Newform orbit 429.2.bm.a

• 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: $8$

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.

\(\operatorname{Tr}(f)(q) = \) \( 416 q - 3 q^{3} - 54 q^{4} - 15 q^{6} - 30 q^{7} - 9 q^{9}+O(q^{10}) \)

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

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 \(S_{2}^{\mathrm{new}}(429, [\chi])\).