Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7,7,Mod(3,7)] 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("7.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 7.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.61037858534\)
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: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 3.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 7.3
Dual form 7.7.d.a.5.1

$q$-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+(6.00000 + 10.3923i) q^{2} +(-10.5000 - 6.06218i) q^{3} +(-40.0000 + 69.2820i) q^{4} +(157.500 - 90.9327i) q^{5} -145.492i q^{6} -343.000 q^{7} -192.000 q^{8} +(-291.000 - 504.027i) q^{9} +(1890.00 + 1091.19i) q^{10} +(-739.500 + 1280.85i) q^{11} +(840.000 - 484.974i) q^{12} +484.974i q^{13} +(-2058.00 - 3564.56i) q^{14} -2205.00 q^{15} +(1408.00 + 2438.73i) q^{16} +(-2614.50 - 1509.48i) q^{17} +(3492.00 - 6048.32i) q^{18} +(5953.50 - 3437.25i) q^{19} +14549.2i q^{20} +(3601.50 + 2079.33i) q^{21} -17748.0 q^{22} +(2956.50 + 5120.81i) q^{23} +(2016.00 + 1163.94i) q^{24} +(8725.00 - 15112.1i) q^{25} +(-5040.00 + 2909.85i) q^{26} +15895.0i q^{27} +(13720.0 - 23763.7i) q^{28} +3978.00 q^{29} +(-13230.0 - 22915.0i) q^{30} +(-11098.5 - 6407.72i) q^{31} +(-23040.0 + 39906.5i) q^{32} +(15529.5 - 8965.96i) q^{33} -36227.6i q^{34} +(-54022.5 + 31189.9i) q^{35} +46560.0 q^{36} +(30788.5 + 53327.2i) q^{37} +(71442.0 + 41247.1i) q^{38} +(2940.00 - 5092.23i) q^{39} +(-30240.0 + 17459.1i) q^{40} -110574. i q^{41} +49903.8i q^{42} -17414.0 q^{43} +(-59160.0 - 102468. i) q^{44} +(-91665.0 - 52922.8i) q^{45} +(-35478.0 + 61449.7i) q^{46} +(-26554.5 + 15331.2i) q^{47} -34142.2i q^{48} +117649. q^{49} +209400. q^{50} +(18301.5 + 31699.1i) q^{51} +(-33600.0 - 19399.0i) q^{52} +(30256.5 - 52405.8i) q^{53} +(-165186. + 95370.2i) q^{54} +268979. i q^{55} +65856.0 q^{56} -83349.0 q^{57} +(23868.0 + 41340.6i) q^{58} +(-186826. - 107864. i) q^{59} +(88200.0 - 152767. i) q^{60} +(140942. - 81372.6i) q^{61} -153785. i q^{62} +(99813.0 + 172881. i) q^{63} -372736. q^{64} +(44100.0 + 76383.4i) q^{65} +(186354. + 107592. i) q^{66} +(134388. - 232768. i) q^{67} +(209160. - 120759. i) q^{68} -71691.3i q^{69} +(-648270. - 374279. i) q^{70} +101922. q^{71} +(55872.0 + 96773.1i) q^{72} +(275090. + 158823. i) q^{73} +(-369462. + 639927. i) q^{74} +(-183225. + 105785. i) q^{75} +549961. i q^{76} +(253648. - 439332. i) q^{77} +70560.0 q^{78} +(-181116. - 313701. i) q^{79} +(443520. + 256066. i) q^{80} +(-115780. + 200538. i) q^{81} +(1.14912e6 - 663445. i) q^{82} +216783. i q^{83} +(-288120. + 166346. i) q^{84} -549045. q^{85} +(-104484. - 180972. i) q^{86} +(-41769.0 - 24115.3i) q^{87} +(141984. - 245924. i) q^{88} +(-1.15577e6 + 667282. i) q^{89} -1.27015e6i q^{90} -166346. i q^{91} -473040. q^{92} +(77689.5 + 134562. i) q^{93} +(-318654. - 183975. i) q^{94} +(625118. - 1.08274e6i) q^{95} +(483840. - 279345. i) q^{96} +1.51409e6i q^{97} +(705894. + 1.22264e6i) q^{98} +860778. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{2} - 21 q^{3} - 80 q^{4} + 315 q^{5} - 686 q^{7} - 384 q^{8} - 582 q^{9} + 3780 q^{10} - 1479 q^{11} + 1680 q^{12} - 4116 q^{14} - 4410 q^{15} + 2816 q^{16} - 5229 q^{17} + 6984 q^{18} + 11907 q^{19}+ \cdots + 1721556 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.00000 + 10.3923i 0.750000 + 1.29904i 0.947822 + 0.318800i \(0.103280\pi\)
−0.197822 + 0.980238i \(0.563387\pi\)
\(3\) −10.5000 − 6.06218i −0.388889 − 0.224525i 0.292790 − 0.956177i \(-0.405416\pi\)
−0.681679 + 0.731652i \(0.738750\pi\)
\(4\) −40.0000 + 69.2820i −0.625000 + 1.08253i
\(5\) 157.500 − 90.9327i 1.26000 − 0.727461i 0.286926 − 0.957953i \(-0.407367\pi\)
0.973074 + 0.230492i \(0.0740334\pi\)
\(6\) − 145.492i − 0.673575i
\(7\) −343.000 −1.00000
\(8\) −192.000 −0.375000
\(9\) −291.000 − 504.027i −0.399177 − 0.691395i
\(10\) 1890.00 + 1091.19i 1.89000 + 1.09119i
\(11\) −739.500 + 1280.85i −0.555597 + 0.962323i 0.442259 + 0.896887i \(0.354177\pi\)
−0.997857 + 0.0654356i \(0.979156\pi\)
\(12\) 840.000 − 484.974i 0.486111 − 0.280656i
\(13\) 484.974i 0.220744i 0.993890 + 0.110372i \(0.0352042\pi\)
−0.993890 + 0.110372i \(0.964796\pi\)
\(14\) −2058.00 − 3564.56i −0.750000 − 1.29904i
\(15\) −2205.00 −0.653333
\(16\) 1408.00 + 2438.73i 0.343750 + 0.595392i
\(17\) −2614.50 − 1509.48i −0.532160 − 0.307242i 0.209736 − 0.977758i \(-0.432740\pi\)
−0.741895 + 0.670516i \(0.766073\pi\)
\(18\) 3492.00 − 6048.32i 0.598765 − 1.03709i
\(19\) 5953.50 − 3437.25i 0.867984 − 0.501131i 0.00130600 − 0.999999i \(-0.499584\pi\)
0.866678 + 0.498869i \(0.166251\pi\)
\(20\) 14549.2i 1.81865i
\(21\) 3601.50 + 2079.33i 0.388889 + 0.224525i
\(22\) −17748.0 −1.66679
\(23\) 2956.50 + 5120.81i 0.242993 + 0.420877i 0.961566 − 0.274576i \(-0.0885373\pi\)
−0.718572 + 0.695452i \(0.755204\pi\)
\(24\) 2016.00 + 1163.94i 0.145833 + 0.0841969i
\(25\) 8725.00 − 15112.1i 0.558400 − 0.967177i
\(26\) −5040.00 + 2909.85i −0.286755 + 0.165558i
\(27\) 15895.0i 0.807551i
\(28\) 13720.0 − 23763.7i 0.625000 − 1.08253i
\(29\) 3978.00 0.163106 0.0815532 − 0.996669i \(-0.474012\pi\)
0.0815532 + 0.996669i \(0.474012\pi\)
\(30\) −13230.0 − 22915.0i −0.490000 − 0.848705i
\(31\) −11098.5 − 6407.72i −0.372545 − 0.215089i 0.302025 − 0.953300i \(-0.402338\pi\)
−0.674570 + 0.738211i \(0.735671\pi\)
\(32\) −23040.0 + 39906.5i −0.703125 + 1.21785i
\(33\) 15529.5 − 8965.96i 0.432131 − 0.249491i
\(34\) − 36227.6i − 0.921727i
\(35\) −54022.5 + 31189.9i −1.26000 + 0.727461i
\(36\) 46560.0 0.997942
\(37\) 30788.5 + 53327.2i 0.607832 + 1.05280i 0.991597 + 0.129365i \(0.0412938\pi\)
−0.383765 + 0.923431i \(0.625373\pi\)
\(38\) 71442.0 + 41247.1i 1.30198 + 0.751696i
\(39\) 2940.00 − 5092.23i 0.0495625 − 0.0858448i
\(40\) −30240.0 + 17459.1i −0.472500 + 0.272798i
\(41\) − 110574.i − 1.60436i −0.597082 − 0.802180i \(-0.703673\pi\)
0.597082 − 0.802180i \(-0.296327\pi\)
\(42\) 49903.8i 0.673575i
\(43\) −17414.0 −0.219025 −0.109512 − 0.993985i \(-0.534929\pi\)
−0.109512 + 0.993985i \(0.534929\pi\)
\(44\) −59160.0 − 102468.i −0.694497 − 1.20290i
\(45\) −91665.0 − 52922.8i −1.00593 − 0.580772i
\(46\) −35478.0 + 61449.7i −0.364490 + 0.631315i
\(47\) −26554.5 + 15331.2i −0.255767 + 0.147667i −0.622402 − 0.782698i \(-0.713843\pi\)
0.366635 + 0.930365i \(0.380510\pi\)
\(48\) − 34142.2i − 0.308722i
\(49\) 117649. 1.00000
\(50\) 209400. 1.67520
\(51\) 18301.5 + 31699.1i 0.137967 + 0.238966i
\(52\) −33600.0 − 19399.0i −0.238962 − 0.137965i
\(53\) 30256.5 − 52405.8i 0.203232 − 0.352007i −0.746336 − 0.665569i \(-0.768189\pi\)
0.949568 + 0.313562i \(0.101522\pi\)
\(54\) −165186. + 95370.2i −1.04904 + 0.605663i
\(55\) 268979.i 1.61670i
\(56\) 65856.0 0.375000
\(57\) −83349.0 −0.450066
\(58\) 23868.0 + 41340.6i 0.122330 + 0.211881i
\(59\) −186826. − 107864.i −0.909667 − 0.525196i −0.0293430 − 0.999569i \(-0.509342\pi\)
−0.880324 + 0.474373i \(0.842675\pi\)
\(60\) 88200.0 − 152767.i 0.408333 − 0.707254i
\(61\) 140942. − 81372.6i 0.620940 − 0.358500i −0.156295 − 0.987710i \(-0.549955\pi\)
0.777235 + 0.629211i \(0.216622\pi\)
\(62\) − 153785.i − 0.645268i
\(63\) 99813.0 + 172881.i 0.399177 + 0.691395i
\(64\) −372736. −1.42188
\(65\) 44100.0 + 76383.4i 0.160583 + 0.278137i
\(66\) 186354. + 107592.i 0.648197 + 0.374237i
\(67\) 134388. − 232768.i 0.446825 − 0.773924i −0.551352 − 0.834273i \(-0.685888\pi\)
0.998177 + 0.0603486i \(0.0192212\pi\)
\(68\) 209160. − 120759.i 0.665199 − 0.384053i
\(69\) − 71691.3i − 0.218232i
\(70\) −648270. − 374279.i −1.89000 − 1.09119i
\(71\) 101922. 0.284769 0.142385 − 0.989811i \(-0.454523\pi\)
0.142385 + 0.989811i \(0.454523\pi\)
\(72\) 55872.0 + 96773.1i 0.149691 + 0.259273i
\(73\) 275090. + 158823.i 0.707140 + 0.408267i 0.810001 − 0.586428i \(-0.199466\pi\)
−0.102861 + 0.994696i \(0.532800\pi\)
\(74\) −369462. + 639927.i −0.911748 + 1.57919i
\(75\) −183225. + 105785.i −0.434311 + 0.250750i
\(76\) 549961.i 1.25283i
\(77\) 253648. − 439332.i 0.555597 − 0.962323i
\(78\) 70560.0 0.148688
\(79\) −181116. − 313701.i −0.367345 − 0.636261i 0.621804 − 0.783173i \(-0.286400\pi\)
−0.989150 + 0.146912i \(0.953067\pi\)
\(80\) 443520. + 256066.i 0.866250 + 0.500130i
\(81\) −115780. + 200538.i −0.217861 + 0.377347i
\(82\) 1.14912e6 − 663445.i 2.08413 − 1.20327i
\(83\) 216783.i 0.379133i 0.981868 + 0.189567i \(0.0607083\pi\)
−0.981868 + 0.189567i \(0.939292\pi\)
\(84\) −288120. + 166346.i −0.486111 + 0.280656i
\(85\) −549045. −0.894028
\(86\) −104484. − 180972.i −0.164269 − 0.284521i
\(87\) −41769.0 − 24115.3i −0.0634302 − 0.0366215i
\(88\) 141984. − 245924.i 0.208349 − 0.360871i
\(89\) −1.15577e6 + 667282.i −1.63946 + 0.946541i −0.658437 + 0.752636i \(0.728782\pi\)
−0.981020 + 0.193905i \(0.937885\pi\)
\(90\) − 1.27015e6i − 1.74231i
\(91\) − 166346.i − 0.220744i
\(92\) −473040. −0.607483
\(93\) 77689.5 + 134562.i 0.0965858 + 0.167292i
\(94\) −318654. − 183975.i −0.383651 − 0.221501i
\(95\) 625118. − 1.08274e6i 0.729106 − 1.26285i
\(96\) 483840. − 279345.i 0.546875 − 0.315738i
\(97\) 1.51409e6i 1.65896i 0.558535 + 0.829481i \(0.311364\pi\)
−0.558535 + 0.829481i \(0.688636\pi\)
\(98\) 705894. + 1.22264e6i 0.750000 + 1.29904i
\(99\) 860778. 0.887127
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 7.7.d.a.3.1 ✓ 2
3.2 odd 2 63.7.m.a.10.1 2
4.3 odd 2 112.7.s.a.17.1 2
7.2 even 3 49.7.d.b.19.1 2
7.3 odd 6 49.7.b.a.48.1 2
7.4 even 3 49.7.b.a.48.2 2
7.5 odd 6 inner 7.7.d.a.5.1 yes 2
7.6 odd 2 49.7.d.b.31.1 2
21.5 even 6 63.7.m.a.19.1 2
21.11 odd 6 441.7.d.a.244.1 2
21.17 even 6 441.7.d.a.244.2 2
28.19 even 6 112.7.s.a.33.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.a.3.1 ✓ 2 1.1 even 1 trivial
7.7.d.a.5.1 yes 2 7.5 odd 6 inner
49.7.b.a.48.1 2 7.3 odd 6
49.7.b.a.48.2 2 7.4 even 3
49.7.d.b.19.1 2 7.2 even 3
49.7.d.b.31.1 2 7.6 odd 2
63.7.m.a.10.1 2 3.2 odd 2
63.7.m.a.19.1 2 21.5 even 6
112.7.s.a.17.1 2 4.3 odd 2
112.7.s.a.33.1 2 28.19 even 6
441.7.d.a.244.1 2 21.11 odd 6
441.7.d.a.244.2 2 21.17 even 6