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Properties

Label 49.7.h.a.3.19
Level $49$
Weight $7$
Character 49.3
Analytic conductor $11.273$
Analytic rank $0$
Dimension $324$
Inner twists $2$

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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] = [49,7,Mod(3,49)] 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("49.3"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.h (of order \(42\), degree \(12\), 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: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(324\)
Relative dimension: \(27\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label 3.19
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.19

$q$-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+(2.29501 - 5.84759i) q^{2} +(-46.2118 - 3.46309i) q^{3} +(17.9881 + 16.6905i) q^{4} +(-95.8768 + 140.625i) q^{5} +(-126.307 + 262.280i) q^{6} +(-87.9175 - 331.541i) q^{7} +(501.105 - 241.319i) q^{8} +(1402.68 + 211.419i) q^{9} +(602.282 + 883.385i) q^{10} +(248.191 - 37.4088i) q^{11} +(-773.459 - 833.591i) q^{12} +(-626.337 + 499.487i) q^{13} +(-2140.49 - 246.785i) q^{14} +(4917.63 - 6166.52i) q^{15} +(-143.735 - 1918.01i) q^{16} +(1255.50 - 4070.23i) q^{17} +(4455.45 - 7717.07i) q^{18} +(3683.95 - 2126.93i) q^{19} +(-4071.74 + 929.349i) q^{20} +(2914.66 + 15625.6i) q^{21} +(350.850 - 1537.17i) q^{22} +(17744.9 - 5473.57i) q^{23} +(-23992.7 + 9416.42i) q^{24} +(-4874.70 - 12420.5i) q^{25} +(1483.35 + 4808.89i) q^{26} +(-31152.1 - 7110.27i) q^{27} +(3952.12 - 7431.17i) q^{28} +(5888.93 + 25801.1i) q^{29} +(-24773.3 - 42908.5i) q^{30} +(38772.7 + 22385.4i) q^{31} +(22468.8 + 6930.70i) q^{32} +(-11598.9 + 869.217i) q^{33} +(-20919.7 - 16682.9i) q^{34} +(55052.4 + 19423.7i) q^{35} +(21702.7 + 27214.4i) q^{36} +(15037.4 - 13952.7i) q^{37} +(-3982.72 - 26423.6i) q^{38} +(30673.9 - 20913.1i) q^{39} +(-14108.7 + 93605.0i) q^{40} +(-48526.5 - 100766. i) q^{41} +(98061.1 + 18817.1i) q^{42} +(-96246.3 - 46349.8i) q^{43} +(5088.84 + 3469.52i) q^{44} +(-164215. + 176982. i) q^{45} +(8717.48 - 116327. i) q^{46} +(161818. + 63508.7i) q^{47} +89132.4i q^{48} +(-102190. + 58296.5i) q^{49} -83817.7 q^{50} +(-72114.5 + 183745. i) q^{51} +(-19603.3 - 1469.06i) q^{52} +(-53706.7 - 49832.5i) q^{53} +(-113072. + 165847. i) q^{54} +(-18535.1 + 38488.6i) q^{55} +(-124063. - 144921. i) q^{56} +(-177608. + 85531.4i) q^{57} +(164389. + 24777.7i) q^{58} +(-82403.4 - 120864. i) q^{59} +(191381. - 28846.0i) q^{60} +(213108. + 229676. i) q^{61} +(219885. - 175352. i) q^{62} +(-53225.5 - 483632. i) q^{63} +(168844. - 211723. i) q^{64} +(-10189.4 - 135968. i) q^{65} +(-21536.8 + 69820.5i) q^{66} +(-34609.2 + 59944.9i) q^{67} +(90518.2 - 52260.7i) q^{68} +(-838977. + 191491. i) q^{69} +(239927. - 277346. i) q^{70} +(48818.8 - 213889. i) q^{71} +(753907. - 232550. i) q^{72} +(499663. - 196103. i) q^{73} +(-47078.6 - 119954. i) q^{74} +(182255. + 590856. i) q^{75} +(101767. + 23227.6i) q^{76} +(-34222.9 - 78996.6i) q^{77} +(-51894.4 - 227364. i) q^{78} +(196500. + 340348. i) q^{79} +(283502. + 163680. i) q^{80} +(426813. + 131654. i) q^{81} +(-700610. + 52503.4i) q^{82} +(-679250. - 541684. i) q^{83} +(-208369. + 329721. i) q^{84} +(452005. + 566796. i) q^{85} +(-491921. + 456436. i) q^{86} +(-182786. - 1.21271e6i) q^{87} +(115342. - 78639.0i) q^{88} +(61193.7 - 405994. i) q^{89} +(658042. + 1.36644e6i) q^{90} +(220667. + 163743. i) q^{91} +(410552. + 197711. i) q^{92} +(-1.71423e6 - 1.16874e6i) q^{93} +(742747. - 800490. i) q^{94} +(-54105.0 + 721981. i) q^{95} +(-1.01432e6 - 398091. i) q^{96} +449664. i q^{97} +(106367. + 731357. i) q^{98} +356040. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 324 q - 13 q^{2} - 11 q^{3} + 819 q^{4} - 179 q^{5} + 770 q^{6} + 392 q^{7} + 828 q^{8} - 1160 q^{9} - 2594 q^{10} - 5305 q^{11} + 7497 q^{12} - 14 q^{13} - 11403 q^{14} - 6196 q^{15} + 27903 q^{16} - 5107 q^{17}+ \cdots - 4449616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{1}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.29501 − 5.84759i 0.286876 − 0.730949i −0.712689 − 0.701480i \(-0.752523\pi\)
0.999566 − 0.0294692i \(-0.00938171\pi\)
\(3\) −46.2118 − 3.46309i −1.71155 − 0.128263i −0.817410 − 0.576057i \(-0.804591\pi\)
−0.894137 + 0.447794i \(0.852210\pi\)
\(4\) 17.9881 + 16.6905i 0.281063 + 0.260789i
\(5\) −95.8768 + 140.625i −0.767014 + 1.12500i 0.221689 + 0.975117i \(0.428843\pi\)
−0.988703 + 0.149886i \(0.952109\pi\)
\(6\) −126.307 + 262.280i −0.584756 + 1.21426i
\(7\) −87.9175 − 331.541i −0.256319 − 0.966592i
\(8\) 501.105 − 241.319i 0.978721 − 0.471327i
\(9\) 1402.68 + 211.419i 1.92411 + 0.290013i
\(10\) 602.282 + 883.385i 0.602282 + 0.883385i
\(11\) 248.191 − 37.4088i 0.186470 − 0.0281058i −0.0551435 − 0.998478i \(-0.517562\pi\)
0.241613 + 0.970373i \(0.422324\pi\)
\(12\) −773.459 − 833.591i −0.447604 − 0.482402i
\(13\) −626.337 + 499.487i −0.285088 + 0.227350i −0.755583 − 0.655052i \(-0.772647\pi\)
0.470496 + 0.882402i \(0.344075\pi\)
\(14\) −2140.49 − 246.785i −0.780062 − 0.0899362i
\(15\) 4917.63 − 6166.52i 1.45708 − 1.82712i
\(16\) −143.735 − 1918.01i −0.0350915 − 0.468264i
\(17\) 1255.50 − 4070.23i 0.255547 − 0.828462i −0.733349 − 0.679852i \(-0.762044\pi\)
0.988896 − 0.148610i \(-0.0474799\pi\)
\(18\) 4455.45 − 7717.07i 0.763966 − 1.32323i
\(19\) 3683.95 − 2126.93i 0.537098 − 0.310094i −0.206804 − 0.978382i \(-0.566306\pi\)
0.743902 + 0.668289i \(0.232973\pi\)
\(20\) −4071.74 + 929.349i −0.508968 + 0.116169i
\(21\) 2914.66 + 15625.6i 0.314725 + 1.68724i
\(22\) 350.850 − 1537.17i 0.0329498 − 0.144363i
\(23\) 17744.9 − 5473.57i 1.45844 − 0.449870i 0.538735 − 0.842475i \(-0.318902\pi\)
0.919707 + 0.392605i \(0.128426\pi\)
\(24\) −23992.7 + 9416.42i −1.73558 + 0.681165i
\(25\) −4874.70 − 12420.5i −0.311981 − 0.794914i
\(26\) 1483.35 + 4808.89i 0.0843962 + 0.273606i
\(27\) −31152.1 − 7110.27i −1.58269 − 0.361239i
\(28\) 3952.12 − 7431.17i 0.180034 − 0.338519i
\(29\) 5888.93 + 25801.1i 0.241458 + 1.05790i 0.939690 + 0.342027i \(0.111113\pi\)
−0.698232 + 0.715872i \(0.746029\pi\)
\(30\) −24773.3 − 42908.5i −0.917528 − 1.58921i
\(31\) 38772.7 + 22385.4i 1.30149 + 0.751416i 0.980660 − 0.195720i \(-0.0627045\pi\)
0.320831 + 0.947136i \(0.396038\pi\)
\(32\) 22468.8 + 6930.70i 0.685693 + 0.211508i
\(33\) −11598.9 + 869.217i −0.322756 + 0.0241872i
\(34\) −20919.7 − 16682.9i −0.532253 − 0.424458i
\(35\) 55052.4 + 19423.7i 1.28402 + 0.453030i
\(36\) 21702.7 + 27214.4i 0.465165 + 0.583298i
\(37\) 15037.4 − 13952.7i 0.296872 − 0.275457i −0.517647 − 0.855594i \(-0.673192\pi\)
0.814518 + 0.580138i \(0.197001\pi\)
\(38\) −3982.72 − 26423.6i −0.0725819 − 0.481550i
\(39\) 30673.9 − 20913.1i 0.517101 − 0.352554i
\(40\) −14108.7 + 93605.0i −0.220448 + 1.46258i
\(41\) −48526.5 − 100766.i −0.704089 − 1.46206i −0.878678 − 0.477415i \(-0.841574\pi\)
0.174589 − 0.984641i \(-0.444140\pi\)
\(42\) 98061.1 + 18817.1i 1.32358 + 0.253983i
\(43\) −96246.3 − 46349.8i −1.21054 − 0.582965i −0.283877 − 0.958861i \(-0.591621\pi\)
−0.926662 + 0.375896i \(0.877335\pi\)
\(44\) 5088.84 + 3469.52i 0.0597394 + 0.0407297i
\(45\) −164215. + 176982.i −1.80209 + 1.94219i
\(46\) 8717.48 − 116327.i 0.0895607 − 1.19510i
\(47\) 161818. + 63508.7i 1.55859 + 0.611702i 0.979119 − 0.203289i \(-0.0651632\pi\)
0.579473 + 0.814992i \(0.303258\pi\)
\(48\) 89132.4i 0.805957i
\(49\) −102190. + 58296.5i −0.868601 + 0.495512i
\(50\) −83817.7 −0.670542
\(51\) −72114.5 + 183745.i −0.543641 + 1.38517i
\(52\) −19603.3 − 1469.06i −0.139418 − 0.0104479i
\(53\) −53706.7 − 49832.5i −0.360745 − 0.334723i 0.478888 − 0.877876i \(-0.341040\pi\)
−0.839634 + 0.543153i \(0.817230\pi\)
\(54\) −113072. + 165847.i −0.718084 + 1.05324i
\(55\) −18535.1 + 38488.6i −0.111406 + 0.231336i
\(56\) −124063. − 144921.i −0.706446 − 0.825214i
\(57\) −177608. + 85531.4i −0.959041 + 0.461850i
\(58\) 164389. + 24777.7i 0.842539 + 0.126992i
\(59\) −82403.4 − 120864.i −0.401226 − 0.588491i 0.571456 − 0.820633i \(-0.306379\pi\)
−0.972682 + 0.232142i \(0.925426\pi\)
\(60\) 191381. − 28846.0i 0.886022 − 0.133546i
\(61\) 213108. + 229676.i 0.938882 + 1.01187i 0.999906 + 0.0137383i \(0.00437317\pi\)
−0.0610232 + 0.998136i \(0.519436\pi\)
\(62\) 219885. − 175352.i 0.922614 − 0.735760i
\(63\) −53225.5 − 483632.i −0.212862 − 1.93417i
\(64\) 168844. − 211723.i 0.644087 − 0.807660i
\(65\) −10189.4 − 135968.i −0.0371030 − 0.495105i
\(66\) −21536.8 + 69820.5i −0.0749115 + 0.242857i
\(67\) −34609.2 + 59944.9i −0.115071 + 0.199309i −0.917808 − 0.397024i \(-0.870043\pi\)
0.802737 + 0.596333i \(0.203376\pi\)
\(68\) 90518.2 − 52260.7i 0.287878 − 0.166207i
\(69\) −838977. + 191491.i −2.55389 + 0.582909i
\(70\) 239927. − 277346.i 0.699497 − 0.808590i
\(71\) 48818.8 − 213889.i 0.136399 − 0.597605i −0.859810 − 0.510614i \(-0.829418\pi\)
0.996209 − 0.0869902i \(-0.0277249\pi\)
\(72\) 753907. − 232550.i 2.01986 − 0.623043i
\(73\) 499663. − 196103.i 1.28442 − 0.504099i 0.377740 − 0.925912i \(-0.376701\pi\)
0.906683 + 0.421813i \(0.138606\pi\)
\(74\) −47078.6 − 119954.i −0.116179 − 0.296020i
\(75\) 182255. + 590856.i 0.432012 + 1.40055i
\(76\) 101767. + 23227.6i 0.231827 + 0.0529131i
\(77\) −34222.9 − 78996.6i −0.0749625 − 0.173036i
\(78\) −51894.4 − 227364.i −0.109355 − 0.479114i
\(79\) 196500. + 340348.i 0.398548 + 0.690306i 0.993547 − 0.113421i \(-0.0361808\pi\)
−0.594999 + 0.803727i \(0.702848\pi\)
\(80\) 283502. + 163680.i 0.553714 + 0.319687i
\(81\) 426813. + 131654.i 0.803123 + 0.247731i
\(82\) −700610. + 52503.4i −1.27067 + 0.0952239i
\(83\) −679250. − 541684.i −1.18794 − 0.947353i −0.188546 − 0.982064i \(-0.560377\pi\)
−0.999397 + 0.0347114i \(0.988949\pi\)
\(84\) −208369. + 329721.i −0.351557 + 0.556299i
\(85\) 452005. + 566796.i 0.736015 + 0.922933i
\(86\) −491921. + 456436.i −0.773392 + 0.717603i
\(87\) −182786. − 1.21271e6i −0.277579 − 1.84161i
\(88\) 115342. − 78639.0i 0.169255 − 0.115396i
\(89\) 61193.7 − 405994.i 0.0868034 − 0.575903i −0.902574 − 0.430534i \(-0.858325\pi\)
0.989378 − 0.145369i \(-0.0464368\pi\)
\(90\) 658042. + 1.36644e6i 0.902663 + 1.87440i
\(91\) 220667. + 163743.i 0.292828 + 0.217289i
\(92\) 410552. + 197711.i 0.527236 + 0.253903i
\(93\) −1.71423e6 − 1.16874e6i −2.13118 − 1.45302i
\(94\) 742747. − 800490.i 0.894246 − 0.963768i
\(95\) −54105.0 + 721981.i −0.0631054 + 0.842083i
\(96\) −1.01432e6 − 398091.i −1.14647 − 0.449955i
\(97\) 449664.i 0.492689i 0.969182 + 0.246345i \(0.0792295\pi\)
−0.969182 + 0.246345i \(0.920771\pi\)
\(98\) 106367. + 731357.i 0.113013 + 0.777054i
\(99\) 356040. 0.366939
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 49.7.h.a.3.19 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.19 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.19 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.19 yes 324 49.33 odd 42 inner