Newform orbit 888.2.w.a

• Label: 888.2.w.a
• Level: $888$
• Weight: $2$
• Character orbit: 888.w
• Analytic conductor: $7.091$
• Analytic rank: $0$
• Dimension: $296$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	888.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.09071569949\)
Analytic rank:	\(0\)
Dimension:	\(296\)
Relative dimension:	\(148\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 296 * q + 2 * q^6 - 16 * q^7 - 8 * q^9 - 8 * q^10 - 16 * q^12 + 8 * q^15 + 24 * q^16 - 22 * q^18 - 12 * q^22 - 4 * q^24 - 8 * q^31 + 16 * q^33 + 8 * q^34 - 16 * q^39 - 14 * q^42 + 32 * q^46 + 200 * q^49 - 52 * q^54 + 32 * q^55 - 16 * q^57 + 44 * q^60 - 144 * q^63 - 30 * q^66 - 48 * q^70 + 68 * q^72 - 16 * q^76 - 72 * q^79 - 8 * q^81 - 12 * q^82 - 4 * q^84 + 64 * q^87 + 56 * q^88 + 8 * q^90 - 20 * q^94 + 24 * q^96 + 8 * q^97

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
413.1	−1.41412	−	0.0159981i	1.64523	+	0.541495i	1.99949	+	0.0452466i	−1.59147	−	1.59147i	−2.31790	−	0.792061i	−3.18158			−2.82680	−	0.0959723i	2.41357	+	1.78177i	2.22508	+	2.27600i
413.2	−1.41256	+	0.0684143i	−1.42104	+	0.990274i	1.99064	−	0.193278i	2.35556	+	2.35556i	1.93955	−	1.49604i	−3.86845			−2.79867	+	0.409205i	1.03871	−	2.81444i	−3.48852	−	3.16621i
413.3	−1.41183	+	0.0821000i	0.216139	+	1.71851i	1.98652	−	0.231822i	−0.833876	−	0.833876i	−0.446241	−	2.40850i	−0.0353703			−2.78559	+	0.490387i	−2.90657	+	0.742875i	1.24575	+	1.10883i
413.4	−1.41137	−	0.0896546i	1.47198	+	0.912833i	1.98392	+	0.253071i	0.953606	+	0.953606i	−1.99567	−	1.42031i	3.31308			−2.77736	−	0.535045i	1.33347	+	2.68735i	−1.26039	−	1.43138i
413.5	−1.41005	−	0.108491i	−1.70774	−	0.289172i	1.97646	+	0.305955i	1.18494	+	1.18494i	2.37662	+	0.593021i	1.30313			−2.75371	−	0.645840i	2.83276	+	0.987663i	−1.54227	−	1.79938i
413.6	−1.40952	−	0.115108i	−0.389206	−	1.68776i	1.97350	+	0.324493i	2.64757	+	2.64757i	0.354321	+	2.42373i	2.84941			−2.74434	−	0.684545i	−2.69704	+	1.31377i	−3.42705	−	4.03656i
413.7	−1.40431	−	0.167044i	−1.72468	−	0.159642i	1.94419	+	0.469165i	−3.02718	−	3.02718i	2.39532	+	0.512286i	−0.206159			−2.65188	−	0.983622i	2.94903	+	0.550663i	3.74543	+	4.75678i
413.8	−1.39861	+	0.209491i	−1.51391	+	0.841480i	1.91223	−	0.585994i	−1.01345	−	1.01345i	1.94108	−	1.49405i	2.91337			−2.55170	+	1.22017i	1.58382	−	2.54784i	1.62973	+	1.20511i
413.9	−1.39820	+	0.212227i	0.212857	−	1.71892i	1.90992	−	0.593470i	−2.55078	−	2.55078i	0.0671846	+	2.44857i	−3.23545			−2.54450	+	1.23513i	−2.90938	−	0.731769i	4.10784	+	3.02516i
413.10	−1.38738	−	0.274196i	1.43384	−	0.971656i	1.84963	+	0.760828i	0.395753	+	0.395753i	−2.25570	+	0.954902i	0.173105			−2.35752	−	1.56272i	1.11177	−	2.78639i	−0.440545	−	0.657572i
413.11	−1.37909	+	0.313233i	−1.50997	−	0.848530i	1.80377	−	0.863952i	0.277234	+	0.277234i	2.34816	+	0.697226i	−3.47425			−2.21694	+	1.75647i	1.55999	+	2.56250i	−0.469169	−	0.295491i
413.12	−1.37634	−	0.325115i	0.316105	−	1.70296i	1.78860	+	0.894934i	2.11950	+	2.11950i	−0.988724	+	2.24108i	−3.59024			−2.17076	−	1.81323i	−2.80016	−	1.07663i	−2.22806	−	3.60622i
413.13	−1.34999	+	0.421339i	1.27452	−	1.17286i	1.64495	−	1.13761i	−0.834141	−	0.834141i	−1.22641	+	2.12036i	0.123762			−1.74134	+	2.22884i	0.248782	−	2.98967i	1.47754	+	0.774625i
413.14	−1.34620	+	0.433296i	0.864977	−	1.50060i	1.62451	−	1.16661i	1.33121	+	1.33121i	−0.514225	+	2.39491i	4.30095			−1.68143	+	2.27438i	−1.50363	−	2.59598i	−2.36888	−	1.21527i
413.15	−1.33020	+	0.480180i	−0.612714	+	1.62006i	1.53885	−	1.27747i	−0.861836	−	0.861836i	0.0371117	−	2.44921i	0.655765			−1.43356	+	2.43821i	−2.24916	−	1.98526i	1.56025	+	0.732576i
413.16	−1.32670	+	0.489758i	−0.607484	+	1.62202i	1.52027	−	1.29953i	2.82070	+	2.82070i	0.0115503	−	2.44946i	2.87730			−1.38050	+	2.46865i	−2.26193	−	1.97071i	−5.12368	−	2.36076i
413.17	−1.32247	+	0.501071i	1.57495	+	0.720781i	1.49786	−	1.32530i	1.79094	+	1.79094i	−2.44399	−	0.164049i	0.532373			−1.31680	+	2.50321i	1.96095	+	2.27039i	−3.26586	−	1.47108i
413.18	−1.32240	−	0.501253i	−1.05949	−	1.37021i	1.49749	+	1.32571i	−0.831585	−	0.831585i	0.714256	+	2.34304i	2.20909			−1.31577	−	2.50375i	−0.754945	+	2.90346i	0.682855	+	1.51652i
413.19	−1.31638	−	0.516866i	−0.841743	+	1.51376i	1.46570	+	1.36078i	−2.06381	−	2.06381i	1.89046	−	1.55761i	−2.85921			−1.22607	−	2.54887i	−1.58294	−	2.54839i	1.65004	+	3.78346i
413.20	−1.31558	−	0.518881i	0.190377	+	1.72156i	1.46152	+	1.36526i	−0.108025	−	0.108025i	0.642827	−	2.36364i	4.26322			−1.21435	−	2.55448i	−2.92751	+	0.655490i	0.0860642	+	0.198169i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.b	even	2	1	inner
24.h	odd	2	1	inner
37.d	odd	4	1	inner
111.g	even	4	1	inner
296.m	odd	4	1	inner
888.w	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	888.2.w.a	✓	296
3.b	odd	2	1	inner	888.2.w.a	✓	296
8.b	even	2	1	inner	888.2.w.a	✓	296
24.h	odd	2	1	inner	888.2.w.a	✓	296
37.d	odd	4	1	inner	888.2.w.a	✓	296
111.g	even	4	1	inner	888.2.w.a	✓	296
296.m	odd	4	1	inner	888.2.w.a	✓	296
888.w	even	4	1	inner	888.2.w.a	✓	296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
888.2.w.a	✓	296	1.a	even	1	1	trivial
888.2.w.a	✓	296	3.b	odd	2	1	inner
888.2.w.a	✓	296	8.b	even	2	1	inner
888.2.w.a	✓	296	24.h	odd	2	1	inner
888.2.w.a	✓	296	37.d	odd	4	1	inner
888.2.w.a	✓	296	111.g	even	4	1	inner
888.2.w.a	✓	296	296.m	odd	4	1	inner
888.2.w.a	✓	296	888.w	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(888, [\chi])\).