# Properties

 Label 245.6.a.d Level $245$ Weight $6$ Character orbit 245.a Self dual yes Analytic conductor $39.294$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.2940358542$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.577880.1 Defining polynomial: $$x^{3} - 98 x - 232$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{1} ) q^{2} + ( -9 - \beta_{2} ) q^{3} + ( 38 + 2 \beta_{2} ) q^{4} + 25 q^{5} + ( 34 - 17 \beta_{1} + 6 \beta_{2} ) q^{6} + ( -44 + 22 \beta_{1} - 12 \beta_{2} ) q^{8} + ( 166 - 24 \beta_{1} + 9 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta_{1} ) q^{2} + ( -9 - \beta_{2} ) q^{3} + ( 38 + 2 \beta_{2} ) q^{4} + 25 q^{5} + ( 34 - 17 \beta_{1} + 6 \beta_{2} ) q^{6} + ( -44 + 22 \beta_{1} - 12 \beta_{2} ) q^{8} + ( 166 - 24 \beta_{1} + 9 \beta_{2} ) q^{9} + ( -50 + 25 \beta_{1} ) q^{10} + ( -67 - 8 \beta_{1} - 7 \beta_{2} ) q^{11} + ( -998 + 48 \beta_{1} - 38 \beta_{2} ) q^{12} + ( -629 + 32 \beta_{1} + 5 \beta_{2} ) q^{13} + ( -225 - 25 \beta_{2} ) q^{15} + ( 516 - 96 \beta_{1} + 52 \beta_{2} ) q^{16} + ( 61 + 96 \beta_{1} - \beta_{2} ) q^{17} + ( -2060 + 190 \beta_{1} - 102 \beta_{2} ) q^{18} + ( -418 - 152 \beta_{1} - 42 \beta_{2} ) q^{19} + ( 950 + 50 \beta_{2} ) q^{20} + ( -282 - 139 \beta_{1} + 26 \beta_{2} ) q^{22} + ( 1082 - 168 \beta_{1} + 58 \beta_{2} ) q^{23} + ( 4684 - 662 \beta_{1} + 132 \beta_{2} ) q^{24} + 625 q^{25} + ( 3290 - 525 \beta_{1} + 34 \beta_{2} ) q^{26} + ( -2643 + 624 \beta_{1} - 19 \beta_{2} ) q^{27} + ( -3781 + 72 \beta_{1} - 11 \beta_{2} ) q^{29} + ( 850 - 425 \beta_{1} + 150 \beta_{2} ) q^{30} + ( -3036 - 200 \beta_{1} + 92 \beta_{2} ) q^{31} + ( -6792 + 36 \beta_{1} - 120 \beta_{2} ) q^{32} + ( 2771 - 32 \beta_{1} + 35 \beta_{2} ) q^{33} + ( 6230 + 245 \beta_{1} + 198 \beta_{2} ) q^{34} + ( 12980 - 1728 \beta_{1} + 704 \beta_{2} ) q^{36} + ( 1890 - 1256 \beta_{1} - 372 \beta_{2} ) q^{37} + ( -8524 - 1058 \beta_{1} - 52 \beta_{2} ) q^{38} + ( 4533 - 424 \beta_{1} + 757 \beta_{2} ) q^{39} + ( -1100 + 550 \beta_{1} - 300 \beta_{2} ) q^{40} + ( -3236 - 1176 \beta_{1} + 734 \beta_{2} ) q^{41} + ( 9222 + 1544 \beta_{1} - 446 \beta_{2} ) q^{43} + ( -6882 - 96 \beta_{1} - 210 \beta_{2} ) q^{44} + ( 4150 - 600 \beta_{1} + 225 \beta_{2} ) q^{45} + ( -14180 + 1210 \beta_{1} - 684 \beta_{2} ) q^{46} + ( -769 - 1856 \beta_{1} + 23 \beta_{2} ) q^{47} + ( -23236 + 2880 \beta_{1} - 900 \beta_{2} ) q^{48} + ( -1250 + 625 \beta_{1} ) q^{50} + ( 1315 - 1656 \beta_{1} + 323 \beta_{2} ) q^{51} + ( -21646 + 1488 \beta_{1} - 1414 \beta_{2} ) q^{52} + ( 2604 + 1552 \beta_{1} + 622 \beta_{2} ) q^{53} + ( 46774 - 1547 \beta_{1} + 1362 \beta_{2} ) q^{54} + ( -1675 - 200 \beta_{1} - 175 \beta_{2} ) q^{55} + ( 15106 + 1576 \beta_{1} - 190 \beta_{2} ) q^{57} + ( 12490 - 3725 \beta_{1} + 210 \beta_{2} ) q^{58} + ( -8272 - 64 \beta_{1} - 1056 \beta_{2} ) q^{59} + ( -24950 + 1200 \beta_{1} - 950 \beta_{2} ) q^{60} + ( -2568 + 120 \beta_{1} + 1018 \beta_{2} ) q^{61} + ( -8600 - 2700 \beta_{1} - 952 \beta_{2} ) q^{62} + ( 1368 - 4608 \beta_{1} - 872 \beta_{2} ) q^{64} + ( -15725 + 800 \beta_{1} + 125 \beta_{2} ) q^{65} + ( -8214 + 2987 \beta_{1} - 274 \beta_{2} ) q^{66} + ( -32308 - 384 \beta_{1} + 648 \beta_{2} ) q^{67} + ( -1410 + 5232 \beta_{1} - 666 \beta_{2} ) q^{68} + ( -31450 + 4248 \beta_{1} - 1754 \beta_{2} ) q^{69} + ( -17072 - 2944 \beta_{1} + 1600 \beta_{2} ) q^{71} + ( -85352 + 9076 \beta_{1} - 4416 \beta_{2} ) q^{72} + ( -10658 - 560 \beta_{1} + 2896 \beta_{2} ) q^{73} + ( -80724 - 3598 \beta_{1} - 280 \beta_{2} ) q^{74} + ( -5625 - 625 \beta_{2} ) q^{75} + ( -38572 - 6192 \beta_{1} - 460 \beta_{2} ) q^{76} + ( -49162 + 9741 \beta_{1} - 5390 \beta_{2} ) q^{78} + ( -23953 + 4376 \beta_{1} - 313 \beta_{2} ) q^{79} + ( 12900 - 2400 \beta_{1} + 1300 \beta_{2} ) q^{80} + ( -335 - 5232 \beta_{1} + 2952 \beta_{2} ) q^{81} + ( -82888 + 284 \beta_{1} - 6756 \beta_{2} ) q^{82} + ( 7236 - 4912 \beta_{1} + 788 \beta_{2} ) q^{83} + ( 1525 + 2400 \beta_{1} - 25 \beta_{2} ) q^{85} + ( 90596 + 8742 \beta_{1} + 5764 \beta_{2} ) q^{86} + ( 38789 - 1488 \beta_{1} + 4069 \beta_{2} ) q^{87} + ( 19812 - 4306 \beta_{1} + 236 \beta_{2} ) q^{88} + ( 64972 + 4680 \beta_{1} + 2294 \beta_{2} ) q^{89} + ( -51500 + 4750 \beta_{1} - 2550 \beta_{2} ) q^{90} + ( 84540 - 11856 \beta_{1} + 4668 \beta_{2} ) q^{92} + ( -6052 + 5608 \beta_{1} + 2236 \beta_{2} ) q^{93} + ( -121326 - 4297 \beta_{1} - 3850 \beta_{2} ) q^{94} + ( -10450 - 3800 \beta_{1} - 1050 \beta_{2} ) q^{95} + ( 101064 - 3492 \beta_{1} + 6936 \beta_{2} ) q^{96} + ( -37135 + 32 \beta_{1} + 4915 \beta_{2} ) q^{97} + ( -20650 + 3328 \beta_{1} - 1198 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{2} - 26 q^{3} + 112 q^{4} + 75 q^{5} + 96 q^{6} - 120 q^{8} + 489 q^{9} + O(q^{10})$$ $$3 q - 6 q^{2} - 26 q^{3} + 112 q^{4} + 75 q^{5} + 96 q^{6} - 120 q^{8} + 489 q^{9} - 150 q^{10} - 194 q^{11} - 2956 q^{12} - 1892 q^{13} - 650 q^{15} + 1496 q^{16} + 184 q^{17} - 6078 q^{18} - 1212 q^{19} + 2800 q^{20} - 872 q^{22} + 3188 q^{23} + 13920 q^{24} + 1875 q^{25} + 9836 q^{26} - 7910 q^{27} - 11332 q^{29} + 2400 q^{30} - 9200 q^{31} - 20256 q^{32} + 8278 q^{33} + 18492 q^{34} + 38236 q^{36} + 6042 q^{37} - 25520 q^{38} + 12842 q^{39} - 3000 q^{40} - 10442 q^{41} + 28112 q^{43} - 20436 q^{44} + 12225 q^{45} - 41856 q^{46} - 2330 q^{47} - 68808 q^{48} - 3750 q^{50} + 3622 q^{51} - 63524 q^{52} + 7190 q^{53} + 138960 q^{54} - 4850 q^{55} + 45508 q^{57} + 37260 q^{58} - 23760 q^{59} - 73900 q^{60} - 8722 q^{61} - 24848 q^{62} + 4976 q^{64} - 47300 q^{65} - 24368 q^{66} - 97572 q^{67} - 3564 q^{68} - 92596 q^{69} - 52816 q^{71} - 251640 q^{72} - 34870 q^{73} - 241892 q^{74} - 16250 q^{75} - 115256 q^{76} - 142096 q^{78} - 71546 q^{79} + 37400 q^{80} - 3957 q^{81} - 241908 q^{82} + 20920 q^{83} + 4600 q^{85} + 266024 q^{86} + 112298 q^{87} + 59200 q^{88} + 192622 q^{89} - 151950 q^{90} + 248952 q^{92} - 20392 q^{93} - 360128 q^{94} - 30300 q^{95} + 296256 q^{96} - 116320 q^{97} - 60752 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 98 x - 232$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 4 \nu - 66$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + 4 \beta_{1} + 66$$

## 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
 −8.38673 −2.53323 10.9200
−10.3867 −27.9421 75.8842 25.0000 290.227 0 −455.813 537.760 −259.668
1.2 −4.53323 15.7249 −11.4498 25.0000 −71.2847 0 196.968 4.27317 −113.331
1.3 8.91996 −13.7828 47.5657 25.0000 −122.942 0 138.845 −53.0335 222.999
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 245.6.a.d 3
7.b odd 2 1 35.6.a.c 3
21.c even 2 1 315.6.a.i 3
28.d even 2 1 560.6.a.q 3
35.c odd 2 1 175.6.a.e 3
35.f even 4 2 175.6.b.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.c 3 7.b odd 2 1
175.6.a.e 3 35.c odd 2 1
175.6.b.e 6 35.f even 4 2
245.6.a.d 3 1.a even 1 1 trivial
315.6.a.i 3 21.c even 2 1
560.6.a.q 3 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{3} + 6 T_{2}^{2} - 86 T_{2} - 420$$ $$T_{3}^{3} + 26 T_{3}^{2} - 271 T_{3} - 6056$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-420 - 86 T + 6 T^{2} + T^{3}$$
$3$ $$-6056 - 271 T + 26 T^{2} + T^{3}$$
$5$ $$( -25 + T )^{3}$$
$7$ $$T^{3}$$
$11$ $$-3144468 - 15583 T + 194 T^{2} + T^{3}$$
$13$ $$171071930 + 1087501 T + 1892 T^{2} + T^{3}$$
$17$ $$-132716862 - 896603 T - 184 T^{2} + T^{3}$$
$19$ $$140259040 - 2369180 T + 1212 T^{2} + T^{3}$$
$23$ $$125424384 - 1476572 T - 3188 T^{2} + T^{3}$$
$29$ $$51669601050 + 42201805 T + 11332 T^{2} + T^{3}$$
$31$ $$-8818293376 + 19282768 T + 9200 T^{2} + T^{3}$$
$37$ $$1043894647208 - 190556324 T - 6042 T^{2} + T^{3}$$
$41$ $$-4748673370848 - 404569184 T + 10442 T^{2} + T^{3}$$
$43$ $$4763987701360 - 99225644 T - 28112 T^{2} + T^{3}$$
$47$ $$1072310433384 - 337915327 T + 2330 T^{2} + T^{3}$$
$53$ $$515502653472 - 368369248 T - 7190 T^{2} + T^{3}$$
$59$ $$-7000620748800 - 362727680 T + 23760 T^{2} + T^{3}$$
$61$ $$1590789613952 - 485040544 T + 8722 T^{2} + T^{3}$$
$67$ $$26595134017984 + 2939620080 T + 97572 T^{2} + T^{3}$$
$71$ $$-77522711777280 - 1397406976 T + 52816 T^{2} + T^{3}$$
$73$ $$-11550435576632 - 3859440948 T + 34870 T^{2} + T^{3}$$
$79$ $$-40598762145400 - 279253695 T + 71546 T^{2} + T^{3}$$
$83$ $$529374252288 - 2697140848 T - 20920 T^{2} + T^{3}$$
$89$ $$31660963259040 + 8081767840 T - 192622 T^{2} + T^{3}$$
$97$ $$-117394067785166 - 7473101907 T + 116320 T^{2} + T^{3}$$