↑

Properties

Label 230.3.i.a
Level $230$
Weight $3$
Character orbit 230.i
Analytic conductor $6.267$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.i (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240q + 48q^{4} - 8q^{6} + 96q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240q + 48q^{4} - 8q^{6} + 96q^{9} + 154q^{15} - 96q^{16} + 44q^{20} + 16q^{24} - 84q^{25} + 32q^{26} - 100q^{29} - 352q^{30} + 124q^{31} + 28q^{35} - 192q^{36} + 72q^{39} + 116q^{41} - 148q^{46} - 188q^{49} + 144q^{50} + 324q^{54} + 796q^{55} - 264q^{56} + 400q^{59} + 176q^{60} - 616q^{61} + 192q^{64} + 462q^{65} - 176q^{66} + 120q^{69} - 504q^{70} + 464q^{71} - 528q^{74} - 934q^{75} - 968q^{79} - 264q^{80} + 664q^{81} - 352q^{84} - 1196q^{85} + 396q^{86} + 376q^{94} + 126q^{95} - 32q^{96} - 3300q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.28641 + 0.587486i −1.51772 − 5.16888i 1.30972 − 1.51150i −4.19163 + 2.72584i 4.98906 + 5.75768i 0.512235 + 3.56267i −0.796860 + 2.71386i −16.8426 + 10.8241i 3.79078 − 5.96909i
19.2 −1.28641 + 0.587486i −1.02272 − 3.48306i 1.30972 − 1.51150i −2.37251 − 4.40127i 3.36189 + 3.87983i −1.08475 − 7.54462i −0.796860 + 2.71386i −3.51448 + 2.25862i 5.63771 + 4.26804i
19.3 −1.28641 + 0.587486i −0.859820 − 2.92828i 1.30972 − 1.51150i 2.44584 − 4.36095i 2.82640 + 3.26184i 1.96650 + 13.6773i −0.796860 + 2.71386i −0.264236 + 0.169814i −0.584368 + 7.04688i
19.4 −1.28641 + 0.587486i −0.810314 − 2.75968i 1.30972 − 1.51150i 4.95827 − 0.644601i 2.66367 + 3.07404i −0.976678 − 6.79294i −0.796860 + 2.71386i 0.612076 − 0.393358i −5.99970 + 3.74214i
19.5 −1.28641 + 0.587486i −0.424421 − 1.44544i 1.30972 − 1.51150i 3.70278 + 3.35997i 1.39516 + 1.61010i 1.33136 + 9.25983i −0.796860 + 2.71386i 5.66211 − 3.63882i −6.73725 − 2.14698i
19.6 −1.28641 + 0.587486i −0.00972304 − 0.0331137i 1.30972 − 1.51150i −4.99585 + 0.203788i 0.0319617 + 0.0368857i 0.869641 + 6.04849i −0.796860 + 2.71386i 7.57028 − 4.86512i 6.30700 − 3.19714i
19.7 −1.28641 + 0.587486i 0.271363 + 0.924178i 1.30972 − 1.51150i −2.58664 + 4.27894i −0.892026 − 1.02945i −1.64158 − 11.4174i −0.796860 + 2.71386i 6.79082 − 4.36419i 0.813673 − 7.02410i
19.8 −1.28641 + 0.587486i 0.540708 + 1.84148i 1.30972 − 1.51150i 4.67844 + 1.76413i −1.77742 − 2.05125i −0.0656966 − 0.456930i −0.796860 + 2.71386i 4.47258 − 2.87436i −7.05482 + 0.479116i
19.9 −1.28641 + 0.587486i 0.576935 + 1.96486i 1.30972 − 1.51150i 1.83597 − 4.65072i −1.89650 − 2.18868i −0.457318 − 3.18071i −0.796860 + 2.71386i 4.04347 − 2.59858i 0.370417 + 7.06136i
19.10 −1.28641 + 0.587486i 0.725559 + 2.47103i 1.30972 − 1.51150i −2.80253 − 4.14075i −2.38506 − 2.75251i 0.102488 + 0.712820i −0.796860 + 2.71386i 1.99174 − 1.28002i 6.03785 + 3.68027i
19.11 −1.28641 + 0.587486i 1.23261 + 4.19789i 1.30972 − 1.51150i −2.21082 + 4.48467i −4.05185 − 4.67609i 0.828200 + 5.76026i −0.796860 + 2.71386i −8.53169 + 5.48298i 0.209350 − 7.06797i
19.12 −1.28641 + 0.587486i 1.62857 + 5.54639i 1.30972 − 1.51150i 4.18474 − 2.73642i −5.35343 − 6.17819i 0.543186 + 3.77794i −0.796860 + 2.71386i −20.5389 + 13.1996i −3.77570 + 5.97864i
19.13 1.28641 − 0.587486i −1.62857 − 5.54639i 1.30972 − 1.51150i 0.750729 − 4.94332i −5.35343 − 6.17819i −0.543186 − 3.77794i 0.796860 − 2.71386i −20.5389 + 13.1996i −1.93838 − 6.80020i
19.14 1.28641 − 0.587486i −1.23261 − 4.19789i 1.30972 − 1.51150i −3.16099 + 3.87403i −4.05185 − 4.67609i −0.828200 − 5.76026i 0.796860 − 2.71386i −8.53169 + 5.48298i −1.79041 + 6.84065i
19.15 1.28641 − 0.587486i −0.725559 − 2.47103i 1.30972 − 1.51150i 4.93077 + 0.829144i −2.38506 − 2.75251i −0.102488 − 0.712820i 0.796860 − 2.71386i 1.99174 − 1.28002i 6.83012 − 1.83014i
19.16 1.28641 − 0.587486i −0.576935 − 1.96486i 1.30972 − 1.51150i 3.46776 − 3.60204i −1.89650 − 2.18868i 0.457318 + 3.18071i 0.796860 − 2.71386i 4.04347 − 2.59858i 2.34483 − 6.67097i
19.17 1.28641 − 0.587486i −0.540708 − 1.84148i 1.30972 − 1.51150i −3.54821 − 3.52282i −1.77742 − 2.05125i 0.0656966 + 0.456930i 0.796860 − 2.71386i 4.47258 − 2.87436i −6.63406 − 2.44728i
19.18 1.28641 − 0.587486i −0.271363 − 0.924178i 1.30972 − 1.51150i −2.81773 + 4.13042i −0.892026 − 1.02945i 1.64158 + 11.4174i 0.796860 − 2.71386i 6.79082 − 4.36419i −1.19820 + 6.96881i
19.19 1.28641 − 0.587486i 0.00972304 + 0.0331137i 1.30972 − 1.51150i 1.88998 + 4.62904i 0.0319617 + 0.0368857i −0.869641 − 6.04849i 0.796860 − 2.71386i 7.57028 − 4.86512i 5.15079 + 4.84452i
19.20 1.28641 − 0.587486i 0.424421 + 1.44544i 1.30972 − 1.51150i −4.59453 − 1.97239i 1.39516 + 1.61010i −1.33136 − 9.25983i 0.796860 − 2.71386i 5.66211 − 3.63882i −7.06921 + 0.161910i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.d odd 22 1 inner
115.i odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.i.a ✓ 240
5.b even 2 1 inner 230.3.i.a ✓ 240
23.d odd 22 1 inner 230.3.i.a ✓ 240
115.i odd 22 1 inner 230.3.i.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.i.a ✓ 240 1.a even 1 1 trivial
230.3.i.a ✓ 240 5.b even 2 1 inner
230.3.i.a ✓ 240 23.d odd 22 1 inner
230.3.i.a ✓ 240 115.i odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(230, [\chi])\).