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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 33.1
Character \(\chi\) \(=\) 49.33
Dual form 49.7.h.a.3.1

$q$-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+(-5.64737 - 14.3893i) q^{2} +(39.0389 - 2.92556i) q^{3} +(-128.243 + 118.992i) q^{4} +(88.5491 + 129.878i) q^{5} +(-262.564 - 545.219i) q^{6} +(-318.627 + 126.988i) q^{7} +(1545.11 + 744.086i) q^{8} +(794.620 - 119.770i) q^{9} +(1368.77 - 2007.62i) q^{10} +(2171.64 + 327.322i) q^{11} +(-4658.34 + 5020.49i) q^{12} +(361.549 + 288.326i) q^{13} +(3626.67 + 3867.65i) q^{14} +(3836.82 + 4811.22i) q^{15} +(1144.33 - 15270.1i) q^{16} +(-346.911 - 1124.66i) q^{17} +(-6210.91 - 10757.6i) q^{18} +(1453.01 + 838.897i) q^{19} +(-26810.2 - 6119.24i) q^{20} +(-12067.3 + 5889.65i) q^{21} +(-7554.14 - 33096.8i) q^{22} +(12463.6 + 3844.52i) q^{23} +(62496.3 + 24528.0i) q^{24} +(-3318.81 + 8456.18i) q^{25} +(2106.99 - 6830.70i) q^{26} +(2847.06 - 649.823i) q^{27} +(25751.0 - 54199.3i) q^{28} +(-6505.97 + 28504.5i) q^{29} +(47562.0 - 82379.8i) q^{30} +(-8560.88 + 4942.63i) q^{31} +(-121307. + 37418.3i) q^{32} +(85736.2 + 6425.04i) q^{33} +(-14223.8 + 11343.1i) q^{34} +(-44707.0 - 30137.8i) q^{35} +(-87652.6 + 109913. i) q^{36} +(3326.79 + 3086.81i) q^{37} +(3865.41 - 25645.3i) q^{38} +(14958.0 + 10198.2i) q^{39} +(40178.0 + 266564. i) q^{40} +(4779.34 - 9924.40i) q^{41} +(152896. + 140379. i) q^{42} +(-81872.2 + 39427.6i) q^{43} +(-317446. + 216431. i) q^{44} +(85918.2 + 92597.8i) q^{45} +(-15066.9 - 201054. i) q^{46} +(134188. - 52664.9i) q^{47} -599474. i q^{48} +(85397.0 - 80923.7i) q^{49} +140421. q^{50} +(-16833.3 - 42890.5i) q^{51} +(-80674.4 + 6045.71i) q^{52} +(114656. - 106385. i) q^{53} +(-25428.9 - 37297.3i) q^{54} +(149785. + 311032. i) q^{55} +(-586804. - 40874.8i) q^{56} +(59178.2 + 28498.7i) q^{57} +(446900. - 67359.4i) q^{58} +(15337.2 - 22495.5i) q^{59} +(-1.06454e6 - 160454. i) q^{60} +(-34663.8 + 37358.7i) q^{61} +(119467. + 95271.9i) q^{62} +(-237978. + 139069. i) q^{63} +(612452. + 767990. i) q^{64} +(-5432.23 + 72488.1i) q^{65} +(-391732. - 1.26996e6i) q^{66} +(-21617.5 - 37442.7i) q^{67} +(178314. + 102949. i) q^{68} +(497814. + 113623. i) q^{69} +(-181183. + 813500. i) q^{70} +(-127314. - 557798. i) q^{71} +(1.31689e6 + 406208. i) q^{72} +(-459018. - 180151. i) q^{73} +(25629.3 - 65302.4i) q^{74} +(-104824. + 339830. i) q^{75} +(-286160. + 65314.2i) q^{76} +(-733509. + 171479. i) q^{77} +(62271.0 - 272827. i) q^{78} +(-15902.7 + 27544.3i) q^{79} +(2.08457e6 - 1.20353e6i) q^{80} +(-450549. + 138976. i) q^{81} +(-169796. - 12724.4i) q^{82} +(-449830. + 358727. i) q^{83} +(846727. - 2.19122e6i) q^{84} +(115349. - 144643. i) q^{85} +(1.02970e6 + 955418. i) q^{86} +(-170594. + 1.13182e6i) q^{87} +(3.11187e6 + 2.12164e6i) q^{88} +(-136498. - 905608. i) q^{89} +(847202. - 1.75923e6i) q^{90} +(-151813. - 45955.8i) q^{91} +(-2.05584e6 + 990038. i) q^{92} +(-319748. + 218000. i) q^{93} +(-1.51562e6 - 1.63345e6i) q^{94} +(19708.9 + 262997. i) q^{95} +(-4.62622e6 + 1.81566e6i) q^{96} +96272.4i q^{97} +(-1.64670e6 - 771793. i) q^{98} +1.76483e6 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 \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{41}{42}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.64737 − 14.3893i −0.705921 − 1.79866i −0.593239 − 0.805027i \(-0.702151\pi\)
−0.112682 − 0.993631i \(-0.535944\pi\)
\(3\) 39.0389 − 2.92556i 1.44589 − 0.108354i 0.671470 − 0.741032i \(-0.265663\pi\)
0.774415 + 0.632678i \(0.218044\pi\)
\(4\) −128.243 + 118.992i −2.00379 + 1.85925i
\(5\) 88.5491 + 129.878i 0.708393 + 1.03902i 0.996675 + 0.0814757i \(0.0259633\pi\)
−0.288283 + 0.957545i \(0.593084\pi\)
\(6\) −262.564 − 545.219i −1.21557 − 2.52416i
\(7\) −318.627 + 126.988i −0.928941 + 0.370228i
\(8\) 1545.11 + 744.086i 3.01780 + 1.45329i
\(9\) 794.620 − 119.770i 1.09001 − 0.164293i
\(10\) 1368.77 − 2007.62i 1.36877 − 2.00762i
\(11\) 2171.64 + 327.322i 1.63159 + 0.245922i 0.900020 − 0.435848i \(-0.143551\pi\)
0.731567 + 0.681770i \(0.238789\pi\)
\(12\) −4658.34 + 5020.49i −2.69580 + 2.90538i
\(13\) 361.549 + 288.326i 0.164565 + 0.131236i 0.702310 − 0.711872i \(-0.252152\pi\)
−0.537745 + 0.843108i \(0.680724\pi\)
\(14\) 3626.67 + 3867.65i 1.32167 + 1.40949i
\(15\) 3836.82 + 4811.22i 1.13684 + 1.42555i
\(16\) 1144.33 − 15270.1i 0.279378 − 3.72804i
\(17\) −346.911 − 1124.66i −0.0706107 − 0.228914i 0.913388 − 0.407090i \(-0.133457\pi\)
−0.983999 + 0.178176i \(0.942980\pi\)
\(18\) −6210.91 − 10757.6i −1.06497 − 1.84458i
\(19\) 1453.01 + 838.897i 0.211840 + 0.122306i 0.602166 − 0.798371i \(-0.294304\pi\)
−0.390326 + 0.920677i \(0.627638\pi\)
\(20\) −26810.2 − 6119.24i −3.35127 − 0.764905i
\(21\) −12067.3 + 5889.65i −1.30303 + 0.635962i
\(22\) −7554.14 − 33096.8i −0.709442 − 3.10827i
\(23\) 12463.6 + 3844.52i 1.02438 + 0.315979i 0.761027 − 0.648721i \(-0.224696\pi\)
0.263353 + 0.964700i \(0.415172\pi\)
\(24\) 62496.3 + 24528.0i 4.52086 + 1.77431i
\(25\) −3318.81 + 8456.18i −0.212404 + 0.541196i
\(26\) 2106.99 − 6830.70i 0.119879 − 0.388638i
\(27\) 2847.06 − 649.823i 0.144646 − 0.0330144i
\(28\) 25751.0 − 54199.3i 1.17306 − 2.46899i
\(29\) −6505.97 + 28504.5i −0.266758 + 1.16874i 0.647001 + 0.762489i \(0.276023\pi\)
−0.913760 + 0.406255i \(0.866834\pi\)
\(30\) 47562.0 − 82379.8i 1.76156 − 3.05110i
\(31\) −8560.88 + 4942.63i −0.287365 + 0.165910i −0.636753 − 0.771068i \(-0.719723\pi\)
0.349388 + 0.936978i \(0.386389\pi\)
\(32\) −121307. + 37418.3i −3.70200 + 1.14191i
\(33\) 85736.2 + 6425.04i 2.38573 + 0.178786i
\(34\) −14223.8 + 11343.1i −0.361893 + 0.288600i
\(35\) −44707.0 − 30137.8i −1.04273 − 0.702922i
\(36\) −87652.6 + 109913.i −1.87870 + 2.35581i
\(37\) 3326.79 + 3086.81i 0.0656780 + 0.0609403i 0.712329 − 0.701846i \(-0.247640\pi\)
−0.646651 + 0.762786i \(0.723831\pi\)
\(38\) 3865.41 − 25645.3i 0.0704441 − 0.467366i
\(39\) 14958.0 + 10198.2i 0.252162 + 0.171921i
\(40\) 40178.0 + 266564.i 0.627781 + 4.16506i
\(41\) 4779.34 − 9924.40i 0.0693452 − 0.143997i −0.863415 − 0.504495i \(-0.831679\pi\)
0.932760 + 0.360498i \(0.117393\pi\)
\(42\) 152896. + 140379.i 2.06371 + 1.89476i
\(43\) −81872.2 + 39427.6i −1.02975 + 0.495900i −0.870930 − 0.491408i \(-0.836483\pi\)
−0.158818 + 0.987308i \(0.550768\pi\)
\(44\) −317446. + 216431.i −3.72659 + 2.54075i
\(45\) 85918.2 + 92597.8i 0.942861 + 1.01616i
\(46\) −15066.9 − 201054.i −0.154793 − 2.06556i
\(47\) 134188. − 52664.9i 1.29247 − 0.507257i 0.383277 − 0.923633i \(-0.374796\pi\)
0.909192 + 0.416377i \(0.136700\pi\)
\(48\) − 599474.i − 5.42059i
\(49\) 85397.0 − 80923.7i 0.725862 − 0.687840i
\(50\) 140421. 1.12337
\(51\) −16833.3 − 42890.5i −0.126899 − 0.323333i
\(52\) −80674.4 + 6045.71i −0.573754 + 0.0429969i
\(53\) 114656. − 106385.i 0.770138 − 0.714584i −0.193703 − 0.981060i \(-0.562050\pi\)
0.963841 + 0.266476i \(0.0858595\pi\)
\(54\) −25428.9 − 37297.3i −0.161490 − 0.236863i
\(55\) 149785. + 311032.i 0.900286 + 1.86946i
\(56\) −586804. − 40874.8i −3.34140 − 0.232751i
\(57\) 59178.2 + 28498.7i 0.319549 + 0.153887i
\(58\) 446900. − 67359.4i 2.29048 − 0.345234i
\(59\) 15337.2 − 22495.5i 0.0746773 − 0.109532i −0.787083 − 0.616847i \(-0.788410\pi\)
0.861760 + 0.507315i \(0.169362\pi\)
\(60\) −1.06454e6 − 160454.i −4.92843 − 0.742842i
\(61\) −34663.8 + 37358.7i −0.152717 + 0.164590i −0.804776 − 0.593578i \(-0.797715\pi\)
0.652059 + 0.758168i \(0.273905\pi\)
\(62\) 119467. + 95271.9i 0.501272 + 0.399751i
\(63\) −237978. + 139069.i −0.951732 + 0.556172i
\(64\) 612452. + 767990.i 2.33632 + 2.92965i
\(65\) −5432.23 + 72488.1i −0.0197805 + 0.263953i
\(66\) −391732. − 1.26996e6i −1.36257 − 4.41733i
\(67\) −21617.5 − 37442.7i −0.0718757 − 0.124492i 0.827848 − 0.560953i \(-0.189565\pi\)
−0.899723 + 0.436461i \(0.856232\pi\)
\(68\) 178314. + 102949.i 0.567098 + 0.327414i
\(69\) 497814. + 113623.i 1.51537 + 0.345874i
\(70\) −181183. + 813500.i −0.528231 + 2.37172i
\(71\) −127314. − 557798.i −0.355713 − 1.55848i −0.763748 − 0.645515i \(-0.776643\pi\)
0.408034 − 0.912967i \(-0.366214\pi\)
\(72\) 1.31689e6 + 406208.i 3.52820 + 1.08831i
\(73\) −459018. − 180151.i −1.17994 − 0.463094i −0.307377 − 0.951588i \(-0.599451\pi\)
−0.872567 + 0.488494i \(0.837546\pi\)
\(74\) 25629.3 − 65302.4i 0.0632473 − 0.161151i
\(75\) −104824. + 339830.i −0.248471 + 0.805522i
\(76\) −286160. + 65314.2i −0.651881 + 0.148788i
\(77\) −733509. + 171479.i −1.60670 + 0.375612i
\(78\) 62271.0 − 272827.i 0.131221 − 0.574915i
\(79\) −15902.7 + 27544.3i −0.0322545 + 0.0558665i −0.881702 − 0.471807i \(-0.843602\pi\)
0.849448 + 0.527673i \(0.176935\pi\)
\(80\) 2.08457e6 − 1.20353e6i 4.07142 − 2.35064i
\(81\) −450549. + 138976.i −0.847788 + 0.261508i
\(82\) −169796. − 12724.4i −0.307953 − 0.0230779i
\(83\) −449830. + 358727.i −0.786709 + 0.627379i −0.932185 − 0.361983i \(-0.882100\pi\)
0.145476 + 0.989362i \(0.453529\pi\)
\(84\) 846727. − 2.19122e6i 1.42858 − 3.69699i
\(85\) 115349. − 144643.i 0.187827 − 0.235527i
\(86\) 1.02970e6 + 955418.i 1.61888 + 1.50210i
\(87\) −170594. + 1.13182e6i −0.259064 + 1.71877i
\(88\) 3.11187e6 + 2.12164e6i 4.56640 + 3.11332i
\(89\) −136498. − 905608.i −0.193623 − 1.28461i −0.848631 − 0.528985i \(-0.822573\pi\)
0.655008 − 0.755622i \(-0.272665\pi\)
\(90\) 847202. − 1.75923e6i 1.16214 − 2.41322i
\(91\) −151813. − 45955.8i −0.201458 − 0.0609840i
\(92\) −2.05584e6 + 990038.i −2.64013 + 1.27142i
\(93\) −319748. + 218000.i −0.397519 + 0.271024i
\(94\) −1.51562e6 − 1.63345e6i −1.82476 − 1.96663i
\(95\) 19708.9 + 262997.i 0.0229875 + 0.306747i
\(96\) −4.62622e6 + 1.81566e6i −5.22893 + 2.05220i
\(97\) 96272.4i 0.105484i 0.998608 + 0.0527420i \(0.0167961\pi\)
−0.998608 + 0.0527420i \(0.983204\pi\)
\(98\) −1.64670e6 − 771793.i −1.74959 − 0.820017i
\(99\) 1.76483e6 1.81886
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.33.1 yes 324
49.3 odd 42 inner 49.7.h.a.3.1 ✓ 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.1 ✓ 324 49.3 odd 42 inner
49.7.h.a.33.1 yes 324 1.1 even 1 trivial