Properties

Label 115.4.a.f.1.5
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [115,4,Mod(1,115)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("115.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.78521965066\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.16661\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.16661 q^{2} +8.59146 q^{3} -3.30580 q^{4} +5.00000 q^{5} +18.6143 q^{6} +9.86011 q^{7} -24.4953 q^{8} +46.8131 q^{9} +10.8331 q^{10} +12.1995 q^{11} -28.4016 q^{12} +20.6010 q^{13} +21.3630 q^{14} +42.9573 q^{15} -26.6253 q^{16} -7.09451 q^{17} +101.426 q^{18} -155.809 q^{19} -16.5290 q^{20} +84.7127 q^{21} +26.4315 q^{22} +23.0000 q^{23} -210.450 q^{24} +25.0000 q^{25} +44.6344 q^{26} +170.224 q^{27} -32.5955 q^{28} -213.296 q^{29} +93.0717 q^{30} -95.7095 q^{31} +138.275 q^{32} +104.811 q^{33} -15.3710 q^{34} +49.3006 q^{35} -154.755 q^{36} +174.131 q^{37} -337.578 q^{38} +176.993 q^{39} -122.476 q^{40} +122.549 q^{41} +183.539 q^{42} +398.567 q^{43} -40.3290 q^{44} +234.066 q^{45} +49.8320 q^{46} -490.851 q^{47} -228.750 q^{48} -245.778 q^{49} +54.1653 q^{50} -60.9522 q^{51} -68.1028 q^{52} -161.490 q^{53} +368.808 q^{54} +60.9974 q^{55} -241.526 q^{56} -1338.63 q^{57} -462.130 q^{58} -560.053 q^{59} -142.008 q^{60} -321.451 q^{61} -207.365 q^{62} +461.583 q^{63} +512.591 q^{64} +103.005 q^{65} +227.085 q^{66} +280.311 q^{67} +23.4530 q^{68} +197.604 q^{69} +106.815 q^{70} +649.370 q^{71} -1146.70 q^{72} +426.808 q^{73} +377.275 q^{74} +214.786 q^{75} +515.074 q^{76} +120.288 q^{77} +383.474 q^{78} -161.318 q^{79} -133.127 q^{80} +198.514 q^{81} +265.516 q^{82} +801.673 q^{83} -280.043 q^{84} -35.4725 q^{85} +863.540 q^{86} -1832.53 q^{87} -298.830 q^{88} +1100.38 q^{89} +507.129 q^{90} +203.128 q^{91} -76.0334 q^{92} -822.284 q^{93} -1063.48 q^{94} -779.046 q^{95} +1187.99 q^{96} -1179.42 q^{97} -532.506 q^{98} +571.096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 42 q^{4} + 40 q^{5} + 11 q^{7} + 3 q^{8} + 158 q^{9} + 30 q^{10} + 41 q^{11} + 48 q^{12} + 28 q^{13} + 161 q^{14} + 98 q^{16} + 71 q^{17} + 84 q^{18} + 177 q^{19} + 210 q^{20} + 292 q^{21}+ \cdots + 1319 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.16661 0.766013 0.383006 − 0.923746i \(-0.374889\pi\)
0.383006 + 0.923746i \(0.374889\pi\)
\(3\) 8.59146 1.65343 0.826713 − 0.562623i \(-0.190208\pi\)
0.826713 + 0.562623i \(0.190208\pi\)
\(4\) −3.30580 −0.413225
\(5\) 5.00000 0.447214
\(6\) 18.6143 1.26655
\(7\) 9.86011 0.532396 0.266198 − 0.963918i \(-0.414233\pi\)
0.266198 + 0.963918i \(0.414233\pi\)
\(8\) −24.4953 −1.08255
\(9\) 46.8131 1.73382
\(10\) 10.8331 0.342571
\(11\) 12.1995 0.334389 0.167195 − 0.985924i \(-0.446529\pi\)
0.167195 + 0.985924i \(0.446529\pi\)
\(12\) −28.4016 −0.683237
\(13\) 20.6010 0.439515 0.219758 − 0.975555i \(-0.429473\pi\)
0.219758 + 0.975555i \(0.429473\pi\)
\(14\) 21.3630 0.407822
\(15\) 42.9573 0.739435
\(16\) −26.6253 −0.416020
\(17\) −7.09451 −0.101216 −0.0506080 − 0.998719i \(-0.516116\pi\)
−0.0506080 + 0.998719i \(0.516116\pi\)
\(18\) 101.426 1.32813
\(19\) −155.809 −1.88132 −0.940660 − 0.339350i \(-0.889793\pi\)
−0.940660 + 0.339350i \(0.889793\pi\)
\(20\) −16.5290 −0.184800
\(21\) 84.7127 0.880278
\(22\) 26.4315 0.256146
\(23\) 23.0000 0.208514
\(24\) −210.450 −1.78991
\(25\) 25.0000 0.200000
\(26\) 44.6344 0.336674
\(27\) 170.224 1.21332
\(28\) −32.5955 −0.219999
\(29\) −213.296 −1.36580 −0.682899 − 0.730513i \(-0.739281\pi\)
−0.682899 + 0.730513i \(0.739281\pi\)
\(30\) 93.0717 0.566416
\(31\) −95.7095 −0.554514 −0.277257 − 0.960796i \(-0.589425\pi\)
−0.277257 + 0.960796i \(0.589425\pi\)
\(32\) 138.275 0.763871
\(33\) 104.811 0.552888
\(34\) −15.3710 −0.0775327
\(35\) 49.3006 0.238095
\(36\) −154.755 −0.716457
\(37\) 174.131 0.773704 0.386852 − 0.922142i \(-0.373563\pi\)
0.386852 + 0.922142i \(0.373563\pi\)
\(38\) −337.578 −1.44112
\(39\) 176.993 0.726706
\(40\) −122.476 −0.484130
\(41\) 122.549 0.466803 0.233402 − 0.972380i \(-0.425014\pi\)
0.233402 + 0.972380i \(0.425014\pi\)
\(42\) 183.539 0.674304
\(43\) 398.567 1.41351 0.706755 − 0.707458i \(-0.250158\pi\)
0.706755 + 0.707458i \(0.250158\pi\)
\(44\) −40.3290 −0.138178
\(45\) 234.066 0.775388
\(46\) 49.8320 0.159725
\(47\) −490.851 −1.52336 −0.761681 − 0.647952i \(-0.775626\pi\)
−0.761681 + 0.647952i \(0.775626\pi\)
\(48\) −228.750 −0.687859
\(49\) −245.778 −0.716555
\(50\) 54.1653 0.153203
\(51\) −60.9522 −0.167353
\(52\) −68.1028 −0.181619
\(53\) −161.490 −0.418535 −0.209268 − 0.977858i \(-0.567108\pi\)
−0.209268 + 0.977858i \(0.567108\pi\)
\(54\) 368.808 0.929416
\(55\) 60.9974 0.149543
\(56\) −241.526 −0.576344
\(57\) −1338.63 −3.11063
\(58\) −462.130 −1.04622
\(59\) −560.053 −1.23581 −0.617904 − 0.786253i \(-0.712018\pi\)
−0.617904 + 0.786253i \(0.712018\pi\)
\(60\) −142.008 −0.305553
\(61\) −321.451 −0.674714 −0.337357 − 0.941377i \(-0.609533\pi\)
−0.337357 + 0.941377i \(0.609533\pi\)
\(62\) −207.365 −0.424765
\(63\) 461.583 0.923078
\(64\) 512.591 1.00116
\(65\) 103.005 0.196557
\(66\) 227.085 0.423519
\(67\) 280.311 0.511126 0.255563 − 0.966792i \(-0.417739\pi\)
0.255563 + 0.966792i \(0.417739\pi\)
\(68\) 23.4530 0.0418249
\(69\) 197.604 0.344763
\(70\) 106.815 0.182384
\(71\) 649.370 1.08544 0.542719 − 0.839914i \(-0.317395\pi\)
0.542719 + 0.839914i \(0.317395\pi\)
\(72\) −1146.70 −1.87694
\(73\) 426.808 0.684303 0.342152 − 0.939645i \(-0.388844\pi\)
0.342152 + 0.939645i \(0.388844\pi\)
\(74\) 377.275 0.592667
\(75\) 214.786 0.330685
\(76\) 515.074 0.777408
\(77\) 120.288 0.178028
\(78\) 383.474 0.556666
\(79\) −161.318 −0.229744 −0.114872 − 0.993380i \(-0.536646\pi\)
−0.114872 + 0.993380i \(0.536646\pi\)
\(80\) −133.127 −0.186050
\(81\) 198.514 0.272310
\(82\) 265.516 0.357577
\(83\) 801.673 1.06018 0.530091 − 0.847941i \(-0.322158\pi\)
0.530091 + 0.847941i \(0.322158\pi\)
\(84\) −280.043 −0.363753
\(85\) −35.4725 −0.0452651
\(86\) 863.540 1.08277
\(87\) −1832.53 −2.25825
\(88\) −298.830 −0.361992
\(89\) 1100.38 1.31056 0.655280 − 0.755386i \(-0.272551\pi\)
0.655280 + 0.755386i \(0.272551\pi\)
\(90\) 507.129 0.593957
\(91\) 203.128 0.233996
\(92\) −76.0334 −0.0861633
\(93\) −822.284 −0.916848
\(94\) −1063.48 −1.16691
\(95\) −779.046 −0.841352
\(96\) 1187.99 1.26300
\(97\) −1179.42 −1.23455 −0.617276 − 0.786747i \(-0.711764\pi\)
−0.617276 + 0.786747i \(0.711764\pi\)
\(98\) −532.506 −0.548890
\(99\) 571.096 0.579771
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 115.4.a.f.1.5 ✓ 8
3.2 odd 2 1035.4.a.r.1.4 8
4.3 odd 2 1840.4.a.v.1.1 8
5.2 odd 4 575.4.b.k.24.11 16
5.3 odd 4 575.4.b.k.24.6 16
5.4 even 2 575.4.a.n.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.f.1.5 ✓ 8 1.1 even 1 trivial
575.4.a.n.1.4 8 5.4 even 2
575.4.b.k.24.6 16 5.3 odd 4
575.4.b.k.24.11 16 5.2 odd 4
1035.4.a.r.1.4 8 3.2 odd 2
1840.4.a.v.1.1 8 4.3 odd 2