Properties

Label 115.4.a.e.1.1
Level $115$
Weight $4$
Character 115.1
Self dual yes
Analytic conductor $6.785$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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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 = [5,6] 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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92 \) 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 1.1
Root \(-3.93900\) 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)
 
\(f(q)\) \(=\) \(q-2.93900 q^{2} -3.85751 q^{3} +0.637693 q^{4} -5.00000 q^{5} +11.3372 q^{6} -23.5932 q^{7} +21.6378 q^{8} -12.1196 q^{9} +14.6950 q^{10} -58.2387 q^{11} -2.45991 q^{12} +68.5380 q^{13} +69.3404 q^{14} +19.2876 q^{15} -68.6949 q^{16} +101.409 q^{17} +35.6194 q^{18} -7.02478 q^{19} -3.18846 q^{20} +91.0112 q^{21} +171.163 q^{22} -23.0000 q^{23} -83.4681 q^{24} +25.0000 q^{25} -201.433 q^{26} +150.904 q^{27} -15.0452 q^{28} +206.062 q^{29} -56.6861 q^{30} +54.8201 q^{31} +28.7917 q^{32} +224.657 q^{33} -298.040 q^{34} +117.966 q^{35} -7.72857 q^{36} -241.694 q^{37} +20.6458 q^{38} -264.386 q^{39} -108.189 q^{40} -122.510 q^{41} -267.482 q^{42} -320.358 q^{43} -37.1384 q^{44} +60.5979 q^{45} +67.5969 q^{46} +107.032 q^{47} +264.991 q^{48} +213.641 q^{49} -73.4749 q^{50} -391.186 q^{51} +43.7062 q^{52} +127.122 q^{53} -443.507 q^{54} +291.194 q^{55} -510.506 q^{56} +27.0982 q^{57} -605.615 q^{58} +693.766 q^{59} +12.2995 q^{60} -899.524 q^{61} -161.116 q^{62} +285.940 q^{63} +464.941 q^{64} -342.690 q^{65} -660.265 q^{66} +110.435 q^{67} +64.6677 q^{68} +88.7228 q^{69} -346.702 q^{70} +225.992 q^{71} -262.241 q^{72} +746.207 q^{73} +710.336 q^{74} -96.4378 q^{75} -4.47965 q^{76} +1374.04 q^{77} +777.030 q^{78} -1094.02 q^{79} +343.474 q^{80} -254.887 q^{81} +360.055 q^{82} +1289.81 q^{83} +58.0372 q^{84} -507.045 q^{85} +941.531 q^{86} -794.887 q^{87} -1260.16 q^{88} +1208.07 q^{89} -178.097 q^{90} -1617.03 q^{91} -14.6669 q^{92} -211.469 q^{93} -314.567 q^{94} +35.1239 q^{95} -111.064 q^{96} +903.864 q^{97} -627.890 q^{98} +705.829 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{2} + 4 q^{3} + 22 q^{4} - 25 q^{5} + 19 q^{6} - 3 q^{7} + 138 q^{8} + 77 q^{9} - 30 q^{10} + 23 q^{11} + 47 q^{12} + 132 q^{13} + 93 q^{14} - 20 q^{15} + 282 q^{16} + 23 q^{17} - 15 q^{18} - 161 q^{19}+ \cdots + 2021 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.93900 −1.03909 −0.519546 − 0.854443i \(-0.673899\pi\)
−0.519546 + 0.854443i \(0.673899\pi\)
\(3\) −3.85751 −0.742379 −0.371189 − 0.928557i \(-0.621050\pi\)
−0.371189 + 0.928557i \(0.621050\pi\)
\(4\) 0.637693 0.0797116
\(5\) −5.00000 −0.447214
\(6\) 11.3372 0.771400
\(7\) −23.5932 −1.27392 −0.636958 − 0.770899i \(-0.719807\pi\)
−0.636958 + 0.770899i \(0.719807\pi\)
\(8\) 21.6378 0.956264
\(9\) −12.1196 −0.448874
\(10\) 14.6950 0.464696
\(11\) −58.2387 −1.59633 −0.798165 − 0.602439i \(-0.794196\pi\)
−0.798165 + 0.602439i \(0.794196\pi\)
\(12\) −2.45991 −0.0591762
\(13\) 68.5380 1.46223 0.731116 − 0.682253i \(-0.239000\pi\)
0.731116 + 0.682253i \(0.239000\pi\)
\(14\) 69.3404 1.32371
\(15\) 19.2876 0.332002
\(16\) −68.6949 −1.07336
\(17\) 101.409 1.44678 0.723391 − 0.690439i \(-0.242583\pi\)
0.723391 + 0.690439i \(0.242583\pi\)
\(18\) 35.6194 0.466421
\(19\) −7.02478 −0.0848208 −0.0424104 − 0.999100i \(-0.513504\pi\)
−0.0424104 + 0.999100i \(0.513504\pi\)
\(20\) −3.18846 −0.0356481
\(21\) 91.0112 0.945728
\(22\) 171.163 1.65873
\(23\) −23.0000 −0.208514
\(24\) −83.4681 −0.709910
\(25\) 25.0000 0.200000
\(26\) −201.433 −1.51939
\(27\) 150.904 1.07561
\(28\) −15.0452 −0.101546
\(29\) 206.062 1.31947 0.659737 − 0.751497i \(-0.270668\pi\)
0.659737 + 0.751497i \(0.270668\pi\)
\(30\) −56.6861 −0.344980
\(31\) 54.8201 0.317612 0.158806 − 0.987310i \(-0.449236\pi\)
0.158806 + 0.987310i \(0.449236\pi\)
\(32\) 28.7917 0.159053
\(33\) 224.657 1.18508
\(34\) −298.040 −1.50334
\(35\) 117.966 0.569712
\(36\) −7.72857 −0.0357804
\(37\) −241.694 −1.07390 −0.536948 − 0.843615i \(-0.680423\pi\)
−0.536948 + 0.843615i \(0.680423\pi\)
\(38\) 20.6458 0.0881366
\(39\) −264.386 −1.08553
\(40\) −108.189 −0.427654
\(41\) −122.510 −0.466653 −0.233327 − 0.972398i \(-0.574961\pi\)
−0.233327 + 0.972398i \(0.574961\pi\)
\(42\) −267.482 −0.982698
\(43\) −320.358 −1.13614 −0.568071 − 0.822979i \(-0.692310\pi\)
−0.568071 + 0.822979i \(0.692310\pi\)
\(44\) −37.1384 −0.127246
\(45\) 60.5979 0.200742
\(46\) 67.5969 0.216666
\(47\) 107.032 0.332175 0.166088 − 0.986111i \(-0.446887\pi\)
0.166088 + 0.986111i \(0.446887\pi\)
\(48\) 264.991 0.796838
\(49\) 213.641 0.622860
\(50\) −73.4749 −0.207818
\(51\) −391.186 −1.07406
\(52\) 43.7062 0.116557
\(53\) 127.122 0.329463 0.164732 − 0.986338i \(-0.447324\pi\)
0.164732 + 0.986338i \(0.447324\pi\)
\(54\) −443.507 −1.11766
\(55\) 291.194 0.713901
\(56\) −510.506 −1.21820
\(57\) 27.0982 0.0629692
\(58\) −605.615 −1.37105
\(59\) 693.766 1.53086 0.765429 − 0.643521i \(-0.222527\pi\)
0.765429 + 0.643521i \(0.222527\pi\)
\(60\) 12.2995 0.0264644
\(61\) −899.524 −1.88807 −0.944034 − 0.329847i \(-0.893003\pi\)
−0.944034 + 0.329847i \(0.893003\pi\)
\(62\) −161.116 −0.330028
\(63\) 285.940 0.571827
\(64\) 464.941 0.908087
\(65\) −342.690 −0.653930
\(66\) −660.265 −1.23141
\(67\) 110.435 0.201369 0.100685 − 0.994918i \(-0.467897\pi\)
0.100685 + 0.994918i \(0.467897\pi\)
\(68\) 64.6677 0.115325
\(69\) 88.7228 0.154797
\(70\) −346.702 −0.591983
\(71\) 225.992 0.377752 0.188876 − 0.982001i \(-0.439516\pi\)
0.188876 + 0.982001i \(0.439516\pi\)
\(72\) −262.241 −0.429242
\(73\) 746.207 1.19640 0.598198 − 0.801348i \(-0.295884\pi\)
0.598198 + 0.801348i \(0.295884\pi\)
\(74\) 710.336 1.11588
\(75\) −96.4378 −0.148476
\(76\) −4.47965 −0.00676120
\(77\) 1374.04 2.03359
\(78\) 777.030 1.12797
\(79\) −1094.02 −1.55807 −0.779033 − 0.626983i \(-0.784290\pi\)
−0.779033 + 0.626983i \(0.784290\pi\)
\(80\) 343.474 0.480020
\(81\) −254.887 −0.349639
\(82\) 360.055 0.484896
\(83\) 1289.81 1.70572 0.852862 − 0.522137i \(-0.174865\pi\)
0.852862 + 0.522137i \(0.174865\pi\)
\(84\) 58.0372 0.0753854
\(85\) −507.045 −0.647020
\(86\) 941.531 1.18056
\(87\) −794.887 −0.979550
\(88\) −1260.16 −1.52651
\(89\) 1208.07 1.43883 0.719413 − 0.694582i \(-0.244411\pi\)
0.719413 + 0.694582i \(0.244411\pi\)
\(90\) −178.097 −0.208590
\(91\) −1617.03 −1.86276
\(92\) −14.6669 −0.0166210
\(93\) −211.469 −0.235788
\(94\) −314.567 −0.345161
\(95\) 35.1239 0.0379330
\(96\) −111.064 −0.118078
\(97\) 903.864 0.946119 0.473059 − 0.881031i \(-0.343150\pi\)
0.473059 + 0.881031i \(0.343150\pi\)
\(98\) −627.890 −0.647209
\(99\) 705.829 0.716551
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.e.1.1 ✓ 5
3.2 odd 2 1035.4.a.k.1.5 5
4.3 odd 2 1840.4.a.n.1.4 5
5.2 odd 4 575.4.b.i.24.3 10
5.3 odd 4 575.4.b.i.24.8 10
5.4 even 2 575.4.a.j.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.4.a.e.1.1 ✓ 5 1.1 even 1 trivial
575.4.a.j.1.5 5 5.4 even 2
575.4.b.i.24.3 10 5.2 odd 4
575.4.b.i.24.8 10 5.3 odd 4
1035.4.a.k.1.5 5 3.2 odd 2
1840.4.a.n.1.4 5 4.3 odd 2