Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7,4,Mod(2,7)] 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("7.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 7.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.413013370040\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 7.2
Dual form 7.4.c.a.4.1

$q$-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+(-1.00000 + 1.73205i) q^{2} +(-3.50000 - 6.06218i) q^{3} +(2.00000 + 3.46410i) q^{4} +(-3.50000 + 6.06218i) q^{5} +14.0000 q^{6} +(14.0000 - 12.1244i) q^{7} -24.0000 q^{8} +(-11.0000 + 19.0526i) q^{9} +(-7.00000 - 12.1244i) q^{10} +(2.50000 + 4.33013i) q^{11} +(14.0000 - 24.2487i) q^{12} -14.0000 q^{13} +(7.00000 + 36.3731i) q^{14} +49.0000 q^{15} +(8.00000 - 13.8564i) q^{16} +(10.5000 + 18.1865i) q^{17} +(-22.0000 - 38.1051i) q^{18} +(-24.5000 + 42.4352i) q^{19} -28.0000 q^{20} +(-122.500 - 42.4352i) q^{21} -10.0000 q^{22} +(79.5000 - 137.698i) q^{23} +(84.0000 + 145.492i) q^{24} +(38.0000 + 65.8179i) q^{25} +(14.0000 - 24.2487i) q^{26} -35.0000 q^{27} +(70.0000 + 24.2487i) q^{28} +58.0000 q^{29} +(-49.0000 + 84.8705i) q^{30} +(-73.5000 - 127.306i) q^{31} +(-80.0000 - 138.564i) q^{32} +(17.5000 - 30.3109i) q^{33} -42.0000 q^{34} +(24.5000 + 127.306i) q^{35} -88.0000 q^{36} +(-109.500 + 189.660i) q^{37} +(-49.0000 - 84.8705i) q^{38} +(49.0000 + 84.8705i) q^{39} +(84.0000 - 145.492i) q^{40} +350.000 q^{41} +(196.000 - 169.741i) q^{42} -124.000 q^{43} +(-10.0000 + 17.3205i) q^{44} +(-77.0000 - 133.368i) q^{45} +(159.000 + 275.396i) q^{46} +(-262.500 + 454.663i) q^{47} -112.000 q^{48} +(49.0000 - 339.482i) q^{49} -152.000 q^{50} +(73.5000 - 127.306i) q^{51} +(-28.0000 - 48.4974i) q^{52} +(-151.500 - 262.406i) q^{53} +(35.0000 - 60.6218i) q^{54} -35.0000 q^{55} +(-336.000 + 290.985i) q^{56} +343.000 q^{57} +(-58.0000 + 100.459i) q^{58} +(52.5000 + 90.9327i) q^{59} +(98.0000 + 169.741i) q^{60} +(206.500 - 357.668i) q^{61} +294.000 q^{62} +(77.0000 + 400.104i) q^{63} +448.000 q^{64} +(49.0000 - 84.8705i) q^{65} +(35.0000 + 60.6218i) q^{66} +(-207.500 - 359.401i) q^{67} +(-42.0000 + 72.7461i) q^{68} -1113.00 q^{69} +(-245.000 - 84.8705i) q^{70} -432.000 q^{71} +(264.000 - 457.261i) q^{72} +(556.500 + 963.886i) q^{73} +(-219.000 - 379.319i) q^{74} +(266.000 - 460.726i) q^{75} -196.000 q^{76} +(87.5000 + 30.3109i) q^{77} -196.000 q^{78} +(51.5000 - 89.2006i) q^{79} +(56.0000 + 96.9948i) q^{80} +(419.500 + 726.595i) q^{81} +(-350.000 + 606.218i) q^{82} +1092.00 q^{83} +(-98.0000 - 509.223i) q^{84} -147.000 q^{85} +(124.000 - 214.774i) q^{86} +(-203.000 - 351.606i) q^{87} +(-60.0000 - 103.923i) q^{88} +(164.500 - 284.922i) q^{89} +308.000 q^{90} +(-196.000 + 169.741i) q^{91} +636.000 q^{92} +(-514.500 + 891.140i) q^{93} +(-525.000 - 909.327i) q^{94} +(-171.500 - 297.047i) q^{95} +(-560.000 + 969.948i) q^{96} -882.000 q^{97} +(539.000 + 424.352i) q^{98} -110.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 7 q^{3} + 4 q^{4} - 7 q^{5} + 28 q^{6} + 28 q^{7} - 48 q^{8} - 22 q^{9} - 14 q^{10} + 5 q^{11} + 28 q^{12} - 28 q^{13} + 14 q^{14} + 98 q^{15} + 16 q^{16} + 21 q^{17} - 44 q^{18} - 49 q^{19}+ \cdots - 220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 + 1.73205i −0.353553 + 0.612372i −0.986869 − 0.161521i \(-0.948360\pi\)
0.633316 + 0.773893i \(0.281693\pi\)
\(3\) −3.50000 − 6.06218i −0.673575 − 1.16667i −0.976883 − 0.213774i \(-0.931424\pi\)
0.303308 − 0.952893i \(-0.401909\pi\)
\(4\) 2.00000 + 3.46410i 0.250000 + 0.433013i
\(5\) −3.50000 + 6.06218i −0.313050 + 0.542218i −0.979021 − 0.203760i \(-0.934684\pi\)
0.665971 + 0.745977i \(0.268017\pi\)
\(6\) 14.0000 0.952579
\(7\) 14.0000 − 12.1244i 0.755929 − 0.654654i
\(8\) −24.0000 −1.06066
\(9\) −11.0000 + 19.0526i −0.407407 + 0.705650i
\(10\) −7.00000 − 12.1244i −0.221359 − 0.383406i
\(11\) 2.50000 + 4.33013i 0.0685253 + 0.118689i 0.898252 − 0.439480i \(-0.144837\pi\)
−0.829727 + 0.558169i \(0.811504\pi\)
\(12\) 14.0000 − 24.2487i 0.336788 − 0.583333i
\(13\) −14.0000 −0.298685 −0.149342 − 0.988786i \(-0.547716\pi\)
−0.149342 + 0.988786i \(0.547716\pi\)
\(14\) 7.00000 + 36.3731i 0.133631 + 0.694365i
\(15\) 49.0000 0.843450
\(16\) 8.00000 − 13.8564i 0.125000 − 0.216506i
\(17\) 10.5000 + 18.1865i 0.149801 + 0.259464i 0.931154 − 0.364626i \(-0.118803\pi\)
−0.781353 + 0.624090i \(0.785470\pi\)
\(18\) −22.0000 − 38.1051i −0.288081 − 0.498970i
\(19\) −24.5000 + 42.4352i −0.295826 + 0.512385i −0.975177 − 0.221429i \(-0.928928\pi\)
0.679351 + 0.733813i \(0.262261\pi\)
\(20\) −28.0000 −0.313050
\(21\) −122.500 − 42.4352i −1.27294 − 0.440959i
\(22\) −10.0000 −0.0969094
\(23\) 79.5000 − 137.698i 0.720735 − 1.24835i −0.239971 − 0.970780i \(-0.577138\pi\)
0.960706 − 0.277569i \(-0.0895287\pi\)
\(24\) 84.0000 + 145.492i 0.714435 + 1.23744i
\(25\) 38.0000 + 65.8179i 0.304000 + 0.526543i
\(26\) 14.0000 − 24.2487i 0.105601 − 0.182906i
\(27\) −35.0000 −0.249472
\(28\) 70.0000 + 24.2487i 0.472456 + 0.163663i
\(29\) 58.0000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −49.0000 + 84.8705i −0.298205 + 0.516505i
\(31\) −73.5000 − 127.306i −0.425838 − 0.737574i 0.570660 − 0.821186i \(-0.306687\pi\)
−0.996498 + 0.0836128i \(0.973354\pi\)
\(32\) −80.0000 − 138.564i −0.441942 − 0.765466i
\(33\) 17.5000 − 30.3109i 0.0923139 − 0.159892i
\(34\) −42.0000 −0.211851
\(35\) 24.5000 + 127.306i 0.118322 + 0.614817i
\(36\) −88.0000 −0.407407
\(37\) −109.500 + 189.660i −0.486532 + 0.842698i −0.999880 − 0.0154821i \(-0.995072\pi\)
0.513348 + 0.858181i \(0.328405\pi\)
\(38\) −49.0000 − 84.8705i −0.209180 − 0.362311i
\(39\) 49.0000 + 84.8705i 0.201187 + 0.348466i
\(40\) 84.0000 − 145.492i 0.332039 − 0.575109i
\(41\) 350.000 1.33319 0.666595 − 0.745420i \(-0.267751\pi\)
0.666595 + 0.745420i \(0.267751\pi\)
\(42\) 196.000 − 169.741i 0.720082 − 0.623610i
\(43\) −124.000 −0.439763 −0.219882 − 0.975527i \(-0.570567\pi\)
−0.219882 + 0.975527i \(0.570567\pi\)
\(44\) −10.0000 + 17.3205i −0.0342627 + 0.0593447i
\(45\) −77.0000 − 133.368i −0.255077 − 0.441807i
\(46\) 159.000 + 275.396i 0.509636 + 0.882716i
\(47\) −262.500 + 454.663i −0.814671 + 1.41105i 0.0948921 + 0.995488i \(0.469749\pi\)
−0.909564 + 0.415565i \(0.863584\pi\)
\(48\) −112.000 −0.336788
\(49\) 49.0000 − 339.482i 0.142857 − 0.989743i
\(50\) −152.000 −0.429921
\(51\) 73.5000 − 127.306i 0.201805 − 0.349537i
\(52\) −28.0000 − 48.4974i −0.0746712 − 0.129334i
\(53\) −151.500 − 262.406i −0.392644 − 0.680079i 0.600153 − 0.799885i \(-0.295106\pi\)
−0.992797 + 0.119806i \(0.961773\pi\)
\(54\) 35.0000 − 60.6218i 0.0882018 − 0.152770i
\(55\) −35.0000 −0.0858073
\(56\) −336.000 + 290.985i −0.801784 + 0.694365i
\(57\) 343.000 0.797043
\(58\) −58.0000 + 100.459i −0.131306 + 0.227429i
\(59\) 52.5000 + 90.9327i 0.115846 + 0.200651i 0.918118 − 0.396308i \(-0.129709\pi\)
−0.802272 + 0.596959i \(0.796375\pi\)
\(60\) 98.0000 + 169.741i 0.210862 + 0.365224i
\(61\) 206.500 − 357.668i 0.433436 − 0.750734i −0.563730 − 0.825959i \(-0.690634\pi\)
0.997167 + 0.0752252i \(0.0239676\pi\)
\(62\) 294.000 0.602226
\(63\) 77.0000 + 400.104i 0.153986 + 0.800132i
\(64\) 448.000 0.875000
\(65\) 49.0000 − 84.8705i 0.0935031 − 0.161952i
\(66\) 35.0000 + 60.6218i 0.0652758 + 0.113061i
\(67\) −207.500 − 359.401i −0.378361 − 0.655340i 0.612463 − 0.790499i \(-0.290179\pi\)
−0.990824 + 0.135159i \(0.956845\pi\)
\(68\) −42.0000 + 72.7461i −0.0749007 + 0.129732i
\(69\) −1113.00 −1.94188
\(70\) −245.000 − 84.8705i −0.418330 − 0.144914i
\(71\) −432.000 −0.722098 −0.361049 − 0.932547i \(-0.617581\pi\)
−0.361049 + 0.932547i \(0.617581\pi\)
\(72\) 264.000 − 457.261i 0.432121 − 0.748455i
\(73\) 556.500 + 963.886i 0.892238 + 1.54540i 0.837186 + 0.546919i \(0.184199\pi\)
0.0550526 + 0.998483i \(0.482467\pi\)
\(74\) −219.000 − 379.319i −0.344030 − 0.595878i
\(75\) 266.000 − 460.726i 0.409534 − 0.709333i
\(76\) −196.000 −0.295826
\(77\) 87.5000 + 30.3109i 0.129501 + 0.0448603i
\(78\) −196.000 −0.284521
\(79\) 51.5000 − 89.2006i 0.0733443 − 0.127036i −0.827021 − 0.562171i \(-0.809966\pi\)
0.900365 + 0.435135i \(0.143299\pi\)
\(80\) 56.0000 + 96.9948i 0.0782624 + 0.135554i
\(81\) 419.500 + 726.595i 0.575446 + 0.996701i
\(82\) −350.000 + 606.218i −0.471354 + 0.816409i
\(83\) 1092.00 1.44413 0.722064 − 0.691827i \(-0.243194\pi\)
0.722064 + 0.691827i \(0.243194\pi\)
\(84\) −98.0000 − 509.223i −0.127294 − 0.661438i
\(85\) −147.000 −0.187581
\(86\) 124.000 − 214.774i 0.155480 − 0.269299i
\(87\) −203.000 − 351.606i −0.250160 − 0.433289i
\(88\) −60.0000 − 103.923i −0.0726821 − 0.125889i
\(89\) 164.500 − 284.922i 0.195921 − 0.339345i −0.751281 − 0.659982i \(-0.770564\pi\)
0.947202 + 0.320637i \(0.103897\pi\)
\(90\) 308.000 0.360734
\(91\) −196.000 + 169.741i −0.225784 + 0.195535i
\(92\) 636.000 0.720735
\(93\) −514.500 + 891.140i −0.573668 + 0.993623i
\(94\) −525.000 − 909.327i −0.576060 − 0.997765i
\(95\) −171.500 − 297.047i −0.185216 − 0.320804i
\(96\) −560.000 + 969.948i −0.595362 + 1.03120i
\(97\) −882.000 −0.923232 −0.461616 − 0.887080i \(-0.652730\pi\)
−0.461616 + 0.887080i \(0.652730\pi\)
\(98\) 539.000 + 424.352i 0.555584 + 0.437409i
\(99\) −110.000 −0.111671
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 7.4.c.a.2.1 ✓ 2
3.2 odd 2 63.4.e.b.37.1 2
4.3 odd 2 112.4.i.c.65.1 2
5.2 odd 4 175.4.k.a.149.1 4
5.3 odd 4 175.4.k.a.149.2 4
5.4 even 2 175.4.e.a.51.1 2
7.2 even 3 49.4.a.d.1.1 1
7.3 odd 6 49.4.c.a.18.1 2
7.4 even 3 inner 7.4.c.a.4.1 yes 2
7.5 odd 6 49.4.a.c.1.1 1
7.6 odd 2 49.4.c.a.30.1 2
8.3 odd 2 448.4.i.a.65.1 2
8.5 even 2 448.4.i.f.65.1 2
21.2 odd 6 441.4.a.d.1.1 1
21.5 even 6 441.4.a.e.1.1 1
21.11 odd 6 63.4.e.b.46.1 2
21.17 even 6 441.4.e.k.361.1 2
21.20 even 2 441.4.e.k.226.1 2
28.11 odd 6 112.4.i.c.81.1 2
28.19 even 6 784.4.a.r.1.1 1
28.23 odd 6 784.4.a.b.1.1 1
35.4 even 6 175.4.e.a.151.1 2
35.9 even 6 1225.4.a.c.1.1 1
35.18 odd 12 175.4.k.a.74.1 4
35.19 odd 6 1225.4.a.d.1.1 1
35.32 odd 12 175.4.k.a.74.2 4
56.11 odd 6 448.4.i.a.193.1 2
56.53 even 6 448.4.i.f.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.c.a.2.1 ✓ 2 1.1 even 1 trivial
7.4.c.a.4.1 yes 2 7.4 even 3 inner
49.4.a.c.1.1 1 7.5 odd 6
49.4.a.d.1.1 1 7.2 even 3
49.4.c.a.18.1 2 7.3 odd 6
49.4.c.a.30.1 2 7.6 odd 2
63.4.e.b.37.1 2 3.2 odd 2
63.4.e.b.46.1 2 21.11 odd 6
112.4.i.c.65.1 2 4.3 odd 2
112.4.i.c.81.1 2 28.11 odd 6
175.4.e.a.51.1 2 5.4 even 2
175.4.e.a.151.1 2 35.4 even 6
175.4.k.a.74.1 4 35.18 odd 12
175.4.k.a.74.2 4 35.32 odd 12
175.4.k.a.149.1 4 5.2 odd 4
175.4.k.a.149.2 4 5.3 odd 4
441.4.a.d.1.1 1 21.2 odd 6
441.4.a.e.1.1 1 21.5 even 6
441.4.e.k.226.1 2 21.20 even 2
441.4.e.k.361.1 2 21.17 even 6
448.4.i.a.65.1 2 8.3 odd 2
448.4.i.a.193.1 2 56.11 odd 6
448.4.i.f.65.1 2 8.5 even 2
448.4.i.f.193.1 2 56.53 even 6
784.4.a.b.1.1 1 28.23 odd 6
784.4.a.r.1.1 1 28.19 even 6
1225.4.a.c.1.1 1 35.9 even 6
1225.4.a.d.1.1 1 35.19 odd 6