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 33.17
Character \(\chi\) \(=\) 49.33
Dual form 49.7.h.a.3.17

$q$-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+(1.55186 + 3.95409i) q^{2} +(1.47718 - 0.110699i) q^{3} +(33.6888 - 31.2586i) q^{4} +(-51.2166 - 75.1210i) q^{5} +(2.73010 + 5.66911i) q^{6} +(-331.151 - 89.3766i) q^{7} +(420.812 + 202.652i) q^{8} +(-718.688 + 108.325i) q^{9} +(217.554 - 319.093i) q^{10} +(-2476.89 - 373.332i) q^{11} +(46.3042 - 49.9040i) q^{12} +(1811.95 + 1444.98i) q^{13} +(-160.498 - 1448.10i) q^{14} +(-83.9721 - 105.298i) q^{15} +(71.5377 - 954.604i) q^{16} +(-1701.58 - 5516.40i) q^{17} +(-1543.63 - 2673.65i) q^{18} +(-3472.86 - 2005.06i) q^{19} +(-4073.61 - 929.775i) q^{20} +(-499.064 - 95.3672i) q^{21} +(-2367.62 - 10373.2i) q^{22} +(10342.6 + 3190.27i) q^{23} +(644.049 + 252.770i) q^{24} +(2688.43 - 6850.01i) q^{25} +(-2901.68 + 9407.01i) q^{26} +(-2102.45 + 479.871i) q^{27} +(-13949.9 + 7340.33i) q^{28} +(6955.57 - 30474.3i) q^{29} +(286.043 - 495.441i) q^{30} +(-41091.9 + 23724.4i) q^{31} +(32449.8 - 10009.4i) q^{32} +(-3700.15 - 277.288i) q^{33} +(19171.7 - 15288.9i) q^{34} +(10246.4 + 29454.0i) q^{35} +(-20825.6 + 26114.5i) q^{36} +(27532.4 + 25546.3i) q^{37} +(2538.76 - 16843.6i) q^{38} +(2836.53 + 1933.92i) q^{39} +(-6329.12 - 41991.0i) q^{40} +(-16084.4 + 33399.6i) q^{41} +(-397.389 - 2121.34i) q^{42} +(34844.9 - 16780.4i) q^{43} +(-95113.4 + 64847.2i) q^{44} +(44946.2 + 48440.5i) q^{45} +(3435.71 + 45846.3i) q^{46} +(18829.3 - 7389.95i) q^{47} -1418.04i q^{48} +(101673. + 59194.3i) q^{49} +31257.6 q^{50} +(-3124.21 - 7960.36i) q^{51} +(106210. - 7959.37i) q^{52} +(13077.3 - 12133.9i) q^{53} +(-5160.17 - 7568.58i) q^{54} +(98813.1 + 205188. i) q^{55} +(-121240. - 104719. i) q^{56} +(-5352.00 - 2577.39i) q^{57} +(131292. - 19789.1i) q^{58} +(112897. - 165590. i) q^{59} +(-6120.38 - 922.500i) q^{60} +(-274770. + 296131. i) q^{61} +(-157578. - 125664. i) q^{62} +(247676. + 28362.0i) q^{63} +(51737.1 + 64876.3i) q^{64} +(15746.5 - 210122. i) q^{65} +(-4645.71 - 15061.0i) q^{66} +(-233896. - 405120. i) q^{67} +(-229760. - 132652. i) q^{68} +(15631.0 + 3567.68i) q^{69} +(-100563. + 86223.6i) q^{70} +(-49256.6 - 215807. i) q^{71} +(-324384. - 100059. i) q^{72} +(-172062. - 67529.5i) q^{73} +(-58285.9 + 148510. i) q^{74} +(3213.01 - 10416.3i) q^{75} +(-179672. + 41008.9i) q^{76} +(786858. + 345005. i) q^{77} +(-3244.96 + 14217.1i) q^{78} +(131003. - 226905. i) q^{79} +(-75374.7 + 43517.6i) q^{80} +(503249. - 155232. i) q^{81} +(-157026. - 11767.5i) q^{82} +(-552296. + 440441. i) q^{83} +(-19793.9 + 12387.2i) q^{84} +(-327248. + 410356. i) q^{85} +(120426. + 111739. i) q^{86} +(6901.15 - 45786.1i) q^{87} +(-966649. - 659050. i) q^{88} +(-197983. - 1.31353e6i) q^{89} +(-121788. + 252895. i) q^{90} +(-470880. - 640452. i) q^{91} +(448153. - 215819. i) q^{92} +(-58074.0 + 39594.2i) q^{93} +(58441.0 + 62984.4i) q^{94} +(27246.3 + 363577. i) q^{95} +(46826.2 - 18377.9i) q^{96} +516078. i q^{97} +(-76277.0 + 493884. i) q^{98} +1.82055e6 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.55186 + 3.95409i 0.193983 + 0.494261i 0.994462 − 0.105101i \(-0.0335165\pi\)
−0.800479 + 0.599361i \(0.795421\pi\)
\(3\) 1.47718 − 0.110699i 0.0547104 − 0.00409998i −0.0473468 − 0.998879i \(-0.515077\pi\)
0.102057 + 0.994779i \(0.467458\pi\)
\(4\) 33.6888 − 31.2586i 0.526388 − 0.488416i
\(5\) −51.2166 − 75.1210i −0.409733 − 0.600968i 0.564796 − 0.825231i \(-0.308955\pi\)
−0.974529 + 0.224263i \(0.928003\pi\)
\(6\) 2.73010 + 5.66911i 0.0126394 + 0.0262459i
\(7\) −331.151 − 89.3766i −0.965454 − 0.260573i
\(8\) 420.812 + 202.652i 0.821898 + 0.395805i
\(9\) −718.688 + 108.325i −0.985854 + 0.148594i
\(10\) 217.554 − 319.093i 0.217554 − 0.319093i
\(11\) −2476.89 − 373.332i −1.86093 − 0.280490i −0.880015 − 0.474945i \(-0.842468\pi\)
−0.980911 + 0.194455i \(0.937706\pi\)
\(12\) 46.3042 − 49.9040i 0.0267964 − 0.0288796i
\(13\) 1811.95 + 1444.98i 0.824737 + 0.657706i 0.942081 − 0.335385i \(-0.108867\pi\)
−0.117344 + 0.993091i \(0.537438\pi\)
\(14\) −160.498 − 1448.10i −0.0584907 − 0.527733i
\(15\) −83.9721 − 105.298i −0.0248806 − 0.0311993i
\(16\) 71.5377 − 954.604i 0.0174652 − 0.233058i
\(17\) −1701.58 − 5516.40i −0.346343 − 1.12282i −0.946437 − 0.322888i \(-0.895346\pi\)
0.600094 − 0.799929i \(-0.295130\pi\)
\(18\) −1543.63 − 2673.65i −0.264683 − 0.458445i
\(19\) −3472.86 − 2005.06i −0.506322 − 0.292325i 0.224999 − 0.974359i \(-0.427762\pi\)
−0.731320 + 0.682034i \(0.761096\pi\)
\(20\) −4073.61 − 929.775i −0.509201 − 0.116222i
\(21\) −499.064 − 95.3672i −0.0538888 − 0.0102977i
\(22\) −2367.62 − 10373.2i −0.222353 − 0.974194i
\(23\) 10342.6 + 3190.27i 0.850052 + 0.262206i 0.689020 − 0.724743i \(-0.258041\pi\)
0.161033 + 0.986949i \(0.448518\pi\)
\(24\) 644.049 + 252.770i 0.0465892 + 0.0182849i
\(25\) 2688.43 − 6850.01i 0.172060 − 0.438401i
\(26\) −2901.68 + 9407.01i −0.165093 + 0.535219i
\(27\) −2102.45 + 479.871i −0.106816 + 0.0243800i
\(28\) −13949.9 + 7340.33i −0.635471 + 0.334381i
\(29\) 6955.57 − 30474.3i 0.285193 − 1.24951i −0.605845 − 0.795583i \(-0.707165\pi\)
0.891038 − 0.453929i \(-0.149978\pi\)
\(30\) 286.043 − 495.441i 0.0105942 − 0.0183497i
\(31\) −41091.9 + 23724.4i −1.37934 + 0.796362i −0.992080 − 0.125608i \(-0.959912\pi\)
−0.387260 + 0.921970i \(0.626579\pi\)
\(32\) 32449.8 − 10009.4i 0.990289 − 0.305464i
\(33\) −3700.15 − 277.288i −0.102962 − 0.00771594i
\(34\) 19171.7 − 15288.9i 0.487780 − 0.388991i
\(35\) 10246.4 + 29454.0i 0.238982 + 0.686973i
\(36\) −20825.6 + 26114.5i −0.446366 + 0.559725i
\(37\) 27532.4 + 25546.3i 0.543549 + 0.504340i 0.903475 − 0.428641i \(-0.141007\pi\)
−0.359926 + 0.932981i \(0.617198\pi\)
\(38\) 2538.76 − 16843.6i 0.0462669 − 0.306961i
\(39\) 2836.53 + 1933.92i 0.0478183 + 0.0326020i
\(40\) −6329.12 − 41991.0i −0.0988925 − 0.656109i
\(41\) −16084.4 + 33399.6i −0.233375 + 0.484607i −0.984463 − 0.175595i \(-0.943815\pi\)
0.751088 + 0.660202i \(0.229529\pi\)
\(42\) −397.389 − 2121.34i −0.00536374 − 0.0286327i
\(43\) 34844.9 − 16780.4i 0.438262 − 0.211056i −0.201724 − 0.979442i \(-0.564654\pi\)
0.639986 + 0.768386i \(0.278940\pi\)
\(44\) −95113.4 + 64847.2i −1.11656 + 0.761261i
\(45\) 44946.2 + 48440.5i 0.493237 + 0.531583i
\(46\) 3435.71 + 45846.3i 0.0352974 + 0.471011i
\(47\) 18829.3 − 7389.95i 0.181359 − 0.0711783i −0.272924 − 0.962036i \(-0.587991\pi\)
0.454283 + 0.890857i \(0.349895\pi\)
\(48\) − 1418.04i − 0.0128223i
\(49\) 101673. + 59194.3i 0.864203 + 0.503143i
\(50\) 31257.6 0.250061
\(51\) −3124.21 − 7960.36i −0.0235521 − 0.0600098i
\(52\) 106210. − 7959.37i 0.755366 − 0.0566068i
\(53\) 13077.3 − 12133.9i 0.0878395 − 0.0815031i −0.635031 − 0.772487i \(-0.719013\pi\)
0.722871 + 0.690983i \(0.242822\pi\)
\(54\) −5160.17 − 7568.58i −0.0327705 − 0.0480655i
\(55\) 98813.1 + 205188.i 0.593918 + 1.23328i
\(56\) −121240. − 104719.i −0.690368 − 0.596296i
\(57\) −5352.00 − 2577.39i −0.0288996 − 0.0139173i
\(58\) 131292. − 19789.1i 0.672907 − 0.101424i
\(59\) 112897. − 165590.i 0.549702 − 0.806265i −0.446366 − 0.894850i \(-0.647282\pi\)
0.996069 + 0.0885853i \(0.0282346\pi\)
\(60\) −6120.38 − 922.500i −0.0283351 − 0.00427083i
\(61\) −274770. + 296131.i −1.21054 + 1.30465i −0.270852 + 0.962621i \(0.587305\pi\)
−0.939689 + 0.342031i \(0.888885\pi\)
\(62\) −157578. − 125664.i −0.661179 − 0.527273i
\(63\) 247676. + 28362.0i 0.990517 + 0.113427i
\(64\) 51737.1 + 64876.3i 0.197362 + 0.247484i
\(65\) 15746.5 − 210122.i 0.0573382 − 0.765125i
\(66\) −4645.71 − 15061.0i −0.0161592 − 0.0523869i
\(67\) −233896. − 405120.i −0.777677 − 1.34698i −0.933278 − 0.359156i \(-0.883065\pi\)
0.155601 − 0.987820i \(-0.450269\pi\)
\(68\) −229760. − 132652.i −0.730713 − 0.421877i
\(69\) 15631.0 + 3567.68i 0.0475818 + 0.0108602i
\(70\) −100563. + 86223.6i −0.293185 + 0.251381i
\(71\) −49256.6 − 215807.i −0.137622 − 0.602964i −0.995954 − 0.0898675i \(-0.971356\pi\)
0.858331 − 0.513096i \(-0.171502\pi\)
\(72\) −324384. − 100059.i −0.869086 − 0.268077i
\(73\) −172062. − 67529.5i −0.442301 − 0.173590i 0.133731 − 0.991018i \(-0.457304\pi\)
−0.576031 + 0.817427i \(0.695399\pi\)
\(74\) −58285.9 + 148510.i −0.143836 + 0.366489i
\(75\) 3213.01 − 10416.3i 0.00761602 − 0.0246905i
\(76\) −179672. + 41008.9i −0.409298 + 0.0934195i
\(77\) 786858. + 345005.i 1.72355 + 0.755707i
\(78\) −3244.96 + 14217.1i −0.00683793 + 0.0299589i
\(79\) 131003. − 226905.i 0.265706 − 0.460216i −0.702042 − 0.712135i \(-0.747728\pi\)
0.967748 + 0.251919i \(0.0810616\pi\)
\(80\) −75374.7 + 43517.6i −0.147216 + 0.0849953i
\(81\) 503249. − 155232.i 0.946953 − 0.292096i
\(82\) −157026. − 11767.5i −0.284793 − 0.0213423i
\(83\) −552296. + 440441.i −0.965912 + 0.770289i −0.973265 − 0.229684i \(-0.926231\pi\)
0.00735361 + 0.999973i \(0.497659\pi\)
\(84\) −19793.9 + 12387.2i −0.0333959 + 0.0208995i
\(85\) −327248. + 410356.i −0.532869 + 0.668197i
\(86\) 120426. + 111739.i 0.189332 + 0.175675i
\(87\) 6901.15 − 45786.1i 0.0104801 − 0.0695306i
\(88\) −966649. − 659050.i −1.41847 − 0.967098i
\(89\) −197983. − 1.31353e6i −0.280840 − 1.86325i −0.471873 − 0.881667i \(-0.656422\pi\)
0.191033 − 0.981584i \(-0.438816\pi\)
\(90\) −121788. + 252895.i −0.167061 + 0.346906i
\(91\) −470880. − 640452.i −0.624865 − 0.849889i
\(92\) 448153. − 215819.i 0.575523 − 0.277157i
\(93\) −58074.0 + 39594.2i −0.0721992 + 0.0492246i
\(94\) 58441.0 + 62984.4i 0.0703613 + 0.0758315i
\(95\) 27246.3 + 363577.i 0.0317788 + 0.424059i
\(96\) 46826.2 − 18377.9i 0.0529267 − 0.0207722i
\(97\) 516078.i 0.565458i 0.959200 + 0.282729i \(0.0912397\pi\)
−0.959200 + 0.282729i \(0.908760\pi\)
\(98\) −76277.0 + 493884.i −0.0810430 + 0.524743i
\(99\) 1.82055e6 1.87628
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.33.17 yes 324
49.3 odd 42 inner 49.7.h.a.3.17 ✓ 324
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.7.h.a.3.17 ✓ 324 49.3 odd 42 inner
49.7.h.a.33.17 yes 324 1.1 even 1 trivial