# Properties

 Label 1014.4.b.e.337.1 Level $1014$ Weight $4$ Character 1014.337 Analytic conductor $59.828$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,4,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1014 = 2 \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1014.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

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

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

 Self dual: no Analytic conductor: $$59.8279367458$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1014.337 Dual form 1014.4.b.e.337.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +20.0000i q^{5} +6.00000i q^{6} -32.0000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} -3.00000 q^{3} -4.00000 q^{4} +20.0000i q^{5} +6.00000i q^{6} -32.0000i q^{7} +8.00000i q^{8} +9.00000 q^{9} +40.0000 q^{10} +50.0000i q^{11} +12.0000 q^{12} -64.0000 q^{14} -60.0000i q^{15} +16.0000 q^{16} +30.0000 q^{17} -18.0000i q^{18} +120.000i q^{19} -80.0000i q^{20} +96.0000i q^{21} +100.000 q^{22} +20.0000 q^{23} -24.0000i q^{24} -275.000 q^{25} -27.0000 q^{27} +128.000i q^{28} +82.0000 q^{29} -120.000 q^{30} +44.0000i q^{31} -32.0000i q^{32} -150.000i q^{33} -60.0000i q^{34} +640.000 q^{35} -36.0000 q^{36} -306.000i q^{37} +240.000 q^{38} -160.000 q^{40} -108.000i q^{41} +192.000 q^{42} +356.000 q^{43} -200.000i q^{44} +180.000i q^{45} -40.0000i q^{46} -178.000i q^{47} -48.0000 q^{48} -681.000 q^{49} +550.000i q^{50} -90.0000 q^{51} +198.000 q^{53} +54.0000i q^{54} -1000.00 q^{55} +256.000 q^{56} -360.000i q^{57} -164.000i q^{58} +94.0000i q^{59} +240.000i q^{60} -62.0000 q^{61} +88.0000 q^{62} -288.000i q^{63} -64.0000 q^{64} -300.000 q^{66} +140.000i q^{67} -120.000 q^{68} -60.0000 q^{69} -1280.00i q^{70} +778.000i q^{71} +72.0000i q^{72} +62.0000i q^{73} -612.000 q^{74} +825.000 q^{75} -480.000i q^{76} +1600.00 q^{77} -1096.00 q^{79} +320.000i q^{80} +81.0000 q^{81} -216.000 q^{82} +462.000i q^{83} -384.000i q^{84} +600.000i q^{85} -712.000i q^{86} -246.000 q^{87} -400.000 q^{88} +1224.00i q^{89} +360.000 q^{90} -80.0000 q^{92} -132.000i q^{93} -356.000 q^{94} -2400.00 q^{95} +96.0000i q^{96} -614.000i q^{97} +1362.00i q^{98} +450.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 8 q^{4} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 $$2 q - 6 q^{3} - 8 q^{4} + 18 q^{9} + 80 q^{10} + 24 q^{12} - 128 q^{14} + 32 q^{16} + 60 q^{17} + 200 q^{22} + 40 q^{23} - 550 q^{25} - 54 q^{27} + 164 q^{29} - 240 q^{30} + 1280 q^{35} - 72 q^{36} + 480 q^{38} - 320 q^{40} + 384 q^{42} + 712 q^{43} - 96 q^{48} - 1362 q^{49} - 180 q^{51} + 396 q^{53} - 2000 q^{55} + 512 q^{56} - 124 q^{61} + 176 q^{62} - 128 q^{64} - 600 q^{66} - 240 q^{68} - 120 q^{69} - 1224 q^{74} + 1650 q^{75} + 3200 q^{77} - 2192 q^{79} + 162 q^{81} - 432 q^{82} - 492 q^{87} - 800 q^{88} + 720 q^{90} - 160 q^{92} - 712 q^{94} - 4800 q^{95}+O(q^{100})$$ 2 * q - 6 * q^3 - 8 * q^4 + 18 * q^9 + 80 * q^10 + 24 * q^12 - 128 * q^14 + 32 * q^16 + 60 * q^17 + 200 * q^22 + 40 * q^23 - 550 * q^25 - 54 * q^27 + 164 * q^29 - 240 * q^30 + 1280 * q^35 - 72 * q^36 + 480 * q^38 - 320 * q^40 + 384 * q^42 + 712 * q^43 - 96 * q^48 - 1362 * q^49 - 180 * q^51 + 396 * q^53 - 2000 * q^55 + 512 * q^56 - 124 * q^61 + 176 * q^62 - 128 * q^64 - 600 * q^66 - 240 * q^68 - 120 * q^69 - 1224 * q^74 + 1650 * q^75 + 3200 * q^77 - 2192 * q^79 + 162 * q^81 - 432 * q^82 - 492 * q^87 - 800 * q^88 + 720 * q^90 - 160 * q^92 - 712 * q^94 - 4800 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 2.00000i − 0.707107i
$$3$$ −3.00000 −0.577350
$$4$$ −4.00000 −0.500000
$$5$$ 20.0000i 1.78885i 0.447214 + 0.894427i $$0.352416\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 6.00000i 0.408248i
$$7$$ − 32.0000i − 1.72784i −0.503631 0.863919i $$-0.668003\pi$$
0.503631 0.863919i $$-0.331997\pi$$
$$8$$ 8.00000i 0.353553i
$$9$$ 9.00000 0.333333
$$10$$ 40.0000 1.26491
$$11$$ 50.0000i 1.37051i 0.728305 + 0.685253i $$0.240308\pi$$
−0.728305 + 0.685253i $$0.759692\pi$$
$$12$$ 12.0000 0.288675
$$13$$ 0 0
$$14$$ −64.0000 −1.22177
$$15$$ − 60.0000i − 1.03280i
$$16$$ 16.0000 0.250000
$$17$$ 30.0000 0.428004 0.214002 0.976833i $$-0.431350\pi$$
0.214002 + 0.976833i $$0.431350\pi$$
$$18$$ − 18.0000i − 0.235702i
$$19$$ 120.000i 1.44894i 0.689306 + 0.724471i $$0.257916\pi$$
−0.689306 + 0.724471i $$0.742084\pi$$
$$20$$ − 80.0000i − 0.894427i
$$21$$ 96.0000i 0.997567i
$$22$$ 100.000 0.969094
$$23$$ 20.0000 0.181317 0.0906584 0.995882i $$-0.471103\pi$$
0.0906584 + 0.995882i $$0.471103\pi$$
$$24$$ − 24.0000i − 0.204124i
$$25$$ −275.000 −2.20000
$$26$$ 0 0
$$27$$ −27.0000 −0.192450
$$28$$ 128.000i 0.863919i
$$29$$ 82.0000 0.525070 0.262535 0.964923i $$-0.415442\pi$$
0.262535 + 0.964923i $$0.415442\pi$$
$$30$$ −120.000 −0.730297
$$31$$ 44.0000i 0.254924i 0.991843 + 0.127462i $$0.0406830\pi$$
−0.991843 + 0.127462i $$0.959317\pi$$
$$32$$ − 32.0000i − 0.176777i
$$33$$ − 150.000i − 0.791262i
$$34$$ − 60.0000i − 0.302645i
$$35$$ 640.000 3.09085
$$36$$ −36.0000 −0.166667
$$37$$ − 306.000i − 1.35962i −0.733386 0.679812i $$-0.762061\pi$$
0.733386 0.679812i $$-0.237939\pi$$
$$38$$ 240.000 1.02456
$$39$$ 0 0
$$40$$ −160.000 −0.632456
$$41$$ − 108.000i − 0.411385i −0.978617 0.205692i $$-0.934055\pi$$
0.978617 0.205692i $$-0.0659446\pi$$
$$42$$ 192.000 0.705387
$$43$$ 356.000 1.26255 0.631273 0.775561i $$-0.282533\pi$$
0.631273 + 0.775561i $$0.282533\pi$$
$$44$$ − 200.000i − 0.685253i
$$45$$ 180.000i 0.596285i
$$46$$ − 40.0000i − 0.128210i
$$47$$ − 178.000i − 0.552425i −0.961097 0.276212i $$-0.910921\pi$$
0.961097 0.276212i $$-0.0890793\pi$$
$$48$$ −48.0000 −0.144338
$$49$$ −681.000 −1.98542
$$50$$ 550.000i 1.55563i
$$51$$ −90.0000 −0.247108
$$52$$ 0 0
$$53$$ 198.000 0.513158 0.256579 0.966523i $$-0.417405\pi$$
0.256579 + 0.966523i $$0.417405\pi$$
$$54$$ 54.0000i 0.136083i
$$55$$ −1000.00 −2.45164
$$56$$ 256.000 0.610883
$$57$$ − 360.000i − 0.836547i
$$58$$ − 164.000i − 0.371280i
$$59$$ 94.0000i 0.207420i 0.994608 + 0.103710i $$0.0330713\pi$$
−0.994608 + 0.103710i $$0.966929\pi$$
$$60$$ 240.000i 0.516398i
$$61$$ −62.0000 −0.130136 −0.0650679 0.997881i $$-0.520726\pi$$
−0.0650679 + 0.997881i $$0.520726\pi$$
$$62$$ 88.0000 0.180258
$$63$$ − 288.000i − 0.575946i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ −300.000 −0.559507
$$67$$ 140.000i 0.255279i 0.991821 + 0.127640i $$0.0407401\pi$$
−0.991821 + 0.127640i $$0.959260\pi$$
$$68$$ −120.000 −0.214002
$$69$$ −60.0000 −0.104683
$$70$$ − 1280.00i − 2.18556i
$$71$$ 778.000i 1.30045i 0.759744 + 0.650223i $$0.225324\pi$$
−0.759744 + 0.650223i $$0.774676\pi$$
$$72$$ 72.0000i 0.117851i
$$73$$ 62.0000i 0.0994048i 0.998764 + 0.0497024i $$0.0158273\pi$$
−0.998764 + 0.0497024i $$0.984173\pi$$
$$74$$ −612.000 −0.961399
$$75$$ 825.000 1.27017
$$76$$ − 480.000i − 0.724471i
$$77$$ 1600.00 2.36801
$$78$$ 0 0
$$79$$ −1096.00 −1.56088 −0.780441 0.625230i $$-0.785005\pi$$
−0.780441 + 0.625230i $$0.785005\pi$$
$$80$$ 320.000i 0.447214i
$$81$$ 81.0000 0.111111
$$82$$ −216.000 −0.290893
$$83$$ 462.000i 0.610977i 0.952196 + 0.305488i $$0.0988197\pi$$
−0.952196 + 0.305488i $$0.901180\pi$$
$$84$$ − 384.000i − 0.498784i
$$85$$ 600.000i 0.765637i
$$86$$ − 712.000i − 0.892755i
$$87$$ −246.000 −0.303149
$$88$$ −400.000 −0.484547
$$89$$ 1224.00i 1.45779i 0.684623 + 0.728897i $$0.259967\pi$$
−0.684623 + 0.728897i $$0.740033\pi$$
$$90$$ 360.000 0.421637
$$91$$ 0 0
$$92$$ −80.0000 −0.0906584
$$93$$ − 132.000i − 0.147180i
$$94$$ −356.000 −0.390623
$$95$$ −2400.00 −2.59195
$$96$$ 96.0000i 0.102062i
$$97$$ − 614.000i − 0.642704i −0.946960 0.321352i $$-0.895863\pi$$
0.946960 0.321352i $$-0.104137\pi$$
$$98$$ 1362.00i 1.40391i
$$99$$ 450.000i 0.456835i
$$100$$ 1100.00 1.10000
$$101$$ −1058.00 −1.04233 −0.521163 0.853457i $$-0.674502\pi$$
−0.521163 + 0.853457i $$0.674502\pi$$
$$102$$ 180.000i 0.174732i
$$103$$ −1768.00 −1.69132 −0.845661 0.533720i $$-0.820794\pi$$
−0.845661 + 0.533720i $$0.820794\pi$$
$$104$$ 0 0
$$105$$ −1920.00 −1.78450
$$106$$ − 396.000i − 0.362858i
$$107$$ −1808.00 −1.63351 −0.816757 0.576982i $$-0.804230\pi$$
−0.816757 + 0.576982i $$0.804230\pi$$
$$108$$ 108.000 0.0962250
$$109$$ 1886.00i 1.65730i 0.559765 + 0.828652i $$0.310891\pi$$
−0.559765 + 0.828652i $$0.689109\pi$$
$$110$$ 2000.00i 1.73357i
$$111$$ 918.000i 0.784979i
$$112$$ − 512.000i − 0.431959i
$$113$$ 1246.00 1.03729 0.518645 0.854990i $$-0.326437\pi$$
0.518645 + 0.854990i $$0.326437\pi$$
$$114$$ −720.000 −0.591528
$$115$$ 400.000i 0.324349i
$$116$$ −328.000 −0.262535
$$117$$ 0 0
$$118$$ 188.000 0.146668
$$119$$ − 960.000i − 0.739521i
$$120$$ 480.000 0.365148
$$121$$ −1169.00 −0.878287
$$122$$ 124.000i 0.0920199i
$$123$$ 324.000i 0.237513i
$$124$$ − 176.000i − 0.127462i
$$125$$ − 3000.00i − 2.14663i
$$126$$ −576.000 −0.407255
$$127$$ −1624.00 −1.13470 −0.567349 0.823477i $$-0.692031\pi$$
−0.567349 + 0.823477i $$0.692031\pi$$
$$128$$ 128.000i 0.0883883i
$$129$$ −1068.00 −0.728931
$$130$$ 0 0
$$131$$ −2072.00 −1.38192 −0.690960 0.722893i $$-0.742812\pi$$
−0.690960 + 0.722893i $$0.742812\pi$$
$$132$$ 600.000i 0.395631i
$$133$$ 3840.00 2.50354
$$134$$ 280.000 0.180510
$$135$$ − 540.000i − 0.344265i
$$136$$ 240.000i 0.151322i
$$137$$ − 756.000i − 0.471456i −0.971819 0.235728i $$-0.924253\pi$$
0.971819 0.235728i $$-0.0757474\pi$$
$$138$$ 120.000i 0.0740223i
$$139$$ 172.000 0.104956 0.0524779 0.998622i $$-0.483288\pi$$
0.0524779 + 0.998622i $$0.483288\pi$$
$$140$$ −2560.00 −1.54542
$$141$$ 534.000i 0.318943i
$$142$$ 1556.00 0.919554
$$143$$ 0 0
$$144$$ 144.000 0.0833333
$$145$$ 1640.00i 0.939273i
$$146$$ 124.000 0.0702898
$$147$$ 2043.00 1.14628
$$148$$ 1224.00i 0.679812i
$$149$$ − 1272.00i − 0.699371i −0.936867 0.349686i $$-0.886288\pi$$
0.936867 0.349686i $$-0.113712\pi$$
$$150$$ − 1650.00i − 0.898146i
$$151$$ 1404.00i 0.756662i 0.925670 + 0.378331i $$0.123502\pi$$
−0.925670 + 0.378331i $$0.876498\pi$$
$$152$$ −960.000 −0.512278
$$153$$ 270.000 0.142668
$$154$$ − 3200.00i − 1.67444i
$$155$$ −880.000 −0.456021
$$156$$ 0 0
$$157$$ −2170.00 −1.10309 −0.551544 0.834146i $$-0.685961\pi$$
−0.551544 + 0.834146i $$0.685961\pi$$
$$158$$ 2192.00i 1.10371i
$$159$$ −594.000 −0.296272
$$160$$ 640.000 0.316228
$$161$$ − 640.000i − 0.313286i
$$162$$ − 162.000i − 0.0785674i
$$163$$ 248.000i 0.119171i 0.998223 + 0.0595855i $$0.0189779\pi$$
−0.998223 + 0.0595855i $$0.981022\pi$$
$$164$$ 432.000i 0.205692i
$$165$$ 3000.00 1.41545
$$166$$ 924.000 0.432026
$$167$$ 102.000i 0.0472635i 0.999721 + 0.0236317i $$0.00752291\pi$$
−0.999721 + 0.0236317i $$0.992477\pi$$
$$168$$ −768.000 −0.352693
$$169$$ 0 0
$$170$$ 1200.00 0.541387
$$171$$ 1080.00i 0.482980i
$$172$$ −1424.00 −0.631273
$$173$$ −682.000 −0.299720 −0.149860 0.988707i $$-0.547882\pi$$
−0.149860 + 0.988707i $$0.547882\pi$$
$$174$$ 492.000i 0.214359i
$$175$$ 8800.00i 3.80124i
$$176$$ 800.000i 0.342627i
$$177$$ − 282.000i − 0.119754i
$$178$$ 2448.00 1.03082
$$179$$ 612.000 0.255548 0.127774 0.991803i $$-0.459217\pi$$
0.127774 + 0.991803i $$0.459217\pi$$
$$180$$ − 720.000i − 0.298142i
$$181$$ 66.0000 0.0271035 0.0135518 0.999908i $$-0.495686\pi$$
0.0135518 + 0.999908i $$0.495686\pi$$
$$182$$ 0 0
$$183$$ 186.000 0.0751340
$$184$$ 160.000i 0.0641052i
$$185$$ 6120.00 2.43217
$$186$$ −264.000 −0.104072
$$187$$ 1500.00i 0.586582i
$$188$$ 712.000i 0.276212i
$$189$$ 864.000i 0.332522i
$$190$$ 4800.00i 1.83278i
$$191$$ 608.000 0.230332 0.115166 0.993346i $$-0.463260\pi$$
0.115166 + 0.993346i $$0.463260\pi$$
$$192$$ 192.000 0.0721688
$$193$$ 1370.00i 0.510957i 0.966815 + 0.255479i $$0.0822331\pi$$
−0.966815 + 0.255479i $$0.917767\pi$$
$$194$$ −1228.00 −0.454460
$$195$$ 0 0
$$196$$ 2724.00 0.992711
$$197$$ 4908.00i 1.77503i 0.460781 + 0.887514i $$0.347569\pi$$
−0.460781 + 0.887514i $$0.652431\pi$$
$$198$$ 900.000 0.323031
$$199$$ 328.000 0.116841 0.0584204 0.998292i $$-0.481394\pi$$
0.0584204 + 0.998292i $$0.481394\pi$$
$$200$$ − 2200.00i − 0.777817i
$$201$$ − 420.000i − 0.147386i
$$202$$ 2116.00i 0.737036i
$$203$$ − 2624.00i − 0.907235i
$$204$$ 360.000 0.123554
$$205$$ 2160.00 0.735907
$$206$$ 3536.00i 1.19595i
$$207$$ 180.000 0.0604390
$$208$$ 0 0
$$209$$ −6000.00 −1.98578
$$210$$ 3840.00i 1.26183i
$$211$$ 1316.00 0.429371 0.214685 0.976683i $$-0.431127\pi$$
0.214685 + 0.976683i $$0.431127\pi$$
$$212$$ −792.000 −0.256579
$$213$$ − 2334.00i − 0.750812i
$$214$$ 3616.00i 1.15507i
$$215$$ 7120.00i 2.25851i
$$216$$ − 216.000i − 0.0680414i
$$217$$ 1408.00 0.440467
$$218$$ 3772.00 1.17189
$$219$$ − 186.000i − 0.0573914i
$$220$$ 4000.00 1.22582
$$221$$ 0 0
$$222$$ 1836.00 0.555064
$$223$$ 1932.00i 0.580163i 0.957002 + 0.290081i $$0.0936824\pi$$
−0.957002 + 0.290081i $$0.906318\pi$$
$$224$$ −1024.00 −0.305441
$$225$$ −2475.00 −0.733333
$$226$$ − 2492.00i − 0.733475i
$$227$$ − 4998.00i − 1.46136i −0.682720 0.730680i $$-0.739203\pi$$
0.682720 0.730680i $$-0.260797\pi$$
$$228$$ 1440.00i 0.418273i
$$229$$ − 78.0000i − 0.0225082i −0.999937 0.0112541i $$-0.996418\pi$$
0.999937 0.0112541i $$-0.00358237\pi$$
$$230$$ 800.000 0.229350
$$231$$ −4800.00 −1.36717
$$232$$ 656.000i 0.185640i
$$233$$ 1282.00 0.360458 0.180229 0.983625i $$-0.442316\pi$$
0.180229 + 0.983625i $$0.442316\pi$$
$$234$$ 0 0
$$235$$ 3560.00 0.988208
$$236$$ − 376.000i − 0.103710i
$$237$$ 3288.00 0.901175
$$238$$ −1920.00 −0.522921
$$239$$ − 294.000i − 0.0795702i −0.999208 0.0397851i $$-0.987333\pi$$
0.999208 0.0397851i $$-0.0126673\pi$$
$$240$$ − 960.000i − 0.258199i
$$241$$ − 4962.00i − 1.32627i −0.748501 0.663134i $$-0.769226\pi$$
0.748501 0.663134i $$-0.230774\pi$$
$$242$$ 2338.00i 0.621043i
$$243$$ −243.000 −0.0641500
$$244$$ 248.000 0.0650679
$$245$$ − 13620.0i − 3.55163i
$$246$$ 648.000 0.167947
$$247$$ 0 0
$$248$$ −352.000 −0.0901291
$$249$$ − 1386.00i − 0.352748i
$$250$$ −6000.00 −1.51789
$$251$$ −744.000 −0.187095 −0.0935475 0.995615i $$-0.529821\pi$$
−0.0935475 + 0.995615i $$0.529821\pi$$
$$252$$ 1152.00i 0.287973i
$$253$$ 1000.00i 0.248496i
$$254$$ 3248.00i 0.802353i
$$255$$ − 1800.00i − 0.442041i
$$256$$ 256.000 0.0625000
$$257$$ 1026.00 0.249028 0.124514 0.992218i $$-0.460263\pi$$
0.124514 + 0.992218i $$0.460263\pi$$
$$258$$ 2136.00i 0.515432i
$$259$$ −9792.00 −2.34921
$$260$$ 0 0
$$261$$ 738.000 0.175023
$$262$$ 4144.00i 0.977165i
$$263$$ −5532.00 −1.29703 −0.648513 0.761204i $$-0.724609\pi$$
−0.648513 + 0.761204i $$0.724609\pi$$
$$264$$ 1200.00 0.279753
$$265$$ 3960.00i 0.917966i
$$266$$ − 7680.00i − 1.77027i
$$267$$ − 3672.00i − 0.841658i
$$268$$ − 560.000i − 0.127640i
$$269$$ −3534.00 −0.801010 −0.400505 0.916294i $$-0.631165\pi$$
−0.400505 + 0.916294i $$0.631165\pi$$
$$270$$ −1080.00 −0.243432
$$271$$ 2392.00i 0.536176i 0.963394 + 0.268088i $$0.0863918\pi$$
−0.963394 + 0.268088i $$0.913608\pi$$
$$272$$ 480.000 0.107001
$$273$$ 0 0
$$274$$ −1512.00 −0.333370
$$275$$ − 13750.0i − 3.01511i
$$276$$ 240.000 0.0523417
$$277$$ −6102.00 −1.32359 −0.661794 0.749686i $$-0.730204\pi$$
−0.661794 + 0.749686i $$0.730204\pi$$
$$278$$ − 344.000i − 0.0742149i
$$279$$ 396.000i 0.0849746i
$$280$$ 5120.00i 1.09278i
$$281$$ − 7540.00i − 1.60071i −0.599528 0.800354i $$-0.704645\pi$$
0.599528 0.800354i $$-0.295355\pi$$
$$282$$ 1068.00 0.225527
$$283$$ 2756.00 0.578895 0.289447 0.957194i $$-0.406528\pi$$
0.289447 + 0.957194i $$0.406528\pi$$
$$284$$ − 3112.00i − 0.650223i
$$285$$ 7200.00 1.49646
$$286$$ 0 0
$$287$$ −3456.00 −0.710806
$$288$$ − 288.000i − 0.0589256i
$$289$$ −4013.00 −0.816813
$$290$$ 3280.00 0.664166
$$291$$ 1842.00i 0.371065i
$$292$$ − 248.000i − 0.0497024i
$$293$$ 968.000i 0.193007i 0.995333 + 0.0965037i $$0.0307660\pi$$
−0.995333 + 0.0965037i $$0.969234\pi$$
$$294$$ − 4086.00i − 0.810545i
$$295$$ −1880.00 −0.371043
$$296$$ 2448.00 0.480700
$$297$$ − 1350.00i − 0.263754i
$$298$$ −2544.00 −0.494530
$$299$$ 0 0
$$300$$ −3300.00 −0.635085
$$301$$ − 11392.0i − 2.18147i
$$302$$ 2808.00 0.535041
$$303$$ 3174.00 0.601787
$$304$$ 1920.00i 0.362235i
$$305$$ − 1240.00i − 0.232794i
$$306$$ − 540.000i − 0.100882i
$$307$$ − 6436.00i − 1.19649i −0.801314 0.598244i $$-0.795865\pi$$
0.801314 0.598244i $$-0.204135\pi$$
$$308$$ −6400.00 −1.18401
$$309$$ 5304.00 0.976485
$$310$$ 1760.00i 0.322456i
$$311$$ −7932.00 −1.44625 −0.723123 0.690719i $$-0.757294\pi$$
−0.723123 + 0.690719i $$0.757294\pi$$
$$312$$ 0 0
$$313$$ 10358.0 1.87051 0.935254 0.353978i $$-0.115171\pi$$
0.935254 + 0.353978i $$0.115171\pi$$
$$314$$ 4340.00i 0.780001i
$$315$$ 5760.00 1.03028
$$316$$ 4384.00 0.780441
$$317$$ 2820.00i 0.499643i 0.968292 + 0.249822i $$0.0803720\pi$$
−0.968292 + 0.249822i $$0.919628\pi$$
$$318$$ 1188.00i 0.209496i
$$319$$ 4100.00i 0.719611i
$$320$$ − 1280.00i − 0.223607i
$$321$$ 5424.00 0.943110
$$322$$ −1280.00 −0.221527
$$323$$ 3600.00i 0.620153i
$$324$$ −324.000 −0.0555556
$$325$$ 0 0
$$326$$ 496.000 0.0842666
$$327$$ − 5658.00i − 0.956844i
$$328$$ 864.000 0.145446
$$329$$ −5696.00 −0.954500
$$330$$ − 6000.00i − 1.00088i
$$331$$ 4180.00i 0.694120i 0.937843 + 0.347060i $$0.112820\pi$$
−0.937843 + 0.347060i $$0.887180\pi$$
$$332$$ − 1848.00i − 0.305488i
$$333$$ − 2754.00i − 0.453208i
$$334$$ 204.000 0.0334203
$$335$$ −2800.00 −0.456658
$$336$$ 1536.00i 0.249392i
$$337$$ 5026.00 0.812414 0.406207 0.913781i $$-0.366851\pi$$
0.406207 + 0.913781i $$0.366851\pi$$
$$338$$ 0 0
$$339$$ −3738.00 −0.598880
$$340$$ − 2400.00i − 0.382818i
$$341$$ −2200.00 −0.349374
$$342$$ 2160.00 0.341519
$$343$$ 10816.0i 1.70265i
$$344$$ 2848.00i 0.446378i
$$345$$ − 1200.00i − 0.187263i
$$346$$ 1364.00i 0.211934i
$$347$$ −7332.00 −1.13430 −0.567150 0.823614i $$-0.691954\pi$$
−0.567150 + 0.823614i $$0.691954\pi$$
$$348$$ 984.000 0.151575
$$349$$ − 8162.00i − 1.25187i −0.779876 0.625934i $$-0.784718\pi$$
0.779876 0.625934i $$-0.215282\pi$$
$$350$$ 17600.0 2.68788
$$351$$ 0 0
$$352$$ 1600.00 0.242274
$$353$$ − 1244.00i − 0.187568i −0.995593 0.0937839i $$-0.970104\pi$$
0.995593 0.0937839i $$-0.0298963\pi$$
$$354$$ −564.000 −0.0846787
$$355$$ −15560.0 −2.32631
$$356$$ − 4896.00i − 0.728897i
$$357$$ 2880.00i 0.426963i
$$358$$ − 1224.00i − 0.180699i
$$359$$ 9558.00i 1.40516i 0.711605 + 0.702579i $$0.247968\pi$$
−0.711605 + 0.702579i $$0.752032\pi$$
$$360$$ −1440.00 −0.210819
$$361$$ −7541.00 −1.09943
$$362$$ − 132.000i − 0.0191651i
$$363$$ 3507.00 0.507079
$$364$$ 0 0
$$365$$ −1240.00 −0.177821
$$366$$ − 372.000i − 0.0531277i
$$367$$ −11032.0 −1.56912 −0.784558 0.620055i $$-0.787110\pi$$
−0.784558 + 0.620055i $$0.787110\pi$$
$$368$$ 320.000 0.0453292
$$369$$ − 972.000i − 0.137128i
$$370$$ − 12240.0i − 1.71980i
$$371$$ − 6336.00i − 0.886654i
$$372$$ 528.000i 0.0735901i
$$373$$ 5474.00 0.759874 0.379937 0.925012i $$-0.375946\pi$$
0.379937 + 0.925012i $$0.375946\pi$$
$$374$$ 3000.00 0.414776
$$375$$ 9000.00i 1.23935i
$$376$$ 1424.00 0.195312
$$377$$ 0 0
$$378$$ 1728.00 0.235129
$$379$$ 7040.00i 0.954144i 0.878864 + 0.477072i $$0.158302\pi$$
−0.878864 + 0.477072i $$0.841698\pi$$
$$380$$ 9600.00 1.29597
$$381$$ 4872.00 0.655118
$$382$$ − 1216.00i − 0.162869i
$$383$$ 1830.00i 0.244148i 0.992521 + 0.122074i $$0.0389545\pi$$
−0.992521 + 0.122074i $$0.961045\pi$$
$$384$$ − 384.000i − 0.0510310i
$$385$$ 32000.0i 4.23603i
$$386$$ 2740.00 0.361301
$$387$$ 3204.00 0.420849
$$388$$ 2456.00i 0.321352i
$$389$$ −10158.0 −1.32399 −0.661994 0.749509i $$-0.730289\pi$$
−0.661994 + 0.749509i $$0.730289\pi$$
$$390$$ 0 0
$$391$$ 600.000 0.0776044
$$392$$ − 5448.00i − 0.701953i
$$393$$ 6216.00 0.797852
$$394$$ 9816.00 1.25513
$$395$$ − 21920.0i − 2.79219i
$$396$$ − 1800.00i − 0.228418i
$$397$$ − 12658.0i − 1.60022i −0.599854 0.800109i $$-0.704775\pi$$
0.599854 0.800109i $$-0.295225\pi$$
$$398$$ − 656.000i − 0.0826189i
$$399$$ −11520.0 −1.44542
$$400$$ −4400.00 −0.550000
$$401$$ 15720.0i 1.95765i 0.204689 + 0.978827i $$0.434382\pi$$
−0.204689 + 0.978827i $$0.565618\pi$$
$$402$$ −840.000 −0.104217
$$403$$ 0 0
$$404$$ 4232.00 0.521163
$$405$$ 1620.00i 0.198762i
$$406$$ −5248.00 −0.641512
$$407$$ 15300.0 1.86337
$$408$$ − 720.000i − 0.0873660i
$$409$$ − 7654.00i − 0.925345i −0.886529 0.462672i $$-0.846891\pi$$
0.886529 0.462672i $$-0.153109\pi$$
$$410$$ − 4320.00i − 0.520365i
$$411$$ 2268.00i 0.272195i
$$412$$ 7072.00 0.845661
$$413$$ 3008.00 0.358387
$$414$$ − 360.000i − 0.0427368i
$$415$$ −9240.00 −1.09295
$$416$$ 0 0
$$417$$ −516.000 −0.0605962
$$418$$ 12000.0i 1.40416i
$$419$$ −1848.00 −0.215467 −0.107734 0.994180i $$-0.534359\pi$$
−0.107734 + 0.994180i $$0.534359\pi$$
$$420$$ 7680.00 0.892251
$$421$$ 12542.0i 1.45192i 0.687735 + 0.725962i $$0.258605\pi$$
−0.687735 + 0.725962i $$0.741395\pi$$
$$422$$ − 2632.00i − 0.303611i
$$423$$ − 1602.00i − 0.184142i
$$424$$ 1584.00i 0.181429i
$$425$$ −8250.00 −0.941609
$$426$$ −4668.00 −0.530905
$$427$$ 1984.00i 0.224854i
$$428$$ 7232.00 0.816757
$$429$$ 0 0
$$430$$ 14240.0 1.59701
$$431$$ 5238.00i 0.585396i 0.956205 + 0.292698i $$0.0945530\pi$$
−0.956205 + 0.292698i $$0.905447\pi$$
$$432$$ −432.000 −0.0481125
$$433$$ 8258.00 0.916522 0.458261 0.888818i $$-0.348472\pi$$
0.458261 + 0.888818i $$0.348472\pi$$
$$434$$ − 2816.00i − 0.311457i
$$435$$ − 4920.00i − 0.542290i
$$436$$ − 7544.00i − 0.828652i
$$437$$ 2400.00i 0.262718i
$$438$$ −372.000 −0.0405818
$$439$$ 6304.00 0.685361 0.342681 0.939452i $$-0.388665\pi$$
0.342681 + 0.939452i $$0.388665\pi$$
$$440$$ − 8000.00i − 0.866784i
$$441$$ −6129.00 −0.661808
$$442$$ 0 0
$$443$$ 12744.0 1.36678 0.683392 0.730051i $$-0.260504\pi$$
0.683392 + 0.730051i $$0.260504\pi$$
$$444$$ − 3672.00i − 0.392490i
$$445$$ −24480.0 −2.60778
$$446$$ 3864.00 0.410237
$$447$$ 3816.00i 0.403782i
$$448$$ 2048.00i 0.215980i
$$449$$ − 11776.0i − 1.23774i −0.785495 0.618868i $$-0.787591\pi$$
0.785495 0.618868i $$-0.212409\pi$$
$$450$$ 4950.00i 0.518545i
$$451$$ 5400.00 0.563805
$$452$$ −4984.00 −0.518645
$$453$$ − 4212.00i − 0.436859i
$$454$$ −9996.00 −1.03334
$$455$$ 0 0
$$456$$ 2880.00 0.295764
$$457$$ − 2134.00i − 0.218434i −0.994018 0.109217i $$-0.965166\pi$$
0.994018 0.109217i $$-0.0348343\pi$$
$$458$$ −156.000 −0.0159157
$$459$$ −810.000 −0.0823694
$$460$$ − 1600.00i − 0.162175i
$$461$$ − 2724.00i − 0.275205i −0.990488 0.137602i $$-0.956060\pi$$
0.990488 0.137602i $$-0.0439396\pi$$
$$462$$ 9600.00i 0.966737i
$$463$$ − 5648.00i − 0.566922i −0.958984 0.283461i $$-0.908517\pi$$
0.958984 0.283461i $$-0.0914826\pi$$
$$464$$ 1312.00 0.131267
$$465$$ 2640.00 0.263284
$$466$$ − 2564.00i − 0.254882i
$$467$$ 18224.0 1.80579 0.902897 0.429856i $$-0.141436\pi$$
0.902897 + 0.429856i $$0.141436\pi$$
$$468$$ 0 0
$$469$$ 4480.00 0.441081
$$470$$ − 7120.00i − 0.698768i
$$471$$ 6510.00 0.636868
$$472$$ −752.000 −0.0733339
$$473$$ 17800.0i 1.73033i
$$474$$ − 6576.00i − 0.637227i
$$475$$ − 33000.0i − 3.18767i
$$476$$ 3840.00i 0.369761i
$$477$$ 1782.00 0.171053
$$478$$ −588.000 −0.0562646
$$479$$ 9066.00i 0.864794i 0.901683 + 0.432397i $$0.142332\pi$$
−0.901683 + 0.432397i $$0.857668\pi$$
$$480$$ −1920.00 −0.182574
$$481$$ 0 0
$$482$$ −9924.00 −0.937813
$$483$$ 1920.00i 0.180876i
$$484$$ 4676.00 0.439144
$$485$$ 12280.0 1.14970
$$486$$ 486.000i 0.0453609i
$$487$$ − 8948.00i − 0.832593i −0.909229 0.416296i $$-0.863328\pi$$
0.909229 0.416296i $$-0.136672\pi$$
$$488$$ − 496.000i − 0.0460100i
$$489$$ − 744.000i − 0.0688034i
$$490$$ −27240.0 −2.51138
$$491$$ −8720.00 −0.801483 −0.400741 0.916191i $$-0.631247\pi$$
−0.400741 + 0.916191i $$0.631247\pi$$
$$492$$ − 1296.00i − 0.118756i
$$493$$ 2460.00 0.224732
$$494$$ 0 0
$$495$$ −9000.00 −0.817212
$$496$$ 704.000i 0.0637309i
$$497$$ 24896.0 2.24696
$$498$$ −2772.00 −0.249430
$$499$$ − 6604.00i − 0.592456i −0.955117 0.296228i $$-0.904271\pi$$
0.955117 0.296228i $$-0.0957289\pi$$
$$500$$ 12000.0i 1.07331i
$$501$$ − 306.000i − 0.0272876i
$$502$$ 1488.00i 0.132296i
$$503$$ 3404.00 0.301743 0.150872 0.988553i $$-0.451792\pi$$
0.150872 + 0.988553i $$0.451792\pi$$
$$504$$ 2304.00 0.203628
$$505$$ − 21160.0i − 1.86457i
$$506$$ 2000.00 0.175713
$$507$$ 0 0
$$508$$ 6496.00 0.567349
$$509$$ 76.0000i 0.00661815i 0.999995 + 0.00330908i $$0.00105331\pi$$
−0.999995 + 0.00330908i $$0.998947\pi$$
$$510$$ −3600.00 −0.312570
$$511$$ 1984.00 0.171755
$$512$$ − 512.000i − 0.0441942i
$$513$$ − 3240.00i − 0.278849i
$$514$$ − 2052.00i − 0.176089i
$$515$$ − 35360.0i − 3.02553i
$$516$$ 4272.00 0.364466
$$517$$ 8900.00 0.757102
$$518$$ 19584.0i 1.66114i
$$519$$ 2046.00 0.173043
$$520$$ 0 0
$$521$$ 12054.0 1.01362 0.506809 0.862058i $$-0.330825\pi$$
0.506809 + 0.862058i $$0.330825\pi$$
$$522$$ − 1476.00i − 0.123760i
$$523$$ 276.000 0.0230758 0.0115379 0.999933i $$-0.496327\pi$$
0.0115379 + 0.999933i $$0.496327\pi$$
$$524$$ 8288.00 0.690960
$$525$$ − 26400.0i − 2.19465i
$$526$$ 11064.0i 0.917136i
$$527$$ 1320.00i 0.109108i
$$528$$ − 2400.00i − 0.197816i
$$529$$ −11767.0 −0.967124
$$530$$ 7920.00 0.649100
$$531$$ 846.000i 0.0691399i
$$532$$ −15360.0 −1.25177
$$533$$ 0 0
$$534$$ −7344.00 −0.595142
$$535$$ − 36160.0i − 2.92212i
$$536$$ −1120.00 −0.0902549
$$537$$ −1836.00 −0.147540
$$538$$ 7068.00i 0.566400i
$$539$$ − 34050.0i − 2.72103i
$$540$$ 2160.00i 0.172133i
$$541$$ 13778.0i 1.09494i 0.836825 + 0.547470i $$0.184409\pi$$
−0.836825 + 0.547470i $$0.815591\pi$$
$$542$$ 4784.00 0.379134
$$543$$ −198.000 −0.0156482
$$544$$ − 960.000i − 0.0756611i
$$545$$ −37720.0 −2.96467
$$546$$ 0 0
$$547$$ −10844.0 −0.847634 −0.423817 0.905748i $$-0.639310\pi$$
−0.423817 + 0.905748i $$0.639310\pi$$
$$548$$ 3024.00i 0.235728i
$$549$$ −558.000 −0.0433786
$$550$$ −27500.0 −2.13201
$$551$$ 9840.00i 0.760795i
$$552$$ − 480.000i − 0.0370112i
$$553$$ 35072.0i 2.69695i
$$554$$ 12204.0i 0.935917i
$$555$$ −18360.0 −1.40421
$$556$$ −688.000 −0.0524779
$$557$$ 20544.0i 1.56280i 0.624033 + 0.781398i $$0.285493\pi$$
−0.624033 + 0.781398i $$0.714507\pi$$
$$558$$ 792.000 0.0600861
$$559$$ 0 0
$$560$$ 10240.0 0.772712
$$561$$ − 4500.00i − 0.338663i
$$562$$ −15080.0 −1.13187
$$563$$ −6988.00 −0.523107 −0.261553 0.965189i $$-0.584235\pi$$
−0.261553 + 0.965189i $$0.584235\pi$$
$$564$$ − 2136.00i − 0.159471i
$$565$$ 24920.0i 1.85556i
$$566$$ − 5512.00i − 0.409340i
$$567$$ − 2592.00i − 0.191982i
$$568$$ −6224.00 −0.459777
$$569$$ 706.000 0.0520159 0.0260080 0.999662i $$-0.491720\pi$$
0.0260080 + 0.999662i $$0.491720\pi$$
$$570$$ − 14400.0i − 1.05816i
$$571$$ 17532.0 1.28492 0.642462 0.766318i $$-0.277913\pi$$
0.642462 + 0.766318i $$0.277913\pi$$
$$572$$ 0 0
$$573$$ −1824.00 −0.132982
$$574$$ 6912.00i 0.502616i
$$575$$ −5500.00 −0.398897
$$576$$ −576.000 −0.0416667
$$577$$ 14814.0i 1.06883i 0.845222 + 0.534415i $$0.179468\pi$$
−0.845222 + 0.534415i $$0.820532\pi$$
$$578$$ 8026.00i 0.577574i
$$579$$ − 4110.00i − 0.295001i
$$580$$ − 6560.00i − 0.469637i
$$581$$ 14784.0 1.05567
$$582$$ 3684.00 0.262383
$$583$$ 9900.00i 0.703287i
$$584$$ −496.000 −0.0351449
$$585$$ 0 0
$$586$$ 1936.00 0.136477
$$587$$ − 14170.0i − 0.996352i −0.867076 0.498176i $$-0.834003\pi$$
0.867076 0.498176i $$-0.165997\pi$$
$$588$$ −8172.00 −0.573142
$$589$$ −5280.00 −0.369369
$$590$$ 3760.00i 0.262367i
$$591$$ − 14724.0i − 1.02481i
$$592$$ − 4896.00i − 0.339906i
$$593$$ − 11744.0i − 0.813269i −0.913591 0.406634i $$-0.866702\pi$$
0.913591 0.406634i $$-0.133298\pi$$
$$594$$ −2700.00 −0.186502
$$595$$ 19200.0 1.32290
$$596$$ 5088.00i 0.349686i
$$597$$ −984.000 −0.0674580
$$598$$ 0 0
$$599$$ −15076.0 −1.02836 −0.514181 0.857682i $$-0.671904\pi$$
−0.514181 + 0.857682i $$0.671904\pi$$
$$600$$ 6600.00i 0.449073i
$$601$$ 20230.0 1.37304 0.686522 0.727109i $$-0.259137\pi$$
0.686522 + 0.727109i $$0.259137\pi$$
$$602$$ −22784.0 −1.54254
$$603$$ 1260.00i 0.0850931i
$$604$$ − 5616.00i − 0.378331i
$$605$$ − 23380.0i − 1.57113i
$$606$$ − 6348.00i − 0.425528i
$$607$$ −28056.0 −1.87604 −0.938021 0.346577i $$-0.887344\pi$$
−0.938021 + 0.346577i $$0.887344\pi$$
$$608$$ 3840.00 0.256139
$$609$$ 7872.00i 0.523792i
$$610$$ −2480.00 −0.164610
$$611$$ 0 0
$$612$$ −1080.00 −0.0713340
$$613$$ − 27446.0i − 1.80837i −0.427136 0.904187i $$-0.640478\pi$$
0.427136 0.904187i $$-0.359522\pi$$
$$614$$ −12872.0 −0.846045
$$615$$ −6480.00 −0.424876
$$616$$ 12800.0i 0.837219i
$$617$$ − 8804.00i − 0.574450i −0.957863 0.287225i $$-0.907267\pi$$
0.957863 0.287225i $$-0.0927328\pi$$
$$618$$ − 10608.0i − 0.690480i
$$619$$ 3508.00i 0.227784i 0.993493 + 0.113892i $$0.0363318\pi$$
−0.993493 + 0.113892i $$0.963668\pi$$
$$620$$ 3520.00 0.228011
$$621$$ −540.000 −0.0348945
$$622$$ 15864.0i 1.02265i
$$623$$ 39168.0 2.51883
$$624$$ 0 0
$$625$$ 25625.0 1.64000
$$626$$ − 20716.0i − 1.32265i
$$627$$ 18000.0 1.14649
$$628$$ 8680.00 0.551544
$$629$$ − 9180.00i − 0.581925i
$$630$$ − 11520.0i − 0.728520i
$$631$$ 22084.0i 1.39326i 0.717428 + 0.696632i $$0.245319\pi$$
−0.717428 + 0.696632i $$0.754681\pi$$
$$632$$ − 8768.00i − 0.551855i
$$633$$ −3948.00 −0.247897
$$634$$ 5640.00 0.353301
$$635$$ − 32480.0i − 2.02981i
$$636$$ 2376.00 0.148136
$$637$$ 0 0
$$638$$ 8200.00 0.508842
$$639$$ 7002.00i 0.433482i
$$640$$ −2560.00 −0.158114
$$641$$ 7342.00 0.452405 0.226202 0.974080i $$-0.427369\pi$$
0.226202 + 0.974080i $$0.427369\pi$$
$$642$$ − 10848.0i − 0.666879i
$$643$$ − 2996.00i − 0.183749i −0.995771 0.0918746i $$-0.970714\pi$$
0.995771 0.0918746i $$-0.0292859\pi$$
$$644$$ 2560.00i 0.156643i
$$645$$ − 21360.0i − 1.30395i
$$646$$ 7200.00 0.438514
$$647$$ −9344.00 −0.567775 −0.283888 0.958858i $$-0.591624\pi$$
−0.283888 + 0.958858i $$0.591624\pi$$
$$648$$ 648.000i 0.0392837i
$$649$$ −4700.00 −0.284270
$$650$$ 0 0
$$651$$ −4224.00 −0.254304
$$652$$ − 992.000i − 0.0595855i
$$653$$ −16686.0 −0.999960 −0.499980 0.866037i $$-0.666659\pi$$
−0.499980 + 0.866037i $$0.666659\pi$$
$$654$$ −11316.0 −0.676591
$$655$$ − 41440.0i − 2.47205i
$$656$$ − 1728.00i − 0.102846i
$$657$$ 558.000i 0.0331349i
$$658$$ 11392.0i 0.674934i
$$659$$ 31356.0 1.85350 0.926750 0.375679i $$-0.122590\pi$$
0.926750 + 0.375679i $$0.122590\pi$$
$$660$$ −12000.0 −0.707726
$$661$$ 590.000i 0.0347176i 0.999849 + 0.0173588i $$0.00552576\pi$$
−0.999849 + 0.0173588i $$0.994474\pi$$
$$662$$ 8360.00 0.490817
$$663$$ 0 0
$$664$$ −3696.00 −0.216013
$$665$$ 76800.0i 4.47846i
$$666$$ −5508.00 −0.320466
$$667$$ 1640.00 0.0952040
$$668$$ − 408.000i − 0.0236317i
$$669$$ − 5796.00i − 0.334957i
$$670$$ 5600.00i 0.322906i
$$671$$ − 3100.00i − 0.178352i
$$672$$ 3072.00 0.176347
$$673$$ −5938.00 −0.340109 −0.170054 0.985435i $$-0.554394\pi$$
−0.170054 + 0.985435i $$0.554394\pi$$
$$674$$ − 10052.0i − 0.574464i
$$675$$ 7425.00 0.423390
$$676$$ 0 0
$$677$$ 9486.00 0.538518 0.269259 0.963068i $$-0.413221\pi$$
0.269259 + 0.963068i $$0.413221\pi$$
$$678$$ 7476.00i 0.423472i
$$679$$ −19648.0 −1.11049
$$680$$ −4800.00 −0.270694
$$681$$ 14994.0i 0.843717i
$$682$$ 4400.00i 0.247045i
$$683$$ 26162.0i 1.46568i 0.680400 + 0.732841i $$0.261806\pi$$
−0.680400 + 0.732841i $$0.738194\pi$$
$$684$$ − 4320.00i − 0.241490i
$$685$$ 15120.0 0.843366
$$686$$ 21632.0 1.20396
$$687$$ 234.000i 0.0129951i
$$688$$ 5696.00 0.315637
$$689$$ 0 0
$$690$$ −2400.00 −0.132415
$$691$$ 17348.0i 0.955064i 0.878614 + 0.477532i $$0.158468\pi$$
−0.878614 + 0.477532i $$0.841532\pi$$
$$692$$ 2728.00 0.149860
$$693$$ 14400.0 0.789337
$$694$$ 14664.0i 0.802072i
$$695$$ 3440.00i 0.187751i
$$696$$ − 1968.00i − 0.107179i
$$697$$ − 3240.00i − 0.176074i
$$698$$ −16324.0 −0.885204
$$699$$ −3846.00 −0.208110
$$700$$ − 35200.0i − 1.90062i
$$701$$ −30.0000 −0.00161638 −0.000808191 1.00000i $$-0.500257\pi$$
−0.000808191 1.00000i $$0.500257\pi$$
$$702$$ 0 0
$$703$$ 36720.0 1.97002
$$704$$ − 3200.00i − 0.171313i
$$705$$ −10680.0 −0.570542
$$706$$ −2488.00 −0.132630
$$707$$ 33856.0i 1.80097i
$$708$$ 1128.00i 0.0598769i
$$709$$ 31466.0i 1.66676i 0.552703 + 0.833378i $$0.313596\pi$$
−0.552703 + 0.833378i $$0.686404\pi$$
$$710$$ 31120.0i 1.64495i
$$711$$ −9864.00 −0.520294
$$712$$ −9792.00 −0.515408
$$713$$ 880.000i 0.0462220i
$$714$$ 5760.00 0.301908
$$715$$ 0 0
$$716$$ −2448.00 −0.127774
$$717$$ 882.000i 0.0459399i
$$718$$ 19116.0 0.993597
$$719$$ 28892.0 1.49859 0.749297 0.662234i $$-0.230391\pi$$
0.749297 + 0.662234i $$0.230391\pi$$
$$720$$ 2880.00i 0.149071i
$$721$$ 56576.0i 2.92233i
$$722$$ 15082.0i 0.777415i
$$723$$ 14886.0i 0.765721i
$$724$$ −264.000 −0.0135518
$$725$$ −22550.0 −1.15515
$$726$$ − 7014.00i − 0.358559i
$$727$$ −13384.0 −0.682786 −0.341393 0.939921i $$-0.610899\pi$$
−0.341393 + 0.939921i $$0.610899\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 2480.00i 0.125738i
$$731$$ 10680.0 0.540375
$$732$$ −744.000 −0.0375670
$$733$$ − 7130.00i − 0.359280i −0.983732 0.179640i $$-0.942507\pi$$
0.983732 0.179640i $$-0.0574934\pi$$
$$734$$ 22064.0i 1.10953i
$$735$$ 40860.0i 2.05054i
$$736$$ − 640.000i − 0.0320526i
$$737$$ −7000.00 −0.349862
$$738$$ −1944.00 −0.0969643
$$739$$ − 29268.0i − 1.45689i −0.685105 0.728444i $$-0.740244\pi$$
0.685105 0.728444i $$-0.259756\pi$$
$$740$$ −24480.0 −1.21608
$$741$$ 0 0
$$742$$ −12672.0 −0.626959
$$743$$ 9898.00i 0.488725i 0.969684 + 0.244362i $$0.0785786\pi$$
−0.969684 + 0.244362i $$0.921421\pi$$
$$744$$ 1056.00 0.0520361
$$745$$ 25440.0 1.25107
$$746$$ − 10948.0i − 0.537312i
$$747$$ 4158.00i 0.203659i
$$748$$ − 6000.00i − 0.293291i
$$749$$ 57856.0i 2.82245i
$$750$$ 18000.0 0.876356
$$751$$ 15120.0 0.734669 0.367335 0.930089i $$-0.380270\pi$$
0.367335 + 0.930089i $$0.380270\pi$$
$$752$$ − 2848.00i − 0.138106i
$$753$$ 2232.00 0.108019
$$754$$ 0 0
$$755$$ −28080.0 −1.35356
$$756$$ − 3456.00i − 0.166261i
$$757$$ −5454.00 −0.261861 −0.130931 0.991392i $$-0.541797\pi$$
−0.130931 + 0.991392i $$0.541797\pi$$
$$758$$ 14080.0 0.674682
$$759$$ − 3000.00i − 0.143469i
$$760$$ − 19200.0i − 0.916391i
$$761$$ − 11988.0i − 0.571044i −0.958372 0.285522i $$-0.907833\pi$$
0.958372 0.285522i $$-0.0921670\pi$$
$$762$$ − 9744.00i − 0.463239i
$$763$$ 60352.0 2.86355
$$764$$ −2432.00 −0.115166
$$765$$ 5400.00i 0.255212i
$$766$$ 3660.00 0.172639
$$767$$ 0 0
$$768$$ −768.000 −0.0360844
$$769$$ − 1338.00i − 0.0627432i −0.999508 0.0313716i $$-0.990012\pi$$
0.999508 0.0313716i $$-0.00998753\pi$$
$$770$$ 64000.0 2.99532
$$771$$ −3078.00 −0.143776
$$772$$ − 5480.00i − 0.255479i
$$773$$ 14408.0i 0.670401i 0.942147 + 0.335200i $$0.108804\pi$$
−0.942147 + 0.335200i $$0.891196\pi$$
$$774$$ − 6408.00i − 0.297585i
$$775$$ − 12100.0i − 0.560832i
$$776$$ 4912.00 0.227230
$$777$$ 29376.0 1.35632
$$778$$ 20316.0i 0.936200i
$$779$$ 12960.0 0.596072
$$780$$ 0 0
$$781$$ −38900.0 −1.78227
$$782$$ − 1200.00i − 0.0548746i
$$783$$ −2214.00 −0.101050
$$784$$ −10896.0 −0.496356
$$785$$ − 43400.0i − 1.97326i
$$786$$ − 12432.0i − 0.564166i
$$787$$ − 10660.0i − 0.482831i −0.970422 0.241415i $$-0.922388\pi$$
0.970422 0.241415i $$-0.0776117\pi$$
$$788$$ − 19632.0i − 0.887514i
$$789$$ 16596.0 0.748838
$$790$$ −43840.0 −1.97438
$$791$$ − 39872.0i − 1.79227i
$$792$$ −3600.00 −0.161516
$$793$$ 0 0
$$794$$ −25316.0 −1.13153
$$795$$ − 11880.0i − 0.529988i
$$796$$ −1312.00 −0.0584204
$$797$$ −1974.00 −0.0877323 −0.0438662 0.999037i $$-0.513968\pi$$
−0.0438662 + 0.999037i $$0.513968\pi$$
$$798$$ 23040.0i 1.02206i
$$799$$ − 5340.00i − 0.236440i
$$800$$ 8800.00i 0.388909i
$$801$$ 11016.0i 0.485932i
$$802$$ 31440.0 1.38427
$$803$$ −3100.00 −0.136235
$$804$$ 1680.00i 0.0736928i
$$805$$ 12800.0 0.560423
$$806$$ 0 0
$$807$$ 10602.0 0.462464
$$808$$ − 8464.00i − 0.368518i
$$809$$ 31734.0 1.37912 0.689560 0.724229i $$-0.257804\pi$$
0.689560 + 0.724229i $$0.257804\pi$$
$$810$$ 3240.00 0.140546
$$811$$ 38824.0i 1.68100i 0.541808 + 0.840502i $$0.317740\pi$$
−0.541808 + 0.840502i $$0.682260\pi$$
$$812$$ 10496.0i 0.453617i
$$813$$ − 7176.00i − 0.309561i
$$814$$ − 30600.0i − 1.31760i
$$815$$ −4960.00 −0.213179
$$816$$ −1440.00 −0.0617771
$$817$$ 42720.0i 1.82936i
$$818$$ −15308.0 −0.654317
$$819$$ 0 0
$$820$$ −8640.00 −0.367954
$$821$$ 16736.0i 0.711438i 0.934593 + 0.355719i $$0.115764\pi$$
−0.934593 + 0.355719i $$0.884236\pi$$
$$822$$ 4536.00 0.192471
$$823$$ −42096.0 −1.78296 −0.891479 0.453062i $$-0.850332\pi$$
−0.891479 + 0.453062i $$0.850332\pi$$
$$824$$ − 14144.0i − 0.597973i
$$825$$ 41250.0i 1.74078i
$$826$$ − 6016.00i − 0.253418i
$$827$$ 24858.0i 1.04522i 0.852572 + 0.522610i $$0.175042\pi$$
−0.852572 + 0.522610i $$0.824958\pi$$
$$828$$ −720.000 −0.0302195
$$829$$ −922.000 −0.0386277 −0.0193139 0.999813i $$-0.506148\pi$$
−0.0193139 + 0.999813i $$0.506148\pi$$
$$830$$ 18480.0i 0.772832i
$$831$$ 18306.0 0.764173
$$832$$ 0 0
$$833$$ −20430.0 −0.849769
$$834$$ 1032.00i 0.0428480i
$$835$$ −2040.00 −0.0845474
$$836$$ 24000.0 0.992892
$$837$$ − 1188.00i − 0.0490601i
$$838$$ 3696.00i 0.152358i
$$839$$ − 14294.0i − 0.588181i −0.955778 0.294090i $$-0.904983\pi$$
0.955778 0.294090i $$-0.0950167\pi$$
$$840$$ − 15360.0i − 0.630917i
$$841$$ −17665.0 −0.724302
$$842$$ 25084.0 1.02666
$$843$$ 22620.0i 0.924169i
$$844$$ −5264.00 −0.214685
$$845$$ 0 0
$$846$$ −3204.00 −0.130208
$$847$$ 37408.0i 1.51754i
$$848$$ 3168.00 0.128290
$$849$$ −8268.00 −0.334225
$$850$$ 16500.0i 0.665818i
$$851$$ − 6120.00i − 0.246523i
$$852$$ 9336.00i 0.375406i
$$853$$ 37966.0i 1.52395i 0.647605 + 0.761976i $$0.275771\pi$$
−0.647605 + 0.761976i $$0.724229\pi$$
$$854$$ 3968.00 0.158996
$$855$$ −21600.0 −0.863982
$$856$$ − 14464.0i − 0.577534i
$$857$$ −39038.0 −1.55602 −0.778012 0.628249i $$-0.783772\pi$$
−0.778012 + 0.628249i $$0.783772\pi$$
$$858$$ 0 0
$$859$$ 20564.0 0.816804 0.408402 0.912802i $$-0.366086\pi$$
0.408402 + 0.912802i $$0.366086\pi$$
$$860$$ − 28480.0i − 1.12926i
$$861$$ 10368.0 0.410384
$$862$$ 10476.0 0.413937
$$863$$ − 39866.0i − 1.57248i −0.617918 0.786242i $$-0.712024\pi$$
0.617918 0.786242i $$-0.287976\pi$$
$$864$$ 864.000i 0.0340207i
$$865$$ − 13640.0i − 0.536155i
$$866$$ − 16516.0i − 0.648079i
$$867$$ 12039.0 0.471587
$$868$$ −5632.00 −0.220233
$$869$$ − 54800.0i − 2.13920i
$$870$$ −9840.00 −0.383457
$$871$$ 0 0
$$872$$ −15088.0 −0.585945
$$873$$ − 5526.00i − 0.214235i
$$874$$ 4800.00 0.185769
$$875$$ −96000.0 −3.70902
$$876$$ 744.000i 0.0286957i
$$877$$ − 30990.0i − 1.19322i −0.802530 0.596612i $$-0.796513\pi$$
0.802530 0.596612i $$-0.203487\pi$$
$$878$$ − 12608.0i − 0.484623i
$$879$$ − 2904.00i − 0.111433i
$$880$$ −16000.0 −0.612909
$$881$$ 4458.00 0.170481 0.0852405 0.996360i $$-0.472834\pi$$
0.0852405 + 0.996360i $$0.472834\pi$$
$$882$$ 12258.0i 0.467969i
$$883$$ 3164.00 0.120586 0.0602928 0.998181i $$-0.480797\pi$$
0.0602928 + 0.998181i $$0.480797\pi$$
$$884$$ 0 0
$$885$$ 5640.00 0.214222
$$886$$ − 25488.0i − 0.966463i
$$887$$ −32512.0 −1.23072 −0.615359 0.788247i $$-0.710989\pi$$
−0.615359 + 0.788247i $$0.710989\pi$$
$$888$$ −7344.00 −0.277532
$$889$$ 51968.0i 1.96057i
$$890$$ 48960.0i 1.84398i
$$891$$ 4050.00i 0.152278i
$$892$$ − 7728.00i − 0.290081i
$$893$$ 21360.0 0.800431
$$894$$ 7632.00 0.285517
$$895$$ 12240.0i 0.457138i
$$896$$ 4096.00 0.152721
$$897$$ 0 0
$$898$$ −23552.0 −0.875212
$$899$$ 3608.00i 0.133853i
$$900$$ 9900.00 0.366667
$$901$$ 5940.00 0.219634
$$902$$ − 10800.0i − 0.398670i
$$903$$ 34176.0i 1.25948i
$$904$$ 9968.00i 0.366738i
$$905$$ 1320.00i 0.0484843i
$$906$$ −8424.00 −0.308906
$$907$$ −10500.0 −0.384396 −0.192198 0.981356i $$-0.561562\pi$$
−0.192198 + 0.981356i $$0.561562\pi$$
$$908$$ 19992.0i 0.730680i
$$909$$ −9522.00 −0.347442
$$910$$ 0 0
$$911$$ 9840.00 0.357864 0.178932 0.983861i $$-0.442736\pi$$
0.178932 + 0.983861i $$0.442736\pi$$
$$912$$ − 5760.00i − 0.209137i
$$913$$ −23100.0 −0.837348
$$914$$ −4268.00 −0.154456
$$915$$ 3720.00i 0.134404i
$$916$$ 312.000i 0.0112541i
$$917$$ 66304.0i 2.38773i
$$918$$ 1620.00i 0.0582440i
$$919$$ −35040.0 −1.25774 −0.628870 0.777511i $$-0.716482\pi$$
−0.628870 + 0.777511i $$0.716482\pi$$
$$920$$ −3200.00 −0.114675
$$921$$ 19308.0i 0.690793i
$$922$$ −5448.00 −0.194599
$$923$$ 0 0
$$924$$ 19200.0 0.683586
$$925$$ 84150.0i 2.99117i
$$926$$ −11296.0 −0.400874
$$927$$ −15912.0 −0.563774
$$928$$ − 2624.00i − 0.0928201i
$$929$$ − 44172.0i − 1.56000i −0.625782 0.779998i $$-0.715220\pi$$
0.625782 0.779998i $$-0.284780\pi$$
$$930$$ − 5280.00i − 0.186170i
$$931$$ − 81720.0i − 2.87676i
$$932$$ −5128.00 −0.180229
$$933$$ 23796.0 0.834990
$$934$$ − 36448.0i − 1.27689i
$$935$$ −30000.0 −1.04931
$$936$$ 0 0
$$937$$ −54018.0 −1.88334 −0.941671 0.336535i $$-0.890745\pi$$
−0.941671 + 0.336535i $$0.890745\pi$$
$$938$$ − 8960.00i − 0.311892i
$$939$$ −31074.0 −1.07994
$$940$$ −14240.0 −0.494104
$$941$$ 1672.00i 0.0579231i 0.999581 + 0.0289616i $$0.00922004\pi$$
−0.999581 + 0.0289616i $$0.990780\pi$$
$$942$$ − 13020.0i − 0.450334i
$$943$$ − 2160.00i − 0.0745910i
$$944$$ 1504.00i 0.0518549i
$$945$$ −17280.0 −0.594834
$$946$$ 35600.0 1.22353
$$947$$ 5238.00i 0.179738i 0.995954 + 0.0898691i $$0.0286449\pi$$
−0.995954 + 0.0898691i $$0.971355\pi$$
$$948$$ −13152.0 −0.450588
$$949$$ 0 0
$$950$$ −66000.0 −2.25402
$$951$$ − 8460.00i − 0.288469i
$$952$$ 7680.00 0.261460
$$953$$ 50042.0 1.70096 0.850482 0.526004i $$-0.176310\pi$$
0.850482 + 0.526004i $$0.176310\pi$$
$$954$$ − 3564.00i − 0.120953i
$$955$$ 12160.0i 0.412030i
$$956$$ 1176.00i 0.0397851i
$$957$$ − 12300.0i − 0.415468i
$$958$$ 18132.0 0.611501
$$959$$ −24192.0 −0.814599
$$960$$ 3840.00i 0.129099i
$$961$$ 27855.0 0.935014
$$962$$ 0 0
$$963$$ −16272.0 −0.544505
$$964$$ 19848.0i 0.663134i
$$965$$ −27400.0 −0.914028
$$966$$ 3840.00 0.127899
$$967$$ − 37676.0i − 1.25293i −0.779452 0.626463i $$-0.784502\pi$$
0.779452 0.626463i $$-0.215498\pi$$
$$968$$ − 9352.00i − 0.310521i
$$969$$ − 10800.0i − 0.358045i
$$970$$ − 24560.0i − 0.812963i
$$971$$ −17364.0 −0.573880 −0.286940 0.957949i $$-0.592638\pi$$
−0.286940 + 0.957949i $$0.592638\pi$$
$$972$$ 972.000 0.0320750
$$973$$ − 5504.00i − 0.181346i
$$974$$ −17896.0 −0.588732
$$975$$ 0 0
$$976$$ −992.000 −0.0325340
$$977$$ 14904.0i 0.488046i 0.969769 + 0.244023i $$0.0784673\pi$$
−0.969769 + 0.244023i $$0.921533\pi$$
$$978$$ −1488.00 −0.0486513
$$979$$ −61200.0 −1.99792
$$980$$ 54480.0i 1.77582i
$$981$$ 16974.0i 0.552434i
$$982$$ 17440.0i 0.566734i
$$983$$ − 18038.0i − 0.585272i −0.956224 0.292636i $$-0.905467\pi$$
0.956224 0.292636i $$-0.0945325\pi$$
$$984$$ −2592.00 −0.0839735
$$985$$ −98160.0 −3.17527
$$986$$ − 4920.00i − 0.158909i
$$987$$ 17088.0 0.551081
$$988$$ 0 0
$$989$$ 7120.00 0.228921
$$990$$ 18000.0i 0.577856i
$$991$$ 46176.0 1.48015 0.740075 0.672524i $$-0.234790\pi$$
0.740075 + 0.672524i $$0.234790\pi$$
$$992$$ 1408.00 0.0450646
$$993$$ − 12540.0i − 0.400750i
$$994$$ − 49792.0i − 1.58884i
$$995$$ 6560.00i 0.209011i
$$996$$ 5544.00i 0.176374i
$$997$$ 55838.0 1.77373 0.886864 0.462030i $$-0.152879\pi$$
0.886864 + 0.462030i $$0.152879\pi$$
$$998$$ −13208.0 −0.418930
$$999$$ 8262.00i 0.261660i
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 1014.4.b.e.337.1 2
13.5 odd 4 1014.4.a.d.1.1 1
13.8 odd 4 78.4.a.d.1.1 1
13.12 even 2 inner 1014.4.b.e.337.2 2
39.8 even 4 234.4.a.f.1.1 1
52.47 even 4 624.4.a.e.1.1 1
65.34 odd 4 1950.4.a.h.1.1 1
104.21 odd 4 2496.4.a.r.1.1 1
104.99 even 4 2496.4.a.i.1.1 1
156.47 odd 4 1872.4.a.r.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.d.1.1 1 13.8 odd 4
234.4.a.f.1.1 1 39.8 even 4
624.4.a.e.1.1 1 52.47 even 4
1014.4.a.d.1.1 1 13.5 odd 4
1014.4.b.e.337.1 2 1.1 even 1 trivial
1014.4.b.e.337.2 2 13.12 even 2 inner
1872.4.a.r.1.1 1 156.47 odd 4
1950.4.a.h.1.1 1 65.34 odd 4
2496.4.a.i.1.1 1 104.99 even 4
2496.4.a.r.1.1 1 104.21 odd 4