}$$ $$+ ( -21793896 \beta_{1} + 583992 \beta_{2} + 583992 \beta_{3} ) q^{94}$$ $$+ ( -20183500 - 36078750 \beta_{1} - 800850 \beta_{2} + 241650 \beta_{3} ) q^{95}$$ $$+ ( -5767168 - 131072 \beta_{1} + 131072 \beta_{2} - 131072 \beta_{3} ) q^{96}$$ $$+ ( 8470809 + 8470809 \beta_{1} + 115080 \beta_{2} ) q^{97}$$ $$+ ( -13653752 + 12691784 \beta_{1} - 961968 \beta_{3} ) q^{98}$$ $$+ ( 98005883 \beta_{1} - 422885 \beta_{2} - 422885 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 32q^{2}$$ $$\mathstrut +\mathstrut 86q^{3}$$ $$\mathstrut -\mathstrut 870q^{5}$$ $$\mathstrut +\mathstrut 1376q^{6}$$ $$\mathstrut +\mathstrut 5726q^{7}$$ $$\mathstrut -\mathstrut 4096q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 32q^{2}$$ $$\mathstrut +\mathstrut 86q^{3}$$ $$\mathstrut -\mathstrut 870q^{5}$$ $$\mathstrut +\mathstrut 1376q^{6}$$ $$\mathstrut +\mathstrut 5726q^{7}$$ $$\mathstrut -\mathstrut 4096q^{8}$$ $$\mathstrut -\mathstrut 4640q^{10}$$ $$\mathstrut -\mathstrut 14732q^{11}$$ $$\mathstrut +\mathstrut 11008q^{12}$$ $$\mathstrut +\mathstrut 45576q^{13}$$ $$\mathstrut -\mathstrut 115090q^{15}$$ $$\mathstrut -\mathstrut 65536q^{16}$$ $$\mathstrut +\mathstrut 3616q^{17}$$ $$\mathstrut +\mathstrut 60032q^{18}$$ $$\mathstrut +\mathstrut 37120q^{20}$$ $$\mathstrut +\mathstrut 877268q^{21}$$ $$\mathstrut -\mathstrut 117856q^{22}$$ $$\mathstrut -\mathstrut 456794q^{23}$$ $$\mathstrut -\mathstrut 913600q^{25}$$ $$\mathstrut +\mathstrut 729216q^{26}$$ $$\mathstrut -\mathstrut 889240q^{27}$$ $$\mathstrut -\mathstrut 732928q^{28}$$ $$\mathstrut +\mathstrut 421920q^{30}$$ $$\mathstrut +\mathstrut 4672348q^{31}$$ $$\mathstrut -\mathstrut 524288q^{32}$$ $$\mathstrut -\mathstrut 4553788q^{33}$$ $$\mathstrut +\mathstrut 2956030q^{35}$$ $$\mathstrut +\mathstrut 960512q^{36}$$ $$\mathstrut -\mathstrut 5554884q^{37}$$ $$\mathstrut -\mathstrut 3932480q^{38}$$ $$\mathstrut +\mathstrut 1187840q^{40}$$ $$\mathstrut +\mathstrut 10738708q^{41}$$ $$\mathstrut +\mathstrut 7018144q^{42}$$ $$\mathstrut +\mathstrut 4913286q^{43}$$ $$\mathstrut -\mathstrut 12173390q^{45}$$ $$\mathstrut -\mathstrut 7308704q^{46}$$ $$\mathstrut -\mathstrut 5448474q^{47}$$ $$\mathstrut -\mathstrut 1409024q^{48}$$ $$\mathstrut -\mathstrut 1827200q^{50}$$ $$\mathstrut -\mathstrut 1827812q^{51}$$ $$\mathstrut +\mathstrut 5833728q^{52}$$ $$\mathstrut +\mathstrut 20290316q^{53}$$ $$\mathstrut -\mathstrut 9506940q^{55}$$ $$\mathstrut -\mathstrut 11726848q^{56}$$ $$\mathstrut -\mathstrut 2593360q^{57}$$ $$\mathstrut -\mathstrut 4236800q^{58}$$ $$\mathstrut +\mathstrut 21482240q^{60}$$ $$\mathstrut -\mathstrut 43572012q^{61}$$ $$\mathstrut +\mathstrut 37378784q^{62}$$ $$\mathstrut +\mathstrut 37877126q^{63}$$ $$\mathstrut +\mathstrut 15450660q^{65}$$ $$\mathstrut -\mathstrut 72860608q^{66}$$ $$\mathstrut +\mathstrut 9518486q^{67}$$ $$\mathstrut -\mathstrut 462848q^{68}$$ $$\mathstrut +\mathstrut 52077760q^{70}$$ $$\mathstrut +\mathstrut 20406908q^{71}$$ $$\mathstrut +\mathstrut 7684096q^{72}$$ $$\mathstrut -\mathstrut 11608364q^{73}$$ $$\mathstrut -\mathstrut 21302450q^{75}$$ $$\mathstrut -\mathstrut 62919680q^{76}$$ $$\mathstrut -\mathstrut 110066908q^{77}$$ $$\mathstrut -\mathstrut 60769056q^{78}$$ $$\mathstrut +\mathstrut 14254080q^{80}$$ $$\mathstrut +\mathstrut 96218224q^{81}$$ $$\mathstrut +\mathstrut 85909664q^{82}$$ $$\mathstrut +\mathstrut 64264686q^{83}$$ $$\mathstrut -\mathstrut 12424120q^{85}$$ $$\mathstrut +\mathstrut 78612576q^{86}$$ $$\mathstrut +\mathstrut 8501600q^{87}$$ $$\mathstrut +\mathstrut 15085568q^{88}$$ $$\mathstrut -\mathstrut 79432480q^{90}$$ $$\mathstrut -\mathstrut 70189812q^{91}$$ $$\mathstrut -\mathstrut 58469632q^{92}$$ $$\mathstrut +\mathstrut 16706132q^{93}$$ $$\mathstrut -\mathstrut 82819000q^{95}$$ $$\mathstrut -\mathstrut 22544384q^{96}$$ $$\mathstrut +\mathstrut 34113396q^{97}$$ $$\mathstrut -\mathstrut 52691072q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$301$$ $$x^{2}\mathstrut +\mathstrut$$ $$22500$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 151 \nu$$$$)/150$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 250 \nu^{2} + 401 \nu + 37650$$$$)/50$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 375 \nu^{2} + 526 \nu - 56475$$$$)/75$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1506$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$151$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$151$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2255$$ $$\beta_{1}$$$$)/10$$

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
 − 11.7577i 12.7577i 11.7577i − 12.7577i
8.00000 8.00000i −39.7883 39.7883i 128.000i −401.365 479.094i −636.612 144.447 144.447i −1024.00 1024.00i 3394.79i −7043.67 621.836i
3.2 8.00000 8.00000i 82.7883 + 82.7883i 128.000i −33.6352 + 624.094i 1324.61 2718.55 2718.55i −1024.00 1024.00i 7146.79i 4723.67 + 5261.84i
7.1 8.00000 + 8.00000i −39.7883 + 39.7883i 128.000i −401.365 + 479.094i −636.612 144.447 + 144.447i −1024.00 + 1024.00i 3394.79i −7043.67 + 621.836i
7.2 8.00000 + 8.00000i 82.7883 82.7883i 128.000i −33.6352 624.094i 1324.61 2718.55 + 2718.55i −1024.00 + 1024.00i 7146.79i 4723.67 5261.84i
 $$n$$: 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 $$T_{3}^{4}$$ $$\mathstrut -\mathstrut 86 T_{3}^{3}$$ $$\mathstrut +\mathstrut 3698 T_{3}^{2}$$ $$\mathstrut +\mathstrut 566568 T_{3}$$ $$\mathstrut +\mathstrut 43401744$$ acting on $$S_{9}^{\mathrm{new}}(10, [\chi])$$.