Properties

Label 547.2.h.a
Level 547
Weight 2
Character orbit 547.h
Analytic conductor 4.368
Analytic rank 0
Dimension 540
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.h (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.36781699056\)
Analytic rank: \(0\)
Dimension: \(540\)
Relative dimension: \(45\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 540q - 13q^{2} - 10q^{3} + 33q^{4} - 15q^{5} + 10q^{6} - 2q^{7} - 26q^{8} - 96q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 540q - 13q^{2} - 10q^{3} + 33q^{4} - 15q^{5} + 10q^{6} - 2q^{7} - 26q^{8} - 96q^{9} - 4q^{10} - 8q^{11} - 18q^{12} - 11q^{13} + 5q^{14} - 7q^{15} + 53q^{16} - 10q^{17} - 59q^{18} - 12q^{19} - 18q^{20} + 20q^{21} + 70q^{22} - 15q^{23} - 46q^{24} + 26q^{25} - 39q^{26} - q^{27} - 56q^{28} - 40q^{29} - 100q^{30} + 59q^{31} - 5q^{32} - 15q^{33} - 8q^{34} + 36q^{35} + 43q^{36} - 32q^{37} - 108q^{38} + 34q^{39} - 27q^{40} + 39q^{41} + 46q^{42} - 16q^{43} - 110q^{44} - 103q^{45} + 12q^{46} - 60q^{47} + 42q^{48} + 57q^{49} - 168q^{50} + 57q^{51} - 46q^{52} - 30q^{53} + 113q^{54} + 38q^{55} + 64q^{56} + 22q^{57} - 88q^{58} + 55q^{59} + 9q^{60} + 26q^{61} - 32q^{62} + 6q^{63} - 50q^{64} + 73q^{65} + 72q^{66} - 104q^{67} - 181q^{68} - 125q^{69} - 50q^{70} + 39q^{71} - 434q^{72} + 46q^{73} - 178q^{74} + 248q^{75} - 126q^{76} - 52q^{77} - 82q^{78} + 86q^{79} + 208q^{80} - 4q^{81} + 177q^{82} + 39q^{83} + 124q^{84} - 12q^{85} + 2q^{86} + 162q^{87} - 34q^{88} + 63q^{89} + 121q^{90} + 21q^{91} + q^{92} - 57q^{93} + 67q^{94} - 143q^{95} + 489q^{96} + 119q^{97} - 10q^{98} - 120q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −0.208129 + 2.77729i 1.56284 1.95974i −5.69234 0.857982i −1.65592 + 1.53647i 5.11748 + 4.74833i 4.14244 0.624372i 2.32812 10.2002i −0.730547 3.20074i −3.92258 4.91876i
13.2 −0.198483 + 2.64858i −1.46769 + 1.84043i −4.99791 0.753313i 1.33165 1.23559i −4.58320 4.25259i −1.49769 + 0.225741i 1.80518 7.90900i −0.565489 2.47757i 3.00825 + 3.77223i
13.3 −0.190322 + 2.53967i −0.0433855 + 0.0544037i −4.43605 0.668627i 1.43084 1.32763i −0.129910 0.120539i 0.795840 0.119954i 1.40894 6.17298i 0.666485 + 2.92006i 3.09942 + 3.88655i
13.4 −0.186459 + 2.48813i 0.892735 1.11945i −4.17835 0.629785i 1.41905 1.31669i 2.61888 + 2.42997i −1.35379 + 0.204051i 1.23565 5.41374i 0.211361 + 0.926033i 3.01149 + 3.77629i
13.5 −0.180942 + 2.41450i −0.674981 + 0.846399i −3.81943 0.575687i −2.60674 + 2.41870i −1.92150 1.78289i −2.23266 + 0.336520i 1.00353 4.39673i 0.406770 + 1.78218i −5.36829 6.73162i
13.6 −0.165537 + 2.20894i 1.50621 1.88872i −2.87435 0.433239i −1.13679 + 1.05478i 3.92274 + 3.63977i −4.69622 + 0.707841i 0.446983 1.95836i −0.631053 2.76482i −2.14177 2.68570i
13.7 −0.151474 + 2.02129i −0.549391 + 0.688914i −2.08499 0.314262i −0.809812 + 0.751396i −1.30927 1.21483i 2.89077 0.435714i 0.0489578 0.214498i 0.494790 + 2.16782i −1.39612 1.75068i
13.8 −0.143270 + 1.91180i −1.79610 + 2.25224i −1.65680 0.249723i −0.457736 + 0.424717i −4.04852 3.75648i −1.65842 + 0.249967i −0.138427 + 0.606487i −1.17905 5.16574i −0.746396 0.935950i
13.9 −0.136994 + 1.82806i 1.82832 2.29264i −1.34539 0.202784i 1.13848 1.05635i 3.94062 + 3.65637i 3.78047 0.569815i −0.260833 + 1.14278i −1.24589 5.45859i 1.77511 + 2.22592i
13.10 −0.136375 + 1.81980i −1.75329 + 2.19855i −1.31542 0.198268i 2.91595 2.70561i −3.76183 3.49046i 3.22982 0.486817i −0.271960 + 1.19153i −1.09205 4.78460i 4.52601 + 5.67544i
13.11 −0.135474 + 1.80778i 1.13667 1.42534i −1.27204 0.191730i 2.14226 1.98773i 2.42271 + 2.24794i 0.411640 0.0620448i −0.287858 + 1.26119i −0.0720109 0.315501i 3.30315 + 4.14202i
13.12 −0.126616 + 1.68957i −0.122192 + 0.153224i −0.860954 0.129768i −0.755939 + 0.701409i −0.243412 0.225853i 4.32246 0.651506i −0.425776 + 1.86544i 0.659016 + 2.88734i −1.08937 1.36602i
13.13 −0.112068 + 1.49544i 1.39883 1.75407i −0.246126 0.0370975i −3.24687 + 3.01266i 2.46635 + 2.28844i 1.61075 0.242782i −0.584341 + 2.56016i −0.452496 1.98251i −4.14138 5.19313i
13.14 −0.110194 + 1.47043i −1.01232 + 1.26941i −0.172368 0.0259803i 0.340700 0.316124i −1.75503 1.62843i −3.67813 + 0.554389i −0.599043 + 2.62458i 0.0809576 + 0.354698i 0.427296 + 0.535812i
13.15 −0.0868751 + 1.15927i 0.609920 0.764815i 0.641308 + 0.0966616i −0.403768 + 0.374642i 0.833639 + 0.773504i −1.69500 + 0.255480i −0.685140 + 3.00179i 0.454623 + 1.99183i −0.399233 0.500623i
13.16 −0.0690615 + 0.921562i −1.06743 + 1.33851i 1.13315 + 0.170796i 1.32465 1.22909i −1.15981 1.07614i 1.26635 0.190872i −0.646940 + 2.83443i 0.0153482 + 0.0672447i 1.04120 + 1.30563i
13.17 −0.0447565 + 0.597235i −1.37304 + 1.72174i 1.62298 + 0.244624i −2.26785 + 2.10426i −0.966831 0.897088i 2.04652 0.308464i −0.485277 + 2.12614i −0.411584 1.80327i −1.15524 1.44862i
13.18 −0.0388648 + 0.518615i −0.378079 + 0.474096i 1.71021 + 0.257773i −2.53370 + 2.35093i −0.231179 0.214503i −3.39754 + 0.512096i −0.431604 + 1.89098i 0.585740 + 2.56629i −1.12076 1.40538i
13.19 −0.0370969 + 0.495024i −0.0486951 + 0.0610617i 1.73399 + 0.261357i 2.75036 2.55197i −0.0284205 0.0263704i 1.13879 0.171645i −0.414627 + 1.81660i 0.666205 + 2.91884i 1.16125 + 1.45617i
13.20 −0.0325004 + 0.433687i 1.85358 2.32431i 1.79063 + 0.269895i 2.96965 2.75543i 0.947784 + 0.879415i −5.07657 + 0.765170i −0.368796 + 1.61580i −1.29912 5.69182i 1.09848 + 1.37745i
See next 80 embeddings (of 540 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 508.45
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
547.h even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 547.2.h.a 540
547.h even 21 1 inner 547.2.h.a 540
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
547.2.h.a 540 1.a even 1 1 trivial
547.2.h.a 540 547.h even 21 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(547, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database