# Properties

 Label 35.5.l.a Level $35$ Weight $5$ Character orbit 35.l Analytic conductor $3.618$ Analytic rank $0$ Dimension $56$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 35.l (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.61794870793$$ Analytic rank: $$0$$ Dimension: $$56$$ Relative dimension: $$14$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56 q - 2 q^{2} - 2 q^{3} + 16 q^{5} - 144 q^{6} + 46 q^{7} + 108 q^{8} - 66 q^{10} + 296 q^{11} - 358 q^{12} - 8 q^{13} - 68 q^{15} + 468 q^{16} + 28 q^{17} - 868 q^{18} - 1032 q^{20} + 1280 q^{21} + 56 q^{22} + 1618 q^{23} + 1644 q^{25} + 44 q^{26} + 1228 q^{27} - 4798 q^{28} - 5696 q^{30} - 2096 q^{31} - 4166 q^{32} - 2488 q^{33} + 8840 q^{35} + 7280 q^{36} + 528 q^{37} + 4824 q^{38} + 3722 q^{40} + 5888 q^{41} - 6298 q^{42} + 6852 q^{43} - 9284 q^{45} + 5588 q^{46} + 4708 q^{47} - 15460 q^{48} - 22700 q^{50} - 856 q^{51} + 21104 q^{52} - 4172 q^{53} + 24888 q^{55} + 11412 q^{56} - 9352 q^{57} + 17750 q^{58} + 1102 q^{60} - 8860 q^{61} - 37744 q^{62} - 39484 q^{63} - 19448 q^{65} - 46580 q^{66} - 5722 q^{67} - 4924 q^{68} + 42800 q^{70} + 20792 q^{71} + 7508 q^{72} + 22368 q^{73} - 2130 q^{75} + 45904 q^{76} + 16108 q^{77} + 82704 q^{78} - 12884 q^{80} + 22664 q^{81} + 59766 q^{82} - 57908 q^{83} - 71056 q^{85} - 14728 q^{86} + 35030 q^{87} + 25584 q^{88} + 60360 q^{90} - 69064 q^{91} - 80948 q^{92} + 31192 q^{93} - 19308 q^{95} - 116352 q^{96} - 13048 q^{97} - 78606 q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −6.91084 + 1.85175i 0.723742 + 0.193926i 30.4743 17.5943i 20.7271 + 13.9781i −5.36076 −44.7877 + 19.8761i −97.0771 + 97.0771i −69.6619 40.2193i −169.126 58.2188i
2.2 −5.71329 + 1.53087i −15.4449 4.13845i 16.4417 9.49261i −22.7561 10.3519i 94.5767 −15.2970 + 46.5511i −12.4854 + 12.4854i 151.270 + 87.3360i 145.859 + 24.3067i
2.3 −5.55079 + 1.48733i 15.0503 + 4.03273i 14.7427 8.51169i −2.87000 24.8347i −89.5393 40.0414 + 28.2434i −4.15850 + 4.15850i 140.102 + 80.8879i 52.8682 + 133.584i
2.4 −5.20038 + 1.39344i 2.56504 + 0.687301i 11.2458 6.49278i −19.5926 + 15.5283i −14.2969 20.1640 44.6589i 11.4758 11.4758i −64.0410 36.9741i 80.2514 108.054i
2.5 −3.07295 + 0.823393i −9.80023 2.62596i −5.09139 + 2.93952i 24.9768 + 1.07725i 32.2778 46.2809 16.0958i 49.2180 49.2180i 19.0008 + 10.9701i −77.6393 + 17.2554i
2.6 −1.93554 + 0.518627i 0.952096 + 0.255113i −10.3791 + 5.99235i 3.12714 24.8036i −1.97513 −47.7288 11.0890i 39.6520 39.6520i −69.3067 40.0142i 6.81113 + 49.6304i
2.7 −0.495094 + 0.132660i 15.4296 + 4.13434i −13.6289 + 7.86864i 12.6360 + 21.5715i −8.18755 −36.5078 32.6831i 11.5027 11.5027i 150.831 + 87.0822i −9.11771 9.00363i
2.8 −0.167922 + 0.0449946i 2.09038 + 0.560116i −13.8302 + 7.98489i −20.4734 + 14.3471i −0.376224 20.9813 + 44.2807i 3.92997 3.92997i −66.0921 38.1583i 2.79240 3.33039i
2.9 2.34674 0.628807i −11.5823 3.10347i −8.74462 + 5.04871i 11.4928 + 22.2017i −29.1322 −36.0021 + 33.2392i −44.8336 + 44.8336i 54.3705 + 31.3908i 40.9311 + 44.8749i
2.10 3.05908 0.819678i −10.5242 2.81996i −5.17031 + 2.98508i −19.9856 15.0191i −34.5059 9.03266 48.1603i −49.2000 + 49.2000i 32.6590 + 18.8557i −73.4485 29.5629i
2.11 3.33928 0.894758i 6.96176 + 1.86540i −3.50620 + 2.02430i 21.4576 12.8286i 24.9163 48.0380 + 9.66157i −49.0093 + 49.0093i −25.1617 14.5271i 60.1743 62.0376i
2.12 5.58177 1.49563i 11.2270 + 3.00826i 15.0628 8.69652i −22.0427 11.7949i 67.1656 −38.6780 + 30.0834i 5.69217 5.69217i 46.8474 + 27.0474i −140.678 32.8685i
2.13 6.17220 1.65384i 2.01200 + 0.539113i 21.5045 12.4156i 3.38702 + 24.7695i 13.3101 2.59467 48.9313i 39.9028 39.9028i −66.3906 38.3306i 61.8701 + 147.281i
2.14 7.18170 1.92433i −11.0263 2.95448i 34.0173 19.6399i 13.9161 20.7688i −84.8726 −4.26291 + 48.8142i 122.391 122.391i 42.7013 + 24.6536i 59.9749 175.934i
18.1 −6.91084 1.85175i 0.723742 0.193926i 30.4743 + 17.5943i 20.7271 13.9781i −5.36076 −44.7877 19.8761i −97.0771 97.0771i −69.6619 + 40.2193i −169.126 + 58.2188i
18.2 −5.71329 1.53087i −15.4449 + 4.13845i 16.4417 + 9.49261i −22.7561 + 10.3519i 94.5767 −15.2970 46.5511i −12.4854 12.4854i 151.270 87.3360i 145.859 24.3067i
18.3 −5.55079 1.48733i 15.0503 4.03273i 14.7427 + 8.51169i −2.87000 + 24.8347i −89.5393 40.0414 28.2434i −4.15850 4.15850i 140.102 80.8879i 52.8682 133.584i
18.4 −5.20038 1.39344i 2.56504 0.687301i 11.2458 + 6.49278i −19.5926 15.5283i −14.2969 20.1640 + 44.6589i 11.4758 + 11.4758i −64.0410 + 36.9741i 80.2514 + 108.054i
18.5 −3.07295 0.823393i −9.80023 + 2.62596i −5.09139 2.93952i 24.9768 1.07725i 32.2778 46.2809 + 16.0958i 49.2180 + 49.2180i 19.0008 10.9701i −77.6393 17.2554i
18.6 −1.93554 0.518627i 0.952096 0.255113i −10.3791 5.99235i 3.12714 + 24.8036i −1.97513 −47.7288 + 11.0890i 39.6520 + 39.6520i −69.3067 + 40.0142i 6.81113 49.6304i
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 32.14 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.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.5.l.a 56
5.c odd 4 1 inner 35.5.l.a 56
7.c even 3 1 inner 35.5.l.a 56
35.l odd 12 1 inner 35.5.l.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.l.a 56 1.a even 1 1 trivial
35.5.l.a 56 5.c odd 4 1 inner
35.5.l.a 56 7.c even 3 1 inner
35.5.l.a 56 35.l odd 12 1 inner

## Hecke kernels

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