Properties

Label 5.12.a.b
Level 5
Weight 12
Character orbit 5.a
Self dual Yes
Analytic conductor 3.842
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(3.8417159028\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{151}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -10 + 3 \beta ) q^{2} \) \( + ( -110 + 16 \beta ) q^{3} \) \( + ( 3488 - 60 \beta ) q^{4} \) \( -3125 q^{5} \) \( + ( 30092 - 490 \beta ) q^{6} \) \( + ( 28950 + 528 \beta ) q^{7} \) \( + ( -123120 + 4920 \beta ) q^{8} \) \( + ( -10423 - 3520 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -10 + 3 \beta ) q^{2} \) \( + ( -110 + 16 \beta ) q^{3} \) \( + ( 3488 - 60 \beta ) q^{4} \) \( -3125 q^{5} \) \( + ( 30092 - 490 \beta ) q^{6} \) \( + ( 28950 + 528 \beta ) q^{7} \) \( + ( -123120 + 4920 \beta ) q^{8} \) \( + ( -10423 - 3520 \beta ) q^{9} \) \( + ( 31250 - 9375 \beta ) q^{10} \) \( + ( -309088 - 26400 \beta ) q^{11} \) \( + ( -963520 + 62408 \beta ) q^{12} \) \( + ( 1707130 - 12864 \beta ) q^{13} \) \( + ( 667236 + 81570 \beta ) q^{14} \) \( + ( 343750 - 50000 \beta ) q^{15} \) \( + ( 3002816 - 295680 \beta ) q^{16} \) \( + ( 658970 + 126528 \beta ) q^{17} \) \( + ( -6274010 + 3931 \beta ) q^{18} \) \( + ( 2662660 + 274560 \beta ) q^{19} \) \( + ( -10900000 + 187500 \beta ) q^{20} \) \( + ( 1918092 + 405120 \beta ) q^{21} \) \( + ( -44745920 - 663264 \beta ) q^{22} \) \( + ( 29471970 + 33456 \beta ) q^{23} \) \( + ( 61090080 - 2511120 \beta ) q^{24} \) \( + 9765625 q^{25} \) \( + ( -40380868 + 5250030 \beta ) q^{26} \) \( + ( -13384580 - 2613920 \beta ) q^{27} \) \( + ( 81842880 + 104664 \beta ) q^{28} \) \( + ( 47070190 + 2298240 \beta ) q^{29} \) \( + ( -94037500 + 1531250 \beta ) q^{30} \) \( + ( 122271732 - 7207200 \beta ) q^{31} \) \( + ( -313650560 + 1889088 \beta ) q^{32} \) \( + ( -221129920 - 2041408 \beta ) q^{33} \) \( + ( 222679036 + 711630 \beta ) q^{34} \) \( + ( -90468750 - 1650000 \beta ) q^{35} \) \( + ( 91209376 - 11652380 \beta ) q^{36} \) \( + ( 10501610 + 19033728 \beta ) q^{37} \) \( + ( 470876120 + 5242380 \beta ) q^{38} \) \( + ( -312101996 + 28729120 \beta ) q^{39} \) \( + ( 384750000 - 15375000 \beta ) q^{40} \) \( + ( -372871658 - 22651200 \beta ) q^{41} \) \( + ( 714896520 + 1703076 \beta ) q^{42} \) \( + ( 314975050 - 13909104 \beta ) q^{43} \) \( + ( -121362944 - 73537920 \beta ) q^{44} \) \( + ( 32571875 + 11000000 \beta ) q^{45} \) \( + ( -234097428 + 88081350 \beta ) q^{46} \) \( + ( -701030770 - 20505072 \beta ) q^{47} \) \( + ( -3187761280 + 80569856 \beta ) q^{48} \) \( + ( -970838707 + 30571200 \beta ) q^{49} \) \( + ( -97656250 + 29296875 \beta ) q^{50} \) \( + ( 1150279892 - 3374560 \beta ) q^{51} \) \( + ( 6420660800 - 147297432 \beta ) q^{52} \) \( + ( 569160290 - 186753984 \beta ) q^{53} \) \( + ( -4602577240 - 14014540 \beta ) q^{54} \) \( + ( 965900000 + 82500000 \beta ) q^{55} \) \( + ( -1995276960 + 77426640 \beta ) q^{56} \) \( + ( 2360455240 + 12400960 \beta ) q^{57} \) \( + ( 3693708980 + 118228170 \beta ) q^{58} \) \( + ( 3658757780 + 175817280 \beta ) q^{59} \) \( + ( 3011000000 - 195025000 \beta ) q^{60} \) \( + ( -758212838 - 53568000 \beta ) q^{61} \) \( + ( -14282163720 + 438887196 \beta ) q^{62} \) \( + ( -1424316090 - 107407344 \beta ) q^{63} \) \( + ( 409765888 - 354289920 \beta ) q^{64} \) \( + ( -5334781250 + 40200000 \beta ) q^{65} \) \( + ( -1487732096 - 642975680 \beta ) q^{66} \) \( + ( 7867145070 - 91691472 \beta ) q^{67} \) \( + ( -2286887360 + 401791464 \beta ) q^{68} \) \( + ( -2918597916 + 467871360 \beta ) q^{69} \) \( + ( -2085112500 - 254906250 \beta ) q^{70} \) \( + ( 16469235772 - 54804000 \beta ) q^{71} \) \( + ( -9177033840 + 382101240 \beta ) q^{72} \) \( + ( -14991424430 + 339617856 \beta ) q^{73} \) \( + ( 34384099036 - 158832450 \beta ) q^{74} \) \( + ( -1074218750 + 156250000 \beta ) q^{75} \) \( + ( -662696320 + 797905680 \beta ) q^{76} \) \( + ( -17367374400 - 927478464 \beta ) q^{77} \) \( + ( 55178185400 - 1223597188 \beta ) q^{78} \) \( + ( -1651411560 - 575636160 \beta ) q^{79} \) \( + ( -9383800000 + 924000000 \beta ) q^{80} \) \( + ( -21942215899 + 696935360 \beta ) q^{81} \) \( + ( -37315257820 - 892102974 \beta ) q^{82} \) \( + ( 6649551210 - 1100818224 \beta ) q^{83} \) \( + ( -7991243904 + 1297973040 \beta ) q^{84} \) \( + ( -2059281250 - 395400000 \beta ) q^{85} \) \( + ( -28353046948 + 1084016190 \beta ) q^{86} \) \( + ( 17032470460 + 500316640 \beta ) q^{87} \) \( + ( -40397437440 + 1729655040 \beta ) q^{88} \) \( + ( -6337385430 - 1455281280 \beta ) q^{89} \) \( + ( 19606281250 - 12284375 \beta ) q^{90} \) \( + ( 45318929532 + 528951840 \beta ) q^{91} \) \( + ( 101585785920 - 1651623672 \beta ) q^{92} \) \( + ( -83100271320 + 2749139712 \beta ) q^{93} \) \( + ( -30144882764 - 1898041590 \beta ) q^{94} \) \( + ( -8320812500 - 858000000 \beta ) q^{95} \) \( + ( 52757708032 - 5226208640 \beta ) q^{96} \) \( + ( -1540351870 + 4545870528 \beta ) q^{97} \) \( + ( 65103401470 - 3218228121 \beta ) q^{98} \) \( + ( 59350136224 + 1363156960 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 220q^{3} \) \(\mathstrut +\mathstrut 6976q^{4} \) \(\mathstrut -\mathstrut 6250q^{5} \) \(\mathstrut +\mathstrut 60184q^{6} \) \(\mathstrut +\mathstrut 57900q^{7} \) \(\mathstrut -\mathstrut 246240q^{8} \) \(\mathstrut -\mathstrut 20846q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut -\mathstrut 220q^{3} \) \(\mathstrut +\mathstrut 6976q^{4} \) \(\mathstrut -\mathstrut 6250q^{5} \) \(\mathstrut +\mathstrut 60184q^{6} \) \(\mathstrut +\mathstrut 57900q^{7} \) \(\mathstrut -\mathstrut 246240q^{8} \) \(\mathstrut -\mathstrut 20846q^{9} \) \(\mathstrut +\mathstrut 62500q^{10} \) \(\mathstrut -\mathstrut 618176q^{11} \) \(\mathstrut -\mathstrut 1927040q^{12} \) \(\mathstrut +\mathstrut 3414260q^{13} \) \(\mathstrut +\mathstrut 1334472q^{14} \) \(\mathstrut +\mathstrut 687500q^{15} \) \(\mathstrut +\mathstrut 6005632q^{16} \) \(\mathstrut +\mathstrut 1317940q^{17} \) \(\mathstrut -\mathstrut 12548020q^{18} \) \(\mathstrut +\mathstrut 5325320q^{19} \) \(\mathstrut -\mathstrut 21800000q^{20} \) \(\mathstrut +\mathstrut 3836184q^{21} \) \(\mathstrut -\mathstrut 89491840q^{22} \) \(\mathstrut +\mathstrut 58943940q^{23} \) \(\mathstrut +\mathstrut 122180160q^{24} \) \(\mathstrut +\mathstrut 19531250q^{25} \) \(\mathstrut -\mathstrut 80761736q^{26} \) \(\mathstrut -\mathstrut 26769160q^{27} \) \(\mathstrut +\mathstrut 163685760q^{28} \) \(\mathstrut +\mathstrut 94140380q^{29} \) \(\mathstrut -\mathstrut 188075000q^{30} \) \(\mathstrut +\mathstrut 244543464q^{31} \) \(\mathstrut -\mathstrut 627301120q^{32} \) \(\mathstrut -\mathstrut 442259840q^{33} \) \(\mathstrut +\mathstrut 445358072q^{34} \) \(\mathstrut -\mathstrut 180937500q^{35} \) \(\mathstrut +\mathstrut 182418752q^{36} \) \(\mathstrut +\mathstrut 21003220q^{37} \) \(\mathstrut +\mathstrut 941752240q^{38} \) \(\mathstrut -\mathstrut 624203992q^{39} \) \(\mathstrut +\mathstrut 769500000q^{40} \) \(\mathstrut -\mathstrut 745743316q^{41} \) \(\mathstrut +\mathstrut 1429793040q^{42} \) \(\mathstrut +\mathstrut 629950100q^{43} \) \(\mathstrut -\mathstrut 242725888q^{44} \) \(\mathstrut +\mathstrut 65143750q^{45} \) \(\mathstrut -\mathstrut 468194856q^{46} \) \(\mathstrut -\mathstrut 1402061540q^{47} \) \(\mathstrut -\mathstrut 6375522560q^{48} \) \(\mathstrut -\mathstrut 1941677414q^{49} \) \(\mathstrut -\mathstrut 195312500q^{50} \) \(\mathstrut +\mathstrut 2300559784q^{51} \) \(\mathstrut +\mathstrut 12841321600q^{52} \) \(\mathstrut +\mathstrut 1138320580q^{53} \) \(\mathstrut -\mathstrut 9205154480q^{54} \) \(\mathstrut +\mathstrut 1931800000q^{55} \) \(\mathstrut -\mathstrut 3990553920q^{56} \) \(\mathstrut +\mathstrut 4720910480q^{57} \) \(\mathstrut +\mathstrut 7387417960q^{58} \) \(\mathstrut +\mathstrut 7317515560q^{59} \) \(\mathstrut +\mathstrut 6022000000q^{60} \) \(\mathstrut -\mathstrut 1516425676q^{61} \) \(\mathstrut -\mathstrut 28564327440q^{62} \) \(\mathstrut -\mathstrut 2848632180q^{63} \) \(\mathstrut +\mathstrut 819531776q^{64} \) \(\mathstrut -\mathstrut 10669562500q^{65} \) \(\mathstrut -\mathstrut 2975464192q^{66} \) \(\mathstrut +\mathstrut 15734290140q^{67} \) \(\mathstrut -\mathstrut 4573774720q^{68} \) \(\mathstrut -\mathstrut 5837195832q^{69} \) \(\mathstrut -\mathstrut 4170225000q^{70} \) \(\mathstrut +\mathstrut 32938471544q^{71} \) \(\mathstrut -\mathstrut 18354067680q^{72} \) \(\mathstrut -\mathstrut 29982848860q^{73} \) \(\mathstrut +\mathstrut 68768198072q^{74} \) \(\mathstrut -\mathstrut 2148437500q^{75} \) \(\mathstrut -\mathstrut 1325392640q^{76} \) \(\mathstrut -\mathstrut 34734748800q^{77} \) \(\mathstrut +\mathstrut 110356370800q^{78} \) \(\mathstrut -\mathstrut 3302823120q^{79} \) \(\mathstrut -\mathstrut 18767600000q^{80} \) \(\mathstrut -\mathstrut 43884431798q^{81} \) \(\mathstrut -\mathstrut 74630515640q^{82} \) \(\mathstrut +\mathstrut 13299102420q^{83} \) \(\mathstrut -\mathstrut 15982487808q^{84} \) \(\mathstrut -\mathstrut 4118562500q^{85} \) \(\mathstrut -\mathstrut 56706093896q^{86} \) \(\mathstrut +\mathstrut 34064940920q^{87} \) \(\mathstrut -\mathstrut 80794874880q^{88} \) \(\mathstrut -\mathstrut 12674770860q^{89} \) \(\mathstrut +\mathstrut 39212562500q^{90} \) \(\mathstrut +\mathstrut 90637859064q^{91} \) \(\mathstrut +\mathstrut 203171571840q^{92} \) \(\mathstrut -\mathstrut 166200542640q^{93} \) \(\mathstrut -\mathstrut 60289765528q^{94} \) \(\mathstrut -\mathstrut 16641625000q^{95} \) \(\mathstrut +\mathstrut 105515416064q^{96} \) \(\mathstrut -\mathstrut 3080703740q^{97} \) \(\mathstrut +\mathstrut 130206802940q^{98} \) \(\mathstrut +\mathstrut 118700272448q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 15973.7 −244036. 76086.0 261654.
1.2 63.7292 283.223 2013.42 −3125.00 18049.6 41926.3 −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.

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 20 T_{2} \) \(\mathstrut -\mathstrut 5336 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(5))\).