Newform orbit 644.2.z.a

• Label: 644.2.z.a
• Level: $644$
• Weight: $2$
• Character orbit: 644.z
• Analytic conductor: $5.142$
• Analytic rank: $0$
• Dimension: $1840$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	644.z (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.14236589017\)
Analytic rank:	\(0\)
Dimension:	\(1840\)
Relative dimension:	\(92\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1840 q - 11 q^{2} - 11 q^{4} - 22 q^{5} - 28 q^{6} - 32 q^{8} - 102 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.40792	−	0.133298i	1.46742	−	1.86598i	1.96446	+	0.375344i	2.23434	−	2.34331i	−2.31473	+	2.43154i	1.80479	+	1.93462i	−2.71577	−	0.790312i	−0.621268	−	2.56090i	−3.45813	+	3.00136i
11.2	−1.40627	−	0.149694i	−1.00593	+	1.27915i	1.95518	+	0.421019i	−0.227649	+	0.238752i	1.60609	−	1.64824i	1.25961	−	2.32667i	−2.68649	−	0.884745i	0.0829587	+	0.341961i	0.355876	−	0.301671i
11.3	−1.40424	−	0.167663i	0.176448	−	0.224372i	1.94378	+	0.470878i	1.27999	−	1.34242i	−0.285394	+	0.285488i	−2.57126	−	0.623385i	−2.65058	−	0.987125i	0.688068	+	2.83625i	−2.02249	+	1.67047i
11.4	−1.40209	−	0.184791i	2.02892	−	2.57999i	1.93170	+	0.518188i	−2.25933	+	2.36952i	−3.32149	+	3.24244i	−1.02753	+	2.43807i	−2.61266	−	1.08351i	−1.83252	−	7.55376i	3.60565	−	2.90477i
11.5	−1.40114	+	0.191837i	−1.46695	+	1.86538i	1.92640	−	0.537582i	2.70401	−	2.83589i	1.69756	−	2.89507i	−2.12383	+	1.57776i	−2.59603	+	1.12278i	−0.620417	−	2.55739i	−3.24468	+	4.49221i
11.6	−1.40037	+	0.197402i	−1.70435	+	2.16726i	1.92206	−	0.552872i	−1.34977	+	1.41560i	1.95890	−	3.37140i	−2.18652	−	1.48967i	−2.58246	+	1.15364i	−1.08492	−	4.47209i	1.61074	−	2.24881i
11.7	−1.38098	−	0.304777i	−0.581860	+	0.739895i	1.81422	+	0.841782i	−1.43907	+	1.50925i	1.02904	−	0.844445i	1.26798	+	2.32212i	−2.24885	−	1.71542i	0.498393	+	2.05440i	2.44732	−	1.64566i
11.8	−1.37950	−	0.311398i	0.414460	−	0.527029i	1.80606	+	0.859150i	−2.50580	+	2.62800i	−0.735865	+	0.597977i	−2.45782	−	0.979347i	−2.22393	−	1.74761i	0.601294	+	2.47857i	4.27511	−	2.84504i
11.9	−1.37387	+	0.335383i	0.888562	−	1.12990i	1.77504	−	0.921546i	−1.82184	+	1.91069i	−0.841819	+	1.85034i	2.05207	−	1.67003i	−2.12960	+	1.86140i	0.220149	+	0.907466i	1.86216	−	3.23606i
11.10	−1.35848	+	0.393097i	1.71597	−	2.18204i	1.69095	−	1.06803i	1.33049	−	1.39538i	−1.47337	+	3.63880i	−1.30173	−	2.30337i	−1.87728	+	2.11561i	−1.10944	−	4.57319i	−1.25893	+	2.41861i
11.11	−1.34249	+	0.444646i	0.261884	−	0.333012i	1.60458	−	1.19387i	−0.194026	+	0.203488i	−0.203505	+	0.563513i	1.21873	+	2.34834i	−1.62329	+	2.31623i	0.664963	+	2.74101i	0.169998	−	0.359455i
11.12	−1.31387	+	0.523198i	−0.980378	+	1.24665i	1.45253	−	1.37483i	2.51587	−	2.63857i	0.635847	−	2.15087i	2.55915	+	0.671363i	−1.18913	+	2.56632i	0.114277	+	0.471055i	−1.92504	+	4.78305i
11.13	−1.31299	−	0.525413i	−1.76866	+	2.24903i	1.44788	+	1.37972i	−0.0335700	+	0.0352072i	3.50390	−	2.02367i	0.00178818	+	2.64575i	−1.17613	−	2.57230i	−1.22271	−	5.04008i	0.0625753	−	0.0285885i
11.14	−1.28805	−	0.583895i	−0.651254	+	0.828137i	1.31813	+	1.50417i	1.53469	−	1.60954i	1.32239	−	0.686416i	1.67022	−	2.05192i	−0.819540	−	2.70709i	0.445598	+	1.83678i	−2.91656	+	1.17706i
11.15	−1.22277	−	0.710520i	0.651254	−	0.828137i	0.990323	+	1.73760i	1.53469	−	1.60954i	−1.38474	+	0.549890i	−1.67022	+	2.05192i	0.0236662	−	2.82833i	0.445598	+	1.83678i	−3.02018	+	0.877662i
11.16	−1.22214	+	0.711604i	1.28867	−	1.63868i	0.987239	−	1.73936i	−0.421644	+	0.442208i	−0.408843	+	2.91972i	−2.01525	+	1.71428i	0.0311903	+	2.82826i	−0.317320	−	1.30801i	0.200630	−	0.840482i
11.17	−1.20905	+	0.733621i	−1.12290	+	1.42788i	0.923599	−	1.77397i	−2.79050	+	2.92659i	0.310114	−	2.55015i	−1.57275	+	2.12755i	0.184745	+	2.82239i	−0.0706633	−	0.291278i	1.22684	−	5.58557i
11.18	−1.18960	−	0.764760i	1.76866	−	2.24903i	0.830285	+	1.81951i	−0.0335700	+	0.0352072i	−3.82396	+	1.32284i	−0.00178818	−	2.64575i	0.403785	−	2.79946i	−1.22271	−	5.04008i	0.0668598	−	0.0162094i
11.19	−1.13742	+	0.840407i	−0.238501	+	0.303279i	0.587433	−	1.91179i	0.712343	−	0.747084i	0.0163975	−	0.545393i	−2.63482	+	0.240290i	0.938522	+	2.66818i	0.672181	+	2.77077i	−0.182376	+	1.44840i
11.20	−1.06580	+	0.929555i	0.350872	−	0.446170i	0.271854	−	1.98144i	2.03388	−	2.13307i	0.0407807	+	0.801682i	2.13749	−	1.55921i	1.55211	+	2.36452i	0.631320	+	2.60234i	−0.184899	+	4.16402i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.c	even	3	1	inner
23.d	odd	22	1	inner
28.g	odd	6	1	inner
92.h	even	22	1	inner
161.p	odd	66	1	inner
644.z	even	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	644.2.z.a	✓	1840
4.b	odd	2	1	inner	644.2.z.a	✓	1840
7.c	even	3	1	inner	644.2.z.a	✓	1840
23.d	odd	22	1	inner	644.2.z.a	✓	1840
28.g	odd	6	1	inner	644.2.z.a	✓	1840
92.h	even	22	1	inner	644.2.z.a	✓	1840
161.p	odd	66	1	inner	644.2.z.a	✓	1840
644.z	even	66	1	inner	644.2.z.a	✓	1840

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
644.2.z.a	✓	1840	1.a	even	1	1	trivial
644.2.z.a	✓	1840	4.b	odd	2	1	inner
644.2.z.a	✓	1840	7.c	even	3	1	inner
644.2.z.a	✓	1840	23.d	odd	22	1	inner
644.2.z.a	✓	1840	28.g	odd	6	1	inner
644.2.z.a	✓	1840	92.h	even	22	1	inner
644.2.z.a	✓	1840	161.p	odd	66	1	inner
644.2.z.a	✓	1840	644.z	even	66	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(644, [\chi])\).