Properties

Label 2.30.a.b
Level 2
Weight 30
Character orbit 2.a
Self dual Yes
Analytic conductor 10.656
Analytic rank 0
Dimension 1
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 30 \)
Character orbit: \([\chi]\) = 2.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(10.6556084766\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 16384q^{2} + 4782996q^{3} + 268435456q^{4} + 6065841750q^{5} + 78364606464q^{6} + 904018883432q^{7} + 4398046511104q^{8} - 45753326628867q^{9} + O(q^{10}) \) \( q + 16384q^{2} + 4782996q^{3} + 268435456q^{4} + 6065841750q^{5} + 78364606464q^{6} + 904018883432q^{7} + 4398046511104q^{8} - 45753326628867q^{9} + 99382751232000q^{10} + 2348011244715132q^{11} + 1283925712306176q^{12} + 16003222193389886q^{13} + 14811445386149888q^{14} + 29012896826883000q^{15} + 72057594037927936q^{16} - 535853837930780718q^{17} - 749622503487356928q^{18} - 4500449383992496540q^{19} + 1628286996185088000q^{20} + 4323918703379722272q^{21} + 38469816233412722688q^{22} + 6054145763195418936q^{23} + 21035838870424387584q^{24} - 149470078787052640625q^{25} + 262196792416499892224q^{26} - 547096798667290275000q^{27} + 242670721206679764992q^{28} - 2361414770177543957490q^{29} + 475347301611651072000q^{30} + 4580859651056014465952q^{31} + 1180591620717411303424q^{32} + 11230528391427497495472q^{33} - 8779429280657911283712q^{34} + 5483635485910208886000q^{35} - 12281815097136855908352q^{36} - 50376596834828206261258q^{37} - 73735362707333063311360q^{38} + 76543347738095051178456q^{39} + 26677854145496481792000q^{40} + 157191196896822750812682q^{41} + 70843084036173369704448q^{42} - 210053142195379460736484q^{43} + 630289469168234048520192q^{44} - 277532438866768203797250q^{45} + 99191124184193743847424q^{46} - 885237004192860923541648q^{47} + 344651184053033166176256q^{48} - 2402655614211539722738983q^{49} - 2448917770847070464000000q^{50} - 2562986763407572451071128q^{51} + 4295832246951934234198016q^{52} + 19545079942227702370848966q^{53} - 8963633949364883865600000q^{54} + 14242664637662514542361000q^{55} + 3975917096250241269628928q^{56} - 21525631401838574980833840q^{57} - 38689419594588880199516160q^{58} + 4147355257120207357648620q^{59} + 7788090189605291163648000q^{60} - 63157257621786600629654098q^{61} + 75052804522901741010157568q^{62} - 41361871252327937999231544q^{63} + 19342813113834066795298816q^{64} + 97073013315190944526540500q^{65} + 184000977165148118965813248q^{66} + 15782714250030816425172692q^{67} - 143842169334299218472337408q^{68} + 28956954968780635989212256q^{69} + 89843883801152862388224000q^{70} - 458578117673329990520126808q^{71} - 201225258551490247202439168q^{72} + 382939334311990038717870506q^{73} - 825370162541825331384451072q^{74} - 714914788958157631898812500q^{75} - 1208080182596944909293322240q^{76} + 2122646503733154141554493024q^{77} + 1254086209340949318507823104q^{78} + 4055656551380978437159131920q^{79} + 437089962319814357680128000q^{80} + 523306272599437657015977561q^{81} + 2575420569957543949314981888q^{82} - 12017293371822907068052905564q^{83} + 1160693088848664489237676032q^{84} - 3250404582018263289339376500q^{85} - 3441510681729097084706553856q^{86} - 11294637400100112038498840040q^{87} + 10326662662852346650954825728q^{88} + 10442412205106441731707520890q^{89} - 4547091478393130251014144000q^{90} + 14467215058582526712761768752q^{91} + 1625147378633830299196194816q^{92} + 21910233387562312966590552192q^{93} - 14503723076695833371306360832q^{94} - 27299013767183467199062545000q^{95} + 5646764999524895394631778304q^{96} + 108412279645050844119569076962q^{97} - 39365109583241866817355497472q^{98} - 107429325407703998959302915444q^{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
0
16384.0 4.78300e6 2.68435e8 6.06584e9 7.83646e10 9.04019e11 4.39805e12 −4.57533e13 9.93828e13
\(n\): 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
\(2\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{3} - 4782996 \) acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(2))\).