Properties

 Label 98.4.a.h.1.2 Level $98$ Weight $4$ Character 98.1 Self dual yes Analytic conductor $5.782$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [98,4,Mod(1,98)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(98, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("98.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$98 = 2 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 98.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: $$5.78218718056$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 22$$ x^2 - 22 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$4.69042$$ of defining polynomial Character $$\chi$$ $$=$$ 98.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +9.38083 q^{3} +4.00000 q^{4} -9.38083 q^{5} +18.7617 q^{6} +8.00000 q^{8} +61.0000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} +9.38083 q^{3} +4.00000 q^{4} -9.38083 q^{5} +18.7617 q^{6} +8.00000 q^{8} +61.0000 q^{9} -18.7617 q^{10} +20.0000 q^{11} +37.5233 q^{12} -65.6658 q^{13} -88.0000 q^{15} +16.0000 q^{16} -56.2850 q^{17} +122.000 q^{18} -9.38083 q^{19} -37.5233 q^{20} +40.0000 q^{22} +48.0000 q^{23} +75.0467 q^{24} -37.0000 q^{25} -131.332 q^{26} +318.948 q^{27} -166.000 q^{29} -176.000 q^{30} +206.378 q^{31} +32.0000 q^{32} +187.617 q^{33} -112.570 q^{34} +244.000 q^{36} -78.0000 q^{37} -18.7617 q^{38} -616.000 q^{39} -75.0467 q^{40} -393.995 q^{41} +436.000 q^{43} +80.0000 q^{44} -572.231 q^{45} +96.0000 q^{46} -206.378 q^{47} +150.093 q^{48} -74.0000 q^{50} -528.000 q^{51} -262.663 q^{52} +62.0000 q^{53} +637.897 q^{54} -187.617 q^{55} -88.0000 q^{57} -332.000 q^{58} +666.039 q^{59} -352.000 q^{60} -272.044 q^{61} +412.757 q^{62} +64.0000 q^{64} +616.000 q^{65} +375.233 q^{66} +580.000 q^{67} -225.140 q^{68} +450.280 q^{69} -544.000 q^{71} +488.000 q^{72} +600.373 q^{73} -156.000 q^{74} -347.091 q^{75} -37.5233 q^{76} -1232.00 q^{78} -680.000 q^{79} -150.093 q^{80} +1345.00 q^{81} -787.990 q^{82} -196.997 q^{83} +528.000 q^{85} +872.000 q^{86} -1557.22 q^{87} +160.000 q^{88} +1500.93 q^{89} -1144.46 q^{90} +192.000 q^{92} +1936.00 q^{93} -412.757 q^{94} +88.0000 q^{95} +300.187 q^{96} +656.658 q^{97} +1220.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 122 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 122 * q^9 $$2 q + 4 q^{2} + 8 q^{4} + 16 q^{8} + 122 q^{9} + 40 q^{11} - 176 q^{15} + 32 q^{16} + 244 q^{18} + 80 q^{22} + 96 q^{23} - 74 q^{25} - 332 q^{29} - 352 q^{30} + 64 q^{32} + 488 q^{36} - 156 q^{37} - 1232 q^{39} + 872 q^{43} + 160 q^{44} + 192 q^{46} - 148 q^{50} - 1056 q^{51} + 124 q^{53} - 176 q^{57} - 664 q^{58} - 704 q^{60} + 128 q^{64} + 1232 q^{65} + 1160 q^{67} - 1088 q^{71} + 976 q^{72} - 312 q^{74} - 2464 q^{78} - 1360 q^{79} + 2690 q^{81} + 1056 q^{85} + 1744 q^{86} + 320 q^{88} + 384 q^{92} + 3872 q^{93} + 176 q^{95} + 2440 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 + 8 * q^4 + 16 * q^8 + 122 * q^9 + 40 * q^11 - 176 * q^15 + 32 * q^16 + 244 * q^18 + 80 * q^22 + 96 * q^23 - 74 * q^25 - 332 * q^29 - 352 * q^30 + 64 * q^32 + 488 * q^36 - 156 * q^37 - 1232 * q^39 + 872 * q^43 + 160 * q^44 + 192 * q^46 - 148 * q^50 - 1056 * q^51 + 124 * q^53 - 176 * q^57 - 664 * q^58 - 704 * q^60 + 128 * q^64 + 1232 * q^65 + 1160 * q^67 - 1088 * q^71 + 976 * q^72 - 312 * q^74 - 2464 * q^78 - 1360 * q^79 + 2690 * q^81 + 1056 * q^85 + 1744 * q^86 + 320 * q^88 + 384 * q^92 + 3872 * q^93 + 176 * q^95 + 2440 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 0.707107
$$3$$ 9.38083 1.80534 0.902671 0.430331i $$-0.141603\pi$$
0.902671 + 0.430331i $$0.141603\pi$$
$$4$$ 4.00000 0.500000
$$5$$ −9.38083 −0.839047 −0.419524 0.907744i $$-0.637803\pi$$
−0.419524 + 0.907744i $$0.637803\pi$$
$$6$$ 18.7617 1.27657
$$7$$ 0 0
$$8$$ 8.00000 0.353553
$$9$$ 61.0000 2.25926
$$10$$ −18.7617 −0.593296
$$11$$ 20.0000 0.548202 0.274101 0.961701i $$-0.411620\pi$$
0.274101 + 0.961701i $$0.411620\pi$$
$$12$$ 37.5233 0.902671
$$13$$ −65.6658 −1.40096 −0.700478 0.713674i $$-0.747030\pi$$
−0.700478 + 0.713674i $$0.747030\pi$$
$$14$$ 0 0
$$15$$ −88.0000 −1.51477
$$16$$ 16.0000 0.250000
$$17$$ −56.2850 −0.803007 −0.401503 0.915858i $$-0.631512\pi$$
−0.401503 + 0.915858i $$0.631512\pi$$
$$18$$ 122.000 1.59754
$$19$$ −9.38083 −0.113269 −0.0566345 0.998395i $$-0.518037\pi$$
−0.0566345 + 0.998395i $$0.518037\pi$$
$$20$$ −37.5233 −0.419524
$$21$$ 0 0
$$22$$ 40.0000 0.387638
$$23$$ 48.0000 0.435161 0.217580 0.976042i $$-0.430184\pi$$
0.217580 + 0.976042i $$0.430184\pi$$
$$24$$ 75.0467 0.638285
$$25$$ −37.0000 −0.296000
$$26$$ −131.332 −0.990625
$$27$$ 318.948 2.27339
$$28$$ 0 0
$$29$$ −166.000 −1.06295 −0.531473 0.847075i $$-0.678361\pi$$
−0.531473 + 0.847075i $$0.678361\pi$$
$$30$$ −176.000 −1.07110
$$31$$ 206.378 1.19570 0.597849 0.801609i $$-0.296022\pi$$
0.597849 + 0.801609i $$0.296022\pi$$
$$32$$ 32.0000 0.176777
$$33$$ 187.617 0.989693
$$34$$ −112.570 −0.567812
$$35$$ 0 0
$$36$$ 244.000 1.12963
$$37$$ −78.0000 −0.346571 −0.173285 0.984872i $$-0.555438\pi$$
−0.173285 + 0.984872i $$0.555438\pi$$
$$38$$ −18.7617 −0.0800933
$$39$$ −616.000 −2.52920
$$40$$ −75.0467 −0.296648
$$41$$ −393.995 −1.50077 −0.750386 0.661000i $$-0.770132\pi$$
−0.750386 + 0.661000i $$0.770132\pi$$
$$42$$ 0 0
$$43$$ 436.000 1.54626 0.773132 0.634245i $$-0.218689\pi$$
0.773132 + 0.634245i $$0.218689\pi$$
$$44$$ 80.0000 0.274101
$$45$$ −572.231 −1.89562
$$46$$ 96.0000 0.307705
$$47$$ −206.378 −0.640497 −0.320249 0.947334i $$-0.603766\pi$$
−0.320249 + 0.947334i $$0.603766\pi$$
$$48$$ 150.093 0.451335
$$49$$ 0 0
$$50$$ −74.0000 −0.209304
$$51$$ −528.000 −1.44970
$$52$$ −262.663 −0.700478
$$53$$ 62.0000 0.160686 0.0803430 0.996767i $$-0.474398\pi$$
0.0803430 + 0.996767i $$0.474398\pi$$
$$54$$ 637.897 1.60753
$$55$$ −187.617 −0.459968
$$56$$ 0 0
$$57$$ −88.0000 −0.204489
$$58$$ −332.000 −0.751616
$$59$$ 666.039 1.46968 0.734838 0.678243i $$-0.237258\pi$$
0.734838 + 0.678243i $$0.237258\pi$$
$$60$$ −352.000 −0.757383
$$61$$ −272.044 −0.571011 −0.285506 0.958377i $$-0.592162\pi$$
−0.285506 + 0.958377i $$0.592162\pi$$
$$62$$ 412.757 0.845486
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ 616.000 1.17547
$$66$$ 375.233 0.699819
$$67$$ 580.000 1.05759 0.528793 0.848751i $$-0.322645\pi$$
0.528793 + 0.848751i $$0.322645\pi$$
$$68$$ −225.140 −0.401503
$$69$$ 450.280 0.785613
$$70$$ 0 0
$$71$$ −544.000 −0.909309 −0.454654 0.890668i $$-0.650237\pi$$
−0.454654 + 0.890668i $$0.650237\pi$$
$$72$$ 488.000 0.798769
$$73$$ 600.373 0.962580 0.481290 0.876561i $$-0.340168\pi$$
0.481290 + 0.876561i $$0.340168\pi$$
$$74$$ −156.000 −0.245063
$$75$$ −347.091 −0.534381
$$76$$ −37.5233 −0.0566345
$$77$$ 0 0
$$78$$ −1232.00 −1.78842
$$79$$ −680.000 −0.968430 −0.484215 0.874949i $$-0.660895\pi$$
−0.484215 + 0.874949i $$0.660895\pi$$
$$80$$ −150.093 −0.209762
$$81$$ 1345.00 1.84499
$$82$$ −787.990 −1.06121
$$83$$ −196.997 −0.260521 −0.130261 0.991480i $$-0.541581\pi$$
−0.130261 + 0.991480i $$0.541581\pi$$
$$84$$ 0 0
$$85$$ 528.000 0.673760
$$86$$ 872.000 1.09337
$$87$$ −1557.22 −1.91898
$$88$$ 160.000 0.193819
$$89$$ 1500.93 1.78762 0.893812 0.448441i $$-0.148021\pi$$
0.893812 + 0.448441i $$0.148021\pi$$
$$90$$ −1144.46 −1.34041
$$91$$ 0 0
$$92$$ 192.000 0.217580
$$93$$ 1936.00 2.15864
$$94$$ −412.757 −0.452900
$$95$$ 88.0000 0.0950380
$$96$$ 300.187 0.319142
$$97$$ 656.658 0.687356 0.343678 0.939088i $$-0.388327\pi$$
0.343678 + 0.939088i $$0.388327\pi$$
$$98$$ 0 0
$$99$$ 1220.00 1.23853
$$100$$ −148.000 −0.148000
$$101$$ −121.951 −0.120144 −0.0600721 0.998194i $$-0.519133\pi$$
−0.0600721 + 0.998194i $$0.519133\pi$$
$$102$$ −1056.00 −1.02509
$$103$$ 1369.60 1.31020 0.655101 0.755541i $$-0.272626\pi$$
0.655101 + 0.755541i $$0.272626\pi$$
$$104$$ −525.327 −0.495313
$$105$$ 0 0
$$106$$ 124.000 0.113622
$$107$$ −260.000 −0.234908 −0.117454 0.993078i $$-0.537473\pi$$
−0.117454 + 0.993078i $$0.537473\pi$$
$$108$$ 1275.79 1.13670
$$109$$ 1882.00 1.65379 0.826894 0.562358i $$-0.190106\pi$$
0.826894 + 0.562358i $$0.190106\pi$$
$$110$$ −375.233 −0.325246
$$111$$ −731.705 −0.625679
$$112$$ 0 0
$$113$$ −1286.00 −1.07059 −0.535295 0.844665i $$-0.679800\pi$$
−0.535295 + 0.844665i $$0.679800\pi$$
$$114$$ −176.000 −0.144596
$$115$$ −450.280 −0.365120
$$116$$ −664.000 −0.531473
$$117$$ −4005.62 −3.16512
$$118$$ 1332.08 1.03922
$$119$$ 0 0
$$120$$ −704.000 −0.535551
$$121$$ −931.000 −0.699474
$$122$$ −544.088 −0.403766
$$123$$ −3696.00 −2.70941
$$124$$ 825.513 0.597849
$$125$$ 1519.69 1.08741
$$126$$ 0 0
$$127$$ 2312.00 1.61541 0.807704 0.589588i $$-0.200710\pi$$
0.807704 + 0.589588i $$0.200710\pi$$
$$128$$ 128.000 0.0883883
$$129$$ 4090.04 2.79154
$$130$$ 1232.00 0.831181
$$131$$ 253.282 0.168927 0.0844633 0.996427i $$-0.473082\pi$$
0.0844633 + 0.996427i $$0.473082\pi$$
$$132$$ 750.467 0.494846
$$133$$ 0 0
$$134$$ 1160.00 0.747826
$$135$$ −2992.00 −1.90748
$$136$$ −450.280 −0.283906
$$137$$ −1114.00 −0.694711 −0.347356 0.937733i $$-0.612920\pi$$
−0.347356 + 0.937733i $$0.612920\pi$$
$$138$$ 900.560 0.555513
$$139$$ −1378.98 −0.841466 −0.420733 0.907185i $$-0.638227\pi$$
−0.420733 + 0.907185i $$0.638227\pi$$
$$140$$ 0 0
$$141$$ −1936.00 −1.15632
$$142$$ −1088.00 −0.642978
$$143$$ −1313.32 −0.768007
$$144$$ 976.000 0.564815
$$145$$ 1557.22 0.891862
$$146$$ 1200.75 0.680647
$$147$$ 0 0
$$148$$ −312.000 −0.173285
$$149$$ −946.000 −0.520130 −0.260065 0.965591i $$-0.583744\pi$$
−0.260065 + 0.965591i $$0.583744\pi$$
$$150$$ −694.182 −0.377865
$$151$$ 832.000 0.448392 0.224196 0.974544i $$-0.428024\pi$$
0.224196 + 0.974544i $$0.428024\pi$$
$$152$$ −75.0467 −0.0400466
$$153$$ −3433.38 −1.81420
$$154$$ 0 0
$$155$$ −1936.00 −1.00325
$$156$$ −2464.00 −1.26460
$$157$$ −2879.92 −1.46396 −0.731982 0.681324i $$-0.761404\pi$$
−0.731982 + 0.681324i $$0.761404\pi$$
$$158$$ −1360.00 −0.684783
$$159$$ 581.612 0.290093
$$160$$ −300.187 −0.148324
$$161$$ 0 0
$$162$$ 2690.00 1.30461
$$163$$ 636.000 0.305616 0.152808 0.988256i $$-0.451168\pi$$
0.152808 + 0.988256i $$0.451168\pi$$
$$164$$ −1575.98 −0.750386
$$165$$ −1760.00 −0.830399
$$166$$ −393.995 −0.184216
$$167$$ 656.658 0.304274 0.152137 0.988359i $$-0.451385\pi$$
0.152137 + 0.988359i $$0.451385\pi$$
$$168$$ 0 0
$$169$$ 2115.00 0.962676
$$170$$ 1056.00 0.476421
$$171$$ −572.231 −0.255904
$$172$$ 1744.00 0.773132
$$173$$ −666.039 −0.292705 −0.146353 0.989232i $$-0.546753\pi$$
−0.146353 + 0.989232i $$0.546753\pi$$
$$174$$ −3114.44 −1.35692
$$175$$ 0 0
$$176$$ 320.000 0.137051
$$177$$ 6248.00 2.65327
$$178$$ 3001.87 1.26404
$$179$$ −3228.00 −1.34789 −0.673944 0.738782i $$-0.735401\pi$$
−0.673944 + 0.738782i $$0.735401\pi$$
$$180$$ −2288.92 −0.947812
$$181$$ 2823.63 1.15955 0.579776 0.814776i $$-0.303140\pi$$
0.579776 + 0.814776i $$0.303140\pi$$
$$182$$ 0 0
$$183$$ −2552.00 −1.03087
$$184$$ 384.000 0.153852
$$185$$ 731.705 0.290789
$$186$$ 3872.00 1.52639
$$187$$ −1125.70 −0.440210
$$188$$ −825.513 −0.320249
$$189$$ 0 0
$$190$$ 176.000 0.0672020
$$191$$ −2136.00 −0.809191 −0.404596 0.914496i $$-0.632588\pi$$
−0.404596 + 0.914496i $$0.632588\pi$$
$$192$$ 600.373 0.225668
$$193$$ 1658.00 0.618370 0.309185 0.951002i $$-0.399944\pi$$
0.309185 + 0.951002i $$0.399944\pi$$
$$194$$ 1313.32 0.486034
$$195$$ 5778.59 2.12212
$$196$$ 0 0
$$197$$ −978.000 −0.353704 −0.176852 0.984237i $$-0.556591\pi$$
−0.176852 + 0.984237i $$0.556591\pi$$
$$198$$ 2440.00 0.875774
$$199$$ 4934.32 1.75771 0.878855 0.477088i $$-0.158308\pi$$
0.878855 + 0.477088i $$0.158308\pi$$
$$200$$ −296.000 −0.104652
$$201$$ 5440.88 1.90930
$$202$$ −243.902 −0.0849547
$$203$$ 0 0
$$204$$ −2112.00 −0.724851
$$205$$ 3696.00 1.25922
$$206$$ 2739.20 0.926453
$$207$$ 2928.00 0.983140
$$208$$ −1050.65 −0.350239
$$209$$ −187.617 −0.0620943
$$210$$ 0 0
$$211$$ 1556.00 0.507675 0.253838 0.967247i $$-0.418307\pi$$
0.253838 + 0.967247i $$0.418307\pi$$
$$212$$ 248.000 0.0803430
$$213$$ −5103.17 −1.64161
$$214$$ −520.000 −0.166105
$$215$$ −4090.04 −1.29739
$$216$$ 2551.59 0.803766
$$217$$ 0 0
$$218$$ 3764.00 1.16940
$$219$$ 5632.00 1.73779
$$220$$ −750.467 −0.229984
$$221$$ 3696.00 1.12498
$$222$$ −1463.41 −0.442422
$$223$$ −2889.30 −0.867630 −0.433815 0.901002i $$-0.642833\pi$$
−0.433815 + 0.901002i $$0.642833\pi$$
$$224$$ 0 0
$$225$$ −2257.00 −0.668741
$$226$$ −2572.00 −0.757022
$$227$$ 1979.36 0.578742 0.289371 0.957217i $$-0.406554\pi$$
0.289371 + 0.957217i $$0.406554\pi$$
$$228$$ −352.000 −0.102245
$$229$$ −2767.35 −0.798565 −0.399282 0.916828i $$-0.630741\pi$$
−0.399282 + 0.916828i $$0.630741\pi$$
$$230$$ −900.560 −0.258179
$$231$$ 0 0
$$232$$ −1328.00 −0.375808
$$233$$ −6490.00 −1.82478 −0.912391 0.409321i $$-0.865766\pi$$
−0.912391 + 0.409321i $$0.865766\pi$$
$$234$$ −8011.23 −2.23808
$$235$$ 1936.00 0.537407
$$236$$ 2664.16 0.734838
$$237$$ −6378.97 −1.74835
$$238$$ 0 0
$$239$$ −4296.00 −1.16270 −0.581350 0.813654i $$-0.697475\pi$$
−0.581350 + 0.813654i $$0.697475\pi$$
$$240$$ −1408.00 −0.378692
$$241$$ −4521.56 −1.20854 −0.604272 0.796778i $$-0.706536\pi$$
−0.604272 + 0.796778i $$0.706536\pi$$
$$242$$ −1862.00 −0.494603
$$243$$ 4005.62 1.05745
$$244$$ −1088.18 −0.285506
$$245$$ 0 0
$$246$$ −7392.00 −1.91584
$$247$$ 616.000 0.158685
$$248$$ 1651.03 0.422743
$$249$$ −1848.00 −0.470330
$$250$$ 3039.39 0.768911
$$251$$ −5581.59 −1.40361 −0.701807 0.712367i $$-0.747623\pi$$
−0.701807 + 0.712367i $$0.747623\pi$$
$$252$$ 0 0
$$253$$ 960.000 0.238556
$$254$$ 4624.00 1.14227
$$255$$ 4953.08 1.21637
$$256$$ 256.000 0.0625000
$$257$$ 1500.93 0.364302 0.182151 0.983271i $$-0.441694\pi$$
0.182151 + 0.983271i $$0.441694\pi$$
$$258$$ 8180.09 1.97391
$$259$$ 0 0
$$260$$ 2464.00 0.587734
$$261$$ −10126.0 −2.40147
$$262$$ 506.565 0.119449
$$263$$ −400.000 −0.0937835 −0.0468917 0.998900i $$-0.514932\pi$$
−0.0468917 + 0.998900i $$0.514932\pi$$
$$264$$ 1500.93 0.349909
$$265$$ −581.612 −0.134823
$$266$$ 0 0
$$267$$ 14080.0 3.22727
$$268$$ 2320.00 0.528793
$$269$$ 272.044 0.0616610 0.0308305 0.999525i $$-0.490185\pi$$
0.0308305 + 0.999525i $$0.490185\pi$$
$$270$$ −5984.00 −1.34879
$$271$$ −6904.29 −1.54762 −0.773812 0.633416i $$-0.781652\pi$$
−0.773812 + 0.633416i $$0.781652\pi$$
$$272$$ −900.560 −0.200752
$$273$$ 0 0
$$274$$ −2228.00 −0.491235
$$275$$ −740.000 −0.162268
$$276$$ 1801.12 0.392807
$$277$$ −6770.00 −1.46848 −0.734242 0.678888i $$-0.762462\pi$$
−0.734242 + 0.678888i $$0.762462\pi$$
$$278$$ −2757.96 −0.595006
$$279$$ 12589.1 2.70139
$$280$$ 0 0
$$281$$ 1878.00 0.398691 0.199345 0.979929i $$-0.436118\pi$$
0.199345 + 0.979929i $$0.436118\pi$$
$$282$$ −3872.00 −0.817639
$$283$$ −384.614 −0.0807878 −0.0403939 0.999184i $$-0.512861\pi$$
−0.0403939 + 0.999184i $$0.512861\pi$$
$$284$$ −2176.00 −0.454654
$$285$$ 825.513 0.171576
$$286$$ −2626.63 −0.543063
$$287$$ 0 0
$$288$$ 1952.00 0.399384
$$289$$ −1745.00 −0.355180
$$290$$ 3114.44 0.630641
$$291$$ 6160.00 1.24091
$$292$$ 2401.49 0.481290
$$293$$ 3742.95 0.746299 0.373149 0.927771i $$-0.378278\pi$$
0.373149 + 0.927771i $$0.378278\pi$$
$$294$$ 0 0
$$295$$ −6248.00 −1.23313
$$296$$ −624.000 −0.122531
$$297$$ 6378.97 1.24628
$$298$$ −1892.00 −0.367787
$$299$$ −3151.96 −0.609641
$$300$$ −1388.36 −0.267191
$$301$$ 0 0
$$302$$ 1664.00 0.317061
$$303$$ −1144.00 −0.216901
$$304$$ −150.093 −0.0283172
$$305$$ 2552.00 0.479105
$$306$$ −6866.77 −1.28283
$$307$$ −722.324 −0.134284 −0.0671420 0.997743i $$-0.521388\pi$$
−0.0671420 + 0.997743i $$0.521388\pi$$
$$308$$ 0 0
$$309$$ 12848.0 2.36536
$$310$$ −3872.00 −0.709403
$$311$$ −7279.53 −1.32728 −0.663640 0.748052i $$-0.730989\pi$$
−0.663640 + 0.748052i $$0.730989\pi$$
$$312$$ −4928.00 −0.894209
$$313$$ −1519.69 −0.274435 −0.137218 0.990541i $$-0.543816\pi$$
−0.137218 + 0.990541i $$0.543816\pi$$
$$314$$ −5759.83 −1.03518
$$315$$ 0 0
$$316$$ −2720.00 −0.484215
$$317$$ 2358.00 0.417787 0.208893 0.977938i $$-0.433014\pi$$
0.208893 + 0.977938i $$0.433014\pi$$
$$318$$ 1163.22 0.205127
$$319$$ −3320.00 −0.582709
$$320$$ −600.373 −0.104881
$$321$$ −2439.02 −0.424089
$$322$$ 0 0
$$323$$ 528.000 0.0909557
$$324$$ 5380.00 0.922497
$$325$$ 2429.64 0.414683
$$326$$ 1272.00 0.216103
$$327$$ 17654.7 2.98565
$$328$$ −3151.96 −0.530603
$$329$$ 0 0
$$330$$ −3520.00 −0.587181
$$331$$ 2372.00 0.393888 0.196944 0.980415i $$-0.436898\pi$$
0.196944 + 0.980415i $$0.436898\pi$$
$$332$$ −787.990 −0.130261
$$333$$ −4758.00 −0.782993
$$334$$ 1313.32 0.215154
$$335$$ −5440.88 −0.887365
$$336$$ 0 0
$$337$$ −250.000 −0.0404106 −0.0202053 0.999796i $$-0.506432\pi$$
−0.0202053 + 0.999796i $$0.506432\pi$$
$$338$$ 4230.00 0.680715
$$339$$ −12063.7 −1.93278
$$340$$ 2112.00 0.336880
$$341$$ 4127.57 0.655485
$$342$$ −1144.46 −0.180951
$$343$$ 0 0
$$344$$ 3488.00 0.546687
$$345$$ −4224.00 −0.659167
$$346$$ −1332.08 −0.206974
$$347$$ 9540.00 1.47589 0.737945 0.674861i $$-0.235796\pi$$
0.737945 + 0.674861i $$0.235796\pi$$
$$348$$ −6228.87 −0.959490
$$349$$ 5712.93 0.876235 0.438117 0.898918i $$-0.355645\pi$$
0.438117 + 0.898918i $$0.355645\pi$$
$$350$$ 0 0
$$351$$ −20944.0 −3.18492
$$352$$ 640.000 0.0969094
$$353$$ −4390.23 −0.661950 −0.330975 0.943640i $$-0.607378\pi$$
−0.330975 + 0.943640i $$0.607378\pi$$
$$354$$ 12496.0 1.87614
$$355$$ 5103.17 0.762953
$$356$$ 6003.73 0.893812
$$357$$ 0 0
$$358$$ −6456.00 −0.953101
$$359$$ 1840.00 0.270506 0.135253 0.990811i $$-0.456815\pi$$
0.135253 + 0.990811i $$0.456815\pi$$
$$360$$ −4577.85 −0.670205
$$361$$ −6771.00 −0.987170
$$362$$ 5647.26 0.819927
$$363$$ −8733.55 −1.26279
$$364$$ 0 0
$$365$$ −5632.00 −0.807650
$$366$$ −5104.00 −0.728935
$$367$$ 2964.34 0.421628 0.210814 0.977526i $$-0.432389\pi$$
0.210814 + 0.977526i $$0.432389\pi$$
$$368$$ 768.000 0.108790
$$369$$ −24033.7 −3.39063
$$370$$ 1463.41 0.205619
$$371$$ 0 0
$$372$$ 7744.00 1.07932
$$373$$ 3982.00 0.552762 0.276381 0.961048i $$-0.410865\pi$$
0.276381 + 0.961048i $$0.410865\pi$$
$$374$$ −2251.40 −0.311276
$$375$$ 14256.0 1.96314
$$376$$ −1651.03 −0.226450
$$377$$ 10900.5 1.48914
$$378$$ 0 0
$$379$$ 2676.00 0.362683 0.181342 0.983420i $$-0.441956\pi$$
0.181342 + 0.983420i $$0.441956\pi$$
$$380$$ 352.000 0.0475190
$$381$$ 21688.5 2.91636
$$382$$ −4272.00 −0.572185
$$383$$ −7035.62 −0.938652 −0.469326 0.883025i $$-0.655503\pi$$
−0.469326 + 0.883025i $$0.655503\pi$$
$$384$$ 1200.75 0.159571
$$385$$ 0 0
$$386$$ 3316.00 0.437254
$$387$$ 26596.0 3.49341
$$388$$ 2626.63 0.343678
$$389$$ 8658.00 1.12848 0.564239 0.825611i $$-0.309170\pi$$
0.564239 + 0.825611i $$0.309170\pi$$
$$390$$ 11557.2 1.50057
$$391$$ −2701.68 −0.349437
$$392$$ 0 0
$$393$$ 2376.00 0.304970
$$394$$ −1956.00 −0.250106
$$395$$ 6378.97 0.812558
$$396$$ 4880.00 0.619266
$$397$$ 9052.50 1.14441 0.572207 0.820109i $$-0.306088\pi$$
0.572207 + 0.820109i $$0.306088\pi$$
$$398$$ 9868.63 1.24289
$$399$$ 0 0
$$400$$ −592.000 −0.0740000
$$401$$ −5706.00 −0.710584 −0.355292 0.934755i $$-0.615619\pi$$
−0.355292 + 0.934755i $$0.615619\pi$$
$$402$$ 10881.8 1.35008
$$403$$ −13552.0 −1.67512
$$404$$ −487.803 −0.0600721
$$405$$ −12617.2 −1.54804
$$406$$ 0 0
$$407$$ −1560.00 −0.189991
$$408$$ −4224.00 −0.512547
$$409$$ −2420.25 −0.292601 −0.146301 0.989240i $$-0.546737\pi$$
−0.146301 + 0.989240i $$0.546737\pi$$
$$410$$ 7392.00 0.890402
$$411$$ −10450.2 −1.25419
$$412$$ 5478.41 0.655101
$$413$$ 0 0
$$414$$ 5856.00 0.695185
$$415$$ 1848.00 0.218590
$$416$$ −2101.31 −0.247656
$$417$$ −12936.0 −1.51913
$$418$$ −375.233 −0.0439073
$$419$$ 1510.31 0.176095 0.0880473 0.996116i $$-0.471937\pi$$
0.0880473 + 0.996116i $$0.471937\pi$$
$$420$$ 0 0
$$421$$ −16770.0 −1.94138 −0.970689 0.240341i $$-0.922741\pi$$
−0.970689 + 0.240341i $$0.922741\pi$$
$$422$$ 3112.00 0.358981
$$423$$ −12589.1 −1.44705
$$424$$ 496.000 0.0568111
$$425$$ 2082.54 0.237690
$$426$$ −10206.3 −1.16080
$$427$$ 0 0
$$428$$ −1040.00 −0.117454
$$429$$ −12320.0 −1.38652
$$430$$ −8180.09 −0.917392
$$431$$ 1336.00 0.149311 0.0746553 0.997209i $$-0.476214\pi$$
0.0746553 + 0.997209i $$0.476214\pi$$
$$432$$ 5103.17 0.568348
$$433$$ 11163.2 1.23896 0.619479 0.785013i $$-0.287344\pi$$
0.619479 + 0.785013i $$0.287344\pi$$
$$434$$ 0 0
$$435$$ 14608.0 1.61011
$$436$$ 7528.00 0.826894
$$437$$ −450.280 −0.0492902
$$438$$ 11264.0 1.22880
$$439$$ 3602.24 0.391630 0.195815 0.980641i $$-0.437265\pi$$
0.195815 + 0.980641i $$0.437265\pi$$
$$440$$ −1500.93 −0.162623
$$441$$ 0 0
$$442$$ 7392.00 0.795479
$$443$$ 6348.00 0.680818 0.340409 0.940277i $$-0.389434\pi$$
0.340409 + 0.940277i $$0.389434\pi$$
$$444$$ −2926.82 −0.312839
$$445$$ −14080.0 −1.49990
$$446$$ −5778.59 −0.613507
$$447$$ −8874.27 −0.939012
$$448$$ 0 0
$$449$$ 7170.00 0.753615 0.376808 0.926292i $$-0.377022\pi$$
0.376808 + 0.926292i $$0.377022\pi$$
$$450$$ −4514.00 −0.472871
$$451$$ −7879.90 −0.822727
$$452$$ −5144.00 −0.535295
$$453$$ 7804.85 0.809501
$$454$$ 3958.71 0.409232
$$455$$ 0 0
$$456$$ −704.000 −0.0722979
$$457$$ 6866.00 0.702796 0.351398 0.936226i $$-0.385706\pi$$
0.351398 + 0.936226i $$0.385706\pi$$
$$458$$ −5534.69 −0.564671
$$459$$ −17952.0 −1.82555
$$460$$ −1801.12 −0.182560
$$461$$ −1378.98 −0.139318 −0.0696590 0.997571i $$-0.522191\pi$$
−0.0696590 + 0.997571i $$0.522191\pi$$
$$462$$ 0 0
$$463$$ 2648.00 0.265795 0.132897 0.991130i $$-0.457572\pi$$
0.132897 + 0.991130i $$0.457572\pi$$
$$464$$ −2656.00 −0.265736
$$465$$ −18161.3 −1.81120
$$466$$ −12980.0 −1.29032
$$467$$ 12335.8 1.22234 0.611170 0.791500i $$-0.290699\pi$$
0.611170 + 0.791500i $$0.290699\pi$$
$$468$$ −16022.5 −1.58256
$$469$$ 0 0
$$470$$ 3872.00 0.380004
$$471$$ −27016.0 −2.64295
$$472$$ 5328.31 0.519609
$$473$$ 8720.00 0.847666
$$474$$ −12757.9 −1.23627
$$475$$ 347.091 0.0335276
$$476$$ 0 0
$$477$$ 3782.00 0.363031
$$478$$ −8592.00 −0.822153
$$479$$ 13339.5 1.27244 0.636221 0.771507i $$-0.280497\pi$$
0.636221 + 0.771507i $$0.280497\pi$$
$$480$$ −2816.00 −0.267775
$$481$$ 5121.93 0.485530
$$482$$ −9043.12 −0.854570
$$483$$ 0 0
$$484$$ −3724.00 −0.349737
$$485$$ −6160.00 −0.576724
$$486$$ 8011.23 0.747730
$$487$$ 13936.0 1.29672 0.648358 0.761336i $$-0.275456\pi$$
0.648358 + 0.761336i $$0.275456\pi$$
$$488$$ −2176.35 −0.201883
$$489$$ 5966.21 0.551741
$$490$$ 0 0
$$491$$ −12276.0 −1.12833 −0.564163 0.825663i $$-0.690801\pi$$
−0.564163 + 0.825663i $$0.690801\pi$$
$$492$$ −14784.0 −1.35470
$$493$$ 9343.31 0.853553
$$494$$ 1232.00 0.112207
$$495$$ −11444.6 −1.03919
$$496$$ 3302.05 0.298924
$$497$$ 0 0
$$498$$ −3696.00 −0.332574
$$499$$ −2220.00 −0.199160 −0.0995800 0.995030i $$-0.531750\pi$$
−0.0995800 + 0.995030i $$0.531750\pi$$
$$500$$ 6078.78 0.543703
$$501$$ 6160.00 0.549318
$$502$$ −11163.2 −0.992505
$$503$$ 11294.5 1.00119 0.500594 0.865682i $$-0.333115\pi$$
0.500594 + 0.865682i $$0.333115\pi$$
$$504$$ 0 0
$$505$$ 1144.00 0.100807
$$506$$ 1920.00 0.168685
$$507$$ 19840.5 1.73796
$$508$$ 9248.00 0.807704
$$509$$ 15881.7 1.38300 0.691499 0.722377i $$-0.256951\pi$$
0.691499 + 0.722377i $$0.256951\pi$$
$$510$$ 9906.16 0.860102
$$511$$ 0 0
$$512$$ 512.000 0.0441942
$$513$$ −2992.00 −0.257505
$$514$$ 3001.87 0.257600
$$515$$ −12848.0 −1.09932
$$516$$ 16360.2 1.39577
$$517$$ −4127.57 −0.351122
$$518$$ 0 0
$$519$$ −6248.00 −0.528433
$$520$$ 4928.00 0.415591
$$521$$ −11613.5 −0.976575 −0.488287 0.872683i $$-0.662378\pi$$
−0.488287 + 0.872683i $$0.662378\pi$$
$$522$$ −20252.0 −1.69810
$$523$$ −12617.2 −1.05490 −0.527450 0.849586i $$-0.676852\pi$$
−0.527450 + 0.849586i $$0.676852\pi$$
$$524$$ 1013.13 0.0844633
$$525$$ 0 0
$$526$$ −800.000 −0.0663149
$$527$$ −11616.0 −0.960154
$$528$$ 3001.87 0.247423
$$529$$ −9863.00 −0.810635
$$530$$ −1163.22 −0.0953343
$$531$$ 40628.4 3.32038
$$532$$ 0 0
$$533$$ 25872.0 2.10252
$$534$$ 28160.0 2.28203
$$535$$ 2439.02 0.197099
$$536$$ 4640.00 0.373913
$$537$$ −30281.3 −2.43340
$$538$$ 544.088 0.0436009
$$539$$ 0 0
$$540$$ −11968.0 −0.953742
$$541$$ 1798.00 0.142887 0.0714437 0.997445i $$-0.477239\pi$$
0.0714437 + 0.997445i $$0.477239\pi$$
$$542$$ −13808.6 −1.09433
$$543$$ 26488.0 2.09339
$$544$$ −1801.12 −0.141953
$$545$$ −17654.7 −1.38761
$$546$$ 0 0
$$547$$ 1276.00 0.0997401 0.0498700 0.998756i $$-0.484119\pi$$
0.0498700 + 0.998756i $$0.484119\pi$$
$$548$$ −4456.00 −0.347356
$$549$$ −16594.7 −1.29006
$$550$$ −1480.00 −0.114741
$$551$$ 1557.22 0.120399
$$552$$ 3602.24 0.277756
$$553$$ 0 0
$$554$$ −13540.0 −1.03837
$$555$$ 6864.00 0.524974
$$556$$ −5515.93 −0.420733
$$557$$ 2694.00 0.204934 0.102467 0.994736i $$-0.467326\pi$$
0.102467 + 0.994736i $$0.467326\pi$$
$$558$$ 25178.2 1.91017
$$559$$ −28630.3 −2.16625
$$560$$ 0 0
$$561$$ −10560.0 −0.794730
$$562$$ 3756.00 0.281917
$$563$$ 15769.2 1.18045 0.590223 0.807240i $$-0.299040\pi$$
0.590223 + 0.807240i $$0.299040\pi$$
$$564$$ −7744.00 −0.578158
$$565$$ 12063.7 0.898276
$$566$$ −769.228 −0.0571256
$$567$$ 0 0
$$568$$ −4352.00 −0.321489
$$569$$ 12606.0 0.928772 0.464386 0.885633i $$-0.346275\pi$$
0.464386 + 0.885633i $$0.346275\pi$$
$$570$$ 1651.03 0.121323
$$571$$ 6852.00 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −5253.27 −0.384004
$$573$$ −20037.5 −1.46087
$$574$$ 0 0
$$575$$ −1776.00 −0.128808
$$576$$ 3904.00 0.282407
$$577$$ −14371.4 −1.03690 −0.518449 0.855108i $$-0.673491\pi$$
−0.518449 + 0.855108i $$0.673491\pi$$
$$578$$ −3490.00 −0.251150
$$579$$ 15553.4 1.11637
$$580$$ 6228.87 0.445931
$$581$$ 0 0
$$582$$ 12320.0 0.877458
$$583$$ 1240.00 0.0880884
$$584$$ 4802.99 0.340324
$$585$$ 37576.0 2.65569
$$586$$ 7485.90 0.527713
$$587$$ −18977.4 −1.33438 −0.667191 0.744887i $$-0.732503\pi$$
−0.667191 + 0.744887i $$0.732503\pi$$
$$588$$ 0 0
$$589$$ −1936.00 −0.135435
$$590$$ −12496.0 −0.871953
$$591$$ −9174.45 −0.638556
$$592$$ −1248.00 −0.0866427
$$593$$ −8217.61 −0.569067 −0.284534 0.958666i $$-0.591839\pi$$
−0.284534 + 0.958666i $$0.591839\pi$$
$$594$$ 12757.9 0.881253
$$595$$ 0 0
$$596$$ −3784.00 −0.260065
$$597$$ 46288.0 3.17327
$$598$$ −6303.92 −0.431081
$$599$$ −19104.0 −1.30312 −0.651559 0.758598i $$-0.725885\pi$$
−0.651559 + 0.758598i $$0.725885\pi$$
$$600$$ −2776.73 −0.188932
$$601$$ −21538.4 −1.46185 −0.730923 0.682460i $$-0.760910\pi$$
−0.730923 + 0.682460i $$0.760910\pi$$
$$602$$ 0 0
$$603$$ 35380.0 2.38936
$$604$$ 3328.00 0.224196
$$605$$ 8733.55 0.586892
$$606$$ −2288.00 −0.153372
$$607$$ −13733.5 −0.918331 −0.459166 0.888351i $$-0.651852\pi$$
−0.459166 + 0.888351i $$0.651852\pi$$
$$608$$ −300.187 −0.0200233
$$609$$ 0 0
$$610$$ 5104.00 0.338779
$$611$$ 13552.0 0.897308
$$612$$ −13733.5 −0.907100
$$613$$ 28034.0 1.84712 0.923558 0.383458i $$-0.125267\pi$$
0.923558 + 0.383458i $$0.125267\pi$$
$$614$$ −1444.65 −0.0949532
$$615$$ 34671.6 2.27332
$$616$$ 0 0
$$617$$ −8258.00 −0.538824 −0.269412 0.963025i $$-0.586829\pi$$
−0.269412 + 0.963025i $$0.586829\pi$$
$$618$$ 25696.0 1.67256
$$619$$ 5131.31 0.333191 0.166595 0.986025i $$-0.446723\pi$$
0.166595 + 0.986025i $$0.446723\pi$$
$$620$$ −7744.00 −0.501623
$$621$$ 15309.5 0.989291
$$622$$ −14559.1 −0.938529
$$623$$ 0 0
$$624$$ −9856.00 −0.632301
$$625$$ −9631.00 −0.616384
$$626$$ −3039.39 −0.194055
$$627$$ −1760.00 −0.112101
$$628$$ −11519.7 −0.731982
$$629$$ 4390.23 0.278299
$$630$$ 0 0
$$631$$ 912.000 0.0575375 0.0287687 0.999586i $$-0.490841\pi$$
0.0287687 + 0.999586i $$0.490841\pi$$
$$632$$ −5440.00 −0.342392
$$633$$ 14596.6 0.916527
$$634$$ 4716.00 0.295420
$$635$$ −21688.5 −1.35540
$$636$$ 2326.45 0.145047
$$637$$ 0 0
$$638$$ −6640.00 −0.412038
$$639$$ −33184.0 −2.05436
$$640$$ −1200.75 −0.0741620
$$641$$ −890.000 −0.0548407 −0.0274203 0.999624i $$-0.508729\pi$$
−0.0274203 + 0.999624i $$0.508729\pi$$
$$642$$ −4878.03 −0.299876
$$643$$ 29352.6 1.80024 0.900120 0.435642i $$-0.143479\pi$$
0.900120 + 0.435642i $$0.143479\pi$$
$$644$$ 0 0
$$645$$ −38368.0 −2.34223
$$646$$ 1056.00 0.0643154
$$647$$ −11876.1 −0.721637 −0.360818 0.932636i $$-0.617503\pi$$
−0.360818 + 0.932636i $$0.617503\pi$$
$$648$$ 10760.0 0.652304
$$649$$ 13320.8 0.805680
$$650$$ 4859.27 0.293225
$$651$$ 0 0
$$652$$ 2544.00 0.152808
$$653$$ −21526.0 −1.29001 −0.645006 0.764178i $$-0.723145\pi$$
−0.645006 + 0.764178i $$0.723145\pi$$
$$654$$ 35309.4 2.11118
$$655$$ −2376.00 −0.141737
$$656$$ −6303.92 −0.375193
$$657$$ 36622.8 2.17472
$$658$$ 0 0
$$659$$ 23452.0 1.38628 0.693141 0.720802i $$-0.256226\pi$$
0.693141 + 0.720802i $$0.256226\pi$$
$$660$$ −7040.00 −0.415199
$$661$$ 26669.7 1.56934 0.784668 0.619916i $$-0.212833\pi$$
0.784668 + 0.619916i $$0.212833\pi$$
$$662$$ 4744.00 0.278521
$$663$$ 34671.6 2.03097
$$664$$ −1575.98 −0.0921082
$$665$$ 0 0
$$666$$ −9516.00 −0.553660
$$667$$ −7968.00 −0.462552
$$668$$ 2626.63 0.152137
$$669$$ −27104.0 −1.56637
$$670$$ −10881.8 −0.627462
$$671$$ −5440.88 −0.313030
$$672$$ 0 0
$$673$$ −13858.0 −0.793739 −0.396870 0.917875i $$-0.629904\pi$$
−0.396870 + 0.917875i $$0.629904\pi$$
$$674$$ −500.000 −0.0285746
$$675$$ −11801.1 −0.672924
$$676$$ 8460.00 0.481338
$$677$$ −32448.3 −1.84208 −0.921041 0.389466i $$-0.872660\pi$$
−0.921041 + 0.389466i $$0.872660\pi$$
$$678$$ −24127.5 −1.36668
$$679$$ 0 0
$$680$$ 4224.00 0.238210
$$681$$ 18568.0 1.04483
$$682$$ 8255.13 0.463498
$$683$$ −27812.0 −1.55812 −0.779060 0.626949i $$-0.784304\pi$$
−0.779060 + 0.626949i $$0.784304\pi$$
$$684$$ −2288.92 −0.127952
$$685$$ 10450.2 0.582895
$$686$$ 0 0
$$687$$ −25960.0 −1.44168
$$688$$ 6976.00 0.386566
$$689$$ −4071.28 −0.225114
$$690$$ −8448.00 −0.466101
$$691$$ 1303.94 0.0717859 0.0358929 0.999356i $$-0.488572\pi$$
0.0358929 + 0.999356i $$0.488572\pi$$
$$692$$ −2664.16 −0.146353
$$693$$ 0 0
$$694$$ 19080.0 1.04361
$$695$$ 12936.0 0.706029
$$696$$ −12457.7 −0.678462
$$697$$ 22176.0 1.20513
$$698$$ 11425.9 0.619592
$$699$$ −60881.6 −3.29435
$$700$$ 0 0
$$701$$ 22906.0 1.23416 0.617081 0.786900i $$-0.288315\pi$$
0.617081 + 0.786900i $$0.288315\pi$$
$$702$$ −41888.0 −2.25208
$$703$$ 731.705 0.0392557
$$704$$ 1280.00 0.0685253
$$705$$ 18161.3 0.970204
$$706$$ −8780.46 −0.468069
$$707$$ 0 0
$$708$$ 24992.0 1.32663
$$709$$ −15086.0 −0.799107 −0.399553 0.916710i $$-0.630835\pi$$
−0.399553 + 0.916710i $$0.630835\pi$$
$$710$$ 10206.3 0.539489
$$711$$ −41480.0 −2.18793
$$712$$ 12007.5 0.632021
$$713$$ 9906.16 0.520321
$$714$$ 0 0
$$715$$ 12320.0 0.644394
$$716$$ −12912.0 −0.673944
$$717$$ −40300.1 −2.09907
$$718$$ 3680.00 0.191276
$$719$$ 20544.0 1.06559 0.532797 0.846243i $$-0.321141\pi$$
0.532797 + 0.846243i $$0.321141\pi$$
$$720$$ −9155.69 −0.473906
$$721$$ 0 0
$$722$$ −13542.0 −0.698035
$$723$$ −42416.0 −2.18184
$$724$$ 11294.5 0.579776
$$725$$ 6142.00 0.314632
$$726$$ −17467.1 −0.892927
$$727$$ 7223.24 0.368494 0.184247 0.982880i $$-0.441015\pi$$
0.184247 + 0.982880i $$0.441015\pi$$
$$728$$ 0 0
$$729$$ 1261.00 0.0640654
$$730$$ −11264.0 −0.571095
$$731$$ −24540.3 −1.24166
$$732$$ −10208.0 −0.515435
$$733$$ −29427.7 −1.48286 −0.741430 0.671031i $$-0.765852\pi$$
−0.741430 + 0.671031i $$0.765852\pi$$
$$734$$ 5928.69 0.298136
$$735$$ 0 0
$$736$$ 1536.00 0.0769262
$$737$$ 11600.0 0.579771
$$738$$ −48067.4 −2.39754
$$739$$ 32668.0 1.62613 0.813066 0.582171i $$-0.197797\pi$$
0.813066 + 0.582171i $$0.197797\pi$$
$$740$$ 2926.82 0.145395
$$741$$ 5778.59 0.286480
$$742$$ 0 0
$$743$$ −37056.0 −1.82968 −0.914840 0.403816i $$-0.867684\pi$$
−0.914840 + 0.403816i $$0.867684\pi$$
$$744$$ 15488.0 0.763196
$$745$$ 8874.27 0.436413
$$746$$ 7964.00 0.390862
$$747$$ −12016.8 −0.588586
$$748$$ −4502.80 −0.220105
$$749$$ 0 0
$$750$$ 28512.0 1.38815
$$751$$ −19608.0 −0.952738 −0.476369 0.879246i $$-0.658047\pi$$
−0.476369 + 0.879246i $$0.658047\pi$$
$$752$$ −3302.05 −0.160124
$$753$$ −52360.0 −2.53400
$$754$$ 21801.1 1.05298
$$755$$ −7804.85 −0.376222
$$756$$ 0 0
$$757$$ 19378.0 0.930390 0.465195 0.885208i $$-0.345984\pi$$
0.465195 + 0.885208i $$0.345984\pi$$
$$758$$ 5352.00 0.256456
$$759$$ 9005.60 0.430675
$$760$$ 704.000 0.0336010
$$761$$ 13977.4 0.665810 0.332905 0.942960i $$-0.391971\pi$$
0.332905 + 0.942960i $$0.391971\pi$$
$$762$$ 43377.0 2.06218
$$763$$ 0 0
$$764$$ −8544.00 −0.404596
$$765$$ 32208.0 1.52220
$$766$$ −14071.2 −0.663727
$$767$$ −43736.0 −2.05895
$$768$$ 2401.49 0.112834
$$769$$ −8536.56 −0.400307 −0.200154 0.979765i $$-0.564144\pi$$
−0.200154 + 0.979765i $$0.564144\pi$$
$$770$$ 0 0
$$771$$ 14080.0 0.657690
$$772$$ 6632.00 0.309185
$$773$$ 29296.3 1.36315 0.681576 0.731748i $$-0.261295\pi$$
0.681576 + 0.731748i $$0.261295\pi$$
$$774$$ 53192.0 2.47022
$$775$$ −7636.00 −0.353927
$$776$$ 5253.27 0.243017
$$777$$ 0 0
$$778$$ 17316.0 0.797955
$$779$$ 3696.00 0.169991
$$780$$ 23114.4 1.06106
$$781$$ −10880.0 −0.498485
$$782$$ −5403.36 −0.247089
$$783$$ −52945.4 −2.41649
$$784$$ 0 0
$$785$$ 27016.0 1.22833
$$786$$ 4752.00 0.215647
$$787$$ −13780.4 −0.624167 −0.312084 0.950055i $$-0.601027\pi$$
−0.312084 + 0.950055i $$0.601027\pi$$
$$788$$ −3912.00 −0.176852
$$789$$ −3752.33 −0.169311
$$790$$ 12757.9 0.574566
$$791$$ 0 0
$$792$$ 9760.00 0.437887
$$793$$ 17864.0 0.799961
$$794$$ 18105.0 0.809222
$$795$$ −5456.00 −0.243402
$$796$$ 19737.3 0.878855
$$797$$ 34868.6 1.54970 0.774848 0.632148i $$-0.217826\pi$$
0.774848 + 0.632148i $$0.217826\pi$$
$$798$$ 0 0
$$799$$ 11616.0 0.514324
$$800$$ −1184.00 −0.0523259
$$801$$ 91556.9 4.03871
$$802$$ −11412.0 −0.502459
$$803$$ 12007.5 0.527689
$$804$$ 21763.5 0.954652
$$805$$ 0 0
$$806$$ −27104.0 −1.18449
$$807$$ 2552.00 0.111319
$$808$$ −975.606 −0.0424774
$$809$$ 14034.0 0.609900 0.304950 0.952368i $$-0.401360\pi$$
0.304950 + 0.952368i $$0.401360\pi$$
$$810$$ −25234.4 −1.09463
$$811$$ 6632.25 0.287164 0.143582 0.989638i $$-0.454138\pi$$
0.143582 + 0.989638i $$0.454138\pi$$
$$812$$ 0 0
$$813$$ −64768.0 −2.79399
$$814$$ −3120.00 −0.134344
$$815$$ −5966.21 −0.256426
$$816$$ −8448.00 −0.362425
$$817$$ −4090.04 −0.175144
$$818$$ −4840.51 −0.206900
$$819$$ 0 0
$$820$$ 14784.0 0.629609
$$821$$ 28622.0 1.21670 0.608352 0.793667i $$-0.291831\pi$$
0.608352 + 0.793667i $$0.291831\pi$$
$$822$$ −20900.5 −0.886847
$$823$$ 24688.0 1.04565 0.522825 0.852440i $$-0.324878\pi$$
0.522825 + 0.852440i $$0.324878\pi$$
$$824$$ 10956.8 0.463226
$$825$$ −6941.82 −0.292949
$$826$$ 0 0
$$827$$ −30756.0 −1.29322 −0.646609 0.762822i $$-0.723813\pi$$
−0.646609 + 0.762822i $$0.723813\pi$$
$$828$$ 11712.0 0.491570
$$829$$ −23236.3 −0.973499 −0.486750 0.873542i $$-0.661818\pi$$
−0.486750 + 0.873542i $$0.661818\pi$$
$$830$$ 3696.00 0.154566
$$831$$ −63508.2 −2.65111
$$832$$ −4202.61 −0.175119
$$833$$ 0 0
$$834$$ −25872.0 −1.07419
$$835$$ −6160.00 −0.255300
$$836$$ −750.467 −0.0310472
$$837$$ 65824.0 2.71829
$$838$$ 3020.63 0.124518
$$839$$ 24033.7 0.988957 0.494479 0.869190i $$-0.335359\pi$$
0.494479 + 0.869190i $$0.335359\pi$$
$$840$$ 0 0
$$841$$ 3167.00 0.129854
$$842$$ −33540.0 −1.37276
$$843$$ 17617.2 0.719773
$$844$$ 6224.00 0.253838
$$845$$ −19840.5 −0.807731
$$846$$ −25178.2 −1.02322
$$847$$ 0 0
$$848$$ 992.000 0.0401715
$$849$$ −3608.00 −0.145850
$$850$$ 4165.09 0.168072
$$851$$ −3744.00 −0.150814
$$852$$ −20412.7 −0.820807
$$853$$ 23574.0 0.946260 0.473130 0.880993i $$-0.343124\pi$$
0.473130 + 0.880993i $$0.343124\pi$$
$$854$$ 0 0
$$855$$ 5368.00 0.214715
$$856$$ −2080.00 −0.0830525
$$857$$ −24484.0 −0.975912 −0.487956 0.872868i $$-0.662257\pi$$
−0.487956 + 0.872868i $$0.662257\pi$$
$$858$$ −24640.0 −0.980415
$$859$$ 32954.9 1.30897 0.654485 0.756075i $$-0.272885\pi$$
0.654485 + 0.756075i $$0.272885\pi$$
$$860$$ −16360.2 −0.648694
$$861$$ 0 0
$$862$$ 2672.00 0.105579
$$863$$ 40872.0 1.61217 0.806083 0.591803i $$-0.201584\pi$$
0.806083 + 0.591803i $$0.201584\pi$$
$$864$$ 10206.3 0.401883
$$865$$ 6248.00 0.245593
$$866$$ 22326.4 0.876075
$$867$$ −16369.6 −0.641222
$$868$$ 0 0
$$869$$ −13600.0 −0.530896
$$870$$ 29216.0 1.13852
$$871$$ −38086.2 −1.48163
$$872$$ 15056.0 0.584702
$$873$$ 40056.2 1.55292
$$874$$ −900.560 −0.0348534
$$875$$ 0 0
$$876$$ 22528.0 0.868893
$$877$$ −12006.0 −0.462273 −0.231137 0.972921i $$-0.574244\pi$$
−0.231137 + 0.972921i $$0.574244\pi$$
$$878$$ 7204.48 0.276924
$$879$$ 35112.0 1.34732
$$880$$ −3001.87 −0.114992
$$881$$ −35722.2 −1.36607 −0.683037 0.730383i $$-0.739341\pi$$
−0.683037 + 0.730383i $$0.739341\pi$$
$$882$$ 0 0
$$883$$ 19588.0 0.746533 0.373267 0.927724i $$-0.378238\pi$$
0.373267 + 0.927724i $$0.378238\pi$$
$$884$$ 14784.0 0.562488
$$885$$ −58611.4 −2.22622
$$886$$ 12696.0 0.481411
$$887$$ 40243.8 1.52340 0.761699 0.647931i $$-0.224366\pi$$
0.761699 + 0.647931i $$0.224366\pi$$
$$888$$ −5853.64 −0.221211
$$889$$ 0 0
$$890$$ −28160.0 −1.06059
$$891$$ 26900.0 1.01143
$$892$$ −11557.2 −0.433815
$$893$$ 1936.00 0.0725485
$$894$$ −17748.5 −0.663982
$$895$$ 30281.3 1.13094
$$896$$ 0 0
$$897$$ −29568.0 −1.10061
$$898$$ 14340.0 0.532886
$$899$$ −34258.8 −1.27096
$$900$$ −9028.00 −0.334370
$$901$$ −3489.67 −0.129032
$$902$$ −15759.8 −0.581756
$$903$$ 0 0
$$904$$ −10288.0 −0.378511
$$905$$ −26488.0 −0.972918
$$906$$ 15609.7 0.572404
$$907$$ 15868.0 0.580913 0.290457 0.956888i $$-0.406193\pi$$
0.290457 + 0.956888i $$0.406193\pi$$
$$908$$ 7917.42 0.289371
$$909$$ −7439.00 −0.271437
$$910$$ 0 0
$$911$$ 39832.0 1.44862 0.724310 0.689474i $$-0.242158\pi$$
0.724310 + 0.689474i $$0.242158\pi$$
$$912$$ −1408.00 −0.0511223
$$913$$ −3939.95 −0.142818
$$914$$ 13732.0 0.496952
$$915$$ 23939.9 0.864949
$$916$$ −11069.4 −0.399282
$$917$$ 0 0
$$918$$ −35904.0 −1.29086
$$919$$ −30528.0 −1.09578 −0.547892 0.836549i $$-0.684570\pi$$
−0.547892 + 0.836549i $$0.684570\pi$$
$$920$$ −3602.24 −0.129089
$$921$$ −6776.00 −0.242429
$$922$$ −2757.96 −0.0985127
$$923$$ 35722.2 1.27390
$$924$$ 0 0
$$925$$ 2886.00 0.102585
$$926$$ 5296.00 0.187945
$$927$$ 83545.7 2.96009
$$928$$ −5312.00 −0.187904
$$929$$ 16604.1 0.586396 0.293198 0.956052i $$-0.405280\pi$$
0.293198 + 0.956052i $$0.405280\pi$$
$$930$$ −36322.6 −1.28071
$$931$$ 0 0
$$932$$ −25960.0 −0.912391
$$933$$ −68288.0 −2.39619
$$934$$ 24671.6 0.864324
$$935$$ 10560.0 0.369357
$$936$$ −32044.9 −1.11904
$$937$$ −29943.6 −1.04399 −0.521993 0.852950i $$-0.674811\pi$$
−0.521993 + 0.852950i $$0.674811\pi$$
$$938$$ 0 0
$$939$$ −14256.0 −0.495449
$$940$$ 7744.00 0.268704
$$941$$ −5375.22 −0.186214 −0.0931068 0.995656i $$-0.529680\pi$$
−0.0931068 + 0.995656i $$0.529680\pi$$
$$942$$ −54032.0 −1.86885
$$943$$ −18911.8 −0.653077
$$944$$ 10656.6 0.367419
$$945$$ 0 0
$$946$$ 17440.0 0.599390
$$947$$ 45212.0 1.55142 0.775709 0.631091i $$-0.217393\pi$$
0.775709 + 0.631091i $$0.217393\pi$$
$$948$$ −25515.9 −0.874174
$$949$$ −39424.0 −1.34853
$$950$$ 694.182 0.0237076
$$951$$ 22120.0 0.754248
$$952$$ 0 0
$$953$$ 34218.0 1.16310 0.581548 0.813512i $$-0.302447\pi$$
0.581548 + 0.813512i $$0.302447\pi$$
$$954$$ 7564.00 0.256702
$$955$$ 20037.5 0.678950
$$956$$ −17184.0 −0.581350
$$957$$ −31144.4 −1.05199
$$958$$ 26679.1 0.899752
$$959$$ 0 0
$$960$$ −5632.00 −0.189346
$$961$$ 12801.0 0.429694
$$962$$ 10243.9 0.343322
$$963$$ −15860.0 −0.530718
$$964$$ −18086.2 −0.604272
$$965$$ −15553.4 −0.518842
$$966$$ 0 0
$$967$$ 14464.0 0.481004 0.240502 0.970649i $$-0.422688\pi$$
0.240502 + 0.970649i $$0.422688\pi$$
$$968$$ −7448.00 −0.247301
$$969$$ 4953.08 0.164206
$$970$$ −12320.0 −0.407806
$$971$$ −37832.9 −1.25038 −0.625188 0.780474i $$-0.714978\pi$$
−0.625188 + 0.780474i $$0.714978\pi$$
$$972$$ 16022.5 0.528725
$$973$$ 0 0
$$974$$ 27872.0 0.916916
$$975$$ 22792.0 0.748644
$$976$$ −4352.71 −0.142753
$$977$$ 42062.0 1.37736 0.688681 0.725065i $$-0.258190\pi$$
0.688681 + 0.725065i $$0.258190\pi$$
$$978$$ 11932.4 0.390140
$$979$$ 30018.7 0.979980
$$980$$ 0 0
$$981$$ 114802. 3.73634
$$982$$ −24552.0 −0.797847
$$983$$ 43020.5 1.39587 0.697935 0.716161i $$-0.254102\pi$$
0.697935 + 0.716161i $$0.254102\pi$$
$$984$$ −29568.0 −0.957920
$$985$$ 9174.45 0.296774
$$986$$ 18686.6 0.603553
$$987$$ 0 0
$$988$$ 2464.00 0.0793424
$$989$$ 20928.0 0.672873
$$990$$ −22889.2 −0.734816
$$991$$ 21272.0 0.681864 0.340932 0.940088i $$-0.389257\pi$$
0.340932 + 0.940088i $$0.389257\pi$$
$$992$$ 6604.11 0.211372
$$993$$ 22251.3 0.711102
$$994$$ 0 0
$$995$$ −46288.0 −1.47480
$$996$$ −7392.00 −0.235165
$$997$$ −121.951 −0.00387384 −0.00193692 0.999998i $$-0.500617\pi$$
−0.00193692 + 0.999998i $$0.500617\pi$$
$$998$$ −4440.00 −0.140827
$$999$$ −24878.0 −0.787892
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 98.4.a.h.1.2 yes 2
3.2 odd 2 882.4.a.w.1.2 2
4.3 odd 2 784.4.a.z.1.1 2
5.4 even 2 2450.4.a.bs.1.1 2
7.2 even 3 98.4.c.g.67.1 4
7.3 odd 6 98.4.c.g.79.2 4
7.4 even 3 98.4.c.g.79.1 4
7.5 odd 6 98.4.c.g.67.2 4
7.6 odd 2 inner 98.4.a.h.1.1 2
21.2 odd 6 882.4.g.bi.361.1 4
21.5 even 6 882.4.g.bi.361.2 4
21.11 odd 6 882.4.g.bi.667.1 4
21.17 even 6 882.4.g.bi.667.2 4
21.20 even 2 882.4.a.w.1.1 2
28.27 even 2 784.4.a.z.1.2 2
35.34 odd 2 2450.4.a.bs.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.h.1.1 2 7.6 odd 2 inner
98.4.a.h.1.2 yes 2 1.1 even 1 trivial
98.4.c.g.67.1 4 7.2 even 3
98.4.c.g.67.2 4 7.5 odd 6
98.4.c.g.79.1 4 7.4 even 3
98.4.c.g.79.2 4 7.3 odd 6
784.4.a.z.1.1 2 4.3 odd 2
784.4.a.z.1.2 2 28.27 even 2
882.4.a.w.1.1 2 21.20 even 2
882.4.a.w.1.2 2 3.2 odd 2
882.4.g.bi.361.1 4 21.2 odd 6
882.4.g.bi.361.2 4 21.5 even 6
882.4.g.bi.667.1 4 21.11 odd 6
882.4.g.bi.667.2 4 21.17 even 6
2450.4.a.bs.1.1 2 5.4 even 2
2450.4.a.bs.1.2 2 35.34 odd 2