# Properties

 Label 507.4.b.f.337.2 Level $507$ Weight $4$ Character 507.337 Analytic conductor $29.914$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 49$$ x^4 + 49 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.2 Root $$-1.87083 - 1.87083i$$ of defining polynomial Character $$\chi$$ $$=$$ 507.337 Dual form 507.4.b.f.337.3

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.74166i q^{2} -3.00000 q^{3} +0.483315 q^{4} +19.4833i q^{5} +8.22497i q^{6} -7.48331i q^{7} -23.2583i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.74166i q^{2} -3.00000 q^{3} +0.483315 q^{4} +19.4833i q^{5} +8.22497i q^{6} -7.48331i q^{7} -23.2583i q^{8} +9.00000 q^{9} +53.4166 q^{10} -22.8999i q^{11} -1.44994 q^{12} -20.5167 q^{14} -58.4499i q^{15} -59.8999 q^{16} -67.0334 q^{17} -24.6749i q^{18} +16.5167i q^{19} +9.41657i q^{20} +22.4499i q^{21} -62.7836 q^{22} +175.600 q^{23} +69.7750i q^{24} -254.600 q^{25} -27.0000 q^{27} -3.61680i q^{28} +291.800 q^{29} -160.250 q^{30} +117.283i q^{31} -21.8418i q^{32} +68.6997i q^{33} +183.783i q^{34} +145.800 q^{35} +4.34983 q^{36} +154.766i q^{37} +45.2831 q^{38} +453.150 q^{40} -251.716i q^{41} +61.5501 q^{42} +502.566 q^{43} -11.0679i q^{44} +175.350i q^{45} -481.434i q^{46} +281.733i q^{47} +179.700 q^{48} +287.000 q^{49} +698.025i q^{50} +201.100 q^{51} +366.999 q^{53} +74.0247i q^{54} +446.166 q^{55} -174.049 q^{56} -49.5501i q^{57} -800.015i q^{58} +79.6663i q^{59} -28.2497i q^{60} -194.865 q^{61} +321.550 q^{62} -67.3498i q^{63} -539.082 q^{64} +188.351 q^{66} +400.082i q^{67} -32.3982 q^{68} -526.799 q^{69} -399.733i q^{70} +528.299i q^{71} -209.325i q^{72} +734.366i q^{73} +424.316 q^{74} +763.799 q^{75} +7.98276i q^{76} -171.367 q^{77} +113.266 q^{79} -1167.05i q^{80} +81.0000 q^{81} -690.118 q^{82} -933.466i q^{83} +10.8504i q^{84} -1306.03i q^{85} -1377.86i q^{86} -875.399 q^{87} -532.613 q^{88} -1190.91i q^{89} +480.749 q^{90} +84.8699 q^{92} -351.849i q^{93} +772.415 q^{94} -321.800 q^{95} +65.5253i q^{96} +557.165i q^{97} -786.856i q^{98} -206.099i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} - 28 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9 $$4 q - 12 q^{3} - 28 q^{4} + 36 q^{9} + 64 q^{10} + 84 q^{12} - 112 q^{14} - 60 q^{16} - 328 q^{17} - 760 q^{22} - 16 q^{23} - 300 q^{25} - 108 q^{27} + 808 q^{29} - 192 q^{30} + 224 q^{35} - 252 q^{36} - 208 q^{38} + 1184 q^{40} + 336 q^{42} + 1232 q^{43} + 180 q^{48} + 1148 q^{49} + 984 q^{51} - 328 q^{53} + 288 q^{55} + 112 q^{56} + 1256 q^{61} + 1376 q^{62} + 388 q^{64} + 2280 q^{66} + 2744 q^{68} + 48 q^{69} + 1368 q^{74} + 900 q^{75} - 1344 q^{77} - 864 q^{79} + 324 q^{81} - 5664 q^{82} - 2424 q^{87} + 3048 q^{88} + 576 q^{90} + 5488 q^{92} + 1144 q^{94} - 928 q^{95}+O(q^{100})$$ 4 * q - 12 * q^3 - 28 * q^4 + 36 * q^9 + 64 * q^10 + 84 * q^12 - 112 * q^14 - 60 * q^16 - 328 * q^17 - 760 * q^22 - 16 * q^23 - 300 * q^25 - 108 * q^27 + 808 * q^29 - 192 * q^30 + 224 * q^35 - 252 * q^36 - 208 * q^38 + 1184 * q^40 + 336 * q^42 + 1232 * q^43 + 180 * q^48 + 1148 * q^49 + 984 * q^51 - 328 * q^53 + 288 * q^55 + 112 * q^56 + 1256 * q^61 + 1376 * q^62 + 388 * q^64 + 2280 * q^66 + 2744 * q^68 + 48 * q^69 + 1368 * q^74 + 900 * q^75 - 1344 * q^77 - 864 * q^79 + 324 * q^81 - 5664 * q^82 - 2424 * q^87 + 3048 * q^88 + 576 * q^90 + 5488 * q^92 + 1144 * q^94 - 928 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.74166i − 0.969322i −0.874702 0.484661i $$-0.838943\pi$$
0.874702 0.484661i $$-0.161057\pi$$
$$3$$ −3.00000 −0.577350
$$4$$ 0.483315 0.0604143
$$5$$ 19.4833i 1.74264i 0.490715 + 0.871320i $$0.336736\pi$$
−0.490715 + 0.871320i $$0.663264\pi$$
$$6$$ 8.22497i 0.559638i
$$7$$ − 7.48331i − 0.404061i −0.979379 0.202031i $$-0.935246\pi$$
0.979379 0.202031i $$-0.0647540\pi$$
$$8$$ − 23.2583i − 1.02788i
$$9$$ 9.00000 0.333333
$$10$$ 53.4166 1.68918
$$11$$ − 22.8999i − 0.627689i −0.949474 0.313844i $$-0.898383\pi$$
0.949474 0.313844i $$-0.101617\pi$$
$$12$$ −1.44994 −0.0348802
$$13$$ 0 0
$$14$$ −20.5167 −0.391665
$$15$$ − 58.4499i − 1.00611i
$$16$$ −59.8999 −0.935936
$$17$$ −67.0334 −0.956352 −0.478176 0.878264i $$-0.658702\pi$$
−0.478176 + 0.878264i $$0.658702\pi$$
$$18$$ − 24.6749i − 0.323107i
$$19$$ 16.5167i 0.199431i 0.995016 + 0.0997155i $$0.0317933\pi$$
−0.995016 + 0.0997155i $$0.968207\pi$$
$$20$$ 9.41657i 0.105280i
$$21$$ 22.4499i 0.233285i
$$22$$ −62.7836 −0.608433
$$23$$ 175.600 1.59196 0.795979 0.605324i $$-0.206956\pi$$
0.795979 + 0.605324i $$0.206956\pi$$
$$24$$ 69.7750i 0.593449i
$$25$$ −254.600 −2.03680
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ − 3.61680i − 0.0244111i
$$29$$ 291.800 1.86848 0.934239 0.356648i $$-0.116080\pi$$
0.934239 + 0.356648i $$0.116080\pi$$
$$30$$ −160.250 −0.975249
$$31$$ 117.283i 0.679505i 0.940515 + 0.339753i $$0.110343\pi$$
−0.940515 + 0.339753i $$0.889657\pi$$
$$32$$ − 21.8418i − 0.120660i
$$33$$ 68.6997i 0.362396i
$$34$$ 183.783i 0.927013i
$$35$$ 145.800 0.704133
$$36$$ 4.34983 0.0201381
$$37$$ 154.766i 0.687661i 0.939032 + 0.343830i $$0.111724\pi$$
−0.939032 + 0.343830i $$0.888276\pi$$
$$38$$ 45.2831 0.193313
$$39$$ 0 0
$$40$$ 453.150 1.79123
$$41$$ − 251.716i − 0.958815i −0.877592 0.479407i $$-0.840852\pi$$
0.877592 0.479407i $$-0.159148\pi$$
$$42$$ 61.5501 0.226128
$$43$$ 502.566 1.78234 0.891170 0.453669i $$-0.149885\pi$$
0.891170 + 0.453669i $$0.149885\pi$$
$$44$$ − 11.0679i − 0.0379214i
$$45$$ 175.350i 0.580880i
$$46$$ − 481.434i − 1.54312i
$$47$$ 281.733i 0.874361i 0.899374 + 0.437181i $$0.144023\pi$$
−0.899374 + 0.437181i $$0.855977\pi$$
$$48$$ 179.700 0.540363
$$49$$ 287.000 0.836735
$$50$$ 698.025i 1.97431i
$$51$$ 201.100 0.552150
$$52$$ 0 0
$$53$$ 366.999 0.951154 0.475577 0.879674i $$-0.342239\pi$$
0.475577 + 0.879674i $$0.342239\pi$$
$$54$$ 74.0247i 0.186546i
$$55$$ 446.166 1.09384
$$56$$ −174.049 −0.415328
$$57$$ − 49.5501i − 0.115141i
$$58$$ − 800.015i − 1.81116i
$$59$$ 79.6663i 0.175791i 0.996130 + 0.0878955i $$0.0280142\pi$$
−0.996130 + 0.0878955i $$0.971986\pi$$
$$60$$ − 28.2497i − 0.0607837i
$$61$$ −194.865 −0.409016 −0.204508 0.978865i $$-0.565559\pi$$
−0.204508 + 0.978865i $$0.565559\pi$$
$$62$$ 321.550 0.658660
$$63$$ − 67.3498i − 0.134687i
$$64$$ −539.082 −1.05289
$$65$$ 0 0
$$66$$ 188.351 0.351279
$$67$$ 400.082i 0.729519i 0.931102 + 0.364759i $$0.118849\pi$$
−0.931102 + 0.364759i $$0.881151\pi$$
$$68$$ −32.3982 −0.0577774
$$69$$ −526.799 −0.919117
$$70$$ − 399.733i − 0.682532i
$$71$$ 528.299i 0.883065i 0.897245 + 0.441532i $$0.145565\pi$$
−0.897245 + 0.441532i $$0.854435\pi$$
$$72$$ − 209.325i − 0.342628i
$$73$$ 734.366i 1.17741i 0.808347 + 0.588706i $$0.200362\pi$$
−0.808347 + 0.588706i $$0.799638\pi$$
$$74$$ 424.316 0.666565
$$75$$ 763.799 1.17594
$$76$$ 7.98276i 0.0120485i
$$77$$ −171.367 −0.253625
$$78$$ 0 0
$$79$$ 113.266 0.161309 0.0806545 0.996742i $$-0.474299\pi$$
0.0806545 + 0.996742i $$0.474299\pi$$
$$80$$ − 1167.05i − 1.63100i
$$81$$ 81.0000 0.111111
$$82$$ −690.118 −0.929400
$$83$$ − 933.466i − 1.23447i −0.786778 0.617236i $$-0.788252\pi$$
0.786778 0.617236i $$-0.211748\pi$$
$$84$$ 10.8504i 0.0140937i
$$85$$ − 1306.03i − 1.66658i
$$86$$ − 1377.86i − 1.72766i
$$87$$ −875.399 −1.07877
$$88$$ −532.613 −0.645191
$$89$$ − 1190.91i − 1.41839i −0.705012 0.709195i $$-0.749059\pi$$
0.705012 0.709195i $$-0.250941\pi$$
$$90$$ 480.749 0.563060
$$91$$ 0 0
$$92$$ 84.8699 0.0961771
$$93$$ − 351.849i − 0.392313i
$$94$$ 772.415 0.847538
$$95$$ −321.800 −0.347536
$$96$$ 65.5253i 0.0696630i
$$97$$ 557.165i 0.583211i 0.956539 + 0.291606i $$0.0941895\pi$$
−0.956539 + 0.291606i $$0.905811\pi$$
$$98$$ − 786.856i − 0.811066i
$$99$$ − 206.099i − 0.209230i
$$100$$ −123.052 −0.123052
$$101$$ 286.766 0.282518 0.141259 0.989973i $$-0.454885\pi$$
0.141259 + 0.989973i $$0.454885\pi$$
$$102$$ − 551.348i − 0.535211i
$$103$$ 1911.36 1.82847 0.914234 0.405187i $$-0.132794\pi$$
0.914234 + 0.405187i $$0.132794\pi$$
$$104$$ 0 0
$$105$$ −437.399 −0.406531
$$106$$ − 1006.19i − 0.921975i
$$107$$ 834.334 0.753814 0.376907 0.926251i $$-0.376988\pi$$
0.376907 + 0.926251i $$0.376988\pi$$
$$108$$ −13.0495 −0.0116267
$$109$$ − 1077.66i − 0.946986i −0.880798 0.473493i $$-0.842993\pi$$
0.880798 0.473493i $$-0.157007\pi$$
$$110$$ − 1223.23i − 1.06028i
$$111$$ − 464.299i − 0.397021i
$$112$$ 448.250i 0.378175i
$$113$$ −166.065 −0.138248 −0.0691241 0.997608i $$-0.522020\pi$$
−0.0691241 + 0.997608i $$0.522020\pi$$
$$114$$ −135.849 −0.111609
$$115$$ 3421.26i 2.77421i
$$116$$ 141.031 0.112883
$$117$$ 0 0
$$118$$ 218.418 0.170398
$$119$$ 501.632i 0.386424i
$$120$$ −1359.45 −1.03417
$$121$$ 806.595 0.606007
$$122$$ 534.254i 0.396468i
$$123$$ 755.147i 0.553572i
$$124$$ 56.6847i 0.0410519i
$$125$$ − 2525.03i − 1.80676i
$$126$$ −184.650 −0.130555
$$127$$ −1296.16 −0.905637 −0.452819 0.891603i $$-0.649581\pi$$
−0.452819 + 0.891603i $$0.649581\pi$$
$$128$$ 1303.24i 0.899934i
$$129$$ −1507.70 −1.02903
$$130$$ 0 0
$$131$$ −197.201 −0.131523 −0.0657617 0.997835i $$-0.520948\pi$$
−0.0657617 + 0.997835i $$0.520948\pi$$
$$132$$ 33.2036i 0.0218939i
$$133$$ 123.600 0.0805823
$$134$$ 1096.89 0.707139
$$135$$ − 526.049i − 0.335371i
$$136$$ 1559.09i 0.983018i
$$137$$ 546.915i 0.341066i 0.985352 + 0.170533i $$0.0545490\pi$$
−0.985352 + 0.170533i $$0.945451\pi$$
$$138$$ 1444.30i 0.890921i
$$139$$ 609.666 0.372023 0.186012 0.982548i $$-0.440444\pi$$
0.186012 + 0.982548i $$0.440444\pi$$
$$140$$ 70.4672 0.0425397
$$141$$ − 845.199i − 0.504813i
$$142$$ 1448.42 0.855974
$$143$$ 0 0
$$144$$ −539.099 −0.311979
$$145$$ 5685.23i 3.25609i
$$146$$ 2013.38 1.14129
$$147$$ −861.000 −0.483089
$$148$$ 74.8009i 0.0415446i
$$149$$ − 2165.08i − 1.19040i −0.803576 0.595202i $$-0.797072\pi$$
0.803576 0.595202i $$-0.202928\pi$$
$$150$$ − 2094.07i − 1.13987i
$$151$$ 846.549i 0.456233i 0.973634 + 0.228116i $$0.0732567\pi$$
−0.973634 + 0.228116i $$0.926743\pi$$
$$152$$ 384.151 0.204992
$$153$$ −603.300 −0.318784
$$154$$ 469.830i 0.245844i
$$155$$ −2285.06 −1.18413
$$156$$ 0 0
$$157$$ 1653.60 0.840581 0.420291 0.907390i $$-0.361928\pi$$
0.420291 + 0.907390i $$0.361928\pi$$
$$158$$ − 310.536i − 0.156360i
$$159$$ −1101.00 −0.549149
$$160$$ 425.550 0.210267
$$161$$ − 1314.07i − 0.643248i
$$162$$ − 222.074i − 0.107702i
$$163$$ 2866.51i 1.37744i 0.725027 + 0.688720i $$0.241827\pi$$
−0.725027 + 0.688720i $$0.758173\pi$$
$$164$$ − 121.658i − 0.0579262i
$$165$$ −1338.50 −0.631526
$$166$$ −2559.24 −1.19660
$$167$$ − 729.066i − 0.337825i −0.985631 0.168913i $$-0.945974\pi$$
0.985631 0.168913i $$-0.0540255\pi$$
$$168$$ 522.148 0.239789
$$169$$ 0 0
$$170$$ −3580.69 −1.61545
$$171$$ 148.650i 0.0664770i
$$172$$ 242.898 0.107679
$$173$$ 3834.83 1.68530 0.842650 0.538462i $$-0.180995\pi$$
0.842650 + 0.538462i $$0.180995\pi$$
$$174$$ 2400.05i 1.04567i
$$175$$ 1905.25i 0.822990i
$$176$$ 1371.70i 0.587476i
$$177$$ − 238.999i − 0.101493i
$$178$$ −3265.08 −1.37488
$$179$$ 283.862 0.118530 0.0592649 0.998242i $$-0.481124\pi$$
0.0592649 + 0.998242i $$0.481124\pi$$
$$180$$ 84.7492i 0.0350935i
$$181$$ −2363.60 −0.970634 −0.485317 0.874338i $$-0.661296\pi$$
−0.485317 + 0.874338i $$0.661296\pi$$
$$182$$ 0 0
$$183$$ 584.596 0.236145
$$184$$ − 4084.15i − 1.63635i
$$185$$ −3015.36 −1.19835
$$186$$ −964.650 −0.380277
$$187$$ 1535.06i 0.600291i
$$188$$ 136.166i 0.0528240i
$$189$$ 202.049i 0.0777616i
$$190$$ 882.265i 0.336875i
$$191$$ 2514.26 0.952491 0.476246 0.879312i $$-0.341997\pi$$
0.476246 + 0.879312i $$0.341997\pi$$
$$192$$ 1617.25 0.607889
$$193$$ − 2420.73i − 0.902839i −0.892312 0.451420i $$-0.850918\pi$$
0.892312 0.451420i $$-0.149082\pi$$
$$194$$ 1527.55 0.565320
$$195$$ 0 0
$$196$$ 138.711 0.0505508
$$197$$ − 4633.65i − 1.67581i −0.545819 0.837903i $$-0.683781\pi$$
0.545819 0.837903i $$-0.316219\pi$$
$$198$$ −565.053 −0.202811
$$199$$ −3054.17 −1.08796 −0.543980 0.839098i $$-0.683083\pi$$
−0.543980 + 0.839098i $$0.683083\pi$$
$$200$$ 5921.56i 2.09359i
$$201$$ − 1200.25i − 0.421188i
$$202$$ − 786.215i − 0.273851i
$$203$$ − 2183.63i − 0.754979i
$$204$$ 97.1947 0.0333578
$$205$$ 4904.26 1.67087
$$206$$ − 5240.30i − 1.77237i
$$207$$ 1580.40 0.530653
$$208$$ 0 0
$$209$$ 378.230 0.125181
$$210$$ 1199.20i 0.394060i
$$211$$ −4031.60 −1.31539 −0.657694 0.753285i $$-0.728468\pi$$
−0.657694 + 0.753285i $$0.728468\pi$$
$$212$$ 177.376 0.0574634
$$213$$ − 1584.90i − 0.509838i
$$214$$ − 2287.46i − 0.730689i
$$215$$ 9791.66i 3.10598i
$$216$$ 627.975i 0.197816i
$$217$$ 877.666 0.274562
$$218$$ −2954.59 −0.917935
$$219$$ − 2203.10i − 0.679779i
$$220$$ 215.638 0.0660834
$$221$$ 0 0
$$222$$ −1272.95 −0.384841
$$223$$ 3784.95i 1.13659i 0.822826 + 0.568294i $$0.192396\pi$$
−0.822826 + 0.568294i $$0.807604\pi$$
$$224$$ −163.449 −0.0487539
$$225$$ −2291.40 −0.678932
$$226$$ 455.292i 0.134007i
$$227$$ 2013.83i 0.588821i 0.955679 + 0.294411i $$0.0951233\pi$$
−0.955679 + 0.294411i $$0.904877\pi$$
$$228$$ − 23.9483i − 0.00695620i
$$229$$ 3050.73i 0.880340i 0.897915 + 0.440170i $$0.145082\pi$$
−0.897915 + 0.440170i $$0.854918\pi$$
$$230$$ 9379.93 2.68910
$$231$$ 514.101 0.146430
$$232$$ − 6786.78i − 1.92058i
$$233$$ −5587.49 −1.57103 −0.785513 0.618846i $$-0.787601\pi$$
−0.785513 + 0.618846i $$0.787601\pi$$
$$234$$ 0 0
$$235$$ −5489.09 −1.52370
$$236$$ 38.5039i 0.0106203i
$$237$$ −339.798 −0.0931317
$$238$$ 1375.30 0.374570
$$239$$ − 1335.69i − 0.361501i −0.983529 0.180750i $$-0.942147\pi$$
0.983529 0.180750i $$-0.0578526\pi$$
$$240$$ 3501.15i 0.941658i
$$241$$ 571.558i 0.152769i 0.997078 + 0.0763845i $$0.0243376\pi$$
−0.997078 + 0.0763845i $$0.975662\pi$$
$$242$$ − 2211.41i − 0.587416i
$$243$$ −243.000 −0.0641500
$$244$$ −94.1813 −0.0247104
$$245$$ 5591.71i 1.45813i
$$246$$ 2070.36 0.536590
$$247$$ 0 0
$$248$$ 2727.81 0.698452
$$249$$ 2800.40i 0.712723i
$$250$$ −6922.76 −1.75134
$$251$$ −4088.60 −1.02817 −0.514084 0.857740i $$-0.671868\pi$$
−0.514084 + 0.857740i $$0.671868\pi$$
$$252$$ − 32.5512i − 0.00813703i
$$253$$ − 4021.21i − 0.999254i
$$254$$ 3553.64i 0.877854i
$$255$$ 3918.10i 0.962199i
$$256$$ −739.607 −0.180568
$$257$$ −3050.23 −0.740342 −0.370171 0.928964i $$-0.620701\pi$$
−0.370171 + 0.928964i $$0.620701\pi$$
$$258$$ 4133.59i 0.997466i
$$259$$ 1158.17 0.277857
$$260$$ 0 0
$$261$$ 2626.20 0.622826
$$262$$ 540.659i 0.127489i
$$263$$ 5770.99 1.35306 0.676530 0.736415i $$-0.263483\pi$$
0.676530 + 0.736415i $$0.263483\pi$$
$$264$$ 1597.84 0.372501
$$265$$ 7150.35i 1.65752i
$$266$$ − 338.868i − 0.0781102i
$$267$$ 3572.74i 0.818908i
$$268$$ 193.365i 0.0440734i
$$269$$ −2079.40 −0.471314 −0.235657 0.971836i $$-0.575724\pi$$
−0.235657 + 0.971836i $$0.575724\pi$$
$$270$$ −1442.25 −0.325083
$$271$$ − 6012.00i − 1.34761i −0.738908 0.673807i $$-0.764658\pi$$
0.738908 0.673807i $$-0.235342\pi$$
$$272$$ 4015.29 0.895084
$$273$$ 0 0
$$274$$ 1499.45 0.330603
$$275$$ 5830.30i 1.27847i
$$276$$ −254.610 −0.0555279
$$277$$ 735.201 0.159473 0.0797364 0.996816i $$-0.474592\pi$$
0.0797364 + 0.996816i $$0.474592\pi$$
$$278$$ − 1671.50i − 0.360610i
$$279$$ 1055.55i 0.226502i
$$280$$ − 3391.06i − 0.723767i
$$281$$ 1902.92i 0.403981i 0.979387 + 0.201990i $$0.0647410\pi$$
−0.979387 + 0.201990i $$0.935259\pi$$
$$282$$ −2317.25 −0.489326
$$283$$ −2125.71 −0.446502 −0.223251 0.974761i $$-0.571667\pi$$
−0.223251 + 0.974761i $$0.571667\pi$$
$$284$$ 255.335i 0.0533498i
$$285$$ 965.399 0.200650
$$286$$ 0 0
$$287$$ −1883.67 −0.387420
$$288$$ − 196.576i − 0.0402200i
$$289$$ −419.527 −0.0853913
$$290$$ 15586.9 3.15620
$$291$$ − 1671.49i − 0.336717i
$$292$$ 354.930i 0.0711325i
$$293$$ 1641.03i 0.327200i 0.986527 + 0.163600i $$0.0523107\pi$$
−0.986527 + 0.163600i $$0.947689\pi$$
$$294$$ 2360.57i 0.468269i
$$295$$ −1552.16 −0.306341
$$296$$ 3599.61 0.706835
$$297$$ 618.297i 0.120799i
$$298$$ −5935.91 −1.15389
$$299$$ 0 0
$$300$$ 369.155 0.0710439
$$301$$ − 3760.86i − 0.720174i
$$302$$ 2320.95 0.442237
$$303$$ −860.299 −0.163112
$$304$$ − 989.348i − 0.186655i
$$305$$ − 3796.62i − 0.712767i
$$306$$ 1654.04i 0.309004i
$$307$$ 3373.27i 0.627111i 0.949570 + 0.313555i $$0.101520\pi$$
−0.949570 + 0.313555i $$0.898480\pi$$
$$308$$ −82.8242 −0.0153226
$$309$$ −5734.09 −1.05567
$$310$$ 6264.86i 1.14781i
$$311$$ 868.525 0.158359 0.0791793 0.996860i $$-0.474770\pi$$
0.0791793 + 0.996860i $$0.474770\pi$$
$$312$$ 0 0
$$313$$ −4343.19 −0.784319 −0.392159 0.919897i $$-0.628272\pi$$
−0.392159 + 0.919897i $$0.628272\pi$$
$$314$$ − 4533.59i − 0.814794i
$$315$$ 1312.20 0.234711
$$316$$ 54.7431 0.00974537
$$317$$ − 3277.65i − 0.580730i −0.956916 0.290365i $$-0.906223\pi$$
0.956916 0.290365i $$-0.0937767\pi$$
$$318$$ 3018.56i 0.532302i
$$319$$ − 6682.18i − 1.17282i
$$320$$ − 10503.1i − 1.83482i
$$321$$ −2503.00 −0.435215
$$322$$ −3602.72 −0.623515
$$323$$ − 1107.17i − 0.190726i
$$324$$ 39.1485 0.00671271
$$325$$ 0 0
$$326$$ 7859.00 1.33518
$$327$$ 3232.99i 0.546743i
$$328$$ −5854.49 −0.985550
$$329$$ 2108.30 0.353295
$$330$$ 3669.70i 0.612153i
$$331$$ 5589.62i 0.928197i 0.885784 + 0.464099i $$0.153622\pi$$
−0.885784 + 0.464099i $$0.846378\pi$$
$$332$$ − 451.158i − 0.0745798i
$$333$$ 1392.90i 0.229220i
$$334$$ −1998.85 −0.327461
$$335$$ −7794.92 −1.27129
$$336$$ − 1344.75i − 0.218340i
$$337$$ −901.544 −0.145728 −0.0728638 0.997342i $$-0.523214\pi$$
−0.0728638 + 0.997342i $$0.523214\pi$$
$$338$$ 0 0
$$339$$ 498.194 0.0798176
$$340$$ − 631.225i − 0.100685i
$$341$$ 2685.77 0.426518
$$342$$ 407.548 0.0644376
$$343$$ − 4714.49i − 0.742153i
$$344$$ − 11688.9i − 1.83204i
$$345$$ − 10263.8i − 1.60169i
$$346$$ − 10513.8i − 1.63360i
$$347$$ −812.318 −0.125670 −0.0628350 0.998024i $$-0.520014\pi$$
−0.0628350 + 0.998024i $$0.520014\pi$$
$$348$$ −423.093 −0.0651730
$$349$$ − 4437.96i − 0.680683i −0.940302 0.340342i $$-0.889457\pi$$
0.940302 0.340342i $$-0.110543\pi$$
$$350$$ 5223.54 0.797743
$$351$$ 0 0
$$352$$ −500.174 −0.0757368
$$353$$ 7115.35i 1.07284i 0.843952 + 0.536419i $$0.180223\pi$$
−0.843952 + 0.536419i $$0.819777\pi$$
$$354$$ −655.253 −0.0983794
$$355$$ −10293.0 −1.53886
$$356$$ − 575.587i − 0.0856911i
$$357$$ − 1504.90i − 0.223102i
$$358$$ − 778.253i − 0.114894i
$$359$$ − 4693.98i − 0.690081i −0.938588 0.345040i $$-0.887865\pi$$
0.938588 0.345040i $$-0.112135\pi$$
$$360$$ 4078.35 0.597077
$$361$$ 6586.20 0.960227
$$362$$ 6480.17i 0.940857i
$$363$$ −2419.79 −0.349878
$$364$$ 0 0
$$365$$ −14307.9 −2.05181
$$366$$ − 1602.76i − 0.228901i
$$367$$ 9243.98 1.31480 0.657400 0.753542i $$-0.271656\pi$$
0.657400 + 0.753542i $$0.271656\pi$$
$$368$$ −10518.4 −1.48997
$$369$$ − 2265.44i − 0.319605i
$$370$$ 8267.09i 1.16158i
$$371$$ − 2746.37i − 0.384324i
$$372$$ − 170.054i − 0.0237013i
$$373$$ −4311.99 −0.598569 −0.299285 0.954164i $$-0.596748\pi$$
−0.299285 + 0.954164i $$0.596748\pi$$
$$374$$ 4208.60 0.581876
$$375$$ 7575.09i 1.04314i
$$376$$ 6552.64 0.898741
$$377$$ 0 0
$$378$$ 553.951 0.0753760
$$379$$ − 2382.73i − 0.322936i −0.986878 0.161468i $$-0.948377\pi$$
0.986878 0.161468i $$-0.0516229\pi$$
$$380$$ −155.531 −0.0209962
$$381$$ 3888.49 0.522870
$$382$$ − 6893.25i − 0.923271i
$$383$$ 4845.81i 0.646499i 0.946314 + 0.323250i $$0.104775\pi$$
−0.946314 + 0.323250i $$0.895225\pi$$
$$384$$ − 3909.73i − 0.519577i
$$385$$ − 3338.80i − 0.441976i
$$386$$ −6636.81 −0.875142
$$387$$ 4523.10 0.594113
$$388$$ 269.286i 0.0352343i
$$389$$ −9561.50 −1.24624 −0.623120 0.782127i $$-0.714135\pi$$
−0.623120 + 0.782127i $$0.714135\pi$$
$$390$$ 0 0
$$391$$ −11771.0 −1.52247
$$392$$ − 6675.14i − 0.860066i
$$393$$ 591.604 0.0759350
$$394$$ −12703.9 −1.62440
$$395$$ 2206.79i 0.281103i
$$396$$ − 99.6107i − 0.0126405i
$$397$$ 7440.11i 0.940575i 0.882513 + 0.470287i $$0.155850\pi$$
−0.882513 + 0.470287i $$0.844150\pi$$
$$398$$ 8373.48i 1.05458i
$$399$$ −370.799 −0.0465242
$$400$$ 15250.5 1.90631
$$401$$ 8687.80i 1.08192i 0.841050 + 0.540958i $$0.181938\pi$$
−0.841050 + 0.540958i $$0.818062\pi$$
$$402$$ −3290.66 −0.408267
$$403$$ 0 0
$$404$$ 138.598 0.0170681
$$405$$ 1578.15i 0.193627i
$$406$$ −5986.76 −0.731818
$$407$$ 3544.13 0.431637
$$408$$ − 4677.26i − 0.567546i
$$409$$ 2556.10i 0.309024i 0.987991 + 0.154512i $$0.0493805\pi$$
−0.987991 + 0.154512i $$0.950619\pi$$
$$410$$ − 13445.8i − 1.61961i
$$411$$ − 1640.74i − 0.196915i
$$412$$ 923.790 0.110466
$$413$$ 596.168 0.0710303
$$414$$ − 4332.90i − 0.514374i
$$415$$ 18187.0 2.15124
$$416$$ 0 0
$$417$$ −1829.00 −0.214788
$$418$$ − 1036.98i − 0.121340i
$$419$$ −3347.46 −0.390296 −0.195148 0.980774i $$-0.562519\pi$$
−0.195148 + 0.980774i $$0.562519\pi$$
$$420$$ −211.402 −0.0245603
$$421$$ − 1854.48i − 0.214684i −0.994222 0.107342i $$-0.965766\pi$$
0.994222 0.107342i $$-0.0342340\pi$$
$$422$$ 11053.3i 1.27503i
$$423$$ 2535.60i 0.291454i
$$424$$ − 8535.79i − 0.977675i
$$425$$ 17066.7 1.94789
$$426$$ −4345.25 −0.494197
$$427$$ 1458.24i 0.165267i
$$428$$ 403.246 0.0455412
$$429$$ 0 0
$$430$$ 26845.4 3.01069
$$431$$ − 14043.1i − 1.56945i −0.619844 0.784725i $$-0.712804\pi$$
0.619844 0.784725i $$-0.287196\pi$$
$$432$$ 1617.30 0.180121
$$433$$ −3086.47 −0.342555 −0.171278 0.985223i $$-0.554790\pi$$
−0.171278 + 0.985223i $$0.554790\pi$$
$$434$$ − 2406.26i − 0.266139i
$$435$$ − 17055.7i − 1.87990i
$$436$$ − 520.851i − 0.0572116i
$$437$$ 2900.32i 0.317486i
$$438$$ −6040.14 −0.658925
$$439$$ −2837.68 −0.308508 −0.154254 0.988031i $$-0.549297\pi$$
−0.154254 + 0.988031i $$0.549297\pi$$
$$440$$ − 10377.1i − 1.12434i
$$441$$ 2583.00 0.278912
$$442$$ 0 0
$$443$$ 18309.4 1.96367 0.981834 0.189744i $$-0.0607658\pi$$
0.981834 + 0.189744i $$0.0607658\pi$$
$$444$$ − 224.403i − 0.0239858i
$$445$$ 23203.0 2.47174
$$446$$ 10377.0 1.10172
$$447$$ 6495.24i 0.687281i
$$448$$ 4034.12i 0.425433i
$$449$$ − 13861.2i − 1.45690i −0.685098 0.728451i $$-0.740241\pi$$
0.685098 0.728451i $$-0.259759\pi$$
$$450$$ 6282.22i 0.658104i
$$451$$ −5764.26 −0.601837
$$452$$ −80.2614 −0.00835217
$$453$$ − 2539.65i − 0.263406i
$$454$$ 5521.23 0.570758
$$455$$ 0 0
$$456$$ −1152.45 −0.118352
$$457$$ − 8990.36i − 0.920243i −0.887856 0.460122i $$-0.847806\pi$$
0.887856 0.460122i $$-0.152194\pi$$
$$458$$ 8364.05 0.853333
$$459$$ 1809.90 0.184050
$$460$$ 1653.55i 0.167602i
$$461$$ − 3406.90i − 0.344198i −0.985080 0.172099i $$-0.944945\pi$$
0.985080 0.172099i $$-0.0550549\pi$$
$$462$$ − 1409.49i − 0.141938i
$$463$$ 7498.45i 0.752662i 0.926485 + 0.376331i $$0.122814\pi$$
−0.926485 + 0.376331i $$0.877186\pi$$
$$464$$ −17478.8 −1.74878
$$465$$ 6855.19 0.683660
$$466$$ 15319.0i 1.52283i
$$467$$ −7711.38 −0.764112 −0.382056 0.924139i $$-0.624784\pi$$
−0.382056 + 0.924139i $$0.624784\pi$$
$$468$$ 0 0
$$469$$ 2993.94 0.294770
$$470$$ 15049.2i 1.47695i
$$471$$ −4960.79 −0.485310
$$472$$ 1852.91 0.180693
$$473$$ − 11508.7i − 1.11875i
$$474$$ 931.608i 0.0902747i
$$475$$ − 4205.14i − 0.406200i
$$476$$ 242.446i 0.0233456i
$$477$$ 3302.99 0.317051
$$478$$ −3662.01 −0.350411
$$479$$ 9439.82i 0.900451i 0.892915 + 0.450226i $$0.148656\pi$$
−0.892915 + 0.450226i $$0.851344\pi$$
$$480$$ −1276.65 −0.121398
$$481$$ 0 0
$$482$$ 1567.02 0.148082
$$483$$ 3942.20i 0.371380i
$$484$$ 389.839 0.0366115
$$485$$ −10855.4 −1.01633
$$486$$ 666.223i 0.0621821i
$$487$$ − 6156.20i − 0.572821i −0.958107 0.286411i $$-0.907538\pi$$
0.958107 0.286411i $$-0.0924622\pi$$
$$488$$ 4532.25i 0.420420i
$$489$$ − 8599.54i − 0.795265i
$$490$$ 15330.6 1.41340
$$491$$ −3842.74 −0.353198 −0.176599 0.984283i $$-0.556510\pi$$
−0.176599 + 0.984283i $$0.556510\pi$$
$$492$$ 364.974i 0.0334437i
$$493$$ −19560.3 −1.78692
$$494$$ 0 0
$$495$$ 4015.49 0.364612
$$496$$ − 7025.24i − 0.635973i
$$497$$ 3953.43 0.356812
$$498$$ 7677.73 0.690858
$$499$$ − 12842.4i − 1.15211i −0.817410 0.576056i $$-0.804591\pi$$
0.817410 0.576056i $$-0.195409\pi$$
$$500$$ − 1220.38i − 0.109154i
$$501$$ 2187.20i 0.195043i
$$502$$ 11209.5i 0.996627i
$$503$$ 8580.11 0.760573 0.380287 0.924869i $$-0.375825\pi$$
0.380287 + 0.924869i $$0.375825\pi$$
$$504$$ −1566.45 −0.138443
$$505$$ 5587.16i 0.492327i
$$506$$ −11024.8 −0.968599
$$507$$ 0 0
$$508$$ −626.455 −0.0547135
$$509$$ − 43.5957i − 0.00379635i −0.999998 0.00189818i $$-0.999396\pi$$
0.999998 0.00189818i $$-0.000604209\pi$$
$$510$$ 10742.1 0.932681
$$511$$ 5495.49 0.475746
$$512$$ 12453.7i 1.07496i
$$513$$ − 445.951i − 0.0383805i
$$514$$ 8362.68i 0.717630i
$$515$$ 37239.7i 3.18636i
$$516$$ −728.693 −0.0621684
$$517$$ 6451.66 0.548827
$$518$$ − 3175.29i − 0.269333i
$$519$$ −11504.5 −0.973008
$$520$$ 0 0
$$521$$ 11368.1 0.955939 0.477969 0.878377i $$-0.341373\pi$$
0.477969 + 0.878377i $$0.341373\pi$$
$$522$$ − 7200.14i − 0.603719i
$$523$$ −5229.53 −0.437230 −0.218615 0.975811i $$-0.570154\pi$$
−0.218615 + 0.975811i $$0.570154\pi$$
$$524$$ −95.3103 −0.00794590
$$525$$ − 5715.75i − 0.475154i
$$526$$ − 15822.1i − 1.31155i
$$527$$ − 7861.88i − 0.649846i
$$528$$ − 4115.10i − 0.339180i
$$529$$ 18668.2 1.53433
$$530$$ 19603.8 1.60667
$$531$$ 716.997i 0.0585970i
$$532$$ 59.7375 0.00486832
$$533$$ 0 0
$$534$$ 9795.24 0.793786
$$535$$ 16255.6i 1.31363i
$$536$$ 9305.24 0.749860
$$537$$ −851.586 −0.0684333
$$538$$ 5701.01i 0.456855i
$$539$$ − 6572.27i − 0.525209i
$$540$$ − 254.247i − 0.0202612i
$$541$$ 6567.99i 0.521959i 0.965344 + 0.260980i $$0.0840455\pi$$
−0.965344 + 0.260980i $$0.915954\pi$$
$$542$$ −16482.9 −1.30627
$$543$$ 7090.79 0.560396
$$544$$ 1464.13i 0.115393i
$$545$$ 20996.5 1.65026
$$546$$ 0 0
$$547$$ −13675.7 −1.06897 −0.534487 0.845177i $$-0.679495\pi$$
−0.534487 + 0.845177i $$0.679495\pi$$
$$548$$ 264.332i 0.0206053i
$$549$$ −1753.79 −0.136339
$$550$$ 15984.7 1.23925
$$551$$ 4819.57i 0.372632i
$$552$$ 12252.5i 0.944745i
$$553$$ − 847.604i − 0.0651786i
$$554$$ − 2015.67i − 0.154581i
$$555$$ 9046.09 0.691865
$$556$$ 294.661 0.0224755
$$557$$ − 4527.96i − 0.344445i −0.985058 0.172222i $$-0.944905\pi$$
0.985058 0.172222i $$-0.0550947\pi$$
$$558$$ 2893.95 0.219553
$$559$$ 0 0
$$560$$ −8733.39 −0.659023
$$561$$ − 4605.17i − 0.346578i
$$562$$ 5217.15 0.391588
$$563$$ −18441.8 −1.38051 −0.690256 0.723566i $$-0.742502\pi$$
−0.690256 + 0.723566i $$0.742502\pi$$
$$564$$ − 408.497i − 0.0304979i
$$565$$ − 3235.49i − 0.240917i
$$566$$ 5827.96i 0.432804i
$$567$$ − 606.148i − 0.0448957i
$$568$$ 12287.4 0.907687
$$569$$ 13553.5 0.998578 0.499289 0.866436i $$-0.333595\pi$$
0.499289 + 0.866436i $$0.333595\pi$$
$$570$$ − 2646.79i − 0.194495i
$$571$$ −14815.5 −1.08583 −0.542915 0.839788i $$-0.682679\pi$$
−0.542915 + 0.839788i $$0.682679\pi$$
$$572$$ 0 0
$$573$$ −7542.79 −0.549921
$$574$$ 5164.37i 0.375534i
$$575$$ −44707.6 −3.24249
$$576$$ −4851.74 −0.350965
$$577$$ 21596.2i 1.55816i 0.626923 + 0.779081i $$0.284314\pi$$
−0.626923 + 0.779081i $$0.715686\pi$$
$$578$$ 1150.20i 0.0827716i
$$579$$ 7262.19i 0.521254i
$$580$$ 2747.75i 0.196714i
$$581$$ −6985.42 −0.498802
$$582$$ −4582.66 −0.326387
$$583$$ − 8404.23i − 0.597029i
$$584$$ 17080.1 1.21024
$$585$$ 0 0
$$586$$ 4499.13 0.317163
$$587$$ − 918.801i − 0.0646047i −0.999478 0.0323024i $$-0.989716\pi$$
0.999478 0.0323024i $$-0.0102839\pi$$
$$588$$ −416.134 −0.0291855
$$589$$ −1937.13 −0.135514
$$590$$ 4255.50i 0.296943i
$$591$$ 13900.9i 0.967527i
$$592$$ − 9270.49i − 0.643606i
$$593$$ − 19816.0i − 1.37226i −0.727481 0.686128i $$-0.759309\pi$$
0.727481 0.686128i $$-0.240691\pi$$
$$594$$ 1695.16 0.117093
$$595$$ −9773.45 −0.673399
$$596$$ − 1046.42i − 0.0719175i
$$597$$ 9162.50 0.628134
$$598$$ 0 0
$$599$$ −5141.86 −0.350736 −0.175368 0.984503i $$-0.556111\pi$$
−0.175368 + 0.984503i $$0.556111\pi$$
$$600$$ − 17764.7i − 1.20873i
$$601$$ 12380.9 0.840312 0.420156 0.907452i $$-0.361975\pi$$
0.420156 + 0.907452i $$0.361975\pi$$
$$602$$ −10311.0 −0.698081
$$603$$ 3600.74i 0.243173i
$$604$$ 409.150i 0.0275630i
$$605$$ 15715.1i 1.05605i
$$606$$ 2358.65i 0.158108i
$$607$$ −23717.0 −1.58590 −0.792951 0.609286i $$-0.791456\pi$$
−0.792951 + 0.609286i $$0.791456\pi$$
$$608$$ 360.754 0.0240633
$$609$$ 6550.89i 0.435887i
$$610$$ −10409.0 −0.690901
$$611$$ 0 0
$$612$$ −291.584 −0.0192591
$$613$$ − 26157.1i − 1.72345i −0.507373 0.861726i $$-0.669383\pi$$
0.507373 0.861726i $$-0.330617\pi$$
$$614$$ 9248.36 0.607872
$$615$$ −14712.8 −0.964677
$$616$$ 3985.71i 0.260696i
$$617$$ 23613.9i 1.54077i 0.637576 + 0.770387i $$0.279937\pi$$
−0.637576 + 0.770387i $$0.720063\pi$$
$$618$$ 15720.9i 1.02328i
$$619$$ − 23345.4i − 1.51588i −0.652324 0.757940i $$-0.726206\pi$$
0.652324 0.757940i $$-0.273794\pi$$
$$620$$ −1104.40 −0.0715387
$$621$$ −4741.19 −0.306372
$$622$$ − 2381.20i − 0.153501i
$$623$$ −8911.99 −0.573116
$$624$$ 0 0
$$625$$ 17371.0 1.11174
$$626$$ 11907.5i 0.760258i
$$627$$ −1134.69 −0.0722730
$$628$$ 799.207 0.0507832
$$629$$ − 10374.5i − 0.657645i
$$630$$ − 3597.60i − 0.227511i
$$631$$ − 15245.7i − 0.961841i −0.876764 0.480921i $$-0.840302\pi$$
0.876764 0.480921i $$-0.159698\pi$$
$$632$$ − 2634.38i − 0.165807i
$$633$$ 12094.8 0.759439
$$634$$ −8986.20 −0.562914
$$635$$ − 25253.6i − 1.57820i
$$636$$ −532.128 −0.0331765
$$637$$ 0 0
$$638$$ −18320.3 −1.13684
$$639$$ 4754.69i 0.294355i
$$640$$ −25391.5 −1.56826
$$641$$ −10192.7 −0.628063 −0.314032 0.949413i $$-0.601680\pi$$
−0.314032 + 0.949413i $$0.601680\pi$$
$$642$$ 6862.37i 0.421863i
$$643$$ − 5506.31i − 0.337710i −0.985641 0.168855i $$-0.945993\pi$$
0.985641 0.168855i $$-0.0540070\pi$$
$$644$$ − 635.108i − 0.0388614i
$$645$$ − 29375.0i − 1.79324i
$$646$$ −3035.48 −0.184875
$$647$$ 13297.5 0.808005 0.404003 0.914758i $$-0.367619\pi$$
0.404003 + 0.914758i $$0.367619\pi$$
$$648$$ − 1883.93i − 0.114209i
$$649$$ 1824.35 0.110342
$$650$$ 0 0
$$651$$ −2633.00 −0.158518
$$652$$ 1385.43i 0.0832171i
$$653$$ −12440.2 −0.745519 −0.372760 0.927928i $$-0.621588\pi$$
−0.372760 + 0.927928i $$0.621588\pi$$
$$654$$ 8863.76 0.529970
$$655$$ − 3842.14i − 0.229198i
$$656$$ 15077.7i 0.897389i
$$657$$ 6609.29i 0.392470i
$$658$$ − 5780.23i − 0.342457i
$$659$$ −9562.87 −0.565276 −0.282638 0.959227i $$-0.591209\pi$$
−0.282638 + 0.959227i $$0.591209\pi$$
$$660$$ −646.915 −0.0381533
$$661$$ − 2409.69i − 0.141795i −0.997484 0.0708973i $$-0.977414\pi$$
0.997484 0.0708973i $$-0.0225863\pi$$
$$662$$ 15324.8 0.899722
$$663$$ 0 0
$$664$$ −21710.9 −1.26889
$$665$$ 2408.13i 0.140426i
$$666$$ 3818.85 0.222188
$$667$$ 51239.9 2.97454
$$668$$ − 352.368i − 0.0204095i
$$669$$ − 11354.9i − 0.656209i
$$670$$ 21371.0i 1.23229i
$$671$$ 4462.40i 0.256735i
$$672$$ 490.346 0.0281481
$$673$$ −7929.02 −0.454147 −0.227074 0.973878i $$-0.572916\pi$$
−0.227074 + 0.973878i $$0.572916\pi$$
$$674$$ 2471.72i 0.141257i
$$675$$ 6874.19 0.391982
$$676$$ 0 0
$$677$$ −2628.26 −0.149206 −0.0746030 0.997213i $$-0.523769\pi$$
−0.0746030 + 0.997213i $$0.523769\pi$$
$$678$$ − 1365.88i − 0.0773690i
$$679$$ 4169.44 0.235653
$$680$$ −30376.1 −1.71305
$$681$$ − 6041.48i − 0.339956i
$$682$$ − 7363.46i − 0.413433i
$$683$$ − 10021.5i − 0.561437i −0.959790 0.280719i $$-0.909427\pi$$
0.959790 0.280719i $$-0.0905728\pi$$
$$684$$ 71.8448i 0.00401616i
$$685$$ −10655.7 −0.594356
$$686$$ −12925.5 −0.719385
$$687$$ − 9152.18i − 0.508264i
$$688$$ −30103.7 −1.66816
$$689$$ 0 0
$$690$$ −28139.8 −1.55256
$$691$$ 23987.2i 1.32057i 0.751014 + 0.660286i $$0.229565\pi$$
−0.751014 + 0.660286i $$0.770435\pi$$
$$692$$ 1853.43 0.101816
$$693$$ −1542.30 −0.0845415
$$694$$ 2227.10i 0.121815i
$$695$$ 11878.3i 0.648303i
$$696$$ 20360.3i 1.10885i
$$697$$ 16873.4i 0.916964i
$$698$$ −12167.4 −0.659801
$$699$$ 16762.5 0.907032
$$700$$ 920.835i 0.0497204i
$$701$$ 3763.71 0.202787 0.101393 0.994846i $$-0.467670\pi$$
0.101393 + 0.994846i $$0.467670\pi$$
$$702$$ 0 0
$$703$$ −2556.23 −0.137141
$$704$$ 12344.9i 0.660890i
$$705$$ 16467.3 0.879707
$$706$$ 19507.8 1.03993
$$707$$ − 2145.96i − 0.114155i
$$708$$ − 115.512i − 0.00613163i
$$709$$ 36047.8i 1.90946i 0.297479 + 0.954728i $$0.403854\pi$$
−0.297479 + 0.954728i $$0.596146\pi$$
$$710$$ 28219.9i 1.49166i
$$711$$ 1019.39 0.0537696
$$712$$ −27698.7 −1.45794
$$713$$ 20594.9i 1.08174i
$$714$$ −4125.91 −0.216258
$$715$$ 0 0
$$716$$ 137.195 0.00716090
$$717$$ 4007.08i 0.208713i
$$718$$ −12869.3 −0.668910
$$719$$ 3944.18 0.204580 0.102290 0.994755i $$-0.467383\pi$$
0.102290 + 0.994755i $$0.467383\pi$$
$$720$$ − 10503.4i − 0.543667i
$$721$$ − 14303.3i − 0.738812i
$$722$$ − 18057.1i − 0.930770i
$$723$$ − 1714.68i − 0.0882012i
$$724$$ −1142.36 −0.0586402
$$725$$ −74292.1 −3.80571
$$726$$ 6634.22i 0.339145i
$$727$$ 20447.8 1.04315 0.521573 0.853206i $$-0.325345\pi$$
0.521573 + 0.853206i $$0.325345\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 39227.3i 1.98886i
$$731$$ −33688.7 −1.70454
$$732$$ 282.544 0.0142666
$$733$$ − 13536.2i − 0.682089i −0.940047 0.341045i $$-0.889219\pi$$
0.940047 0.341045i $$-0.110781\pi$$
$$734$$ − 25343.8i − 1.27447i
$$735$$ − 16775.1i − 0.841851i
$$736$$ − 3835.40i − 0.192085i
$$737$$ 9161.83 0.457911
$$738$$ −6211.07 −0.309800
$$739$$ − 15839.1i − 0.788433i −0.919018 0.394217i $$-0.871016\pi$$
0.919018 0.394217i $$-0.128984\pi$$
$$740$$ −1457.37 −0.0723972
$$741$$ 0 0
$$742$$ −7529.60 −0.372534
$$743$$ − 1664.92i − 0.0822075i −0.999155 0.0411037i $$-0.986913\pi$$
0.999155 0.0411037i $$-0.0130874\pi$$
$$744$$ −8183.43 −0.403252
$$745$$ 42182.9 2.07445
$$746$$ 11822.0i 0.580206i
$$747$$ − 8401.19i − 0.411491i
$$748$$ 741.916i 0.0362662i
$$749$$ − 6243.58i − 0.304587i
$$750$$ 20768.3 1.01113
$$751$$ −22399.1 −1.08835 −0.544177 0.838970i $$-0.683158\pi$$
−0.544177 + 0.838970i $$0.683158\pi$$
$$752$$ − 16875.8i − 0.818346i
$$753$$ 12265.8 0.593613
$$754$$ 0 0
$$755$$ −16493.6 −0.795050
$$756$$ 97.6535i 0.00469792i
$$757$$ 23798.9 1.14265 0.571326 0.820723i $$-0.306429\pi$$
0.571326 + 0.820723i $$0.306429\pi$$
$$758$$ −6532.64 −0.313029
$$759$$ 12063.6i 0.576920i
$$760$$ 7484.53i 0.357227i
$$761$$ − 13693.5i − 0.652285i −0.945321 0.326142i $$-0.894251\pi$$
0.945321 0.326142i $$-0.105749\pi$$
$$762$$ − 10660.9i − 0.506829i
$$763$$ −8064.50 −0.382640
$$764$$ 1215.18 0.0575441
$$765$$ − 11754.3i − 0.555526i
$$766$$ 13285.5 0.626666
$$767$$ 0 0
$$768$$ 2218.82 0.104251
$$769$$ 16299.9i 0.764358i 0.924088 + 0.382179i $$0.124826\pi$$
−0.924088 + 0.382179i $$0.875174\pi$$
$$770$$ −9153.84 −0.428418
$$771$$ 9150.68 0.427437
$$772$$ − 1169.97i − 0.0545444i
$$773$$ 33532.2i 1.56024i 0.625628 + 0.780122i $$0.284843\pi$$
−0.625628 + 0.780122i $$0.715157\pi$$
$$774$$ − 12400.8i − 0.575887i
$$775$$ − 29860.2i − 1.38401i
$$776$$ 12958.7 0.599473
$$777$$ −3474.50 −0.160421
$$778$$ 26214.3i 1.20801i
$$779$$ 4157.51 0.191217
$$780$$ 0 0
$$781$$ 12098.0 0.554290
$$782$$ 32272.1i 1.47577i
$$783$$ −7878.59 −0.359589
$$784$$ −17191.3 −0.783130
$$785$$ 32217.5i 1.46483i
$$786$$ − 1621.98i − 0.0736055i
$$787$$ − 16163.3i − 0.732097i −0.930596 0.366049i $$-0.880710\pi$$
0.930596 0.366049i $$-0.119290\pi$$
$$788$$ − 2239.51i − 0.101243i
$$789$$ −17313.0 −0.781189
$$790$$ 6050.27 0.272480
$$791$$ 1242.71i 0.0558607i
$$792$$ −4793.52 −0.215064
$$793$$ 0 0
$$794$$ 20398.2 0.911720
$$795$$ − 21451.1i − 0.956970i
$$796$$ −1476.12 −0.0657284
$$797$$ 39636.4 1.76160 0.880798 0.473492i $$-0.157007\pi$$
0.880798 + 0.473492i $$0.157007\pi$$
$$798$$ 1016.60i 0.0450969i
$$799$$ − 18885.5i − 0.836197i
$$800$$ 5560.90i 0.245760i
$$801$$ − 10718.2i − 0.472797i
$$802$$ 23819.0 1.04872
$$803$$ 16816.9 0.739048
$$804$$ − 580.096i − 0.0254458i
$$805$$ 25602.4 1.12095
$$806$$ 0 0
$$807$$ 6238.21 0.272113
$$808$$ − 6669.71i − 0.290396i
$$809$$ −23811.2 −1.03481 −0.517403 0.855742i $$-0.673101\pi$$
−0.517403 + 0.855742i $$0.673101\pi$$
$$810$$ 4326.74 0.187687
$$811$$ 27218.6i 1.17851i 0.807946 + 0.589256i $$0.200579\pi$$
−0.807946 + 0.589256i $$0.799421\pi$$
$$812$$ − 1055.38i − 0.0456116i
$$813$$ 18036.0i 0.778045i
$$814$$ − 9716.80i − 0.418395i
$$815$$ −55849.2 −2.40038
$$816$$ −12045.9 −0.516777
$$817$$ 8300.73i 0.355454i
$$818$$ 7007.94 0.299544
$$819$$ 0 0
$$820$$ 2370.30 0.100944
$$821$$ − 43094.8i − 1.83193i −0.401253 0.915967i $$-0.631425\pi$$
0.401253 0.915967i $$-0.368575\pi$$
$$822$$ −4498.36 −0.190874
$$823$$ −26541.1 −1.12414 −0.562068 0.827091i $$-0.689994\pi$$
−0.562068 + 0.827091i $$0.689994\pi$$
$$824$$ − 44455.1i − 1.87945i
$$825$$ − 17490.9i − 0.738127i
$$826$$ − 1634.49i − 0.0688512i
$$827$$ 44898.7i 1.88788i 0.330112 + 0.943942i $$0.392913\pi$$
−0.330112 + 0.943942i $$0.607087\pi$$
$$828$$ 763.829 0.0320590
$$829$$ 7137.48 0.299029 0.149514 0.988760i $$-0.452229\pi$$
0.149514 + 0.988760i $$0.452229\pi$$
$$830$$ − 49862.6i − 2.08525i
$$831$$ −2205.60 −0.0920717
$$832$$ 0 0
$$833$$ −19238.6 −0.800213
$$834$$ 5014.49i 0.208198i
$$835$$ 14204.6 0.588708
$$836$$ 182.804 0.00756270
$$837$$ − 3166.64i − 0.130771i
$$838$$ 9177.59i 0.378323i
$$839$$ 4387.17i 0.180527i 0.995918 + 0.0902634i $$0.0287709\pi$$
−0.995918 + 0.0902634i $$0.971229\pi$$
$$840$$ 10173.2i 0.417867i
$$841$$ 60758.1 2.49121
$$842$$ −5084.36 −0.208098
$$843$$ − 5708.76i − 0.233239i
$$844$$ −1948.53 −0.0794683
$$845$$ 0 0
$$846$$ 6951.74 0.282513
$$847$$ − 6036.01i − 0.244864i
$$848$$ −21983.2 −0.890219
$$849$$ 6377.12 0.257788
$$850$$ − 46791.0i − 1.88814i
$$851$$ 27176.9i 1.09473i
$$852$$ − 766.004i − 0.0308015i
$$853$$ 9328.85i 0.374459i 0.982316 + 0.187230i $$0.0599508\pi$$
−0.982316 + 0.187230i $$0.940049\pi$$
$$854$$ 3997.99 0.160197
$$855$$ −2896.20 −0.115845
$$856$$ − 19405.2i − 0.774833i
$$857$$ 5010.39 0.199710 0.0998552 0.995002i $$-0.468162\pi$$
0.0998552 + 0.995002i $$0.468162\pi$$
$$858$$ 0 0
$$859$$ 30233.4 1.20088 0.600438 0.799672i $$-0.294993\pi$$
0.600438 + 0.799672i $$0.294993\pi$$
$$860$$ 4732.45i 0.187646i
$$861$$ 5651.01 0.223677
$$862$$ −38501.4 −1.52130
$$863$$ 4334.93i 0.170988i 0.996339 + 0.0854940i $$0.0272468\pi$$
−0.996339 + 0.0854940i $$0.972753\pi$$
$$864$$ 589.728i 0.0232210i
$$865$$ 74715.2i 2.93687i
$$866$$ 8462.05i 0.332047i
$$867$$ 1258.58 0.0493007
$$868$$ 424.189 0.0165875
$$869$$ − 2593.78i − 0.101252i
$$870$$ −46760.8 −1.82223
$$871$$ 0 0
$$872$$ −25064.7 −0.973391
$$873$$ 5014.48i 0.194404i
$$874$$ 7951.69 0.307746
$$875$$ −18895.6 −0.730043
$$876$$ − 1064.79i − 0.0410684i
$$877$$ 34683.3i 1.33543i 0.744416 + 0.667716i $$0.232728\pi$$
−0.744416 + 0.667716i $$0.767272\pi$$
$$878$$ 7779.95i 0.299044i
$$879$$ − 4923.08i − 0.188909i
$$880$$ −26725.3 −1.02376
$$881$$ 18269.2 0.698642 0.349321 0.937003i $$-0.386412\pi$$
0.349321 + 0.937003i $$0.386412\pi$$
$$882$$ − 7081.70i − 0.270355i
$$883$$ 14592.0 0.556128 0.278064 0.960563i $$-0.410307\pi$$
0.278064 + 0.960563i $$0.410307\pi$$
$$884$$ 0 0
$$885$$ 4656.49 0.176866
$$886$$ − 50198.0i − 1.90343i
$$887$$ 30459.3 1.15301 0.576507 0.817092i $$-0.304415\pi$$
0.576507 + 0.817092i $$0.304415\pi$$
$$888$$ −10798.8 −0.408091
$$889$$ 9699.60i 0.365933i
$$890$$ − 63614.6i − 2.39592i
$$891$$ − 1854.89i − 0.0697432i
$$892$$ 1829.32i 0.0686662i
$$893$$ −4653.30 −0.174375
$$894$$ 17807.7 0.666196
$$895$$ 5530.57i 0.206555i
$$896$$ 9752.58 0.363628
$$897$$ 0 0
$$898$$ −38002.6 −1.41221
$$899$$ 34223.2i 1.26964i
$$900$$ −1107.47 −0.0410172
$$901$$ −24601.2 −0.909638
$$902$$ 15803.6i 0.583374i
$$903$$ 11282.6i 0.415793i
$$904$$ 3862.39i 0.142103i
$$905$$ − 46050.7i − 1.69147i
$$906$$ −6962.84 −0.255326
$$907$$ 9364.89 0.342840 0.171420 0.985198i $$-0.445164\pi$$
0.171420 + 0.985198i $$0.445164\pi$$
$$908$$ 973.313i 0.0355733i
$$909$$ 2580.90 0.0941727
$$910$$ 0 0
$$911$$ 32479.8 1.18123 0.590616 0.806952i $$-0.298885\pi$$
0.590616 + 0.806952i $$0.298885\pi$$
$$912$$ 2968.04i 0.107765i
$$913$$ −21376.3 −0.774864
$$914$$ −24648.5 −0.892012
$$915$$ 11389.9i 0.411516i
$$916$$ 1474.46i 0.0531851i
$$917$$ 1475.72i 0.0531435i
$$918$$ − 4962.13i − 0.178404i
$$919$$ 295.958 0.0106232 0.00531161 0.999986i $$-0.498309\pi$$
0.00531161 + 0.999986i $$0.498309\pi$$
$$920$$ 79572.9 2.85157
$$921$$ − 10119.8i − 0.362062i
$$922$$ −9340.56 −0.333639
$$923$$ 0 0
$$924$$ 248.473 0.00884649
$$925$$ − 39403.5i − 1.40062i
$$926$$ 20558.2 0.729572
$$927$$ 17202.3 0.609489
$$928$$ − 6373.42i − 0.225450i
$$929$$ − 5620.38i − 0.198492i −0.995063 0.0992458i $$-0.968357\pi$$
0.995063 0.0992458i $$-0.0316430\pi$$
$$930$$ − 18794.6i − 0.662687i
$$931$$ 4740.29i 0.166871i
$$932$$ −2700.52 −0.0949125
$$933$$ −2605.58 −0.0914284
$$934$$ 21142.0i 0.740670i
$$935$$ −29908.0 −1.04609
$$936$$ 0 0
$$937$$ −32583.1 −1.13601 −0.568006 0.823024i $$-0.692285\pi$$
−0.568006 + 0.823024i $$0.692285\pi$$
$$938$$ − 8208.35i − 0.285727i
$$939$$ 13029.6 0.452827
$$940$$ −2652.96 −0.0920532
$$941$$ 8812.99i 0.305308i 0.988280 + 0.152654i $$0.0487821\pi$$
−0.988280 + 0.152654i $$0.951218\pi$$
$$942$$ 13600.8i 0.470422i
$$943$$ − 44201.2i − 1.52639i
$$944$$ − 4772.00i − 0.164529i
$$945$$ −3936.59 −0.135510
$$946$$ −31552.9 −1.08443
$$947$$ − 13426.8i − 0.460732i −0.973104 0.230366i $$-0.926008\pi$$
0.973104 0.230366i $$-0.0739923\pi$$
$$948$$ −164.229 −0.00562649
$$949$$ 0 0
$$950$$ −11529.1 −0.393739
$$951$$ 9832.96i 0.335285i
$$952$$ 11667.1 0.397199
$$953$$ 13394.6 0.455293 0.227647 0.973744i $$-0.426897\pi$$
0.227647 + 0.973744i $$0.426897\pi$$
$$954$$ − 9055.67i − 0.307325i
$$955$$ 48986.2i 1.65985i
$$956$$ − 645.560i − 0.0218398i
$$957$$ 20046.5i 0.677129i
$$958$$ 25880.7 0.872828
$$959$$ 4092.74 0.137812
$$960$$ 31509.3i 1.05933i
$$961$$ 16035.7 0.538273
$$962$$ 0 0
$$963$$ 7509.00 0.251271
$$964$$ 276.243i 0.00922944i
$$965$$ 47163.8 1.57332
$$966$$ 10808.2 0.359986
$$967$$ 45590.8i 1.51613i 0.652178 + 0.758066i $$0.273856\pi$$
−0.652178 + 0.758066i $$0.726144\pi$$
$$968$$ − 18760.1i − 0.622904i
$$969$$ 3321.51i 0.110116i
$$970$$ 29761.8i 0.985149i
$$971$$ 264.763 0.00875041 0.00437521 0.999990i $$-0.498607\pi$$
0.00437521 + 0.999990i $$0.498607\pi$$
$$972$$ −117.445 −0.00387558
$$973$$ − 4562.32i − 0.150320i
$$974$$ −16878.2 −0.555248
$$975$$ 0 0
$$976$$ 11672.4 0.382812
$$977$$ 610.521i 0.0199921i 0.999950 + 0.00999606i $$0.00318190\pi$$
−0.999950 + 0.00999606i $$0.996818\pi$$
$$978$$ −23577.0 −0.770868
$$979$$ −27271.8 −0.890308
$$980$$ 2702.56i 0.0880918i
$$981$$ − 9698.98i − 0.315662i
$$982$$ 10535.5i 0.342363i
$$983$$ 57829.7i 1.87638i 0.346121 + 0.938190i $$0.387499\pi$$
−0.346121 + 0.938190i $$0.612501\pi$$
$$984$$ 17563.5 0.569007
$$985$$ 90278.8 2.92033
$$986$$ 53627.7i 1.73210i
$$987$$ −6324.89 −0.203975
$$988$$ 0 0
$$989$$ 88250.4 2.83741
$$990$$ − 11009.1i − 0.353427i
$$991$$ −56780.7 −1.82008 −0.910039 0.414522i $$-0.863949\pi$$
−0.910039 + 0.414522i $$0.863949\pi$$
$$992$$ 2561.67 0.0819890
$$993$$ − 16768.9i − 0.535895i
$$994$$ − 10838.9i − 0.345866i
$$995$$ − 59505.3i − 1.89592i
$$996$$ 1353.47i 0.0430587i
$$997$$ 18616.6 0.591369 0.295684 0.955286i $$-0.404452\pi$$
0.295684 + 0.955286i $$0.404452\pi$$
$$998$$ −35209.4 −1.11677
$$999$$ − 4178.69i − 0.132340i
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 507.4.b.f.337.2 4
13.5 odd 4 39.4.a.b.1.1 2
13.8 odd 4 507.4.a.f.1.2 2
13.12 even 2 inner 507.4.b.f.337.3 4
39.5 even 4 117.4.a.c.1.2 2
39.8 even 4 1521.4.a.s.1.1 2
52.31 even 4 624.4.a.r.1.2 2
65.44 odd 4 975.4.a.j.1.2 2
91.83 even 4 1911.4.a.h.1.1 2
104.5 odd 4 2496.4.a.bc.1.1 2
104.83 even 4 2496.4.a.s.1.1 2
156.83 odd 4 1872.4.a.t.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 2 13.5 odd 4
117.4.a.c.1.2 2 39.5 even 4
507.4.a.f.1.2 2 13.8 odd 4
507.4.b.f.337.2 4 1.1 even 1 trivial
507.4.b.f.337.3 4 13.12 even 2 inner
624.4.a.r.1.2 2 52.31 even 4
975.4.a.j.1.2 2 65.44 odd 4
1521.4.a.s.1.1 2 39.8 even 4
1872.4.a.t.1.1 2 156.83 odd 4
1911.4.a.h.1.1 2 91.83 even 4
2496.4.a.s.1.1 2 104.83 even 4
2496.4.a.bc.1.1 2 104.5 odd 4