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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 3.12
Character \(\chi\) \(=\) 49.3
Dual form 49.7.h.a.33.12

$q$-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+(-1.34783 + 3.43420i) q^{2} +(-1.28182 - 0.0960589i) q^{3} +(36.9382 + 34.2736i) q^{4} +(103.584 - 151.930i) q^{5} +(2.05755 - 4.27255i) q^{6} +(-113.036 - 323.839i) q^{7} +(-380.217 + 183.103i) q^{8} +(-719.224 - 108.406i) q^{9} +(382.145 + 560.504i) q^{10} +(48.9320 - 7.37532i) q^{11} +(-44.0557 - 47.4808i) q^{12} +(2757.90 - 2199.35i) q^{13} +(1264.48 + 48.2909i) q^{14} +(-147.370 + 184.796i) q^{15} +(124.653 + 1663.38i) q^{16} +(2579.08 - 8361.19i) q^{17} +(1341.67 - 2323.85i) q^{18} +(-3791.76 + 2189.17i) q^{19} +(9033.41 - 2061.82i) q^{20} +(113.784 + 425.961i) q^{21} +(-40.6235 + 177.983i) q^{22} +(7691.75 - 2372.59i) q^{23} +(504.957 - 198.181i) q^{24} +(-6644.61 - 16930.2i) q^{25} +(3835.85 + 12435.5i) q^{26} +(1825.07 + 416.560i) q^{27} +(6923.81 - 15836.2i) q^{28} +(3896.52 + 17071.8i) q^{29} +(-435.999 - 755.172i) q^{30} +(41317.6 + 23854.7i) q^{31} +(-31689.1 - 9774.79i) q^{32} +(-63.4304 + 4.75345i) q^{33} +(25237.9 + 20126.5i) q^{34} +(-60909.7 - 16371.1i) q^{35} +(-22851.4 - 28654.7i) q^{36} +(2631.41 - 2441.59i) q^{37} +(-2407.43 - 15972.3i) q^{38} +(-3746.39 + 2554.24i) q^{39} +(-11565.6 + 76733.0i) q^{40} +(-19639.6 - 40782.0i) q^{41} +(-1616.20 - 183.365i) q^{42} +(-42487.0 - 20460.7i) q^{43} +(2060.24 + 1404.65i) q^{44} +(-90970.3 + 98042.6i) q^{45} +(-2219.18 + 29612.9i) q^{46} +(-50203.3 - 19703.3i) q^{47} -2144.13i q^{48} +(-92094.8 + 73210.9i) q^{49} +67097.5 q^{50} +(-4109.08 + 10469.8i) q^{51} +(177252. + 13283.2i) q^{52} +(119199. + 110600. i) q^{53} +(-3890.43 + 5706.21i) q^{54} +(3948.05 - 8198.22i) q^{55} +(102274. + 102432. i) q^{56} +(5070.63 - 2441.89i) q^{57} +(-63879.8 - 9628.32i) q^{58} +(-44229.8 - 64873.2i) q^{59} +(-11777.2 + 1775.13i) q^{60} +(-200091. - 215647. i) q^{61} +(-137611. + 109741. i) q^{62} +(46192.1 + 245167. i) q^{63} +(9719.20 - 12187.5i) q^{64} +(-48472.9 - 646826. i) q^{65} +(69.1687 - 224.240i) q^{66} +(-90123.1 + 156098. i) q^{67} +(381835. - 220453. i) q^{68} +(-10087.3 + 2302.36i) q^{69} +(138317. - 187111. i) q^{70} +(15339.5 - 67206.9i) q^{71} +(293311. - 90474.3i) q^{72} +(-28472.9 + 11174.8i) q^{73} +(4838.23 + 12327.6i) q^{74} +(6890.88 + 22339.7i) q^{75} +(-215092. - 49093.3i) q^{76} +(-7919.49 - 15012.4i) q^{77} +(-3722.32 - 16308.5i) q^{78} +(324415. + 561904. i) q^{79} +(265630. + 153362. i) q^{80} +(504380. + 155581. i) q^{81} +(166524. - 12479.3i) q^{82} +(177781. + 141775. i) q^{83} +(-10396.3 + 19634.0i) q^{84} +(-1.00316e6 - 1.25793e6i) q^{85} +(127531. - 118332. i) q^{86} +(-3354.73 - 22257.2i) q^{87} +(-17254.4 + 11763.8i) q^{88} +(-42335.6 + 280878. i) q^{89} +(-214086. - 444555. i) q^{90} +(-1.02398e6 - 644511. i) q^{91} +(365437. + 175985. i) q^{92} +(-50670.1 - 34546.3i) q^{93} +(135330. - 145852. i) q^{94} +(-60165.0 + 802846. i) q^{95} +(39680.6 + 15573.5i) q^{96} +590161. i q^{97} +(-127293. - 414948. i) q^{98} -35992.6 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.34783 + 3.43420i −0.168478 + 0.429275i −0.990030 − 0.140859i \(-0.955014\pi\)
0.821552 + 0.570134i \(0.193109\pi\)
\(3\) −1.28182 − 0.0960589i −0.0474747 − 0.00355774i 0.0509716 − 0.998700i \(-0.483768\pi\)
−0.0984463 + 0.995142i \(0.531387\pi\)
\(4\) 36.9382 + 34.2736i 0.577159 + 0.535526i
\(5\) 103.584 − 151.930i 0.828673 − 1.21544i −0.145437 − 0.989368i \(-0.546459\pi\)
0.974110 − 0.226073i \(-0.0725888\pi\)
\(6\) 2.05755 − 4.27255i 0.00952570 − 0.0197803i
\(7\) −113.036 − 323.839i −0.329550 − 0.944138i
\(8\) −380.217 + 183.103i −0.742611 + 0.357623i
\(9\) −719.224 − 108.406i −0.986590 − 0.148704i
\(10\) 382.145 + 560.504i 0.382145 + 0.560504i
\(11\) 48.9320 − 7.37532i 0.0367634 − 0.00554119i −0.130635 − 0.991431i \(-0.541702\pi\)
0.167398 + 0.985889i \(0.446463\pi\)
\(12\) −44.0557 − 47.4808i −0.0254952 − 0.0274773i
\(13\) 2757.90 − 2199.35i 1.25530 − 1.00107i 0.255893 − 0.966705i \(-0.417631\pi\)
0.999409 − 0.0343648i \(-0.0109408\pi\)
\(14\) 1264.48 + 48.2909i 0.460817 + 0.0175987i
\(15\) −147.370 + 184.796i −0.0436652 + 0.0547545i
\(16\) 124.653 + 1663.38i 0.0304330 + 0.406100i
\(17\) 2579.08 − 8361.19i 0.524951 − 1.70185i −0.172319 − 0.985041i \(-0.555126\pi\)
0.697270 − 0.716808i \(-0.254398\pi\)
\(18\) 1341.67 − 2323.85i 0.230054 − 0.398465i
\(19\) −3791.76 + 2189.17i −0.552815 + 0.319168i −0.750257 − 0.661147i \(-0.770070\pi\)
0.197442 + 0.980315i \(0.436737\pi\)
\(20\) 9033.41 − 2061.82i 1.12918 − 0.257727i
\(21\) 113.784 + 425.961i 0.0122863 + 0.0459951i
\(22\) −40.6235 + 177.983i −0.00381513 + 0.0167152i
\(23\) 7691.75 − 2372.59i 0.632181 − 0.195002i 0.0379285 − 0.999280i \(-0.487924\pi\)
0.594253 + 0.804278i \(0.297448\pi\)
\(24\) 504.957 − 198.181i 0.0365276 − 0.0143360i
\(25\) −6644.61 − 16930.2i −0.425255 − 1.08353i
\(26\) 3835.85 + 12435.5i 0.218244 + 0.707529i
\(27\) 1825.07 + 416.560i 0.0927232 + 0.0211635i
\(28\) 6923.81 − 15836.2i 0.315407 − 0.721401i
\(29\) 3896.52 + 17071.8i 0.159765 + 0.699978i 0.989823 + 0.142301i \(0.0454501\pi\)
−0.830058 + 0.557677i \(0.811693\pi\)
\(30\) −435.999 − 755.172i −0.0161481 − 0.0279693i
\(31\) 41317.6 + 23854.7i 1.38692 + 0.800736i 0.992966 − 0.118396i \(-0.0377752\pi\)
0.393949 + 0.919132i \(0.371109\pi\)
\(32\) −31689.1 − 9774.79i −0.967074 − 0.298303i
\(33\) −63.4304 + 4.75345i −0.00176504 + 0.000132272i
\(34\) 25237.9 + 20126.5i 0.642119 + 0.512073i
\(35\) −60909.7 − 16371.1i −1.42063 − 0.381833i
\(36\) −22851.4 − 28654.7i −0.489784 − 0.614170i
\(37\) 2631.41 − 2441.59i 0.0519497 − 0.0482022i −0.653771 − 0.756693i \(-0.726814\pi\)
0.705720 + 0.708490i \(0.250623\pi\)
\(38\) −2407.43 − 15972.3i −0.0438737 − 0.291083i
\(39\) −3746.39 + 2554.24i −0.0631566 + 0.0430595i
\(40\) −11565.6 + 76733.0i −0.180713 + 1.19895i
\(41\) −19639.6 − 40782.0i −0.284958 − 0.591721i 0.708527 − 0.705683i \(-0.249360\pi\)
−0.993485 + 0.113963i \(0.963646\pi\)
\(42\) −1616.20 − 183.365i −0.0218145 − 0.00247496i
\(43\) −42487.0 − 20460.7i −0.534381 − 0.257344i 0.147175 − 0.989110i \(-0.452982\pi\)
−0.681556 + 0.731766i \(0.738696\pi\)
\(44\) 2060.24 + 1404.65i 0.0241858 + 0.0164896i
\(45\) −90970.3 + 98042.6i −0.998302 + 1.07591i
\(46\) −2219.18 + 29612.9i −0.0227991 + 0.304233i
\(47\) −50203.3 − 19703.3i −0.483547 − 0.189778i 0.111030 − 0.993817i \(-0.464585\pi\)
−0.594577 + 0.804039i \(0.702680\pi\)
\(48\) − 2144.13i − 0.0193877i
\(49\) −92094.8 + 73210.9i −0.782793 + 0.622282i
\(50\) 67097.5 0.536780
\(51\) −4109.08 + 10469.8i −0.0309766 + 0.0789271i
\(52\) 177252. + 13283.2i 1.26061 + 0.0944695i
\(53\) 119199. + 110600.i 0.800654 + 0.742898i 0.970117 − 0.242638i \(-0.0780128\pi\)
−0.169463 + 0.985537i \(0.554203\pi\)
\(54\) −3890.43 + 5706.21i −0.0247068 + 0.0362382i
\(55\) 3948.05 − 8198.22i 0.0237298 − 0.0492755i
\(56\) 102274. + 102432.i 0.582373 + 0.583273i
\(57\) 5070.63 − 2441.89i 0.0273802 − 0.0131856i
\(58\) −63879.8 − 9628.32i −0.327400 − 0.0493477i
\(59\) −44229.8 − 64873.2i −0.215357 − 0.315871i 0.703325 − 0.710868i \(-0.251698\pi\)
−0.918682 + 0.394998i \(0.870745\pi\)
\(60\) −11777.2 + 1775.13i −0.0545242 + 0.00821820i
\(61\) −200091. − 215647.i −0.881531 − 0.950065i 0.117544 − 0.993068i \(-0.462498\pi\)
−0.999075 + 0.0430030i \(0.986307\pi\)
\(62\) −137611. + 109741.i −0.577401 + 0.460462i
\(63\) 46192.1 + 245167.i 0.184734 + 0.980482i
\(64\) 9719.20 − 12187.5i 0.0370758 − 0.0464916i
\(65\) −48472.9 − 646826.i −0.176506 − 2.35531i
\(66\) 69.1687 − 224.240i 0.000240590 − 0.000779975i
\(67\) −90123.1 + 156098.i −0.299648 + 0.519006i −0.976055 − 0.217522i \(-0.930203\pi\)
0.676407 + 0.736528i \(0.263536\pi\)
\(68\) 381835. − 220453.i 1.21436 − 0.701114i
\(69\) −10087.3 + 2302.36i −0.0307064 + 0.00700853i
\(70\) 138317. − 187111.i 0.403257 − 0.545512i
\(71\) 15339.5 − 67206.9i 0.0428585 − 0.187775i −0.948967 − 0.315374i \(-0.897870\pi\)
0.991826 + 0.127599i \(0.0407271\pi\)
\(72\) 293311. − 90474.3i 0.785833 − 0.242397i
\(73\) −28472.9 + 11174.8i −0.0731918 + 0.0287257i −0.401654 − 0.915791i \(-0.631565\pi\)
0.328463 + 0.944517i \(0.393469\pi\)
\(74\) 4838.23 + 12327.6i 0.0119397 + 0.0304217i
\(75\) 6890.88 + 22339.7i 0.0163339 + 0.0529533i
\(76\) −215092. − 49093.3i −0.489985 − 0.111836i
\(77\) −7919.49 − 15012.4i −0.0173470 − 0.0328836i
\(78\) −3722.32 − 16308.5i −0.00784385 − 0.0343662i
\(79\) 324415. + 561904.i 0.657991 + 1.13967i 0.981135 + 0.193325i \(0.0619270\pi\)
−0.323143 + 0.946350i \(0.604740\pi\)
\(80\) 265630. + 153362.i 0.518809 + 0.299535i
\(81\) 504380. + 155581.i 0.949080 + 0.292752i
\(82\) 166524. − 12479.3i 0.302020 − 0.0226333i
\(83\) 177781. + 141775.i 0.310921 + 0.247951i 0.766500 − 0.642244i \(-0.221996\pi\)
−0.455579 + 0.890195i \(0.650568\pi\)
\(84\) −10396.3 + 19634.0i −0.0175404 + 0.0331261i
\(85\) −1.00316e6 − 1.25793e6i −1.63348 − 2.04832i
\(86\) 127531. − 118332.i 0.200503 − 0.186040i
\(87\) −3354.73 − 22257.2i −0.00509448 − 0.0337997i
\(88\) −17254.4 + 11763.8i −0.0253192 + 0.0172624i
\(89\) −42335.6 + 280878.i −0.0600532 + 0.398427i 0.938529 + 0.345200i \(0.112189\pi\)
−0.998582 + 0.0532270i \(0.983049\pi\)
\(90\) −214086. − 444555.i −0.293671 − 0.609814i
\(91\) −1.02398e6 − 644511.i −1.35883 − 0.855275i
\(92\) 365437. + 175985.i 0.469298 + 0.226002i
\(93\) −50670.1 − 34546.3i −0.0629946 − 0.0429490i
\(94\) 135330. − 145852.i 0.162934 − 0.175601i
\(95\) −60165.0 + 802846.i −0.0701735 + 0.936400i
\(96\) 39680.6 + 15573.5i 0.0448502 + 0.0176024i
\(97\) 590161.i 0.646629i 0.946292 + 0.323315i \(0.104797\pi\)
−0.946292 + 0.323315i \(0.895203\pi\)
\(98\) −127293. − 414948.i −0.135247 − 0.440875i
\(99\) −35992.6 −0.0370944
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.3.12 ✓ 324
49.33 odd 42 inner 49.7.h.a.33.12 yes 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.12 ✓ 324 1.1 even 1 trivial
49.7.h.a.33.12 yes 324 49.33 odd 42 inner