# Properties

 Label 1911.4.a.h.1.1 Level $1911$ Weight $4$ Character 1911.1 Self dual yes Analytic conductor $112.753$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1911,4,Mod(1,1911)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1911, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1911.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1911.a (trivial)

## 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: yes Analytic conductor: $$112.752650021$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 14$$ x^2 - 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.74166$$ of defining polynomial Character $$\chi$$ $$=$$ 1911.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.74166 q^{2} +3.00000 q^{3} -0.483315 q^{4} -19.4833 q^{5} -8.22497 q^{6} +23.2583 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.74166 q^{2} +3.00000 q^{3} -0.483315 q^{4} -19.4833 q^{5} -8.22497 q^{6} +23.2583 q^{8} +9.00000 q^{9} +53.4166 q^{10} +22.8999 q^{11} -1.44994 q^{12} +13.0000 q^{13} -58.4499 q^{15} -59.8999 q^{16} -67.0334 q^{17} -24.6749 q^{18} -16.5167 q^{19} +9.41657 q^{20} -62.7836 q^{22} -175.600 q^{23} +69.7750 q^{24} +254.600 q^{25} -35.6415 q^{26} +27.0000 q^{27} +291.800 q^{29} +160.250 q^{30} -117.283 q^{31} -21.8418 q^{32} +68.6997 q^{33} +183.783 q^{34} -4.34983 q^{36} -154.766 q^{37} +45.2831 q^{38} +39.0000 q^{39} -453.150 q^{40} +251.716 q^{41} -502.566 q^{43} -11.0679 q^{44} -175.350 q^{45} +481.434 q^{46} +281.733 q^{47} -179.700 q^{48} -698.025 q^{50} -201.100 q^{51} -6.28309 q^{52} +366.999 q^{53} -74.0247 q^{54} -446.166 q^{55} -49.5501 q^{57} -800.015 q^{58} +79.6663 q^{59} +28.2497 q^{60} +194.865 q^{61} +321.550 q^{62} +539.082 q^{64} -253.283 q^{65} -188.351 q^{66} +400.082 q^{67} +32.3982 q^{68} -526.799 q^{69} +528.299 q^{71} +209.325 q^{72} +734.366 q^{73} +424.316 q^{74} +763.799 q^{75} +7.98276 q^{76} -106.925 q^{78} +113.266 q^{79} +1167.05 q^{80} +81.0000 q^{81} -690.118 q^{82} +933.466 q^{83} +1306.03 q^{85} +1377.86 q^{86} +875.399 q^{87} +532.613 q^{88} -1190.91 q^{89} +480.749 q^{90} +84.8699 q^{92} -351.849 q^{93} -772.415 q^{94} +321.800 q^{95} -65.5253 q^{96} -557.165 q^{97} +206.099 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} + 54 q^{8} + 18 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 6 * q^3 + 14 * q^4 - 24 * q^5 + 6 * q^6 + 54 * q^8 + 18 * q^9 $$2 q + 2 q^{2} + 6 q^{3} + 14 q^{4} - 24 q^{5} + 6 q^{6} + 54 q^{8} + 18 q^{9} + 32 q^{10} - 44 q^{11} + 42 q^{12} + 26 q^{13} - 72 q^{15} - 30 q^{16} - 164 q^{17} + 18 q^{18} - 48 q^{19} - 56 q^{20} - 380 q^{22} + 8 q^{23} + 162 q^{24} + 150 q^{25} + 26 q^{26} + 54 q^{27} + 404 q^{29} + 96 q^{30} - 40 q^{31} - 126 q^{32} - 132 q^{33} - 276 q^{34} + 126 q^{36} - 100 q^{37} - 104 q^{38} + 78 q^{39} - 592 q^{40} - 200 q^{41} - 616 q^{43} - 980 q^{44} - 216 q^{45} + 1352 q^{46} + 324 q^{47} - 90 q^{48} - 1194 q^{50} - 492 q^{51} + 182 q^{52} - 164 q^{53} + 54 q^{54} - 144 q^{55} - 144 q^{57} - 268 q^{58} - 140 q^{59} - 168 q^{60} - 628 q^{61} + 688 q^{62} - 194 q^{64} - 312 q^{65} - 1140 q^{66} - 472 q^{67} - 1372 q^{68} + 24 q^{69} + 428 q^{71} + 486 q^{72} + 900 q^{73} + 684 q^{74} + 450 q^{75} - 448 q^{76} + 78 q^{78} - 432 q^{79} + 1032 q^{80} + 162 q^{81} - 2832 q^{82} + 1388 q^{83} + 1744 q^{85} + 840 q^{86} + 1212 q^{87} - 1524 q^{88} - 960 q^{89} + 288 q^{90} + 2744 q^{92} - 120 q^{93} - 572 q^{94} + 464 q^{95} - 378 q^{96} + 532 q^{97} - 396 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 6 * q^3 + 14 * q^4 - 24 * q^5 + 6 * q^6 + 54 * q^8 + 18 * q^9 + 32 * q^10 - 44 * q^11 + 42 * q^12 + 26 * q^13 - 72 * q^15 - 30 * q^16 - 164 * q^17 + 18 * q^18 - 48 * q^19 - 56 * q^20 - 380 * q^22 + 8 * q^23 + 162 * q^24 + 150 * q^25 + 26 * q^26 + 54 * q^27 + 404 * q^29 + 96 * q^30 - 40 * q^31 - 126 * q^32 - 132 * q^33 - 276 * q^34 + 126 * q^36 - 100 * q^37 - 104 * q^38 + 78 * q^39 - 592 * q^40 - 200 * q^41 - 616 * q^43 - 980 * q^44 - 216 * q^45 + 1352 * q^46 + 324 * q^47 - 90 * q^48 - 1194 * q^50 - 492 * q^51 + 182 * q^52 - 164 * q^53 + 54 * q^54 - 144 * q^55 - 144 * q^57 - 268 * q^58 - 140 * q^59 - 168 * q^60 - 628 * q^61 + 688 * q^62 - 194 * q^64 - 312 * q^65 - 1140 * q^66 - 472 * q^67 - 1372 * q^68 + 24 * q^69 + 428 * q^71 + 486 * q^72 + 900 * q^73 + 684 * q^74 + 450 * q^75 - 448 * q^76 + 78 * q^78 - 432 * q^79 + 1032 * q^80 + 162 * q^81 - 2832 * q^82 + 1388 * q^83 + 1744 * q^85 + 840 * q^86 + 1212 * q^87 - 1524 * q^88 - 960 * q^89 + 288 * q^90 + 2744 * q^92 - 120 * q^93 - 572 * q^94 + 464 * q^95 - 378 * q^96 + 532 * q^97 - 396 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.74166 −0.969322 −0.484661 0.874702i $$-0.661057\pi$$
−0.484661 + 0.874702i $$0.661057\pi$$
$$3$$ 3.00000 0.577350
$$4$$ −0.483315 −0.0604143
$$5$$ −19.4833 −1.74264 −0.871320 0.490715i $$-0.836736\pi$$
−0.871320 + 0.490715i $$0.836736\pi$$
$$6$$ −8.22497 −0.559638
$$7$$ 0 0
$$8$$ 23.2583 1.02788
$$9$$ 9.00000 0.333333
$$10$$ 53.4166 1.68918
$$11$$ 22.8999 0.627689 0.313844 0.949474i $$-0.398383\pi$$
0.313844 + 0.949474i $$0.398383\pi$$
$$12$$ −1.44994 −0.0348802
$$13$$ 13.0000 0.277350
$$14$$ 0 0
$$15$$ −58.4499 −1.00611
$$16$$ −59.8999 −0.935936
$$17$$ −67.0334 −0.956352 −0.478176 0.878264i $$-0.658702\pi$$
−0.478176 + 0.878264i $$0.658702\pi$$
$$18$$ −24.6749 −0.323107
$$19$$ −16.5167 −0.199431 −0.0997155 0.995016i $$-0.531793\pi$$
−0.0997155 + 0.995016i $$0.531793\pi$$
$$20$$ 9.41657 0.105280
$$21$$ 0 0
$$22$$ −62.7836 −0.608433
$$23$$ −175.600 −1.59196 −0.795979 0.605324i $$-0.793044\pi$$
−0.795979 + 0.605324i $$0.793044\pi$$
$$24$$ 69.7750 0.593449
$$25$$ 254.600 2.03680
$$26$$ −35.6415 −0.268842
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ 291.800 1.86848 0.934239 0.356648i $$-0.116080\pi$$
0.934239 + 0.356648i $$0.116080\pi$$
$$30$$ 160.250 0.975249
$$31$$ −117.283 −0.679505 −0.339753 0.940515i $$-0.610343\pi$$
−0.339753 + 0.940515i $$0.610343\pi$$
$$32$$ −21.8418 −0.120660
$$33$$ 68.6997 0.362396
$$34$$ 183.783 0.927013
$$35$$ 0 0
$$36$$ −4.34983 −0.0201381
$$37$$ −154.766 −0.687661 −0.343830 0.939032i $$-0.611724\pi$$
−0.343830 + 0.939032i $$0.611724\pi$$
$$38$$ 45.2831 0.193313
$$39$$ 39.0000 0.160128
$$40$$ −453.150 −1.79123
$$41$$ 251.716 0.958815 0.479407 0.877592i $$-0.340852\pi$$
0.479407 + 0.877592i $$0.340852\pi$$
$$42$$ 0 0
$$43$$ −502.566 −1.78234 −0.891170 0.453669i $$-0.850115\pi$$
−0.891170 + 0.453669i $$0.850115\pi$$
$$44$$ −11.0679 −0.0379214
$$45$$ −175.350 −0.580880
$$46$$ 481.434 1.54312
$$47$$ 281.733 0.874361 0.437181 0.899374i $$-0.355977\pi$$
0.437181 + 0.899374i $$0.355977\pi$$
$$48$$ −179.700 −0.540363
$$49$$ 0 0
$$50$$ −698.025 −1.97431
$$51$$ −201.100 −0.552150
$$52$$ −6.28309 −0.0167559
$$53$$ 366.999 0.951154 0.475577 0.879674i $$-0.342239\pi$$
0.475577 + 0.879674i $$0.342239\pi$$
$$54$$ −74.0247 −0.186546
$$55$$ −446.166 −1.09384
$$56$$ 0 0
$$57$$ −49.5501 −0.115141
$$58$$ −800.015 −1.81116
$$59$$ 79.6663 0.175791 0.0878955 0.996130i $$-0.471986\pi$$
0.0878955 + 0.996130i $$0.471986\pi$$
$$60$$ 28.2497 0.0607837
$$61$$ 194.865 0.409016 0.204508 0.978865i $$-0.434441\pi$$
0.204508 + 0.978865i $$0.434441\pi$$
$$62$$ 321.550 0.658660
$$63$$ 0 0
$$64$$ 539.082 1.05289
$$65$$ −253.283 −0.483322
$$66$$ −188.351 −0.351279
$$67$$ 400.082 0.729519 0.364759 0.931102i $$-0.381151\pi$$
0.364759 + 0.931102i $$0.381151\pi$$
$$68$$ 32.3982 0.0577774
$$69$$ −526.799 −0.919117
$$70$$ 0 0
$$71$$ 528.299 0.883065 0.441532 0.897245i $$-0.354435\pi$$
0.441532 + 0.897245i $$0.354435\pi$$
$$72$$ 209.325 0.342628
$$73$$ 734.366 1.17741 0.588706 0.808347i $$-0.299638\pi$$
0.588706 + 0.808347i $$0.299638\pi$$
$$74$$ 424.316 0.666565
$$75$$ 763.799 1.17594
$$76$$ 7.98276 0.0120485
$$77$$ 0 0
$$78$$ −106.925 −0.155216
$$79$$ 113.266 0.161309 0.0806545 0.996742i $$-0.474299\pi$$
0.0806545 + 0.996742i $$0.474299\pi$$
$$80$$ 1167.05 1.63100
$$81$$ 81.0000 0.111111
$$82$$ −690.118 −0.929400
$$83$$ 933.466 1.23447 0.617236 0.786778i $$-0.288252\pi$$
0.617236 + 0.786778i $$0.288252\pi$$
$$84$$ 0 0
$$85$$ 1306.03 1.66658
$$86$$ 1377.86 1.72766
$$87$$ 875.399 1.07877
$$88$$ 532.613 0.645191
$$89$$ −1190.91 −1.41839 −0.709195 0.705012i $$-0.750941\pi$$
−0.709195 + 0.705012i $$0.750941\pi$$
$$90$$ 480.749 0.563060
$$91$$ 0 0
$$92$$ 84.8699 0.0961771
$$93$$ −351.849 −0.392313
$$94$$ −772.415 −0.847538
$$95$$ 321.800 0.347536
$$96$$ −65.5253 −0.0696630
$$97$$ −557.165 −0.583211 −0.291606 0.956539i $$-0.594189\pi$$
−0.291606 + 0.956539i $$0.594189\pi$$
$$98$$ 0 0
$$99$$ 206.099 0.209230
$$100$$ −123.052 −0.123052
$$101$$ 286.766 0.282518 0.141259 0.989973i $$-0.454885\pi$$
0.141259 + 0.989973i $$0.454885\pi$$
$$102$$ 551.348 0.535211
$$103$$ 1911.36 1.82847 0.914234 0.405187i $$-0.132794\pi$$
0.914234 + 0.405187i $$0.132794\pi$$
$$104$$ 302.358 0.285084
$$105$$ 0 0
$$106$$ −1006.19 −0.921975
$$107$$ 834.334 0.753814 0.376907 0.926251i $$-0.376988\pi$$
0.376907 + 0.926251i $$0.376988\pi$$
$$108$$ −13.0495 −0.0116267
$$109$$ −1077.66 −0.946986 −0.473493 0.880798i $$-0.657007\pi$$
−0.473493 + 0.880798i $$0.657007\pi$$
$$110$$ 1223.23 1.06028
$$111$$ −464.299 −0.397021
$$112$$ 0 0
$$113$$ −166.065 −0.138248 −0.0691241 0.997608i $$-0.522020\pi$$
−0.0691241 + 0.997608i $$0.522020\pi$$
$$114$$ 135.849 0.111609
$$115$$ 3421.26 2.77421
$$116$$ −141.031 −0.112883
$$117$$ 117.000 0.0924500
$$118$$ −218.418 −0.170398
$$119$$ 0 0
$$120$$ −1359.45 −1.03417
$$121$$ −806.595 −0.606007
$$122$$ −534.254 −0.396468
$$123$$ 755.147 0.553572
$$124$$ 56.6847 0.0410519
$$125$$ −2525.03 −1.80676
$$126$$ 0 0
$$127$$ 1296.16 0.905637 0.452819 0.891603i $$-0.350419\pi$$
0.452819 + 0.891603i $$0.350419\pi$$
$$128$$ −1303.24 −0.899934
$$129$$ −1507.70 −1.02903
$$130$$ 694.415 0.468494
$$131$$ 197.201 0.131523 0.0657617 0.997835i $$-0.479052\pi$$
0.0657617 + 0.997835i $$0.479052\pi$$
$$132$$ −33.2036 −0.0218939
$$133$$ 0 0
$$134$$ −1096.89 −0.707139
$$135$$ −526.049 −0.335371
$$136$$ −1559.09 −0.983018
$$137$$ −546.915 −0.341066 −0.170533 0.985352i $$-0.554549\pi$$
−0.170533 + 0.985352i $$0.554549\pi$$
$$138$$ 1444.30 0.890921
$$139$$ −609.666 −0.372023 −0.186012 0.982548i $$-0.559556\pi$$
−0.186012 + 0.982548i $$0.559556\pi$$
$$140$$ 0 0
$$141$$ 845.199 0.504813
$$142$$ −1448.42 −0.855974
$$143$$ 297.699 0.174090
$$144$$ −539.099 −0.311979
$$145$$ −5685.23 −3.25609
$$146$$ −2013.38 −1.14129
$$147$$ 0 0
$$148$$ 74.8009 0.0415446
$$149$$ −2165.08 −1.19040 −0.595202 0.803576i $$-0.702928\pi$$
−0.595202 + 0.803576i $$0.702928\pi$$
$$150$$ −2094.07 −1.13987
$$151$$ −846.549 −0.456233 −0.228116 0.973634i $$-0.573257\pi$$
−0.228116 + 0.973634i $$0.573257\pi$$
$$152$$ −384.151 −0.204992
$$153$$ −603.300 −0.318784
$$154$$ 0 0
$$155$$ 2285.06 1.18413
$$156$$ −18.8493 −0.00967404
$$157$$ −1653.60 −0.840581 −0.420291 0.907390i $$-0.638072\pi$$
−0.420291 + 0.907390i $$0.638072\pi$$
$$158$$ −310.536 −0.156360
$$159$$ 1101.00 0.549149
$$160$$ 425.550 0.210267
$$161$$ 0 0
$$162$$ −222.074 −0.107702
$$163$$ −2866.51 −1.37744 −0.688720 0.725027i $$-0.741827\pi$$
−0.688720 + 0.725027i $$0.741827\pi$$
$$164$$ −121.658 −0.0579262
$$165$$ −1338.50 −0.631526
$$166$$ −2559.24 −1.19660
$$167$$ −729.066 −0.337825 −0.168913 0.985631i $$-0.554026\pi$$
−0.168913 + 0.985631i $$0.554026\pi$$
$$168$$ 0 0
$$169$$ 169.000 0.0769231
$$170$$ −3580.69 −1.61545
$$171$$ −148.650 −0.0664770
$$172$$ 242.898 0.107679
$$173$$ 3834.83 1.68530 0.842650 0.538462i $$-0.180995\pi$$
0.842650 + 0.538462i $$0.180995\pi$$
$$174$$ −2400.05 −1.04567
$$175$$ 0 0
$$176$$ −1371.70 −0.587476
$$177$$ 238.999 0.101493
$$178$$ 3265.08 1.37488
$$179$$ −283.862 −0.118530 −0.0592649 0.998242i $$-0.518876\pi$$
−0.0592649 + 0.998242i $$0.518876\pi$$
$$180$$ 84.7492 0.0350935
$$181$$ −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
$$184$$ −4084.15 −1.63635
$$185$$ 3015.36 1.19835
$$186$$ 964.650 0.380277
$$187$$ −1535.06 −0.600291
$$188$$ −136.166 −0.0528240
$$189$$ 0 0
$$190$$ −882.265 −0.336875
$$191$$ 2514.26 0.952491 0.476246 0.879312i $$-0.341997\pi$$
0.476246 + 0.879312i $$0.341997\pi$$
$$192$$ 1617.25 0.607889
$$193$$ 2420.73 0.902839 0.451420 0.892312i $$-0.350918\pi$$
0.451420 + 0.892312i $$0.350918\pi$$
$$194$$ 1527.55 0.565320
$$195$$ −759.849 −0.279046
$$196$$ 0 0
$$197$$ −4633.65 −1.67581 −0.837903 0.545819i $$-0.816219\pi$$
−0.837903 + 0.545819i $$0.816219\pi$$
$$198$$ −565.053 −0.202811
$$199$$ −3054.17 −1.08796 −0.543980 0.839098i $$-0.683083\pi$$
−0.543980 + 0.839098i $$0.683083\pi$$
$$200$$ 5921.56 2.09359
$$201$$ 1200.25 0.421188
$$202$$ −786.215 −0.273851
$$203$$ 0 0
$$204$$ 97.1947 0.0333578
$$205$$ −4904.26 −1.67087
$$206$$ −5240.30 −1.77237
$$207$$ −1580.40 −0.530653
$$208$$ −778.699 −0.259582
$$209$$ −378.230 −0.125181
$$210$$ 0 0
$$211$$ −4031.60 −1.31539 −0.657694 0.753285i $$-0.728468\pi$$
−0.657694 + 0.753285i $$0.728468\pi$$
$$212$$ −177.376 −0.0574634
$$213$$ 1584.90 0.509838
$$214$$ −2287.46 −0.730689
$$215$$ 9791.66 3.10598
$$216$$ 627.975 0.197816
$$217$$ 0 0
$$218$$ 2954.59 0.917935
$$219$$ 2203.10 0.679779
$$220$$ 215.638 0.0660834
$$221$$ −871.434 −0.265244
$$222$$ 1272.95 0.384841
$$223$$ −3784.95 −1.13659 −0.568294 0.822826i $$-0.692396\pi$$
−0.568294 + 0.822826i $$0.692396\pi$$
$$224$$ 0 0
$$225$$ 2291.40 0.678932
$$226$$ 455.292 0.134007
$$227$$ −2013.83 −0.588821 −0.294411 0.955679i $$-0.595123\pi$$
−0.294411 + 0.955679i $$0.595123\pi$$
$$228$$ 23.9483 0.00695620
$$229$$ 3050.73 0.880340 0.440170 0.897915i $$-0.354918\pi$$
0.440170 + 0.897915i $$0.354918\pi$$
$$230$$ −9379.93 −2.68910
$$231$$ 0 0
$$232$$ 6786.78 1.92058
$$233$$ 5587.49 1.57103 0.785513 0.618846i $$-0.212399\pi$$
0.785513 + 0.618846i $$0.212399\pi$$
$$234$$ −320.774 −0.0896139
$$235$$ −5489.09 −1.52370
$$236$$ −38.5039 −0.0106203
$$237$$ 339.798 0.0931317
$$238$$ 0 0
$$239$$ −1335.69 −0.361501 −0.180750 0.983529i $$-0.557853\pi$$
−0.180750 + 0.983529i $$0.557853\pi$$
$$240$$ 3501.15 0.941658
$$241$$ 571.558 0.152769 0.0763845 0.997078i $$-0.475662\pi$$
0.0763845 + 0.997078i $$0.475662\pi$$
$$242$$ 2211.41 0.587416
$$243$$ 243.000 0.0641500
$$244$$ −94.1813 −0.0247104
$$245$$ 0 0
$$246$$ −2070.36 −0.536590
$$247$$ −214.717 −0.0553122
$$248$$ −2727.81 −0.698452
$$249$$ 2800.40 0.712723
$$250$$ 6922.76 1.75134
$$251$$ −4088.60 −1.02817 −0.514084 0.857740i $$-0.671868\pi$$
−0.514084 + 0.857740i $$0.671868\pi$$
$$252$$ 0 0
$$253$$ −4021.21 −0.999254
$$254$$ −3553.64 −0.877854
$$255$$ 3918.10 0.962199
$$256$$ −739.607 −0.180568
$$257$$ −3050.23 −0.740342 −0.370171 0.928964i $$-0.620701\pi$$
−0.370171 + 0.928964i $$0.620701\pi$$
$$258$$ 4133.59 0.997466
$$259$$ 0 0
$$260$$ 122.415 0.0291996
$$261$$ 2626.20 0.622826
$$262$$ −540.659 −0.127489
$$263$$ 5770.99 1.35306 0.676530 0.736415i $$-0.263483\pi$$
0.676530 + 0.736415i $$0.263483\pi$$
$$264$$ 1597.84 0.372501
$$265$$ −7150.35 −1.65752
$$266$$ 0 0
$$267$$ −3572.74 −0.818908
$$268$$ −193.365 −0.0440734
$$269$$ 2079.40 0.471314 0.235657 0.971836i $$-0.424276\pi$$
0.235657 + 0.971836i $$0.424276\pi$$
$$270$$ 1442.25 0.325083
$$271$$ −6012.00 −1.34761 −0.673807 0.738908i $$-0.735342\pi$$
−0.673807 + 0.738908i $$0.735342\pi$$
$$272$$ 4015.29 0.895084
$$273$$ 0 0
$$274$$ 1499.45 0.330603
$$275$$ 5830.30 1.27847
$$276$$ 254.610 0.0555279
$$277$$ −735.201 −0.159473 −0.0797364 0.996816i $$-0.525408\pi$$
−0.0797364 + 0.996816i $$0.525408\pi$$
$$278$$ 1671.50 0.360610
$$279$$ −1055.55 −0.226502
$$280$$ 0 0
$$281$$ −1902.92 −0.403981 −0.201990 0.979387i $$-0.564741\pi$$
−0.201990 + 0.979387i $$0.564741\pi$$
$$282$$ −2317.25 −0.489326
$$283$$ −2125.71 −0.446502 −0.223251 0.974761i $$-0.571667\pi$$
−0.223251 + 0.974761i $$0.571667\pi$$
$$284$$ −255.335 −0.0533498
$$285$$ 965.399 0.200650
$$286$$ −816.187 −0.168749
$$287$$ 0 0
$$288$$ −196.576 −0.0402200
$$289$$ −419.527 −0.0853913
$$290$$ 15586.9 3.15620
$$291$$ −1671.49 −0.336717
$$292$$ −354.930 −0.0711325
$$293$$ 1641.03 0.327200 0.163600 0.986527i $$-0.447689\pi$$
0.163600 + 0.986527i $$0.447689\pi$$
$$294$$ 0 0
$$295$$ −1552.16 −0.306341
$$296$$ −3599.61 −0.706835
$$297$$ 618.297 0.120799
$$298$$ 5935.91 1.15389
$$299$$ −2282.79 −0.441530
$$300$$ −369.155 −0.0710439
$$301$$ 0 0
$$302$$ 2320.95 0.442237
$$303$$ 860.299 0.163112
$$304$$ 989.348 0.186655
$$305$$ −3796.62 −0.712767
$$306$$ 1654.04 0.309004
$$307$$ 3373.27 0.627111 0.313555 0.949570i $$-0.398480\pi$$
0.313555 + 0.949570i $$0.398480\pi$$
$$308$$ 0 0
$$309$$ 5734.09 1.05567
$$310$$ −6264.86 −1.14781
$$311$$ 868.525 0.158359 0.0791793 0.996860i $$-0.474770\pi$$
0.0791793 + 0.996860i $$0.474770\pi$$
$$312$$ 907.075 0.164593
$$313$$ 4343.19 0.784319 0.392159 0.919897i $$-0.371728\pi$$
0.392159 + 0.919897i $$0.371728\pi$$
$$314$$ 4533.59 0.814794
$$315$$ 0 0
$$316$$ −54.7431 −0.00974537
$$317$$ −3277.65 −0.580730 −0.290365 0.956916i $$-0.593777\pi$$
−0.290365 + 0.956916i $$0.593777\pi$$
$$318$$ −3018.56 −0.532302
$$319$$ 6682.18 1.17282
$$320$$ −10503.1 −1.83482
$$321$$ 2503.00 0.435215
$$322$$ 0 0
$$323$$ 1107.17 0.190726
$$324$$ −39.1485 −0.00671271
$$325$$ 3309.79 0.564906
$$326$$ 7859.00 1.33518
$$327$$ −3232.99 −0.546743
$$328$$ 5854.49 0.985550
$$329$$ 0 0
$$330$$ 3669.70 0.612153
$$331$$ 5589.62 0.928197 0.464099 0.885784i $$-0.346378\pi$$
0.464099 + 0.885784i $$0.346378\pi$$
$$332$$ −451.158 −0.0745798
$$333$$ −1392.90 −0.229220
$$334$$ 1998.85 0.327461
$$335$$ −7794.92 −1.27129
$$336$$ 0 0
$$337$$ 901.544 0.145728 0.0728638 0.997342i $$-0.476786\pi$$
0.0728638 + 0.997342i $$0.476786\pi$$
$$338$$ −463.340 −0.0745633
$$339$$ −498.194 −0.0798176
$$340$$ −631.225 −0.100685
$$341$$ −2685.77 −0.426518
$$342$$ 407.548 0.0644376
$$343$$ 0 0
$$344$$ −11688.9 −1.83204
$$345$$ 10263.8 1.60169
$$346$$ −10513.8 −1.63360
$$347$$ −812.318 −0.125670 −0.0628350 0.998024i $$-0.520014\pi$$
−0.0628350 + 0.998024i $$0.520014\pi$$
$$348$$ −423.093 −0.0651730
$$349$$ −4437.96 −0.680683 −0.340342 0.940302i $$-0.610543\pi$$
−0.340342 + 0.940302i $$0.610543\pi$$
$$350$$ 0 0
$$351$$ 351.000 0.0533761
$$352$$ −500.174 −0.0757368
$$353$$ −7115.35 −1.07284 −0.536419 0.843952i $$-0.680223\pi$$
−0.536419 + 0.843952i $$0.680223\pi$$
$$354$$ −655.253 −0.0983794
$$355$$ −10293.0 −1.53886
$$356$$ 575.587 0.0856911
$$357$$ 0 0
$$358$$ 778.253 0.114894
$$359$$ 4693.98 0.690081 0.345040 0.938588i $$-0.387865\pi$$
0.345040 + 0.938588i $$0.387865\pi$$
$$360$$ −4078.35 −0.597077
$$361$$ −6586.20 −0.960227
$$362$$ 6480.17 0.940857
$$363$$ −2419.79 −0.349878
$$364$$ 0 0
$$365$$ −14307.9 −2.05181
$$366$$ −1602.76 −0.228901
$$367$$ −9243.98 −1.31480 −0.657400 0.753542i $$-0.728344\pi$$
−0.657400 + 0.753542i $$0.728344\pi$$
$$368$$ 10518.4 1.48997
$$369$$ 2265.44 0.319605
$$370$$ −8267.09 −1.16158
$$371$$ 0 0
$$372$$ 170.054 0.0237013
$$373$$ −4311.99 −0.598569 −0.299285 0.954164i $$-0.596748\pi$$
−0.299285 + 0.954164i $$0.596748\pi$$
$$374$$ 4208.60 0.581876
$$375$$ −7575.09 −1.04314
$$376$$ 6552.64 0.898741
$$377$$ 3793.40 0.518223
$$378$$ 0 0
$$379$$ −2382.73 −0.322936 −0.161468 0.986878i $$-0.551623\pi$$
−0.161468 + 0.986878i $$0.551623\pi$$
$$380$$ −155.531 −0.0209962
$$381$$ 3888.49 0.522870
$$382$$ −6893.25 −0.923271
$$383$$ −4845.81 −0.646499 −0.323250 0.946314i $$-0.604775\pi$$
−0.323250 + 0.946314i $$0.604775\pi$$
$$384$$ −3909.73 −0.519577
$$385$$ 0 0
$$386$$ −6636.81 −0.875142
$$387$$ −4523.10 −0.594113
$$388$$ 269.286 0.0352343
$$389$$ 9561.50 1.24624 0.623120 0.782127i $$-0.285865\pi$$
0.623120 + 0.782127i $$0.285865\pi$$
$$390$$ 2083.25 0.270485
$$391$$ 11771.0 1.52247
$$392$$ 0 0
$$393$$ 591.604 0.0759350
$$394$$ 12703.9 1.62440
$$395$$ −2206.79 −0.281103
$$396$$ −99.6107 −0.0126405
$$397$$ 7440.11 0.940575 0.470287 0.882513i $$-0.344150\pi$$
0.470287 + 0.882513i $$0.344150\pi$$
$$398$$ 8373.48 1.05458
$$399$$ 0 0
$$400$$ −15250.5 −1.90631
$$401$$ −8687.80 −1.08192 −0.540958 0.841050i $$-0.681938\pi$$
−0.540958 + 0.841050i $$0.681938\pi$$
$$402$$ −3290.66 −0.408267
$$403$$ −1524.68 −0.188461
$$404$$ −138.598 −0.0170681
$$405$$ −1578.15 −0.193627
$$406$$ 0 0
$$407$$ −3544.13 −0.431637
$$408$$ −4677.26 −0.567546
$$409$$ −2556.10 −0.309024 −0.154512 0.987991i $$-0.549381\pi$$
−0.154512 + 0.987991i $$0.549381\pi$$
$$410$$ 13445.8 1.61961
$$411$$ −1640.74 −0.196915
$$412$$ −923.790 −0.110466
$$413$$ 0 0
$$414$$ 4332.90 0.514374
$$415$$ −18187.0 −2.15124
$$416$$ −283.943 −0.0334650
$$417$$ −1829.00 −0.214788
$$418$$ 1036.98 0.121340
$$419$$ 3347.46 0.390296 0.195148 0.980774i $$-0.437481\pi$$
0.195148 + 0.980774i $$0.437481\pi$$
$$420$$ 0 0
$$421$$ −1854.48 −0.214684 −0.107342 0.994222i $$-0.534234\pi$$
−0.107342 + 0.994222i $$0.534234\pi$$
$$422$$ 11053.3 1.27503
$$423$$ 2535.60 0.291454
$$424$$ 8535.79 0.977675
$$425$$ −17066.7 −1.94789
$$426$$ −4345.25 −0.494197
$$427$$ 0 0
$$428$$ −403.246 −0.0455412
$$429$$ 893.096 0.100511
$$430$$ −26845.4 −3.01069
$$431$$ −14043.1 −1.56945 −0.784725 0.619844i $$-0.787196\pi$$
−0.784725 + 0.619844i $$0.787196\pi$$
$$432$$ −1617.30 −0.180121
$$433$$ −3086.47 −0.342555 −0.171278 0.985223i $$-0.554790\pi$$
−0.171278 + 0.985223i $$0.554790\pi$$
$$434$$ 0 0
$$435$$ −17055.7 −1.87990
$$436$$ 520.851 0.0572116
$$437$$ 2900.32 0.317486
$$438$$ −6040.14 −0.658925
$$439$$ −2837.68 −0.308508 −0.154254 0.988031i $$-0.549297\pi$$
−0.154254 + 0.988031i $$0.549297\pi$$
$$440$$ −10377.1 −1.12434
$$441$$ 0 0
$$442$$ 2389.17 0.257107
$$443$$ 18309.4 1.96367 0.981834 0.189744i $$-0.0607658\pi$$
0.981834 + 0.189744i $$0.0607658\pi$$
$$444$$ 224.403 0.0239858
$$445$$ 23203.0 2.47174
$$446$$ 10377.0 1.10172
$$447$$ −6495.24 −0.687281
$$448$$ 0 0
$$449$$ 13861.2 1.45690 0.728451 0.685098i $$-0.240241\pi$$
0.728451 + 0.685098i $$0.240241\pi$$
$$450$$ −6282.22 −0.658104
$$451$$ 5764.26 0.601837
$$452$$ 80.2614 0.00835217
$$453$$ −2539.65 −0.263406
$$454$$ 5521.23 0.570758
$$455$$ 0 0
$$456$$ −1152.45 −0.118352
$$457$$ −8990.36 −0.920243 −0.460122 0.887856i $$-0.652194\pi$$
−0.460122 + 0.887856i $$0.652194\pi$$
$$458$$ −8364.05 −0.853333
$$459$$ −1809.90 −0.184050
$$460$$ −1653.55 −0.167602
$$461$$ 3406.90 0.344198 0.172099 0.985080i $$-0.444945\pi$$
0.172099 + 0.985080i $$0.444945\pi$$
$$462$$ 0 0
$$463$$ −7498.45 −0.752662 −0.376331 0.926485i $$-0.622814\pi$$
−0.376331 + 0.926485i $$0.622814\pi$$
$$464$$ −17478.8 −1.74878
$$465$$ 6855.19 0.683660
$$466$$ −15319.0 −1.52283
$$467$$ −7711.38 −0.764112 −0.382056 0.924139i $$-0.624784\pi$$
−0.382056 + 0.924139i $$0.624784\pi$$
$$468$$ −56.5478 −0.00558531
$$469$$ 0 0
$$470$$ 15049.2 1.47695
$$471$$ −4960.79 −0.485310
$$472$$ 1852.91 0.180693
$$473$$ −11508.7 −1.11875
$$474$$ −931.608 −0.0902747
$$475$$ −4205.14 −0.406200
$$476$$ 0 0
$$477$$ 3302.99 0.317051
$$478$$ 3662.01 0.350411
$$479$$ 9439.82 0.900451 0.450226 0.892915i $$-0.351344\pi$$
0.450226 + 0.892915i $$0.351344\pi$$
$$480$$ 1276.65 0.121398
$$481$$ −2011.96 −0.190723
$$482$$ −1567.02 −0.148082
$$483$$ 0 0
$$484$$ 389.839 0.0366115
$$485$$ 10855.4 1.01633
$$486$$ −666.223 −0.0621821
$$487$$ −6156.20 −0.572821 −0.286411 0.958107i $$-0.592462\pi$$
−0.286411 + 0.958107i $$0.592462\pi$$
$$488$$ 4532.25 0.420420
$$489$$ −8599.54 −0.795265
$$490$$ 0 0
$$491$$ 3842.74 0.353198 0.176599 0.984283i $$-0.443490\pi$$
0.176599 + 0.984283i $$0.443490\pi$$
$$492$$ −364.974 −0.0334437
$$493$$ −19560.3 −1.78692
$$494$$ 588.680 0.0536153
$$495$$ −4015.49 −0.364612
$$496$$ 7025.24 0.635973
$$497$$ 0 0
$$498$$ −7677.73 −0.690858
$$499$$ −12842.4 −1.15211 −0.576056 0.817410i $$-0.695409\pi$$
−0.576056 + 0.817410i $$0.695409\pi$$
$$500$$ 1220.38 0.109154
$$501$$ −2187.20 −0.195043
$$502$$ 11209.5 0.996627
$$503$$ −8580.11 −0.760573 −0.380287 0.924869i $$-0.624175\pi$$
−0.380287 + 0.924869i $$0.624175\pi$$
$$504$$ 0 0
$$505$$ −5587.16 −0.492327
$$506$$ 11024.8 0.968599
$$507$$ 507.000 0.0444116
$$508$$ −626.455 −0.0547135
$$509$$ 43.5957 0.00379635 0.00189818 0.999998i $$-0.499396\pi$$
0.00189818 + 0.999998i $$0.499396\pi$$
$$510$$ −10742.1 −0.932681
$$511$$ 0 0
$$512$$ 12453.7 1.07496
$$513$$ −445.951 −0.0383805
$$514$$ 8362.68 0.717630
$$515$$ −37239.7 −3.18636
$$516$$ 728.693 0.0621684
$$517$$ 6451.66 0.548827
$$518$$ 0 0
$$519$$ 11504.5 0.973008
$$520$$ −5890.94 −0.496798
$$521$$ −11368.1 −0.955939 −0.477969 0.878377i $$-0.658627\pi$$
−0.477969 + 0.878377i $$0.658627\pi$$
$$522$$ −7200.14 −0.603719
$$523$$ 5229.53 0.437230 0.218615 0.975811i $$-0.429846\pi$$
0.218615 + 0.975811i $$0.429846\pi$$
$$524$$ −95.3103 −0.00794590
$$525$$ 0 0
$$526$$ −15822.1 −1.31155
$$527$$ 7861.88 0.649846
$$528$$ −4115.10 −0.339180
$$529$$ 18668.2 1.53433
$$530$$ 19603.8 1.60667
$$531$$ 716.997 0.0585970
$$532$$ 0 0
$$533$$ 3272.31 0.265927
$$534$$ 9795.24 0.793786
$$535$$ −16255.6 −1.31363
$$536$$ 9305.24 0.749860
$$537$$ −851.586 −0.0684333
$$538$$ −5701.01 −0.456855
$$539$$ 0 0
$$540$$ 254.247 0.0202612
$$541$$ −6567.99 −0.521959 −0.260980 0.965344i $$-0.584046\pi$$
−0.260980 + 0.965344i $$0.584046\pi$$
$$542$$ 16482.9 1.30627
$$543$$ −7090.79 −0.560396
$$544$$ 1464.13 0.115393
$$545$$ 20996.5 1.65026
$$546$$ 0 0
$$547$$ −13675.7 −1.06897 −0.534487 0.845177i $$-0.679495\pi$$
−0.534487 + 0.845177i $$0.679495\pi$$
$$548$$ 264.332 0.0206053
$$549$$ 1753.79 0.136339
$$550$$ −15984.7 −1.23925
$$551$$ −4819.57 −0.372632
$$552$$ −12252.5 −0.944745
$$553$$ 0 0
$$554$$ 2015.67 0.154581
$$555$$ 9046.09 0.691865
$$556$$ 294.661 0.0224755
$$557$$ 4527.96 0.344445 0.172222 0.985058i $$-0.444905\pi$$
0.172222 + 0.985058i $$0.444905\pi$$
$$558$$ 2893.95 0.219553
$$559$$ −6533.36 −0.494332
$$560$$ 0 0
$$561$$ −4605.17 −0.346578
$$562$$ 5217.15 0.391588
$$563$$ −18441.8 −1.38051 −0.690256 0.723566i $$-0.742502\pi$$
−0.690256 + 0.723566i $$0.742502\pi$$
$$564$$ −408.497 −0.0304979
$$565$$ 3235.49 0.240917
$$566$$ 5827.96 0.432804
$$567$$ 0 0
$$568$$ 12287.4 0.907687
$$569$$ −13553.5 −0.998578 −0.499289 0.866436i $$-0.666405\pi$$
−0.499289 + 0.866436i $$0.666405\pi$$
$$570$$ −2646.79 −0.194495
$$571$$ 14815.5 1.08583 0.542915 0.839788i $$-0.317321\pi$$
0.542915 + 0.839788i $$0.317321\pi$$
$$572$$ −143.882 −0.0105175
$$573$$ 7542.79 0.549921
$$574$$ 0 0
$$575$$ −44707.6 −3.24249
$$576$$ 4851.74 0.350965
$$577$$ −21596.2 −1.55816 −0.779081 0.626923i $$-0.784314\pi$$
−0.779081 + 0.626923i $$0.784314\pi$$
$$578$$ 1150.20 0.0827716
$$579$$ 7262.19 0.521254
$$580$$ 2747.75 0.196714
$$581$$ 0 0
$$582$$ 4582.66 0.326387
$$583$$ 8404.23 0.597029
$$584$$ 17080.1 1.21024
$$585$$ −2279.55 −0.161107
$$586$$ −4499.13 −0.317163
$$587$$ 918.801 0.0646047 0.0323024 0.999478i $$-0.489716\pi$$
0.0323024 + 0.999478i $$0.489716\pi$$
$$588$$ 0 0
$$589$$ 1937.13 0.135514
$$590$$ 4255.50 0.296943
$$591$$ −13900.9 −0.967527
$$592$$ 9270.49 0.643606
$$593$$ −19816.0 −1.37226 −0.686128 0.727481i $$-0.740691\pi$$
−0.686128 + 0.727481i $$0.740691\pi$$
$$594$$ −1695.16 −0.117093
$$595$$ 0 0
$$596$$ 1046.42 0.0719175
$$597$$ −9162.50 −0.628134
$$598$$ 6258.64 0.427985
$$599$$ −5141.86 −0.350736 −0.175368 0.984503i $$-0.556111\pi$$
−0.175368 + 0.984503i $$0.556111\pi$$
$$600$$ 17764.7 1.20873
$$601$$ −12380.9 −0.840312 −0.420156 0.907452i $$-0.638025\pi$$
−0.420156 + 0.907452i $$0.638025\pi$$
$$602$$ 0 0
$$603$$ 3600.74 0.243173
$$604$$ 409.150 0.0275630
$$605$$ 15715.1 1.05605
$$606$$ −2358.65 −0.158108
$$607$$ 23717.0 1.58590 0.792951 0.609286i $$-0.208544\pi$$
0.792951 + 0.609286i $$0.208544\pi$$
$$608$$ 360.754 0.0240633
$$609$$ 0 0
$$610$$ 10409.0 0.690901
$$611$$ 3662.53 0.242504
$$612$$ 291.584 0.0192591
$$613$$ −26157.1 −1.72345 −0.861726 0.507373i $$-0.830617\pi$$
−0.861726 + 0.507373i $$0.830617\pi$$
$$614$$ −9248.36 −0.607872
$$615$$ −14712.8 −0.964677
$$616$$ 0 0
$$617$$ 23613.9 1.54077 0.770387 0.637576i $$-0.220063\pi$$
0.770387 + 0.637576i $$0.220063\pi$$
$$618$$ −15720.9 −1.02328
$$619$$ −23345.4 −1.51588 −0.757940 0.652324i $$-0.773794\pi$$
−0.757940 + 0.652324i $$0.773794\pi$$
$$620$$ −1104.40 −0.0715387
$$621$$ −4741.19 −0.306372
$$622$$ −2381.20 −0.153501
$$623$$ 0 0
$$624$$ −2336.10 −0.149870
$$625$$ 17371.0 1.11174
$$626$$ −11907.5 −0.760258
$$627$$ −1134.69 −0.0722730
$$628$$ 799.207 0.0507832
$$629$$ 10374.5 0.657645
$$630$$ 0 0
$$631$$ 15245.7 0.961841 0.480921 0.876764i $$-0.340302\pi$$
0.480921 + 0.876764i $$0.340302\pi$$
$$632$$ 2634.38 0.165807
$$633$$ −12094.8 −0.759439
$$634$$ 8986.20 0.562914
$$635$$ −25253.6 −1.57820
$$636$$ −532.128 −0.0331765
$$637$$ 0 0
$$638$$ −18320.3 −1.13684
$$639$$ 4754.69 0.294355
$$640$$ 25391.5 1.56826
$$641$$ 10192.7 0.628063 0.314032 0.949413i $$-0.398320\pi$$
0.314032 + 0.949413i $$0.398320\pi$$
$$642$$ −6862.37 −0.421863
$$643$$ 5506.31 0.337710 0.168855 0.985641i $$-0.445993\pi$$
0.168855 + 0.985641i $$0.445993\pi$$
$$644$$ 0 0
$$645$$ 29375.0 1.79324
$$646$$ −3035.48 −0.184875
$$647$$ 13297.5 0.808005 0.404003 0.914758i $$-0.367619\pi$$
0.404003 + 0.914758i $$0.367619\pi$$
$$648$$ 1883.93 0.114209
$$649$$ 1824.35 0.110342
$$650$$ −9074.32 −0.547576
$$651$$ 0 0
$$652$$ 1385.43 0.0832171
$$653$$ −12440.2 −0.745519 −0.372760 0.927928i $$-0.621588\pi$$
−0.372760 + 0.927928i $$0.621588\pi$$
$$654$$ 8863.76 0.529970
$$655$$ −3842.14 −0.229198
$$656$$ −15077.7 −0.897389
$$657$$ 6609.29 0.392470
$$658$$ 0 0
$$659$$ −9562.87 −0.565276 −0.282638 0.959227i $$-0.591209\pi$$
−0.282638 + 0.959227i $$0.591209\pi$$
$$660$$ 646.915 0.0381533
$$661$$ −2409.69 −0.141795 −0.0708973 0.997484i $$-0.522586\pi$$
−0.0708973 + 0.997484i $$0.522586\pi$$
$$662$$ −15324.8 −0.899722
$$663$$ −2614.30 −0.153139
$$664$$ 21710.9 1.26889
$$665$$ 0 0
$$666$$ 3818.85 0.222188
$$667$$ −51239.9 −2.97454
$$668$$ 352.368 0.0204095
$$669$$ −11354.9 −0.656209
$$670$$ 21371.0 1.23229
$$671$$ 4462.40 0.256735
$$672$$ 0 0
$$673$$ 7929.02 0.454147 0.227074 0.973878i $$-0.427084\pi$$
0.227074 + 0.973878i $$0.427084\pi$$
$$674$$ −2471.72 −0.141257
$$675$$ 6874.19 0.391982
$$676$$ −81.6802 −0.00464726
$$677$$ 2628.26 0.149206 0.0746030 0.997213i $$-0.476231\pi$$
0.0746030 + 0.997213i $$0.476231\pi$$
$$678$$ 1365.88 0.0773690
$$679$$ 0 0
$$680$$ 30376.1 1.71305
$$681$$ −6041.48 −0.339956
$$682$$ 7363.46 0.413433
$$683$$ 10021.5 0.561437 0.280719 0.959790i $$-0.409427\pi$$
0.280719 + 0.959790i $$0.409427\pi$$
$$684$$ 71.8448 0.00401616
$$685$$ 10655.7 0.594356
$$686$$ 0 0
$$687$$ 9152.18 0.508264
$$688$$ 30103.7 1.66816
$$689$$ 4770.99 0.263803
$$690$$ −28139.8 −1.55256
$$691$$ −23987.2 −1.32057 −0.660286 0.751014i $$-0.729565\pi$$
−0.660286 + 0.751014i $$0.729565\pi$$
$$692$$ −1853.43 −0.101816
$$693$$ 0 0
$$694$$ 2227.10 0.121815
$$695$$ 11878.3 0.648303
$$696$$ 20360.3 1.10885
$$697$$ −16873.4 −0.916964
$$698$$ 12167.4 0.659801
$$699$$ 16762.5 0.907032
$$700$$ 0 0
$$701$$ −3763.71 −0.202787 −0.101393 0.994846i $$-0.532330\pi$$
−0.101393 + 0.994846i $$0.532330\pi$$
$$702$$ −962.322 −0.0517386
$$703$$ 2556.23 0.137141
$$704$$ 12344.9 0.660890
$$705$$ −16467.3 −0.879707
$$706$$ 19507.8 1.03993
$$707$$ 0 0
$$708$$ −115.512 −0.00613163
$$709$$ −36047.8 −1.90946 −0.954728 0.297479i $$-0.903854\pi$$
−0.954728 + 0.297479i $$0.903854\pi$$
$$710$$ 28219.9 1.49166
$$711$$ 1019.39 0.0537696
$$712$$ −27698.7 −1.45794
$$713$$ 20594.9 1.08174
$$714$$ 0 0
$$715$$ −5800.15 −0.303376
$$716$$ 137.195 0.00716090
$$717$$ −4007.08 −0.208713
$$718$$ −12869.3 −0.668910
$$719$$ 3944.18 0.204580 0.102290 0.994755i $$-0.467383\pi$$
0.102290 + 0.994755i $$0.467383\pi$$
$$720$$ 10503.4 0.543667
$$721$$ 0 0
$$722$$ 18057.1 0.930770
$$723$$ 1714.68 0.0882012
$$724$$ 1142.36 0.0586402
$$725$$ 74292.1 3.80571
$$726$$ 6634.22 0.339145
$$727$$ 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
$$730$$ 39227.3 1.98886
$$731$$ 33688.7 1.70454
$$732$$ −282.544 −0.0142666
$$733$$ 13536.2 0.682089 0.341045 0.940047i $$-0.389219\pi$$
0.341045 + 0.940047i $$0.389219\pi$$
$$734$$ 25343.8 1.27447
$$735$$ 0 0
$$736$$ 3835.40 0.192085
$$737$$ 9161.83 0.457911
$$738$$ −6211.07 −0.309800
$$739$$ 15839.1 0.788433 0.394217 0.919018i $$-0.371016\pi$$
0.394217 + 0.919018i $$0.371016\pi$$
$$740$$ −1457.37 −0.0723972
$$741$$ −644.151 −0.0319345
$$742$$ 0 0
$$743$$ −1664.92 −0.0822075 −0.0411037 0.999155i $$-0.513087\pi$$
−0.0411037 + 0.999155i $$0.513087\pi$$
$$744$$ −8183.43 −0.403252
$$745$$ 42182.9 2.07445
$$746$$ 11822.0 0.580206
$$747$$ 8401.19 0.411491
$$748$$ 741.916 0.0362662
$$749$$ 0 0
$$750$$ 20768.3 1.01113
$$751$$ 22399.1 1.08835 0.544177 0.838970i $$-0.316842\pi$$
0.544177 + 0.838970i $$0.316842\pi$$
$$752$$ −16875.8 −0.818346
$$753$$ −12265.8 −0.593613
$$754$$ −10400.2 −0.502325
$$755$$ 16493.6 0.795050
$$756$$ 0 0
$$757$$ 23798.9 1.14265 0.571326 0.820723i $$-0.306429\pi$$
0.571326 + 0.820723i $$0.306429\pi$$
$$758$$ 6532.64 0.313029
$$759$$ −12063.6 −0.576920
$$760$$ 7484.53 0.357227
$$761$$ −13693.5 −0.652285 −0.326142 0.945321i $$-0.605749\pi$$
−0.326142 + 0.945321i $$0.605749\pi$$
$$762$$ −10660.9 −0.506829
$$763$$ 0 0
$$764$$ −1215.18 −0.0575441
$$765$$ 11754.3 0.555526
$$766$$ 13285.5 0.626666
$$767$$ 1035.66 0.0487556
$$768$$ −2218.82 −0.104251
$$769$$ −16299.9 −0.764358 −0.382179 0.924088i $$-0.624826\pi$$
−0.382179 + 0.924088i $$0.624826\pi$$
$$770$$ 0 0
$$771$$ −9150.68 −0.427437
$$772$$ −1169.97 −0.0545444
$$773$$ −33532.2 −1.56024 −0.780122 0.625628i $$-0.784843\pi$$
−0.780122 + 0.625628i $$0.784843\pi$$
$$774$$ 12400.8 0.575887
$$775$$ −29860.2 −1.38401
$$776$$ −12958.7 −0.599473
$$777$$ 0 0
$$778$$ −26214.3 −1.20801
$$779$$ −4157.51 −0.191217
$$780$$ 367.246 0.0168584
$$781$$ 12098.0 0.554290
$$782$$ −32272.1 −1.47577
$$783$$ 7878.59 0.359589
$$784$$ 0 0
$$785$$ 32217.5 1.46483
$$786$$ −1621.98 −0.0736055
$$787$$ −16163.3 −0.732097 −0.366049 0.930596i $$-0.619290\pi$$
−0.366049 + 0.930596i $$0.619290\pi$$
$$788$$ 2239.51 0.101243
$$789$$ 17313.0 0.781189
$$790$$ 6050.27 0.272480
$$791$$ 0 0
$$792$$ 4793.52 0.215064
$$793$$ 2533.25 0.113441
$$794$$ −20398.2 −0.911720
$$795$$ −21451.1 −0.956970
$$796$$ 1476.12 0.0657284
$$797$$ 39636.4 1.76160 0.880798 0.473492i $$-0.157007\pi$$
0.880798 + 0.473492i $$0.157007\pi$$
$$798$$ 0 0
$$799$$ −18885.5 −0.836197
$$800$$ −5560.90 −0.245760
$$801$$ −10718.2 −0.472797
$$802$$ 23819.0 1.04872
$$803$$ 16816.9 0.739048
$$804$$ −580.096 −0.0254458
$$805$$ 0 0
$$806$$ 4180.15 0.182679
$$807$$ 6238.21 0.272113
$$808$$ 6669.71 0.290396
$$809$$ −23811.2 −1.03481 −0.517403 0.855742i $$-0.673101\pi$$
−0.517403 + 0.855742i $$0.673101\pi$$
$$810$$ 4326.74 0.187687
$$811$$ −27218.6 −1.17851 −0.589256 0.807946i $$-0.700579\pi$$
−0.589256 + 0.807946i $$0.700579\pi$$
$$812$$ 0 0
$$813$$ −18036.0 −0.778045
$$814$$ 9716.80 0.418395
$$815$$ 55849.2 2.40038
$$816$$ 12045.9 0.516777
$$817$$ 8300.73 0.355454
$$818$$ 7007.94 0.299544
$$819$$ 0 0
$$820$$ 2370.30 0.100944
$$821$$ −43094.8 −1.83193 −0.915967 0.401253i $$-0.868575\pi$$
−0.915967 + 0.401253i $$0.868575\pi$$
$$822$$ 4498.36 0.190874
$$823$$ 26541.1 1.12414 0.562068 0.827091i $$-0.310006\pi$$
0.562068 + 0.827091i $$0.310006\pi$$
$$824$$ 44455.1 1.87945
$$825$$ 17490.9 0.738127
$$826$$ 0 0
$$827$$ −44898.7 −1.88788 −0.943942 0.330112i $$-0.892913\pi$$
−0.943942 + 0.330112i $$0.892913\pi$$
$$828$$ 763.829 0.0320590
$$829$$ 7137.48 0.299029 0.149514 0.988760i $$-0.452229\pi$$
0.149514 + 0.988760i $$0.452229\pi$$
$$830$$ 49862.6 2.08525
$$831$$ −2205.60 −0.0920717
$$832$$ 7008.06 0.292020
$$833$$ 0 0
$$834$$ 5014.49 0.208198
$$835$$ 14204.6 0.588708
$$836$$ 182.804 0.00756270
$$837$$ −3166.64 −0.130771
$$838$$ −9177.59 −0.378323
$$839$$ 4387.17 0.180527 0.0902634 0.995918i $$-0.471229\pi$$
0.0902634 + 0.995918i $$0.471229\pi$$
$$840$$ 0 0
$$841$$ 60758.1 2.49121
$$842$$ 5084.36 0.208098
$$843$$ −5708.76 −0.233239
$$844$$ 1948.53 0.0794683
$$845$$ −3292.68 −0.134049
$$846$$ −6951.74 −0.282513
$$847$$ 0 0
$$848$$ −21983.2 −0.890219
$$849$$ −6377.12 −0.257788
$$850$$ 46791.0 1.88814
$$851$$ 27176.9 1.09473
$$852$$ −766.004 −0.0308015
$$853$$ 9328.85 0.374459 0.187230 0.982316i $$-0.440049\pi$$
0.187230 + 0.982316i $$0.440049\pi$$
$$854$$ 0 0
$$855$$ 2896.20 0.115845
$$856$$ 19405.2 0.774833
$$857$$ 5010.39 0.199710 0.0998552 0.995002i $$-0.468162\pi$$
0.0998552 + 0.995002i $$0.468162\pi$$
$$858$$ −2448.56 −0.0974272
$$859$$ −30233.4 −1.20088 −0.600438 0.799672i $$-0.705007\pi$$
−0.600438 + 0.799672i $$0.705007\pi$$
$$860$$ −4732.45 −0.187646
$$861$$ 0 0
$$862$$ 38501.4 1.52130
$$863$$ 4334.93 0.170988 0.0854940 0.996339i $$-0.472753\pi$$
0.0854940 + 0.996339i $$0.472753\pi$$
$$864$$ −589.728 −0.0232210
$$865$$ −74715.2 −2.93687
$$866$$ 8462.05 0.332047
$$867$$ −1258.58 −0.0493007
$$868$$ 0 0
$$869$$ 2593.78 0.101252
$$870$$ 46760.8 1.82223
$$871$$ 5201.06 0.202332
$$872$$ −25064.7 −0.973391
$$873$$ −5014.48 −0.194404
$$874$$ −7951.69 −0.307746
$$875$$ 0 0
$$876$$ −1064.79 −0.0410684
$$877$$ 34683.3 1.33543 0.667716 0.744416i $$-0.267272\pi$$
0.667716 + 0.744416i $$0.267272\pi$$
$$878$$ 7779.95 0.299044
$$879$$ 4923.08 0.188909
$$880$$ 26725.3 1.02376
$$881$$ 18269.2 0.698642 0.349321 0.937003i $$-0.386412\pi$$
0.349321 + 0.937003i $$0.386412\pi$$
$$882$$ 0 0
$$883$$ −14592.0 −0.556128 −0.278064 0.960563i $$-0.589693\pi$$
−0.278064 + 0.960563i $$0.589693\pi$$
$$884$$ 421.177 0.0160246
$$885$$ −4656.49 −0.176866
$$886$$ −50198.0 −1.90343
$$887$$ −30459.3 −1.15301 −0.576507 0.817092i $$-0.695585\pi$$
−0.576507 + 0.817092i $$0.695585\pi$$
$$888$$ −10798.8 −0.408091
$$889$$ 0 0
$$890$$ −63614.6 −2.39592
$$891$$ 1854.89 0.0697432
$$892$$ 1829.32 0.0686662
$$893$$ −4653.30 −0.174375
$$894$$ 17807.7 0.666196
$$895$$ 5530.57 0.206555
$$896$$ 0 0
$$897$$ −6848.38 −0.254917
$$898$$ −38002.6 −1.41221
$$899$$ −34223.2 −1.26964
$$900$$ −1107.47 −0.0410172
$$901$$ −24601.2 −0.909638
$$902$$ −15803.6 −0.583374
$$903$$ 0 0
$$904$$ −3862.39 −0.142103
$$905$$ 46050.7 1.69147
$$906$$ 6962.84 0.255326
$$907$$ −9364.89 −0.342840 −0.171420 0.985198i $$-0.554836\pi$$
−0.171420 + 0.985198i $$0.554836\pi$$
$$908$$ 973.313 0.0355733
$$909$$ 2580.90 0.0941727
$$910$$ 0 0
$$911$$ 32479.8 1.18123 0.590616 0.806952i $$-0.298885\pi$$
0.590616 + 0.806952i $$0.298885\pi$$
$$912$$ 2968.04 0.107765
$$913$$ 21376.3 0.774864
$$914$$ 24648.5 0.892012
$$915$$ −11389.9 −0.411516
$$916$$ −1474.46 −0.0531851
$$917$$ 0 0
$$918$$ 4962.13 0.178404
$$919$$ 295.958 0.0106232 0.00531161 0.999986i $$-0.498309\pi$$
0.00531161 + 0.999986i $$0.498309\pi$$
$$920$$ 79572.9 2.85157
$$921$$ 10119.8 0.362062
$$922$$ −9340.56 −0.333639
$$923$$ 6867.89 0.244918
$$924$$ 0 0
$$925$$ −39403.5 −1.40062
$$926$$ 20558.2 0.729572
$$927$$ 17202.3 0.609489
$$928$$ −6373.42 −0.225450
$$929$$ 5620.38 0.198492 0.0992458 0.995063i $$-0.468357\pi$$
0.0992458 + 0.995063i $$0.468357\pi$$
$$930$$ −18794.6 −0.662687
$$931$$ 0 0
$$932$$ −2700.52 −0.0949125
$$933$$ 2605.58 0.0914284
$$934$$ 21142.0 0.740670
$$935$$ 29908.0 1.04609
$$936$$ 2721.23 0.0950278
$$937$$ 32583.1 1.13601 0.568006 0.823024i $$-0.307715\pi$$
0.568006 + 0.823024i $$0.307715\pi$$
$$938$$ 0 0
$$939$$ 13029.6 0.452827
$$940$$ 2652.96 0.0920532
$$941$$ −8812.99 −0.305308 −0.152654 0.988280i $$-0.548782\pi$$
−0.152654 + 0.988280i $$0.548782\pi$$
$$942$$ 13600.8 0.470422
$$943$$ −44201.2 −1.52639
$$944$$ −4772.00 −0.164529
$$945$$ 0 0
$$946$$ 31552.9 1.08443
$$947$$ 13426.8 0.460732 0.230366 0.973104i $$-0.426008\pi$$
0.230366 + 0.973104i $$0.426008\pi$$
$$948$$ −164.229 −0.00562649
$$949$$ 9546.76 0.326555
$$950$$ 11529.1 0.393739
$$951$$ −9832.96 −0.335285
$$952$$ 0 0
$$953$$ −13394.6 −0.455293 −0.227647 0.973744i $$-0.573103\pi$$
−0.227647 + 0.973744i $$0.573103\pi$$
$$954$$ −9055.67 −0.307325
$$955$$ −48986.2 −1.65985
$$956$$ 645.560 0.0218398
$$957$$ 20046.5 0.677129
$$958$$ −25880.7 −0.872828
$$959$$ 0 0
$$960$$ −31509.3 −1.05933
$$961$$ −16035.7 −0.538273
$$962$$ 5516.11 0.184872
$$963$$ 7509.00 0.251271
$$964$$ −276.243 −0.00922944
$$965$$ −47163.8 −1.57332
$$966$$ 0 0
$$967$$ 45590.8 1.51613 0.758066 0.652178i $$-0.226144\pi$$
0.758066 + 0.652178i $$0.226144\pi$$
$$968$$ −18760.1 −0.622904
$$969$$ 3321.51 0.110116
$$970$$ −29761.8 −0.985149
$$971$$ −264.763 −0.00875041 −0.00437521 0.999990i $$-0.501393\pi$$
−0.00437521 + 0.999990i $$0.501393\pi$$
$$972$$ −117.445 −0.00387558
$$973$$ 0 0
$$974$$ 16878.2 0.555248
$$975$$ 9929.38 0.326148
$$976$$ −11672.4 −0.382812
$$977$$ 610.521 0.0199921 0.00999606 0.999950i $$-0.496818\pi$$
0.00999606 + 0.999950i $$0.496818\pi$$
$$978$$ 23577.0 0.770868
$$979$$ −27271.8 −0.890308
$$980$$ 0 0
$$981$$ −9698.98 −0.315662
$$982$$ −10535.5 −0.342363
$$983$$ 57829.7 1.87638 0.938190 0.346121i $$-0.112501\pi$$
0.938190 + 0.346121i $$0.112501\pi$$
$$984$$ 17563.5 0.569007
$$985$$ 90278.8 2.92033
$$986$$ 53627.7 1.73210
$$987$$ 0 0
$$988$$ 103.776 0.00334165
$$989$$ 88250.4 2.83741
$$990$$ 11009.1 0.353427
$$991$$ −56780.7 −1.82008 −0.910039 0.414522i $$-0.863949\pi$$
−0.910039 + 0.414522i $$0.863949\pi$$
$$992$$ 2561.67 0.0819890
$$993$$ 16768.9 0.535895
$$994$$ 0 0
$$995$$ 59505.3 1.89592
$$996$$ −1353.47 −0.0430587
$$997$$ −18616.6 −0.591369 −0.295684 0.955286i $$-0.595548\pi$$
−0.295684 + 0.955286i $$0.595548\pi$$
$$998$$ 35209.4 1.11677
$$999$$ −4178.69 −0.132340
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 1911.4.a.h.1.1 2
7.6 odd 2 39.4.a.b.1.1 2
21.20 even 2 117.4.a.c.1.2 2
28.27 even 2 624.4.a.r.1.2 2
35.34 odd 2 975.4.a.j.1.2 2
56.13 odd 2 2496.4.a.bc.1.1 2
56.27 even 2 2496.4.a.s.1.1 2
84.83 odd 2 1872.4.a.t.1.1 2
91.34 even 4 507.4.b.f.337.2 4
91.83 even 4 507.4.b.f.337.3 4
91.90 odd 2 507.4.a.f.1.2 2
273.272 even 2 1521.4.a.s.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.1 2 7.6 odd 2
117.4.a.c.1.2 2 21.20 even 2
507.4.a.f.1.2 2 91.90 odd 2
507.4.b.f.337.2 4 91.34 even 4
507.4.b.f.337.3 4 91.83 even 4
624.4.a.r.1.2 2 28.27 even 2
975.4.a.j.1.2 2 35.34 odd 2
1521.4.a.s.1.1 2 273.272 even 2
1872.4.a.t.1.1 2 84.83 odd 2
1911.4.a.h.1.1 2 1.1 even 1 trivial
2496.4.a.s.1.1 2 56.27 even 2
2496.4.a.bc.1.1 2 56.13 odd 2