# Properties

 Label 245.12.a.b Level 245 Weight 12 Character orbit 245.a Self dual yes Analytic conductor 188.244 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$188.244079237$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{151})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{151}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -10 + 3 \beta ) q^{2} + ( 110 - 16 \beta ) q^{3} + ( 3488 - 60 \beta ) q^{4} + 3125 q^{5} + ( -30092 + 490 \beta ) q^{6} + ( -123120 + 4920 \beta ) q^{8} + ( -10423 - 3520 \beta ) q^{9} +O(q^{10})$$ $$q + ( -10 + 3 \beta ) q^{2} + ( 110 - 16 \beta ) q^{3} + ( 3488 - 60 \beta ) q^{4} + 3125 q^{5} + ( -30092 + 490 \beta ) q^{6} + ( -123120 + 4920 \beta ) q^{8} + ( -10423 - 3520 \beta ) q^{9} + ( -31250 + 9375 \beta ) q^{10} + ( -309088 - 26400 \beta ) q^{11} + ( 963520 - 62408 \beta ) q^{12} + ( -1707130 + 12864 \beta ) q^{13} + ( 343750 - 50000 \beta ) q^{15} + ( 3002816 - 295680 \beta ) q^{16} + ( -658970 - 126528 \beta ) q^{17} + ( -6274010 + 3931 \beta ) q^{18} + ( -2662660 - 274560 \beta ) q^{19} + ( 10900000 - 187500 \beta ) q^{20} + ( -44745920 - 663264 \beta ) q^{22} + ( 29471970 + 33456 \beta ) q^{23} + ( -61090080 + 2511120 \beta ) q^{24} + 9765625 q^{25} + ( 40380868 - 5250030 \beta ) q^{26} + ( 13384580 + 2613920 \beta ) q^{27} + ( 47070190 + 2298240 \beta ) q^{29} + ( -94037500 + 1531250 \beta ) q^{30} + ( -122271732 + 7207200 \beta ) q^{31} + ( -313650560 + 1889088 \beta ) q^{32} + ( 221129920 + 2041408 \beta ) q^{33} + ( -222679036 - 711630 \beta ) q^{34} + ( 91209376 - 11652380 \beta ) q^{36} + ( 10501610 + 19033728 \beta ) q^{37} + ( -470876120 - 5242380 \beta ) q^{38} + ( -312101996 + 28729120 \beta ) q^{39} + ( -384750000 + 15375000 \beta ) q^{40} + ( 372871658 + 22651200 \beta ) q^{41} + ( 314975050 - 13909104 \beta ) q^{43} + ( -121362944 - 73537920 \beta ) q^{44} + ( -32571875 - 11000000 \beta ) q^{45} + ( -234097428 + 88081350 \beta ) q^{46} + ( 701030770 + 20505072 \beta ) q^{47} + ( 3187761280 - 80569856 \beta ) q^{48} + ( -97656250 + 29296875 \beta ) q^{50} + ( 1150279892 - 3374560 \beta ) q^{51} + ( -6420660800 + 147297432 \beta ) q^{52} + ( 569160290 - 186753984 \beta ) q^{53} + ( 4602577240 + 14014540 \beta ) q^{54} + ( -965900000 - 82500000 \beta ) q^{55} + ( 2360455240 + 12400960 \beta ) q^{57} + ( 3693708980 + 118228170 \beta ) q^{58} + ( -3658757780 - 175817280 \beta ) q^{59} + ( 3011000000 - 195025000 \beta ) q^{60} + ( 758212838 + 53568000 \beta ) q^{61} + ( 14282163720 - 438887196 \beta ) q^{62} + ( 409765888 - 354289920 \beta ) q^{64} + ( -5334781250 + 40200000 \beta ) q^{65} + ( 1487732096 + 642975680 \beta ) q^{66} + ( 7867145070 - 91691472 \beta ) q^{67} + ( 2286887360 - 401791464 \beta ) q^{68} + ( 2918597916 - 467871360 \beta ) q^{69} + ( 16469235772 - 54804000 \beta ) q^{71} + ( -9177033840 + 382101240 \beta ) q^{72} + ( 14991424430 - 339617856 \beta ) q^{73} + ( 34384099036 - 158832450 \beta ) q^{74} + ( 1074218750 - 156250000 \beta ) q^{75} + ( 662696320 - 797905680 \beta ) q^{76} + ( 55178185400 - 1223597188 \beta ) q^{78} + ( -1651411560 - 575636160 \beta ) q^{79} + ( 9383800000 - 924000000 \beta ) q^{80} + ( -21942215899 + 696935360 \beta ) q^{81} + ( 37315257820 + 892102974 \beta ) q^{82} + ( -6649551210 + 1100818224 \beta ) q^{83} + ( -2059281250 - 395400000 \beta ) q^{85} + ( -28353046948 + 1084016190 \beta ) q^{86} + ( -17032470460 - 500316640 \beta ) q^{87} + ( -40397437440 + 1729655040 \beta ) q^{88} + ( 6337385430 + 1455281280 \beta ) q^{89} + ( -19606281250 + 12284375 \beta ) q^{90} + ( 101585785920 - 1651623672 \beta ) q^{92} + ( -83100271320 + 2749139712 \beta ) q^{93} + ( 30144882764 + 1898041590 \beta ) q^{94} + ( -8320812500 - 858000000 \beta ) q^{95} + ( -52757708032 + 5226208640 \beta ) q^{96} + ( 1540351870 - 4545870528 \beta ) q^{97} + ( 59350136224 + 1363156960 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 20q^{2} + 220q^{3} + 6976q^{4} + 6250q^{5} - 60184q^{6} - 246240q^{8} - 20846q^{9} + O(q^{10})$$ $$2q - 20q^{2} + 220q^{3} + 6976q^{4} + 6250q^{5} - 60184q^{6} - 246240q^{8} - 20846q^{9} - 62500q^{10} - 618176q^{11} + 1927040q^{12} - 3414260q^{13} + 687500q^{15} + 6005632q^{16} - 1317940q^{17} - 12548020q^{18} - 5325320q^{19} + 21800000q^{20} - 89491840q^{22} + 58943940q^{23} - 122180160q^{24} + 19531250q^{25} + 80761736q^{26} + 26769160q^{27} + 94140380q^{29} - 188075000q^{30} - 244543464q^{31} - 627301120q^{32} + 442259840q^{33} - 445358072q^{34} + 182418752q^{36} + 21003220q^{37} - 941752240q^{38} - 624203992q^{39} - 769500000q^{40} + 745743316q^{41} + 629950100q^{43} - 242725888q^{44} - 65143750q^{45} - 468194856q^{46} + 1402061540q^{47} + 6375522560q^{48} - 195312500q^{50} + 2300559784q^{51} - 12841321600q^{52} + 1138320580q^{53} + 9205154480q^{54} - 1931800000q^{55} + 4720910480q^{57} + 7387417960q^{58} - 7317515560q^{59} + 6022000000q^{60} + 1516425676q^{61} + 28564327440q^{62} + 819531776q^{64} - 10669562500q^{65} + 2975464192q^{66} + 15734290140q^{67} + 4573774720q^{68} + 5837195832q^{69} + 32938471544q^{71} - 18354067680q^{72} + 29982848860q^{73} + 68768198072q^{74} + 2148437500q^{75} + 1325392640q^{76} + 110356370800q^{78} - 3302823120q^{79} + 18767600000q^{80} - 43884431798q^{81} + 74630515640q^{82} - 13299102420q^{83} - 4118562500q^{85} - 56706093896q^{86} - 34064940920q^{87} - 80794874880q^{88} + 12674770860q^{89} - 39212562500q^{90} + 203171571840q^{92} - 166200542640q^{93} + 60289765528q^{94} - 16641625000q^{95} - 105515416064q^{96} + 3080703740q^{97} + 118700272448q^{99} + O(q^{100})$$

## 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
 −12.2882 12.2882
−83.7292 503.223 4962.58 3125.00 −42134.4 0 −244036. 76086.0 −261654.
1.2 63.7292 −283.223 2013.42 3125.00 −18049.6 0 −2204.06 −96932.0 199154.
 $$n$$: 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 245.12.a.b 2
7.b odd 2 1 5.12.a.b 2
21.c even 2 1 45.12.a.d 2
28.d even 2 1 80.12.a.j 2
35.c odd 2 1 25.12.a.c 2
35.f even 4 2 25.12.b.c 4
105.g even 2 1 225.12.a.h 2
105.k odd 4 2 225.12.b.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.a.b 2 7.b odd 2 1
25.12.a.c 2 35.c odd 2 1
25.12.b.c 4 35.f even 4 2
45.12.a.d 2 21.c even 2 1
80.12.a.j 2 28.d even 2 1
225.12.a.h 2 105.g even 2 1
225.12.b.f 4 105.k odd 4 2
245.12.a.b 2 1.a even 1 1 trivial

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{2} + 20 T_{2} - 5336$$ $$T_{3}^{2} - 220 T_{3} - 142524$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 20 T - 1240 T^{2} + 40960 T^{3} + 4194304 T^{4}$$
$3$ $$1 - 220 T + 211770 T^{2} - 38972340 T^{3} + 31381059609 T^{4}$$
$5$ $$( 1 - 3125 T )^{2}$$
$7$ 1
$11$ $$1 + 618176 T + 245194892966 T^{2} + 176372827291625536 T^{3} +$$$$81\!\cdots\!21$$$$T^{4}$$
$13$ $$1 + 3414260 T + 6398662197390 T^{2} + 6118901546944767620 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$
$17$ $$1 + 1317940 T + 59308395866630 T^{2} + 45168303019681836020 T^{3} +$$$$11\!\cdots\!89$$$$T^{4}$$
$19$ $$1 + 5325320 T + 194538827137638 T^{2} +$$$$62\!\cdots\!80$$$$T^{3} +$$$$13\!\cdots\!61$$$$T^{4}$$
$23$ $$1 - 58943940 T + 2773540471931410 T^{2} -$$$$56\!\cdots\!80$$$$T^{3} +$$$$90\!\cdots\!29$$$$T^{4}$$
$29$ $$1 - 94140380 T + 23426350431097358 T^{2} -$$$$11\!\cdots\!20$$$$T^{3} +$$$$14\!\cdots\!41$$$$T^{4}$$
$31$ $$1 + 244543464 T + 34393316207729486 T^{2} +$$$$62\!\cdots\!84$$$$T^{3} +$$$$64\!\cdots\!61$$$$T^{4}$$
$37$ $$1 - 21003220 T + 137126715218410590 T^{2} -$$$$37\!\cdots\!60$$$$T^{3} +$$$$31\!\cdots\!69$$$$T^{4}$$
$41$ $$1 - 745743316 T + 929792912462405846 T^{2} -$$$$41\!\cdots\!56$$$$T^{3} +$$$$30\!\cdots\!81$$$$T^{4}$$
$43$ $$1 - 629950100 T + 1840945003918927050 T^{2} -$$$$58\!\cdots\!00$$$$T^{3} +$$$$86\!\cdots\!49$$$$T^{4}$$
$47$ $$1 - 1402061540 T + 5181805952108806370 T^{2} -$$$$34\!\cdots\!20$$$$T^{3} +$$$$61\!\cdots\!09$$$$T^{4}$$
$53$ $$1 - 1138320580 T - 2203723231625575330 T^{2} -$$$$10\!\cdots\!60$$$$T^{3} +$$$$85\!\cdots\!09$$$$T^{4}$$
$59$ $$1 + 7317515560 T + 55027608950440780118 T^{2} +$$$$22\!\cdots\!40$$$$T^{3} +$$$$90\!\cdots\!81$$$$T^{4}$$
$61$ $$1 - 1516425676 T + 85869525433683691566 T^{2} -$$$$65\!\cdots\!36$$$$T^{3} +$$$$18\!\cdots\!21$$$$T^{4}$$
$67$ $$1 - 15734290140 T +$$$$30\!\cdots\!30$$$$T^{2} -$$$$19\!\cdots\!20$$$$T^{3} +$$$$14\!\cdots\!89$$$$T^{4}$$
$71$ $$1 - 32938471544 T +$$$$73\!\cdots\!26$$$$T^{2} -$$$$76\!\cdots\!24$$$$T^{3} +$$$$53\!\cdots\!41$$$$T^{4}$$
$73$ $$1 - 29982848860 T +$$$$78\!\cdots\!10$$$$T^{2} -$$$$94\!\cdots\!20$$$$T^{3} +$$$$98\!\cdots\!29$$$$T^{4}$$
$79$ $$1 + 3302823120 T +$$$$12\!\cdots\!58$$$$T^{2} +$$$$24\!\cdots\!80$$$$T^{3} +$$$$55\!\cdots\!41$$$$T^{4}$$
$83$ $$1 + 13299102420 T +$$$$18\!\cdots\!30$$$$T^{2} +$$$$17\!\cdots\!40$$$$T^{3} +$$$$16\!\cdots\!89$$$$T^{4}$$
$89$ $$1 - 12674770860 T +$$$$43\!\cdots\!78$$$$T^{2} -$$$$35\!\cdots\!40$$$$T^{3} +$$$$77\!\cdots\!21$$$$T^{4}$$
$97$ $$1 - 3080703740 T +$$$$18\!\cdots\!70$$$$T^{2} -$$$$22\!\cdots\!20$$$$T^{3} +$$$$51\!\cdots\!09$$$$T^{4}$$