Newform orbit 69.4.e.b

• Label: 69.4.e.b
• Level: $69$
• Weight: $4$
• Character orbit: 69.e
• Analytic conductor: $4.071$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07113179040\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(6\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 4 q^{2} + 18 q^{3} - 28 q^{4} - 6 q^{5} + 21 q^{6} - 4 q^{7} - 52 q^{8} - 54 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−1.62255	−	3.55289i	2.87848	−	0.845198i	−4.75146	+	5.48347i	2.37866	+	1.52867i	−7.67337	−	8.85554i	−5.05798	−	35.1790i	−2.78946	−	0.819058i	7.57128	−	4.86577i	1.57171	−	10.9315i
4.2	−0.936192	−	2.04997i	2.87848	−	0.845198i	1.91295	−	2.20766i	17.1881	+	11.0461i	−4.42744	−	5.10954i	3.15549	+	21.9469i	−23.6153	−	6.93407i	7.57128	−	4.86577i	6.55289	−	45.5764i
4.3	−0.468580	−	1.02605i	2.87848	−	0.845198i	4.40568	−	5.08442i	−16.0845	−	10.3369i	−2.21601	−	2.55741i	1.00849	+	7.01421i	−15.9396	−	4.68029i	7.57128	−	4.86577i	−3.06925	+	21.3471i
4.4	0.675509	+	1.47916i	2.87848	−	0.845198i	3.50729	−	4.04763i	3.16001	+	2.03082i	3.19462	+	3.68679i	−2.13282	−	14.8341i	20.8382	+	6.11864i	7.57128	−	4.86577i	−0.869282	+	6.04599i
4.5	1.45666	+	3.18964i	2.87848	−	0.845198i	−2.81307	+	3.24646i	−1.51957	−	0.976567i	6.88885	+	7.95015i	4.44617	+	30.9238i	12.4631	+	3.65950i	7.57128	−	4.86577i	0.901405	−	6.26941i
4.6	2.23641	+	4.89704i	2.87848	−	0.845198i	−13.7407	+	15.8576i	9.26837	+	5.95642i	10.5764	+	12.2058i	−1.89824	−	13.2025i	−67.0611	−	19.6909i	7.57128	−	4.86577i	−8.44102	+	58.7086i
13.1	−2.63888	−	3.04543i	−2.52376	−	1.62192i	−1.17243	+	8.15446i	−4.73327	+	10.3644i	1.72045	+	11.9660i	10.4190	+	3.05928i	0.807865	−	0.519183i	3.73874	+	8.18669i	44.0546	−	12.9356i
13.2	−2.23546	−	2.57986i	−2.52376	−	1.62192i	−0.519874	+	3.61580i	8.61254	−	18.8588i	1.45744	+	10.1367i	−2.20230	−	0.646654i	−12.4835	+	8.02266i	3.73874	+	8.18669i	−67.9062	+	19.9391i
13.3	−0.298405	−	0.344378i	−2.52376	−	1.62192i	1.10897	−	7.71304i	−4.80158	+	10.5140i	0.194549	+	1.35312i	−20.2653	−	5.95042i	−6.05385	+	3.89057i	3.73874	+	8.18669i	5.05360	−	1.48387i
13.4	1.56552	+	1.80671i	−2.52376	−	1.62192i	0.325181	−	2.26169i	1.13914	−	2.49437i	−1.02066	−	7.09886i	33.1193	+	9.72471i	20.6842	−	13.2929i	3.73874	+	8.18669i	6.28995	−	1.84690i
13.5	1.56668	+	1.80804i	−2.52376	−	1.62192i	0.323978	−	2.25332i	6.20398	−	13.5848i	−1.02142	−	7.10410i	−24.5496	−	7.20842i	20.6825	−	13.2918i	3.73874	+	8.18669i	34.2816	−	10.0660i
13.6	2.95596	+	3.41136i	−2.52376	−	1.62192i	−1.76116	+	12.2491i	−6.65677	+	14.5763i	−1.92717	−	13.4038i	1.88458	+	0.553362i	−16.6135	+	10.6769i	3.73874	+	8.18669i	−69.4021	+	20.3783i
16.1	−2.63888	+	3.04543i	−2.52376	+	1.62192i	−1.17243	−	8.15446i	−4.73327	−	10.3644i	1.72045	−	11.9660i	10.4190	−	3.05928i	0.807865	+	0.519183i	3.73874	−	8.18669i	44.0546	+	12.9356i
16.2	−2.23546	+	2.57986i	−2.52376	+	1.62192i	−0.519874	−	3.61580i	8.61254	+	18.8588i	1.45744	−	10.1367i	−2.20230	+	0.646654i	−12.4835	−	8.02266i	3.73874	−	8.18669i	−67.9062	−	19.9391i
16.3	−0.298405	+	0.344378i	−2.52376	+	1.62192i	1.10897	+	7.71304i	−4.80158	−	10.5140i	0.194549	−	1.35312i	−20.2653	+	5.95042i	−6.05385	−	3.89057i	3.73874	−	8.18669i	5.05360	+	1.48387i
16.4	1.56552	−	1.80671i	−2.52376	+	1.62192i	0.325181	+	2.26169i	1.13914	+	2.49437i	−1.02066	+	7.09886i	33.1193	−	9.72471i	20.6842	+	13.2929i	3.73874	−	8.18669i	6.28995	+	1.84690i
16.5	1.56668	−	1.80804i	−2.52376	+	1.62192i	0.323978	+	2.25332i	6.20398	+	13.5848i	−1.02142	+	7.10410i	−24.5496	+	7.20842i	20.6825	+	13.2918i	3.73874	−	8.18669i	34.2816	+	10.0660i
16.6	2.95596	−	3.41136i	−2.52376	+	1.62192i	−1.76116	−	12.2491i	−6.65677	−	14.5763i	−1.92717	+	13.4038i	1.88458	−	0.553362i	−16.6135	−	10.6769i	3.73874	−	8.18669i	−69.4021	−	20.3783i
25.1	−4.69841	−	3.01948i	0.426945	+	2.96946i	9.63444	+	21.0965i	−6.04862	−	1.77604i	6.96029	−	15.2409i	10.0657	−	11.6164i	12.0753	−	83.9856i	−8.63544	+	2.53559i	23.0562	+	26.6083i
25.2	−2.39601	−	1.53982i	0.426945	+	2.96946i	0.0464818	+	0.101781i	−1.22325	−	0.359177i	3.54948	−	7.77227i	−1.84765	+	2.13230i	−3.19731	+	22.2377i	−8.63544	+	2.53559i	2.37784	+	2.74417i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	69.4.e.b	✓	60
3.b	odd	2	1		207.4.i.b		60
23.c	even	11	1	inner	69.4.e.b	✓	60
23.c	even	11	1		1587.4.a.w		30
23.d	odd	22	1		1587.4.a.v		30
69.h	odd	22	1		207.4.i.b		60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
69.4.e.b	✓	60	1.a	even	1	1	trivial
69.4.e.b	✓	60	23.c	even	11	1	inner
207.4.i.b		60	3.b	odd	2	1
207.4.i.b		60	69.h	odd	22	1
1587.4.a.v		30	23.d	odd	22	1
1587.4.a.w		30	23.c	even	11	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^60 - 4*T2^59 + 46*T2^58 - 108*T2^57 + 783*T2^56 - 2648*T2^55 + 30060*T2^54 - 178312*T2^53 + 1245136*T2^52 - 3654021*T2^51 + 16638238*T2^50 - 26663422*T2^49 + 234981441*T2^48 - 748054404*T2^47 + 7124741271*T2^46 - 17448631956*T2^45 + 222579318145*T2^44 - 903915048291*T2^43 + 6123080869884*T2^42 - 17441394527012*T2^41 + 111460950837279*T2^40 - 305388050828394*T2^39 + 1358733920103630*T2^38 - 3159970591147286*T2^37 + 12151625790813153*T2^36 - 16773840562654462*T2^35 + 154515640504809536*T2^34 + 119889074046939126*T2^33 + 2226143418943853179*T2^32 + 3999386383253921094*T2^31 + 28154994648813834335*T2^30 + 43016222743025848355*T2^29 + 317700075227607556017*T2^28 + 435388646745068079588*T2^27 + 2957944204166374347278*T2^26 + 4701913435947461454122*T2^25 + 22597666426431515908064*T2^24 + 30820717115830473695034*T2^23 + 167635018688153913788990*T2^22 + 171576119323935935873081*T2^21 + 968049321290932332396611*T2^20 + 888355222035920236161514*T2^19 + 3598554090575967376682344*T2^18 + 5002452229812179659169368*T2^17 + 9124730703749721037047104*T2^16 + 18648889412517210230219584*T2^15 + 16034578296285255902986432*T2^14 + 22380757087465660986973824*T2^13 + 47889326703097583110731520*T2^12 - 51953639173218074546029568*T2^11 + 87392816940722201995679744*T2^10 - 135481377018480884645249024*T2^9 + 80268629373309975539970048*T2^8 - 99837047207759086535786496*T2^7 + 83808611555929798687621120*T2^6 + 33034400033738661874696192*T2^5 - 96248970687405080707072*T2^4 - 12801185084628479328649216*T2^3 + 1711022333270815929008128*T2^2 + 2956913850326314770235392*T2 + 867699003790550191046656