Properties

Label 3.11.b.a
Level 3
Weight 11
Character orbit 3.b
Analytic conductor 1.906
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 3.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.90607175802\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 12\sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + ( -27 + 9 \beta ) q^{3} \) \( + 304 q^{4} \) \( -106 \beta q^{5} \) \( + ( -6480 - 27 \beta ) q^{6} \) \( + 17234 q^{7} \) \( + 1328 \beta q^{8} \) \( + ( -57591 - 486 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + ( -27 + 9 \beta ) q^{3} \) \( + 304 q^{4} \) \( -106 \beta q^{5} \) \( + ( -6480 - 27 \beta ) q^{6} \) \( + 17234 q^{7} \) \( + 1328 \beta q^{8} \) \( + ( -57591 - 486 \beta ) q^{9} \) \( + 76320 q^{10} \) \( -6962 \beta q^{11} \) \( + ( -8208 + 2736 \beta ) q^{12} \) \( -169654 q^{13} \) \( + 17234 \beta q^{14} \) \( + ( 686880 + 2862 \beta ) q^{15} \) \( -644864 q^{16} \) \( -12792 \beta q^{17} \) \( + ( 349920 - 57591 \beta ) q^{18} \) \( -949462 q^{19} \) \( -32224 \beta q^{20} \) \( + ( -465318 + 155106 \beta ) q^{21} \) \( + 5012640 q^{22} \) \( + 99044 \beta q^{23} \) \( + ( -8605440 - 35856 \beta ) q^{24} \) \( + 1675705 q^{25} \) \( -169654 \beta q^{26} \) \( + ( 4704237 - 505197 \beta ) q^{27} \) \( + 5239136 q^{28} \) \( + 118594 \beta q^{29} \) \( + ( -2060640 + 686880 \beta ) q^{30} \) \( -29793118 q^{31} \) \( + 715008 \beta q^{32} \) \( + ( 45113760 + 187974 \beta ) q^{33} \) \( + 9210240 q^{34} \) \( -1826804 \beta q^{35} \) \( + ( -17507664 - 147744 \beta ) q^{36} \) \( -60811846 q^{37} \) \( -949462 \beta q^{38} \) \( + ( 4580658 - 1526886 \beta ) q^{39} \) \( + 101352960 q^{40} \) \( + 6770372 \beta q^{41} \) \( + ( -111676320 - 465318 \beta ) q^{42} \) \( + 107419706 q^{43} \) \( -2116448 \beta q^{44} \) \( + ( -37091520 + 6104646 \beta ) q^{45} \) \( -71311680 q^{46} \) \( -9987608 \beta q^{47} \) \( + ( 17411328 - 5803776 \beta ) q^{48} \) \( + 14535507 q^{49} \) \( + 1675705 \beta q^{50} \) \( + ( 82892160 + 345384 \beta ) q^{51} \) \( -51574816 q^{52} \) \( + 7158798 \beta q^{53} \) \( + ( 363741840 + 4704237 \beta ) q^{54} \) \( -531339840 q^{55} \) \( + 22886752 \beta q^{56} \) \( + ( 25635474 - 8545158 \beta ) q^{57} \) \( -85387680 q^{58} \) \( -24192682 \beta q^{59} \) \( + ( 208811520 + 870048 \beta ) q^{60} \) \( + 1030793642 q^{61} \) \( -29793118 \beta q^{62} \) \( + ( -992523294 - 8375724 \beta ) q^{63} \) \( -1175146496 q^{64} \) \( + 17983324 \beta q^{65} \) \( + ( -135341280 + 45113760 \beta ) q^{66} \) \( + 1876742474 q^{67} \) \( -3888768 \beta q^{68} \) \( + ( -641805120 - 2674188 \beta ) q^{69} \) \( + 1315298880 q^{70} \) \( + 100003596 \beta q^{71} \) \( + ( 464693760 - 76480848 \beta ) q^{72} \) \( -2846528494 q^{73} \) \( -60811846 \beta q^{74} \) \( + ( -45244035 + 15081345 \beta ) q^{75} \) \( -288636448 q^{76} \) \( -119983108 \beta q^{77} \) \( + ( 1099357920 + 4580658 \beta ) q^{78} \) \( + 1488647618 q^{79} \) \( + 68355584 \beta q^{80} \) \( + ( 3146662161 + 55978452 \beta ) q^{81} \) \( -4874667840 q^{82} \) \( + 47175562 \beta q^{83} \) \( + ( -141456672 + 47152224 \beta ) q^{84} \) \( -976285440 q^{85} \) \( + 107419706 \beta q^{86} \) \( + ( -768489120 - 3202038 \beta ) q^{87} \) \( + 6656785920 q^{88} \) \( -224371428 \beta q^{89} \) \( + ( -4395345120 - 37091520 \beta ) q^{90} \) \( -2923817036 q^{91} \) \( + 30109376 \beta q^{92} \) \( + ( 804414186 - 268138062 \beta ) q^{93} \) \( + 7191077760 q^{94} \) \( + 100642972 \beta q^{95} \) \( + ( -4633251840 - 19305216 \beta ) q^{96} \) \( -1592948926 q^{97} \) \( + 14535507 \beta q^{98} \) \( + ( -2436143040 + 400948542 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 54q^{3} \) \(\mathstrut +\mathstrut 608q^{4} \) \(\mathstrut -\mathstrut 12960q^{6} \) \(\mathstrut +\mathstrut 34468q^{7} \) \(\mathstrut -\mathstrut 115182q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 54q^{3} \) \(\mathstrut +\mathstrut 608q^{4} \) \(\mathstrut -\mathstrut 12960q^{6} \) \(\mathstrut +\mathstrut 34468q^{7} \) \(\mathstrut -\mathstrut 115182q^{9} \) \(\mathstrut +\mathstrut 152640q^{10} \) \(\mathstrut -\mathstrut 16416q^{12} \) \(\mathstrut -\mathstrut 339308q^{13} \) \(\mathstrut +\mathstrut 1373760q^{15} \) \(\mathstrut -\mathstrut 1289728q^{16} \) \(\mathstrut +\mathstrut 699840q^{18} \) \(\mathstrut -\mathstrut 1898924q^{19} \) \(\mathstrut -\mathstrut 930636q^{21} \) \(\mathstrut +\mathstrut 10025280q^{22} \) \(\mathstrut -\mathstrut 17210880q^{24} \) \(\mathstrut +\mathstrut 3351410q^{25} \) \(\mathstrut +\mathstrut 9408474q^{27} \) \(\mathstrut +\mathstrut 10478272q^{28} \) \(\mathstrut -\mathstrut 4121280q^{30} \) \(\mathstrut -\mathstrut 59586236q^{31} \) \(\mathstrut +\mathstrut 90227520q^{33} \) \(\mathstrut +\mathstrut 18420480q^{34} \) \(\mathstrut -\mathstrut 35015328q^{36} \) \(\mathstrut -\mathstrut 121623692q^{37} \) \(\mathstrut +\mathstrut 9161316q^{39} \) \(\mathstrut +\mathstrut 202705920q^{40} \) \(\mathstrut -\mathstrut 223352640q^{42} \) \(\mathstrut +\mathstrut 214839412q^{43} \) \(\mathstrut -\mathstrut 74183040q^{45} \) \(\mathstrut -\mathstrut 142623360q^{46} \) \(\mathstrut +\mathstrut 34822656q^{48} \) \(\mathstrut +\mathstrut 29071014q^{49} \) \(\mathstrut +\mathstrut 165784320q^{51} \) \(\mathstrut -\mathstrut 103149632q^{52} \) \(\mathstrut +\mathstrut 727483680q^{54} \) \(\mathstrut -\mathstrut 1062679680q^{55} \) \(\mathstrut +\mathstrut 51270948q^{57} \) \(\mathstrut -\mathstrut 170775360q^{58} \) \(\mathstrut +\mathstrut 417623040q^{60} \) \(\mathstrut +\mathstrut 2061587284q^{61} \) \(\mathstrut -\mathstrut 1985046588q^{63} \) \(\mathstrut -\mathstrut 2350292992q^{64} \) \(\mathstrut -\mathstrut 270682560q^{66} \) \(\mathstrut +\mathstrut 3753484948q^{67} \) \(\mathstrut -\mathstrut 1283610240q^{69} \) \(\mathstrut +\mathstrut 2630597760q^{70} \) \(\mathstrut +\mathstrut 929387520q^{72} \) \(\mathstrut -\mathstrut 5693056988q^{73} \) \(\mathstrut -\mathstrut 90488070q^{75} \) \(\mathstrut -\mathstrut 577272896q^{76} \) \(\mathstrut +\mathstrut 2198715840q^{78} \) \(\mathstrut +\mathstrut 2977295236q^{79} \) \(\mathstrut +\mathstrut 6293324322q^{81} \) \(\mathstrut -\mathstrut 9749335680q^{82} \) \(\mathstrut -\mathstrut 282913344q^{84} \) \(\mathstrut -\mathstrut 1952570880q^{85} \) \(\mathstrut -\mathstrut 1536978240q^{87} \) \(\mathstrut +\mathstrut 13313571840q^{88} \) \(\mathstrut -\mathstrut 8790690240q^{90} \) \(\mathstrut -\mathstrut 5847634072q^{91} \) \(\mathstrut +\mathstrut 1608828372q^{93} \) \(\mathstrut +\mathstrut 14382155520q^{94} \) \(\mathstrut -\mathstrut 9266503680q^{96} \) \(\mathstrut -\mathstrut 3185897852q^{97} \) \(\mathstrut -\mathstrut 4872286080q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
2.23607i
2.23607i
26.8328i −27.0000 241.495i 304.000 2844.28i −6480.00 + 724.486i 17234.0 35634.0i −57591.0 + 13040.7i 76320.0
2.2 26.8328i −27.0000 + 241.495i 304.000 2844.28i −6480.00 724.486i 17234.0 35634.0i −57591.0 13040.7i 76320.0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{11}^{\mathrm{new}}(3, [\chi])\).