# Properties

 Label 78.4.a.a.1.1 Level $78$ Weight $4$ Character 78.1 Self dual yes Analytic conductor $4.602$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,4,Mod(1,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("78.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 78.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: $$4.60214898045$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 78.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.0000 q^{5} +6.00000 q^{6} +28.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -16.0000 q^{5} +6.00000 q^{6} +28.0000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +32.0000 q^{10} +34.0000 q^{11} -12.0000 q^{12} -13.0000 q^{13} -56.0000 q^{14} +48.0000 q^{15} +16.0000 q^{16} +138.000 q^{17} -18.0000 q^{18} +108.000 q^{19} -64.0000 q^{20} -84.0000 q^{21} -68.0000 q^{22} -52.0000 q^{23} +24.0000 q^{24} +131.000 q^{25} +26.0000 q^{26} -27.0000 q^{27} +112.000 q^{28} -190.000 q^{29} -96.0000 q^{30} -176.000 q^{31} -32.0000 q^{32} -102.000 q^{33} -276.000 q^{34} -448.000 q^{35} +36.0000 q^{36} +342.000 q^{37} -216.000 q^{38} +39.0000 q^{39} +128.000 q^{40} +240.000 q^{41} +168.000 q^{42} -140.000 q^{43} +136.000 q^{44} -144.000 q^{45} +104.000 q^{46} +454.000 q^{47} -48.0000 q^{48} +441.000 q^{49} -262.000 q^{50} -414.000 q^{51} -52.0000 q^{52} +198.000 q^{53} +54.0000 q^{54} -544.000 q^{55} -224.000 q^{56} -324.000 q^{57} +380.000 q^{58} -154.000 q^{59} +192.000 q^{60} +34.0000 q^{61} +352.000 q^{62} +252.000 q^{63} +64.0000 q^{64} +208.000 q^{65} +204.000 q^{66} -656.000 q^{67} +552.000 q^{68} +156.000 q^{69} +896.000 q^{70} +550.000 q^{71} -72.0000 q^{72} +614.000 q^{73} -684.000 q^{74} -393.000 q^{75} +432.000 q^{76} +952.000 q^{77} -78.0000 q^{78} +8.00000 q^{79} -256.000 q^{80} +81.0000 q^{81} -480.000 q^{82} +762.000 q^{83} -336.000 q^{84} -2208.00 q^{85} +280.000 q^{86} +570.000 q^{87} -272.000 q^{88} -444.000 q^{89} +288.000 q^{90} -364.000 q^{91} -208.000 q^{92} +528.000 q^{93} -908.000 q^{94} -1728.00 q^{95} +96.0000 q^{96} +1022.00 q^{97} -882.000 q^{98} +306.000 q^{99} +O(q^{100})$$

## 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
$$3$$ −3.00000 −0.577350
$$4$$ 4.00000 0.500000
$$5$$ −16.0000 −1.43108 −0.715542 0.698570i $$-0.753820\pi$$
−0.715542 + 0.698570i $$0.753820\pi$$
$$6$$ 6.00000 0.408248
$$7$$ 28.0000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ 9.00000 0.333333
$$10$$ 32.0000 1.01193
$$11$$ 34.0000 0.931944 0.465972 0.884799i $$-0.345705\pi$$
0.465972 + 0.884799i $$0.345705\pi$$
$$12$$ −12.0000 −0.288675
$$13$$ −13.0000 −0.277350
$$14$$ −56.0000 −1.06904
$$15$$ 48.0000 0.826236
$$16$$ 16.0000 0.250000
$$17$$ 138.000 1.96882 0.984409 0.175893i $$-0.0562813\pi$$
0.984409 + 0.175893i $$0.0562813\pi$$
$$18$$ −18.0000 −0.235702
$$19$$ 108.000 1.30405 0.652024 0.758199i $$-0.273920\pi$$
0.652024 + 0.758199i $$0.273920\pi$$
$$20$$ −64.0000 −0.715542
$$21$$ −84.0000 −0.872872
$$22$$ −68.0000 −0.658984
$$23$$ −52.0000 −0.471424 −0.235712 0.971823i $$-0.575742\pi$$
−0.235712 + 0.971823i $$0.575742\pi$$
$$24$$ 24.0000 0.204124
$$25$$ 131.000 1.04800
$$26$$ 26.0000 0.196116
$$27$$ −27.0000 −0.192450
$$28$$ 112.000 0.755929
$$29$$ −190.000 −1.21662 −0.608312 0.793698i $$-0.708153\pi$$
−0.608312 + 0.793698i $$0.708153\pi$$
$$30$$ −96.0000 −0.584237
$$31$$ −176.000 −1.01969 −0.509847 0.860265i $$-0.670298\pi$$
−0.509847 + 0.860265i $$0.670298\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ −102.000 −0.538058
$$34$$ −276.000 −1.39216
$$35$$ −448.000 −2.16359
$$36$$ 36.0000 0.166667
$$37$$ 342.000 1.51958 0.759790 0.650169i $$-0.225302\pi$$
0.759790 + 0.650169i $$0.225302\pi$$
$$38$$ −216.000 −0.922101
$$39$$ 39.0000 0.160128
$$40$$ 128.000 0.505964
$$41$$ 240.000 0.914188 0.457094 0.889418i $$-0.348890\pi$$
0.457094 + 0.889418i $$0.348890\pi$$
$$42$$ 168.000 0.617213
$$43$$ −140.000 −0.496507 −0.248253 0.968695i $$-0.579857\pi$$
−0.248253 + 0.968695i $$0.579857\pi$$
$$44$$ 136.000 0.465972
$$45$$ −144.000 −0.477028
$$46$$ 104.000 0.333347
$$47$$ 454.000 1.40899 0.704497 0.709707i $$-0.251173\pi$$
0.704497 + 0.709707i $$0.251173\pi$$
$$48$$ −48.0000 −0.144338
$$49$$ 441.000 1.28571
$$50$$ −262.000 −0.741048
$$51$$ −414.000 −1.13670
$$52$$ −52.0000 −0.138675
$$53$$ 198.000 0.513158 0.256579 0.966523i $$-0.417405\pi$$
0.256579 + 0.966523i $$0.417405\pi$$
$$54$$ 54.0000 0.136083
$$55$$ −544.000 −1.33369
$$56$$ −224.000 −0.534522
$$57$$ −324.000 −0.752892
$$58$$ 380.000 0.860284
$$59$$ −154.000 −0.339815 −0.169908 0.985460i $$-0.554347\pi$$
−0.169908 + 0.985460i $$0.554347\pi$$
$$60$$ 192.000 0.413118
$$61$$ 34.0000 0.0713648 0.0356824 0.999363i $$-0.488640\pi$$
0.0356824 + 0.999363i $$0.488640\pi$$
$$62$$ 352.000 0.721033
$$63$$ 252.000 0.503953
$$64$$ 64.0000 0.125000
$$65$$ 208.000 0.396911
$$66$$ 204.000 0.380465
$$67$$ −656.000 −1.19617 −0.598083 0.801434i $$-0.704071\pi$$
−0.598083 + 0.801434i $$0.704071\pi$$
$$68$$ 552.000 0.984409
$$69$$ 156.000 0.272177
$$70$$ 896.000 1.52989
$$71$$ 550.000 0.919338 0.459669 0.888090i $$-0.347968\pi$$
0.459669 + 0.888090i $$0.347968\pi$$
$$72$$ −72.0000 −0.117851
$$73$$ 614.000 0.984428 0.492214 0.870474i $$-0.336188\pi$$
0.492214 + 0.870474i $$0.336188\pi$$
$$74$$ −684.000 −1.07451
$$75$$ −393.000 −0.605063
$$76$$ 432.000 0.652024
$$77$$ 952.000 1.40897
$$78$$ −78.0000 −0.113228
$$79$$ 8.00000 0.0113933 0.00569665 0.999984i $$-0.498187\pi$$
0.00569665 + 0.999984i $$0.498187\pi$$
$$80$$ −256.000 −0.357771
$$81$$ 81.0000 0.111111
$$82$$ −480.000 −0.646428
$$83$$ 762.000 1.00772 0.503858 0.863787i $$-0.331914\pi$$
0.503858 + 0.863787i $$0.331914\pi$$
$$84$$ −336.000 −0.436436
$$85$$ −2208.00 −2.81754
$$86$$ 280.000 0.351083
$$87$$ 570.000 0.702419
$$88$$ −272.000 −0.329492
$$89$$ −444.000 −0.528808 −0.264404 0.964412i $$-0.585175\pi$$
−0.264404 + 0.964412i $$0.585175\pi$$
$$90$$ 288.000 0.337310
$$91$$ −364.000 −0.419314
$$92$$ −208.000 −0.235712
$$93$$ 528.000 0.588721
$$94$$ −908.000 −0.996309
$$95$$ −1728.00 −1.86620
$$96$$ 96.0000 0.102062
$$97$$ 1022.00 1.06978 0.534889 0.844923i $$-0.320354\pi$$
0.534889 + 0.844923i $$0.320354\pi$$
$$98$$ −882.000 −0.909137
$$99$$ 306.000 0.310648
$$100$$ 524.000 0.524000
$$101$$ −1190.00 −1.17237 −0.586185 0.810177i $$-0.699371\pi$$
−0.586185 + 0.810177i $$0.699371\pi$$
$$102$$ 828.000 0.803767
$$103$$ −224.000 −0.214285 −0.107143 0.994244i $$-0.534170\pi$$
−0.107143 + 0.994244i $$0.534170\pi$$
$$104$$ 104.000 0.0980581
$$105$$ 1344.00 1.24915
$$106$$ −396.000 −0.362858
$$107$$ −640.000 −0.578235 −0.289117 0.957294i $$-0.593362\pi$$
−0.289117 + 0.957294i $$0.593362\pi$$
$$108$$ −108.000 −0.0962250
$$109$$ −1934.00 −1.69948 −0.849741 0.527200i $$-0.823242\pi$$
−0.849741 + 0.527200i $$0.823242\pi$$
$$110$$ 1088.00 0.943061
$$111$$ −1026.00 −0.877330
$$112$$ 448.000 0.377964
$$113$$ −418.000 −0.347983 −0.173992 0.984747i $$-0.555667\pi$$
−0.173992 + 0.984747i $$0.555667\pi$$
$$114$$ 648.000 0.532375
$$115$$ 832.000 0.674647
$$116$$ −760.000 −0.608312
$$117$$ −117.000 −0.0924500
$$118$$ 308.000 0.240286
$$119$$ 3864.00 2.97657
$$120$$ −384.000 −0.292119
$$121$$ −175.000 −0.131480
$$122$$ −68.0000 −0.0504625
$$123$$ −720.000 −0.527807
$$124$$ −704.000 −0.509847
$$125$$ −96.0000 −0.0686920
$$126$$ −504.000 −0.356348
$$127$$ −1040.00 −0.726654 −0.363327 0.931662i $$-0.618359\pi$$
−0.363327 + 0.931662i $$0.618359\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 420.000 0.286658
$$130$$ −416.000 −0.280659
$$131$$ −568.000 −0.378827 −0.189414 0.981897i $$-0.560659\pi$$
−0.189414 + 0.981897i $$0.560659\pi$$
$$132$$ −408.000 −0.269029
$$133$$ 3024.00 1.97153
$$134$$ 1312.00 0.845817
$$135$$ 432.000 0.275412
$$136$$ −1104.00 −0.696082
$$137$$ 528.000 0.329271 0.164635 0.986355i $$-0.447355\pi$$
0.164635 + 0.986355i $$0.447355\pi$$
$$138$$ −312.000 −0.192458
$$139$$ −1556.00 −0.949483 −0.474742 0.880125i $$-0.657459\pi$$
−0.474742 + 0.880125i $$0.657459\pi$$
$$140$$ −1792.00 −1.08180
$$141$$ −1362.00 −0.813483
$$142$$ −1100.00 −0.650070
$$143$$ −442.000 −0.258475
$$144$$ 144.000 0.0833333
$$145$$ 3040.00 1.74109
$$146$$ −1228.00 −0.696096
$$147$$ −1323.00 −0.742307
$$148$$ 1368.00 0.759790
$$149$$ 1524.00 0.837926 0.418963 0.908003i $$-0.362394\pi$$
0.418963 + 0.908003i $$0.362394\pi$$
$$150$$ 786.000 0.427844
$$151$$ −3024.00 −1.62973 −0.814866 0.579649i $$-0.803190\pi$$
−0.814866 + 0.579649i $$0.803190\pi$$
$$152$$ −864.000 −0.461050
$$153$$ 1242.00 0.656273
$$154$$ −1904.00 −0.996290
$$155$$ 2816.00 1.45927
$$156$$ 156.000 0.0800641
$$157$$ 2198.00 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ −16.0000 −0.00805628
$$159$$ −594.000 −0.296272
$$160$$ 512.000 0.252982
$$161$$ −1456.00 −0.712726
$$162$$ −162.000 −0.0785674
$$163$$ −268.000 −0.128781 −0.0643907 0.997925i $$-0.520510\pi$$
−0.0643907 + 0.997925i $$0.520510\pi$$
$$164$$ 960.000 0.457094
$$165$$ 1632.00 0.770006
$$166$$ −1524.00 −0.712562
$$167$$ 702.000 0.325284 0.162642 0.986685i $$-0.447998\pi$$
0.162642 + 0.986685i $$0.447998\pi$$
$$168$$ 672.000 0.308607
$$169$$ 169.000 0.0769231
$$170$$ 4416.00 1.99230
$$171$$ 972.000 0.434682
$$172$$ −560.000 −0.248253
$$173$$ 2066.00 0.907948 0.453974 0.891015i $$-0.350006\pi$$
0.453974 + 0.891015i $$0.350006\pi$$
$$174$$ −1140.00 −0.496685
$$175$$ 3668.00 1.58443
$$176$$ 544.000 0.232986
$$177$$ 462.000 0.196192
$$178$$ 888.000 0.373924
$$179$$ −276.000 −0.115247 −0.0576235 0.998338i $$-0.518352\pi$$
−0.0576235 + 0.998338i $$0.518352\pi$$
$$180$$ −576.000 −0.238514
$$181$$ −3474.00 −1.42663 −0.713316 0.700843i $$-0.752808\pi$$
−0.713316 + 0.700843i $$0.752808\pi$$
$$182$$ 728.000 0.296500
$$183$$ −102.000 −0.0412025
$$184$$ 416.000 0.166674
$$185$$ −5472.00 −2.17465
$$186$$ −1056.00 −0.416289
$$187$$ 4692.00 1.83483
$$188$$ 1816.00 0.704497
$$189$$ −756.000 −0.290957
$$190$$ 3456.00 1.31960
$$191$$ −3920.00 −1.48503 −0.742516 0.669828i $$-0.766368\pi$$
−0.742516 + 0.669828i $$0.766368\pi$$
$$192$$ −192.000 −0.0721688
$$193$$ 2186.00 0.815294 0.407647 0.913140i $$-0.366349\pi$$
0.407647 + 0.913140i $$0.366349\pi$$
$$194$$ −2044.00 −0.756447
$$195$$ −624.000 −0.229157
$$196$$ 1764.00 0.642857
$$197$$ −1368.00 −0.494751 −0.247376 0.968920i $$-0.579568\pi$$
−0.247376 + 0.968920i $$0.579568\pi$$
$$198$$ −612.000 −0.219661
$$199$$ −1072.00 −0.381870 −0.190935 0.981603i $$-0.561152\pi$$
−0.190935 + 0.981603i $$0.561152\pi$$
$$200$$ −1048.00 −0.370524
$$201$$ 1968.00 0.690607
$$202$$ 2380.00 0.828991
$$203$$ −5320.00 −1.83936
$$204$$ −1656.00 −0.568349
$$205$$ −3840.00 −1.30828
$$206$$ 448.000 0.151523
$$207$$ −468.000 −0.157141
$$208$$ −208.000 −0.0693375
$$209$$ 3672.00 1.21530
$$210$$ −2688.00 −0.883284
$$211$$ 5444.00 1.77621 0.888105 0.459640i $$-0.152022\pi$$
0.888105 + 0.459640i $$0.152022\pi$$
$$212$$ 792.000 0.256579
$$213$$ −1650.00 −0.530780
$$214$$ 1280.00 0.408874
$$215$$ 2240.00 0.710543
$$216$$ 216.000 0.0680414
$$217$$ −4928.00 −1.54163
$$218$$ 3868.00 1.20172
$$219$$ −1842.00 −0.568360
$$220$$ −2176.00 −0.666845
$$221$$ −1794.00 −0.546052
$$222$$ 2052.00 0.620366
$$223$$ 96.0000 0.0288280 0.0144140 0.999896i $$-0.495412\pi$$
0.0144140 + 0.999896i $$0.495412\pi$$
$$224$$ −896.000 −0.267261
$$225$$ 1179.00 0.349333
$$226$$ 836.000 0.246061
$$227$$ 198.000 0.0578930 0.0289465 0.999581i $$-0.490785\pi$$
0.0289465 + 0.999581i $$0.490785\pi$$
$$228$$ −1296.00 −0.376446
$$229$$ 5922.00 1.70889 0.854447 0.519538i $$-0.173896\pi$$
0.854447 + 0.519538i $$0.173896\pi$$
$$230$$ −1664.00 −0.477047
$$231$$ −2856.00 −0.813468
$$232$$ 1520.00 0.430142
$$233$$ −5114.00 −1.43789 −0.718947 0.695065i $$-0.755376\pi$$
−0.718947 + 0.695065i $$0.755376\pi$$
$$234$$ 234.000 0.0653720
$$235$$ −7264.00 −2.01639
$$236$$ −616.000 −0.169908
$$237$$ −24.0000 −0.00657792
$$238$$ −7728.00 −2.10476
$$239$$ −5226.00 −1.41440 −0.707200 0.707013i $$-0.750042\pi$$
−0.707200 + 0.707013i $$0.750042\pi$$
$$240$$ 768.000 0.206559
$$241$$ −762.000 −0.203671 −0.101836 0.994801i $$-0.532472\pi$$
−0.101836 + 0.994801i $$0.532472\pi$$
$$242$$ 350.000 0.0929705
$$243$$ −243.000 −0.0641500
$$244$$ 136.000 0.0356824
$$245$$ −7056.00 −1.83996
$$246$$ 1440.00 0.373216
$$247$$ −1404.00 −0.361678
$$248$$ 1408.00 0.360516
$$249$$ −2286.00 −0.581805
$$250$$ 192.000 0.0485726
$$251$$ 3240.00 0.814769 0.407384 0.913257i $$-0.366441\pi$$
0.407384 + 0.913257i $$0.366441\pi$$
$$252$$ 1008.00 0.251976
$$253$$ −1768.00 −0.439341
$$254$$ 2080.00 0.513822
$$255$$ 6624.00 1.62671
$$256$$ 256.000 0.0625000
$$257$$ −1386.00 −0.336406 −0.168203 0.985752i $$-0.553796\pi$$
−0.168203 + 0.985752i $$0.553796\pi$$
$$258$$ −840.000 −0.202698
$$259$$ 9576.00 2.29739
$$260$$ 832.000 0.198456
$$261$$ −1710.00 −0.405542
$$262$$ 1136.00 0.267871
$$263$$ 3300.00 0.773714 0.386857 0.922140i $$-0.373561\pi$$
0.386857 + 0.922140i $$0.373561\pi$$
$$264$$ 816.000 0.190232
$$265$$ −3168.00 −0.734372
$$266$$ −6048.00 −1.39409
$$267$$ 1332.00 0.305307
$$268$$ −2624.00 −0.598083
$$269$$ 4290.00 0.972364 0.486182 0.873858i $$-0.338389\pi$$
0.486182 + 0.873858i $$0.338389\pi$$
$$270$$ −864.000 −0.194746
$$271$$ 2452.00 0.549625 0.274813 0.961498i $$-0.411384\pi$$
0.274813 + 0.961498i $$0.411384\pi$$
$$272$$ 2208.00 0.492205
$$273$$ 1092.00 0.242091
$$274$$ −1056.00 −0.232830
$$275$$ 4454.00 0.976677
$$276$$ 624.000 0.136088
$$277$$ −42.0000 −0.00911024 −0.00455512 0.999990i $$-0.501450\pi$$
−0.00455512 + 0.999990i $$0.501450\pi$$
$$278$$ 3112.00 0.671386
$$279$$ −1584.00 −0.339898
$$280$$ 3584.00 0.764946
$$281$$ −2288.00 −0.485732 −0.242866 0.970060i $$-0.578088\pi$$
−0.242866 + 0.970060i $$0.578088\pi$$
$$282$$ 2724.00 0.575219
$$283$$ 1156.00 0.242816 0.121408 0.992603i $$-0.461259\pi$$
0.121408 + 0.992603i $$0.461259\pi$$
$$284$$ 2200.00 0.459669
$$285$$ 5184.00 1.07745
$$286$$ 884.000 0.182769
$$287$$ 6720.00 1.38212
$$288$$ −288.000 −0.0589256
$$289$$ 14131.0 2.87625
$$290$$ −6080.00 −1.23114
$$291$$ −3066.00 −0.617636
$$292$$ 2456.00 0.492214
$$293$$ −8684.00 −1.73148 −0.865742 0.500491i $$-0.833153\pi$$
−0.865742 + 0.500491i $$0.833153\pi$$
$$294$$ 2646.00 0.524891
$$295$$ 2464.00 0.486304
$$296$$ −2736.00 −0.537253
$$297$$ −918.000 −0.179353
$$298$$ −3048.00 −0.592503
$$299$$ 676.000 0.130749
$$300$$ −1572.00 −0.302532
$$301$$ −3920.00 −0.750648
$$302$$ 6048.00 1.15240
$$303$$ 3570.00 0.676868
$$304$$ 1728.00 0.326012
$$305$$ −544.000 −0.102129
$$306$$ −2484.00 −0.464055
$$307$$ −7552.00 −1.40396 −0.701979 0.712197i $$-0.747700\pi$$
−0.701979 + 0.712197i $$0.747700\pi$$
$$308$$ 3808.00 0.704484
$$309$$ 672.000 0.123718
$$310$$ −5632.00 −1.03186
$$311$$ 2652.00 0.483541 0.241770 0.970334i $$-0.422272\pi$$
0.241770 + 0.970334i $$0.422272\pi$$
$$312$$ −312.000 −0.0566139
$$313$$ −4426.00 −0.799273 −0.399636 0.916674i $$-0.630864\pi$$
−0.399636 + 0.916674i $$0.630864\pi$$
$$314$$ −4396.00 −0.790066
$$315$$ −4032.00 −0.721198
$$316$$ 32.0000 0.00569665
$$317$$ −4944.00 −0.875971 −0.437985 0.898982i $$-0.644308\pi$$
−0.437985 + 0.898982i $$0.644308\pi$$
$$318$$ 1188.00 0.209496
$$319$$ −6460.00 −1.13383
$$320$$ −1024.00 −0.178885
$$321$$ 1920.00 0.333844
$$322$$ 2912.00 0.503973
$$323$$ 14904.0 2.56743
$$324$$ 324.000 0.0555556
$$325$$ −1703.00 −0.290663
$$326$$ 536.000 0.0910623
$$327$$ 5802.00 0.981197
$$328$$ −1920.00 −0.323214
$$329$$ 12712.0 2.13020
$$330$$ −3264.00 −0.544477
$$331$$ −6088.00 −1.01096 −0.505478 0.862839i $$-0.668684\pi$$
−0.505478 + 0.862839i $$0.668684\pi$$
$$332$$ 3048.00 0.503858
$$333$$ 3078.00 0.506527
$$334$$ −1404.00 −0.230010
$$335$$ 10496.0 1.71181
$$336$$ −1344.00 −0.218218
$$337$$ 6638.00 1.07298 0.536491 0.843906i $$-0.319750\pi$$
0.536491 + 0.843906i $$0.319750\pi$$
$$338$$ −338.000 −0.0543928
$$339$$ 1254.00 0.200908
$$340$$ −8832.00 −1.40877
$$341$$ −5984.00 −0.950298
$$342$$ −1944.00 −0.307367
$$343$$ 2744.00 0.431959
$$344$$ 1120.00 0.175542
$$345$$ −2496.00 −0.389508
$$346$$ −4132.00 −0.642016
$$347$$ −2292.00 −0.354585 −0.177293 0.984158i $$-0.556734\pi$$
−0.177293 + 0.984158i $$0.556734\pi$$
$$348$$ 2280.00 0.351209
$$349$$ −9866.00 −1.51322 −0.756612 0.653865i $$-0.773147\pi$$
−0.756612 + 0.653865i $$0.773147\pi$$
$$350$$ −7336.00 −1.12036
$$351$$ 351.000 0.0533761
$$352$$ −1088.00 −0.164746
$$353$$ 2368.00 0.357042 0.178521 0.983936i $$-0.442869\pi$$
0.178521 + 0.983936i $$0.442869\pi$$
$$354$$ −924.000 −0.138729
$$355$$ −8800.00 −1.31565
$$356$$ −1776.00 −0.264404
$$357$$ −11592.0 −1.71853
$$358$$ 552.000 0.0814919
$$359$$ 5070.00 0.745360 0.372680 0.927960i $$-0.378439\pi$$
0.372680 + 0.927960i $$0.378439\pi$$
$$360$$ 1152.00 0.168655
$$361$$ 4805.00 0.700539
$$362$$ 6948.00 1.00878
$$363$$ 525.000 0.0759101
$$364$$ −1456.00 −0.209657
$$365$$ −9824.00 −1.40880
$$366$$ 204.000 0.0291346
$$367$$ −8584.00 −1.22093 −0.610465 0.792043i $$-0.709017\pi$$
−0.610465 + 0.792043i $$0.709017\pi$$
$$368$$ −832.000 −0.117856
$$369$$ 2160.00 0.304729
$$370$$ 10944.0 1.53771
$$371$$ 5544.00 0.775822
$$372$$ 2112.00 0.294360
$$373$$ 4994.00 0.693243 0.346621 0.938005i $$-0.387329\pi$$
0.346621 + 0.938005i $$0.387329\pi$$
$$374$$ −9384.00 −1.29742
$$375$$ 288.000 0.0396593
$$376$$ −3632.00 −0.498155
$$377$$ 2470.00 0.337431
$$378$$ 1512.00 0.205738
$$379$$ 1300.00 0.176191 0.0880957 0.996112i $$-0.471922\pi$$
0.0880957 + 0.996112i $$0.471922\pi$$
$$380$$ −6912.00 −0.933100
$$381$$ 3120.00 0.419534
$$382$$ 7840.00 1.05008
$$383$$ −4590.00 −0.612371 −0.306185 0.951972i $$-0.599053\pi$$
−0.306185 + 0.951972i $$0.599053\pi$$
$$384$$ 384.000 0.0510310
$$385$$ −15232.0 −2.01635
$$386$$ −4372.00 −0.576500
$$387$$ −1260.00 −0.165502
$$388$$ 4088.00 0.534889
$$389$$ 3510.00 0.457491 0.228746 0.973486i $$-0.426538\pi$$
0.228746 + 0.973486i $$0.426538\pi$$
$$390$$ 1248.00 0.162038
$$391$$ −7176.00 −0.928148
$$392$$ −3528.00 −0.454569
$$393$$ 1704.00 0.218716
$$394$$ 2736.00 0.349842
$$395$$ −128.000 −0.0163048
$$396$$ 1224.00 0.155324
$$397$$ 6230.00 0.787594 0.393797 0.919197i $$-0.371161\pi$$
0.393797 + 0.919197i $$0.371161\pi$$
$$398$$ 2144.00 0.270023
$$399$$ −9072.00 −1.13827
$$400$$ 2096.00 0.262000
$$401$$ −7500.00 −0.933995 −0.466998 0.884259i $$-0.654664\pi$$
−0.466998 + 0.884259i $$0.654664\pi$$
$$402$$ −3936.00 −0.488333
$$403$$ 2288.00 0.282812
$$404$$ −4760.00 −0.586185
$$405$$ −1296.00 −0.159009
$$406$$ 10640.0 1.30063
$$407$$ 11628.0 1.41616
$$408$$ 3312.00 0.401883
$$409$$ 8254.00 0.997883 0.498941 0.866636i $$-0.333722\pi$$
0.498941 + 0.866636i $$0.333722\pi$$
$$410$$ 7680.00 0.925093
$$411$$ −1584.00 −0.190105
$$412$$ −896.000 −0.107143
$$413$$ −4312.00 −0.513752
$$414$$ 936.000 0.111116
$$415$$ −12192.0 −1.44212
$$416$$ 416.000 0.0490290
$$417$$ 4668.00 0.548185
$$418$$ −7344.00 −0.859346
$$419$$ −14808.0 −1.72653 −0.863267 0.504747i $$-0.831586\pi$$
−0.863267 + 0.504747i $$0.831586\pi$$
$$420$$ 5376.00 0.624576
$$421$$ 10354.0 1.19863 0.599315 0.800513i $$-0.295440\pi$$
0.599315 + 0.800513i $$0.295440\pi$$
$$422$$ −10888.0 −1.25597
$$423$$ 4086.00 0.469665
$$424$$ −1584.00 −0.181429
$$425$$ 18078.0 2.06332
$$426$$ 3300.00 0.375318
$$427$$ 952.000 0.107893
$$428$$ −2560.00 −0.289117
$$429$$ 1326.00 0.149230
$$430$$ −4480.00 −0.502430
$$431$$ −15486.0 −1.73071 −0.865353 0.501163i $$-0.832906\pi$$
−0.865353 + 0.501163i $$0.832906\pi$$
$$432$$ −432.000 −0.0481125
$$433$$ −2018.00 −0.223970 −0.111985 0.993710i $$-0.535721\pi$$
−0.111985 + 0.993710i $$0.535721\pi$$
$$434$$ 9856.00 1.09010
$$435$$ −9120.00 −1.00522
$$436$$ −7736.00 −0.849741
$$437$$ −5616.00 −0.614759
$$438$$ 3684.00 0.401891
$$439$$ 8792.00 0.955853 0.477926 0.878400i $$-0.341389\pi$$
0.477926 + 0.878400i $$0.341389\pi$$
$$440$$ 4352.00 0.471531
$$441$$ 3969.00 0.428571
$$442$$ 3588.00 0.386117
$$443$$ −2760.00 −0.296008 −0.148004 0.988987i $$-0.547285\pi$$
−0.148004 + 0.988987i $$0.547285\pi$$
$$444$$ −4104.00 −0.438665
$$445$$ 7104.00 0.756768
$$446$$ −192.000 −0.0203844
$$447$$ −4572.00 −0.483777
$$448$$ 1792.00 0.188982
$$449$$ 9532.00 1.00188 0.500939 0.865483i $$-0.332988\pi$$
0.500939 + 0.865483i $$0.332988\pi$$
$$450$$ −2358.00 −0.247016
$$451$$ 8160.00 0.851972
$$452$$ −1672.00 −0.173992
$$453$$ 9072.00 0.940927
$$454$$ −396.000 −0.0409366
$$455$$ 5824.00 0.600073
$$456$$ 2592.00 0.266188
$$457$$ 12862.0 1.31654 0.658270 0.752782i $$-0.271288\pi$$
0.658270 + 0.752782i $$0.271288\pi$$
$$458$$ −11844.0 −1.20837
$$459$$ −3726.00 −0.378899
$$460$$ 3328.00 0.337323
$$461$$ −6744.00 −0.681344 −0.340672 0.940182i $$-0.610654\pi$$
−0.340672 + 0.940182i $$0.610654\pi$$
$$462$$ 5712.00 0.575208
$$463$$ −9572.00 −0.960796 −0.480398 0.877051i $$-0.659508\pi$$
−0.480398 + 0.877051i $$0.659508\pi$$
$$464$$ −3040.00 −0.304156
$$465$$ −8448.00 −0.842509
$$466$$ 10228.0 1.01674
$$467$$ 9104.00 0.902105 0.451052 0.892498i $$-0.351049\pi$$
0.451052 + 0.892498i $$0.351049\pi$$
$$468$$ −468.000 −0.0462250
$$469$$ −18368.0 −1.80843
$$470$$ 14528.0 1.42580
$$471$$ −6594.00 −0.645086
$$472$$ 1232.00 0.120143
$$473$$ −4760.00 −0.462717
$$474$$ 48.0000 0.00465129
$$475$$ 14148.0 1.36664
$$476$$ 15456.0 1.48829
$$477$$ 1782.00 0.171053
$$478$$ 10452.0 1.00013
$$479$$ −18870.0 −1.79998 −0.899992 0.435906i $$-0.856428\pi$$
−0.899992 + 0.435906i $$0.856428\pi$$
$$480$$ −1536.00 −0.146059
$$481$$ −4446.00 −0.421456
$$482$$ 1524.00 0.144017
$$483$$ 4368.00 0.411493
$$484$$ −700.000 −0.0657400
$$485$$ −16352.0 −1.53094
$$486$$ 486.000 0.0453609
$$487$$ −1744.00 −0.162276 −0.0811378 0.996703i $$-0.525855\pi$$
−0.0811378 + 0.996703i $$0.525855\pi$$
$$488$$ −272.000 −0.0252313
$$489$$ 804.000 0.0743520
$$490$$ 14112.0 1.30105
$$491$$ 13360.0 1.22796 0.613980 0.789322i $$-0.289568\pi$$
0.613980 + 0.789322i $$0.289568\pi$$
$$492$$ −2880.00 −0.263903
$$493$$ −26220.0 −2.39531
$$494$$ 2808.00 0.255745
$$495$$ −4896.00 −0.444563
$$496$$ −2816.00 −0.254924
$$497$$ 15400.0 1.38991
$$498$$ 4572.00 0.411398
$$499$$ 17368.0 1.55811 0.779057 0.626954i $$-0.215698\pi$$
0.779057 + 0.626954i $$0.215698\pi$$
$$500$$ −384.000 −0.0343460
$$501$$ −2106.00 −0.187803
$$502$$ −6480.00 −0.576129
$$503$$ −5828.00 −0.516616 −0.258308 0.966063i $$-0.583165\pi$$
−0.258308 + 0.966063i $$0.583165\pi$$
$$504$$ −2016.00 −0.178174
$$505$$ 19040.0 1.67776
$$506$$ 3536.00 0.310661
$$507$$ −507.000 −0.0444116
$$508$$ −4160.00 −0.363327
$$509$$ 10744.0 0.935598 0.467799 0.883835i $$-0.345047\pi$$
0.467799 + 0.883835i $$0.345047\pi$$
$$510$$ −13248.0 −1.15026
$$511$$ 17192.0 1.48832
$$512$$ −512.000 −0.0441942
$$513$$ −2916.00 −0.250964
$$514$$ 2772.00 0.237875
$$515$$ 3584.00 0.306660
$$516$$ 1680.00 0.143329
$$517$$ 15436.0 1.31310
$$518$$ −19152.0 −1.62450
$$519$$ −6198.00 −0.524204
$$520$$ −1664.00 −0.140329
$$521$$ −12234.0 −1.02875 −0.514377 0.857564i $$-0.671977\pi$$
−0.514377 + 0.857564i $$0.671977\pi$$
$$522$$ 3420.00 0.286761
$$523$$ 1812.00 0.151498 0.0757488 0.997127i $$-0.475865\pi$$
0.0757488 + 0.997127i $$0.475865\pi$$
$$524$$ −2272.00 −0.189414
$$525$$ −11004.0 −0.914769
$$526$$ −6600.00 −0.547098
$$527$$ −24288.0 −2.00759
$$528$$ −1632.00 −0.134515
$$529$$ −9463.00 −0.777760
$$530$$ 6336.00 0.519280
$$531$$ −1386.00 −0.113272
$$532$$ 12096.0 0.985767
$$533$$ −3120.00 −0.253550
$$534$$ −2664.00 −0.215885
$$535$$ 10240.0 0.827502
$$536$$ 5248.00 0.422909
$$537$$ 828.000 0.0665379
$$538$$ −8580.00 −0.687565
$$539$$ 14994.0 1.19821
$$540$$ 1728.00 0.137706
$$541$$ 6098.00 0.484609 0.242305 0.970200i $$-0.422097\pi$$
0.242305 + 0.970200i $$0.422097\pi$$
$$542$$ −4904.00 −0.388644
$$543$$ 10422.0 0.823666
$$544$$ −4416.00 −0.348041
$$545$$ 30944.0 2.43210
$$546$$ −2184.00 −0.171184
$$547$$ −18332.0 −1.43294 −0.716471 0.697616i $$-0.754244\pi$$
−0.716471 + 0.697616i $$0.754244\pi$$
$$548$$ 2112.00 0.164635
$$549$$ 306.000 0.0237883
$$550$$ −8908.00 −0.690615
$$551$$ −20520.0 −1.58654
$$552$$ −1248.00 −0.0962290
$$553$$ 224.000 0.0172250
$$554$$ 84.0000 0.00644191
$$555$$ 16416.0 1.25553
$$556$$ −6224.00 −0.474742
$$557$$ −20004.0 −1.52172 −0.760859 0.648917i $$-0.775222\pi$$
−0.760859 + 0.648917i $$0.775222\pi$$
$$558$$ 3168.00 0.240344
$$559$$ 1820.00 0.137706
$$560$$ −7168.00 −0.540899
$$561$$ −14076.0 −1.05934
$$562$$ 4576.00 0.343464
$$563$$ 10988.0 0.822538 0.411269 0.911514i $$-0.365086\pi$$
0.411269 + 0.911514i $$0.365086\pi$$
$$564$$ −5448.00 −0.406741
$$565$$ 6688.00 0.497993
$$566$$ −2312.00 −0.171697
$$567$$ 2268.00 0.167984
$$568$$ −4400.00 −0.325035
$$569$$ 11062.0 0.815014 0.407507 0.913202i $$-0.366398\pi$$
0.407507 + 0.913202i $$0.366398\pi$$
$$570$$ −10368.0 −0.761873
$$571$$ −708.000 −0.0518895 −0.0259447 0.999663i $$-0.508259\pi$$
−0.0259447 + 0.999663i $$0.508259\pi$$
$$572$$ −1768.00 −0.129237
$$573$$ 11760.0 0.857384
$$574$$ −13440.0 −0.977308
$$575$$ −6812.00 −0.494052
$$576$$ 576.000 0.0416667
$$577$$ −2094.00 −0.151082 −0.0755410 0.997143i $$-0.524068\pi$$
−0.0755410 + 0.997143i $$0.524068\pi$$
$$578$$ −28262.0 −2.03381
$$579$$ −6558.00 −0.470710
$$580$$ 12160.0 0.870546
$$581$$ 21336.0 1.52352
$$582$$ 6132.00 0.436735
$$583$$ 6732.00 0.478235
$$584$$ −4912.00 −0.348048
$$585$$ 1872.00 0.132304
$$586$$ 17368.0 1.22434
$$587$$ −17854.0 −1.25539 −0.627695 0.778460i $$-0.716001\pi$$
−0.627695 + 0.778460i $$0.716001\pi$$
$$588$$ −5292.00 −0.371154
$$589$$ −19008.0 −1.32973
$$590$$ −4928.00 −0.343869
$$591$$ 4104.00 0.285645
$$592$$ 5472.00 0.379895
$$593$$ 23948.0 1.65839 0.829196 0.558958i $$-0.188799\pi$$
0.829196 + 0.558958i $$0.188799\pi$$
$$594$$ 1836.00 0.126822
$$595$$ −61824.0 −4.25973
$$596$$ 6096.00 0.418963
$$597$$ 3216.00 0.220473
$$598$$ −1352.00 −0.0924538
$$599$$ −18068.0 −1.23245 −0.616226 0.787570i $$-0.711339\pi$$
−0.616226 + 0.787570i $$0.711339\pi$$
$$600$$ 3144.00 0.213922
$$601$$ 19942.0 1.35350 0.676748 0.736215i $$-0.263389\pi$$
0.676748 + 0.736215i $$0.263389\pi$$
$$602$$ 7840.00 0.530788
$$603$$ −5904.00 −0.398722
$$604$$ −12096.0 −0.814866
$$605$$ 2800.00 0.188159
$$606$$ −7140.00 −0.478618
$$607$$ 26376.0 1.76370 0.881852 0.471526i $$-0.156296\pi$$
0.881852 + 0.471526i $$0.156296\pi$$
$$608$$ −3456.00 −0.230525
$$609$$ 15960.0 1.06196
$$610$$ 1088.00 0.0722161
$$611$$ −5902.00 −0.390785
$$612$$ 4968.00 0.328136
$$613$$ −19426.0 −1.27995 −0.639975 0.768396i $$-0.721055\pi$$
−0.639975 + 0.768396i $$0.721055\pi$$
$$614$$ 15104.0 0.992749
$$615$$ 11520.0 0.755335
$$616$$ −7616.00 −0.498145
$$617$$ −8024.00 −0.523556 −0.261778 0.965128i $$-0.584309\pi$$
−0.261778 + 0.965128i $$0.584309\pi$$
$$618$$ −1344.00 −0.0874816
$$619$$ −20648.0 −1.34073 −0.670366 0.742031i $$-0.733863\pi$$
−0.670366 + 0.742031i $$0.733863\pi$$
$$620$$ 11264.0 0.729634
$$621$$ 1404.00 0.0907256
$$622$$ −5304.00 −0.341915
$$623$$ −12432.0 −0.799482
$$624$$ 624.000 0.0400320
$$625$$ −14839.0 −0.949696
$$626$$ 8852.00 0.565171
$$627$$ −11016.0 −0.701653
$$628$$ 8792.00 0.558661
$$629$$ 47196.0 2.99178
$$630$$ 8064.00 0.509964
$$631$$ 12280.0 0.774737 0.387369 0.921925i $$-0.373384\pi$$
0.387369 + 0.921925i $$0.373384\pi$$
$$632$$ −64.0000 −0.00402814
$$633$$ −16332.0 −1.02550
$$634$$ 9888.00 0.619405
$$635$$ 16640.0 1.03990
$$636$$ −2376.00 −0.148136
$$637$$ −5733.00 −0.356593
$$638$$ 12920.0 0.801736
$$639$$ 4950.00 0.306446
$$640$$ 2048.00 0.126491
$$641$$ −15878.0 −0.978383 −0.489191 0.872176i $$-0.662708\pi$$
−0.489191 + 0.872176i $$0.662708\pi$$
$$642$$ −3840.00 −0.236063
$$643$$ −21520.0 −1.31985 −0.659927 0.751330i $$-0.729413\pi$$
−0.659927 + 0.751330i $$0.729413\pi$$
$$644$$ −5824.00 −0.356363
$$645$$ −6720.00 −0.410232
$$646$$ −29808.0 −1.81545
$$647$$ 7312.00 0.444304 0.222152 0.975012i $$-0.428692\pi$$
0.222152 + 0.975012i $$0.428692\pi$$
$$648$$ −648.000 −0.0392837
$$649$$ −5236.00 −0.316689
$$650$$ 3406.00 0.205530
$$651$$ 14784.0 0.890062
$$652$$ −1072.00 −0.0643907
$$653$$ 3090.00 0.185178 0.0925889 0.995704i $$-0.470486\pi$$
0.0925889 + 0.995704i $$0.470486\pi$$
$$654$$ −11604.0 −0.693811
$$655$$ 9088.00 0.542134
$$656$$ 3840.00 0.228547
$$657$$ 5526.00 0.328143
$$658$$ −25424.0 −1.50628
$$659$$ −13428.0 −0.793749 −0.396875 0.917873i $$-0.629905\pi$$
−0.396875 + 0.917873i $$0.629905\pi$$
$$660$$ 6528.00 0.385003
$$661$$ 22598.0 1.32974 0.664872 0.746958i $$-0.268486\pi$$
0.664872 + 0.746958i $$0.268486\pi$$
$$662$$ 12176.0 0.714854
$$663$$ 5382.00 0.315263
$$664$$ −6096.00 −0.356281
$$665$$ −48384.0 −2.82143
$$666$$ −6156.00 −0.358168
$$667$$ 9880.00 0.573546
$$668$$ 2808.00 0.162642
$$669$$ −288.000 −0.0166438
$$670$$ −20992.0 −1.21044
$$671$$ 1156.00 0.0665080
$$672$$ 2688.00 0.154303
$$673$$ 6178.00 0.353855 0.176927 0.984224i $$-0.443384\pi$$
0.176927 + 0.984224i $$0.443384\pi$$
$$674$$ −13276.0 −0.758713
$$675$$ −3537.00 −0.201688
$$676$$ 676.000 0.0384615
$$677$$ 22398.0 1.27153 0.635764 0.771883i $$-0.280685\pi$$
0.635764 + 0.771883i $$0.280685\pi$$
$$678$$ −2508.00 −0.142064
$$679$$ 28616.0 1.61735
$$680$$ 17664.0 0.996152
$$681$$ −594.000 −0.0334246
$$682$$ 11968.0 0.671962
$$683$$ 11410.0 0.639226 0.319613 0.947548i $$-0.396447\pi$$
0.319613 + 0.947548i $$0.396447\pi$$
$$684$$ 3888.00 0.217341
$$685$$ −8448.00 −0.471214
$$686$$ −5488.00 −0.305441
$$687$$ −17766.0 −0.986631
$$688$$ −2240.00 −0.124127
$$689$$ −2574.00 −0.142325
$$690$$ 4992.00 0.275423
$$691$$ 32488.0 1.78857 0.894285 0.447498i $$-0.147685\pi$$
0.894285 + 0.447498i $$0.147685\pi$$
$$692$$ 8264.00 0.453974
$$693$$ 8568.00 0.469656
$$694$$ 4584.00 0.250729
$$695$$ 24896.0 1.35879
$$696$$ −4560.00 −0.248342
$$697$$ 33120.0 1.79987
$$698$$ 19732.0 1.07001
$$699$$ 15342.0 0.830168
$$700$$ 14672.0 0.792214
$$701$$ 5094.00 0.274462 0.137231 0.990539i $$-0.456180\pi$$
0.137231 + 0.990539i $$0.456180\pi$$
$$702$$ −702.000 −0.0377426
$$703$$ 36936.0 1.98160
$$704$$ 2176.00 0.116493
$$705$$ 21792.0 1.16416
$$706$$ −4736.00 −0.252467
$$707$$ −33320.0 −1.77246
$$708$$ 1848.00 0.0980962
$$709$$ 25418.0 1.34639 0.673197 0.739463i $$-0.264921\pi$$
0.673197 + 0.739463i $$0.264921\pi$$
$$710$$ 17600.0 0.930305
$$711$$ 72.0000 0.00379777
$$712$$ 3552.00 0.186962
$$713$$ 9152.00 0.480708
$$714$$ 23184.0 1.21518
$$715$$ 7072.00 0.369899
$$716$$ −1104.00 −0.0576235
$$717$$ 15678.0 0.816605
$$718$$ −10140.0 −0.527049
$$719$$ −20428.0 −1.05958 −0.529788 0.848130i $$-0.677729\pi$$
−0.529788 + 0.848130i $$0.677729\pi$$
$$720$$ −2304.00 −0.119257
$$721$$ −6272.00 −0.323969
$$722$$ −9610.00 −0.495356
$$723$$ 2286.00 0.117590
$$724$$ −13896.0 −0.713316
$$725$$ −24890.0 −1.27502
$$726$$ −1050.00 −0.0536765
$$727$$ −38336.0 −1.95571 −0.977857 0.209276i $$-0.932889\pi$$
−0.977857 + 0.209276i $$0.932889\pi$$
$$728$$ 2912.00 0.148250
$$729$$ 729.000 0.0370370
$$730$$ 19648.0 0.996171
$$731$$ −19320.0 −0.977532
$$732$$ −408.000 −0.0206012
$$733$$ −166.000 −0.00836473 −0.00418237 0.999991i $$-0.501331\pi$$
−0.00418237 + 0.999991i $$0.501331\pi$$
$$734$$ 17168.0 0.863328
$$735$$ 21168.0 1.06230
$$736$$ 1664.00 0.0833368
$$737$$ −22304.0 −1.11476
$$738$$ −4320.00 −0.215476
$$739$$ −25248.0 −1.25678 −0.628392 0.777897i $$-0.716286\pi$$
−0.628392 + 0.777897i $$0.716286\pi$$
$$740$$ −21888.0 −1.08732
$$741$$ 4212.00 0.208815
$$742$$ −11088.0 −0.548589
$$743$$ −4442.00 −0.219329 −0.109664 0.993969i $$-0.534978\pi$$
−0.109664 + 0.993969i $$0.534978\pi$$
$$744$$ −4224.00 −0.208144
$$745$$ −24384.0 −1.19914
$$746$$ −9988.00 −0.490197
$$747$$ 6858.00 0.335905
$$748$$ 18768.0 0.917414
$$749$$ −17920.0 −0.874209
$$750$$ −576.000 −0.0280434
$$751$$ −19848.0 −0.964399 −0.482200 0.876061i $$-0.660162\pi$$
−0.482200 + 0.876061i $$0.660162\pi$$
$$752$$ 7264.00 0.352248
$$753$$ −9720.00 −0.470407
$$754$$ −4940.00 −0.238600
$$755$$ 48384.0 2.33228
$$756$$ −3024.00 −0.145479
$$757$$ −29166.0 −1.40034 −0.700169 0.713977i $$-0.746892\pi$$
−0.700169 + 0.713977i $$0.746892\pi$$
$$758$$ −2600.00 −0.124586
$$759$$ 5304.00 0.253653
$$760$$ 13824.0 0.659802
$$761$$ −6240.00 −0.297240 −0.148620 0.988894i $$-0.547483\pi$$
−0.148620 + 0.988894i $$0.547483\pi$$
$$762$$ −6240.00 −0.296655
$$763$$ −54152.0 −2.56938
$$764$$ −15680.0 −0.742516
$$765$$ −19872.0 −0.939181
$$766$$ 9180.00 0.433012
$$767$$ 2002.00 0.0942478
$$768$$ −768.000 −0.0360844
$$769$$ −39750.0 −1.86401 −0.932004 0.362449i $$-0.881941\pi$$
−0.932004 + 0.362449i $$0.881941\pi$$
$$770$$ 30464.0 1.42577
$$771$$ 4158.00 0.194224
$$772$$ 8744.00 0.407647
$$773$$ 9764.00 0.454317 0.227158 0.973858i $$-0.427057\pi$$
0.227158 + 0.973858i $$0.427057\pi$$
$$774$$ 2520.00 0.117028
$$775$$ −23056.0 −1.06864
$$776$$ −8176.00 −0.378223
$$777$$ −28728.0 −1.32640
$$778$$ −7020.00 −0.323495
$$779$$ 25920.0 1.19214
$$780$$ −2496.00 −0.114578
$$781$$ 18700.0 0.856772
$$782$$ 14352.0 0.656300
$$783$$ 5130.00 0.234140
$$784$$ 7056.00 0.321429
$$785$$ −35168.0 −1.59898
$$786$$ −3408.00 −0.154656
$$787$$ −36016.0 −1.63130 −0.815649 0.578547i $$-0.803620\pi$$
−0.815649 + 0.578547i $$0.803620\pi$$
$$788$$ −5472.00 −0.247376
$$789$$ −9900.00 −0.446704
$$790$$ 256.000 0.0115292
$$791$$ −11704.0 −0.526102
$$792$$ −2448.00 −0.109831
$$793$$ −442.000 −0.0197930
$$794$$ −12460.0 −0.556913
$$795$$ 9504.00 0.423990
$$796$$ −4288.00 −0.190935
$$797$$ −22290.0 −0.990655 −0.495328 0.868706i $$-0.664952\pi$$
−0.495328 + 0.868706i $$0.664952\pi$$
$$798$$ 18144.0 0.804875
$$799$$ 62652.0 2.77405
$$800$$ −4192.00 −0.185262
$$801$$ −3996.00 −0.176269
$$802$$ 15000.0 0.660434
$$803$$ 20876.0 0.917432
$$804$$ 7872.00 0.345304
$$805$$ 23296.0 1.01997
$$806$$ −4576.00 −0.199979
$$807$$ −12870.0 −0.561395
$$808$$ 9520.00 0.414496
$$809$$ −25578.0 −1.11159 −0.555794 0.831320i $$-0.687586\pi$$
−0.555794 + 0.831320i $$0.687586\pi$$
$$810$$ 2592.00 0.112437
$$811$$ 29900.0 1.29461 0.647306 0.762230i $$-0.275895\pi$$
0.647306 + 0.762230i $$0.275895\pi$$
$$812$$ −21280.0 −0.919682
$$813$$ −7356.00 −0.317326
$$814$$ −23256.0 −1.00138
$$815$$ 4288.00 0.184297
$$816$$ −6624.00 −0.284174
$$817$$ −15120.0 −0.647469
$$818$$ −16508.0 −0.705610
$$819$$ −3276.00 −0.139771
$$820$$ −15360.0 −0.654140
$$821$$ 16412.0 0.697665 0.348832 0.937185i $$-0.386578\pi$$
0.348832 + 0.937185i $$0.386578\pi$$
$$822$$ 3168.00 0.134424
$$823$$ 18552.0 0.785762 0.392881 0.919589i $$-0.371478\pi$$
0.392881 + 0.919589i $$0.371478\pi$$
$$824$$ 1792.00 0.0757613
$$825$$ −13362.0 −0.563885
$$826$$ 8624.00 0.363278
$$827$$ −28662.0 −1.20517 −0.602585 0.798055i $$-0.705863\pi$$
−0.602585 + 0.798055i $$0.705863\pi$$
$$828$$ −1872.00 −0.0785706
$$829$$ −3686.00 −0.154427 −0.0772136 0.997015i $$-0.524602\pi$$
−0.0772136 + 0.997015i $$0.524602\pi$$
$$830$$ 24384.0 1.01974
$$831$$ 126.000 0.00525980
$$832$$ −832.000 −0.0346688
$$833$$ 60858.0 2.53134
$$834$$ −9336.00 −0.387625
$$835$$ −11232.0 −0.465508
$$836$$ 14688.0 0.607650
$$837$$ 4752.00 0.196240
$$838$$ 29616.0 1.22084
$$839$$ 13370.0 0.550159 0.275080 0.961421i $$-0.411296\pi$$
0.275080 + 0.961421i $$0.411296\pi$$
$$840$$ −10752.0 −0.441642
$$841$$ 11711.0 0.480175
$$842$$ −20708.0 −0.847559
$$843$$ 6864.00 0.280437
$$844$$ 21776.0 0.888105
$$845$$ −2704.00 −0.110083
$$846$$ −8172.00 −0.332103
$$847$$ −4900.00 −0.198779
$$848$$ 3168.00 0.128290
$$849$$ −3468.00 −0.140190
$$850$$ −36156.0 −1.45899
$$851$$ −17784.0 −0.716366
$$852$$ −6600.00 −0.265390
$$853$$ 11398.0 0.457515 0.228757 0.973483i $$-0.426534\pi$$
0.228757 + 0.973483i $$0.426534\pi$$
$$854$$ −1904.00 −0.0762922
$$855$$ −15552.0 −0.622067
$$856$$ 5120.00 0.204437
$$857$$ 7990.00 0.318475 0.159238 0.987240i $$-0.449096\pi$$
0.159238 + 0.987240i $$0.449096\pi$$
$$858$$ −2652.00 −0.105522
$$859$$ 7652.00 0.303938 0.151969 0.988385i $$-0.451439\pi$$
0.151969 + 0.988385i $$0.451439\pi$$
$$860$$ 8960.00 0.355271
$$861$$ −20160.0 −0.797969
$$862$$ 30972.0 1.22379
$$863$$ −1022.00 −0.0403120 −0.0201560 0.999797i $$-0.506416\pi$$
−0.0201560 + 0.999797i $$0.506416\pi$$
$$864$$ 864.000 0.0340207
$$865$$ −33056.0 −1.29935
$$866$$ 4036.00 0.158371
$$867$$ −42393.0 −1.66060
$$868$$ −19712.0 −0.770817
$$869$$ 272.000 0.0106179
$$870$$ 18240.0 0.710798
$$871$$ 8528.00 0.331757
$$872$$ 15472.0 0.600858
$$873$$ 9198.00 0.356592
$$874$$ 11232.0 0.434700
$$875$$ −2688.00 −0.103853
$$876$$ −7368.00 −0.284180
$$877$$ −15546.0 −0.598576 −0.299288 0.954163i $$-0.596749\pi$$
−0.299288 + 0.954163i $$0.596749\pi$$
$$878$$ −17584.0 −0.675890
$$879$$ 26052.0 0.999673
$$880$$ −8704.00 −0.333422
$$881$$ 11310.0 0.432513 0.216256 0.976337i $$-0.430615\pi$$
0.216256 + 0.976337i $$0.430615\pi$$
$$882$$ −7938.00 −0.303046
$$883$$ 17260.0 0.657809 0.328904 0.944363i $$-0.393321\pi$$
0.328904 + 0.944363i $$0.393321\pi$$
$$884$$ −7176.00 −0.273026
$$885$$ −7392.00 −0.280768
$$886$$ 5520.00 0.209309
$$887$$ 832.000 0.0314947 0.0157474 0.999876i $$-0.494987\pi$$
0.0157474 + 0.999876i $$0.494987\pi$$
$$888$$ 8208.00 0.310183
$$889$$ −29120.0 −1.09860
$$890$$ −14208.0 −0.535116
$$891$$ 2754.00 0.103549
$$892$$ 384.000 0.0144140
$$893$$ 49032.0 1.83739
$$894$$ 9144.00 0.342082
$$895$$ 4416.00 0.164928
$$896$$ −3584.00 −0.133631
$$897$$ −2028.00 −0.0754882
$$898$$ −19064.0 −0.708434
$$899$$ 33440.0 1.24059
$$900$$ 4716.00 0.174667
$$901$$ 27324.0 1.01032
$$902$$ −16320.0 −0.602435
$$903$$ 11760.0 0.433387
$$904$$ 3344.00 0.123031
$$905$$ 55584.0 2.04163
$$906$$ −18144.0 −0.665336
$$907$$ 31740.0 1.16197 0.580986 0.813913i $$-0.302667\pi$$
0.580986 + 0.813913i $$0.302667\pi$$
$$908$$ 792.000 0.0289465
$$909$$ −10710.0 −0.390790
$$910$$ −11648.0 −0.424316
$$911$$ −23568.0 −0.857127 −0.428563 0.903512i $$-0.640980\pi$$
−0.428563 + 0.903512i $$0.640980\pi$$
$$912$$ −5184.00 −0.188223
$$913$$ 25908.0 0.939134
$$914$$ −25724.0 −0.930935
$$915$$ 1632.00 0.0589642
$$916$$ 23688.0 0.854447
$$917$$ −15904.0 −0.572733
$$918$$ 7452.00 0.267922
$$919$$ −18864.0 −0.677112 −0.338556 0.940946i $$-0.609938\pi$$
−0.338556 + 0.940946i $$0.609938\pi$$
$$920$$ −6656.00 −0.238524
$$921$$ 22656.0 0.810576
$$922$$ 13488.0 0.481783
$$923$$ −7150.00 −0.254978
$$924$$ −11424.0 −0.406734
$$925$$ 44802.0 1.59252
$$926$$ 19144.0 0.679385
$$927$$ −2016.00 −0.0714284
$$928$$ 6080.00 0.215071
$$929$$ −19536.0 −0.689941 −0.344971 0.938613i $$-0.612111\pi$$
−0.344971 + 0.938613i $$0.612111\pi$$
$$930$$ 16896.0 0.595744
$$931$$ 47628.0 1.67663
$$932$$ −20456.0 −0.718947
$$933$$ −7956.00 −0.279172
$$934$$ −18208.0 −0.637884
$$935$$ −75072.0 −2.62579
$$936$$ 936.000 0.0326860
$$937$$ 18174.0 0.633638 0.316819 0.948486i $$-0.397385\pi$$
0.316819 + 0.948486i $$0.397385\pi$$
$$938$$ 36736.0 1.27876
$$939$$ 13278.0 0.461460
$$940$$ −29056.0 −1.00819
$$941$$ 51172.0 1.77275 0.886376 0.462966i $$-0.153215\pi$$
0.886376 + 0.462966i $$0.153215\pi$$
$$942$$ 13188.0 0.456145
$$943$$ −12480.0 −0.430970
$$944$$ −2464.00 −0.0849538
$$945$$ 12096.0 0.416384
$$946$$ 9520.00 0.327190
$$947$$ 3726.00 0.127855 0.0639275 0.997955i $$-0.479637\pi$$
0.0639275 + 0.997955i $$0.479637\pi$$
$$948$$ −96.0000 −0.00328896
$$949$$ −7982.00 −0.273031
$$950$$ −28296.0 −0.966362
$$951$$ 14832.0 0.505742
$$952$$ −30912.0 −1.05238
$$953$$ −40498.0 −1.37656 −0.688279 0.725447i $$-0.741633\pi$$
−0.688279 + 0.725447i $$0.741633\pi$$
$$954$$ −3564.00 −0.120953
$$955$$ 62720.0 2.12521
$$956$$ −20904.0 −0.707200
$$957$$ 19380.0 0.654615
$$958$$ 37740.0 1.27278
$$959$$ 14784.0 0.497810
$$960$$ 3072.00 0.103280
$$961$$ 1185.00 0.0397771
$$962$$ 8892.00 0.298014
$$963$$ −5760.00 −0.192745
$$964$$ −3048.00 −0.101836
$$965$$ −34976.0 −1.16675
$$966$$ −8736.00 −0.290969
$$967$$ 28568.0 0.950036 0.475018 0.879976i $$-0.342442\pi$$
0.475018 + 0.879976i $$0.342442\pi$$
$$968$$ 1400.00 0.0464852
$$969$$ −44712.0 −1.48231
$$970$$ 32704.0 1.08254
$$971$$ −8676.00 −0.286742 −0.143371 0.989669i $$-0.545794\pi$$
−0.143371 + 0.989669i $$0.545794\pi$$
$$972$$ −972.000 −0.0320750
$$973$$ −43568.0 −1.43548
$$974$$ 3488.00 0.114746
$$975$$ 5109.00 0.167814
$$976$$ 544.000 0.0178412
$$977$$ −2796.00 −0.0915578 −0.0457789 0.998952i $$-0.514577\pi$$
−0.0457789 + 0.998952i $$0.514577\pi$$
$$978$$ −1608.00 −0.0525748
$$979$$ −15096.0 −0.492819
$$980$$ −28224.0 −0.919982
$$981$$ −17406.0 −0.566494
$$982$$ −26720.0 −0.868299
$$983$$ −406.000 −0.0131733 −0.00658667 0.999978i $$-0.502097\pi$$
−0.00658667 + 0.999978i $$0.502097\pi$$
$$984$$ 5760.00 0.186608
$$985$$ 21888.0 0.708030
$$986$$ 52440.0 1.69374
$$987$$ −38136.0 −1.22987
$$988$$ −5616.00 −0.180839
$$989$$ 7280.00 0.234065
$$990$$ 9792.00 0.314354
$$991$$ −23232.0 −0.744691 −0.372346 0.928094i $$-0.621446\pi$$
−0.372346 + 0.928094i $$0.621446\pi$$
$$992$$ 5632.00 0.180258
$$993$$ 18264.0 0.583676
$$994$$ −30800.0 −0.982814
$$995$$ 17152.0 0.546487
$$996$$ −9144.00 −0.290902
$$997$$ 6110.00 0.194088 0.0970440 0.995280i $$-0.469061\pi$$
0.0970440 + 0.995280i $$0.469061\pi$$
$$998$$ −34736.0 −1.10175
$$999$$ −9234.00 −0.292443
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 78.4.a.a.1.1 1
3.2 odd 2 234.4.a.k.1.1 1
4.3 odd 2 624.4.a.f.1.1 1
5.4 even 2 1950.4.a.o.1.1 1
8.3 odd 2 2496.4.a.g.1.1 1
8.5 even 2 2496.4.a.q.1.1 1
12.11 even 2 1872.4.a.o.1.1 1
13.5 odd 4 1014.4.b.a.337.2 2
13.8 odd 4 1014.4.b.a.337.1 2
13.12 even 2 1014.4.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.a.1.1 1 1.1 even 1 trivial
234.4.a.k.1.1 1 3.2 odd 2
624.4.a.f.1.1 1 4.3 odd 2
1014.4.a.i.1.1 1 13.12 even 2
1014.4.b.a.337.1 2 13.8 odd 4
1014.4.b.a.337.2 2 13.5 odd 4
1872.4.a.o.1.1 1 12.11 even 2
1950.4.a.o.1.1 1 5.4 even 2
2496.4.a.g.1.1 1 8.3 odd 2
2496.4.a.q.1.1 1 8.5 even 2