Properties

Label 805.2.bs.a
Level $805$
Weight $2$
Character orbit 805.bs
Analytic conductor $6.428$
Analytic rank $0$
Dimension $3680$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.bs (of order \(132\), degree \(40\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(3680\)
Relative dimension: \(92\) over \(\Q(\zeta_{132})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{132}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3680q - 18q^{2} - 18q^{3} - 22q^{5} - 160q^{6} - 44q^{7} - 56q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 3680q - 18q^{2} - 18q^{3} - 22q^{5} - 160q^{6} - 44q^{7} - 56q^{8} - 22q^{10} - 44q^{11} + 14q^{12} - 72q^{13} - 88q^{15} - 196q^{16} - 22q^{17} - 62q^{18} - 88q^{20} - 88q^{21} - 44q^{23} - 30q^{25} - 28q^{26} - 48q^{27} + 132q^{28} - 22q^{30} - 40q^{31} - 50q^{32} - 22q^{33} - 40q^{35} - 416q^{36} - 110q^{37} - 22q^{38} - 22q^{40} - 192q^{41} - 264q^{42} - 88q^{43} - 44q^{47} + 24q^{48} - 44q^{51} - 374q^{52} - 22q^{53} - 32q^{55} - 220q^{56} + 88q^{57} - 58q^{58} - 22q^{60} - 220q^{61} + 8q^{62} - 44q^{63} - 22q^{65} - 44q^{66} - 22q^{67} - 96q^{70} - 136q^{71} - 46q^{72} + 18q^{73} - 34q^{75} - 176q^{76} - 116q^{77} + 80q^{78} + 176q^{80} - 236q^{81} - 2q^{82} - 88q^{83} - 448q^{85} - 86q^{87} - 110q^{88} - 88q^{90} - 228q^{92} + 4q^{93} - 236q^{95} - 332q^{96} - 88q^{97} + 84q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −2.37524 + 1.44778i 0.0889339 + 0.183078i 2.62922 5.09998i −1.40279 1.74131i −0.476296 0.306097i 1.44280 + 2.21773i 0.741747 + 10.3710i 1.82887 2.32560i 5.85301 + 2.10509i
37.2 −2.30557 + 1.40532i −1.13353 2.33346i 2.42427 4.70243i −0.533908 + 2.17139i 5.89267 + 3.78699i −0.0145566 + 2.64571i 0.633836 + 8.86218i −2.30569 + 2.93193i −1.82053 5.75660i
37.3 −2.26511 + 1.38065i 1.14262 + 2.35218i 2.30804 4.47698i 1.24242 + 1.85914i −5.83570 3.75038i −1.39693 2.24691i 0.574713 + 8.03553i −2.37270 + 3.01713i −5.38104 2.49578i
37.4 −2.24779 + 1.37010i 0.546593 + 1.12521i 2.25894 4.38172i 2.23051 0.157581i −2.77028 1.78035i −1.70764 + 2.02089i 0.550193 + 7.69270i 0.887145 1.12810i −4.79781 + 3.41023i
37.5 −2.22577 + 1.35668i −0.128329 0.264176i 2.19703 4.26165i −2.05851 + 0.873231i 0.644033 + 0.413895i −1.75315 1.98153i 0.519683 + 7.26612i 1.80116 2.29036i 3.39708 4.73635i
37.6 −2.22496 + 1.35619i 1.31748 + 2.71216i 2.19477 4.25726i −2.13549 + 0.663098i −6.60955 4.24770i 2.64575 + 0.00158949i 0.518576 + 7.25065i −3.76556 + 4.78829i 3.85210 4.37149i
37.7 −2.19061 + 1.33525i −1.35414 2.78762i 2.09945 4.07236i 0.904680 2.04488i 6.68858 + 4.29849i −2.20529 1.46175i 0.472496 + 6.60636i −4.08266 + 5.19152i 0.748626 + 5.68753i
37.8 −2.08232 + 1.26924i 0.0668818 + 0.137682i 1.80864 3.50827i 0.447044 2.19092i −0.314021 0.201809i 0.109828 2.64347i 0.338730 + 4.73606i 1.83999 2.33974i 1.84992 + 5.12962i
37.9 −2.03883 + 1.24273i 0.879859 + 1.81127i 1.69598 3.28975i 1.72696 1.42043i −4.04479 2.59943i 1.66548 2.05577i 0.289777 + 4.05161i −0.652056 + 0.829157i −1.75576 + 5.04215i
37.10 −1.96942 + 1.20042i −1.19833 2.46688i 1.52115 2.95061i −2.19848 0.408248i 5.32132 + 3.41981i 2.13293 1.56544i 0.217132 + 3.03590i −2.79499 + 3.55412i 4.81981 1.83510i
37.11 −1.96536 + 1.19795i −0.713439 1.46868i 1.51109 2.93111i 1.59357 + 1.56861i 3.16156 + 2.03181i −2.41121 1.08907i 0.213081 + 2.97927i 0.206463 0.262539i −5.01104 1.17387i
37.12 −1.90357 + 1.16029i −0.663296 1.36545i 1.36087 2.63972i 2.13191 0.674517i 2.84695 + 1.82963i 2.60528 + 0.460985i 0.154245 + 2.15663i 0.429976 0.546759i −3.27561 + 3.75762i
37.13 −1.89442 + 1.15471i 0.806631 + 1.66052i 1.33901 2.59732i −1.78921 1.34116i −3.44551 2.21430i −2.54843 + 0.710975i 0.145954 + 2.04070i −0.252199 + 0.320697i 4.93817 + 0.474704i
37.14 −1.88830 + 1.15098i −0.812204 1.67199i 1.32447 2.56911i 0.424637 2.19538i 3.45811 + 2.22240i 0.926181 + 2.47834i 0.140475 + 1.96410i −0.281410 + 0.357843i 1.72499 + 4.63428i
37.15 −1.88161 + 1.14690i −0.180822 0.372238i 1.30862 2.53836i −0.543974 + 2.16889i 0.767156 + 0.493021i 2.09714 1.61308i 0.134538 + 1.88109i 1.74861 2.22354i −1.46396 4.70489i
37.16 −1.85265 + 1.12925i 1.44589 + 2.97649i 1.24065 2.40653i 0.806258 2.08565i −6.03992 3.88162i −0.440107 + 2.60889i 0.109510 + 1.53114i −4.91443 + 6.24921i 0.861507 + 4.77444i
37.17 −1.75718 + 1.07105i −0.388807 0.800393i 1.02406 1.98639i 1.91892 + 1.14793i 1.54047 + 0.989998i −1.91797 + 1.82247i 0.0344743 + 0.482013i 1.36502 1.73576i −4.60138 + 0.0381534i
37.18 −1.59878 + 0.974510i 0.214397 + 0.441355i 0.689990 1.33839i −1.58263 + 1.57964i −0.772880 0.496700i −0.922277 + 2.47980i −0.0660143 0.923000i 1.70565 2.16891i 0.990910 4.06780i
37.19 −1.52376 + 0.928783i −0.164334 0.338297i 0.542767 1.05282i −2.23602 0.0147772i 0.564611 + 0.362854i 1.55128 + 2.14325i −0.103819 1.45157i 1.76704 2.24697i 3.42089 2.05426i
37.20 −1.47228 + 0.897400i 1.05789 + 2.17776i 0.445823 0.864775i −1.05740 + 1.97026i −3.51183 2.25691i −2.36683 1.18243i −0.126334 1.76638i −1.76902 + 2.24949i −0.211323 3.84967i
See next 80 embeddings (of 3680 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 802.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
23.d odd 22 1 inner
35.l odd 12 1 inner
115.l even 44 1 inner
161.p odd 66 1 inner
805.bs even 132 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.bs.a 3680
5.c odd 4 1 inner 805.2.bs.a 3680
7.c even 3 1 inner 805.2.bs.a 3680
23.d odd 22 1 inner 805.2.bs.a 3680
35.l odd 12 1 inner 805.2.bs.a 3680
115.l even 44 1 inner 805.2.bs.a 3680
161.p odd 66 1 inner 805.2.bs.a 3680
805.bs even 132 1 inner 805.2.bs.a 3680
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.bs.a 3680 1.a even 1 1 trivial
805.2.bs.a 3680 5.c odd 4 1 inner
805.2.bs.a 3680 7.c even 3 1 inner
805.2.bs.a 3680 23.d odd 22 1 inner
805.2.bs.a 3680 35.l odd 12 1 inner
805.2.bs.a 3680 115.l even 44 1 inner
805.2.bs.a 3680 161.p odd 66 1 inner
805.2.bs.a 3680 805.bs even 132 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(805, [\chi])\).