# Properties

 Label 429.2.bm.a Level $429$ Weight $2$ Character orbit 429.bm Analytic conductor $3.426$ Analytic rank $0$ Dimension $416$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.bm (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$416$$ Relative dimension: $$52$$ over $$\Q(\zeta_{30})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$416q - 3q^{3} - 54q^{4} - 15q^{6} - 30q^{7} - 9q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$416q - 3q^{3} - 54q^{4} - 15q^{6} - 30q^{7} - 9q^{9} - 36q^{12} - 20q^{13} - 9q^{15} + 14q^{16} - 30q^{19} - 28q^{22} + 15q^{24} - 84q^{25} - 24q^{27} - 30q^{28} - 5q^{30} - 27q^{33} - 73q^{36} - 18q^{37} - 65q^{39} - 120q^{40} - 25q^{42} + 36q^{45} + 30q^{46} - 41q^{48} + 14q^{49} + 60q^{51} + 20q^{52} + 18q^{55} - 126q^{58} - 30q^{61} + 105q^{63} - 56q^{64} + 170q^{66} - 33q^{69} - 195q^{72} + 77q^{75} + 4q^{78} - 13q^{81} + 36q^{82} - 60q^{84} - 30q^{85} + 38q^{88} - 190q^{90} - 56q^{91} + 24q^{93} - 90q^{94} + 54q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −0.555113 + 2.61160i 1.41025 1.00558i −4.68522 2.08599i 0.0414819 0.127668i 1.84333 + 4.24122i −3.24627 1.44533i 4.90990 6.75789i 0.977616 2.83624i 0.310391 + 0.179205i
17.2 −0.548083 + 2.57853i 1.55935 + 0.753941i −4.52132 2.01302i −0.573201 + 1.76413i −2.79871 + 3.60761i 2.49601 + 1.11130i 4.56973 6.28970i 1.86314 + 2.35132i −4.23470 2.44491i
17.3 −0.544551 + 2.56191i −1.01360 + 1.40450i −4.43977 1.97671i −1.15305 + 3.54872i −3.04625 3.36157i −2.16772 0.965133i 4.40285 6.06001i −0.945233 2.84720i −8.46362 4.88647i
17.4 −0.524846 + 2.46921i 0.0967708 + 1.72935i −3.99443 1.77843i 0.945056 2.90858i −4.32090 0.668693i −2.33841 1.04113i 3.52021 4.84515i −2.98127 + 0.334700i 6.68589 + 3.86010i
17.5 −0.521222 + 2.45216i −1.63920 + 0.559477i −3.91432 1.74277i 0.413155 1.27156i −0.517537 4.31120i 4.33375 + 1.92951i 3.36669 4.63385i 2.37397 1.83419i 2.90272 + 1.67589i
17.6 −0.516841 + 2.43155i −1.38291 1.04286i −3.81820 1.69997i 0.603851 1.85846i 3.25051 2.82362i −1.58507 0.705718i 3.18465 4.38329i 0.824883 + 2.88437i 4.20684 + 2.42882i
17.7 −0.450414 + 2.11903i 0.480989 1.66393i −2.46033 1.09541i −1.18878 + 3.65868i 3.30927 + 1.76869i 3.06864 + 1.36625i 0.882652 1.21487i −2.53730 1.60066i −7.21741 4.16697i
17.8 −0.450345 + 2.11870i −1.68191 0.413738i −2.45901 1.09482i −0.711758 + 2.19057i 1.63403 3.37715i −0.204771 0.0911700i 0.880673 1.21214i 2.65764 + 1.39174i −4.32063 2.49452i
17.9 −0.435266 + 2.04776i 1.43664 0.967498i −2.17679 0.969169i 0.305012 0.938731i 1.35589 + 3.36303i 0.679317 + 0.302451i 0.471041 0.648332i 1.12789 2.77990i 1.78954 + 1.03319i
17.10 −0.390743 + 1.83830i 0.153413 1.72524i −1.39958 0.623134i 1.08132 3.32797i 3.11157 + 0.956146i −1.04855 0.466846i −0.516948 + 0.711518i −2.95293 0.529349i 5.69530 + 3.28818i
17.11 −0.375289 + 1.76559i 1.32300 + 1.11789i −1.14939 0.511742i −0.439759 + 1.35344i −2.47024 + 1.91635i −3.76852 1.67785i −0.787069 + 1.08331i 0.500649 + 2.95793i −2.22459 1.28437i
17.12 −0.361366 + 1.70009i −0.634525 + 1.61164i −0.932645 0.415240i 0.654561 2.01453i −2.51064 1.66114i 2.75978 + 1.22873i −1.00026 + 1.37673i −2.19475 2.04525i 3.18836 + 1.84080i
17.13 −0.358071 + 1.68459i 0.834638 + 1.51769i −0.882537 0.392931i −0.368818 + 1.13510i −2.85554 + 0.862583i 1.13679 + 0.506130i −1.04666 + 1.44060i −1.60676 + 2.53344i −1.78012 1.02775i
17.14 −0.305648 + 1.43796i 0.244883 1.71465i −0.147221 0.0655471i −0.592286 + 1.82287i 2.39076 + 0.876212i −3.94686 1.75725i −1.58894 + 2.18699i −2.88006 0.839778i −2.44018 1.40884i
17.15 −0.305300 + 1.43633i −1.57840 + 0.713204i −0.142730 0.0635477i 0.895304 2.75546i −0.542507 2.48483i −3.37251 1.50154i −1.59137 + 2.19034i 1.98268 2.25144i 3.68440 + 2.12719i
17.16 −0.300888 + 1.41557i 1.73162 0.0388118i −0.0862042 0.0383806i 0.702761 2.16288i −0.466082 + 2.46290i 3.04220 + 1.35448i −1.62101 + 2.23112i 2.99699 0.134414i 2.85024 + 1.64559i
17.17 −0.247814 + 1.16587i −1.16286 1.28365i 0.529239 + 0.235632i 0.0759465 0.233739i 1.78475 1.03764i 1.74216 + 0.775657i −1.80706 + 2.48720i −0.295527 + 2.98541i 0.253690 + 0.146468i
17.18 −0.211457 + 0.994829i −0.464497 + 1.66860i 0.882120 + 0.392745i −1.09667 + 3.37519i −1.56176 0.814934i 2.59393 + 1.15489i −1.77286 + 2.44014i −2.56849 1.55012i −3.12584 1.80470i
17.19 −0.175413 + 0.825254i 1.67006 0.459246i 1.17682 + 0.523953i −0.503937 + 1.55096i 0.0860448 + 1.45878i −1.56138 0.695173i −1.63064 + 2.24438i 2.57819 1.53394i −1.19154 0.687934i
17.20 −0.169208 + 0.796062i −1.73164 + 0.0376535i 1.22201 + 0.544073i 0.0210213 0.0646970i 0.263033 1.38487i −0.681329 0.303347i −1.59662 + 2.19756i 2.99716 0.130405i 0.0479459 + 0.0276816i
See next 80 embeddings (of 416 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.52 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
13.e even 6 1 inner
33.f even 10 1 inner
39.h odd 6 1 inner
143.v odd 30 1 inner
429.bm even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.bm.a 416
3.b odd 2 1 inner 429.2.bm.a 416
11.d odd 10 1 inner 429.2.bm.a 416
13.e even 6 1 inner 429.2.bm.a 416
33.f even 10 1 inner 429.2.bm.a 416
39.h odd 6 1 inner 429.2.bm.a 416
143.v odd 30 1 inner 429.2.bm.a 416
429.bm even 30 1 inner 429.2.bm.a 416

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.bm.a 416 1.a even 1 1 trivial
429.2.bm.a 416 3.b odd 2 1 inner
429.2.bm.a 416 11.d odd 10 1 inner
429.2.bm.a 416 13.e even 6 1 inner
429.2.bm.a 416 33.f even 10 1 inner
429.2.bm.a 416 39.h odd 6 1 inner
429.2.bm.a 416 143.v odd 30 1 inner
429.2.bm.a 416 429.bm even 30 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(429, [\chi])$$.