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 = [4] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 3.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 7.3
Dual form 7.7.d.b.5.2

$q$-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+(0.121320 + 0.210133i) q^{2} +(32.0772 + 18.5198i) q^{3} +(31.9706 - 55.3746i) q^{4} +(-143.566 + 82.8879i) q^{5} +8.98729i q^{6} +(-197.286 - 280.583i) q^{7} +31.0437 q^{8} +(321.463 + 556.790i) q^{9} +(-34.8350 - 20.1120i) q^{10} +(86.5410 - 149.893i) q^{11} +(2051.05 - 1184.17i) q^{12} +1963.14i q^{13} +(35.0250 - 75.4969i) q^{14} -6140.25 q^{15} +(-2042.35 - 3537.45i) q^{16} +(3199.04 + 1846.96i) q^{17} +(-78.0000 + 135.100i) q^{18} +(3953.11 - 2282.33i) q^{19} +10599.9i q^{20} +(-1132.05 - 12654.0i) q^{21} +41.9967 q^{22} +(7895.75 + 13675.8i) q^{23} +(995.795 + 574.922i) q^{24} +(5928.30 - 10268.1i) q^{25} +(-412.520 + 238.168i) q^{26} -3188.14i q^{27} +(-21844.6 + 1954.25i) q^{28} -23782.2 q^{29} +(-744.938 - 1290.27i) q^{30} +(1785.57 + 1030.90i) q^{31} +(1488.96 - 2578.95i) q^{32} +(5551.98 - 3205.44i) q^{33} +896.298i q^{34} +(51580.6 + 23929.6i) q^{35} +41109.4 q^{36} +(-24255.5 - 42011.7i) q^{37} +(959.185 + 553.786i) q^{38} +(-36356.8 + 62971.8i) q^{39} +(-4456.82 + 2573.15i) q^{40} -26437.9i q^{41} +(2521.69 - 1773.07i) q^{42} +68471.6 q^{43} +(-5533.53 - 9584.36i) q^{44} +(-92302.3 - 53290.8i) q^{45} +(-1915.83 + 3318.32i) q^{46} +(-121557. + 70180.9i) q^{47} -151295. i q^{48} +(-39805.2 + 110711. i) q^{49} +2876.89 q^{50} +(68410.7 + 118491. i) q^{51} +(108708. + 62762.6i) q^{52} +(127065. - 220083. i) q^{53} +(669.934 - 386.786i) q^{54} +28692.8i q^{55} +(-6124.50 - 8710.36i) q^{56} +169073. q^{57} +(-2885.27 - 4997.43i) q^{58} +(-84178.0 - 48600.2i) q^{59} +(-196307. + 340014. i) q^{60} +(15062.4 - 8696.26i) q^{61} +500.276i q^{62} +(92805.9 - 200044. i) q^{63} -260698. q^{64} +(-162720. - 281840. i) q^{65} +(1347.14 + 777.770i) q^{66} +(60340.8 - 104513. i) q^{67} +(204550. - 118097. i) q^{68} +584910. i q^{69} +(1229.37 + 13741.9i) q^{70} -339555. q^{71} +(9979.41 + 17284.8i) q^{72} +(96432.8 + 55675.5i) q^{73} +(5885.36 - 10193.7i) q^{74} +(380326. - 219581. i) q^{75} -291869. i q^{76} +(-59131.0 + 5289.95i) q^{77} -17643.3 q^{78} +(307919. + 533332. i) q^{79} +(586424. + 338572. i) q^{80} +(293390. - 508167. i) q^{81} +(5555.47 - 3207.45i) q^{82} +383668. i q^{83} +(-736904. - 341869. i) q^{84} -612364. q^{85} +(8307.00 + 14388.1i) q^{86} +(-762866. - 440441. i) q^{87} +(2686.56 - 4653.25i) q^{88} +(-668299. + 385843. i) q^{89} -25861.0i q^{90} +(550824. - 387300. i) q^{91} +1.00973e6 q^{92} +(38184.0 + 66136.6i) q^{93} +(-29494.6 - 17028.7i) q^{94} +(-378355. + 655329. i) q^{95} +(95523.0 - 55150.2i) q^{96} -292263. i q^{97} +(-28093.1 + 5067.06i) q^{98} +111279. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 18 q^{3} + 60 q^{4} - 150 q^{5} + 280 q^{7} - 928 q^{8} + 624 q^{9} - 1200 q^{10} + 1882 q^{11} + 4284 q^{12} - 1820 q^{14} - 16500 q^{15} + 248 q^{16} + 13458 q^{17} - 312 q^{18} + 18078 q^{19}+ \cdots + 157872 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.121320 + 0.210133i 0.0151650 + 0.0262666i 0.873508 − 0.486809i \(-0.161839\pi\)
−0.858343 + 0.513076i \(0.828506\pi\)
\(3\) 32.0772 + 18.5198i 1.18804 + 0.685917i 0.957861 − 0.287231i \(-0.0927346\pi\)
0.230182 + 0.973148i \(0.426068\pi\)
\(4\) 31.9706 − 55.3746i 0.499540 − 0.865229i
\(5\) −143.566 + 82.8879i −1.14853 + 0.663103i −0.948528 − 0.316693i \(-0.897428\pi\)
−0.200000 + 0.979796i \(0.564094\pi\)
\(6\) 8.98729i 0.0416078i
\(7\) −197.286 − 280.583i −0.575179 − 0.818028i
\(8\) 31.0437 0.0606323
\(9\) 321.463 + 556.790i 0.440964 + 0.763773i
\(10\) −34.8350 − 20.1120i −0.0348350 − 0.0201120i
\(11\) 86.5410 − 149.893i 0.0650195 − 0.112617i −0.831683 − 0.555251i \(-0.812622\pi\)
0.896703 + 0.442633i \(0.145956\pi\)
\(12\) 2051.05 − 1184.17i 1.18695 − 0.685286i
\(13\) 1963.14i 0.893553i 0.894646 + 0.446777i \(0.147428\pi\)
−0.894646 + 0.446777i \(0.852572\pi\)
\(14\) 35.0250 − 75.4969i 0.0127642 − 0.0275134i
\(15\) −6140.25 −1.81933
\(16\) −2042.35 − 3537.45i −0.498621 − 0.863636i
\(17\) 3199.04 + 1846.96i 0.651137 + 0.375934i 0.788892 − 0.614532i \(-0.210655\pi\)
−0.137755 + 0.990466i \(0.543989\pi\)
\(18\) −78.0000 + 135.100i −0.0133745 + 0.0231653i
\(19\) 3953.11 − 2282.33i 0.576339 − 0.332749i −0.183338 − 0.983050i \(-0.558690\pi\)
0.759677 + 0.650301i \(0.225357\pi\)
\(20\) 10599.9i 1.32499i
\(21\) −1132.05 − 12654.0i −0.122238 − 1.36638i
\(22\) 41.9967 0.00394410
\(23\) 7895.75 + 13675.8i 0.648948 + 1.12401i 0.983375 + 0.181589i \(0.0581241\pi\)
−0.334427 + 0.942422i \(0.608543\pi\)
\(24\) 995.795 + 574.922i 0.0720338 + 0.0415887i
\(25\) 5928.30 − 10268.1i 0.379411 − 0.657160i
\(26\) −412.520 + 238.168i −0.0234706 + 0.0135508i
\(27\) − 3188.14i − 0.161974i
\(28\) −21844.6 + 1954.25i −0.995106 + 0.0890237i
\(29\) −23782.2 −0.975121 −0.487561 − 0.873089i \(-0.662113\pi\)
−0.487561 + 0.873089i \(0.662113\pi\)
\(30\) −744.938 − 1290.27i −0.0275903 − 0.0477878i
\(31\) 1785.57 + 1030.90i 0.0599365 + 0.0346044i 0.529669 − 0.848205i \(-0.322316\pi\)
−0.469732 + 0.882809i \(0.655650\pi\)
\(32\) 1488.96 − 2578.95i 0.0454393 − 0.0787032i
\(33\) 5551.98 − 3205.44i 0.154492 − 0.0891960i
\(34\) 896.298i 0.0228042i
\(35\) 51580.6 + 23929.6i 1.20305 + 0.558125i
\(36\) 41109.4 0.881117
\(37\) −24255.5 − 42011.7i −0.478855 − 0.829401i 0.520851 − 0.853648i \(-0.325615\pi\)
−0.999706 + 0.0242462i \(0.992281\pi\)
\(38\) 959.185 + 553.786i 0.0174804 + 0.0100923i
\(39\) −36356.8 + 62971.8i −0.612903 + 1.06158i
\(40\) −4456.82 + 2573.15i −0.0696379 + 0.0402054i
\(41\) − 26437.9i − 0.383597i −0.981434 − 0.191799i \(-0.938568\pi\)
0.981434 − 0.191799i \(-0.0614320\pi\)
\(42\) 2521.69 − 1773.07i 0.0340364 − 0.0239320i
\(43\) 68471.6 0.861202 0.430601 − 0.902542i \(-0.358302\pi\)
0.430601 + 0.902542i \(0.358302\pi\)
\(44\) −5533.53 − 9584.36i −0.0649597 − 0.112514i
\(45\) −92302.3 − 53290.8i −1.01292 − 0.584810i
\(46\) −1915.83 + 3318.32i −0.0196826 + 0.0340913i
\(47\) −121557. + 70180.9i −1.17081 + 0.675967i −0.953870 − 0.300219i \(-0.902940\pi\)
−0.216938 + 0.976185i \(0.569607\pi\)
\(48\) − 151295.i − 1.36805i
\(49\) −39805.2 + 110711.i −0.338338 + 0.941024i
\(50\) 2876.89 0.0230152
\(51\) 68410.7 + 118491.i 0.515719 + 0.893252i
\(52\) 108708. + 62762.6i 0.773128 + 0.446366i
\(53\) 127065. − 220083.i 0.853489 − 1.47829i −0.0245502 − 0.999699i \(-0.507815\pi\)
0.878039 − 0.478588i \(-0.158851\pi\)
\(54\) 669.934 − 386.786i 0.00425452 − 0.00245635i
\(55\) 28692.8i 0.172459i
\(56\) −6124.50 − 8710.36i −0.0348744 − 0.0495989i
\(57\) 169073. 0.912954
\(58\) −2885.27 − 4997.43i −0.0147878 − 0.0256131i
\(59\) −84178.0 − 48600.2i −0.409867 − 0.236637i 0.280866 − 0.959747i \(-0.409378\pi\)
−0.690732 + 0.723110i \(0.742712\pi\)
\(60\) −196307. + 340014.i −0.908830 + 1.57414i
\(61\) 15062.4 − 8696.26i 0.0663596 − 0.0383127i −0.466453 − 0.884546i \(-0.654468\pi\)
0.532813 + 0.846233i \(0.321135\pi\)
\(62\) 500.276i 0.00209911i
\(63\) 92805.9 − 200044.i 0.371154 − 0.800027i
\(64\) −260698. −0.994485
\(65\) −162720. − 281840.i −0.592518 − 1.02627i
\(66\) 1347.14 + 777.770i 0.00468576 + 0.00270532i
\(67\) 60340.8 − 104513.i 0.200626 − 0.347494i −0.748105 − 0.663581i \(-0.769036\pi\)
0.948730 + 0.316087i \(0.102369\pi\)
\(68\) 204550. − 118097.i 0.650538 − 0.375588i
\(69\) 584910.i 1.78050i
\(70\) 1229.37 + 13741.9i 0.00358418 + 0.0400639i
\(71\) −339555. −0.948713 −0.474357 − 0.880333i \(-0.657319\pi\)
−0.474357 + 0.880333i \(0.657319\pi\)
\(72\) 9979.41 + 17284.8i 0.0267367 + 0.0463093i
\(73\) 96432.8 + 55675.5i 0.247888 + 0.143118i 0.618797 − 0.785551i \(-0.287620\pi\)
−0.370909 + 0.928669i \(0.620954\pi\)
\(74\) 5885.36 − 10193.7i 0.0145237 − 0.0251558i
\(75\) 380326. − 219581.i 0.901514 − 0.520489i
\(76\) − 291869.i − 0.664886i
\(77\) −59131.0 + 5289.95i −0.129522 + 0.0115872i
\(78\) −17643.3 −0.0371788
\(79\) 307919. + 533332.i 0.624533 + 1.08172i 0.988631 + 0.150362i \(0.0480440\pi\)
−0.364098 + 0.931361i \(0.618623\pi\)
\(80\) 586424. + 338572.i 1.14536 + 0.661274i
\(81\) 293390. − 508167.i 0.552065 − 0.956205i
\(82\) 5555.47 − 3207.45i 0.0100758 − 0.00581727i
\(83\) 383668.i 0.670999i 0.942040 + 0.335499i \(0.108905\pi\)
−0.942040 + 0.335499i \(0.891095\pi\)
\(84\) −736904. − 341869.i −1.24329 − 0.576796i
\(85\) −612364. −0.997133
\(86\) 8307.00 + 14388.1i 0.0130602 + 0.0226209i
\(87\) −762866. − 440441.i −1.15849 − 0.668852i
\(88\) 2686.56 − 4653.25i 0.00394228 − 0.00682823i
\(89\) −668299. + 385843.i −0.947983 + 0.547318i −0.892454 − 0.451139i \(-0.851018\pi\)
−0.0555294 + 0.998457i \(0.517685\pi\)
\(90\) − 25861.0i − 0.0354746i
\(91\) 550824. − 387300.i 0.730951 − 0.513953i
\(92\) 1.00973e6 1.29670
\(93\) 38184.0 + 66136.6i 0.0474714 + 0.0822229i
\(94\) −29494.6 − 17028.7i −0.0355107 − 0.0205021i
\(95\) −378355. + 655329.i −0.441294 + 0.764344i
\(96\) 95523.0 − 55150.2i 0.107968 − 0.0623352i
\(97\) − 292263.i − 0.320227i −0.987099 − 0.160113i \(-0.948814\pi\)
0.987099 − 0.160113i \(-0.0511860\pi\)
\(98\) −28093.1 + 5067.06i −0.0298485 + 0.00538367i
\(99\) 111279. 0.114685
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.b.3.2 ✓ 4
3.2 odd 2 63.7.m.b.10.1 4
4.3 odd 2 112.7.s.b.17.1 4
7.2 even 3 49.7.d.c.19.2 4
7.3 odd 6 49.7.b.b.48.2 4
7.4 even 3 49.7.b.b.48.1 4
7.5 odd 6 inner 7.7.d.b.5.2 yes 4
7.6 odd 2 49.7.d.c.31.2 4
21.5 even 6 63.7.m.b.19.1 4
21.11 odd 6 441.7.d.b.244.4 4
21.17 even 6 441.7.d.b.244.3 4
28.19 even 6 112.7.s.b.33.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.7.d.b.3.2 ✓ 4 1.1 even 1 trivial
7.7.d.b.5.2 yes 4 7.5 odd 6 inner
49.7.b.b.48.1 4 7.4 even 3
49.7.b.b.48.2 4 7.3 odd 6
49.7.d.c.19.2 4 7.2 even 3
49.7.d.c.31.2 4 7.6 odd 2
63.7.m.b.10.1 4 3.2 odd 2
63.7.m.b.19.1 4 21.5 even 6
112.7.s.b.17.1 4 4.3 odd 2
112.7.s.b.33.1 4 28.19 even 6
441.7.d.b.244.3 4 21.17 even 6
441.7.d.b.244.4 4 21.11 odd 6