# Properties

 Label 304.7.e.f Level $304$ Weight $7$ Character orbit 304.e Analytic conductor $69.936$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 304.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$69.9364414204$$ Analytic rank: $$0$$ Dimension: $$30$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$30q + 720q^{7} - 8670q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q + 720q^{7} - 8670q^{9} + 2524q^{11} + 9700q^{17} - 4014q^{19} + 39376q^{23} + 110742q^{25} + 19976q^{35} + 266500q^{39} + 106788q^{43} - 91360q^{45} - 222756q^{47} + 593586q^{49} - 540936q^{55} - 545972q^{57} - 242640q^{61} + 377716q^{63} + 545964q^{73} - 272356q^{77} + 2189926q^{81} - 1542652q^{83} - 826908q^{85} + 2729572q^{87} - 2139912q^{93} - 2142716q^{95} + 293012q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
113.1 0 50.6470i 0 207.318 0 −216.532 0 −1836.12 0
113.2 0 48.7582i 0 −200.006 0 536.711 0 −1648.37 0
113.3 0 47.0707i 0 −22.4613 0 196.082 0 −1486.65 0
113.4 0 43.2780i 0 −100.391 0 −640.715 0 −1143.99 0
113.5 0 36.8625i 0 9.31317 0 −333.649 0 −629.840 0
113.6 0 33.4715i 0 71.5877 0 −51.8109 0 −391.339 0
113.7 0 33.1604i 0 225.972 0 256.290 0 −370.612 0
113.8 0 28.4003i 0 32.4520 0 263.602 0 −77.5754 0
113.9 0 27.9511i 0 −141.368 0 −302.893 0 −52.2651 0
113.10 0 20.4841i 0 −148.208 0 486.917 0 309.402 0
113.11 0 17.1584i 0 166.964 0 600.271 0 434.588 0
113.12 0 11.4888i 0 −77.3334 0 256.225 0 597.007 0
113.13 0 9.57785i 0 39.7533 0 24.5800 0 637.265 0
113.14 0 9.20554i 0 −204.194 0 −264.373 0 644.258 0
113.15 0 7.05423i 0 140.601 0 −450.706 0 679.238 0
113.16 0 7.05423i 0 140.601 0 −450.706 0 679.238 0
113.17 0 9.20554i 0 −204.194 0 −264.373 0 644.258 0
113.18 0 9.57785i 0 39.7533 0 24.5800 0 637.265 0
113.19 0 11.4888i 0 −77.3334 0 256.225 0 597.007 0
113.20 0 17.1584i 0 166.964 0 600.271 0 434.588 0
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 113.30 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.7.e.f 30
4.b odd 2 1 152.7.e.a 30
19.b odd 2 1 inner 304.7.e.f 30
76.d even 2 1 152.7.e.a 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.7.e.a 30 4.b odd 2 1
152.7.e.a 30 76.d even 2 1
304.7.e.f 30 1.a even 1 1 trivial
304.7.e.f 30 19.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{7}^{\mathrm{new}}(304, [\chi])$$:

 $$10\!\cdots\!99$$$$T_{3}^{22} +$$$$18\!\cdots\!10$$$$T_{3}^{20} +$$$$22\!\cdots\!17$$$$T_{3}^{18} +$$$$20\!\cdots\!20$$$$T_{3}^{16} +$$$$12\!\cdots\!52$$$$T_{3}^{14} +$$$$53\!\cdots\!12$$$$T_{3}^{12} +$$$$15\!\cdots\!28$$$$T_{3}^{10} +$$$$30\!\cdots\!52$$$$T_{3}^{8} +$$$$37\!\cdots\!96$$$$T_{3}^{6} +$$$$27\!\cdots\!08$$$$T_{3}^{4} +$$$$10\!\cdots\!80$$$$T_{3}^{2} +$$$$16\!\cdots\!68$$">$$T_{3}^{30} + \cdots$$ $$20\!\cdots\!35$$$$T_{5}^{9} -$$$$44\!\cdots\!50$$$$T_{5}^{8} +$$$$26\!\cdots\!00$$$$T_{5}^{7} +$$$$51\!\cdots\!00$$$$T_{5}^{6} -$$$$15\!\cdots\!00$$$$T_{5}^{5} -$$$$12\!\cdots\!00$$$$T_{5}^{4} +$$$$34\!\cdots\!00$$$$T_{5}^{3} -$$$$28\!\cdots\!00$$$$T_{5}^{2} -$$$$15\!\cdots\!00$$$$T_{5} +$$$$14\!\cdots\!00$$">$$T_{5}^{15} - \cdots$$