Newform orbit 210.4.m.a

• Label: 210.4.m.a
• Level: $210$
• Weight: $4$
• Character orbit: 210.m
• Analytic conductor: $12.390$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	210.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.3904011012\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 32 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−1.41421	+	1.41421i	−2.12132	+	2.12132i		−	4.00000i	−10.8946	+	2.51150i		−	6.00000i	1.05175	−	18.4904i	5.65685	+	5.65685i		−	9.00000i	11.8555	−	18.9591i
13.2	−1.41421	+	1.41421i	−2.12132	+	2.12132i		−	4.00000i	−3.01009	+	10.7675i		−	6.00000i	−13.9348	+	12.1992i	5.65685	+	5.65685i		−	9.00000i	−10.9706	−	19.4845i
13.3	−1.41421	+	1.41421i	−2.12132	+	2.12132i		−	4.00000i	−1.84883	−	11.0264i		−	6.00000i	−18.1918	+	3.47276i	5.65685	+	5.65685i		−	9.00000i	18.2083	+	12.9791i
13.4	−1.41421	+	1.41421i	−2.12132	+	2.12132i		−	4.00000i	4.48606	−	10.2409i		−	6.00000i	16.8923	−	7.59280i	5.65685	+	5.65685i		−	9.00000i	8.13851	+	20.8270i
13.5	−1.41421	+	1.41421i	−2.12132	+	2.12132i		−	4.00000i	5.97814	+	9.44785i		−	6.00000i	18.4168	+	1.95445i	5.65685	+	5.65685i		−	9.00000i	−21.8156	−	4.90690i
13.6	−1.41421	+	1.41421i	−2.12132	+	2.12132i		−	4.00000i	10.2391	+	4.49016i		−	6.00000i	−17.1840	−	6.90718i	5.65685	+	5.65685i		−	9.00000i	−20.8303	+	8.13017i
13.7	1.41421	−	1.41421i	2.12132	−	2.12132i		−	4.00000i	−8.75067	+	6.95886i		−	6.00000i	−15.2212	−	10.5506i	−5.65685	−	5.65685i		−	9.00000i	−2.53400	+	22.2166i
13.8	1.41421	−	1.41421i	2.12132	−	2.12132i		−	4.00000i	−7.55614	−	8.24043i		−	6.00000i	−10.7401	+	15.0881i	−5.65685	−	5.65685i		−	9.00000i	−22.3397	−	0.967740i
13.9	1.41421	−	1.41421i	2.12132	−	2.12132i		−	4.00000i	−4.67136	+	10.1577i		−	6.00000i	8.17414	+	16.6188i	−5.65685	−	5.65685i		−	9.00000i	7.75882	+	20.9714i
13.10	1.41421	−	1.41421i	2.12132	−	2.12132i		−	4.00000i	−2.62344	−	10.8682i		−	6.00000i	17.9665	+	4.49483i	−5.65685	−	5.65685i		−	9.00000i	−19.0801	−	11.6598i
13.11	1.41421	−	1.41421i	2.12132	−	2.12132i		−	4.00000i	8.69040	−	7.03399i		−	6.00000i	10.1557	−	15.4875i	−5.65685	−	5.65685i		−	9.00000i	2.34251	−	22.2376i
13.12	1.41421	−	1.41421i	2.12132	−	2.12132i		−	4.00000i	9.96147	+	5.07633i		−	6.00000i	−13.3854	−	12.7997i	−5.65685	−	5.65685i		−	9.00000i	21.2667	−	6.90864i
97.1	−1.41421	−	1.41421i	−2.12132	−	2.12132i			4.00000i	−10.8946	−	2.51150i			6.00000i	1.05175	+	18.4904i	5.65685	−	5.65685i			9.00000i	11.8555	+	18.9591i
97.2	−1.41421	−	1.41421i	−2.12132	−	2.12132i			4.00000i	−3.01009	−	10.7675i			6.00000i	−13.9348	−	12.1992i	5.65685	−	5.65685i			9.00000i	−10.9706	+	19.4845i
97.3	−1.41421	−	1.41421i	−2.12132	−	2.12132i			4.00000i	−1.84883	+	11.0264i			6.00000i	−18.1918	−	3.47276i	5.65685	−	5.65685i			9.00000i	18.2083	−	12.9791i
97.4	−1.41421	−	1.41421i	−2.12132	−	2.12132i			4.00000i	4.48606	+	10.2409i			6.00000i	16.8923	+	7.59280i	5.65685	−	5.65685i			9.00000i	8.13851	−	20.8270i
97.5	−1.41421	−	1.41421i	−2.12132	−	2.12132i			4.00000i	5.97814	−	9.44785i			6.00000i	18.4168	−	1.95445i	5.65685	−	5.65685i			9.00000i	−21.8156	+	4.90690i
97.6	−1.41421	−	1.41421i	−2.12132	−	2.12132i			4.00000i	10.2391	−	4.49016i			6.00000i	−17.1840	+	6.90718i	5.65685	−	5.65685i			9.00000i	−20.8303	−	8.13017i
97.7	1.41421	+	1.41421i	2.12132	+	2.12132i			4.00000i	−8.75067	−	6.95886i			6.00000i	−15.2212	+	10.5506i	−5.65685	+	5.65685i			9.00000i	−2.53400	−	22.2166i
97.8	1.41421	+	1.41421i	2.12132	+	2.12132i			4.00000i	−7.55614	+	8.24043i			6.00000i	−10.7401	−	15.0881i	−5.65685	+	5.65685i			9.00000i	−22.3397	+	0.967740i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
35.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	210.4.m.a	✓	24
5.c	odd	4	1		210.4.m.b	yes	24
7.b	odd	2	1		210.4.m.b	yes	24
35.f	even	4	1	inner	210.4.m.a	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
210.4.m.a	✓	24	1.a	even	1	1	trivial
210.4.m.a	✓	24	35.f	even	4	1	inner
210.4.m.b	yes	24	5.c	odd	4	1
210.4.m.b	yes	24	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T13^24 - 112*T13^23 + 6272*T13^22 + 143712*T13^21 + 45527936*T13^20 - 5137855104*T13^19 + 300215126528*T13^18 + 8267816985344*T13^17 + 444944772313728*T13^16 - 42703014055208960*T13^15 + 2710830944880197632*T13^14 + 112127092926883706880*T13^13 + 3845204533101837875200*T13^12 - 115439329600076799221760*T13^11 + 1508699550149484432228352*T13^10 + 13396039062273713059348480*T13^9 + 1576536449471237873849438208*T13^8 - 55217382951950249164247597056*T13^7 + 905419206796740608274003918848*T13^6 - 5220015477883023984042959437824*T13^5 + 4325149891644799369885276307456*T13^4 + 154112234444355431844735737659392*T13^3 + 770991464315572710960242635046912*T13^2 + 1411896964884276764878992098459648*T13 + 1292785414439748622346292529463296