Newform orbit 775.2.bk.a

• Label: 775.2.bk.a
• Level: $775$
• Weight: $2$
• Character orbit: 775.bk
• Analytic conductor: $6.188$
• Analytic rank: $0$
• Dimension: $624$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.bk (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.18840615665\)
Analytic rank:	\(0\)
Dimension:	\(624\)
Relative dimension:	\(78\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 2 q^{2} + 3 q^{3} - 154 q^{4} - 3 q^{5} + 7 q^{6} - 34 q^{7} + 6 q^{8} - 293 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
196.1	−2.22159	+	1.61408i	−0.566274	+	0.980815i	1.71217	−	5.26951i	−1.97287	+	1.05251i	−0.325085	−	3.09297i	−3.78809	−	0.805183i	3.00452	+	9.24697i	0.858668	+	1.48726i	2.68407	−	5.52261i
196.2	−2.16256	+	1.57119i	0.153742	−	0.266290i	1.58999	−	4.89349i	1.62578	−	1.53520i	0.0859151	+	0.817427i	2.75534	+	0.585665i	2.59811	+	7.99617i	1.45273	+	2.51620i	−1.10374	+	5.87438i
196.3	−2.12774	+	1.54590i	1.58541	−	2.74601i	1.51946	−	4.67643i	−1.58666	+	1.57560i	0.871701	+	8.29369i	−1.20227	−	0.255551i	2.37079	+	7.29655i	−3.52705	−	6.10903i	0.940291	−	5.80528i
196.4	−2.12438	+	1.54345i	0.682598	−	1.18229i	1.51270	−	4.65563i	2.18814	−	0.460487i	0.374717	+	3.56519i	−4.44509	−	0.944833i	2.34929	+	7.23038i	0.568120	+	0.984013i	−3.93769	+	4.35553i
196.5	−2.10408	+	1.52870i	−1.36283	+	2.36050i	1.47217	−	4.53089i	2.15946	+	0.580292i	−0.740989	−	7.05003i	1.10424	+	0.234713i	2.22143	+	6.83687i	−2.21464	−	3.83587i	−5.43076	+	2.08019i
196.6	−2.07149	+	1.50503i	0.340837	−	0.590347i	1.40794	−	4.33318i	0.832941	+	2.07514i	0.182447	+	1.73587i	3.82500	+	0.813028i	2.02255	+	6.22476i	1.26766	+	2.19565i	−4.84857	−	3.04504i
196.7	−2.03676	+	1.47979i	−0.884367	+	1.53177i	1.34056	−	4.12583i	−1.77090	−	1.36525i	−0.465456	−	4.42852i	2.72905	+	0.580077i	1.81902	+	5.59837i	−0.0642087	−	0.111213i	5.62718	+	0.160131i
196.8	−2.00333	+	1.45551i	1.44191	−	2.49746i	1.27681	−	3.92961i	−0.235037	−	2.22368i	0.746445	+	7.10195i	1.49035	+	0.316784i	1.63129	+	5.02059i	−2.65821	−	4.60416i	3.70744	+	4.11267i
196.9	−1.95988	+	1.42394i	−1.31292	+	2.27405i	1.19551	−	3.67939i	0.859188	−	2.06441i	−0.664931	−	6.32639i	−2.61916	−	0.556720i	1.39895	+	4.30553i	−1.94754	−	3.37324i	1.25569	+	5.26943i
196.10	−1.86405	+	1.35431i	0.453070	−	0.784741i	1.02249	−	3.14689i	−1.45639	+	1.69674i	0.218238	+	2.07639i	0.973717	+	0.206970i	0.931897	+	2.86808i	1.08945	+	1.88699i	0.416868	−	5.13521i
196.11	−1.69604	+	1.23224i	−0.436256	+	0.755617i	0.740082	−	2.27774i	1.25499	+	1.85068i	−0.191198	−	1.81913i	−2.31626	−	0.492337i	0.255863	+	0.787466i	1.11936	+	1.93879i	−4.40898	−	1.59237i
196.12	−1.63781	+	1.18994i	−1.54575	+	2.67732i	0.648435	−	1.99568i	−1.04777	+	1.97539i	−0.654199	−	6.22429i	−2.29239	−	0.487263i	0.0615450	+	0.189416i	−3.27869	−	5.67885i	−0.634540	−	4.48210i
196.13	−1.60185	+	1.16381i	0.904275	−	1.56625i	0.593430	−	1.82639i	−2.12605	−	0.692771i	0.374308	+	3.56130i	−0.328253	−	0.0697723i	−0.0487153	−	0.149930i	−0.135426	−	0.234565i	4.21186	−	1.36460i
196.14	−1.59831	+	1.16124i	1.01109	−	1.75126i	0.588082	−	1.80993i	1.14689	+	1.91955i	0.417598	+	3.97318i	−2.58723	−	0.549933i	−0.0591731	−	0.182116i	−0.544610	−	0.943292i	−4.06213	−	1.73622i
196.15	−1.59057	+	1.15562i	−0.697000	+	1.20724i	0.576435	−	1.77409i	−2.13884	+	0.652184i	−0.286480	−	2.72567i	2.58692	+	0.549866i	−0.0817851	−	0.251709i	0.528381	+	0.915182i	2.64831	−	3.50904i
196.16	−1.58081	+	1.14853i	1.39135	−	2.40989i	0.561815	−	1.72909i	1.90515	+	1.17064i	0.568359	+	5.40757i	3.44866	+	0.733035i	−0.109851	−	0.338085i	−2.37170	−	4.10791i	−4.35619	+	0.337559i
196.17	−1.56578	+	1.13761i	−0.654365	+	1.13339i	0.539487	−	1.66037i	1.31298	−	1.81000i	−0.264763	−	2.51905i	1.41312	+	0.300368i	−0.152021	−	0.467874i	0.643614	+	1.11477i	0.00323209	+	4.32771i
196.18	−1.50723	+	1.09507i	−0.477379	+	0.826845i	0.454540	−	1.39893i	−1.03107	−	1.98416i	−0.185931	−	1.76901i	−4.45084	−	0.946056i	−0.304596	−	0.937451i	1.04422	+	1.80864i	3.72685	+	1.86150i
196.19	−1.38882	+	1.00903i	−1.24543	+	2.15715i	0.292628	−	0.900616i	0.681860	+	2.12957i	−0.446963	−	4.25257i	3.70149	+	0.786776i	−0.558615	−	1.71924i	−1.60220	−	2.77509i	−3.09579	−	2.26956i
196.20	−1.37965	+	1.00238i	0.370294	−	0.641368i	0.280647	−	0.863743i	2.16688	−	0.551917i	0.132015	+	1.25604i	−0.501118	−	0.106516i	−0.575360	−	1.77078i	1.22576	+	2.12309i	−2.43632	+	2.93348i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
775.bk	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	775.2.bk.a	✓	624
25.d	even	5	1		775.2.bo.a	yes	624
31.g	even	15	1		775.2.bo.a	yes	624
775.bk	even	15	1	inner	775.2.bk.a	✓	624

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
775.2.bk.a	✓	624	1.a	even	1	1	trivial
775.2.bk.a	✓	624	775.bk	even	15	1	inner
775.2.bo.a	yes	624	25.d	even	5	1
775.2.bo.a	yes	624	31.g	even	15	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(775, [\chi])\).