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Properties

Label 49.7.h.a.3.18
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 3.18
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.18

$q$-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+(2.25296 - 5.74046i) q^{2} +(-30.3956 - 2.27784i) q^{3} +(19.0383 + 17.6650i) q^{4} +(25.7577 - 37.7795i) q^{5} +(-81.5560 + 169.353i) q^{6} +(83.9581 + 332.566i) q^{7} +(499.884 - 240.732i) q^{8} +(197.847 + 29.8207i) q^{9} +(-158.841 - 232.977i) q^{10} +(181.782 - 27.3992i) q^{11} +(-538.443 - 580.304i) q^{12} +(2469.44 - 1969.31i) q^{13} +(2098.23 + 267.301i) q^{14} +(-868.976 + 1089.66i) q^{15} +(-131.475 - 1754.41i) q^{16} +(1436.25 - 4656.21i) q^{17} +(616.927 - 1068.55i) q^{18} +(9968.20 - 5755.14i) q^{19} +(1157.76 - 264.250i) q^{20} +(-1794.43 - 10299.8i) q^{21} +(252.264 - 1105.24i) q^{22} +(-18145.3 + 5597.08i) q^{23} +(-15742.6 + 6178.53i) q^{24} +(4944.62 + 12598.7i) q^{25} +(-5741.19 - 18612.5i) q^{26} +(15717.7 + 3587.45i) q^{27} +(-4276.35 + 7814.61i) q^{28} +(-5366.28 - 23511.2i) q^{29} +(4297.38 + 7443.28i) q^{30} +(38927.3 + 22474.7i) q^{31} +(23564.2 + 7268.60i) q^{32} +(-5587.79 + 418.747i) q^{33} +(-23492.9 - 18735.0i) q^{34} +(14726.7 + 5394.22i) q^{35} +(3239.90 + 4062.70i) q^{36} +(29684.2 - 27542.9i) q^{37} +(-10579.2 - 70188.1i) q^{38} +(-79545.8 + 54233.4i) q^{39} +(3781.12 - 25086.1i) q^{40} +(48369.6 + 100441. i) q^{41} +(-63168.3 - 12904.2i) q^{42} +(61007.8 + 29379.8i) q^{43} +(3944.83 + 2689.54i) q^{44} +(6222.70 - 6706.47i) q^{45} +(-8750.87 + 116772. i) q^{46} +(-133310. - 52320.5i) q^{47} +53625.8i q^{48} +(-103551. + 55843.2i) q^{49} +83462.2 q^{50} +(-54261.8 + 138257. i) q^{51} +(81801.7 + 6130.19i) q^{52} +(-80448.8 - 74645.6i) q^{53} +(56004.9 - 82144.1i) q^{54} +(3647.15 - 7573.39i) q^{55} +(122028. + 146033. i) q^{56} +(-316099. + 152225. i) q^{57} +(-147055. - 22165.0i) q^{58} +(-27792.1 - 40763.5i) q^{59} +(-35792.7 + 5394.87i) q^{60} +(-175721. - 189383. i) q^{61} +(216717. - 172826. i) q^{62} +(6693.55 + 68301.0i) q^{63} +(165017. - 206925. i) q^{64} +(-10792.7 - 144019. i) q^{65} +(-10185.3 + 33019.9i) q^{66} +(-16839.4 + 29166.6i) q^{67} +(109595. - 63275.0i) q^{68} +(564286. - 128795. i) q^{69} +(64144.1 - 72385.3i) q^{70} +(-2937.90 + 12871.8i) q^{71} +(106080. - 32721.2i) q^{72} +(171899. - 67465.2i) q^{73} +(-91231.5 - 232454. i) q^{74} +(-121597. - 394208. i) q^{75} +(291442. + 66519.7i) q^{76} +(24374.1 + 58154.1i) q^{77} +(132111. + 578815. i) q^{78} +(167447. + 290026. i) q^{79} +(-69667.3 - 40222.4i) q^{80} +(-608956. - 187838. i) q^{81} +(685550. - 51374.8i) q^{82} +(-267581. - 213389. i) q^{83} +(147783. - 227789. i) q^{84} +(-138915. - 174194. i) q^{85} +(306102. - 284021. i) q^{86} +(109557. + 726861. i) q^{87} +(84274.2 - 57457.2i) q^{88} +(-91943.6 + 610006. i) q^{89} +(-24478.7 - 50830.6i) q^{90} +(862254. + 655911. i) q^{91} +(-444328. - 213977. i) q^{92} +(-1.13203e6 - 771802. i) q^{93} +(-600687. + 647387. i) q^{94} +(39330.8 - 524833. i) q^{95} +(-699692. - 274609. i) q^{96} -513613. i q^{97} +(87268.7 + 720243. i) q^{98} +36782.2 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.25296 − 5.74046i 0.281620 − 0.717557i −0.718145 − 0.695894i \(-0.755008\pi\)
0.999765 − 0.0216636i \(-0.00689627\pi\)
\(3\) −30.3956 − 2.27784i −1.12576 − 0.0843643i −0.501195 − 0.865334i \(-0.667106\pi\)
−0.624569 + 0.780970i \(0.714725\pi\)
\(4\) 19.0383 + 17.6650i 0.297474 + 0.276015i
\(5\) 25.7577 − 37.7795i 0.206061 − 0.302236i −0.709271 − 0.704936i \(-0.750976\pi\)
0.915332 + 0.402700i \(0.131928\pi\)
\(6\) −81.5560 + 169.353i −0.377574 + 0.784041i
\(7\) 83.9581 + 332.566i 0.244776 + 0.969580i
\(8\) 499.884 − 240.732i 0.976337 − 0.470179i
\(9\) 197.847 + 29.8207i 0.271396 + 0.0409063i
\(10\) −158.841 − 232.977i −0.158841 − 0.232977i
\(11\) 181.782 − 27.3992i 0.136576 − 0.0205855i −0.0803986 − 0.996763i \(-0.525619\pi\)
0.216974 + 0.976177i \(0.430381\pi\)
\(12\) −538.443 − 580.304i −0.311599 − 0.335824i
\(13\) 2469.44 − 1969.31i 1.12400 − 0.896363i 0.128559 − 0.991702i \(-0.458965\pi\)
0.995445 + 0.0953389i \(0.0303935\pi\)
\(14\) 2098.23 + 267.301i 0.764663 + 0.0974128i
\(15\) −868.976 + 1089.66i −0.257474 + 0.322862i
\(16\) −131.475 − 1754.41i −0.0320983 − 0.428323i
\(17\) 1436.25 − 4656.21i 0.292336 − 0.947732i −0.683633 − 0.729826i \(-0.739601\pi\)
0.975970 − 0.217906i \(-0.0699226\pi\)
\(18\) 616.927 − 1068.55i 0.105783 − 0.183222i
\(19\) 9968.20 − 5755.14i 1.45330 − 0.839064i 0.454635 − 0.890678i \(-0.349770\pi\)
0.998667 + 0.0516140i \(0.0164366\pi\)
\(20\) 1157.76 − 264.250i 0.144720 − 0.0330313i
\(21\) −1794.43 − 10299.8i −0.193762 − 1.11217i
\(22\) 252.264 − 1105.24i 0.0236912 − 0.103798i
\(23\) −18145.3 + 5597.08i −1.49135 + 0.460021i −0.930001 − 0.367557i \(-0.880194\pi\)
−0.561351 + 0.827578i \(0.689718\pi\)
\(24\) −15742.6 + 6178.53i −1.13879 + 0.446942i
\(25\) 4944.62 + 12598.7i 0.316455 + 0.806315i
\(26\) −5741.19 − 18612.5i −0.326649 − 1.05897i
\(27\) 15717.7 + 3587.45i 0.798540 + 0.182261i
\(28\) −4276.35 + 7814.61i −0.194804 + 0.355986i
\(29\) −5366.28 − 23511.2i −0.220029 − 0.964009i −0.957455 − 0.288581i \(-0.906816\pi\)
0.737427 − 0.675427i \(-0.236041\pi\)
\(30\) 4297.38 + 7443.28i 0.159162 + 0.275677i
\(31\) 38927.3 + 22474.7i 1.30668 + 0.754412i 0.981541 − 0.191254i \(-0.0612553\pi\)
0.325140 + 0.945666i \(0.394589\pi\)
\(32\) 23564.2 + 7268.60i 0.719123 + 0.221820i
\(33\) −5587.79 + 418.747i −0.155489 + 0.0116523i
\(34\) −23492.9 − 18735.0i −0.597724 − 0.476669i
\(35\) 14726.7 + 5394.22i 0.343481 + 0.125813i
\(36\) 3239.90 + 4062.70i 0.0694423 + 0.0870778i
\(37\) 29684.2 − 27542.9i 0.586030 − 0.543757i −0.330457 − 0.943821i \(-0.607203\pi\)
0.916487 + 0.400064i \(0.131012\pi\)
\(38\) −10579.2 − 70188.1i −0.192797 − 1.27912i
\(39\) −79545.8 + 54233.4i −1.34098 + 0.914267i
\(40\) 3781.12 − 25086.1i 0.0590800 − 0.391970i
\(41\) 48369.6 + 100441.i 0.701812 + 1.45733i 0.880806 + 0.473478i \(0.157002\pi\)
−0.178993 + 0.983850i \(0.557284\pi\)
\(42\) −63168.3 − 12904.2i −0.852611 − 0.174174i
\(43\) 61007.8 + 29379.8i 0.767326 + 0.369525i 0.776242 − 0.630435i \(-0.217124\pi\)
−0.00891557 + 0.999960i \(0.502838\pi\)
\(44\) 3944.83 + 2689.54i 0.0463095 + 0.0315733i
\(45\) 6222.70 − 6706.47i 0.0682875 − 0.0735964i
\(46\) −8750.87 + 116772.i −0.0899037 + 1.19968i
\(47\) −133310. − 52320.5i −1.28402 − 0.503939i −0.377460 − 0.926026i \(-0.623203\pi\)
−0.906556 + 0.422087i \(0.861298\pi\)
\(48\) 53625.8i 0.484898i
\(49\) −103551. + 55843.2i −0.880170 + 0.474659i
\(50\) 83462.2 0.667698
\(51\) −54261.8 + 138257.i −0.409057 + 1.04226i
\(52\) 81801.7 + 6130.19i 0.581771 + 0.0435977i
\(53\) −80448.8 − 74645.6i −0.540371 − 0.501391i 0.362101 − 0.932139i \(-0.382059\pi\)
−0.902472 + 0.430748i \(0.858250\pi\)
\(54\) 56004.9 − 82144.1i 0.355668 − 0.521669i
\(55\) 3647.15 − 7573.39i 0.0219213 − 0.0455200i
\(56\) 122028. + 146033.i 0.694859 + 0.831548i
\(57\) −316099. + 152225.i −1.70686 + 0.821981i
\(58\) −147055. − 22165.0i −0.753696 − 0.113601i
\(59\) −27792.1 − 40763.5i −0.135321 − 0.198479i 0.752597 − 0.658482i \(-0.228801\pi\)
−0.887918 + 0.460002i \(0.847849\pi\)
\(60\) −35792.7 + 5394.87i −0.165707 + 0.0249763i
\(61\) −175721. − 189383.i −0.774168 − 0.834354i 0.215683 − 0.976463i \(-0.430802\pi\)
−0.989851 + 0.142109i \(0.954612\pi\)
\(62\) 216717. − 172826.i 0.909322 − 0.725160i
\(63\) 6693.55 + 68301.0i 0.0267691 + 0.273153i
\(64\) 165017. − 206925.i 0.629492 − 0.789358i
\(65\) −10792.7 − 144019.i −0.0392999 − 0.524420i
\(66\) −10185.3 + 33019.9i −0.0354276 + 0.114853i
\(67\) −16839.4 + 29166.6i −0.0559888 + 0.0969755i −0.892661 − 0.450728i \(-0.851164\pi\)
0.836673 + 0.547703i \(0.184498\pi\)
\(68\) 109595. − 63275.0i 0.348551 − 0.201236i
\(69\) 564286. − 128795.i 1.71772 − 0.392058i
\(70\) 64144.1 − 72385.3i 0.187009 − 0.211036i
\(71\) −2937.90 + 12871.8i −0.00820845 + 0.0359636i −0.978866 − 0.204501i \(-0.934443\pi\)
0.970658 + 0.240465i \(0.0772999\pi\)
\(72\) 106080. − 32721.2i 0.284207 − 0.0876662i
\(73\) 171899. − 67465.2i 0.441879 − 0.173425i −0.133962 − 0.990987i \(-0.542770\pi\)
0.575841 + 0.817562i \(0.304675\pi\)
\(74\) −91231.5 − 232454.i −0.225138 − 0.573643i
\(75\) −121597. − 394208.i −0.288230 − 0.934418i
\(76\) 291442. + 66519.7i 0.663913 + 0.151534i
\(77\) 24374.1 + 58154.1i 0.0533896 + 0.127382i
\(78\) 132111. + 578815.i 0.278391 + 1.21971i
\(79\) 167447. + 290026.i 0.339622 + 0.588242i 0.984362 − 0.176160i \(-0.0563676\pi\)
−0.644740 + 0.764402i \(0.723034\pi\)
\(80\) −69667.3 − 40222.4i −0.136069 − 0.0785594i
\(81\) −608956. − 187838.i −1.14586 − 0.353450i
\(82\) 685550. − 51374.8i 1.24336 − 0.0931770i
\(83\) −267581. − 213389.i −0.467973 − 0.373196i 0.360926 − 0.932595i \(-0.382461\pi\)
−0.828899 + 0.559398i \(0.811032\pi\)
\(84\) 147783. − 227789.i 0.249336 − 0.384322i
\(85\) −138915. − 174194.i −0.226200 − 0.283645i
\(86\) 306102. − 284021.i 0.481250 − 0.446535i
\(87\) 109557. + 726861.i 0.166372 + 1.10381i
\(88\) 84274.2 − 57457.2i 0.123665 − 0.0843133i
\(89\) −91943.6 + 610006.i −0.130422 + 0.865295i 0.823947 + 0.566667i \(0.191767\pi\)
−0.954369 + 0.298629i \(0.903471\pi\)
\(90\) −24478.7 − 50830.6i −0.0335785 − 0.0697264i
\(91\) 862254. + 655911.i 1.14422 + 0.870403i
\(92\) −444328. − 213977.i −0.570611 − 0.274792i
\(93\) −1.13203e6 − 771802.i −1.40737 − 0.959527i
\(94\) −600687. + 647387.i −0.723210 + 0.779435i
\(95\) 39330.8 − 524833.i 0.0458735 − 0.612139i
\(96\) −699692. − 274609.i −0.790849 − 0.310385i
\(97\) − 513613.i − 0.562757i −0.959597 − 0.281378i \(-0.909208\pi\)
0.959597 − 0.281378i \(-0.0907916\pi\)
\(98\) 87268.7 + 720243.i 0.0927214 + 0.765246i
\(99\) 36782.2 0.0379081
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.3.18 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.18 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.18 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.18 yes 324 49.33 odd 42 inner