Properties

Label 61.6.i.a
Level $61$
Weight $6$
Character orbit 61.i
Analytic conductor $9.783$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [61,6,Mod(12,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.12");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 61.i (of order \(15\), degree \(8\), 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: \(9.78341300859\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 5 q^{2} + 12 q^{3} + 307 q^{4} - 96 q^{5} - 287 q^{6} - 407 q^{7} - 454 q^{8} - 3654 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 5 q^{2} + 12 q^{3} + 307 q^{4} - 96 q^{5} - 287 q^{6} - 407 q^{7} - 454 q^{8} - 3654 q^{9} + 949 q^{10} + 142 q^{11} - 5419 q^{12} + 1576 q^{13} + 2731 q^{14} + 66 q^{15} + 2327 q^{16} - 4098 q^{17} + 80 q^{18} - 8919 q^{19} + 4954 q^{20} + 4273 q^{21} - 13217 q^{22} - 1922 q^{23} - 50431 q^{24} - 6442 q^{25} + 4104 q^{26} + 4587 q^{27} + 34654 q^{28} - 6430 q^{29} + 8478 q^{30} - 12059 q^{31} - 22506 q^{32} - 20173 q^{33} + 43018 q^{34} + 53262 q^{35} - 10354 q^{36} - 23410 q^{37} + 15958 q^{38} + 47857 q^{39} - 26311 q^{40} + 51321 q^{41} + 65639 q^{42} - 46512 q^{43} - 142551 q^{44} - 40168 q^{45} + 187223 q^{46} + 77472 q^{47} + 97241 q^{48} + 146645 q^{49} + 181534 q^{50} - 72017 q^{51} + 114787 q^{52} + 8557 q^{53} - 110121 q^{54} - 6964 q^{55} - 54862 q^{56} - 73954 q^{57} + 53750 q^{58} - 110131 q^{59} - 531924 q^{60} + 6229 q^{61} - 555354 q^{62} + 194838 q^{63} - 104302 q^{64} - 164735 q^{65} + 288458 q^{66} - 10997 q^{67} + 722742 q^{68} - 20428 q^{69} - 63238 q^{70} + 145991 q^{71} + 215262 q^{72} - 185414 q^{73} + 400008 q^{74} + 323433 q^{75} - 462279 q^{76} - 165953 q^{77} + 429727 q^{78} - 184673 q^{79} - 989825 q^{80} + 94688 q^{81} + 71588 q^{82} - 210025 q^{83} - 509336 q^{84} + 256857 q^{85} - 217486 q^{86} + 149273 q^{87} - 112370 q^{88} + 165694 q^{89} - 365157 q^{90} - 219042 q^{91} - 929837 q^{92} - 254389 q^{93} - 237484 q^{94} - 578981 q^{95} + 186576 q^{96} - 444981 q^{97} + 880033 q^{98} - 996159 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1 −9.79271 4.36000i −15.4981 + 11.2600i 55.4755 + 61.6118i 6.26551 1.33178i 200.862 42.6946i 0.114956 1.09373i −168.629 518.985i 38.3119 117.912i −67.1629 14.2759i
12.2 −9.08233 4.04371i 6.29058 4.57037i 44.7249 + 49.6720i 58.9191 12.5236i −75.6144 + 16.0723i −6.33279 + 60.2525i −107.036 329.424i −56.4081 + 173.606i −585.765 124.508i
12.3 −8.64454 3.84880i 19.9923 14.5252i 38.5027 + 42.7615i −16.6634 + 3.54192i −228.729 + 48.6178i 11.8339 112.592i −74.6857 229.859i 113.617 349.678i 157.680 + 33.5159i
12.4 −7.88700 3.51152i −2.78668 + 2.02464i 28.4618 + 31.6100i −83.1717 + 17.6787i 29.0881 6.18288i 2.26926 21.5905i −28.1071 86.5049i −71.4247 + 219.823i 718.054 + 152.627i
12.5 −5.95610 2.65183i −19.9269 + 14.4778i 7.03076 + 7.80845i 44.7677 9.51567i 157.079 33.3882i 23.4605 223.212i 43.3017 + 133.269i 112.386 345.888i −291.875 62.0399i
12.6 −5.67334 2.52593i 5.86202 4.25901i 4.39427 + 4.88033i 85.0195 18.0715i −44.0152 + 9.35572i 0.479274 4.55999i 48.8075 + 150.214i −58.8670 + 181.174i −527.992 112.228i
12.7 −5.63033 2.50678i −14.7894 + 10.7451i 4.00446 + 4.44740i 11.5874 2.46298i 110.205 23.4247i −20.5234 + 195.267i 49.5470 + 152.490i 28.1768 86.7193i −71.4152 15.1798i
12.8 −5.06561 2.25535i 17.4423 12.6726i −0.838416 0.931156i −40.1020 + 8.52395i −116.937 + 24.8557i −23.2306 + 221.025i 56.9790 + 175.363i 68.5486 210.971i 222.366 + 47.2653i
12.9 −3.70015 1.64741i 3.10452 2.25556i −10.4350 11.5893i −37.7596 + 8.02605i −15.2030 + 3.23150i 17.5205 166.696i 59.5706 + 183.339i −70.5407 + 217.102i 152.938 + 32.5081i
12.10 −1.54751 0.688997i 20.6348 14.9920i −19.4921 21.6482i 23.0911 4.90816i −42.2620 + 8.98307i 13.0756 124.406i 31.9996 + 98.4845i 125.942 387.609i −39.1154 8.31423i
12.11 −0.861070 0.383373i −17.8098 + 12.9396i −20.8177 23.1204i −71.0225 + 15.0963i 20.2962 4.31409i 0.980358 9.32748i 18.3823 + 56.5749i 74.6654 229.797i 66.9428 + 14.2291i
12.12 −0.483266 0.215164i −1.06047 + 0.770480i −21.2249 23.5727i 49.2240 10.4629i 0.678271 0.144171i 4.69268 44.6479i 10.4164 + 32.0582i −74.5602 + 229.473i −26.0396 5.53488i
12.13 0.0806955 + 0.0359279i 5.44969 3.95943i −21.4070 23.7748i −35.3565 + 7.51525i 0.582019 0.123712i −14.0977 + 134.130i −1.74674 5.37591i −61.0691 + 187.951i −3.12312 0.663839i
12.14 1.40588 + 0.625940i −13.5362 + 9.83466i −19.8275 22.0206i 80.0991 17.0256i −25.1863 + 5.35351i −7.01227 + 66.7173i −29.3093 90.2048i 11.4183 35.1419i 123.267 + 26.2012i
12.15 3.16532 + 1.40929i 18.5890 13.5057i −13.3790 14.8589i 78.3972 16.6638i 77.8737 16.5526i −18.1331 + 172.525i −55.6710 171.338i 88.0559 271.008i 271.637 + 57.7382i
12.16 3.76503 + 1.67630i 19.6312 14.2629i −10.0467 11.1580i −90.7852 + 19.2970i 97.8211 20.7925i 10.0715 95.8241i −59.8760 184.280i 106.863 328.891i −374.156 79.5294i
12.17 4.36383 + 1.94290i 1.90174 1.38170i −6.14401 6.82362i −54.7109 + 11.6292i 10.9834 2.33459i −6.40689 + 60.9575i −60.7895 187.091i −73.3836 + 225.851i −261.344 55.5503i
12.18 4.75842 + 2.11858i −7.64829 + 5.55681i −3.25804 3.61842i 11.7544 2.49848i −48.1663 + 10.2381i 25.2257 240.006i −59.3440 182.642i −47.4729 + 146.107i 61.2256 + 13.0139i
12.19 5.94341 + 2.64618i −21.9531 + 15.9499i 6.90970 + 7.67399i −1.09371 + 0.232475i −172.683 + 36.7048i −4.23773 + 40.3193i −43.5732 134.105i 152.450 469.192i −7.11554 1.51246i
12.20 6.70832 + 2.98674i 11.4876 8.34622i 14.6688 + 16.2913i 53.7990 11.4353i 101.990 21.6787i 11.6118 110.479i −22.8683 70.3813i −12.7859 + 39.3511i 395.055 + 83.9716i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.i even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.6.i.a 192
61.i even 15 1 inner 61.6.i.a 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.6.i.a 192 1.a even 1 1 trivial
61.6.i.a 192 61.i even 15 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(61, [\chi])\).