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

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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:

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