Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [87,2,Mod(7,87)] 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("87.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(14)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 87.g (of order \(7\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-4] 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: no
Analytic conductor: \(0.694698497585\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 18 x^{16} - 37 x^{15} + 71 x^{14} - 83 x^{13} + 225 x^{12} - 237 x^{11} + 485 x^{10} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label 7.2
Root \(1.03105 - 0.496527i\) of defining polynomial
Character \(\chi\) \(=\) 87.7
Dual form 87.2.g.a.25.2

$q$-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+(0.0321271 + 0.140758i) q^{2} +(-0.900969 - 0.433884i) q^{3} +(1.78316 - 0.858723i) q^{4} +(-0.345850 - 1.51527i) q^{5} +(0.0321271 - 0.140758i) q^{6} +(1.37625 + 0.662766i) q^{7} +(0.358196 + 0.449164i) q^{8} +(0.623490 + 0.781831i) q^{9} +(0.202175 - 0.0973624i) q^{10} +(-0.478737 + 0.600317i) q^{11} -1.97916 q^{12} +(-1.63237 + 2.04693i) q^{13} +(-0.0490748 + 0.215011i) q^{14} +(-0.345850 + 1.51527i) q^{15} +(2.41625 - 3.02988i) q^{16} -3.51434 q^{17} +(-0.0900182 + 0.112879i) q^{18} +(-4.72829 + 2.27702i) q^{19} +(-1.91790 - 2.40497i) q^{20} +(-0.952393 - 1.19426i) q^{21} +(-0.0998798 - 0.0480996i) q^{22} +(-1.89083 + 8.28425i) q^{23} +(-0.127839 - 0.560098i) q^{24} +(2.32842 - 1.12131i) q^{25} +(-0.340565 - 0.164008i) q^{26} +(-0.222521 - 0.974928i) q^{27} +3.02320 q^{28} +(3.40207 - 4.17443i) q^{29} -0.224397 q^{30} +(0.315006 + 1.38013i) q^{31} +(1.53932 + 0.741300i) q^{32} +(0.691794 - 0.333151i) q^{33} +(-0.112906 - 0.494672i) q^{34} +(0.528293 - 2.31460i) q^{35} +(1.78316 + 0.858723i) q^{36} +(-1.87409 - 2.35004i) q^{37} +(-0.472416 - 0.592391i) q^{38} +(2.35885 - 1.13596i) q^{39} +(0.556722 - 0.698107i) q^{40} +5.79055 q^{41} +(0.137505 - 0.172425i) q^{42} +(-0.955986 + 4.18845i) q^{43} +(-0.338157 + 1.48156i) q^{44} +(0.969050 - 1.21515i) q^{45} -1.22682 q^{46} +(1.10631 - 1.38727i) q^{47} +(-3.49158 + 1.68146i) q^{48} +(-2.90963 - 3.64856i) q^{49} +(0.232638 + 0.291719i) q^{50} +(3.16631 + 1.52482i) q^{51} +(-1.15303 + 5.05176i) q^{52} +(-1.50686 - 6.60199i) q^{53} +(0.130080 - 0.0626432i) q^{54} +(1.07521 + 0.517795i) q^{55} +(0.195276 + 0.855561i) q^{56} +5.24800 q^{57} +(0.696884 + 0.344757i) q^{58} +14.9605 q^{59} +(0.684491 + 2.99895i) q^{60} +(-12.3585 - 5.95153i) q^{61} +(-0.184144 + 0.0886793i) q^{62} +(0.339905 + 1.48922i) q^{63} +(1.66981 - 7.31591i) q^{64} +(3.66621 + 1.76555i) q^{65} +(0.0691190 + 0.0866725i) q^{66} +(-4.54833 - 5.70342i) q^{67} +(-6.26662 + 3.01785i) q^{68} +(5.29798 - 6.64345i) q^{69} +0.342772 q^{70} +(-2.99945 + 3.76119i) q^{71} +(-0.127839 + 0.560098i) q^{72} +(-2.04705 + 8.96872i) q^{73} +(0.270578 - 0.339294i) q^{74} -2.58435 q^{75} +(-6.47595 + 8.12058i) q^{76} +(-1.05673 + 0.508894i) q^{77} +(0.235679 + 0.295532i) q^{78} +(0.340352 + 0.426788i) q^{79} +(-5.42674 - 2.61338i) q^{80} +(-0.222521 + 0.974928i) q^{81} +(0.186034 + 0.815067i) q^{82} +(-10.4943 + 5.05376i) q^{83} +(-2.72381 - 1.31172i) q^{84} +(1.21543 + 5.32517i) q^{85} -0.620271 q^{86} +(-4.87638 + 2.28493i) q^{87} -0.441122 q^{88} +(-3.06588 - 13.4325i) q^{89} +(0.202175 + 0.0973624i) q^{90} +(-3.60319 + 1.73520i) q^{91} +(3.74224 + 16.3958i) q^{92} +(0.315006 - 1.38013i) q^{93} +(0.230813 + 0.111154i) q^{94} +(5.08558 + 6.37712i) q^{95} +(-1.06525 - 1.33578i) q^{96} +(1.60034 - 0.770683i) q^{97} +(0.420086 - 0.526771i) q^{98} -0.767834 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{2} - 3 q^{3} - 6 q^{4} - q^{5} - 4 q^{6} - 4 q^{7} - 15 q^{8} - 3 q^{9} - 14 q^{10} + 26 q^{11} + 22 q^{12} + 9 q^{13} - 10 q^{14} - q^{15} - 14 q^{16} + 4 q^{17} - 4 q^{18} - 10 q^{19} - q^{20}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(e\left(\frac{3}{7}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0321271 + 0.140758i 0.0227173 + 0.0995310i 0.985016 − 0.172465i \(-0.0551733\pi\)
−0.962298 + 0.271996i \(0.912316\pi\)
\(3\) −0.900969 − 0.433884i −0.520175 − 0.250503i
\(4\) 1.78316 − 0.858723i 0.891579 − 0.429362i
\(5\) −0.345850 − 1.51527i −0.154669 − 0.677649i −0.991491 − 0.130174i \(-0.958446\pi\)
0.836822 − 0.547475i \(-0.184411\pi\)
\(6\) 0.0321271 − 0.140758i 0.0131158 − 0.0574642i
\(7\) 1.37625 + 0.662766i 0.520173 + 0.250502i 0.675499 − 0.737361i \(-0.263928\pi\)
−0.155326 + 0.987863i \(0.549643\pi\)
\(8\) 0.358196 + 0.449164i 0.126641 + 0.158803i
\(9\) 0.623490 + 0.781831i 0.207830 + 0.260610i
\(10\) 0.202175 − 0.0973624i 0.0639334 − 0.0307887i
\(11\) −0.478737 + 0.600317i −0.144345 + 0.181002i −0.848748 − 0.528797i \(-0.822643\pi\)
0.704404 + 0.709800i \(0.251215\pi\)
\(12\) −1.97916 −0.571333
\(13\) −1.63237 + 2.04693i −0.452739 + 0.567716i −0.954851 − 0.297086i \(-0.903985\pi\)
0.502112 + 0.864803i \(0.332557\pi\)
\(14\) −0.0490748 + 0.215011i −0.0131158 + 0.0574641i
\(15\) −0.345850 + 1.51527i −0.0892981 + 0.391241i
\(16\) 2.41625 − 3.02988i 0.604063 − 0.757471i
\(17\) −3.51434 −0.852353 −0.426176 − 0.904640i \(-0.640140\pi\)
−0.426176 + 0.904640i \(0.640140\pi\)
\(18\) −0.0900182 + 0.112879i −0.0212175 + 0.0266059i
\(19\) −4.72829 + 2.27702i −1.08474 + 0.522385i −0.888831 − 0.458236i \(-0.848481\pi\)
−0.195913 + 0.980621i \(0.562767\pi\)
\(20\) −1.91790 − 2.40497i −0.428856 − 0.537768i
\(21\) −0.952393 − 1.19426i −0.207829 − 0.260610i
\(22\) −0.0998798 − 0.0480996i −0.0212945 − 0.0102549i
\(23\) −1.89083 + 8.28425i −0.394264 + 1.72739i 0.255106 + 0.966913i \(0.417890\pi\)
−0.649371 + 0.760472i \(0.724968\pi\)
\(24\) −0.127839 − 0.560098i −0.0260950 − 0.114330i
\(25\) 2.32842 − 1.12131i 0.465684 − 0.224261i
\(26\) −0.340565 − 0.164008i −0.0667904 − 0.0321645i
\(27\) −0.222521 − 0.974928i −0.0428242 − 0.187625i
\(28\) 3.02320 0.571331
\(29\) 3.40207 − 4.17443i 0.631749 − 0.775173i
\(30\) −0.224397 −0.0409692
\(31\) 0.315006 + 1.38013i 0.0565768 + 0.247879i 0.995306 − 0.0967760i \(-0.0308531\pi\)
−0.938729 + 0.344655i \(0.887996\pi\)
\(32\) 1.53932 + 0.741300i 0.272117 + 0.131044i
\(33\) 0.691794 − 0.333151i 0.120426 − 0.0579941i
\(34\) −0.112906 − 0.494672i −0.0193631 − 0.0848355i
\(35\) 0.528293 − 2.31460i 0.0892978 − 0.391239i
\(36\) 1.78316 + 0.858723i 0.297193 + 0.143121i
\(37\) −1.87409 − 2.35004i −0.308099 − 0.386344i 0.603542 − 0.797331i \(-0.293756\pi\)
−0.911641 + 0.410987i \(0.865184\pi\)
\(38\) −0.472416 − 0.592391i −0.0766360 − 0.0960984i
\(39\) 2.35885 − 1.13596i 0.377718 − 0.181899i
\(40\) 0.556722 − 0.698107i 0.0880254 − 0.110380i
\(41\) 5.79055 0.904332 0.452166 − 0.891934i \(-0.350651\pi\)
0.452166 + 0.891934i \(0.350651\pi\)
\(42\) 0.137505 − 0.172425i 0.0212174 − 0.0266058i
\(43\) −0.955986 + 4.18845i −0.145787 + 0.638733i 0.848242 + 0.529609i \(0.177661\pi\)
−0.994028 + 0.109123i \(0.965196\pi\)
\(44\) −0.338157 + 1.48156i −0.0509790 + 0.223354i
\(45\) 0.969050 − 1.21515i 0.144458 − 0.181144i
\(46\) −1.22682 −0.180885
\(47\) 1.10631 − 1.38727i 0.161372 − 0.202355i −0.694571 − 0.719424i \(-0.744406\pi\)
0.855943 + 0.517070i \(0.172977\pi\)
\(48\) −3.49158 + 1.68146i −0.503967 + 0.242698i
\(49\) −2.90963 − 3.64856i −0.415661 − 0.521223i
\(50\) 0.232638 + 0.291719i 0.0329000 + 0.0412553i
\(51\) 3.16631 + 1.52482i 0.443372 + 0.213517i
\(52\) −1.15303 + 5.05176i −0.159897 + 0.700552i
\(53\) −1.50686 − 6.60199i −0.206983 − 0.906853i −0.966561 − 0.256437i \(-0.917451\pi\)
0.759578 − 0.650417i \(-0.225406\pi\)
\(54\) 0.130080 − 0.0626432i 0.0177016 − 0.00852466i
\(55\) 1.07521 + 0.517795i 0.144982 + 0.0698194i
\(56\) 0.195276 + 0.855561i 0.0260949 + 0.114329i
\(57\) 5.24800 0.695115
\(58\) 0.696884 + 0.344757i 0.0915054 + 0.0452688i
\(59\) 14.9605 1.94769 0.973845 − 0.227213i \(-0.0729614\pi\)
0.973845 + 0.227213i \(0.0729614\pi\)
\(60\) 0.684491 + 2.99895i 0.0883674 + 0.387163i
\(61\) −12.3585 − 5.95153i −1.58234 − 0.762016i −0.583594 − 0.812046i \(-0.698354\pi\)
−0.998748 + 0.0500301i \(0.984068\pi\)
\(62\) −0.184144 + 0.0886793i −0.0233864 + 0.0112623i
\(63\) 0.339905 + 1.48922i 0.0428240 + 0.187624i
\(64\) 1.66981 − 7.31591i 0.208726 − 0.914489i
\(65\) 3.66621 + 1.76555i 0.454737 + 0.218990i
\(66\) 0.0691190 + 0.0866725i 0.00850796 + 0.0106686i
\(67\) −4.54833 − 5.70342i −0.555666 − 0.696784i 0.422084 − 0.906557i \(-0.361299\pi\)
−0.977750 + 0.209773i \(0.932727\pi\)
\(68\) −6.26662 + 3.01785i −0.759939 + 0.365968i
\(69\) 5.29798 − 6.64345i 0.637801 − 0.799778i
\(70\) 0.342772 0.0409690
\(71\) −2.99945 + 3.76119i −0.355969 + 0.446371i −0.927283 − 0.374360i \(-0.877862\pi\)
0.571314 + 0.820731i \(0.306434\pi\)
\(72\) −0.127839 + 0.560098i −0.0150659 + 0.0660082i
\(73\) −2.04705 + 8.96872i −0.239589 + 1.04971i 0.701796 + 0.712378i \(0.252382\pi\)
−0.941386 + 0.337332i \(0.890476\pi\)
\(74\) 0.270578 − 0.339294i 0.0314540 − 0.0394421i
\(75\) −2.58435 −0.298415
\(76\) −6.47595 + 8.12058i −0.742842 + 0.931495i
\(77\) −1.05673 + 0.508894i −0.120426 + 0.0579939i
\(78\) 0.235679 + 0.295532i 0.0266853 + 0.0334624i
\(79\) 0.340352 + 0.426788i 0.0382926 + 0.0480174i 0.800609 − 0.599187i \(-0.204509\pi\)
−0.762316 + 0.647204i \(0.775938\pi\)
\(80\) −5.42674 − 2.61338i −0.606728 − 0.292185i
\(81\) −0.222521 + 0.974928i −0.0247245 + 0.108325i
\(82\) 0.186034 + 0.815067i 0.0205440 + 0.0900090i
\(83\) −10.4943 + 5.05376i −1.15189 + 0.554723i −0.909602 − 0.415481i \(-0.863613\pi\)
−0.242292 + 0.970203i \(0.577899\pi\)
\(84\) −2.72381 − 1.31172i −0.297192 − 0.143120i
\(85\) 1.21543 + 5.32517i 0.131832 + 0.577596i
\(86\) −0.620271 −0.0668856
\(87\) −4.87638 + 2.28493i −0.522803 + 0.244970i
\(88\) −0.441122 −0.0470238
\(89\) −3.06588 − 13.4325i −0.324982 − 1.42384i −0.828565 − 0.559893i \(-0.810842\pi\)
0.503583 − 0.863947i \(-0.332015\pi\)
\(90\) 0.202175 + 0.0973624i 0.0213111 + 0.0102629i
\(91\) −3.60319 + 1.73520i −0.377717 + 0.181899i
\(92\) 3.74224 + 16.3958i 0.390155 + 1.70938i
\(93\) 0.315006 − 1.38013i 0.0326646 − 0.143113i
\(94\) 0.230813 + 0.111154i 0.0238065 + 0.0114646i
\(95\) 5.08558 + 6.37712i 0.521770 + 0.654278i
\(96\) −1.06525 − 1.33578i −0.108721 − 0.136332i
\(97\) 1.60034 − 0.770683i 0.162490 − 0.0782510i −0.350871 − 0.936424i \(-0.614114\pi\)
0.513361 + 0.858173i \(0.328400\pi\)
\(98\) 0.420086 − 0.526771i 0.0424351 − 0.0532119i
\(99\) −0.767834 −0.0771702
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 87.2.g.a.7.2 ✓ 18
3.2 odd 2 261.2.k.c.181.2 18
29.5 even 14 2523.2.a.o.1.6 9
29.24 even 7 2523.2.a.r.1.4 9
29.25 even 7 inner 87.2.g.a.25.2 yes 18
87.5 odd 14 7569.2.a.bm.1.4 9
87.53 odd 14 7569.2.a.bj.1.6 9
87.83 odd 14 261.2.k.c.199.2 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.a.7.2 ✓ 18 1.1 even 1 trivial
87.2.g.a.25.2 yes 18 29.25 even 7 inner
261.2.k.c.181.2 18 3.2 odd 2
261.2.k.c.199.2 18 87.83 odd 14
2523.2.a.o.1.6 9 29.5 even 14
2523.2.a.r.1.4 9 29.24 even 7
7569.2.a.bj.1.6 9 87.53 odd 14
7569.2.a.bm.1.4 9 87.5 odd 14