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

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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:

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 \(S_{9}^{\mathrm{new}}(10, [\chi])\).