# Properties

 Label 95.11.m.a Level $95$ Weight $11$ Character orbit 95.m Analytic conductor $60.359$ Analytic rank $0$ Dimension $392$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,11,Mod(7,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("95.7");

S:= CuspForms(chi, 11);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$95 = 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 95.m (of order $$12$$, 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: $$60.3589390040$$ Analytic rank: $$0$$ Dimension: $$392$$ Relative dimension: $$98$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8}+O(q^{10})$$ 392 * q - 2 * q^2 - 2 * q^3 + 2220 * q^5 + 4856 * q^6 + 7780 * q^7 - 142356 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8} - 2 q^{10} + 32064 q^{11} - 232108 q^{12} - 818490 q^{13} - 1141388 q^{15} + 49026272 q^{16} + 2072126 q^{17} + 4464336 q^{18} - 30123952 q^{20} - 1448284 q^{21} + 4905706 q^{22} + 131242 q^{23} - 5318128 q^{25} + 35863504 q^{26} - 40337048 q^{27} + 35162792 q^{28} - 626212 q^{30} - 64672896 q^{31} + 101446340 q^{32} + 16944736 q^{33} + 97780658 q^{35} - 1696907108 q^{36} + 188603000 q^{37} + 316760754 q^{38} + 440672472 q^{40} - 428613964 q^{41} - 774149798 q^{42} - 203275548 q^{43} + 445246440 q^{45} - 1583809336 q^{46} + 827682184 q^{47} + 93769212 q^{48} + 3444668164 q^{50} - 375042204 q^{51} - 1680459778 q^{52} - 312762144 q^{53} - 922476300 q^{55} + 1245970416 q^{56} - 1399302072 q^{57} + 5178550404 q^{58} + 2790379518 q^{60} + 2729471636 q^{61} + 1463639316 q^{62} - 5716734310 q^{63} - 5741012704 q^{65} - 14278935268 q^{66} - 1123450394 q^{67} + 2843511172 q^{68} + 5385951110 q^{70} + 3202662556 q^{71} - 10998992704 q^{72} + 4692302722 q^{73} + 35410377548 q^{75} - 4934546772 q^{76} + 13472322668 q^{77} + 3477052916 q^{78} - 3697722930 q^{80} + 61033508952 q^{81} - 14484700660 q^{82} + 6779152124 q^{83} - 20028400922 q^{85} + 24296217656 q^{86} - 17074613364 q^{87} + 9785438184 q^{88} - 10206419488 q^{90} + 46291684236 q^{91} - 17646129964 q^{92} - 14290061476 q^{93} - 9651375010 q^{95} - 147008832576 q^{96} - 12701510184 q^{97} - 535132524 q^{98}+O(q^{100})$$ 392 * q - 2 * q^2 - 2 * q^3 + 2220 * q^5 + 4856 * q^6 + 7780 * q^7 - 142356 * q^8 - 2 * q^10 + 32064 * q^11 - 232108 * q^12 - 818490 * q^13 - 1141388 * q^15 + 49026272 * q^16 + 2072126 * q^17 + 4464336 * q^18 - 30123952 * q^20 - 1448284 * q^21 + 4905706 * q^22 + 131242 * q^23 - 5318128 * q^25 + 35863504 * q^26 - 40337048 * q^27 + 35162792 * q^28 - 626212 * q^30 - 64672896 * q^31 + 101446340 * q^32 + 16944736 * q^33 + 97780658 * q^35 - 1696907108 * q^36 + 188603000 * q^37 + 316760754 * q^38 + 440672472 * q^40 - 428613964 * q^41 - 774149798 * q^42 - 203275548 * q^43 + 445246440 * q^45 - 1583809336 * q^46 + 827682184 * q^47 + 93769212 * q^48 + 3444668164 * q^50 - 375042204 * q^51 - 1680459778 * q^52 - 312762144 * q^53 - 922476300 * q^55 + 1245970416 * q^56 - 1399302072 * q^57 + 5178550404 * q^58 + 2790379518 * q^60 + 2729471636 * q^61 + 1463639316 * q^62 - 5716734310 * q^63 - 5741012704 * q^65 - 14278935268 * q^66 - 1123450394 * q^67 + 2843511172 * q^68 + 5385951110 * q^70 + 3202662556 * q^71 - 10998992704 * q^72 + 4692302722 * q^73 + 35410377548 * q^75 - 4934546772 * q^76 + 13472322668 * q^77 + 3477052916 * q^78 - 3697722930 * q^80 + 61033508952 * q^81 - 14484700660 * q^82 + 6779152124 * q^83 - 20028400922 * q^85 + 24296217656 * q^86 - 17074613364 * q^87 + 9785438184 * q^88 - 10206419488 * q^90 + 46291684236 * q^91 - 17646129964 * q^92 - 14290061476 * q^93 - 9651375010 * q^95 - 147008832576 * q^96 - 12701510184 * q^97 - 535132524 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −16.0969 + 60.0744i 123.006 + 32.9594i −2463.02 1422.02i −211.163 3117.86i −3960.03 + 6858.97i 8225.62 + 8225.62i 80041.1 80041.1i −37093.8 21416.1i 190703. + 37502.3i
7.2 −15.7465 + 58.7666i 388.673 + 104.145i −2318.75 1338.73i −1999.78 + 2401.36i −12240.4 + 21201.1i 10291.6 + 10291.6i 71132.1 71132.1i 89082.5 + 51431.8i −109630. 155333.i
7.3 −15.7051 + 58.6121i −383.064 102.642i −2301.92 1329.02i 896.038 2993.78i 12032.1 20840.2i −11158.5 11158.5i 70111.4 70111.4i 85064.6 + 49112.1i 161400. + 99536.3i
7.4 −15.6303 + 58.3330i 213.494 + 57.2056i −2271.63 1311.52i 2345.18 + 2065.38i −6673.95 + 11559.6i −12299.4 12299.4i 68283.7 68283.7i −8830.70 5098.41i −157135. + 104519.i
7.5 −15.4368 + 57.6108i −202.636 54.2963i −2193.90 1266.65i 3032.06 + 756.447i 6256.10 10835.9i 13207.4 + 13207.4i 63653.0 63653.0i −13024.5 7519.68i −90384.7 + 163002.i
7.6 −15.2296 + 56.8375i −76.4669 20.4892i −2111.75 1219.22i −1708.34 + 2616.72i 2329.12 4034.15i 14139.6 + 14139.6i 58852.1 58852.1i −45710.6 26391.0i −122710. 136949.i
7.7 −14.8503 + 55.4219i 143.595 + 38.4762i −1964.25 1134.06i −3040.04 723.714i −4264.85 + 7386.94i −18580.5 18580.5i 50476.0 50476.0i −31998.7 18474.5i 85255.1 157738.i
7.8 −14.5104 + 54.1536i −304.466 81.5815i −1835.25 1059.58i 415.173 + 3097.30i 8835.86 15304.2i −19458.5 19458.5i 43415.6 43415.6i 34906.3 + 20153.2i −173754. 22459.9i
7.9 −14.4462 + 53.9139i −259.520 69.5381i −1811.20 1045.70i −3015.66 819.415i 7498.14 12987.2i −366.409 366.409i 42127.8 42127.8i 11377.0 + 6568.54i 87742.6 150748.i
7.10 −13.5993 + 50.7533i 410.062 + 109.876i −1504.15 868.420i 1812.79 2545.47i −11153.1 + 19317.8i −7833.14 7833.14i 26484.9 26484.9i 104940. + 60587.2i 104538. + 126622.i
7.11 −13.3807 + 49.9376i 47.8041 + 12.8091i −1427.91 824.405i 2760.52 1464.63i −1279.31 + 2215.83i −9929.92 9929.92i 22841.1 22841.1i −49016.8 28299.8i 36202.1 + 157452.i
7.12 −12.8353 + 47.9020i 337.958 + 90.5556i −1243.04 717.671i 2913.63 + 1129.76i −8675.58 + 15026.5i 20232.3 + 20232.3i 14424.4 14424.4i 54877.4 + 31683.5i −91515.1 + 125068.i
7.13 −12.5417 + 46.8064i −107.569 28.8232i −1146.74 662.069i 2321.76 2091.66i 2698.22 4673.45i 4809.27 + 4809.27i 10284.1 10284.1i −40397.5 23323.5i 68784.1 + 134907.i
7.14 −12.4652 + 46.5207i 325.085 + 87.1063i −1121.98 647.777i −2409.61 1989.82i −8104.49 + 14037.4i 8414.06 + 8414.06i 9247.91 9247.91i 46954.8 + 27109.4i 122604. 87293.3i
7.15 −12.3979 + 46.2697i −443.419 118.814i −1100.37 635.297i −850.116 + 3007.15i 10995.0 19043.8i 11578.3 + 11578.3i 8352.55 8352.55i 131366. + 75844.1i −128600. 76617.0i
7.16 −12.1090 + 45.1916i 62.4601 + 16.7361i −1008.84 582.454i −1109.16 2921.54i −1512.66 + 2620.01i 16010.4 + 16010.4i 4661.56 4661.56i −47516.8 27433.8i 145460. 14747.5i
7.17 −11.6784 + 43.5845i 29.8802 + 8.00639i −876.413 505.997i −2369.61 + 2037.30i −697.909 + 1208.81i −349.071 349.071i −383.062 + 383.062i −50309.2 29046.0i −61121.4 127071.i
7.18 −11.5840 + 43.2320i −420.625 112.706i −848.005 489.596i 471.452 3089.23i 9745.02 16878.9i 19755.7 + 19755.7i −1418.10 + 1418.10i 113085. + 65289.7i 128092. + 56167.4i
7.19 −11.5704 + 43.1812i 28.4153 + 7.61384i −843.934 487.246i 81.7868 + 3123.93i −657.550 + 1138.91i −7638.87 7638.87i −1565.02 + 1565.02i −50388.5 29091.8i −135841. 32613.4i
7.20 −11.3887 + 42.5031i 182.520 + 48.9060i −790.000 456.107i 1136.08 + 2911.18i −4157.31 + 7200.67i 4854.66 + 4854.66i −3478.16 + 3478.16i −20216.3 11671.9i −136672. + 15132.3i
See next 80 embeddings (of 392 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.98 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
19.c even 3 1 inner
95.m odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 95.11.m.a 392
5.c odd 4 1 inner 95.11.m.a 392
19.c even 3 1 inner 95.11.m.a 392
95.m odd 12 1 inner 95.11.m.a 392

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.11.m.a 392 1.a even 1 1 trivial
95.11.m.a 392 5.c odd 4 1 inner
95.11.m.a 392 19.c even 3 1 inner
95.11.m.a 392 95.m odd 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{11}^{\mathrm{new}}(95, [\chi])$$.