# Properties

 Label 35.5.i.a Level $35$ Weight $5$ Character orbit 35.i Analytic conductor $3.618$ Analytic rank $0$ Dimension $28$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 35.i (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.61794870793$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 94 q^{4} - 30 q^{5} - 222 q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28 q + 94 q^{4} - 30 q^{5} - 222 q^{9} + 390 q^{10} + 34 q^{11} - 258 q^{14} + 60 q^{15} - 206 q^{16} + 366 q^{19} - 1116 q^{21} - 288 q^{24} + 40 q^{25} + 474 q^{26} + 856 q^{29} - 1620 q^{30} - 5520 q^{31} - 2620 q^{35} + 11892 q^{36} + 1656 q^{39} + 6510 q^{40} - 1082 q^{44} + 7920 q^{45} - 5442 q^{46} + 8356 q^{49} - 15960 q^{50} - 2976 q^{51} + 4392 q^{54} + 11640 q^{56} - 13920 q^{59} - 5700 q^{60} - 24960 q^{61} - 45596 q^{64} + 5290 q^{65} + 10332 q^{66} - 26400 q^{70} + 38272 q^{71} + 31806 q^{74} + 41310 q^{75} + 22028 q^{79} + 48000 q^{80} - 29238 q^{81} + 62784 q^{84} - 33820 q^{85} - 21180 q^{86} - 12516 q^{89} + 4448 q^{91} - 89418 q^{94} - 28890 q^{95} - 27900 q^{96} - 3108 q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −6.61273 3.81786i −0.705380 1.22175i 21.1521 + 36.6365i −23.8771 + 7.40838i 10.7722i 43.2397 + 23.0505i 200.851i 39.5049 68.4245i 186.177 + 42.1698i
19.2 −5.18283 2.99231i 1.73039 + 2.99712i 9.90784 + 17.1609i 24.7118 + 3.78477i 20.7114i −18.5361 45.3587i 22.8354i 34.5115 59.7757i −116.752 93.5614i
19.3 −4.29750 2.48116i −5.01068 8.67875i 4.31236 + 7.46922i 4.41022 24.6079i 49.7293i −32.5834 + 36.5967i 36.5986i −9.71376 + 16.8247i −80.0093 + 94.8102i
19.4 −4.03096 2.32728i 7.86751 + 13.6269i 2.83245 + 4.90594i −24.7840 3.27931i 73.2395i −45.1600 19.0150i 48.1053i −83.2953 + 144.272i 92.2715 + 70.8980i
19.5 −3.03175 1.75038i −7.77475 13.4663i −1.87234 3.24299i −2.74588 + 24.8487i 54.4351i 37.0452 32.0727i 69.1214i −80.3936 + 139.246i 51.8195 70.5288i
19.6 −2.10231 1.21377i 3.73387 + 6.46725i −5.05353 8.75297i 11.6512 + 22.1190i 18.1282i 22.9051 + 43.3169i 63.3759i 12.6165 21.8524i 2.35294 60.6428i
19.7 −0.575938 0.332518i 2.14828 + 3.72093i −7.77886 13.4734i −2.48237 24.8765i 2.85737i 47.2932 12.8202i 20.9870i 31.2698 54.1608i −6.84217 + 15.1527i
19.8 0.575938 + 0.332518i −2.14828 3.72093i −7.77886 13.4734i −22.7848 + 10.2884i 2.85737i −47.2932 + 12.8202i 20.9870i 31.2698 54.1608i −16.5437 1.65087i
19.9 2.10231 + 1.21377i −3.73387 6.46725i −5.05353 8.75297i 24.9812 0.969268i 18.1282i −22.9051 43.3169i 63.3759i 12.6165 21.8524i 53.6947 + 28.2837i
19.10 3.03175 + 1.75038i 7.77475 + 13.4663i −1.87234 3.24299i 20.1467 14.8024i 54.4351i −37.0452 + 32.0727i 69.1214i −80.3936 + 139.246i 86.9895 9.61266i
19.11 4.03096 + 2.32728i −7.86751 13.6269i 2.83245 + 4.90594i −15.2320 19.8239i 73.2395i 45.1600 + 19.0150i 48.1053i −83.2953 + 144.272i −15.2637 115.358i
19.12 4.29750 + 2.48116i 5.01068 + 8.67875i 4.31236 + 7.46922i −19.1060 + 16.1233i 49.7293i 32.5834 36.5967i 36.5986i −9.71376 + 16.8247i −122.113 + 21.8850i
19.13 5.18283 + 2.99231i −1.73039 2.99712i 9.90784 + 17.1609i 15.6336 + 19.5087i 20.7114i 18.5361 + 45.3587i 22.8354i 34.5115 59.7757i 22.6504 + 147.891i
19.14 6.61273 + 3.81786i 0.705380 + 1.22175i 21.1521 + 36.6365i −5.52270 24.3824i 10.7722i −43.2397 23.0505i 200.851i 39.5049 68.4245i 56.5683 182.319i
24.1 −6.61273 + 3.81786i −0.705380 + 1.22175i 21.1521 36.6365i −23.8771 7.40838i 10.7722i 43.2397 23.0505i 200.851i 39.5049 + 68.4245i 186.177 42.1698i
24.2 −5.18283 + 2.99231i 1.73039 2.99712i 9.90784 17.1609i 24.7118 3.78477i 20.7114i −18.5361 + 45.3587i 22.8354i 34.5115 + 59.7757i −116.752 + 93.5614i
24.3 −4.29750 + 2.48116i −5.01068 + 8.67875i 4.31236 7.46922i 4.41022 + 24.6079i 49.7293i −32.5834 36.5967i 36.5986i −9.71376 16.8247i −80.0093 94.8102i
24.4 −4.03096 + 2.32728i 7.86751 13.6269i 2.83245 4.90594i −24.7840 + 3.27931i 73.2395i −45.1600 + 19.0150i 48.1053i −83.2953 144.272i 92.2715 70.8980i
24.5 −3.03175 + 1.75038i −7.77475 + 13.4663i −1.87234 + 3.24299i −2.74588 24.8487i 54.4351i 37.0452 + 32.0727i 69.1214i −80.3936 139.246i 51.8195 + 70.5288i
24.6 −2.10231 + 1.21377i 3.73387 6.46725i −5.05353 + 8.75297i 11.6512 22.1190i 18.1282i 22.9051 43.3169i 63.3759i 12.6165 + 21.8524i 2.35294 + 60.6428i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 24.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.5.i.a 28
5.b even 2 1 inner 35.5.i.a 28
5.c odd 4 2 175.5.i.e 28
7.c even 3 1 245.5.c.a 28
7.d odd 6 1 inner 35.5.i.a 28
7.d odd 6 1 245.5.c.a 28
35.i odd 6 1 inner 35.5.i.a 28
35.i odd 6 1 245.5.c.a 28
35.j even 6 1 245.5.c.a 28
35.k even 12 2 175.5.i.e 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.i.a 28 1.a even 1 1 trivial
35.5.i.a 28 5.b even 2 1 inner
35.5.i.a 28 7.d odd 6 1 inner
35.5.i.a 28 35.i odd 6 1 inner
175.5.i.e 28 5.c odd 4 2
175.5.i.e 28 35.k even 12 2
245.5.c.a 28 7.c even 3 1
245.5.c.a 28 7.d odd 6 1
245.5.c.a 28 35.i odd 6 1
245.5.c.a 28 35.j even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(35, [\chi])$$.