Properties

Label 630.2.cd.a
Level 630
Weight 2
Character orbit 630.cd
Analytic conductor 5.031
Analytic rank 0
Dimension 192
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 630.cd (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{12})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192q + 8q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 192q + 8q^{6} + 24q^{11} + 12q^{15} - 192q^{16} - 36q^{17} - 8q^{18} + 24q^{23} + 36q^{27} - 12q^{30} + 40q^{33} + 12q^{41} - 4q^{42} + 8q^{45} - 12q^{46} - 48q^{50} + 24q^{51} - 12q^{56} - 48q^{57} - 12q^{58} - 16q^{60} - 24q^{61} - 52q^{63} - 36q^{68} + 8q^{72} + 8q^{75} + 96q^{77} - 16q^{78} + 16q^{81} + 76q^{87} - 20q^{90} - 24q^{92} - 8q^{96} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.707107 0.707107i −1.72986 + 0.0870655i 1.00000i −1.30535 1.81551i 1.28476 + 1.16163i 0.500654 2.59795i 0.707107 0.707107i 2.98484 0.301222i −0.360736 + 2.20678i
23.2 −0.707107 0.707107i −1.72848 + 0.111237i 1.00000i 1.72320 1.42499i 1.30087 + 1.14356i 2.49778 + 0.872417i 0.707107 0.707107i 2.97525 0.384541i −2.22610 0.210865i
23.3 −0.707107 0.707107i −1.66933 + 0.461890i 1.00000i −1.22527 + 1.87049i 1.50700 + 0.853788i 0.471054 + 2.60348i 0.707107 0.707107i 2.57332 1.54209i 2.18903 0.456238i
23.4 −0.707107 0.707107i −1.66709 0.469915i 1.00000i 0.917891 + 2.03899i 0.846529 + 1.51109i −1.29507 2.30712i 0.707107 0.707107i 2.55836 + 1.56678i 0.792736 2.09083i
23.5 −0.707107 0.707107i −1.48287 0.895033i 1.00000i 1.60407 1.55787i 0.415666 + 1.68143i −0.441381 + 2.60867i 0.707107 0.707107i 1.39783 + 2.65444i −2.23583 0.0326704i
23.6 −0.707107 0.707107i −1.26995 1.17780i 1.00000i −2.02633 + 0.945500i 0.0651609 + 1.73082i −2.61782 + 0.383406i 0.707107 0.707107i 0.225564 + 2.99151i 2.10140 + 0.764265i
23.7 −0.707107 0.707107i −0.913950 + 1.47129i 1.00000i 0.187599 + 2.22818i 1.68662 0.394099i 1.77346 1.96337i 0.707107 0.707107i −1.32939 2.68937i 1.44291 1.70822i
23.8 −0.707107 0.707107i −0.863799 + 1.50128i 1.00000i 0.347488 2.20890i 1.67237 0.450769i −2.23833 + 1.41062i 0.707107 0.707107i −1.50770 2.59361i −1.80764 + 1.31622i
23.9 −0.707107 0.707107i −0.853456 1.50719i 1.00000i −2.22267 + 0.244427i −0.462258 + 1.66923i 2.34249 1.22994i 0.707107 0.707107i −1.54323 + 2.57264i 1.74450 + 1.39883i
23.10 −0.707107 0.707107i −0.717442 + 1.57648i 1.00000i 2.15148 0.609219i 1.62204 0.607429i −0.673710 2.55854i 0.707107 0.707107i −1.97055 2.26206i −1.95211 1.09054i
23.11 −0.707107 0.707107i −0.568379 + 1.63614i 1.00000i −2.22302 0.241222i 1.55883 0.755019i 2.52346 + 0.795090i 0.707107 0.707107i −2.35389 1.85989i 1.40134 + 1.74248i
23.12 −0.707107 0.707107i −0.299381 1.70598i 1.00000i 2.04921 + 0.894851i −0.994616 + 1.41801i −1.82080 + 1.91955i 0.707107 0.707107i −2.82074 + 1.02148i −0.816252 2.08176i
23.13 −0.707107 0.707107i 0.176209 1.72306i 1.00000i 2.05547 0.880375i −1.34299 + 1.09379i 1.34585 2.27787i 0.707107 0.707107i −2.93790 0.607238i −2.07595 0.830914i
23.14 −0.707107 0.707107i 0.386634 + 1.68835i 1.00000i −1.36579 + 1.77049i 0.920450 1.46723i −2.64536 0.0453818i 0.707107 0.707107i −2.70103 + 1.30554i 2.21768 0.286164i
23.15 −0.707107 0.707107i 0.623928 + 1.61577i 1.00000i 2.08758 + 0.801264i 0.701339 1.58371i 0.153256 + 2.64131i 0.707107 0.707107i −2.22143 + 2.01625i −0.909560 2.04272i
23.16 −0.707107 0.707107i 0.638220 1.61018i 1.00000i −1.09810 1.94787i −1.58986 + 0.687278i 0.498510 + 2.59836i 0.707107 0.707107i −2.18535 2.05530i −0.600878 + 2.15382i
23.17 −0.707107 0.707107i 0.970368 + 1.43471i 1.00000i −1.84064 1.26967i 0.328338 1.70065i −1.20135 2.35728i 0.707107 0.707107i −1.11677 + 2.78439i 0.403736 + 2.19932i
23.18 −0.707107 0.707107i 1.18452 + 1.26369i 1.00000i 0.379698 2.20359i 0.0559834 1.73115i 2.56402 0.652527i 0.707107 0.707107i −0.193831 + 2.99373i −1.82666 + 1.28969i
23.19 −0.707107 0.707107i 1.27109 1.17657i 1.00000i −1.89933 + 1.18006i −1.73076 0.0668357i 2.22881 1.42562i 0.707107 0.707107i 0.231353 2.99107i 2.17746 + 0.508599i
23.20 −0.707107 0.707107i 1.27573 1.17154i 1.00000i −0.562088 + 2.16427i −1.73048 0.0736737i −2.58208 + 0.576951i 0.707107 0.707107i 0.254982 2.98914i 1.92783 1.13291i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 527.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
63.j odd 6 1 inner
315.bv even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.cd.a yes 192
5.c odd 4 1 inner 630.2.cd.a yes 192
7.c even 3 1 630.2.bt.a 192
9.d odd 6 1 630.2.bt.a 192
35.l odd 12 1 630.2.bt.a 192
45.l even 12 1 630.2.bt.a 192
63.j odd 6 1 inner 630.2.cd.a yes 192
315.bv even 12 1 inner 630.2.cd.a yes 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bt.a 192 7.c even 3 1
630.2.bt.a 192 9.d odd 6 1
630.2.bt.a 192 35.l odd 12 1
630.2.bt.a 192 45.l even 12 1
630.2.cd.a yes 192 1.a even 1 1 trivial
630.2.cd.a yes 192 5.c odd 4 1 inner
630.2.cd.a yes 192 63.j odd 6 1 inner
630.2.cd.a yes 192 315.bv even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(630, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database