# Properties

 Label 10.12.a.d Level 10 Weight 12 Character orbit 10.a Self dual yes Analytic conductor 7.683 Analytic rank 0 Dimension 2 CM no Inner twists 1

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## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 10.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.68343180560$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{1969})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 5$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 10\sqrt{1969}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 32 q^{2} + ( 302 - \beta ) q^{3} + 1024 q^{4} + 3125 q^{5} + ( 9664 - 32 \beta ) q^{6} + ( 7046 + 177 \beta ) q^{7} + 32768 q^{8} + ( 110957 - 604 \beta ) q^{9} +O(q^{10})$$ $$q + 32 q^{2} + ( 302 - \beta ) q^{3} + 1024 q^{4} + 3125 q^{5} + ( 9664 - 32 \beta ) q^{6} + ( 7046 + 177 \beta ) q^{7} + 32768 q^{8} + ( 110957 - 604 \beta ) q^{9} + 100000 q^{10} + ( 210792 + 1254 \beta ) q^{11} + ( 309248 - 1024 \beta ) q^{12} + ( 865262 - 1452 \beta ) q^{13} + ( 225472 + 5664 \beta ) q^{14} + ( 943750 - 3125 \beta ) q^{15} + 1048576 q^{16} + ( -3161814 + 5028 \beta ) q^{17} + ( 3550624 - 19328 \beta ) q^{18} + ( -14448700 - 7404 \beta ) q^{19} + 3200000 q^{20} + ( -32723408 + 46408 \beta ) q^{21} + ( 6745344 + 40128 \beta ) q^{22} + ( -22618038 - 39081 \beta ) q^{23} + ( 9895936 - 32768 \beta ) q^{24} + 9765625 q^{25} + ( 27688384 - 46464 \beta ) q^{26} + ( 98938220 - 116218 \beta ) q^{27} + ( 7215104 + 181248 \beta ) q^{28} + ( 29113110 + 107592 \beta ) q^{29} + ( 30200000 - 100000 \beta ) q^{30} + ( 20706692 - 113442 \beta ) q^{31} + 33554432 q^{32} + ( -183253416 + 167916 \beta ) q^{33} + ( -101178048 + 160896 \beta ) q^{34} + ( 22018750 + 553125 \beta ) q^{35} + ( 113619968 - 618496 \beta ) q^{36} + ( 188627726 - 303192 \beta ) q^{37} + ( -462358400 - 236928 \beta ) q^{38} + ( 547207924 - 1303766 \beta ) q^{39} + 102400000 q^{40} + ( -392635518 + 1168812 \beta ) q^{41} + ( -1047149056 + 1485056 \beta ) q^{42} + ( -726493618 + 986667 \beta ) q^{43} + ( 215851008 + 1284096 \beta ) q^{44} + ( 346740625 - 1887500 \beta ) q^{45} + ( -723777216 - 1250592 \beta ) q^{46} + ( -644063874 - 2086863 \beta ) q^{47} + ( 316669952 - 1048576 \beta ) q^{48} + ( 4240999473 + 2494284 \beta ) q^{49} + 312500000 q^{50} + ( -1944881028 + 4680270 \beta ) q^{51} + ( 886028288 - 1486848 \beta ) q^{52} + ( -15245418 - 9183012 \beta ) q^{53} + ( 3166023040 - 3718976 \beta ) q^{54} + ( 658725000 + 3918750 \beta ) q^{55} + ( 230883328 + 5799936 \beta ) q^{56} + ( -2905659800 + 12212692 \beta ) q^{57} + ( 931619520 + 3442944 \beta ) q^{58} + ( 4338551220 - 1261416 \beta ) q^{59} + ( 966400000 - 3200000 \beta ) q^{60} + ( 557749382 - 22983984 \beta ) q^{61} + ( 662614144 - 3630144 \beta ) q^{62} + ( -20268382178 + 15383605 \beta ) q^{63} + 1073741824 q^{64} + ( 2703943750 - 4537500 \beta ) q^{65} + ( -5864109312 + 5373312 \beta ) q^{66} + ( -6336884854 - 22637439 \beta ) q^{67} + ( -3237697536 + 5148672 \beta ) q^{68} + ( 864401424 + 10815576 \beta ) q^{69} + ( 704600000 + 17700000 \beta ) q^{70} + ( 6899916492 + 7768566 \beta ) q^{71} + ( 3635838976 - 19791872 \beta ) q^{72} + ( -8921039758 + 29730708 \beta ) q^{73} + ( 6036087232 - 9702144 \beta ) q^{74} + ( 2949218750 - 9765625 \beta ) q^{75} + ( -14795468800 - 7581696 \beta ) q^{76} + ( 45188770632 + 46145868 \beta ) q^{77} + ( 17510653568 - 41720512 \beta ) q^{78} + ( -6318465160 - 30074724 \beta ) q^{79} + 3276800000 q^{80} + ( 33106966961 - 27039268 \beta ) q^{81} + ( -12564336576 + 37401984 \beta ) q^{82} + ( 20993244462 - 77912265 \beta ) q^{83} + ( -33508769792 + 47521792 \beta ) q^{84} + ( -9880668750 + 15712500 \beta ) q^{85} + ( -23247795776 + 31573344 \beta ) q^{86} + ( -12392705580 + 3379674 \beta ) q^{87} + ( 6907232256 + 41091072 \beta ) q^{88} + ( 6604370010 + 35835480 \beta ) q^{89} + ( 11095700000 - 60400000 \beta ) q^{90} + ( -44507451548 + 142920582 \beta ) q^{91} + ( -23160870912 - 40018944 \beta ) q^{92} + ( 28590150784 - 54966176 \beta ) q^{93} + ( -20610043968 - 66779616 \beta ) q^{94} + ( -45152187500 - 23137500 \beta ) q^{95} + ( 10133438464 - 33554432 \beta ) q^{96} + ( -30893731414 - 175966980 \beta ) q^{97} + ( 135711983136 + 79817088 \beta ) q^{98} + ( -125746362456 + 11821710 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 64q^{2} + 604q^{3} + 2048q^{4} + 6250q^{5} + 19328q^{6} + 14092q^{7} + 65536q^{8} + 221914q^{9} + O(q^{10})$$ $$2q + 64q^{2} + 604q^{3} + 2048q^{4} + 6250q^{5} + 19328q^{6} + 14092q^{7} + 65536q^{8} + 221914q^{9} + 200000q^{10} + 421584q^{11} + 618496q^{12} + 1730524q^{13} + 450944q^{14} + 1887500q^{15} + 2097152q^{16} - 6323628q^{17} + 7101248q^{18} - 28897400q^{19} + 6400000q^{20} - 65446816q^{21} + 13490688q^{22} - 45236076q^{23} + 19791872q^{24} + 19531250q^{25} + 55376768q^{26} + 197876440q^{27} + 14430208q^{28} + 58226220q^{29} + 60400000q^{30} + 41413384q^{31} + 67108864q^{32} - 366506832q^{33} - 202356096q^{34} + 44037500q^{35} + 227239936q^{36} + 377255452q^{37} - 924716800q^{38} + 1094415848q^{39} + 204800000q^{40} - 785271036q^{41} - 2094298112q^{42} - 1452987236q^{43} + 431702016q^{44} + 693481250q^{45} - 1447554432q^{46} - 1288127748q^{47} + 633339904q^{48} + 8481998946q^{49} + 625000000q^{50} - 3889762056q^{51} + 1772056576q^{52} - 30490836q^{53} + 6332046080q^{54} + 1317450000q^{55} + 461766656q^{56} - 5811319600q^{57} + 1863239040q^{58} + 8677102440q^{59} + 1932800000q^{60} + 1115498764q^{61} + 1325228288q^{62} - 40536764356q^{63} + 2147483648q^{64} + 5407887500q^{65} - 11728218624q^{66} - 12673769708q^{67} - 6475395072q^{68} + 1728802848q^{69} + 1409200000q^{70} + 13799832984q^{71} + 7271677952q^{72} - 17842079516q^{73} + 12072174464q^{74} + 5898437500q^{75} - 29590937600q^{76} + 90377541264q^{77} + 35021307136q^{78} - 12636930320q^{79} + 6553600000q^{80} + 66213933922q^{81} - 25128673152q^{82} + 41986488924q^{83} - 67017539584q^{84} - 19761337500q^{85} - 46495591552q^{86} - 24785411160q^{87} + 13814464512q^{88} + 13208740020q^{89} + 22191400000q^{90} - 89014903096q^{91} - 46321741824q^{92} + 57180301568q^{93} - 41220087936q^{94} - 90304375000q^{95} + 20266876928q^{96} - 61787462828q^{97} + 271423966272q^{98} - 251492724912q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 22.6867 −21.6867
32.0000 −141.734 1024.00 3125.00 −4535.49 85586.9 32768.0 −157058. 100000.
1.2 32.0000 745.734 1024.00 3125.00 23863.5 −71494.9 32768.0 378972. 100000.
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.12.a.d 2
3.b odd 2 1 90.12.a.l 2
4.b odd 2 1 80.12.a.g 2
5.b even 2 1 50.12.a.f 2
5.c odd 4 2 50.12.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.a.d 2 1.a even 1 1 trivial
50.12.a.f 2 5.b even 2 1
50.12.b.f 4 5.c odd 4 2
80.12.a.g 2 4.b odd 2 1
90.12.a.l 2 3.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 604 T_{3} - 105696$$ acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(10))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 32 T )^{2}$$
$3$ $$1 - 604 T + 248598 T^{2} - 106996788 T^{3} + 31381059609 T^{4}$$
$5$ $$( 1 - 3125 T )^{2}$$
$7$ $$1 - 14092 T - 2164380498 T^{2} - 27864488462356 T^{3} + 3909821048582988049 T^{4}$$
$11$ $$1 - 421584 T + 305428208086 T^{2} - 120282835342867824 T^{3} +$$$$81\!\cdots\!21$$$$T^{4}$$
$13$ $$1 - 1730524 T + 3917874059118 T^{2} - 3101376573730485388 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$17$ $$1 + 6323628 T + 73563074016262 T^{2} +$$$$21\!\cdots\!24$$$$T^{3} +$$$$11\!\cdots\!89$$$$T^{4}$$
$19$ $$1 + 28897400 T + 430951545856038 T^{2} +$$$$33\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!61$$$$T^{4}$$
$23$ $$1 + 45236076 T + 2116464952736398 T^{2} +$$$$43\!\cdots\!52$$$$T^{3} +$$$$90\!\cdots\!29$$$$T^{4}$$
$29$ $$1 - 58226220 T + 22969270731722158 T^{2} -$$$$71\!\cdots\!80$$$$T^{3} +$$$$14\!\cdots\!41$$$$T^{4}$$
$31$ $$1 - 41413384 T + 48711797584420926 T^{2} -$$$$10\!\cdots\!04$$$$T^{3} +$$$$64\!\cdots\!61$$$$T^{4}$$
$37$ $$1 - 377255452 T + 373315553507530302 T^{2} -$$$$67\!\cdots\!76$$$$T^{3} +$$$$31\!\cdots\!69$$$$T^{4}$$
$41$ $$1 + 785271036 T + 985831391781991606 T^{2} +$$$$43\!\cdots\!76$$$$T^{3} +$$$$30\!\cdots\!81$$$$T^{4}$$
$43$ $$1 + 1452987236 T + 2194695988642931238 T^{2} +$$$$13\!\cdots\!52$$$$T^{3} +$$$$86\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 1288127748 T + 4501637759066496382 T^{2} +$$$$31\!\cdots\!44$$$$T^{3} +$$$$61\!\cdots\!09$$$$T^{4}$$
$53$ $$1 + 30490836 T + 1934178302201224318 T^{2} +$$$$28\!\cdots\!92$$$$T^{3} +$$$$85\!\cdots\!09$$$$T^{4}$$
$59$ $$1 - 8677102440 T + 78821502141035647318 T^{2} -$$$$26\!\cdots\!60$$$$T^{3} +$$$$90\!\cdots\!81$$$$T^{4}$$
$61$ $$1 - 1115498764 T - 16676167592870147154 T^{2} -$$$$48\!\cdots\!04$$$$T^{3} +$$$$18\!\cdots\!21$$$$T^{4}$$
$67$ $$1 + 12673769708 T +$$$$18\!\cdots\!82$$$$T^{2} +$$$$15\!\cdots\!64$$$$T^{3} +$$$$14\!\cdots\!89$$$$T^{4}$$
$71$ $$1 - 13799832984 T +$$$$49\!\cdots\!06$$$$T^{2} -$$$$31\!\cdots\!64$$$$T^{3} +$$$$53\!\cdots\!41$$$$T^{4}$$
$73$ $$1 + 17842079516 T +$$$$53\!\cdots\!18$$$$T^{2} +$$$$55\!\cdots\!32$$$$T^{3} +$$$$98\!\cdots\!29$$$$T^{4}$$
$79$ $$1 + 12636930320 T +$$$$13\!\cdots\!58$$$$T^{2} +$$$$94\!\cdots\!80$$$$T^{3} +$$$$55\!\cdots\!41$$$$T^{4}$$
$83$ $$1 - 41986488924 T +$$$$18\!\cdots\!78$$$$T^{2} -$$$$54\!\cdots\!08$$$$T^{3} +$$$$16\!\cdots\!89$$$$T^{4}$$
$89$ $$1 - 13208740020 T +$$$$53\!\cdots\!78$$$$T^{2} -$$$$36\!\cdots\!80$$$$T^{3} +$$$$77\!\cdots\!21$$$$T^{4}$$
$97$ $$1 + 61787462828 T +$$$$91\!\cdots\!02$$$$T^{2} +$$$$44\!\cdots\!84$$$$T^{3} +$$$$51\!\cdots\!09$$$$T^{4}$$
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