# Properties

 Label 32.8.g.a Level $32$ Weight $8$ Character orbit 32.g Analytic conductor $9.996$ Analytic rank $0$ Dimension $108$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,8,Mod(5,32)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("32.5");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$32 = 2^{5}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 32.g (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$9.99632081549$$ Analytic rank: $$0$$ Dimension: $$108$$ Relative dimension: $$27$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10})$$ 108 * q - 4 * q^2 - 4 * q^3 - 4 * q^4 - 4 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$108 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9} - 13004 q^{10} - 4 q^{11} + 30668 q^{12} - 4 q^{13} - 26196 q^{14} - 52784 q^{16} + 102056 q^{18} - 4 q^{19} + 81996 q^{20} - 4 q^{21} - 374240 q^{22} - 143420 q^{23} + 380880 q^{24} - 4 q^{25} - 363984 q^{26} + 476084 q^{27} - 195464 q^{28} - 4 q^{29} + 870468 q^{30} - 714992 q^{31} + 535656 q^{32} - 8 q^{33} - 201072 q^{34} + 816500 q^{35} - 1669712 q^{36} - 4 q^{37} - 2533100 q^{38} - 283948 q^{39} + 2746344 q^{40} - 4 q^{41} + 4924936 q^{42} + 366180 q^{43} - 1585404 q^{44} + 8744 q^{45} - 2715812 q^{46} - 5044368 q^{48} - 1653804 q^{50} + 3002072 q^{51} + 8882212 q^{52} - 1815636 q^{53} + 6815776 q^{54} - 4191012 q^{55} - 3655000 q^{56} - 4 q^{57} - 12448736 q^{58} - 1835940 q^{59} - 6364808 q^{60} - 4559780 q^{61} + 12948600 q^{62} + 10001872 q^{63} + 16915544 q^{64} - 8 q^{65} - 11246068 q^{66} - 1940684 q^{67} - 8895536 q^{68} + 9580892 q^{69} - 8149480 q^{70} - 12348708 q^{71} + 12884564 q^{72} - 4 q^{73} + 10931436 q^{74} + 11573008 q^{75} + 14443516 q^{76} - 11914452 q^{77} - 38252364 q^{78} - 11333192 q^{80} + 15827196 q^{82} - 9565884 q^{83} + 8101984 q^{84} + 312496 q^{85} - 14812992 q^{86} + 40789108 q^{87} + 5716448 q^{88} - 4 q^{89} + 5335400 q^{90} - 3406996 q^{91} + 27562664 q^{92} - 8752 q^{93} - 14227032 q^{94} - 54872008 q^{95} - 25080448 q^{96} - 8 q^{97} + 57598104 q^{98} - 9738480 q^{99}+O(q^{100})$$ 108 * q - 4 * q^2 - 4 * q^3 - 4 * q^4 - 4 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 - 4 * q^9 - 13004 * q^10 - 4 * q^11 + 30668 * q^12 - 4 * q^13 - 26196 * q^14 - 52784 * q^16 + 102056 * q^18 - 4 * q^19 + 81996 * q^20 - 4 * q^21 - 374240 * q^22 - 143420 * q^23 + 380880 * q^24 - 4 * q^25 - 363984 * q^26 + 476084 * q^27 - 195464 * q^28 - 4 * q^29 + 870468 * q^30 - 714992 * q^31 + 535656 * q^32 - 8 * q^33 - 201072 * q^34 + 816500 * q^35 - 1669712 * q^36 - 4 * q^37 - 2533100 * q^38 - 283948 * q^39 + 2746344 * q^40 - 4 * q^41 + 4924936 * q^42 + 366180 * q^43 - 1585404 * q^44 + 8744 * q^45 - 2715812 * q^46 - 5044368 * q^48 - 1653804 * q^50 + 3002072 * q^51 + 8882212 * q^52 - 1815636 * q^53 + 6815776 * q^54 - 4191012 * q^55 - 3655000 * q^56 - 4 * q^57 - 12448736 * q^58 - 1835940 * q^59 - 6364808 * q^60 - 4559780 * q^61 + 12948600 * q^62 + 10001872 * q^63 + 16915544 * q^64 - 8 * q^65 - 11246068 * q^66 - 1940684 * q^67 - 8895536 * q^68 + 9580892 * q^69 - 8149480 * q^70 - 12348708 * q^71 + 12884564 * q^72 - 4 * q^73 + 10931436 * q^74 + 11573008 * q^75 + 14443516 * q^76 - 11914452 * q^77 - 38252364 * q^78 - 11333192 * q^80 + 15827196 * q^82 - 9565884 * q^83 + 8101984 * q^84 + 312496 * q^85 - 14812992 * q^86 + 40789108 * q^87 + 5716448 * q^88 - 4 * q^89 + 5335400 * q^90 - 3406996 * q^91 + 27562664 * q^92 - 8752 * q^93 - 14227032 * q^94 - 54872008 * q^95 - 25080448 * q^96 - 8 * q^97 + 57598104 * q^98 - 9738480 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −11.2389 + 1.29892i −18.4581 44.5619i 124.626 29.1969i 118.503 + 49.0855i 265.332 + 476.851i 519.116 + 519.116i −1362.73 + 490.020i −98.6180 + 98.6180i −1395.60 397.741i
5.2 −11.1568 + 1.87795i 26.2524 + 63.3790i 120.947 41.9036i −452.646 187.492i −411.914 657.803i 210.637 + 210.637i −1270.68 + 694.640i −1781.26 + 1781.26i 5402.17 + 1241.76i
5.3 −10.5410 4.10950i 1.43825 + 3.47225i 94.2241 + 86.6361i −25.4606 10.5461i −0.891385 42.5113i −192.841 192.841i −637.183 1300.44i 1536.45 1536.45i 225.040 + 215.796i
5.4 −10.3072 + 4.66502i 9.79359 + 23.6438i 84.4752 96.1662i 215.611 + 89.3089i −211.243 198.013i −1104.78 1104.78i −422.082 + 1385.28i 1083.33 1083.33i −2638.96 + 85.3073i
5.5 −9.92396 5.43278i 32.0253 + 77.3160i 68.9698 + 107.829i 390.991 + 161.954i 102.223 941.267i 314.329 + 314.329i −98.6407 1444.79i −3405.70 + 3405.70i −3000.32 3731.39i
5.6 −9.18285 6.60873i −32.0943 77.4825i 40.6495 + 121.374i −324.954 134.601i −217.343 + 923.612i −942.234 942.234i 428.849 1383.20i −3427.05 + 3427.05i 2094.47 + 3383.55i
5.7 −8.01849 + 7.98147i −15.4489 37.2970i 0.592237 127.999i −402.357 166.662i 421.562 + 175.761i 70.8947 + 70.8947i 1016.87 + 1031.08i 394.043 394.043i 4556.50 1875.03i
5.8 −7.13171 + 8.78286i 15.6913 + 37.8822i −26.2773 125.274i 221.433 + 91.7205i −444.620 132.350i 1084.35 + 1084.35i 1287.66 + 662.626i 357.601 357.601i −2384.77 + 1290.69i
5.9 −6.08567 9.53754i 8.84135 + 21.3449i −53.9292 + 116.085i −278.144 115.211i 149.772 214.223i 413.550 + 413.550i 1435.36 192.100i 1169.01 1169.01i 593.864 + 3353.95i
5.10 −6.01533 9.58206i −22.2952 53.8254i −55.6316 + 115.278i 416.314 + 172.443i −381.645 + 537.412i 881.560 + 881.560i 1439.25 160.372i −853.657 + 853.657i −851.908 5026.45i
5.11 −5.32488 + 9.98226i −29.4208 71.0282i −71.2912 106.309i 373.571 + 154.738i 865.684 + 84.5302i −760.180 760.180i 1440.82 145.566i −2632.97 + 2632.97i −3533.86 + 2905.12i
5.12 −3.36715 10.8010i 10.2484 + 24.7417i −105.325 + 72.7373i 148.342 + 61.4454i 232.728 194.002i −729.643 729.643i 1140.28 + 892.698i 1039.32 1039.32i 164.184 1809.14i
5.13 −2.49503 + 11.0352i 26.3289 + 63.5636i −115.550 55.0661i −124.019 51.3705i −767.126 + 131.951i −765.180 765.180i 895.963 1137.72i −1800.68 + 1800.68i 876.314 1240.40i
5.14 0.362596 + 11.3079i −6.67386 16.1121i −127.737 + 8.20039i −71.5848 29.6514i 179.774 81.3095i 108.936 + 108.936i −139.046 1441.46i 1331.38 1331.38i 309.339 820.225i
5.15 1.61553 11.1978i −17.7725 42.9066i −122.780 36.1807i 97.0019 + 40.1795i −509.170 + 129.695i −891.771 891.771i −603.498 + 1316.41i 21.3266 21.3266i 606.631 1021.29i
5.16 2.25547 11.0866i 29.8161 + 71.9825i −117.826 50.0111i −87.5694 36.2724i 865.292 168.205i 226.593 + 226.593i −820.206 + 1193.49i −2746.04 + 2746.04i −599.648 + 889.036i
5.17 2.26269 11.0851i −17.9118 43.2428i −117.760 50.1645i −367.733 152.320i −519.882 + 100.709i 951.213 + 951.213i −822.535 + 1191.88i −2.66939 + 2.66939i −2520.56 + 3731.72i
5.18 5.84536 + 9.68668i 3.37541 + 8.14896i −59.6635 + 113.244i 482.880 + 200.015i −59.2059 + 80.3301i −141.209 141.209i −1445.72 + 84.0120i 1491.43 1491.43i 885.122 + 5846.66i
5.19 6.15006 + 9.49615i −35.2038 84.9894i −52.3536 + 116.804i −114.898 47.5925i 590.567 856.990i 874.813 + 874.813i −1431.16 + 221.192i −4437.45 + 4437.45i −254.687 1383.79i
5.20 7.18642 + 8.73815i 25.5010 + 61.5648i −24.7106 + 125.592i −98.9895 41.0028i −354.702 + 665.262i 436.145 + 436.145i −1275.02 + 686.633i −1593.48 + 1593.48i −353.092 1159.65i
See next 80 embeddings (of 108 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.27 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.8.g.a 108
32.g even 8 1 inner 32.8.g.a 108

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.8.g.a 108 1.a even 1 1 trivial
32.8.g.a 108 32.g even 8 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(32, [\chi])$$.